Aerosol Science and Technology 36 1045ndash1060 (2002)cdeg 2002 American Association for Aerosol ResearchPublished by Taylor and Francis0278-6826=02=$1200 C 00DOI 10108002786820290092203

Three-Dimensional Modeling of MicrosphereContactImpact with Smooth Flat Surfaces

W Cheng R M Brach and P F DunnParticle Dynamics Laboratory Department of Aerospace and Mechanical EngineeringUniversity of Notre Dame Notre Dame Indiana

A three-dimensional (3D) simulation model of a microspherein contact with a nominally-smooth at surface is establishedThe model is based on Hertzian contact stresses an idealized ringforce distribution of adhesion nonlinear damping and frictionAlthough originally developed for the study of oblique impact themodel also is shown to describe the motion of a microsphere insustained contact with a at surface including nonlinear normaloscillatorymotion in the presence of sliding and rolling Nonlinearnormal oscillatory motion is illustrated using conventional phase-plane techniques Methods for the determination of damping co-ef cients associated with normal motion from impact experimentsare discussed The signi cance and modeling of rolling resistanceand implicationsof asymmetriccontactstressdistributionsare pre-sentedSimulations show that energyexchangebetween translationand rotationcan play an important role during oblique impactTheeffects of complex initial conditions on the rebound and captureof a microsphere are signi cant as are the effects of contact fric-tion Results show that the inability to measure and the failure toaccount for rotationalvelocitiesin experimentalmeasurements canlimit the intepretation of the results

INTRODUCTIONMicrosphere contactimpact with surfaces has many applica-

tions and much theoretical signi cance The modeling of mi-crosphere contactimpact is important in predicting particle de-position on and rebound from surfaces The results can be usedin contaminationmonitoring and control especially in environ-mental particle sampling and collection On the other hand theissue of microsphere contactimpact is closely related to ad-vances in micro-scale contact mechanics for example in theunderstanding and modeling of various surface forces The re-sults can be useful in an even wider range of areas Microsphere

Received 12 September 2001 accepted 14 March 2002This research was supported in part during its early phase by the

Center for Indoor Air Research (contract 96-06) and the Electric PowerResearch Institute (contract RP 8034-03)

Address correspondence to Patrick F Dunn 107 Hessert CenterNotre Dame IN 46556 E-mail patrickfdunn1undedu

contact with at surfaces relates to the interpretation of the out-put of surface force apparatus such as Atomic Force Micro-scope (AFM) (Burnham et al 1997) More recent applicationsare found in the area of Micro Electro Mechanical Systems(MEMS) and nano-technologywhich involvesmicrometer- andnanometer-sized components (Rollot et al 1999) Also the mi-crosphere contactimpact model is fundamental in granular owand geomechanics modeling The Hertzian elastic contact the-ory for spheres is used with discrete element approaches for theconstitutive model of granular materials Fundamental aspectsin the modeling of discrete element systems are summarized byCundall and Hart (1992) More applicationscan be easily foundfor single and multimicrosphere contactimpact models

Classical studies on the contactimpact problem may be di-vided into 3 chronological stages In the rst stage modelswere restricted to rigid bodies The basic ideas were based onNewtonrsquos third law and Coulombrsquos friction law for rigid bod-ies Some of the results are still used widely today In the sec-ond stage local elasticity over the contact area was includedThe geometry and deformation of a contact body was assumedsuch that a closed-form solution could be found by using theavailable mathematical and mechanical tools Application ofHertzrsquos work on the static contact problem of 2 spheres is re-garded as a milestone in the eld (Hertz 1881 1882) Followingthis work many researchers studied contact problems betweenelastic bodies of different shapes under different circumstanceswith or without friction One of the leading contributions is theJohnson-Kendall-Roberts (JKR) model (Johnson 1985) Exam-ples can also be found in Goldsmith (1960) and Gladwell (1980)The third stage is numerical simulationComputationalmethodsincluding nite-elementmethods are widely used in simulationsin contact mechanics in which contact bodies are discretized asthe collection of nite elements More complicated geometryboundary conditions and loading can be considered The con-tact bodies may deform in an arbitrary way Examples can befound in Zhong (1993) and Quesnel et al (1998) Except forthe advantages of third-stage methods the modeling of surfaceinteractive forces such as adhesion and friction on the contact

1045

1046 W CHENG ET AL

boundary still depends on the results of the second stage Theroughness of the nominal smooth surface is still modeled by thecontact of microspherical asperities with or without friction andadhesion The sphere contactimpact model remains the funda-mental case for interface analysis

Most studies have focused on the normal attraction actingbetween solid surfaces (detailed explanations can be found inIsraelachvili and Tabor (1972)) The theoretical adhesive con-tact area of spherical surfaces under elastic deformation is de-rived by Johnson et al (1971) and Derjaguin et al (1975) Thetangential friction problem for obliquecontactimpact still is notfully solved Some effort has been made to model the tangen-tial adhesion contact to extend the JKR model for the case oftangential loading (Savkoor and Briggs 1977 Thornton 1991)Criteria for tangential separation have been proposed includingthat a tangential force is capable of causing normal separationofthe surface which is called ldquopeelingrdquo Skinner and Gane (1972)and Ando et al (1995) measured the relationship between thedry friction and the normal attraction forces between surfacesunder micro-loads when tangential motion occurs Several im-pact models have been developed recently to predict the obliquerebound velocity of the microsphere (Brach and Dunn 1995Dahneke1975Wall et al 1990Tsai et al 1991Xu and Willeke1993)

Experiments and numerical simulations have focused on theaccurate prediction of the rebound velocities of spheres in pla-nar motion However three-dimensional (3D degrees of free-dom) measurements and simulationshave not been reportedYetthe full nonlinear nature of the problem can only be visualizedthrough 3D simulations with a wide range of initial conditions3D simulationsre ect the propertiesof the simulationmodel andgenerate useful informationneeded to improve understandingofcontactimpact problems Another reason for 3D simulations isthat in reality microparticle impacts are almost always 3D with 6degrees of freedom In this paper a 3D simulation model for mi-crosphere impactwith a at surface is developedThen an exper-imental data tting method is introduced to determine the damp-ing coef cients required as known parameters in the simulation

The focus of this paper is the oblique impact of microspheresHowever some aspects of dynamics of established contact arealso covered The model for friction and rolling resistance inthis paper is different from those used by other researchers suchas Wang (1990) and Ziskind et al (1997) Some existing mod-els of rolling are fundamentally limited when external or inertiaforces are not included in the balance of moments leading toincomplete moments for rolling In the current simulation thesurfaces of the microsphere and the substrate are considered tobe smooth Even though all surfaces are rough at the micro-scopic level there are compelling reasons to assume that thesurfaces are smooth In reality roughness has scale levels In or-der to keepa nominal at geometry roughnessmust have a muchsmaller wavelength and height than the curvature or the radiusof the contact area Any surface irregularity with approximatelythe same order in height and wavelength as the contact area

is considered as a large geometry problem and contradicts theassumptionofa single sphere Hertziancontactwith a at surfaceA result of this reasoning is that no moment is generated from thenormal contactAnotherdifference relates to the tangential load-ing or tangential inertia force In Ziskind et al (1997) becausethe tangential loading is assumed to be much smaller than thedry friction force (in other words sliding does not occur) thetangential contact was modeled as a linear elastic spring Forimpact there is little or no experimental evidence to indicatethat tangential elasticity plays a signi cant role On the otherhand ample evidence exists (Brach et al 2000) that a tangentialcontact force composed primarily of dry friction is appropriate

Some argue that in particle removal when external forcessuch as gravity are weak an energy method based on vibrationalresonance may be used This may not be a productive approachbecause the concept of resonance is linear whereas Hertziancontact and sphere vibration is a nonlinear phenomenon Theapproach in this paper considers the nonlinear properties of thenormalcontactthroughBrach and Dunnrsquos modelThe possibilityof resonance is discussed

Several cases of 3D oblique impact are simulated to identifythe effects of changes in initial conditionsand surface friction onthe energy dissipation and rebound velocitiesThe sensitivityofparameters and some potential sources of errors in experimentsare identi ed through the numerical simulations The resultsshow that 3D simulation and nonlinear analysis are importanttools to understand the mechanism of particle impact and depo-sition with surfaces

SIMULATION MODELThe current simulation model was proposed by Brach and

Dunn (1995) and was based on Hertzian contact theory underadhesion loadingFigure 1 shows the coordinate system pertain-ing to the 3D impact between a microsphere and a at surfacewhere Ccediln lt 0 initially Typically the contact radius a is smallcompared to the sphere radius r and the radius of curvatureof the contact area Because the particle is small the elasticforce and the adhesion force are the dominant forces Gravity isnegligible and not considered Energy is dissipated through thematerial and adhesion damping

According to Newtonrsquos second law the equation of normalmotion is

mn D FH C FH D C FA C FAD [1]

The rst term on the right side of Equation (1) is the Hertzianelastic force FH It is calculated from the Hertzian theory

FH Dp

r K (iexcln)3=2 n middot 0 [2]

where n is the relative displacement of the mass center andn middot 0 during contact The original radius of the microsphere isr The effective material stiffness is K which is de ned by thefollowing equations

K D4

3frac14 (k1 C k2) [3]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1047

Figure 1 Schematic of a sphere in contact with a at surface

and

ki Diexcl1 iexcl ordm2

i

cent

frac14 Ei i D 1 2 [4]

where ordmi and Ei are Poissonrsquos ratio and Youngrsquos modulus re-spectively for the sphere and the substrate material The adhe-sion force FA in Equation (1) is modeled as a ring force actingon the periphery of the circular contact area where the intensityof the adhesion ring force is f0

FA D iexcl2frac14a f0 [5]

As predicted by Hertzian theory the contact radius a is

a Dp

iexclrn [6]

The calculation of the intensity of the adhesion ring force f0uses a relationship between Brach and Dunnrsquos model and theJKR model It is assumed that both models produce the samecontact radius at the equilibrium position Thus the followingequation was used to calculate f0 (Li et al 1999)

f0 Dsup3

92frac14

Krw2A

acute1=3

[7]

In Equation (7) wA is the combined surface energy between thesphere and the surface materials wA can be obtained throughDuprersquos equation

wA D F1 C F2 iexcl F12 [8]

where F1 and F2 are surface free energies of body 1 and body 2respectively and F12 is the interfacial free energy In reality thesurface tensions of the 2 bodies deg1 and deg2 and the interfacialtension deg12 are used instead

wA D deg1 C deg2 iexcl deg12 [9]

The interfacial tension can be estimated by

deg12 D deg1 C deg2 iexcl 2812(deg1deg2)1=2 [10]

Note that the interaction parameter 812 is not exactly at unityPractically however it is near unity The uncertainty in using812 D 1 is lt2 This small correction is a second-order effectthat is negligible in the current model Thus the surface energycan be estimated as follows

waraquoD 2

pdeg1deg2 [11]

The values of f0 for different cases used in the simulationare listed in Table 1 Dynamic damping is included using rst-power velocity dependence for the material damping FH D andthe adhesion damping FAD

FH D D iexclFH CH Ccediln [12]

and

FAD D FACA Ccediln [13]

The damping coef cients CH and CA can be put into

1048 W CHENG ET AL

Table 1The basic parameters for numerical simulation

DensityReference r (sup1m) (kgm3) K (pound109 Pa) f0(N=m) sup3A sup3H f Descriptions

Wall et al (1990) 129ndash345 1350 1776 5340ndash7390 290 050 015 Figure 3Wall et al (1990) 345 1350 1776 7422 290 050 015 Figures 5 7ndash12Wall et al (1990) 345 1350 1776 7422 290 050 010ndash090 Figures 13ndash14

nondimensional forms

sup3A D CA

sup3r 3K

pvn

m

acute2=5

[14]

sup3H D CH

sup3r3 K

pvn

m

acute2=5

[15]

where vn is a reference velocity typically the initial normal ve-locityThe nondimensionaldampingcoef cients are determinedthrough an experimentaldata tting procedure that will be intro-duced in the following section Two tangential directions in theCartesian coordinate system t and t 0 are considered to includethe ldquooff-planerdquo spin and sliding in the simulation The equationof motion in the t direction is

mt D Ft [16]

The equation of motion in the t 0 direction is

mt 0 D Ft 0 [17]

A form of Coulomb friction with a coef cient f is used for thetangential friction At the beginning and the end stage of con-tact the normal force is negative due to the fact that for smalldisplacement n the adhesion force is larger than the Hertzianelastic force Thus a friction for ldquopullingrdquo force contact has tobe modeled In the current simulation model an assumption ismade that the contact is tangentially rigid and the tangential de-formation is neglected The tangential friction force is assumedto be proportional to the magnitude of the ldquopullingrdquo force basedon the fact that in the ldquosnap-onrdquo and ldquosnap-offrdquo procedure theparticle is in ldquocontactrdquo with the substrate surface and the fric-tion still exists due to the relative motion at the interface Thisfriction model agrees with the measurement results of Skinnerand Gane (1972) The frictional force has components Ft andFt 0 that oppose the relative tangential contact velocity and areproportional to the resultant normal force Fn Except for an iso-lated special case the initial relative tangential contact velocityis nonzero for oblique microsphere impacts This means that afriction force exists initially If during contact friction causessliding to end a sphere continues to move under the conditionof pure rolling and sliding cannot reoccur

It shouldbe mentioned that the behaviorof tangential frictionin the presence of adhesion is still not fully understood particu-

larly when tangential elastic deformation is signi cant An elas-tic model and an extension of the JKR model in the tangentialdirection have been discussed (Ziskind et al 2000 Savkoor andBriggs 1977 Thornton 1991) However for impact problemsin which the initial relative tangential contact velocity is largeseparation likely occurs over the whole contact area The tran-sition from shear traction to sliding friction can be neglectedThe model is tangentially rigid in the current simulation Eventhough this condition is simpli ed it appears suitable for thesimulation of impact problems

As a result of the tangential friction moments torque androtation are also produced about the mass center The micro-sphere is considered as a rigid body in modeling the spin androtation The spin about the normal axis n is described in thefollowing rigid-body equation

I sbquon D 2frac14

Z a

0f jpnjfrac122dfrac12 D iexcl

75frac14 f aFn

64r 2 [18]

where I is the centroidal moment of inertia of the sphere and pn

is thenormal contactpressure over the contactareaThe torsionalmoment caused by friction in Equation (18) is calculated by theintegration of Coulomb friction about the center of the contactarea The equations of rotation about the tangential axes are thefollowing

I sbquot D iexclr Ft 0 [19]

I sbquot 0 D r Ft [20]

In order to describe and compare the simulation results sev-eral coef cients are de ned The restitution coef cient en isthe ratio of the magnitude of rebound (subscript r ) to initial(subscript i ) normal (subscript n) velocities

en Dvrn

vin [21]

The tangential impulse ratios sup1t and sup1t 0 are de ned as the ratioof tangential impulse to normal impulse after impact in t and t 0

directions

sup1t Dvtr iexcl vti

vnr iexcl vni

[22]

sup1t 0 Dvt 0

riexcl vt 0

i

vnr iexcl vni

[23]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1049

In the absence of adhesion the upper limit of the tangential im-pulse ratio is the Coulomb friction coef cient f However inthe current nonlinearmodel when the ldquosnap-onrdquo and ldquosnap-offrdquoperiod is long the tangential impulse ratio can be larger than f The normalized total kinetic energy loss coef cient K t is thepercentage of total energy loss (total energy means the summa-tion of translational and rotational kinetic energy) after impact

K t Dm

iexclv2

i iexcl v2r

centC I

iexcl2

i iexcl 2r

centiexclmv2

i C I 2i

cent [24]

The normalized translationalkinetic energy loss coef cient KL is de ned as the ratio of translational kinetic energy loss to theinitial translational kinetic energy

KL Dv2

i iexcl v2r

v2i

[25]

A reason for de ning 2 different kinetic energy loss termsis because experimentally the rotational kinematics typicallyare not measured Consequently experimental kinetic energyloss values reported in the past correspond to Equation (25)yet the total kinetic energy loss is given by Equation (24) Thesimulationallows both to be determined(Brach and Dunn 1996)

Figure 2 Schematic for the analysis of the rolling resistance

ROLLING RESISTANCEThe equations for rolling in the current simulation (Equa-

tions (19) and (20)) are different from those used by other re-searchers It is suggested (for example see Wang (1990)) that amoment or couple due to adhesion over the contact area pro-duces a resistance to rotation The magnitude of this resistanceis estimated as the product of a ldquopull-offrdquo force and the con-tact radius Although this theory has been commonly accepted(Ziskind et al 1997) it can overestimate rolling resistance sig-ni cantly Figure 2 illustrates schematically the conditions thatoccur when a microsphere in contact with a surface rolls inthe presence of adhesion Both the Hertzian and adhesion con-tact forces develop an asymmetry causing their lines of ac-tion to move away from the normal axis n In the case of theadhesion force this asymmetry is due to the leading edge ofthe contact area establishing new contact and the trailing edgepulling away This causes a moment Mt 0 that resists rollingwhere

Mt 0 D xa Fa C xh Fh [26]

The force Fa includes the adhesion force FA and the corre-sponding adhesion damping FAD and the force Fh includes theHertzian force FH and the corresponding Hertzian damping

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1046 W CHENG ET AL

boundary still depends on the results of the second stage Theroughness of the nominal smooth surface is still modeled by thecontact of microspherical asperities with or without friction andadhesion The sphere contactimpact model remains the funda-mental case for interface analysis

Most studies have focused on the normal attraction actingbetween solid surfaces (detailed explanations can be found inIsraelachvili and Tabor (1972)) The theoretical adhesive con-tact area of spherical surfaces under elastic deformation is de-rived by Johnson et al (1971) and Derjaguin et al (1975) Thetangential friction problem for obliquecontactimpact still is notfully solved Some effort has been made to model the tangen-tial adhesion contact to extend the JKR model for the case oftangential loading (Savkoor and Briggs 1977 Thornton 1991)Criteria for tangential separation have been proposed includingthat a tangential force is capable of causing normal separationofthe surface which is called ldquopeelingrdquo Skinner and Gane (1972)and Ando et al (1995) measured the relationship between thedry friction and the normal attraction forces between surfacesunder micro-loads when tangential motion occurs Several im-pact models have been developed recently to predict the obliquerebound velocity of the microsphere (Brach and Dunn 1995Dahneke1975Wall et al 1990Tsai et al 1991Xu and Willeke1993)

Experiments and numerical simulations have focused on theaccurate prediction of the rebound velocities of spheres in pla-nar motion However three-dimensional (3D degrees of free-dom) measurements and simulationshave not been reportedYetthe full nonlinear nature of the problem can only be visualizedthrough 3D simulations with a wide range of initial conditions3D simulationsre ect the propertiesof the simulationmodel andgenerate useful informationneeded to improve understandingofcontactimpact problems Another reason for 3D simulations isthat in reality microparticle impacts are almost always 3D with 6degrees of freedom In this paper a 3D simulation model for mi-crosphere impactwith a at surface is developedThen an exper-imental data tting method is introduced to determine the damp-ing coef cients required as known parameters in the simulation

The focus of this paper is the oblique impact of microspheresHowever some aspects of dynamics of established contact arealso covered The model for friction and rolling resistance inthis paper is different from those used by other researchers suchas Wang (1990) and Ziskind et al (1997) Some existing mod-els of rolling are fundamentally limited when external or inertiaforces are not included in the balance of moments leading toincomplete moments for rolling In the current simulation thesurfaces of the microsphere and the substrate are considered tobe smooth Even though all surfaces are rough at the micro-scopic level there are compelling reasons to assume that thesurfaces are smooth In reality roughness has scale levels In or-der to keepa nominal at geometry roughnessmust have a muchsmaller wavelength and height than the curvature or the radiusof the contact area Any surface irregularity with approximatelythe same order in height and wavelength as the contact area

is considered as a large geometry problem and contradicts theassumptionofa single sphere Hertziancontactwith a at surfaceA result of this reasoning is that no moment is generated from thenormal contactAnotherdifference relates to the tangential load-ing or tangential inertia force In Ziskind et al (1997) becausethe tangential loading is assumed to be much smaller than thedry friction force (in other words sliding does not occur) thetangential contact was modeled as a linear elastic spring Forimpact there is little or no experimental evidence to indicatethat tangential elasticity plays a signi cant role On the otherhand ample evidence exists (Brach et al 2000) that a tangentialcontact force composed primarily of dry friction is appropriate

Some argue that in particle removal when external forcessuch as gravity are weak an energy method based on vibrationalresonance may be used This may not be a productive approachbecause the concept of resonance is linear whereas Hertziancontact and sphere vibration is a nonlinear phenomenon Theapproach in this paper considers the nonlinear properties of thenormalcontactthroughBrach and Dunnrsquos modelThe possibilityof resonance is discussed

Several cases of 3D oblique impact are simulated to identifythe effects of changes in initial conditionsand surface friction onthe energy dissipation and rebound velocitiesThe sensitivityofparameters and some potential sources of errors in experimentsare identi ed through the numerical simulations The resultsshow that 3D simulation and nonlinear analysis are importanttools to understand the mechanism of particle impact and depo-sition with surfaces

SIMULATION MODELThe current simulation model was proposed by Brach and

Dunn (1995) and was based on Hertzian contact theory underadhesion loadingFigure 1 shows the coordinate system pertain-ing to the 3D impact between a microsphere and a at surfacewhere Ccediln lt 0 initially Typically the contact radius a is smallcompared to the sphere radius r and the radius of curvatureof the contact area Because the particle is small the elasticforce and the adhesion force are the dominant forces Gravity isnegligible and not considered Energy is dissipated through thematerial and adhesion damping

According to Newtonrsquos second law the equation of normalmotion is

mn D FH C FH D C FA C FAD [1]

The rst term on the right side of Equation (1) is the Hertzianelastic force FH It is calculated from the Hertzian theory

FH Dp

r K (iexcln)3=2 n middot 0 [2]

where n is the relative displacement of the mass center andn middot 0 during contact The original radius of the microsphere isr The effective material stiffness is K which is de ned by thefollowing equations

K D4

3frac14 (k1 C k2) [3]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1047

Figure 1 Schematic of a sphere in contact with a at surface

and

ki Diexcl1 iexcl ordm2

i

cent

frac14 Ei i D 1 2 [4]

where ordmi and Ei are Poissonrsquos ratio and Youngrsquos modulus re-spectively for the sphere and the substrate material The adhe-sion force FA in Equation (1) is modeled as a ring force actingon the periphery of the circular contact area where the intensityof the adhesion ring force is f0

FA D iexcl2frac14a f0 [5]

As predicted by Hertzian theory the contact radius a is

a Dp

iexclrn [6]

The calculation of the intensity of the adhesion ring force f0uses a relationship between Brach and Dunnrsquos model and theJKR model It is assumed that both models produce the samecontact radius at the equilibrium position Thus the followingequation was used to calculate f0 (Li et al 1999)

f0 Dsup3

92frac14

Krw2A

acute1=3

[7]

In Equation (7) wA is the combined surface energy between thesphere and the surface materials wA can be obtained throughDuprersquos equation

wA D F1 C F2 iexcl F12 [8]

where F1 and F2 are surface free energies of body 1 and body 2respectively and F12 is the interfacial free energy In reality thesurface tensions of the 2 bodies deg1 and deg2 and the interfacialtension deg12 are used instead

wA D deg1 C deg2 iexcl deg12 [9]

The interfacial tension can be estimated by

deg12 D deg1 C deg2 iexcl 2812(deg1deg2)1=2 [10]

Note that the interaction parameter 812 is not exactly at unityPractically however it is near unity The uncertainty in using812 D 1 is lt2 This small correction is a second-order effectthat is negligible in the current model Thus the surface energycan be estimated as follows

waraquoD 2

pdeg1deg2 [11]

The values of f0 for different cases used in the simulationare listed in Table 1 Dynamic damping is included using rst-power velocity dependence for the material damping FH D andthe adhesion damping FAD

FH D D iexclFH CH Ccediln [12]

and

FAD D FACA Ccediln [13]

The damping coef cients CH and CA can be put into

1048 W CHENG ET AL

Table 1The basic parameters for numerical simulation

DensityReference r (sup1m) (kgm3) K (pound109 Pa) f0(N=m) sup3A sup3H f Descriptions

Wall et al (1990) 129ndash345 1350 1776 5340ndash7390 290 050 015 Figure 3Wall et al (1990) 345 1350 1776 7422 290 050 015 Figures 5 7ndash12Wall et al (1990) 345 1350 1776 7422 290 050 010ndash090 Figures 13ndash14

nondimensional forms

sup3A D CA

sup3r 3K

pvn

m

acute2=5

[14]

sup3H D CH

sup3r3 K

pvn

m

acute2=5

[15]

where vn is a reference velocity typically the initial normal ve-locityThe nondimensionaldampingcoef cients are determinedthrough an experimentaldata tting procedure that will be intro-duced in the following section Two tangential directions in theCartesian coordinate system t and t 0 are considered to includethe ldquooff-planerdquo spin and sliding in the simulation The equationof motion in the t direction is

mt D Ft [16]

The equation of motion in the t 0 direction is

mt 0 D Ft 0 [17]

A form of Coulomb friction with a coef cient f is used for thetangential friction At the beginning and the end stage of con-tact the normal force is negative due to the fact that for smalldisplacement n the adhesion force is larger than the Hertzianelastic force Thus a friction for ldquopullingrdquo force contact has tobe modeled In the current simulation model an assumption ismade that the contact is tangentially rigid and the tangential de-formation is neglected The tangential friction force is assumedto be proportional to the magnitude of the ldquopullingrdquo force basedon the fact that in the ldquosnap-onrdquo and ldquosnap-offrdquo procedure theparticle is in ldquocontactrdquo with the substrate surface and the fric-tion still exists due to the relative motion at the interface Thisfriction model agrees with the measurement results of Skinnerand Gane (1972) The frictional force has components Ft andFt 0 that oppose the relative tangential contact velocity and areproportional to the resultant normal force Fn Except for an iso-lated special case the initial relative tangential contact velocityis nonzero for oblique microsphere impacts This means that afriction force exists initially If during contact friction causessliding to end a sphere continues to move under the conditionof pure rolling and sliding cannot reoccur

It shouldbe mentioned that the behaviorof tangential frictionin the presence of adhesion is still not fully understood particu-

larly when tangential elastic deformation is signi cant An elas-tic model and an extension of the JKR model in the tangentialdirection have been discussed (Ziskind et al 2000 Savkoor andBriggs 1977 Thornton 1991) However for impact problemsin which the initial relative tangential contact velocity is largeseparation likely occurs over the whole contact area The tran-sition from shear traction to sliding friction can be neglectedThe model is tangentially rigid in the current simulation Eventhough this condition is simpli ed it appears suitable for thesimulation of impact problems

As a result of the tangential friction moments torque androtation are also produced about the mass center The micro-sphere is considered as a rigid body in modeling the spin androtation The spin about the normal axis n is described in thefollowing rigid-body equation

I sbquon D 2frac14

Z a

0f jpnjfrac122dfrac12 D iexcl

75frac14 f aFn

64r 2 [18]

where I is the centroidal moment of inertia of the sphere and pn

is thenormal contactpressure over the contactareaThe torsionalmoment caused by friction in Equation (18) is calculated by theintegration of Coulomb friction about the center of the contactarea The equations of rotation about the tangential axes are thefollowing

I sbquot D iexclr Ft 0 [19]

I sbquot 0 D r Ft [20]

In order to describe and compare the simulation results sev-eral coef cients are de ned The restitution coef cient en isthe ratio of the magnitude of rebound (subscript r ) to initial(subscript i ) normal (subscript n) velocities

en Dvrn

vin [21]

The tangential impulse ratios sup1t and sup1t 0 are de ned as the ratioof tangential impulse to normal impulse after impact in t and t 0

directions

sup1t Dvtr iexcl vti

vnr iexcl vni

[22]

sup1t 0 Dvt 0

riexcl vt 0

i

vnr iexcl vni

[23]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1049

In the absence of adhesion the upper limit of the tangential im-pulse ratio is the Coulomb friction coef cient f However inthe current nonlinearmodel when the ldquosnap-onrdquo and ldquosnap-offrdquoperiod is long the tangential impulse ratio can be larger than f The normalized total kinetic energy loss coef cient K t is thepercentage of total energy loss (total energy means the summa-tion of translational and rotational kinetic energy) after impact

K t Dm

iexclv2

i iexcl v2r

centC I

iexcl2

i iexcl 2r

centiexclmv2

i C I 2i

cent [24]

The normalized translationalkinetic energy loss coef cient KL is de ned as the ratio of translational kinetic energy loss to theinitial translational kinetic energy

KL Dv2

i iexcl v2r

v2i

[25]

A reason for de ning 2 different kinetic energy loss termsis because experimentally the rotational kinematics typicallyare not measured Consequently experimental kinetic energyloss values reported in the past correspond to Equation (25)yet the total kinetic energy loss is given by Equation (24) Thesimulationallows both to be determined(Brach and Dunn 1996)

Figure 2 Schematic for the analysis of the rolling resistance

ROLLING RESISTANCEThe equations for rolling in the current simulation (Equa-

tions (19) and (20)) are different from those used by other re-searchers It is suggested (for example see Wang (1990)) that amoment or couple due to adhesion over the contact area pro-duces a resistance to rotation The magnitude of this resistanceis estimated as the product of a ldquopull-offrdquo force and the con-tact radius Although this theory has been commonly accepted(Ziskind et al 1997) it can overestimate rolling resistance sig-ni cantly Figure 2 illustrates schematically the conditions thatoccur when a microsphere in contact with a surface rolls inthe presence of adhesion Both the Hertzian and adhesion con-tact forces develop an asymmetry causing their lines of ac-tion to move away from the normal axis n In the case of theadhesion force this asymmetry is due to the leading edge ofthe contact area establishing new contact and the trailing edgepulling away This causes a moment Mt 0 that resists rollingwhere

Mt 0 D xa Fa C xh Fh [26]

The force Fa includes the adhesion force FA and the corre-sponding adhesion damping FAD and the force Fh includes theHertzian force FH and the corresponding Hertzian damping

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1047

Figure 1 Schematic of a sphere in contact with a at surface

and

ki Diexcl1 iexcl ordm2

i

cent

frac14 Ei i D 1 2 [4]

where ordmi and Ei are Poissonrsquos ratio and Youngrsquos modulus re-spectively for the sphere and the substrate material The adhe-sion force FA in Equation (1) is modeled as a ring force actingon the periphery of the circular contact area where the intensityof the adhesion ring force is f0

FA D iexcl2frac14a f0 [5]

As predicted by Hertzian theory the contact radius a is

a Dp

iexclrn [6]

The calculation of the intensity of the adhesion ring force f0uses a relationship between Brach and Dunnrsquos model and theJKR model It is assumed that both models produce the samecontact radius at the equilibrium position Thus the followingequation was used to calculate f0 (Li et al 1999)

f0 Dsup3

92frac14

Krw2A

acute1=3

[7]

In Equation (7) wA is the combined surface energy between thesphere and the surface materials wA can be obtained throughDuprersquos equation

wA D F1 C F2 iexcl F12 [8]

where F1 and F2 are surface free energies of body 1 and body 2respectively and F12 is the interfacial free energy In reality thesurface tensions of the 2 bodies deg1 and deg2 and the interfacialtension deg12 are used instead

wA D deg1 C deg2 iexcl deg12 [9]

The interfacial tension can be estimated by

deg12 D deg1 C deg2 iexcl 2812(deg1deg2)1=2 [10]

Note that the interaction parameter 812 is not exactly at unityPractically however it is near unity The uncertainty in using812 D 1 is lt2 This small correction is a second-order effectthat is negligible in the current model Thus the surface energycan be estimated as follows

waraquoD 2

pdeg1deg2 [11]

The values of f0 for different cases used in the simulationare listed in Table 1 Dynamic damping is included using rst-power velocity dependence for the material damping FH D andthe adhesion damping FAD

FH D D iexclFH CH Ccediln [12]

and

FAD D FACA Ccediln [13]

The damping coef cients CH and CA can be put into

1048 W CHENG ET AL

Table 1The basic parameters for numerical simulation

DensityReference r (sup1m) (kgm3) K (pound109 Pa) f0(N=m) sup3A sup3H f Descriptions

Wall et al (1990) 129ndash345 1350 1776 5340ndash7390 290 050 015 Figure 3Wall et al (1990) 345 1350 1776 7422 290 050 015 Figures 5 7ndash12Wall et al (1990) 345 1350 1776 7422 290 050 010ndash090 Figures 13ndash14

nondimensional forms

sup3A D CA

sup3r 3K

pvn

m

acute2=5

[14]

sup3H D CH

sup3r3 K

pvn

m

acute2=5

[15]

where vn is a reference velocity typically the initial normal ve-locityThe nondimensionaldampingcoef cients are determinedthrough an experimentaldata tting procedure that will be intro-duced in the following section Two tangential directions in theCartesian coordinate system t and t 0 are considered to includethe ldquooff-planerdquo spin and sliding in the simulation The equationof motion in the t direction is

mt D Ft [16]

The equation of motion in the t 0 direction is

mt 0 D Ft 0 [17]

A form of Coulomb friction with a coef cient f is used for thetangential friction At the beginning and the end stage of con-tact the normal force is negative due to the fact that for smalldisplacement n the adhesion force is larger than the Hertzianelastic force Thus a friction for ldquopullingrdquo force contact has tobe modeled In the current simulation model an assumption ismade that the contact is tangentially rigid and the tangential de-formation is neglected The tangential friction force is assumedto be proportional to the magnitude of the ldquopullingrdquo force basedon the fact that in the ldquosnap-onrdquo and ldquosnap-offrdquo procedure theparticle is in ldquocontactrdquo with the substrate surface and the fric-tion still exists due to the relative motion at the interface Thisfriction model agrees with the measurement results of Skinnerand Gane (1972) The frictional force has components Ft andFt 0 that oppose the relative tangential contact velocity and areproportional to the resultant normal force Fn Except for an iso-lated special case the initial relative tangential contact velocityis nonzero for oblique microsphere impacts This means that afriction force exists initially If during contact friction causessliding to end a sphere continues to move under the conditionof pure rolling and sliding cannot reoccur

It shouldbe mentioned that the behaviorof tangential frictionin the presence of adhesion is still not fully understood particu-

larly when tangential elastic deformation is signi cant An elas-tic model and an extension of the JKR model in the tangentialdirection have been discussed (Ziskind et al 2000 Savkoor andBriggs 1977 Thornton 1991) However for impact problemsin which the initial relative tangential contact velocity is largeseparation likely occurs over the whole contact area The tran-sition from shear traction to sliding friction can be neglectedThe model is tangentially rigid in the current simulation Eventhough this condition is simpli ed it appears suitable for thesimulation of impact problems

As a result of the tangential friction moments torque androtation are also produced about the mass center The micro-sphere is considered as a rigid body in modeling the spin androtation The spin about the normal axis n is described in thefollowing rigid-body equation

I sbquon D 2frac14

Z a

0f jpnjfrac122dfrac12 D iexcl

75frac14 f aFn

64r 2 [18]

where I is the centroidal moment of inertia of the sphere and pn

is thenormal contactpressure over the contactareaThe torsionalmoment caused by friction in Equation (18) is calculated by theintegration of Coulomb friction about the center of the contactarea The equations of rotation about the tangential axes are thefollowing

I sbquot D iexclr Ft 0 [19]

I sbquot 0 D r Ft [20]

In order to describe and compare the simulation results sev-eral coef cients are de ned The restitution coef cient en isthe ratio of the magnitude of rebound (subscript r ) to initial(subscript i ) normal (subscript n) velocities

en Dvrn

vin [21]

The tangential impulse ratios sup1t and sup1t 0 are de ned as the ratioof tangential impulse to normal impulse after impact in t and t 0

directions

sup1t Dvtr iexcl vti

vnr iexcl vni

[22]

sup1t 0 Dvt 0

riexcl vt 0

i

vnr iexcl vni

[23]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1049

In the absence of adhesion the upper limit of the tangential im-pulse ratio is the Coulomb friction coef cient f However inthe current nonlinearmodel when the ldquosnap-onrdquo and ldquosnap-offrdquoperiod is long the tangential impulse ratio can be larger than f The normalized total kinetic energy loss coef cient K t is thepercentage of total energy loss (total energy means the summa-tion of translational and rotational kinetic energy) after impact

K t Dm

iexclv2

i iexcl v2r

centC I

iexcl2

i iexcl 2r

centiexclmv2

i C I 2i

cent [24]

The normalized translationalkinetic energy loss coef cient KL is de ned as the ratio of translational kinetic energy loss to theinitial translational kinetic energy

KL Dv2

i iexcl v2r

v2i

[25]

A reason for de ning 2 different kinetic energy loss termsis because experimentally the rotational kinematics typicallyare not measured Consequently experimental kinetic energyloss values reported in the past correspond to Equation (25)yet the total kinetic energy loss is given by Equation (24) Thesimulationallows both to be determined(Brach and Dunn 1996)

Figure 2 Schematic for the analysis of the rolling resistance

ROLLING RESISTANCEThe equations for rolling in the current simulation (Equa-

tions (19) and (20)) are different from those used by other re-searchers It is suggested (for example see Wang (1990)) that amoment or couple due to adhesion over the contact area pro-duces a resistance to rotation The magnitude of this resistanceis estimated as the product of a ldquopull-offrdquo force and the con-tact radius Although this theory has been commonly accepted(Ziskind et al 1997) it can overestimate rolling resistance sig-ni cantly Figure 2 illustrates schematically the conditions thatoccur when a microsphere in contact with a surface rolls inthe presence of adhesion Both the Hertzian and adhesion con-tact forces develop an asymmetry causing their lines of ac-tion to move away from the normal axis n In the case of theadhesion force this asymmetry is due to the leading edge ofthe contact area establishing new contact and the trailing edgepulling away This causes a moment Mt 0 that resists rollingwhere

Mt 0 D xa Fa C xh Fh [26]

The force Fa includes the adhesion force FA and the corre-sponding adhesion damping FAD and the force Fh includes theHertzian force FH and the corresponding Hertzian damping

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1048 W CHENG ET AL

Table 1The basic parameters for numerical simulation

DensityReference r (sup1m) (kgm3) K (pound109 Pa) f0(N=m) sup3A sup3H f Descriptions

Wall et al (1990) 129ndash345 1350 1776 5340ndash7390 290 050 015 Figure 3Wall et al (1990) 345 1350 1776 7422 290 050 015 Figures 5 7ndash12Wall et al (1990) 345 1350 1776 7422 290 050 010ndash090 Figures 13ndash14

nondimensional forms

sup3A D CA

sup3r 3K

pvn

m

acute2=5

[14]

sup3H D CH

sup3r3 K

pvn

m

acute2=5

[15]

where vn is a reference velocity typically the initial normal ve-locityThe nondimensionaldampingcoef cients are determinedthrough an experimentaldata tting procedure that will be intro-duced in the following section Two tangential directions in theCartesian coordinate system t and t 0 are considered to includethe ldquooff-planerdquo spin and sliding in the simulation The equationof motion in the t direction is

mt D Ft [16]

The equation of motion in the t 0 direction is

mt 0 D Ft 0 [17]

A form of Coulomb friction with a coef cient f is used for thetangential friction At the beginning and the end stage of con-tact the normal force is negative due to the fact that for smalldisplacement n the adhesion force is larger than the Hertzianelastic force Thus a friction for ldquopullingrdquo force contact has tobe modeled In the current simulation model an assumption ismade that the contact is tangentially rigid and the tangential de-formation is neglected The tangential friction force is assumedto be proportional to the magnitude of the ldquopullingrdquo force basedon the fact that in the ldquosnap-onrdquo and ldquosnap-offrdquo procedure theparticle is in ldquocontactrdquo with the substrate surface and the fric-tion still exists due to the relative motion at the interface Thisfriction model agrees with the measurement results of Skinnerand Gane (1972) The frictional force has components Ft andFt 0 that oppose the relative tangential contact velocity and areproportional to the resultant normal force Fn Except for an iso-lated special case the initial relative tangential contact velocityis nonzero for oblique microsphere impacts This means that afriction force exists initially If during contact friction causessliding to end a sphere continues to move under the conditionof pure rolling and sliding cannot reoccur

It shouldbe mentioned that the behaviorof tangential frictionin the presence of adhesion is still not fully understood particu-

larly when tangential elastic deformation is signi cant An elas-tic model and an extension of the JKR model in the tangentialdirection have been discussed (Ziskind et al 2000 Savkoor andBriggs 1977 Thornton 1991) However for impact problemsin which the initial relative tangential contact velocity is largeseparation likely occurs over the whole contact area The tran-sition from shear traction to sliding friction can be neglectedThe model is tangentially rigid in the current simulation Eventhough this condition is simpli ed it appears suitable for thesimulation of impact problems

As a result of the tangential friction moments torque androtation are also produced about the mass center The micro-sphere is considered as a rigid body in modeling the spin androtation The spin about the normal axis n is described in thefollowing rigid-body equation

I sbquon D 2frac14

Z a

0f jpnjfrac122dfrac12 D iexcl

75frac14 f aFn

64r 2 [18]

where I is the centroidal moment of inertia of the sphere and pn

is thenormal contactpressure over the contactareaThe torsionalmoment caused by friction in Equation (18) is calculated by theintegration of Coulomb friction about the center of the contactarea The equations of rotation about the tangential axes are thefollowing

I sbquot D iexclr Ft 0 [19]

I sbquot 0 D r Ft [20]

In order to describe and compare the simulation results sev-eral coef cients are de ned The restitution coef cient en isthe ratio of the magnitude of rebound (subscript r ) to initial(subscript i ) normal (subscript n) velocities

en Dvrn

vin [21]

The tangential impulse ratios sup1t and sup1t 0 are de ned as the ratioof tangential impulse to normal impulse after impact in t and t 0

directions

sup1t Dvtr iexcl vti

vnr iexcl vni

[22]

sup1t 0 Dvt 0

riexcl vt 0

i

vnr iexcl vni

[23]

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1049

In the absence of adhesion the upper limit of the tangential im-pulse ratio is the Coulomb friction coef cient f However inthe current nonlinearmodel when the ldquosnap-onrdquo and ldquosnap-offrdquoperiod is long the tangential impulse ratio can be larger than f The normalized total kinetic energy loss coef cient K t is thepercentage of total energy loss (total energy means the summa-tion of translational and rotational kinetic energy) after impact

K t Dm

iexclv2

i iexcl v2r

centC I

iexcl2

i iexcl 2r

centiexclmv2

i C I 2i

cent [24]

The normalized translationalkinetic energy loss coef cient KL is de ned as the ratio of translational kinetic energy loss to theinitial translational kinetic energy

KL Dv2

i iexcl v2r

v2i

[25]

A reason for de ning 2 different kinetic energy loss termsis because experimentally the rotational kinematics typicallyare not measured Consequently experimental kinetic energyloss values reported in the past correspond to Equation (25)yet the total kinetic energy loss is given by Equation (24) Thesimulationallows both to be determined(Brach and Dunn 1996)

Figure 2 Schematic for the analysis of the rolling resistance

ROLLING RESISTANCEThe equations for rolling in the current simulation (Equa-

tions (19) and (20)) are different from those used by other re-searchers It is suggested (for example see Wang (1990)) that amoment or couple due to adhesion over the contact area pro-duces a resistance to rotation The magnitude of this resistanceis estimated as the product of a ldquopull-offrdquo force and the con-tact radius Although this theory has been commonly accepted(Ziskind et al 1997) it can overestimate rolling resistance sig-ni cantly Figure 2 illustrates schematically the conditions thatoccur when a microsphere in contact with a surface rolls inthe presence of adhesion Both the Hertzian and adhesion con-tact forces develop an asymmetry causing their lines of ac-tion to move away from the normal axis n In the case of theadhesion force this asymmetry is due to the leading edge ofthe contact area establishing new contact and the trailing edgepulling away This causes a moment Mt 0 that resists rollingwhere

Mt 0 D xa Fa C xh Fh [26]

The force Fa includes the adhesion force FA and the corre-sponding adhesion damping FAD and the force Fh includes theHertzian force FH and the corresponding Hertzian damping

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1049

In the absence of adhesion the upper limit of the tangential im-pulse ratio is the Coulomb friction coef cient f However inthe current nonlinearmodel when the ldquosnap-onrdquo and ldquosnap-offrdquoperiod is long the tangential impulse ratio can be larger than f The normalized total kinetic energy loss coef cient K t is thepercentage of total energy loss (total energy means the summa-tion of translational and rotational kinetic energy) after impact

K t Dm

iexclv2

i iexcl v2r

centC I

iexcl2

i iexcl 2r

centiexclmv2

i C I 2i

cent [24]

The normalized translationalkinetic energy loss coef cient KL is de ned as the ratio of translational kinetic energy loss to theinitial translational kinetic energy

KL Dv2

i iexcl v2r

v2i

[25]

A reason for de ning 2 different kinetic energy loss termsis because experimentally the rotational kinematics typicallyare not measured Consequently experimental kinetic energyloss values reported in the past correspond to Equation (25)yet the total kinetic energy loss is given by Equation (24) Thesimulationallows both to be determined(Brach and Dunn 1996)

Figure 2 Schematic for the analysis of the rolling resistance

ROLLING RESISTANCEThe equations for rolling in the current simulation (Equa-

tions (19) and (20)) are different from those used by other re-searchers It is suggested (for example see Wang (1990)) that amoment or couple due to adhesion over the contact area pro-duces a resistance to rotation The magnitude of this resistanceis estimated as the product of a ldquopull-offrdquo force and the con-tact radius Although this theory has been commonly accepted(Ziskind et al 1997) it can overestimate rolling resistance sig-ni cantly Figure 2 illustrates schematically the conditions thatoccur when a microsphere in contact with a surface rolls inthe presence of adhesion Both the Hertzian and adhesion con-tact forces develop an asymmetry causing their lines of ac-tion to move away from the normal axis n In the case of theadhesion force this asymmetry is due to the leading edge ofthe contact area establishing new contact and the trailing edgepulling away This causes a moment Mt 0 that resists rollingwhere

Mt 0 D xa Fa C xh Fh [26]

The force Fa includes the adhesion force FA and the corre-sponding adhesion damping FAD and the force Fh includes theHertzian force FH and the corresponding Hertzian damping

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1050 W CHENG ET AL

FH D The distances xa and xh are the offsets of the lines ofaction of the adhesion and Hertzian forces respectively Thisphenomenon was modeled for the process of impact by Brachet al (1999b) where it was found that rolling resistance has anegligibleeffect during impact and attachmentof microspheresSome evidence exists to indicate that this rolling resistance mayalso be negligible for particle resuspension (Dunn et al 2001)For this reason it is not included in the model presented in theprevious section

DETERMINATION OF THE DAMPING COEFFICIENTSThe Runge-Kutta-Gill (R-K-G) method was used to solve the

previously described governing equations In the 3D simulation4 parameters are required the effective material stiffness K theadhesion line force f0 the nondimensional Hertzian dampingcoef cient sup3H and the nondimensional adhesion damping co-ef cient sup3A The estimation of the material stiffness and theadhesion line force are introduced in the previous section Thedampingcoef cientssup3H and sup3A can be determinedby the ttingof the 2D microsphere impact data For each set of experimen-tal conditions an optimal set of sup3H and sup3A is found to t thecoef cient of restitution en Figure 3 shows an example of theleast-square regression of the experimental data of Wall et al(1990) After tting the values of sup3H and sup3A are xed and theset with the least-square error are selected for 3D simulationThe values used in the simulation are listed in Table 1

NONLINEAR DYNAMICS OF NORMALOSCILLATORY MOTION

Most if not all previous studies of microsphere impact haveconsidered only before-and-after contact conditionsNumericalsimulation offers a way to understand the dynamics of particle

Figure 3 Numerical simulation results (solid lines) and Wall et al data (1990) sup3A D 290 sup3H D 050

impact and the effects of parameters during contact as well Thisis particularlypertinentafter attachmentwhen it occurs Ziskindet al (2000) estimate a natural frequency of vertical oscillationusing a simpli ed contact model The system is linearized nearthe equilibrium position and damping is neglected Howevernonlinear systems do not possess a single natural frequency be-cause of a strong dependence on initial conditions Even for alinear system an over-damped system has no natural frequencyThe model covered in this paper can be used to simulate verticaloscillatory motion of an attached microsphere

Because the tangential and rotational motion is driven by thenormal contact for a Coulomb frictional system proper repre-sentation of the nonlinear dynamics of the normal process iscrucial By changing the coordinate system x D iexcln the equa-tion of motion at the normal direction can be rewritten as

mx D iexclp

r K x 3=2(1 C CH sbquox) C 2frac14 f0p

rx1=2(1 iexcl CA sbquox ) [27]

Using the same symbols for the displacementof the mass centerx and the time t the nondimensional form of Equation (27) is

x D iexclx3=2(1 C C1 sbquox) C x1=2(1 iexcl C2 sbquox) [28]

The following length X and time scales T are used in nondi-mensionalization

X D2frac14 f0

K [29]

T Dm

pr K

Xiexcl1=2 Dm

p2frac14r f0K

[30]

The length scale X is also the equilibrium displacement andthe time scale T is set so that the Hertzian force is at the same

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1051

Figure 4 Bifurcation on the parameter plane

order of the adhesion force which is true for adhesive impactproblems The nondimensionaldamping parameters C1 and C2

(different from sup3H and sup3A) are

C1 D CHX

TD CH

rr

K

(2frac14 f0)3=2

m [31]

and

C2 D CAX

TD CA

rr

K

(2frac14 f0)3=2

m [32]

The system form of Equation (28) is

raquosbquox D y

sbquoy D iexclx3=2(1 C C1y) C x1=2(1 iexcl C2y)

frac14 [33]

The Jacobian of Equation (33) is

J (x y)

Dsup3

0 1

iexcl 32 x1=2(1 C C1 y) C 1

2 xiexcl1=2(1 iexcl C2y) iexclx3=2C1iexcl x 1=2C2

acute

[34]

Because translation of the coordinates x and y does not affectthe Jacobian J the frequency of the solution at any point (x y)in the phase portrait is the imaginary part of the eigenvalueof theJacobian J Because the eigenvaluesof J are functions of bothx and y (the displacement and the impact velocity) it is clearthat no single natural frequency exists for the contactimpactproblem based on the model with adhesion and damping

Equation (33) has 2 equilibrium points The nonlinear prop-erties of the contact system can be studied at the equilibriumposition of the adhesion contact (x D 1 y D 0) The Jacobianat 1 0 is

J (1 0) Dsup3

0 1iexcl1 iexclC

acute [35]

where C D C1 C C2 The eigenvalues of J (1 0) are

cedil DiexclC sect

pC2 iexcl 4

2 [36]

A bifurcationon the parameter plane occurs with a critical pointC D 2 Figure 4 shows the parameter plane When C gt 2 botheigenvalues are real It corresponds to the over-damped systemand no frequency is possible at all When C D 2 a double non-zero eigenvalueappears this corresponds to a critically-dampedcase in which the frequency is zero When C lt 2 a damped os-cillation occurs The frequency is

p4 iexcl C2 and the point (1 0)

is a spiral sink because the real part of the eigenvalue is nega-tive (represents an amplitude decay) Physically these differentproperties of the equilibrium position correspond to 3 kinds ofsolutions in the simulation rebound (spiral sink large incidentvelocityFigure 5a) capture with oscillation (spiral sink low in-cident velocityFigure 5b) and capture without oscillation (puresink Figure 5c)

The nonlinear analysis of the nondimensional system can beprojected back to the dimensional system In the dimensionalsphere contactimpact problem 3 types of numerical solutionsfrom the 3D simulation model are shown in Figure 5 that corre-spond to the experiments of Wall et al (1990) for microsphereswith a radius of 345 sup1m

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1052 W CHENG ET AL

Figure 5 Phase portraits for variations of incident velocities

When the forces on the right side of Equation (1) balance themicrosphere is attracted to the surface and reaches the equilib-rium position The velocity at the equilibrium position is zeroand the displacement at the equilibrium position is the equilib-rium displacement ne D X Physically a microsphere sittingon the surface with no external force is at its static equilibriumposition or an equilibrium position can be approached after amicrosphere is attached on the surface

De ne a constant CT where CT D CH C CA a total damp-ing coef cient The property of the equilibrium position is de-termined by the following conditions

1 When CT lt 2(2frac14 f0)iexcl5=4r iexcl1=4 K 3=4m1=2 the equilibriumposition is a spiral sink The solution around the equilib-rium position is a damped oscillation with an asympoticfrequency

f D

q4X1=2 iexcl C2

TXT

2X3

2T [37]

It can be shown from Equations (37) (14) and (15) that f depends not only on the material properties and theradius of the sphere but also on the initial velocity vn f cannot be considered as a single natural frequencyof the system because the Hertzian damping and adhesion

dampingforces are nonlinear Thedampedoscillationnearthe equilibrium position identi es the energy dissipationmechanismfor particlecapture It also indicatesa potentialmechanism for particle resuspension

2 When CT D 2(2frac14 f0)iexcl5=4r iexcl1=4K 3=4m1=2 the solutionaround the equilibrium position is a critical dampingoscillation

3 When CT gt 2(2frac14 f0)iexcl5=4riexcl1=4K 3=4m1=2 the system is anover-damped system The equilibriumpoint is a sink Thesphere is captured by the surface without oscillation

If all the material parameters are xed and the damping co-ef cients CH and CA have the form as shown in Equations(15) and (14) the property of the solution is determined by theincident velocity The critical velocity vc is de ned as

vc D 25=4frac14 25=4 f025=4riexcl19=4 K iexcl23=4miexcl1=2(sup3H C sup3A)5 [38]

The critical velocity for the case in Figure 5 is vc D 225 pound10iexcl5 ms When the incident velocity is larger than the criticalvelocity (see Figures 5a and 5b) the equilibriumpoint is a spiralsink In the case of vn(0) D 10 ms the maximum displacementnmax is far away from the equilibrium displacement ne Thusthe inertia of the sphere in the rst cycle is large enough to over-come the attraction of the spiral sink and the sphere rebounds

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1053

Figure 6 Schematic of the incident angles

after impact In Figure 5a the impact begins with zero normaldisplacement and a negative velocity iexcl1 ms and ends witha zero normal displacement and a positive impact velocity Inthe case of vn(0) D 05 ms Figure 5b the incident velocityis not large enough and nmax is near ne The microsphere fallsinto the path of the spiral sink and is captured The velocityand displacement then form a damped cyclic path near the nalequilibrium point This phase portrait indicates a vibrationalcontactThe frequencyof vibrationnear theequilibriumpositionin Figure 5b according to Equation (37) is 20pound107 Hz The nalvelocity of the particle is zero but the nal static equilibriumdisplacement is negative This type of motion exists whether theinitial conditions are due to impact or external forces

When the incidentvelocityis smaller than thecritical velocity(the case vn(0) D 10 pound 10iexcl5 ms Figure 5c) the equilibriumpoint is a pure sink and the particle is capturedand retainedby thesurface without oscillation Detailed theoretical analysis offers

Table 2The initial conditions used to simulate their effects on 3D impact

Case number Conditions Descriptions

No 21 v D 02 10 ms reg D 0plusmn 30plusmn Figures 7 8No 22 v D 02 10 ms reg D 0plusmn t 0 D 0 18 pound 105 rads Figures 9 10No 23 v D 02 10 ms reg D 0plusmn 30plusmn t D 29 pound 105 Figures 11 12No 24 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 105 rads f D 0 09 Figure 13No 25 v D 02 10 ms reg D micro D 0plusmn t D t 0 D 29 pound 106 rads f D 0 09 Figure 14

the accurate expressions for the capture velocity as a functionof material parameters (Cheng et al 2000)

EFFECTS OF INITIAL CONDITIONSIn order to understand the energy dissipation and the rela-

tionship between rotation and tangential motions 3D obliqueimpacts are simulatedFigure 6 is a schematic diagram illustrat-ing the geometry and the incident direction The values for theinitial conditions used in the example cases are listed in Table 2and the values of the correspondingbasic parameters in Table 1The normalized translationalkineticenergy loss coef cient KL normalized total kinetic energy loss coef cient K t restitutioncoef cient en and tangential impulse ratio sup1 are compared byvariations of the initial conditions over the range listed in Table2 The magnitudeof the initialvelocitiesin Table 2 is representedby v

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1054 W CHENG ET AL

Figure 7 Energy loss for case No 21 (reg D 5plusmn increment)

An oblique impact is complicatedbecause the friction is non-linear and rolling and sliding of the microsphere are dif cult topredict In order to understand the energy loss by friction andenergy transport between rotation and tangential translation thetotal energy loss coef cients and translational energy loss coef- cients are plotted as a series of curves related to the incidentangles and incidentvelocitiesThe curves show the sensitivityofthe energy loss to the incident velocitiesand the incident angles

The sensitivity of energy loss to the magnitude of incidentvelocities and impact angles is shown in Figure 7 For the sameimpact anglereg bydecreasing the magnitudeof the impact veloc-ity both KL and K t increase For the same magnitude of impactvelocity when the impact angle reg increases KL and K t in-crease When the incident velocity is smaller than the capturevelocity the particle is captured by the surface and the energyloss is the maximum In Figure 7 the normalized energy lossesfor captured particles are constant For oblique impacts follow-ing capture microspheres continue to roll In the current simu-lation rolling friction is neglected and the energy loss ratios forcaptured particles are middot10 In reality because of rolling frictionall of the initial kinetic energy eventually would be dissipatedThe difference between KL and K t shows that the translationalenergy loss KL is larger than the total energy loss K t in theoblique impact without initial rotational velocities For nonzeroincident angles in addition to the energy dissipation part of the

translational energy transforms into rotational energy It is alsoclear in Figure 7 that the increase of the impact angle reg causesthe increase of the difference KLiexclK t This reveals that for ashallow impact angle (a large reg) the rotational energy becomessigni cant and measurement of rotationalvelocities is crucial inan experiment

Figure 8 shows the sensitivity of the tangential impulse ratiosup1 and the coef cient of restitution en to impact velocities andimpact angles It is clear that sup1 can be sensitive to the increasein approach angle This means that near capture a small errorin the measurement of the impact angle reg can cause a largedifference between the experimental and simulation results intangential velocities A high accuracy in angle measurement isneeded in oblique impact experiments The numerical simula-tions show that en drops at low incident velocities and when theincident velocity is smaller than the capture velocity en is zeroThe effects of the incident angle on en are signi cant at lowvelocities (depending on the physical properties of the sphereand the surface) This means that the normal rebound velocityis more sensitive to the incident angles for low velocity obliqueimpacts

The effects of initial rotational velocities on normal impactsare shown in Figures 9 and 10 Figure 9 shows that neither KL

nor K t is sensitive to the initial rotationalvelocities for the rangeof initial angular velocitieschosen In order to make a difference

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1055

Figure 8 en and sup1 for case No 21 (reg D 5plusmn increment)

Figure 9 Energy loss for case No 22

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1056 W CHENG ET AL

Figure 10 en and sup1 for case No 22

in KL and K t an initial rotational velocity on the order of105 rads is needed However in reality the rotation of the par-ticle may still be important because it is possible to have suchhigh initial rotational velocities for microspheres For a radiusof 689 sup1m a velocity of 10 ms at the center of mass andzero velocity at the contact area the rotational velocity is 29 pound105 rads The difference KLiexclK t is associated with the mag-nitude of incident velocitiesThe magnitudeof KLiexclK t is largerat low incident velocities The rotational energy dissipation isimportant for low velocity impact In Figure 9 the values ofKL iexcl K t are negative because the initial rotational velocity ofthe particle increases the initial tangential contact velocityof theparticle and in uences the translational energy loss

Figure 10 shows the effects of initial particle rotationalveloc-ity on sup1 and en The particle rotation has no effect on the normalimpact which is characterized by the coef cient of restitutionen All the values for en under different initial angular veloci-ties fall onto a single curve However the effects of the particlerotation on the tangential impulse ratio sup1 are quite obviousIn order to explain the experimental results of particle impactthe rotational rate of the particle which is extremely dif cult tomeasure for small particles must be includedThe values for en-ergy losses and sup1 are not calculated when the incidentvelocitiesare smaller than the capture velocity in Figures 9 and 10

Figures 11 and 12 show the combined effect of initial ldquooff-planerdquo spin t and the incident angle reg on the impact re-sults The initial rotational velocity t is 29 pound 105 rads From

Figure 11 by changing the value of reg both KL and K t showsimilar behavior as in Figures 7 and 9 The difference KLiexclK t

is the combinationof Figures 7 and 9 At low velocities (lt4 msin the simulated cases) the effects of initial rotation are moresigni cant The values of KLiexclK t are negative At high veloci-ties the effect of incident angle becomes more important Thevalues of KLiexclK t are positive In Figure 12 the changing of reg

has signi cant effects on the tangential impact along the t direc-tion which is represented by sup1t It has less effect on the normalrestitution coef cient en and tangential impulse ratio along theldquooff-planerdquo directionsup1t 0 The variationsof en and sup1t 0 are causedby the variation of normal impact velocity at different incidentangles Near the capture velocity the tangential impulse ratiosup1t 0 is much larger than the friction coef cient f because ofthe signi cant ldquosnap-onrdquo and ldquosnap-offrdquo effects This result in-dicates that the coupled effects of adhesion and friction can besigni cant near capture

Simulation results also show that the friction coef cient af-fects the tangential motion and rotation of the microsphere Theeffects are closely related to the initial rotation and the rela-tive tangential velocities as shown in Figures 13 and 14 It isreasonable that the initial spin has no effect on the restitutioncoef cient In Figure 13 because the initial rotational velocityis small for all the values of the friction coef cients shown inTable 2 the particle is rolling at the end of impact (except in thecase of zero friction) All the curves for sup1t and sup1t 0 fall onto a sin-gle curve However in Figure 14 the initial rotational velocity

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1057

Figure 11 Energy loss for case No 23 (reg D 5plusmn increment)

Figure 12 en and sup1 for case No 23 (reg D 5plusmn increment)

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

1058 W CHENG ET AL

Figure 13 en and sup1 for case No 24 ( f D 0 09 increment)

Figure 14 en and sup1 for case No 25 ( f D 01 increment)

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

38Johnson K L (1985) Contact Mechanics Cambridge University Press

Cambridge

MICROSPHERE CONTACTIMPACT WITH FLAT SURFACES 1059

is large The effects on tangential impulse ratios in both t andt 0 directions are signi cant and this is even more so for smallernormal impact velocities For very low velocity and high spinimpact or contact the determination of friction coef cient isvery important On the other hand in order to measure the fric-tion coef cient accurately the experiment should be conductedin the very low velocity and high rotational velocity range Theabove-sensitivity analysis can be valuable in the design of theexperiment of particle impact

The results can be summarized as follows

1 By changing the incident angle reg at low velocities theimpulse ratio sup1 and the restitution coef cient en changesigni cantly The changes of the coef cient of restitutionen at high velocities and energy loss KL and K t are notas signi cant

2 The effects of initial angular velocities are signi cant onall the calculations including sup1 and energy losses Theeffects are more signi cant for low initial velocities

3 When the adhesion and damping forces increase the par-ticle is captured by the surface followed by damped vi-bration with a decayingsolutionOn the contrary reboundoccurs for low adhesion and low damping cases

CONCLUSIONSThis study clearly demonstrates that 3D simulations can pro-

vide important information for microsphere contactimpactThrough numerical simulation the parameter sensitivity prop-erties of the forces and factors that control the particle deposi-tion and rebound are understoodThe simulation shows that therolling and translationalenergy transport plays an important roleduring attachment including oblique impact The results offera better understanding of the friction and adhesion model Thetangential motion of a particle during impact is quite sensitiveto the incident angles and the initial rotation

The nonlinear analysis shows that any one of 3 types of mo-tion can occur for a microsphere impact rebound capture withdamped oscillation and capture without oscillation Which oc-curs is determined by the material parameters and the normalincident velocity Two characteristic velocities the capture ve-locity and the critical velocity are de ned in determining theconditions for capture and different types of motions after thecapture respectively Capture occurs when the incident veloc-ity of the microsphere is lower than the capture velocity suchthat the phase path of the microsphere falls into the sink on thephase portrait The occurrence of oscillation after the capture isdetermined by the normal incident velocity and the critical ve-locity If the normal incident velocity is smaller than the criticalvelocity no oscillation occurs after the capture The asymptoticfrequency of the oscillation near the equilibrium position is de-termined by the material properties and the radius of the sphereBecause of the nonlinearity of the system the asymptotic fre-quency is also dependenton the incident velocity if the damping

coef cients CH and CA have the form shown in Equations (15)and (14)

The simulation results also identify potential sources of er-rors in experiments and the best incident velocities for the mea-surement of different parameters It is clear from the simulationresults that experimental measurements of the rotational veloc-ity of microspheres would greatly increase the understandingof low velocity oblique impacts Analysis and experimental re-sults show that some rolling models are not able to characterizerolling friction accurately In order to simulate rolling resistanceduring established contact the asymmetric distribution of thenormal contact pressure over the contact area must be modeledproperly However for impact problems the rolling resistancecan be neglected because the contact radius is very small andthe duration of microsphere-substrate contact is very short

REFERENCESAndo Y and Ino J (1998) Friction and Pull-Off Forces on Submicron-Size

Asperities WEAR 216115ndash122Ando Y Ishikawa Y and Kitahara T (1995) Friction Characteristics and

Adhesion Force under Low Normal Load Transactions of the ASME J ofTribology 117569ndash574

Brach R M and DunnP F (1995) Macrodynamics of Microparticles AerosolSci Technol 2351ndash71

Brach R M and Dunn P F (1996) Simulation of Microparticle Impact withAdhesion Proc of Adhesion Soc Symposium on Particle Adhesion MyrtleBeach SC

Brach R M DunnP F and ChengW (1999b) RotationalDissipation DuringMicrosphere Impact Aerosol Sci Technol 301321ndash1329

Brach R M Dunn P F and Li X (2000) Experiments and EngineeringModels of Microparticle Impact and Deposition J Adhesion 74227ndash282

BurnhamN A Behrend O P OuleveyF Gremaud G Gallo P-J GourdonD Dupas E Kulik A J Pollock H M and Briggs G A D (1997) HowDoes a Tip Tap Nanotechnol 867ndash75

Cheng W Brach R M and Dunn P F (2000) Numerical Studies on 3DParticle Impact With Surfaces Hessert Center Technical Report Universityof Notre Dame Notre Dame IN

Cundall P A and Hart R D (1992) Numerical Modeling of DiscontinuaEngineering Computations 9101ndash113

Dahneke B (1975) Further Measurements of the Bouncing of Small LatexSpheres J Colloid Interface Sci 5158ndash65

Derjaguin B V Muller V N and Toporov Yu P (1975) Effect of ContactDeformations on the Adhesion of Particles J Colloid Interface Sci 53314ndash

325Dunn P F Ibrahim A and Brach R M (2001) Microparticle Resuspension

Experiment 20th Annual Meeting of the American Association for AerosolResearch Portland OR

Gladwell G M L (1980) Contact Problems in The Classical Theory ofElasticity Sijithoff amp Noordhoff Alphen aan den Rijin

GoldsmithW (1960) ImpactmdashTheTheoryandPhysicalBehaviourofCollidingSolids Edward Arnold London

Hertz H (1881) Uber die Beruhrung fester elastischer Korper J fur die Reineund Angewandte Mathematik 92156ndash171

Hertz H (1882) Uber die Beruhrung fester elastischer Korper und Uberdie Harte Verhandlungen des Vereins zur Beforderung des Gewerb eifsesBerlin

Israelachvili J N and Tabor D (1972) The Measurement of van der WaalsDispersion Forces in the Range 15 to 130 nm Proc R Soc Lond A 33119ndash

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Johnson K L Kendall K and Roberts A D (1971) Surface Energy and theContact of Elastic Solid Proc R Soc Lond A 324301ndash313

Li X Dunn P F and Brach R M (1999) Experimental and NumericalStudies on the Normal Impact of Microspheres with Surfaces J Aerosol Sci30439ndash449

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Quesnel D J Rimai D S Gady B and DeMejo L P (1998) Exploring theJKR Formalism with Finite Element Analysis Proc 21st Annual MeetingAdhesion Society Savannah GA pp 290ndash292

Rollot Y Regnier S and Guinot J (1999) Simulation of Micro-ManipulationsAdhesionForces andSpeci c Dynamic Models InternationalJ Adhesion and Adhesives 1935ndash48

Savkoor A R and Briggs G A D (1977) Effect of Tangential Force on theContact of Elastic Solids in Adhesion Proc R Soc Lond A 356103ndash114

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Thornton C (1991) Interparticle Sliding in the Presence of Adhesion J PhysD 241942ndash1946

Tsai C J Pui D Y H and Liu B Y H (1991) Elastic Flattening and ParticleAdhesion Aerosol Sci Technol 15239ndash255

Wall S John W Wang H C and Goren S L (1990) Measurements ofKinetic Energy Loss for Particles Impacting Surfaces Aerosol Sci Technol12926ndash946

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