hysteretic effects in droplet motions on heterogeneous substrates: direct numerical simulation

Hysteretic Effects in Droplet Motions on HeterogeneousSubstrates: Direct Numerical Simulation

Leonard W. Schwartz

Departments of Mechanical Engineering and Mathematical Sciences, The University ofDelaware, Newark, Delaware 19716

Received December 22, 1997. In Final Form: April 1, 1998

Amethodof calculation is introduced that allows the simulation of the time-dependent three-dimensionalmotion of liquid droplets on solid substrates for systems with finite equilibrium contact angles. Thecontact angle is a prescribed function of position on the substrate. An evolution equation is given, usingthe lubrication assumption, that includes viscous, capillary, and disjoining forces. Motion to and from drysubstrate regions is made possible by use of a thin energetically stable wetting layer. We simulate motionon heterogeneous substrates with periodic arrays of high-contact-angle patches. For a homogeneoussubstrate, calculated droplet spreading rates are compared to published experimental results, revealingthe need for a time-scale correction. Two different problems are treated. The first is spontaneous motiondriven only by wetting forces. If the contact-angle difference is sufficiently high, the droplet can findseveral different stable positions, depending on the previous history of the motion. A second simulationtreats a forced cyclical motion. Energy dissipation per cycle for a heterogeneous substrate is found to belarger than that for a uniform substrate with the same total energy. Comparison is made with simplequasi-static theories.

1. IntroductionWetting and capillary considerations during the slow

motion of liquids on solid substrates are important in boththe technologicalandnaturalworlds. Applications includethe spreading behavior of liquid coatings, as well as flowsin oil reservoirs, chemical reactors, and heat exchangers.Some biological application areas are motions in the tearfilm on the cornea of the eye, flows on liquid-coveredmembranes in the lungs, and the general area of cellmotility, where cell motions have many of the features ofinert liquid droplets.1 The spreading properties of agro-chemicals suchaspesticidesand insecticidesare importantdeterminants of their effectiveness. Capillary forces areoften dominant in problems of small physical dimensions;however, they can also be important for large-scalephenomena when body forces are very weak, as in themicrogravity environment of orbiting satellites in space,for example.Most substrates are of intermediate energy in that a

liquiddroplet,whenplaceduponthem,ultimatelyachievesa static shape with a nonzero value of contact angle.Systemsexhibitingnonzero contact anglesarealso termed“partially wetting”. The contact angle is the angle,measured within the liquid, at the so-called wetting orcontact line where the three phases, solid (S), liquid (L),and vapor (V), meet. The Young-Laplace equation,derived using either force or energy considerations,predicts a unique value for the contact angle as a functionof the three pairwise interfacial energies γSL, γSV, and γLV.The latter is often termed the surface tension, or surfaceenergy density σ. The equation is

There are serious experimental difficulties in measuringa single reproducible value of equilibrium contact angleθe in particular systems, and often two different valuesare measured, depending on whether the liquid has

recently advanced onto a previously dry substrate orrecently receded, leaving behind a sensibly dry substrate.These angles are termed the advancing contact angle θaand the receding angle θr. Systems for which θa and θrare different are said to display contact angle hysteresis(CAH). Invariably the receding angle is smaller than theadvancing one.Contact angle hysteresis has been attributed to the

chemical nonhomogeniety of the substrate and the pos-sibility that the liquid front may be trapped in differentconfigurations, each configuration corresponding toa localminimum of the total free energy of the system. Oftenthe scale of substrate heterogeneity ismuch smaller thanmacroscopicmeasurements candiscern. Theproblemmaybe simplified by assuming that a single parametercharacterizes the geometry of the system. For example,JohnsonandDettre2 consideranaxisymmetric sessiledropthat is placed at the center of a patterned substratecomposed of alternating concentric rings of differentmaterials. The equilibrium contact angle takes on adifferent constant value on eachmaterial. Their analysislooked for local minima in the functional describing thefreeenergyof the system. Forgivenvaluesofdropvolume,ringwidth, and contact angles, they found a range of dropradii over which the free energy versus radius curve hasa set of localminima. Assuming that the dropmoves veryslowly, its edge may be trapped at the first local energyminimum that is encountered. Although the scale of theheterogeneity, in this case the ring width, may be small,the range of energyminima, termed themetastable range,may bemuch larger, leading to quite large values of CAH.Similar calculations were performed by Neumann andGood3 for horizontal stripes of differentmaterials appliedto a plate that is held vertically in a reservoir of liquid.If the contact line is also assumed to be horizontal, ametastable energy range and correspondingly differentvalues of advancing and receding angle can be predicted,

(1) Greenspan, H. P. J. Fluid Mech. 1978, 84, 125-143.

(2) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744.(3) Neumann, A. W.; Good, R. J. J. Colloid Interface Sci. 1972, 38,

341.

γSL + σ cos θe ) γSV (1.1)

3440 Langmuir 1998, 14, 3440-3453

S0743-7463(97)01407-8 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 06/09/1998

depending onwhether the plate ismoved into or out of theliquidbath. Theanalysis includedgravitational potentialenergy in the free energy of the system. Physicalroughness is believed to play a role similar to that ofchemical heterogeneity. Dussan4 demonstrates that pat-terns of alternating surface inclination on a chemicallyhomogeneous substrate may lead to different “apparent”values of contact angle.The assumption that the contact line shape is known

a priori is highly restrictive. Schwartz and Garoff5considered the vertical plate geometry with horizontalwettability stripes and allowed the contact line shape tobe found as part of the energy minimization procedure.They found that straight three-phase lines correspond tothe minimum energy solutions only in special circum-stances. As the fluid advances and recedes along thesubstrate, the contact line alternates between straightand nonrectilinear configurations. The extra freedomallowed to the system leads to a dramatic lowering of theheights of the energy barriers. The metastable range isconsiderably reduced, as is the degree of CAH. Wavesmoving transversely to the direction of platemotion werepredicted by the analysis when the contact line was nearawettability boundary. Thesewaveswere then observedexperimentally using a silvered plate whose lower halfwas coated with a surfactant monolayer. In later work6more general patterns ofwettabilitywere considered, alsofor theproblemof capillary rise ontoapartially submergedvertical plate. It was found that CAH was a strongfunction of the details of the “patch” pattern. Predictionsof the progression of the three-phase line as it transversesthe wettability pattern include alternating continuousdeformationsanddiscrete jumps toother solution families.The phrase “sticking, stretching, and jumping” wasintroduced to describe these motions.Missing in the above analyses is any consideration of

dynamiceffects. Evenonaperfectlyhomogeneoussurface,apparent contact angles while a liquid is advancing aredifferent fromtheapparentangleswhenthe liquidrecedes.These so-called dynamic contact angles are not accessibleto the previous quasi-static formulations. Onemust alsodistinguish between spontaneous and forced motions.Spontaneousmotionsare thosedrivensolelyby theenergydifference between a given nonequilibrium configurationand a final statically stable configuration. The quasi-static analyses can give no indication of the time scalesin spontaneous motion. Spreading of a drop on a plate isan example of such a spontaneous motion. The responseof a liquid to the motion of a vertical plate, as discussedabove, is an example of a forced motion. While the timescale is set, in such cases as the period of an imposedoscillation, the quasi-static theories require that themotion be so slow that the system has time to relax to astable equilibrium configuration at each instant. Thiscertainly need not be the case. Also the energy balancefor the quasi-static theories is incomplete. There is noconsideration of dissipative viscous effects which, evenfor slowmotions, can be comparable to changes in the freeenergy of the system. A principal purpose of the presentpaper is to resolve these inadequacies by formulating andsolving a mathematical model for the direct time-depend-ent numerical simulation of either forced or spontaneousspreading of a liquid droplet on amixed-wettable surface.Hysteretic effectswill be explored. These include contact-angle hysteresis, as defined above, as well as hysteresis

in the traditional physical sense. Traditional hysteresishere refers to the energy dissipated per cycle in a forcedperiodic motion. This energy loss is approximatelyproportional to the speed of an imposed motion and ispresent even for an ideally homogeneous substrate.The following sectiondiscusses themathematicalmodel

for droplet motion on a heterogeneous substrate. It isbased on the long-wave or lubrication approximation forthe unsteady motion of a liquid coating layer on a flatsubstrate. The approximation requires that the inclina-tion of the liquid free surface, relative to the substrate,be small. Inertial effects are neglected; thus, a suitablydefinedReynoldsnumbermust also be small. Systematicderivations of lubrication theory, as an expansion in theinclination of the free surface, are given by Benney7without surface tension and by Atherton and Homsy,8where surface tension is included. Because integrationacross the thin dimension is performed analytically, thedimensionality in space is reduced from three to two inthe time-dependentnumerical simulation. Forapartiallywetting system, the shape of statically stable liquiddroplets can be represented mathematically using theFrumkin-Deryaguin model.9-11 The substrate regionnear themargin of themacroscopic droplet is coveredwitha submicroscopic layer of wetting liquid. This wettinglayer ismaintained inastateof stable equilibriumthroughthe action of intermolecular forces. There is excess or“disjoining” pressure in the thin film that arises from avariety of causes; these contributions are often dividedinto molecular, ionic-electrostatic, and structural com-ponents. A useful survey of the substantial theoreticalandexperimental literatureondisjoiningpressure is givenby Mohanty.12 In the contact region, the condition ofconstancy of total pressure, including capilllary anddisjoiningcomponents,determines theshapeof the liquid-vapor interface. Theapparent contact angle, ormaximumslope of the interface, may be found as a function of thesurface tensionand theparameters in the expressionusedfor the disjoining pressure.The existence of the wetting layer serves another

purpose for dynamic simulations. The lubrication for-mulation, like the fullNavier-Stokesproblem fromwhichit is derived, does not allow liquid motion to or fromperfectly dry regions of the substrate. The difficulty liesin the incompatibility between theusualno-slip condition,where the moving liquid meets the substrate, and theboundaryconditionsonthe liquid-vapor interface, leadingto a nonintegrable force singularity where these twointerfaces meet at the contact line.13 The thin wettinglayer alleviates this problem and is an alternative to theslip models otherwise used in dynamic studies.1,14 As-ymptotic analyses of spreading using a thinwetting layerare given by Tanner15 and Tuck and Schwartz.16 Ex-perimentaldata for the speedof spreadingdrops invarioussystems suggests that thewetting layer thickness,h* say,lies in the broad range 1-100 nm.17-19 A number ofdynamic studies, using various forms of the disjoining

(4) Dussan, V. E. B. Annu. Rev. Fluid Mech. 1979, 11, 371.(5) Schwartz, L. W.; Garoff, S. J. Colloid Interface Sci. 1985, 106,

422.(6) Schwartz, L. W.; Garoff, S. Langmuir 1985, 1, 219.

(7) Benney, D. J. J. Math. Phys. 1966, 45, 150.(8) Atherton, R.W.; Homsy, G.M.Chem. Eng. Commun. 1976, 2, 57.(9) Frumkin, A. N. Zh. Fiz. Khim. 1938, 12, 337.(10) Derjaguin, B. V. Zh. Fiz. Khim. 1940, 14, 137.(11) Churaev, N. V.; Sobolev, V. D. Adv. Colloid Interface Sci. 1995,

61, 1.(12) Mohanty, K. K. Ph.D. Dissertation, University of Minnesota,

1981. See also: Sharma, A. Langmuir 1993, 9, 861.(13) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85.(14) Hocking, L. M. Q. J. Mech. Appl. Math. 1981, 34, 37.(15) Tanner, L. J. Phys. (D) 1979, 12, 1473.(16) Tuck, E. O.; Schwartz, L. W. SIAM Rev. 1990, 32, 453.(17) Lelah, M. D.; Marmur, A. M. J. Colloid Interface Sci. 1981, 82,

518.

Hysteretic Effects in Droplet Motions Langmuir, Vol. 14, No. 12, 1998 3441

pressure term in a one-dimensional or axisymmetriclubrication model for wetting/dewetting, have alsoappeared.18,20-23 A fully two-dimensional spreading prob-lem, with disjoining pressure effect, has recently beenconsidered by us.24 That paper treated the spontaneousmotion of a droplet placed on a horizontal glass slide nearthe center of a “cross” of high-contact-anglematerial. Thedroplet ultimately splits into four unequal segments asthe liquidwithdrawsonto the lower-contact-angle regions.The numerical simulation was compared with an experi-ment using glycerin as the spreading liquid and a Tefloncross. All qualitative features of the observedmotionwerereproduced by the simulation. However a time-scalecorrection needed to be applied due primarily to the too-large value of the wetting layer thickness used in thesimulation.Section 2 also considers the energetics of droplet

spreading. The free energy of the system is taken to bethe sum of the appropriate interfacial energy densitiesintegrated over the instantaneous areas of the corre-sponding interfaces. Gravitational contributions areomitted here but could have been included withoutdifficulty; thus,we consider only relatively small droplets.For the glycerin droplet treated previously,24 the averageradius R0 on the substrate was about 4 mm. Thecorresponding Bond number Bo ) FgR0

2/σ is 2.5, whereF and g are the liquid density and the acceleration ofgravity, respectively. We performed simulations withgravity effects both includedandomitted. Thedifferenceswere limited primarily to the speed of motion, with thepredicted speed greater by about 5% when gravity isincluded. Thus for drops of this size or smaller, gravi-tational effects for a drop placed upon a horizontalsubstrate are relatively unimportant. For spontaneousmotions, it is shown that the rate of decrease of the totalfree energy of the system is equal to the rate of viscousdissipation. For forced motions, on the other hand, theenergy balance also must include the rate of working byliquid injection. Some details of the numerical procedureare given in subsection 2.2.Sections 3 and 4 discuss calculations and results for

dropletmotiononsubstrateswithdiscretepatches ofhigh-contact-anglematerial or “grease”. A schematic diagramis shown in Figure 1, where the blackened areas are thegrease patches. Two different patch patterns, shown inFigure 2, are used in this study. They are both doublyperiodic on the substrate. The pattern denoted “isolatedspots” is a simple square array where the fractionalcoverage of grease is a small fraction of the total substratearea. The checkerboardpattern is a staggered arraywith

each of the two substrate materials occupying an equalfraction of the substrate. The mathematical procedurefor generating these patterns is given in the Appendix.For the isolated spot pattern, results are given for a

number of simulations of spontaneous spreading. In eachcase, the substrate-average wettability is the same. Twodifferent types of simulations are performed: ones wherethedropletadvancesandotherswhere thedroplet recedes.Inboth cases thedroplets eventually stabilize. Byvaryingthewettability contrast, we establish the level of contrastnecessary for hysteresis, that is when the final positionsin advance and recede are not the same. For a particularcase where CAH occurs, the detailed calculations arecomparedwith the results of a simple theory that assumesthedroplet shape isaxisymmetric. While thesimplemodelgives a good estimate of the final free energy, it does notpredict any hysteresis. When the wettability contrast isnot sufficiently great to cause different final positions, itsinfluence is still felt. The time required to reach equi-librium is increased, in proportion to the degree ofheterogeneity.Results for forced spreading are given in section 4. A

prototypical experiment is simulatedwhere the liquiddropis forced to expand and then retract by cyclical pumpingthroughahole in the substrate lyingunderneath thedrop.The cycle time is varied, and the total work per cycle iscalculated. The viscous losses for a uniform substrateareapproximately inverselyproportional to the cycle time.The additional dissipations on a patterned substrate arerelatively insensitive to speed. Both the checkerboardand isolated-spot patterns are used in this section. Whilethe isolated spots are more effective for “pinning” thecontact line, the checkerboard, usinga smallerwettabilitycontrast, can dissipate more energy. The results suggestthat a wettability pattern may be optimized to satisfy aparticular purpose.Further remarks are given in the final section. The

need for additional theoretical and experimental work toaddress unexplored areas is explained. Implications forterrestrial and microgravity applications are also dis-cussed.

2. Mathematical ModelWe consider a layer of liquid, or an isolated droplet, on

a plane substrate. x, y, and z form a right-handed triadwith the x and y axes lying on the substrate. The liquidsurface corresponds to z ) h(x,y,t) where t is time. Thestatement of mass conservation is

(18) Teletzke, G.; Davis, H. T.; Scriven, L. E. Chem. Eng. Comm.1987, 55, 41.

(19) Starov, V. M.; Kalinin, V. V.; Chen, J.-D. Adv. Colloid InterfaceSci. 1994, 50, 187.

(20) Williams, M. B.; Davis, S. H. J. Colloid Interface Sci. 1982, 90,220.

(21) Kheshgi, H. S.; Scriven, L. E. Chem. Eng. Sci. 1991, 46, 519.(22) Mitlin, V. S.; Petviashvili, N. V. Phys. Lett. A 1994, 192, 323.(23) Jameel, A. T.; Sharma, A. J. Colloid Interface Sci. 1994, 164,

416.(24) Schwartz, L. W.; Eley, R. R. J. Colloid Interface Sci., in press.

Figure 1. Schematic diagram of a liquid droplet on a mixed-wettable substrate.

Figure 2. Substrate wettability patterns used in this study.The dark regions have larger values of contact angle than thesurrounding field: (a) isolated spot pattern; (b) checkerboardpattern.

ht ) -∇‚Q + wi(x,y,t) (2.1)

3442 Langmuir, Vol. 14, No. 12, 1998 Schwartz

Here Q is the two-dimensional areal flux vector definedas

where u and v are velocity components in the x and ydirections, respectively. Likewise∇ is a two-dimensionaloperator with respect to x and y. wi is a local injectionrate that is an input function of substrate position andtime. It will be used for the forced-motion simulations insection 4.Under the assumptions of lubrication theory, that is,

the motion is sufficiently slow that inertial forces may beneglected and that the free surface is inclined at a smallangle relative to the substrate, the momentum equationfor an assumed Newtonian liquid may be integrated toyield

Here µ is the viscosity and the pressure p is independentof z. Theno-slip condition for thevelocityhasbeenappliedon the substrate, and the liquid free surface is stress-free.The pressure in the liquid is given by

and the pressure above the liquid is taken to be zerowithout loss of generality. The first term on the right isan approximation of the free-surface curvature κ whenthe surface slope is small. The error in this curvatureapproximation is proportional to the square of the surfaceinclination. σ is surface tension, and the so-calleddisjoining pressure is given by the two-term model

B and the exponents n andm are positive constants withn > m > 1. The local disjoining energy density

has a single stable energy minimum at the thin wetting-layer thickness h ) h*. Because the disjoining pressureis assumed to depend only on the local interfacialseparation h, the validity of expressions such as eq 2.4also requires the small-slope approximation.A diagram of the edge of a drop at static equilibrium is

shown in Figure 3. It is useful to review the basic forcebalanceand its relationship to the equivalent line tensionsusing thepresentdisjoining-pressuremodel. Theanalysis,including disjoining effects, follows from the pioneeringwork of Frumkin andDeryaguin9-11 and has been furtheramplified by Sharma and Mitlin and co-workers.22,23 Ona macroscopic scale, the thin wetting layer is effectivelyinvisible and the line-tension balance is as shown in partb of the figure. Because the analysis is local to the dropedge, only a two-dimensional problem needs to be con-sidered. The point labeled A in Figure 3a is assumed tobe sufficiently far from the substrate that its height h,when measured in units of h*, is effectively infinite andΠ is zero there. The inclination atAhas become constantat theequilibriumcontactangleθe; thus, the total pressure

there, from eq 2.3, is zero. Similarly, at point B, on thewetting layer, the inclination θ and its rate of changealong the surface are both zero, and Π is also zero there.Performing an integrated force balance in the x directionon the region enclosed by the dashed line and recognizingthat the total pressure is zero on the vertical faces at Aand B, we have

where s is arc length measured along the free surface.But dh/ds ) -sin θ; thus,

or

This is the disjoining-model equivalent of the Young-Laplace eq 1.1. It is sometimes referred to as theaugmented Young-Laplace equation.12 Evidently

and e(*d)(∞) is equivalent to the so-called spreadingcoefficient as usually employed.25

Using the two-term disjoiningmodel, the constantB ineq 2.4 may be replaced in favor of θe using eqs 2.5 and 2.6and

and the small-argument approximation to cos θe is usedfor the approximate equality. Plots of disjoining pressureand disjoining energy density are given as functions ofh/h* in Figure 4. Several different exponent pairs (n,m)are compared.

(25) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.

Q )∫0h(x,y)(u,v) dz

Q ) - 13µ(h3∇p) (2.2)

p ) -σκ - Π ≈ -σ∇2h - Π (2.3)

Π ) B[(h*h )n - (h*h )m] (2.4)

e(d)(h) ) -∫h*hΠ(h′) dh′ (2.5)

Figure3. Contact region, at the edge of a drop, in equilibrium.A thin wetting layer of thickness h* is used in a disjoiningpressure term to specify the equilibrium contact angle θe: (a)Microscopic picture; (b) equivalent force balance on a macro-scopic scale.

0 ) ∫h*∞p(h) dh ) σ∫h*∞ dθds

dh - ∫h*∞Π dh

0 ) σ cos θ|0θe - ∫h*∞Π dh

σ cos θe ) σ - e(d)(∞) (2.6)

e(d)(∞) ) σ + γSL - γSV

B )(n - 1)(m - 1)h*(n - m)

σ(1 - cos θe) ≈(n - 1)(m - 1)2h*(n - m)

σθe2

(2.7)


The equilibrium contact angle is taken to be anarbitrarily prescribed function of position, and the evolu-tion equation for flow over a heterogeneous substratebecomes

It is possible to augmented this equationwith gravity andother force terms as additional driving mechanisms.24

In the following we will present results using dimen-sionless variables. The reference state is a stationaryparaboloidal drop with central height h0 and radius R0

lying on a flat substrate characterized by the constantequilibrium contact angle θe0. Within the small-slopeassumption, the reference contact angle satisfies θe0 ≈2h0/R0. The substrate coordinates are measured in unitsof R0, while film thickness h and wetting-layer thicknessh* are measured in units of h0. The time unit is T* )3µR0

4/(σh03). The evolution equation becomes, in dimen-sionless variables,

The injection function wi is now measured in units of h0/T*. All information concerning the substrate energetics

is included in the function

We will refer to G(x,y) as the wettability.2.1. Global Energetics. Each pressure component

on the right of eq 2.3may be identifiedwith an integratedenergycomponent. The free-surfaceenergy isproportionalto the area of the liquid surface and is given by

where A is the total area of the substrate and γ is theangle between the normal to the surface and the normalto the substrate. Over portions of the substrate whereonly the thin, flatdisjoining layerof thicknessh* ispresent,there is no contribution to E1. With the small-slopeassumption, eq 2.11 becomes

The total disjoining energy is

It may be verified that the integrand is zero when h) h*.For (n,m) ) (3,2), which is the set of exponents used inthis report, eq 2.13 takes the simpler form

Using the reference quantities introduced above, the unitof energy isσh02, and theenergy componentsare rewritten,in dimensionless form, as

The global energy change equation is now shown to be

which is the statement that the rate of viscous workingis equal to the rate of decrease of the stored, or potential,energy componentsplus the rate ofworkingon the system.The left side is the rate of viscous dissipation over theentire substrate

using eq 2.1. The divergence theoremwas also usedhere,and we require, for example, that either p or the normalcomponent of areal flux Q‚n is zero at each point on thesubstrate boundary.

Figure 4. (a) Nondimensional disjoining pressure π ) πh*/(σθe

2), and (b) disjoining energy density e(d) ) e(d)/(σθe2) plotted

versusdimensionless layerheighth/h* for three exponent pairsin eq 2.4.

ht ) - σ3µ∇‚[h3(∇∇2h +

(n - 1)(m - 1)2h*(n - m)

×

∇[θe2(x,y)(h* n

hn- h* m

hm )])] + wi(x,y,t) (2.8)

ht ) -∇‚[h3(∇∇2h +(n - 1)(m - 1)2h*(n - m)

×

∇[C(h* m

hn- h* m

hm )])] + wi(x,y,t) (2.9)

C(x,y) ) 4( θeθe0)

2

) 4G (2.10)

E1 ) E(σ) ) σ∫∫( 1cos γ

- 1) dA (2.11)

E1 ) E(σ) ≈ σ2∫∫∇h‚∇h dA (2.12)

E2 ) E(d) ) ∫∫e(d) dA ) σ2∫∫θe

2[1 + m - 1n - m(h*h )n-1

-

n - 1n - m(h*h )m-1] dA (2.13)

E2 ) E(d) ) ∫∫e(d) dA ) σ2∫∫θe

2(1 - h*h )2 dA (2.14)

E1 ) 12∫∫∇h‚∇h dA (2.12a)

E2 ) 12∫∫C(x,y)[1 + m - 1

n - m(h*h )n-1-

n - 1n - m(h*h )m-1] dA (2.13a)

E(µ) ) -(E1 + E2) + W (2.15)

E(µ) ) -∫∫Q‚∇p dA ) ∫∫p∇‚Q dA )

-∫∫p(ht - wi) dA (2.16)


The integral in eq 2.16 can be expressed as the sum ofthree terms, corresponding to the right side of eq 2.15.The first of these is

Here, againusing thedivergence theoremand taking∇h‚n) 0 on the edges of the domain to eliminate the first termon the right, we obtain

Similarly, the disjoining energy term becomes

where E(d) is given for general values of (n,m) in eq 2.13.Finally

may be recognized as the rate of working on the systembymeans of the injection. For the cyclical motion treatedbelow in section 4, the drop shape returns to its originalconfiguration and there is no change, over a cycle, in E1orE2. In that case the total viscous work done is the timeintegral of W or

where V is drop volume, dV ) wi dA, and the specialintegral sign denotes a full cycle.2.2. Numerical Techniques and Computational

Issues. Equation2.9 canbe solvedusing finite-differencemethods. It has the character of a higher-order diffusionequation and may be solved by marching in time.Equationsof this typecanbe treatedbyexplicit techniques,as was done by us in early work.26 Much more efficientmethods can be constructed, however. The stabilityrequirement for an explicit method is quite severe; it iseasily shown, for the model equation

that the maximum permissible time step for stability ∆tmust be no larger than order∆4, where∆ is the space stepormeshsize. Thus thecomputational requirementquicklybecomesmore severe as themesh size becomes small. Wehave found that ∆ must be comparable in size to thewetting-layer thickness h*, in dimensionless units, tomaintain accuracy in the “contact region” where themacroscopic drop meets the wetting layer.For the two-dimensional problems treated here, nu-

merical solutions use finite difference methods and analternating-direction-implicit (ADI) technique is imple-mented. Developedoriginally for second-order elliptic andparabolic systems,27 the ADI technique uses alternatingsweeps in each direction and only a banded system ofequations needs to be solved to update the discrete set of

hi,j values in a row or column. A version of the ADItechnique called “time-splitting” has been applied tohigher-order elliptic problems by Yanenko;28 similarlyanother variant, due to Conte and Dames,29 employs apredictor-corrector approach. We have used adaptationsof both techniques, and they have similar performance.They each produce a high degree of stability; maximumpermissible time steps can be as much as a factor of 105larger than the characteristicmaximum step for stabilityfor the explicit method. For the spatially fourth-ordersystem eq 2.9, pentadiagonal systems need to be solvedto update each row or column. Apparent contact lines arecaptured by themethod, and theirmotion appears as partof the evolving solution. There is no need to track or fittheir positions. This benefit is notwithout cost, however;the computation is also performed over portions of thedomain where only the thin, stagnant wetting layer ispresent. Nonlinear prefactors in eq 2.9 are evaluated atthe old time level. The method may be made second-order-accurate in time by use of a predictor-correctortechnique. Alternatively, it has been found that anadaptive time-stepping procedure, where the time step isadjusted dynamically on the basis of a preset maximumpermissible change in any h value, greatly increasescomputational efficiency. Temporal convergence can beverified by reducing the allowed maximum change in h.Further details of the unsteady simulation methodsemployed here are given by Moriarty and Schwartz30 foraxisymmetrical cases and by Weidner et al.,31 where atwo-dimensional problem is solved and an ADImethod isimplemented.Here the standarddiscretization andparameter values

are ∆ ) h* ) 0.02 for the spontaneous spreadingcalculations of section 3 and (n,m) ) (3,2). As explainedin theAppendix, the isolated square defect pattern shownin Figure 2a requires specification of three lengths: thepattern wavelength λ, the defect size w, and a length dthat characterizes the small length scale over which thewettability changes on the edges of each patch occur. Acubic polynomial is used over the smoothing length 2d, sothat the wetting function is continuously differentiable,as required by the field equation. The final inputparameter is the minimum value Gmin of the wettabilityG(x,y). For Gmin ) 1, the substrate is uniform. As Gminis reduced, the contact angle on the patches increases asthe field contact angle is reduced.The cyclical calculations of section 4 used a finer mesh

∆ ) 0.01 andh*) 0.01 or 0.02. Since the calculation timeis nominally proportional to the number of mesh points,these casesweremore time-consuming. Calculationsweremade using the isolated square pattern as well as thecheckerboard pattern of Figure 2b in order to discern theeffect of the wettability pattern. The formulas forgenerating the checkerboard pattern are also given in theAppendix.Thenumerical codesused to solve theevolutionequation

were written in the Fortran language. Runs were madeon a workstation with a PentiumPro 200 processor usingLinux, apublic-domainUnixoperating system. Graphicaloutput of evolving drop shapes can be stored and theresults later displayed sequentially as a “movie.” Run

(26) Schwartz, L. W. Phys. of Fluids 1989, A1, 443.(27) Peaceman, D. W.; Rachford, H. H. SIAM J. 1955, 3, 28.

(28) Yanenko, N. N. The Method of Fractional Steps; Springer-Verlag: New York, 1971.

(29) Conte, S. D.; Dames, R. T.Math. Tables Aids Comput. 1958, 12,198.

(30) Moriarty, J. A.; Schwartz, L. W. J. Colloid Interface Sci. 1993,161, 335-342.

(31) Weidner, D. E.; Schwartz, L.W.; Eres,M.H. J. Colloid InterfaceSci. 1997, 187, 243.

-E1 ) σ∫∫ht∇2h dA ) σ∫∫[∇‚(ht∇h) - ∇ht‚∇h] dA

-E1 ) - σ2ddt ∫∫∇h‚∇h dA ) - dE(σ)

dt(2.17)

-E2 ) ∫∫Πht dA ) -∫∫ de(d)dhht dA )

- ddt ∫∫e(d) dA ) - dE(d)

dt(2.18)

W ) ∫∫pwi dA (2.19)

Ip dV

ht ) -hxxxx


times varied from a fewminutes for coarse-mesh cases tomore than 24 h for fine meshes when the wettabilitycontrast is relatively large.

3. Simulation of Contact Angle Hysteresis inSpontaneous Spreading

We consider a droplet on a heterogeneous substratecomposed of a uniform field upon which a square patternof “grease,” that is, high-contact-anglematerial, has beensuperimposed. The contact-angle variation is given byspecifying the wettability function G(x,y), from eq 2.10,which is proportional to the square of the local equilibriumcontact angle. The particular pattern used in this sectionhasawavelength λ equal to 0.4R0,whereR0 is thenominaldrop radius, that is, the radius of a static drop if thesubstrate was uniform at the area-average wettability.The size of each grease patch is 0.1R0. The pattern isshown in Figure 2a. The smoothing length at the patchboundaries is given by d/λ ) 0.05. A variable is thewettability contrast, that is, the ratio of the wettabilitieson the grease patches and the surrounding field. Valuesof Gmin used here range from 0.1 to 1, corresponding toequilibrium contact angle ratios θe,max/θe,min ranging from12.04 down to 1. Many of the graphical results are forGmin ) 0.5; the contact-angle ratio for this value is 4.12.For all cases considered, the average value of G is thesame, that is,

where A is the substrate area.Figures 5 and 6 show drop central height versus time

over a range of values of Gmin. The starting profiles, att ) 0, are given by

corresponding to a paraboloidal drop superposed upon athin wetting layer of thickness h*. The figures showadvancing cases, where the initial central height is hc,0 )2, and receding cases with hc,0 ) 0.5. All calculations areperformed for 1/4 of a quandrant-symmetric drop. Sincethevolumeof the referenceparaboloidaldrop is (1/2)πh0R0

2,the liquid volume for a quarter-drop, above the wettinglayer, is π/8 in dimensionless units. As time proceeds,the drops move toward the nominal equilibrium positioncorresponding to central height h(0,t) ≈ 1. The actualfinal positions depend on the strength of the wettabilitypattern, characterized by the value of Gmin. The calcula-tions were allowed to proceed until the drops ceased to

move. For small wettability contrast, shown in Figure 6,the advancing and receding cases converged to the sameprofile for each value ofGmin. The curves forGmin ) 1 arethe uniform substrate cases. Also in Figure 6 are curvesforGmin)0.9and0.8. In these casesaswell, theadvancingand recedingdropsultimately find the samestableprofile.These stableprofilesavoid thehigh-contact-anglepatches.Since thearea-averagewettability is the same forall cases,as Gmin is reduced, the contact angle on the surroundingfield also goes down. This is the reason the final dropradii increasesomewhat, and thecentralheight is reduced,as Gmin is decreased. Figure 5, on the other hand, forlargerwettability contrast, shows that the advancing andreceding drops do not stabilize in the same positions. Thedifference in the final values of central height also reflectsdifferences in average drop radius or effective contactangle. This difference tends to increase somewhat asGminis reduced. Theprincipal cause is the inability of the dropto find a way of passing the stronger wettability defects.The result, when the wettability contrast is sufficientlystrong, that different statically stable final positions areachieved is the numerical demonstration of contact-anglehysteresis. For this particular substrate pattern, thecritical value of Gmin for CAH to disappear is about 0.75,corresponding to a contact-angle ratio of about 2.5 usingeq A2 in the Appendix.Figure 7 shows contour plots of the final positions for

the advancing and receding (one-quarter) droplets whenGmin ) 0.5. Also shown is the wettability pattern. Thedrop contours are drawn for h ) 0.026 and 0.1. For theadvancing case the contour h ) 1 is also shown. Noticehow, in advance, the drop margins are held up againstthe pattern. In the recede case, themargins are also heldup, in this case against the next “circle” of defects. Forthis value of Gmin, the contact angle on the field is lowerthan the average value by a factor of Gmin

1/2 ≈ 0.707. Thecorresponding equilibrium radius, using the field value ofcontact angle is thus larger than the nominal radius bya factor ofGmin

-1/2 )1.122. Thedrop is seen to be “resting”slightlyagainst thedefect in the (2,2)position, even thoughthis defect lies outside the nominal unit-radius circle.Three-dimensional pictures of the two stable droplets areshown in Figure 8. Indentations at the dropmargins areapparent, more so for the advancing droplet, where thepinning is stronger. The wetting layer thickness used inthe simulations, h* ) 0.02, is also visible in this figure.Because h* is taken to be constant over the entiresubstrate, values ofh on the thinwetting filmarevirtuallyinsensitive to the local contact angle. The final centralheights of the two drops are ha ) 1.04 and hr ) 0.80.

Figure 5. Central drop height versus time for substrateswithlargewettability contrast.Curvesare labeledwith theminimumvalue of wettability Gmin. Different limiting values in advanceand recede indicate contact angle hysteresis.

1A∫∫G(x,y) dA ) 1 (3.1)

h(r,0) ) max[h* + hc,0(1 - hc,0r2),h*] (3.2)

Figure 6. Central drop height versus time for substrateswithsmall wettability contrast. Curves are labeled with the mini-mum value of wettability. The curves labled 1.0 are for asubstrate of uniform energy. Each set of curves ultimatelymeets, indicating lack of hysteresis.


Because the volumes of the two drops are the same, theaverage or effective contact angle ratio is

which, in this case, is 1.14.

Figure 9 compares the configuration energies forassumed paraboloidal drops with the minimum energypositions found by the numerical algorithm forGmin ) 0.5.The simple theory used for calculating the curves in thefigure assumes the drops are represented by eq 3.2; theindependent variable is taken as the normalized radiusR ) (h(0) - h*)-1/2, where h(0) is the center height of theparaboloid. The decreasing curve in the figure is thedimensionless surface energy

and A is the area of the quarter-circle of radius R. Theundulating increasing function is the substrate energyE(d), calculated from eq 2.13a by integrating the actualpattern over the quarter-circle. The total energy retainsthe undulating character that it inherits from E(d). ForGmin ) 0.5, the amplitude of the undulations is notsufficient to produce a second local minimum in the totalenergy curve. For comparison, the energies for the twostable shapes, as calculated by the time-dependentprogram and shown in Figures 7 and 8, are shown hereas large dots. The radii values for these two points usethe actual center heights found by the calculation. Thesingle energy minimum predicted by the simple theory isclose to the calculated value for the receded case. Thedimensionless total energy for this case is E ) 2.065 foran equivalent radius of R ) 1.125. The other calculatedstable position is found by the advancing drop and has anenergyE) 2.064 atR) 0.994. The two calculated stablepositions are quite distinct; since their total energies arevirtually the same, however, it is not possible for one toevolve to the other, even over a long time period, since noexcess stored energy is available to overcome viscousdissipation. The simple theory used here, and its use inpredicting contact anglehysteresis by looking formultiplestable energy minima, is modeled after the early work ofJohnson and Dettre2 and Neumann and Good,3 who firstcalculated the hysteretic range assuming perfectly two-dimensional or axisymmetric geometries. Because thedrop can realize lower total energy by avoiding patchesof contamination, as for the recently advanced staticpicture in Figure 7a, other stable positions, that are notpredicted by the simple theory, are possible. Thus thesimple theory can underpredict the amount of hysteresisover complex patterns of contamination. This was rec-

Figure 7. Contour plots of final static shapes for advanced(top) and receded (bottom) drops forGmin ) 0.5 in spontaneousmotion. The wettability pattern is also shown. The differencebetween the two cases is ameasure of contact angle hysteresis.

Figure 8. Wire cage pictures of drops corresponding to thecontour plots in Figure 7; static advanced (a) and receded (b)drops for Gmin ) 0.5.

θaθr

) (hahr)1/2

(3.3)

Figure9. Plot of energyE versus drop radiusRusing a simplequasi-static theory. The total energy (solid line) is composed oftwo components (dashed lines). Here Gmin ) 0.5. According tothe simple theory, there is only one energyminimum,andhencenohysteresis. The time-dependent theorypredicts twodifferentstable positions shown by the dots.

E(σ) ) 1/2∫∫∇h‚∇h dA ) π4∫0Rhr2r dr ) π

4 R4(3.4)


ognized in our earlier quasi-static analyses,5,6 wherecontact-line shapes were calculated by variational meth-ods. Other calculations made with the simple modelindicate that Gmin must be smaller than about 0.1 for thetotal energy curve, as in Figure 9, to have a secondminimum.Figure 10 is similar to Figures 5 and 6 in that it shows

the variation of the drop central heighth(0,t) versus time.However here the initial paraboloidal drop shape is verymuch steeper; for the calculated results shown in Figure10, the initial central height h(0,0) ) 10. Log scales areused to demonstrate power-law behavior as the dropletexpands. A second objective for the calculations shownin Figure 10 is to determine a time-scale correction thatmust be applied to the present calculation because thewetting-layer thickness used is overly large.Using a lubrication-theory model for axisymmetric

droplet expansion on a perfectly wetting substrate,Tanner15 predicted that the droplet central height shoulddecrease as a function of time to the -1/5 power. He alsoconfirmed this prediction experimentally. The straightline shown in the figure fits the equation

withK ) 0.75, which is representative of spreading ratesfor water drops on high-energy substrates. Experimentsof Lelah and Marmur17 examined the spreading of waterdrops on clean glass and found that the droplet area Aincreased according to the law

where V is droplet volume and t is dimensional time. Cis a dimensional constant with units of speed to the one-fifth power. Values of C, obtained from curve fits to thedata, varied from 11 to about 15 in cgs units, with thelarger values obtained when the relative humidity washigh. It can be shown that the constants C and K satisfy

Using the values for water, µ ) 0.01 P and σ ) 72 dyn/cm,the value C ) 15 corresponds to K ) 0.75. Previous

calculations, using an axisymmetric form of the presentalgorithm that allows much larger spatial resolution,24produced spreading withK) 0.75 when h*/h0 ) 5× 10-4.Using the present program, with h*/h0 ) 0.02, yields theresults shown in Figure 10 by the solid curve and thelines-points curve. In the straight-line range, they areclosely approximated with K taken as 0.62. Thus, as h*is varied, spreading follows the same power law, but theconstant K in eq 3.5 is weakly dependent on the actualvalue of h*.Spreading speeds must be compared at the same drop

position rather than at the same time. Differentiating eq3.5 with respect to time, one obtains

Since the droplet expansion rate ∂R/∂t is proportional to-∂hc/∂t, the spreading speed is proportional toK-5. Usingthe calculated results above, for example, and changingh* from 2 × 10-2 to 5 × 10-4 reduce the spreading rateby a factor of about 2.6. This is a correction factor thatshould be applied to the time scale due to our presentinability to run a sufficiently fine spatial mesh to resolvevery thin wetting-layer thicknesses h* in multidimen-sional problems. Using values for water and assuming adrop with central height hc ) 0.02 cm and radius R ) 0.5cm, theunit of time isT*)3.25 s. Using currently feasiblevalues of h*, we estimate that the time-scale correctionshould increase T* by a factor of between 2 and 4. Thiswill of course depend on thenature of the spreading liquidand the experimental conditions such as the relativehumidity. This is consistent with local asymptotic analy-ses of spreading; those results15,16 indicate the speed ofadvance should scale as 1/log(h1/h*), where h1 is amacroscopic measure of droplet thickness.The calculated results shown in Figure 10 are found

using thepresent programwith finite equilibriumcontactangles. It is significant that they follow the same powerlawas for spreading onperfectlywetting substrates,whenthey are far from the equilibrium value hc ≈ 1. This isconsistent with the experimental observations of Zosel,32who noted that the nature of the substrate only appearedto influence spreading rates quite close to equilibrium. Itis also noteworthy that the results for the nonuniformsubstrate (lines-with-points) are essentially identical tothose for the uniform-substrate case (solid curve) untilthe drop nears its final position.Figure 11 illustrates an interesting transient effect

during the receding calculation forGmin )0.5. At an earlytime in the calculation, before the drop has contracted anappreciable amount, dry patches appear near the dropperiphery. These dry areas appear entirely within thereceding contact line. They are caused by high values ofdisjoining pressure immediately over the patches ofcontamination in the relatively thin border region of thedrop. Thisdewetting instability is absent in theadvancingcalculations, since the high interior capillary pressuremakes thedrop expand too rapidly for interior drypatchesto form. The receding contact line ultimately passes theinterior dry patches via a slow drainage process, and thefinal receding drop configuration is the one shown inFigure 8b.Figure 12 is a plot of total energy versus time for the

advancingandreceding caseswithGmin)0.5. Bothcurvesultimately reach the values E ≈ 2.06, but the receding

(32) Zosel, A. Colloid Polym. Sci. 1993, 271, 680.

Figure 10. Drop central height hc versus time, showing thesimilarity range for axisymmetric spreading. The initial dropheight is 10.The straight line on this log-logplot ishc)0.75t1/5,which matches experiments. The solid curve is the calculatedbehavior for finite contact angle on a uniform substrate, whilethe lines-points curve is calculated for the square wettabilitypattern with Gmin ) 0.5. The large-time height asymptote isalso shown.

hc ) h(0,t) ) Kt1/5

(3.5)

A ) CV3/5t1/5 (3.6)

K ) 2C(π2)

5/2( σ3µ)

1/5(3.7)

∂hc∂t

) - K5t6/5

) -hc6

5K5(3.8)


case proceeds at a much slower rate, due primarily to thedevelopment of dry patches as noted above. Also visible,on each curve, are “jumping events” that correspond tothe drop being temporarily hung up on, and finallybreaking free of, the patches of contamination.

4. Simulation of Cyclical Motions onContaminated Surfaces

The previous section dealt with spontaneous motionsin that the only driving mechanism is the energy storedin the free-surface shape and the substrate wettabilitypattern. In each case, the motion ultimately stops whenthe drop has found a stable position. Here we treat aclass of forced motions where a sessile drop is made tospread by slowly increasing its volume by injection, aswith a syringe. A full cycle is simulated by first injectingliquid into the drop and thenwithdrawing the liquid untilthe drop returns to its original shape and position. Thenotional experiment is shown schematically in Figure 13.This is the geometry analyzed in this section. Fordefiniteness, the injection is taken to be through thesubstrate, so that the liquid free surface remains smooth.Again we will calculate motions on one-quarter of aquandrant-symmetric drop. We note, however, thatsyringe injection through the free surface may be morelikely toensure that thedropremains centered inanactualexperiment.

The governing time-dependent lubrication equation (eq2.1) now includes the injection term wi(x,y,t). The dropisassumedtobecenteredover thesyringeandthe injectionprofile is taken to be axisymmetric,

where r) (x2 + y2)1/2. HereRi is ameasure of the injectionradius. All results below useRi ) 0.4R0. tc may be calledthe characteristic time for injection. Starting with aparaboloid of normalized central height and radius bothequal to 1, if wi is given by eq 4.1 and Q ) 1, then thevolume will be doubled over the time interval ∆t ) tc.Starting froman initial paraboloidal shape, adrop ismadeto expand and then contract according to the followingschedule:

The total period is 3tc and consists of a relaxation period,followed by a period of constant injection at unit injectionrate, a second relaxation period, and a final withdrawalperiod also at unit injection rate. For most of the resultsin this section the substrate wettability pattern is takento be a “checkerboard” rather than the square pattern ofdiscrete high-contact-angle spots. The checkerboardfunction is doubly periodic and is given by

where a is the input amplitude. For the results givenbelow, a will be 0.5 or, for a homogeneous substrate, a )0. The function f1 is a periodic power-law that variesbetween +1 and -1 and whose specific form is given inthe Appendix. The power-law exponent used here is 6;thisproducesawettability function that is almosta squarewave, but the discontinuities at patch boundaries aresmoothed.A contour plot for a statically stable drop on the

checkerboard is shown in Figure 14. The contours of

Figure 11. Frame during recede at t ) 0.014 for Gmin ) 0.5.A dewetting instability leads to interior dry patches near thereceding periphery. This is a transient effect, and the dropcopntinues to recede until the final configuration, shown inFigure 8b, is attained.

Figure 12. Total energy versus time: a comparison of theadvancing (A) and receding (R) cases; Gmin ) 0.5. Steep-sloperegions correspond to rapid events as themeniscii pass patchesof high-contact-angle material.

Figure 13. Schematic diagram of a notional experiment toinvestigate energy dissipation in a cyclic motion of a drop. Thepump piston slowly moves in and out while the pressure isbeing monitored in order to calculate the work done on, or by,the system.

wi(r,t) ) Q2tcRi

2e-r2/Ri

2(4.1)

0 < t < tc/2 Q ) 0

tc/2 < t < 3/2tc Q ) 1

3/2tc < t < 2 tc Q ) 0

2tc < t < 3tc Q ) -1

G(x,y) ) 1 + af1(x) f1(y) (A.3)


surface wettability G(x,y) are drawn for a value of Gslightly greater than the average value, so that the high-contact-angle regionsappearenclosed. Theratioof contactangles on the checkerboard elements is [(1+ a)/(1- a)]1/2

)x3. The drop edge contours are seen to deform to avoida portion of the high-contact-angle material. The defor-mation is much less, however, than that for the strongerisolated-spot pattern of Figure 7a, for example.Figure 15 is a pressure P versus volume V plot for an

injection-withdrawal cycle using tc ) 5. The rate ofinjection work, from eq 2.19, is

while the rate of volume increase is

The pressure p in eq 4.2 includes capillary and disjoiningcomponents as in eq 2.3. Since p is effectively zero in the

wetting layer outside the drop, the area integral in eq 4.2may be taken over the entire substrate. An averagepressure P is then

Pwill be the average pressure at the piston face in Figure13, provided themotion is sufficiently slow so that viscouswork in the syringe, between the piston and the dropopening, can be neglected. Note that P is defined in allphases of the motion including the relaxation periods. Ingeneral, positive work is done on the system during theinjection phase, while the system does work on the pistonduringwithdrawal. Nowork isdoneduring the relaxationperiods. In these periods, the drop seeks an equilibriumconfiguration as some of the stored energy is dissipatedbyviscosity according to eq2.15. Since thedrop is allowedto return to its equilibriumshape after a full cycle, the network done is equal to the total energy consumed in themotion. This work corresponds to the enclosed area foreach closed loop inFigure 15, in accordancewith theusualdefinition of hysteresis in mechanical systems.Threedifferent loopsare shown inFigure15. The range

of V is from π/8, the original volume of the quarter-dropin dimensionless units, up to twice that value. Motion ona homogeneous substrate with the same average wetta-bility, shownas thesolid-line loop, is comparedwithmotionon the checkerboard, denoted by the dashed loop. Thearrows indicate the direction of motion on each of theloops. The checkerboard loop shows significant irregular-ity, corresponding to the various contact-line distortionsduring the cycle. Energy is temporarily stored in thesedistortions and then is quickly released as the injectionor withdrawal proceeds. The enclosed area for thehomogeneous loop is 0.106; this is the dissipated energymeasured in units of σh02. For the checkerboard, thecalculated dissipation is 0.143. The difference may beconsidered to be the additional work done due to thedistortion caused by the pattern of wettability.Also shown in Figure 15 is a single curve denoted by

symbols. This is the result of a simple quasi-static theoryappropriate to a very slow cycle on a homogeneoussubstrate. Thedrop is takentobeaparaboloid thatalwaysmeets the substrateat theequilibriumcontact angle≈2h0/R0 as the volume is varied. The pressure within the dropis independent of positionand is equal to 4hc/R2,measuredin units of σh0/R0

2 and using the small-slope approxima-tion. The resultant dimensionless quasi-static pressure-volume relation is

The dynamic loop for the homogeneous substrate is seemto form an envelope around the quasi-static curve,indicating, for example, that the advancing contact angleexceeds the equilibrium value, while the contact angle inrecede is smaller than the equilibrium value. Both sidesof the loop approximately follow the V-1/3 law, implyingthat the dynamic angles are essentially constant duringthe appropriate phases. Since the speed of the dropletedges is not constant during either advance or recede (infact the “contact-line” speed is proportional to V-2/3), theoften-used ansatz that the dynamic contact angle is aunique function of speed is not supported by these results.Figure 16 is the pressure-volume plot, for the same

substrates, for amotion that is five times as fast. For the

Figure 14. Contour plot of a statically stable drop on thecheckerboard pattern. The ratio Gmax/Gmin is 3. The wettinglayer thickness is h*) 0.01h0. Contour levels shown are 0.013,0.1, and 0.7. On a uniform substrate with the same averagewettability, the drop central height h(0,0)/h0 and radius R/R0are each about equal to 1. The checkerboard is drawn so thatthe high-contact-angle regions are enclosed.

Figure15. DimensionlesspressurePversusvolumeV showinghysteresis loops in cyclical motion. The cycle time tc ) 5. Thesolid line is for a uniform substrate, and the dashed line is fora checkerboard wettability pattern. The ratio Gmax/Gmin is 3.The symbol curve is for the nondissipative quasi-static theorywith the equation P ) 2π1/3V-1/3. Arrows show the direction ofmotion around the loops. The energy dissipation for the fullcycle, corresponding to theenclosedarea, is0.106 for theuniformsubstrate and 0.143 for the checkerboard.

dWdt

) ∫∫pwi dA ) Q2tcRi

2∫∫p(x,y)e-r2/Ri2dA (4.2)

dVdt

) ∫∫wi dA ) π8Qtc

(4.3)

P ) dWdV

) 4πRi

2∫∫pe-r2/Ri2dA (4.4)

P ) 2π1/3

V1/3(4.5)


homogeneous substrate, the dissipation is 0.480, aboutfive times as large as that for the previous slow motion.That thedissipationshouldbeapproximatelyproportionalto the speed, in a forced motion, follows from elementaryconsiderations; dynamic pressure differences are propor-tional to speed when inertial effects are negligible, as ineq 2.2. In advance the V-1/3 law is still approximatelyvalid, indicating that theeffectiveadvancing contact angleis almost constant. The receding pressure ismore nearlyconstant, representing a large deviation from the quasi-static law. The calculateddissipationon the checkerboardpattern is larger than that for thehomogeneous substrate;however, the additionalwork is only a bit larger than thatfor theslowcycle showninFigure15. Thepattern-crossingundulations in the pressure are considerably reducedcompared with those for the slow case. This indicatesthat the drop does not have enough time to deformappreciably at any given location on the pattern.Results for another slow cycle and shown in Figure 17.

Thedifference is that the pattern is now the isolated spotswithGmin )0.8. The corresponding ratio of contact anglesis x5 ≈ 2.236 while, for the checkerboard of Figure 15,the contact angle ratiowasx3≈1.732. Because the ratiois larger, stronger “pinning” can be expected as the dropmargins become temporarily trapped near the edges ofthegreasepatches. Sincevolumechangesareproportionalto time changes, either in injection or withdrawal, thelarge slope regions on the pressure-volume curve areindicative of “jumping”motions. By comparisonwith theresults of section 3, it is seen that Gmin ) 0.8 is not smallenough for contact-angle hysteresis in spontaneous mo-tions. Thus, for slow enoughmotions, a closed pressure-volume loop is attained on the very first cycle. While theisolated pattern shown here is clearly more effective forpinning the drops, it actually dissipates less energy percycle than the checkerboard. Thenumerical value is 0.131compared with 0.143 in Figure 15.

5. Discussion and ConclusionsThe feasibility of calculating the unsteady motion of a

liquid droplet on particular patterns of wettability, usinga disjoining pressuremodel to characterize the substrate,hasbeendemonstrated. Calculationsareperformedusinga stable and efficient alternating-direction-implicit algo-

rithm. Estimates of the degree of heterogeneity requiredto produce contact angle hysteresis on various patternscan be made using this procedure. It is found that evenwhen theheterogeneity is insufficient to result inmultiplestable configurations, the motion can be slowed signifi-cantly by the substrate defects. We suggest, therefore,that, in the interpretation of experimentalmeasurementsof contact angle hysteresis, care must be taken to ensurethat sufficient timehaselapsedbefore thenominal steady-state angle is measured. Slow relaxation to equilibriumis especially likely for receding contact lines, as has beenshown in the calculations of Figure 6. Receding contactlines lead to relatively thin liquid layers; these thin layersaresusceptible to interiordewettingoversubstratedefects,as shown in Figure 11.A basic issue in industrial applications is the level of

cleanliness thatmustbemaintained for successful coating.The speed of spreading motions on imperfect, that is,contaminated, surfaces is generally relevant here. Thetime scale required to overcome substrate defects mustbe compared with the characteristic time for drying of acoating. While there may be no hysteresis in a quasi-static sense for an assumed nonvolatile liquid, weakdefects still impede spreading and may result in inad-equate coverage in an actual dry coating.Theresultsof thepresentdynamicstudiesare consistent

withpreviousquasi-static analysesofCAHin thatmotionscease when the system finds a local minimum of totalenergy. However, the very simplest static theories thatassume the shape of the liquid surface to be of a simpleform can fail to predict CAH when it is present. Even ifsystemenergies are calculated indetail, as in ourpreviouswork,5,6 the necessary assumption, that the systemprogresses fromone energyminimumto another in forcedmotions,may lead to incorrect results at all but the slowestpossible speeds. Because the present work is a directnumerical simulation, there are no inherent restrictionson thegeometryof the liquid regionor thepossiblepatternsof wettability. The quasi-static analyses are essentiallytheoretical methods that employ rather sophisticatedgeometric and variational techniques. The dynamicsimulation is conceptually simpler and is easier to

Figure16. DimensionlesspressurePversusvolumeV showinghysteresis loops in cyclical motion. The cycle time tc ) 1. Thismotion is 5 times as fast as the one in the previous figure. Thesolid line is for a uniform substrate, and the dashed line is fora checkerboard wettability pattern. The ratio Gmax/Gmin is 3.The symbol curve is the same as that in the previous figure.Arrows show the direction of motion around the loops. Theenergy dissipation for the full cycle is 0.480 for the uniformsubstrate and 0.534 for the checkerboard.

Figure17. DimensionlesspressurePversusvolumeV showinghysteresis loops in cyclical motion for the isolated squarepattern. The cycle time tc ) 5. The uniform substrate (solid)and the isolated square pattern (dashed line) with Gmin ) 0.8are compared. h* ) 0.01, and ∆ ) 0.01 for both. Arrows showthedirection ofmotionaround the loops.The energydissipationfor the full cycle is 0.106 for the uniform substrate and 0.131for the isolated square pattern. While the isolated pattern ismore effective for pinning the drop edges, since the wettabilitycontrast is higher, 5 versus 3 for the checkerboard, thecheckerboard is more effective for dissipating energy.


implement. Itmakesuse of computational resources thatwere not available when the quasi-static theories wereformulated.Forced cyclicalmotions onhomogeneous andpatterned

substrates have been investigated. On a perfectly ho-mogeneous substrate, low-speed axisymmetric motionscompare well with the result of a simple quasti-statictheory. For the patterned substrates, the substrate-averaged wettability is the same as that for the homo-geneous case. When a pattern is used, the total energydissipation per cycle increases. By comparing cyclicalmotionsat different speeds, it hasbeendemonstrated thatthe relative importance of the pattern increases as thespeed is reduced. Isolated-spot patterns cause greaterdeformations of the drop and more rapid changes in thetotal system energy. It cannot be concluded, however,that these greater distortions lead to greater overalldissipation. Comparing the cyclical motion on two dif-ferent substrate patterns, we find that the dissipation isactually greater for a pattern with a smaller wettabilitycontrast. The more dissipative pattern is the “checker-board” shown in Figures 2 and 14. While the wettabilitycontrast is smaller for this pattern, it is noted that thetotal arc length of the wettability boundaries is larger bya factor of 4 for the checkerboard, when compared withthe spot pattern. Clearly both the wettability contrastand the length of the boundaries contribute to the addeddissipation caused by the pattern.Applicationofawettabilitypattern to thewalls ofvessels

containing liquid is a possible strategy for controlling theposition of the liquid in the microgravity environment oforbiting space vehicles. In the absence of gravity, liquidin a partially filled container can assume many differentconfigurations. Small imposed accelerations can causelarge displacements of the liquid, which is undesirablefor a variety of reasons. Contact line dynamics will playan important role, either by “pinning” the location of theliquid on the wall or by dissipating the kinetic energyimparted to the liquid. The present results suggest thatdifferentwall wettability patternsmay accomplish one orthe other of these objectives.The disjoining model used here has a single stable

energy minimum. A thickness less than the equilibriumthickness h* will be repelled from the substrate while athicker layerwill be attracted to it. The local shape of theenergy “well” is given by its second derivative at h ) h*,which, using the present model, is

The choice of exponent pairs (n,m) determines the depthof the energywell; this is illustrated inFigure 4. Previouswork24,33 has shown that a choice such as (4,3) or (9,3)increases the “stiffness” of a drop, relative to that for the(3,2) pair used here. The deformation of droplet edges atwettability boundaries can be expected to be smaller thanwhat is calculated here when the other pairs are used.Larger gradients in e(d)(h) require somewhat finer nu-merical meshes, resulting in longer computation times.This is the primary reason for the exponent choice usedhere. More complicated functional forms for e(d)(h) havealso been used in static and dynamic studies.11,18,22 Theeffect of this choice of function on the time-dependentdynamics ofmotiononmixed-wettable substrates remains

to be explored. Replacement of the present disjoining lawwith other forms is easily done in the numerical simula-tion.Wetting-line speeds in spontaneous motions are con-

trolled by the thin equilibrium layer thickness h*. Forspreading on homogeneous substrates, spreading ratesare virtually independent of the value of the static contactangle until the drop has almost stopped. In this regimethe disjoining function also is unimportant. This inde-pendence has been observed experimentally32 and con-firmed by calculation.24 The speed dependence on h* isknown to follow a logarithmic law for the homogeneouscase.15,16,25 Forone-dimensional oraxisymmetricproblemsit is possible to use realistically small values of h* so thatgood agreement with experiment can be expected. Ac-curate simulation results require that the scaled meshsize ∆ be comparable with or smaller than h*; thus,realistic valuesofh* cannotyetbeused in two-dimensionalsimulations. The logarithmic law predicts that reducingh* by a factor of order 102 to 103 will lead to a speedreduction factor between 2 and 4. Thus the spreadingrates calculated here are expected to be larger than whatwill be observed experimentally.The dependence on h* is somewhat different for forced

motions. For very slowmotions onemay assume that thelocal speed of the contact line U is directly proportionalto the injectionrateand thus is independentofh*. Assumethat a local force F, per unit contact-line length, drivesthe motion. F is balanced by the integrated shear stresson the substrate

usingdh/dx≈θe,which isappropriate to thewedge-shapedcontact region where most of the viscous work is done. Ifthe limits of integration on h are h* and some h1, oneobtains25

Here h1 is a macroscopic drop thickness that is assumedconstant as h* f 0. Identifying h1 with the drop centralheight, which is approximately 1 in dimensionless units,the local rate ofworkingFUwill vary logarithmicallywithh*. Equation 5.3 predicts that replacing h* ) 0.01 by h*) 0.02 will reduce the work done per cycle by 15%. Thefour cases of forced motion whose results are shown inFigures 15 and 16 were rerun with h* ) 0.02. In eachcase the totalworkdonewasreduced; thereductionsvariedfrom seven to 15%.There is a need for experimental measurements that

canbe comparedwith the calculated results. The presentmathematical model employs a variety of simplifyingassumptions; thus, experimental validation is requiredfor specific features of themotions predicted. A candidateor “notional” experiment is outlined in section 4. Energytransfer to and from that system can be obtained if theinjectionpressures canbeaccuratelymeasured. Thiswillrequire pressure resolution of 1 mm of H2O or less.Regular patterns of wettability on scales of 10-100 µmcan be produced using microfabrication techniques, asemployed in the electronics industry, for example. Low-power microscopy can be used to observe the dropletdistortions near the moving contact lines.Naturally occurring heterogeneous surfaces will have

characteristic lengths for the inhomogeneity that can be(33) Schwartz, L. W. Proc. 3rd Micrograv. Fluid Phys. Conf. 1996,

3338, 609.

[d2e(d)dh2 ]h)h*

)σθe

2

2h*2(m - 1)(n - 1) (5.1)

∫τ(0) dx ) 3µU∫ dxh ) 3µUθe∫ dhh (5.2)

F ) 3µUθe

log(h1h*) (5.3)


significantly smaller than this. It will be useful toinvestigate to what extent the hysteretic behavior on anaturally heterogeneous surface, that must also exhibita degree of randomness, is similar to that on a preparedpatterned surface. Some aspects of randomness and itsinfluence on hysteresis have recently been discussed byCollet et al.34The principal purpose of the present work has been to

demonstrate the feasibility of time-dependent simulation.Relatively few defects have been traversed by themovingboundary of the drop. It is possible, at greater compu-tational cost, to considermoredefectsanddefectsarrangedin more complicated patterns in order to more closelysimulate behavior on a natural material. For relativelylarge drops, gravity can be included in the simulation.This has been done for a related problem24 and can beincluded here without difficulty.

Acknowledgment. This work is supported by theNASAMicrogravity Program, the ICI Strategic ResearchFund, and the State of Delaware.

Appendix: Substrate Wettability Functions(i) Isolated-Spot Patterns. We consider a planar

substratewith a constant averagewettability. Aperiodicsquare pattern of wettability with wavelength λ isgenerated by the following procedure. The wettabilityfunction C(x,y) in eq 2.9 is given by

where

Here 0 e Gmin e 1 is the minimum value on the pattern,corresponding to the “clean” or high-energy areas. w isthe fraction of λ occupied by the patch of contamination.The function f is a periodic piecewise smooth function ofone variable consisting of constant values connected bycubic polynomials. It is given by

where the square brackets signify the “integer part”function and F(ê) is

Here we have defined the constants

and the linear functions

The small quantity 2d < w is the width of the regionbetween the two constant plateau values. d is alsoexpressedasa fractionofλ. It isused toprovidecontinuousderivatives forC(x,y), asused ineq2.9. Secondderivativesare bounded but discontinuouswhere the cubicmeets theconstant values. It is easily verified that

thus the average value of wettability is constant, ir-respective of the value of Gmin. If Gmax is the maximumvalue ofG on the pattern, the ratio of equilibrium contactangles is

since G(x,y) is proportional to θe2, under the small-slope

approximation.(ii) Checkerboard Patterns. The checkerboard pat-

tern of Figure 2b is generated by

where 0 e a e 1 is an input amplitude parameter. Thefunction f1 is odd with period λ. Let

where again the square brackets signify the integer-partfunction. The range of η is (-λ/2,λ/2). f1 is given by

and

Itmay be verified that f1 has a continuous first derivativeand a bounded, but discontinuous, second derivative at η) 0. The input parameter m is a positive even integer.Since the range of f1 is [-1,1], themaximumandminimumvalues of G are Gmax ) 1 + a and Gmin ) 1 - a. It maybe verified that the area-average wettability is also equalto 1 for the checkerboard.

LA971407T(34) Collet, P.; De Coninck, J.; Dunlop, F.; Regnard, A. Phys. Rev.

Lett. 1997, 79, 3704.

C(x,y) ) 4G(x,y)

G(x,y) ) Gmin +1 - Gmin

w2f(x) f(y) (A.1)

f(x) ) F(x/λ - [x/λ];w,d)

1 0 e ê e ê1

12

- 34(R1 -

R13

3 ) ê1 < ê e ê2

0 ê2 < ê e 1 - ê2

12

+ 34(R2 -

R23

3 ) 1 - ê2 < ê e 1 - ê1

1 1 - ê1 < ê e 1

ê1 ) w2

- d, ê2 ) w2

+ d

R1(ê) )ê - w

2d

R2(ê) )ê - 1 + w

2d

∫01∫01G(x,y) dx dy ) 1

θe,maxθe,min

) 1w(1 - Gmin(1 - w2)

Gmin )1/2 (A.2)

G(x,y) ) 1 + af1(x) f1(y) (A.3)

η ) x + λ4

- [x + 3λ/4λ ]λ

1 - (4ηλ

- 1)m η > 0 (A.4a)

-[1 - (4ηλ

+ 1)m] η < 0 (A.4b)


hysteretic effects in droplet motions on heterogeneous substrates: direct numerical simulation

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