an energy balance approach to modeling the hydrodynamically driven spreading of a liquid drop

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 182 (2001) 109–122

An energy balance approach to modeling thehydrodynamically driven spreading of a liquid drop

David Erickson, Byron Blackmore, Dongqing Li *Department of Mechanical and Industrial Engineering, Uni6ersity of Toronto, Toronto, Ontario, Canada M5S 3G8

Received 12 April 2000; accepted 11 December 2000

Abstract

This paper extends the overall energy balance (OEB) approach to the modeling of liquid drop spreading to caseswhere the primary motive force is hydrodynamic in nature. In such cases the mass, and therefore total energy, of thesystem varies with time thus invalidating previous OEB models. By relating the change in internal energy of the dropwith the energy entering the system, changes in potential energy and work done by the system, a non-linear first-orderordinary differential equation has been developed which describes the drop spreading process for the case of aspherical cap of constant dynamic contact angle and constant mass flow rate. To provide validation data for themodel, an experimental study has also been undertaken. In these experiments the Axisymmetric Drop Shape Analysistechnique has been used to monitor the changing contact angle and contact radius of the drop as it advances acrossa solid surface. The solution of the ODE is shown to be in very good agreement with the experimental results, untilsuch time as the drop grows to beyond the range where the spherical cap approximation is valid. A comparison ofthe magnitude of the energetic terms has revealed that the major deterrent to drop spreading is the work required toincrease the surface area of the drop. In the case of slow speed spreading, it is shown that the viscous dissipation atthe three-phase line is negligible, leading to a model completely free of any empirically determined curve fittingfactors. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Liquid drop spreading; Moving contact line; Overall energy balance

www.elsevier.nl/locate/colsurfa

1. Introduction

The spreading of a viscous liquid drop on asolid surface is a problem of both fundamentalconcern and practical interest. Applications rangefrom large scale industrial lubrication to ink-jetprinting, from spray cooling to road wetting, and

adhesion to metal forming in material processing.A large number of reviews have been written onthe subject of drop spreading and wettability andare available elsewhere [1–4].

Generally, previous papers have addresses thestudy of drop spreading in one of two-ways, as aclassical fluid mechanics problem [5–7] orthrough a surface physics approach [8–10]. Twomajor difficulties arise from the fluid mechanicsapproach: modeling the slip condition at the

* Corresponding author.E-mail address: [email protected] (D. Li).

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

PII: S0927 -7757 (00 )00834 -7

D. Erickson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 182 (2001) 109–122110

three-phase line and determining the relationshipbetween the dynamic contact angle and the veloc-ity of the advancing contact radius [6]. The sur-face physics approach has revealed that thespreading process is dependent largely on theinterfacial tensions and contact angles as well asvarious chemical, physical and quantum mechani-cal parameters, which may or may not be possibleto determine [11]. Models resulting from the com-bination of the two approaches have been success-ful, however they are generally largely dependenton empiricism to quantify the solution [11].

Other researchers [12–16] have studied the rela-tionship between dynamic contact angle and theadvancing speed of the three-phase line. A num-ber of these studies have developed simple, semi-empirical equations, which generally relate thedynamic contact angle to the capillary numberplus a ‘shift factor’ [12]. For example Kalliadasisand Chang [16] use a precursor film model andscaling arguments to develop the relationshiptanud= [tan3 ue−9 log hCa]1/3, where Ca is thecapillary number, h is a scaled Hamaker constant,and ue and ud are the equilibrium and dynamiccontact angles.

In principal a general theoretical model fordrop spreading must account for not only themotion of the bulk phase but also the continuouschanges in the surface and line phases, viscousdissipation, gravitational potential energy andwettability. Mathematically this kind of motionover a solid surface is described by high orderdifferential equations to even the crudest ofapproximations.

Madejski [17] is likely the first to employ anoverall energy balance (OEB) approach to modeldrop spreading, specifically considering the so-lidification of impinging drops on a cold surface.In that paper Madejski proposed that the changein the total surface and kinetic energy of the dropmust be completely balanced by the internal vis-cous dissipation, thus yielding Eq. (1) (using thenomenclature of this paper):

ddt

(Ek+Es+W6)=0, (1)

where Ek, Es and W6 are the internal kineticenergy, surface potential energy and work of vis-

cous dissipation, respectively. Madejski deter-mined the internal kinetic energy by assuming asimple velocity profile within the drop and thenused this profile to estimate the internal viscousdissipation using Newton’s law. Surface energywas calculated by considering only the change inarea of the liquid–vapour interface. San Marchiet al. [18] used Madejski’s manipulation of theenergy components in Eq. (1) as a basis for theirmodel of impinging drops with partialsolidification.

More recently Gu and Li have used the OEBapproach to model both spontaneous [19] and lowspeed impact [20] drop spreading. Their analysisyielded a similar OEB equation to that of Made-jski’s (for completeness the gravitational potentialenergy was also considered, but found to be negli-gible), however their calculation of the energycomponents was dramatically different. In theirapproach a numerical simulation of the impactingdrop was used to determine the internal velocityprofile and thus the internal kinetic energy. Dissi-pation work was calculated by considering theviscous force at the three-phase line. A more exactapproach to the surface energy term was taken byconsidering the changes in the surface areas of allthree interfaces (liquid–vapor, liquid–solid andsolid–vapor). Their model proved very successfulin modeling spontaneous spreading, and wasfound to slightly overestimate the growth rate forhigher speed impacting.

Generally, dynamic wetting can be classifiedinto two broad categories [15], spontaneous andforced, the latter of which can be further classifiedby the primary motive force. Spontaneous spread-ing can be thought of as the migration of liquiddrop over a solid surface as it approaches thermo-dynamic equilibrium. It is this type of spreadingthat the original model by and Gu and Li [19] wasoriginally designed to simulate.

The more general and industrially applicablecase is that of forced spreading. In the case of theMadejski [17] and second Gu and Li paper [20]the primary driving force was the initial kineticenergy of a drop of fixed mass. In many applica-tions the primary driving force is hydrodynamicin nature, meaning that drop spreading is accom-plished through the addition of liquid (mass) to

D. Erickson et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 182 (2001) 109–122 111

the system. This is the essential mechanism insome industrial coating processes, polymer pro-cessing technologies (such as mold filling) andwelding. A common method of failure in thesecases is the inability of the fluid to displace suffi-cient surface area quick enough, thus limiting thespeed at which the process can be done [15]. As aresult simple and reliable models of predicting thedrop spreading speed are required.

In this paper we will extend the OEB model ofdrop spreading to account for the more generalcase of hydrodynamically-driven drop spreading.As will be shown the overall energy balance isdramatically different for this case than thoseprevious examples, due to the addition of mass,and therefore energy, to the system. Since thegeometry of the drop spreading process can varydramatically from application to application, ageneral derivation using a spherical cap approxi-mation will be presented. By using a similar evalu-ation of the applicable energy terms as the Guand Li model, we will show how this drop spread-ing phenomena can be modeled using a numericalsolution to an ordinary differential equation withlittle or no degree of empiricism. A comparison ofthe magnitudes of the various energetic terms willbe conducted and their effect on the overall dropspreading process will be discussed. An experi-mental study has also been undertaken to providevalidation data for the OEB model. The experi-ments use the Axisymmetric Drop Shape Analysis[21] technique to measure the changes in contactradius and dynamic contact angle as the dropadvances across the surface.

2. Formulation of the OEB drop spreading model

Fig. 1 shows a schematic of a growing non-volatile droplet on a smooth, rigid and homoge-neous solid surface. The primary driving forcebehind drop spreading in this case is the quasi-steady addition of liquid to the system through ahole in the surface (at point O). In the followinganalysis it has been assumed that the drop spreadsaxisymmetrically about this point and that it is inthermal equilibrium with its surroundings. In gen-eral the dynamic contact angle, ud, the contact

radius, R, and the advancing speed of the threephase line (or drop spreading speed), dR/dt, allvary as a function of time.

As mentioned Section 1, previous energy bal-ance approaches to drop spreading have consid-ered only the case of a drop of fixed massimpinging on a solid surface. There are manyengineering applications where the addition ofenergy to the system (in the form of mass trans-fer) is the primary motive force behind spreading,thus invalidating these previous models. A moregeneral OEB equation would consider the amountof energy transferred to the system and the totalinternal energy of the system in addition to theapplicable potential energy and work terms,which are the basis of the previous energy balancemodels. Thus for the case shown in Fig. 1, theenergy entering the system (Ein) is balanced withthe change in internal energy of the system (Esys),increase or decrease in surface (Es) and gravita-tional (Eg) potential energies, boundary move-ment work (Wb), and the viscous dissipation work(W6). Thus the OEB equation for the systemshown in Fig. 1 is given by Eq. (2):

dEin

dt=

ddt

(Esys+Es+Eg) +ddt

(W6+Wb). (2)

In general Eq. (2) may contain a heat transferterm on the right hand side to account for anyheat transferred from or to the drop during thespreading process. However, for the case of aliquid drop in thermal equilibrium with its sur-roundings and negligible internal heat generation,such as the case here and in previous OEB models[17,19,20], this term can be ignored. Each of theterms in Eq. (2) will be examined separately in thefollowing sections.

Fig. 1. Schematic diagram of the hydrodynamically forcedspreading of a sessile drop on a solid surface.


2.1. Energy transferred to the system, Ein

For the case of interest in this study, the energytransferred to the system for an arbitrary changein mass Dm is defined by the enthalpy and kineticenergy of the liquid entering the system at O, asshown in Eq. (3) [22]:

DEin=Dm�

h(To)+62

2n

, (3)

where h(To) is the enthalpy evaluated at the sys-tems equilibrium temperature To and 6 is theentrance velocity. Redefining the enthalpy termaccording to its definition yields:

DEin=Dm�

u(To)+Pr

+62

2n

, (4)

where P is the absolute pressure of the fluid at thejet entrance, r is its density and u(To) its internalenergy. In general both u and h are functions ofboth pressure and temperature however for in-compressible liquids, such as those examined here,the dependence on pressure is negligible.

In principal an exact value for the entrancepressure can be determined through a numericalsolution to the Laplace equation of capillarity.For small drop sizes however (such as those con-sidered here) or small differences between thedensities of the drop and its surrounding medium,surface forces will dominate and the droplet willassume a basically spherical shape. In such a casethe pressure difference will be uniform throughoutthe drop and equal to the sum of the Laplace andatmospheric pressures, as given by Eq. (5):

P=2glv sin (ud)

R+Po (5)

where glv is surface tension of the liquid–vaporinterface. By combining Eq. (4) and Eq. (5) andconsidering an infinitesimal change in mass withrespect to time, Eq. (6) is obtained which is thesought after rate of change of incoming energy asa function of time:

dEin

dt=�

u(To)+1r

�2glv sin (ud)R

+Po

�+62

2n dm

dt.

(6)

2.2. Internal energy of the system, Esys

Analogous to the previous section the totalinternal energy of the system is defined as the sumof the internal energy and kinetic energy terms. Inprincipal the total internal kinetic energy of thedrop should be evaluated by integrating the veloc-ity profile over its volume, as is done in previousOEB models. By an order of magnitude analysis(in Appendix A), it is shown that the internalkinetic energy of the drop can be ignored whenthe surface area of the sessile drop is significantlylarger than the area of the liquid jet. As this is thecase for most examples of hydrodynamicallydriven drop spreading, the internal kinetic energywill be ignored and only the internal energy u willbe considered. Although the temperature of thesystem remains constant, the total internal energychanges with the amount of mass in the system asper Eq. (7):

DEsys= [(m+Dm) u(To)−mu(To)]. (7)

Reducing the above equation and consideringan infinitesimal change in mass with respect totime yields Eq. (8):

dEsys

dt=u(To)

dmdt

. (8)

2.3. Surface potential energy, Es

The total surface energy of the system is equiv-alent to the sum of the grand canonical freeenergy of the three interfacial phases (liquid–solid, liquid–vapor and solid–vapor):

Es=glsAls+gsvAsv+glvAlv, (9)

where g and A represent the surface tension andarea of the three interfacial phases describedabove. Differentiating with respect to time andrecognizing the equality between the change insolid–vapor and solid–liquid surface areas yieldsthe change in surface potential energy with time,Eq. (10):

dEs

dt= (gsl−gsv)

dAsl

dt+glv

dAlv

dt. (10)


For a homogeneous, rigid and smooth solidsurface the surface tensions can be related usingYoung’s equation:

glv cos ue=gsv−gsl, (11)

where ue is the equilibrium contact angle which isin general not equivalent to the dynamic contactangle. The change in the solid–liquid and liquid–vapor surface areas of an axisymmetric and spher-ical drop can be estimated from Eqs. (12a) and(12b) and Eq. (12c):

dAsl

dt=2pR

dRdt

, (12a)

dAlv

dt=4pRh(ud)

dRdt

+2pR2 dh(ud)dt

, (12b)

h(ud)=1−cos ud

sin2 ud

. (12c)

Combining Eqs. (10), (11) and 12 results in thetotal change in the surface potential energy, givenby Eq. (13) below:

dEs

dt=2pRglv [2h(ud)−cos ue]

dRdt

+2pR2glv

dh(ud)dt

. (13)

2.4. Gra6itational potential energy, Eg

The change in the gravitational potential energyfor an arbitrary change of mass, Dm, is dependenton two variables, the total mass in the system andthe change in the height of the center of gravity,Dz, as described by Eq. (14):

DEg=g [(m+Dm) (z+Dz)−mz ], (14)

where g is the acceleration due to gravity and z isthe location of the center of mass. In evaluatingthis term it is important to realize that buoyancyeffects have been neglected. In the case where thedensity difference between the drop and its sur-rounding medium is small, the buoyancy effect onthis term would have to be considered. ExpandingEq. (14), eliminating the higher order terms andagain considering an infinitesimal change withrespect to time yields the general gravitationalpotential energy term, Eq. (15):

dEg

dt=g

�m

dzdt

+zdmdtn

. (15)

To evaluate this term, assuming dm/dt isknown, requires knowledge of the total mass ofthe system and location of its center of gravity.The former of these can be determined throughevaluation of Eq. (16):

m=mo+& t

0

dmdt

dt, (16)

where mo is the initial mass of the system, whichcan be estimated from the initial radius, Ro, andcontact angle as below:

mo=prRo

3 (1−cos ud)2 (2+cos ud)3sin3 ud

. (17)

For a spherical drop it can be shown that thecenter of mass is given by Eq. (18a) Eq. (18b):

z=R4

f(ud), (18a)

f(ud)

=6 sin2 ud−3(1−cos ud)2−8(1−cos ud)cos ud

(2+cos ud)sin ud

.

(18b)

Thus the general result for the change in thecenter of gravity with respect to time is given byEq. (19):

dzdt

=f(ud)

4dRdt

+R4

df(ud)dt

. (19)

Combining Eqs. (15), (18) and (19) yields thedesired relation for the change in gravitationalpotential energy with respect to time:

dEg

dt=g

�m�f(ud)

4dRdt

+R4

df(ud)dt

�+

R4

f(ud)dmdtn

.

(20)

2.5. Boundary mo6ement work, Wb

In the case of the fixed mass, or more appropri-ately fixed volume, droplets considered in theprevious OEB treatments obviously the boundarymovement work must be identically zero. In thepresent treatment however, work must be done todisplace the atmospheric pressure as the dropgrows beyond its original volume. For an arbi-trary change in volume, DV, against a constant


atmospheric pressure, this is expressed by the wellknow equation shown below:

DWb=PoDV. (21)

Again we consider an infinitesimal change involume with respect to time to obtain the desiredresult, Eq. (22):

dWb

dt=

Po

r

dmdt

. (22)

2.6. Viscous dissipation work, W6

The de Gennes framework for dissipative losseshas recently been employed successfully to modelwetting behavior in a number of cases [19,20,23].In his work [24] de Gennes concluded that forlow-speed spreading the hydrodynamic lossesdominate over molecular forces and derived thefollowing formula for the viscous force per unitlength of three-phase line:

F6(t)=3m

ud

ln(o−1)dRdt

, (23)

where m is the liquid viscosity and od=Ld/L is theratio of the microscopic to macroscopic cut-offlengths. Ld is the cut-off length below which thecontinuum theory breaks down and is though tovary from 1 to 5 mm. L is proportional to thehorizontal length scale of the liquid drop, thusL=R. Detailed information on the application ofthis ratio to drop spreading is available elsewhere[19]. By multiplying Eq. (23) with the velocity ofthe advancing front (dR/dt) and integrating overthe three-phase line, the viscous dissipation workper unit time is obtained as below:

dWv

dt=6pm ln(o−1)

Rud

�dRdt

�2

. (24)

2.7. General OEB equation

Combining the results of Sections 2.1, 2.2, 2.3,2.4, 2.5 and 2.6 with the general OEB Eq. (2)yields the following general differential equationdescribing the spreading of a viscous drop on ahomogeneous, smooth, solid surface:

�6pm ln(o−1)

Rud

n �dRdt

�2

+�

2pRglv [2h(ud)−cos ue]+mgf(ud)

4n dR

dt

+dmdt

�gR4

f(ud)−2glv sin(ud)

rR−62

2n

+2pR2glv

dh(ud)dt

+mgR4

df(ud)dt

=0, (25a)

where f(ud) and h(ud) are as defined above. Inprincipal Eq. (25a) relates the speed of the ad-vancing contact radius to the material propertiesand the dynamic contact angle. It should be notedthat the effect of the solid surface on which thedrop is spreading, is accounted for in the abovemodel by the solid–vapor and solid–liquid freeenergies, related to the equilibrium contact anglethrough Young’s Eq. (11). Since the total mass ofthe drop, m as calculated from Eq. (16), appearsin the above equation, Eq. (25a) is in general anon-linear, ordinary, integro-differential equationwhich must be solved numerically. By subjectingit to an appropriate initial condition, i.e. R(t=0)=Ro, the above equation can be solved using aRunge–Kutta technique for the contact radius asa function of time, assuming that the dynamiccontact angle and its time derivatives are eitherknown or can be estimated from one of therelations/techniques described elsewhere [12–16].Additionally a description of dm/dt must be avail-able, such that at each time step, m can be recal-culated using Eq. (16). It is also important to notethat in Eq. (25a) the only curve-fitting term is theratio of macroscopic to microscopic cut-offlengths, o.

In the case of a constant dynamic contact angleand constant mass flow rate, dm/dt=m; , Eq. (25a)can be simplified to the form shown below:�

6pm ln(o−1)Rud

n �dRdt

�2

+�

2pRglv [2h(ud)−cos ue]+ (mo+m; t)g

f(ud)4n dR

dt+m; �gR

4f(ud)−

2glvsin(ud)rR

−62

2n

=0.

(25b)


Table 1Comparison of energy balance methods

Madejski [17]Energetic terms Gu and Li [19,20] Current model

N/A N/AEnergy transferred to the Enthalpy and kinetic energyof entering fluidsystem (dEin/dt)

Numerical calculation ofAssumed internal velocity Change of total internalInternal kinetic and internalprofile to determine kinetic energy due to the addition ofenergy of the system internal velocity to determine

(dEsys/dt) mass dominating overenergy kinetic energyinternal kinetic energy

Change in l–6 interfacial area Change in l–6, l–s, s–6Surface potential Energy Change in l–6, l–s, s–6interfacial areas(dEs/dt) interfacial areas

Not considered Mass center approach (foundGravitational Potential Energy Mass center approach(dEs/dt) to be negligible)

Boundary movement work N/A N/A Work done to displace the(dWb/dt) environmental pressure

Assumed internal velocity de Gennes framework forViscous dissipation work de Gennes framework forprofile, Newton’s Law(dW6/dt) dissipation at three phase linedissipation at three phase line

It is this form of Eq. (25a) that will be used tovalidate the OEB method in Section 3 by model-ing the hydrodynamically driven spreading of anoil droplet over a solid surface.

2.8. Summary of OEB approaches

To emphasize the considerable differences be-tween this and previous OEB approaches to dropspreading, Table 1 presents a comparison of thismethod with the two previous models outlined inSection 1. As mentioned the previous methodsonly consider the spreading of a drop of fixedmass, whereas here a hydrodynamically drivensystem of variable mass is considered.

3. Experimental study of drop spreading

In order to validate the above model severalexperiments were conducted using MCT-30 oil(Imperial Oil product) advancing on a glass slidecoated with FC-725 (a fluorocarbon coating, 3Mproduct). The experimental fluid has the followingphysical properties; viscosity m=0.286 N s m−2,density r=879.2 kg m−3, surface tension glv=31.16 mJ m−2 and equilibrium contact angle=78.6°. Five tests were conducted at two differentmass flow rates, 2.76 and 3.11 mg min−1.

A schematic of the experimental set-up isshown in Fig. 2. As mentioned above the spread-ing surface was a glass slide coated with FC-725to ensure that roughness and surface inhomogene-ity effects would be negligible. A 250 mm diameterhole was drilled in the center of the plate, whichwas supported on an anti-vibration table andinsured to be level prior to beginning all tests.One end of a 200 mm (outside diameter) capillarytube was inserted and fixed into the hole while theother end was fitted with a flexible nylon tubingwhich connected to the syringe pump. The motor-ized syringe was driven by an Anaheim Automa-tion Miniangle stepper motor (Model c

Fig. 2. Experimental set-up for dynamic contact angle andcontact radius measurements using the Axisymmetric DropShape Analysis technique.


23PM-C402) and controlled by an Anaheim Au-tomation (Model c DPF72) Controller.

The critical aspect of this experiment is theability to accurately measure the changes in con-tact radius and dynamic contact angle as the dropadvances across the surface. The automatedAxisymmetric Drop Shape Analysis (ADSA) tech-nique [21] combined with a digital image analysisand processing system is a powerful tool for mak-ing such measurements. In previous studies, thistechnique has been applied to the study of manycapillary phenomena including the drop size de-pendence of contact angles and surface tensionageing [21].

The ADSA computer image system is depictedin Fig. 2. A Cohu 4910 CCD monochrome cam-era is mounted upon a Leica wild M3B micro-scope. The video signal generated is transmittedfirst to a Sanyo B/W Video Monitor and then toa VideoPix digital video processor. The videoprocessor was used to record the signal and digi-tize the image to a resolution of 640×480 pixelswith 256 gray levels. A Sun Sparc 10 Unix com-puter was used to operate the digital video proces-sor and to analyze the data. The AxisymmetricDrop Shape Analysis software package was uti-lized to acquire the drop images at different times,digitize the recorded images and to determine thedrop’s contact diameter and the advancing con-tact angle as a function of time.

To conduct a test the syringe plunger was firstadvanced at a slow rate until a drop of suitablesize was visible on the surface. After allowing thedrop to settle to equilibrium, the syringe was setto advance (at a constant speed) once again andthe image analysis system began recording andsaving an image every 10 s, which was laterprocessed using the ADSA technique outlinedabove. In each run a nearly constant dynamiccontact angle was observed as shown in Table 2.The error associated with assuming a constant ud

was 91° over the course of any single run. Allexperiments were conducted at room temperature,22°C.

This process was repeated several times and fortwo different syringe plunger speeds (correspond-ing to the high and low mass flow rates mentionedearlier). At each trial, a freshly prepared glass

Table 2Measured dynamic contact angles

Mass flow rate Dynamic contact angleRun(mg min−1) (°)

c12.76 81918291c2

c3 7991

79913.11 c1c2 8091

slide was used to ensure that the surface was notprewetted with the experimental fluid.

4. Results and discussion

Fig. 3(a and b) show the measured values of thecontact radius, using the ADSA technique de-scribed in Section 3, as the drop advanced acrossthe surface for the five runs at the two separatemass flow rates (2.76 and 3.11 mg min−1). Arange of initial contact radii has been selected inorder to test the model over a variety of initialconditions. In each case the drop was allowed toreach a stable dynamic contact angle first.

4.1. Comparison of OEB relation withexperimental results

Using the starting contact radii shown in Fig.3(a and b) as an initial condition and the dynamiccontact angles shown in Table 2, the OEB rela-tion, Eq. (25b), was solved using a forth orderRunge–Kutta technique. The results are shown assolid lines superimposed over the raw data. Theinitial mass of the drop was estimated by thespherical cap approximation outlined in Section2.5 and the ratio of the microscopic to macro-scopic cut-off lengths, o, was taken as a constantvalue of 0.005 in accordance with the value usedby Gu and Li [19].

As can be seen in the two figures the solution tothe OEB equation models the experimental datawell over the range of tested values until such timeas the drop becomes so large such that the spher-ical cap approximation is no longer representative


of the true drop shape. In each case once the dropgrows beyond a certain limit, the model tends topredict a slower advancing velocity than is actu-ally observed.

As mentioned in Section 2.1 the model calcu-lates the pressure at point O (see Fig. 1) byassuming a spherical shape and therefore constantpressure field within the drop, defined by Eq. (5).

Since this value varies with 1/R and the hydro-static pressure varies approximately with R, as thedrop grows the Laplace pressure will become lesssignificant and the hydrostatic, which the modelignores, more significant. As a result the value ofdEin/dt in Eq. (6) is likely to be underestimated atthese larger drop sizes resulting in the slower thanobserved advancing velocity shown at the tail end

Fig. 3. (a) Comparison of measured contact radius and model prediction for slow spreading (mass flow rate=2.76 mg min−1) ofMCT-30 oil on FC-725 surface. Solid lines represent model prediction and hollow symbols represent experimental data; (b)Comparison of measured contact radius and model prediction for the fast spreading (mass flow rate=3.11 mg min−1) of MCT-30oil on FC-725 surface. Solid lines represent model prediction and hollow symbols represent experimental data.


Fig. 4. (a) Predicted speed of advancing contact radius (dR/dt) for the slow spreading case (mass flow rate=2.76 mg min−1); (b)Predicted speed of advancing contact radius (dR/dt) for the fast spreading case (mass flow rate=3.11 mg min−1).

of the results. Additionally as the drop grows andthe surface forces become less significant, the dropwill begin to flatten out and the spherical assump-tion will likely overestimate the work done in agravitational field and underestimate the surfacearea growth. In order to model larger drop sizes amore complex and accurate model of the truedrop shape [19,20] would have to be used.

4.2. Drop spreading 6elocity

Of interest in most of the engineering applica-tions, mentioned earlier, is the speed at which thedrop spreads over the surface. Fig. 4(a and b)show the energy model prediction of the dropspreading speed for the two different mass flowrates. Intuitively it would be expected that the


advancing front would slow down as the dropsize grows since mass is being added to the sys-tem at a constant rate. This is trend is indeedreflected by the OEB model, as shown in Fig.4(a and b), and is well established in the Fig.3(b) for high mass flow rate case. In the lowmass flow rate case it is likely that the totalamount by which the drop grows is to small tonotice this trend.

Knowledge of the drop spreading velocity iscritical in the determining the applicability ofthe lubrication theory approximation, used inthe derivation of the viscous force formula atthe three-phase line, to this situation. Batchelor[25] gives the following constraint where such anapproximation is valid:

ud

rRm

dRdt�1. (26)

In this study the value of this parameter isgenerally of the order 10−3, thus validating theapplicability of this model in predicting the vis-cous dissipation at the three-phase line.

4.3. Comparison of energetic components

Critical to the modeling of any phenomenathrough thermodynamic methods is a firm un-derstanding of the relative magnitude of each ofthe energetic terms being considered. Gu and Li[19] showed a comparison of the relative impor-tance of the applicable terms for the case ofspontaneous spreading. Their results showedthat for all practical purposes the viscous dissi-pation work was completely balanced by the re-duction in the total surface energy and that thechange in gravitational potential energy wasnegligible.

Fig. 5 shows an analogous plot of energeticconsumption by or contribution to the systemby the terms outlined in Section 2 for a repre-sentative run. Terms on the right hand side ofEq. (2) can be though of as prohibitive to dropspreading and are thus given a negative magni-tude. For ease of comparison with the remain-ing terms, the energy entering the system, Ein,change in total internal energy, Esys, andboundary movement work, Wb, have been com-

Fig. 5. Comparison of the energy consumption and delivery by the effective energy and work terms for hydrodynamically drivendrop spreading. Run c2 of the fast spreading case (mass flow rate=3.11 mg min−1) is presented as a representative example.


bined and can be though of as the effectiveamount of usable energy delivered to the system,Eeff.

In this case the effective energy delivered to thesystem is nearly completely balanced by the workdone in increasing the total surface energy and thework done against gravity. Over the range oftested values the gravitational term varies from 35to 65% of the magnitude of the surface energyterm. The general downward trend in the effectiveenergy delivered to the system is caused by thedecrease in the Laplace pressure, and thus Eq. (6),as the drop grows. Physically this can be inter-preted as a decrease in the total flow work done inpumping the liquid into the system. The surfaceand gravitational potential terms each show op-posite trends with the magnitude of the formerdecreasing and the latter increasing slightly as thedrop grows. Of course as the total mass of thedrop increases the work required to raise it in agravitational field should also. The decreasingtrend in the magnitude of the surface energyconsumption is the result of the slowing of theadvancing contact radius velocity, observed inSection 4.2.

The negligible effect of the viscous dissipationat the three-phase line, shown in Fig. 5, is aconsequence of the slow speed of the advancingcontact radius. As can be seen in Eq. (25b) itvaries with the square of dR/dt while all otherterms are linear, thus the relative importance ofthis term will decrease with the decreasing spread-ing velocity. A consequence of this is that whenthe spreading velocity is slow enough, such as thecase examined here, the viscous term can be omit-ted from Eq. (25b), leaving a completely generalmodel for drop spreading which contains no curvefitting factors.

When Fig. 5 is compared with the results forspontaneous or surface tension dominated spread-ing mentioned earlier, there are several differencesof note. Most immediately is the fact that for thehydrodynamically-driven drop spreading, thework required to change its surface areas is thelargest energy consumer. In the case of sponta-neous spreading, this change in surface area, tolower to total free energy of the system, is themain driving force behind drop spreading. The

other significant difference is the relative magni-tude of the gravitational potential energy term. InEq. (20) it is shown that dEg/dt is dependent onboth the total mass of the drop and the mass flowrate (dm/dt). Obviously for the fixed mass casedm/dt would be zero, thus this term would beomitted all together. Additionally the drops con-sider here are much more massive than thoseconsidered by Gu and Li increasing the relativeimportance of the first term.

5. Conclusions

The study of the drop spreading on a flat solidsurface is an excellent example of the interplaybetween a classical fluid mechanics and a surfacephysics problem. In general previous models ofthis phenomena have involved complex numericalsolutions which may or may not be physicallyintuitive or been highly dependent on empiricismto quantify the model. In this paper an overallenergy balance (OEB) approach was undertakento model the spreading of a hydrodynamically-driven drop of variable mass and energy over asolid surface. The model balances the amount ofenergy delivered to the system with the change ininternal energy, changes in surface and gravita-tional potential energy, and boundary movementand viscous dissipation work. The resulting firstorder ODE was solved for the case of a sphericalcap using a simple Runge–Kutta technique andthe results compared with those obtained from anexperimental study.

In general it was shown that the OEB solutiondid successfully model the spreading phenomenonvery well until such time as the drop grew beyondthe range where the spherical cap approximationwas valid. As intuitively expected the model pre-dicted a decrease in the velocity of the advancingfront as the drop grew in volume. A comparisonof the energetic magnitude of the various termswas conducted and it was shown that in this casethe effective energy delivered to the system isbalanced by the increase in total surface energyand gravitational potential energy, with the for-mer having approximately twice the magnitude ofthe latter. Additionally it was shown that when


the drop spreading is slow enough the viscousdissipation at the three-phase line is negligible andthe OEB approach yielded a model completelyfree of any empirically determined curve fittingfactors. Thus in such a case the drop spreadingphenomena can be entirely modeled with knowl-edge of the material properties and an estimationof the dynamic contact angle. The method itself isnot explicitly constrained to small droplets, how-ever a more flexible final solution would have toinvolve a numerical computation of the true dropshape at each step.

Acknowledgements

The authors wish to thank the financial supportof the Natural Sciences and Engineering ResearchFund through scholarships to David Ericksonand Byron Blackmore, and through a researchgrant to D. Li.

Appendix A. Order of magnitude analysis forkinetic energy terms

As a method of simplifying the analysis it wasproposed in Section 2.2 that the change in theinternal kinetic energy of the drop was negligiblecompared with the kinetic energy of the fluidentering the system, and thus it could be effec-tively ignored. By an order of magnitude analysisthe conditions under which this assumption isvalid will be investigated.

From Eq. (6) the kinetic energy transferred tothe system (Ein,k) over an infinitesimal change intime is given by:

dEin,k

dt=62

2dmdt

, (A1)

where 6 is the velocity of the liquid entering thedrop. Recognizing that 6 is of the order of m; /rAin, where Ain is the area of the entrance, andsubstituting m; for dm/dt yields the following ex-pression for the order of magnitude of dEin,k/dt :

dEin,k

dt=O

�m; 3

2r2

1A in

2

n. (A2)

The change in the internal kinetic energy of thedrop due to the addition of mass is given by Eq.(A3):

DEsys,k=& V+DV

0

r62

2dV−

& V

0

r62

2dV, (A3)

where 6 is the internal velocity of the fluid and Vis the volume of the drop. If fluid is being addedat a constant rate, by continuity the velocityprofile over a given region in a spherically grow-ing drop must remain constant, thus Eq. (A3)reduces to:

DEsys,k=& V+DV

V

r62

2dV. (A4)

Recognizing that 6 in the region formed by theaddition of mass is of the order m; /rAlv, where Alv

is the area of the liquid–vapor interface, Eq. (A5)is obtained:

DEsys,k=O�r

2� m;

rAlv

�2

DVn

. (A5)

Reducing the above to an infinitesimal changewith respect to time and recognizing that dV/dt=m; /r, yields the following estimate for the order ofmagnitude of dEsys,k/dt :

dEsys,k

dt=O

�m; 3

2r2

1A lv

2

n. (A6)

Comparing the two results, as is done in Eq.(A7), suggests that when the surface area of thedrop is much greater than the area through whichthe fluid enters, the kinetic energy of the fluidentering the drop will dominate. For the case of aspherically growing drop on a flat surface whereinitial drop radius is larger than the entranceradius, such as that considered here, this condi-tion is always met thus it is justifiable to ignorethe internal kinetic energy of the system.

dEin,k/dtdEsys,k/dt

=O��Alv

Ain

�2n. (A7)

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an energy balance approach to modeling the hydrodynamically driven spreading of a liquid drop

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