A study of the advancing interface

A Study of the Advancing Interface II. Theoretical

Prediction of the Dynamic Contact Angle in Liquid-Gas

RICHARD Monsanto

Company,

Systems

L. HOFFMAN

Indian

Orchard,

Massachusetts

01151

Received June 1, 1982; accepted January 3 1, 1983 A molecular model has been developed which predicts the dependence of the dynamic contact angle upon the capillary number when a liquid moves to displace a gas. As a basis for this analysis we have assumed that molecules at the contact line between the advancing liquid and the solid substrate move forward by two different mechanisms. One involves the diffusion of liquid molecules across the solid substrate. This process is dominant when the dynamic contact angle is less than 120”, and the rate of surface diffusion increases with increasing values of the dynamic contact angle. The other process involves a “tank tread” motion of the liquid molecules normal to the solid substrate; this latter mechanism predominates when the dynamic contact angle is 180”. At contact angles between 120 and 180”, both processes come into play. From the model used in this analysis one finds that there are several molecular parameters which have some influence on the relationship between the dynamic contact angle and the capillary number. Although the effect of these parameters may be secondary in nature, it is likely that a universal correlation for the dynamic contact angle which is valid for all fluids and solid substrates must include these variables along with the capillary number and the static contact angJe. INTRODUCTION

In the first paper of this series ( 1), the shape of the advancing liquid-air interface was studied in a glass capillary over the range in which viscous and interfacial forces are the dominant factors controlling the system. Data were obtained for the full range of contact angles from 0 to 180”, and out of this study we found that a good correlation of the data is obtained if the apparent contact angle is plotted as a function of the capillary number plus a shift factor which is determined solely by the static contact angle between the liquid and the solid substrate. With these data to use as a guide, one can propose and test models which will characterize the process by which the interface advances across the solid substrate. Having this purpose in mind we present in this paper the results obtained from a molecular model which has as its basis the forces of interaction

between molecules in the region of the contact line located at the juncture of the liquid, gas, and solid substrate. The results obtained with this model are noteworthy from several standpoints. First one finds that the model predicts the complex response of the apparent contact angle to capillary number which is found experimentally. Secondly, one finds that several molecular parameters are important, and changes in these parameters must be accounted for if one is to obtain a fully adequate correlation of the data for all possible systems. In addition to these results the concepts used to develop the model also give one a sound basis for understanding mechanisms by which the molecules at the contact line can advance across the solid substrate, and when properly incorporated into continuum models for flow near the advancing front, the prediction of infinite stress at the contact line should not occur (2-5). Before considering the details of the model

470 0021-9797183 $3.00 Copyright 0 1983 by Academic Press. Inc. All rights of reproduction m any form reserved.

Journal of C&id

and Interface Science. Vol. 94, No. 2, August 1983

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developed in this study, it is appropriate to gain some perspective on the present state of knowledge with respect to the advancing interface by reviewing the literature on this subject. One area of study is concerned with the shape of the liquid-gas interface, and much of the available literature has been cited in the first paper of this series ( 1) and in a review by Dussan V. (5). Experimental systems that have been used in these studies include liquids flowing into a capillary filled with air (1, 6-9), drops spreading on a solid substrate (IO-14), fibers, belts, and plates which are plunging into a bath of liquid (15-20), rotating cylinders partly submerged in a liquid bath with the axis of the cylinder parallel to the plane of the surface of the undisturbed liquid (21, 22), and liquid moving in radial flow between two flat, parallel plates (23-26). Perusing the data obtained in these studies one finds that in all cases except in Refs. (1) and (15), only limited parts of various flow regimes have been studied, and in a few cases several flow regimes were tested inadvertently. As one step in alleviating this problem, Hoffman (1) studied the shape of the advancing liquid-air interface over the full range in which viscous and interfacial forces are the dominant factors controlling the system, and these data form the basic core of experimental information which will be used to test the model presented in this study. A second area of study is concerned with the development of models that will predict the response of the dynamic contact angle to changes in the variables which affect it. Approaches that have been used in these studies include dimensional analysis ( 17, 1X,20,22), models based on continuum concepts (8,2729), models based on molecular concepts (30, 31), and one notable effort to wed the latter two approaches (32, 33). Results obtained from these studies have a direct bearing on a third area of interest which is the prediction of flow fields in the region of the advancing interface (2,8,32-44). One connection which is quite obvious is that studies in this third area require a knowledge of the shape of the

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interface which constitutes one boundary in the system to be analyzed. There is a second connection, however, which has to do with an understanding of the process by which the fluid advances across the solid substrate in the region of the contact line. The importance of this information was not appreciated until Huh and Striven pointed out that the solutions obtained for the flow fields were giving an infinite force at the contact line when a no-slip boundary condition was used (2). Since that time, numerous authors have been concerned with this problem and ways in which boundary conditions near the contact line can be altered to eliminate this problem in a continuum analysis. Various slip conditions have been suggested and considered as a means of eliminating the prediction of infinite force (2, 40-44), but it seems evident that an analysis of molecular motions near the contact line is needed to elucidate the process(es) which must be modeled by any continuum analysis. Thus, it is the purpose of this study not only to develop a molecular model which predicts the shape of the advancing liquid-gas interface, but also to develop an understanding of the kinds of molecular motions that may occur near the contact line. As a first step in understanding the kinds of molecular motions that may occur near the contact line, it is appropriate to consider the results of recent studies concerned with the behavior of adsorbed molecules on a solid substrate. Using field-ion microscopy, surface scientists are now able to locate the position of individual adatoms relative to the positions of the molecules in the solid substrate, and this technique can also be used to follow the movement of individual adatoms with time (45-47). One striking result that has come from these studies is the discovery of a large number of well-ordered periodic arrangements which can be formed when layers of adsorbed atoms develop on well-defined surfaces (48). Preferred adsorption sites are quite evident from this work, and results such as these from field-ion miJournal oj’Co//oid and Inter/h?

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croscopy provide direct evidence for the idea that “. . . forces between a single (adsorbed) atom and a solid substrate can be thought of in terms of a potential-energy surface with valleys and hills, which defines the equilibrium position and the binding energy for adsorption, as well as the barriers against lateral migration. The details of the energy contours are usually not known, but they must reflect the spacing and symmetry of the substrate structure so that the adatom site will be simply related to the positions of the substrate atoms” (45). As a result of these energy contours, the dynamics of atomic motion appears to be quite simple; atoms hop from one preferred site to the next, and the rate of jumping is customarily modeled by an Eyring type of relationship (47). THEORY

Having this knowledge of the process by which atoms diffuse across the surface of a solid, it seems quite reasonable for one to

HOFFMAN

presume that molecules of a liquid at the contact line of an advancing front will move out across the solid substrate by a similar mechanism. Thus, we can use an Eyring type of relationship (49, 50) to characterize the process by which a molecule jumps by a Brownian process from one preferred site to another as shown in Fig. 1 for molecule c. This approach was first proposed by Cherry and Holmes (30). Taking the frequency of the jumps as i per molecule one can write

ill

j = (K-T/h) exp(-AGi,,/RT)

for the static contact angle where K and h are the Boltzmann and Plank constants, R is the molar gas constant, AG& is the molar “free energy of activation” of a molecule such as c located at the contact line of a static interface, and T is the absolute temperature. Now when the interface is driven across the solid substrate the contact angle increases to (I,, its dynamic value. As a result, the force field on molecule c is altered in a way that distorts the potential energy barrier as shown

Liquid molecules O+the liquid-gas interface Y Gas phase

-Solid

21 P t w

!

-‘-Dynamic

substrate molecules

interface

FIG. 1. Illustration of the surface diffusion process for molecule c at the contact line. Molecule pass through the potential energy barrier to reach the next preferred adsorption site. Journal of Co/hid and Interface Science. Vol. 94, No. 2, August 1983

c must

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in Fig. 1. This makes it easier for molecule frequency of the backward jumps (in the c to advance across the solid substrate, and -x direction), one finds that the net velocity, thus the speed at which the contact line uc, at which the liquid molecules at the conmoves across the solid substrate is enhanced tact line will move out over the solid subas ed increases. The equation that is use to strate is just the distance traveled per jump, characterize the change in the energy barrier a,, times the net frequency of forward jumps. Thus, when we replace -AG& in Eq. [ 11 by with a change in the contact angle is obtained in the following way. Under static (equilib-AG* for the dynamic case (Eq. [5]) we find rium) conditions we assume that the energy that barriers to motion are the same in the plus and minus x directions as shown in Fig. 1, vc = a&F - jB) and an energy balance yields the Young = a,KT/h) ew(-AG~,J~~)l equation: X [2 sinh {(cos 0, - cos 19~) ~sv= UsI+ CT\”cos e,, PI where u, is the solid-vapor interfacial free energy, (T,~is the solid-liquid interfacial free energy, aI, is the liquid-vapor interfacial tension, and BSthe static contact angle. When we drive the fluid across the solid substrate, however, the contact angle increases to 13~and this creates an imbalance of energy which is given by: E = -cq,(cos Od- cos O,),

[31

where E is the energy per unit area in the region of the contact line. Now, if the number of molecules per unit area of interface is given by n in the region of the contact line then the energy per mole that is added by increasing the contact angle to 0, is given by: E(N/n) = -qv(cos 0, - cos B,)N/n,

[4]

where N is Avogadro’s number. It is this imbalance which provides more energy for the molecules of liquid at the contact line to move across the solid substrate as shown in Fig. 1, and so when the interface is moving we must write -AG*

[61

At this point it is expedient for us to define AG,& as the sum of two components, i.e., $ = AGo$ + AGO/A> $ AGo,c ]71 where AG8 is the molar free energy of activation of a molecule in the bulk liquid phase and AGi,a is the difference between AG& and AG:. When this is done we can make use of the fact that the viscosity, p, of a Newtonian fluid described by the Eyring model is given by (50): p = (S2Nh/azV) exp(AG$RT), where 6 is nominally uid molecules and V of liquid. Using this and rearranging one

PI

the diameter of the liqis the volume of a mole relationship in Eq. [6] obtains

cLv,/alv = 2(82N/naCV)(nRT/a,,N) X {w(-A&/RT)} X sinh [(q,N/nRT)(cos

8, - cos O,)]

or

= -AG& f qv(cos 8, - cos &)N/n,

x dV~~~}l.

c&D = (K/k)(lY - e-y, [5]

where the energy in the positive x direction is given by the (+) sign and the energy in the negative x direction is given by the (-) sign. Defining jr as the frequency of forward jumps (in the +x direction), and j, as the

[91

where we let

k = qvN/nRT

[loal [lob1

u = k(cos8, - cose,>

r1w

Ca,, = ~vdfa,

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RICHARD

and K =

(62N/naGV) exp(-AGi,@T).

[lOd]

Equation [9] is an important one to have in our model building process because it establishes a quantitative relationship between the dynamic contact angle and the parameters which control it when surface diffusion is the only mechanism by which molecules at the contact line advance across the solid substrate. As indicated by this equation, changes in the contact angle alter the force balance on molecules at the contact line, and as a result, the speed of the contact line increases with increasing contact angle. Over the range from O-90” this mechanism will provide us with a reasonable accounting of the process by which molecules at the contact line advance across the solid substrate, but once the contact angle exceeds 90”, another form of molecular motion comes into play which we must also consider. Forces of interaction with the solid substrate are not only acting on molecule c at the contact line but also on liquid molecules such as i in Fig. 2. The net effect is different, however, in that molecule c is effectively restricted to motion in the x direction because of the repulsive barrier afforded by the solid substrate through Born repulsion, while molecule i can translate both in the x and z directions. Thus, once the contact angle is greater than 90”, forces which move molecule i in the z direction can advance the con-

L.

HOFFMAN

tact line by a mechanism which differs from surface diffusion. Molecules such as i moving toward the solid substrate in the z direction will produce a “tank tread” sort of motion which acts in concert with the surface diffusion mechanism to advance the contact line, and as & increases beyond 90” this processbecomes more and more important until at ed = 180” the motion of the contact line is completely dominated by this “tank tread” mechanism. To establish the effective velocity of the contact line, V, when both surface diffusion and the tank tread motion are important, we presume that the combined effects of these two types of motion are additive. Once this is done we can write v= 21,

for

19~< 90”

[l la]

= v,[ 1 + (u,/v,) cot ( 180 - Q] for

BdZ 90”,

[llb]

where U, is the velocity of molecule i in the z direction and n, is the velocity of molecule c in the x direction as given by Eq. 161. In Eq. [ 11b], V, cot ( 180 - Bd)gives the effective x component of velocity for the contact line which results from the motion of molecule i normal to the solid substrate. As indicated by Eq. [ 1 la], V effectively equals V, when f& < 90” because the attractive forces from the solid substrate cannot pull molecule i down in front of molecule c.

Liquid molecules at the llquld- gas Interface

-Solid

FIG. 2. Molecule i can move the x direction. The neighboring direction. Journal of C&id

both in the x and z directions while solid boundary effectively prevents

and Inlerface Science, Vol. 94, No. 2, August 1983

substrate

the molecule c can only molecule c from moving

move in in the z

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Combining Eqs. [ 1 la] and [ 11 b] with Eq. [9] one obtains Ca = (~/k)(e’ - emu) for

ed < 90”

[12a]

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the frequency of jumps as ji for molecule i in the z direction, one can write ii = (KT/h) exp(-AG&/RT),

[141

where AGi,i is the molar free energy of activation of molecule i when it is unaffected x (~/k)(e” - eP> for f& B 90”, [12bl by the solid substrate; AG;f,i represents the barrier to motion and RT the Brownian where Ca = pV/q, [131 forces which tend to overcome it. Now as the dynamic contact angle, Bd, increases beyond and k, u, and K are defined by Eqs. [lob], 90”, attractive forces between molecule i and [ 1Oc], and [ 1Od], respectively. Ca differs from the solid substrate will act to reduce the poCusn when Bd > 90” because the velocity of tential energy barrier to motion toward the the contact line, V, becomes greater than the solid substrate and so we must write velocity, o,, at which molecules can advance AG: = AGi,i +- @ziN, [I51 by surface diffusion only. Equations [ 12a] and [ 12b] give us the basic where AG’ represents the modified barrier relationships required to predict 19~as a func- to motion. In this equation the energy for tion of Cu over the full range of contact an- motion toward the solid is given by the (+) gles from 0 to 180”, and they account for the sign and the energy for motion away from motion caused by the “tank tread” effect as the solid is given by the (-) sign. 4jzi reprewell as surface diffusion. Equation [ 12b] is sents the attractive potential from the solid also important, however, because it estab- which acts on molecule i in the z direction, lishes pertinent dimensionless groups which and it will be a strong function of 19~.Interinfluence the dynamic contact angle, &, at actions with molecules in the gas phase are an advancing liquid-gas interface. Two of neglected because of the relatively low conthese groups, Ca and B,, only involve con- centration of molecules in that phase. tinuum parameters but k, J2N/na,V, and Defining j, as the frequency of jumps toAGi,JRT contain molecular variables as ward the solid, and jai as the frequency of well. backward jumps, one finds that the net veThe task remaining before us at this stage locity in the z direction, o,, at which liquid in the model building process is to calculate molecules such as i will jump to the solid V, so that we can obtain the relative mag- surface is just the distance traveled per jump, nitudes of u, and 2), as a function of &. To a,, times the net frequency of forward jumps. do this we will use an approach analogous Thus, when we replace AGi,i in Eq. [ 141 by to the approach that was used to derive the AG: we find equation for 0,. Brownian forces will tend to make molecule i jump about, and as a result, molecule i can move from its position in the = a,[(KT/h) exp(-AG&/RT)] liquid-gas interface (Fig. 2) to some preferred adsorption site on the solid substrate. To do X [2 sinh (aziN/RT)]. [ 161 this, however, molecule i must have suffiA relationship is required to quantify the cient energy to overcome the barrier to modependence of +zi upon the distance from the tion which is established by the attractive potential of the surrounding liquid mole- solid substrate, and as a model to characterize these interactions, we use the classical cules. As with molecule c we use the Eyring Leonard-Jones 6-12 potential (50, 51). Use model to characterize this process. Taking of additional terms to account for dipoleCa = [ 1 + { zl&}

cot (180 - &)I

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quadrupole, and quadrupole-quadrupole interactions is deemed unnecessary in this model since dipole interactions are by far the dominant factor in molecular attraction (5 1). In addition to these factors we will neglect the repulsion term, since we are concerned principally about the effect of the solid substrate on molecule i at distances where repulsion is not a major factor. When this is done the equation for characterizing the interaction potential, $ij, between two molecules i and j becomes 4ij Z 4t(C4!/&j)6,

HOFFMAN

the distance of a few molecular diameters, and the effects of curvature will not be significant in the cases of interest to us. Now starting with Eq. [ 171 for the interaction between liquid molecule i and a molecule j a the surface of the solid, we can integrate over the plane of centers of the solid molecules from -cc S x S co and from -co $ y S co for any given contact angle to establish the total potential generated by the surface molecules of the solid acting on liquid molecule i; i.e., the total potential is

[I71

where e is a characteristic energy of interaction between the molecules (the maximum energy of attraction between a pair of molecules), (Y is a characteristic diameter of the molecules (the “collision diameter”), and Sij is the distance between molecules i and j. To get the total attractive potential resulting from the interaction of molecule i in the liquid-gas interface with all the j molecules in the solid substrate, one must sum or integrate over all the j molecules. This task is less difficult than it might appear, however, since it is possible to approximate the interaction quite well by considering the interaction of molecule i only with molecules in the solid which are at the surface. In doing so, we make use of the fact that the first layer of molecules dominate the interactions (5 1). At this point let us focus our attention on molecule i so that we can establish the field of force acting on i from the solid substrate. A coordinate system is needed for reference in the integration process, and Cartesian coordinates are most conveniently used as shown in Fig. 3. The origin is taken as the point where a line drawn from the center of molecuie i intersects a plane defined by the centers of the surface molecules of the solid substrate when the line is drawn normal to that plane. Although the surface of the solid may be either curved or flat, we will approximate it by a flat surface in all cases. This we can do since the interaction potential between molecules falls off dramatically over Journal of Co/laid and Interface Science, Vol. 94, No. 2, August ,983

@i = 4EN

s

(Ck!/Sij)6&l,

where N is the number of molecules per unit area of solid surface and A is the area of the solid surface involved in interactions with molecule i. From this equation one finds that the z component of the total potential on molecule i is 9, = 4tN = 4&N

s

(a/Sij)6(Zi/(X2 + .~z)“~}dA m s -m

{ZidX/(X2

+

Zf)“‘}

co

X

s

dy/(zf + x2 + Y*)~, -CC

[ 181

where 3 is shown in Figs. 2 and 3. Integrating with respect to y we find azi = (3&N/z;)

co (dX/(X2 + l)‘} s -co

X tan-’ {co/(X”

+ 1)‘12},

[19]

where Xz

X/Zi.

For the purposes of simplifying the integration it is useful to assume that tan-’ {cc/(X2 + l)“*} = n/2. This we can do because the rest of the integrand, i.e., l/(X* + 1)3, falls off very rapidly with increasing 1x1 and as a result, significant contributions to the integral only occur in the range where tan-’ {co/(X2 + 1)“2) z 7T/2. When this approximation is used, Eq. [ 191

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Center of liquid molecule I -Sii --.../

Center of soled molecule 1 located I” surface layer

FIG. 3. Cartesian coordinate system used for reference. The z axis is defined by the normal from the center of liquid molecule i to the plane of centers of the molecules in the solid surface. The x and y axes lie in the plane of centers; the x axis runs at right angles to the contact line of the advancing interface.

can be integrated to give the final result where Ds and DL in Eq. [2 l] are the diameters which is of the hard sphere solid and liquid molecules @zi= 9a2~c@N/16z4. PO1 respectively; their ratio along with &jNN/ RTD: establish two more dimensionless groups in this model which relate to molecValues of Zi relate to the dynamic contact ular parameters. angle, and by considering hard sphere molHaving the desired relationships at this ecules, one finds that point which relate 9, to the dynamic contact angle, we can use Eqs. [6], [ 161, and [20] to Zi = DL{ [( 1 + (W~L))/~I + COScod - 90)) calculate the ratio of vz,/vC which is required for 8d 2 90”, [21] in Eq. [ 12b]. The result is

sVC

a,,[exp(- AGQRT)] sinh (9~2~a6hW/ 16RTz:) a,[exp(-AG&JRT)] sinh ((cos 8, - cos 8d)fl,1,NjnRT)

where Zi is specified by Eq. [21]. At this point we are in a position to predict the dynamic contact angle as a function of the capillary number, Ca, by use of Eqs. [12a] and [12b] along with Eqs. [21] and [22]. To compare these predictions with experimental data from Ref. (l), however, we must assume that the true contact angle and the apparent contact angle, as calculated from the shape of the advancing front, are essentially the same. This we do although it is possible that they may not be the same (8, 43, 44). Values for various molecular parameters must be estimated or assumed; they are k, K, DL, Ds/DL, LY, t, a,/a,, and the ratio of (exp[-AG&/RT])/exp[-AGQRT]. Those needed for surface diffusion effects alone,

for

&j 2 90”,

1221

when ed d 90”, are k and K. In K there are two dimensionless groups, see Eq. [ 1Od]. One of these is a2N/naCV and within this dimensionless group GN/nV corresponds closely to the ratio of the molecular volume of a liquid molecule at the liquid-gas interface divided by the molecular volume of the same molecule in the bulk liquid phase. This ratio should be of the order of 1.0, and in the calculations which follow we will assume that it is equal to one. Following general practice we will also assume that 6/a, is equal to one (50). The other dimensionless group in K is AGi,JRT where AG:,, is the difference between the molar free energy of activation of a molecule at the contact line of the static Journai of Co&id and Interfar

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interface, and the molar free energy of activation of a molecule in the bulk liquid phase. Although a precise value is not known for this parameter, one can judge whether the value needed to obtain the best fit between theory and experiment is reasonable. Data obtained by Lau and Burns (11) give us one source of information to make this judgment, and as a further guide one can compare values of the activation energy obtained for the bulk liquid phase from viscosity data (52) with the activation energy associated with surface diffusion (47). This latter quantity should allow one to make a first-order estimate of the activation energy of a liquid molecule at the contact line of a static interface. In the work which follows we have taken the value of AGi,a to be equal to 2.71 kcal/ g mole. Estimates based on the work of Lau and Burns (11) indicate that this is a reasonable value for AGi,a. See the Appendix. The data which will be used for comparison with the theory were taken principally on silicone fluids in glass tubes (1). With these polymeric fluids, small segments of the molecules are likely to be involved in the Brownian-driven jumping processes of the model which cause the contact line to advance across the solid substrate. This being so, it is reasonable to consider the motion of these segments in terms of the motion of the basic monomeric units of the polymer. As one approach to represent this motion one could use a bead-spring model similar to those used to model the flow behavior of polymeric fluids. To avoid the complexities involved in such a model, however, we have chosen a rigid sphere model. Although this approach is rather crude, it does allow one to model the salient features of the process. A rationale that can be used for this approach is that the motions modeled are only over distances which can be allowed even if the spheres are joined, and the effects of the bonding energy between the monomeric units will be represented in a crude way by the activation energies given in the model.

L. HOFFMAN

Taking this approach we considered monomeric units of dimethyl siloxane, (-0-Si(CH&-), moving over the silicon dioxide molecules of glass or adsorbed molecules of water. To calculate Ds/DL for this situation, we estimated the molecular diameter of monomeric units of the test fluid in the bulk phase by using the density of the liquid and the molecular weight of the monomeric unit. For dimethyl polysiloxane the number obtained is 5 A. This diameter should change when a molecule moves from the bulk phase to the gas-liquid interface near the contact line, but the change should not be drastic. A similar calculation gives 3.1 A for water which may be adsorbed on the glass surface and 3.4 A for glass. Thus, we used a value of 0.65 for Ds/DL in our calculations. Another parameter which must be estimated is k which is equal to a&/nRT. For silicone fluids the surface tension, ur”, is roughly 21 dyn/cm at 25°C so that k = 5.10 x 10-2/n,

where n is the number of molecules/A’ at the liquid-gas interface near the contact line and ( l/n)“2 is nominally the diameter DL of the liquid molecules. As before, we presume that with polymeric molecules this diameter corresponds to the diameter of the monomeric unit. For the bulk phase the value estimated was 5 A. In our model a value of 6.3 8, gives the best fit with the data and this value seems reasonable since one expects some change when a molecule moves from the bulk phase to the interface near the contact line. Values of the average diameters of the liquid and solid molecules and their collision diameters are interrelated. Knowing values of DL and Ds, one can calculate the collision diameter of these molecules by using the Leonard-Jones 6-12 potential. This is done in the following way. The Leonard-Jones relationship for a pair of molecules is d, = 4E[(Q/S)6 - (a/sy2], where s is the distance between them. Now

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at the position s = s,, where the attractive and repulsive forces are just equal, &/ds = 0, and from this information we find &xl = 1.12a. Taking S, to be equal to DL for the liquid and equal to Ds = 0.65& for the solid, we calculate a collision diameter of 3.6 A for solid molecules and 5.6 A for liquid molecules. We are, however, interested in the collision diameter for the interaction of a liquid and a solid molecule, and this we can estimate as the linear average of the values given above, i.e., 4.6 A. Values of this parameter have a critical effect on the predictions of our model at large f&, and the best fit between the predictions of the theory and the experimental results is obtained when a value of 4.8 1 w is used for a. This agreement is quite good. Other molecular parameters to be estimated are the maximum energy of attraction

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between a liquid and a solid molecule, 6, the ratio of jump distances, ~,/a,, for molecules i and c, and the activation energies, AGi,i and which characterize the jumps. As a A&, first approximation we assume that a,/a, = 1 and that AGi,i = AGi,C. A value of E can be crudely estimated by reference to tables of values for various molecules. See, for example, Table B-l, p. 744 of (50). For these calculations we used a value of c/K of 360°K. Taking Ds/DL = 0.65; 62Nfna,V, a=,/ a,, and (exp[-AGiJRT])/exp[-AG&JRT] equal to one; (l/nj”2 z DL = 6.3 A; (l/ N)1’2 z Ds = 0.65 DL; (y. = 4.81 A; c/K

= 360°K; AG:, = 2.71 kcal/g mole; & = 0; gl, = 21 dyn/cm; and T = 298”K, we predict the dashed curve shown in Fig. 4. Data from (1) are also given in this figure. Agreement between the values predicted by the model and the experimental values is good for reasonable values of the model parameters. We

NOl?O1601501401301202 z

IlO-

P 2

IOO-

a? ca”-

goBO706050-

FIG. 4. Comparison of experimentally measured values of the apparent contact angle, 8~, as a function of Cu + F(0,) (see Ref. (1)) versus values of the contact angle, 8,, predicted by the theory developed in this paper. A solid line is drawn through the data; the theory values are given by the dashed line. Journal ofCo/hd

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believe that this agreement is a good indication that the basic concepts of the model are correct. Having a complete development of the model for the motion of the contact line at this point, it is helpful to consider the contribution of each factor that leads to its movement. As modeled, two kinds of molecular motion are presumed to be important. One is surface diffusion which moves the molecules at the contact line parallel to the solid surface, and the other is the motion of molecules in the liquid-gas interface normal to the solid substrate. The contribution of each can be shown rather nicely by contrasting the predictions of the full model with the predictions based on the surface diffusion mechanism only. This is done in Fig. 5, and as one can see, surface diffusion predominates for dynamic contact angles less than 120”. For dynamic contact angles greater than 120”, however, the “tank tread” motion of

L. HOFFMAN

the molecules normal to the solid surface begins to play a significant role and at 180” this mode of motion predominates. DISCUSSION

Having established a model to predict dynamic contact angles at an advancing liquidgas interface, it is appropriate for us to address several problems which have arisen in studies concerned with dynamic contact angles at the advancing front. One is concerned with a definition of all the variables which influence the shape of the advancing front, and the other is the closely allied problem of how one compares data obtained from different experimental geometries. Let us consider first the definition of all variables which influence the shape of the advancing front. Studying the shape of an advancing interface in a pipe, we found in a previous study (1) that all values of the apparent contact

180l70170l60160l50150l40140l30130120;;

IIO-

3 P 3

IOO-

a?

go80706050 403020-

10-5

10-4

10-3

10-z

I q

IO

102

tF(eS)

FIG. 5. Predictions of the model. (- - -) Assuming that the molecules advance only by surface diffusion (i.e., Eq. [12a] for all 0,). (--) Assuming that molecules advance both by surface diffusion and the tank tread mode of motion (Eqs. [12a] and [ 12b]). Journal of C&id

and Inter~ce Science, Vol. 94, No. 2, August 1983

STUDY

OF

THE

ADVANCING

angle could be correlated as a function of the capillary number plus a shift factor which relates to the static contact angle. The geometry of the apparatus (e.g., tube diameter) was not considered as a factor in that study. Since that time, Dussan V. and co-workers have published several papers concerned with dynamic contact angles (5, 44, 54, 58), and in these studies they suggest that geometry is also a factor whenever one is concerned with any of several apparent contact angles. Considering this suggestion and the results of the study presented in this paper, one can make several observations with regard to the effect of the apparatus geometry. As derived, the model presented in this paper is concerned with the effect of specific molecular forces on the true dynamic contact angle which occurs at the junction between the solid substrate, the advancing liquid, and the gas which is being displaced. The result obtained shows that the geometry of the apparatus is not a factor. This condition agrees with the views of Dussan V. and co-workers who only maintain that any of several apparent dynamic contact angles are a function of geometry. We also find that our model for the true dynamic contact angle, 0,,, predicts the general response obtained for the apparent dynamic contact angle, &, when reasonable values of the model parameters are used. In other words it appears that 0, and 0, are closely related (See Fig. 4). This conclusion can also fit within the framework of the theoretical study presented by Kafka and Dussan V. (44). See, for example, their Eqs. [6.1]-[6.4]. One question which remains, however, is just how important the geometry of the apparatus is in affecting oM. Data from a study by Ngan and Dussan V. (58) with glass plates separated by varying distances lead one to believe that the effect is quite strong, but other sources of data should be considered. Experimental data taken in different geometries can also help determine the effect of geometry on the dynamic contact angle. It is important, however, to select the data

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481

used with care because problems have arisen in some studies. One problem mentioned in the introduction to this paper is that data have been taken which inadvertently span from one flow regime to another. In this study we are only interested in data taken when the dynamic contact angle, as measured, is unaffected by inertial and gravitational effects. Another problem arises with multicomponent fluids which has not been appreciated up to this time. From the model developed in this study, one finds that intermolecular forces, and thus the rate of molecular diffusion across the solid substrate, are important factors affecting the dynamic contact angle. This being the case, one anticipates that mixed molecule systems, such as polymer solutions, may yield anomalous data unless differences in the molecular interactions and their effect on the dynamic contact angle can be properly accounted for. We will presently avoid this problem by examining data on single-component fluids. This is no guarantee, however, that all variation will be removed when data on the dynamic contact angle are plotted as a function of Ca + F(B,) because the forces of interaction between the liquid and the solid molecules will change from one combination to the next. Model parameters affected by these changes are Ds, DL, cy, e, a,, a,, and the various activation energies. We expect these effects to be secondary in nature for many cases, however, and for this reason we can still explore the effect of substrate geometry. Understanding these limitations let us consider data from several different studies. One set of data that can be used was taken by Lau and Bums who examined the spreading rate of low-molecular-weight polystyrene on soda lime glass microscope slides ( 10, 11). Taking information from both of these papers, one finds most of the data needed to calculate the necessary parameters for sample PS4000 at 174°C. At this temperature the surface tension aI,, is 30.3 dyn/cm, the static contact angle B, is 37.4”, and the apparent contact angle can be calculated as a function Journal o/Colloid and Inter/ace Science, Vol. 94, No. 2, August I983

482

RICHARD

of the velocity of the advancing contact line from Fig. 6 of this paper. For these calculations we will use values predicted by the line through the data, and we observe that the velocity of the contact line is given by the relation V = dA/2ardt. Viscosity data are also required, but Lau and Burns have not listed this information in their papers. Fortunately it appears that one can circumvent this problem by using data obtained by Plazek and O’Rourke (55). In their study Plazek and O’Rourke measured the flow behavior of various narrow molecular weight distribution polystyrene samples of low molecular weight. One of these, sample PC-l 1[2], was obtained from Pressure Chemicals Company. This company also supplied the samples used by Lau and Bums, and sample PC-l 1[2] appears to have the same molecular weight as the sample PS4000 used by Lau and Bums. Presuming that viscosity data on sample PC-l 1[2] are representative of the flow be0.7

I

I

I

L. HOFFMAN

havior of sample PS4000, we extrapolated the data on sample PC- 11[2] (see Figs. 14 and 15 of Ref. (55)) to 174°C. By this procedure we estimate that the viscosity of sample PS4000 was 52 P at 174°C. With this value for viscosity and the other material constants previously listed, we used Fig. 6 to calculate the dynamic contact angle as a function of Cu + F(&). I;(f),) has a value of 4 X low3 as determined by the procedures given in (1). The results obtained are plotted in Fig. 7 along with a line representing the capillary data from (1). The agreement between the drop spreading data and the capillary data is good. Another set of data that can be used for comparison was taken by Inverarity (15) who drew filaments of glass into pools of various liquids. Major portions of his data will not be considered because they were taken on mixed fluid systems such as polystyrene in xylene and glycerine in water. Let us consider, however, data which he obtained on I

I

I

!

0.6 -

os-

OAa? E 0.3 -

0.2-

0.1 -

FIG. 6. Plot showing the linear relationship between cos 0d and dA/rdt for a drop of polystyrene, sample PS4000, spreading on a glass microscope slide at 174°C. Data used in this plot were obtained from (10). Journal o/Colloid

and Inter.ce

Science. Vol. 94, No. 2, August 1983

STUDY

OF THE ADVANCING

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483

FIG. 7. A comparison of the dynamic contact angle versus Ca + F(B,) as determined in different experimental geometries. (- - -) Advancing front in capillary (1). (---) Drop of sample PS4000 spreading on glass plate at 174°C (10, 1 l), (- - - -) Glass filament drawn into bath of polypropylene glycol (15).

glass filaments drawn into polypropylene glycols ranging in viscosity from 1.8 to 10.6 P. All the necessary data are available from Fig. 5 of that study except for values of the surface tension. Estimates must be made for this parameter, and this can be done by utilizing the work of Rastogi and St. Pierre (56) along with a study of Barlow and Erginsav (57). From the work of Barlow and Erginsav one finds that polypropylene glycol samples ranging in viscosity from 1.8 to 10.6 P will have molecular weights which fall somewhere between 400 and 4000. This being the case, it is evident from the work of Rastogi and St. Pierre that the surface tension of the samples tested by Inverarity should have been roughly 31 dyn/cm. Using this number and taking the static contact angle, B,, to be zero, one obtains the results shown in Fig. 7. As with the drop spreading data, one finds generally good agreement with the capillary data al-

though there is some deviation at low values of Cu. A deviation of this sort will occur if the assumption that 0, = 0 is incorrect. This aside, we believe that the agreement of data from the three radically different experimental geometries considered is good enough for one to conclude that apparatus geometry is not a major factor. Large differences which might be expected on the basis of the work by Ngan and Dussan V. (58) are not apparent. CONCLUSIONS

In this study a model based on molecular concepts has been developed which predicts that the dynamic contact angle, in advancing liquid-gas systems, depends primarily upon the capillary number when the static contact angle is zero. Several parameters which characterize interactions between liquid and solid molecules near the contact line will also have Journal o/Coilord and Inlerface Scmce. Vol. 94, No. 2, August 1983

484

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L.

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some effect, but for many systems, the effect profound and lead to phenomena such as of these parameters will probably be second- channelling (7) which are not observed when ary in nature. Comparing the predictions of a liquid advances to displace a gas in a pipe. this model with experimental results one As a result the liquid-liquid system warrants finds reasonable agreement between the two. careful consideration as another case. FurAlthough the importance of parameters char- ther consideration of the advancing liquidacterizing the interaction between liquid and liquid interface is not given in this paper. solid molecules near the contact line should APPENDIX not be as great as the capillary number, one’s attention is drawn to the need to be careful Estimate of the Valueof AC& in comparing data taken on different liquid Estimates for AGi,& are not easily oband solid substrates without a knowledge of tained, but an order-of-magnitude value can changes in the pertinent molecular paramebe extracted from the work of Lau and Burns ters. This is particularly the case with fluids (11). To do this one must recognize that the which contain more than one molecular spemodel proposed by these authors, as well as cies such as a mixture containing both a polymodels proposed by Cherry and Holmes (30) mer and a solvent. As modeled in this study, we presume that and Blake and Haynes (3 l), embody one of the motion of liquid molecules in the region the concepts used to develop the model given of the contact line is determined by their in- in this paper. The concept is that molecules teraction with nearby molecules in the solid at the contact line can advance by molecular substrate. When the dynamic contact angle diffusion, and the rate at which they advance is 120” or less, molecular forces cause a net will depend upon the dynamic contact angle. motion of the liquid molecules parallel to the This being so, one expects that the equation developed by Lau and Burns (IO, 11) can be surface of the solid. As the dynamic contact angle increases beyond 120”, however, at- obtained as one limiting case for our model. tractive forces normal to the solid substrate To show this, we begin with Eq. [12a] which also become important. These forces effec- applies when the surface diffusion mechtively introduce a velocity component to the anism is dominant. Now we know that molecules in the liquid-gas interface near the e” - e-u 3 2 sinh u = 2[u + (u3/3!) contact line which is normal to the solid sub+ (U5/5!) + - - -1 for u2 < co strate. This motion is analogous to the motion of a tank tread, and it becomes predomso that inant when the contact angle is 180”. e” - e+ 2 2u In this paper the model proposed has been when u is small. Over the range of data taken applied only to the advancing liquid-gas inby Lau and Bums (10) this approximation terface, and in the model used, the effect of the gas phase molecules upon the surface seems adequate. For example, if k = 2 then diffusion and tank tread motion of the ad- u 5 0.64 for their sample PS4000 (see Fig. 6 of Ref. ( lo)), and the maximum error made vancing liquid molecules has been neglected. by using this approximation is 23%. For most This is perfectly acceptable for the case conof the data the error is much less. With this sidered. The same approach cannot be used, approximation Eq. [ 1Za] becomes however, to explain the behavior of the dynamic contact angle for liquid-liquid sys- V g K,{exp(-[AGi,A + AG&RT)) tems because molecules of liquid in the disX qv(cos OS- cos O,), [23] placed phase will have an important effect on both the surface diffusion and tank tread where mechanisms. In many cases these effects are KI = 2aJnh. Journal o?Colloid and Interface Science, Vol. 94, No. 2, August 1983

STUDY

OF THE ADVANCING

To obtain this equation we make use of the Newtonian viscosity relationship given by Eq. [8]. At the very low rates of shear involved in the spreading of a drop of polystyrene, the problem of interest to Lau and Burns, the Newtonian viscosity relationship is quite adequate as long as we incorporate a term to account for the dependence of viscosity on molecular weight. This we can do by saying that K,

= (Icf@,‘.5)/2T,

where A&, is the weight-average molecular weight of the polymer and K is a numerical constant. Thus, for a spreading drop, the velocity of the contact line is given by V = (l/ 2ar)dAldt, where r-‘(dA/dt)

= K{exp(-[AGi,*

-t- AG&RT))

X A?,i.5a,v(cos 13,- cos 0,).

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485

g mole represents a reasonable value for the value of AGi,&. In their work, Lau and Bums examined the effect of temperature upon the spreading rate, and from these data they extracted values of E (their notation) or AG& as given by our Eq. [7]. Evaluating slopes they found that E = AG:,, = 25.2 rfr 3 kcal/g mole (11). Values of AG:, the activation energy for the viscosity of polystyrene melts, generally range from 20-30 kcal/ g mole (53). Using these numbers in Eq. [7] to evaluate AGi,* for polystyrene, one finds that this parameter should lie in the range of -5.0 to 5.0 kcal/g mole. Lacking better information for dimethyl polysiloxane at this time, we suggest that it is likely to have a similar range, and thus a value of 2.7 1 kcal/ g mole was used to generate the predicted curve in Fig. 4.

[24]

Equation [24] is very close to Eq. [6] as given by Lau and Bums (11). In Eq. [24] dA/dt is the rate of increase of the liquid-solid contact area, and the sum of AGi,A + AG: is equal to E which Lau and Bums call the “activation energy of spreading.” It is equivalent to AG& as given by Eq. [7] in this study. The only difference between Eq. [24] as given above and Eq. [6] of Lau and Bums is the occurrence of r, the radius of the wetted area, in our equation. This difference is not trivial, however, and it should be resolved. Using Lau and Bums’ equation one infers that cos 19~should be a linear function of dA/dt, and this appears to be what they find experimentally. See Fig. (6) of Ref. (10). Reworking the data of Figs. 5 and 6 of that study for sample PS4000 at 174”C, however, one finds that plotting cos Od versus dA/rdt also gives a linear relationship, see Fig. 6 of this study. In our view Eq. [24] is the correct form for the linear relationship because the velocity of the contact line, V, should be a function of the dynamic contact angle and V = d4/ 2 wdt. This aside, one can still use the data obtained by Lau and Bums to see if 2.71 kcal/

II

ACKNOWLEDGMENTS I am grateful to W. R. Schowalter for helpful comments on this work and to Q. A. Trementozzi for his continuing support. REFERENCES Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975). Huh, C., and Striven, L. E., J. Colloid Interface Sci. 35, 85 (1971). Dussan V., E. B., and Davis, S. H., J. Fluid Mech. 65, 71 (1974).

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