Influence of surface forces on hydrodynamics of wetting

Colloids and Surfaces A:Physicochemical and Engineering Aspects 91( 1994) 149-154

ELSEVIER

COLLOIDS AND SURFACES

A

Influence of surface forces on hydrodynamics of wetting * Victor M. Starov a,*, Vasiliy V. Kalinin b, Vladimir I. Ivanov b a Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712-1062, USA b Moscow Academy of Food Industry, 11 Volokolamskoe sh., Moscow 125080, Russian Federation Received 20 December

1993; accepted

16 March 1994

Abstract It is well known that properties of thin liquid films deviate from those of a bulk liquid. Disjoining pressure was introduced by Derjaguin to describe thin liquid film behavior. The motion of a long oil drop or bubble under the action of an imposed pressure difference in a thin capillary filled with another immiscible liquid is investigated theoretically, taking into account the action of disjoining pressure. The velocity of the drop motion as a function of the pressure difference and the thickness of the film intervening between the drop and the capillary walls are calculated. The solution obtained coincides with Bretherton’s equation at high velocities and deviates substantially at low velocities, which is a manifestation of disjoining pressure action in the thin film. In the same way, a problem concerning the thickness of a thin liquid film covering a thin solid thread drawing out from a liquid container is considered. The thickness of the film is calculated as a function of the thread velocity. The theoretical results obtained are in good agreement with known experimental data.

Keywords:

Surface force; Long drop; Motion

1. Introduction

The studies published in Refs. [l-4] on the capillary motion of two immiscible fluids did not take into account the action of disjoining pressure in thin liquid layers. This approach can be justified only in the case when the thickness of the film h, formed between the surface of the drop and the capillary wall is sufficiently large (h, > 10e5 cm). At the same time, experimental studies [S-7] showed that at low drop velocities, the thickness of the intervening film becomes comparable to the

* A short version of this paper was presented at International Congress on Emulsions, Paris, October 1993. * Corresponding author. Present address: Mathematical Dept., Moscow Academy of Food Industry, Volokolamskoe sh. 11, Moscow

125080, Russian

Federation.

0927-7757/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0927-7757(94)02876-T

range of the surface forces action. Wetting phenomena, which occur in this case, cannot be explained without the effect of disjoining pressure, and it turns out that the latter is the most important factor. The first attempt to include disjoining pressure in hydrodynamic equations was undertaken in Refs. [S and 91. In Ref. [S] the advancing of the meniscus over an equilibrium liquid film was considered in the case of complete wetting. An equation of hydrodynamic motion taking into account disjoining pressure action was derived, and it was shown that two different regimes of flow in thin liquid films exist, depending on a parameter CI=nAl/” x p$“-‘)/[y”3( 3~Lu)~‘~]. If a < 3/4113 z 1.89 (corresponding to a “high” velocity of the advancing meniscus), damped waves are present ahead of the meniscus; if c(> 3/41’3 (“low” velocity), the liquid

150

V.M. Starov et al./Colloids Surfaces A: Physicochem. Eng. Aspects 91 (1994) 149-154

profile is quite different, because now there are not damped waves but the liquid profile has only one minimum and that minimum moves ahead of the advancing meniscus. In Ref. [9] the motion of a long bubble under the action of an imposed temperature gradient was considered. The interplay between three fluxes was taken into account. Those fluxes are a vapor flux, Qv, a thermocapillary flux in the thin liquid film, QT, and a pressure-driven flux in the thin liquid film, Qp. It was shown that the last flux is directed in the opposite direction to the first two fluxes and is negligibly small in a cylindrical capillary. The situation is different in conical capillaries. It was shown that the flux QP increases drastically with the increase of taper of the conical capillary; its effect becomes decisive and at some specific value of the taper the total flux becomes zero, i.e. the bubble sticks in the conical capillary. The shape of the disjoining pressure isotherm depends on the interaction of the fluids in the film and the drop with the solid capillary walls. The S-shaped disjoining pressure isotherm most commonly reported for aqueous solutions [lo] determines a whole range of hysteresis effects (advancing and receding contact angles, equilibrium film thickness, shape of film in the transition region, etc.) [ 111. A numerical analysis of the motion of long oil drops in thin capillaries, taking into account disjoining pressure action, has been undertaken

cm The problem examined in Ref. [ 131 relating to the stability of a fluid film between the droplet and capillary walls for an S-shaped disjoining pressure isotherm is also based to a significant extent on the shape of the isotherm (the locations of the c(and B-branches of the isotherm). Non-polar liquids are characterized by monotonically decreasing disjoining pressure isotherms having the form 17(h) = A/h”, (n 3 2, where A>0 is usually referred to as the Hamaker constant at n = 3), and h is the film thickness. Such isotherms pertain to the case of complete wetting and have been subject to closer theoretical and experimental investigations. We will restrict ourselves below to the isotherm of the latter type. The filtration motion of a drop in a capillary without allowing for the effect of disjoining pres-

sure was investigated in Ref. [l] and the following dependence of the film thickness on the drop velocity U was deduced: h, = 0.643R( 3Ca)213, where R is the radius of the capillary and Ca = plU/y is the capillary number (pi is the viscosity of the fluid in the capillary and y is the surface tension at the interface between the fluids). The latter equation (usually referred to as Bretherton’s equation) yields an infinite reduction in film thickness with a decrease in the velocity of the drop, U. However, it is known that the thickness of the film approaches an equilibrium value h, as the drop velocity decreases [ 5-7,181. This equilibrium thickness can be found from the condition ZZ(h,) = pe where pe is the excess equilibrium pressure in the drop.

2. Theory As in Ref. [ 141, we examined a system consisting of a cylindrical capillary of radius R with a fluid containing a drop of another fluid (fluid 2 in Fig. 1). Assuming the fluids to be immiscible and limiting ourselves to the axisymmetric case, we introduce a coordinate system connected with the drop. The x axis of the system coincides with the axis of the capillary, while the r axis is perpendicular to the capillary axis. Let 1 be the length of the drop, L+ and L- the lengths of the drop-free sections of the capillary, and L its overall length. The pressures p- > p+ are assigned at the ends of the capillary. We examined the steady motion of the drop along the capillary in the positive direction of the x axis

Fig. 1. Schematic representation of a drop in a capillary: 1, liquid in a capillary of length L; 2, drop of length 1.

151

V.M. Starov et al./Colloids Surfaces A: Physicochem. Eng. Aspects 91 (1994) 149-154

at a velocity U (to be determined as a function of the applied pressure difference) under the following restrictions: (i) low Reynolds numbers, Re = U R/p1 <<1; (ii) a long drop, i.e. R/l <<1; (iii) a sufficiently small slope of the surface of the drop in the transition zone [4]; (iv) a small capillary number, Ca <<1. As in Ref. [4] we divided the flow field in the capillary into the following regions: A’A and FF’ (those parts of the capillary not containing the drop, where Poiseuille flow is obtained); AB and EF (spherical menisci at the ends of the drop); CD (a region with a constant film thickness h,,, which ii11 be determined below as a function of the capillary number Ca); and BC and DE (transition regions from the film of constant thickness film to the menisci). An equation which describes the profile of the surface of the drop in the transition regions BC and DE with the assumption that the viscosity ratio is on the order of 1, i.e. pLz/pl z 1 and omitting higher order terms of the small quantity h/R cc 1, has the following form [ 18,191: h3 d( pZ - p,)/dx = 3Ca(h - h,) where of the fluids to the P2

-P1

(1)

h(x) is the thickness of the film. In the case absence of disjoining pressures, p1 and pZ in 1 and 2 differ by an amount corresponding Laplacian pressure: =

(2)

YKW

where K(x) is the curvature of the surface of the drop K(x)=d2h/dx2+

l/(R-h)

(3)

With an allowance for the effect of the surface forces, Eq. (3) should be rewritten to include the action of disjoining pressure [11,15,16]: ~2

-

~1

=

YKW

+

W4

(4)

where 17(h) is the disjoining pressure isotherm. In this case, Eq. (1) for the profile of the surface has the form

IT(h) = A/h

n=2,3

(5)

We will restrict ourselves from now on to isotherms

(6)

Let us introduce the dimensionless variables H= h/ho, y=x/x*, where x* is the characteristic scale of regions BC and DE, which will be chosen as the scale of the transition region in the equilibrium state. Integrating Eq. (5) at U = 0, we obtain y d=h/dx= + A/h” = pe = y/R,

(7)

where R e zR are the radii of the equilibrium menisci at the ends of the drop. In the transition region, y d2hldx2 z A/h” z y/R; hence yh,/x: z y/R and therefore x.+ = (hOR)“’

(8)

without the higher order terms, Eq. (5) can be written in the dimensionless form d3H/dy3 - (a dHldy)/H”+’ = V(H - 1)/H3

(9)

where a = nRA/yh,“, V=3Ca(R/h,)213. The case a =0 corresponds to the case of filtration motion at a “high” velocity, which was examined in Refs. [1,4,14], when it is possible to ignore the effect of disjoining pressure. We therefore set a # 0. Eq. (9) has the solution H = 1 corresponding to a film of constant thickness in the zone CD. This solution should ensure the joining of this zone with the surfaces of the spherical menisci in regions AB and EF at the ends of the drop. The matching conditions for the leading end of the drop have the form d’H/dy= -R/R+ H-+1

aty++oo

sty+-cc

(10)

while for the trailing end d2H/dy2 -+ R/R H-+1

at+-cc

aty-++cc

(11)

The satisfaction of the conditions Eqs. (10) and (11) is assured by the existence of asymptotic solutions of Eq. (9) having the form H-+F’(Ca)y2/2 + G’(Ca)

h3[d3h/dx3 + R - 2 dhldx + (dl7jdh dh/dx)/y] = 3Ca(h - h,)

of the disjoining pressure of the type

aty+fcc

(12)

The plus sign denotes the leading meniscus, while the minus sign denotes the trailing meniscus. The parameters F’ and G’ depend on Ca and are

V.M. Starov et al./ColloidF Surfaces A: Physicochem. Eng. Aspects 91 (1994) 149-154

152

determined from the numerical solution of Eq. (9). Comparing Eq. (12) with Eqs. (10) and (ll), we obtain R’ CR/F’ ho =R(F+-l)/F+G+

=R(F--l)/FG

(13)

Assuming in Eq. (9) that V = 0 (Ca = 0), we obtain an equation for the transition region between the film and the meniscus for the equilibrium case studied in Ref. [ 161. Here, for the isotherm of the form of Eq. (6), the equilibrium meniscus has the radius R, = R - h,n/(n - 1). Comparing these relationships with Eq. (13) to within the leading terms, we obtain the following limit relationships: Lim F’(V)= v-0

Ap = (2y/R) [R/R+ -R/R+ 4Cu (L+ + L- + vl)/R]

1 + h,n/R(n-1)

Lim G’(V) = n/(nv-0

1)

(14)

Fig. 2 shows the dependence of the film thickness ho on the capillary number Cu. We adopted for the calculations the values R = 10m2cm and y = 30 dyn cm-‘. It can be seen from Fig. 2 that the thickness of the film in the region CD can be considered as being equal to the equilibrium thickness h, at (~CU)“~
“high” velocities, the graph of the relationship h,(Cu) coincides with Bretherton’s equation. Fig. 3 shows the calculated dependences of the parameters F’, G’, and the radii of the leading and trailing meniscii on the capillary number. In each case we took R= 10m2cm and y= 30 dyne cm-‘. To finish the solution of the problem, we need to establish the relationship between the pressure difference dp = p- - p+ at the ends of the capillary and the velocity of the drop U. As was shown in Ref. [ 141, this relationship can be represented in the form

10’

where v = p2/p1. With the help of Eqs. (13), we find from Eq. ( 15) that Ap = (2y/R)[F+(Cu) - F(Cu) +4Cu(L+ + L- +vl)/R]

(16)

The graph of Eq. (16) obtained from a numerical calculation is shown in Fig. 4. Here, we adopted the values R=10p2cm, y=30dynecm-‘, v=O (gas bubble), .I,+ + L- =4 cm, 1= 1 cm and p1 = 10m3 Pas. Fig. 4 shows that the presence of the single drop substantially changes the velocity of the motion at Ap < 0.5 Pa, and U(Ap) is highly non-linear. Let us examine the motion of a sequence of k drops in the capillary. Let Ii, where i= 1,2, . . .. k, be the lengths of the drops. In accordance with Ref. [ 171, the effects of drops on one another will W/R

Fig. 2. Dependence of h, on the capillary number: curve 1, Bretherton’s equation (II(h) = 0 or a = 0); curve 2, n(h) = A/h3, A= lo-l4 erg; curve 3, 17(h) = A/h*, A= lo-* erg cn-‘. The broken lines correspond to equilibrium thicknesses.

(15)

0’

Fig. 3. Dependence of the parameters F+ (curve l), F (curve 2), G+ (curve 3) G (curve 4) and the radii of the leading R+ (curve 5) and trailing R (curve 6) meniscii on the capillary number, at n(h) = A/h’.

153

V.M. Starov et al.lColloids Surfaces A: Physicochem. Eng. Aspects 91 (1994) 149-154

0.1

a.5

0.3

APO Pa Fig. 4. Dependence of velocity U on the pressure difference at the motion of a single drop; curve 1, at IT(h)=O; curve 2, at L’(h) = Alh2.

be negligibly small if the distance between them exceeds 2R. For a chain of k drops, Eq. (16) is then rewritten in the form dp = (21)/R) [k(F+ -F-) +4Ca L’+L-+Vi:li (

/R] 1

calculated. The solution obtained coincides with Bretherton’s equation at high velocities but deviates substantially at low velocities, which is a manifestation of disjoining pressure action in the thin film. In the same way, a problem concerning the thickness of a thin liquid film covering a thin solid thread drawing out from a liquid container is considered. The thickness of the film is calculated as a function of the thread velocity. Theoretically obtained results are in good agreement with known experimental data. It is necessary to emphasize that in the problems considered above, there are no fitting parameters used to adjust the theoretical calculations and experimental data. The solutions obtained are among a few exact solutions of hydrodynamic equations that include the action of surface forces.

List of symbols

>

The calculation shows that in the case k= 10, the influence of the chain of drops (each of them is identical to the single drop from the previous case) is significant at dp < 15 Pa. The same method was applied to the problem of coating a thin fiber with a thin liquid film. The thickness of the thin liquid film on the surface of the fiber, which moves out of a liquid with a constant velocity, was calculated as a function of the velocity and the calculated dependence is very similar to curve 3 in Fig. 2. The results obtained are in good quantitative agreement with experimental results [ 19,201.

3. Conclusion The motion of a long oil drop or bubble under the action of an imposed pressure difference in a thin capillary filled with another immiscible liquid is investigated theoretically in this paper, taking into account the action of disjoining pressure. The velocity of the drop as a function of the pressure difference, and the thickness of a film intervening between the drop and the capillary walls are

nRA/yh,” “A Hamaker constant Ca capillary number F, G dimensionless functions Eq. (13)) film thickness h H dimensionless thickness curvature K drop/bubble length 1 length L exponent n pressure P Q flux R capillary radius Re Reynolds number velocity of filtration u dimensionless velocity V Greek letters M.

Y cc V 17

a parameter surface tension viscosity =

/b/PI

disjoining pressure

(see

definition,

V.M. Starov et al./Colloids Surfaces A: Physicochem. Eng. Aspects 91 (1994) 149-154

154

Subscripts

[S]

0

[6]

e P T V 1 2 *

flat liquid film equilibrium pressure driven thermocapillary vapor outside the drop inside the drop characteristic scale

Superscripts +

-

in front of the drop and leading meniscus behind the drop and trailing meniscus

References

Cl1 F.P. c21 A.L.

Bretherton, J. Fluid Mech., 10 (1961) 166. Frenkel, A.J. Babchin, B.G. Levich, T. Shlang and G.I. Sivashinsky, J. Colloid Interface Sci., 115(l) (1987) 225. c31 L.W. Schwartz, H.M. Princen and A.D. Kiss, J. Fluid Mech., 172 (1986) 259. c41 C.W. Park and G.M. Homsy, J. Fluid Mech., 139 (1984) 291.

E.E. Voznaya, A.Yu. Komissarova, E.D. Taevere and K.P. Tikhomolova, Colloid J. USSR, 42(4) (1980) 620.

D.V. Tikhomolov, Ph.D. Thesis, Leningrad State University, Leningrad, 1986. [7] E.E. Voznaya, K.P. Tikhomolova and E.D. Taevere, Colloid J. USSR, 41(5) (1979) 827; 41(6) (1979) 982. [S] V.M. Starov, N.V. Churaev and A.G. Khvorostjanov, Colloid J. USSR, 39( 1) (1977) 176. [9] O.A. Kiseleva, V.M. Starov and N.V. Churaev, Colloid J. USSR, 39(6) (1977) 1021. [lo] B.V. Derjaguin and N.V. Churaev, J. Colloid Interface Sci., 62(3) (1977) 369. [ 1 l] V.M. Starov, Adv. Colloid Interface Sci., 39 (1992) 147. [12] G.F. Teletske, Ph.D. Thesis, University of Minnesota, MN. 1983. [13] V.V. Kalinin and V.M. Starov, Colloid J. USSR, 51(6) (1989) 949. [14] V.I. Ivanov, V.V. Kalinin and V.M. Starov, Colloid J. USSR, 53(l) (1991) 25. [15] B.V. Derjaguin and N.V. Churaev, J. Colloid Interface Sci., 66( 3) (1978) 389. [ 161 B.V. Derjaguin, V.M. Starov and N.V. Churaev, Colloid J. USSR, 38(5) (1976) 786. [17] P.K. Volkov and E.A. Chinnov, Modelling Heat and Mass Transfer Processes in Chemical and Biochemical Reactors, Proc. 6th Int. Summer Conf., Varna, Bulgaria, 1989. [ 181 V.M. Starov, Colloid J. USSR, 45(6) (1983) 1009. [ 191 V.I. Ivanov, D. Quiere, J.-M. di Meglio and V.M. Starov, Colloid J. Russ. Acad. Sci., 54(3) (1992) 346. [20] D. Quere, J.-M. di Meglio and F. Brochard-Wyart, Science, 249 (1990) 1256.