fluid contact lines

Notes Comments on the Stability of Solid/Liquid/Fluid Contact Lines Solid/liquid/fluid contact fines are encountered in such phenomena as flotation, detergency, adhesion, and oil recovery. Usually, the parameter of fundamental interest is the angle of contact which the liquid/fluid interface makes with the solid surface at the line of three phase intersection. Consequently, much effort has centered around measurement of contact angles, their dependence on the nature of the solid and liquid phases and their relation to the surface free energy of the solid surface. Frequently, it is not only of interest to determine the magnitude of the contact angle in a particular system, but also under what conditions the contact line itself will be stable. One such well-known condition is that the interfacial tension at each of the interfaces be less than the sum of the interfadal tensions of the remaining intersecting surfaces (1). However, contact line stability does not depend solely on the intensive state of the system, but also on its capillary configuration, e.g., consider the flow of raindrops down a window pane. Capillary configuration stability of a variety of configurations has been studied recently by several workers, e.g., (2-7), and the approaches used have been diverse. Herein, we will apply Gibbs' method of constrained equilibrium states to the problem of contact line stability. Two situations wilI be considered, namely, a somewhat general configuration not subjected to external fields and a specific configuration in the gravity field. The latter configuration is the axisymmetric dry patch whose stability has already been rigorously studied by Huh (4) and Taylor and Michael (7). In the following discussion, it will always be assumed that the solid surface is ideal in that it is smooth, homogeneous, and in a constant state of strain independent of the presence of the other phases. Furthermore, the solid is mutually insoluble with the liquid and fluid phases.

cal, and thermal equilibrium, except those conditions relating to equilibrium of the solid/fluid/liquid contact line. If this state happens not to satisfy the latter conditions, then the position of the contact line is regarded as being constrained in a manner such that the former conditions are entirely satisfied. Furthermore, we only consider states for which the contact angle is uniform aiong the perimeter of the contact line. Let us compute the virtual change in free energy accompanying an isothermal change in the position of the contact line among the collection of constrained equilibrium states as defined above. The change is envisioned to occur in a closed system in which the chemical potentials of all the components remain uniform throughout the system, so that dF = "YsLdAsn q- "Y~FdAsF -[- VLFdALF --PLdVL

- -

PvdVv,

[1]

where v z j and A H are the interracial tension and area of the I J interface; P~ and VI are the pressure in and ~;olume of phase I. Since the system is closed dVF= --dVL and dAs~ = --dAsh. Furthermore, dAsL, dVL, and dA sL are related by dVL =

fA~

6,dALF

[2]

and dALF = cosOdAsL +

[3]

~ + LF

K2/

A. CONFIGURATIONS I N T H E ABSENCE OF E X T E R N A L FIELDS Consider some initial state of the system in the configuration shown in Fig. 1. This state may or may not be at total equilibrium. However, we assume that this state satisfies all the conditions for mechanical, chemi-

FIO. 1. A configuration in the absence of external fields.

489 Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, VoL 53, No. 3, December 1975

490

NOTES

where 0 is the contact angle at the constrained contact line (regarded as uniform along its perimeter) and 1/R1 and 1/R2 are the principal curvatures of an element of the liquid/fluid interface and ~ is the normal displacement of this element. Eq. [-3] is a form of the Gauss relationship (10-12). Substituting the conditions of closure and Eqs. [2] and [3] into Eq. [1] yields:

dF = (3"sL -- 3"sF + ~'z~ cosO)dAsL

By assumption, all constrained states to which Eq. [4] applies are states of semi-equilibrium and in particular the Laplace (13) equation of capillarity is obeyed. Thus, the second term in Eq. [4] vanishes. I t should be noted that the angle in each constrained state is not arbitrary. Now, if the initial state is not at total equilibrium, then when the constraint is relaxed, the system will adjust so as to minimize the free energy over all states and consequently, also over the manifold of constrained equilibrium states. However, if the initial constrained state by chance happens to be one of stable total equilibrium, then the system will not readjust when the constraint is lifted, as it would already be a state of minimal free energy in the absence of the constraint. For this to be the case requires that d F = O and d 2 F > O (neglecting higher-order differentials), at least over the collection of constrained equilibrium states. Defining ~- as ( 3 ' s L - 3'sF + 3%F cos0) and considering r = r (Asr., VL), d2F becomes:

d'F = ( O-'-2-r"~ (dAsr.) ~ OA s L/ 2",V.~,ar.vL + -

dAsLdVL.

~-52

OVL T,V,g,N,AsI~

Thus, ior the constrained equilibrium state also to be one of stable equilibrium, at least with respect to the collection of constrained states of uniform contact angle, requires: r =

0

force. The physical implication is evident. If the configuration were stable with respect to fluctuations in the contact line, then, if this contact line were "pulled out" slightly, "held" and "let go," the contact line would return to its initial state. B. T H E AXISYMMETRIC DRY PATCH AS AN E X A M P L E OF A CONFIGURATION I N T H E GRAVITY FIELD The profile of the axisymmetric dry patch configuration is shown in Fig. 2. Following Huh and Scriven (15) we will regard the liquid/vapor interface as unbounded, i.e., as extending to infinity with condition that dZ/ dR--~O as R--*~ (see Fig. 2). The static profile of this configuration has been studied in detail by Huh and Scriven (15), Padday (16), and Kovitz (17) and its stability has also already been rigorously studied by Huh (4) and Taylor and Michael (7). The results that will be used here to apply Gibbs' method are those due to Huh and Scriven (15) and are reproduced in Fig. 3. The variables x0 and y0 are related to the contact line radius R and water layer thickness Z by

x0= (\ (pL C- ~pv)g),,"½ R

r8l

y0 = \( (p~ ~- v p v ) g/~. . Z

[9]

and

In this case, we define constrained states as those states of uniform contact angle and identical water film thickness that collectively satisfy all the conditions for mechanical, chemical, and thermal equilibrium, save those conditions relating to equilibrium at the solid/liquid/vapor contact line. At the outset, it is assumed that the liquid is incompressible. Following a similar route as taken above, the conditions that also render a particular constrained state to be stable in the absence of the constraint, at least

[6-1

and

"\'OA-~L/T.V,,.N,~ > 0. E73 Eq. [-6] is the familiar Young equation [-14]. The requirement in Eq. [-7] has a particularly simple form when the liquid phase happens to be incompressible, namely, (Or/OAsL)vz.v.~, u.N > 0 or (O0/OAsL)Vz.v.~r,,.~r <0. This relationship is essentially a form of the Le Chatelier principle (19) if r is interpreted as a restoring

FLUID D -- R ~

SOLID

FIG. 2. The profile of an axisymmetric dry patch in the gravity field.

Journal of Colloid and Interface Science, Vol. 53, No. 3, December 1975

491

NOTES C. LIMITATIONS 2 150 °

120" 90 °

Yo

6o0

1

30 °

0

~

-

~

- -

5° I

Xo

FIG. 3. Dependence of the liquid layer thickness (at R - - ~ ) on the contact line radius and contact angle for the axisymmetric dry patch, y0 and x0 are defined in the text. These results are taken from the work of Huh and Scriven (15). with respect to the collection of states defined above, are:

l-lO]

r = 0

and T,~,N,Z

Neglecting the possible influence of curvature and gravity on the interracial tensions, the condition in Eq. [11] reduces to O0/Olsn < 0. For the axisymmetric mode (which by virtue of the requirement of uniform contact angle is essentially the only one being considered here), the requirement reduces further to

(00/0-8) > O. Finally, given some contact angle 0, which satisfies Eq. ~10], inspection of Fig. 3 reveals that the unbounded dry patch generally can not be at stable equilibrium (at least over the range of x0 values plotted in Fig. 3) when the equilibrium contact angle is regarded as unique. This conclusion follows from the fact that (O0/O-8)T,~,N,Z is not positive for any 0. This conclusion is in agreement with the earlier results of Huh (4) and Taylor and Michael (7). I t might appear from Fig. 3 that for large R (or x0) the dry patch eventually could be considered as a configuration of neutral equilibrium in the sense that d2F = 0. In other words, (00/0-8) = 0. However, to a good approximation for very large R [-see (15)'1:

{ 0_00~

ko~0/ T,~.N,Z

=

yo ~

--

y~,,~

2X0 sin(0)

(12)

where yeyl is the limiting two dimensional solution (cylindrical) for the unbounded interface ~yoyl = 2.sin(0/2)3. As long as x0 is finite, y < yoyl (15) and thus, O0/OXo is strictly always less than zero for large

X0 (or _8).

Several assumptions have been made in the above discussion of contact line stability that clearly limit its generality. The first concerns the very existence of constrained equilibrium states. T h a t is, given a state at total equilibrium, it has been assumed that there are neighboring states of different solid/liquid and solid/ fluid interracial areas, but, nevertheless, states that satisfy all the remaining conditions for equilibrium (particularly the Laplace equation), provided the contact line is suitably constrained. If this is not the case, then the Gibbs method can not be applied in the present form. However, ff the assumption is valid for the configuration of interest and if the solid surface is ideal, then the conditions derived from the Gibbs method will always be necessary for contact line stability, e.g., Eqs. [61 and ET"I for a system in the absence of external fields. For, if the assumption is met and the conditions are not met, then there will be some state of identical volume, temperature, and mass, but of lower free energy. In this discussion, only constrained states of uniform contact angle have been considered. Thus~ the conditions in Eqs. E6"] and [7-] or [107 and [11"] are only necessary for stability, but are obviously not sufficient. However, the utility of the Gibbs method as employed herein is not so much in establishing stability (a considerably more complicated task frequently employing variational calculus), but rather in recognizing obvious instances of instability. As will be shown in a future communication, this will prove quite useful in the study of the properties of partially submerged drops for which a complete analysis of stability would indeed be formidable. ACKNOWLEDGMENTS The authors wish to acknowledge the helpful comments made by Dr. H. M. Princen of Lever Brother Company and Dr. James C. Melrose of Mobil Research and Development Corporation.

REFERENCES

I. DEFAY, R., PRIOOGINE~ I., BELLE~t~NS, A., AND EVERETT, D. H.~ ¢~Surface Tension and Adsorption," p. 16. John Wiley, N e w York, (1966). 2. CoNcus, P., ldvan. Astronom. Sci. 14, 21 (1963). 3. GILLETTE, R. D. AND DYSON, D. C., Chem. Eng. J. (London) 3, 196 (1972). 4. HuH, CnVN, Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota, 1969. 5. PADDAY, J. F. AND PITT, A. R., Philos. Trans. R. Soc. London Set. A. 275, 489 (1973). 6. MELROSE, J. C., Soc. Chem. [nd. London, Monograph No. 25, (1967).

Journal of Colloid and Interface Science, VoL 53, No. 3, December 1975

492

NOTES

7. TAYLOR,G. I., AZCOMICHAEL,D. H., Y. Fhdd Mech. 58, 625 (1973). 8. GIBBS, J. WILLA~D, "Thermodynamics," Vol. I, p. 246. Dover, Publications, New York, (1961). 9. CALLA•, H. B., "Thermodynamics," John Wiley, New York, 1960. 10. GAuss, C. F., "Theorie de Gestalt yon Flussigkeiten," Verlag, Von Wilhem Engelmann, Leipzig, 46, (1903). 11. HWA~C, Su~-TAK, 48th National Colloid Symposium, p. 54. Austin, Texas, 1974. 12. MELI~OSE,J. C., A.I.C.H.E. J. 12, 986 (1966). 13. DE LAPLACE,P. S., "Mechanique Celeste," Suppl., Vol. 10, 1806. 14. YoI~ND, T., "Miscellaneous Works," Vol. 1, p. 418. (G. Peacock, Ed.), J. Murray, London, 1855.

15. Hun, C~tuz,,rAI,,~ SCglVEI% L. E., Y. Colloid Interface Sci. 30, 323 (1969). 16. PADDAY,J. F., Philos. Trans. R. Soc. London Ser. A 269, 265 (1971). 17. KovITz, A. A., J. Colloid Interface Sci. 50, 125 (1975). 18. WILKINSON,M. C., ARONSON, M. P., AND ZETTLE~OYER, A. C., Y. Colloid Interface Sci. 37, 498 (1971). M, P. ARONSONAND A. C. ZETTLEMOYER Lever Brothers Company, Research Division JEdgewater, New Jersey 07020, and Center for Surface and Coating Research Lehigh University, Bethlehem, Pennsylvania 18015 Received November 6, lP74; accepted July 8, 1975

Journal of Colloidand Interface Science, VoI. 53, No. 3, December 1975