Langmuir 1991, 7, 430-433
430
Notes Surface Energy of Fluid Dispersions in Hydrostatic Equilibrium
AMBIENT PHASE
T. L. Crowley
C 0 N T I NU0 US PHASE
Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4WT, England Received March 26, 1990. In Final Form: June 11, 1990
Introduction Dispersions of fluid particles in hydrostatic equilibrium include high volume fraction foams and emulsions and, probably of equal importance, high-volume fraction dispersions of “film-forming”latices. The latter provide the base of latex-based paints. In addition, small clusters of fluid droplets or even individual droplets, may be so considered. The clusters may be either freely suspended in an ambient phase, supported on a substrate, or suspended at a fluid interface, e.g. a bubble raft at the air/liquid interface. At high volume fractions a “polyhedral foam” type structure is adopted in which adjacent fluid particles are distorted from their initially spherical shape with the formation of, typically curved, interfaces between them. The equilibrium properties of such systems and, in particular, their geometry, internal pressures and solvent osmotic pressure, and interfacial surface energy, has been discussed in detail by Princen’. These systems comprise fluid particles i, and an ambient phase, a, which is taken to provide the ambient pressure P. We write ni for the number of moles of material, Vi for the volume, and pi for internal pressures of the particles. In addition there may be present a continuous solvent phase, s, which is imbibed into the dispersion at a positive osmotic pressure, ir (Figure 1). Such a phase has an internal pressure, ps,below ambient, ir = P - ps. We will generally treat this solvent phase equally with the fluid particles, with i = s and by the dispersion we will mean both the fluid particles and any imbibed solvent. Vis the total volume of the dispersion and N the total number of phases present in the dispersion. For a dispersion at a flat fluid interface, there will be two ambient phases present. The equilibrium configuration of a fluid dispersion arises from a tendency to minimize the interfacial surface energy. This tendency is resisted by (i) the development of pressure differentials across individual curved surfaces, (ii) the surface tension applied by other fluid surfaces, for example at three-phase boundaries, and (iii) the normal tensions provided by rigid substrates. These effects are described by the equation of Young and Laplace, Neumann’s triangle, and Young’s equation, respectively. The equilibrium conditions for individual surfaces are thus well understood. When considering the overall stability of fluid dispersions, it is sufficient to combine these results. However, as there will frequently be a large or very large number of individual interfaces present, the more usual approach has been to investigate the total equilibrium surface energy of the dispersion, U , in particular the relationship between U and the mechanical and compositional variables of the dispersion. With this approach the situation appears less (1) Princen, H. M. Langmuir 1988, 4, 164.
‘
DISPERSION
Figure 1. Fluid dispersion in neighborhood of ambient phase, showing fluid particles and continuous phase. AMBIENT PHASE
1 I I I I I I
Figure 2. General view of fluid dispersion showing container walls and imaginary test volume, VT, enclosing dispersion.
clear. Although the basic concepts and equations are present in the literature, it does seem that there is uncertainty as to the range of validity and the interpretation of the fundamental results. In particular our ultimate objective is a general proof of the energyequation, (3) below. When applicable, the energy equation provides a rigorous relationship between the surface energy and the internal pressures and volumes and, thus, provides a consistency check that can be applied to models of the equilibrium structure of fluid dispersions and small clusters of fluid particles. It is thus important to be able to prove its validity for as wide a range of systems as possible. It is stated that it has so far only been proved for dispersions of monodisperse particles.1*2 Morrison and Ross2discuss in some detail previous attempts to provide more generally valid proofs of eq 3.
Surface Energy and Equilibrium The fundamental results we have in mind are the following: (i) The criterion of surface energy minimization. That is, the necessary and sufficient condition for a configuration to be an equilibrium configuration is that its surface energy should be a minimum, a t fixed Vi. (ii) The equilibrium equation, e.g., see ref 2. This is the expression for the total differential of the equilibrium surface energy
(iii) That the equilibrium configuration and surface energy is, a t least locally, completely determined by the N volumes Vi, that is U is a unique function of Vi, i.e. u(Vi). (iv) The similarity principle. That is, that if all the Vi, including the solvent phase, are increased in proportion, by a factor c3,then the resulting equilibrium configuration (2) Morrison, I. D.; Ross, S. J. Colloid Interface Sci. 1983, 95, 97.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 2, 1991 431
Notes is that which is obtained by a similarity transform, in which all of the linear dimensions of the system are scaled by a factor c. In particular Uincreases by a factor c2and hence U(C3VJ = C2U(VJ (2) (v) The energy e q ~ a t i o n . l - ~ This expresses U directly in terms of the mechanical variables of the system i
Especially for foams, when the equation of state of the gaseous bubbles is substituted, this is often referred to as the "equation of state" of the dispersiona3p4This leads to the equation of Princen' in the form
nRT - PV = 'l3U + TV,
similarity principle is valid for a dispersion (a) on flat substrate, (b)between inclined flat plates,and (c)at flat fluid-fluid interface.
(4)
where n is the total number of moles of gas. This can in principle be applied as a practical method of determining U. For incompressible particles there is however no simple direct method of determining the average of the internal pressures. The criterion of surface energy minimization and the equilibrium equation are obtained directly from the principle of virtual work below. We have not previously seen the equilibrium equation stated as being the total differential of U. This is important as once this is established, then (iii-v) follow in easy succession. Also its immediate consequence (iii) and the similarity principle, as we have stated it (iv), are not obvious in the literature. The use of the equilibrium equation appears to have been unnecessarily restricted to considering experimentally achievable processes such as compressing a foam by varying the ambient pressures, P, e.g. refs 1 and 2. As noted there, for such processes, the similarity principle is only applicable to monodisperse "dry" foams. This is because only then do the volumes all decrease in proportion; otherwise the initial premise of the similarity principle is not satisfied, and hence its conclusion does not then follow. In their case where the similarity principle can be applied, the subsequent proofsly2 are essentialy identical to ours. The similarity principle is always applicable to the more abstract "scaling process" in which all the Vi are increased in proportion, and this is sufficient to provide the proof of (3)-the only difficulty is the physical one of achieving this process as it would involve varying the n, independently and in a complicated manner. This is unimportant as the equilibrium equation is better interpreted as describing the difference in U between two slightly different systems, rather than as referring to a genuine physical transformation between one system and another.
Conditions on the Similarity Principle The validity of the similarity principle is crucial to the the proof of the energy equation. Previous workers have only considered systems that are either unconstrained or for which wall effects are sufficiently weak that they can be considered unconstrained. However, as we show below the similarity principle and hence the energy equation may readily be extended to encompass an important range of constrained systems. In particular we shall show that the similarity principle is rigorously valid in the following cases: (a) unconstrained systems, that is freely suspended dispersions in an ambient phase; (b) dispersions supported (3) Derjaguin, B.V. Kolloid-Z. 1933, 64,l. (4) Ross, S. Ind. Eng. Chem. 1969, 61, 48.
Figure 3. Examples of constrained dispersions for which the
DISPERSION
/
Figure 4. Examples of constrained dispersions for which the similarity principle is not valid: for a dispersion (a) between parallel flat plates and (b) in a cylindrical capillary.
on a flat substrate or more generally a dispersion bounded by self-similar walls, for example, between nonparallel flat plates, in the vertex of a polyhedron or in the vertex of a cone, or a generalized cone (Figure 3); (c) dispersions suspended a t a flat fluid interface (Figure 3). For cases b and c it is necessary to interpret U as a surface excess energy. Examples of bounding walls for which the similarity principle and the energy equation do not work include parallel flat plates, cylindrical capillaries, and, typically, curved walls-such walls are not self-similar (Figure 4). The failure of (3) in such cases can easily be shown by considering a single fluid particle. When considering whether the walls follow case b, it is only necessary to consider those walls which the dispersion is actually in contact with. For example, if a dispersion is between two flat plates but is only in contact with one of them, then it agrees with case b and the similarity principle holds, although in this case only over a restricted range of scale parameters-but this is still sufficient for (3) to hold. We now proceed to prove the above assertions. With Princen we assume that the fluid particles are sufficiently small that gravity may be neglected and that they are sufficiently large that the effect of curvature on surface tension may be neglected.
Equilibrium Configurations and System Variables By a configuration, C, of the dispersion we simply mean a shape of the fluid particles, or equivalently a shape of the fluid interaces that bound the individual components of the system. Of all possible configurations only a certain number of configurations can correspond to an equilibrium configuration. The surface energy only depends on the geometrical configuration, and U(C) may be simply evaluated even if C is a nonequilibrium configuration. If Aij is the area of the interface between i and j and bij is its surface tension, then its contribution to the surface energy is Aijuij; similar contributions must be added for the interfaces between the dispersion and the ambient
Notes
432 Langmuir, Vol. 7, No. 2, 1991 phase and for the interfaces of the dispersion and the ambient phase with any rigid walls. For an equilibrium configuration, the pressure difference across Aij is given by the equation of Young and Laplace pi - pj = 2gijHij, where Hij is the mean curvature with respect to i, taking the sign convention such that a convex body has positive curvature. By evaluation of the pressure differential from particle to particle until the ambient phase is reached, the pressure pi relative to ambient may be determined. That is, pi - P solely depends on the configuration C and the surface tensions gij. The amount of each phase ni is obtained from its internal pressure and volume, using the equation of state of phase i, i.e. Vi = niVi(pi),where Vi is the partial molar volume of component 1.
We consider a particular configuration in which there is mechanical equilibrium. In general for a given composition of the dispersion there will be many possible stable equilibrium configurations, even for a monodisperse dispersion, corresponding to the various possible packing schemes, which may be ordered or disordered. However, in the neighborhood of a particular equilibrium configuration there will be a single, continuous N 1 dimensional, family of equilibrium configurations, corresponding to systems with slightly different system variables, such as ni, Vi, pi, and P. By system variables we simply mean parameters of the system and do not imply that it would be a practical proposition to experimentally vary these variables. Obviously as long as equilibrium is preserved, all of these variables cannot be varied independently. Useful independent sets of variables for equilibrium states are the N + 1 variables (ni,P) or (Vi,P). Although (ni,P) is the most useful set for experimental purposes, we shall find that the set (Vi,P) is more useful when analyzing the stability. We then write C(ni,P) and U(ni,P) or equivalently C(Vi,P) and U(Vi,P), for the corresponding equilibrium configuration and surface energy. We prove below the stronger assertion that locally theNvolumes Vi provide a complete set of independent variables for equilibrium surface energy, U(Vi), with P being redundant.
+
The Principle of Virtual Work and the Equilibrium Equation We consider a fixed test volume VT completely enclosing a fluid dispersion, where we allow the possibility of rigidwall constraints being present (Figure 2). With this prescription, we do not allow any fluid interface to cross VT. We take the “system” to be the surface itself and the “surroundings”to be the fluid particles and solvent, i, and the ambient phase, a. The principle of virtual work then states that dU2dw (5) where dw is the work done by the surroundings on the system. The equality corresponds to changes at equilibrium and the inequality to distortions of the surface from equilibrium. If V, is the volume of the ambient phase enclosed in VT, then VT = V + V,. dw is given by dw = dVi + P dVa
Cpi i
where
Substitution into (5) results in i
In particular, for perturbations a t constant Vi, d U 2 0,
where d U = 0 corresponds to equilibrium. Hence the necessary and sufficient condition for equilibrium is that U is a minimum a t constant Vi. P i s a subsidiary variable which may be varied ad lib. For definiteness P may be considered to be held constant. We have excluded the possibility of fluid interfaces crossing VT to avoid having to consider the contribution to dw arising from fluid interfaces moving with respect to the boundary of VT.We can however relax this condition, to allow a flat fluid interface, between two ambient fluid phases, to be included, as in case c above. This can be done simply by fixing VT relative to the flat interface. Both ambient phases will have the same internal pressure, P. The combined contribution of the ambient phases will still be -P dV as before, and hence again (6) follows. We have thus proved the criterion of surface-energy minimization and the equilibrium equation (1)discussed above. Note that the minimization criterion is purely geometric, being a minimization at constant Vi. The form of (6) shows that, a t least over a range of (Vi,P), U is solely a function of Vi independent of P. Hence locally we can consider U(Vi). The equilibrium equation (1)provides explicit expressions for the partial derivatives of U(Vi) with respect to Vi, when all the other Vj,are heldconstant, remembering that the continuous .solvent is included as i or j = s.
(7)
The Similarity Principle We now prove the similarity principle for any unconstrained equilibrium state, case a, or for the particular constrained cases b and c. Formally this states that the equilibrium configuration specified by (c3Vi,P)is identical with that obtained by applying a similarity transform with a linear factor c to the equilibrium configuration specified by ( Vi ,P). This assertion is possibly intuitively obvious, as equilibrium states are determined by the volume fractions of the individual particles and the solvent and by contact angles, which are each scale invariant. It may be proved simply by noting that the condition for equilibrium is that U should be a minimum with respect to perturbations a t constant volumes, V,. As U scales as c2 not only for the equilibrium state but also for all perturbed nonequilibrium states in its vicinity, then the scaled state is also a minimum energy state. Hence as the configuration is uniquely determined by the Vi, this equilibrium state is identical with that specified by (c3Vi,P). Then, when applied to surface energy, the similarity principle states that U is a homogeneous function of the Vi, including V,, of order 213, that is, eq 2 holds. For the constrained cases b and c, in order for the scaling arguments to hold, U must be interpreted as an excess surface energy, that is, as the total surface energy less the surface energy of the substratelambient interface or ambient/ambient fluid interface in the absence of the dispersion. The requirement for case b that the walls should be self-similar arises as follows: we are assuming that the walls are fixed, if they were not, there would be an additional contribution to dw in the principle of virtual work. The similarity transform is to be applied to the dispersion and not to the walls. If the walls are self-similar, then, a t least if a suitable origin for scaling is chosen, the resultant scaled system is still similar to the original system. However if the walls are not self-similar, for example, parallel plates, then scaling the dispersion alone will not
Notes lead to a similar configuration and typically will lead to
a system incompatible with the rigid constraints. The Energy Equation By applying the equilibrium equation to the particular equilibrium path specified by the scaling transformation, we can simply prove the energy equation for general polydisperse systems. For the scaling path we have dVi/ Vi = dV/Vand d U / U = 2/3dV/V. Substituting for dVi and d U in the equilibrium equation (1)results in
from which (3) follows. Equivalently, Euler’s theorem on homogeneous functions can be applied directly to Ugiven by (2), using (7) for the partial derivatives of U. It has been suggested in the literature that the previous absence of a rigorous proof of the equation of state is analogous to the situation with the absence of proof of Fermat’s last theorem.’P2 However, it can be seen that the proof is actually extremely simple. The results that we have derived may also be derived directly by “patching together” the equations of Young and Laplace and Neumann’s triangle as applied to the individual interfaces in the dispersion and using Young’s equation at rigid boundaries; this direct approach has the disadvantage of being less intuitive and more lengthy and relying on results of differential geometry. In particular the surface divergence theorem5 is then used to relate the pressure and tensions applied to each interface to its ( 5 ) Weatherburn, C. E. Differential Geometry of Three Dimensions; Cambridge University Press: Cambridge, 1927;Vol. 1.
Langmuir, Vol. 7, No. 2, 1991 433 surface energy. We shall use this approach in a subsequent publication to generalize the above results to (i) arbitrary external constraints and (ii) arbitrary subvolumes, VT,of the dispersion. With this point of view the energy equation may be seen to be the integrated form of the equation of Young and Laplace, Neumann’s triangle, and the equation of Young.
Nomenclature A , , area of interface between i and j c, scaling constant C, configuration of dispersion Hii, mean curvature of interface between i and j i, index for particle i (or solvent i = s) j , index for particle j (or solvent j = s) N , number of fluid components of dispersion n,,number of moles of component i pi, internal pressure of particle i p s , internal pressure of imbibed solvent R , gas constant s, index of solvent T, temperature U , surface energy of dispersion V, volume of dispersion, including solvent V,, volume of ambient phase Vi, volume of particle i ys,solvent volume Vi, partial molar volume of component i VT, test volume enclosing dispersion w ,work done on surface by fluid phases T , solvent osmotic pressure u+ surface tension between i and j Acknowledgment. We thank Professor D. G. Hall for detailed discussions on this problem.
