Langmuir 1993,9, 101-104
101
Thermodynamics of Fluid Dispersions in Equilibrium T.L.Crowley' Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4WT, England
D.G. Hall Unilever Research, Port Sunlight Laboratory, Quarry Road Eost, Bebbington, The Wirral, Merseyside L63 3JW, England Received June 8,1992. In Final Form: October 23,1992 For a fluid dispersion, D,with fluid phases, i, of internal pressures pi ,and volumes Vi, and interfacee s of eurface tensione u8 and areas A,, in contact with an ambient phase, I = 0, of pressure po and a rigid container wall w,the equilibrium equation is From Gibbs'general criterionfor equilibrium,we show that thisequationis valid for an arbitrarycontinuous infiiteaimal geometric displacement of the interfacial network, as long as it is consistent with the rigid wall constraints. In particular it is not nec88882y that the dispacement preserves equilibrium. Thie geometrical interpretation of the equilibrium equation allowe a eimple proof of Princen's virial equation for freely euspended diepersion, and ita extension to a dispersion on a flat rigid substrate
When applied to "dry" foams thisbecomea Derjaguin's equation of state for foams. T h e proof is obtained by applying the equilibrium equation to an infinitesimal scaling transformation. Ruivalent expressions for a fluid dispersion suspended at a flat fluid interface are also obtained. Introduction
AMBIENT PHASE CONTINUOUS PHASE
There has recently been interest in the thermodynamics and mechanics of fluid dispersionsin equilibrium,centered around Derjaguin's "equation of etate" for foams1S and its extension to "wet" foams and emulsions by Princen.3 Initially proofs of these equations were r e s t r i d to ideal monodisperse dispersions,or special cases such as a single or a double fluid drop.' More recently generalproofs have appeared which establish the result for freely suspended diepersions.516 A static fluid diepersion in thermodynamicequilibrium is considered, such as a foam or a high volume fraction emulsi~n.~ The dispersion consists of fluid phases i, of volume Vi and internal pressure pi and an interfacial network with surface components8 of area A, and surface tension u8. One phase, e.g. the atmosphere, is identified as an ambient phase, of pressure po and volume VO. A wet foam or an emulsion also includes a continuous, solvent phase, i = c, which is imbibed into the dispersion at a pressure below ambient, Figure 1. This continuous phase may be treated equally with the discrete phases. The discrete p h together with the continuous phase comprise the dispersion proper, D, of volume VD; D excludes the ambient phase. Typically the dispersion will be in contact with the rigid walls of a container, Figure 2a,b; if the dispersion is completely surrounded by the ambient phase, then we say that the dispersion is freely suspended, and the container walls are effectively absent, Figure 2c. For a freely suspended dispersion Gibbs' equilibrium equation is
CiWi -Po>SVi
DACRETE PHASE
DISPERSION
Figure 1. Section through fluid dispersion, showing ambient phase and continuow phase.
(a)
(C)
not neceeearily preserve equilibrium. As pointed out by Gibbs? the equilibrium equation is "a purely geometrical condition,since the preseures, and tenaions are constant". That is the given preseures, pi, and tensions, u,, are multipliers of the purely geometrical infinitesimal variations SA, and SVi.
C~U~SA, (1)
We use S to indicate an infinitesimalvariation, which need
(4) Morrieon, I. D.;Roan, S. J. J. Colloid Interface Sci. lSM,96,W. (6)Crowley, T.L. L a n g d r 1991,7,430. (6) Hollinger, H.B.J. Colloid Interface Sci. 1991,143,278. (7) Gibbs, J: W. The Scientific Papers of J. W. Gibbr Vol. l.,
* Author to whom correspondence rhould be addressed. (1)hi-, B. V. Kolloid-2. l V l , 64,l. ( 2 ) €+, 9. Ind. Eng. Che? 1969,61,48. (3)Pnncen, H. M.Langmurr 1988,4,164,
0743-7463/93/2409-0101$04.00/0
(b)
Figure 2. Boundary conditione for dispersions (a) encloeed container; (b) open container; (c) free dispersion.
Thermodynamics; Dover Pubhatiom, Inc.: New York, 1961;Chnptm 111. (6
1993 American Chemical Society
Crowley and Hall
102 Langmuir, Vol. 9, No. 1, 1993
Princen’s equation is
AMBIENT PHASE 0
’ ~
This equation relates the virial function or grand potential of the dispersion to that of the individual fluid phases and the individual interfaces. As the nomenclature “grand potential”arises from statistical mechanics,while the above equation is a purely mechanical equation, we prefer the older, mechanical term “virial” and we will refer to eq 2 and its analogues as the virial equation for the dispersion. The virial equation when applied to foams is referred to as the equation of state for Then CipiVi = ngRT+ p,V, where ngis the total number of moles of gas, R the gas constant, and T the temperature. p c and Vcare the internal pressure and volume of the continuous phase, c; the internal pressure of the continuous phase will be below ambient and may be expressed as an osmotic pressure, ll = ( p o - pc). Then for foams eq 2 may be rewritten as
.
-
L-
SUBSTRATE
AMBIENT PHASE 1
(b)
(a)
Figure 3. Dispersionsat flat interfaces: (a) rigid substrate, (b) fluid-fluid interface.
manner than in ref 5, the virial equation for dispersions at a flat solid interface, Figure 3a, and a flat fluid interface, Figure 3b.
Derivation of the Equilibrium Equation According to Gibbs’ general criterion for equilibrium,’ for any infinitesimalvariation whatsoever of an equilibrium State 6~ 2 6w
The familiar form of Derjaguin’s equation for dry foams is obtained when Vc = 0. As we shall show below, the virial equation is the integrated form of the equilibrium equation. It is then perhaps surprising that it has taken so long after Derjaguin’s original formulation for proofs of the virial equation to be obtained. It is possible that the reason for this is that the degree of generality of the equilibrium equation is not widely realized. In proving eq 2 from eq 1, it is necessary to consider an alteration of the system in which all the phase volumes change in proportion, under a similarity or scaling transformation. By applying eq 1and noting that if the linear scale factor is A, then the volumes grow as X3 and the areas grow as X2 the virial equation is obtained. The earlier restricted proof~l-~ have applied eq 1 to readily achievable physical processes, such as compressing the system by increasing the ambient pressure, while keeping the composition of the discrete phases constant. This only results in a similarity transformation if the dispersion is monodisperse. The more recent proofs5v6have applied eq 1to more general transformations between equilibrium states, where the composition of the system has been allowed to change. This has required a detailed discussion of which state variables are required to specifyan equilibriumstate. (Further in both proofs a finite scaling has been introduced, while the surfacetensions are to be held constant, whereas without further control the surface tension would be expected to vary-albeit by relatively little for large particles.) All of the above technical difficulties may be avoided, once the geometrical nature of the equilibrium equation (1)is realized. The proof of the virial equation is then purely geometricaland is obtained directly by considering an infinitesimal scaling transformation, A = 1 + 6X. The aim of this present publication is to investigate the nature of the equilibrium equation and its range of validity for a variety of special cases and to show how it leads to direct proofs of the virial equation. In particular from Gibbs’originalformulationfor a dispersiontotally enclosed within a rigid container, we obtain the equivalent formulations for dispersions partly in contact with a rigid wall and partly surrounded by an ambient phase and also for dispersions suspended at a flat fluid interface, e.g. a bubble raft. We use these to obtain the virial equation for a freely suspended dispersion, eq 2, and, in a more explicit
DISPERSION
DISPERSION ,
+ TJS+ &6nr
(4)
6w is the work done on the system. E is the intemal energy,
S is the entropy, and pr and n, are the chemical potential and number of moles of component r. For the multiphase systems which we consider, it is typical that the interfaces may be impermeable to one or more components. In this case the chemicalpotential of a particular chemicalspecies can be different in different phases. Then these independently variable constituents should be treated as distinct chemical components, according to the phase or phases in which they are contained. (Equation 4 is a paraphrase of eqs 1 and 12 of ref 7.) As long as the opposite variation is possible, then for the opposite variation all the signs in eq 4 are reversed, and the equality sign in eq 4 applies. Hence as long as the opposite variation is possible 6~ = 6~ - T ~ -SC+r6nr
(5)
It is not necessarythat the variation preserves equilibrium. Variations as normally considered are such that the opposite variation is possible. These include continuous material displacements of the system and the use of a continuous distribution of matter sourcesand heat sources. The opposite variation is obtained by reversing the displacement and replacing sources by sinks. Variations for which the opposite variation is not possible are variations at the natural boundariesof the parameter space used to describe the system, such as variations which change the topology of the system, by creating new phases and interfaces or which introduce novel components which had not previously been present. For our purposes it is sufficient to consider continuous infinitesimal material displacements for which eq 5 always holds. Now the internal energy of our system is the sum of the internal energies of the fluid phases and interfaces that comprise it Forinfhitesimaldisplacementsaboutequilibriumwe have, for a fluid phase i
6Ei = T6Si + &6n,
- pi6Vi
(7)
and for a surface s (cf. eq 494 of ref 7) and hence adding these contributions
Thermodynamics of Fluid Dispersions in Equilibrium
By comparing this with eq 5, we obtain the equilibrium equation (10) The work term, 6w, arises from displacement of the boundary of the dispersion, e.g. the container walls or the interface between the dispersion and an ambient phase. If the container walls are held fixed then
6~ u,6AWD - p$VD (11) where the fmt term on the right is the work of expansion of the dispersion-container interface, in the direction of the ambient-wall surface tension, and the second term is the work of expansion of the dispersion volume against the ambient pressure. Equating these two expressions for 6w results in the equilibrium equation for a fluid dispersion surrounded by an ambient phase and by rigid Walls We may introduce an effective surface tension, uBo:for interfaces between the dispersion and the wall, usois the surface tension measured relative to the ambient-wall surface tension, uso = u, - UWO;for the fluid interfaces there is no difference between the two surface tensions, uno= u8. For interfaces between fluid phases and rigid walls uno, unlike us, is an experimentally measurable quantity. It is determined, accordingto Young's equation, by (i) the surface tension of the interface between the fluid phase and the ambient phase and (ii) the contact angle of this interface with the rigid wall. The equilibrium equation may then be rewritten as In order to interpret the equilibrium equation, it is sufficient to consider infinitesimal displacements, described by an infinitesimalcontinuousvector field 6r,such that a point of the system, r, is displaced to a point r + 6r; continuity of 6r is required to preserve the topology. Further it is sufficient to know the behavior of 6r on the interfacial network; the behavior of 6r inside the fluid phases is not needed. The only other restriction on 6r is that it should be consistent with the rigid wall boundary conditions. That is, at the wall 6r is restricted to having no componentnormal to the wall, tangential displacements are allowed, corresponding to the dispersion sliding along the wall. To summarize, eq 13 is valid for arbitrary continuous infinitesimal displacements of the interfacial network, consistentwith the rigid wall conditions. In effect it is valid as long as it can be sensibly interpreted and may thus be regarded as completely general. A somewhat different case, which can be analyzed in a similar manner, is that of a fluid dispersion suspended at a flat fluid interface between two ambient phases, i = 0 and i = 1, e.g. a bubble raft, Figure 3b. Because of the flatness of the interface, the internal pressures of the ambient fluids are identical, that is p1= PO. If Awl is the area of the flat fluid-fluid interface occupied by the dispersion, then the resulting equilibrium equation is
This equation is valid for arbitrary infiiitseimal displacements, whether or not the displacements preserve planarity. The correction for the displacement of the fluidfluid interface is similarto that for the displacementof the
Langmuir, Vol. 9, No. 1, 1999 103 interface between the ambient phase and the container wall; the difference is that the fluid-fluid interface is deformable and is penetrated by the dispersion.
The Virial Equation We shall now prove the virial equation for a number of special cases by applying the appropriate equilibrium equationtoaninfiniteeimalscalingtransform. Byascaling transformation with scaleparameter X we mean a mapping of each point r to Xr; for an infmiiteeimalscalingtransform X = 1 + 6A where 16x1
