164
Langmuir 1988,4 , 164-169
Pressure/Volume/Surface Area Relationships in Foams and Highly Concentrated Emulsions: Role of Volume Fraction H.M. Princent CollGge de France, Physique de la MatiGre Condens&e,11 Place Marcelin Berthelot, 75231 Paris, Cedex 05, France Received March 17, 1987. In Final Form: August 17, 1987 Various pressures in and around a foam or concentrated emulsion are discussed in some detail. These include the internal pressure inside the dispersed cells, the pressure in the interstitial continuous phase, the disjoining pressure in the films separating the cells, the osmotic pressure, and the vapor pressures of the dispersed and continuous phases. The effect of the volume fraction, 4, of the dispersed phase is investigated explicitly, not only on some of the above parameters, but also on the equation of state and the compressibility of a foam. Earlier results, valid for C$ 1 only, are corrected for 4 < 1. The area of the intercellular films as a fraction of the total surface area and its dependence on 4 have been evaluated also. the films and the adjacent Plateau borders is zero. Introduction We shall further assume that the cell size is R > a. Weaire,’ Khan and Armstrong? Kraynik and Hansen? etc. The dispersed phase is phase 1; the continuous phase Often this dry foam approximation is appropriate, e.g., in is phase 2. the top region of a very tall, equilibrated foam or emulsion column, where the shape of the bubbles or drops indeed Pressures in and around the System approaches that of fully developed polyhedra with sharp There are a number of interrelated pressures one may edges and corners. In many other cases, however, this is distinguish in a concentrated fluidlfluid dispersion of 4 not so, and a more general treatment must incorporate the 2 40. volume fraction as a variable in the range $o I4 I 1,where Internal Pressure pi. This is the pressure inside the $o is the volume fraction of the close-packed, undeformed, cells. If the system is monodispersed and gravity is absent, spherical bubbles or drops (“Kugelschaum”). For monop i is identical in all cells. If the system is polydisperse, disperse systems, 4o = 0.7405; for typical polydisperse the internal pressure will vary from cell to cell, since the systems, it has been found to be only slightly different and cells are then generally separated by films of nonzero smaller’+12 (not larger as one might expect for some very curvature. In that case one may define an average internal specific multimodal size distributions, which, in fact, are pressure given by rarely, if ever, encountered in practice). In a series of recent papers, we have specifically investigated the dependence of a variety of static and dynamic properties on 4, as well as on the drop size and interfacial Thus, we have shown that the osmotic where p i and vi are the pressure and volume of cell i , V1 pressure,12J4 shear modulus,” and yield s t r e s ~ ~ all ~J~J~ depend strongly on 4. The same is true for the viscosity, (1)Derjaguin, B.V. Kolloid-Z. 1933,64, 1. mostly through the yield stress.l6p2l (2)Ross, S. Ind. Eng. Chem. 1969,61, 48. In the present study, we investigate the effect of C$ on (3)Ross, S. Am.J.Phys. 1978,46,513. a number of other properties, such as the “internal (4)Nishioka, G.;Ross, S. J. Colloid Interface Sci. 1981,81, 1. (5)Nishioka, G.;Ross, S.; Whitworth, M. J. Colloid Interface Sci. pressure“ in a foam, the equation of state, the compres1983,95,435. sibility, and the film area per unit volume of the dispersed (6) Morrison, I. D.; Ross, S. J. Colloid Interface Sci. 1983, 95,97. phase. We shall also take this opportunity to establish (7)Weaire, D.; Kermode, J. P. Philos. Mag., [Part]B 1983,48, 245; 1984,50, 379. links between various properties that seem not to have (8)Khan, S. A,; Armstrong, R. C. J. Non-Newtonian Fluid Mech. been made before. 1986,22,1. I t will be assumed that the thickness of the films be(9)Kraynik, A. M.; Hansen, M. G. J. Rheol. 1986,30,409. tween the bubbles or drops (the “cells”) is negligible com(10)Princen, H. M. J. Colloid Interface Sci. 1985,105, 150. (11)Princen, H. M.; Kiss, A. D. J. Colloid Interface Sci. 1986,112,427. pared to the cell size (finite film thickness simply gives rise (12)Princen, H.M.; Kiss, A. D. Langmuir, 1987,3, 36. to an upward shift in the effective volume fraction, as (13)Princen, H. M. J. Colloid Interface Sci. 1983,91, 160. indicated beforel0J’ and that the contact angle between (14)Princen, H. M. Langmuir, 1986,2, 519. (15)Yoshimura, A,; Prud’homme, R. K.; Princen, H. M.; Kiss, A. D. ‘Current address: General Foods Corp., Technical Center, 555 South Broadway, Tarrytown, New York 10591.
J. Rheol., in press. (16)Schwartz, L.W.; Princen, H. M. J. Colloid Interface Sci. 1987, 118, 201.
0143-7463/88/2404-0164$Q1.50/00 1988 American Chemical Society
Langmuir, Vol. 4, No. 1, 1988 165
PressurelVolurnelSurface Area in Foams and Emulsions ATM 0 SPHERE P
~ C O N T.PHASE(
n:
u
MEMBRANE +
i XiZ
DISPERSION
(4
(b)
Figure 2. Semipermeablemembrane separating dispersion from continuous phase: (a) general view; (b)detailed view of contact
between deformed cells and membrane.
.c
Figure 1. Fluid/fluid dispersion in contact with atmosphere.
is the volume of the dispersed phase, V is the total volume, and the summations are taken over all cells. For a foam, filled with (ideal) gas, pirepresents the pressure inside the gaseous volume Vl, if all foam lamellae would rupture and total phase separation would 0ccur.l We shall return to this pressure in more detail later. Border Pressure p b. This is the pressure inside the continuous phase in the plateau borders. I t is related to p i of a given cell by Pb = Pi - u121Kbil
(3)
where Kbi is the curvature of the free surface of cell i , i.e., the surface outside the films. In the absence of gravity, pb is constant everywhere. In a gravitational field, Pb varies linearly with height, z , according to Pb
=
c +ZPg
(4)
’
where C is a constant, p 2 is the density of the continuous phase, and z is pointing downward. If, as in most practical situations, the system is in “contact” with a gaseous atmosphere of pressure P at some level z = 0, then the value of C in eq 4 is determined by the curvature, Kt, of the free continuous phaselgas interface between the dome-shaped films at the top of the highest layer of cells (Figure l),i.e.,
c = P - U21Ktl
(5)
In a foam, u2 = u12is the continuous phaselgas interfacial tension. (Similarly, for an emulsion in contact with a layer of dispersed (liquid) phase, u2 = u12is the liquid/liquid interfacial tension.) For an emulsion in contact with air, on the other hand, u2 in eq 5 is the continuous phaselgas interfacial tension. It is important to note that, however complex the detailed shape of the top surface of the system (particularly in a polydisperse system), Kt is constant everywhere, even in gravity, provided the surface is horizontally flat on the scale of the capillary length, a. Combining eq 4 and 5 yields Pb = p - UZIKtl + Z P g
.
(6)
The second term on the right-hand side is often over10oked.~J’ It is obvious from Figure 1that the curvature Kfi at the top of cell i in the first layer is given by (412
+ u2)lKfil
=
cl2bbil
- u2lKtl
(7)
and that the internal pressure in that cell is
Pi = P + IKfil(Ul2 + U Z )
(8) provided that, in the case of an emulsion in contact with a gaseous atmosphere, the continuous phase spreads over (17) Khristov, K. I.; Exerowa, D. R.; Krugljakov, P. M. J. Colloid Interface Sci. 1981, 79, 584.
the dispersed-phasedroplets in the top layer. If not, a bare surface of the dispersed phase with tension c1 is in contact with the atmosphere over some of its area and there will be a three-phase contact line with the continuous phase between the drops. Using Neumann’s triangle for the equilibrium between ul,u2, and u12at the contact line, one can readily reformulate the problem for this case. The curvature of the roof of the drops is simply increased by a factor of (al2+ u2)/u1, while Kbi and Kt remain unaffected. I t is suggested by Figure 1 that the average relative internal pressure pi - P inside the system is already fully established in the very first layer of cells. The disjoining pressure, IID,is the extra pressure inside a film due to interaction forces. At equilibrium, it equals the total capillary pressure acting on the film. For any one of the films sumunding any cell i (not just a film facing the atmosphere), it may be written as
where the + and - signs refer to situations where the convex or the concave side of the curved film is directed toward the interior of cell i , respectively. In a monodisperse system, Kfi is zero everywhere, except for the films directly facing the atmosphere. When gravity is present, IKbil and IIDthen vary linearly with the level z. In a polydisperse system, Kfi, IID,and, therefore, the equilibrium film thickness vary from film to film, even in the absence of gravity. This may further complicate the proper description of the process of foam coarsening by gas diffusion, where one usually assumes that the thicknesses and, therefore, the gas permeabilities of all the films are identica1.7J8J9 The osmotic pressure, II, is the pressure that must be applied to a freely movable, semipermeable membrane, separating the fluid-fluid dispersion from a layer of continuous phase, in order to prevent the latter from entering the di~persion.’~J* The membrane is semipermeablein the sense that it is freely permeable to all the components of the continuous phase but impermeable to the drops or bubbles (Figure 2). II equals the pressure exerted on the membrane by the flattened cells and may therefore be represented by
where n is the number of cells per unit area of the membrane and fi is the flattened “contact area” for cell i . Previouslylo we have introduced a total fractional contact area f: n f = C f i i=l
(11)
which can be readily measured optically.1° It is determined (18) Lemlich, R. Ind. Eng. Chem. Fundam. 1978, 17,89. (19) Beenakker, C. W. J. Phys. Reu. L e t t . 1986, 57, 2454.
166 Langmuir, Vol. 4, No. 1, 1988
Princen
primarily by the volume fraction, 4, but may be slightly dependent on the details of the size distribution. An alternative expression for II follows from the process whereby the membrane is displaced downward to squeeze a volume dV2 of continuous phase out of the dispersion at constant Vl. For this process one may write II dV2 = -al2 d S (12) where S is the surface area in the dispersion. Equation 12 may be transformed into12J4
right-hand side equals -11 at the top of the column, while we may write for the third term, using eq 6 (20) P?gH = Po - p + 4 4 where IKtl is the curvature of the free continuous phase/gas interface at the top of the column (Figure 1). Thus, by combining eq 19 and 20, we establish the interesting fact that for the top of the column
n = 821Ktl
(21)
Now, the vapor pressure of the continuous phase above the dispersion is simply given by the Kelvin equation, applied to the concave continuous phase/gas interface at the top, i.e. p V c = (PVC)Oe-uziKtIV2/RT = (p$)oe-"V2/RT (22)
or
where S/Vl is the surface area per unit volume of the dispersed phase, S/So is the ratio of the surface area of the deformed cells and that of the spherical cells of the same volume, and R32 is the surface-volume or Sauter mean radius of these spherical cells:
For small cell size and high volume fraction, II tends to infinity and the reduction in vapor pressure becomes quite pronounced. In the case of an emulsion, one may also consider the vapor pressure of the dispersed phase. It is, of course, increased relative to that of the bulk dispersed phase. It can be shown to be given, to a very good approximation, by pVd
In ref 12 we have experimentally determined fi and S/So as a function of for a typical polydisperse system. It ,was found that S/Sovaries from unity at 4 = c $(where ~ II = 0) to 1.083 as 4 1 and fI a. The vapor pressure,p,", of the continuous phase above a fluidlfluid dispersion has been reported by us,14without proof, to be related to II according to p V c = (PvC) oe-nVz/RT (16)
-
-
where (pVc),,i_s the normal vapor pressure of bulk continuous phase, V2 is ita molar volume (or, rather, the partial molar volume of the solvent), R is the gas constant, and T is temperature. Two different proofs will be given here. First, consider the transfer of dn2 mol of continuous phase from the bulk to the dispersion. The free energy gained is exactly equal to the decrease in surface energy in the dispersion as a result of the accompanying dilution, i.e. P V C
Ap2 dn2 = RT In -dn, =
a12
dS
(17)
(PvC)0
However, according to eq 12, at constant V1 a12d S = -11 dV, = -IIv2dn,
(18) From eq 17 and 18, one immediately obtains eq 16. In the second proof, we consider an equilibrated dispersion column of height H , resting on a bulk layer of continuous phase. We have considered such a column in detail in ref 12. The values of II and 4 change continuously with the height in the column. Relative to the atmospheric pressure, P, above the dispersion, the pressure po = p b ( H ) in the continuous phase just below the dispersion must equal the weight of the column per unit cross sectional area, i.e.
E
(pvd)oe(2~dR)(V~/RT)(S/S~)
where (p:)o is the vapor pressure of the bulk dispersed phase, a12 is the liquidlliquid interfacial tension, R is the drop radius, and Ql is the molar volume of the dispersed phase. Although p$ depends strongly on R when R 0, it is only slightly affected as 4 1 through &'/So. Expressions, similar to the above, may be derived for the mutual solubility of the phases.
-
-
Internal Pressure, Equation of State, and Compressibility For a dry,polyhedral foam (Cui= V; 4 = l),Derjaguinl has shown that the average internal pressure p i and the equation of state are given by 2 s p . - p = -a 3 l2V
(4 = 1)
where pi is as defined in eq 2 and n is the number of moles of gas in the foam. The same results were later obtained by Ross.2 Subsequently, Morrison and Ross have pointed out6 that, although eq 23-24 are strictly valid for monodisperse systems and for small clusters of only a few unequally sized bubbles at 4 = 1, their general validity for multibubble, polydisperse systems has not been rigorously proven, neither by Derjaguin, nor by Ross himself. They seem convinced, however, that the equations are exact even in that more complicated case, and they liken the lack of a rigorous proof to the situation pertaining to Fermat's famous last theorem. To avoid this difficulty, let us assume that the system is indeed monodisperse. We shall remove the restriction on 4, however. It will be allowed to vary from +o to unity. With Derjaguin, we consider an infinitesimal change in the foam volume, dV = dV,. At equilibrium Cpi dui - P dVl = a12dS (25) 1
Since we assume monodispersity, pi and ui are identical for all cells, so that From ref 12 and 14 it follows that the second term on the
Cpi dui = p i c dui = pi dV1 i
I
(26)
PressurelVolumelSurface Area in Foams and Emulsions Therefore
pi - P =
u12(
Langmuir, Vol. 4, No. 1, 1988 167 Table I. Terms Involved in Internal Pressure
")
dV1 v, 0.80 0.85 0.90 0.92 0.93 0.94 0.95 0.96 0.965 0.970 0.975 0.980 0.985 0.990 0.995 0.997 0.999 1.000
For 4 = 1and an isomorphic change in the cells, one may write from simple geometric considerations 2s -d=S - = -2 s dV1 ~ V I 3V which, when inserted in eq 27, leads to eq 23. For doI4 < 1,however, such an isomorphic change is impossible, as the cell shape must change as Vl and, therefore, 4 are varied. Equation 28 no longer holds. Instead, we replace S in eq 27 by
s = (S/SO)SO
(29)
where, as before, So is the surface area of the spheres of the same volume as the deformed cells of surface area S. It is clear that S/Sois a shape parameter that depends only on 4, whereas So simply depends on the cell radius. Hence
0.019 0.030 0.041 0.047 0.050 0.052 0.055 0.056 0.057 0.056 0.055 0.052 0.047 0.039 0.031 0.026 0.016
2.003 2.009 2.021 2.028 2.033 2.038 2.045 2.054 2.059 2.065 2.072 2.080 2.090 2.102 2.117 2.127 2.142 2.166
0
2.022 2.039 2.062 2.075 2.083 2.090 2.100 2.110 2.116 2.121 2.127 2.132 2.137 2.141 2.148 2.153 2.158 2.166
to about 2.8% of the value of the second term. In the limits, we have 2 s 2 s pi - P = -412= -412(4 = 1) 3 V 1 3 V which is identical to Derjaguin's result, and 292
Realizing that
pi-p= (4 = 40) R as expected for spheres. Using the result of eq 32, we may write for the generalized equation of state of the foam piVl = nRT (34) which leads to
we find from eq 30
or
4V = nRT
(36)
Because of eq 14, this may be written in the form
pi-p =
1-4 2 s II + -4124
($0
I4 I1) (32)
Vl where II is the osmotic pressure. Comparing eq 32 and 23, we see that for 4 < 1there are two corrections to Derjaguin's original expression. First, perhaps intuitively obvious, the surface area per unit volume of foam, S / V, is replaced by the surface area per unit volume of the dispersed phase S / VI = S/$V. Second, there is a small correction term resulting from the nonisomorphic nature of the change in cell shape from that of a sphere to that of a fully developed polyhedron. Equation 32 can be written in reduced form 3
where each term depends on 4 only. Although the exact dependence of fI and S / S o on 4 is not yet known for a monodisperse system, we have determined these functions experimentally for a typical polydisperse system.12 The results should not differ by very much and may be used to give excellent estimates of the terms in eq 33. This has been done in Table I. It is seen that the first term is zero at both ends of the range, i.e., at = 4o and 4 = 1. It passes through a maximum at 4 N 0.965 where it amounts
For the limit of 4 = 1,Derjaguin has also evaluated the compression modulus K , which is defined as
His result is
This compares with K = P for a simple, ideal gas. As with the previous expressions, eq 37 has been rigorously proven only for monodisperse systems. With our previous results, it can now be generalized for 4 < 1. We find for the compression process at constant V2
K
= -(P 1
34
z)vz
+ 2p;) + (' ,4)2( -
(38)
where pi is given by eq 32 or 33. It is noted that the derivative in the second term is at constant volume of the continuous phase, which implies a change in the cell radius R. It can be shbwn that
168 Langmuir, Vol. 4, No. 1, 1988
Princen
is contained in a polyhedron of volume up and area sp (up > v; sp> s). This polyhedron tessellates in space. It is clear that u/up = 4
Table 11. Terms Involved in Eauation of State
2 b 0.715 0.75 0.80 0.85 0.90 0.92 0.93 0.94 0.95 0.96 0.965 0.97 0.975 0.98 0.985 0.990 0.995 0.997 0.999 1.000
((1 - b ) / 3b)n 0 0.003 0.006 0.010 0.014 0.016 0.017 0.017 0.018 0.019 0.019 0.019 0.018 0.017 0.016 0.013 0.010 0.009 0.005 0
(1 - b)2.
(dn/db) 0.048 0.050 0.054 0.057 0.062 0.066 0.067 0.067 0.065 0.060 0.057 0.052 0.046 0.038 0.034 0.028 0.020 0.016 0.010 0
C
4/3(S/SJ 1.333 1.334 1.335 1.339 1.347 1.352 1.355 1.359 1.363 1.369 1.373 1.377 1.381 1.387 1.393 1.401 1.411 1.418 1.428 1.444
1.381 1.387 1.395 1.406 1.423 1.434 1.439 1.443 1.446 1.448 1.449 1.448 1.445 1.442 1.443 1.442 1.442 1.443 1.443 1.444
where fI is the reduced osmotic pressure. Hence, one ultimately finds
=
++ &-= 4
I-$-
+ (1-$)Z-
It is clear from geometric considerations that so that
=(:)
dfI
4
’)]
“+- d@ 3 so
2/3 S( I.) /So
s/s,
1.083 -- -(S/S0)-1 @2/3
which, because of eq 32, may be written as ‘12
For the purpose of this discussion, it is not essential to speculate on the exact shape of the polyhedron. At the lower volume fractions, it is probably a rhombic dodecahedron, while Kelvin’s minimal tetrakaide~ahedron~ is more likely at higher 6. Now consider another polyhedron, somewhat smaller than but isomorphous with the first, whose volume is upl = u. Its surface area spl is the same as that of the cell at @ = 1: up’ = u = @up spl = s(1)
(41)
where the terms inside the large parentheses depend on @ only. As before, these may be estimated from the data in ref 12. This has been done in Table 11. As was seen for the internal pressure pi,the “Derjaguin _term”in S / S , is much more important than the terms in II and dfI/d@. The main correction is the factor l/@ in front of the square brackets.
Total Surface and Film Area In ref 12 and 14 we have shown how the total surface area of a fluid/fluid dispersion increases as the volume fraction is raised from @o to 1. This information can be derived from n(@), which, in turn, may be determined experimentally from the variation of the volume fraction with height in an equilibrated emulsion column. Thus we have found for a typical, polydisperse emulsion that the transition from spheres to polyhedra is accompanied by an increase in the specific surface area of the dispersed phase, S / V l , of 8.3%, i.e., S(l)/So= 1.083. A t any stage, a fraction of the total surface is “free”,while the complement, Sf/S, is the fraction that forms part of the films separating the droplets. Obviously, Sf/S varies from zero at @o to unity at @ = 1. Detailed knowledge of its dependence on @ is important, as it clearly plays a role in problems relating to the stability of, and mass transport in, such systems. For example, the rate of coarsening of a foam by gas diffusion will be primarily proportional to Sf, since the gas diffuses almost exclusively through the films. The same may be true for the rate of coalescence in relatively unstable systems. For convenience, again assume the system to be monodisperse. Each cell, of volume u and total surface area s,
where S/Sois known from experiment, albeit for a polydisperse system.12 I t is not expected to be very sensitive to the size distribution, however. To obtain the fraction Sf/S of the total cell area that is locked up in the films, we write
Now, careful consideration of the compressive forces that are responsible for the deformation of the cells leads one to the conclusion that sf/spexactly equals the fraction of a confining wall that is taken up by the flattened parts of the cells pushing against it. This fraction, f , has been discussed above; see eq 11. I t also has been measured experimentally on a typical, polydisperse system.1° Empirically, it was found to depend on @ as 3.20 f(4) = 1 (43) 112 + 7.70)
(A
between 4,, and 4 = 0.975. For @ 2 0.98, we expect, on theoretical grounds,14
f(4) = [l - 1.892(1 - 4)lI2I2 (4 2 0.98) Thus, replacing sf/spby f in eq 42, we obtain
(44)
(45) where f(4) and S/So are known, at least approximately. A plot of Sf/S vs @ is shown in Figure 3.
Discussion We have seen that the volume fraction in concentrated fluidlfluid dispersions has a profound effect on some properties, e.g., the shear modulus, yield stress, osmotic pressure, disjoining pressure, vapor pressure of the continuous phase, specific film area, and fractional contact
PressurelVolumelSurface Area in Foams and Emulsions
Langmuir, Vol. 4, No. 1, 1988 169
Acknowledgment. I thank Professors P. G. de Gennes and A. M. Cazabat for helpful comments.
List of Symbols
Figure 3. Surface area in T i s , as fraction of total surface area.
area of the cells against a confining wall. Other properties are less affected, e.g., the internal pressure (relative to the external atmosphere), the equation of state, and the vapor pressure of the dispersed phase. It appears that the recently introduced concept of osmotic pressure, and its measured dependeme on $, are extremely useful in describing and evaluating other system parameters. If there is one uncertainty remaining in this and previous work, it is the lack of detailed knowledge as to the effects of polydispersity and size distribution. We are inclined to believe, with Morrison and ROSS,~ that at least some results are unaffected by polydispersity, but a rigorous proof is lacking. A closely related question has to do with the proper mean drop radius that is to be used to characterize a polydisperse system. We believe that in most cases it is the surface-volume or Sauter mean radius, since most properties are determined by the specific surface area S/VP In other cases, this is less obvious, however. Since a detailed theoretical analysis of polydisperse systems is extremely complex and unlikely to be forthcoming soon, the more promising avenue is the experimental one, provided one employs systems that are well characterized in terms of their size distribution, volume fraction, and interfacial tension. Of primary importance would be the experimental determination of n($)and f ( $ ) for monodisperse, as well as a variety of well-characterized polydisperse, systems, both unimodal and multimodal. An interesting and important recent finding20 is that S(l)/So = 1.097 for Kelvin's minimal tetrakaidecahedron, which is the ideal cell in a monodisperse system at $ = 1. Since our experimental value for a typical unimodal polydisperse system is S(l)/So = 1.083, this provides strong evidence that, contrary to our previous assumption,12 polydispersity may give rise to a decrease in S(l)/&. (20) Princen, H. M.; Levinson, P. J.Colloid Interface Sci. 1987,120, 172. (21) Princen, H. M.; Kiss,A. D. J.Colloid Interface Sci., submitted.
capillary length, (alz/Apg)l/' numerical constant fractional area of "contact" of cells and adjacent wall contact area for cell i acceleration due to gravity height of foam or emulsion column index for cell i compression modulus for foam, -V(dP/dV),, moles of gas in foam moles of continuous phase ambient pressure above foam or emulsion pressure in continuous phase just below foam or emulsion column pressure in continuous-phase Plateau borders pressure inside cell i average pressure inside cells vapor pressure of continuous phase above dispersion vapor pressure of bulk continuous phase vapor pressure of dispersed phase above dispersion vapor pressure of bulk dispersed phase gas constant radius of sphere of same volume as deformed cell surface-volume or Sauter mean radius of cells in polydisperse system surface area of single cell surface area of single cell at 4 = 1 surface area of circumscribing polyhedron of single cell surface area of polyhedron of same volume as a cell total surface area in system total surface area in system at 4 = 4o total surface area in system at 4 = 1 fraction of S forming part of intercellular films temperature volume of single cell volume of cell i volume of circumscribing polyhedron of single cell volume of polyhedron of same volume as cell (=u) total volume of system, Vl + V, volume of dispersed phase volume of continuous phase molar volume of dispersed phase molar volume of continuous phase height in foam or emulsion column, measured from top down curvature of free surface of a cell curvature of free continuous phase/gas interface at top of system curvature of intercellular films chemical potential of continuous phase osmotic pressure reduced osmotic pressure, rLR32/u12 disjoining pressure in intercellular films density of dispersed phase density of continuous phase P2
- P1
surface tension of dispersed phase surface tension of continuous phase interfacial tension between dispersed and continuous phase volume fraction of dispersed phase volume fraction of close-packed spheres
