36
Langmuir 1987, 3, 36-41
Osmotic Pressure of Foams and Highly Concentrated Emulsions. 2. Determination from the Variation in Volume Fraction with Height in an Equilibrated Column H. M. Princen* and A. D. Kiss Corporate Research Laboratories, Exxon Research and Engineering Company, Clinton Township, Annandale, New Jersey 08801 Received May 30, 1986. In Final Form: September 29, 1986 The osmotic pressure of concentrated fluidlliquid dispersions has been determined as a function of the volume fraction, 4, of the dispersed phase by measuring the volume fraction as a function of height in an equilibrated, fully drained, well-characterized oil-in-water emulsion. The results at high volume fraction are in agreement with a previously derived limiting solution. At "low" and intermediate volume fraction, the results have been expressed in the form of simple, empirical equations. Other parameters that may be derived from these resulta are the specific surface area and the mean capillary pressure inside the drops as a function of volume fraction and the volume of continuous phase below a certain level, z, in a well-drained vertical column. It is found that the surface area of the fully developed polyhedral drops at 4 1exceeds that of the initially spherical drops by a factor of 1.083, while the reduced volume of continuous phase in an infinitely tall column (z a) equals 0.258, which is close to our recent estimate but considerably in excess of previous predictions.
-
-
Introduction In a previous paper,' we discussed the concept of the osmotic pressure, II, of foams and concentrated emulsions. Although we derived a limiting solution for the case of high volume fraction of the dispersed phase ( 4 l),a general solution could not be obtained because of the complex geometry of the deformed bubbles or drops. It was proposed to fill this gap by the direct or indirect measurement of II of well-characterized oil-in-water emulsions. After presenting the necessary background, we shall discuss the experimental approaches and results, followed by calculation of some derived properties, such as the variation in surface area with volume fraction.
-
Background When the volume fraction of the dispersed phase in foams or emulsions exceeds about 0.74, the bubbles or drops are no longer spherical but assume more or less strongly developed polyhedral shapes, with a consequent increase in their surface area. As a result, when the system is separated from a layer of continuous phase via a freely movable, semipermeable membrane that is permeable to all components of the continuous phase but impermeable to the bubbles or drops, the continuous phase will flow into the dispersion in an attempt to restore the spherical shape of the drops, unless a pressure II is applied to the membrane. We derived' two general expressions for II,
d(S/ Vi)
n = a42 d4
Vi
Vi So
3 s R32 So
(4)
or (5)
-
-
Thus, knowledge of fi(4) between (where fi = 0) and 4 1 (where n a) enables one to evaluate S/Soas a function of 4, and vice versa. For monodisperse systems, a reasonable estimate of S/S, as 4 1 is
-
S(l)/S, = 1.10
(3)
(1) Princen, H.M.Longmuir 1986,2, 519.
0743-7463/87/2403-0036$01.50/0
(6)
which is based on considering the transition from a sphere to a rhombic dodecahedron, regular pentagonal dodecahedron, or tetrakaidecahedron of the same volume.'S2 In terms of the reduced osmotic pressure, eq 2 may be written as
(1)
where u is the interfacial tension, S / VI is the surface area per unit volume of the dispersed phase, f is the fraction of the total wall area taken up by the flattened areas of the drops, and K is the mean curvature of the free drop surfaces, i.e., outside the interdroplet films. We may write -s= - -sos =--
where R32is the Sauter or surface-volume mean drop radius and Sois the surface area a t 4 6 4o where the drops are spherical (do= 0.7405 for monodisperse systems but tends to be somewhat smaller for typical polydisperse systems), while S is the surface area of the deformed drops a t 4 > 40When this expression for S / V , is substituted in eq 1, we find for the reduced osmotic pressure
(7) where K,, = 2/R32is the "Sauter meann drop curvature and Apo = 2a/R32is the corresponding excess capillary pressure inside the drops (relative to the pressure in the continuous phase) when the drops are spherical, while Ap(4) is the internal pressure when 4 > dO- Clearly, as 4 increases from c $ to ~ unity, f(4) varies from zero to unity, regardless of whether the system is mono- or polydisperse. The picture is less clear, however, for Ap/Apo. For a monodisperse system, AplAp, is expected to increase monotonically from unity to infinity. This was shown explicitly for a twodimensional system3 and should equally be so for the (2) Princen, H.M.; Kiss, A. D.J. Colloid Interface Sci. 1986,112,427.
0 1987 American Chemical Society
Langmuir, Vol. 3, No. 1, 1987 37
Osmotic Pressure of Foams and Emulsions three-dimensional case. For a polydisperse system,on the other hand, the relative contributions of the drops to f and A p vary with their size. Moreover, the size distribution of the drops contacting a wall may differ from that in the bulk.4 Hence, although Ap/Apo must go to infinity as 4 1, its value a t q50 may well deviate from one. We shall return to this later. We also showed in (1)that fI(4) is related to the variation in the volume fraction in an equilibrated foam or emulsion column in a gravitational field through
To Pipet Bulb
tI
-
where
7AP
3-
Figure 1. Homemade osmometer (schematic; not to scale).
z is the height above the interface between the foam and
the underlying continuous phase, and
-
where Ap is the density difference between the phases and g is the acceleration due to gravity. Provided that R32 4.0). In this range the previously proposed limiting solution applies with sufficient accuracy, i.e., z'
i=
fi = 0.5842
[ l - 1.892(1 - 4)'/']' (1 - 4)1/'
(28)
We may write for the surface area
s
-
1 ' fid4 so +3- $ 0.99
S(O.99)
so
i=-
where the value of the first term (=1.051) may be obtained from eq 26 and the integral from eq 28, which leads to S / S o = 1.014
+ 0.0686[1 - 1.892(1 - 4)'/2]3
(29)
Although the relative difference between z' and fi d_ecreases with increasing 4,the absolute difference (=V) keeps increasing. We write for the volume of continuous phase between the zero level and t
6 = V(O.99) +
(1 - 4)
$@
0.99
2 d4 d4
where the value of the first term (=0.200) is obtained from eq 27 and 22, while the integral may be evaluated by using eq 28 and making suitable approximations. The result is
Q
i=
0.258 - 0.5842(1 - $)l/'
(31)
or
6 0.258 - 0.3412 i=
z'
+ 2.211
Discussion and Concluding Remarks
To our knowledge, this is the first time that such simple but highly informative measurements have been made over a large range of q5 with such accuracy. The results enable one to calculate a variety of important properties of foams
1
-
v,
10
100
2
Figure 8. Reduced volume, of continuous phase between z' = 0 and 2. At infinite height, V approaches a limiting value of 0.258 (dashed line).
and emulsions from simple, albeit mostly empirical, equations. Although the results, strictly speaking, apply only to the particular emulsion used in this study, we believe them to be valid quite generally, unless the drop size distribution deviates strongly from that encountered here. Of particular interest is the variation of the relative surface area with volume fraction, as embodied in eq 16, 26, and 29. The results are plotted in Figure 7 and indicate that as 4 tends to unity, S/S, reaches an ultimate value of 1.083, which, by coincidence, is one of the values arbitrarily selected in ref 1. This value for the transition from spheres to fully developed polyhedra deviates significantly from the value of 1.100 one would predict for a monodisperse system of regular pentagonal dodecahedra, rhombic dodecahedra, or tetrakaidecahedra. It must be realized, however, that none of these "simple" polyhedra are true minimum-surface bodies, as would be Kelvin's "minimal tetrakaidecahedron", some of whose faces and sides are nonplanar and curvilinear, respe~tively.~Therefore, the value of 1.100 for a monodisperse system must be an overestimate. On the other hand, our system is polydisperse, and polydispersity is expected to increase the value of S(l)/S,,since the polyhedra in such a system are bound to be less regular and more asymmetrical than the above types. Apparently, the former effect overshadows the latter. This further suggests that in a truly monodisperse system S(l)/So should not only be smaller than 1.100 but may even be somewhat _smaller than 1.083. The reduced volume V of continuous phase between z' = 0 and z' in an equilibrated column is plotted in Figure 8 according to eq 17,27, and 30. As z' tends to infinity and 4 1, it reaches a limiting value of 0.258. This is reasonably close to our previous estimate,' while it is considerably in excess of thg maximum value of Krotov's proposed range of 'Ig < V < '/6.5 Finally, with our newly acquired knowledge of fI(+), eq 7 permits evaluation of the mean capillary pressure Ap/ Ap, inside the drops, provided that f(4) is known. In the
-
(9) Ross, S. Am. J. Phys. 1978, 46, 513; Colloids Surf., in press.
41
Langmuir 1987, 3,41-45
-
regime of high volume fraction, we have shown before' that
f(4) = [l- 1.892(1 - 4)1/2]2
(4
1)
(33)
and
Thus, a t 4 = 0.99, we find f = 0.66 and Ap/Apo = 2.92. As 4 tends to unity, f approaches one and Ap tends to infinity according to eq 33 and 34. As indicated above, the picture is less clear for intermediate and low volume fraction. Although we measured f(4) between $o and 4 = 0.975: the data are somewhat uncertain for reasons stated elsewhere.2 However, if we accept the previous empirical relationship6 0.70
0.90
0.80
1.00
$
Figure 9. Speculative curve of reduced capillary pressure, APIAP,, vs. 4.
then the derived A p ( 4 ) / A p oappears as in Figure 9. As discussed above, it may not be particularly disturbing that Ap($)/Ap0 drops below unity a t low volume fraction. Because of the uncertainties in the general validity of eq 35, however, it is prudent to consider the data in Figure 9 as being somewhat speculative.
Since submission of this paper, we have become aware of two recent publications by Kann,lo some aspects of which are related to, but clearly less advanced than, our work. (10) Kann, K. B. Colloid J. USSR (Engl. Transl.) 1984,46,397; 1985, 47, 744.
Investigation of Liquid Drop Evaporation by Laser Interferometry R. N. O'Brien* and Paul Saville Department of Chemistry, University of Victoria, Victoria, BC, V8W 2Y2 Canada Received April 24, 1986. In Final Form: September 16, 1986 The evaporation of a sessile drop of several liquids has been studied by two interferometers simultaneously. Liquid-phase interferograms which gave depth contours of the drop during its lifetime and some details of its final evaporation were supplemented by simultaneous vapor-phase interferograms. It appears that convection occurs in the liquid phase and also in the vapor phase; the vapor-phase physical structure appears to be a torus. An advancing foot, in this case a retreating foot, may have been detected.
Introduction Rates of evaporation from drops, sessile or suspended, have been carefully studied by LOU,^ Yang and Nouri,2 and Cammenga et a1.13 but the manner of evaporation has attracted much less interest. Rayleigh4 and others5 have established the conditions under which bouyancy-driven (density difference) convection should occur in pure liquid drops and Marangoni and Pearson6p7have derived the theory for surface-tension-driven (Marangoni) convection (1) Lou, Y. S. J. Appl. Phys. 1978, 49, 2350. (2) Yang, W.-J.; Nouri, A. Lett. Heat Mass Transfer 1981, 8, 115. (3) Cammenga, H. K.; Schreiber, D.; Barnes, G. T.; Hunter, D. S. J. Colloid Interface Sci. 1984, 98, 585. (4) Lord Rayleigh Phil. Mag. 1916, 32, 529. London, A 1940,176, (5) (a) Pellew, A.; Southwell, R. V. Proc. R. SOC. 312. (b) Law, A. R. Proc. R. SOC. London, A 1929,125,180. (c) Sparrow, E. M.; Goldstein, R. J.; Jonsson, V. K. Fluid Mech. 1964, 18, 513. (6) Marangoni, C. Nuouo Cimento 1871, 16, 239; 1878, (3) 3, 97. (7) Pearson, J. R. A. J.Fluid Mech. 1958,4, 489.
0743-7463/87/2403-0041$01.50/0
and there is experimental evidence for the general adequacy of theory.s Cammenga et aL3find Marangoni convection absent in water when theory predicts its presence. Yang and Nouri2 find three stages of drop evaporation by shadow graph studies in several polar liquids including polygonal cells as one stage. The relationship between increasing drop size on a solid substrate (or alternatively evaporation of a sessile drop) and dynamic contact angles is real enough though admittedly obscure a t this time as can be inferred from the work of Schwartz and Tejada? They extensively review (8)(a) Schmidt, R. J.; Milverton, S. W. Proc. R. SOC.London, A 1935, 152,586. (b) Silveston, P. L. Forsch. Ingenieurwes. 1958,24, 29,59. (c) Palmer, H. J.; Berg, J. C. J.Fluid Mech. 1971,47,779. (d) Berg, J. C.; Boudart, M.; Acrevos, A. J. Fluid Mech. 1966,24, 721. (9) Schwartz, A. M.; Tejada, S. B. J. Colloid Interface Sci. 1972,38, 359.
0 1987 American Chemical Society
