Langmuir 1991,7,1846-1848
1846
Concepts of Capillary and Osmotic Pressures in Foam P. M.Kruglyakov,+D.R. Exerowa,*Jand Khr. I. Khristov'J Department of Chemistry, Institute of Ciuil Engineering, Penza 28, USSR, and Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1040, Bulgaria Received March 30, 1990. In Final Form: June 3, 1991 Properties of polyhedral foam with high capillary pressure, obtained by semipermeable porous plates, are analyzed. It is shown that the excess pressure (toward surrounding medium) in the liquid phase of foam is very close in physical sense to the osmotic pressure, which has been investigated by Princen. The physicochemical properties of such foams are compared on the basis of capillary (excess) and osmotic pressure concepts. The structural characteristics of the foam (the shape and sizes of the films, of the Plateau-Gibbs borders and of their cross-points) as well as the pressure in the liquid (i.e. the continuous) phase depend essentially on the expansion ratio n or gas volume fraction 6 in the foam. The latter parameters are connected by the relation @=-
ATMOSPHERE PO
n-1 n
The difference in the properties of high and low expansion ratio foams has been known for quite sometime. However, the formation of foams of very high expansion ratio ( n > lOOO), the measurement of the pressure in the liquid phase, and the study of various physicochemical properties of such foams have been initiated quite At the moment of its formation the foam is usually of low expansion ratio. Even when the foam is generated by the method of mixing solution and gas flows on wire gauzes, the expansion ratio rarely reaches n = 1000values. In the process of drainage in a gravitational field a foam with low n could be transformed into a foam with high n only in the upper layers of a high foam column (at least a meter). When the foam is placed on a liquid phase and is in hydrostatic equilibrium, the pressure PL in the PlateauGibbs borders at height z (Figure 1)from the liquid level is given by
PL = pL,o- Pgz
(2)
where P L ,is~the pressure in the liquid phase at the foam/ liquid interface and usually it approximates the external pressure of the gas phase (the difference between them is equal to the mass of the liquid phase in a foam column of height z on a unit of area, p is the density of liquid phase, and g is the acceleration due to gravity. If the reading of pressure is taken from the top of the highest layer of the
* Author to whom correspondence should be addressed.
+ Institute
of Civil Engineering. Institute of Physical Chemistry. (1)Kruglyakov, P. M.;Exerowa, D. R.;Khristov, Khr. I.; Scheludko, A. D. BG Invention Certificate No19398,Published in Bull. Inst. Inu. Razio,Bulgaria 1976, 1 , 7. (2) Kruglyakov, P. M.;Exerowa, D. R.;Khristov, Khr. I.; Scheludko, A. D. BG Invention Certificate No19399,Published in Bull. Inst. Inu. Razio, Bulgaria 1978, 1 , 7. (3)Kruglyakov, P. M.;Exerowa, D. R.; Khriatov, Khr. I.; Scheludko, A. D. USSR Invention Certificate No 524557, Published in Bull. Inu., t
USSR - - - -.1976. . - . -,30. - -. (4) Khristov, Khr. I.; Kruglyakov, P. M.; Exerowa, D. R.h o c . VIZInt. Con/. Surf. Akt. Subst. 1976, 2,p.I, s "B", 462. (5) Khriitov, Khr. I.; Kruglyakov, P. M.;Exerowa, D. R.Colloid Polym. Sci. 1979, 257, 506.
0743-746319112407-1846$02.50/0
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Figure 1. Schematic illustrationof a foam on the liquid surface:
Po,pressure in the surroundingmedium; Pi pressure in the foam bubble; P, capillary pressure in the Plateau-Gibbs borders; kt, curvature of the free continuousphase/ gas interface;P L ,pressure ~ in the liquid phase at the foamfliquidinterface;z, height; 1, level from the foam highest layer; H, height of the foam column. foam column, PL may be written as6
PL = Po - ukt + pgl (24 where Po is the pressure in the surrounding medium, kt is the curvature of the free continuous phase/gas interface at the top of the foam, u is the surface tension of the continuous phase, 1 = H - z is the level from the top of the foam, and H is the height of the foam column. We have proposed a method'" for obtaining 'dry" polyhedral foams with strongly lowered (toward the surrounding medium) pressure in the liquid phase. According to this method the foam is brought into contact with a porous plate (usually sintered glass filter) under which a reduced pressure is created, not exceeding the capillary pressure in the filter pores, ensuring semipermeability of the plate (Figure 2). At this condition only the liquid passes through the filter pores, but not the gas. Due to the creation of pressure difference APo between the space under the porous plate and above the foam a pressure gradient arises in the Plateau-Gibbs borders, which increases the rate of foam drainage. When hydro-
-
(6) The comparison of eq 2 and 2a gives ukt = pgH - (PI,+,- Po) pgH. The difference between Po - P L =~ pg(H - Jpt$dz) > pgH > pgh
-
PL'= akt AP, = constant
(7)
Recently, Princen7JoJl has introduced the concept of foam osmotic pressure. It is defined as the pressure that has to be applied to a movable semipermeable membrane separating the liquid and the foam in order to prevent the liquid phase from entering the foam. The osmotic pressure might be considered as the elevating force of the gas bubbles in liquid, which equals to the difference between pressure of liquid column and the pressure of liquid in foam at the same height (see for example eq 11 in ref 11)
where V L is ~ the volume of liquid in foam column of height z and A is the cross-sectional area. The osmotic pressure by its physical sense and value is very close to the above-discussed excess pressure PL'(ita influence on various properties of foams has already been studied by us in details in refs 1-5, 8, etc.). Comparison of A and PL'might serve as proof of this concept, which has not been discussed in refs 7, 10, and 11. For a foam in gravitational field at hydrostatic equilibrium, from eqs 2 and 2a and the definition for P~'follow8 (see as well refs 7 and 11):
The comparison of (8) and (9) gives
(5)
the method,4*6allows obtaining foams with practically constant pressure along the column height and equal curvature radius of Plateau-Gibbs borders. The capillary pressure P,,in the foam is determined by the difference Pi - PLbetween the pressure Pi in the foam bubbles and the pressure PLin the liquid phase'
where t = S / V, = s/$vF is the foam specific area, S is the (7) Princen, H. M. Langmuir, 1988,4, 164. (8) Khristov, Khr. I.; Kruglyakov, P. M.; Exerowa, D. R. J. Colloid Interface Sci. 1981, 79, 584. (9) Kruglyakov, P. M.; Mikina, T. V. Kolloidn. Zh. 1981,43, 168.
This means that osmotic pressure differs from PL'by a value equal to the difference between the pressures of the foam column of heights H a n d z on the liquid. When z = H, i.e. at the top of the foam column, A = PL'= ukt. The equivalence of T and PL' at the top of the column comes directly from eq 6 and 21 in ref 7. For polyhedral foams with a high capillary pressure P, = PL'= AP, >> pgH > pg VLH/A because of the neglible value of pg(VLH/A - VLZ/A)compared to AP,. That is why the capillary and osmotic pressure for dry foams might be considered equivalent at each point in the foam column. (10)Princen, H. M. Langmuir 1986,2,619. (11) Princen, H. M.; Kiss A. D. Langmuir 1987, 3, 36.
Letters
1848 Langmuir, Vol. 7, No. 9,1991
Expressing the excess (osmotic) pressure PI,',either by means of the difference between the pressure above the foam and the pressure under the immobile porous plate on which the foam is situated or by means of the difference between the pressure in the borders and the pressure in the surrounding medium, appears more convenient for practical use. Such a definition PI,'corresponds to the experimental method using immobile porous plates for obtaining "dry" foams and for measuring the foam capillary pressure415and is suitable also for the determination of the nonequilibrium capillary pressure during the process of establishment of hydrostatic equilibrium. The pressure 7r (and PI,') is a close analogue to the osmotic pressure of liquid solutions, but that analogy is incomplete. While the osmotic pressure of solutions is created only when they are in contact with the solvent (or another solution) through a semipermeable membrane, the excess (capillary) pressure exists in "dry" foams in the absence of porous plates as well. When such a foam is in contact with another foam or lower capillary pressure or another liquid (e.g. benzene, toluene, etc.), sucking of the continuous phase into the "dry" foam o c c u r ~ . ~ ~ J ~ While the osmosis in liquid solutions of equal hydrostatic pressure takes place up to the complete passage of the solvent through the semipermeable membrane into the solution, in foams the movement of the external continuous phase into the foam stops when the latter becomes spherical (n ru 4). For the evaluation of the excess pressure in the liquid phase of a foam, we constructed several modifications of micromanometers (open and closed t y p e ~ ) ~ fwhich j J ~ allow the precise measurement of various pressure values. These micromanometers are of particularly simple construction and they are suitable for measurement of excess pressure of emulsions as well. We have also used5J5J6 another method for determination of the excess pressure PI,' in the Plateau-Gibbs borders by direct microscopical measurement of their radius r of curvature and subsequent calculation with the help of the formula
PI,' = P , = a/r (11) Later, on the basis of our methods, other constructions of micromanometers were suggested.17J8 Quantitatively, the value of PL' is connected with the expansion ratio n and the dispersity of the foam. Thus, for monodisperse polyhedral foam with cells in the shape of pentagonal dodecahedron, simple structural relations yieldI3 Po = Po- PI, = (0.46n'/2 - 1.8)a/a (12) where a is the length of the dodecahedron edge. A series of other convenient PL'(a,r,$)dependencies were obtained later.'JOJl The calculation of the excess pressure by eq 12 and by eqs 15 and 16 from ref 10, made for c = 30 mN/m, a = (12) Pertaov, A. V.; Chernin, V. N.; Chistyakov, B. E.; Shcukin, E. D. Dokl. Acad. Nauk SSSR 1978,238, 1395. (13) Kachalova, E. I.; Kruglyakov, P. M.; Kuz'min, N. P. Kolloidn. Zh. -.
1988, 50, 575. (14) Levinskij, B. V.; Krualyakov. P. M.: Safonov, V. F. Kolloidn. Zh. __ 1982,44, 696. (15) (a) Lalchev, Zdr.; Khristov, Khr.;Exerowa, D. Colloid Polym.Sci. 1979,257,1248. (b) Lalchev, Zdr.; Khristov, Khr.; Exerowa, D. Comm. Dep. Chem., Bulg. Acad. Sci. 1979, 12, 473. (16) Khristov, Khr.; Exerowa, D.; Kruglyakov, P. Kolloidn. Zh. 1988, 50. 765. (17) Sharovarnikov, A. F.; Tsap, V. N. Kolloidn. Zh. 1983,45, 120. (18) Racz, G.; ErdGs, E.; Kocz6, K. Colloid & Polym.Sci. 1982, 260, 720.
0.03 cm, corresponding to R = 1.223a in ref 10, and n = 1000 (4 = 0.999), yields in both cases similar pressure values, PI,'= 1275 N/m2 and K = 1316 N/m2, respectively. The small difference could be due to the different models used for the derivation of eq 12 (pentagonal dodecahedron) and eq 16 from ref 10 (tetrakaidecahedron). Results of essential importance were obtained by investigating the kinetics of establishment of the excess press~re,4JQ9-~~ the regularities of the solution flow through a foam of unchanging (in size and with time) b0rders,2~*~3 and the dependence of foam stability on the pressure difference AP,,in the foam liquid phase.4J't8-24J5It should be mentioned that for pressure differences APo > (2 to 3) X lo4 N/m2, the corresponding excess and capillary pressures (in foam) are never reached in practice.21 The increase of the pressure difference APo leads to a significant decrease in the lifetime 7,,of the foam column,4.5,24*25 the 7,,( AP)dependence being determined by the type of both the surfactant and the films in the foam. These dependencies were analyzed in detail in refs 8, and 24-26. The investigation of polyhedral foams of high excess pressure also leads to development of series of new experimental methods for determination of the foam disp e r ~ i t y ,stability,8yZ5 ~ ~ , ~ ~ etc., of the theory of the optical density of foams,28experimentally tested in ref 29, of the theory and methods for accumulation in f0ams,~@32 and of the method for chromatographic fractionation in foams.33 The development of new methods for calculation of the osmotic (excess) pressure7J0 of the variation of the film area as a fraction of the total surface area and of a number of other properties dependent on the foam expansion ratio, along with the direct measurements of the excess pressure and the model studies of the shape of the films and of the b0rders,~313~*~~ enables new important information to be obtained about the structure and the physicochemical properties of foams which are not fully polyhedral (4 # 1).
Acknowledgment. We express thanks to Professor
H. Princen for his valuable remarks and fruitful suggestions. (19) Kruglyakov,P. M.; Kuznetaova, L. L.; Khriitov, Kh. I.; Exerowa, D. R. Kolloidn. Zh. 1979, 41, 3, 445. (20) Kuznetaova,L. L.;Kruglyakov,P. M. Kolloidn.Zh. 1983,46,1076. (21) Khristov, Khr. I.; Exerowa, D. R.; Kruglyakov, P. M. Kolloidn. Zh. 1988,50, 765. (22) Kuznetaova, L. L.; Kruglyakov, P. M. Dokl. Akad. Nauk USSR 1981,260,928. (23) Fokina, N. G.; Kruglyakov, P. M. Kolloidn. Zh. 1986, 48,318. (24) Khristov, Khr. I.; Exerowa, D. R.; Kruglyakov, P. M. Kolloidn. Zh. 1981. 1. 101. (25) Khhristov,Khr. I.; Exerowa, D. R.; Kruglyakov,P. M. Colloid Polym. Sci. 1982,261, 265. (26) Kruglyakov, P. M.; Exerowa, D. R. Foam and Foam F i l m ; Izd. Chimia: Moscow, 1990. (27) Kuznetsova, L. L.; Kruglyakov,P. M. Kolloidn. Zh.1979,41,673. (28) Krotov, B. B.; Kruglyakov, P. M. Kolloidn. Zh. 1990,52,479. (29) Kruglyakov, P. M.; Bulavchenko, A. I.; Zemskova, S. M.; Krotov, V. V . Kolloidn. Zh. 1986,48, 62. (30) Kruglyakov, P. M.; Kuznetaova, L. L. Colloid Polym. Sci. 1980, 258, 451. (31) Lalchev, Zdr.; Exerowa, D. Biotechnol. Bioeng. 1981, 23, 669. (32) Lalchev, Zdr.; Dimitrova, L.; Tzvetkova, P.; Exerowa, D. Biotechnol. Bioeng. 1982, 24, 2253. (33) Kruglyakov,P. M.; Kuznetsova, L. L.; USSR Invention Certificate No 882548, Publ. in Bull. Zno. USSR 1981,42. (34) Kruglyakov, P. M.; Exerowa, D. R.; Khristov, Khr. I. h o c . Znt. Cong. Pure Appl. Chem., 31st 1987, 7, 116. (35) Kruglyakov,P. M.; Kuz'min, N. P.; Kachalova, E. I. Kolloidn. Zh. 1988, 50, 460.
