Theoretical Foundations of Chemical Engineering, Vol. 34, No. 3, 2000, pp. 211-226. Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 34, No. 3, 2000, pp. 237-254. Original Russian Text Copyright 9 2000 by Kazenin, Vyaz'min, Polyanin.

Foams as Specific Gas-Liquid Technological Media D. A. Kazenin*, A. V. Vyaz'min**, and A. D. Polyanin*** * Moscow State University of Engineering Ecology, Moscow, Russia ** Karpov Research Institute of Physical Chemistry, Russian State Scientific Center, ul. Vorontsovo pole 10, Moscow, 103064 Russia *** Institute for Problems of Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, Moscow, 117526 Russia Received December 3, 1997

Abstract--Basic physicochemical notions of foams as specific gas-liquid technological media are systematized. Structural properties, main parameters, and models of foam are considered. The adsorption-kinetic nature of the skeleton structure-forming phase of foam is discussed. Particular attention is given to the theoretical description of the internal hydrodynamics of foams, the concept of hydroconduction, and the syneresis phenomenon. Basic principles of rheological models of foams are discussed. Foam is a technological medium widely employed in many processes in the chemical, biotechnological, mineral, and food industries. For example, exchange processes in gas-liquid systems in bubble column reactors are inevitably accompanied by abundant foaming, since the liquid used almost always contains chemical, biological, or mineral surfactants. These surfactants stabilize the interface [1, 2], thus converting the foam layer into a single working medium with unique physicochemical and hydrodynamic properties. The excess surface energy of such a gas-liquid disperse system predetermines its nonequilibrium character. However, owing to the stabilizing effect of surfactants, foam has a metastable structure and can persist for some time (the foam lifetime) [3]. Its properties slowly relax under external actions, provided that these actions do not exceed certain threshold values, above which the foam structure is ruptured. The large specific interface area of foam enables one to efficiently use foam in desorption, flotation purification, separation, gas exchange, etc. At the same time, the role of foam in chemical processes is not unambiguously beneficial. The progressive gravitational dehydration, the excessive strength of foam structures, and the stagnancy of actual interfaces diminish the efficiency of exchange processes and, at a certain stage of the evolution of the foam system, can render a considerable portion of its volume technologically ineffective. That is the reason why the problem of defoaming [4] is an important part of the general problem of foam control. In this work, the basic theoretical notions of physical chemistry, hydrodynamics, and rheology of foam are considered. Here, we do not discuss any numerical or experimental investigations in this field, except the ones that seem to be necessary for the completeness of the presentation of the matter. Many sources in the literature on this subject are not included in the References because they are cited many times in the monographs and articles given therein.

FOAM STRUCTURE AND ITS BASIC PARAMETERS Foam is a specific gas-liquid system that comprises a discontinuous disperse gas phase and a continuous dispersion liquid phase. 1 The dispersion phase is the binding medium, which continuously fills the entire space among disperse gas inclusions. In a monodisperse foam with a gas content of no more than 74%, gas bubbles are undeformed and spherical, and this foam is termed the spherical foam [5]. As the gas content becomes higher, bubbles deform, and fiat faces with rounded edges are produced. In the course of time, the foam cell shape tends to become polyhedral. The shape of bubbles in a real foam is intermediate between the spherical and polyhedral. Such foam is called a cellular foam [5, 6]. The boundary between the cellular and polyhedral foams is quite arbitrary, and lies in the range of a very low (on the order of several tenths of a percentage point) liquid content. Nonetheless, the polyhedral model of a foam cell is employed rather often [1, 4, 7-1 1]. Let us further consider the basic quantitative characteristics of foam.

Foam ratio. One of the most important quantitative characteristics of foam is the foam ratio K, which is the reciprocal of the volumetric liquid content of the foam. If ~ is the volumetric gas content of the foam, then (1) g = (1-~)-1.

(1)

Cellular foams are characterized by 4 < K < 170 [6]. Dispersity. Foam cells are commonly shaped like rounded polyhedra; therefore, as the only linear size describing the internal scale of foam, it is convenient to 1 Sometimes,the disperse phase is not a gas but a liquid immiscible with the liquid of the dispersion phase. Such a foam is then called the liquid-liquid foam.

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KAZENIN

take the radius a of an equivalent bubble, i.e., a spherical bubble whose volume is equal to that of a polyhedral cell. The foams consisting of cells of the same size are referred to as monodisperse. Such foams occur very seldom. As a rule, there is a spectrum of sizes a I..... a,; in this case, the dispersity represents the average linear cell size [6]:

et al.

multimode distribution function tl

f(a)

=

E'~i 2(~i- 1)O~ia i= I ( 1 + t~ia2) fh

(7)

with the normalization condition n

tl

?t= ~_~ai.

(2)

i=1

The foam dispersity is sometimes characterized by the specific interface area e, which is equal to the total surface area of gas bubbles per unit foam volume: n

~.~niai s = 3i=1

(8)

i=1

where o~i, I]i, and Yi are constants chosen so that the function give the best fit to experimental data. When gas exchange in polydisperse foams is considered, the Sauter [ 12] surface-averaged cell radius is introduced, which is determined from (4):

2

K-1

"

K

'

(3)

i---1

where ni is the number of bubbles of size ai per unit foam volume. For a monodisperse foam, 3K-1 a K

E -- - ~

(4)

Polydispersity. If a liquid is added to a polydisperse cellular foam (i.e., if the foam ratio is lowered) until the latter becomes a spherical foam with the same bubble size distribution, then the spherical foam produced is referred to as the equivalent foam. The foam ratio of the equivalent foam is said to be the minimum foam ratio Kmi,. The minimum foam ratio of a polydisperse foam is apparently higher than that of a monodisperse one, since the spaces among closely packed identical large spheres can be occupied by smaller spheres. Thus, Kmi, can serve as a quantitative measure of polydispersity [6]. For the monodisperse foam, Kmin = 3.86, whereas for real polydisperse foams, Kmin is =10-15 and higher but typically no greater than 20 [6].

In [3], the following bubble-size distribution function has been put forward:

f(a) =

,~_,Ti = 1,

6o~a

(5)

(1 + ~ a 2 ) 2'

where o~ is the distribution function parameter. This function agrees well with the distribution in real foams. One can readily check that the functionf(a) is normalized and that the parameter t~ is related to the average bubble radius by the formula t~ = ~(~)-2.

(6)

Generalizing (5), one can obtain the multiparameter

3K-I as . . . . e K

(9)

Recently, it has been experimentally found [13] that foam can be in a steady state, in which the volumes of the phases and the bubble size distribution are timeinvariable. This can occur if the foam is a closed, but not isolated, system (in Prigogine's terms [14]), i.e., a system that exchanges energy, but not matter, with the environment. Such conditions can be produced, e.g., within a closed container when the energy is supplied to the foam by vibration and the heat of dissipation is removed. After a certain threshold of the kinetic energy of vibration has been exceeded, the foam, on having relaxed in some time, attains the steady state (dissipative structure [15]) with a Gaussian bubble size distribution. Capillary pressure. This term means the excess of the pressure within foam bubbles over the atmospheric pressure:

APg = P g - P a .

(10)

This pressure is determined by the curvature of the upper surface of bubbles in the uppermost foam layer (which borders the atmosphere) and is described by the corresponding term in the Laplace law. This is an integrated characteristic of foam, since the pressure within bubbles in lower foam layers is the same due to the flatness of the interfaces between bubbles [7]. The capillary pressure depends on the surface tension a of the foam-forming solution and on the specific interface area e of foam: 2 APg = ~ae.

(11)

For a monodisperse foam, according to (4), we have

APg - 2 a K - 1 a

(12)

K

Note that, in real foams, the capillary pressure is low, usually on the order of 0.001 atm.

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Capillary rarefaction. The interface between a deformed bubble and the liquid phase that fills the space among such bubbles is concave. Therefore, the local pressure Pt in the liquid phase is lower than the pressure P8 in the gas phase and is related to it by the formula Pt = P g - ~ , (13) where ~: is the average interface curvature. The capillary rarefaction is the excess of atmospheric pressure over the pressure in the liquid phase: AP t = Pa-Pt. (14) By combining (10), (13), and (14), one can obtain AP t = aK-APg.

(15)

In contrast to the capillary pressure zSJ'g, the capillary rarefaction APt is a local parameter. The APt distribution over the dispersion medium governs all the internal flows of the liquid in foam. Moreover, the averaged capillary rarefaction determines the ability of foam to absorb a liquid and imparts a certain strength to the foam body. Sometimes, instead of the term capillary rarefaction, the term osmotic pressure of dispersion is used [17, 18]. The osmotic pressure is defined as the additional external pressure that should be applied to a semipermeable membrane that separates foam and free liquid to compensate for the liquid sucked into the foam from the free liquid. The faces of foam cells bordering the membrane are supposed to be flat; therefore, the capillary pressure within foam bubbles is zero. Then, pursuant to (15), the capillary rarefaction is determined by the average curvature ~cof the internal foam surface at the nodes of the foam structure. This curvature has been calculated [17] for the model of a monodisperse foam with pentagondodecahedral cells: K=

0.5842 ~1/3 a (1 _q~)~/2"

(16)

According to [17, 18], the expression for the osmotic pressure differs from that for the capillary rarefaction by the factor g(q~) = ( 1 - 1 . 8 9 2 1,~1--~) 2, which characterizes the fraction of the area of the membrane surface that borders the flat faces of foam cells. For the polydisperse foam, the Sauter radius (9) should be used in (16). Gotovtsev [19], using an approach based on the strain theory [20] and the model of rounded dodecahedral cell, suggested another approximate expression for the average interface curvature, =

9.334 [ 1 - exp(-0.00825K)], a

(17)

which is accurate to within 3% in the foam ratio range 30 < K < 300.

213

POLYHEDRAL FOAM MODEL The polyhedral shape of a foam cell is the limiting shape at an infinitely high foam ratio. At the same time, it is also quite a convenient model of the structure of real foams with finite foam ratios, A polyhedron composed of liquid films should obey two rules formulated by Plateau [21-23]: (1) at any edge, three films should meet, forming equal dihedral angles (120 ~ with one another; (2) at any node, three edges should converge, making equal angles (109~ " ) with one another. In addition, it is natural that, in a foam cell, the number F of faces (films), the number B of edges (the Plateau-Gibbs channels), and the number N of nodes must satisfy the Descartes-Euler fundamental topological formula [24, 25] N-B+F

= 2.

(18)

Investigation of the topological properties of multidimensional convex polyhedra have shown [18] that, along with the Descartes-Euler formula, the linearly independent Den-Sommerville equation is valid in the three-dimensional space: 2 B - 3 N = 0.

(19)

The simplest space body (regular polyhedron) obeying the Plateau rules, the Descartes-Euler formula, and the Dehn-Sommerville equation is a pentagondodecahedron, which is a polyhedron with twelve identical faces, each being a regular pentagon [5]. Drawbacks of this model are the impossibility of partitioning the space into pentagondodecahedra and the disagreement of their parameters with the statistical data on real foams. Experimental and statistical studies have demonstrated [26] that a polyhedral foam cell should have, on average, 13.7 faces, each having 5.1 sides. For this reason, the assumption has been made [25] that a random packing of identical bodies (statistical honeycomb) is a spatial mosaic, with the parameters consistent with the above statistics. As the polyhedral model of a foam cell and a cell of any three-dimensional biological tissue, Kelvin's tetrakaidecahedra have been proposed many times, which are minimum truncated octahedra [22-24] that have eight hexagonal and six tetragonal faces. Noteworthy, it has been revealed statistically [27] that, in three-dimensional biological tissues, Kelvin's tetrakaidecahedra occur among tetrakaidecahedral cells only in N10% of cases. In [11, 28], it has been shown that the polyhedral foam structure is more exactly modeled with compact tetrakaidecahedral cells. The fact is that, if the numbers of faces, edges, and vertices are fixed in accordance with the Descartes-Euler formula (18) and the DehnSommerville equation (19), then the specific interface surface of the tetrakaidecahedral foam and, hence, its free energy are lower the closer the areas of polyhedral faces are to one another and the more circular their

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KAZENIN et al. Gibbs channels as a unified integrated characteristic of foam. This means that the total curvatures of the menisci at the nodes and in the Plateau-Gibbs channels are identical. However, since the curvature of the nodes is spherical, and that of the Plateau-Gibbs channels is cylindrical, the radius of curvature of the PlateauGibbs channels should be half as large as that of the nodal menisci. R n and Rb are related to the average curvature of the internal surface of foam at the nodes and in the PlateauGibbs channels by the formulas

Fig. 1. Modelof a foamcell:(a) pentagondodecahedronand (b) cross sectionof the Plateau-Gibbs channel.

shapes are. Therefore, the shape of the polyhedron should be as close to spherical as possible. Such a tetrakaidecahedron with F = 14, B = 36, and N = 24 is termed the compact tetrakaidecahedron. It is difficult to calculate the geometrical parameters (surface area, edge length, etc.) of this polyhedron; for this reason, the data calculated for a pentagondodecahedron are used after applying necessary corrections (e.g., the total length of edges is raised 1 4 ~ = 1.08 times). Note that these correction factors are all close to unity (the deviation is about 5-10%). Thus, the foam structure can be approximately described by the following polyhedral model [6]: (1) Foam cells are shaped like identical pentagondodecahedra with rounded edges and vertices. (2) Deformed bubbles are separated by thin planeparallel films, each shaped like a regular pentagon. (3) Along each of the lines where three films meet, there is a thickening (the Plateau-Gibbs channel), whose cross section is invariable along the channel length and is shaped like a curvilinear triangle formed by three circular arcs intersecting pairwise in three vertices. (4) The joints of four Plateau-Gibbs channels is one more sort of structural elements of foam, specifically, a node. It is formed by four concave spherical surfaces arranged in a spherical tetrahedron. Up to K = 200, the nodes contain most of the dispersion medium; however, at K > 200, the Plateau-Gibbs channels become the main reservoir of the liquid phase [6]. Figure 1 depicts the pentagondodecahedral model of a foam cell. The radius of curvature of the nodal menisci, Rn, is among the most important quantitative characteristics of foam. Another characteristic related to R n is the radius of the Plateau-Gibbs channels, R b. Note that, although the capillary rarefaction is a local parameter, its equalization among the nodes and the PlateauGibbs channels owing to the liquid inflow and outflow proceeds rapidly enough. This allows one to regard the capillary rarefaction at the nodes and in the Plateau-

-

1

2

Rb

Rn,

(20)

where ~: is computed by (16) and (17). A rough estimate can be obtained by using the simpler relationship Rb= A a ,,fK"

(21)

One should bear in mind that the A values recommended in the literature differ somewhat from one another: A = 1.782 [291, 1.73 [28], and 1.628 [12]. In real foams, the foam film thickness h continuously decreases owing to a slow drainage of the liquid into the Plateau--Gibbs channels. In principle, this process is complete after the pressure difference between the phases (which equals the sum of the capillary pressure and the capillary rarefaction) has been balanced by the so-called wedging pressure H(h), which emerges in thin films owing to the interaction between the boundary layers [30]: A P t + A P g = a~: = t~ R b'

(22)

---ff = l-I(h).

(23)

Rb

In practice, the wedging pressure should be taken into account only in films whose thickness is within the range 10-9 < h < 10-7 m. The wedging pressure is produced by various physical forces: 2 van der Waals attraction, electrostatic repulsion, steric elastic interaction, etc. Approximate formulas for the wedging pressure arising via different mechanisms are reported in [12, 30, 31]. Since, as noted above, there is no sharp distinction between cellular and polyhedral foams, the satisfaction of the following inequality can be taken as the criterion that foam is polyhedral [11]: (24)

h ~ R b ~ a.

Formulas (16), (17), (20), and (21) can be used for assessing the basic geometrical parameters of both polyhedral and cellular foams. 2 This issue is consideredin detail in the section Stability, Evolution, and Ruptureof Foams.

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FOAMS AS SPECIFIC G A S - L I Q U I D TECHNOLOGICAL M E D I A

Other important geometrical characteristics of polyhedral foams are the length and the cross-sectional area of the Plateau-Gibbs channel. Expressions for calculating these quantities are presented in [6, 11]. For example, in the pentagondodecahedral model, the PlateauGibbs channel length is b = 0.818a.

Gas

Wall

(25)

The cross-sectional area Sb of the Plateau-Gibbs channel in the monodisperse foam is Sb = 0.1612R 2.

Continuousfilm of free liquid

(26)

Planar lamellar micelles ,.d

In polydisperse foams, S b is much smaller. STRUCTURE OF THE FOAM CELL WALL AND SURFACTANT ADSORPTION KINETICS Foam cell structure. During foaming while bubbling a gas through a surfactant solution, the complex multilayer structure of foam cells is produced. According to [32], each of the bubbles having emerged has the double-sided wall, which is a solvent layer into which the polar hydrophilic parts of surfactant molecules are immersed (Fig. 2). The nonpolar hydrophobic parts of surfactant molecules on the inner side of the wall are directed toward the bubble, and those on the outer side of the wall are oriented away from the wall. Between two encapsulated bubbles, there is a lamella, which is an interlayer with a complex structure. In the middle of the lamella, there is a liquid layer, which forms a continuous dispersion medium. On each of the two sides of this layer, there is a surfactant monolayer, and the hydrophobic parts of surfactant molecules in this monolayer interact with the hydrophobic parts of surfactant molecules in the wall to produce two planar lamellar micelles [2], which separate the walls, and a continuous liquid film at the center. Thus, in each pair of neighboring bubbles in foam, the bubbles are separated by as many as five layers of dissimilar natures. All this imparts stability to foam cells and differentiates foam from emulsion. This fact has been passed over in many works, e.g., [6, 10]. A foam cell does not lose its individuality even if it is left alone on the liquid surface [32]. A two-sided wall ensures foam strength and elasticity [3, 33]. Surface elasticity of surfactant solutions and elasticity of foam wall elements. As is known [4], pure liquids cannot produce stable foams. Foam-forming properties are imparted to liquids by surfactants. A surfactant molecule comprises a nonpolar hydrophobic part ("tail") of a sufficiently large size and a polar hydrophilic part ("head") [21]. The hydrophobic part is, as a rule, a hydrocarbon chain, and the hydrophilic part is a carboxyl, sulfate, or some other group. Owing to the absence of the dipole forces and hydrogen bonding, the cohesive force in a nonpolar solvent is much weaker than that in a polar solvent such as water; therefore, the rise of surfactant molecules into the near-

Wall

Gas Fig. 2. Structure of a lamella.

surface adsorption layer is favored thermodynamically. The adsorption layer structure is such that the polar hydrophilic parts of surfactant molecules are strongly bound to the solvent, whereas the hydrophobic tails are much less tightly bound to water and are outside the aqueous phase, i.e., in the adjacent boundary layer of gas or some nonpolar liquid. When the surfactant concentration in the adsorption layer is very low, surfactant molecules in this layer are recumbent. As this concentration is raised, the hydrophobic parts of surfactant molecules are detached from the surface and assume, for the most part, an inclined position. After the adsorption layer has been filled in, the tails of surfactant molecules become absolutely upright [21]. Let us define the adsorption F as the adsorbed surfactant weight per unit interface area, and let us refer to the surfactant adsorption F at the limiting occupation of the adsorption layer as the ultimate adsorption F~. Then, the ensemble of surfactant molecules in the adsorption layer can be regarded as a peculiar kind of two-dimensional gas obeying the van Laar equation [34-36] A~ = t~0-~ = RTF.~ln(1-F/F~) -1,

(27)

where Aa is the surface pressure. At low degrees of occupation of the adsorption layer (F "~ F.~), equation (27) takes the form that is similar to the ideal gas law: A~ = RTF.

(28)

The surface tension at the solution interface is lower than the surface tension of a pure solvent by the surface

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pressure of surfactant molecules [21, 37]. This is not inconsistent with the high strength and elasticity of the films that constitute the foam skeleton. The equilibrium surface layer of a pure liquid is perfectly inelastic [38]. The enlargement of the free surface under the action of external forces is not due to the extension (the increase in the distance between molecules in the near-surface layer) but is due to the rise of new molecules from the bulk of the solution. The decrease in the equilibrium surface tension as a result of adding a surfactant is not evidence of a reduction in the elasticity of the surface, since under slow external actions, the surface is far from elastic. Note nonetheless that, under very fast external actions whose characteristic time is shorter than the time of autoadsorption relaxation of the surface layer, the elastic properties (dynamic surface tension [39]) are possessed by the surfaces of even pure liquids [40]. Such a property must not be substantially dependent on the presence of the adsorbed surfactant layer. At the same time, the presence of this layer imparts extra elasticity to the surface under both slow and fast deformations. The work done by external forces during the deformation of the surfactant solution surface changes the free energy of the system. This energy varies owing to changes in both the surface area and the surface tension [38, 41]. The free energy is one of the fundamental notions of thermodynamics. In our case, it coincides with the notion of interfacial energy ~P = oS. The rate of variation of W with S can be represented as

Do

(29)

The first term of the right of this equation is the (equilibrium or dynamic) surface tension o, and the second term is the modulus of elasticity of the adsorption layer,

T,V

which can also be both equilibrium (the Gibbs elasticity [41]) and dynamic (the Marangoni elasticity [34]). The mechanism of the elastic action of the adsorption layer can be represented as follows. Any deformation of the surface, which is accompanied by, for instance, an increase in its area, diminishes the adsorbed surfactant amount per unit area. This decreases the surface pressure of surfactant molecules and, hence, raises the surface tension, which counteracts any further extension of the surface. If the surfactant concentration in the adsorption layer is low, then the two-dimensional gas of surfactant molecules obeys the equation of state that follows from (28): o0-o

=

NsMRT S "

(31)

Let us assume that Ns = const, which is the case when either the surfactant is insoluble, or the films are very thin, or the characteristic time of the external force is substantially shorter than the relaxation time of the adsorption layer. Differentiation of (31) with respect to S gives an estimate of the Marangoni modulus of elasticity:

EM= S(0~

\ o o j T, V , N s

= o0-o.

(32)

Expressing the total number of moles in the system, N, as

CV

N = - ~ - + N s = const,

(33)

one can assess the Gibbs modulus of elasticity: Ef=

S(0~-S)

= E . + RTV(O~-C)

T, V, N

(34, T, V, N

Apparently, Ec < EM, since the equilibrium value of the derivative (OC/OS)r,v,tr < 0. The actual value of the modulus of elasticity at a finite rate of surface deformation lies between E~t and Ec. Since the foam cell wall is a multilayer film [32], its elasticity exceeds the elasticity of the adsorption layer. At higher surfactant concentrations in the adsorption layer, along with the surface pressure, cohesive forces between the adsorbed molecules can also occur. The adsorption layer gains additional strength, whose maximum is attained for the incompletely filled layer, where the adsorbed molecules are in an inclined position, with intertangled hydrophobic tails [21]. In addition, if the surface is highly developed, its elasticity can be lower because of the limited surfactant amount in the system [ 1]. The highest strength is possessed by films produced with the use of colloidal, rather than molecular, foaming agents. In this case, micelles can be adsorbed at the surface [2]. The colloid passes to the gel state, in which its strength characteristics are akin to those of solids [21, 42]. In colloidal solutions, the wall strength increases by raising the concentration and reaches a maximum when the adsorption layer is saturated. In some instances, the irreversible transition (denaturation) of the colloidal substance on the surface to the insoluble solid state is observed. Kinetics of surfactant adsorption in liquid solutions and foams. The unsteady distributions of a soluble surfactant throughout the volume V of a system and over its surface S are described by the respective convective diffusion equations [35]: 0C 2-7 + (v. V)C =

DAC,

(35)

0F 2-7 + (vs. Vs)r = D s A s r - r ( V s . vs) + j,

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F O A M S A S SPECIFIC G A S - L I Q U I D T E C H N O L O G I C A L M E D I A

where the subscript s refers to the value on the surface. The boundary condition on the surface for equation (35) is - D - ~ = j.

(37)

the film thickness occurs virtually instantaneously. Therefore, one can omit the subscript s and consider that Cs = C(t) at t > td. In this case, one can obtain the solution of equation (41) that describes the adsorption during extension (or contraction) of foam film: t

The surfactant flux j from the bulk of the solution toward its surface is given by the kinetic equation (the law of surface action [35, 43]), which is similar to the Langmuir equation [44, 45]: j = 13(1- F/F~)Cls - 13iF.

13 =

131

=

b,~ nlax

C(x)dx

0

t

"~

o

o

1{

S(0)F(0) (42)

(39)

where S(0) and F(0) are the initial values of S and F, respectively, and x and ~ are variables of integration. At the interface with the surfactant solution, where one can suppose that C = const and S = const, relation (42) becomes simpler: r(t)=

+

r(0)expI-(13i + ~s

I (43)

13C

ex,

Herefrom, one can assess the adsorption relaxation time:

13(~max -- ~rnin) ( - 1 ) n+l=r

X

'I

(38)

This equation describes the kinetics of adsorption, which consists in the exchange of surfactant molecules between the surface monolayer and the adjacent solution in the presence of a potential barrier between them. The kinetic coefficients of adsorption and desorption can be estimated [46-49] if the form of the potential of interaction of a particle (a surfactant molecule) with the film surface, E(z), is known. If the plot of the function .~.(z) has the shape of a potential barrier with a potential well, then the saddle-point method [50, 51] yields f ~ (2) . ~ . D ~/ ] -~max [ ~max~ ~ e Xp~,----~--),

r ( t ) = S--~exp -13~t-

(

(40)

t. = @1 + F J

n = z ( n + 1)!(Zmax - Zmin)n+i

For example, in solutions of surfactants such as proteins, this time has been estimated at ta = 103-104 s [52]. where '~max and "max = (n) are the values of the potential and Thus, the adsorption layer is occupied by a kinetic its nth derivative at the maximum point (at z = Zmax), mechanism. respectively, and Emi n is the potential value at the point Note that, if the adsorption layer is in contact with the liquid layer of a large enough thickness, the diffuof minimum (at z = Zmin). In foams, most of the interfaces are films where the sion relaxation time can be comparable to the adsorpliquid is almost stagnant. The film surface is even more tion relaxation time. Then, the kinetics of filling of the stagnant. Therefore, equation (35) describes the molec- adsorption layer, whose rate is given by relations (37) ular (or Brownian) diffusion in the bulk of the liquid, and (38), can be diffusion-controlled (at low surfactant and equation (36) under the additional assumption of volume concentrations in the solution) or proceed insignificance of the surface diffusion characterizes the under the mixed control (at higher surfactant contents) dynamics of the localized [35] (or perfect [43]) adsorp- [53]. The purely kinetic mode of occupation of the adsorption layer can take place only in thin surfactant tion layer: solution layers, to which the liquid elements of foam structures belong. -.~ 13 1 C~ -13 , r r d S = ~ Sdt' (41) At t ,. oo, formula (43) gives an equation for the surfactant adsorption isotherm: where C~ is the instantaneous local surfactant concentration at the foam film boundary. Note that the last F = 13C(131 [3C')-l, (44) term of (41) owes its origin [35, 44] to the term of equa+ro.) tion (36) that contains the divergence of the surface velocity. which has the purely Langmuirian form [38, 44]. Since in real high-ratio foams, the film thickness The dispersion liquid in the foam layer is a surfacusually does not exceed 10 gm, then the time td = h2/D tant solution whose volume and interface area are variof diffusion relaxation of film is fractions of a second, able because of the evolution of the layer. The followand hence, the concentration equalization throughout ing processes occur during the system evolution: the THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

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layer volume changes due to syneresis, and the interface area alters owing to the diffusion gas exchange with the environment. For the system with varying volume V(t) and interface area S(t), one can write the mass balance equation

d (VC + SF) = C dV -~-.

(45)

With regard for (41), from (45), one can deduce the equation for the change in the surfactant concentration:

S(t) V(t)[13(1-FF(~t)~)C(t)-131F(')] '

dC

(46)

whose solution is

[

t

C(t) = exp-13 ~ t

1-

dx

]{

C(O) (47)

T

s(x)

s(;)

r(;) d; dx

At given external actions V(t) and S(t), the set of equations (42) and (47) should be solved simultaneously. In practice, it is easier to simultaneously solve the set of equations (41) and (46). The modulus of elasticity of the adsorption layer in this process is also a function of time and is given by E-

S(~-~) r.

(48)

In view of (27), this yields -1

RTS (dF~(dS~ E ( t ) = 1 ---~F.. k.-d-[J ~,'~ J '

(49)

where F(t) is found by solving the set of (42) and (47). It is noteworthy that the elasticity of a lamella, which separates disperse gas inclusions and is the main topological element of the foam structure, should be considerably higher than the adsorption layer elasticity. The point is that the lamella [32] has a complex multilayer structure (Fig. 2). It comprises two foam cell walls, two planar lamellar micelles [2], and the liquid film, which is a part of the continuous phase of the entire foam volume. Altogether, the lamella is composed of six parallel adsorption layers. Therefore, its modulus of elasticity should be about six times larger than that of the simple adsorption layer. INTERNAL HYDRODYNAMICS OF FOAMS, HYDROCONDUCTION, AND SYNERESIS Foam is a multiphase disperse system with an internal structure. Its disperse phase consists of gas bubbles, each encapsulated in the multilayer elastic wall produced from several adsorbed surfactant layers, with

these bubbles being immersed in the continuous dispersion phase. The walls are in close contact with one another, undergo local deformation, and form a structure with an anisotropic distribution of the dispersion liquid around each of the walls. As distinguished from two-phase bubble liquids, aerosols, and emulsions, foam has at least three phases. Along with the gas and the free dispersion liquid, there exists so-called skeleton phase, which comprises the adsorbed surfactant layers and the liquid bound by them (in encapsulating walls). 3 The volumetric concentration of the skeleton phase is extremely low, even compared with that of the free liquid. Nevertheless, this phase determines the individuality, the structure, and the rheological properties of foam. The skeleton phase is the frame of reference with respect to which, under the action of external force fields and internal inhomogeneities, the diffusion flow of the gas and the hydrodynamic flow of the free liquid can occur. Simultaneously, elements of the skeleton phase can undergo deformations, move relative to one another, and exchange mass with the other phases (by evaporation and condensation of the solvent, and adsorption and desorption of the surfactant). The evolution of the foam system, which consists in the spatial redistribution of phases and extensive properties, proceeds in regions with very intricate geometrical and topological structures. Complete examination of the problems of transfer in the phases with all necessary boundary conditions, though, is extremely difficult to perform, would give additional information on the fields of the microparameters of the system. For practical purposes, it is sufficient to know the averaged parameters and fluxes described within the framework of the mechanics of heterogeneous media [54, 55]. Internal hydrodynamics of foams. The most topical question in studying foam systems is the problem of spatial redistribution of the liquid phase under the action of external fields and internal inhomogeneities. The drainage of liquid from foam under gravity is called syneresis. However, capillary effects, produced mainly by the capillary rarefaction gradient, have also been regarded as the cause of syneresis. The topicality of this phenomenon has already been emphasized in [29], but in that work, only the steady liquid flow through the foam layer has been considered, and the capillary rarefaction gradient has correctly been supposed to be zero. The main role of the capillary rarefaction gradient in the evolution of the foam layer during syneresis has also been stressed in [41, 56, 57]. Indeed, as the liquid flows, the foam channels closed from above become thicker below, thus producing an increasing counteracting gravitational force, which suppresses the drainage until equilibrium is established [6]. This effect is only possible in closed deformable 3 This statementis not generallyaccepted but represents our opinion.

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channels with a negative curvature, which are inherent to foam. In terms of [58], the capillary rarefaction is a compression characteristic of foam, which governs the elastic resistance of foam to deformation during liquid redistribution. The theory of syneresis has been advanced in [11, 28, 59--62]. In that work, which adopted a nonequilibrium thermodynamic approach [14, 15], the following expression has been proposed [28] for the local flux density q of the liquid-phase volume content Vt (an extensive quantity), which is the reciprocal of the foam ratio K: q = H [ ( p / - pg)g + gradAPt].

-2

2

4.64 x 10 as

9 g(K - 1)2Kmin'

(51)

for the high-ratio polyhedral foam [63], H = 7"44x 10-3a'~(K- 1)2 [ 4 [aK 4

3Vt Ot

(52)

~/gmin"

Here, K = 1/Vl, and Vl is the liquid-phase volume content. The factor ~ is introduced to account for the polydispersity, since the hydroconductivity of the polydisperse polyhedral foams is known [65] to be 1.52 times lower than that of monodisperse ones. Generalized syneresis equation. Along with the liquid flow relative to the skeleton phase, there is also translational transfer VtU of the foam (or its skeleton phase) as a whole, which is described by the field of the local velocity U. In this case, the law of conservation of the liquid mass has the form

V ) V 1 ~- V . [S(VI)VVI]

-I-(U.

dH

(53)

Pg)-7~-(g " VVl) aVl

- (P/-

- ( P t - P s ) H ( V I ) V ' g - VtV" U, where

(54)

7S(Vt) = -H dAPt

(50)

Here, the capillary rarefaction APt is found from formulas (10) and (15). The kinetic coefficient H is called the hydroconductivity and has been calculated within the polyhedral foam model [28, 61]. Generally, H is a tensor, which is anisotropic in the near-surface layers; however, the isotropic approximation is commonly applied, in which H is considered a scalar. In [1, 6, 28], various expressions for H have been put forward and refined. For example, different approaches to computing the hydroconduction coefficient have been analyzed in [63]. For the spherical and cellular foam structures, the assumption of liquid filtration through a porous layer within the framework of the Kozeny-Carman model has been used; and for the polyhedral foam structure, the Lemlich-Poiseuille models of channel hydroconduction have been usually employed. Currently, the basic recommended, experimentally checked [64] expressions for the hydroconductivity are the following: for the polydisperse cellular foam [6], H =

and enables one to obtain the following generalized syneresis equation in the laboratory frame of reference:

dVt is the syneresis coefficient [60]. The left side of equation (53) is the substantial derivative of the liquid volume content in foam. The last term of the right side is significant only if the volume compressibility of foam is taken into account. The next to last term is nonzero when the mass force is coordinate-dependent, e.g., during syneresis in centrifugal fields. During gravity syneresis, this term is zero. For the polyhedral foam, the coefficients of hydroconduction and syneresis have been calculated in [61]: .^-3a

2

. 2

H = 3.3XLO - - v t, g

(55)

= 9.5 x 10-4~--~,~/.

(56)

Note that formula (52) overestimates the hydroconductivity more than twofold in comparison with formula (55), since in deriving the latter, the stagnancy of the walls of the Plateau-Gibbs channels has been assumed, which has been experimentally corroborated in [1, 64] for the high-ratio polydisperse foams. Formula (52) overestimates H for the monodisperse foams due to the presence of the empirical factor ~ introduced to account for the polydispersity. As the boundary condition for equation (53) at the surfaces that bound the foam space, the normal component of the liquid volume flux density q is usually taken. This value is nonlinearly related by formulas (50) and (53)-(56) to the values Vls and dVt

d~s

at the

region boundary. At the boundary between the foam and the liquid space, one can also specify the local foam ratio, which is equal to the spherical foam ratio Kmin; and at the interface with a porous filter, one can preset the liquid volume content ensured by this filter [61]:

0Vt 0t + V ' ( q + V t U ) = 0 THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

(Y2

Vts = 0"33a2(p f - pg)2, Vol. 34

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220

Fig. 3. Macrorheologicalmodelof the foambehavior. where ( P f - Pg) is the pressure drop across the filter. Unlike the condition imposed on q, these conditions are linear. Some analytical solutions of the syneresis equation. In [62], equation (53) has been solved for vertical

steady (OVt/Ot - 0) stagnant (U = 0) nonflow (q - 0) foam column. At the boundary with the liquid (at the section Z = 0), the condition that the foam ratio is minimum (Kmin)has been imposed. The quadratic distribution of the foam ratio along the height has been derived: 2

In [62], the solution has been obtained also to the problem of steady (OVllOt - 0) barosyneresis (g = 0) in the stagnant (U -- 0) flow (q # 0) foam column that borders a filter (Vls = const), and the foam ratio distribution over the foam layer in a centrifuge has been determined. By analyzing the equation of syneresis in centrifugal fields, the theory has been developed [66, 671 for a centrifugal plate-type foam breaker, which is a stack of conical plates rotating about a common axis. For the liquid content averaged over the width of the gap between the plates, the following distribution along the plate generatrix length has been obtained: 1

1

3

3 _-1/2

V, = [ - ~ t o + ~ A ( x o - x )J

.

(59)

The dimensionless parameter A characterizes the operating conditions and the geometry of the plates: 2

2 3

A = Tfat~ "2 -sm )'cosT. 18 vQ

(60)

The unsteady syneresis problems are of considerable interest. For example, the unsteady syneresis in a stagnant (U -- 0) foam layer at a constant body force is described by a complex nonlinear parabolic equation. Some self-similar solutions and travelling-wave solutions have been found [68] for particular forms of this equation. For the one-dimensional unsteady barosyneresis (g = 0), equation (53) takes the form

~VI Ot =

lx

(vl/2OVl~

10-3Go ~ -~-b'-Z~, t ~-~j.

(61)

Some known exact solutions of this equation, including wave and self-similar solutions, and also blow-up solutions [69], have been given in [70, 71]. Note, however, that these exact solutions exist only under certain initial and boundary conditions (which follow from the very form of a solution), and are therefore difficult to interpret. If the boundary conditions imposed are natural, numerical or approximate analysis methods should be applied. The solution derived numerically in [68] describes the propagation of a capillary suction wave, and the solution representing the centrifugal syneresis has been obtained [67] by the method of a small parameter. RHEOLOGICAL PROPERTIES OF FOAMS Macrorheological model of foam. Foam is a complex rheological body. To a certain extent, it possesses all of the three main macrorheological properties, namely, elasticity, plasticity, and viscosity. These properties are conveniently described by such mechanical models [72] as Hooke's body (elastic spring); SaintVenant's body (bar lying on solid surface), which models dry friction; and Newton's body (piston in a vessel with viscous liquid), respectively. Various combinations of these elementary rheological models in parallel and in series allow one to describe quite (rheologically) complex cases. The key point in the rheological classification of materials is the question of whether or not a material has a preferred shape or a natural state [73]. If it has, this material is said to be solidlike; if it has not, it is spoken of as liquidlike [74]. The simplest model of viscoelastic solidlike material is Kelvin's [72] (or Voigt's [74]) body composed of Hooke's and Newton's bodies connected in parallel. It describes the delayed deformation and the elastic aftereffect. The classical model of viscoelastic liquidlike material is Maxwell's body [72], made up of Hooke's and Newton's bodies connected in series, which represents the stress relaxation. It is known [1,75] that, for foam, there exists an ultimate shear stress x0 that separates the types of theological behavior of foam: at x < x0, foam is solidlike; and at x _>x0, it is liquidlike. That is why the mechanical model of foam should include Saint-Venant's body. Figure 3 depicts one of the simplest macrorheological models of a foam body. Shear modulus. One of the most important elastic characteristics of foam is the shear modulus G, which is the coefficient of proportionality between the tangential stress 'r in the solidlike foam ('r < %) and the shear strain 8. This quantity has been theoretically obtained by Deryagin [16]:

4 G = ]~oE.

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When estimating the specific interface area e for the monodisperse foam by formula (4), one can rewrite relation (62) as G = 0.8 o K - 1 a K

(63)

Without assuming that strains are small, Khan [76] has assessed theoretically the shear modulus for the twodimensional dry foams whose cells have the shape of a regular hexagon: G =

2~ 1 ',/~ a (82 + 4) 1/2"

(64)

The Princen-Kiss semiempirical formula [77] seems to give the most reliable estimate: G = 1.769(0.288 - 1 ) ( - ~ )

1/3 m a~ a

(65)

Effective viscosity of viscoelastic foam. Foam exhibits viscous properties even when it remains solidlike. The point is that liquid elements of the foam cell are also affected by shear deformation. The energy dissipation rate in these elements depends on the frequency of the applied action or, what is the same, on the deformation rate dS/dt. The effective viscosity, which determines the dissipative properties of such a medium, has been theoretically found in [78] for the two-dimensional foam model: ~eff = 6.7~t(Ca) -1/3,

Ca = ~ad~

dt

is the capillary number. The energy dissipation rate proves to be proportional to com3, where co is the frequency of the action. Thus, at high co, the viscosity of foam is substantial even if foam is solidlike. Ultimate shear stress. An important characteristic that governs the rheological behavior of foam is the ultimate shear stress %. This quantity has been computed in [79] for the two-dimensional foam model: 4/3 X0= 1.28~(~-~--~) ( 0 . 2 8 8 - 1 ) . (67, In [80], experimentation on a setup with a tube (14 mm in diameter) threaded inside to prevent the slipping of the foam column in the plug-flow mode has led to the following relationship for z0: 0.35

4"~0pta2 = 0 . 6 1 K 0 . , 8 ( 2 0 P 2 / a / ~ ~'2

<" la

/

t

The accuracy of this formula is 10%. In the experiments, the foam ratio K was varied within the range 36--322; the foam dispersity a, within the range 0.175-0.50 mm; and the solution viscosity It, within the range 1.5-10.5 Pa s. In all the instances, a 0.4% sulfonol solution in distilled water that also contained 5.2 or 30 wt % glycerol to vary the viscosity was used as a surfactant solution. Pl and ~ varied slightly. In [19, 81], a new approach to studying the rheological properties of foam has been demonstrated by the example of the pressure flow of structured foams in channels. In particular, a number of hypothetical models of the strained and stressed states of foam structures in shear have been put forward. It has been noted that the elastic component, which is inherent to high-ratio foams, imparts thixotropic (time-dependent) properties to such foams and prevents them form being described by a single flow curve. Application of the shearing force redistributes the foam ratio over the channel section owing to the Weissenberg effect. In the core, which is drier, elastic constraints predominate, and the core moves in the plug flow. At the same time, in moister peripheral regions, the capillary rarefaction, which strengthens the foam, vanishes, and the shear sliding of layers becomes possible. In this case, one can calculate such important characteristics of moving foam as the foam ratio distribution over the section and the ultimate shear stress, which depends on the initial uniform foam ratio K0. STABILITY, EVOLUTION, AND RUPTURE OF FOAMS

(66)

where

/

IX

J

(68)

221

Foam lifetime. The metastability of the foam structure causes its continuous evolution under the action of external and internal forces. The main phenomena governing the evolution of the spherical foam are gravity, centrifugation, and barosyneresis, and also the diffusion redistribution of the gas phase. For the mature cellular and polydisperse foams, the lifetime is determined by the thinning of the films because of the liquid drainage therefrom into the Plateau-Gibbs channels under the action of the capillary forces (see (22)). This film thinning is very, but not infinitely, long, and it is this stage that determines the foam lifetime, which can range from several seconds to several days [3, 4]. Wedging pressure. As noted above, the factor that counteracts the thinning of the foam film at the last stage of the foam evolution is the wedging pressure H(h), which is substantial for films whose thickness is in the range 10-9 < h < 10-7 m. The wedging pressure is usually considered to have three components [ 12, 30, 31 ]: H(h) = H,,(h) + He(h) + Hs(h).

(69)

The molecular component I-I,,(h) of the wedging pressure is negative and describes the long-range van der Waals attraction between the boundary layers of the foam film. This attraction is due to the London-van der

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KAZENIN et al. been analyzed in [83-85]. The stability of ultrathin films is controlled by the interplay of the capillary forces and the forces that govern the molecular component of the wedging pressure. Instability can emerge if dI-l/dh > 0, and if the stabilizing capillary forces, which arise when the surface is perturbed, are weak. Linear analysis into the stability of temporal [83, 85] and spatiotemporal [86] perturbations has revealed the critical conditions for losing film stability, and weakly nonlinear and nonlinear analyses [87, 88] have disclosed the wavelength range of unstable perturbations and the wave whose amplitude rises most rapidly.

11

I••

10-7

II

'

III

I

h//.~2 m

'

Fig. 4. Isothermof the wedgingpressurein a foamfilm:(I) ordinary black films, (II) Newtonblack films, and (III) unstablefoamfilms. Waals forces and, in different ranges of h, is represented by various approximate analytical expressions [12, 30, 31], in which the lag of the electromagnetic action is either taken into account or neglected. The ion-electrostatic component Fie(h) of the wedging pressure is negative (long-range repulsion). This component is due to the interaction between the overlapping electrical double layers at the film surface and is essentially dependent on the type of the surfactant and on the ion composition of the liquid in the film. The interaction between lira(h) and Fie(h), and their effect on the film stability is described by the Deryagin-Landau-Fervay-Overbeck theory [30]. The structural component I-Is(h) of the wedging pressure is positive and is governed by the short-range elastic interaction between the closed boundary layers of the film. Critical thickness of a foam film. The sum of the components of the wedging pressure in the range 10 -9 < h <__10 -7 m is a nonmonotonic function of the film thickness. In this range, there is one region (or two regions, depending on the type and the concentration of the surfactant and the electrolyte in the dispersion liquid) in which dI-l/dh < 0.

(70)

This relation is the strictest condition of the hydrodynamic stability of a liquid film [82]. Therefore, it is considered that the instability and the rupture of a liquid film can occur when the film thickness attains, during liquid drainage, a certain critical value her such that d..~RI dh Ih=

= 0.

(71)

her

Destabilization mechanism. The mechanism of hydrodynamic destabilization of thin liquid films has

Black films. In some cases, it has been discovered experimentally that enhancement of perturbations does not bring about film rupture, but the emergence of metastable black spots on the film surface. These are the so-called Newton black films, which are several nanometers in thickness. Their black color in the reflected light is a purely optical effect that is caused by the fact that the thickness of these films is much smaller than a quarter of the visible-light wavelength. The Newton black films can arise if the dependence of the wedging pressure 11 on h has one more interval (at very small h) where dI-I/dh < 0 (Fig. 4); i.e., if at small h, the structural component 1-Is(h) of the wedging pressure dominates over its molecular component Flm(h), which is possible only at the maximum filling of the adsorbed surfactant layers [1]. Strictly speaking, the film can appear black in the reflected light as early as at the first stage of the hydrodynamic stability at the thicknesses on the order of several tens of nanometers, when dl-l/dh < 0 because of the fact that, in this range, the electrostatic component Iie(h ) of the wedging pressure predominates. The Newton black films are metastable. They are ruptured after a certain threshold of fluctuation perturbations has been exceeded. The hole mechanism of rupture of the Newton black films has been described in [1]. Rupture of foams. The rupture of foams (as the final stage of their evolution) is related to the thinning and the rupture of the foam films. In [82], as a foam rupture criterion, the concept of critical film thickness has been advanced, specifically, the polyhedral foam commences to rupture as soon as the thickness of the films that are the faces of foam cells (polyhedra) reaches a critical value. Within the framework of the channel model of foams, e.g., under the assumption that, late in the foam column evolution, virtually all the liquid is within the Plateau-Gibbs channels, and using (21) and (23), one can express the liquid-phase volume content V t in the foam (which is the reciprocal of the foam ratio K) in terms of the wedging pressure, which depends on the film thickness. Hence, assuming that a foam film collapses after its thickness has attained a critical thickness her, the critical volumetric liquid-

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F O A M S AS SPECIFIC G A S - L I Q U I D T E C H N O L O G I C A L M E D I A

phase content of the foam before its rupture is 2 Vt r = 0.33

2

2

(72)

"

a Fl (her) A foam column can collapse at any part of its space, but this collapse is most likely to begin at its upper (driest) part at the boundary with the atmosphere. Rupture of a vertical foam column. Following [82], let us consider the steady problem of rupture of a vertical foam column in gravity syneresis. Let the axis Z be directed vertically along the foam column axis. At the section Z = 0 (at the boundary with the solution), fresh spherical (Vt = 1 / 4) foam enters the column and moves upward at the translational velocity U. Toward it from the top and downward along the Plateau-Gibbs channels, the liquid phase flows under gravity (via the hydroconduction mechanism). Since the problem is steady, the total liquid flux at every section of the foam column is zero. Let us write the generalized syneresis equation (53) for this case in the form

dVt UVt-ApgH(Vt) dZ =

(73) '

where Ap = Pt- Pg is the difference between the densities of the phases, and H(Vt) and ;S(Vt) are the local hydroconductivities and syneresis coefficients, respectively, which are given by formulas (55) and (56). The boundary conditions for equation (73) are Vt = 1/4 at Z = O; (74)

Vt = V~r at Z = Z 0.

The first of boundary conditions (74) is used for finding the constant of integration in solving equation (73), and the second is employed for determining the steady flow foam column height Zo. The exact solution of problem (73), (74) is

VI(Z) = A U tanha(. 2"42 ~, i Pg \,,/A1ApgUZ)

(75)

where A 1 = 3.3 x 10-3a2/l.t, A 2 = 9.5 x 10-4(ya/]/. The height Z0 is Zo =

2A 2

arctanh [

A,f~IApgU

U

cr'

(76)

~]A,ApgV l

Additional conditions at the upper boundary of the foam column in unsteady syneresis have been discussed [82] as well.

223

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project nos. 00-02-18033 and 00-03-32055. NOTATION a--radius of equivalent bubble, m; as--surface-averaged bubble radius, m; --average radius of bubbles in foam, m; B--number of edges (Plateau-Gibbs channels) of a foam cell; b--Plateau-Gibbs channel length, m; C--partial surfactant density in solution, kg/m3; D---diffusion coefficient, m2/s; Ds--surface diffusion coefficient, m2/s; E--modulus of elasticity of an adsorption layer, N/m2; E~ Gibbs modulus of elasticity, N/m2; Eu---Marangoni modulus of elasticity, N/m2; F--number of faces (films) of a foam cell; j--flux of substance toward the surface, kg/(m2 s); f(a)--bubble size distribution function; G--shear modulus, N/mZ; g--vector of the acceleration of gravity, m/s2; H--kinetic hydroconductivity, m2/(Pa s); h--film thickness, m; her--Critical film thickness, m; K--foam ratio; k--Boltzmann constant, J/K; M--molecular weight of a surfactant; N--number of nodes of a foam cell; number of moles of surfactant in the system; Ns--number of moles of a surfactant in an adsorption layer; n--number of characteristic sizes of bubbles in foam; P~--atmospheric pressure, N/m2; Pg--pressure in the gas phase of foam, N/m2; Pt--pressure in the liquid phase of foam, N/m2; Q--volumetric flow rate of foam through the gap between plates, m3/s; q--local volumetric liquid content flux density; Rb - Plateau-Gibbs channel radius, m; R,,--radius of curvature of nodal menisci, m; R--universal gas constant, J/(mol K); rl--inlet radius of a foam breaker, m; S--surface area, m2;

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KAZENIN et al.

Sb---cross-sectional area of a Plateau-Gibbs channel, m2; T--temperature, K; tmtime, s; ta--adsorption relaxation time, s; U--translational velocity of foam in a column, m/s; U---local velocity of foam (or its skeleton phase), m/s; Vmvolume, m3; Vr--mole fraction of the liquid phase; V/0--1iquid content of foam at the inlet of a foam breaker; v--liquid velocity, m/s; vs--surface liquid velocity, m/s; x---coordinate along the plate generatrix, normalized to the inlet radius; Z--vertical coordinate, m; Z0--steady-flow foam-column height, m; z--coordinate counted from the surface deep into the liquid, m; o~--distribution function parameter; [~---kinetic adsorption coefficient, s-l; lit--kinetic desorption coefficient, m/s; F--surfactant adsorption, kg/m2; F**----surfactant adsorption at the total occupation of an adsorption layer, kg/m2; y--angle between the generatrix and axis of rotation, deg; i~----shear strain; emspecific interface area of foam, m-t; ~c--surface curvature, m-t; kt---dynamic viscosity of the liquid, kg/(m s); v--kinematic viscosity of the liquid, m2/s; F,-- potential of interaction of a surfactant molecule with the film surface; ~---coordinate along the normal to the surface, m; Fl--wedging pressure, N/m2; p/--liquid density, kg/m3; ps--gas density, kg/m3; G~surface tension, N/m2; G0msurface tension of a pure solvent, N/m2; x--shear stress, N/m2; x0--uitimate shear stress, N/m2; O~--volumetdc gas content of foam; W--free energy, J; to---angular velocity of the plates of a foam breaker; frequency, s-t.

REFERENCES 1. Kruglyakov, P.M. and Ekserova, D.R., Pena i pennye plenki (Foam and Foam Films), Moscow: Khimiya, 1990. 2. Rusanov, A.I., Mitselloobrazovanie v rastvorakh poverkhnostno-aktivnykh veshchestv (Micelle Formation in Surfactant Solutions), St. Petersburg: Khimiya, 1992. 3. De Vris, K., Foam Stability, Amsterdam: Center, 1957. 4. Tikhomirov, V.K., Peny (Foams), Moscow: Khimiya, 1983. 5. Manegold, E., Schaum, Heidelberg: Strassenbau, Chemie und Technik, 1953. 6. Kann, K.B., Kapillyarnaya gidrodinamika pen (Capillary Fluid Dynamics of Foams), Novosibirsk: Nauka, 1989. 7. Bikerman, J.J., Foams, New York: Springer, 1973. 8. Berkman, S. and Egloff, G., Emulsions and Foams, New York: Reinhold, 1941. 9. Schwarz, H.W., Rearrangements in Polyhedric Foam, Recl. Trav. Chim. Pays-Bas, 1965, vol. 84, no. 5, p. 771. 10. Sheludko, A.D., Kolloidnaya khimiya (Colloid Chemistry), Moscow: Mir, 1984. 11. Krotov,V.V.,Theory of Syneresis of Foams and Concentrated Emulsions. 1. Local Multiplicity of Polyhedral Disperse Systems, Kolloidn. Zh., 1980, vol. 42, no. 6, p. 1081. 12. Foams: Fundamentals and Applications, Prud'homme, R.K. and Khan, S.A., Eds., New York: Marcel Dekker, 1995. 13. Licinio, P. and Figneizedo, J.M., Steady Foam State, Europhys. Lett., 1996, vol. 36, no. 3, p. 173. 14. Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, Springfield (I11): Thomas, 1955. Translated under the title Vvedenie v termodinamiku neobratimykh protsessov, Moscow: Inostrannaya Literatura, 1960. 15. Nicolis, G. and Prigogine, I., Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, New York: Wiley, 1977. Translated under the title Samoorganizatsiya v neravnovesnykh sistemakh, Moscow: Mir, 1979. 16. Deryagin, B.V., Elastic Properties of Foams, Zh. Fiz. Khim., 1931, vol. 2, no. 6, p. 745. 17. Princen, H.M., Osmotic Pressure of Foams and Highly Concentrated Emulsions. 1. Theoretical Consideration, Langmuir, 1986, vol. 2, no. 4, p. 519. 18. Emelichev, V.A. and Kovalev, M.M., Mnogogranniki, grafy, optimizatsiya (Polyhedra, Graphs, and Optimization), Moscow: Nauka, 1981. 19. Gotovtsev, V.M., Viscoelastic Model for the Plug Flow of a Foam in a Cylindrical Channel, Teor. Osn. Khim. Tekhnol., 1996, vol. 30, no. 6, p. 576. 20. Sedov, L.I., Mekhanika sploshnoi sredy (Continuum Mechanics), Moscow: Nauka, 1973. 21. Adamson, A., The Physical Chemistry of Surfaces, New York: Wiley, 1976. Translated under the title Fizicheskaya khimiya poverkhnostei, Moscow: Mir, 1979. 22. Ross, S. and Prest, H.E, On the Morphology of Bubble Clusters and Polyhedral Foams, Colloids Surfi., 1986, vol. 21, p. 179.

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