Modeling of the Generation and Collapse of Aqueous Foams

Langmuir 1996, 12, 3089-3099

3089

Modeling of the Generation and Collapse of Aqueous Foams Ashok Bhakta and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received November 29, 1995. In Final Form: March 15, 1996X Drainage of the continuous phase liquid from a foam plays a pivotal role in determining its stability to collapse. A theoretical model for the drainage of the liquid during generation of a foam by bubbling and its subsequent collapse is presented. The model accounts for drainage in the films as well as in the plateau border channels. Drainage of the films is modeled using Reynold’s equation for the flow between parallel flat circular disks under the influence of van der Waals, electrical double-layer, and plateau border suction forces with film rupture occurring when the film thickness attains a certain critical value. Since this model accounts for collapse at the foam/gas interface during generation, it is able to predict the steady state height attained by pneumatic foams. The model is also able to predict the establishment of a drainage equilibrium when the opposing forces of gravity and plateau border suction gradient balance each other. The effect of various parameters such as superficial gas velocity, electrolyte concentration, and bubble size on the steady state height and collapse half-life (time required for the foam to collapse to half the steady state height) is examined. It is shown that, for a given system, there is an upper limit on the superficial gas velocity beyond which a steady state height will not be attained. With increasing salt concentration, the stability of a foam first increases, attains a maximum, and then decreases. The steady state height and the collapse half-life decrease with a decrease in bubble size due to an increase in the capillary pressure. It is shown that, for a given system, there is an upper limit to the salt concentration and a lower limit to the bubble size beyond which no drainage equilibrium is possible and complete collapse will occur. It is also shown that plots of dimensionless foam height versus dimensionless time practically coincide for most of the period of collapse, a feature which is consistent with some experimental results.

1. Introduction Foams have been used over the years for a variety of processes such as removal of radioactive impurities,1 separation of proteins from dilute solutions,2 enhanced oil recovery, and fire fighting.3 Foams are also a nuisance in refineries and reactors where the emphasis is on the prevention of foam formation and its quick destruction.4 A fundamental understanding of the stability of foams is therefore essential to gain some control over these extremely complex systems. Foams consist of polyhedral bubbles separated by thin liquid films or lamellae. Three adjacent films meet at a plateau border (PB) channel, which resembles a duct of approximately triangular cross section, similar to that formed between three cylinders in mutual contact. The curvature of the PB walls gives rise to a pressure difference in the liquid between the films and the PB’s causing flow out of the films into the PB channels. Thus, the films become thinner with time and finally collapse. On the other hand, the liquid in the PB channels (which form a complex interconnected network) drains out of the foam under the influence of gravity. Thus, at least in the short term (before disproportionation effects, such as Ostwald ripening, take over) the stability of a foam is determined primarily by two factors: the gravity driven flow in the PB channels and the capillary flow in the films. It is clear from the above discussion that drainage in the films and PB channels plays a crucial role in the collapse of foams. Several theoretical models5-13 for foam drainage have * Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, May 15, 1996. (1) Adsorptive Bubble Separation Techniques,;Lemlich, R., Ed.; Academic Press: New York, 1972. (2) Brown, L.; Narsimhan, G.; Wankat, P. C. Biotechnol. Bioeng. 1990, 36, 947. (3) Aubert, J. H.; Kraynik, A. M.; Rand, P. B. Sci. Am. 1986, 254 (5), 74. (4) Kouloheris, A. P. Chem. Eng. 1987, Oct. 26, 88. (5) Krotov, V. V. Colloid J. USSR 1981, 43, 33 (English Translation).

S0743-7463(95)01092-4 CCC: $12.00

appeared in the literature. Each of these models however suffers from several drawbacks. Only a few of them5,6,12,13 account for the important effect of the plateau border suction gradient, first recognized by Krotov,5 which under certain conditions results in a drainage equilibrium when it balances gravity. Narsimhan6 accounted for this effect in his drainage model for pneumatic foams. However, he ignored the inherently unsteady nature of foam formation by bubbling and used a quasi-steady-state approximation to compute the profile of the liquid fraction in the freshly generated foam. Bhakta and Ruckenstein12 used a more appropriate unsteady state model to describe the drainage during foam generation. Another feature that has largely been ignored by most drainage models is the collapse. We have presented a model in an earlier publication13 which accounts for collapse in initially homogeneous foams and concentrated emulsions and provides insights regarding the separation of the two phases at drainage equilibrium. However, it was assumed there that the films are always in drainage equilibrium with the adjacent PB channels. In other words, the assumption was made that the films instantaneously equilibrate with the PB channels and that, as drainage proceeds through the PB channels, they become thinner and finally rupture while always maintaining a quasi-equilibrium. In these circumstances, the time taken for the film to rupture is dependent solely on the drainage through the PB channels. However, in systems of high capillary pressure, the disjoining pressure (6) Narsimhan, G. J. Food Eng. 1991, 14, 139. (7) Bhakta, A.; Khilar, K. C. Langmuir 1991, 7, 1827. (8) Ramani, M. V.; Kumar, R.; Gandhi, K. S. Chem. Eng. Sci. 1993, 48, 455. (9) Haas, P. A.; Johnson, H. F. Ind. Eng. Chem. Fundam. 1967, 6, 225. (10) Jacobi, W. H.; Woodcock, K. E.; Grove, C. S. Ind. Eng. Chem. 1956, 48, 9046. (11) Miles, G. D.; Sheklovsky, L.; Ross, J. J. Phys. Chem. 1945, 49, 93. (12) Bhakta, A. R.; Ruckenstein, E. Langmuir 1995, 11, 1486. (13) Bhakta, A. R.; Ruckenstein, E. Langmuir 1995, 11, 4642.

© 1996 American Chemical Society

3090 Langmuir, Vol. 12, No. 12, 1996

Figure 1. Schematic of foam being generated by bubbling: (a) no collapse; (b) collapse occurs during generation.

will never become as large as the capillary pressure and a film will rupture without ever achieving equilibrium with the PB channels. In these circumstances, the time scale of film drainage becomes important and the earlier model fails. An improvement can therefore be made by accounting for the drainage in the films. In this paper, we present a more complete treatment in which film drainage is modeled using Reynold’s14 equation for radial flow between parallel circular disks. We apply our model to foams produced by bubbling, taking into account the fact that some collapse occurs during foam generation. This feature has been ignored in previous models. Several experimental methods have been devised to evaluate foam stability.15 For relatively short lived foams, a simple method is to monitor the height of the collapsing foam with time. In such experiments, the initial condition of the foam plays an important role in influencing its subsequent collapse behavior and needs to be standardized. For foams produced by bubbling, it has been shown16 that the collapse is well defined when the initial state of the foam is taken as the “equilibrium state” of Bikerman’s classical experiment in which the rate at which gas leaves the foam due to collapse is equal to the rate at which gas enters the foam due to bubbling. Since our model takes into account collapse during generation, we are able to theoretically predict this steady state height attained by the foam. In this paper, we examine the effect of various parameters such as electrolyte concentration, bubble size, and superficial gas velocity on the steady state height and subsequent collapse of the foam. 2. Theoretical Model for Drainage Figure 1 shows a schematic diagram of a typical experimental setup used to produce foam by bubbling. An inert gas is bubbled at a fixed volumetric flow rate through a porous frit into the surfactant solution. Foam is formed and moves up at a rate which depends on the superficial gas velocity. As long as there is no collapse at the top during generation, the motion of the foam/gas interface is determined by the bulk movement of the foam. However, as soon as the thickness of the liquid films at the top decreases to a critical value, collapse starts at the top and a downward component is superimposed on the bulk movement of the foam. As a result, the net upward velocity of the foam/gas interface decreases. The vertical space coordinate ‘z’ is selected to increase in the downward direction, and the plane z ) z1 represents the foam/gas interface at the top. The origin (z ) 0) (see Figure 1) defines the upper boundary of the entire system. In other words, all the gas entering the foam lies between the reference plane (z ) 0) and the foam/liquid interface(z ) z2). Thus, during generation, before collapse starts, the (14) Reynold, O. Philos. Trans. R. Soc. London 1886, A177, 157. (15) Bikerman, J. J. Foams; Springer Verlag: New York, 1973. (16) Iglesias, E.; Anderez, J.; Forgiarini, A.; Salager, J. Colloids Surf., A 1995, 98, 167.

Bhakta and Ruckenstein

reference plane lies at the foam/gas interface and an observer at the origin sees the foam/gas interface to be fixed (z1 ) constant ) 0) and the foam/liquid interface to be moving away (z2 increases). After collapse, the position of the reference plane (z ) 0) is such that all the gas from the collapsed bubbles is contained between z ) 0 and z ) z1. The observer at the origin now sees the foam/gas interface to be moving away (i.e., z1 increases). As in our previous publication,13 coalescence and Ostwald ripening are ignored and the foam is assumed to consist of uniform pentagonal dodecahedra. The dimensions of these polyhedra can be conveniently expressed in terms of the effective bubble radius (R), which is the radius of a sphere with a volume equal to that of a bubble. In the following sections, expressions for the drainage in the individual films and PB channels are presented and the macroscopic equations for the drainage and collapse of bulk foam are derived. 2.1. Drainage in Films. For sufficiently large surfactant concentrations, the walls of the films can be assumed to be immobile and the rate of film thinning can be computed using the Reynolds equation for flow between two circular parallel disks, which is given by

-

dxF 2∆Px3F ) VRe ) dt 3µR2

(1)

F

where xF is the film thickness, t is the time, VRe is the Reynolds velocity, µ is the viscosity of the continuous phase, ∆P is the pressure difference causing the flow, and RF is the radius of the film. Since the films are not actually circular but regular pentagons, the film radius(RF) can be expressed in terms of the bubble radius as

RF ) 0.606R

(2)

and the driving force ∆P, which is a net result of the suction pressure in the adjacent PB channels and the disjoining pressure (Π) in the films, is given as

∆P )

σ -Π rp

(3)

where σ is the interfacial tension and rp is the radius of the plateau border. In the absence of any short range repulsive forces, the disjoining pressure in the film is a result of two forces, viz. the van der Waals attractive forces and the repulsive double-layer forces. The van der Waals force per unit area is given by

ΠVW ) -

Ah 6πx3F

(4)

where Ah is the Hamaker constant and xF is the film thickness. The repulsive pressure (ΠDL) between interacting double layers at moderate potentials is given by17

ΠDL ) 16(6.02 × 1026)ckT[γ2A21 + 2γ4A1(A2 + A31) + γ6(2A1A3 + 8A31A2 + 3A61 + A22)] (5) where for monovalent electrolytes (17) Oshima, H.; Kondo, T. J. Colloid Interface Sci. 1988, 122, 591.

Generation and Collapse of Aqueous Foams

( )

γ ) tanh

eψs ; A1 ) 4kT

1 cosh

( ) ( )

Langmuir, Vol. 12, No. 12, 1996 3091

Since the liquid fractions in the films and PB channels are NnFAFxF and Nnpapl, respectively, the overall liquid fraction () is given by

( ) κxF 2

;

κxF κxF tanh A1 - A31 2 2 ; A3 ) + A2 ) κxF κxF cosh3 4 cosh2 2 2 κxF 2 A1 κxF κxF 3A2 2 1-4 (6) tanh κx 2 2 κxF F 4 cosh2 2 cosh4 2 2 -

[ ( )

( ) ( ) ( )

( )]

and ψs, c, and k refer to the surface potential, ionic strength (mol/L), and Boltzmann constant, respectively. κ is the reciprocal Debye length, which for monovalent electrolytes at 298 K is c0.5/0.304 (nm)-1. 2.2. Flow through a PB Channel. The crosssectional area of a PB channel (ap) can be expressed in terms of the radius of curvature of the walls (rp) and the film thickness (xF) as18

(0.322rp + 1.732xF)2 - 2.721x2F ap ) 0.644

cvap

(Fg - ∂p∂z)

20x3µ

(8)

In eq 8, z refers to the vertical space coordinate (see Figure 1), which increases in the downward direction, and g, F, µ, and p refer to the gravitational acceleration, density, viscosity, and pressure in the continuous phase. The factor cv accounts for the effect of finite surface viscosity (µs) and has been computed by Desai and Kumar19 as a function of the inverse of the dimensionless surface viscosity (γs ) 0.4387µxap/µs). In our calculations, cv was taken as unity, as the walls are assumed to be immobile. Now, if pgas is the pressure in the dispersed gas phase, one has

pgas - p ) σ/rp

u)

cvap

(

20x3µ

)

( ))

∂ 1 Fg + σ ∂z rp

(18) Leonard, R. A.; Lemlich, R. AIChE J. 1965, 11, 18. (19) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982, 37, 1361. (20) Narsimhan, G.; Ruckenstein, E. Langmuir 1986, 2, 230.

(13)

qPB )

3 (NapnpuR) 15

(14)

The PB channels receive liquid from the channels above and drain liquid into the channels below. At the same time, they receive liquid from the surrounding films. Let the rate of decrease of the volume fraction in the films be denoted

QF ) -

∂(NnFAFxF) ∂t

(15)

where AF is the area of the film given by AF ) πRF2. Since all the liquid lost from the films drains into the surrounding PB channels, the volumetric flow rate of liquid from the films into the PB channels in an infinitesimal element of volume A∆z is given by QFA∆z, where A is the cross-sectional area of the foam. Since the volume of continuous phase liquid contained in the PB channels in this infinitesimal element of volume A∆z is NnpaplA∆z, an unsteady state mass balance over the PB channels can be written as

NnpaplA∆z|t+∆t - NnpaplA∆z|t ) [qPB|z qPB|z+∆z]A∆t + QFA∆z∆t (16) Consequently

∂qPB ∂ (Nnpapl) ) + QF ∂t ∂z

(17)

∂ 3 ∂(NnpapuR) (Nnpapl) ) + QF ∂t 15 ∂z

(18)

and

(10)

The terms in the parentheses represent the driving forces due to gravity (Fg) and the gradient of PB suction (σ(∂/ ∂z)(1/rp)). Note that these forces oppose each other when the latter is negative, i.e. when the liquid fraction is smaller at the top. Other needed quantities include np (the number of PB channels per bubble), nF (the number of films per bubble), l (the length of a PB channel), V (the volume of a bubble), R (the radius of a sphere of volume V), and N (the number of bubbles per unit volume). For pentagonal dodecahedra, we have the following relations:6,20

l ) 0.816R; np ) 10; nF ) 6

npapl + nFAFxF V + npapl + nFAFxF

2.3. Conservation Equations. For PB channels oriented randomly in space, the volumetric flow rate per unit area (qPB) due to gravity drainage through the plateau borders is given by6,20-22

(9)

Using eq 9, eq 8 can be rewritten as

(12)

Since N ) (1 - )/V, can be expressed in terms of ap and xF as

(7)

An expression for the average velocity (u) of the liquid in a vertical PB channel has been derived for a triangular PB cross section and is given by19

u)

) Nnpapl + NnFAFxF

(11)

Note that, in eqs 16-18, ap is the “average” cross-sectional area of a plateau border at a given z. The use of an average is a necessary feature of macroscopic models in which balances are written over an infinitesimal element containing a large number of bubbles. Using eq 7, ap can be expressed in terms of xF and rp and all terms in eq 18 can be obtained as a function of these two independent variables which are then computed as a function of time and space by simultaneously solving eqs 1 and 18. Equation 18 being second order in space in terms of rp, two boundary conditions are needed for rp: one each at the top and bottom of the foam. (21) Narsimhan, G.; Ruckenstein, E. Langmuir, 1986, 2, 494. (22) Narsimhan, G.; Ruckenstein, E. In Foams: Theory, Measurements and Applications; Prud’homme, R. K., Khan, S. A., Eds.; Marcell Dekker: New York, 1995; p 99.

3092 Langmuir, Vol. 12, No. 12, 1996

Figure 2. Disjoining pressure as a function of film thickness.

2.4. Boundary Conditions at the Foam/Gas Interface. The boundary condition at the top (z ) z1) is inherently linked to the movement of the foam/gas interface, which is triggered by the rupture of films. The mechanism of film rupture and its influence on foam collapse must therefore be discussed first. 2.4.1. Film Rupture. Rupture of a film occurs due to the unbounded growth23 of random fluctuations on the film surface. Whether a surface wave is damped or undergoes catastrophic growth is however determined by the shape of the disjoining pressure isotherm. If the disjoining force which opposes film thinning increases in response to the local thinning, the wave is damped and no rupture occurs. On the other hand, if the disjoining pressure decreases, the local thinning is accelerated and leads to rupture. Film rupture can therefore occur only when dΠ/dxF is positive. In other words, a thin film with a disjoining pressure isotherm as in Figure 2 will rupture only for xF < xFm. Since a film with an initial thickness xF > xFm can arrive at thicknesses less than xFm only when the capillary pressure in the plateau borders exceeds the maximum disjoining pressure, it can rupture only if the capillary pressure exceeds the maximum disjoining pressure (Πmax). The actual rupture of the film, however, occurs only when the waves on the two surfaces touch. When this occurs, the mean film thickness is smaller than xFm because as an unstable wave grows, there is a decrease in the mean thickness because of the drainage driven by the capillary pressure (σ/rp - Π). Therefore, for a given system, the mean film thickness at the moment of rupture (henceforth referred to as the critical thickness of film rupture xFc) is a unique function of the radii (rp) of the surrounding PB channels. Figure 3 shows a typical curve (curve 1) of xFc versus the PB radius rp obtained using stability theory.23-25 We shall refer to this curve as the criticality curve, and for purposes of discussion, the relation between xFc and rp is written as xFc ) F(rp). (23) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23. (24) Scheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391. (25) Ivanov, I. B.; Radoev, B.; Manev, E.; Scheludko, A. Trans. Faraday Soc. 1970, 66, 1262. (26) Kann, K. B. Colloid J. USSR 1979, 714 (English Translation).


Figure 3. Critical thickness of film rupture (xFc) versus rp. Curve 1 is obtained using linear stability theory with the following parameters: F ) 1000 kg/m3, µ ) 1 cP, ψs ) 19 mV, T ) 298 K, Ah ) 3.7 × 10-20 J, c ) 0.001 M, Πmax ) 283.80 N/m2, and σ ) 40 mN/m. Curve 2 is an arbitrary curve which is consistent with equilibrium.

It was our intention to incorporate the results of the theories of single film rupture into the model for the collapse of bulk foam. However, it was found that the shape of the criticality curve obtained using linear stability theory was inconsistent with the mechanical drainage equilibrium which must necessarily be established. This problem and its resolution are discussed in the following sections. 2.4.2. Movement of the Foam/Gas Interface. When the film thickness and the PB radius at the top are such that they correspond to a point on the criticality curve of Figure 3, i.e. when xF|z1 ) F(rp|z1), the films at the top rupture and the foam starts to collapse. The rate at which the collapsing front moves dz1/dt is given by (see Appendix)

VRe|z1 + dz1 ) dt ∂xF ∂z

|

z1 -

|

dF L drp

z1

|

dF ∂rp drp ∂z

(19) z1

As mentioned earlier, we were unable to use the relation between xFc and rp obtained using stability theory due to some inconsistencies. In order to elaborate further, we need to discuss the experimentally observed phenomenon of drainage equilibrium. In this “equilibrium” state, the velocity of the continuous phase throughout the foam becomes zero, since the opposing forces due to gravity and PB suction balance each other throughout the system; i.e., ∂/∂z (1/rp) ) - Fg/σ. At the same time, there is no drainage in the films; i.e., Π ) σ/rp throughout the foam. It is clear that, under these conditions, there will be no collapse; i.e., dz1/dt ) 0. Rupture of a film can occur only when xF < xF|Πmax. Also, since a film can attain a thickness below xF|Πmax only if σ/rp exceeds Πmax, a collapsing foam must necessarily arrive at an equilibrium with xF ) xF|Πmax and σ/rp ) Πmax at the


Langmuir, Vol. 12, No. 12, 1996 3093

top. However, it is clear from the criticality curve obtained from linear stability theory (curve 1 in Figure 3) that as σ/rp f Πmax, dF/drp diverges and hence dz1/dt given by eq 19 does not become 0 as it should. The only situation in which an equilibrium with the above conditions will be established is if dF/drp f 0 as σ/rp f Πmax. In other words, a physically consistent curve should look like curve 2 in Figure 3. We therefore conclude that some of the assumptions used in obtaining the critical thickness using linear stability theory are inappropriate. As a first approximation, we assume that

xFc ) xF|Πmax ) constant

(20)

This satisfies all the requirements of equilibrium. The accuracy of this approximation depends on the actual shape of the criticality curve. The part of the criticality curve that is relevant to our calculation depends on what the critical thickness is when collapse starts. Consider curve 2 in Figure 3. If collapse starts at point A, we are interested only in the portion of the curve to the right of A. This is because as collapse occurs, the liquid from the collapsed bubbles enters the PB’s below and this increases rp|z1. Thus if the plateau of the criticality curve is large enough, the critical thickness during the entire period of collapse will be very close to xF|Πmax, so that our approximation will be quite valid. Thus, in this approximation dF/drp ) 0 and the conditions at the foam/gas interface during collapse are given by

|

(21)

dz1 dt

(22)

xF|z1 ) xF|Πmax

(23)

dz1 VRe ) dt ∂xF ∂z

qPB|z1 ) |z1

films and the PB’s. Since the bubbles are almost spherical at the bottom, it is reasonable to assume that the PB radius at the bottom equals the bubble radius. In other words:

z1

and

Equation 22 simply states the fact that all the liquid released due to collapse enters the PB channels at the top. Before collapse, there is no movement of the foam/gas interface with respect to the reference plane and the conditions at the upper boundary are

dz1 )0 dt

(24)

qPB|z1 ) 0

(25)

|

dxF dt

z1

) -VRe

Figure 4. Effect of film thickness at the bottom (xF0) on the initial liquid profiles in a freshly generated foam of height 7.5 cm. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, G ) 0.0001 m/s, ψs ) 20 mV, R ) 0.2 mm, T ) 298 K, Ah ) 3.7 × 10-20 J, c ) 0.005 M, Πmax ) 473.79 N/m2, and σ ) 40 mN/m.

(26)

2.5. Conditions at the Foam/Liquid Interface. 2.5.1. Boundary Conditions. In foams formed by bubbling, new bubbles are continuously introduced into the system at the foam/liquid interface. Thus, when the foam is being generated, the boundary condition at the bottom is a statement about the condition of the bubbles as they enter the foam. When films were neglected,12,13 the boundary condition at the bottom was simply the condition for close-packed spheres ( ) 0.26). This was sufficient, since there was only one variable (). However, now there are two independent variables (rp and xF) and information is needed about the distribution of liquid between the

rp|z2 ) R

(27)

Since the differential equation for xF is first order in time, an initial condition is needed. In other words, the value of xF at the foam/liquid interface during formation is required. We have no physical arguments to specify the film thickness. In order to identify the effect of its value, we carried out simulations with several values of xF. It was found that this value has practically no effect on the drainage, since, for thicker films, most of the drainage takes place within a very short distance from the bottom. Figure 4 shows the liquid fraction profiles in a freshly formed foam of height 7.5 cm. In one case, the film thickness at the bottom was 1000 nm, and in the other the value was 100 nm. The curves practically coincide, indicating that the value one takes for the film thickness at the bottom is largely irrelevant. Thus, as long as the foam is being generated (gas is being pumped in), the condition at the bottom is

xF|z2 ) xF0 ) constant

(28)

Once the gas supply is stopped, the films at the bottom simply obey the Reynolds equation with eq 28 as the initial condition. For the present calculations xF0 was taken to be 500 nm. 2.5.2. Movement of the Foam/Liquid Interface. As a foam is generated by bubbling, an observer at the origin sees the foam/liquid interface moving away from him (i.e., z2 increases). In order to completely specify the system, we need an expression for dz2/dt. Depending on the collapse occurring at the top of the foam, part of the gas resides in the foam, while the rest escapes from the top

3094 Langmuir, Vol. 12, No. 12, 1996


due to collapse. Since we have defined the reference plane to be such that all the escaped gas resides between it and the foam/gas interface (i.e. between z ) 0 and z ) z1), a mass balance for the bubbling gas can be written as

GA )

dz

∫zz A(1 - ) dz + A dt1

d dt

2

1

(29)

In eq 29, A is the cross-sectional area and G is the superficial gas velocity. The first term represents the gas in the foam, while the second term corresponds to the gas escaped from the collapsed bubbles. Simplifying eq 29, one obtains

G)

dz2 d dt dt

∫zz dz 2

1

(30)

Applying Leibnitz’s rule and using the fact that

∂qPB ∂ )∂t ∂z

(31)

we have

dz1 dz2 + qPB|z2 + |z1 - qPB|z1 (32) dt dt

G ) (1 - |z2)

Eliminating dz1/dt using eq 21 yields

dz2 G - qPB ) | dt 1 - z2

(33)

The system is now completely specified and can be solved to obtain z1 and z2 as a function of time. 2.6. Steady State Height and Collapse Half-Life. In pneumatic foams, after a certain height is attained, bubbles at the top burst and the foam length increases at a smaller and smaller rate until finally a steady state length is achieved. Under these circumstances, the rate at which the foam is generated at the foam/liquid interface equals the rate at which the bubbles are destroyed at the top. In a recent publication,16 this steady state height and the time required for the foam to collapse to half this height after the gas supply is shut off have been used to characterize the stability of foams. To model this experiment, eqs 1, 18, 21, and 33 are solved with G > 0 till (dz1/dt) - (dz2/dt) f 0, which implies that a steady state is achieved. Then the gas supply is shut off and z1 and z2 are computed as a function of time. 2.7. Solution Method. The boundaries of the “foam” were immobilized using the transformation z f ξ, where ξ ) (z - z1)/(z2 - z1). Thus the boundaries of the foam in ξ space are fixed at ξ ) 0 and ξ ) 1. The system was solved to obtain the two dimensionless variables y ) R/rp and x ) xF/xFd, where xFd is a constant taken here to be 100 nm. The discretization in ξ space was carried out using second-order finite differencing. This gives us a system of ordinary differential equations in time, which are solved along with the expressions for dz1/dt and dz2/ dt, using Gear’s backward difference formula.27 Since the equations are partial differential equations in space and time, in addition to the boundary conditions, an initial condition (i.e. initial values of z1 and z2 and the distribution of xF and rp) is needed. If the initial value of (z2 - z1) is small enough, the initial values of xF and rp have no effect (27) Gear, C. W. Numerical initial value problems in ordinary differential equations; Prentice Hall: New York, 1971.

Figure 5. A comparison of liquid fraction profiles in a freshly generated foam of height 7.5 cm obtained using the current model and the more approximate earlier model. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, G ) 0.0001 m/s, ψs ) 20 mV, R ) 0.2 mm, T ) 298 K, Ah ) 3.7 × 10-20 J, c ) 0.005 M, Πmax ) 473.79 N/m2 and σ ) 40 mN/m.

on the results as long as they are consistent with the boundary conditions. The initial height of the foam (z2 z1) was taken to be 0.5 mm. 3. Results The emphasis in this paper is to theoretically model the effect of various parameters on the steady state height and collapse half-life of pneumatic foams. However, since the present model improves on previous models by including the drainage of films, it would be instructive to make a comparison with results obtained using the earlier models, in which only the drainage through the plateau borders was taken into account. 3.1. Comparison with Previous Model. In a more approximate earlier model,12,13 it was assumed that films were always in equilibrium with the adjacent PB channels and that the fraction of liquid in the films was negligible. Collapse was deemed to start when the capillary pressure at the foam/gas interface exceeded the maximum disjoining pressure. There was thus a critical PB radius (rpc) given by

rpc )

σ Πmax

(34)

at which collapse started, and the movement of the boundary was given by eq 21. In order to compare the results from these two approaches, a sample calculation was carried out. The foam was produced with a superficial gas velocity of 0.0001 m/s, and the gas supply was shut off when a foam of length 7.5 cm was produced. In the previous model, the boundary condition at the bottom was ) 0.26, which corresponded to rp ) 1.01R. For proper comparison, we replace this condition with rp ) R. Figure 5 shows the profiles of liquid fraction () in the foam when the gas supply is just shut off. Figure 6 shows the evolution of foam height (z2 - z1)


Langmuir, Vol. 12, No. 12, 1996 3095

Figure 6. Variation of foam height with time obtained using the current model and the earlier model for the system of Figure 5.

as a function of time. Here the difference is much more significant. The reason for this is that while films do not influence the drainage process much, they play a crucial role in determining the characteristics of foam collapse. The older model predicts an earlier onset of collapse and a quicker decrease in foam length. This can be explained as follows: In the previous model, the collapse starts the moment rp|z1 ) σ/Πmax. In the current model, collapse begins when xF ) xF|Πmax. Under these circumstances, rp|z1 < σ/Πmax. Because collapse begins in the new model at a lower value of rp, it starts later. Also the current model predicts a slower collapse because the smaller PB radius at the collapsing front means that qPB is smaller, which in turn implies a smaller rate of collapse (see eq 21). 3.2. Effect of Superficial Gas Velocity on the Steady State Height. It is practical to use the steady state height as a measure of foam stability only for relatively short lived foams. For extremely stable foams, it is possible that the height of the foam when collapse starts will be too large to be conveniently measured in a laboratory. It is also obvious that the superficial gas velocity used to generate the foam will have a significant impact on the steady state height. Figure 7 shows the steady state height (H0) as a function of the superficial gas velocity (G). It is clear that the steady state height increases with G. The slope of the curve increases and seems to become unbounded for G ) 0.00035 m/s, indicating that there is an upper limit on the superficial gas velocity beyond which a steady state foam height will never be achieved. Some explanation for this can be provided as follows. From eq 21, it can be seen that the rate of foam collapse is determined by qPB|z1. Since the film thickness at the collapsing front is fixed, qPB is determined primarily by the value of the PB radius at the top. However, since collapse can occur only if the capillary pressure at the top exceeds the maximum disjoining pressure, there is an upper limit to the PB radius at the top given by

rp max ) σ/Πmax

(35)

Figure 7. Effect of superficial gas velocity (G) on the steady state height. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, ψs ) 18 mV, R ) 0.2 mm, T ) 298 K, Ah ) 3.7 × 10-20 J, c ) 0.001 M, Πmax ) 227.23 N/m2 and σ ) 40 mN/m.

In other words, there is an upper limit on the value of qPB|z1, given by

qPB max )

apFg 3 | Nn a R 15 p p 20x3µ rp)rp max,xF)xFc

(36)

This means that there is an upper limit to the value of dz1/dt, given by

dz1/dt )

qPB max |rp)rp max,xF)xFc

(37)

Thus for values of G for which dz2/dt (eq 33) is larger than the maximum value given by eq 37, dz2/dt will always be larger than dz1/dt and a steady state height (for which dz1/dt ) dz2/dt) will never be achieved. 3.3. Effect of Electrolyte Concentration on Foam Collapse. The concentration of electrolyte in the surfactant solution plays a crucial role in determining the stability of a foam due to its effect on the repulsive electrical double-layer forces in a film. The primary effect of an increase in electrolyte concentration is to compress the electrical double layer; i.e., the Debye length decreases. The lower the value of Πmax, the smaller is the capillary pressure required to cause collapse. On the other hand, the smaller the critical thickness, the larger is the residence time required for the film to become critical. Figures 8 and 9 show the variation of Πmax and the critical thickness (xFc) with electrolyte concentration. When the double layer is not too compressed at small concentrations and the attractive van der Waals forces are not very large, an increase in electrolyte concentration increases Πmax. However, at very small thicknesses, the van der Waals force increases and Πmax starts decreasing with increasing electrolyte concentration. There is thus a maximum in Πmax with an increase in electrolyte concentration. The position of this maximum however moves to smaller film thicknesses. This means that the critical thickness decreases with an increase in electrolyte concentration.

3096 Langmuir, Vol. 12, No. 12, 1996


Figure 8. Effect of electrolyte concentration (c) on the maximum disjoining pressure (Πmax). The values of the other parameters used are: ψs ) 19 mV, T ) 298 K, and Ah ) 3.7 × 10-20 J.

Figure 10. Effect of the electrolyte concentration on the steady state height. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, ψs ) 19 mV, R ) 0.2 mm, T ) 298 K, Ah ) 3.7 × 10-20 J, σ ) 40 mN/m, and G ) 0.0001 m/s.

Figure 9. Effect of electrolyte concentration (c) on the critical thickness (xFc) for the system of Figure 8.

Figure 11. Effect of electrolyte concentration on the collapse half-life for the system of Figure 10.

It is therefore reasonable to expect that as the electrolyte concentration increases, the stability of the foam as quantified by the steady state height (H0) and collapse half life (t1/2) increases at first, achieves a maximum, and then decreases. Figures 10 and 11, which show the effect of salt concentration on H0 and t1/2, show that this indeed is the case. It has been observed experimentally16 that plots of (z2 - z1)/H0 versus log(t/t1/2) practically coincide and at lower salt concentrations are linear. To check if our model predicts this feature (see Figure 12), such plots were

generated for six different NaCl concentrations. While the curves are not linear, they seem to coincide quite well, especially at higher concentrations. The shape of the curve is more similar to the data provided in the above paper at higher concentrations (labeled by these authors as “not so nice data”). The slope of the linear portion of the theoretical curves is reasonably close to the universal experimental line provided by these authors. A more direct comparison with the above experiments is not possible because there is no information on the bubble size used in the above experiments.


Figure 12. Effect of electrolyte concentration on the dimensionless plots of (z2 - z1)/H0 versus log(t/t1/2). The values of the parameters are the same as in Figure 10.

Figure 13. Variation of foam height with time for a foam with an initial length of 4 cm and G ) 0.0005 m/s. With c ) 0.007 M, complete collapse occurs, while, with c ) 0.003 M, the foam achieves an equilibrium height. The other parameters are the same as in Figure 10.

The concentration of salt also plays an important role in determining whether the foam collapses completely or arrives at a drainage equilibrium. Figure 13 shows the variation in foam height with time for two systems which differ only in the salt concentration. The initial foam height is 4 cm, and the other parameters are the same as above. The foam with c ) 0.003 M equilibrates at a height of about 1.85 cm, while the foam with c ) 0.007 M collapses completely. This can be explained as follows: The

Langmuir, Vol. 12, No. 12, 1996 3097

Figure 14. Effect of the bubble radius R on the steady state foam height. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, ψs ) 19 mV, T ) 298 K, Ah ) 3.7 × 10-20 J, σ ) 40 mN/m, c ) 0.005 M, and G ) 0.0001 m/s.

Figure 15. Effect of bubble radius on the collapse half life for the system of Figure 14.

capillary pressure has its smallest value (σ/R) at the bottom of the foam, which in this case is 200 N/m2. For c ) 0.003 M, Πmax ) 380.85 N/m2 while, for c ) 0.007 M, it is about 150 N/m2. Since in the latter case the smallest capillary pressure possible in the system exceeds the maximum disjoining pressure, the foam collapses completely. 3.4. Effect of Bubble Size. Figures 14 and 15 show the effect of bubble size (R) on H0 and Πmax for a system with the following parameters: ψs ) 19 mV, σ ) 40 mN/ m, G ) 0.0001 m/s, c ) 0.005 M, T ) 298 K.

3098 Langmuir, Vol. 12, No. 12, 1996

Figure 16. Variation of foam height with time for a foam with an initial length of 3 cm. With R ) 0.125 mm, complete collapse occurs, while with R ) 0.3 mm an equilibrium height is attained. The values of the parameters used are F ) 1000 kg/m3, µ ) 1 cP, ψs ) 19 mV, T ) 298 K, Ah ) 3.7 × 10-20 J, σ ) 40 mN/m, c ) 0.005 M, and G ) 0.0005 m/s.

Clearly, H0 and t1/2 increase with bubble size. This can be explained as follows: In general, in a foam column draining under gravity, the curvature of the PB channels (rp) is the largest and the capillary pressure the smallest at the foam/liquid interface, where rp ) R. For a given Πmax, the smaller the bubble size, the smaller is the decrease in rp required for the capillary pressure to rise to Πmax. This means that collapse starts at smaller heights, resulting in smaller values of H0 and t1/2. Figure 16 shows a plot of z2 - z1 versus time for two bubble sizes. It is clear that there is a qualitative difference, since while the foam with R ) 0.3 mm collapses to reach an equilibrium height, the foam with R ) 0.125 mm collapses completely. This is because (σ/R) is less than Πmax for the former and greater than Πmax for the latter, resulting in complete collapse. Figure 17 shows a comparison of the dimensionless plots for three bubble sizes with the universal experimental curve. The electrolyte concentration used here is 0.005 M. The other parameters remain the same. Again the agreement is fair. In an attempt to improve the overlap in the later stages of collapse, we made plots of [(z2 - z1) - (z2 - z1)eq]/[H0 - (z2 - z1)eq] versus log(t/t0.5) (see Figure 18), where t0.5 is the time required for [(z2 - z1) - (z2 z1)eq]/[H0 - (z2 - z1)eq] to become 0.5. The rationale for this was that this would force all the curves to become 0 at long times. It is seen from Figure 18 that there is some improvement. 4. Conclusion A more complete theoretical treatment of drainage and collapse of monodispersed pneumatic foams is presented in which the drainage and collapse of the films are included. The model accounts for the collapse of the foam during generation by bubbling and is therefore able to predict the steady state height attained by such foams. The effect of various parameters such as superficial gas velocity, salt concentration, and bubble size on the steady state height and collapse half-life is examined. It is shown


Figure 17. Effect of bubble radius (R) on the dimensionless plots of (z2 - z1)/H0 versus log(t/t1/2) with G ) 0.0001 m/s. The values of the other parameters are the same as in Figure 16.

Figure 18. Effect of bubble radius (R) on the dimensionless plots of [(z2 - z1) - (z2- z1)eq]/[H0 - (z2 - z1)eq] versus log(t/t0.5). The values of the parameters are the same as in Figure 17.

that, for a given system, there is an upper limit on the superficial gas velocity beyond which a steady state height will not be attained. With increasing salt concentration, the stability of a foam first increases, attains a maximum, and then decreases. The steady state height and the collapse half-life decreases with a decrease in bubble size due to an increase in the capillary pressure. It is shown that for a given system there is an upper limit to the salt concentration and a lower limit to the bubble size beyond which no drainage equilibrium is possible and complete collapse will occur. It is also shown that dimensionless


Langmuir, Vol. 12, No. 12, 1996 3099

dF L| VRe|z1 + drp z1 ∆z dz1 ) ) ∆t dt ∂xF dF ∂rp z1 ∂z drp ∂z z1

plots (z2 - z1)/H0 versus log(t/t1/2) coincide for most of the period of collapse even when the electrolyte concentration and bubble size are changed. Acknowledgment. This work was supported by a grant from the National Science Foundation. Appendix Derivation of the Expression for the Movement of the Foam/Gas Interface. When the film thickness and the PB radius at the top are such that they correspond to a point on the criticality curve 1 of Figure 3, i.e. when xF|z1 ) F(rp|z1), the films at the top rupture and the foam starts to collapse. To specify the problem completely, the rate at which the collapsing front moves is required. By expressing N and ap in terms of xF and rp, the conservation equation (eq 18) can be written in the form

(

)

∂rp ∂rp ∂2rp ∂xF ) L rp, , 2 ,xF, ∂t ∂z ∂z ∂z

(A1)

where L is a complicated nonlinear function of the terms inside the parentheses. Suppose, at a time t, the interface is at a position z ) z1. Since collapse is occurring, the film thickness and PB radius at this interface will always correspond to a point on the criticality curve. Thus

xF|z1,t ) F(rp|z1,t)

(A2)

Suppose in a time ∆t, the front moves by an amount ∆z. This means that, in this time, xF|z1 and rp|z1+∆z have changed such that

xF|z1+∆z,t+∆t ) F(rp|z1+∆z,t+∆t)

(A3)

i.e.

xF|z1+∆z,t +

|

∂xF ∂t

z1+∆z,t∆t

) F(rp|z1+∆z,t) +

|

dF ∂rP drp ∂t

∆t (A4)

z1+∆z,∆t

Thus using eq A1, we get

xF|z1+∆z,t - VRe|z1+∆z,t∆t ) F(rp|z1+∆z,t) +

|

dF L drp

∆t z1+∆z,∆t

(A5) Expanding the terms in Taylor series and neglecting second-order terms, we get:

|

|

|

∂xF dF ∂rp dF ∆z - VRe|z1+∆z,t∆t ) ∆z + L ∂z z1 drp ∂z z1 drp

∆t z1+∆z,∆t

(A6)

|

|

(A7)

Nomenclature ap ) average cross-sectional area of a plateau border channel A ) cross-sectional area of the foam column AF ) area of the surface of a film Ah ) Hamaker constant c ) concentration of electrolyte cv ) coefficient accounting for the mobility of the walls of a plateau border channel p ) pressure in the liquid phase pg ) pressure in the gas phase e ) protonic charge F ) function relating the critical thickness of film rupture to the PB radius G ) superficial gas velocity g ) gravity H0 ) steady state height of a foam k ) Boltzmann constant l ) length of a plateau border channel N ) number of bubbles per unit volume np ) number of PB channels per bubble QF ) volumetric flow rate of liquid from the films per unit foam volume qPB ) flow rate per unit area through the PB channels R ) radius of a bubble RF ) radius of a film rp ) plateau border radius t ) time t1/2 ) time taken by the foam to collapse to half the steady state height t0.5 ) time taken by the foam to collapse to a height which is half the difference between the steady state height and the equilibrium height T ) absolute temperature u ) average velocity in a PB channel V ) volume of a sphere of radius R VRe ) Reynolds velocity xF ) film thickness xF0 ) film thickness at the foam/liquid interface during generation xFc ) critical thickness of film rupture xFm ) film thickness corresponding to the maximum disjoining pressure z ) space coordinate z1 ) coordinate of the foam/gas interface z2 ) coordinate of the foam/liquid interface Greek Letters ) dielectric constant; liquid fraction 0 ) permittivity of free space γs ) inverse of dimensionless surface viscosity µs ) surface viscosity κ ) reciprocal Debye length µ ) bulk viscosity F ) density of the continuous phase σ ) surface tension ψs ) surface potential Π ) disjoining pressure Πmax ) maximum disjoining pressure ΠDL ) electrical double-layer force per unit area ΠVDW ) van der Waals force per unit area LA951092M

Modeling of the Generation and Collapse of Aqueous Foams

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