decay of standing foams: drainage, coalescence and collapsecgay/gdr-mousses... · homogeneous...

ELSEVIER Advances in Colloid and Interface Science

70 (1997) 1-124

ADVANCES IN COLLOID AND INTERFACE SCIENCE

Decay of standing foams: drainage, coalescence and collapse

Ashok Bhakta, Eli Ruckenstein*

Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260, USA

A b s t r a c t

A summary of recent theoretical work on the decay of foams is presented. In a series of papers, we have proposed models for the drainage, coalescence and collapse of foams with time. Each of our papers dealt with a different aspect of foam decay and involved several assumptions. The fundamental equations, the assumptions involved and the results obtained are discussed in detail and presented within a unified framework.

Film drainage is modeled using the Reynolds equation for flow between parallel circular disks and film rupture is assumed to occur when the film thickness falls below a certain critical thickness which corresponds to the maximum disjoining pressure. Fluid flow in the Plateau border channels is modeled using a Hagen-Poiseuille type flow in ducts with t r iangular cross-section.

The foam is assumed to be composed of pentagonal dodecahedral bubbles and global conservation equations for the liquid, the gas and the surfactant are solved to obtain information about the state of the decaying foam as a function of time. Homogeneous foams produced by mixing and foams produced by bubbling (pneumatic foams) are considered. It is shown that a draining foam eventually arrives at a mechanical equilibrium when the opposing forces due to gravity and the Plateau-border suction gradient balance each other. The properties of the foam in this equilibrium state can be predicted from the surfactant and salt concentration in the foaming solution, the density of the liquid and the bubble radius.

For homogeneous foams, it is possible to have conditions under which there is no drainage of liquid from the foam. There are three possible scenarios at equilibrium: separation of a single phase (separation of the continuous phase liquid by drainage or separation of the dispersed phase gas via collapse), separation of both phases (drainage

* To whom correspondence must be addressed.

0001-8686/97/$32.00 © 1997 - - Elsevier Science B.V. All rights reserved. PII: S0001-8686(97)00031-6

2 A. Bhakta, E. Ruckenstein/Adv. Colloid Interface Sci. 70 (1997) 1-124

and collapse occurs) or no phase separat ion (nei ther drainage nor collapse occurs). It is shown tha t the phase behavior depends on a single dimensionless group which is a measure of the re la t ive magni tudes of the gravi ta t ional and capil lary forces. A generalized phase d iagram is presented which can be used to de termine the phase behavior.

For pneumatic foams, the effects of various system parameters such as the superficial gas velocity, the bubble size and the surfactant and salt concentrations on the rate of foam collapse and the evolution of liquid fraction profile are discussed. The steady state height at tained by pneumatic foams when collapse occurs during generation is also evaluated.

Bubble coalescence is assumed to occur due to the non-uniformity in the sizes of the films which const i tute the faces of the polyhedral bubbles. This leads to a non-uniformity of f i lm-drainage rates and hence of film thicknesses wi thin any volume e lement in the foam. Smal le r films drain faster and rupture earlier, causing the bubbles containing them to coalesce. This leads to a bubble size distr ibution in the foam, with the bubbles being larger in regions where grea ter coalescence has occurred.

The formation of very stable Newton black films at high salt and surfactant concen- t ra t ions is also explained.

Keywords: Foams, Surfactants , Drainage, Concentra ted Emulsions, Stabil i ty

C o n t e n t s

1. 2.

Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Theoret ical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Foam geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 F i lm drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Fi lm rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Calculat ion of the disjoining pressure . . . . . . . . . . . . . . . . . . . . 13

2.4.1 van der Waals force . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Electrical double layer force . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 The short range repulsive force . . . . . . . . . . . . . . . . . . . . 18 2.4.4 Effect of surfactant concentrat ion on the disjoining pressure

for ionic surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.5 Effect of salt concentrat ion on disjoining pressure isotherms

of ionic surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.6 Steric repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Flow in P la teau border channel . . . . . . . . . . . . . . . . . . . . . . . 28 Theoret ical model for the drainage and coalescence in foam . . . . . . . . 30 2.6.1 Conservat ion equat ion for drainage . . . . . . . . . . . . . . . . . 30 2.6.2 Conservat ion equat ion for surfactant . . . . . . . . . . . . . . . . 40 2.6.3 Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6.4 Movement of the foam/gas interface . . . . . . . . . . . . . . . . . 49 2.6.5 Movement of the foam/liquid interface . . . . . . . . . . . . . . . . 50 Drainage equi l ibr ium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Init ial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Homogeneous foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.9.1 Ini t ial and boundary conditions for homogeneous foams . . . . . . 52

2.5 2.6

2.7 2.8 2.9

A. Bhakta, E. Ruckenstein/Adv. Colloid Interface Sci. 70 (1997) 1-124 3

2.9.2 D r a i n a g e e q u i l i b r i u m in h o m o g e n e o u s f o a m s . . . . . . . . . . . . 55 2.9.3 Effec t of in i t ia l cond i t ion of p h a s e s e p a r a t i o n . . . . . . . . . . . . 59 2.9.4 S i m u l a t i o n r e s u l t s for h o m o g e n e o u s f o a m s / c o n c e n t r a t e d

e m u l s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.10 P n e u m a t i c f o a m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.10.1 D r a i n a g e a n d co l lapse d u r i n g b u b b l i n g . . . . . . . . . . . . . . . 75 2.10.2 S i m u l a t i o n r e s u l t s for p n e u m a t i c foams . . . . . . . . . . . . . . . 77

3. I n t e r - b u b b l e gas d i f fus ion (Os twa ld r ipen ing ) . . . . . . . . . . . . . . . . . 108

4. C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 N o m e n c l a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A p p e n d i c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

1. I n t r o d u c t i o n

Foams are highly concentrated dispersions of gas (dispersed phase) in a liquid (continuous phase). Since they contain more than 90% gas by volume, the bubbles are usually polyhedral. This is because the highest volume fraction that identical spheres can occupy is 0.74. An increase in the volume fraction of the dispersed phase gas beyond this value is characterized by a departure from spherical geometry. It must be noted, however, that this criterion for determining the onset of polyhedricity is applicable only to monodisperse systems. With polydisperse systems, the bubbles can remain spherical at much higher volume fractions.

Concentrated emulsions (biliquid foams) are a similar class of systems composed of one liquid dispersed in another. In this review, the terms "foams" and "concentrated emulsions" will be used interchange- ably, as much of the principles discussed here are equally applicable to both foams and liquid-liquid concentrated emulsions.

Foams are metastable systems. They show a spontaneous tendency to separate into two distinct bulk phases. The time scale for the disin- tegration, however, varies widely. Foams can persist for a few minutes or several days depending on the conditions.

Foams have several very interesting and unusual properties which make them possible candidates for use in many industrial applications. The large gas/liquid interfacial area available can be exploited in foam fractionation to efficiently separate surface active substances such as proteins from their solutions [1-5]. An often cited example of foam fractionation is beer foam which contains more than 80% of the protein in the beer. Foams are also being considered for use in enhanced oil recovery [6,7], insulation and to reduce the impact of explosions [6]. On


t he nega t ive side [8,9], foams can be a major nu i sance in the chemica l process indus t ry . W h e n they las t long enough, they in te r fe re wi th physical and chemical processes and can adverse ly impac t product iv i ty and efficiency. L iqu id - l iqu id concen t ra t ed emuls ions have been used in our labora tory to p repa re h igh molecula r weigh t po lymers and compos- ites as well as m e m b r a n e s for separa t ion processes [10,11].

To effectively uti l ize foams and concen t ra t ed emuls ions in any of these s i tua t ions , it is i m p o r t a n t to have some control over the i r stabil i ty. A de ta i led u n d e r s t a n d i n g of the m e c h a n i s m s involved in foam persis t - ence and decay is therefore desirable.

Unfo r tuna t e ly , m u c h of the work in this area has been r a t h e r empir i - cal [12] and mos t expe r imen ta l da ta are r endered useless because i m p o r t a n t p a r a m e t e r s such as bubble size have not been measu red . The re is there fore a need for reliable theoret ica l models which can be used to i n t e rp r e t expe r imen ta l resul ts . A fairly comprehens ive review of the theore t ica l work in this a rea up to 1990 has been p re sen t ed by N a r s i m h a n and Ruckens t e in [13]. In this paper , we a t t e m p t to s u m m a - rize the work done in th is labora tory over the las t few years [14-18] on the decay of s t a n d i n g foams.

A s t a n d i n g foam is a s imple and conven ien t sys t em to s tudy foam decay. Typically, a cer ta in vo lume of foam is p r epa red and the changes occurr ing in it are moni to red as a funct ion of t ime. The l eng th of the foam/concen t r a t ed emuls ion usua l ly decreases; the heav ie r phase accu- m u l a t e s at t he bo t tom and the l ighter phase accumula t e s at t he top. At the s a m e t ime, t h e m e a n size of t he bubbles increases r e su l t ing in a decrease in the interfacial area. A sui table model for the decay of s t a n d i n g foam should be able to predic t the foam length , t he bubble size and the composi t ion of the foam (proport ion of the two phases) as a func t ion of t ime.

The con t inuous phase l iquid in a foam is p r e sen t in the l iquid f i lms t h a t are formed be tween the faces of t he polyhedra l bubbles and the channe l s (P la teau border channels ) t h a t are formed where ne ighbor ing f i lms meet . F igure 1 shows a typical foam column, a polyhedra l bubble and t he cross-section of a typical P l a t eau border (PB) channe l formed w h e r e th ree f i lms meet . These P l a t eau border channe l s form a complex in t e r connec ted n e t w o r k t h r o u g h which l iquid flows out of the foam u n d e r t he act ion of gravity. At the s ame t ime, t he l iquid in the f i lms is sucked into the P l a t eau border channels . As a resul t , t hey become t h i n n e r a nd f inally rup tu re . F i lm r u p t u r e at the foam b o u n d a r y causes loss of t he d i spersed phase gas f rom the foam. W h e n a f i lm r u p t u r e s

A. Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124 5

m~ FILM INTERSECTION OF" .__XI~ HORIZONTAL PL~

PFTiM I ~PLATEAU BORDER . . . . x.,~|j'(INTERSECTION OF

A- THREE FILMS)

P P B - - ~ - - r p PGAS BUBBLE . ~ ~ ' ~ F I L M (INTERSECTIDN

OF FACES OF TVO ADJACENT POLYHEDRA)

Fig. 1. A typical foam column, a dodecahedral bubble and the Plateau border cross-section.

w i t h i n a foam, bubbles sha r i ng the fi lm coalesce, l ead ing to an increase in the bubble volume. Dra inage of the con t inuous phase l iquid there fore p lays a pivotal role in foam decay.

Severa l models [19-26] for the d ra inage of s t a n d i n g foams have a p p e a r e d in l i t e ra ture . The mos t notable papers are those of Krotov [22] and N a r s i m h a n [23]. Krotov [22] was the first to recognize the effect of P l a t e a u border suct ion (capil lary pressure) on foam d ra inage and for- m u l a t e d the basic equa t ions and b o u n d a r y condit ions. He showed t h a t d r a inage in a foam even tua l ly comes to a ha l t w h e n the g r ad i en t of t he


capillary pressure, that is set up as drainage proceeds, balances gravity. However, while he formulated the basic equations, he did not at tempt to solve them. Nars imhan [23] recognized the importance of the method of foam generation and numerically solved the drainage equations for foams produced by bubbling using a quasi-steady state approximation to simulate the drainage during bubbling. He studied the effect of various parameters including the bubble size and viscosity on foam drainage. In a series of papers [14-18] over the past couple of years, we have studied the problem of standing-foam decay in detail. For pneumatic foams produced by bubbling, we used an unsteady state model [14] to simulate the drainage during bubbling. The results obtained differed significantly from those obtained by Nars imhan using a quasi- steady state approach and in some cases showed better agreement with experiment. In a later paper [15], we also considered the drainage of homogeneous foams. Most of the above models involved the numerical solution of the relevant differential equations. However, some efforts to obtain analytical solutions have been made [26].

These drainage models provided useful insights into the drainage process. However, they did not deal with the larger questions involving foam stability such as the decrease in the foam volume and the interfacial area with time. These phenomena are a direct consequence of film rupture which has traditionally been ignored in drainage models. Film rupture was included in our model [15-18]. This has enabled us to directly evaluate the effect of drainage on the various phenomena associated with foam decay such as bubble coalescence and foam collapse. Each of our papers dealt with a different aspect of foam decay and the basic equations were improved as we learnt more about the system. The aim in this paper is to present our results within a unified framework. We first present in detail the fundamental equations in their most rigorous form. The results for the various cases are presented towards the end.

2. T h e o r e t i c a l m o d e l

As mentioned in the previous section, drainage of the continuous phase liquid plays a crucial role in the decay of a standing foam. Two mechanisms are responsible for fluid flow in a foam. Flow in the films is driven by the capillary pressure, while the flow in the Plateau border channels occurs due to gravity. After a brief discussion of foam structure, we will discuss the flow in individual films and Plateau border channels and then formulate the bulk conservation equations.


2 .1 . F o a m s t r u c t u r e a n d g e o m e t r y

Any theoret ical model of foam mus t necessari ly involve assumpt ions regard ing foam s t ruc ture and geometry, which, needless to say, is ex t remely complex. Foams are composed of polyhedral bubbles which differ marked ly in shape and size. Even among bubbles hav ing the same volume, the re is a var ia t ion in the number of faces and the n u m b e r of edges per face. There is some order, however, amidst this appa ren t chaos. Almost wi thout exception, foams always obey a few simple rules, referred to as Pla teau ' s laws in honor of the Belgian physicist P la teau who first formula ted them.

P la teau ' s laws state: (1) Three and only th ree films meet at an edge at an angle of 120. (2) Four and only four edges (Plateau border channels) meet at a point

at an angle of 109. These rules have been used as a justif ication for the use of the regular pen tagonal dodecahedraon as the "idealized" foam bubble. While it is not space filling, the regular pentagonal dodecahedron comes remark- ably close to satisfying Plateau 's laws. The angle between the faces is about 116 and the angle between the edges is about 108. Indeed, detai led s tudies of foam s t ruc ture by Matzke [27] indicate t ha t pentagonal films and dodecahedra l bubbles do occur most frequently. Several in teres t ing discussions of foam s t ruc ture can be found in l i te ra ture [27-33].

The size of a bubble is expressed in the model in te rms of the bubble rad ius R, which is the radius of a sphere having the same volume as t ha t of the bubble. All other pa ramete r s which depend on the bubble size such as a rea of the films (faces) and the length of the P la teau border channe l s are expressed in te rms of R as shown below. The volume of a regu la r pentagonal dodecahedron with edge length l is:

V = 7 . 7 l 3 (1)

and the a rea ( A F) of each pentagonal film is:

AF=l .72 l 2 (2)

Since by definit ion V = (4KR3)/3, we have l = 0.816R. In the model, the pentagonal films are represen ted by circular disks

which have the same surface area. I f R F is the radius of these circular disks, we have

A F = x R 2 ~ R E = 0.606R (3)


,q

S Fig. 2. A draining film.

XF

2.2. Fi lm drainage

Figure 2 shows a cross-section of a d ra in ing film. The surface is curved at t he edges w he re ne ighbor ing films come toge the r to form a P l a t e a u border channel . Because of th is curva ture , the p res su re is sma l l e r a t t he edges t h a n at t he cen ter of the film and a radia l flow is i nduced lead ing to a reduc t ion of the fi lm th ickness wi th t ime.

F i lm t h i n n i n g is modeled us ing a Reynolds [33] type equa t ion for the flow be tween two circular paral le l disks, wh ich gives the ra te of f i lm t h i n n i n g as:

dx F 2c f APx 3 dt 3P'R 2 - Vf (4)

In Eq. (4), t is t he t ime, ~t is the viscosity of the con t inuous phase , R E is t he r ad iu s of t he disk and AP is the p res su re difference caus ing the flow. The dr iv ing force AP is a ne t resu l t of t he suct ion p ressure in the ad jacen t P l a t e a u border channe l s and the dis joining p ressu re (H) in the f i lms and is g iven by

Ap =-q-~ _ FI, rp

w h e r e a is t he surface tens ion and rp is the rad ius of cu rva tu re of t he P l a t e a u border channels . The coefficient cf is a correct ion factor which accounts for t he mobi l i ty of the fi lm surfaces and is g iven by [35]"

1 ~ (6~t + Tlsk2asXF)2 - - = 16 ~ + Vlsk2nO~sXF) 4 (5) cf (6p~ + 6p-c¢ s ~'n

n=l


In Eq. (5), Z n is the n t h root of the equa t ion

-3D. 3D. Jo(~'n):O'kn:Zn/RF'a~--~G~[l+'[~c~J DxF j O~Cs)'

J0 is the 0 th order Bessel funct ion of the first k ind, D is the bu lk diffusivi ty and D s is t he surface diffusivi ty of t he sur fac tan t .

The dis joining p re s su re (I]) refers to tbe repuls ive force t h a t ar ises w h e n the fi lm surfaces are close enough (x F < 1000 Angs t roms) to in t e rac t wi th each other. W h e n l-I is posit ive (repulsive), it opposes film th inn ing , whi le w h e n it is nega t ive (at tractive), it increases the dr iv ing force (AP) and accelerates f i lm th inn ing . In the mos t genera l case, 1-I is c o m p u t e d as a s u m of an a t t rac t ive van der Waals force (HvDw) , a repu ls ive force (HDL) due to the in te rac t ion of the electrical double layers on the two surfaces and a shor t range repuls ive force (I-ISR) which could resu l t f rom steric in te rac t ion w h e n long cha ined molecules are adsorbed on the surfaces or due to hydra t ion forces t h a t are set up due to the o rde r ing of t he wa t e r molecules nea r charged surfaces. The former are not observed wi th ionic sur fac tan ts . They arise only w h e n long-chained nonionic s u r f ac t an t s or polymeric molecules such as pro te ins are adsorbed on the fi lm surfaces. T h u s we have:

l-I -- Vivw + I~DL + I-IsR (6)

Un l ike the o the r two forces which are reasonably well unde r s tood wi th in the r e a lm of the DLVO theory, the origin of the shor t r ange forces is still a subject of discussion. It is genera l ly agreed t h a t t he force decays rough ly exponen t ia l ly wi th dis tance.

The plot of FI ve r sus x F is refer red to as the dis joining p re s su re i s o t h e r m and plays a crucial role in d e t e r m i n i n g the ra te of f i lm t h i n n i n g and its s tabi l i ty to rup tu re . F i lm r u p t u r e will be d iscussed next.

2.3. Film rupture

R u p t u r e of a f i lm occurs w h e n waves gene ra t ed on the surface due to mechan ica l and t h e r m a l pe r tu rba t ions grow in an u n b o u n d e d fashion. W h e t h e r a surface wave is d a m p e d or undergoes ca tas t roph ic g rowth is d e t e r m i n e d m a i n l y by the shape of the dis joining p ressu re i so therm. If t he repuls ive dis joining force increases in response to the local t h i n n i n g

10 A. Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

(a)

(b)

Z

a .

o

a

39.0 -

34.0

29.0

24.0

19.0

14.0 I

9.0 f 4,0

-1.0 0.0 2.0 4.0 6.0 8.0 10.0

Film Thickness (nm)

E 8

== == O-

e -

:_e 0 :_="

0.0

-3.0

-6.0

-9.0

-12.0

. j .

, , I i

- 15.00'0 1.0 2.0 Him Thickness(nm)

Fig. 3. V a r i o u s t y p e s of d i s j o i n i n g p r e s s u r e i s o t h e r m s .


(c)

99.0 !

79.0

59.0

39.0

19.0

-1.0 0.0 2.0 4.0 6.0 8.0 10.0

Rim Thickness(nm)

(d) 99.0

A

E, 79.0

T - • ~" 59.0 o.

• ~ 39.0 C O

e~

19.0

-1.0 ='- 0.0 2.0

Film Thickness(nm)

Fig . 3. C o n t i n u e d .

4.0

12 A. Bhakta, E. Ruckenstein /Adv. Colloid Interlace Sci. 70 (1997) 1-124

due to the d i s tu rbance , the wave is d a m p e d and no r u p t u r e occurs. On the o the r hand , if t he dis joining p ressu re decreases , the local t h i n n i n g is accelera ted and leads to rup tu re . F i lm r u p t u r e can therefore occur only if the dis joining p ressure decreases w h e n the f i lm th ickness de- c r ea se s , i.e. w h e n i ts d e r i v a t i v e w i t h r e s p e c t to f i lm t h i c k n e s s (dH/dxF) is positive. F igure 3 shows some typical plots of dis joining p r e s su re (H) ve r sus fi lm th ickness (XF). For the r u p t u r e of a t h in fi lm to occur via the g rowth of instabi l i t ies , d H / d x F m u s t be positive. Thus , a t h in f i lm wi th an i so the rm like the one in Fig. 3a will r u p t u r e only for x y < XFm. A film wi th an ini t ial th ickness x F > Xgm can arr ive at t h i cknesses less t h a n XFm only if the capi l lary p res su re ((~/rp) in the P l a t e a u borders , which drives film t h i n n i n g exceeds the m a x i m u m dis jo ining pressure . I t can therefore r u p t u r e only if the capi l lary pres- su re exceeds the m a x i m u m dis joining p ressu re (l-[max). On the o the r h a n d , a f i lm wi th an i so the rm like t h a t in Fig. 3b will def ini te ly r u p t u r e since d H / d x F is a lways positive. F igure 3c shows an i so the rm in which t he r e are two max ima . The smal le r one to the r igh t is p r imar i ly due to the electr ical double layer forces, while the o ther is due to a shor t r ange repu ls ive force. At h igh electrolyte concent ra t ions , the electrical double layer is compressed and the double layer forces are ove rwhe lmed by the van der Waals forces at all th icknesses . I f the shor t r ange force is s ignif icant , we will e i ther have an i so the rm like Fig. 3a, whe re the m a x i m u m is ma in ly due to the shor t range repuls ive forces or like Fig. 3d i f t he shor t r ange forces are large enough to overcome the van der Waals forces at shor t d is tances (few Angst roms) .

The la t t e r case corresponds to an ex t remely s table f i lm because dH / dx F is a lways negat ive . We will p r imar i ly consider i so the rms of the type shown in Fig. 3a. For this k ind of i so therm, the s i tua t ion r ega rd ing the s tabi l i ty of t he fi lm to r u p t u r e can be s u m m a r i z e d as follows:

dFl - - < 0 {Stable}

X F > XFm ~ d x F

dH X F < XFr n ~ ~ > 0 {Unstable} (7)

~ F

dH - - = 0 {Metastable} X F = XFm ~ d x F

A n u m b e r of papers have appea red in the l i t e r a tu re t h a t a t t e m p t to predic t t he l i fe t ime of a f i lm us ing l inear and non- l inear s tabi l i ty theory

A. Bhakta, E. Ruckenstein/Adv. Colloid Interlace Sci. 70 (1997) 1-124 13

XFm XF~

Fig. 4. Film rupture.

[35-40]. The e m p h a s i s in these papers is on compu t ing the t ime e lapsed be tween the onset of ins tabi l i ty ( the poin t at which d H / d x F = 0 & x F =

XFm) and the ac tua l r u p t u r e of the film which is deemed to occur w h e n the waves on the surface become large enough for the two film surfaces to touch (Fig. 4). The m e a n fi lm th ickness w h e n fi lm r u p t u r e occurs (XFc)

is ac tua l ly smal le r t h a n the m e a n th ickness (XFm), w h e n the ins tabi l i ty begins, i.e. XFc < xF, n. Needless to say, these theor ies are ex t remely complex and it is pract ical ly impossible to incorpora te t h e m into a global model for foam collapse. We therefore a s s u m e t h a t a film r u p t u r e s at t he m o m e n t its surfaces become uns table . In o ther words, we a s s u m e t h a t Xgc ~ XFm. This a s s u m p t i o n is reasonable because the t ime scale for t he g rowth of the ins tabi l i ty is m u c h shor te r t h a n the t ime for film dra inage . For an i so the rm of the type shown in Fig. 3a, XFc is therefore the fi lm th ickness cor responding to the m a x i m u m dis joining p ressu re ([Imax). The dis joining p ressure i so the rm and especial ly the va lue of the m a x i m u m dis joining p ressu re (lima x) therefore plays a critical role in our model for foam collapse.

It m u s t be e m p h a s i z e d here t ha t a film can also r u p t u r e w h e n it is very thick, m u c h before the fi lm surfaces are close enough to in teract . The s tabi l i ty of the film in this s i tua t ion will be i n d e p e n d e n t of the d is jo in ing p re s su re and will be d e t e r m i n e d pr imar i ly by the elast ic i ty of t he fi lm [41] and the n a t u r e of the mechanica l pe r tu rba t i ons involved. This m e c h a n i s m of film r u p t u r e is not considered in our model.

Detai ls r ega rd ing the calculat ion of the dis joining p ressure i so the rm are d iscussed next.

2.4. C a l c u l a t i o n o f the d i s j o i n i n g p r e s s u r e

2 .4 .1 v a n d e r W a a l s force

The a t t rac t ive van der Waals componen t of the dis joining p ressu re (HVD w) can be obta ined us ing the expression:

Ah H V D W - - 6xx~ (8)


The H a m a k e r ' s c o n s t a n t (A h) h a s a w e a k d e p e n d e n c e on f i lm th i ckness , b u t is o f ten t a k e n to be cons tan t . D o n n e r s e t al. [42] h a v e o b t a i n e d a s imple exp re s s ion for Ah: E +cx l A h = 6re 1 + d x F + ex'~ + q (9)

w h e r e b, c, d, e and q a re c o n s t a n t s wh ich d e p e n d on the specif ics of t he p a r t i c u l a r s y s t e m such as the m e d i a involved and the t h i c k n e s s 'p' of t h e s u r f a c t a n t f i lm (see Fig. 5). In some of our ca lcu la t ions , s o d i u m dodecyl su l f a t e w a s used as the mode l su r f ac t an t . Fo r th is sy s t em, t he v a l u e s ava i l ab l e for a i r - w a t e r - a i r s y s t e m s w i t h dodecane m o n o l a y e r s (p = 0.9 nm) a t t he a i r - w a t e r in te r faces we re used . These v a l u e s are:

b = - 3 . 9 6 x 1 0 -23 J , c = - 2 . 0 5 x 1 0 -13 d /m, d = 8.86x107 J / m , e = 6.61x1016 J / m 2, q = - 1 . 8 x 1 0 -22 d.

I t m u s t be e m p h a s i z e d t h a t x F re fe rs to t he t h i cknes s of the a q u e o u s core of t h e film. E x p e r i m e n t a l l y ob t a ined d is jo in ing p r e s s u r e i s o t h e r m s [43,44] a re u s u a l l y p r e s e n t e d in t e r m s of the to ta l f i lm t h i c k n e s s w h i c h inc lude t he t h i c k n e s s e s of the s u r f a c t a n t layer. Thus , a cor rec t ion (2.3 n m in case of s o d i u m dodecyl su l f a t e [43]) m u s t be app l ied to x F to m a k e a c o m p a r i s o n w i t h e x p e r i m e n t a l resu l t s .

Surfactant

Air Water Air

p p

Fig. 5. The three layers involved in the calculation of the van der Waals forces.


2.4.2. Electrical double layer force (I-IDL) When the film surfaces are charged, a repulsive force arises due to

the confinement of ions in the electrical double layer as the film thickness decreases. Calculat ion of [IDL involves the solution of the Poisson- Bol tzmann equat ion to obtain the concentrat ion of ions at the midplane. An approximate power series solution to the Poisson-Bol tzmann equa- t ion at moderate potentials has been obtained by Oshima and Kondo [45,46]. The potential ~g at a distance x from the midplane is given by:

tanh[e~l=YAl(r,x)+y3A2(rx)+?SA3(r,x ) (10)

where:

. l(e~gs ~ cosh(r~x) . Y=~an~-4k-T);Al(X)= cos Kx~--~- / '

(~.x) sinh(~.x) - ~ t anh cosh(r.x)

A2(x) = (11a)

cosh ( 1 I + 1 - 4 ~ t anh -

1 / A3(x) =

A~(x) {/--~/2 - (~zc) 2 }

2cosh4/ - -~/ ( l l b )

The mid plane potent ial (W m) is therefore:

[e~gm 1 = t anh [kBT j yAl(O ) + 73A2(0) + 75A3(0 ) (12)

and the repulsive force per uni t area is given by:

16CelRTY2m [IDa - (1 - Y2m)2 - 16Ce~RTY2m [1 + 2Fern + 3Y4m] (13)


[Te~,~ wher e Ym = t a n h / - - / and cel is the electrolyte concentra t ion. Us ing Eq.

[kBTJ (10), Eq. (13) can therefore be r ewr i t t en as:

[. y2A'~(O) + 274Al(O)(A2(O) + A3(0)) + 7 FIDE = 16(C~ + ce)RGT [y6(2AI(0)A,~(0) + SA13(0)A2(0) + 3A6(0) + A2(0)) j (14)

If the surface po ten t ia l (~s) is fixed and known, Eqs. (10-14) suffice for t he ca lcula t ion of I-IDL. However , the surface potent ia l is often not fixed and de pe nds on the surface charge. For nonionic sur fac tan ts , t he surface charge arises due to the b ind ing of ions p resen t in the w a t e r and is probably i n d e p e n d e n t of su r f ac t an t concentra t ion. On the o ther hand , for ionic sur fac tan t s , the surface charge resul t s f rom the dissociat ion of the adsorbed su r f ac t an t molecules and is l ikely to be s t rongly d e p e n d e n t on the su r f ac t an t concentra t ion. :The calculat ion of the surface charge and the surface potent ia l for ionic su r fac t an t s is d iscussed below.

Calculation of the surface charge and surface potential for ionic surfactants

In f i lms stabi l ized by ionic sur fac tan ts , the charge on the surface of a f i lm is caused by the adsorbed su r fac t an t molecules t h a t are dissociated. Let us consider a model sys t em of sod ium dodecyl sulfate and s o d i u m chloride. I f c~ d is the degree of dissociat ion of the adsorbed s u r f a c t a n t molecules, t he surface charge per un i t a rea ((~e) can be expressed in t e rms of the surface dens i ty (F) as:

(~c = -Fade (15)

whe re e is the protonic charge and the degree of dissociat ion ad is p rovided by the equ i l ib r ium of the following reaction: R - N a I surface --~

R- [ surface + Na+ I aqueous phase- Thus , deno t ing K d as the equi l ib r ium constant , we have

((ZdF) CNa + OCdCNa ÷ = - - - (16) Kd= (1 _ CCd) F 1-C~ d

In Eq. (16), the concent ra t ion CNa+ of the sod ium ions nea r the surface can be expressed in t e rms of the su r f ac t an t concent ra t ion (c S) and the sal t (NaC1) concent ra t ions (c~) in the P l a t eau border channe l s as:

eV. ] CNa + = (C~ + Ce) exp - kBT) (17)


and F can be computed using the F r u m k i n adsorption isotherm:

F exp 2a 1 F~

(18) blCR- = F 1 -

F~

where

CR-= C s exp ( k B T J \ J

is the concentra t ion of the sur fac tant anions nea r the interface, I~ is the surface excess at sa tura t ion and the constants a] and b 1 are empir ical pa ramete r s which are available in l i te ra ture from experi- men t s carr ied out for a i r - w a t e r interfaces in contact wi th a large a m o u n t of water . The exper imenta l resul ts however, re la te F to the bulk sur fac tan t concentra t ion c s and not CR-. At large ionic s t rengths , the double layer is completely compressed and c R- is equal to the c s. The h ighes t sal t (NaC1) concentrat ion for which da ta on sodium dodecyl sulfate are available [47] is 1 M for which F~ = 5x10 -6 mol/m 2, a 1 = -1.53 and b] = 881 m3/mole. These values were used in our calculat ions wi th ionic surfactants . The F r u m k i n i so therm when combined with the Gibbs adsorpt ion equat ion provides the following relat ion for the surface tension:

2

G o - cJ = - F ~ G T | log 1 J

le sl_] N A e cosh [ 2 k B T ) 1 (19)

The surface charge can also be expressed in te rms of the surface potent ia l and the midplane potential as:

( ~ = - I ]2~cosh(e~s ) - c o s h ( e~m I11 k e d e d ° k B T [~ [ [ k B T ) [ kBT)JJ e (20)

Equa t ing the r ight hand sides of Eq. (20) and Eq. (16) and solving with Eq. (13) we can get the midplane (~m) and surface potential (~s) in te rms of the dissociation constant K d and electrolyte concentra t ion Cel. It may


be noted that for ionic surfactants, the electrolyte concentration Cel will depend on the surfactant concentration. Thus, i f c e is the concentration of the monovalent salt (e.g. NaC1) and c s is the concentration of the monovalent ionic surfactant (e.g. NaDS), we have:

Cel = c s + c e (21)

It may be noted that the use of Eq. (17) implies that the effect of the interactions between the adsorbed species has been ignored. Because these interactions provide a positive contribution to the free energy, they decrease the degree of dissociation. These effects have been considered for micelles by Ruckenstein and Beunen [50].

2.4.3 . T h e s h o r t range r epu l s i v e force (1-IsR) [17] As mentioned earlier, large exponentially decaying repulsive forces

arise due to the steric interaction of long-chained molecules or due to the organization of water molecules near charged surfaces. We will first discuss the lat ter in detail. A short discussion of steric forces is provided in a later section.

S h o r t r a n g e r e p u l s i o n d u e to o r g a n i z a t i o n o f w a t e r m o l e c u l e s

It has been experimentally observed [37] that in many systems, the resistance of a foam to collapse increases sharply at sufficiently high surfactant and salt concentrations. Experimental measurements of the disjoining pressure isotherms of single films containing sodium dodecyl sulfate [43] show that qualitatively different isotherms are obtained depending on the concentration of sodium chloride. As the pressure on a film is increased, its thickness decreases and the repulsive disjoining pressure increases. When the thickness is small enough (about 10 nm), black spots appear which eventually cover the entire film giving rise to a common black film. Fur ther increase in pressure, however, gives rise to different phenomena depending on the salt concentration. At lower salt concentrations, the common black film ruptures, while at higher salt concentrations, there is a sudden transition to a very stable Newton black film. In the latter case, the film rupture predicted by the DLVO theory is prevented by the emergence of a large short range repulsive force. We use a simple model proposed by Schiby and Ruckenstein [51] to compute the short range repulsive force. An electric field is generated by the undissociated surfactant molecules via their dipoles. This electric field organizes the water molecules near the surface and gives rise to the repulsive force. Thus, increased counterion binding at high salt


concentrat ion increases the number of surface dipoles and hence the electric field and makes the short range repulsive force stronger. We show tha t most effects of salt and surfactant concentration can be qual i ta t ively accounted for by combining this theory with the DLVO theory. Schiby and Ruckenstein derived the expression for the short range repulsive force between two parallel plates using an idealized model in which the water molecules are ar ranged in a hexagonal close packed configuration in parallel planes separated by a distance 5. They assumed tha t the electric field generated by the surfaces orients the wate r molecules in the direction of the field and gives rise to a profile of mean dipole moment . They obtained a simple expression for the polari- zation (dipole momen t per uni t volume) profile between the plates and used it to obtain the following expression for the force per uni t area (lI h) be tween the plates:

Pj [exp(-2~x F) - exp(-~,x F ) + ctx F exp(-~.x F)] I Ih= (A' exp(--IZrF) + B' )2 I-

PjA' [exp(---ax F) - exp(-3a, x F) - 2m: F exp(-Zm:r) ] + (22)

[A' exp(--CZXF) + ff]3

P2 exp(--(ZXF) P2 A' (exp(--aXF) -- exp(--2C~XF)) 4

[Z' exp(--~CF) +/~] [A' exp(--~p) +/~]2

where A',B" and a are constants which can be computed (see Eqs.(24- 27)) and P1 and P2 are given by:

P1 = nTp [EO]2 ; P2 = 2 P 1 ~ 5 (23)

In Eq.(23), n is the number of water molecules per uni t volume, 7p is the polarizabili ty of wate r and E ° is the electric field genera ted by the surface which acts on the first layer of water molecules.

Calculations indicate tha t the third te rm is dominant , l-I h is therefore pr imari ly an exponential ly decaying force with a decay length of a -1. The constants A',B' and a in Eq.(22) can be expressed in terms of the distance between adjacent water molecules (a) and the distance between the parallel planes of water molecules (5) as follows:

c~ 2 = [1-T,,(2A(5) + A(0))] (24)

7 ,A(5 )5 2

where:

20 A. Bhakta, E, Ruckenstein/Adv. Colloid Interface Sci. 70 (1997) 1-124

2 6 28 2 _ a 2 A ( + 6 ) : 3 , (25) (a 2 8- + . :

Also,

A' = - 1 + 7p[A(8) exlq(Z3) + A(0)] (26)

B ' = 1 - yp[A(3) exp(-c~5) + A(0)] (27)

For 8 = a = 2.8 A, A' = 0.268, B ' --- 2.53 and c~ = 3.58x109 m -1. The electr ic field E ° h o w e v e r d e p e n d s on the cha rac t e r i s t i c s of the su r f ace and is c o m p u t e d as s h o w n below.

Calculat ion of E 0

The inne r su r f ace of the f i lm con ta ins d i s soc ia ted cha ins w h i c h b e h a v e as po in t cha rges and u n d i s s o c i a t e d cha ins wh ich b e h a v e as d ipoles po in t ing into t he l iquid phase . I f we cons ider bo th su r f ace s to be of in f in i te ex ten t , t he electr ic field a t any poin t b e t w e e n the p l anes due to the su r f ace cha rges will be 0. The r ea son for th is is t h a t t he e lectr ic f ield due to a cha rged inf in i te p l ane is i n d e p e n d e n t of d is tance . The f ields f rom the cha rges on the two su r f aces t he re fo re cancel each o the r and E ° d e p e n d s solely on the dipoles p r e s e n t on the surface . The electr ic field a t a d i s t a n c e 'z' f rom a semi- inf in i te p l ane w i th a dipole m o m e n t per u n i t a r e a u.~ is g iven by (see A p p e n d i x B):

2 u s E - ( 2 8 )

3CdCd0Z

Thus , i f g,~ is t he dipole m o m e n t of a s ingle und i s soc i a t ed s u r f a c t a n t mo lecu le we have:

u,~ = F(1 - Otd)NA~ts (29)

w h e r e N A is t he Avogadro n u m b e r . C o m b i n i n g Eqs. (28) and (29), we get:

2F(1 - Oral)NAP s E - (30)

3CdgdOZ

Since w e a re dea l ing w i t h ind iv idua l w a t e r molecules , t he m e d i u m a m o n g t h e m is v a c u u m and we h a v e c d =1. The a s s u m p t i o n in t h a t t h e o r y is t h a t t he e x t e r n a l e lectr ic field is i m p o r t a n t only in the f i rs t l aye r of molecules. Thus, i f d is the d is tance b e t w e e n the p lane con ta in ing


t h e s u r f a c t a n t molecu les a n d the f i rs t l aye r of w a t e r molecules , we need E a t z = d. Since bts/d is not known, we use it as a p a r a m e t e r . T h u s we wri te :

E ° = F(1 - cQ)E d w i t h E d = 2NAgs/(3edod) (31)

Since E d is a p a r a m e t e r to be var ied , it is he lpfu l to h a v e a r e f e r e n c e va lue for E d abou t wh ich it can be var ied . In t he model of Schiby a n d R u c k e n s t e i n , t h e m e a n dipole m o m e n t of t h e w a t e r molecules close to t h e su r f ace is g iven by:

~(0) - 7 pE°( 1 - exp(--CZXF)) = 7 ,,F(1 - (z ~)E~t(1 - exp(--C~CF) ) (32)

A' exp(--0~XF) + /~ A' exp(--0~XF) + B'

Thus :

E(, = g(0)(A' exp(--OCXF) + / ~ ) (33) y ,,F(1 - c¢ j)(! - exp(-(zx F ))

There fo re , i f gw is t he dipole m o m e n t of wa te r ,

btw(A" + B') E d r e f - _ 3.65X1015 N.m2/C

F~yp

se rves as a use fu l r e f e r ence value. It m a y be no ted t h a t we a s s u m e h e r e t h a t t he shor t r a n g e repu l s ion

force a n d t h e e lectr ic double l aye r force a re addi t ive. An a t t e m p t to couple t h e m n o n l i n e a r l y has been m a d e by Schiby and R u c k e n s t e i n [52].

We ca r r i ed out some sample ca lcu la t ions us ing express ions p r e s e n t e d above for a mode l s y s t e m compr i s ing sod ium dodecyl su l fa te and s o d i u m chloride. Some re su l t s a re p r e s e n t e d in t he following sections.

2.4.4 Effect o f surfactant concentration on the disjoining pressure for ionic surfactants [17]

Being an ionic su r f ac t an t , t he bu lk concen t r a t i on of sod ium dodecyl su l f a t e affects t he dis jo ining p r e s s u r e in two ways:

(a) I t affects t h e su r face excess F a n d hence affects t he su r face c h a r g e (see Eq.(16)).

(b) I t affects t h e ionic s t r e n g t h and the degree of d issocia t ion a n d t h e r e f o r e t h e e x t e n t of t he e lect r ica l double layer . F igures 6 a n d 7 show t h e effect of t h e s u r f a c t a n t concen t r a t i on on the dis jo ining p r e s s u r e


i so therm for two different values of the dissociation constant K d viz. K d

= 158 moles/m 3 and K d = 0.316 moles/m 3. In both cases, Ed/EdrefWaS t aken 0.04. It is clear tha t there is a qual i ta t ive difference in the effect of sur fac tan t concentrat ion in the two cases. In Fig. 6, since the degree of dissociation is l a r g e , the number of dipoles which contr ibute to the short range repulsive force is small and it is too weak to overcome the van der Waals forces. Thus, the increase in sur fac tant concentrat ion pr imar i ly affects the disjoining pressure by changing the double layer force. This is clear from the fact tha t the m a x i m u m disjoining pressure is very close to t h a t (l~max,dl) obtained when no short range repulsive force is included (see Table 1). In the sys tem considered in Fig. 7, the smal ler degree of dissociation significantly increases the number of surface dipoles and hence the contr ibution of the short range polariza- t ion force. Thus the increase in sur fac tant concentrat ion causes the

10.0

E.

i n C

.~, ~3

9.0

8.0

7.0

6.0

5.0

4.0

3.0 l

2.0

1.0 l

0.0

; Surfactant Concenb'ation=O.O01M o o Surfactant Concentration=O.OO3M 0 0 ~ Surfactant Concen~'ation=O.OOSM

. , I . I .

0.0 1.0 2.0 3.0 4.0 5.0 Film Thickness(nm)

Fig. 6. Effect of surfactant concentration on the disjoining pressure isotherms with K~ = 158 moles/m 3 and E d = 0 . 0 4 E d r e f .


Tab l e 1

Effec t of s u r f a c t a n t c o n c e n t r a t i o n on s y s t e m p a r a m e t e r s for E d = 0.04Ed,efand c e = 0

C s g d [Imax XFm I - I m a x , d l XFm,d l (M) (moles]m 3) (N/m 2) (nm) (N/m 2) (nm)

0.001 158 217500 2.624 217200 2.64

0.003 158 384000 1.855 378000 1.94

0.005 158 534000 1.402 506000 1.7

0.001 0.316 30990 3.97 30970 3.97

0.005 0.316 A)* 106000 0.828 21900 3.62

B)* 22000 3.56

* T h e r e a r e two m a x i m a .

0.4 i | w t

: : Surfactant Conce~t~atlon=O.O01M o o Surfactant Concentmtion=O.OO5M

0.3

,p-

i 0.2

tt

0.1

I o.%.° 2.o ,o.o

Film Thickness(nm)

Fig. 7. Ef fec t o f s u r f a c t a n t concen t r a t i on on t h e d i s jo in ing p r e s s u r e i s o t h e r m s w i t h K d

= 0.316 m o l e s / m a and E d = 0 . 0 4 / ~ d r e f .

24 A. Bhakta, E. Ruckenstein / Adv. Colloid Interface Sci. 70 (1997) 1-124

i s o t h e r m to unde rgo a qual i ta t ive change from one con ta in ing a single m a x i m u m cor responding to the double layer force to one con ta in ing two peaks. In Fig. 7, the shor te r peak at a la rger f i lm th ickness is due to the electr ical double layer while the s teep peak to the left is due to the shor t r ange hyd ra t i on force.

2.4.5 Effect of salt concentration on the disjoining pressure isotherms of ionic surfactants

The concen t ra t ion of sod ium chloride has the following effects :(a) it compresses the electrical double layer by ra i s ing the ionic s t r eng th , (b) it decreases the degree of dissociat ion of the adsorbed su r f ac t an t s (see Eq. (17)) by caus ing increased b ind ing of the counterions,(c) it affects t he concen t ra t ion of R- ions nea r the in terface via the surface po ten t ia l and hence the surface excess (F). In expe r imen t s wi th single th in l iquid films, it has been observed by Exerowa et al. [43] t h a t a t h igh sal t concent ra t ions , as the capi l lary p ressure is increased, t he r e is, i n s t ead of r u p t u r e , a s u d d e n t r ans i t ion to very s table Newton black films. F igure 8 shows the effect of sal t concent ra t ion on a sy s t em wi th c s -- 0.001 M. Wi th no sal t (c e -- 0), t he re is j u s t one peak cor responding to the electrical double layer. As the sal t concent ra t ion is increased, however , the curve changes first to one in which the re are two m a x i m a and t h e n to one h a v i n g only a shor t r ange repuls ion. This can be expla ined as follows: An increase in the sal t concent ra t ion increases the b ind ing of sod ium ions a nd ra ises the n u m b e r of dipoles on the surface. The shor t r ange repuls ive force therefore increases. At the s ame t ime, the double layer is compressed . At i n t e r m e d i a t e concent ra t ions , the double layer is not too compressed and the increased shor t r ange repuls ive force gives rise to an add i t iona l s t rongly repuls ive force at a smal le r th ickness . In th is case, t he re is a j u m p t rans i t ion w h e n the capi l lary p res su re is ra i sed above t he va lue at po in t A. However , at h igh concent ra t ions , t he electr ical double layer is so compressed t h a t I~DL n o w has a r ange compara- ble to t h a t of t he shor t range polar izat ion force and the two combine to give a s ingle peaked shor t r ange repuls ion. These resu l t s are in quali- t a t ive a g r e e m e n t wi th the observat ions of Exerowa et al., in the sense t h a t a j u m p t r ans i t ion is observed as the sal t concent ra t ion is increased. We would like to poin t out however t h a t Exerowa et al. [43] observed these j u m p t r ans i t ions only at sal t concent ra t ions above 0.165 M, m u c h h ighe r t h a n those used in Fig. 8.

While the ca lcula t ions here have been done for ionic sur fac tan t s , the mode l is appl icable in pr inciple to non-ionic su r fac t an t s wi th shor t h e a d

A. Bhak ta , E. Ruckens t e in /Adv . Colloid Interface Sci. 70 (1997) 1-124 25

E 0.2

8 o 8

ft .

._c

o -~ 0.0

-0 .2 0 .0

= : NaCI Concer~atJon=O o [] NaCI ConcentratJon=O.01M o o NaCI Concentration=O.1M

J 2.0 4.0 6.0 8.0 10.0

Film Thickness(nm)

Fig. 8. Effect of sodium chloride concentration on the disjoining pressure isotherm with c~ = 0.001 M, E d = 0 . 0 5 E d r e f and K d = 0.316 moles/m 3.

groups as well. In these systems, the surface charge arises due to ion-binding r a t h e r t han the dissociation of the adsorbed su r fac tan t molecules as happens with ionic surfactants . The short range force in this case is likely to be independent of electrolyte concentrat ion, since the surface dipole moment would be essential ly unchanged by the ion binding. This could be the reason why Newton black films are formed at much lower ionic s t rengths with non-ionic sur fac tan ts [44]. It m a y be noted, however, t ha t in case of non-ionic sur fac tan ts with large head groups, steric forces are likely to play a significant role. A brief discussion of steric forces is given below.

2.4.6 Steric repulsion Steric forces arise when large molecules (polymers like proteins or

su r fac tan t s wi th large head groups) are adsorbed on the film surfaces.


D e p e n d i n g on the in te rac t ion be tween the adsorbed molecules and the f i lm liquid, t he free ends of the adsorbed molecules ex tend a s ignif icant d i s tance into the liquid. W h e n the fi lm surfaces app roach each other , t he en t ropy of these free ends decreases. This resu l t s in a repuls ive osmotic p re s su re t h a t increases as the cha ins become more confined, i.e. as t he f i lm th ickness decreases. The steric in te rac t ions usua l ly come into play w h e n the f i lm th ickness becomes smal le r t h a n 2 L , where L is t he m e a n th i ckness of the adsorbed layer. The va lue of L d e p e n d s on the conf igura t ion of the chain. W h e n the chain is fully ex tended, L is m a x i m u m and is given by L = n s l s where ns is t he n u m b e r of s e g m e n t s and l~ is t he l eng th of each segment . The ac tua l va lue of L depends on t he in te rac t ion be tween the molecule and the fi lm liquid. In a 0 solvent , in wh ich the re is no in te rac t ion be tween var ious segments , we have:

L Lo---zs For low surface coverage in a 0 solvent, Dolan and E d w a r d s [53] have der ived a s imple express ion for the in te rac t ion energy per un i t area:

f ' k . T[~L~'. r ~ I , [3x~-17 W ( X F ) -- + -- ln/~_--:~/[ for x F < f f3L,,

L6x~ e L8rtL;,jj (34)

= 2 F k B T exp - for x F ~ E L o L ~ 2L,, ) j

The force per un i t a rea is therefore given as:

V2rt 3/~ f i s t = F k . T I . - ~ " 2 for x F < I~,xl~

L 3xb x F (35)

= 12FkBT 55- exp -~75-,2 for x r >_ Loft3

At h igh surface coverages, the adsorbed molecules form a brush . The t h i cknes s of t he b ru sh in a good solvent has been given by Alexander [54] as:

L = F 1/2 R ? / 2 (36)

whe r e R f = l s n s 3/5 is t he Flory radius . The repuls ive p res su re be tween two b r u s h - bea r ing surfaces is given by the A l e x a n d e r - d e Gennes theory [55]:


I( 9 for x F < 2L (37)

The first t e rm in Eq.(37) resul ts from the osmotic repulsion while the second t e r m arises from the elastic energy which opposes stretching. F igure 9 shows the disjoining pressure i so therm obtained us ing Eq. (37) for a typical sur fac tan t wi th a large head group (polyoxyethylene n-dodecanol) wi th 12 ethoxy (CH2-CH2-O) groups. The volume of an ethoxy uni t is approximate ly 63,~3. This gives the l e n g t h o f e a c h uni t as approximate ly 0.49 nm. So we have n s = 12 and l s = 0.49 n m which gives L = 3.04 n m from Eq. (36). The values of the o ther p a r a m e t e r s used are: ~t s = 20 mV, A h =

3.7×10 -20 J and %1 = 1 mM. It is clear tha t for film thicknesses smal ler t h a n 2L, the steric repuls ion overcomes the a t t rac t ive van der Waals forces.

600

5OO

4O0

300

.= :3 == ,0 Q. o)

._.R p.

2OO

100

-100

\

-2000.0 2.0 4 0 6.0 8 0 10.0

Film Thickness (nm)

Fig. 9. Theoretical disjoining pressure isotherm for polyoxyethylene n-dodecanol.

28 A. Bhakta, E. Ruckenstein /Adv. Colloid Interlace Sci. 70 (1997) 1-124

This concludes our discussion of single films. The flow in individual P la teau border channels is considered next.

2.5 F low in a P la t eau border ci~annel

As ment ioned earlier, the flow of the continuous phase liquid out of a foam takes place through the P la teau border channels . The flow of liquid in a single P la teau border channel nmst therefore be unders tood before an a t t empt to model dra inage in a foam is made. An expression for the average velocity (u) of' the liquid in a vertical P la teau border channel has been derived for a t r i angu la r P la teau border cross-section and is given by [56]:

u = 20~-p~ Pc g - ~z (38)

In Eq. (38), z refers to the vertical space co-ordinate (see Fig. 10) which increases in the downward direction and g, Pc, P and p to the gravita- t ional acceleration, density, viscosity and pressure in the continuous phase, ap is the cross-sectional a rea of a P la teau border channel and can be computed from the radius of curva ture of the walls (rp) and the film th ickness (x F) using the expression [571:

(0.322rp + 1.732XF) 2 - 2.721x 2 ap = 0.644 (39)

V

Fig. 10. A plateau border channel,

A. Bhak ta , E. Ruckens te in /Adv . Colloid Interface Sci. 70 (1997) 1-124 29

The factor c v in Eq.(38) accounts for the effect of f inite surface viscosi ty (~s) a nd has been compu ted by Desai and K u m a r [55] as a funct ion of

0.4387~ aq~-p the inverse of t he d imens ion less surface viscosity (Ys = ). I t

~s m u s t be e m p h a s i z e d t h a t the i r resul t s for the calculat ion of cv is val id only for foams since they neglec ted the viscosity of t he d i spersed phase . E q u a t i o n (38) can be used for l iqu id- l iqu id concen t ra t ed emuls ions only t he sur faces are immobile , i.e. w h e n c v = 1. The p ressu re g r ad i en t ( ~ p / ~ z ) can be compu ted as follows:

IfPref is t he p re s su re at a posi t ion Zre f ins ide a droplet /bubble , t he Pi

at any posi t ion z ins ide a drople t is:

P i = Pre f + PD g ( z - Zref) (40)

w h e r e PD is t he dens i ty of the d ispersed phase. Also, i f~ is t he in terfacia l t ens ion be tween the two phases , Pi - P = ~/rp. Thus , if we a s s u m e t h a t t he d i spersed phase is no t compressed, PD is i n d e p e n d e n t of z and Eq. (38) can be r ewr i t t en as:

U-2og -Z °zlr,,JJ=:-i- l 7 w h e r e p = Pc - PD is the dens i ty difference be tween the two phases . For

a foam, in which the d i spersed phase is a gas, Pe >> PD and p = Pc. The

t e r m s in the p a r e n t h e s e s r ep re sen t the dr iv ing forces due to grav i ty (pg)

a nd the g r a d i e n t of P l a t eau border suct ion (-~-~0z/~,,l" These forces oppose

each o the r w h e n the l a t t e r is negat ive , i.e. w h e n the l iquid f ract ion is sma l l e r a t t he top.

Leona rd and Leml ich [57] were the first to consider the effect of sur face viscosi ty on d ra inage in a P l a t eau border channel . E q u a t i o n (38), wh ich differs f rom the i r equa t ion only by a cons tan t factor however has been used more f r equen t ly because Desai and K u m a r have provided a very s imple equa t ion for %. E x p e r i m e n t s [58] on the flow of fluid in ind iv idua l P l a t e a u border channe l s seem to sugges t t h a t Eq. (38) appro- p r ia te ly descr ibes the dependence of the velocity on the viscosi ty and channe l d imens ions . The usefu lness of the factor c v in descr ib ing the effect of surface viscosi ty is, however , still not clear. In mos t of our calcula t ions , we have a s s u m e d c v = 1. For s imula t ions t h a t were car r ied


3.0

2.0

A

== § 0

8. 1.0

8

0.0

-~ ExpedrnentaJ Cubic spline fit

- 1 ° o o ' 11o ' 21o ' 31o ' 4 .0 Surface Density of Surfactant(O.O00001 moles/m.m)

Fig. 11. Piecewise polynomial fit of surface viscosity as a function of the surface dens i ty (F) of surfac tant .

out for sod ium dodecyl sulfate foams, an a t t e m p t was m a d e to re la te the surface viscosi ty to the surface dens i ty (F) of t he sur fac tant . To our knowledge , the re is no avai lable theoret ica l re la t ion be tween the surface viscosi ty (ms) and F. Expe r imen ta l ly m e a s u r e d [59] va lues of TIs as a func t ion of c s and of c S as a funct ion of F are however available. We there fore combined the two sets to get a piecewise cubic polynomial fit of TIs ve r sus F. F igure 11 shows the curve obtained.

2.6. Theoretical model for the drainage and coalescence in foam

2.6.1 Conservation equation for drainage [18] Model ing of the d ra inage in a foam essent ia l ly involves the formula-

t ion of the bu lk conservat ion equa t ions in t e rms of the express ions for f i lm t h i n n i n g (Eq. (4)) and P l a t eau border d ra inage (Eq. (41)). There are


two ways to approach this problem. In the microscopic approach, each Plateau border channel and film in the foam is considered separately and the liquid content in the foam at any time is computed by summing the liquid content in each Plateau border channel and film in the foam. Typically, this would involve the construction of a detailed polyhedral network in which the position and orientation of each channel and film in the foam would need to be specified. An attempt was made to formulate a microscopic model for foam drainage using Voronoi polyhedra to generate the network. In the non-degenerate case, these polyhedra possess two important structural properties seen in actual foams and concentrated emulsions: four and only four edges (Plateau border channels) meet at a point and three and only three faces (films) meet at an edge. The goal was to solve the partial differential equations arising from the mass balances for each channel with appropriate conditions at the junctions and the top and bottom of the foam. The basic balance equation in a single Plateau border channel is:

~ap = - Ouyap (42) Ot ~y

where y is the direction of flow and Uy is the fluid velocity in the y direction. I fa channel is inclined at an angle 0 to the vertical, Uy is given by:

Uy ---- ~ pg cos0 + ~yy (43)

Two conditions need to be satisfied at the junctions in the network: (a) The pressure in the continuous phase is continuous at the junc-

tion. Thus, if rpi a n d rpj are the Plateau border radii of channels i a n d j meeting at a junction, we have:

-- ~ rpi --- rpj (44) rpi rpj

Since there is no accumulation at a junction, the net volumetric flow rate of fluid into the junction is 0. Thus, we have:

~, C i api Uy i = 0 (45)

The summation is over all channels meeting at the junction and c i is a constant which takes the values +1 or -1 depending on whether the


channe l is above or below the junct ion. A serious a t t e m p t was m a d e to mode l foam dra inage us ing this microscopic approach. However , an incredible a m o u n t of cpu t ime would be requi red to get resu l t s of any value. Ins tead , we decided to use the macroscopic approach in which the foam is t r e a t ed as a con t inuous fluid. In th is approach, one does not concern onese l f w i th detai ls of ind iv idua l f i lms and P l a t eau border channe ls . Ra ther , one considers the proper t ies of in f in i tes imal e l emen t s of foam, wi th each e l em en t con ta in ing a large n u m b e r of bubbles. The di f ferent ia l e l e m e n t is large enough to conta in a large n u m b e r of bubbles , yet smal l e n o u g h relat ive to the total vo lume of the foam for a c o n t i n u u m t r e a t m e n t to be valid. This approach is usua l ly appropr i a t e w h e n the bubble size is m u c h smal le r t h a n the l eng th of the foam. Thus , in each e lement , we consider a m e a n P l a t eau border r ad ius (rp), a m e a n

f i lm th ickness (x F) and a m e a n bubble rad ius (/~). The u l t ima te goal is

to ob ta in these quan t i t i e s as a funct ion of t ime and posi t ion wi th in the foam. Since all quan t i t i e s of in te res t such as the l iquid vo lume fract ion (e) can be expressed in t e rm s ofrp, x F and R, the conservat ion equa t ions

are f o r m u l a t e d solely in t e rm s of these quant i t ies . The exact d i s t r ibu t ion ofxF, rp a nd R wi th in an e l emen t is i m p o r t a n t only to the ex ten t t h a t it is r equ i r ed to compute these m e a n quant i t ies . The P l a t eau border r ad ius rp can be a s s u m e d to be un i fo rm wi th in a vo lume e l emen t because of the t e n d e n c y of the capi l lary p ressure (O/rp) to equalize due to the flow of l iquid f rom the th icker to t h i n n e r channels . This is because the p re s su re

(Pi will be smal le r in channe l s wi th smal le r rp and l iquid will flow

f rom regions of h ighe r p res su re ( larger rp) to regions of lower p res su re ( smal le r rp). Thus , a "local" equi l ib r ium of the capi l lary p res su re is e s t ab l i shed in a very shor t t ime and one can a s s u m e rp = rp. However ,

t he f i lm th ickness is not necessar i ly uniform. Even in a monod i spe r se foam, the faces of the polyhedra l bubbles are not ident ical [27] and the re

is a non-un i fo rmi ty in the fi lm areas (A F = xR2). This causes a non-uni-

formi ty of f i lm th icknesses because the ra te of f i lm t h i n n i n g depends on R F (see Eq. (4)). For a given capi l lary p ressure ((~/rp), f i lms wi th smal le r R E dra in fas ter and are therefore th inner . Thus , if REtain ~ R F <_ RFmax a nd f d R F is t he fract ion of fi lms in the e l emen t t h a t have radi i be tween R E and R E + d R f , t he m e a n fi lm th ickness (x F) and t he m e a n fi lm area

(A F) are given b y


RI, mux RF max RF m~x

x--;= I xefeRr;-~e: I AefdRF : IrcR~fdRe (46) RF rain RI." mm RI" mi~l

In an ini t ia l ly monod i spe r sed foam, the bubble size is un i fo rm wi th in an e lement , i.e. all bubbles w i th in the e l emen t have the s a m e rad ius (R) a nd R = R. However , once fi lms s t a r t r up tu r ing , coalescence s t a r t s and gives rise to a size d i s t r ibu t ion wi th in the element, i.e. R ¢ R. The m e a n bubble r ad ius (/~) increases because ne ighbor ing bubbles coalesce to form a la rger bubble. The ra te at which R increases wi th t ime, wh ich in t u r n d e p e n d s on the n u m b e r of f i lms r u p t u r i n g per un i t t ime and hence d e p e n d s on the d i s t r ibu t ion of f i lm areas. This r e la t ionsh ip will be d i scussed af ter we fo rmula te the conservat ion equa t ion equa t ions in t e r m s of the m e a n quant i t ies .

I t m u s t be e m p h a s i z e d here t h a t we ignore the increase in the bubble size due to Os twald r ipening , which occurs due to the diffusion of gas f rom smal l e r to la rger bubbles. In monod i spe r se foams con ta in ing gases l ike n i t rogen , which are spar ing ly soluble in water , the t ime scale for Os twa ld r ipen ing is l ikely to be m u c h la rger t h a n t h a t for d ra inage and collapse, so t h a t foam decay can be said to occur in two stages. In the f irst s tage, d ra inage , coalescence and collapse is dominan t , whi le in the second stage, Os twald r ipen ing domina tes . We consider only the f irst stage. Os twa ld r ipen ing for s t eady s ta te foams has been cons idered by N a r s i m h a n and Ruckens t e in [61].

F igu re 12 shows a schemat ic of a foam w h e n it is f reshly fo rmed (a) and w h e n some decay has t a k e n place (b). The space coordinate z inc reases in the d o w n w a r d direct ion and z 1 and z 2 r ep r e sen t t he u p p e r and lower bounda r i e s of t he foam respectively. The origin (z = 0) def ines the u p p e r b o u n d a r y of the ent i re system. In o the r words, all t he gas e n t e r i n g the foam lies be tween the reference p lane (z = 0) and t he foam/l iquid in ter face (z = z2). Before collapse, all t he gas lies w i th in the foam and z 1 = 0 (see Fig. 12a). After collapse s tar ts , the foam/gas in te r face moves down as gas is re leased from the bubbles at t he top due to f i lm r u p t u r e and z 1 > 0 (see Fig. 12b). The gas re leased f rom the r u p t u r e d bubbles lies be tween z = 0 and z = z 1. The foam vo lume decreases due to the loss of l iquid and the lower b o u n d a r y of the foam moves up as t he d r a inage occurs and z 2 decreases wi th t ime. I t is to be no ted t h a t since wall effects are neglected, all quan t i t i e s a re funct ions of ver t ica l posi t ion z only.

Some var iab les r e l evan t to the macroscopic approach need to be def ined [60,61] before we proceed to formulate the conservat ion equat ions .

34 A. Bhak#a, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

z=O dispersed phase

j z-o

z--L0

Fig. 12. Schematic of a standing foam before and after phase separation.

, ~ m ~ . , 0

As m e n t i o n e d earl ier , we deal wi th groups of bubbles r a t h e r t h a n ind iv idua l bubbles. Cons ider a vo lume e l e m e n t of foam at a level z. Let e be t he vo lume fract ion of l iquid in th is e lement . The vo lume fract ion of gas is t h e n (1 -e ) . I f V i s the m e a n bubble vo lume in th is e lement , t he n u m b e r of bubbles per un i t vo lume (N) in th is e l emen t is g iven by:

1 - ~ N - - - (47)

V

Le t n F be the average n u m b e r of f i lms per bubble. The total n u m b e r of f i lms per un i t vo lume is t h e n Nn F and the a m o u n t of l iquid in the f i lms

RFmax

per un i t vo lume is Nn F f fAl~xFdR F = NnFAI~x F. Similar ly, if np is t he

RF rmn

average n u m b e r of P l a t eau border channe l s per bubble, t he total n u m - ber of P l a t e a u border channe l s per un i t vo lume is Nnp and the a m o u n t of l iquid in the P l a t eau border channe l s per un i t vo lume is Nnpap-[where

ap is t he cross-sect ional a rea of a P l a t eau border channe l in th is e l e m e n t and I is t he m e a n l eng th of a P l a t eau border channel . We a s s u m e t h a t in spi te of coalescence, t he bubbles m a i n t a i n the i r pen t agona l dodeca- h e d r a l g e o m e t r y and t h a t the geometr ica l re la t ionsh ips val id for a single bubble are val id for m e a n va lues as well. T h u s we have:

2r = 0.816R 0.816 ( ~ v / l /3 = (48)


The values ofn F and np follow from the fact t ha t there are 12 films and 30 channe ls in a pentagonal dodecahedral bubble. Since, each film is shared by two bubbles and each P la teau border channel is shared by th ree bubbles, we have n F = 6, n p = 10. The liquid volume fraction is given by the sum of liquid per uni t volume in the P la teau border channe ls and the liquid per uni t volume in the films as:

= NnpapT + Nnt~FXF (49)

Combining Eqs. (47) and (49), e can be expressed as:

npapl + n ~ F x F = ( 5 0 )

V + npap-[ + n ~ T ~

Each of the variables T, ~--FF,F and ap are functions of t ime and vert ical position z.

The var ia t ion of ~ in any volume e lement wi th t ime depends on the net r a te at which liquid flows into the volume element. Hence, the volumetr ic flux of liquid at each z is required in order to formulate the bulk conservat ion equations. This will be considered next.

Calculation of the volumetric flux of liquid Liquid flow in a foam takes place in two ways. Firstly, the re is gravi ty

d ra inage th rough the P la teau border channels . In addition, as the foam becomes drier, the re is a net upward movement of bubbles as the volume of the foam decreases (Z 2 decreases) due to the outflow of liquid from the foam. As these bubbles move, they carry wi th them the liquid in the associated films and P la teau border channels. Thus, an upward convective flow of liquid due to bubble movement is super imposed on the downward gravi ty-dr iven flow through the P la teau border channels . We will discuss each of these components of the flux below:

Gravity drainage through the Plateau border channels [14] The flow ra te of l iquid across any horizontal plane in a foam is given

by the sum of the flow ra tes th rough all the P la teau border channels t ha t in tersect the plane. The flow ra te th rough a P la teau border channel wi th cross-sectional a rea ap will depend on its orientation. It is h ighes t in a ver t ical channe l and lowest in a horizontal channel . Consider a P la t eau border channel inclined 'at an angle 0 with the vertical. The average velocity of l iquid in this channel is given by:


u o = 20q~j_ p pg co~0 + a (51) d z

C o m p a r i n g Eq. (51) wi th Eq. (41), we get:

/ u 0 = cos 0 [20~3-p pg + G ~ = u cos0 (52)

t h u s the vo lumet r i c flow ra te of l iquid t h r o u g h th is channe l is given by:

qo = Uo ap = Uap cos 0 (53)

If we a s s u m e t h a t the P l a t eau border channe l s a re or ien ted r andomly , t he probabi l i ty t h a t a channe l is o r ien ted be tween 0 and 0 + dO is sin(0)d0 and t he average vo lumet r i c flow ra te per channe l is given by:

K

u% = fq0(sin0) dO = 2 (54)

0

The tota l vo lumet r i c flux t h r o u g h the P l a t eau border channe l s across the p l ane is therefore given as

qPB = (number of PB channels per bubble intersected by a horizontal plane) × ( n u m b e r of bubbles in te r sec ted by the hor izonta l p l ane per un i t area) × (mean flow ra te t h r o u g h a PB channel) .

For a p e n t a g o n a l dodecahedra l bubble, the n u m b e r of P l a t e a u border channe l s per bubble in te r sec ted by a hor izonta l p lane is np/5. The n u m b e r of bubbles in te r sec ted by a hor izonta l p lane can be obta ined as shown below us ing a t echn ique common in dea l ing wi th the flow of f luids in porous med ia [60,61]. Let F ( R ) d R be the fract ional n u m b e r of bubbles wi th radi i be tween R a nd R + dR. The n u m b e r of bubbles wi th centers at a d i s tance be tween X and X + dX from the p lane is 2NAF(R)dRdX, where A is the cross-sec- t ional a rea of t he foam. Only those bubbles whose cen ters are at a d i s t ance smal le r t h a n the i r radii (X < R) will in te rsec t t he plane. Therefore, the n u m b e r of bubbles of radius R which intersect the plane is X=R

2 N F ( R ) d R d X = 2 N F ( R ) d R

X=O


w h e r e the factor 2 ar ises due to the fact t h a t bubbles on both sides of t he p lane are considered. The total n u m b e r of spheres per un i t a rea in t e r sec ted by the p lane is t h e n given as

R=oo

f 2 N R F ( R ) d R = 2Ni~

R=O

and the flux of l iquid due to grav i ty d ra inage t h r o u g h the P l a t eau border channe l s is therefore:

Calcu la t ion o f the convective f l u x As m e n t i o n e d earl ier , d ra inage of the l iquid causes a decrease in the

foam volume, which leads to an u p w a r d m o v e m e n t of the bubbles. We need to know the ra te at which the l iquid associated wi th the bubbles moves up. Let qG be the vo lumet r i c flux of gas due to the convect ion of the bubbles. The n u m b e r of bubbles moving per un i t a rea per un i t t ime is t h e n g iven by qa/V. Since the a m o u n t of l iquid per bubble is d N , t he ra te (qe) a t wh ich the l iquid is convected a long wi th the bubbles (qe) is g iven as:

qc = N u m b e r of bubbles moving up per un i t a rea per un i t t ime x Vo lume of l iquid associa ted wi th each bubble (56)

qG ~ g - V N - q G 1 - e

The tota l flow ra te of l iquid per un i t a rea per un i t t ime (qL) is ob ta ined by s u m m i n g the cont r ibu t ion due to the P l a t eau border d ra inage (qPS) and t he convect ion due to bubble m o v e m e n t (qc) and is g iven as:

qL = qPB + qc = qPB + qa (57)

However , we still do not know qv" When the foam dra ins , t he convect ive mot ion of the bubbles occurs in the upward direction as the foam volume decreases, i.e. qv is negative. Also, since this m o v e m e n t occurs due to dra inage in the P la teau border channels, it is na tu ra l to expect qa to be re la ted to qPS. The re la t ion be tween these two quan t i t i e s is der ived as follows.


Since the rate of accumulation of liquid in a volume element equals the net flow rate into the volume element, the conservation equation for the liquid is:

~ ~qL - ( 5 8 )

Ot Oz

If the dispersed gas phase is assumed to be incompressible, a similar conservation equation can be written for the gas phase as:

~qG 0t (1 - a) - 0z (59)

i.e.

D t - Dz (60)

Comparing Eq. (58) and Eq. (60) we have:

-qL = qG (61)

Combining Eqs. (56) and (60), qL and qG can be expressed in terms of

qPB a s :

qG = - ( 1 - - £)qPB (62)

qL = (1 - e)qpB ( 6 3 )

Thus, the total flux (qL) is smaller than the flux (qPB) due to Plateau border drainage alone because bubble convection moves some liquid upward.

Equation (58) can therefore be rewritten as:

~ 2 ( 1 - e)qpB - ( 6 4 )

~t ~z

Equation (64) is the conservation equation for the liquid in which the quantities e and qPB can be expressed in terms of rp, )[-~F and/~ using Eq.(48), Eq. (49) and Eq. (50). It is to be noted that Eq. (64) is valid whether coalescence occurs or not. Coalescence occurring in a volume element does not by itself change the amount of liquid in the element.


It only causes a redis t r ibut ion of liquid from the rup tu red films and P la t eau border channels to the remain ing P la teau border channels wi th in the element . The change in the amount of liquid in the e lement occurs only due to drainage. The effect of coalescence is felt t h rough R. When coalescence occurs, the local R increases. The expressions for the change in /~ due to coalescence are derived later.

The two boundary conditions required for Eq. (64) are presented next.

Boundary conditions dz 1

When the foam is collapsing, i.e. when ~ > 0, the liquid re leased

from the collapsed bubbles reenters the foam. The amoun t of liquid dz 1

re leased depends on the ra te of foam collapse (--~-). Equa t ing the ra te

at which liquid is re leased from the collapsed bubbles (e dz_~ Zl ) to the

flow ra te of l iquid at the foam gas interface (qLI 1) gives the boundary

condition at the foam/gas interface (z = Zl):

°1 I qL zl = a - - ~ zl

(65)

The o ther boundary condition follows from the fact tha t the bubbles are spherical at the foam/liquid interface:

rp]z2 = R] z2 (66)

Since coalescence never occurs at the foam/liquid in te r face , / [ I z = R0, . . . . 2

where R o is the r a d m s of a bubble m the freshly formed monodmperse foam.

It m a y be noted tha t the effect of bubble convection was ignored in our ear l ier papers [14-17]. While this does not affect the computed liquid profiles much, it cannot be ignored when the conservation equat ions for the sur fac tan t are formulated, since bubble convection produces a s ignif icant movemen t of the surface area and hence of the adsorbed surfactant . The balance equat ions for the sur fac tan t are considered next.


2.6.2 Conservation equations for surfactant Par t of the sur fac tan t in the foam is present in dissolved form in the

l iquid in the films and P la teau border channels while the rest is adsorbed on the film surfaces. Thus, the amoun t of sur fac tant per un i t volume of the foam is given by ecs + ( 2 N n F FAR, where the first t e rm (ec~) represen ts the sur fac tan t dissolved in the liquid contained in the films and the P la teau border channels and the second t e rm corresponds to the adsorbed sur fac tan t on the film surfaces. The t e rm

RF max

FAR= ~ WFAI~IRF

RFmin

is the m e a n value of the sur fac tant adsorbed on the surface of a film. The mass flux of the sur fac tant contains three contributions:

(a) from the flow of liquid in the P la teau border channels (qLc~);

~c s (b) from the dispersion (-aD -~-z ) where D is the apparen t diffusivity;

(c) from the movement of the adsorbed surfac tant due to the convec-

t ion of bubbles ( - ~ 2n F FA F) where ~ is the n u m b e r of bubbles per un i t

a rea per uni t t ime moving in the downward direction and 2n F FA F is the

sur fac tan t adsorbed per bubble. It mus t be emphas ized tha t D is different from the molecular diffusiv-

ity and depends on the degree of local mixing. D is actual ly different in different portions of the foam. We have, however, a s sumed D to be cons tant in order to avoid complexity. The net flux of sur fac tan t (qes) is then given as:

~Cs + ~ (2nFFAF) d ) qcs = qLCs = ~ -~z = qLCs - ~Cs qL

- (2nFFAF) (67) ~z F

and the conservat ion equat ion for the sur fac tan t can be wr i t t en as:

~ (~cs + 2NnFF---flffF)__ Dqcs 2 [ ~Cs q--~-L 2nFFAFJ (68) Dt ~ - ~z qLCs-aD ~z - V"

This equat ion is also second order in space and we need two boundary conditions for the sur fac tant concentrat ion c s.


Boundary conditions for surfactant Before collapse starts , there is no surfac tant en te r ing the sys tem at

the top. In o ther words , the su r fac t an t flux at the foam/gas interface (z = z 1) is 0. The boundary condition at the top before collapse is therefore:

qcs [z1 = qLCs - aD 3c~s3z qLv (2nFF---A-FF) Zl : 0 (69)

Now, since from Eq. (65), qL [z 1 becomes:

is 0 before collapse starts , Eq. (69)

bz = 0 z 1

(70)

On the o ther hand, once collapse starts , the sur fac tan t in the collapsed bubbles is re leased into the foam. I fA is the cross-sectional a rea of the foam, the n u m b e r of bubbles collapsing when the front moves with a

d Z l . A dzl velocity ~ is - - . Also, the sur fac tant associated with a bubble

V[z 1 dt

includes the amoun t adsorbed on the film surfaces and tha t dissolved in the liquid in the films and P la teau border channels and is given by (npap7 + n~FF)Cs + 2nF~[zx . Thus, the flux of sur fac tan t into the

sys tem due to collapse is given by:

(number of bubbles collapsing per uni t a rea per uni t t ime) x (amount

1 dzl [(npap-[+ nt~-~-FF) cs + 2nFFAF]z~ of sur fac tan t per bubble) - V[zl dt

Equa t ing this to the flux of sur fac tant at the foam/gas interface from Eq.(67), we get the boundary condition:

-~sD Oz + qLc~ V (2nFP-~r) ~ dt - z l - [(npap-[ + nFAFXF)C s + 2nFFAF]Iz ~ z 1

(71) dz 1

Using Eq. (65) to e l iminate - ~ and simplifying we get:


aD Oz + qL Cs + VE (2nFF~F) = 0 (72) Z 1 Z 1

The o the r b o u n d a r y condit ion is ob ta ined from the fact t h a t a t t he foam/l iquid interface, the concent ra t ion is t h a t of the or iginal foaming solut ion (Cso)

c sl =cs0 (73) Z 2

The basic conservat ion equa t ions for the con t inuous phase l iquid and the s u r f a c t a n t have now been formula ted . However , several addi t iona l deta i l s are r equ i red to specify the sys t em completely. Express ions are needed for the m o v e m e n t of the boundar ie s z 1 and z 2. F u r t h e r m o r e , t he m e a n va lues ~F, ~ a n d / ~ depend on the degree of coalescence. We shal l f i rs t d iscuss coalescence and t h e n fo rmula te express ions for t he m o v e m e n t of t he boundar ies .

2.6.3 Coalescence [18]

W h e n a f i lm r u p t u r e s wi th in a foam, the bubbles sha r ing the film coalesce. Coalescence resu l t s in a decrease in the n u m b e r of bubbles in t he e l e m e n t and an increase in the m e a n bubble vo lume in the e lement . The a s s u m p t i o n m a d e in our model is t h a t coalescence wi th in any v o l u m e e l e m e n t in the foam occurs because the faces (films) of t he po lyhedra l bubbles are not identical . Since the films differ in size (RF) , t hey d r a in at d i f ferent ra tes (see Eq. (4)) and r u p t u r e a t d i f ferent poin ts in t ime. The fi lm radii (R E ) are a s s u m e d to lie be tween two l imi t ing va lues REtain and RFmax and the d i s t r ibu t ion of R F about a m e a n va lue RFO = 0.606R 0 wi th in these l imits is a s s u m e d to be given by the funct ion

expL- 2s J J (74)

S exp - F dR F

RFmin

RFmax

Note t h a t j f dR F = 1. Thus , fdR F is the fract ion of the fi lms in t he

RFmin

e l e m e n t t h a t have radi i be tween R E and R E + dR F. It m u s t be m e n t i o n e d


t h a t th is does not imply a var ia t ion in bubble volumes. The foam is a s s u m e d to be ini t ia l ly monodispersed , i.e. all the bubbles have the s a m e volume. A bubble size d is t r ibu t ion ar ises only af ter coalescence s tar ts .

It is clear f rom Eq. (4) tha t , since all t he fi lms are subjected to the s a m e capi l lary p r e s su re (rp = rp), f i lms wi th smal le r R E will d ra in fas ter a nd hence will r u p t u r e earlier. Thus , if we mon i to r a f reshly fo rmed group of f i lms in a cer ta in vo lume e lement , f i lms wi th the smal les t R E will r u p t u r e first, followed by fi lms t h a t are s l ight ly la rger and so on unt i l all t he fi lms at th is level have e i ther r u p t u r e d or have s topped dra in ing .

Before coalescence s tar ts , RFmin is cons tan t and equal to RFmin 0 ( the va lue Of RFmin in f reshly formed foam; an i n p u t paramete r ) . The distribu t ion in a ny vo lume e l em en t in the foam before coalescence s ta r t s is the re fore the s a m e as t h a t w h e n the foam is j u s t formed and is given by

2

e x p - 2s ) J = (75) f0=f =0 RFm r 2

Sex,_ / R~nin

Once coalescence s ta r t s , however , the va lue of RFmin in an e l emen t is no t a c o n s t a n t and increases as coalescence proceeds (see Fig. 13). As RFmin increases , t he n u m b e r of bubbles decreases and the m e a n bubble vo lume (V) increases . At the s ame t ime, l iquid and su r f ac t an t f rom the r u p t u r e d f i lms are d i s t r ibu ted am ong the r e m a i n i n g P l a t eau border channe ls . I t m a y be no ted t h a t we a s s u m e RFmax to be a cons tant . This is not s t r ic t ly t rue. I t is possible t h a t geometr ica l r e a r r a n g e m e n t s occur r ing d u r i n g coalescence will give rise to films larger t h a n RFmax. I t m a y also be no ted t h a t f and REtain and hence the m e a n bubble vo lume are func t ions of only the vert ical posi t ion (z) and t ime (t). This is because, if wall effects are neglected, t he capi l lary p res su re (~/rp) at a given level (z) is i n d e p e n d e n t of the radial posi t ion and any vo lume e l emen t at a given level will have the s a m e d is t r ibu t ion of f i lm radii. On the o the r hand , a t a ny t ime t and at each z, we need to know the f i lm th ickness x F cor respond ing to each R E so t h a t we can d e t e r m i n e w h e n coalescence s t a r t s a t each level and how quickly coalescence proceeds. T h u s we need to c ompu te x F as a funct ion of t h ree var iables , viz. t, z and R E us ing Eq. (4). This is a mov ing b o u n d a r y problem, since one of the boundar i e s in R E space (R E = RFmin) moves when coalescence s tar ts and both boundar ie s


f fo ~'

I l

R FminO R Fmin R Fmax

Fig. 13. The d is t r ibut ion f of R E before and af ter coalescence.

in z space (z = z 1 and z = z 2) move w h e n foam col lapse s t a r t s . The c o m p u t a t i o n is s impl i f ied cons ide rab ly if t he v a r i o u s b o u n d a r i e s a re immob i l i zed b y ca r ry ing ou t a t r a n s f o r m a t i o n f rom t, RE, z space to t, r,

space where :

Z -- Z 1 RF - RFmin - - - ; r = (76)

Z 2 -- Z 1 R F m a x - RFmin

W i t h th i s t r a n s f o r m a t i o n , t he v a r i a b l e s ~ and r lie b e t w e e n 0 and 1 and t he R e y n o l d s e q u a t i o n (Eq. (4)) can be w r i t t e n in t, r, ~ space as:

- - o[,~XF] ~ ~XF t,~ddt ~ RF, Zq- ~ t,r-~ RF,Z dXF -~- Vf.=-----~ r' + ~XF dR

Rv,z (77)

whe re :

I 1 E o1 -£i z2 - z l - ~ - d ? R~z- ( ~ - I ) -

(78)

a n d


RF,Z

( r - 1) dRFmin z- ( r - 1) VORFmin

(RFmax - RFmin) dt (RFmax RFmin) I 8----'~--

Using Eqs. (78) and (79), Eq. (77) becomes:

I 0RFmin dZl dz 2 - V f = ~x~: +b I ~ I+b2 +b3

0t r,~ ~I ]~ ~ - dt

where

OXF (r -- 1)

bl = Or (RFmax- RFmin

Z 2 -- Z 1 [ 0~ 4- (RFmax_ RFmin)

and

~RFmin d~ 7.

(79)

(80)

(81)

(82)

b3- -~ [Ox F (r- 1) 0RFmin 1 2 2 -- Z ~ [ " ~ - + (RFmax RFmin) - ~ J

(83)

In order to specify the system completely, expressions for the rate of movement of the various boundaries (z I and z 2 in z space and RFmin in R E space) are required. The expression for dRFmin/dt is derived below.

Express ion for dRFmin/dt Before coalescence starts, the value of RFmin is constant throughout

the foam and is given by REtain 0 which is an input parameter. Coales- cence begins at a given z when the thinnest films at that z (those with R E = REtain o) become critical (x F = XFc) and rupture. Once this happens, RFmin starts increasing. Since the capillary pressure and the disjoining pressure isotherm are different in different parts of the foam, the degree of coalescence and hence the value Of RFmin will be different in different portions of the foam. Thus RFmin is a function of time t and vertical position z. As long as coalescence occurs, RFmin increases and the rate of increase is obtained by recognizing that x F I z~ , i.e. x F I ~,r=0 always . . . . ~mi corresponds to the critical thickness (XFc) which m ~;urn is a function of the surfactant concentration (c S) at that ~. Thus coalescence at a given

is characterized by the following conditions

46 A. Bhakta, E. Ruckenstein /Adv. Colloid Inter{ace Sci. 70 (1997) 1-124

XF[ ~,r=O = XFc(C s I ~ )

i.e.

-Vr r=0, = - v r

Different ia t ing Eq. (84) wi th respect to time, we get

~x F

3t ~,r=-O

~XFc dXFc

Ot ~ dc s at

Combining Eq. (86) wi th Eq. (80) for r = 0, we get

- g f r = o , ~ - - xFc(sl 0 d c s Ot ~ + b l l r = ° ' ~ -

dz 1 dz 2 + b2 r=O,~ d--t- + b3 r=-O,~ dt

which can be rewr i t ten as:

0RFmin

dRFmin dt ~-

-Vf[ xpc(cs it ) dc s ~C s

8t dz 1 dz 2

- b2 I r=-0,~ d~- - b3 [r=-0,~ dt

(84)

(85)

( 8 6 )

(87)

(88) b l[ r:-0,~

Before we proceed fur ther , some features of Eq. (88) need to be noted. The ra te (dRFmin[dt) at which coalescence proceeds at any level depends

s t rongly on the film velocity (Vy) corresponding to the critical thickness (XFc) at t ha t level. Since b 1 < 0, dRpmin/dt increases as the film velocity

corresponding to the critical thickness increases. On the other hand ,

dXFc since --=-- < 0, the coalescence ra te will decrease if the local sur fac tan t

dc s

concentra t ion increases, i.e. if~t~ > 0.

As ment ioned earl ier , coalescence gives rise to a bubble size distribu- t ion in an ini t ial ly monodispersed foam. The number of bubbles decreases and the mean bubble volume increases. Owing to the complexity involved in comput ing the evolution of the bubble size distr ibution, we


res t r ic t our t r e a t m e n t to the calculat ion of the m e a n bubble vo lume ]7 wh ich is r equ i red in the fo rmula t ion of the bulk conservat ion equat ions . This is cons idered next .

Calculation of the mean bubble volume Before coalescence begins, ]7 is the s ame t h r o u g h o u t the foam and

4 ~ 3 ) . As equal to the vo lume of the bubble as it is f irst formed (V 0 = coalescence s ta r t s , i.e. as REtain s ta r t s increasing, ]7 increases because t he r u p t u r e of a f i lm causes coalescence of the ne ighbor ing bubbles t h a t s ha r e the f i lm and the s ame vo lume of gas is d i s t r ibu ted a m o n g a smal le r n u m b e r of bubbles. Cons ider a cer ta in vo lume e lement . If/Co is t he d i s t r ibu t ion of f i lm radii in th is e l emen t before coalescence s ta r t s (see Eq. (75)), t he following re la t ionsh ip is a lways satisfied:

N u m b e r of f i lms in the e l emen t at any t ime t = (Init ial n u m b e r of RFmax

f i lms in t he e lement ) x f fodRF

RFmin(t)

(89)

I t m a y be no ted t h a t

RFmax

j f0dRF = 1

RFmm(t=O)

and f0 is only a funct ion of R E and is i n d e p e n d e n t of t ime. As fi lms r u p t u r e a nd d i sappear , RFmin (t) increases and

RFmax

RFmin(t)

decreases (see Fig. 14). Since the n u m b e r of bubbles in this e l emen t is p ropor t iona l to the n u m b e r of films, we have f rom Eq. (89):

N u m b e r of bubbles in the e l e m e n t at any t ime t =

RFmax

In i t ia l n u m b e r of bubbles in the e l emen t × fodRF (90)

RFmin(t)


R FminO R Fmax

BEFORE C O A L E S C E N C E

fo

R FminO R Fmin R Fmax

AFTER COALESCENCE

Fig. 14. The a rea under the fo curve is proport ional to the degree of coalescence


M e a n bubble vo lume in e l emen t = Volume of gas in e l e m e n t

N u m b e r of bubbles in e l e m e n t (91)

Now since the vo lume of gas in th is e l emen t r e m a i n s unchanged , t he m e a n bubble vo lume in the e l emen t at any t ime is inverse ly propor t iona l to the n u m b e r of bubbles in the e lement . In o the r words:

N u m b e r of bubbles in e l em en t before coalescence N u m b e r of bubbles in e l e m e n t af ter coalescence

Mean bubble vo lume af ter coalescence Ini t ia l bubble v o l u m e

(92)

T h u s we have:

v(t) V(t = 0)

RFmax

fodRF RFn~n(t=O)

R Fmax R Fmax

I I roOF RFrain(t) RFmir~t)

(93)

All t he equa t ions dea l ing wi th coalescence have now been fo rmula ted . Express ions for t he m o v e m e n t of the two foam boundar i e s (z 1 and z 2) a re p r e s e n t e d next.

2.6.4 Movement of the foam/gas interface W h e n REtain at any z becomes equal to RFmax, all t he fi lms at th i s z

a re crit ical and r u p t u r e s imul taneous ly . W h e n th is h a p p e n s a t t he foam/gas in te r face (z = Zl), all t he bubbles at the top of the foam collapse and t he in te r face moves d o w n w a r d i.e. z 1 s ta r t s increasing. In o the r words, t he b o u n d a r y s t a r t s moving w h e n the d i s t r ibu t ion funct ion 09 at t he top becomes a Dirac-del ta function. The moving in ter face is, therefore, a lways charac te r ized by REtain = RFmax = cons tant , i.e.

~RFmin = 0

~t ~=0

Us ing th is condi t ion in Eq. (88) wi th ~ = 0, we get:


dXFc 0cs dz 1 dz 2 -VflxFc(csl~ =o) dc s ~t -b2ir=°'~=° dt -b3]r=O'%=° dt

O= ~=o bllr:o,~:o (94)

Since b31%=o, Eq. (94) can be r ewr i t t en as:

dXFc Ocs ~=o dzl -Vf[ Xvc(c,l~:o) dc s Ot

= ( 9 5 ) dt b21 r=O,~=O

Since it is not possible numer ica l ly to deal wi th the Dirac de l ta funct ion, collapse was deemed to occur w h e n the d is t r ibu t ion was na r row enough , i.e. w h e n RFmin became equal to RFc where RFc is s l ightly smal le r t h a n RFmax and is given by RFc = RFmax/1.001.

I t m a y be noted t h a t w h e n the critical th ickness (XFc) is i n d e p e n d e n t of t he su r f ac t an t concentra t ion , as is probably the case w i th non ionic su r fac t an t s , Eq. (95) reduces to

dz l - V r l = (96)

dt b21r__O,%=O

F u r t h e r m o r e , if t he re is no var ia t ion in f i lm area, t he t e rms involved w i th R F space d i sappea r f rom the above equa t ions and we have:

d z I vfl< - ( 9 7 )

dt 1 OXF I

z2 zl 0%

E q u a t i o n (97) was used in our ear l ier papers [16,17] in which coalescence was ignored.

2.6. 5 Movement of the foam / liquid interface The foam/l iquid in terface moves up (z 2 decreases) as l iquid d ra ins out

of t he foam at the bot tom. This u p w a r d m o v e m e n t occurs because the vo lume of the foam decreases as l iquid leaves the foam. Thus , the ra te at wh ich th is in terface moves up is equal to the ra te at which the he igh t of t he l iquid pool at the bo t tom increases , wh ich in t u r n depends on the flow ra te of l iquid t h r o u g h the P l a t eau border channe l s at the bot tom. Therefore , i fA is the cross-sectional a rea of the foam:


d z 2 dz 2 - A ~ = A q p B ]z2 , i.e. dt - qPBIz2 (98)

The sys tem is now completely specified and wi th the appropr ia te init ial conditions, the conservation equat ions can be solved wi th equations for the change of/~, RFmin , z 1 and z 2. Before we proceed wi th the formulat ion of the init ial conditions and the solution of the differential equations, we will briefly discuss the phenomenon of drainage equilibrium.

2.7 Drainage equil ibrium

It has been exper imenta l ly observed tha t a d ra in ing foam usual ly arr ives at a metas tab le mechanical equi l ibr ium in which all processes associated wi th dra inage come to a halt. This equi l ibr ium is character ized by the following properties:

(1) There is no flow in the P la teau border channels , i.e. u = 0 th roughou t the foam. This happens when the gradient in capil lary

~ [~p) Indeed, exper imenta l pressure balances gravity, i.e. when pg = - Ozz

measurements [62] of the pressure in a draining foam indicate tha t after a sufficiently long time, the pressure profile becomes l inear with a slope equal to pg.

dXF (2) There is no film thinning, i.e. dt - 0 th roughout the foam. This

occurs when the repulsive disjoining pressure (H) in the films balances

the capi l lary pressure (_~_G). In other words, ~ = H th roughout the foam. rp rp

(3) The sur fac tan t concentrat ion does not change wi th time. There is

~c s no diffusion (-~-z = 0) and there is no convection (u = 0).

(4) There is no movement of the boundaries , i.e. there is no collapse dz I dz 2

at the top ( -~- = 0) and no dra inage at the bottom (--~- = 0). This actual ly

follows from conditions (1) and (2). If there is no drainage, rp will not change and the capil lary pressure cannot increase to overcome the m a x i m u m disjoining pressure (Hma x) and cause collapse. Thus, in an

equi l ibra ted foam, ~p _< I-lma x and x F > XFc.

The init ial conditions are formulated in the next section.


2.8. Initial conditions

The extent of drainage and collapse that occurs in time will depend on the difference in the amount of liquid in the foam initially and at equilibrium. The larger this difference, the larger is the amount of drainage that will occur. The state of the foam (distribution of liquid) at the start of the experiment therefore plays an important role in determining its drainage and collapse behavior. The initial distribution of liquid, however, depends on the manner in which the foam is produced. Foams can be divided into two categories depending on the method of generation: (1) homogeneous foams, and (2) pneumatic foams.

Homogeneous foams are usually produced by vigorous agitation of surfactant solutions. As the name suggests, the distribution of liquid is uniform in such foams. Pneumatic foams, on the other hand, are produced by bubbling a gas through a surfactant solution. As the bubbling proceeds and the length of the foam increases, there is drainage of liquid in the foam and a profile of liquid fraction develops. The foam is drier (liquid fraction is lower) higher up in the foam where the bubbles are older and more drainage has occurred.

We will first discuss homogeneous foams in detail and then consider pneumatic foams. Before we proceed further, it must be mentioned that several assumptions were involved in obtaining most of the results that will be presented in the following sections:

(1) Coalescence was ignored [14-17]. The films areas were assumed to be the same and the film thickness was only a function of the vertical position 'z'.

(2) The surfactant concentration was assumed to be uniform, i.e. the surface potential and surface tension were assumed to be constant.

(3) In our first two papers [14,15], film drainage was ignored. These assumptions were relaxed as we learned more about the

system. However, most of our results (which were qualitative in any case) remain valid. In fact, an analysis of the results obtained using these simplifying assumptions actually improves our understanding of the system. The results and the simplifying assumptions are presented below.

2.9. Homogeneous foams [15]

2.9.1. Initial and boundary conditions for homogeneous foams As mentioned earlier, homogeneous foams are characterized by a

uniform initial liquid fraction. The initial condition for homogeneous foams therefore is:


At t = 0, e = e o for all z (99)

The o ther d is t inguish ing proper ty of homogeneous foams is t ha t they do not s ta r t d ra in ing immediately . Some t ime usual ly elapses before some cont inuous phase liquid appears at the bottom of the foam. This is in cont ras t to pneumat ic foams which are always in contact wi th the cont inuous phase liquid at the bottom. It has also been noted [63] t ha t in homogeneous foams, phase separat ion occurs only when ~ at the bot tom exceeds a cer ta in m i n i m u m (£b) (the exact value though has not been specified). We postulate tha t for a "monodisperse" foam/concen- t r a t ed emulsion, this value corresponds to the point when the amoun t of l iquid at the bottom becomes large enough for the bubbles at the bot tom to become spherical (rp = Ro). The liquid fraction e at this point corresponds to t ha t for close-packed spheres, i.e. ~ = 0.26 - ~b. The reasoning behind this can be understood as follows: Consider a foam/ concent ra ted emuls ion wi th a uniform init ial cont inuous phase fraction of e 0 < £b (say ~0 = 0.15). The bubbles at the bottom will be polyhedral (see Fig. 15a). As t ime passes, the P la teau border channels at the top dra in into those below, so tha t e decreases at the top and increases at the bottom. Suppose, at a given point in t ime, ~ at the bot tom jus t exceeds 0.26. At this ins tant , (see Fig. 15b) the bubbles at the bot tom will cease to be in contact and become spherical. (Implicit in this a r g u m e n t is the assumpt ion tha t droplet radius is much smal ler t han the capi l lary length (R ~ >> ' ~q-~pg).) However, due to buoyancy, these bubbles will rise and ar range themselves in a compact fashion (~ = 0.26) (see Fig. 15c) causing the excess continuous phase to accumulate at the bottom as a separate phase. As drainage proceeds, this process continues and the interface rises (Z 2 increases), always mainta ining e = 0.26 at the bottom till a d ra inage equi l ibr ium is established. Thus, there is no flow at the

( a ) e=6o ( b ) E>eb (c) Phase Separation

Fig. 15. Mechanism of continuous phase separation.


bottom as long as ~ I z-z < % and E I z=z does not change once it a t ta ins - - 2

the va lue %. The appropria te boundary condition at the bot tom is therefore:

t < t b ~ u l z ~ 2=0, t > t b ~ a = a b (100)

whe re t b is the t ime when ~ at the bottom jus t becomes E b. Thus, for a homogeneous foam, there are two critical conditions: one

for the separa t ion of the dispersed phase (x F = XFc) and the other for the separa t ion of the continuous phase (% = R 0 or ~ = e b = 0.26). Four scenarios are therefore possible for a foam/concentrated emulsion wi th a cer ta in ini t ial uni form continuous phase volume fraction (%) depending on w h e t h e r one or both these critical conditions are achieved before equi- l ibrium:

(a) Only the cont inuous phase separa tes out at the bottom; (b) Only the disperse phase separa tes out at the top; (c) Both phases separa te out; (d) There is no phase separat ion. By analyzing the equi l ibr ium state of a foam in relat ion to its init ial

state, it is possible to de termine , at the outset, which of the above s i tuat ions arise. In the following sections, we will discuss the equilib- r i um state in detail and show tha t by mak ing suitable approximations, it is possible to de te rmine analyt ical ly the phase behavior from the ini t ial conditions.

The sys tem can be simplified considerably, if films are neglected. In this section, we will discuss the equi l ibr ium in detail using simplified equat ions tha t are obtained when films are neglected. Fi lm dra inage is impor t an t in de te rmin ing the kinetics of foam collapse (see Eq. (95)). However , since most of the liquid in a foam is stored in the P la teau border channels , films can be ignored when computing the l iquid volume fraction. Our main concern in this section is wi th the dra inage equilib- r ium. This assumpt ion is therefore valid. This assumpt ion was made in our first two papers [14,15] and proved to be useful in obtaining insights into m a n y features of foam decay. Also, in an equi l ibrated foam, the bubble rad ius is uniform and equal to the init ial bubble radius (R 0) because no coalescence has occurred in foam tha t remains at equilib- r ium. When films are neglected, the liquid fraction e depends only on the local P la teau border radius rp and the various macroscopic variables can be expressed as


V ,ap: - ~ ~-e ,rp 0~3(0.161in~(0.816)~1. s

(101)

Let us re -examine the condition for foam collapse wi thin the f ramework of this assumption.

Since film rup tu re can occur only if the capil lary pressure ((~/rp) exceeds the m a x i m u m disjoining pressure (Hmax) , it is possible to define a critical P la teau border radius (rpe) given by:

G rpc -- l - Imax (102)

For film rup tu r e to occur, the P la teau border radius (rp) mus t be less t h a n rpc. When a collapsing foam arr ives at an equilibrium, i.e. when dzl/dt decreases to 0, rp = rpc at the foam/gas interface. This is because (see Eqs. (95-97)) when collapse comes to a halt:

Vfl = 0 i.e. ~ = r l m a x at z = z a (103) XFc rp

i.e.

E] = E c z 1

(104)

where e c is the liquid volume fraction corresponding to rp = rpc. Let us now consider the equi l ibr ium sta te in more detail.

2.9.2 Drainage equilibrium in homogeneous foams Since at equil ibrium, u = 0 for all z, we have:

dz - -

Using Eq. (101), Eq. (105) can be wr i t ten in t e rms of e at equi l ibr ium

(Eeq) a s :

~ 3(0~161)np0.816 1 d ] 1 - eeq = - pg

R-0d-z~ Eeq (~ (106)

Now let us define:

5 6 A. Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

4rt K-- 3(0.161)np0.816

Integration of Eq. (106) gives:

~ e e q _ KpgRoz e q (:Y

+ c (107)

where "c" is a constant of integration. Equation (107) can be rewritten a s :

1 eeq = (108)

Let z 1 = Zle and z 2 ~- Z2e at equilibrium. The volume of the continuous phase per unit cross-section at equilibrium is then given by:

Z2 e

Veq= ~ eeqdz (109)

Zl e

Substi tut ing the expression for Eeq from Eq. (108) into Eq. (109) and integrating, we get:

Veq- KpgRo [tan-l (c - gPgR°Zle)- tan-l (c g~°z2ell (110)

IfL o is the initial foam length, the volume of liquid per unit area initially present in the foam is L0e 0. The distance moved by the lower boundary by the time equilibrium is established is equal to the loss of continuous phase per unit cross-section. Thus:

L o - Z2e = LoG o - Yeq (111)

Combining Eq. (110) and Eq. (111) gives:

Lo-z2e=Lo~o-KpgRoltan-l(c gP~°Zlel-tan-llc (112)


To accurately de te rmine the conditions at equilibrium, we need to evaluate the constant of integrat ion c, i.e. we require eeq at one of the boundaries. There are four possible scenarios for the state of the foam/ concentra ted emulsion at equilibrium. Each is discussed below.

Case a: The equilibrated foam is in contact with the continuous phase at the bottom

In this case, since there is no collapse, we have Zle = 0 and since the foam is in contact with liquid at the bottom, eeqlz= z = ~b = 0.26. Using these conditions in Eq. (107) gives the integrat ion co~hstant for this case:

~ - ab K p g R o Z2e c = + (113) V E b

Using this expression for c Eq. (110) and Eq. (112) gives:

Veq - KpgRo t a n l Eb- eb I t a n (114)

Lo - Z2e = LO tO c [ l ( ~ - e b

KpgRo [ tan- ~ ~b + KpgR°(T Z2e) _ tan.1 ( ~ b ~ b ) J l - eb

(115)

Case b: The equilibrated foam is in contact with the separated disperse phase at the top

In this case, since no dra inage of liquid from the foam has taken place, the lower boundary does not move and Z2e = L o. Also, since a collapsing foam arrives at equi l ibr ium with rp = rpe at the top, e[z=zl e = ~c. The constant of integrat ion is given by:

~ 1 -a,. KpgRo zl,, c = + (116)

EL. (Y

Subst i tu t ing the expression for c in Eqs. (112) and (110) we get:


Lo eo (~ F 1 ( ~ ~ 1( ~ KpgRo

] t a n | ~ ] - - - - - ~ | - t a n - I~] - - " (Lo-z le ) KpgRo L \ V ec J \ ~1 ac (~ ) = 0

(117)

g e q - - - KpgRo t a n 1 _ t a n - 1 1-acaC KpgR°cr ( L o - Z l e ) ) ]

(118)

Case c: Both phases separate In this case, since both drainage and collapse has t a k e n place, zle > 0

= = e b. T h u s f rom Eq. (107) we have: and Z2e < L o el z=zl e a c and ~lz=z2 e

~1 - el, KpgR,) Z2e c . . . . ÷ (119)

El, (~

Also, we have:

~ - a,. KpgRo zl,, c = ~ (120)

Ec

Combin ing Eqs. (119) and (120), we have:

- ° [ , F Z2e -- Zle KpgRo L V ec V eb J

(121)

In Eq. (121), Z2e --Zle is the l eng th of the foam at equi l ibr ium. Since the r igh t h a n d side of Eq. (121) is a cons tan t for a given sys tem, it is clear t h a t w h e n bo th phases separa te , the final l eng th (Z2e - - Z l e ) of t he equ i l ib ra ted co lumn is fixed and given by Lma x.

An i m p o r t a n t po in t to be noted is t h a t the l eng th of a foam canno t increase. W h e n one or more phases separa te , the l eng th of t he foam has to decrease. I f t he re is no loss of l iquid or g a s , t he re is no change in the foam length. Thus , the following condi t ion a lways holds:

Z2e -- Z le ~ L o (122)


Thus, if L 0 < Lmax, it is not possible to have an equi l ibr ium wi th both phases separat ing. In other words, when L o < Lma x only one phase will separate . The init ial composition (e 0) of the sys tem de te rmines which phase separates . If dur ing drainage, e at the bottom exceeds e b before the films at the top rupture , the continuous phase separa tes and Eqs. (113-115) are valid. On the other hand if film rup tu re occurs first, collapse occurs at the top and Eqs. (116-118) are valid. If, however, u becomes zero before e i ther of the above occur, no phase separa t ion takes place. The init ial s ta te of the foam (L0, e 0) therefore plays a crucial role in de t e rmin ing the phase behavior. The effect of the init ial condition on phase separa t ion is discussed next.

2.9.3. Effect o f initial condition on phase separation Based on the init ial s ta te of a foam/concentrated emulsion, it is

possible to de te rmine at the outset which of the two phases separa tes out. As has been discussed earlier, if L 0 < Lmax, only one phase will separa te out. Two si tuat ions are possible in this case. Each is discussed separately.

Case 1: Separation o f the continuous phase Let us define a critical volume:

(~ l ( f i ' E b K p g R ° L ° } ( ~ b E b ) ] (123) V c - K- - -~0 tan" LVEb + o - t a n - 1

A comparison wi th Eq. (114) indicates tha t V c is the volume of the cont inuous phase per uni t cross-section in an equil ibrated foam/concen- t r a t ed emulsion when Z2e = L o (see Fig. 16). Thus, if eoL0 = Vc, the cont inuous phase will redis t r ibute i tself unti l an equi l ibr ium is estab- l ished wi th E ] = e b. However, if eoL 0 > Vc, some cont inuous phase has

Z ~ 2

to separa te out before an equi l ibr ium can be established. To unde r s t and the reasoning behind this, consider Fig. 17a. The curve HAFB represents the equi l ibr ium profile of e in a column of length Lma x. Consider a column of length L o < Lma x with an initial profile represen ted by the horizontal line EFG. Thus:

L o = length(EG); Area(CDEG) = ~oLo (124)

Also note t h a t for this sys tem, the volume of cont inuous phase per


e~q

e~

Foam~gas

e.=o

q ' L r , , ~ I,

Eb=0.26

Foam/liquid

Fig. 16. Profile of liquid fraction (e@ in an equilibrated foam of length Lma x in contact with the continuous phase at the bottom and the dispersed phase at the top. V c is the volume of continuous phase per unit area in an equilibrated foam of length L 0 with e = 0.26 at the bottom. VD is the volume of the dispersed phase per unit area in an equilibrated foam of length L 0 with ~ = ec at the top.

u n i t c ross -sec t ion in an e q u i l i b r a t e d co lumn of l e n g t h L 0 is g iven by

Area (ABCD) . Thus :

V c = A r e a ( A B C D ) (125)

I t is c l ea r f r om t he f igure t h a t t h e p a r t of t h e e m u l s i o n f rom E to F

h a s an excess of c o n t i n u o u s p h a s e g iven by V 1 = Area (AEF) , whi le t h e p a r t of t h e f o a m f r om F to G ha s a def ic iency of c o n t i n u o u s p h a s e g iven

by V 2 = Area (BFG) . Thus :

EoL 0 - g c = V 1 - g 2 (126)

Now i f V 1 = V 2 , i.e. i f V c = e0L o, t he a m o u n t of c o n t i n u o u s p h a s e t h a t t h e u p p e r p a r t (EF) loses as i t a t t a i n s e q u i l i b r i u m is j u s t equa l to t h e

A. B hakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124 61

amoun t the lower par t needs. Thus, at equil ibrium, the ~ profile is given by curve(AFB) and the length of the column remains unchanged. On the o ther hand, i fV 1 > V2, i.e. ife0L 0 > V c (see Fig. 17b), the upper par t mus t lose a g rea te r amoun t of liquid than the lower par t requires, i.e., the

(a)

0.20

0.10

H

0.%.® o, 5

• , . , .

E

'D i I

0.10 0.15

z(m)

0.20

0.10

(b) 0"0~.00 0.05

V2 E F

i V~

D 0.10 D1 O. 5

z(rn)

Fig. 17. A pseudo-phase diagram for continuous phase separation: (a) V 1 - V 2 and (b) V~ > V 2.

62 A. Bhakta, E. Ruckenstein / Adv. Colloid Interface Sei. 70 (1997) 1-124

lower par t will gain more cont inuous phase liquid than it can hold. As a resul t the excess separa tes out and the column length decreases. The final profile is now given by curve A I F B and the equi l ibr ium length is given by length(DIC). One may note tha t length(DD1) = V 1 - V 2. Pr incen [64-67] has t rea ted the problem of continuous phase separa t ion using osmotic pressure considerations.

Case 2: Separa t ion of the dispersed phase As in the previous case, it is possible to define a critical volume for

disperse phase separation:

o ° [ l- c VD - KpgR-- t a n l - t a n l ec (~ (127)

It may be noted tha t Eq. (127) is a special case of Eq. (118) wi th Zle = 0 (see Fig. 16). Consider Fig. 18. The init ial profile in this case is given by segment EFG. It is clear tha t V D = Area(ABCD). Thus, the excess cont inuous phase in the upper par t is V 1' -- Area(AEF) and the deficiency in the lower par t is V2'= Area(BGF). Now, if V 1" = V 2' we have a0L 0 -- V D

and the cont inuous phase will simply redis t r ibute so as to a t ta in a final equi l ibr ium profile given by cm~e AFB. However, if V 1' < V2' , the lower par t (FG) requires more continuous phase than can be provided by the upper par t (EF), so tha t ~ at the top falls below e e and the disperse phase separa tes out. This causes the length of the foam to decrease and the final profile is given by curve(AFB1). In this case, the change in length is given by length(CC1).

The discussion in the above sections can be summar ized as follows (see Fig. 16):

w h e n L o < Lmax:

LO~ o < V D ~ separat ion of the disperse phase

aoLo > V c ~ separat ion of the continuous phase

If V D < aoL 0 < V c, equi l ibr ium is establ ished before e at e i ther end can reach a c or e b and no phase separat ion takes place.

Separa t ion of both phases As ment ioned earl ier , when both phases separate , the length of the

column at equi l ibr ium is fixed and given by L m a x. Thus, separa t ion of


0.04-

0.02

0.00 0.00

J

I

BI

g

CI C J I , I' ,

0.05 0.10

z(m) Fig. 18. A pseudo-phase diagram for dispersed phase separation.

both phases can occur only if L 0 > Lma x. However, L 0 > Lma x does not necessar i ly imply tha t both phases separate. Using a reasoning s imilar to t ha t used above, it is possible to de te rmine the conditions under which both phases separate .

The volume of cont inuous phase per uni t cross-section in an emulsion in equi l ibr ium wi th two phases(Vce q) is given by (see Fig. 19):

Vceq - KPgRo t an 1 - t an -1 (128)

gDeq -- Lmax - Vceq (129)


e:=l sS ~ s J

s j s ~ s S s S J s SS s~ / s s J s S s

• f s S s ~ i s

J J s S J J

s S s ~ J J sS 7 s~s s~ s

s S f s s s s s I / J J s s

/ / / VDeq r s" " j , ,s • j p

~-'eq s IS ~s j ~s s I S ~P s p J

• f I s J i ," / / " / , eb=0.26

jP i J s j ,'" ,," ," ,,' Foam/liquid

s I e' J

s J J J s~ s S s S i

s • s s p P

r s ~J J J r

s J J ~ s J j ~ t ~

L • ~ • •

",., ,. ,, V eq , . , J Foamlgas 1

e=-O Z

Fig. 19, Profile of liquid fraction (eeq) in an equilibrated foam of length Lma x in contact with the continuous phase at the bottom and the dispersed phase at the top. Vce a is the volume of continuous phase per unit area and VDe q is the volume of the dispersed phase per unit area in an equilibrated foam of length Lma × with e = e¢ at the top and e = 0.26 at the bottom.

The v o l u m e per uni t cross-sect ion of the disperse phase (VDe q) in this s i tuat ion is given by, i.e.:

= ~ I l f~c "ab-tan" |,,-----:/ + tan t~]---~--b ~)J VDeq KpgRo L Eb k ) (130)

Clearly, gce q and YDe q are fLxed for given R0, (~ and p. Thus, i f the v o l ume s per uni t cross-sect ion of the two phases in the ini t ia l e mul s i o n exceed Yce q a n d YDe q i.e. i f%L 0 > Yce q and (1 - ao)L 0 > VDe q both pha s e s separate out. I f any one of these condit ions are not sat isf ied, only one phase wi l l separate . The f inal l ength wi l l then be less than Lma x and the s i tuat ion is the s a m e as that d iscussed earlier.


A generalized "phase" diagram The condition for the separat ion of the continuous phase can be

wr i t t en as:

%L o > V c for L o < Lma x (131)

EoL 0 > Yce q for L o > Lma x (132)

Using Eq. (123), Eq. (131) can be wr i t ten as:

¢Y [ 1( ~-eb KpgRoLo) .1[" 1-f-f~eb~] > t t a n / . I 4 eo KpgRoLo [ [,V eb ~ - t a n (~--~b-~)j (133)

And using Eq. (128), Eq. (132) can be wr i t ten as:

> tan -1 _ tan a eo KpgRo Lo L k V e~ J (134)

Similarly, using Eq. (127), the condition for disperse phase separa t ion can be wr i t t en as:

eo < KpgRo Lo tan-1 _ tan.1 e~ _ Lo k V e~

for L o < Lma x (135)

e0 < 1 - (~ 1 - ~c _ 1 - eb _ t a n . 1 + tan- 1 KpgRo Lo ec Eb k, V Ec J ~ V Cb )J

for L o > Lma x (136)

Let us define a dimensionless number :

G p = KpgRoL o"

The above conditions can then be wr i t ten in t e rms of P as follows: Cont inuous phase separa tes when


ao > P | t a n - |~1 + - t a n -1 for L o < L m ~ (137)

eo > P l t a n - /~--a-~-:-c - ) - tan-1 for Lo < Lmax (138)

Dispersed phase sepa ra te s where

eo < P [ t a n ~ 1 ] - - ~ - ) - , L o < Lma x (139)

c , F LV ee V el, t a n - / ~ ] - ~ . ) + t a n - ~ ] - - ~ b ~ ) J

for L 0 > Lma x (140)

It can be seen from Eqs. (137-140) t h a t for a given eo, t he occurrence of phase sepa ra t ion is d e t e r m i n e d ent i re ly by the va lue of P. F igure 20 shows the r igh t h a n d sides of Eqs. (137-140) p lot ted ve r sus P for e c = 0.00826. The curves AB (Eq. (140)) and BC (Eq. (139)) show the lower l imit of a o below which the disperse phase separa tes and curves DB (Eq.(138)) and BE (Eq. (137)) r ep re sen t the uppe r l imit for e 0 beyond which the con t inuous phase separa tes . Poin t B corresponds to L o = Lma x. At th is po in t the re is no phase separa t ion and at equ i l ib r ium a = e b at t he b o t t o m and e = e c at the top. The region ABD corresponds to those ini t ia l condi t ions (L o > Lma x) for which both phases separa te . F igure 20 is also useful in u n d e r s t a n d i n g the effect of ini t ial co lumn he igh t (L 0) and bubble rad ius (R o) on the s tabi l i ty of a concen t ra ted emuls ion. The inf luence of these two p a r a m e t e r s is d e t e r m i n e d ent i re ly by the i r effect on P. For a given pai r of f luids we have

p ~ _ _ L o R o"

Cons ider an emuls ion wi th e 0 = 0.025. Now if i ts l eng th and bubble size are such t h a t t he ini t ial condit ion is r ep re sen t ed by H it is uns tab le .


0.20

~o

0.10

separation of both phases

continuous phase separation

• / ' no phase separation i

disperse phase separation C

0.00 0.00 0.05 0.10 0,15 0.20

P Fig. 20. A generalized phase diagram.

However, if its length and/or bubble size are decreased so tha t it is character ized by point G, it is stable and no phase separat ion will take place.

2.9.4. Simulation results for homogeneous foams/concentrated emulsions

To d e m o n s t r a t e the resul t s of our model, we carr ied out s imula t ions for a typical oil in w a t e r concen t ra ted emulsion. The following va lues for t he physical p a r a m e t e r s were used:


o = 20 m N / n , PD = 0.7 gm/cc , Pe = 1 gm/cc , ~ = 1 cP, R o = 0.2 m m , L o = 10 cm, H a m a k e r c o n s t a n t (A h) = 0 .41 10 -20 J , ion ic s t r e n g t h (%1) =

0 .01 M, s u r f a c e p o t e n t i a l ( ~ ) = 10 mV.

F o r t h e s e c o n d i t i o n s w e ge t l ima x = 615 .9 5 N / m 2 a n d e e = 0 .00826 . I n t h i s

case , P = 0 . 1 9 0 4 5 a n d Lma x = 17 .65 cm. S i n c e L 0 < Lma x o n l y o n e p h a s e s e p a r a t e s i f a t all. F r o m t h e p h a s e d i a g r a m , i t is c l e a r t h a t fo r ao >

0 .20

0.10

0 . 0 0 t_ . 0 . 0 0

Equilibrium o t=O + t=108 s * t = 8 9 0 s o t=4050 s

oooooooooooooooooooo~$~m~eeece~8~ ++÷++++++~ ~*

/ -F

÷ +

iO000000000(

0.02 0.04 0.06 0.08 0.10 Z (m)

Fig. 21. Evo lu t i on of t h e ~ profi le w h e n the c o n t i n u o u s p h a s e s e p a r a t e s . The s y s t e m

p a r a m e t e r s are: ~t =1 cP, Pc = i gm/cc, PD = 0.7 gm/cc, R o =0.2 m m , A h = 0.41×10 -21 J , L o

= 10 cm, ~o = 0.125 a n d ~s = 10 mV.


0.03

0 . 0 2

e 0 .01

0 . 0 1

Equi l ib r ium

x t=O

o t = 2 0 0 0 s

o t = 1 5 0 0 0 s

• t = 2 0 0 0 0 s

0000 OQO

iO00000000j 00~-

0 . 0 1 , , , i , , , I , 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 O. 10

z (m)

Fig. 22. Evolution of e profile when the dispersed phase separates (L 0 = 10 cm and E o = 0.01). Other parameters are the same as those for Fig. 21.

0 . 0 7 4 6 , t h e c o n t i n u o u s p h a s e s e p a r a t e s ; fo r e 0 < 0 .0157 , t h e d i s p e r s e

p h a s e s e p a r a t e s a n d fo r 0 . 0 7 4 6 < e 0 < 0 . 0 1 5 7 t h e r e is n o p h a s e s e p a r a -

t i on . F i g u r e 21 s h o w s t h e e v o l u t i o n o f e w i t h t i m e fo r e 0 = 0 .125 . T h e

c o n t i n u o u s c u r v e r e p r e s e n t s t h e e q u i l i b r i u m e - p r o f i l e in t h i s s y s t e m .

T h e o t h e r c u r v e s r e p r e s e n t t h e p r o f i l e s a t t = 0, t = 108 s, t = 890 s a n d

t = 4 0 5 0 s. N o t e t h a t fo r t = 890 s a n d t = 4 0 5 0 s t h e l e n g t h o f t h e c o l u m n

d e c r e a s e s i n d i c a t i n g t h a t t h e c o n t i n u o u s p h a s e s e p a r a t e s ou t . F i g u r e 22 s h o w s t h e e v o l u t i o n o f ~ fo r e 0 = 0.01. I n t h i s c a se , t h e i n t e r f a c e w i t h

z I t=o = 0 m o v e s i n d i c a t i n g t h a t t h e d i s p e r s e p h a s e s e p a r a t e s ou t . F i g u r e


0.20 . . . .

0 . 1 5

e 0 . 1 0

0 . 0 5

Equ i l ib r ium

x t=O

o t = 1 0 9 s

[] t = 1 0 9 0 s

• t = l O 0 0 0 s

¢

a °

°a° ~aa

o a aa o

a~la o o° o oo x xxxxxx x ~ 6 ~ e e e ~ e e e e e e e e e e e ooo oo°°° aOa a° D ~ ~ a a a a a a ~ ° a ~ l ° ° a

0 . 0 0 , I , i , | , I , 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0

z(m)

Fig. 23, Evolution of the e profile when there is no phase separation (L o = 10 cm and E o = 0.05), Other parameters are the same as those for Fig. 21.

23 shows t h e evo lu t ion of ~ for a 0 = 0.05. I t is c lea r t h a t t h e pos i t ions of t h e two i n t e r f a c e s do no t c h a n g e w i t h t ime. T h e r e is m e r e l y a r ed i s t r i -

b u t i o n of t h e c o n t i n u o u s p h a s e before e q u i l i b r i u m is e s t ab l i shed , no p h a s e s e p a r a t i o n .

F i g u r e 24 shows t he evo lu t ion of t h e e prof i les for L 0 = 22.5 cm a n d a 0 = 0.09. T h e p h a s e d i a g r a m ind ica t e s t h a t bo th p h a s e s s e p a r a t e out. I nde e d , Fig. 24 shows t h a t bo t h i n t e r f ace s move.


t=0 I o t = 3 9 9 S I

° 22°2 I 0 . 2 0 • t = 2 2 5 2 5 s L "

0 O0 O0

0.10

000000~ / 0 o° a o o o o o o o O O o ° ° U . . . .

a a

o ° aa ° . o ° aaO aa

ioaaaOOaooaaaO oaaaaaDaaaaaaoa ~eeee~ 0 . 0 0 ' ' ' ' ' ' ' '

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0

z(m) Fig. 24. E v o l u t i o n o f t h e ~ prof i le w h e n b o t h p h a s e s s e p a r a t e (L 0 = 22.5 cm a n d ~0 = 0.09). O t h e r p a r a m e t e r s a r e t h e s a m e as t h o s e for Fig. 21.

Lag time in homogeneous foam drainage Kann [63] has presented some exper imenta l da ta on the lag t ime of

homogeneous foams. We made an a t t empt to check if our model quali- ta t ively predicts the exper imenta l trends. There are two main resul ts in t ha t paper:

(a) For a given e 0 the lag t ime for drainage, i.e. the t ime requi red for the first drop of liquid to appear at the bottom decreases as the init ial foam he ight (L o) increases (Fig. 2 in Ref. [63]).

(b) For a given height , the lag t ime increases as the e 0 decreases (Fig. 4 in Ref. [63]).


0.8

0.6

._.R

" 0 '10 , M

• _g" 0.4 ~6

0

0.2 Initial liquid fraction= 0,033 C'~ © Initial liquid fraction=O.02

~ Initial liquid fraction=O.014

0.0 0.0 500.0 1000.0 1500,0

t (secs)

Fig . 25. E f f e c t o f % o n f o a m d r a i n a g e (L 0 = 10 cm, R o = 0 .5 m m , o = 10 m N / m a n d ~ = 1 cP).

S i m u l a t i o n s w e r e ca r r ied ou t w i th the fol lowing v a l u e s of t he pa ra - me te r s :

R 0 = 0.5 mm, o = 10 mN/m, ~ = 1 cP and p = 1 gm/cc.

L o - z 2 F i g u r e 25 s h o w s the s i m u l a t e d d r a i n a g e curves ( - -~ -n ) for L o = 10 cm

and t h r e e v a l u e s of e o. Clear ly , t he lag t ime inc reases as e o dec reases . F i g u r e 26 s h o w s the effect of in i t ia l f oam h e i g h t (L o) on the d r a i n a g e c u r v e s for e o = 1/70. F i g u r e 27 shows a plot of lag t ime (~) v e r s u s L o for th i s sy s t em. One can see t h a t t he lag t ime d e c r e a s e s w i t h an i nc rease in L o for s m a l l e r h e i g h t s b u t r e m a i n s u n c h a n g e d w i t h h e i g h t for l a rge r


0.8

e- a

~6

0

U .

© E]

0.6

0.4

0.2

0.0 0.0

Init/al foam length=4.5cm C) Initial foam length=5cm [] Initial foam length=7.5cm

Initial foam length--lOcm

500.0 1000.0 1500.0 2000.0 t (secs)

Fig. 26. Effect of in i t ia l length (L o) on the foam dra inage (eo = 1/70).

heights . F igure 27 is very s imi lar to Fig. 3 in Ref. [63]. An exp lana t ion for th is t r e n d in lag t imes can be given as follows:

The lag t ime is essent ia l ly the t ime requi red for e at t he bo t tom to become equal to eb. I f a0 is cons tant , the a m o u n t of l iquid t h a t the lower por t ion of the foam m u s t receive for e to become eb is fixed. The t ime lag is the re fore d e t e r m i n e d by the flow ra te of l iquid into the bo t tom P l a t e a u border . The flow ra te is ini t ia l ly the s ame t h r o u g h out the column(s ince

-- e0 t h r o u g h o u t the foam). However , since the re is no i n p u t of l iquid at t he top, the P l a t eau borders at the top contrac t t he reby decreas ing t he flow into the P l a t eau borders below t h e m and so on. For longer foams, t he effect of the contrac t ion at the top t akes a longer t ime to reach the bot tom. Thus , beyond a cer ta in he ight , e a t t he bo t tom becomes equal

74 A. Bhakta, E. Ruckenstein /Adu. Colloid Interface Sci. 70 (1997) 1-124

500 .0 , ', ,

450.0

400.0

(D

E 350.0

300 .0

250 .0

200 .0 ' 0 .0 40 .0

v )( )(

I , I , I

10.0 20.0 30.0 tnitial foam height (cm)

Fig. 27. Effect of the initial foam length (L 0) on the drainage lag time for the system of Fig. 26.

to E b before the inf luence of the uppe r b o u n d a r y becomes signif icant , so t h a t t he flow ra te in the lower por t ion is essent ia l ly i n d e p e n d e n t of foam h e i g h t and the lag t ime r ema ins more or less constant . The sha rp rise in lag t imes is s imply due to the fact t h a t as we decrease L0, we are app r oac h ing the poin t at which no phase separa t ion t akes place (i.e. the lag t ime is infinite).

W h e n L 0 is fixed (Fig. 25), as e 0 decreases, the a m o u n t of l iquid requ i red by the lower pa r t increases and the flow ra te decreases resul t- ing in an increase in lag t ime.

P n e u m a t i c foams will be considered next.


2.10. Pneumatic foams

2.10.1. Drainage and collapse during bubbling Figure 28 shows a schemat ic d i ag ram of a typical expe r imen ta l s e tup

used to p roduce foam by bubbling. An iner t gas is bubbled at a fixed vo lumet r i c flow ra te t h r o u g h a porous frit into the su r f ac t an t solution. F o a m is fo rmed and moves up at a ra te which depends on the superf icial gas velocity. As the foam moves up, d ra inage of l iquid occurs f rom the u p p e r to the lower por t ions of the foam and by the t ime the gas supp ly is s topped, a profile of l iquid fract ion (e) develops. The uppe r por t ions of t he foam are dr ie r (lower ~) because more d ra inage has t aken place there . In order to compute this profile, the d ra inage equa t ions need to be solved d u r i n g foam format ion. The basic equa t ions are the s ame as those p r e s e n t e d earlier. However , as the foam moves up, an observer at t he origin sees the foam/l iquid interface moving away from h i m (i.e. z 2 increases) a t a ra te which depends on the superficial gas velocity. T h u s Eq. (98) is not val id du r ing bubbl ing and a new express ion needs to be fo rmula ted .

z=zt=0

gas

Liouid

z=O

gas >

Z'-Zl

Z=Z2

(a) (b)

Fig. 28. A pneumatic foam being generated by bubbling; (a) no collapse, (b) collapse has occurred.


Depending on the collapse occurring at the top of the foam, par t of the gas resides in the foam, while the rest escapes from the top due to collapse. Since, we have defined the reference plane to be such tha t all the escaped gas resides between it and the foam/gas interface (i.e. be tween z = 0 and z = zl), a mass balance for the bubbling gas can be wr i t t en as:

z 2 d dz~

G A =~-~ ~ A(1 - a)dz + A d-T (141)

z 1

In Eq. (141), A is the cross-sectional area and G is the superficial gas velocity. The first t e rm represents the gas in the foam, while the second t e rm corresponds to the gas escaped from the collapsed bubbles. Simpli- fying Eq. (141), one obtains:

z 2 dz2 d

J adz (142) G - d ~ - dt z 1

Applying Leibnitz 's rule and using the fact that:

~e ~ [(1 - e)qps] 0t ~z (143)

we have:

d z 2 d z 1

G = (1 - e[z2 ) ~ + (1 - e ) q p B [ 2 + e]z, d-t- - (1 - e)qpBlz , (144)

d z 1 El imina t ing ~ using Eq.(65), yields:

dz 2 _ G - (1 - e_)qp s

d t 1 - e I z2

dz2

(145)

- - - which When the gas supply is shut off, G = 0 and we get d t qPBIz2

is the same as Eq. (98). As long as there is no collapse at the top dur ing generat ion, the motion of the foam/gas interface is de te rmined by the bulk movement of the foam. However, as soon as the thickness of the l iquid films at the top decreases to a critical value, collapse s tar ts at the


top and a d o w n w a r d componen t is supe r imposed on the bulk m o v e m e n t of t he foam. As a resul t , t he ne t u p w a r d velocity of the foam/gas in terface decreases . In case of less s table foams, it is possible to have a s teady s t a t e in which the foam leng th does not change wi th t ime because the ra te of foam collapse at the top becomes equal to the ra te of foam

dz I dz 2 generation, i.e. d~- - d~- and the foam length (z 2 - z I) does not change

with time. This steady state has been used to characterize the stability of short-lived foams.

In the sections that follow, we shall present some of our results for pneumatic foams. These results are reproduced from our papers and reflect the improvements with time. We first present results for foam drainage where there is no collapse. Here [14] films are neglected and a comparison is made with some experimental data [68] and the results from the quasi-steady state model for foam drainage proposed by Nar- simhan [23]. Results are then presented for the model in which film drainage and rupture are accounted for but coalescence and the change in surfactant concentration are ignored. The effect of various parameters on the steady-state height are examined. Finally, results are presented for our most detailed model [18] in which coalescence and the variation of surfactant concentration are taken into account.

2.10.2. Simulation results for pneumatic foams N a r s i m h a n [23] has used a quas i - s teady s ta te model to compute the

ini t ia l l iquid fract ion in a p n e u m a t i c foam. In s t ead of solving the d r a inage equa t ions us ing moving boundar ies as we have done, he c o m p u t e s the d i s t r i b u t i o n of l iquid a s s u m i n g t h a t t he d o w n w a r d flow ra te of l iquid per un i t cross-section due to grav i ty d ra inage (qPB) is c o m p e n s a t e d by the l iquid e n t r a i n e d by the r is ing bubbles (qB = Gnpapl/V), so t h a t quas i - s t eady s ta te ma te r i a l balances can be wr i t t en a long the l eng th of the foam. The quas i - s teady s ta te balance is wr i t t en a s :

/ d dz = ~ (qPB) (146)

This o rd ina ry dif ferent ia l equa t ion is solved us ing the b o u n d a r y conditions:

u = 0 fo r z = z I = 0; at ~ = 0.26 fo rz = z 2 (147)


0 . 2 6 " , " l , ~" '~"J~l~llt

illlll 0.24 / / ///

/ ,' ,- !

.o illlllllll I ~- 0.22

.~_

..J

0.20 / / ! / / - - - - - Unsteady

/ /

/ /

0.18 ' ' ' ' / ' ' ' ' ' 0.00 0.01 0.02 0.03 0.04 0.05

z(rn)

Fig . 29. L i q u i d f r a c t i o n p rof i l es in a p n e u m a t i c foam w h e n t h e gas s u p p l y is s h u t off. L~ = 5 cm, ~ = i cP, R 0 =0.2 m m , G = 0 .001 m/s, p = 103 k g / m 3, ~ = oo a n d (~ = 50 m N / m .

P r e s e n t e d be low are compar i sons of our r e su l t s w i t h those f rom the q u a s i - s t e a d y s t a t e model .

F i g u r e 29 shows a compar i son of the profi les of the in i t ia l l iquid f rac t ion (profile of ~ w h e n the gas supp ly is s h u t off) o b t a i n e d f rom the two mode l s for L 0 = 5 cm and G = 10 -3 m/s. It is c lear t h a t t he profi les a re q u a l i t a t i v e l y d i f fe ren t . The to t a l a m o u n t of l iqu id in t h e f o a m per u n i t c r o s s - s e c t i o n (V o) w h e n t he gas s u p p l y is j u s t s h u t off, i.e. w h e n z 2 - z 1 -- L0, is t he a r e a u n d e r the ~ vs z curve and is g iven by:

L o

v o = j ~dz (148)

o


0.20

O } ¢-

E t~

._=

t -

t - o

c .o_.

0 .15

0 .10

0 .05

// I / Q u a s i - s t e a d y

- - - - - U n s t e a d y l/ •

0 . 0 0 ' ' ' ' 0.00 100.00 200 .00 300.00 400 .00 500.00

r (seas)

Fig. 30. Variation of the fractional change in foam length with time (t') for the system of Fig. 29.

It is clear that the quasi-steady state model predicts a higher V 0. Figure 30 shows the evolution of the fractional change in foam height with time t" for the above system. Here t" is the time elapsed after the gas supply is shut off. Note that the curves flatten out indicating that the drainage practically ceases after a given time and the foam height remains almost constant. When the quasi-steady state model is used to compute the initial distribution, the change in the height is about 70% greater than that obtained from our model. Since the two models differ only in the way the initial distribution is computed, it is clear that the initial distribution strongly affects the drainage process. Figures 31-33 show comparisons of the theoretical predictions of the two models with the experimental data


(a) 0.20

0.15

E ,R .=_

~, O.lO =,. .

0

0.05 Quasi-steady

o Unsteady -~ Experimental

0.00 0.00 200.00 400.00 600.00 800.00 1000.00

r(secs)

Fig . 31. F r a c t i o n a l c h a n g e in foam l e n g t h v e r s u s t i m e (t') for two bubb le s izes w i t h G =

0.0578 cm/s, L o = 19.8cm, ~ = 1 cP, p = 1000 kg/m 3 a n d qs = ~: ( a ) R 0 = 0.153 m m .

of Germick et al. [68]. Figure 31 shows the effect of bubble size, while Fig. 32 shows the results for 3 viscosities. It is clear tha t our model shows significantly bet ter ag reement wi th the data. Figure 33 shows the effect of initial foam height (L 0) on the drainage. Our model shows reasonable agreement for the smal les t he ight (L 0 = 10.61 cm). However the theore t ica l d ra inage curves do not change to the same extent wi th he ight as the exper imenta l curves. At larger heights, the foam is drier at the top and the film thickness is very thin. I t is t hus possible t ha t bubble collapse which was ignored in our model causes the discrepancy. Before we proceed fur ther , some quali tat ive features need to be noted in the figures. The extent and ra te of dra inage is greater when the bubble size is smaller and


(b) 0.10

e

E

¢1 O~

o

0

LL

0.08

0.06

0.04

0.02

Quasi-steady c = Unsteady

0.00 0.00 200.00 400.00 600.00 800.00 1000.00

r(secs)

Fig. 31 c o n t i n u e d . (b) R 0 = 0.203 m m .

the viscosity is higher. With smaller bubble sizes (i.e., smaller PB cross-sectional area) and higher viscosities, the downward fluid velocity is smaller. This means that a smaller amount of liquid drains out of the foam during bubbling. A greater amount of liquid is therefore present in the foam when the gas supply is shut off. This in tu rn means tha t the foam is far ther from equilibrium when the gas supply is shut off. The change in foam height is therefore larger. It mus t be emphasized here tha t film drainage was ignored [14] in obtaining the curves shown in Figs. 31-33.

Fi lm drainage was incorporated into the model in a subsequent paper [16]. This, however, brought to the fore, a new issue regarding the boundary conditions at the foam/liquid interface during foam formation.


( a ) 0 .20

e E

.c_

c

t - o

t - O

LL

0.15

Quasi-steady c : Unsteady

-~ Experimental

0.10

0 .05

0 .00 ~ 0.00 200.00 400.00 600.00 800.00 1000.00

t'(secs)

Fig . 32 . F r a c t i o n a l c h a n g e i n f o a m h e i g h t w i t h t i m e (t ') for t h r e e v i s c o s i t i e s w i t h L 0 =

18 .67 c m , G = 0 . 0 5 7 8 ChriS, p = 1 0 0 0 k g / m ~ a n d TIs = oo: (a) ~ = 1.3 cP , ~ = 5 7 . 6 d y n d c m ,

R 0 = 0 . 1 7 1 m m .

In foams formed by bubbling, new bubbles are con t inuous ly intro- duced into the sy s t em at the foam/l iquid interface. Thus , w h e n the foam is be ing genera ted , the b o u n d a r y condi t ion at the bo t tom is a s t a t e m e n t about the condi t ion of the bubbles as they en te r the foam. W h e n fi lms were neglec ted [14,15], t he b o u n d a r y condit ion at the bo t tom was s imply the condi t ion for close-packed spheres (e = 0.26). This was sufficient since the re was only one var iable (~). However , wi th the inclusion of f i lms into the model, t he re are two i n d e p e n d e n t var iables (rp and x F) a nd in fo rmat ion is needed about the d i s t r ibu t ion of l iquid be tween the f i lms and the P l a t e a u border channels . Since the different ia l equa t ion

(b)

A. Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

0.20 . . . .

83

=. e -

E SZ

== ¢ -

o

e - 0

0.15

0.10

0.05 Quasi-steady

o o Unsteady -~ Experimental

0.00 0.00 200.00 400.00 600.00 800.00 1000.00

~(secs)

Fig . 32 c o n t i n u e d . (b) p = 3 .02 cP , a = 55 .8 d y n d c m , R 0 = 0 . 1 7 9 m m .

for x F is f irst order in t ime, an ini t ial condit ion is needed. In o the r words, t he va lue of x F at the foam/l iquid interface du r ing bubbl ing (xFo) is requi red . We have no physical a r g u m e n t s to specify the fi lm th ickness . In order to ident i fy the effect of its value, we carr ied out s imula t ions wi th several va lues ofxF0. I t was found t h a t this va lue has pract ical ly no effect on the d ra inage [16], since, for th icker films, mos t of the d r a inage t akes place wi th in a very shor t d is tance f rom the bot tom. F igure 34 shows the l iquid fract ion profiles in a f reshly formed foam of he igh t 7.5 cm. In one case, the film th ickness at the bo t tom was 1000 n m and in the o the r the va lue was 100 nm. The curves pract ical ly coincide ind ica t ing t h a t the va lue one takes for t he fi lm th ickness at the


(c) 0.20

0.15

=. e-. q3 E ~o

== O.lO e-. o

c o

U .

0.05

' | ! | !

] . ~ ~ Q u a s i - ~ e a d y / / o ou.= dy

~ ~ Experimental

0,00 ~ m , I , = , I

0.00 200.00 400.00 600.00 800.00 t'(secs)

Fig. 32 c o n t i n u e d . (c) ~ = 6.28 cP, ~ -- 54.2 dym/cm, R 0 = 0 .203 ram.

!

1000.00

bottom is largely irrelevant. Thus, as long as the foam is being generated (gas is being pumped in), the condition at the bottom is:

XFI = xFO = constant (149) Z 2

Once the gas supply is stopped, the films at the bottom simply obey Reynolds equation with Eq. (149) as the initial condition. For the present calculations, XF0 was taken to be 500 nm.

The emphasis in that paper [16] was to theoretically model the effect of various parameters on the steady state height and collapse half life of pneumatic foams. However, since the assumption regarding the neglect

A. Bhakta, E. Ruckenstein /Adv. Colloid Interface ScL 70 (1997) 1-124 85

(a) 0.2o

0.15

Quasi-steady o o Unsteady

Experimental

"~ 0.10 ~

=0

0.05

O I J I , I J I i

0"00.00 200.00 400.00 600.00 800.00 1000.00 r(secs)

Fig. 33. F rac t iona l change in foam leng th for three values o f L 0 w i t h I~ = 0.9632 cP, ~ = 59.4 dyrdcm, G = 0.0375 ChriS, p = 1000 kg /m 3 and qs = oo: (a) L o = 22.78 cm, R 0 = 0.203 ram.

of the liquid in the films was first relaxed here, it would be ins t ruct ive to make a comparison with resul ts obtained using the earl ier models in which only the d ra inage th rough the P la teau borders was t aken into account.

In a more approximate earl ier model [15], it was assumed tha t films were a lways in equi l ibr ium with the adjacent P la teau border channels and t ha t the fraction of liquid in the films was negligible. Collapse was deemed to s ta r t when the capil lary pressure at the foam/gas interface exceeded the m a x i m u m disjoining pressure, i.e. when the P la teau border rad ius became smal ler than a critical value rpe given by Eq. (102).

86

(b)

A. Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

0.12 Quasi-steady o o Unsteady

Experimental

e -

0.10 e.- _e

E

g o.o8 e -

c o

0.05 LI .

0.03

0.00 0.00 200.00 400.00 600.00 800.00 1000.00

t'(secs)

Fig . 33 c o n t i n u e d . (b) L 0 = 16.27 cm, R 0 = 0 .203 m m .

In order to compare the resu l t s f rom these two approaches , a sample ca lcula t ion was carr ied out. The foam was produced wi th a superf icial gas velocity of 0.0001 m/s and the gas supply was s h u t o f f w h e n a foam of l eng th 7.5 cm was obtained. F igure 35 shows the profiles of l iquid fract ion (e) in the foam w h e n the gas supply is j u s t s h u t off. F igure 36 shows the evolut ion of foam he igh t (Z 2 --Zl) as a funct ion of t ime. Here t he difference is m u c h more significant. The reason for th is is t h a t whi le f i lms do not inf luence the d ra inage process much , they play a crucial role in d e t e r m i n i n g the character is t ics of foam collapse. The older model predic ts an ear l ier onset of collapse and a quicker decrease in foam length . This can be expla ined as follows. In the previous model , the


(c)

0.12

e- 0 .10

e

E

. c

g 0.08 c e,,- o

c o

0.05 LL

0 .o3

, Quasi-steady : Unsteady -~- Exper imenta l

0 .00 0.00 200.00 400.00 600.00

r(secs)

Fig. 33 c o n t i n u e d . (c) L o = 10.61 cm, R 0 = 0 .193 m m .

800.00 1000.00

i z 1 o In the cu r r en t model, collapse collapse s tar t s t he m o m e n t rp =i]ma x.

G begins w h e n x F = XFI nm ~. U n d e r these c i rcumstances , rp I zl < --l_imax. Be-

cause collapse begins in the new model at a lower va lue of rp, it s t a r t s later . Also the cu r r en t model predic ts a slower collapse because the sma l l e r P l a t e a u border r ad ius at the col lapsing front is smal le r which in t u r n impl ies a smal le r ra te of collapse (see Eq. (65)).

The resu l t s of some s imula t ions done to examine the effect of var ious p a r a m e t e r s on the s t eady-s ta te he igh t are d iscussed below.


0.40 I I I

* * lOOnm ,,, X FO ~

0 0 XFo = lO00nm 0.30

0.20

0.10

0.00 ' ' ' ' '

0.00 0.02 0.04 0.06 0.08

Distance from ~oam/g~s interface (m)

Fig. 34. Effec t of f i lm th i cknes s a t the bo t tom (xF0) on the l iquid profi les in a f resh ly

g e n e r a t e d p n e u m a t i c foam of h e i g h t 7.5 cm, w h e n the gas supp ly is s h u t off .The v a l u e s of t h e p a r a m e t e r s used are: p = 1000 kg /m 3, ~t = 1 cP, G = 0.0001 m/s, ~s = 20 mV, R o =

0.2 m m , T = 298 K, A h = 3.7x10 -z° J , c,l = 0.005 M, r l m a x = 473,79 N / m 2 and o = 40 mN/m.

Effect of superficial gas velocity on the steady-state height [16] It is practical to use the s teady-sta te height as a measure of foam

stabil i ty only for relat ively short-lived foams. For ext remely stable foams, it is possible t ha t the height of the foam when collapse s tar ts will be too large to be convenient ly measured in a laboratory. It is also obvious t ha t the superficial gas velocity used to genera te the foam will have a significant impact on the s teady-sta te height. Figure 37 shows the s teady-s ta te he ight (H 0) as a function of the superficial gas velocity (G). I t is clear t ha t the s teady-sta te he ight increases wi th G. The slope of the curve increases and seems to become unbounded for G = 0.00035 m/s, indicat ing t ha t there is an upper limit on the superficial gas velocity beyond which a s teady-sta te foam height will never be achieved. Some explanat ion for this can be provided as follows. From Eq. (65) it can be seen tha t the ra te of foam collapse is de te rmined by qPB]~. Since the


0 . 4 0

0.30

0.20

0.10

o o old model -~ new model

0 . 0 0 , I , I , i , 0.00 0.02 0.04 0.08 0.08

D i s t a n c e f r o m f o a m / g a s i n t e r f a c e (m)

Fig. 35. A compar i son of t h e l iqu id- f rac t ion profi les in a f resh ly g e n e r a t e d p n e u m a t i c foam of h e i g h t 7.5 cms ob ta ined us ing two models . F i l m d r a i n a g e is inc luded in one (new) and neg l ec t ed in t he o t h e r (old). The v a l u e s of t he p a r a m e t e r s used are: p = 1000 kg /m 3,

= I cP, G = 0.0001 m/s, ~ = 20 mV, R o = 0.2 m m , T = 298 K, A h = 3.7×10 -2° J , cei = 0.005 M, [lm~ x = 473.79 N / m s and o = 40 mN/m.

film thickness at the collapsing front is fixed, qPB is de te rmined pr imar- ily by the value of the P la teau border radius at the top. However, since collapse can occur only if the capil lary pressure at the top exceeds the m a x i m u m disjoining pressure, there is an upper limit to the P la teau border rad ius at the top given by:

(150) Fpmax- [-[max


0.080

E

e , -

o

0.070

0.060

0.080

0.040

0.030

0.020 L_ 0.0

: = old model = ; new model

2000.0 4000.0 6000.0 t ime (s)

Fig. 36. Var ia t ion of foam height wi th t ime for the sys tem of Fig. 35.

In other words, there is an upper l imit on the value ofqpBIzl , given by:

3 NnpapR ° appg qPBmax-- 15 20~-p ( 1 5 1 )

rp=r pmax, X ~ F c

dzl This means t ha t there is an upper limit to the value o f ~ T , given by:

dz 1 qPBmax (1 - alrp=rpmax, XF=Xi~ :) dt eJ (152)

rp=rpma) 2 XF=XFc


30.0

E

._co

"I"

¢n

(D

20.0

10.0

0.0 , I , I ,

0 . 0 0 0 1 0 0.00020 0.00030 0 .00040

G (m/s)

Fig. 37. Effect of super f ic ia l gas ve loc i ty (G) on the s t e a d y s t a t e he ight . The v a l u e s of

t h e p a r a m e t e r s used are: p = 1000 kg/m 3, p = 1 cP, Us = 18 mV, R 0 = 0 .2mm, T = 298 K, At, = 3.7×10 -2o J , ce, = 0.001 M, H, ,~ = 227.23 N / m 2 and g = 40 mN/m.

d z 2 Thus for values of G for which ~ (Eq. (145)) is larger than the

d z 2 d z 1 m a x i m u m value given by Eq. (152), - ~ will always be larger than d-~

d z 1 d z 2 and a s teady-state height (for which d~- - d-t ) will never be achieved.

Effect of electrolyte concentration on foam collapse [16] The concentrat ion of electrolyte in the surfactant solution plays a

crucial role in de te rmining the stability of a foam due to its effect on the

92 A. Bhakta, E. Ruckenstein /Adv. Colloid Inter[&ce Sci. 70 (1997) 1-124

400.0

E ~ 300.0

Q.

._=

._c 0

E

E 200,0

1 00 f~ , I i I i I i

'0.000 0.002 0.004 0.006 0.008 Electrolyte Concentration(M)

Fig. 38. Effect of electrolyte concentra t ion (c~,) on the m a x i m u m disjoining pressure ([Im~). The va lues of the o ther pa r ame te r s used are: ~s = 19 mV, T = 298 K, and A h = 3.7×10 e~*.

repulsive electrical double layer forces in a film. The pr imary effect of an increase in electrolyte concentrat ion is to compress the electrical double layer, i.e. the Debye length decreases. The lower the value of l-Ima X, the smal ler is the capil lary pressure required to cause collapse. On the other hand, the smaller the critical thickness, the larger is the residence t ime required for the film to become critical. Figures 38 and 39 show the var ia t ion of ]-[max and the critical thickness (XFc) with electrolyte concentrat ion. At small concentrat ions, the double layer is not too compressed. The m a x i m u m of the disjoining pressure therefore corresponds to large film thicknesses for which the van der Waals forces are not very large. An increase in the electrolyte concentrat ion therefore


20.0

E ¢--

o~

e ' -

t-

L~

18.0

16.0

14.0

12.0

10 .0 ~ I ~ I , ~ I

0.000 0.002 0,004 0.006 Electrolyte Concentration (M)

i

0.008

Fig. 39. Effect of electrolyte concentration (cei) on the critical thickness (XF~) for the system of Fig. 38.

ra ises ]-IDL wi thou t affect ing FIVD w m u c h and raises []max" At h ighe r e lectrolyte concent ra t ions , the double layer is compressed and Nma x cor responds to very smal l th icknesses at which the van der Waals force is s ignif icant . ]-[max n o w decreases wi th an increase in electrolyte concen t r a t i on because l-IvD w rises sha rp ly wi th a decrease in the film th ickness , l-lma x therefore increases , a t t a ins a m a x i m u m and t h e n decreases as the electrolyte concent ra t ion increases. The posi t ion of this m a x i m u m however moves to smal le r film th icknesses . This m e a n s t h a t t he crit ical t h i ckness decreases wi th an increase in electrolyte concen- t ra t ion . I t is therefore reasonable to expect t ha t as the electrolyte concen t ra t ion increases , the s tabi l i ty of the foam as quant i f ied by the


10.0

9.0

8.0

E 7 .0 t~

.E ,m -r" = 6 .0

" 0

-.~ 5.0

4.0

3.0

2'0.000 0.002 0,004 0.006 0.008 Electrolyte Concentrat ion (M)

Fig, 40. Effec t of t h e e l ec t ro ly t e c o n c e n t r a t i o n (ce,) on t h e s t e a d y s t a t e h e i g h t . T h e v a l u e s

of t h e p a r a m e t e r s u s e d are : p = 1000 kg /m 3, ~ = i cP, ~s = 19 mV, R 0 = 0.2 m m , T = 298 K, A/~ = 3.7×10 .2o J , o = 40 m N / m a n d G = 0 .0001 m/s.

s t eady s ta te he igh t (H 0) increases at first, achieves a m a x i m u m and t h e n decreases . F igure 40, which p resen t s the effect of sal t concen t ra t ion on H 0, shows t h a t th is indeed is the case.

I t has been observed exper imenta l ly [69] t h a t plots of (z 2 - z l ) / H o

ver sus log( t / t l /2 ) pract ical ly coincide and at lower salt concent ra t ions are l inear. To check if our model predicts this fea tu re (see Fig. 41) such plots were ge ne ra t ed for six different va lues of the electrolyte concent ra t ion (Cell While the curves are not l inear, they seem to coincide qui te well, especial ly at h ighe r concentra t ions . The shape of the curve is more s imi la r to the da t a provided in the above paper at h igher concent ra t ions


,,7 t

1.0

0.8

0.6

0.4

%

<>

* Electrolyte Conc.=O.O01M × Electrolyte Conc.=O.OO2M o Electrolyte Conc.=O.OO3M o Elec~'otyte Cortc.=O.OO4M o Electrolyte Conc.=O.OO5M ,', Electrolyte Conc.=O.OO6M

Universal Exptl. Line

0.2 ~ -4 .0 -2 .0 0.0 2.0 4.0

log(t/tin)

Fig. 41. Effect of electrolyte concentrat ion (eel) o n the dimensionless plots of (Ze-Zl)/Ho versus log(t/tu2). The values of the parameters are the same as in Fig. 40.

(labeled by these authors as "not so nice data"). The slope of the l inear portion of the theoret ical curves is reasonably close to the universal exper imenta l line provided by these authors. A more direct comparison with the above exper iments is not possible because there is no information on the bubble size used in the above experiments .

The concentra t ion of the elctrolyte also plays an impor tan t role in de t e rmin ing w he t he r the foam collapses completely or arr ives at a d ra inage equilibrium. Figure 42 shows the variat ion in foam height with t ime for two sys tems which differ only in the electrolyte concentrat ion. The initial foam height is 4 cm and the values of the o ther pa rame te r s used are: 9 = 1000 kg/m 3, p = 1 cP, ~gs = 19 mV, R 0 = 0.2 mm, A h = 3.7×10 -2°, g = 40 mN/m and G = 0.0001 m/s. The foam with cel = 0.003 M equil ibrates at a he ight of about 1.85 cm, while the foam with Cel = 0.007 M collapses completely. This can be explained as follows: The


g O'J C

0 . 0 6 0 , , ' .

0 . 0 5 0

0.040

0.030

0.020

0.010

O'

Electrolyte concentration=O.OO3M o Electrolyte concentration=O.OO7M

° ' ° ° °o .o 1 ooo.o 2000.0 aooo.o t' (s)

Fig. 42. Variation of foam height with time for a foam with an initial length of 4 cm and G = 0.0005 m/s. With cel = 0.007 M, complete collapse occurs, while with cej = 0.003 M, the foam achieves an equilibrium 'height. The other parameters are the same as in Fig. 40.

c a p i l l a r y p r e s s u r e h a s i t s s m a l l e s t v a l u e (c~/R o) a t t h e b o t t o m of t h e f o a m , w h i c h in t h i s c a s e is 200 N / m 2. F o r %1 = 0 .003 M, 1-lma x = 3 8 0 .8 5 N / m 2

w h i l e fo r Cel = 0 .007 M, i t is a b o u t 150 N / m 2. S in ce in t h e l a t t e r c a s e t h e s m a l l e s t c a p i l l a r y p r e s s u r e pos s ib l e in t h e s y s t e m e x c e e d s t h e m a x i m u m d i s j o i n i n g p r e s s u r e , t h e f o a m co l l ap se s c o m p l e t e l y .

Ef f ec t o f b u b b l e s ize

F i g u r e 43 s h o w s t h e e f fec t o f b u b b l e r a d i u s (R 0) on H o for a s y s t e m w i t h t h e f o l l o w i n g p a r a m e t e r s : Vs = 19 mV, c~ = 40 m N / m , G = 0 .0001

m/s , %1 = 0 .005 M, T = 298 K. C l e a r l y , H 0 i n c r e a s e s w i t h b u b b l e s ize .


10.0

8.0

A

E o 6.0

._~ G) "1"

O9

4.0

oo

2.0

i I • i i i

0 . 0 I i I = I i I = I

0.10 0.15 0.20 0.25 0.30 0.35

Bubble Radius (ram)

0.40

Fig. 43. Ef fec t of t h e b u b b l e r a d i u s R] on t h e s t e a d y s t a t e f o a m h e i g h t . T h e v a l u e s of t h e

p a r a m e t e r s u s e d a re : p = 1000 k g / m , ~ = 1 cP, ~s = 19 mV, T = 298K, A h = 3.7×10 -2° J ,

(~ = 40 m N / m , Cej = 0 .005 M a n d G = 0 .0001 m/s.

This can be expla ined as follows: In general , in a foam co lumn d ra in ing u n d e r gravi ty, t he cu rva tu re of the P l a t eau border channe l s (rp) is the la rges t a nd the capi l lary p ressure the smal les t a t the foam/l iquid in te r face whe re rp = R o. For a given I]max, the smal le r the bubble size, t he sma l l e r is the decrease in rp requi red for the capi l lary p r e s su re to rise to []max" This m e a n s t h a t collapse s t a r t s a t smal le r he igh t s r esu l t ing in smal le r va lues o f H 0. F igure 44 shows a plot ofz 2 - z 1 ve r sus t ime for two bubble sizes. I t is clear t h a t the re is a qua l i ta t ive difference since the foam wi th R 0 = 0.3 m m collapses to reach an equ i l ib r ium height , whi le the foam wi th R o = 0.125 m m collapses completely. This is because

98 A. Bhak ta , E. Ruckens te in /Adv. Colloid Interface Sci. 70 (1997) 1-124

0.0,30

0.020

E v

C

,,o

0.010

= = Bubble radius=0.125 mm o o Bubble radius=0.3 mm

0.000 0.0 1000.0 2000.0 3000.0 4000.0

r (s)

Fig. 44, V a r i a t i o n of f oam h e i g h t w i t h t i m e for a f o a m w i t h a n i n i t i a l l e n g t h of 3 cm. W i t h R 0 = 0 .125 m m , c o m p l e t e co l lapse occurs , w h i l e w i t h R 0 = 0.3 m m a n e q u i l i b r i u m

h e i g h t is a t t a i n e d . T h e v a l u e s of t h e p a r a m e t e r s u s e d are : p = 1000 k g / m 3, p = 1 cP, ~

= 19 mV, T = 298K, A h = 3.7×10 -2° J , a = 40 m N / m , c~, = 0 .005 M a n d G = 0 .0005 m/s .

((~/R o) is less t han l ' -[ma x for the former and grea ter t han l i m a x for the lat ter . F igure 45 shows a comparison of the dimensionless plots for th ree bubble sizes wi th the universal exper imenta l curve. The electrolyte concentra t ion used here is 0.005 M. The other pa ramete rs r ema in the same. Again the ag reement is fair. In an a t t empt to improve the overlap in the la ter stages of collapse, we made plots of [(z2-z 1) - (Z2--Zl)eQ] / [ H 0 - ( z 2 - Zl)eq] versus log(t/t05) (see Fig. 46), where t0. 5 is the t ime requi red for [(z2-z 1) - (z2-Zl)eq] / [Ho-(Z2-Zl)eq] to become 0.5. The ra t ionale for this was tha t this would force all the curves to become 0 at long times. It is seen from Fig. 46 tha t there is some improvement .


1,0

0.8

0.6

0.4

0.2

<>

0 • Universal experimental lira

Bubble Radius=O.2mm 0 Bubble Radius=O, 15ram <> Bubble Radius=O.3mm

J I t ,

o'%.o -2.0 o:o 2:0 4.0 Iog(t '/trj2)

Fig. 45. Effec t of bubble r a d i u s (R o) on the d imens ion l e s s plots of (zz-zl) /H o v e r s u s log(t/tvz) w i t h G = 0.0001 m/s. The v a l u e s of t he o the r p a r a m e t e r s a r e t he s a m e as in Fig. 44.

Coalescence and enrichment in foam [18] In all the resul ts discussed thus far, coalescence has been ignored. It

was a s sumed tha t the film thickness was a function of only the vert ical position 'z'. Thus film rup tu re at any level implied complete collapse of the foam at t ha t level. In reality, however, all films at a given level do not rup tu re at the same time. There is usual ly a dis tr ibut ion of film thicknesses. The th inne r films rup ture first leading to a local increase in bubble size due to coalescence of the neighboring bubbles. Complete collapse occurs only when all the films at a given level rupture . The other impor t an t aspect t ha t was ignored thus far was the var ia t ion of the


,.7 ,'T"

.3 ,.7 ,.7

..L.

~7

1.0

0.8

0.6

0.4

0.2

i i

O,

C,

0

o

q~

,0

0 0 0

* Bubble radius=O.15mm o Bubble radius=O.2mm o Bubble radius=O.3mm

<>

,0 0 c,. .,QS,

0 . 0 , I , I

-4 .0 -2 .0 0.0 4.0

k i

21o log t'/t ~s)

Fig. 46. Effect of bubble radius (R o) on the dimensionless plots of [(zz-z 1 ) - (zz-z l)~ql/[H o- (ze-zl)eq] versus log(t/to.5). The values of the parameters are the same as in Fig. 45.

s u r f a c t a n t concen t ra t ion . The s u r f a c t a n t concen t r a t ion was a s s u m e d to be u n i f o r m t h r o u g h o u t t he foam and equa l to t he concen t r a t i on in t h e foaming solution. T h u s t he r e was a s ingle l ~ m a x and a s ingle cr i t ical t h i c k n e s s (XFc) for a given foam. However , coalescence and collapse leads to a local i nc r ea se in t he s u r f a c t a n t concen t r a t ion because t h e adso rbed s u r f a c t a n t on t h e f i lm sur faces is r e l ea sed into t he l iquid in t h e P l a t e a u b o r d e r channe l s . Thus , d e p e n d i n g on the t r a n s p o r t proper t ies , a profile of s u r f a c t a n t concen t r a t i on is set up wh ich in t u r n leads to a v a r i a t i o n o f Xgc a n d Hma x. 1-[ma x is h i g h e r and the f i lms a re more s table in reg ions w h e r e t h e s u r f a c t a n t concen t r a t ion is h igher . T h u s coalescence a n d col lapse can ac tua l ly s tabi l ize t h e r e m a i n i n g films. This effect is prob- ab ly m o r e s ign i f ican t for ionic s u r f a c t a n t s w h e r e t h e dis joining p r e s s u r e


i so therm is more sensitive to the sur fac tan t concentration. Some of the resul ts obtained when coalescence and sur fac tan t t ranspor t are included are presented below.

R e s u l t s for coalescence a n d collapse In this section, resul ts of s imulat ions done to examine the effect of

var ious pa r ame t e r s such as the apparen t diffusivity (D), the concentration o f su r f ac t an t (Cso) and salt (c e) in the foaming solution and the initial dis tr ibut ion of film radii (D will be discussed.

The initial dis tr ibut ion f0 was assumed to be a t runca ted normal dis t r ibut ion given by:

RFminO

(153)

The m e a n film radius was taken to be RFO = 0.606 R 0, where R 0 is the bubble radius ( taken to be 0.2 mm) in the monodispersed foam before coalescence starts. The s t anda rd deviation s was t aken to be 0.5 RFO and RFmin 0 and RFc were taken to be 0.99 RFO and 1.01 RFO , respectively, except when the i r effects on the resul ts were examined. This represents a deviation of about 1% from the mean value. These values are a rb i t r a ry as we have no way of identifying wha t the exact distr ibution is. Unless o therwise specified, the apparen t diffusivity is t aken as D = 5x10 -5 m2/s. This value is apparen t and is much higher than the molecular diffusivity which has a value of 10 -l° m2/s. In reality, D will va ry th roughout the foam depending on the fluid velocity and will be h igher in regions where significant coalescence (which results in g rea te r mixing) occurs. The values of the o ther pa rame te r s used were G = 0.0001 m/s, c e = 8 moles/m 3, R 0 = 0.2 mm, p = 1 cP, T = 298 K, Cso = 0.012 moles/m 3 and K d = 0.156 moles/m 3.

Effect o f d i f f u s i v i t y To examine the effect of the apparen t diffusivity (D) on foam collapse,

s imulat ions were carr ied for two values of D, viz. D = 5x10 -5 m2/s and D = 5x10 -6 m2/s. Figure 47 shows the variat ion in foam length wi th t ime

102 A, Bhakta, E. Ruckenstein /Adv. Colloid Interface Sci. 70 (1997) 1-124

0.050

0 .049

. - . 0 .046 E e -

e -

E 0

u. 0 .047

0.046

- = D = 5 e - 6 m.m/s

: = D = S e - 5 m.m/s

0.045 0.0 500.0 1000.0 1500.0 2000.0 2500.0

t' (sec)

Fig. 47. V a r i a t i o n of foam l e n g t h w i t h t i m e for two v a l u e s of t h e a p p a r e n t d i f fus ion coeff ic ient D. The v a l u e s o f t h e o the r p a r a m e t e r s u sed a re cs0 = 0.012 mo le s /m 3, c e = 8

m o l e s / m 3, R 0 = 0.2 m m , RFO = 0.606 R0, RFmin 0 = 0.99 RE0 and RF~ = 1.01 RFo.

for t hese two cases. Poin ts A and B indica te the onse t of foam collapse. The foam wi th the h ighe r diffusivi ty s ta r t s collapsing ear l ier and col- l apses a t a m u c h h ighe r rate . This can be expla ined as follows: Coales- cence leads to a re lease of su r f ac t an t f rom the r u p t u r e d fi lms and hence leads to a local a c c u m u l a t i o n of sur fac tan t . This local increase of the s u r f a c t a n t concen t ra t ion t ends to stabil ize the r e m a i n i n g fi lms since t he m a x i m u m dis joining p re s su re increases wi th su r f ac t an t concent ra t ion , i.e. a f i lm becomes more s table as the su r f ac t an t concent ra t ion increases. F u r t h e r coalescence depends s t rongly on the ra te at wh ich the s u r f a c t a n t is t r a n s p o r t e d away. Wi th low diffusivity, t he t r a n s p o r t of


A

E. E E o O

g e -

0

e -

¢,,.

8

0.0160

0.0140

t'=0

0.0120 -0.003

: : D=5e-5 m.m/s c : D=5e-6 m.m/s

, t'=1800S

0.017 0.037 0.057 z (m)

Fig. 48. Sur fac tan t concentra t ion (c~) profiles for the sys tem of Fig. 47.

surfactant away from the region is slower and the local accumulation is higher. The films are therefore stabilized to a greater extent and the rate of coalescence is much smaller. Thus it takes longer for REtain to approach RFmax and foam collapse starts much later. A similar explana- tion applies to the rate of foam collapse. A higher D means that the surfactant released due to collapse quickly diffuses away without significantly stabilizing the films and the foam collapses at a much faster rate. Figure 48 shows a comparison of the surfactant concentration profiles at different times. The rise in surfactant concentration in the upper levels of the foam is primarily due to coalescence and collapse. However, the surfactant concentration throughout the foam is always


t/)

_e . Q

_= f -

.o t/) C

._E " 0

2.6

2.2

1.8

1.4

1.0

• D = 5 e - 5 m.m/s o o D = 5 e - 6 m,rn/s

t'=1800s

0.6 0.000 0,005 0.010 0.015 0.020

z (m)

Fig. 49. Profiles of the dimensionless bubble radius (I~/R o) at different times for the system of Fig. 47.

h igher when the diffusivity is smaller. Figure 49 shows the profiles of the dimensionless bubble radius R / R o at th ree different times: when the gas supply is jus t off(t ' = 0), 10 minutes la ter (t" = 600 s) and 30 minutes l a te r (t ' = 1800 s). Here t' represents the t ime elapsed af ter the gas supply is shu t off. At t ' = O, there is no coalescence and the bubble size th roughou t the foam is uniform and equal to the initial bubble radius, i.e. R / R o = 1. At other t imes however there is a sharp increase in the bubble radius in the upper portion of the foam. This can be explained as follows: due to drainage, the liquid content is lowest at the top of the foam. In o ther words, at z = z I = 0, rp is the smal les t and the capil lary pressure (~/rp) is the largest which in t u rn means tha t for a given RF,


f i lms at t he top of the foam dra in mos t rap id ly and r u p t u r e the earl iest . Coalescence there fore s ta r t s f irst a t the top of the foam and p ropaga t e s down the foam wi th the passage of t ime. I t is clear f rom the f igure t h a t for both va lues of D, the re are regions in which coalescence has t a k e n place at t ' -- 600 s and t ' = 1800 s. However , for l a rger D, s ignif icant ly more coalescence ha s occurred and the foam/gas in ter face has moved f a r t he r for reasons a l ready out l ined ear l ier in this section.

Effect of the initial distribution To e xa mine the effect of the ini t ial d i s t r ibu t ion (fo), a s imu la t ion was

car r ied out u s ing a di f ferent f0 wi th a la rger width , i.e. wi th RFmin 0 = 0.95 RFO and RFc -- 1.05 RFO, a var ia t ion of about 5% about the mean . F igures 50 and 51 show the resu l t s for th is case. With the b roader d i s t r ibu t ion , coalescence occurs in a la rger por t ion of the foam and foam collapse s t a r t s later. Both these effects are consequences of a smal le r RFmin 0. Since fi lms wi th a smal le r R E dra in quicker, a smal le r RFmin 0 m e a n s t h a t t he cor responding films are t h i n n e r at any given t ime and hence r u p t u r e earl ier , r e su l t ing in the ear l ier onse t of coalescence. On t he o the r hand , s ince RFmax is larger, it t akes longer for RFmin to app roach RFmax and foam collapse s t a r t s later.

Effect of surfactant concentration An increase in su r f ac t an t concent ra t ion usua l ly stabil izes a foam by

increasing lima x and t he surface viscosity. A la rger ]']max has two consequences: f i rs t ly th is impl ies t h a t a l a rger capi l lary p re s su re is r equ i red to r u p t u r e a film. This m e a n s t h a t g rea t e r d ra inage m u s t occur and a longer t ime m u s t e lapse before fi lms at the top wi th R E = REtain 0 r u p t u r e

and bubbles s t a r t coalescing. A la te r onse t of coalescence m e a n s t h a t foam collapse will s t a r t m u c h la te r since it t akes a longer t ime for RFmin to app roach RFmax. The o ther i m p o r t a n t impac t of t he ini t ial

s u r f a c t a n t concen t ra t ion is on the character is t ics of the foam w h e n a mechanica l dra inage "equilibrium" is achieved. At "equilibrium", the re is

(A(°/_ no flow in the P l a t e a u Border channe l s Dz [rp) - - pg)' no fi lm d ra inage

0c s (--~-~ = 1-I), no d i f fus ion of s u r f a c t a n t (-~--z--0) and no foam col lapse % dz 1

( ~ - = 0). T h u s in an equi l ib ra ted foam, the concen t ra t ion of su r f ac t an t

t h r o u g h o u t the foam is the s ame and equal to t h a t of t he foaming


0.050

0 .049

A 0.048

C

0

u. 0 .047

0.046

= =A o oB

0.045 0.0 500.0 1000.0 1500.0 2000.0 2500.0

t'(sec)

Fig. 50. V a r i a t i o n of foam l e n g t h w i t h t ime for two in i t ia l d i s t r i bu t ions (f0). In bo th cases c~o = 0.012 mo le s /m 3, c e = 8 moles /m 3, R o = 0.2 m m and D = 5×10 -5 m2/s. In case A,

RFinin0 = 0.99 RFo and RFc = 1.01 RFo and in case B, REta in 0 = 0.95 RFO and Rfc = 1.05 RFo.

solution (cso). By integrating the equation ~zz =-pg using the condi-

tions ~ l = 1-Imax(Cs°)andrplz2 = Ro, the length ofthe equilibrated foam Z 1

is given by:

($

- R--o + I-Imax(Cs°) (Z 2 -- Z l ) eq = L m a x - - ( 1 5 4 ) pg


5.0

4.6

4.2

3.8

:~ 3.4

e,,

" 3.0

-~ 2.6 .9 ( / ) r - e

.E_ 2.2 10

1.8

1.4

1.0

c -~B

I

~ / t'=1800s ~ t'=6OOs ~t

t';O

z (rn)

k 0 . 6 = I ~ I ~ I =

0.000 0.005 0.010 0.015 0.020

Fig. 51. Profiles of dimensionless bubble radius (R/R o) for the system of Fig. 50.

The subscript "max" indicates tha t this is the largest length a foam can have wi thout undergoing collapse. Figures 52 and 53 show the resul ts for two values of Cso, viz. 0.012 moles/m 3 and 0.015 moles/m 3. Lma x for the l a t t e r is 3.67 cm as opposed to 1.81 cm for the former. Thus in the l a t t e r case, collapse slows down as the foam length approaches 3.67 cm. Such a slowdown is not in evidence in the former case and collapse cont inues well below (z 2 - z 2) = 3.67 cm. The opposite effect is seen when the sal t concentra t ion is raised to 8.245 moles/m 3. Raising the salt concent ra t ion compresses the electrical double layer and lowers YImax, the reby lowering Lma x from 3.67 cm to 2.07 cm. Figures 54-56 show the effect of sal t concentra t ion on a foam with Cso = 0.015 moles/m 3. With


0 . 0 5 0 . . . . , . . . . . - . • , . . . . , . . . . , . . . . , . . . .

0 . 0 4 - 9

0 . 0 4 - 8

0 . 0 4 - 7

0 . 0 4 - 6

0 . 0 4 - 5 E "~ 0.04-4- E

0.04-3

"J 0 . 0 4 2 E

o ~ o . o ¢ 1 L~

0.04.0

0 . 0 3 9 s . 1 le

0 . 0 3 8

0 . 0 3 7

0 . 0 3 6

0 . 0 3 5 . . . . . . . . . . . . . . . . . . . 0 . 0 5 0 0 . 0 1 0 0 0 . 0 1 5 0 0 . 0 2 0 0 0 . 0 2 5 0 0 . 0 3 0 0 0 . 0 5 5 0 0 . 0

t'(sec)

Fig. 52. V a r i a t i o n of foam l e n g t h w i t h t i m e for two v a l u e s of t h e f o a m i n g so lu t ion

s u r f a c t a n t c o n c e n t r a t i o n (C~o). The v a l u e s of t he o the r p a r a m e t e r s u sed a re D = 5×10 -'~

m2/s, c e = 8 m o l e s / m 3, R 0 = 0.2 mm, R~, o = 0.606 Ro, Rgmia 0 = 0.99 RE0 and Rfc = 1.01 Rfo.

higher salt concentration, coalescence and foam collapse start earlier and occur at a faster rate. This is further emphasized by the profiles of surfactant concentration. With higher salt, greater coalescence results in a much larger amount of surfactant being released into the bulk leading to higher surfactant concentrations.

Before we proceed with the conclusions, we will discuss briefly the phenomenon of inter-bubble gas diffusion (Ostwald r ipening)which has been ignored in our model.

3. Inter-bubble gas diffusion (Ostwald Ripening)

When the pressure in the foam bubbles varies significantly, there is a transfer of gas from bubbles with higher pressure to those with lower pressure. This transfer occurs by diffusion through the liquid films


2.9

1.4

0.9 0.000 0.001

:-5 2.4 o

(D

,.CI

2

~ 1.9 e -

.o_

E ° _

- = c~o=O.O 12moles/m 3 = o C~o=O.O 15moles/m 3

I I

0.002 0.003

z ( m )

I

0.004 0.005

Fig. 53. Profiles of dimensionless bubble radius (-R/R o) at different t imes for the system of Fig. 52.

s e p a r a t i n g the bubbles and is dr iven by the difference in the concentra- t ion of the dissolved gas at t he two fi lm surfaces. Since these concentra- t ions are propor t iona l to the p ressure at the in terfaces (Henry 's law) the t r a n s p o r t of gas is fas ter w h e n the p res su re difference is h igher . The o the r i m p o r t a n t factor is the solubil i ty of the gas in the liquid. In ter- bubble diffusion increases d ramat i ca l ly as the solubil i ty of the gas increases . This is the m a i n reason why CO 2 foams are less s table t h a n N 2 foams. The p re s su re ins ide a bubble is d e t e r m i n e d by its vo lume w h e n it is j u s t formed. Since the bubble is ini t ia l ly spher ical , t he p r e s s u r e ins ide the bubble is g iven by:

P i = Ppool + ~ - (155)

w h e r e R is t he bubble r ad ius and Ppool is t he p ressure in the l iquid pool. Thus , t he p re s su re is h ighe r in the smal le r bubbles. This m e a n s t h a t gas is t r a n s f e r r ed f rom smal le r to la rger bubbles caus ing the smal le r bubbles to get sma l l e r and the la rger bubbles to get larger. This is very


0.050

e ' -

e -

=, E ,,o

0 .049

0 .048

0.047

0.046

0.045 - 0.0

: = salt concentration=O.OO8M o o salt concentrat ion=O.OO825M

500.0 1000.0 1500.0 2000.0 2500.0 t'(sec)

Fig. 54. V a r i a t i o n of foam l e n g t h w i t h t i m e for two v a l u e s of t h e foaming so lu t ion sa l t c o n c e n t r a t i o n (ce). The v a l u e s of the o the r p a r a m e t e r s used are D = 5×10 .5 m2/s, c~o =

0.015 m o l e s / m 3, R 0 = 0.2 m m , RFO = 0.606R0, RFmin o = 0.99 RE0 and RFc = 1.01 Rfo.

s imi la r to the p h e n o m e n o n of Ostwald r ipen ing observed in supe r sa tu - r a t e d solut ions, whe re the la rger par t ic les grow in size at t he expense of smal le r ones. Hence in ter -bubble diffusion is also re fer red to as Os twa ld r ipening.

The first theore t ica l t r e a t m e n t of in ter -bubble diffusion was provided by Cla rk and B l a c k m a n [70], who showed t h a t the ra te of g rowth of a bubble of r ad ius R is given by the expression:

- C Rm - (156)


= : salt concentrat ion=O.OO8M o salt concentTat ion=O.OO825M

2 .6

2 .2 ~ ~ = 1 8 0 0 s

~ '=

~ 1.4

1.0

t';O

0.6 ~ 0.000 0.005 0.010 0.015 0.020 z (m)

Fig. 55. Profiles of dimensionless bubble radius (R/Ro) at different times for the system of Fig. 54.

~.,Ni R 2 . where R m ~_.NiR i is a certain mean radius such that bubbles with R <

R m shrink and those with R > R m grow. Typically, a population balance equation is solved using Eq. (156) to obtain the evolution of the bubble size distribution. A pivotal assumption made in deriving Eq. (156) which has been made in all subsequent papers [61,71-73] dealing with this phenomenon is that the pressure inside a bubble of radius R

2~ is g iven b y p i =Patm + -R-, w h e r e Patm is the a tmosphe r i c p ressure . The

p r e s s u r e d i f fe rence be tw een two bubbles w i t h radi i R] and R 2 is t h e n

112 A. Bhakta, E. Ruckenstein ]Adv. Colloid Interface Sci. 70 (1997) 1-124

0.0200

salt concentration=O.OOSM salt concentration=O.OO825M

E

0

c- O

e-

0.0180 I" ~, A t'=6OOs

oo

0.0160

t' = 1800s

~=0

o.o14o ~ o.o8o z (m)

Fig. 56. Profiles of surfactant concentration (c~) at different times for the system of Fig. 54.

2a - This relationship is true only for spherical bubbles. It is

not clear how the pressure changes with size in polyhedral bubbles. Equation (156) may therefore not be valid for polyhedral foams. Prob- ably the only option for polyhedral foams is to use a microscopic approach in which the curvature and area of each film is a function of time and changes as the pressure in the bubbles sharing the film changes. This approach has been used to simulate the evolution of a two-dimensional foam [31-33]. Using this approach for three dimensional foam and integrating it with drainage and coalescence is likely to be ex- tremely difficult. A model which incorporates drainage, coalescence and


inter bubble gas diffusion has been proposed by Narsimhan and Ruck- enstein [60,61] for steady-state foams. However, drainage is treated assuming pentagonal dodecahedral geometry and inter bubble gas diffusion is treated assuming spherical bubbles. Given the complexity of the system, such assumptions are probably inevitable. Even with these assumptions, extending this model to unsteady state standing foams would be very difficult.

The results of our model are therefore valid only for initially monodispersed foams composed of gases that are sparingly soluble in the continuous phase liquid.

4. C o n c l u s i o n s

A detailed model is formulated for the drainage, coalescence and collapse of aqueous foams. Simulations have provided several insights into the kinetics of foam collapse and drainage. New insights into the metastable equilibrium established in foams have also been obtained.

The foam is assumed to be composed of identical pentagonal dodecahedral bubbles. Drainage of the liquid in the films is modeled using the Reynolds equation for flow between circular parallel disks. This flow is driven by the capillary pressure generated due to the curvature of the adjacent Plateau border channels and is opposed by the disjoining pressure that arises between the two film surfaces. The disjoining pressure isotherm (which expresses the relationship between the disjoining pressure and the film thickness) plays a crucial role in determining the characteristics of film drainage and rupture. Since the disjoining pressure is composed of attractive van der Waals and repulsive electrical double layer and hydration forces, the isotherm often has a maximum. The film thickness corresponding to this maximum is the critical thickness of film rupture. Film with thicknesses smaller than this critical thickness rupture because of the unbounded growth of disturbances on the film surfaces. On the other hand, films with thicknesses larger than this thickness are stable because any waves on their surfaces are damped.

Whether a film arrives to this critical thickness depends on the relative magnitudes of the capillary pressure and the disjoining pressure. If the capillary pressure exceeds the maximum disjoining pressure, the film will drain to the critical thickness and rupture; if the capillary pressure is less than the maximum disjoining pressure, the


film will arrive at a stable equilibrium thickness at which the capillary pressure and the disjoining pressure are equal. The capillary pressure must therefore exceed a critical in order to cause film rupture. In other words, there is a critical Plateau border radius (which depends on the local surface tension) below which a film drains to its critical thickness and ruptures.

The capillary pressure is different in different portions of the foam and varies with time depending on the liquid content in the foam. When the liquid content is larger, the Plateau border radius is larger, the capillary pressure is smaller and hence the foam is more stable. The local liquid content in a foam is largely determined by the gravity driven flow in the network of Plateau border channels. While the liquid flows out of the foam through the channels at the bottom, the channels in the upper portion of the foams drain into the channels below. Thus, with the passage of time, there is a redistribution of liquid in the foam with the upper portions becoming progressively drier. This sets up a pressure gradient within the foam which opposes gravity and hence slows down the flow in the foam leading eventually to a drainage equilibrium in which the liquid content in the various portions of the foam no longer changes with time. If this equilibrated state is reached before the capillary pressure exceeds the maximum disjoining pressure, no foam collapse occurs. If not, foam collapse occurs until an equilibrium height is reached.

Foams can be distinguished into two categories depending on the method employed to generate them. Pneumatic foams are produced by bubbling a gas through a solution of surfactant. Foam is formed at the surface of the liquid and rises as more gas is bubbled into the system. The foam length increases till the gas supply is shut off. The drainage that occurs as the foam moves up leads to a distribution of liquid in the freshly formed foam. On the other hand, in homogeneous foams which are produced by vigorous agitation of the surfactant solution, the liquid content is uniform throughout the foam. The other important difference between the two types is that pneumatic foams are always in contact with the foaming solution, while with homogeneous foams, there is a time lag before some liquid appears at the bottom of the foam. Both kinds of foams are modeled in this work.

The model successfully predicts the trend of lag times observed in the drainage of homogeneous foams. For a foam of a given length, the lag time decreases as the initial volume fraction of continuous phase (e 0) increases and for a given eo, the lag time decreases as the foam length


increases. It is shown that based on certain properties of a homogeneous foam such as the bubble radius (R0), column height (L0), interfacial tension (a) and density difference between the two phases (p), it is possible to determine whether phase separation(separation of the continuous phase liquid via drainage or the separation of the dispersed phase gas via collapse). It is also possible to determine which of the two phases (if not both) will separate out before equilibrium is established. Two-phase separation(collapse as well as drainage) is possible only if the column length exceeds a certain value, which for a given pair of fluids depends only on the bubble radius (R0). A generalized "phase" diagram is presented which makes it possible to determine the phase behavior of a concentrated emulsion based on the value of a single dimensionless

(I

number ( P - pgKRoL0).

In the case of pneumatic foams, the superficial velocity of the bubbling gas plays an important role in determining the liquid content in the foam. Foams produced with a higher superficial gas velocity contain a greater amount of liquid and hence drain at a faster rate. Simulations were carried out to examine the effect of the initial foam length, the bubble radius and the viscosity of the foaming solution on the drainage of liquid from foams in which no film rupture occurs. The results were in good agreement with experimental data.

When foam collapse starts during the generation of a pneumatic foam, the foam often attains a steady state height when the rate at which the foam is produced equals the rate at which the foam is destroyed. 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 (time required for a foam to collapse to half the steady state height) was examined. 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 (as determined by the steady state height and collapse half life) 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. Also, 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 of the foam occurs. Dimensionless plots of (z 2 - Z l ) / H 0 versus log(t/Qi 2) where H 0 is the s teady state height and tl/2 is the ha l f life, coincide for most of the period of foam collapse even w h e n the electrolyte concentra t ion and bubble size are changed.

It has been observed exper imenta l ly t ha t wi th some foams, the stabil i ty of a foam increases sharply at h igh salt concentrat ions due to the formation of very stable Newton black films. An a t t empt is made to explain this phenomenon by considering, in addit ion to the van der Waals and the electrical double layer forces, a short range repulsive force due to the organizat ion of wa te r molecules nea r the surfaces. The electrical double layer is a resul t of the surface charge due to the dissociated adsorbed surfac tant molecules, while the short range repulsive force is caused by the dipole moments of the undissociated adsorbed sur fac tan t molecules. It is shown tha t depending on the values of the pa rame te r s used, it is possible to have a disjoining pressure i so therm wi th two max ima at sufficiently high salt and surfac tant concentrat ions. U n d e r these conditions, ins tead of film rupture , there is a j u m p transi- t ion to very stable Newton black films. These resul ts are in qual i ta t ive ag reemen t wi th exper imenta l observations.

It is exper imenta l ly observed tha t in some foams, there is often signif icant coalescence of bubbles which resul ts in an increase in the bubble volume. A model is presented tha t includes the effect of coalescence in s tand ing foams. Coalescence is assumed to resul t from the non-uniformi ty of film sizes wi thin a volume element. This leads to a non-uniformi ty of d ra inage ra tes and hence of film thicknesses wi th in the element . Smal ler films dra in faster and hence rup tu re first, causing bubble coalescence. Thus a bubble size dis tr ibut ion is set up wi th in the foam, which evolves wi th time. The model is also able to predict the var ia t ion of the sur fac tan t concentra t ion profile as it changes due to coalescence and collapse. Simulat ions were performed to examine the effect of var ious pa ramete r s including the apparen t diffusion coefficient, the dis t r ibut ion of film sizes and the concentrat ions of sur fac tan t and sal t in the foaming solution. A high diffusion coefficient resul ts in faster coalescence and an ear l ier onset of foam collapse. This is because the sur fac tan t re leased due to coalescence and collapse is t ranspor ted away before it can accumula te and stabilize the films. An increase in the wid th of the init ial dis tr ibut ion of film sizes resul ts in a la ter onset of foam collapse, because a grea te r amount of coalescence needs to occur before all the films at the top become critical and rupture . The concentra t ion of sur fac tan t and salt in the foaming solution pr imar i ly affects the

A. Bhakta, E. Ruckenstein / Adv, Colloid Interface Sci. 70 (1997) 1-124 117

equi l ibr ium length at which a collapsing foam mus t necessar i ly arrive. A h igher su r fac tan t concentrat ion increases the surface charge, raises the m a x i m u m disjoining pressure and hence increases the equi l ibr ium foam length. On the other hand, an increase in salt concentra t ion compresses the electrical double layer and has the opposite effect.

N o m e n c l a t u r e

a = Distance be tween adjacent wa te r molecules A' = Cons tan t in the expression for the short range repulsive force

A h = H a m a k e r constant ap= Average cross-sectional a rea of a P la teau border channel a 1 = Empir ical p a r a m e t e r in the F r umkin adsorption equat ion A1, A2, A 3 = Funct ions used in the calculation of the electrical double layer force A r = Area of the surface of a film

RFmax

~F = f fAFdRF = Mean surface area of a film at any level

RFnfin B' = Cons tan t in the expression for the short range repulsive force b 1 = Empir ical p a r a m e t e r in the F rumkin adsorption equat ion bl, b2, b 3 = Variables defined in Eqs. (81-83) b = Empir ical cons tant used in the calculation of the van der Waals force c = Empir ical constant used in the calculation of the van der Waals force; cons tan t of in tegra t ion c f = Correction factor for Reynolds equat ion account ing for the mobili ty

of the film surfaces c~ = Concentra t ion of salt in the foam Cel =Electrolyte concentrat ion c s = Concentra t ion of dissolved sur fac tan t Cso = Sur fac tan t concentra t ion in the foaming solution

c R- = Concent ra t ion of su r fac tan t anions nea r the surface

CNa + = Concentra t ion of Na ÷ ions nea r the surface

c v = Coefficient account ing for the mobility of the walls of a P la teau

border channel C = Cons tan t used in Eq. (156) D = Apparen t diffusivity


D s = Su r f ace d i f fus iv i ty E = Elec t r ic f ield E ° = Elec t r ic field due to the su r face ac t ing on the f i rs t l aye r of w a t e r mo lecu le s E d = P a r a m e t e r u s e d in the ca lcu la t ion of the shor t r a n g e r epu l s ive force

Edre f = R e f e r e n c e v a l u e for E d

e = P ro ton ic charge ; a lso empi r ica l c o n s t a n t u s e d in the ca lcu la t ion of t h e v a n de r W a a l s force f = D i s t r i b u t i o n of f i lm radi i fo = In i t i a l d i s t r i bu t i on of f i lm radi i G = Super f i c i a l gas ve loc i ty g = g r a v i t y

K - %] 4~ 2(0 .161)np0.816

K d = Dissoc ia t ion c o n s t a n t of t he a d s o r b e d s u r f a c t a n t molecu les k B = B o l t z m a n n c o n s t a n t

= A v e r a g e l eng th of a P l a t e a u bo rde r channe l ls = L e n g t h of a s e g m e n t

L = T h i c k n e s s of b r u s h L 0 = In i t ia l l e n g t h of foam/concen t ra t ion ; C h a i n l eng th in a t h e t a so lven t

L m a x = M i n i m u m he igh t of the f o a m / c o n c e n t r a t e d emu l s ion for co l lapse as wel l as d r a i n a g e to occur n = N u m b e r of w a t e r molecu les pe r un i t v o l u m e

n S = N u m b e r s e g m e n t s in molecu le N = N u m b e r of b u b b l e s pe r un i t v o l u m e n p = N u m b e r of PB channe l s pe r b u b b l e

n F = N u m b e r of f i lms per b u b b l e N A = Avogadro ' s n u m b e r p = P r e s s u r e in l iquid

Pref = R e f e r e n c e p r e s s u r e in l iquid P i = P r e s s u r e ins ide b u b b l e / d r o p l e t AP = P r e s s u r e d i f fe rence dr iv ing f i lm d r a i n a g e

P = D i m e n s i o n l e s s n u m b e r g iven by K p g R o L o

P 1 , P 2 = Var iab les used in the calculat ion of the shor t r ange repuls ive force qe = V o l u m e t r i c flow r a t e t h r o u g h a P l a t e a u b o r d e r channe l inc l ined a t an ang le O to t he ve r t i ca l


q = Empir ical cons tant used in the calculation of the van der Waals force q P B = Volumetr ic flux of liquid due to dra inage of liquid th rough the PB channels qc = Volumetr ic flux of liquid due to bulk movement of bubbles q L = Total flux of liquid q v = Volumetr ic flux of gas qcs = Flux of su r fac tan t

r = ( R F - R F m i n ) / ( R F m a x - RFmin) = Dimensionless co-ordinate in R F space

R = Bubble radius R 0 = Bubble radius in the init ially monodispersed foam

1/3 R = ( ~ V) =Mean bubble radius

R m = A m e a n radius used in models for inter-bubble gas diffusion R F = Radius of a film RFmin = Smal les t value of the film radius at a given level

RFmax -- Larges t value of the film radius

R F c = The upper limit for RFmin beyond which the distr ibution is a s sumed

to be Dirac del ta rp = P la t eau border radius R v = Molar gas cons tant s = S t anda rd deviation T = Absolute t e m p e r a t u r e t = t ime t' = t ime elapsed af ter gas supply is shu t off u = Average velocity in a PB channel; also dipole momen t

u e = Average velocity in a PB channel inclined at an angle 0 to the vert ical u s = Dipole m o m e n t per uni t a rea V c = Volume of continuous phase per uni t a rea in an equi l ibrated foam of length L 0 which is in contact with continuous phase at the bottom Yeq ---- Volume of continuous phase per uni t a rea a t equi l ibr ium Vc~ q = Volume of continuous phase per uni t a rea in an equi l ibrated foam

of length Lma x

V D e q = Volume of dispersed phase per uni t a rea in an equi l ibrated foam

of length Lma x V D = Volume of dispersed phase per uni t a rea in an equi l ibrated foam of length L 0 in which the films at the top are critical V = Mean bubble volume


V f = R a t e of f i lm t h i n n i n g W = I n t e r a c t i o n ene rgy x = Dis t ance f rom the mid-p lane

x F = F i l m t h i c k n e s s RFmax - I x F = f x ~ t R F = M e a n fi lm th i cknes s a t a g iven level

RFmin XFc = Cri t ica l t h i cknes s of f i lm r u p t u r e

XFm = F i l m th i cknes s cor responding to the m a x i m u m dis jo in ing p r e s su re

XFm, d l = F i l m th i ckness cor responding to t he m a x i m u m dis jo in ing pres- su re w h e n only electr ical double l aye r forces are p r e sen t

z = Space co-ordinate z 1 = Co-ord ina te of the foam/gas in te r face

zle = Co-ord ina te of t he foam/gas in te r face a t equ i l ib r ium

Z2e = Co-ord ina te of t he foam/l iquid in te r face a t equ i l ib r ium

z 2 = Co-ord ina te of the foam/ l iquid in te r face

G r e e k L e t t e r s

(z = Reciprocal decay l eng th of the shor t r ange repuls ive force

(z d = Degree of d issocia t ion of the adsorbed s u r f a c t a n t molecules

= ( z - z l ) / ( z 2 - z 1) = Dimens ion le s s coord ina te in z-space

5 = Dis t ance be tween ad j acen t p l anes of w a t e r molecules

e d = Dielectr ic cons tan t ; L iqu id f rac t ion

E = L iqu id f rac t ion

Eeq ----- Liqu id f rac t ion a t equ i l ib r ium

e b = L iqu id f rac t ion a t the foam/ l iquid in te r face = 0.26

~c = L iqu id f rac t ion a t the foam/gas in te r face w h e n collapse occurs

e 0 = In i t i a l l iquid f rac t ion

adO = P e r m i t t i v i t y of free space

yp = Po la r i zab i l i ty of w a t e r

Ys = Inve r se of d imens ion les s sur face viscosi ty

ms = Sur face viscosi ty = Reciprocal Debye l eng th

)~n = n t h root of t he equa t ion Jo(~n) = 0

~t = B u l k viscosi ty

~s = Dipole m o m e n t of an adsorbed s u r f a c t a n t molecule

~t w = Dipole m o m e n t of a w a t e r molecule


Pc = D e n s i t y of t he c o n t i n u o u s p h a s e

PD = D e n s i t y of the d i s p e r s e d p h a s e p = D e n s i t y d i f fe rence b e t w e e n the c o n t i n u o u s p h a s e and t he d i s p e r s e d

p h a s e

(~ = S u r f a c e t ens ion

c c = S u r f a c e cha rge pe r un i t a r ea

G O = S u r f a c e t ens ion of p u r e w a t e r

• s = Su r f ace po t en t i a l = P o t e n t i a l

~Vm = M i d - p l a n e po ten t i a l

F = S u r f a c e d e n s i t y of s u r f a c t a n t i

F = M e a n su r f ace d e n s i t y of s u r f a c t a n t a t a g iven level. I t t a k e n to be a

func t ion of the m e a n f i lm th i cknes s and the s u r f a c t a n t c o n c e n t r a t i o n

F~ = Su r f ace d e n s i t y a t s a t u r a t i o n

1-I = Dis jo in ing p r e s s u r e

~ h ---- C o n t r i b u t i o n to the d is jo in ing p r e s s u r e due to o rgan iza t ion of w a t e r mo lecu l e s

] - [ m a x ---- M a x i m u m dis jo in ing p r e s s u r e

[IDL = Elec t r ica l doub le l aye r force pe r un i t a r e a

rlsR = S h o r t r a n g e r epu l s ive force pe r un i t a r e a

l i s t = S te r ic r epu l s ive force pe r un i t a r e a

[IvD w = v a n de r W a a l s force pe r un i t a r e a

limax,dl ---- The m a x i m u m dis jo in ing p r e s s u r e w h e n no sho r t r a n g e repu l - s ive forces a re p r e s e n t

R e f e r e n c e s

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Appendix A

Calculation of velocity coefficient

Desai and K u m a r [56] have expressed the velocity coefficient (c v) as

a funct ion of the d imens ion less surface viscosity (Ys = 0 . 4 3 8 7 ~ a p ) as: Ns

Cv = b i o + b i l ( T s - 7/) + b i2 (Ts - 7/) 2 + b i 3 ( Y s - 7/) 3 (A1)

w h e r e 7 /< 7s < 7i+1 are cons tants . The va lues of the coefficients {bij j = 0-3} can be found e l sewhere in the l i t e ra tu re [56].


Appendix B

Calculat ion of the electric f ield due to dipoles on the surface

The component of the electric field at the point P in a direction perpendicu la r to the plane due to a dipole of moment 'u' at point A is given by:

2ucos2(a) E - (B1)

4Tcea0y 3

If u~ is the dipole momen t per uni t area, the electric field at the point P due to a r ing of radius r and thickness dr is given as:

d E = [ 2Us(2nrdr)c°s2(a) ~ (B2)

[ 4T~E£0Y 3 ]

Since cos(a) = r/y and y2 = r 2 + z 2, we have:

2Ps 2nrdr r 2 d E - 4nae°Y 3 (r 2 +ze)3/2 y2 (B3)

Thus the total electric field at P is given as:

E = us f rz2 us ££00 (r 2 +Z2)5/2 d r - 3aa(~z (B4)

A

Z

Fig. B. 1. Schematic diagram for the computation of the electric field.

decay of standing foams: drainage, coalescence and collapsecgay/gdr-mousses... · homogeneous...

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