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164

Langmuir 1988, , 164-169

Pressure/Volume/Surface Area Relationships in Foams and

Highly Concentrated Emulsions: Role of Volume Fraction

H M. Princent

CollGge de France, Physiqu e de la MatiGre Con dens e,

11

Place Marcelin Berthelot ,

75231 Paris, Cedex 05, France

Received M arc h 17, 1987. In Final Form: Augu st 17, 1987

Various pressures in and around a foam

or

concentrated emulsion are discussed in some detail. These

include the internal pressure inside the dispersed cells, the pressure in the interstitial continuous phase,

the disjoining pressure in the films separating the cells, the osmotic pressure, and the vapor pressures of

the dispersed and continuous phases. The effect of the volume fraction, 4, of the dispersed phase is

investigated explicitly, not only on some of the above parameters, but also on the equation of state and

the compressibility of a foam. Earlier results, valid for

C 1

only, are corrected for

4 < 1.

The area of

the intercellular films as a fraction of the total surface area and its dependence on

4

have been evaluated

also.

Introduction

The study of foams and highly concentrated emulsions

has seen a resurgence in recent years, partly as a result of

additional potential applications, such as tertiary oil re-

covery by means of such systems.

In

most published work,

consideration has been limited to situations in which the

volume fraction of the dispersed phase,

4,

is extremely

close to unity, such as in dry, polyhedral foams, where

there is negligible continuous phase not only in the thin

films between the bubbles but also in the Plateau borders

along the intersections of films. This is the approach taken

in such studies as those of Derjaguin, Ross,28Ross et al.,p6

Weaire, Khan and Armstrong? Kraynik and Hansen? etc.

Often this dry foam approximation is appropriate, e.g., in

the top region of a very tall, equilibrated

foam or

emulsion

column, where the shape of the bubbles or drops indeed

approaches tha t of fully developed polyhedra with sharp

edges and corners. In many other cases, however, this is

not

so,

and a more general treatment must incorporate the

volume fraction asa variable

in

the range $o

I 1,

where

$o

is the volume fraction of the close-packed, undeformed,

spherical bubbles or drops (Kugelschaum). For mono-

disperse systems, 4o = 0.7405; for typical polydisperse

systems,

it

has been found

to

be only slightly different and

smaller+12 (not larger as one might expect for some very

specific multimodal size distributions, which, in fact , are

rarely, if ever, encountered in practice).

In a series of recent papers, we have specifically inves-

tigated the dependence of a variety of static and dynamic

properties on

4,

as well as on the drop size and interfacial

Thus, we have shown that the osmotic

pressure,12J4 shear modulus, and yield s t r e s ~ ~ ~ J ~ J ~ll

depend strongly on

4.

The same

is

true for the viscosity,

mostly through the yield stress.l6p2l

In the present study, we investigate the effect of

C

on

a number of other properties, such as the internal

pressure in a foam, the equation of state, the compres-

sibility, and the film area per unit volume of the dispersed

phase. We shall also take this opportunity to establish

links between various properties that seem not to have

been made before.

I t will be assumed that the thickness of the films be-

tween the bubbles or drops (the cells) is negligible com-

pared

to

the cell size (finite film thickness simply gives rise

to an upward shift in the effective volume fraction, as

indicated beforel0J and that the contact angle between

Curren t address: G eneral

Foods

Corp., Technical Center, 555

South Broadway, Tarrytown, New York 10591.

0143-7463/88/2404-0164 Q1.50/0

the films and the adjacent Plateau borders is zero.

where a is the capillary length

We shall further assume that the cell size is R >

a.

The dispersed phase is phase 1; the continuous phase

is phase

2.

Pressures in and around the System

There are a number of interrelated pressures one may

distinguish in a concentrated fluidlfluid dispersion of

4

2

40.

Internal Pressure

pi.

This is the pressure inside the

cells. If the system is monodispersed and gravity is absent,

p

is identical in all cells. If the system is polydisperse,

the internal pressure will vary from cell to cell, since the

cells are then generally separated by films of nonzero

curvature. In that case one may define an average internal

pressure given by

where

p

and vi are the pressure and volume of cell

i

V1

(1)

Derjaguin,B.V.Kolloid-Z.

1933,

4, 1.

(2)Ross, S. nd. Eng. Chem. 1969,

1,

48.

(3)

Ross,

S.

Am. .

Phys. 1978, 6, 13.

(4)Nishioka, G.;Ross, S.J.Colloid Interface Sci . 1981, 1 1.

(5) ishioka,

G ;

oss,

S.;

Whitworth, M. J. olloid Interface Sci .

(6) Morrison, I. D.; Ross, S.J. olloid Interface Sci . 1983, 95, 97.

(7)

Weaire, D.; Kermode, J . P. Philos. Mag. , [P ar t ]B

1983,

8 ,

245;

(8)Khan, S. A,; Armstrong, R. C. J.

Non-New ton ian

Fluid

Mech.

(9)Kraynik, A.M.; Hansen, M . G. J.Rheol. 1986, 0,409.

(10)

rincen, H. M. J.Colloid Interface Sci .

1985,

05,

150.

(11)

rincen, H. M.; Kiss, A. D.

J.

Colloid Interface Sci.

1986,112,427.

(12) rincen, H M.; Kiss, A. D . Langmuir , 1987,

,

36.

(13)

rincen, H. M.

J. Colloid

Interface Sci .

1983,

1

160.

(14) rincen, H. M. Langmuir , 1986, , 519.

(15) oshimura, A,; Prudhomme, R. K .; Princen, H. M.; Kiss,

A.

D.

(16)

chwartz,

L.

W.; Princen, H. M. J.

Colloid

Interface Sci .

1987,

1983, 5, 35.

1984,

0,

379.

1986,22,

.

J.

Rheol . , in press.

118

201.

1988 American Chemical Society


2/6

PressurelVolurnelSurface Area i n Foams and Emulsions

A T M S P H E R E

P

DISPERSION

i Z

c

Figure 1.

Fluid/fluid dispersion in contact

with

atmosphere.

is the volume of the dispersed phase,

V

is the tot volume,

and the summations are taken over all cells. For a foam,

filled with (ideal) gas,p i epresents the pressure inside the

gaseous volume Vl, if

all

foam lamellae would rupture and

total phase separation would 0ccur.l

We shall return to this pressure in more detail later.

Border Pressure p b.

This is the pressure inside the

continuous phase in the plateau borders. I t is related to

pi f a given cell by

Pb

=

Pi

u121Kbil

(3)

where Kbi is the curvature of t he free surface of cell i , i.e.,

the surface outside the films. In the absence of gravity,

pb

is

constant everywhere.

In

a gravitational field,

Pb

varies

linearly with height,

z ,

according to

P b = c +

Z P g

4)

where

C

is a constant,

p 2

is the density of the continuous

phase, and z is pointing downward. If, as in most practical

situations, the system is in contact with a gaseous at-

mosphere of pressure

P

at some level

z

=

0,

then the value

of C n eq 4 is determined by the curvature, Kt of the free

continuous phaselgas interface between the dome-shaped

films at the top of the highest layer of cells (Figure

l ) ,

.e.,

(5)

In a foam,

u2

= u12 s the continuous phaselgas interfacial

tension. (Similarly, for an emulsion in contact with a layer

of dispersed (liquid) phase, u2 = u12 s th e liquid/liquid

interfacial tension.) For an emulsion in contact with air,

on the other hand, u2in eq 5 is the continuous phaselgas

interfacial tension. It is important to note that , however

complex the detailed shape of the top surface of the system

(particularly in a polydisperse system), Kt is constant ev-

erywhere, even in gravity, provided the surface is hori-

zontally flat on the scale of the capillary length, a.

c

= P

U21Ktl

Combining eq 4 and 5 yields

Pb = p UZIKtl

+ Z P g

. (6)

The second term on the right-hand side is often over-

10oked.~J

It

is obvious from Figure

1

hat the curvature

Kfi

at the

top of cell i in the first layer is given by

4 1 2 + u2)lKfil = c l 2 b b i l u 2 l K t l (7)

Pi

=

P +

IKfil(Ul2

+ U Z

(8)

provided that, in the case of an emulsion in contact with

a gaseous atmosphere, the continuous phase spreads over

and that the internal pressure in th at cell is

Langmuir, Vol. 4 ,

No. 1,

1988

165

~ C O N TP H A S E

n

MEMBRANE

(4 b)

Figure 2. Semipermeable membrane separating dispersion from

continuous phase: (a) general

view;

(b)

detailed

view of

contact

between deformed

cells

and membrane.

the dispersed-phasedroplets in the top layer. If not, a bare

surface of the dispersed phase with tension

c1

s in contact

with the atmosphere over some of i ts area and there will

be a three-phase contact line with the continuous phase

between the drops. Using Neumanns triangle for th e

equilibrium between ul,u2,and u12at the contact line, one

can readily reformulate the problem for this case. Th e

curvature of the roof of the drops is simply increased by

a factor of al2+ u2)/u1, while Kbi and Kt remain unaffected.

It is suggested by Figure

1

that the average relative

internal pressure

pi P

nside the system is already fully

established in the very first layer of cells.

The disjoining pressure,

IID,

s the extra pressure

inside a film due to interaction forces. A t equilibrium, it

equals the total capillary pressure acting on the film. For

any one of the films sumunding any cell (not ust a film

facing the atmosphere), it may be written as

where the + and signs refer to situations where th e

convex

or

the concave side of t he curved film is directed

toward the interior of cell

i

respectively. In a monodis-

perse system, Kfi is zero everywhere, except for the films

directly facing the atmosphere. When gravity is present,

IKbil

and

IID

hen vary linearly with the level z . In a po-

lydisperse system,

Kfi, IID,

nd, therefore, the equilibrium

film thickness vary from film to film, even in the absence

of

gravity. This may further complicate the proper de-

scription of the process of foam coarsening by gas diffusion,

where one usually assumes that the thicknesses and,

therefore, the gas permeabilities of all the films are iden-

The osmotic pressure,

II, s the pressure that must be

applied to a freely movable, semipermeable membrane,

separating the fluid-fluid dispersion from a layer of con-

tinuous phase, in order

to

prevent the latter from entering

the di~persion.~J* he membrane is semipermeable n the

sense tha t i t is freely permeable to all the components of

the continuous phase but impermeable to the drops or

bubbles (Figure

2). II

equals the pressure exerted on the

membrane by the flattened cells and may therefore be

represented by

tica1.7J8J9

where n is the number of cells per unit area of the mem-

brane and

fi

is the flattened contact area for cell i .

Previouslylo we have introduced a total fractional contact

area

f:

n

i = l

f = C f i 11)

which can be readily measured optically.1 It is determined

(17) Khristov, K.

I.;

Exerowa, D. R.; Krugljakov,

P.

M. J .

Colloid

Interface Sci.

1981, 79, 584.

18) Lemlich, R.

I nd . E ng . C hem. F undam.

1978, 17,89.

19)

Beenakker,

C. W. J.

Phys. R eu. L e t t .

1986, 57, 2454.


3/6

166 Langmuir, Vol.4, No. 1, 1988

primarily by the volume fraction,

4,

but may be slightly

dependent on the details of the size distribution.

A n

alternative expression for

II

follows from the process

whereby the membrane is displaced downward

to

squeeze

a volume dV2 of continuous phase out of the dispersion

at constant Vl. For this process one may write

(12)

where

S

is the surface area in the dispersion. Equation

12

may be transformed into12J4

II

dV2

= -al2

d S

Princen

right-hand side equals

-11

at the top of the column, while

we may write for the third term, using eq 6

(20)

where

IKt l

is the curvature of the free continuous phase/gas

interface at t he top of the column (Figure

1) .

Thus, by combining eq 19 and 20, we establish the in-

teresting fact that for the top of the column

P?gH

=

Po p + 4 4

n = 8 2 1 K t l (21)

Now, the vapor pressure of th e continuous phase above

the dispersion is simply given by the Kelvin equation,

applied to the concave continuous phase/gas interface a t

the top, i.e.

p V c = (PVC)Oe-uziKtIV2/RT = (p$)oe- V2/RT

(22)

For small cell size and high volume fraction, II tends to

infinity and the reduction in vapor pressure becomes quite

pronounced.

In the case of an emulsion, one may also consider the

vapor pressure of t he dispersed phase.

It

is, of course,

increased relative to tha t of the bulk dispersed phase. It

can be shown to be given, to a very good approximation,

by

p V d ( p v d ) o e ( 2 ~ d R ) ( V ~ / R T ) ( S / S ~ )

where (po is the vapor pressure of the bulk dispersed

phase,

a12

is the liquidlliquid interfacial tension, R is the

drop radius, and

Ql

is the molar volume of the dispersed

phase. Although

p$

depends strongly on R when R

- ,

it is only slightly affected

as 4 -

through '/So.

Expressions, similar

to

the above, may be derived for

the mutual solubility of the phases.

Internal Pressure, Equation of State, and

Compressibility

For a

dry

olyhedral foam Cui

= V; 4 = l ,

Derjaguinl

has shown that the average internal pressure p i and the

equation of sta te are given by

or

where S/Vl is the surface area per unit volume of the

dispersed phase,

S/So

is the ratio of the surface area of

the deformed cells and tha t of t he spherical cells of t he

same volume, and R32 s the surface-volume or Sauter

mean radius of these spherical cells:

In ref 12 we have experimentally determined

fi

and S / S o

as

a function of for a typical polydisperse system. I t ,was

found that

S / S o

varies from unity at 4 =

c ~

where

II

=

0)

to 1.083 as 4 - and fI

-

The vapor pressure,p, ,

of the continuous phase above

a fluidlfluid dispersion has been reported by us,14 without

proof, to be related to II according to

p V c =

(P

C)

oe

- n V z / R T (16)

where (pVc),,

s

th e normal vapor pressure of bulk contin-

uous phase, V2 is ita molar volume

(or,

rather, the partial

molar volume of the solvent), R is the gas constant, and

T

is temperature.

Two different proofs will be given here. First , consider

the transfer of dn2 mol of continuous phase from the bulk

to the dispersion. The free energy gained is exactly equal

to the decrease in surface energy in the dispersion as a

result of the accompanying dilution, i.e.

P V C

( P v C ) 0

Ap2 dn2

= R T

In n,

=

a12

dS

(17)

However, according to eq 12, at constant V1

(18)

From eq 17 and 18, one immediately obtains eq 16.

In the second proof, we consider an equilibrated dis-

persion column of height

H ,

resting on a bulk layer of

continuous phase. We have considered such a column in

detail in ref 12. The values of II and 4 change continuously

with the height in the column. Relative

to

the atmospheric

pressure,

P

above the dispersion, the pressurepo = p b ( H )

in the continuous phase just below the dispersion must

equal the weight of the column per unit cross sectional

area, i.e.

a12

d S

= -11

dV,

=

-IIv2dn,

From ref 12 and 14 it follows that the second term on the

2 s

p . - p

= -a -

3 l 2 V

4

= 1)

where pi

isas

defined in eq

2

and

n

is the number of moles

of gas in the foam. The same results were later obtained

by Ross.2 Subsequently, Morrison and

Ross

have pointed

out6 hat, although eq 23-24 are strictly valid for mono-

disperse systems and for small clusters of only a few

unequally sized bubbles at

4

=

1,

their general validity for

multibubble, polydisperse systems has not been rigorously

proven, neither by Derjaguin, nor by

Ross

himself. They

seem convinced, however, that the equations are exact even

in tha t more complicated case, and they liken the lack of

a rigorous proof to the situation pertaining to Fermat's

famous last theorem. To avoid this difficulty, let us as-

sume tha t the system is indeed monodisperse. We shall

remove the restriction on

4,

however.

It

will be allowed

to vary from

+o

to unity.

With Derjaguin, we consider an infinitesimal change in

the foam volume, dV

=

dV,.

A t

equilibrium

Cpi dui

P

dVl = a12dS

(25)

Since we assume monodispersity, pi and

ui

are identical

for all cells, so that

Cpi

dui

= p i c

dui = pi dV1

(26)

1

i


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PressurelVolumelSurface Area i n Foams and Em ulsions

Therefore

Langmuir, Vol.

4 , No. 1, 1988

167

Table I. Term s Involved

in

Internal Pressure

pi P

= u12( )

V1 v

For 4 = 1and an isomorphic change in the cells, one may

write from simple geometric considerations

d S

2 s 2 s

- = - = -

dV1 ~ V I 3V

which, when inserted in eq 27, leads to eq 23.

For do < 1,

however, such an isomorphic change is

impossible, as the cell shape must change as Vl and,

therefore,

4

are varied. Equation 28 no longer holds.

Instead, we replace

S

in eq 27 by

s = S/SO)SO

(29)

where, as before,

So

is the surface area of the spheres of

the same volume as the deformed cells of surface area

S.

It is

clear that

S/So

s a shape parameter that depends only

on

4,

whereas

So

simply depends on the cell radius.

Hence

Realizing that

we find from eq 30

Because of eq 14, this may be written in the form

1 - 4

2

s

p i - p

= I + -412-

4 3 Vl

( 0 )

(32)

where

II

is the osmotic pressure.

Comparing eq 32 and 23, we see that for 4

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