rheology of liquid foams

Physica A 205 (1994)565-576 [~1 North-Holland U ~

SSDI 0 3 7 8 - 4 3 7 1 ( 9 3 ) E 0 5 1 7 - 1

Rheology of liquid foams

K.D. Pi th ia a and S.F. E d w a r d s b aSouth Bank University, Wandsworth Road, London SW8 2JZ, UK ~ Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK

Received 1 December 1993

This paper further develops the model of the preceding paper, basing itself on the fact that liquid films are of fundamental importance in explaining the behaviour of the liquid foam. The first step to bring this to importance is illustrated by considering the liquid foam as composed of straight liquid films. The liquid foam is then modelled by either a Kelvin or Maxwell fluid. The simplicity of the model is emphasized when considering large extensions or large compressions. The analysis is based on modelling the T1 process and taking into account the orientation of the films. The key in determining the viscosity is the movement of ceils when the T1 process occurs. The rate of T1 processes determines the motion of the cells which in turn will determine the viscosity. The calculation to determine the viscosity is performed using the Boltzmann equation and the viscosity is found to be in the ideal case to have the form "O~ ~'Oe6~ ~/2.

1. Viscoelasticity of foams

As in the preceding paper we confine ourselves to 2D liquid foams for simplicity but without loss of generality. As a means of introduction into the rheology of 2D liquid foams, we consider the ideal 2D hexagonal liquid foam [1,2]. In this model there is no evolution, no drainage, and there is a film thickness s and all films are of length l.

We now create a very simple model in two dimensions based on the assumptions above to illustrate the importance of liquid films in 2D liquid foams. We consider the 2D liquid foam to be composed of straight liquid films and that any one film is a representative of the whole 2D liquid foam, no orientation of the films will be considered.

We now apply a force. To determine the response of the films and thus the material, we assume the liquid films to be of polymeric nature and allowing a

0378-4371/94/$07.00 © 1994 - Elsevier Science B.C. All rights reserved

566 K.D. Pithia, S.F. Edwards / Rheology o f liquid foams

viscoelastic behaviour. We model this behaviour on the basis of a Kelvin material [3], where we have a dash pot and spring in parallel. Thus for strain e,

de ~r = qo e + ql dt ' (1)

where qo and ql are constants and or is the stress. For constant stress O-o,

= ~r° [1 - exp(- ,~/ ) ] , (2) e q0

where A = q o / q l and

= ~o °/[1 - exp(- ,~t)] . (3) AI r

The applied stress is o- o, the macroscopic deflection is Aly, and strain is e. If Ef is the elastic modulus of the foam then

o- o = E~e , (4)

1 1 Et q0 [1 exp(-At)] (5)

and elastic compliance of the foam is given by

1 1 E--~ = q--~ [1 - exp( -At ) ] . (6)

Thus the material creeps and approaches a final elastic modulus. For a Maxwell fluid [3] the strain is given by

~0 (t + p l) (7) 6 = q l

and the differential equation governing Maxwell behaviour is

d~r o + p , - - ~ = q , e , (8)

where ql and Pi constants which are properties of the material forming the liquid film. This system is a dash pot and spring in series. The stress is given by

o- o = E e e . (9)

K.D. Pithia, S.F. Edwards / Rheology of liquid foams 567

Thus

1 1 E f q l (t + P l ) ' (10)

The elastic compliance increases with time; the material creeps. Thus we see that the properties of liquid films control the motion. This is particularly more striking when considering the compression of the films. The films can go to zero length and we see from Khan and Armstrong [1] that at this point a T1 process occurs and there will be movement of the ceils i.e. viscoelastic behaviour i.e., the liquid foam can flow under a constant stress.

There are two points to notice here: the first is, that the lengths must go to zero before any flow occurs and secondly the modelling by dash pots and spring represents a useful picture of the film length. The behaviour of the film when the T1 occurs is reminiscent of the behaviour of springs, the dash pot is a modification to bring in the fluid behaviour of the film.

Since the liquid films are of crucial importance it is worthwhile fixing attention on their behaviour and thus using the films to explain rheological properties. That is, in order to explain the behaviour of the 2D liquid foam as a whole we need to focus attention on the behaviour of 2D liquid films that make up the 2D liquid foam.

2. Probability density distributions

So far the 2D liquid foam is characterized by the number of sides and the area that the cell possesses. When considering the behaviour of 2D liquid foams to external forces there is as seen above the possibility of flow as well as deformation. The characterisation of such properties is not fully described by the parameters that describe 2D liquid foam behaviour used so far.

The deformation of a cell of n sides as seen from above, requires a knowledge of the behaviour of the film lengths and their orientation. The stresses or strains that may act on a system will concentrate themselves at these films. If we define the cell centre to exist at a point d in space, we need to define probability density distributions for the parameters which describe the cells. In particular we wish to address the question of defining a probability density distribution P ( n , 11, . . . , In, r , d; t ) , such that P ( n , 11, . . . , In, r , d; t ) ×

dn d l l . - , dl n dd is the probability of finding a cell lying in the ranges n to n + d n , 11 t o l l + d l 1 . . . . . l n t o l n+d /n , r to r + d r , a n d d t o d + d d .

Let the number of cells in a 2D liquid foam be N. Then 3N is the number of

568 K.D. Pithia, S.F. Edwards / Rheology of liquid foams

sides of cells [5]. Let the total area occupied be A and let ~b be the area fraction of gas occupied. Then

fraction of liquid = 1 - ~b (11)

and

area of liquid = A(1 - ~b). (12)

For a volume fraction we have to multiply by a plate separation d (or some equivalent length), i.e.

volume fraction = Ad(1 - qb). (13)

Since the number of films is 3N, then average area of fihn is

(AL) = A(1 - 05) (14) 3N

But from the conservation of gas fraction we have

N ( A g ) = A 6 , (15)

so that

( 1 - 6 ) ( A L ) - - 36 ( A g ) . (16)

Assume that the thickness, s, is constant for simplicity, then

(1 - 6 ) (1) 3~bs (Ag), (17)

that is, the average length scales in the same manner as the average area (ensemble average). We construct the Fokker-Planck equation for the lengths of the sides of a cell. Thus if P(l, t) dl at time t is the probability of finding a length of side of a cell in range l to 1 + dl then, we have for P(l, t)

0e(l, t) 0 ( 0e(l, t) ) Ot Ol A Ol ~ B ( l - ( l))P(l , t) , (18)

where ( l ) is the ensemble average at the time t of all the lengths and A and B are constants whose nature we left undetermined [6]. Thus the equilibrium solution, with initial condition


P(l ,O) = 6 ( l - a ) , (19)

where a is the length's of the hexagons at time t = 0, is

1 [exp( ( 1 _ ( / ) ) 2 ~ exp( (1+_ ( / ) )2~] (20) (2'rr/.Le) 1/2 -2~e ] - - 2/Ze / J '

with ( l ) = a at t = 0, where ( l ) = the equilibrium value of the ensemble average a n d / z e is the second moment of the lengths in the equilibrium state.

From [6] the final distribution in the asymptotic state may be approximated by the following distribution:

P(x) = cx 2 exp( -bx2) , (21)

where c and b are constants related by c = 4b3/Z/'rr 1/2 and x = l.

3. Flow and deformation

In the deformation of foam, we shall assume that the gas is incompressible. Any external forces imposed on the system will result in the forces being concentrated at the films.

The usual assumptions regarding the make up of these films are made [1,2]. The behaviour of the foam under external forces is determined solely by the behaviour of the films. The process of T1 transformations is an underlying feature. We assume the T1 process occurs when the length of the film is zero. We assume that the T1 process is still the dominant mechanism that will occur in order for the foam to relax against any external forces. Initially the deformation of the foam is governed by the deformation of the liquid films. The flow of the cells occurs at the point where the T1 process occurs, i.e. at the point where the length participating in the process is zero. At this point there is a sudden reformation of the film length with the result that the ceils have moved along [1,2].

We now call the film length a rod, the term film length will be dropped. Let the rod be of length of 21 and be tilted at some angle 0 measured counter clockwise from the x-axis. Let the rod be a distance r from the centre a cell. When the rod is of zero length the T1 process occurs and movement of the cells O c c u r s .

When the rod contracts to zero and forms it returns to 2l but the angle at which it does so is not defined [4]. We will assume it is ,tr/2, which certainly is a


good approximation. The rod lying at 0 = "rr has the same orientation as at 0 = 0, i.e., the orientation of the rod is confined to the region between 0 and -rr. Thus the probability density for the angular distributions should be such that P(O ) and P(O + ~r) are the same.

Before considering this point further let us characterise the changes in the probability density distributions by the use of the Boltzmann equation. We define the P(l ,O, t ) as the probability density distribution such that P(l, 0, t) dl dO is the probability of finding a length in the range l + dl to l and in the orientation 0 + dO to 0 at a time t. We define the Boltzmann equation as

~e(t, o, t) f ot - j [W(l,O;O') P(l, o ; t ) - W(l, 0';0) P(l, o';t)] dO', (22)

where W(l,O;O') is the transition probability for the T1 process from the orientation 0 to 0'. We assume this to be the same in either direction in an equilibrium situation. It must be remembered that the probability density distribution given in (20) for the film lengths is for the film length 21, i.e. we know from (20) the probability density distribution is P(2/) , such that P(2/) d(2/) is the probability of finding a film length in the range 21 to 2l + d(2/). It must be noted here also that we are considering the T1 process at zero length. The effect of length changes will be seen later.

We now need to consider the dependence of the transition probability on the length, since the transition occurs only at zero length. We write the Boltzmann equation as

aP(l, o, t) Ot ~(t) f [w(o; o') e(z , o, t) - P(Z, o', t) w(o ' ; o)] dO', (23)

where w(O; 0 ') is now a new transition probability and is no longer a function of length but a function of the orientation only. Now

0 ' - - ~ 0 = 0 ' - w / 2 and 0 - - - ~ 0 ' = 0 + w / 2 . (24)

We write these transitions as

6(0' - (0 + ~r/2)) and 6(0 - (0' - 7r/2)) (25)

and the Boltzmann equation becomes

OP(I, O, t) Ot

f - - k 6 ( l ) J [6(0' - (0 + -rr/2)) P(I, 0 ', t)

- 6 ( 0 - ( 0 ' - Ir/2)) P ( I , 0, t)] dO', (26)


where k 6 ( l ) now represents W,

OP(l, O, t) - 1,a(t) [e ( t , 0 + t) - P(t , o, t ) ] . ( 2 7 )

Ot

The rod is invariant under a rotation of xr that the angular probability density distribution for the rod is such that a change in variable of 0 to 0 + ~r leaves the angular probability distribution unchanged,

P(I, 0 + ~ , t) = P(1, O, t) . (28)

4. The Boltzmann equation for the probability density distribution of the film length

The general equation for evolution of the probability density distribution of the rod is given by the Boltzmann equation which we write as

OP a-T + div(VP) + Ka(/) [P(I, 0 + ,rr/2, t) - P(I , O, t)] = A P , (29)

where V is the velocity of the rod A is an operator defined in eq. (18), and the constant K is a strain rate term. The probability density distribution is now a function of the length and the orientation of the rod. The inducement of the T1 process is due to the application to the system of external stresses or strains. Thus eq. (29) is the general equation of motion of the probability density distribution of the rod. The third term on the left hand side expresses the changes in the orientation of the rod and the right hand side of (29) term expresses the changes in length. Notice in this equation we have left out the position coordinate of the centre of the rod, for we are able to consider the behaviour of the rod at a particular position in space.

We make another strong simplifying assumption. The rod is connected to four other rods. The behaviour of this one rod effects the behaviour of the others connected to it.

Thus for a richer understanding and a more complex model we should create a set of equations each coupled to four other lengths through the geometry and the incompressibility of the 2D liquid foam. However since we wish to work at the level of a model which contains the basic physics and is soluble, we only retain the behaviour of just one rod rather than create five differential equations which have to be solved simultaneously, subject to the geometrical constraints and incompressibility of the 2D liquid foam.


The equation for an incompressible flow ~rx~ = - o ' y y and O'xy = O'y x = O, is given by

OP OP kcrx.l-- ~ cos 20 - kO'~x sin 20 ~ + 6(l) ko'~x[P(l, 0 + -rr/2) - P(1, 0)] = A P

(30)

for the steady case. The solution must take the form

P = Poo(l) + Po(l) + P,(1) cos 20 (31)

where Poo is the equilibrium solution and is given by (20) Poo >> Po, and >>P1- Substituting this into (30) we have

O Poo ko'xxl ~ cos 20 + 2ko-xx sin 20 P1 sin 20 - 26(0 kO'~xP I cos 20

= A P o + A P 1 cos 20 , (32)

(kaxx laP°° AP~) --if/- - 26(/) ko'xxP 1 - cos 20

+ 2ko'xx sin 20 P~ sin 20 - A P o = 0 . (33)

Choose the space variable '20' and integrate sin 20 d(20) over 0 to -rr so that we can use the orthogonality relations of Legendre polynomials. We have

OPoo ko'xxl - - ~ - 26(/) ko'xxP 1 - A P 1 = 0 , (34)

2 ko',~P, - A P o = 0. (35)

Thus f rom (35) we can find Po, i.e.,

no = ] ~Go(l, l ' ) koxxPl(l ' ) d l ' , (36) o

where G0(l, l ' ) is the Green function for the A operator . Now define G~(l, l ' ) as

a O , ( l , l ' ) + 26(l) korxxGl( l , l ' ) = a( l - 1 ' ) . (37)

By definition

AGo(l, l ' ) = 6(l - l ' ) , (38)

K.D. Pithia, S.F. Edwards / Rheology of liquid foams

G1(l,l ')= f Go( l , l " )3 ( l ' - l " )d l " - 2 i G°(l'l")k°'xx3(l")G1(l"l')dl"' o o

GI(I , 1') = Go(l , l') - 2k~rGo(l , O) GI(O , l ' ) .

Let l = O, then

G I ( 0 , l ' ) = G o ( 0 , l ' ) - 2 k o ' x G ~ (0, l ' ) G o ( 0 , 0 ) ,

rearranging

Co(0, r) 61(°' It) = 1 + 2ko-xxGo(O , O)

and

ol(t, r ) = Co(t, r ) ¢ koGGo(O, 1 ) Go(l, O)

1 + 2ko'xxGo(O , O)

Thus the solution is given by

PI(I) = i Gl(l' l') ko-xxl' 3P°°(Z') Ol' 0

i d l ' .

Now

Po =2 f Go(l ' l') kO-~xPl(l' ) dl ' , o

thus substitution of Pl(l) gives

0 0

OPoo(l" ) Ol" dl" dl ' ,

zv

Po - 2 k 2 - 2 f f Go(I,I ' )GI(I ' , I ' )I" - - ~ ° - x x

0 0

OPoo(I" ) Ol" dl"dl ' .

Thus, given Poo, the functions Po and P1 are determined.

57!

(391

(40]

(41)

(42)

(43)

(44)

(45)

(46)


5. Calculation of the viscosity

To calculate the viscosity we define the velocity of the cell as

E . r = velocity, (47)

where r is the position vector of the cell centre and E is the strain rate tensor. The rate of change of the film length is given by taking the first moment of eq. (29):

Ot - I div(VP) dl - 16(l) K[P( l , 0 + "rr/2; t) - P(l , O; t)] dl 0 0

+ f l A P d l , (48) 0

where P = P(l, O; t) , and is given by

P = P o o ( l ) --~ P o ( l ) ol- P I ( / ) cos 20 , (49)

with Po(l) and Pl(1) given by eqs. (46) and (44) respectively. Substitution of this, and the assumptions for the stress field, lead, in the very first approximation, to

at - 2k°xx<l) cos 20 . (50)

This is the magnitude of the rate of change of the film length. The magnitude of the radial velocity of the cell is given by

v x cos 01 + vy sin 01 , (51)

where v x and Vy are the velocity components of the cell and 01 is the orientation of the cell as measured from the film length centre. These can be written as

ox = ex~r x , (52)

Oy = Eyyry , (53)

with r~ = r cos 01 and ry = r sin 01 and substitution of this into the magnitude of the radial velocity equation (51) and equating with the rate of change of the film length gives

K.D. Pithia, S.F. Edwards Rheology of liquid foams 575

1 O(l) Ex~=rcos201 0t ' (54)

where d( l ) /d t is given by (50). Writing the strain field component as

exert, = o-~,, (55)

where ,/f is the 2D liquid foam viscosity,

1 (r) cos 201 T/r- 2k (l) cos 20 ' (56)

where we have introduced (r) instead of r, the average linear intercept of the cell. We write ~e as l / k ,

(r) cos 201 n~-n~ (l) c o s 2 0 ' (57)

for the ideal structure [1,2] we approximate (57) giving

n f - ~e4~-1/2, (58)

where ~b is the gas fraction.

6. Conclusion

One of the main thrusts of the preceding argument is that for an understanding of the behaviour of 2D liquid foams an understanding of the liquid film must be taken into account and for those liquid foams with a high gas fraction, typically approaching unity then the behaviour of thin liquid films must be taken into account. This is seen particularly when considering the 2D liquid foam composed in the first instance as comprising of liquid films which are all parallel and thus having a simple structure. In this model the fundamental point emerges that it is the straight liquid films which determine the behaviour of the 2D liquid foam as a whole. This model is obviously, rather limited and although it emphasises the importance of the liquid films it neglects the pattern of behaviour generally associated with 2D liquid foams that of the T1 process. It is this process that makes the flow of the 2D liquid foam under a stress possible and leads to the phenomena of viscoelasticity.

To model this and to determine the visoelastic behaviour of the 2D liquid foam requires a mathematical expression of the T1 process. The crucial point


that emerges in determining the expression is that the T1 process occurs with film lengths and that when these lengths become zero the lengths reform at an angle normal to the original. This deformation of the film lengths occurs for certain orientations. This reformation procedure is vital for the cells to move and thus the rate of this process determines the motion of the cells which in turn determine the viscosity.

This process has been modelled by a Boltzmann equation that expresses the behaviour of the angles (29). The differential equation for the film lengths is then coupled with this Boltzmann equation for the angles to give the behaviour of the liquid film. However this must be solved simultaneously with all the other films being deformed. Given that for a single T1 process five liquid films are involved it will then require five differential equations to be solved simultaneously. For simplicity and to give a feeling for the solution a single differential equation has been described and solved.

The resulting distributions for the angles are then averaged out and using the structure of the T1 process which gives rise to motion of the cells determine the viscosity in this ideal simple case.

This result (58) is very simple and comes from the neglect of four other differential equations which have to be coupled to it. However the result does indicate that not only is the film length and its orientation important but also that the size of the cell and its orientation are crucial factors in determining the

viscosity.

References

[1] S.A. Khan and R.C. Armstrong, J. Non-Newtonian Fluid Mech. 22 (1986) 1. [2] A.M. Kraynik and M.G. Hansen J. Rheol. 31 (1987) 175. [3] W. Flugge, Viscoelasticity (Springer, Berlin, 1975). [4] M.F. Ashby and R.A. Verrall, Acta Metall. 21 (1973) 149. [5] C.S. Smith, Grain Shape and other Metallurgical Applications of Topology (Metal Interfaces,

Ohio, Cleveland, 1952). [6] In preparation.

rheology of liquid foams

Documents