Hydrodynamics of thin liquid films bounded by viscoelastic interfaces

Hydrodynamics of Thin Liquid Films Bounded by Viscoelastic Interfaces

DAVID E. TAMBE AND MUKUL M. SHARMA 1

Center for Petroleum and Geosystems Engineering, University of Texas at Austin, CPE 2.502-Austin, Texas 78712

Received December 3, 1990; accepted April 15, 1991

A hydrodynamic model has been developed to describe the rate of drainage of an axisymmetric plane- parallel horizontal film bounded by viscoelastic interfaces. The effects of interracial tension gradients, interfacial viscosity, and interfacial mobility are discussed. The viscoelastic nature of the interfaces is shown to have a pronounced effect on the rate of film drainage. © 1991 Academic Press, Inc.

INTRODUCTION

Coalescence of liquid drops plays an important role in many engineering processes. When two drops separated by an immiscible liquid approach one another, a liquid film of the continuous phase is formed between the two drops (see Fig. 1 ). This liquid film must drain for the two drops to coalesce. The rate of coalescence of these drops will thus depend on the rate of drainage of the liquid films separating them. Meanwhile, the rate of drainage of the thin liquid film depends strongly on the rheological properties of the interfacial layer separating the film from the dispersed phase.

BACKGROUND

Reynolds ( 1 ) originally posed the problem of the drainage of a liquid film between two identical drops. In his model, he approximated the deformed area between the drops with two rigid parallel discs. Several investigators have since tried to modify and improve on Reyn- olds' theory. Boussinesq (2), Radoev et al. ( 3 ), Traykov and Ivanov (4), Ivanov (5), and Malhotra and Wasan (6) investigated the effect of the presence of surfactant in either the continuous or the dispersed phase. Johannes and Whitaker (7) and Lee and Hodgson (8) studied the effect of interfacial tension gradients.

1 TO whom correspondence should be addressed.

The influence of interfacial viscosities on film drainage was investigated by Ivanov et al. (9), Barber and Hartland (10), and Ivanov and Dimitrov ( 11 ). Good (12) studied the effects of the bulk viscosities of both phases on drainage. The role of flow in both the dispersed and the continuous phases has been investigated by Levich ( 13 ), Jeffreys and Hawksley (14), Lucassen et al. ( 15 ), Murdoch and Leng (16), Berg (17), Woods and Bun'ill (18), Liem and Woods ( 19 ), and Jones and Wilson (20). Ra- doev et al. (3) studied the effect of bulk and interfacial diffusion. The effect of interfacial mobility was also investigated by Hartland (21) and Ivanov (5). Mackay and Mason (22), Hartland (21), Lang and Wilke (23), and Ivanov and Traykov (24) studied the drainage of pure films (systems without surfactant), while Traykov and Ivanov (4), and Zapryanov et al. ( 2 5 ) studied the thinning of films with surfactant soluble in both phases. Traykov and Ivanov (4) and Ivanov (5) reported that the presence of surfactant in the dispersed phase only has no significant influence on the rate of film thinning.

However, in all of these investigations, the fluid-fluid interfaces were assumed to be either Newtonian or rigid. The assumption of a Newtonian interface may not be valid when there is a substantial amount of surfactant adsorbed at the interface. The assumption of a rigid interface is seldom reasonable since most

137

Journal of Colloid and Interface Science, Vol. 147, No. 1, November 1991

0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All fights of reproduction in any form reserved.

138 TAMBE AND SHARMA

~ op

I Draining h film

Drop

Vz

Dispersed / C phase

ontinuous Ur / phase

LVr

Dispersed phase \

FIG. 1. The film thinning process during drop-drop coalescence.

E* = E'(w) + iE"(w), [2]

where E' and E" are the real and imaginary components at a frequency w. The complete frequency spectrum of E' and E" is obtained by a Fourier transform of the dynamic interfacial tension data (a Fourier transform of the numerator and denominator of Eq. [ 1 ]). This gives

f o A'r(t)e-i~tdt E* = [3]

AA f o ~ (t) e-i'°'dt

surfactants do not render the interface immobile. When interfaces are slightly mobile, the rate of film thinning can be several orders of magnitude greater than the rate predicted by immobile interfaces (21 ). Under the conditions described above, fluid-fluid interfaces will tend to exhibit both viscous and elastic (viscoelastic) properties.

Measurements of the rheologlcal properties of interfaces have indeed shown that some fluid-fluid interfaces do exhibit viscoelastic behavior. Callaghan et al. (26), Mukherjee et al. (27), Neustadter et al. (28), Clint et al. (29), and Eley et aL (30) have shown through measurements that crude oil-brine interfaces are viscoelastic. They also reported that the dilational modulus (the measured viscoelastic parameter) is a key fundamental property governing the stability of crude oil foams and emulsions.

The complex interfacial dilational modulus is the interfacial tension increment per unit fractional change in interfacial area, i.e.,

E* =A d~r [11

where E* is the dilational modulus, 3, is the interfacial tension, and A is the area of the interface. Variations in interfacial tension with time as the interface is expanded can be measured using the pulse drop technique (29) or the trough method (28). E* is a complex function of frequency and can be written as

For an instantaneous change in area,

fo °~ AA e_i,otdt _ A A / A iw [4]

With this, Eq. [ 3 ] becomes

E* = iw fo ~ A ~ A AT(t)

× [cos wt - i sin wt]dt. [5]

The real part of Eq. [ 5 ] is the dilational elasticity and the imaginary part is the dilational loss factor, i.e.,

Ofo A A / A A-r(t)sin wtdt = G(w) [6]

A A / A A'v(t)cos o~tdt = o~d(w), [71

where G is the dilational elasticity and #d is the dilational viscosity. These parameters strongly influence the drainage of thin films during drop-drop coalescence and should be included in film thinning models.

The objective of this work is to develop a hydrodynamic model to describe the drainage of thin films bounded by viscoelastic interfaces. The importance of interfacial rheology relative to the effects ofinterfacial tension gradients, interfacial mobility, and the effect of the presence of surfactant in the continuous phase is also evaluated.


HYDRODYNAMICS OF THIN LIQUID FILMS 139

PROBLEM FORMULATION

Consider two identical drops colliding sym- metrically about the z-axis (see Fig. 1). The drops are forced together by a constant applied force F. To solve the problem, the following assumptions were made:

1. The draining film is Newtonian. 2. The interfacial layer is viscoelastic. 3. The film remains plane-parallel and

horizontal during the entire drainage period.

4. The interfaces are partially mobile. 5. Surfactant is soluble in the continuous

phase only. 6. Flow in the film obeys the simplified

Navier-Stokes equation. 7. There is creeping flow in the film hence

a quasi steady state approach can be used.

Because of the natural symmetry of the system, we used cylindrical coordinates (r, 0, z) to solve the problem. The 0 components of all quantities are identically zero. In addition, the problem was solved only for z >~ 0.

HYDRODYNAMIC EQUATIONS AND BOUNDARY CONDITIONS

Equations of Change in the Film

Assuming isothermal flow and constant physical properties for the film, the Navier- Stokes and continuity equations for the flow ofa Newtonian film in cylindrical coordinates can be simplified to obtain

02V~ 10P - 0 [81 Oz 2 # Or

OVz 10(rVr) - - + - - - 0 . [9] Oz r Or

If we also assume negligible hydrostatic effects in the thin film, then we have

OP - - = 0 . [101 Oz

These equations are solved subject to the boundary conditions,

Vr= Ur at z = h / 2 [11]

0Vr Oz 0 at z = 0 [12]

V z - ~- at z = ~ V = - , [13]

where V~ is the radial component of the film velocity, Vz is the normal component of the film velocity, U~ is the radial component of the interfacial velocity, V is the rate of film thinning, and h is the thickness of the film.

With these boundary conditions, Eqs. [8 ] to [10] can be integrated and rearranged to give

VF - -

1 0 r Or

v~ -

2 ~ d r ~ + Ur [141

Or~ 4 z_~) Urrz) [15]

V - dh 1 2 u r h - - - ~ r [16] dt r 6tt

3tt dh P(r) = Pb + r(h) + -~3 --~ (R2(t) - r2)

12~ fret) h2 Urdr, [171

where P(r) is the pressure distribution in the film, Pb is the pressure in the bulk away from the film, a-(h) is the disjoining pressure, and R(t) is the radius of the film region (see Fig. 1 ). Since the interface is mobile, the radius of the film region will either be increasing or de- creasing depending on the velocity of the interface. The disjoining pressure term is a function of h only since the draining film is considered plane parallel. The different components of the disjoining pressure are

r(h) = 7relec(h) + rvw(h) + rs(h), [18]

where ~'elec(h) is the electrostatic component, rvw(h) is the component due to attractive Van der Waals forces, and rs(h) is the structural


140 T A M B E A N D S H A R M A

component (31 ). These surface forces also affect the rates of drop--drop coalescence at short distances and should be considered (32, 33). The explicit expressions for the three components of the disjoining pressure are

r~elcc( h ) = 64nKTy 2 e x p ( - r h ) , [19]

where

and

e x p ( z / 2 ) - 1

~' exp (z /2 ) + 1

ev~ Z -

K T A

~r,~(h) = 67rh 3 [20]

its(h) = K s e x p ( - h / l ) . [21]

In the above expressions n is the concentration of ions in the film, K is the Boltzmann's constant, T is the temperature, K is the inverse of the Debye length, h is the thickness of the film, e is the charge on an electron, v is the valence of the ions, ~p is the electric potential of the interfaces, A is the Hamakar constant, Ks is an empirical constant, and l is an empirical correlation length. The effects of the electrostatic component are negligible for an oil film, and are neglected in this study.

Since the hydrodynamic equations of the film are coupled with the mobility of the interface, we need to derive equations that gov- ern the movement of the interface.

Surfactant Balance at the Interface

When surfactant is present in the continuous phase and at the interface, the radial flow of liquid sweeps away surfactant from the interface towards the periphery, thereby dis- turbing the equilibrium distribution of surfactant at the interface. As a result, reverse fluxes are generated in an attempt to restore the equilibrium state of the interface (see Fig. 2). In this figure the small bars represent surfac- tam molecules, jc is the convective flux, js is the surface diffusive flux, andj f i s the flux from

s

Drop

'oh l ,ll I I I I I I I I -- - - - - d

_ _ _ |

J J - - - - _ c ~ ~ j

FIG. 2. Surfactant balance at the film interface.

the film to the interface. The fluxes j r and js are reverse fluxes generated by f t . The above phenomenon enters the film thinning problem as a boundary condition. In the absence of any reactions at the interface, a general boundary condition that describes the transport of surfactant to and from the interface is (13)

OF + Us.VsF - 2 H U s . n P Ot

- FVs. Us - V s . j s + r / ' ( j F _jD), [22]

where F(r , t) is the surfactant concentration at the interface, Us is the velocity of the interface, jD, jF, andjS are the net diffusive fluxes to and from the drop phase, the film phase, and along the interface, respectively. H is the mean curvature of the interface, X7s is a surface gradient operator, and n is a unit normal vec- tor perpendicular to the interface. In this equation the terms on the left hand side account for surface accumulation and convection. The first term on the right hand side accounts for variations in F due to the expansion of the interface in the normal direction, the second term is a surface convection term, the third term accounts for surface diffusion, and the last term accounts for diffusion to and from the continuous (film) and dispersed (drop) phases. If we let H = 0 (plane interfaces), and also assume that there is no surfactant in the dispersed phase, then the above equation reduces to

Journal of Colloid and Interface Science. Vol. 147, No. I, November 1991


OF - - + Us. VsI' + FVs. Us Ot

+ V s . j s - n . j r=O. [23]

Equation [23 ] is the surfactant balance equation at the interface. When expanded and rearranged, this equation becomes

0---~ ~- Ur - - g -- Ds Or---- f

+ r +--~-r ] + DF-~'z = 0, [241

where Ur is the radial component (the only nonzero component) of the interface velocity.

Conditions for Mechanical Equilibrium at the Interface

Because of the nonuniform distribution of surfactant caused by the draining film, variations will exist in the local values of the interfacial tension. These variations give rise to interracial tension gradients. In addition, the flow of the film usually causes surfactants at the interface as well as the interface itself to be stretched. This gives rise to extensional, compressional, and shear stresses. A general boundary condition that accounts for these surface stresses can be written as (34)

n(pV_ pD) + n . ( r v _ r D)

= -Vs3"-- 2Hn3" + V s . r s, [25]

where r s is the surface stress tensor, P is pressure, and 3" is interracial tension. The super- scripts F and D denote quantities in the film and in the drop, respectively. On the left hand side of this equation the first term accounts for stresses at the interface due to pressures exerted by the film and by the drop, the second term accounts for stresses at the interface due to flow in the film and in the drop, the first term on the right hand side accounts for the effects of surface tension gradients, the second term accounts for curvature effects, and the third term accounts for surface stresses that are generated when the interface is stretched. If we let H = 0, neglect the effects of pD and

pF on the interface, and also neglect the effects of flow in the drop, the above equation reduces to,

Vs3' + r E. n - V s . r s = 0. [26]

To expand the first term of this equation, we need an adsorption isotherm and an equation of state that will relate the interracial tension to the surfactant concentration at the interface. The Langmuir isotherm (35) and the Szyszkowski equation (36) have been found to be appropriate for surfactants adsorbed at aqueous solution/hydrocarbon interfaces (37). These equations are

F=C F - (Langmuir) and

a + C

3 ,0- 3" = F m R T l n ( 1 + C ) (Szyszkowski),

where C is the concentration of surfactant in the bulk, F is the concentration of surfactant at the interface, I'm is the concentration at sat- uration, a is a constant, 3'o is the interfacial tension in the absence of any surfactant, R is the ideal gas constant, and T is the absolute temperature. If we combine these equations we obtain

3"0-3" = - R T F m l n ( 1 - ~m) . [27]

From Eq. [ 27 ]

Fm OF Vs3" = - R T - -

F m -- F Or

OF - K

Or '

Fm where K = R T - - [28]

Fm - F

The second term in Eq. [ 26 ] can be simplified as follows:

F _gF OVr [29] r F- n = r~r = 0--7"

When evaluated at the interface we obtain

Journal of Colloid and Interface Science. Vol. 147, No. I, November 1991


F rV'n=rzr=-U~h-7-~+6 . [30]

The superscript F denotes film. To evaluate the interfacial stress term in Eq.

[26 ], we need an explicit expression for the surface stress tensor, r s, for a viscoelastic interface. A Maxwell fluid with a single relaxation time is the simplest model for the viscoelastic interfacial layer. The constitutive equation for a Maxwell fluid is

1 u . i + ~ r = - ~ 3 ' [311

where r is the stress tensor, ~- is the time de- rivative of r, + is the strain rate tensor, u is the viscosity, and ~, (~, = u/G) is the relaxation time of the fluid. Figure 3 shows the mechanical analog of a Maxwell fluid, a series com- bination of a Hookean solid and a Newtonian liquid. G is the elastic modulus of the spring, and u is the viscosity of the liquid. The viscoelastic model thus accounts for the viscous and elastic components of the fluid. The relaxation time ~,, is a measure of the elasticity of the fluid. Note that when G is infinite (i.e.,

= 0) the spring becomes a rigid connection, and the model reduces to a Newtonian fluid. Conversely, when u is infinite (i.e., X infinite) the dashpot becomes rigid and the model becomes that of a Hookean solid. These conclusions follow directly from Eq. [31].

Equation [31] is a first order differential equation which can be integrated to give

rS = -do(t k -~g[us e_U_,,)/~s)+s(t,)dt, ' [321

where t is the present time, the superscript S denotes a surface quantity, and the integration is performed over the history of the interface.

FIG. 3. Mechanical analog of a Maxwell fluid.

~x

Y A

/ to

~ t

B Y

to

FIG. 4. Creep and recoil test. (A) Newtonian fluid response, (B) Viscoelastic fluid response.

~ t

In this equation the quantity within brackets is referred to as the relaxation modulus. The surface viscosity u s in this equation is the sum of the shear and dilational surface viscosities. Johannes and Whitaker (7) investigated the role of the shear and dilational surface viscosities on the drainage of a vertical liquid film, and pointed out that the surface dilational viscosity may be several orders of magnitude larger than the corresponding shear viscosity. Barber (38) made measurements to establish the fact that it is principally the dilational surface viscosity that controls the drainage of Newtonian films.

When written in the form of Eq. [32], the model reveals that the stress r s depends on the rate of strain at the present time t, and also on the rate of strain at all past times t', with a weighting factor (the relaxation modulus) that decays exponentially as one goes back in time. This equation thus exhibits a fading memory. This is a characteristic of viscoelastic fluids. Because of this fading memory, viscoelastic fluids always attempt to return to their original state after they have been strained. Note that a Newtonian fluid has no memory, while the Hookean solid has a perfect memory. Figure 4 illustrates the responses of a Newtonian and a viscoelastic fluid when



subjected to a constant stress from to and Ii (39). Note that the strain is linear (constant strain rate) for the Newtonian fluid and nonlinear for the viscoelastic fluid. In terms of film thinning hydrodynamics, this implies that the mobility of the viscoelastic interface will be less than the mobility of the Newtonian interface for the same U. As a result, films with viscoelastic interfaces will thin slower than films with Newtonian interfaces. The degree of interfacial immobility will depend on the degree of elasticity (resistance to flow) of the interface.

The general behavior of a viscoelastic fluid may not be properly represented by one characteristic relaxation time. One way of approx- imating the behavior of viscoelastic fluids bet- ter is by using multiple Maxwell elements in parallel as shown in Fig. 5. Such a model is called the generalized Maxwell model, and has the following constitutive equation:

=_£, , ,3, p=, k s

The above equation now provides for a continuous distribution of relaxation times and will in general more closely mimic real systems. Studies on the molecular theory of poly- mer solutions performed by Rouse (40) sug- gest the expressions for Xp and up

J

G it 1 1

FIG. 5. Mechanical analog of a generalized Maxwell Model. Model contains N elements in parallel.

k , [34] up = u0 Z ~ kp

~0 kp p 2 , [35]

where ko is the longest relaxation time and #0 is the zero strain rate viscosity.

With the surface stress expressed as shown in Eq. [33], the r-component of XTs-r s becomes,

£ t [02Ur 1 OUr Or 7 ( V s - = _ +

1 ~pS e ] .~. [36]

With all quantities expressed as given above, Eq. [26] can now be written as,

K OF £t { O2Ur lOUt (Jr) Or ~ Or 2 + r Or -~

+ , / p - ~ + 6 = 0 . [37]

Equation [ 371 is the stress balance equation for a viscoelastic interface. This equation can be reduced to the stress balance equation for a Newtonian interface. If we assume that the velocity of the interface is a function of the present time only, then Eq. [37 ] will become

KOI' 132U~ I OUr Ur) or - -gP-r r Tr 7

×f:(~Spe-(t-'"/XOdt'xSp

v{3rdh -~) + # ~-~-d-7 + 6 = 0 . [381

Performing the integration and setting X = 0 we obtain,

Journal of Colloid and Interface Science, Vol. 147, No. 1, November 199t


KOP [ a2U, l OUr Ur) #s -17 r + ; Tr 7

F{3rdh ~ ) + # ~- i~-~ + 6 = 0 . [391

This is the stress balance equation for a New- tonian interface.

Equations [24] and [37] can be used to solve for Ur. TO do so we need two additional equations involving the dependent variables h and C. We can obtain an equation for C by performing a surfactant balance in the film. If we do, we obtain the equation

O f Vr OC Vz OC %-? + -g;r + Oz

(1 0 (rOCI 02C~ - D r r Or !~ -~r ] + -~S ] = 0' [401

where C is the concentration of surfactant in the film, V~ and Vr are the normal and radial components of the film velocity, and D z is the coefficient of diffusion in the film.

For a small Peclet number flow, and assuming negligible changes in surfactant concentration in the film with time, the above equation reduces to

l d ( O C ) O2C rOr r-~r + - ~ z 2 =0" [41]

The final equation is obtained by performing a force balance on the draining film. If F is the force acting on the drops, then

~R(t)

V = J0 27rr(P(r) - Pb)dr. [421

P ( r ) is given in Eq. [ 171. If we substitute P( r ) in Eq. [42 ], we obtain the following force balance equation:

F = 7r(h)TrR 2 6#VTrR4(t) dh 4h 3 dt

- 247r S fomt) ( fmt) Urdr)rdr. [43]

This equation will reduce to a modified Reyn-

olds equation if we set Ur = 0. Reynolds original equation did not account for the effects of surface forces.

A summary of the equations derived thus far is provided in Table I. These equations completely describe the film thinning process for the problem addressed in this paper. As shown earlier, previous models that assume

TABLE I

Summary of Equations

1. Equations of change in the film

'=2-#~-~r~ --4-] + /Jr [14]

I O { I__OP z-z3~ - U:z) [151 Vz = r Or k2# Or ( h24 3]

V= dh l (2u h h3 Oe~ dt rk ~ -~#~r] [16]

3# dh 2 P(r) = Pb + ~r(h) + ~3 --~ (R (t) - r:)

12# ~R(,) Urdr [17] ~ J r

2. Surfactant balance at the interface

Ot -~r ' Or z

r(Ur + aU, I + ~ OC 0 ]24] + t 7 W;/ ,9~ =

3. Tangential stress balance at the interface

ar- ~o (-~r2 +r Or r2]~t h~ ] r[3rdh ~)

+U [ ~ + 6 = 0 [37]

4. Surfactant balance in the film

1 o {rOC : c 7 ~r ~ Tr / + -g~: = ° I411

5. Force balance in the film

F = 7r(h)IrR 2 - 6#FlrR4(t) dh 4h 3 dt

-~R~°[12#F (Rt°U'dr] [43]


H Y D R O D Y N A M I C S O F T H I N L I Q U I D F I L M S 145

Newtonian or rigid interfaces can be obtained as special cases of the equations derived above. The Newtonian limit is achieved if we let Ur be a function of the present t ime only (see Eq. [37]) and let A = 0, and the rigid interface approximation is achieved by setting us = ( U r = 0 ) .

From these equations, the film thinning profile can be obtained in several ways. In this model we obtained this profile by solving for h as a function of t ime from Eq. [43] subject to the conditions given in Eqs. [24] and [37]. The transport of surfactant from the film to the interface was assumed to be adsorption controlled. The last term of Eq. [24] was thus replaced by K0(F0 - P(r ) ) , where K0 is an adsorption constant, and Fo is the equilibrium surfactant concentration at the interface. Equations [24], [37], and [43] were solved numerically subject to the following initial and boundary conditions:

h = H o , R ( t ) = R o at t = 0

0P Ur=O, ~ r = 0 at r = 0

F ( r , t ) = P 0 at r = R ( t ) . [44]

The data set used in the numerical analysis is provided in Table II.

R E S U L T S A N D DISCUSSION

It is evident from the equations developed that the rate of film thinning depends on the applied force F, the viscosity of the draining film #F, the concentration of surfactant at the interface F, the diffusion coefficients DE and Ds, the surface viscosity #s, the elasticity of the interface G, and the surfactant concentration gradients at the interface. The results obtained from this model clearly demonstrate the relative importance of some of these parameters. The following dimensionless variables are used in the results presented. The dimensionless time, tD = t/X (X = #/G); the dimensionless radial position, rD = r/R(t); the dimensionless force, FD = F/FG (FG denotes

TABLE II

Model Parameters

1. Force: Force due to gravity, FG. 2. Aqueous phase: Water. 3. Oleic Phase: Decane. 4. Oil viscosity = 5 cp. 5. Ho = 2000 A 6. Ro = 0.1 cm. 7. Fo = 1.5E-11 moles/cm 2. 8. Dr = 1.0E-4 cm2/sec. 9. Ds = 1.0E-4 cm2/sec.

10. Ko = 1.0E-5 sec-1 11. Fm= 6.0E- 10 moles/cm 2. 12. Hamakar constant, A = 1.0E-14 13. Empirical constant, K, = 4.0E+10 14. Correlation length, l = 2.0E-7 15. Dimensionless force = force/FG 16. Dimensionless thickness = h/Ho 17. Dimensionless conc. = F/U0 18. Dimensionless radius = r/Ro 19. Dimensionless time = t/~

force due to gravity); the dimensionless surfactant concentration, FD = F/F0 and the dimensionless thickness, hD = h/Ho.

Figure 6 illustrates the interface velocity profiles at times tDO, to1, and to2 with to2 > tDl > tD0. These profiles are slightly nonlinear and depend on the rheology of the interface. The profiles will be linear for a Newtonian interface and are nonlinear when the interface exhibits both viscous and elastic properties. When the velocity profile is a linear function of r (see Refs. (4), (9), and (24)) then,

02Ur l OUr Ur Or 2 + O. [45] r Or r 2

If this is the case then the surface viscosity term (the second term ofEq. [ 39 ]) will vanish. This implies that all models that assume linear interfacial velocity profiles automatically eliminate the effects of the interfacial viscosity. This assumption may not be correct, since measurements (26-30, 41-44) have shown that the surface viscosity plays a fundamental role in emulsion and foam stability. It is also clear from Fig. 6 that the velocity of the in-


5.0e-3 ' 1.1

t~

I n t e r f a c e P a r a m e t e r s

40e-a ~ = 0,1g/sec

G= 0,1g/se& ~ .0e-3

~: 2.0e-3'

l.Oe-a'

0 . 0 e + 0 • , • , • , - , - , -

0.0 0 .2 0 .4 0.6 0 .8 1.0 1.2

N o r m a l i z e d r a d i u s

FIG. 6. Velocity profiles of the interface at different times.

1.0 ̧

tOO

:D1

0.9. E

8 c oo

0 .8"

E

0.7


0.6 • , - , • , - , - , • 0 . 0 0.2 0 .4 0.6 0 .8 1,0 1.2

I n t e r f a c e P a r a m e t e r s

G=O.lg/sec 2 =0.1g/sec

N o r m a l i z e d r a d i u s

FIG. 7. Surfactant concentration profiles at the interface at different times.

terface increases with time. This is probably because the amount of surfactant at the interfaces in the film region decreases as the film drains.

Figure 7 illustrates the interfacial surfactant concentration profiles at times tDO, tD1, and tm with tD2 > tD~ > tD0. These profiles reveal a nonlinear increase in Pg with rb. The profiles also illustrate that the amount of surfactant in the interfacial region decreases with time. This implies that at some point in time sufficient concentration gradients in surfactant concentration may be set up to cause reverse diffusive fluxes to become equal to the convective flux out. When this happens stable films will result.

Figures 8, 10 to 16 are film thinning profiles. These profiles exhibit an interesting pattern. There is first a region of slow thinning AB then a region of rapid thinning BC, and finally a region where the thickness of the film becomes stable (see Fig. 8). The increasing film thinning rate between points B and C is a result of increasing surface velocity. The velocity of the interface increases as the film thins due to several factors. As the film thins, it carries away surfactant from the interface to the periphery.

As the amount of surfactant at the interface is reduced the mobility of the interface increases, thereby increasing the film thinning rate. The region of slow thinning beyond point C is due to the presence of surface forces which become significant at small separations

1 . 2 "

1.0"

0 . 8 "

E . o .

E 0 . 6 -

E ~6

0.4 E

i T

0.2

0 . 0

A ~ B

C

10 20 30

D i m e n s i o n l e s s t i m e

FIG. 8. Film thinning profile. General profile observed.

Journal of Colloid and Interface Science, Vol. 147, No. I, November 1991


(~<300A). These forces, when significant, can influence the rate of thin film drainage. In all cases they influence the equilibrium film thickness attained at point C. Further film thinning and subsequent film rupture is not well understood and is not considered in this study. There is little doubt that the disjoining pressure will play a role in this process as well.

As mentioned above, the rate of film thinning depends on several factors. Figures 10 to 16 illustrate the effects of some of these factors on film drainage.

The Role of lnterfacial Rheology

It is evident from Eqs. [14 ] to [17] that the hydrodynamic equations of draining films are coupled with the mobility of the interfaces bounding these films. Film thinning rates are fastest when these interfaces are fully mobile, and slowest when the interfaces are immobile. Hartland (21 ) reported that when interfaces are slightly mobile, the rate of film thinning can be several orders of magnitude greater than the rates predicted by immobile interfaces. It is therefore important that the mobility of the film interfaces be properly estimated. From Eqs. [24] and [37] it is also clear that the mobility of the interfacial layer depends on the rheological properties (the interfacial viscosity and the dilational elasticity) of the interface.

(a,b)

Viscous J

Viscoelastic ~ i resp°nse

g

FIG. 9. Paths along which interfacial rheological parameters can be varied. Paths 1, 2, and 3 (see Figs. 10, 1 1, and 12).

1.2"

6 1.0"

0.8" g

g 0.6. Q)

~ 0.4"

0.2"

0.0 0 10 20 30

Dimensionless time

FIG. 10. The effects of increasing both interfacial viscosity and elasticity on the drainage of films bounded by Newtonian ( 1 ), viscoelastic (2, 3, 4, 5), and rind (6) interfaces.

Interface Parameters

g(g/sec) G(g/s~ 1. Newtonian Interf. 2. 0.1 0.1 3. 0.3 0.3 4. 0.7 0.7 5. 0.9 0.9 6. Rigid Interfaces

The importance of the interfacial viscosity (both shear and dilational) of Newtonian interfaces in film thinning has been previously investigated (6-11 ). These studies have shown that film thinning is retarded when the viscosity of a Newtonian interfacial layer is increased. Studies on emulsion stability (41-44) have also shown that the stability of an emulsion depends on the viscosity of the interfacial region.

When the interfacial layer exhibits both viscous and elastic properties, the mobility oftliis layer will also depend on its dilational elasticity. Although fluid-fluid interfaces containing surfactants will usually exhibit some degree of elasticity, the role of dilational elasticities on the drainage of liquid films has been neglected in film thinning models. However, experi- ments (26-30) have shown that the dilational elasticity of the interfacial layer plays a major role in foam and emulsion stability. These studies have indicated that high surface dilational viscosities and elasticities lead to rigid interfacial layers and slow drainage.

Journal of Colloid and Interface Science, VoL 147, No. 1, November 1991

1 4 8 TAMBE A N D SHARMA

The effects o f the rheology o f the interface on film drainage as predicted by this mode l are shown in Figs. 10 to 12. Figure 9 illustrates the different paths along which the interfacial rheological parameters (#s and G) can be varied and the implicat ions. The interface becomes elastic as #s _.. ~ (for a finite G), purely viscous as G --* ~ (with #s finite), and viscoelastic otherwise. These conclusions follow f rom Eq. [ 31 ]. Figure 10 illustrates the effects of increasing both the interfacial viscosity and the modulus of elasticity on film thinning. The relaxation time remains constant when both parameters are increased by the same factor. The results (Fig. 10) show that drainage rates decrease as bo th interfacial elasticities and viscosities are increased. Figure 11 illustrates the effects of increasing the modulus of elasticity while holding the viscosity constant. This figure also reveals that an increase in the elasticity alone will slow drainage. Finally, Fig. 12 illustrates the effects o f increasing the interfacial viscosity while holding the modu lus o f elasticity constant. It is also evident f rom this fig-

Interface Parameters

}~(g/sec) G(g/sed) 1. Newtonian interf. 2. 0.1 3.0 3. 0.1 30.0 4. Rigid interfaces

,2! 1.0 4

0.2

0 . 0 i - , - ,

0 10 20 30

1.2

Dimensionless t i m e

F I G . 1 l. The effects of increasing the interfacial elasticity only on the thinning of films bounded by Newtonian ( 1 ), viscoelastic (2, 3), and rigid (4) interfaces.

12 __ j 1.0' 5" ' Interface Parameters

ec) G(g/~ \ ' ~ X / 1. Newtonian interf.

~o 0 8 \ \ \ \ / 2 0.1 0.1

"~ 4 3. 0.3 0.1 4 0.9 0.1 ~5 5. Rigid interfaces =~" 0.6'

~_ 0.4

0.2

0.0 , , '

10 20 30

Time(secs)

FIG. 12. The effects of increasing the interfacial viscosity only on film drainage. Films are bounded by Newtonian ( 1 ), viscoelastic (2, 3, 4) , and rigid (5) interfaces.

ure that high interfacial viscosities retard drainage. The trends observed here agree with exper imenta l observations. It should be

1.0

g 0.8 m

~" 0.6"

x::

E ~_ o.4-

0.2"

0.0 • 1~ 2~ 3~ ,0

0 . 8 ~o == eo E ~5 ~" 0.6

~2 E 0.4

iT_

Parameters

1. Force = FG 2. Force = 2FG

G=010/sed l~=01g/s~

Dimensionless t i m e

FIG. 13. The effect of the applied force on film drainage. FG denotes force due to gravity.



pointed out that the effect of #s on film thin- 1.2- ning is much more pronounced when using a linear viscoelastic model (LVE) with a finite 1.o- elasticity. If the modulus of elasticity is set to zero, the film thinning rates are less sensitive ~ 0.8- to #s, as is the case for Newtonian interfaces. In addition to the observations discussed ~

.~- 0.6- above, Figs. 10-12 clearly il lustrate the ira- of using the correct rheological model portance

for the interface when modeling the drainage _~ 0.4- of thin films. Model predictions can be off by ~: several orders of magnitude when proper as-

0.2"

sumptions are not made. These figures reveal that predictions can be faulty when a viscoelastic interface is assumed to be either New- tonian or rigid. Interfaces containing surfactants will usually exhibit some degree of elasticity, hence deviating from Newtonian behavior. The immobile interface approximation is also unlikely since surfactants seldom render the interface rigid. Under these conditions, the choice of a viscoelastic interface is appropriate. In cases where the interface exhibits both viscous and elastic behavior, it

1

\ 0.0

0 10 20 30

Parameters

1. IFTG = 0 2. IFTG > 0 3. IFTG = 2xIFTG in (2)

#=0,1g/sec G = 0.1g~ec 2

Dimensionless l ime

FIG. 15. The influence of interfacial tension gradients (IFTG) on film thinning. The first curve neglects the effects of IFI'G. IFrG is as defined in Eq. [24].

is evident from Figs. 10-12 that film thinning predictions are very sensitive to both the interfacial viscosity and elasticity.

4o t 30

a~ 20

"3 Q

10 1 2

Inter face Parameters

g(g/sec) G(g/sec~ 1, Newton ian 2. 0.1 0.1

Fo rce (d imens ion less )

FIG. 14. The effects of the applied force on the drainage of films bounded by Newtonian (1), and viscoelastic (2) interfaces.

The Influence of the Applied Force

The applied force is one of the more important factors affecting liquid film drainage. Figure 13 illustrates how the rate of film thinning will increase if the applied force is increased. This effect is also seen on Fig. 14. Fig- ure 14 also compares the drainage times for films bounded by Newtonian and viscoelastic interfaces as the applied force is varied. For both cases the drainage time decreases as the force is increased. Note, however, that viscoelastic interfaces show a smaller dependence of film thinning on the applied force. This implies that the effect of the applied force on drainage times can vary significantly depending on the rheological properties of the interface.

In this study we assumed the drop below the film is stationary (see Fig. 1 ), and the force acting on the drop above the film is the force due to gravity, FG.



1.2"

1.0"

o.8.

E

~ 0.6"

g g E ~_ 0.4"

0.2-

0.0 I 0 20 30

Surface Diffusion Coefficients

1. D = .0002 sq cm/sec 2. D=.0001 sq cm/sec 3. D ~ .00005 sq era/see

Interlace Parameters

= 0.1gtsec G= 0.1g/sec 2

Dimensionless time

FIG. 16. The effect of surface diffusion on the rate of film thinning.

The Influence o f Interfacial Tension Gradients

Interfacial tension gradients arise when variations exist in the local values of surfactant concentration and therefore interfacial tension. These forces act to oppose the drainage of liquid films. The extent to which drainage is retarded will depend on the ability of the surfactant to lower the interfacial tension. Figure 15 illustrates the effects of interfacial tension gradients on film thinning. The first curve neglects the effects of these gradients, and the second and third curves are profiles obtained for 3"o - 3' greater than zero (see Eq. [27]). Note that (3'o - 3") will depend on the surfactant's ability to reduce interfacial tension. It can be seen from this figure that as surface tension gradients become significant the rate of film thinning decreases significantly.

The Role o f Surface and Bulk Diffusion

The flow of liquid from the film region to the bulk usually disturbs the equilibrium distribution of surfactant at the interface. As a result reverse fluxes (both convective and dif-

fusive) are generated to restore the equilibrium state of the system. The magnitude of surface diffusive fluxes generated depends on the surface diffusion coefficient of the surfactant. A high surface diffusivity favors the rapid res- toration of equilibrium at the interface thereby eliminating interfacial tension gradients. A high surface diffusivity will therefore favor rapid film drainage. This conclusion is also evident from Fig. 16. This figure clearly shows that, as might be expected, less stable films result when surface diffusion coefficients are high. Any nonequilibrium distribution of surfactant at the interface will also result in the transport of surfactant from the film to the interface. When surfactant adsorption is the rate-controlling step, the equilibrium adsorption isotherm can be used as is done in this paper. However, if the rate of diffusion of surfactant from the film to the interface is slow, it could result in a nonequilibrium diffusion controlled process.

CONCLUSION

A model has been developed to describe the rate of drainage of plane-parallel films bounded by viscoelastic interfaces. It is clear from the results obtained that film thinning rates are sensitive to both the dilational elasticity and viscosity of the interfacial layer. The effect of the surface viscosity is more pronounced when the interface exhibits a finite elasticity. Other important factors that affect drainage include the applied force and interfacial tension gradients.

REFERENCES

1. Reynolds, O., Philos. Trans. R. Soc. London A 177, 157 (1886).

2. Boussinesq, J., Ann. Chim, Phys. 29, 349 (1913). 3. Radoev, B. P., Dimitrov, D. S., and Ivanov, I. B.,

Colloid Polym. ScL 252, 50 (1974). 4. Traykov, T. T., and Ivanov, I. B., Int. J. Multiphase

blow 3, 471 (1977). 5. Ivanov, I. B., PureAppl. Chem. 52, 1241 (1980). 6. Malhotra, A. K., and Wasan, D. T., Chem. Eng.

Commun. 55, 95 (1986).

Journal of Colloid and Interface Science, Vol. 147, No. I, November 1991


7. Johannes, W., and Whitaker, S., J. Phys. Chem. 69, 1471 (1965).

8. Lee, J. C., and Hodgson, T. D., Chem. Eng. Sci. 23, 1375 (1968).

9. Ivanov, I. B., Jain, R. K., Somasundaran, P., and Traykov, T., in "Solution Chemistry of Surfac- tants" (K. L. Mittal, Ed.), Vol. 2. Plenum, New York, 1979.

10. Barber, A. D., and Hartland, S., Can. J. Chem. Eng. 54, 279 (1976).

l l. Ivanov, I. B., and Dimitrov, D. S., Colloid Polym. Sci. 252, 982 (1974).

12. Good, P., Ph.D. Thesis, Univ. of Minnesota, Min- neapolis, (1974).

13. Levich, V. G., "Physico-chemical Hydrodynamics." Prentice Hall, Englewood Cliffs, New Jersey, 1962.

14. Jeffreys, G. V., and Hawksley, J. L., AIChEJ. 11,413 (1965).

15. Lucassen, J., van den Temple, M., Vrij, A., and Hes- selink, F., Proc. K. Ned. Akad. Wet. Set. B: Palae- ontol., Geol., Phys., Chem. 73, 109 (1970).

16. Murdoch, P. G., and Leng, D. E., Chem. Eng. Sci. 26, 1881 (1971).

17. Berg, J. C. in "Recent Developments in Separation Science," (N. N. Li, Ed.), Vol. 2, p. 1. CRC Press, New York, 1972.

18. Woods, D. R., and Burrill, K. A., J. Electroanal. Chem. 37, 191 (1972).

19. Liem, A. J. S., and Woods, D. R., A1ChE Symp. Ser. 70, 8 (1974).

20. Jones, A. F., and Wilson, S. D., J. Fluid Mech. 87, 263 (1978).

21. Hartland, S., Trans. Inst. Chem. Eng. 45, TI02 (1967).

22. Mackay, G., and Mason, S., Can. J. Chem. Eng. 41, 203 (1963).

23. Lang, S. B., and Wilke, C., lnd. Eng. Chem. Fundam. 10, 341 (1971).

24. Ivanov, I. B., and Traykov, T., Int. J. Multiphase Flow 2, 397 (1976).

25. Zapryanov, Z., Malhotra, A., Aderangi, N., and Wasan, D. T., Int. J. Multiphase Flow 9, 105 (1983).

26. Callaghan, I. C., and Neustadter, E. L., Chem. Ind (London) 17, 53 (1981).

27. Mukherjee, S., and Kushnick, A. P., ACS Symp. Ser. 364 (1989).

28. Neustadter, E., Whittingham, K., and Graham, D., in "Surface Phenomena in Enhanced Oil Recov- ery," (D. Shah, Ed.), p. 307. Plenum, New York, 1981.

29. Clint, J. H., Neustadter, E., and Jones, T., Dev. Pet. Sci. 13, 135 (1981).

30. Eley, D. D., Hey, M. J., and Lee, M. A., Colloids Surf 24, 173 (1987).

31. Churaev, N. V., Kolloidn. Zh. 46(2), 302 (1984). 32. Scheludko, A. D., "Colloid Chemistry." Elsevier,

Amsterdam, 1966. 33. Toshev, B. V., and Ivanov, I. B., Colloid. Polym. Sci.

253 (1975). 34. Scriven, L. E., Chem. Eng. Sci. 22, 98 (1960). 35. Langmuir, I., J. Am. Chem. Soc. 39, 1848 (1917). 36. Szyszkowski, B., Z. Phys. Chem. 64, 385 (1908). 37. Lucassen-Reynders, E. H., J. Phys. Chem. 70, 1777

(1966). 38. Barber, A. D., Ph.D. Thesis, University of Nottingham

(1973). 39. Darby, R., "Viscoelastic Fluids: An Introduction to

Their Properties and Behavior." Dekker, New York, 1976.

40. Rouse, P. E., J. Chem. Phys. 21, 1272 (1953). 41. Berridge, S. A., Thew, M. A., and Loriston-Clarke, J.

Inst. Pet. 54, 333 (1968). 42. Berger, P. D., Hsu, C., and Arendell, J., SPE 16285. 43. Blair, C. M., Chem. lnd. (London) 538 (1960). 44. Johansen, E. J., Skjarvo, M., Toreirlund, and Sjoblom,

J., Colloids Surf. 34, 353 (1988).


Hydrodynamics of thin liquid films bounded by viscoelastic interfaces

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