Effect of aging on surface shear viscosity of surfactant solutions

Effect of Aging on Surface Shear Viscosity of Surfactant Solutions V. M O H A N , L. GUPTA, AND D. T. W A S A N Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

Received February 1i, 1976; accepted April 27, 1976 The adsorption and accumulation of surfactants at fluid-fluid interfaces results in additional, intrinsic, hydrodynamic resistance to flow of which surface shear viscosity is a measure. Surface theological behavior is often dependent on the rapid aging of surfactant solutions. This paper deals with the interpretation of interfacial shear flow data in the viscous traction surface viscometer in terms of aging effects on surface viscosity. It was found that the surface shear viscosity increased considerably with aging of dilute aqueous surfactant solutions. For the same age of the interface, the surface shear viscosity increased with the concentration of the surfactant. 1. INTRODUCTION Surface active agents in solution tend to adsorb and accumulate at interfaces between their solutions and the adjacent solid, liquid, or gaseous phases. Specifically, such accumulation at fluid-fluid interfaces, results in additional, intrinsic, hydrodynamic resistance to flow. Interfacial shear viscosity is a measure of this resistance. The significance of interfacial shear viscosity has been recognized in several instances such as foam fractionation, emulsion stability, suspension polymerization, film permeability, tanning, Marangoni instability, lung surfactant systems, films flowing down solid walls, bubble and drop behavior, and mass transfer operations. Surface aging is a phenomenon that is commonly encountered in surfactant solutions. This aging effect can be described as a timedependent variation in surface concentration which leads to an age-dependent variation in surface tension and surface shear viscosity. The age of the surface, therefore, becomes important in controlling dynamic rate processes involving fluid surfaces. In the present study we are interested in surface shear viscosity which m a y vary under one or more of the following conditions.

(1) when surfactant molecules undergo time-dependent adsorption at the interface; (2) when surfactant molecules are subject to time-dependent desorption from the interface ; (3) when surfactant molecules adsorbed at the surface undergo time-dependent chemical reaction ; (4) when surfactant molecules in the bulk of the solution react chemically and consequent time-dependent adsorption of the product or products alters the character of the adsorbed layer of molecules. This paper deals with the effect of surface aging on surface shear viscosity determined using the viscous traction interfacial viscometer. This viscometer has been used previously and described in (1-7). While the unsteady state operation has been used to extract information on viscoelasticity (8), an analysis to interpret quantitatively the effect of surface aging on surface viscosity is presented here for the first time. 2. PROBLEM FORMULATION The geometry which we wish to consider is sketched in Fig. 1. The viscometer consists

496 Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976

Copyright ~ 1976by AcademicPress, Inc. All rights of reproductionin any form reserved.

497

S H E A R V I S C O S I T Y OF S U R F A C T A N T S

of a flat-bottomed dish containing the fluid, and two stationary, concentric cylinders. The cylinders are held vertically and so placed that they almost touch the bottom of the dish. The dish is rotated, causing the fluids in the channel between the cylinders to rotate. The stationary channel walls tend to shear the fluids, thereby increasing the effect of the interface. The mathematical model to be used here assmnes that the circular channel can be approximated by an infinitely long rectilinear channel. This assumption has been shown (4, 9) to be valid as long as the channel width (0.871 cm) is small compared to the radius (5.494 cm) of the stationary outer cylinder. It will be assumed that the velocity of the base of the rectilinear channel corresponds to that in the original geometry. It will also be assumed that there is no gap between the channel walls and the rotating dish. This assumption has been found to be valid even though a small gap is actually present (4). The fluids in the channel will be assumed to be Newtonian and incompressible. The bulk viscosity of the gas, air in most cases, will be considered negligible with respect to the bulk viscosity of the liquid. The flow in the channel will be assumed to be laminar. The liquid-gas interface shall be considered flat and assumed to exhibit Newtonian surface shear viscosity. With the above assumptions, and placing the origin of coordinates at the floor and one of the channel walls, with the x-axis taken along the channel wall and the y-axis along the floor, the x and y components of the velocity in the channel are zero. The z component of velocity is given by the following initial and boundary value problem. Ov p-

O~v~

( 02v = ~ - - + - -

Ot

\Ox 2

,

[-13

Oy2/

where p and ~ are the density and absolute viscosity of the liquid and v(t, x, y) is the fluid velocity dependent on the time t and the spacial coordinates x and y. The initial

STAT ION ARY PARALLEL CHANNEL WALLS

I

INNER

OUTER

GAS

--LIQUID

MOVING FLOOR

v=(ritY)COo V = (RitY) / ( Ri! 0.5 }

FIG. 1. Cartesian representation of channel in viscous traction interfacial viscometer.

condition is v(0, x, y) --= (ri -[- y)~0o, x = 0, = 0, otherwise,

[-2-]

where r~ is the outer radius of the inner channel and w0 is the angular velocity of the floor of the channel. Here time t = 0 is reckoned at the instant the floor starts moving. The boundary conditions are

~,(t, x, 0)

o,

[3-]

vq, x, y0) = o,

[4-]

=

v(/, O, y) = (r~ + y)~oo,

[-57

0 02 - - v (l, xo, y) = e - - v (/, Xo, y), Ox Oy2

[-6-]

where x0 is the liquid depth in the channel, y0 is the channel width, and the surface shear viscosity (e) is a function of the surface age. Boundary conditions [-3-] and [-4-] result from the no-slip condition at the stationary channel walls. Boundary condition [-53 follows from the no-slip condition at the rotating floor of the channel. Equation [-6-] results from the interfacial momentum balance at the liquid-gas interface under the assumptions

Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976

498

MOHAN, GUPTA AND WASAN

of negligible mass of the interface and negligible mass transfer across it. The initial and boundary value problem defined by Eqs. [1]-[6] can be nondimensionalized to yield

OV

02V -

OT

Taking the Laplace transform of the differential equation [7] and the boundary- conditions, the general solution in the transform domain can be shown to be

02V

V (X, Y) = F. fn(X) sin (turF),

+ --

OX 2

[7]

OY 2'

where

V(O, X, Y) = [R~ + Y]/[R~ + 0.5],

.;?~(X) = A ,, sinh [ (s + n2~-2)~X] +B,, cosh [(s + n2~r2)~X], [16]

X=0,

otherwise,

= o,

V (T, X, O) = O,

V(T, X,

[83

1) = 0,

c ----cosh[(s +r~)~D]},/

11 =

[-93

S~"

[103

sinh [ (s + ~r2)}D],

V(T, O, Y) = (R, + Y ) / ( R , + 0.5), Ell-] 0 a2 - - - V(T, D, Y) = E - - - V(T, D, V), OX

[153

n

L173

4 A k = - - - - - coth [(s + k2~'2)½D], sk~r

[123

O y2

k odd # 1,

where V is the ratio of the fluid velocity v to the centerline velocity 9b of the channel floor, X=x/yo, Y = y / y o , T = ~ t / p y o 2, and E

sk~'(Ri + 0.5)

coth [(s + k2~'2)~D],

= ~/~yo.

k even, and

3. SOLUTION Equations [9] and [10], representing conditions at Y = 0 and Y = 1, are time-independent. Hence the Y-dependence of the velocity can be taken to be the same as that under steady state conditions. For deep channels (D > 2/x-), the steady state velocity at the liquid-gas interface is related to the interfacial centerline velocity (Ve) by (4)

V(D, Y) = V~ sin ~-Y.

[183

[13]

4 k odd,

Bk ~ - -

skit'

skzr(R~ + 0.5)'

k even.

[193

The inverse of Eq. El5] yields V(T, X, Y) in the form

V(T, X, Y) = ~_. f,,(T, X) sin (torY).

[20]

n

Thus the time-dependent interfacial velocity in our problem can be expressed as

V(T, D, Y) = V,(T) sin TrY.

[-14]

It is to be noted that the interfacial centerline velocity here is a function of time. The objective now is to solve for the velocity field in terms of Ve using Eq. [14] as opposed to a solution in terms of E involved in boundary condition [12]. Once such a solution is obtained, however, E can be determined by application of Eq. [12]. Journal of Colloid and Interface Science,

Using Eqs. [12] and [20] it can be shown that (10) E

=

-

[213

-

T~2

X ~ D"

From Eq. [16] we have 0fl = {A1 cosh [(s + 7r2)~D] aX X=D +B1 sinh [-(s + 7r~)~D]}(s + 7r~)½. [223

Vol. 57, No. 3, December 1976

SHEAR

VISCOSITY

Substituting for A a and B~, and rearranging,

0~

cosh =

f'~

OX X=D

[,(s + ~)~D]

sinh [(s + 7r2)~D] 4

101-08 -

(s + ~2)~

~-2)~

(s +

o6 i-

bJ

sr sinh [,(s 4- 7r2)~D]

E23]

499

OF S U R F A ( : T A N T S

L ta_ O4

Using the deep channel approximation, that is, 02 i

sinh [(s + r2)lD] = cos], [(s + 7r2)'~D]

O0 001 002

e(s+r2)½D ,.2_

0.05

0.1

02

O5

1,0

SURFACEAGE(T),DIMENSIONLESS

2

we have FIG. 2. D e p e n d e n c e of K t on s u r f a c e age. =

(sV~)

-

OX X=D X

_

_

S

e-(.+~

),o

r

The first term on the right side of Eq. [-24-] can be inverted as follows. We observe that 6

(s + ~-9~ -

(s + ~-9~ T

7~

[24]

7r

/~/

The second term on the right side of Eq. [-24] can be inverted using (11) [ e-'~.' [ __

slYe - Vd0) = Sf:e

[27]

and from (11)

e -a2/4T

~-'ts:+b/

beab+b2T

(~r)~

Xerfc

£-1

+ bT½ ;

_

s

(~T)~

+ r erf (7r%)"

so that

Re a _> 0, R e s > 0 if Reb>__0, R e s > max0, Reb 2 if R e b < 0 , X /e-'"'[ 0rr) l + ~" err

to yield ,~--1

_

_

e--(s+r )~D (s

•r

+

r2)~ -

r

= 8 K l e -'~o,

+ (s + ~'~)~+

[-25]

where D

K, = [4{e--'D erfc(2&fi

rT½)

(~-%'],/T, E28-I

where V / is the time rate of change of centerline velocity. The integral in Eq. [-28] exists if V / is bounded, a condition which holds. Since the polynomials form a set of complete functions (12) we shall express the centerline velocity in the form Vc(T) = £ aiT'

~ e "~Derfc - 8

_7r(~rT)~ _ e-~Te-DW4T

]/

[-29]

i~3

+ lrT½ so that 8e-~°. [26]

Vo'(T) = ~ b,r',

b, = (i + 1)a,.~,.

[30]

i=O

Journal of C~lloid and Interface Science, Vol.

57, No. 3, December 1976

500

MOHAN, GUPTA AND WASAN

/- /

[313,

Combining Eqs. [24], [2S], [28], and [33], we have

1.0

j

0.8

all 0,6

= ~ ( _ l ) k r ( k +½, r2T)

Ox I .= D

~ o4 w 123

X 0.2

0 0.01

I

~ I 0.2 0.5 SURFACE AGE (T), DIMENSIONLESS

0.02

I

I

0.05

0.i

I'O

~-2k+~k!

I ~ dkVe

dk+tV~1 + - dT k dTk+~ J

[

x=.

It follows that,

r(k + ½, ~r2T) (2k) l

= E (-1)* *~

¢2*-lr(k + ½)(k~)2 2~

X I ~Ve 1 dk+IVc1 + dT k+l-i -- 8 K , e - " . [ dT k 7r2

e--r~r

[35]

- r] (,r,)! dr

1L T

Furthermore, since fl(D) = Vo fl(T, D) = Vc(T), Eq. [21] yields

e--r*r

V** • -E = K t - -

V©

--b, (-t?[)~ ~rl i~o k=0 \ k l lr

1 "l-

-

-

Xf dO

kffiO

71"2k+t

~r2~tk !

X

0k-, e-odO

oo

E

( - - 1 ) k+/

F(k + {)Tr2. 22k(k!)2

X I dkV* 1 d~We 1 -F - [ dT k 7r2 dTk+l l,

i=k k - - .

4--0

that

1~(k + ½, r2T) (2k)!

[36]

where

(_1) r(k + ½,~'r) dk+lV, =

so

Ve k='O

~4T

[31]

Vo* = 8 e-'D/lr.

dT ~+t

[37]

In the-linear portion of the plot of Vc versus T, Eq. [36] simplifies to the con-

Furthermore, on integrating by parts, 7r

[34]

8Kle -'D.

On rearrangement of Eq. [34], we obtain

FIG. 3. Dependence of K~ on surface age.

~0TV ' E T

k=o

Ve'[T -- r] erf (~r%)Idr 1.4

=~r~

VeET- r ] , - ~ e - " ' d r .

[32] I.Z hl

Substituting [29] into [32] it can be readily shown that LT 7r VJET -- r] err (Tr2r)½dr = ~. (--1) k k~O

~r2k-~k!

Journalof Colloidand InterfaceScience,Vol.

dT k

hl

8

08 OGoo~

0;2

I

0.85 o., &

o~

1.0

SURFACE AGE ff),DIMENSIONLESS

r(k + ½ #-2T) dAVe X

_o

1.2- - ~ 1.0

[33]

57, No. 3, D e c e m b e r 1976

FIo. 4. Dependence of Ka on surface age.

SHEAR VISCOSITY OF SURFACTANTS

501

oz 500

~ 200 ~- I00 z

Ve~ =0 I

50

Vc~ , 0 2 V~=03 V~=04

oz Z0

fit g 0.1

[ Ill [11 I T IIIir!l 1 I [ll[lll I I TIEI: 02 05 I 2 5 I0 20 50 I00 200 500 UNCORRECTED SURFACEVISCOSITY {'/TE ) ,DIMENSIONLESS

FIG. 5. Correction required in the calculated value of surface viscosity under the pseudo-steady-state assumption.

venient form V e*

[393

Mannheimer (13) used Eq. [43-] to obtain a qualitative idea of the aging of air-oil surfaces containing oil soluble surfactants. Comparing Eqs. [-42-] and [43-], the correction term CE to the value of the surface viscosity (rE) obtained from pseudo-steady state assumption is given by

[40-]

CB = B/2r~Vc.

B

~rE = K1 - - -- K2 + K 3 - - - , Vc 27rWe

[-38]

where

B

-dV~/dT,

=

r (½, 7r2T) K2 ~--- -

-

7

r(½) and

1' (1, r2T)

1' (~, ~r2T)

r(½)

r(~)

K~ = 2

1-41-1

Figures 2, 3, and 4 indicate the variation of

K1, Ks, and Ks with T. 4. SPECIAL CASE

A special case arises when the centerline velocity V~ varies linearly with time T. If such a state corresponds to T > 0.6 (which corresponds to approximately 60 sec for aqueous systems), K1 = K2 = K3 = 1 and hence V** B ~rE = - - -- 1 -at- - V, 2r ~Vc"

[-42-]

Assuming pseudo-steady state behavior, Eq. [42-] reduces to

rE

Vc* -

V~

1.

1-43-]

[-44"]

Figure 5 depicts the correction required in the calculated value of surface viscosity under the pseudo-steady state assumption when actually unsteady state flow conditions prevail in the viscometer. The correction is plotted as percent of the uncorrected value (Vc*/ V c - 1) per unit dimensionless deceleration versus the uncorrected value, with Vc* as a parameter. The percentage correction is seen to decrease with an increase in the uncorrected value and reach an asymptotic value for each Vc*. Furthermore, at a given value of surface velocity, the percentage correction decreases with increasing values of Vc* or equivalently with decreasing values of the depth of the liquid. However, the lower limit on the percentage correction is set by t h e deep channel approximation which requires that D be greater than 2/7r. 5. E X P E R I M E N T A L APPARATUS AND PROCEDURE

It has been shown (2, 4, 9, 14) that curved interfaces in the channel result in an error

Journal of Colloid and Interface Science, Vol. 57. No. 3, December 1976

502

MOHAN, GUPTA AND WASAN tance on the microscope scale. The time taken by the rotating dish to complete one revolution was also recorded. The depth of the liquid in the channel was determined from the difference in the micrometer readings taken for two positions of the probe: one when the tip of the probe just penetrated the surface and the other when it made contact with the floor of the dish.

~_o × ~-...j

6. EXPERIMENTAL RESULTS AND DISCUSSION

5

I--

0

2000 4000 SURFACE A6E ( 1 ) , SEC

6000

FIO. 6. Time-dependent interfacial flow data for mixed solutions of sodium lauryl surface and laurie acid. of less than 10% in the calculated values of interfacial velocities and shear viscosities. Furthermore, maintenance of flat interfaces in the channel requires the construction of a step or ledge (9) in the channel walls. Such a ledge was machined into the channel walls of the interfacial viscometer built in our laboratory, but it made experimentation more difficult. Liquids have to be introduced extremely slowly and consequently, no timedependent data can be taken at low surface ages. Also, evaporation of liquid will cause the liquid level to fall below the ledge and the interfacial profile to assume a curved shape. On the basis of these considerations, it was decided to ignore curvature effects in .this work. The description of the viscous traction interfacial viscometer and its operation are described elsewhere (7). Interracial centerline velocities were determined by "following" a teflon particle ( d i a m e t e r - 0.01 cm) in the field of view of a microscope and noting the time taken to traverse a predetermined dis-

Interracial viscometric data are obtained as particle time (&) required for one revolution versus surface age (t) at a given depth of liquid (x0) and a given time for one revolution of the dish (to). For our purpose, the data are most conveniently plotted as the reciprocal (1/to) of particle time versus surface age (t). The dimensionless deceleration (B) can be calculated from the slope of such a graph. The surfactant solutions investigated include : (1) aqueous solutions of a commercial anionic surfactant, Duponol RA, (2) a mixed aqueous solution of anionic surfactant sodium lauryl sulfate and nonionic lauryl alcohol, (3) mixed aqueous solutions of anionic surfactants sodium lauryl sulfate and lauric acid. Duponol RA is a commercial grade of fortified alkoxyalkyl sodium sulfate manufactured by E. I. DuPont de Nemours and Company. Sodium lauryl sulfate, lauric acid, and lauryl alcohol were laboratory grade chemicals purchased from Fisher Scientific Company. The surface tension of the solutions investigated ranged from 28 to 37 dynes/cm as determined by the Wilhelmy plate method. The densities and bulk viscosities of the solutions were essentially the same as those of distilled water (0.997 g/cm 3 and 0.9 cP, respectively). Figure 6 is a typical plot of the experimental viscometric data as the reciprocal

Journal of Colloid and Interface Science, Vol. 57, N o . 3, D e c e m b e r 1976

503

SHEAR VISCOSITY OF S U R F A C T A N T S

(l/G) of particle time (which is directly proportional to the centerline velocity vc), versus surface age. The reciprocal of particle time decreases slowly at first, then very rapidly, and finally extremely slowly. Now the correction term given by Eq. [44] is directly proportional to the dimensional deceleration (B) and hence, we shall first determine the correction term for the time interval in which the velocity changes most rapidly. Note that the plot of (l/G) versus t is linear in this region. Table I summarizes the results so obtained. For the systems investigated, the maximum value of the error in surface viscosity incurred by the pseudo-steady state assumption is 11.42%. This occurs for the 0.025% Duponol RA solution at a surface age of 214 sec. It is

to be noted that the dimensionless deceleration (B) is also maximum in this case. For all the other systems, the maximum error is less than 4%. This suggests that the pseudosteady state approximation can be used to interpret aging behavior at fluid surfaces. It is to be noted, however, that in some situations the use of Eq. 1-38] or 1-42] may be necessary. Figure 7 depicts the corrected time-dependent surface viscosity for a typical system, namely, adsorbed surfaces in mixed aqueous solutions of sodium lauryl sulfate and lauric acid. The plot reveals a considerable increase in the surface shear viscosity with aging. Furthermore, it is seen in combination with Table I that for the same age of the inter-

TABLE I Corrections to Surface Viscosity Calculated under the Pseudo Steady State Assump tion (to = 40.1 sec, to* = 241 sec) System

O.OLO% Duponol RA

0.025% Duponol RA 0.010% sodium lauryl sulfate + 0.000167% lauryl alcohol

0.005% sodium lauryl sulfate W 0.00125% lauric acid

0.010% sodium lauryl sulfate -4- 0.00125% lauric acid

O.OLO% sodium lauryl sulfate + 0.00025% laurie acid

(sec)

t

(sec)

tc

d ( I/t~) dt

B

,l~g

1242 1738 214 250 294 336 553 1029

332 1448 283 349 576 1048 336 1850

--0.496 --0.496 --1.860 --1.860 --1.860 -1.860 --0.516 --0.516

1.68 1.68 6.30 6.30 6.30 6.30 1.75 1.75

X X X X X X X X

10-z 10-2 10-2 10-2 10-2 10-2 10-2 10-2

1048 1099 1148 1230 1289 1348 492 526 567 679 737 1945 229l

315 377 476 844 1300 2100 330 389 489 1138 2525 427 1731

-0.900 --0.900 --0.900 --0.900 --0.900 -0.900 --1.120 -1.120 --1.120 --1.120 --1.120 --0.510 --0.510

3.05 3.05 3.05 3.05 3.05 3.05 3.8 3.8 3.8 3.8 3.8 1.73 1.73

X X X X X X X X X X X X X

10-2 10-2 10-3 10-2 10-2 10-2 10-2 10-2 10-2 10-2 l0 -2 10-2 10-2

(10-~ sec-"-)

~r*/nyo

(10-~ s.p.)

Uncor- Corrected rected

Error (%)

0.0070 0.0307 0.0225 0.0278 0.0458 0.0834 0.0074 0.0409

0.378 5.008 0.174 0.448 1.390 3.349 0.394 6.676

0.385 5.039 0.197 0.476 1.436 3.423 0.401 6.717

1.82 0.6l 11.42 5.84 3.19 2.44 1.85 0.61

0.096 1.258 0.049 0.119 0.358 0.854 1.001 1.676

0.0121 0.0145 0.0183 0.0325 0.050l 0.0809 0.0158 0.0187 0.0235 0.0546 0.1212 0.0093 0.050l

0.307 0.564 0.975 2.502 4.394 7.714 0.369 0.614 1.029 3.721 9.477 0.772 6.183

0.319 0.579 0.993 2.535 4.444 7.795 0.385 0.633 1.053 3.776 9.598 0.781 6.233

3.79 2.51 1.84 1.28 1.13 1.04 4.11 2.96 2.23 1.45 1.26 1.19 0.80

0.080 0.145 0.248 0.633 1.109 1.946 0.096 0.158 0.263 0.942 2.396 0.195 1.556

Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976

504

MOHAN, GUPTA AND WASAN

I000

SODIUM LAURYL SULFATE,WT% o

E]

U ,oo

A

0005 0.010 0010

viscosity was larger at higher surfactant concentrations. At lower surfactant concentrations, the time taken for the interfacial shear viscosity to build-up to a specific value was larger.

LAU RIG ACID,WT % 0.00125 0,00125 000025

/

ACKNOWLEDGMENTS

I0

This work was supported by the National Science Foundation under grants GK-30028X1 and GK-43135, and in part by the American Petroleum Institute under Research Project 133. We are especially grateful to Professor F. C. Goodrich for his useful comments on this work.

[3 >I.O o

to = o

O.I t.o

REFERENCES 1. BURTON., R. A. AND MAN.NIIEIMER, R. J., in O.OI iO0

200

500 I000 2000 SURFACE AGE ( t ) . SEC

5000

FIG. 7. Time-dependent surface viscosity for mixed solutions of sodium lauryl sulfate and lauric acid. face, the surface shear viscosity increases with the concentration of the surfactant. Also, at lower concentrations of the surfactant, the aging effects are less pronounced, i.e., the time taken for the interfacial shear viscosity to build-up to a specific value is larger. 7. CONCLUSIONS AND SIGNIFICANCE An analysis has been developed for timedependent velocity fields in the deep channel interfacial viscometer. An expression has been presented which permits the interpretation of interracial flow data in terms of aging effects on the surface viscosity of surfactant solutions. For the systems investigated, it was found that the surface shear viscosity increased considerably with surface aging. I t is therefore apparent that in any analysis involving fluid interfaces, the age of the interface which influences the surface shear viscosity is of paramount importance and enters in the boundary condition. Furthermore, for the same age of the interface, the surface shear

"Ordered Fluids and Liquid Crystals," p. 315, Advances in Chemistry, Series, No. 3, 1967. 2. PINTAR, A. J., "The Measurement of Surface Viscosity," Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1968. 3. MANNHEIMER, R. J. AND SCHECTER, R. S., J . Colloid Interface Sci. 32, 212 (1970). 4. PINTAR,A. J., ISRAEL,A. B., AND WASAN',D. T., J. Colloid Interface Sci. 37, 52 (1971). 5. HEGDE, M. G. AND SLATTERY, J. C., J. Colloid Interface Sci. 35, 593 (1971). 6. WASAN, D. T., GUPTA, L., AND VORA, ~V[. K., A.I.Ch.E. 17, 1287 (1971). 7. GUPTA, L. AND WASAN,D. T., Ind. Eng. Chem. Fundam. 13, 26 (1974). 8. MANNHEIMER, R. J. AND SCHECHTER, R. S., J. Colloid Interface Sci. 32, 225 (1970). 9. MANNHEIMER, R. J. AND SCHECHTER, R. S., Y. Colloid Interface Sci. 32, 195 (1970). 10. GUPTA, L., "Interfacial Shear Viscosity of Films Adsorbed from Surfactant Solutions," Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1972. 11. ROBERTS, G. E. AND KAUYMAN, H., "Table of

Laplace Transforms," Saunders, Philadelphia, 1966. 12. FINLAYSON.,B. A., "The Method of Weighted Residuals and Variational Principles," Academic Press, New York/London, 1972. 13. MAN~InEIMER,R. J., A.I.Ch.E.J. 15, 88 (1969). 14. GUPTA, L., "The Role of Curved Interfaces in Surface Viscometric Measurements," M. S. Thesis, Illinois Institute of Technology, Chicago. 1970.

Journal of Colloid and Interface Science, Vol. 57, No. 3, December 1976