Electrohydrodynamic lubrication with thin double layers

Electrohydrodynamic Lubrication with Thin Double Layers S T A C Y G. B I K E l AND D E N N I S C. P R I E V E 2

Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received February 22, 1989; accepted August 1, 1989 Sliding of one charged body past another in an electrolyte solution induces a streaming potential in the fluid between the surfaces which in turn generates an electrokinetic force on the bodies. In this paper, we extend our previous lubrication analysis for bodies bearing thin double layers from two-dimensional to three-dimensional flows and to squeezing as well as sliding motion. A general differential equation for the streaming potential is derived which is analogous to Reynolds' equation for pressure. When the double layer is thin compared to the minimum distance separating the two bodies, this electrokinetic force tends to be small compared to viscous forces. However, for sliding motion, one component of the electrokinetic force acts to push the bodies apart; this component has no viscous counterpart. This "electrokinetic lift" is always repulsive, regardless of the relative signs or magnitudes of the ~-potentials and can be comparable to double-layer repulsion for fluids of low conductivity. Electrokinetic lift may reduce the elution volume of particles in chromatographic columns, inhibit the capture of particles from moving fluids, or cause detachment by shear of particles from solid surfaces. © 1990AcademicPress,Inc. fluids o f low conductivity. In the case o f a particle sliding along a plane wall, we call the force " e l e c t r o k i n e t i c lift." Electrokinetic lift p r o v i d e s an e x p l a n a t i o n for the a n o m a l y o f A l e x a n d e r a n d Prieve ( 2 ) , w h o o b s e r v e d lateral m i g r a t i o n o f a m i c r o scopic sphere a w a y from a wall in linear shear flow o f glycerol, b u t n o t in water. This lateral m i g r a t i o n c a n n o t be inertial in origin because it o n l y occurs in the m o r e viscous fluid. Elect r o k i n e t i c lift is o f p o t e n t i a l i m p o r t a n c e for n o n a q u e o u s fluids in a p p l i c a t i o n s such as hyd r o d y n a m i c c h r o m a t o g r a p h y , tribology, det a c h m e n t by shear o f particles f r o m solid surfaces, a n d the capture o f particles from m o v i n g fluids. A related b u t m o r e familiar p h e n o m e n o n is the electroviscous effect which occurs during pressure-driven flow t h r o u g h charged capillaries (3, 4 ) . F l o w o f an electrolyte solution t h r o u g h a capillary p r o d u c e s a s t r e a m i n g current due to shear o f the c o u n t e r i o n cloud. Charge s e p a r a t i o n resulting from this c u r r e n t induces a s t r e a m i n g p o t e n t i a l along the length o f the capillary, in t u r n p r o d u c i n g an elect r o o s m o t i c flow which o p p o s e s the pressure-

1. I N T R O D U C T I O N

Sliding or squeezing m o t i o n b e t w e e n two charged bodies s e p a r a t e d by a thin film o f electrolyte solution causes c o n v e c t i o n of: charge w i t h i n the c o u n t e r i o n c l o u d s u r r o u n d ing each body. T o conserve charge, this convection induces a s t r e a m i n g p o t e n t i a l profile in the fluid b e t w e e n the bodies which in t u r n generates an electrokinetic force on the bodies. W e have p r e v i o u s l y p r e d i c t e d this electrokinetic force in the l u b r i c a t i o n limit for twod i m e n s i o n a l sliding m o t i o n between two parallel cylinders b e a r i n g thin d o u b l e layers ( 1 ). Owing to the a n t i s y m m e t r i c pressure profile which arises in the absence o f charge, S t o k e s ' e q u a t i o n predicts no a t t r a c t i o n or r e p u l s i o n b e t w e e n the cylinders for p u r e sliding m o t i o n . W i t h surface charges, a n electrokinetic force arises which is always repulsive, regardless o f w h e t h e r the surfaces b e a r charges o f like or unlike sign. M o r e i m p o r t a n t l y , this force can be c o m p a r a b l e to d o u b l e - l a y e r r e p u l s i o n for 1Present address: Chemical Engineering Dept., University of Michigan, Ann Arbor, MI 48109-2136. 2 To whom all correspondence should be addressed. 95

0021-9797/90 $3.00 Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

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BIKE AND PRIEVE

driven flow. When the capillary diameter is comparable to the Debye length, a measurable reduction in net flow occurs, which can be mistaken for an increase in fluid viscosity. Electroviscous effects also occur in suspensions of charged particles. Resistance to the distortion of the counterion cloud by an applied shear field gives rise to the primary electroviscous effect ( 5 - 9 ) . The ensuing electrical stresses increase the intrinsic viscosity above that predicted from Einstein's analysis. The secondary electroviscous effect describes the increase in the suspension viscosity resulting from the electrostatic interactions between adjacent particles ( 10, 11 ). In this paper, we extend our previous analysis from two- to three-dimensional flows and to squeezing as well as sliding motion. A general differential equation for streaming potential is derived which is analogous to Reynolds' equation for pressure. As two examples, solutions for the streaming potential profile and the electrokinetic force are obtained for a sphere sliding parallel to a flat plate and for squeezing between two unequal spheres. We first review traditional lubrication theory in the absence of charge and hydrostatic equilibrium for an isolated electrical double layer; then we present our analysis for electrohydrodynamic lubrication.

Let Rl,max and Rl,min be the principal radii of curvature of body 1. For distances hi(x, y) much less than both Rl,max and Rl,min, hi can be approximated by X2

y2

hi(x, y) = 61 + 2R1 . . . . + 2Rl,min If ~ represents the angle between the x-axis and the direction of principal curvature of body 2, having a radius of curvature R2 . . . . . then h2 can be approximated by (cos2~b

sin2qS) x 2

h 2 ( x , y) = 62 + \R2,max -1- R2,min

(,

,)

+ sin q~ cos 4~ R2~max + -~2.min xy

+(sin \e2,max

R2,min] 2 - •

The total distance between the two surfaces is

h(x, y) = hi(x, y) + h2(x, y). Two vectors (not necessarily of unit magnitude) which are normal to the lower and upper surfaces are

ni =- V f ,

[la]

f l ( x , y, z) =- hi(x, y) + z

[lb]

2.1 System Geometry

f 2 ( x , y , Z) =-- h 2 ( x , y ) - z.

[lc]

Consider two bodies of arbitrary (but smooth) shape moving slowly through a viscous fluid in the near vicinity of each other (Fig. 1 ). A rectangular Cartesian coordinate system is chosen such that the z-axis coincides with the shortest line connecting the two surfaces. The origin is located at some arbitrary point along this line. 6i represents the distance (along the z-axis) from the origin to the surface of body i (i = 1, 2), while z = - h i ( x , y) and z = h z ( x , y ) describe portions of each surface near the origin. The x- and y-axes are oriented so that they coincide with the directions of principal curvature of the surface of body 1.

Note that f l vanishes on the lower surface while f2 vanishes on the upper surface.

2. GENERAL ANALYSIS

Journal of Colloid and Interface Science,

Vol. 136,No. 1, April 1990

where

and

2.2 Translation of Electrically Neutral Bodies In the absence of electrical forces, Stokes' equation describes the balance of viscous and pressure forces within the fluid: ~VZv = ~Tp.

[2 ]

Requiring no slip between the fluid and solid at the boundaries provides the boundary conditions

ELECTROHYDRODYNAMIC LUBRICATION

ILy

52

t

97

h2(x'Y)

-x

51

hl(X,y)

FIG. 1. Schematicshowingtwo bodies of arbitrary shape moving at differentvelocities.

at z = - h i :

v

= ¥1

[3a]

at z

v = v2,

[3b]

=

+h2:

where vi --- Uiex + Viey + Wiez are the translational velocities of the two bodies. In the usual lubrication approximation (which is applied when hi/Ri.min~ 1), the partial derivatives of Vx and vy with respect to z are very large compared to derivatives with respect to x and y, whereas Op/Oz is negligible compared to Op/Ox and Op/Oy. Then Eq. [2] can be approximated by ["IOZVs/OZ2 ~- ~ s P ,

[4]

where Vs-= Vxex+ vyey is constituted just from the x and y components of v, p = p(x, y) and Vs is exO/Ox + eyO/Oy. Integration of Eqs. [3] and [4 ] yields VS ~--- Vsl ~- (VS2 - -

Vsl)fl/h - (flf2/2g)Vsp ==-v~0 [5]

and

f vsdz = (vsl + v~2)(h/2) - (h3/12g)V~p, where Vs,--~ U~ex+ Vieyis to vi as v~ is to v and f dz is the definite integral evaluated on the interval ( - h , , h2). The first two terms in Eq. [ 5 ] represent linear shear flow caused by sliding motion between the two surfaces, while

the third term represents the parabolic pressure-driven flow. Still unknown in Eq. [5] is the pressure profile p(x, y), which develops such that fluid is conserved (see Appendix A):

v s . f vsdz = nl"

Vl ~- n2" v2.

[6]

Substituting Eq. [ 5 ] into Eq. [ 6 ] yields the differential equation for pressure

Vs" (h 3Vsp) = -6/~(v2 - Vl )" (n2 - nl ), [ 7a] where the ni are defined by Eqs. [1]. Outside of the film, the pressure must tend to that of the bulk fluid: a s x 2 + y 2 " ~ o0:

p "~ 0.

[7b]

Equation [7a] is Reynolds' lubrication equation for an incompressible fluid with constant viscosity. The solution to Eqs. [ 7 ] yields the pressure profile in the film for any prescribed translation of the two bodies. A more general form of Reynolds' equation which also includes the effect of an arbitrary rotation of either body was derived by Cox (12). For further information on the study of hydrodynamic lubrication, the reader is referred to Ref. (13). Since only the difference of the translational velocities appears in Eqs. [ 7 ], the solution will be independent of the reference frame used to describe speed. This result differs from our previous analysis ( 1 ) for the pressure profile in two-dimensional flow in the xz-plane, which Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

98

BIKE A N D PRIEVE

was incorrectly calculated by requiring Vs" f vsdz = 0 instead of Eq. [6]. For two-dimensional flow in the reference frame chosen in Ref. 1, the integration of Eqs. [ 7 ] leads to a pressure profile which differs in sign from Eq. [ 31 ] in that work. A similar sign error was made in the electrostatic potential profile although the conclusions with regard to the normal c o m p o n e n t of the electrokinetic force remain correct. For the special case of pure sliding motion between two identical spheres or any two bodies whose surfaces are mirror images of each other (i.e., h~ = h2), n2 - nl has only a zcomponent whereas in the absence of squeezing motion v2 - Vl has no z-component. Then Eqs. [7] yields the trivial solution p = 0 everywhere.

2.3 Hydrostatic Equilibrium with Surface Charges Now let both of the surfaces bear a charge without any relative motion, so that hydrostatic equilibrium occurs. When a single charged surface comes to equilibrium with an infinite reservoir of electrolyte solution, a cloud ofcounterions will form next to the surface with a thickness on the order of the Debye length, g-I = (ekT/87rCe2)l/2,

where e is the dielectric constant, kT is the thermal energy, e is the protonic charge, and C is the ionic strength. The ion concentration profiles within this counterion cloud will not be altered by the presence of another charged surface provided the clouds from the two surfaces do not overlap, or provided that Kh >> 1. The presence of a charge on the fluid elements gives rise to electrostatic body forces. At hydrostatic equilibrium, a force balance requires that V p e q = peqEeq,

[8 ]

where Peq is the space charge density and Eeq = -Vffeq is the electric field. The subscript "eq" is added to remind us that these relations Journalof ColloidandlnterfaceScience,Vol. 136,No. 1, April 1990

apply for an equilibrium double layer. The electric field depends on the distribution of charges according to C o u l o m b ' s law, which for continua is represented by Gauss's equation ~ 7 . Ee q = 47i-peq/e.

[9]

Eliminating p oa between Eqs. [ 8 ] and [ 9 ], then integrating from some arbitrary reference point outside the counterion cloud, where the pressure vanishes and Eeq = O, to some arbitrary point inside the cloud yields Peq = (E/87r)EeZq,

[10]

where Eeq is the magnitude of Eeq.

2.4 Translation of Charged Bodies 2.4.1 Velocity and pressure fields. Any relative motion between the charged surfaces perturbs the pressure and potential profiles established in the absence of motion; in particular, motion induces a streaming potential profile within the film between the surfaces. Whereas p and ¢ vary only along the normal to the surface in the absence of flow, both will also vary along the surface in the presence of flow. Since the fluid between the two surfaces but outside the clouds is electrically neutral, no forces act in a direction normal to the surfaces. Then, outside the clouds, we expect the pressure and streaming potential to be independent of z, as with flow in the absence of charge, so that p = po(x, y)

and ¢ = Co(X, y), which should apply for a range of z values which excludes those within several Debye lengths of either surface. When the counterion clouds are very thin, diffusion of the ions will rapidly establish equilibrium with the fluid just outside the cloud. Neglecting the slight difference in orientation between the z-axis and the local normal, equilibrium in the cloud means that Eq.

ELECTROHYDRODYNAMIC

[10] is still the increase in pressure over the reference state, but the new reference state is now chosen to be the fluid just outside the counterion cloud, but at the same x and y. Mathematically, this corresponds to a solution in which the pressure and potential profiles have the form

p ( x , y, z) = po(x, y) + Peq(Z)

[1 la]

~ ( x , y, z) : ~o(X, y) + ~oq(z).

[1 l b ]

In effect, we have assumed that the distribution of ions and charge within the cloud are not perturbed significantly by the flow. Then, including electrostatic body forces, Stokes' equation becomes ~z~72v = Vp + OeqXT~,

Peq = - ( e / 4 7 r ) d 2 ~ P e q / d

z2.

[12]

Substituting Eqs. [ 11 ] and applying the lubrication approximations, Stokes' equation reduces to I-LO2¥s/OZ 2 =

VsPo +

Peq~s~0

profile is determined such that fluid is conserved. Substituting Eq. [ 15 ] into Eq. [6 ] leads to the modified Reynolds equation for pressure V s " (h3VsPo)

= -6#(v2

[13]

after the contributions from XTpeqand p eq~t/eq are cancelled (see Eq. [ 8 ] ). Substituting Eq. [ 12 ] and integrating, subject to the boundary conditions of Eqs. [3] and at

z = -hi:

@eq = ~'1

[14a]

at

z = +h2:

~t%q = ~'2,

[14b]

[ 13 ] yields Vs = Vso + (E/47r#)

-

v1 ) . ( n 2 -

nl)

+ (3e~'/~r)Vs'(hXTs~bo), [16] where ~"-= (~'2 + ~'~)/2. In comparison with Eqs. [7a], the last term in Eq. [16] represents the contribution from the induced electroosmotic flow. 2.4.2 Streaming potential. Just as the pressure outside the counterion cloud (P0) arises to conserve fluid, the streaming potential profile (~0) arises to conserve charge. Continuity of charge requires (see Appendix A)

vs'f

where the local space charge density was assumed to be the same as without flow, which is known and given by Eq. [9] as

99

LUBRICATION

isdz : 0,

where is consists of just the x and y components of the local current density i. The local current density has contributions from both convection and conduction of charge, i = PeqV q- KEo,

[18]

where K is the specific conductance of the solution. Owing to the steep gradient of ion concentrations near the charged interface, one might also expect a contribution from diffusion of the ions. However, this contribution is neutralized by conduction through the equilibrium double layer (KE~q). Instead, Eq. [ 18 ] represents the result after the contributions from diffusion and conduction in the equilibrium double layer have been cancelled. Substituting Eq. [ 18 ] for the current and Eq. [ 15 ] for the velocity profile into Eq. [ 17 ] gives (see Appendix B)

[151

XTs. (hVs~o) = (e~/47rK#)Vs" (hVsp0)

where ~'iis the potential drop across the counterion cloud next to surface i, A~" is ~'2 - ~~, and vs0 represents the linear shear flow plus pressure-driven flow in the absence of charge (defined by Eq. [ 5 ]). The new term represents the electroosmotic contribution to flow generated by the induced streaming potential. As in the absence of charge, the pressure

- (~A~/47rKhZ)(vs2 - vsl)'Vsh.

× [~'2 - ~eq - ( M ) A / h l V s ~ 0 ,

[17]

[19]

Variations in the specific conductance within the counterion clouds have been neglected together with the electroosmotic contribution to the velocity profile. At the outer edge of the film, the streaming potential must vanish: asx 2+72~:

~P0--~0.

[20]

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100

BIKE AND PRIEVE

Equation [19 ] is to the streaming potential what the Reynolds equation is to the pressure: the solution to Eq. [ 19 ] gives the streaming potential in the film (when the pressure is known). When A~" = 0, Eqs. [19] and [20] integrate to Smoluchowski's equation for streaming potential, ~Po(X, y) = (E~/47rK#)po(x, y),

[21]

except that the proportionality between streaming potential and pressure applies locally as well as across the opposite ends of the film. More generally, we expect Eq. [ 19 ] to be satisfied by

~0(x, y) = ~ +

po(x,y)

~x~(~UYx(x, y) + AvYy(x, y))]. [22]

The first term represents the contribution from pressure-driven flow while the second term (in parentheses) represents the contribution from linear shear flow. 2.4.3 Electrokinetic contribution to the force. The streaming potential induced by

motion of the surfaces generates an electrical stress which contributes to the force felt by either body. To calculate this electrical contribution to the force, first consider a fluid element having charge density pe in an electric field E. The body force felt by this fluid element is p~E. The charge density is related to the electric field by Eq. [9], XT" E

=

47rpe/e ,

where we have dropped the subscript "eq" here since this particular equation applies whether or not the system is at electrochemical equilibrium. Thus, the body force on the fluid element can be written entirely in terms of the electric field, peE = (e/47r)(V. E ) E = ~7. SE,

is the Maxwell stress, E is the scalar magnitude of the vector E, and I is the identity tensor. The second equality above, in which the body force is related to the divergence of the stress, is a mathematical identity: no additional physics have been introduced (except the assumption that c is spatially uniform). By applying the Divergence Theorem to an arbitrary volume of fluid, it can be shown that n. SE is the local electric force per unit area exerted on the body for which n is the outward unit normal. Let F~ and F2 denote the external forces applied to bodies 1 and 2 to cause the sliding or squeezing motion described in previous sections. To calculate the force F2, consider any closed surface, A, which completely includes body 2 and completely excludes body 1. When mechanical equilibrium is achieved, the net force on the system (consisting of the region enclosed by A) must vanish F2+~An-Sda=O, where S is the total stress S = - I p + #[~Tv + (Vv) z] + ~

1 [EE - ~ E2I]

and da is the area of a differential element of surface A. For the present problem, it is convenient to choose A to be As + Ap, where As is the surface of a hemisphere and Apis a disk lying in the xy-plane having the same radius as the hemisphere (see Fig. 2). As the radius of the hemisphere increases, p, v, E, and the stress on As all tend to vanish, leaving F2 = --fA n. Sda. p

The outward normal to A v is - e z and - n . S becomes - n . S = izOvs/Oz + (e/47r)EzE

where SE = (~/47r)[EE - ( 1 / 2 ) E 2 I ] Journal of Colloid and Interface Science, Vol. i36, No. 1, April 1990

- [p + (e187r)E2]ez

ELECTROHYDRODYNAMIC LUBRICATION AS

/

/

/

/

/

\

\

\

I//

\\\

//

F2

\

\\

pulsion "electrokinetic lift." In general, there will also be an electrokinetic contribution to the "shear" force required to cause sliding motion. We will now explore the importance of the electrokinetic contributions to the force by means of two examples: (1) sliding motion between a sphere and a flat plate and (2) squeezing motion between two spheres. The first case is relevant, for example, to tribology, the capture of particles from moving fluids, and hydrodynamic chromatography; while the second is relevant to the rheology of concentrated dispersions.

FIG. 2. Dotted line indicates hemisphere on which a force balance is made. after neglecting IVvzl compared to IOvs/Ozl (the usual lubrication approximation). Substituting Eqs. [10] and [ l l a ] and E = Eeq + E0 (the gradient of Eq. [1 l b ] ) , - n . S becomes - n . S = #OvslOz - (e147r)(d~peqldz)Eo

3. EXAMPLE NO. 1: SLIDING MOTION BETWEEN A SPHERE AND A PLATE 3.1 Analysis Consider a sphere of radius R being pulled without rotation parallel to a stationary flat plate (the xy-plane) in the x-direction at a speed U. In this case hi (x, y) = 0

- [P0 + (e/87r)Eglez, where Ez has been replaced by - d~eq / dz and Eeq" E 0 vanishes (see Eq. [11 b ] ). Substituting Eqs. [ 5 ] and [ 15 ], then evaluating the result at z = 0, yields -n.

Slz

0 = (#/h)(vs2

-

Ysl)

- ( 1 / 2 ) ( h 2 - hl)V~p0 + (e2x~'/4~-h)V~ff0 - [P0 + (e/8~r)Eg]ez.

[231

The integral of Eq. [23] over the plane Ap yields the external force F2 which must be applied to body 2 to move it at the velocity prescribed by the boundary condition of Eqs. [ 3 ]. On the other hand, - F 2 can be thought of as the force exerted by the fluid on body 2. In particular, the last term gives the force of attraction or repulsion between the two bodies. Owing to the appearance of E 2 in this term, any streaming potential induced by the relative motion between the two bodies generates a repulsion between the bodies. We call this re-

101

and h2(x, y) = 6 + (x 2 + y 2 ) / 2 R

Vl = O

and

v 2 = Uex.

3.1.1 Force on neutral bodies. G o l d m a n et al. (14) determined the force for electrically neutral bodies using matched asymptotic expansions as 6 / R --~ O. In particular, the pressure profile for the inner expansion has the form: pi= (i.tU/R)(~/R)-3/2[pi(p) + O(b/R)]cosO,

[24a]

with p i ( p ) = 6oH-2 '

[24b]

where o, O, ~ are dimensionless cylindrical coordinates: p ~ r / ( R ~ ) 1/2 = [(x 2 + y 2 ) / R r l l / 2 0 ~ tan-1 ( y / x ) ~ z/6

[25a] [25b] [25c1

Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

102

BIKE AND PRIEVE

and

as r--~ 0: H ( p ) =- h/b = 1 + 02/2

[25d]

is the dimensionless profile of film thickness, where the superscript " i " has been added to denote the inner solution. Owing to the cos 0 dependence of the pressure profile, its integral over the surface A v will vanish, leaving x as the only nonzero component of F2. The xcomponent of Eq. [ 23 ] for this problem is S~zx]z=0 = ( # U / f ) { H -l - ( H / 2 ) × [ ( d p i / d p ) c o s 2 0 + p-lpisin2 0] }. Multiplying this stress by da = (Rb)odOdp and integrating over a circular disk of radius 0~ yields po~

H ( dpi

pi

For the pressure profile of Eq. [24], this integral becomes Fi2x = !~zcI~UR ln(1 + p ~ / 2 ) ,

[26]

which diverges as p~ ~ ~ . Goldman et al. reasoned that the inner expansion is valid only within a finite inner region, 0 ~< 0 ~< 0o~. Outside of the inner region, the pressure profile takes on a different form:

h ° --~ r 2 / 2 R .

[29a]

Thus the integral in Eq. [28 ] becomes singular at the lower limit as r~ ~ 0. To obtain the leading term in its singular behavior, only the asymptotic form of po as r --~ 0 is needed. This is easily obtained by matching the inner limit of the outer expansion for pO with the outer limit of the inner expansion for pi: as r--~ 0:

po__~ ~ ( R / r ) 3 .

[29b]

Substituting Eqs. [29] into Eq. [28] and integrating: as r~ --~ 0:

F~x = - ~ T r u U R

× ln(roo/R) + O [ ( 6 / R ) ° ] .

[30]

Now r~ is defined as the boundary between the inner and outer regions. Its value changes with 6 / R . Goldman et al. note that as 6 / R --~ 0:

ro~/R oc ( 6 / R ) k.

Using Eq. [25a] to relate p and r, it follows that p~ oc ( b / R ) k-1/2. Thus i f 0 < k < ½, then r~ ~ 0 and oo~ --~ oo as b / R --~ O. Neglecting unity in the argument of the logarithm of Eq. [ 26 ] as b / R --~ 0:

pO = ( # U / R ) [ p O ( r ) + O ( r / R ) ] c o s O. [271 Using this outer expansion, denoted by the superscript "o", instead of the inner expansion [24a ], the x-component of Eq. [ 23 ] multiplied by da = rdOdr and integrated over the outer region becomes

tt rco

h°

whereas Eq. [30] can be rewritten as F~x = - ~ ~r~UR [ In p + 1 In(b/R)] + O[(b/R)°]. Adding the inner and outer contributions to the force results in cancellation of the terms involving the unknown quantity (0oo), leaving

F~x = ~r~U fo~

×

Fi2x = ~Tr~UR In p~ + O [ ( b / R ) ° ] ,

2R \ dr +

rdr,

[281

where r~ is related to 0~ by Eq. [25a]. The differential equations describing the leading terms in the pressure and velocity profiles in the outer region satisfy boundary conditions corresponding to the sphere being in intimate contact with the plate ( 14): Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

F2x = ~TrtzUR I n ( R / b ) + O [ ( b / R ) ° ] . [31] Goldman et al. (14) confirmed this result by comparing it to the numerical solution of O'Neill ( 15 ). 3.1.2 Force on charged bodies. When the sphere and plate are charged, the sliding motion will also generate a streaming potential profile, whose form is given by Eq. [ 22 ]. In

103

ELECTROHYDRODYNAMIC LUBRICATION

terms of the inner variables defined by Eq. [25], the functional form for the streaming potential resembles that for pressure as given by Eq. [24a], 3/2[~'P(0 )

~Po = ( e U / 4 r c K R ) ( b / R )

[32]

+ A~B(o) + O(6/R)]cosO,

where we have dropped the superscript "i", since we will have no further need to distinguish between inner and outer expansions. When A~-= 0, this reduces to Smoluchowski's equation [21]. More generally, the first term inside brackets gives the contribution to the streaming potential from pressure-driven flow while the second term gives the contribution from linear shear flow. A differential equation for B ( O ) is obtained by substituting Eq. [32] into Reynolds' equation [19] for streaming potential, `£1B = -oH

-e,

asp~

B is finite,

oo:

B~0,

K = KZekT/247r2ai#,

[ 35 ]

where ai is an average of the hydrodynamic radii of the ions. For binary univalent electrolytes, ai is the harmonic mean of the cation and anion radii: a i I = ½(a~ 1 + a - l ) .

[33a]

Substituting Eqs. [24a] (dropping the superscript "i"), [32], and [35] into Reynolds' equation [ 16 ] yields

[33b]

`£3P = -6p + 9AX2Z

with boundary conditions given by as p ~ 0:

This problem was solved by numerical integration (see Appendix C). The result is shown in Fig. 3. In preparation for determining the pressure profile, we will express the conductivity in terms of the Debye length. For solutions of univalent electrolytes, both K and K2 are proportional to the ionic strength of the solution. If the specific conductance is written in terms of the hydrodynalrfic radii of the cations and anions (a+ and a_), then

× (Z`£1P + AZ`£1B),

[33c]

where

[36]

where `£1 and -£3 are differential operators defined by Eq. [ 34 ], =-- -

oH

n

-

.

P

[341

A =- 2 e k T a i / e 2

10° 10-1 10-2 -1-'~

10-3 0.3 0.2 10-4

10-5

10-2

10-a

10°

101

102

FIG. 3. Contribution of linear shear flow to the streaming potential between a sphere and a plate in sliding motion (B), compared with the contribution from pressure-driven flow (P). See Eq. [ 32].

Journal of Colloidand InterfaceScience, Vol. 136,No. 1, April 1990

104

BIKE AND

is the ratio of the average ion radius to the Bjerrum length, X --- (K~) -1 is the ratio of the Debye length to the minim u m distance separating the two bodies,

PRIEVE

p~ --~ ~ . The singularity can again be resolved by adding the contribution from the outer region. Applying the same procedure which led to Eq. [ 31 ] yields Fz~ = ~rr#UR In(R~6)[1 + O(X2)]

Z =- e ~ / k T

+ O [ ( 6 / R ) ° I + O(X2).

is the dimensionless average ~"potential, and A Z =- e A ~ / k T is the dimensionless ~"potential difference. For thin double layers, we seek the asymptotic solution as X --~ 0. On the basis of the form of Eq. [ 36 ], we anticipate the pressure profile will have the asymptotic form P(p; X) = eo(p) + X2pI(P)

+ O()k4) •

[37]

Substituting Eq. [ 37 ] into Eq. [ 36 ] and equating terms of like order in X: X°:

£3Po = - 6 p

[38a]

~2: A~3PI = 9 A Z ( Z f - I P o + A Z f . IB). [38b] Observing that Eq. [38a] is satisfied by Eq. [ 24b ], we conclude that the leading term in the pressure profile is that which applies in the absence of charge, whereas Pl gives the correction for electroosmotic flow through the film. The force F2 required to drag the sphere along the plate is again obtained by integrating Eq. [ 23 ] over the disk Ap. By use of Eq. [ 24a] for the pressure and Eq. [32] for the streaming potential, the x-component of the force becomes Fl2x = ~r#UR

- -~ 2DP

for the contribution from the inner region, where :D =- d / d p + p-i is a differential operator. Like the integral before Eq. [26], this integral is singular as

JournalofColloidandInterfaceScience,Vol. 136, No. I, April 1990

[39]

For thin double layers, the electrokinetic contribution to the x-component of the force is O(X 2) and is negligibly small. More interesting is the z-component of the force, which is obtained by integrating the zcomponent of Eq. [23 ]: S=lz-0 = -[Po + (e/8r)E2]-

[40]

Owing to the cos 0 dependence in P0 (see Eq. [24a]), the integral of the pressure over the disk Av is zero, and consequently neither the viscous nor the electroosmotic contributions to pressure contributes to the normal force. However, there is a nonzero contribution from the normal electric stress. Substituting Eq. [ 32 ] for the streaming potential and integrating over the disk, the normal component of the force becomes F2z

_(~3

U27rR ~p [t2fl(p)

k4~r ] 2K263 ¢u = + ~'A~'f2(o) + (A~)2f3(P)]pdp J

f l ( p ) =- p,2 + (p/p)2 f2(P) =-- P'B' + P B / p 2 f 3 ( p ) ~- B '2 + ( B / P ) 2

[41]

after the integration with respect to 0 has been performed, where the prime (') denotes differentiation with respect to p- The f ( p ) represent three contributions to the normal electric stress. Figure 4 shows the profiles of these three stresses, all of which decay to zero more rapidly than 0-34 as O becomes large. Consequently, the integral converges as the radius of the disk Ap becomes arbitrarily large. Numerical evaluation of this integral yields

ELECTROHYDRODYNAMIC LUBRICATION 102

/~

/q //f2

100

105

10-2 10-4 10-6

10-8 10q0

0

\\\

;

10-12 10-2

10"1

100

101

10 2

P FIG. 4. Three contributions to the normal electric stress. See Eq. [41].

2z= ( ) 2.E03840 2 + 0.1810~-~x~- + 0.0242(~x~-) 2]

m

[42]

This represents the force which must be applied externally to body 2 (see Fig. 2 ) to keep the bodies from moving apart. The force exerted by the fluid on the sphere is equal in magnitude but opposite in sign to that given above. In particular, the expression inside brackets can be shown to be positive or zero for all values of the two ~"potentials. Thus the z-component of the force exerted by the fluid on the sphere is always positive (repulsive) or zero, regardless of the relative signs or magnitudes of the ~"potentials, and so is called "electrokinetic lift." 3.2 Discussion

Alexander and Prieve (2) used the results of Goldman et al. (14) to infer separation distance (6) from measurements of translation speed of a freely moving sphere in linear shear flow along a plane. Do electrokinetic effects alter the relationship between speed and sep-

aration distance in this situation? Goldman et aL computed the speed in linear shear flow by solving three simpler hydrodynamic problems: they determined the force and torque on a nonrotating translating sphere in a quiescent fluid, on a nontranslating rotating sphere in a quiescent fluid, and on a stationary sphere in linear shear flow. To answer the question above, we must evaluate the electrokinetic effect in each of these three problems separately. Although a complete answer is beyond the scope of the present paper, we can address the first problem. For a non-rotating translating sphere, the force required to pull the sphere is given by Eq. [39 ]. The electrokinetic contribution to this force is O(X2), which is negligible compared to the viscous contribution. Thus, electrokinetic effects do not contribute significantly to the force required to translate the sphere without rotation. We have shown that this is also true for the second and third problems (16). Thus the relationship between translation speed and elevation of the sphere is not significantly altered by electrokinetic effects induced by the flow--at least when the double layer is thin compared to the separation distance. Journal of Colloid and Interface Science, Vol. 136,No. 1, April 1990

106

BIKE AND PRIEVE

Instead, the m a i n electrokinetic effect in sliding m o t i o n is a n o r m a l force which acts to push the surfaces apart even when the surfaces bear charges o f opposite sign. Since sliding does not produce any other force n o r m a l to the direction o f relative motion, we must gauge the significance o f electrokinetic lift by comparing it with other kinds o f forces such as double-layer repulsion or gravity. As an example, let's try to estimate the relative importance o f electrokinetic lift under conditions similar to those f o u n d in the experiments o f Alexander and Prieve (2), who report an a n o m a l o u s lift which increased with shear rate. While developing a h y d r o d y n a m i c technique to measure colloidal forces, they observed the m o t i o n o f polystyrene spheres ( 8 - 1 8 # m diameter) in linear shear flow o f glycerol/water mixtures very near a glass plate. While they observed lift in nearly pure glycerol, no lift was seen when the glycerol was diluted with water. In these experiments, the shear rates were on the order o f 5 to 10 s -l,

and the separation distances between the sphere and the plate were between 0.2 and 1.2 # m with a Debye length o f 25 n m in the nearly pure glycerol. Typical translational speeds o f the sphere along the plate were approximately 50 ~ m / s . Figure 5 compares the predicted electrokinetic lift on a 10-~m sphere (being dragged parallel to the plate without rotation) to double-layer repulsion and to the sphere's net weight when immersed in distilled deionized water. Pure water has the lowest conductivity, and therefore would p r o m o t e the largest electrokinetic lift, o f any aqueous solution. Yet, over the range o f separation distances encountered in the experiments, electrokinetic lift on the sphere in pure water (/z = 0.001 k g / m s) is at least an order o f magnitude weaker than the force due to its net weight or doublelayer repulsion. Also shown in Fig. 5 is the predicted electrokinetic lift on the sphere in hypothetical fluids having larger viscosities; all other prop-

10 6

10 3 ~

~

.

.

~

~

/ Gravity u b l e

~

Layer

~ k g / m - s

o-6

10 0

101

10 2

~c8 FIG. 5. A comparison of electrokinetic lift (the four parallel straight lines), double-layer repulsion (the curve), and gravity (horizontal line) as a function of sphere-plate distance for four hypothetical fluids having an ionic strength of 10 -4 rnol/m 3 but different viscosities. Values of other parameters used in this sample calculation include R = 5 ~zm, AU = 50 ~m/s, ~'2= 75 mV, ~'~ - 125 mV, E = 78.54, ai = 0.145 nm, and T = 298°K. Fluid properties for the lowest line (t~ = 0.001 kg/m s) coincide with deionized water, while the viscosity of the upper line (~t = 1 kg/m s) is that of pure glycerol. All forces have been scaled such that the net weight of the sphere equals unity. lournal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

107

ELECTROHYDRODYNAMIC LUBRICATION

erties are kept the same as those in pure water. Increasing the viscosity decreases the specific conductance (assuming the concentration and hydrodynamic radii of the ions are constant), thus increasing the relative importance of electrokinetic lift but not affecting the other forces. When the viscosity is comparable to that of glycerol ( 1 k g / m s), electrokinetic lift is predicted to be orders of magnitude stronger than the other forces. This suggests that electrokinetic lift can be important in fluids which have a much lower conductivity than aqueous solutions. These predictions also appear to explain the observations of Alexander and Prieve: electrokinetic lift would be more active in the nearly pure glycerol than in glycerol/ water solutions which have higher conductivity. Owing to a larger number of ionic carriers, the actual conductivity of the glycerol used in these experiments is more nearly equal to that ofdeionized water in the above example. Correcting the Debye length for the higher apparent ionic strength, the predicted electrokinetic lift turns out to be an order of magnitude weaker than gravity over the range of separation distances encountered. This casts doubt on whether electrokinetic lift is causing the phenomenon observed in this particular experiment. However, translation without rotation is but one of three hydrodynamic problems which must be solved in order to obtain the solution for a freely moving sphere. Moreover, we must also determine if the conditions of this experiment meet the criteria for the above predictions to be valid. Two major assumptions have been made to obtain the predicted values of the electrokinetic lift: the lubrication approximation and a thin equilibrium double layer. The lubrication approximation is essentially a perturbation expansion which becomes valid in the limit 6/R ~ O. Comparing the tangential force required to produce a given tangential speed without rotation through a quiescent fluid as computed by the lubrication approximation (14) to that computed numerically ( 15 ), one finds that the difference exceeds 10% when 6/

R > 0.1. For comparison, separation distances encountered in the experiments were in the range of 6/R between 0.04 and 0.3. Nonequilibrium double layers have been thoroughly investigated in the associated problem of electrophoresis of an insulating sphere through an infinite fluid (17). By comparison of the electrophoretic mobility computed using the thin equilibrium double layer approximation of Smoiuchowski with that computed numerically (18), one finds that the criteria for the two to agree to within 10% is KR > 25 c o s h ( Z / 2 ) . For a typical ~'potential, say ~" = 50 mV or Z = 2, this translates into KR > 40. While KR is about 180 in these experiments, K6 ranges from 8 to 50. Although it is not clear that K6 should be subject to the same criteria as KR, we do need to have K6 >> 1 since we have ignored a number of terms because they were O[(Kh) -1] compared to others which we retained. In summary, neither the lubrication approximation nor the thin equilibrium doublelayer approximation are clearly applicable for all of the conditions encountered in the experiments showing the anomalous lift. Until we relax these approximations and solve the remaining two hydrodynamic problems, we will not be able to make a compelling test as to whether electrokinetic lift is responsible for the anomalous behavior observed. 4. EXAMPLE NO. 2: SQUEEZING M O T I O N BETWEEN T W O SPHERES

4.1 Analysis Consider two spheres of radii Rl and R2 moving apart at speed A W = W2 - WI. In this case, h i ( x , y ) = 6i + ( x 2 Av y2)/2Ri,

i = 1, 2,

and ¥i

:

W/ez.

Hence A W > 0 if the spheres are moving apart and A W < 0 if they are moving together. Again, cylindrical coordinates (r, 0, z) are more convenient. The total thickness of the film is Journalof Colloidand InterfaceScience, Vol. 136,No. 1, April 1990

108

BIKE AND PRIEVE

h ( r ) = hi + h2 = 6 + r2/R, where R is the harmonic mean of R~ and R2: R-I

= ½ ( R I 1 q- R ~ - I ) .

Since vi has only a z-component while 7sh has only an r-component, the second term on the right-hand side of Reynolds' equation [ 19 ] for streaming potential vanishes, leaving the streaming potential to be given by Smoluchowski's equation [ 21 ], even when the two spheres have different ~"potentials. Thus the streaming potential and pressure profiles are mathematically similar. In terms of the dimensionless variables defined by Eqs. [25a][25c], these profiles have the form Po = - ( # , S W / R ) ( b / R ) - 2 [ p ( P ) + O(6/R)] [43a] ~o = - ( ¢~/ 4~rK# )(IzA W / R )( 6 / R ) -2 × [P(p) + O ( 6 / R ) ] .

[43b]

after applying Eq. [ 35 ] to relate the conductivity to the Debye length. By contrast, the pressure is O[(6/R) -2] according to Eq. [43a]. Hence the electrical stress is higher order in 6 / R and can be neglected. Integrating the pressure over the disk Ap yields the force F2z = 37r#AWR(R/~) × [1 + 3AZZ)k2

-J- O ( ) k 4 ) ]

[46]

which must be externally applied to body 2 to cause the squeezing motion. Note that this squeezing problem was also solved by Goddard and Huang (9) in the context of the primary electroviscous effect. However, their analysis becomes valid in the limit of small potentials, and the resulting expression for the force is in the form of an integral which must be evaluated numerically. Our result, while applicable only for thin double layers, is not limited to small potentials and can be written in terms of elementary functions.

Reynolds' equation [ 16 ] for pressure becomes ~ 3 P = - 1 2 + 9AZ2?~2~lP,

[44]

where

and H ( p ) = 1 + p2. Note that this expression for H(p) differs from that in Eq. [25d], although H still represents the ratio of the local film thickness to the minimum film thickness. The solution to Eq. [ 44 ] is P(p) -

1

6AZ2)t 2 In( 1 - 9AZ2X2H -2) [45a]

= 3H-2 + 2--]AZ2)~2H-4 + O(X4).

[45b]

The z-component of the stress on Ap is given by Eq. [23] as Po + (e/87r)E02. Using Eq. [43b], the normal electric stress is

4.2 Discussion Figure 6 compares the total pressure profile for squeezing flow between two spheres with that which would be calculated if electroviscous effects were neglected. The total pressure was calculated from Eq. [45a] taking 2, = 0.1, Z = 2, and A = 1. Although the two agree very well as p --~ oo, electrokinetic effects account for 19% of the total pressure near p = 0. Once integrated over the disk Ap, the resulting force has a 6% contribution from electroviscous effects. Note that, according to Eq. [46 ], electrokinetic effects always increase the squeezing force in absolute magnitude and thus act to increase the apparent viscosity of the fluid. Although negligible for X ~ 1, this electrokinetic contribution to the pressure profile might be significant compared to viscous contributions for larger X in sheared suspensions (9). SUMMARY

8--~E°2 =

27~r3K265 \ d o ]

= O[(6/R)-~X 4] Journal of Colloid and Interface Science, Vol. 136,No. 1, April 1990

A general lubrication analysis is presented which predicts the velocity, pressure, and

ELECTROHYDRODYNAMIC

3

0

LUBRICATION

109

P

1

2

3

FIG. 6. Importance of electroviscous effects in squeezing flow between two spheres. Comparison of total pressure (or streaming potential) profile P(p) given by Eq. [45a] with the leading term of Eq. [45b], in which electroviscous effects are neglected. Calculations were performed using A Z 2~ 2 = 0.04 (e.g., X = 0.1, A = 1, a n d Z = 2 ) .

electrostatic potential profiles in the film of electrolyte solution held between two charged bodies of arbitrary shape undergoing arbitrary translational motion. This analysis becomes valid when the thickness of the counterion cloud (the Debye length) is much thinner than the minimum film thickness, which in turn is much thinner than the smallest radius of curvature of either body. The main result of this analysis is the Reynolds equation [19] for streaming potential, which can be solved once the profile of film thickness is specified. In addition, a new Reynolds' equation for pressure was derived [16] which includes the effect of electroosmotic flow in the film. When the two bodies have equal ~" potentials, the local streaming potential is proportional to the local pressure and the proportionality constant is the same as in Smoluchowski's equation [ 21 ]. Once the streaming potential profile has been determined, the external force needed to cause the translation can be computed by integrating [23] over the z = 0 plane separating the two bodies.

In one example, we solved these equations for a sphere being pulled along a plane wall. When the sphere and plate are not charged, our solution is identical with that of Goldman el al. (14). For charged bodies the electrokinetic contributions to the pressure [37] and the component of the force which is parallel to the plate [39] are O(X2), where ~, is the ratio of the Debye length to the minimum film thickness. More important is electrokinetic lift, the component of the force normal to the plate [42], which always acts to push the sphere away from the plate regardless of the relative signs of the two ~'potentials. Electrokinetic lift is inversely proportional to the square of the specific conductance of the solution and its magnitude can be comparable to double-layer repulsion in nonnaqueous fluids. In a second example, we solved the Reynolds equations for squeezing of the film between two unequal spheres. The streaming potential is proportional to pressure. The eleetrokinetic contribution to the squeezing force is O( X2) and acts to increase the apparent viscosity of the film. Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990

1 10

BIKE A N D PRIEVE

APPENDIX A: DERIVATION OF 2D F O R M OF C O N T I N U I T Y E Q U A T I O N

In obtaining this result, we have also used Eq. [A2] to eliminate the boundary terms.

Conservation of fluid mass or charge requires

Application to Conservation of Mass

Op/Ot + ~7. N = 0,

[A1]

where p represents either mass density or space charge density and N represents the flux of mass or charge. Impenetrability of the surfaces give rise to the following boundary conditions at f = 0 :

ni'(N-pv)=0,

f
where f dz is the definite integral evaluated on the interval ( - h i , h2) and pi is the density evaluated next to surface of body i. Any squeezing motion will cause hi and h2 to vary with time. In addition, when the surfaces are inclined, sliding motion causes hi and h2 to vary with time for a fixed x and y. By use of the Chain Rule one can show in general that (Ohi/Ot)x,y = - n i ° v i.

f V. Ndz = V,. f N~dz nl

° NI

--

n2" N2.

[A5]

Substituting Eq. [A4] into Eq. [A3], adding the result to Eq. [ A5 ], then using Eq. [ A 1] to set the result equal to zero, yields

(O/Ot)f odz+V~'f N~dz=O.

By use of Eq. [A4], the 2D continuity equation for mass becomes

v~.fvsdz=nl.vl

+ n2" v2.

[A7]

Application to Conservation of Charge When applied to conservation of charge, p in Eq. [A6] is the space charge density and N is the current density i. Overall electroneutrality of the equilibrium double layer requires that f

p d z = - - ( f f l ~- O'2),

where ai is the surface charge density borne by body i. Assuming the surface charge densities are uniform over the two bodies and constant in time, the derivative in Eq. [A6] vanishes, leaving

[A4]

Decomposing the flux into components normal to and tangent to the xy-plane, then applying Leibnitz' rule to the integral of the second term in Eq. [A1] yields

--

Oh/ Ot + ~7~.f vsdz = O.

[A2]

for i = l, 2 w h e r e f and ni are defined by Eqs. [ 1]. The 2D form of the continuity equation is obtained by integrating Eq. [ A 1] across the film and then switching the order of integration and differentiation. Applying Leibnitz' rule to the integral of the first term yields

- 02(Oh2/Ot) - Ol(Ohl/Ot),

When applied to conservation of mass, p in Eq. [A6] is the mass density and Ns is ors. When applied to an incompressible fluid, o is constant and can be divided out, leaving

[a6]

Journal of Colloidand InterfaceScience, Vol. 136,No. 1, April 1990

XTs.f isdz = 0. d

[A81

APPENDIX B: DERIVATION OF EQ. [19]

The integral appearing in Eq. [ 17 ] can be decomposed into contributions from conduction and convection:

f isdz = -(V~Po) f Kdz + f pcVsdZ. [B1] The specific conductance K differs from the bulk value only inside the counterion clouds. When the clouds are thin compared to the width of the gap, this variation in K will con-

ELECTROHYDRODYNAMIC

tribute only O[(Kh) -1] to the integral and can consequently be neglected. Thus, the first integral on the right-hand side of Eq. [ B 1] is

f Kdz = Kh + O[(Kh)-l]. Using Poisson's Eq. [12] for the space charge density, the second integral on the right-hand side of Eq. [B1] can be integrated by parts:

LUBRICATION

of the asymptotic behavior of this function at large or small values of its argument, which can be determined analytically. Keeping only leading terms in the coefficients as p --~ 0, Eq. [ 33a ] becomes

B" + p-IB' - p-2B = - p whose general solution is asps0:

oe,,vsaz

OZ

Vs

"J hi

-hx

F

02v~ dz]

-- J-h1 ~eq OZ---~

On surface z = - h i , we have ¢/eq = ~'~ and (e/ 4~r)(d~peq/dz) = -tTl whereas on surface z = h2, we have ~Peq = ~'2 and (c/47r)(d~eq/dz) = +~2. Moreover, since ~beqvanishes over most of the range of the remaining integral, the remaining integral is O[(Kh) -~] compared to the boundary terms. Ignoring this contribution, Eq. [ B 1] becomes

f isdz = -KhVsCJO + alvsl + O'2Vs2 + (~/47r)[~2(OVs/OZ)2

-- f l ( O V s / O Z ) l ] . [ B 2 ]

By ignoring the electroosmotic contribution to the velocity profile along with the other O[(Kh) -~] terms, the shear rates at the two surfaces are calculated directly from Eq. [ 5 ]:

(Ovs/Oz)l = (v~2 - Vsl)/h + (h/2g)Vspo [B3a] ( O V s / O Z ) 2 = (Vs2 -- V s l ) / h

- (h/2#)Vspo.

[B3b]

Finally, we obtain Eq. [ 19 ] by substituting Eqs. [ B2 ] and [ B3 ] into Eq. [ 17 ]. Since the charge density and velocity of either body are uniform, we have taken Vs" (aivsi) = 0 for i = 1, 2.

111

B=

Clp i +Czp+O(p3).

In the opposite extreme as p ~ oo, Eq. [33a] becomes

B" + 3p I B ' - p - 2 B = - 8 p -5 whose general solution is - 4 p 3 + C3p-l-~ + C4p -1+v2. To meet boundary condition [33c], we must select C4 = 0, leaving as p ~

Go:

B -- - 4 p 3 +

C3p-l-~.

To solve Eqs. [33] for arbitrary O, an initial guess was provided for the value of C3 (say C3 = 1 ). Starting at p = 25, Eqs. [ 33 ] was integrated numerically in the direction of smaller p using the algorithm of Nordsieck (19). The corresponding value of C1 was evaluated as the limit ofpB(p) as p --~ 0. A second guess for C3 (C3 = 2) similarly leads to a second value for C1. Owing to the linearity of Eqs. [33], C~ should be a linear function of C3. This provides the basis for determining the value of C3 corresponding to C~ = 0 (CI must vanish in order to meet boundary condition [33b]). However, owing to truncation and round-off errors, a fourth trial was needed to obtain a tolerably small value for C1. Thus we determined that C3 = 3.010 leads to Cl = 0 and C2 -- 0.2356. Doubling the value of p at which integration begins changes C3 by less than 0.1%.

REFERENCES APPENDIX C: INTEGRATION OF EQ. [33]

Equation [33a] is an ordinary differential equation for B(p) which was solved by numerical integration. Our solution makes use

1. Pfieve, D. C., and Bike, S.G., Chem. Eng. C o m m . 55, 149 (1987). 2. Alexander, B. A.,and Pfieve, D. C., L a n g m u i r 3 , 7 8 8 (1987). Journal of Colloid and Interface Science, Vol. 136,No. 1, April 1990

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BIKE AND PRIEVE

3. Kharin, S., KolloidZh. 6, 51 (1940). 4. Elton, G. A. H., Proc. R. Soc. London A 194, 259 (1948). 5. Booth, H., Proc. R. Soc. LondonA 203, 533 (1950). 6. Chan, F. S., and Goring, D. A. I., J. Colloidlnterface Sci. 22, 371 (1966). 7. Russel, W. B., J. FluidMech. 85, 209 (1978). 8. Hinch, E. J., and Sherwood, J. D., J. Fluid Mech. 132, 337 (1983). 9. Goddard, J. D., and Huang, L.-C., in "Advances in Rheology" (Mena, B., Garcia-Rejon, A., and Rangel-Naxaile, C., Eds.). Elsevier Science, Amsterdam, 1984. 10. Chan, F. S., Blachford, J., and Goring, D. A. I., 9.. Colloid Interface Sci. 22, 378 (1966).

Journalof Colloidand InterfaceScience, Vol.136,No. 1, April1990

11. Russel, W. B., J. ColloidlnterfaceSci. $5, 590 (1976). 12. Cox, R. G., Int. J. Multiphase Flow 1, 343 (1974). 13. Cameron, A., "The Principles of Lubrication." Wiley, New York, 1966. 14. Goldman, A. J., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 22, 637 (1967). 15. O'Neill, M. E., Mathematika 11, 67 (1964). 16. Bike, S., PhD dissertation, Carnegie Mellon University, 1988. 17. Derjaguin, B. V., and Dukhin, S. S., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 7, pp. 273-335. Wiley, New York, 1974. 18. O'Brien, R. W., and White, L. R., J. Chem. Soc. Faraday Trans. 2 "/2, 1607 (1978). 19. Nordsieck, A., Math. Comput. 6, 22 (1962).