A micromechanical derivation of Fick's law for interfacial diffusion of surfactant molecules

A Micromechanical Derivation of Fick's Law for Interfacial Diffusion of Surfactant Molecules H. B R E N N E R Department of Chemical Engineering, University of Rochester, Rochester, New York 14627 AND

L. G. L E A L Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Received September 27, 1976; accepted October 13, 1977 A theory is presented for the diffusion of surfactant molecules, modeled as noninteracting Brownian spheres in proximity to, or straddling, the interface between two immiscible fluids. The spheres are assumed to be physicochemically inert except for the existence of short-range forces, which may be attractive and/or repulsive with respect to the interface. The number density of such particles is assumed small. Two alternative views of the diffusion process are presented; one, termed microscopic, in which variations in concentration are resolvable down to the scale 1 imposed by the short-range attractive or repulsive forces between the particles and the interface, and the other, termed macroscopic, in which variations are only discernible at the much larger scale, L, of gradients in the bulk surfactant concentration distribution, which are assumed to exist parallel to the interface. It is demonstrated that a rigorous theory can be developed, at the microscale, for prediction of the concentration profiles and diffusive flux of surfactant molecules parallel to the interface using only the well-established Stokes-Einstein theory of bulk-phase Brownian diffusion, and low Reynolds number hydrodynamics for the motion of a torque-free particle in close proximity, or straddling, a fluid-fluid interface. On the other hand, a macroscopic description of the same phenomena requires the introduction of new concepts, such as "adsorption" and "surface diffusion," which are specifically associated with the interface, as seen from the macroscale of O(L). In the present paper, constitutive relationships for these macrosurface processes are derived in a rigorous manner from the more fundamental and complete microscopic description of the system. It is shown that Fick's law is applicable to surface diffusion, with the driving gradient based on a surface-excess concentration for surfactant particles. Furthermore, the surface diffusion coefficient is found to depend upon the Stokesian hydrodynamic resistance of a torque-free sphere translating parallel to the interface, as a function of the distance of its center from the interface, and upon the "adsorptive-potential" energy function which tends to cause the particles to accumulate there. Numerical values await the solution of the low Reynolds number hydrodynamics problem thereby posed, as well as the acquisition of knowledge relating to the potential energy function. Two examples of such potential energy functions are discussed for Brownian particles, one based upon a difference in area-specific, surface-free energies (i.e., solid/ liquid interfacial tensions) for the two fluids, and the other on a simple density differential between the particle and the two fluids. Simple geometric models of surfactant molecules are also discussed in the context of potential energy functions for interaction between the molecules and the interface. The possibility of using such models in the analysis of other equilibrium and surface transport phenomena is pointed out.

t r a n s p o r t o f s u r f a c t a n t m o l e c u l e s in proximity to fluid/solid and fluid/fluid interfaces. The first (1) considered the steadystate diffusion of molecules, modeled as

I. INTRODUCTION

This paper is the second in a series concerned with the diffusive and convective 191

Journal of Colloidand Interface Science, Vol.65, No. 2, June 15, 1978

0021-9797/78/0652-0191 $02.00/0 Copyright© 1978by AcademicPress, Inc. All rightsof reproductionin any formreserved.

192

BRENNER AND LEAL

spherical Brownian particles in a viscous, Newtonian solvent, moving in close proximity to a plane solid surface, when shortrange attractive and repulsive forces exist between the particle and wall, and the concentration of particles is small. The present contribution is the natural extension of this work to diffusive transport at a fluid/fluid interface in the presence of a bulk concentration gradient maintained parallel to the interface. Following the precedent of our earlier paper (1), we adopt the micromechanical point of view in which a rigorous, predictive theory is developed for a simple, but physically realizable, model system. In particular, we assume the existence of two widely differing length scales; a macroscopic scale of O(L), which characterizes the bulk solute concentration gradient that is maintained parallel to the interface, and a microscopic scale of O(l), which is typical of the equilibrium concentration profile normal to the interface in the presence of the short-range interaction forces. Two distinct descriptions of the system are then possible, which we shall term microscopic and macroscopic. In the former, variations on the scale O(l) are resolvable, and the presence of an "adsorptive" interaction potential is seen as contributing to a nonuniform solute concentration distribution normal to the interface. This microscopic point of view is the more fundamental of the two that have been mentioned, and a completely rigorous theory for the various equilibrium and transport processes requires only the classical bulk-phase theories of low Reynolds hydrodynamics, and Brownian motion, modified as required for the presence of the interface. In contrast, when one adopts the macroscopic point of view, characterized by the scale L, the w h o l e " surface" region of thickness O(l) becomes indistinguishable from the interface between the pure bulk fluids, and it is necessary to introduce " n e w " concepts such as adsorption, or surface diffusion, which are specifically associated with the Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

interface (as distinct from the bulk phases). Although these new macrosurface phenomena require the introduction of additional surface constitutive laws, the macroscopic point of view is not without motivation, for it is this approach which is normally adopted for continuum-mechanical treatments of transport processes in the presence of a fluid/fluid interface. Most previous investigators have simply guessed at the appropriate (and necessary) constitutive forms. In the micromechanical approach which we have adopted, the constitutive relationships for these macrosurface processes are derived in a rigorous manner from the more fundamental and complete microscopic description of the system, albeit for a simple, model configuration. The present paper is concerned only with diffusive transport, in the absence of any bulk motion in the two fluid phases. It is shown, as suggested above, that a self-consistent macroscopic view of this process actually requires that the overall flux of solute particles parallel to the interface be separated into "surface" and " b u l k " contributions. Although Fick's law is often assumed to apply to interfacial "surface" diffusion, at least at low surfactant surface concentrations, no satisfactory theoretical demonstration of this fact appears to exist. Even empirical experimental evidence for its applicability appears to be nonexistent because of difficulties encountered in accurate measurements of "surface flux" and "surface concentration gradients" (2, 3). However, the present micromechanical theory provides a rigorous demonstration of the applicability of Fick's law to surface diffusion for the simple model system which we have studied, as well as a definite formula relating the surface diffusion coefficient, Ds, to the micromechanical parameters of the system. Although the model system is, perhaps, too simple for direct application to systems of technological interest, it does retain at least some of the essential physical features and, at a minimum, will therefore

INTERFACIAL DIFFUSION serve as a testing ground, and source of general insight, for the development of phenomenological theories which will apply to these systems. A sequel (4) to this present paper will apply arguments, analogous to those of Einstein (5), for diffusive transport in a bulk fluid, to a derivation of Gibbs equation, and to the macroscopic constitutive relationship for diffusive transport at an interface between nonideal solutions. Subsequent contributions will treat nonspherical particles, where additional considerations of particle orientation lead to algebraic and computational complexities. However, the fundamental physical considerations advanced here remain essentially unaltered in this geometric generalization. II. MICROSCOPICDESCRIPTION-FUNDAMENTAL EQUATIONS We begin, as suggested in the Introduction, with a description of the model system, and an analysis of its behavior at the microscopic level. The system in which we are interested consists of two immiscible solvents, and a surface-active solute which is diffusing under the action of a bulk concentration gradient maintained parallel to the interface. Locally, the interface may be regarded as planar, and we thus introduce a system of rectangular Cartesian coordinates (x,y,z), as illustrated in Fig. I, with the plane y = 0 corresponding to the undisturbed interface between the two solvents. The solute molecules are modeled as rigid spheres of radius a, which undergo both translational and rotational Brownian motions. The two solvents are viewed as Newtonian fluids, i.e., as continua with respect to the microscale, l = O(a). Furthermore, the transition in bulk properties from pure fluid (solvent) " 1 " to fluid " 2 " is assumed to occur over a length scale which is small when compared to the particle radius a, so that the interface between pure solvents may be approximated as a singular surface, even from the microscopic point of view.

193

/-dS Fluid "1" Interface,

.y

i

~ X

t

,

/Brownian

'

,,, ~_

~,,~.~particles

FIG. 1. Definition sketch. The particles are acted on by a body force, F, which describes the physicochemical attraction to, or repulsion from, the interface. This force is assumed to act in a direction normal to the interface, and to derive from a potential energy function V = V(y)(-oo < y < oo), per particle. Thus, F = -VV

= -ndV/dy,

[2.1]

where n is a unit normal vector in they direction. It is the presence of this interaction potential between the particle and interface which distinguishes the present problem from ordinary "hindered diffusion," for which the appropriate theoretical description is well known. At a later stage in the development we shall consider two specific illustrative examples of interaction potentials which tend to cause particle migration toward the fluid-fluid interface. For the present, however, it is advantageous to leave the functional form of V unspecified. In order to provide a focus for subsequent discussion we simply note that the general appearance of the potential energy function for surfactant m o l e c u l e s is likely to be of the form sketched in Fig. 2, wherein V varies between the two bulk limits, Vo~1 and V=2. At any instant, Brownian particles will generally be found which are either wholly immersed in one of the fluids, or else straddling the interface, and " b a t h e d " by both fluids simultaneously. For convenJournal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

194

BRENNER AND LEAL V

lot: I!

from the Fokker-Planck to the Smoluchowski equation, i.e., to a Fickian form for the diffusion flux in the direction normal to the interface. Thus, in the general case, we may write

c = C(x) exp[-V(y)/kT]

[2.2]

jz = GDll(y) exp[- V(y)/kT],

[2.3]

and

L=L=0, FIG. 2. Idealized potential energy function for a surfactant molecule. ience, the center of each sphere will be chosen to serve as its "locator point," specifying the instantaneous position of the particle in space. The independent variable y then denotes the distance of the sphere center from the undisturbed interface. It is both unnecessary and inappropriate to attempt to distinguish formally between the configurations with the particle in fluid" 1" or fluid" 2" or at the interface, and our choice of notation will generally be such as to avoid affices peculiar to fluid 1 or fluid 2. Instead, wherever possible, our terminology will be chosen to emphasize the continuous variation in properties associated with the Brownian particles in the region -oo
with j the microscale flux density vector of sphere centers, and c = c(x,y,z) the singleparticle probability density, or equivalently (for a dilute suspension) the number density of sphere centers at a point. C ( x ) is a linear function of x, given by

C ( x ) = - G x + constant,

[2.4]

in which

G-

--dxdC~--,= = constant.

[2.5]

The scalar Dll(Y) is the diffusion coefficient for torque-free translational motion of the sphere center parallel to the interface. Owing to "boundary effects" engendered by the presence of the interface in proximity to the sphere, this diffusivity is dependent upon the distance y of the sphere center from the interface. As per the subsequent discussion of Section IIIC, the calculation of D~(y) can be made via the Nernst-Planck-Einstein relation,

Mll = Dii/kT,

[2.6]

by solving the linearized Navier-Stokes equations for torque-free motion of the sphere in either of the two fluids, or straddling the interface. Here M~ is the hydrodynamic mobility for translation of the sphere center parallel to the interface, and kT is the Boltzmann temperature. In view of the geometric symmetry of the sphere and plane interface configuration, M~(y) is related to the full mobility dyadic M by the expression M = nnM.(y) + (I - nn)Mll(y),

[2.7]

INTERFACIAL DIFFUSION

with I the dyadic idemfactor. The most important feature of this asymptotically exact microscale description (i.e., Eqs. [2.1]-[2.7]) is that there is no reference to "surface" or "interface" transport processes; in effect all diffusion is "bulk diffusion," albeit with corrections to the hydrodynamic resistance formulas to account for the presence of the interface. From [2.2] it may be seen that the surfactant concentrations, c~(x) (i = 1,2), prevailing at large distances, ly [ from the interface are, respectively, def c~ 1 = lim c = C ( x ) y---*+oo

× exp(-V®l/kT),

[2.8a]

def c~2 = lim c = - C ( x ) y---*-- 0o

x exp(-V~2/kT).

[2.8b]

The ratio of these bulk concentrations is cooVco~2 = K,

[2.9]

in which K def exp[_(V~ 1 _ V 2 ) / k T ]

[2.10]

represents the thermodynamic equilibrium phase-distribution coefficient. A value of K other than unity clearly arises from the fact that the interaction potential function possesses different values in the two fluids. We shall discuss this point, in the context of a specific potential function, in Section IV. Equations [2.1] to [2.7] apply, irrespective of whether a Brownian particle is wholly immersed in one or the other of the fluids (ly[ > a), or straddles the interface < a). At the present "microscopic" level of description, the dependent variables j, c, and D are continuous functions o f y in the range - ~ < y < oo. That c is a continuous function o f y can, perhaps, be seen most readily by adopting a single particle, rather than a multiparticle, view of the Brownian motion, and returning to the initial definition of c as the probability density for finding the sphere center at the position

(lyl

195

(x,y,z). It would be unnatural to attempt to write down equations of the forms [2.2] and [2.3] separately for fluids 1 and 2, since a particle straddling the interface cannot then be unambiguously assigned to either fluid. 1 Only at a macroscopic level of description, where the particles can be viewed as points, would it appear appropriate to distinguish between the two fluids by affixing subscripts 1 and 2 to the dependent variables. III. M A C R O S C O P I C D E S C R I P T I O N

A. Adsorption

We now turn to the macroscale point of view, characterized by the length scale L of the bulk concentration gradient, which is the normal domain of continuum-mechanical descriptions of interfacial phenomena. As we have noted earlier, the whole region of extent l on either side of y = 0, which encompasses the variations in concentration associated with V (cf. Fig. 2), is indistinguishable from the undisturbed interface y = 0 when examined from this macroscopic viewpoint. In particular, the solute (particle) concentration will appear to be independent of y, with values c~ 1 and c~ z on the two sides of the interface, and a discontinuity between these two values at y = 0. The depth of the potential well (cf. Fig. 2), or equivalently, the magnitude of the concentration peak (cf. Eq. [2.2]), will not be evident; in fact it is only through an overall solute mass balance that any distinction would be apparent between an active and an inert interface for a particular chemical species. From the microscale point of view, there is neither a need, nor an operationally satisfactory way, for distinguishing "surface" and " b u l k " phenomena. Macroscopically, on the other hand, one must ascribe i Of course the center of the sphere can be unambiguously assigned to a given fluid. H o w e v e r , the choice o f the sphere center as the locator point for the particle is purely a matter o f m a t h e m a t i c a l convenience. In particular, it cannot affect the physics o f the basic problem. Journal of Colloid and Interface Science, Vol. 65, N o . 2, J u n e 15, 1978

196

BRENNER AND LEAL

all of the " e x c e s s " solute, which "accumulates" near the interface due to the particle/ interface potential, to the interface itself, i.e., y = 0. The number of solute particles which are assigned to the surface per unit of surface area defines a surface concentration of "adsorbed" surfactant, and this is related to the corresponding bulk concentration through an adsorption coefficient. It is evident from this discussion (and we have presented a much more detailed discussion of the same point in Ref. (1)) that "adsorption" is a macroscopic concept, which may be viewed as playing the role of reconciling the microscopic and macroscopic descriptions of the concentration profiles near a phase boundary. To determine the surface concentration of "adsorbed" species, and thus also the adsorption coefficient, we carry out the solute mass balance which is implied in the preceding paragraph. Thus, referring to Fig. 1, the total number of sphere centers contained within the right rectangular prismatic volume element of cross-sectional area dA = dxdz and height 2L is

dN = f cdV = dAC(x) x

exp[-V(y)/kT]dy.

[3.1]

L

For an inert interface, on the other hand, the number of sphere centers in the same volume element would be

d N ' = f c®dV=dA[c=2(x) xf~Ldy+c®1(x) I~cly ] . [3.2a] With use of [2.8a] and [2.8b], this may be written equivalently as

dN' = dA C( x ) [ e x p ( - Vo~2/kT) ×

dy + exp(-V~l/kT)

The surface " e x c e s s " (6-9) of Brownian particles, dNs, in the volume element is defined as z

dNs def lim (dN - dN').

This quantity represents the number of solute particles"assigned" to the interfacial area element dA. By definition, the quantity

Fs de=fdNs/dA

-L Journal o f CoUoid and Interface Science, Vol. 65, No. 2, June 15, 1978

[3.4]

represents the (local) number of "adsorbed" surfactant molecules per unit of superficial interfacial area; that is, it is the surface " e x c e s s " areal number density. Contrary to a common misconception, it is evident that F~ does not generally correspond to the number of particles actually physically straddling a unit area of interface. Rather, it simply represents the number of particles assigned to the unit area. Some of the latter number of molecules may merely be in proximity to the interface, but not actually intersected by it. This point of view is consistent with the general principles of continuum mechanics (10), w h e r e - - a t least when dealing with volumetric quantities--one has to distinguish between the true contents of some volume and what is merely assigned to it. Upon combining the preceding results it follows that 3 rs(x)

= KaC(x),

[3.5]

This calculation follows the original choice o f Gibbs (cf. Refs. (6, 9)) in defining surface-excess quantities as the difference between the true contents and those that would be calculated by imagining that bulk values prevailed right down to the interface itself. a In view of [2.8] one could write either Fs = Kalc® 1 or F s = Ka~c® ~, where Ka ~• d=e f Ka exp(V®i/kT)

dy . [3.2b]

[3.3]

L--~oo

(i = 1, 2)

represents the adsorption coefficient based upon either o f the bulk concentrations, c®l, prevailing in the respective fluids on each side of the interface.

INTERFACIAL

where the position-independent constant,

Ka = I° [exp(- V/kT) - e x p ( - V=~/kT)]dy J-

+

~o

[exp(- WkT)

- exp(-V~l/kT)]dy,

[3.6]

is an adsorption coefficient, relating the local surface density to the local bulk density of surfactant molecules in its proximity. The linear adsorption isotherm [3.5] corresponds to Henry's law. Each of the preceding integrands tends to zero as lYl ~. The rate at which the integrands vanish for large [y [ is assumed sufficiently rapid to secure convergence of the respective integrals. Equation [3.6] may be written more compactly and suggestively as

197

DIFFUSION

Similarly, the diffusive flux of particles (molecules) parallel to the fluid/fluid interface will be greater than that calculable for a physicochemically inert interface by an amount which depends on the existence and strength of the attractive force potential. This " e x c e s s " diffusive flux must be assigned to the interface as "surface diffusion" in order to reconcile the macroscale description of the transport process with that rigorously calculable at the microscale. The surface "excess" diffusion flux is calculated in the same manner as that used in the preceding section to determine the "excess" solute concentration for"adsorption." With reference to Fig. 1, the element of area lying normal to the x-direction is dS = dydz. Equation [2.3] then shows that the number, d/~/x, of sphere centers per unit time crossing the vertical surface of width dz and height 2L is

Ka = f ~ [exp{-V(y)/kT}

- exp{-V=(y)/kT}]dy,

[3.7]

where V=(y) is the discontinuous function,

step

W=(y) = Wa¢1 for 0 < y < 0%

[3.8]

= V = z for

x

D,(y) exp[-V(y)/kT]dy,

[3.10]

-L

0>y>-~.

Note that in this notation, [2.2] may be written in the form

c/c~(y) = exp[-{V(y) - V=(y)}/kT],

dNx = f jxdS = dzG

in the x-direction. The comparable transport rate, dN'x, that would be predicted based upon an inert interface is

dfV'z = f (jx)~dS

[3.9]

where c and ca pertain to the same value ofx.

= - f/'

Dij(y) ~dc=(x) dydz

=-L

B. Surface Diffusion It was shown in the preceding section that the concept of surface adsorption, in the context of surface-active molecules in a system of two immiscible fluids, can be viewed as a macroscopic construct, required to account for the accumulation of surfactant in the vicinity of the interface, l yl -- o(l), due to a particle/interface interaction force.

dx

[ dc=2 I~ D,,(y)dy - d z L dx L dx

Dil(y)dy

.

[3.11]

(i = 1, 2).

[3.12]

However, from [2.8],

dc~i/dx = (dC/dx ) e x p ( - V~i/kr)

Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

198

BRENNER AND LEAL

With use of [2.5], Eq. becomes

[3.11] thereby

in [3.7],

Ds = f~_®Dtl(y) [exp{-V(y)/kT} dN'x = dzG exp(- Vo~2/kT)

DLl(y)dy L

- exp{-

V~'/kT)

+ exp(-

Dll(y)dy .

[3.13]

The surface-excess surfactant transport rate, (dNs)x, which is to be assigned to the interface is thus (dNs)x def lim (dNz - d~/'~).

[3.14]

L--*oo

The local surface excess flux density, (Js)x def

(dfVs)jdz,

[3.15]

represents the number of "adsorbed" particles crossing a line of unit length drawn in the interface and lying normal to the x-direction at the point (x,O,z). Explicitly,

(J~)z = GI{ I° DII(y)exp(-V/kT)dy - exp(- V=2/kT) I°oo Dll(y)dy t o

+

Dll(y) e x p ( - V/k T)dy

- exp(-V=l/kT) j; D,,(y)dy}].

V=(y)/kT}]dy/

[3.16]

× I~o [exp{-

V(y)/kT}

- exp{-V®(y)/kT}]dy.

Since x denotes any arbitrary direction lying parallel to the interface, and since the sphere/plane configuration is transversely isotropic, 4 Eq. [3.18] can be written quite generally in the invariant, vector form, as = -DsVsFs, [3.20] in which Vs = (I - n n ) . V

G-

1 dF~

Ka dx

[3.17]

Introduction of this expression for G into [3.16], and subsequent use of [3.6], thereby yields

(J~)x = -OflFs/dx,

[3.18]

in which, in the succinct notation employed Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

[3.21]

is the surface gradient operator (11, 12). This represents Fick's law of diffusion in the interface, with Ds the surface diffusivity. This derivation simultaneously furnishes a plausible demonstration of the applicability of Fick's law to interfacial surface diffusion phenomena, as well as a prescription for calculating the surface diffusion coefficient D~ appearing therein from fundamental bulk parameters of the system. As in Ref. (1), beginning with [3.19], it appears possible to construct a proof of the "thermodynamic" inequality, Ds -> 0,

However, from [2.4] and [3.5],

[3.19]

[3.22]

by making use of the positivity of D~ for all y, and invoking physically plausible assumptions about the position dependence of the potential energy function V (cf. Fig. 2). However, we shall refrain from attempting 4 That is, all directions in a plane perpendicular to the y-axis are physically indistinguishable. Hence, all directions parallel to the interface are physically equivalent. This leads to the conclusion that the surface diffusivityis isotropic, and therefore a scalar quantity.

INTERFACIALDIFFUSION

199

a formal proof at this time, since in the ab- singular behavior 5 at y = O, the parameter sence of more detailed knowledge of the y- ym]a may be set equal to zero, and [3.23] dependence of Dll and V, especially their approximated as domains of monotonicity, such a proof Ds = Dil(0) [3.24] would be speculative. The mechanism advanced here of surface without incurring significant error. To the extent that the preceding assumpdiffusion in interfaces departs from the tions are correct, calculation of the surface traditional view, which envisions the difdiffusivity does not therefore require exfusion process as actually occurring in the plicit knowledge of the potential energy plane of the interface (13). However, in at least some plausible physical circumstances function. Rather, the surface diffusivity is the theory will now be shown to lend itself simply equal to the value of the bulk diffusivity for a sphere which is centered in to just such an interpretation. From [2.1] the adsorptive force on a the plane of the interface. This result might particle vanishes at that distance, Ym, say, have been anticipated intuitively on physiat which the potential energy function at- cal grounds. In such circumstances the tains its minimum value. As discussed in sphere centers are constrained to lie in the Section IV, and as is clearly true in general, interface, but are otherwise free to diffuse Ym represents an accumulation point for in this plane, animated by thermal molecular the sphere centers. Analogous to the re- motion. The mode of analysis employed in the marks made in Ref. (1), in connection with surface diffusion along solid surfaces, this derivation of [3.20] and [3.19] can obviously distance may be of the order of molecular be extended to nonspherical surfactant dimensions in some real situations; more- molecules by inclusion of particle orientaover, the shape of the V vs y curve may be tion (17-19) as an additional independent expected to display a deep, narrow mini- variable. These results will be reported elsemum in the neighborhood ofym, as sketched where. So far as we are aware, this represents in Fig. 2. In such circumstances the dominant contribution to the integral [3.19] arises the first ab initio demonstration of the from those points in the neighborhood Ofym. applicability of Fick's law to the interfacial Laplace's asymptotic integration scheme diffusion of adsorbed species, albeit for ideal solutions and, hence, low surface cov(14-16) then yields erages. The sequel (4) to the present paper Ds = Dll(ym). [3.23] extends this proof to more general situations. In words, the surface diffusivity is equal in value to the bulk diffusivity for a particle C. Discussion whose center is situated at a distance Ym Calculation of the surface diffusivity from from the inlierface. Since Ym can often be [3.19] requires, inter alia, knowledge of the expected to be of the order of molecular position-dependent bulk diffusion coeffidimensions, it follows that ym/a < l, and the cient Dtt for translational diffusion of the surface diffusivity, as calculated by (3.23), sphere center parallel to the interface. The corresponds to the ordinary bulk diffusivity latter can be calculated by considering the for a sphere which straddles the interface, with its center almost precisely in the interAt least, if the hydrodynamicresistanceis singular face. Assuming that the variation of hydro- when the sphere straddles the interface, it will be dynamic resistance with y for a particle no more singular at y = 0 than it is for any other straddling the interface does not manifest value, lY[ < a. Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

200

BRENNER AND LEAL

......

~rodius=o 2

~.

Fired I

,0,_~%

u

Fluid "2" FIG. 3. Translation of the center of a sphere parallel to a fiat interface.

purely hydrodynamic problems depicted in Figs. 3a and b. In each, the sphere center translates parallel to the interface with a prescribed velocity U and, at the same time, the particle rotates about the z-axis with an angular velocity II which is determined by the condition of zero net hydrodynamic torque. For simplicity, it is supposed that the interface is fiat, and that no interfacial tension forces act on the sphere in the direction, x, of its motion. The development of more accurate representations of the shape of the interface, or of capillary forces acting on the particle, is a difficult and unresolved problem, which requires much more extensive investigation than is possible here. At small Reynolds number, Stokes equations (20), describing the local hydrodynamic fields generated by the sphere's motion, are linear. Accordingly, it may be anticipated that the hydrodynamic force F on the sphere, in the x-direction and torque T (about the sphere center) in the z-direction will each be linear functions of U and fl, of the forms (21) F = -1~2(aKt U + a2KeD.),

[3.25]

T = -1~2(a2KeU + a~KrDO.

[3.26]

Here, /x is the viscosity, and Kt, Kr, and Ke are, respectively, the dimensionless, position-specific, translational, rotational, and coupling hydrodynamic resistance coefficients for motion of the sphere center. Each is necessarily of the functional form Ki - K~(y/a,lxl/lX2),

[3.27]

Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

and can be obtained numerically by solution (21-23) of the separate translational and rotational Stokes flow problems (assuming no difficulties arise from the "contact line" (24) problem). Putting T = 0 gives II = - ( K c / K r a ) U for the angular velocity of the translating sphere. Introduction of this into [3.25] yields [3.28]

F = -lz2aK, U,

with K~ = gt

--

Kr-lKe 2.

[3.29]

By definition, the mobility coefficient, M,, of the sphere for motion of its center parallel to the interface is M, d e f _ U / F =- 1/l~2aK,.

[3.30]

However, from [2.6], D, = kTMll,

[3.31]

D, = kT/lx2aK,.

[3.32]

whence It may be noted that Eqs. [3.23], [3.27], and [3.32] lead to the approximate functional form for the surface diffusivity, Ds =

kT (ym /~1) function , . /z2a a /~2/

[3.33]

Ifym/a ~ 1, as would often be the case, this expression reduces further to Os =

k T function(/Zl) .

t~za

[3.34]

\/zz/

Thus, under the conditions cited earlier in connection with [3.24], the surface diffusivity depends only on the hydrodynamic resistance to translational motion of a sphere which is centered at the interface, and moving parallel to it. In particular, Ds is independent of the details of the potential function, V(y). The more general form [3.19] combined with [3.31] permits the surface diffusivity to be calculated in any case from a knowledge of V(y), and the hydrodynamic resistance of a sphere mov-

INTERFACIAL

ing along, or in proximity to, the interface. The required fluid-mechanical calculations will be presented elsewhere. IV. I L L U S T R A T I V E E X A M P L E S O F "ADSORPTIVE" POTENTIALS

Subsequent papers in this series will furnish numerics, enabling the surface diffusivity to be calculated from [3.19]. Whereas the hydrodynamic portion of the problem required for the ultimate computation of D~ is relatively straightforward in principle, determination of realistic expressions for V, and concomitant numerical values for the phenomenological coefficients appearing therein, is rather more difficult (25). Accordingly, it is instructive to provide simple illustrative examples of interaction (force) potential functions which are relevant to Brownian particles in proximity to the interface. Though somewhat artificial in the context of the molecular body or surface forces acting on real surfactant molecules, such illustrative examples might nevertheless provide the basis for " m o d e l " surface diffusion experiments that could be performed with actual Brownian particles. Indeed, monodisperse polystyrene and latex particles in the Brownian size range, and having well-defined physical properties, are commercially available. Moreover, accurate bulk Brownian motion experiments have already been carried out with such particles (26, 27).

A. Surface Free Energy Difference The first example is one in which the particle/interface "interaction" force derives from a difference in surface free energy between the particle and fluids 1 and 2. We denote by gs a and gs~ the surface free energies or, equivalently, surface chemical potentials, of a Brownian particle (with respect to the same datum) when wholly bathed by fluids 1 and 2, respectively.

201

DIFFUSION

These position-independent constants are given by (6-9). gs~ = Aso'si

(i = 1,2),

[4.1]

where As = 4~ra2 is the surface area of a Brownian sphere of radius a, and o's~ is the interfacial tension existing between the surface of the solid sphere and fluid i. Thus, at large distances from the interface, it is required that (cf. [3.8]) V-->gs 1 as

y---> +~,

-->gs~ as

y--->-~.

[4.2]

The particle surface free energy gs is a real potential in that, if it varies with the position y of the sphere center, its gradient will give rise to a force tending to move the particle from a region of high to low potential. If the particle is wholly immersed in either fluid, the value of gs in that fluid will be constant, independent ofy. Its gradient is then zero, whence no force arises from the existence of the solid-fluid interfacial tension o's~.It is only when the particle straddles the interface that a force arises (28-32), tending to move the particle into that fluid possessing the smaller of the two o-s~values, thus driving the system toward its equilibrium state of minimum free energy. For when the particle is partly immersed in each fluid, its surface free energy will be a function of its depth of penetration into the interface, and hence, of y. It may be noted that, as a result of [4.2], the majority of the analysis and discussion of Sections II and III which involves V=~ and V=2 can be repeated with V=1 and V=2 replaced by gs 1 and gs2 or, equivalently, by Asos1 and Aso'sz. However, the description of the preceding paragraph makes it clear that the potential function is not of the qualitative form which we have sketched in Fig. 2 (and utilized, implicitly, for the approximations [3.23], [3.24], [3.33], and [3.34]). Instead, it simply changes monotonically from gs2 to gs 1, for - 2 a _
Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

202

BRENNER

AND LEAL

sphere radius). Rather than pursue this point further we choose to concentrate only on the (ideal) interphase distribution coefficient [2.10], which may now be rewritten as K = exp[-(g~ ~ - gs2)/kT] --- exp[-As(o-sl - o-~)/kT].

[4.3]

Thus, from the point of view of [4.1] and [4.2], the existence of a value of the constant K = c=i/c= 2 different from unity arises from the differences in surface free energies, or interfacial tensions, for the particle in fluids 1 and 2. We believe that this identification of the interphase distribution coefficient with a surface free energy difference is novel, and hence worthy of further elaborat i o n ~ e s p e c i a l l y so as it forms the basis for a possible experimental test of the existence of solid-fluid interfacial tension. The Brownian particles alone may be regarded as constituting a thermodynamic system or phase. The criterion of equilibrium in such a system (at constant temperature and pressure) in the presence of a potential is that (9, 33) g + V = constant,

[4.4]

for all -oo < y < oo. Here, V and g are, respectively, the potential energy and Gibbs free energy, or chemical potential, per particle. This relation applies even to those particles straddling the interface. Applied to the bulk interphase equilibria existing in the two fluids, the preceding equation adopts the form go01 + V ~ 1 =

go~2 + V=2,

[4.5]

where Vo~~ denotes the potential of the Brownian particle when wholly immersed in fluid i. Inasmuch as g and V are each separately determinate only to within an additive constant, it is purely a matter of convention as to whether the constant surface potentials, A~o-~, should be assigned to g or V. If we elect to assign it to g, there results, at least in ideal solutions, Journal of Colloid and Interface Science,

Vol. 65, No. 2, June 15, 1978

g i = gs~ + k T In eoJ + constant,

[4.6]

and V~ i = 0,

[4.7]

with gs i = Asosi. The constant in [4.6] depends only upon the i n t e r n a l state of the Brownian particle (that is, upon the pressure and temperature), and hence is the same whether the particle is bathed by fluid 1 or 2. Alternatively, upon assigning the surface potential to V, there results go/ = k T In c= ~ + constant

[4.8]

V~ ~ = A~o-s~.

[4.9]

and Either of these two choices leads to the same interphase equilibrium relation, namely,

AsOrsl +

k T In c= 1 = Ascrs2 + k T l n c o o 2.

[4.10]

In turn, the latter leads immediately to Eqs. [2.9] and [2.10]. The form of the equilibrium interphase distribution coefficient, K = exp[-A~(o-s~ - o ~ ) / k T ] ,

[4.11]

has surprising implications with regard to the distribution of Brownian particles between two liquid phases. This result is, of course, not limited to surface-active agents, and this is the main point of the following discussion. The solid/liquid interfacial tension difference, def A°rs = o'~1 - ~2,

[4.12]

for a given solid and pair of immiscible liquids 1 and 2 is readily measured (6, 7) by classical contact angle experiments of the type shown in Fig. 4, wherein Ao-~ = o-12 cos 0,

[4.13]

with tr12 the interfacial tension between the pair of liquids. Thus, independent m a c r o s c o p i c measurements of tr12 and of the contact angle 0 against the solid material of

INTERFACIAL

which the Brownian particles are composed, suffices to determine Ao.s. Knowledge of the radii a of the Brownian particles furnishes the surface area, As = 47ra 2. En toto, such information permits calculation of K. Typical values of IAo.sl appear to be of the order of 10 dyn/cm. For Brownian particles of 1-/zm radius this yields def /3 = a s l a o . s l / k T

= 3.11 x l0 T

[4.14]

at room temperature, whence K = 4.5 x 10-l°'°°°'°°2.

[4.15]

This value is so close to zero as to be experimentally indistinguishable from it. Thus, for all practical purposes, at equilibrium, all the particles will end up in only one of the phases. This phenomenon is well known (34), at least with regard to the distribution of macroscopic liquid droplets in systems composed of three immiscible liquids. The droplets may be likened to the solid particles under discussion. It is interesting to note that it is only for "molecules" of true molecular size, viz., a = 10-8 to 10-7 cm, that the dimensionless group /3, and hence K, is of order unity. This can hardly be a coincidence. Application of macroscopic concepts like particle size and surface tension to individual molecules appears to us no more absurd than are the applications of macroscopic concepts like particle size and viscosity to the determination of molecular diffusivity via the Stokes-Einstein equation, D = kT/ 6rrtm. It is known that the latter equation is quite accurate when applied to small molecules, despite the fact that it was originally derived strictly for Brownian particles, obeying macroscopic hydrodynamics. Solid/liquid interfacial tension and particle size also appear as parameters in the equation (Ref. (7, p. 10)) for the effect of

203

DIFFUSION I

c~2

/I

I

~12 c~- ~ -

~'~ c~

..........

'sl i°'s2

--

~crs[- °'s2 ~

FI~. 4. The contact angle 0.

particle size on the solubility of finely divided solids in liquids. This solubility plays a role similar to that of the interphase distribution coefficient K. However, in that case, the effects of particle size and interfacial tension on solubility are miniscule. Finally, in this same context, we note that in systems where IAo.sl is very small, experimental measurement of K affords a method for the determination of particle size. Conversely, if the particle size be known independently, experimental measurements of K and 0 in conjunction with [4.13] offer the possibility of determining the liquid/liquid interracial tension o12 in systems of exceptionally low interracial tension. Since Icos 01 <- 1, systems where o'12 is small necessarily give rise to small IAo.sl values.

B. Density Differences Until now, gravity effects have been consistently ignored in the present paper. That is, the particles and fluids have all effectively been envisioned as possessing the same densities. However, in the commonly occurring case where the "upper" fluid, 1, is air, or some other low density vapor, it is clearly necessary to consider--at least implicitly--the effects of gravity on the potential energy function. For it is this factor alone which is responsible for the fact that none of the surfactant molecules is able to escape from the liquid into the surrounding vapor. Gravity has the effect of making V = oofory > a. In general, the relative importance of gravity and surface effects is Journal of CoUoid and Interface Science, Vol. 65, No. 2, June 15, 1978

204

BRENNER AND LEAL /-rodius=o

Fl°id"l"0

wherein def (P2

interface ~

"Y

Y

=

--

Ps)/(P2

--

Pl)

(0
gravity

1).

[4.18]

Comparison of [4.16] with [2.1] yields the potential energy function,

FIG. 5. Non-neutrally buoyant sphere floating at the interface between two immiscible fluids of respective densities greater, and less, than that of the sphere.

V(y) = (Ps - p , ) % a g ~ ( ' 0 , y ) ,

xt, = "0 governed by the magnitude of the dimensionless parameter, IAplga2/IA~, I. if ]Aplg/[Ao's[ is of O(1), gravity effects may only be neglected if a "~ 1. The second example of a potential function to be considered here is therefore that corresponding to a homogeneous sphere (of radius a) whose density p~ is intermediate between the densities px and/92 of fluids 1 and 2, respectively. For definiteness it will be supposed that P2 > Ps > Pl, so that in Fig. 1 the gravity force acts vertically downward, normal to the horizontal interface. Inasmuch as our calculation is purely illustrative, it will be supposed, for simplicity, that the interface is fiat. As in Fig. 5, the algebraically signed scalar y (-oo < y < oo) denotes the distance of the sphere center a b o v e the interface. From elementary considerations of the buoyant forces acting upon the spherical segments immersed in each of the fluids, the net external force exerted on the sphere is found to be F

=

-n(ps

-

pl)gTpX('0),

[4.16]

with g the acceleration of gravity, % = 47ra3/3 the sphere volume, '0 = y/a, and X('0) = 1

for

1 -< '0 < 0%

= - y ( 1 - y)-~ for

- 1 -> "0 > -0%

= 1 - ¼(1 - "y)-1(2 - 3"0 + '03) for

- 1 - < ' 0 - < 1,

[4.17]

Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

[4.19]

with

= -y(l

=

'0 +

-

for

1-<'0 < ~ ,

for

- 1 > ' 0 > -0%

y)-1'0

(1/,6)(1

-

y)-,

x [3 - '0(8 - 6"0 + "03)] for

- 1 - < ' 0 - < 1.

[4.20]

The nondimensional force function, h, as well as the potential energy function, ~ , and its derivative are continuous at the end points "0 = ___1. The force on the particle vanishes when the sphere center is situated at that position, "am = 'am(Y) (-- 1 < 'ore < 1), defined by the equation X('0m) = 0,

[4.21]

where X is the function appearing in the last of Eqs. [4.17]. (In the special case where the sphere density is exactly midway between the two fluid densities, corresponding to 3' = ½, "ore = 0. In this symmetrical case the force is zero when the sphere center lies in the plane of the interface.) If the sphere center is situated at any point other than '0 = "ore, an " a d s o r p t i v e " force acts, tending to move it to that position. The point "0m also corresponds to the minimum, dxIr('0m)/d'r I = 0 , in the potential energy function. A sketch of the dimensionless potential energy function, g' vs '0, for the asymmetrical case where 3' = aA, is given in Fig. 6. (For the case where y = ½, the curve is symmetric about "0 = 0.) That is positive for all "0 is without physical

205

INTERFACIAL DIFFUSION

significance, since the potential energy function is defined only to within an additive constant. In the absence of (vertical) Brownian movement, all particles in the system would eventually accumulate with their centers at the position ~m. In this case, corresponding to kT---> O, the number-density distribution function [2.2] would adopt the Dirac delta function form, c = aC(x)8(y

- Ym),

[4.23]

for in terms of this function, V / k T -- (47r/3)X~.

84

Z l

[4.22]

with a a normalization constant, and Ym = a'0m- Of course, the thermal molecular motion smoothes out this sharp maximum in c, and reduces its magnitude. However, the maximum value of c necessarily continues to occur at the value Ym, since this clearly represents the most probable position. The extent to which the sharp peak is diminished depends, for a fixed value of y, upon the dimensionless Langevin parameter, X = (Ps - Pl)g a4/kT,

9~

[4.24]

For a unit density difference in [4.23], which would occur, for example, in the case of air as the upper fluid, and for Brownian particles of 1-/zm radius, with g = 980.7 cm sec -2 a n d J = 4.184 x 107 ergs ca1-1 as the mechanical equivalent of heat, this yields X = 2.43 at room temperature. This is of order unity, and hence of real physical interest, since potential energy and thermal energy effects are then of the same order of magnitude. Since V tends to infinity, rather than a constant, with ly[, the absorption coefficient K a is infinite in the present case. However, it can be shown that expression [3.19] for Ds tends to a definite limit, by retaining a finite length parameter L in [3.3] and passing to the limit L = oo at the conclusion of the analysis.

L iX` I

-

-2

-I

0 I

2

5 -~7

FIG. 6. Variation of the potential energy function of a non-neutrally buoyant sphere with distance of the sphere center from the interface. The nondimensional plot shown is for a density difference ratio o f 3' = aA. For Ir/I > 1 the sphere lies wholly in one o f the two fluids, in which case the potential energy function increases linearly with distance from the interface.

These calculations suggest the possibility of performing some pertinent model surface diffusion experiments with non-neutrally buoyant Brownian particles. V. M O D E L S O F S U R F A C T A N T M O L E C U L E S

The general appearance of the potential energy function for surfactant molecules is likely to be of the form shown in Fig. 2, wherein V varies between the two bulk limits, V~1 =Aso'sl and V~2 =Asors2. As a crude model, which appears, however, to capture the essence of the phenomena, one could visualize a surfactant molecule as an inhomogeneous sphere, as in Fig. 7, one portion, " O " , of its surface being hydrophobic, and the other, " W " , hydrophilic. The spherical caps, O ("oil" loving) and W ("water" loving), are permanently fixed on the sphere surface. They represent the nonpolar and polar ends, respectively, of a real surfactant, molecule. The respecJournal of Colloid and Interface Science,

Vol. 65, No. 2, June 15, 1978

206

BRENNER AND LEAL

~

FIG. 7. Model of a hydrophobic/hydrophilic spherical surfactant "molecule."

sphere are governed by the ratio h/a, with h ( - a < h < a ) the distance from the sphere center to the circle of intersection of the caps, reckoned as being positive in the direction from W toward O. When totally immersed in either the o or the w phase, the surface free energies or potentials of such spherical surfactant molecular models are constant, being given respectively by the expressions

tive cap areas, A o and Aw, are such that

and

zlo

AW~s :o

V®O_=g®O = Aoo-oo + Awo-wo [5.3] [5.1]

V®w -g®W = Ao(row + Awo-ww" [5.4]

represents the total sphere area. Each cap is regarded as possessing its own separate solid/fluid interfacial tensions. These differ, according as the cap material is wholly immersed in oil (o) or water (w). There thus exist four possible intrinsic solid/ fluid interfacial tensions: troo, (row, trwo and trww. For example, trow represents the interfacial tension existing between the hydrophobic portion O of the particle and " w a t e r . " From the definitions of the words hydrophobicity and hydrophilicity, it follows that

Each of these is independent of the distance y of the sphere center above the interface, as well as of the orientation,

As = Ao + Aw

(too < O'ow and

(rww < O'wo, [5.2]

0 = cos-1(n'e)

(0 -- 0 < 7r),

[5.5]

of the particle relative to the normal n to the interface. The bulk potential difference, V= ° - V® w, represents the driving force for interphase transport of the particle. On the other hand, when the sphere straddles the interface--which will be assumed flat for simplicity,7 as in Fig. 8--the potential energy function will be of the form V = Aooo'oo + Aowo'Ow

as a consequence of the tendency of the cap material to enter that phase in which its free energy is a minimum, that is, the phase possessing the smaller of the two interfacial tensions: The orientation of such a " m o l e c u l e " can be represented by means of a unit vector e, drawn normal to the plane of the circle of intersection of the two caps in, say, the direction from W toward O, and passing through the sphere center. As in Fig.7, the relative areas, Ao/Aw, of the hydrophobic and hydrophilic portions of the

Here, for example, Aoo is the area of the hydrophobic portion O of the sphere in contact with the o phase. These four areas can obviously be uniquely expressed in terms of the variables y and 0 for fixed values of the parameters h and a. By these means one may model the orientation-specific potential energy function, V(y,0), in terms of h, a, and the four material constants, o-o. The orientation-averaged potential energy function, V(y), may

6 It is, of course, possible to physically create particles of these shapes by coating a portion of the surface o f a hydrophilic sphere with an adhering hydrophobic substance (e.g., paraffin wax) or conversely.

Thus, we are neglecting liquid/liquid interracial forces on the sphere. Furthermore, we ignore capillary fluctuations in the shape of the interface due to thermal agitation.

Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978

+ Awoo'wo + Awwo'ww. [5.6]

207

INTERFACIAL DIFFUSION

then be computed from the expression

~oo= V(y,O)f(y,O) sin OdO V(y) =

0

,

[5.7]

Ii r f(y,O) sin OdO

o-phase AW°~

w-p;se

Oo

TY

--

=0

with f the positional-orientational distribution function. This probability density is defined such thatfdy sin 0 d0 is the joint probability of finding the sphere center in the region between y and y + dy while, at the same time, the particle orientation lies in the angular region between 0 and 0 + d0. At steady state, this distribution is necessarily of the Boltzmann form,

f(y,O) = C exp[-V(y,O)/kT],

[5.8]

with C a normalization constant. The peak of the distribution must obviously occur at the value 0 = 0°, corresponding to a vertical orientation, with the circle of intersection between O and W lying parallel to the plane of the interface. In this orientation the particle experiences no couple due to the solid/ fluid interfacial tensions. The y value at the peak will correspond to that position at which, for 0 = 0°, the particle experiences no net force due to these interfacial tensions. This surfactant sphere model interacting with a fiat interface clearly offers opportunities for the quantitative modeling of a variety of equilibrium and interfacial transport phenomena. If, for example, the O and W ends of the molecule are charged, so that the sphere possesses a permanent electric dipole (35-37), the action of an external electric field will have an effect upon the interfacial phenomena owing to the orienting effect of the field upon the orientational distribution function, f. In the case where convective shear is imposed, such fluid motions necessarily affect f and, hence, the surface transport properties of the surfactant. This obviously possesses implications for surface rheology,

~Ww

FIG. 8. A spherical surfactant "molecule" straddling the interface. The dividing circle between the hydrophobic and hydrophilic ends of the sphere is inclined at an angle to the interface, by which it is intersected.

akin to those for bulk suspension rheology (19, 38). In particular, it is evident that one may anticipate nonlinear and elastic surface rheological effects due to the interactions between: (i) the hydrodynamic forces and torques on the sphere arising from the shear; (ii) the forces and torques on the sphere arising from the interfacial tension forces acting over its surface; (iii) the Brownian forces and torques on the sphere. Additionally, other forces and torques may act in consequence of any external fields, e.g., electrical, centrifugal, gravitational, etc. Each of these affects f. The present model offers a rational and simple approach to the analysis of surface transport processes. Another potentially useful model of a surfactant molecule is the hydrophilic/hydrophobic dumbbell shown in Fig. 9. For lo + lw >> max (ao,aw) the spheres may be regarded as acting independently of one another. Each then separately experiences only a Stokes law force, corrected for the "wall effect" arising from the proximity of each sphere to the interface, including the case where the sphere may straddle the interface. Such a dumbbell possesses more complex physical attributes than those of the spherical surfactant "molecule" of Fig. 7, even in the absence of interfacial tension effects resulting from the different character of the two ends. The hydrodynamic properJournal of CoUoid and Interface Science, Vol. 65, No. 2, June 15, 1978

208

BRENNER AND LEAL

~ ~

radius:°O

o-phase -----'- w-phasep/

FIG. 9. A hydrophobic/hydrophilic dumbbell model of a surfactant molecule. The connecting rod between the spheres is assumed offer negligible hydrodynamic resistance, and to otherwise possess no physical attributes. The "locator point" P for the dumbbell is most conveniently chosen to lie at the centroid of the dumbbell. This is also its center of buoyancy and mass, the latter only if the densities of the two spheres are identical.

ties of such models are currently being calculated. Computations with either the sphere or the dumbbell model require development of a surface transport theory of orientable particles, analogous to that for such particles in bulk (17-19). While such surface calculations are clearly more difficult than comparable bulk computations, they obviously have the capacity to furnish considerable insight into both equilibrium and surface transport properties. Moreover, they can serve as a stimulus and guide for the development of purely continuummechanical approaches to surface and interface phenomena. ACKNOWLEDGMENTS This work was performed while H.B. was on sabbatical leave with the Chemical Engineering Department at the California Institute of Technology as a Sherman Mills Fairchild Distinguished Scholar. Thanks are due them for providing a congenial and hospitable atmosphere. REFERENCES 1. Brenner, H., and Leal, L. G., J. Colloid Interface Sci. 62, 238 (1977). 2. Imahori, K., Bull. Chem. Soc. Japan 25, 13 (1952).

Journalof Colloidand InterfaceScience, Vol.65, No. 2, June 15, 1978

3. Sakata, E. K., and Berg, J. C., Ind. Eng. Chem. Fundam. 8, 570 (1969). 4. Brenner, H., and Leal, L. G.,J. Colloid Interface Sci. (to appear). 5. Einstein, A., "Investigations on the Theory of the Brownian Movement," (R. Fiirth, Ed.). Dover, New York, 1956. 6. Adam, N. K., "The Physics and Chemistry of Surfaces," pp. 107, 204. Dover, New York, 1968. 7. Davies, J. T., and Rideal, E. K., "Interfacial Phenomena," 2nd ed., p. 196. Academic, New York, 1963. 8. Harper, J. F., Advan. Appl. Mech. 12, 59 (1972). 9. Guggenheim, E. A., "Thermodynamics," 3rd ed. North-Holland, Amsterdam, 1957. 10. Dahler, J. D., and Scriven, L. E., Proc. Roy. Soc. (London) A 275, 504 (1963). 11. Aris, R., "Vectors, Tensors, and the Basic Equations of Fluid Mechanics." Prentice-Hall, Englewood Cliffs, N.J., 1962. 12. Gibbs, J. W., and Wilson, E. B., "Vector Analysis." Dover, New York, 1960. 13. Saffman, P. G., J. Fluid Mech. 73, 593 (1976). 14. Erd61yi, A., "Asymptotic Expansions," p. 36. Dover, New York, 1956. 15. De Bruijn, N. G., "Asymptotic Methods in Analysis," p. 60. North-Holland, Amsterdam, 1958. 16. Sirovich, L., "Techniques of Asymptotic Analysis," p. 80, Springer-Verlag, New York, 1971. 17. Condiff, D. W., and Brenner, H., Phys. Fluids 12, 539 (1969). 18. Brenner, H., and Condiff, D. W., J. Colloidlnterface Sci. 41, 228 (1972). 19. Brenner, H., and Condiff, D. W.,J. Colloidlnterface Sci. 47, 199 (1974). 20. Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics," Noordhoff, Leyden (Netherlands), 1973. 21. Goldman, A. J., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 22, 637, 653 (1967). 22. Schneider, J. C., O'Neill, M. E., and Brenner, H., Mathematika 20, 175 (1973). 23. Majumdar, S. R., O'Neill, M. E., and Brenner, H., Mathematika 21, 147 (1974). 24. Dussan V., E. B., and Davis, S. H., J. Fluid Mech. 65, 71 (1974). 25. Kimuzuka, H., Abood, L. G., Tahara, T., and Kaibara, K., J. Colloid Interface Sci. 40, 27 (1972). 26. Vadas, E. G., Cox, R. G., Goldsmith, H. L., and Mason, S. G., "The Microrheology of Colloidal Dispersions II. Brownian Diffusion of Doublets of Spheres," J. Colloid Interface Sci. 57,

INTERFACIAL DIFFUSION

27.

28. 29. 30. 31.

308 (1976); see also Vadas, E. G., "The Microrheology of Colloidal Dispersions," Ph.D. thesis, McGill University, Montreal, 1975. Angus, J. C., and Dunning, J. W., "Diffusion Measurements Using Scattered Laser Light." Paper presented at AIChE Meeting, Tampa Fla. (May 19-22, 1968). Huh, C., and Mason, S. G., J. Colloid Interface Sci. 47, 271 (1974). Huh, C., and Mason, S. G., Canad. J. Chem. 54, 969 (1976). Sheludko, A. D., and Nikolov, A. D., Colloid Polymer Sci. 253, 396 (1975). Padday, J. F., Pitt, A. R., and Pashley, R. M., J. Chem. Soc. (Farad. Trans. 1) 71, 1919 (1975).

209

32. Maru, J. C., Wasan, D. T., and Kintner, R. C., Chem. Eng. Sci. 26, 1615 (1971). 33. Denbigh, K., "The Principles of Chemical Equilibrium," p. 85, Cambridge, London, 1955. 34. Torza, S., and Mason, S. G., Kolloid-Z. u. Z. Polymere 246, 593 (1971); ibid., J. Colloid Interface Sci. 33, 67 (1970); ibid., Science 163, 813 (1969). 35. Brenner, H., J. Colloid Interface Sci. 32, 141 (1970). 36. Brenner, H., and Weissman, M. H., J. Colloid Interface Sci. 41, 499 (1972). 37. Leal, L. G., J. Fluid Mech. 46, 395 (1971). 38. Brenner, H., Intern. J. Multiphase Flow 1, 195 (1974).

Journal of Colloid and Interface Science, Vol. 65, No. 2, June 15, 1978