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Ind. Eng. Chem. Fundam. 1986, 25, 662-668
Effect of Surfactants on Interphase Solute Transport. A Theory of Interfacial Resistance Rajendra P. Borwankar and Darsh T. Wasan* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 606 16
A theoretical investigation into the nature of interfacial resistance is carried out by a detailed resolution of the interfacial region, achieved through the use of a microscopic length scale in addition to the macroscopic one. While the solute concentration and flux fields are continuous, their expansions are not uniformly valid over the entire domain. The method of matched asymptotic expansions is, therefore, employed. An asymptotically rigorous derivation of the conventional macrocontinuum formulation of interphase transport is given. The investigation reveals that a significant lowering of the solute diffusivity in the interfacial region due to the high concentration of surfactant may be responsible for the macroscopically significant interfacial resistance. Its precise value is also shown to be determined by the physicochemical interaction forces acting on the solute molecules in the interfacial region.
Introduction Interphase solute transport is of fundamental importance in a variety of industrial operations and, hence, is a subject of intense research. In several such systems, surfactants are added to the system for a specific purpose, e.g., t o increase the interfacial area of mass transfer in dispersed-phase systems by decreasing the size of the drops. In several other systems, surfactants enter the system as impurities, not by fabrication or design. The importance of the effect of surfactants on mass transfer has been recognized and, consequently, has led to many investigations over several years. According to the extended two-film theory, the overall resistance to mass transfer consists of the individual resistances of the two phases and the resistance of the interface itself (Davies and Rideal, 1961). Thus, the overall resistance to transfer of solute from phase 1 to phase 2 is given by the sum of partial resistances R‘ = R,‘ + RI’ + R2’ (1) where R’is the overall resistance, R1’ and R2’ are the resistances in phases 1 and 2, respectively, and RI’ is the interfacial resistance. The reduction in R‘due to the presence of surfactants can, in general, occur due to changes in R,’ and R2’ as well as changes in RI‘. The changes in R,’ and R i are due to the alteration of the flow fields near the interface due to positive adsorption of surfactants. The increase of Rr’ is usually interpreted in terms of a barrier to transport across the interface. Several earlier investigators had used the term “interfacial resistance” in the context of the reduction in overall specific (per unit interfacial area) transfer rates in the presence of surfactants (e.g., Davies and Myers, 1961; Brown, 1965). This may be cited as a cause of the controversy regarding the mechanism of interfacial resistance. Two mechanisms were propounded: the hydrodynamic mechanism and the barrier mechanism. It is now clear that, while the reduction in the overall transfer rates may be a simultaneous manifestation of both mechanisms, the term R1’ refers to the barrier mechanism. In this work the term interfacial resistance will be used exclusively to refer to RI‘ and is, thus, a measure of the so-called barrier effect. There is sizable, but seemingly controversial, literature on the interfacial resistance in the absence of foreign surface-active agents. In the special case when the soIute itself exhibits surface activity, adsorptive accumulation of the solute must be taken into account (England and Berg. 0196-4313/86/1025-0662$01 5 0 / 0
19711, in addition to adsorption/desorption barriers. We exclude this case from our discussion. For a solute with negligible surface activity, Ward and Brooks (1952) and Ward and Quinn (1965) have reported negligible interfacial resistance. Davies (1963) has pointed out that the interfacial resistance of a clean interface is very small. Yet, reports of nonzero interfacial resistance, correct or not, regularly appear in literature. In most cases they are later withdrawn or contested. The presence of soluble or insoluble surfactants or polymeric materials, on the other hand, leads to measurable and high interfacial resistance in both gas-liquid and liquid-liquid systems (Hutchinson, 1948; Lindland and Terjesen, 1956; Cullen and Davidson, 1956; Whitaker and Pigford, 1966; Mudge and Heideger, 1970). Retardation of mass transfer by insoluble monolayers has in fact been put to practical use for reducing evaporation from lakes (Davies, 1963). Several investigators have attempted to provide theories for interfacial resistance due to insoluble monolayers a t surfaces (Blank, 1964; Bockman, 1969; Dickinson, 1978; Milliken et al., 1980). The monolhyer permeation is related to the probability of holes appearing in the film. In this paper, we attempt to develop a theory of interfacial resistance to transfer of solute due to the presence of soluble surfactants or macromolecules. We use the microscopic approach of Brenner and Leal (1982). The interfacial resistance is thereby shown to be a macroscopic concept arising from the low diffusivity in the interfacial region. The low diffusivity is traced to the high concentration of surfactant in the interfacial region. Consequently, the problem becomes nonlinear if simultaneous surfactant transport should also be considered. Hence, we assume here that the surfactant is at its equilibrium distribution throughout the system.
The Model System Description. We are concerned here with determining the interfacial resistance to transfer of solute-which is assumed to have negligible surface activity-due to the presence of a soluble surfactant a t equilibrium distribution in the system. To this end, consider one-dimensional unsteady transport of solute in the model system shown in Figure 1. The solvents, 1 and 2, which are essentially immiscible, are bounded by surfaces whose instantaneous positions are at y’ = -L,(t 3 and y’ = L 2 ( t3,respectively. The solvents are in contact a t a flat iaterface at y ’= 0. \Ve define the macroscopic length scale, L, such that L , i L and 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 663
\\\\ \ \
d
y’
I
y’=o
y’=-L1(t)
ounding \ \ \ \ \ I Bsurface \ \ _ continuous, inhomogeneous phase, the field variables being
Phase 2
continuous functions of the normal coordinate in this region, approaching the bulk-phase values asymptotically at the two extremities. Thus, the interfacial region can be treated as a microcontinuum with the surfactant and the solvents regarded as a “composite” solvent with local-average properties. Since the compositions of the surfactant and solvents vary continuously in the region, the composite solvent has position-dependent properties. Specifically, both shear viscosity and solute diffusivity then are functions of position. The very high concentration of the surfactant in the interfacial region causes the shear viscosity of this region to be higher than that of the bulk phases. In the macroscopic approach, the viscosity manifests itself as the twodimensional interfacial shear viscosity-an excess property which is assigned to the interface (Goodrich, 1981a, 1981b). Just as the viscosity is higher in the interfacial region than in either of the bulk phases, the solute diffusivity is lower in this region than in either of the bulk phases. In addition to the spatially varying diffusivity, the solute experiences physicochemical force in the interfacial region due to its inhomogeneous nature. This force (assumed conservative) is represented in terms of a position-dependent interaction potential energy. This interaction potential energy is responsible for yielding a continuous concentration profile for the solute, at equilibrium through its uniform chemical potential. Formulation of the Problem. Let the position-dependent interaction energy and the position-dependent solute diffusivities be denoted by E(y? and D’(y?, respectively. In order for the solute concentration in the interfacial region to asymptotically approach the uniform bulk-phase values a t equilibrium, the potential energy function, E (assumed dimensionless), must approach constant values asymptotically in the two bulk phases. Similarly, the position-dependent solute diffusivity must also approach the constant values asymptotically in the two bulk phases. We assume that the solute concentration is low. The exact transport equation governing the unsteady-state solute transport can then be written as (Brenner and Leal, 1982)
1 Interface
Phase 1
\\\\\\\\\\\\\\\\\\
Bounding
Surface
Figure 1. Schematic diagram of one-dimensional (unsteady) interphase solute transport in a model system with a thin planar interface containing surfactant at uniform distribution.
L , / L are both of order unity. On this macroscale the interface between the two solvents appears as a surface of discontinuity. We now follow the microscopic approach of Brenner and Leal (1982) and adopt the methodology of multiple length scales. We already have the macroscopic length scale, L , which is unable to resolve the interfacial region. Thus, all field variables appear to suffer discontinuity a t y ’ = 0 when based on this macroscale. Now we invoke a microscopic length scale to resolve the interfacial region. When viewed on this length scale, the field variables are seen as continuous across the interfacial region. This microscale is large compared to sizes of molecules. Thus, continuum transport equations can be used, even on the microscale. The surfactant is a t its uniform distribution in the system. In the macroscopic description, the surfactant concentration has uniform values in the two bulk phases, all the way to the interface. In the microscopic description, on the other hand, the surfactant concentration varies rapidly but smoothly in the interfacial region. The concentration of the surfactant in the interfacial region is necessarily very large, and to compensate for this, surface excess concentration is assigned to the interface (Gibbs, 1906). We assume the simplest form of the equilibrium concentration profile for the surfactant in the interfacial region-one with a single maximum. We assume that, due to high concentrations of the surfactant in the interfacial region the solvents are rendered miscible over the interfacial region. This assumption represents a departure from the work of Brenner and Leal (1978a, 1978b, 1982), and also of Larson (1982) and Shaeiwitz and Raterman (1982), who had examined the case of solute transfer across a clean interface using the model of Brenner and Leal (1978a). Brenner and Leal had assumed that solvents are miscible only cn a molecular scale so that, even on the microscale, the interface may be approximated as a singular surface (Brenner and Leal, 1978b). Statistical mechanical theories of fluid interfaces (see, for example, Carey et al., 1980; Falls et al., 1983) yield estimates of interfacial or surface thickness in the absence of surfactants to be of the order of molecular dimensions a t temperatures below critical temperatures. We believe that surfactant adsorption renders the solvents miscible over larger dimensions, so that interfacial thickness becomes of the order of the microscale. (Large amounts of surfactants are known to render oil-water bulk phases to become completely miscible (Healy and Reed, 1974).) We, therefore, assume that on the microscale, l , the interfacial region is a single,
a c r aj’ -at’+ - / =ayo with
(3) where c’is the concentration of the solute and j‘is its flux in the y’ direction. All of the above variables except E are dimensional and, hence, with the exception of dimensional length scales, L and 4, are denoted with a prime. The molar physicochemical interaction energy is made dimensionless by dividing it by the thermal energy, R T , to yield the dimensionless interaction energy, E. It is convenient to introduce dimensionless variables in place of the above dimensional ones. We use
where Di) is either D,’ or D2/, whichever is appropriate, and
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c,,' is any characteristic solute concentration.
Nondimensionalization now permits us to discuss the scaling relevant to D and E in detail. To this end we define the dimensionless small parameter
6 = E/L
(5)
As discussed above, the cause of position dependencies of the interaction potential energy, E , and the solute diffusivity, D, is the surfactant adsorption. Significant departures of the surfactant concentration (and, consequently, the concentrations of the solvents) from their respective bulk values are confined to the interfacial region characterized by the microscale, [. In fact, this statement may be taken as a definition for the microscopic length scale, t. Therefore, significant departures of the interaction potential energy and the solute diffusivity from their respective bulk values, which are consequences of this nonuniform environment, are also confined to the interfacial region. In other words, E and D scale with the characteristic length, (, i.e.
of surfactants, we have assumed that the interfacial region is a microcontinuum and that the position dependence of diffusivity stems from the position-dependent surfactant concentration. The interaction potential energy for the surfactant, which is responsible for its nonuniform concentration in the interfacial region, satisfies an equation similar to eq 14 (Brenner and Leal, 1982). Therefore, it is reasonable to expect that m is greater than one. We assume this to be the case. Equations 14 and 15 with m > 1 should be regarded as sufficient for the existence of jump conditions connecting concentrations and fluxes across the interface in the macroscopic theory (see later). Before we begin our asymptotic investigation, it is necessary to rewrite the transport equations given by eq 2 and 3 in nondimensional forms
and
and D = D(y:'/E)
(7)
The solute concentration, c, and its flux, j , in the y direction are continuous everywhere.
In terms of dimensionless quantities we have
E = E(6-l~)
(8)
D = D(6-l~)
(9)
and Since E and D must possess asymptotic forms as discussed above, we have lim E ( 6 - l ~ =) E , for y < 0 6-'1)
I--
for y > 0
= E,
(10)
and lim D ( 6 - l ~ = ) D , for y < 0 6-'1)1--
=
D, for y > 0
(11)
For a solute to have negligible surface activity we also have
The thermodynamic (equilibrium) partition coefficient of the solute between the two solvents is represented as
K = exp[E,
-
E2]
(13)
As pointed out by Brenner and Leal (1982), the rate of approach to the asymptotic limits is of crucial importance in the convergence of the various integrals that will arise in the course of
It is also necessary to know the functional approach of diffusivity to the respective bulk values. Following Brenner and Leal (1982), we assume that D ( 6 - l ~behaves ) as
Asymptotic Investigation of Interfacial Resistance In the conventional macrocontinuum theory of interphase transport of a non-surface-active solute, the effects of the interface are encompassed in jump conditions connecting solute concentrations and fluxes on either side of the interface, viz., the fluxes on either side are equal and the concentrations are connected through a rate expression for the flux (see, for example, Scott et al., 1951). When the rate constant tends to infinity, there is no interfacial resistance and the concentrations on the two sides of the interface are connected through the equilibrium partition coefficient. It is also assumed that the effects of the interface are sufficiently localized so that transport processes in the bulk fluid are unaffected by the presence of an interface. In the strict sense, it is incorrect to accept the above macroscopic description as a foregone conclusion. Our microscopic investigation will be shown to validate the above macroscopic description and at the same time to give insights into the nature of the interfacial resistance. Toward this end we carry out the asymptotic investigation as follows. The ratios, of the microscale, t, to the macroscale, L , as defiied in eq 5, is used as a small parameter. Perturbation expansions are then sought in terms of 6. It will be shown that the uniform perturbation expansions are not valid over the entire domain. The method of matched asymptotic expansions (Nayfeh, 1980) is used, therefore, t o solve the singular perturbation problem. We have the inner domain which is the interfacial region with the characteristic length scale, E, bounded by two outer domains, each with a characteristic length, L. Outer Domains. The governing equations in the outer domains are obtained from eq 16 and 17 by invoking the limiting process 6
-
0, y = y ' / L = O(1)
(18)
The resulting equations with i = 1, 2 are where i denotes 1 for y < 0 m-d 2 for y > 0. In the presence
-dci+ - =aj, o at
ay
(19)
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 665
j@,t;6) = j(y=6g,t;6)
and
(30)
We again employ the straightforward perturbation expansions for concentration F and flux as F = F(0) + F&l) + &(2) + ... (31)
7
Note that D and E are position dependent only over the microscale, [, and are constant in the outer domain. We assume that straightforward perturbation expansions in 6 are applicable for concentrations Ci and fluxes ji. Thus c 1. =
C.(O) 1
+ 6ci(1) + &i(2) + ...
(21)
...
(22)
and J ’i
= J’i (0) + 6ji(l)+ 6zji(2) +
respectively. Using eq 21 and 22 we obtain the zerothorder outer fields from eq 19 and 20 as
and
and
-
-
j = j ( 0 ) + @l) + 6
Note that these outer fields are identical in form with the equations of the normal macrocontinuum theory. These, and similar equations for each higher order in 6, are to be solved subject to initial conditions and boundary conditions at external boundaries. Any nonhomogeneous initial and boundary conditions are necessarily independent of 6 and are, therefore, reflected in the zerothorder fields. Higher order fields have homogeneous initial and boundary conditions. The outer fields provide the macroscopic description of the system. From the macroscopic point of view, both D and E appear to suffer discontinuities at y = 0. These discontinuities must manifest in the outer concentration and flux fields. Thus, these outer solutions do not represent uniformly valid approximations of the exact solution ( c j ) of eq 16 and 17 which are necessarily continuous over the entire domain; they must be supplemented by the solutions in the “inner” domain near y = 0. For our purposes, only zeroth-order outer and inner fields need to be considered. Inner Domain. In order to obtain the govern’ cg equations in the inner domain, it is necessary to resc e the governing equations (16) and (17) to account for t,+e local gradients in the interfacial region. We rescale y in these equations using g = (y’y and invoke the limiting process 6 + 0, g = 6-’y =
Equations 16 and 17 then become
o
c
(32)
(33)
and
Matching Conditions, According to the basic premise of the method of matched asymptotic expansions, the domains of validity of neighboring expansions overlap and, thus, the neighboring solutions can be matched (Nayfeh, 1980). The matching conditions that are required to obtain the solutions are lim ~@=b-’y,t)= lim ci(y=6g,t) 9 fixed
y fixed
6-0
6-0
6-0
6-0
(35)
and
where i = 1 for y , g < 0 and i = 2 for y , 9 > 0. Note that the right-hand limits in eq 35 and 36 are merely ci(O,t) and ji(O,t). Thus, from knowledge of the inner concentration and flux fields, the appropriate lefthand limits can be evaluated in terms of parameters characterizing c and 3. These parameters can, in principle, be eliminated to connect the outer fields across the interface. Since governing equations of the outer domain are identical with those of the “normal macrocontinuum” description, this matching procedure yields the asymptotically rigorous jump conditions which connect the macroscopic fields across the interface. For our purposes we are interested in the zeroth-order fields only. Zeroth-Order Fields. Equation 33 is easily integrated to yield
J‘O’(g,t) = J y t ) and
+ ...
respectively. We note here that the diffusivity in the inner domain can become very low as the viscosity becomes very high. We emphasize that it is this very low diffusivity in the interfacial region, a consequence of surfactant adsorption, that gives rise to the macroscopically significant interfacial resistance. (We shall see later how the physicochemical factors also enter the picture.) Thus, in obtaining the zeroth-order inner fields, we note that D@)/6 should be of order unity for interfacial resistance to be macroscopically significant. In view of this, the zeroth-order inner fields are governed by aj(o)/ay =
(24)
2 p
(37)
where J(O)(t)is the constant of integration. By use of eq 37, eq 34 can be solved to yield exp[EG)] =
where E and 3 are the inner concentration and flux fields defined as F@,t;6) = c(y=69,t;6) (29) and
where c(O)(t) is again the integration constant. Before we apply the matching conditions, it is convenient to rearrange eq 38 to
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Ind. Eng. Chem. Fundam., Vol. 25, No. 4 , 1986
[ & &] -
with i = 1 for 3 < 0 and 2 for 9 > 0. Using the mean-value theorem, we can simplify the second integral in eq 39, and eq 39 becomes
dm
+ c(O)(t)exp[-E,]
(44b)
and eliminating c o ( t ) and $(t) we have the jump condition c,'O)(o,t) exp[E,] - ~ ~ ( ~ ) (exp[E,] o,t) =
or For non-surface-active solutes we know from eq 12 that are of order unity. We also know that the first term of the right-hand side is of order unity because the diffusivity in the interfacial region is of order 6. However, the second term on the right-hand side is of order 6. Retaining only the terms of order unity, we have
E G ) and E,,,,
where r
or, in shorter notation
or
The matching process may now be carried out using eq 37 and 41. Using eq 36 and 37, we have j((O'(O,t)= lim j1(O)(y=69,t)= lim f0)G=6-'y,t) = Jco)(t)
t fixed
3 fixed 6-0
6-0
i.e.
where i = 1 for a < 0 and i = 2 for > 0. Note that the integral in eq 46 exists because of eq 14 and 15 with m > 1. The existence of the integral is necessary for the matching hypothesis to work; therefore, eq 14 and 15 are the sufficient conditions for the macroscopic transport theory to be possible. In view of eq 12, it is readily apparent that the order of RI is critically dependent on the order of the diffusivity in the interfacial region. Unless the diffusivity is significantly lowered in the interfacial region, RI will be of order less than one. Equation 45 then becomes
j,(O)(O,t) = J(O)(t)
(424
c,'O'(O,t) = c2'0)(0,t)/K
j2'O'(0,t)= P ( t )
(4%)
Le., the zeroth-order fields are in equilibrium across the interface. When the diffusivity is significantly small in the interfacial region and becomes 0(6),then D G ) / S = O(1) (48)
and eliminating Jco)(t)we have the jump condition j,(O)(o,t) = j,(O)(o,t)
(43)
This implies that RI is O(l), and eq 45 can be written as
Using eq 35 and 41, we have
j!(O)(o,t)= h [ ~ , ( ~ ) ( O - ,~t ,)( ~ ' ( o /, tK) ]
c , ( O ) ( ~ , t=) lim c,(0)(y=6jji,t) = lim c(')(J=6-'y,tj > fixed 6 4
(47)
\
fixed
0
4
(49)
where h = 1/RI
i.e.
and
Equation 43 and either eq 47 or 49 represent the asymptotically rigorous jump conditions connecting the zeroth-order outer fields across the interface. Since the governing equations for the outer fields are identical with those of the normal continuum theory, eq 43 and either eq 47 or 49 represent the "normal" continuum interfacial boundary conditions. These are, indeed, the equations used by all the previous investigators who, in fact, assumed their validity. Our microscopic investigation, on the other hand, provides a rigorous derivation of these macroscopic jump conditions which have been normally assumed in the description of interphase solute transport. Furthermore, it is readily seen that RI is the (dimensionless) interfacial resistance to the transfer of solute, arising due to the
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
presence of adsorbed surfactant.
The Nature of Interfacial Resistance Our foregoing microscopic investigation shows that it is rigorously possible to derive the interfacial boundary conditions which have been commonly employed in the continuum description of transport of non-surface-active solutes across a fluid-fluid interface. However, the primary objective of our work was not merely to provide a justification for the commonly used description, but to gain insight into the nature of interfacial resistance. Interfacial resistance is shown about to be the macroscopic manifestation of low solute diffusivity in the interfacial region. In fact, the (dimensionless) diffusion coefficient in the interfacial region has to be of order 6 for interfacial resistance to be macroscopically significant. Such low diffusion coefficients may arise because of the high surfactant concentration in the interfacial region. Although significant reduction of diffusivity in the interfacial region is a necessary condition for nonzero interfacial resistance, its precise value is also determined by the physicochemical interaction forces acting on the solute in the interfacial region. This is seen from the appearance of E(y) in eq 46. Besides lending insights into the origin of interfacial resistance, the preceding theory provides a theoretical route for a priori calculation of the same. From eq 46 it is clear that two functions are necessary to complete the interfacial resistance. These are E ( y ) , the interaction potential energy, and D b ) , the solute diffusivity. For a three-component system, Shaeiwitz and Raterman (1982) performed a similar calculation. They computed interfacial resistance form the formula of Brenner and Leal (1978a) for steady-state solute transfer across a clean interface. The solutes considered were only moderately surface active, a t best. Therefore, the assumption by Brenner and Leal (1978a, 1978b, 1982) that the solvents are miscible only over a molecular scale of distance is valid and the interface can be approximated as a singular surface, even on the microscale. Thus, Shaeiwitz and Raterman considered that the diffusivity reduction is due to the increased hydrodynamic drag on the solute molecule in the vicinity of the interface. They employed the Stokes-Einstein equation for the diffusivity in the interfacial region and the hydrodynamic drag function given by Lee et al. (1979) in order to obtain the function D(3). To obtain E@),they assumed that only dispersion energies may be considered They could then use the expression for self-energy of a molecule near the interface of two media derived from the Lifshitz theory of dispersion forces (Mahanty and Ninham, 1976). Schaeiwitz and Raterman’s calculations are again consistent with the derivation of the self-energy profile, in that an infinitely sharp interface is assumed. For the calculation of the self-energy profile, dielectric data are needed over the entire electromagnetic spectrum. Such data are necessarily scant and often unreliable. The exercise of Shaeiwitz and Raterman (1982) thus underlines the voluminous effort that should go into the calculation of interfacial resistances. Additional complications arise in our case where we are dealing with a four-component system: two solvents, the surfactant, and the solute. This is not only because of the need to consider an extra component but also because the physical nature of the system is drastically altered. We have said earlier that the presence of the surfactant renders the solvents miscible over dimensions much larger than the molecular dimensions. Thus, the assumption of an infinitely sharp interface is not applicable, and the interaction potential energy function, E @ ) ,is not known.
667
Furthermore, in the interfacial region, which is an inhomogeneous microcontinuum, the diffusivity function, DG), is also not known. It is hoped that advances in statistical mechanics of fluid interfaces will, in the future, throw light on the density profiles of the three-component system in which one component is a surfactant. Then it may be possible to derive the interaction potential energy function, E G ) ,for the solute and also the diffusivity function, DG).
Concluding Remarks A theoretical investigation into the nature of interfacial resistance due to surfactants is carried out by a detailed resolution of the interfacial region. This is accomplished by using a microscopic length scale, characteristic of the thickness of the interfacial region, in addition to the usual macroscopic length scale. The solvents are assumed to be completely miscible within the interfacial region, whose thickness is much larger than molecular dimensions. The composition of the interfacial region varies continuously within the region. Therefore, the transferring solute experiences a (conservative) physicochemical force in the interfacial region which is expressed in terms of positiondependent interaction energy. The diffusion coefficient of the solute also varies continuously. Once the above mathematical apparatus is set up, it is seen that the concentration and the flux fields are not uniform over the entire region. The method of matched asymptotic expansions is therefore employed. This methodology allows the conventional, continuum jump conditions connecting the solute concentrations and fluxes across the interface to be rigorously derived. Whitaker and Pigford (1966) and also Ly e t al. (1979) developed continuum descriptions of interphase solute transport which are different from that verified above. Whitaker and Pigford assumed that the interface is of infinitely small thickness, but with finite capacity for the solute, and considered a nonequilibrium adsorption/desorption mechanism-a case which can be derived from the microscopic approach. Ly et al. (1979) considered the gasliquid interfacial region to be of finite thickness, having a capacity for dissolved gases greater than their solubility in water but with much lower diffusion coefficients. They also assumed local equilibrium. While the inclusion of higher order t e r m in the expansion may provide asymptotically rigorous justification of such a model through the above microscopic approach, it would call for a detailed knowledge of the diffusivity and interaction energy profiles that is not available a t the present time. Besides providing asymptotically rigorous verification of the interfacial boundary conditions which have been used in the theory of interphase solute transport, this exercise, more importantly, provides insights into the nature of interfacial resistance. For macroscopically significant interfacial resistance to exist, the solute diffusivity must be significantly lower in the interfacial region then in either of the bulk phases. While the interfacial resistance arises primarily from the very low solute diffusivity in the interfacial region, its precise value is also governed by physicochemical factors. An a priori calculation o f the interfacial resistance is curiently ruled out for solute transport across interfaces with adsorbed surfactant because of the lack of detailed knowledge regarding the structure and composition of the interfacial region. Acknowledgment This study was supported in part by a grant from the National Science Foundation and in part b y the Department of Energy.
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Ind. Eng. Chem. Fundam. 1986, 2 5 , 668-677
Nomenclature c = solute concentration, dimensionless F =
solute concentration in the inner domain, dimensionless
c, = solute concentration in the outer domain in phase i (i = 1, 2), dimensionless c’ = solute concentration, mol/cm3
c( = characteristic solute concentration, mol/cm3 D = solute diffusivity, dimensionless D’ = solute diffusivity, cmz/s D, = solute diffusivity in the outer domain, dimensionless D,’ = solute diffusivity in phase i (i = 1, 2), cm2/s E = interaction energy divided by RT, dimensionless E, = asymptotic limits of E in the phase i (i = 1, 2), dimension 1ess i = solute flux, dimensionless j = solute flux in the inner domain, dimensionless j , = solute flux in the outer domain in phase i (i = 1, 2), dimensionless j ‘ = solute flux, mol/(cm2 s) J(O)= integration constant, dimensionless K = equilibrium partition coefficient, dimensionless L = macroscopic length scale, cm L, = distance of the bounding surface from the interface in phase i (i = 1, 2), cm m = exponent in eq 15, dimensionless R = gas constant, erg/(mol K) R’ = total resistance, s/cm R,’ = resistance in phase i (i = 1, 2), s/cm R{ = interfacial resistance, s/cm RI = interfacial resistance, dimensionless t = time, dimensionless t’ = time, s T = temperature, K LZ = dummy integration variable, dimensionless y = coordinate normal to the interface, dimensionless y’ = coordinate normal to the interface, cm = coordinate normal to the interface in the inner domain, dimensionless Greek Letters d = parameter defined in eq 5 , dimensionless K~ = constant in eq 15 for phase i (i = 1, 2), dimensionless = microscopic length scale, cm
Superscripts
’ = dimensional quantities k = perturbation quantities ( k = 1, 2, ...)
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Received for review September 13, 985 Revised manuscript received July 10, 986 Accepted July 17, 986
Surface Fractionation of Multicomponent Oil Mixtures James W. Peterson and John C. Berg’ Department of Chemical Engineering, BF- 10, University of Washington, Seattle, Washington 9 8 195
An investigation is made of fractionation which occurs in oil mixtures as they spread over water under the control of surface forces. Its Occurrence is suggested in the literature, but the evidence is scant and fragmentary. Carefully controlled experiments using mixtures of poly(dimethylsi1oxane) and tetradecane were undertaken to verify or disprove the existence of the phenomenon. The results provide positive evidence of fractionation and indicate that a preferential spreading mechanism is involved. Additional fractionation measurements are made of a mixture of toluene, octane, and decane. This system also exhibits fractionation trends, indicating preferential spreading. A mathematical model is developed which describes the process as a multistage, batch-charged separation and yields good fits to the fractionation data. Oil film thickness profiles needed in the model were obtained by a new method using polychromatic interference fringes.
Introduction
It has been suggested (e.g., Phillips and Groseva, 1975; Fazal, 1975) that a separation process occurs during the
* To whom correspondence should be addressed. 0196-4313/86/1025-0668$01.50/0
surface-tension-controlled spreading of oil mixtures on water. This “surface fractionation” occurs as some components of the mixture apparently spread faster and farther than others. There are numerous possible causes for the existence of variations in the local composition of a spreading oil film. By limiting our interest to calm water 0 1986 American Chemical Society
