396
Langmuir 1988,4, 396-406
be related to any physical quantity characterizing the interaction between adsorbates and adsorbents. We inferred that the heat of adsorption a t zero coverage would be the most pertinent quantity to be compared with the affinity coefficient. Presented in Figure 5 is a plot of Pij against the initial heat of adsorption qoi.; values of qoijwere obtained by applying the Clausiusdlapeyron equation to the low-pressure regions of adsorption isotherms. As we can see in Figure 5, the affinity coefficient Pij has a linear relation with the initial heat of adsorption qoij. In addition, the straight line passes through the origin. This implies that the affinity coefficient can be represented by (7)
where qoij is the heat of adsorption of the i-adsorbate upon the j-adsorbent a t zero coverage, q0*,$is the heat of adsorption of reference adsorbate (*, nitrogen) on the reference adsorbent (§, ZSM-5-A) at zero coverage, and n (-1.0) is a correction factor. Table I1 summarizes the values for goij and n obtained in this work. The relation expressed by eq 7 appears to be worth comment. Firstly, it must be pointed out that eq 7 with
n = 1 can theoretically be derived by using the method reported earlier,12with the very reasonable assumption that the adsorption potential a t the adsorbent surface (ei j ) r = O is proportional to the initial heat of adsorption q0:lj ( z denotes the distance from the adsorbent surface). Secondly, it must be recognized that the initial heat of adsorption (qoij) is one of the fundamental quantities associated with adsorption. Equation 7 demonstrates that such a fundamental quantity as qostcan be used as a basis of generalizing adsorption data obtained under a very wide range of experimental conditions, though there exists a limitation. Thus we have to pay much attention to theory and experiments regarding the heat of adsorption. In principle, the initial heat of adsorption qoiij relates the molecular property of adsorbate to the physical property of solid adsorbent. Therefore, eq 7 appears to suggest a possibility of finding a good generalization method based on modern theories of materials. Registry No. Methane, 74-82-8. Supplementary Material Available: Numerical data of adsorption isotherms used in this work, Tables I and I1 (12 pages). Ordering information is given on any current masthead page.
Diffusiophoresis of Latex Particles in Electrolyte Gradients J. P. Ebel,t J. L. Anderson,* and D. C. Prieve Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received June 17, 1987 Transport rates of 0.1-hm diameter polystyrene latex spheres through porous membranes separating aqueous solutions differing in electrolyte concentration were measured to determine the diffusiophoretic velocities of the particles. Two types of experiments were performed. In the first the initial particle concentration was the same on both sides of the membrane; after an electrolyte concentration difference was created, the accumulation of particles on one side and the depletion on the other side were measured as a function of time. These experiments convincingly demonstrate the phenomenon of diffusiophoresis because there is no other obvious explanation for the creation of a particle gradient from a system initially uniform in particle concentration. In the second type of experiment particles were placed only on one side of the membrane, and the electrolyte gradient was established such that it augmented the particle transport rate down the particle concentration gradient. Enhancements of 50 times Brownian diffusion of the particles were ohrved when there was a 70% change in electrolyte concentration across the membrane. The experiments were performed with membranes of different pore size, and the results were extrapolated to infinite pore size to obtain the diffusiophoretic velocity of the particles in free solution. Comparison between these experimental velocities and theoretical predictions shows very good agreement when the electrslyte was lithium, sodium, or potassium chloride in the concentration range 0.01-0.1 M. Particle fluxes (based on pore area) decreased as pore size decreased, as expected from a model based on steric and charge repulsion from the pore wall. The experimental velocities for diffusiophoresisin gradients of potassium acetate were an order of magnitude greater than predicted from theory; much of this discrepancy might be due to very large uncertainty in the theoretical prediction for this electrolyte. Introduction Diffusiophoresis is the movement of a rigid, colloidal particle caused by a gradient of a molecular solute.' The origin of this transport is in the interaction between the solute and the surface of the particle, which occurs over distances of order 10-100 A in most cases. Derjaguin and co-workers2* were the first to recognize this transport mechanism and develop theories that connect the flow within the interfacial region a t the particle's surface to forces, either attractive (leading to solute adsorption) or 'Present address: Procter and Gamble, Miami Valley Laboratory, Cincinnati, OH 45247.
0743-7463/88/2404-0396$01.50/0
repulsive (leading to solute exclusion), between the solute molecules and the particle. Subsequent improvements in the theory have been made to account for surface curvature and convective mass transfer in the case of a particle (1) Anderson, J. L.; Prieve, D. C. Sep. Purif. Methods 1984, 13, 67. (2) Derjaguin, B. V.; Sidorenko, G. P.; Zubashchenko, E. A.; Kiseleva, E. V. Kolloidn. Zh. 1947, 9, 335. (3) Derjaguin, B. V.; Dukhin, S. S.; Korotkova, A. A. Kolloidn. Zh. 1961, 23, 53. (4) Derjaguin, B. V.; Dukhin, S. S. In Research i n Surface Forces;
Derjaguin, B. V., Ed.; Consultants Bureau: New York, 1971; p 269. (5) Derjaguin, B. V.; Dukhin, S. S.; Koptelova, M. M. J. Colloid In-
terface Sci. 1572, 38, 584. (6) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7.
0 1988 American Chemical Society
Langmuir, Vol. 4, No. 2, 1988 397
Diffusiophoresis of Latex Particles migrating through a quiescent f l u i d l ~ ~and - ~ for micromixing effects when a particle is suspended in a laminar flow.1° The goal of our work is to demonstrate unequivocally the existence of diffusiophoresis in gradients of electrolytes and to test the theory quantitatively by performing experiments with well-defined colloidal particles and electrolyte gradients. The theory for diffusiophoresis of spherical particles in electrolyte gradients is based on the classical equations of electrokinetic~.~~~ Some of the more important assumptions are that the surface of the particle is uniformly charged and smooth over length scales comparable to the Debye screening length, the electrolyte is completely dissociated and forms an ideal solution, and the transport of the ions is described by the Nernst-Planck equation. In the absence of a macroscopic electrolyte gradient, the equilibrium double layer is described by the GouyChapman model. With a macroscopic electrolyte gradient, the solution is assumed to be only slightly nonuniform so that the particle’s diffusiophoretic velocity (UD) is proportional to the macroscopic electrolyte concentration gradient. The diffusiophoretic mobility (kD)is defined by UD = kDv In (1) where C, is the concentration of the electrolyte. A semianalytical solution to the governing equations has been obtained8correct to O(l/Ku) where K is the Debye screening parameter and a is the particle radius:
c,
uo is a function of the {potential (made dimensionless by kT/ze) and the difference between ion diffusion coefficients of the electrolyte (p = (D+- DJ/(D+ + DJ): uo = 2 p { - 4 In (1 - y2) y = tanh ({/4) (3) u1 is also a function of t and p but in general must be evaluated numerically through successive quadratures; an expansion in powers of {is possible for small {potentials! The complete set of equations describing the fluid dynamics and electrolyte transport was numerically solved to obtain the diffusiophoretic velocity over broad ranges of { and KU for particles in gradients of KC1 and NaCLg This theory of diffusiophoresis identifies two mechanisms that contribute to particle motion, “electrophoresis” and “chemiophoresis”. The electrophoretic mechanism is based on the electric field that is generated spontaneously, in the absence of electrical current, when a concentration gradient of an electrolyte is established.” This field is caused by unequal diffusion coefficients of the ions and is proportional to 0. The chemiphoretic mechanism is analogous to the mechanism driving diffusiophoresis in gradients of nonelectrolyte^.^ The diffusiophoretic velocity is the sum of the electrophoretic (fie))and chemiphoretic (flC))parts:
up the gradient toward higher electrolyte concentration if /3 and {have the same sign and down the gradient if their signs are opposite. fit) is an even function of { and is usually positive (Le., particles move up the electrolyte gradient); however, recent calculationsg indicate that fit) is negative at certain values of { and KU. Note that for an electrolyte such as KC1, for which = 0, the chemiphoretic mechanism dominates, while for LiCl the large negative value of /3 means the electrophoretic mechanism is the primary determinant of a particle’s velocity. Because fie) and F(C)depend on different ways on K U , with K Cell2, it is possible that the direction of diffusiophoresis, either up or down the electrolyte gradient, could change as C, changes. Most previous experiments on diffusiophoresis have been concerned with its role in depositing latex films, a process in which other phenomena (e.g., coagulation) are also likely to participate. Derjaguin et al.12and Smith and Prieve13studied the formation of latex films on nonporous metallic surfaces undergoing attack by acids. Dukhin and Zueva14 and Derjaguin et al.15 visually observed the deposition of latex films onto a porous cellophane membrane through which CaC12was diffusing from the opposite side. The authors concluded that diffusiophoresis was the rate-limiting step in film formation; however, the complexity of the coating process makes the electrolyte gradient difficult to access. Using a sterically stabilized, commercial latex suspenion that was not susceptible to coagulation by increases in ionic strength, Lin and Prieve16 gravimetrically measured the rate of deposition of either a cationic or anionic latex onto a porous polycarbonate membrane through which one of a dozen different electrolytes was diffusing. From their data they concluded that the rate of latex deposition was determined by electrophoresis of the particles in the electric field produced by the concentration gradient of the electrolyte in the boundary layer. A particularly curious result was that with KC1, whose gradient produces a negligible electric field (p = 0), there was no deposition of the latex even though the chemiphoretic contribution was predicted from eq 4 to be significant. Microscopic observations of single particles moving in a salt gradient were attempted by Derjaguin et al.17 Looking downward along a line tangent to a vertical cellophane membrane, they noted the speed and direction of motion. When a solution of KCl or CaC1, was placed on the opposite side of the membrane from the latex, the anionic particles migrated toward the membrane; placing distilled water on one side and salt plus latex on the other side reversed the direction of particle movement. Unfortunately, these directions also coincide with those expected by natural convection caused by the more dense salt solution. No quantitative comparisons between theory and experiment were made by the authors. Lechnick and Shaeiwitzl8Jg were the first to use a membrane with very large pores through which latex particles could be transported. By measuring turbidity
-
(4)
P)is an odd function of C this term drives the particle (7)Anderson, J. L.; Lowell, M. E.; Prieve, D. C. J. Fluid Mech. 1982, 117, 107. (8)Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech. 1984,148, 247. (9)Prieve, D. C.; Roman, R. J. Chem. Soc., Faraday Trans. 2 1987, 83,1287. (10)Anderson, J. L.; Prieve, D. C.; Ebel, J. P.Chem. Eng. Commun. 1987,55, 211. (11)Newman, J. S. Electrochemical Systems; Prentice-Hall Englewood Cliffs, NJ,1973.
(12)Derjaguin, B.V.;Ul’berg, Z. R.; Dvornichenko, G . L.; Dukhin, S. S. Kolloidn. Zh. 1980,42, 838. (13) Smith, R. E.; Prieve, D. C. Chem. Eng. Sci. 1982,37, 1213. (14)Dukhin, S. S.; Zueva, T. I. Kolloidn. Zh. 1962,24, 444. (15)Derjaguin, B. V.;Dukhin, S. S.; Korotkova, A. A. Kolloidn. Zh. 1978,40, 643; (16)Lin, M.J.; Prieve, D. C. J. Colloid Interface Sci. 1983,95, 327. (17)Deriaeuin, B.V.;Dukhin, S. S.: Ul’bere, Z. R.; Kuznatsova, T. V. Kolloidn. Zh: 1980,42, 464. (18)Lechnick, W.J.; Shaeiwitz, J. A. J. Colloid Interface Sci. 1984, 102, 71. (19)Lechnick, W.J.; Shaeiwitz, J. A. J.Colloid Interface Sci. 1985, 104, 456.
398 Langmuir, Vol. 4,No. 2, 1988 changes on the salt side of the membrane due to increases in particle concentration, much lower rates of particle transport could be detected than with the previously mentioned gravimetric methods. Using lower latex concentrations permits use of a latex other than those made for commercial applications (which contain polymeric stabilizers). In contrast to the results of Lin and Prieve,16 Lechnick and Shaeiwitz observed that a KC1 gradient enhanced the rate of particle migration through the membrane; however, particle transport rates without an electrolyte gradient were 100 times greater than expected from Brownian diffusion of the particles, perhaps indicating that convection of fluid through the membrane was occurring. Another difficulty in interpreting their data arises because the diameter of the pores exceeded the pore length by a factor of 3, so that pore end effects on the electrolyte gradient and particle transport rates were probably not negligible. In the work reported in this paper, a new technique based on fluorescence spectroscopy was used to obtain results that (1) confirm the existence of diffusiophoresis with due regard for eliminating the possibility of particle transport by fluid convection and (2) agree quantitatively with a priori theoretical predictions of diffusiophoretic velocities under a variety of experimental conditions. Like Lechnick and Shaeiwit~'~J~ we used a membrane with large pores, which allows accurate measurements of particle transport rates using very small quantities of well-characterized latex. Unlike previous investigations, we monitored the transport rates continuously and in situ using fluorescence methods, and we used track-etch membranes known to have straight pores of uniform cross section. Furthermore, the length/diameter ratio of the pores was always greater than 5 so that end effeds, though not always negligible, were small and accountable in interpreting the measured transport rates. The monodisperse latex particles were synthesized in our laboratory with a fluorescent dye trapped inside; they were characterized by electron microscopy and electrophoresis. By using membranes with three different pore sizes, we determined the effects of pore size on diffusiophoresis. More details of the experiments are given elsewhere.20
Ebel et al. (a) o NaCl A KCzH30z
c-41i
0 O
-3
O
0
0
0
0 AQ A
O
O
0
I
1
- 2I 3
5
10
30
50
100
30
50
100
KO
(b) -10 NaCl
v
3
5
10 KO
Figure 1. (a) {potential for the B2 particles (a = 57 nm). {(made dimensionless by k T / e ) was determined from our electrophoretic mobility measurements and the numerical solution of O'Brien and White.n (b) Surface charge of the B2 particles as calculated from { and the numerical solution of Wiersema et al.23
transport cell or in the dialysis bags. Two batches of dye-loaded particles were synthesized at different conditions; they are referred to as batches B1 and B2. Using Latex Particles. The particles were polystyrene latexes scanning electron microscopy we found the mean particle diamprepared in our laboratory by surfactant-free emulsion polymeters of B1 and B2 to be 100 & 9 nm sad. and 114 f 7 nm, erization following the technique of Juang and Krieger.21 The respectively. The B2 particles were used in the "quantitative" initiator was potassium persulfate, and the charged groups also experiments, described later, because they contained a higher included sodium styrenesulfonate, a comonomer. A fluorescent density of the dye and hence could be detected accurately at lower dye, nile red (9-(diethylamino)-5H-benzo[aJphenoxazin-5-one), concentrations (2 ppm). was incorporated into the particles by mixing it with the styrene The (potential of the particles was determined by measuring monomer phase. We believe the dye was trapped inside the their electrophoretic mobilities at dilute concentrations in a Model particles. Unreacted styrene, initiator, and other unwanted 501 Lazer-Z apparatus (Pen Kem Corp.). Particle velocities were molecular species were removed from the particles by dialysis measured as a function of depth within the channel at a constant against deionized, ultrafiltered water for 10 days. The particles electric field, and the true electrophoreticvelocity was determined were stored in deionized water at a concentration of about 8.5% by plotting these velocities in a manner that avoided the usual by weight. The latex solutions used in the transport experiments complications of convection by electroosmosis and pressure-driven were made from this stock solution after it had been further f l o ~ The . ~ experimental mobilities were converted to {potentials dialyzed against a solution of the desired electrolyte concentration by applying the numerical technique of O'Brien and White,22 for a t least 8 days. which accounts for finite K a . From these {potentials we calculated The wavelengths of the excited (520 nm) and fluoresced (560 the corresponding surface charge densities (6)using the numerical nm) light were approximately the same for the dyeloaded particles solutions of the Poisson-Boltzmann equation by Loeb et aLZ3 as for the free dye in a solution of styrene. There was no detectable Figure 1 shows { and u for the B2 particles as a function of loss of fluorescence activity of the particles after storage in electrolyte concentration. The pH was 5.4 for the chloride sodeionized water for more than 1.5 years, the time period of this lutions and 7.6 for the potassium acetate solutions. The temstudy. Furthermore, there was no evidence of photobleaching of the particles nor loss of particles by coagulation in the stirred (22) O'Brien, R. W.; White, L. R. J. Chem. SOC.,Faraday Trans. 2
Experimental Section
(20) Ebel, J. P. Dijfusiophoretic Transport of Colloidal Particles; Ph.D. Thesis, Carnegie Mellon University: Pittsburgh, P A 1986. (21) Juang, M. S.;Krieger, I. M. J. Polym. Sci. 1976, 14, 2089.
1978, 72,1607. (23) Loeb, A. L.; Overbeek, J. Th. G.; Wiersema, P. H. The Electrical Double Layer Around a Spherical Colloidal Particle; M.I.T. Press: Cambridge, MA, 1961.
Langmuir, Vol. 4, No. 2,1988 399
Diffuiophoresis of Latex Particles
Table I. Geometric and Electrical Promrties of Three Membranes Used in the Quantitative Exoerimentsn electrolyte membrane pore radius, pm pore length, pm pore density, pores/cm2 f a,pC/cm2 concn, M -1.39' 4.28 7.93 5.70 x 104 0.001 QZ 0.770 -1.44 0.01 -2.05 -1.83 -3.94 0.1 84 0.489 7.48 13.0 x 104 0.001 -1.23 4.24 -1.07 -1.63 0.01
85
0.318
7.22
23.1 x 104
n.1 ..
-1.77
-3.75
0.001 0.01 0.1
4.77 -2.02
-0.15 -1.41
-1.88
-4.06
'Other membranes were ala0 uasd,but each of these had wentially the aame properties as one of the three membranes listed above. The f values listed here were obtained with KCI solutions; results for the other electrolytes were very similar. 'Made dimensionless hy kTle. PLATINUM
t
- 1).
conduction experimentsto test electrokinetic theory is that the desired experimentalquantity is the difference between two conductance measureniests, which could lead to magnification of experimental errors if these two measurements are close as is often the case with systems having thin double layers. At the present time we have no sound explanation for why we obtained agreement between theory and experiment while others have not found such agreement in different experiments. It is possible that processes contributing to the extra conduction of charge in the double layer are susceptible to flaws in the assumptions of the classical electrokinetic theory (for example, zero mobility of the "fixed charge^"^^^^^) while with diffusiophoresis such flaws are of little consequence. Pore size effects on diffusiophoresis are significant even when a / R is as small as 0.1. We recognize three effects of pore size that would hinder the transport of particles. First, there is a weak hydrodynamic effect of the pore wall tending to slow the particles. In the case of thin double layers (Ka a) this hydrodynamic effect is O ( a / R ) 3and thus would be insignificant in our experiment^.^^ How-
-
(32)Van der Put, A. G.; Bijsterbosch, B. H.J . Colloid Interface Sci. 1983, 92,499.
(30)Zukoski, C.F., IV; Saville, D. A. J. Colloid Interface Sci. 1985, 107, 322.
(31)OBrien, R. W.J. Colloid Interface Sci. 1986, 110, 477.
(33)Zukoski, C . F.,IV; Saville, D. A. J. Colloid Interface Sci. 1986,
114, 32.
(34)Simonova, T.S.;Shilov, V. N. Kolloidn. Zh. 1986, 48, 319.
404 Langmuir, Vol. 4, No. 2, 1988
Ebel et al.
ever, when the electrolyte concentration is 0.01 M or lower the thin double layer assumption may break down, and the hydrodynamic retardation of diffusiophoresis by the pore wall could be substantially greater than predicted. The second pore size effect is a reduction of the transport area due to exclusion of particle centers from regions near the pore wall. For a purely hard sphere-hard wall interaction this would lead to a factor equal to (1- a/R)2that must multiply the diffusiophoretic velocity in an unbounded fluid. Electrostatic repulsion between the charged particles and the pore wall would enhance this effect. Third, an osmotic flow of solution caused by interaction between the electrolyte gradient and the pore wall* and the resulting pressure flow needed to nullify the osmosis (see Appendix A) together reduce the transport rate of the particles. We consider the second and third effects in more detail below. Assume that the distribution of particles within the crow section of a pore is determined by equilibrium statistics: P(r) = Bo exp(-E/kT)
(6)
where E is the pair potential between a particle located at radial position r and the pore wall and Bo is a normalization constant that equals the particle concentration just outside the pore (Cp(B))if the pore entrance is considered. Neglecting diffusion, the flux of particles is given by C,‘B’ R Np = Up(r)P(r)2rdr (7)
FJ
The local particle velocity (Up) is the sum of diffusiophoresis plus convection by the osmotic and pressure flow discussed in Appendix A:
Up = UD + Uo - 2Uo[l - (r/RP]
(8)
Note that the integral of the second and third terms over the pore cross section sum to zero, meaning the average fluid velocity through the pores is zero. By combining eq 7 with 8, using the definition of U* in eq 5, and dropping terms of order higher than a/R, we obtain
u*
UD - 2 x [ u + ~ Uo](a/R)
X = 1 + a - l A m [ 1- exp(-E/ltT)] dh
(9)
(10)
where h is the distance from the pore wall to the center of a particle. For purely hard sphere-hard wall interactions
E = w
forha
Equation 10 gives X = 1. Attractive van der Waals interactions would decrease X, and electrostatic repulsions would increase it. The pores of our membranes were 60’ rhomboidal cylinders. For this shape eq 9 must be modified by using a pore radius based on an inscribed circle (R11.21) rather than a circle of equivalent area (R).36 Keeping eq 10 for the calculation of X,eq 9 becomes
U* = UD - 2(1.21)X(UD + Uo)(a/R)
(11)
Because the t potential of the pores does not vary greatly among different membranes, as seen in Table I, Uoshould be about the same for membranes having different pore (35)Keh, H.J.; Anderson, J. L. J. Fluid Mech. 1985,153,417. (36)Malone, D.M.; Anderson, J. L. Chem. Eng. Sci. 1978, 33, 1429.
Table IV. Values of X Parameter (Equation Experiment and Theory concn, M electrolyte 0.1 KCl 1.18 NaCl 1.30 LiCl 1.13 0.01 KC1 2.68 NaCl 1.36 LiCl 2.72 X R X n
9) from Xth 1.11 1.11 1.11 1.35 1.35 1.34
sizes as long as the electrolyte driving force is constant. Thus, estimates of X can be made from the experimental values of S*: = 2(1.21)x(uD + UO) (12)
-s*
By using this expression with Table I11 and the values of Uocalculated from eq A1 and A2, we have determined the “experimental”values of X listed in Table IV. With the exception of the one point for NaCl at 0.01 M, the experimental X values for different electrolytes are consistent with each other at each electrolyte concentration. Theoretical predictions of X were made from eq 10 and the following expression for the electrostatic energy between a sphere and a flat surface:37 hx,(h) =
B = 16ro(
eXP[-K(h - a ) ]
Fr (:) tanh
tanh
(:)
(13)
where subscripts p and w represent particle and pore wall, respectively. The required p potentials were obtained from Figure l a and Table I. The theoretical predictions of X are in reasonable agreement with the experimental estimates at 0.1 M electrolyte, but the differences are significant at 0.01 M. Figures 8 and 9 indicate the pore size effect becomes even more pronounced at larger a/R. Intuitively we expect the relation between U* and a/R to flatten as a/R increases, as indeed it must since U* 0 as a/R 1. The data for all three chloride salts at both 0.01 and 0.1 M concentration show instead an acceleration of the pore size effect at larger a/R; that is, U* at a/R i= 0.21 falls below the straight line connecting the data for the two larger pore sizes. Furthermore, data taken at electrolyte concentrations less than 0.01 M show such a strong dependence on a l R that we cannot infer values of UD from them.m Thus, we must be cautious in claiming the extent of our understanding of how pore size affects diffusiophoresis. The data for KC2H302,summarized in Table 111,agree with the theory in the sense that the diffusiophoretic transport was from high to low electralyte concentration (kD < 0) and UD was proportional to the dqiving force, Ce1/C,0- 1. However, the magnitudes of kD from experiment and theory differ by an order of magnitude. Experimental errors in determining UDcould not account €or this discrepancy because the experimental results were reproducible. Furthermore the acetate had no unusual effect on the f potential of the particles. Much of the discrepancy between theory and experiment could be the result of small errors in the theory for the electrophoretic and chemiphoretic effects, F(e)and Fee) of eq 4, since they have opposite signs and nearly equal magnitudes in the case of KC2H302.If the theory underpredicts F(e)by 10% and overpredicts Fee) by lo%, then we could account for a factor of 7 error in kD (theoretical prediction of k D goes from -0.04 to -0.27). Furthermore, the fact that acetic acid
-
-
(37)Alexander, B. M.; Prieve, D. C . Langmuir 1987,3,788.
Langmuir, Vol. 4, No. 2, 1988 405
Diffusiophoresis of Latex Particles is a weak acid is not accounted for in the theoretical predictions. In retrospect, KC2H302was not a good choice of electrolyte to test the theory for the case of kD < 0.
Summary The diffusiophoretic velocity of polystyrene latex particles in electrolyte concentration gradients has been measured. The membrane cell, while introducing additional complexity into the analysis of the data, was capable of establishing steep and stable electrolyte gradients. Because of our ability to detect and eliminate spurious fluid convection, the experiments demonstrating particle migration in KCl gradients represent an unambiguous confirmation of the existence of the chemiphoretic mechanism of diffusiophoresis. As shown by Figures 10-13, our experimental results with LiCl, NaCl and KC1 gradients at concentrations equal to 0.01 and 0.1 M are in excellent quantitative agreement with theoretical predictions of diffusiophoresis based on electrokinetic theory and independently measured [ potentials of the particles. The effect of the membrane, specifically the pore size, is significant and becomes more pronounced as the electrolyte concentration decreases. In fact, it is the very strong dependence of particle flux on pore size, even at a / R = 0.1, that precludes quantitative interpretation of our data below 0.01 M electrolyte. Experiments with gradients of KC2H302,for which kD < 0, show particle transport from high to low electrolyte concentration as predicted, but the magnitude of the diffusiophoretic velocity at each electrolyte driving force is an order of magnitude greater than predicted from the theory. However, much of this discrepancy could be due to small (=lo%) errors in predicting the magnitudes of the competing electrophoretic and chemiphoretic contributions to the particle's velocity. Acknowledgment. This work was supported by NSF Grant CBT-8513673. We also appreciate the financial support of the Westvaco Corp. J. P. Ebel was the recipient of a Procter and Gamble Fellowship. Finally, the advice of Professor A. Waggoner in selection of fluorescent dyes and optics was invaluable to the success of this work. Appendix A. Osmotic Flow Because the pore wall is charged, an electrolyte gradient produces flow of solvent through the pores. For thin double layers the "slip velocity" of the fluid a t the outer edge of the double layer is given byS
where uo is calculated from eq 3 with 1: equal to the dimensionless t potential of the pore wall (tw). The predicted magnitude of Vf&p) for our experimental conditions is small relative to the diffusional velocity of the electrolytes; hence, the solute concentration field is linear as given by eq B3. P P ) is not constant along the pore because d In Ce/dx varies and both [ and K depend on Ce(x). The mean flow velocity, defined as the flow rate divided by pore area, is independent of x . An exact solution to the Stokes equations within a long pore gives the following result?
sw
By use of eq Al, A2, and B3, along with the data for in Table I, mean osmotic velocities can be calculated given the electrolyte gradient.
(38)Anderson, J. L.;Idol, W.K.Chem. Eng. Commun. 1985,38, 93.
Solvent flow by osmosis would cause a deflection in the membrane because our transport cell was a closed system. Pressure in the side receiving the osmotic flow rises until it forces fluid in the reverse direction at a total rate equal in magnitude to the osmotic flow. The pressure difference at this zero net flow condition is determined from
where the factor 0.68 corrects Poisueille's equation for the rhomboidal shape of the pores. For our largest gradients of KC1 we have Uoi= -10 pm/s, which would give APo = 35 dyn/cm2 if R = 0.5 pm. Mica membranes are quite stiff, and a pressure this small would produce a very small deflection (
