J . Phys. Chem. 1988, 92, 517-525 clathrate-like water is proposed to contribute to this high-frequency relaxation (within the gigahertz range). This high-frequency dielectric relaxation exhibits a normal phase transition, which indicates an increasing disorder on raising temperature. Thus considering the behavior of these two relaxations (one arises from the polypentapeptide backbone and the other one from the clathrate-like water or hydrogen-bonded water), there emerges an interesting picture of the temperature dependence of the water-polypentapeptide system. When the temperature is increased, the polypentapeptide derivative becomes more ordered and the clathrate-like water becomes less ordered. This inverse behavior is demonstrated in Figure 9. Thus hydrophobicity plays a role in strucutre formation in these bioelastomers, which correlates with development of elastomeric force as has been rep~rted.~~,~~ The high frequency (1 GHz) of the dielectric permittivity of these polypentapeptide derivatives seems to correlate with the
517
hydrophobicity of the polypentapeptide. Perhaps one of the most interesting applications of this two-component system could be the establishment of a hydrophobicity scale based on the relative amounts of observable clathrate-like water. Acknowledgment. This work was supported in part by the Department of the Navy, Office of Naval Research Contract N00014-86-K-0402. Ren&Buchet was supported by Swiss National Science Foundation Fellowship, and Chi-Hao Luan held a National Education Commission of the People’s Republic of China Scholarship, each for a part of the research period. Registry No. I, 83610-44-0; 11, 83610-64-4; 111, 96847-92-6; IV, 111583-44-9; V, 111583-45-O;VI, 111615-37-3; VII, 111583-48-3; VIII, 11 1583-46-1; LPGVG, homopolymer, 11 1583-50-7; LPGVG, SRU, 11 1583-51-8; IPGVG, homopolymer, 106855-57-6; IPGVG, SRU, 106855-24-7; BNPC, 5070-13-3; VPGVG, 52231-42-2; VPGVG, SRU, 11 1793-53-4; BOC-Leu-OH, 13139-15-6; H-Pro-OBzLHCl, 16652-71-4.
Electrokinetic Salt Rejection by a Hypothetlcai One-Dimensional Inhomogeneous Charged Membrane Scott A. Kuehl* and Ronald D. Sanderson Institute f o r Polymer Science, University of Stellenbosch, Stellenbosch, South Africa (Received: May 19, 1987)
Rejection of ionic salt by a hypothetical one-dimensional charged-gelmembrane operating in a reverse osmosis configuration is evaluated by means of different empirical forms for the spatial charge distribution. The charge density is specified while the resulting electrical potential is determined by means of a model that represents an inhomogeneous and continuous mass density of variable wavelength. The three usual sets of coupled partial differential equations that describe a convective transport process, namely, the Navier-Stokes flow, Nemst-Planck flux, and Poisson-Boltzmann equations,reduce to only two simultaneous differential equations when the flow is treated according to Darcy’s law with a Debye-Bueche permeability at the high-pressure limit. Without restricting the problem to low potentials, the exact Poisson-Boltzmann equation containing the source term representing the fixed space charge is solved numerically simultaneously with an ion flux equation derived from a zero net current condition. The membrane performance for different density profiles is evaluated and discussed.
Introduction Two approaches have been used in the theoretical modeling of solvent flow and selective ion transport through charged membranes in osmosis and reverse osmosis (RO) processes. One of these treats the membrane statically as a continuous homogeneous ion-exchange body in which it is assumed that solute rejection follows a Donnan’” equilibrium (electroneutrality) at both surfaces rather than a dynamic or convective transport process throughout. Although it is expected that an electroneutrality condition should exist at both surfaces where the total charge density must reverse sign, it is not clear that a state of electroneutrality should exist everywhere else, even in a homogeneous body, nor is it obvious that membrane potential gradient should be a constant, derived from the simple Donnan potentials at the feed and permeate sides. The alternative approach is the capillary-tube model,&” which Teorell, T. Prog. Biophys. 1953, 3, 305. Glueckauf, E.; Watts, R.E. Proc. R . SOC.London A 1962, 268, 339. Glueckauf, E. Proc. R. SOC.London A 1962, 268, 350. Spencer, H. G. Desalination 1984, 52, 1. (5) Hoffer, E.; Kedem, 0. Desalination 1967, 2, 25. (6) Dresner, L.; Johnson,J. S. Principles of Desalination; 2nd ed.;Spiegler, K. S.; Laird, A. D. K., Eds.; Academic: New York, 1980; Part B, Chapter 8. (7) Dresner, L.; Kraus, K. A. J . Phys. Chem. 1963, 67, 990. (8) Dresner, L. J . Phys. Chem. 1963, 67, 1635. (9) Morrison, F. A,; Osterie, J. F. J . Chem. Phys. 1965, 43, 2111. (10) Gross, R. J.; Osterle, J. F. J . Chem. Phys. 1968, 49, 228.
(1) (2) (3) (4)
0022-3654/88/2092-0517$01.50/0
correctly incorporates the dynamic coupling between the ion distribution, potential field, and hydrodynamic flow. This approach in particular is motivated in.part by the simplifying advantages developed on applying the fundamental set of differential equations that describe the process, namely, the Nernst-Planck flux, Navier-Stokes flow, and the Poisson-Boltzmann equations. Other advantages of the above model, the various approximate schemes used to broaden the scope of the capillary model, and other aspects of the problem have been discussed in the literat~re.~-’’ The capillary-tube hypothesis approaches its objective when it is applied to two-phase porous solids, such as compacted clays and porous glasses. These solids consist of continuous networks of insoluble or hydrophobic regions of zero permeability that form the boundaries of definite channels, the unobstructed flow of solvent through which can be adequately described by the Navier-Stokes equation. In this model the tortuous pathways and (11) Fair, J. C.; Osterle, J. F. J . Chem. Phys. 1971, 54, 3307. (12) Jacazio, G.; Probstein, R. F.; Sonin, A. A,; Young, D. J . Phys. Chem. 1972, 76, 4015. (13) Kobatake. Y. J . Chem. Phvs. 1958. 28. 146. (14j Kobatake; Y.; Takeguchi,”.; Toyoshima, Y.; Fujita, H. J . Phys. Chem. 1965.69, 3981. (15) Kobatake, Y.; Toyoshima, Y.; Takeguchi, N. J . Phys. Chem. 1966, 70 _ ,1187 -.-.
(16) Sidhar, V. S.; Ruckenstein, E. J . Colloid Interface Sci. 1981,82, 439; J . Colloid Interface Sci. 1982, 85, 332. (17) Neogi, P.; Ruckenstein, E. J . Colloid Interface Sci. 1981, 79, 159.
0 1988 American Chemical Society
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geometrically complicated bounding surfaces are replaced by a long, narrow cylindrical tube having a stick boundary condition. Simultaneously, the fixed real charge or dipole charge associated with the membrane structure spreads uniformly over the exposed surfaces inside the tubes, and the electrical potential gradient arising from this surface charge and that due to the membrane potential become orthogonal. On the basis of the easily interpretable single-tube solution, an entire membrane is then built from a bank of these tubes with effective pore diameters and lengths reflecting the macroscopic porosity, permeability, and membrane thickness as might be approximated by means of the Kozeny-Carmen relations. Few attempts have been made to remedy the deficiencies of the capillary model. Mehta and Morse,18 however, suggested a cell model based on a lattice of uniformly charged spheres in which the lattice constant and sphere radii are variable, thus specifying the porosity. They showed that the cell model rejected salt more effectively than the capillary model did and that at the limit of high porosity the two models gave identical results.
Development of the Model A critical examination of the actual situation that exists in swollen polyelectrolyte gels when these are used as membranes indicates that the modeling, on a microscopic scale, of dynamic flow and the electrical potential on the basis of the capillary model is a poor one. Indeed, structural studies have that a swollen cross-linked polyelectrolyte gel, functioning here as the active layer of an R O membrane, is approximately equivalent to a dense solution of stationary expanded and overlapping polymeric coils. In actual fact, the cross-linking sites22,23and the Gaussian nature of the segment distribution give rise to regions of higher average polymer-chain-segment concentration, causing spatial permeability and charge fluctuations. The inhomogeneous distribution of electrolyte resulting from fixed-charge inhomogeneities3s4is known to be reflected in the concentration dependence of salt rejection in RO. This same phenomenon has also been proposed as being responsible for the high current efficiencies observed in electrodialysis applications of the phase-separatedZ4 cation-exchange Nafion membrane. In this regard, Reiss et a1.25*26 have employed an inhomogeneous charge model combined with the Nernst-Planck-Poisson triad in order to prove this positively. These fixed-charge inhomogeneities, however, have not been adequately addressed in the design of any dynamic model treating convective transport. A more accurate treatment of R O applications of charged-gel membranes would be a discussion of the process in terms of the actual topography and polymeric nature of the membranes and of their structure and properties. Much preliminary work has been devoted to determining, from a vis~ o m e t r i c ~or? -sedimentation ~~ coefficient3e36 viewpoint, the hy-
(18) Metha, G. D.; Morse, T. F. J. Chem. Phys. 1975, 63, 1878. (19) Pines, E.; Prins, W. Macromolecules 1973, 6 , 888. (20) De Gennes, P. G.; Pincus, P.; Velasco, R. M. J. Phys. (Paris) 1976, 37, 1461. (21) Nystrom, B.; Roots, J. Prog. Polym. Sci. 1982, 8, 333. (22) Nagy, M. Colloid Polym. Sci. 1985, 263, 245. (23) Weiss, N.; Silberberg, A. Br. Polym. 1977, 9, 144. Weiss, N.; van Vliet, T.; Silberg, A. J. Poiym. Sci., Polym. Phys. Ed. 1979, 17, 2229. (24) Hsu, W. Y.; Gierke, T. D. J. Membr. Sci. 1983, 13, 307. (25) Reiss, H.; Bassignana, I. C. J. Membr. Sci. 1982, 11, 219. (26) Selvey, C.; Reiss, H. J . Membr. Sci. 1985, 23, 11. (27) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca and London, 1978. (28) Kirkwood, J. G.; Riseman, J. J . Chem. Phys. 1948, 16, 565. (29) Felderhof, B. 0.; Deutsch, J. M. J. Chem. Phys. 1975, 62, 2391. (30) Brinkman, H. C. Physica (Amsterdam) 1947, 13, 447. (31) Debye, P. Phys. Rev. 1947, 71, 486. (32) Debye, P.; Bueche, A. M. J. Chem. Phys. 1948, 16, 573. (33) Wiegel, F. W. Lect. Notes Phys. 1980, 121, 102pp. (34) Nystrom, B.; Roots, J. J. Macromol. Sci., Rev. Macromol. Chem. 1980, C19, 35. (35) Mijnlieff, P. F.; Jaspers, W. J. M. Trans. Faraday SOC.1971, 67, 1837.
Kuehl and Sanderson drodynamic flow field in and around an isolated free or partially draining macromolecular coil. A related but separate problem is that of the electrical potential in and around a charged polyelectrolyte coi137-39or rod as related to its expansion coefficient or conformation.3743 In this context, one characteristic function is required to describe both the spatial distribution of the mass and the charge on the polymer chain. In the active layer of an actual charged polyelectrolyte membrane, the entire volume, excluding that occupied by the polymer itself, is accessible by solvent,37 usually water, in its bound and unbound forms. As the local volume fraction of polymer and water may change in space, the local dielectric constant and viscosity of that fraction may also change. The topography of the network structure is such that there are no macroscopic channel^^^,^^ traversing the membrane free of monomer in which Poiseuille-like flow can be assumed to occur. Neither are there boundaries nor bounding surfaces to create a situation in which a charged-wall boundary condition could realistically replace what the network structure insures as being present as a space charge. Many workers model infinitely dilute polyelectrolytes in the absence of added salt as semi-infinite parallel aligned charged rods. Since the situation that is known to exist under actual operating conditions of polyelectrolyte membranes cross-linked in their hypercoiled state (like the dynamic composite membranes) differ, such a model may justifiably be ignored. The trajectory of a particle traversing the active layer of such a membrane will be ultimately determined by the spatial mass distribution of resisting points it encounters as well as the pressure gradient. The mass-distribution function characterizing the membrane as experienced by an ion or water molecule traveling through it will be quasi-onedimensional and ideally will be a series of random &function peaks corresponding to chain obstructions. Regions within a real membrane in which concentration gradients of mobile ions have been created which are perpendicular to the flow direction should spatially cancel out, thus minimizing the contour length trajectory of an ion as compared with a neutral solvent molecule, for instance. All particles entering different parts of the membrane will, on average, experience the same distribution function as they travel through it. A one-dimensional model of transport based on an inhomogeneous permeability and charge density is reasonable in that all particles are confined to flow in the same average direction and because the average extent to which off-axis convective or diffusive flow occurs in a highly swollen polyelectrolyte gel relative to on-axis flow is small. In other words, the tortuosity of a polyelectrolyte gel membrane can be treated as negligible in this model. In the further development of the model, the continuity of the threadlike polymer chain dictates that neighboring charges along the length of the chain will be in close proximity and that it can be expected that distances between charges located on distant segments of the same chain or on the loops of different but overlapping coils will be much greater. This permits a visualization of the Coulomb potential as a sheath or double layer enveloping the contour of the polymer chain. Common counterion concentrations dictate that the inverse Debye screening lengths should be quite small. The potential, therefore, will be high near any chain segment and approximately constant along the length of (36) Wiegel, F. W.; Mijnlieff, P. F. Physica A (Amsterdam) 1976, 85A, 207. (37) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (38) Rice, S . A.; Nagasawa, M. Polyelectrolyte Solutions; Academic: London and New York, 1961. (39) Harris, F. E.; Rice, S.H. J. Chem. Phys. 1956, 25, 955. (40) Kimball, G. E.; Cutler, M.; Samelson, H. J. Phys. Chem. 1952, 56, 57. (41) Wall, F. T.; Berkowitz, J. J. Chem. Phys. 1957, 26, 114. (42) Nagasawa, M.; Kagawa, I. Bull. Chem. SOC.Jpn. 1957, 30, 961. (43) Fixman, M. J. Chem. Phys. 1964, 41, 3772. (44) Peterlin, A.; Yasuda, H.; Olf, H. G. J. Appl. Polym. Sci. 1972, 16, 865
(45) Yasuda, H.; Lamaze, C. E.; Peterlin, A. J. Appl. Polym. Sci., Part A-2 1971, 9, 1117.
Electrokinetic Salt Rejection the chain; along the flow axis, on the other hand, the potential to which an ion is subjected will vary rapidly with minima between distant loops of the same chain or segments from overlapping coils. The tendency for ions to travel perpendicularly to the flow direction in an effort to follow a minimal potential surface is regarded as negligible. Unlike the capillary model, this one-dimensional model assumes the internal potential gradient due to the space charge and screened by a spatially decreasing electrolyte concentration to be parallel to the axis of flow and therefore parallel to the direction of the gradient of the generated membrane potential. Also, the membrane will begin to differentiate between ions before solute actually enters the membrane physically. These features are two of the main ones distinguishing this model. This definition of the structure of a polyelectrolyte gel questions the applicability of the simple uniform or smeared charge model used commonly in the treatment of the potential withiqcoiled polyelectrolytes. It also questions the related Donnan potential approximation which is its equivalent in the membrane field. This work evaluates theoretically the electrokinetic salt rejection by a hypothetical one-dimensional nonhomogeneous membrane having different empirical mass distributions. The model membrane operates under the usual hyperfiltration conditions typified in cross-flow RO. A semi-infinite sheet of membrane material separates two aqueous reservoirs, one containing a binary and symmetrical fully ionized electrolyte a t a feed concentration C1 and a pressure PI while the permeate side is at a pressure Pzand a concentration C,, this latter concentration being determined by the rejection efficiency of the membrane. The high cross-flow velocity usually causes a turbulence that allows concentration polarization to be neglected. The solution that enters the membrane undergoes one-dimensional viscous flow under the pressure gradient acting perpendicularly to the membrane surface. Salt rejection is assumed to be due entirely to electrokinetic phenomena, and exclusion due to ion size (sieving) is absent from the model. The effect of the chemical environment on the propagation of the electrical potential is introduced by letting the local dielectric constant vary with the polymer volume fraction. The amount of bound water not able to flow, which presumably excludes ions, is introduced by adding a 1-8, sheath of water around the fictitious mass distribution of which the membrane is composed (this can be varied). The ions take up a Boltzmann distribution within the potential field created by the fixed-charge distribution and that due to the membrane potential inside the membrane. The electrical potential everywhere is determined by numerical solution of the Poisson-Boltzmann equation containing the source term representing the space charge. The Poisson-Boltzmann equation, in addition, is solved simultaneously with the first-order nonhomogeneous differential equation arising from the NernstPlanck flux equations which describe the concentration profile of ions through the membrane. The porosity, permeability, and charge density are all related to a principal function p , the mass density profile through the membrane. Because it is not possible to use an actual spatial, unidirectional distribution function taken from a real polyelectrolyte gel, various synthetic forms that might justifiably be equivalent to a real mass distribution or at least be instructive are used. In this regard, models based on the actual random placement of masses are abandoned, and conveniently easily generated continuous and periodic functions are adopted instead. With the same average mass density, this continuous mass is distributed in different ways with different degrees of inhomogeneity, and the levels of membrane effectiveness are then compared and contrasted. For real one-dimensional flow to be possible, the mass density must be imaginary, as if it had nearly zero volume, so that as a source of friction the mass density will be fictitious from a hydrodynamics viewpoint. A statistical argument is invoked whereby the charge density is related to the mass density by only a multiplicative constant factor in that all sites in time are neither fully charged nor completely ion paired, but continuously fractionally charged. This constant factor becomes the principal parameter in the model and must be determined self-consistently. Ion condensation of
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988 519 these charged sites due to pH changes, different local electrolyte concentration, and local electrical potential will be considered in future studies.
Mathematical Formalism A. Hydrodynamic Equations. In continuation of the discussion of the previous section, the pressure-driven flow through the model membrane is treated by invoking the concept of macromolecular permeability developed by D e b ~ e , ~Bl Jr i~n l ~ n a nand , ~ ~ other^.^^^^^ Before sacrificing generality, first consider the complete Navier-Stokes equation describing flow in macromolecular solutions:
a3+ po(3.b)ii= -bP
dt
+ qoa23+ P
(1)
and subject to the incompressible flowfequirement b.? = 0. Here, of the solution and Vis the mass-centered fluid velocity at a point 7 relative to a coordinate system fixed to a macromolecule and equal in these circumstances to the velocity of the solvent. P is the fluid pressure in the membrane phase. In the high-pressure limit it equals the hydrostatic pressure, which is myth greater than the osmotic pressure. The viscosity is qo, and F represents the vector components of forces resisting the flow. Applied to polyelectrolytes, these forces will, for our purposes, be a viscous (v) and an electrical (e) resisting force. To illustrate the flow through a polyelectrolyte gel membrane, imagine a partially free-draining macromolecule acting as an immobile porous plug through which solvent is driven. Effectively, we then consider the flat sheet gel as being a giant macromolecule with solvent flowing through it in one direction and apply some simple hydrodynamic formulas developed for macromolecules and other porous media.33*46Assign to this porous body an isotropic (scalar) but nonhomogeneous permeability k(r) even though more complex representations are possible. In creeping-flow situations characteristic of flow through porous materials as dense as macromolecular coils, the viscous force per unit volume resisting the flow can be written as Darcy's law: po is the density
Taking a continuum approach, we represent the macromolecule as a cloud of spherical segments of radius a, a Stokes-Einstein a , a number density p(r). In both friction factor f = 6 ~ ~ oand micro- and macroscopic approaches to this problem,29 the permeability has been shown to be related to the segment density by
(3) For the electrical body force, we define pt as being the difference in local molar densities between mobile cations and anions wjich can xetard the flow. This electrical body force becomes F, = -ep,V@, where @ is the total local electrical potential inside the gel, and is due to both the fixed and mobile charge distributions and the generated membrane potential, as formally expressed by the Poisson-Boltzmann equation. Operationally, a steady-state flow condition through the membrane is assumed and the nonlinear convective term which is small for low Reynolds numbers, Le., creeping-flow situations, are neglected. The Navier-Stokes equation, after insertion of the body force terms, becomes qob2ii(i)=
b~ +fp(qC(i)
+ eptao(q
(4)
In this one-dimensional model, the segment density is specified as being nonhomogeneous, varying along the flow axis perpendicular to the membrane surface denoted by a coordinate z. We may then ignore the Laplacian and write the velocity as a modified Debye-Brinkman equation:
(46) Bear, J. Dynamics of Fluids in Porous Media; American Elsevier, New York, 1972.
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The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
This flow profile satisfies the irrotational flow criterion identically. Application of the incompressible flow criterion yields a complicated conservation equation that would be quite intractable without further simplification. By a dimensional analysis using some actual results from this study (to be described later), it was found that the second term on the right of eq 5 expressing the contribution of the electrical potential to the velocity can be neglected in most practical situations for pressures over 2.0 mPa ( X lo6 Pa) by using usually encountered values for up)-’ and the volume fraction of polymer, the values of which are assigned later. Such an assignment specifies a solvent flux that is much higher than usually observed in R O processes; however, it is justified as the intention here to expose trends. This simplification decouples the velocity from the electrical resistance and vice v_er:a. The boundary conditions are applied under the condition V.V = 0 to yield a velocity constant through the membrane: where ( p ) is the average segment density and 1 is the membrane thickness, while A P = PI - P2is the absolute pressure difference across the membrane. B. Flux Conservation Equations. The influx and outflux densities of mobile cations and anions across any point in the membrane are determined through a balance of the diffusive flux, convective flux, and the flux associated with the internal and the membrane potential gradients at that point. Remembering that the internal and membrane potential gradients are parallel in one dimension, a suitable Nernst-Planck flux equation for the mobile ions, which will describe this process, is written j* = Vi*(Z) - D*VF*(z)
Kuehl and Sanderson d In (1 - a ( z ) ) dz
dz
where $(z) is the normalized potential e@(z). It has been assumed that the ions are geometrically symmetric (D+= D- = D). C. Poisson-Boltzmann Equation. The electrical potential in the membrane phase satisfies the Poisson-Boltzmann equation in terms of the total charge density pt:
v@= -(4X/€€o)pt
The charge density pt in the membrane phase is the sum of both the fixed and mobile-charge densities. Employing eq 8 gives Pt = el(c(z)/2)(1
- &))bp(-+(z))
- exp(d(z))l + pf(z)l (1 1)
The fixed charge density p&) is related to the density of segments p(z) by a factor p’ so that pdz) = Ofep(z). Normalizing and letting the local dielectric constant vary linearly with polymer volume ) ~ (1 a ( z ) ) in one dimension gives fraction as ~ ( z =
Equation 12 together with eq 9 determines the simultaneous problem to be solved numerically. The normalized Poisson-Boltzmann equation applicable to either side of the membrane is written in terms of the solute concentration in either reservoir C d2+(z)
(7)
4se2P =C sinh +(z)
dz2
Herej* is the flux of singly charged positive or negative ions ( e ) per unit area and D* represents their diffusion constants, while i&) represents the concentration of mobile cations or anions inside the membrane phase relative to a position outside either side of the membrane where the potential is zero. The distribution of cations and anions, subject to the zero net current stipulation, varies across the membrane, decreasing from the feed side to permeate side and determines the final flux that specifies the “performance” of the membrane. The profile of ions partially determines, and is partially determined by, the potential O(z). The local volume fraction of the gel that consists of polymer segments is inaccessible to solute and solvent and is directly related to the segment density and the segmental volume. Denoting this 7ra~ local volume fraction as a(z),we have a(z) = p ( ~ ) ( ~ / ~ ) where a as a radius might incorwrate a sheath of bound water associated with each segment. The actual concentration of ions inside the gel membrane is corrected by the fraction of free water available (1 - a(z)). The Boltzmann equation for this situation is written
where x represents higher order Coulombic or non-Coulombic contributions to the potential, and although treated explicitly by Harris and Rice39and by F i ~ m a in n ~their ~ studies of polyelectrolytes, this term will be omitted here. Equation 8, through the function c(z) satisfies the conditions that, outside the membrane where a and @ are zero, the total ion concentration c+ + c- must equal C,or C2,the feed or permeate concentrations. The concentration of solute changes from the feed side to permeate side. In regions where the potential is not zero, c(z) for convenience represents a fictitious concentration of solute at that point relative to a situation where a equals zero at that same position. The solute flux, j = j + + j - , must therefore correspond to the flux of this fictitious solute and must self-consistently satisfy the boundary conditions mentioned above. In accordance with the precedent contained in the l i t e r a t ~ r e , ~it” is assumed that this flux can be written as a product of the solvent velocity and the mobile ion concentration, J = Vc(z). Coupled with the zero net current stipulation applied everywhere, I = 0 = e(j+ - j - ) , the flux equations yield a nonlinear first-order differential equation written in terms of this concentration c(z) as
(10)
+(z) = 2 In
di(z) dz
‘€0
1
+ tanh (+w/4) exp(-4rz)
1 - tanh (+,/4)
exp(-4xz)
1
(14)
47re2/3
-= -(c(z) sinh d(z) - et0
1 - a(z)
(47) International Mathematical and Statistical Libraries, Inc., Houston,
TX .
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988 521
Electrokinetic Salt Rejection
100
90
Numerical values for the constants which are not universal were set as a = 5 A, t = 78, T = 25 OC,and viscosity qo = 0.02 P. The diffusion constant D = 2.0 X 10” cm2/s was used, supposedly representing the isotropic value of a concentration and electrical potential independent diffusion constant of a simple binary and symmetrical salt like KCl. The above viscosity value is assigned a figure that is indicative of a high fraction of bound water, similar to water near its freezing point. For all calculations a reference condition was chosen arbitrarily such that the average volume fraction of polymer plus the bound water in the membrane was 10.0%. Furthermore, when the parameter is allowed to behave as shown in eq 15c,d, p’ will take on a value such that when given a feed concentration C1of 0.01 mol/L, a membrane thickness of 1000 A, and an absolute pressure difference AP of 2.0 mPa, a permeate concentration C, of 0.002 mol/L, i.e., a rejection of 80%, will be obtained if a completely homogeneous mass density ( p ) = 0.1/(4/37ra3) is assumed. These concentrations are defined as being the sum of the concentrations of anions and cations a great distance from the membrane where the potential is zero. Two initial and two final boundary conditions may then be specified. Near the membrane surface ions accumulate to concentration levels greater than that in the feed whereas within the membrane, this ion density is again reduced, corrected by the volume fraction of space available to them. From eq 8 we have at the boundaries ~(0= ) C,( 1 - a(0)) cash $(O)
~ ( 1= )
C2.1 - .(I))
(16)
cash $ ( I )
Next, the electrostatic boundary conditions must be satisfied. The magnitude of the potential upon leaving the membrane phase on either side must be continuous and undergo a point of inflection at the point where the charge density reverses sign. At the surface of either side of the membrane, the charge density is equal in magnitude but opposite in sign to the charge density of the electrolyte solution in contact with it. From eq 12 and 13 47re2@
--
‘‘0
C sinh 4(z) =
47re2/3
1
B’p(z) (17) -c(z) sinh d(z) - -
‘
e0
1 - .(z)
where C = C, and C2 when z = 0 and I , respectively, invoking eq 16. Problems arose in the use of this boundary condition in that the computation converged to the other, unique, but also unphysical, solution where the potential approached the point of inflection on the permeate side with positive curvature instead of the correct negative curvature. To avoid this ambiguity, a condition for the electric field at each surface was substituted, derived from the exact solution to the potential outside the membrane. At the points of inflection, the electric field -d+(z)/dz has a maximum value, a positive slope on the feed side, and a negative slope on the permeate side of the membrane. The first derivative of eq 14 is written
-dJi(z) - - *2( dz
z C ) ” 2 sinh (Ji(z)/2)
(18)
which is positive for z = 0 and c = C1and negative for z = 1 and c = C2,respectively. The electric field is discontinuous at the interface where the dielectric constant changes abruptly. In order to be used as a boundary condition within the membrane, eq 18 was corrected with the dielectric constant inside the membrane phase. When this procedure was followed, a unique solution to eq 15 was arrived at which yielded a value of 0.1 12 689 6 for p’. The tolerance used in all calculations was set at 1.0 X This value of p’ implies that if the specified reference condition is to be met, each segment should have a fraction of the fundamental charge corresponding to p’. In the homogeneous case, this value gives a fuced-charge density of 3.4435 X lo3C / L which is approximately
80 70
E
60 50 40 2.2
30
20
,
,
,
10
20
30
,
#
40
50
#
60
#
70
,
80
.
30
180
n Figure 1. Rejection versus n for pressures between 2.0 and 3.6 mPa.
-
equal to the concentration of electrolyte in the feed. According C,) would lead to the Donnan principle, this situation ( p ’ ( p ) to a reasonably rejecting membrane, in this case about 86.27%. Under this reference condition, the second term on the right of eq 5 was well below 1.0% of the pressure gradient term across most of the membrane thickness. Once a value of @’ had been determined, the function p’(z) in eq 15c was replaced by this constant value and eq 15d was eliminated, thus reducing the problem to one of three coupled firstorder, nonlinear differential equations requiring two initial and one final boundary condition. The set of two on the feed side becomes the equation for c(0) as in eq 16 and 18 modified for the internal dielectric constant evaluated at z = 0. As it is known that on the permeate side the potential must approach z = 1 with a negative curvature, eq 18 is used but with the external concentration replaced by c([)/[(l - .(I)) cosh 4(I)]. The potential, concentration, and charge-density distributions obtained in this manner were identical with the results obtained in the calculation by which p’ was determined. Once this algorithm had been established, the membrane performance could be easily evaluated for charge-density fluctuations about the mean value ( p ) defined previously. Initially a function was chosen having a single Fourier component giving an antisymmetric fixed-charge density. The mass density became P(Z) = (P)[l + sin (27r(n/l)z)l (19) where n is an integer, such that the charge density will fluctuate between 0 and ( p ) for all wavelengths. To facilitate the computation of this highly nonlinear problem, eq 19 was parametrized with a factor forming a product with the trigonometric function such that this fluctuating term starts at a small value, in which case the numerical solution is approximately that of the homogeneous case, and when the factor tends to one, the desired solution is arrived at. Data from previous computations were passed on as initial values for subsequent calculations until the chosen problem was solved. A similar procedure was required when the pressure was varied.
Results Data were accumulated for absolute pressures of 2.0-3.6 mPa and wavelengths (lOOO/n) of 14.28- A. Due to the large amount of information produced, we have chosen to present in graphical form the results of only a few of the calculations of this series. Rejection, 1 - C2/C, versus n for different pressures, for instance, is plotted in Figure 1. The data here and in subsequent graphs were supplemented with interpolated points generated through a cubic spline algorithm supplied by ISML routine ICSCCU. This enabled the number of calculations to be reduced considerably below that which would have been required to display actual calculated data graphically with a continuous smooth curve drawn through all points. The total charge density pt, the fictitious concentration of ions c(z), the normalized potential 4(z), and the total concentration of mobile cations and anions c(z) cosh $(z) for a homogeneous distribution (n = 0) are displayed in Figures 2, 3, 4, and 5 ,
Kuehl and Sanderson
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
522
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respectively, for different pressures. In the plots of total charge density, normalized potential, and total mobile ion concentration the values of these parameters as plotted represent also what occurs outside the membrane surfaces on both the feed and permeate sides in order to show the situation that exists before solution actually enters or leaves the membrane. Discontinuities in the total charge densities and total mobile ion concentrations are evident a t the boundaries due to both the presence and absence, respectively, of the fixed charge within and outside the membrane, the abrupt change in polymer volume fraction, and dielectric constant. Figures 6-9 show these parameters for three different wavelengths, m, 40, and 14.28 A at a single pressure of 2.0 mPa. Figures 10-13 show the pressure dependence at a wavelength of 40 A, for the same four variables.
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988 523
Electrokinetic Salt Rejection
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Discussion and Conclusion As with most theoretical treatments of purely hypothetical models, an analysis of the results of such a procedure can be based solely on a discussion of trends observed when the various parameters in the problem are varied in importance relative to some standard condition. Because of the coupling between, and the abstruse nature of, the variables that define the equations that in turn define this simple model, identification or isolation, however desirable, of specific factors that might be regarded as essential in controlling any trend, is difficult. In this context, it is advantageous to discuss the observed trends that characterize membrane performance in terms relative to the behavior of the simplified case which defines, in part, the reference
Figure 13. Concentration of mobile anions and cations c(z) cosh @ ( z ) for a wavelength of 40 8, for different pressures.
condition around which more complicated situations can be built. If the discussion depends on the results of the homogeneous case, there will also be provided a framework for internal consistency. In opening, it should be recognized that this defect-free model predicts that a 100%rejection will ultimately be attained and quite rapidly as the pressure is increased from 2.0 mPa to only about 3.0 mPa as shown in Figure 1. This maximum rejection, moreover, is not approached asymptotically with pressure. It is significant that in this model the electrical potential gradient becomes strongly coupled to the pressure to the extent that it is able to exceed the diffusive and convective terms in the flux equation and totally prevent ions from reaching the permeate side characterizing a "perfect" membrane. Unlike the capillary model, this is undoubtably a result of the membrane potential gradient being parallel to and coupled with the potential gradient due to the fixed charge. A minimum in the rejection coefficient with n is observed for each pressure; this minimum becomes shallower as the pressure approaches about 3.0 mPa. The significance of the value of n at 25), can be better unwhich this minimum occurs, n* (n* derstood by inspecting Figure 8 in which normalized potential is plotted for n = 0, 25, and 50, two of these values being less than, and one greater than, this value of n*. As an example of the previously delineated difficulty of decomposing certain trends observed in the data in an attempt to offer a sound physical explanation, consider the wave forms of the calculated potentials displayed in Figure 8. Although they would not be expected to resemble a simple trigonometric relation such as that employed for the fixed-charge density, it is apparent that these wave forms exhibit many characteristics. Although the wavelengths of the oscillatory potential for the nonhomogeneous cases are consistent with the wavelength of the charge density, as is demonstrated more directly in Figure 10, for example, where the second derivative of the potential is plotted for n = 25, the amplitude is aperiodic while the profile is, predictably, asymmetrical. This aperiodicity is also evident from the plot
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524
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
(Figure 11) of the fictitious concentrations for the same wavelength. In addition, the crests in the potential, being regions of negative charge density, are smaller than the troughs whereas the widths of the crests are also smaller than the widths of the troughs, and the sum of both widths conforms to the wavelength of the imposed charge-density profile. The oscillation of the potential is also slightly out of phase relative to the charge density, and this phase shift changes with the value of n. A mathematical deconvolution or decomposition of this wave form, however desirable it would be, would first require identification of the wave form as a member of a particular class. It is questionable whether such a procedure would be successful or justifiable. If the membrane performance, as far as achieving a high rejection coefficient is concerned, is based on the criterion that the average electrostatic potential within the membrane should be high, it becomes obvious from Figure 8 that, as the membrane becomes less homogeneous, the potential oscillates with a large amplitude, the average value of which is less than that for the homogeneous case, up to n* for any pressure, with a corresponding decrease in the rejection. As the wavelength of the oscillation is reduced further below n*, an amplitude-damping effect is apparent, with the additional consequence that the average magnitude approaches the homogeneous case, corresponding to an increase in rejection. The physical origin might in part be due to a reduced tendency of the Coulomb potential to effectively screen itself over the intermediate wavelengths of the fixed-charge fluctuation. This tendency is decreased by the compensating effect of increasing convective flow through pressure. This relationship, however, is too heavily obscured to permit analysis. It should be noted that a similar trend was observed in the theoretical modeling26of the current efficiency of inhomogeneous Nafion membranes using a symmetric fixed-charge distribution of variable cluster size. An enhancement of the current efficiency found in this case was above that expected for a homogeneous membrane or that based on a Donnan equilibrium. The explanation for the critical wavelength at which their current efficiency reached a maximum is similar to that that for why the rejection has a minimum. It can be assumed that as the wavelength of the charge distribution approaches zero, the potential will oscillate with a vanishingly small amplitude about the homogeneous case. Although the average potential for a rapidly varying fixed-charge density approaches the potential envelope resembling that of a homogeneous distribution, the total charge density has a tendency to vary rapidly about electroneutrality. As shown in Figure 5, the total charge density fluctuates out of phase with the fluctuation of the fixed charge density. Thus a fixed-charge fluctuation of the order of the separation of charges on a polymer chain is almost indistinguishable from a homogeneous distribution. It is important to note also that, in all circumstances, the normalized potential is everywhere very much larger than what is permitted in the use of the Debye-Huckel approximation. The trends that occur as a result of pressure changes are more definite. Increasing the magnitude of the flow velocity through the membrane alters the distribution of ions, causing them to be rejected more rapidly than at lower pressures; the rate of change is apparently linear with respect to the applied pressure (Figures 3, 7, and 11) where the fictitious concentration is monitored. Because the ions are rejected more effectively through the membrane, the potential becomes less screened, and this is reflected in higher magnitudes biased toward the permeate side, as shown in Figures 4 and 12. The effect of the flow velocity on the charge density and the distribution of cations and anions is much less pronounced (Figures 2, 5, 10, and 13). At this stage, the opportunity is presented where the Goldman constant field and the Donnan/electroneutrality approximations applied to nonconvective membrane transport may be evaluated for this more complex description of membrane processes. Using a perturbative approach to the Nernst-Planck-Poisson system of equations, M a ~ G i l l i v r a yshowed ~ ~ ~ ~ ~that, for nonconvective (48) MacGillivray, A. D. J . Chem. Phys. 1968, 48, 2903.
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transport through completely homogeneously charged membranes, these two assumptions represent two different limiting cases of the full system. The constant field limit assumes that the potential is either constant or a linear function of distance through the membrane. It is valid when the expansion parameter, the product of the fiied-charge concentration and the square of the membrane thickness (diffusion barrier), is small. In the opposite limit, a very thick membrane, an electroneutrality condition is approached. It is clear from the plots of normalized potential and charge density that, for convective transport through a membrane described by our reference condition, a state of virtual electroneutrality exists throughout most of the membrane's interior in addition to the two surfaces for any pressure but only for a homogeneous fiied-charge distribution. At the lowest pressures, only in these homogeneous cases is the potential nearly linear in most of the interior; however, the membrane at 1000 A is too thick to allow the other limiting case to be tested. To further compare the Constant field/electroneutrality condition, the normalized potential and charge density for this model membrane operating at its reference pressure are plotted for different thicknesses (Figures 14 and 15). In accordance with MacGillivray's findings for nonconvective transport, it was found that in the case of convective transport the electroneutrality assumption does indeed become poor for thin membranes. For thick membranes only a thin region at the surfaces carries any substantial charge. For thin membranes, although the potential becomes almost constant, the charge density becomes very high. Thus for convective flow through this model membrane, an electroneutral state is most compatible with a constant field, but linear potential for thick membranes. For thin membranes, less than about 100.0 A, a zero field-constant potential state exists with large deviations from electroneutrality. (49) MacGillivray, A. D.; Hare, D. J . Theor. Eiol. 1969, 25, 113.
J. Phys. Chem. 1988, 92, 525-532
In addition, the rejection is strongly dependent upon membrane thickness and reaches a maximum for thicknesses of about 1000.0
A.
Investigations in which the more practical aspects and applications of the onedimensional model presented here will be studied are being carried out in order to gain insight into real R O membrane processes. This paper is intended to serve as a theoretical foundation. Future studies will include specific modifications to the model, omitted here, such as membrane defects, the suppression of charge density of the Manning type due to the interrelation of the degree of ionization with electrolyte concentration
525
and with local potential, applied pressures nearer to commonly encountered osmotic pressures, an ion velocity possibly less than the velocity of the solvent, different sized anions and cations, and a possible modification for an environmentally dependent diffusion constant. Acknowledgment. We thank N. Dowler and A. van Reenen for interesting discussions in the dynamic membrane field and Dr. J. Smit and Dr. J. Buys from the Mathematics Department. Financial support from the Water Research Commission is gratefully acknowledged.
Thermal Modulation of Rotating Disk Electrodes: Steady-State Response J. L. Valdes and B. Miller* AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received: May 12, 1987)
Absorption of heat from a laser beam on the back of a thin rotating disk mounted on a hollow tube affords a ready way to modulate the electrode temperature. Steady-state response of the electrochemical reaction parameters to such heat input is a function of the surface temperature and thermal gradients developed under convective cooling. This interplay is dependent upon the heat input level, the rotation speed, and the physical properties of the electrode and solution. A theoretical analysis of this process and its experimental verification are presented using the two important electrochemical cases of zero-current potentiometry and mass transport limited currents.
Introduction constant potential were obtained, along with those of temperature. A limitation in this method is the lack of control over natural Temperature is an important factor in determining electrode convection during a relatively long transient (seconds). potentials and charge-transfer rates in electrochemical systems. Miller’ described a method called thermal modulation volThere are a number of major contributions to this temperature tammetry (TMV) in which an argon ion laser beam was used to dependence of a given electrode process, including the temperature heat the absorptive back of a thin disk electrode of hollow tube coefficients of standard potentials (entropy changes), double-layer design. Chopping the beam at 5-10 H z produced a periodic effects (surface potential and concentration distribution), the temperature change at the convectively cooled rotating disk kinetics of charge transfer (activation energies), and mass transfer electrode (RDE) and a corresponding current modulation which and conductivity (mobility factors). In the typical electrochemical was extracted and recorded under controlled potential scan. experiment, temperature is fixed at one or a series of values while Electrochemical processes respond to such temperature modulation other fundamental characteristics (viz. potential, current, mass according to their respective rate or transport dependencies on transfer) are programmed or measured in some manner. this variable. However, some efforts have been made where temperature is This mode of interface heating (nonsolution side of the e!ecthe forcing variable. Heating the electrodesolution interface in trode) has been used by Smalley, Krishnan, Goldman, and single or periodic events is a mode of introducing thermal perFeldbergs to apply submicrosecond perturbations in order to study turbation to an electrochemical experiment. One limit is a narrow electron-transfer rates through open circuit potential relaxation. (in time) temperature jump, with observation of the relaxation In such experiments, the heating power is about 105-106 times of the system to the ambient, as was well-established for homohigher than in TMW, which allows probing of fast events, though geneous chemical reactions by Eigen.’ A scheme for diode laser the temperature changes involved in both these different schemes irradiation of a dropping mercury electrode was suggested by are in the range of a few degrees kelvin. This works also points Barker and Gardner2 in 1975. Nanosecond dye laser irradiation out that the temperature differential leading to these open circuit of the mercury surface to study the electrical double layer was performed by Benderskii and Veli~hko.~Harima and A o y a g ~ i ~ , ~ potential excursions may include contributions from the Soret effect, double-layer reorganization, and metal-metal junctions. used resistive heating (A1 film coated with Mylar and then a gold In the present paper we wish to begin to interpret the response electrode) and a IO-ms voltage pulse to observe redox couple of TMV in a quantitative manner in order to utilize the inforrelaxations in the few milliseconds time scale. mation contained in thermal perturbations of an electrode system More recently Gabrielll”, Keddam, and Lizee6 utilized highwith rigorously understood hydrodynamic conditions. We have frequency (250 kHz) heating of a wire electrode, which is also theoretically analyzed this system and experimentally studied the a leg of a Wheatstone bridge, allowing its temperature excursion cases of open circuit potentials and mass transfer limited currents and relaxation to be directly measured. Current transients at for their steady-state response to laser heating input. In a subsequent paper,g we will offer the same for the transient and fre-
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(1) Eigen, M. Discuss. Faraday SOC.1954, 17, 194. (2) Barker, G. C.; Gardner, A. W. J. Electroanal. Chem. 1975, 65, 95. (3) Benderskii, V. A.; Velichko, G. I. J . Electroanal. Chem. 1982, 140, 1. (4) Harima, Y.;Aoyagui, S. J. Electroanal. Chem. 1976, 69, 419. (5) Harima, Y.; Aoyagui, S. J. Electroanal. Chem. 1977, 81, 47. (6) Gabrielli, C.; Keddam, M.; Lizee, J. F. J. Electroanal. Chem. 1963, 148. 293.
0022-3654/88/2092-0525$01.50/0
(7) Miller, B. J. Electrochem. SOC.1983, 130, 1639. (8) Smalley, J. F.; Krishnan, C. V.; Goldman, M.; Feldberg, S. W. Ab-
stracts of Papers, Electrochemical Society Meeting, Philadelphia, PA, May 1987. Feldberg, S. W., private communication. (9) Valdes, J. L.; Miller, B., to be submitted for publication.
0 1988 American Chemical Society
