K
~A. A. Sonin, ~ and &8.~ Y ~ n g ~
~
i
~
~
~
I)~:iar!n'?entof Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachuseris 02 139 iPncs;veci M a y 30, 1972) Pi/Slic,ztiori costs
assisfed ~y the Office of Saline IpJater, ii. S. Department of the Interior
A theory
i s presented for the salt rejecting characteristics of porous membranes, whose pore size Is large coinpared v~ithmolecu!ar dimensions, under reverse osmosis conditions where t h saline solution is moved through the membrane pores by a n applied pressure gradient. The salt rejection mechanisn; i s assumed to be an electrokinetic one resulting from charge bui1.t up on the interior surfaces of the materia i wh.en in contact with the salt solution. The performance of the porous material is shown to depend on three parameters: the ratio of Bebye length to effective pore radius; a dimensionless wall potentia! related to the j- potential; and a Peclet mumbler based on the filtration velocity through .the pore, the membrane thickness! and the diffusion coefficient of the salt in the water. W universal correlation is given for the 6ractioaal salt rejection in terms of the three membrane parameters. Experiments were carried out on the salt rejecting properties of compacted clay, through which saline solutions were forced under high pressures. Absoiute comparisons between the experimentally determined and theoretically pred.icted rejactiomn characteristics are shown to be excellent. A comparison of the theory with experimental data of Micirelsen and Harriott on ceilolphane also shows very good agreement.
It has long lkreen known that clays2,3 and other porous solids have the property of partially rejecting salt when a saline solution f i l t m ~through them. The same phenomenon has been ea7id.eneed by many relatively porous materials including callopkiane membranes4 and porous glass,5 as well as by finely porous materials like ion exchange resins.6.7 Porous t?ynan;rically formed meinbranes developed idge Xational Laboratory group,8 as, for example, thosc formed from colloidal dispersions of hyof hydrolyzable salts, and solutions drous oxides, s~~iutioiis containing small concentrations of organic polyelectrolytes, have bwn proposed for and successfully used in brackish water desalting applications. In many such niacsoporous nraterieils, where the pore size is large cornpared with molecu?;ir dimensions, the rejection mechanism has been assumxi to be an electrokinetic one, resulting from charge buill up OIL the interior surfaces of the porous material arjneir in contfict with the salt solution. on phenomenon in reverse osmosis (hyIigh macroporous media can be analyzed in terms of a relatively simpie physical model, where the f h i d is assumed t o fiow through the porous materialdoia a series of zlniformly dist,ributed, straight, cylindrical pores. The prohlern c m f h t be solved for a single capillary and back to apply to an actual porous meditha effective pore radius arid pore length alXy measurable bulk q.uantities like the permeability, p:)rosit:y, and -Lbi.cknessof the porous bed.1° The physicai 1Tielel:hanis sakt rejection in a riow through. :I. capillary with a ed interior wall is qualitatively ciaar. Tlie mc.f:rce charge gives rise to a potential fkld. which exttinds a distance approximately equal to the Debye length Into the liquid within the pore. Within this region, there is XI excess of ions (counkerions) having a 3 9
charge opposite to that of the wall. If a pressure gradient is applied to make the salt solution flow t h r o ~ g hthe pore, then because of an excess of charge of on& sign within the pore liquid there results a net transport of charge and the buildup of the streaming potentiai. The effect of the potential is to set up an electric field parallel to the surface which will increase the transport of the coions and reduce that of the counterions until there is an equal steady-state transport of positive and negative ions, the feed and effluent solutions being insulated from each other so that there is no current flow, In this steady-state salt and water are transported tlirough the capillary9 but not charge. However, the net effect of the eoion exclusion and the axial field i s scch as to cause the ratio of the molar salt flux to the volume flus of water to be less f;han the molar salt concentration on the upsirearn 4 d e of the membrane. In other words, the membrane tends t,o reject salt. Related electrokinetic phenomena in a simple capillary tube have been analyzed in a Barge iiurnber of evorks.ll In (1) Presenl address: Poiitecnico di Torino. Toiino, Italy. (2) A . S. Michaels and C. S. Lin, lnd. Eiig. Cirem., 47, I249 (1955). (3) J. 6 .McKelvey and I . H. Milne, "Clays and Clay Minerais," Vol. 9, Pergamon Press. New York, N. Y., "!962, p 248, (4) D. F. Michelsen and P. Harriolt, "Membranes from Cellulose and Cellulose 3er,vatives," A. F. Turbak, Ed., 'Siiley, N e w York, N. Y , 1970, p 27. (5) K. A, Kraus, A . E. Marcinkowsky, J. S, Joi'nscn, and A. J. Shor, Science, 151, 194 (1966). (a) J. C . McKelvey, K. S. Spiegler, an(! M . W . ,J. Wyllie, Chem. Eng. Progr. Symp. Ser., 55, No. 24, 199 (1959) (7) S. %. Sachs, E. Hoffer, and 0 . Kedem. OSW (Off. Saime Vi'arer) Res Develop. Progr Rep.. No. 324 (1968). (8) A E. Marcinkowsky, K. A. Kraus, H ,C . Phillips, J. S. Johnson, Jr,, and A . J. Shor, d. Amer. Chem. Soc.. (9) K . A . Kraus, A . A . Shor, and J. S. Johnson, J r . , Desaiinaiion, 2 , 243 (1967). (10) A . E. Scheidegger, "The Physics of F!ow tbroiigli ForOlJS Modia," University of Toronto Press, Toronto. 1963. (11) J. T. navies ana E. K. Rideal. "Interfacial Phenomena." Academic Press, New York. N. Y., 1963. The Journal of Physical Chemistry. V d 76, No. 26, 1972
4
general the earlier treatments were approximate and applied to the case where the Debye length was small compared with the tube radius and/or the surface charge was sufficiently small that the Poisson equation could be linearized. Dresner12 solved the capillary model without either of these assumptions, but he considered only the special case of electro osmosis where the flow of the electrolyte is associated with gradients in pressure and electric potential, acrom the membrane but with no concentration gradient. He showed the dependence of various eiectrokinetic quantities s11ch as streaming potential on both e length to radius ratio. Howthe surface charge and D ever, because no calculations were carried out with an axial concentration difference the corresponding information on the salt rejection characteristics of the capillary was not obtained. A paper on the capillary membrane problem was published in 1968 by I h s s and for the general case. assu.niing only that the Deb t h can be described by the C,hapman-Gouy model. er, as will be noted in connection with the model t opted, an approximate form for the axial. ion concentration distribution was implicitly introduced. Their formulation was the usual phenomenological one in which the transport of ions and water molecules ue83uitjs fro? the effects of gradients across the membrane in pressure, concentration, and electrical potential. T:he authors applied their modal to some fundamental elecrrolcinetic problems, but restricted tliemselves to smaE c o ~ ~ e n t r a t i ogradients n and did iiot work out explicit solutions fwr the problem of interest here, In the present paper we apply the capillary membrane model to derive explicit solutions for salt rejection for a viscoins flow driven by a pressure gradient through a macroporous medium with charged interior surfaces. The analysis i s carried out for the case of a dilute fully dissociated salt solution, and we assume the potential of the pore wall to have a fixed value independent of sait concentration. In order to check out the theoretical results derived for th.e salt rejection characteristics, hyperfiltration experiments were carried out on the salt rejecting properties of compacted clay, easurements were made of the wall potentia! ,ar.d all of the bulk properties which determine the parameters appearing in the theoretical solution. In what follows we shall describe the general experimental setup alcxg with the results and comparisons with the theoretical predictions. Comparison of the theory with experimental data of Michelsea and Harriott4 on cellophane will also be :>resented. For the capillary membrane model described in the Introduction the perfommance of a single cylindrical capiiiary pore characterizes the membrane performance as a whole. This is a consequence of relating the membrane permeability, porosity, and thickness to an equivalent uniform distributjo2 of cylindrical pores having the same radius and length, The interior wail of the capillary is assumed to become charged when in contact with a saline sohlion, as a result, for example, of ion adsorption, m r face c.nemica! reac'cion? or other reasons.14 This surface charge leads to the foxnation of a Debye screening layer of opposite charge in the iiquid adjacent to the wall. The ratio of this thickness (AD) to the pore radius ( a ) will be one measure of the salt rejection, with the salt rejection being greater the larger is :his ratio. The Journal of Physiciil Chemistry. Vu/. 76, No. 26, 7972
Jacazio, Probstein, Ssnin, and Y g n g In the analysis to follow we adopt a cylindrical coordinate system ( x , r ) with x positive in the direction of flow and r the radial coordinate with origin at the axis of symmetry. Following Gross and Osterle13 we split the electric potential ZP into two parts @ = $ 4 ~ 1f
I&(x,r?
(1)
where the total potential ip is connected with the ion concentration distribution c(, x, r) through the Poisson equation, the subscript -i- referring to cations and the subscript - to anions. Because of the cylindrical symmetry of the problem, within the pore thew is no net ion flux or bulk flow in the radial direction. The integration of the radial ion flux equations leads to the radial ion concentration distribution being in Boltzmann equilibrium with respect to the radial component oftfie field, $
cI(x,r) = f*(n)e x p [ ~ $ ( x , r ) l
(24
Here $ = ZF+/(RT is the dimensionless radial potential, Z is the absolute value of the charge number which we shall take equal for the positive and negative ions, F and &the i Faraday and gas constants, respectively, T the absolute temperature. With f,(xf undetermined functions, eq 2a applies a t any station uithin the pore. For the case when the Debye length is very small compared with the capiilary radius, the potential in the central core of the capillary depends essentially on x alone, and the positivi: and negative ion concentrations are equal except within the thin Debye sheaths next to the wall. For this case we have c s . = c - in the core of the capillary where we set = 0 . Equation 2a then takes the approximate form given by eq 2b: with c(x) the salt concensation in the core where the total potential i s defined by d ( x ) . Although eq 2b was derived only for conditions within the pore it can also be taken outside the pore, since it contains the conditions of equilibrfhiiri for the ionized species at the pore entrance and exit. We emphasize again that it is generally applicable only in the limit when the Debye length is sniall. However, when the Debye length is comparable with the pore radius or larger, eq 2b still describes accuratdy the situation within the pore near either end of it, but inside the rest of t h ~pore it implies a certain axial distribution of charge either on the pore wall or in the d i d wall material. Throughout their formulation for arbitrary Debye lengths, Gross and Osterlel3 used eq 2b racher than eq 2a. We shall adopt this approach, since t!ie use of eq 2b greatly simpiifies the analysis and the calculations for arbitrary Debye length. The results obtained for integrated quantities, such as, salt rejection and streaming potential, are rigorous for small Debye lengths, and irr the general case they satisfy al: the boundary conciitioris and represent a good approximation to the exact sojution. ith a denoting the radius: of the c y ~ i n d r k apore, ~ the potential $(x,a)= $w is termed the wal! potential, In the present development we shall assume the wail potential & to be ~onstarnt,largely because it i s a reasonable assumptinn for a cumber of materials in typiczl aqueous so(72) L. Dresner.J. Phys. Chem., 6 7 , 1635 (1963) (13) R . J. Gross and J. F. Osterle, J , Chem. Phys.. 491, 228 (1968). (14) P. Sennit and J. P. Oiivier, "Chemistry and Physics o i interfaces," American Chemical Society, Washington, D C 1965 p 7 3 .
401 1
!utiami. However, the salt concentration varies along the
ta behave ih
ii
manner intermedi-
(3)
Eere i' given :,y
./a
a n d X = ~ i ~ where / a the Debye length is 0
(4) with t the permitlivity, We note that X is a local value which depends on x i;hrough the dependence of AD on the ss!t concentration c ~ x . ) Fql~ation . 3 Is solved subject to the constant waI.1 pootentiaB condition $ = $, at F = 1 and to the symmetry :on.diticm on the pore axis = o at F = 0. Although we shall discuss later analytic soliations of t h i s equation whlch are available for large and small values of the Debye mtio h, V J would ~ only note here that in gecerai eq 4 can be solved numerically when ccupled appropriately to tix equations for the concentration distribution. 'To iliustrare the 'behavior of li'/ we have shown in Figure 1 the poieratital distribution across the pore calculatecl far a wall pieraLial $(I) = 2.73 for different constant values of h. It c a n be seen that, the influence of the potential arid hence the saPt rejecting characteristic, increases markedly with h, mtl for a I?ebye ratio greater than about 2 the potential d i e biition is c!osely characterized by th.e constant value It a s ! ~ ~ : mwhen es A m. *.. l o complete t k e ! ~ t a ~ , e ~ of n e the ~ t problem we must wzite th.e Nernst-Plan& equations for the ion fluxes which couple the ion cancen~rafiondistribu"io1a to that of the potential. Emp?rrying eq 1 and 2 the flux equations may be written ;ID
-p
1
r/a
(&1'/2cZ"3)1.'2
igure 9 ,, Dimensionless radial potential cylindric31 pore .for different Debye ratios tential (&, = 2 . 7 9 ) .
distribution across a arid a fixed wall po-
present problem there is no net current through the pore so that from eq 2b, 5 , and 6
Js = -2cQK1 i -?AD&(dc/dx) 0
z=
4-
~
~
~ 1 7 4c
~ c Q -t K 2ADKd(dC/dX) ~ - 2AD-~&(d$/dxj (7b)
The K's are definite integrals of fnnctaons of $ over the cross section defined by
I
-
j* ;= uc,,; :exp(r uj)(dc/bx)=F e4c,(diqax) I_
=
4JlsiniiFil - - F ~ ~K,~=> 1 which is a weaker one than X >> 1. For this limit it is readily apparent from Figure 1 and eq 3 that $(r) is constant across the capillary and the solution of eq 3 satisfying the boundary conditions is simply $ = GW.It follows that
K1= K3 = cash$,, Kz = K4 I- sinh J W
(16)
from which COSh $W
KI
= 1,K2
KQ
= 1,K 4 = tanh
t 17)
.JW
From eq 13 and 14 we find for h >> 1 the foliowing limiting forms for the rejection coefficient acd streaming potential
A& = - t a n h & i n
[(l-cash$,) 4- e P @ ~ o s h $ ~(191 ]
Figures 2 and 3 are plots of the rejection coefficient R as a function of Peclet number for different values of the wall potential. The theoretical curves were calculated using eq 18. The data points shown in the figures will be discussed later in connection with the experimental results. For a fixed wall potential the rejection is seen to increase with the Peclet number, as it should, since the Peclet number is a measure of the ratio of the convective to diffusive effects. At small Pe diffusion is dominant with the result that there is a lower rejection, while a t large values of Pe convection is dominant and the rejection is increased (see, e.g., eq 7a). The maximum possible rejection for a fixed wall potential is given by the asymptotic limit (201 R,,, = 1 - (l/cosh obtained from eq 18 for Pe >> 1. The approach to this limit is characterized by a typical relaxation behavior so that the rejection is already an appreciable fraction of the asymptotic value for Peclet numbers as low as 2. That the &I)
401
0 b!-
l 0 4--
F i 0 3 -
c M'ChELSEN - H A R f ? l O ~ T(1970) CELLOPHANE, o = 2 0 A , X1=17
0
!
-/
Figure 3.
Salt rejection coefficient of a cylindrical pore with a
constan't suirface potsnli,al for large Debye ratio as a function of
Peciel nurnber, and comparison with experiments of Michelsen and Harriott4on cellophane membranes. rejection increases with increasing wall potential for all Pe is a direct consequence of the increased effect of t,he wall region on the charge distribution across the pore and hence the rejection, as described earlier. The opposite limit of X < 1 (or really X 2 > 1 (Figures 2 end 3) and the similarity of the family of rejection curves as a function of the Debye ratio for different values of gVvwith F e $ 1 (Figilre 4) suggest that it might b e possible to empirically correlate these solutions in the general case with a single universal correlation for the rejection coefficient R in terms of the three parameters XI, ,$, and Pz. To C R P T ~3i1t this correlation we use as a guide the limiting asymptotic solutions which have been derived. Equation 131 which is valid for both sniail and large r-
(15) E. J. W. Verwey and J. Th. G. Qverbeek, "Theory of the Stability of Lyophobic Colloids, Elsevier, New York, N. Y . , 7948, p 07. The Journal ot Physical Chemistry Vol 76. No 26, 1972
Jacazio, Probstein, Sonin, and Yung
4020
\
*,ru UMERICAL
1
ALCULATIONS
~
122
0. I
\
I I 0 4 IO 01-201
I IO
A
T o scale out the dependence on $+, of the family of solutions of EZIRmax as a function of X I it is necessary to determine what is the “proper” scale measuring the sheath thickness. As in all nonconstant density boundary layersheath type problems, the appropriate normal scale is a length which is measured by the integral of the density over the normal coordinate, and not the value of the coordinate itself, in this case X I when measured in pore radii. For the present problem the appropriate “density” is the counter ion charge distribution in the sheath, which is proportional to exp $. Consistent with the derived asymptotic relations we may estimate a mean charge strength over the sheath as exp($,XI/2) and conclude that a proper normal scale, denoted by XI, is Xi
XI
Figure 5 . Empiirical cient. R and X are
universal correlation of salt rejection coeffidefined in eq 33-35. Points represent exact numericai solutions.
Debye ratios, serves as a guide for the scaling of the family of solutions of .R ,BS a function of )\I for different values of Pe with gVvconstant. The parameters K I and K Z of X I and GW appearing .In eq 13 were evaluated only in the large Debye ratio limit, where they become K I = 1 and K 2 = cosh qW.We recognize, however, from eq 9 for the axial salt concentration distribution that Pe/K1 should represent the ratio of the time for the salt to diffuse across the length of‘ the capillary in the sheath region compared to the time for it to be convected across (or alternatively the ratio of the axially convected to the axially diffused flux within the sheath). If we observe that the average streamwise velocity along the capillary in the sheath is given approximately by V ( S o / a ) , where 61, is the “sheath thickness,” then in order of magnitude
-
-
(p(?/Kl) (VL/D)(sD/a)
(24)
with K I a/Su. I t follows that for large Debye ratios, when the sheath region fills the capillary, K I = 1 consistent with the previously obtained result. From numerical solutions of the Poisson equation the qualitative behavior lof d D / u is of the form bD/ti
- I - exp(
--cd~/$~)
(25)
where a is a constant. It is to be noted that the calcula1 2) tions were carried oiut over a wide range of XI( ~ 0 . to but for only the relatively narrow range of $w of interest ( ~ 1 . 2to 2.8). The linear function of $w should be interpreted in this context. In any case, the empirical form of eq 25 is physically reasonable, approaches the proper limits for small and large XI, and is consistent with all the derived asymptotic Emits such as eq 23. From eq 13 with K I = cosh GW and eq 24 and 2 5 the Peclet number dependence can be shown to be scaled out of the rejection R, referenced with respect to the maximum value of eq 20 by the relations
= XI ~XP(XP&/B)
Figure 5 presents numerical calculations for the rejection coefficient for various ranges of the parameters ItW, Pe, and XI. These results are plotted using the scalings of eq 26-28. It can be seen that ail of the solutions collapse onto a single curve which to good approximation is given by R/Rmax
= exp( -0.4 XI -1.05)
A slightly more accurate representation for small
= pe[1
- exp( -5h1/&)]
(27)
Were we note that the scaled rejection coefficient is denoted by R and that tht: value of CY = 5 was determined empirically. The Journaiof Physical Chemistry, Voi. 76, No. 26, 1972
(29)
XI
is
given by EZlRmax
= 2.6);1~
( K I ~
v=q
/ y m
(36)
With these relations we can express the two dimensionless parameters Pe and X I in QUY thecry in terms of th,0 measurable bulk properties of the porous medium Pe = VL/D = ( l . / 2 K y ) ( q h / D = ) - k A p / 2 K y p R (37) XI = [ ( h ~ ) i j a=](c-CRTKy/8Z2E2h~1)1/2
(38)
where eq 4 was used ~QGC,XD, In all the work reported here, the Kozeny constant was given the value determined emof K = 115. pirically by We note that the above matching procedure ignores the fact that the electrostatic effects associated with the salt rejection process actually influence the permeability somewhat. 33 ‘The resu-lting permeability change was, however, computed to be quite small in our experiments, and it can in fact be shown that even though the effect i s sometimes measurable2 it is never a dominant one. e
Experiments were performed on salt rejection in potassium chloride (KCI) solutions filtered through thin beds or “membranes” of clay (Wyoming bentonite, 325 and 625 mesh). The clay was chosen mainly because it is a substance where electrostatic effects are known to be significant, and where salt rejection has previously been observed by other En~estigators.~ Clay beds are also relatively easy to prepare in a wide range of thicknesses and porosities. The apparatus is shown schematically in Figure 6. The clay sample, 10 em in diameter and typically 0.15 to 0.4 em thick, was held between two porous disks which were compressed by pistons. The compaction stress exerted by the pistons was always larger than the applied fluid pressure drop across the membrane. The side walls of the chamber were lined with Teflon and the feed and the product solutions were in electrical contact only through the clay membrane itself. The feed was pressurized via an inflatable balloon in the supply reservoir and circulated
Figure 6. Experimental setup for measuiement of salt rejection by
compacted clay.
(by convection: one of the connecting tubes was heated slightly, the other cooled) over, the feed side of the clay membrane so as to prevent concentration polarization. Silver-silver chloride reversible electrodes were fitted on either side of the sample for the measwement of streaming potential, All metal parts of the system, including the porous disks which supported the ciay, were made of stainless steel. The KCI feed concentration in these tests was in the relatively narrow racge between 0.005 and 0.012 N. The pressure drop appiied across the membrane ranged up to 2000 psi, and the porosity y was usually. between 0.45 and 0.75. Typical product flow rates were measured,in cm3/hr. For any given applied pressure difference A p , steady-state measurements were made of the volume flow rate q per unit bed area, and the feed and product concentrations cr and C I I . The latter were determined by the stmdard titration method. The permeability was calculated from A p , q, and the bed thickness h. The porosity of the bed was derived from the dry weight of clay and the final wet volume of the porous bed, which could be controlled by the stress exerted by the compressing pistons. This stress was always larger than A p , so that the c1a.y filled the compartment available to it. A clay grain density of 2.80 gm/cm3, quoted from pycnometer measurements by the clay supplier (Nispel, Inc.), was used in the calculation ofporosity, The aim of the experiments was to map out the dependence of the salt rejection coefficient R = ( C I -. c r r ) / c ~(eq 11) on the three dimensioniess parameters which characterize the problem theoretically. These are the Peciet number Pe, the Debye ratio XI, and the dimensionless wall potential $ w ~ For the clay membrane the Pecl.et number is given by eq 37. Its value can easily be varied by means of the applied pressure difference A p 9 and to a certain extent through y which also controls h. The Debye zation i s expressed in terms of the bed properties by ‘eq 38. Experimental control of this quantity is through CI and y. The remaining parameter which must be determined in the experiments is the wall potential &. The wail potential qWis defhed in the same way as the classical 1’ POtential, in the sense that it represents the potential a t the radius where the flow velocity goes to zero (this is the wall in our macroscopic, continuum formulation). Although this potential cannot be measured directly, it can be in(16) P. C. Carman, Trans. inst. Chem. Eng, 15, .I50 (1937); 16, 168 ( 1 9 3 8 ) ; J .SOC.Cheni. Ind., 57, 225 (1938). The Journal of Physical Chemistry, Yo/. 76, No. 26, 1972
Jacazio, Probstein, Sonin, and Yung
4022
CELLOPHANE !(MICHiLSEN$30 HARRIOTT 1970jl
i'
i__
0 01
._ .
, 023-3131
17
I
____L--L'
A
IO
x: Figure 7. Compatisoi with experiment of theoretical universal correiating funclior! io: salt rejection coefficient.
ferred from measurements of classical electrokinetic quantities like the streaming potential. For flow through a capillary there is a classical relation between the wall potentiai (here termed the t potential) and the streaming potent:al Ad (39)
Here g is the conductivity of the solution i.n the capillary and gWis the surfa,:e conductance of the capillary material. The classical relation given by eq 39 correctly represents the wall potential only when the Debye length is negligible compared with the capillary radius, that is X I 0. However, Oldham, Young, and Osterlel7 have derived a correction which enables one to calculate a value of $ w from t in the more general case where X! is finite. There are, however, several implicit assumptions in ref 16, not noted by the authors, which still restrice the correction to small (but finite) values of XI. In the present experiments qWwas obtained from measurements of streaming potential, with X I in the range 0.1-0.2. The correction of ref 16 was applied to the { POtential given by eq 39 to account for the finite value of XI. The value of ibw obtained in this way was quite sensitive to the magnitude of the surface conductance term in eq 39 which dominated. Streetfg has shown that in the concentration range 0.005--6.12 N of our experiments, the surface cond.uctance aC bentonite in a KBr solution is about 4 X 10--s'ohm-1, with .the data scattering about + I 0 to 15%. Since for bentnriite the current associated with the surface S assumed that cond.uctance iri carried by the ~ a t i o n s , ~we 5trei3"5s value held also for our KC1 solution. In any case, this value is q u i t e consistent with other estimates.lg The effective capillary radiiris appearing in eq 39 was evaluated from eq 36. The value of li;w was found to be somewhat dependect on the p l l of the feed solution. A t a feed concentration of 0.00!i1 M KCl.: $\.Jwas 30 mV at the normal pH value of 5 i : 0.3 found in the feed, while a t a pI4 of 4 it rose to 40 mV. These values a n based on the assumption that rw = 4 x 10-9 ohm--I, and are subject to an absolute error equal a t least t o the scatter in Street's data for gW, as mentioned abwve: that is f l 8 to 15%. Whether $w was in fact constant, as assumed in our theory, or depended on salt concentratio-n c o d d not be conclusiveiy established from t h e present iimited data, mainly because the uncertsinl.y in the! sixface conductance was large enough to
-
?he Journai of Physics/ Chemistry, Voi. 76, No. 26, 1972
mask a possible variation with salt concentration. In any case, the feed concentration was varied only by about a factor of 2 in these preliminary experiments (indeed, most data are for a fixed feed concentration of 0.0051 N KC1) and it seems likely that we w o d d not, incur a gross error by approximating the wall potential as being independent of concentration. Figure 2 shows data for the rejection coefficient as a function of the Peclet number. We note that the Peclet number is proportional to A p , as expressed in eq 37, and hence this figure can be read as showing the rejection us. Ap, as is commonly done when characterizing hyperfiltration membranes. In these data the Debye ratio was kept re!atively fixed about a sufficiently large value, for which the theory indicates the dependence of R on Pe should be essentially indepeadent of XI. For a = 1C.9 and 16.3 A,the experimental wall potential was 30 mV, while for the a = 17.2 A data the wall potential was increased to 40 mV by decreasing the feed pH from 5 to 4, as men.tioned previously. The data for the two larger pore radii are for e: -0.0051 N, while for a = 10.9 fi the feed concentration is 0.01 N . The agreement with theory is excellent. The theoretical trend with both Pe and $w is well confirned. This agreement is all the more remarkable because the effective pore radius, as defined by eq 36, was only between 10 and 20 A, which is not very large compared wil-E, molecular dimensions. Our macroscopic theory, which i s based entirely on continuum concepts, nevertheless s e e m to predict the results very accurately. Figure 3 shows a similar plot of data for cellophane obtained by Michelsen and Harriott.4 These authors did not measure the streaming potential so that the value of GW for their data is unknown. However, with an assumed = 1.30 our theory provides an excellent match value of with these experimental results. This corresponds to a wall potential of 33.4 m\J, a quite reasonable value for ceiiophane. In Figure 4 we plot our data for clay to show the rejection coefficient as a function of the Debye ratio XI. Here, the value of Pe is maintained high enough in all the data points so that its precise magnitude should not influence R critically, as discussed earlier in. connection with the theoretical analysis (cf. Figures 2 and 3 for Pe >> I). Again, two experimental values of were obtained by changing the feed pH. For $w = 30 mV all the points except two, and for &, = 40 mV all except one, were taken with CI = 0.0051 N. Again, the results are in quite good agreement with theory, with what differences that exist ascribable to the fact that the Peclet number of the experiments was somewhat lower than required to compare with the large ?e theory. The results dearly show the increase in R with XI predicted by theory and the tendency of R to ieve? off a t higher values of XI. The dependence on $Jw, with R increasing with qW,is also verified quantitatively. The final Figure 7 shows ali of the previous data, including that of Michelsen and Harriott4 for cellophane, plotted as the reduced rejection I? us. the reduced Debye ratio (see eq 26-28). In terms of these variables, the $sr
(17) !. B. Oldham, F. J. Young, and 2 . F. Ostsrle. J. Colioid Sci., 38, 328 (1963). (18) N. Street, Aus?. J. Chsm., 1 0 , 207 (1957). (19) H . van Olphen and M. H. Waxman, "Clays and Clay Minerals," A. Swineford, Ed., Publication No. 566, National Academy of Sciences-National Research Council, Washington, D. C . , 1958, p 61.
Kinetics of Acid D;ssorciation-lon Recombination theoretical curves for ail $w and Pe collapse into the single curve shown, as discussed earlier. Again, the agreement is very good. We would note that the apparent larger scatter of the data is misleading in the sense that it is due to the fact that the rejection coordinate has been stretched by ref,mrrcin.g it with respect t~ the maximum rejecticm Rmax.
A theory of salt rejection by hyperfiltration through rmcroporous membranes has been presented. The theory is based on a relatively simple capillary model, wherein the membrane is assuined to consist of a uniform distribution of cylindrical pores whose interior surfaces acquire a constant potential when in contact with the saline solution. Although the Bow phenomena in actual porous membranes is undoubtedly more complex, the simple model appears to provide a theory in excellent agreement with sait rejectkm data from experiments on bentonite clay a d cellophane. Furthermore, the theoretical characterization of the rejectioa in terms of a dimensionless wall potential, the r a t i o of the Debye length to an effective pore radius, and a Peelet number based on the filtration velocity and an effective membrane thickness seems appropriate for the materials and operating conditions tested so far. Our present macroscopic, continuum model requires that both the Debye length and the effective pore size be large in comparison with the moiecular dimensions. Although the Debye iicxigth always satisfied this requirement in the present tesrs, the same was not true of the effective pore radius, which was in the range 10-20 A. An explana-
Kinetics of
4023
tion is required as to why the theory holds so well down to geometrical dimensions of the order of molecular size. In addition, although the assumption of a constant wall potential independent of salt concentration was made in the present work this is known not always to be true, depen jng on the so!id-solution interface, so that the effect of re., laxing this assumption should be determined. Another" assumption in the present work is that the axial distribution of charge om the pore wall is such that the ion dlstrihuti.ons in the pore fo'ollow the functional ~ e ~ a t ~ o (eq n s 2b) h~~ suggested by Gross and Osterle.13 Alth.ougR the effect of this assumption is not expected to be iarge on such integrated characteristics as fractional salt rejection and streaming potemtiai, nevertheless a detailed study should be made of the rejection characteristics corresponding to various physically realistic wal.1 charge or potential specifications. I f the theory i s to be useful for c h a r ~ c t e r ~ z ~porous ng membranes for desalination or d e ~ ~ n e ~ ~ ~of i water ~ a t i o ~ by reverse osmosis it should be extended to the case where the fluid contains several ionic solutes of the same sign having different valences agld diffusion coefficients. It would also be of interest to extend the analysis to model geometries more complicated than a straight cylindrical pore. Finally, it is evident that a pnilch wider range of membrane materials and operating conditiions must be investigated experimentaliy before the full utility of the theory can be properly assessed.
Achnoltledgmnent This research wa6 sponsored by the ater, 1.; S.Department of the Interior, under Grant No. 14-30-2575.
issociation- Ion Recombination of andberg, Gary H. Henderson, Robert D. White, and an war^ Oepartment of Chemistry, Universityof Utah, Salt Lake City, Urah 841 12
(Received May 25, 7972)
Publication sosts essisted by the Air Force Office of Scienfific Research
In -&10-5 M acidic aqueous solutions of Methyl Orange the rate constant a t 25" for the recombination of a proton with the monoanion of the indicator is 2.9 =k 0.4 X 109 M - I sec-I. This specific rate, determined by a spectrophotometric electric field jump relaxation technique, is significantly smaller than that expected for a diffusion-controlled ion recombination. An explanation for this result foOiluws from a comparison with similar kinetic data for aqueous Methyl Red.
Methyl Orange (I) is one of the most frequently used acid-base indicators a t acidic pH values near its reported1 pKa = 3.47 in water a t 25". In general, when Methyl Qrange is used as the indicator in temperature jump or other re:.axation experiments exact numerical values for the dissociation and ion recombination specific rates are unnecessary if the relaxation time ( 7 ) of the coupled system is long campared to that of Methyl Orange. However, in
studying very fast reactions (e.g., metal hydrolysis) characterized by relaxation times of the order of microseconds, values for the recombinatiom and dissociation specific rates of the indicator should be known in order to reliably interpret the observed relaxation data. The dissociation kinetics of Methyl Red (IT) in dilute aqueous solution were reported2 in terms of a one-step (1) R. L. Reeves, J. Amer. Chem. Soc., 88, 2240 ('1986). ( 2 ) i. P. Holmes, A . Silzars, D. L. Cole, L, D. Rich, and E. M. Eyring, J. Phys. Chem., 13,737 (1989). The Journal of Physical Chemistry dol 76 No 26, 7972
