EFFECTS OF ~ D S O R P ' r I O SON
2779
DIFFUSION I N POROUS GLASS
Effects of Adsorption on Diffusion in Porous (Vycor) Glass'
by Robert L. Cleland, Jeffrey K. Brinck, and Richard K. Shaw Department of Chemistry, Dartmouth CoZli,ge, Hanovel., .Vtw Hamphire
(Received Febrziary 26, 1964)
Apparent diffusion coefficients were measured in porous (Vycor) glass disks hy ail imniersion method2 as a function of composition and temperature for the CSH8-CC1, system. An estimate of the pore diffusion coefficient indicates that diffusion in pores of about 50 '1. diameter is somewhat, but not greatly, slower than free diffusion. .4 treatment of pore diffusion leads to expressions for diffusion coefficients in a n interfacial region at, the pore surface and in the bulk solution phase in the pore interior. When free diffusion is assumed to occur in the bulk phase, the surface diffusion coefficient is estimated to be one-third to one-half the free diffusion coefficient for a n interfacial region thickness of 4 ,I.
Introduction Liquid diffusim coefficients may be determined with rather good accuracy by the disk immersion method described by Wall, Grieger, and Childers.2a Thc method involves the observation as a function of time of the weight W of a porous disk suspended in a bath liquid composed of two components having different densities. The porous disk initially contains a mixture of the same components differing in composition from the bath liquid. Solutions of the time-dependent differential equation for diffusion couched in a form adapted to this method h a w been given by Wall and co-workers for the ease (i) t h a t the diffusion coefficient D is independent of concentration,28 and (ii) that D depends in quadratic fashion on concentration. 21' According to thcir rcsults the coefficient Docorresponding to the composition of the bath liquid may be dcterinined from the slope of a plot of the experiniental data in thc form log (V7 - W,) against time, where W , is the w i g h t of the immersed disk after equilibrium with t,he bath liquid is reached. The slope must be taken at times sufficiently long after the heginning of tho experiincrit to permit neglect of trarisicnt terms occurring in the solutions of the equation. The result does not depend on the shape of the pnrous body3 nor on it8sinternal pore geometry fcr pores which are siifficicntly wide so that effects of the pore walls on the diffusion process arc negligible. In thc present work WP, have been iritercsted in a case where such effects cannot h neglected. The disk innnrrsion method is well suited t o this case when thc porous
material is available as a n immersible body of such dimensions that accurate dcteriiiiriation of weight change during a reasonable time period is possible. The material studied was porous (T'ycor) glass (hercafter termed simply porous glass), an intermediate in the manufacture of coinniercial Vycor glass, whose preparation and properties have been described by n'0rdbe1-g.~ The specific surface area of porous glass has been studied+R with results consistent with an equivalent capillary model composrd of capillai+s approsimat~ely50 A. in diariieter. I n their intcrpretation of anonialics in t h c flow behavior of paraffin hydrocarbons with rcspect to iiiacroscopic viscosity, Debye and Cleland9 postulated H flow nicchanisni which required introduction of a friction coefficient between a inovirig liquid layer and the pore boundary. .In investigation of binary liquid diffusion in porous glass was carried out under thcl assuinption ~~
~
~
~~
(1) This work was supported in part b y a grant from the Research
Corporation. (2) (a) F. T. Wall, P. E'. Crieger, and C.If7.(:hilders, .I. Am. Chi,m. Soc.. 74, 3562 (1952); (h) 1:. T. Wall and I t . C . Wencit, J . P h y s . Chem., 62, 1581 (1958). (3) F. Gr(in and C. Rlat,ter, J . Am. Chem. Soc.. 8 0 , 3838 (1958). (4) M .E. Nordberg, ,J. A m . Ceram. SOC.,27, 299 (1044). ~ 5 P. ) H. Emmett and XI. Cines, J . Phjjs. Colloid C'hcm., 51, 1248 1947). (6) I?. M . Barrer and J. A. Bnrrie. I'roc. IT'o?~. S o c . (Lorition). A 2 1 3 ,
250 (195.2). ( 7 ) 1%.Rrurnherger and P. I)ebye, .I. Phxp. C ' h m . , 6 1 , 162.7 (195:). (8) A. R o n , 21. Folinciri. and 0. Schncpp, .I. Chcnr. /'h?/s.. 3 6 , 2149 (1962). (9) 1'. Debye and It.
L.Cleland, J .
A p p l . rhus., 3 0 , 843 (1959,
2780
that this phcnomcrioii would also be affected by such frictional interactions, an assuniption which has been confirmed both by the present investigation and by a study of binary liquid flow in porous glass.1° Values of D12*, the diffusion coefficient for free diffusion, are available as a function of tcinperature for the C6H6CC1, systeni.” The density difference betweeii the cornponeiits niade the latter system useful for cur purposes. .\pparent and estimated true pore diffusion coefficients of this liquid system as a function of composition at 10, 25, and 40” are presented in this work.
Experimental The experimental method was described by Wall, Grieger, and Childers” and has been reviewed in a recent laboratory manual. l 2 Three similar disks of porous Vycor (supplied by Corning Glass Works) , all 4.81 (*O.Ol) cm. in diameter had the following thicknesses: disk 2, 0.36 cm.; disk 3, 0.365 cm.; disk 4, 0.35 cin. (all +00.0C)5 cm.) The disks were all heated prior to use in air at 530” for at least 24 hr. to remove adsorbed organic matter This treatment also causes some surface dehydration.l3 Before a diffusion experiment the disks were soaked in the desired initial mixture of CeIIa and C(’14. This soaking procedure was normally an automatic result of the previous experiment. The soaking time was sufficiently long (usually 24 hr.) to assure attainment of equilibrium between porc liquid and bath liquid. Bath liquids were originally prepared volumetrically; the same liquid was reused in further experiments for which the same composition was required, evaporation losses being made up when necessary. Compositions were checked occasionally by pycnometry, but changes in voluine fraction never exceeded I(%. T h i s resulted from use of a largc bath liquid volunie (ca. 800 ml.) compared to pore volume (ca. 2 nil.) and use of successive experiments in which either larger or smaller concentrations of a given component were prcsent in the porc liquid than in thc bath. Bath liquids which wex initially purr, solvents were rcplaccd when their cornposition approached 1% of the other componcnt. I‘ixpcriincnts w ~ conducted e with the bath liquid containvd in 1-1. crlerinicyer flasks clamped in a thcrinostatcd water bath. The disks were attached to fine copper nire by itwan$ of cadmium-plated steel clips. A loop in thc wire was attac1it.d to t h c pan of an analyt ical balance. ‘l’wo balances were used in this work, first a chain-equipped balance (Christian Bccker Chain-o-inatic) and later an analytical balance with optical scale (ltettlcr JTodd 111.5) fitted with a suspen-
R. 1,. CLELAKD, J. IC. BRIXCK, AND R.
I(.SIIAW
sion for below-balance weighing. Both balances were supported over thermostatcd water baths by framcworks made of 2 X 4 in. wooden planks. T o bcgin a n experiment the porous glass disk was lifted out of the liquid in which it had soaked, was quickly wiped dry with a cotton towel, and was immediately dropped into the new bath liquid. The wire loop was attached to the balance pari and the bath flask was covered with a piece of heavy aluminum foil fitted with a slot to avoid touching the wire. The time was observed on an electric wall clock with a sweep second hand or, more conveniently, on a laboratory electric tinier provided with digital indication of elapsed time to 0.01 min. (Lab-Line Model 1401). Readings mere taken after about 0.5 hr. at 10- or 20-min. intervals either of time as weight balance was obtained (Chain-0-matic balance) or of weight at desired time mtervals (Mettler balance). The Mettler balance proved to be particularly well suited to this experimental use. Weight readings were usually continued for about 4 hr. A value of W , was obtained as the wcight reading some 24 hr. after the beginning of the experiment. The readings of W , in a given bath liquid appeared to increase by about 0.1 g. o \ w the course of the first 2 weeks of use of a given disk, and then to become reproducible thereafter within 0.01 g. The bath liquid was stirred slightly by the motion of the suspended disk and this was supplemented occasionally by manual raising and dipping of the disk. The extra stirring did not appear to be necessary in view of good reproducibility of results in stirred and unstirred cxperiments. The reagents used in the bath liquids were reagent grade benzene and carbon tetrachloride (JTallinckrodt).
Results For the description of their mixtures we shall designate benzene as coniponcnt 1 and ( X I , as component 2. Compositions will be cxpressed as volunic fraction (p? or molar conecntration c? (niolcs/cm.d) when they refer to the bath liquid, and cp,’ or e,‘ when they refer to the liquid in which the disk had initially been soaked The raw data obtained from the measurement of the weight W of the suspended disk (with suspension) and W,, the equilibrium weight after at least 24 hr., could be plottcd in the forin log (W - W,) against timcx I An example of such a plot is given in Fig 1. When (IO) R I, Cleland “Binary Liquid I’low in Porous (Vycor) Claqs,” to be published. (11) C S. Caldwell and A L. Dabb, .I P h y s Chem.. 6 0 , 51 (1956). (12) D P. Shoemaker and W Garland, “1:upeiirnents i n I’hywcal Chemistry,” 1IcGrrtw-IIill Rook Co , New Pork, S . Y , 1962, p 150. (13) R. L. Cleland, .I Phys Chem , 68, 1432 (1964)
2781
EFFECTS O F A n s o w n o s os DIFFUSIOS I N POROUS GLASS
I I I I I I I I I cpz' > pL0, the absolute magnitude of the slopes of such plots became steady after falling during the first 30 or 40 min. (at 25'). When cpz' < p Z o , the slopes tcnded to decrease throughout. This behavior confornis to that predicted by the calculations of Wall and W ~ n d t ~ ~ for the case that D decreases with increasing cp2, as it does in this system.l4 The slopes were consequently corrected by the usc of the calculations of these authors for the case that D varies quadratically with concentration. The paranicters required for the correction procedure were obtained from a quadratic fit of the actual diffusion data. In treating our data the steady uncorrected slopes were usually obtained with suficient accuracy for cp2' > cp20 by direct calculation from the values of log (W 1.5 I I I I I I I I I 0.0 0.5 10 W,) and t without making the plot of Fig. 1. The correction procedure was applied to the slope thus ob9; tained to provide a corrected slope na which corFigure 2. T h e apparent diffusion coefficient D,,, for CeHeresponded thcoretically t o the apparent diffusion coCC1, in porous glass as a function of bath composition at 28'. efficient, D,,,, a t the composition of the bath liquid. Numbers indicate disk designations. The dashed line (vertical bar points) is n plot of D,,, obtained from flow experiments The value of D,,, was calculated fromzb
21-
nap,= whew
111
with a porous glam membrane.'a
-2.303r2m/a2
(1)
is the valuc of lini d log (W - W,)/dt t--r
m
obtained by correction of the cxperiment,al value of d log (W - W,),'dt. 'rhc disk thickness r is the only other experiinental quantity required in the evaluation I
I
P\
I
I
I
of Dapp. Values of D,,, obtained a t 23' for the three disks studied are shown in Fig. 2 as a function of composition. The deviation of values of the slope m from the mean for duplicate cxpcrinicrits was most often within ly0 and seldom greater than 2yo of the mean. Also shown in Fig. 2 are valucs of D,,, estimated from flow experinients'o in which a concentration gradient was applied bctwccri two cell compartments across a membrane. The diffusion was observed as a time ratc of change of concentration in onc cell compartment. Since the esperin~cnt,altechnique in the flow case is diffewnt and the calculation of D,,, involves an estimate of the cross-sectional area of nieinbranc pores available for diffusion, the two methods givc rather good agreement.
0
3
I 5 O--
Treatment of Pore Diffusion I n order to estiniate the rnagnitudc of the supposed frictional eff ects of thc hundaries of microscopic pores on oi*dinai*yliquid diffusion in the pore, a theoretical investigation of a simple capillary pore model was undertaken. iVc shsll discuss in this ewtion a single pore having a circular cross scction of uniform dianicter. Later (sec Discussion) we shall suppose that the porous Incdium is composed of a set of such porcs of equal diameter Diffusion in a pore may coni-eniently be trcatcd as
Figure 1. Plot for deterniinution of d log (M' - V,)/dt. The d a t a are for disk 4 a t 25', (p2' = 1.0, ea= 0.75.
(14) This ststement c o r r e ~ p o n dto s l'ig 1 rn Wall and Wendt'* paper The sign of the coefficient @ of theie authors lius evidently bern reversed in their I'rg 2
3
Y
(3
0
-I
+
Y \l
50
IO0
TIME (minutes)
R. L. CLELAND, J. K. BRTXCK, AND R. K. SHAW
2782
a frictional process in which the frictional forces exerted at any point on component i (per mole) by component j are assumed proportional to the product of c J , the molar concentration of j , and the relative velocity uj - ui of the two components a t that point. When mechanical equilibrium exists in a macroscopic phase so that no net volume flow occurs, a force balance between the gradient of the chemical potential (per mole) at constant temperature, and the sum of the frictional forces Fij exerted by other components j may be written16for an isothermal system of n coniponents
vT,iii =
Fij =
-E
j=1
Ri,cj(ui
- uj)
(2a)
j = l
VrP1 = V1Vp
Rij
+ Rji
VT,PPI
(2b)
(2c)
where the u1are the local component velocities, PI are partial molar volumes, V p is the pressure gradient, VT,,Ji, are the chemical potential gradients a t constant temperature and pressure, and the R,, form a symmetric matrix of friction coefficients. I n the absence of a pressure gradient eq. 2a-2c reduce to the diffusion equations discussed by LaityI6 as implicit in Onsager's treatment of diffusion.17 Treatments of diffusion in membranes in terms of eq. 2a-2c have been made for biological membranes by Kirkwood'8 and for ion-exchange membranes by Spiegler. l 9 In these treatments the membrane mas taken as a component of the diffusing system, while concentrations and velocities of the various coniponents were supposed uniform over a membrane cross section. The membrane and the solution contained in it were, thus, treated as a single macroscopic phase. I n our discussion we wish to include the possibility that concentrations and velocities may differ in the interfacial region at the pore surface from those in the bulk solution phase in a given pore cross section. The model chosen to approximate the physical situation is illustrated in Fig. 3. The coordinate T represents radial distance from the pore center. The pore boundary situated a t r = a consists of the structural material of the membrane with any immobile adsorbed matter; the boundary is assumed smooth on a molecular scale. An interfacial film s, assumed to be of molecular thickness 6 , forms an annular ring (a - 6 < r < a) adjacent to the pore boundary. The solution phase occupies the region of the pore cross section r < a - 6 . The model resembles that employed by Cuggenheimz0 to discuss the thermodynamics of the interfacial region. We shall limit our discussion to the simple case for which the concentration cl* and thc velocity ul*of each component are uniform throughout the The Jovrnul of Phz~sicalChemistru
Figure 3.
Diagram for model of capillary pore cram section.
-
solution phase portion of the cross section (0 < r < a 6), but may be different from the concentration cLB and the velocity u,'of that component in the interfacial region. The concentrations cl* may thus be taken to represent the concentrations of each component far enough from the pore boundary so that adsorption effects become negligible. Experimental evidence on adsorption from solution21 supports the assumption that concentration effects occur only in a surface monolayer. Although eq. 2a-2c are valid strictly only for a macroscopic phase, we shall assume their applicability to the solution phase of the microscopic pore, regardless of its dimensions. We shall also assume that a force balance similar to (2a) may be written for the interfacial region s. For this purpose the frictional forces on component i molecules in s may be divided into forces exerted by other molecules in (a) the solution phase and (b) the interfacial region, as well as forces exerted by (c) the pore boundary. The principal contributions to the integrals'; which express the coefficients Ri, in moleculax terms come from nearest-neighbor interactions for relatively ideal nonelectrolyte solutions. I n applying eq. 2a-2c in regions of molecular dimensions we shall make the approximation that the coefficients R,, depend solely on nearestneighbor interactions, As a further approximation the values of R,, observed in the macroscopic solution will be assumed to represent frictional interactions between molecules in the interfacial region and their nearest neighbors when the latter arc solution com(15) R. 3. Bearman and J. G. Kirkwood, J . Chem. P h y s . , 28, 136
(1956). (16) R. W. Laity, J . Phys. Chem., 6 3 , 80 (1959). (17) L. Onsager, A n n . N . Y . Acad. Sci., 46, 241 (1945). (18) 3. G . Kirkwood, in "Ion Transport Across Membranes," H. T. Clarke, Ed., Academic Press, New York, N . Y., 1954, p . 119 R . (19) K. S. Spiegler, Trans. Faraday Soc., 54, 1408 (1958). (20) E. A. Guggenheim, ibid.,41, 160 (1945). (21) J. J. Kipling and D . A . Tester, J . Chem. SOC., 4123 (1962).
ponents. This assumption ignores the possible effects on K1, which niay result from changes in orientation, distribution functions, intermolecular potentials, etc., due t o the proximity of the pore boundary. The frictional force I“,,* exerted per inole of coniporient i in region s by j molecules in the bulk solution phase and the force FlJ8exerted by j niolecules in the interfacial region niay then be written in a form analogous to (2a)
I”,,*
= -
F,,s
R l 1 ~ * ~ , * (-u lul*) s
= -zZlrZSCls(U,s
-
UjS)
The quantities z* and zs represent for a given molecule in s the fraction of its nearest neighbors which lie in the solution phase and in the interfacial region, respectively. These fractions are independent of which moleculc in s is chosen when a random arrangement is assunicd, on the averagc, in that part of its nearestneighbor shell lying in a given region. We have also assumed arrangement of nearest neighbors on a lattice of equal-volume sites. The frictional force I“,@exerted per mole of component i in region s by the pore boundary may be written F1O
=
- R ~ O ’ ( U~ ’UO)
= -R12Z*Cp*(Uls
- uZ*)
-
+ Rzo’~z’ -
Rlz’(Ula
-
UZ’)
(3)
RI?‘
= Rl?(zscs
+ z*c*),
C‘
=
=
- R ~ ~ c * ( u ,*
uZ*)
(4)
I n experiments of the kind described in this work the volume fluxes JV8and J,* in the interfacial region and the solution phase, respectively, may each be taken equal t o zero. Thus
J,,
=
J,* =
ClSPlSUlS
cI*F‘~*u~*
+ 0 + cZ*P~*N*= 0 C:PZsu28 =
(5)
(6)
A relation between eq. 3 and 4 may be obtained by assuming that lateral equilibrium exists across a pore cross section in the sense that p 1 is constant over the cross section. Tv\hen V p = 0, this assumption implies that V T , p p l * = V T , , p l S . The Gibbs -Duhcni equation for the bulk solution phase may be written in the form
+
CZ*vT,pp2
=
0
(7)
where we drop the superscript notation on p,. The differences occurring in (3) and (4) may then be written with use of (7) V T . ~ PI
V T , ~ P=Z (c*
’c2*)VT.pFil
(8)
The molar flux J1’ (moles cm.? sec.-l) = clsuls with respect to the pore boundary may be found from (3) and ( 5 ) and the flux J1* = cI*ul*from (4)and (6) with use of (8) to give
- ~ 1 * )- u:) - Rio’ui’
A similar equation may be written for coniponent 2 . Subtraction then gives = -Rlo’lLIR
- VTp2*
Rllz*cl*(uls
R12zSC:(Uis
V,,T~’ - VTp;
VTp1*
Cl*VT,ppl
where RIO’is a boundary friction coefficient and uo is the velocity of the boundary. We shall set uo = 0, so that all component velocities are measured with respect to the pore boundary. For consistency with the previous approximation limiting contributions to R , , to ncarest-ncighbor interactions, we shall neglect disect frictional interactions of the boundary with the bulk solution phase since thesc are assumed not to be in mutual contact. The thermodynamic driving force VT,iils in the interfacial region may now be assumed equal t o the sum of frictional forces exerted per mole of component i in that region. For example, for component 1 of a binary solution MY’ write VTplS
sents the differences between frictional forces exerted per mole of component 1 in s by molecules of the same kind in the solution phase and those exerted in analogous fashion per niole of 2 in s. I n the solution phase we have assumed the applicability of eq. 2s-2c, which implies neglect of this quantity for that phase as well. For the binary solution we then obtain
~ 1 ‘
+
CZ’,
c*
=
c1*
+
c2*
whcn we neglect, as a first approximation, the quantity z*(Rllc,*(ul’- u,*)- R 2 2 ~ 2 * (-~ u 22 S * ) ]which , rcpre-
where cp18 = c , ’ ~ , ‘ ,the volume fraction in region s. Diffusion coefficients D, and D* arc’ defined formally in (9) for the interfacial region and the bulk solution, respectively . The relation of these coefficients to experinir~ntmay be discussed as follows. Equation 1 mas derived2 for the caw in which a concentration C in a porous disk was a function of the spatial coordiriatcs and the time. If we neglect diffusion in thc radial direction, C may be taken as a function of x and t , whcw x drnotcs the Volumr, GR, Number 10 Octoher, ll)Ci4
R. L. CLELAND, J. I
