Znd. Eng. Chem. Res. 1995,34, 2692-2699
2692
Molecular Diffusion in Microporous Materials: Formalisms and Mechanisms Paul B. Weisz Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6392
In a large variety of microporous materials such as high-surface area catalytic solids, natural or polymer fibers, zeolites, porous sorbents, and other media, the diffusional mass transport of molecules is accompanied by immobilization on internal surface sites. The standard formalisms for diffusion, such as the Fick equation, require modification in order to distinguish between effects of the stochastic diffusion phenomenon (the molecular diffisivity) and consequences related to the immobilization kinetics, This can lead to corrections of several orders of magnitude to the apparent diffusivities derived from non-steady state techniques, such as uptake rate measurements. The modifications to the classical formalisms are relatively simple when the diffusion process is slow compared to the kinetics of ad-/desorption (immobilization). This is the case in many situations and technologies. The results have important implications in zeolite technology, molecular shape-selective catalysis, and many technologies employing microporous substances. Classical considerations of physical chemistry also provide guidance to conditions and systems where interpretation of transport in terms of diffusion coefficients may not be valid.
Introduction In a review of theories of ion exchange F. G. Hemerich (1983) once commented on “the general progression of attitude toward phenomena of the world we live in from wonder to speculation to basic theory t o struggle with the complexities of reality”. He noted that often theories get accepted and used ahead of observation and verification. This comment strikes multiple dij& uu reactions to encounters in other problem areas. Although pursued in several and segregated “disciplines” of science and technology, they relate to the common theme of diffusional mass transport phenomena: They are processes in which restless molecules in a fluid are penetrating into and interacting with a microporous solid structure. Examples are the dyeing of natural or synthetic fibers, the molecular shape-dependent uptake by zeolites, and catalytic processes in porous solids. Other seemingly unrelated processes are the metabolism of biological cells, the uptake of pharmacological agents by tissues, the preparation of catalysts by “impregnation”, the oxidation of carbon (burning) or of metals (corrosion), and many others. Reflection on a number of experiences will illustrate some basic mechanistic aspects of diffusion coefficients applicable to microporous materials and will indicate possible stumbling blocks in the analysis, understanding, treatment and application of measured “diffusion coefficients” or “diffusivities” in a variety of research endeavors and technologies.
Diffusion: Nature and Imagery Diffusivity is a property intended to characterize the degree of statistical motion of a molecular species in a medium. In a mathematical formalism, it is a coefficient that relates the change of the concentration profile of a species that must result from currently existing spatial differences in concentration. It describes the ability of the molecular species to undergo that stochastic progression in the specific medium. The diffusion coefficient, in contrast to any overall transport coefficient, is concerned only with this stochastic mechanism, Le., with the natural random walk of the 0888-5885/95/2634-2692$09.00/0
molecules. It is a phenomenon of entropic thermodynamics. As such it is not concerned with any mechanism responsive to a force, be it physical or electrical, although such forces may provide additional motion, i.e., an additional contribution t o the mass transport coefficient. Formalisms Model but Do Not Identify Mechanisms. A mathematical formalism can, of course, describe the behavior of a particular mechanism. Conversely, however, adherence t o that formalism does not imply existence of that mechanism. For example, the rate of flow F of a viscous gas through a narrow tube between two vessels at different pressures, i.e., concentrations c (pressures) is mathematically described by
F = -m dddx just as the first of Fick’s laws; yet m is not a diffusion coefficient. Here the flow is driven by a physical force. There are more subtle examples of the need for differentiation between formalism and mechanism. Experience with the Fick mathematical formalisms have generated familiarity with the square root of time dependence of the progression of a concentration front or of the initial uptake rate of sorbate into a sorbent volume. In fact, the slope of this is used to derive the diffusivity. It has led to a tendency to equate a square root of time dependence with the existence of diffusion as the transport mechanism. Such a conclusion, made blindly, is unwarranted. Capillary Spreading and Diffusion Are Separate Mechanisms. Helfferich pointed out that a process of ion exchange may initially involve only the ions contained in the wetting solution that initially penetrates a dry exchanger by the capillary mechanism. Only thereafter will further exchange occur with ions transported by diffusion. The progression of a fluid along a capillary of radius r by wetting or spreading along length x , dxldt x rdyfqx
also follows the square root of time dependence (Bikerman, 1958). The amount transported due to the “pulling“ force exerted by the surface tension y around the 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2693 meniscus r is inversely related to the “drag” exerted by the viscosity 7 along the length x . The applicable coefficient, of course, is not a diffisivity. The need to differentiate between diffision and capillary spreading is not a rare occurrence. In the dyeing of natural or synthetic fibers, immersion of the fiber into the dye bath results initially in capillary spreading of the dye solution into the fiber, thereby carrying an initial quantity of dye. It is followed by dye diffusion into the fiber. Generally, the first step is fast and provides a minor amount of dye. In processes for impregnation of porous catalysts, we must also recognize the existence of both capillary wetting and diffusion processes. The result, i.e., the resulting spatial distribution or uniformity of the impregnant can depend decisively on the manipulation of these two distinct processes (Maatman and Prater, 1957). The capillary transport process has to be considered even when a gaseous sorbate is applied t o a microporous solid a t certain temperatures and/or pressures due to the possible occurrence of capillary condensation at the boundary within the pores. This is due to the vapor pressure reduction PIP, a t very small pore radii r, according t o Kelvin’s equation
PIP, = exp[-2yVlrRTl
n-BUTANE (at 1 atm)
0-XYLENE ( a t 7 tor)
2
3
10
5
2
3
P o r e Radius
- x
5
10
Figure 1. Critical temperature below which “capillary condensation” would occur according to the classical Kelvin equation (see text) and the relationship between vapor pressure and temperature (log P = B - 0.2185A/T, from Handbook of Chemistry and Physics, 52nd ed.; CRC: Boca Raton, FL, 1972); for o-xylene a t 7 Torr and for n-butane at 1 atm pressure. 240
180
140 120 100
80
60
(1)
with y = liquid’s surface tension and V = molar volume. It is a classical approximation for cylindrical pores. (For other geometries, conditions for at least partial condensation would tend to be more stringent since deviation from a circular cross section would result in portions of still greater surface curvature). Awareness of this phenomenon becomes important as zeolites have attained scientific and technological prominence. Transition of Mass Transport Regimes in Zeolites. The relation (1)that forecasts the condensation phenomenon has been found to be applicable to pores as small as four molecular diameters (Pierce et al., 1949). For considering transport of gases into zeolites having pores that further approach molecular dimensions, we are inclined t o say that the classical relationship for the capillary would surely “break down”. However, we cannot dismiss the existence of the same elementary forces and, in fact, must expect them to have a still greater impact than is predicted by the “simple” macroscopic model; Le., the conditions for a condensation phenomenon will be, if anything, more stringent. Figure 1 shows the temperatures below which classical capillary condensation would be predicted against “pore radius”. It is derived from (1) and the vapor PressureItemperature relationship for two cases: oxylene vapor a t 7 Torr and n-butane at 760 Torr. (a) corresponds t o the nominal “pore diameter” of ZSM-5 zeolite and (b) to that for zeolite X. In attempts to derive diffusion coefficients from uptake rates of o-xylene on zeolite ZSM-5 by conventional Fickian methods (initial slope), a transition of the mechanism of entry is observed (Garcia and Weisz, 1990) at conditions consistent with the onset of condensation at the particle boundary. An upward discontinuity of the uptake rate occurs on lowering the temperature, at 120-140 “C at an applied vapor pressure of 5-7 Torr, see Figure 2. Frequently, this transition is accompanied by instability (i.e., scatter of data, see shaded
1.8
2.0
2.2
2.4
1000/T(oK) M
+
HZSM-5 NaZSM-5
SVAI SVAI
2.6
-
2.8
3.0
2600 40
Figure 2. Change of transport mechanism at critical temperature region (vertical shaded area). Uptake diffusivities (cm2/s) for o-xylene in ZSM-5 zeolite are derived from the initial uptake slope using the classical Fick equation. vs
area). Indeed, the observations are consistent with the classical physical chemistry represented by Figure 1for o-xylene. “he relationship for butane in Figure 1illustrates the need for caution in attempts to probe diffusional mass transport into zeolites even for a gas such as butane at too low a temperature. Such is always a temptation for instrumental reasons, because of convenience and difficulties such as mounting of zeolite membranes, or for in situ operation within instruments (e.g., NMR). We may thus encounter results that are not representative of truly diffusive transport. We may observe strange behavioral effects such as instabilities, or preferentially faster transport of a larger (condensable) molecular component (see, e.g., Wernick and Osterhuber, 1985). “he measurement of zeolite diffisivities clearly must insure pressureItemperature conditions that preserve a stochastic diffusion mechanism. Fick’s Equations: Intent and Universality of Application. The fact that the course of the process fits a Fick diffusion equation does not assure existence of a diffision mechanism. Alternatively, when observations have not “fit”the solution of the classical Fick formalism, this has often led to abandonment of basic,
2694 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
simple, and classical concepts of diffusion. It has led to invoking new names and concepts such as “nonFickian diffusion”, concentration-dependent diffusivities, diffusion driven by a “potential” instead of stochastically meaningful concentration gradients, etc. It is well to remember the simple intent and meaning of Fick’s second law. It is no more than an assertion of a mass balance. In any volume element of a continuous fluid medium, the difference between what flows in and out shall constitute and be equal to the enrichment or depletion of molecules in that volume element,
6 1 6 ~[ D ( d c l d ~= ) ] -6cl6t
(2a)
or
D d2c/6x2= -&I&
(2b)
When external molecules diffuse into the spaces of a porous structure, the relationship needs some geometric corrections: Only the pore (empty) fraction P of the structure is involved, and the length parameter seen by the traveling molecules is larger than x by a tortuosity factor b (usually of the order of 1.5-2). This leads to a correction of the D observed for the volume of porous substance to obtain the diffusivity in the pore space per se, Do,of
Do = (b/P)D
(3)
Fick’s equation deals with cases where all the molecules that exist in any volume element are also participating in the concentration gradient that is subject t o statistical motion. In that case the parameter c on the right side is identical with the c of the left. If, however, some or all molecules of c that enter are immobilized at internal sites, they are removed from the mobile assembly c that generates the gradient; yet they become part of the accumulating mass balance on the right. Now there are two different Concentration variables:
D d2c/6x2= -del&
(4)
At each point, C is the concentration of immobilized plus remaining (if any) mobile molecules at (x,t). To obtain the true stochastic diffusion coefficient D of the molecules in motion, the solution of this modified Fick equation now requires knowledge of the relationship between C and c! Fortunately, there are many processes and technologies in which that relationship is not complex, and no new analytical or numerical solutions need be sought. Yet the correct solution may differ by orders of magnitudes in significance and utility.
Diffusion and Immobilization Re-examining Fick’s Equation. In many technologies, the sorptionfdesorption equilibrium between the mobile phase and the adsorbed concentration on the framework will often occur rapidly compared to the time taken by the overall transport, i.e., diffusion process. In that case the missing information for the solution of (4) is simply the sorption isotherm C = Ac), which is readily attainable. If the adsorption isotherm is linear, C = Ac (Henry’s law), Fick’s equation (4) becomes
Since A is immediately available as the final equilibrium uptake C ffor the applied concentration eo, A = Cd co, the applicable Fick equation is now
Dappd2c/6x2= -6cldt; Dapp= DcJC,
(5a)
This result is physically easy to picture: If the final concentration inside is A times the concentration c applied to the boundary, then it will take A times as long to do that transport than would be needed to bring in c. It will take A times as long, which leads t o the apparent diffusivity being A times smaller. What if the isotherm is not linear? Fortunately, the solutions are not greatly affected by this. The most drastic deviation from linear behavior is the case of very strong adsorption, sometimes referred to as “irreversible adsorption”. It leads to what has been called “shell progressive” or “shrinking core” mechanism. The analytical solution for the progression of sorption is mathematically simple (Crank, 1957; Weisz and Goodwin, 1963), and the diffusivity derived from it differs by no more than a factor of 1.6 from the Fick type response, if just as in the case of the Henry’s law isotherm, the cdCf factor is included. Furthermore, numerical solutions are available for all intermediate Langmuir isotherms between weak (linear) and strong (“irreversible”) sorption isotherms (Weisz and Hicks, 1967) and the applicable correction factors (between 1.0 and 1.6) (Garcia and Weisz, 19901, in case more accuracy is desired. Thus, regardless of the type of the adsorption (immobilization) isotherm, the true mechanistic diffusivity D is obtainable from the uptake-derived apparent diffusivity by
D = (1.0- l . 6 ) D a p p ( C ~ ~ 0 )
(6)
In most practical cases of research or technology, the numerical correction is minor compared to the need for recognizing the other factors (cdCf)(and P, b to obtain Do if diffusivity within the pore space alone is desired). Sorption and Immobilization of Solute Molecules in Porous Particles. As Helfferich has pointed out, ion exchange between solid media and incoming ions in solution can be considered t o be reactions in which the incoming solute ion species is immobilized by its new site association, sometimes in a shell progressive manner. There are many processes to which similar analysis and the relation (6) would apply. Examples are purification processes employing a variety of porous sorbents like charcoal, alumina, special synthetic resins, etc; the technology of catalyst impregnation, such as the manufacture of platinum on alumina catalysts; the technology of dyeing of natural or synthetic fibers, etc. Such processes are well modeled by the sorption of dye molecules into porous silica particles (Weisz and Zollinger, 1967). Figure 3 exemplifies diffusivity results derived from the uptake rates of four different dye molecules (Figure 41, in two different solvents, on three different spherical aluminosilicate particles (3-4 mm diameter) having structures with about 45% porosity and average pore diameters ranging from 4.7 to 10.5 nm. The diffusivities resulting from the “routine” Fick t o 1.2 x interpretation Dapprange from 9 x cm2/s. The true molecular difisivity D, however, found by (6) is some 2 orders of magnitude greater and the same for all cases. Most significantly, D is equal to the
Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 2695 cmqsec
IO-^ c 0 =?I
10-6
c 3
F
U
n- 11
10
9
DIFFUSIVITIES
lo-g
I
Due Silica
I
j
S
I
DSC DSC PAN PAN PADP RhB RhB RhB JME W200 W200 W450 W450 W450 W450 W450 Solvent ----------benzene water-----Temp deg C 20 70 100 100 c ( w t X) ---------------------o 2--------------------------C f ( w t W ) 160 8 6 18 51 2.2 101 154 9 9
_____________ ____________ _________________
Figure 3. Apparent ( 0 )and true ( 0 )diffusion coefficients, Dapp and Do, of several dyes in various porous silica-alumina particles, derived from uptake rate measurements. The shaded area corresponds to the magnitude of the diffusion coefficients in the liquid medium (water or benzene). The dyes are pictured in Figure 4.
' "M 1\
PAN
N=N
NHz
0coo-
RhB (C,H,)
k-
u:;()N (C,H,)
Figure 4. The dyes used in the determination of dye diffusivities in silica beads shown in Figure 3.
molecular diffusivity in the solvent for all solids, dye, molecules and solvents! The mechanistic implications of this unifying result are obvious: The mass transport occurs by free molecules in the pore solution, greatly unhindered by the solid structure. Nevertheless, the bulk of all molecules within the microporous solid is immobilized at any one time.
Dye Diffusion in Natural or Synthetic Fibers The technology of dyeing of fibers provides an ideal area for observation (almost literally) and for analysis of diffusiodimmobilization processes. By the middle of the 1960s, a large number of diffusion coefficients had been derived from uptake measurements and routine Fick evaluation had been reported by many authors, for many dyes and many fibers. They were found (Weisz and Zollinger, 1967) to span a range of 4 orders of magnitude, ranging between some lo-' and lo-" cm2/ s, see Figure 5. Yet, the large spread of Dappreported for a wide variety of fibers and for various dyes, when recalculated as Do, reduced to a narrow range of values as shown in Figure 5. The diffusivities in all cases approached the magnitude of free diffusion in water within the pore space of the fibers! Within the remaining spread, there was a tendency for Do t o be smaller for very large dye molecules.
7
0
-
10-
"
6
5
(cm2/sec)
Figure 5. Diffusivities Dappas reported in the literature for various dyes in a number of fibers (see text), and recalculated actual diffusivities within the fiber pore space, Do. The latter all converge on the magnitude of dye diffusivity in water (Do* 10-6 cmZ/s).
The consistent and common denominator reached by the same simple transformation (6) applied to all these cases is a compelling force for a simple and classical mechanistic conclusion. It established the "pore diffusion model" as one basic mechanistic process operating in dyeing technology. An additional source of dye transport cannot be discarded for more recent synthetic polymers of negligible pore volume (water uptake) dyed near or above the glass transition temperature: Segment mobility, i.e., the contribution of motion of polymer segments. However, the pore mechanism remains a basic element in the treatment of dyeing phenomena and procedures (Rys, 1973; Rohner and Zollinger, 1986; Rys and Sperb, 1989).
Molecular Shape Selective Diffusion in Zeolites Molecular Shape Selectivity. The development of molecular shape selective catalysis began with the demonstration that the induction of acidic sites in zeolite A of about 5 A "pore" diameter would lead to the ability to selectively acid catalyze the cracking of C-C bonds, the dehydration of linear alcohols, and other reactions only when the reactants were linear; in other words, as long as the reactant could diffuse into the zeolite structure (Weisz and Frilette, 1960; Weisz et al., 1962; Weisz, 1965; Chen and Weisz, 1967). The development proceeded from this yes/no differentiation of a diffusion capability to achieving shape selectivity based on finite but large differences in the diffusivity between molecules. Differences of diffusivities by 2 or more orders of magnitude are needed to produce decisive selectivities between respective reactant species (Weisz, 1980). Examples are the pair trans- vs cis-butene in zeolite A (Chen and Weisz, 1967) and reactions of p-xylene in contrast to 0- and m-xylenes in zeolite ZSM-5 (Haag et al., 1984). In some important catalytic processes, the shape or size selectivity is achieved by differentiation against a reaction complex which cannot be spatially accommodated at the catalytic site (Csicsery, 1984, 1986). However, large differences in diffusion coefficients dependent on molecular shape parameters are and will be the basis for many current and future shape selective catalytic processes. Thus the quest for knowledge concerning that dependence remains an important objective for science and the technologies of separation and catalysis. Reported Zeolite Diffusivities. The diffusivity values reported for zeolites span an astounding range of nearly 5 orders of magnitude. Even focusing on one
2696 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
zeolite type, ZSM-5, and one molecular species, benzene, provides a similar spread of data (Garcia and Weisz, 1990). Some variations were explained by structural differences or inhomogeneitiesof samples; an observed strong temperature dependence has led to the immediate conclusion that diffusion in zeolites is an “activated diffusion” process; a concentration dependence of diffusivity has been seen and explained on the basis of a concept of irreversible thermodynamics, correctable by the Darken correction. Inconsistencies remained, especially the need to know which of the reported diffusivities will determine catalytic selectivity. The state of affairs is a dija vu of the dyeing-of-fibers experience. It presented a challenge for examination and mechanistic inquiry. Examination of the Diffusiodmmobilization Mechanism. In zeolite science strong adsorption of externally applied molecules is routinely acknowledged, with heats of sorption of organic molecules on zeolite sites ranging from 10 to 30 kcaymol. The specific localized sites of such adsorption have been investigated and assigned (e.g., Habgood,l967; Neddenriep, 1968; Reischman et al., 1988). These circumstances suggest the applicability of the diffusiodimmobilization mechanism for deriving the real intra-zeolite molecular diffusivities. In a systematic study on zeolite ZSM-5 (Garcia and Weisz, l990,1993b), the Dappfrom uptake rates and the diffusivities D corrected by applying (6) for o-xylene were determined for a wide variety of concentrations and temperatures. Needless to say, such studies had to be performed at temperatures above the likely “capillary condensation” regime (see above, Figure 2). While Dappvaried with temperature the corrected values of D for each molecular species remained constant! Figure 6 shows this for o-xylene. Dapp ranges between and cm2/sfor various concentrations and temperatures; yet, for all conditions of concentration cm2h was and temperature, D = (2.2 f 0.2) x constant, which represents a dramatic unification of the apparent complexity of uptake observations. Similar constancy of D with temperature was observed for p-xylene, P-methylnaphthalene, and toluidine. The resulting diffisivities D are larger depending sensitively on smaller effective minimum diameters of these molecules. Activated Diffusion? The temperature dependence observed above is characteristic of the methodology of “uptake”, requiring accumulation time for the immobilized species. On the other hand, in any catalytic conversion or membrane separation process we deal with a steady state; the accumulation has long been completed, leaving only the mobile concentration to carry the diffusional rate process characterized by D. In examining the molecular propagation process, we have the following elementary steps: 1. The immobilization process a. adsorption rate b. desorption rate 2. The stochastic (diffusion) process a. probability (shape, orientation, etc.) of entering a passage channel b. energy requirement, if any, t o enter the passage channel c. availability of free adsorption “landing“ sites The “activated”, i.e., temperature dependent, uptake (Dapp)behavior was shown t o be due to the thermody-
300 2 6 0 220
10.8
180
1
10’”
100
140
Ess
6 O°C
= 0.0 kcallmol
1
1o’lJ-
:
lo*16-
1 8
Po= 0.1 mmHg I
I
1.8
2.0
t
I
8
2.2
I
2.4
I
I
2.6
I
I
2.8
I
3.0
1000/T(K)
Figure 6. Apparent diffusivities Dappdetermined from uptake studies of o-xylene on ZSM-5 zeolite at various pressures and temperatures (below); and the diffisivities D aRer correcting each observation by the accumulation factor Clc,. A uniform value of the diffisivity is attained for all cases of pressure and temperature.
namic equilibrium between the diffusing (mobile) and immobilized species ( l a vs lb). On the other hand, the absence of a temperature dependence of D for these cases indicated that the role of molecular shape on diffusivity reflects the probability of successful channel entry (2a) due to size and shape and not due to an energy of activation for “squeezing into” the channel (2b). It is important to remember that Dappderived from a transient process involves the (often very large) time to arrive at the final pool of equilibriated immobilized species. This has little to do with the net diffusional, i.e., stochastic transport, process in the steady-state catalytic or membrane processes. Any inquiry into the effect of molecular shape on that diffusional mobility in such a processes must properly eliminate the effect of the accumulation phenomenon involved in nonsteady-state measurements. When very strong (thermodynamic) sorption of a sorbate creates a high fractional occupation of the total available jump sites (2c), then an inhibition of the number of stochastic transitions can take place, and a temperature dependence of D will occur, as was shown for a noncatalytic Na-ZSM-5 and a polar sorbate (Garcia and Weisz, 1993b). But again, this is due to the thermodynamics of sorption, and is not connected with the energetics of diffusional passage (2b). It is important to differentiate between the large bulk of total immobilization sites (”jumpsites”) and the usually much smaller number of catalytic sites and to recognize that in most gas phase catalytic uses the temperature is high enough to have only a relativly small fraction of jump sites occupied.
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2697 While there may well be an influence of molecular shape on the thermodynamics of catalytic site occupancy, such effects would fall into the realm of kinetics and not mass transport. Such effects would, for example, be unaffected by particle (zeolite crystallite) size. In long term operation, accumulations of internal pore deposits (e.g. “coke”)can ultimately inhibit molecular diffusion (reduce D ) . But this has little to do with the subject of this discussion. One might have (steric) effects of molecular shape on the strength and therefore coverage of sites, but such factors of “shape selectivity” would have to be treated as part of site reactivity, not of mass transport. Such factors would therefore be independent of crystallite size for example. Challenges Regarding Shape Selective Diffusion in Zeolites. 1. Activation Energies of D. The molecules above were aromatics of rigid structure (except for rotating methyl groups). Would other structures need an energy of activation (2a above) t o achieve certain bending or distorted structures to enter the passageways? Or, could it be that since the repulsive forces of the channel atoms vary so critically, i.e., as (llr)7with linear distance, few molecules would be found that would not experience either no repulsion or such high repulsion that “no” useful diffusion rate would be observed? 2. The ‘Window Effect”. Much speculation has been dedicated t o the stochastic reasons for the unusual behavior of Dapp for n-paraffins of various chain length in the specific zeolites T (Gorring, 1973), erionite (Chen et al., 1969),and ferrierite (Gianetti and Perrotta, 1975). It was named the window effect. It now remains to be seen if this curious selectivity is maintained when D is determined by applying the Cdcocorrection, i.e., is the effect really related t o the stochastic diffusion process at all, or is this a case of a sterically induced effect on adsorbability on available sites?
Other Aspects of Immobilization Modified Diffision There are many aspects and variants of this class of mass transport that cannot be adequately discussed in this article. Only a few should be mentioned. “Non-Fickian Diffusion”. When adsorption (immobilization) by Henry’s law (linear isotherm) occurs, the functional shape of solutions to (4) remains perfectly “Fickian”, but the coefficient D changes t o AD. If we insist that whatever coefficient precedes d2c/dx2 in the Fick formalism should be a “diffusivity”, then it has changed by A compared to the free molecular species. Yet, it is the number of diffusing molecules and not their diffusion capability that has changed. When any other isotherm form, C = f f c ) is applicable, Fick’s equation becomes
D [l/f’(c)]
d2c/dx2= -&/dt
(7)
and the shape of solutions will be different. Figure 7 provides two examples of concentration profiles observed in the course of dye diffusion into a fiber (after Peters et al., 1961): In part A,the dye Duranol Red 2B diffuses into cellulose acetate, providing a t all times of penetration the concave concentration profile typical of unmodified Fick behavior. The immobilization isotherm is linear. In part B, penetration of nylon by the dye Scarlet 4R is an example of the S-shaped concentration profiles attained during strong immobilization, wherein
Z
I
.1.
I
A
I
I
x = 2 m
OL
4
1.0
C
0 0
U
0
10
10
30
x (microns)
Figure 7. The concentration profiles during diffusion of dyes into fibers: (A) when the immobilization-isotherm is linear and (B) when the isotherm is nonlinear (typically of the Langmuir form).
the isotherm can be described by the Langmuir form. Such behavior (part B) has been termed as “nonFickian” for many years Formally, it has been attributed to “concentration-dependent diffusion coefficients”. As a phenomenon it has been explained or accepted by concepts of activity or chemical potential driven diffusion or the existence of thermodynamic nonideality (Peters et al., 1961). It seems appropriate now to accept the more elementary and classic explanation: Immobilization occurs according to an independently measurable isotherm determining the relationship of mobile and immobile species, and D is the classical characteristic property of the freely moving molecules. As a matter of fact, the so-called Darken correction for the ”concentration dependence” of diffusion (Dapp) in the unmodified Fick equation-used by proponents of concepts of irreversible thermodynamics or the use of a chemical potential - is identical with the factor [l/f(C)l in (7) applicable to the Langmuir isotherm functionality, but it fails to provide the large constant correction factor Cdco, which is of magnitude significance to the applied science of diffusion (Garcia and Weisz, 1990). The Equilibrium Assumption between Mobile and Immobilized Species. In order t o create the simplification and order in the determination and use of the molecular diffusivities in these various microporous environments, the assumption of near-equilibrium conditions between free and immobilized molecules in any volume element was essential. The assumption is valid when the ad-/desorption kinetics is fast compared to the diffusive transport process studied. It is an important point to recognize when such an assumption is or is not likely to be valid. In any case of appreciably strong adsorption (i.e., appreciable Cdc0,or large negative energy of sorption, -Q),the molecular desorption rate is slowest. For orientation, we can get an estimate of the molecular
.
2698 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 T sec 106
/I hour
min.
sec
meec 10
20 30 P , kcal/mol
40
Figure 8. Classical relationship for the magnitude of characteristic desorption time t of molecules immobilized by an adsorption energy of Q kcavmol, at two temperatures.
desorption time constants theory
Zd
from elementary kinetic
Figure 8 shows calculated values of Zd against Q for two temperatures, 27 and 227 “C. In order to secure applicability of the assumption of near equilibrium of the immobilization process, the time taken by the diffusion process should be long compared to Zd. Thus we find that for non-steady-state techniques such as in sorption or desorption techniques that require many minutes to hours (e.g., in zeolites), the assumption of near equilibrium for gashurface interchange will be amply satisfied when Q 30 kcal/mol a t elevated temperature, but near room temperature Q must be below about 20 kcaymol. For narrow pore materials, such as zeolites, this adds another cause for concern in interpretation of the problem of transition as a “mechanism of capillary spread”. For experimental techniques involving pulsing, such as in NMR techniques that rely on observations over intervals as short as milliseconds, there obviously exist limitations to the interpretation of diffusion measurements, when immobilization energies Q are appreciable and/or temperature is low. (It was noted as a possible explanation for discrepancies between NMR and “macroscopic” diffusion measurements by Karger and Ruthven, 1989.)
Summary and Conclusions Mathematical formalisms can model the behavior of a mechanism; but the nature of mechanisms does not
necessarily follow from adherence to a specific formalism. Fick’s classical formalism for diffision must be modified when molecular diffusion in and through microporous media is accompanied by immobilization (adsorption on surface sites). The diffisivity D,,, derived from a transient (i.e., uptake) experiment is related t o the time of reaching saturation of the pool of equilibrium immobilized species. It involves the thermodynamics of immobilization in addition to the diffusivity D characteristic of molecular motion. In steadystate processes such as catalysis or membrane separation the net diffusivityD after fillage of the pool is operative. D and Dappcan differ by many orders of magnitude. The
transformation of Dappto D is relatively simple, when the equilibration between immobilized and mobile molecules is rapid. It may then be accomplished by a simple but important correction factor (Cdc,). The assumption of near equilibration between free and immobilized species is applicable to many studies and technologies. An elementary theory of molecular ad- and desorption kinetics provides guidance as to certain conditions of low temperature or energy of sorption, when kinetic considerations modify or can invalidate observed “diffision coefficients”. This is most prone to occur in short time observations, such as pulsed probing of the transport behavior. Caution is also indicated in situations where mass transport processes other than diffusion can take place. In micropores approaching molecular dimensions, capillary condensation (or its more sophisticated equivalent) of a gas or vapor may occur a t the particle boundary, followed by a mechanism of “capillary spreading” which follows a square root of time dependence similar to diffusion. Classical laws of capillary condensation can provide guidance as to the operating conditions that require such caution, even for pore dimensions of zeolites. Low (near room) temperature operations are particularly suspect of involvement of such other mechanisms of transport. A strong (exponential)temperature dependence of the (unmodified) Fick diffusion coefficient, Dapp, derived from uptake experiments is often interpreted in terms of the existence of “activated diffusion”, as in the case of zeolites. Such behavior can be the result of the thermodynamics of the immobilization process rather than a property of the diffusive mobility of a molecular species: The uptake time from which the Fick diffisivity coefficient is derived, in case A may be shorter than in case B because a smaller amount of total A molecules is sorbed even though their diffusional mobility is the same or even slower. Determination of D is important for the use in steadystate processes and particularly in relation to such questions as the role of the shape and size of a molecule relative to the size of zeolite passages in shape selective catalysis or shape selective membrane separation processes. The appearance of activated “diffusion” must be subjected to more critical mechanistic examination.
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Received for review November 4, 1994 Accepted February 10, 1995 @
IE940646Y Abstract published in Advance ACS Abstracts, J u n e 15, 1995. @
