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Calculation of the “surface flow” of a dilute gas in model pores from first principles
Calculation of the "Surface Flow" of a Dilute Gas in Model Pores from First Principles III. Molecular Gas Flow in Single Pores and Simple Model Porous Media D. N I C H O L S O N *
AND J. H. P E T R O P O U L O S t
*Department of Chemistry, Imperial College of Science and Technology, London SW7 2AY, England, and tChemistry Department, Democritos Nuclear Research Center. Aghia Paraskevi Attikis, Athens, Greece Received September 10, 1984; accepted January 15, 1985 An earlier theoretical investigation of the transport of a dilute (collisionless) adsorbable gas through narrow model pores has been extended to include model porous media, and to a more detailed examination of the strong adsorption region. Certain characteristics of the pore structure of real porous media have also been modeled in a simple manner. The strength of the gas-solid interaction is characterized by the parameter U0, normalized with respect to temperature. At high U0 the behavior of ~ (= flux at a given Uo/flux at U0 = 0) was similar to that predicted by conventional surface flow theory, with the important difference that the dependence of ¢ on the radius of the model pore was stronger and varied with pore geometry, but not with the form of the adsorption potential function. Porous media were modeled by considering simple examples of the extreme cases of a previously investigated network model, namely parallel or serial arrays of pores of varying radius. In all the cases examined the minimum in the 4~ vs U0 plot, previously found for single pores was retained, thus confirming that the main conclusions from our earlier investigations of single pores are relevant to the flow behavior of real porous media. © 1985AcademicP~ss,Inc. 1. INTRODUCTION In p r e v i o u s papers o f this series (1, 2) we have d e v e l o p e d a n d a p p l i e d m e t h o d s o f calc u l a t i o n o f the free m o l e c u l a r ( K n u d s e n ) flow o f a n a d s o r b a b l e gas in a m o d e l p o r e w h i c h a v o i d the m o r e q u e s t i o n a b l e a s s u m p tions o f c o n v e n t i o n a l "surface flow" theory. C a l c u l a t i o n s were p e r f o r m e d for different p o r e shapes (cylindrical p o r e or t w o - d i m e n sional slit) a n d a d s o r p t i o n p o t e n t i a l fields ( n a m e l y " m o d e l t r i a n g u l a r " or 9 - 3 p o t e n tials). T h e results were expressed in t e r m s o f dp = J/Jg, where J is the flux o f the a d s o r b a b l e gas a n d Jg is the flux o f an identical, but n o n a d s o r b e d , gas u n d e r identical e x p e r i m e n tal c o n d i t i o n s . T h e b e h a v i o r o f ~b was investigated (2) over a wide range o f p o r e widths as a f u n c t i o n o f the a d s o r b a b i l i t y o f the gas a n d the t e m p e r a t u r e , w h i c h were c o m b i n e d in a single dimensionless variable Uo d e n o t i n g
t h e m a x i m u m d e p t h o f the a d s o r p t i o n p o t e n tial well n e a r the (structureless) solid surface in units o f kT. It was f o u n d (2) that, at sufficiently high Uo, the "surface flow c o m p o n e n t , " ¢, = ¢ - 1, decreases e x p o n e n t i a l l y (or very nearly so) with decreasing Uo, in general a g r e e m e n t with the r e q u i r e m e n t s o f c o n v e n t i o n a l "surface flow" theory; b u t a radical d e p a r t u r e from this t h e o r y occurs in the low U0 region. Here, in c o n t r a s t to the m o n o t o n i c decrease o f Cs to zero (i.e., 4~ - ~ l ) as U0 - - ' 0 p r e d i c t e d b y c o n v e n t i o n a l theory, ~b was f o u n d to decrease b e l o w u n i t y a n d t h e n pass t h r o u g h a m i n i m u m before r e t u r n i n g to u n i t y at Uo = O. These theoretical results (based on m o l e c u l a r p a t h tracing c a l c u l a t i o n carefully crossc h e c k e d in Ref. (1) against two i n d e p e n d e n t M o n t e C a r l o procedures) leave n o d o u b t a b o u t the validity o f similar earlier p r e l i m i n a r y c a l c u l a t i o n s (4) o r o f yet earlier a n a l o -
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0021-9797/85 $3.00 Copyright © 1985 by Academic Press, Inc. All rights of reproducfioD in any form reserved.
Jtna'nal of Colloid and Imeff'ace Science, Vol. 106, No. 2, August 1985
CALCULATION OF THE "SURFACE FLOW" OF A DILUTE GAS gous conclusions drawn from a simpler less rigorous theory (5) and permit the interpretation of experimental observations (3) for which no satisfactory alternative explanation has been found. Further detailed calculations at the high end of the U0 range for cylindrical pores showed that the decline of q~ with decreasing Up was predicted by our approach (2) to be appreciably steeper than that required by conventional theory. A much more marked dependence of 4~s on the pore radius R was also found amounting approximately to 4~s R-4(2) as against the conventional ~s R -2 (5). A question naturally arises concerning the extent to which these features of the behavior of ¢hs are affected by pore shape or by the functional form of the adsorption potential. Accordingly, we report here the results of similar computations of 4~s for the two-dimensional slit using both the 9-3 and the "model triangular" potential functions of Ref. (2). A further important point is that the proper application to real porous media of these theoretical conclusions, based exclusively on the behavior of single pores, requires prior knowledge of the likely effect of some basic features of the complex pore structure of such media. The problem of devising models for this purpose, which are not only realistic but also relatively simple, has been examined by us in previous papers (6-8), where a network model, consisting of a regular array o f nodes joined together by capillary tubes of randomly variable radius, was investigated. The characteristic structural parameters of this model are the frequency distribution of pore radius and the connectivity nx (i.e., the number of tubes meeting at a junction). The flow behavior of networks built in this manner was found in all cases to be intermediate between that of parallel and that of serial arrays of capillary tubes, characterized by the same radius distributions (7). These simpler model arrays therefore bracket the behavior of networks. Furthermore, they have the advantage of being amenable to analytical
539
treatment and thus offer a simple and convenient means of investigating the effect of pore structure on the behavior of q~ which is of interest here. Their application to this problem is described in the next section and the results of appropriate model calculations are reported in Section 3. 2. THEORY The values of q~ = J/Jg reported in Ref. (2) refer to a single model pore (in the form of either a cylinder or a two-dimensional slit), which represents a straight section of length 2l in a very long, repeatedly kinked tube whose total length is L. As long as the pore radius (half-width) is the same for all sections of this long tube, it is characterized by the same value of q~. Here, it is convenient to modify somewhat the symbolism of Ref. (2) and define R as the effective pore radius (i.e., that "seen" by the gas molecules) designated by Refr in Ref. (2). R , / , and L are in units of a standard distance (r0 or rm defined in the following section). Denoting by ACg the external gas concentration difference between the ends of the long tube and by ~Cgi the corresponding quantity for a single section i we have
Ji = d;iJgi
[ 1]
where,
Jgi = 7rR~Ogi~Cgi/2li = 7rRZOgiACg/L
[2]
4
2
1 Ln(~,-1 ) ////
/
v
-2
FIG. 1. Dependenceof Ss on U0 for two-dimensional slits with (0 model triangular adsorption potential, I/2R = 10 and: 2R = 1 (O); 2R = 2 (I]);2R = 3 (A); 2R = 5 (V) (in units of r0 see text); or (ii) 9-3 adsorption potential,/]2R = 5 and: 2R = 2.5 (O); 2R = 3.7 (11);2R = 5 (A) (in units of r0, see text). Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
540
NICHOLSON AND PETROPOULOS TABLE I
Dependence o f the Slope of the Linear Part of the In $, vs Uo(Um)Plots of Fig. 1 on Pore Width and Adsorption Potential Function Um Uo
2R
d In ¢, dUo
d In @, dU=
Model Triangular Potential
1 2 3
1 1 1
0.7s 0.79 0.83
0.7s 0.79 0.83
9-3 Potential
2,09 3.7 5
1.07 1.03 1.01
0.79 0.79 0.82
0.74 0.76
0.8n
or, Jv = 2R~Dv(~Cv/21~ = 2 R ~ D v A C J L
[3]
for a cylinder or a two-dimensional slit, respectively. In a single cylinder the diffusion coefficient is given by Dg i = 2 ~)g(li/Ri)R i = B3Ri,
[4]
where ~ is the mean gas molecular speed and g ( l i / R i ) is a complicated function (9) which
reduces to unity when li ~> Rg. For a twodimensional slit (1),
Ov =
2vRi rsinh_ 1 li Ir
L
2Ri
The parallel (P) and serial (S) capillary models to be considered here introduce the effects due to variable pore radius at the extremes of high and low network connectivity as already mentioned. In the parallel model, the pore radius is uniform along any given long tube; but varies from one such tube to another, there being a fraction f = N i / N of long tubes of radius Ri. In the serial model, on the other hand, the pore radius varies from one section to the next; but since L ~>/, all long tubes are statistically identical. Thus, if the total length of sections of radius Ri in any given long tube in the serial model is Li, then f / = L i / L is the same for all long tubes. For purposes of easier comparison with the results of Part II, and to avoid making the calculations unduly complicated, it is advantageous to adjust the length of the individual sections of the long tubes in order to keep a constant lt/Rj ratio, thus ensuring a constant B, in Eqs. [4] and [5]. The quantities of interest here for the above model porous media can be calculated as follows: (a) Parallel (P) Model Here, 6Cv/2li = A C v / L = ACe~L; whereas J, is a function of Ri. We have
4R] _
In contrast to B3 in Eq. [4], B2 in Eq. [5] remains a function of ldRi even when l~ Ri.
A bundle of N long tubes of the type described above is the simplest available model of a porous medium, when Ri is chosen to be equal to the mean effective pore radius of the given medium Re = 2~/A. Here, c is the porosity and A is the specific surface area per unit volume of the porous medium. The fluxes J, Jg through such a model porous medium (which may conveniently be used as a standard of reference) are simply given by J = NJi, Jg = N J v, so that for this model, @ = 0;. Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
T tn(,p-1 )
d ~.n 2R
,,
FIG. 2. Dependence of 4~, on pore width for twodimensional slits with (i) model triangular adsorption potential (l/2R = 10) at: Uo = 5 (XT); Uo = 6 (0); Uo = 6.5 (O); Uo = 7.5 (&); U0 = 8 (I-I); or (ii) 9-3 adsorption potential (I/2R = 5) at: U0 = 6.16 (O); Uo 6.93 (ml).
CALCULATION OF THE "'SURFACE FLOW" OF A DILUTE GAS
541
1.4
,-tr
$2
/ / A
9-
/
08 r~ ~'~ P2 ,
/
1
IR2
a4
9- *'~
5 R3
~4 o
R1
4
2
0
5
FIG. 3. Dependence of ~b on U0 for parallel (P) and serial (S) pore models (full lines P and S, respectively) with radius distributions Fa~, F~, Fa3 (full lines 1, 2, and 3, respectively) in comparison with the corresponding lines (broken lines labeled R) for the reference medium of radius Re (see text).
changed. W e have 6Cgi/2li = ACgdLi, where ACg; represents the total concentration drop (in terms o f external gas phase concentration) in all sections o f radius Ri in any given long tube. Thus,
J~ = E NiJ~ = N E fJ~4~i i
i
4 = E N~4i = N E ~rB,fR7 ACg i i L
J = NJi = NJ~49i
Therefore,
4~ = ~ f R 7 4 ~ d ~ f R 7 i
Jg = NJgi = NTrB,,RT AC~i/Li
[6]
i
where n = 3 for cylinders and n = 2 for slits.
(b) Serial (S) Model Here J~ = const. The concentration gradient is the same in all tube sections with the same radius R~ but changes when R~ is
=N
B.R7 AC
df, L.
But
f. = Z ACgi =
i
Z NTrB, i R i
JL N ~ B , ~ RT~i" Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
542
NICHOLSON
AND
PETROPOULOS
1.6
1.~
T 9c
0~
0.4
$2 1.2
0.8 P2
T
0¢;
R2
S3 J° .
9t--
P3
0.~
P/ R1
2
4
Uo FIG. 4. AS Fig. 3 for radius distributions Fb~, Fb2, Fb3.
Hence
4a = Z ( f l R ~ ) I Z i
(f/R~cbi),
[7]
sorption potential function used previously (1, 2), namely:
i
(i) 9 - 3 Adsorption Potential where n = 3 for cylinders and n = 2 for slits. The above model porous media are characterized by a mean effective radius
Re = 2,/A = Z f R T - 1 / Z f R 7 i
-2
[8]
U(r) --- UoKI [(Ro + r) -9 + (Ro - r) -9 - (R0 + r) -3 - (Ro - r)-3l
[91
i
where n = 3 (cylinders) or 2 (slits). 3. RESULTS AND DISCUSSION 3.1 SINGLE PORES Calculations o f ~ have been performed for two-dimensional slits with both types o f adJournal of Colloid and Interface Science, Vol. 106, No, 2, August 1985
where Kt = 2.598 and - R 0 ~< r ~< Ro measures distance from the center o f the slit, at right angles to the slit walls, in units o f ro, the separation o f an admolecule from the surface layer o f adsorbent atoms. At the potential energy m i n i m u m near to the slit wall, r = __+R and the solid surfaces are located at r = +Ro.
CALCULATION
OF
THE
"SURFACE
FLOW"
OF
A DILUTE
GAS
543
2
1.6
s2/
//"/1 / /
(38 0.4
....
/
-0,8
0.4
-
1-
9c
I
JR3
J
P1 $1 0.4
0
-
0-4 /
o
FIG. 5. As Fig. 3 for radius distributions Fc~, F¢2, Fc3.
(ii) Model Triangular Potential U(r) = -Uo(1 - R + Ir[) ra ~< Irl < R U(r)
=
0
0
<
Irl
<
ra
[lO] where - R ~< r ~ R; ra = R - 1 if R > 1, ra = 0 if R ~< 1; and r, R are now in units of rm, the m a x i m u m width of the potential well (1). This is, or course, a purely artificial adsorption potential function; but one which is useful here for the purpose of (a) saving computer time and (b) studying the effect of the functional form of U(r). The results obtained for various values of
R have been plotted in Fig. 1 in the same way as was done for cylinders in Fig. 5 of Ref. (2). The behavior of In @s as a function of U0 is found to be very similar, the relevant plots becoming linear, or very nearly so, at sufficiently high Uo. The slopes of these linear sections are given in Table I together with those of the corresponding lines in terms of Urn, the actual depth of the potential well m i n i m u m in a pore (cf. Uo which is the depth of the potential well near an isolated solid surface; U m = U0 by definition in the case of the model triangular potential). C o m parison of these results with the corresponding ones of Ref. (2) has brought to our attention Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
544
NICHOLSON
3
sa 2 $1
?
AND
PETROPOULOS
9-3 potential), Fig. 6 of Ref. (2) yields m 4 at constant U0; but the value of m is found to be very close to that for slits with 9-3 potential when evaluated at constant Um. By comparison the conventional theory (5) yields m = 2.
1
3.2 MODEL POROUS MEDIA
.En($-1)
Calculations have been performed on parallel and serial combinations of pores of two • / different radii using Eqs. [6] and [7] (corre-1 sponding to radius distribution F in our -2 I I I ~ I earlier works (7, 8)). To reduce computer time, slit-shaped pores (I/R = 10) with the U O model triangular adsorption potential were FIG. 6. Dependenceof $~ on U0 for parallel and serial chosen. Three basic sets of pore widths were pore models (lines P and S, respectively)with radius employed, namely 2R1 = 2, 2RI = 5 (desigdistributions F~, F~2,F~3 (lines 1, 2, 3 respectively). nated as F0; 2R1 = 1, 2R2 = 5 (F2); and 2RI = 2, 2R2 = 10 (F3). Each of these sets was used in three forms, which had (i) equal the fact that the values of U0(Um) and of 1 proportions of RI, R2 (f~ = J2 = 0.5, desig+ Rerr for cylinders reported there were not in fact normalized in the manner described. nated as D~); (ii) a higher proportion of the The properly normalized values are respec- larger radius (j~ = 0.1, f2 = 0.9, designated tively 1.491 and 1.570 times higher than as Fb); (iii) a higher proportion of the smaller those shown (Figs. 2, 5, and 6 and Table I radius (f~ = 0.9, J~ = 0.1, designated as Fc). o f Ref. (2)). The definition of r0 should also These nine combinations cover a wide range be corrected to that given in the present of effective mean pore widths, namely 2R¢ paper. It should be stressed that these changes = 1.4 (Fc2), 2.3 (Fc0, 2.8 (Fc3), 2 (F~2), 3.5 do not affect the basic conclusions in (2). (Fa~), 4.6 (Fb2), 4.7 (Fbl), 6 (Fa3), 9.2 (Fb3) However the values of 0 In cks/OUoand of 0 and may be expected to include a wide In Cks/OUmof Table I therein are now reduced variety of possible flow behaviors. by a factor of 1.491. This brings these values into line with those recorded in Table I here. 3 Hence it appears that pore shape or the functional form of the adsorption potential 2 have little effect on the slope of the linear part of the In ~bs versus Um plot, which I however, turns out to be significantly less steep than the unit slope implied by conven0 tional theory (5). In the region of the linear In $~ versus Um plot the dependence of ~bs -I o on pore size is expressible, at least approximately, as an inverse power law 4~s ~ R -m, as indicated by the results obtained for both slits (Fig. 2) and cylinders (Fig. 6 of Ref. (2)). These results yield m = 2.1 or m = 2.8-3.0 I./O for slits with model triangular potential or FIG. 7. AS Fig. 6 for radius distributions Fb~,Fb2,Fb3. 9-3 potential, respectively. For cylinders (with 0
?
Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
545
C A L C U L A T I O N OF THE "SURFACE F L O W " O F A D ILU TE GAS
The resulting values of @are shown as a function of U0 in Figs. 3-5 with particular emphasis on the lower Up region. The main point to note is that the behavior of @ is in no case radically different from that found previously for single pores. In particular, ~b always falls below unity as U0 decreases before returning to unity at U0 ~ 0. However, the position, depth, and width of the minimum can differ quite markedly from what is expected on the basis of the mean effective pore size of the model porous medium (indicated by the broken reference lines in Figs. 3-5). Some general pore structure effects are easily discernible. Thus, S-type structure (constricted pores) causes a shift of the minimum to lower U0, whereas P-type structure generally leads to a shallower minimum. These effects reflect the fact that flow through the model porous medium tends to be ratecontrolled by the narrow pores (constrictions) and wide pores in the S and P models, respectively. Reference to Fig. 4 shows that the presence of a small proportion of small radii has no noticeable effect on ~ in a Ptype structure; but the effect becomes very strong in an S-type structure. This agrees with our earlier study of the influence of pore structure on adsorbable gas flow based on conventional "surface flow" theory (7). Such a pore size distribution is not realistic from the practical point of view, however. In practice, there is usually a preponderance of small radii. This case is represented by F¢ here. Reference to Fig. 5 shows that both P and S models with distribution F~ tend to produce appreciable deviations from the reference line R. Perhaps the most striking effect to be noted in Fig. 5 is the marked flattening of the region of the minimum in curve P3 by comparison with R 3. It will be recalled (cf. Introduction) that Figs. 3-5 depict extreme limits of behavior for the corresponding random networks, which should, therefore, exhibit considerably smaller deviations from the relevant reference curves at realistic values of the connectivity nx. On the other hand, it is most probable
that the deviations found here would have been considerably more marked (especially for cylinders) if the calculations had been carried out with a 9-3 adsorption potential, because @then depends more strongly on R (see section 3.1). Another point of practical interest is that a P model with radius distribution F¢ should be representative of bidisperse pore networks, such as would be found in a porous solid consisting of relatively lightly compacted microporous particles. In such a case the intraparticle micropore network and the interparticle network are effectively in parallel. When Up is large, interest centers on the form of the In @~vs U0 plot. The results shown in Figs. 6-8 are for the model porous media studied here. The plots for radius distributions Fa and F¢ (Figs. 6 and 8) do not differ materially from the corresponding ones for single pores (Fig. 1), apart from a tendency for the slope of the linear part to be higher or lower in the case of the P or S model, respectively, which is discernible in Fig. 6 (distribution Fa). Particularly low values for this slope appear to be obtainable in the case of the S model with radius distribution Fb ('~0.54 for line $3 in Fig. 7; however the quasilinear part of these plots shows a tendency to curve slightly upward (cf. line $2 of Fig. 7), so that the slope tends to become higher at still higher Up. 4. CONCLUSION
The present paper complements our previous study on single pores (2) by a detailed 4
SI, $3
31 2 ¢.n~-I )
I 0
o
/a ,/a
-I -2 4
6
Uo~
B
FIG. 8. As Fig. 6 for radiusdistributionsFct, Fc2,F¢3. Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
546
NICHOLSON AND PETROPOULOS
examination of the quantitative predictions of our theory about the flow of more strongly adsorbed gases in two-dimensional slits. The main conclusion to be drawn from these results, in conjunction with those of Ref. (2), is that the functional dependence of 4~s on Um and R for strongly adsorbed gases flowing in narrow pores closely approximates to that predicted by the conventional surface flow theory, although there may be significant differences in the values of the relevant parameters. A second important objective of the present paper is to obtain some information bridging the gap between single model pores and real porous solids by investigating the behavior of suitably chosen simple model porous media. For simplicity and ease of computation, these model porous media were constructed from slits with a triangular adsorption potential of only two different widths; but care was taken to include examples of different pore size distribution shape, width, and mean value. The main conclusion drawn from the results is that there is no fundamental departure from the behavior previously found in single pores. In particular, the ~b vs Uo plot always exhibits a minimum. It is shown, however, that various pore structure effects can markedly shift, flatten, or (occasionally) deepen this minimum. In the high U0 region,
Journal of Colloidand InterfaceScience, Vol. 106,No. 2, August1985
the linearity of the In ~bs vs Uo plot appears to be largely preserved, at least for pore size distributions which are reasonably realistic from the practical point of view, although the slope may be modified. The above results strongly support the main conclusions of our theory in its application to the flow of dilute adsorbable gases in porous solids. REFERENCES 1. Nicholson, D., Petrou, J., and Petropoulos, J. H., J. Colloid Interface Sci. 71, 570 (1979). 2. Nicholson, D., and Petropoulos, J. H., J. Colloid Interface Sci. 83, 420 (1981). 3. Hwang, S. T., and Kammermeyer, K., Canad. J. Chem. 44, 1982 (1966); Havredaki, V., and Petropoulos, J. H., J. Membrane Sci. 12, 303 (1983); Whiting, D. E. G., "'Isothermal Transport of Sorbable Gases in Graphitized Carbon Membranes: The Effect of Temperature," Thesis. London University, 1981. 4. Nicholson, D., and Petropoulos, J. H., Ber. BunsenGes. Phys. Chem. 79, 796 (1975). 5. Nicholson, D., and Petropoulos, J. H., J. Colloid Interface Sci. 45, 459 (1973). 6. Nicholson, D., and Petropoulos, J. H., J. Phys. D 4, 181 (1971). 7. Nicholson, D., and Petropoulos, J. H., J. Phys. D 6, 1737 (1973). 8. Nicholson, D., and Petropoulos, J. H., J. Phys. D 8, 1430 (1975). 9, Berman, A. S., J. Appl. Phys. 17, 1901 (1966).
AND J. H. P E T R O P O U L O S t
*Department of Chemistry, Imperial College of Science and Technology, London SW7 2AY, England, and tChemistry Department, Democritos Nuclear Research Center. Aghia Paraskevi Attikis, Athens, Greece Received September 10, 1984; accepted January 15, 1985 An earlier theoretical investigation of the transport of a dilute (collisionless) adsorbable gas through narrow model pores has been extended to include model porous media, and to a more detailed examination of the strong adsorption region. Certain characteristics of the pore structure of real porous media have also been modeled in a simple manner. The strength of the gas-solid interaction is characterized by the parameter U0, normalized with respect to temperature. At high U0 the behavior of ~ (= flux at a given Uo/flux at U0 = 0) was similar to that predicted by conventional surface flow theory, with the important difference that the dependence of ¢ on the radius of the model pore was stronger and varied with pore geometry, but not with the form of the adsorption potential function. Porous media were modeled by considering simple examples of the extreme cases of a previously investigated network model, namely parallel or serial arrays of pores of varying radius. In all the cases examined the minimum in the 4~ vs U0 plot, previously found for single pores was retained, thus confirming that the main conclusions from our earlier investigations of single pores are relevant to the flow behavior of real porous media. © 1985AcademicP~ss,Inc. 1. INTRODUCTION In p r e v i o u s papers o f this series (1, 2) we have d e v e l o p e d a n d a p p l i e d m e t h o d s o f calc u l a t i o n o f the free m o l e c u l a r ( K n u d s e n ) flow o f a n a d s o r b a b l e gas in a m o d e l p o r e w h i c h a v o i d the m o r e q u e s t i o n a b l e a s s u m p tions o f c o n v e n t i o n a l "surface flow" theory. C a l c u l a t i o n s were p e r f o r m e d for different p o r e shapes (cylindrical p o r e or t w o - d i m e n sional slit) a n d a d s o r p t i o n p o t e n t i a l fields ( n a m e l y " m o d e l t r i a n g u l a r " or 9 - 3 p o t e n tials). T h e results were expressed in t e r m s o f dp = J/Jg, where J is the flux o f the a d s o r b a b l e gas a n d Jg is the flux o f an identical, but n o n a d s o r b e d , gas u n d e r identical e x p e r i m e n tal c o n d i t i o n s . T h e b e h a v i o r o f ~b was investigated (2) over a wide range o f p o r e widths as a f u n c t i o n o f the a d s o r b a b i l i t y o f the gas a n d the t e m p e r a t u r e , w h i c h were c o m b i n e d in a single dimensionless variable Uo d e n o t i n g
t h e m a x i m u m d e p t h o f the a d s o r p t i o n p o t e n tial well n e a r the (structureless) solid surface in units o f kT. It was f o u n d (2) that, at sufficiently high Uo, the "surface flow c o m p o n e n t , " ¢, = ¢ - 1, decreases e x p o n e n t i a l l y (or very nearly so) with decreasing Uo, in general a g r e e m e n t with the r e q u i r e m e n t s o f c o n v e n t i o n a l "surface flow" theory; b u t a radical d e p a r t u r e from this t h e o r y occurs in the low U0 region. Here, in c o n t r a s t to the m o n o t o n i c decrease o f Cs to zero (i.e., 4~ - ~ l ) as U0 - - ' 0 p r e d i c t e d b y c o n v e n t i o n a l theory, ~b was f o u n d to decrease b e l o w u n i t y a n d t h e n pass t h r o u g h a m i n i m u m before r e t u r n i n g to u n i t y at Uo = O. These theoretical results (based on m o l e c u l a r p a t h tracing c a l c u l a t i o n carefully crossc h e c k e d in Ref. (1) against two i n d e p e n d e n t M o n t e C a r l o procedures) leave n o d o u b t a b o u t the validity o f similar earlier p r e l i m i n a r y c a l c u l a t i o n s (4) o r o f yet earlier a n a l o -
538
0021-9797/85 $3.00 Copyright © 1985 by Academic Press, Inc. All rights of reproducfioD in any form reserved.
Jtna'nal of Colloid and Imeff'ace Science, Vol. 106, No. 2, August 1985
CALCULATION OF THE "SURFACE FLOW" OF A DILUTE GAS gous conclusions drawn from a simpler less rigorous theory (5) and permit the interpretation of experimental observations (3) for which no satisfactory alternative explanation has been found. Further detailed calculations at the high end of the U0 range for cylindrical pores showed that the decline of q~ with decreasing Up was predicted by our approach (2) to be appreciably steeper than that required by conventional theory. A much more marked dependence of 4~s on the pore radius R was also found amounting approximately to 4~s R-4(2) as against the conventional ~s R -2 (5). A question naturally arises concerning the extent to which these features of the behavior of ¢hs are affected by pore shape or by the functional form of the adsorption potential. Accordingly, we report here the results of similar computations of 4~s for the two-dimensional slit using both the 9-3 and the "model triangular" potential functions of Ref. (2). A further important point is that the proper application to real porous media of these theoretical conclusions, based exclusively on the behavior of single pores, requires prior knowledge of the likely effect of some basic features of the complex pore structure of such media. The problem of devising models for this purpose, which are not only realistic but also relatively simple, has been examined by us in previous papers (6-8), where a network model, consisting of a regular array o f nodes joined together by capillary tubes of randomly variable radius, was investigated. The characteristic structural parameters of this model are the frequency distribution of pore radius and the connectivity nx (i.e., the number of tubes meeting at a junction). The flow behavior of networks built in this manner was found in all cases to be intermediate between that of parallel and that of serial arrays of capillary tubes, characterized by the same radius distributions (7). These simpler model arrays therefore bracket the behavior of networks. Furthermore, they have the advantage of being amenable to analytical
539
treatment and thus offer a simple and convenient means of investigating the effect of pore structure on the behavior of q~ which is of interest here. Their application to this problem is described in the next section and the results of appropriate model calculations are reported in Section 3. 2. THEORY The values of q~ = J/Jg reported in Ref. (2) refer to a single model pore (in the form of either a cylinder or a two-dimensional slit), which represents a straight section of length 2l in a very long, repeatedly kinked tube whose total length is L. As long as the pore radius (half-width) is the same for all sections of this long tube, it is characterized by the same value of q~. Here, it is convenient to modify somewhat the symbolism of Ref. (2) and define R as the effective pore radius (i.e., that "seen" by the gas molecules) designated by Refr in Ref. (2). R , / , and L are in units of a standard distance (r0 or rm defined in the following section). Denoting by ACg the external gas concentration difference between the ends of the long tube and by ~Cgi the corresponding quantity for a single section i we have
Ji = d;iJgi
[ 1]
where,
Jgi = 7rR~Ogi~Cgi/2li = 7rRZOgiACg/L
[2]
4
2
1 Ln(~,-1 ) ////
/
v
-2
FIG. 1. Dependenceof Ss on U0 for two-dimensional slits with (0 model triangular adsorption potential, I/2R = 10 and: 2R = 1 (O); 2R = 2 (I]);2R = 3 (A); 2R = 5 (V) (in units of r0 see text); or (ii) 9-3 adsorption potential,/]2R = 5 and: 2R = 2.5 (O); 2R = 3.7 (11);2R = 5 (A) (in units of r0, see text). Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
540
NICHOLSON AND PETROPOULOS TABLE I
Dependence o f the Slope of the Linear Part of the In $, vs Uo(Um)Plots of Fig. 1 on Pore Width and Adsorption Potential Function Um Uo
2R
d In ¢, dUo
d In @, dU=
Model Triangular Potential
1 2 3
1 1 1
0.7s 0.79 0.83
0.7s 0.79 0.83
9-3 Potential
2,09 3.7 5
1.07 1.03 1.01
0.79 0.79 0.82
0.74 0.76
0.8n
or, Jv = 2R~Dv(~Cv/21~ = 2 R ~ D v A C J L
[3]
for a cylinder or a two-dimensional slit, respectively. In a single cylinder the diffusion coefficient is given by Dg i = 2 ~)g(li/Ri)R i = B3Ri,
[4]
where ~ is the mean gas molecular speed and g ( l i / R i ) is a complicated function (9) which
reduces to unity when li ~> Rg. For a twodimensional slit (1),
Ov =
2vRi rsinh_ 1 li Ir
L
2Ri
The parallel (P) and serial (S) capillary models to be considered here introduce the effects due to variable pore radius at the extremes of high and low network connectivity as already mentioned. In the parallel model, the pore radius is uniform along any given long tube; but varies from one such tube to another, there being a fraction f = N i / N of long tubes of radius Ri. In the serial model, on the other hand, the pore radius varies from one section to the next; but since L ~>/, all long tubes are statistically identical. Thus, if the total length of sections of radius Ri in any given long tube in the serial model is Li, then f / = L i / L is the same for all long tubes. For purposes of easier comparison with the results of Part II, and to avoid making the calculations unduly complicated, it is advantageous to adjust the length of the individual sections of the long tubes in order to keep a constant lt/Rj ratio, thus ensuring a constant B, in Eqs. [4] and [5]. The quantities of interest here for the above model porous media can be calculated as follows: (a) Parallel (P) Model Here, 6Cv/2li = A C v / L = ACe~L; whereas J, is a function of Ri. We have
4R] _
In contrast to B3 in Eq. [4], B2 in Eq. [5] remains a function of ldRi even when l~ Ri.
A bundle of N long tubes of the type described above is the simplest available model of a porous medium, when Ri is chosen to be equal to the mean effective pore radius of the given medium Re = 2~/A. Here, c is the porosity and A is the specific surface area per unit volume of the porous medium. The fluxes J, Jg through such a model porous medium (which may conveniently be used as a standard of reference) are simply given by J = NJi, Jg = N J v, so that for this model, @ = 0;. Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
T tn(,p-1 )
d ~.n 2R
,,
FIG. 2. Dependence of 4~, on pore width for twodimensional slits with (i) model triangular adsorption potential (l/2R = 10) at: Uo = 5 (XT); Uo = 6 (0); Uo = 6.5 (O); Uo = 7.5 (&); U0 = 8 (I-I); or (ii) 9-3 adsorption potential (I/2R = 5) at: U0 = 6.16 (O); Uo 6.93 (ml).
CALCULATION OF THE "'SURFACE FLOW" OF A DILUTE GAS
541
1.4
,-tr
$2
/ / A
9-
/
08 r~ ~'~ P2 ,
/
1
IR2
a4
9- *'~
5 R3
~4 o
R1
4
2
0
5
FIG. 3. Dependence of ~b on U0 for parallel (P) and serial (S) pore models (full lines P and S, respectively) with radius distributions Fa~, F~, Fa3 (full lines 1, 2, and 3, respectively) in comparison with the corresponding lines (broken lines labeled R) for the reference medium of radius Re (see text).
changed. W e have 6Cgi/2li = ACgdLi, where ACg; represents the total concentration drop (in terms o f external gas phase concentration) in all sections o f radius Ri in any given long tube. Thus,
J~ = E NiJ~ = N E fJ~4~i i
i
4 = E N~4i = N E ~rB,fR7 ACg i i L
J = NJi = NJ~49i
Therefore,
4~ = ~ f R 7 4 ~ d ~ f R 7 i
Jg = NJgi = NTrB,,RT AC~i/Li
[6]
i
where n = 3 for cylinders and n = 2 for slits.
(b) Serial (S) Model Here J~ = const. The concentration gradient is the same in all tube sections with the same radius R~ but changes when R~ is
=N
B.R7 AC
df, L.
But
f. = Z ACgi =
i
Z NTrB, i R i
JL N ~ B , ~ RT~i" Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
542
NICHOLSON
AND
PETROPOULOS
1.6
1.~
T 9c
0~
0.4
$2 1.2
0.8 P2
T
0¢;
R2
S3 J° .
9t--
P3
0.~
P/ R1
2
4
Uo FIG. 4. AS Fig. 3 for radius distributions Fb~, Fb2, Fb3.
Hence
4a = Z ( f l R ~ ) I Z i
(f/R~cbi),
[7]
sorption potential function used previously (1, 2), namely:
i
(i) 9 - 3 Adsorption Potential where n = 3 for cylinders and n = 2 for slits. The above model porous media are characterized by a mean effective radius
Re = 2,/A = Z f R T - 1 / Z f R 7 i
-2
[8]
U(r) --- UoKI [(Ro + r) -9 + (Ro - r) -9 - (R0 + r) -3 - (Ro - r)-3l
[91
i
where n = 3 (cylinders) or 2 (slits). 3. RESULTS AND DISCUSSION 3.1 SINGLE PORES Calculations o f ~ have been performed for two-dimensional slits with both types o f adJournal of Colloid and Interface Science, Vol. 106, No, 2, August 1985
where Kt = 2.598 and - R 0 ~< r ~< Ro measures distance from the center o f the slit, at right angles to the slit walls, in units o f ro, the separation o f an admolecule from the surface layer o f adsorbent atoms. At the potential energy m i n i m u m near to the slit wall, r = __+R and the solid surfaces are located at r = +Ro.
CALCULATION
OF
THE
"SURFACE
FLOW"
OF
A DILUTE
GAS
543
2
1.6
s2/
//"/1 / /
(38 0.4
....
/
-0,8
0.4
-
1-
9c
I
JR3
J
P1 $1 0.4
0
-
0-4 /
o
FIG. 5. As Fig. 3 for radius distributions Fc~, F¢2, Fc3.
(ii) Model Triangular Potential U(r) = -Uo(1 - R + Ir[) ra ~< Irl < R U(r)
=
0
0
<
Irl
<
ra
[lO] where - R ~< r ~ R; ra = R - 1 if R > 1, ra = 0 if R ~< 1; and r, R are now in units of rm, the m a x i m u m width of the potential well (1). This is, or course, a purely artificial adsorption potential function; but one which is useful here for the purpose of (a) saving computer time and (b) studying the effect of the functional form of U(r). The results obtained for various values of
R have been plotted in Fig. 1 in the same way as was done for cylinders in Fig. 5 of Ref. (2). The behavior of In @s as a function of U0 is found to be very similar, the relevant plots becoming linear, or very nearly so, at sufficiently high Uo. The slopes of these linear sections are given in Table I together with those of the corresponding lines in terms of Urn, the actual depth of the potential well m i n i m u m in a pore (cf. Uo which is the depth of the potential well near an isolated solid surface; U m = U0 by definition in the case of the model triangular potential). C o m parison of these results with the corresponding ones of Ref. (2) has brought to our attention Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
544
NICHOLSON
3
sa 2 $1
?
AND
PETROPOULOS
9-3 potential), Fig. 6 of Ref. (2) yields m 4 at constant U0; but the value of m is found to be very close to that for slits with 9-3 potential when evaluated at constant Um. By comparison the conventional theory (5) yields m = 2.
1
3.2 MODEL POROUS MEDIA
.En($-1)
Calculations have been performed on parallel and serial combinations of pores of two • / different radii using Eqs. [6] and [7] (corre-1 sponding to radius distribution F in our -2 I I I ~ I earlier works (7, 8)). To reduce computer time, slit-shaped pores (I/R = 10) with the U O model triangular adsorption potential were FIG. 6. Dependenceof $~ on U0 for parallel and serial chosen. Three basic sets of pore widths were pore models (lines P and S, respectively)with radius employed, namely 2R1 = 2, 2RI = 5 (desigdistributions F~, F~2,F~3 (lines 1, 2, 3 respectively). nated as F0; 2R1 = 1, 2R2 = 5 (F2); and 2RI = 2, 2R2 = 10 (F3). Each of these sets was used in three forms, which had (i) equal the fact that the values of U0(Um) and of 1 proportions of RI, R2 (f~ = J2 = 0.5, desig+ Rerr for cylinders reported there were not in fact normalized in the manner described. nated as D~); (ii) a higher proportion of the The properly normalized values are respec- larger radius (j~ = 0.1, f2 = 0.9, designated tively 1.491 and 1.570 times higher than as Fb); (iii) a higher proportion of the smaller those shown (Figs. 2, 5, and 6 and Table I radius (f~ = 0.9, J~ = 0.1, designated as Fc). o f Ref. (2)). The definition of r0 should also These nine combinations cover a wide range be corrected to that given in the present of effective mean pore widths, namely 2R¢ paper. It should be stressed that these changes = 1.4 (Fc2), 2.3 (Fc0, 2.8 (Fc3), 2 (F~2), 3.5 do not affect the basic conclusions in (2). (Fa~), 4.6 (Fb2), 4.7 (Fbl), 6 (Fa3), 9.2 (Fb3) However the values of 0 In cks/OUoand of 0 and may be expected to include a wide In Cks/OUmof Table I therein are now reduced variety of possible flow behaviors. by a factor of 1.491. This brings these values into line with those recorded in Table I here. 3 Hence it appears that pore shape or the functional form of the adsorption potential 2 have little effect on the slope of the linear part of the In ~bs versus Um plot, which I however, turns out to be significantly less steep than the unit slope implied by conven0 tional theory (5). In the region of the linear In $~ versus Um plot the dependence of ~bs -I o on pore size is expressible, at least approximately, as an inverse power law 4~s ~ R -m, as indicated by the results obtained for both slits (Fig. 2) and cylinders (Fig. 6 of Ref. (2)). These results yield m = 2.1 or m = 2.8-3.0 I./O for slits with model triangular potential or FIG. 7. AS Fig. 6 for radius distributions Fb~,Fb2,Fb3. 9-3 potential, respectively. For cylinders (with 0
?
Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
545
C A L C U L A T I O N OF THE "SURFACE F L O W " O F A D ILU TE GAS
The resulting values of @are shown as a function of U0 in Figs. 3-5 with particular emphasis on the lower Up region. The main point to note is that the behavior of @ is in no case radically different from that found previously for single pores. In particular, ~b always falls below unity as U0 decreases before returning to unity at U0 ~ 0. However, the position, depth, and width of the minimum can differ quite markedly from what is expected on the basis of the mean effective pore size of the model porous medium (indicated by the broken reference lines in Figs. 3-5). Some general pore structure effects are easily discernible. Thus, S-type structure (constricted pores) causes a shift of the minimum to lower U0, whereas P-type structure generally leads to a shallower minimum. These effects reflect the fact that flow through the model porous medium tends to be ratecontrolled by the narrow pores (constrictions) and wide pores in the S and P models, respectively. Reference to Fig. 4 shows that the presence of a small proportion of small radii has no noticeable effect on ~ in a Ptype structure; but the effect becomes very strong in an S-type structure. This agrees with our earlier study of the influence of pore structure on adsorbable gas flow based on conventional "surface flow" theory (7). Such a pore size distribution is not realistic from the practical point of view, however. In practice, there is usually a preponderance of small radii. This case is represented by F¢ here. Reference to Fig. 5 shows that both P and S models with distribution F~ tend to produce appreciable deviations from the reference line R. Perhaps the most striking effect to be noted in Fig. 5 is the marked flattening of the region of the minimum in curve P3 by comparison with R 3. It will be recalled (cf. Introduction) that Figs. 3-5 depict extreme limits of behavior for the corresponding random networks, which should, therefore, exhibit considerably smaller deviations from the relevant reference curves at realistic values of the connectivity nx. On the other hand, it is most probable
that the deviations found here would have been considerably more marked (especially for cylinders) if the calculations had been carried out with a 9-3 adsorption potential, because @then depends more strongly on R (see section 3.1). Another point of practical interest is that a P model with radius distribution F¢ should be representative of bidisperse pore networks, such as would be found in a porous solid consisting of relatively lightly compacted microporous particles. In such a case the intraparticle micropore network and the interparticle network are effectively in parallel. When Up is large, interest centers on the form of the In @~vs U0 plot. The results shown in Figs. 6-8 are for the model porous media studied here. The plots for radius distributions Fa and F¢ (Figs. 6 and 8) do not differ materially from the corresponding ones for single pores (Fig. 1), apart from a tendency for the slope of the linear part to be higher or lower in the case of the P or S model, respectively, which is discernible in Fig. 6 (distribution Fa). Particularly low values for this slope appear to be obtainable in the case of the S model with radius distribution Fb ('~0.54 for line $3 in Fig. 7; however the quasilinear part of these plots shows a tendency to curve slightly upward (cf. line $2 of Fig. 7), so that the slope tends to become higher at still higher Up. 4. CONCLUSION
The present paper complements our previous study on single pores (2) by a detailed 4
SI, $3
31 2 ¢.n~-I )
I 0
o
/a ,/a
-I -2 4
6
Uo~
B
FIG. 8. As Fig. 6 for radiusdistributionsFct, Fc2,F¢3. Journal of Colloid and Interface Science, Vol. 106, No. 2, August 1985
546
NICHOLSON AND PETROPOULOS
examination of the quantitative predictions of our theory about the flow of more strongly adsorbed gases in two-dimensional slits. The main conclusion to be drawn from these results, in conjunction with those of Ref. (2), is that the functional dependence of 4~s on Um and R for strongly adsorbed gases flowing in narrow pores closely approximates to that predicted by the conventional surface flow theory, although there may be significant differences in the values of the relevant parameters. A second important objective of the present paper is to obtain some information bridging the gap between single model pores and real porous solids by investigating the behavior of suitably chosen simple model porous media. For simplicity and ease of computation, these model porous media were constructed from slits with a triangular adsorption potential of only two different widths; but care was taken to include examples of different pore size distribution shape, width, and mean value. The main conclusion drawn from the results is that there is no fundamental departure from the behavior previously found in single pores. In particular, the ~b vs Uo plot always exhibits a minimum. It is shown, however, that various pore structure effects can markedly shift, flatten, or (occasionally) deepen this minimum. In the high U0 region,
Journal of Colloidand InterfaceScience, Vol. 106,No. 2, August1985
the linearity of the In ~bs vs Uo plot appears to be largely preserved, at least for pore size distributions which are reasonably realistic from the practical point of view, although the slope may be modified. The above results strongly support the main conclusions of our theory in its application to the flow of dilute adsorbable gases in porous solids. REFERENCES 1. Nicholson, D., Petrou, J., and Petropoulos, J. H., J. Colloid Interface Sci. 71, 570 (1979). 2. Nicholson, D., and Petropoulos, J. H., J. Colloid Interface Sci. 83, 420 (1981). 3. Hwang, S. T., and Kammermeyer, K., Canad. J. Chem. 44, 1982 (1966); Havredaki, V., and Petropoulos, J. H., J. Membrane Sci. 12, 303 (1983); Whiting, D. E. G., "'Isothermal Transport of Sorbable Gases in Graphitized Carbon Membranes: The Effect of Temperature," Thesis. London University, 1981. 4. Nicholson, D., and Petropoulos, J. H., Ber. BunsenGes. Phys. Chem. 79, 796 (1975). 5. Nicholson, D., and Petropoulos, J. H., J. Colloid Interface Sci. 45, 459 (1973). 6. Nicholson, D., and Petropoulos, J. H., J. Phys. D 4, 181 (1971). 7. Nicholson, D., and Petropoulos, J. H., J. Phys. D 6, 1737 (1973). 8. Nicholson, D., and Petropoulos, J. H., J. Phys. D 8, 1430 (1975). 9, Berman, A. S., J. Appl. Phys. 17, 1901 (1966).