J. Phys. Chem. 1994, 98, 2403-2411
2403
Theory of Adsorption of Trace Components Shaoyi Jiang and Keith E. Gubbins' School of Chemical Engineering, Cornel1 University, Ithaca, New York 14853
Perla B. Balbuena Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712 Received: September 8, 1993; In Final Form: November 24, 1993"
W e report a systematic theoretical study of the influence of pore width, intermolecular potential parameters, and state condition on the selective adsorption of trace components (e.g., pollutants or odorants) from simple fluid mixtures. The pores are of slit shape, and the carrier fluid is taken to be methane. The range of fluid potential parameters chosen embraces common constituents of natural gas. Calculations are based on the nonlocal density functional theory of Kierlik and Rosinberg and show the influence of the relevant variables on the selectivity for the trace component a t infinite dilution in the bulk fluid phase. Fluid pressures and pore sizes that lead to a maximum in the selectivity are found, suggesting optimal designs for adsorbents and adsorption processes for trace removal. At these optimum values, extremely large selectivities are possible, particularly a t low temperatures.
1. Introduction
Physical adsorption provides an attractive method for the removal of trace components (e.g., pollutants or odorants), since it is economical and can be highly selective. However, there appear to have been no systematic studies of the influence of the various factors determining selectivity in such cases. Direct experimental studies are difficult because it is necessary to go to extremely low concentrations before the infinite dilution limit is reached; we are aware of only one such investigation.' This is because the concentration of the trace component in the adsorbed phase is often much higher than that in the bulk phase, and it is necessary to reach a sufficiently dilute condition that tracetrace intermolecular interactions can be ignored. Molecular simulation faces the same difficulty of reaching sufficiently low trace concentrations. In this paper we report a theoretical study of the influence of the relevant variables (pore size, intermolecular interactions, pressure, and temperature) on adsorption selectivity for trace components in simple fluid mixtures (spherical neutral molecules) in pores of slit geometry. For convenience we assume a 10-4-3 wall modeled on carbon and take the carrier fluid (gas or liquid) to be methane. All intermolecular interactions are taken to be given by the Lennard-Jones (LJ) potential model. The aim of our work is to provide an understanding of the importance of the different variables in determining the selectivity for infinitely dilute constituents and to show how such modeling techniques can be used to suggest optimal designs for particular separations. A longer range goal of our work is to couple theoretical modeling such as that presented here with an ability to tailor optimal microporous structures for specific applications. As far as we are aware, there have been no previous theoretical or simulation studies of adsorption of trace components and relatively few such investigations of mixture adsorption at finite concentrations. Among the latter are studies of argon-krypton mixtures in cylindrical24 and slit536 pores, methane-ethane mixtures in slit pores,7J methane-X mixtures in slit pores,9 argonmethanelo on a graphite surface, and cyclohexane-octamethylcyclotetrasiloxane (OMCTS) mixtures in slit pores." In some of our previous work6J2 on adsorption of mixtures, we have used the extension to mixtures12 of the density functional
* Author to whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, January 1, 1994.
0022-3654/94/2098-2403%04.50/0
theory (DFT) of Tarazona.*3 However, the form of D I T presented by Kierlik and Rosinberg14 is more convenient for mixtures, and we used it in this work to study the influence of pore size, temperature, bulk fluid density (or pressure), and intermolecular interaCtion parameters on the selectivity at infinite dilution, S;, for methane (1)/X(2) mixtures in slit carbon pores. The fluid potential parameters are so chosen that X covers constituents commonly found in natural gas.
2. Model and Theory Potential Model. The LJ potential is used in this work to represent the interactions between a pair of molecules,
with r the interparticle distance, unthe point at which the potential is zero, and tff the well depth. The cross-parameters for fluidfluid interactions between a pair of molecules of species i and j , tij and uij, are calculated using a geometric mean for ~g and an arithmetic mean for uij (the Lorentz-Berthelot rules). Numerical calculations imply some truncation of the potentials. In our DFT calculations, the numercial integrations were done to sufficiently large distances so that they are numercially equivalent to infinity (the full potential) within negligible error. The LJ parameters's of common constituents of natural gas along with their critical temperatures are listed in Table 1. The solid-fluid interactions are represented by the 10-4-3 solid-fluid potential,16
where A = 2.rrp,e,~cr,~A, z is the distance between a fluid particle and the substrate surface, A is the separation between lattice planes, p, is the solid density, and usf and esf are the crossparameters for solid-fluid interaction, which are again calculated using a geometric mean for e,f and an arithmetic mean for usf. The graphite parameters are taken from Steelel6 and are us = 0.340nm, t,,/k = 28.0 K, and A = 0.335nm. Equation 2 results from integrating the solid-fluid LJ potential over the positions of the solid atoms in a given graphite plane and then summing over the planes. Thus, this solid-fluid potential neglects the atomic 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
2404
TABLE 1: U Parameters’s cii
(nm)
elilk
(K)
~sf.ilcsf,l €sf,i/ €sf,I
Tc (K)
0.3817 148.2 1.Ooo 1.ooo 190.4
(CUul) and
Jiang et al.
Critical Temperatures ( Te)of Common Constituents of Natural Cas.
0.3954 243 1.019 1.28 1 305.4
0.5637 242. 1.252 1.278 369.8
0.375 95.2 0.99 0.802 126.2
0.4486 189. 1.092 1.130 304.1
0.2641 809.1 0.837 2.337 647.3
0.41 12 335.4 1.040 1.505 430.8
0.3623 301.1 0.973 1.426 373.2
0.4163 224.7 1.048 1.232 282.4
and a,fd are the energy and size parameters for the intermolecular potential between a fluid molecule of species i and the carbon wall. Methane is always component 1. 0
structure of the graphite planes. This neglect is known to have little influence on the adsorption of methane at temperatures above 80 K, because the methane molecule is relatively large compared to the carbon-carbon lattice spacing. Thus we do not expect the neglect of wall structure to be an important factor in the calculations presented here, since the trace molecules are similar in size to, or larger than, methane molecules. Theory. We use the nonlocal DFT due to Kierlik and Rosinberg,I4 which follows the general formalism of the density functional theories.” It is based on the usual separation of fluidfluid potential &,+) into repulsive and attractive contributions. According to Weeks, Chandler, and Andersen (WCA),I8thefluidfluid potential can be split at the minimum r,in = 21/6uninto a repulsive and an attractive potential,
(3) The repulsive part is modeled by an equivalent hard-sphere potential with an appropriate hard-sphere (HS) diameter (d)in this theory. As the hard sphere serves as the reference system, the Helmholtz free energy F can be expressed as
F = p + Wttr
(4) where Fs is the free energy of a hard-sphere reference fluid and A P r is the attractive free energy. The attractive contribution A P r can be estimated from the mean-field approximation
Wttr = ’JJdr dr’ pi(r) pi(r’) 2
$y(lr- r‘l)
(5)
where pi(r) is the number density of component i at position rand is the attractive part of the fluid-fluid potential. In eq 5 , the correlations between molecules due to attractive forces are neglected, Le., g(r,f) is set equal to unity. The repulsive contribution is modeled by the free energy functional of a reference hard-sphere fluid,
$g,
p
= F!!=l+
z-
(6)
The ideal gas part of the free energy is
the scaled particle theory (SPT),21
+ = -no In( 1 - n3) + n1n2/(1 - n3) + (1/24?r)n;/( 1 - n3)2 (10) where no, nl, n2, and n3 are the reduced variables of the SPT: n, = piR?, with Rjo) = 1 , RI’) = R,, RIZ)= 4?rR?, and R13)= 4/3~R?. The four weight functions oy)are scalars that are simply related to the Heaviside step function and its derivatives and, hence, are independent of density:
v 2 ) ( r )= 6(Ri - r )
q o ) ( r )= -(1/8?r)6’(R,-r)
+ (1/27rr)6”(Ri-r)
The introduction of the four density-independent weight functions is the main difference from the other nonlocal approximations proposed in the current literature. This is a great simplification. On the other hand, by construction, this functional generates the Percus-Yevick pair direct correlation function in the uniform limit and also predicts good values for higher order correlation functions. Therefore, this version of nonlocal DFT has several advantages over the others, especially for application to mixtures. The grand potential functional of the system is given by
where H~ is the chemical potential of component i and 4?‘(r) is the external fluid-wall potential for a molecule of species i at r. Therefore, the grand potential functional of a multicomponent fluid is described with a pertubational approach by substituting eqs 4 and 5 into eq 12
(7) The ideal gas part of the Helmholtz energy density is local, Le., depends only on the density p(r) at the point r of interest. The excess part of the free energy density is nonlocal and depends on the density at other points near the position r of interest. For the excess contribution to the free energy functional of the reference hard-sphere fluid, Kierlik and Rosinberg14 follow Percus19 and RosenfeldZO in writing
where kT+ is the Helmholtz free energy density of the uniform hard-sphere fluid for some ‘smoothed” or weighted density pa(r) and pa(r) = Jdr‘ pi(r’) o?)(r - r’)
(9)
where or‘“) (a = 1-4) are weighting functions.
+ is taken from
The equilibrium density distribution is thus a solution to the following Euler-Lagrange equation for the unconstrained minimization of Q:
The equation of state of the homogeneous fluid can be obtained from the generic free energy functional. The pressure of the homogeneous fluid is composed of a hard-sphere pressure and a mean-field attractive contribution:
The chemical potential in the homogeneous fluid given by the
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2405
Theory of Adsorption mean-field equation of state is CL
= kT In P
+ &(P) + P h J P ) + PJdr
4attr(lrl) (16)
Selectivity at Infinite Dilution. For a binary mixture, the selectivity at infinite dilution of component 2 (X)relative to component 1 (methane) is defined as
(17) where xfis mole fraction and p i density. The values labeled "pore" are averaged over the pore volume, and values labeled "bulk" are values for the bulk gas in equilibrium with gas adsorbed by the porous material. We note that ST is the inverse of S;; Le., S; = (S;)-',S; is strongly dependent on pore width (H), temperature (T),bulk fluid number density ( p b = Nb/V), or pressure (P)and on intermolecular potential parameter ratios (tsn/tsfl and usn/asfl). In this work, we have varied all the parameters to study their effects on S;. We note that in the most general case other potential parameter ratios (cfo/tss, ufa/uSs, tfn/tsn, ufn/u,n) would need to be considered. In order to keep the study within reasonable bounds, we have kept the solid parameters fixed (carbon) and assumed the Lorentz-Berthelot combining rules, thus reducing the number of independent parameter ratios to two. For a structureless slit pore, the average density in the pore for component i at the limit of zero pressure can be expressed as P,j
= PbifexP[-r#J?t(z)/kTI
dz
(P-0)
(18)
Substituting eq 18 into eq 17, one obtains'
JoHexP [-4$(z) lim pb4
/ TI dz
S; =
(19)
JoHexp[-@(z)/kr]
dz
The zero-pressure selectivity is a function of temperature and pore width only. It does not depend on the composition. Equation 19 is a good approximation to estimate the selectivity at low density. We note that the external potential ~J?'(Z) experienced by a fluid molecule of species i a t z for a given slit width H is calculated as the superposition of 4sf,ifor the two walls ~ Y ' ( Z > 4sf,j(z) + +sf,i(H-z)
(20)
All variables are reduced using methane parameters in performing our calculations and showing results. Component 1 is always methane and component 2 is always infinitely dilute. Thus, H* = H / u ~ TI . = T/tm, pb* = p b ~ f i land ~ , z* = Z / a f f i , where all1 = 0.3817 nm and tffl/k = 148.2 K.
3. Results and Discussion
To study the effect of all the variables on the limiting selectivity, we first examine the effect of temperature, bulkdensity, and pore width for two particular systems, methane (l)/propane (2) and methane (l)/nitrogen (2). We follow this by studying the effect of the parameter ratio tsn/tlfl a t fixed u8n/atfl,temperature, pore width, and bulk density. Finally, we investigate the effect of the parameter ratio a,n/asfl at fixed tsn/tsfl, temperature, pore width, and bulk density. Before showing the results, we note from eqs 2 and 19 that we expect S; to be very sensitive to both asfz/asfland (4sn,min 4sfl,min)/kT, and that the dependence on the difference in the well depths for the solid-fluid interaction potential for the two components will have a particularly strong influence, a t least in the low-density limit. Equations 2 and 19 suggest that S; will
H' = 5.0
- 30
0
I
I
I
1
1
2
3
4
Z'
5
(b)
10
0
$.;-lo
-20
-30
'0
I
I
1
2
3
Z' F m e 1. Solid-fluid interaction potentials for (a) HS = 5 and (b) HS = 3, for methane, nitrogen, and propane in slit carbon pores.
vary roughly exponentially with (&n,min - &f,,min)/kT, or equivalently with (tun- t,fl)/kT, so that rather small differences in the well depths may have a large effect. Although eq 19 does not hold at higher densities, where fluid-fluid interactions become significant for the pore fluid, we find that S; still varies strongly (very roughly exponentially) with the difference in well depths. Thus, selectivity can be greatly enhanced by either increasing the difference in well depths (e.g., by a suitable choice of material, modification of the solid surfaces, or changing the pore size or shape) or by lowering the temperature. In Figure 1 we show typical plots of the solid-fluid potentials for methane, propane, and nitrogen. We note that for the CHI (l)/C3Hs (2) mixtures the difference (&n,min - &fl,dn)/kT is large and positive, whereas for CH4 ( l ) / N z (2) it is small and negative. Thus, we can expect S; to be very large for methane/propane mixtures, with propane strongly favored by the adsorbent, whereas for methane/nitrogen mixtures S; will be smaller than unity but relatively close to it, so that the adsorbent favors methane over nitrogen. Since the differences, (4sn,min - &fl,min)/kT,become larger as the pore size decreases (at least, down to some minimum value of H), we can expect the preferential adsorption of the favored component to increase as H decreases. Thus, (&n,dn- &fl,dn)/kTdecreases from -12 at H+ = 5 to about -16 a t 9= 3 for methane/propane mixtures. These qualitative features help to understand the selectivity results presented below. We also note from Figure 1 that even the molecules in the center of the pore will experience a strong interaction with the walls (solid-fluid potentials substantially below zero), so we can expect the fluid throughout the pore to be highly inhomogeneous. Only for pore widths above about 5 or 6 nm will the fluid in the center of the pore approach uniformity for the molecules considered here. Selectivity for U Model of Metbane/Propane Mixtures. We first discuss the behavior of the selectivity for the model of methane/propane mixtures, where the ratios asn/asfland e,n/ tsfl are greater than unity. Figure 2 shows the effect of
2406 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
L
4000
B
2000 1 500 -
s',
v
1
1000500
2 40
220
200
T
260
280
300
300
Figure2. Effect of temperatureon selectivity at infinite dilution of propane for CH4 (l)/C& (2) mixtures at H* = 5 and pb* = 10-4,
200
S,' 100
1
s,"1 1
0
1
lo-s lo-'
1
lo-'
lo-'
loo
Pb.
1 1
7
4
10
H' (b) I
1
I
I
I
I
4
7
H'
Figure 4. Effect of bulk gas density pb on selectivity at infinite dilution of propane for CH4 (l)/C4H8 (2) mixtures at H* = 5 for (a) T = 230 K and (b) T = 300 K.
in solid-fluid interactions is increasing rapidly in magnitude for this H* range; the two solid-fluid potential wells merge in this region. For higher bulk fluid densities, the selectivity oscillates with the pore width for H*below 6 due to packing effects.' Thus, as H* is decreased whole layers of molecules are squeezed out of the pore at certain specific pore widths, leading to a drop in pore fluid density and selectivity. The optimal pore sizes corresponding to the first and second peaks in Figure 3, gpt and qPt, can be estimated by
10
Fipre3. Effect of pore width on selectivityat infinite dilution of propane for CH4 (l)/C3H* (2) mixtures with pb* = 10-4 (- -) and pb* = 10-1 (-) at (a) T = 230 K and (b) T = 300 K.
-
temperature on the limiting selectivity for CH4 (l)/C3Hs (2). The limiting selectivity for the most strongly adsorbed component decreases with increasing temperature, as expected. Over the 100 K interval from 200 to 300 K, S; decreases by a factor of about 19-fold. The effect of pore width is shown in Figure 3. For lower densities, the limiting selectivity increases smoothly with decreasing pore width, reaches a maximum, and afterward decreases rapidly when pore width is further decreased. For H* values below the maximum, propane molecules are excluded from the pores due to the molecular sieve effect. The maximum in the curve corresponds to a pore width of about H* = 2.37 (H= 0.904 nm), for which one propane molecule can just fit between the two carbon walls. The location of the maximum indicates the pore width that optimizes the selectivity. The sharp rise in S; in the region of H* from about 4 to 2.37 arises because the difference
where a%is the diameter of a carbon atom, aff,strongcr the diameter of the more strongly adsorbed molecule, and bff,w&r the diameter of the less strongly adsorbed molecule. From eqs 21 and 22, we estimate that the first and second peaks in Figure 3 occur at 0.904 and 1.285 nm, which are in good agreement with 0.904 and 1.290 nm from Figure 3. The effect of pore width at 300 K is similar to that at 230 K except that the maximum selectivity decreases by about 2 orders of magnitude for the lower density case and by about 1 order of magnitude for the higher density case. The effect on S; of increasing the bulk gas density (or pressure) is shown in Figure 4; it is interesting to see that there are maxima at pb* = 4 X 10-3 and pb* = 1.2 X 10-2 for T = 230 and 300 K, respectively. The density profiles corresponding to the four points marked in Figure 4a are given in Figure Sa for methane and Figure 5b for propane. In Figure 5b the results are plotted as limx2+(pp2*/x2) instead of as pp!* versus z*. This is because as propane approaches infinite dilution the absolute quantity pp2*varies with x2. However, the relativequantity ppz*/ x2 is convergent as long as x2 is small enough. The adsorption
Theory of Adsorption
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2401
n N W
W
h
Q c
0
0.0
0.5
1.5
1.0
2.0
2.5
Z'
0 0
I
I
8 *
8 n
x"
-0
Ne
'E
>Odo
-8
.--E
W
Qo W
V
Q
.E -a
8 c
0
0
0.0
0.5
1.0
1.5
2.0
2.5
Z'
Figure 5. Density profiles corresponding to points A, B, C, and D in Figure 5 for (a) methane and (b) propane.
isotherms of methane and propane for these same conditions are shown in Figure 6. Here the adsorption is plotted as the mean density of i in the pore, ppi*, given by
At lower densities, the percentage increase of the adsorption of methane and propane in the monolayer is almost the same as the bulk density increase, so the selectivity is almost constant. With increasing bulk densities (A, B), the percentage increase of the adsorption of propane in the monolayer is greater than that of methane, so the selectivity curve rises to peak B. Peak B corresponds to the inflection point in the monolayer. At this point, the slope of the adsorption isotherm of methane is largest, as shown in Figure 6a. As pb* is further increased (C, D), the percentage increase in methane adsorption outweighs that of propane, so the selectivity for propane falls after the peak. In this region molecules are being added to the second adsorbed layer, where the difference in solid-fluid potential for the two components is much weaker. As can be seen from Figure 6b, the adsorption of propane exhibits a maximum a t point C and then decreases. This indicates that propane molecules are squeezed out of the pores and replaced by methane molecules at relatively high bulk densities. This can also be seen from Figure 5b, where the peak of the density profile for the monolayer decreases after point C. Thus, the limiting selectivity curve sharply decreases after point C in Figure 4a. The selectivity at T = 300 K shown in Figure 4b is about 1 order of magnitude lower and occurs a t higher bulk fluid density than at 230 K. The effect on S l of increasing the bulk gas density (or pressure) is shown in Figure 7 for three pore widths, two of which (H*= 2.37 and 3.40) correspond to first and second maxima in Figure 3, respectively. In the same way as in Figure 4a, the maxima in Figure 4b and
Figure 6. Adsorption isotherms for CH, (l)/CSHa (2) mixtures when component 2 is at infinite dilution in the bulk gas, T = 230 K and H* = 5: (a) methane, (b) propane. The points A, B, C, and D correspond to those in Figure 5. Here ppr is the mean density of component i in the pore, averaged over z . lo8
I
I
. . 1 . . 1 ,
r
1 . 1 1 1 1 ,
I
I 1 . 1 1 1 . ,
.
. I . . . - (
r
. . I
io4
s: 10'
IO-^
1 0-3 10" 10-1 Pb( Figure 7. Same as Figure 4, but at T = 230 K and Zf' = 2.37 (-), 3.40 (- -), and 5.0 (- -).
-
10-~
lo-'
-
Figure 7 correspond to the inflection points in the monolayer. At these points the slope of the adsorption isotherm of methane is largest. The effect of bulk density on the limiting selectivity a t a low temperature, 100 K, is shown in Figure 8. The vertical jumps in S; are due to layering transitions, as seen from the adsorption isotherms of Figure 9. The adsorption of the dilute component (propane) grows rapidly in the contact layer a t low densities and then decreases sharply as the bulk fluid density is further increased, as shown in Figure 9b. The decrease in adsorbed propane a t higher densities is similar to that found for smaller pore widths and higher temperatures shown in Figure 6b. The density profiles
Jiang et al.
2408 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
loo
B 2
0 10-7
10-3
;
4
2
8
E
10
2' 10"
io-'
' ' s . . . . b l
io-6
''.l.l*.l
io-'
''...llll
'
10-~
'.."..l
lo-'
'
'*..J
10-2
(b)
10- 1
Pb.
Figure 8. Same as Figure 4, but at T = 100 K and H* = 10.
r E
4
1 0 - 3 r
8
10
2' Figure 10. Density profiles for CH4 (1)/C3Hs (2) mixtures when component 2 is at infinite dilution in the bulk gas, T = 100 K and H+ = 1 0 (a) methane, (b) propane.
Pb.
2
1
io+
io-'
10-~
io-'
io-*
Pb. Figure 9. Adsorption isotherms for CH4 (l)/CjHs (2)mixtures when component 2 is at infinite dilution in the bulk gas, T = 100 K and'H = 10: (a) methane, (b) propane. The numbers 1,2,3in part a show the
number of adsorbed layers on each wall. of methane and propane corresponding to these adsorption isothermsareshowninFigure 10. AporeofH* = lO(3.817nm) can accommodate about six layers of propane molecules for pure propane. However, at infinite dilution, propane molecules do not necessarily lie on a line along the direction perpendicular to the walls. Therefore, more than six layers are observed at high bulk densities. Selectivity for U Model of Methane/Nitrogen Mixtures. The effect of temperature on the limiting selectivity for CH4 (l)/Nz (2) is shown in Figure 11. Since CH4 is the preferentially adsorbed component, S; is less than unity and increases with decreasing temperature. The trend is opposite that of Figure 2 for CH4 (l)/C3Hs (2). The limiting selectivity increases by about 80% over the 100 K interval shown. The effect of pore width on the
0.2 200
I
I
I
I
220
240
260
280
300
T
Figure 11. Same as Figure 2, but for CHI (l)/Nz (2).
limiting selectivity for CH4 (l)/Nz (2) is shown in Figure 12. For lower densities, the limiting selectivity decreases sharply and smoothly a t smaller pore widths, reaches a minimum, and then increases when the pore width is further decreased. The minimum in the curves in figure 12 occurs at H* = 1.89 (H= 0.722 nm), the pore width that can accommodate one layer of adsorbed methane molecules. For pore widths below H1= 1.89, methane molecules are excluded from the pores due to the molecular sieve effect. For higher densities, the selectivity oscillatesslightly with the pore width as a result of packing effects. The smaller oscillations for CH,/N* are due to the fact that the sizes of methane and nitrogen molecules do not differ much. From eqs 21 and 22, we estimate that the first and second minima in figure 12 occur at 0.722 and 1.097 nm, which are in good agreement with 0.722 and 1.095 nm from Figure 12. The effect of bulk density (or pressure) on the limiting selectivity is shown in Figure 13. It is interesting to see that there are two minima in the higher density region in Figure 13. This behavior is due to competing
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2409
Theory of Adsorption
10'1
I
I
I
0.4
0.3 S',
0.2
0.1
3
1
5
7
1o-'I
9
0.5
H'
I
' """'1
'
*.'"''I
'
'
"""'1
I
I
I
I
1 .o
1.5
2.0
2.5
&dZ/&dl
Figure 12. Same as Figure 3a, but for CHI (l)/Nz (2).
0.4
'
* ' I -
Figure 14. Effect of the parameter ratio c&/c,fl on selectivity at infinite dilution of component 2 with pb* = lo" and u&/u,fi = 0.95 at T = 230 K (-) and 300 K (- - -): 0,H+ = 3 (1.15 nm); A,H+ = 5 (1.91 nm). A and B are crossing points with H* = 3.0 and H* = 5.0 for different
temperatures,respectively, while A'and B'arecrossingpoints for different pore widths at T = 300 and 230 K, respectively.
S; 0.3
& I
0.2 10-0
10-5
.
,
.
..,..I
io-'
. . ......
I
lo-'
, ,
. ...,.
I
10-1
, , ,. Y
lo-'
Pb. Figure 13. Same as Figure 4a, but for CHI (l)/Nz (2).
adsorption between methane and nitrogen in the first and second layers. Thus, the percentage increase of the adsorption of nitrogen in the monolayer is smaller than that of methane at low densities (e.g., A and below), so the limiting selectivity of nitrogen relative to methane is smaller. By point B the monolayer is nearing completion for N2, but the density of methane in this layer continues to increase quite rapidly up to point C and somewhat beyond, leading to a minimum in S; at B. The second minimum in S; at D apparently arises from a similar competition between N2 and CHI adsorption in the second adsorbed layer. Effect of e & / s a Ratio. The effects of the ratio e,n/e,fl on the limiting selectivity at T = 230 and 300 K, Hs = 3 (1.15 nm) and 5(1.91 nm),andpb* = 1WareshowninFigures 14-16forthree usn/usflratios: 0.95,l .OO,and 1.25, respectively. In these figures, the bulkdensity and the ratio u,n/u,fl areconstant. These figures illustrate the effect of e,n/Gfl on the limiting selectivityfor different temperatures and pore widths. The slope of the plots S; vs esn/ elfl increases as both the temperature and pore width decrease, as expected. For a given pore width there is a particular value of c,n/tsflfor which two isotherms cross. These points are marked A for H* = 3 and B for H* = 5 in the figures. For e,n/e,fl values above that at the crossing point S; increases on lowering the temperature, since 2 is preferentiallyadsorbed, whereas the reverse is the case for e s ~ / e avalues below that at the crossing point, since for these conditions 1 is preferentially adsorbed. for u , ~ / usflvalues less than 1 (Figure 14) this crossing occurs at esn/esfl values greater than 1, whereas for u,n/u,f1 values greater than 1 (figure 16) the crossing point is for eSn/e,flvalues smaller than 1. A second kind of crossing point (A', B') occurs for a particular temperature on changing the pore width. For esn/esflvalues above that for these crossing points, reducing the pore size will cause
10''
'
0.5
I
I
1 .o
1.5
I
I
2.0
2.5
&dd&dI
Figure 15. Same as Figure 14, but with u,n/u,fl = 1.0. 1' 0
10 ' 10' 10' 1 oo
I"
0.0
0.5
1 .o
1.5
2.0
2.5
&dZ/&dI
Figure 16. Same as Figure 14, but with
usn/trdI =
1.25.
the selectivity for component 2 to rise, whereas for e,n/esfl values below the crossing point the selectivity for 2 will decrease on reducing the pore width. For usn/usflratios less than 1 (Figure 14) the points A' and B' occur at esn/edl values above those for A and B, whereas the reverse is the case when u,n/udl is greater than 1 (Figure 16). In general, increasing the ratio u,n/u,fl, keeping the other variable fixed, causes both S; and the slope S; vs eaR/esfl to increase. An understanding of the location of
Jiang et al.
2410 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 lo’
I
’
10-2 0.7
I
I
I
I
I
I
0.8
0.9
1.0
1.1
1.2
1.3
1
Ual2/U,fI
Figure 17. Effect of the parameter ratio u,n/u,fl on selectivity at infinite dilution of component 2 with pb* = 10-4 and c,fl/c,fj = 0.707 at T = 230 K (-) and 300 K (- -): 0, H* = 3; A, H* = 5 . A and B are crossing points with H* = 3.0 and H* = 5.0 for different temperatures,respectively, while A’ and B’ are crossing points for different pore widths at T = 300 and 230 K, respectively.
-
10’
P
102
s‘,
10’
Figure 20. Variation in S 2 as X2,b approaches zero at H* = 5.0, Pb* = lo“,150 K (-), 230 K (- - -), and 300 K (- .-) for (a) CH4 (1)/C3H8 (2) and (b) CH4 (l)/Nz (2).
1 oo
lo-‘
Figure 18. Same as Figure 17 with c,n/c,fl = 1.0.
10‘
10’
10’ 1 OD
1 0-‘
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Gi.2/U.,l
Figure 19. Same as Figure 17 with
c,n/c,fl =
1.5.
these crossing points, and how they vary with changes in the system parameters and state conditions, will be important in applications to trace removal. Effect of a,fi/odj Ratio. The effects of the ratio usn/uPflon the limiting selectivity for T = 230 and 300 K, H* = 3 and 5, and pb* = 1 v are shown in Figures 17-19 for three esn/e,fl ratios: 0.707,1.00,and 1.5, respectively. The behavior is similar to that of Figures 14-16. The crossings A’ and B’ occur a t lower
u,n/u,fl ratios for larger e,n/esfl ratios. The crossing points A and B (intersection of two isotherms for same pore width) occur a t lower u,n/usfl values for higher esn/esflvalues. Approach to Infinite Dilution Limit. For finite concentrations S2varies with the bulk phase mole fraction X2,b of the trace component, and it is of importance to know how small X2.b must be before S 2 reaches its limiting value, S.; For the thermodynamic properties of bulk fluid mixtures, the Henry law limit is usually reached when X2,b falls below a few percent (unless ions, associated species, or chain molecules are present). In adsorbed phases, however, the concentration of component 2 may be many orders of magnitude higher than that in the bulk phase (see, e.g., Figure 2-47, and 8), so the infinite dilution limit S; is reached only at much smaller X2.b values. We show some typical results for the variation in S2 as X2,b approaches zero in Figures 20-22. The mole fraction below which S, reaches the infinite dilution limit becomes lower for higher values of S;. Thus, as the temperature and pore size are decreased or when one component is much more strongly adsorbed than the other, as is the case for propane in methane/propanemixtures, greater dilution is needed to reach the infinitely dilute value (cf. the case for nitrogen in methane/nitrogen mixtures). Thus, in Figures 20 and 21 we see that for low temperatures or smaller pores it is necessary to go to X2,b = 10-8or 10-9 for propane/methane mixtures, while X2,b = 10-2 is sufficient to reach the limiting value for nitrogen/ methane mixtures. For higher values of S l such as in Figure 8, extreme dilution (e.g., X2.b = or lower) is needed. One example showing the extreme dilution needed is given in Figure 22. These results suggest that it will be difficult to determine these limiting selectivities by experiment, at least for mixtures where the trace component is strongly adsorbed, since accurate concentration measurements down to parts per trillon or lower will be needed. Molecular simulation will not be possible for such cases with current computers.
Theory of Adsorption
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2411 temperature, there is an optimum pore width that gives the maximum separation of the two components (a maximum in 27; when 2 is the strongly adsorbed component, or a minimum when 1 is preferentially adsorbed). This optimum pore width is that which just accommodates one layer of the strongly adsorbed molecules with larger diameter. We also find that for a given system, temperature and pore width, there is an optimum bulk gas density (pressure) at which S; is maximized. This optimum bulk gas density corresponds to the inflection point in the monolayer. At this point the slope of the adsorption isotherm of methane is largest. Thus, the theory can be used to predict (i) optimal pore sizes (and shapes) for removal of specific trace components and (ii) the optimum pressure at which to operate. Our results show that it should be possible to design microporous adsorbents to effectively separate trace components in a mixture when there are significant differences in fluid-wall interactions or size between the trace and other components of the mixture. For such cases, adsorption offers a highly selective method for trace removal, provided the optimal pore size and operating conditions are used. The models used here are highly simplified (spherical LJ interactions, slit pores, smooth walls) but serve to illustrate the role of the independent variables and qualitative trends, as well as the most promising pore designs and operating conditions for more precise and detailed investigations. Further work is needed to study heats of adsorption and diffusion rates, as well as the role of pore shape, and we are beginning to carry out such studies. More sophisticated models for the intermolecular potentials and pore structures will be needed to obtain reliable results for specific mixtures.
s2
0.0 10-l~
io-'
io-*
IO-^
IO-^
loo
x2
Figure 21. Variation in Sz as X2,b approaches zero at T
-
l W , H * = 2.5 (-) and 5.0 (- -)for (a) CHI (l)/C& (1)/Nz (2). 1""'V
.
1.1.110,
1
""",
I"""?
230 K, Pb* = (2) and (b) CHI
-111111.)
I
I .
p ; = lo4
s2
1lo' o*
1
10 ' lo-14
p; = 3x10-'
lo-lo
lo-lJ
,o-s
x 2
Figure 22. Variation in S2 as X2.b approaches zero for CH4 (l)/CpHs (2) at H* 10.0, T 100 K,pb* = 3 X 1 W (-), and pb* Iod (- - -).
4. Conclusions
The influence of pore size, temperature, bulk density (or pressure), and intermolecular interaction parameters on limiting (infinite dilution) selectivity S; for a trace component X in model methane (1)/X (2) mixtures in slit carbon pores has been studied using nonlocal density functional theory. Our results show that when X is the preferentially adsorbed component, the selectivity S; decreases with increasing temperature and pore width (at low density); at higher densities the selectivity oscillates as the pore width is reduced due to packing effects. The effect of temperature and pore width has been reflected in the crossing of lines in the plots of S; versus the parameter ratios usfi/us~land t s ~ / t s f land more directly in the plots of S; versus T and H.Of particular importance is that for a given system, bulk density and
Acknowledgment. We thank E. Kierlik and M.L. Rosinberg for providing the computer program for the nonlocal density functional theory. This work was supported by a grant from the National Science Foundation (No. CTS-9122460) and a contract from the Gas Research Institute (No. 5086-260-1254). References and Notes (1) Joseph, J. C.; Myers, A. L.; Golden, T. C.; Sircar, S . J . Chem. SOC., Faraday Trans. 1993, 89, 3491. (2) Tan, 2.;van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 62, 1213. (3) Heffelfinger, G. S.;Tan, 2.;Marini Marconi, U.; van Swol, F.; Gubbins, K. E. Mol. Sim. 1989, 2, 393. (4) Marini Bettolo Marconi, U.; van Swol, F. Mol. Phys. 1991,72,1081. (5) Sokolowski, S.; Fischer, J. Mol. Phys. 1990, 71, 393. Sokolowski, S.;Fischer, J. Mol. Phys. 1990, 70, 1097. (6) Tan, 2.;Gubbins, K. E.; van Swol, F.; Marini Marconi, U. Proc. Third Internal. Conf. on Fund. Ads.; Mersmam, A. B., Scholl, S.E., Eds.; Engineering Foundation: New York, 1991. (7) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (8) Cracknell, R. F.; Nicholson, D.; Quirke, N. In Characterization of Porous Solids IIfi IUPAC, Ed.; Elsevier: Amsterdam, 1993. (9) Balbuena, P. B.;Gubbins, K. E. AIChE Annual Meeting, Maimi Beach, FL, 1992. (10) Finn, J. E.; Monson, P. A. Mol. Phys. 1991, 72, 661. Kierlik, E.; Rosinberg, M. L.; Finn, J. E.; Monson, P. A. Mol. Phys. 1992, 75, 1435. (11) Somers, S. A.; McCormick, A. V.; Davis, H. T., submitted for publication in Europhys. Lett. (12) Tan, 2.;Marini Marconi, U.; van Swol, F.; Gubbins, K.E. J. Chem. Phys. 1989,90, 3704. (1 3) Tarazona, P. Phys. Rev. A 1985,31,2672; 1985,32,3 148. Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 573. (14) Kierlik, E.; Rosinberg, M. L. Phys. Rev. 1990, A42.3382. Kierlik, E.; Rosinberg, M. L. Phys. Rev. 1991, A44, 5025. (15) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B.Molecular Theory of Gases and Liquids, 2nd ed.; J. Wiley and Sons: New York, 1964. (16) Steele, W. A.Surf.Sci. 1973,36,317. Steele, W. A . The Interaction of Gases with Solid Surfaces; Pergman: Oxford, 1974. (17) Evans, R. In Inhomogeneous Fluids; Henderson, D., Ed.; Dekker: New York, 1992. (18) Weeks, J. D.; Chandler, D.; Andersen, H. L. J. Chem. Phys. 1971, 54, 5237. (19) Percus, J. J . Stat. Phys. 1988, 52, 1157. (20) Rosenfeld, Y. Phys. Rev. Lett. 1989,63,980. Rosenfeld, Y.; Levesque, D.; Weis, J. J. J . Chem. Phys. 1990,92,6918. Rosenfeld, Y. Phys. Rev. 1990, A42, 5978. Rosenfeld, Y. J. Chem. Phys. 1990, 93, 4305. (21) Reiss, H.; Frisch, H.; Lebowitz, J. L. J . Chem. Phys. 1959.31.369, Helfand, E.; Frisch, H. L.; Lebowitz, J. L. J . Chem. Phys. 1961, 34, 1037.
