Langmuir 1993,9, 1801-1814
1801
Theoretical Interpretation of Adsorption Behavior of Simple Fluids in Slit Pores Perla B. Balbuenat and Keith E. Gubbins' School of Chemical Engineering, Cornell University, Ithaca, New York 14853 Received December 30,1992. In Final Form: March 29, 1993 Nonlocal density functional theory is used to interpret and classify the adsorption behavior of simple fluids in model materials having slit pores. A systematicstudy is reported for a wide range of the variables involved temperature, pressure, pore width H, and the intermolecularparameter ratios td/tfi and u d / u ~ . Adsorption isotherms, isosteric heats of adsorption, and phase diagrams are calculated. The isotherms are related to those of the 1985 IUPAC classification;the range of variables corresponding to each of the six isotherm types is determined, and the underlying factors leading to each of the types are elucidated. In addition to the six types of the 1985classification,a seventhtype is identified,corresponding to capillary evaporation. A similarstudy and classificationis reported for the heata of adsorptionand phase transitions (capillary condensation and layering transitions) in pores. Since the materials studied do not exhibit either heterogeneity or networking, the conditions leading to phase transitions are clearly seen. Where possible,qualitative comparisons with experimentalobservations are made. The theoretical classification reported here should provide a useful framework against which to interpret experimental data. 1. Introduction
The first classification of physical adsorption isotherms for pure fluids was presented by Brunauer et a1.12 They proposed five isotherm types, based on known experimental behavior. In 1985, the IUPAC Commission on Colloid and Surface Chemistry3 proposed a modification of this classification; in addition to the original five types of Brunauer et al. they added a sixth type, the stepped isotherm. These six types are shown schematically in Figure 1. Type I (the Langmuir isotherm) is typical of many microporous adsorbents (pore widths below 2 nm); at relative pressures approaching unity the curve may reach a limiting value or rise if large pores are present. Types I1 and I11 are typical of nonporous materials with strong (type 11) or weak (type 111) fluid-wall attractive forces. Types IV and V occur for strong and weak fluid-wall forces, respectively,when the material is mesoporous (porewidths from 2 to 50 nm) and capillary condensation occurs; these types exhibit hysteresis loops. Type VI occurs for some materials with relatively strong fluid-wall forces, usually when the temperature is near the melting point for the adsorbed gas. The interpretation of experimental adsorptionisotherms is complicated in practice by uncertainties concerning the morphology of the adsorbing material. Materials studied are frequently heterogeneous,having not only an unknown range of pore sizes but a range of pore shapes, active adsorption sites, and blocked and networked pores. For suchmaterialsthe measured isotherm is a weighted average over the adsorption (and any phase transitions that occur) due to these various effects. The interpretation has been further clouded by the use of methods based on the Kelvin equation,which is known to give large errors for micropores and the smaller mesopores. The latter difficulty can be largely overcome by the use of modern statistical mechanical theories, particularly density functional theory, + Present address: Department of Chemical Engineering, University of Texas, Austin, TX 78712-1062. On leave from INTEC, Univereidad Nacional del Litoral, Santa Fe,Argentina. (1) Brunauer,S.;Deming,L.S.;Deming,W. E.;Teller,E. J.Am. Chem.
SOC.1940,62, 1723. (2)Brunauer, S. The Adsorption of Guses und Vupours; Oxford University Press: London, 1945; pp 149-151. (3) Sing, K. S. W.; Everett,D. H.; Haul, R. A. W.; Moecou, L.; Pierotti, R. A.; RouquBrol,J.; Siemineieweka,T. Acre Appl. Chem. 1988,57,603.
1
In
I
MICROPORES
Y
SUBSTRATE
WEAK
LAYER ING
Relative pressure, P/ Po
Figure 1. The six types of adsorption isotherm according to the 1986 IUPAC classification.
to analyze isotherm data: but the difficulties of accounting for heterogeneity of various kinds, networking etc., is still not resolved. If one neglectspore blockingand networking and assumes the heterogeneity is due only to a distribution of pore sizes and chemically heterogeneous sites on the surface, one can approach the problem by writing the adsorption rs in the form
where H is the pore width, ed is the attractive energy between an adsorbed molecule and a chemically heterogeneous site on the surface, I'(H,ed) is the adsorption isotherm for a material in which all pores are of width H with energy tsf, as calculated by some accurate theory or molecular simulation, and P(H,ed) is the probability distribution for H and for the real material, as (4) Laetoekie, C.; Gubbine, K. E.; Quirke, N.Langmuir, in prese.
0743-7463/93/2409-1001$04.00/00 1993 American Chemical Society
Balbuena and Cubbins
1802 Langmuir, Vol. 9,No. 7, 1993
determined experimentally. The difficulty with even this simplified approach is that we do not yet have reliable methods for determining the probability distribution function P(H,ed)or even the simpler pore size distribution P(H)except for some rather easily characterized materials. In view of this rather unsatisfactorysituation, we believe it is useful to analyze the behavior, I'(H,e,f), for single pores of simple geometry. Once a sound understanding of this simpler case is achieved, the additional effects due to chemical heterogeneity, networking, etc. can be evaluated. Accordingly, in this paper we use density functional theory to determine the effect of the molecular and state variables on adsorption isotherms, heats of adsorption, and phase transitions for simple fluids adsorbed in pores of slit geometry. The variables involved are temperature and pressure, pore width (H),and the ratios of the intermolecular potential parameters, e d e e and o,f/uft, where e and u are parameters in the Lennard-Jones potential and sf and ff subscripts indicate values for the solid-fluid and fluid-fluid interactions,respectively. Since heterogeneity is absent in our model, the relation between adsorption type, phase transitions, etc. and the underlying molecular and pore properties can be clearly seen. We first determine the range of parameter space (kTleff,H/uft, e,f/eft, and uduft) corresponding to each of the adsorption types of the W A C classification; in doing this we also introduce a new type, VII, which corresponds to capillary evaporation (drying). This is followed by a similar analysis of heats of adsorption and phase transitions (layering transitions and capillary condensation) in terms of these sameparameters. Where possible we relate these findings in a qualitative way to experimental results. 2. Theory
2.1. Model. The system consists of a single slit pore having two semi-infiiite parallel walls separated by a distance H. The pore is open and immersed in a very large reservoir containinga single-componentfluid at fixed chemical potential p and temperature T, the totalvolume of the system being V. The fluid inside the pores feels the presence of the solid surfaces as an external potential, and on reaching equilibrium its chemical potential equals the bulk chemical potential. For the fluid-fluid intermolecular pair potential energy we use the cut and shifted Lennardpotential, given by Jones (U)
~ , ~ ( ifr r, < ) r, if r > rc (2) where r, = 2 . 5 is ~ the cutoff radius and uuis the full LJ potential, u&) = uU&) =O
-~
uU,ff= 4e[(uff/r)12- ( u , / ~ ) ~ I (3) The advantage of using the cut and shifted potential is that comparisons ofthe theoretical resulta with molecular simulations can readily be made. Where comparisons of the theoretical results with experimental data are made, the potential parameters (eff, uft) used should be those fitted to the cut and shifted potential, rather than those for the full U potential. When theoretical calculations are made for the adsorptionisotherm with the two potential models of eqs 2 and 3 using the samepotential parameters, the isotherm for the full U model is shifted to lower pressures than for the cut and shifted U and exhibits a higher adsorption on pore filling. The results for the two models are compared for a typical case in Figure 2, using the theory described below. The shift of the capillary condensation to lower pressures in the case of the full U
2.0r
1
r: 1.5 .
t
'.O
0 10-1
10-2
10-3
IO0
P/ Po
FYgum 2. Adsorption isotherms for fluids in a slit pore of width
H*= 6 at Tz = 0.8, d e n = 0.3,a d a n = 0.9462. Resulta for both the full LJ (rc* = -) and the cut and shifted LJ potential (re* = 2.6) are shown. Vertical lines are capillary condensation; the approximate extent of thermodynamichyekreeis is ale0 shown.
model is similar to the shift in the condensationtransition found for bulk fluids for the two modelsa6 For fluids in pores this shift becomes smaller for smaller pores and vice versa. For the solid-fluid interaction the full U model is used. We neglect the lateral solid structureof the wall and obtain the external potential due to the solid by integrating the LJ potential between one fluid molecule and each of the molecules of the solid over the lateral solid structure.6 A sum is then performed over the planes of molecules in the solid, the separation between planes being A. This yields the 10-4-3 potential,
-(
tp&)lkT = A[ 2 -)lo Usf 5 2
y:(
-
-
02 3A(0.61A
+z ) ~
where A = 2.rrps(cdkT)(usr)2(A) and is the solid density. The cross-parameters are calculated according to the Lorentz-Berthelot rules, esf = ( ~ ~ e f t ) lad / ~= ; (uw + q ) / 2 . The externalpotential involves severalinputs, two of which are characteristic of the surface itself: the solid density and the separation between layers. In all our calculations, we have used the values corresponding to a graphite surface, ps = 114 nm3, A = 0.335 nm. The other two variables are the relative strength of the solid-fluid to fluid-fluid interactions, ed/eft, and the relative range of the solid-fluid and fluid-fluid potentials, a,f/uff. Since eq 4 is the potential exertad by one wall, the external potential for the slit geometry is
V*&m= 9,(d + 4#
-2)
(5)
The total adsorption per unit area, Fa*, is calculated according to
where Fa* = I',uf?, p* = puf?, H* = Hiaft, and z* = daft. Here FBis the number of molecules adsorbedper unit area and p is the number density. Throughout this paper we adopt the convention of defining dimensionless quantities by using the fluid-fluid parameters, uff and eft. (5) Powlee, J. G. Physica 1984,126A, 289. (6) Steele, W . A. Surf. Sci. 1973,36, 317; The Interaction of Gosee
with Solid Surfaces; Pergamon: Oxford, 1974.
Langmuir, Vol. 9, No. 7, 1993 1803
Adsorption Behavior of Simple Fluids
The adsorption behavior depends on the independent reduced variables T* = kBT/eft,H* (for pores), edetf, and usst/uff. In most of our calculationswe have fixed the value of uBf/uR= 0.9462, which corresponds to the U model for methane on graphite. We vary the ratio e,f/etf in order to change the value of A in eq 3. With the values we have adopted for the solid density, the separation between layers, A, and the ratio u8f/uff,the value of A is given by 31.18(eSf/eff)/!P. Changing the value of pa, the density of the solid, represents a mathematically equivalent modification to changing the solid-fluid interaction parameter tsf. Changing the b S f / b f f ratio has additional effects, since it is raised to various powers, as shown by eq 4. This ratio gives also the relative range of the potentials, which has been shown7to be crucial in determining the order of the wetting transitions. Moreover, we have found that it plays an important role for solvation forces.8 2.2. Density Functional Theory. Several theories have been used for inhomogeneous fluids, particularly integralequation and density functionaltheories. We have adopted the latter approach, since it is more tractable and describes a wide range of surface-drivenphase transitions; moreover, it provides results that are in good agreement with molecular simulation for a wide range of condition^.^ Within this theory, the thermodynamic grand potential, il, the free energy appropriate to the grand canonical (T,V,p)ensemble, is a functionalof the one-particle density distribution, p(r). The equilibrium density profile is obtained by minimizing this functional. When more than one minimum exists, the one with the lower free energy is the stable one. A phase transition occurs when two minima have the samevalue for the free energy. We adopt the nonlocal mean field version of this theory due to Tarazona.'OJ1 The grand potential energy functional Q[p(r)] is the sum of the intrinsic Helmholtz free energy functional F[p(r)l and two other terms corresponding to the contributions of the bulk chemical potential p and the external potential V,a(r), il[p(r)l = F[p(r)l- J d r p ( W - Vext(r))
(7)
where p(r) is the fluid number density at point r. The Helmholtz free energy is expanded about a WCA reference system of molecules with purely repulsive forqes, and this is replaced by the free energy of a fluid of hard spheres of diameter d in the usual ~ a y . The ~ J perturbation ~ term involves the attractive potential ua+,t(r- rl).
where Fh[p(r)l is the free energy functional for an inhomogeneous hard sphere fluid, pW,r') is the pair distribution function, and uatt is given by
- uLJVC) uLJ(r)- uLJ(rc)
-eff uatt=
(0
r < r, rm< r < rc r > re
(9)
thereby neglecting correlations due to attractive forces, so that Tarazona's model expresses the hard sphere free energy as a contribution of an ideal gas and an excess part. The ideal term is exactly a functional of the local density p(r), while the excess part is considered a functional of a smoothed density, p , which is defined as
p(r) = Jdr'w(lr - $1; p(r))p(r') (11) where w(lr - r'l) is a weighting function chosen to give a good description of the hard sphere direct pair correlation function in the uniform fluid over a wide range of densities. In Tarazona's theory, this is carried out by expandingthe uniform fluid hard sphere direct correlation function c(r1,rz) given by the density functional theory in powers of the number density and matching the coefficients in this expansion to the ones in the expansion of the PercusYevick expression for the hard sphere c(r1,rz). Terms up to second order in the density are retained. The inputs to the model are the intermolecular potentials and an equation of state for the excess Helmholtz free energy for hard spheres. The equivalent hard sphere diameter, d, is calculated as a function of temperature, as suggestad by Lu et The explicit form is the one that approximates the B a r k e ~ H e n d e r s o ddiameter ~ dlqf = ( a l P + q ) / ( a r 3 T * + a& (12) where the constants a1 a 4 were obtained by requiring good agreement between theory and simulation at low temperature^.'^ Equation 7 is solved numerically for the density profile, given the conditions of bulk density, temperature, and separation between walls. A simple iteration scheme is used. The Carnahan-starling e x p r e sionI6is used for the hard sphere excess free energy. This theory has been s u c d u l l yapplied to calculateadmrption properties of hard sphere (HS) and W fluids near hard walls and U ~ a l l s . ~It~has J ~ also been used to study layeringtransitions18and prewettingefor particular values of the intermolecular potential parameters and to predict the bulk freezing transiti~n,'~ but a parametrized form for the density had to be imposed. In the presence of a structuredwall, at sufficientlylow temperatures, the fluid might experience a fluid-solid transition, the ordering being influencedby the wdordering. Mederoe, Tarazona, and NavascuW' have applied the same model to understand phase transitions of submonolayer fie. Again, a parametrized form for the density is used. In thie paper we take the walls to be structureless, and no parametrized form is used for the density profile; hence, no solid p k are predicted. However, surface transitions among fluid phases are found with the model; the most important for the description of adsorption behavior are wetting, layering transitions, and capillary condensation.
...
where rmis the value of the U potential at the minimum. The attractive term is treated in mean field approximation,
3. Adsorption Isotherm Behavior In this section we use the theory to explore the effecte of P ,H*, cat/ctt, and usfluffon the form of the adsorption
(7) Teletzke, G. F.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1983, 78, 1431. (8)Balbuena, P. B.; Berry, D.; Gubbms, K. E. J. Phys.Chem. 1993, 97, 937. (9) Evans, R. In Inhomogeneous Fluids; Henderson, D., Ed.; Dekker: New York, 1991; Chapter 5. (10) Tarazona, P. Phys.Rev. A 1988,31, 2672. (11) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 673. (12) Haneen,J. P.;McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986, pp 184-192.
(13) Lu, B. Q.; Evans, R.; Telo da Gama, M. M. Mol. Phys. 1986,66, 1319. (14) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967,47,4714. (16) Telo da Gama, M. M. Private communication. (16) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969,51,636. (17) Peterson,B. K.; Gubbins,K. E.; Heffelfimger,G. 5.; Marini Bettolo Marconi, U.; van Swol, F. J. Chem. Phys. 1988,88,6487. (18) Ball, P. C.; Evans, R. J. Chem. Phys.1988,89,4412. (19) Tarazona, P. Mol. Phys. 1984,52, 81. (20) Mederoe, L.; Tarazona, P.; Navasculs, G. Phys.Rev. B 1987,96, 3376.
Balbuena and Gubbina
1804 Langmuir, Vol. 9, No. 7,1993
I
t
/ I
f ,
pre-wetting
0.9
transition
I
I
/
0
1
0
0.2
0.4
0.6
0.8
I .o
06
08
10
P/PO
I
0 P/ PO
Figure 3. (a) Classificationof adsorption isothermsfor a single surface for Q/UE = 0.9462. (b) Schematic vapor-liquid phase diagram,showing the prewetting line, wetting temperature, and surface critical temperature, T,c(these temperatures, and also the prewetting line, will depend on the value of Ea/w). (c) Schematicadsorption isotherms for various temperatures for a value of €$en of around 0.10, where prewetting can occur; isotherms corresponding to partial wetting, complete wetting, and a prewetting transition are shown.
isotherms. The range of values of these variables that correspond to the various isotherm types (Figure 1) is determined. We first consider the case of a single planar surface (nonporousmaterials) and then consider pores of slit geometry. A brief preliminary discussion of some of these results has been reported previously.21 3.1. Single Planar (Nonporoua) Surfaces. For a = m), only types 11,111,and VI adsorption single surface (H* isotherms of Figure 1are found. The ranges of T* and est/cff corresponding to each of these classes are shown in Figure 3 for usduff = 0.946, the value corresponding to methane on graphite. The dashed lines in the figure dividing the types correspond to qualitative changes in the adsorption behavior (they are not phase transitions) and are approximate and are accurate to about 0.02 in e,f/cnand 0.05 in T*. The subscript f on some of the classes in Figure 3 refers to the fact that the adsorption remains PO; such behavior is observed below the finite as P wetting temperature, where there is only partial wetting. For values of e d e f f below about 0.18, only class I11is found. For higher values of this ratio class 11is found at the higher temperatures, while layering transitions (class VI) occur at temperatures below about 0.8. The wettingtemperature as a function of esf/eff is also shown. This was calculated in the usual way22by solving Young's equation, rs = ra1 + yc cos 0, to find the temperature for which cos B = 1; here the y's are the surface tensions for the interfaces indicated and B is the contact angle. The liquid-gas surface tensions were interpolated from values calculated by Lu et aZ.,13and the solid-gas and solid-liquid surface tensions were calculated from the theory. The bulk phase diagram shown in Figure 3b illustrates the different wetting regimes schematically, for some fiied value of the ratios e d e f f and u,f/uff. The prewetting line (shown dashed) represents a coexistence between thin and thick adsorbed films and
-
(21) Balbuena, P. B.; Gubbine, K. E. Fluid Phase Equilib. 1992, 76, 21. (22) Tarazona, P.; Evans, R. Mol. Phys. 1983,48,799.
0
02
0 4
P/P"
Figure 4. Adsorption isotherms calculated from the theory for a single wall with = 0.9462: (a, top) for = 0.10;(b, bottom) for CJJCR = 0.20.
ends at the surface critical temperature Tuc,whose value depends on the values of e,fIeff and ussfluff. Prewetting transitions were found in our calculations at temperatures above the wetting value, for a small range of cudenvalues near 0.10-0.12, for reduced temperatures around 1.101.14. Schematicadsorption isotherms for the single surface for c,f/eR values in the range where prewetting occurs are shown in Figure 3c. Several theoretically calculated isotherms are shown in Figure 4 for two different fluidwall strengths. The weaker fluid-wall strength used in the results of Figure 4a results in prewetting transitions over an intermediate temperature range. The isotherm for T* = 0.7 in Figure 4b shows layering, but this temperature is above the critical temperature for layering transitions; at somewhat lower temperatures the steps shown become discontinuities, i.e. first-order layering transitions. 3.2. Slit Pores. We present results at three different and 1.4,corresponding temperatures, T* = kT/etf= 0.5,0.8, to a low, intermediate, and supercritical regime. The critical temperature of the bulk fluid has been estimated by Powles6 to be 1.12. Thus, we can expect phase transitions to occur in pores at the temperatures T* = 0.5 and 0.8, but not at the supercritical temperature of T* = 1.4. The ranges of H*and e d e g corresponding to each of the adsorption types of the IUPAC classification scheme are shown in Figure 5, for a uduff value of 0.9462 (corresponding to methane on graphite) for the three temperatures. The divisions between the different types of isotherms in Figure 5 are not always clear-cut; we expect the divisions shown to be accurate to about 0.02 in e d e and to 1unit in H*.We note that classes I, N,and V are
~
Adsorption Behavior of Simple Fluids
Langmuir, Vol. 9, No. 7, 1993 1806
/ i 1-
4
12
I
I
H* = 7.5
3-
i
I
(VI
I I
I
I CP/Pc3 0
0
0.2 Csf/@ff
a4
0.6
Figure 5. Classificationof adsorption isotherms for slit pores for a d u =~ 0.9462: (a) !P = 0.5; (b) !P = 0.8; (c) !P = 1.4.The linea in the f i i are approximateand refer to qualitativechanges in the adsorption behavior. specific to pores and do not occur on single surfaces. Moreover, the classes referred to as 11,111,and VI in Figure 5differ from those for a single surface in that the adsorption does not divergeto infinity as PapproachesPObut remains finite because of the confinement of the fluid. All six of the IUPAC types are found, and in addition we find a new type, which we call VII, at the two lower temperatures. In this type the fluid-wall forces are very weak and capillary evaporation takes place, i.e. the gas-liquid transition in the pore occurs at a pressure higher than the normal vapor pressure, PIP" greater than 1. Isotherms showing negligible adsorption for all P < PO and finite adsorption at P = PO have been reported experimentally, e.g., for Kr on Na and on Na2023and for water on graphite;24however, they correspond to class 111. Several typical isotherms for the lowest temperature of T* = 0.5 are shown in Figures 6 and 7. In Figure 6 the effect of variation in the strength of the fluid-wall forces, e,f/efi, at a fixed pore width of H* = 7.5 is shown. For very weak walls type VI1 is found with capillary evaporation, and as the wall strength increasesthe behavior passes first to type V and then to VIf;in the latter case there is a single layering transition from 0 to 1 layers at PIP of about 0.26,followed by capillary condensation at PIP" = 0.46. In the figure the regions of thermodynamichysteresis are also shown. These show metastable regions of the isotherms for which solutionsto the density functionaltheory equations are found, even though they lie at higher (adsorption)or lower (desorption)pressures than the value for the true thermodynamic transitions; such solutions correspond to localminimain the grand free energy,rather than to the global minimum. The dashed lines show the true thermodynamictransitions, calculated by evaluating ~~
(23)Pierotti, R.A.; Halsey, G.D.J. Plays. Chem. 1959,63,680. (24) Avgul, N.N.;Berezin,G.I.; Kiselev, A. V.;Lygine, I. A. Zzu. Akud. Nuuk SSR Otd. Khim. Nuuk 1961,2,205.
I
rY
I
I I
2-
I I
I
0
I
02
H* = 5 ( V I
I I
I
,
06
04
0 8
I O
P/ P O
Figure 7. Typical isothermsfor !P = 0.6,ed/ee = 0.18, and a d / a ~ = 0.9462,showing the effect of varying pore width. the grand free energy for each phase and determining the point where they are equal. In Figure 7 we show the effect of varying H*,keeping constant the fluid wall strength. At this particular value of elJeff the behavior passes from type I to V as the pore size increases. For somewhat stronger fluid-wall strengths (see Figure Sa) the behavior passes from type I to IV and then to VI. We note that in the region marked type I in Figure 5a the isotherme generally show a first-order phase transition from a gaslike adsorbed phase to a liquidlike one; this occurs at low pressures (see Figure 7). The pressure at which this transition occurs depends strongly on both H*and e d / e ~ . As H* decreases, or as e d e f t increases, the transition pressure decreases. For H* = 2.5, !P = 0.6,and csf/ea= 0.3 (with uduff= 0.94621, for example, the transition occurs at PIP 4 X 106, while for a pore of this size and temperature when edeff= 0.4348 (the value for methane on graphite, and close to the value for N2 on graphite4)the transition is at a P I P value of 2.5 X IO-". When the adsorption isotherm is plotted against PIP" on the usual s d e of 0 to 1 such low pressure transitions are not visible, and the isotherm has the usual Langmuir form. For H* values below about 1.6,the adsorption drops rapidly to zero for all relative pressures; the pores are now too small to admit molecules. As H*increases, for e,$~valueslarger than about 0.2, the behavior passes from type I to VIf when the pore becomes large enoughto accommodatemore than two layers of adsorbate. For the intermediate temperature of !P = 0.8 the range of edlefffor which type VI1 is found is smaller than at the
Balbuena and Gubbins
1806 Langmuir, Vol. 9, No. 7, 1993
O0.2 I I I
0-
1
I
I
I
I
I
I
I
I
"0
0.02
0.06
0.04
0.08
PO3/ kT
Figure 9. Adsorption isotherm for P = 1.4, udus = 0.9462, and H* = 2.5. The behavior is type I", but for values of u/qt slightly below 0.1 it changes to type 111".
I
I I
2
IC
I
/
I
I
0 IIV) 0 0
I I
0.2
06
04
08
.
I O
P/PO
Figure 8. Adsorption isotherms for T* = 0.8, u,dun = 0.9462: (a, top) H* = 3; (b, bottom) H* = 10. For H* = 3 there is a fiit-order phase transition at PIP" = 0.0035, which is not visible on the scale of the plot shown here.
lower temperature, and the range for class V is correspondingly larger (Figure 5b). For values of d e f f larger than about 0.2, on increasing H*the behavior first passes from type I to IV, and at larger H* to type VIf. Typical isotherms for !P = 0.8 are shown in Figure 8. At this temperature when type I occursthe sharprise in adsorption at very low relative pressures, correspondingto pore filling, is sometimes continuous and is not always a first-order transition; this is becauee for very smallpores the capillary critical temperature lies below !P = 0.8. For somewhat larger pores the capillary critical temperature is above 0.8 and class I with first-order transitions is found, while for larger H*values class IV is found with capillary condensation; for such pores the transition usually occurs at PIP" values of 0.03 or above. In the region of Figure 5b corresponding to type VIfthe layering involves continuous transitions at this temperature, rather than the sharp firstorder transitions seen at !P = 0.5 (see, for example, the isotherm for eden = 0.2 in Figure 6, as compared to the one for e d e f f = 0.435 in Figure 8b). For the supercritical temperature of !P = 1.4 the vapor pressure PO is undefined, and the IUPAC classification scheme of Figure 1 is not strictly applicable. It is, nevertheless, convenient to adopt an analogous scheme. We therefore adopt the IUPAC schemewith the following modifications: We plot against another dimensionless (other chaica, e.g. PIP,, where P, is the pressure, P a g l k ~ critical pressure for the bulk gas, could equally well be used) and add the superscript sc to the type numbers to remind ourselves that the temperature is supercritical. At these high temperatures only three types are found, IW,
IIp, and IIIp; types IV, V, and VI, which involves firstorder transitions or are at temperatures close to such transitions, cannot occur; in effect, classes IVt and Vf, present at lower temperatures, shift to IIP and IIIP, respectively. Some typical isotherms for micropores are shown at this temperature in Figure 9. The results shown in Figures 5-9 are for a fixed value of a B f / aof~ 0.9462. Changing this ratio (equivalent to changing the diameter of the adsorbate molecule, keeping the substrate fixed) affects both the constant A in eq 3 and the terms (usf/z) raised to the powers 10,4, and 3. The effect of increasing usduffis qualitatively rather similar to decreasing edeaand is shown in Figure 10for some typical cases at the intermediate temperature. For a carbon surface, the size ratios shown, usduff = 0.90,0.9462, and 0.98, correspond to uff values of 0.425, 0.382, and 0.354 nm, respectively. These latter three values of uffareabout the values for ethane, methane, and argon, respectively. 4. Heat of Adsorption
The isosteric heat of adsorption, qat,is the heat released (per molecule) on transferring an infinitesimal amount of the adsorbate from the coexisting bulk gas phase to the adsorbed phase at some constant temperature, pressure (and hence constant total adsorption Fa*),surface area A, and pore width H,as defined, qnt is minus the enthalpy change for such a transfer from the bulk to adsorbate phases and so is positive since heat is released on adsorption. It can be related to the entropies (SI, internal energies (LO,and volumes (V) of the two phases by8,26 Qat
- #")) = T(,S"@
= LI(@- p)+ p(Vg) - lr(B)) (13)
where superscripts (a) and (8) refer to the values for the adsorbed and bulk gas phases, respectively. In this equation Sb),IJ(B),and v(B)are the values per mole for the coexisting gas phase; fi(a),S(a),and v[B) are the partial molal quantities for the adsorbed phase, e.g. = (ab'/ a n ( a ) ) ~ p , ~where p, V is the total volume and n") is the number of moles in the adsorbed phase at a particular T, P, A, and H. The isosteric heat obeys the Clapeyron equation (25) Nicholson, D.; Pareonage, N. C. Computer Simulation and the StatisticaZMechonics of Adsorption;AcndemicPreaa: London, 1982;pp 34-35.
Adsorption Behavior of Simple Fluids
Langmuir, Vol. 9, No. 7,1993 1807 21
3
* qcond 6
P/ P O
I
I
,
I
I
0
3'0
P/P0
Figure 10. Effectof variation in adusonthe adsorptionisotherm for T+ = 0.8,u/f/tff = 0.30 (a, top) H*= 2.5 (type I); (b,bottom) H* = 7.5 (types w ,VI&
s@) - #a) E !(d) T r.,w = p)- p")
(14)
(15) Over small temperature intervals it is usually possible to neglect the temperature dependence of qat, so that (15) can then be integrated to give
+ constant
(16) Equations 15and 16provide a convenient relation between the isoeteric heat of adsorption and measurablequantities. The assumptions implicit in (15) and (16) are usually innocuous; in the work reported here we have checked the ideal gas and W >> Va)approximations by including higher order virial coefficient terms and f i d that the approximations lead to errors of no more than a few percent. An equation that is equivalent to (15) is2s
+
qst = fi)-l[U@) Pv(B)- u""' - n c a ) ( d ~ ) / d n ' a ' ) , ] (17)
where 2%) = P P ) / R Tis the compressibility factor of the (26)W&, 1988,63,49.
. , ....
, , , ,,...I
16-4
,
10-3
,
, ,,
,
1'0-2
, ,,
10-1
100
P/ Po
whereP is the pressure of the bulk gas phase in equilibrium with the adsorbed phase and the derivative on the lefthand side of the equation is taken with the adsorption, re, kept constant. If we further assume that vCa)
