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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 reportedfor 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; he range of variables correspondingto each of the

six isotherm types is determined, and the underlying factors leadingto each of the types are elucidated.

In additionto the six ypes of the 1985classification,a seventh ype is identified, corresponding to capillary

evaporation. A similarstudy and classification s reported for the heata of adsorption and 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 experi-

mental 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 elative pressures approaching unity the curve may reach

a limiting value or rise if large pores are present. Types

I1

and I11are 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 adsorption sotherms

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

such materials he 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 me-

chanical theories, particularly density functional theory,

+ Present address: Department of Chemical Engineering,Uni-

versity 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. 940,62, 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.

0743-7463/93/2409-1001 04.00/0

I

I n

1

MICROPORES

SUBSTRATE

WEAK Y

LAYER ING

Relative

pressure,

P/ Po

Figure

1. The six types

of

adsorption isotherm accordingto he

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 neglects pore blockingandnetworking

and assumes the heterogeneity isdue onlytoadistribution

of pore sizes and chemically heterogeneous sites on the

surface, one can approach the problem by writing the

adsorption

rs

n the form

where H s the pore

width,

ed is the attractive energy

between an adsorbed molecule and a chemically hetero-

geneous 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 th e real material, as

(4)

Laetoekie, C.;

Gubbine,

K.

E.; Quirke,

N.

angmuir, in

prese.

0

1993

American Chemical Society

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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

functionP(H,ed)or even the simpler pore size distribution

P(H)

xcept for some rather easily characterized materials.

Inview of this rather unsatisfactory situation, 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 eval-

uated. Accordingly, in

h i s

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),nd 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, espectively. Since

heterogeneity is absent in our model, the relation between

adsorptiontype,phase transitions, etc. and the underlying

molecular and pore properties can be clearly seen. We

firstdetermine the range of parameter space ( k

Tleff,

H/

ft,

e,f/eft, and uduft)correspondingto each of the adsorption

types of the W A C classification; in doing this we

also

introduce a new type, VII,which correspondstocapillary

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

same parameters. 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-component luid at fixed

chemical potential p and temperatureT,the totalvolume

of the system being

V.

The fluid inside the pores feels the

presence of the solid surfacesasan 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 Lennard-

Jones U)otential, given by

u )

= uU&)

- ~ ~ , ~ ( r , )

f r < r ,

= O if r > rc (2)

where r , = 2 . 5 ~s the cutoff radius and uu s the full LJ

potential,

(3)

The advantage of using the cut and shifted potential is

thatcomparisonsof he theoretical resultawithmolecular

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 fullU otential. When theoretical calculations

are made for the adsorption sotherm with the two potential

modelsof eqs 2 and 3 using the same potential parameters,

the isotherm for the full U model is shifted t o lower

pressures than for the cut and shifted

U

nd 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

uU,ff

= 4e[(uff/r)12- (u , /~ )~ I

Balbuena and Cubbins

2.0 1

r:

1.5

.

'.O

0

10-3

10-2

10-1

IO0

P/ Po

FYgum 2.

Adsorption isotherms for

fluids

n a slitpore of width

H*

6 at

Tz

= 0.8, d e n =

0.3, d a n = 0.9462. Resulta forboth

the full LJ (rc*= -) and the cut and shifted LJ

potential (re*

= 2.6) are shown. Vertical lines

are

capillary condensation; the

approximate exten t of thermodynamic hyekree isisale0 shown.

model is similar

to

the shift in the condensation ransition

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 fullU model isused

We neglect the lateral solid structure of 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 isthen performed over the planes of molecules in the

solid, the separation between planes being A. Thisyields

the 10-4-3 potential,

tp&)lkT

=

A[ -(-)losf - ( -

02

3A(0.61A+ z ) ~

2

where A = 2.rrps(cdkT)(usr)2(A)nd isthe solid density.

The cross-parameters are calculated according

to

the

Lorentz-Berthelot rules, esf = ( ~ ~ e f t ) l / ~ ;d

=

uw + q ) / 2 .

The external potential involves several nputs, two of which

are characteristic of the surface itself: the solid density

and the separation between layers. Inall 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

isthe potential exertad by one wall, the external potential

for the slit geometry is

(5)

The total adsorption per unit area, Fa*, is calculated

according to

V* m

=

9,(d

+

4#

-

2 )

where Fa*

=

I',uf?, p*

=

puf?, H*

=

Hiaft ,and z*

=

daft .

HereFB s the numberofmolecules adsorbed per 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. 19 73,36 , 317; The Interaction of Gosee

with Solid Surfaces; Pergamon: Oxford,

1974.

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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 calculations we have fixed the value

of uBf/uR

= 0.9462,

which corresponds

to

theU 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,

he 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 modi-

fication

to

changing the solid-fluid interaction parameter

tsf. Changing the bSf/bff ratio has additional effects, since

it is raised to various powers, asshown by eq

4.

This ratio

gives also the relative range of the potentials, which has

been shown7

o

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 Func tiona l Theory . Several theories

have been used for inhomogeneous fluids, particularly

integral equation and density functional heories. We have

adopted the latter approach, since i t 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 functional of 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 same value 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 - Jd rp (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 . ~ J ~he perturbation term

involves the attractive potential ua+,t(r

rl).

Langmuir, Vol. 9, No. 7,

1993

1803

thereby neglecting correlations due to attractive forces,

so that

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

r

re

where rm s the value of the

U

otential a t the minimum.

The attractive term is treated in mean field approximation,

-eff

-

uLJVC)

uatt= uLJ(r ) uLJ(rc)

rm

< r