A Pore-Size-Dependent Equation of State for MultilayerAdsorption in Cylindrical Mesopores

H. Y. Zhu, L. A. Ni, and G. Q. Lu*

Department of Chemical Engineering, The University of Queensland, Qld 4072, Australia

Received October 27, 1998. In Final Form: March 1, 1999

The chemical potential of adsorbed film inside cylindrical mesopores is dependent on the attractiveinteractions between the adsorbed molecules and adsorbent, the curvature of gas/adsorbed phase interface,and surface tension. A state equation of the adsorbed film is proposed to take into account the above factors.Nitrogen adsorption on model adsorbents, MCM-41, which exhibit uniform cylindrical channels, are usedto verify the theoretical analysis. The proposed theory is capable of describing the important features ofadsorption processes in cylindrical mesopores. According to this theory, at a given relative pressure, thesmaller the pore radius is, the thicker the adsorbed film will be. The thickening of adsorbed films in thepores as the vapor pressure increases inevitably causes an increase in the interface curvature, whichconsequently leads to capillary condensation. Besides, this study confirmed that the interface tensiondepends substantially on the interface curvature in small mesopores. A quantitative relationship betweenthe condensation pressure and the pore radius can be derived from the state equation and used to predictthe pore radius from a condensation pressure, or vice versa.

Introduction

Adsorption in cylindrical mesopores consists of tworegimes.1 In the low-pressure regime, an adsorbate layerlining on the pore walls forms and thickens as theadsorbate vapor pressure increases, similar to adsorptionon a flat surface. The thickening of the adsorbed film incylindrical pores inevitably leads to an increase in thecurvature of the concave interface between the gas andadsorbed phases. According to the well-known Kelvinequation, increases in the curvature of the interface willreduce the chemical potential of the adsorbed phase. Thesmaller the mean radius of the curvature is, the lower thechemical potential will be. Therefore, vapor at equilibriumwith the adsorbed layer will condense at a relative pressurewell below unity, filling the mesopores spontaneously.Such a capillary condensation is the second regime.Experimental observations that capillary condensationoccurs in a larger pore at higher relative pressure issuccessfully explained by the Kelvin equation whichrelates the radius of the interface curvature to the loweringof the vapor pressure at equilibrium with the adsorbedphase. On the basis of the Kelvin equation, variousmethods have been developed to calculate pore sizedistributions of porous solids from adsorption data.2,3 Forcylindrical pores, the pore radius, R, should be the sumof the radius of the interface curvature rc, and the criticalthickness of the adsorbed film tpc. The interface curvature,rc, can also be understood as the radius of the condensationcore and derived by the Kelvin equation. In conventionaltheoretical analysis, the critical thickness, tp, is notevaluated, but the mean thickness of an adsorbed film ona nonporous solid of a surface nature similar to that of thesample under consideration and at the same vaporpressure, t, is usually used instead.1-3 Although thisapproach provides convenience for calculation, it is

accepted only as an approximation. For many years, suchan approximation has been widely used in cylindricalpores.

On the other hand, the Kelvin equation does not providean explanation of why the adsorbed phase in cylindricalpores can exist prior to the condensation. The stability ofthe adsorbed phase is due to attractive surface forces, theinteraction between adsorbate molecules and the mol-ecules or atoms of the solid.4 For multilayer adsorption ona flat surface, the surface force results in the differencebetween the chemical potentials of the adsorbed phaseand bulk liquid adsorbate. The well-known Frenkel-Halsey-Hill (FHH) equation5,6 which is proposed on solidthermodynamic arguments can give the adsorption iso-therm on flat surfaces. However, the properties of theadsorbed film in a cylindrical pore could be highly sensitiveto the interface curvature which increases as adsorptionproceeds. For instance, Broekhoff and De Boer7 arguedthat the thickness of the adsorbed film increases sub-stantially with the curvature of the gas-adsorbed-phaseinterface, even at moderate relative pressures. To ac-curately describe the chemical potential of the adsorbedlayer in a cylindrical mesopore, both the surface force andthe curvature of the gas-adsorbed-phase interface haveto be taken into account. This has been recognized for afew decades by Broekhoff and De Boer,7,8 Everett andHaynes,9 Steele,10 Saam and Cole,11,12 and Findenegg etal.13 Effort has been made to develop theories taking bothfactors into account. Nevertheless, because of the absenceof suitable model adsorbents to verify the arguments andequations, the above theories have not been well-

* Corresponding author. Phone: 61 7 33653735. Fax: 61 733654199. E-mail: maxlu@cheque.uq.edu.au.

(1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity,2nd ed.; Academic Press: New York, 1982; Chapter 3.

(2) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc.1951, 73, 373.

(3) Dollimore, D.; Heal, G. R. J. Appl. Chem. 1970, 14, 109.

(4) Steele, W. A. Chem. Rev. 1993, 93, 2355.(5) Hill, T. L. Adv. Catal. IV 1949, 17, 668.(6) Halsey, G. D., Jr. J. Am. Chem. Soc. 1951, 73, 2693.(7) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 15.(8) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 8.(9) Everett, D. H.; Haynes, J. M. Colloid Science, Volume 1, Specialist

Periodical Reports; Chemical Society: London, 1973; pp 123-172.(10) Steele, W. A. The interaction of gases with solid surfaces;

Pergamon Press: Oxford, 1974.(11) Cole, M. W.; Saam, W. F. Phys. Rev. Lett. 1974, 32 (18), 985.(12) Saam, W. F.; Cole, M. W. Phys. Rev. B 1975, 11 (3), 1086.(13) Findenegg, G. H.; Gross, S.; Michalski, Th. Stud. Surf. Sci. Catal.

1994, 87, 71.

3632 Langmuir 1999, 15, 3632-3641

10.1021/la981515v CCC: $18.00 © 1999 American Chemical SocietyPublished on Web 04/24/1999

established and accepted as the FHH equation for the flatsurface. Besides, Defay et al.14 predicted theoretically asubstantial dependence of the interface tension on theinterface curvature, particularly when the radius of thecurvature is several nanometers or below. It will benecessary to verify this prediction and incorporate thisfactor into models for adsorption in cylindrical mesopores.

In 1992, a novel class of silicates and aluminosilicatesof highly uniform mesopore structures was reported byMobil researchers,15,16 known as M41S. For instance, oneof the M41S family members, MCM-41, having hexagonalarrays of uniform cylindrical pores,17 can serve as a modelsystem to verify the predictions by various theories andmethods for adsorption in cylindrical mesopores. Thesematerials are of great significance for theoretical studiesof adsorption,18 and indeed, have encouraged great interestin these studies. A number of articles have been publishedon developing and verifying various theories and meth-ods.18-24

In this study, we start from the formalism of Steele10

in which both interface curvature and surface force ofpore walls are taken into account in terms of chemicalpotential. By application of the 6-12 relation of Lennard-Jones potential to the surface force and the Yong-Laplaceequation to the interface curvature, we arrive at a relationof relative vapor pressure, P/P0, radius of cylindrical pore,R, and thickness of the adsorbed film in the pores.Meanwhile, the impact of interface curvature on theinterface tension is also considered. With a comparison ofthe results derived from approaches in which differentfactors are incorporated into consideration as well as thosederived from adsorption of the model adsorbents, theinfluence of these factors can be illustrated. This providesa comprehensive understanding of multilayer adsorptionin mesopores and can be used to derive pore size fromadsorption data.

Theory

Chemical Potential of an Adsorbate in a Cylin-drical Pore. Consider the work required to transfer dnmoles of adsorbate from the bulk liquid to the interfaceof the gas-adsorbed phase of an adsorbed film inside acylindrical pore at a fixed temperature. The film is of athickness, tp, lining on the wall and its chemical potentialis µ(tp). The chemical potential of the bulk liquid is µ(l)].This work is [µ(tp) - µ(l)] dn and has two contributingterms.10 One is the work required to change the potentialenergy of dn moles by u(tp) and the second term representsthe work required to change the surface area of theinterface A when dn moles are added at the thickness tp:

where γl,g is the surface tension of the gas-liquid interface.Suppose the cylindrical pore is very long and the amountof adsorbate transferred is 1 mol, we can write

Contribution of the Interface Curvature. This isexpressed by the second term on the right-hand side of eq2. We have dn ) dV/Vh , where Vh is the molar volume of thebulk liquid adsorbate. Substituting it into the second term,we have

The area of the gas/adsorbed-phase interface of the filmis 2π(R - tp)l, where l is the length of the pore and R theradius of the cylindrical pore. The volume of the adsorbatein the film is π[2Rtp - tp

2]l. Thus, eq 3 becomes

Influence of the Interface Curvature on SurfaceTension. The surface tension for a curved interface iscurvature-dependent, according to theoretical investiga-tion of Defay et al.14 The interface curvature brings aboutan increase in the surface tension. For the mean radiusof the curvature being around several nanometers, theincrease becomes important. Defay et al. suggested arelation between the surface tension of curved interface,γc, and mean radius, rm, as14,24

For the interface in a cylindrical pore, rm ) 2(R - tp).According to Bhatia and Sonwane,24 when an adsorbateis nitrogen at liquid nitrogen temperature, λ is 2.97 Å.Substituting the expression of rm and the value of λ intoeq 5, we obtain an expression for the surface tension, takinginto account the curvature, as

Equation 6 predicts a substantial increase in surfacetension, especially when R is small. In such circumstances,the value was regarded only as a general indication of thetrend of the influence of curvature on the surface tensionsince the value has not been supported yet by anyexperimental observation.14 It will be very interesting toverify the theoretical prediction of Defay in this range.

Gas-Solid Potential Energy. The change in thepotential energy of the transferred adsorbate u(tp) is dueto the interaction between the adsorbed species and theadsorbent. When the Lennard-Jones (12:6) potential isapplied for the interaction, the potential between anadsorbed molecule and one atom or molecular species inthe adsorbent is

(14) Defay, R.; Prigogene, I.; Bellemans, A.; Everett, D. H. SurfaceTension and Adsorption; Longmans: London, 1966; pp 257-258.

(15) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck,J. S. Nature 1992, 359, 710.

(16) Beck, J. S.; Vartuli, C.; Roth, W. J.; Leonowicz, M. E.; Kresge,C. T.; Schmitt, K. D.; Chu, C. T.-W.; Olson, D. H.; Sheppard, E. W.;McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992,114, 10834.

(17) Zhao, X. S.; Lu, G. Q.; Millar, G. J. Ind. Eng. Chem. Res. 1996,35, 2075.

(18) Branton, P. J.; Hall, P. G.; Sing, K. S. W.; Reichert, H.; Schuth,F.; Unger, K. K. J. Chem. Soc., Chem. Faraday Trans. 1994, 90, 2965.

(19) Branton, P. J.; Hall, P. G.; Sing, K. S. W. J. Chem. Soc. Chem.Commun. 1993, 1257.

(20) Branton, P. J.; Hall, P. G.; Treguer, M.; Sing, K. S. W. J. Chem.Soc., Chem. Faraday Trans. 1995, 91, 2041.

(21) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101(4), 583.

(22) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267.(23) Maddox, M. W.; Oliver, J. P.; Gubbins. K. E. Langmuir 1997,

13, 1737.(24) Bhatia, S. K.; Sonwane, C. G. Langmuir 1998, 14, 1521.

[µ(tp) - µ(l)] dn ) u(tp) dn + γl,g (∂A∂n) dn (1)

µ(tp) - µ(l) ) u(tp) + γl,g (∂A∂n) (2)

γl,g (∂A∂n) ) γl,gVh (∂A

∂V) (3)

γl,gVh (∂A∂V) ) γl,gVh

dAdtp

/dVdtp

) -γl,gVh

R - tp(4)

γc )γl,g

1 - λ/rm(5)

γc )γl,g

1 - 1.486/(R - tp)(6)

ε12 ) 4εgs[( σ0

F12)12

- ( σ0

F12)6] (7)

EOS for Adsorption in Cylindrical Mesopores Langmuir, Vol. 15, No. 10, 1999 3633

where F12 denotes the position of the gas atom relative toan atom in the solid, εgs is the depth of the energy well,and σ0 is the distance at which ε12(F12) ) 0. The interactionpotential of the adsorbed molecule with an adsorbentcomprising j atoms is

If a large number of atoms in the solid is involved, thesummation in eq 8 can be replaced by an integral, so that

where F is the distance between the gas atom and anysolid atom, ki ) 4εgsσ0

inc and nc is the number of interactingcenters per unit volume of adsorbent; meanwhile Ii )∫∫V∫ F-i dV (i ) 6, 12) and V is the solid region. Theanalytical expressions of Ii can be derived and the detailsare given in the Appendix.

In general, Ii can be expressed as an absolutelyconvergent power series (0 < a < R). In practice, we cantake the partial sum of the series as the value of Ii. Twosituations of pore wall are considered here: the pore wallis thick (consisting of infinitive layers of atoms) and thepore wall is thin (consisting of a single layer of atoms).For the first case, I6,T is

where

In the case where the pore wall consists of a single layerof atoms (or for the pore wall of a thickness larger thana single layer, the interaction between the adsorbatemolecules and the atoms which are located beyond thesurface layer of the pore wall is ignored), the term I6,R, is

where

The coefficients, en*, in eqs 10 and 11 are given from n )0 to n ) 11 in Table 1.

If multilayer adsorption (after monolayer adsorption iscompleted) is concerned, the distance between the inter-acting pair under our consideration is definitely largerthan the diameter of the adsorbate molecules. The term

with a power of 12, I12, is negligible. The omission of thiswill bring about substantial simplification for the analysis.

The contribution from the change in the potential energyu(tp) to the chemical potential of the adsorbed phase isequal to NAεj,c, where NA is Avogadro’s constant. The termu(tp) is

where I6 ) I6,T or I6,R depending on the situation we areconcerned with.

A Pore-Size-Dependent Equation for a StableAdsorbed Film inside a Cylindrical Pore. The termson the left-hand side of eq 2, [µ(tp) - µ(l)], can be substitutedwith R0T ln P/P0 (R0 being the universal gas constant).1,10

The first term on the right-hand side of eq 2 is given byeq 12 and the second term by eq 4, and then a generalexpression for a stable adsorbed film lining inside acylindrical pore can be written as

Rearranging theequationandsubstituting theparametersof nitrogen, the adsorbate used in this work (T ) 77.4 K,γl,g ) 8.72 × 10-3 N/m, Vh ) 34.68 cm3/mol)1 and eq 6 intothe second term on the right-hand side of the equation,we have

where k is the Boltzmann constant and the unit for σ0, R,and tp is Å. The analytical expressions of I6 are shownabove (eqs 10 and 11).

Since the amount adsorbed on the unit surface area ofthe pore wall, nu, can be expressed as nu ) tpc (c is aconverting constant),1 the isotherm of nitrogen adsorbedin the pores of R in a form of nu ) f(P/P0)T,R,g,s must havethe shape and features very similar to that of tp ) f(P/P0)T,R,g,s which can be derived from eq 13. Therefore, eq 13is a state equation for the multilayer adsorbed film insidecylindrical pores just as the FHH equation is for multilayeradsorbed film on flat surfaces.

Specific Cases. To illustrate the effects of severalimportant factors such as pore wall thickness (infinitiveor single layer) and surface tension, we consider severalspecial cases. Equation 13 can be rewritten as

Table 1. Coefficients of the First 12 Terms in Equations10 and 11

termen

(×R-3)en*

(×R-4)

0 2.47 2.471 77.7 52.52 102.7 97.13 251.1 305.34 312.2 463.95 527.6 926.66 628.4 1273.57 908.3 2086.08 1050.8 2697.29 1392.4 3953.5

10 1579.1 4906.711 1983.2 6697.9

u(tp) ) -4NAncεgsσ06I6 (12)

R0T ln P/P0 ) -4NAncεgsσ06I6 -

γcVhR - tp

(13)

ln P/P0 ) -k6

kTI6 - 4.7

R - tp - 1.486(14)

ln P/P0 ) -k6

kTI6 -

γcVhR - tp

(15)

εj,c ) ∑j

4εgs[( σ0

F1,j)12

- ( σ0

F1,j)6] (8)

εj,c(F) ) ∫∫∫V

4εgs[(σgs

F )12

- (σgs

F )6] dV ) k12I12 - k6I6

(9)

I6,T ) ∑n)0

∞

en(a2

R2)n

(10)

en )

3π2

4R3( dn,0cn

2n + 3+

dn-1,1cn-1

2(n - 1) + 3+ ‚‚‚ +

d1,n-1c1

5+

d0,nc0

3 )(n ) 0, 1, 2, ...)

I6,R ) ∑n)0

∞

en/(a2

R2)n

(11)

en/ )

3π2

4R4(dn,0

/ cn + dn-1,1/ cn-1 + ‚‚‚ + d1,n-1

/ c1 + d0,n/ c0)

(n ) 0, 1, 2, ...)

3634 Langmuir, Vol. 15, No. 10, 1999 Zhu et al.

For thin pore walls consisting of a single layer of solidatoms, the surface tension is independent of curvature(no Defay’s effect). Equation 15 becomes

For thick pore walls (infinitive layers of atoms), we have

When the effect of interface curvature on the surfacetension (Defay’s effect) is considered in the case of a thickpore wall, the equation is

The calculation results using these equations will becompared to the experimental results of the modeladsorbents.

Model Adsorbents. A series of MCM-41 samples wereprepared as model adsorbents of cylindrical mesopores,to validate the proposed theoretical calculations. Prepa-ration and characterization of MCM-41 with various poresizes have been extensively reported in the literatures.15-17

The framework of the MCM-41 samples is hexagonalarrays of uniform cylindrical tubes, according to X-raydiffraction patterns of the samples. In general, a sharpd100 peak and secondary diffraction at larger angles wereobserved, which is characteristic of MCM-41 solids. Theinner radius of the pure silica tube, that is, the pore radius,varies from sample to sample in the range between 20and 50 Å. Pore radii can be estimated from the positionof the [100] peak: Rxrd ) 1/2[a0 - wall thickness] wherea0 ) 2d100/x3, and the wall thickness for silicate is assumedto be 10 Å.16 The wall thickness, however, might deviatefrom 10 Å, as a wide range of thicknesses is reported inthe literature. Moreover, the wall thickness is not neces-sarily identical for all samples studied. However, the poreradius can be calculated by an improved MP method27

using the adsorption data of the sample only. If such apore radius is close to that derived from X-ray diffractionpatterns with the assumption that the thickness of thepore wall is 10 Å, the wall thickness of the sample isregarded as being close to 10 Å. We chose such samplesin this study. Nitrogen adsorption was measured at liquidnitrogen temperature (77.4 K) using an automated surfacearea and pore structure analyzer, NOVA 1200 (Quanto-chrome, USA), and the samples were degassed at 573 Kfor 3 h prior to adsorption analysis.

The isotherms of nitrogen adsorption on three MCM-41 samples are shown in Figure 1 as representatives. Thestepwise shape is typically characteristic of a N2 isothermof MCM-41.18 It has been found that the relative pressure(P/P0) at which the adsorption reaches the last plateaushifts to the right for samples with a larger pore diameter.This characteristic relative pressure (P/P0) has been usedto qualitatively compare the pore dimensions of differentsamples.15,17 A higher value of the characteristic (P/P0)indicates a larger pore size.

Results and Discussion

Multilayer Adsorption in Cylindrical Mesopores.If the pore radius R is known, the thickness of the adsorbednitrogen film in pores of radius R at various relativepressures (P/P0) can be derived from eq 15 and itsderivatives. Figure 2 shows four curves of tp against P/P0

(lines), calculated using eq 18. The R value obtained fromX-ray diffraction measurements were used for these curvesso that we can compare them conveniently with the dataobtained from adsorption data. Besides, a k6,T/kT value of36 was used, with which the calculated results are in thebest agreement with those experimentally obtained fromthe model adsorbents. The trend shows that the thicknessincreases with increasing P/P0. Evidently, at a givenrelative pressure P/P0, the smaller the radius of the pore

(25) Lippens, B. C.; De Boer, J. H. J. Catal. 1965, 4, 319.(26) Payne, D. A.; Sing, K. S. W.; Turk, D. H. J. Colloid Interface Sci.

1973, 43, 287.(27) Zhu, H. Y.; Zhao, X. S.; Lu, G. Q.; Do, D. D. Langmuir, 1996, 26

(12), 6513.

ln P/P0 ) -k6,R

kTI6,R -

γl,gVhR - tp

) -k6,R

kTI6,R - 4.7

R - tp

(16)

ln P/P0 ) -k6,T

kTI6,T - 4.7

R - tp(17)

ln P/P0 ) -k6,T

kTI6,T - 4.7

R - tp - 1.486(18)

Figure 1. N2 adsorption isotherms of three representativeMCM-41 samples.

Figure 2. The thickness (tp) of the adsorbed nitrogen film inpores against P/P0. The lines were derived from eq 18. Thesymbols are the thickness obtained from the adsorption dataof four MCM-41 samples. From left to right, pore radius Rincreases and the lowest plot is for reference TK 800.

EOS for Adsorption in Cylindrical Mesopores Langmuir, Vol. 15, No. 10, 1999 3635

(R), the thicker the adsorbed film. This is consistent withthe prediction of Broekhoff and De Boer.7

The above results can be verified with the thickness ofadsorbed nitrogen film in the model adsorbents. Toestimate the thickness of an adsorbed nitrogen film inpores of MCM-41 samples, De Boer’s t-plot method 25 wereused. t is the statistical thickness of nitrogen film adsorbedon a nonporous solid. In this study, we use the data of thenonporous hydroxylated silica, TK-800.26 The adsorptionamount on a sample, Va, is plotted as a function of t at thesame relative pressure (Figure 3).

Three linear stages are observed in the t-plot and theassociated adsorption regimes were discussed in detail inthe previous work.27 The porosity of MCM-41 samplesconsists of two parts: uniform framework-confined chan-nels and nonuniform interparticle voids.28 From the slopeof the last plateau section, the surface area from non-uniform interparticle voids Se is calculated. Then, theadsorption over the surface area Se can be estimatedaccording to the adsorption amount on the nonporous solid,TK 800, by

where Ae and ATK are the adsorption amounts on theinterparticle void surface of a MCM 41 sample and thenonporous reference, respectively, and STK is the BETsurface area of TK 800, being 164 m2/g. An impliedassumption in eq 19 is that the adsorbed film on theinterparticle void surface is the same as that on thenonporous reference surface. Since the interparticle voidsare macropores,27 such an assumption should not causesubstantial error. The amount of nitrogen adsorbed byuniform framework-confined mesopores is then calculatedby deducting Ae from the overall adsorption measured atthe same P/P0. With known pore radius R, one can convertthis amount to the thickness of the adsorbed film on the

pore wall. The radius obtained from X-ray diffractionmeasurements can be used to derive the film thicknesstpe, as

In Figure 2, the thickness tpe thus obtained (symbols) forfour MCM-41 samples is compared well to that obtainedfrom eq 18 (lines). Since we are interested in the multilayeradsorption prior to capillary condensation, only data abovea thickness of 3.64 Å (the kinetic diameter of nitrogenmolecules) until capillary condensation are illustrated inFigure 2.

The data from MCM-41 samples are of the samecharacteristics as those predicted using eq 18 and thecorresponding R. The prediction based on eq 18 isconsistent with the results from MCM-41 samples. Asteeper increase of tpe at the end of high P/P0 is observedfor the curves of the model adsorbents, indicating thatcapillary condensation has occurred in the pores. Thelowest line in the figure is for the reference TK 800. Thechemical potential of the adsorbed phase over a flat surfaceis wholly due to the adsorption potential u(t). For theadsorbed layer inside the pore, however, the curvature ofthe vapor-adsorbed interface has an additional contribu-tion (the second term on the right-hand side of eqs 13-18)to lowering the chemical potential of the adsorbed phase.Therefore, if in equilibrium with a vapor at the samerelative pressure, the adsorbed film on a curved solidsurface is thicker than that on a flat surface of the samenature. Moreover, continuous thickening of the film insidea cylindrical pore with increasing P/P0 will lead conse-quently to the loss of thermodynamic stability of theadsorbed film.

Thermodynamic Stability of the Adsorbed Filmin Mesopore. To verify the thermodynamic stability ofthe adsorbed film, let us consider the transfer of a smallamount of material from the vapor to the adsorbed phaseat a fixed temperature, T, and pressure, P. If the transferleads to a decrease in the chemical potential, the adsorbedstate is unstable and spontaneous condensation will occuruntil there is no such decrease. This argument could beexpressed as a criteria of stability of the adsorbed film,as

It is evident that tp will vary proportionally to n, and thecriteria may be rewritten as

Combining eqs 2 and 4 and substituting them into eq 22,we have

According to eq 23, thickening of the adsorbed film affectsthe chemical potential of the adsorbed phase in two ways.First, it inevitably leads to an increase in the curvatureof the gas-adsorbed film interface, which lowers thepotential (more negative) as indicated by the second termof eq 23. Second, it is expected that the force emanating(28) Tanev, P. T.; Pinnavaia, T. J. Chem. Mater. 1996, 8, 2068.

Figure 3. The t-plot of an MCM-41 sample. The amount ofnitrogen adsorbed by the sample, Va, is plotted against t whichis calculated from the adsorption of the nonporous referenceTK 800 at the same relative pressure.

Ae ) ATK

Se

STK(19)

tpe ) R - xR2 -2VmesoR

SBET - Sext(20)

[∂[µ(tp) - µ(l)∂n ]T,P

g 0 (21)

[∂[µ(tp) - µ(l)∂tp

]T,P

g 0 (22)

[∂u(tp)∂tp

- ∂

∂tp( γcVhR - tp

)]T,P

g 0 (23)

3636 Langmuir, Vol. 15, No. 10, 1999 Zhu et al.

from the solid surface declines steeply as the distancefrom the solid surface (and thus tp) increases. Theadsorption potential term up will bring about an increaseto the chemical potential of the adsorbed phase. As tpincreases, the contribution from the second term on theleft-hand side of eq 23 will become more dominant thanthe first. Consequently, the film is thermodynamicallyunstable and capillary condensation occurs spontaneously,filling the pores. The condition 23 should be satisfied foradsorption prior to capillary condensation.

The transition from multilayer adsorption to capillarycondensation can be reflected on the curves of the thicknessof the adsorbed film against its chemical potential. Thedifference in the chemical potential of the adsorbed filmfrom that of pure bulk liquid nitrogen at standard state(P0 ) 1 atm) can be expressed as

where P is the pressure of the vapor at equilibrium withthe adsorbed film. Apparently, ln P/P0 is an indication of∆µfilm since the difference between them is only a constantfactor of R0T. In Figure 4, ln P/P0, as a function of the filmthickness tp, is shown. The curves in Figure 4 wereconstructed for the case where the thickness of the porewall is supposed to be infinite, using eq 18. The chemicalpotential of the adsorbed film increases as the thicknessof the adsorbed film increases and reaches a maximum.The dashed lines merely reflect the postmaximum trendfor the chemical potential of the adsorbed phase accordingto the calculation. It reveals that a further increase in thethickness will lead to a decrease in the chemical potentialand a spontaneous process, capillary condensation, sub-sequently takes place, filling the pore. The adsorptionsystem will follow the course illustrated by the solid lines.

As R increases, the maximum shifts obviously to largertp values. This is consistent with the experimentalobservation that capillary condensation in larger poresoccurs at higher P/P0. After the maximum is reached, aspontaneous condensation in the pores occurs and theregime of multilayer adsorption in the pores ends.

Effects of Pore Wall Thickness and Surface Ten-sion. The results calculated using eqs 16-18 and thoseobtained from the adsorption of model adsorbents arecompared in Figure 5.

Curve A in Figure 5a-d is calculated using eq 16, curveB using eq 17, and curve C using eq 18. The symbols aredata obtained from adsorption on MCM-41 adsorbents.The trends and shapes of curves A and B are similar. Thisis because the contribution in the interaction potentialfrom the pore wall is mainly from the first layer adjacentto the adsorbed film. The adsorption force is a short-rangeinteraction (described by the term of distance with a powerof -6). However, the assumption of a single layer of solidatoms seems to result in a larger deviation. With theDefay’s effect taken into account, the results of eq 18 areapparently close to the real situation shown by dataobtained from N2 adsorption of the model adsorbents.Compared to eq 17, the contribution from the second termon the right-hand side of eq 18 is greater. In particular,when the pore radius is small, Defay’s effect is so importantthat the calculated curves deviate substantially from thereal situation if it is neglected, as can be seen in Figure5. As the pore radius increases, the difference betweencurves B and C decreases. For the smallest R, the increasein surface tension strongly affects the chemical potentialof the adsorbed film. The prediction of Defay is stronglysupported by the data of the model adsorbents, asillustrated in Figure 5.

The relation between the Condensation VaporP/P0 and Pore Radius R. The thickness correspondingto the maximum in Figures 4 and 5 is the critical thickness,tpc. Let the derivative, with respect to tp, of eq 15 (or itsderivatives, eq The relation of tpc and R for mesopores canbe obtained. When the values of tpc and R are substitutedinto eq 15, the P/P0 at which capillary condensation occursin the pores of R can be predicted. Experimentally, thecapillary condensation will bring about a steep increasein adsorption. Particularly, in samples of uniform porestructures, such as MCM-41, capillary condensation occursin all uniform-framework mesopores at almost the samerelative pressure. The increase in adsorption can beobserved clearly on their isotherms or even more clearlyon their comparison plots (t- and Rs-plots).25,29 Therefore,we can compare the predicted condensation P/P0 to thoseobserved on the isotherms of the model adsorbents. Therelationship between the pore radius R and the relativepressure at which the pore is filled is illustrated in Figure6.

The denotation of the curves is the same as that inFigure 5. The solid line, C, is obtained according to eq 18,line B, according to eq 17, and line A, to eq 16. The symbolsrepresent the experimental data of MCM-41 samples. Rvalues of these samples were obtained from X-ray dif-fraction measurements and the P/P0 from their t-plot. Itis evident that the calculated results based on eq 18 bestfit the experimental data. The assumptions of a singlelayer of solid atoms and constant surface tension appearto result in overestimation of the vapor pressure at whichcapillary condensation occurs in the pore radius.

Figure 6 also suggests an important application of eq18 in practice: estimating pore radius R of a sampledirectly from the condensation pressure P/P0 observedon its adsorption isotherm. In conventional methods ofcalculating pore size using adsorption data, the mean

(29) Sing, K. S. W. In Surface Area Determination, Proceedings of theInternational Symposium on Surface Area Determination, Universityof Bristol, U.K., July 16-18, 1969; Everett, D. H., Ottewill, R. H., Eds.;Butterworths: London, 1970.

Figure 4. Chemical potential of the adsorbed phase as afunction of the thickness of the adsorbed film in mesopores ofvarious R. They were derived from eq 18 for thick pore walls.

∆µfilm ) R0T ln P/P0 (24)

EOS for Adsorption in Cylindrical Mesopores Langmuir, Vol. 15, No. 10, 1999 3637

thickness of the adsorbed film on a nonporous solid of asurface nature similar to the sample under test, t, isusually used to make the correction for the adsorbedfilm.2,30 As discussed above, this mean thickness is smallerthan that in the pore because of the curvature. Therefore,conventional methods may underestimate the pore size.

Constant k6. Constant k6, before I6 in the expressionof the interaction term u(tp), has two important meanings.First, it is indicative of interaction between the adsorbateand adsorbent since it contains parameters related to themolecular properties of the adsorbate and the propertiesof the species in adsorbent, such as σ0 and εgs. For thesame adsorbent, k6/kT will be greater for the adsorbatewith a larger molecular entity since σ0 is larger, indicatinga stronger interaction between the adsorbate and adsor-bent. εgs is related to the polarity of the interactingspecies.31-34 Adsorbate molecules of a high polarity gener-ally have a stronger interaction with the adsorbent. Asthe adsorption temperature T increases, k6/kT decreasesand the interaction weakens. Second, it is independent ofthe pore geometry. This feature provides, in principle, an

opportunity to derive the constant term from adsorptionon a flat surface of a solid. Consider the adsorption on aflat surface, the difference between the chemical potentialfor the adsorbed phase and the bulk liquid phase of theadsorbate is due to the interaction between molecules inthe adsorbed phase and the solid. Everett and Powl havegiven such an interaction potential,35 which contains theproduct of the parameters ncσ06 εgs/(kT). Therefore, wecan estimate this product by fitting adsorption data of aknown nonporous solid to the following equation:

The adsorption data of nitrogen on nonporous silica TK-800 was fitted into eq 25. However, the value of k6,T thusobtained is 82, being about twice the value of 36 whichgives the best fit for all model adsorbents used.

One may suggest derivation of the values of nc, σ0, andεgs from physical parameters of the interacting species fora given adsorption system. However, there are uncertain-ties in the data thus obtained. Discrepancy in theparameters, such as polarities, magnetic susceptibilities,densities, and diameters of the interacting species, havebeen reported in the literature.32,33

(30) Dollimore, D.; Heal, G. R. J. Colloid Interface Sci. 1970, 33, 508.(31) Walker, P. L., Jr., Chemistry and Physics of Carbon; Walker, P.

L., Jr., Ed.; A Series of Advances; Marcel Dekker: New York 1966; Vol.2.

(32) Saito, A.; Foley, H. C. AIChE J. 1991, 37 (3), 429.(33) Baksh, M. S. A.; Yang, R. T. AIChE J. 1991, 37 (6) 923.(34) Baksh, M. S. A.; Yang, R. T. AIChE. J. 1992, 38 (9), 1357.

(35) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1 1976,72, 619.

Figure 5. Comparison of the relations between ln P/P0 and tp. Curve A is calculated using eq 16, curve B, using eq 17, and curveC using eq 18. Symbols are from the data of adsorption on MCM-41 samples. (a) for the case R ) 11.9 Å; (b) for R ) 15.3 Å; (c)for R ) 18.2 Å; (d) for R ) 22.4 Å.

ln P/P0 ) -2πncεgsσ0

6

kT t3(25)

3638 Langmuir, Vol. 15, No. 10, 1999 Zhu et al.

There are a number of reasons as to the cause of thediscrepancy between the k6,T value of 82 and the best-fitvalue of 36. We have found that the surface of the porewall of the model adsorbents is heterogeneous.36 However,it is very difficult to take the surface heterogeneity intoour calculation. Besides, in theoretical analysis of adsorp-tion on oxides, the adsorbent is assumed to be an assemblyof oxide ions.32,33 The function of countercations of oxygenanions in real solids is usually not specified to avoidintractable complexity in relevant calculations. Since theinteraction between each adsorbate molecule and eachspecies in the solid is generally assumed to be additive,it is possible to describe the interaction between theadsorbate and silica adsorbent with two terms. One termis for interaction with all the oxygen species and anotheris the interaction with the silicon species. Both terms canbe expressed in a form similar to that of the first term onthe right-hand side of eq 18. Nevertheless, the numberdensity of silicon species is half the density of the oxygenspecies. Compared to the assembly of oxygen species, theassembly of silicon species is shifted away from theadsorbed phase by about 0.85 Å (with a Si-O bond lengthof 1.62 Å). The k6 constant in each term then depends onlyon the properties of the corresponding species.

where k6,O and k6,Si are constants for the assemblies ofoxygen and silicon species, respectively. In principle, suchan approach appears more reasonable compared to thatgiven by eq 18. However, great difficulty arises fromdetermining the parameters such as εgs and σ0, for thespecies of oxygen and silicon in this approach. Moreover,the critical thickness, tpc, in small mesopores corresponds

to a thickness below two layers of the adsorbate molecules.Under this circumstance, the adsorption force of the porewall may still affect the surface tension.

Discrepancies between the theoretical analysis and theadsorption of the model adsorbents may arise from suchsimplifications. Moreover, continuum approaches are usedin the theoretical analysis, for instance, the integralintroduced to replace summation and using surfacetension, while the real systems are discrete on a molecularlevel. The approximate approaches result in simplifica-tions of the sophisticate situation. On the other hand, thevalidity of the analysis could also be affected because oftheir deviation from the real system. However, the resultswe obtained do indicate clearly the influences of theimportant factors, as shown above. The relationships suchas those in eq 15 and its derivatives provide a compre-hensive understanding of multilayer adsorption in me-sopores.

ConclusionThe main factors influencing the chemical potential of

the adsorbed film inside small cylindrical mesoporesinclude the attractive interaction between the adsorbedspecies and adsorbent, the curvature of the gas/adsorbed-phase interface, and the dependence of surface tension onthe interface curvature. The contribution from thesefactors have been taken into account in a pore-size-dependent state equation for the multilayer film inmesopores. According to the equation, at a given P/P0,the smaller the pore size, the thicker the adsorbed film.Thickening of the film brings about an increase in theinterface curvature, which causes decreases in the chemi-cal potential of the adsorbed phase. Consequently, thefilm becomes thermodynamically unstable at a certainP/P0, leading to capillary condensation in the pores. Thesurface tension of the interface strongly depends on theradius of the curvature when the pore radius is small (inan order of nanometers), as predicted by Defay.14 Theproposed theory is well-supported by nitrogen adsorptiondata of the model adsorbents, MCM-41 samples. Apartfrom providing a comprehensive description of multilayeradsorption in cylindrical mesopores, the proposed equationcan also be used to quickly determine the sizes ofcylindrical pores from adsorption isotherms.

Appendix: Analytical Expressions of Ii

Ii is expressed as

V is the solid region; in rectangular coordinates (x, y, z),it can be shown as follows:

Supposing the coordinate of any solid point M is (x, y, z),the coordinate of the adsorbate molecule M0 is (a, 0, 0) (0< a < R) (Figure 7), we have

Transforming rectangular coordinates (x, y, z) into cy-lindrical coordinates (r, θ, z), one has

and(36) Zhao, X. S.; Lu, G. Q.; Whittaker, A. K.; Millar, G. J.; Zhu, H.

Y. J. Phys. Chem., B 1997, 101, 6525.

Figure 6. Relationship between the pore radius R and therelative condensation pressure (curves have the same denota-tion as those in Figure 5).

ln P/P0 ) - 1kT

(k6,OI6,O + k6,SiI6,Si) -

4.7R - tp - 1.486

(26)

Ii ) ∫∫∫V

F-1 dV (i ) 6, 12) (A-1)

V: x2 + y2 g R2 - ∞ < z < +∞ (A-2)

F ) x(x - a)2 + y2 + z2

x ) r cos θ, y ) r sin θ, z ) z (A-3)

EOS for Adsorption in Cylindrical Mesopores Langmuir, Vol. 15, No. 10, 1999 3639

therefore,

Since 0 < a < R, we have |(2ar/(r2 + a2)) cos θ| < 1 (R er < +∞, 0 e θ e 2π). Therefore, we can write

Noting ∫02π cos2k-1 θ dθ ) 0, and ∫0

2π cos2k θ dθ )(2π(2k-1)!!/(2k)!!, (k ) 1, 2, 3, ...), where (2k - 1)!! )(2k - 1)(2k - 3)‚‚‚3‚1, (2k)!! ) (2k)(2k - 2)‚‚‚4‚2, we obtain

Substituting eq A-7 into eq A-5 yields

where

(i, k are any positive integers);

therefore, we have

where

Substituting eq A-9 into eq A-8 yields

where

Since 0 < a < R, J0,n can be given as follows

where

Substituting eq A-11 into eq A-10, we get

where

Equation A-12 is an absolutely convergent power series(0 < a < R). In practice, we can take the partial sum ofthe series as the value of I6. The coefficients of eq A-12 areshown in Table 1 from n ) 0 to n ) 11.

In the case where the pore wall consists of a single layerof atoms, it is not necessary to integrate in the r direction.

Figure 7. Coordinate of an adsorbate molecule on the interfacein a cylindrical pore.

V: 0 e θ e 2π, R e r < +∞, - ∞ < z < +∞ (A-4)

I6 ) ∫∫∫V

F-6 dV )

∫0

2πdθ∫R

+∞rdr∫-∞

+∞(r2 + a2 - 2ar cos θ + z2)-3 dz

) 38

π∫R

+∞ r dr(r2 + a2)5/2

dr∫0

2π(1 - 2arr2 + a2

cos θ)-5/2dθ

(A-5)

(1 -2ar

r2 + a2cos θ)-5/2

)

1 + ∑n)1

∞ (5 + 2n - 2)!!

3[(2n)!] ( ar

r2 + a2)n

cosn θ (A-6)

∫0

2π(1 -2ar

r2 + a2cos θ)-5/2

dθ )

2π[1 +2

3π∑

n)1

∞ (5 + 4n - 2)!!

[(2n)!!]2 ( ar

r2 + a2)2n] (A-7)

I6 )3

4π2 ∑

n)0

∞

bna2n Jn,2n (A-8)

Ji,k ) ∫R

+∞ r2i+1 dr(r2 + a2)k+5/2

)

∫R

+∞ r2i+1 + r2i-1a2 - r2i-1a2

(r2 + a2)k+5/2dr ) Ji-1,k-1 - a2 Ji-1,k

bn ) 2π3

(5 + 4n - 2)!!

[(2n)!!]2(n ) 1, 2, 3, ...), b0 ) 1

Jn,2n ) ∑i)0

n

(-1)i(ai )a2i J0,n+i (n ) 0, 1, 2, ...)

(A-9)

(ni )) n!i!.(n - i)!

and 0! ) 1

I6 )3

4π2 ∑

n)0

∞

cna2n J0,n (A-10)

c2k-1 ) (2k - 10 )b2k-1 - (2k - 2

1 )b2k-2 + ‚‚‚ +

(-1)k-1(kk - 1 )bk

c2k ) (2k0 )b2k - (2k - 1

1 )b2k-1 + ‚‚‚ +

(-1)k(kk )bk (k ) 1,2,3, ...)

J0,n )

∫R

+∞ r dr(r2 + a2)n+5/2

) 12n + 3

1(R2 + a2)n+3/2

(n ) 0, 1, 2, ...)

J0,n )1

2n + 3

1

(R2 + a2)n+3/2)

1

2n + 3

1

R2n+3∑j)0

∞

dn,j(a2

R2)j

(A-11)

dn,J )

(-1)j (n + 3/2)(n + 1 + 3/2)‚‚‚(n + j - 1 + 3/2)j!

(j ) 1, 2, 3, ...); dn,0 ) 1

I6 ) ∑n)0

∞

en(a2

R2)n

(A-12)

en )

3π2

4R3( dn,0cn

2n + 3+

dn-1,1cn-1

2(n - 1) + 3+ ‚‚‚ +

d1,n-1c1

5+

d0,nc0

3 )(n ) 0, 1, 2, ...)

3640 Langmuir, Vol. 15, No. 10, 1999 Zhu et al.

Therefore, we can obtain

where

Substituting J 0,n/ into eq A-13 yields

where

Similarly, we can calculate I12. However, the term witha power of 12 is negligible except for the situation wherethe interacting species are in very short distance.

Acknowledgment. Financial Support from the Aus-tralian Research Council (ARC) is gratefully acknowledged.

LA981515V

I6 )3

4π2 ∑

n)0

∞

cna2n J0,n/

J0,n/ )

1

R2n+4∑j)0

∞

dn,j/ (a2

R2)j

(n ) 0, 1, 2, ...) (A-13)

dn,j/ )

(-1)j (n + 5/2)(n + 1 + 5/2)‚‚‚(n + j - 1 + 5/2)j!

(j ) 1, 2, 3, ...), dn,0/ ) 1

I6,R ) ∑n)0

∞

en/(a2

R2)n

(A-14)

en/ )

3π2

4R4(dn,0

/ cn + dn,1,1/ cn-1 + ‚‚‚ + d1,n-1

/ c1 + d0,n/ c0)

(n ) 0, 1, 2, ...)

EOS for Adsorption in Cylindrical Mesopores Langmuir, Vol. 15, No. 10, 1999 3641