Molecular Theory of Adsorption in Pore SpacesPart 2.-Thermodynamic and Molecular Lattice Model Descriptionsof Capillary CondensationBY DAVID NICHOLSONPhysical Chemistry Laboratories, Department of Chemistry,Imperial College, London SW7 2AYReceived 3rd March, 1975A new thermodynamic analysis, which avoids some of the limitations of previous treatments,is made of the process of adsorption and filling in cylinders. Expressions for the surface tensionare derived for a molecular lattice model and relationships with the Gibbs and Frenkel, Halsey,Hill equation examined. The special nature of the latter equation is discussed and compared withthe lattice model. Results for a nitrogen-like system at 77.4 K are compared with predictions fromvarious thermodynamic equations.It is found that aq adsorption cgrrectim to the Cohan equationis necessary for the best agreement but that none of the thermodynamic expressions tested were ableto predict the molecular results.There is considerable uncertainty concerning the part played by adsorption forcesin capillary condensation in mesopores. Nevertheless extensive use is made of thisphenomenon in pore size distribution analysis where a simplified picture of isolatedcylinders is often ad0pted.l A common procedure is to employ either the Kelvinequation for desorption or the Cohan equation for adsorption, corrected for adsorbatethickness. Broekhoff and de Boer showed that the Cohan equation, which appliesto a film of adsorbate on the cylinder wall, does not correspond to thermodynamicallystable conditions and they introduced an additional term which restores this stabilityand which depends on the adsorbent field.The derivation of this corrected equationhas been criticised however and the correction has been largely ignored in inakingpore size distribution analyses.Apart from the question of the rale played by the adsorbent field, there is additionaluncertainty because the bulk fluid approximations used in thermodynamic treatmentsmay not be applicable to systems of small radius or to thin films of adsorbate. Theselimitations are expected to apply outside the micropore regime, i.e., even whenmesopore behaviour (adsorption plus capillary condensation) is found.An evaluation of the problem is possible in terms of the lattice model developedin Part 1 and enables some insight to be gained into the process of adsorption incylinders.In this paper calculations based on the lattice model are compared withthe thermodynamic approach using as a reference point the relative pressure at whichcondensation occurs.EQUATIONS BASER ON THERMODYNAMICSPLANAR ADSORBENTConsider a planar adsorbent of area A in the presence of X molecules of adsorbatecontained in a system of volume V (bounded by the adsorbent as one face), entropy S,230 DESCRIPTIONS OF CAPILLARY CONDENSATIONtemperature T and chemical potential p. The total differential of the internal energyU isdU = TdS-p dV+y dA+p dX. (1)The adsorbent is assumed to be unperturbed by the adsorbate. y can be definedin terms of the diagonal components of the pressure tensor bywhere the z axis is perpendicular to the surface and z = h is in the homogeneous gasphase at a great distance from the adsorbent surface.The tangential component ofthe pressure tensor pT is equal to pxx = pry and the normal component p equals pzz.For the adsorptive interface at T, pzz = p o , the saturation vapour pressure, and ybecomesS hY o = 1 [Po-P;(z)l dz. (3)LR = U+pV-TS-yA (4)(5)- hA free energy C2 can be defined byand the Maxwell relations derived from (4) lead to a total differential for the chemicalpotentialwhere the bars denote partial derivatives with respect to X at constant T, p , y. At alarge distance from the surface in the homogeneous gas phasedp = - S dT+ Vdp-A dydp = - S , dT+ V, dp.(6)Eqn (5) at constant temperature is the Gibbs isothermAd?- Vdp = -dp. (7)A more familiar form of this equation is obtained by introducing (6) with v 4 7,and an ideal gas phase assumption to giveA dy = k,Td lnp. (8)WPIPo) = -Pf(t,), (9)A dy = -flf’(t,) dt,. (10)The adsorption isotherm can be expressed in terms of the adsorbate thickness onthe plane surface t, asin whicb p/po is the relative pressure, p = l / k J and tp and the function f will bediscussed more .fully below. Eqn (9) with (8) givesThe spreading pressure # ( X ) can be found from (7), (8) or (10) by integratingbetween limits which correspond to X = 0 and X . Alternatively integration of (8)from X to X = oc) at p = po givesBA(y -yb) = pv@ -PO) - 1 In (pko)? (1 1)where the ideal gas condition has been used with eqn (6), A = U / d X = A/X andAh = V.In eqn (1 1) yb applies to the solid covered with a Duplex film of adsorbatemany molecular diameters thick andwith ysL the interfacial tension between adsorbent and adsorbate.Y b = YO+YSL (12D. NICHOLSON 31ANNULAR ADSORBATE IN A CYLINDERHere the adsorbate density decreases radially towards the centre of the cylinder,the pressure in the radial direction is a function of radius r. For a bulk fluidapproximation the pressure can be divided into two parts :dense phase p(r) = ps = constant ; r (rp- t )rarefied phase p(r) = pa = constant ; r < (rp - t ) (1 3)where rp is the radius of the cylinder and t the thickness of the adsorbate on thecylinder walls; as will be seen below t can be determined in more than one way.Eqn (1) is nowwhere R1, R2 are principal radii of curvature and C1, C2 the conjugate potentials;for an annular fluid R1 = - r and C2 dR2 = 0.Eqn (14) leads toand y is now dependent both on position and curvature of the interface. It is usualto choose y to have its minimum value with respect to r which leads to the Laplaceequation; in the present context this iswhere ym is not necessarily identified with y defined for the plane surface. The negativesign appears in (16) because the area of the cc-b interface decreases with increasingvolume.The difference between the chemical potential of the adsorbate film at ps and thesame phase with a plane surface under saturated vapour pressure ps is expressed bythe conditionIf phase a is an ideal gas and vp is constant for an incompressible dense phase,eqn (17) givesd U = TdS-pSdVg-p,dV,+YdA+Cl dRi+CzdR2+pd-Y (14)dp = -S dT+ Vj dpp+ V, dpa-A dy-C dr (1 5)(PB -Pa) = - ~ m / ( r p - t), (16)Pa) - PO) = P ~ T , PP) - P~(T, PO).(17)where the final equality is obtained by introducing eqn (1 6).Eqn (1 8) would be the Kelvin equation for the adsorbate film in a cylinder, if thefirst term on the right-hand side was neglected. Melrose has given a detaileddiscussion of the importance of this term (usually small) and of the consequencesof neglecting the compressibility of P (which can be significant).The foregoing discussion emphasises the fact that both ym and t involve someuncertainties in the present context.Even if the curvature dependence of ym isneglected it must be expected to vary with the thickness according to eqn (8)-(10).For example the left-hand side of eqn (9) can be integrated between y& and ymgivingand A for a film of thickness t in the cylinder is given byaA at av, 32 DESCRIPTIONS OF CAPILLARY CONDENSATIONEqn (19) with (20) giveswhich, after substitution in (1 8) and rearrangement givesTHE SLAB THEORY OR FHH EQUATIOKThe Frenkel, Halsey, Hill (FHH) equation can be writtenIn(p/pO) = -KF/lS,in which KF is a positive constant. The equation can be derived both from thermo-dynamic reasoning and from a perturbation theory * to give a value of s = 3.0.Several experimental investigations have indicated that s is less than 3 for realadsorbents and this diminution is attributed to surface heterogeneity effects.A rnodi-fied form of eqn (23), obtained by fitting experimental isotherm data, has been usedby de Boer and co-workers.2The FHH equation is of special interest here because the adsorbate is treated asa slab of fluid in the field of the adsorbent. This description is more likely to becorrect for thick films of adsorbate (t > 2 layers) and accords rather well with theassumptions leading to the derivation of eqn (22). In particular the value of y foran infinitely thick film will simply be yo and eqn (22) becomesThis equation would be the same as that derived by Broekhoff and de Boer if thesecond term on the left-hand side was neglected and the second term on the right-handside was replaced by their t-curve.The equation also carries the implication that onlythe adsorbate/vapour interface is relevant to capillary condensation.When the second terms on both sides of eqn (24) are neglected the remainingexpression is the Cohan equation.To conclude this section a derivation of the FHH equation from the lattice modelof Part 1 is given. A comparison with calculated isotherms is made in the Resultsand Discussion section.In the slab theory the adsorbate consists of iz layers with fractional occupationequal to that of the reference state. When this condition is imposed on eqn (17)of Part 1 the entropy term and adsorbate interaction term vanish leavingIn (PlPo) -C E mi JT- (25)The adsorbent-adsorbate interaction function mi = - I/$ for a planar continuumadsorbent and inverse sixth power attraction between individual atoms.The distancezi of the ith layer from the surface is measured in units of the adsorbate moleculardiameter ro and to a close appr~ximation,~ zi = i-3, thus (25) isThe number of layers n is related to the thickness t, by 6’a = t, where 6’ is the(= 0,816 5 for a cubic close packed lattice). thickness of each layer in ro unitD. NICHOLSON 33After carrying out the summation in eqn (26), rearranging and substituting for n theslab theory result is obtained :The constant 4.35 in eqn (27) is for a cubic close packed lattice and E is definedas 2E(Zo)/3kB where E(zo) is the potential energy minimum.MOLECULAR EQUATIONS BASED ON THE LATTICE MODELPLANAR SYSTEMSIn the lattice model the surface tension can be derived by several routes; onewhich is convenient and direct is to integrate eqn (1) to givewhere Z is the grand partition function,and for the most probable occupation set chosen from the sum over all occupationswhere Q is the canonical partition function given by eqn (9) and (1 1) of Part 1 andj?p is given by eqn (12) of Part 1.When these equations are substituted into (30)the result for a planar lattice of k layers isP(U-TS-pN) = P(-pV-yA) = -In= (28)/ M y = -lnZ+flpV (29)I n 3 = lnQ+XPp (30)where M is the number of sites in a layer ; @& is the adsorbate interaction term andE* is the well depth of the 6-12 interaction.The equation of state for the reference fluid is given by eqn (18) of Part 1 whichfor a lattice with k layers can be writtenwhere 8 is the fractional occupation of the lattice for a bulk homogeneous referencestate at (po,T).For the adsorbate a similar expression describes the gas phase at a large distancefrom the surface, in this case p o is replaced by p and 6 by eg, the fractional occupationof the gas phase at (p,T).In eqn (32) the term a is given byppo V = Mk[ - ln (1 - 0) + 36’a-j (32)and by expanding the interaction term in (31) it can be shown that this becomes #12awhen = 8, = 8 and k is large compared with the range of the interaction. Anexpression for yo is now obtained from (31) and (32) as a sum of contributions fromeach layer.where 6 is the area per m01ecule.~ A comparison with eqn (3) shows that for thelattice model Asp+(z) dz is replaced by the sum over i in eqn (31).Eqn (31)can therefore be interpreted as a sum of pV contributions, one from each layer.When Og replaces 8 and (0,’) for adsorbate in contact with adsorbent is used,eqn (33) gives y.1-34 DESCRIPTIONS OF CAPILLARY CONDENSATIONCYLINDRICAL SYSTEMSFor adsorbate in a cylinder eqn (31) becomeswhere the (Oil is now a solution for the lattice model equations in cylindrical geo-r n e t r ~ . ~ The external gas phase equation of state for a fractional occupation eg isand V here is the cylinder volume. Eqn (34) and (35) can be substituted in (29) togive a lattice model equation for pAy, (where the subscript c emphasises that theresult is for cylindrical geometry).Clearly the product Aye is invariant to placementof the interface for a given cylinder.Some further insight into the link between the thermodynamic and molecularequations can be gained by writing pa, ps from eqn (13) as the mean values,P/? = P(l)pdr/Sm r,-2 r dr; pa = J::ot p(r)r dr/S?' 0 r dr. (36)It can then be shown, after some algebraic manipulation, that(37)2(rp - t>ri(Pa - Pi)t k2 Ym = c l < i < kwhere p i , given by flZpiV = In Z, can be found from (34). Although eqn (37) canbe considered as the cylindrical analogue of eqn (3) it cannot be evaluated withoutassigning values to t and pa. The latter is usually identified with the external gasphase pressure, but eqn (36) shows that pQ would equal this pressure only when thecylinder of radius rp- t lies totally within a region where the adsorbent field is zero.THICKNESS OF THE ADSORBATE LAYERSeveral definitions of thickness of the adsorbate layer are possible; three, whichare of particular interest, are considered here.A natural definition of thickness can be given in terms of the Gibbs excess withrespect to the homogeneous gas and dense phases pg, pz at (p,T).In generalised formthe defining equation iswhich can be solved for V(t) the volume enclosed by a surface at thickness t. Inplanar geometry eqn (38) giveswhere GM = E(Oi - Og) is the number of filled layers in the surface excess. It is worthnoting that neither (38) nor (39) implies that the adsorbate necessarily has the densitypz at any point.t, = sw,/(e-eg) (39)For an adsorbate in a cylinder eqn (38) can be solved to givet,, = rp 1- - [ (BgT;)+]where 0' is the fractional filling of the cylinder = XCxi/CMi for the lattice model.If it is assumed that an adsorbed rnonolayer fills to the same fraction 0 as thD. NICHOLSON 35I.C30.90.-(2.8s-0.70C.6 0dense reference state then eqn (39) defines the number of such layers in the surfaceexcess.A formal thickness for the adsorbate can be defined in an analogous way.An adsorbate thickness can also be found 2 * lo by solving eqn (24) and in particularthe value at which this equation predicts filling can be found from the conditiona(p/po)/at = 0 ; some further detail is given below, the result of this calculation givesa value of t = tE.--RESULTS AND DISCUSSIONIn this section the properties of eqn (24) are compared with results for the mole-cular theory based on the lattice model.Isotherms were calculated for spherical nitrogen-like molecules using the para-meters; r0 = 4.15, E* = 95 K, E = 750 K, T = 77.4 K and the range of radii2.0 < rp < 33.0 in ro units.The isotherms are similar to those obtained for Ar at80 K except that step structure is very much less evident. There is a two phaseco-existence region for Y, > 4.0. A plot of (p/p0)*, the relative pressure at which thephase transition ocurs, against rp is shown in fig. 1.*040 t G2.30 - I 0 . 2 dc\. I 00 -4- - .w.01 I I I IC 10 2 0 3 0rPFIG.1.-Relative prcssure at condenstion as a function of cylinder radius for a nitrogen-like adsor-bate at 77.4 I<.An isotherm for the same system with a plane surface was calculated and the dataused to evaluate y from the appropriate version of eqn (33) as explained in 3.1.A calculation of Byo for the reference fluid was also made. /?y is shown as a functionof p/po in fig. 2. It may seem surprising at first sight that y < 0, but analysis ofeqn (33) and the calculated (&> shows that a large negative contribution comes fromthe first one or two layers where Oi > (1 - Og). This means that these layers aredenser than the bulk reference fluid wouid be at (p,T) with a consequently largenegative contribution from the logarithmic (i.e., entropy) term in eqn (33).Th36 DESCRIPTIONS OF CAPILLARY CONDENSATIONmagnitude of this term very probably highlights the limitations of the model, never-theless the abnormally high density of the layers near to the wall is a realistic feature.It is clear from eqn (33) that non-zero contributions to y or yo can only come fromlayers in which 13, is not equal to a homogeneous bulk phase value. For the referencestate, zero contributions occur when Oi = I3 or 0; in the liquid or gas-like phases eitherside of the interface. For an adsorbate phase, contributions to y fall to zero at avalue of i which increases with p and for which Oi + Og. However when p is veryclose to p o a second region of zero contributions to y appears which separates thefirst few layers with Qi > 8, due to proximity of the adsorbent, from a region withthe structure of the reference state interface. This is illustrated in fig.3.15.0 --13.01 ' 4*0r(PIPO)FIG. 2.-Surface energy as a function of relative pressure for adsorption at a plane surface.iFIG. 3.-Adsorbate density profile over 40 layers at p/po = 1.00.In fig. 4, eqn (23) is tested for the plane surface results from the lattice model; anexcellent fit was found for 1.8 < OM < 6.0 but the value of s (= 2.87) was less than 3.A similar result was obtained for Ar-like molecules at 80 K with s = 2.88. In viewof the approximations necessary to derive eqn (27) it is not surprising that s # 3 butit is of interest that the result is not much higher than that found for experimentD.NICHOLSON 37isotherms and that surface heterogeneity effects do not occur here. The value ofthe constant KF (= 15.3) was likewise much lower than would be predicted fromeqn (27). An expression for y from eqn (23) can be found by substitution in eqn (8)and integration to givewhich predicts y > yo and positive. The apparent contradiction between this resultand y calculated for the lattice model (fig. 2) can be resolved by the conjecture thatan isotherm of the form of eqn (23) accounts only for the field effect on a film ofadsorbate caused by replacing the adsorbate fluid by adsorbent, but includes noallowance for the entropy change. This model certainly underlies all derivations ofeqn (27) and must be presumed to extend to the general empirical form (23).It might, , ,\ - 3.00 . 2 0.4 0 . 6 0 . 8 1.0 1.2 1.4 1.6 1.8la OMFIG. 4.-Test of the FHH equation for nitrogen-like adsorbate at 77.4 K.also be conjectured that eqn (41) is only relevant to the adsorbate/vapour interface[cf. eqn (12)] and is thus particularly suited to representf(t) as in eqn (24). At thesame time the foregoing discussion of y makes it clear that its components in eqn (12)are not distinguishable at p/po < 1 .O which adds to the difficulties of finding a reliablethermodynamic equation to describe adsorption in mesopores.The table compares lattice model calculations for the left-hand side of eqn (24)with various estimates based on the right-hand side38 DESCRIPTIONS OF CAPILLARY CONDENSATIONThe following parameters were used for these calculations : vb was equated withthe molar volume of the dense reference state (poO0)-l where po, the number densityof the filled C.C.P.lattice = J2 and 8 was found to be 0.949 6 for the energy para-meters &* = 95 K, T = 77.4 K. As in Part 1 the reduced temperature was used inthe calculations. From the equation of state, eqn (31), flVBpo = 0.045. The con-stant was found to be 0.508 from eqn (33) (about half the experimental valuefor N2 at 77.4 K). Together with the FHH isotherm parameters given in the preced-ing paragraph these results, inserted into eqn (24), giveln(p/po) + 0.045(1- p/po) = - [;;::) - +l:.:'i] -TABLE 1 .-THERMODYNAMIC AND LATTICE MODEL PREDICTIONS FOR THE RELATIVE PRESSURE OFCONDENSATION IN CYLINDERS'D4.134.955.766.587.398.219.851 1.4813.1114.7516.3819.6422.9126.1829.4432.71(A)W P b e ) '3- 0.045-p/pO)* tGC-1.270 2.22-0.911 2.59-0.715 2.89-0.528 3.30-0.467 3.39-0.385 3.59-0.299 4.00-0.219 4.4-0.170 4.7-0.145 4.9-0.133 5.2-0.103 5.6-0.087 6.0-0.075 6.3-0.053 6.6-0.043 6.8tE2.763.163.543.904.244.575.195.776.326.837.338.259.119.9210.6711.40(B)-0.SOSl(rp- t c 3- 0.266-0.215- 0.177-0.155-0.127-0.110- 0.087- 0.072- 0.060 - 0.052 - 0.045- 0.037- 0.030- 0.026- 0.022- 0.020(C)epn (40)wlth t = tGC- 0.752- 0.528- 0.405- 0.31 1- 0.271- 0.233-0.177-0.140-0.117-0.102- 0.087- 0.071- 0.065- 0.054-0.043-0.040(D)epn (40)- 0.63 1- 0.460-0.356- 0.286- 0.237- 0.201-0.152- 0.120- 0.099- 0.084- 0.072- 0.056 - 0.046- 0.038- 0.033- 0.040with t = tE (A)/@) (A)/(C) (A)/@)4.77 1.69 2.014.24 1.73 1.984.04 1.77 2.003.41 1.70 1.853.68 1.72 1.973.51 1.65 1.923.44 1.69 1.973.06 1.56 1.832.81 1.45 1.722.80 1.45 1.742.96 1.53 1.852.81 1.45 1.842.90 1.34 1.922.94 1.38 1.982.40 1.28 1.632.20 1.10 15.2Values of tE were found by iterative solution of the equation obtained from thecondition 8(p/po)/8t = 0; lo which, froni eqn (42) givesA zeroth approximation was obtained from the solution lo for s = 3 which with theparameters used here isDirect iteration of (43) is slow but is accelerated by setting t(') = +(t(f-1)+P-2)) andis easily performed on a modern desk calculating machine.The Gibbs thickness defined by eqn (40) was estimated at the transition value(p/po)* from the calculated isotherms, the uncertainty in each value can be quite largeas indicated in fig.5 ; the tabulated values were estimated from the best line as drawnin the figure.The ability of the thermodynamic equations (24) or (42) to predict the molecularlattice model results is tested by calculating ratios of the right-hand side to the left-hand side of eqn (42) ; the latter [(A) in the table] having been obtained from calculatedisothcrms. The poorest and most variable prediction is that given by the Cohan(43)(44)t(i+l) = 2.345 (r, - t(1))0-517,t ( O ) = 2.59 [(1+0.769 rp)'- 11I>.NICHOLSON 39equation (€3 in the table). The correction for adsorption makes a marked improve-ment with t = tcc (C) giving a slightly better result than t = tE (D). In all cases thereis a slight improvement towards higher rp but this may be only apparent due to un-certainties in (p/po)*.The results summarised in the last three columns of the table are consistent withthe view that an equation of the form of (24) is more suitable for the description ofadsorption in a cylinder than is the uncorrected Cohan equation. It has been arguedelsewhere lo that an analysis of adsorption branch data should be preferable todesorption branch analysis in networks because of freedom from the pore blocking01 t I I I 1 J0 10 20 3 0rPFIG. 5 . 4 i b b s thickness of the adsorbate prior to condensation as a function of pore radius.effects which contribute to hysteresis. The present work furthermore indicates thatit is unlikely that any thermodynamically based treatment can properly represent theadsorption process in the mesopore size range. However the lattice model itselfinvolves too much approximation to allow any quantitative as opposed to qualitativereliance to be placed on the results presented here.We thank Dr. J. H. Petropoulos for discussions, in particular of the thermodynamicsection of this work.S. J. Grcgg and K. S. W. Sing, Aclsorption, Swfme Area and Porosity (Academic Press, London,1967).J. C. P. Broekhoff and J. H. de Boer, J. Catalysis, 1967, 9, 8.J. C. Melrose, J. Colloid Interface Sci., 1972, 38, 312.D. Nicholson, J.C.S. Fwaday I, 1975,71,238.F. C. Goodrich, in Swface and Colloid Science, ed. E. Matijevic (Wiley, New York, 1971),vol. 1.D. H. Everett, Soc. Chem. I d . Monograph No. 25,1967, 157.J. C. Melrose, A.Z.Ch.E.J., 1966, 12,986.K. S. W. Sing in Colloid Science (Spec. Periodical Rep., Chemical Society, London, 1973),vol. 1.a W. A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974).lo D. Nicholson, Trans. Farahy SOC., 1968,64,3416