J. Phys. Chem. 1993,97,661-665
661
Adsorption in Zeolites, Dispersion Self-Energy, and Gaussian Curvature Zoltan Blum' Inorganic Chemistry 2, University of Lund,Chemical Center, P.O. Box 124,S-22100 Lund, Sweden Stephen T. Hyde and Barry W. Ninham Department of Applied Mathematics Australian National University, Institute of Advanced Studies, Research School of Physical Sciences, GPO Box 4,Canberra, ACT 2601,Australia Received: July 1, 1992;In Final Form: October 8, I992
Zeolite performance in various areas of chemistry is a much debated and largely unsettled issue. Interpretations based on traditional chemical reasoning have of late been challenged by the contention that the framework geometry per se enables the multifarious interactive sites, present in a typical zeolite framework, to focus and act cooperatively and thereby consummate the diverse chemical transformations which are observed. The present work clearly shows, from first principles, that such a conjecture is well founded and that the nonEuclidian geometry common to most zeolite frameworks is essential in the administration of dispersive forces.
Introduction In attempts to account for the extraordinary performance of zeolites in areas like catalysis, cracking, and adsorption, workers in the field have conventionally resorted to ad hoc effects such as superacid sites, ill-defined structural defects, or the occasional "hot spot". There is an altemativeviewpointthat associates zeolite phenomena with the non-Euclidian, Le., hyperbolic, geometry traced out by the surface of the atoms in these periodic open networks. Thisviewpoint has not been received with acclamation. Indeed the notion that heats of adsorption in zeolites are related to the product of polarizability and average Gaussian curvature seems well enough established experimentally.' Yet the proponents of what we might call the "curvature" argument have had to resort to what appear to be equally ill-defined assertions, e.g., that a molecule "feels" forces from a surface element and its integral curvature and that the totalvan der Waals force exerted by the various structural elements establishing the surface are "focused" into a force field that runs parallel to the original surface, and speakof a quasi-liquid state of aggregation within channeh2 We shall attempt to show that the champions of curvature have no need to apologize. The mathematical and physical basis for their theories was already established over 20 years ago through the concept of dispersion self-energy. That it has not yet permeated to the chemical literature is an example of the tyranny of disciplinary boundaries. Dispersion Self-Energy The dispersion self-energy of an atom or molecule is exactly analogous to the Born electrostatic self-energy of an ion and takes on the same order of magnitude in comparable situations. The theory, and applications, have been extensively developed elsewhere.3d The familiar dispersionor van der Waals interaction energies between two atoms or molecules or between a molecule and a surface are simply the change in dispersion self-energy due to the presence of another object; in exactly the same way that the Debye-Hiickel result for the free energy of interaction in an electrolyte solution is the change of the Born self-energy of the individual ions due to the presence of all the others. Similarly, the OnsagerSamaris result for the change in interfacial tension of water when a salt is dissolved in it is the change in Born selfenergy due to the presence of an interface. Confusion arises because most discussions of interaction dispersion energiesassume point molecules. Once it is recognized that molecules have size and shape, which can be defined, and 0022-3654/58/2097-0661$04.00/0
emerge from quantum mechanics in a semiclassical theory, the (infinite) self-energy of a point molecule becomes finite and depends on the geometry of a cavity in which the molecule finds itself. We omit technical discussions of how molecular size is characterized and refer to references for details. Suffice it to say that the response of an atom in its nth state due to an incident electric field of the form E(r,t) = E(k,w)ei('*-") (1) can be derived in terms of a polarization density Pn(r,o) where integration over all space gives the polarization developed by the atom centered at R where P,(r,o)
3 a,(r-R;w)
E(r,w)
= a,(r-R;u)
E(R,o)
(2)
The polarizability tensor an(r-R) is a peaked function with a peak at R and a range of the order of the atomic size. (Explicitly, it is the Fourier transform of the k-dependent polarizabilitytensor a,(k,w) in the nth state of the atom given by a,(k,w) =
where u for a single electron atom is the electron coordinate measured from the nucleus at R, (nldm)(mlue'I*/*(n)stands for the dyadic formed out of two vector matrix elements, and w, = (En - Em)/h).
For a point atom, the term in &y12 does not occur in the matrix element, so that a,(k,o) is independent of k. This leads to an(rR;o) becoming a delta function a(w)b(r-R). In any event, the detailed analysis shows that size, of an atom or molecule, can be taken account of through a well-defined polarizabilitytensor un(rR). The dispersion self-energy of an atomic system, which is the change in zero-point energy of the electromagnetic field due to its coupling with the atomic system can then be calculated from the secular equation for the perturbed frequencies of the field. If we assume for illustration that the atomic system is isotropic and that a ( r , w ) has the form a(r,w) = la(o)f(r) (4) where [denotes the unit tensor and wherefir) is a peaked function, 0 1993 American Chemical Society
Blum et al.
662 The Journal of Physical Chemistry, Vol. 97, No. 3, 1993
TABLE I: Physical Data Used in Plot of Figure 1 refractive index zeolite Y
silicalite
a
DroDane butHne pentane hexane Octane cyclohexane butane hexane benzene I-butene cyclopentane
1.2898 1.3326 1.3575 1.3751 1.3974 1.4266 1.3326 1.3751 1.5011 1.3962 1.4065
number density/102’ 6.833 I 5.9972 5.2267 4.6141 3.7035 5.5706 5.9972 4.6141 6.7568 6.3859 6.4025
a a ( ~x ) 1024
ionizn potential (ev)
w,/ 10‘6
M e $
(A-3)
(rad s-I)
(J
6.33 8.18 10.0 11.8 15.5 11.0 8.18 11.8 10.4 8.99 9.17
11.07 10.63 10.35 10.18 10.03 9.88 10.63 10.18 9.24 9.58 10.90
1.68 1.61 1.57 1.55 1.52 1.50 1.61 1.55 1.40 1.46 1.66
g-l)
14.8 19.0 23.4 27.4 35.4 25.9 27.4 36.5 28.6 25.6 29.0
From ref 1.
Lorenz formula can be expanded in eq 10 or 11, and we have
e.g., a Gaussian of width o
E,(vacuum) - ,??,(medium) = with a the radius of the atom, then we find for the self-energy
For the simple classical form a ( w ) = e2/m(uo2 - w2), eq 6 reduces to E, z he2/d/2a2mwo. If we make the identification that hwo is the ground-state energy (IEcl), we have hwo IEG) = e2/2ao, where a. is the Bohr radius for a hydrogen atom, and ifwetotakea = ao,weseethatE,= ( 2 / ~ ~ / ~ ) ( e ~=/ (a 4o /)d 2 ) R Y ; Le., the self-energy of an atom is of the same order of magnitude as the binding energy but of opposite sign. If we consider two atoms centered at RI, Rz, with R12 = IR2 - RII, the formalism gives
=
where .,(it) is the polarizability of the adsorbing medium. That is, theenthalpyof transfer is just the sumof the painviseinteraction energies of the adsorbed molecules with its neighbors. (Note, however, that pairwise summation will give quantitatively erroneous estimates. The collective interaction of the whole medium are built into the (in principle) measurable dielectric response function). In the open (pseudo-two-dimensional) network formed by the aluminum silicate framework of zeolites, the Lorentz-Lorenz formula for the frequency-dependent dielectric constant of the composite medium takes the form (derived in the Appendix)
Jw,,)
E = E,(1) + + (7) where the interaction energy V(R12)reduces to the usual van der Waals interaction energy
At small distances R12 reduces to
+
0, the energy does not diverge but
which is of the order of the binding energy of a molecule formed from the two atomic systems. The important result we note is the removal of the divergence in the van der Waals interaction as R12 0 due to the introduction of finite size. Effect of Media. If the adsorbate molecule or atom (“a”) is immersed in a medium of dielectric constant e,(@), instead of a vacuum, the dispersion self-energy is modified and now takes the form corresponding to eq 6
-.
The expressionimmediatelygives us an estimate of the enthalpy of adsorption in taking an atom from the gas (vacuum) state to the (compositemedium) zeolite (characterized by its measurable dielectric constant ez(w)). It is (exactly as the Born energy of an ion from vacuum to water) E,(vacuum) - E,(medium) =
Pairwise Interactions. If we like, we can view the process differently. For a medium of uniform density p z , the Lorentz-
per adsorbed molecule, where a is a typical dimension (radius) of the chemical group adsorbed into the zeolite (e.g., CH2 or CH3 for hydrocarbons), n, and p , denote the refractive index and number density of a silicate standard, p z is the number density of the zeolite, and wa and wz are the UV relaxation frequencies of the adsorbate and the zeolite, respectively. The applicability of this expression to the adsorption of organic molecules in zeolites can be directly tested. We have taken integrated adsorption enthalpies for hydrocarbon adsorption in the hydrophobic zeolites Y and silicalite, for adsorption up to 0.5 mmol g-I of zeolite. [This corresponds to less than a monolayer of adsorbate surface coverage-about one adsorbate molecule for every 30 T-atoms-consistent with our assumption that all adsorbed molecules interact directly with the zeolite framework.] The adsorbate polarizabilities and UV frequencies (assumed to be equal to the first ionization potential) have been calculated from standard data (CRC Handbook, 66th ed., 1985). These data are tabulated in Table I. The UV frequency for the zeolites is assumed to be the same as for fused quartz, 2.02 X 1016 rad and we have taken fused quartz as our refractive index and density standards: n, = 1.45, p s = 2.2. The density of dehydrated hydrophobic zeolite Y is equal to 1.35 g ~ m - while ~ , that for silicalite is 1.76 g ~ m - ~ .
The Journol of Physical Chemistry, Vol. 97, No. 3, 1993 663
Adsorption in Zeolites
TABLE II: Geometric Data for Zeolites Lying on Periodic Minimal Surfaces
O
Y
0
'
I
.
200
I
400
'
I
600
7
I
800
AH,$
'
1
loo0
zeolite/framework silicate
periodic minimal surface
Si/A1 ratio
silicalite zeolite Y gmelinite Linde A gismondine sodalite
genus 9 surfaceD D-surface (genus 3) H-surface (genus 3) P-surface (genus 3) T-surface (genus 3) P-surface (genus 3)
m
>3 2 1 1
1
Q (A21 12.2 12.3 13.9 15.2 15.5 15.8
This surface has yet to be explicitly parametrized.
A
(JA3g.l)
z axis
Figure 1. Calculated (eq 13) versus measured (ref 1) heats of adsorption on zeolite Y and silicalite. Hydrocarbons as per Table I.
Figure 1 plots our calculated heat of adsorptionvs the measured value. A single undetermined parameter remains in our equation: the effective 'radius" of a single CHJCH2 group, o in eq 13. A straight line fits the data well, and we estimate this dimension to be -2.9 A from the slope of the line. Clearly this is a plausible value. We conclude from this data that our model adequately accounts for adsorption energies in these zeolites, provided we restrict the adsorbed volume to submonolyer coverage of the zeolite framework. A more sophisticated model could be developed for higher adsorbed quantities, by accounting for the effectivepolarizability and UV frequency of zeolite together with adsorbed layers. Framework Curvature. Equation 1 3 is-to leading order-proportional to the framework density of the zeolite. Thomasson et al. have suggested that the adsorption enthalpy is related to the (Gaussian) curvature of the zeolite framework. Assume for now that the framework tessellates a (necessarily hyperbolic) surface. If each TO2 group occupies an area fl on the surface, and a single unit cell of the surface occupies an area A, the number density of the framework is given by p = A / Q V , where Vis the volume occupied by a unit cell of the framework. The area of a curved surface is related to the (mean) Gaussian curvature ( K ) , by JJYnil,,11Kda = ( K ) A . Further, if x denotes the Euler-Poincar6 characteristic of an unit cell of the surface (related to the surface topology or number of handles),I0we know thatJJunit,,IIKda = 2rx,sothatp = 2rlxl/l(K)lflV. Weintroduce a dimensionless global surface index, u, relating the surface area to the volume u = A / W . Further, we define the parameter C = which implies that VZ = A3/2aJxIC2, or V = 2rIxI/((K)11/2C. The framework density is thus related to the (mean) Gaussian curvature of the surface, the area per TO2 group and the global parameter C by
Following the observations of Mackay and Anders~on,*,~ numerous examples support the observation that many zeolite frameworks lie on, or close to, periodic minimal surfaces.I0 For such surfaces, we can derive a good estimate of the global parameters, C = 3/4." Further, the area occupied by a single TOz group on these surfaces appear to be constant (for a given Si/Alratio),varying between 12.3A2for pureSiO2 frameworks and 15.5 A2 for Si/AI zeolites. These values have been verified for a range of zeolites whose surface topology is known. These data are summarized in Table 11. The prefactors to I( K)J1/2in eq 14 are thus constant (for a fixed Si/AI ratio), so that the frameworkdensity scales with the square root of the Gaussian curvature. It can be seen from Table I that the frequency contributions to the adsorption energy vary little for hydrocarbons. It follows from eq 13that the adsorptionenergy for hydrocarbonsin different zeolites depends only on the polarizability of the hydrocarbon
-
-
-
pl2
Figure 2. Two molecules inside a planar slit.
and the curvature of the zeolite framework. Since the polarizability scales with the effective number of electrons contributing we have the remarkably simple to the dispersion interaction, form
where N is the effective number of electrons contributing to the dispersion interaction. The relation derived by Thomasson et a1.I requires modification of the curvature term. Note that the equation rests on the requirement that the aluminosilicate framework lies close to an "ideal" periodic minimal surface of perfect homogeneity, so that the surface to volume factor C in eq 14 is constant. Such a requirement is fulfilled by a range of zeolites, detailed in Table 11. Interactions within Zeolites. Self-Focusingand Quasi-Liquid States. Having dealt with the gross features of adsorption, we turn now to interactions within the zeolite medium. Here one can get some insights by considering the effects of boundaries and of the confinement of molecules in slits on the van der Waals interaction energies between them. These arevery different from those that we obtain in free space and are quite drastically modified. It will be seen to be reassuring that the theory of such interactions does indeed predict effects that are actually observed. Dispersion Forces Operating within Zeolites. One factor not taken into account in discussions of the peculiar adsorptive properties of zeolites-in the arguments on force field focusing and quasi-liquidityof adsorbate molecules-is the effect provided by the zeolitecavitystructureon thevan der Waals forcesbetween the adsorbed atoms or molecules. (At low adsorbate loadings it seems fairly clear what is going on, as discussed above.) The van der Waals interactions between molecules are modified by the presence of the zeolite bounding geometry. There are a number of different results that bear on the problem that havebeenderived. Consider two molecules in vacuum a distance Rlz apart. The potential of interaction is V(Rl2) = -c/R1z6, where cis given by eq 8. Now put them inside a planar slit of width 1 (Figure 2). The results quoted have been derived in ref 4. Suppose it is a conducting slit (zero tangential Weld at the surfaces). Then the following cases are germane:
Blum et al.
664 The Journal of Physical Chemistry, Vol. 97, No. 3, 1993 (i) plz/l >> 1, distance apart much greater than slit width:
The interaction energy is much weakened compared to the free space value. (ii) At very large distances, the retarded regime, the free space value goes over to
Appendix: Derivntion of the Adsorption Enthalpy in Hydrophobic Zeolites We use eq 11 of the main text, plus the Clausius-Mossotti relation for the zeolite framework, (e - l)/(c + 2) = ( 4 ~ / 3 ) p a , where e = n2.
where w, is the UV relaxation frequency of the zeolite, we obtain for the zeolite part of the adsorption enthalpy But, in the slit, the interaction is vastly enhanced. Here
(iii) When the molecules are close to one of the conducting plates the interaction is reduced by about 2/3 (assuming the other plate is far away). (iv) For dielectric surfaces, and it is not clear yet which surface boundary condition is appropriate, the interaction is much weakened. The reason is that the molecules interact not just with each other, but with their images. (The net result is to make the interaction more like an induced multipole-multipole (weaker) interaction.) (v) At very large distances,there will be an increased attraction. (vi) These effects will be even further enhanced in the zeolite topology. They will act to give very peculiar localized but still gaslike properties to a system that ought to condense to form a liquid in free space at such densities as exist inside the zeolites. The entropy is in any event much reduced because of the confined intersecting pseudo-one-dimensional geometry of the zeolite. In addition, the weakened interaction potentials between adsorbate molecules will tend to localize molecules at points of maximum Gaussian curvature. (vii) There are some very peculiar effects on the resonance energy. That is, if one molecule is in an excited state and the other is in the ground state, the interaction energy is different. It is basically the energy transferred from the one in the excited state to the one in the ground state. In free space this behaves as E(r,r) 0: 1/Rlz3(nonretarded) and at very large distances E(r,r) a cos (wo(R/c)/Rl2. This interaction is very much enhanced if the molecules are confined in slits or the zeolite geometry.j With a catenoid type focusing arrangement energy will be transferred and focused on a molecule in the cavity with very long range and extreme efficiency.
Conclusion We have verified, with substantial mathematical rigor, that what can be called host-guest interactions are distinctively dependent on the geometry of the interacting species. The contention that the true shape, as measured by K,will influence theinteractions should also hold true for finite hosts, i.e., molecules. Thus, an appreciation of the accurate shape of the host is obviously imperative,contrary to the common implementation of assuming that the shape or geometry of the host is to an extent superfluous and can be approximated with planes, cylinders, or, spheres. As it seems, the only provision to be made is to ascertain that the assembly is sterically tolerable. This is plainly inadequate, or at any rate constitutes a gross oversimplification. The dielectric constant e, of any host, convoluted with size and composition of the host in question, contains the information of its inclination to engage in interactions. Hence, irrespective of the chemical nature of the host, ranging from inorganic infinite crystals over proteins to organic molecules, the interaction will be governed by how is sampled by the guest, the sampling efficiency, and hence the profundity of the interaction, will be proportional to ((Khost)l’’2.
It will prove to be useful to express the zeolite refractive indices in terms of a standard silicate refractive index, rg. Since the polarizability of the silica network (a,)is essentially independent of the global framework geometry, we can express the zeolite indices as follows: 1 -Fpp, -t, + 2 ts-
+ 2 -- T p z a z
5-1 4T and e,
(A3)
so the refractive index functions are modified by the framework
density of the zeolite (assuming equal polarizabilities for our standard silicate and the zeolite):
Substituting eqs A2 and A4 into eq 11 of the main text gives EJvacuum) - EJzeolite) =
where ( ~ ~ (and 0 ) wa refer to the static polarizability and the UV relaxation frequency of the incoming adsorbate. The UV frequency has been rescaled for notational convenience to
Integrating (A5) gives
AE, =
Details of the validity of this procedure are discussed in detail in ref 4.
References and Notes (1) Thomasson, R.; Lidin, S.;Andersson, S. Angew. Chcm., Inf. Ed. Engl. 1987, 26, 1017.
Adsorption in Zeolites (2) Blum, Z.; Lidin, S.;Thomasson, R. J . Solid Srare Chem. 1988, 74, 353. ( 3 ) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976; p 71. (4) Mahanty, J.; Ninham, B. W. J . Phys. 1973,115,1140. (5) Mahanty, J.; Ninham, B. W. Phys. Len. 1973,434 495. (6) Chan, D.;Ninham, B. W. J . Chem. Soc., Faraday Trans. 2 1974, 70, 586.
The Journal of Physical Chemistry, Vol. 97, No. 3, 1993 665 (7) Hunter,R. J..Ed. FoundationsofColloidScfencel;OxfodUnivenity Press: Oxford, U.K., 1987; p 586. (8) Mackay, A. IUC Copenhagen Meeting, Poster, 1979. (9) Andersson, S. Angew. Chem., Inr. Ed. Engl. 1983, 22, 69. (10) Andersson, S.;Hyde, S.T.; Larsson, K.; Lidin, S.Chem. Reo. 1988, 88, 221. ( I 1) Hyde, S.T. J. Physique (Colhq.) 1990, C-7, 209.
