J. Phys. Chem. 1995, 99, 7089-7095
7089
Temperature Dependence of 129XeNMR of Xenon in Microporous Solids T. T. P. Cheung Phillips Research Center, Phillips Petroleum Company, Bartlesville, Oklahoma 74004 Received: December 22, 1994; In Final Form: February 14, 1995@
The temperature dependence of the 129XeNMR chemical shift of xenon trapped inside a micropore is shown to be sensitive to the shape of the potential experienced by the xenon atom within the pore. If the pore is about the same size as the xenon atom, the potential has a single minimum at the center region of the pore. This leads to an increasing xenon chemical shift with temperature. For larger pores, the potential has a hump at the center surrounded by a low-energy trough. Consequently, the xenon chemical shift decreases with increasing temperature. Numerical calculations of the chemical shift of a xenon atom trapped between two parallel layers are used to illustrate these points. It is shown that the reciprocal of the chemical shift varies linearly with the pore size when the pore is much larger than the xenon atom. Furthermore, the slope of the linear dependence increases with increasing temperature. This phenomenon is explained in terms of a squarewell potential model, which turns out to be quite useful in organizing 129XeNMR data of xenon in different types of zeolites and can be used to extract from experiments information like the strength and the range of the interaction between xenon and the wall of the pore.
I. Introduction The high sensitivity of the chemical shift of lZ9Xenuclear magnetic resonance (NMR) to the local electronic environment of the xenon atom has been well documented in the studies of microporous materials in the past decade (for general reviews, see refs 1 and 2). The variations in the 129Xechemical shift with different types of cations in zeolite^,^,^ the correlation of the chemical shift with the size of the micr~pore,~-'and the discontinuity in the slope of the chemical shift at the glass transition temperatures in polymers*,9are some examples. The 129XeNMR chemical shift at the limit of zero adsorption of xenon a, reflects solely the interaction between the xenon atom and the surface of the adsorbent. The interaction can be represented by a potential U(r), which obviously depends on the position of the xenon atom r in relation to the surface. The potential U(r) gives rise to a 129Xechemical shift u(r). In a confined volume like that in a micropore, u,, is given by the canonical ensemble average
that the model calculations should reproduce the results of the ab initio calculations, at least qualitatively. In this paper, the temperature dependence of u, is examined in terms of a model in which a(r)is proportional to the van der Waals polarization energy and U(r) is given by the sum of the Lennard-Jones potentials between a xenon atom and the adsorption surface. We compute the chemical shift for a xenon atom trapped between two parallel layers as a function of the temperature as well as the separation between the two layers. With the convention of a resonance downfield (less shielded) from that of an isolated xenon atom having a positive chemical shift, we shall show that the sign of the slope of a, as a function of T depends critically on the layer separation. When the separation is small, about the size of a xenon atom, duddT is positive. However, it is negative for large separations. In the regime of negative slopes, l/u, is shown to increase linearly with the layer separation.
11. Relation between the Shape of U(r) and the Temperature Dependence of a,(T) The surface of a micropore consists of atoms. The interaction of the adsorbed xenon with a surface atom is modeled by a painvise Lennard-Jones 6- 12 potential where Vis the volume of the pore, k the Boltzmann's constant, and T the temperature. In this paper, we investigate the temperature dependence of a0 and show how it is related to the size of the micropore. The temperature dependence of a, has been studied by others1°-14 both theoretically and experimentally. In principle, a(r) and U(r) can be determined theoretically by ab initio c a l c ~ l a t i o n sfollowing ~~ a procedure similar to that used for xenon interacting with other molecules in the gas phase. The chemical shift a, is then obtained by computing the ensemble average in eq 1. However, calculations based on simple models of a(r) and U(r) have the advantage of highlighting the physical picture behind the numerical results. The important point is @
Abstract published in Advance ACS Abstracts, April 15, 1995.
where c is the depth of the potential well, and a, the position of the zero potential energy, is the sum of the van der Waals radii of the xenon atom and the surface atom. The total potential U experienced by the xenon is the sum of the painvise 6-12 potentials over all the surface atoms. Consider the case of a xenon atom trapped between two infinite parallel layers where the layer separation is R. Referring to the schematic in Figure 1, one sees that U is a function of the coordinate y only. Let n be the number of surface atoms in an area of a*. By integrating over both layers and assuming that there is only one type of surface atom, one has
0022-3654/95/2099-7089$09.Q0/0 0 1995 American Chemical Society
7090 . I . Phys. Chem., Vol. 99, No. 18, 1995
Cheung Rla=2.0
2.2
2.4
2.6
2.8
3.0
8
4
w
d
so -4
Figure 1. Schematic showing a xenon atom trapped inside a layerlike pore. The xenon is confined between two infinite parallel layers. The xenon and the surface atoms are represented respectively by the large and small circles. The sum of their van der Waals radii is a and the separation of the layers, measured from center to center of the opposite surface atoms, is R. The y coordinate which is perpendicular to the layers extends from -R/2 to R12.
U(Y)=
-( 2 n d 5 )
5(R/2a - y/a)6 - 2 (Rl2a - y/a)'O
+ y/a)6 - 2 } (3) + 5(R/2a (R/2a + y/a)'O
with
0.0
0.2
Yla
originated from the sum of the second term of the LennardJones potential, becomes the determining factor of the shape of U. It is clear that U will have a single minimum at the center such as that portrayed in Figure 2. To a fiist approximation, the shape of the potential near the minimum can be represented by a parabola
(6)
where e is the coordinate of the xenon atom whose origin is at the minimum. equals y for a layer-like pore. For a spherical pore, e is the displacement of the xenon from the center of the pore. The coefficient b is always positive. Since exp(-be2/ kT) is nonzero only over a narrow region of @ [le1 5 (kT/b)1'2], one can expand a(@) into a Taylor series about e = 0 and retain only to the second order,
4e) =
+
