Microscopic Approaches to Physisorption: Theoretical and Experimental Aspects

1 Introduction

The writing of scientific reviews can be regarded as a service to the scientific community as well as a beneficial, though time-consuming, undertaking for the authors. There can be several purposes of such a review. First, it can be viewed as a collective reference source to aid workers in the field being reviewed and in related subjects in obtaining information about current developments. Secondly, it can be used as a means of forming a coherent unified approach to a certain set of problems, thus bringing together various aspects of the subject under study. Thirdly, it can be constructed such as to introduce the field to workers in other areas and hence serve as a cross-fertilization agent. We have here attempted the impossible, namely, to achieve all three purposes.

The study of adsorption phenomena is one of the oldest branches of physical chemistry (for a comprehensive review of early literature see ref. 1), starting with the observations by Scheele (1773) and Fontana (1777) of the uptake of gases by charcoal, and the discovery by Lowitz (1785) of the discoloration of solutions by charcoal. The distinction between regimes of adsorption phenomena, namely physical and chemical adsorption, was recognized early on and was often associated with the conditions leading to one adsorption class or the other. Among the names used to describe physical adsorption are van der Waals adsorption, low-temperature adsorption (versus high-temperature adsorption), secondary adsorption (versus primary adsorption), and capillary condensation, implying that capillary forces are responsible for physical adsorption. More modern definitions are based on the theory of chemical binding, identifying physisorption as the state of interaction between a n atom or molecule and a sur face, where no chemical bonds arc formed by charge rearrangement or sharing of electrons. Related to the above is the distinction of classes of adsorption according to the magnitude of the interaction energy (electron volts for chemisorption versus fractions of electron volts for physisorption). Modern spectroscopic techniques, especially ultraviolet photoemission spectroscopy (UPS), provide new diagnostic methods for distinguishing between the above adsorption regimes, as discussed in Section 3 (p. 34).

The traditional, and until quite recently the only, methods used for the study of adsorption phenomena were thermodynamical in nature. In this respect, theoretical efforts towards microscopic models of the interaction preceded the development and application of microscopic experimental tools. The main microscopic theoretical approaches are reviewed in Section 2. These studies are complemented by statistical mechanical treatments of the interaction of gases with surfaces, which have been reviewed extensively recently. Section 3 covers microscopic experimental approaches for the study of physisorption. In order to satisfy our 'third review criterion', the discussion of the results obtained by each of the experimental techniques is preceded by a brief exposition of the physical principles underlying the methods and the conventions and terminology employed. We have attempted throughout to review the methodology and results of recent studies and to indicate the link between the data and microscopic theoretical models.

Section 2 contains three main parts in which we discuss the general theory of van der Waals forces and physical adsorption (p. 5), semi-empirical calculations (p. 19), and the underlying principles and results of microscopic theories of physisorption (p. 21). In the last we have concentrated mainly on a discussion of our own theoretical studies.

Section 3 is divided into four main parts. In the first (p. 33) a classification of the methods is followed by a discussion of the use of u.v. photocmission in establishing a spectroscopic criterion for the definition of a physisorption system. Experimental methods for the study of atomic arrangement (LEED, neutron scattering, molecular beam scattering) are discussed and the results of recent studies demonstrated in the second part (p. 39). In addition the use of the above techniques for investigations of dynamical properties of physisorbed atoms (p. 61) and the atom-surface interaction potential (p. 77) are discussed. In the third subsection (p. 81) experimental techniques for the study of electronic structure are discussed (UPS, XPS, and field emission techniques). The review of the results is accompanied by theoretical arguments. Finally, we discuss the application of electron spectroscopy methods (LEED and Auger) for the measurement of adsorption isotherms (p. 97).

We conclude the review, in Section 4, with a brief summary and prognosis.

2 Theoretical Approaches for the Study of Physisorption

Introduction. — The basic theoretical problem in adsorption studies is the calculation of the wavefunctions of molecules which are interacting with a solid surface as well as with each other. The procedure which offers the greatest opportunity for insight into the physics of the process is that of first calculating the potential energy of a single molecule interacting with the solid surface and then determining its quantum and statistical mechanical properties in this potential field. The larger problem of interaction among adsorbed molecules is much more complicated and much less progress has been made. In general, it is obvious that the potential energy of interaction is fundamental.

We are concerned with the calculation of the potential energy [??] of a neutral molecule at position r which is interacting with a solid surface through forces associated with physical adsorption. The definition of these forces as distinct from those involved in chemical adsorption (which are caused by chemical interactions) is vague. It is known, however, that a neutral molecule far from a solid surface interacts through attractive van der Waals, or dispersion, forces. Dispersion forces may be described as physical in that they involve no chemical interactions such as charge transfer or rearrangement. From a theoretical viewpoint, therefore, it is natural to define the forces involved in physical adsorption as physical, producing no significant changes in electronic densities. This information provides the basis of the conventional picture of physical adsorption. Indeed, physical adsorption is usually described as arising mainly from van der Waals forces, even though it is not clear that the many-body forces which are responsible for attraction assume the character of dispersion forces at the relatively small distance from the surface corresponding to binding.

In analysing experiments, one must substitute operational definitions for these a priori ones. The most common is that binding energies associated with chemisorption are of the order of electron volts (i.e., chemical binding energies) whereas those linked to physisorption are at least a factor of 103 smaller. Actually, there is no operational definition which clearly separates physical from chemical adsorption. For this reason, most experimental and theoretical physical adsorption studies involve rare gases, whose chemical inertness presumably precludes chemical binding. Even in this case, the distinction is not always clear. Large negative work-function variations have been observed for noble gas adsorption on t ransition-metal surfaces. These indicate a strong interaction energy, which may be correlated with the empty d levels in these metals (see p. 90).

Despite these cautionary words, we take the point of view here that physical adsorption is a separate phenomenon from chemical adsorption, and that the definition given above in terms of physical forces is meaningful. This point of view is in accord with that of theoretical studies of the potential energy, most of which, in fact, assume that van der Waals forces are responsible for binding in physisorption.

The potential energy can be Fourier-analysed in components of momentum parallel to the surface, which we assume to be planar. In the case of a single crystal surface, we can write equation (1), where [??] represents a reciprocal lattice

[MATHEMATICAL EXPRESSION OMITTED] (1)

vector of the surface Bravais lattice, [??] the component of [??] parallel to the surface, z is the component perpendicular to the surface (positive z is defined outward from the solid), and the primed sum is taken over all [??]. We have thus decomposed W into terms dependent only on z and terms dependent upon the translational symmetry of the lattice. This decomposition serves as a convenient means by which to classify theoretical treatments of W. Those studies which involve actual calculations of the potential energy from a more or less fundamental basis neglect all [??]-dependence and, therefore, correspond to the first term in equation (1). In addition, these 'first-principles' studies are devoted mainly to adsorption on metal surfaces. On the other hand, treatments which involve the crystal structure of the surface [which include the summation in equation (1)) are of a semi-empirical nature: that is, the potential energy is calculated by summing pairwise interaction energies between the adsorbate molecule and the constituent atoms of the solid. Since this type of treatment neglects the collective nature of interactions in condensed matter, these semi-empirical calculations are generally applied to interactions with insulators where the locality of electronic orbitals minimizes collective effects.

van der Waals Forces and Physical Adsorption. — The general theory of van der Waals forces has been exhaustively reviewed recently. This discussion, therefore, is intended to provide an insight into the physical mechanisms responsible for the phenomenon and into the calculational methods most commonly employed. We start with a treatment of the van der Waals interaction between two spherically symmetric atoms originated by London, which has become a standard text-book example of the use of second order perturbation theory (the early history of van der Waals studies is well reviewed in ref. 58).

Suppose that we have two atoms in their ground states, labelled by subscripts 1 and 2, which are separated by a displacement [??] so great that the electronic clouds of the atoms do not interpenetrate; the normal condition for dispersion interactions is that R [??] a, where a characterizes atomic separations (in a molecule). The dipole-dipole interact ion potential, V, between the atoms is given by equations (2) and (3). The quantity [??]ik denotes the position of the kth electron of

[MATHEMATICAL EXPRESSION OMITTED] (2)

[MATHEMATICAL EXPRESSION OMITTED] (3)

the ith atom. The Hamil tonian of the isolated system, H0 = H1 + H2, is given as the sum of the individual atom Hamiltonians, such that, in Dirac notation, equations (4) — (7) can be written. The operator × in equation (5) denotes a direct

[MATHEMATICAL EXPRESSION OMITTED] (4)

[MATHEMATICAL EXPRESSION OMITTED] (5)

[MATHEMATICAL EXPRESSION OMITTED] (6)

[MATHEMATICAL EXPRESSION OMITTED] (7)

product of the individual atom manifolds.

Because of reflection symmetry, the first-order energy correction, <10,20 V 10,20> vanishes. In second order, the energy correction W2 is given by equation (8), and

W2(R) = <10.20 V σΨ>(8)

the first-order eigenstate correction, δΨ>, is given by equation (9). The final

[MATHEMATICAL EXPRESSION OMITTED] (9)

result, equations (10) and (11), is in terms of the oscillator strengths. The quantity M

[MATHEMATICAL EXPRESSION OMITTED] (10)

[MATHEMATICAL EXPRESSION OMITTED] (11)

is the free electron mass and i represents either 1 or 2 in equation (11). The appearance of oscillator strengths, f, in these equations for the interaction energy resulting from van der Waals forces manifests the intimate relation between the optical properties of matter and dispersion forces (hence the name): classically, the oscillator strength represents the number of electrons participating in an optical transition. Furthermore, we can see from equations (8) and (9) that the characteristic R-6 dependence of van der Waals interaction energies in equation (10) arises from a combination of two factors: the R-3 dependence of the classical dipole-dipole interaction, V, and the R-3 dependence of the dipole induced in one atom by the other. In other words, each atom is momentarily a dipole which induces a dipole in the other atom [corresponding to the state correction δΨ>in equation (9)] which has an R-3 dependence, and these two dipoles interact through V. Therefore, from this simple example, we can see that van der Waals inter actions arise from quantum fluctuations. Later we shall see that the phenomenon of van der Waals forces can be attributed to quantum fluctuations of the electro-magnetic field: this is the reason for the appearance of oscillator strengths in equation (10).

The first treatment of the van der Waals interaction between a molecule and a solid surface was that of Lennard-Jones, who considered atom–metal interactions. He assumed that the metal corresponds to a perfect conductor. Consequently, he considered the interaction between the molecule and its image in the metal. The potential energy of a molecule with Z electrons whose centre is located at a distance l from the surface, resulting from this interaction is given by equations (12) and (13). The quantity IT[??] denotes the position of the ith electron relative to the

[MATHEMATICAL EXPRESSION OMITTED] (12)

[MATHEMATICAL EXPRESSION OMITTED] (13)

centre of the molecule (zi, is the component of [??] in the direction perpendicular to the surface, where positive z is defined outward from the surface). In Lennard-Jones' treatment, it is assumed that all of the electrons are in spherical orbits, so that = (r2 >/3, where the bracket signifies quantum mechanical averaging. The result for the van der Waals interaction energy is given in equation (14). This

[MATHEMATICAL EXPRESSION OMITTED] (14)

expression represents the attractive energy of interaction between a molecule and a surface. As in the other fundamental calculations discussed in this section, all short-range repulsive interactions are neglected. Therefore, it is important to note that these theories cannot provide information about the binding of molecules without some extra assumption about the equilibrium position at which binding takes place.

The result in equation (14) is the first to show that the van der Waals interaction between a molecule and surface has an l-3 dependence. In later work, however, Bardeen demonstrated that the image method provides a lower limit for the true interaction energy, which is negative. In deriving his result, Lennard-Jones assumed that the metal responds instantaneously to the charge density fluctuations in the molecule. Bardeen, on the other hand, pointed out that corrections arising from the finite kinetic energy of the metallic electrons reduce the magnitude of the binding energy. He considered two systems, A and B, connected by a perturbation, V. Defining WA as the change of the energy of system A caused by V when the electrons of B are fixed (with a similar definition for WB), he was able to show, employing second-order perturbation theory, that |W|, the magnitude of the exact interaction energy, is smaller than either |WA| or |WB| and is closer to the smaller of the two. He derived the approximate expression (15). In applying this

W [??] WAWB/(WA + WB)(15)

formula to the interaction of a continuum metal (i.e., system A) and a one-electron atom (i.e., system B), it follows that WA is given by equation (14). WB was calculated to second order by assuming that the metallic electrons obey Fermi–Dirac statistics and by accounting for their mutual repulsion in an approximate way. Application of this method yields, from equation (15), the expressions (16) and (17)

W(I) [??] WL-J(I)FA/(1 + FA)(16)

FA [??] Ce2/2rsΔA (17)

The factor C in equation (17) results from the approximate treatment of the electronic repulsion; for the specific approximation made, C = 2.6 and it is expected that C varies slowly with rs at ordinary densities (i.e., rs is the radius of a sphere of a volume sufficient to contain one electron). The quantity ΔA is an average excitation energy of the atom. Bardeen was able to show that the Lennard-Jones image force result applies only for rs -> 0 (high densities). In other words, the quantity FA in equation (17) represents the correction produced by the finite kinetic energy of the metallic electrons.