Colloid Science Volume 4

A Review of the Literature published 1977-1981

By D. H. Everett

The Royal Society of Chemistry

Copyright © 1983 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-538-6

Contents

Chapter 1 Adsorption on Heterogeneous Surfaces By W. A House, 1,
Chapter 2 The Dubinin-Radushkevich (DR) Equation: History of a Problem and Perspectives for a Theory By G. F. Cerofolini, 59,
Chapter 3 Adsorption from Solution By J. Davis and D. H. Everett, 84,
Chapter 4 Statistical Mechanics of Colloidal Suspensions By E. Dickinson, 150,
Chapter 5 Micellar Structure and Catalysis By J. M. Brown and R. L. Elliott, 180,
Chapter 6 A Bibliography of Gas-Liquid Surface Tensions for Binary Fluid Mixtures By I. A. McLure, I. L. Pegg, and V. A. M. Soaves, 238,

CHAPTER 1

Adsorption on Heterogeneous Surfaces

BY W. A. HOUSE

1 Introduction

This Report is concerned with the surface heterogeneity of gas-solid and liquid-solid systems. It presents a historical development of the subject with particular emphasis placed on work published in the period 1970 to July 1981.

Properties of the solid/gas and solid/liquid interfaces are dependent on the chemical structure and topology of the solid surface. The chemical potential of the adsorbate, whether vapour, gas, or liquid, is dependent upon the nature of the surface and the spatial variation of the adsorption potential energy. Generally the development of physical adsorption theory has tended to con centrate on homogeneous surfaces, although the subsequent applications have been made in a haphazard manner to a wide class of absorbents commonly termed heterogeneous; since the details of surface heterogeneity are invariably unknown, the entire development and results derived there from are disputable.

In many instances the macroscopic surface topology may be examined using electron microscopy and such techniques as ultraviolet and X-ray photoelectron spectroscopy, Auger electron spectroscopy, secondary ion mass spectroscopy, and infrared spectroscopy;2 these are commonly adopted to investigate qualitatively and sometimes quantitatively, the chemical nature of the solid surface. Other methods using ultra-high-vacuum surface techniques such as RHEED and LEED u are available to probe the microscopic structure of the surface and have proved invaluable in elucidating the structure of metal surfaces with and without adlayers. Unfortunately these methods are not generally applicable to the macroscopic analysis of particulate surfaces.

It is interesting to reflect on the situation described over thirty years ago by Roginskii and Todes: ‘The inhomogeneity of the surfaces of most solids is borne out by a number of independent facts. The achievement of electron microscopy justifies the hope that the time is not very far off when the peculiar structure of an active surface will probably be open to direct observation and measurement. At present however, only integral and indirect methods of studying surface inhomogeneity are at our disposal.’ As yet no direct and quantitative method of assessing surface heterogeneity is available. The theoretical development and applications of physical adsorption to heterogeneous surfaces has been orientated to deriving information about surface heterogeneity of particular systems (i.e. gas/solid combinations) from gas adsorption isotherm data.

The Development and Early Solutions of the Integral Equation of Adsorption. — Langmuir derived the classical isotherm equation which bears his name by assuming the adsorbate to be localized on the surface without lateral interactions and that adsorption was on a homogeneous crystal surface exhibiting only one kind of ‘elementary space’ (i.e. a region of uniform adsorption energy, U). The fractional monolayer coverage of the total surface was expressed as:

θ = p/(K + p), (1)

where

K = Ao exp(-U/RT). (2)

Here Ao is a constant related to the partition function for the adsorbate and usually assumed to be temperature dependent and independent of the adsorption energy; this approximation will be discussed in Section 3.

Langmuir himself was the first to generalize his equation: 5 ‘Let us assume that the surface contains several different kinds of elementary spaces representing the fractions (β1, β2, β3, etc., of the surface so that (β1+β2+β3 + … = 1′. This approach led to an equation describing the total adsorption, θT,on a heterogeneous surface:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where K1, K2 …, are the constants governing adsorption on sites of types l, 2,… This equation was also written in an integral form to describe adsorption on a surface divided into infinitesimal fractions dβ (3. Equation (3) found limited application owing to the number of unknowns required to describe a real surface and the uncertainty of the effects of ignoring lateral interactions. In spite of this, the model was developed to describe multilayer adsorption on a surface with one elementary space producing the well known BET equation.

During the same period the analysis of experimental data, particularly at low adsorbate concentrations, led to the popular application of an empirical equation usualJy attributed to Freundlich:

θT = cp1/n, (4)

where c is a constant and n an integer (n> 1). The success of this equation for surf aces that were not expected to be homogeneous led a number of researchers to investigate what distribution of adsorption energies the overall isotherm equation (4) represented. Zeldowich in 1935 was perhaps the most successful; he adopted the Langmuir integral equation for adsorption in a form equivalent to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where F'(K) is the distribution function for the elementary spaces on the surf ace. Zeldowich was not able to solve equation (5) exactly for F'(K) but formulated an approximate method of solution by approximating the Langmuir equation for the individual ‘elementary spaces’ as shown in Figure l(a) i.e.

θ(p, K) = p/K 0<p : Henry’s Law, θ(p, K) = 1 K<p: Surface completely covered. (6)

This method produced the result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

which when applied to the Freundlich isotherm [equation (4)] gave:

F'(K) = AFK(l/n-1), (8)

where

AF = -c/n2 + c/n (9)

Equation (8) may be rewritten by substituting equation (2);

F'(U) = B exp(-α U/RT), (10)

where

B = AFAo(1/n-1), α = 1/n – 1 (11)

When this distribution was adopted to generate θT(P) using equation (5), the Freundlich isotherm at low pressures was obtained.

Although more exact solutions of equation (5) were not forthcoming until later years, much effort using approximate methods was spent, including formidable pioneering work by many Russian investigators (see Tolpin, John, and Field for a review of this work prior to 1952). Roginskii’s ‘control band method’ was particularly enlightening but restricted to distribution functions that do not change rapidly between the limits of the adsorption energy. Once again the Langmuir equation was employed to describe adsorption on the ‘elementary spaces’ and led to the formulation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Roginskii and Todes were fully aware of the problems of determining a more rigorous solution to the integral equation, particularly the demands made upon the quality of experimental adsorption data.

Sips in 1948 was able to solve the integral equation for F'(U) using a Stieltjes transform method (see Section 2). He assumed that the internal partition function for the adsorbed phase (incorporated in K) is independent of the adsorption energy and obtained a solution of the form shown in equation (10). However the normalization integral:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

was nonconvergent. This result led Sips to suggest a generalized Freundlich

isotherm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

and the subsequent analysis produced an exponential distribution function. At the suggestion of Honig and Hill, Sips later modified his theory by limiting the adsorption energies to positive values.

Extensions of the Adsorption Model to Include Adsorbate Lateral Interactions. — The assumptions implicit in the Langmuir model throw some doubt upon evaluations of the adsorption energy distributions obtained via equation (5). It is not surprising that later developments were primarily concerned with adopting more sophisticated local isotherms (sometimes referred to as model isotherms) to describe adsorption on homogeneous surfaces. This however complicates the analysis since once lateral adsorbate interactions are allowed, the spatial distribution of adsorption energies must be specified. Tompkins 16 in 1950 clarified this situation by recognizing two different types of surface characterized by: ‘(a) sites of equal energy grouped in patches, i.e. the adsorbed film is a polyphase system; or (b) sites of different energy distributed individually over the surface such that any small element of area contains a representative array of sites, i.e. the adsorbed film is a monophase system.’

The work discussed in this report is concerned with these two groups. The surface of type (a) will be referred to as ‘patchwise heterogeneous’ after Ross and Olivier and type (b) as ‘random heterogeneous’ after Hill. The development of the patchwise heterogeneous model followed an extension of the concepts discussed by Langmuir. It was realized that equation (5) could be written in a more general form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where θ(p, T, U) is the local isotherm equation, which for a patchwise heterogeneous model must account for adsorbate lateral interactions within individual patches. Uh and U1 are the upper and lower limits respectively of the adsorption energy.

Terminology. — It is relevant here to clarify the terminology now associated with surface heterogeneity and adopted throughout this report.

Unisorptic Surface. A hypothetical surface with an adsorption potential energy U(τ, z) which is independent of τ (where τ is a surface plane vector). This surface is sometimes referred to as a structureless surface.

Homotattic Surface. A microscopically uniform and homogeneous surface. This includes ideal crystal surfaces where the adsorption potential energy, U(τ, x) is a periodic function of τ, e.g. the ideal (100) surface of sodium chloride or an ideal basal surface plane of graphitized carbon black.

Intrinsic Heterogeneity. That periodic surface heterogeneity possessed by a single homotattic surface.

Patch. An area of a surface that is either energetically unisorptic or homotattic.

Residual Heterogeneity. A term used when describing a patch-wise heterogeneous surface; it is that heterogeneity that cannot be ascribed to homotattic surfaces and is generally caused by impurities and defects on the surface.

2 Solutions of Fredholm Equations of the First Kind

Introduction. — The generalized integral equation for a heterogeneous surface (equation 15) is a linear Fredholm equation of the first kind defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The solution, f(y), given the kernel, K(x, y) and function g(x), poses special difficulties which have been discussed in an excellent review by Miller. The most obvious difficulty concerns the character and range of the operator K(x, y) as not all functions, g(x), can be written in a form given by equation (16), e.g. if K(x, y) = cos(x) cos(y) and f(y) is any integrable function, then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

i.e. g(x) must be a multiple of cos(x). In this example even if a solution did exist, it would not be unique as there are an infinity of functions, φ(y) such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The greatest problem however concerns the ill-posed nature of equation (16). A small perturbation in the data function g(x) leads to a large perturbation in the solution, f(y). This may be demonstrated by considering f(y) in equation (16) as an exact and unique solution. Applying a perturbation δg(x) to g(x) and examining the subsequent perturbation on the solution δf(y) we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

<->

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

It is convenient to write the perturbation

δg(x) = chn(x), (21)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Substituting equation (21) into (20) leads to:

δf(y) = c cos(ny), (24)

<->

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

hence if n is large [the perturbation in g(x) is small], this ratio will be large. The absolute value of the fraction expressed in equation (25) depends on n and the form of the kernel in equation (22). Generally the instability is greater for flat and smooth kernels than for sharply peaked ones. This may be demonstrated by allowing K(x, y) = δ(y-yo), where δ(y-yo) is a Dirac-delta function; following the argument above, hn(x) = cos(nyo), δg(x) = c cos(nyo), and δf(y) = c cos(nyo) thus yielding the fraction δf(y)/δg(x) = 1. Hence the degree of success achieved in solving equation (16) depends to a large extent on the accuracy of g(x) and the shape of the kernel.

General Numerical Methods of Solution. — A comprehensive discussion and comparison of the numerical methods of solution of the Fredholm equation will not be presented here. (See Miller and the references cited by Dormant and Adamson. However, it is pertinent to examine the mathematical basis of some of the methods that have been successful and to discuss their limitation when applied to the solution of equation (15).

The Matrix Regularization Approach. For a data set {g(x1), g(x2), …, g(xi), …, g(xn)} a system of linear equations may be constructed [cf. equation (16)].

Af = g, (26)

where A is the matrix of quadrature coefficients for the tabular points {y1, y2, …, yi, …, ym} and {x1, x2, …, xj, …, xn}i.e. the elements aij of A are aij = Wi kji;, where wi are quadrature weights and kji elements of the kernel matrix. The ill-conditioned nature of equation (16) means that a physically realistic solution for the matrix equation (26) is impossible unless allowance is made for inherent errors in the data set g(x)24i.e.

Af = g + E, (27)

where the elements {e1, e2, …, ej, …, en} are the absolute errors for the data set g. The values of these errors are of course unknown but a solution to equation (27) is possible if some prior information concerning its general form is available. Twomey has shown that if a ‘smooth’ solution is required, the minimization of the curvature, i.e. Σi(fi-1-2fi +fi+1)2, with the conditions Af = g+E and Σj ej = constant, gives a general solution:

f = (ATA + γH)-1ATg, (28)

where γ is a Lagrange multiplier and H the expansion coefficient matrix:

[TABLE OMMITTED]

Figure 2(a, b) show the results obtained by Phillips and House (unpublished) using this matrix method. The decreasing oscillations in the solution with increasing damping constant, γ, is evident but when the solution itself has a number of peaks, increasing γ tends to reduce their definition and at the same time smooth the shorter-range oscillations. The choice of γ can be guided by a comparison between the anticipated error in the gj values and the deviations between the values in the dataset g and the generated values obtained using the calculated fi values for a particular γ. Unfortunately when the technique is applied to the solution of equation (15), physically unrealistic negative frequencies may arise as permitted solutions. This problem has been discussed in detail by Merz who postulates a ‘concavity criterion’, which he claims is a necessary condition, but not a sufficient condition, for the use of a particular kernel i.e. local isotherm. In other words some adsorption data and local isotherms are incompatible if negative frequencies are to be avoided. This aspect of the problem needs further study. The regularization method is particularly promising but needs further improvement to incorporate a nonnegativity constraint and a mathematical criterion for the choice of γ. Merz has already suggested methods for the choice of γ. It is also desirable to limit the analysis to some finite range of adsorption energies, e.g. the condensation approximation (see Section below) may be used to define the adsorption energy limits if the adsorption is in the region of the two-dimensional critical temperature.

(Continues…)Excerpted from Colloid Science Volume 4 by D. H. Everett. Copyright © 1983 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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