Chapter 12. Multilayer adsorption as a tool to investigate the fractal nature of porous adsorbents

w. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.

625

Multilayer Adsorption as a Tool to Investigate the Fractal Nature of Porous Adsorbents Peter Pfeifer and Kuang-Yu Liu Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, U.S.A.*

This chapter surveys the use of physical adsorption, from a monolayer upward, as an experimental method to study the fractal surface structure found in many porous and irregular adsorbents. The fractal structure leads to power laws of the Frenkel-Halsey-Hill (FHH) type for the adsorption isotherm, with exponents depending on the fractal dimension of the surface and on whether the dominant force is the substrate potential (van der Waals wetting, low coverage) or the film-vapor surface tension (capillary wetting, high coverage). We derive the power laws from a unifying framework which treats the two forces as competing effects and automatically identifies well-defined coexistence lines in the pressure-dimension diagram between the submonolayer regime, the van der Waals wetting regime, and the capillary wetting regime. We compute the resulting phase diagram for several adsorbate/adsorbent pairs, predicting which of the two power laws will be observed in what pressure range for a given surface geometry and adsorbate. A detailed comparison of the adsorption isotherm on a fractal surface with that in a single pore exhibits many parallels and differences between the two, which we also discuss in terms of t-plots and comparison plots. The aim of the presentation is to provide a simple, but complete set of guidelines for the interpretation of experimental adsorption isotherms, with a minimum number of parameters, in a thermodynamically and geometrically consistent way. A variety of recent experimental studies using multilayer adsorption for fractal analysis are reviewed as illustrations. The examples include some important test cases and range from metal films to carbon blacks, activated carbons, carbon fibers, pyrogenic silicas, silica xerogels and aerogels, porous glasses, and cements.

1. I N T R O D U C T I O N Much of our understanding of how solid surfaces interact with their surroundings, physically or chemically, depends on quantitative models for their structure. Until quite recently, most of these models have been based on Euclidean geometry, such as planar surfaces, straight-line step edges, cylindrical or slit-shaped pores, etc. However, many systems of practical importance (colloidal aggregates, adsorbents, catalyst supports, electrodes, hightemperature superconductors, reinforced materials, and other micro-engineered or naturally porous solids) have a complex structure which cannot be adequately described in such terms. * Research supported in part by the Petroleum Research Fund, administered by the American Chemical Society, Grant No. 28052-ACS

626 Typical of these complex geometries is that they exhibit structural features (departures from a planar surface) over a whole range of length scales----often several decadesmrather than features of one characteristic size only. Often the coexistence of features at different length scales leads to surface geometries where iteratively small pores (or other features) are subpores of larger pores. Such nested pore hierarchies render the concept of individual pores, cylindrical or otherwise, as underlying the conventional notion of pore-size distribution, meaningless. Similarly, other Euclidean descriptors, such as terrace-width distribution or surface height fluctuation, are inapplicable. This creates an eminent need for models in which structural features span a whole range of length scales, i.e., that treat surface irregularities as recurrent and nested, rather than isolated, entities. The simplest such model is that of a fractal surface [ 1-11 ]. A fractal surface has the s a m e structural features (Fig. 1) at length scales between s (inner cutoff) and s (outer cutoff) and is characterized by the fractal dimension D, 2 < D < 3, describing the irregular surface geometry in terms of its space-filling ability in the interval [gmin, s ]- At the low end, D = 2, the surface is planar, at length scales between s and gmax. At the high end, D = 3, the surface is maximally convoluted and fills a volume, at length scales between s and gmaxAt intermediate values, 2 < D < 3, the surface interpolates in a natural way between a plane and a volume. These basic properties have made fractal geometry, since its first application to surface problems [12-18], a highly appealing and successful new tool to study a wide range of phenomena associated with complex and disordered surfaces [4, 5, 7-11, 19-35]. The appeal comes from the fact that (i) the complexity is captured by a single number, the fractal dimension (for phenomena involving connectivity properties of the surface, also the spectral dimension is important [36]); (ii) the fractal dimension identifies the recurrence of the same features at different levels of resolution as a hidden symmetry (self similarity or self affinity) in an otherwise irregular structure; (iii) the resolution analysis automatically focuses on a range of features and is indifferent to whether small features are subfeatures of larger features or not; (iv) the metric properties of a fractal surfacemsuch as the number of pixels required to digitize the surface at a prescribed level of resolution, the number of surface sites present within a prescribed radius from a given site, or the number of surface sites as a function of the surface diameter--are as simple as for a planar surface (power laws with D-dependent exponents); (v) the metric properties, combined with the irregular structure of the surface, generate a wealth of explicitly D-dependent properties (Table 1.A);

Figure 1. Two fractal surfaces (cross sections), with small features being replicas of large features and small pores being subpores of large pores. The two cross sections have the same fractal dimension by construction.

627 (vi) a given value of D can be realized in many different ways (Fig. 1), giving the properties in (iv, v) a high degree of universality and making the description in terms of D stripped of all redundancy. The practical success comes from the fact that (vii) many of the random processes that produce complex surfaces (polymerization, aggregation, vapor deposition, electrolytic deposition, phase separation, drying, leaching, decomposition, corrosion, fracture, laser ablation) give rise to fractal structures by fairly well-understood mechanisms [ 1-5, 7, 8, 11, 22, 31, 34, 35]; (viii) when a surface has features at length scales between s and s and the ratio s is not very large, one may always approximate the features of different size as being in lowest order the same (least-biased guess), which makes the fractal model a good approximation even if the surface is not genuinely fractal; (ix) the D-dependent properties in (iv, v) generate explicit relations, usually in the form of power laws, between a wide variety of experimentally measurable quantities, offering many experimental methods of fractal surface analysis (structural analysis, Table 1.B) and predictions of how the structure controls the performance of the surface in physical and chemical processes (structure-function relations, Table 1.C). Common to all methods of fractal analysis is that the surface is subjected to the interaction with probes ("yardsticks") of different size. Depending on the method, the probes may be molecules of different size, electromagnetic waves diffracted at different angles, energy transfer from a donor molecule to acceptor molecules at different distances, liquid menisci with variable radius of curvature, films of variable thickness, molecules diffusing to the surface starting from different locations in the pore space, etc. In multilayer adsorption, the subject of this chapter, the probes are films which vary both in thickness and radius of curvature. Multilayer adsorption plays a special role among the methods listed in Table 1. First of all, in many applications one is interested in the morphology of the surface starting at atomic length scales, i.e., one would like to know whether the surface is fractal in an interval [gmin, gmax] with ~min of the order of a few ,~,ngstroms and s of the order of hundreds of .&ngstroms (nanostructured surfaces). Films of adsorbed N 2, Ar, Kr, or other inert gases can easily span this range and offer a structural resolution down to atomic length scales by virtue of the atomic size of the adsorbate particles. Second, no matter how convoluted and porous the surface is, gas diffusion through pore space and complete wetting of the surface by the adsorbate guarantee that the film probes the entire surface. Multilayer adsorption is therefore an important tool whenever the surface is too tortuous for methods like scanning tunneling microscopy, atomic force microscopy, or reflectometry to be applicable. The only other methods which compete with multilayer adsorption, in terms of length scales and applicability, are molecular tiling, small-angle X-ray or neutron scattering, electronic energy transfer, and preadsorbed films (Table 1). Combined, these other methods have provided some of the most extensively investigated case studies of fractal surfaces [43, 80, 81 ]. But they are quite time-consuming or require instrumentation that is not readily available in most laboratories. Thus, an important rationale for investigating fractal surfaces by multilayer adsorption is the wide availability of gas adsorption instruments and the ease with which adsorption isotherms can be measured up to quite high relative pressures. The first experimental study of a fractal surface by multilayer adsorption was published in 1989 [44] and has been followed by numerous investigations since. The purpose of this

628 Table 1 Geometric quantities controlled by the fractal dimension, experimental methods of fractal analysis, and applications controlled by the fractal dimension. References A. Geometric quantities: Pore-size distribution Chord-length distribution Porosity Density-density correlation function Height-height correlation function Fourier transform of surface cross sections Dimension of cross sections and projections of the surface

[14,20,21] [19,32] [19,32] [8,23] [8,11,31] [17] [ 1, 6, 30]

B. Experimental methods: Molecular tiling and monolayer capacity Small-angle X-ray and neutron scattering Electronic energy transfer Kelvin porosimetry Thermoporosimetry Hg porosimetry Surface area of preadsorbed films Scanning tunneling microscopy X-ray reflectivity NMR spin relaxation Multilayer adsorption

[12-16,29] [21,29,37] [29,38] [29,39] [40, 41] [42] [43] [44-46] [47, 48] [29,49] this chapter

C. Applications: Surface-diffusion controlled reactions Pore-diffusion controlled reactions Catalysis Dissolution and combustion Chromatography Electrochemical impedance Debye-Hfickel screening Electrical conductivity Hydrodynamic flow Magnetic phase transitions 4He phase transitions Vibrations Thermal conductivity Scattering and absorption of light Compaction Sintering

[20,50] [18,20,51-55] [ 18, 20, 25, 51-55]

[55-57] [58] [53, 54, 59-61 ] [62-64] [19,28,32,33] [19,28,32,33] [65-67] [21, 68, 69] [70-73] [73] [74-76] [77] [78, 79]

629 chapter is to give an account of these developments and to describe the various aspects which have placed multilayers on fractal surfaces at the crossroads of several different areas of surface science. One of the distinguishing features is that an adsorbed multilayer is not a "premanufactured" probe which is brought to the surface and interacts with it like a rigid object, but is a "self-assembled" probe formed by complicated substrate-adsorbate and adsorbate-adsorbate interactions (repulsive at short range and attractive at long range). These interactions result in a layer that is controlled by a strong interplay between energetic and geometric factors when the surface is irregular. At temperatures above the triple point and below the critical point of the adsorptive, i.e., at which gas and bulk liquid of the adsorptive coexist, the adsorbed multilayer is a liquid. So our first task will be to find an adequate description of the energy and thermodynamics of a liquid film, in equilibrium with its own vapor, on an arbitrarily shaped solid (Sect. 2). The description we shall use models the film as homogeneous liquid with sharp liquid-gas interface. The substrate-adsorbate interaction enters through the substrate potential, dependent on the surface geometry, and the adsorbate-adsorbate interaction enters through the liquid-gas surface tension. This describes the two interactions with a minimum number of parameters and provides a framework to calculate adsorption isotherms in various approximations. The key is that equilibrium shape of the film-gas interface is determined by a variational principle (minimization of the grand potential). In Sect. 3, we apply this variational principle to derive the adsorption isotherm for multilayers on fractal, self-similar surfaces. The genetic form of the isotherm is N or [_In(p/p0)] -(3-D)/3 N 0r [_In(p/p0)] -(3-D)

for low p, for p ---) P0,

(1 a) (lb)

where N is the number of adsorbed particles adsorbed at gas pressure p, and P0 is the coexistence pressure for gas and bulk liquid. These are the power laws that most experimental studies have used to infer fractal surface properties from multilayer data. The power law (1 a), called van der Waals wetting regime, is characteristic of multilayers in which the substrate potential is the dominant interaction. The power law (lb), called capillary wetting regime, results when the liquid-gas surface tension is the dominant interaction. Together, the two regimes provide a unified description of the transition from substrate-controlled adsorption to capillary condensation. The description includes the form of the crossover from (la) to (lb), explicit expressions for the prefactors in (1), the connection between gas pressure and length scales probed by the film, and refinements of (1) when the length scales probed approach the inner or outer cutoff of the fractal regime. Section 4 analyzes the transition from van der Waals to capillary wetting as a function of the fractal dimension. For every D value, there is a pressure which defines this transition in a natural way. Similarly, there is a pressure which defines the transition from submonolayer to multilayer adsorption. The resulting transition lines divide the D-p plane into a phase diagram with three distinct regions: submonolayer adsorption, van der Waals wetting, and capillary wetting. For example, when D decreases, capillary wetting is restricted to progressively higher pressures and is completely absent at D = 2. By explicitly predicting in what pressure intervals the power laws (1) occur for a given solid and adsorbate, the phase diagram offers important consistency tests for the interpretation of experimental data of the form (1). Armed with these theoretical prerequisites, we review experimental results in Sect. 5. The examples illustrate the wide applicability of the power laws (1) and amplify earlier accounts of the pervasiveness of fractal surfaces at nanoscales [29]. Our main focus, however, is on

630 studies in which the surface structure inferred from (1) has been tested for internal and external consistency. Internal consistency includes meaningful values for D, groin, and grnax, consistency with the phase diagram, etc. External consistency means agreement with results from other experimental analyses (fractal or otherwise). These tests form an important experimental confirmation of the adequacy of the liquid-film framework used in Sect. 2 to describe multilayers adsorbed on arbitrarily shaped solids. Indeed, the ultimate goal is to develop a general framework from which it will be possible infer structural properties of solids with arbitrarily shaped surfaces (fractal or not), without geometric model assumptions. This is the inverse problem of the program that wishes to predict the isotherm for an arbitrary given surface geometry. The common view is that both the direct problem and the inverse problem are intractably difficult. We therefore would like to point out that, in Sect. 3.6, we will present a solution of both the direct and the inverse problem. The solution will, of course, not be free of approximations; but the approximation will be one in the variational determination of the equilibrium film, not in the surface geometry. In this sense, this chapter may be viewed as fractal illustration of a much more general framework.

2. LIQUID-GAS E Q U I L I B R I U M ON NONPLANAR SURFACES When analyzing the 3-phase equilibrium between a liquid, its vapor, and a solid, we first must determine whether the liquid completely wets the solid or not. If the surface of the solid is planar, the macroscopic condition for complete wetting is that the contact angle between the liquid and the solid, 0, be zero (Fig. 2). Partial wetting corresponds to 0 < 0 < r~. The contact angle is determined by Young's equation, (2)

COS 0 = (O'sg -- O'sl)/(Ylg,

where the o's are the solid-gas, solid-liquid, and liquid-gas surface tension. Thermodynamic equilibrium ensures that the right-hand side of (2) always lies between -1 and 1 [82]. Thus complete wetting is characterized by the condition Csg- Cysl= c]g. Microscopically, complete vs. incomplete wetting can be distinguished by considering the high-pressure behavior of the adsorption isotherm: If N --->o0 for p --->P0, one has complete wetting; if N approaches a finite limit for p --->P0 (i.e., if there is a discontinuous jump to N = oo at p = P0), wetting is incomplete [83-85]. From this criterion and a large body of adsorption data, one concludes that nitrogen and other inert adsorptives, at their normal boiling temperature, completely wet most solids. Consequently, we assume the liquid to be completely wetting in the sequel.

0=0

0
0=n

Figure 2. Macroscopic contact angle of a completely wetting (0 = 0), partially wetting (0 < 0 < ~), and completely nonwetting (0 = ~) liquid on a planar surface. In the partially wetting case, a microscopic layer of liquid covers the surface outside the macroscopic drop.

631 Under these circumstances, the 3-phase equilibrium reduces to a 2-phase equilibrium, namely that between the liquid film, which completely covers the surface and can have any thickness, and the gas phase. A convenient starting point to treat this problem is to construct the grand potential of the film, f~, as a function of the chemical potential It. The potential f2, sometimes also called Landau potential, is the Legendre transform of the Helmholtz free energy F with respect to the particle number N in the film (see, e.g., [86]),

f~ = F - rtN,

(3)

and is natural whenever the system (here the film) exchanges particles with a reservoir (here the gas phase). Equilibrium then requires that ~t be equal to the chemical potential of the reservoir and that ~ , for given It, be a minimum. As reference state for the film, we take N particles of bulk liquid at chemical potential kt0. This turns (3) into

(4) (5)

ZKQ = zKF-AItN, Ait = It - Ito = kT In(p/p0),

where AF is the difference in free energy between N particles in the f'dm and N particles in bulk liquid. The last part of (5) equates It and ]ao to the chemical potential of the coexisting gas phase, at pressure p and P0, for the film and bulk liquid, respectively. The treatment of the gas as ideal gas, k being Boltzmann's constant and T the temperature, is for simplicity. The chosen reference state of the film eliminates the self-interaction of the film (transferring it to the gas pressure P0 instead), i.e., makes AF contain only the interaction of the liquid withthe solid and the gas. To complete the construction of the grand potential, we let the surface S of the solid have arbitrary shape and treat the film as a homogeneous liquid (continuum) with sharp liquid-gas interface I. The grand potential then becomes a function of I and Ait, given by Aft[I, AIx] = n [

. , f (s,i)

= I[nr

(U(x) - Ait)d3x + ~ [_ d2x -'1

(U(xll, x• - Ait)dx• + (~ ~/1 + IVI(x,)I 2 ]d2xll

(general)

(6a)

(no overhangs). (6b)

"1S(xj[)

We first discuss the general case, (6a). The function U(x) is the potential energy of an adsorbate particle at position x in the f'tlm, due to the interaction with the solid, relative to that of bulk liquid; it is referred to as the substrate potential. The integration domain f(S, I) is the set of points enclosed between the surface S and the interface I (Fig. 3). The prefactor n is the number density of the film, taken to be equal to the value for bulk liquid (incompressible film). Thus, the first term represents the solid-liquid interaction, obtained by summing the substrateadsorbate energies of all the particles in the film. The second term is the t e r m - A ~ N in (4). The last term, in which c is the liquid-gas surface tension and the integral denotes the area of the interface I, represents the liquid-gas interaction, i.e., the work needed to create a liquid-gas interface of area ~I d2x (free energy of the interface). We drop the liquid-gas subscript in a because no other surface tensions remain in the problem, and take (~ to be equal to the value for bulk liquid. Equation (6b) describes the special case where the surface has no overhangs, i.e., where S and I can be parametrized as height above some reference plane, at position xll in the reference

632

gas

I(x I) i

"

~

S(x u)

solid xll

Figure 3. A liquid film configuration with liquid-gas interface I, adsorbed on a surface S. Shown is the special case in which the surface has no overhangs.

plane (Fig. 3). In this case, the position x is specified by (xll, x.u) where x• is the coordinate perpendicular to the plane, and the square root expresses the surface element of I in terms of the gradient of I. The substrate potential U(x) is the interaction energy between the solid and a particle at position x outside the solid, minus the corresponding energy if the solid is replaced by bulk liquid. The reference to bulk liquid in U(x) comes from the fact that the free energy of the film was chosen relative to bulk liquid. It makes U(x) coincide with the substrate potential in the Frenkel-Halsey-Hill (FHH) theory of multilayer adsorption [87-90]. In good approximation, U(x) may be taken as U(x) = - e~/[dist(x, S)] 3

(7)

where a is a positive constant independent of S, and dist(x, S) denotes the distance between the point x and the surface S (shortest distance between x and any point on S). When the surface is planar, the potential (7) reduces to the familiar inverse distance cubed law for the long-range van der Waals attraction [87]. The constant ~ describes the strength of the attraction and is known from dielectric properties of the solid and adsorptive for a variety of substrate-adsorbate pairs [88-90]. The repulsive interaction at short distances, neglected in (7), may be added to (7) to make the integral in (6) finite, but makes negligible contributions to the multilayer adsorption situation at issue. The expression (6) for A,Q, often referred to as effective-interface or capillary-wave Hamiltonian, may be interpreted and used in several ways. The different interpretations differ in how they interpret the interface I. While we will use exclusively the interpretation as equilibrium interface in later sections, it is useful to compare it with other conceptual frameworks. The comparison identifies the conditions under which the equilibrium interpretation is justified and provides corrections when these conditions are not satisfied.

2.1 Interpretation as Effective Interface If one starts from a description of the liquid and gas in terms of a spatially variable fluid density, determined by the pair interaction between the fluid particles (density-functional theory), there is no sharp liquid-gas interface and the density representing the liquid may vary

633 as a function of position. Nevertheless, one may, in this description, evaluate the grand potential for a density ("trial density") that is equal to the density of bulk liquid on one side of some dividing surface I and equal to the gas density on the other side. The density drop at I identifies I as sharp, effective liquid-gas interface. The resulting grand potential, as a function of I, can be expanded in terms of derivatives of I (Taylor expansion around a planar interface). This gradient expansion has been carried out by Napi6rkowski and Dietrich [91 ]. The leadingorder term gives the expression (6b). The surface tension ~, in this framework, is a function of the pair interaction and the density of the liquid and gas (see also [82, 92, 93]). Higher-order terms in the expansion, which involve the curvature of I and represent bending energies of the interface, can be accounted for by letting ~ become curvature-dependent (see also [82, 92]).

2.2 Interpretation as Fluctuating Interface For fixed substrate potential and fixed parameters n and cy, one may interpret (6) as the energy of a microscopic film configuration, specified by I, in the grand canonical ensemble. This interpretation allows each I as possible interface, occurring in the film with relative probability e-&O[l,A~ t]/(kT). It views the I's as thermal excitations, or thermally fluctuating interface, of the film. The particle number N in the film then is the average of the particle number in the individual configurations, given in terms of the grand canonical partition function by 0 l n ( ~ I e_Z~ti, aM/(kT)). N = kT ~)(A~)

(8)

The sum over all I in Eq. (8) is a functional integral if I is taken in the continuum representation stipulated in (6), a sum over lattice walks if I is discretized on a lattice, or a sum over wave numbers if I in (6b) is decomposed into Fourier components (capillary waves). Equation (8), together with the expression (5) for the chemical potential, is the adsorption isotherm. It takes into account that the thermal motion of particles in the film creates thermal fluctuations of the interface I, making the interface effectively diffuse at T > 0. The importance of such departures from a sharp interface can be estimated by analyzing the width of the interface, w(T, A~t), in the special case where the surface S is planar. The width in this case is defined as standard deviation, thermally and spatially averaged, of the film height relative to S: (9)

w(T, A~) = ff (([I(xll) - (I(xll))Xll]2)i)xll .

Here (---)x.. is the average over all points in the reference plane, chosen as S, and (---)I denotes II the grand canomcal average, (f(I)) I = ( Z I f(I) e -af~[I' AI't]/(kT))/(Z I e -Af~[I' Alx]/(kT)).

(10)

The width is zero at zero temperature, w(0, AB) = 0, corresponding to a sharp interface, and increases with increasing temperature because the fluctuations increase. The attractive substrate potential, on the other hand, tends to reduce the fluctuations. Therefore, at fixed temperature, the width increases with increasing A~t because the effect of the substrate potential diminishes as the film grows in thickness. Upon adding to (6b) a term that accounts for the gravitational potential, which dominates over the substrate potential when AIx --> 0, this leads to the bound

w(T,A~) < w(T,O) =

lkT ~

In Apg a2 +1

1

(11)

634 where Ap is the difference in mass density between bulk liquid and gas, g is the gravitational acceleration, and ao is the thickness of a monolayer. The right-hand side of (11) is the width, due to the thermal excitation of capillary waves, of a free liquid-gas interface (no substrate) in the approximation where the gradient VI(xll) in (6b) is small [92-95]. When T approaches the liquid-gas critical temperature T c, the quantities cy and Ap go to zero and w(T, 0) diverges, in agreement with the fact that the liquid-gas fluctuations become macroscopic at T c. At temperatures not close to T c, the width w(T, 0) is very small, however, typically between ao and 2ao [94]. It follows that the width for finite film thickness is even smaller. Measurements of the width w(T, A~t) by X-ray scattering confirm this [96, 97].

2.3 Interpretation as Equilibrium Interface The foregoing analysis leads to the following conclusions if the film is sufficiently thick and the temperature is well below T c. (i) Departures of the film density from the density of bulk liquid are negligible because they occur at the surface and at the interface at most. (ii) Bending energies of the interface are negligible because the curvature of the interface decreases with increasing film thickness (Fig. 3). (iii) Thermal fluctuations are negligible since the width of the interface is of the order of a molecular diameter only. Under these conditions, the energy of a microscopic film configuration specified by I is well described by (6) and the number of particles in the film is well approximated by neglecting all thermal excitations in (8). This amounts to evaluating (8) in the limit T ~ 0 (keeping n and ff fixed), which gives N = - D(A~t)

A~2[Imi n, AlX] = n fff

(S,Imin)

d3x,

(12)

where Imin is the interface that minimizes z~,Q[I, A~t] for given A~t (ground-state interface). The second part follows from (6). The zero-temperature limit leading to (12) does not imply that the actual physical temperature goes to zero; it is simply a device to suppress the configurations in (8) that correspond to thermal fluctuations. In fact, the interface Imi n depends on the physical temperature through the temperature dependence of AFt, n, and o in (5) and (6). The interface Imin is the equilibrium interface, viewed as microscopic ground-state configuration in the grand canonical ensemble. Alternatively, A~[I, All.] as given by (6) may be viewed as grand potential of a macroscopic, thermally averaged trial configuration I of the film. In this case, the grand potential must be a minimum at equilibrium by the thermodynamic variational principle. This again singles out the equilibrium interface as the interface, Imi n, that minimizes z~[I, A~t] for given A~t. It again leads to (12) for the number of particles in the film, now by the inverse Legendre transform of the grand potential, or directly from N = n • film volume. A third way of constructing the equilibrium interface starts from the Helmholtz free energy, zkF[I], of a macroscopic trial configuration I of the film. The expression for zkF[I] is the righthand side of (6) without the chemical-potential term. At equilibrium and fixed particle number, the Helmholtz free energy is a minimum. Hence the equilibrium interface is the interface that minimizes ZkF[I] subject to the constraint that the number of particles in the trial configuration be equal to N. This yields the equilibrium interface Imin as a function of N. The chemical potential of the film, Ag, is obtained by differentiating the minimized Helmholtz free energy with respect to N. When the minimization of ZkF[I] is carried out using the Lagrange method, the Lagrange parameter for the constraint is AFt and the unconstrained quantity to be minimized is the grand potential zS~Q[I,A~]. Thus, the two procedures of minimizing the grand potential at fixed A~t and minimizing the Helmholtz free energy at fixed N are completely equivalent (Table 2).

635 Table 2 Variational principles for the grand potential and Helmholtz free energy, with Ag given by (5). Grand potential

Helmholtz free energy

Functional

A.Q[I, Ag] = AF[I] - Ag n ~f(s, I) d3x

AF[I] = n ~f(s, I) V(x)d3x + cr Si d2x

Interface Imin

Minimize A.Q[I, Ag] with respect to I, for fixed Ag

Minimize AF[I] with respect to I, subject to n ~f(s, I) d3x = N

Imin is function of

Chemical potential At.t

Number of particles, N

Adsorption isotherm

N =

A[.t = d-~ AF[Imin(N)]

Explicit formula

N = n If(S, imin(Ag)) d3x

Solve (*) for N as function of At.t

Strategy

Put N particles, N arbitrary, close to the surface (makes the potential energy and film area low for large N) while keeping the film volume small (makes the Ag-term small)

Put N particles, N given, close to the surface (makes the potential energy low) while keeping the film area small

d

d(Ag)

A~[Imin(A~), A[.t]

(*)

Equation (12) reduces the computation of the adsorption isotherm to the minimization of (6) with respect to I. The Euler equation for the minimizing interface Imin is n [ U ( x ) - Ag] +

o(

1 + 1 )=0 Ri(x) R2(x)

n[U(xll, Imin(xll)) - A[.t] - oV-

Vlmin(Xil)

= 0

for all x on Imin (general)

(13a)

(no overhangs)

(13b)

41 + IVImln(Xll)l2 from Eq. (6a) and (6b), respectively [87, 95, 98-105]. Here Rl(x) and R2(x) are the principal radii of curvature of I at position x, taken to be positive if the tangent circle lies on the liquid side of the interface, and negative if it lies on the gas side. The boundary conditions necessary if the interface is not closed are described in [ 101]. The surface geometry S enters through the substrate potential U(x), Eq. (7). Equation (13) is a nonlinear partial differential equation for the equilibrium interface. In the absence of the substrate potential, it reduces to the Kelvin equation. In the presence of the substrate potential, it is nonlinear both via the substrate potential and the curvature term, as seen in (13b). For general surface geometry, it has no simple solution. It may have several solutions (capillary instabilities, metastable states, unstable states [87, 95, 98, 106]), in which case the minimizing solution must be determined by additional evaluation of (6). Thus, an exact calculation of the adsorption isotherm (12b) requires considerable numerical effort in general and does not lead to results from which the geometry of S is easily reconstructed (interpretation of

636 experimental isotherms). In Sect. 3, we will present an approximate calculation of the isotherm which provides a simple, general one-to-one correspondence between experimental data and geometric information about S. Before we turn to that treatment, however, it is useful to recall two examples in which (13) can be solved exactly.

2.4 Van der Waals Wetting and Capillary Wetting The examples serve to illustrate the fundamental role of the substrate potential, even in situations where nU(x) is small compared to the curvature term in (13) (capillary condensation in large pores), in which case the folk wisdom is that the substrate potential can be neglected. At the same time, they illustrate the basic two adsorption mechanisms which compete with each other whenever the surface is nonplanar and which, for a fractal surface, will separate the adsorption isotherm into two distinct regimes. Example 1. The first example is that of a planar surface. Choosing the reference plane equal to S and substituting (7) into (13b), one verifies that the function Irnin(Xll ) - (-IX/A~I,) 1/3 solves (13b) and yields [ IX /1/3 N = nAl-~--~]

(14)

upon substitution into (12), where A is the area of S. This, together with Ag = kT In(p/p0), is the classical FHH isotherm on a planar surface [87]. It depends only on the substrate potential and is the paradigm of van der Waals wetting. It is independent of the surface tension because the last term in (6) is the same for all interfaces parallel to S. Example 2. The simplest case of a nonplanar surface is a spherical pore of radius R. Assuming the minimizing interface to be spherical and concentric with the pore, at distance z from the pore wall (0 < z < R), one finds from (7) and (13a) that z, the film thickness, must satisfy g(z)-A~t = 0 IX 2~ g(z) := - - - v 9 n(m z) z"

(15) (16)

For low A~, this equation has two solutions, 0 < z 1 < Z2 < R, with asymptotic behavior [ tX /1/3 z 1 -- ~-~--~] ,

2~ z 2 -- R + nAg

(17)

as Al,t -~ -,,o (Fig. 4). The expression for z 2 represents the Kelvin solution, i.e., the solution

0

z1 I l

ZXlac -J . . . .

Ag

z2 ........

i f

I

R Z i I

I

I

I

I

I

I

I

I

I

I

I

I

, I I I I

Figure 4. Graphical determination of the solutions z 1 and z 2 of Eq. (15) [schematic].

637 of (16) when z is so large that the substrate potential can be neglected. However, Z 2 cannot be the equilibrium solution because it would give the unphysical result that the pore is full at zero pressure. A simple calculation and Fig. 4 show that the grand potentials of the interfaces 11,12, 13 corresponding to z = z 1, z 2, R (full pore) satisfy A~[I 1, Ag] - A.Q[I2, Ag] = 4rm "Jz2 [Ag - g(z)l(R- z)2dz < 0,

(18a)

z1

A~[I 1, Agl - A.Q[I3, Ag] = 4rtn I R [Ag - g(z)](R- z)2dz,

(18b)

z1

i.e., 12 is unstable and the equilibrium state is 11 or 13 depending on whether the integral (18b) is negative (low Ag; metastable I3) or positive (high Ag; metastable I1). When Ag reaches the value Al.tc := max0 Ag c, in which case 13 is the equilibrium state. This gives the adsorption isotherm (partly metastable) N = (4rff3)n[R 3 - (R - zl) 3] N = (4/ff3)nR 3 Agc -- g(zr

Z1 = Zc "=

[ 4/ 1

-2__~_~ 1 + nR

~ 8(Y L ~

+

3~n 2(5R2

-

]

2R/-~ 3om8(5- 1

for Ag _< Ag c, for Ag > Ag c,

(19a) (19b)

for R ---) oo,

(20)

for Ag = Ag c.

(21)

For low Ag, one may use (17) to evaluate (19a), which in the limit of large R leads back to the FHH isotherm, Eq. (14). So adsorption at low Al.t is dominated by the substrate potential and, in a large pore, is indistinguishable from adsorption on a planar surface. As Ag increases, however, the film thickness grows faster than on a planar surface because the decrease of the energy (5~I d2x' resulting from a small interface radius, outweighs the increase of the particlenumber t e r m - n A g If(S,I) d3x" At Ag = Ag c, the enhancement of adsorption due to surface tension leads to spontaneous pore filling, resulting in a jump of the isotherm (Fig. 5). The jump describes capillary condensation, corresponding to a first-order phase transition ("thin-f'dm/fullpore" coexistence), in a highly idealized setting. The film thickness jumps from z c [Eq. (21)] to R. During desorption, the pore remains full at all Ag because 13 never turns unstable. Thus neither the adsorption nor desorption isotherm jumps at the point Ag c' < Ag e at which the integral (18b) vanishes (coexistence of 11 and I 3 as equilibrium states, "Maxwell construction"). The example illustrates that capillary condensation is controlled by surface tension, but that a thermodynamically consistent construction of the equilibrium interface is not possible without taking the substrate potential into account. This point has already been emphasized by Broekhoff and Linsen [ 107], and the example here has in fact been analyzed along similar, somewhat less explicit, lines before [ 108]. The example will serve as a useful comparison frame to appreciate similarities and differences in adsorption on a fractal surface. On a fractal surface, the adsorption regime controlled by surface tension will no longer exhibit any jump because the surface will have a hierarchy of pores and voids of many sizes. Instead, above some characteristic value Agc of Ag, one will have a continuous sequence of first-order transitions in each of which the "next" larger void in the hierarchy fills by capillary condensation. This turns the first-order transition at Al.t = Ag e in a single pore into what resem-

638

A~t

Al.t A~tc Figure 5. Film shape and adsorption isotherm in Examples 1 and 2 (schematic).

bles a second-order transition at A~t = Al.tc on a fractal surface. The change from a single pore to a fractal surface spreads the event of capillary condensation from a single chemical potential, Al.tc, to a whole range of chemical potentials, A~t > A~te. Since capillary condensation is often equated to an isolated jump in the isotherm, we refer to the regime Al.t > A~tc on a fractal surface as capillary wetting instead. Accordingly, we refer to the transition at A~tc as transition from van der Waals wetting to capillary wetting. This transition is worked out in the next section.

3. ADSORPTION ISOTHERM ON FRACTAL SURFACES The variational principle for the grand potential provides a powerful tool to construct approximations of the isotherm (12) when the Euler equation (13) has no simple solution, as is the case for a general fractal surface. We note that such approximations may be more desirable than the exact isotherm: The two fractal surfaces shown in Fig. 1 give isotherms that certainly differ in their exact details, but in practice will be indistinguishable from each other. So, an approximation well may capture this indistinguishability, while the exact isotherm does not. One chooses a family of trial interfaces and minimizes the grand potential with respect to this family. If the interfaces are sufficiently flexible, the one with the lowest grand potential should be a good approximation of the exact equilibrium interface. A simple and natural choice are equidistant interfaces I z [36, 43-45, 81,101,109, 110], defined as the locus of all points x outside the solid whose distance from the surface, dist(x, S), equals z (Fig. 6). The distance z is the variational parameter (0 < z < oo), making the trial family a one-parameter family. In comparison, the exact equilibrium interface may be regarded as member of an inf'mite-parameter family. The equidistant interfaces form a natural trial family for two reasons. First, they are equipotential surfaces of the substrate potential (7) because the latter depends only on dist(x, S). This makes them automatically solutions of the Euler equation (13) in the absence of surface tension. Second, they offer a convenient characterization of fractal surfaces. There are several, essentially equivalent ways of measuring the space-filling ability of a fractal surface [36]. The most precise one, well-defined for any surface S, is the function V(z) := If(S, Iz) d3x'

(22)

639 which is the volume of all points outside the solid whose distance from the surface is less than or equal to z. The derivative dV/dz equals the area of the interface I z, by the equidistance property of I z. We shall refer to f(S, I z) as film of thickness z, and to dV/dz as the corresponding film area. Equivalenfly, dV/dz may be interpreted as the surface area of the solid measured by tiling the surface with spheres of diameter z [36]. A self-similar surface with fractal dimension D, inner cutoff gmin, and outer cutoff gmax then is characterized by the power law dV dz

(23)

oc z 2-D

for gmin < z < s It describes how the film area on a fractal surface decreases with increasing film thickness by filling progressively larger pores and voids (D > 2), and remains constant on a planar surface (D = 2). The film area dV/dz plays a special role among different ways of assessing fractality because it depends only on the structure of the surface at length scale z, whereas the film volume V(z) and other interrogator functions depend cumulatively on all structure below z, including the regime below the inner cutoff, which may lead to departures from a pure power law for V(z) as z approaches s [43]. If the fractal regime starts at the smallest resolvable scale, i.e., at z = ao where a0 is the thickness of a monolayer (diameter of an adsorbate particle) as in (11), then (23) can be integrated and expressed entirely in terms of a0 and the number of particles in a monolayer, N m. The result, in its simplest form, is V(z) = Nma~ (z/ao) 3-D

(2 < D < 3),

(24a)

V(z) = Nma~ [1 + ln(z/ao)]

(D = 3, nonuniform space filling),

(24b)

V(z) = Nma~

(D = 3, uniform space filling),

(24c)

for ao < z < s Here the unspecified prefactor in (23) and the integration constant have been calibrated by the condition V(z)/V(%) -- (z/ao) 3-D or V(z)/V(a o) -- ln(z/a o) for large z/a o (scale invariance), and by equating the monolayer volume V(a 0) to Nma~. Other calibrations yield similar results [43, 110].

89 liquid

Figure 6. Equidistant interface, I z, at distance z from a self-similar fractal surface.

640

The two ways in which a three-dimensional surface can fill space, nonuniformly or uniformly, are discussed in [36, 43, 81, 111 ]. In the nonuniform case, the surface visits only certain regions in a compact manner and therefore, by leaving other regions empty, contains pores of many sizes. Examples for such surfaces, satisfying (23) with D = 3 and a nonzero prefactor, are certain silica and alumina xerogels [12, 39, 80, 81, 111 ]. A uniformly space-filling surface, by contrast, contains pores of one size only. A good example is a zeolite with channel width w and comparable wall thickness, which is uniformly space-filling at length scales above w. In this case, the volume V(z) trivially remains constant for all z > ao, if we choose a0 = w. This illustrates (24b) and the mechanism by which the prefactor in (23) may be zero. The expressions (24) show that the film volume grows, with increasing film thickness, increasingly slowly as the fractal dimension increases from two to three. They quantify that on an increasingly convoluted surface there is less and less space for a film to grow. The slow growth on a high-dimensional surface may seem to contradict the intuition that a highly irregular surface should have a large surface area and hence should be able to support a large film volume. There is no contradiction, however: In (24), we compare surfaces with variable D and fixed number of surface sites (adsorption sites), N m. In the surface-area consideration, one compares surfaces with variable D and fixed diameter L (largest distance between two points on the surface). Thus, the two opposed conclusions are due to different comparison frames. To switch from one frame to the other, one simply uses the relation Nm =

(L/gmax)3(gmax/ao)D,

(25)

in which the first factor counts how many fractal, identical "pieces" the surface consists of and the second factor counts the number of surface sites on each piece. For example, if the surface is fractal over the entire range of length scales between a0 and L ( g m a x = L), then N m = (L/ao) D and substitution into (24) shows that the film volume indeed increases with increasing D, for fixed L and' z. Similar differences in the performance at constant N m and at constant L, as a function of fractal surface dimension, exist also for diffusion-controlled reactions [20, 52-54], electrochemical response [53, 54, 59, 61 ], and other applications. This completes our analysis of the equidistant interface I z and the associated film volume V(z), in which we have treated the thickness z as a variable that can take arbitrary values. The variational determination of the equilibrium value for z and the construction of the isotherm is now easy: Substitution of (7) and (22) into the grand potential, Eq. (6a), gives A.Q[I z, Al.t] = n rjz [_tx(z,)_3 _ Akt]dV(z') + ~dV(z)/dz, a0

(26)

where we have used that dV(z') is the volume of a layer of thickness dz' at distance z' from the surface (equidistance property of Iz,), and have set the lower integration limit equal to the monolayer thickness to ensure a finite integral. Minimization with respect to z yields the algebraic equation (replacing the Euler equation (13)) n[--txz - 3 - Al.t]dV(z)/dz + t~dV2(z)/dz 2 = 0

(general surface)

(27)

n[-o~z -3 - A~t] - ( D - 2)~z -1 = 0

(fractal surface )

(28)

for the equilibrium value of z. The use of (24) in arriving at (28) is limited to the cases (24a, b), of course. In the case (24c), there is no isotherm to construct and we shall disregard it in the

641 sequel. The left-hand side of (28) increases with increasing z, so that the equation has a unique positive solution z for every value of Ag. The equation is equivalent to a cubic equation for z, which can be solved exactly. But the basic features of the solution can be deduced directly from the low- and high-Ag regime in (28): For zig ---) ..oo, the surface-tension term can be neglected, while for zig ~ 0, the substrate-potential term can be neglected. This gives z - ( ~__~)1/3

(Ag --+ _._~),

(29a)

z ~ - (D - 2).._.____~ nag

(Ag ~ 0).

(29b)

-

The crossover between the two regimes occurs at the chemical potential and film thickness, Ag c and zc, at which the asymptotes (29a, b) intersect, i.e., at

A~I,c 1= - 0~-1/2 [(D zc :=

2)6/n] 3/2,

(30)

(D - 2)6 "

(31)

Substitution of the equilibrium film thickness z as a function of Ag into N = nV(z) [recall Eqs. (12, 22)] yields the adsorption isotherm. Thus, by extrapolating the asympotic expressions (29) to where they intersect, inserting them into (24), and using ao - n-u3 to eliminate %, we obtain the results displayed in Table 3.

Table 3 Multilayer adsorption on a fractal surface, with Ag and Agc given by (5) and (30). Van der Waals wetting

Capillary wetting

Range of Ag 9 D --~ 2 9 D --, 3

Ag < Ag c range increases range decreases

Ag >__Ag c range vanishes range increases

Isotherm: 9 2
N =Nrn [ - - ~ g 1j

(32a)

N=N m

N ~ [-ln(p/p0)]-(3-D)/3

(33a)

N ~ [-ln(p/p0)]-(3-~

(33b)

N = N m [ l + l l n ( -~,I x nAg)J )13

(34a)

N = N m I l + l n (-n2/36)lAg

(34b)

N ~ const- ln(-ln(p/p0))

(35a)

N ~ c o n s t - ln(-ln(p/p0))

(35b)

9 D=3

Interaction

__ (3-D)/3

Substrate potential pulls liquid-gas interface I to the surface S; f'tim grows slowly.

[ (D--2)6n2/3A] -t jq3_D

(32b)

Surface tension pushes liquid-gas interface I from the surface S; f'flm grows fast.

642 The results in Table 3 are the comerstones of multilayer adsorption on a self-similar fractal surface, as function of the chemical potential A~t or gas pressure p. They are subject to the condition that the adsorbed film is at least a monolayer (N > Nm) and that fractality starts at the length scale of the thickness of a monolayer. The regimes A~t < A~tc and All, > A~tc are identified as van der Waals wetting and capillary wetting because the isotherm depends only on the substrate potential strength a and surface tension ~, respectively The chemical potential A~tc given by Eq. (30) is therefore the transition point anticipated in Sect. 2.4 for a fractal surface. It is the fractal analog of the capillary-condensation point A~tc, Eq. (20), in a single pore. We divide the discussion into several subsections. The transition from van der Waals wetring to capillary wetting, which for simplicity we treat as sharp transition in this section, will be analyzed further in Sect. 4. Experimental examples will be presented in Sect. 5. 3.1 Earlier Derivations The power law for van der Waals wetting was first obtained in [44] (see also [ 109]). It was derived by assuming that films are sufficiently thin that surface tension can be ignored. The whole set of isotherms (32-35), including the logarthmic dependence for D = 3 and the coexistence point (30, 31) of the two types of wetting, were obtained in [101] (see also [45]). In fact, our derivation here is a streamlined version of the one in [ 101]. A more refined analytic calculation, in which the interface was not equidistant and the substrate potential was not simply a function of dist(x, S), was performed in [112] and reproduces the results (30-33) within a factor of order one. Other analytic treatments give similar agreement [ 113, 114]. The power law for capillary wetting or equivalents thereof, on the other hand, has been discovered and rediscovered in several different contexts, such as pore filling under hydrostatic pressure [ 115-117], third sound in superfluid 4He films [ 118], adsorption in micropores [ 119], and capillary condensation in mesopores [120-124]. Even the fractal generalization of the BET isotherm [109, 110, 125-128], which is unphysical beyond a few layers, coincides with (33b, 35b) in the limit p ~ P0- Most of these contexts neglect the substrate potential or model it inadequately. As a result, they give no estimate of the pressure above which (33b) would be valid, give prefactors which differ from the one in (32b), or give no prefactor at all. We therefore will focus our discussion on the present framework. 3.2 Nonclassical FHH Isotherms The classical FHH isotherm, Eq. (14), is of the form

N o, [-ln(p/p0)]-Y,

(36)

with y = 1/3. It is included in the fractal isotherm as the special case in which the surface irregularity vanishes. Indeed, in the planar-surface limit D -~ 2 the transition point Al.tc approaches zero and the associated film thickness zc diverges, so that for D = 2 the capillary wetting regime is absent in the isotherm (32) and only the van der Waals regime with y = 1/3 exists. We call (36) with y ~ 1/3 a nonclassical FHH isotherm because a departure from the value 1/3 signals a departure from the classical planar surface geometry, a departure from the classical inversedistance-cubed law for the energy of a particle at some distance from the surface, or both. Experimentally, it is weN-known that many adsorption data can be fitted to (36), but rarely with an exponent equal to 1/3. Typical experimental exponents y range between 0.4 and 0.7 [85, 87, 129-131]. Following Halsey [132], the discrepancy has been rationalized for a long time by a model in which the surface is planar, but patchwise energetically heterogeneous.

643 Each patch supports a film obeying Eq. (14), and the substrate potential strength ot varies from patch to patch. This model, under suitable assumptions for the distribution of or, leads to isotherms that can be fitted to (36) with an exponent y = 0.4 [87, 131,132]. The nonclassical FHH isotherms in Table 3, covering the whole range 0 < y < 1 by virtue of 2 < D < 3, offer a very different interpretation of experimental isotherms with y ~: 1/3. They replace Halsey's hypothesis that the surface is geometrically homogeneous (planar) and energetically heterogeneous by the hypothesis that the surface is geometrically heterogeneous (fractal) and energetically homogeneous. The merits of the fractal interpretation, as a working hypothesis, are considerable: (i) Halsey's interpretation requires the specification of a whole function, namely the distribution of ~ values, which is difficult to test by independent experiments and therefore must be regarded as phenomenological. The fractal interpretation requires the specification of a single parameter, D, which can be tested by independent experiments (Table 1) and by internal thermodynamic consistency (Sect. 4). (ii) The variable o~ values in Halsey's model may be attributed to variable-index domains in a polycrystalline surface or other defects in a planar surface. But such local energy differences are rapidly averaged out as the adsorbed film grows beyond a few layers and cannot account for y ~ 1/3 for thick films. By the same argument, the local energy differences (departures from the substrate potential (7)) that may exist on a fractal surface as a result of the geometric heterogeneity, are averaged out in films exceeding a few layers. Exact calculations of the potential U(x) for a fractal surface confirm this [133]. Even at coverages less than a layer, energetic heterogeneities are irrelevant on a fractal surface at sufficiently high temperatures [ 125] (see also [29]). Thus, under a wide range of circumstances, energetic heterogeneities of Halsey's type can be disregarded, leaving geometric heterogeneity as the most likely source of experimental exponents y ~: 1/3. (iii) Halsey's model is designed to explain exponents y > 1/3. But experimentally, one finds also y < 1/3, although less frequently. The fractal model is capable of explaining both deviations and covers to the best of our knowledge all experimental exponents that have ever been observed. The correspondence between y and the fractal dimension is illustrated in Fig. 7. It shows that, for 1/3 < y < 1, the dimension can be uniquely reconstructed from y as D = 3 - y, that the resulting D value lies between 2 and 8/3, and that the value of y necessarily implies capillary wetting. This agrees remarkably well with the earlier suggestion, predating the fractal concept, that exponents y > 1/3 are "probably explicable by some reversible capillary condensation" [ 129], and converts that suggestion into a quantitative statement about surface irregularity. For 0 < y < 1/3, the correspondence between the isotherm exponent and the fractal dimension is no longer one-to-one. That is, the fractal dimension may be either D = 3 - y (capillary wetting), giving 8/3 < D < 3, or else D = 3 - 3y (van der Waals wetting), giving 2 < D < 3.

Zyy

= 3 - D [capillary wetting]

.

.

.

.

.

.

.

.

= (3 - D)/3 [van der Waals wetting] 1/3

-ilj~176 2

8/3

3

D

Figure 7. Isotherm exponent y as function of the fractal dimension D, as given by Eq. (33).

644 Which of the two possibilities is the appropriate assignment when only y is known, is a question that has been raised in several experimental studies using the isotherms in Table 3. We will answer that question systematically in Sect. 4. A partial answer is provided by the inequality 3-3y

< D < 3-y,

(37)

which translates the two possibilities into upper and lower bounds for the sought-after dimension (Fig. 7). These bounds, which are tight when y is small, may be taken as substitute for the missing one-to-one correspondence between y and D for 0 < y < 1/3. If the y value under consideration comes from the low-coverage regime in the isotherm, i.e., from the regime starting at N/N m = 1, one additionally has the role of thumb that the closer to 1/3 the y value lies, the more likely is the choice D = 3 - 3y (van der Waals wetting) the correct one. Indeed, the only way how a value y -- 1/3 can originate from capillary wetting at low coverage is if D -- 8/3 and simultaneously the film thickness z c at which capillary wetting sets in, Eq. (31), is of the order of a monolayermwhich would be an unlikely coincidence. Together, this rule and the above bounds give fairly strong guidelines for the interpretation of exponents in the range 0 < y < 1/3 in terms of fractal dimension [ 134].

3.3 Reduced Isotherm and Law of Corresponding States The isotherms (32, 34) offer a fully developed theory of multilayer adsorption on irregular surfaces with a minimum number of parameters: The number of particles in the monolayer, N m, describes the sample size; the fractal dimension D describes the surface geometry; the potential strength ~ describes the substrate-adsorbate interaction; the surface tension ~ describes the adsorbate-adsorbate interaction; and the number density n describes the adsorbate particle size (recall a0 = n-1/3). Interpreted in this way, the five parameters are clearly independent, none of them is dispensable, and none of them is adjustable. One gains additional insight into the analytic structure of the isotherms by expressing them in reduced variables. The reduced form is N/N m = [max{~, (D-2)T~3}] 3.D

(2 < D < 3),

(38a)

N/N m = 1 + In(max{ ~, y~3 })

(D = 3),

(38b)

/ an/l/3

; := ~-S-ff] y :=

'

t~ . t~n5/3

(39) (40)

The expression max {....... } selects the larger of the two arguments and automatically reproduces the van der Waals regime for ~ < ~c [N/Nrn < (N/Nm)e], and the capillary wetting regime for ~ ___~e [N/Nm > (N/Nm)c], where ~c := [(D-2)Y] -1/2,

(41)

(N/Nm) c := [(D-2)y] -(3-D)/2.

(42)

The variables ~, ~c, and T are all unitless and take the role of Ag, Ag e, and the triple (~, tj, n) in Table 3. The variable ~ is the film thickness z, measured in units of the thickness of a monolayer, one would have at chemical potential Al.t if the surface were planar. In agreement, the isotherm (38) for D = 2 reads N/N m = ~. The value ~c predicts the film thickness z c at which

645 capillary wetting sets in, Eq. (31), in units of the monolayer thickness; and the value (N/Nm) c predicts the associated coverage. By the remarks at the beginning of this subsection, the constant 3' measures the strength of the adsorbate-adsorbate interaction relative to the substrateadsorbate interaction. Since ao = n -1/3, it may be interpreted as the ratio of the liquid-gas free energy, ca~, to the potential energy, ~a~ 3, of a particle in a monolayer. We therefore call 3' the capillary-energy ratio. In wetting dynamics (spreading of liquids), a similar energy ratio, namely surface tension divided by the product of liquid velocity and viscosity, is known as the inverse of the capillary number [95, 133]. The reduced form of the isotherm displays the full D dependence of the adsorption process (note that ~ and 3' do not involve D), encapsulates the competition between the substrate potential and surface tension in the single number q,, and describes the dependence on the chemical potential through the single variable ~, the number of equivalent layers on a planar surface. As a result, it reduces the dependence on four parameters in Table 3 (D, ~, (~, n) to a dependence on only two parameters (D, 3'). The fact that 3' enters exclusively via the combination (D - 2)3' [Eqs. (38, 41, 42)] says that the liquid filling the hierarchy of small and large pores of the fractal surface has an "effective surface tension" equal to ( D - 2)~. Thus, compared to the liquid-gas equilibrium in a single pore, the fractal surface lowers the effective liquid-gas free energy by a factor of D - 2. A similar lowering of effective energy exists in chemical reactions on a fractal surface, for stationary diffusion of reactant molecules from an external source to the surface: There the effective activation energy is lowered, by a factor of 1/(D - 1), compared to the reaction on a planar surface [52, 136, 137]. The reduced isotherm leads to the following simple way of testing an experimental isotherm for fractality of the underlying surface. The idea is to employ the coverage N/N m, rather than the chemical potential A~t, as calibration scale, i.e., as variable to distinguish van der Waals wetting from capillary wetting and to determine absolute film thicknesses. Assuming that the monolayer value N m is known (say from a BET analysis of the low-pressure part of the isotherm), all one needs to do is to plot the coverage N/N m as a function of In(p/p0); test whether the data, starting at N/N m = 1, obeys one of the power laws (33a, b); andmin the event that the data obeys (33a) at low coverage and (33b) at high coverage (with the same D)---test whether the crossover from (33a) to (33b) occurs at the coverage predicted by Eq. (42). The start of the power law at N/N m = 1 reflects the start of the fractal regime at the molecular scale. The test for D = 3 proceeds similarly. Since the expression max {....... } in the reduced isotherm is the film thickness, in units of the monolayer thickness, on the fractal surface (both for van der Waals and capillary wetting), one can convert any experimental coverage into a corresponding film thickness z: / N ~l/(3-D) z = /N~-~) a~

(2 < D < 3),

(43a)

z = exp

(D = 3).

(43b)

- 1 ao

This gives the length scale of surface irregularities probed at coverage N/N m [ 134]. The smallest and largest film thickness computed from (43), as the coverage varies over the range in which the data obeys (33) or (35), gives the inner cutoff s (-- %) and outer cutoff s respectively, of the fractal regime of the surface. Representative values of the parameter 3' and related data are listed in Table 4. The film thickness ~/o~n/o is the smallest possible film thickness at which capillary wetting may set in. It

646 Table 4 Substrate potential strength or, film thickness ~/~n/c~ at which capillary wetting sets in for D = 3 [Eq. (31)], and capillary-energy ratio 3' [Eq. (40)], for nitrogen on different solids. The surface tension and number density of liquid nitrogen, in coexistence with its vapor at T = 77.347 K, are o = 8.85.10 -16 erg,~ -2 and n = 1.738-10 -2 ,~-3 [138]. The resulting monolayer thickness is a 0 = n -1/3 = 3.86 ,~. Solid

c~ (erg,~ 3)

SiO 2 (quartz) C (graphite) Si A1 Ag Au

6.19-10 -13 1.42.10 -12 1.59.10 -12 2.13-10 -12 2.35-10 -12 2.62.10 -12

[89] [89] [89] [89] [112] [89]

qo~n/o (.~)

7

3.49 5.28 5.58 6.46 6.80 7.18

1.23 0.534 0.478 0.357 0.323 0.289

measures the competition between the substrate potential and surface tension in terms of a length and is a characteristic length also for multilayer adsorption on other nonplanar surfaces (see Sect. 2.4 and [ 100-104]). When the substrate potential is so weak or surface tension is so strong that qom/c < ao, which is equivalent to ~/> 1, and if D > 2 + 1/3', then capillary wetting sets in at a film thickness that is nominally less than the thickness of a monolayer [Eqs. (31, 41)]. In this case, adsorption obeys the capillary-wetting isotherm for all N/N m > 1, i.e., the regime of van der Waals wetting is absent. This means that capillary forces may dominate already at the stage of a monolayer and conforms with the observation that the monolayer has an effective liquid-gas free energy exceeding the substrate-potential energy whenever (D - 2)7 > 1. On a planar surface, the situation 3' > 1 expresses that the lateral attraction among adparticles in a monolayer is stronger than their attraction to the solid. An instance of such a weak substrate potential is SiO 2 in Table 4. The weak substrate potential of SiO 2 can be attributed to its low polarizability (electric insulator). Accordingly, the increase of the substrate potential, as we go down in the table, reflects the increase in polarizability with increasing metallic character of the solid. If one views the capillary-energy ratios in Table 4 as significantly different, then the variety of ratios predicts a corresponding variety of different behaviors of the isotherm (38). But if one regards the capillary-energy ratio as only weakly varying, T-- 1, then the reduced isotherm has the status of a law of corresponding states for films on a fractal surface, quite analogous to the law of corresponding states for bulk fluids. Indeed, just as bulk fluids satisfy, at least semiquantitatively, a universal equation of state that makes no reference to material constants if expressed relative to the pressure Pc, temperature T e, and number density n c at the liquid-gas critical point [139], the films on a fractal surface satisfy a universal isotherm making no reference to material constants (other than the fractal dimension) if expressed in the reduced form (38). The independence of material constants manifests itself in the relation pe/(kTenc) = 0.27 + 0.04

(normal bulk fluids),

(44)

o/(otn 5/3) = 0.76 + 0.47

(adsorbed films),

(45)

respectively, where the estimate (45) is based on Table 4 and hence is preliminary.

647

3.4 Does the Roughness of the Substrate Enhance Wetting? In the study of general 3-phase equilibria between a liquid, its vapor, and a solid (Fig. 2), this question [ 140] has been investigated from many different angles (interfacial energies, contact angle, spreading kinetics, transition to complete wetting) and for several types of roughness [83, 85, 135, 140-152]. The genetic answer is yes, in the sense that the contact angle decreases or increases with increasing roughness (enhanced or diminished wetting), on a surface without overhangs, depending on whether the contact angle on the planar surface is less than or larger than rff2 [85,140]. Here, for completely wetting films on a fractal surface with Overhangs (Fig. 6), we address the question in terms of whether the coverage N/N m grows faster or slower than on a planar surface as A~t ~ 0 (enhanced or diminished wetting). We assume for simplicity that the surface is fractal at all length scales above a 0, i.e., that the outer cutoff is formally infinite. The surface then has voids of all sizes and, based on the picture that capillary condensation in voids of ever larger size systematically enhances adsorption [ 129], one might expect that the coverage always grows faster than on a planar surface. However, this is not the case. The isotherms (32, 34), or equivalently (38a, b), show that the coverage grows faster in the limit Akt ~ 0 if and only if 1/3 < 3 - D < 1. We therefore have enhanced wetting if 2 < D < 8/3, diminished wetting if 8/3 < D < 3

(46a) (46b)

[ 112]. In Fig. 7, the situation (46a) corresponds to 1/3 < y < 1 and identifies the isotherm exponents which give enhanced wetting as those from which D can be uniquely reconstructed. To explain how (46) arises, including the paradoxical consequence that the coverage grows fastest on a nearly planar surface (D ~ 2), we begin at low A~ and note that the film thickness at low A~t is the same as on a planar surface, Eq. (29a), so that the coverage depends on the fractal dimension only through the film volume, which decreases with increasing D. Hence, for van der Waals wetting the coverage always decreases with increasing D. When we enter the capillary wetting regime, the film thickness becomes D-dependent and grows faster than on a planar surface, Eq. (29b). The question then is, can this fast growth of the film thickness compensate for the slow growth of the film volume, so as to make the isotherm for D > 2 overtake the classical FHH isotherm at sufficiently high Al.t? The answer is yes if the film volume V(z) does not grow too slowly with increasing film thickness z, i.e., if the fractal dimension is not too high. Thus, a surface with 2 < D < 8/3 is sufficiently porous to induce capillary condensation and, at the same time, sufficiently "open" for the condensate to grow in excess of the growth on a planar surface. A surface with D _>8/3 is also sufficiently porous to induce capillary condensation, but too "closed" for the condensate to grow as fast as on a planar surface. On a nearly planar surface, with D very close to 2, the van der Waals regime will be very extended and the coverage will be close to that on a planar surface up to very high values of Alx; but no matter how close to zero A~tc may be, once Akt exceeds the value Al.tc, the nonplanarities of the surface at those large length scales will be sufficiently magnified to induce capillary condensation and make the coverage eventually grow as (-A[.t) -(3-o). Thus the paradox of fastest growth on a nearly planar surface is resolved by the observation that the limits Al.t -~ 0 and D --~ 2 are not interchangeable [ 112]. A convenient way to analyze the growth of the coverage in more detail and compare it with the growth on a planar surface is to plot the reduced isotherm (38) as a function of ~. Since ~ is the coverage one would have at chemical potential Akt if the surface were planar, such a plot is in fact equivalent to a t-plot or comparison plot [ 129], using the classical FHH isotherm as

648

4

3 -

~

N/N m 2

-

. ~

(b)

1

0 0

1

I ,, 2

I 3

I 4

J

5

Figure 8. Reduced isotherm (38) as a function of the unitless variable ~: (a) for D = 2; (b) for D = 2.40 and T = 0.323; (c) for D = 2.40 and y = 2.50. Depending on whether ~ is interpreted as film thickness (in units of the monolayer thickness) or amount adsorbed (in units of a monolayer) on the planar reference surface, the curves represent the t-plot or the comparison plot of the adsorption isotherm on the respective substrates.

standard isotherm. Such plots for selected D values and capillary-energy ratios y are shown in Fig. 8. The straight line (a) is the isotherm on the planar reference surface. Curves (b) and (c) are the isotherms on a 2.4-dimensional surface for two different values of y. They both lie above the planar-surface isotherm for sufficiently large ~, in agreement with (46a). In (b), the value of T, representing nitrogen on silver, is low enough for the van der Waals regime to extend up to ~ = 2.78 [Eq. (41), second dotted line] and for capillary wetting to make the isotherm cross the planar-surface isotherm only at ~ = 4.64 [third dotted line]. In (c), the value of y has been chosen high enough for the van der Waals regime to be absent, i.e., for capillary wetting to set in at ~ = 1 [Eq. (41), first dotted line]. This makes the isotherm (c) lie above the planarsurface isotherm for all ~ > 1. We note that the break points at ~ = 2.78 and ~ = 1 in (b) and (c) result from our treating the crossover from van der Waals wetting to capillary wetting as a sharp transition in this section. They are smoothed out if one solves Eq. (28) for the film thickness exactly and puts the resulting isotherm into reduced form. Likewise, the submonolayer regime in the isotherms in Fig. 8 (N/N m < 1) should not be taken at face value, because the underlying expression (32a) does not describe submonolayer adsorption properly. Figure 8 gives a fine-tuned picture of how a surface with 2 < D < 8/3 enhances wetting, by displaying the value ~e at which the fractal isotherm intersects and overtakes the planar-surface isotherm. We call this point the onset of enhanced wetting and define it generally as ~e := min{~'" (~---m-m)(D,~)>(~mm}(2,~)for all ~> ~'; ~'> 1} = max{ 1, [ ( D - 2)T]-'(3-D)/(8-3D)},

(47a) (47b)

where the condition ~'> 1 serves to exclude the submonolayer regime from consideration. The result (47b) is simply the solution of [ ( D - 2)]t~3}] 3-D = ~ if this solution is > 1, and equal to

649

20. 15. I0.

2 1.5 ~ ~

~

~

'

~

I

~

I

~ - ~

I 1

2

2 2

2 4

,

,

,

i

2.6

,

I,

~ ,

,

~

~

...._._.

_........

I

9

2.8

3

D

Figure 9. Onset of capillary wetting, ~c [Eq. (41), dashed curves], and of enhanced wetting, [Eq. (47), full curves], as function of the fractal dimension. The curves from top to bottom are for y = 0.323, y = 1.50, and 3t = 2.50. For fixed D and y, the regimes ~ < ~c, ~ > ~c, and ~ > ~e amount to van der Waals wetting, capillary wetting, and enhanced wetting. To the right of the vertical line at D = 8/3, enhanced wetting no longer exists.

one else. The significance of ~e is that it distinguishes whether in (46a) the onset of enhanced wetting occurs at low or high chemical potential (low or high ~e)" We illustrate this in Fig. 9. For D ~ 2, all curves ~ diverge in agreement with our discussion of Eq. (46a). For D ~ 8/3, the curves ~e diverge, too, if'f < 3/2. In this case capillary condensation is not strong enough to overcome, at some low chemical potential, the slow growth of the film volume as D ~ 8/3. Accordingly, the curve ~e for nitrogen on silver first drops with increasing D until it reaches a minimum of ~e = 4.41 at D = 2.32, and then rises again. If 3t > 3/2, however, capillary condensation is strong enough to make the onset of enhanced wetting drop all the way down to the chemical potential at which the planar reference surface supports a monolayer, as D increases. For example, the fact that the isotherm (c) in Fig. 8 lies above the planar-surface isotherm for all > 1 is mapped into the value ~e = 1 for D = 2.40 and y = 2.50 in Fig. 9. We note that Fig. 9 also clarifies the status of the borderline case D = 8/3 in (46): For D = 8/3, wetting is enhanced if 3t > 3/2 (~e < oo), and diminished if T < 3/2 (~e = oo). These results for the onset of enhanced wetting on a fractal surface lead to guidelines for the interpretation of t-plots and comparison plots which differ substantially from the traditional guidelines. The traditional wisdom [129] is that (i) capillary condensation, with or without hysteresis, manifests itself in an "upward swing" of the t-plot or comparison plot, i.e., in a plot that lies above the straight line representing the standard isotherm; and that (ii) the presence of micro- and/or mesopores manifests itself in an upward swing of the plot, with or without the adsorption equilibrium being controlled by capillary forces. The presence of an upward swing, also called enhanced adsorption in [ 129], amounts to a finite value of ~e in Fig. 9. Thus, Fig. 9 confirms the validity of statement (i) for 2 < D < 8/3, but shows that the statement is not valid in the range 8/3 < D < 3 because in that range capillary

650 condensation does not enhance adsorption. The validity of statement (ii) is even more restricted. Indeed, if we interpret the statement as saying that enhanced adsorption starts at ~ = dmin/a0 where dmi n is the smallest pore diameter, enhanced adsorption should start at ~ = 1 on a fractal surface with inner cutoff a 0. But since van der Waals wetting never enhances adsorption, the statement is valid only if ~e = 1. Hence statement (ii) is valid only if 2 + 1/3, < D < 8/3 [Eq. (47b)]. It is easy to see how the guidelines need to be modified to be valid without restrictions. All one has to do is to weaken them from "if-and-only-if" statements to "if" statements. That is, capillary condensation is operative and micro- or mesopores are present if there is an upward swing in the plot. The converse, however, is not true: As fractal surfaces show, pore filling may occur, by capillary condensation or otherwise, without an upward swing.

3.5 Comparison with Adsorption in a Single Pore Having obtained the transition from van der Waals wetting to capillary wetting in a single pore (Sect. 2.4) and on a fractal surface (Table 3), we now wish to highlight the parallels and differences between the two transitions. We compare the transitions by analyzing their dependence on the energy parameters and surface geometry. The chemical potential Agc, Eqs. (20) and (30), sets the energy scale for the transition. The film thickness z c, Eqs. (21) and (31), sets the associated length scale. As in Sect. 3.3, we express Agc in units of the potential energy per particle in a monolayer, czn, and z c in units of the monolayer thickness n -1/3. This reduces the dependence on o~ and ~ to a dependence on the capillary-energy ratio 3, and leads to the results in Table 5, the fractal entries being familiar from Sect. 3.3. We begin with the dependence on the energy parameters. Since ABc and z c predict the point at which capillary forces become stronger than the substrate potential, their value must decrease with increasing 3,. Lines 1 and 2 in Table 5 confirm this both for the fractal surface and the single pore. The specific dependence on 3, is quite different, however: The magnitude of the exponent of 7 is consistently larger in fractal case (by a factor of 3/2 for Agc, and 2 for Zc). Thus the onset of capillary wetting depends strongly on the competition between cz and o in the fractal case, and by comparison weakly in the single-pore case.

Table 5 Onset of capillary wetting on a fractal surface and in a single pore, with 3, given by (40). The first two entries in the last column are the leading terms of the asymptotic expansion. Fractal surface

Single pore

Agc/(om)

- [ ( D - 2)T] 3/2

-27/(Rn I/3) [R --4 oo]

Zcn 1/3

[(D - 2)3,]-1/2

(23'/3)-1/41/Rnl/3 [R --4 oo]

Dependence on energy

strong

weak

Dependence on geometry

D

R

Dependence on size

none

R

Planar-surface limit

z c ~ oo [D ~ 2]

z c --~ oo [R ~ oo]

Maximally nonplanar surface

z c = n-1/3T-1/2

Zc= n-1/3(l+ ~/l+V8T/3 ) -1

= 3-7 ~, [ D = 3 ]

= 1-2 ,~ [R = (1/2)n -1/3]

651 Next we consider the transition as function of surface geometry, restricting attention to the film thickness z c. Just as z c in the single pore depends on the surface geometry via the pore radius R, the value of z c on the fractal surface depends on the surface geometry via D. However, while in the single-pore case this dependence on geometry involves simultaneously a dependence on size, R, there is no size dependence in the fractal case because a fractal surface is scaleinvariant (D is a size-independent measure of surface irregularity [12]). Capillary effects should become weak and z c should increase, as the degree of nonplanarity of the surface decreases. This is indeed the case, as the planar-surface limits in Table 5 show; both for the fractal surface and the single pore, there is no capillary condensation at any finite film thickness in this limit. Conversely, capillary effects should be strongest and z c should be smallest when the surface is maximally nonplanar. The notion of maximally nonplanar is naturally defined as D = 3 in the fractal case, and as a pore that can hold exactly one particle in the single-pore case. The resulting values for z c in Table 5 (based on the values from Table 4 for n and 7) are of atomic size in both cases, reinforcing the conclusion that capillary forces may control the formation of already the first layer. It might be argued that the onset of capillary consensation at a film thickness of a few ,~mgstroms, on a maximally nonplanar surface, is not really surprising because, if pores with diameter comparable to the diameter of a single adsorbate particle are present, these pores are automatically full after the first or second layer. But capillary condensation at the level of one or two layers is not trivial because in the fractal case a surface with D = 3 has room for an unlimited number of layers [Eq. (34)]. In the single-pore case, it is not trivial because condensation occurs at z c < R (Table 5). In this light, we consider it as a signature of remarkable internal consistency that nonplanar surface geometries of such different types yield similarly low values for z c in the limit of maximum nonplanarity. This consistency suggests that the model of capillary condensation in a single pore, when treated including the effects of the substrate potential and applied to micropores, may in fact perform considerably better than what recent model studies of adsorption in micropores seem to suggest [ 153, 154]. A somewhat different picture of the transition from van der Waals wetting to capillary wetting emerges when we compare the fractal and single-pore case away from the transition point. On the fractal surface, the isotherm gives the full structural information about the surface (i.e., the dimension D) both below and above the transition point, as shown by the D dependence of Eqs. (32, 34) or (38). In principle, therefore, the D value can be obtained in three different ways, namely from the isotherm exponent in the van der Waals regime, from the isotherm exponent in the capillary regime, and from the transition point itself. The multiple manifestation of D is naturally a consequence of the recurrence of the same structure at all length scales. In a single pore, by contrast, the isotherm exhibits only little information about R below the transition point (the film thickness z 1 in Fig. 4 depends only weakly on R for Ag << Agc), and no information above the transition point, when analyzed in terms of incremental adsorption dN/d~g. Thus in a single pore, the structural information revealed by the isotherm is restricted to the transition point, reflecting that the surface has structure only at one length scale, R. Finally, adsorption on a fractal surface and in a single pore can also be compared from the viewpoint of decomposing the fractal surface into single pores (hierarchy of ever larger voids) and regarding adsorption on the fractal surface as the sum of adsorption events in single pores. A decomposition of a fractal surface into nominally single pores is shown in Fig. 10. The decomposition and the results for a single pore imply that, at any given chemical potential Ag, some pores are full (capillary condensation in small pores) and some pores are partly filled (van der Waals wetting in large pores). This leads to the conclusion that van der Waals and capillary

652

Figure 10. Decomposition of a fractal surface into spherical pores of variable radius R (from Ref. [44]).

wetting coexist at any Ag, and one may ask how such coexistence can be reconciled with two separate regimes as predicted by Table 3 [155]. The answer is as follows. The film thickness z c at which capillary wetting sets in on the fractal surface, Eq. (30), divides the pore hierarchy into small and large pores according to R < zc R > zc

(small pores), (large pores).

(48a) (48b)

The filling of small pores, taking place when Ag < Ag c, is governed by the substrate potential, in the sense that the isotherm jump in a single pore due to capillary condensation is small. In this sense capillary condensation is negligible in small pores. The filling of large pores, taking place when Ag > Ag c, is governed by capillary condensation, in the sense that most of the increase in adsorption comes from the jump in pores with radius R-- 2~/(-nAg) [Eq. (20)], while the contributions from van der Waals wetting in larger pores (where capillary condensation has not occurred yet) are negligible. The reason why van der Waals wetting is negligible in large pores is because the film thickness is at most R1/E(6om/tJ) TM prior to capillary condensation [Eq. (21)], which is small compared to R for large R. Thus small and large pores are indeed filled by van der Waals and capillary wetting, in agreement with Table 3. Explicit calculations of the isotherm on a fractal surface in terms of single-pore adsorption have been carried out in Refs. [44, 112].

3.6 Effective Potential and Extension to Arbitrary Surface Geometries Can the variational isotherm calculation, as underlying the results in Table 3, be extended to other irregular surface geometries? If so, can the calculation be inverted to extract information about the geometry from a general experimental isotherm? The answer is yes. To see that and gain additional insight into the nature of the adsorption equilibrium, we return to Eq. (28) which determines the equilibrium film thickness z on a fractal surface and can be written as Ueff(z) - Ag = 0 Ueff(z) := - ~ z 3 - (D - 2)~/(nz).

(49) (50)

The function Ueff(z), which we call effective potential has a simple and important physical meaning. It is the difference between the energy of adding a particle to an equilibrium film of

653 thickness z on the fractal surface and the energy of adding a particle to bulk liquid, all effects included. It separates the adsorption isotherm into two regimes, van der Waals wetting and capillary wetting, by decomposing the energy into a sum of the substrate potential -ot/z 3, dominant at short distances, and the "capillary p o t e n t i a l " - ( D - 2)cy/(nz), dominant at large distances. The slow growth of the capillary potential at large distances induces the fast film growth indicated in Table 3. More generally, the lower the effective potential is as a function of distance z, the larger is the equilibrium film thickness at given chemical potential Ag. Remarkably, the capillary potential in (50) acts like a Coulomb potential, with "electric charge" ( D - 2)~/n. The electrostatic analogy makes clear that the capillary potential is overwhelmingly stronger than the substrate potential at large distances. In the spherical pore, the equilibrium (or metastable) interface is an equidistant interface, too, and the associated film thickness z is determined by Eq. (49), too, but now with the effective potential

Ueff(z) =

- ot/z 3 Al.tc

2(y/[n(R

- z)]

if z < z c

(51)

if z c < z < R

where z c and A~tc are defined as in Eq. (21) [recall Eqs. (15, 16) and Fig. 4]. The constancy of the effective potential (51) at distances larger than z c, which implies that the film can have any thickness between z c and R at A~t = A~tc, describes the vertical jump in the isotherm (Fig. 5). This extends the effective potential to include all physically realizable states on the isotherm, regardless of whether they are equilibrium states, metastable states, or even unstable states (vertical jump in the isotherm). Under this extension, constancy of the effective potential is the most extreme form of slow growth of the effective potential and entails the most rapid growth of the film thickness. For adsorption on a general surface, in terms of equidistant trial interfaces, the condition (27) for the equilibrium film thickness z leads again to Eq. (49), with effective potential Ueff(z )

=

o~ z3

~

(3"~VP'(z) . n V'(z)

(52)

The functions V'(z) and V"(z) denote the first and second derivative of the film volume V(z), Eq. (22), and the expression is subject to the condition that the right-hand side of (52) is monotone increasing with increasing z. Under this condition, Eq. (49) has a unique solution z for every A~ and the solution represents the equilibrium state. If the fight-hand side of (52) is not monotone increasing, Eq. (49) with (52) has more than one solution z and the physically realized film configuration, depending on whether one moves along the adsorption or desorption branch of the isotherm, is found by a stability analysis similar to Eqs. (18a, b). The stability analysis yields

Ueff(z) = nde+_. - ~ +

W(z)1

n V'(z)

(53+)

for the adsorption (+) and desorption (-) isotherm, respectively. Here nde_ denotes what we call the upper and lower nondecreasing envelope, defined as

654 nde+ qo(z) := max{ qo(t)" t ___z; qo'(t) _>0 }, nde_ q)(z) := min{qo(t): t _>z; ~o'(t) > O}

(54+)

(54-)

for a function q~(z) with derivative cp'(z). The construction of the two envelopes is illustrated in Fig. 11. As in the spherical-pore potential (51), which is a special case of (53+), the potentials (53+) lead to a vertical jump in the isotherm where they are constant. Clearly, they reduce to (52) if the function on which nde_+ acts is monotone increasing. We note that V'(z) is always positive, V"(z) is zero if and only if the surface is planar, V"(z) is negative for the spherical pore and every fractal surface, and V"(z) is positive for every convex solid. Thus, to compute the adsorption/desorption isotherm on an arbitrary surface S, one has the following simple steps: 9 Compute the function V(z), i.e., the volume of points outside the solid whose distance from the surface is less than or equal to z; this is the only surface-geometric input needed. If no analytic expression for V(z) is available, the computation requires only the sorting of distances dist(x, S) of points x close to the surface [ 156]. 9 Compute the effective potential Ueff(z) according to (53). If no analytic expression for V(z) is available, use V"(z)/V'(z) = (d/dz) [In V'(z)] for robust numerical differentiation. 9 Solve Eq. (49) for z as a function of Ag and substitute the result, z = z(Ag), into N = nV(z); this is the desired isotherm, N(Ag) = nV(z(Ag)).

(55)

The procedure is remarkable because, unlike the original problem (12, 13), it does not require to solve any differential equation. It operates with a minimum of geometric input, namely a function of one variable, which automatically maps two surfaces with "similar degree of irregularity" (Fig. 1) onto the same function V(z). The presence of the second derivative V"(z) in (53) corresponds to the Euler equation (13) being a second-order differential equation. The bifurcation generated by the operator nde+_in (53) automatically selects, from the several solutions of the Euler equation, the ones that are realized during adsorption and desorption, respectively. It may lead to hysteresis loops with any number of steps in either isotherm (Fig. 11). For the case in which there is no hysteresis, the procedure has been described before [43, 157].

Ueff

(+)

f f

S Figure 11. Construction of the effective potential Ueff(z) given by (53). The solid curve is the f u n c t i o n - ~ z 3 + a V"(z)/(nV'(z)) [schematic]. The dashed lines, together with the adjoining monotone parts of the solid curve, represent Ueff(z) for adsorption (+) and desorption (-).

655 When there is no hysteresis and therefore the form (52) of the effective potential applies, it is possible to invert the procedure and deduce the function V(z) from the adsorption isotherm N(A~t): By differentiation of (55) with respect to A~t one obtains V'(z(A~I,))

= n -1N'(A~I,)/z'(A].I,)

g"(z(A].l,)) = n-l[N"(Akl,)-

N'(A].t)z"(A~l,)[z'(Akt)]][z'(A~l,)] 2,

(56)

(57)

where N', N", z', z" are the derivatives of the functions N(A~t) and z(A~). Putting (56, 57) into (52) and using the equilibrium condition (49) yields

(58)

This is a second-order nonlinear differential equation for the film thickness z(Als), in terms of the given isotherm N(AI.t), subject to Akt > A~trn where Aktm is the chemical potential at monolayer coverage. The initial data for (58) is z(A~l,m) = n-l/3

z'(Aktm) = n-l/3 N'(A~m)/N(AILtm)'

(59a) (59b)

expressin~ that the monolayer thickness is n- 1/3 and that the area of th e monolayer, V'(n-l/3), equals n-2/3N(Alarn), which after substitution into (56) gives (59b). The solution of the initialvalue problem (58, 59) gives the film thickness z(A~t) at any chemical potential A~t > Atxm. Since z(A~t) is a monotone increasing function, it can be inverted to obtain the chemical potential as function of the film thickness, Akt(z), which yields the sought-after surface geometry as V(z) = n-lN(A~t(z)),

(60)

for any z > n-1/3. If desired, the volume V(z) can be converted into the pore-size distribution Vpore(Z), the cumulative volume of pores with radius less than or equal to z, which is defined-free of any model assumptions about the surface geometry--as the volume of space outside the solid that is inaccessible to spheres of radius z [21, 36, 43, 109, 128]. The connection between the two volumes is given by Vpor~(Z) = V(z) - z V ' ( z ) ,

(61)

for a surface that is neither convex nor concave [43]. This concludes our demonstration, by explicit construction, of the existence of a general one-to-one correspondence between surface geometry and experimental adsorption isotherms as announced in Sect. 2.3. More details will be published elsewhere. The inverse part of the correspondence, which transforms the isotherm N(AI.t) into the function V(z), is optimal in the sense that from a scalar function of one variable, N(Atx), one cannot expect to get more information about the surface than another scalar function of one variable. The correspondence shows that there is nothing special about fractal surfaces with regard to the computability and invertibility, as a matter of principle, of the isotherm. What does make fractal surfaces special is that their isotherms are power laws from which all geometric information (D, groin, gmax) carl be obtained without having to solve the differential equation (58).

656 4. W E T T I N G P H A S E D I A G R A M In our discussion in Sect. 3, we treated the transition from van der Waals wetting to capillary wetting on a fractal surface as a sharp transition. The sharp transition resulted from extrapolation of the asymptotic film thicknesses (29) to the point where they are equal to each other. But in reality, when Eq. (28) for the film thickness is solved exactly, the transition is smooth. This raises the question in what sense the notion of two distinct wetting regimes is well-defined independently of any approximation, how accurate the simplification of the transition between the two regimes as a sharp transition is, and in what sense the transition is a phase transition, similar to that for capillary condensation in a single pore. The same question arises for the transition from submonolayer adsorption to multilayer adsorption, which we treated as sharp, but which in fact is smooth, too. We address these questions in an operational way. Our goal is to map the qualitative regimes of submonolayer adsorption, van der Waals wetting, and capillary wetting onto quantitative intervals of chemical potential, (-oo, A~tm)' (Aktrn' A~tc)' and (A~tc, 0), so that the intervals, when used to analyze experimental isotherms, predict the pressure ranges in which one should, or should not, expect to find one of the power laws (33a, b). The challenge is to identify A~n (transition to multilayer adsorption) and Al.tc (transition to capillary wetting) in a way that is conceptually independent of whether the surface is fractal or not and independent of whether the isotherm computation ( 12, 13) is carried out exactly or approximately. With such an identification at hand, one may then compute A].I.m and A~c as function of D, which will be the desired wetting phase diagram for fractal surfaces. Depending on what approximations are used to compute the isotherm, the resulting phase diagram is approximate, too. We begin by examining how well the adsorption isotherm on a fractal surface, computed from Eq. (24) by solving Eq. (28) for z exactly, is approximated by the power laws (32a, b) [Eq. (24) with z from (29a, b)] which describe the transition as sharp. This is done in Fig. 12, for a choice of parameters that corresponds to one of the experimental examples in Sect. 5. The isotherm is plotted over a deliberately wide range of A~t values, including very low values where the multilayer framework no longer applies (recall Sect. 3.4), so as to exhibit the full asymptotic behavior for A~ --~ _.oo and A~t --~ 0. The figure shows that the isotherm rapidly approaches the power laws (32) as Akt moves in either direction from the point of intersection of the power laws. At the point of intersection, where the difference between the isotherm and the power laws is largest, the difference is about 20 %, which is small for a power-law crossover. The crossover interval, which we take as the range of N/N m values for which the isotherm differs by more than 10 % from the respective power law, spans a coverage ratio of 2.5, which is again small for a power-law crossover. In units of film thickness, Eq. (43a), the interval spans a ratio of 3.8; and in units of A~t, the interval spans a ratio of 10. Thus we conclude that the description of the transition from van der Waals wetting to capillary wetting as a sharp transition, as given by the power laws (32), is a good approximation for most practical purposes and can easily be replaced by an exact evaluation of Eqs. (24, 28) if need arises. The evaluation of Eqs. (24, 28) for other D values and substrates yields similar results for the rate of approach to the asymptotic power laws and the behavior in the crossover region. That is, there is no significant dependence of the crossover behavior, in relative units, on the fractal dimension and the material constantsmin agreement with the universal form of Eqs. (24, 28) when expressed in terms of the reduced variables ~ and ~t introduced in Eqs. (39, 40). The only thing that depends on D and the material constants is the position of the transition point on the Al.t and N/N m axis.

657

1000

100

10

zX~t/(kT)

[mL1tl I

[illl,tt

I

hillllll

-10000

],lltili~

-10

hiil|itl

'.htli~il i

Ir

-0.01

h,,t, ll t

]u,,~,~,

N/Nm

0.1

-0.00001

Figure 12. Adsorption isotherm for nitrogen on a silver surface with D = 2.30, computed from Eqs. (24, 28) and the constants in Table 4. The dashed straight lines are the asymptotic power laws (32a, b) for van der Waals wetting and capillary wetting. They intersect at Al.t = A~tc and N/N m = (N/Nm) c, given by Eqs. (30, 42), and have slopes (3 - D)/3 and 3 - D, respectively.

Figure 12 contains an important caveat, however: If one selects a sufficiently small portion of the isotherm from the crossover region and fits a power law to it, one may find a nonclassical FHH behavior, Eq. (36), but with an exponent y neither equal to (3 - D)/3 nor to 3 - D. Thus, an experimental exponent y obtained from a small pressure range must be interpreted with care. If the underlying surface is known to be fractal, one can use Fig. 12 to conclude that the exponent must satisfy (3-D)/3 < y < D-3,

(62)

which leads back to the bounds (37) for the fractal dimension. The exponent in this case, even though it is nonasymptotic, still provides valuable information. If nothing is known about the surface and the analyzed pressure range is small (e.g., in the sense that the range of film thicknesses calculated from Eqs.(43a) and (37) spans less than a factor of two [36, 158]), a fractal interpretation of the exponent is not meaningful, as a rule. Figure 12 depicts A~tc, the transition to capillary wetting, as the point of intersection of the asymptotic power laws (32a, b). More generally, we define h~tc as follows. We write the adsorption isotherm (12) on a surface with arbitrary geometry, computed from (13) exactly or approximately, in the form

N(AB) = NmO(A~t; n, o~, <3)

(63)

where N m is the number of particles in the monolayer as before. The function | which is the coverage and is viewed as the result of the computation (12, 13), displays both the dependence

658 on the chemical potential and on the material constants. If O as a function of A~t has a singularity at one or several values of the chemical potential (--oo < Akt < 0), A~tc by definition is the lowest of these values, and the transition is a genuine phase transition. The transition is firstorder if the singularity at Alttc is a jump, and second-order else (e.g., if some derivative is discontinuous). A case where | has several singularities is illustrated by the effective potential in Fig. 11. If O has no singularity, we define A~tc as the lowest chemical potential for which O(A~c; n, o~, 0) = O(A~c; n, 0, o),

(64)

where the two sides are the coverage in the limit t~ ~ 0 (pure van der Waals wetting) and o~~ 0 (pure capillary wetting), respectively. The motivation for identifying the transition to capillary wetting through (64), when the isotherm has no singularity, is simple: In a single spherical pore, the isotherm jumps at Agc as described by Eqs. (19, 20). The jump describes a first-order phase transition. But if instead of a single pore a whole set of pores is present, described by the pore-size distribution Vpore(R) introduced in Sect. 3.6, the isotherm changes from (19) to N(A).t) = nVpore(Rc(Al.t)) + nf. ~ [1 - (1 - Zl(R, Akt)/R)3]V~ore(R)dR. Rc(A~t)

(65)

Here Rc(A g) is the radius of the pore filled at chemical potential Al.t (inverse of Al.tc in Fig. 4 as a function of R), and zl(R, Ag) is the film thickness in a pore of radius R at chemical potential Ag (solution z 1 in Fig. 4 as a function of R and Ag). The first term is the contribution from the full pores, and the integral is the contribution from the large, partly filled pores.t Equation (65) shows that, no matter how strongly peaked the derivative Vpore(R) may be, the isotherm is analytic in Al.t whenever Vpore(R) is analytic in R. In the framework of (65), the isotherm for a single pore of radius R 0 corresponds to the nonanalytic function

Vp~

=

0 ifR R 0.

(66)

Thus the slightest variation in pore size removes the singularity in (19) and erases the phase transition present in the single pore. Instead of jumping at A~tc, the isotherm now rises steeply. The rise is what is left over of the phase transition in the single pore, and the chemical potential at which the rise occurs represents the transition to capillary wetting. Taking the change of power law in Fig. 12 as paradigm for the rise, we are then led to the def'mition (64) of the transition point A~tc. The definition (64) is natural in several respects. It implements the notion that capillary wetting occurs whenever adsorption is controlled by surface tension, as opposed to being controlled by the substrate potential. It identifies the "rise point" in the isotherm as the point at which adsorption in the absence of the substrate potential exceeds adsorption in the absence of t The right-hand side of (65) can be evaluated for an arbitrarily shaped surface (recall the definition of Vpore(R)) and thus seems to offer an isotherm expression that competes with Eq. (55). But the two expressions are quite complementary: (65) approximates the surface as collection of independent spherical pores, and (55) approximates the film interface as equidistant. In the limit o~~ 0, Eq. (65) reduces to N(A~t) = nVpore(-2t~/(nA~t)), i.e., to the traditional formula for Kelvin porosimetry, which neglects the substrate potential.

,659 surface tension. If the surface has a geometry such that V~ore(R) is peaked at R, Eq. (64) recovers Ag c - -2cy/(nl~)

(67)

for R ~ o~, in agreement with Eq. (20) for the phase transition in a single large pore. Thus, Eq. (64) automatically locates what may be regarded as approximate first-order phase transition if the distribution of pores is narrow, and what may be regarded as approximate second-order phase transition if the distribution of pores is wide as on a fractal surface. If the distribution of pores is narrow, Ag c essentially coincides with the inflection point of N(AB) [which might be considered as an alternative definition of the transition point]; and if the distribution of pores is wide, Ag e is well-defined also in the absence of an inflection point (Fig. 12). We now turn to the transition point Al,tm from submonolayer adsorption to multilayer adsorption, which we define as the chemical potential for which

O(Agm; n, ct, or) = 1.

(68)

This is the lowest chemical potential for which the grand potential (6), together with the substrate potential (7), provides a meaningful description of the adsorbed film. Indeed, the expression (6) assumes that the adsorbed particles are densely packed as in bulk liquid, which leads to the condition that the number of particles in the film f(S, I) must be larger than or equal to the number of particles in a monolayer, n If(s,i ) d3x >- n If(S,in_l/3)d3x.

(69)

A slightly stronger condition would be dist(x, S) > n-1/3 for all x on I, requiring the local film thickness to equal at least the diameter of an adsorbate particle. The chemical potentials obeying Ag > AB~ are precisely those for which the minimizing interface satisfies condition (69). When AB < ABe, the minimizing interface no longer satisfies (69), and Eq. (12) no longer describes the adsorption isotherm properly. To get the adsorption isotherm in the submonolayer regime properly (from first principles), one has to replace (12) by a full-fledged statistical mechanical calculation, ~) ln(~ c e-[E(c)N = kT ~)(Ag)

AgN(c)]/(kT)),

(70)

similar to Eq. (8). Here the sum is over film configurations c that include submonolayer configurations (e.g., modeled as occupied sites on a lattice representation of the adsorption space), with respective energy E(c) and particle number N(c). Such an extension of the adsorption isotherm to submonolayer coverage is outside the scope of this chapter and is mentioned here only to illustrate why ABrn, as defined in (68), is the natural lower limit of applicability of the multilayer isotherm (12). A simple theoretical treatment of submonolayer adsorption on fractal surfaces can be found in Ref. [125]. The chemical potentials ABm and Ag e completely specify the three adsorption regimes under consideration (Table 6). If ABm < Ag e, the expression max {Ag m, Agc } takes Age as the onset of capillary wetting, leaving room for van der Waals wetting in the interval (Agrn, Agc)- If AB~ > Ag e, the expression max{ABe, Ag c } selects ABTnas the onset of capillary wetting (for

660

Table 6 Submonolayer adsorption, van der Waals wetting, and capillary wetting on a solid with arbitrary surface geometry. The chemical potentials Agrn and Aktc are defined by Eqs. (68, 64). Regime

Range of Ag

Regime exists

Submonolayer adsorption van der Waals wetting Capillary wetting

--oo < A~t < A~n A~m < A~I,< max {Ap~n, Aktc } max {All.m, AILtc } < A[.t < 0

always if and only if A[.I,m < A[.t c if and only if A~tc < 0

A~tc lies in the regime where the isotherm (12) is no longer meaningful), leaving no room for van der Waals wetting. This completes the construction of the different adsorption regimes from the isotherm (12) or from approximations thereof. To implement the construction for a fractal surface, we first consider the isotherm expression (32) which approximates the isotherm in terms of pure power laws. Equations (68) and (64) in this case give Al.tm

=

-

max

{ t~n, (D

-

2)o'n -2/3 },

A[U,c = _ ~-1/2 [(D - 2 ) ~ / n ] 3/2.

(71) (72)

The result (71) is the lower of the two chemical potentials which yield N = N m in (32a, b) The result (72) agrees with (30) because the expression (30) was defined as the point where the power laws (32a, b) intersect. The results also apply to the logarithmic isotherm (34) for D = 3. A simple calculation shows that Ala~n< Al.tc holds if and only D < 2 + otnS/3c-1,

(73)

which is precisely the condition we found in Sect. 3.3 for the van der Waals regime to be realized. The three regimes in Table 6, upon insertion of the expressions (71, 72), divide the set of all pairs (D, A~t) into three respective regions of the D-A~ plane. The regions correspond to different phases of the adsorbed film, as a function of D, and constitute the wetting phase diagram for self-similar surfaces (Fig. 13): The submonolayer region lies below the line Alarn and may be thought of as a D-dimensional gas. The region of van der Waals wetting lies between the lines Ala~nand Al.tc and corresponds to a 3-dimensional, slowly growing liquid film ("thin film") on a D-dimensional surface. The region of capillary wetting lies above the line A~tc and describes a 3-dimensional, rapidly growing liquid film ("thick film") on a D-dimensional surface. The lines are coexistence lines of the respective phases. The line which separates van der Waals wetting and capillary wetting may be viewed as analog of the prewetting line in a thermal wetting transition on a planar surface (the prewetting line, too, separates thin from thick films [83, 84, 146]). In this analogy, the fractal dimension plays the role of temperature, in agreement with the picture that the transition on the fractal surface is driven by quenched disorder (geometric disorder), while the transition on the planar surface is driven by thermal disorder. Just as the prewetting line joins the coexistence line for bulk liquid and gas (A~t = 0 for all T < T c) tangentially at the wetting temperature T w, the capillary wetting line joins the coexistence line for bulk liquid and gas (A~t = 0 for all 2 <_D _<3) tangentially at D = 2. The tangential behavior of the capillary wetting line is not visible on the logarithmic scale in Fig. 13,

661

2

2.2

2.4

2.6

2.8

3

D

_ 1 0 -4

_10 -3 [..,

_10 -2

=1.

<1

_10 -1 _10 o _101

Figure 13. Wetting phase diagram for the power-law isotherm (32, 34). The full, dashed, and dashed-dotted lines are for nitrogen on SiO 2, C, and Ag, respectively. The lower and upper three lines are Aktrn and Al.tc, respectively, calculated from (71, 72) and Table 4.

but is obvious from Eq. (72). The region of van der Waals wetting shrinks with increasing D, as discussed in Sect. 3, and disappears when condition (73) is no longer satisfied. At the point where (73) ceases to be satisfied, several things happen: (i) the lines A ~ and Alsc merge; (ii) the line A~tm switches from A~tm = - txn to AlXrn= - ( D - 2)~n-2/3; (iii) the line AlXrn switches from being coexistence line for submonolayer adsorption and van der Waals wetting to becoming coexistence line for submonolayer adsorption and capillary wetting. In Fig. 13, this occurs at D = 2.82 for nitrogen on SiO 2, in agreement with the discussion of the weak substrate potential of SiO 2 in Sect. 3.3. Other earlier instances of D > 2 + txnS/3o-1 have been encountered in Figs. 8 and 9 (curves with Y > 1). In fact, the curves for the onset of capillary wetting in Fig. 9 may be regarded as phase diagram in which the chemical potential At.t has been replaced by the reduced film thickness ~ and submonolayer adsorption corresponds to the region ~ < 1. A somewhat different phase diagram is obtained if we use the equidistant-interface isotherm, Eqs. (24, 28), instead of the approximation (32, 34). Equations (68, 64) then give At.tm = - o m - (D - 2)on -2/3, A~c = - Ix-1/2 [(D - 2)a/n] 3/2.

(74) (75)

The expression for Apt n results from noting that, for equidistant trial interfaces, the condition O = 1 is equivalent to z = n -1/3, so that the equilibrium condition (49) yields Ala~n = Ueff(n-1/3 ) for arbitrary surface geometry, which leads to (74) for the special case (50). The expression for Aktc is still the same as before because the isotherm (24, 28) coincides with (32, 34) for a = 0 and tx = 0. Here Alxm < Aktc holds if and only if D < 2 + 2.147...-omS/3t~ -1.

(76)

Thus, the two isotherms give the same Al.tc'S, but different APTn'S and different intersection points for A~tc and Aktm. The phase diagram for (74, 75) is plotted in Fig. 14. It shows that the line Apt n is shifted to lower values compared to Fig. 13 and no longer meets the line At.tc for

662

2

2.2

2.4

2.6

2.8

3

D

_10 -4 _10 -3 ~"

_10 2

<1

_10 -1

~

~-.-.... ----

~---.._.-

---..__._

~_._.

_ ..~

_10 ~ _101 Figure 14. Wetting phase diagram similar to Fig. 13, but for the equidistant-interface isotherm (24, 28). Here the lines A;lm and Abtc are given by Eqs. (74, 75). any of the three substrate-adsorbate pairs. The reason for the shift is clear from Fig. 12: The equidistant-interface isotherm lies above the power-law approximation and therefore reaches the monolayer value at a lower chemical potential. In experimental applications, the wetting phase diagram, together with Eq. (5), predicts what adsorption behavior one should find in what pressure range, for a given substrate-adsorbate pair and fractal dimension. But which of the two diagrams, Fig. 13 or Fig. 14, should be used? Will other approximations, say for the substrate potential or calculation of the isotherm, yield a significantly different phase diagram? These questions reflect the contrast between the isotherm exponents ( 3 - D)/3 and 3 - D, which are independent of the approximations used to find the isotherm and independent of the material constants, and the transition points, which do not share this independence. The answer depends on which model isotherm one seeks to fit to the experimental data. If the data is to be analyzed in terms of the power laws (33a, b), Fig. 13 is the appropriate phase diagram. For example, on a sample of SiO 2 that is known to have substantial surface irregularity at atomic length scales, suggesting a high-D surface, one may expect to find the power law (33b), but not (33a), by virtue of the narrowness or absence of the van der Waals regime at high D. Conversely, when an experimental isotherm exponent y [Eq. (36)] is translated into a D value according to the criteria in Sect. 3.2, the obtained D value can be tested for consistency by comparing the experimental pressure interval from which y was obtained with the interval predicted from Fig. 13. This test is important when the test for fractality in Sect. 3.3 cannot be carried out because the monolayer value N m is not known. When N m is not known and the test is successful, one can convert the experimental pressure interval into an interval of film thickness z by means of Eq. (29a) or (29b), depending on whether D 3 - 3y or D - 3 - y. If, on the other hand, the data is to be fitted to the full isotherm (24, 28), rather than to individual power laws, then Fig. 14 is the appropriate phase diagram. In that case, the line A~lrnpredicts the low end of the pressure interval to be analyzed, and the line A~tc may be used to decide whether a departure from power-law behavior in the data is consistent with a crossover from van der Waals wetting to capillary wetting, such as in Fig. 12, or signals the upper end of the fractal regime.

663 5. E X P E R I M E N T A L EXAMPLES

The experimental observation of nonclassical FHH exponents, y ~:1/3, and their interpretation has a long history, as mentioned in Sect. 3.2. The discovery that they have a natural interpretation in terms of van der Waals wetting and capillary wetting of fractal surfaces is much more recent [44, 101 ]. Experimental studies of exponents y ~ 1/3 can therefore be divided into "pre-fractar' investigations and "fractal" investigations. We restrict ourselves to the latter here and leave it for future research to revisit the former in terms of fractal analysis. We give particular attention to those case studies where the fractal interpretation of the isotherm exponent has been tested for consistency with the wetting phase diagram (PD) or for consistency with independent structural data. Methods which have provided independent structural data include scanning tunneling microscopy (STM), small-angle X-ray or neutron scattering (SAXS, SANS), X-ray reflectivity (XR), thermoporometry (TP), isotherm steps at low temperature (IS, test for surface planarity), adsorption-desorption hysteresis (ADH), molecular tiling (MT), and measurement of active surface areas (ASA). A survey of experimental isotherm studies is presented in Table 7. For each example we list the experimental exponent y (in the event that the isotherm exhibited more than one power law, we quote the exponents as separate entries), its interpretation in terms of van der Waals wetting or capillary wetting as given by the authors, the range of film thickness z probed by the power law (the low-end and high-end thickness, Zmin and Zmax, give the bounds s <- Zmin and s > Zmaxfor the inner and outer cutoff of the fractal regime), and performed consistency tests. Consistency with structural data from other methods than listed above is designated by OM. The examples include cases that were analyzed in terms of what sometimes is called the fractal Dubinin-Radushkevich isotherm [ 119], which assigns the fractal dimension D = 3 - y to the exponent y, just as the capillary-wetting exponent does. We list those examples as instances of capillary wetting, believing that the interpretation in terms of capillary wetting has a better understood physical basis and offers more opportunities for consistency tests. Similarly, we list cases that were analyzed by the "thermodynamic method" [ 120], which is equivalent to the analysis in terms of capillary wetting (see below), as instances of capillary wetting. We group the examples according to the chemical composition of the substrate, in the order of increasing values of the capillary-energy ratio T for nitrogen. This makes metals, with their tendency to favor van der Waals wetting, come first and silicas, with their tendency to favor capillary wetring, come near the end of the table. In each group, we order the examples by increasing fractal dimension. Case Study 1: Classical FHH exponent on planar surfaces. An important test of the theoretical framework described in the previous sections is to verify that planar surfaces experimentally exhibit the predicted exponent y = 1/3. One of the most systematic and striking verifications has been carried out on vapor-deposited gold films (first four entries in Table 7). For seven different gases, the chemical potential was varied by variation of temperature (triple-point wetting), which tests the exponent 1/3 via the temperature dependence in Eq. (5) [nonisothermal adsorption]. Direct tests, by varying the chemical potential isothermally, were done with nitrogen and argon. In all instants, not only the exponent 1/3 but also the prefactor in Eq. (14), with A taken as the geometric surface area of the sample, was confirmed with remarkable precision. This experimental confirmation is important because earlier Monte Carlo simulations of multilayer films on a planar surface, based on Lennard-Jones potentials for the adsorbateadsorbate and substrate-adsorbate interaction, exhibited significant departures from Eq. (14) [ 179]. The confirmation also corroborates the calculated values of the constant ot in the sub-

664

Table 7 E x p e r i m e n t a l determination of the fractal dimension of solid surfaces from multilayer adsorption. T h e labels 'vdW' and 'cw' specify whether D is obtained from van der W a a l s wetting or capillary wetting (Fig. 7). Substrates are specified by crystallographic indices of surface, temperature of deposition, sample name, or other labels from the references. System

y

D(vdW)

D(cw)

z (,~)

Tests

Ref.

Metals: N2/Au (111) A r / A u ( l l 1) .../Au(111)b N2/Au(300 K) N2/Ag(293 K) N2/Ag(Crystek) N2/Ag(80 K)

0.33 0.33 0.33 0.33 0.33 0.23 0.21

2.00 + 0.05 2.00 2.00 2.00 2.00 2.30 + 0.02 2.36

-

4-20 a 6-60 7-20 a 8-45 4-70

XR XR XR XR, S T M STM, PD XR

[159] [1601 [1601 [481 [161] [44, 112] [48, 1611

0.33 0.33 0.33 0.33 0.33 0.30 0.27 0.27 0.38 0.30 0.28 0.27 0.34 0.40 0.40 0.43 0.42 0.62 0.57 0.44

2.00 2.02 + 0.05 2.00+0.05 2.01 + 0.05 2.01 + 0 . 0 5 2.11 + 0 . 0 5 2.19+0.10 2.18 + 0 . 1 0 1.85 + 0 . 1 0 2.10+0.10 2.15 + 0 . 1 0 2.20+_0.10 1.97 _+ 0.09 -

< 2.60 < 2.60 < 2.57 < 2.58 2.38 _+ 0.03 2.43 2.56 _+ 0.03

7-30 a 6-24 6-16 a 7-14 a 6-14 a 4-16 a

Carbons: Ar/graphite c N2/carbon film N2/acetylene black

N2/TB#4500HT d N2/TB#4500 d N2/TB#5500 d N2 / N 1 1 0 d N2/N220 d

N2/N300 d NE/N550d N2/N762 d N2/N787 d N2/N990 d N2/N762 d N2/N550 d

N2/N330 d NE/V3 d

NE/VSB-32 GRo N2/VSB-32 A R e N2/WCA e N2/T-300 G R e N2/T-300 A R e N2/CEL e

0.56 0.41 0.40 N2/PIT e 0.38 N2/PAN e 0.29 0.57 N2/natural coal N2/activated carbon 0.28

-

IS (~ OM (~ C~ OM MT MT MT MT MT MT 4-18 NIT MT MT MT MT 16-190 ASA, STM, PD ASA 4-360 ASA, STM, PD, SAXS 2.44 ASA 2.59 ASA, STM 2.60 _ 0.01 f MT, S A X S 2.62 + 0.01 f MT, S A X S 2.71 + 0.02 f MT, S A X S 2.43g 25-500 ADH 2.72 + 0.01g 10-200 ADH

[162] [163] [164] [164]

[164] [164] [165] [165] [165] [165] [165] [165] [134] [134] [134] [134] [134] [134] [134] [134, 166] [134] [134] [167] [167] [167] [168] [120]

665

Table 7----continued System

y

D(vdW)

D(cw)

z (/k)

Tests

Ref.

Silicas: N2/FK320DS h N2/Tixosil h NE/Sipemat h NE/Aerosil 200 i N2/aerogel B N2/aerogel A Nz/Aerosi1200 i N2/Aerosil 150 i N2/Aerosil 5 0 i Ne/quartz Nz/cristobalite Nz/Aerosi1200 i Nz/aerogel s NE/aerogel s Nz/aerogel s N2/CPG-240 n Nz/Si 300 h Nz/Si 300 h P B D Nz/Si 60 h

0.33 0.33 0.32 0.32 0.23 0.06 0.40 0.37 0.39 0.32 0.35 0.36 k 1 m 0.88 0.79 o 0.78

2.01 + 0 . 0 5 2.01 + 0.05 2.03 + 0.05 2.03 + 0.05 2.30 + 0.05 2.83 + 0.05 2.0-2.1 2.0-2.1 2.0-2.1 2.04 + 0.10i 1.95 +_ 0 . 1 0 J 1.92 + 0.10J 2.0 2.0-2.2 _ _ -

-

4-12 a 5-12 a 4-12 a 5-12 a 6-18 a -

TP TP TP TP, S A N S TP, S A N S TP, S A N S Mr MT MT OM

-

-

-

2.5-2.6 2.12 + 0.02g 2.21 +_ 0.01g 2.21-2.28g 2.22

5-12 a 10-200 15-150 15-150 9-20 a

MT SAXS, OM SAXS, OM ADH ADH ADH SAXS

[164,169] [164, 169] [164,169] [164, 169] [164, 169] [164, 169] [134] [134] [134] [170,171] [170,171] [170, 171] [172] [173] [173] [168] [168] [174] [175]

0.20 P q 0.33 0.04

2.40 + 0.05

2.25-2.55 2 . 3 4 - 2 . 7 lg 2.68 + 0.03g 2.96 + 0.02 f

5-24 10-230 17-800 -

(~I ADH ADH (~I

[163] [176, 177] [178] [168] [170,171]

Others: N2/magnetic film

N2&H20/cement N2/CaO N2/apatite N2&CC14/zeolite Y a b c d e f g h i J k

Our estimate. Adsorbates: Ar, Kr, Xe, N 2, 02, CH4, C2H 6. Variation of Al.t by variation of temperature. Isotherm measured by ellipsometry. Carbon black. Carbon fiber. Analysis according to the fractal Dubinin-Radushekevich isotherm (see text). Analysis according to the "thermodynamic method" (see text). Precipitated silica (xerogel). Pyrogenic silica. Our reinterpretation of the experimental y value (see text). y = 0.34-0.35 for four acid-catalyzed samples. 1 y = 0.25-0.33 for nine samples, base-catalyzed samples at high water concentration. m y = 0.38-0.43 for six samples, base-catalyzed samples at low water concentration. n Controlled-pore glass. o y = 0.72-0.79 for five samples coated with bolybutadiene. P y = 0.45-0.75 for three samples of variable composition. q y = 0.29-0.66 for six samples calcinated at different temperatures.

666 strate potential. An example of the nitrogen adsorption isotherm on a planar gold surface, with y = 1/3, is shown in Fig. 15. Of all isotherms on planar gold films quoted in Table 7, it spans the largest film thickness range (6-60.4,). Measurements up to ln(p0/p) = 10-2, on annealed gold films, show an even better agreement with the predicted behavior [159]. The other two data sets in the figure illustrate the behavior of the isotherm on nonplanar substrates (the silver data describes entry 7 in Table 7). They show that different exponents y can be easily distinguished experimentally, and that not every experimental isotherm follows an FHH-type power law. The exponent y -- 1/3 on a planar surface has been confirmed also for many other solids, of course. Noteworthy examples in Table 7 are graphite, where planarity was established from the stepwise growth of solid Ar films at low temperature; various graphitized carbon blacks; several silicas; and crystalline SiO 2 (quartz and cristobalite). In none of these cases have the isotherms been measured up to high enough pressures, however, to span a film thickness range comparable to that in Fig. 15. The case of quartz, cristobalite, and Aerosil 200 studied by Lefebvre et al. [ 170, 171] deserves some explanation. Those authors measured N 2 and CC14 isotherms on six SiO 2 samples of various origins (three quartz and two crystobalite samples) and interpreted the observed exponents y in terms of the fractal Dubinin-Radushkevich isotherm. This led them to conclude that all six samples have fractal dimension between 2.55 and 2.68. These values are much larger than any previously reported D values for quartz (D = 2.02.2 [158]) and Aerosils (D = 2.0 [14]) determined by other methods. In Table 7, we therefore interpret the three quoted y values as instances of van der Waals wetting, which makes the resulting D values agree with all other known structural properties of these samples. Case Study 2: Van der Waals wetting of fractal silver surfaces. The first fractal surface analysis by multilayer adsorption was done on two silver electrodes plated on quartz crystals employed in microprocessor circuits [44]. The corresponding isotherms are shown in Fig. 16.

~

.. !.

~'x.~./,..,..i.

,

"

,

,

,

w

~ .

..

"%%

ioO

.:

I# Xx

N.

~

X =

_z 10

~g(a0K)~Au(S00K)

10- I

m..

"'"e

io-2

"

% \

#.

10-3

40

~

40

' ''"

'

'

'X"" ' ~ "

100 400 QUANTITY ADSORBED(hE/era')

lOO

Figure 15. N 2 isotherms on Au and Ag films, vapor-deposited under various conditions. The data for Au(300 K) follows the prediction (14) for adsorption on a planar surface (solid line) without adjustable parameters. Planarity of the Au(300 K) surface over large lateral distances was confirmed by scanning tunneling microscopy and X-ray scattering. (From Ref. [48].)

667

0r " ~ , substrate ~ 2 substrate 1 +##§ ~Ii~

\

I0 0

\\

~

~+'~

' . Zmln-8A

10-1 ,,,-,

10-2

_

,,,+, . \[ I

I

I

50

I

I

I

I

I

i

100

i

~I

I

"

500

J

Quantity Adsorbed {ng/cm 2)

Figure 16. N 2 adsorption isotherms on two rough Ag electrodes. The dashed line represents the prediction (14) for adsorption on a planar Ag electrode. In all three isotherms the reference area is the macroscopic area of the electrode (geometric surface area). (From Ref. [44].)

Substrate 2 corresponds to entry 6 in Table 7. From the values y = 0.23 for both isotherms in Fig. 16 and the assumption that van der Waals wetting prevails, it was concluded that both surfaces have fractal dimension D = 2.30 + 0.02, over an approximate range of 8-20 ,~ and 8-45 .&, respectively. This is consistent with the roughness factor of 2 and 3, respectively, implied by the isotherms, and agrees with the value D = 2.30 + 0.10 over a range of 5-50 .A obtained from STM [44]. The surface topography as seen by STM is shown in Fig. 17.

20

10 0-I0 nm

10

Figure 17. STM image of a Ag electrode from the same manufacturer as substrate 2 in Fig. 16. Note the different length scale in the horizontal and vertical direction. (From Ref. [ 180].)

668 The STM topography suggests, by virtue of the absence of overhangs in the image, that the surface is self-affine rather than self-similar (introductory expositions of self affinity can be found in Refs. [2, 6-8, 11, 31, 36, 101 ]). This means that the surface height S(x,) at position x,, as illustrated in Fig. 3, satisfies ([S(x, + y , ) -

S(yll)]2)Yll

=

b(lx,l/b) H

(77)

where the average {...)., is defined as in Eq. (9). The exponent H, 0 < H < 1, is called roughall ness (or H61der) exponent of the self-affine surface, and the length b is called crossover length. In the regime Ix,! << b, called steep or local regime, the surface behaves like a self-similar surface with fractal dimension D = 3 - H. For example, the volume of a film of thickness z, V(z), on a self-affine surface is proportional to z H [Eq. (24a)] for z << b. In the regime Ix,I >> b, called shallow or global regime, the surface behaves like a surface with D = 2, however, in the sense that film volume V(z) is in leading order propertional to z for z >> b (only the correction to the leading-order term depends on H). These properties make the analysis of multilayer adsorption on a self-affine surface potentially quite different from the self-similar case treated in Sect. 3. As a result, it was suggested that the observed isotherm exponent y = 0.23 might be interpreted in terms of capillary wetting in the shallow regime of a self-affine surface [113, 114] instead of van der Waals wetting of a self-similar surface or in the steep regime of a self-affine surface [44, 45, 101, 110, 112]. This controversy was resolved when (i) a reexamination of the STM topography of the Ag surface gave H = 0.70 _+0.10 and b = 1 A, for 5 ,~ < Ix,I < 50 .~, as best estimate from various methods of analysis [ 180]; and (ii) an analytic isotherm expression for self-affine surfaces with b _
~ ~ ~

(-A~t) -1/3 (-A~I.)-1/3 (-A~) -H/(2-H)

for low A~t (van der Waals wetting), for A~t --~ 0, 0 < H < 1/2 (van der Waals wetting), for Al.t -~ 0, 1/2 < H < 1 (capillary wetting),

(78a) (78b) (78c)

earlier obtained by scaling arguments [45, 114], but found that capillary wetting on a surface with H = 0.7 sets in only at film thicknesses in excess of 103 A [ 104]. The STM analysis rules out van der Waals wetting in the steep regime because the steep regime is absent (as expected); the isotherm expression rules out capillary wetting in the shallow regime because the films in Fig. 16 are much too thin to follow the power law (78c); and the experimental isotherm rules out van der Waals wetting in the shallow regime because it does not follow the power law (78a). Isotherm calculations based on the numerical solution of the Euler equation (13b) conf'trm these conclusions [ 181 ]. Thus, we are led back to the original interpretation that Fig. 16 represents van der Waals wetting of a self-similar surface with D = 2.3. The observed van der Waals regime is somewhat larger than predicted by the phase diagram in Fig. 13, but consistent with the phase diagram calculated for N 2 on Ag without using the equidistant-interface approximation [ 112]. But a self-similar surface with D > 2 necessarly has overhangs. This implies that Fig. 17 does not image the surface faithfully. Indeed, there exists considerable experimental and theoretical evidence that solid films grown by vapor deposition, as is the case for the Ag film, often contain overhangs and voids. Columnar growth of sputtered films is an experimental example [ 182]. Molecular dynamics simulations illustrate this at the atomic level (Fig. 18). Consequently, the combined adsorption and STM study of the Ag surface highlights the generic overhang problem

669

(a)

E = 0.05~

(b)

E = 0.3~ (c)

o

~

~

~

E = 1.5r

~ ) )

('~X ...............................................

:

Figure 18. Microstructure of overhangs and voids, obtained in molecular dynamics simulations of random deposition under normal incidence at different incident kinetic energies (from Ref. [183]).

of STM. The problem is that STM may not give a faithful image of the surface because it cannot detect overhangs, crevices, or deep pits: When the STM is operated at constant current and the tip moves across an overhang, there comes the point where the tip is about to lose its tunneling current and another part of the tip, is close to the surface elsewhere, begins to draw current, with no indication that the "new" current comes from another location. In this way, the STM image always looks as if overhangs were absent, and the adsorption isotherm provides a strong tool to detect the presence of surface regions invisible to STM. Case Study 3: Capillary wetting of fractat carbon surfaces. An interesting example of capillary wetting of a fractal surface was analyzed by Neimark [120-122]. The example is interesting because it has an isotherm exponent of y = 0.28, which in the absence of other information might be interpreted as van der Waals wetting of a surface with D = 2.16. However, the isotherm exhibited hysteresis in the pressure regime where y = 0.28 was observed, showing that capillary effects must be involved. Interestingly, the adsorption branch and the desorption branch gave the same exponent y within the experimental accuracy. The experimental data and analysis are reproduced in Fig. 19. Neimark's analysis, which he calls the "thermodynamic method," starts from the observation that, when the substrate potential can be neglected, the change in the surface area of the liquid-gas interface of the adsorbed film, dI, is related to the

N mmole/g

tnS, m2/g

2O

6'

12.~nm

Z _

dts,,,Z? 3 ~ ] ,

ZOnm ~

~.

~

Inm

I 0

I 1

2

-/ . . . . o

I o, z5

,, I 0.5 a

, I o.7~

I o/e 6 f

-6~ -6

I -5

I -4

I

I

-3

-2 b

~ -7

t.t.t~/P)

Figure 19. (a) Adsorption and desorption isotherm for N 2 on an activated charcoal. (b) Surface area of the liquid-gas interface as function of pressure. (From Ref. [ 120].)

670 change in the number of adsorbed particles, dN, by t~dI = AktdN.

(79)

Upon use of Eq. (5) and integration, this yields

I(p) = (kT/o) [ N(p) ln(p(N,)/P0)dN ,

(80)

" Nmax

for the surface area of the liquid-gas interface as function of the gas pressure p. In (80), the function N(p) is the number of adsorbed particles at pressure p, p(N') is the pressure at which N' particles are adsorbed, and Nmax is the number of adsorbed particles at complete pore filling (i.e., at the pressure Pmax for which I(Pmax) = 0). In Fig. 19, the area I(p) is denoted as S(p). Under the assumption that at every pressure p the liquid-gas interface has a constant radius of curvature r(p), given by r(p) = 2g/[-kT In(p/p0)] (Kelvin equation), it follows for a fractal surface that

I(p)

o~ [ r ( p ) ] 2 - D

or

[_ln(p/p0)]D-2.

(81)

Equations (80) and (81) together imply N(p) 0r [_In(p/p0)] -(a-D),

(82)

which is the power law for capillary wetting derived in Eq. (33b). Thus, Neimark's analysis using Eq. (81), and the direct analysis of the experimental data using Eq. (82) are completely equivalent. The virtue of the derivation in (33b) is, of course, that it also predicts the pressure at which the transition to van der Waals wetting occurs. The analysis gives D = 2.73 from the adsorption branch and D = 2.71 from the desorption branch (Fig. 19). Case Study 4: Transition from van der Waals wetting to capillary wetting on fractal carbon fibers. Our final case study reviews an investigation in which the transition predicted in Table 3 was experimentally observed [ 134]. This may also shed useful light on investigations of carbon samples in which the interpretation of observed exponents y either as van der Waals wetting or capillary wetting was ambiguous [184]. The samples studied in Ref. [134] were carbon fibers, designated as VSB-32 GR and WCA, whose adsorption isotherms are shown in Fig. 20. The isotherm for WCA exhibits a single power law with y = 0.44, starting at monolayer coverage. Since the interpretation of this exponent as van der Waals wetting would imply an absurdly low fractal dimension of 1.68, the exponent was taken as capillarywetting exponent, yielding D = 2.56 + 0.03 over a range of 4-360 ,~. For this fractal dimension, the phase diagram in Fig. 13 predicts the onset of capillary wetting at a coverage of N/N m = 1.4, which is in very good agreement with the observed low end of the power-law regime at N/Nrn = 1.1. Recent SAXS studies of the sample gave a dimension of D = 2.50 + 0.05 up to even larger length scales [ 166]. Thus, the agreement and consistency of all data is very good and makes the system a remarkable case where capillary wetting sets in already at the first or second adsorbed layer. The transition from van der Waals wetting to capillary wetting is exhibited by the sample VSB-32 GR in Fig. 20: At a coverage of N/N m - 1.5, the power law at low pressure (which is too limited to lend itself to a separate fractal analysis) changes to a power law with exponent y = 0.62, which gives D - 2.38 _+ 0.03 over a range of 16-190/~,

671

2.4 2.1

.~,,' ' ' i . . . . i "x~ i "~ "x

I ....

t ....

I ....

I .... t .... I .... I .... ~ WCA- As-received - - 4 - V S B - 3 2 G R l o w e r end ~ V S B - 3 2 G R u p p e r end

-

1.8 1.5 >

1.2

g

.9 R E G I M E II

3.8

-3.3

-2.8

-2.3

i

-1.8 -1.3 ln(ln(P~

"lk~

-.8

,.,

"

-.3

.2

.7

Figure 20. N 2 adsorption isotherms on two carbon fibers (from Ref. [ 134]).

upon interpretation as capillary wetting. Here the phase diagram predicts the transition to occur at N/N m = 1.8, which again is in good agreement with the observed change in power law. The power law at low pressure is therefore interpreted as crossover regime from van der Waals wetting to capillary capillary wetting (recall the crossover in Fig. 12).

ACKNOWLEDGMENT We thank D. Avnir, M.W. Cole, S. Dietrich, F. Ehrburger-Dolle, J. Krim, W. Rudzinski, and W.A. Steele for stimulating discussions which have influenced the presentation in this chapter in many different ways.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). J. Feder, FractaIs (Plenum, New York, 1988). M. Barnsley, Fractals Everywhere (Academic Press, Boston, 1988). D. Avnir (ed.), The Fractal Approach to Heterogeneous Chemistry (Wiley, 1989). A.J. Hurd (ed.), Fractals--Selected Reprints (American Association of Physics Teachers, College Park, 1989). K. Falconer, Fractat Geometry (Wiley, Chichester, 1990). F. Family and T. Vicsek (eds.), Dynamics of FractaI Surfaces (World Scientific, Singapore, 1991). T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1992). K.S. B irdi, Fractals in Chemistry, Geochemistry, and Biophysics (Plenum, New York, 1993).

672 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

A. Harrison, Fractals in Chemistry (Oxford University Press, Oxford, 1995). A.L. Barab~isi and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995). D. Avnir and P. Pfeifer, Nouv. J. Chim. 7, 71 (1983). P. Pfeifer, D. Avnir, and D. Farin, Surf. Sci. 126, 569 (1983). P. Pfeifer and D. Avnir, J. Chem. Phys. 79, 3558 (1983); 80, 4573 (1984). D. Avnir, D. Farin, and P. Pfeifer, J. Chem. Phys. 79, 3566 (1983). D. Avnir, D. Farin, and P. Pfeifer, Nature 308, 261 (1984). P. Pfeifer, Applic. Surf. Sci. 18, 146 (1984). P. Pfeifer, D. Avnir, and D. Farin, J. Stat. Phys. 36, 699 (1984); 39, 263 (1985). A.H. Thompson, A.J. Katz, and C.E. Krohn, Adv. Phys. 36, 625 (1987). P. Pfeifer, in: Preparative Chemistry Using Supported Reagents, ed. P. Laszlo (Academic Press, New York, 1987), p. 13. P. Pfeifer, in: Chemistry and Physics of Solid Surfaces VII, eds. R. Vanselow and R. Howe (Springer, Berlin, 1988), p. 283. P. Meakin, in: Phase Transitions and Critical Phenomena, Vol. 12, eds. C. Domb and J.L. Lebowitz (Academic Press, London, 1988), p. 335. P. Meakin, Prog. Solid St. Chem. 20, 135 (1990). Y.T. Cheng, Mater. Sci. Rep. 5, 45 (1990). W.G. Rothschild, Catal. Rev.~Sci. Eng. 33, 71 (1991). A. Bunde and S. Havlin (eds.), Fractals and Disordered Systems (Springer, Berlin, 1991). K. Kaneko (ed.), Fractal and Physically Adsorbed Molecular States (Chiba University, Chiba 1992). P.M. Adler, Porous Media (Butterworth-Heinemann, Boston, 1992). D. Avnir, D. Farin, and P. Pfeifer, New J. Chem. 16, 439 (1992). M.B. Isichenko, Rev. Mod. Phys. 64, 961 (1992). P. Meakin, Phys. Rep. 235, 189 (1993). M. Sahimi, Rev. Mod. Phys. 65, 1393 (1993). M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH, Weinheim, 1995). T. Halpin-Healy and Y.C. Zhang, Phys. Rep. 254, 215 (1995). (a) A. Erzan, L. Pietronero, and A. Vespignani, Rev. Mod. Phys. 67, 545 (1995). (b) D. Hamburger, O. Biham, and D. Avnir, Phys. Rev. E 53, 3342 (1996). P. Pfeifer and M. Obert, in Ref. 4, p. 11. P.W. Schmidt, in Ref. 4, p. 67. P. Evesque, in Ref. 4, p. 81. P. Pfeifer, in: Fractals in Physics, eds. L. Pietronero and E. Tosatti (Elsevier, Amsterdam, 1986), p. 47. F. Ehrburger and R. Jullien, in: Characterization of Porous Solids, eds. K.K. Unger, J. Rouquerol, K.S.W. Sing, and H. Kral (Elsevier, Amsterdam, 1988), p. 441. F. Ehrburger-Dolle, S. Misono, and J. Lahaye, J. Colloid Interface Sci. 135, 468 (1990). F. Ehrburger-Dolle, A. Lavanchy, and F. Stoeckli, J. Colloid Interface Sci. 166, 451 (1994). P. Pfeifer, G.P. Johnston, R. Deshpande, D.M. Smith, and AJ. Hurd, Langmuir 7, 2833 (1991).

673 44. 45. 46. 47. 48. 49 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.

P. Pfeifer, Y.J. Wu, M.W. Cole, and J. Krim, Phys. Rev. Lett. 62, 1997 (1989). P. Pfeifer, M.W. Cole, and J. Krim, Phys. Rev. Lett. 65, 663 (1990). J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Brujnseraede, Phys. Rev. Lett. 70, 57 (1993). S.K. Sinha, E.B. Sirota, S. Garoff, and H.B. Stanley, Phys. Rev. B 38, 2297 (1988). R. Chiarello, V. Panella, J. Krim, and C. Thompson, Phys. Rev. Lett. 67, 3408 (1991). F. Devreux, J.P. Boilot, F. Chaput, and B. Sapoval, Phys. Rev. Lett. 65, 614 (1990). R. Kopelman, in Ref. 4, p. 295. B. Sapoval, C.R. Acad. Sci. Paris II 312, 599 (1991). P. Pfeifer and B. Sapoval, Mat. Res. Soc. Syrup. Proc. 366, 271 (1995). W.T. Wu, A.J. McEvoy, and M. Gr~itzel, J. Electroanal. Chem. 291, 235 (1990). D. Avnir, J.J. Carberry, O.Citri, D. Farin, M. Gr/itzel, and A.J. McEvoy, Chaos 1, 397 (1991). D. Avnir, R. Guffraind, and D. Farin, in: Fractals in Science, eds. A. Bunde and S. Havlin (Springer, Berlin, 1994), p. 229. M. Sahimi, G.R. Gavalas, and T.T. Tsotis, Chem. Eng. Sci. 45, 1443 (1990). P. Salatino and F. Zimbardi, Carbon 32, 51 (1994). M. Semetz, H.R. Bittner, H. Willems, and C. Baumhoer, in Ref. 4, p. 361. P. Meakin and B. Sapoval, Phys. Rev. A 43, 2993 (1991). T.C. Halsey and M. Leibig, Ann. Phys. 219, 109 (1992). B. Sapoval, Phys. Rev. Lett. 73, 3314 (1994). R. Blender and W. Dieterich, J. Phys. A 19, L785 (1986). R.E. Goldstein, T.C. Halsey, and M. Leibig, Phys. Rev. Lett. 66, 1551 (1991). B. Duplantier, Phys. Rev. Lett. 66, 1555 (1991). O. Penrose, J. Stat. Phys. 45, 69 (1986). R.B. Stinchcombe, Phys. Rev. B 41, 2510 (1990). T. Korea and H. Tasaki, Phys. Rev. Lett. 74, 3916 (1995). M.H.W. Chan, K.I. Blum, S.Q. Murphy, G.K.S. Wong, and J.D. Reppy, Phys. Rev. Lett. 61, 1950 (1988). G.A. Williams, Phys. Rev. Lett. 64, 978 (1990). B. Sapoval, T. Gobron, and A. Margolina, Phys. Rev. Lett. 67, 2974 (1991). B. Sapoval and T. Gobron, Phys. Rev. E 47, 3013 (1993). S. Alexander, E. Courtens, and R. Vacher, Physica A 195, 286 (1993). A. Bernasconi, T. Sleator, D. Posselt, J.K. Kjiems, and H.R. Ott, Phys. Rev. B 45, 10363 (1992). M.V. Berry and I.C. Percival, Opt. Acta 33, 577 (1986). D.L. Jaggard, in: Recent Advances in Electromagnetic Theory, eds. H.N. Kritikos and D.L. Jaggard (Springer, New York, 1990), p. 183. R. Botet, P. Rannou, and M. Cabane, J. Phys. A 28, 297 (1995). F. Ehrburger and J. Lahaye, J. Phys. France 50, 1349 (1989). P. Pfeifer, in: Materials Research Society 1992 Fall Meeting (Materials Research Society, Pittsburgh, 1992), p. 511. R. Semp6r6, D. Bourret, T. Woignier, J. Phalippou, and R. Jullien, Phys. Rev. Lett. 71, 3307 (1993). D. Rojanski, D. Huppert, H.D. Bale, X. Dacai, P.W. Schmidt, D. Farin, A. Seri-Levy, and D. Avnir, Phys. Rev. Lett. 56, 2505 (1986). P. Pfeifer, D. Avnir, and D. Farin, in: Large-Scale Molecular Systems, eds. W. Gans,

674

82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104.

105. 106. 107. 108. 109. 110. 111. 112.

A. Blumen, and A. Amann (Plenum, New York, 1991), NATO Adv. Study Inst. B258, p. 215. J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982). S. Dietrich, in: Phase Transitions and Critical Phenomena, Vol. 12, eds. C. Domb and J.L. Lebowitz (Academic Press, London, 1988), p. 2. M. Schick, in: Liquids at Interfaces, eds. J. Charvolin, J.F. Joanny, and J. ZinnJustin (Elsevier, Amsterdam, 1990), p. 415. A.W. Adamson, Physical Chemistry of Surfaces, 5th ed. (Wiley, New York, 1990). H.S. Robertson, Statistical Thermophysics (Prentice-Hall, Englewood Cliffs, 1993). W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon Press, Oxford, 1974). S. Rauber, J.R. Klein, M.W. Cole, and L.W. Bruch, Surf. Sci. 123, 173 (1982). E. Cheng and M.W. Cole, Phys. Rev. B 38, 987 (1988). E. Cheng and M.W. Cole, Langmuir 5, 616 (1989). M. Napi6rkowski and S. Dietrich, Phys. Rev. E 47, 1836 (1993). R. Evans, in: Liquids at Interfaces, eds. J. Charvolin, J.F. Joanny, and J. Zinn-Justin (Elsevier, Amsterdam, 1990), p. 1. J.R. Henderson, Heterog. Chem. Rev. 2, 233 (1995). M. Plischke and B. Bergerson, Equilibrium Statistical Physics, 2nd ed. (World Scientific, Singapore, 1994). S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (Addison-Wesley, Reading, 1994). D.K. Schwartz, M.L. Schlossman, E.H. Kawamoto, G.J. Kellogg, P.S. Pershan, and B.M. Ocko, Phys. Rev. A 41, 5687 (1990). I.M. Tidswell, T.A. Rabedeau, P.S. Pershan, and S.D. Kosowsky, Phys. Rev. Lett. 66, 2108 (1991). R. Finn, Equilibrium Capillary Surfaces (Springer, New York, 1986). L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1987). D. Andelman, J.F. Joanny, and M.O. Robbins, Europhys. Lett. 7, 731 (1988). P. Pfeifer and M.W. Cole, New J. Chem. 14, 221 (1990). E. Cheng and M.W. Cole, Phys. Rev. B 41, 9650 (1990). M.O. Robbins, D. Andelman, and J.F. Joanny, Phys. Rev. A 43, 4344 (1991). P. Pfeifer and K.Y. Liu, in: Surface Disordering: Growth, Roughening, and Phase Transitions, eds. R. Jullien, J. Kert6sz, P. Meakin, and D.E. Wolf (Nova Science, New York, 1992), p. 295. H.T. Davis, Statistical Mechanics of Phases, Interfaces, and Thin Films (VCH, New York, 1996). D.H. Everett and J.M. Haynes, J. Colloid Interface Sci. 38, 125 (1972). J.C.P. Broekhoff and B.G. Linsen, in: Physical and Chemical Aspects of Adsorbents and Catalysts, ed. B.G. Linsen (Academic Press, London, 1970), p. 1. J.C.P. Broekhoff and J.H. de Boer, J. Catalysis 10, 153 (1968). P. Pfeifer, in: IUPAC Symposium on Characterization of Porous Solids, ed. K.K. Unger (DECHEMA, Frank~rt, 1987), p. 48. P. Pfeifer, M. Obert, and M.W. Cole, Proc. R. Soc. Lond. A 423, 169 (1989). P. Pfeifer and P.W. Schmidt, Phys. Rev. Lett. 60, 1345 (1988). P. Pfeifer, J. Kenntner, and M.W. Cole, in: Fundamentals of Adsorption, eds. A.B. Mersmann and S.E. Scholl (Am. Inst. Chem. Eng., New York, 1991), p. 689.

675 113. M. Kardar and J.O. Indekeu, Europhys. Lett. 12, 161 (1990). 114. M. Kardar and J.O. Indekeu, Phys. Rev. Lett. 65, 662 (1990). 115. P.G. de Gennes, in: Physics of Disordered Materials, eds. D. Adler, H. Fritzsche, S.R. Ovshinsky (Plenum, New York, 1985), p. 227. 116. F. Brochard-Wyart, C.R. Acad. Sci. Paris II 304, 785 (1987). 117. H.T. Davis, Europhys. Lett. 8, 629 (1989). 118. S.M. Cohen, R.A. Guyer, and J. Machta, Phys. Rev. B 34, 6522 (1986). 119. D. Avnir and M. Jaroniec, Langmuir 5, 1431 (1989). 120. A.V. Neimark, JETP Lett. 51, 607 (1990). 121. A.V. Neimark, Russ. J. Phys. Chem. 64, 1397 (1990). 122. A.V. Neimark, Ads. Sci. Tech. 7, 210 (1991). 123. Y. Yin, Langmuir 7, 216 (1991). 124. A.V. Neimark, Physica A 191, 258 (1992). 125. M.W. Cole, N.S. Holter, and P. Pfeifer, Phys. Rev. B 33, 8806 (1986). 126. J.J. Fripiat, L. Gatineau, and H. Van Damme, Langmuir 2, 562 (1986). 127. P. Levitz, H. Van Damme, and J.J. Fripiat, Langmuir 4, 781 (1988). 128. J.J. Fripiat, in Ref. 4, p. 331. 129. S.J. Gregg and K.S.W. Sing, Adsorption, Surface Area, and Porosity, 2nd ed. (Academic Press, London, 1982). 130. P.J.M. Carrott, A.I. McLeod, and K.S.W. Sing, in: Adsorption at the Gas-Solid and Liquid-Solid Interface, eds. J. Rouquerol and K.S.W. Sing (Elsevier, Amsterdam, 1982), p. 403. 131. W. Rudzinski and D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces (Academic Press, London, 1992). 132. G.D. Halsey, J. Am. Chem. Soc. 73, 2693 (1951). 133. E. Cheng, M.W. Cole, and A.L. Stella, Europhys. Lett. 8, 537 (1989). 134. I.M.K. Ismail and P. Pfeifer, Langmuir 10, 1532 (1994). 135. L. Leger and J.F. Joanny, Rep. Prog. Phys. 55, 431 (1992). 136. P. Pfeifer and B. Sapoval, in: Proceedings of the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, eds. A. Brunovska, W. Rudzinski, B.W. Wojciechowski (University of Lublin, 1995), p. 246. 137. P. Pfeifer and P.J. Hagerty, in: Fractals and Chaos in Chemical Engineering, ed. A.R. Giona (University of Rome, 1996), in press. 138. L. Medard, Gas Encyclopedia (Elsevier, Amsterdam, 1976). 139. H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985). 140. C. Borgs, J. De Coninck, R. Kotecky, and M. Zinque, Phys. Rev. Lett. 74, 2292 (1995). 141. P.G. de Gennes, Rev. Mod. Phys. 57, 827 (1985). 142. R. Lipowsky and M.E. Fisher, Phys. Rev. Lett. 56, 472 (1986). 143. M.E. Fisher, J. Chem. Soc., Faraday Trans. 2 82, 1569 (1986). 144. G. Forgacs, J.M. Luck, T.M. Nieuwenhuizen, Phys. Rev. Lett. 57, 2184 (1986). 145. H. Li and M. Kardar, Phys. Rev. B 42, 6546 (1990). 146. G. Forgacs, R. Lipowsky, and T.M. Nieuwenhuizen, in: Phase Transitions and Critical Phenomena, Vol. 14, eds. C. Domb and J.L. Lebowitz (Academic Press, London, 1991), p. 135. 147. J.O. Indekeu, Physica Scripta T35, 31 (1991).

676 148. E.H. Hauge, Phys. Rev. A 46, 4994 (1992). 149. G. Langie, M.J.P. Nijmeijer, Y.J. Nikas, and J.O. Indekeu, Physica A 192, 391 (1993). 150. M. Napiork6wski, W. Koch, and S. Dietrich, Ber. Bunsenges. Phys. Chem. 98, 411 (1994). 151. V. Andr6 and A. Zosel, Ber. Bunsenges. Phys. Chem. 98, 429 (1994). 152. W. Koch, S. Dietrich, and M. Napi6rkowski, Phys. Rev. E 51, 3300 (1995). 153. C. Lastoskie, K.E. Gubbins, and N. Quirke, J. Phys. Chem. 97, 4786 (1993). 154. C. Lastoskie, K.E. Gubbins, and N. Quirke, in: Characterization of Porous Solids III, eds. J. Rouquerol, F. Rodrfguez-Reinoso, K.S.W. Sing, and K.K. Unger (Elsevier, Amsterdam, 1994), p. 51. 155. F. Ehrburger-Dolle, personal communication. 156. B. Dubuc, J.F. Quiniou, C. Roques-Carmes, C. Tricot, and S.W. Zucker, Phys. Rev. A 39, 1500 (1989). 157. P. Pfeifer, in: 3rd IUPAC Symposium on Characterization of Porous Solids, ed. J. Rouquerol (University of Marseille, 1993), p.46. 158. D. Avnir, D. Farin, and P. Pfeifer, J. Colloid Interface Sci. 103, 112 (1985). 159. (a) J. Krim and E.T. Watts, in: Fundamentals of Adsorption, eds. A.B. Mersmann and S.E. Scholl (Am. Inst. Chem. Eng., New York, 1991), p. 445. (b) J. Krim, Thin Solid Films 137, 297 (1986). 160. J. Krim, J.G. Dash, and J. Suzanne, Phys. Rev. Lett. 52, 640 (1984). 161. J. Krim and V. Panella, in: Characterization of Porous Solids II, eds. F. RodfiguezReinoso, J. Rouquerol, K.S.W. Sing, and K.K. Unger (Elsevier, Amsterdam, 1991), p. 217. 162. H.S. Youn and G.B. Hess, Phys. Rev. Lett. 64, 918 (1990). 163. C.L. Wang, J. Krim, and M.F. Toney, J. Vac. Sci. Technol. A 7, 2481 (1989). 164. F. Ehrburger-Dolle, M. Holz, and J. Lahaye, Pure & Appl. Chem. 65, 2223 (1993). 165. T.W. Zerda, H. Yang, and M. Gerspacher, Rubber Chem. Technol. 65, 130 (1992). 166. P.W. SchmJdt, P. Pfeifer, and I.M.K. Ismail, unpublished. 167. K. Kaneko, M. Sato, T. Suzuki, Y. Fujiwara, K. Nishikawa, and M. Jaroniec, J. Chem. Soc. Faraday Trans. 87, 179 (199 I). 168. A.V. Neimark and K.K. Unger, J. Colloid Interface Sci. 158, 412 (1993). 169. F. Ehrburger-Dolle, M. Holz, C. Mauzac, J. Lahaye, and G.M. Pajonk, J. Non-Cryst. Solids 145, 185 (1992). 170. Y. Lefebvre and C. Jolicoeur, Colloids Surfaces 63, 67 (1992). 171. Y. Lefebvre, C. Lacelle, and C. Jolicoeur, J. Mater. Res. 7, 1888 (1992). 172. F. Ehrburger-Dolle, J. Dallamo, E. Elaloui, and G. Pajonk, J. Non-Cryst. Solids 186, 9 (1995). 173. A.B. Jarzebski, J. Lorenc, Y.I. Aristov, and N. Lisitza, J. Non-Cryst. Solids 190, 198 (1995). 174. A.V. Neimark, M. Hanson, and K.K. Unger, J. Phys. Chem. 97, 6011 (1993). 175. J.M. Drake, L.N. Yacullo, P. Levitz, and J. Klafter, J. Phys. Chem. 98, 380 (1994). 176. G.A. Niklasson, Cem. Concr. Res. 23, 1153 (1993). 177. G.A. Niklasson, Physica Scripta T49, 659 (1993). 178. A. Venkatraman, L.T. Fan, and W.P. Walawender, J. Colloid Interface Sci., in press. 179. W.A. Steele, J. Colloid Interface Sci. 75, 13 (1980). 180. P. Pfeifer, S.R. Lautenschlager, J.L. Wragg, J. West, and H.W. White, unpublished.

677 181. 182. 183. 184.

K.Y. Liu, Ph.D. Thesis, University of Missouri, 1996, unpublished. R. Messier, A.P. Giri, and R.A. Roy, J. Vac. Sci. Technol. A 2, 500 (1984). K.H. Mtiller, Surface Sci. 184, L375 (1987). H. Darmstadt, C. Roy, S. Kaliaguine, B. Sahouli, S. Blacher, R. Pirard, and F. Brouers, Rubber Chem. Technol. 68, 330 (1995).