4.2 Statistical physics of adsorption at low coverage	109

physisorbed molecules on many substrates. Bulk solid F2 is, however, quite dangerous, and has an alarming tendency to blow up by reacting dissociatively with its container.

This chapter starts by considering adsorption at low coverage, where the statistical mechanics of adsorption can be worked out precisely in terms of simple models; two of these limiting models are considered in some detail in section 4.2. The next section 4.3 discusses the application of thermodynamic reasoning to the adsorbed state of matter, including how to describe phases and phase transitions. The ®nal two sections 4.4 and 4.5 discuss the application of thermodynamic and statistical models to ®rst physisorption and then chemisorption, with experimental examples and literature references.

4.2Statistical physics of adsorption at low coverage

4.2.1General points

We have already discussed, in section 1.3.1, the sublimation of a pure solid at equilibrium, given by the condition mv5ms, with

m	5m	1kT ln (p), and the standard free energy m	52 kT ln (kT/l3).	(4.1)
v	0	0

Now we wish to consider adsorbed layers in more detail, with a corresponding chemical potential ma. Thus we have two possible conditions: ma5mv for equilibrium with the vapor, and ma5ms for equilibrium with the solid. The ®rst case is discussed in the following sections 4.2.2 and 4.2.3. The second case was the subject of problem 1.2(b) in chapter 1.

4.2.2Localized adsorption: the Langmuir adsorption isotherm

In the Langmuir picture, each adatom is adsorbed at a well-de®ned adsorption site on

the surface. The canonical partition function for the adsorbed atoms is Za5oi exp (2 Ei/kT), and in general the Helmholtz free energy F52 kTln(Z), where Ei represents the energies of all the quantized states of the system. For Na adsorbed atoms distributed over N0 sites, each of which have the same adsorption energy Ea, Za5 Q(Na,N0)exp(NaEa/kT), where Q represents the con®gurational (and vibrational) degeneracy. The new element is the con®gurational entropy, since there are many ways,

at low coverage, to arrange the adatoms on the available adsorption sites. This Q is given by (e.g. Hill 1960, chapter 7.1) as

Q5N0!/(Na !)((N0 2 Na)!),

(4.2)

multiplied by a factor qNa if vibrational eVects are included, as discussed below. The expression for ln(Q) is evaluated using Stirling's approximation for ln(N!)5N ln(N)2 N, valid for large N, to give

ma5F/Na52 (kT/Na) ln (Za)5kT ln (u/(12 u))2 Ea 2 kT ln(q),

(4.3)

where u5Na/N0.

110 4 Surface processes in adsorption

The ®rst term is the con®gurational contribution in terms of the adatom coverage u, the second the adsorption energy (measured positive with the vacuum level zero), and the last term is the (optional) vibrational contribution. We can now see that if ma 5ms, the density of adatoms in ML units is determined, in the high temperature Einstein model, by

ms53kT ln(hn/kT)2 L05ma.	(4.4a)
Using the form of ma in (4.3), and rearranging to ®nd u gives, at low coverage,
u5C exp{(2 L01Ea)/kT},	(4.4b)

where the pre-exponential function C depends on vibrations in both the solid and the adsorbed layer, and the important exponential term depends on the diVerence between the sublimation and the adsorption energy.

The Langmuir adsorption isotherm results from putting ma5mv, using this to calculate the vapor pressure p in equilibrium with the adsorbed layer. We now have

p5C1u/(12 u) exp (2 Ea/kT), or	21exp(E		(4.5a)
u5x(T)p/(11x(T)p), with x(T)5C		/kT);	(4.5b)
1	a

the constant C1 can be shown by direct substitution to be kT/ql3. The form of this isotherm is shown in ®gure 4.1(a), using parameters appropriate to xenon adsorbed on graphite. The coverage starts out linearly proportional to p, but goes to 1 as p → `.

The internal partition function q is the product of vibrational functions for the three dimensions, i.e. q5qxqyqz. If the Einstein model is chosen, then we can think of the z- direction, perpendicular to the surface, having a vibrational frequency na; this is the frequency appropriate to desorption, and in the high temperature limit qz5(kT/hna). The other two (x,y) frequencies, in the plane of the surface, will be the same on the square (or triangular, hexagonal) lattice, and correspond to diVusion frequencies nd. Thus q is inversely proportional to an `eVective' value ne3, for the adsorbed state, namely nand2. As we will see in section 4.4.4, this model is very good for the z-vibrations, but is certainly not exact for vibrations in the surface plane.

The pre-exponential constant in (4.5) is

C	5kT/ql35(2pmn 2)3/2	(kT)21/2 .	(4.6)
1	e

It is instructive to note that this is in exactly the same form as that for the vapor pressure in section 1.3.1. Moreover, the value of Ea includes the zero-point motion, analogously to the sublimation energy L0. Inserting reasonably realistic values for the vibration frequencies in (4.6) gives the full curve of ®gure 4.1(a), to be compared with the dashed curve in which vibrational eVects are neglected.

4.2.3The two-dimensional adsorbed gas: Henry law adsorption

If the entropy due to vibrations in the adsorbed layer becomes even more important, the adsorbate can eventually translate freely in two dimensions. This case is appropriate to a very smooth substrate, with shallow potential wells, and/or at high temperatures. Thus,

4.2 Statistical physics of adsorption at low coverage	111

(a)	1.0
		Ea = 1925 K
	0.8
)	0.6
θ		νa = 1, νd = 0.2 THz
(
Coverage			T=60 K, Einstein vibrations
	0.4		No vibrations
	0.2
	0.0	2x10±7	4x10±7	6x10±7	8x10±7	1x10±6
	0
(b)	1.0
		Ea±E0 = 36 K
		T=80 K, Einstein vibrations
	0.8	No vibrations (x10)
		2D gas
±2	0.6
)x10
θ
(
Coverage	0.4
	0.2
	0.0	2.0x10±74.0x10±76.0x10±78.0x10±71.0x10±6
	0.0

Pressure (Torr)

Figure 4.1. Vapor pressure isotherms of a monolayer using parameters approximating to xenon on graphite, with Ea51925 K/atom (166 meV/atom), but ignoring lateral interactions.

(a) Langmuir isotherms for T560 K: full line, Einstein model for vibration frequencies

na51,nd50.2 THz; dashed line, without vibrational eVects so that q51. (b) Comparison of Langmuir with 2D gas isotherm at T580 K: full and dashed lines as (a), dot-dash line, 2D gas with average adsorption energy E051889 K/atom. Note the lower coverage scale (4100 with respect to (a)) and the extra factor of 10 for the dashed curve without vibrational eVects.

112 4 Surface processes in adsorption

in contrast to the previous section, the other limit of isolated adatom behavior is the 2D gas. The mobile adatoms see the average adsorption energy E0, rather than the maximum energy Ea at the bottom of the potential wells. In compensation, they gain additional entropy from the gaseous motion. The chemical potential is now

m	5 2 E	1kT ln(N	l2/Aq	),	(4.7)
a	0	a	z

where this expression is valid at suYciently low density for the distinction between classical, Bose±Einstein and Fermi±Dirac statistics not to be important. The derivation involves evaluating the partition function by summing over 2D momenta, analogously to a 3D gas, while retaining the z-motion partition function qz. The diVerence between 2D and 3D accounts for l2 rather than l3, and the Na/A, the number of adsorbed atoms per unit area, is the 2D version of the 3D density N/V, as in pV5NkT.

In fact, there is a 2D version of the perfect gas law of the form FA5NakT, where F is known as the spreading pressure. This means that we could write

m	52E	1kT ln(Fl2/kTq ),	(4.8a)
a	0	z
	52E0	1m21kT ln(F ),	(4.8b)

where m25 2 kT ln (kTqz /l2) is the standard free energy of a 2D gas. This makes the correspondence between 3D gases and 2D adsorption clear: p↔F, m0 ↔m2, and the energy is lower in the 2D case by E0. Note that it is easy to forget the qz term, as is often done, since the various qs are dimensionless: this doesn't make them unimportant numerically.

By equating ma5mv we get Henry's law for 2D gas adsorption:

p5C2(Na/A)exp(2 E0/kT), or	(4.9a)
(Na/A)5x9(T)p,	(4.9b)

with x9(T)5C221exp(E0/kT), and C25kT/qzl.

You may feel that detailed discussion of these constants is rather laboring the point, but it is instructive if we stick with it for a while. Note that the 2D gas form has (Na/A) proportional to p, whereas the localized form has the coverage u5Na/N0 proportional to p. These can be reconciled if we write (Na/A)5u (N0/A). Here we have de®ned the monolayer coverage (N0/A), and then de®ned (Na/A), the areal density of adsorbed atoms in terms of this, rather arti®cial, constant. (Both N0 and Na are numbers here, not areal densities, though we can think of them as densities by choosing A51). This is the identical problem we discussed in section 2.1.4, emphasizing the need for consistency in the de®nition of the ML unit. If, however, we do make this de®nition, we can recast the 2D gas equation as

p5(kTN0/Aqzl)u exp(2 E0/kT),

(4.10)

which can be compared directly with the corresponding equation for localized adsorption.

This comparison shows that there is a transition from localized to 2D gas-like behavior as the temperature is raised, because Ea.E0, whereas the pre-exponential (entropic) term is larger for the 2D gas. The ratio of coverages at a given p for the two states is