Concentration fluctuations in adsorbed layers

Surface Science 55 (1976) 494-508 0 North-Holland Publishing Company

CONCENTRATION FLUCTUATIONS IN ADSORBED LAYERS

Brian BELL and Robert GOMER The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, U.S.A.

and

Howard REISS ‘Department of Chemistry, The University of California at Los Angeles, Los Angeles, California, U.S. A.

Received 7 October 1975; manuscript received in final form 12 December 1975

The temperature and coverage dependence of the mean square concentration fluctuations in a small open domain of an adsorbed layer is discussed for various situations. It is shown that fluctuations decrease with increasing temperature and reach a limiting value when attractive interactions predominate, but increase and reach a limiting value when repulsive interactions predominate, if a single non-ideal two -dimensional phase exists. De- viations from ideal gas behavlor are strongest at half coverage. At very low coverage (low particle concentration) and very high coverage (low hole concentration) ideal behavior is approached. If the layer consists of a two phase system, for instance a two-dimensicnal liquid or solid in equilibrium with a two-dimensional gas, fluctuations far below the critical temperature are dominated by fluctuations in the partition between phases. As the critical temperature is approached fluctuations fist decrease because the mean concentra- tions in the two phases approach each other, and then increase very sharply near T,. Detailed calculations for the single phase situation are given for several approximations: a dilute gas; a mean field approximation; a lattice gas in both the Bragg-Williams and the Bethe- Peierls-Weiss approximations. The latter which takes some account of correlations between adsorbate particle positions seems to explain reasonably the presently available experimental observations on chemisorbed layers.

1. Introduction

In a previous publication [l] it was shown that the time correlation function of current fluctuations in field emission can be utilized for the determination of surface diffusion coefficients of adsorbates on single planes of field emitters. In a typical ex- periment one would examine a region of 20-50 A in,radius contained, in the case of a close-packed plane, within a plane of perhaps 100-300 A radius. Explicit expres- sions for the correlation function were worked out with the assumption of no interactions between the adsorbate particles. The current correlation function for f = 0,

B. Bell et al./Concentration fluctuations in adsorbed layers 495

-1 ,(y 600K 500K

I", I' 'I

(AI)' .

7- \ i .

IO67

I I I I I lo' 1.6 1.8 20 2.2

‘/l ( x b’l

Fig. 1. Relative mean square current fluctuation in field emission from art oxygen covered (110) plane of a tungsten field emitter versus l/T [2]. The radius of the probed region is roughly 50 A; coverage 0 - 0.4. The mean square current fluctuation is proportional to the absolute concentration fluctuation.

is the mean square fluctuation and is roportional to the absolute mean square adsorbate concentration fluctuation, (An) P

(Anj2 = (n2) - (nj2. (1) For non-interacting adsorbate particles

(Anj2 =(n), (2)

where (n) z n’ is the mean number of adsorbate particles in the region of observation. Experiments by Chen and Comer [2] indicate that strong deviations from such

ideal behavior can occur. For 0 or CO on the (110) plane of W, for instance, (A.nj2 first increases with T and then levels off (fig. 1). The present paper explores various possible deviations of (Anj2 from ideal gas behavior that might be encountered in chemisorption or physisorption.

It can be shown from general considerations [3] that

(An,2/ii = (E/A) K kT, (3)

where A is the area of the probed region and K the two-dimensional compressibility

K = -( 1 /A) aA/aps, (4)

with p, the two-dimensional pressure. By analogy to p-V diagrams for 3-dimensional phases K decreases with increasing T if there are short range repulsive and long range

496 3. Bell et al./Concentrution fluctuations in adsorbed layers

attractive interactions. The reason for this is, of course, that increasing temperature increases the importance of kinetic relative to potential energy. In the case of an imperfect gas with attractive interactions predominating at the mean intermolecular distance, the compressibility exceeds that of an ideal gas and an increase in Twill, therefore, decrease K; similarly K will increase with T if repulsive interactions predominate. Thus, an increase of K with T implies the predomin~ce of repulsive interactions at all separations. This could come about if dipole-dipole repulsions predominate.

2. Single phase of an imperfect gas

We shall now consider the adsorbate as an imperfect two-dimensional gas in various approximations. The first to be presented, the dilute gas and mean field approxima-

tions, ignore the structure of the substrate completely, and are therefore most ap propriate to situations where binding is relatively non-specific, for instance physisorp-

tion. Next, lattice gas approximations, which emphasize the existence of discrete binding sites will be discussed. These probably apply to most cases of chemisorption.

2.1. Dilutegas

An equation of state can be obtained by standard methods [4] and then leads to

{AF~~~/~ = (1-81)-1, (5)

where t? = E/A and

7 1=2nOJ [exp(-up)- l] rdr.

Here /_I = IjkT, and U(T) is the pair potential, r being the separation between adsorbate particles. If u(r) is taken to be

u(x) = m, x4 1, (7)

u(x) = uox-s’ x2 1, (8)

with x = r/r,, r. being the distance of closest approach, one obtains -1

s I [exp (+-$3x-“) - I] x . (9)

If u. < 0, corresponding to long range attractive forces, the integral in eq. (9) is positive and approaches 0 as T-+ 00. Thus, (Anj2 decreases as T increases. On the other hand, if u0 > 0, i.e. if the potential is everywhere repulsive, the integral will be

B. Bell et al/Concentration fluctuations in adsorbed layers 491

Table 1

ValuesofP=Jy[exp(-cWr-3)- l] xdx

0.1 -9.875 x 16’

0.5 -4.714 x 10-l

1.0 -8.950

5.0 -3.416

7.0 -4.400

10.0 -5.715

20.0 -9.365

x = r/r,,; rO is minimum separation of 2 ad-particles; OL = uo/kT; uo.is (repulsive) interaction en-

ergy at ro.

negative, approaching 0 as T increases, and thus (Anj2 will increase and then level off as T increases. Theoretical considerations [5] indicate that a part of U(I) is in fact oscillatory for adsorbed particles because of substrate-mediated interactions. A cal- culation using a damped sinusoidal interaction energy showed the same qualitative features: the repulsive part of the potential must dominate, even at long range, to account for an increase in (Anj2 with increasing T. Table 1 lists numerically computed values of

1’ = s 1

{exp [-(uo/kT)x3] - l} x dx (10)

for various positive values of uo/kT, corresponding to repulsive interaction. The ex- ponent s = 3 corresponds to dipole-dipole repulsion. Fig. 2 shows a plot of (Anj2/E versus u,/kT for t)rrri = 1, i.e. high adsorbate coverage, for which eq. (9) is not expected to be quantitatively applicable, having been derived specifically for the dilute gas case.

Fig. 2. tAn)E versus kT/uo for the dilute gas approximation, eq. (9); em-i = 1.

498 B. Bell et al./Concentration fluctuations in adsorbed layers

2.2. Mean field

It is also possible to treat the case of high adsorbate concentration in the mean field approximation. We assume that the effective potential energy U felt by an adsorbate particle because of the presence of all other adsorbate is given by

U = 2nqc&i/A = (Y A$, (11)

where q is the adsorbate charge, P the dipole moment per ad-particle and cy a coverage dependent factor [6] < 1 given by

ff = I+&,)/@=) = V(d,)/A$, (12)

where V is the dipole layer potential, d, the adsorbate-surface separation and A$ the work function change caused by the ad-layer.

The partition function is given approximately [4] by

In Z = 5 In (A - A,,) - FiU/kT, (13)

where A, = &~ri ‘is the excluded area, so thatthe two-dimensional pressure p, be- comes

p, = kT(a lnZ/aA) = iikT/(A-Ao) t z2 q2nP/A2,

and K is given by

(14)

K= 1

AiikT/(A-A,)2 t 4Fi27rPcuj/A2 ’ (IS)

Thus

(Anj2 1 ~ = i? [I/( 1 - (-&rr;)] 2 + 2(O/kT) 27~Pcr4

(1-3&r@ =

1 + (1 - &rr;)2(O/kT)4~Pcxq ’ (16)

Eq. (16) shows that in the mean field approximation 1 /(An j2 varies inversely with T and linearly with coverage for a gas of point particles. The finite size of the adsorbate tends to decrease fluctuations as 0 increases. This corresponds to the fact that one may consider fluctuations to involve particles at low 0 and holes at high 0. Al- though the diffusion coefficient of holes may differ from that of particles, a dilute “hole gas” should again be nearly perfect, just like a dilute particle gas. E % (16) contains this feature qualitatively. If we define maximum coverage err, = 2/nro the number of holes is given by

nrr=nrrr - n =+,/n - 1) = n(e,/e-i),

where nr,, is the maximum number of adsorbate particles. Consequently,

tnnj2 + n(e/em)2(e,/e-i)2 = t2,(e/emj2(i-eie,)

(17)

08)

B. Bell et al./Concentration fluctuations in adsorbed layers 499

I I I I I I I I 300 400 500 600 700 800 900 1000

Fig. 3. (AM2/3 versus T for the mean field approximation, eq. (16). For specificity the following values have been assumed: A@ = 1 eV; 4 = 0.1 e-; (Y = 0.7; erri = 1.

as8+0,. It is not entirely obvious why this expression contains the extra factor

(0/0m)2 (1 - e/e,). A plot of (An)2/Z versus Tbased on eq. (16) is shown in fig. 3.

2.3. Lattice gas

Possibly a somewhat better approximation, at least for chemisorption, may be provided by the lattice gas model. Even in its simplest form this gives the expected linear behavior with concentration both for particles and holes. We start by assuming

that the Helmholtz free energy, F, of a lattice gas is given by

F=+zenx+kT[nlnxtm(l -x)ln(l -x)]. (19)

Here E is the nearest neighbor interaction energy (all other being neglected), z is the number of nearest neighbors of a site; m is the number of sites, n the number of particles distributed on the m sites and x = n/m the fraction of filled sites. In this approximation we ignore the fact that the existence of an interaction leads, at least for e 9 kT, to a non-random distribution of nearest neighbor occupancy. Let the area per site be a so that A = ma. We then obtain

P, = (-l/a)@F/am), = (-kT/a) [ln (1 -x) - vx2],

where 1) = ze/2kT, and straightforwardly

(20)

CAnj2/Z = l-x

1 + 2QX(l - x) . (21)

Eq. (21) has the same T-dependence as eq. (16), but shows a more plausible coverage dependence: at very small x, &zj2 = n(l - x) 1. n, while at very high x it assumes the form (1 - x)n = nh. This behavior is insured by the factor x( 1 - x) in the denominator which approaches 0 as x + 0 and as x + 1. We also see that attractive inter-

actions (r) < 0) lead to a decrease in fluctuations with increasing T while repulsive interactions (77 > 0) lead to increasing values which approach (1 - x) asymptotically.

For the experimental cases studied so far the fluctuations increase far more rapidly with T than is predicted by any of the models treated so far. It is possible that this results from neglect of correlation, i.e. in the lattice model from the assumption of


completely random nearest neighbor occupancy. It is of course extremely difficult to treat this problem correctly, even in the lattice gas model, but an approximate

treatment based on the Bethe-Peierls-Weiss model [7] is possible. This model con- siders in detail a pair of adjacent sites, and treats the environment of this pair in the mean field approximation. If again we consider only nearest neighbor interactions the canonical partition function is

Z = C &,Y) exp (G&W, J

(22)

where g(n,y) is the number of ways of putting II atoms on m sites, such that there are y pairs of adjacent atoms. The equilibrium number of pairs isJ and can be found from Z in the usual way. We now define an effective number of pairs, yo, such that

Z = exp (-yoe/kT) [m!/n!(m - n)!] . (23)

The last factor is just the number of ways of putting natoms randomly on m sites, and the effects of interactions are thus contained inyo. It can be shown [7] that

7 = ~(&Jl~& (24)

where fl= l/kT, and further that

2F(zm - 2zn + 27) = (zn - 2J)2 exp (-e/kT).

Eq. (25) is quadratic in 7 and has the solution

-7 = +z12 [ 1 - 2( 1 - x)/( 1 + IX)] ,

where

(25)

(2’3

CY= (1 -4x(1 -x)[l - exp(-e/kT)]}1’2. (27)

Thus we have an expression for 7, the mean number of pairs, and, in eq. (24) a rela- tion between7 and yo, the effective number of pairs. We can now use eq. (23) for the canonical partition function, expressed in terms ofYo, to calculate the fluctuations. Proceeding as before, that is, finding the compressibility from Z, we obtain

(Anj2 = x2kT(a2Z/am2).

which yields, using eq. (23)

(28)

(Anj2 _ l-x ii 1 + (m/x)(a2yo/am2)(I - X)TJ '

(29)

We shall evaluate eq. (29) for three cases: low coverage, x = 0.5, and high coverage. (1) Low coverage. For IZ 4 rn,y < n,m, eq. (25) reduces to

7 = +znx exp (--E/kT). (30)

B. Bell et al/Concentration fluctuations in adsorbed layers 501

E/kT

Fig. 4. (Ad as a function of e/kT, divided by the infinite temperature value, <&z(O))~ versus q’kT, plotted according to eqs. (33) and (40) for x = 0.1 and OS, respectively; z = 6 in both cases.

From eq. (24) we obtain

@y. = &XX j exp (-0’~) do’, 0

or

(31)

y. = (2Xx/2/k) [ 1 - exp (-e/CT)] .

Using the result (32) in eq. (29) finahy yields

(32)

CAnj2 l-x -= Ti 1 + zx( 1 - x) [ 1 - exp (-e/U)] ’ (33)

Eq. (33) is reminiscent of eq. (21) for the simple lattice gas model and reduces to it for e < kT. A plot of (An)* versus e/kT based on eq. (33) is shown in fig. 4.

(2) I? = 0.5. The general intermediate’coverage case is rather messy, but simplifica- tions occur for half coverage, x = 0.5. For this case

01= exp (-c/2kT), (34)

&Y/am = 0, (35)

a2cu/am2 = 4 [l - exp (-e/kT)] exp (e/2kT)(aJ/am)2, (36)

where CK is defined by eq. (27). Differentiating eq. (26) twice with respect to m yields

a*js _ zx exp (42kT~ + exp (e/2kT) [exp (q’kT) - 1 J

am2 m 2 [l + exp (e/2kT)] 2 . 1 + exp (e/2kT) (37)

We can now find a2~o~am2 from

(38)

502 B. Bell et al. fConcentration fluctuations in adsorbed layers

using eq. (37) for a2~/&n2. The result, after some straightforward manipulation, is

a2-vo kT zx ~ =T m [l - exp (- e/2kT)] exp (e/2kT), am2

(39)

which leads to

(Anj2 l-x - = z 1 + z( 1 - x) exp (e/2kT) [ 1 - exp (-e/2kT)] ’ (40)

where x = 0.5, for this special case. Eq. (40) shows the strongest variation of (Anj2 with temperature of any of the cases considered so far because of the exponential exp (e/2kT) in the denominator. It also reduces to eq. (21) for E Q kT. It is interesting that an intermediate limiting form of eq. (40) is

(Anj2 l-x __ = 1 + z(1 1 x)(e/2kT) exp (e/2kT) . ii (41)

This is the form one would obtain from the simple lattice gas model by arbitrarily

multiplying the energy term in eq. (19) by a Boltzmann factor exp (e/2kT). A plot of (A~r)~/ii versus e/kT based on eq. (40) is shown in fig. 4.

(3) High coverage. By analogy to the arguments advanced for the simple lattice

gas, we should expect fluctuations to decrease at high particle coverage, corresponding to low hole coverage. It can in fact be shown that

(Anj2 (42)

where cp = x( 1 - x) and g(p) is a function of up. Thus the absolute mean square particle fluctuation is a symmetric function of mean coverage about x = 0.5, since the factor x( 1 - x) is symmetric about this point. The arguments leading to eq. (42) are tedious and will only be outlined here: We note from eqs. (26) and (27) that _F can be written as

jj=3zn[l -2(1-x).@)], (43)

and that intergration with respect to /3 leads to

Lo = :zn P - 2(1 - xhydl > (44)

where f and f. are different functions of cp. Double differentiation of y. with respect to m leads to

a2yOiai722 = (zx~/vz)~(~), (45)

where g(cp) is a function of cp only, i.e. does not depend on x or (1 - x) separately, SO that eq. (42) results.

B. Bell et al./Concentration jhctuations in adsorbed layers 503

3. Two coexisting phases

So far we have considered only a single phase of the adsorbed layer. While it seems probable that the results on chemisorbed layers obtained to date by Chen and Gomer [2] can be explained in these terms, it is worthwhile to explore other situations as well, since these could occur for instance in physisorption, which may also be amen- able to study by the field emission technique outlined at the beginning of this paper. Specifically, we consider here two adsorbed phases in equilibrium. This implies attractive interactions and a temperature below the (two-dimensional) critical T, T,. We carry out the analysis on the basis of the simple lattice gas model, whose free en-

ergy is given by eq. (19). For equilibrium between two phases, i = 1,2, we require that ~_l, = p2 and also that

pI = pZ, where the chemical potential p is given by

P = (aFlani)~,~ it (46)

and the two-dimensional pressures p, by eq. (20). One obtains from eqs. (19) (46) and (20)

2@, + In [Z1/(l -X1)] = 2$X$ t In [Z2/(1 - Z2)], (47)

-T# + In (1 - XI) = -7$i$ t In (1 - X,), (48)

for the conditions of equilibrium in a closed system; all Xi are average values, as usual. r~ is defined as before as zc/2kT. Eqs. (47) and (48) are also obtained by minimizing the total free energy for the two phase system as a function of ~1 and ml.

Eqs. (47) and (48) are satisfied by the trivial solution Z1 =X2, but also by

x1 +x1 = 1, (49)

which leads to

rj (1 - 23) = In [.Y2/(1 - Z2)], W)

which can be solved numerically for Z, for given r) or trivially for Q, for given x2. It is also possible to obtain the critical temperature, or simply the critical value of Q, 9, from the requirement that

ap/am=a2p/am2=0' (51) at qC. This leads straightforwardly to 71, = 2.

We consider now the fluctuations to be expected in an open subregion of the (in~nitej closed system. We define the total number of sites in the open subregion A

as m, and the number of sites occupied by phase 1 in A as no l , those occupied by phase 2 as m2 and the number of ad-particles in A belonging to phase 1 as tzl, those belonging to phase 2 as n2, and define x1 = nlfml, x2 = n2/m2, It is to be noted that

ml +m2=m, (52)


since the number of sites in A is fixed. but that nI, x1, n2, and x2 obey no conser- vation rules, other than xi G 1, ni =G m. However the average values, El + X, = 1 as before.

We now define the following quantities:

Qm, is the probability of a partition into ml, m - m, sites for phases 1 and 2 respectively dividing m sites into ml for phase 1 and m - ml for phase 2.

Since there are m ways of dividing m sites into two groups,

QWI, = I/m, (53)

pl(n,) is the probability of there being pzl molecules in phase 1 in A, if there are m1 sites belonging to phase 1 in A. Pm-“‘1 (n2) is the probability of there being fz2

molecules in phase 2 in A if there are m2 = m - ml sites belonging to phase 2 in A.

It is to be noted that Pml(nl) andP-m1(tz2) are independent. Then Q,, pm1 (n l)~-mr (n2) is the probability of a partition with m 1 sites assigned to phase 1, and n1 and n2 ad-particles in phases 1 and 2, respectively. Finally we define (A,, tz 1 j2 as the mean square fluctuation of nl under the partition nzl, m - m 1,

(n 1 &, as the mean number of ad-particles in phase 1 in A, under the partition m 1,

m - ml, and (Am-m, n2j2 and (n2)m-m, as the corresponding quantities, under the

same partition, for n2.

We can then write for the total fluctuation in region A

(Anj2 = gee Q,, Pml(nl) Pmmml (n2) (tzl + 1~~)~

--[cFc Q,, P~1(nl)Pm-m1(n2)(t+ +n2)12.

Carrying out the summations over nl ,n2 and noting that

C Pml(nl)nl 2 = (n;), 4

1’

and so forth, we have

(Anj2 = c Qm, oI~~m, + c Qm, hz~)ml + 2 c Q,, hl)m, (tz2jm, ml 1111 ml

_ (C Qm,(nl)m,)2 -(g Qml %)m,)2

ml

_

( 2 CQml (nl),

w )( C Qm, %‘m,) . 1 ml

(54)

(56)

B. Bell et al/Concentration fluctuations ill adsorbed lavers

Since

(?Z;& 1

= (Amln1j2 + (nl)fn I

with an analogous expression for (ni),, we obtain

(Anj2 = c Q,, (A,,,r n,j2 + c Q,, (AmI n2j2 ml *‘l

+2x Q,, (nl)m,(n2)ml - ml

~Qml(nl)m ml

1

505

(57)

(58)

We note that

(Am1n1j2 = kTKIEf/mIa = (kT/a) KIZfml,

tA,r n2j2 = (kT/a) K2Eg(m - ml).

Replacing Zml by Jo” dm, we finally obtain

(59)

(60)

(61)

The first term in eq. (61) represents the contribution from fluctuations within the two phases. It is interesting that on average our model predicts half the sites in A to be covered by each phase which accounts for the factor of 2 in the denominator of the first term. The second term in eq. (61) corresponds to fluctuations in the partition between phases, and is proportional to the square of the number of sites in A, ortoA2.

K, and K2 can be evaluated explicitly by means of eq. (20) and lead to the result, already familiar from eq. (2 l),

X1(1 - 21) (Anj2 = m II-

2

2nx,(l -2) + 1 +% (Z1 - Y2)2.

3.1. Fluctuations in the dilute phase only

(62)

Up to this point we have assumed the open subregion to contain both dilute and dense phases and have calculated the total fluctuation in number density. It is easy, of course, to calculate also the fluctuations in the dilute phase alone, when the region of observation contains both phases and when it contains only the dilute phase.

506 B. Bell et al./Concentration fluctuations in adsorbed la)‘ers

The latter situation might conceivably occur if the dense phase is anchored to the

edge of a plane of the field emitter. The results are given by eq. (2 1) with the following modification: If the region of observation contains both dense and dilute phases, the dilute phase occupies, on average, half of the region and hence (An)” dilute is given by the result of eq. (2 1) divided by 2. If the observation region contains only the dilute phase, eq. (2 1) applies without modification.

3.2. P/lase 2 ahva_vs deme

The analysis given here has omitted line tension, or better line energy effects, i.e. the two-dimensional analogue of surface energy effects. These are twofold. First, they will tend to prevent the segregation of the dense phase into many small patches. Second, they will have the effect of keeping the density of the dense phase high. We simulate this by postulating that X2 = 1 at all T. An analysis analogous to that leading to eq. (SO) then shows that

n(1 - 2X,) = ln[ZI/(l - .?I)], (63)

which is identical in form to eq. (50). However, Fyl + .Y2 # 1 here. Proceeding by steps analogous to those leading from eq. (54) to eq. (62) we find that (Anj2 is identical to that obtained by letting X2 = 1 in eq. (62). Since the solution of eq. (62) for Z1 is identical to that obtained for ?, from eq. (50) and since for that case ff,(l - 21) = X7( 1 - X2), we need not make arzy change in eq. (62) other than to set Z2 = 1 in the second term. It further follows that the fluctuations in phase 1 alone are still given by both the form and numerical values, for given XI, of eq. (2 l), if only the dilute phase exists in the region of observation, or by half that value if both phases occur in it.

Table2 showsvaluesofZI(1 -X,)/(1 +2nZZ1(1 -X,)),(m/12)(Z1 -“2)2 and

Table 2 Fluctuations in a 2 phase ad-layer

-t7 x(1-x)

1 + 2qx(l ~ x,

2.0005

2.01

2.5

3.0

5.0

10

16

499

24.8

0.33

0.11

0.007

4.5 x 100

1o-7

6 x 16’

1.2 x 1o-3

0.042

0.07 1

0.08 1

0.083

0.083

(A,z)~/TH ___~~~ ~~~ -.-. ~_~__

For ,,I = 100 For WI = 1000

~~~ _-

499 499

24.9 26

4.52 42.3

7.19 70.9

8.10 80.9

8.33 83.3

8.33 83.3

B. Bell et al./Concentrution fluctuations in adsorbed layers 507

Fig. 5. Various quantities relevant to the two-phase situation plotted versus Q. Solid lines: (AM2/m for m = 100, 1000, plotted according to the unrestricted case of eq. (62); for M = 1000 the curve corresponding to X2 = 1 is also shown; dashed lines: PI, Fz; dotted line: Xl (l-X,)/ [I+ 2qQ,(l-K,].

(Anj2/m as functions of n according to the unrestricted form of eq. (62). These data are also displayed in fig. 5. Several interesting conclusions can be drawn from these results. First, we note that far below the critical temperature (In I 9 l), fluctuations in the partition between the phases, i.e. the second term in eq. (62), dominate and are effectively constant since X2 3 Z1. As the temperature rises, X2 -+ Z1 and this term decreases. The fluctuations thus decrease with increasing temperature. This is, of course, a result we might have anticipated since the two phase system is in fact a special case of attractive interactions. However, very near the critical temperature, i.e. as I ?I I --f 2, fluctuations within each phase become very large and dominate the total fluctuations. Thus the curves of fig. 4 go through a minimum and then rise steeply. To date this behavior has not in fact been observed but should be expected for physisorption systems near 7’e.

4. Equilibrium between singlets and doublets

We conclude by discussing very briefly a special case of attractive interactions. Suppose that a given adsorbate B can exist as single ad-particles B and also as doublets B2:, and suppose that only the single particles B are mobile. For this arguments B2 need not be an adsorbed molecule; it suffices that there be an attractive interaction between two B particles, and that an adjacent filled pair of sites constitute an immo- bile entity. It might be thought at first that the fluctuations in such a system will in-

508 B. Bell et aLlConcentration fluctuations in adsorbed layers

crease with increasing T, since the number of mobile particles, B, increases with increasing T. However, the observed fluctuations refer to both B and B,, and fluctuations in the latter are caused via the equilibrium

B2 = 2B

with B mobile. A straightforward analysis, based for simplicity on the classical form of the free energy

Fi = nikT In xi t Eni ~ kTni

and equilibrium between B and B, with an energy of dissociation E leads to the result

(A,$ [ 1 + 4X exp (-e/kT)] l/2 __ =-

ii 1 + [l t 4si exp (-e/kT)] ‘I2 ’ (65)

where X is the mean total adsorbate concentration in terms of B. Thus (Arrj2 decreases as T increases.

5. Conclusion

The foregoing indicates that a study of mean square concentration fluctuations in small open domains of adsorbed layers can provide valuable information on the ther-

modynamics and statistical mechanics of such layers, including the magnitude and sign of ad-particle interactions, and the occurence of phase transitions and critical

points.

Acknowledgements

We wish to thank Mr. Brice Carnahan of the Department of Materials and Metal- lurgical Engineering of the University of Michigan for carrying out the computer calculations used in this work. This research was supported in part by NSF Grant, NSF DMR 73-07589-A01 and by the Advanced Research Projects Agency of the Depart- ment of Defense under Contract DAHC 15-71-C-0253.

References

[l] R. Gomer, Surface Sci 38 (1973) 373.

[2] J.R. Chen and R. Gomer, to be published.

[3] N. Davidson, Statistical Mechanics (McGraw-Hill, New York, 1962).

[4] F. Reif, Fundamentals of Statistical and Thermal Physics’(McCraw-Hill, New York, 1965).

[5] T.L. Einstein and J.R. Schrieffer, Phys. Rev. B7 (1973) 3629;

T.B. Grimley, Proc. Phys. Sot. (London) 90 (1967) 751.

[6] L.D. Schmidt and R. Gomer, J. Chem. Phys. 45 (1966) 1605. [7] R. Fowler and E.A. Guggenheim, Statistical Thermodynamics (Cambridge Univ. Press, Cam-

bridge, 1952) pp. 429-441.

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Concentration fluctuations in adsorbed layers