Langmuir 1992,8, 222-228
222
Ideal Adsorption on a Lattice with Exclusion of Nearest Neighbors Edoardo Garrone* and Piero Ugliengo Dipartimento di Chimica Inorganica, Chimica Fisica e Chimica dei Materiali, Universith di Torino, via P. Giuria 7, 10125 Torino, Italy Received February 25, 1991. I n Final Form: August 8, 1991 An old subject in absorption theory is revisited from the standpoint of both analytical and Monte Carlo solutions. Three cases are studied in the ideal approximation: (A) the five-exclusionpattern on the square lattice; (B) the seven-exclusionpattern on the triangular lattice; (C) the nine-exclusion pattern on the square lattice. The starting point is the calculation of o(0), the average occupation number per particle at coverage 0, from which the configurationalentropy is computed. In casesA and B analytical considerations allow the a priori determination of a fourth-degree polynomial for ir(0), which reproduces satisfactorily the MC results. In case C, analytical considerations are restricted by the presence of a residual disorder at monolayer coverage: the corresponding configurational entropy is calculated from MC results. Introduction The subject of the present paper goes back to the early days of the theory of adsorption. Soon after the development of the Langmuir model of ideal adsorption (all sites equivalent and noninteracting, 1:l correspondence between site and possibly adsorbed particle), the case was considered of the ideal adsorption of particles larger than the distance between adjacent sites, so that one particle occupies more than one site, or at least inhibits adsorption on the sites neighboring an occupied one. Curiously,the reverse problem of the adsorption of more than one particle on the same site has been treated only recently,' although it corresponds to situations often encountered in surface chemistry, e.g. the formation of polycarbonyls or polynitrosyls on dispersed atoms or cations. Langmuir himself? R ~ b e r t s ,and ~ Tonks4 have attempted different analytical solutions to the problem under consideration. The method adopted by Roberts3has been proved to be inc~nsistent.~ The results by Langmuir and Tonks are in agreement. The statistical method employed by Tonks, though more general than Langmuir's kinetic approach, only allowed him to propose analytical expressions for the isotherm of adsorption on a square lattice of noninteracting particles with a five- or nine-exclusion pattern at either low or high coverage, an overall expression for the isotherm being outside the capability of his method. In the absence of interactions among particles, the difficulty resides in the computation of the configurational entropy of the adsorbed phase. In the mid 1960s Baker5 has calculated the differential configurational entropy for particles with a nine-exclusionpattern on the square lattice and with a seven-exclusionpattern on the triangular lattice by a Monte Carlo (MC) method. The full range of coverage was however not explored, nor was any equation proposed for the adsorption isotherms. We have become recently interested in this old problem when considering the adsorption of NO (or CO) at the (100) face of Ni0.6-s Sites of adsorption form a square (1) Garrone, E.; Ugliengo, P. J. Chem. Soc., Faraday Trans. 1 1989, 85, 585. (2) Langmuir, I. J. Chem. SOC.1940, 511. (3) Roberts, J. K. Proc. Cambridge Philos. Soc. 1938, 34, 577. (4) Tonks, L. J. Chem. Phys. 1940,8, 477. ( 5 )Baker, B. G. J. Phys. Chem. 1966,45, 2694. (6) Escalona Platero, E.; Coluccia, S.;Zecchina,A. Surf. Sci. 1986,171, 465.
lattice; owing to the strong interrepulsion, adsorption on adjacent sites occurs with much difficulty, and indeed an adsorbed molecule blocks neighbor sites. Baker's treatment deals with W ) ,the average number of sites blocked by one particle at a given coverage 8, which is calculated via the MC method up to a limit coverage. We have (i) extended MC calculations of D up to full coverage; (ii) considered also the case of a five-exclusion pattern on the square lattice; (iii) found an analytical approach to the calculation of D(8) different from those already p r o p o ~ e d ,which ~ ~ ~ is capable of yielding the adsorption isotherm in the whole coverage range. We are aware that the problem dealt with in the present paper is, on the one hand, the oversimplification of much more realistic models envisaging the interaction of adparticles, which constitute an active field of research, where powerful techniques (e.g. real space renormalization group) are currently applied. On the other hand, much work has been done on the nonequilibrium distribution in blocking interactions, which bears much similarity with the present paper; see, e.g., the work by Evans.+ll Nonetheless, it is our opinion that something can still be learned from the analytical approach to simple problems like the one studied here. General Features Consider the adsorption on a regular array of M equivalent sites of N particles of circular symmetry, large enough to prevent adsorption on neighboring sites. A site is then blocked to adsorption of other particles if it is occupied or a neighbor of an occupied site. The number and disposition of sites blocked per particle at vanishing N is the exclusion pattern, the nature of which is dictated by the nature of the lattice and by the ratio between the radius of the particle and the distances between sites. The simplest cases are the five-exclusion pattern on the square lattice and the seven-exclusion pattern on the triangular one, which will be referred to hereafter as case A and case B, respectively (Chart I). By increasing the particle dimension, one gets the 9exclusion pattern on the square lattice and the 13-exclusion
~~
0743-7463/92/2408-0222$03.00/0
(7) Escalona Platero, E.; Coluccia, S.; Zecchina, A. Langmuir 1985,1, 407. ( 8 ) Garrone, E.; Fubini, B.; Escalona Platero, E.; Zecchina, A. Langmuir 1989, 5, 240. (9) Evans, J. W.;Hoffman, D. K.; Pak, H. Surf. Sci. 1987, 192, 475. (10) Evans, J. W.;Pak, H. Surf. Sci. 1988, 199, 28. (11) Evans, J. W. Surf. Sci 1989, 215, 319.
0 1992 American Chemical Society
Langmuir, Vol. 8, No. 1, 1992 223
Ideal Adsorption on a Lattice Chart I
Chart IV
0
0
O
O
0
0
0
0
Q
O
O
o
O
Q
O
O
0
0
0
0
0
0
Q
0
0
@ @ e
o
0
O
0
0
.a.
0
Case C
Case B
Case A
O
Q
0
o
empty site 8 inhibited S l i p
0
e
0
e\.
Q
0
o
e
0
0
O
0
occupied
SltQ
Chart I1 Case D
Case C
0
0
O
.De 0
0
0 0
0
0
Q
0
Q
)
Q
0
.
0
0 8
0
0
Q
O 0
8
e
0
0
e
0
o
0
0 empty site
e
o
OCCuplQd Site
0 inhibited site
Chart 111 Case B
CaSQ A .
0
.
@
0
.
e
.
e
.
.
0
occupied site
8
inhibited site
8
e
e
e
e
e
thermodynamics: in the present case residual freedom concerns the translation of either rows or columns only. It has to be noted, however, that a vanishingly small number of vacancies (actually, one per column or row, respectively) introduces a further degree of freedom in the location of the adsorbed particle. In conclusion, we consider s(1) in case C as unknown, and regard its evaluation as one goal of the present work. The average occupation number D is defined as the ratio between BN (number of sites blocked), and N (number of particles)
w )= BN/N
OCCUplPd
SltQ
Q I n h i b i t e d site
pattern on the triangular one. In the following, the former will be referred to as case C; the latter (case D)will not be treated (Chart 11). The number of sites blocked per adsorbed particle will be denoted as u ( 0 ) a t vanishing coverage and as u(1) at the monolayer capacity. It has to be noted that in both cases A and B, at the monolayer capacity an ordered overlayer is obtained. Case A yields a ~ ( 2 x 2structure, ) where u(1) = 2; case B yields a d 3 X 4 3 structure with u(1) = 3. When case C is considered, a fully ordered structure is not arrived at, as already noted by Tonks4 (Charts I11 and IV). Equivalent configurations are obtained by the translation of rows of particles either horizontally or vertically, but not simultaneously. A similar situation is encountered in case D and whenever larger particles, i.e. higher values of u(O), are considered. In cases A and B the molar configurational entropy of the adsorbed phase s(6) vanishes at full coverage, i.e. s(1) = 0. It is not straightforward to decide whether the residual disorder at the monolayer coverage in case C gives rise to a nonzero value of s(1). This would require some residual degree of freedom for each particle, like in the well-known cases of molecular crystals violating the third principle of
(1)
0 is a decreasing function of coverage, since BN increases more slowly than N because of multiple blocking of the same site: it obviously varies between u(0) and ~ ( 1 ) . As to the coverage, the most straightforward definition is
r=N/M (2) which at monolayer capacity has the value llu(1): it is thus more convenient to adopt as a coverage scale the quantity
e = u(i) N I M
(3)
which ranges between zero and unity. Theory
In the absence of interactions among particles, the energy of the adsorbed phase is simply
E=-Ne (e>O) (4) where e is the interaction energy between the adsorbed particle and the site. For the sake of simplicity, any internal degree of freedom of the particle is ignored. In particular, the entropy of the adsorbed phase S is only given by the configurational term, which is related to the number, W, of distinkuishable arrangements of the N particles over the lattice by the Boltzmann equation 5' = k In W. To estimate W, following Baker's idea, a random filling of the lattice is carried out, which yields
Garrone and Ugliengo
224 Langmuir, Vol. 8, No. 1, 1992 N-l
W = (N!)-'M[M - b,] [ M - b,]
... [M - x b k l
(5)
k=l
+
bk being the number of sites blocked by the ( k 1)th particle. The term N! takes as usual into account the undistinguishability of the particles. Posing bo = 0, eq 5 may be written as I
N-1
sidlk= -In [eI(l- e)]
(10) Given the function 0 = O(S), the differential and molar entropies of the adsorbed phase are computable, as well as the adsorption isotherm. Indeed, the equilibrium condition between gas and adsorbed phase may be written as Pads = PgaS,i.e. 6E/6N -T 6S/6N = e - TS = po + kT In p (11) In the absence of interaction, as in the present case, t? = From eq 9, we obtain
-e.
Defining as BI = Ci=Obk,the number of sites blocked when Z particles are adsorbed, eq 5 becomes N-l
N-1
A = exp[(p"
Again, in the case of Langmuir adsorption, eq 12 reduces to the well-known isotherm
According to eq 1, BI = D(0) I, so that one gets N- 1
In using eq 1, we are assuming that random filling leads to an average value of u which is the same, irrespective of the order followed in the filling procedure. Baker has shown that MC calculations support this crucial assumption up to a limit coverage, below which different filling procedures lead indeed to a narrow distribution of D(e) values at the same coverage. At high coverages, a random filling cannot lead to ordered structures such as in Charts I11 and IV. As shown below, narrow distributions of u(8) values at high coverages may be obtained by random emptying of the ordered structures. The entropy of the adsorbed phase is
S / k = I n W = -Nln N
+N +Nln M +
N- 1
~p = e/(i - e) (13) In formulas concerning the molar and differential entropy, and the adsorption isotherm, the D(e) function (which determines the adsorption features, together with the energy factor, t, included in the A parameter) appears in the form f ( 0 ) = u(1) - e D ( 0 )
(14) Being f(1) = 0, it is possible to put f(e) = (1- 8 ) g(@,so defining the nonzero g(6) function g(e) = - e ~ ( e ) i i (-i e) (15) g(0) varies with coverage between u(1) and unity; in the case of Langmuir adsorption, on the other hand, it coincides at any coverage with unity: g(0) is thus a measure of the deviation from the Langmuir behavior. Through g(6), it is indeed possible to reformulate eq 7, 9, and 12 in this sense. The molar entropy, the differential entropy, and the adsorption isotherm respectively become
Because N is very large, the sum may be substituted by the corresponding integral
M l l n 11- D ( @ ] d7
s / k = sid/k+ (1/@JOo In g(6) dB
(16)
S/k = sid/k + In g(0)
(17)
ei(i - 8) = A P g(e) Being sid(l)= 0, from eq 14 the result is
so that
s(1) = k
S / k = N[1- In 71 + M s,'ln 11- O(0) 71 d7
(6)
The molar entropy s = SIN is
slk = 1- In 7 + (UT)JOT In [l - o(8) (1/8)
71
d7 = 1-In 0
+ e)/kTl
Jol
(8) By differentiation of eq 7, the differential entropy of the adsorbed phase S = 6S/6N is arrived at
slk = -In [7/(1- Dr)l = -In [B/(u(l)- 0 D(0))l (9) Equation 9 coincides with that obtained by Baker in a slightly different way. For a Langmuir system, eq 9 becomes
In g(8) dB = 0
(20)
From eq 15, one gets
In [ ~ (-lu(0) ) 01 dB (7)
sid/k = -(1/8)[0 In 0 + (1- 0) In (1- e)]
(19)
As already stated, the molar entropy at full coverage vanishes for cases A and B, so that
+
It is easily checked that, for a 1:l correspondence between site and adsorbed particle, u(1) = D ( 6 ) = 1,and eq 7 reduces to the molar entropy of a Langmuir phase
lng(0) dB
Jol
(18)
c s dB = I s i ddB +
k l In g(B) dB
(21)
or - sid)dB = s(1)
(22)
where s(1) = 0 for cases A and B and s(1) # 0 for case C. MC Calculation of V ( 0 ) Only cases A and C were considered (square lattice). As for case B (triangular lattice) we rely on Baker's data, although these only concern coverages up to 0 = 0.68. The first step in each case was the building up of a starting configuration of the system. This can be done,
Langmuir, Vol. 8, No. 1, 1992 225
Ideal Adsorption on a Lattice
Table I. Average Occupation Numbers and Related Standard Deviations as a Function of Coverage case A
coverage
case Ba
0.05 0.10
4.6842 0.0124 4.5137 0.0136 4.3457 0.0145 3.9779 0.0128 3.6090 0.0119 3.2393 0.0100 2.8747 0.0106 2.5534 0.0065 2.3009 0.0037 2.2103 0.0018 2.1215 0.0021 2.0528 0.0002
0.15 0.20 0.30 0.40 0.50 0.60 0.70 0
0.2
Ob
06
21
0.8
0.80
1.0
Figure 1. Average occupation number as a function of coverage 0 for case A (square lattice, five-exclusion pattern), case B (triangular lattice, seven-exclusionpattern); case C (square lattice, nine-exclusion pattern): symbols, MC results; solid curves, analytic approach (fourth degree polynomials with coefficients as in upper Table I11 for cases A and B and in lower Table I11 for case C).
at low or intermediate coverage, by random sequential filling of the lattice. There is, however, an upper limit to the coverage which can be reached, because the final configuration is an ordered one and the filling procedure is intrinsically unable to produce order. For high coverages, we have created the initial configuration by randomly emptying the monolayer configuration. The second step was the performance of a large number of random moves in which a particle was displaced from an occupied site to an empty one, to disorder the system. The third step consisted in performing random moves and computing b after each particle had, as an average, moved once. The sequence of b obtained allowed checking whether the system had forgotten the initial configuration, the evaluation of ( b ) , and the relative variance uv. The whole procedure is typical of the Metropolis method.I2 The only difference is that, having all configurations the same energy, every transition has to be accepted. A 100 X 100 array was usually used for computer experiments with periodic boundary conditions to reduce size effects. Preliminarily, we have studied the role of the various parameters affecting the calculations, namely the size of the array, the number of preliminary moves in the second step, and the number of moves necessary to obtain reliable (0) and u,. The results are as follows: (i) considering arrays of increasing extent, a definite decrease in uvis observed, which thus appear to be related basically to the finiteness of the array; (ii) after some lo4moves the system has no memory of the initial configuration; (iii) averages carried out over 2 X lo3are thoroughly sufficient to ensure accurate estimates of (0) and u,, although usually 4-fold sets have been used. The results for all three cases are reported in Figure 1 and in Table I, which also gives the corresponding u, values. These are observed to decrease markedly along the coverage. (12) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J . Chem. Phys. 1953, 21, 1087.
0.85 0.90 0.95 a
6.672 0.032 6.324 0.029 5.954 0.027 5.545 0.024 5.102 0.020 4.622 0.011
case C 8.7942 0.0114 8.5797 0.0154 8.3611 0.0196 8.1262 0.0208 7.6357 0.0220 7.1140 0.0191 6.5760 0.0141 6.0220 0.0105 5.4574 0.0070 4.9147 0.0044 4.6624 0.0034 4.4235 0.0020
From ref 5, where data at intermediate coveragesare also reported.
Analytical Approach to t = o(0)
The f(B) function defined above appears implicitly in Tonk’s treatment in a polynomial form of third or fourth degree, different for 0 = 0 and B = 1. For instance, the adsorption isotherm for the five-exclusion pattern on the square lattice at low coverage is proposed to have the form4 0 = A p [ 2 - 58 3B2 + 3/2B31. On the other hand, it is useful to express the D(8) data from MC calculations in an analytical form, and to this purpose, a polynomial is highly advisable. We show in the following that some analytical features of b can be established a priori and that on this basis a minimal fourth degree polynomial for b(B) may be proposed for cases A and B. Two trivial conditions are
+
b ( 0 ) = u(0);
O(1) = u(1)
(23)
Two other conditions concern the slope of the curve at B = 0 and B = 1, respectively. Let us calculate b’(8) = [do/dB]. Recalling that 0 = B N / Nand B = u ( 1 ) NIM, one
gets
which, for B = 1 becomes
At B = 1,the removal of any particle only liberates one site in all three cases: all surrounding sites are blocked by nearby particles. Thus ( ~ B N / C W ) B== ~1 for all three lattices considered, and thus 0’(l) = 1- u ( 1 )
(25)
Equation 25 is meaningless at B = 0: (~BN/CW)S=O is equal to u(O), and an undefined 0/0 ratio is obtained. The value of b’(0) can be obtained as follows. The decrease in 0 is caused by multiple blocking of the same site. A t low coverages, the contact of more than two
Garrone and Ugliengo
226 Langmuir, Vol. 8, No. 1, 1992
Table 11. Analytical Conditions for v = v(8)
Chart V Case A
0
0
0
0
u(0) u(1)
0
"u U'(1)"
Intb
0
0
0
e
0
0
0
Q,
0
e
0
0
e
0
0
0
0
Q
0
0
0
e
0
e
0
e
0
0
0
0
0
8
0
0
0
?
for v(e)
Analytical Considerations Only case A ao
0
0
case C 9 4 -4 -3
In the text it is shown that ~'(1) = 1 - ~(1). * Int is the integral So1In g(8) de. Table 111. Coefficients for the Fourth Degree Polynomial
0
0
case B 7 3 -3 -2 0
a
a1 a2 a3
0
case A 5 2 -3 -1 0
04
do)
5 -3 -3.1400 4.2800 -1.1400 0.0344
case B 7 -3 -0.8462 -3.3075 3.1538 0.0248
Analytical Considerations and Least-Squares Fit case A case B case C 5 7 9 -3 -4 -3 -2.3801 -0.4866 -0.5833 2.7603 -4.0268 -3.8335 -0.3801 3.5134 3.4167 0.0198 0.0334 0.0158 0.0589 0.8910 0.3302 -0.050 -0.034 0.339 a
Int a~ defined in Table 11, in R units.
calculated by means of the five conditions listed above in the form A
It results in
0 0 0 0 empty cas4 @occupied case einhibitml case 0
0
particles can be neglected, and the decrease in D can be ascribed to the formation of pairs of nearby particles. Consider Chart V, which refers to case A. Because of the absence of interactions among adparticles, it is equally probable that the two types of sites (labeled K and L) adjacent to an occupied site P will be occupied. K sites have only one blocked site in common with site P, whereas L sites share two blocked sites with P sites. Simple considerations show that the decrease in tr is thus given by 38, so that D'(0) = -3. Similarly, one gets that O'(0) is -4 in case B and -3 in case C. In summary, up to five analytic conditions have been found for the D function, namely the values of NO), ~ ( l ) , O'(O), O'(1) and the condition of thermodynamic consistency (19) ~ ( 1=)hJol In g ( 8 ) d8 = 0
These analytical constraints are gathered in Table 11. As previously stated, the last condition is not valid for case C. This fact leads us t,o treat cases A and B separately from case C. Let us assume now O to be expressed by a polynomial. In cases A and B, fourth degree expression may be
the latter relationship arising from eq 25. a0 and a1 are immediately known; a2 and a3 may be expressed as a function of a4. It is readily shown that the g(8) function becomes A
d
so that the fifth analytical condition (eq 19) actually becomes an implicit equation in a4 which is solved numerically. The resulting values of coefficients ai are given in the upper part of Table 111. The visual comparison between the MC values and the o(8) functions calculated analytically is reported in Figure 1. The standard deviations between the two sets of data are (upper part of Table 111) 0.0099 and 0.024 in cases A and B, respectively. In both cases, the agreement between the two types of data is good, and the analytical expressions may be considered as representative of u(8). The standard deviations of u(8) are, however, higher than those of MC results in Table I, so that a further level of treatment is desirable. In principle, this involves the assumption of a higher degree for the polynomial (e.g. fifth), whose coefficients
Langmuir, Vol.8, NO. 1, 1992 227
Ideal Adsorption on a Lattice
6
d2
0’4
-r
06
~~
08
10
0.2
0:4
0.6
,-f
0.8
Figure 3. Differentialentropy of the adsorbedphase as a function of the coverage 0 for the three cases and Langmuir adsorption. Shadowed areas are equal (see text).
must obey first of all to conditions such as eqs 27. A linear least-squares fit of MC data then allows determination of the independent variables, e.g. a4 and a5. It has to be noted, however, that eq 19 constitutes a nonlinear constraint between the independent variables, to be dealt with by means of the method of Lagrange multipliers: this renders the computations rather cumbersome. To improve the fit of the analytical expression to MC data, we have therefore chosen the simpler alternative to ignore the constraint of eq 19 and have carried out linear regressionsin the ai variables of a fourth degree polynomial, subjected to eqs 27. Results are reported in the lower part of Table 111. Noteworthy features are (i) a marked decrease in the standard deviations of u(8) with respect to the analytical set of ai coefficients, (ii) a large standard deviation on a4, and (iii) a moderate value of the integral over In g(@, showing a weak violation of the condition of thermodynamic consistency. In conclusion, although surely better descriptions are furnished by higher order polynomials, a fourth degree polynomial is sufficient to account for the main features of the adsorption process. For the sake of simplicity, in the following the analytical set of ai coefficients has been used. In case C, where it is not possible to make use of eq 19, in the first instance a similar procedure has been adopted, i.e. a best-fit evaluation of the coefficients ai of a fourth degree polynomial: results are reported in the lower section of Table I11and the corresponding curve in Figure 1. Owing to the delicacy of case C, caused both by a standard deviation on u(8) somewhat higher than in cases A and B, and the presence of residual disorder at the monolayer coverage, the polynomial fit has been extended to the fifth and sixth order, always taking into account the constraints of eqs 27. The standard deviation on u decreases to 0.0166 and 0.0108, respectively; the latter values compare now with the standard deviation in MC data. The integral over In g(0) becomes 0.258 and 0.212, respectively, definitely higher than in cases A and B. For coherence, calculations are given in the followingfor the fourth degree polynomial.
of t9 for the three cases under study. The use of the other sets of coefficients does not alter the curves too much. For comparison, the molar entropy of the Langmuir phase is also reported (eq 8). A t very low coverages, in the Henry region, the molar entropy curves parallel each other and are displaced from the Langmuir curve by the amount In ~ ( 1 ) .Indeed, for low 8, eq 7 reduces to s = k [ l - In 8 + In u(l)I, whereas eq 9 becomes s = k [ l -In 81. For 8 = 1, the curves for cases A and B and the ideal one tend to coincide. The molar entropies of the adsorbed phase at any 8 are larger than that of the Langmuir case, because a larger number of sites is always available to a single particle at the same 0 value. This is obviously not true if comparison is made at same 7 coverage values. As expected, s(1) vanishes in cases A and B and does not vanish in case C. A residual value is observed, around O A R , which decreases to 0.25-0.20R when increasing the degree of the polynomial for u(8), remaining definitely different from zero. I t may be noted that in cases A and B, when condition 19 is not imposed, and an analytical fit is carried out on the v(8) values, 4 1 ) turns out to be -0.05 and -0.03, respectively (lower part of Table 111). We conclude that s(1) in case C is very likely to be different from zero. The differential entropies for cases A, B, and C and Langmuir adsorption are reported as a function of 8 in Figure 3, as calculated from eqs 9 and 10, respectively. The followingfeatures are noteworthy. Whereas the curve sid(8)is symmetrical around 8 = 0.5, which is also a point of inflection, such a symmetry is lost in the other cases. As a consequence of eq 22, the whole area between the sid(8)curve and that of curve A (or B) must vanish. This means that the two curves must cross and that the areas between the two curves before and after the intersection point must be equal: In Figure 3 this is illustrated for case A by appropriate shadowing. In case C, the two areas are clearly not equal.
Thermodynamic Results for the Adsorbed Phase The molar entropy of the adsorbed phase, calculated as stated above from u(8) values represented by a fourth degree polynomial, is reported in Figure 2 as a function
Application of standard statistical thermodynamics (Boltzmann’s formula) to the problem under investigation leads to studying the average occupation number 0 as a function of coverage, by means of which one may express
Conclusions
228 Langmuir, Vol. 8, No. 1, 1992
all relevant thermodynamic entities (adsorption isotherm and molar and differential entropy of adsorption). Some properties of D may be established a priori, and this allows the calculation, entirely analytical in cases A and B, of a fourth degree polynomial which represents satisfactorily the MC results: it appears that in these two cases MC calculations are almost useless. In case C , recourse to a fourth degree polynomial requires considering both analytical constraints and least-squares fit of MC
Garrone and Ugliengo
results concerning D: the main result here is that at full coverage a residual disorder is still possible and, accordingly, s(1) is different from zero. On the basis of the above treatment, it is straightforward to propose a Bragg-Williams model to take into account the interaction among adparticles, as already outlined by Baker;4 this is not reported here because any such treatment is overcome by the more powerful ones now available.
