Electroanalytical Chemistry and Interj~tcial Electrochemistry, 57 (1975) 117 © Elsevier Sequoia S.A., Lausanne
Printed in The Netherlands
ADSORPTION ISOTHERMSA CRITIQUE II. T H E AO 2 A P P R O A C H A N D T H E F U N C T I O N A L E X P A N S I O N S. K. RANGARAJAN
Indian Institute of Science, Bangalore 560012 (India)* and International Centre for Theoretical Physics, Trieste (Italy) (Received 1st March 1974; in revised form 25th July 1974)
NOTATIONS
A A(p)
Aa, A 2 B c
C(r, p)
c*(p)
= the constant coefficient of 0 2 in the approximation for (6P) = coefficient of pZ in the expression for (3P) = the operator coefficients in the functional expansion for (6P) in "powers" of Ua(r) = defined in the adsorption isotherm through Bc = f ( F ) = "bulk" concentration/activity = the OrnsteinZernike's "direct" correlation function =
C (r, p) dr 0
d E f(rij) f F AF
Go(r, ro) g(r, p)
= "width" of the compact double layer = the potential difference in the compact layer = exp (  ur (ro)/kT) 1
=  AF/Vk T
(7) (1) (18) (14), (18)
(1.) (14') (20) (20) (12), (13) (A.7)
= Helmholtz free energy = !'excess" Helmholtz free energy = Green's function for the potential distribution
(19)
= radial distribution functions (r.d.f.)
(lO)
0o (r, p)
= r.d.f, for the potential = terms in the expansion for 9 as a power series in p p) = radial distribution function for the "test" potential Ur+ 2Ua H , / / k i n , Hpo t = Hamiltonians of the system k = Boltzmann constant g l , g2
K
= ~ (g(r, p )  l ) d r 3o
(10) (7) (21)
~14~
* Visiting Professor, Inorganic and Physical Chemistry Department; permanent address: Materials Science Division, National Aeronautical Laboratory, Bangalore 560017, India.
2
~1)(.)
S~K. RANGARAJAN = integral operator over the radial coordinates defined by its action on any function f ( r ) as . . . . ( f ( r )) = 'fo~ U a ( r ) ( f ( r ) ) d r ~(1)
~2~(.)
(7)
= integral operator on (2, r) defined as ff~,z,)a (F (2, r)) =
fo d 2
Ua(r) F (2, r) dr
(7)
pressure in the lattice gas model (15) total pressure when the interaction potential is ur (r) + b/a(r) (1) = the "basis" pressure due to the twoparticle potential ur(r) (1) 6P = the "excess" pressure attributed to the superposition of the potential Ua(r) on "the basis" (6) r = radial distance r, dr = configurational coordinate vector; volume/ area element T = temperature U = the total twoparticle potential Ua ~  superposed potential, usually but not necessarily attractive (1) Up particleparticle short range potential (21) Ur basis potential usually but not necessarily repulsive (10) bl 2 Ur + ~ Ua V = the long range (usually coulombic) potential (20) (also used to denote volume or area of the system c f (A.7), (A.23)) X exp (  e / k T ) (15) y = fugacity (15) Z1 distance of the inner plane from the electrode (2C0 Zij = distances of "the images" from the electrode occurring in the adsorption isotherm /~c=F (8), (14), (A.8), (A.9) ~, ~o, ~ , ~ . , (4) 7 = "reciprocal range" of the potential u~(Tr) F = surface excess, molar A = coefficient of p 2 / 2 in the van der Waals limit = IUa(r)dr (5) 6(0 = Dirac's delta function interaction between the nearest neighbours in the Ising model (16) 0, 0g, 01 = coverages (3) 2 = "coupling" coefficient for the potentials u r and /~a (7), (A.lS) "dimensionality" occurring in the representation of potentials ( e l the van der Waals limit) (4) p
P
= the
=
ADSORPTION ISOTHERMSA CRITIQUE. II
p Pi, P~
= density of particles = coordinates of the charges in the cylindrical frame = twoparticle density functions= p eg~. = exp ( ur/kT ) = exp(  ur/kT)[exp(  ua/kT) 1] = grand canonical partition function
p~2) ¢Pr Aq) =
(5) (20) (A.20) (A.2) (A.17)
(1) INTRODUCTION
The essential motivation for the study of electrochemical adsorption is to obtain information on the adsorbate/electrode and the adsorbate/adsorbate interactions. Two lines of theoretical investigations then suggest themselves. Firstly one proposes a model for the interactions and finds what equation of state or pressure this corresponds to. For example, one may work out the relevant virial functions for the various models and compare them with the appropriate experimental results. In contrast to this "direct problem" we have an "inverse problem" formulated by Frisch and Lebowitz as follows. "Given, as a starting point, a model class of functional relations between the values of macroscopic properties of a hypothetical Newtonian molecular system, what is the functional class of Hamiltonians which is consistent with the given information?" Understanding "the inverse" problem is more involved and complex than "the direct" problem. In an earlier communication 1, we discussed certain aspects of the direct problem. The emphasis therein was on the question of separability of the (i) particle/ particle, (ii) the particle/electrode interactions and (iii) the influence of external fields, besides relating these questions via adsorption isotherms directly to the pressure. Simple linear additivity of forces in the Hamiltonian does not, as a rule, result in any superposition of what one may term as corresponding pressure terms. Such a consequence is not surprising in view of the nonlinear functional dependence of the partition function (or free energy) on the interaction potentials. Certain heuristic methods of modelling the free energy of adsorption and accounting for "the electrical nature" of the interface were also sketched briefly 1. We revert, in this paper, to a more detailed discussion of the so called "direct problem", alluded to earlier. The aim of the present exposition is not "to solve" the problem as understood in the usual parlance but to provide a perspective and to emphasise certain inadequacies in tackling the complex problem of deriving adsorption isotherms. For this purpose, we ignore in the following discussions the multicomponent nature of our system and draw freely inferences based on asinglecomponent discussions. If anything, the multicomponent nature will only worsen the situation! (2) THE
AO 2
APPROACH
Even simple models like those of hard spheres/disks elude exact analytical solutionsnot to speak of those Hamiltonians including general attractive potentials. Without much exaggeration one may state that the elegant results available for long range potentials are only of a semiquantitative nature either assurin~ the convergence of grand partition functions and the existence of thermodynamic limits or deducing
4
s.K. RANGARAJAN
the expressions for the limiting models (e.g. the van der Waals limit). Thus, it is clear how futile it is, with our present knowledge of equilibrium statistical mechanics, to look for exact analytical expressions valid for adsorption of ions/dipoles on charged electrode/solution interfaces. For, there are several things to reckon with under these conditions. Firstly, one models the essential short range repulsive part, thus ensuring some semblance of validity for high densities. Secondly, the nearest neighbour interactions (attractive/repulsive) are to be included. Thirdly, any long range effects like the coulombic part of the particleparticle potential should be taken care of. Then, it is also necessary to look into the problem of the twodimensional layer of the particles interacting with the neighbourhood (substrate, for example). Lastly, the effect of the external.field should also be introduced. Difficulties accumulate from all directions; (i) for, it is difficult to include properly both the short range and long range repulsive/attractive potentials and solve for them satisfactorily; moreover (ii) the inclusion of the external field parameter (X, say) in the Hamiltonian would mean the evaluation of local "potentials" dependent upon the positions (rl...r,) of all the adsorbate entities. With subsequent configurational averaging, no simple functional form in X is to be expected in the end! The approximations which become a sine qua non for any further discussion, have usually two forms: (I) Let us take as a "basis" or "a reference" a well studied "particleparticle" model like that of a Langmuir lattice, hard sphere/disk etc. Assume that the superposition on this basis of the extra interactionsusually those due to the nearest neighbours, be they repulsive or attractiveis adequately accounted for by the addition of the term (A0 2) to the pressure Pr of the basis alone. Table 1 of ref. 1 is illustrative of this approach. Similar simple macroscopic approaches to account for ion/ion or ion/dipole interactions with slightly more general functions (than AO 2) are also not unknown. (II) The electrical effects due to the external fields are usually considered through a naive "congruence" hypothesis 2. Indeed such an addition of "A0 2'' to Pr, preserves the simplicity of the expressions and has been found to be highly useful in analysing the experimental data too. But what deters one from employing such approaches is its ad hoc nature: the coefficient "A" mixes within itself information as regards the perturbation potential Ua(r) and certain characteristics of the "basis" itself, to an unknown extent. No doubt, A can be taken as a measure of the superimposed potential Ua. But, can one say that all the effects in P due to the introduction of ua(r) are contained in the "AO 2'' correction? The answer is "no". If one expands P as functionals of Ua(r) and its higher powers, even the linear correction has 0 2 and other powers of 0! Thus one may term the "AO 2'' method as a mathematical approximation while the alternative method which expresses the perturbation 6P in P due to ua (r) in the form 6P ~ X 1 u~(r) thereby linking 6P directly to the potential u,, may be termed a "physical" approximation. In general, P is a nonlinear functional in the potential ua (r). It is possible to develop P as P = P,(O, T ) + X , ua(r)+fiz(u~(r)) 2 + ... (1)
ADSORPTION I S O T H E R M S ~ CRITIQUE. II
(Appendix I suggests an even more general procedure) in a Volterralike fashion. The operator coefficients/~i's depend only upon the "basis". Ignoring considerations of convergenceour concern is with the first or second terms in (l)we may then offer as art approximation, the following (also see Appendix I, equation (A.6) and equation (7) below) e '~ Pr (0, T) q A1/3a (r),
(2)
where ,41 is an appropriate operator. In general A1 is an integral operator and its kernel is an explicit function of the radial distribution function pertinent to the "basis" and also of the density O. Here, the perturbation in P that we are seeking is A1 (u,(r)). We term such a procedure as the "physical" approximation since this immediately links our approximation to the physics of the problem viz. the interaction potential. In general/~1 Ua is not of the form A02; A, a constant. On the contrary, one may view the perturbation ( P  P~) as a general function of 0 and can, in a certain range of 0< 0", express p ~ p~ + AO 2 " (3) Such an approach is essentially mathematical and one is then, as observed earlierl at a loss to interpret A in terms of the interaction Ua or the distribution functions arising from the "basis". At this juncture, let us ask ourselves if it is possible at all to have these two 1Lypesof approximations becoming identical which if it happens, is a very welcome situation indeed ! (3) THE VAN DER WAALS LIMIT
Kac et al. 3 have shown, while treating their one dimensional fluid where the pair :interaction is an exponential attraction superimposed on the hard core repulsion, that the perturbation in P is  AO2, only in the van der Waals limit. Subsequently a similar limiting result has been obtained 4 for higher dimensions and other forms of potential functions. To be precise, if one considers the potential : u (r) = u r (r) + ?~ua (Tr),
(4)
v is the dimensionality of the space; and passes over to the van der Waals limit viz. 7~0, the pressure P is then* Lt P(7, P)+ Pr(P)+Ap2/2
y~0
(5)
where Pr (P) is that pressure appropriate to "the basis" viz. with u (r)= Ur(r) only, and A =j'Ua(P) d =
The above interesting result which unites the mathematical and the perturbation approaches under limiting conditions illustrates at once the strength and the weakness of the "A02'' approach. The latter approximation is no doubt sound under certain limiting conditions discussed above (where it is also exact) while in the general case 7%0, it is at best a mathematical approximation. Asymptotic expansions in terms of powers of ? would then be worthwhile. Thus, in the van der Waals limit when the range goes over to infinity simultaneously with the strength of the potential approaching zero, the credibility of the AO 2 approach is established. How should one then tackle the more general potentials?
6
S.K. RANGARAJAN
(4) THE PERTURBATION APPROACH
Let us now look at the functional approach where we express the modifications in the pressure due to the presence of an attractive potential Ua(r), say, over and above "a basis". The basis may be for example a hard core potential Ur(r) with its pressure given as Pr' (In this section we shall be using the number density p rather than coverage 0 or molar surface concentration F while discussing equations of state). Let the radial distribution function appropriate to the potential Ur alone ("the basis") be go(r, p) and that due to the "test" potential (Ur+2Ua) be g4(r, p). Then for the potential u=ur +ua, the pressure P defined as p2 a (FI~)IOp = P is the sum Of Pr for u=ur alone (i.e. 2 = 0 ) and a perturbation term 6P due to the superposition of the potential ua. We can show that (Appendix, also ref. 5)
P = Pr+dP where
,
6P = y
E ....
(Pg°(r' p))
) + """
(0(04
0o,,)]
= A (p) p2, say.
,6, (7)
The integral operators 5~r(1) and _(o(2) are respectively (the integration over the , ga r, ua "appropriate surface") S Ua(r)(') dr and S+ d2 S Ua(V)(') dr. Equation (7) is an exact identity. In the above representations v is a vector. The suffixes r and u, for the operators ~ indicate that ~ depends on these. We are not concerned here in any derivations or explicit formulation of the equations of state under various potentials. Our object is only to convey a simple message viz. that one may develop the pressure (or equation of state) as a functional of u, (r) (if the gfunction for the basis alone is "known") rather than as a power series in the density or coverage. In the former case, the new terms for pressure are directly and linearly related to the powers of interaction potential u, (r). As a rule, this perturbation in pressure, however, is not a simple quadratic in p. Since the radial distribution functions go (r, p) and g4 (r, p) are dependent on p2, the simplicity of eqn. (7) is somewhat deceptive. An approximate form of (7) when used in the adsorption isotherm leads to the expression (see Appendix) , 2 go, ,tP' r ) + 2 P90 (p, r)) 2 In flpert =   ~(1) r,UatP
(g'o(P)= (Ogo/t3P)r)
(8)
The familiar " A 0 2'' approach would of course have resulted in In flpert ~   2 A 0 "
(9)
Obviously if eqns. (8) and (9) are to be compatible in their dependence on p or 0, we require that (t?9o/C3p)r ~O, for all r   a result that can be true only for a basis with no interactions (but see the preceding section on van der Waals limit)! The conclusion is that, in general, the mathematical and physical approximations are not identical for a basis which assumes interactions. The "correction" 6P is not only characteristic of the superposed potential Ua(r) but is also dictated by the "basis" radial functions. Since we are supposed to use as basis some model whose
ADSORPTION ISOTHERMS~
C R I T I Q U E . II
functional parameters like 90 (r, p) are tractable albeit approximately, there should be no problem in employing expressions for go(r, p). Of several possible schemes, a simple perturbation for go (r, p) may be (e.g. ref. 6) g (r, p) = exp (  u r (r)/k T) (1 + gl (r) p + g2 (r) p2 +...)
(10)
a result due to Yvon, Kirkwood, de Boer et al. 6 gl (r) = gl (rl  r2) = g(rx2) = Sf(rlz)f(r23)dr3
(11)
g2(r12) = ½{gl(r12)}2+q)(r12)+20(r12)+½z(r12)
(12)
where f(r)
= exp(ur(r)/kT)
1
o(r) = lf(?12)f(r24)f(r34)dradr4 (r12) lf(r13)f(r23)f(r24)f(r34)dr3 dr4 z(r12) = .If(r12)f(r23)f(fl )f(rz4)f(r34) dr3 dr4 =
(13)
The use of (10) in conjunction with (8) essentially means that we may be able to write the pressure 6P in the form
AO 2 + BO 3 +
CO 4 + . . . .
Such a partial* virial expansion, which we alluded to earlier 1, is not entirely "mathematical" since the coefficients A, B, C etc. are now directly (and linearly in this case) related to the perturbation potential ua(r) and the basis functions. We mention (see Appendix) that the adsorption isotherm may also be written in terms of the radial distribution function instead of pressure as tic = F
l + p K
In fl = 
dp
(14)
where K = 
(1g(P, p))dr 0
(Alternatively, In fl = I r C* dF where C* = I C(r12, r) dr2, see Appendix). Equation (14) follows immediately from the OrnsteinZernike's relation for relative compressibility and using eqn.(9) of ref. (1). One may wonder whether eqn. (14) can be employed directly for the potential u(r) = Ur(r)+ u,(r), without defining a basis state and therefore a perturbation state. It is noAoubt valid to do this but one then has to know the radial distribution function g (r, p) for the total potential u(r)= ur (r)+ u a (r)a problem which is in principle no less difficult! Maybe an interesting hybrid approach viz. to model for g (r, p) based partly on theoretical and partly on experimental considerations and then use (14) for checking consistency with the optical isotherm studies, can prove useful. * "Partial" because, that part of the pressure due to the basis viz. Pr is supposed to have been obtained "globally".
8
S.K. RANGARAJAN
(5) ISING AND LATTICEGAS MODELS There is another class of modelsLattice modelsthat are widely used in describing the adsorption phenomena. Let us briefly touch upon the " A 0 2 approach" with reference to these lattice systems. In particular the Ising twodimensional lattice 7 with nearest neighbourhood interactions only had been (and continues to be) the subject of detailed theoretical discussions. It is not out of place to mention that the model possesses certain interesting (in some ways, even unique) features recommending this for detailed attention. For example it is one of the "few" exactly solvable models and can be solved exactly even in the phase transition region. Its capability to simulate some simple spin systems and "th~ identity 7'' with the lattice 9as models are also worthy of mention. For example, the series expansion for pressure p is given in terms of fugacity y as:
p/kT = In (1 + y)+ 2 (1/x 2  1)(2y+ y2 (...) +...) 0 = y (c3/•y)(p/kT)
;
(15)
the above expansion being valid for y < x 4. In the region y > x 4
p/k T = In (y/x 4) + In (1 + y1 x 8) + 2 (1/x z  1) ((2xS/y) + yZ (...) +...) 0 = y ~yy (p/kT)
(16)
In (15), (16) x = e x p (  e / k T ) where e is the nearest neighbour interaction. We also note from the above expansions that (a) eqn. (15) is Mayer's virial expansion and eqn. (16) is an expression valid in "liquid phase" (and also in the "critical region") where virial expansion fails. (b) The case of attraction is more instructive (and interesting in view of the occurrence of phase transitions) than that of repulsion in (15), (16) and, in this case, we assume e > 0. (c) When e*0 or x*I, the Langmuir's equation is recoveredLas is to be expected. When y = x 4, the phase transition is attained (a square lattice is assumed) and in the transition region,
p / k T = ln(1 + x2)+(1/2rc) i~ In (~(1+ ( l  K 1 z sin 2 go)~))dgo du
with
(17)
/(1 = 4 x ( 1  x2)(1 + x2)2. The corresponding coverage in the gas phase is 0g  1 _ 1 ((1 + x 2) (1  6x z + x4)~/(1  x2)2)~ and in the transition region 0g[01 = 1 .
These are exact results. An approach based on "AO 2'' addition when applied to Langmuir basis purporting to account for nearest neighbour interactionsis obviously a "gross" effect and a mathematical approximation, as may be seen from (15), which contains all the powers of O.
ADSORPTION ISOTHERMSA CRITIQUE. II
9
Such an analysis of lattice gas may be extended to other systems like hard core lattice gas with exclusion shells but we shall not discuss them here. The "A0 2 isotherm" corresponding to Langmuir model is well known as Frumkin's isotherm though a nonlattice origin has been claimed z~ for the latter. (6) THE COULOMBIC EFFECTS The next problem in obtaining the adsorption isotherms from statistical analysis pertains to the effect of electrical/coulombic interactions and the effect of the external field. These are two different aspects even though they are usually lumped together in the parameter B of the isotherm 2 Bc = f ( F ) .
(18)
It is impliedwhile writing the adsorption relation in the form (18)that f(F) takes care of all particleparticle interactions that are nonelectrical in nature. The congruence hypothesis 2 attempts to approximate the effects due to all the coulombic interactions (including particleelectrode) by writing B as a function of the potential difference E (Econgruence) or of the charge q (qcongruence) alone and not of any other variable like F. We had earlier argued 2c on the naivity of these attempts and suggested possible "pseudocongruence'. The latter arises when B is a function of both F (or 0) and E, say, but is "product separable" 1,2c. In other words B ~ B ° (E) ~p(F) so that what is truly a pseudocongruence arising out of this particular form of B go camouflaged as "Econgruence'! We sketch here a simple formal analysis of how this may actually occur under certain idealisations. Let us assume for simplicity an isotropic dielectric medium as a model for the inner (double) layer. Let us denote the positions of the ions adsorbed as {Pi}, i= +_1, _+2. . . . in the inner Helmholtz plane (i.H.p.). {The charges and their images form a threedimensional array with their locations given a s {Pi, Zij}, i , j = + 1, +_2.... }. The "metal surface" is assumed parallel to the i.H.p, and the normal to these planes is the zaxis, with the origin on the metal surface, and the i.H.p, itself being at a distance zI from the surface. Consider now the Green's function Go(r, ro) of Laplace's equation in this domain i.e. V2 Go (r, ro) = 6 ( r  ro)
(19)
with homogeneous conditions on the boundaries; (in our case, the boundaries are the metal surface and the outer Helmholtz plane (o.H.p.)). Let r = (p, z), r 0  (Po, z0). It is easy to see then that the actual potentialunder our idealised conditions concerning the symmetry and location of the planes and also the polarisability of the mediumat the locations {Pi} in the inner layer are given by
v(p,, {pj})
Ez I = Z ' Co (p,, p j) +  d
(20)
Let us also assume in (20) that E is the potential drop and d the distance between the metal and the outer Helmholtz plane. The (') on the summation sign indicates a self energy correction.
10
S.K. RANGARAJAN
Inserting the various contributions of the interactions in the Hamiltonian we have for our two dimensional system, H = Hki.+Hpo, = Hki. + Z ij
up(Ioipjl) + • V(p,,{pi})
(21)
ij
where Up is the two body central potential accounting for the short range particleparticle interactions. It will be worthwhile ensuring that V(p~, pj) too is of the same type i.e. dependent upon (p~pj) only. This is by no means obvious because of the contributions to the Green's function Go arising from the images extending over "the three dimensional" domain. But a simple analysis shows that V(p~, P.i) is a function of fPi Pjl and z~, d, ~ etc. (i.e.) of the form (p(( [Pi Pj I); z~, d...). For example consider a single charge at (Pi, z~) between the two "conducting", the metal and the o.H.p. The imagearray induced is {ri} = {pi, z~} where zo's are the zcoordinates of the images. Then the potential at any other location (Pk, Z~) in the i.H.p, is proportional to
Z ([Pk P,I2 + z~) ~. i)
Since the zij's are entirely governed by the distances z~, d (rather the geometry) and the dielectric constants, the V(pi, p j) is of the form V(p~pj), as we desired. With the above remarks and eqn. (20) it becomes clear that the partition function, when evaluated has its external field dependence separated out conveniently from the rest, leaving no doubt a function like q~(I p~pj[...) to take care of the mutual coulombic interactions with the "substrate" (of images). We have laboured hard thus far to illustrate the following features for this simple model which assumes among other things that the dielectric constant is not a function of E: (a) the external field dependence can, under these circumstances, separate out i.e. fl in (14) can become "productseparable"; but then, (b) as a rule, the particleparticle noncoulombic interactions (as given by ~o) are inseparable in view of the averaging over the positions {p~}, resorted to while obtaining the partition function and hence the free energy/pressure. Expressing this contention formally, we may state that it is not always right to separate a Bpart and f(F) part in the isotherm as in (18). As a matter of fact, all the characteristic functions like the radial distribution function of the "basis" ur are altered by the presence of the coulombic interactions and the external field. It becomes then obvious that conventional discrete calculations for this model superimposed on a "Langmuir basis" can not easily be extended to any other basissay a hard core. It is possible to discuss in similar vein the problem of adsorption of dipoles in the presence of electric field. We shall consider the details of this problem later. Our scope, in this and the earlier article 1, was restricted to examine physical basisif anyfor certain approximations frequently resorted to in the area of adsorption isotherms. The aim has been to emphasise the need for a more rigorous discussion of this problemnot losing track of the essential motivation governing the study of adsorption isotherms. With a view to eliminate any deviations from this limited objective, we had avoided working out actual details or results for several models. We shall revert to some of these questions later.
11
ADSORPTION ISOTHERMSA CRITIQUE. II TABLE 1 SUMMARY OF RESULTS
(A ) Functional approach (perturbation in Atp) 6P= PPr 2
P~2kTSdr(1e "/kr) ~p (pgo (r, p)) (linear in 5¢p)
f
~
u.(r) (pgo)dr
(linear in Ua)
, Ua(r)dr 2 . (zero strengthinfinite range potentials 7~u~(?r): van der Waals limitl (For higher order approximations see Appendix).
( B) "Coupling" approach 6P = PPr~ A(p)p 2 (exact) A ( p ) ~ } od2
u~(r)
(i)
(pgz)dr
(dr=2nrdr for two dimensions). (ii)
A(p) = ~ Ua(r) ~pp (pgo) dr (weak coupling linear in u,).
(C) Adsorption isotherms tic = F In flpo~,= In f l  In fir

~ T ( 2 A ( p ) p  fl
OAp~pdp)
(exact)
(AO2 approximation) ½
u~(r) (p2oo)dr
("linear" in u~)
0
(van der Waals limit).
~  A p
(D) Isotherms visa vis correlationfunctions P K lnfl=
o l+~pK dp cO
K
(o(r, p) 1)dr 0
In fl
•
b dp
C (r, p) dr
Ornstein Zerl ike direct correlation function).
(continued)
S. K. RANGARAJAN
12 TABLE 1 ( E)
(continued)
LatticeModels with nearest neighbour interaction
~P kT
P kT
Pr kT
y +Jl P = l~y (~7
P kT
In (I + y)
3 I) O Y 2 + 6 (  X S   5 ) y a ? 4 f 3 ( x ) Y 4 + ' " )
y is the fugacity and the above equation for p is itself the isotherm in the presence o/interactions. There is only one parameter e and
1966
:, = e~/k~, f3 (x) = U + 2s  ~ + 5
)
and e >0 for attraction.
The "strength of interaction"= (x2_ 1) and ~ 0 in the Langmuir case. For validating the "'AO2" form viz. P RTFm
l n ( l  0 ) + A 0 2 A must be small, ,~ 1 .
With the above approximation, one should identify A with 2(x 21). Even "the nearest neighbonr interaction alone" induces a power series in 0 over the Langmuir pressure and, then, P~ ~_,  ln (1 O)+ AO2 + B03 + C04 + ... kT A=2(xl~
 1);
B=2(~
1)2;
C=(xl~l)2(~x2~82+36),
etc.
(cf Frumkin's isotherm and termination of the above series with "A0 2" especially when A is 3~ 1). (7) ACKNOWLEDGEMENT The author wishes to thank the International Centre for Theoretical Physics, Trieste, for their hospitality. SUMMARY The well k n o w n method of appending to the expression for pressure deduced for a "basis model" a term like "AOz'', to account for the nearest neighbour interaction is examined. A distinction between such a mathematical approximation and what we term as "a physical approximation" is made. The latter invokes a perturbation expansion as functionals of the perturbation potential. It is indicated how these two may "converge in the case of limiting models" like that of Kac's in the "van der Waals limit". An analysis of the problems arising in the presence of coulombic interactions and the external field is briefly sketched. APPENDIX We indicate in this appendix h o w one expands the pressure AP (excess pressure) of the actual system as compared with that for an ideal gas at the same temperature, density, volume, etc. as a functional series in (e ~u,/kv)_ 1), and hence in Ua(r ).
13
ADSORPTION ISOTHERMSA CRITIQUE. II
Let the "initial" or "basis" potential be u r for which the pressure AP turns out to be APr, say. We assume, for the time being that the problem of evaluating APr has been solved. What concerns us now is the question : if ur is altered to a new interaction potential Ur+G, what is the corresponding change in AP? (note that u~ need not necessarily be "attractive" in its nature). A Taylor like series for AP with functionals replacing the more familiar power terms, reads as (~i F (q)r)iq) (r) b~
P = APr + I d r
+
drdr' 626P(°3
6q~(r)g~p(r')
+ ~ j" . . . .
(A.1)
In eqn. (1), APr is the "excess" pressure corresponding to the potential Ur. The functional derivatives 6AP/bqo, 62AP/&o (r) 6~o(v'), etc. are evaluated at qor exp (  G/kT). By definition Aq~ = exp (  uJkT)(exp (  G / k T )  1).
(A.2)
It is only natural (in view of the dependence of the partition function on H) that the q~functions rather than {u} themselves should occur in the expansion procedure adopted in (A.1). The functional derivative 6Y(r)/bZ(r) is itself defined in a straight forward manner. If
Y(r) = l K(r')Z(r')dr', then
5Y(r)/6Z(r) = K(r).
(A.3)
Observing that
Aqo 6(AP/kT) _ 5~o
p2 c3 (pgo(r, p))(exp(_G/kT)_l), 2 8p
(A.4)
the first order approximation for the pressure AP is
P(Ur+Ua; T, p) ~ P(u ..... ) + ~ kT j" dk
(pgo (r, p))[1  e x p (  u d k r ) ] .
(A.5)
(A.5) is the linear version (in A~o and not u) of (A.7). If one wishes to "linearise" (A.5) further With respect to Ua, (A.5) can be further approximated as P(ur}Ua; T, P) ~ P(u . . . . .
p2 c? ) + ~[. dru,(r) ~pp (pgo(r, p)) p2 ~f(1) . g9
=
P(u ..... ) + ~ . . . . .
where ~(1) r,Ua is the integral operator ~ dru,(r).
~p (pgo(r, p))
(A.6)
14
S. K. RANGARAJAN
It must be emphasised that (A.5) is more general and there is no need to simplify it as in (A.6). A similar remark applies to eqns. (2) and (8) of the text. The second order functional derivative
b2(Ap/kT) _ _p2 ~ &o(r, fq),r') ~p(1A'2))
,1.7)
where
A ~2) = [  bzf/bq~ (r) 6~o(r')], f being simply related to "excess" Helmholtz free energy AF as
a = AF/VkT. An explicit form for A (2) had been derived byAnderson et al. 5 and is not reproduced here.
Expressions for In fi Recalling that the fifactor in the isotherm fla=F is related to the "excess pressure" AP through eqns. (9), ref. 1 f l = flo exp (I~ _ d(AP/kT)) ,
(1.8)
a substitution of the linear form (A.6) in (A.8) gives In fl ~ In fir " 5 °  ( 1.... ) , (P 2 go(P, r)+ 2pgo(P, r)).
(A.9)
Note : r d(AP/RT)
In fl = In rio 
f
= In
Is
d(APJk T)
do
P
= ln
o

F

o •
P
In flo 
fl O(AP/kT) dp Op
p
f~ d((6P)/kT)p 1 kT
A(p)p 
o
p ~fip dp
A (p) is defined in (7) and is approximately equal to ±2 ~~ 'rm O(~pp(Pgo )) if gZ ~ g0 which is eqn. (8) of the text. rio in (A.8) is the value for fl in Henry's limit and fir in (A.9) is that fl corresponding to the potential u = ur. If go  1, a constant, g~ = 0
(A:10)
and hence In fi ~ In flrZp ~ dr(u,(r)/kT) = In flr2Aop, say
(A.ll)
The constant A o is now immediately related to the perturbation potential u a. But note that the above approximation (A.10) is true only if there are no interactions. But if ur ~ 0, then, the go for the basis is
ADSORPTION
ISOTHERMSA
15
CRITIQUE. II
go(r, p) ~ exp(ur/kV )
(A.12)
the "Zeroth" approximation. Then from (A.9) and the general formula (A.5) In fl = In flrZp ~ e(ur/kT)(e u"/kT
1)dr
(A.13)
(A.13), under the limit expressed by (A.12), no doubt gives an identity with the "A0 2approximation" but now, ,
A = A 1oc kT~ e ur/kr (eu~/kT
1)dr
(A.14)
One notes that, already certain weighting due to the "reference potential u~" is present in Aa. The next approximation for go (r, p) gives
go(r, p)~ e x p (  u J k T ) ( l +gl(r)p)
(A.15)
where gl(r) is given by eqn. (ll). Now the expression for In fl is In fl ~ In fir2pA 1 (~ dr(e u"(r)/kr 1)eu~)krg1(r))p2
(A.16)
Thus "A" is no longer "a constant" and is linearly varying with p, as is evident through the presence of the last term in the right hand side of (A.16). Higher order approximations for go expose a more basic dependence of "A" on p. Let us also remember, at this juncture, that we have all these results only after assuming a simple expansion for 6P with u a (or with Atp)!
An alternative method Use eqn. (7) directly by approximating gz~ g0 and then using eqn. (9) of ref. 1. However, this seems to be more restricted than, say, that obtained from (A.5). A scheme for relating the pressure with the perturbation potential ua (v) which is different from that outlined in the Appendix, above is as follows. Consider the grand partition function S and the distribution function p(rl, rE) for a system with a "test" two particle potential u ~  ur (r)+ 2/,/a (r). 2 is some coupling constant so that 4=0 gives us the "reference" i.e. u ~ ur, 2~ 0 while 2 = 1 is the "actual" system with the superimposed potential u~ (r)(uz ~Ur + Ua, 2 ~ 1). By definition,
yN ZN ~ = ~ N>~O N! where
(g.17)
(
Ur(rii)+~Ua(riJ!~ drldr2...drN / kT
(A.lS)
We now differentiate In ~ with respect to the parameter 2 to obtain
yN OZu 0 In Z 02
 (''~)1
02 N '
2 N>~O
(A.19)
yN
=  (S)I ~ ~
j....j exp (  (u~+ 2Ua)/kr ) Z Ua (rlj)(1/kr) dr 1 ... drN ij
16
S.K. RANGARAJAN
yN
=  1/(2kTZ) Z
 [ ... [ exp (NZ)!
1
~, 2kT "~Ua(rl2i drxd r2r'z
((ur+2Ua)/kT) [ .u,(ra2)dr,
dr 2
(A.20)
(rl, r2; ;.)
Integrating now equation (A.20) with respect to 2 in the interval (0, 1), we deduce that 3 In S =  (ln S)ref + (ln ~),~tual 
2kT o
u~(rlz)dr~drzp~2)(rl, rz)
d2
(A.21)
The subscript 2 for p(2) indicates that the distribution function is to be evaluated for the test system i.e. with uz. Defining p(~2)(rl, r 2 ) = p2g(2)(l"1, r2, p ) = p292(r12 , /0), (A.22) equation (A.21) is further simplified as 31nS
p2 2kTV flo d 2 Iua(r)gz(r, p)dr Np 2kT
f~l)~ Iu~(,)g~(r, p)dr.
(A.23)
Observing that
kT =  p \
"~p
)z,r =
N
/Z,T
(A.24)
we rewrite (A.23), after differentiating with respect to p, AP=AP r+~~pp
p
0
d2
ua(r)gz(r,p)d
p2 f 0(poo(r, p)) + = aPr + 2 ua(0 0p + 2
o d2
tUa(r)~p(p(gzgo))dr
(A.25)
Equation (A.25) is (7) of the text and is exact. Since, in general, gz is a functional (nonlinear) of Ua(r), AP may be expanded as a Volterrafunctional series. The first term in the right hand side of (A.25) is the "basis" pressure due to the potential Ur(r ) alone while the second term is the linear functional correction due to the presence of ua (r). The last term forms the nucleus from which higher order corrections are recovered. REFERENCES 1 S. K. Rangarajan, J. Electroanal. Chem., 45 (1973) 283. 2 (a) B. B. Damaskin, O. A. Petrii and V. V. Batrakov, Adsorption of Organic Compounds on Electrodes,
ADSORPTION 1SOTHERMSA CRITIQUE. II
3 4 5 6 7
17
Plenum Press, New York, London, 1971, Chap. III ; (b) R. Parsons, J. Electroanal. Chem., 7 (!964) 136; (c) S. K. Rangarajan, J. Electroanal. Chem., 45 (1973) 279. M. Kac, G. E. Uhlenbeck and P. C. Hemmer, J. Math. Phys., 4 (1963) 216. N. G. Van Kampan, Phys. Rev., 135 (1964) 362; J. L. Lebowitz and O. Penrose, J. Math. Phys., 7 (1966) 98. H. C. Anderson, J. D. Weeks and D. Chandler, Phys. Rev., A4 (1971) 1597; J. D. Weeks, D. Chandler and H. C. Anderson, J. Chem. Phys., 54 (1971) 5237. B. R. A. NijBoer, L. Van Hove, Phys. Rev., 85 (1952) 777. T. D. Lee and C, N. Yang, Phys. Rev., 87 (1952) 410.