Langmuir 2008, 24, 2569-2572
2569
Statistical Mechanics of Molecular Adsorption: Effects of Adsorbate Interaction on Isotherms Je Hyun Bae,† Yu Rim Lim,† and Jaeyoung Sung* Department of Chemistry, Chung-Ang UniVersity, Seoul 156-756 Korea ReceiVed October 29, 2007. In Final Form: NoVember 29, 2007 We investigate a simple, exactly solvable model for interacting adsorbates. From the model study, we find that (1 - θ)-1 as a function of density of molecules in bulk media can have a positive curvature only in the presence of attractive interaction between adsorbed molecules. We propose a novel experimental observable, χ. Positive χ is the sufficient condition for the presence of attractive interactions between adsorbate molecules.
I. Introduction Molecular adsorption onto a finite number of receptor units is ubiquitous in nature, which manifests its characteristics not only in the adsorption isotherms of gas molecules on a solid surface,1 but also in many other systems of interest in chemistry and biology such as chemo-sensor systems of various kind,2-4 catalytic reaction systems including enzymatic reaction systems,5,6 and gene regulation systems,7 to name a few. To understand the behaviors of the latter systems, it is crucial to know the relationship between the amount of the adsorbed molecules and the density of the molecules in bulk media in equilibrium with the adsorbent. Among the numerous equations describing the latter relation, the Langmuir isotherm for monolayer adsorption8 and the Brunauer-Emmett-Teller (BET) isotherm for multilayer adsorption9 are the most famous ones, which provide explanations on the key features of a large number of adsorption experiments. On the basis of the latter models, generalizations have been made to take into account the effects of geometric confinement posed by environments, heterogeneity in adsorbent sites, lateral interactions, multisite occupancy, and other factors10-17 neglected in the Langmuir and the BET equations. For these purposes, a number of different theoretical approaches have been developed. * Corresponding author. E-mail:
[email protected]. † These two authors contributed equally to this work. (1) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, 1974. (2) Janata, J. Principles of Chemical Sensors; Plenum Press: New York & London, 1989. (3) Swager, T. M. Acc. Chem. Res. 1998, 31, 201. (4) Sung, J.; Silbey, R. J. Anal. Chem. 2005, 77, 6169. (5) Vannice, M. A. Kinetics of Catalytic Reactions; Springer: New York, 2005. (6) Bisswanger, H. Enzyme Kinetics Principle and Methods: Wiley-VCH: Weinheim, Germany, 2002. (7) Goodrich, J. A.; Kugel, J. F. Binding and Kinetics for Molecular Biologists; Cold Spring Harbor: New York, 2007. (8) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848. (9) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (10) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (11) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley and Sons: New York, 1990. (12) Anderson, R. B. J. Am. Chem. Soc. 1946, 68, 686. (13) de Boer, J. H. The Dynamical Character of Adsorption, 2nd ed.; Clarendon Press: Oxford, 1968. (14) Guggenheim, E. A. Application of Gases on Heterogeneous Surfaces; Oxford University Press: Oxford, 1966; Chapter 11. (15) Riccardo, J. L.; Ramirez-Pastor, A. J.; Roma´, F. Phys. ReV. Lett. 2004, 93, 186101. (16) Hill, T. J. Chem. Phys. 1946, 14, 263. (17) Ayappa, K. G. J. Chem. Phys. 1999, 111, 4736. Ayappa, K. G.; Kamala, C. R.; Abinandanan, T. A. J. Chem. Phys. 1999, 110, 8714.
In the present work, we first introduce a general relationship of the adsorption isotherm, the dependence of receptor coverage θ(F) on the substrate density F in bulk media, to the partition functions of the molecules, the receptor units, and their complexes, which is obtained by considering the grand canonical ensemble of adsorbed molecules. The general result provides the theoretical basis for a simple statistical mechanical derivation of the wellknown results such as the BET isotherms and their generalizations to more complicated models. Application of the general result is made to a simple, exactly solvable model for interacting adsorbates. From the model study, we find that [1 - θ(F)]-1 as a function of density F of molecules in bulk media can have a positive curvature only in the presence of attractive interaction between adsorbed molecules. We also propose a novel experimental observable, χ, which can serve as an indicator of the sign of the interaction potential between adsorbed molecules.
II. Grand Canonical Approach to General Molecular Adsorption Problems The grand canonical approach provides a much simpler description for molecular adsorption than the canonical approach does. Grand canonical approaches were previously used in investigating several models of molecular adsorption.17-20 In the present work, we introduce a unified description of molecular adsorption based on grand canonical approaches, which provides bases for very simple derivations of well-known results and their generalizations to more complex situations. Let an adsorbent material be composed of a number, ST, of binding sites, each of which can adsorb one or more substrate molecules. If sj denotes the number of binding sites with j M adsorbate molecules, we have ST ) ∑j)1 sj, where M denotes the maximum number of adsorbate molecules that a binding site can host. In this work, a binding site is composed of a number NR of receptor units, in which the substrate molecules can interact with each other. For the case with monolayer adsorption, NR should be equal to M; however, for the case with multilayer adsorption, NR is smaller than M. Let qj denote the canonical partition function of the single binding site with j substrate molecules adsorbed. Then the partition function of the grand canonical ensemble of the substrate molecules adsorbed onto a single binding site is given by (18) Presber, M.; Du¨nweg, B.; Landau, D. P. Phys. ReV. E 1998, 58, 2616. (19) Celestini, F.; Passerone, D.; Ercolessi, F.; Tosatti, E. Phys. ReV. Lett. 2000, 84, 2203. (20) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Surf. Sci. 2005, 583, 213.
10.1021/la703372t CCC: $40.75 © 2008 American Chemical Society Published on Web 02/16/2008
2570 Langmuir, Vol. 24, No. 6, 2008 M
Ξ(µ,β,M) )
exp(jβµ)qj(β) ∑ j)0
(1)
[ ( )]
(2)
Ξ(µ,T,M)
j exp(jβµ)qj(β) ∑ j)0
M
〈j〉eq({Kj}) )
( ) ∑ (∏ )
(j ) 0)
k)1 j
K l Fj ∏ l)1 M
k
k)1
l)1
(7) (j * 0)
Kl Fk
jPeq ∑ j ({Kj}) j)1
M ∂ ) F ln 1 + ∂F j)1
j
K l Fj
l)1
(8)
β,{Kj }
M
exp(jβµ)qj(β) ∑ j)0 1 ∂ [ln Ξ(µ,β,M)]β,M β ∂µ
)
(3)
At chemical equilibrium, the chemical potential µ in the binding site is the same as the chemical potential µb in bulk media surrounding the binding site, i.e., µ ) µb. If we can neglect interactions between the substrate molecules in bulk media, µb is given by µb ) -β-1 ln(qb/N), where qb is the molecular partition function of a substrate molecule in bulk media, and N is the number of substrate molecules in bulk media, respectively. Therefore, we have
exp(βµ) ) exp(βµb) ) F/q/b
(4)
where F denotes the density, N/V, of substrate molecules in bulk media, and q/b is defined by q/b ) (qb/V), with V being the volume of the bulk media. Equations 2-4 contain the complete information about the statistics of molecular adsorption onto binding sites with a homogeneous binding affinity to adsorbate molecules. From eqs 3 and 4, one can obtain the equilibrium probability of sj and the quantitative relationship between the average number 〈j〉 of adsorbed molecules and the density F of substrate molecules in bulk as a functional of molecular partition functions, q/b and {qj}:
(F/q/b)jqj(β)
)
(5)
M
∑ j)0
(F/q/b)jqj(β)
M
〈j〉eq )
and
∑ ∏ l)1
K l Fk
[ ∑ (∏ ) ]
M
Peq j
Peq j ({Kj}) )
-1
k
M
1+
1+
exp(jβµ)qj(β)
and the average number of the adsorbed molecules in a single binding site is given by
〈j〉 )
{
In terms of binding constants defined by Kj ≡ qj/(q/bqj-1), eqs 5 and 6 can also be rewritten as
where µ denotes the chemical potential in the binding site, and β denotes the inverse temperature defined by (kBT)-1 with kB and T being the Boltzmann constant and absolute temperature, respectively. The probability Pj(≡ sj/ST) of the binding site sj with j substrate molecules adsorbed is given by
Pj )
Bae et al.
j(F/q/b)jqj(β) ∑ j)0
)F
∂ ∂F
[∑ M
ln
j)0
Peq j )
∫dK1dK2‚‚‚dKMf({Kj})Peqj ({Kj})
(9)
〈j〉eq )
∫dK1dK2‚‚‚dKMf({Kj})〈j〉eq({Kj})
(10)
The upper bar on the left-hand side (LHS) of eqs 9 and 10 represents the average over the heterogeneity in the binding affinity of binding sites. Peq j ({Kj}) and 〈j〉eq({Kj}) in eqs 9 and 10 are given eqs 7 and 8. When there is not any heterogeneity M in the binding affinities, f({Kj}) becomes f 0({Kj}) ) ∏j)1 δ(Kj -K h j), and eqs 9 and 10 reduce to eqs 7 and 8. The total volume Vads of adsorbate molecules is related to the average number 〈j〉eq of molecules adsorbed to each binding site by Vads ) V0〈j〉eqST with V0 being the volume of a single adsorbate molecule. Since NR denotes the number of receptors in each binding site, the maximum volume V1 of adsorbate molecules in the first monolayer is given by V1 ) V0NRST. Therefore, the ratio θ of Vads to V1 is given by
θ)
M
(F/q/b)jqj(β) ∑ j)0
If Cj denotes the binding site with j adsorbates adsorbed, Kj in eqs 7 and 8 is nothing but the equilibrium constant for the k(j) association-dissociation reaction: Cj-1 + A f Cj with A a k(j) r eq denoting the adsorbate molecule, which is the same as Peq j /FPj-1 (i) (i) or kf /kr . Derivations of eqs 7 and 8 based on phenomenological rate equations are easy, but it is mute to the relation of 〈j〉eq to microscopic properties of adsorbates, receptors, and their complexes. In the presence of heterogeneity in the binding affinity, Kj would be different from site to site. The effects of such heterogeneity in the binding affinity can be described by introducing the probability density function f({Kj}) of binding constants, {Kj}:
]
(F/q/b)jqj(β)
(6) β, M
Vads 〈j〉eq ) V1 NR
(11)
Equations 9, 10, and 11 are general results that can describe molecular adsorption phenomena of a variety of complicated models. However, in many cases, key features of the adsorption isotherm observed in experiments can be explained by such simple models as the Langmuir model for monolayer adsorption and the
Statistical Mechanics of Molecular Adsorption
Langmuir, Vol. 24, No. 6, 2008 2571
θBET ) 〈j〉BET )
K1F [1 + (K1 - KL)F](1 - KLF)
(15)
The simplicity of the present derivation is remarkable, compared with that of previously reported ones.16 It is also straightforward to rederive previously reported generalizations of the Langmuir or BET models, which cannot be discussed completely here. Instead, we will investigate a new simple model of interacting adsorbates, starting from the general formulation presented in Section II.
IV. Simple Model for Interacting Adsorbates
Figure 1. Effects of interaction potential on the curvature of [1 θ(F)]-1. The value of c(≡ KA/KB) is set to be 1. In the absence of the interaction between adsorbate, (1 - θ)-1 is a linear function of density F of molecules in bulk media. The curvature (1 - θ)-1 becomes positive (negative) for the case with attractive (repulsive) interaction potentials.
Let an adsorbent material contain a number of independent binding sites, each of which is composed of two receptors, A and B. Each receptor can adsorb one adsorbate molecule with different binding affinities from each other, and the adsorbate molecule adsorbed on receptor A can interact with that adsorbed on receptor B. For this model, the partition function qj of the binding site with j adsorbate molecules is given by
BET model for multilayer adsorption. Starting from eqs 3 and 4, one can easily derive the isotherms of the latter models, and generalize these results into more complicated models.
III. Rederivation of the Well-Known Results IIIa. Langmuir Isotherm. If the adsorbate molecules are not interacting with each other, we can regard one receptor as one binding site in the present formulation, i.e., NR ) 1. In the absence of multi-molecule adsorption to a single binding site, we have qj ) 0 for any j greater than 1. Therefore, the grand canonical partition function given in eq 1 becomes
ΞL ) q0 + q1 exp(βµ)
(12)
(13)
with K1 ) q1/(q/bq0). IIIb. BET Isotherm. In the BET model, a single binding site can adsorb many molecules to form a multilayer of adsorbent molecules on the site. For this model also, interaction between the adsorbate in one receptor and that in another one is assumed to be zero, and one receptor can be regarded as one binding site, i.e., NR ) 1. The interactions of an adsorbate molecule with a bare receptor are different from those with a receptor-adsorbent complex. However, the interactions of an adsorbate molecule with a receptor-adsorbate complex in site Cj+1 are assumed to be the same as those of a receptor-adsorbate complex in site Cj for all positive j. For this model, the partition function of the receptor-adsorbate complex becomes qj+1 ) q1qjL (j g 0), where qL denotes the molecular partition function of an adsorbate molecule interacting with the receptor-adsorbate complex so that the grand canonical partition function given in eq 1 becomes
ΞBET ) q0 +
exp(βµ)q1 1 + exp(βµ)qL
(14)
From eqs 3, 4, and 14, one can immediately obtain the BET isotherm as follows:
(16a)
(B) (A) (B) q1 ) q(A) 1 q0 + q0 q1
(16b)
(B) q2 ) q(A) 1 q1 exp(-βUAB)
(16c)
qj ) 0 (j g 3)
(16d)
where q(X) j denotes the molecular partition function of receptor X with j molecules adsorbed, and UAB denotes the interaction potential between adsorbate molecule bound to receptor A and that bound to receptor B in a binding site. Substituting eq 16 into eq 6, one can obtain
〈j〉 )
for the Langmuir monolayer adsorption model. Substituting eq 11 into eq 3, and using the equilibrium condition (eq 4), one can obtain
K1F θL ) 〈j〉L ) 1 + K1F
(B) q0 ) q(A) 0 q0
F(KA + KB) + 2F2KAKB exp(-βUAB) 1 + F(KA + KB) + F2KAKB exp(-βUAB)
(17)
/ (X) with KX ) q(X) 1 /qbq0 (X ) A,B). Since each binding site can absorb two adsorbate molecules, the surface coverage θ is given by 〈j〉/2 for this model, which can be written as
θ)
(F/F0A)(1 + c)/2 + (F/F0A)2c exp(-βUAB) 1 + (F/F0A)(1 + c) + (F/F0A)2c exp(-βUAB)
(18)
where F0A and c are defined by F0A t KA-1 and c ) KB/KA, respectively. Note that eq 18 reduces to the Langmuir isotherm when the binding affinity of receptor A is the same as that of receptor B, and there is no interaction between adsorbate molecules, i.e., when c ) 1 and UAB ) 0. In this case, we have (1 - θ)-1 ) 1 + (F/F0A). However, in general, (1 - θ)-1 is nonlinear in (F/F0A). In the low-density limit, its asymptotic behavior is given by (1 - θ)-1 = 1 + (1 + c)/2 (F/F0A) + O[(F/F0A)2]; in comparison, the high density asymptotic behavior of (1 - θ)-1 is given by (1 - θ)-1 = (2c exp(-βUAB))/(1 + c) (F/F0A){1 + O[(F/F0A)-1]}. Note that the slope changes from (1 + c)/2 (≡ Sl) in the low-density limit to (2c exp(-βUAB))/(1 + c) (≡ Sh) in the high-density limit. This tells us that (1 - θ)-1 as a function of F will have a positive (negative) curvature when
χ≡
4KAKB Sh -1) exp(-βUAB) - 1 > (
