Clustering as a mechanism of enhanced adsorption
The role of adsorbate clustering in increasing the adsorption on crystalline surfaces is analyzed in terms of the lattice gas model combined with the results of the associative theory. The effect of the cluster size and architecture in equilibrium and transport properties is studied. Some conject...
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Інститут фізики конденсованих систем НАН України
2003
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| Цитувати: | Clustering as a mechanism of enhanced adsorption / E.V. Vakarin, M.F. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 119-126. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1206982017-06-13T03:06:03Z Clustering as a mechanism of enhanced adsorption Vakarin, E.V. Holovko, M.F. The role of adsorbate clustering in increasing the adsorption on crystalline surfaces is analyzed in terms of the lattice gas model combined with the results of the associative theory. The effect of the cluster size and architecture in equilibrium and transport properties is studied. Some conjectures related to HCl-ice interfaces are drawn. В рамках моделі ґраткового газу, доповненої результатами асоціативної теорії, аналізується роль кластеризації адсорбату на зростання адсорбції на кристалічній поверхні. Вивчається вплив розмірів і архітектури кластера на рівноважні та транспортні властивості. Зроблені певні передбачення відносно міжфазної області НСІ-лід. 2003 Article Clustering as a mechanism of enhanced adsorption / E.V. Vakarin, M.F. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 119-126. — Бібліогр.: 14 назв. — англ. 1607-324X PACS: 68.45.-v DOI:10.5488/CMP.6.1.119 http://dspace.nbuv.gov.ua/handle/123456789/120698 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The role of adsorbate clustering in increasing the adsorption on crystalline
surfaces is analyzed in terms of the lattice gas model combined with the
results of the associative theory. The effect of the cluster size and architecture
in equilibrium and transport properties is studied. Some conjectures
related to HCl-ice interfaces are drawn. |
| format |
Article |
| author |
Vakarin, E.V. Holovko, M.F. |
| spellingShingle |
Vakarin, E.V. Holovko, M.F. Clustering as a mechanism of enhanced adsorption Condensed Matter Physics |
| author_facet |
Vakarin, E.V. Holovko, M.F. |
| author_sort |
Vakarin, E.V. |
| title |
Clustering as a mechanism of enhanced adsorption |
| title_short |
Clustering as a mechanism of enhanced adsorption |
| title_full |
Clustering as a mechanism of enhanced adsorption |
| title_fullStr |
Clustering as a mechanism of enhanced adsorption |
| title_full_unstemmed |
Clustering as a mechanism of enhanced adsorption |
| title_sort |
clustering as a mechanism of enhanced adsorption |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
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http://dspace.nbuv.gov.ua/handle/123456789/120698 |
| citation_txt |
Clustering as a mechanism of
enhanced adsorption / E.V. Vakarin, M.F. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 119-126. — Бібліогр.: 14 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vakarinev clusteringasamechanismofenhancedadsorption AT holovkomf clusteringasamechanismofenhancedadsorption |
| first_indexed |
2025-07-08T18:25:38Z |
| last_indexed |
2025-07-08T18:25:38Z |
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1837104251142144000 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 119–126
Clustering as a mechanism of
enhanced adsorption
E.V.Vakarin 1,2 , M.F.Holovko 1
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Laboratoire de Electrochimie et Chimie Analytique, Ecole Nationale
Superieure de Chimie de Paris – Universite Pierre et Marie Curie,
11 rue P. et M. Curie, 75231 Cedex 05, Paris, France
Received February 17, 2003
The role of adsorbate clustering in increasing the adsorption on crystalline
surfaces is analyzed in terms of the lattice gas model combined with the
results of the associative theory. The effect of the cluster size and architec-
ture in equilibrium and transport properties is studied. Some conjectures
related to HCl-ice interfaces are drawn.
Key words: adsorption, clustering, chemical diffusion
PACS: 68.45.-v
1. Introduction
Adsorption from gaseous or condensed phases onto solid surfaces is a phenomen-
on, which has numerous technologically important applications [1,2]. The process is
driven by several factors which are usually competing. These include the adsorbate-
substrate, the substrate-substrate and adsorbate-adsorbate interactions, which were
shown [3,4] to result in a remarkable interface restructuring, especially when the
substrate is not absolutely rigid. It is important that at fluid-solid interfaces the
fluid (adsorbate) acts as a thermodynamic subsystem [5] whose properties are de-
termined from the requirement of the solid-fluid equilibrium. This means that the
adsorbate has an additional (energetic and entropic) impact from its bulk, which
induces a correlation for the adsorbed species. The correlation through the bulk is
particularly pronounced when the pair interaction in the fluid is attractive enough
to induce the clustering [6,7]. Then, the intra-cluster correlation, arising due to the
cluster connectivity, enhances the adsorption in comparison with the monomeric
case. The effect takes place at a rather low adsorbate density in the interfacial re-
gion, that is, when the cluster-cluster repulsion is negligible. This mechanism has
c© E.V.Vakarin, M.F.Holovko 119
E.V.Vakarin, M.F.Holovko
been shown [8] to be responsible for a remarkable increase of the surface coverage
at low polymer concentrations.
In this paper we study the clustering as a mechanism of increasing the adsorption,
focusing on how this effect depends on the adsorbate density, cluster size and cluster
shape (linear chains or network-like aggregates). We first discuss the equilibrium
characteristics and then analyze the kinetic properties.
2. Adsorption of clusters
2.1. General formulation
We consider a model of a clustering fluid-solid interface. The fluid consists of
associating hard spheres with the diameter σ and bulk number density ρ, so that
the packing fraction is η = πρσ3/6. The association is mediated by the site-site
interaction responsible for a formation of clusters. Depending on the specificity of
the site-site attraction, different shape clusters can be formed (e.g. dimers, linear
polymer chains, network-like aggregates, star polymers). In this study we restrict
ourselves to the cases of linear chains and four-fold coordinated network-like clusters.
Details of the related potential models can be found in [6–8]. Therefore, the fluid is
characterized by its packing η and mean cluster size S. The latter is the number of
hard sphere monomers belonging to a cluster.
The fluid is near a hard wall (located at z = 0) on which there is a lattice of N
sticky sites (lattice spacing is d), modelling a crystalline structure. The adsorption
potential Us(ri) is defined by
exp(−βUs(ri)) = 1 + λ
∑
Rm
δ(Ri − Rm)δ(z), (1)
where Ri is a projection of ri onto the wall, Rm is a position of a lattice site and λ
is the stickiness parameter. This singular one-body potential allows one to perform
an exact integration in the partition function
Z = Zref
∑
n=0
λn
n!
∑
Rn
ρref
n (R1, ...,Rn), (2)
where Zref is the partition function for the same system but without the adsorbing
potential, and ρref
n (R1, ...,Rn) is the n-body distribution function taken at the po-
sitions of the lattice sites. In such a way we have an infinite series on λ including
the correlations of all orders for the reference state. If only pair interactions are
important, then the problem can be mapped onto the lattice gas model [9,10]
Ξ = Z/Zref =
∑
ti
exp(−βHLG) (3)
with the Hamiltonian
HLG =
∑
ij
W (Ri,Rj)titj −
∑
i
µ(Ri)ti (4)
120
Clustering as a mechanism of enhanced adsorption
in which ti is a set of occupation numbers. The chemical potential µ(Ri) and pair
interaction W (Ri,Rj) are closely connected with the properties of the fluid in the
bulk phase. Namely,
βµ(Ri) = ln(λρref
1
(Ri, σ/2)), (5)
βW (Ri,Rj) = − ln(gref
2
(Ri,Rj)), (6)
where ρref
1
(Ri) and gref
2
(Ri,Rj) are respectively the one-body and pair correlation
functions for the reference state (i.e., a state without specific adsorption) – the
hard wall problem. The latter is independent of the adsorption site position, and if
σ ≈ d then we deal with the contact values ρ1(σ/2) and g2(σ). In the Percus-Yevick
approximation they are given by [6,8]
ρ1(σ/2) = ρ
[
1 + 2η
(1 − η)2
−
S − 1
S
1
1 − η
]
, (7)
g2(σ) =
1 + η/2
(1 − η)2
−
S − 1
S
1
1 − η
+
C
η
[
S − 1
S
]2
. (8)
For both functions, the first terms correspond to the hard sphere packing effects
which dominate with the increase of η. The second terms describe the depletion
with the increase of S due to the cluster-wall or cluster-cluster repulsion. The last
term in g2(σ) is responsible for the intramolecular correlation inside a cluster. At
fixed η and S the intensity of this correlation is given by the coefficient C. For chains
C = 1/24 and for networks C = 1/16. Therefore, in the approximation considered
here, the network-like clusters differ from linear chains only by the strength of the
intramolecular correlation. The latter dominates at low densities when the cluster-
cluster contacts are rather rare and the monomers correlate mainly through the
cluster connectivity. Note, however, that the mean cluster size S is chosen as an
external variable. This means that we do not consider the clustering process itself,
assuming that the cluster composition remains fixed in the course of adsorption.
This assumption is valid if the adsorption is not dissociative and the adsorption
rate dominates the rate of dissociation in the fluid bulk. In general, S should be
considered as a function of density and temperature. Then, equations (7) and (8)
remain applicable with S = S(η, T ). In this case the difference between the chains
and network clusters would be remarkable. In particular, for networks, S could
diverge signalling the percolation threshold [6,7].
2.2. Mean-field solution
Having determined the bulk input to the lattice gas Hamiltonian we can find the
mean-field approximation for the free energy, which, after minimization, gives the
relation for the chemical potential at the equilibrium
βµ = qβWθ + ln
(
θ
1 − θ
)
, (9)
121
E.V.Vakarin, M.F.Holovko
Figure 1. Surface coverage θ as a function of the bulk packing fraction η for
chains and network-like clusters of the same mean size S.
where θ = 〈
∑
i
ti〉/N is the coverage and q is the lattice coordination number.
Equation (9) has been solved numerically in order to determine the coverage θ as
a function of density η and cluster size S. Depending on η and S, g2(σ) > 1 or
< 1 and consequently the lateral interaction βW = − ln g2(σ) is either attractive
or repulsive. For negative βW the system exhibits two-dimensional criticality of
the gas-liquid type under the conditions qβW |c = −4 and 2µ|c = qW |c. Then,
for given surface activity λ and S the coverage becomes discontinuous at some
critical density ηc . Note that g2(σ) is a nonmonotonic function of η as a result
of the competition between the inter- and intramolecular correlations. Therefore,
the critical condition can be reached in two ways. First is the case of low surface
activity λ and high density η. Then we deal with a cluster condensation induced by
the packing effects. This effect has been extensively analyzed [9,10] and we focus on
the second way, namely – the case of high surface activity and low density. Then the
cluster condensation is induced by the intramolecular correlation. In other words, at
low densities when the cluster-cluster contacts are negligible, the adsorption of one
bead promotes the adsorption of the entire cluster. This cooperative effect has been
confirmed experimentally [8]. The role of clustering is remarkable as the coverage
increases by several orders of magnitude in comparison with the case of monomeric
adsorption (S = 1).
In figure 1 the coverage is plotted as a function of density. As is discussed above,
θ is a nonmonotonic function of η. Three distinct domains are clearly seen. At low
η the coverage grows sharply exhibiting the cluster condensation transition due to
the intramolecular correlation. Then, at intermediate densities the cluster-cluster
repulsion leads to a decrease of θ. At high densities the coverage increases again due
122
Clustering as a mechanism of enhanced adsorption
Figure 2. Surface coverage θ as a function of the mean cluster size S for chains
and network-like clusters.
to the packing effects. It is seen that the network-like clusters are more efficient in
increasing θ than the chains of the same size.
In figure 2 θ is plotted as a function of S at η = 0.03, which belongs to the
region where the cluster-cluster repulsion decreases the coverage. For network-like
clusters θ grows with S due to the increasing role of the intramolecular correlations.
On the contrary, the chain coverage decreases in a broad range of cluster sizes since
the intramolecular correlation is weaker than that in the network. This once again
emphasizes the efficiency of the network in increasing the coverage.
3. Kinetic properties
Based on the equilibrium properties we can investigate the kinetic character-
istics with the purpose to realize the role of the clustering effects. Starting from
equation (9) the chemical diffusion coefficient can be found as [11]
D/D0 = θ(1 − θ)
d(βµ)
dθ
= 1 + qβWθ(1 − θ), (10)
where D0 is the single site hopping. It is seen that for attractive interaction (βW < 0)
the diffusion coefficient decreases. Taking θ as a solution of equation (9) we analyze
the behavior of D/D0 as a function of η at different S for the case of polymer
chains. This is depicted in figure 3. As expected, at low densities the diffusion is
suppressed due to the cluster condensation effect. At intermediate densities the
diffusion coefficient D/D0 is even higher than that of monomers (S = 1) due to
the increasing role of the cluster-cluster repulsion. This may seem rather strange.
123
E.V.Vakarin, M.F.Holovko
Figure 3. The surface chemical diffusion coefficient for monomers and chains as
a function of the bulk packing fraction.
In this respect we have to point out that we deal with a “single-site” transport in
the presence of the clustering, but not with the motion of clusters. Moreover, the
monomer coverage is much higher in this domain of densities. Hence, the monomer
transport is essentially blocked by the factor 1−θ, while there is enough free space for
the clusters to diffuse. Note also that we operate with the single site hopping D0, but
the cluster hopping should decrease with S (e.g. like D0/S) such that the absolute
value of D (at the same coverage) for clusters would be reduced in comparison with
that for monomers. Nevertheless, it is clear that the functional dependence of D vs
η in the presence of clustering is different from that of monomers.
As we have seen above the clustering significantly enhances the adsorption, espe-
cially at low densities. The clustering itself is a balance of two competing processes
evolving in time, i.e. formation of bonds between the monomers and their destruction
(spontaneously or under external fields). Obviously, the stability of such a cluster
adlayer depends upon the average number of bonds n(t) per monomer. In the mean-
field spirit such a balance is given by the following kinetic equation
τ
∂n(t)
∂t
= a[nm − n(t)] − bn(t), (11)
where τ is the characteristic time scale, the coefficients a, and b are the intensities
of the bond formation and destruction, respectively, nm is the maximal number of
bonds per monomer (e.g., nm = 4 for network-like clusters and nm = 2 for polymer
chains). The solution with the initial condition n(t = 0) = n0 is given by
n(t) = nm
a
a + b
[
1 − exp
(
−
(a + b)t
τ
)]
+ n0 exp
(
−
(a + b)t
τ
)
. (12)
124
Clustering as a mechanism of enhanced adsorption
It is clear that the asymptotic behavior does not depend on the initial condition
na = n(t → ∞) = nm
a
a + b
.
Nevertheless, depending upon whether n0 < na or n0 > na, n(t) approaches na either
from below (partial self-healing) or from above (partial destruction). Therefore, the
clustering-mediated enhancement of the adsorption could decrease with the time
increase depending upon which of the two cases is realized. This might be essential
for analyzing the adsorption kinetics.
4. Conclusion and implications
It is shown that the clustering in the interfacial region enhances the adsorption by
several orders of magnitude due to an increasing role of the intracluster correlation.
This effect occurs at low adsorbate densities in the interfacial region, that is, when
the cluster-cluster repulsion is negligible. It is demonstrated that the network-like
clusters are more efficient in increasing the coverage than the linear chains of the
same size. The functional dependence of the chemical diffusion coefficient on the
adsorbate interfacial density in the presence clustering is found to be different from
that for monomers.
Similar enhancement effect could take place in the course of HCl adsorption on
ice surfaces [12]. It is known that HCl forms a hydrogen bond with ice. Therefore, one
can expect an attraction (with the hydrogen bond strength ≈ 20 kJ/mole) between
HCl and water during the HCl-vapor co-adsorption [13]. As we have shown [14],
such an assumption is compatible with experimental data on the HCl coverage and
adsorption energy. At stratospheric conditions (low temperatures and pressures),
resembling the low density region in our model, the attraction could result in HCl-
water clustering over the surface of ice. Then the adsorption would increase through
the two-dimensional condensation transition. This conjecture is partially coherent
with the encapsulation scenario [13] studied in a dynamic context. Therefore, a
detailed study of the HCl-water interaction would be much helpful in analyzing
the clustering. The latter enhances the adsorption and also is essential for the HCl
ionization, providing a link between these two processes.
5. Acknowledgements
This work was supported in part by INTAS (Grant INTAS99-01162).
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E.V.Vakarin, M.F.Holovko
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