Entropic solvation force between surfaces modified by grafted chains: a density functional approach
The behavior of a hard sphere fluid in slit-like pores with walls modified by grafted chain molecules composed of hard sphere segments is studied using density functional theory. The chains are grafted to opposite walls via terminating segments forming pillars. The effects of confinement and of &quo...
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irk-123456789-320442012-04-07T12:20:42Z Entropic solvation force between surfaces modified by grafted chains: a density functional approach Pizio, S. Sokołowski, O. The behavior of a hard sphere fluid in slit-like pores with walls modified by grafted chain molecules composed of hard sphere segments is studied using density functional theory. The chains are grafted to opposite walls via terminating segments forming pillars. The effects of confinement and of "chemical" modification of pore walls on the entropic solvation force are investigated in detail. We observe that in the absence of adsorbed fluid the solvation force is strongly repulsive for narrow pores and attractive for wide pores. In the presence of adsorbed fluid both parts of the curve of the solvation force may develop oscillatory behavior dependent on the density of pillars, the number of segments and adsorption conditions. Also, the size ratio between adsorbed fluid species and chain segments is of importance for the development of oscillations. The choice of these parameters is crucial for efficient manipulation of the solvation force as desired for pores of different width. Поведінку плину твердих сфер у щілиноподібних порах зі стінками, модифікованими розгалуженими ланцюговими молекулами, які складаються із твердосферних сегментів, досліджено з використанням теорії функціонала густини. Ланцюги є розгалуженими до протилежних стінок через скінченні сегменти, що формують опори колони. Досліджено вплив обмеження та "хімічної" модифікації стінок пори на ентропійну силу сольватації. Спостережено, що за відсутності адсорбованого плину сила сольватації є сильно відштовхувальною для вузьких пор і притягальною для широких. Коли адсорбований плин є присутнім, обидві частини кривої сили сольватації можуть розвивати осциляційну поведінку залежно від густини опор колон, числа сегментів та умов адсорбції. Також, розмірний коефіцієнт між частинками адсорбованого плину та сегментами ланцюгів є важливим для розвитку осциляцій. Вибір цих параметрів є дуже суттєвим для ефективного керування силою сольватації для пор різної ширини. 2010 Article Entropic solvation force between surfaces modified by grafted chains: a density functional approach / O. Pizio, S. Sokołowski // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13602: 1-12. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 68.08.-p, 68.15.+e, 82.35.Gh, 68.43.-h http://dspace.nbuv.gov.ua/handle/123456789/32044 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The behavior of a hard sphere fluid in slit-like pores with walls modified by grafted chain molecules composed of hard sphere segments is studied using density functional theory. The chains are grafted to opposite walls via terminating segments forming pillars. The effects of confinement and of "chemical" modification of pore walls on the entropic solvation force are investigated in detail. We observe that in the absence of adsorbed fluid the solvation force is strongly repulsive for narrow pores and attractive for wide pores. In the presence of adsorbed fluid both parts of the curve of the solvation force may develop oscillatory behavior dependent on the density of pillars, the number of segments and adsorption conditions. Also, the size ratio between adsorbed fluid species and chain segments is of importance for the development of oscillations. The choice of these parameters is crucial for efficient manipulation of the solvation force as desired for pores of different width. |
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Pizio, S. Sokołowski, O. |
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Pizio, S. Sokołowski, O. Entropic solvation force between surfaces modified by grafted chains: a density functional approach Condensed Matter Physics |
| author_facet |
Pizio, S. Sokołowski, O. |
| author_sort |
Pizio, S. |
| title |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach |
| title_short |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach |
| title_full |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach |
| title_fullStr |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach |
| title_full_unstemmed |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach |
| title_sort |
entropic solvation force between surfaces modified by grafted chains: a density functional approach |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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| citation_txt |
Entropic solvation force between surfaces modified by grafted chains: a density functional approach / O. Pizio, S. Sokołowski // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13602: 1-12. — Бібліогр.: 40 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT pizios entropicsolvationforcebetweensurfacesmodifiedbygraftedchainsadensityfunctionalapproach AT sokołowskio entropicsolvationforcebetweensurfacesmodifiedbygraftedchainsadensityfunctionalapproach |
| first_indexed |
2025-07-03T12:31:39Z |
| last_indexed |
2025-07-03T12:31:39Z |
| _version_ |
1836628993903689728 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13602: 1–12
http://www.icmp.lviv.ua/journal
Entropic solvation force between surfaces modified by
grafted chains: a density functional approach
O. Pizio1, S. Sokołowski2
1 Instituto de Quı́mica de la UNAM, Coyoacán 04510, México
2 Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University,
20-031 Lublin, Poland
Received October 21, 2009, in final form January 27, 2010
The behavior of a hard sphere fluid in slit-like pores with walls modified by grafted chain molecules composed
of hard sphere segments is studied using density functional theory. The chains are grafted to opposite walls
via terminating segments forming pillars. The effects of confinement and of “chemical” modification of pore
walls on the entropic solvation force are investigated in detail. We observe that in the absence of adsorbed
fluid the solvation force is strongly repulsive for narrow pores and attractive for wide pores. In the presence of
adsorbed fluid both parts of the curve of the solvation force may develop oscillatory behavior dependent on the
density of pillars, the number of segments and adsorption conditions. Also, the size ratio between adsorbed
fluid species and chain segments is of importance for the development of oscillations. The choice of these
parameters is crucial for efficient manipulation of the solvation force as desired for pores of different width.
Key words: density functional approach, grafted chains, solvation force, density profiles
PACS: 68.08.-p, 68.15.+e, 82.35.Gh, 68.43.-h
1. Introduction
Thin fluid films confined by solid surfaces are of much interest for basic research and several
applications. The microscopic structure, thermodynamic, electric and other properties of such
systems have been extensively studied in experiments, computer simulations and using various
theoretical methods. One of the important properties of this type of systems is the solvation force,
i.e. the force acting between solid surfaces due to the intervening fluid between them. Experiments
using surface force apparatus yield the solvation force that indirectly describes the fluid structure
in confined films [1–4]. Several important conclusions for numerous applications have been made
by analyzing experimental results for the solvation force.
The solvation force is determined by different factors, e.g. it is effected by the properties of
molecules of confined fluid [5–9]. Simple fluids, associating and hydrogen-bonded fluids, fluids of
complex molecules as well as some mixtures and solutions were investigated experimentally in this
aspect. On the other hand, computer simulation and theoretical methods have been extensively
applied to the study of the solvation force [10–16]. The dependence of solvation force on the
characteristics of confining solid substrates, namely the shape of the surface, its in-plane symmetry,
energetic and geometric heterogeneity, was investigated in many publications, see e.g. [17–21]. In
the last decade, much research has been focused on synthetic and chemically modified substrates
for specific applications [22–26].
From a general theoretical perspective, one of the powerful and popular tools of studying
inhomogeneous fluids is the density functional theory. Modern versions of this theory are capable
of accurately describing the microscopic structure, thermodynamic properties and phase behavior of
confined fluid systems. This is proved by comparison of theoretical results with computer simulation
data. In particular, investigations of the solvation force for some model fluid systems by means
of density functional approaches have been undertaken [27–29]. In the recent works, Cao and Wu
[30], as well as Jiang et al. [31] examined the solvation force between plane walls in the presence of
c© O. Pizio, S. Sokołowski 13602-1
http://www.icmp.lviv.ua/journal
O. Pizio, S. Sokołowski
telechelic polymers in the pore space. We give a few details regarding their modelling in order to
stress the differences between that approach and the present study. The telechelic polymers were
considered [30, 31] to be built up of the jointed tangent-sphere chains with “sticky” ends adhered
to a solid surface. It was assumed that the terminating segments of each chain were attracted to the
hard wall via narrow square well potential. On the other hand, the “sticky” segments themselves
interacted via square well attractive potential. The solvent effects were totally neglected; the solvent
was treated as a continuum without any specification. A polymer density functional theory was
used to obtain theoretical predictions. Excluded volume effects in the excess free energy and the
effects of chain connectivity were considered using the fundamental measure and the first-order
perturbation theories, respectively [32]. The attractive interactions between terminating segments
were accounted for at the level of a mean field theory. The solvation force between substrates coming
from the presence of telechelic polymers and their grafting was explored for a model permitting the
formation of brushes at each pore wall (due to a single adhesive segment on chain molecules) and a
model in which each surface was covered by loops (due to two adhesive segments per chain). Most
recently, the effects of energetic parameters characterizing attractive interactions in the model,
in addition to the density changes of grafted species were studied within the same theoretical
framework [33, 34].
In the present work we use a methodologically similar approach, however, with some modifi-
cations described below. Our principal objective is to investigate the solvation force for a model
system that explicitly includes a simple fluid confined by solid walls. These walls are modified by
grafted chain molecules. Each chain molecule in our study consists of segments of equal diame-
ter (homopolymer), all the molecules being pinned to the opposite walls of a pore through the
first and the last segment forming flexible pillars. The solvation force comes from the presence of
chain particles and fluid species. The fluid species are taken into account explicitly, in contrast
to the model by Cao and Wu [30, 31]. However, to elucidate the underlying phenomena in such
a class of systems, at this stage of investigation we restrict ourselves to the model in which only
the repulsive forces are taken into account. Namely, the fluid and segments of chain species are
hard spheres. The solid substrates solely provide geometric confinement. Thus, we are able to de-
scribe the underlying entropic contributions to the solvation force between pore walls joined by
pillars.
Entropic effects in confined fluids are common. They arise either due to a molecular shape
or internal architecture of molecules or, on the other hand, can be of combined nature, namely
involving the former factors and specific form of interaction between complex molecules and sub-
strates. Entropic effects can lead to various phase transformations in confined fluids in the absence
of attractive interactions determining energetic aspects of adsorption and surface phase transitions.
Our setup seems to be simple, but permits various physically interesting extensions and generali-
zations. Namely, one can extend the present techniques to explore the adsorption of mixtures in
which entropic effects under confinement may induce different phase separations, e.g. nonadditive
hard sphere mixtures, hard sphere mixtures with rather big difference of diameters of particles
belonging to different components, the Widow-Rowlinson model, the Asakura-Oosawa model for
colloid-polymer mixtures and some other complex fluids. Also, it is of interest to formulate the
problem of swelling at the level of present modelling. However, before comprehensively studying
such models under confinement, one need to have a rather complete understanding of a compara-
tively simpler system in question. Even in the present formulation, the model is characterized by
various parameters. Investigation of the role of each of them is one of our primary objectives. As
for the subsystem of chains, most important is to capture the effect of grafted density and length of
chains that form pillars between walls. On the other hand, the density of fluid species in the pore
is not of less importance. Finally, the ratio of diameters of chain segments and of fluid particles
affects adsorption and microscopic structure, i.e. both the distribution of fluid species and the way
segments of chains accommodate in the pore.
The theoretical approach has been described in detail in our recent works [35–40]. Therefore,
hereinafter we will give only some essential ingredients. Our approach is based on the method
developed by Yu and Wu [32] and extended later by us to describe a particular mixture of spherical
13602-2
Entropic solvation force
and chain molecules in which some chains are pinned to the walls. We expect that the insights
obtained in the present study may be useful in considering the novel setups for nano-systems and
devices with desired interfacial properties.
2. The model and the method
We study adsorption of hard spheres of species S in a slit-like pore of the width H with identical
walls that have been modified by pre-adsorbed chain molecules, C, composed of M tangentially
bonded hard spheres (segments) of identical diameter σ(C). The chain connectivity is ensured by
the binding potential at contact between nearest-neighbor segments. The total binding potential
Vb is given by [32]:
exp[−βVb(R)] =
M−1
∏
i=1
δ(|ri − ri+1|) − σ(C))/4π(σ(C))2. (1)
In the above R = r1, r2, . . . , rM denotes the vector of coordinates of all the segments, β = 1/kT .
We would like to study the model in which the first, J = 1, and the last, J = M , segments of
chains are “chemically” bonded to the opposite pore walls i.e. they interact with the surface via
very strong, but short-ranged potentials of the form,
exp[−βv
(C)
J (z)] = Cδ(κJ), (2)
where κ1 = z − σ(C)/2 and κM = z − (H − σ(C)/2). Here C is a constant, the precise value of
which is irrelevant if the total number of grafted chains per unit surface area is fixed [35–39]. The
external potential acting on all the remaining segments is
v
(C)
i (z) =
{
∞, z 6 σ(C)/2, z > (H − σ(C)/2),
0, otherwise,
(3)
for i = 2, 3, . . . , M − 1. It just provides the state that non-terminating segments are in the pore
space between two hard walls. The total external potential acting on a chain particle is V (C)(R) =
∑M
i=1 v
(C)
i (z).
Adsorption of hard spheres with the diameter σ(S) into the pore with modified walls takes place
at a certain chemical potential, µS . Adsorption relies on chemical equilibrium between the fluid
particles in the pore and in a reservoir containing solely one-component fluid of S-species. During
adsorption the structure of the pre-adsorbed chain molecules changes, but their amount inside the
pore remains unaltered.
The fluid species S interact with the wall via hard sphere potential
v(S)(z) =
{
∞, z 6 σ(S)/2, z > (H − σ(S)/2),
0, otherwise.
(4)
Again, all the segments of chains and spherical molecules interact via hard sphere potentials
U (ij)(r) =
{
∞, r 6 σ(ij)/2,
0, r > σ(ij)/2,
(5)
where i, j = S, C, σ(ij) = 0.5(σ(i) + σ(j)).
In order to proceed, let us introduce the notation, ρ(C)(R) and ρ(S)(r), for the density distri-
bution of chains and spherical species, respectively. However, the theory is constructed in terms of
the density of particular segments of chains, ρ
(C)
sj (r), that yield auxiliary quantity describing the
total segment density of chains, ρ
(C)
s (r). The segment densities are introduced via commonly used
relations, see the original development in [32]
ρ(C)
s (r) =
M
∑
j=1
ρ
(C)
sj (r) =
M
∑
j=1
∫
dRδ(r− rj)ρ
(C)(R) . (6)
13602-3
O. Pizio, S. Sokołowski
To simplify the equations below we also use the notation ρ
(S)
s (r) ≡ ρ
(S)
s1 (r) ≡ ρ(S)(r).
The system is studied in a grand canonical ensemble with the constraint of constancy of the
number of chain molecules, i.e.
RC =
∫
ρ
(C)
s1 (z)dz =
∫
ρ
(C)
sM (z)dz, (7)
where RC is the number of chain molecules per area of the surface. The thermodynamic potential
appropriate to the description of the system is
Y = F [ρ(C)(R), ρ(S)(r)] +
∫
drρ(S)(r)(v(S)(r) − µS) +
M
∑
i=1
∫
drρ
(C)
si (r)v
(C)
i (r), (8)
where F [ρ(C)(R), ρ(S)(r)] is the Helmholtz free energy functional.
The expression for F [ρ(C)(R), ρ(S)(r)] is chosen according to the theory of Yu and Wu [32].
The density profile ρ(S)(r) and the segment density profiles ρ
(C)
si (r) are obtained by minimizing
the functional Y under the constraint (7). For the sake of brevity we do not present the resulting
density profile equations, (they can be found in our recent works [35–39]), but you should only
note that the configurational chemical potential µS in terms of the bulk density ρb of species S is
µS/kT = ln ρb + βµS
(hs) , (9)
where µS
(hs) is the excess chemical potential of pure hard sphere fluid at the density ρb resulting
here from the Carnahan-Starling equation of state.
For two planar surfaces separated by a film of adsorbed fluid, the solvation force per unit area
is calculated from
fs/kT =
∂(Y/AkT )
∂H
− p/kT, (10)
where p is the pressure of the bulk system, i.e. the fluid in a reservoir. We choose the diameter of a
segment of chain species as a length unit, σ(C) = 1. Also, for the sake of convenience the reduced
solvation force as f∗ = fs(σ
(C))3/kT is introduced.
We would like to finish this section by the following comment. The first and the last segment
of each chain are fixed at z = σ(C)/2 and z = H − σ(C)/2, respectively. The parameter RC
determines the surface density of chains at each of the walls. Formally, the situation is symmetric
with respect to the pore center for these terminating segments. However, one should remember
that the terminating segments can slip within the plain separated by σ(C)/2 from the pore walls.
The configuration of the first and the last segments in the x, y plane is not fixed. It is formed
dependent on RC and on the density of an adsorbed fluid. Also, the connectivity and the chain
length effect this configuration. Physically, one can design a model permitting some sort of phase
separation of terminating segments and adsorbed fluid species in the monolayer on the solid surface.
However, we are not able to follow changes of the configuration of terminating segments in the
framework of the effective one-dimensional density functional used in this study. Application of
the full three-dimensional functional is not feasible numerically at present.
3. Results and discussion
Let us proceed to the discussion of the results. Figure 1 shows the reduced solvation force
obtained for the pores with different density of grafted chains with eight segments (M = 8) that
serve as connectors of the walls, or, in other words, form bridges between the walls in the absence
of adsorbed fluid, i.e. ρ∗
b = ρb(σ
(C))3 = 0. The pore width is given everywhere below in the reduced
units, H∗ = H/σ(C). For the pores with non-modified walls the solvation force exhibits common
trends. Namely it approaches zero as the wall-to-wall distance increases. Usually, for non-modified
walls the solvation force, f∗, exhibits oscillatory behavior at small interparticle separations and
tends to infinity for H∗ → 1. The presence of even a small amount of connecting chain molecules
13602-4
Entropic solvation force
causes the state that for H∗ → 8 (i.e. when the wall-to-wall distance equals the length of extended
chains) a strong attraction between the walls appears, see figure 1. The plot of the reduced solvation
force significantly differs from the plots shown in [30, 31]. According to the model of this study
none of the bonds connecting segments of chains can be broken and the existing bridges cause the
state at which the maximum value of H cannot exceed Mσ(C). Therefore, even for a small value
of RC strong effective attraction between two walls appears at wall-to-wall distance close to 8.
0 2 4 6 8
H*
-4
-2
0
2
4
6
f*
0.05
0.1
0.2
0.4
RC
* =
M=8
ρb
* = 0
Figure 1. The reduced solvation force, f∗(H∗), for the model pore with different density of
grafted chains as connectors in the absence of adsorbed fluid, ρ∗b = 0. The calculations are for
chains with eight segments M = 8, and for the values of R∗
C given in the figure.
When the number of connecting molecules (one can call them pillars) increases, then effective
attraction between two walls becomes stronger at wall-to-wall separation larger than one half of
the nominal chain length, c.f. the curves for R∗
C = RC(σ(C))2 = 0.05 and R∗
C = 0.2 in figure 1.
The attractive branch for each chain density is rather smooth, i.e. we do not observe an oscillatory
behavior. At small distances, strong repulsion between walls develops. For the case of a pore with
non-modified walls, as well as when R∗
C is small, the solvation force diverges close to H∗ → 1.
However, when R∗
C increases, the strong repulsion between the walls develops at larger distances,
e.g. for R∗
C = 0.2 at H∗ ≈ 2. Obviously, for R∗
C high enough a narrow pore is almost completely
filled by the chain segments and there is no room for fluid species to be adsorbed. In the absence of
adsorbed fluid, one does not observe oscillations of the reduced solvation force at small distances
between the walls. Rather, there is a nonmonotonous decay of the repulsion for an interval of the
length σ(C), see e.g. the behavior of f∗ between H∗ = 1 and H∗ = 2 terminating with a not very
sharp cusp for R∗
C = 0.1 in figure 1. This feature actually can be attributed to the accommodation
of segments of the grafted chains due to a slightly augmenting free volume when the pore width
increases in this range. The cusp describing the fall of repulsion occurs when the neighbors of the
grafted segment gain room for their movement in the pore space. Under such circumstances, say for
H∗ > 2, farther segments contribute into the overall repulsion but they do not form any layer-type
structure as it happens during the filling of a pore with non-modified walls by certain fluid species.
In the pores with non-modified walls, the solvation force is at maxima if the pore width ap-
proximately matches the integer number of layers of fluid species, i.e. for nσ(S), and it is at minima
for (n + 1/2)σ(S). A complementary insight into the interpretation of the solvation force can be
obtained by analyzing the density profiles of chain segments given in three panels of figure 2. In the
narrow pore, the segments are distributed close to the pore walls and have room for their move-
ment only if the density of grafted chain molecules is not very high, e.g. R∗
C = 0.1, figure 2 (a).
Consequently, the solvation force is repulsive and quite high. Increasing room for chain segments
to accommodate, by passing to a wider pore, H∗ = 4, leads to a low value of repulsive solva-
tion force, cf. figure 2 (b) and figure 1. In another extreme, for a wide pore close to the nominal
13602-5
O. Pizio, S. Sokołowski
0 0.5 1 1.5 2
z*
0
0.05
0.1
0.15
0.2
ρ(C
)* sj
(z
)
j=2
j=3
j=4
M=8
H*=2
RC
* =0.1
ρb
* = 0
a
0 1 2 3 4
z*
0
0.05
0.1
0.15
0.2
ρ(C
)* sj
(z
)
j=2
j=3
j=4
M=8
H*=4
RC
* =0.1
ρb
* = 0
b
0 2 4 6
z*
0
0.5
1
ρ(C
)* sj
(z
)
M=8
H*=7.5
RC
* =0.1
ρb
* = 0
cj=2 j=7
j=3 j=6
j=4 j=5
Figure 2. Density profiles of segments of chain particles, M = 8, in the pores of different width,
H∗, in the absence of adsorbed fluid, ρ∗b = 0, and at fixed density of grafted chains, R∗
C = 0.1. In
all panels, the index j indicates the consecutive segment number. Parts (a), (b) and (c) are for
H = 2, H = 4 and H = 7.5, respectively. The nomenclature of lines is explained in the figures.
length of extended chains H∗ = 7.5 the segments are regularly distributed showing a layered struc-
ture. However, due to the bonding constraints accounted for in the free energy expression for the
grand thermodynamic potential, the solvation force becomes attractive but without pronounced
oscillations for high values of H∗, cf. figure 2 (c) and figure 1.
The behavior of the solvation force with the bulk fluid density, ρ∗
b , is shown in figure 3. The
amount of chain connectors is constant and not high, R∗
C = 0.1. An increase of the density of
fluid species results in more pronounced oscillations of the solvation force. However, only at the
highest bulk density considered, ρ∗
b = 0.7, the oscillations alter the repulsive branch (H∗ < 5) of
the curve to develop a set of attractive minima. This behavior is contrary to lower values of fluid
density where the oscillations develop but still lead to repulsive solvation force. In the region of
attraction, i.e. for wider pores, the solvation force is monotonous. Some density profiles of fluid
species describing trends for their accommodation in layers and consequently maxima or minima of
the reduced solvation force are shown in figure 4 (a). On the other hand, the corresponding profiles
for the total segment density are given in figure 4 (b). We observe that the presence of connecting
chain molecules in this case does not prohibit the development of fluid layer-like structure at high
fluid density and leads to a common behavior of the solvation force for not very wide pores. These
trends can be altered by increasing the density of chains or if only the fluid-segment attraction
13602-6
Entropic solvation force
were present in the model.
2 4 6 8
H*
-4
-2
0
2
4
6
f*
0.1
0.4
0.6
0.7
ρb
* =
M=8
RC
* =0.1
σ(S)=σ(C)
Figure 3. The reduced solvation force f∗(H∗), for the model pore at different density of adsorbed
fluid, ρ∗b , at a fixed density of chain connectors, R∗
C = 0.1, M = 8. The nomenclature of lines is
given in the figure.
0 1 2 3
z*
0
1
2
3
4
ρ(S
)*
(z
)
H*=2.30
H*=2.84
H*=3.30
H*=3.82
M=8
RC
* =0.1
ρb
* = 0.7
a
σ(S)=σ(C)
0 1 2 3
z*
0
1
2
ρ(C
)* s
(z
)
H*=2.30
H*=2.84
H*=3.30
H*=3.82
M=8
RC
* =0.1
ρb
* = 0.7
b
σ(S)=σ(C)
Figure 4. The density profiles of fluid species (a) and chain connectors (b) for slit-like pore of
different width. The calculations are for R∗
C = 0.1, ρ∗b = 0.7, and M = 8. The nomenclature of
lines is given in the figures.
In order to capture these possibilities at least partially, we have performed calculations at a
constant high fluid density and varied the amount of chain connectors. Moreover, for the sake of
generality, we would like to get insight into the effects of fluid species and of chain segments on
the oscillations of the solvation force separately by considering the model with unequal diameters.
In the restricted case of equal diameters two species may contribute to the solvation force at
the same distance scale. First, we studied the case in which fluid species have slightly smaller
diameter than the segments of chains, i.e. σ(S) = 0.8σ(C). One set of results concerning such a
model is given in figure 5. The fluid density is chosen such to correspond to 0.7, if the diameters of
segments and fluid particles are equal. From this figure we learn that the scale of oscillations of the
solvation force is determined by the diameter of fluid species. Most importantly, we observe that
13602-7
O. Pizio, S. Sokołowski
the attractive branch of the solvation force becomes oscillatory, if the density of connector chain
molecules is sufficiently high. This chain density should provide an overall attraction between pore
walls whereas the fluid density should be high enough to provide the formation of layers of fluid
species. Then, the total picture is like the one we show in figure 5. The density profiles illustrating
distribution of both species in the pore are shown in figure 6 (a and b).
0 2 4 6 8
H*
-4
-2
0
2
4
6
8
f*
0.1
0.2
0.4
M=8
ρb
* =1.3672
RC
* =
σ(S)=0.8σ(C)
Figure 5. The reduced solvation force as a function of the density of chain connectors R∗
C
(M = 8) at a fixed bulk fluid density, ρ∗b = 1.3672. The fluid particles are smaller than chain
segments, σ(S) = 0.8σ(C).
0 0.5 1 1.5 2
z*
0
2
4
ρ(S
)*
(z
),
ρ(C
)* s
(z
)
chain
fluid
M=8
RC
* =0.1
ρb
* = 1.3672
a
H*=2.08
σ(S)=0.8σ(C)
0 0.5 1 1.5 2 2.5
z*
0
1
2
3
4
5
6
ρ(S
)*
(z
),
ρ(C
)* s
(z
)
chain
fluid
M=8
RC
* =0.1
ρb
* = 1.3672
b
H*=2.90
σ(S)=0.8σ(C)
Figure 6. The density profiles of fluid species ρ(S)∗(z) and the total segment density profiles
ρ
(C)∗
s (z) (panels a and b) for selected pore width, H∗ as an illustration of the behavior of the
solvation force shown in figure 5. The nomenclature of lines is given in the figures.
If the fluid particles are of larger diameter than the segments of chain connectors, the behavior
of the reduced solvation force differs from what was discussed concerning similar curves in figure 5.
Some examples of the solvation force for the model in which σ(S) = 1.6σ(C) are given in figure 7 (a).
The bulk fluid density is chosen to coincide with 0.7, if the diameters of segments and fluid particles
were equal. In this case we observe that the solvation force behaves similarly to the model without
13602-8
Entropic solvation force
fluid species, cf. figure 1, if the density of chain pillars is not high, e.g. R∗
C = 0.1 and R∗
C = 0.2.
At a higher density of chain connectors, R∗
C = 0.4, the attractive branch becomes just slightly
oscillatory indicating some sort of ordering of fluid species in wider pores. In the intermediate
pore, H∗ = 4, that cannot accommodate an integer number of fluid layers, the solvation force is
weakly attractive, and the distribution of species in such a pore can be imagined by analyzing the
density profiles shown in figure 7 (b).
0 2 4 6 8
H*
-4
-2
0
2
4
6
f*
ρb
* = 0.1
ρb
* = 0.1709
σ(S)=1.6σ(C)
RC
* =0.1
RC
* =0.2
RC
* =0.4
M=8
a
0 1 2 3 4
z*
0
0.2
0.4
0.6
ρ(S
)*
(z
),
ρ(C
)* s
(z
)
chain
fluid
RC
* =0.1
ρb
* = 0.1709
M=8
σ(S)=1.6σ(C)
b
Figure 7. (a) The reduced solvation force as a function of the density of chain connectors, R∗
C ,
at a fixed bulk fluid density, ρ∗b = 0.1709. The fluid particles are of larger diameter than chain
segments, σ(S) = 1.6σ(C). (b) The density profiles of fluid species ρ(S)∗(z) and the total segment
density profiles ρ
(C)∗
s (z) for H∗ = 4 as an illustration of the behavior of the solvation force
shown in (a).
0 2 4 6 8 10
H*
-4
-2
0
2
4
6
8
f*
0.8
1
1.6
RC
* =0.1
ρb
*= 0.7
M=10
σ(S)/σ(C)=
Figure 8. A comparison of the reduced solvation force for the models characterized by the
different ratio of diameters of fluid species and chain segments.
A summarizing insight into the effects of size ratio of segments and fluid particles at low density
of chain connectors with M = 10 and fixed reduced density of the bulk fluid, ρ∗
b , is presented in
figure 8. The fragment of the solvation force curves corresponding to rather narrow pores is very
sensitive to the type of fluid one adsorbs into the pores. Smaller species more easily develop a
layer-type structure which is manifested in the oscillations of the solvation force. Only for a set of
13602-9
O. Pizio, S. Sokołowski
specific pore widths one can find either repulsive maxima or attractive minima in the adsorption
of particular fluid species. Another important factor effecting the adsorption of a given fluid into
narrow pores is the bulk fluid density or equivalently the chosen value of the fluid chemical potential
in the bulk reservoir (or pressure), figure 9 (a). For wider pores, the value of an attractive force
hardly depends on fluid density and is predominantly determined by the density of chain connectors.
Thus, there exists a certain window of parameters of fluid density, chain pillars density and the
number of segments in chains that permit to manipulate adsorption and the solvation force most
efficiently for pores of a given width.
2 4 6 8 10
H*
-2
0
2
4
6
8
f*
0.2
0.4
0.6
0.7
RC
*=0.1
ρb
*=
M=10
σ(S)=σ(C)
a
5 10
H*
-2
0
2
4
6
8
10
f*
0.1
0.2
0.4
0.6
0.7
RC
*=0.1 RC
*=0.4 RC
*=0.6
ρb
*=
M=16
σ(S)=σ(C)
b
0 5 10
H*
-2
0
2
4
6
8
10
f*
σ(S)=0.8σ(C)
σ(S)=1.6σ(C)
RC
*=0.1
M=16
RC
*=0.4
c
Figure 9. (a) The reduced solvation force f∗(H∗), for the model pore at different density of
adsorbed fluid, ρ∗b , at a fixed density of chain connectors, R∗
C = 0.1, M = 10. The nomenclature
of lines is given in the figure. (b) The same as in (a), but for the model with M = 16. (c) The
same as in figure 8 but for the model with M = 16, ρ∗b = 0.7.
Our previous discussion has been focused on rather short chains M = 8 and M = 10. We have
not studied much longer chains so far. However, for the sake of better illustration and to confirm
our previous discussion, we give only two results concerning the effect of bulk fluid density and of
size ratio together with the effect of chain pillars density in figures 9 (b) and 9 (c) for chains with
sixteen segments, M = 16. These figures confirm our above mentioned conclusions and in some
sense provide estimates for the range of parameters that are necessary to be chosen in order to
effect the entropic solvation force between pore walls as desired.
A brief summary of this study is as follows. We have investigated the entropic solvation forces
13602-10
Entropic solvation force
induced by short polymers serving as connectors of the pore walls (pillars) via a density functional
approach. Our focus was on the origin of attraction/repulsion in these systems. The pillars could
not be broken, therefore at the wall-to-wall separation approaching the pillar length we observed
the development of a strong, diverging attraction. On the other hand, in narrow pores an overall
repulsion is observed. Both parts of the solvation force curve can be modulated in order to yield
an oscillatory behavior under a certain choice of the density of the pillars, their length (number
of segments), external pressure for desired adsorbing conditions for specific fluid systems. It seems
that there exists a window of the values of parameters permitting to manipulate the solvation
force most efficiently. Some possible extensions of the present study have been already noted in the
introduction. Nevertheless, it is important to mention that additional theoretical and computer
simulation work is necessary to clearly understand the complex systems of this study.
Acknowledgements
O.P. has been supported by the grant IN223808 of the National University of Mexico. S.S.
acknowledges partial support from the grant Number N N204 123737 of the Ministry of Science of
Poland.
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Ентропiйна сила сольватацiї мiж поверхнями,
модифiкованими розгалуженими ланцюжками: метод
функцiоналу густини
О. Пiзiо1, С. Соколовський2
1 Iнститут хiмiї УНАМ, Койокан 04510, Мексика
2 Унiверситет Марiї Склодовської-Кюрi, Люблiн, Польща
Поведiнка плину твердих сфер у щiлиноподiбних порах зi стiнками, модифiкованими розгалужени-
ми ланцюговими молекулами, якi складаються iз твердосферних сегментiв, дослiджується iз ви-
користанням теорiї функцiоналу густини. Ланцюги є розгалуженими до протилежних стiнок через
скiнченi сегменти, що формують опори колони. Детально дослiджено вплив обмеження i “хiмiчної”
модифiкацiї стiнок пори на ентропiйну силу сольватацiї. Ми спостерiгаємо, що при вiдсутностi ад-
сорбованого плину сила сольватацiї є сильно вiдштовхувальною для вузьких пор i притягальною для
широких пор. Коли адсорбований плин є присутнiм, обидвi частини кривої сили сольватацiї можуть
розвивати осциляцiйну поведiнку в залежностi вiд густини опор колон, числа сегментiв i умов ад-
собцiї. Також, розмiрний коефiцiєнт мiж частинками адсорбованого плину i сегментами ланцюгiв є
важливим для розвитку осциляцiй. Вибiр цих параметрiв є дуже суттєвим для ефективного керува-
ння силою сольватацiї для пор рiзної ширини.
Ключовi слова: метод функцiоналу густини, розгалуженi ланцюжки, сила сольватацiї, профiлi
густини
13602-12
Introduction
The model and the method
Results and discussion
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