A density functional study of the structure of tethered chains in a binary mixture
A density functional study of the structure of a layer formed by chain molecules pinned to a solid surface is presented. The chains are modeled as freely joined spheres. Segments and all components interact via Lennard-Jones (12-6) potential. The interactions of fluid molecules with the wall are des...
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| Zitieren: | A density functional study of the structure of tethered chains in a binary mixture / M. Borówko, T. Staszewski // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23603:1-12. — Бібліогр.: 57 назв. — англ. |
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| citation_txt | A density functional study of the structure of tethered chains in a binary mixture / M. Borówko, T. Staszewski // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23603:1-12. — Бібліогр.: 57 назв. — англ. |
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| description | A density functional study of the structure of a layer formed by chain molecules pinned to a solid surface is presented. The chains are modeled as freely joined spheres. Segments and all components interact via Lennard-Jones (12-6) potential. The interactions of fluid molecules with the wall are described by the Lennard-Jones (9-3) potential. We analyze how different parameters of the model affect the dependence of the brush height upon the mixture composition. We consider the effect of grafting density and the parameters characterizing the interactions of fluid molecules with the substrate and with the chains as well as interactions within the mixture. The changes in the brush height correlate with the adsorption of particular components.
Метод функцiоналу густини використовується з метою дослiдження структури шару, утвореного ланцюж-ковими молекулами, якi прикрiпленi до твердої поверхнi. Ланцюжки моделюються як вiльно зв’язанi сфе-ри. Сегменти ланцюжкiв i всi компоненти взаємодiють за допомогою потенцiалу Леннарда-Джонса (12–6).
Взаємодiї молекул плину зi стiнкою описуються потенцiалом Леннарда-Джонса (9–3). Ми аналiзуємо як рi-знi параметри моделi впливають на залежнiсть висоти щiтки вiд концентрацiї сумiшi. Ми розглядаємо вплив густини прищеплення i параметрiв, що характеризують взаємодiї молекул плину з пiдкладкою i з ланцюжками, як i з взаємодiями в сумiшi. Змiни висоти щiток корелюють з адсорбцiєю окремих компо-нент.
|
| first_indexed | 2025-12-07T13:16:23Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 2, 23603: 1–12
DOI: 10.5488/CMP.15.23603
http://www.icmp.lviv.ua/journal
A density functional study of the structure of
tethered chains in a binary mixture∗
M. Borówko, T. Staszewski
Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University,
20-031 Lublin, Poland
Received January 20, 2012, in final form March 20, 2012
A density functional study of the structure of a layer formed by chain molecules pinned to a solid surface is
presented. The chains are modeled as freely joined spheres. Segments and all components interact via Lennard-
Jones (12–6) potential. The interactions of fluid molecules with the wall are described by the Lennard-Jones
(9–3) potential. We analyze how different parameters of the model affect the dependence of the brush height
upon the mixture composition. We consider the effect of grafting density and the parameters characterizing the
interactions of fluid molecules with the substrate and with the chains as well as interactions within the mixture.
The changes in the brush height correlate with the adsorption of particular components.
Key words: brushes, adsorption, density functional theory
PACS: 68.47.Mn, 61.25.H, 68.47.Pe, 82.35.Gh
1. Introduction
The solid surfaces modified with tethered chains have been extensively studied in recent years due
to their importance for a wide range of technological processes, such as adhesion, colloidal stabilization,
lubrication, drug delivery, nanotechnology and chromatography [1–3]. The theoretical studies included
scaling theories [4, 5], self-consistent field methods [6–13], single chain mean-field methods [14–16], den-
sity functional theories [17–25] and computer simulations [26–41]. The structure of the polymer layer can
play a fundamental role in these processes.
When free segments of tethered chains are not very strongly attracted by the solid surface, the struc-
ture of an end-grafted chain can be a “mushroom” or a “brush”, depending on the conditions. In good
solvents, the grafting density controls the structure of a polymer film. At low grafting densities, the chains
are isolated and assume nearly unperturbed configuration. As the surface coverage increases, the chains
overlap and the interchain repulsion causes the chains to extend away from the wall. The configuration
switches in the brush-like structure. In a real system the structure of the bonded phase is the result of
a very complicated interplay between the entropic repulsion and attractive forces acting upon all the
molecules. The problem of a change of the configuration of grafted chains with their length, the grafting
density and the quality of solvent has been investigated [1, 3, 4, 41]. The structure of a polymer brush
immersed into a multicomponent solvent has been studied using an analytical self-consistent field the-
ory [10–13]. The properties of the brush were determined based on the phase diagram of a polymer
dissolved in a mixed solvent. A binary solvent with partially miscible components has been in the focus
of these investigations. There has been found the effect of selective solvation of the immersed chains and
big changes in their conformations. The studies have shown that the brush height depends on the solvent
content in a complicated way. However, this approach has some restricting assumptions. One of them is
the infinities of polymer chains. As a result, this theory does not describe the brush features on the scale
less than the brush height [12].
∗Dedicated to Prof. Orest Pizio on the occasion of his 60th birthday.
© M. Borówko, T. Staszewski, 2012 23603-1
http://dx.doi.org/10.5488/CMP.15.23603
http://www.icmp.lviv.ua/journal
M. Borówko, T. Staszewski
The surfaces modified with end-grafted chains are used as stationary phases in reversed-phase liq-
uid chromatography. The most popular chemically bonded-phase consists of relatively short chains. Nu-
merous studies have shown that the stationary phase takes an active part in chromatographic separa-
tions [3]. This process is affected by the chemical nature of the bonded chains, their length, the num-
ber of chains attached to the supporting surface and the composition of hydro-organic mobile phases.
The solvent-induced conformational changes of the bonded phases have been investigated using nuclear
magnetic resonance [42, 43], fluorescence [44], vibrational [45, 46] and Raman [47, 48] spectroscopic tech-
niques. Martire and Boehm [49] introduced the model of “breathing” surface in which the alkyl chains
are swollen by solvent penetration in the presence of nonpolar solvents and collapse toward the wall
in the presence of more polar solvents. The chromatographic stationary phases have been also studied
using the molecular dynamics [26–28, 38] and Monte Carlo method [32–36]. The computer simulations do
not support the hypothesis that the bonded chains collapse or elongate considerably with changes in the
liquid phase composition. Under the assumed conditions only more subtle solution-induced changes in
the structure of the bonded-phase were observed. However, these simulations have been performed for a
fewmodel systems only. The effective screening of the space of the model parameters that alter the struc-
ture of the bonded phase would require long-standing computer simulations. Such model calculations
can be efficiently carried out using the density theory.
In our previous paper [50] the effect of strong adsorption of one-component fluid on the height of the
bonded layer has been investigated. In this work we employ the density functional theory to describe the
short grafted chains immersed by a mixture of two liquids. The main goal of this study is to investigate
how the thickness of the bonded phase changes with the composition of the mixture. We consider the
solvents with different affinities to the solid surface and the grafted chains.
2. Theory
We consider a binarymixture of spherical molecules in contact with a solid surface covered by grafted
chains. We introduce the model described in our previous papers [22–24]. The chains are tangentially
joined M spherical segments of the same diameter,σc. The chain connectivity is enforced by the following
bonding potential
exp
[
−βVB(R)
]
=
M−1
∏
i=1
δ(|ri+1 −ri |−σc)/4π(σc)2, (1)
where Rk ≡ (r1,r2, · · · ,rM ) is the vector specifying positions of segments, the symbol δ denotes the Dirac
function, σc is the polymer segment diameter and β−1 = kBT . Each chain has a surface-binding segment
located at its end (indexed as i = 1) that is pinned to the wall by the following potential
exp
[
−βv
(c)
s1 (z)
]
=Cδ(z −σc/2), (2)
where z is a distance from the surface, C is a constant. The remaining segments of the tethered chains
(i = 2,3, · · · , M) are “neutral” with respect to the surface and they interact with the surface via the hard-
wall potential
vsi =
{
∞ z <σc/2,
0 otherwise.
(3)
The fluid molecules, however, interact with the surface via the Lennard-Jones (9–3) potential
vks(z)= 4εks
[
(z0/z)9
− (z0/z)3
]
, (4)
where εks characterizes the strength of interaction between the k-th component and the adsorbent (k =
1,2) and z0 =σk /2.
Interactions between segments and all fluid molecules are described by the Lennard-Jones (12–6)
potential
ukl (r ) =
{
4εkl
[
(σkl /r )12 − (σkl /r )6
]
r < r
(kl )
cut ,
0 otherwise,
(5)
23603-2
Tethered chains in a binary mixture
where σkl = 0.5(σk +σl ) and the parameter εkl characterizes the interactions between species k and l
(k = c,1,2), r
(kl )
cut = 5σ.
We split these interactions into the repulsive and attractive parts according to the Weeks-Chandler-
Anderson scheme [51]
u
(kl )
att (r )=
{
−εkl r < 21/6σkl ,
ukl (r ) r Ê 21/6σkl .
(6)
We define the total segment density of the grafted chains as follows:
ρ(c)
s (r) =
M
∑
i=1
ρ(c)
s,i
(r)=
M
∑
i=1
∫
dRδ(r−ri )ρ(c)(R) , (7)
where ρ(c)
s,i
(r) is the density of the individual i-th segment and ρ(c)(R) is the total density of grafted chains.
In the system the condition of constancy of the number of grafted molecules is fulfilled
(M+1/2)σc
∫
0
dzρ(c)
s,i
(z) = ρc , (8)
where ρc is the grafting density defined as ρc = Nc/A, Nc is the number of chains, A denotes the surface
area. For such a system the thermodynamic potential is given by
Y = F
[
ρ(c)(R),ρ1(r1),ρ2(r2)
]
+
∫
dRρ(c)(R)v
(c)(R)+
∑
k=1,2
∫
drkρk (rk )[vk (rk)−µk ], (9)
where µk is the chemical potential of the k-th component.
We describe the system in the framework of the density functional theory proposed by Yu andWu [52–
54]. In our previous papers we applied the theory to the study of adsorption on the bonded-phases [22–
24]. Since the details of computational method have been already described elsewhere we outline here
only most important steps of the calculations.
As usual, we split the free energy functional into the sum, F = Fid+Fhs+Fc+Fatt. The ideal part of the
free energy (Fid) is known exactly [55]. The hard sphere contribution (Fhs) is calculated according to the
fundamental measure theory of Rosenfeld [56]. The excess free energy due to the chain connectivity (Fc)
follows from the first order perturbation theory of Wertheim [57]. The attractive interactions between
molecules of the liquid mixture and their interactions with the chain segments are described using a
mean field approximation. One can find appreciable formulas in reference [25] (equations (7), (10), (12),
(14)). Next, we minimize the thermodynamic potential (9) under the constraint (8). As a result, we obtain
a set of Euler-Lagrange equations (equation (17) in [25]) and solve it numerically. This procedure provides
the density profiles of all species. In our model all densities depends only on the distance from the wall,
i.e, ρ(c)
s = ρ(c)
s (z) and ρk = ρk (z).
We calculate the brush height from the first moment of the total segment density profile [41]
h = 2
∫
dzzρ(c)
s (z)
∫
dzρ(c)
s (z)
. (10)
Next, we can evaluate the excess adsorption isotherms of particular components
Γk =
∫
[
ρk (z)−ρkb
]
dz (11)
and the relative excess adsorption isotherm defined as
N
e
k
=
∫
[xk (z)− xkb] dz, (12)
where xk = ρk /ρF is the local molfraction k-th component and xkb is its value in the bulk phase whereas
ρF = ρ1 +ρ2 is the total density of the fluid.
23603-3
M. Borówko, T. Staszewski
The brush height is usually scaled by the radius of gyration of the chainmolecules [41]. Unfortunately,
we cannot obtain the radius of gyration within the considered theory. Therefore, we used reduced units
which are defined below.
We have assumed that the segments of the chains and all spherical molecules have the same diame-
tersσk =σ, (k = c,1,2) and takeσ as the unit of length. The energy parameter characterizing interactions
betweenmolecules of the component 2 has been treated as the unit of energy, ε22 = ε. We have introduced
the usual reduced unit as ε∗
kl
= εkl /ε and ε∗
ks
= εks/ε, (k, l = c,1,2). Similarly, the reduced densities are
defined as ρ∗
k
(z) = ρk (z)σ3, ρ∗
kb
(z) = ρkb(z)σ3, ρ(c)∗
s,i
(z) = ρ(c)
s,i
(z)σ3, ρ(c)∗
s = ρ(c)
s σ3 and ρ∗
c = ρcσ
2. The
reduced temperature is defined as T
∗ = kT /ε.
3. Results and discussion
In our model the parameter ε∗2s characterizes the interactions of the k-th component with the solid
surface. The affinity of the k-th component to the grafted chains is altered by adjusting the value of the
parameter ε∗
kc
. The interactions in the bulk mixture are characterized by the parameters ε∗11, ε
∗
22 and
ε∗12. Depending on the values of these parameters the bulk mixture can undergo a demixing transition.
However, all the fluids considered here exhibit a complete mixing in the bulk phase.
The system in question depends on numerous parameters. In order to reduce the number of parame-
ters to a minimum, we assume that σkl =σ and ε∗22 = ε∗cc = ε∗2s = 1. All calculations have been carried out
at T
∗ = 1 and for the fluid bulk density ρ∗
F
= 0.66. This density nearly coincides with the liquid density on
the liquid coexistence curve at T
∗ = 1.
The goal of the study is to estimate the effect of the selected parameters on the height of the brush.We
have focused our attention on the role of the following factors: (i) the interactions of the both components
with the bare surface (the surface effects), (ii) the interactions of the fluid molecules with segments of the
grafted chains and (iii) the interactions in a bulk mixture (the solvent effects).
Figure 1. The dependence of the brush height on the grafting density of tethered 18-mers for two values
of the molfraction: x1b = 0.2 (solid lines) and x1b = 0.8 (dashed lines). In all cases ε∗1s = 15, the parameter
characterizing interactions with the chains ε∗1c = 1 (part (a)) and ε∗1c = 1.1 (part (b)).
In figure 1 we demonstrate the effect of the selected parameters of the model on the relationship
of the logarithm of the brush height upon the logarithm of the grafting density. The curves shown in
part (a) have been obtained for a simple “adsorption system”, for which interactions of molecules of the
both components in the liquid and their interactions with the brush are identical, ε∗
kl
= ε∗
kc
= 1,(k, l =
c,1,2) whereas the strengths of interactions with the wall are different and the energy parameters are
23603-4
Tethered chains in a binary mixture
equal to ε∗1s = 15 and ε∗1s = 1. In this system the driving forces for adsorption are the interactions of fluid
molecules with the underlying surface. In part (b) we present the relationships ln(h) vs ln(ρ∗
c ) for the
model systems in which the first component has additionally a high affinity to the grafted chains, ε∗1c =
1.1. The remaining parameters are the same as in part (a). In the both figures we show the log-log plots
of the brush height as a function of the grafting density calculated for different mixture compositions,
x1b = 0.2 (solid lines) and x1b = 0.8 (dashed lines). All curves have the same shapes. When the number
of tethered chains is small, i.e. at low surface coverages, the brush height is almost unaffected by the
grafting density. Under such conditions the grafted chains do not practically effect one another and they
assume unperturbed configurations. For mediate-covered surfaces a minimum on the curve is observed
for certain parameters (see, the solid line in part (b)). With a further increase of the grafting density
the brush height increases markedly. In this region the logarithm of the brush height changes with lnρ∗
c
almost linearly. A similar behavior was observed in computer simulations [38].
One sees in the figure 1 that the composition of the liquid mixture strongly affects the brush height
at low surface coverages. The brush height increases with an increase of the molfraction x1b. The same
effect gives an increase of the parameter εc1 (cf., dashed lines in parts (a) and (b)). Accumulation of the
fluid molecules close to the wall causes an increase in the brush height because the liquid molecules
compete for space inside the bonded-phases and the chains are pushed away from the surface.
In the previous paper [50] we have studied the changes in the structure of tethered chains in contact
with a one-component fluid. We have shown that in the region of mediate grafting coverages, a minimum
on the curve ln(h) vs ln(ρ∗
c ) can develop. For a strongly attractive surface this minimum disappeared.
Theminimum also vanishes when segment-segment attraction is lowered or when temperature is raised.
The similar effects are observed for the grafted chains immersed by the liquid mixture. In part (b) one
sees the minimum on the curve corresponding to the low molfraction of the first component (solid line).
However, when the concentration of the first component in the bulk mixture increases, the minimum
disappears. In this case molecules of the first component are strongly attracted by the substrate as well
as by the grafted chains. Thus, we observe more molecules 1 in the mixture, higher accumulation of the
fluid molecules in the surface layer which inhibits the chain coiling.
Figure 2. The dependence of the brush height on
the molfraction of the first mixture component for
different grafting densities ρ∗c : 0.05 (double dotted-
dashed line), 0.1 (dot-dashed line), 0.15 (dotted line),
0.2 (dashed line), 0.25 (solid line). The grafted chains
are 18-mers. The calculations were carried out for
ε∗1s = 15 and ε∗1c = 1.
At high grafting densities the following rela-
tion is fulfilled
ln(h) ∼ γ ln(ρ∗
c ). (13)
We have found that exponent γ depends on
the mixture composition and on the parame-
ters characterizing the system. For the “adsorp-
tion” system (part (a)) we have obtained γ(x =
0.2) = 0.26 and γ(x = 0.8) = 0.24. However, for a
stronger attractive interaction of the first compo-
nent with the chains (part (b)) the exponents are
γ(x = 0.2) = 0.48, γ(x = 0.8) = 0.21
Figure 2 illustrates how the brush height
changes with the mixture composition for se-
lected grafting densities. The remaining system
parameters are the same as in figure 1. For suf-
ficiently dense bonded-phases the brush height is
a monotonically increasing function of the mole
fraction x1b in the whole concentration region.
When x1b > 0.15 the brush height is almost a lin-
ear function of the mixture composition. An inter-
esting feature is observed for low concentrations
of the first component. For high grafting densities
the brush height rapidly increases. The opposite
effect is observed at low surface coverages. After
adding a small amount of the first component to a
23603-5
M. Borówko, T. Staszewski
pure component 2, the brush height decreases until a minimum value is attained. A further increase of
the molfraction x1b causes an increase of the brush height. The analysis of density profiles in a pure fluid
2 and in a mixture allows us to understand this effect (see figure 3).
Figure 3. The segment density profiles and the density profile of the mixture components. Part (a) is for
the brush in contact with the pure component 2 (x1b = 0) and two grafting densities ρ∗c : 0.05 (the segment
density profile – solid line, the fluid density profile – dot-dashed line) and 0.2 (the segment density profile
– dashed line, the fluid density profile – dotted line). Part (b) is for the mixture, in which x1b = 0.0667
and the grafting density ρ∗c = 0.05 . The density profiles are plotted for: the brush (solid line), the first
component (dashed line), the second component (dotted line) and for the total density of the fluid (dot-
dashed line). In the inset the density profiles of end-segments are shown for x1b = 0 (dashed line) and
x1b = 0.0667 (solid line). The calculations were carried out for 18-mers and ε∗1s
= 15 ε∗1c
= 1.
Figure 3 (a) shows an effect of grafting density on the structure of the surface layer in a pure compo-
nent 2 that weakly interacts with the solid surface (ε∗2s = 1). We start with a comparison of the segment
density profiles of the grafted chains calculated at two grafting densities ρ∗
c = 0.05 and ρ∗
c = 0.2. At very
low density, ρ∗
c = 0.05, only two local peaks are observed on the segment density profile. However, for
a denser bonded-phase there are four well-pronounced peaks, which correspond to successive layers of
polymer segments. An extended region, where the segment density smoothly tends to zero, follows these
peaks. For a higher grafting density the chains are more stretched, so the brush height is greater than
that for a low grafting density. Also, density profiles of the fluid in the considered systems are different.
In the both cases, depletion in the fluid density is observed in the surface layer. At low grafting density,
the fluid density gradually decreases when the distance from the wall decreases. There is a local mini-
mum close to the surface where the minimum fluid density is much lower than the bulk fluid density. A
distinct behavior is observed for a higher coverage ρ∗
c = 0.2. There are three peaks at local density of the
fluid. Their positions are correlated with the positions of the peaks at the segment density profiles of the
grafted chains. The height of the local density peak adjacent to the wall is lower than the bulk density of
the fluid. The grafted chains attract molecules of the fluid. However, the fluid density in the middle part
of the surface layer is considerably lower for the denser bonded-phase. On the contrary, close to the wall
the fluid density for a low grafting density is lower than that for a dense brush. The grafted chains are
always a space barrier for the fluid molecules. However, in the dense bonded-phase there are “tunnels”
between the stretched chains, and fluid molecules can flow through them to the wall. Due to the balance
between attractive interactions and the entropic repulsion, the primary adsorption, i.e., adsorption near
the surface is preferred.
Now, we analyze the structure of the surface layer for ρ∗
c = 0.05 after adding a small amount of the
component 1 to the pure component 2. The density profiles presented in Figure 3b are calculated at the
molfraction of the first component x1b = 0.0667, which corresponds to the minimum brush height (see
double dot-dashed line in figure 2). The density profile of the second component is almost the same as
23603-6
Tethered chains in a binary mixture
that in the pure fluid 2. However, the presence of the molecules 1 changes the total density profile of
the fluid. The molecules 1 are strongly attracted by the solid surface. For this reason, close to the wall
the density of the first component is markedly greater than in the bulk mixture. As a consequence, also
the total density of the fluid is high in this region, whereas the fluid density far away from the surface
only slightly differs from the bulk value. Notice that at such a surface coverage the tethered chains are
isolated from one another but the molecules 1, which have been adsorbed near the wall, attract them.
This causes the chain segments to cumulate somewhat closer the surface and we observe a decrease of
the brush height. The results shown in the inset confirm this conclusion. We present here the density
profile of the end-segment of the grafted chains for x1b = 0 and x1b = 0.0667. One sees that the end-
segment density close to the surface is lower for x1b = 0. This means that the chains are less coiled in
the pure component 2 than in the considered mixture. Moreover, the end-segment density profile has a
maximum at the distance from the surface at the points z = 4.80 and z = 4.45, respectively for x1b = 0 and
x1b = 0.0667. The situation changes when concentration of the first component increases. Then, the fluid
density considerably increases in the vicinity of the surface. The fluid molecules start to compete with the
grafted chains for place in the surface layer. The entropic repulsion prevails the attraction between fluid
molecules and chain segments. The chains are pushed away from the surface by fluid molecules and the
brush height increases.
Figure 4. The dependence of the brush height on the
molfraction of the first mixture component for dif-
ferent values of the parameter ε∗1c characterizing
the interactions of the molecules 1 with the brush.
From the top: 1.13, 1.1, 1.05, 1.0, 0.95, 0.9, 0.8, 0.7.
The grafted chains are 18-mers. The calculations are
carried out for ε∗1s = 1.0 and ρ∗c = 0.1.
Next, we discuss an impact of the solvent na-
ture on the structure of the surface layer. In this
order we have carried out the calculations assum-
ing that interactions of the both components with
the surface are the same and rather weak, namely
ε∗1s = ε∗2s = 1. First, we have studied the effects of
the brush-fluid interactions. We have varied the
parameter ε∗1c and kept the remaining parameters
fixed ε∗2c = ε∗
kl
= 1 for k, l = 1,2. These results are
presented in figure 4. One sees here that when
the component 1 is a better solvent for the chains
than the component 2, ε∗1c > ε∗2c, the brush height
monotonically increases with the molfraction x1b.
However, if the component 1 is a worse solvent
than the component 2 (ε∗1c < ε∗2c) the brush height
decreases with an increase of the concentration of
the first component. Of course, under the assumed
conditions, for ε∗1c = ε∗2c the brush height is inde-
pendent of the fluid composition.
We have also investigated the effect of interac-
tions in the liquid mixture on the brush structure.
We have assumed that ε∗1s = ε∗
ks
= ε∗
kc
= ε∗22 = 1,
(k, l = 1,2) and we have varied the parameters ε∗11
and ε∗12.
We start with the analysis of the structure of
the brush immersed by a pure component 1 (see
figure 5). Initially, i.e., for low values of ε∗11, the
brush height decreases with an increase of the parameter ε∗11 to a certain minimal value. Then, the brush
height rapidly increases (figure 5 (a)). These effects can be explained as follows. For ε∗11 < 1 the interac-
tions in the bulk fluid are weaker than the interactions of fluid molecules with the brush segments and
with the underlying substrate. Therefore, the fluid molecules penetrate the brush and inhibit the coiling
of the chains. Thus, the lower is ε∗11 the higher is the brush. When ε∗11 > 1, the interactions in the bulk
fluid are stronger than those with a modified adsorbent, and the fluid molecules are sucked from the sur-
face layer. In figure 1 (b) we display the density profiles of the fluid 1 for different values of the energy
parameters ε∗11. When ε∗11 < 1, the fluid molecules penetrate the brush to a great extent. The correspond-
ing density profiles have two well-pronounced peaks near the surface. A decrease of the parameter ε∗11
causes an accumulation of fluidmolecules in the vicinity of the surface. That is why the chains are pushed
23603-7
M. Borówko, T. Staszewski
Figure 5. Part (a) shows the dependence of the brush height on the parameter ε∗11 characterizing 11-
interactions. Part (b) presents the density profiles of the fluid for selected values of the parameter ε∗11:
0.8 (solid line), 0.9 (dashed line), 1.0 (dotted line), 1.1 (dot-double dashed line), 1.15 (dot-dashed line), 1.18
(double dotted-dashed line). The calculations were carried out for the pure component 1 and M = 18,
ε∗1s = 1.0 and ρ∗c = 0.1.
from the surface and the brush height increases. A situation changes for ε∗11 > 1. In this case, depletion
in the fluid density is observed close to the wall. The “pushing effect” gradually diminishes and the mini-
mum height of the brush is attained for ε∗11 ≈ 1.10. A different phenomenon is observed for higher values
of the parameter ε∗11. Namely, the fluid molecules located on top of the brush attract the chains and pull
them in the direction of the bulk phase. This leads to the brush expansion.
Figure 6. The dependence of the brush height on the molfraction of the first mixture component for
different values of the parameter ε∗11 characterizing interactions of the molecules: 0.8 (solid line), 0.9
(dashed line), 1.0 (dotted line), 1.1 (dot-double dashed line), 1.15 (dot-dashed line), 1.18 (double dotted-
dashed line). In the inset the corresponding relative excess isotherms of the first component are shown.
The calculations were carried out for ε∗12 =
√
ε∗
11
ε∗
22
, M = 18, ε∗1s = 1.0 ρ∗c = 0.1.
23603-8
Tethered chains in a binary mixture
Figure 7. Part (a) – The dependence of the brush
height on the molfraction of the first mixture com-
ponent for different values of the parameters ε∗1c
and ε∗11. The presented results correspond to: ε∗1c
=
1.0 and ε∗11 = 0.9 (dashed line), 0.95 (short dashed
line), 1.0 (dotted line), 1.05 (dot-short dashed line),
1.1 (solid line); ε∗1c
= 1.1 and ε∗11 = 0.9 (dot-
double dashed line), 1.0 (dot-dashed line), 1.1 (dou-
ble dotted-dashed line). Part (b) – The relative ex-
cess adsorption isotherms of the first component for
ε∗1c
= 1.0. The nomenclature of the lines is the same
as in the part (a). All calculation were carried out for
M = 18, ε∗1s
= 1.0, ε∗12 = 1.0 and ρ∗c = 0.1.
The results obtained for the model mixtures
are shown in figures 6–8. We begin with the dis-
cussion of the behavior of the mixtures for which
the Berthelot rule is fulfilled, i.e., when ε∗12 =
√
ε∗11ε
∗
22 (see figure 6). In this case, at a given mol-
fraction x1b, the parameter ε∗
1b
, characterizing in-
teractions between molecules of the component,
is only a factor that controls the system behav-
ior. For ε∗11 < 1, the brush height is an increas-
ing function of the molfraction of the first com-
ponent. In such systems the interactions of the
fluid molecules with the solid surface and with
the grafted chains are stronger than the interac-
tions between molecules in the bulk mixture. The
tethered chain is much more extended in the first
component than in the second. The brush height
is a monotonically increasing function of the mol-
fraction x1b. If ε
∗
11 = 1, the brush height does not
depend upon the mixture composition. The effect
of the parameter ε∗11 on the brush height is more
complicated for ε∗11 > 1. When the 11-interactions
are slightly stronger than the interactions with the
bare surface and with segments of grafted chains,
the brush height decreases with an increase of the
molfraction x1b. However, for ε∗11 = 1.10 and ε∗11 =
1.18 there is a minimum on the curve h vs x1b.
At high concentrations of the first component, the
brush height rapidly increases. Such systems be-
have similarly to the grafted chains in contact
with the pure component 1 described above.
The interactions in the mixture can be also
characterized by the exchange parameter ω∗
12 =
ε∗12 −0.5
(
ε∗11 +ε∗22
)
. In the studied systems ω∗
12 < 0
for ε∗11 < 1 and ω∗
12 > 0 for ε∗11 > 1. In the inset we
show the relative excess isotherms corresponding
to the curves h vs x1b. We see that the relative ex-
cess adsorption of the first component is positive for ω∗
12 < 0 and negative for ω∗
12 > 0.
We have also considered the model mixtures for which the parameters ε∗11 and ε∗12 can be changed
independently. This allows us to separate the effects connected with interactions between the same
molecules (ε∗11) and between unlike molecules (ε∗12) in the mixture. First, we have calculated the brush
height for ε∗12 = 1, two values of the parameter ε∗1c = 1 and ε∗1c = 1.1 and different values of the parameter
ε∗11. In the both cases, if ε∗11 < 1, the brush height increases with an increase of the concentration of the
first component in the mixture and the opposite effect is observed for ε∗11 > 1 (figure 7 (a)). Notice that the
affinity of the first component to the grafted chains ε∗c1 affects the brush height more strongly than the
parameter ε∗11. The relative excess isotherms corresponding to the curves h vs x1b calculated for ε∗c1 = 1
are shown in part (b). These results are analogous to those presented in figure 6.
Now, we discuss the results obtained for symmetrical mixtures. We have assumed that ε∗11 = ε∗22 and
varied the parameter ε∗12. The selected results are shown in figure 8. In this case the plots h vs x1b have
regular shapes, namely, there is one extreme at x1b = 0.5 on each curve. This is a maximum for ε∗12 < 1
(ω∗
12 < 0) and a minimum for ε∗12 > 1 (ω∗
12 > 0) (see figure 8 (a)). The relative excess isotherm N
e
1 has an
azeotropic point, i.e., the point at which the relative adsorption isotherm is equal to zero and 0 < xaz < 1,
just at x1b = 0.5. The relative excess adsorption isotherm changes the sign at the azeotropic point. When
x1b < xaz, the relative excess isotherm N
e
1 is positive for ω∗
12 < 0 and negative for ω∗
12 > 0. In the case
23603-9
M. Borówko, T. Staszewski
of stronger deviation of the exchange parameter from ω∗
12 = 0 the shapes of the curves h vs x1b can be
different. This problem will be a subject of investigations in the future.
Figure 8. Part (a) — The dependence of the brush
height on the molfraction of the first mixture com-
ponent for different values of the parameter ε∗12: 0.9
(solid line), 0.95 (dashed line), 1.0 (dotted line), 1.05
(dot-dashed line), 1.1 (double dotted-dashed line).
Part (b) — The relative excess adsorption isotherms
of the first component. The nomenclature of the
lines is the same as in the part (a). All calculations
were carried out for M = 18, ε∗1s = 1.0, ε∗11 = 1.0 and
ρ∗c = 0.1.
In summary, the structure of the surface layer
varies considerably with the change of the com-
position of the mixture. The grafted chains can
be more or less extended in the mixtures of dif-
ferent compositions. The curves describing the
brush height versus the molfraction of a given
component have various shapes depending on
the relation between the parameters characteriz-
ing the system. When the components have dif-
ferent affinities to the underlying surface and
to the grafted chains, the mixture composition
strongly affects the brush height. We conclude
that stronger interactions of the fluid molecules
with the wall cause an increase of the brush
height. The same trend is observed in the case
of the interactions with the segments of grafted
chains. The nature of liquid mixture also plays
a significant role in the process of coiling of the
grafted chains. Interactions between molecules of
particular components of the mixture effect their
adsorption on the bonded-phase, which, in turn,
affects the height of the brush. The different de-
pendences of the brush height on the mixture
composition are observed for different sets of pa-
rameters of the model. The grafting density is an
important parameter that can decide about the
brush height. We have analyzed the dependence
of the brush height on the grafting density for
different mixtures. We have found that the scal-
ing relation describing the changes in the brush
height with the changes of the grafting density
is approximately satisfied within the region of
stretched chains and the exponent γ depends on
the type of mixture and its compositions. Our find-
ings are qualitatively consistent with the computer simulation carried out for selected systems [32, 33, 41].
Unfortunately, there are no systematic simulation studies of the effect of mixture composition on the
structure of the layer of the grafted chains. The problem of comparing the theoretical predictions with
the simulation data requires further investigations.
Acknowledgements
This work is supported by the Ministry of Science of Poland under the Grant No. N N204151237 and
by EC under the Grant No. PIRES–GA2010–268498 (MB).
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Дослiдження методом функцiоналу густини структури
прив’язаних ланцюжкiв у бiнарнiй сумiшi
М. Борувко, T. Сташевскi
Вiддiл моделювання фiзико-хiмiчних процесiв, унiверситет Марiї Кюрi-Склодовської,
20031 Люблiн, Польща
Метод функцiоналу густини використовується з метою дослiдження структури шару, утвореного ланцюж-
ковими молекулами, якi прикрiпленi до твердої поверхнi. Ланцюжки моделюються як вiльно зв’язанi сфе-
ри. Сегменти ланцюжкiв i всi компоненти взаємодiють за допомогою потенцiалу Леннарда-Джонса (12–6).
Взаємодiї молекул плину зi стiнкою описуються потенцiалом Леннарда-Джонса (9–3). Ми аналiзуємо як рi-
знi параметри моделi впливають на залежнiсть висоти щiтки вiд концентрацiї сумiшi. Ми розглядаємо
вплив густини прищеплення i параметрiв, що характеризують взаємодiї молекул плину з пiдкладкою i з
ланцюжками, як i з взаємодiями в сумiшi. Змiни висоти щiток корелюють з адсорбцiєю окремих компо-
нент.
Ключовi слова: щiтки, адсорбцiя, теорiя функцiоналу густини
23603-12
http://dx.doi.org/10.1103/PhysRevLett.63.980
http://dx.doi.org/10.1063/1.453326
Introduction
Theory
Results and discussion
|
| id | nasplib_isofts_kiev_ua-123456789-120280 |
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| issn | 1607-324X |
| language | English |
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| publishDate | 2012 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Borówko, M. Staszewski, T. 2017-06-11T14:31:41Z 2017-06-11T14:31:41Z 2012 A density functional study of the structure of tethered chains in a binary mixture / M. Borówko, T. Staszewski // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23603:1-12. — Бібліогр.: 57 назв. — англ. 1607-324X PACS: 68.47.Mn, 61.25.H, 68.47.Pe, 82.35.Gh DOI:10.5488/CMP.15.23603 arXiv:1207.3264 https://nasplib.isofts.kiev.ua/handle/123456789/120280 A density functional study of the structure of a layer formed by chain molecules pinned to a solid surface is presented. The chains are modeled as freely joined spheres. Segments and all components interact via Lennard-Jones (12-6) potential. The interactions of fluid molecules with the wall are described by the Lennard-Jones (9-3) potential. We analyze how different parameters of the model affect the dependence of the brush height upon the mixture composition. We consider the effect of grafting density and the parameters characterizing the interactions of fluid molecules with the substrate and with the chains as well as interactions within the mixture. The changes in the brush height correlate with the adsorption of particular components. Метод функцiоналу густини використовується з метою дослiдження структури шару, утвореного ланцюж-ковими молекулами, якi прикрiпленi до твердої поверхнi. Ланцюжки моделюються як вiльно зв’язанi сфе-ри. Сегменти ланцюжкiв i всi компоненти взаємодiють за допомогою потенцiалу Леннарда-Джонса (12–6). Взаємодiї молекул плину зi стiнкою описуються потенцiалом Леннарда-Джонса (9–3). Ми аналiзуємо як рi-знi параметри моделi впливають на залежнiсть висоти щiтки вiд концентрацiї сумiшi. Ми розглядаємо вплив густини прищеплення i параметрiв, що характеризують взаємодiї молекул плину з пiдкладкою i з ланцюжками, як i з взаємодiями в сумiшi. Змiни висоти щiток корелюють з адсорбцiєю окремих компо-нент. This work is supported by the Ministry of Science of Poland under the Grant No. N N204151237 and by EC under the Grant No. PIRES–GA2010–268498 (MB). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics A density functional study of the structure of tethered chains in a binary mixture Дослiдження методом функцiоналу густини структури прив’язаних ланцюжкiв у бiнарнiй сумiшi Article published earlier |
| spellingShingle | A density functional study of the structure of tethered chains in a binary mixture Borówko, M. Staszewski, T. |
| title | A density functional study of the structure of tethered chains in a binary mixture |
| title_alt | Дослiдження методом функцiоналу густини структури прив’язаних ланцюжкiв у бiнарнiй сумiшi |
| title_full | A density functional study of the structure of tethered chains in a binary mixture |
| title_fullStr | A density functional study of the structure of tethered chains in a binary mixture |
| title_full_unstemmed | A density functional study of the structure of tethered chains in a binary mixture |
| title_short | A density functional study of the structure of tethered chains in a binary mixture |
| title_sort | density functional study of the structure of tethered chains in a binary mixture |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120280 |
| work_keys_str_mv | AT borowkom adensityfunctionalstudyofthestructureoftetheredchainsinabinarymixture AT staszewskit adensityfunctionalstudyofthestructureoftetheredchainsinabinarymixture AT borowkom doslidžennâmetodomfunkcionalugustinistrukturiprivâzanihlancûžkivubinarniisumiši AT staszewskit doslidžennâmetodomfunkcionalugustinistrukturiprivâzanihlancûžkivubinarniisumiši AT borowkom densityfunctionalstudyofthestructureoftetheredchainsinabinarymixture AT staszewskit densityfunctionalstudyofthestructureoftetheredchainsinabinarymixture |