Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes
We study self-assembly of a binary mixture of components A and B confined in a slit-like pore with the walls modified by the stripes of tethered brushes made of beads of a sort A. The emphasis is on solvent mediated transitions between morphologies when the composition of the mixture varies. For cer...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes / J.M. Ilnytskyi, S.Sokolowski, T. Patsahan // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13606:1–14. — Бібліогр.: 53 назв. — англ. |
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irk-123456789-1210722017-06-14T03:05:25Z Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes Ilnytskyi, J.M. Sokolowski, S. Patsahan, T. We study self-assembly of a binary mixture of components A and B confined in a slit-like pore with the walls modified by the stripes of tethered brushes made of beads of a sort A. The emphasis is on solvent mediated transitions between morphologies when the composition of the mixture varies. For certain limiting cases of the pore geometry we found that an effective reduction of the dimensionality may lead to a quasi one- and two-dimensional demixing. The change of the environment for the chains upon changing the composition of the mixture from polymer melt to a good solvent conditions provides explanation for the mechanism of development of several solvent mediated morphologies and, in some cases, for switching between them. We found solvent mediated lamellar, meander and in-lined cylinder phases. Quantitative analysis of morphology structure is performed considering brush overlap integrals and gyration tensor components. Розглянуто процес самоформування фаз у порi iз стiнками декорованими полосками полiмерної щiтки (з мономерiв сорту А), яка заповнена бiнарною сумiшшю iз компонент А i В. Акцент зроблено на дослiдженнi ролi розчинника при фазових переходах спричинених змiною складу сумiшi. Знайдено граничнi випадки квазиодновимiрного та квазидвовимiрного незмiшування. Показано, що механiзм формування морфологiй (та, в деяких випадках, переключення мiж ними) при змiнi складу сумiшi полягає у змiнi локального середовища для ланцюжкiв полiмерних щiток. Знайдено сформованi за посередництва розчинника ламеларнi, меандро-подiбнi та порядковi цилiндричнi фази. Структури вивчено за допомогою аналiзу iнтегралiв перекриття щiток та величин компонент тензора гiрацiї. 2013 Article Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes / J.M. Ilnytskyi, S.Sokolowski, T. Patsahan // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13606:1–14. — Бібліогр.: 53 назв. — англ. 1607-324X PACS: 62.23.St, 36.20.Ey, 61.20.Ja DOI:10.5488/CMP.16.13606 arXiv:1303.6061 http://dspace.nbuv.gov.ua/handle/123456789/121072 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study self-assembly of a binary mixture of components A and B confined in a slit-like pore with the walls modified by the stripes of tethered brushes made of beads of a sort A. The emphasis is on solvent mediated transitions between morphologies when the composition of the mixture varies. For certain limiting cases of the pore geometry we found that an effective reduction of the dimensionality may lead to a quasi one- and two-dimensional demixing. The change of the environment for the chains upon changing the composition of the mixture from polymer melt to a good solvent conditions provides explanation for the mechanism of development of several solvent mediated morphologies and, in some cases, for switching between them. We found solvent mediated lamellar, meander and in-lined cylinder phases. Quantitative analysis of morphology structure is performed considering brush overlap integrals and gyration tensor components. |
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Article |
| author |
Ilnytskyi, J.M. Sokolowski, S. Patsahan, T. |
| spellingShingle |
Ilnytskyi, J.M. Sokolowski, S. Patsahan, T. Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes Condensed Matter Physics |
| author_facet |
Ilnytskyi, J.M. Sokolowski, S. Patsahan, T. |
| author_sort |
Ilnytskyi, J.M. |
| title |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
| title_short |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
| title_full |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
| title_fullStr |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
| title_full_unstemmed |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
| title_sort |
dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/121072 |
| citation_txt |
Dissipative particle dynamics study of solvent mediated transitions in pores decorated with tethered polymer brushes in the form of stripes / J.M. Ilnytskyi, S.Sokolowski, T. Patsahan // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13606:1–14. — Бібліогр.: 53 назв. — англ. |
| series |
Condensed Matter Physics |
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AT ilnytskyijm dissipativeparticledynamicsstudyofsolventmediatedtransitionsinporesdecoratedwithtetheredpolymerbrushesintheformofstripes AT sokolowskis dissipativeparticledynamicsstudyofsolventmediatedtransitionsinporesdecoratedwithtetheredpolymerbrushesintheformofstripes AT patsahant dissipativeparticledynamicsstudyofsolventmediatedtransitionsinporesdecoratedwithtetheredpolymerbrushesintheformofstripes |
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2025-07-08T19:08:48Z |
| last_indexed |
2025-07-08T19:08:48Z |
| _version_ |
1837106965696741376 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 1, 13606: 1–14
DOI: 10.5488/CMP.16.13606
http://www.icmp.lviv.ua/journal
Dissipative particle dynamics study of solvent
mediated transitions in pores decorated with
tethered polymer brushes in the form of stripes
Ja.M. Ilnytskyi1, S. Sokołowski2 , T. Patsahan1
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Department for the Modelling of Physico-Chemical Processes, Maria Curie-Skłodowska University,
20031 Lublin, Poland
Received October 29, 2012, in final form February 1, 2013
We study self-assembly of a binary mixture of components A and B confined in a slit-like pore with the walls
modified by the stripes of tethered brushes made of beads of a sort A. The emphasis is on solvent mediated
transitions between morphologies when the composition of the mixture varies. For certain limiting cases of
the pore geometry we found that an effective reduction of the dimensionality may lead to a quasi one- and
two-dimensional demixing. The change of the environment for the chains upon changing the composition of
the mixture from polymer melt to a good solvent conditions provides explanation for the mechanism of devel-
opment of several solvent mediated morphologies and, in some cases, for switching between them. We found
solvent mediated lamellar, meander and in-lined cylinder phases. Quantitative analysis of morphology structure
is performed considering brush overlap integrals and gyration tensor components.
Key words: dissipative particle dynamics, mixtures, pores, nanostructures
PACS: 62.23.St, 36.20.Ey, 61.20.Ja
1. Introduction
In recent years investigations of polymer films on solid surfaces have become one of the most rapidly
growing research area in physics, chemistry, and material science. The reason for such sustained growth
is due to the availability of a wealth of fundamentally interesting information in thermodynamics and
kinetics, such as long and short range forces, interfacial interactions, flow, and instability phenomena
[1–8]. Moreover, polymer thin films are widely used as an industrial commodity in coatings and lubri-
cants and have become an integral part of the development process in modern hi-tech applications such
as optoelectronics, biotechnology, nanolithography, novel sensors and actuators [9–16]. Most of these ap-
plications have been connected with an intrinsic property of polymer films to exhibit a variety of surface
morphologies, the size of which ranges from a few tens of nanometers to hundreds of micrometers. Since
many of the modern technologically relevant phenomena occur at the nanoscale, the behavior of poly-
mer thin films deposited on a flat surface that exhibit morphologies at the nanoscale has been one of
major focuses in recent years [6, 8, 17–22].
The development of several new techniques [23] in material science permits now the productions
of solid substrates whose surface is “decorated” with precisely characterized surface structures on the
length scales ranging from nanometers to microns. In particular, advances in nanotechnology have per-
mitted the establishment of methods for obtaining functional polymeric films on solid surfaces exhibiting
quite complex topographic nanostructures. Such chemically decorated substrates allow for manipulation
of fluid at very short length scales and can play an important role in a variety of contexts [24–30].
© Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan, 2013 13606-1
http://dx.doi.org/10.5488/CMP.16.13606
http://www.icmp.lviv.ua/journal
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
The importance of systems involving brushes tethered at structured surfaces urged the develop-
ment of methods for theoretical description of such systems. Theoretical studies of fluids in contact with
brushes on patterned surfaces have been mainly based on different simulation methods. They include
Monte Carlo [31–35] and molecular dynamics [36, 37], as well as dissipative particle dynamics (DPD) sim-
ulations [17, 38–43]. The studies performed so far indicated that heterogeneity of tethered layers has a
great impact on the structure of the confined fluid and on thermodynamic and dynamic properties of the
systems.
In numerous previous studies [38–43], including our work [44], the simulations have been carried
out assuming constant composition in the confined system. However, the demixing phenomena and the
formation of different morphologies strongly depend on the composition [45]. Thus, the aim of this work
is to determine the effect of the change in the fluid composition on the structure of a confined system.
Similarly to the previous work [44], we use DPD to investigate the behavior of a binarymixture composed
of beads A and B confined in slit-like pores with walls modified by the stripes of tethered chains made
of beads A. The stripes at the opposing pore walls are placed “in-phase” (face-to-face), or “out-of-phase”.
Species A and B are assumed to exhibit a demixing in a bulk phase. Simulations are carried out for dif-
ferent compositions of the fluid. Our interest is to determine possible morphologies that can be formed
inside the pore depending on the fluid composition as well as on the geometrical parameters charac-
terizing the system (the size of the pore and the width of the stripes, the arrangement of the stripes).
In particular, we analyze special limiting cases, where geometry of the pore leads to the reduction of
the effective dimensionality. Special emphasis is made on the cases of stripes between which the sepa-
ration distance, either within the pore wall or across the pore, is small. The crossover for the polymer
chains from the regime of polymer melt to the regime of a good solution provides a basis for solvent
mediated morphology formation and morphology switching. We performed complementary simulations
where the solvent in a form of a binary mixture is replaced by a one-component solvent of variable qual-
ity. Moreover, we also consider how the arrangement of the stripes (“in-” versus “out-of-phase”) effects
the observed phenomena. Quantitative analysis of morphologies is performed by means of overlap in-
tegrals of polymer chains belonging to different stripes or surfaces. Conformational properties of chains
are studied via gyration tensor components. The simulation method was described in our previous work
[44] and for the sake of brevity is omitted here.
The paper is organized as follows. In the next section we will briefly recall the model and the simula-
tionmethod. A number of limiting cases for the pore geometry that show the effects of a reduced effective
dimensionality are considered in section 3. Quantitative analysis of solvent mediated morphologies for
the case of weakly separated stripes is performed in section 4. The summary of the results is presented
in section 5.
2. The model
To simulate the pore, we use the box of the dimensions Lx , Ly and Lz . All the dimensions are in
reduced units measured with respect to the cutoff distance rc for the repulsive interaction, which is set to
rc = 1. The planes z = 0 and z = Lz ≡ d are impenetrable walls. Periodic boundary conditions are applied
in both X and Y directions. Each wall is divided into stripes of equal width, w , alternating those with
and without polymers attached. The stripes at the opposite walls can be placed either in- or out-of-phase,
see figure 1. The width of the stripes with attached polymers is the same as the width of the polymer
free spaces. Therefore, the term “out-of-phase” means that the stripe with polymer on one wall faces the
polymer free stripe at the other wall. The rest of the pore interior is filled with single beads representing
the fluid components.
We use the DPDmethod in the form outlined by Groot andWarren [46]. This is amesoscopic approach
in which one considers beads that interact via soft repulsive potential. Each bead can be interpreted as a
fragment of a polymer chain or a collection of solvent particles, etc. The strength of the repulsive inter-
action can be tuned to match the compressibility of e.g. water, whereas the difference in repulsion for
various species reflects their miscibility and can be related to the Flory-Huggins parameter [46]. The pair-
wise dissipative (friction) and random forces are added in a spirit of Langevin dynamics to compensate
for averaging over a small-scale structure being neglected in this approach. The method proved itself to
13606-2
DPD study of solvent mediated transitions in pores
Figure 1. Geometry of the pore with in-phase (a) and out-of-phase (b) arrangement of the stripes with
tethered polymer chains. The pore size is d and the stripes width is w . The length of polymer chains is
L = 20 beads.
be fast and effective for the study of microphase-separation driven phenomena [38–43].
The method is extended to include the smooth repulsive walls via the reflection algorithm that pre-
serves the total momentum [44]. The reduced number density for beads is equal to ρ = 3.0. Polymer
chains are built of L = 20 beads linked via harmonic bonds. The grafting points have been fixed on each
wall and are distributed randomly inside the stripeswith a reduced grafting density ρg = 1.0. More details
are provided elsewhere [44].
Two bead sorts, A and B, are considered. Polymer chains aremade of beads of sort A. The total number
of polymer beads in the system is N
p
A
. A binary fluid mixture (which fills the interior of the pore) contains
N s
A
beads of sort A and NB beads of sort B. The fraction of the beads of sort B is fB = NB/N , where
N = N
p
A
+ N s
A
+ NB is the total number of the beads. Repulsion parameters for A–A and B–B pairs are
defined to be the same, aAA = aBB = 25, the choice being based on matching the compressibility of the
model system (at ρ = 3.0) to that of water, as discussed in [46]. Poor miscibility between A and B beads
is set by the higher value of aAB for the A–B interaction as compared to aAA. The choice of aAB value is
quite flexible and we use the value of aAB = 40 which corresponds to the regime of moderate to strong
segregation and ensures firm separation between species.
To visualize the morphologies, we employ the following density grid approach. Simulation box is
split into the grid of cubic cells with the linear dimension of 0.75÷2.0, depending on the simulation box
size. Local densities for polymer A beads, ρ
p
A
(x, y, z), solvent A beads, ρs
A
(x, y, z), and solvent B beads,
ρB(x, y, z) are evaluated in each cell centered at (x, y, z). They are averaged over 10−15 configurations
(the subsequent configurations are separated by 5000 simulation steps). The averaging is carried out after
the particular morphology is stabilized (typically, after 2 ·105 simulation steps).
We found that presenting both local densities of A and B beads in the same snapshot is not very
informative. Instead, we show separate snapshots for local density of A or of B beads. In the first case, we
represent the cells with low local density of A beads [ρ
p
A
(x, y, z)+ρs
A
(x, y, z)]/ρ < 0.15 as dots. All other
cells are space-filled with the color saturation proportional to the value of [ρ
p
A
(x, y, z)+ρs
A
(x, y, z)]/ρ. The
color tint is blueish if ρ
p
A
(x, y, z) > ρs
A
(x, y, z) and greenish otherwise. Similar approach is used in the
second case, when the density of B beads is plotted. In this case, ρB(x, y, z) is color coded with red tint.
3. Solvent-mediated transitions in special geometries
3.1. Wide stripes geometry
In the case of wide stripes that are in-phase arranged, the periodic pattern of alternating sub-regions
is formed within the pore. The sub-regions free of polymer brush provide an environment for a bulk,
quasi-3D demixing of confined A and B beads. On the contrary, the sub-regions dominated by a brush,
reduce the volume accessible for the demixing of A and B beads to quasi-2D slabs, especially for moderate
values of pore size d . This is demonstrated in figure 2, where we show examples of the morphologies
observed for wide stripes with w = 90 in a pore of size of d = 20 when the fraction fB is varied.
The first effect to mention is that with an increase of the fraction of A beads, fA = 1− fB , more of
them are adsorbed by the stripes of polymer brush causing their considerable swelling. We will discuss
this effect quantitatively in section 4. The remaining A and B beads are spread within the rest of the
accessible volume.
We will concentrate here on the phenomena that occur within quasi-3D sub-regions. The morpholo-
gies formed here repeat those observed for diblock copolymers at various composition (or in similar
13606-3
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
Figure 2. (Color online) The sequence of morphologies obtained for the case of wide stripes, w = 90, at
pore size d = 20 by varying fraction of B beads, fB (indicated on the right). Color-coded density of B beads
in the selected part of the pore is shown only for the sake of clarity.
systems) [44, 45]. To relate both cases, one should look at the local fractions of A and B beads, f ′
A
and f ′
B
,
that differ from their global counterparts, fA and fB, due to adsorption of some of A beads into the brush
mentioned above. We found that cylinders made of B beads are formed at fB = 0.16 ( f ′
B = 0.31, lowest
frame in figure 2). Similarly, cylinders made of A beads are observed inside the interval from fB = 0.40
( f ′
A = 0.26) to fB = 0.50 ( f ′
A = 0.17), first and second frames from the top of figure 2. Lamellar-like phase
(in the OY Z plane; alternating blocks of A and B beads along the X -axis) is observed in the interval from
fB = 0.26 ( f ′
A
= 0.50) to fB = 0.30 ( f ′
A
= 0.42), second frame from the bottom in figure 2. These boundaries
of morphologies (in terms of the values f ′
U for minor fraction U ) correlate well with the phase diagram
for diblock copolymers (see, e.g., [44, 45]).
Morphology transformations observed within the quasi-2D regions literally repeat those found for
the narrow stripes geometry, and this case is considered in detail in the following subsection.
3.2. Narrow stripes geometry
For narrow stripes, w < 5, the polymer chains from adjacent stripes bridge themselves into lamellae
that envelope each wall [44]. It is quite obvious that the same scenario holds for the out-of-phase arrange-
ment of stripes, as far as both surfaces are decoupled. This is confirmed in our simulations (not shown
for the sake of brevity). Two flat lamellae, formed at each wall, reduce the region accessible to free A and
B beads to a slab, which turns into a quasi-2D one for moderate pore size d ∼ 14÷20. This situation also
represents the sub-regions dominated by a brush for the case of wide stripes (considered in the previous
subsection).
In figure 3, we display the sequence of quasi-2D morphologies that appear in this central slab as the
result of a micro-phase separation between A and B beads. The pore size is d = 13.333, the stripes width is
w = 4 and only color-coded density of B beads is shown (in red tint). This pore size is a special one, as far
as for the parameters being used here (polymer length, bulk and grafting densities), no solvent A beads
are present at fB = 0.5 [44]. At this fraction fB, solvent B beads fill-in all the slab-like accessible volume in
the middle of the pore (first frame from the left in figure 3). The slab stays continuous but thins out with a
Figure 3. (Color online) Sequence of morphologies obtained for narrow stripes, w = 4, in a pore of size
d = 13.333 at various fractions fB (indicated at the bottom). From left to right: lamellar, thinned lamellar,
perforated lamellar, “sausage” and “cake”, single “sausage”, and hexagonally packed “cakes” morpholo-
gies are shown (only color-coded density of B beads is shown).
13606-4
DPD study of solvent mediated transitions in pores
Figure 4. (Color online) Sequence ofmorphologies obtained for d = 13.333 and w = 10 at various fractions
fB (indicated at the bottom). Blocks of B beads filling the space between pillars of A beads ( fB = 0.50),
thinned blocks ( fB = 0.26), hexagonally arranged columns ( fB = 0.20), and random columns ( fB = 0.14)
are shown. Only color-coded density of B beads is displayed.
decrease of fB down to fB = 0.28 (the next frame). Then, it turns into a perforated lamellar one and then
into disjointed prolate and/or oblate objects made of beads B (see respective frames in figure 3). For still
lower value of fB, fB ≈ 0.12, there appear hexagonally distributed “cookies” of B beads. One can easily see
that this morphology possesses the same symmetry as the perforated lamellar one, if beads A and B are
interchanged. Further decrease of fB causes the loss of hexagonal order of the “cookies”, and for fB < 0.1
the “cookies” of B beads become randomly arranged (not shown). This sequence of the morphologies
is reminiscent of the one observed in partial mixing of the two-dimensional fluids [47]. Therefore, the
case of narrow stripes can be classified as geometry-driven dimensional crossover from 3D to 2D for a
confined fluid.
3.3. Pillar geometry
In the case of wider stripes (w > 6), bridging of polymer chains that belong to adjacent stripes is
prohibited due to high penalty in conformational entropy. Instead, the in-phase arranged stripes can
bridge themselves across the pore to form pillars, providing that the pore width d is not too big (cf. the
sketch phase diagram presented in the previous work [44]). Such pillars are observed, for example, at
d = 13.333 and w = 10. Similarly to the case shown in figure 2, the system again displays (periodic along
X axis) the pattern of sub-regions, one being pillars of merged brushes and another being polymer-free
sub-regions. At fB = 0.5, the latter are in a form of blocks filled exclusively with solvent B beads. For
0.5 > fB > 0.26, these blocks become thinner first (as far as solvent A beads are adsorbed into pillars
causing the latter to swell). Then, for fB ∼ 0.20, the blocks split into rounded “columns” spanning across
Figure 5. (Color online) Left bottom frame shows the density profiles ρ(i)(y) along each i -th slab region
shown in the top frame. Right bottom frame shows the dependence of the integral profile I (averaged
over four slabs) on fB . The results for simulation set I are shown as blue disks and for the set II — as red
asterisks.
13606-5
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
the pore and arranged almost hexagonally. At still lower values of fB, we arrive at randomly arranged
columns of B beads of random thickness. All these morphologies are displayed in figure 4.
The regions accessible to the micro-phase separation of solvent beads have a shape of slabs extended
along Y -axis. The important point here is that all morphologies observed within these slabs at various
fB are uniform within the X dimension of each slab. Thus, the behavior of the system can be interpreted
as being quasi-1D along Y axis within each slab. One can quantify the observed morphology changes by
the density profiles of B beads, ρ(y), along Y axis. It can be evaluated within a thin cross-section slabs
located in the middle of each region (marked as 1−4 in the top frame in figure 5). The thickness of each
cross-section slab in X direction is equal to 2 and the averaging of the density is made in Z -direction.
The histograms for the density profiles ρ(i)(y) in each i -th slab obtained in this way for the morphology
shown in the top frame in figure 5, are presented in the left bottom frame of figure 5. The local density
inside each column of B beads is equal to the bulk density, ρ(y) ≈ ρ = 3, whereas it drops down to zero
outside the column. Therefore, the integrated density, I =
∫Ly
0 dyρ(y)/Ly is a good measure for “block
continuity”. Right hand frame in figure 5 shows the values of I averaged over all four slabs, 1−4. Two sets
of simulations were performed. For the set I we started the simulation from the morphology equilibrated
at fB = 0.5 and converted the required number of B beads, chosen randomly across the system, into A
beads. In the set II simulation, the initial configuration involved linearly stretched polymer chains and
solvent A and B beads randomly distributed within the pore.
One can observe the transformation from continuous into discontinuous block morphology that oc-
curs at f ∗
B
≈ 0.23 (for the simulations set I; blue open disks) or at f ∗
B
≈ 0.28 (for the simulation set II; red
asterisks). The former value is lower which indicates the presence of the “underconcentrated” continu-
ous block morphology at 0.28 > fB > 0.23 in the simulations set I. It is also interesting to note that for
small fB, the integral I changes nearly linearly with fB.
4. Competition between solvent mediated morphologies for closely
arranged stripes
So far we considered several special cases when both the geometry restrictions imposed by the pore
geometry and the composition of the mixture confined within the accessible sub-regions promote essen-
tial morphological changes. In both cases of a narrow pore (subsection 3.2) and of a pillar (subsection 3.3),
the stripes of brushes are bridged in one of the directions by purely geometrical means, literally by bring-
ing stripes close enough to form lamellae (pillars). The role of the solvent is restricted to the swelling of the
already formed lamellae (pillars) and to a micro-phase separation inside the polymer-free sub-regions.
However, one can envisage the situation when the lamellar (pillar) bridges are formed exclusively
due to the action of a solvent. Apparently, this is possible in the geometries where the stripes of brush
are brought sufficiently close in one of X or Z dimensions, but not close enough to bridge over by them-
selves. Even a more promising case can be designed by provoking the competition between the bridging,
which may happen when the stripes are positioned sufficiently close in both X and Z directions. Possible
technological applications could involve the use of a solvent mixture which contains A component in the
form of a short chain. This component could be first used for demixing with the B component and for the
formation of certain morphology, and then to be used as a crosslinker to permanently fix the structure.
4.1. Mechanism for solvent-mediated morphologies and quantitative characterization
of brush bridging and deformation of chains
The mechanism for the solvent-mediated morphologies lies in the swelling of a polymer brush due
to adsorption of a good solvent, the effect already mentioned above. Here, we will discuss this effect in
more detail. Let us consider the environment within the stripes of polymer chains. Due to relatively low
grafting density ρg =
1
3
ρ (where ρ = 3 is bulk number density), the stripes are far from the regime of a
dense brush [48] and are exposed to the available solvent. For the case of poor solvent, the brush collapses
and each chain is in a regime of polymer melt, whereas for a good solvent, the chains are expected to be
in the regime of a good solution. Two cases are well distinguished by their respective scaling laws [49].
13606-6
DPD study of solvent mediated transitions in pores
Table 1. Results of fitting the average bond length l0 and radius of gyration Rg to the scaling law, equa-
tion 1, for the assumed values of the exponent ν. Output: prefactor R in equation 1 is numerically consis-
tent for all the cases considered.
pore geometry solvent l0 Rg R
d = 22, in-phase aAC = 40 (ν= 0.50) 0.893 1.909 0.463
aAC = 25 (ν= 0.59) 0.937 2.557 0.468
d = 20, out-of-phase aAC = 40 (ν= 0.50) 0.892 1.909 0.464
aAC = 25 (ν= 0.59) 0.936 2.558 0.468
For instance, the radius of gyration scales as:
Rg = R [l0(N −1)]ν , (1)
where R = const, l0 is the equilibrium bond length, N is the number of monomers and ν= 0.5 for the case
of polymer melt and ν ≈ 0.59 (Flory exponent) for the case of a good solution [49]. As it was discussed
previously [50], the softness of the potentials employed in typical DPD simulations does not violate the
correct value for the exponent ν= 0.59 for a single chain in a good solvent.
To check whether this scenario holds, we performed a set of simulations for the pore geometry with
well separated stripes. Polymer chains are made of A beads and a one-component solvent of C beads is
used. The quality of the solvent is tuned via the repulsion parameter aAC between A and C beads ranging
from 25 (good solvent) up to 40 (bad solvent). For each simulation run, the average bond length l0 and
the radius of gyration Rg were evaluated and the scaling law (1) was employed for the assumed values
of the exponent ν, ν = 0.5 or 0.59. Next, the prefactor R (see equation (1) was estimated. The results are
collected in table 1. As it follows from the enclosed data, the difference between the estimated prefactors
R does not exceed 1% which proves that the scaling law (1) holds consistently. These simulations confirm
a crossover from the regime of polymer melt to a good solution regime for each chain which is driven
by the quality of the solvent. Typical chain extension ratio is estimated as [l0(N −1)]0.09 ∼ 1.3 which is
of the order of 2÷4 unit lengths for our particular model, and this provides the length scale for brush
separations at which one expects solvent-mediated bridging.
To quantitatively characterize the level of bridging between stripes we introduce the overlap integrals
Ix and Iz between polymer chains in X and Z directions (by analogy to the case of uniform brushes [51]):
Ix =
∑
〈ik〉
Lx
∫
0
ρi (x)ρk (x)dx ·
∑
〈ik〉
Lx
∫
0
dx
−1
Iz =
d
∫
0
ρbot(z)ρtop(z)dz ·
d
∫
0
dz
−1
, (2)
where ρi (x) is the density profile along X axis for polymer beads belonging to i -th stripe (averaged over
Y and Z directions); ρbot(z) and ρtop(z) are density profiles for polymer beads belonging to the bottom
or top wall, respectively (averaged over X and Y directions). These density profiles are illustrated in
figure 6. The evaluation of Ix overlap integral is performed over all 〈i k〉 pairs that are located at the
same wall and are adjacent in X direction (with the account of the periodic boundary conditions).
The bridging of brushes involves a certain amount of bending and/or anisotropic stretching of poly-
mer chains. These can be quantified via the components of gyration tensor, Gxx , Gy y and Gzz . The com-
ponents are evaluated for each k-th chain
G [k]
αβ
=
1
N
N
∑
i=1
(
r [k]
i ,α
−R[k]
α
)(
r [k]
i ,β
−R[k]
β
)
, (3)
where α,β denote Cartesian axes, r [k]
i ,β
are the positions of individual monomers and R[k] is the center of
mass position of the k-th chain. Then, G [k]
αβ
are averaged over the chains and over the time trajectory after
the morphology stabilizes itself, providing the estimates for Gxx , Gy y and Gzz . One should mention that
due to the symmetry of the pore in Y direction, the Gy y component is found to be unchanged and it is not
considered in our analysis. The average radius of gyration Rg is defined as R2
g =Gxx +Gy y +Gzz , and the
average extension of chains can be found from the maximal eigenvalue of the gyration tensor, σ2
max.
13606-7
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
Figure 6. (Color online) Left hand frame illustrates the density distributions ρi (x) of the polymer beads
in each i -th stripe along X axis averaged over Y and Z axes (the distributions 1− 4 belonging to the
bottom surface are shown only). Right hand frame illustrates the density distributions of the polymer
beads along Z axis that belong to bottom and top surfaces (abbreviated as “bot” and “top”, respectively),
the distributions are averaged in X and Y directions.
4.2. Solvent-mediated morphological changes
The properties introduced above are used to characterize solvent-mediated bridging between stripes
of brushes. At first we will consider the case of one-component solvent of variable quality (tuned via
aAC parameter, see above). The results are provided for the pore size of d = 20 and stripes of width
w = 13 arranged out-of-phase (see figure 7). With a decrease of aAC towards the case of a good solvent
(aAC = 25), one observes a monotonous increase of all the metric properties, Gxx , Gzz and σ2
max. This
quantifies the effect of polymer chains swelling due to the crossover from the polymer melt to polymer
in a good solvent regime. The overlap integrals Ix and Iz have been found to become non-zero at a certain
threshold value of aAC < 28. This indicates a solvent-mediated bridging between the droplets of polymer
chains. The bridging happens almost simultaneously in both X and Z directions for this geometry of the
pore. In view of our analysis, it is important to mention that all the characteristics, Ix , Iz , Gxx , Gzz and
σ2
max increase monotonously with a decrease of aAC down to 25 with no peculiarities observed either in
the behavior of metric properties and overlap integrals or in the snapshots.
The case of a one-component solvent of the variable quality provides a suitable reference point for
a more complex case of a two-component solvent composed of A and B beads. The repulsion parameter
aAB is chosen to be always equal to 40 in our study. Therefore, the effective “goodness” of such a mixture
can be characterized by the fraction fB of B beads (the mixture is a good solvent for fB = 0). The analogy
with a one-component solvent is, however, not exact, as far as the mixture is prone to segregation (see,
e.g. figure 4 in [44]). Local inhomogeneities may effect the way in which the brushes are bridged and, as
a result, the sequence of solvent-mediated morphologies may differ from the case of a one-component
solvent. Hereafter, we will consider the fixed stripe width of w = 13 and both: in- and out-of phase ar-
rangements. The pore size will be fine-tuned in each case and ranges from d = 19 to d = 24.
In the case of in-phase arrangement of the stripes, we select three pore sizes, namely d = 20,22 and 24
(the distance between brushes in X and Z directions is approximately the same for d = 22). The evolution
of Ix , Iz , Gxx , Gzz and σ2
max upon the change of the fraction of fB from 0.5 down to 0 is shown for each
Figure 7. (Color online) Behavior of the overlap integrals Ix and Iz (left hand frame) and the components
Gxx and Gzz and maximal eigenvalue σ2
max of the gyration tensor (right hand frame) upon the changes
of the quality of the one-component solvent. The quality of the solvent is defined via the repulsion pa-
rameter aAC between polymer (A) and solvent (C) beads, the value of aAC = 25 represents the case of a
good solvent.
13606-8
DPD study of solvent mediated transitions in pores
Figure 8. (Color online) Behavior of the overlap integrals Ix and Iz (top row of frames) and the compo-
nents Gxx and Gzz and themaximal eigenvalue σ2
max of the gyration tensor (bottom row of frames) upon
the changes of the effective quality of the two-component solvent of A and B beads. Effective quality of
the solvent is defined via the fraction fB of “bad solvent” beads B. The case of in-phase arrangement of
stripes of the width w = 13 is shown at the pore sizes d = 20,22 and 24.
pore size in figure 8. One can compare these plots with the curves displayed in figure 7 and observe that
the metric properties of chains at f = 0.5 subject to the two-component solvent, match approximately
their counterparts for the case of a one-component solvent at aAC = 28. This provides the estimate for
an effective “goodness” of the two-component solvent. Metric properties coincide for the case of good
solvent ( f = 0 and aAC = 25, respectively).
When one moves away from fB = 0.5 towards 0, the behavior of all the properties Ix , Iz , Gxx , Gzz and
σ2
max differs much from the case of a one-component solvent. For the pore of d = 20, the bridging in Z
direction is observed first. This is indicated by a large hill at Iz centered around the value of fB = 0.34
(top left hand frame in figure 8). To be able to form the pillar morphology, the chains need to rearrange
themselves preferentially in Z direction as indicated by an increase of Gzz at the expense of Gxx values
(bottom left hand frame in figure 8). The value of Gzz = 3.69 found at fB = 0.34 is of the same order and
even exceeds that observed at fB = 0, namely Gzz = 3.56. This indicates that the chains within pillars are
in the regime of a good solvent. This is confirmed by the snapshots shown in top-middle and bottom-
middle columns of figure 9, where the droplets merge together by the good solvent beads. Therefore, as a
result of a different miscibility of A and B components, the solvent A nucleates by filling the gap between
polymer droplets, and the solvent mediated transition from separate droplets to pillar morphology oc-
Figure 9. (Color online) Sequence of morphologies obtained for d = 20 and for in-phase arranged stripes
of the width w = 13 at various fractions fB (indicated at the bottom). Top row: color-coded densities of
brush prevailing (blue) and good solvent prevailing (green) regions, bottom row: bad solvent prevailing
regions (red). Morphologies: separate droplets ( fB = 0.50); solvent mediated pillars ( fB = 0.36) and in-
lined cylinders of B beads ( fB = 0.22).
13606-9
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
Figure 10. (Color online) The same properties as in figure 8 are shown for the case of out-of-phase ar-
rangement of stripes of the width w = 13 at the pore sizes d = 19,20 and 21.
curs. With a further decrease of fB, the pillars are also bridged in X direction. This is indicated by an
increase of Gxx values and is seen in the top-right and bottom-right columns of figure 9.
With an increase of the pore size d to 22, the bridging capability in Z direction is lessened giving a
way for bridging along the walls. This is demonstrated by the existence of local maxima and minima of
Ix , Iz and Gxx and Gzz components (see top-center and bottom-center frames in figure 8). With a further
increase of d to 24, the situation is reversed with respect to the case of d = 20. Now, the droplets are
farther in Z direction and their bridging is observed in X direction: Ix and Gxx possess maxima centered
around fB = 0.4 (top-right and bottom-right frames of figure 8).
Let us switch now to the case of out-of-phase arrangement of stripes. The set of pore sizes d = 19,20
and d = 21 are analyzed in this case (the distance between brushes in X and Z directions is approximately
the same for d = 20). The behavior of Ix , Iz , Gxx , Gzz and σ2
max is shown in figure 10. The limiting cases of
d = 19 and d = 21 bear similarities to their respective counterparts (d = 20 and d = 24) for the in-phase
arrangement of stripes. Indeed, at d = 19, the polymer-rich droplets are bridged in Z direction first with a
decrease of fB. With a further decrease of fB, the droplets are bridged in X direction. Situation is reversed
at d = 21.
The intermediate case of the pore size d = 20 is more interesting because it demonstrates a solvent
mediated switching between various morphologies. When fB decreases away from 0.5, first the bridges
in X direction are formed. Both Ix and Gxx exhibit maxima centered around fB = 0.36, cf. top-middle
and bottom-middle frames of figure 10. This indicates the formation of a solvent mediated lamellar mor-
phology, see snapshots in top-middle and bottom-middle frames of figure 11. With a further decrease of
fB, the bridges are formed in Z direction at the expense of those that have been previously formed in X
Figure 11. (Color online) Sequence of morphologies obtained for d = 20, for out-of-phase arranged stripes
of the width of w = 13 at various fractions of fB. Separate droplets ( fB = 0.50); modulated lamellar ( fB =
0.36) and meander ( fB = 0.24) morphologies.
13606-10
DPD study of solvent mediated transitions in pores
Figure 12. (Color online) Distributions of the relevant gyration tensor components, Gxx and Gzz for var-
ious fractions fB (indicated in each frame). The case of d = 20, w = 13 and out-of-phase arrangement of
stripes is shown.
direction. This is indicated by a large value of Iz and by practically zero value of Ix within the interval
of fB ∈ [0.23,0.3]. The structure of this morphology is of simple meander (see snapshots in top-right and
bottom-right hand frames of figure 11).
One can see that when the fraction of good solvent in a system increases, the chains within each stripe
swell and acquire certain bistability properties. The chains can redistribute their mass either in X or in
Z direction (as indicated by the behavior of Gxx and Gzz components) and form the bridges (as indicated
by non-zero values of the overlap integrals Ix and Iz ). The histograms for the corresponding components
of the gyration tensor, f (Gxx ) and f (Gzz ) provide additional insight into spatial redistribution of the
polymer chains in various morphologies. These histograms are shown in figure 12 for the case of pore
size d = 20 and out-of-phase arrangement of the stripes of the width of w = 13. The distribution function
f (Gzz ) is found to be essentially stretched towards larger values of Gzz for meander ( fB = 0.26) morphol-
ogy and at still lower values of fB. It exhibits a quite irregular shape, whereas the distribution of f (Gxx )
is similar to the Lhuillier form [50, 52, 53] that was found as a typical distribution of experimental radii
of gyration for long polymers. Broad distributions of a gyration tensor of both components indicate the
existence of weakly and highly deformed chains within each stripe. It is plausible to assume that highly
deformed chains are found within the bridges that connect polymer droplets in a solvent mediated pil-
lar, meander and possibly other morphologies. Ameasure for the average stretch of chains is provided by
the maximal eigenvalue of the gyration tensor, σ2
max. As it follows from the set of plots shown in figures 8
and 10, the values of σ2
max exhibit local maxima for certain morphologies. The values of these maxima
are higher compared to the case of one-component solvent mapped into an effective value of aAC (see fig-
ure 7). This indicates that the loss of conformation entropy due to an excessive average stretch of chains
is compensated by a decrease of the enthalpy caused by an increase of contacts between similar beads of
type A.
5. Summary
In this work we consider the formation and transitions between the nanostructures that occur inside
a pore modified by stripes of tethered polymer brushes filled with a binary mixture. The beads A of filler
solvent are identical to those of chain monomers, while the beads B exhibit a partial mixing with beads
A. Compared to our previous study [44], a number of extensions are made. In particular, two options
of in- and out-of-phase arrangement of polymer stripes are considered. Apart from that, we examine in
detail the effect that the composition of A and B beads produces on the formation of the morphologies
in a broad range of pore geometries. At last, for some characteristic cases we undertook supplementary
13606-11
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
simulations with the one-component solvent of variable quality which replaces the mixture to serve as a
reference.
The change of the composition of the fluid inside the pores has a great impact on the developing struc-
tures. For pore geometries with narrow stripes, the problem of the description of microphase separation
inside the pore reduces to quasi two-dimensional; in the case of moderately wide stripes and narrow
pore, one faces quasi one-dimensional demixing; whereas for very wide stripes, the system is split into
quasi two-dimensional and bulk regions.
The most interesting effects, in our view, that demonstrate solvent mediated changes in the nanos-
tructures occur for the geometries with weakly separated stripes of polymer chains. With an increase of
A component, the latter dissolves the polymer chains and causes their swelling. As a result, the system
acquires a bistability in terms of either bridging the adjacent stripes along the wall or bridging the op-
posite stripes across the pore. Stable morphology is formed due to competition between segregation of
A and B beads and the deformation of the chains. In our simulations we observe morphology switching
due to subtle changes in pore geometry and/or in the composition of a mixture.
We found the following solvent mediated morphologies: in-lined cylinders (made of one component),
meander structure and wave-shaped modulated internal channels. The suggested applications of such
structures (possibly, after making the structure permanent via crosslinking) involve nanopatterning for
manufacturing of nanochannels, nanorods and similarly sized objects.
So far our studies concentrated on the equilibrium DPD simulations. However, it would be of interest
to check how the morphologies change if the fluid undergoes a pressure-driven flow along the pore axis.
This problem is currently under study in our laboratories.
Acknowledgements
This work was supported by EC under the Grant No. PIRSES 268498. J. I. is thankful to T. Kreer, S. San-
ter and D. Neher for fruitful and stimulating discussions.
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http://dx.doi.org/10.1063/1.2954022
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http://dx.doi.org/10.1021/la102666g
http://dx.doi.org/10.1063/1.3592562
http://dx.doi.org/10.1021/ma801045z
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http://dx.doi.org/10.1063/1.458147
Ja.M. Ilnytskyi, S. Sokołowski, T. Patsahan
Фазовi переходи за посередництва розчинника у порах
iз стiнками декорованими полiмерними щiтками.
Комп’ютерне моделювання методом
дисипативної динамiки
Я.М. Iльницький1, С.Соколовскi2, Т. Пацаган1
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Вiддiл моделювання фiзико-хiиiчних процесiв, Унiверситет Марiї Кюрi-Склодовської,
20031 Люблiн, Польща
Розглянуто процес самоформування фаз у порi iз стiнками декорованими полосками полiмерної щiтки
(з мономерiв сорту А), яка заповнена бiнарною сумiшшю iз компонент А i В. Акцент зроблено на дослi-
дженнi ролi розчинника при фазових переходах спричинених змiною складу сумiшi. Знайдено граничнi
випадки квазиодновимiрного та квазидвовимiрного незмiшування. Показано, що механiзм формуван-
ня морфологiй (та, в деяких випадках, переключення мiж ними) при змiнi складу сумiшi полягає у змiнi
локального середовища для ланцюжкiв полiмерних щiток. Знайдено сформованi за посередництва роз-
чинника ламеларнi, меандро-подiбнi та порядковi цилiндричнi фази. Структури вивчено за допомогою
аналiзу iнтегралiв перекриття щiток та величин компонент тензора гiрацiї.
Ключовi слова: дисипативна динамiка, сумiшi, пори, наноструктури
13606-14
Introduction
The model
Solvent-mediated transitions in special geometries
Wide stripes geometry
Narrow stripes geometry
Pillar geometry
Competition between solvent mediated morphologies for closely arranged stripes
Mechanism for solvent-mediated morphologies and quantitative characterization of brush bridging and deformation of chains
Solvent-mediated morphological changes
Summary
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