Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study
We develop a simulational methodology allowing for simulation of the pressure-driven flow in the pore with flat and polymer-modified walls. Our approach is based on dissipative particle dynamics and we combine earlier ideas of fluid-like walls and reverse flow. As a test case we consider the oligome...
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Інститут фізики конденсованих систем НАН України
2016
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Ilnytskyi, J.M. Bryk, P. Patrykiejew, A. 2019-06-17T12:10:38Z 2019-06-17T12:10:38Z 2016 Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study / J.M. Ilnytskyi, P. Bryk, A. Patrykiejew // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13609: 1–14 . — Бібліогр.: 65 назв. — англ. 1607-324X DOI:10.5488/CMP.19.13609 arXiv:1603.02432 PACS: 68.03.Cd, 68.08.-p, 68.03.-g, 68.35.Md, 68.47.Mn https://nasplib.isofts.kiev.ua/handle/123456789/155772 We develop a simulational methodology allowing for simulation of the pressure-driven flow in the pore with flat and polymer-modified walls. Our approach is based on dissipative particle dynamics and we combine earlier ideas of fluid-like walls and reverse flow. As a test case we consider the oligomer flow through the pore with flat walls and demonstrate good thermostatting qualities of the proposed method. We found the inhomogeneities in both oligomer shape and alignment across the pore leading to a non-parabolic velocity profiles. The method is subsequently applied to a nano-channel decorated with a polymer brush stripes arranged perpendicularly to the flow direction. At certain threshold value of a flow force we find a pillar-to-lamellar morphological transition, which leads to the brush enveloping the pore wall by a relatively smooth layer. At higher flow rates, the flow of oligomer has similar properties as in the case of flat walls, but for the narrower effective pore size. We observe stretching and aligning of the polymer molecules along the flow near the pore walls. Розвинуто симуляцiйний метод, спрямований на моделювання потоку у порi iз гладкими та полiмермодифiкованими стiнками. Пiдхiд грунтується на методi дисипативної динамiки i реалiзує iдеї “рiдинних стiнок” та реверсних потокiв. Як тест розглянуто потiк олiгомерiв крiзь пору iз гладкими стiнками i продемонстровано добре термостатування системи при застосуваннi цього методу. Отримано неоднорiдностi як форми плинних олiгомерiв, так i ступеня їх вирiвнювання вздовж пори, якi призводять до непараболiчних профiлiв швидкостей. Метод застосовано до наноканалу, декорованого смугами полiмерних щiток, розташованих перпендикулярно до напрямку потоку. При певному граничному значеннi сили потоку зафiксовано перехiд iз стовпцевої до ламеларної морфологiї, який призводить до обгортання стiнок пори гладким шаром iз полiмерних щiток. При вищих швидкостях потiк олiгомерiв набуває властивостей аналогiчних до випадку гладких стiнок, але iз вужчим ефективним розмiром пори. Спостережено розтяг i вирiвнювання полiмерних молекул вздовж напрямку потоку поблизу стiнок пори. This work was supported by the EU under IRSES Project STCSCMBS 268498. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study Потiк олiгомерного флюїду в нано-каналi iз комплексною структурою пiд впливом зовнiшнього тиску. Дослiдження методом дисипативної динамiки Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study |
| spellingShingle |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study Ilnytskyi, J.M. Bryk, P. Patrykiejew, A. |
| title_short |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study |
| title_full |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study |
| title_fullStr |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study |
| title_full_unstemmed |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study |
| title_sort |
pressure-driven flow of oligomeric fluid in nano-channel with complex structure. a dissipative particle dynamics study |
| author |
Ilnytskyi, J.M. Bryk, P. Patrykiejew, A. |
| author_facet |
Ilnytskyi, J.M. Bryk, P. Patrykiejew, A. |
| publishDate |
2016 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Потiк олiгомерного флюїду в нано-каналi iз комплексною структурою пiд впливом зовнiшнього тиску. Дослiдження методом дисипативної динамiки |
| description |
We develop a simulational methodology allowing for simulation of the pressure-driven flow in the pore with flat and polymer-modified walls. Our approach is based on dissipative particle dynamics and we combine earlier ideas of fluid-like walls and reverse flow. As a test case we consider the oligomer flow through the pore with flat walls and demonstrate good thermostatting qualities of the proposed method. We found the inhomogeneities in both oligomer shape and alignment across the pore leading to a non-parabolic velocity profiles. The method is subsequently applied to a nano-channel decorated with a polymer brush stripes arranged perpendicularly to the flow direction. At certain threshold value of a flow force we find a pillar-to-lamellar morphological transition, which leads to the brush enveloping the pore wall by a relatively smooth layer. At higher flow rates, the flow of oligomer has similar properties as in the case of flat walls, but for the narrower effective pore size. We observe stretching and aligning of the polymer molecules along the flow near the pore walls.
Розвинуто симуляцiйний метод, спрямований на моделювання потоку у порi iз гладкими та полiмермодифiкованими стiнками. Пiдхiд грунтується на методi дисипативної динамiки i реалiзує iдеї “рiдинних
стiнок” та реверсних потокiв. Як тест розглянуто потiк олiгомерiв крiзь пору iз гладкими стiнками i продемонстровано добре термостатування системи при застосуваннi цього методу. Отримано неоднорiдностi
як форми плинних олiгомерiв, так i ступеня їх вирiвнювання вздовж пори, якi призводять до непараболiчних профiлiв швидкостей. Метод застосовано до наноканалу, декорованого смугами полiмерних щiток,
розташованих перпендикулярно до напрямку потоку. При певному граничному значеннi сили потоку зафiксовано перехiд iз стовпцевої до ламеларної морфологiї, який призводить до обгортання стiнок пори
гладким шаром iз полiмерних щiток. При вищих швидкостях потiк олiгомерiв набуває властивостей аналогiчних до випадку гладких стiнок, але iз вужчим ефективним розмiром пори. Спостережено розтяг i
вирiвнювання полiмерних молекул вздовж напрямку потоку поблизу стiнок пори.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155772 |
| citation_txt |
Pressure-driven flow of oligomeric fluid in nano-channel with complex structure. A dissipative particle dynamics study / J.M. Ilnytskyi, P. Bryk, A. Patrykiejew // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13609: 1–14 . — Бібліогр.: 65 назв. — англ. |
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2025-11-25T22:33:40Z |
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2025-11-25T22:33:40Z |
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| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13609: 1–14
DOI: 10.5488/CMP.19.13609
http://www.icmp.lviv.ua/journal
Pressure-driven flow of oligomeric fluid in
nano-channel with complex structure. A dissipative
particle dynamics study∗
J.M. Ilnytskyi1, P. Bryk2, A. Patrykiejew2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University,
20–031 Lublin, Poland
Received November 2, 2015, in final form January 26, 2016
We develop a simulational methodology allowing for simulation of the pressure-driven flow in the pore with flat
and polymer-modified walls. Our approach is based on dissipative particle dynamics and we combine earlier
ideas of fluid-like walls and reverse flow. As a test case we consider the oligomer flow through the pore with flat
walls and demonstrate good thermostatting qualities of the proposed method. We found the inhomogeneities
in both oligomer shape and alignment across the pore leading to a non-parabolic velocity profiles. The method
is subsequently applied to a nano-channel decorated with a polymer brush stripes arranged perpendicularly to
the flow direction. At certain threshold value of a flow force we find a pillar-to-lamellar morphological transition,
which leads to the brush enveloping the pore wall by a relatively smooth layer. At higher flow rates, the flow of
oligomer has similar properties as in the case of flat walls, but for the narrower effective pore size. We observe
stretching and aligning of the polymer molecules along the flow near the pore walls.
Key words: Pouiseuille flow, polymer brush, oligomers, dissipative particle dynamics
PACS: 68.03.Cd, 68.08.-p, 68.03.-g, 68.35.Md, 68.47.Mn
1. Introduction
Understanding the behavior of polymers attached to surfaces is of importance inmany research areas
including biophysics, polymer-induced effective interactions in colloidal suspensions, chromatographic
separation, catalysis, and drug delivery [1]. Grafting polymer chains can significantly alter the properties
of the surface and make it, for example, biocompatibile or responsive to external stimuli [2]. Due to the
large field of potential applications, polymer brushes have been the subject of many theoretical studies.
In the seminal works Alexander [3], de Gennes [4, 5] have calculated the brush profile and explored
analytically the impact of grafting density and molecular weight. Since then, the properties of tethered
chains have been investigated by means of self-consistent field theory [6–10], polymer density functional
theory [11–16], and computer simulation [17–21]. Many theoretical predictions have been confirmed by
experiment [22–25].
Polymer brushes can be used to tailor static properties of surfaces, such as wettability, as well as
dynamical, such as hydrodynamic boundary conditions and friction. Fluid flow in polymer grafted nano-
pores can be described via continuum hydrodynamic equations (e.g., the Brinkman equation [26]) with
a priori assumed permeability related to the monomer density profile. The resulting velocity profile is
sensitive to the assumed form of the monomer profile [27]. However, the continuum hydrodynamic de-
scription of a flow has not been firmly established on the nanoscale [28]. This is important in the context
∗It is our pleasure to dedicate this paper to Professor Stefan Sokołowski, our Colleague and Mentor for many years.
© J.M. Ilnytskyi, P. Bryk, A. Patrykiejew, 2016 13609-1
http://dx.doi.org/10.5488/CMP.19.13609
http://www.icmp.lviv.ua/journal
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
of micro- and nanofluidic devices [29]. Downsizing a channel to the nanoscale, increases the surface-to-
volume ratio and introduces new physical phenomena not observed in the macroscale [30]. Covering the
surface by a polymer brushmay introduce a pronounced reduction of friction, which lowers the pressure
difference required to maintain the flow through a nanochannel [31]. Flow in polymer brushes has been
the subject of numerous simulational studies in recent years [32–38].
Recently, the equilibrium properties of binary mixture confined in a slit-like pore decorated with
polymer brush stripes were studied bymeans of dissipative particle dynamics (DPD) [39, 40]. It was found
that, depending on the geometrical parameters characterizing the system (the size of the pore and the
width of the stripes), several different structures (or morphologies) inside the pore can be formed. Such
patterned brushes can be fabricated experimentally by means of electron beam litography [41]. In the
present paper, we wish to study nonequilibrium properties of such system by considering the pressure
driven oligomer flow inside a channel with either flat or brush-modified walls. In particular, we focus on
three features such as: (i) the microstructure of a flow depending on the molecular mass of an oligomer
and the magnitude of a bulk flow force; (ii) flow-induced morphology changes; and (iii) the effect of the
patterned brush decoration of the walls on the properties of the flow. Our paper is arranged as follows: In
section 2 we introduce a new simulation method which combines the ideas of fluid-like walls and reverse
flow to minimise the near-wall artefacts and maintain constant temperature under flow condition. As a
simple test, we apply themethod to the case of oligomer flow through the porewith flat walls. In section 3,
the analysis is extended to the case when the walls are modified by a polymer brush arranged in a form
of stripes. Conclusions are provided in section 4.
2. Flow of oligomeric fluid through a channel with flat walls
Let us first consider the simulational approach employed in this study. We use the non-equilibrium
extension of the DPD technique in a form discussed by Groot and Warren [42]. This is a mesoscopic
method that operates at a level of coarse-grained beads, each representing either a fragment of a polymer
chain or a collection of solvent particles. The force acting on i th bead due to its pairwise interaction with
j th bead can be written as
Fi j = F
C
i j +F
D
i j +F
R
i j , (2.1)
where F
C
i j
, F
D
i j
and F
R
i j
denote the conservative, dissipative and random contribution, respectively. These
have the following form [42]
F
C
i j =
{
a(1− ri j )r̂i j , ri j < 1,
0, ri j Ê 1,
(2.2)
F
D
i j =−γwD(ri j )(r̂i j ·vi j )r̂i j , (2.3)
F
R
i j =σwR(ri j )θi j∆t−1/2
r̂i j . (2.4)
Here, vi j = vi −v j , vi and v j are the velocities of the beads, θi j is Gaussian random variable, 〈θi j (t)〉 = 0,
〈θi j (t)θkl (t ′)〉 = (δikδi l +δi lδ j k )δ(t − t ′) and ∆t is the time-step of the integrator. As already discussed in
references [28, 65], the effective range of friction between beads can be modified by adjusting the shape
of the weight functions wD(ri j ) and wR(ri j ). We use the following general form for wR(ri j ):
wR(ri j ) =
{
(1− ri j )β, ri j < 1,
0, ri j Ê 1,
(2.5)
where the exponent β is adjusted, and the weight function wD(ri j ) is set equal to [wR(ri j )]2 according to
Español and Warren [43] arguments. Likewise, it is required that σ2 = 2γ.
The oligomers and tethered polymer chains (if any) are represented as necklaces of beads bonded to-
gether via harmonic springs, the force acting on i th bead from the interaction with its bonded neighbour,
j th bead, is
F
B
i j =−kri j r̂i j , (2.6)
13609-2
Flow of oligomeric fluid
where ri j = |ri j |, ri j = ri −r j is the vector connecting the centers of i -th and j -th beads, r̂i j = ri j /ri j and
k is the spring constant. The same bonding force is used to tether the end polymer bead to the surface.
The length, mass, time and energy (expressed via T ∗ ≡ kBT ) units are all normally set equal to unity.
Let us now turn to the case where the fluid (or a mixture of fluids) is confined within a slit-like pore.
In order to commence a simulation of the pressure-driven flow, it is required to provide a set of rules
defining the behaviour of the fluid particles at walls, and a prescription for the construction of the walls.
These rules should recover the well known cases of hydrodynamic flow such as the Poiseuille flow (i.e.,
a flow of a Newtonian fluid with no-slip boundary conditions and a parabolic velocity profile). On the
other hand, for the flow of a polymeric fluid (i.e., a non-Newtonian fluid) the set of rules should lead to
the slip boundary conditions. The simplest set of rules comprise elastic reflections off the wall [44, 45].
Unfortunately, they give rise to a hydrodynamic slip for Newtonian fluids, as well as suffer from near-
wall density artifacts at higher density. This can be traced back to the fact that the atoms repelled each
other strongly but did not interact with the wall until they attempted to cross [46].
A number of more sophisticated set of rules have been suggested. One option is to form the crystalline
walls of a few layers of frozen (or having large mass) particles [47–53]. The interaction between the
bulk fluid particles and those of the wall creates the near-wall drag which leads to the formation of the
Poiseuille flow. A drawback of this approach is the propagation of the crystalline order into the near-
wall regions of bulk fluid. This effect is perfectly physical for the atomic molecular dynamics simulation,
where the solid wall mimics a real crystalline structure. However, for the mesoscopic DPD simulations,
each soft bead is assumed to represent a meso-scale portion of the material, on which scale the atomic
crystalline structure is smeared-out.
This drawback can be avoided by using the structureless fluid-like walls [46]. The walls in this case
are made of the fluid confined in the slabs adjacent to the pore boundary, and the elastic reflections
are applied on both sides of the boundary. Therefore, bulk and wall fluid particles are immiscible. Still,
the interaction between near-wall beads on the opposite sides of the boundary creates a near-wall drag
ensuring no-slip boundary condition for Newtonian fluids.
Another important issue in flow simulation is to avoid the system overheating due to the presence of
the body force. This problem was addressed in several studies, cf. for example references [54, 55]. In the
molecular dynamics simulation, the excessive energy is absorbed by an external thermostat, in either
bulk or near-wall form [54]. In DPD simulations, the thermostat is “internal”, provided by the balance
between interparticle friction and random forces. For the case of a flow, some means for dissipation of
additional energy related to the body force should be provided. One of the elegant ways to do this is the
concept of a reverse flow [56–58]. In this approach, the simulation box contains two sub-flows driven
oppositely. The total force applied to the system is equal to zero and a no-slip boundary is formed at the
interface between two opposite flows of Newtonian fluids.
In our study, we combine both concepts by employing the fluid-like walls on both boundaries of
a pore and initiating contraflows (reverse flows) within them. Separation between the main pore and
contraflow-containing walls prevents intermixing between the beads from both regions. This is impor-
tant both in the case when the flow of a mixture is considered, or in the case of polymer modified walls,
where polymer chains are tethered to the boundary between the main pore and fluid-like wall. However,
the existence of the reflective boundaries does not prevent a friction between the beads located on the
opposite sides of the boundary, enabling the formation of the no-slip boundary condition for Newtonian
fluids.
In this section, we consider the pressure-driven flow of oligomeric one-component fluid through the
pore with flat walls. The oligomers of length Lo = 1,4,10 and 20 beads are considered. The aim is twofold.
Firstly, we would like to test to what typical values of bulk force the approach outlined above can be
stretched without violation of temperature conservation. Secondly, we aim to study the flow microstruc-
ture depending on molecular length of the flowing oligomer and the magnitude of a flow force. The
geometry of the system is illustrated in figure 1. Here, X -axis runs from left to right, Z -axis — from bot-
tom to top, Y -axis coincides with the viewing direction. The simulation box is of dimensions Lx = 80,
Ly = 50 and Lz = 26.667 with the periodic boundary conditions applied along X and Y axes, the pore size
is d = 13.333, the size of the contraflow regions is c = d/2 = 6.667. The chains in contraflow regions are
of the same length Lo as in the main pore. Therefore, the total number of main and contraflow chains is
the same. All beads are assumed to be of the same type, which is reflected in the fact that the parameter
13609-3
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
d
c
c
Figure 1. (Color online) Simulation box geometry showing the pressure-driven flow of the oligomer of
Lo = 3 beads in a slit-like pore of size d (gray beads). Contraflow containing an oligomer of the same
length is contained in two fluid-like walls of size c (yellow beads). The direction of the applied flow force
is shown via arrows of different length reflecting the force distribution biased towards the middle bead,
see text for details.
a in equation (2.2) that controls the bead repulsion is the same for all pairwise interactions, a = 25. The
pore and the fluid-likewalls are separated via the internal boundaries (shown as dashed lines in figure 1),
impenetrable for the beads on both sides by applying the reflection algorithm described in detail in ref-
erence [39]. The same reflection algorithm is used at the external walls (solid lines in the same figure)
but, alternatively, the periodic boundary conditions can be used in Z direction, similarly to the original
reverse flow setup [56–58].
Each i th bead within a pore is subjected to the flow force of certain amount fi
F
FL
i = fi x̂, x̂ = {1,0,0}, (2.7)
applied along X -axis, where fi > 0, this is indicated by the right-hand side directed arrows in figure 1.
The beads in the contraflow regions are subjected to the force F
FL
i
=− fi x̂, indicated as reversely directed
arrows in the same figure. Several options are available for choosing the amount of fi . The simplest one
would be to choose fi ≡ f , the same amount for each bead. However, such an algorithm could lead to less
than optimal match of the micro-fluctuations of the applied pressure in real systems, since the polymer
molecules tend to form coils with varying distribution of the density. Another, rather extreme option
would be to apply the amount f Lo to the middle bead only. The other beads feel this force indirectly
and are delayed via the elastic spring forces. The latter approach might suffer from large fluctuations
of bond lengths and slower relaxation of the intra-chain vibrations, due to the soft nature of the model.
In our view, a reasonable compromise can be achieved by applying a fixed amount of the force f Lo to
each oligomer, but biasing it towards the middle bead of the chain. Namely, assuming that the beads of
an oligomer are numbered sequentially as l = 1, . . . ,Lo, then the amount of the force applied to the bead
number l is found according to the Gaussian distribution:
f (l) = f wG(l), wG(l) = 2exp
[
−
(l − l̄ )2
σ2
]
. (2.8)
Here, l̄ = (Lo + 1)/2 is the mid-index of the chain, and the breadth of the distribution is given by σ =
Lo/(2
p
π). The distribution is normalized to Lo:
No
∑
l=1
wG(l) =
+∞
∫
−∞
wG(l)dl = Lo. (2.9)
The shape of the weight function wG(l) is shown in figure 2 for the cases of Lo = 4, 10 and 20. As a result,
the total force applied to the oligomer of Lo beads is equal to f Lo, but it is biased towards the middle
beads (illustrated by arrows of different length in figure 1).
The acceleration of the fluid beads due to applying the bulk force affects the accuracy of the integra-
tor, as far as the expression for the coordinates at the time instance t +∆t contains the term proportional
to v(t)∆t , where v(t) is the velocity of the particle at the time instance t . The only way to keep the same
13609-4
Flow of oligomeric fluid
0 5 10 15 20
0
1
2
Lo=4 Lo=10 Lo=20
l
wG(l)
Figure 2. (Color online) Weight function wG(l ) for the amount of bulk force applied to l th bead of the
oligomer of length Lo, equation (2.8).
numerical accuracy of the integrator is to reduce the time-step∆t of the integrator. In our simulations, we
use the value ∆t = 0.001, about 40 times smaller compared to the values typically used in the case of equi-
librium simulation. Temperature conservation is one of the most important indicators of the accuracy
of the integrator. Following reference [55], we consider the transverse temperature, which is evaluated
from the two components of the velocity perpendicular to the flow direction: T ∗
⊥ = m〈v2
y〉/2+m〈v2
z 〉/2.
The profiles of T ∗
⊥ for two extreme cases of oligomer length Lo = 1 (simple fluid) and 20 with respect to
z coordinate are shown in figure 3 (a) and (b), respectively. We allow the maximum deviation of these
profiles from the required value 1 not to exceed 3−4%. As follows from figure 3, this is achieved for all
Lo = 1− 20 if the flow force magnitude is restricted to f É 0.2. At larger values, f > 0.2, the system is
prone to local heating near the internal walls, which signals a breakdown of this thermostatting method.
We should also remark that for the setup with no contraflow regions, no thermostatting can be achieved
at all: the temperature was found to rise monotonously even for the smallest considered values of f .
The profiles for the velocity components vx of individual beads along the flow direction are built
by binning the pore along the Z -axis. These are shown in figure 4 for the cases of Lo = 1 and Lo = 20
oligomer length obtained at various flow force amplitudes f = 0.05,0.1 and 0.2. For the case of simple
fluid (a), almost perfect parabolic shape is achieved inside the flow region indicating the properties of a
Newtonian fluid. The velocity drops to zero exactly at the pore walls giving rise to the no-slip boundary
condition. In this case, the Stokes formula can be used to estimate the viscosity of the fluid. With an
increase of the oligomer length, Lo, the shape of the velocity profile gradually diverges from a parabolic
one and turns into a bell-like shape at Lo = 20, as seen in (b). This indicates the non-Newtonian fluid
behaviour. The models describing such non-parabolic profiles exist (see, e.g., reference [58]) and involve
an analogue for the viscosity and a number of additional parameters. We found, however, a numerical
fitting to these forms impractical. The set of rules defining the behaviour of the particles at walls, as
imposed in our simulation, leads here to the slip boundary conditions. The discontinuity of the velocity
profile at the wall boundary is clearly visible in figure 4 (c) and is a characteristic feature of the flows
of polymeric fluids. In figure 4 (d), we compare two velocity profiles of the flows obtained with applying
equation (2.8)–(2.9), i.e., the Gaussian distribution of the bulk force, and a uniform distribution of the bulk
force. We note that even for such an extremely large value of the bulk force, the profiles are practically
identical. We expect that for very long polymers, the Gaussian distribution of the bulk force would prove
Figure 3. (Color online) T∗
⊥ profile at various amplitudes of bulk force f indicated in the figure. (a) simple
liquid, Lo = 1; (b) longest oligomer, Lo = 20.
13609-5
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
Figure 4. (Color online) Profiles for the velocity component along the flow direction, vx , evaluated bead-
wise at several values of f (indicated in the figure). (a) the case of simple fluid, Lo = 1; (b) the case of the
longest oligomer considered, Lo = 20; (c) near-wall behaviour of the velocity profiles for Lo = 20; (d) the
velocity profiles for f = 0.4 and for Lo = 20 evaluated with the Gaussian distribution of the bulk force,
[cf. equation (2.8)–(2.9)], (symbols), and a uniform distribution of the bulk force (solid line).
beneficiary and could lead to better stability of the integration of the equations of motion.
An alternative route is to concentrate on the details of the microstructure of the oligomer flow, be-
cause these must be responsible for its non-Newtonian behaviour. In particular, comparing to the case
of a simple fluid, oligomers have additional conformational degrees of freedom which will affect their
flow properties. Therefore, we build the profiles for the average shape anisotropy and the molecular
orientation for the oligomers in a flow. The components of the gyration tensor
Gα,β =
1
Lo
Lo
∑
i=1
(ri ,α−Rα)(ri ,β−Rβ) (2.10)
are evaluated for each oligomer of length Lo at a given time instance t . Here, α,β denote the Cartesian
axes, ri ,α are the coordinates of i th monomer, and Rα are the coordinates for the center of mass of the
oligomer. In the equivalent ellipsoid representation, the eigenvalues λ1 > λ2 > λ3 of this tensor provide
the squared lengths of its semiaxes, whereas the respective eigenvectors u1, u2 and u3 — the orientation
of these axes in space.
The shape anisotropy of an individual oligomer can be defined as
κ2 =
3
2
λ2
1 +λ2
2 +λ2
3
[λ1 +λ2 +λ3]2
−
1
2
. (2.11)
It is zero for a spherically symmetric body, where λ1 = λ2 = λ3 > 0 and is equal to 1 for an infinitely long
thin rod, where λ1 > 0, λ2 = λ3 = 0. The average profile is built for the shape anisotropy in a steady state.
It is obtained by first binning the system in Z -axis and averaging κ2 for individual oligomers found in
each bin. Then, time averaging within the steady state is performed.
The orientation of each oligomer in space is defined by that for the longest axis of its equivalent
ellipsoid. The latter is characterised by the eigenvector u1 associated with the largest eigenvalue λ1. The
level of alignment of the oligomer along the flow axis X can be characterised by the order parameter:
Sx = P2(u1 · x̂), (2.12)
where x̂ is defined in equation (2.7) and P2(x) is the second Legendre polynomial. The alignment profile is
built then in a steady state by averaging Sx in each bin and then performing time averaging. It is obvious
that both κ2 and Sx can be defined for the case Lo > 1 only.
13609-6
Flow of oligomeric fluid
Figure 5. (Color online) Profiles for average shape anisotropy [(a) and (b)] and oligomer alignment [(c)
and (d)]. Respective oligomer length Lo and the magnitude of the flow force are indicated in each plot.
Average shape anisotropy and oligomer alignment profiles are shown in figure 5 for the shortest
Lo = 4 and longest Lo = 20 oligomer considered in this study. The case Lo = 4 is characterised by flat
anisotropy profile with the value κ2 ≈ 0.6 independent of the magnitude of the flow force [see, frame (a)].
One can conclude that for the oligomer being this short, the flow does not change its shape (at least for the
flow force magnitude range used here). For the longest oligomer, Lo = 20 [frame (b)], the average value
of κ2 over the profile is close to that for Lo = 4, but the profile exhibits distinct shoulders with higher κ2
values near both channel edges and a well in its center. The channel edges, therefore, promote a stronger
anisotropy for the adjacent oligomers, presumably due to entropic effects.
The alignment profile for the shortest oligomer [frame (c)] indicates no orientational order in the
middle of the channel (Sx ≈ 0), whereas a relatively weak alignment is observed near the channel edges,
which rises to Sx ∼ 0.35 with the increase of the flow force magnitude f . For the longest oligomer [frame
(d)], the alignment profile has a cosine-like shape which moves up almost unchanged with an increase of
f . It is non-zero in the middle of a channel for all f being considered. Therefore, at least for longer
oligomers, Lo > 4, there is a variation of the oligomer shape and alignment across the channel: the
molecules are found to be much more elongated and aligned near the edges as compared to the mid-
dle part.
The flow-induced deformation of the polymer molecules renders their shape to be more similar to
liquid crystals. The effect is detected for longer oligomers Lo > 4 and stronger flows, where the effective
length-to-breadth ratio of oligomer exceeds a certain threshold. Similar effect is well known for the sys-
tems of anisotropic hard bodies, where the orientationally ordered phases are also found above certain
threshold length-to-breadth ratio [59–62]. Using this liquid crystal analogy, we recall the results obtained
by Mazza et al. [63, 64] reporting the high self-diffusivity of the Gay-Berne-Kihara fluid along the director
in the “supernematic” phase. Following these findings, one expects an essential reduction of the friction
between the aligned oligomers near the channel edges, as compared to that in the central part. Larger
friction between oligomers in the middle of a channel is seen as the reason for the suppression of the ve-
locity profile here and, as a result, its non-parabolic, bell-like shape [cf. figure 4 (b)], and the appearance
of the slip boundary condition [cf. figure 4 (c)].
13609-7
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
3. Flow of oligomeric fluid through a channel with polymer modified
walls
We turn now to the case when the pore walls are modified by polymer brushes arranged in the form
of stripes (see, figure 6). Each chain of a brush is of length L = 20 beads of type A, the pore interior is filled
with the oligomer fluid of the length Lo beads of type B , the contraflow regions contain an oligomer fluid
of the length Lo beads of type A. The difference between the bead types is in the value of the repulsion
amplitude a in equation (2.2) being set to aA A = aBB = 25 and aAB = 40 for the interaction of similar
and dissimilar beads, respectively. Therefore, the oligomer acts as a bad solvent for the brush. The good
solvent case, aA A = aBB = aAB = 25, is briefly discussed in the end of this section.
The equilibrium properties of the setup depicted in figure 6 for the case of Lo = 1 and no contraflow
regions are studied in detail in reference [39, 40]. Equilibrium morphology was found to depend on the
parameters d and w , and is formed as a result of an interplay between the enthalpy and the entropy
of the system. In particular, at small w ≪ L and any d , the adjacent brush stripes belonging to the same
wall merge and form a homogeneous “coat” on the wall resulting in the lamellar morphology. In this case,
the chains are stretched and aligned along the X -axis. With an increase of w , the adjacent brush stripes
are incapable of merging any more. Instead, they either stay separately (at relatively large d ∼ L) or
merge across the pore with their counterparts grafted to the opposite wall to form a pillar phase (at small
enough d < L). In this case, the brush chains are stretched and aligned in Z direction. This demonstrates
a strong correlation between the alignment direction of brush chains and the topology of the equilibrium
morphology. Therefore, it looks plausible that the change of the alignment of the brush chain by means
of an external stimulus could result in a morphology change in the system.
This is the case, indeed, when a flow force above certain threshold value is applied to the fluid in the
pore. Let us consider first the visual representation of morphology changes in the form of a snapshot se-
quence. The case of d = 13.333, w = 10, c = 4, at various values of the force f is presented in figure 7. For
this geometry of a pore, a stable pillar phase is observed when no or weak flow force is applied [cf. refer-
ence [39] and figure 7 (a)]. With an increase of f above the threshold value of f ≈ 0.04−0.06, the pillars
break and the morphology switches to the modulated lamellar morphology [see figure 7 (b)]. The layers,
formed of brush chains bent along the flow, gradually flatten as f increases further, as shown in fig-
ure 7 (c). One should remark that a perfect stationary lamellar morphology is also aided by a microphase
separation between the A beads of tethered chains and B beads of the flowing oligomer.
Let us check the quality of temperature conservation, similarly to the analysis performed in section 2
for the pore with flat boundaries. As follows from figure 8 (a), maximum deviation of the temperature
profile from the required value 1 does not exceed 4% if the force amplitude is restricted to f É 0.4 for
both cases of Lo = 1 and Lo = 20. It is worth mentioning that the maximum usable value for f = 0.4
here is twice as large as its counterpart for the case of flat boundaries, see figure 3. This relation can
be attributed to the fact that the total amount of the force applied inside a pore with polymer modified
d
c
c
w w . . .
Figure 6. (Color online) Extension of the simulation box geometry of figure 1 to the case of a slit-like
pore with its internal walls modified by stripes of polymer brushes (displayed in blue). The stripes are of
width w and are periodic along the X -axis (direction of flow). Pore interior is filled with oligomeric fluid
(displayed in green). The flow force is applied to the oligomers only.
13609-8
Flow of oligomeric fluid
(a)
(b)
(c)
Figure 7. (Color online) Sequence of snapshots showing the flow-induced transitions from pillar (a)
through modulated lamellar (b) into flatten lamellar (c) morphology. System geometry: d = 13.333,
w = 10, c = 4, Lo = 1, the flow force magnitude is f = 0.02, 0.1, and 0.4 for (a), (b) and (c), respectively.
Colours follow these in figure 6, contraflow regions not shown.
Figure 8. (Color online) T∗
⊥ profile at various amplitudes of the bulk force f indicated in the figure. Ge-
ometry d = 13.333, w = 10, c = 4 is considered with polymer-modified pore boundaries. (a) simple liquid,
Lo = 1; (b) longest oligomer simulated, Lo = 20.
boundaries (figure 8) is twice less compared to the case of the pore with flat boundaries (figure 3). This is
so due to the fact that no force is applied to the polymer brush beads (which are half of all the beads in
the system).
The average profiles for the velocity component along the flow direction, vx are shown in figure 9 at
various force amplitudes f = 0.05, 0.1, 0.2 and 0.4 for the cases of Lo = 1 and Lo = 20. Comparing these
profiles with their counterparts for the case of flat internal walls (figure 4), one can make the following
observations. Firstly, at f Ê 0.1, the profiles exhibit two “shoulders” near each internal wall which are
characterized by zero values for vx . These are, obviously, the regions occupied by the polymer brush
which “envelopes” the internal walls (see, figure 7). The flow is completely suppressed within these lay-
ers, rendering the walls thicker and reducing the pore size accessible to the flow. As a consequence, the
maxima for vx decrease compared to the case of flat walls. Secondly, the shape of the central part of
Figure 9. (Color online) Profile for velocity component along the flow direction, vx , evaluated bead-wise
at several values of f (indicated in the figure) for the setup depicted in figure 6, the other parameters are
the same as in figure 8.
13609-9
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
0 2 4 6 8 10 12 14 16 18 20
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
rrrrrrrrrrrrrrrrrrrr
r
r
r
r
r
r
rrr
r
rr
r
r
rr
r
r
r
r
r
rrrrrrrrrrrrrrrrrrrrrr
r
r
r
r
r
r
r
r
r
r
rr
r
r
rrr
r
r
rrr
rrrrrrrrrrrrrrrrrrr❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜
❜
❜
❜
❜
❜
❜
❜
❜
❜
❜❜
❜
❜
❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜
z
✛ bh
✲
Lo = 10
f = 0.2
z1 z2
(a)
✛ bh
✲
ρ′s
vx
ρ′pρ′p
0.0 0.1 0.2 0.3 0.4
4
5
6
7
r
r r r
❜ ❜ ❜ ❜
f
rdeff
Lo = 20
Lo = 10
Lo = 4
❜ Lo = 1
lamellar: 〈deff〉 ≈ 6.4
(b)
Figure 10. (Color online) (a) Schema explaining the estimates for the effective pore size deff in stationary
lamellar morphology for Lo = 10, and f = 0.2, ρ′p and ρ′s are reduced density profiles for brush and
oligomer beads, respectively. bh is brush thickness (3.1), z1 and z2 are the intersection points of ρ′p
and ρ′s . The velocity profile vx is also shown (not in scale). (b) deff at various Lo and f . Solid legends:
estimates from the intersection points z1 and z2 , open legends: esimates from the brush height bh . The
dashed vertical line marks the formation of the lamellar morphology.
each velocity profile follows closely their counterparts in figure 4. It is of parabolic shape for Lo = 1 and
bell-like for Lo = 20, suggesting qualitative similarities between both flows.
This interpretation brings up the possibility to treat a fluid flowwithin a stationary lamellar morphol-
ogy similarly to the case of the pore with flat boundaries, discussed in section 2, except for the smaller
effective pore size deff [65]. To evaluate the latter, one can use the expression for an average brush thick-
ness
bh = 2
∫
z̃ρp (z̃)dz̃
∫
ρp (z̃)dz̃
, (3.1)
where ρp (z̃) is the density profile of the beads that belong to the tethered chains and z̃ is the distance
from the nearest pore boundary along the Z -axis. In this case, one obtains deff = d −2bh . Alternatively,
the effective pore size can be estimated as the distance between the intersection points z1 and z2 for
ρ′
p = ρp /ρ and ρ′
s = ρs /ρ, the reduced density profiles for the polymer and the flowing oligomer beads,
respectively. This is illustrated in figure 10 (a). It is evident that, for this particular case, both estimates
for deff are extremely close. To check how this observation holds for other oligomer lengths Lo and flow
forces f , we performed both types of estimates for deff in each case. The results are presented in fig-
ure 10 (b), where the estimates for deff made from the intersection points z1 and z2 are presented via
solid legends, whereas the estimates performed via the evaluation of bh are shown via open symbols.
One can make several conclusions from this plot. First, the value of deff, evaluated by both approaches,
are similar to each other for f Ê 0.1. This threshold correlates well with the value of f , at which the
stationary lamellar morphology is formed (marked with the dashed line in figure 10). While deff can be
also calculated at smaller values of f , these results would carry no physical significance due to the pillar
morphology. Second, the difference between the values for deff estimated by means of two alternative
methods at the same Lo and at the same f , does not exceed 4%. Therefore, either of the estimates for deff
can be used. Third, there is a trend for an increase of deff with the growth of oligomer length Lo, although
it is rather modest. For example, for the case of f = 0.4 the value of deff for oligomer length of Lo = 20 is
only 10% higher than its counterparts for Lo = 1 and 4, and this increase is of the order of the error in the
estimates of deff mentioned above. A slight increase of the brush height with an increasing flow can be
attributed to the fact that there is some residual flow of oligomers inside the brush. As the flow increases,
the flow-induced elongation of the oligomers leads to an increase of their effective size and this will lead
to a slight increase of the brush height.
As was discussed in section 2, flat walls of the setup depicted in figure 1 act as effective “stretchers”
and “aligners” for the adjacent oligomer molecules, which results in characteristic profiles for κ2 and Sx
shown in figure 5. It is, therefore, of interest to see whether or not the effective walls formed by a flat-
tened polymer brush, as pictured in figure 7 (b) and (c), have a similar impact on the adjacent oligomer
molecules. We examine the aligning capabilities of such flattened brushes more in detail, considering
both cases of bad and good oligomer solvent. For the former case, the repulsion parameter a in equa-
13609-10
Flow of oligomeric fluid
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=4
(a)
κ2
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=20
(b)
κ2
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=4
(c)
Sx
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=20
(d)
Sx 0.10
0.20
0.40
Figure 11. (Color online) Profiles of the average shape anisotropy [(a) and (b)] and oligomer alignment [(c)
and (d)] for the geometry depicted in figure 6 and the case of bad oligomer solvent. Respective oligomer
length Lo and the magnitude of the flow force are indicated in each plot.
tion (2.2) is set to aAB = 40 for the interaction between oligomer and brush monomers, whereas for the
latter case we set aAB = 25. The repulsion parameter between similar beads is equal to aA A = aBB = 25
in both cases.
For the bad solvent case, the flattened brush and oligomer flow are strongly demixed with oligomers
being expelled from the brush-rich regions. The profiles for κ2 and Sx are shown in figure 11 for the
oligomer lengths of Lo = 4 and Lo = 20. Here, we make use of our estimates for the effective pore size,
deff, indicated on each plot by vertical dashed lines. If restricted to this region, then the profiles depicted
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=4
(a)
κ2
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=20
(b)
κ2
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
Lo=4
(c)
Sx
0.10
0.20
0.40
4 6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
z
(d)
Lo=20
Sx 0.10
0.20
0.40
Figure 12. (Color online) The same as in figure 11 but for the case of good oligomer solvent.
13609-11
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
in figure 11 are extremely close to their respective counterparts for the case of flat walls shown in figure 1
in both shape and absolute values, save for being “squeezed” into the smaller pore size deff. This indicates
that the existing roughness of the flattened polymer wall does not reduce its impact on the adjacent
oligomer molecules.
For the good solvent case, the brush and oligomer mix well if no flow force is applied, but a flow-
driven lamellarization of the system takes place at about f = 0.2, similarly to the bad solvent case. The
profiles for κ2 and Sx are shown in figure 12 for the oligomer lengths of Lo = 4 and Lo = 20. The estimated
effective pore size deff is of the same order but a fraction smaller than that for the bad solvent case. This is
indicated in figure 12 by vertical dashed lines. One should remark that despite the strong alignment of the
polymer brush and good mixing between the brush and the oligomers, the latter are found less elongated
and less aligned along a flow compared to the bad solvent case shown in figure 11. The respective curves
for κ2 are lower by about 0.1 compared to their counterparts for the bad solvent case, whereas these for
Sx are about 0.2 lower. One should attribute this to the fact that the flattening of the brush in the case
of a good solvent requires a higher flow force compared to the case of a bad solvent. In the latter case,
lamellarization is also aided by the microphase separation between the brush and the oligomer fluid.
Despite these quantitative differences, the qualitative picture emerging for both cases of the bad and
good solvent is essentially the same. Namely, at a certain value of the flow force f , the stationary lamellar
phase is formed with the flowing oligomer occupying the center part of the pore. The oligomer is found
essentially elongated and aligned along the flow near the walls of this channel andmuch less in the center
of the pore. This effect is detected for the oligomer lengths Lo > 4 and, due to its impact on the distribution
of the local friction across the pore, affects the behaviour of the fluid turning it into a non-Newtonian one.
4. Conclusions
In this paper, we developed the simulation approach, which allows one to simulate the pressure-
driven flow in the pore with flat and polymer-modified walls. It combines the earlier ideas of fluid-like
walls and reverse flow. The former enables to avoid highly structured solid walls that usually lead to
near-wall artefacts. The latter introduces friction between oppositely flowing streams which makes it
possible to conserve the total momenta and keep the temperature constant. Our system geometry con-
tains the central main pore “enveloped” by two fluid-like pores on each side. The flow force is introduced
in the main pore and oppositely directed contra-flow of the same magnitude — in both fluid-like walls.
Simulation of the oligomer flow through the pore with flat walls is used as a check for the credibility
of the method and reproduction of the hydrodynamic boundary conditions. Good thermostatting of the
system is achieved when the flow force magnitude does not exceed a certain threshold. For the case of
the oligomer length Lo > 4, we found the molecules adjacent to the central pore boundaries essentially
stretched and aligned along the flow, whereas their shape ismore spherical and less aligned in themiddle
of a pore. This provides the basis for variation of the local friction across the pore and, as a consequence,
the non-parabolicity of the velocity profile for the oligomer fluid and the slip boundary condition.
The case of polymer-modified walls is also consideredwhen the polymer brush has the form of stripes
arranged perpendicularly to the flow direction. In this case, at a certain threshold value of the flow force,
one observes the pillar-to-lamellar transition induced by the flow which leads to the brush enveloping
the pore wall with a relatively smooth layer. At higher flow rates, the flow of oligomer is similar to the
case of flat walls, although for the narrower effective pore size. The latter is estimated both from the
intersection of density profiles for the brush and the flowing oligomer and from the integral equation for
the average brush thickness. The effect of local stretching and alignment of oligomers near the walls of
the effective pore is detected the same as for the case of flat walls.
The method can be extended to more complex systems, namely: the flow of mixtures and their flow-
induced separation; the flow of amphiphilic molecules; the flow of complex macromolecules or their
solutions. Combined with the fine-tunable structure of the brush, this opens up a possibility to study
various problems of transport of oligo- and macromolecules through a complex structured environment.
13609-12
Flow of oligomeric fluid
Acknowledgements
This work was supported by the EU under IRSES Project STCSCMBS 268498.
References
1. Zhao B., Brittain W.J., Prog. Polym. Sci., 2000, 25, 677; doi:10.1016/S0079-6700(00)00012-5.
2. Minko S., J. Macromol. Sci., Polymer Rev., 2006, 46, 397; doi:10.1080/15583720600945402.
3. Alexander S.J., J. Phys. (Paris), 1977, 38, 983; doi:10.1051/jphys:01977003808098300.
4. De Gennes P.-J., J. Phys. (Paris), 1976, 37, 1445; doi:10.1051/jphys:0197600370120144500.
5. De Gennes P.-J., Macromolecules, 1980, 13, 1069; doi:10.1021/ma60077a009.
6. Milner S.T., Witten T.A., Cates M.E., Europhys. Lett., 1988, 5, 413; doi:10.1209/0295-5075/5/5/006.
7. Scheutjens J.M.H.M., Fleer G.J., Cohen Stuart M.A., Colloids Surf., 1986, 21, 285; doi:10.1016/0166-6622(86)80098-1.
8. Wijmans C.M., Scheutjens J.M.H.M., Zhulina E.B., Macromolecules, 1992, 25, 2657; doi:10.1021/ma00036a016.
9. Netz R.N., Schick M., Macromolecules, 1998, 31, 5105; doi:10.1021/ma9717505.
10. Carigano M.A., Szleifer I., J. Chem. Phys., 1995, 102, 8662; doi:10.1063/1.468968.
11. McCoy J.D., Teixeira M.A., Curro J.G.J., J. Chem. Phys., 2002, 117, 2975; doi:10.1063/1.1491242.
12. Cao D.P., Wu J., Langmuir, 2006, 22, 2712; doi:10.1021/la0527588.
13. Xu X.F., Cao D.P., J. Chem. Phys., 2009, 130, 164901; doi:10.1063/1.3119311.
14. Xu X.F., Cao D.P., Soft Matter, 2010, 6, 4631; doi:10.1039/C0SM00034E.
15. Milchev A., Egorov S.A., Binder K., J. Chem. Phys., 2010, 132, 184905; doi:10.1063/1.3414996.
16. Borówko M., Sokołowski S., Staszewski T., J. Phys. Chem. B, 2013, 117, 10293; doi:10.1021/jp4027546.
17. Grest G.S., J. Chem. Phys., 1996, 105, 5532; doi:10.1063/1.472395.
18. Weinhold J.D., Kumar S.K., J. Chem. Phys., 1994, 101, 4312; doi:10.1063/1.467481.
19. Pastorino C., Binder K., Keer T., Müller M., J. Chem. Phys., 2006, 124, 064902; doi:10.1063/1.2162883.
20. MacDowell L.G., Müller M., J. Chem. Phys., 2006, 124, 084907; doi:10.1063/1.2172597.
21. Fouqueau A., Meuwly M., Bemish R.J., J. Phys. Chem. B, 2007, 111, 10208; doi:10.1021/jp071721o.
22. Hadziioannou G., Patel S., Granick S., Tirrell M., J. Am. Chem. Soc., 1986, 108, 2869; doi:10.1021/ja00271a014.
23. Taunton H.J., Toprahcioglu C., Fetters L.J., Klein J., Nature, 1988, 332, 712; doi:10.1038/332712a0.
24. Taunton H.J., Toprahcioglu C., Fetters L.J., Klein J., Macromolecules, 1990, 23, 571; doi:10.1021/ma00204a033.
25. Auroy P., Auvray L., Leger L., Phys. Rev. Lett., 1991, 66, 719; doi:10.1103/PhysRevLett.66.719.
26. Brinkman H.C., Appl. Sci. Res., 1947, 1, 27; doi:10.1007/BF02120313.
27. Milner S.T., Macromolecules, 1991, 24, 3704; doi:10.1021/ma00012a036.
28. Servantie J., Müller M., Phys. Rev. Lett., 2008, 101, 026101; doi:10.1103/PhysRevLett.101.026101.
29. Squires T.M., Quake A.R., Rev. Mod. Phys., 2005, 77, 977; doi:10.1103/RevModPhys.77.977.
30. Schoch R.B., Han J., Renaud P., Rev. Mod. Phys., 2008, 80, 839; doi:10.1103/RevModPhys.80.839.
31. Klein J., Perahia D., Warburg S., Nature, 1991, 352, 143; doi:10.1038/352143a0.
32. Lai P.Y., Binder K., J. Chem. Phys., 1993, 98, 2366; doi:10.1063/1.464164.
33. Doyle P.S., Shaqfeh E.S.G., Gast A.P., Phys. Rev. Lett., 1997, 78, 1182; doi:10.1103/PhysRevLett.78.1182.
34. Grest G.S., Phys. Rev. Lett., 1996, 76, 4979; doi:10.1103/PhysRevLett.76.4979.
35. Adiga S.P., Brenner D.W., Nano Lett., 2005, 5, 2509; doi:10.1021/nl051843x.
36. Huang J., Wang Y., Laradji M., Macromolecules, 2006, 39, 5546; doi:10.1021/ma060628f.
37. Masoud H., Alexeev A., Chem. Commun., 2011, 47, 472; doi:10.1039/c0cc02165b.
38. Dimitrov D.I., Milchev A., Binder K., Macromol. Theory Simul., 2008, 17, 313; doi:10.1002/mats.200800038.
39. Ilnytskyi J.M., Patsahan T., Sokołowski S., J. Chem. Phys., 2011, 134, 204903; doi:10.1063/1.3592562.
40. Ilnytskyi J.M., Sokołowski S., Patsahan T., Condens. Matter Phys., 2013, 16, 13606; doi:10.5488/CMP.16.13606.
41. Paik M.Y., Xu Y., Rastogi A., Tanaka M., Yi Y., Ober C.K., Nano Lett., 2010, 10, 3873; doi:10.1021/nl102910f.
42. Groot R.D., Warren P.B., J. Chem. Phys., 1997, 107, 4423; doi:10.1063/1.474784.
43. Español P., Warren P., Europhys. Lett., 1995, 30, 191; doi:10.1209/0295-5075/30/4/001.
44. Ashurst W.T., Hoover W.G., Phys. Rev. Lett., 1973, 31, 206; doi:10.1103/PhysRevLett.31.206.
45. Levesque D., Ashurst W.T., Phys. Rev. Lett., 1974, 33, 277; doi:10.1103/PhysRevLett.33.277.
46. Ashurst W.T., Hoover W.G., Phys. Rev. A, 1975, 11, 658; doi:10.1103/PhysRevA.11.658.
47. Koplik J., Banavar J.R., Willemsen J.F., Phys. Rev. Lett., 1988, 60, 1282; doi:10.1103/PhysRevLett.60.1282.
48. Goujon F., Malfreyt P., Tildesley D.J., ChemPhysChem, 2004, 5, 457; doi:10.1002/cphc.200300901.
49. Goujon F., Malfreyt P., Tildesley D.J., Mol. Phys., 2005, 103, 2675; doi:10.1080/00268970500134706.
50. Visser D.C., Hoefsloot H.C.J., Iedema P.D., J. Comput. Phys., 2005, 205, 626; doi:10.1016/j.jcp.2004.11.020.
51. Pivkin I.V., Karniadakis G.E., J. Comput. Phys., 2005, 207, 114; doi:10.1016/j.jcp.2005.01.006.
13609-13
http://dx.doi.org/10.1016/S0079-6700(00)00012-5
http://dx.doi.org/10.1080/15583720600945402
http://dx.doi.org/10.1051/jphys:01977003808098300
http://dx.doi.org/10.1051/jphys:0197600370120144500
http://dx.doi.org/10.1021/ma60077a009
http://dx.doi.org/10.1209/0295-5075/5/5/006
http://dx.doi.org/10.1016/0166-6622(86)80098-1
http://dx.doi.org/10.1021/ma00036a016
http://dx.doi.org/10.1021/ma9717505
http://dx.doi.org/10.1063/1.468968
http://dx.doi.org/10.1063/1.1491242
http://dx.doi.org/10.1021/la0527588
http://dx.doi.org/10.1063/1.3119311
http://dx.doi.org/10.1039/C0SM00034E
http://dx.doi.org/10.1063/1.3414996
http://dx.doi.org/10.1021/jp4027546
http://dx.doi.org/10.1063/1.472395
http://dx.doi.org/10.1063/1.467481
http://dx.doi.org/10.1063/1.2162883
http://dx.doi.org/10.1063/1.2172597
http://dx.doi.org/10.1021/jp071721o
http://dx.doi.org/10.1021/ja00271a014
http://dx.doi.org/10.1038/332712a0
http://dx.doi.org/10.1021/ma00204a033
http://dx.doi.org/10.1103/PhysRevLett.66.719
http://dx.doi.org/10.1007/BF02120313
http://dx.doi.org/10.1021/ma00012a036
http://dx.doi.org/10.1103/PhysRevLett.101.026101
http://dx.doi.org/10.1103/RevModPhys.77.977
http://dx.doi.org/10.1103/RevModPhys.80.839
http://dx.doi.org/10.1038/352143a0
http://dx.doi.org/10.1063/1.464164
http://dx.doi.org/10.1103/PhysRevLett.78.1182
http://dx.doi.org/10.1103/PhysRevLett.76.4979
http://dx.doi.org/10.1021/nl051843x
http://dx.doi.org/10.1021/ma060628f
http://dx.doi.org/10.1039/c0cc02165b
http://dx.doi.org/10.1002/mats.200800038
http://dx.doi.org/10.1063/1.3592562
http://dx.doi.org/10.5488/CMP.16.13606
http://dx.doi.org/10.1021/nl102910f
http://dx.doi.org/10.1063/1.474784
http://dx.doi.org/10.1209/0295-5075/30/4/001
http://dx.doi.org/10.1103/PhysRevLett.31.206
http://dx.doi.org/10.1103/PhysRevLett.33.277
http://dx.doi.org/10.1103/PhysRevA.11.658
http://dx.doi.org/10.1103/PhysRevLett.60.1282
http://dx.doi.org/10.1002/cphc.200300901
http://dx.doi.org/10.1080/00268970500134706
http://dx.doi.org/10.1016/j.jcp.2004.11.020
http://dx.doi.org/10.1016/j.jcp.2005.01.006
J.M. Ilnytskyi, P. Bryk, A. Patrykiejew
52. Pivkin I.V., Karniadakis G.E., J. Chem. Phys., 2006, 124, 184101; doi:10.1063/1.2191050.
53. Jian F., Yongmin H., Honglai L., Ying H., Front. Chem. Eng. China, 2007, 1, 132; doi:10.1007/s11705-007-0025-5.
54. Ghosh A., Paredes R., Luding S., In: Proceedings of Congres on Particle Technology, PARTEC 2007,
(Nürnberg, 2007), Peukert W., Schreglmann C. (Eds.), Nürnberg Messe, Nürnberg, 2007.
55. Pastorino C., Kreer T., Müller M., Binder K., Phys. Rev. E, 2007, 76, 026706; doi:10.1103/PhysRevE.76.026706.
56. Müller-Plathe F., Phys. Rev. E, 1999, 59, 4894; doi:10.1103/PhysRevE.59.4894.
57. Backer J.A., Lowe C.P., Hoefsloot H.C.J., Iedema P.D., J. Chem. Phys., 2005, 122, 154503; doi:10.1063/1.1883163.
58. Fedosov D.A., Karniadakis G.E., Caswell B., J. Chem. Phys., 2010, 132, 144103; doi:10.1063/1.3366658.
59. Vieillard-Baron J., Mol. Phys., 1974, 28, 809; doi:10.1080/00268977400102161.
60. Frenkel D., Lekkerkerker H.N.W., Stroobants A., Nature, 1988, 332, 822; doi:10.1038/332822a0.
61. McGrother S.C., Williamson D.C., Jackson G., J. Chem. Phys., 1966, 104, 6755; doi:10.1063/1.471343.
62. Cuetos A., Dijkstra M., Phys. Rev. Lett., 2007, 98, 095701; doi:10.1103/PhysRevLett.98.095701.
63. Mazza M.G., Greschek M., Valiullin R., Kärger J., Schoen M., Phys. Rev. Lett., 2010, 105, 227802;
doi:10.1103/PhysRevLett.105.227802.
64. MazzaM.G., GreschekM., Valiullin R., SchoenM., Phys. Rev. E, 2011, 83, 051704; doi:10.1103/PhysRevE.83.051704.
65. Pastorino C., Binder K., Müller M., Macromolecules, 2009, 42, 401; doi:10.1021/ma8015757.
Потiк олiгомерного флюїду в нано-каналi iз комплексною
структурою пiд впливом зовнiшнього тиску.
Дослiдження методом дисипативної динамiки
Я.М. Iльницький1, П. Брик2, А. Патрикєєв2
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Вiддiл моделювання фiзико-хiмiчних процесiв, Унiверситет Марiї Кюрi-Склодовської,
20–031 Люблiн, Польща
Розвинуто симуляцiйний метод, спрямований на моделювання потоку у порi iз гладкими та полiмер-
модифiкованими стiнками. Пiдхiд грунтується на методi дисипативної динамiки i реалiзує iдеї “рiдинних
стiнок” та реверсних потокiв. Як тест розглянуто потiк олiгомерiв крiзь пору iз гладкими стiнками i проде-
монстровано добре термостатування системи при застосуваннi цього методу. Отримано неоднорiдностi
як форми плинних олiгомерiв, так i ступеня їх вирiвнювання вздовж пори, якi призводять до непараболi-
чних профiлiв швидкостей. Метод застосовано до наноканалу, декорованого смугами полiмерних щiток,
розташованих перпендикулярно до напрямку потоку. При певному граничному значеннi сили потоку за-
фiксовано перехiд iз стовпцевої до ламеларної морфологiї, який призводить до обгортання стiнок пори
гладким шаром iз полiмерних щiток. При вищих швидкостях потiк олiгомерiв набуває властивостей ана-
логiчних до випадку гладких стiнок, але iз вужчим ефективним розмiром пори. Спостережено розтяг i
вирiвнювання полiмерних молекул вздовж напрямку потоку поблизу стiнок пори.
Ключовi слова: Пуазейлевий потiк, полiмерна щiтка, олiгомери, метод дисипативної динамiки
13609-14
http://dx.doi.org/10.1063/1.2191050
http://dx.doi.org/10.1007/s11705-007-0025-5
http://dx.doi.org/10.1103/PhysRevE.76.026706
http://dx.doi.org/10.1103/PhysRevE.59.4894
http://dx.doi.org/10.1063/1.1883163
http://dx.doi.org/10.1063/1.3366658
http://dx.doi.org/10.1080/00268977400102161
http://dx.doi.org/10.1038/332822a0
http://dx.doi.org/10.1063/1.471343
http://dx.doi.org/10.1103/PhysRevLett.98.095701
http://dx.doi.org/10.1103/PhysRevLett.105.227802
http://dx.doi.org/10.1103/PhysRevE.83.051704
http://dx.doi.org/10.1021/ma8015757
Introduction
Flow of oligomeric fluid through a channel with flat walls
Flow of oligomeric fluid through a channel with polymer modified walls
Conclusions
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