Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study
We study the effect of the molecular architecture of amphiphilic star polymers on the shape of aggregates they form in water. Both solute and solvent are considered at a coarse-grained level by means of dissipative particle dynamics simulations. Four different molecular architectures are considere...
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| Cite this: | Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study / O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13802: 1–10. — Бібліогр.: 36 назв. — англ. |
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irk-123456789-1569692019-06-20T01:26:04Z Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study Kalyuzhnyi, O.Y. Ilnytskyi, J.M. C. von Ferber We study the effect of the molecular architecture of amphiphilic star polymers on the shape of aggregates they form in water. Both solute and solvent are considered at a coarse-grained level by means of dissipative particle dynamics simulations. Four different molecular architectures are considered: the miktoarm star, two different diblock stars and a group of linear diblock copolymers, all of the same composition and molecular weight. Aggregation is started from a closely packed bunch of Na molecules immersed into water. In most cases, a single aggregate is observed as a result of equilibration, and its shape characteristics are studied depending on the aggregation number Na. Four types of aggregate shape are observed: spherical, rod-like and disc-like micelle and a spherical vesicle. We estimate “phase boundaries” between these shapes depending on the molecular architecture. Sharp transitions between aspherical micelle and a vesicle are found in most cases. The pretransition region shows large amplitude oscillations of the shape characteristics with the oscillation frequency strongly dependent on the molecular architecture. В роботi вивчається ефект молекулярної архiтектури амфiфiльних зiркових полiмерiв на форму агрегатiв, якi вони формують у водi. Полiмери та вода описуються на мезоскопiчному рiвнi, використовуючи метод дисипативної динамiки. Розглянутi чотири молекулярнi архiтектури: мiктозiрки, два типи дiблок-зiрок i група лiнiйних дiблок кополiмерiв, усi з однаковою композицiєю та молекулярною масою. Ми розглядаємо початкову конфiгурацiю у виглядi густого клубка iз Na молекул, помiщеного у воду. В рiвноважному станi формується агрегат, характеристики форми якого дослiджуються при рiзних значеннях Na. Знайдено чотири форми агрегатiв: сферична, стержнеподiбна та дископодiбна мiцелла та сферична весiкула. Оцiненi “фазовi границi” мiж цими формами залежно вiд молекулярної архiтектури. У бiльшостi випадкiв знайдено рiзкий перехiд мiж асферичною мiцеллою та весiкулою. Передперехiдна область характеризується осциляцiями властивостей форми iз великою амплiтудою, частота яких суттєво залежить вiд молекулярної архiтектури. 2017 Article Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study / O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13802: 1–10. — Бібліогр.: 36 назв. — англ. 1607-324X PACS: 82.70.Uv, 78.67.Ve, 61.20.Ja DOI:10.5488/CMP.20.13802 arXiv:1703.10401 http://dspace.nbuv.gov.ua/handle/123456789/156969 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study the effect of the molecular architecture of amphiphilic star polymers on the shape of aggregates they
form in water. Both solute and solvent are considered at a coarse-grained level by means of dissipative particle
dynamics simulations. Four different molecular architectures are considered: the miktoarm star, two different
diblock stars and a group of linear diblock copolymers, all of the same composition and molecular weight. Aggregation is started from a closely packed bunch of Na molecules immersed into water. In most cases, a single
aggregate is observed as a result of equilibration, and its shape characteristics are studied depending on the aggregation number Na. Four types of aggregate shape are observed: spherical, rod-like and disc-like micelle and
a spherical vesicle. We estimate “phase boundaries” between these shapes depending on the molecular architecture. Sharp transitions between aspherical micelle and a vesicle are found in most cases. The pretransition
region shows large amplitude oscillations of the shape characteristics with the oscillation frequency strongly
dependent on the molecular architecture. |
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Article |
| author |
Kalyuzhnyi, O.Y. Ilnytskyi, J.M. C. von Ferber |
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Kalyuzhnyi, O.Y. Ilnytskyi, J.M. C. von Ferber Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study Condensed Matter Physics |
| author_facet |
Kalyuzhnyi, O.Y. Ilnytskyi, J.M. C. von Ferber |
| author_sort |
Kalyuzhnyi, O.Y. |
| title |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| title_short |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| title_full |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| title_fullStr |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| title_full_unstemmed |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| title_sort |
shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156969 |
| citation_txt |
Shape characteristics of the aggregates formed by amphiphilic stars in water: dissipative particle dynamics study / O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber // Condensed Matter Physics. — 2017. — Т. 20, № 1. — С. 13802: 1–10. — Бібліогр.: 36 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kalyuzhnyioy shapecharacteristicsoftheaggregatesformedbyamphiphilicstarsinwaterdissipativeparticledynamicsstudy AT ilnytskyijm shapecharacteristicsoftheaggregatesformedbyamphiphilicstarsinwaterdissipativeparticledynamicsstudy AT cvonferber shapecharacteristicsoftheaggregatesformedbyamphiphilicstarsinwaterdissipativeparticledynamicsstudy |
| first_indexed |
2025-07-14T09:19:44Z |
| last_indexed |
2025-07-14T09:19:44Z |
| _version_ |
1837613485908819968 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 1, 13802: 1–10
DOI: 10.5488/CMP.20.13802
http://www.icmp.lviv.ua/journal
Shape characteristics of the aggregates formed by
amphiphilic stars in water: dissipative particle
dynamics study∗
O.Y. Kalyuzhnyi1,4, J.M. Ilnytskyi1,4, C. von Ferber2,3,4
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, United Kingdom
3 Heinrich-Heine Universität Düsseldorf, D-40225 Düsseldorf, Germany
4 Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (L4),
D-04009 Leipzig, Germany
Received January 20, 2017, in final form February 16, 2017
We study the effect of the molecular architecture of amphiphilic star polymers on the shape of aggregates they
form in water. Both solute and solvent are considered at a coarse-grained level by means of dissipative particle
dynamics simulations. Four different molecular architectures are considered: the miktoarm star, two different
diblock stars and a group of linear diblock copolymers, all of the same composition and molecular weight. Ag-
gregation is started from a closely packed bunch of Na molecules immersed into water. In most cases, a single
aggregate is observed as a result of equilibration, and its shape characteristics are studied depending on the ag-
gregation number Na . Four types of aggregate shape are observed: spherical, rod-like and disc-like micelle and
a spherical vesicle. We estimate “phase boundaries” between these shapes depending on the molecular archi-
tecture. Sharp transitions between aspherical micelle and a vesicle are found in most cases. The pretransition
region shows large amplitude oscillations of the shape characteristics with the oscillation frequency strongly
dependent on the molecular architecture.
Key words: star-like polymer, amphiphiles, micelle, vesicle, dissipative particle dynamics
PACS: 82.70.Uv, 78.67.Ve, 61.20.Ja
1. Introduction
Star polymers represent one of the simplest branched polymeric architectures utilising several arms
in the form of linear chains linked to a central core. Branching imposes intramolecular constraints on the
monomers and modifies mechanical, viscoelastic, and solution properties of such molecules in bulk and
in a solution. A relatively recent review of the synthesis, properties and application of star polymers is
given in [1].
Recently, one observes a growing interest towards amphiphilic and polyphilic star polymers. There
are two main types of such star polymers, namely, the miktoarm and the diblock stars. In the miktoarm
star, all the arms are of a homopolymer type but their properties differ from arm to arm. In the di-
block star, each arm is a linear diblock copolymer itself. Both cases generate interest in both synthetic
methodologies and in the self-assembly of such molecules in solution [2–4]. In particular, depending on
the details of the molecular architecture, a number of morphologies are observed, such as: Archimedean
tiling patterns and cylindrical microdomains for miktoarm star copolymers as well as asymmetric lamel-
lar microdomains for diblock stars, which have not been reported for linear block copolymers, for more
details see [4].
∗This paper is written in honor of Yurij Holovatch on the occasion of his 60th birthday.
© O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber, 2017 13802-1
https://doi.org/10.5488/CMP.20.13802
http://www.icmp.lviv.ua/journal
O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber
Applications of star polymers range from nano- and micropatterning [1, 4] in gel and bulk state to
drug delivery via their micellisation in solution (see, e.g. [5]). In the latter case, both the size and the
shape of a micelle plays an important role from the requirements of low toxicity and efficient transport
of the drug. For example, it has been reported that smaller nanoparticles (∼ 25 nm) travel through the
lymphatic more readily than the larger particles (∼ 100 nm) and accumulate in lymph node resident
dendritic cells [6, 7]. Besides that, the shape of the particles also affect their bio-application: e.g., worm-
shaped filamentous micelles show less phagocytosis as compared to spheres, rod-like and red blood cell
discoidal particles boasted longer blood circulation time than spheres, which reduced their clearance
from the bloodstream (see [5] and references therein).
Besides experimental studies, the aggregation of polyphilic star polymers has been a subject of theo-
retical investigations [8] as well as computer simulations using various approaches. In particular, Monte
Carlo simulations [9, 10] performed on a lattice revealed a spontaneous formation of roughly spherical
aggregates of diblock stars in a selective solvent. Significant changes in the micellar properties, such as
the critical micellar concentration and aggregation number are reported upon changing the fraction of
the solvophobic part of the diblock while keeping the total arm length constant.
Coarse-grained approaches, such as the dissipative particle dynamics (DPD) [11, 12], make it possi-
ble to consider both a variety of molecular architectures and chemical compatibility of the particular
groups, combined with the computational efficiency. As a result, one can study microphase separation
driven effects in amphi- and polyphilic molecules, self-assembly, adsorption, etc. (see, e.g. [13–16]). With
respect to the aggregation of the polyphilic star molecules, there are a number of studies of polymero-
somes [17], ABC star block copolymers [18–20], as well as related systems, e.g., a mixture of diblock and
homopolymers [21]. A number of micellar shapes have been observed, such as spherical micelles, worm-
like micelles, hamburgers and others.
Therefore, coarse-grained DPD simulations can be considered as a useful instrument to study the size
and the shape of aggregates formed of amphi- and polyphilic molecules of star architecture and, there-
fore, are capable of sheding more light on their applicability for micelle-based drug delivery systems.
The aim of this work is to study the effect of the molecular architecture of amphiphilic star polymers on
the type of aggregates they form in water. We consider three types of stars: the miktoarm star and two
different diblock stars. These are compared against the equivalent set of linear diblock copolymers. The
molecular weight and the fraction of hydrophilic monomers are chosen to be the same in all cases. The
outline of the study is as follows. The simulation approach and the properties of interest are described in
section 2, the results are presented in section 3 followed by conclusions in section 4.
2. Simulation approach and properties of interest
We employ themesoscopic method of DPD [11, 12] simulations, which describes polymer molecules at
the level of coarse-grained beads each representing a fragment of the chain. The same applies to thewater
solvent, in which case a single solvent bead is assumed to contain several molecules of water. All beads
representing both a polymer and water are spheres of the same diameter which provides the length-
scale of the problem, whereas the energy scale is assumed to be ǫ∗ = kBT = 1, where kB is the Boltzmann
constant, T is the temperature, and time is expressed in t∗ = 1 [22]. The monomers are connected via
harmonic springs, which results in a force
FB
i j =−kxi j , (2.1)
where xi are the coordinates of i th bead, xi j = xi −x j and k is the spring constant. The non-bonded forces
contain three contributions
Fi j = F
C
i j +F
D
i j +F
R
i j , (2.2)
where FC
i j
is the conservative force resulting from the repulsion between i th and j th soft beads, FD
i j
is the
dissipative force that occurs due to the friction between soft beads, and random force FR
i j
that works in
pair with a dissipative force to thermostat the system. The expressions for all these three contributions
13802-2
Shape characteristics
are given below [22]
FC
i j =
a(1− xi j )
xi j
xi j
, xi j < 1,
0, xi j Ê 1,
(2.3)
F
D
i j =−γw D(xi j )(xi j ·vi j )
xi j
x2
i j
, (2.4)
FR
i j =σw R(xi j )θi j∆t−1/2
xi j
xi j
, (2.5)
where xi j = |xi j |, vi j = vi −v j , vi is the velocity of i th bead, a is the amplitude for the conservative repul-
sive force. The dissipative force has an amplitude γ and decays with the distance according to the weight
function w D(xi j ). The amplitude for the random force isσ and the respective weight function is w R(xi j ).
θi j is the Gaussian random variable and ∆t is the time-step of the simulations. As was shown by Español
and Warren [12], to satisfy the detailed balance requirement, the amplitudes and weight functions for
the dissipative and random forces should be interrelated: σ2 = 2γ and w D(xi j ) = [w R(xi j )]2.
We will denote the beads representing water as of type A. The compressibility of such coarse-grained
solvent at a number density of beads equal to ρ = 3 matches that for water at normal condition if the re-
pulsion parameter a in equation (2.3) is chosen equal to aAA = 25 [22]. Similarly, the level of hydrophilicity
of a polymer fragment composed of beads of type P can be controlled by the value aPA for the repulsion
interaction between beads P and A, where the difference aPA−aAA is related to the Flory-Huggins param-
eter [23]. In our study, we assume the hydrophilic fragments of a polymer chain to be composed of the
beads A, the same as for water, whereas the hydrophobic fragments — of the beads B, characterised by
the repulsion parameter aAB = 40, the value already used in a number of studies [14, 15, 24]. The other pa-
rameters are as follows: γ= 6.75, σ=
√
2γ= 3.67 and the time-step ∆t = 0.04. Another parametrisation
of the repulsive potential in DPD studies aimed at polymerosomes can be found in [17].
We use the simulation box with the periodic boundary conditions. Its linear dimension L is chosen at
least twice the linear dimension of the largest aggregate formed out of Nmol molecules each containing n
beads. This requirement appears from the need to restore the integrity of an aggregate if it is split across
one or several walls of the simulation box. A rough estimate for L can be achieved by assuming the for-
mation of a single spherical aggregate with the diameter Da. As far as each bead on average occupies the
volume of 1/ρ, the volume of an aggregate is nNmol/ρ = πD3
a/6, hence Da ∼ (6nNmol/πρ)1/3 and, there-
fore, the condition for the linear dimension of the simulation box reads L > 2Da. For highly aspherical
aggregates, the value of L should be accordingly increased.
To identify the existing aggregates at each time instance, we use the following algorithm:
(1) for each (imol, jmol) pair of molecules, find the number nc of close bead-bead intermolecular con-
tacts evaluated for hydrophobic beads (type B), and if nc > nmin, then register the link between the
molecules imol and jmol;
(2) using the linking list for molecular pairs, build neighbour lists for linked molecules;
(3) apply a “snowball” cluster identification algorithm, in which the neighboring molecules are added
to the existing ones within each aggregate until no neighbours remain;
(4) final information contains: the lists of molecules that belong to a particular kth aggregate, and the
index array to determine the aggregate to which any molecule belongs.
A close bead-bead contact is registered when the distance xi j between them is less than 1.5, and the
threshold number of close contacts nmin is chosen to be equal to 10.
After the aggregates have been identified, the lists of molecules for each of it is used to rejoin the
aggregates split by the periodic boundary conditions. Then, we proceed to the analysis of their size and
shape properties. The radius of gyration and the shape characteristics of each aggregate are derived from
the components of the gyration tensor Q defined as in [25, 26]:
Qαβ =
1
N
N
∑
n=1
(
xα
n −X α
)
(
x
β
n −X β
)
, α,β= 1,2,3. (2.6)
13802-3
O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber
Here, N is the total number of beads in an aggregate, xα
n denotes the coordinates of nth bead: xn =
(x1
n , x2
n , x3
n ), and X α = 1
N
∑N
n=1 xα
n are the coordinates of the center of mass for the aggregate. Its eigenvec-
tors define the axes of a local frame of the chain and the mass distribution of the latter along each axis is
given by the respective eigenvalue λi , i = 1,2,3, respectively. The trace of Q is an invariant with respect
to rotations and is equal to an instantaneous squared gyration radius of the chain
R2
g = TrQ= 3λ̄. (2.7)
Here, the average over three eigenvalues, λ̄, is introduced to simplify the following expressions. The
hydrodynamic radius Rh is found according to the following expression
R−1
h =
〈 1
ri j
〉
, (2.8)
where the averaging is performed over all pairs of beads that belong to the aggregate.
The shape properties of aggregates are characterised by the asphericity A (sometimes also referred to
as “the relative shape anisotropy”) and prolateness S [27–30], as well as the shape descriptor B introduced
in this study:
A =
1
6
∑3
i=1
(λi − λ̄)2
λ̄2
, S =
∏3
i=1
(λi − λ̄)
λ̄3
, B =−
λ2 − λ̄
λ̄
. (2.9)
For a spherical shape: A = S = B = 0, whereas for the ideal rod (λ1 = λ, λ2,3 = 0): A = 1, S = 2, B = 1 and
for the ideal disc (λ1,2 = λ, λ3 = 0): A = 1/4, S = −1/4, B = −1. The convenience of the use of the shape
descriptor B is that its range is symmetric spanning from −1 to 1 and it reverses its sign when the shape
changes from the disc-like to the rod-like.
At each time instant t , the largest aggregate, or “the giant component” in percolation terminology, is
identified and the set of its characteristics, R2
g (t), R−1
h (t), A(t), S(t) and B(t) are saved. Based on their
behaviour with time t , we estimate the time tequ needed for the aggregate to forget its initial state and
to start displaying stationary oscillatory behaviour in terms of its shape properties. The conservative
estimate reads t∗
ini
∼ 8000 (200 · 103 DPD steps). Then, the average values 〈R2
g 〉, 〈R
−1
h 〉, 〈A〉, 〈S〉 and 〈B〉
are evaluated by simple averaging of the instant values over the time interval tequ < t < tfin, where
t∗
fin
∼ 10t∗equ is the total duration of each run. One should note that the other type of averaging is also
possible, where the nominators and denominators in equations (2.9) are averaged separately [27–32].
Besides the simple averages, we also consider the histograms of probability distributions for the shape
characteristics, where the p(B), for the shape descriptor B , is found to be the most informative.
3. Aggregates and their properties
We consider four types of molecular architectures, all shown in figure 1. All of them can be inter-
preted as four linear diblock copolymers, each of eight hydrophobic beads (type B) and of two hydrophilic
beads (type A) bonded in a different way. In particular, architecture (a) is just a set of four non-bonded
diblocks and serves as some kind of a reference system. Architecture (b) represents the bonding of four
(a) (b) (c) (d)
Figure 1. (Color online) Molecular architectures used in this study: (a) linear diblock copolymers, (b) mik-
toarm star-polymer, (c) diblock 1 star-copolymer and (d) diblock 2 star-copolymer. Colour coding: hy-
drophilic beads (type A) are shown in blue, hydrophobic (type B) — in yellow, central bead is shown in
red.
13802-4
Shape characteristics
3
4
5
6
20 40 60 80 100 Na
〈R2
g〉1/2
(a)
(b)
(c)
(d)
1.15N
1/3
a
3
4
5
6
20 40 60 80 100 Na
〈R-1
h 〉-1
(a)
(b)
(c)
(d)
1.22N
1/3
a
(i) (ii)
Figure 2. (Color online) Comparison of the estimates for the average radius or gyration [frame (i)] and
hydrodynamic radius [frame (ii)] as functions of the aggregation number Na performed for themolecular
architectures listed in figure 1. The collapse-like scaling law, N1/3
a , is provided in both cases as a guide for
the eye.
diblocks in the form of an asymmetric miktoarm star of eight arms: four hydrophilic and four hydropho-
bic. The (c) and (d) architectures are of the diblock star type with the arms being bonded by either their
hydrophobic ends, as in case (c), or by their hydrophilic ends, case (d). Molecular mass in all four cases is
practically the same, save for an additional central bead in the cases (b)–(d), as compared to the case (a).
The aggregation transition in a solution of polymers is of much interest and is the subject of numer-
ous recent works [10, 33–35]. In this study, we consider the setup that mimics a drop of a highly concen-
trated polymer solution immersed into water. The drop ismodelled via a bunch of Nmol molecules closely
packed inside a central part of the simulation box. Namely, a central bead of each star is positioned ran-
domly with all its coordinates falling into the interval of [L/3; 2L/3]. Each arm is then generated as a
randomwalk. This algorithm results in inevitable overlaps of beads in the initial configuration. The over-
laps, however, do not produce forces of large magnitude, as it would occur in atomic simulations, due
to the soft nature of the interaction potential (2.3). In the course of simulation, this bunch of molecules
looses thememory concerning its initial state during the time tequ. After that it starts to display stationary
oscillations in its shape properties. The values for tini and for the simulation duration tfin have already
been provided in section 2.
In most cases being studied, the initial setup equilibrates into the single aggregate state. This, how-
ever, might be the result of a finite system size and also a finite simulation time. In a case of a single aggre-
gate, the aggregation number Na coincides with the total number of star molecules Nmol in the solution.
The increase of the latter results in a growth of the average aggregate size characterised by the estimates
for the gyration and hydrodynamic radii, as shown in frames (i) and (ii) of figure 2, respectively. We find
the respective data points obtained for the molecular architectures (a)–(c) to be very close for both size
properties being considered, especially at larger Na. Therefore, the difference in intramolecular binding
in these cases has a minor impact on the average aggregate size. The aggregate size is dominated by a
collapse of the hydrophobic part of stars characterised by the scaling laws 〈R2
g 〉
1/2, 〈R−1
h 〉−1 ∼ (nhNa)1/3,
where nh is the number of hydrophobic beads in each star. Data points obtained for the case (d) are found
to be higher than the respective values for the cases (a)–(c), especially in frame (i). This deviation, as well
as local deviations from the scaling law observed at specific values of Na in cases (a)–(c) are related to the
peculiarities of the shape oscillations discussed below.
On a classification note, we found four types of aggregate shape: spherical, rod-like and disc-like mi-
celles, as well as a spherical vesicle, all visualised infigure 3. Formost molecular architectures considered
here, the increase of Na results in the following sequence of shapes: spherical micelle → aspherical mi-
celle (rod-like or oscillating between the rod- and disc-like)→ spherical vesicle.
This sequence can be understood in terms of the competition between the enthalpic and entropic
contributions to the free energy related to the presence of the aggregate. The enthalpic contribution is
minimal when the number of contacts between the hydrophobic and hydrophilic beads are minimised,
which is achieved in the case of a spherical shape. The entropic contribution is minimised when the
hydrophobic chains within the aggregate display coil-like conformations. Both factors do not compete in
13802-5
O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber
spherical micelle
rod-like micelle
disc-like micelle
top view
side view
spherical vesicle
Figure 3. (Color online) Basic shapes observed for aggregates of amphiphilic stars (see respective caption
next to each frame). Spherical vesicle is shown sliced in the middle to reveal its internal void, colours of
beads follow these in figure 1.
the case of a spherical micelle until its radius Ra is smaller than the average end-to-end distance re of
the hydrophobic chains in a coiled state, see frame (i) in figure 4. With the growth of Na, there are two
options: sperical micelles with Ra > re shown in frame (ii); or aspherical micelles shown in frame (iii) of
the same figure. For the case (ii), the need to fill-in the center of the micelles inevitably leads to stretching
some of the hydrophobic chains beyond the re distance (these chains are shown in red), thus reducing the
entropy (i.e., increasing the free energy). The effect strengthens with the increase of Na and, hence, of Ra.
Therefore, at a certain value of Na, the aspherical aggregate becomes more favourable, when the penalty
of aspherical shape is compensated by the gain from a coiled state for all chains, as shown in frame (iii).
A further increase of Na forces a higher asphericity of this shape and a spherical vesicle with the internal
void, shown in frame (iv), becomes more favourable. It combines both overall spherical shape and coiled
conformations for the chains within a hydrophobic shell, which is about ra for a single layer vesicle.
Let us consider now how the details of the molecular architecture affect the “phase boundaries” for
(i)
(ii)
(iii)
(iv)
Figure 4. (Color online) Schematic representation of the conformations of polymer chains for the case
of (i) spherical micelle, (ii) overgrown spherical micelle, (iii) rod-like micelle and (iv) spherical vesicle.
Colours of beads follow these in figure 1.
13802-6
Shape characteristics
0
0.1
0.2
0.3
20 40 60 80 100 Na
〈A〉 (a)
(b)
(c)
(d)
0
0.1
0.2
0.3
20 40 60 80 100 Na
〈S〉 (a)
(b)
(c)
(d)
0
0.1
0.2
0.3
0.4
20 40 60 80 100 Na
〈B〉 (a)
(b)
(c)
(d)
Figure 5. (Color online) Comparison of the simple averages for the shape characteristics 〈A〉, 〈S〉 and
〈B〉 as functions of the aggregation number Na. (a)–(d) follow the notations introduced for molecular
architectures in figure 1.
the shapes shown in figure 3. These can be traced from the dependencies of simple averages for the
shape characteristics 〈A〉, 〈S〉 and 〈B〉 as functions of the aggregation number Na shown in figure 5.
For molecular architectures (a)–(c), the behaviour for all three properties is quite similar. In particular,
at small Na ∼ 20, the aggregate is a spherical micelle. With an increase of Na, the asphericity grows
monotonously until Na < N∗
a , where the N∗
a ≈ 70 for the cases (a) and (c) and N∗
a ≈ 60 for the case (b).
At Na ∼ N∗
a , a sharp transition occurs between the spherical micelle (Na < N∗
a ) and a spherical vesicle
(Na > N∗
a ), where the shape characteristics drop to zero. A decline for the shape properties is the smallest
for the case (b), whereas for the case (d) no sharp transition is detected at all and the entire interval 20 É
Na É 120 is characterised by non-zero values for all three shape characteristics. Their gradual decrease
is observed with an increase of Na.
To shed more light on the behaviour of the shape characteristics, we examine the probability distri-
bution p(B) for the shape descriptor B in the pretransitional region Na ∼ N∗
a . The respective histograms
for all four cases (a)–(d) are shown in figure 6. First, let us note that the cases (a) and (c) are very similar
-0.2 0 0.2 0.4 0.6 B
p(B) Na=20
Na=40
Na=50
Na=64
Na=74
-0.2 0 0.2 0.4 0.6 B
p(B)
Na=20
Na=40
Na=60
Na=64
Na=80
-0.2 0 0.2 0.4 0.6 B
p(B) Na=20
Na=50
Na=60
Na=71
Na=74
-0.2 0 0.2 0.4 0.6 B
p(B)
Na=20
Na=40
Na=50
Na=60
Na=120
(a) (b)
(c) (d)
Figure 6. (Color online) Probability distribution p(B) for the shape descriptor B shown for selected values
of aggregation number Na for the molecular architectures (a)–(d) listed in figure 1.
13802-7
O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber
showing rather sharp peaks centered around zero for Na = 20 with the distribution getting broader and
shifting towards larger values of B for larger Na < N∗
a . In this interval both disc-like (B < 0) and rod-like
(B > 0) shapes are found, albeit the disc-like tendencies are rather weak not reaching the values below
−0.25. Above the transition, Na > N∗
a , the distribution is back to a single sharp peak centered around
zero indicating a spherical vesicle. Its shape practically overlaps with the one for the spherical micelle
(Na = 20). The case (b) shows similar tendencies for the probability distribution p(B), but the effect is
much weaker. The case (d) is rather different. At Na = 20, the distribution p(B) shows a series of peaks at
relatively large values of B indicating that the spherical shape is not the single favourable one, presum-
ably due to specific intramolecular constraints of the case (d). The tail of the distribution which spreads
towards the larger values of B persists even for the largest value of Na = 120 being considered. It lifts the
simple average 〈B〉 to the nonzero values, as seen earlier in figure 5.
The time evolution for the instant values of the shape descriptor B(t) is shown in figure 7 in a pre-
transition region for selected characteristic cases (indicated in each frame). In each case, one can observe
a certain level of oscillations of the aggregate shape occurring between a weak disc-like (−0.20 < B < 0),
spherical (B = 0) and rod-like (B > 0) shapes. The period of oscillations is about 2 time units for the
case (a) and about 5 for the case (c). Case (b) is mainly characterised by values B ∼ 0, with rare outbursts
into B > 0 region. This explains that the simple averages for this case shown in figure 5 are rather small.
The case (d) is characterised by fast oscillations between the weakly disc-like and highly rod-like shapes,
resulting in essentially non-zero simple averages for this case in figure 5.
One should note that the aspherical shapes of aggregates observed for the amphiphilic stars in this
study, are reminiscent of the worm-like prolate and burger-like oblate multicompartment micelles ob-
served for the case of the ABC miktoarm stars [18, 20, 36]. In the latter case, however, these shapes are
stabilised by adding the third component, whereas for the two-component amphiphilic stars they are
prone to large scale oscillations observed in our work. It is plausible to suggest that these oscillations
occur due to a delicate balance between the enthalpic and entropic contributions, whereas the addition
of the hydrophobic drug agent [19] may stabilise the aggregate shape. This will be a subject of subsequent
studies.
-0.2
0
0.2
0.4
0.6
20 25 30 35 t ×10
-3
B(t) at Na=64
-0.2
0
0.2
0.4
0.6
20 25 30 35 t ×10
-3
B(t) at Na=60
-0.2
0
0.2
0.4
0.6
30 35 40 45 t ×10
-3
B(t) at Na=71
-0.2
0
0.2
0.4
0.6
20 25 30 35 t ×10
-3
B(t) at Na=50
(a) (b)
(c) (d)
Figure 7. (Color online) Time evolution of the shape descriptor B shown for selected characteristic cases.
(a)–(d) stay for the architectures listed in figure 1.
13802-8
Shape characteristics
4. Conclusions
The simulation study presented in this work reveals strong dependencies of the shapes of aggregates
formed by amphiphilic stars in water on the details of their molecular architecture. It has a prospect of
the application in the drug delivery systems, where both the size and the shape of the aggregate is known
to play an important role for the flow of the enveloped agent through the vessels.
Four molecular architectures have been examined: (a) four disjoint linear diblocks, (b) asymmetric
miktoarm polymer, (c) diblock star 1 (hydrophilic parts pointing outwards) and (c) diblock star 2 (hy-
drophilic parts next to a central bead). For all cases, the same general sequence of shapes is found with
an increase of the aggregation number, namely: spherical micelle, aspherical micelle and a spherical
vesicle. The “phase boundaries” between these are found to depend on the details of the molecular ar-
chitecture. For the case (a)–(c), the transformation between a spherical and aspherical micelle occurs
gradually, whereas the transition from an aspherical micelle into a spherical vesicle is in a form of a
sharp transition. In the case (b), aspherical micelle is less stable and transition to a vesicle occurs at a
lower aggregation number. The case (d) is characterised by gradual transitions between all the shapes.
Histograms for the probability distributions of the shape descriptor are relatively narrow for both
spherical micelle and spherical vesicle regimes but become wider next to the micelle-vesicle transition,
indicating that a broad range of shapes are possible. The shape of the aggregate is found to oscillate be-
tween the rod-, disc-like and spherical with the period of oscillations strongly dependent on themolecular
architecture. Both effects of slowing down and acceleration of these oscillations are found. These findings
are relevant for the case of aggregates filled with water-insoluble drug agent, which will be a topic of the
further studies.
Acknowledgements
This work was supported in part by FP7 EU IRSES projects No. 612707 “Dynamics of and in Com-
plex Systems” and No. 612669 “Structure and Evolution of Complex Systems with Applications in Physics
and Life Sciences”, and by the Doctoral College for the Statistical Physics of Complex Systems, Leipzig-
Lorraine-Lviv-Coventry (L4).
References
1. Hadjichristidis N., Pitsikalis M., Iatrou H., Driva P., Sakellariou G., Chatzichristidi M., In: Polymer Science: A
Comprehensive Reference, Elsevier, Oxford, 2012, 29–111; doi:10.1016/b978-0-444-53349-4.00161-8.
2. Khanna K., Varshney S., Kakkar A., Polym. Chem., 2010, 1, No. 8, 1171; doi:10.1039/c0py00082e.
3. Moughton A.O., Hillmyer M.A., Lodge T.P., Macromolecules, 2012, 45, No. 1, 2; doi:10.1021/ma201865s.
4. Park J., Jang S., Kim J.K., J. Polym. Sci., Part B: Polym. Phys., 2014, 53, No. 1, 1; doi:10.1002/polb.23604.
5. Kozlovskaya V., Xue B., Kharlampieva E., Macromolecules, 2016, 49, No. 22, 8373;
doi:10.1021/acs.macromol.6b01740.
6. Reddy S.T., van der Vlies A.J., Simeoni E., Angeli V., Randolph G.J., O'Neil C.P., Lee L.K., Swartz M.A., Hubbell J.A.,
Nat. Biotechnol., 2007, 25, No. 10, 1159; doi:10.1038/nbt1332.
7. Zolnik B.S., González-Fernández Á., Sadrieh N., Dobrovolskaia M.A., Endocrinology, 2010, 151, No. 2, 458;
doi:10.1210/en.2009-1082.
8. Ma J.W., Li X., Tang P., Yang Y., J. Phys. Chem. B, 2007, 111, No. 7, 1552; doi:10.1021/jp067650v.
9. Binder K., Paul W., Macromolecules, 2008, 41, No. 13, 4537; doi:10.1021/ma702843z.
10. Patti A., Colloids Surf. A, 2010, 361, No. 1–3, 81; doi:10.1016/j.colsurfa.2010.03.022.
11. Hoogerbrugge P.J., Koelman J.M.V.A., Europhys. Lett., 1992, 19, No. 3, 155; doi:10.1209/0295-5075/19/3/001.
12. Español P., Warren P., Europhys. Lett., 1995, 30, No. 4, 191; doi:10.1209/0295-5075/30/4/001.
13. Ilnytskyi J.M., Patsahan T., Holovko M., Krouskop P.E., Makowski M.P., Macromolecules, 2008, 41, No. 24, 9904;
doi:10.1021/ma801045z.
14. Ilnytskyi J.M., Patsahan T., Sokołowski S., J. Chem. Phys., 2011, 134, No. 20, 204903; doi:10.1063/1.3592562.
15. Ilnytskyi J.M., Sokołowski S., Patsahan T., Condens. Matter Phys., 2013, 16, No. 1, 13606;
doi:10.5488/cmp.16.13606.
16. Ilnytskyi J.M., Bryk P., Patrykiejew A., Condens. Matter Phys., 2016, 19, No. 1, 13609; doi:10.5488/cmp.19.13609.
13802-9
https://doi.org/10.1016/b978-0-444-53349-4.00161-8
https://doi.org/10.1039/c0py00082e
https://doi.org/10.1021/ma201865s
https://doi.org/10.1002/polb.23604
https://doi.org/10.1021/acs.macromol.6b01740
https://doi.org/10.1038/nbt1332
https://doi.org/10.1210/en.2009-1082
https://doi.org/10.1021/jp067650v
https://doi.org/10.1021/ma702843z
https://doi.org/10.1016/j.colsurfa.2010.03.022
https://doi.org/10.1209/0295-5075/19/3/001
https://doi.org/10.1209/0295-5075/30/4/001
https://doi.org/10.1021/ma801045z
https://doi.org/10.1063/1.3592562
https://doi.org/10.5488/cmp.16.13606
https://doi.org/10.5488/cmp.19.13609
O.Y. Kalyuzhnyi, J.M. Ilnytskyi, C. von Ferber
17. Ortiz V., Nielsen S.O., Discher D.E., Klein M.L., Lipowsky R., Shillcock J., J. Phys. Chem. B, 2005, 109, No. 37, 17708;
doi:10.1021/jp0512762.
18. Cui Y., Zhong C., Xia J., Macromol. Rapid Commun., 2006, 27, No. 17, 1437; doi:10.1002/marc.200600326.
19. Xia J., Zhong C., Macromol. Rapid Commun., 2006, 27, No. 19, 1654; doi:10.1002/marc.200600411.
20. Xia J., Zhong C., Macromol. Rapid Commun., 2006, 27, No. 14, 1110; doi:10.1002/marc.200600187.
21. Zhao Y., You L.Y., Lu Z.Y., Sun C.C., Polymer, 2009, 50, No. 22, 5333; doi:10.1016/j.polymer.2009.09.014.
22. Groot R.D., Warren P.B., J. Chem. Phys., 1997, 107, No. 11, 4423; doi:10.1063/1.474784.
23. Groot R.D., Madden T.J., J. Chem. Phys., 1998, 108, No. 20, 8713; doi:10.1063/1.476300.
24. Ilnytskyi J.M., Holovatch Yu., Condens. Matter Phys., 2007, 10, No. 4, 539; doi:10.5488/cmp.10.4.539.
25. Šolc K., J. Chem. Phys., 1971, 54, No. 6, 2756; doi:10.1063/1.1675241.
26. Šolc K., Stockmayer W.H., J. Chem. Phys., 1971, 55, No. 1, 335; doi:10.1063/1.1675527.
27. Aronovitz J., Nelson D., J. Phys., 1986, 47, No. 9, 1445; doi:10.1051/jphys:019860047090144500.
28. Zifferer G., J. Chem. Phys., 1999, 110, No. 9, 4668; doi:10.1063/1.478350.
29. Zifferer G., Macromol. Theory Simul., 1999, 8, No. 5, 433;
doi:10.1002/(SICI)1521-3919(19990901)8:5<433::AID-MATS433>3.0.CO;2-C.
30. Blavatska V., von Ferber C., Holovatch Yu., Condens. Matter Phys., 2011, 14, No. 3, 33701;
doi:10.5488/cmp.14.33701.
31. Jagodzinski O., Eisenriegler E., Kremer K., J. Phys. I, 1992, 2, No. 12, 2243; doi:10.1051/jp1:1992279.
32. Kalyuzhnyi O., Ilnytskyi J.M., Holovatch Yu., von Ferber C., J. Phys.: Condens. Matter, 2016, 28, No. 50, 505101;
doi:10.1088/0953-8984/28/50/505101.
33. Zierenberg J., Mueller M., Schierz P., Marenz M., Janke W., J. Chem. Phys., 2014, 141, No. 11, 114908;
doi:10.1063/1.4893307.
34. Zierenberg J., Janke W., Phys. Rev. E, 2015, 92, 012134; doi:10.1103/physreve.92.012134.
35. Zierenberg J., Schierz P., Janke W., Preprint arXiv:1607.08355, 2017.
36. Li Z., Science, 2004, 306, No. 5693, 98; doi:10.1126/science.1103350.
Властивостi форми агрегатiв iз амфiфiльних зiркових
полiмерiв у водi: дослiдження методом дисипативної
динамiки
О.Ю. Калюжний1,4, Я.М. Iльницький1,4, К. фон Фербер2,3,4
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
2 Дослiдницький центр прикладної математики, Унiверситет Ковентрi, Ковентрi, CV1 5FB, Великобританiя
3 Дюссельдорфський унiверситет iм. Г. Гайне, D-40225 Дюссельдорф, Нiмеччина
4 Докторський коледж статистичної фiзики складних систем, Ляйпцiґ-Лотарингiя-Львiв-Ковентрi (L4),
D-04009 Ляйпцiґ, Нiмеччина
В роботi вивчається ефект молекулярної архiтектури амфiфiльних зiркових полiмерiв на форму агрегатiв,
якi вони формують у водi. Полiмери та вода описуються на мезоскопiчному рiвнi, використовуючи метод
дисипативної динамiки. Розглянутi чотири молекулярнi архiтектури: мiктозiрки, два типи дiблок-зiрок i
група лiнiйних дiблок кополiмерiв, усi з однаковою композицiєю та молекулярною масою. Ми розгляда-
ємо початкову конфiгурацiю у виглядi густого клубка iз Na молекул, помiщеного у воду. В рiвноважному
станi формується агрегат, характеристики форми якого дослiджуються при рiзних значеннях Na . Знайде-
но чотири форми агрегатiв: сферична, стержнеподiбна та дископодiбна мiцелла та сферична весiкула.
Оцiненi “фазовi границi” мiж цими формами залежно вiд молекулярної архiтектури. У бiльшостi випад-
кiв знайдено рiзкий перехiд мiж асферичною мiцеллою та весiкулою. Передперехiдна область характе-
ризується осциляцiями властивостей форми iз великою амплiтудою, частота яких суттєво залежить вiд
молекулярної архiтектури.
Ключовi слова: зiрковi полiмери, амфiфiли, мiцелли, весiкули, дисипативна динамiка
13802-10
https://doi.org/10.1021/jp0512762
https://doi.org/10.1002/marc.200600326
https://doi.org/10.1002/marc.200600411
https://doi.org/10.1002/marc.200600187
https://doi.org/10.1016/j.polymer.2009.09.014
https://doi.org/10.1063/1.474784
https://doi.org/10.1063/1.476300
https://doi.org/10.5488/cmp.10.4.539
https://doi.org/10.1063/1.1675241
https://doi.org/10.1063/1.1675527
https://doi.org/10.1051/jphys:019860047090144500
https://doi.org/10.1063/1.478350
https://doi.org/10.1002/(SICI)1521-3919(19990901)8:5%3C433::AID-MATS433%3E3.0.CO;2-C
https://doi.org/10.5488/cmp.14.33701
https://doi.org/10.1051/jp1:1992279
https://doi.org/10.1088/0953-8984/28/50/505101
https://doi.org/10.1063/1.4893307
https://doi.org/10.1103/physreve.92.012134
http://arxiv.org/abs/1607.08355
https://doi.org/10.1126/science.1103350
Introduction
Simulation approach and properties of interest
Aggregates and their properties
Conclusions
|