Star polymers: From conformations to interactions to phase diagrams
We review recent progress achieved in the theoretical description of the interactions, correlations, and phase behavior of concentrated solutions of star polymers, sterically stabilized colloids, and micelles. We show that the theoretical prediction of an ultrasoft, logarithmically diverging effe...
Gespeichert in:
| Datum: | 2002 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
2002
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/120565 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Star polymers: From conformations to interactions to phase diagrams / C.N. Likos, H.M. Harreis // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 173-200. — Бібліогр.: 112 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120565 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1205652017-06-13T03:05:45Z Star polymers: From conformations to interactions to phase diagrams Likos, C.N. Harreis, H.M. We review recent progress achieved in the theoretical description of the interactions, correlations, and phase behavior of concentrated solutions of star polymers, sterically stabilized colloids, and micelles. We show that the theoretical prediction of an ultrasoft, logarithmically diverging effective interaction between the star centers, which has been confirmed by SANSexperiments and computer simulations, lies in the core of a host of unusual phenomena encountered in such systems. These include anomalous structure factors, reentrant melting behavior, as well as a variety of exotic crystal phases. Extensions to polydisperse stars and the role of many-body forces are also discussed. A particular ‘mean-field’ character of star polymer fluids is presented and it is shown that it manifests itself in the shape and structure of sedimentation profiles of these systems. Здійснено огляд недавніх досягнень у теоретичному описі взаємодій, кореляцій і фазової поведінки концентрованих розчинів зіркових полімерів, просторово стійких колоїдів і міцел. Ми покажемо, що теоретично передбачена надм’яка логарифмічно розбіжна ефективна взаємодія між центрами зірок, що була підтверджена SANS-експериментами і комп’ютерними симуляціями, потрапляє в множину незвичних явищ, які спостерігаються в таких системах. Сюди відносяться аномальні структурні фактори, поведінка зворотнього плавлення, множини екзотичних кристалічних фаз. Також обговорено узагальнення на випадок полідисперсних зірок і роль сил багатьох тіл. Представлено особливу поведінку типу “cереднього поля” плинів зіркових полімерів і показано, що вона проявляється у формі і структурі профілів осаджування цих систем. 2002 Article Star polymers: From conformations to interactions to phase diagrams / C.N. Likos, H.M. Harreis // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 173-200. — Бібліогр.: 112 назв. — англ. 1607-324X PACS: 82.70.Dd, 61.25.Hq, 61.20.-p, 64.70.Dv DOI:10.5488/CMP.5.1.173 http://dspace.nbuv.gov.ua/handle/123456789/120565 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We review recent progress achieved in the theoretical description of the
interactions, correlations, and phase behavior of concentrated solutions of
star polymers, sterically stabilized colloids, and micelles. We show that the
theoretical prediction of an ultrasoft, logarithmically diverging effective interaction
between the star centers, which has been confirmed by SANSexperiments
and computer simulations, lies in the core of a host of unusual
phenomena encountered in such systems. These include anomalous structure
factors, reentrant melting behavior, as well as a variety of exotic crystal
phases. Extensions to polydisperse stars and the role of many-body forces
are also discussed. A particular ‘mean-field’ character of star polymer fluids
is presented and it is shown that it manifests itself in the shape and
structure of sedimentation profiles of these systems. |
| format |
Article |
| author |
Likos, C.N. Harreis, H.M. |
| spellingShingle |
Likos, C.N. Harreis, H.M. Star polymers: From conformations to interactions to phase diagrams Condensed Matter Physics |
| author_facet |
Likos, C.N. Harreis, H.M. |
| author_sort |
Likos, C.N. |
| title |
Star polymers: From conformations to interactions to phase diagrams |
| title_short |
Star polymers: From conformations to interactions to phase diagrams |
| title_full |
Star polymers: From conformations to interactions to phase diagrams |
| title_fullStr |
Star polymers: From conformations to interactions to phase diagrams |
| title_full_unstemmed |
Star polymers: From conformations to interactions to phase diagrams |
| title_sort |
star polymers: from conformations to interactions to phase diagrams |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120565 |
| citation_txt |
Star polymers: From conformations to
interactions to phase diagrams / C.N. Likos, H.M. Harreis // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 173-200. — Бібліогр.: 112 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT likoscn starpolymersfromconformationstointeractionstophasediagrams AT harreishm starpolymersfromconformationstointeractionstophasediagrams |
| first_indexed |
2025-07-08T18:08:39Z |
| last_indexed |
2025-07-08T18:08:39Z |
| _version_ |
1837103183365668864 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 1(29), pp. 173–200
Star polymers: From conformations to
interactions to phase diagrams
C.N.Likos, H.M.Harreis
Institut für Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf,
Universitätsstraße 1, D-40225 Düsseldorf, Germany
Received October 9, 2001
We review recent progress achieved in the theoretical description of the
interactions, correlations, and phase behavior of concentrated solutions of
star polymers, sterically stabilized colloids, and micelles. We show that the
theoretical prediction of an ultrasoft, logarithmically diverging effective in-
teraction between the star centers, which has been confirmed by SANS-
experiments and computer simulations, lies in the core of a host of unusual
phenomena encountered in such systems. These include anomalous struc-
ture factors, reentrant melting behavior, as well as a variety of exotic crystal
phases. Extensions to polydisperse stars and the role of many-body forces
are also discussed. A particular ‘mean-field’ character of star polymer flu-
ids is presented and it is shown that it manifests itself in the shape and
structure of sedimentation profiles of these systems.
Key words: polymers, colloids, liquid-state theory, phase transitions
PACS: 82.70.Dd, 61.25.Hq, 61.20.-p, 64.70.Dv
1. Introduction
What are star polymers and why study them? In order to answer this question,
we begin from its last part and consider the typical physical systems that are com-
monly classified under the name ‘soft matter’. This term describes a huge variety of
substances, the most prominent of which are solutions and suspensions of particles
with mesoscopic dimensions, typically 1 nm to 1µm [1], diffusing into an atomic or
molecular solvent. These solute, colloidal particles can have various compositions
and molecular architectures. The simplest and most common examples are solid,
spherical particles, such as, polystyrene, polymethylmethacrylate (PMMA) or sili-
ca spheres, or ‘fractal’ objects, such as polymer chains. The former are hard and
impenetrable, the latter flexible and interpenetrable. Being faced with these two
extremes, it is desirable, for both theoretical reasons and practical applications, to
design systems that smoothly interpolate between the two, by displaying a tunable
‘hardness’ in their internal conformations and associated effective interactions [2].
c© C.N.Likos, H.M.Harreis 173
C.N.Likos, H.M.Harreis
Star polymers offer a natural way for bridging this gap. They are macromolecular
entities consisting of f polymeric chains chemically anchored to a common center.
The number of arms f is also called functionality of the stars. When the degree of
polymerizationN is the same for all chains, one talks about regular stars. In the limit
N ≫ 1, the microscopic dimensions of the central core where the arms are attached
become irrelevant when compared with the overall size of the star and hence we will
be making the further simplification of treating the core of the stars as vanishingly
small. Related systems are sterically stabilized colloidal particles or micelles, which
have star-like behavior at the limit where the size of the core particle becomes much
smaller than the extent of the tethered polymeric brush [3–5].
Star polymers have attracted considerable interest in the recent past [6–8]. The
reason for that is threefold. First, from a technical point of view, star polymers
are important in several industrial applications [6]. One example are hydrogenated
polyisoprene star polymers,1 which are used as viscosity index modifiers in oil in-
dustry applications due to their excellent shear stability. Further, commercial star
polymers are brought into action in coating materials, such as binders in toners
for copying machines, and in several pharmaceutical and medical applications [6].
Second, from an experimental point of view, the recent synthesis of regular star poly-
mers with various possible numbers of arms by Roovers and coworkers [9,10] made
it possible to explore the physics of well-defined model systems, which are monodis-
perse in both the number of arms and in the degree of polymerization. Important
examples are polyisoprene stars with f = 8, 18 [11] and polybutadiene stars with
f = 32, 64, 128 [10], both synthesized by anionic polymerization. Thereafter, the
static properties of stars can be studied by small-angle neutron scattering (SANS),
light scattering and small-angle X-ray scattering (SAXS). Quasi-elastic neutron scat-
tering experiments using neutron spin-echo spectroscopy [12], as well as dynamical
light scattering are used to study the collective and single-chain dynamics of arms
of the star polymers, for a recent review see [6]. Third, star polymers constitute an
important soft-condensed matter system, linking the fields of polymer physics and
colloid physics, thus attracting interest from a purely theoretical point of view as
well. Star polymers with small arm numbers (f = 1, 2) resemble linear polymers.
Thus, their chain-averaged configurations show a considerable asphericity [13–15],
although their averaged number density of monomers, ϕ(r), is spherically symmet-
ric around the center of mass of the polymer. With increasing arm number f , the
asphericity of the stars has been shown to decrease considerably [16–23]. This evo-
lution of the shape of the star with increasing functionality is shown in figure 1. It
is essentially in the limit f ≫ 1 where a description of star polymers as sterical-
ly stabilized colloidal particles holds. This polymer-colloid hybrid character of star
polymers has been explored in a number of publications dealing with the structural
[4,6,11,24–38] and dynamical [4,27,28,39–48] properties of the same.
The rest of this paper is organized as follows: in section 2 we discuss the con-
formations of isolated stars and in section 3 we present recent results from theory,
experiment, and simulation regarding the effective interaction between them. In sec-
1“Shellvis”, distributed by Shell Chemical Co. [6].
174
Star polymers. . .
(a) (b)
Figure 1. Snapshots of star polymers in good solvents as obtained from MD
simulations employing the model of Grest et al. [49] with: (a) f = 10, N = 50,
and (b) f = 50, N = 50. For small f , the star looks like a fractal, aspherical
object whereas for large f it resembles a spherical, colloidal particle. (Courtesy
of A.Jusufi).
tions 5 and 6 we show that star polymer fluids satisfy a certain ‘mean-field’ limit
which manifests itself in the form of their sedimentation profiles. Finally, in section 7,
we summarize and conclude.
2. Conformations of star polymers
The polymer chains that are attached on the center of the star undergo constantly
thermal fluctuations, akin to those of free polymers in a solvent. The molecular
architecture of the macromolecule, however, restricts the freedom of the chains as,
on the one hand, the anchored monomer is forced to remain fixed and, on the other
hand, many chains are forced to converge at the star center. One expects, therefore,
an interplay between three factors: the solvent quality, which determines the value
of the excluded-volume parameter v of the chains, the degree of polymerization N
and the functionality f . Intuitively, it is anticipated that the crowding of f chains
in the star interior will cause them to stretch, with the stretching becoming more
pronounced as f increases.
Quantitatively, the conformations of single stars are described within the frame-
work of the ‘blob model’ of Daoud and Cotton [51], schematically depicted in figure 2.
In this model, the inner of the star is regarded as a succession of concentric shells of
blobs, each blob in the shell having size ξ(r). The spherical blobs are closely packed
and within each one every chain behaves as if it were free, i.e., the effects of the
neighboring chains are not present. This model yields scaling predictions for the
monomer concentration around the star center ϕ(r) and the corona radius Rc of the
175
C.N.Likos, H.M.Harreis
star, defined through the conservation law [51]:
4π
∫ Rc
0
dr r2ϕ(r) = Nf. (2.1)
Rc
r
r
2
r
1
ξ(r)
X
0
Figure 2. The Daoud-Cotton [51] blob model of a star.
By purely geometrical arguments, it follows that the blob size ξ(r) scales as
ξ(r) ∼ rf−1/2. (2.2)
Let n(r) be the number of monomers inside a blob centered at r. A local swelling
parameter α(r) can then be defined as the ratio between the swollen and ideal blob
size ξ0(r):
α(r) =
ξ(r)
ξ0(r)
. (2.3)
The ideal and swollen sizes are given by ξ0(r) ∼ an1/2(r) and ξ(r) ∼ v̄a3n3/5(r),
respectively, yielding the result
α5(r) ∼ v̄n1/2(r), (2.4)
where v̄ ≡ va−3 is a dimensionless excluded volume parameter and a is the monomer
(Kuhn) length. Depending on the value of α(r), one distinguishes three separate
regions of the star, namely the following.
I. The swollen region. Here, the blobs are swollen, i.e., their size ξ(r) scales
with the number of monomers n(r) with the Flory exponent 3/5, yielding the fol-
176
Star polymers. . .
lowing scaling laws for the various quantities of interest:
n(r) ∼
(r
a
)5/3
v̄−1/3f−5/6, (2.5)
α(r) ∼
(r
a
)1/6
f−1/12v̄1/6, (2.6)
ϕ(r) ∼ a−3
(r
a
)
−4/3
v̄−1/3f 2/3. (2.7)
The last equation, combined with the known scaling law for the osmotic pressure
Π ∼ c9/4 in a semidilute polymer solution [2,52,53], yields the scaling of the local
osmotic pressure inside the star as
Π(r) ∼ f 3/2r−3. (2.8)
This last equation has been employed by Jusufi et al. [54,55] for the determination
of accurate effective potentials acting between star polymers and hard, colloidal
particles of arbitrary curvature.2
The extent of the swollen region ranges from the outermost of the star, r = Rc,
down to a distance r1 from the center (see figure 2), at which the local scaling factor
α(r) attains the unity value. From equation (2.6) we obtain this value as
r1 ∼ f 1/2v̄−1a. (2.9)
II. The unswollen region. The results for the swollen region are based on the
assumption that the blob size ξ(r) exceeds the size of the thermal blob lc ∼= av̄−1
[52], where the latter is a measure of the distance over which the behavior of the
chain is ideal; on longer scales the chain is self-avoiding. However, as ξ(r) becomes
smaller with approaching the star center, [see equation (2.2)], at distances r < r1
from the center, the blobs have their ideal size, α(r) = 1, and this leads to the
scaling relations:
n(r) ∼
(r
a
)2
f−1, (2.10)
ϕ(r) ∼ a−3
(r
a
)
−1
f 1/2. (2.11)
The unswollen region persists down to a distance r2 from the center (see figure 2),
where the local concentration ϕ(r) reaches the unity value and the star resembles a
melt. From equation (2.11) we obtain this value as
r2 ∼ af 1/2. (2.12)
III. The core. At distances 0 < r < r2, the concentration is unity, the blob size
coincides with the monomer size and the chains are completely stretched. In this
2For details see also the article by J.Dzubiella and A.Jusufi in the next issue.
177
C.N.Likos, H.M.Harreis
core region of the star there are Nc monomers. With Rcore denoting the core size,
the scaling laws read as
fNc ∼
(r2
a
)3
, (2.13)
Nc ∼ f 1/2, (2.14)
Rcore ∼ Nc a, (2.15)
ϕ(r) ∼ a−3. (2.16)
Combining the results of this section with the definition of the corona radius R c,
equation (2.1), the scaling of the latter is obtained as
Rc ∼
[
Nf +
1
10
f 3/2
v̄2
+
1
6
f 3/2
]3/5
v̄1/5f−2/5a. (2.17)
Integrating in equation (2.1) from a lower limit equal to zero, is tantamount to ignor-
ing the size of the microscopic particle on which the chains are attached. Depending
on the degree of polymerization N (length of the chains), the strength of the exclud-
ed volume interactions v̄ and the functionality f , one can distinguish three regimes
for the N -, f - and v̄-dependencies of the star size, as follows:
Regime Rc
N ≫ f 1/2v̄−2 → N3/5v̄1/5f 1/5a, (2.18)
f 1/2v̄−2 ≫ N ≫ f 1/2 → N1/2f 1/4a, (2.19)
f 1/2 ≫ N → (Nf)1/3 a. (2.20)
If the conditions of (2.18) hold, then most of the star is in the swollen region, so
that the core and the unswollen part of the star can be ignored. As can be seen from
equations (2.18) and (2.19) above, the spatial extension of stars is larger than that
of an isolated chain with the same degree of polymerization, due to stretching of the
chains caused by the star architecture. At the same time, stars are smaller than a
single chain with a degree of polymerizationNf ; the latter has an extent which scales
as f 3/5N3/5 in good solvents and as f 1/2N1/2 in Θ-solvents. The above equations
demonstrate that the quality of the solvent alone is not sufficient to determine the
scaling behavior of the stars; if the chains are short, then the star can show Θ-type
behavior even in a good solvent. In this case, the whole of the star will be in the
unswollen region and then the scaling relation (2.19) will be satisfied.
We are going to be considering the case of stars with long chains in a good
solvent, so that the swollen region will dominate throughout the extent of the star
and the scaling relation (2.18) will hold. Results on isolated star polymers in Θ-
solvent conditions can be found in [6,21–23,39,56,57]. An effective pair potential
describing accurately experimental SANS data for star polymer solutions in the
neighborhood of the Θ point was introduced by Likos et al. [32].
The conformations of isolated stars and the associated predictions of the scaling
theory have been tested in a number of microscopic Monte Carlo and Molecular
178
Star polymers. . .
Dynamics simulations, both on-lattice [18,21,23,58–60] and off-lattice [8,19,49,57,
61–63]. Whereas in on-lattice simulations a coarse-grained model of description is
employed automatically, in off-lattice simulations a model microscopic interaction
between the beads is used, for a recent review see [60].
A common model for the monomer-monomer interactions in off-lattice simula-
tions in good solvents has been introduced by Grest et al. [49]. All monomers are
assumed to interact by means of the purely repulsive truncated and shifted Lennard-
Jones (LJ) potential, V0(r), given by
V0(r) =
4ε
[
(
r
σLJ
)12
−
(
r
σLJ
)6
+ 1
4
]
if r < 21/6σLJ,
0 if r > 21/6σLJ,
(2.21)
where ε sets an energy and σLJ a length scale. Good solvent quality is modeled by
a purely repulsive nature of interaction. On the other hand, the connectivity of the
chains is modeled by an additional interaction between monomers on the same chain,
which is given by the so-called finite extendible nonlinear elastic (FENE) potential
VFENE(r):
VFENE(r) =
−15ε
(
R0
σLJ
)2
ln
[
1−
(
r
R0
)2
]
if r < R0,
∞ if r > R0,
(2.22)
with R0 = 1.5σLJ. For small f , f . 6, the interior of the star is not too dense and
standard MC simulations are efficient. However, as f increases, it becomes dense
and hence one has to employ either MD simulations or local stochastic MC methods
[64,65]. In addition, it is necessary to attach the chains to a core of microscopic
dimensions, so that the problems with overlaps of the innermost monomers can be
avoided [63]. In this case, the size of the core has to be taken into account in testing
the predictions of the scaling theory regarding the size of the star, as far as in the
latter the core size is assumed to be vanishingly small [63]. A detailed study of the
chains tethered on a spherical particle of nonnegligible extent can be found in [66].
The scaling law regarding the behavior of the monomer density ϕ(r), equa-
tion (2.7), is tested by averaging the latter quantity during the simulation run and
plotting ln[f−2/3ϕ(r)] against ln r. In this way, the exponent −4/3 has been con-
firmed in a number of simulations, for a broad range of arm numbers, ranging from
f = 3 up to f = 270 [6]. The scaling of the star size with N and f is difficult
to be verified in a simulation if the corona radius Rc is used as a measure; there
always exists a diffuse layer of polymer beyond the scaling regime, whereas in the
Daoud-Cotton model an artificially sharp drop of the monomer density to zero at
a distance r = Rc from the center is assumed. Therefore, the gyration radius Rg is
used to characterize the star size. The predictions of the Daoud-Cotton model have
been confirmed, both in on-lattice [18,21,23,58,63,67] and off-lattice [8,22,49,68] sim-
ulations.
Experimentally, information about the star conformations is obtained by per-
forming scattering from very dilute solutions, where the effects of star-star inter-
actions are vanishingly small. In this way, the only contribution to scattering is
179
C.N.Likos, H.M.Harreis
the form factor P (Q) of the stars, which is related to the Fourier transform of the
monomer density ϕ(r) [6,69,70]. A common and successful fit for the form factor,
which is consistent with the predictions of the Daoud-Cotton theory, is provided by
the Dozier expression [11]:
VWP (Q) = VW exp
[
−(QRg)
2
3
]
+
4πα
Qξ
sin [µ tan−1(Qξ)]
[1 + (Qξ)2]µ/2
Γ(µ), (2.23)
where VW denotes the molar volume, µ = 1/ν − 1, ξ is the average blob size,
α ∼ v̄(ξ/a)1/ν and Γ(µ) is the gamma function. According to Dozier et al. [11],
the quantities VW , α, ξ and Rg are to be treated as free fit parameters, whereas the
exponent ν is fixed at its Flory value 3/5, yielding µ = 2/3. The Dozier form factor
yielded excellent fits for an array of data obtained for stars with f = 8 − 128 arms
[30].
3. Effective interactions between star polymers
We now consider a concentrated solution ofNs star polymers enclosed in a macro-
scopic volume V . The questions that arise are what kinds of spatial correlations,
structure and phases will be displayed by this system, as a function of the concen-
tration, functionality, and temperature. To simplify matters, we focus now on the
case of athermal solvents, for which temperature is an irrelevant thermodynamic pa-
rameter, as the monomer-monomer interactions are purely of the excluded-volume
type. Hence, no independent energy scale appears in the problem other than the
thermal energy kBT .
A powerful idea to analyze such complex fluids is that of introducing an effective
interaction v(r) between the stars, which formally results in canonically tracing out
all monomer degrees of freedom while keeping their centers held at a distance r
apart [2]. Once such an effective interaction has been derived, it is then possible to
treat the complicated star-polymer fluid in a drastically simplified fashion: the stars
become ‘point particles’ interacting via the potential v(r) and the known techniques
from liquid-state theory can be employed in order to quantitatively describe the
fluid [71]. It can be shown that the process of ‘partial elimination’ of degrees of
freedom, which is employed in the derivation of the effective interaction, exactly
preserves both the overall thermodynamics and the correlations of the star centers
in the effective picture [2].
3.1. Theory
Formally, the effective potential between the star centers (considered for sim-
plicity to be point particles featuring no direct interaction with one another) is
proportional to the logarithm of the ratio of the partition function Q2(r) of two
stars at distance r over the product of the two partition functions of the isolated
stars, Q2
1 [2]:
βv(r) = − ln
[
Q2(r)
Q2
1
]
, (3.1)
180
Star polymers. . .
where β ≡ (kBT )
−1.
The pioneering work in establishing scaling relations for the ratio on the rhs of
equation (3.1) above was carried out by Witten and Pincus [25]. Consider first the
case of single chains (f = 1), for which the ‘center’ has to be interpreted as the
end-monomer (alternatively one can look at single chains as ‘stars’ with f = 2 if
one chooses the central monomer to be the effective degree of freedom.) It is known
that the partition function Q1 of a single chain is a power-law of the number of
monomers N [52]:
Q1 ∝ τNNγ−1, (3.2)
where γ = 7/6 is a universal exponent, whereas τ is a parameter the value of which
depends on the microscopic details of the chain. Let us consider the case where r is
of the order of the monomer length, a. Then, the bulk of the system will look like
a single fluctuating chain with 2N monomers. Accordingly, the partition function
Q2(r ∼ a) will be given by the general result, equation (3.2), with N → 2N , namely
Q2(r ∼ a) ∝ τ 2N (2N)γ−1 . (3.3)
Equations (3.2) and (3.3) yield
Q2(r ∼ a)
Q2
1
∝ N1−γ . (3.4)
When the chains are looked upon at scales where the microscopic details are not
visible, the ratio of the partition functions can only depend on the dimensionless
ratio r/R, where R denotes the typical chain size.3 At the same time, it is a power
law in N and the latter enters the combination r/R only through the dependence
of the chain size on the degree of polymerization, R ∝ N ν . The consequence is that
Q2(r ∼ a)/Q2
1 must be a power law in r/R,
Q2(r ∼ a)
Q2
1
∝
( r
R
)x
∝ N−xν . (3.5)
Equations (3.4) and (3.5) imply x = (γ−1)/ν. Though the derivation is based on the
assumption r ∼ a, the result of equation (3.5) can be extended up to r ∼ R. Indeed,
at these distances, one expects the effective interaction to have fallen practically to
zero, as the two chains do not overlap, and equation (3.5) yields Q2(r ∼ R)/Q2
1 = 1
there, with the implication that βv(r) from equation (3.1) vanishes. The scaling
argument yields the correct physical behavior at both extreme cases, r ∼ a and
r ∼ R, hence we can write, using equation (3.1),
βv(r) = −γ − 1
ν
ln
( r
R
)
(r 6 R). (3.6)
Note the ultrasoft, logarithmic divergence of the effective interaction as r → 0.
3It is at this point irrelevant whether R stands for the gyration radius, the end-to-end radius
or the corona radius, since all these quantities simply differ by factors of order unity.
181
C.N.Likos, H.M.Harreis
The extension of these ideas to large functionalities f is straightforward and is
based on geometrical considerations of counting the number of Daoud-Cotton blobs
in a star [25]. The main result is the generalization of equation (3.2) to arbitrary f
which reads as:
Q1 ∝
[
Nf (ν−1)/(2ν)
]−νf3/2
. (3.7)
Following the same steps as for the case f = 1 we obtain
βv(r) = −αf 3/2 ln
( r
R
)
(r 6 R), (3.8)
with the unknown proportionality factor α. The logarithmic divergence is recovered
again.
Likos et al. [31], introduced thereafter a complete effective interaction potential
between star centers for arbitrary f . By matching the expression (3.6), valid for
f = 1, with the expression (3.8), valid for f ≫ 1, it was postulated that α =
(γ − 1)/ν = 5/18 for all f . This assumption evidently gives the exact result for
f = 1. For f = 2, it yields for the prefactor αf 3/2 the value 0.786, which is very
close to the exact value 0.8 calculated by means of renormalization group techniques
by des Cloizeaux [72]. Moreover, recent renormalization group calculations show that
the functional form (5/18)f 3/2 yields excellent agreement with the ‘exact’ value, at
least up to f = 6 [73–76].
The logarithmic interaction was assumed to set in whenever significant overlap
between the blobs of the two stars takes place. Thereafter, the length scale R was
fixed as the length at which the outermost blobs of the two stars fully overlap. Hence,
by defining σ/2 as the distance from the star center to the center of the outermost
blob (see figure 2), the logarithmic interaction was assumed to be valid for r 6 σ.
For distances r > σ, the interaction was postulated to have a Yukawa form, with
the decay length given by the diameter of the outermost blob, ξmax = 2σ/
√
f , as
dictated by geometry. This is intuitively clear, as the blob size is the only relevant
length scale at weak overlaps between the chains. This kind of decay is valid only
for sufficiently large functionalities, f & 10, for which the blob picture applies. For
small functionalities, Jusufi et al. [54] established that the Yukawa decay must be
replaced by a Gaussian decay.
By combining the logarithmic with the Yukawa form and matching them at r = σ
by requiring that both the potential and its first derivative (the force) be continuous
there, the effective star-star interaction was put forward [31]. For f & 10, it reads
as:
βv(r) =
5
18
f 3/2
[
− ln
( r
σ
)
+
1
1 +
√
f/2
]
for r 6 σ,
=
5
18
f 3/2 1
1 +
√
f/2
(σ
r
)
exp
[
−
√
f(r − σ)
2σ
]
for r > σ. (3.9)
Comparisons with simulations established that σ ∼= 1.32Rg [63]. The potential is
shown in figure 3 for various values of the functionality f . It can be seen that it
182
Star polymers. . .
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r/σ
0
20
40
60
80
100
120
140
βv
(r
)
f = 18
f = 32
f = 64
f = 128
f = 256
Figure 3. The effective star-star potential of equation (3.9) for various f -values.
becomes harder with increasing f , tending eventually to a hard-sphere interaction
which formally obtains in the limit f → ∞.
The effective potential valid for f . 10 has the form [54]:
βv(r) =
5
18
f 3/2
[
− ln
( r
σ
)
+
1
2τ 2σ2
]
for r 6 σ,
=
5
18
f 3/2
[
1
2τ 2σ2
exp
[
−τ 2(r2 − σ2)
]
]
for r > σ. (3.10)
Here, σ and τ(f) were determined by comparison to simulation. Again, it turns out
that σ ∼= 1.32Rg and τ(f) is of the order of 1/Rg and is obtained by fitting to
computer simulation results. For f = 2, the value τσ = 1.03 was obtained, which,
together with the potential in equation (3.10) above, yields, for the second virial
coefficient of polymer solutions, the value B2/R
3
g = 5.59, in agreement with the
estimate 5.5 < B2/R
3
g < 5.9 from renormalization group and simulations [77]. For
f = 5, it was found that τσ = 1.12, which leads to B2/R
3
g = 11.48, in accordance
with Monte Carlo simulation results [78,79].
3.2. Simulations
Jusufi et al. [63] performed extensive Molecular Dynamics simulations for a large
variety of combinations of arm numbers and degrees of polymerization of the stars.
The simulations involved two star polymers, the centers of which were kept fixed
at a distance r, while the monomeric degrees of freedom in the arms were moved.
The monomer-monomer interactions used were the truncated and shifted Lennard-
Jones potential of equation (2.21) between all monomers and, in addition, the FENE
potential of equation (2.22) for monomers along the same chain. Due to the large
number of arms f , which could not be accommodated in the small region at the
center of the star [49,57], the arms had to be attached on a spherical particle of
microscopic size Rd ∼ σLJ.
183
C.N.Likos, H.M.Harreis
0.0 0.5 1.0 1.5 2.0 2.5 3.0
(r−2Rd)/Rg
0
40
80
120
β
|F
|R
g
f = 50
f = 10
f = 5
N = 100
(a)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
(r−2Rd)/Rg
0
200
400
600
β
|F
|R
g
f = 50
f = 30
f = 18
N = 50
(b)
Figure 4. Simulation results (symbols) and the theoretical prediction (lines) for
the magnitude of the reduced effective force β|F|Rg between the centers of two
star polymers, plotted against the normalized distance (r − 2Rd)/Rg.
The results are shown in figure 4, where an excellent agreement between theory
and simulation can be seen.4 It should be noted that the kink in the theoretical
force curves appearing at r = σ stems from the fact that, by construction, the
potential is continuous there only up to its first derivative. The second derivative
of βv(r) has a jump there, however, and this shows up as a kink in the force vs.
distance curves. Moreover, the Yukawa tail of the potential was carefully checked by
plotting the logarithm of the force against distance for r > σ [63] and finding that
the simulation data fell on a straight line, the slope of which was given by the decay
length of the effective potential of equation (3.9).
Further Monte Carlo simulations were carried out by Rubio and Freire [79] for
star polymers of functionalities f = 4 − 18. In this work, the second virial coef-
ficient was measured and compared with the one predicted from the potential of
equation (3.9). Discrepancies at large star separations were found, which have a no-
ticeable effect on the second virial coefficient but not on the distribution functions.
However, in view of the fact that the functionalities simulated in [79] were small,
it is not a surprise that the potential of equation (3.9) turns out to be inaccurate
for large separations: the Yukawa decay of this potential should be replaced with
a Gaussian decay, and the resulting, modified interaction of equation (3.10) indeed
yields the correct values of the second virial coefficients. Shida et al. [80] recently
performed monomer-resolved Monte Carlo simulations and measured the radial dis-
tribution function g(r) of two stars, which is directly connected to the ‘potential of
mean force’ v(r) via g(r) = exp[−βv(r)]. The results of these simulations matched
4In this article, we focus our attention on the case f & 10, for which the logarithmic-Yukawa
potential of equation (3.9) is valid, because it is for such functionalities that we obtain several
phase transitions for the system. For small f , the system is in the fluid state for all densities, as
will be shown in section 4. For a comparison between theory and simulation regarding the potential
of equation (3.10), see [54].
184
Star polymers. . .
very well with the theory for the values f = 6, 12, 18, and 24 studied there. Further
corroboration for the validity of the interaction was offered by the recent work of
Striolo et al. [81].
3.3. Experiments
A detailed comparison between theory and SANS data from a series of solutions
of 18-arm polyisoprene (PI) stars was performed by Likos et al. [31], and for 57-arm
solutions by Stellbrink et al. [82]. In [31], samples with volume fractions φ in the
range 5 · 10−4 6 φ 6 0.3 were investigated. The theoretically obtained structure
factor of the stars, calculated by employing the potential of equation (3.9), was
multiplied by the form factor of the stars and convoluted with the known resolution
function of the experimental apparatus in order to obtain the theoretical prediction
for the total scattering intensity I(Q) at momentum transfer Q. The star size was
taken from the experiment and no adjustable parameters were used. This was then
compared to the measured intensities. The results are shown in figure 5.
0.01 0.10
Q [A
−1
]
1e+02
1e+03
1e+04
I(
Q
)/
φ
[
c
m
3
/m
o
l]
φ = 2%
φ = 8%
(a)
0.01 0.10
Q [A
−1
]
1e+02
1e+03
1e+04
I(
Q
)/
φ
[
c
m
3
/m
o
l]
φ = 15%
φ = 30%
(b)
Figure 5. Experimental (symbols) and theoretical (lines) results for the scattering
intensities I(Q) scaled with polymer volume fraction φ. From top to bottom (a)
φ = 2% and 8%; (b) φ = 15% and 30%. The overlap volume fraction is φ∗ = 10%
for this sample. (Redrawn from [31].)
It can be seen that the agreement is quite satisfactory for the whole range of
concentrations. In particular, the osmotic compressibility of the solution, being pro-
portional to I(Q → 0) is given correctly for all concentrations, as well as the general
shape and wavenumber Qmax at which the scattering intensity displays a maximum.
The height of the peak is underestimated by the theory, and the agreement worsens
somewhat as the concentration grows. However, at high values of φ, the decoupling
between form- and structure factors implied in writing down I(Q) ∝ P (Q)S(Q)
becomes questionable and this is a possible source of discrepancies between theory
and experiment. The logarithmic-Yukawa potential is the first that gave quantitative
agreement between theory and experiment for such a wide range of concentrations.
185
C.N.Likos, H.M.Harreis
Earlier attempts to fit the experimental results with a hard sphere-Yukawa inter-
action, for example, failed at and beyond the overlap concentration [83,84]. Indeed,
the existence of a ‘soft core’, such as the logarithmic term in the potential is crucial
at high concentrations where the stars start interpenetrating. A similar comparison
performed recently for f = 57-arm samples again yielded a very satisfying agree-
ment between theory and experiment [82]. Further experimental support for the
interaction potential is offered by shear moduli measurements on micelles [85] and
osmotic second virial coefficient measurements for star polysterenes [81].
3.4. Polydispersity effects and many-body forces
The chemical synthesis of monodisperse chains, i.e., of chains with a fixed number
of monomers N is nowadays relatively straightforward. However, experimental star
polymer samples display inevitably polydispersity in the arm number f which reflects
itself into a size polydispersity by means of equation (2.18) as well. The distribution
of arm numbers of a sample around its mean value is rather sharp for arm numbers
f ∼= 30 but grows with the increasing f . Typical values of the relative polydispersity
are of the order of 5% for f ∼= 60. Hence, it is desirable to extend the theory of
effective interactions to polydisperse samples. This was done by von Ferber et al. [37],
using a combination of scaling ideas and computer simulations. It was established
that the effective interaction vf1f2(r) between two stars of functionalities f1, f2 and
respective sizes σf1 , σf2 is given by the form [37]:
βvf1f2(r) = Θf1f2
[
− ln
( r
σ
)
+
1
1 + κσ
]
for r 6 σ,
= Θf1f2
1
1 + κσ
(σ
r
)
exp [−κ(r − σ)] for r > σ, (3.11)
with the scaling prefactor
Θf1f2 =
5
36
1√
2− 1
[
(f1 + f2)
3/2 −
(
f
3/2
1 + f
3/2
2
)]
, (3.12)
and the mean size σ and decay length κ
σ =
σf1 + σf2
2
, κ−1 =
κ−1
f1
+ κ−1
f2
2
. (3.13)
Here κ−1
fi
= 2σfi/
√
fi is the decay length of the monodisperse interaction, equa-
tion (3.9).
On the other hand, the process of tracing out the monomers in order to derive
effective interactions between the star centers always has the effect of generating a hi-
erarchy of forces, commencing at pair interactions and proceeding to include triplet,
quadruplet and higher-order terms [86]. It is usual practice to truncate this series
at the pair level, introducing thereby solely pair interactions, such as the ones we
have presented in the preceding subsections. Nevertheless, the missing higher-order
terms should be estimated, in order to establish the validity of the pair potential
186
Star polymers. . .
approximation. For stars, this was done by von Ferber et al. [36]. As a first result,
it was found that triplet forces become relevant only when three stars start to have
triple overlaps within their coronae, yielding a packing fraction which is much higher
than the highest shown in figure 6. Hence, the star polymer phase diagram shown
there remains unaffected by the presence of three-body forces. Moreover, the triplet
force was found to be attractive, i.e., the total force turns out to be smaller than
the sum of the two body forces. This can be understood intuitively, if one imagines
that the force arises from an overlap between the stars’ coronae. In a geometry with
a triple overlap, the sum of the three two-body overlaps overestimates the total
overlap volume, as it counts the triplet overlap three times. On a quantitative basis,
it was found that independently of the arm numbers and of r, in the full region
described by the logarithmic potential, the relative deviation of the force caused
by the triplet forces is equal to (33/2 − 3)/(23/2 − 2) ∼= −0.11 [36]. At very high
concentrations, where the system looks like a semidilute polymer solution, the pair
potential approximation breaks down, however. This was established in the work
of von Ferber et al. [36], through the investigation of the general M-body forces in
the limit when the interstar distances go to zero. It was found that the contribution
from a cluster of M stars increases with M and even diverges for M → ∞.
4. The phase diagram of star polymer solutions
Taking the pair potential of equation (3.9) for granted, it is then possible to
trace out in detail the phase diagram of the system. Clearly, as the temperature
plays no role, the relevant parameters are the functionality f and the density ρ =
Ns/V of stars, which was expressed through the dimensionless ‘packing fraction’
η = (π/6) ρσ3. The phase diagram was calculated by Watzlawek et al. [35,38].
For the fluid phase, the very accurate and thermodynamically consistent Rogers-
Young closure [87] was solved to obtain pair correlations and structure factors. The
free energy of the fluid phase was then calculated by means of the compressibility
or virial routes [71]. In order to measure the fluid free energy in a simulation, the
technique of thermodynamic integration was employed [88,89].
As far as the solid phases are concerned, Watzlawek et al. [35,38] considered
a large number of candidate crystal structures and the competition among them.
The structures were the fcc, bcc and simple cubic (sc) lattices, as well as the more
‘exotic’ diamond (diam), face-centered orthogonal (fco), body-centered orthogonal
(bco), orthogonal (orth) and diamond orthogonal (diao) structures [90]. In the last
four, ‘orthogonal’ structures, produced by stretching the conventional unit cells of
the original ones by an arbitrary amount and keeping the conventional cell volume
fixed, the ratios of the lattice constants a, b and c in the three spatial directions act
as additional degrees of freedom which have to be relaxed so that a minimum of the
free energy can be achieved.5 The analysis was carried out for a wide range of arm
5We note that the bco structure reduces to the bcc for a : b : c = 1 : 1 : 1 and to the fcc for
a : b : c = 1 : 1 :
√
2 [90].
187
C.N.Likos, H.M.Harreis
numbers, 18 6 f 6 512 and packing fractions, 0 6 η 6 1.5. The common tangent
construction yielded then the phase diagram shown in figure 6.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
η
0.00
0.01
0.02
0.03
96
64
48
40
34
f f
-1
fluid
bcc
fcc bco diam.
(ηc, fc)
X
Figure 6. Bulk phase diagram of star polymers interacting with potential of
equation (3.9), calculated in [35]. The squares indicate the phase boundaries;
solid lines are guide to the eye. The black cross denotes the point with critical
arm number fc ≃ 34 and corresponding density ηc ≃ 0.43. The arrow indicates
a path through the phase diagram that is roughly equivalent to a change in the
altitude z when the solution is under the influence of a gravitational field, see
section 6.
For f . fc = 34, the fluid phase is the only stable one and freezing is impossible
at all densities. We note that this feature had been predicted by Witten et al. [24]
using scaling arguments. These authors had estimated fc
∼= 100. The detailed phase
diagram also confirms the reentrant melting scenario put forward in [24], with the bcc
crystal being nested between a low- and a high-density liquid at about the overlap
packing fraction η∗ ∼= 0.50. For f > fc, four stable crystal structures exist and,
depending on f , a given system will have at least one stable crystalline phase. In the
domain of intermediate densities, 0.2 . η . 0.7, the bcc phase is stable for fc < f .
54, whereas for f & 70 the fcc crystal wins. In the regime 54 . f . 70, the system
will first freeze into a bcc lattice but then it will undergo a structural phase transition
into a fcc crystal. The interplay between bcc and fcc can be easily understood here,
as at these densities only the Yukawa tail of the interaction is visible to the particles.
Hence, for large f (strong screening), packing effects dominate and stabilize the fcc,
whereas for smaller f , energy plays an important role stabilizing the more open, bcc
structure. This is completely analogous to the phase behavior observed in charge-
stabilized colloids [91–93] as well as in copolymer micelles [4,5,94,95]. The HS-type
behavior, formally corresponding to f → ∞ sets in at very high f -values, typically
f ∼= 10 000, due to an interplay between the steeply increasing logarithmic core and
the Yukawa tail, for details see [38].
Quite more unexpected phases show up for high packing fractions, η & 0.7. A bco
phase with a strong anisotropy along its three principal space axes is stable in the
188
Star polymers. . .
region 0.70 . η . 1.1 and a diamond lattice becomes stable for 1.1 . η . 1.4. The
ratios of the lattice constants in the bco crystal turned out to be nearly independent
of f and increasing from b/a ∼= 2.24 and c/a ∼= 1.32 at η = 0.7 to b/a ∼= 3.14
and c/a ∼= 1.81 at η = 1.0. This means that, throughout the range of stability
of the bco phase, the anisotropy in the conventional unit cells leads to a strong
interpenetration of the particle coronae along one of the three lattice axes (the one
with the smallest lattice constant). In this way, the nearest-neighbor distance is
smaller than σ whereas all other neighbors are kept at distances larger than σ. This
can be intuitively understood from the form of the interaction potential (3.9). Due
to the very weak divergence at small r, the energy penalty for overlaps within the
logarithmic core is not prohibitively large. At the same time, the potential falls
rapidly enough at distances r > σ, so that it becomes energetically favorable to
have two neighbors very close to a central particle and all remaining ones outside
the σ-range rather than twelve (fcc) or eight (bcc) nearest neighbors, all of them
at distances larger than the nearest neighbor distance in the bco but nevertheless
within the σ-range.
The same argumentation leads to an understanding of the stability of the dia-
mond lattice. As the density increases further, the bco becomes unstable with respect
to a lattice in which two more neighbors enter into the corona and the diamond lat-
tice with four nearest neighbors is stabilized. Again, the simulations revealed that
in the domain of stability of the diamond, the second neighbors are kept outside
the corona. Hence, both the ultrasoft character of the logarithmic potential and the
crossover to a Yukawa tail at r = σ are necessary to stabilize the diamond crystal.
This phenomenon is unknown for the usual, hard pair potentials and, in fact, there
exists a widespread belief in the literature that three-body interactions are necessary
for such a stabilization [96–100].
The information about the stability of the bco and diamond phases is already
hidden in the radial distribution function g(r) of a stable liquid with f = 40 arms,
lying slightly above the domains of stability of these phases. Indeed, this g(r) shows
a pronounced peak close to the origin and the fluid state coordination number is
about two above the stability domain of the bco and four above that of the diamond.
The same is seen in the angle-averaged g(r)’s of the solid phases [38]. A very similar
effect was found by Broughton and Li [101] for the bond-angle dependent Stillinger-
Weber potential [96], which was constructed with the goal of stabilizing the diamond
lattice. Here, however, the latter has been stabilized for the first time by means of
a simple, radially symmetric pair interaction alone. Moreover, the second freezing
observed in experiments on starlike copolymeric micelles [5,102,103], can now be
understood as the freezing of the system into the anisotropic bco phase, see figure 6.
5. The mean-field character of the star polymer fluid
The ultrasoft, logarithmic divergence of the star-star potential of equation (3.9)
gives rise to some novel features of the macroscopic star polymer solution. On the
one hand, it opens up the possibility that for low enough f , the system does not
189
C.N.Likos, H.M.Harreis
freeze at any density, a feature unknown for the usual hard or power-law diverging
interactions. In this way, the correlation functions of the fluid can be studied at
arbitrarily high densities. On the other hand, the ultrasoftness of the interaction
renders it integrable in three dimensions, meaning that the integral
∫
d3r v(r) = 4π
∫
∞
0
dr r2v(r) = ṽ(0), (5.1)
is finite and equal (by definition) to the value of the Fourier transform of the potential
ṽ(k) at k = 0.
Before discussing the ramifications of these properties on the thermodynamics of
the star polymer fluid, we point out that recently quite a bit of attention has been
focussed on a related but distinct class of fluids, namely the ones whose constituent
particles interact by means of positive-definite, purely repulsive potentials that do
not diverge at the origin [104–110]. It was shown that for such systems, a ‘mean-
field limit’ is satisfied [109]: at sufficiently high densities for all temperatures and at
sufficiently high temperatures for all densities, the direct correlation function c(r)
of the fluid [71] is approximated extremely well by its large-r asymptotic behavior,
c(r) ∼ −βv(r) at all r. Hence, for bounded potentials and under the condition
of high densities and/or temperatures, we can write at all separations and for all
densities:
c(r; ρ) = −βv(r). (5.2)
To characterize this whole class of systems, Louis coined the term ‘mean-field fluids’
[110]. Among the many elegant relations satisfied by the thermodynamic quantities
of these models [105], here we single out the quadratic scaling of the excess Helmholtz
free energy density with the particle density ρ:
βFex(ρ)
V
≡ f(ρ) =
βṽ(0)
2
ρ2, (5.3)
which is a direct consequence of the exact compressibility sum rule [71]:
f ′′(ρ) = −c̃(k = 0; ρ) = −4π
∫
∞
0
dr r2c(r; ρ), (5.4)
in conjunction with equation (5.2).
We will argue in what follows that the star polymer fluids satisfy approximately
the thermodynamical scaling relation (5.3) although they do not satisfy the more
stringent structural relation (5.2). Thereby, we introduce the terminology ‘strong
mean-field fluids’ for those discussed in the preceding paragraph and ‘weak mean-
field fluids’ for those that, like the star polymer fluid, feature an interaction that
diverges but only weakly so. The divergence of the interaction is responsible for the
impossibility to satisfy the strong mean-field condition of equation (5.2). Indeed,
the direct correlation function c(r) has to remain finite at r = 0, whereas the pair
potential diverges. Hence, as shown in figure 7(a), there will always exist a region in
the neighborhood of the origin in which equation (5.2) cannot be satisfied. At the
190
Star polymers. . .
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
r/σ
−120
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
c
(r
)
c(r), η = 1.0
c(r), η = 2.0
c(r), η = 10.0
−βv(r)
(a)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
r/σ
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
r2
c(
r)
r
2
c(r), η = 1.0
r
2
c(r), η = 2.0
r
2
c(r), η = 10.0
−βr
2
v(r)
(b)
Figure 7. (a) Comparison between the direct correlation function of a f = 32 star
polymer fluid at various densities with the mean-field result, −βv(r). (b) Same
as in (a) but for the quantities r2c(r) and −βr2v(r).
same time, it can be seen in this figure that the extent of this region shrinks with
the increasing density, hence the fluid becomes more ‘strong mean-field’-like as it
gets denser.
The discrepancies between c(r) and −βv(r) become innocuous when we turn
our attention to the thermodynamics. To obtain the excess Helmholtz free energy,
one needs only the integral of r2c(r), see equation (5.4). As demonstrated in fig-
ure 7(b), upon multiplication with the geometrical factor r 2, the deviations of c(r)
from−βv(r) become suppressed, so that we can write, to a very good approximation:
∫
∞
0
dr r2c(r; ρ) ∼= −
∫
∞
0
dr r2βv(r). (5.5)
Equation (5.5) together with equation (5.4) yield an approximate scaling of the
excess free energy of the weak mean-field fluids with density that is identical to that
of the strong mean-field fluids, equation (5.3). The accuracy of the approximation
for the f = 32 star fluid is shown in figure 8. The line labeled as exact free energy
there was obtained by solving the Rogers-Young closure for the fluid at a wide
density range and subsequently utilizing the compressibility sum rule [equation (5.4)]
to obtain the excess free energy. Comparisons with simulations [33] have indeed
demonstrated that this procedure delivers an essentially exact numerical result.
Clearly, the mean-field approximation improves with the increasing density, since
the number of particles effectively interacting with one another grows. The crossover
density ρ× above which the quadratic scaling of the free energy holds is f -dependent
and grows with the increasing f . Indeed, the functionality acts as a prefactor that
controls the strength of the logarithmic divergence of the potential at the origin.
Formally, the mean-field approximation also becomes better at a growing spatial
dimension d, since the geometrical prefactor rd−1 multiplying c(r) and −βv(r) sup-
presses the small-r discrepancies of the two more efficiently. However, we reiterate
191
C.N.Likos, H.M.Harreis
10
−1
10
0
10
1
10
2
ρσ3
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
βσ
3 F
ex
/V
Exact
βv(0)/2*(ρσ3)2
(MFA)
Slope = 2
~
ρ
Figure 8. Comparison of the mean-field approximation (dashed line) with the
exact result (solid line) concerning the excess free energy density of the f = 32
star fluid. The slope of the straight line is 2, indicating the quadratic dependence
of the excess free energy density on particle density. The arrow indicates the
location of the crossover density ρ×, above which the scaling of equation (5.3)
holds with a relative error of less than 10%.
that the mean-field idea holds only for the thermodynamics of weak mean-field flu-
ids: if one näıvely applies the strong mean-field relation, equation (5.2), to weak
mean-field fluids, one obtains results for the structure factor S(k) = [1 − ρc̃(k)]−1
that are seriously in error for finite k-values. Only at k = 0 and at sufficiently high
densities is it a reasonable approximation to set S(0) = [1 + ρβṽ(0)]−1.
6. Star polymers in gravitational fields
The mean-field character of star polymer fluids presented in the preceding section
manifests itself in the sedimentation profiles of the same. Furthermore, the sedimen-
tation profiles provide an independent check for the bulk phase diagram since the
effects stemming from the reentrant melting behavior are prominent. Sedimentation
profiles of star polymer solutions were investigated by Dzubiella et al. [111] using
MC simulations as well as by applying different theoretical approaches. The MC
simulations were carried out in a gravitational field of strength
α =
mgσ
kBT
, (6.1)
withm denoting the star polymer mass and g denoting the gravitational acceleration.
Apart from α, the following two thermodynamic variables characterize the system:
the arm number f of the star polymers and the number density per unit surface, τ ,
192
Star polymers. . .
given by the normalization condition of the density profile ρ(z):
τ =
∫
∞
0
ρ(z)dz. (6.2)
The simulations carried out by the authors of [111] revealed the density profiles
shown in figure 9. Let us first turn our attention to figures 9a and 9b, where density
0
1
2
3
profile
Ψ4
(a)
0
1
2
3
ρ
σ
3
(b)
0 5 10 15 20 25 30 35
z/σ
0
1
2
3 (c)
0
1
2
ρ
σ
3
(e)
0 10 20 30 40 50 60
z/σ
0
1
2
(f)
0
1
2 profile
Ψ4
(d)
Figure 9. Sedimentation profiles of star polymers for an arm number f = 39 and
a density τσ2 = 48.87. The gravitational strength α is decreased from (a) to (f)
with (a) α = 30.0, (b) α = 17.0, (c) α = 16.0, (d) α = 8.0, (e) α = 6.0 and
(f) α = 4.0. In plots (c)-(f) the order parameter Ψ4 is also shown (dashed line)
using the same y-scale as the profiles. In (a) and (b) a straight line the equation
of which is derived within the LDA [see equation (6.7)] is superimposed on the
plots (dotted line).
profiles for strong gravitational fields α > α∗ are shown. Beyond the layering on
the wall due to packing effects, a fluid regime with the density decaying as a linear
function of altitude z can be discerned. At some height (z ≃ 25σ in (a)) the density
rapidly decays to zero. At this strong inhomogeneity, oscillations in density with
wavelength σ can be distinguished in the sedimentation profile, which is smooth
elsewhere in the linear regime. These could be reproduced [111] in the framework
of the hybrid-weighted-density-approximation (HWDA) density functional theory
[112]. By lowering the gravitational strength α further, see figures 9c to 9f, a crit-
ical strength α∗ in the range 16.0 < α∗ < 17.0 is discovered. Below α∗ the density
profiles qualitatively change, exhibiting strong density oscillations, a clear indica-
tion for a crystalline phase. They extend over 10 to 20 star diameters, equivalent
to several crystalline layers. The length of the crystal grows as α decreases. Com-
paring with the bulk phase diagram, it can be presumed that the reentrant melting
behavior is mirrored in the density profiles, with a bcc solid intervening in a fluid.
193
C.N.Likos, H.M.Harreis
To complement the indications of the density profiles with an additional check for
crystalline order, the local order parameter Ψ4 that checks for fourfold symmetry in
two dimensions around a given particle was calculated. It is defined as
Ψ4(z) =
∣
∣
∣
∣
∣
〈
1
4Nl
Nl
∑
j=1
∑
<k>
e4iφjk
〉
∣
∣
∣
∣
∣
, (6.3)
where the k-sum includes the four nearest neighbors of the given particle and the
j-sum extends over Nl particles in the corresponding layer. The angular brackets
indicate a canonical ensemble average. φjk is the polar angle of the interparticle
distance vector with respect to a fixed reference frame. For ideal fourfold symmetry,
i.e., for a particle contained in a bcc-solid layer, Ψ4 = 1. For practical purposes,
values of Ψ4 > 0.8 are taken to be conclusive evidence for a crystalline phase with
fourfold-in-layer-symmetry. Thus, obviously, a bcc solid is stable and the sedimen-
tation profiles present a confirmation of the bulk phase diagram: the gravitational
field induces a scan of all densities as a function of the height z, thereby mapping
the freezing/reentrant melting succession on the height and producing a solid sheet
at intermediate heights intercalated in a fluid at lower and at higher elevations.
The linear dependence of the density profile on z, on the other hand, is a conclu-
sive evidence for the star polymer fluids belonging to the class of ‘weak mean-field
fluids’, see equation (5.3), as we demonstrate below. We work in the grand canonical
ensemble and introduce a variational grand potential per unit area, Σ̃(T, µ; [ρ(z)]),
which is a functional of the density profile and a function of the chemical potential µ
and temperature T . Introducing the ideal and excess per unit area contributions to
the intrinsic Helmholtz free energy of the system, F id[ρ(z)] and Fex[ρ(z)] respective-
ly, we find that in the local density approximation, the expression for Σ̃(T, µ; [ρ(z)])
reads as:
Σ̃(T, µ, [ρ(z)]) = Fid[ρ(z)] + Fex[ρ(z)] +
∫
dz Φext(z)ρ(z) − µ
∫
dz ρ(z)
= kBT
∫
∞
0
dz ρ(z)
[
ln(ρ(z)λ3)− 1
]
+
∫
∞
0
dz [f(ρ(z)) + (mgz − µ)ρ(z)] , (6.4)
where λ =
√
h2/2πmkBT is the thermal de Broglie wavelength and f(ρ(z)) is the
Helmholtz free energy density of the bulk fluid. The minimization of Σ̃ with respect
to ρ(z) yields the equilibrium profile ρ0(z); the value of the functional at equilibrium,
Σ̃(T, µ, [ρ0(z)]) is then the Gibbs free energy per unit area, Σ(T, µ) of the system.
Setting δΣ̃(T, µ, [ρ(z)])/δρ(z)|ρ0(z) = 0 in equation (6.4), leads to:
kBT ln[ρ0(z)σ
3] + f ′(ρ0(z)) = µ′ −mgz, (6.5)
where f ′(x) denotes the derivative of f(x) and µ′ = µ−3 ln(λ/σ) is a shifted chemical
potential.
194
Star polymers. . .
At this point, we resort to the the property equation (5.3), which, together with
the use of the dimensionless variables x ≡ z/σ, ρ̄(x) ≡ ρ(z)σ3, B ≡ βṽ(0)/σ3,
µ̄ ≡ βµ′ and introducing equation (5.3) into equation (6.5), yields the equilibrium
profile through:
ln[ρ̄0(x)] +Bρ̄0(x) = µ̄− αx. (6.6)
For star functionality f = 39, we obtain B = 250.4 [see equation (3.9)]. Hence, the
second term in the lhs of equation (6.6) above dominates over the logarithmic term
for densities ρ̄(x) & 0.10. As far as almost the entire simulation density profile fulfills
this condition, we finally omit the logarithmic term from equation (6.6) above. Using
the normalization condition
∫ µ̄/α
0
dxρ̄0(x) = τσ2 ≡ τ̄ , the chemical potential µ̄ can
be evaluated. Finally, we thereby obtain a linear density profile:
ρ̄0(x) =
0 for x < 0,
√
2ατ̄
B
− α
B
x for 0 < x <
√
2Bτ̄
α
,
0 for
√
2Bτ̄
α
< x.
(6.7)
In figures 9a and 9b, this prediction, which crucially depends on the mean-field
approximation used, is checked against simulation. Excellent agreement is found,
thereby confirming again the mean-field character of star polymer solutions. More-
over, a phenomenological Landau theory revealed a host of scaling relations satisfied
by the thickness of the intercalating solid layer, which have been explicitly verified
by Monte Carlo simulations [111].
7. Summary and concluding remarks
In this paper, we have presented a concise review of a recent progress achieved in
understanding the properties of star polymer solutions in terms of effective interac-
tions between the stars’ centers. This constitutes a coarse-grained description of the
system, in which all degrees of freedom at atomic length scales ( ∼ 1 Å) have been
traced out and hence they drop out of the picture. In our mesoscopic description,
the only relevant length scales are the size of the stars σ (typically of the order of
100 Å or more) and the structural length scale a = ρ−1/3 set by the star density. It
has been demonstrated that this approach is indeed fruitful. By integrating out the
details of the monomers at the one- and two-star level, an enormous simplification of
the many-star problem is achieved: the total number of degrees of freedom is reduced
from Ns × N × f to Ns, i.e., the number of stars in the system. Though the effec-
tive interactions presented here have been derived strictly for star polymers with a
vanishingly small core, they should remain approximately valid for macromolecular
aggregates with a core-brush (shell) structure, such as sterically stabilized colloids
and micelles, provided that the ratio of core size to brush height is sufficiently small.
195
C.N.Likos, H.M.Harreis
8. Acknowledgements
We thank J.Dzubiella, A.Jusufi, C. von Ferber, A.Lang, H.Löwen, and M.Watzla-
wek for fruitful collaboration and J.Dzubiella for a critical reading of the manuscript.
This work has been supported by the Deutsche Forschungsgemeinschaft through the
SFB 237.
References
1. Hunter R.J. Foundations of Colloid Science. Vol. I. Oxford, Clarendon Press, 1986.
2. Likos C.N. // Phys. Rep., 2001, vol. 348, p. 267.
3. Gast A.P., Russel W.B. // Physics Today, 1998, vol. 51, p. 24.
4. Gast A.P. // Langmuir, 1996, vol. 12, p. 4060.
5. Gast A.P. // Curr. Opinion Coll. Int. Sci., 1997, vol. 2, p. 258.
6. Grest G.S., Fetters L.J., Huang J.S., Richter D. // Adv. Chem. Phys., 1996,
vol. XCIV, p. 67.
7. Burchard W. // Adv. Polym. Sci., 1999, vol. 143, p. 113.
8. Freire J.J., Pla J., Rey A., Prats R. // Macromolecules, 1986, vol. 19, p. 452.
9. Zhou L.-L., Roovers J. // Macromolecules, 1993, vol. 26, p. 963.
10. Roovers J., Zhou L.-L., Toporowski P.M., van der Zwan M., Iatrou H., Hadjichris-
tidis N. // Macromolecules, 1993, vol. 26, p. 4324.
11. Dozier W.D., Huang J.S., Fetters L.J. // Macromolecules, 1991, vol. 24, p. 2810.
12. Mezei F. – In: Neutron Spin Echo. ed. by Mezei F., Berlin, Springer Verlag, 1979.
13. Solc K., Stockmayer W.H. // J. Chem. Phys., 1971, vol. 54, p. 2756.
14. Solc K. // J. Chem. Phys., 1971, vol. 55, p. 2335.
15. Tanaka G., Mattice W.L. // Macromol. Theory Simul., 1996, vol. 5, p. 499.
16. Solc K. // Macromolecules, 1973, vol. 6, p. 378.
17. Solc K. // Macromolecules, 1980, vol. 13, p. 506.
18. Batoulis J., Kremer K. // Macromolecules, 1989, vol. 22, p. 4277.
19. Zifferer G. // J. Chem. Phys., 1995, vol. 102, p. 3720.
20. Zifferer G. // Macromol. Theory Simul., 1997, vol. 6, p. 381.
21. Zifferer G. // J. Chem. Phys., 1999, vol. 110, p. 4668.
22. Forni A., Ganazzoli F., Vacatello M. // Macromolecules, 1997, vol. 30, p. 4737.
23. Sikorski A., Romiszowski P. // J. Chem. Phys., 1998, vol. 109, p. 6169.
24. Witten T.A., Pincus P.A., Cates M.E. // Europhys. Lett., 1986, vol. 2, p. 137.
25. Witten T.A., Pincus P.A. // Macromolecules, 1986, vol. 19, p. 2509.
26. Richter D., Jucknischke O., Willner L., Fetters L.J., Lin M., Huang J.S., Roovers J.,
Toporowski C., Zhou L.L. // J. Physique IV Suppl., 1993, vol. 3, p. 3.
27. Roovers J. // Macromol. Symp., 1997, vol. 121, p. 89.
28. Adam M., Fetters L.J., Graessley W.W., Witten T.A. // Macromolecules, 1991,
vol. 25, p. 2434.
29. Willner L., Jucknischke O., Richter D., Farago B., Fetters L.J., Huang J.S. // Euro-
phys. Lett., 1992, vol. 19, p. 297.
30. Willner L., Jucknischke O., Richter D., Roovers J., Zhou L.-L., Toporowski P.M., Fet-
ters L.J., Huang J.S., Lin M.Y., Hadjichristidis N. // Macromolecules, 1994, vol. 27,
p. 3821.
196
Star polymers. . .
31. Likos C.N., Löwen H., Watzlawek M., Abbas B., Jucknischke O., Allgaier J.,
Richter D. // Phys. Rev. Lett., 1998, vol. 80, p. 4450.
32. Likos C.N., Löwen H., Poppe A., Willner L., Roovers J., Cubitt B., Richter D. //
Phys. Rev. E, 1998, vol. 58, p. 6299.
33. Watzlawek M., Löwen H., Likos C.N. // J. Phys.: Condens. Matter, 1998, vol. 10,
p. 8189.
34. Stellbrink J., Abbas B., Allgaier J., Monkenbusch M., Richter D., Likos C.N.,
Löwen H., Watzlawek M. // Progr. Coll. Polym. Sci., 1998, vol. 110, p. 25.
35. Watzlawek M., Likos C.N., Löwen H. // Phys. Rev. Lett., 1999, vol. 82, p. 5289.
36. von Ferber C., Jusufi A., Likos C.N., Löwen H., Watzlawek M. // Eur. Phys. J. E,
2000, vol. 2, p. 311.
37. von Ferber C., Jusufi A., Watzlawek M., Likos C.N., Löwen H. // Phys. Rev. E,
2000, vol. 62, p. 6949.
38. Watzlawek M. Phase Behavior of Star Polymers. Aachen, Shaker-Verlag, 2000.
39. Roovers J. // Macromolecules, 1994, vol. 27, p. 5359.
40. Seghrouchni R., Petekidis G., Vlassopoulos D., Fytas G., Semenov A.N., Roovers J.,
Fleischer G. // Europhys. Lett., 1998, vol. 42, p. 271.
41. Stellbrink J., Allgaier J., Richter D. // Phys. Rev. E, 1997, vol. 56, p. R3772.
42. Vlassopoulos D., Pakula T., Fytas G., Roovers J., Karatasos K., Hadjichristidis N.
// Europhys. Lett., 1997, vol. 39, p. 617.
43. Nommensen P.A., Duits M.H.G., Lopulissa J.S., van den Ende D., Mellema J. //
Progr. Coll. Polym. Sci., 1998, vol. 110, p. 144.
44. Fleischer G., Fytas G., Vlassopoulos D., Roovers J., Hadjichristidis N. // Physica A,
2000, vol. 280, p. 266.
45. Semenov A.N., Vlassopoulos D., Fytas G., Vlachos G., Fleischer G., Roovers J. //
Langmuir, 1999, vol. 15, p. 358.
46. Vlassopoulos D., Fytas G., Roovers J., Pakula T., Fleischer G. // Faraday Discuss.,
1999, vol. 112, p. 225.
47. Kapnistos M., Semenov A.N., Vlassopoulos D., Roovers J. // J. Chem. Phys., 1989,
vol. 111, p. 1753.
48. Vlassopoulos D., Pakula T., Fytas G., Pitsikalis M., Hadjichristidis N. // J. Chem.
Phys., 1999, vol. 111, p. 1760.
49. Grest G.S., Kremer K., Witten T.A. // Macromolecules, 1987, vol. 20, p. 1376.
50. Jusufi A. Diploma thesis. Düsseldorf, Heinrich-Heine-Universität, 1998.
51. Daoud M., Cotton J.P. // J. Phys. (Paris), 1982, vol. 43, p. 531.
52. de Gennes P.G. Scaling Concepts in Polymer Physics. Ithaca, Cornell University
Press, 1979.
53. Vilgis T.A. // Phys. Rep., 2000, vol. 336, p. 167.
54. Jusufi A., Dzubiella J., Likos C.N., von Ferber C., Löwen H. // J. Phys.: Condens.
Matter, 2001, vol. 13, p. 6177.
55. Dzubiella J., Jusufi A., Likos C.N., von Ferber C., Löwen H., Stellbrink J., Allgaier J.,
Richter D., Schofield A.B., Smith P.A., Poon W.C.K., Pusey P.N. // Phys. Rev. E,
2001, vol. 64, p. 010401(R).
56. Batoulis J., Kremer K. // Europhys. Lett., 1988, vol. 7, p. 683.
57. Grest G.S. // Macromolecules, 1994, vol. 27, p. 3493.
58. Shida K., Ohno K., Kimura M., Kawazoe Y., Nakamura Y. // Macromolecules, 1998,
vol. 31, p. 2343.
197
C.N.Likos, H.M.Harreis
59. Molina L.A., Freire J.J. // Macromolecules, 1999, vol. 32, p. 499.
60. Grest G.S., Murat M. – In: Monte Carlo and Molecular Dynamics Simulations in
Polymer Science. ed. by Binder K., New York, Oxford University Press, 1995.
61. Forni A., Ganazzoli F., Vacatello M. // Macromolecules, 1996, vol. 29, p. 2994.
62. Rubio A.M., Freire J.J. // Macromolecules, 1996, vol. 29, p. 6946.
63. Jusufi A., Watzlawek M., Löwen H. // Macromolecules, 1999, vol. 32, p. 4470.
64. Su S.-J., Denny M.S., Kovac J. // Macromolecules, 1991, vol. 24, p. 917.
65. Su S.-J., Kovac J. // J. Chem. Phys., 1992, vol. 96, p. 3931.
66. Toral R., Chakrabarti A. // Phys. Rev. E, 1993, vol. 47, p. 4240.
67. Barrett A.J., Tremain D.L. // Macromolecules, 1987, vol. 20, p. 1687.
68. Rey A., Freire J.J., de la Torre J.G. // Macromolecules, 1987, vol. 20, p. 342.
69. Gratter O., Kratky O. Small Angle X-Ray Scattering. New York, Academic Press,
1982.
70. Higgins J.S., Benôıt H.C. Polymers and Neutron Scattering. Oxford, Clarendon Press,
1994.
71. Hansen J.-P., McDonald I.R. Theory of Simple Liquids. 2nd ed. London, Academic
Press, 1986.
72. des Cloizeaux J. // J. Phys. (Paris), 1980, vol. 41, p. 223.
73. von Ferber C., Holovatch Yu. // Theor. Math. Phys., 1996, vol. 109, p. 34.
74. von Ferber C., Holovatch Yu. // Europhys. Lett., 1997, vol. 39, p. 31.
75. von Ferber C., Holovatch Yu. // Phys. Rev. E, 1997, vol. 56, p. 6370.
76. von Ferber C., Holovatch Yu., Jusufi A., Likos C.N., Löwen H., Watzlawek M. //
J. Mol. Liquids, 2001, vol. 93, p. 151.
77. Bolhuis P.G., Louis A.A., Hansen J.-P., Meier E.J. // J. Chem. Phys., 2001, vol. 114,
p. 4296.
78. Ohno K., Shida K., Kimura M., Kawazoe Y. // Macromolecules, 1996, vol. 29,
p. 2269.
79. Rubio A.M., Freire J.J. // Comp. Theor. Polymer Sci., 2000, vol. 10, p. 89.
80. Shida K., Ohno K., Kimura M., Kawazoe Y. // Macromolecules, 2000, vol. 33,
p. 7655.
81. Striolo A., Prausnitz J.M., Bertucco A. // Macromolecules, 2000, vol. 33, p. 9583.
82. Stellbrink J., Allgaier J., Monkenbusch M., Richter D., Lang A., Likos C.N., Wat-
zlawek M., Löwen H., Ehlers G., Schleger P. // Progr. Coll. Polym. Sci., 2000, vol. 115,
p. 88.
83. Jucknischke O. Ph.D. thesis. Münster, Westfälische Wilhelms-Universität, 1995.
84. Abbas B. Diploma thesis. Münster, Westfälische-Wilhelms Universität, 1996.
85. Buitenhuis J., Förster S. // J. Chem. Phys., 1997, vol. 107, p. 262.
86. Dijkstra M., van Roij R., Evans R. // Phys. Rev. E, 1999, vol. 59, p. 5744.
87. Rogers F.A., Young D.A. // Phys. Rev. A, 1984, vol. 30, p. 999.
88. Frenkel D., Smit B. Understanding Molecular Simulation. San Diego, Academic
Press, 1996.
89. Allen M.P., Tildesley D.J. Computer Simulation of Liquids. Oxford, Clarendon Press,
1987.
90. Ashcroft N.W., Mermin N.D. Solid State Physics. Philadelphia, Holt Saunders, 1976.
91. Sirota E.B., Ou-Yang H.D., Sinha S.K., Chaikin P.M. // Phys. Rev. Lett., 1989,
vol. 62, p. 1524.
92. Kremer K., Robbins M.O., Grest G.S. // Phys. Rev. Lett., 1986, vol. 57, p. 2694.
198
Star polymers. . .
93. Robbins M.O., Kremer K., Grest G.S. // J. Chem. Phys., 1988, vol. 88, p. 3286.
94. McConnell G.A., Gast A.P., Huang J.S., Smith S.D. // Phys. Rev. Lett., 1993, vol. 71,
p. 2102.
95. McConnell G.A., Lin E.K., Gast A.P., Huang J.S., Lin M.Y., Smith S.D. // Faraday
Discuss. Chem. Soc., 1994, vol. 98, p. 121.
96. Stillinger F.H., Weber T.A. // Phys. Rev. B, 1985, vol. 31, p. 5262.
97. Tersoff J. // Phys. Rev. Lett., 1986, vol. 56, p. 632.
98. Biswas R., Hamann D.R. // Phys. Rev. Lett., 1985, vol. 55, p. 2001.
99. Phillpot R.S., Lutsko J.F., Wolf D., Yip S. // Phys. Rev. B, 1989, vol. 40, p. 2831.
100. Likos C.N., Ashcroft N.W. // Phys. Rev. E, 1995, vol. 52, p. 5714.
101. Broughton J.Q., Li X.P. // Phys. Rev. B, 1987, vol. 35, p. 9120.
102. McConnell G.A., Gast A.P. // Phys. Rev. E, 1996, vol. 54, p. 5447.
103. McConnell G.A., Gast A.P. // Macromolecules, 1997, vol. 30, p. 435.
104. Likos C.N., Watzlawek M., Löwen H. // Phys. Rev. E, 1998, vol. 58, p. 3135.
105. Lang A., Likos C.N., Watzlawek M., Löwen H. // J. Phys.: Condens. Matter, 2000,
vol. 12, p. 5087.
106. Louis A.A., Bolhuis P.G., Hansen J.-P., Meier E.J. // Phys. Rev. Lett., 2000, vol. 85,
p. 2522.
107. Louis A.A., Bolhuis P.G., Hansen J.-P. // Phys. Rev. E, 2000, vol. 62, p. 7961.
108. Likos C.N., Schmidt M., Löwen H., Ballauff M., Pötschke D., Lindner P. // Macro-
molecules, 2000, vol. 34, p. 2914.
109. Likos C.N., Lang A., Watzlawek M., Löwen H., Phys. Rev. E, 2001, vol. 63, p. 031206.
110. Louis A.A. // Philos. Trans. Ros. Soc. A, 2001, vol. 359, p. 939.
111. Dzubiella J., Harreis H.M., Likos C.N., Löwen H. // Phys. Rev. E, 2001, vol. 64,
p. 011405.
112. Leidl R., Wagner H. // J. Chem. Phys., 1993, vol. 98, p. 4142.
199
C.N.Likos, H.M.Harreis
Зіркові полімери: від конформацій через взаємодії
до фазових діаграм
К.Н.Лікос, Г.М.Харрайс
Інститут теоретичної фізики II, Університет Гайнріха Гайне,
Університатштрасе 1, D-40225 Дюссельдорф, Німеччина
Отримано 9 жовтня 2001 р.
Здійснено огляд недавніх досягнень у теоретичному описі взаємо-
дій, кореляцій і фазової поведінки концентрованих розчинів зірко-
вих полімерів, просторово стійких колоїдів і міцел. Ми покажемо,
що теоретично передбачена надм’яка логарифмічно розбіжна ефек-
тивна взаємодія між центрами зірок, що була підтверджена SANS-
експериментами і комп’ютерними симуляціями, потрапляє в мно-
жину незвичних явищ, які спостерігаються в таких системах. Сюди
відносяться аномальні структурні фактори, поведінка зворотнього
плавлення, множини екзотичних кристалічних фаз. Також обговоре-
но узагальнення на випадок полідисперсних зірок і роль сил багатьох
тіл. Представлено особливу поведінку типу “cереднього поля” плинів
зіркових полімерів і показано, що вона проявляється у формі і струк-
турі профілів осаджування цих систем.
Ключові слова: полімери, колоїди, теорія рідкого стану, фазові
переходи.
PACS: 82.70.Dd, 61.25.Hq, 61.20.-p, 64.70.Dv
200
|