Field-theoretical renormalization group analysis for the scaling exponents of star polymers
We review recent results of the field theoretical renormalization group analysis on the scaling properties of star polymers. We give a brief account of how the numerical values of the exponents governing the scaling of star polymers were obtained as well as provide some examples of the phenomena...
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irk-123456789-1205822017-06-13T03:05:06Z Field-theoretical renormalization group analysis for the scaling exponents of star polymers von Ferber, Ch. Holovatch, Yu. We review recent results of the field theoretical renormalization group analysis on the scaling properties of star polymers. We give a brief account of how the numerical values of the exponents governing the scaling of star polymers were obtained as well as provide some examples of the phenomena governed by these exponents. In particular we treat the interaction between star polymers in a good solvent, the Brownian motion near absorbing polymers, and diffusion-controlled reactions involving polymers. Зроблено огляд недавніх результатів аналізу методом теоретикопольової ренормалізаційної групи масштабних (скейлінґових) властивостей зіркових полімерів. Коротко пояснено, як були отримані чисельні значення показників скейлінґу зіркових полімерів. Приведено приклади явищ, які описуються цими показниками. Зокрема, розглядається взаємодія міз зірковими полімерами у доброму розчиннику, бровнівський рух біля полімерного абсорбера, керовані дифузією реакції за участю полімерів. 2002 Article Field-theoretical renormalization group analysis for the scaling exponents of star polymers / Ch. von Ferber, Yu. Holovatch // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 117-136. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh DOI:10.5488/CMP.5.1.117 http://dspace.nbuv.gov.ua/handle/123456789/120582 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We review recent results of the field theoretical renormalization group analysis
on the scaling properties of star polymers. We give a brief account of
how the numerical values of the exponents governing the scaling of star
polymers were obtained as well as provide some examples of the phenomena
governed by these exponents. In particular we treat the interaction
between star polymers in a good solvent, the Brownian motion near absorbing
polymers, and diffusion-controlled reactions involving polymers. |
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Article |
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von Ferber, Ch. Holovatch, Yu. |
| spellingShingle |
von Ferber, Ch. Holovatch, Yu. Field-theoretical renormalization group analysis for the scaling exponents of star polymers Condensed Matter Physics |
| author_facet |
von Ferber, Ch. Holovatch, Yu. |
| author_sort |
von Ferber, Ch. |
| title |
Field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| title_short |
Field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| title_full |
Field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| title_fullStr |
Field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| title_full_unstemmed |
Field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| title_sort |
field-theoretical renormalization group analysis for the scaling exponents of star polymers |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2002 |
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http://dspace.nbuv.gov.ua/handle/123456789/120582 |
| citation_txt |
Field-theoretical renormalization group
analysis for the scaling exponents of
star polymers / Ch. von Ferber, Yu. Holovatch // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 117-136. — Бібліогр.: 44 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vonferberch fieldtheoreticalrenormalizationgroupanalysisforthescalingexponentsofstarpolymers AT holovatchyu fieldtheoreticalrenormalizationgroupanalysisforthescalingexponentsofstarpolymers |
| first_indexed |
2025-07-08T18:10:19Z |
| last_indexed |
2025-07-08T18:10:19Z |
| _version_ |
1837103286531915776 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 1(29), pp. 117–136
Field-theoretical renormalization group
analysis for the scaling exponents of
star polymers
Ch. von Ferber 1 , Yu.Holovatch 2,3
1 Theoretische Polymerphysik, Universität Freiburg,
D-79104 Freiburg, Germany
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
3 Ivan Franko National University of Lviv,
79005 Lviv, Ukraine
Received September 18, 2001
We review recent results of the field theoretical renormalization group anal-
ysis on the scaling properties of star polymers. We give a brief account of
how the numerical values of the exponents governing the scaling of star
polymers were obtained as well as provide some examples of the phenom-
ena governed by these exponents. In particular we treat the interaction
between star polymers in a good solvent, the Brownian motion near ab-
sorbing polymers, and diffusion-controlled reactions involving polymers.
Key words: star polymers, star exponents, renormalization group
PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh
1. Star exponents in polymer theory
It is well known that the asymptotic properties of a flexible polymer chain in a
good solvent are universal in the limit of infinite chain length. Lowering the temper-
ature in a system of a polymer in a good solvent the so called Θ temperature may
be reached below which the polymer collapses. At the transition temperature, the
effective two point attractive and repulsive interactions between different monomers
compensate each other and as a result the polymer chain may be described by a
random walk (up to higher order corrections): the mean square distance between the
chain endpoint 〈R2〉 scales with the number of monomers N as 〈R2〉 ∼ N . Above the
Θ temperature the effective interaction between the monomers is repulsive resulting
in a swelling of the polymer coil which in the asymptotics again obeys a universal
c© Ch. von Ferber, Yu.Holovatch 117
Ch. von Ferber, Yu.Holovatch
scaling law:
〈R2〉 ∼ N2ν , N → ∞ (1)
with the universal exponent ν depending on the dimension of space d only. The
number of configurations Z of a polymer chain scales with N like
Z ∼ eµNNγ−1, N → ∞ (2)
with a non-universal fugacity eµ and universal scaling exponent γ.
In the early 70-ies, following the work of de Gennes [1], the analogy between the
asymptotic properties of long polymer chains and the long-distance correlations of
magnetic systems in the vicinity of the 2nd order phase transition was recognized
and elaborated in detail. This mapping allows to define the above exponents (1),
(2) as m → 0 limits of the correlation length critical exponent ν and the magnetic
susceptibility critical exponent γ of the O(m)–symmetric model. In this way, the
powerful tools of the field theoretical renormalization group approach to the de-
scription of critical phenomena [2] can be applied in polymer theory (see e.g. [3]).
In particular, this allows to calculate the exponents at d = 3 with record accuracy
[4]:
ν(d = 3) = 0.5882± 0.0011 and γ(d = 3) = 1.1596± 0.0020. (3)
In dimension d = 2, the exact values ν = 3/4 and γ = 43/32 are known due to a
mapping of two-dimensional (2D) polymers to the 2D Coulomb gas problem [5].
Under similar conditions (i.e. in good solvents above the Θ temperature and
long polymer chains), star polymers [6] also obey universal scaling laws. For the
homogeneous star polymer, the asymptotic properties are uniquely defined by the
number of its constituting chains and by the dimension of space [7]. For the partition
function (number of configurations) Zf of a polymer star of f chains, each consisting
of N monomers, one finds:
Zf ∼ eµNfNγf−1 ∼ (R/ℓ)ηf−fη2 , N → ∞. (4)
The second part shows the scaling in terms of the size R ∼ N ν of the isolated
coil of N monomers on some microscopic length ℓ, omitting the fugacity factor. The
exponents γf , ηf , f = 1, 2, 3, . . . constitute families of ‘star exponents’, which depend
on the number of arms f in a nontrivial way. The case of linear polymer chains is
included in this family with the exponent γ = γ1 = γ2 defined in (2).
Apart from ηf and γf other sets of star exponents my be defined that govern the
scaling of different physical properties of polymer stars. However, all these sets of
exponents can be expressed in terms of a given set by familiar scaling relations [7].
Moreover, the scaling properties of polymer networks of arbitrary but fixed topology
are uniquely defined by its constituting stars [7], as long as the statistical ensemble
respects some conditions on the chain length distribution [8]. Thus, the knowledge of
the set of star exponents γf or ηf allows to obtain the power laws corresponding to
(4) also for any polymer network of arbitrary topology [7–9]. Recently, the theory has
118
Scaling exponents of star polymers
also been generalized to the case of star polymers and networks that contain chains
of different species, introducing additional independent families of star exponents
[9,11,12].
As it is usual in the theory of critical phenomena, the space dimension d is
crucial for the scaling laws (4). At d = 2, there exist exact formulas expressing
values of the scaling exponents of the star polymers as functions of the number of
arms [10–12]. In d = 3 dimensions on the other hand the problem does not allow for
an exact solution and different approximate schemes are used. The so-called cone
approximation takes into account the fact that for large f each chain of the star is
restricted approximately to a cone of space angle Ωf = 4π/f . In this approximation
one finds for large f [13]
γf − 1 ∼ −f 3/2. (5)
However, the most accurate values of critical exponents for d = 3 are obtained
by the field theoretical renormalization group approach refined by a resummation of
asymptotic series. Just in this way high precision results for the exponents governing
different critical phenomena are derived [2], and this is the way the scaling exponents
(3) for a polymer chain in d = 3 were obtained. In this review, we give a brief account
on how the numerical values of the star exponents were obtained for d = 3 by means
of these techniques as well as we provide some examples for phenomena governed
by the star exponents.
The paper is arranged as follows. In the next section 2 we give a field-theoretical
description of a star polymer in d dimensions, the exponents governing scaling of
a homogeneous star polymer are evaluated numerically in section 3. The theory
is generalized to the case of a polymer star consisting of two species of polymers
(a copolymer star) in section 4. The following sections of the review are devoted
to the numerical description of different phenomena where star exponents come
to play: the interaction between polymer stars in a good solvent (section 5), the
Brownian motion near absorbing polymers (section 6), diffusion-controlled reactions
in presence of polymers (section 7). We finally give some conclusions and an outlook
in section 8.
2. Renormalization of polymer stars
To give a field-theoretical description of a star polymer let us introduce an Ed-
wards model of continuous polymer chain [3] generalized to describe a set of f
polymer chains of varying composition, possibly tied together at their end points.
The configuration of one polymer is given by a path ra(s) in d-dimensional space IRd
parametrized by a surface variable 0 6 s 6 Sa. Let us allow for a symmetric matrix
of excluded volume interactions uab between chains a, b = 1, . . . , f . The Hamiltonian
H is then given by
1
kBT
H(ra) =
f
∑
a=1
∫ Sa
0
ds
(
dra(s)
2ds
)2
+
1
6
f
∑
a,b=1
uab
∫
ddrρa(r)ρb(r), (6)
119
Ch. von Ferber, Yu.Holovatch
with densities ρa(r) =
∫ Sa
0
dsδd(r − ra(s)). In this formalism the partition function
is calculated as a functional integral:
Zf{Sa} =
∫
D[ra(s)] exp
{
−
1
kBT
H(ra)
}
. (7)
Here, the symbol D[ra(s)] includes normalization such that Z{Sa} = 1 for all uab =
0. To make the exponential of δ-functions in (7) and the functional integral well-
defined in the bare theory, a cutoff s0 is introduced such that all simultaneous
integrals of any variables s and s′ on the same chain are cut off by |s − s′| > s0.
Let us note that the continuous chain model (6) may be understood as a limit of
discrete self-avoiding walks, when the length of each step is decreasing ℓ0 → 0 while
the number of steps Na is increasing keeping the ‘Gaussian surface’ Sa = Naℓ
2
0 fixed.
The continuous chain model (7) can be mapped onto a corresponding field theory
by a Laplace transform in the Gaussian surface variables Sa to conjugate chemical
potentials (“mass variables”) µa [14]:
Z̃f{µa} =
∫ ∞
0
∏
b
dSbe
−µbSbZf{Sa}. (8)
The Laplace-transformed partition function Z̃f{µa} can be expressed as the m = 0
limit of the functional integral over vector fields φa, a = 1, . . . , f with m components
φα
a , α = 1, . . . , m :
Z̃f{µb} =
∫
D[φa(r)] exp[−L{φb, µb}] |m=0. (9)
The Landau-Ginzburg-Wilson Lagrangian L of f interacting fields φb each with m
components reads
L{φb, µb} =
1
2
f
∑
a=1
∫
ddr
(
µaφ
2
a + (∇φa(r))
2
)
+
1
4!
f
∑
a,a′=1
ua,a′
∫
ddrφ2
a(r)φ
2
a′(r), (10)
here φ2
a =
∑m
α=1(φ
α
a )
2. The limit m = 0 in (9) can be understood as a selection
rule for the diagrams contributing to the perturbation theory expansions which can
be easily checked diagrammatically. A formal proof of (9) using the Stratonovich-
Hubbard transformation to linearize terms in (6) is given for the multi-component
case in [14]. The one particle irreducible (1PI) vertex functions Γ(L)(qi) of this theory
are defined by:
δ
(
∑
qi
)
Γ(L)
a1...aL
(qi) =
∫
eiqiridr1 . . .drL〈φa1(r1) . . . φaL(rL)〉
L
1PI,m=0. (11)
The average 〈· · ·〉 in (11) is understood with respect to the Lagrangian (10) keeping
only those contributions which correspond to one-particle-irreducible graphs and
which have non-vanishing tensor factors in the limit m = 0. The partition function
120
Scaling exponents of star polymers
Z∗f{Sa} of a polymer star consisting of f polymers of different species 1, . . . , f
constrained to have a common end point is obtained from (7) by introducing an
appropriate product of δ-functions ensuring the “star-like” structure. It reads:
Z∗f{Sa} =
∫
D[ra] exp
{
−
1
kBT
H(ra)
} f
∏
a=2
δd(~ra(0)− ~r1(0)). (12)
The vertex part of its Laplace transformation may be defined by:
δ(p+
∑
qi)Γ
(∗f)(p, q1 . . . qf) =
=
∫
ei(pr0+qiri)ddr0d
dr1 . . .d
drf〈φ1(r0) . . . φf(r0)φ1(r1) . . . φf (rf)〉
L
1PI,m=0 , (13)
where all a1, . . . af are distinct. The vertex function Γ(∗f) is thus defined by an
insertion of the composite operator
∏
a φa. Its scaling properties are governed by
the scaling dimension of this operator.
In the following we will be mainly interested in two different cases: (i) a polymer
star consisting of chains of one species with the single interaction u between them
(i.e. a homogeneous polymer star) and (ii) a polymer star constituted by two species
of polymers, with interactions u11, u22 between the polymers of the same species
and u12 = u21 between the polymers of different species (i.e. a copolymer star). In
the first case (i) one can also define Γ(∗f) by the insertion of a composite operator of
traceless symmetry [15]. In the case (ii) the composite operator in (13) reduces to
the product of two power-of-field operators with appropriate symmetry (φ)f1(φ′)f2
each corresponding to a product of fields of the same ‘species’. Nevertheless, the
results are easily generalized to the case of any number of polymer species.
As is well known, ultraviolet divergences occur when the vertex functions (11),
(13) are evaluated naively [2]. We apply renormalization group (RG) theory to make
use of the scaling symmetry of the system in the asymptotic limit to extract the
universal content and at the same time remove divergences which occur for the eval-
uation of the bare functions in this limit [2]. The theory given in terms of the initial
bare variables is mapped to a renormalized theory. This is achieved by a controlled
rearrangement of the series for the vertex functions. Several asymptotically equiv-
alent procedures serve to this purpose. The results reviewed in the next sections
were obtained by two somewhat complementary approaches: zero mass renormaliza-
tion (see [2] for instance) with successive ε-expansion [18] and the fixed dimension
massive RG approach [19]. The first approach is performed directly for the critical
point. Results for critical exponents at physically interesting dimensions d = 2 and
d = 3 are calculated in an ǫ = 4−d expansion. The second approach renormalizes off
the critical limit but calculates the critical exponents directly in space dimensions
d = 2, d = 3.
Let us formulate the relations for a renormalized theory in terms of the corre-
sponding renormalization conditions. Though they are different in principle for the
two procedures, we may formulate them simultaneously using the same expressions.
121
Ch. von Ferber, Yu.Holovatch
Note that the polymer limit of zero component fields leads to essential simplifica-
tion. Each field φa, mass ma and coupling uaa renormalizes as if the other fields were
absent. First we introduce renormalized couplings gab by:
uaa = µεZ−2
φa
Zaagaa, a = 1, 2, (14)
u12 = µεZ−1
φ1
Z−1
φ2
Z12g12. (15)
Here, µ is a scale parameter equal to the renormalized mass at which the mas-
sive scheme is evaluated and sets the scale of the external momenta in the massless
scheme. The renormalization factors Zφa
, Zab are defined as power series in the renor-
malized couplings which fulfil the following RG conditions:
Zφa
(gaa)
∂
∂k2
Γ(2)
aa (uaa(gaa)) = 1, (16)
Z2
φa
(gaa)Γ
(4)
aaaa(uaa(gaa)) = µεgaa, (17)
Zφ1(g11)Zφ2(g22)Γ
(4)
1122(uab(gab)) = µεg12. (18)
These formulas are applied perturbatively while the corresponding loop integrals
are evaluated for zero external momenta in the massive approach and for external
momenta at the scale of µ in the massless approach [2,9]. In the massive case the
RG condition for the vertex function Γ(2) reads
Zφa
(gaa)Γ
(2)
aa (uaa(gaa))|k2=0 = µ2, a = 1, 2. (19)
In the case of massless renormalization, the corresponding condition reads [2]:
Zφa
(gaa)Γ
(2)
aa (uaa(gaa))|k2=0 = 0, a = 1, 2. (20)
In order to renormalize the star vertex functions we introduce renormalization factors
Z∗f1,f2 by
Z
f1/2
φ1
Z
f2/2
φ2
Z∗f1,f2Γ
(∗f1f2)(uab(gab)) = µδf1+f2 . (21)
With the same formalism one can also describe a star of f mutually avoiding
walks [16]. In this case all interactions on the same chain uaa vanish and only those
uab with a 6= b remain. Then we define the appropriate renormalization for the vertex
function of mutually avoiding walks (MAW):
Z
f/2
φ1
Z(MAWf)Γ
∗f
MAW(u12(gab)) = µδf . (22)
The powers of µ absorb the engineering dimensions of the bare vertex functions.
These are given by δf = f(ε/2− 1)+ 4− ε. The renormalized couplings gab defined
by the relations (14), (15) depend on the scale parameter µ. By their dependence
on gab also the renormalization Z-factors implicitly depend on µ. This dependence
is expressed by the RG functions defined by the following relations:
µ
d
dµ
gab = βab(ga′b′), (23)
122
Scaling exponents of star polymers
µ
d
dµ
lnZφa
= ηφa
(gaa), (24)
µ
d
dµ
lnZ∗f1f2 = η∗f1f2(gab), (25)
µ
d
dµ
lnZMAWf = ηMAW
f (gab). (26)
The function ηφa
describes the pair correlation critical exponent, while the functions
η∗f1f2 and ηMAW
f (gab) define the set of exponents for copolymer stars and the stars of
mutually avoiding walks. Note that Z∗20 renormalizes the vertex function with a φ2
insertion which coincides with Γ(∗20). Consequently, the usually defined correlation
length critical exponent ν is expressed in terms of functions η∗20 and ηφ.
Expressions for the β and η functions will be discussed in the next sections
together with a study of the RG flow and the fixed points of the theory.
3. Homogeneous star polymer in three dimensions
First we give numerical results for a homogeneous polymer star constituted by f
polymer chains of the same species. The scaling of a star is governed by the laws (4).
For one species of polymers (a, b = 1), corresponding RG function (23)–(25) depend
on a single coupling gab = g and reduce to a single functions βab(ga′b′) ≡ β(g),
ηφa
(gaa) ≡ ηφ(g) and a set of functions η∗f1f2(gab) ≡ η∗(f1+f2)(g) with f1 + f2 = f .
Here, the function β(g) is a standard β-function of the O(m = 0)-symmetric theory.
In the infrared-stable fixed point g∗:
β(g∗) = 0,
dβ(g)
dg
|g=g∗ > 0; (27)
the function ηφ(g) gives the pair correlation function critical exponent η:
η = η(g∗), (28)
whereas the functions η∗f)(g) give the scaling exponents ηf (4):
ηf = η∗f (g
∗). (29)
The RG functions (27)–(29) are obtained in the form of perturbation theory series.
In the massive RG scheme, these are the series in a renormalized coupling g for fixed
space dimension d whereas the zero-mass renormalization expansion is performed
both in g and in ε = 4− d. Increasing the order of perturbation theory corresponds
to increasing the number of loops when the expansion is written in terms of Feyn-
man diagrams. That is why sometimes the expansions are classified in successive
approximations in number of loops. The functions (27), (28) are the standard RG
functions of the O(m = 0) symmetric theory. They are known by now in high orders
of perturbation theory, whereas the functions (29) have been obtained only up to
three loops. In the field theory they are related to the anomalous dimension of com-
posite operators of traceless symmetry [15]. In polymer language the star exponents
123
Ch. von Ferber, Yu.Holovatch
Table 1. Values of the star exponent γf obtained in three-dimensional theory
(columns 2,3,4,5) in comparison with the results of ε-expansion (columns 6,7,8)
and Monte-Carlo simulations (column 9). See the text for a full description.
f pseudo-ε expansion [22] ε-expansion MC
Padé-Borel conf.mapping [8] [25,26]
3 1.06 1.05 1.06 1.06 1.05 1.05 1.07 1.09
4 0.86 0.86 .86 .83 0.84 0.83 0.85 0.88
5 0.61 0.61 .58 .56 0.53 0.52 0.55 0.57
6 0.32 0.32 .24 .22 0.14 0.18 0.16
7 –0.02 –0.01 –.17 –.17 –0.33 –0.20
8 –0.40 –0.36 –.63 –.62 –0.88 –0.60 (–0.99, –0.30)
9 –0.80 –0.72 –1.14 –1.11 –1.51 –1.01
were originally calculated to order ε in [20], the ε2 results have been obtained in
[7,21], whereas ε3 expansions are given in [8]. The ε-expansion for the exponent ηf
reads [15,8]:
ηf = −
ε
8
f(f−1)
[
1−
ε
32
(8f−25)+
ε2
64
[(28f−89)ζ(3)+8f 2−49f+
577
8
]+O(ε3)
]
. (30)
In the massive RG scheme, the three-loop expansions for the d = 3 star expo-
nents were obtained in the form of pseudo-ε expansion in [22]. However, as already
noted above, the perturbation theory expansions of the renormalized field theory
have zero radius of convergence and are asymptotic at best. Therefore, in order to
make numerical estimates for the expansions one should apply an appropriate re-
summation procedure. Here, we will not describe the different ways of resummation
which were applied to the series for the star exponents referring rather to the papers
[8,9,22,24] where they are given in detail. For the purpose of the present review we
just mention that a Borel transform was applied to get rid of the (conjectured) fac-
torial divergence of the perturbation theory terms [17]. Subsequently, the expansion
for the Borel transformed function was refined by a conformal mapping from the ini-
tial variable defined on the cut-plane to a disk keeping the origin invariant. Another
scheme of resummation exploits the analytic continuation of the series for the Borel
transformed function by means of Padé approximant (Padé-Borel resummation).
In table 1 we give the numerical results for the star exponents γf as obtained by
the fixed d = 3 RG technique and in the ε-expansion in comparison with the data
of the numerical simulation. First, we give the values of the exponent γ f obtained
in the fixed d = 3 technique by Padé-Borel resummation [22]: the second column
contains the value of the γf obtained directly from the resummation of the series for
γf , while the third column gives the γf based on the resummed series for the star
exponents x
′
f . These are connected with the γf via the scaling relation:
γf = 1− νx
′
f + [ν(2− η/2)− 1]f (31)
124
Scaling exponents of star polymers
substituting the well-known values of the exponents ν(d = 3) = 0.588, η(d = 3) =
0.027. Columns 4 and 5 give γf obtained by the resummation using the conformal
mapping technique [22]: the resummation of the series γf(τ) (fourth column) and the
resummation of the series x
′
f (τ) (fifth column). The next columns show the results
obtained by the ε3-expansion using: simple Padè approximation (the 6th column)
and Padè-Borel analysis neglecting or exploiting the exact results for d = 2 (the
7th and 8th columns, respectively) [8]. The last column contains Monte-Carlo data
[25,26]. For low numbers of arms F 6 5 the results of the different approaches agree
reasonably well and are also close to the values obtained by MC simulation.
The data collected in table 1 was obtained using two different renormalization
schemes as well as different procedures for the resummation of the resulting asymp-
totic series. Thus, table 1 gives a test for the stability of the results under the
changes of the calculational scheme. Obviously, for higher numbers f > 5 of arms,
the coincidence of the results is no longer good. The main reason for this is that
calculating the exponents, combinatorial factors lead to an expansion in fε for the
ε expansion and of fg when directly expanding in a renormalized coupling g. For
such large values of the expansion parameter, even resummation of the series fails.
For larger numbers of arms, other approaches to the theory of polymer stars, like a
self consistent field approximation, might be more useful.
4. Copolymers and copolymer stars
Now let us pass to the case of a polymer star constituted by two species of
polymers. It is described by the Lagrangian (10) with interactions u11, u22 between
the polymers of the same species and u12 = u21 between the polymers of different
species. Such a Lagrangian is used to describe the system of polymers of two species
immersed in a solvent: the so-called ternary solution [27]. A comprehensive analysis
of the fixed point behaviour of a ternary solution was given in [14] where the RG flow
of the theory was calculated within the massless renormalization and is known by
now to the third loop order of the ε-expansion. Note, that for the diagonal coupling
gaa the corresponding expressions are also found in the polymer limit m = 0 of the
O(m)-symmetric φ4 model. They are known in even higher orders of perturbation
theory. In the massive RG approach, the corresponding expressions were obtained
in [9]. The equations for the fixed points (FP) P({g∗
11, g
∗
22, g
∗
12}) of the β-functions
read:
βgaa(g
∗
aa) = 0, a = 1, 2,
βg12(g
∗
11, g
∗
22, g
∗
12) = 0. (32)
As is well known, the first equation has two solutions g∗
aa = 0, g∗S. For the sec-
ond equation, one finds a total of 8 FPs depending on the choice for g∗
aa. The
trivial FPs are G0(0, 0, 0),U0(g
∗
S, 0, 0),U
′
0(0, g
∗
S, 0), S0(g
∗
S, g
∗
S, 0), all corresponding to
g∗12 = 0. The non-trivial FPs are found as G(0, 0, g∗
G), U(g
∗
S, 0, g
∗
U), U
′(0, g∗S, g
∗
U), and
S(g∗S, g
∗
S, g
∗
S). In the three dimensional space of couplings g11, g22, g12 these FPs are
placed at the corners of a cube that is deformed in the g12 direction (see figure 1).
125
Ch. von Ferber, Yu.Holovatch
✉
✉
✉✉
✉
✉
✉ ✉
G0 U0
g11
g12 g22
U ′
0 S0
SU
G U ′
�
�
�
✲
✻
✒�
�
�
Figure 1. Fixed points (FPs) of ternary
polymer solution. The trivial FPs G0,
U0, U
′
0, S0 correspond to a vanishing mu-
tual interaction. The non-trivial FPs G,
U , U ′, S correspond to a non-vanishing
mutual interaction (g12 6= 0).
Looking for the stability of the above
described fixed points one finds that on-
ly the fixed point S is stable [14]. In
the excluded volume limit of infinitely
long chains, the behaviour of a system
of two polymer species is thus described
by the same scaling laws as a solution of
only one polymer species. Nevertheless,
taking into account that real polymer
chains are not infinitely long, one may
also find crossover phenomena which are
governed by the unstable fixed points.
Knowing the complete RG flow, allows
one to describe crossover phenomena in
the whole accessible region [14]. Howev-
er, for the purpose of our review we are
interested only in the values of the fixed
points and in the properties of the star
vertex functions at these points.
Let us define the asymptotic values
of the copolymer star exponents and the
mutually avoiding walk (MAW) star ex-
ponents by:
ηSf1f2 = η∗f1f2(gab)|S, (33)
ηGf1f2 = η∗f1f2(gab)|G, (34)
ηUf1f2 = η∗f1f2(gab)|U = η∗f2f1(gab)|U ′, (35)
ηMAW
f = ηMAW
f (gab)|G. (36)
Taking into account the nature of the fixed points, where the exponents (32)–(36)
are defined, one arrives at the following physical interpretation:
• the exponent ηGf1f2 describes a star of f1 random walks of the 1st type and f2
random walks of the 2nd type all of which are not selfavoiding; only the chains
of the 1st type avoid those of the 2nd type and vice versa;
• the exponent ηUf1f2 describes a star of f1 self and mutually avoiding polymer
chains walks of the 1st type and f2 random walks which only avoid those of
the 1st type;
• the exponent ηMAW
f1f2
describes a star of f1 + f2 random walks all of which are
not selfavoiding; however, each chain avoids any other chain of the star;
• exponent ηSf1f2 describes a star of f1 + f2 self and mutually avoiding chains.
The exponent can also be expressed by ηSf1f2 = ηUf1+f2,0
(This relation is broken
126
Scaling exponents of star polymers
for the exact results known in d = 2 where a finite effect remains in the zero
walk limit). This situation has already been discussed in the previous section.
Starting from the expressions for the fixed points (available both in the form of
a three-loop ε expansion [14] or in the form of pseudo-ε expansion at fixed d [9])
and the relations (33)–(36), one can find the series for the star exponents. In the
ε-expansion the following expansions for ηf1f2 are obtained [9]:
ηGf1f2(ε) = −f1 f2
ε
2
+ f1 f2
(
f2 − 3 + f1
)ε2
8
− f1 f2
(
f2 − 3 + f1
)(
f1 + f2 + 3 ζ(3)− 3
) ε3
16
, (37)
ηUf1f2(ε) = f1
(
1− f1 − 3f2
)ε
8
+ f1
(
25− 33f1 + 8f1
2 − 91f2 + 42f1f2 + 18f2
2
) ε2
256
+ f1
(
577− 969f1 + 456f1
2 − 64f1
3 − 2463f2 + 2290f1f2
− 492f1
2f2 + 1050f2
2 − 504f1f2
2 − 108f2
3 − 712ζ(3) + 936f1ζ(3)
− 224f1
2ζ(3) + 2652f2ζ(3)− 1188f1f2ζ(3)− 540f2
2ζ(3)
) ε3
4096
,
(38)
ηMAW
f1f2
(ε) = −(f1 − 1)f1
ε
4
+ f1(f1 − 1)(2f1 − 5)
ε2
16
− (f1 − 1)f1(4f1
2 − 20f1 + 8f1ζ(3)− 19ζ(3) + 25)
ε3
32
. (39)
Here ζ(3) ≃ 1.202 is the Riemann ζ-function. The above formulas reproduce the
3rd order calculations [8] of the scaling exponents of homogeneous polymer stars
ηf = ηUf,0 given in formula (35) of the previous section. The corresponding pseudo-ε
expansion for the exponents ηf1f2 obtained in the massive scheme may be found in
[9]. We do not give it here explicitly although in what follows below we will give
numerical values of the exponents obtained in both approaches. It has been pointed
out in [28] that for the exponent ηG12 an exact estimate equal to our first order con-
tribution may be found. It is indeed remarkable that all higher order contributions
to ηG12 vanish in both approaches, ηG
12 = −ε being an exact result [23].
With these exponents one can describe the scaling behaviour of polymer stars
and networks of two components, generalizing the relation for single component
networks [7]. In the notation of (4) one finds for the number of configurations of a
network G of F1 and F2 chains of species 1 and 2
ZG ∼ (R/ℓ)ηG−F1η20−F2η02 , with ηG = −dL +
∑
f1+f2>1
Nf1f2ηf1f2 , (40)
where L is the number of loops and Nf1f2 is the number of vertices with f1 and f2
arms of species 1 and 2 in the network G. To receive an appropriate scaling law we
127
Ch. von Ferber, Yu.Holovatch
Table 2. Values of the copolymer star exponent ηGf1f2 for d = 3 obtained by the
ε-expansion (ε) and by fixed dimension technique (3d).
f1 1 2 3 4 5 6
f2 ε 3d ε 3d ε 3d ε 3d ε 3d ε 3d
1 –0.56 –0.58 –1.00 –1.00 –1.33 –1.35 –1.63 –1.69 –1.88 –1.98 –2.10 –2.24
2 –1.77 –1.81 –2.45 –2.53 –3.01 –3.17 –3.51 –3.75 –3.95 –4.28
3 –3.38 –3.57 –4.21 –4.50 –4.94 –5.36 –5.62 –6.15
4 –5.27 –5.71 –6.24 –6.84 –7.12 –7.90
5 –7.42 –8.24 –8.50 –9.54
6 –9.78 –11.07
Table 3. The values of the copolymer star exponent ηUf1f2 at d = 3 obtained by
the ε-expansion (ε) and by fixed dimension technique (3d).
f1 1 2 3 4 5 6
f2 ε 3d ε 3d ε 3d ε 3d ε 3d ε 3d
0 0 0 –0.28 –0.28 –0.75 –0.76 –1.36 –1.38 –2.07 –2.14 –2.88 –3.01
1 –0.43 –0.45 –0.98 –0.98 –1.64 –1.67 –2.39 –2.47 –3.21 –3.38 –4.11 –4.40
2 –0.79 –0.81 –1.58 –1.60 –2.44 –2.52 –3.33 –3.50 –4.28 –4.57 –5.29 –5.73
3 –1.09 –1.09 –2.13 –2.19 –3.16 –3.30 –4.20 –4.48 –5.28 –5.71 –6.41 –7.03
4 –1.35 –1.37 –2.61 –2.71 –3.82 –4.04 –5.02 –5.40 –6.24 –6.81 –7.48 –8.28
5 –1.60 –1.64 –3.05 –3.21 –4.44 –4.75 –5.80 –6.30 –7.15 –7.89 –8.51 –9.50
6 –1.81 –1.89 –3.46 –3.68 –5.01 –5.42 –6.53 –7.15 –8.02 –8.92 –9.50 –10.69
assume the network to be built of chains which for both species will have the same
coil radius R when isolated.
In order to calculate numerical values for the exponents ηG
f1f2
, ηUf1f2 and ηMAW
f
in [9,29–31] the Borel resummation refined by conformal mapping was applied to
the series (37)–(39) as well as to the appropriate series in pseudo-ε expansion. The
resummation showed that the two schemes yield consistent numerical estimates. Our
tables 2, 3, and 4 list the results.
Comparing the numerical values listed in the above tables it is convincing that
the two approaches and the different resummation procedures all lead to the results
which lie within a bandwidth of consistency, which is broadening for larger values
of number of chains f1, f2 > 1. This is not surprising. One can see, e.g. from the
formulas (37)–(39), that the expansion parameters are multiplied by f1 and f2.
Rather, it is remarkable that even for a total number of chains of the order of 10
(see tables 2, 3) one still receives the results which are comparable to each other.
In the next three sections we will briefly review several phenomena where star
exponents become important.
128
Scaling exponents of star polymers
Table 4. The values of the ηMAW
f exponents of stars of mutually avoiding walks
for d = 3, d = 2 obtained by the ε-expansion (ε) and by the fixed dimension
technique (3d, 2d). The last column gives the exact conjecture for d = 2 [16].
d = 3 d = 2
f ε 3d ε 2d exact
1 0 0 0 0 –.250
2 –.56 –.56 –1.20 –1.19 –1.250
3 –1.38 –1.36 –2.71 –2.60 –2.916(6)
4 –2.36 –2.34 –4.36 –4.07 –5.250
5 –3.43 –3.43 –6.04 –5.61 –8.250
6 –4.58 –4.64 –7.78 –7.17 –11.916(6)
5. Colloids with polymer stars: The interaction
Let us now consider the effective interaction between the cores of two star poly-
mers at a small distance (small on the scale of the size Rg of the star). Let us analyse
the general case of two stars of different functionalities f1 and f2. The power law for
the partition sum Z
(2)
f1f2
of such two star polymers at a distance r [7],
Z
(2)
f1f2
(r) ∼ r
Θ
(2)
f1f2 , (41)
is governed by the contact exponent Θ
(2)
f1f2
. As we have seen in the section 2 in the
formalism of the m = 0 component model, the core of a star polymer corresponds to
a local product of f spin fields φ1(x) · · ·φf(x), each representing the endpoint of one
polymer chain. The probability of approach of the two cores of the star polymers
at a small distance r in these terms is described by a short distance expansion for
the composite fields. The short distance expansion provides the scaling relations
between the exponents Θf1f2 and the star exponents γf (or ηf):
νΘf1f2 = (γf1 − 1) + (γf2 − 1)− (γf1+f2 − 1) ,
Θf1f2 = ηf1 + ηf2 − ηf1+f2 . (42)
The mean force F
(2)
f1f2
(r) between two star polymers at a short distance r is easily
derived from the effective potential V eff
f1f2
(r) = logZ
(2)
f1f2
(r) as
F
(2)
f1f2
(r) =
Θ
(2)
f1f2
r
, with Θ
(2)
ff ≈
5
18
f 3/2. (43)
The factor 5/18 is found by matching the cone approximation for Θ
(2)
ff (see formula
(5)) to the known values of the contact exponents for f = 1, 2 [34]. This matching,
in turn, proposes an approximate value for the ηf exponents,
ηf ≈ −
5
18
(23/2 − 2)f 3/2 − fη1, (44)
129
Ch. von Ferber, Yu.Holovatch
Table 5. The prefactor Θf1f2 of the force between two star polymers at a short
distance calculated in non-resummed 1-loop and resummed 3-loop RG analysis
in comparison to the result of the cone approximation.
Θf1f2 f2 = 1 f2 = 2 f2 = 3 f2 = 4 f2 = 5 app.
f1 = 1 –.21 –.42 –.63 –.85 –1.06 1-loop
f1 = 1 –.28 –.48 –.62 –.76 –.87 3-loop
f1 = 1 –.27 –.45 –.60 –.73 –.84 cone
f1 = 2 –.85 –1.27 –1.70 1-loop
f1 = 2 –.82 –1.10 –1.35 3-loop
f1 = 2 –.78 –1.05 –1.29 cone
f1 = 3 –1.91 1-loop
f1 = 3 –1.49 3-loop
f1 = 3 –1.44 cone
where −fη1 is introduced for consistency with the exact result η1 = 0. This assump-
tion nicely reproduces the contact exponents as derived from 3-loop perturbation
theory [9] as is displayed in table 5 taken from the [33].
In table 5 we use the approximate values of ηf to calculate the cone estimation
of the contact exponents and compare these with the corresponding values of a
renormalization group calculation. A perturbation series is taken in terms of an ε-
expansion. The result labelled ‘1-loop’ corresponds to optimal truncation of the series
by simply inserting ε = 1 or d = 3 in the first order term of the expansion. The 3-
loop result includes a resummation procedure that takes into account the asymptotic
nature of the series [9]. The large f result corresponds to the cone approximation (5).
The above data convincingly shows that the large f approximation for the short
distance force between two star polymers can be consistently fitted to the results
of perturbation theory for low values of the functionality f of the stars. Note, that
increasing the number of chains f , the polymer stars interpolate between the proper-
ties of linear polymers and polymeric micelles [35]. This approach is general enough
to describe the interaction between two stars of different functionalities f1 and f2.
This is essential in extending the theory of colloidal solutions of star polymers to
general polydispersity in f as it appears naturally in any real experiment.
6. Brownian motion near an absorbing polymer star
Now, let us turn to another physical phenomenon that is described in terms
of the star exponents. To this end we consider a model proposed by Cates and
Witten [28] to describe the flow of diffusing particles near an absorbing object, which
may be a polymer chain ( = a selfavoiding walk (SAW)) or a random walk (RW).
The flux of particles on the absorbing chain is defined by the density of incoming
130
Scaling exponents of star polymers
particles close to the absorber (see figure 2). Taken ρ(r) to be the probability density
of incoming particles, the problem is to solve the steady-state diffusion equation
(Laplace equation)
∇2ρ(r) = 0 (45)
with boundary conditions for a given absorbing polymer
ρ(r) = 0 on the surface of the absorber,
ρ(r) = ρ∞ = const for |r| → ∞. (46)
One of the motivations for introducing this model was to gain insight for the problem
of diffusion limited aggregation (DLA) [39]. The latter phenomenon is much more
complicated because of the fact that for DLA the boundary conditions are given on
the surface of the growing aggregate itself. The process described by equations (45),
(46) may be rather considered as a diffusion limited catalysis, when particles of one
type interact with a prescribed fractal (a catalyzing polymer) and transform into the
other type [38] (see section 7 for more details).
s
x
R
s
l
1 2 s s s l − 1
Figure 2. Star of l random walks at dis-
tance x from an absorbing polymer of
size R.
Let the absorbing polymer of size R
be chosen from the well defined ensem-
ble of SAWs. Then one may introduce
exponents that govern the scaling of the
moments of the field ρ(r) close to the
surface of the absorber (see figure 2).
For distances R ≫ r > a (a being a
cut off) the averaged moments 〈ρ(r)n〉
are expected to scale as [28]
〈ρ(r)n〉 ∼ (R/r)−λ(n), (47)
where 〈. . .〉 denotes the average over
the ensemble of polymers. Taking that
the flux φ(x) onto any randomly chosen
point x of the absorber is proportional
to the field ρ(r) at a point that is as close as a cut off length a from the absorber,
one finds for the averaged moments of the flux:
〈φn〉 ∼ (R/a)−λ(n). (48)
There is a natural way to associate the exponents λ(n) (47), (48) with the copoly-
mer star exponents ηf1f2 (34), (35). The central idea that allows one to calculate
the properties of the solutions of equation (45) with the boundary conditions (46)
is that a path integral representation both for the field ρ(r) and for the absorber
(being a RW or a SAW) is possible [28,38]. In terms of the path integral solution of
the Laplace equation, one finds that ρ(r) at point r near the absorber is proportional
to the number of RW that end at point r and avoid the absorber. The nth power of
this field is proportional to the nth power of the above mentioned number, i.e. it is
131
Ch. von Ferber, Yu.Holovatch
defined by the partition function of a star polymer with n arms (12) (the latter is
shown by dashed lines in figure 2). Furthermore, introducing the mutual avoidance
conditions between the “n-arm star” and the 2-arm polymer (representing the ab-
sorber), one has to calculate the partition function of a co-polymer star consisting
of chains of two different species that avoid each other. These correspond to the
trajectories of diffusing particles (being RW) and the absorbing polymer (which is
chosen to be a RW or SAW). Making use of the theory of copolymer stars and net-
works [9], one may relate [24] the spectrum of exponents (47), (48) to the exponents
that define the scaling properties of co-polymer stars. In particular, as we have seen
in the section 4 for a co-polymer star consisting of f1 chains of species 1 and f2
chains of species 2, the scaling of the number of configurations Z∗ is governed by the
exponents ηf1f2 (40). By means of a short-chain expansion [36] it is clear that only
the (smaller) length scale of the absorbing polymer remains and the set of exponents
ηf1f2 can be related to the exponents λ(n) (47) that govern the scaling of the nth
moment of the flux onto an absorbing linear chain. Considering the absorber to be
either a RW or a SAW the exponents read:
λG
RW(n) = −ηG2n, (49)
λU
SAW(n) = −ηU2n + η20. (50)
Numerical values for the exponents λ(n) may be easily extracted from the nu-
merical values of the exponents η2n given in tables 2, 3. Moreover, one can consider
an absorber in the form of an f1-arm star and generalize the above considerations
looking for the scaling laws of diffusing particles in the vicinity of the core of such a
polymer star. Again, the scaling may be expressed in terms of the co-polymer star
exponents [24].
It is well known that polymer chains are fractals with a fractal dimension de-
fined by a correlation length exponent: df = 1/ν. It appears that the behaviour of
Brownian motion in the vicinity of an absorbing star polymer possesses multifractal
[37] features. These are analyzed in detail in [24,30].
7. Diffusion-controlled reactions in presence of polymers
As mentioned in the preceding section, one of the processes described by equa-
tions (45), (46) may also be considered as diffusion limited catalysis. In this section,
we will comment on a general phenomenon which can again be described in terms
of the star exponents: chemical reactions between diffusing reactants. Examples of
such reactions can be found in different systems, ranging from biological systems to
nuclear reactors (see e.g. [40] and references therein). One more place where these
reactions appear as a limiting stage is the aggregation models [41]. Of particular
interest are the reactions between reactants of different nature: particles A which
freely diffuse in a solution and particles B which are attached to polymer chains im-
mersed in the same solvent [38] (the concentration of polymers being low enough to
allow to neglect the inter-chain interaction). Such a process may be also considered
132
Scaling exponents of star polymers
as a trapping reaction of n particles of A type and traps B:
An +B → B. (51)
The reaction rate kn of (51) in the vicinity of a certain trap on the polymer of size R
is proportional to the averaged moments of the concentration ρ of diffusing particles
near this trap. From (47) we get that the reaction rate scales with R as [28]:
kn ∼ 〈ρn〉 ∼ (R/l)−λn , (52)
with l being a characteristic length scale.
The scaling exponents λn (52) were considered in the previous section (see for-
mulas (49) and (50)). Now, considering the absorber to be either a RW star or a
SAW star of m chains, we define the exponents λmn which can be related to the
familiar copolymer star exponents ηmn (35) via scaling laws:
λRW
mn = −ηGmn,
λSAW
mn = −ηUmn + ηUm0. (53)
Let us note that the case m = 2 corresponds to a trap located on the chain polymer,
whereas m = 1 corresponds to a trap attached at the polymer extremity.
Numerical values for the exponent λmn as well as perturbation theory expansions
for them follow from tables 2, 3 and are analyzed in detail in [9,42–44]. Here, let us
analyze several particular cases for an absorbing m-star with a reactant trap at its
core and simply absorbing traps along all chains:
• For a given m-star absorber of size R, the reaction rate (52) scales as kmn ∼
(R/l)−λmn . The increase of the size R by a factor of a results in k ′
mn ∼
(aR/l)−λmn leading to:
k′
mn/kmn ∼ a−λmn (54)
with λmn positive: increasing R by a factor of a the reaction rate decreases
a−λmn times due to the increase of absorbing traps.
• For a given reaction type (51) (i.e. for a fixed number n of particles which
are trapped simultaneously) attaching m1 additional arms to an m-arm star
absorber results in a decrease of the reaction rate:
k(m+m1),n/kmn ∼ (R/l)−(λ(m+m1),n
−λmn), (55)
as far as λm2n > λm1n for m2 > m1.
• For a given m-star absorber the change of the type of reaction (51) to An1 +
B → B, n1 > n results in a decrease of the reaction rate:
km,(n+n1)/kmn ∼ (R/l)−(λm,(n+n1)
−λmn), (56)
as far as λmn2 > λmn1 for n2 > n1; now, more particles need to find the trap
at the core simultaneously for a reaction to take place.
133
Ch. von Ferber, Yu.Holovatch
8. Conclusions and outlook
The notion of critical exponents is one of the central notions in the theory of crit-
ical phenomena. The precise determination of the set of critical exponents governing
an equilibrium thermodynamic second order phase transitions provides a challenge
in numerous studies. The exponents usually govern the power-law temperature be-
haviour which manifests itself in the vicinity of the critical point. In polymer theory,
the direct “relatives” of the exponents governing temperature behaviour at the 2nd
order phase transitions are the exponents describing the configurational properties
of long flexible polymer chains. They govern the scaling laws appearing in the limit
of infinite chain length. These exponents are important in defining the entropy of a
polymer solution, the radius of gyration of a polymer coil, the osmotic pressure of
the polymer component in a solution, its rheologic properties, etc [1,3]. Both the 2nd
order phase transition critical exponents and the polymer chain scaling exponents
are closely related to each other and, finally, correspond to the second derivative of
the appropriate free energy. However, in polymer theory there exist sets of exponents
which do not have direct “relatives” in the theory of 2nd order phase transitions.
These exponents describe the scaling of star polymers and polymer networks [7].
Whereas the scaling exponents for chain polymers are calculated by now in different
techniques with the standard accuracy of modern theory of critical phenomena, this
is not the case for the star exponents.
In the present paper we have reviewed some recent calculations of the critical
exponents governing the scaling behaviour of star polymers. To calculate the criti-
cal exponents it is today standard to rely on the renormalization group technique
improved by a resummation of asymptotic series [2,3]. The numbers provided in
tables 1–4 of the paper provide reliable data which may be used in different prob-
lems where the star exponents become important. Some of the problems were men-
tioned in the sections 5–7: the interaction between star polymers in a good solvent,
the Brownian motion near absorbing polymers, diffusion-controlled reactions in the
presence of polymers.
References
1. see e.g.: de Gennes P.-G. Scaling Concepts in Polymer Physics. Ithaca and London,
Cornell University Press, 1979.
2. Zinn-Justin J. Quantum Field Theory and Critical Phenomena. Oxford University
Press, 1996; Kleinert H., Schulte-Frohlinde V. Critical Properties of φ4-Theories. Sin-
gapore, World Scientific, 2001.
3. des Cloizeaux J., Jannink G. Polymers in Solution. Oxford, Clarendon Press, 1990;
Schäfer L. Excluded Volume Effects in Polymer Solutions. Berlin, Springer, 1999.
4. Guida R., Zinn Justin J. // J. Phys. A, 1998, vol. 31, p. 8104.
5. Nienhuis B. // Phys. Rev. Lett., 1982, vol. 49, p. 1062.
6. We neither give here examples of star polymers nor emphasize their importance in
applications: rather we refer to the bulk of papers constituting this special issue of
Condens. Matter Phys. and devoted to the subject.
134
Scaling exponents of star polymers
7. Duplantier B. // J. Stat. Phys., 1989, vol. 54, p. 581–680.
8. Schäfer L., von Ferber C., Lehr U., Duplantier B. // Nuc. Phys., 1992, vol. 374,
p. 473–495.
9. von Ferber C., Holovatch Yu. // Phys. Rev. E, 1997, vol. 56, p. 6370–6386.
10. Duplantier B. // Phys. Rev. Lett., 1986, vol. 57, p. 941–945.
11. Duplantier B. // Phys. Rev. Lett., 1999, vol. 82, p. 880; Physica A, 1999, vol. 263,
p. 452.
12. von Ferber C., Holovatch Yu. // Phys. Rev. E, 2002 (in print).
13. Ohno K., Binder K. // J. Phys. France, 1988, vol. 49, p. 1329.
14. Schäfer L., Lehr U., Kappeler C. // J. Phys. Paris I, 1991, vol. 1, p. 211–233.
15. Wallace D.J., Zia R.K.P. // J. Phys. C, 1975, vol. 8, p. 839.
16. Duplantier B. // Commun. Math. Phys., 1988, vol. 117, p. 279; Duplantier B.,
Kwon K.-H. // Phys. Rev. Lett., 1988, vol. 61, p. 2514–2517.
17. Hardy G. Divergent Series. Oxford University Press, 1948.
18. Wilson K.G., Fisher M.E. // Phys. Rev. Lett., 1972, vol. 28, p. 240.
19. Parisi G. // 1973 (unpublished); J. Stat. Phys., 1980, vol. 23, p. 49.
20. Miyake A., Freed K.F. // Macromolecules, 1983, vol. 16, p. 1228–1241.
21. Ohno K. // Phys. Rev. A, 1989, vol. 40, p. 1524–1538.
22. von Ferber C., Holovatch Yu. // Condens. Matter Phys., 1995, No. 5, p. 8–22;
Theor. Math. Phys. (Moscow), 1996, vol. 109, p. 34–50 (Engl. ed.: p. 1274–1286).
23. Semenov A.N. // J. Phys. France, 1988, vol. 49, p. 1353.
24. von Ferber C., Holovatch Yu. // Phys. Rev E, 1999, vol. 59, p. 6914–6923.
25. Batoulis J., Kremer K. // Macromolecules, 1989, vol. 22, p. 4277.
26. Barret A.J., Tremain D.L. // Macromolecules, 1987, vol. 20, p. 1687.
27. Joanny J.-F., Leibler L., Ball R. // J. Chem. Phys., 1984, vol. 81, p. 4640; Douglas J.F.,
Freed K.F. // J. Chem. Phys., 1987, vol. 86, p. 4280; Schäfer L., Kappeler Ch. //
J. Phys. Paris, 1985, vol. 46, p. 1853; Colloid Polym. Sci., 1990, vol. 268, p. 995.
28. Cates M.E., Witten T.A. // Phys. Rev. Lett., 1986, vol. 56, p. 2497; Phys. Rev. A,
1987, vol. 35, p. 1809–1824.
29. von Ferber C., Holovatch Yu. // Condens. Matter Phys., 1997, No. 10, p. 9–40.
30. von Ferber C., Holovatch Yu. // Europhys. Lett., 1997, vol. 39, p. 31–36.
31. von Ferber C., Holovatch Yu. // Physica A, 1998, vol. 249, p. 327–331.
32. von Ferber C., Holovatch Yu., Schäfer L. // Condens. Matter Phys., 1996, No. 7,
p. 15–25.
33. von Ferber C., Holovatch Yu., Jusufi A., Likos C.N., Löwen H., Watzlawek M. //
J. Mol. Phys., 2001, vol. 93, p. 155–158.
34. Likos C.N., Löwen H., Watzlawek M., Abbas B., Jucknischke O., Allgaier J., Richter D.
// Phys. Rev. Lett., 1998, vol. 80, p. 4450; Watzlawek M., Likos C.N., Löwen H.
// Phys. Rev. Lett., 1999, vol. 82, p. 5289; Jusufi A., Watzlawek M., Löwen H. //
Macromolecules, 1999, vol. 32, p. 4470.
35. Gast A.P. // Langmuir, 1996, vol. 12, p. 4060; Grest G.S., Fetters L.J., Huang J.S.,
Richter D. // Adv. Chem. Phys., 1996, vol. 94, p. 67.
36. von Ferber C. // Nucl. Phys. B, 1997, vol. 490, p. 511–542.
37. Hentschel H.G.E., Procaccia I. // Physica D, 1983, vol. 8, p. 435–444; Halsey T.C.,
Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.I. // Phys. Rev. A, 1986,
vol. 33(2), p. 1141–1151.
38. Burlatsky S.F., Oshanin G.S., Likhachev V.N. // Sov. J. Chem. Phys., 1991, vol. 7,
135
Ch. von Ferber, Yu.Holovatch
p. 1680–1698; Burlatsky S.F., Oshanin G.S. // Phys. Lett. A, 1990, vol. 145, p. 61–65;
Oshanin G., Moreau M., Burlatsky S. // Anv. Coll. Int. Sci., 1994, vol. 49, p. 1–46.
39. Witten Jr., T.A., Sander L.M. // Phys. Rev. Lett., 1981, vol. 47, p. 1400–1403; Phys.
Rev. A, 1983, vol. 27, p. 5686–5697.
40. Rice S.A. Diffusion-limited reactions. – In: Bamford C.H., Tipper C.F.H., Comp-
ton R.G. (eds). Comprehensive Chemical Kinetics. Vol. 25. Amsterdam, Elsevier, 1985.
41. Meakin P. – In: Domb C., Lebowitz J.L. (eds). Phase Transitions and Critical
Phenomena. Vol. 12. New York, Academic Press, 1988, p. 336–442; Simulation
of Non-equilibrium Growth and Aggregation Processes. Landau D.P., Moon K.K.,
Schüttler H.-B. (eds). Springer Proceedings in Physics. Computer Studies in Con-
densed Matter Physics. Vol. 33. Berlin, Springer-Verlag, 1988.
42. von Ferber C., Holovatch Yu. – In: “Renormalization Group’96” 3rd Int. Conf., Dubna,
Russia, August 1996, Ed. by: Shirkov D.V., Priezzhev V.B. JINR, Dubna, 1997, p.
123–138.
43. von Ferber C., Holovatch Yu. // Condens. Matter Phys., 1999, vol. 2, p. 303–312
44. von Ferber C., Holovatch Yu. // J. Mol. Phys., 2001, vol. 93, p. 151–154.
Теоретико-польовий ренормгруповий аналіз
критичних показників зіркових полімерів
К. фон Фербер 1 , Ю. Головач 2,3
1 Теоретична фізика полімерів, Університет Фрайбурґу,
D-79104 Фрайбурґ, Німеччина,
2 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
3 Львівський національний університет ім. І. Франка,
79005 Львів
Отримано 18 вересня 2001 р.
Зроблено огляд недавніх результатів аналізу методом теоретико-
польової ренормалізаційної групи масштабних (скейлінґових) влас-
тивостей зіркових полімерів. Коротко пояснено, як були отримані чи-
сельні значення показників скейлінґу зіркових полімерів. Приведено
приклади явищ, які описуються цими показниками. Зокрема, розгля-
дається взаємодія міз зірковими полімерами у доброму розчинни-
ку, бровнівський рух біля полімерного абсорбера, керовані дифузією
реакції за участю полімерів.
Ключові слова: зіркові полімери, зіркові показники,
ренормалізаційна група
PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh
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