Disorder effects on the static scattering function of star branched polymers
We present an analysis of the impact of structural disorder on the static scattering function of f-armed star branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good solvent in the presence of structural defects, correlated at large distances r acco...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Disorder effects on the static scattering function of star branched polymers / V. Blavatska, C. von Ferber, Yu. Holovatch // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33603:1-17. — Бібліогр.: 47 назв. — англ. |
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irk-123456789-1201742017-06-12T03:02:35Z Disorder effects on the static scattering function of star branched polymers Blavatska, V. von Ferber, C. Holovatch, Yu. We present an analysis of the impact of structural disorder on the static scattering function of f-armed star branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good solvent in the presence of structural defects, correlated at large distances r according to a power law ~r-a. In particular, we are interested in the ratio g(f) of the radii of gyration of star and linear polymers of the same molecular weight, which is a universal experimentally measurable quantity. We apply a direct polymer renormalization approach and evaluate the results within the double ϵ = 4 - d, δ = 4 - a-expansion. We find an increase of g(f) with an increasing δ. Therefore, an increase of disorder correlations leads to an increase of the size measure of a star relative to linear polymers of the same molecular weight. Представлено аналiз впливу структурного безладу на статичну функцiю розсiяння f -гiлкового зiркового полiмера у d-вимiрному просторi. Розглянуто модель зiркового полiмера у хорошому розчиннику в при-сутностi структурних дефектiв, скорельованих на великих вiддалях r згiдно степеневого закону ∼ r −a. Зокрема, ми цiкавимось вiдношенням g(f ) iнтенсивностей розсiяння зiркового та лiнiйного полiмерiв однакової молекулярної маси, що є унiверсальною, експериментально спостережуваною величиною. Ми застосовуємо метод прямого полiмерного перенормування i використовуємо подвiйний ε = 4 − d, δ = 4−a-розклад. Знайдено зростання величини g(f ) iз зростанням параметра δ. Таким чином, зростання кореляцiй безладу приводить до зменшення вiдмiнностi мiж розмiром зiркових та лiнiйних полiмерiв iз однаковою молекулярною вагою. 2012 Article Disorder effects on the static scattering function of star branched polymers / V. Blavatska, C. von Ferber, Yu. Holovatch // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33603:1-17. — Бібліогр.: 47 назв. — англ. 1607-324X PACS: 61.41.+e, 61.25.hp, 64.60.ae DOI:10.5488/CMP.15.33603 arXiv:1207.2881 http://dspace.nbuv.gov.ua/handle/123456789/120174 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present an analysis of the impact of structural disorder on the static scattering function of f-armed star branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good solvent in the presence of structural defects, correlated at large distances r according to a power law ~r-a. In particular, we are interested in the ratio g(f) of the radii of gyration of star and linear polymers of the same molecular weight, which is a universal experimentally measurable quantity. We apply a direct polymer renormalization approach and evaluate the results within the double ϵ = 4 - d, δ = 4 - a-expansion. We find an increase of g(f) with an increasing δ. Therefore, an increase of disorder correlations leads to an increase of the size measure of a star relative to linear polymers of the same molecular weight. |
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Article |
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Blavatska, V. von Ferber, C. Holovatch, Yu. |
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Blavatska, V. von Ferber, C. Holovatch, Yu. Disorder effects on the static scattering function of star branched polymers Condensed Matter Physics |
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Blavatska, V. von Ferber, C. Holovatch, Yu. |
| author_sort |
Blavatska, V. |
| title |
Disorder effects on the static scattering function of star branched polymers |
| title_short |
Disorder effects on the static scattering function of star branched polymers |
| title_full |
Disorder effects on the static scattering function of star branched polymers |
| title_fullStr |
Disorder effects on the static scattering function of star branched polymers |
| title_full_unstemmed |
Disorder effects on the static scattering function of star branched polymers |
| title_sort |
disorder effects on the static scattering function of star branched polymers |
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Інститут фізики конденсованих систем НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/120174 |
| citation_txt |
Disorder effects on the static scattering function of star branched polymers / V. Blavatska, C. von Ferber, Yu. Holovatch // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33603:1-17. — Бібліогр.: 47 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT blavatskav disordereffectsonthestaticscatteringfunctionofstarbranchedpolymers AT vonferberc disordereffectsonthestaticscatteringfunctionofstarbranchedpolymers AT holovatchyu disordereffectsonthestaticscatteringfunctionofstarbranchedpolymers |
| first_indexed |
2025-07-08T17:22:15Z |
| last_indexed |
2025-07-08T17:22:15Z |
| _version_ |
1837100261857820672 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 3, 33603: 1–17
DOI: 10.5488/CMP.15.33603
http://www.icmp.lviv.ua/journal
Disorder effects on the static scattering function of
star branched polymers
V. Blavatska1, C. von Ferber2,3, Yu. Holovatch1
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 79011 Lviv, Ukraine
2 Applied Mathematics Research Centre, Coventry University, CV1 5FB Coventry, UK
3 Institut für Theoretische Physik II, Heinrich-Heine Universität Düsseldorf, D–40225 Düsseldorf, Germany
Received May 24, 2012, in final form June 26, 2012
We present an analysis of the impact of structural disorder on the static scattering function of f -armed star
branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good
solvent in the presence of structural defects, correlated at large distances r according to a power law ∼ r−a .
In particular, we are interested in the ratio g ( f ) of the radii of gyration of star and linear polymers of the
same molecular weight, which is a universal experimentally measurable quantity. We apply a direct polymer
renormalization approach and evaluate the results within the double ε = 4−d , δ = 4−a-expansion. We find
an increase of g ( f ) with an increasing δ. Therefore, an increase of disorder correlations leads to an increase of
the size measure of a star relative to linear polymers of the same molecular weight.
Key words: polymers, structural disorder, universality, renormalization group
PACS: 61.41.+e, 61.25.hp, 64.60.ae
1. Introduction
Scattering experiments have been commonly used in investigations of the structure properties of
condensed matter for more than a century (see, e.g. [1]). For polymer systems, the quantity of interest is
the static structure function S(k) as a function of the wave vector ~k , representing the Fourier transform
of the monomer-monomer correlation function [2–7]. The scattering intensity I (k) ≡ S(k)/S(0) at small k
gives the radius of gyration Rg of a single macromolecule:
I (k) = 1−k2
〈R2
g 〉
d
+ . . . , (1.1)
where d is the space dimension and 〈. . .〉 denotes the average over the ensemble of all conformations that
a macromolecule in a solvent is capable of attaining.
In this paper, we intend to derive some theoretical predictions and quantitatively describe the pecu-
liarities of scattering experiments with star-like polymers. Star-like polymers are the simplest represen-
tatives of a class of branched polymer structures that are in a close relationship to complex systems such
as gel, rubber, micellar and other polymeric and surfactant systems [8–10]. In particular, some confor-
mational properties of star polymers could be easily generalized to determine the behavior of polymer
networks of a more complicated structure [11, 12]. The star polymer can be viewed as f linear polymer
chains (arms) linked together at the central core (see figure 1). For f = 1(2), one restores the polymer
chain of linear architecture, whereas it has been shown that in another limiting situation ( f ≫ 1), the
star polymer attains the features of a soft colloidal particle [13, 14].
© V. Blavatska, C. von Ferber, Yu. Holovatch, 2012 33603-1
http://dx.doi.org/10.5488/CMP.15.33603
http://www.icmp.lviv.ua/journal
V. Blavatska, C. von Ferber, Yu. Holovatch
. . .
1
2f
f−1
Figure 1. Schematic presentation of an f -armed star polymer.
A convenient parameter for comparing the size measure of a star consisting of f arms (each of length
N ) and a linear polymer chain having the same molecular weight f N is the ratio:
g ( f ) ≡
〈R2
g star〉
〈R2
g chain
〉
, (1.2)
here, 〈R2
g star〉 is the mean square radius of gyration of the star polymer. Note, that the value (1.2) is an
experimentally measurable quantity. Indeed, recalling (1.1), one has at small k:
I ′(k)star
I ′(k)chain
= g ( f )+ . . . . (1.3)
Therefore, the ratio of the derivatives I ′(k) of the corresponding scattering intensities for small values of
the wave vector permits to experimentally define the g ( f ) ratio. Moreover, it is well established that the
gyration radius of a star polymer scales with its total number of segments according to the scaling law:
〈R2
g star〉 ∼ ( f N )2ν, (1.4)
with an universal exponent ν that also governs the scaling of a single polymer chain of N monomers:
〈R2
g chain
〉 ∼ N 2ν. Therefore, the ratio g ( f ) of a star and linear chain of the same total molecular weight is
N -independent.
In the pioneering work by Zimm and Stockmayer [15], an estimate for the size ratio g ( f ) was found
analytically:
g ( f ) =
3 f −2
f 2
. (1.5)
Inserting f = 1 or f = 2 in this relation, one restores the trivial result g = 1. For any f Ê 3, ratio (1.5)
is smaller than 1, reflecting the fact that the size of a branched polymer is always smaller than the size
of a linear polymer chain of the same molecular weight. Note that in deriving the expression (1.5) the
excluded volume effect was neglected (restricting to the idealized Gaussian case, where (1.4) holds with
size exponent ν = 1/2). In this limit, the result holds for any space dimension. Dimensional dependence
of the size ratio g ( f ) is found by introducing the concept of excluded volume applied to polymer macro-
molecules by P. Flory. It refers to the idea that any segment (monomer) of macromolecule is not capable
of occupying the space that is already occupied by another segment; this causes a swelling of a polymer
in a solution with size exponent ν(d) Ê 1/2. Later on, analytical [16–19] and numerical [20–24] studies
have found the value of g ( f ) to increase if the excluded volume effect is taken into consideration.
While reliable estimates are known for the value of g ( f ) for polymers in a good solvent, there are
no similar estimates for the case when the polymer is immersed in a good solvent in the presence of
structural impurities or a porous environment. However, such estimates are of a great importance for
understanding the behavior of macromolecules in colloidal solutions [25] or near microporous mem-
branes [26]. The density fluctuations of such obstacles lead to large spatial inhomogeneities which of-
ten produce pore-like fractal structures [27]. Such a disordered (porous) environment may be found, in
particular, in a biological cell composed of many different kinds of biochemical species [28–30]. It has
33603-2
Disorder effects on the static scattering function of star branched polymers
been proven both analytically [31] and numerically [32, 33] that the presence of uncorrelated point-like
defects of weak concentration does not change the universality class of polymers. Here, however, we ad-
dress a case where the structural obstacles of the environment are spatially correlated on a mesoscopic
scale [34]. Following reference [35], this case may be described by assuming the defects to be correlated
at large distances r according to a power law resulting in a pair correlation function:
h(r ) ∼ r−a . (1.6)
For a < d , specific situations that give rise to such a correlation function include the defects extended
in space, e.g., the cases a = d −1 (a = d −2) may be represented by lines (planes) of defects of random
orientation, whereas non-integer values of a may include obstacles of fractal structures (see [35–37] for
further details). The impact of long-range-correlated disorder on the scaling of linear and star branched
polymers has been analyzed in previous works [36, 37] by means of the field-theoretical renormalization
group (RG) approach. The aim of the present paper is to analytically evaluate an experimentally measur-
able ratio (1.3) for a star polymer in a solvent in the presence of structural defects correlated according
to (1.6).
The layout of the reminder of the paper is as follows. In the next section, we develop a description of
the problem in the frames of the Edwards continuous chain model. In section 3, a direct polymer renor-
malization method is briefly described; the results of its application to the problem under consideration
are presented in section 4. Conclusions and an outlook are given in section 5.
2. The model
We consider star polymers with f arms in a solution in the presence of structural obstacles. In the
frames of the Edwards continuous chain model [1], each arm of the star is presented by a path ri (s),
parameterized by 0 É s É Si , i = 1,2, . . . , f . The central branching point of the star is fixed in space, so
that: ~r1(0) = . . . =~r f (0) = 0. We take the contour length of all arms to be equal: S1 = . . . = S f = S. The
partition function of the system reads:
Z f (S) =
∫
D{r }
f
∏
i=1
δ(~ri (0))exp
−
1
2
f
∑
i=1
S
∫
0
ds
[
d~ri (s)
ds
]2
−
b0
2
f
∑
i , j=1
S
∫
0
ds′
S
∫
0
d s′′δ
[
~ri (s′′)−~r j (s′)
]
+
f
∑
i=1
S
∫
0
ds V [~ri (s)]
. (2.1)
Here, a multiple path integral is performed for the paths r1, . . . ,r f and the product of δ-functions reflects
the star-like configuration of f chains. The first term in the exponent represents the chain connectivity,
the second term describes the short range excluded volume interaction with a bare coupling constant b0,
and the last term contains a random potential V [~ri (s)] arising due to the presence of structural disorder.
Denoting by (. . .) the average over different realizations of disorder, the first moment of the distribution
is:
V [~r (s)]= ρ0
with ρ0 being the density of obstacles. Let us introduce a notation for the second moment:
V [~r (s)]V [~r (s′)]≡ h
[
~r (s)−~r (s′)
]
. (2.2)
Note that dealing with systems that display randomness of structure, one usually encounters two
types of ensemble averaging treated as quenched and annealed disorder [38, 39]. The annealed case
amounts to averaging the partition sum of a system over the random variables, whereas in the quenched
case, the free energy (or the logarithm of the partition sum) is to be averaged; the replica formalism is
usually applied in the last situation. In principle, the behavior of systems with quenched and annealed
disorder is quite different. However, as it has been shown in a number of works [40–43], the distinction
between quenched and annealed averages for an infinitely long single polymer chain is negligible, and in
33603-3
V. Blavatska, C. von Ferber, Yu. Holovatch
performing analytical calculations for quenched polymer systems one may thus restrict the problem to
the simpler case of annealed averaging. To average the partition function of a system over different
realizations of obstacles, we make use of the relation:
eax = exp
[
∞
∑
n=1
an Mn (x)
n!
]
, (2.3)
where Mn(x) are nth cumulants of the random variable x: M1(x) = x, M2(x) = (x − x)2 etc. Noticing that
only the last term in (2.1) contains random variables and taking into account (2.2), we obtain:
Z f (S) =
∫
D{r }
f
∏
i=1
δ[~ri (0)]e−Hdis (2.4)
with an effective Hamiltonian:
Hdis =
1
2
f
∑
i=1
S
∫
0
ds
[
d~ri (s)
ds
]2
+
b0
2
f
∑
i , j=1
S
∫
0
ds′
S
∫
0
d s′′δ
[
~ri (s′′)−~r j (s′)
]
−
1
2
f
∑
i=1
S
∫
0
ds′
S
∫
0
d s′′ h
[
~ri (s′′)−~r j (s′)
]
−ρ0 f S −
1
2
ρ2
0 f S2. (2.5)
The last two terms in (2.5) correspond to a trivial constant shift which will be omitted in the following
analysis. Note also, that in (2.5) we do not take into account the terms generated by higher-order corre-
lations of the type (2.2), because for the problem under consideration these terms are irrelevant in the
renormalization group sense.
The case of structural disorder in the form of point-like uncorrelated defects corresponds to h[~r (s′′)−
~r (s′)] = v0δ[~r (s′′)−~r (s′)] where v0 is some constant. One immediately reveals that in this case one can
adsorb the effect of disorder into the excluded volume coupling constant passing to the coupling: b0 ≡
b0−v0. This conclusion was obtained for the case of polymers in quenched disorder by Kim [31] based on
a refined field-theoretical study; in the present case of annealed disorder, this is a straightforward result.
We address the model where the structural obstacles are spatially correlated at large distances r
according to (1.6). Taking into account that the Fourier transform of the correlation function at small k is
related to its large-r behaviour via:
h
[∣
∣~ri (s′′)−~r j (s′)
∣
∣
]
�
∣
∣~ri (s′′)−~r j (s′)
∣
∣
−a
�wo
∫
dk ka−d ei~k
[
~ri (s ′′)−~r j (s ′)
]
, (2.6)
one is left with a model with two couplings b0 and w0. Note that coupling b0 should be positive, which
corresponds to an effective mutual repulsion of the monomers due to the excluded volume effect. The
coupling w0 is positive as results from the Fourier image of the correlation function.
Performing dimensional analysis for the terms in (2.5), one finds the dimensions of the couplings in
terms of a dimension of contour length S: [b0]= [S]db0 , [w0] = [S]dw0 with db0
= (4−d)/2, dw0 = (4−a)/2.
Note that for the exponent in (2.1) to be dimensionless, the contour length needs to have units of surface.
The “upper critical” values of the space dimension (dc = 4) and the correlation parameter (ac = 4), at
which the couplings are dimensionless, play an important role in the renormalization scheme, as outlined
below.
3. The method
To study the universal properties of polymer macromolecules in solutions, it is convenient to apply
the direct renormalization method, as developed by des Cloizeaux [1]. The efficiency of this approach
comes, on the one hand, from its close relation to the concepts of field theory [45–47], and, on another
hand, from providing a considerably simpler treatment of a variety of complex polymer systems.
33603-4
Disorder effects on the static scattering function of star branched polymers
In the asymptotic limit of an infinite linear measure of a continuous polymer curve (corresponding
to an infinite number of configurations), one observes various divergences. All these divergences can
be eliminated by introducing corresponding renormalization factors directly associated with physical
quantities. This postulates the existence of a limiting theory that describes the sets of very long polymers.
As a first step within this theory, we consider the size measure of an f -arm star polymer given by
the mean square end-to-end distance of its individual arm. When evaluated in terms of a perturbation
theory series in bare coupling constants {λ0}, this reads:
〈R2
e 〉 = 〈[~r (S)−~r (0)]2
〉 =χ0({λ0})S. (3.1)
Here, the averaging is performed with respect to a corresponding effective Hamiltonian, and χ0({λ0})
is the so-called swelling factor that reflects the impact of interactions on the effective size of macro-
molecules. For the case of a Gaussian chain (all couplings λ0 = 0), one has χ0({0}) = 1. Recalling the
scaling of a polymer size with its molecular weight:
〈R2
e〉 ∼ N 2ν
∼ S2ν, (3.2)
one finds an estimate for the effective critical exponent ν({λ0}):
2ν({λ0})−1= S
∂
∂S
lnχ0({λ0}). (3.3)
The second renormalization factor χ1({λ0}) is introduced via:
Z f (S)
Z
0
f
(S)
=
[
χ1({λ0})
]2
. (3.4)
Here, Z f (S) is the partition function of an f -arm star polymer and Z
0
f
(S) is the partition function of an
idealized Gaussian model. It is established that the number of all possible conformations of an f -armed
star polymer scales with the weight of a macromolecule parametrised by S as:
Z f (S)∼µ f S ( f S)γ f −1. (3.5)
Here, the γ f are additional universal critical exponents depending only on the space dimension d and
the number of arms f (exponents γ1 = γ2 ≡ γ restore the value for the single polymer chain), µ is a non-
universal fugacity. In a similar way as for the size measure, from the scaling assumption (3.5) one finds
an estimate for an effective critical exponent γ f ({λ0}) governing the scaling behavior of the number of
possible configurations as:
γ f ({λ0})−1
2
= S
∂ lnχ1({λ0})
∂S
. (3.6)
The critical exponents (3.3) and (3.6) presented in the form of series expansions in the coupling constants
{λ0} are, however, divergent in the asymptotic limit of large S. To eliminate these divergences, renormal-
ization of the coupling constants is performed. Subsequently, the critical exponents attain finite values
when evaluated at a stable fixed point (FP) of the renormalization group transformation. Note that the FP
coordinates are universal. In particular, the scaling of a single polymer chain and that of a polymer star
is governed by the same unique FP. Therefore, to evaluate the FP coordinates in the following analysis,
we restrict ourselves to a simpler case of a single chain polymers ( f = 1). To define the coupling constant
renormalization, one considers the second virial coefficient of a polymer solution given by the relation:
Πβ=C −
1
2
C 2
∑
λ0
Zλ0
(S,S)
[Z1(S)]2
+ . . . , (3.7)
here, Π is the osmotic pressure, β = 1/kBT , C is the number of monomers per unit volume, and Z1(S)
is the partition function of a single polymer chain. Zλ0
(S,S) are contributions into a partition function
of two interacting chain polymers having dimensions Zλ0
(S,S) ∼ [S]2[λ0]. The renormalized coupling
constants λR are thus defined by:
λR({λ0}) =−
[
χ1({λ0})
]−4
Zλ0
(L,L)
[
2πχ0({λ0})L
]−(2−dλ0
)
, (3.8)
33603-5
V. Blavatska, C. von Ferber, Yu. Holovatch
therefore:
Πβ=C +
1
2
∑
λR
λRC 2
[
2πχ0({λ0})L
](2−dλ0
)
+ . . . . (3.9)
In the limit of infinite linear size of macromolecules, the renormalized theory remains finite, such that:
lim
S→∞
λR({λ0})= λ∗
R . (3.10)
Moreover, for negative dλ0
É 0, macromolecules are expected to behave like Gaussian chains in spite of
the interactions betweenmonomers, thus each λ∗
R
= 0 for corresponding dλ0
É 0. It is, therefore, proper to
choose {λR} as expansion parameters which remainfinite for S →∞ andwhich are also rather small close
to the critical dimensions of the corresponding couplings. The concept of expansion in small deviations
from the upper critical dimensions of the coupling constants thus naturally arises.
The flows of the renormalized coupling constants are governed by functions βλR
:
βλR
= 2S
∂λR({λ0})
∂S
. (3.11)
Reexpressing {λ0} in terms of renormalized couplings λR according to (3.8), the fixed points of renormal-
ization group transformations are given by common zeros of the β-functions. Stable fixed points govern
the asymptotical scaling properties of macromolecules in solutions and make it possible, e.g., to obtain
reliable asymptotical values of the critical exponents (3.3) and (3.6).
4. Results
We start by evaluating the partition function (2.4) of the model with an effective Hamiltonian (2.5),
performing an expansion in coupling constants b0, w0:
Z f (S) =
∫
Dr exp
−
1
2
f
∑
i=1
S
∫
0
ds
[
dri (s)
ds
]2
1−
b0
2
f
∑
i , j=1
S
∫
0
ds′
S
∫
0
ds′′
∫
dk ei~k
[
~ri (s ′′)−~r j (s ′)
]
+
w0
2
f
∑
i , j=1
S
∫
0
ds′
S
∫
0
d s′′
∫
dk ka−d ei~k
[
~ri (s ′′)−~r j (s ′)
]
+ . . .
, (4.1)
here, the Fourier-transform of the δ-function is exploited and the last term originates from the Fourier
transform of the function h at small k [see equation (2.6)]. Below, we will consider the one-loop approxi-
mation, keeping only the first-order terms in b0, w0 in the expansions. One may rewrite:
ei~k[~ri (s ′′)−~ri (s ′)] = exp
i~k
s ′′
∫
s ′
d~ri (s)
d s
ds
, (4.2)
ei~k
[
~ri (s ′′)−~r j (s ′)
]
= exp
i~k
s ′′
∫
0
d~ri (s)
ds
ds − i~k
s ′
∫
0
d~r j (s)
ds
ds
, (4.3)
taking into account that ri (0) = r j (0) = 0 in our model of a star-shaped polymer. Making use of the iden-
tity:
exp
−
1
2
s ′′
∫
s ′
ds
[
dri (s)
ds
]2
+ i~k
s ′′
∫
s ′
d~ri (s)
d s
ds
= exp
−
1
2
s ′′
∫
s ′
ds
{[
dri (s)
ds
− i~k
]2
+k2
}
(4.4)
and taking into account that:
∞
∫
−∞
dxe−A(x−ik)2
=
∞
∫
−∞
dxe−Ax2
, (4.5)
33603-6
Disorder effects on the static scattering function of star branched polymers
we receive:
Z f (S) = Z
0
f (S)
1−b0(2π)−
d
2
f
S
∫
0
ds′′
s ′′
∫
0
d s′
(
s′′− s′
)−d/2
+
f ( f −1)
2
S
∫
0
ds′′
S
∫
0
d s′
(
s′+ s′′
)−d/2
+ w0(2π)−
a
2
f
S
∫
0
ds′′
s ′′
∫
0
d s′
(
s′′− s′
)−a/2
+
f ( f −1)
2
S
∫
0
ds′′
S
∫
0
d s′
(
s′+ s′′
)−a/2
. (4.6)
In the last equation, the Gaussian integration over k is performed and the notation Z
0
f
(S) is introduced
for the partition function of the “unperturbed” Gaussian model:
Z
0
f (S)=
∫
Dr exp
−
1
2
f
∑
i=1
S
∫
0
ds
[
dri (s)
ds
]2
. (4.7)
In what follows, we will use the diagrammatic representation of the perturbation theory series (see fig-
ure 2). Performing the integrals in (4.6) and introducing dimensionless couplings
b = b0(2π)−d/2S2−d/2, w = w0(2π)−a/2S2−a/2 (4.8)
we obtain:
Z f (S) = Z
0
f (S)
{
1−
4b
(2−d)(4−d)
[
f +
f ( f −1)
2
(22−d/2
−2)
]
+
4w
(2−a)(4−a)
[
f +
f ( f −1)
2
(22−a/2
−2)
]}
. (4.9)
Finally, one may perform a double ε= 4−d , δ= 4−a-expansions:
Z f (S) = 1+b
[
f (3− f )
ε
+
f (3− f )
2
+
f ( f −1)
2
ln(2)
]
−w
[
f (3− f )
δ
+
f (3− f )
2
+
f ( f −1)
2
ln(2)
]
. (4.10)
The averaged squared end-to-end distance 〈R2
e 〉 of a single arm of a star polymer may be calculated
using the identity:
〈R2
e〉 = 〈[~r (S)−~r (0)]2〉 =−2d
∂
∂q2
〈ei~q [~r (S)−~r (0)]〉
∣
∣
q=0 , (4.11)
Figure 2. Diagram contributions to the partition function of a 4-arm star up to the first order in the cou-
pling constants. Dotted lines denote possible interactions between points s′ , s′′ , governed by couplings
b0 and w0. Integrations are to be performed over all positions of the segment end points, i.e., over all
mutual interaction points within a single arm and between different arms.
33603-7
V. Blavatska, C. von Ferber, Yu. Holovatch
where:
〈. . .〉 =
∫
Dr e−Hdis (. . .)
∏ f
i=1
δ[~ri (0)]
Z f (S)
. (4.12)
Following the same scheme as described above for the partition function, we find:
〈R2
e〉 = Sd
[
1+
4b
(4−d) (6−d)
−
4w
(4−a) (6−a)
]
. (4.13)
We may, therefore, define a swelling factor χ0(b0, w0) [cf. (3.1)] as:
χ0(b0, w0) =
[
1+
4b
(4−d) (6−d)
−
4w
(4−a) (6−a)
]
=
[
1+
b
ε
(2−ε)−
w
δ
(2−δ)
]
. (4.14)
Figure 3. Diagrammatic presentation of contributions into radius of gyration of star polymer in zeros
order of perturbation theory.
Now we return to the calculation of the gyration radius. The gyration radius of a star polymer in a
solvent in the presence of correlated defects is defined by:
〈R2
g star〉 =
1
2( f S)2
S
∫
0
ds1
S
∫
0
ds2
〈
f
∑
i , j=1
[
~ri (s2)−~r j (s1)
]2
〉
. (4.15)
We rewrite:
〈
f
∑
i , j=1
[
~ri (s2)−~r j (s1)
]2
〉
=−2d
∂
∂|q|2
〈
e
i~q
∑ f
i , j=1
[~ri (s2)−~r j (s1)]
〉
∣
∣
∣
q=0
, (4.16)
where d is the space dimension. First, let us consider the zero-loop order of the expansion of (4.16) in
coupling constants. A diagrammatic representation is given in figure 3. The analytic expression, corre-
sponding to the diagram (a) reads:
f
〈
ei~q[~r1(s2)−~r1(s1)]
〉
= f e−
q2
2 (s2−s1), (4.17)
whereas the diagram (b) gives:
f ( f −1)
2
〈
ei~q[~r2(s2)−~r1(s1)]
〉
=
f ( f −1)
2
e−
q2
2 (s1+s2). (4.18)
Note, that here and in what follows the Gaussian integral in (4.7) cancels against the numerator when
averaging (4.12) is performed. One thus distinguishes between two types of contributions: one resulting
from insertions s1, s2 along the same arm of the star, and the second corresponding to insertions lo-
cated on two different arms. Taking the derivative with respect to q , and evaluating for q = 0 according
to (4.16), we have for the radius of gyration of a star polymer in the unperturbed (Gaussian) case:
〈R2
g star〉 =
d
( f S)2
f
S
∫
0
ds2
s2
∫
0
ds1 (s2 − s1)+
f ( f −1)
2
S
∫
0
ds1
S
∫
0
ds2 (s1 + s2)
=
dS
f
3 f −2
6
. (4.19)
33603-8
Disorder effects on the static scattering function of star branched polymers
Figure 4. Diagram contribution into the gyration radius at the one-loop level.
Now, let us perform calculations up to the first order of the perturbation theory expansion in the
couplings b0, w0. In figure 4, we present diagram contributions to (4.16), see appendix for details. We
have:
〈
f
∑
i , j=1
[
~ri (s2)−~r j (s1)
]2
〉
=−2d
∂
∂q2
{
f
[
I1(d)+ I1(a)+ I2(d)+ I2(a)+ I3(d)
+I3(a)+ I4(d)+ I4(a)+ I5(d)+ I5(a)+ I6(d)+ I6(a)+ I7(d)+ I7(a)
]
+
f ( f −1)
2
[
I8(d)+ I8(a)+ I9(d)+ I9(a)+ I10(d)+ I10(a)+ I11(d)+ I11(a)
+I12(d)+ I12(a)+ I13(d)+ I13(a)+ I14(d)+ I14(a)+ I15(d)+ I15(a)
+I16(d)+ I16(a)+ I17(d)+ I17(a)
]
+
f ( f −1)( f −2)
6
[
I18(d)+ I18(a)
+I19(d)+ I19 + (a)I20(d)+ I20(a)
]
+
f ( f −1)( f −2)( f −3)
24
[
I21(d)+ I21(a)
]}∣
∣
∣
q=0
. (4.20)
Here, Ii (d), Ii (a) are integrals listed in the appendix. Performing the integration according to (4.15) and
evaluating the double ε, δ-expansion, we find for the gyration radius:
〈R2
g star〉 =
dS
6 f
(3 f −2)
{
1+b
2
ε
−w
2
δ
− (b −w)
[
13
12
+
13
2
( f −1)( f −2)
3 f −2
− ln(2)
4( f −1)(3 f −5)
3 f −2
]}
. (4.21)
The result for the radius of gyration of a single chain of total length f S is straightforward:
〈R2
g chain
〉 =
d(S f )
6
[
1+b f ε/2
(
2
ε
−
13
12
)
−w f δ/2
(
2
δ
−
13
12
)]
=
d(S f )
6
[
1+b
(
2
ε
−
13
12
)
−w
(
2
δ
−
13
12
)
+ (b −w) ln( f )
]
. (4.22)
Finally, we need to perform renormalization of coupling constants. To this end, we need the contribu-
tions to the partition function Z (S,S) of a system of two interacting polymer chains of the same length L.
In the diagrammatic representation of this function, one takes only those terms into account which con-
tain at least one interaction line (see figure 5). In general, performing a thorough dimensional analysis of
the contributions produced by different diagrams, we find two distinct classes of the diagrams. The first
33603-9
V. Blavatska, C. von Ferber, Yu. Holovatch
Figure 5. Diagrammatic presentation of contributions into the partition function of two interacting
chains.
class of graphs produces the terms which behave as [S]2[b0], the sum of all such terms contributing to
the function denoted by Zb0
(S,S); the second class of diagrams behaves as [S]2[w0] and thus contributes
to the function Zw0 (S,S).
We find the contributions to functions Zb0
(S,S) and Zw0 (S,S) as:
Zb0
(S,S) = −b0S2
[
1−
8b
(2−d)(4−d)
+
8w
(2−a)(4−a)
−32b
d +24−d/2 −10
(d −2)(d −4)(d −6)(d −8)
−32
w2
b
2a −d +24−(2a−d)/2 −10
(2a −d −2)(2a −d −4)(2a −d −6)(2a −d −8)
+ 64w
a +24−a/2 −10
(a −2)(a −4)(a −6)(a −8)
]
,
Zw0 (S,S) = w0S2
[
1−
8b
(2−d)(4−d)
+
8w
(2−a)(4−a)
]
. (4.23)
We may, therefore, define the dimensionless renormalized coupling constants bR, wR by (3.8):
bR =−
Zb(S,S)
〈Z (S)〉
2
[
2πχ0(b0, w0)S
]− d
2 ,
wR =−
Zw (S,S)
〈Z (S)〉
2
[
2πχ0(b0, w0)S
]− a
2 . (4.24)
The RG flows of the renormalized coupling constants are governed by corresponding β-functions:
βbR
(b, w) = 2S
∂bR
∂S
, βwR (b, w) = 2S
∂wR
∂S
. (4.25)
Reexpressing in (4.25) b and w in terms of the renormalized couplings (4.24), we finally have:
βbR
(bR, wR) = (4−d)bR −b2
R
2d
6−d
+wRbR
2d
6−a
+32b2
R
d +24−d/2 −10
(d −2)(d −6)(d −8)
+32w2
R
2a −d +24−(2a−d)/2 −10
(2a −d −2)(2a −d −6)(2a −d −8)
−64bR wR
a +24−a/2 −10
(a −2)(a −6)(a −8)
,
βwR (bR, wR) = −(4−a)wR −w2
R
2a
6−a
+bRwR
2a
6−d
. (4.26)
Performing a double ε, δ expansion and keeping the terms up to linear in these parameters, we then have
the RG functions:
βb = εbR −8b2
R −4w2
R +12bR wR ,
βw =−δwR −4w2
R +4bR wR . (4.27)
33603-10
Disorder effects on the static scattering function of star branched polymers
The fixed points b∗
R , w∗
R of the renormalization group transformations are defined as the common zeros
of the RG functions (4.27). We find three distinct fixed points that determine the scaling behavior of the
system in various regions of the a and d plane:
Gaussian: b∗
R = 0, w∗
R = 0 stable at ε,δ< 0, (4.28)
Pure: b∗
R =
ε
8
, w∗
R = 0 stable at δ< ε/2, (4.29)
LR: b∗
R =
δ2
4(ε−δ)
, w∗
R =
δ(ε−2δ)
4(δ−ε)
stable at ε/2 < δ< ε. (4.30)
Here and below, the index LR means that the corresponding quantity is evaluated in the region of d , a
plane, where the effect of long-range-correlated disorder is relevant.
We can also find the estimates for critical exponents that govern the scaling of star polymers in solu-
tions in the presence of a correlated disorder. Making use of definition (3.3), recalling the expression or
renormalized scale (4.14) and passing to the renormalized couplings, we have:
ν(bR, wR) =
1
2
+
bR
2
−
wR
2
. (4.31)
Evaluating this expression at fixed points of the renormalization group transformation (4.28)–(4.30), we
find the corresponding estimates for the size exponents:
νGaussian
=
1
2
, (4.32)
νPure
=
1
2
+
ε
16
, (4.33)
νLR
=
1
2
+
δ
8
. (4.34)
Note that these exponents govern the scaling behavior of macromolecules in the regions of stability of
the corresponding fixed points (4.28)–(4.30).
Similarly, evaluating (3.6) and taking into account (4.10), one finds in terms of renormalized cou-
plings:
γ(bR, wR) = 1+
bR f (3− f )
2
−
wR f (3− f )
2
. (4.35)
Again, evaluating this expression at the fixed points (4.28)–(4.30), we find:
γGaussian
f = 1, (4.36)
γPure
f =
1
2
+
ε
16
f (3− f ), (4.37)
γLR
f =
1
2
+
δ
8
f (3− f ). (4.38)
At f = 1, one restores the corresponding exponents for a single polymer chain. Note, that the first-order
expressions for fixed point coordinates (4.28)–(4.30) and for critical exponents (4.32)–(4.34) and (4.36)–
(4.38) were obtained here for an annealed system. Thus, with νLR and γLR
f
as obtained in the regime
of annealed disorder, we restore the corresponding exponents that govern the scaling of polymers in
solutions in the presence of a quenched long-ranged correlated disorder studied by us earlier [36, 37].
This supports the statement of equivalence between the annealed and quenched averaging when dealing
with polymer systems.
Finally, we obtain estimates for the size ratio g = 〈R2
g star〉
/
〈R2
g chain
〉 of star and linear polymers, re-
calling (4.21) and (4.22):
g ( f ) =
g Gaussian, ε,δ< 0,
g Pure, δ< ε/2,
g LR, ε/2 < δ< ε
(4.39)
33603-11
V. Blavatska, C. von Ferber, Yu. Holovatch
with:
g Gaussian
=
3 f −2
f 2
, (4.40)
g Pure
=
3 f −2
f 2
{
1−
ε
8
[
13
2
( f −1)( f −2)
3 f −2
− ln(2)
4( f −1)(3 f −5)
3 f −2
+ ln( f )
]}
, (4.41)
g LR
=
3 f −2
f 2
{
1−
δ
4
[
13
2
( f −1)( f −2)
3 f −2
− ln(2)
4( f −1)(3 f −5)
3 f −2
+ ln( f )
]}
. (4.42)
With g Pure, we restore the size ratio for the case of polymers in a pure solvent evaluated previously in
references [17–19].
5. Conclusions and outlook
In the present paper, we have studied the impact of structural disorder on the static scattering func-
tion of f -arm star branched polymers in d dimensions. To this end, we consider the model of a star
polymer immersed in a good solvent in the presence of structural defects, correlated at large distances
according to (1.6) with parameter a [35]. The impact of such long-range-correlated disorder on the scal-
ing of linear and star branched polymers has been analyzed in our previous works [36, 37]. Here, we
are interested, in particular, in the ratio g ( f ) of scattering intensities of star and linear polymers of the
same molecular weight, which is a universal experimentally measurable quantity [see (1.3)]. We used
a direct polymer renormalization approach [1] and evaluated the results within the double ε = 4−d ,
δ= 4−a-expansion.
Let us analyze the expressions obtained for the size ratio of star and linear polymers in a solution
in the presence of structural defects. First of all, as far as the δ = 4− a-prefactor in (4.42) is positive
for all f > 2, one immediately concludes that the stronger the correlations of disorder (i.e., the larger is
parameter δ), the larger is the size ratio g ( f ) and thus, the smaller is the distinction between the size
measure of a star and linear polymers of the same molecular weight. In figure 6 (a), we present the
estimates for the size ratio (4.42) as a function of the number of arms f obtained by a direct evaluation at
several fixed values of the correlation parameter δ. Besides an expected increase of g LR( f ) with growing
δ at each fixed f , we also notice a decrease of the size ratio with f for any fixed δ. As can be seen from
(4.40), this occurs already at the gaussian approximation: re-arranging the monomers of a linear chain
into the shape of a star naturally leads to a smaller size and this effect becomes stronger when the star
has more arms.
Another interesting aspect results from a qualitative comparison of the impact of a long-range-correlated
disorder on the size ratio with the impact of the excluded volume effect in a pure solvent. We consider
the ratio:
g Pure( f )
g LR( f )
= 1−
(
ε
8
−
δ
4
)[
13
2
( f −1)( f −2)
3 f −2
− ln(2)
4( f −1)(3 f −5)
3 f −2
+ ln( f )
]
. (5.1)
In figure 6 (b), we present the evaluation of this ratio as a function of f at three dimensions (taking ε= 1)
and several fixed values of correlation parameter δ. Note, that a correlated disorder with δ = 0.5 plays
the role of “marginal”, separating a region where the ratio (5.1) increases with f (at δ< 0.5) and a region
where it decreases with f (any δ> 0.5).
Let us recall that with the critical exponents νLR (4.34) and γLR
f
(4.38) that we obtained in the present
paper in the regime of annealed disorder, we restore the corresponding exponents that govern the scaling
of polymers in solutions in the presence of a quenched long-ranged correlated disorder, studied by us ear-
lier [36, 37]. This supports the statement of equivalence between the annealed and quenched averaging
when dealing with polymer systems. Thus, our qualitative estimates for the size ratio of star and linear
polymers in annealed correlated disorder (4.42) also holds for the case of quenched systems.
Note that our results are based on the first-order perturbation theory expansions and provide a qual-
itative description of the impact of structural disorder on the quantities of interest. To obtain reliable
quantitative estimates, the higher order analysis would be worthwhile, which is the subject of our forth-
coming studies.
33603-12
Disorder effects on the static scattering function of star branched polymers
Acknowledgements
This work was supported in part by the FP7 EU IRSES project N269139 “Dynamics and Cooperative
Phenomena in Complex Physical and Biological Media” and Applied Research Fellowship of Coventry
University.
Appendix
Here, as an example we evaluate the analytic expression corresponding to diagram (9) in figure 4
(shown more in detail in figure 7) presenting contributions into the radius of gyration of a star polymer.
We have for this contribution:
〈e−i~q[~r1(s2)−~r1(s1)]〉(9) = −b0
∫
Dr exp
−
1
2
f
∑
i=1
S
∫
0
ds
[
dri (s)
ds
]2
×e−i~q[~r1(s2)−~r1(s1)]
S
∫
s2
d s′
S
∫
0
d s′′
∫
d~k e−i~k[~r1(s ′)−~r2(s ′′)] .
Rewriting the last exponent:
− i~k
[
~r1(s′)−~r2(s′′)
]
=−i~k
{[
~r1(s′)−~r1(s2)
]
+ [~r1(s2)−~r1(s1)]+
[
~r1(s1)−~r2(s′′)
]}
, (5.2)
Figure 6. Left: the size ratio (4.42) as a function of f at different values of correlation parameter δ. From
below: δ = 0.1, δ = 0.5, δ = 1.0. Right: the ratio (5.1) as a function of f at different values of correlation
parameter δ. From above: δ= 0.1, δ= 0.5, δ= 1.0. In both (a) and (b) we fix the value of parameter ε= 1.
Figure 7. Example of diagrammatic contribution into the radius of gyration of a star polymer (4.15).
33603-13
V. Blavatska, C. von Ferber, Yu. Holovatch
and making use of (4.3)–(4.5), one arrives at:
〈e−i~q[~r1(s2)−~r1(s1)]〉(9) =−b0
∫
Dr exp
−
1
2
f
∑
i=1
S
∫
0
ds
[
dri (s)
ds
]2
I9(d) (5.3)
with:
I9(d) ≡
S
∫
0
ds′′
S
∫
s2
ds′
∫
d~k e
−
k2
2
(
s
′
−s2+s1+s
′′
)
−
(~q+~k)2
2 (s2−s1)
= e−
q2
2 (s2−s1)
S
∫
0
ds′′
S
∫
s2
ds′
∫
d~k e
−
k2
2
(
s
′
+s
′′
)
−~q~k(s2−s1)
= (2π)−d/2e−
q2
2 (s2−s1)
S
∫
0
ds′′
S
∫
s2
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s2−s1)2
(s′+s′′) . (5.4)
Taking the derivative over q according to (4.16) we get:
∂I9(d)
∂q2
∣
∣
∣
∣
q=0
= (2π)−d/2d(s2 − s1)
S
∫
0
ds′′
S
∫
s2
ds′
(
s′′+ s′
)−d/2
− (2π)−d/2d(s2 − s1)2
S
∫
0
ds′′
S
∫
s2
ds′
(
s′′+ s′
)−d/2−1
= (2π)−d/2 d(s2 − s1)
(1−d/2)(2−d/2)
[
(2S)2−d/2
− (S + s2)2−d/2
−S2−d/2
+ s2−d/2
2
]
− (2π)−d/2 d(s2 − s1)2
(1−d/2)(−d/2)
[
(2S)1−d/2
− (S + s2)1−d/2
−S1−d/2
+ s1−d/2
2
]
. (5.5)
Finally, the contribution into the gyration radius (4.15) can be found by integrating over s1, s2. Note, that
the Gaussian integral in (5.3) cancels against the numerator when averaging (4.12) is performed.
The expressions corresponding to other diagrams in figure 4 are listed below (note, that I (d) and I (a)
arise, correspondingly, when treating interactions with couplings b0 and w0, respectively). The factors
(2π)−d/2 and (2π)−a/2 in front of each integral are omitted.
I1(d) = e−
q2
2 (s2−s1)
S
∫
s2
ds′′
s2
∫
s1
ds′
(
s′′− s′
)−d/2
e
q2
2
(s2−s′ )2
(s′′−s′) ,
I2(d) = e−
q2
2 (s2−s1)
s2
∫
s1
ds′′
s1
∫
0
ds′
(
s′′− s′
)−d/2
e
q2
2
(s′′−s1)2
(s′′−s′) ,
I3(d) = e−
q2
2 (s2−s1)
S
∫
s2
ds′′
s1
∫
0
ds′
(
s′′− s′
)−d/2
e
q2
2
(s2−s1)2
(s′′−s′) ,
I4(d) = e−
q2
2 (s2−s1)
S
∫
s2
ds′′
s ′′
∫
s2
ds′
(
s′′− s′
)−d/2
,
I5(d) = e−
q2
2 (s2−s1)
s1
∫
0
ds′′
s ′′
∫
0
ds′
(
s′′− s′
)−d/2
,
33603-14
Disorder effects on the static scattering function of star branched polymers
I6(d) = e−
q2
2 (s2−s1)
s2
∫
s1
ds′′
s ′′
∫
s1
ds′
(
s′′− s′
)−d/2
e
q2
2 (s ′′−s1),
I7(d) = e−
q2
2 (s2−s1)
S
∫
0
ds′′
s1
∫
0
ds′
(
s′′+ s′
)−d/2
,
I8(d) = e−
q2
2 (s2−s1)
S
∫
0
ds′′
s2
∫
s1
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s′−s1)2
(s′+s′′) ,
I9(d) = e−
q2
2 (s2−s1)
S
∫
0
ds′′
S
∫
s2
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s2−s1)2
(s′+s′′) ,
I10(d) = e−
q2
2 (s2−s1)
S
∫
0
ds′′
S
∫
0
ds′
(
s′′+ s′
)−d/2
,
I11(d) = e−
q2
2 (s2−s1)
S
∫
0
ds′′
s ′′
∫
0
ds′
(
s′′− s′
)−d/2
,
I12(d) = e−
q2
2 (s2+s1)
S
∫
s1
ds′′
s ′′
∫
s1
ds′
(
s′′− s′
)−d/2
,
I13(d) = e−
q2
2 (s2+s1)
S
∫
s1
ds′′
s1
∫
0
ds′
(
s′′− s′
)−d/2
e
q2
2
(s1−s′)2
(s′′−s′) ,
I14(d) = e−
q2
2 (s2+s1)
s1
∫
0
ds′′
s ′′
∫
0
ds′
(
s′′− s′
)−d/2
e
q2
2 (s ′′−s ′),
I15(d) = e−
q2
2 (s2+s1)
S
∫
s2
ds′′
S
∫
s1
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s1+s2)2
(s′′+s′) ,
I16(d) = e−
q2
2 (s2+s1)
s2
∫
0
ds′′
S
∫
s1
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s1+s′′)2
(s′′+s′) ,
I17(d) = e−
q2
2 (s2+s1)
s2
∫
0
ds′′
s1
∫
0
ds′
(
s′′+ s′
)−d/2
e
q2
2 (s ′′+s ′),
I18(d) = e−
q2
2 (s2+s1)
S
∫
0
ds′′
s2
∫
0
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s′ )2
(s′′+s′) ,
I19(d) = e−
q2
2 (s2+s1)
S
∫
s2
ds′′
S
∫
0
ds′
(
s′′+ s′
)−d/2
e
q2
2
(s2)2
(s′′+s′) ,
I20(d) = e−
q2
2 (s2+s1)
S
∫
0
ds′′
s ′′
∫
0
ds′
(
s′′− s′
)−d/2
,
I21(d) = e−
q2
2 (s2+s1)
S
∫
0
ds′′
S
∫
0
ds′
(
s′′+ s′
)−d/2
.
33603-15
V. Blavatska, C. von Ferber, Yu. Holovatch
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33603-16
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http://dx.doi.org/10.1021/ma00134a026
http://dx.doi.org/10.1021/ma00029a024
http://dx.doi.org/10.1021/ma00158a050
http://dx.doi.org/10.1021/ma00172a035
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http://dx.doi.org/10.1002/9780470141533.ch6
Disorder effects on the static scattering function of star branched polymers
Вплив безладу на статичну функцiю розсiяння зiркових
полiмерiв
В. Блавацька1 , К. фон Фербер2,3, Ю. Головач1
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, Україна
2 Дослiдницький центр прикладної математики, Унiверситет Ковентрi, CV1 5FB Ковентрi, Великобританiя
3 Iнститут теоретичної фiзики II, Унiверситет iм. Гайнрiха Гайне, D-40225 Дюссельдорф, Нiмеччина
Представлено аналiз впливу структурного безладу на статичну функцiю розсiяння f -гiлкового зiркового
полiмера у d -вимiрному просторi. Розглянуто модель зiркового полiмера у хорошому розчиннику в при-
сутностi структурних дефектiв, скорельованих на великих вiддалях r згiдно степеневого закону ∼ r−a .
Зокрема, ми цiкавимось вiдношенням g ( f ) iнтенсивностей розсiяння зiркового та лiнiйного полiмерiв
однакової молекулярної маси, що є унiверсальною, експериментально спостережуваною величиною.
Ми застосовуємо метод прямого полiмерного перенормування i використовуємо подвiйний ε = 4−d ,
δ= 4−a-розклад. Знайдено зростання величини g ( f ) iз зростанням параметра δ. Таким чином, зростан-
ня кореляцiй безладу приводить до зменшення вiдмiнностi мiж розмiром зiркових та лiнiйних полiмерiв
iз однаковою молекулярною вагою.
Ключовi слова: полiмери, структурний безлад, унiверсальнiсть, ренормалiзацiйна група
33603-17
Introduction
The model
The method
Results
Conclusions and outlook
|