Conformational properties of polymers in anisotropic environments
We analyze the conformational properties of polymer macromolecules in solutions in presence of extended structural obstacles of (fractal) dimension ϵd causing the anisotropy of environment. Applying the pruned-enriched Rosenbluth method (PERM), we obtain numerical estimates for scaling exponents and...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1534842019-06-15T01:27:18Z Conformational properties of polymers in anisotropic environments Haydukivska, K. Blavatska, V. We analyze the conformational properties of polymer macromolecules in solutions in presence of extended structural obstacles of (fractal) dimension ϵd causing the anisotropy of environment. Applying the pruned-enriched Rosenbluth method (PERM), we obtain numerical estimates for scaling exponents and universal shape parameters of polymers in such environments for a wide range 0<ϵd<2 in space dimension d=3. An analytical description of the model is developed within the des Cloizeaux direct polymer renormalization scheme. Both numerical and analytical studies qualitatively confirm the existence of two characteristic length scales of polymer chain in directions parallel and perpendicular to the extended defects. Проаналiзовано конформацiйнi властивостi полiмерних макромолекул у розчинах в присутностi протяжних структурних домiшок (фрактальної) вимiрностi εd , що спричиняють просторову анiзотропiю. Застосовуючи збiднено-збагачений алгоритм Розенблюта (PERM), отримано чисельнi оцiнки для скейлiнгових показникiв та унiверсальних характеристик форми полiмерiв у таких середовищах у широкому спектрi 0 < εd < 2 при вимiрностi простору d = 3. Аналiтичний опис моделi розвинено в рамках пiдходу прямого полiмерного перенормування де Клуазо. Як чисельнi, так i аналiтичнi дослiдження пiдтверджують iснування двох характеристичних масштабiв довжини полiмерного ланцюжка в паралельному та перпендикулярному напрямках до протяжних дефектiв. 2014 Article Conformational properties of polymers in anisotropic environments / K. Haydukivska, V. Blavatska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23301:1-15. — Бібліогр.: 64 назв. — англ. 1607-324X DOI:10.5488/CMP.17.23301 arXiv:1407.3117 PACS: 36.20.-r, 89.75.Da, 64.60.ae, 07.05.Tp http://dspace.nbuv.gov.ua/handle/123456789/153484 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We analyze the conformational properties of polymer macromolecules in solutions in presence of extended structural obstacles of (fractal) dimension ϵd causing the anisotropy of environment. Applying the pruned-enriched Rosenbluth method (PERM), we obtain numerical estimates for scaling exponents and universal shape parameters of polymers in such environments for a wide range 0<ϵd<2 in space dimension d=3. An analytical description of the model is developed within the des Cloizeaux direct polymer renormalization scheme. Both numerical and analytical studies qualitatively confirm the existence of two characteristic length scales of polymer chain in directions parallel and perpendicular to the extended defects. |
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Haydukivska, K. Blavatska, V. |
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Haydukivska, K. Blavatska, V. Conformational properties of polymers in anisotropic environments Condensed Matter Physics |
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Haydukivska, K. Blavatska, V. |
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Haydukivska, K. |
| title |
Conformational properties of polymers in anisotropic environments |
| title_short |
Conformational properties of polymers in anisotropic environments |
| title_full |
Conformational properties of polymers in anisotropic environments |
| title_fullStr |
Conformational properties of polymers in anisotropic environments |
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Conformational properties of polymers in anisotropic environments |
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conformational properties of polymers in anisotropic environments |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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| citation_txt |
Conformational properties of polymers in anisotropic
environments / K. Haydukivska, V. Blavatska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23301:1-15. — Бібліогр.: 64 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT haydukivskak conformationalpropertiesofpolymersinanisotropicenvironments AT blavatskav conformationalpropertiesofpolymersinanisotropicenvironments |
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2025-07-14T04:37:43Z |
| last_indexed |
2025-07-14T04:37:43Z |
| _version_ |
1837595742886166528 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23301: 1–15
DOI: 10.5488/CMP.17.23301
http://www.icmp.lviv.ua/journal
Conformational properties of polymers in anisotropic
environments
K. Haydukivska, V. Blavatska
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitski St., 79011 Lviv, Ukraine
Received November 28, 2013, in final form April 21, 2014
We analyze the conformational properties of polymer macromolecules in solutions in presence of extended
structural obstacles of (fractal) dimension εd causing the anisotropy of environment. Applying the pruned-enriched Rosenbluth method (PERM), we obtain numerical estimates for scaling exponents and universal shape
parameters of polymers in such environments for a wide range 0 < εd < 2 in space dimension d = 3. An ana-
lytical description of the model is developed within the des Cloizeaux direct polymer renormalization scheme.
Both numerical and analytical studies qualitatively confirm the existence of two characteristic length scales of
polymer chain in directions parallel and perpendicular to the extended defects.
Key words: polymers, scaling, disorder, renormalization group, computer simulations
PACS: 36.20.-r, 89.75.Da, 64.60.ae, 07.05.Tp
1. Introduction
Many physical objects are characterized by anisotropy of structure: real magnetic crystals often con-
tain extended defects in the form of linear dislocations, disclinations or planar regions of different phases
[1–5]; in polymer systems, the understanding of the behavior of macromolecules in solutions having spa-
tial anisotropy caused by the presence of fibrous obstacles is of great importance, e.g., in gels [6], intra-
and extracellular environment [7, 8], or in the vicinity of planes (membranes) [9].
The analytical description of crystalline materials with extended defects attracts a lot of interest [1–5,
10–14]. In particular, Dorogovtsev [1] proposed the model of a d -dimensional spin system with quenched
nonmagnetic defects in the form of εd -dimensional objects, which are randomly distributed over theremaining d − εd dimensions. The anisotropy of the system brings about two different characteristiclength scales (correlation lengths ξ|| and ξ⊥), reflecting the macroscopic properties of the system alongthe directions “parallel” to the εd -dimensional defect and along the “perpendicular” directions:
ξ|| ∼ |t |−ν|| , ξ⊥ ∼ |t |−ν⊥ , (1.1)
where t is the reduced distance to the critical temperature t = (T −Tc)/Tc, ν⊥ and ν|| are universal criticalexponents.
A number of conformational properties of long flexible polymer chains in solutions can also be de-
scribed within the critical exponent formalism: for example, the averaged mean-squared distance be-
tween two ends of a chain obeys the scaling law:
〈R2〉 ∼ N 2ν, (1.2)
where N is the number of monomers in the chain and ν is an universal quantity that does not depend on
chemical properties of the macromolecule, but only on space dimension d (e.g., the phenomenological
Flory theory [15] gives ν(d) = 3/(d +2)). Thus, for polymers in d = 1-dimensional space, this exponent
takes on the maximal value of 1 (completely stretched chain), in d = 2, one restores the exact value
© K. Haydukivska, V. Blavatska, 2014 23301-1
http://dx.doi.org/10.5488/CMP.17.23301
http://www.icmp.lviv.ua/journal
K. Haydukivska, V. Blavatska
3/4 [16] and in the 3-dimensional case, the Flory theory nicely agrees, e.g., with the analytical result
ν= 0.5882±0.0011 [17]. For space dimension d Ê 4, the polymer behaves as an idealized Gaussian chain
with ν = 1/2. The relation of the polymer size exponent (1.2) to the correlation length critical index of
the m-component spin vector model in the formal limit m → 0 was provided by P.-G. de Gennes (the
well-known de Gennes limit [15]).
Note that the Flory theory is applicable only in the case of polymers in a pure environment, but
in reality most of the solutions contain impurities (obstacles), that interact with the macromolecules.
These obstacles can be very small or penetrate through the whole space; randomly distributed or obey-
ing some correlations on amesoscopic scale [18, 19]. It was shown [20–22] that in the environment having
a weak concentration of quenched point-like obstacles, themacromolecules behave like in pure solutions,
namely the value of the critical exponent ν in (1.2) is the same as in the idealized case of a pure solvent.
Only when the concentration of defects is close to the percolation threshold where an incipient percola-
tion cluster of fractal structure emerges in the system, the scaling properties of polymers are modified
in a non-trivial way [20–40]. It is appropriate to mention that according to a simple generalization of the
Flory formula ν(d f ) = 3/(d f +2) [21], there should be a different behavior for polymers on spaces with
fractal dimension d f < d , e.g., on percolation clusters.
The conformational properties are also influenced in a non-trivial way when the position of one ob-
stacle particle affects the other, i.e., there are correlations in the spatial distribution of impurities. In
particular, these correlations often express themselves by a power law behavior ∼ |r |−a [41] with a < d ,
where r is the distance between two obstacles. This type of disorder has a direct interpretation for integer
values of a, namely, the cases a = d −1 (a = d −2) describe straight lines (planes) of impurities of random
orientation, whereas non-integer values of a are interpreted in terms of impurities organized in fractal
structures. This type of disorder leads to a new universality class of polymers [42].
Herein above we were speaking about polymers in disordered but isotropic environment. It means
that all the observable statistical properties of the molecules are the same when explored in different
spatial directions. An interesting question concerns a quantitative change of these properties when there
are some selected directions in the system: the case of spatial anisotropy. In this concern, a widely dis-
cussedmodel of directed self-avoiding walks (DSAW) [43] may describe the properties of macromolecules
in an applied external field. This causes an elongation of the molecule along the field direction [44, 45]
and leads to the existence of two characteristic length scales in parallel and perpendicular directions.
The anisotropic properties of the environment can also be caused by the presence of extended obsta-
cles correlated in εd -dimensions, similar to those discussed previously for the spin systems. These maybe ordered colloid particles, gel fibers or biological species in a cell environment. In these systems we
should also expect a different behavior along the chosen direction and perpendicular to it. Baumgartner
and Muthukumar [46] discussed a model of polymer chains in an environment having impurities in the
form of absorbing parallel cylinders. They predicted that the polymer chain is elongated in the direction
parallel to the cylinders and, correspondingly, the parallel component of the end-to-end distance is gov-
erned by an exponent ν|| = 1, whereas in normal direction there will be no dependence of the end-to-end
distance component on the number of monomers at all (exponent ν⊥ = 0). This gives a reason to expect
an anisotropic behavior for polymers in solution having impurities correlated in εd dimensions [47].In the present paper, we study the conformational properties of polymers in the environment where
an anisotropy is caused by the presence of εd -dimensional defects of parallel orientation, which arerandomly distributed in the remaining d − εd dimensions, both using numerical simulations based ona discrete lattice model (Section II) and an analytical description within des Cloizeaux direct polymer
renormalization scheme (Section III). We end up by giving conclusions and an outlook.
2. Numerical studies
2.1. The model
Our goal is to investigate the conformational properties of polymers in anisotropic environments in
the presence of obstacles that are ordered in some subspace. For this purpose, we choose a lattice model
of a long flexible polymer chain— the model of self-avoiding random walks (SAW), which is established
23301-2
Conformational properties of polymers in anisotropic environments
Figure 1. (Color online) Schematic representation of polymer chain in the environment having structural
obstacles in the form of lines (a), partially penetrable lines (b) and partially penetrable planes (c).
to perfectly capture the universal properties of polymers in good solvent with the excluded volume effect.
We deal with the cubic lattice since it is known that universal properties do not depend on the lattice type
[51] and the cubic one is the most simple and easy to realize.
The simplest types of extended obstacles that can be chosen for our purposes are spacial objects in
the form of lines (εd = 1) of parallel orientation (see figure 1 (a)), spreading throughout the lattice in
some chosen direction, since they should obviously lead to a different behavior in directions parallel and
perpendicular to them. The case of homogeneous planes is not of interest since they divide the space into
small restricted regions, and thus the problem is reduced to that of polymers in confined geometries.
We start with an x y -plane of a lattice with concentration c of randomly chosen sites containing point-
like defects and then we build lines perpendicular to this plane in z-direction through chosen sites. In
figure 2 one can see a projection of the SAW trajectory on this plane for different concentrations of lines.
As one can see at high concentrations c of such obstacles, the SAW trajectory appears to collapse in small
restricted regions. In this case, a long polymer chain will behave like a one-dimensional rod, extended
Figure 2. (Color online) Projections of SAW trajectories on the x y -plane of a lattice at different concentra-
tions c of impurity lines extended throughout the system in z direction.
23301-3
K. Haydukivska, V. Blavatska
in the direction parallel to the lines of defects. Thus, we expect a crossover to an extended regime for an
increasing defect concentration.
We also consider more interesting situations, namely a set of partially penetrable lines (figure 1 (b))
and planes (figure 1 (c)). To this end, with some fixed probability p , we randomly choose the sites on the
constructed lines (planes) of defects and treat them as “open” (allowed for SAW trajectory). Note that one
can, roughly speaking, treat these objects as fractals. For example, the fractal (cluster) dimension of such
a line with concentration p of the lacking sites can be estimated from the well-known relation between
the linear size of an object and “the number of particles”: εd � ln[(1−p) ·L]/ ln(L) (here, L is the length of
a line). Indeed, at p = 0, one has a Euclidian line (like in figure 1 (a)) and simply restores εd = 1, whereas
with increasing p , the line can be treated as a set of disconnected sites (points) and the fractal dimension
of this object gradually tends to 0.
2.2. The method
For our purposes we use the Pruned-Enriched Rosenbluth Method (PERM) [52]. It is based on the
original Rosenbluth-Rosenbluth algorithm of growing chains with population control parameters [53].
On each step n, the chain has a weightWn given by:
Wn =
n∏
i=1
mi , (2.1)
wheremi is the number of possibilities to perform the next step, which varies from 0 to 2d −1 due to the
fact that the chain may not cross itself. This value is also reduced by the presence of impurities.
When the chain of total length N is constructed, a new one is started from the same starting point,
until the desired number of chain configurations are obtained. In this manner, all observables should be
averaged over an ensemble of different chain configurationsM :
〈(. . .)〉 = 1
ZN
M∑
k=1
W k
N (. . .) , ZN =
M∑
k=1
W k
N , (2.2)
here,W k
N is the weight of the k-th configuration of the N -step trajectory.
It is also necessary to average over different configurations of disorder:
(. . .) = 1
p
p∑
k=1
(. . .), (2.3)
where p is the number of replicas (the number of different realizations of the disorder). For our purposes,
we consider about 105 chains for each of the 400 replicas.
Weight fluctuations are reduced by using population control (pruning and enrichment). It means that
with probability of 1/2 we reject the chains having low weight and enrich the statistics by replication in-
creasing the number of high weighted configurations. To do this, we use lower and upper bound weights,
that are updated at each step according to [54–56]: W >
n = C (Zn/Z1)(cn/c1)2 and W <
n = 0.2W >
n , where
cn is the number of chains of length n created, and the parameter C controls the pruning-enrichment
statistics; it is chosen in such a way to allow one to receive in average 10 chains of total length N in each
tour.
2.3. Results
We concentrate on the conformational properties of long flexible polymer chains, in particular, in the
critical exponents governing the behavior of the effective linear size of the macromolecule with respect
to the number of monomers (1.2). Let us note that in the anisotropic case we expect to have two different
exponents rather than one:
〈R2
||〉 = (zN − z0)2 ∼ N 2ν|| ,
〈R2
⊥〉 = (xN −x0)2 + (yN − y0)2 ∼ N 2ν⊥ , (2.4)
23301-4
Conformational properties of polymers in anisotropic environments
Figure 3. (Color online) Parallel (a) and perpendicular (b) components of the end-to-end distance of SAW
as a function of chain length in a double logarithmic scale at various concentrations of defects in the form
of parallel lines.
so that R2 = R2
‖ +R2
⊥. The parallel and perpendicular components are expected to behave in a differentway [46].
We start by considering the case of the anisotropy caused by the presence of obstacles in the form of
impenetrable parallel lines [47]. In this case, we took chains up to N = 300monomers and find the com-
ponents of the end-to-end distance vector, performing double averaging according to (2.3). The results
are presented in figure 3. It is clearly observed that the behavior of the two components differ from each
other: the parallel component grows with an increase of concentration of obstacles, whereas the perpen-
dicular component collapses indicating the stretching of the polymer chain in longitudinal direction. As
one can see, there is a crossover between the two types of behavior: one for short chains (less than 40
monomers) and then the other one for long polymers. We expect this crossover to take place when the
averaged end-to-end distance R2 of the SAW trajectory is comparable to the distance between the lines
of impurities. This is similar to the polymer behavior in a restricted space, for example a cylinder, when
short chains (with end-to-end distance smaller than the radius of cylinder) behave like 3-dimensional,
and long chains as 1-dimensional [15]. In our case, we observe a crossover to such a behavior when the
concentration of lines that penetrate the system is close to a percolative concentration of point-like de-
fects on a simple square lattice. At smaller concentrations, applying the least-square fitting of the data
observed, we obtain the estimates for the two critical exponents ν‖ and ν⊥ (see figure 4), which coincide
0,0 0,2 0,4 0,6 0,8 1,0
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
C
rit
ic
al
e
xp
on
en
ts
Concentrations of defects
perp
Figure 4. (Color online) Critical exponents ν‖ and ν⊥ governing the components of the end-to-end dis-tance of SAWs, parallel and perpendicular to the defects in the form of parallel lines, as a function of the
concentration of defects.
23301-5
K. Haydukivska, V. Blavatska
at c = 0, where we restore the corresponding value of the SAW exponent on a pure lattice. As one can
see, the exponent governing the scaling of a parallel component of the end-to-end distance is larger than
the pure one and gradually reaches the maximal value of 1 with an increasing concentration of disorder,
while ν⊥ is smaller and gradually tends to zero. It gives us the right to say that polymers in anisotropicspace are more elongated than polymers in isotropic environments.
Let us check the fact of possible elongation by analyzing the shape properties. All information con-
cerning the shape measure of a chain is given by the gyration tensor with its components defined by:
Qαβ =
1
N 2
N∑
i=1
N∑
i=1
(xαi −xαj )(xβi −xβj ), α,β= 1, . . . ,3, (2.5)
where xα, xβ are the components of the position vector ~ri of the i -th monomer. Eigenvalues of this tensor
provide a full information on the shape of the polymer. To receive these in simulations we need to solve a
cubic equation for each of the 105 chains. Thus, we are interested in calculating the rotationally invariant
shape characteristics, such as asphericity and prolateness, defined in d = 3 as combinations of gyration
tensor components according to [48–50]:
A3 = 3
2
TrQ̂2
(TrQ)2 , S3 = 27
detQ̂2
(TrQ)2 , (2.6)
with Q̂ =Q − ÎTrQ/3 (Î being a unity matrix). Asphericity is normalized in such a way that it changes the
value from 0 for spherical configuration to 1 for a completely stretched rod-like structure. Prolateness
obeys the inequality: −0.25 É S É 2, it is negative for oblate configurations and positive for prolate ones;
a value 2 corresponds to a rod-like (completely prolate) state.
In figures 5 and 6 we present our data for 〈A3〉 and 〈S3〉, averaged over realizations of disorder atvarious fixed concentrations of defects. At c = 0, in both cases we restore the corresponding values on
a pure lattice. One can easily see in figures 5 (a) and 6 (a) that both quantities are growing with an
increasing defect concentration and gradually reach the corresponding values of rod-like structures.
Next, we consider an interesting situation, where point-like defects in the lattice are aligned in some
particular direction (say, z) forming partially penetrable lines (see figure 1 (b)). To investigate the influ-
ence of such a type of space anisotropy on the conformational properties of flexible polymer chains, we
randomly choose some concentration p of sites on lines of defects (constructed as described previously),
and treat them as “open” (allowed to SAW trajectory). As a result, we obtain aligned fractal-like objects
with dimension 0 < εd < 1. In such problems, we have two parameters: the concentration c of obstacles
in the form of parallel lines, and the probability p of a SAW trajectory to penetrate through this line.
Figure 5. (Color online) Asphericity of SAW trajectories as a function of the concentration of defects at var-
ious numbers of monomers (a). Asphericity of SAW trajectories as a function of the number of monomers
at various fixed concentrations of defects (b).
23301-6
Conformational properties of polymers in anisotropic environments
Figure 6. (Color online) Prolateness of SAW trajectories as a function of the concentration of defects at var-
ious numbers of monomers (a). Prolateness of SAW trajectories as a function of the number of monomers
at various fixed concentrations of defects (b).
Performing simulations for chains up to N = 100 steps and applying the least-square fits for the data
obtained for parallel and perpendicular components of the end-to-end distances of polymer chains, we
received the estimates for critical exponents ν‖ and ν⊥ (see figure 7 (a)) as functions of these two param-eters. Again, the exponent governing the scaling of a parallel component of the end-to-end distance is
larger than the pure one and gradually reaches the maximal value of 1, while the other one is lower and
gradually tends to zero. This tendency is kept even at a rather high probability of a growing trajectory
to penetrate the line (up to the concentration p of “open” sites close to a critical percolation concentra-
tion). When p is close to 1, the spatial anisotropy disappears and both exponents gradually reach the
corresponding value of the pure lattice case.
Finally, we consider another possible model of anisotropic environment caused by the presence of
structural defects in the form of partially penetrable planes of parallel orientation (see figure 1 (c)). We
start with homogeneous planes of concentration c , randomly distributed in z-direction, and again ran-
domly choose some concentration p of sites on these planes and treat them as “open”(allowed to SAW
trajectory). As a result, we obtain fractal-like objects having dimension 1 < εd < 2. Performing simula-
Figure 7. (Color online) Critical exponents ν‖ and ν⊥ of SAW on a lattice with partially penetrable linesof defects as a function of the concentration of lines at various probabilities to penetrate through these
lines (a). Critical exponents ν‖ and ν⊥ of SAW on a lattice with partially penetrable planes of defects as afunction of probabilities to penetrate these lines at various concentrations of planes (b).
23301-7
K. Haydukivska, V. Blavatska
tions for chains up to N = 100 steps and applying the least-square fits for data obtained for parallel and
perpendicular components of the end-to-end distances of polymer chains, we receive estimates for the
critical exponents ν‖ and ν⊥ (see figure 7 (b)) as functions of these two parameters. The exponent gov-erning the scaling of the parallel component of the end-to-end distance gradually changes from the value
on three-dimensional pure lattice (at c = 0) to that in two dimensions with a growing concentration of
impurity planes. This can be treated as a crossover to a restricted geometry regime of polymers confined
between two planes. The exponent ν⊥ gradually tends to zero.
3. Analytical approach
3.1. The model
We deal with flexible polymers in an environment with extended impurities correlated in εd -dimen-sions and randomly distributed in the remaining space. Let us start with a continuous x‘model, where a
polymer chain is presented as a path of length (or surface) S parameterized by~r (s), with s = 0. . .S [57].
An effective Hamiltonian of the system is given by:
H = 1
2
S∫
0
(
d~r (s)
ds
)2
ds +u
S∫
0
ds′
s′∫
0
ds′′δ
(
~r (s′)−~r (s′′)
)
ds +
S∫
0
V (~r (s))ds. (3.1)
Here, the first term describes the chain connectivity, the second term reflects the short-range repulsion
between monomers due to the excluded volume effect with coupling constant u, and the last term arises
due to the interaction between the monomers of the polymer chain and the structural defects in the
environment given by potential V . We work in the formalism of partition functions:
Z =
∫
D~r e−H ,
where ∫ D~r denotes integration over different paths.
Dealing with systems that display randomness of structure, one usually encounters two types of en-
semble averaging, treated as annealed and quenched disorder [58, 59]. In general, the critical behavior of
disordered systems with annealed and quenched averaging is quite different. However, for the polymer
systems it has been shown [28, 60–63], that the distinction between quenched and annealed averages
for an infinitely long single polymer chain is negligible, and in performing analytical calculations for
quenched polymer systems one may restrict the problem to the simpler case of annealed averaging. In
this paper, we deal with annealed averaging over disorder because it is technically simpler. After av-
eraging the partition sum over realizations of disorder and including only up to the second moment of
cumulant expansion, we receive:
exp
S∫
0
V (~r (s′))ds′
= exp
S∫
0
V (~r (s′))ds′
×exp
1
2
S∫
0
s′∫
0
V (~r (s′))V (~r (s′′))−V (~r (s′))
2
ds′ds′′
,
where V (~r (s′)) gives the average concentration of impurities ρ, and for the second moment we assume
[1, 46]:
V (~r (s′))V (~r (s′′)) = v δd−εd
(
rd−εd
(s′)− rd−εd
(s′′)
)
, (3.2)
which reflects the fact that the impurities are correlated in εd dimensions and uncorrelated in the re-maining space. Omitting the terms ρS+ 1
2ρ
2S2, which give a trivial constant shift, we obtain an averaged
23301-8
Conformational properties of polymers in anisotropic environments
partition function Z = ∫
d~r e−Heff with an effective Hamiltonian:
Heff = 1
2
S∫
0
(
d~r (S)
ds
)2
ds +u
S∫
0
ds′
s′∫
0
ds′′δ
(
~r (s′)−~r (s′′)
)
ds
−v
S∫
0
ds′
s′∫
0
ds′′δd−εd
(
rd−εd
(s′)− rd−εd
(s′′)
)
. (3.3)
Note that the last term in (3.3) describes an effective attractive interaction between monomers in the
direction perpendicular to the extended obstacles governed by a coupling constant v .
3.2. The method
Within the model, all the parameters depend on the polymer area S and on dimensionless coupling
constants {za} = a(2π)−(da )/2S4−(da )/2 (here da is dimension of coupling constant a) in a way that when
{za} = 0, one restores the case of idealized Gaussian chain without any interactions between monomers.
To calculate the universal properties of the model we need to find such values of parameters that lead
to physical values of the universal characteristics. For that reason, we use the direct renormalization
technique proposed by des Cloiseaux [51].
Within this scheme, renormalized coupling constants are defined by:
ga({za}) =−[χ1({za})]−4Zza (S,S)[2πχ0{za}S]−(2−εa /2) , (3.4)
where εa is the deviation from the upper space dimension for the coupling constant za . χ0 = R2/Sd is the
swelling factor that governs the behavior of the end-to-end distance of the polymer in solution. It can be
presented as a perturbation theory series over the coupling constants:
χ0({za}) = 1+∑
a
za · fa(da), (3.5)
where the first term (1) corresponds to an ideal Gaussian chain and the others give corrections caused by
interactions in the system. fa(da) is the factor that depends only on the dimension of the corresponding
coupling constant. This factor allows us to estimate the critical exponent ν using the relation:
2ν−1 = S
∂
∂S
χ0({za}).
The factor χ1 is connected with the partition function Z (S)/Z 0(S) = [χ1({za})]2. Here, Z 0(S) is the parti-
tion function of an idealized Gaussian chain, Zza (S,S) is the partition function of two interacting polymer
chains.
Renormalized coupling constants given by the equation (3.4) tend to constant values or the so-called
fixed points as the polymer area tends to infinity. The fixed points are defined as common zeros of the
flow equations:
Wa = 2S
∂
∂S
z∗
a ({za}). (3.6)
To find the fixed points of the model, one need to express za in terms of ga and find the common zeros of(3.6) and then choose those that are stable and physical in the region of interest for the parameters εa .
3.3. Results
We start with the restricted partition function
Z̃ (~k,S) =
〈
ei~k(~r (S)−~r (0)
〉
, (3.7)
23301-9
K. Haydukivska, V. Blavatska
Figure 8. Diagrammatic presentation of the contributions to the restricted partition function (3.7) up to
the first order of perturbation theory expansion in the coupling constants.
where 〈(. . .)〉 means averaging with the hamiltonian (3.3). We consider the evaluation of the expression
(3.7) by performing the perturbation theory expansion in coupling constants u, v . The terms in this ex-
pansion can be presented diagrammatically as shown in figure 8. The first diagram describes the ze-
roth order approach corresponding to the idealized Gaussian chain without any interaction between the
monomers. Solid line on the diagrams presents the polymer chain, the dashed line describes the excluded
volume interaction between monomers governed by the coupling u, and the wavy line presents the at-
tractive interaction caused by the presence of the impurities governed by the coupling v . Let us consider
the expressions corresponding to the second and third diagram:
D2 =−u
∫
dd~q
S∫
0
ds′
s′∫
0
ds′′e−
q2
2 (s′′−s′)e−
k2
2 (S−s′′+s′),
D3 = v
∫
dd−εd~q
S∫
0
ds′
s′∫
0
ds′′e−
q2
2 (s′′−s′)e−
k2
2 (S−s′′+s′). (3.8)
It is necessary to point out that in the expression for D3, the integration is performed only in subspace
d − εd due to the fact that the interaction v acts only in this subspace. Using the Poisson formula to
integrate over the wave vector ~q we receive:
D2 =−u
1
(2π)d/2
S∫
0
ds′
s′∫
0
ds′′(s′′− s′)
d
2 e−
k2
2 (S−s′′+s′),
D3 = v
1
(2π)(d−εd )/2
S∫
0
ds′
s′∫
0
ds′′(s′′− s′)
d−εd
2 e−
k2
2 (S−s′′+s′). (3.9)
Expanding the exponents over~k and then integrating over areas we finally receive:
D2 =−zu
1(
1− d
2
)(
2− d
2
) + zu
k2S
2
2(
1− d
2
)(
2− d
2
)(
3− d
2
) ,
D3 = zv
1(
1− d−εd
2
)(
2− d−εd
2
) − zv
k2
d−εd
S
2
2(
1− d−εd
2
)(
2− d−εd
2
)(
3− d−εd
2
) , (3.10)
where zu = u(2π)−d/2S2−d/2 and zv = v(2π)−(d−εd )/2S2−(d−εd )/2 are dimensionless coupling constants.
Collecting all contributions from the considered diagrams one receives an expression for the partition
function of the model by keeping terms that do not depend on~k:
Z (S) = 1− zu(
1− d
2
)(
2− d
2
) − zv(
1− d−εd
2
)(
2− d−εd
2
) .
The expressions for the components of the end-to-end distance of the polymer chain can be estimated
using the identities:
〈R2
d−εd
〉 =−2
1
Z (S)
[
d
d~kd−εd
Z̃ (~k,S)
]
~k=0
, 〈R2
εd
〉 =−2
1
Z (S)
[
d
d~kεd
Z̃ (~k,S)
]
~k=0
.
23301-10
Conformational properties of polymers in anisotropic environments
Figure 9. Diagramatic presentation of the contributions into the partition function Z (S,S) of two interact-
ing polymer chains up to the second order of expansion in the coupling constants.
We distinguish between the components in subspaces εd and d −εd , corresponding to components of theend-to-end distance in directions parallel and perpendicular to extended defects:
〈R2
d−εd
〉 = S(d −εd )
1+ zu(
2− d
2
)(
3− d
2
) − zv(
2− d−εd
2
)(
3− d−εd
2
)
, (3.11)
〈R2
εd
〉 = Sεd
1+ zu(
2− d
2
)(
3− d
2
)
. (3.12)
References (3.11) and (3.12) confirm the existence of two characteristic lengths for polymers in aniso-
tropic environments. The presence of extended defects makes the polymer radius shrink in transverse
direction due to the attractive interactions between monomers governed by the coupling v , whereas in
parallel direction, the increase of the effect of repulsive interactions (as consequence of the increase of
monomer density) is responsible for the elongation of the polymer chains.
Calculating contributions to the partition function of two interacting polymer chains one may use a
diagrammatic representation (see figure 9). Note that only those diagrams are taken into account which
contain at least one interaction line. The first few diagrams, those with one interaction acting between
two polymers (G1−G4), can be gathered and presented as −uS2Z (S)2 − vS2Z (S)2. Performing the di-
mensional analysis of the contributions, produced by different diagrams, we find two distinct classes
of graphs. The first class of graphs produces terms which behave like [S]
4−d
2 , the sum of all such terms
gives contributions into the function denoted by Zu(S,S). The diagrams of the second class behave like
[S]
4−d+εd
2 and thus give contributions into the function Zv (S,S). As a result, the “two polymer function”
can be presented in the form: Z (S,S) = Zu(S,S)+ Zv (S,S), where Zu(S,S) and Zv (S,S) are given by the
expressions:
Zu(S,S) = −uS2
[
1+2
zu(
1− d
2
)(
2− d
2
) −2
zv(
1− d−εd
2
)(
2− d−εd
2
)
+2zu
24−d/2 −10+d(
1− d
2
)(
2− d
2
)(
3− d
2
)(
4− d
2
) −4zv
24−(d−εd )/2 −10+d(
1− d−εd
2
)(
2− d−εd
2
)(
3− d−εd
2
)(
4− d−εd
2
)]
,
23301-11
K. Haydukivska, V. Blavatska
Zv (S,S) = vS2
[
1+2
zu(
1− d
2
)(
2− d
2
) −2
zv(
1− d−εd
2
)(
2− d−εd
2
)
−2zv
24−(d−εd )/2 −10+d(
1− d−εd
2
)(
2− d−εd
2
)(
3− d−εd
2
)(
4− d−εd
2
)]
. (3.13)
The swelling factor in our model reads:
χ0 = R2
Sd
= εd
d
R2
εd
+ d −εd
d
R2
d−εd
=
1+ zu(
2− d
2
)(
3− d
2
) + d −εd
d
zv(
2− d−εd
2
)(
3− d−εd
2
)
. (3.14)
The renormalized coupling constants can be presented in the form:
gu =χ−4
1 χ−2+ε/2
0 Zu(S,S),
gv =χ−4
1 χ−2+δ/2
0 Zv (S,S).
The corresponding flow equations read:
W [gu] = εgu −8g 2
u +12gu gv ,
W [gv ] =−δgv −8g 2
v +4gu gv ,
here, ε = 4−d , δ = ε+ εd . The coordinates of fixed points can be found as common zeros of functions
W [gu],W [gv ]:
g∗
u = 0, g∗
v = 0, (3.15)
g∗
u = ε/8, g∗
v = 0, (3.16)
g∗
u = 0, g∗
v =−δ/8, (3.17)
g∗
u = ε/2− (3/4)δ, g∗
v = ε/4−δ/2. (3.18)
The first fixed point describes the case of an idealized Gaussian chain without any interactions between
monomers. Expression (3.16) corresponds to the case of a polymer chain with short-range excluded vol-
ume interactions in a pure solvent. The fixed points (3.17) and (3.18) describe, correspondingly, the Gaus-
sian chain and the chain with excluded volume effect in the anisotropic environment. However, since
both of them are associated with attractive interactions between monomers due to the presence of de-
fects, these fixed points appear to be unstable in the physical region of the parameters (ε> 0 and εd > 0)
and thus cannot provide estimates of scaling exponents. Note that a similar problem of the absence of sta-
ble and physically accessible fixed points also exists in the case of uncorrelated point-like impurities [24].
The latter was solved by absorbing the interaction with disorder into the excluded volume interaction
due to a special symmetry [64]. However, this does not work in the present case of extended defects.
4. Conclusions
We analyzed the influence of anisotropy of the environment caused by the presence of impurities
correlated in εd dimensions, on conformational size and shape characteristics of long flexible polymerchains. The integer values of εd have direct physical interpretation and describe extended defects, e.g.,in the form of lines or planes of parallel orientation (εd = 1 or 2, correspondingly). In this case, it is
obvious that one should distinguish between two characteristic length scales, in directions parallel and
perpendicular to such extended defects. Non-integer values of εd may correspond to complex defects offractal nature.
23301-12
Conformational properties of polymers in anisotropic environments
Applying the numerical simulations based on the model of self-avoiding random walks on a regu-
lar cubic lattice, we considered three cases: impurities in the form of parallel lines (εd = 1), fractal-like
structures with 0 < εd < 1 (which can be treated as partially penetrable lines) and fractal-like structures
with 1 < εd < 2 (partially penetrable planes). In the first case, we found that components of the effective
linear size of polymer chain, that are either parallel or perpendicular to the lines of impurities, behave
differently and their scaling is governed by two distinct scaling exponents ν‖ and ν⊥ (see equation (2.4)).The exponent governing the scaling of a parallel component of the end-to-end distance gradually reaches
the maximal value of 1 with increasing of concentration of defects, while ν⊥ gradually tends to zero.Analyzing the influence of disorder in the form of partially penetrable lines on scaling properties of poly-
mers, we again found the existence of two distinct exponents ν‖ and ν⊥ (see figure 7 (a)). This tendency(and thus the anisotropy) surprisingly persists even at high probability of the polymer chain to penetrate
through such “line” (which corresponds to εd close to 0). Considering structural defects in the form of
partially penetrable planes of parallel orientation (see figure 1 (c)), we found that the exponent ν‖ gradu-ally changes from the value found earlier for the three-dimensional pure lattice to that in two dimensions
with growing concentration of impurity planes. This can be treated as a crossover to a restricted geome-
try regime of the polymer confined between two homogeneous planes. The exponent ν⊥ gradually tendsto zero.
Our analytical studies were performed within the frames of the direct polymer renormalization ap-
proach using the double ε, ε+εd expansion. In particular, we found expressions for the components ofthe end-to-end distance of polymer chain (3.11), (3.12). The presence of extended defects makes the poly-
mer radius shrink in transverse direction due to attractive interactions between monomers governed
by the coupling v , whereas in parallel direction the increase of the effect of repulsive interactions (as a
consequence of the increase of monomer density) is responsible for the elongation of the polymer chain.
We conclude that the presence of extended defects correlated in εd dimensions makes the polymer chainelongated in the direction parallel to these extended impurities, which confirms the existence of two
characteristic lengths for polymers in anisotropic environments.
Acknowledgements
This work was supported in part by the FP7 EU IRSES projects N269139 “Dynamics and Cooperative
Phenomena in Complex Physical and Biological Media” and N295302 “Statistical Physics in Diverse Real-
izations”.
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Conformational properties of polymers in anisotropic environments
Конформацiйнi властивостi полiмерiв в анiзотропних
середовищах
К. Гайдукiвська, В. Блавацька
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Проаналiзовано конформацiйнi властивостi полiмерних макромолекул у розчинах в присутностi протя-
жних структурних домiшок (фрактальної) вимiрностi εd ,що спричиняють просторову анiзотропiю. Засто-совуючи збiднено-збагачений алгоритм Розенблюта (PERM), отримано чисельнi оцiнки для скейлiнгових
показникiв та унiверсальних характеристик форми полiмерiв у таких середовищах у широкому спектрi
0 < εd < 2 при вимiрностi простору d = 3. Аналiтичний опис моделi розвинено в рамках пiдходу пря-
мого полiмерного перенормування де Клуазо. Як чисельнi, так i аналiтичнi дослiдження пiдтверджують
iснування двох характеристичних масштабiв довжини полiмерного ланцюжка в паралельному та пер-
пендикулярному напрямках до протяжних дефектiв.
Ключовi слова: полiмери, заморожений безлад, скейлiнг, ренормалiзацiйна група, чисельнi симуляцiї
23301-15
Introduction
Numerical studies
The model
The method
Results
Analytical approach
The model
The method
Results
Conclusions
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