Randomly charged polymers in porous environment
We study the conformational properties of charged polymers in a solvent in the presence of structural obstacles correlated according to a power law ~x-a. We work within the continuous representation of a model of linear chain considered as a random sequence of charges qi=± q₀. Such a model captures...
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nasplib_isofts_kiev_ua-123456789-1208242025-02-09T10:07:01Z Randomly charged polymers in porous environment Випадково зарядженi полiмери в пористому середовищi Blavatska, V. von Ferber, C. We study the conformational properties of charged polymers in a solvent in the presence of structural obstacles correlated according to a power law ~x-a. We work within the continuous representation of a model of linear chain considered as a random sequence of charges qi=± q₀. Such a model captures the properties of polyampholytes~-- heteropolymers comprising both positively and negatively charged monomers. We apply the direct polymer renormalization scheme and analyze the scaling behavior of charged polymers up to the first order of an ε=6-d, δ=4-a-expansion. Дослiджуються конформацiйнi властивостi заряджених полiмерiв в розчинi у присутностi структурних неоднорiдностей, скорельованих згiдно степеневого закону ∼ x−a. Використовується модель, в якiй полiмерний ланцюжок представлено як випадкову послiдовнiсть зарядiв qi = ±q₀. Така модель описує властивостi полiамфолiтiв – гетерополiмерiв, що мiстять як позитивно, так i негативно зарядженi групи моно-мерiв. Застосовується пiдхiд прямого полiмерного перенормування i аналiзується скейлiнгова поведiнка заряджених полiмерiв до першого порядку подвiйного ² = 6−d, δ = 4−a-розкладу. This work was supported in part by the FP7 EU IRSES project N269139 “Dynamics and Cooperative Phenomena in complex Physical and Biological Media”. 2013 Article Randomly charged polymers in porous environment / V. Blavatska, C. von Ferber // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 34601:1-6. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 61.25.hp, 11.10.Hi, 64.60.ae, 89.75.Da DOI:10.5488/CMP.16.34601 arXiv:1307.2878 https://nasplib.isofts.kiev.ua/handle/123456789/120824 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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We study the conformational properties of charged polymers in a solvent in the presence of structural obstacles correlated according to a power law ~x-a. We work within the continuous representation of a model of linear chain considered as a random sequence of charges qi=± q₀. Such a model captures the properties of polyampholytes~-- heteropolymers comprising both positively and negatively charged monomers. We apply the direct polymer renormalization scheme and analyze the scaling behavior of charged polymers up to the first order of an ε=6-d, δ=4-a-expansion. |
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Blavatska, V. von Ferber, C. Randomly charged polymers in porous environment Condensed Matter Physics |
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Blavatska, V. von Ferber, C. |
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Randomly charged polymers in porous environment |
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Randomly charged polymers in porous environment |
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Randomly charged polymers in porous environment |
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Randomly charged polymers in porous environment |
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Randomly charged polymers in porous environment |
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randomly charged polymers in porous environment |
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Randomly charged polymers in porous environment / V. Blavatska, C. von Ferber // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 34601:1-6. — Бібліогр.: 42 назв. — англ. |
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Condensed Matter Physics |
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AT blavatskav randomlychargedpolymersinporousenvironment AT vonferberc randomlychargedpolymersinporousenvironment AT blavatskav vipadkovozarâdženipolimerivporistomuseredoviŝi AT vonferberc vipadkovozarâdženipolimerivporistomuseredoviŝi |
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2025-11-25T15:46:11Z |
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Condensed Matter Physics, 2013, Vol. 16, No 3, 34601: 1–6
DOI: 10.5488/CMP.16.34601
http://www.icmp.lviv.ua/journal
Rapid Communication
Randomly charged polymers in porous environment
V. Blavatska1, C. von Ferber2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Applied Mathematics Research Centre, Coventry University, CV1 5FB Coventry, UK
Received June 11, 2013, in final form July 11, 2013
We study the conformational properties of charged polymers in a solvent in the presence of structural obstacles
correlated according to a power law ∼ x−a . We work within the continuous representation of a model of linear
chain considered as a random sequence of charges qi =±q0 . Such a model captures the properties of polyam-
pholytes – heteropolymers comprising both positively and negatively charged monomers. We apply the direct
polymer renormalization scheme and analyze the scaling behavior of charged polymers up to the first order of
an ǫ= 6−d , δ= 4−a-expansion.
Key words: polymers, quenched disorder, scaling, renormalization group
PACS: 61.25.hp, 11.10.Hi, 64.60.ae, 89.75.Da
1. Introduction
Many polymer macromolecules encountered in chemical and biological physics can be considered
as long flexible chains. The conformations of individual macromolecules are in general controlled by
the type of monomer-monomer interactions. In good solvents, where interactions between monomers
are mainly steric, N -monomer homogeneous polymer chains form coil-like structures with the mean-
squared end-to-end distance Re obeying a scaling law [1–3]:
〈R2
e 〉 ∼ N 2νcoil , (1.1)
with a universal exponent depending on space dimension d only [e.g., the phenomenological Flory theory
[1] gives νcoil(d) = 3/(d+2)]. Note that at d = 4, the intrachain steric interactions are rendered irrelevant,
and the polymer behaves like an idealized Gaussian chain with νGauss = 1/2.
The long-range nature of the electrostatic Coulomb interaction between charged monomers produces
more complicated effects on polymer conformations. Of particular interest in this context are polyam-
pholytes (PAs) [4] (for a recent review, see e.g.: [5]): heteropolymers comprising both positively and neg-
atively charged monomers. Examples of polyampholytes are proteins and synthetic copolymers bearing
acidic and basic repeat groups. The interaction between anionic and cationic groups leads to additional
complications in their physical behavior. These polymers usually dissolve only when there is a sufficient
amount of salt added, which screens the interactions between oppositely charged segments. The proper-
ties of polyampholytes are successfully capturedwithin the frames of a randomly charged polymermodel
[6, 7, 12]: a linear chain composed of a random sequence of N monomers carrying a charge ±q0 with a
fixed overall charge Q . It is established that the conformational properties of PAs strongly depend both
on Q and the quality of the solvent (and thus on temperature T ) [6–19]. If positive and negative charges
are nearly balanced (Q is small), the attractive Coulomb interaction dominates, and the polymer collapses
into a globular (sphere-like) state, as predicted by the Debye-Hückel theory [20]. For polyampholytes with
a considerable disbalance between positive and negative charges (Q > Qc ∼ q0N 1/2), Coulomb repul-
sion will be predominant. They are expected to attain the properties of polyelectrolytes (homogeneously
charged polymers) with a stretched configuration governed by the size exponent value ν≃ 1.
© V. Blavatska, C. von Ferber, 2013 34601-1
http://dx.doi.org/10.5488/CMP.16.34601
http://www.icmp.lviv.ua/journal
V. Blavatska, C. von Ferber
In polymer physics, comprehension of the behavior of macromolecules in the presence of structural
disorder is of great importance, e.g., in colloidal solutions [21], near microporous membranes [22], or in
the crowded environment of biological cells [23–25]. The density fluctuations of obstacles often lead to
spatial inhomogeneity and create pore spaces of fractal structure [26–30]. In the present study we ad-
dress a model where the structural obstacles of the environment are spatially correlated on a mesoscopic
scale [31]. Following reference [32], this case can be described by assuming the defects to be correlated
at large distances r according to a power law with a pair correlation function g (r ) ∼ r−a . Such a corre-
lation function describes the defects extended in space, e.g., the cases a = d −1 (a = d −2) correspond to
lines (planes) of defects of random orientation, whereas non-integer values of a can describe obstacles of
fractal structures (see [32–35] for further details). The impact of long-range-correlated disorder on con-
formational properties of polymer chains has been analyzed in previous works [33–35] by means of the
field-theoretical renormalization group (RG) approach. The question how the characteristics of a poly-
mer with long-range Coulomb interaction are effected by the presence of such a long-range-correlated
structural disorder remains, however, still unresolved.
In the present study, we aim to develop a direct renormalization group approach to analyze the con-
formational properties of a random charge model, proposed in references [6, 7, 12] in solution in the
presence of long-range-correlated structural obstacles. The layout of the paper is as follows. In the next
section, we develop a continuous chain representation of the model and discuss its general properties
in detail. In section 3, the direct renormalization procedure is described and the results for the scaling
properties of charged polymers are discussed. We close by giving conclusions and an outlook.
2. The model
Following reference [6, 7], we consider a flexible linear chain consisting of N monomers, each car-
rying a charge ±q0. If we restrict the ensemble of random charged sequences to have a fixed overall
non-zero charge Q , correlations within the positive and negative charge distributions are induced. De-
noting by (. . .) the average over the distribution of charges along the chain, one finds [12]:
qi =
Q
N
, q2
i
= q2
0 , qi q j =
Q2 −q2
0 N
N 2
=
Q2 −Q2
c
N 2
, i , j , (2.1)
with Qc =
√
∑N
i=1 q2
i
= q0N 1/2.
Let us pass to a continuous model, where the polymer chain in a porous environment is represented
by a path r (s), parameterized by 0 É s É L. The probability of each path configuration is given by the
Boltzmann distribution function:
P ({~r }) = exp
−
1
2
L
∫
0
ds
[
dr (s)
ds
]2
−
u0
2
L
∫
0
ds′
L
∫
0
ds′′δ
[
~r (s′)−~r (s′′)
]
−
1
2
L
∫
0
ds′
L
∫
0
ds′′
q(s′)q(s′′)
|~r (s′)−~r (s′′)|d−2
−
L
∫
0
ds V
[
~r (s)
]
. (2.2)
Here, the first term in the exponent represents the chain connectivity, the second term describes the short
range excluded volume interaction with the coupling constant u0, the third term gives the unscreened
electrostatic interaction in d dimensions (Coulomb potential), where the function q(s) represents the
charges along the chain in a particular configuration, and the last term arises due to the steric interactions
between the polymer chain and the structural defects in the environment given by the potential V
[
~r (s)
]
.
Following [32], we assume the pair correlation function of defects to decay with distance according to the
scaling law:
〈
V
[
~r (s′)
]
V
[
~r (s′′)
]〉
= w0|~r (s′)−~r (s′′)|−a , (2.3)
here, 〈. . .〉 denotes the average over different realizations of the disordered environment).
34601-2
Randomly charged polymers
Taking into account that only the two last terms in (2.2) include random variables, we find the aver-
aged partition function 〈Z (L)〉 =
∫
Dr exp(−Heff) with an effective Hamiltonian:
Heff =
1
2
L
∫
0
ds
[
dr (s)
ds
]2
+
u0
2
L
∫
0
ds′
L
∫
0
d s′′δ
[
~r (s′)−~r (s′′)
]
+
v0
2
L
∫
0
ds′
L
∫
0
ds′′
1
|~r (s′)−~r (s′′)|d−2
+
w0
2
L
∫
0
ds′
L
∫
0
ds′′|~r (s′)−~r (s′′)|−a . (2.4)
Here, we introduce the notation v0 ≡ q(s)q(s′) ∼ (Q2 −Q2
c ). We thus have a model with three types of
intrachain interactions governed by coupling constants u0, v0 and w0. From a dimensional analysis of
the couplings [u0] = L−(4−d)/2, [v0] = L−(6−d)/2, [w0] = L−(4−a)/2 one easily concludes that for Coulomb
interaction, the upper critical dimension dc = 6, whereas for the excluded volume interaction dc = 4, and
thus the latter may be neglected in the renormalization group scheme [40–42]. In what follows, we thus
restrict our considerations to a model with only two couplings v0 and w0. The sign of the coupling v0
depends on the overall charge Q of a given polyampholyte: a positive value of v0 (i.e., Q2 > Qc) corre-
sponds to the case, where the Coulomb repulsion is predominant in determining the conformation of the
polymer chain (and thus the polymer is expected to expand).
Note that in deriving equation (2.4) we restrict the consideration to a simpler case of annealed disor-
der averaging, taking into account that for an infinitely long single polymer chain in random disorder,
the distinction between quenched and annealed averages is negligible [36–39].
3. Renormalization and results
We follow the direct polymer renormalization scheme, developed by des Cloizeaux [2], generalizing
it to the case of two intrachain interactions. Working within the frames of the continuous polymer chain
model, one encounters problems with various divergences as the polymer length diverges (and thus the
number of configurations tends to infinity). However, all divergencies can be eliminated by introducing
renormalization factors, allowing to define and directly estimate the physical quantities of interest. The
renormalization procedure is related to the existence of universal critical indices and critical factors.
Expanding the theory in the couplings v0 and w0 and passing to the Fourier transform according to
∣
∣~r (s′)−~r (s′′)
∣
∣
−a
≃
∫
d~k |k|a−d exp
{
i~k
[
~r (s′)−~r (s′′)
]
}
,
we find for the partition function of the system:
〈Z (L)〉 = Z 0(L)
[
1−
4 v0(2π)−
(d−2)
2 L3− d
2
(4−d) (6−d)
−
4 w0(2π)−
a
2 L2− a
2
(2−a) (4−a)
]
. (3.1)
Here,
Z 0(L) =
∫
Dr exp
−
1
2
L
∫
0
ds
[
dr (s)
ds
]2
denotes the partition sum of the unperturbedmodel (Gaussian chain). Note that only the first order terms
are kept in the above relation (we restrict ourselves to the so-called “one-loop approximation” of pertur-
bation theory).
The averaged squared end-to-end distance R2
e = [r (L)−r (0)]2 of a typical polymer chain configuration
may be calculated using the identity:
(〈R2
e 〉)H =
{
−2d
∂
∂q2
ei~q [r (L)−r (0)]2
}
H
(3.2)
34601-3
V. Blavatska, C. von Ferber
with:
(. . .)H =
∫
Dr (. . .)e−Heff
〈Z (L)〉
. (3.3)
For the one-loop approximation we find:
(〈R2
e 〉)H =Ld
[
1+
4 v0(2π)−
d−2
2 L3− d
2
(6−d) (8−d)
+
4 w0(2π)−
a
2 L2− a
2
(4−a) (6−a)
]
. (3.4)
Wemay thus define a new (renormalized) scale LR and introduce a swelling factor χ2 via: (〈R2
e 〉)H = dLR ,
χ2(v0, w0) = LR/L. Remembering that (〈R2
e 〉)H ∼ L2ν and thus χ2 ∼ L2ν−1 one has:
(2ν−1) = L
∂χ2(v0, w0)
∂L
. (3.5)
The final renormalization step can be performed by analyzing the virial expansion for the osmotic
pressure of a solution of polymers [2]. To this end, we need the contributions to the partition functions
〈Z (L,L)〉 = 〈Zv0 (L,L)〉+〈Zw0 (L,L)〉 of the system of two interacting polymer chains of the same length L.
The dimensionless renormalized coupling constants v , w are thus defined by:
v =−
〈Zv0 (L,L)〉
〈Z 2(L)〉
L
−
d−2
2
R
, w =−
〈Zw0 (L,L)〉
〈Z 2(L)〉
L
−
a
2
R
. (3.6)
The RG flows of renormalized coupling constants are governed by the functions: βu = L∂ ln v/∂L, βw =
L∂ ln w/∂L. The corresponding expressions read:
βv = (6−d)v −
2v2(d −2)
(8−d)
−
2v w(d −2)
6−a
= ǫv −4v2
−4v w,
βw = (6−a)w −
2w2a
(6−a)
−
2v w a
8−d
= δw −4w2
−4v w. (3.7)
Above, we have performed a double ǫ = 6−d , δ = 4− a expansion, keeping terms up to linear order
in these parameters. The fixed points u∗, v∗ of the renormalization group transformations are defined
as common zeros of the RG functions (3.7). We find three distinct fixed points determining the scaling
behavior of a system at different a and d :
Gaussian: v∗
= 0, w∗
= 0, stable for ε,δ< 0, (3.8)
Coulomb: v∗
=
ǫ
4
, w∗
= 0, stable for δ< ǫ, (3.9)
Disorder: v∗
= 0, w∗
= δ/4, stable for δ> ǫ. (3.10)
Let us analyze the above results more in detail. The Gaussian fixed point corresponds to the situation,
where any monomer-monomer interaction is irrelevant. This happens when we are above the upper
critical dimensions of all interactions (d > 6, a > 4). In the case when a > d −2, the presence of correlated
defects plays no role, and the Coulomb fixed point is stable (only the electrostatic interaction is relevant
in this case). The fixed point obtained has a positive value, and thus we restore the behavior for Q > Qc
(polyelectrolyte limit). Finally, in the case where a < d − 2, the strongly correlated disorder causes the
main effect on the polymer behavior, and the Coulomb interaction is now irrelevant (the Disorder fixed
point is stable). The absence of a stable fixed point where both long range interactions are present can
be explained using dimensional analysis [see explanation after (2.4)]: at a > d −2, coupling w0 becomes
dimensionless for d < 6, and thus it is irrelevant in the renormalization group sense, whereas at a < d −2
it is dimensionless for d > 6 and thus the interaction v0 is irrelevant. The critical exponents ν, governing
the size measure of a charged polymer in all three situations, described above, are found by evaluating
(3.5), rewriting the corresponding expressions in terms of renormalized couplings and finally substituting
34601-4
Randomly charged polymers
the values of fixed points listed above. We find:
νGauss =
1
2
, (3.11)
νCoulomb =
1
2
+
ǫ
8
, (3.12)
νDisorder =
1
2
+
δ
8
. (3.13)
Note that νCoulomb coincides with the critical exponent governing the stretching of a polyelectrolyte chain
estimated previously within the RG scheme in references [40, 41]. Thus, we conclude that PAs with any
Q >Qc belong to the universality class of polyelectrolyts. For the physically interesting case d = 3 (ε= 3)
we may estimate νCoulomb ≃ 0.88. According to (3.13), any neutral chain in solution in the presence of
long-range-correlated defects, governed by correlation function with a = d −2, obeys exactly the same
scaling behaviour as a charged polymer with unscreened Coulomb interaction in a pure solvent.
4. Conclusions
We studied the scaling properties of charged polymers (polyampholytes) in a solvent in the pres-
ence of structural obstacles spatially correlated on a mesoscopic scale according to a power law ∼ x−a
[32]. Such correlations can describe extended pore-like defects of fractal structure. Within the randomly
charged polymer model, a polyampholyte is considered as a linear chain composed of a random sequence
of N monomers, each carrying a charge ±q0 [6, 7, 12]. The model predicts a contraction of the charged
polymer size when the total charge Q < Qc and an expansion for Q > Qc (with Qc ∼ N 1/2). The presence
of long-range correlated disorder leads to additional steric interaction between monomers in addition to
the long-range Coulomb potential.
Passing to the continuous chain limit with two types of interactions (electrostatic and steric), we de-
veloped a direct renormalization group approach by generalizing the scheme of des Cloizeaux [2] and
analyzed the peculiarities of conformational transitions, which can be observed in charged polymers in
a disordered environment. In the case where a > d −2, the presence of correlated defects plays no role,
and the Coulomb fixed point is stable (only electrostatic interaction is relevant in this case). The fixed
point obtained has a positive value, and thus we restore the behavior at Q > Qc (polyelectrolyte limit
[40, 41]). Thus, PAs with any Q > Qc belong to the universality class of polyelectrolytes. At a < d −2, the
strongly correlated disorder causes the main effect on the polymer behavior, while Coulomb interaction
is irrelevant. In particular, one may conclude that any neutral chain in solution in the presence of long-
range-correlated defects, governed by correlation function with a = d−2, follows exactly the same scaling
behaviour, as a charged polymer with unscreened Coulomb interaction in a pure solvent.
Acknowledgement
This work was supported in part by the FP7 EU IRSES project N269139 “Dynamics and Cooperative
Phenomena in complex Physical and Biological Media”.
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Випадково зарядженi полiмери в пористому середовищi
В. Блавацька1 , К. фон Фербер2
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
2 Дослiдницький центр прикладної математики, Унiверситет Ковентрi, CV1 5FB Ковентрi, Англiя
Дослiджуються конформацiйнi властивостi заряджених полiмерiв в розчинi у присутностi структурних не-
однорiдностей, скорельованих згiдно степеневого закону ∼ x−a . Використовується модель, в якiй полi-
мерний ланцюжок представлено як випадкову послiдовнiсть зарядiв qi = ±q0 . Така модель описує вла-
стивостi полiамфолiтiв – гетерополiмерiв, що мiстять як позитивно, так i негативно зарядженi групи моно-
мерiв. Застосовується пiдхiд прямого полiмерного перенормування i аналiзується скейлiнгова поведiнка
заряджених полiмерiв до першого порядку подвiйного ǫ= 6−d , δ= 4−a-розкладу.
Ключовi слова: полiмери, заморожений безлад, скейлiнг, ренормалiзацiйна група
34601-6
http://dx.doi.org/10.1080/00150198008209479
http://dx.doi.org/10.1021/ma00058a023
http://dx.doi.org/10.1209/0295-5075/24/4/005
http://dx.doi.org/10.1103/PhysRevE.50.R3322
http://dx.doi.org/10.1103/PhysRevE.52.835
http://dx.doi.org/10.1021/ma9507958
http://dx.doi.org/10.1016/0378-4371(96)00107-0
http://dx.doi.org/10.1103/PhysRevE.56.5798
http://dx.doi.org/10.1209/epl/i1997-00143-x
http://dx.doi.org/10.1021/ma970947u
http://dx.doi.org/10.1021/ma981818w
http://dx.doi.org/10.1103/PhysRevLett.85.4305
http://dx.doi.org/10.1063/1.460012
http://dx.doi.org/10.1038/320340a0
http://dx.doi.org/10.1021/ma60078a046
http://dx.doi.org/10.1016/S0968-0004(98)01207-9
http://dx.doi.org/10.1074/jbc.R100005200
http://dx.doi.org/10.1038/425027a
http://dx.doi.org/10.1007/BF01055706
http://dx.doi.org/10.1023/A:1023032307964
http://dx.doi.org/10.1103/PhysRevB.67.094404
http://dx.doi.org/10.1103/PhysRevB.72.064417
http://dx.doi.org/10.1103/PhysRevB.27.413
http://dx.doi.org/10.1016/S0167-7322(01)00179-9
http://dx.doi.org/10.1103/PhysRevE.64.041102
http://dx.doi.org/10.1016/j.physleta.2010.03.037
http://dx.doi.org/10.1063/1.458349
http://dx.doi.org/10.1063/1.462469
http://dx.doi.org/10.1103/PhysRevE.57.3656
http://dx.doi.org/10.1103/PhysRevE.68.051802
http://dx.doi.org/10.1051/jphyslet:019770038010500
http://dx.doi.org/10.1088/0305-4470/14/6/014
Introduction
The model
Renormalization and results
Conclusions
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