How does the scaling for the polymer chain in the dissipative particle dynamics hold?
We performed a series of simulations for a linear polymer chain in a solvent using dissipative particle dynamics to check the scaling relations for the end-to-end distance, radius of gyration and hydrodynamic radius in three dimensions. The polymer chains of up to 80 beads in explicit solvent of var...
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| Zitieren: | How does the scaling for the polymer chain in the dissipative particle dynamics hold? / J.M. Ilnytskyi, Yu. Holovatch // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 539-551. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1188992017-06-02T03:03:36Z How does the scaling for the polymer chain in the dissipative particle dynamics hold? Ilnytskyi, J.M. Holovatch, Yu. We performed a series of simulations for a linear polymer chain in a solvent using dissipative particle dynamics to check the scaling relations for the end-to-end distance, radius of gyration and hydrodynamic radius in three dimensions. The polymer chains of up to 80 beads in explicit solvent of various quality are studied. To extract the scaling exponent , the data are analyzed using linear fits, correction-to-scaling forms and analytical fits to the histograms of radius of gyration distribution. For certain combinations of the polymer characteristics and solvent quality, the correction-to-scaling terms are found to be essential while for the others these are negligibly small. In each particular case the final value for the exponent ν was chosen according to the best least-squares fit. The values of ν obtained in this way are found within the interval ν = 0.55 ÷ 0.61 but are concentrated mostly around 0.59, which is very close to the best known theoretical result ν = 0.588. The existence of this interval is attributed both to the peculiarities of the method and to the moderate chain lengths being simulated. Within this shortcoming, the polymer chain in this kind of modeling is found to satisfy the scaling relations for all three radii being considered. Використовуючи метод дисипативної динамiки ми виконали низку комп’ютерних симуляцiй лiнiйного полiмерного ланцюга в розчиннику. Головною метою наших дослiджень була перевiрка спiввiдношень скейлiнґу для вiдстанi мiж кiнцями ланцюга, радiусу ґiрацiї та гiдродинамiчного радiусу в тривимiрному просторi. Аналiзувались полiмернi ланцюги розмiром до 80 частинок в розчинниках рiзної якостi. Для отримання показника скейлiнґу ν результати апроксимувались лiнiйною функцiєю, функцiєю, що враховувала поправки до скейлiнґу та вiдомою аналiтичною залежнiстю гiстограм розподiлу радiусу ґiрацiї. Було виявлено, що для одних комбiнацiй характеристик полiмеру i якостi розчинника, поправки до скейлiнґу є суттєвими, в той час як для iнших комбiнацiй вони нехтовно малi. В кожному окремому випадку остаточна оцiнка для значення показника ν здiйснювалась методом найменших квадратiв. Отриманi в такий спосiб значення ν лежать в iнтервалi ν = 0.55÷0.61 i в основному зосередженi бiля величини 0.590, що є дуже близько до найточнiшої теоретичної оцiнки ν = 0.588. Iснування такого iнтервалу можна пояснити як особливостями методу так i помiрними довжинами симульованих ланцюгiв. Беручи до уваги згаданi неточностi, можна зробити висновок про те, що спiввiдношення скейлiнґу в застосованому нами способi моделювання виконуються для всiх трьох розглянутих характеристик полiмерного ланцюга. 2007 Article How does the scaling for the polymer chain in the dissipative particle dynamics hold? / J.M. Ilnytskyi, Yu. Holovatch // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 539-551. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.25.Hq, 61.20.Ja, 89.75.Da DOI:10.5488/CMP.10.4.539 http://dspace.nbuv.gov.ua/handle/123456789/118899 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We performed a series of simulations for a linear polymer chain in a solvent using dissipative particle dynamics to check the scaling relations for the end-to-end distance, radius of gyration and hydrodynamic radius in three dimensions. The polymer chains of up to 80 beads in explicit solvent of various quality are studied. To extract the scaling exponent , the data are analyzed using linear fits, correction-to-scaling forms and analytical fits to the histograms of radius of gyration distribution. For certain combinations of the polymer characteristics and solvent quality, the correction-to-scaling terms are found to be essential while for the others these are negligibly small. In each particular case the final value for the exponent ν was chosen according to the best least-squares fit. The values of ν obtained in this way are found within the interval ν = 0.55 ÷ 0.61 but are
concentrated mostly around 0.59, which is very close to the best known theoretical result ν = 0.588. The existence of this interval is attributed both to the peculiarities of the method and to the moderate chain lengths being simulated. Within this shortcoming, the polymer chain in this kind of modeling is found to satisfy the scaling relations for all three radii being considered. |
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Ilnytskyi, J.M. Holovatch, Yu. |
| spellingShingle |
Ilnytskyi, J.M. Holovatch, Yu. How does the scaling for the polymer chain in the dissipative particle dynamics hold? Condensed Matter Physics |
| author_facet |
Ilnytskyi, J.M. Holovatch, Yu. |
| author_sort |
Ilnytskyi, J.M. |
| title |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? |
| title_short |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? |
| title_full |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? |
| title_fullStr |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? |
| title_full_unstemmed |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? |
| title_sort |
how does the scaling for the polymer chain in the dissipative particle dynamics hold? |
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Інститут фізики конденсованих систем НАН України |
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| citation_txt |
How does the scaling for the polymer chain in the dissipative particle dynamics hold? / J.M. Ilnytskyi, Yu. Holovatch // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 539-551. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-07-08T14:51:44Z |
| last_indexed |
2025-07-08T14:51:44Z |
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1837090792685961216 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 4(52), pp. 539–551
How does the scaling for the polymer chain in the
dissipative particle dynamics hold?
J.M.Ilnytskyi1,2, Yu.Holovatch1,3
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Institut für Physik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Deutschland
3 Institut für Theoretische Physik, Johannes Kepler Universität Linz, 69 Altenbergerstr., 4040 Linz, Austria
Received October 9, 2007
We performed a series of simulations for a linear polymer chain in a solvent using dissipative particle dynamics
to check the scaling relations for the end-to-end distance, radius of gyration and hydrodynamic radius in three
dimensions. The polymer chains of up to 80 beads in explicit solvent of various quality are studied. To extract
the scaling exponent ν, the data are analyzed using linear fits, correction-to-scaling forms and analytical fits
to the histograms of radius of gyration distribution. For certain combinations of the polymer characteristics
and solvent quality, the correction-to-scaling terms are found to be essential while for the others these are
negligibly small. In each particular case the final value for the exponent ν was chosen according to the best
least-squares fit. The values of ν obtained in this way are found within the interval ν = 0.55 ÷ 0.61 but are
concentrated mostly around 0.59, which is very close to the best known theoretical result ν = 0.588. The
existence of this interval is attributed both to the peculiarities of the method and to the moderate chain lengths
being simulated. Within this shortcoming, the polymer chain in this kind of modeling is found to satisfy the
scaling relations for all three radii being considered.
Key words: polymers, scaling, scaling exponents, dissipative particle dynamics
PACS: 61.25.Hq, 61.20.Ja, 89.75.Da
1. Introduction
Variety of physical, chemical, and biological phenomena which involve polymers as well as
variety of polymer characteristics one is interested in have been naturally reflected in numerous
theoretical methods and models used for polymer description [1]. In particular, it is generally
recognized by now, that the scaling properties of a long flexible polymer chain immersed in a good
solvent are perfectly described by the model of self-avoiding walks (SAW) [2]. A textbook example
is given by the mean square end-to-end distance R1N of a SAW of N steps and that of a linear
polymer chain, either of which scales in an asymptotic limit N → ∞ as:
〈R2
1N 〉 ∼ N2ν , (1)
with a universal exponent ν. The value of the exponent depends on the space dimension only and
is the same for all SAW on different three dimensional (3d) lattices. Relation (1) is violated for
the finite N and an approach to the asymptotics is governed by the corrections to scaling. With
an account of the first correction equation (1) reads:
〈R2
1N 〉 = AN2ν
(
1 +
B
N∆
+ · · ·
)
. (2)
Here A, B are non-universal amplitudes and ∆ is an universal correction-to-scaling exponent.
Theoretical estimates for the exponents follow from the field theoretical renormalization group
calculations [3]:
ν(3d) = 0.5882 ± 0.0011, ∆(3d) = 0.478 ± 0.010. (3)
c© J.M.Ilnytskyi, Yu.Holovatch 539
J.M.Ilnytskyi, Yu.Holovatch
Whereas the SAW on the discrete lattice is a perfect model to study scaling of long flexible
polymer chains in a good solvent [4], for obvious reasons it fails to describe most of the other
properties of the chain [1,5]. A more realistic model would be defined in a continuous space and
involves both monomer-monomer and monomer-solvent interactions. Brownian dynamics is one of
the candidates, as it operates on the mesoscale and incorporates the random force acting on each
monomer which mimics the effect of a solvent [6]. However, the method lacks correct hydrodynamics
limit. Similar but more rigorous technique, i.e., the dissipative particles dynamics (DPD), was
introduced by Hoogerbrugge and Koelman [7] and later refined by Español and Warren to satisfy
the detailed balance [8]. The method has correct hydrodynamics limit [9] and quickly gained much
attention as a powerful technique for mesoscale simulations of various kinds of macromolecules [6].
Therefore, an important issue of analysis is to check how the scaling laws (e.g. of equation (1)
form) hold in the case of the DPD-based simulations. This issue was partly considered in [10–14].
However, some issues remain unclear (see the next section 2 for details). We see a good reason
to discuss in details the scaling laws for the polymer chain in a solvent in the DPD method. As
far as the method was successfully used for the simulations of branched, amphiphilic and other
complex molecules, the validation of correct scaling laws on such a well-studied system as linear
chain polymer would be a strong supporting factor for extending the subject to the scaling laws in
more complex systems (e.g. star-like, branched, dendritic systems, etc.).
The set-up of our paper is as follows: in the next section 2 we give a brief review of former
simulation results, section 3 contains the details of our simulations. Our main results are presented
in section 4, section 5 gives conclusions and outlook.
2. Previous simulations
A number of various simulation techniques have been used to numerically verify the scaling
laws of the polymer in solution, namely the molecular dynamics (MD), Monte Carlo (MC), DPD
and others. Here we shall provide a rather brief account of references relevant to the present study.
By using the MC method enriched by special techniques, one can move very efficiently through
the conformational space (see, e.g. [6]). As the result, the method is quite suitable for the simu-
lations of random walks (RW) and SAW [15,16]. Li et al. [16] used a pivot algorithm and other
techniques applied to the chain of N = 100 ÷ 1000 monomers. The value for the scaling exponent
obtained is ν = 0.588 and it agrees very well with the theoretical estimate (3). To study dynami-
cal properties, however, one should turn to true dynamical methods. Pierleone and Ryckaert [17]
performed a comprehensive analysis of the scaling laws for both static (radius of gyration, Rg and
end-to-end distance R1N ) and dynamical (diffusion coefficient, D) properties using MD simulati-
ons. One of the important outcomes is that the static properties are unaffected by the finiteness
of the box size L providing that L/Rg > 3. The scaling exponent values ν = 0.59 and 0.568 (for
the R1N and Rg, respectively) were obtained. On the contrary, dynamical properties were found
to be strongly dependent on L. This fact was explained by hydrodynamic interactions of the chain
with its own images due to the periodic boundary conditions (PBC). If, following Dunweg and
Kremer[18] one uses the Kirkwood formula, then a correct dynamical scaling is also recovered [17].
A number of simulations of linear polymers using DPD technique are available. Schlijper et al.
[10] were the first to study static and dynamic scaling laws for the DPD chain in athermal solvent.
Later their results for this case were confirmed and extended to include the solvents of variable
quality by Kong et al. [11]. These authors studied Rg (among the other properties) and have found
for the athermal solvent ν = 0.52 which is closer to the polymer at the θ-condition. With the
increase of solvent quality, the value ν ≈ 0.6 close to a SAW exponent was recovered. This problem
was further addressed by Spenley [12] who studied the R1N in polymer melt and solution. In the
latter case, relevant to our study, the exponent ν = 0.58 was obtained. The previous result ν = 0.52
by Kong et al.[11] was commented as being influenced by the type of spring potential used there.
In our opinion, this discrepancy could also be due to insufficient statistics.
It has also been debated whether the soft repulsion inherent to the DPD method is sufficient
to model the self-avoidance of the chains. In particular, Symeonidis et al. [13] used additional
540
Scaling of the DPD polymer chain
intramolecular interactions (e.g. Lennard-Jones-like repulsion term) to increase the chain self-
avoidance and experimented with different forms of the bond springs (i.e. FENE, Hookean and
WLC). One of the conclusions made by the authors is, that the presence of FENE potential alone
ensures a correct value for the exponent ν ≈ 0.6 and no extra interactions for the improvement of
self-avoidance are required.
Jiang et al. [14] have carried out an extensive study of the hydrodynamical properties obtained
via the DPD method. In particular, scaling laws for both Rg and R1N for the dilute athermal
solution are relevant to our study. The authors found the values for the scaling exponent ν weakly
dependent on the box size with the averages of ν = 0.58 for the radius of gyration and ν = 0.595 for
the end-to-end distance[14] (the box sizes of L = 15÷30 DPD length units were used and a range of
chain lengths was N = 10÷ 100 DPD beads). The other conclusion given in [14] is the one already
mentioned above, claiming that the original soft interactions are sufficient for self-avoidance of the
chains in the DPD method. Similarly to the MD study of Pierleone and Ryckaert [17], the diffusion
coefficient was found to be highly sensitive to the simulational box size. However, after the appro-
priate corrections to the system size were introduced, the scaling law for the diffusion coefficient
D ∼ N−ν was obtained. Since, according to Kirkwood theory, the diffusion coefficient D is related
to the hydrodynamic radius Rh [18], the latter is also shown to obey the correct scaling law.
Despite the availability of simulational studies that concentrate on static and dynamical scaling
of the polymer in DPD method, there are still certain points that need to be clarified, namely:
(i) do all the relevant radii, R1N , Rg and Rh scale with the same scaling exponent ν?
(ii) are the corrections to scaling relevant and can these improve the results or make these more
consistent?
(iii) in case of a good solvent, will the changes in the polymer-solvent interaction potential lead
to any detectable drift of the scaling exponent ν?
These three points will be addressed in the current study.
3. Simulational details
In our study we closely follow the DPD method as discussed by Groot and Warren [19]. Polymer
molecule (see, figure 1) is modeled as a chain of soft beads connected via harmonic springs. The
bonding force acting on the i-th bead from its bond neighbour j is:
~FB
ij = −krij r̂ij , (4)
where rij = |~rij |, ~rij = ~ri − ~rj , r̂ij = ~rij/rij and k = 4 is the spring constant. Here and thereafter,
the length, mass, time and energy (kBT ) units are chosen to be equal to unity. The solvent is
Figure 1. The polymer molecule in a solvent, as modeled in the DPD method. Both monomers
constituting a polymer chain and solvent molecules are considered as interacting soft beads (grey
and hollow discs, correspondingly).
described in an explicit way, in a form of isolated beads of the same origin as the polymer ones
541
J.M.Ilnytskyi, Yu.Holovatch
(see, figure 1). Each i-th bead (polymer or solvent one) is subject to a pairwise non-bonded force
from its j-th counterpart. The force acting on the i-th bead due to the interaction with the j-th
bead consists of three contributions:
~Fij = ~FC
ij + ~FD
ij + ~FR
ij , (5)
where the conservative ~FC
ij , dissipative ~FD
ij and random ~FR
ij contributions are of the following form:
~FC
ij =
{
a(1 − rij)r̂ij , rij < 1,
0, rij > 1,
(6)
~FD
ij = −γwD(rij)(r̂ij · ~vij)r̂ij , ~FR
ij = σwR(rij)θij∆t−1/2r̂ij . (7)
Here ~vij = ~vi−~vj , ~vi, ~vj being velocities of the beads, θij is Gaussian random variable: 〈θij(t)〉 = 0,
〈θij(t)θkl(t
′)〉 = (δikδil + δilδjk)δ(t− t′). According to Español and Warren [8], the dissipative and
random force amplitudes are interrelated, wD(rij) = (wR(rij))
2 and σ2 = 2γ, to satisfy the detailed
balance. The frequently used analytical form is:
wD(rij) = (wR(rij))
2 =
{
(1 − rij)
2, rij < 1,
0, rij > 1.
(8)
We choose the value γ = 6.75. Parameter a in the conservative force ~FC
ij defines maximum repulsion
between two beads which occurs at complete overlap rij = 0. Three different a parameters can be
distinguished, app for the polymer-polymer, ass for the solvent-solvent and aps for the polymer-
solvent interactions and all these can be varied. As was shown in [19], these parameters can be
related to the Flory-Huggins parameter χ. To integrate the equations of motion, we used the
modified velocity – Verlet algorithm by Groot and Warren [19]. Let us note that the structural
properties of the chain in a solvent are defined solely by the conservative forces (6), while the
dissipative and random forces (7) govern the phase space trajectory.
In our DPD simulations we considered the following set of chain lengths: N = 5, 7, 10, 14, 20,
28, 40, 56 and 80 beads in four types of solvent:
athermal solvent 1, app = ass = 25, aps = 25,
athermal solvent 2, app = ass = 33, aps = 33,
good solvent, app = ass = 25, aps = 20,
very good solvent, app = ass = 25, aps = 10.
(9)
The terms “good” and “very good” are indicative rather than exact, since only repulsive forces are
taken into account in DPD. Hence, “good solvent” means less repulsion for the polymer-solvent
interaction than that for the polymer-polymer one. The last two cases in equations (9) correspond
to the values ξ = −0.2 and ξ = −0.6, respectively, in notation of Kong et al. [11].
All the simulations were performed in NV T ensemble. At first, the initial chain conformation
was generated as a RW. Then, it was immersed into a relatively large box filled by solvent and short
preliminary runs were performed to estimate the approximate Rg for the polymer. Thereafter, the
final run was performed using the cubic box of the linear size L ≈ 5Rg. This size is considered to
be sufficiently large to eliminate the effects of PBC [14,17]. The number of solvent beads Ns was
chosen to keep the total reduced density of the solution equal to ρ∗ = (N + Ns)/V = 3. The time
step was chosen to be ∆t = 0.04 in reduced units. We performed from 3 · 106 (for N = 5) up to
15 · 106 (for N = 80) DPD steps for the final run. We did not concentrate on the relaxation times
in detail, but found the behaviour similar to Jiang et al. [14], τ ≈ 0.13N1.81 who used similar DPD
parameters for athermal solvent. This brings us to the following estimates for the simulation runs:
about 5 · 104 τ for N = 5 and 1.7 · 103 τ for N = 80 with the estimates for intermediate values
of N falling in between. The chain conformations were saved after performing every 1000 DPD
steps and last 3/4 of these were used for the subsequent analysis. As far as all the simulations were
performed using PBC, the unwrapping of the chain at each time step was performed based on the
connectivity information.
542
Scaling of the DPD polymer chain
4. Results
We concentrated our analysis on the three following metric characteristics of the linear polymer.
The instantaneous squared end-to-end distance at a given time t is defined as:
R2
1N (t) = (~r1 − ~rN )2, (10)
where ~r1 and ~rN are the radius-vectors for the first and last bead of the chain, respectively. The
squared radius of gyration at time t is defined as:
R2
g(t) =
1
N
N
∑
i=1
(~ri − ~RCOM)2, (11)
where ~RCOM is the radius-vector of the polymer center of mass. The inverse hydrodynamic radius
is defined as follows:
1
Rh(t)
=
1
N2
∑
i6=j
1
rij
, (12)
where the sum runs over all different bead pairs i, j. All three properties are time averaged (denoted
hereafter as 〈· · · 〉) and then the first two can be square rooted while the third one is inverted to
provide the estimates for R1N , Rg and Rh, respectively.
One should consider Rg and Rh as more useful characteristics as compared to R1N . First of all,
R1N is meaningful for the linear polymer or for a certain subchain of the branched molecule (e.g.
backbone, side chain, star-polymer arm, etc.), while both Rg and Rh are universally defined. Apart
from that, both these properties are directly measurable via small-angle scattering experiments
and via small-angle dynamic light scattering experiments, respectively (for more discussion on
this, see [20]).
4.1. Linear fit and first correction-to-scaling fit
As was already mentioned above, a number of universal scaling laws have been observed for the
polymer properties [2] in the limit of infinite polymer length, e.g. equations (1). To reproduce these
properties, one needs to consider rather long chains in the simulations [16]. Due to the polymer coil
self-similarity, one would expect that such a coarse-grained approach as DPD should be capable
of reproducing correct scaling laws for much shorter chain lengths. For the moderate polymer
lengths, usually the scaling is written in terms of a number of bonds, N −1, instead of the number
of monomers N . Also, as is known, the correction-to-scaling terms are important for moderate
system sizes, as is indicated by the studies of the phase transitions in spin lattice models [21–23].
Taking into account the first correction-to-scaling term, equations (2), one has for the 〈R2
1N 〉:
〈R2
1N 〉 = A(N − 1)2ν
(
1 +
B
(N − 1)∆
+ · · ·
)
. (13)
We shall work with square rooted values
R1N =
√
〈R2
1N 〉 ≈ A1/2(N − 1)ν
(
1 +
1
2
B
(N − 1)∆
+ · · ·
)
, (14)
and keep the linear term only in the correction-to-scaling expansion above. Taking a logarithm of
both sides of equations (14), we obtain
lnR1N =
1
2
lnA + ν ln(N − 1) + ln
(
1 +
1
2
B
(N − 1)∆
+ · · ·
)
= A′ + ν ln(N − 1) + ln
(
1 +
B′
(N − 1)∆
+ · · ·
)
, (15)
543
J.M.Ilnytskyi, Yu.Holovatch
where A′ and B′ are self-explanatory. The same formulae can also be applied for Rg =
√
〈R2
g〉 and
Rh = 1/〈1/Rh〉 (in the latter case one has B′ ≈ −B). As a result, the same fit (15) can be used
for all three quantities under interest, R1N , Rg and Rh. In the case of linear fit the correction-to-
scaling amplitude B′ is assumed to be zero and one recovers the linear fit form. To perform the
fits we used the least-squares routine mrqmin from the Numerical recipes book [24]. The accuracy
of the fit was monitored by the cumulative squared deviation per one data point,
χ2 =
1
ndata
ndata
∑
k=1
[
R1N
[k] − F (N [k])
]2
, (16)
where (N [k],R1N
[k]) is the k-th out of ndata data points and F (N) is the fitting function.
It is generally known that very good statistics for each data point are required to ensure robust
results for the scaling exponent ν. Otherwise the latter depends essentially on the selection of data
points being used for the fit. This was also found to be the case in the course of the present study.
Hence, relatively lengthly simulations are performed to improve statistics. To check the consistency
of the results, we performed a series of fits using a gradually increasing range of intervals in the
chain length N , e.g. N = 5 ÷ 28, N = 5 ÷ 40, · · ·N = 5 ÷ 80, which are presented below.
As was anticipated, both cases of “athermal 1” and “athermal 2” solvents were found to show
the same averaged properties within the accuracy of the simulations. Therefore, the data for both
cases have been averaged at each N and marked as “athermal” further in the text. At first glance,
the data points for both R1N and Rg are well fitted by the line in log-log scale (see, figure 2), but,
in fact, the best fits were achieved using the first correction-to-scaling form (15). We should remark
here that we were unable to achieve stable results for both R1N and Rg using the four-parameter
fits, in which all A′, ν, B′ and ∆ in (15) are fitted simultaneously. Instead, we were forced to fix
the correction-to-scaling exponent ∆ at certain value and to perform the fit over three remaining
parameters. We used two choices for ∆, namely the best theoretical value ∆ = 0.478 [3] and ∆ = 1.
1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.5
1.0
1.5
2.0
2.5
b
b
b
b
b
b
b
b
b
r
r
r
r
r
r
r
r
r
r very good
good
b athermal
ln(N−1)
ln R1N
1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.5
0.0
0.5
1.0
1.5
2.0
b
b
b
b
b
b
b
b
b
r
r
r
r
r
r
r
r
r
r very good
good
b athermal
ln(N−1)
ln Rg
Figure 2. Data points for R1N (on the left) and Rg (on the right) and their best fits using the
form (15) obtained from the DPD simulations on a set of chain lengths N = 5 ÷ 80 at different
solvent conditions, see notations on both plots.
The results presented in table 1 indicate that the correction-to-scaling term for R1N is essential
in the case of athermal solvent only. In the table, we also show the linear fit performed over the
largest chain lengths, N = 20÷ 80, for the sake of comparison. One can see that the latter leads to
almost the same result ν = 0.586 as the correction-to-scaling fit ν = 0.591, which is indicative of
the crossover quickly decreasing at N > 20. In the cases of good and very good solvents very small
amplitudes B′ are found and, as the result, linear fits are almost equally good (see, table 1). For
Rg the situation is exactly reversed. The correction to scaling is important for the cases of good
and very good solvents (see, table 2). We cannot suggest some definite explanation for this effect,
except the conclusion that preference should be given to the general form (15), rather than to the
544
Scaling of the DPD polymer chain
linear fit, otherwise the exponent ν could be essentially under- or overestimated (see, tables 1, 2).
Table 1. Linear fit and first correction-to-scaling fits to the form (15) for R1N with fixed exponent
∆ = 0.478 and ∆ = 1 when using various sets of data points. The best fit for the interval
N = 5 ÷ 80 is underlined.
fitting linear fit fit with ∆ = 0.478 fit with ∆ = 1
range, N ν χ2 ν [B′] χ2 ν [B′] χ2
athermal solvent
5 ÷ 28 0.564 1.7e–5 0.617 [0.370] 9.4e–7 0.588 [0.219] 1.3e–6
5 ÷ 40 0.569 3.1e–5 0.621 [0.399] 9.0e–7 0.592 [0.252] 1.8e–6
5 ÷ 56 0.573 5.5e–5 0.627 [0.447] 1.3e–6 0.596 [0.295] 3.6e–6
5 ÷ 80 0.574 5.2e–5 0.611 [0.318] 8.2e–6 0.591 [0.248] 6.9e–6
20 ÷ 80 0.586 8.6e–6
good solvent
5 ÷ 28 0.607 8.6e–6 0.591 [–0.099] 6.4e–6 0.600 [–0.066] 6.9e–6
5 ÷ 40 0.607 7.7e–6 0.595 [–0.076] 5.7e–6 0.601 [–0.056] 6.0e–6
5 ÷ 56 0.607 7.3e–6 0.604 [–0.026] 7.0e–6 0.605 [–0.023] 7.0e–6
5 ÷ 80 0.605 1.7e–5 0.594 [–0.087] 1.1e–5 0.600 [–0.073] 1.2e–5
20 ÷ 80 0.601 1.5e–5
very good solvent
5 ÷ 28 0.614 6.3e–7 0.610 [–0.021] 5.4e–7 0.612 [–0.019] 5.0e–7
5 ÷ 40 0.613 1.7e–6 0.605 [–0.052] 8.3e–7 0.609 [–0.041] 8.4e–7
5 ÷ 56 0.613 2.1e–6 0.612 [–0.009] 2.1e–6 0.612 [–0.011] 2.1e–6
5 ÷ 80 0.613 2.0e–6 0.611 [–0.012] 1.9e–6 0.612 [–0.012] 1.8e–6
20 ÷ 80 0.613 2.8e–6
The four-parameter fit, mentioned above, does not work for either R1N or Rg but is supposed
to perform better for Rh, where essentially larger correction to scaling is to be expected [20].
Indeed, this is found to be the case in our simulations. First of all, this can be noticed from the
arrangement of the data points, see figure 3. The results based on numerical fits are presented in
table 3, where both the cases of fixed ∆ = 0.478 and of the four-parameter fits are shown. The
latter give essentially better results in terms of fitting accuracy and provide independent estimates
for both exponents, ν ≈ 0.56 ÷ 0.58 and ∆ ≈ 0.65 ÷ 0.70. As was demonstrated by Dunweg et al.
[20] there are two comparable correction-to-scaling terms for the Rh, one is proportional to N−∆
and another, the so-called “analytic” correction, is proportional to N−(1−ν). Therefore, if one uses
the form (15) with a single correction term, then a rather effective exponent ∆ will be obtained.
This is how we should interpret the value ∆ obtained in our study.
4.2. The fits based on the distribution of Rg values
The averaging of the metric properties of interest, e.g. Rg =
√
〈R2
g〉, throughout this study was
performed as the arithmetic mean over discrete set of time points. One can also be interested in
the distribution p̄(Rg) of the Rg values themselves and, for instance, in examination of the scaling
law for the most probable values with the chain length N . The following analytical form has been
postulated by Lhuillier in 3d [25]
p̄(Rg) ∼ exp
[
−A1(N
ν/Rg)
3α − A2(Rg/N
ν)δ
]
, (17)
where α = 1/(3ν −1), δ = 1/(1−ν) and A1, A2 are constants. Subsequently, this has been verified
in a number of lattice MC simulations [26,27]. One can rewrite this distribution in terms of R2
g,
using the expressions for α and δ and incorporating the N -dependent factors into the constants,
545
J.M.Ilnytskyi, Yu.Holovatch
Table 2. Linear fit and first correction-to-scaling fits to the form (15) for Rg with fixed exponent
∆ = 0.478 and ∆ = 1 when using various sets of data points. The best fit for the interval
N = 5 ÷ 80 is underlined.
fitting linear fit fit with ∆ = 0.478 fit with ∆ = 1
range, N ν χ2 ν [B′] χ2 ν [B′] χ2
athermal solvent
5 ÷ 28 0.601 2.7e–6 0.584 [–0.106] 2.0e–7 0.592 [–0.080] 2.5e–7
5 ÷ 40 0.600 3.1e–6 0.587 [–0.090] 3.0e–7 0.593 [–0.073] 2.6e–7
5 ÷ 56 0.600 2.8e–6 0.593 [–0.052] 1.5e–6 0.597 [–0.047] 1.3e–6
5 ÷ 80 0.599 8.0e–6 0.587 [–0.087] 2.9e–6 0.593 [–0.077] 3.1e–6
20 ÷ 80 0.596 6.0e–6
good solvent
5 ÷ 28 0.641 4.9e–5 0.573 [–0.368] 2.7e–6 0.605 [–0.318] 4.2e–6
5 ÷ 40 0.638 5.3e–5 0.586 [–0.314] 5.4e–6 0.610 [–0.283] 5.0e–6
5 ÷ 56 0.636 5.1e–5 0.597 [–0.260] 9.4e–6 0.615 [–0.244] 7.3e–6
5 ÷ 80 0.631 9.7e–5 0.588 [–0.306] 1.4e–5 0.609 [–0.299] 1.4e–5
20 ÷ 80 0.618 2.8e–5
very good solvent
5 ÷ 28 0.651 6.9e–6 0.626 [–0.144] 1.9e–6 0.639 [–0.109] 2.2e–6
5 ÷ 40 0.647 1.6e–5 0.618 [–0.185] 2.6e–6 0.633 [–0.151] 3.6e–6
5 ÷ 56 0.645 2.4e–5 0.616 [–0.194] 2.4e–6 0.631 [–0.168] 3.7e–6
5 ÷ 80 0.642 3.4e–5 0.615 [–0.201] 2.2e–6 0.629 [–0.184] 3.9e–6
20 ÷ 80 0.632 2.3e–6
Table 3. Linear fit and first correction-to-scaling fits to the form (15) for Rh with fixed exponent
∆ = 0.478 and from the four-parameter fits when using various sets of data points. The best fit
for the interval N = 5 ÷ 80 is underlined.
fitting fit with ∆ = 0.478 four-parameter fit
range, N ν [B′] χ2 ν ∆ [B′] χ2
athermal solvent
5 ÷ 28 0.596 [3.008] 4.9e–5 0.497 0.836 [1.324] 1.9e–7
5 ÷ 40 0.611 [3.275] 6.9e–5 0.534 0.732 [1.683] 4.7e–7
5 ÷ 56 0.625 [3.555] 9.5e–5 0.572 0.666 [2.195] 1.0e–6
5 ÷ 80 0.635 [3.786] 1.0e–4 0.552 0.699 [1.912] 1.2e–6
good solvent
5 ÷ 28 0.623 [2.953] 2.0e–5 0.476 1.017 [0.925] 4.8e–7
5 ÷ 40 0.639 [3.274] 3.1e–5 0.563 0.654 [1.636] 2.3e–6
5 ÷ 56 0.654 [3.592] 5.1e–5 0.641 0.577 [2.993] 3.0e–6
5 ÷ 80 0.658 [3.710] 4.7e–5 0.565 0.664 [1.663] 5.5e–6
very good solvent
5 ÷ 28 0.625 [3.052] 5.9e–5 0.652 0.643 [3.159] 1.5e–7
5 ÷ 40 0.639 [3.284] 7.8e–5 0.580 0.703 [1.925] 6.1e–7
5 ÷ 56 0.651 [3.533] 9.7e–5 0.583 0.698 [1.963] 5.4e–7
5 ÷ 80 0.662 [3.779] 1.1e–4 0.577 0.710 [1.885] 5.1e–7
546
Scaling of the DPD polymer chain
1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.5
1.0
1.5
b
b
b
b
b
b
b
b
b
r
r
r
r
r
r
r
r
r
r very good
good
b athermal
ln(N−1)
ln Rh
Figure 3. Data points for Rh and their best fits using the form (15) obtained from the DPD
simulations on a set of chain lengths N = 5 ÷ 80 at different solvent conditions, see notations
on the plot.
and the result reads:
p(R2
g) = C exp
[
−
a1
[R2
g]
3
2(3ν−1)
− a2[R
2
g]
1
2(1−ν)
]
. (18)
This distribution has a maximum at
[R2
g]
max =
[
3a1(1 − ν)
a2(3ν − 1)
](1−ν)(3ν−1)
(19)
which can be square rooted to obtain an estimate for the most probable value of the radius of
gyration Rg
max =
√
[R2
g]
max. We prefer to work with the distribution of R2
g (18) and then to
square root the maximum to match the averaging procedure for simple averaging, Rg =
√
〈R2
g〉.
From the point of view of numerical fitting, the form (18) has four parameters: one exponent
ν and three coefficients, a1, a2 and C. Therefore, one can have a first estimate for the exponent
ν right from the fit of the data to the form (18). The second estimate for ν can be obtained from
the anticipated scaling of Rg
max according to the form (15).
0 10 20 30 40 50
bb
bb
bb
b
bb
b
b
b
bbb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bb
bb
bb
b
bb
b
b
b
b
b
b
b
b
bbb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
bbb
b
b
b
b
bbbbbb
bb
bbb
bbbbbbbbbbbbb
b
b
bbbbb
b
bbbbbbb
bbbbbbbbbbbbbbbbb bbbb bb b
R2
g
N = 20
b N = 40
N = 80
p(R2
g)
Figure 4. Mapping of the distribution of radius of gyration values p(R2
g) obtained from the
DPD simulations in athermal solvent, onto Lhuillier form (18), a few indicated chain lengths are
shown only for the sake of clarity.
The distributions p(R2
g) were built in a form of histograms and were found to map very well on
the Lhuillier form (18), see samples in figure 4 obtained for the case of athermal solvent. These fits,
as was mentioned above, provide both the estimate for Rg
max and an independent estimate for
the exponent ν. The latter was found to be scattered in the interval ν = 0.53÷ 0.58 depending on
N and solvent type. We do not expect high accuracy from such a fitting, as far as dependence on
547
J.M.Ilnytskyi, Yu.Holovatch
ν in equations (18) is rather complex. Nevertheless, the values of ν found are very reasonable and
indicate a self-consistency of this numerical approach. The next step is to use the maxima positions
Rg
max of the distributions (18), found via equations (19), and to find a fit to the scaling law (15)
similarly to the analysis of Rg data in the previous subsection. The data points alongside with the
best fits are shown in figure 5 and the results for the exponent ν, correction-to-scaling amplitude
B′ and fitting error χ2 are presented in table 4. We would like to stress that, contrary to the results
given in table 2, no dependence on solvent quality is observed in table 4 and the same value ν =
0.582−0.585 is found essentially in all three cases. This might indicate that the effect of small drift
of ν observed in a very good solvent in table 2 could be an artefact of the numerical method used.
1.0 1.5 2.0 2.5 3.0 3.5 4.0
-0.5
0.0
0.5
1.0
1.5
2.0
b
b
b
b
b
b
b
b
b
r
r
r
r
r
r
r
r
r
r very good
good
b athermal
ln(N−1)
ln Rmax
g
Figure 5. Data points for Rg
max and their best fits using the form (15) obtained from the DPD
simulations on a set of chain lengths N = 5 ÷ 80 at different solvent conditions, see notations
on the plot.
Table 4. Linear fit and first correction-to-scaling fits to the form (15) for Rmax
g with fixed
exponent ∆ = 0.478 and ∆ = 1 when using various sets of data points. The best fit for the
interval N = 5 ÷ 80 is underlined.
fitting linear fit fit with ∆ = 0.478 fit with ∆ = 1
range, N ν χ2 ν [B′] χ2 ν [B′] χ2
athermal solvent
5 ÷ 28 0.602 6.4e–5 0.539 [–0.345] 2.6e–5 0.568 [–0.306] 2.4e–5
5 ÷ 40 0.601 5.6e–5 0.562 [–0.242] 3.2e–5 0.579 [–0.222] 2.8e–5
5 ÷ 56 0.601 4.9e–5 0.576 [–0.169] 3.4e–5 0.587 [–0.165] 3.0e–5
5 ÷ 80 0.598 5.6e–5 0.574 [–0.183] 3.1e–5 0.585 [–0.181] 2.7e–5
20 ÷ 80 0.598 2.3e–5
good solvent
5 ÷ 28 0.646 4.6e–5 0.594 [–0.291] 2.1e–5 0.618 [–0.247] 2.0e–5
5 ÷ 40 0.641 6.6e–5 0.588 [–0.316] 1.8e–5 0.613 [–0.283] 1.9e–5
5 ÷ 56 0.640 5.9e–5 0.606 [–0.231] 2.7e–5 0.622 [–0.217] 2.5e–5
5 ÷ 80 0.633 1.8e–4 0.582 [–0.352] 6.2e–5 0.607 [–0.344] 6.9e–5
20 ÷ 80 0.609 9.4e–5
very good solvent
5 ÷ 28 0.646 2.6e–4 0.517 [–0.626] 6.0e–5 0.571 [–0.634] 5.5e–5
5 ÷ 40 0.636 3.4e–4 0.522 [–0.610] 5.2e–5 0.570 [–0.637] 4.7e–5
5 ÷ 56 0.628 3.8e–4 0.529 [–0.582] 4.9e–5 0.572 [–0.624] 4.2e–5
5 ÷ 80 0.625 3.7e–4 0.549 [–0.503] 7.8e–5 0.582 [–0.551] 5.9e–5
20 ÷ 80 0.607 3.9e–5
The cumulative outcome of our analysis for the scaling exponent ν which governs the scaling
law (2) is presented in figure 6. Here we show a histogram of the values of ν obtained for R1N ,
548
Scaling of the DPD polymer chain
Rg and Rh of a polymer chain in different solvents considered in the current study, namely, in
athermal, good and very good solvents. To make this histogram, the best fits (underlined results in
tables 1–4) have been used. One can see from the histogram that the value of ν is scattered in an
interval ν = 0.55÷0.61 but in fact is found predominantly in much narrower interval ν = 0.58÷0.60
centered around the best theoretical estimate ν = 0.5882 ± 0.0011 [3]. This allows us to conclude
that within the accuracy of our analysis, the scaling law (2) holds for all polymer characteristics
and in all solvents considered in this study. Our conservative estimate for the scaling exponent
therefore is
ν ≈ 0.59 ± 0.01. (20)
Figure 6. A histogram of values of ν obtained for different characteristic radii of a polymer
chain in different solvents (9). The data are collected according to best fits results (underlined
in tables 1–4). The value of ν is found predominantly in the interval ν = 0.55 ÷ 0.61 and
concentrate around the best theoretical estimate ν = 0.5882 ± 0.0011 [3].
5. Conclusions and outlook
In this study we performed DPD simulations of the polymer chain in a solvent of various quality
with the aim to reexamine how well the scaling laws hold for various polymer characteristics. Chains
of up to 80 beads were considered and simulation runs from 1600 up to 50000 relaxation times
were performed. Three metric properties were considered, end-to-end distance, radius of gyration
and hydrodynamic radius. For the analysis we used the linear fits, first correction-to-scaling fits
and fitting the distribution of the radius of gyration. As an outcome, within the accuracy of the
simulations and of the data analysis technique, the following conclusions can be drawn:
(i) All three metric properties obey the scaling law with very close values for ν found within
the interval 0.55÷ 0.61 (depending on the method of analysis). Most values are concentrated
around the average ν = 0.59 being very close to the theoretical estimate ν = 0.5882± 0.0011
[3], see, figure 6.
(ii) Corrections to scaling are found to be important in all combinations except a few prop-
erty/solvent ones and thus the conclusion (i) is valid only when the correction-to-scaling
terms are taken into account.
(iii) No or very small (up to 4%) drift of the scaling exponent ν is observed when changing the
solvent quality (remaining, however, in a good solvent regime), but this effect depends on a
numerical technique being used.
The method of analysis can be extended to the study of the scaling laws in more complex
molecular architectures, e.g. star-like polymers, branched and hyperbranched molecules (including
amphiphiles).
549
J.M.Ilnytskyi, Yu.Holovatch
Acknowledgements
We thank Prof. Myroslav Holovko for the invitation to contribute to the Festschrift dedicated
to Prof. Fumio Hirata 60th birthday. Work of Yu.H. was supported in part by the Austrian Fonds
zur Förderung der wissenschaftlichen Forschung under Project P 19583 and work of J.I. by DFG
grant NE410/8-2.
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6. Frenkel D., Smit B. Understanding molecular simulation: from algorithms to applications. Academic
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25. Lhuillier D., J. Phys. France, 1988, 49, 705.
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550
Scaling of the DPD polymer chain
Як виконуються закони скейлiнґу для полiмерного ланцюга в
методi дисипативної динамiки?
Я.М.Iльницький1,2, Ю.Головач1,3
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького 1, 79011 Львiв, Україна
2 Iнститут фiзики, Унiверситет Потсдама, Нiмеччина
3 Iнститут теоретичної фiзики унiверситету Йогана Кеплера, Альтенберґерштрасе 69, 4040, Лiнц,
Австрiя
Отримано 9 жовтня 2007 р.
Використовуючи метод дисипативної динамiки ми виконали низку комп’ютерних симуляцiй лiнiй-
ного полiмерного ланцюга в розчиннику. Головною метою наших дослiджень була перевiрка спiв-
вiдношень скейлiнґу для вiдстанi мiж кiнцями ланцюга, радiусу ґiрацiї та гiдродинамiчного радiусу в
тривимiрному просторi. Аналiзувались полiмернi ланцюги розмiром до 80 частинок в розчинниках
рiзної якостi. Для отримання показника скейлiнґу ν результати апроксимувались лiнiйною функцi-
єю, функцiєю, що враховувала поправки до скейлiнґу та вiдомою аналiтичною залежнiстю гiстограм
розподiлу радiусу ґiрацiї. Було виявлено, що для одних комбiнацiй характеристик полiмеру i якостi
розчинника, поправки до скейлiнґу є суттєвими, в той час як для iнших комбiнацiй вони нехтовно
малi. В кожному окремому випадку остаточна оцiнка для значення показника ν здiйснювалась ме-
тодом найменших квадратiв. Отриманi в такий спосiб значення ν лежать в iнтервалi ν = 0.55÷ 0.61 i
в основному зосередженi бiля величини 0.590, що є дуже близько до найточнiшої теоретичної оцiн-
ки ν = 0.588. Iснування такого iнтервалу можна пояснити як особливостями методу так i помiрними
довжинами симульованих ланцюгiв. Беручи до уваги згаданi неточностi, можна зробити висновок
про те, що спiввiдношення скейлiнґу в застосованому нами способi моделювання виконуються для
всiх трьох розглянутих характеристик полiмерного ланцюга.
Ключовi слова: полiмери, скейлiнґ, показники скейлiнґу, дисипативна динамiка
PACS: 61.25.Hq; 61.20.Ja; 89.75.Da
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