Statistics of linear polymer chains in the self-avoiding walks model
A strict statistics of self avoiding random walks in d-dimensional discrete (lattice) and continuous space is proposed. Asymptotic analytical expressions for the distribution and distribution density of corresponding random values characterizing a conformational state of polymer chain have been...
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irk-123456789-1204272017-06-13T03:04:45Z Statistics of linear polymer chains in the self-avoiding walks model Medvedevskikh, Yu.G. A strict statistics of self avoiding random walks in d-dimensional discrete (lattice) and continuous space is proposed. Asymptotic analytical expressions for the distribution and distribution density of corresponding random values characterizing a conformational state of polymer chain have been obtained and their quantitative estimation has been given. It is shown that conformation of polymer chain possesses a structure of spherical or, more commonly, of elliptical shell diffusely blurred both outside and inside the polymer coil, which nucleus is statistically void and has a radius of about half of Flory radius. Statistics of self-avoiding walks describes completely an effect of excluded volume and meets the terms of Flory method in Pietronero’s concepti. Запропонована точна статистика випадкових блукань із самоуниканням полімерного ланцюга у d-вимірному дискретному (гратка) і в неперервному просторах. Одержані асимтотичні аналітичні вирази для розподілу і густини розподілу відповідних випадкових величин, що характеризують конформаційний стан полімерного ланцюга, дана їх кількісна оцінка. Показано, що конформація полімерного ланцюга має структуру сферичної або, у більш загальному випадку, еліпсоїдної оболонки, що дифузно розмита як назовні, так і всередину полімерного клубка, ядро якого з радіусом порядка половини радіуса Флорі статистично пусте. Статистика випадкових блукань без перетинів повністю описує ефект вилученого об’єму і збігається з результатами методу Флорі у концепції П’єтронеро. 2001 Article Statistics of linear polymer chains in the self-avoiding walks model / Yu.G. Medvedevskikh // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 209-218. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 05.40.Fb DOI:10.5488/CMP.4.2.209 http://dspace.nbuv.gov.ua/handle/123456789/120427 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
A strict statistics of self avoiding random walks in d-dimensional discrete
(lattice) and continuous space is proposed. Asymptotic analytical expressions
for the distribution and distribution density of corresponding random
values characterizing a conformational state of polymer chain have been
obtained and their quantitative estimation has been given. It is shown that
conformation of polymer chain possesses a structure of spherical or, more
commonly, of elliptical shell diffusely blurred both outside and inside the
polymer coil, which nucleus is statistically void and has a radius of about
half of Flory radius. Statistics of self-avoiding walks describes completely
an effect of excluded volume and meets the terms of Flory method in
Pietronero’s concepti. |
| format |
Article |
| author |
Medvedevskikh, Yu.G. |
| spellingShingle |
Medvedevskikh, Yu.G. Statistics of linear polymer chains in the self-avoiding walks model Condensed Matter Physics |
| author_facet |
Medvedevskikh, Yu.G. |
| author_sort |
Medvedevskikh, Yu.G. |
| title |
Statistics of linear polymer chains in the self-avoiding walks model |
| title_short |
Statistics of linear polymer chains in the self-avoiding walks model |
| title_full |
Statistics of linear polymer chains in the self-avoiding walks model |
| title_fullStr |
Statistics of linear polymer chains in the self-avoiding walks model |
| title_full_unstemmed |
Statistics of linear polymer chains in the self-avoiding walks model |
| title_sort |
statistics of linear polymer chains in the self-avoiding walks model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120427 |
| citation_txt |
Statistics of linear polymer chains in the self-avoiding walks model / Yu.G. Medvedevskikh // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 209-218. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT medvedevskikhyug statisticsoflinearpolymerchainsintheselfavoidingwalksmodel |
| first_indexed |
2025-07-08T17:51:55Z |
| last_indexed |
2025-07-08T17:51:55Z |
| _version_ |
1837102129738678272 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 209–218
Statistics of linear polymer chains in
the self-avoiding walks model
Yu.G.Medvedevskikh
The Department of the L.V. Pisarzhevsky Institute of Physical Chemistry
of the National Academy of Sciences of Ukraine,
3a Naukova Str., 79053 Lviv, Ukraine
Received December 20, 2000
A strict statistics of self avoiding random walks in d-dimensional discrete
(lattice) and continuous space is proposed. Asymptotic analytical expres-
sions for the distribution and distribution density of corresponding random
values characterizing a conformational state of polymer chain have been
obtained and their quantitative estimation has been given. It is shown that
conformation of polymer chain possesses a structure of spherical or, more
commonly, of elliptical shell diffusely blurred both outside and inside the
polymer coil, which nucleus is statistically void and has a radius of about
half of Flory radius. Statistics of self-avoiding walks describes complete-
ly an effect of excluded volume and meets the terms of Flory method in
Pietronero’s concepti.
Key words: Flory method, statistics, self-intersection, polymer chain,
random walks, lattice, fatigue function, conformation
PACS: 05.40.Fb
1. Flory method and Pietronero’s concepti
One of the peculiarities of long flexible polymer chains is that their thermody-
namic and dynamic properties in solution are to a great extent determined not by
the nature of links but by their quantity, i.e. by conformation of macromolecules.
The basis of its description is a model of free-joined Kuhn’s chain and statistics
of random walks (RW) of its growing end [1]. In terms of those statistics, the dis-
tance distribution between the chain ends is described by Gaussian function having
two features: distribution mode corresponds to zero distance between chain ends,
and root-mean-square radius R0 of polymer coil and chain length N correlate as
R0 ∼ Nν , where ν = 1/2. Experimental data show the values close to 3/5.
Taking into account the effects of short range ordering, e.g. of fixed valent angles
and hinderance of internal rotation [2,3], do not change the shape of distribution
function and the correlation R0 ∼ N1/2 but increase R0 that can be interpreted in
terms of the increase of a statistical length of equivalent Kuhn’s chain segment [4].
c© Yu.G.Medvedevskikh 209
Yu.G.Medvedevskikh
It was several times noticed [2,3] that soft polymer chain configuration differs in
one important relation from RW trajectory: it must not intersect. This limitation
known as long range ordering effect or excluded volume effect demands new statis-
tics, i.e., statistics of self-avoiding walks (SAW). The attempts made (see [2]) have
not permitted to solve this problem completely.
That is why the authors of references [5,6] find the success of Flory method
(after de Gennes [7]) to be especially unexpected. In this method, the free energy F
of polymer chain conformation in kT units is presented by a sum
F = vN2/Rd + dR2/2a2N, (1)
where v = ad(1 − 2χ) is an excluded or an effective volume of monomer link, a is
its statistical length, χ is a Flory-Huggins parameter, R is a distance between chain
ends and d is the Euclidean space dimension.
The augend in (1) represents a pairwise repulsion energy of chain links in the self-
consistent field approximation, and the addend represents the polymer chain elastic
energy determined via entropy (F = −TS) of its conformation in RW statistics.
Minimization of F by R permits to find the equilibrium or the most probable
radius Rf of Flory coil:
Rf = afN
ν , (2)
ν =
3
(d+ 2)
, (3)
where af = a(1− 2χ)1/(d+2).
Formula (3) appears to be universal: it is in good accordance with the results of
physical experiment and computer simulation of SAWs on a lattice [5–8].
Flory conception put in expression (1) cannot give a strict explanation for those
results. That is why, Pietronero [5] has proposed another concepti of Flory method,
in accordance with which, both components of expression (1) possess an entropic
nature but the conformation entropy is determined by SAW statistics.
Pietronero’s concepti makes explainable a good correlation of Flory method for-
mula (3) with the results of SAW computer simulation on the lattice and gives a
heuristic landmark for the construction of strict SAW statistics.
2. Statistics of SAWs on a d -dimensional lattice
Let us incorporate the numbers ni of chain end steps along the i direction of
d-dimensional lattice with cell size equal to statistical length of chain link. Besides:
∑
i
ni = N, i = 1, d. (4)
The quantity of realization variants of random steps along the i direction is equal
to ni!/n
+
i !n
−
i !, where the numbers of steps in positive n+
i and negative n−
i directions
of wandering are correlated by n+
i + n−
i = ni. Considering that the probability of
210
Statistics of linear polymer chains in the SAW model
wandering positive or negative direction choice is assumed to be the same and equal
to 1/2, the probability ω(ni) that at given ni, n
+
i positive and n−
i negative steps will
be made is determined by Bernoulli distribution:
ω(ni) =
(
1
2
)ni
ni!/n
+
i !n
−
i ! . (5)
Then, incorporating the quantity of effective steps si = n+
i −n−
i along i direction
of wandering, we obtain n+
i = (ni + si)/2, n
−
i = (ni − si)/2. Then (5) can look like
the following:
ω(ni) =
(
1
2
)ni
ni!/((ni + si)/2)!((ni − si)/2)! . (6)
For d-dimensional wandering we have
ω(n) =
(
1
2
)N
∏
i
ni!/((ni + si)/2)!((ni − si)/2)! . (7)
One can see that change of sign at si does not change the value ω(n). That is
why ω(n) represents the probability that the trajectory of random wandering after
ni steps along i directions will be finished in one of 2d cells Mp(s), which coordinates
are determined by vectors s = (si), i = 1, d having a distinction in signs of their
components si only. These cells or states of chain end are equiprobable.
The condition of self-avoidance of a RW trajectory on d-dimensional lattice de-
mands the step not to fall twice in the same cell. From the point of view of chain
link distribution over cells it means that every cell cannot contain more than one
chain link. Chain links are inseparable. They cannot be torn off one from another
and placed to cells in random order. Consequently, the numbering of chain links
corresponding to wandering steps is their significant distinction. That is why, the
quantity of different variants of N distinctive chain links placement in Z identical
cells under the condition that one cell cannot contain more than one chain link is
equal to Z!/(Z −N)! .
Considering the identity of cells, a priori probability that the given cell will be
filled is equal to 1/Z, and that it will not be filled is (1 − 1/Z). Respectively, the
probability ω(z) that N given cells will be filled and Z − N cells will be empty,
considering both the above mentioned condition of placement of N distinctive links
in Z identical cells and the quantity of its realization variants, will be determined
by the following expression
ω(z) = Z!(1/Z)N(1− 1/Z)Z−N/(Z −N)! . (8)
Probability ω(s) of simultaneous event, meaning that the RW trajectory is also
the SAW trajectory and, at given Z, N and ni, it will fall with its last step into one
of 2d equiprobable cells Mp(s), will be equal to
ω(s) = ω(z)ω(n). (9)
211
Yu.G.Medvedevskikh
Let us find the asymptotic (9), assuming Z ≫ 1, N ≫ 1, ni ≫ 1 under the
condition si ≪ ni, N ≪ Z. Using the approximated Stirling formula lnx! ≈ x lnx−
x+ln(2π)1/2 for all x ≫ 1 and expansion ln(1−1/Z) ≈ −1/Z, ln(1−N/Z) ≈ −N/Z,
ln(1 ± si/ni) ≈ ±si/ni − (si/ni)
2/2, and assuming also N(N − 1) ≈ N 2, we will
obtain
ω(s) ≈ β exp
{
−N2/Z − 1
2
∑
i
s2i /ni
}
, (10)
where β−1 = e(2π)d/2.
With the growth of the quantity si of effective steps the chain end moves away
from its origin increasing the conformational volume where those SAW trajectories
are localized at that end in one of 2d equiprobable cells Mp(s). That is why the cells
quantity Z, allowed for a SAW trajectory, is not a fixed parameter of distribution
(10), but it is a function of vector s = (si) : Z = Z(s). This function choice can
be made based on different geometrical estimations – see figure 1. For instance,
if one considers that conformational volume of SAW trajectories is localized in d-
dimensional rectangle with vertexes Mp(s), then Z = 2d
∏
i |si|. If one assumes that
conformational volume is localized in the sphere with radius of R s = (
∑
i s
2
i )
1/2
then the expression Z = (
∑
i s
2
i )
d/2 can be used for a definition. At least, if one
assumes that fiducial cells Mp(s) belong to an ellipsoid surface, which semi-axes in
accordance to canonical equation
∑
i s
2
i /b
2
i = 1 must be equal to bi = dd/2|si|, it can
be laid: Z = dd/2
∏
i |si|. The analysis has shown that this definition is the most
suitable (it will be confirmed below), that is why we assume
Z = dd/2
∏
i
|si|. (11)
Putting together (10) and (11), we have
ω(s) = β exp
{
−N2
/
dd/2
∏
i
|si| −
1
2
∑
i
s2i /ni
}
, 1 6 si 6 ni. (12)
Function ω(s) determines the probability of a complicated event meaning that RW
trajectory is also a SAW trajectory at the same time and with it last step it falls into
one of 2d equiprobable cells Mp(s) or one can say that it realizes the state Mp(s).
This implies that it is numerically equal to the part of those SAW trajectories from
the whole quantity of RW trajectories (2d)N , which realize the stateMp(s). Quantity
L(s) of such trajectories define the thermodynamical probability of a realization of
the state Mp(s)
L(s) = (2d)Nω(s). (13)
Summing L(s) over the whole set Ω of chain end states, we find the total quantity
L of SAW trajectories
L = (2d)N
∑
Ω
ω(s). (14)
Designating
c(s) =
∑
Ω
exp
{
−N2
/
dd/2
∏
i
|si| −
1
2
∑
i
s2i /ni
}
, (15)
212
Statistics of linear polymer chains in the SAW model
Figure 1. Geometrical variants of determination of RWEI trajectories conforma-
tional volume at given coordinates of equiprobable states Mp(S) of chain end for
a two-dimensional lattice.
the function
w(s) = 1/c(s) exp
{
−N2
/
dd/2
∏
i
|si| −
1
2
∑
i
s2i /ni
}
(16)
is normalized to unity and define the mathematical probability of chain end distri-
bution over the states Mp(s) of d-dimensional lattice. It equals to the relation of
quantity L(s) of those SAW trajectories, which realize the state Mp(s), to the total
quantity of SAW trajectories L : w(s) = L(s)/L.
In turn, the ratio L/(2d)N equals to the part of total number of SAW trajectories
from the total number of RW trajectories and is a function g(N) of SAW trajectories
fatigue, in accordance to terminology [5,6], and quantitatively, it can be determined
via the normalizing constant of distribution (16):
g(N) = L/(2d)N = βc(s). (17)
3. Statistics of SAWs in continuous space
Expressions (12)–(17) represent the SAW statistics at d-dimensional lattice. Let
us work with a metric space, incorporating the variable of shifting the x i-semi-axis
of conformational ellipsoid, with the states Mp(s) belonging to its surface
xi = a|si|d1/2 (18)
213
Yu.G.Medvedevskikh
and parameter σi is the standard deviation of the Gaussian part of the distribution
σ2
i = a2nid. (19)
In accordance with (4), the coupling is laid on the value σi
∑
i
σ2
i = a2Nd. (20)
Since s2i /ni = x2
i /σ
2
i , d
d/2
∏
i |si| =
∏
i xi/a
d , the distribution (12) can be rewritten
as follows
w(x) = 1/c(x) exp
{
−adN2
/
∏
i
xi −
1
2
∑
i
x2
i /σ
2
i
}
, a 6 xi 6 ani. (21)
If ximin = a ≪ ximax = ani, the distribution (21) can be considered as a continuous
one with a normalizing constant
c(x) =
xmax
∫
xmin
exp
{
−adN2
/
∏
i
xi −
1
2
∑
i
x2
i /σ
2
i
}
dx, a 6 xi 6 ani, (22)
where dx = Πidxi, and the integral is d-aliquot. In this case: c(x) = addd/2c(s).
Substitution of (18) induces an essential distinction between w(x) and w(s): w(x)
determines the probability w(x)dx that the SAW trajectory at given parameters N
and σi will be ended in elementary volume dx, laying on an ellipsoid surface with
semi-axes xi, i = 1, d. In the other case, all the surface of ellipsoid is a geometrical
place of points or chain end states with the same corresponding distribution density
w(x).
Maximum of w(x) at given N and σi corresponds to the most probable or equilib-
rium state of a polymer chain. Semi-axes x0
i of equilibrium conformational ellipsoid
can be found from the condition
∂ lnw(x)/∂xi = adN2
/
xi
∏
i
xi − xi/σ
2
i = 0 when xi = x0
i . (23)
Solving the system of algebraic equations (23), we obtain
x0
i = σi
(
adN2
/
∏
i
σi
)1/(d+2)
. (24)
Let us continue to consider the situation when all the directions of chain steps
are equiprobable, i.e.
ni = N/d, (25)
σ2
i = σ2
0 = a2N. (26)
Substitution of (26) into (24) makes the semi-axes of equilibrium ellipsoid equal,
and equal to the radius of Flory coil
x0
i = Rf = aNν , (27)
where ν is defined by (3).
214
Statistics of linear polymer chains in the SAW model
4. Some properties of distribution in SAW statistics
To avoid the utilization of parameter a, i.e., the length of chain link in calcula-
tions, we will use a dimensionless variable
λi = xi/Rf . (28)
Besides, let us consider only the distribution corresponding to the condition
(25) of RW equiprobability over the axes of d-dimensional space. In this case the
expressions (21) and (22) can be rewritten as follows
w(λ) = 1/c(λ) exp
{
−(Rf/σ0)
2
(
1
/
∏
i
λi +
1
2
∑
i
λ2
i
)}
, (29)
c(λ) = I0, (30)
I0 =
λmax
∫
λmin
exp
{
−(Rf/σ0)
2
(
1
/
∏
i
λi +
1
2
∑
i
λ2
i
)}
dλ, (31)
where dλ =
∏
i dλi, λmin = ad1/2/Rf , λmax = aN/Rfd
1/2. Normalizing constants of
distributions (21) and (29) are correlated with c(λ) = c(x)/Rd
f , that is why we have
a coupling
c(s) = (Rf/a)
dc(λ)/dd/2 = c(x)/addd/2. (32)
In accordance to (26) and (27)
(Rf/σ0)
2 = N (4−d)/(d+2). (33)
Quantitative estimation of I0 (and of the following ones) is performed using the
Romberg’s algorithm from Mathcad software with accuracy parameter of Tol =
10−4. Values I0 are listed in the table.
Values λ0
i = 1, i = 1, d, relating to maximal distribution density, correspond to
the equilibrium state of polymer coil
w(λ0) = 1/c(λ) exp{−(Rf/σ0)
2(d+ 2)/2}. (34)
Distribution density decreases at any deviation of values λi at λ
0 = 1: it this case
it tends to zero not only at λi → λmax, but also at λi → λmin. Consequently, the
probability of chain end placement close to its origin is negligibly small. The latter
makes a radical distinction between SAW statistics and gaussian RW statistics.
Figure 2 represents the illustrations for several variants of function w(λ) be-
haviour for a three-dimensional space d = 3 at (Rf/σ0)
2 = N1/5, equal to 3 and 4,
and changing of one variable λz at fixed values of λx = λy = 1, λx = λy = 0.8 and
λx = λy = 1.25. The two latter variants clearly show what will happen at the defor-
mation of polymer coil: at compression along x and y axes, polymer chain stretches
along z axis transforming to oblong ellipsoid of revolution; on the contrary, at com-
pression along z axis, polymer chain stretches along x and y axes transforming to
oblate ellipsoid of revolution.
215
Yu.G.Medvedevskikh
Figure 2. Behaviour of distribution density w(λ) at the variation of λ = λz and
fixed values of λx and λy for d = 3. N1/5 = 3 (1, 2, 3), N1/5 = 4 (4), λx = λy =
1 (1, 4), λx = λy = 0.8 (2), λx = λy = 1.25 (3).
Starting moments M(λk) (k = 1, 2) of distribution (29) are numerically calcu-
lated (as it was mentioned before) using the correlation
M(λk) =
λmax
∫
λmin
λk
iw(λ)dλ. (35)
Values M(λk) for Flory coil do not depend on the index i. Root-mean-square
value λ0 = (M(λ2))1/2, dispersion D(λ) = M(λ2)− (M(λ))2 and standard deviation
σ(λ) = (D(λ))1/2 of distribution (29) are calculated via the values M(λ) and M(λ2).
A part of the data calculated is presented in the table.
Table 1. Some characteristics of distribution density w(λ)
N1/5 I0 I0(L) M(λ) M(λ2) λ0 D(λ) σ(λ)
2 7.72 · 10−3 8.39 · 10−3 1.139 1.487 1.219 0.191 1.437
4.0 1.91 · 10−5 2.00 · 10−5 1.073 1.246 1.116 0.094 0.307
6.0 7.11 · 10−8 7.33 · 10−8 1.048 1.163 1.078 0.065 0.255
8.0 3.13 · 10−10 3.21 · 10−10 1.035 1.125 1.060 0.053 0.230
10.0 1.52 · 10−12 1.55 · 10−12 1.026 1.099 1.048 0.045 0.217
12.0 7.79 · 10−15 7.9 · 10−15 1.024 1.083 1.041 0.034 0.185
216
Statistics of linear polymer chains in the SAW model
5. The fatigue function
The fatigue function g(N) is defined by (17) and expresses the part of the total
quantity of SAW trajectories from the total quantity of RW trajectories. The latter
quantity (2d)N is strictly defined for the lattice only, i.e. for the discrete space of
wanderings. The total quantity of RW trajectories for continuous space is unknown,
however the fatigue function can still be defined by (17) via normalizing constant
c(x). Considering coupling (32), we can express the fatigue function via normalizing
constant c(λ):
g(N) = β(Rf/a)
dd−d/2c(λ). (36)
One can use the quantitative values c(λ) = I0 presented in the table for calcu-
lations. However, in case of d < 4 space, one can give an approximated analytical
expression I0, using the Laplace’s method [10]. Then, at d = 3, the fatigue function
can be written as follows
g(N) ≈ g0N
3/2 exp{−5/2N1/5}, (37)
where g0 = (e2
√
533/2)−1. Function (37) possesses a maximum at N 1/5 = 3. Its
graphical representation is given in figure 3.
Figure 3. Dependence of fatigue function g(N) by (37) on a chain length N .
217
Yu.G.Medvedevskikh
References
1. Kuhn W. // Koll. Zs., 1934, vol. 68. p. 2.
2. Volkenshtein M.V. Configurational Statistics of Polymer Chains. Moscow-Leningrad,
USSR Academy of Sciences, 1959 (in Russian).
3. Flory P. Statistical Mechanics of Chain Molecules. Moscow, Mir, 1971 (in Russian).
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Статистика лінійних полімерних ланцюгів у моделі
випадкових блукань із самоуниканням
Ю.Г.Медведевських
Відділення Інституту фізичної хімії
ім. Л.В.Писаржевського НАН України
79053 Львів, вул. Наукова, 3а
Отримано 20 грудня 2000 р.
Запропонована точна статистика випадкових блукань із самоуникан-
ням полімерного ланцюга у d-вимірному дискретному (гратка) і в не-
перервному просторах. Одержані асимтотичні аналітичні вирази для
розподілу і густини розподілу відповідних випадкових величин, що
характеризують конформаційний стан полімерного ланцюга, дана їх
кількісна оцінка. Показано, що конформація полімерного ланцюга
має структуру сферичної або, у більш загальному випадку, еліпсоїд-
ної оболонки, що дифузно розмита як назовні, так і всередину по-
лімерного клубка, ядро якого з радіусом порядка половини радіуса
Флорі статистично пусте. Статистика випадкових блукань без пере-
тинів повністю описує ефект вилученого об’єму і збігається з резуль-
татами методу Флорі у концепції П’єтронеро.
Ключові слова: метод Флорі, статистика, випадкові блукання,
полімерний ланцюг, конформація, функція виживання
PACS: 05.40.Fb
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