Internal links of the polymer chain in the self−avoiding random walks statistics Текст научной статьи по специальности «Биологические науки»

UDC 544.1:519.246

Yu. G. Medvedevskikh, A. M. Kochnev, G. E. Zaikov,

Kh. S. Abzaldinov

INTERNAL LINKS OF THE POLYMER CHAIN IN THE SELF-AVOIDING RANDOM

WALKS STATISTICS

Key words: macromolecule, internal link, conformation, distribution, solution.

Within the frame of the self-avoiding random walks statistics (SARWS), the derivation of the internal n-link (1 <<n<<N) distribution of the polymer chain with respect to the chain ends is suggested. The analysis of the obtained expressions shows, that the structure of the conformational volume of the polymer chain is heterogeneous; the largest density of the number of links takes place in conformational volumes nearby the chain ends. It can create the effect of blockage of the active center of the growing macroradical and manifest itself as a linear chain termination. The equation for the most probable distance between two internal links of the polymer chain was obtained as well. The polymer chain sections, separated by fixing the internal links, are interactive subsystems. Their total conformational volume is smaller than the conformational volume of undeformed Flory coil. Therefore, total free energy of the chain sections conformation equals to free energy of the conformation of deformed (i. e. compressed down to the total volume of the chain sections) Flory coil.

Ключевые слова: макромолекула, внутренняя связь, конформация, распределение, решение.

В рамках статистики случайных блужданий без самопересечения (СББС) предлагается вывод распределения внутренней п-связи (1<<п<<Ы) полимерной цепи относительно ее концов. Анализ полученных выражений показывает, что структура конформационного объема полимерной цепи является гетерогенной; наибольшее количество связей имеет место в конформационном объеме около концов цепи. Это может создать эффект блокирования активного центра растущего макрорадикала и проявляться в виде линейного обрыва цепи. Также было получено уравнение для наиболее вероятного расстояния между двумя внутренними связями полимерной цепи. Участки полимерной цепи, разделенные фиксированием внутренних связей, являются интерактивными подсистемами. Их общий конформационный объем меньше конформационного объема недеформированного клубка Флори. Таким образом, общая свободная энергия конформации участков цепи равна свободной энергии конформации деформированного (т. е. сжатого до общего объема участка цепи) клубка Флори.

Introduction

In Gaussian random walks statistics, the mean-square end-to-end distance R for a polymer chain, as well as mean-square distance between two not very closely located internal links obey general dependence[1]:

R = an12 n >> 1 (1)

where a is the mean length of the chain link according to Kuhn [2]; n is the chain length or the length of a given chain section, expressed by the number of links in it.

Self-avoiding random walks statistics

(SARWS) determines the conformational radius RNf of the undeformed Flory coil as the most probable end-to-end distance of the polymer chain[3,4]:

RNf = aN3/(d+2)

yN,f u" (2)

Here N is the total chain length, d is the Euclidian space

dimension.

According to (2), Flory coil is a fractal, i. e. an object, possessing the property of the scale invariance in dimensionality space df = (d + 2 ) 3.

At derivation[3] of (2), however, the distribution of the internal polymer chain links in its conformational space remains unknown, therefore, it can not be indicated in advance that the distances

between the terminal and internal chain links or

between the internal ones obey the same dependence (2) at the value of N as the length of the selected section of a

polymer chain.

Study of the problem of internal polymer chain links' distribution is based mainly on the analysis [5,6] of the scale distribution function Pij(r) of distance r between two links with ordinal numbers i and j:

P (r)= |/ - j]-dof

(r4- j\ 1 (3)

Function f (r /1/ - j| ) = f (x) is usually

written in the form of power or exponential dependence on the only variable x:

f(x) ~ x f(x) ~ exp{-xs}

at x << 1, at x >> 1.

(4)

Studying the correlations between two arbitrary points i and j of a polymer chain, Des Cloizeaux[7] suggested dividing the scale function Pij(r) into three classes, that describe the distribution of distances between two terminal points of a polymer chain (P(o)j(r) with exponents 8o and So at i = 1, j = N), between the initial and internal points (P(1)ij(r) with exponents 0\ and S1 at i = 1, 1 << j << N) and between two internal points (P(2)j(r) with exponents 02 and S2 at 1 << i << j << N), respectively.

Using the method of the second order e-expansion within the range x << 1 for the space d = 3, Des Cloizeaux[7] has obtained in particular: 0O = 0,273, ex = 0,459, e2 = 0,71.

To evaluate the exponents dt and Si some other methods were used as well. Let us present some of the obtained results: [8R = 0,27; [8,9,1O]01 = 0,55, 0,61,

0,70; [8,9]02 = 0,9, 0,67; [8,11]So = 2,44, 2,5; [8]S1 = 2,6; [8]S2 = 2,48.

In spite of the spread in exponent values, they unambiguously indicate (especially when comparing the values of do, 81 and 02), that distribution function Pj(r), retaining their scale universality, quantitatively significantly depends on whether we consider the distance between terminal points, a terminal and internal one or between two internal points of a polymer chain. Whereas the proposed methods of analysis establish this fact, they however do not reveal the reason of the above-mentioned difference. Reference to strengthening the effects of the volume interaction between the internal links of a polymer chain can not be absolutized, since these effects can not be taken into account at computer simulation of self-avoiding random walks, but the results of the calculations according to them give the same estimations of exponents 8i and Si as the analytic methods that take into account the volume interaction.

The shortcoming of the proposed approaches is also the fact that the scale distribution function Pij(r) is approximate and does not enclose the most significant region of parameter x changing between x << 1 and x >> 1, where Pij(r) takes on maximal values. Finally, it should be noted that the role of the length of the second section of a polymer chain (at evaluating 01 and S1 the length of the second section is extrapolated to ®) or the lengths of its two sections (at evaluating 02 and S2) is outside of the analysis.

Hence, the suggested approaches do not allow to solve the problem of the internal links distribution for a polymer chain completely. In the present work we propose its analytic solution in terms of SARW strict statistics, i. e. without taking into account of the so-called volume interaction.

Initial statements

Preliminary let us briefly introduce the main statements of SARW statistics, that are necessary for the subsequent analysis[3,4]. The Gaussian random walks in N steps are described by the density of the Bernoulli distribution:

(6)

a(N,s) = (-) n

n,!

(5)

v2J V [(n, + S / 2] [(n, - s, )/2]!

where ni is a number of the random walk steps in

/'-direction of d-dimensional lattice space with the step length a, which is equal to the statistical length of Kuhn link; s/ is the number of effective steps in -direction:

s i = s i +-s i - where s i +, s i - are numbers of positive

and negative steps in i-direction. Numbers of n i steps are limited by the following correlation:

The condition of self-avoidance of a random walk trajectory on d-dimensional lattice demands the step not to fall twice into the same cell. From the point of view of chain link distribution over cells it means that every cell can not contain more than one chain link. Chain links are inseparable. They can not 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 can not 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 a>(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

a

(Z / =

(7)

Probability density a>(N)of the fact that random walk trajectory is at the same time SARW statistics trajectory and at given Z, N, ni will get the last step in one of the two equiprobable cells, which coordinates are set by vectors s = (s t), differentiated only by the signs of their components si, is equal to

a(N) = a>(Z)a(N,s) .

(8)

Let us find the asymptotic limit (8) assuming Z

>> 1, N >> 1, nt >> 1 under the condition si << ni, N << Z. Using the approximated Stirling formula ln x! » x ln x - x + ln (2^)12 for all x >> 1 and expansion ln(1-1/Z) » -1/Z, ln(1-N/Z) » -N/Z, ln(1±si/ni) » +s,/n, - (s/n)2^, and assuming also N(N - 1) we will obtain[3,4]:

(N) = exp{-N2 /Z - (1/ 2)£ s2 /nj •

(9)

Transition to the metric space can be realized by introduction of the displacement variable

xi = a\s,\d

1 / 2

(10)

and also the parameter cr(- - the standard deviation of Gaussian part of distribution (9):

g, = a nd.

(11)

Then

N

s2/n, = x2/af ,

Z = ]Ix7ad

and for the metric space expression (9) becomes:

(o(N) = exp

adN2 -1 v x2

n x, 2 / af

(12)

(13)

(14)

Here J_ lx, is the volume of conformational ellipsoid

i

with the semiaxes of xi, to the surface of which the states of the chain end belong.

A maximum of a(N) at the set values of a, and N corresponds to the most probable, i. e. equilibrium state of the polymer chain. From the condition of

o f

semiaxes x of

a

(N)/d xi = 0 at x, = x 0 we find

the equilibrium conformational ellipsoid [3]: x0 =a(adN2 Uai)1(d+2).

(15)

In the absence of external forces, all directions of random walks of the chain end are equiprobable, that allows to write:

a2 = aN = a 2N .

(16)

(17)

The substitution of (17) into (15) makes the semiaxes of the equilibrium conformational ellipsoid identical and equal to the undeformed Flory coil radius:

X0 = Rn f. Let us underline two important

circumstances. First, SARW statistics leads to the same result, i. e. to formula (2), that Flory method, which takes into account the effect (repulsion) of the volume interaction between monomer links in the self-consistent field theory. However, as it was explained by De Gennes[6], accuracy of formula (2) in Flory method is provided by excellent cancellation of two mistakes: top-heavy value of repulsion energy as a result of neglecting of correlations and also top-heavy value of elastic energy, written for ideal polymer chain, that is in Gaussian statistics. Additionally, one must note, that formula (2) is only a special case of formula (15), which represents conformation of polymer chain in the form of ellipsoid with semiaxes x° 4 Rf, allowing to consider this conformation as deformed state of Flory coil.

Second, obtained expression (14) for density of distribution of the end links of polymer chain is not only more detailed but also more general than scale dependencies (4) at 6o and So, which are approximately correct only at the limits xi /R,- << 1 and xi /Rf >> 1.

Free energy FN of the equilibrium

conformation is determined by the expression

Fn =-kTlna(N)

at X, = XJ

(18)

From here for undeformed Flory coil we have:

Fn,f = (1 + d/2)kT(RNf/aN)2. (19)

For the deformed one -

Fn = Fn, f /A , (20)

where Av is the repetition factor of Flory coil's volume deformation:

Av = № R =№i • <21>

/ /

where Xi is a repetition factor of linear deformation,

A = X°/Rf. (22)

At any deformations the conformational volume diminishes, therefore in general case A <1 [3].

Sarw statistics for the internal links of a chain

The internal n-link (1<<n<<N) divides polymer chain into two sections with the lengths of n and N-n links, respectively. This situation is illustrated with high quality in Figure 1.

Fig. 1 - The scheme, explaining the necessity to enter new conformational volume Z’ at fixing of the position of the polymer chain internal n-link

Let us assume, as it is shown in Figure 1, that the most probable position of n-link with respect to the chain end is the surface of sphere with radius R n and chain's n-link accidentally appeared at point O n on this sphere. Then with respect to this point N-link of the chain over N-n steps with the highest probability should appear on the surface of sphere with radius

R N-n .

As it can be seen from Figure 1, Rn + Rn n < Rn . Consequently, fixing of n-link position diminishes the polymer chain's conformational volume regardless of where specifically point O n is situated on the surface of sphere with radius R n .

This means that for the analysis of SARW statistics of the chain's internal links in the lattice space,

a new number of cells Z' < Z needs to be introduced. Then the probability density of the random walk trajectory's self-avoiding for the polymer chain with fixed position of the internal link can be described by the Bernoulli distribution in the same form (7), but with a new value of cells number:

<z^ ( (1 - or-

(23)

The Gaussian random walks in n and N-n steps of the first and second chain sections can be described thereby by the Bernoulli distribution (5), but here in expressions for the distribution density O (n, s) for the first chain section and O (N-n, s) for the second one, respectively, the following conditions of normalization must be implemented:

Zft = n,

i

Z ni = N - n

(24)

(25)

and in place of the factor (1/2) factors (1/2)” and (1/2)N-n respectively should be used.

As o(Z') applies to the whole polymer chain, the distribution densities m(ri) and m(N-ri) of the SARW statistics trajectories for the first and second chain sections can be determined by the following expressions:

a

(n) = (a(Z'))nNa(n, s),

(26)

a(N - n) = (a(Z'))

' ))(N-n)/N

a(N - n,S) . (27)

In an asymptotic limit the expressions (26) and (27) can be written:

a

(n) = exp{-aN 2/Z'-(12 )Z sf/nj, (28)

i

(N - n) = exp{-(1 - a)Nj/Z' - (1/2)Zsj/n,}- (29)

The lengths of every section fractions of the total chain length are introduced here as:

a = n/N, 1 -a = (N - n)/N. (30)

Defining the variables of the metric displacement of Xj and Vi in the form (10) and standard deviations an, and aN-n, of the Gaussian

part of distribution (28) and (29) in the form (11), instead of (28) and (29) we obtain:

O(n) = exp{-aNyZ' - (1/2)ix2 jal), (31)

a(N - n) = exp{-(1 -a)NyZ' - (12ivl/*l-nj). (32) Owing to the normalization (24) and (25) we

have:

ZG ji = a2 nd, Z gN-n,i = a2 (N - n /. (33)

The values

nxi and IIy,

are the

volumes of the conformational ellipsoids with the semiaxes x, and y, of the first and second sections of the polymer chain, respectively. Hence, as laid down earlier (13), it is possible to write:

Z' = (nx, + nYl)lad .

(34)

Entering the volume fractions of the proper conformational ellipsoids

p = nx, / nx, + nv,

i V / /

1-p = n v,/ in x +n Vi

, V , ,

we obtain:

(35)

a(n) = exp{-edajSNyn xi - ( 12)Z x2 jaj,}, (36)

ofN-n) =exp-a (1-a)(1-^)N/nY -(Î2)ZYf/ GSLn, }(37)

These expressions are sought densities of distribution of internal links of the chain from its ends. Parameter ft will be determined later.

The most probable states of the polymer chain sections meet the conditions da(n)jSx, =0 at

xi = x°, Sa(N - n)/Sy, =0 at y j =y ° . Using them and assuming that values P and 1- P do not depend

on specific realizations of x i and y i , i. e. these are functions of n and N-n only, we find

1/ (d+f )

x 0 =a„,(adapN 7 Ua„)

y0=&N-M(*d(1-a)(1-0)iN / n<W’

(38)

(d+f)

(39)

In the absence of external forces, all directions of random walks are equiprobable, therefore according to (31) it is possible to write:

j j j 2 Gn,i = Gn = a n = aNa>

(40)

al-nj = aN-n = a2 (N - n) = a2N (1 -a). (41)

Using (40) and (41) in (38) and (39), we will obtain expressions for the equilibrium conformational radii of both polymer chain sections:

Rn=RNf a(d+2 ^(d+2 ^,

(42)

Rn-n = Rw f( 1 - a)j (d+2)(1 - ^ (d+2). (43)

The conformational volumes here are equal to

nx- = Rd, n Vi = Rn n, therefore expression (35) i i may be rewritten in the form

p (1 -p)=Rd/Rin (44)

From (42) - (44) it follows:

P = ad /[ad + (1 - a)d ], 1 -P = (1 -a)d /[ad + (1 -a)d]. (45)

Excepting P from (42) and (43), we get

finally

Rn = Ria /[ad + (1 - a)d J7 (d+2 (, (46)

Rn-n = Rn, f(1 -a)/[ad + (1 -a)d( ^+2'. (47)

Eqns. (46) and (47) together determine the most probable, that is the equilibrium distances of the internal link from the polymer chain ends.

As one can see, although between R n and R N-n a simple correlation

RJ Rn - n = a /(1 - a) is observed, each of these values depends by complicated way not only on its own section length but also on the length of another one.

Formulas (46) and (47) are correct at d <4 [2], including at d = 1. For one-dimensional space from (46) and (47) follows physically expected result Rn = an, Rl-n = a(N - n).

Structure of the polymer chain conformational space

From eqns. (46) and (47) follows that

RJ Rn f >a and Rn-n / Rn f > 1 - a, and

signs of equality are achieved only on the chain ends, i. e. at a = 1 and a = 0, respectively. This gives

evidence to heterogeneity of the polymer chain

conformational volume. In addition, because of interconnection between Rn and RN-n, both chain sections are not fractals. Let us comment both

circumstances, confronting the values Rn and Rl-n from (46) and (47) with those values of Rn f and Rl-n f, which these chain sections would have, if they were free and submitted to fractal correlation of the type (2):

Rnf = an3 (d+2) = RNfa3 (d+2), (48)

Rn_n: f = a(N - n)3 (+2) = Riff(1 - af (+2). (49)

From comparison of (46) - (49) it follows:

RJ Rn_f = a(d-1)/(d+2) /[ + [1 - a)d J1 (d+2), (50)

Rln/ Flnf =(1-a)(d-1)l(d+2)/{cd +(1-a)dJ1 d2) (51) Dependencies (50), (51) are illustrated on Figure 2 for the option of d = 3.

a

Fig. 2 - Ratios RJ Rn f and Rn_J Rn_nf

calculated on the eqns. (50) and (51) depending on a and 1-a

As one can see, only in the area a >0.5 and correspondingly 1-a >0,5, ratios Rn/ Rn f and

RN n/ RN n f, though are more than 1, but

insignificantly. It allows to consider of these chain sections as the fractals objects with a small error and to describe them by fractal dependences (48) and (49). However, for short chain sections, i. e. at a <0,5 or

1-a <0,5 ratios RJ Rn f and RN-J Rl-n f

become less than 1 and sharply diminish towards the chain ends, which indicates the compression of the conformational volume space nearby the ends of a chain.

Yet even more evidently heterogeneity of the structure of polymer chain's conformational volume becomes apparent at the analysis of volume density p ,

i. e. the numbers of links in the unit of conformational volume for given chain section. Let us be limited to considering only the first chain section n in length, for which

pn = n/Rd . (52)

Using (46), we get pj Pn =[ad + (1 -a)d ]d/rd+22 / ad1, (53)

where pN = N / RNd f is an average links' density in

conformational volume of the whole polymer chain.

The correlation between local and average density of the chain links is illustrated on Figure 3 at d = 3. Evidently, ratio pj pN in the range of a <0.2 at a ^ 0 sharply increases, and for example at

4

a =0.01 achieves the value of 10 order. As the

dependence of the ratio pN n/ pN is similar, but asymmetric, it can be concluded that the conformational volumes near the chain ends are strongly compressed, so that the density of links in them considerably exceeds the average one over the conformational volume of the whole chain. With some caution one can suppose that the conformational volumes near the chain ends have a globular structure.

Fig. 3 - Ratio between local density of polymer chain link pn and average one pN depending on a calculated on (53)

To support this point, we propose also considerations based on experimental research of dymethacrylates postpolymerization kinetics, i. e. dark, after turning off UV irradiation, process of polymerization[4]. It was found, that the chain termination is linear, and its kinetics submits to the law of stretched Kohlrausch exponent:

y/(t) = gexp{- t/to Y

(54)

Here g and 0 < y < 1 are constants; 10 is

characteristic time of linear chain termination.

A theoretical derivation[12] was based on the idea, that linear chain termination is the act of «self-burial» of macroradical's active center and manifests itself as the act of chain propagation, leading into a trap. Taking into account the fractal properties of polymer chain and assuming that a set of traps in its conformational volume is a fractal as well, we obtain the expression similar to (54):

where df = (d + 2 )j 3 and dl are fractal dimensions of the conformational volume of macroradical and a set of traps in it. From experimental data the value of y = 1/ £ = 0.6, so we can accept £ = df. Then for the dimension of the traps set fractal the expression (57) follows from (56):

dL =

df + d = 2d +1

(57)

2 3

which not only satisfactorily coincides with experimental value of dL at d = 3, but also presents

physically justified value of dL = 1 at d = 1.

Correlation (57) shows that df < dL < d . Therefore, in the reaction zone of growing macroradical, there are both «strange» traps formed by polymer chains, external for given macroradical, with fractal dimension close to d and «own» ones with fractal dimension of polymer chain df = (d + 2)3 .

This derivation in[12] is based only on kinetic researches of dymethacrylates postpolyme-rization, but, apparently, it coordinates well with the results of present work, according to which «own» traps for a growing macroradical are caused by high density of links in the conformational volumes near the polymer chain ends, that can screen or even block up macroradical active center.

Free energy of conformation of polymer chain sections

Let us determine free energies Fn and FN-n of the conformation of polymer chain sections, separated by fixing of internal n-link, on type (18) by expressions:

Fn = -kT lna(n) at xi = x°, (58)

Fn - n = -kT lna(N - n) at y = y/. (59)

For the equilibrium state in the absence of external forces, i. e. at all x,0 = Rn and y° = Rn- n, using (36), (37), (45)-(47) in (58) and (59), we get

\d l2/ (+2!

y/(t) = gexp{-t/ Tp(pm/P0)£Y

£ Y1/£

(55)

Fn = FN , f al\ttd + (1 -af J

Fn-n = Fn ff (1 -a)l[ad +(1 -a)df7+2)

(60)

(61)

where Tp is characteristic time of chain propagation

act, pm and p0 are monomer and traps concentrations

in the macroradical's conformational volume, respectively.

According to the derivation of expression (55)

£ = df/(f + dL -d),

(56)

From here it follows:

Fn, + Fn-n = Fn ,/ [«d + (1 -a)d J2

>./(d + 2)

(62)

Hence, Fn + Fn n >Fn f. It is related to the

fact that two chain sections are thermodynamic subsystems, which interact with each other. Thus, fixing

a

the position of a polymer chain internal link increases its negative entropy and positive free energy of conformation due to diminishment of the polymer chain conformational volume. Therefore the sum Fn + Fn-n of free energies of the chain sections conformation must be compared not to FN f, representing free energy of undeformed Flory coil conformation with the volume of RN f, whereas for free energy of conformation Fn of

the Flory coil deformed to the volume Rd + Rl- n , determined by the expression (20).

As the multiple (repetition factor) of the volume deformation AV here is equal to

\d 12 (+2)

a=(+Rd-n )Rdf=[+d_ad f

we have identically

Fn + FN- n = FN, f /A .

(63)

(64)

It is now possible to accomplish the reverse transition and write the expression

[(n )o(N - n ')fV = a(N),

which was not obvious in the beginning.

(65)

The most probable distance between two internal links of a polymer chain

If two internal links of a chain are selected according to the condition 1<< k << n << N, the polymer chain is divided into three sections with the lengths of k, n-k and N-n, to which the fractions of the total chain length ak = k/N, an-k = (n - k)/ N and aN - n = (N - n)/N correspond.

Let us suggest that j = 1, 3 are numbers of sections with quantities of links k, n - k and N - n, to which fractions a1 = k /N, a2 = (n-k) / N and a3 = (n-k) / N from general number N of links in a chain are corresponded.

Extending the above-mentioned procedure of analysis of two chain sections to three sections, we get the general expression for the distribution density of the end of the given section regarding to its beginning:

Oj) = exp{-ad api2/nx,2 - (1/2)i x,2 laf),

j = 13, i = 1,d .

(66)

Here ftj is a fraction of conformational volume of the given section in the sum of conformational volumes of

all sections; xjt , i = 1,d are semiaxes of conformational ellipsoid j with center in the beginning of the given section. The surfaces of this section involve the states of its end.

The square deviations ffji of Gaussian part of distribution (66) obey the normalization conditions of the form:

= a2ajNd.

(67)

At equiprobability of walks in all directions of d-dimensional space we have:

Cj = <jj0 = a ajN.

(68)

In this case the most probable distance between the beginning and the end of the chain in the given section will be equal to:

Rj =

RN,faj

d \1/( d+2 )

(69)

and for ßj the following expression will be

correct:

ß, =

a

(70)

According to (69) the value of Rj depends not only on the length of the given chain section but also on where this section has been chosen. Thus, we can note again that selected three sections of polymer chain are not independent, but are interactive subsystems. Therefore, total free energy of conformation of the chain three sections exceeds free energy of conformation of undeformed Flory coil. But the following equality holds identically:

Fk + Fn-k + FN-k = FN,f / \

(71)

where the multiplicity of Flory coil's volume deformation at the division of the chain into three sections is determined by the expression:

d i Qd Qd

k r'n-k ^ r'N-n

)/ .

(72)

Conclusions

Fixation of the position of polymer chain internal links separates its conformational volume into interacting subsystems. Their total conformational volume is smaller, and free energy is larger than the conformational volume and free energy of Flory coil, respectively. From expressions, which determine the probable distance between the polymer chain's internal link and its ends, as well as between any internal links, it follows that the structure of the polymer chain conformational volume is heterogeneous: the largest density of the number of links is observed near the chain ends. This can result in blockage of the macroradical's active centre and appear as a linear chain termination.

d

References

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© Yu. G. Medvedevskikh - L. M. Lytvynenko Institute of Physical-Organic Chemistry & Coal Chemistry of Ukraine National Academy of Sciences, Physical Chemistry of Combustible Minerals Department; A. M. Kochnev - Kazan National Research Technological University, eleonora@kstu.ru; G. E. Zaikov - N. N. Emanuel Institute of Biochemical Physics of Russian Academy of Sciences; Kh. S. Abzaldinov - Kazan National Research Technological University.