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ORGANIC AND PHYSICAL CHEMISTRY
USING CHEMICAL KINETICS:
PROSPECTS AND DEVELOPMENTS

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ORGANIC AND PHYSICAL CHEMISTRY
USING CHEMICAL KINETICS:
PROSPECTS AND DEVELOPMENTS

Y.G. MEDVEDEVSKIKH
ARTUR VALENTE
ROBERT A. HOWELL
AND

G.E. ZAIKOV
EDITORS

Nova Science Publishers, Inc.
New York

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Copyright © 2007 by Nova Science Publishers, Inc.

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AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Organic and physical chemistry using chemical kinetics : prospects and developments / Y.G.
Medvedevskikh ... [et al.], editors.
p. cm.

Includes bibliographical references and index.
ISBN-13: 978-1-60692-749-6
1. Chemical kinetics. 2. Chemistry, Organic. 3. Chemistry, Physical and theoretical. I.
Medvedevskikh, Y. G.
QD502.O74 2007
541'.394--dc22
2007017506

Published by Nova Science Publishers, Inc.

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New York


CONTENTS
Preface
Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7


ix
Conformation and Deformation of Linear
Macromolecules in Concentrated Solutions and
Melts in the Self–Avoiding Random Walks Statistics
Yu. G. Medvedevskikh

1

Thermodynamics of Osmotic Pressure
of Polymeric Solutions
Yu. G. Medvedevskikh, L. I. Bazylyak and G. E. Zaikov

23

Generalization of Data Concerning to the Coal
Swelling in Organic Solvents and Their Extraction
Using the Linear Multiparametric Equations
L. I. Bazylyak, D. V. Bryk, R. G. Makitra,
R. Ye. Prystansky and G. E. Zaikov

35

New Silazane Oligomers and Polymers with
Organic-Inorganic Main Chains: Synthesis,
Properties and Application
N. Lekishvili, Sh. Samakashvili,
G. Lekishvili and G. Zaikov

51


To Question about Influence of Solvent on Interaction
Propanethiole by Chlorine Dioxide
R. G. Makitra, G. E. Zaikov and I. P. Polyuzhyn

65

Mathematical Modelling of Thermo-Mechanical
Destruction of Polypropylene
G. M. Danilova-Volkovskaya,
E. A. Amineva and B. M. Yazyyev
Energy Criterions of Photosynthesis
G. А. Коrablev and G. Е. Zaikov

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73


vi

Contents

Chapter 8

Spatial-Energy Interactions of Free Radicals
G. А. Коrablev and G. Е. Zaikov


Chapter 9

Poly(Vinyl Alcohol)[PVA]-Based Polymer Membranes:
Synthesis and Applications
Silvia Patachia, Artur J. M. Valente,
Adina Papancea and Victor M. M. Lobo

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

89

103


The Research on the Process of Thermo-Mechanical
Destruction in Polypropylene
G.M. Danilova-Volkovskaya and E. A. Amineva

167

Zinccontaining Polymer - Inorganic Composite as
Vulcanization Active Component for Rubbers of
General and Special Assignment
V .I. Ovcharov, I. A. Kachkurkina,
O. V. Okhtina and B. I. Melnikov

173

Formation of Carbon Nanostructures and
Spatial-Energy Stabilization Criterion
G. А. Korablev and G. E. Zaikov

187

The Structural Treatment of Limiting Conversion
Degree for Solid-State Imidization
L. Kh. Naphadzokova, G. V. Kozlov and M. A. Tlenkopachev

201

A Solid-State Imidization and Heterogeneity
of Reactive Medium
L. Kh. Naphadzokova, G. V. Kozlov and G. E. Zaikov


207

Fractal-Like Kinetics of Reesterification
Reaction in Catalyst Presence
L. Kh. Naphadzokova, G. V. Kozlov and G. E. Zaikov

217

Description of the Model Reesterification Reaction
within the Framework of a Strange Diffusion Concept
L. Kh. Naphadzokova, G. V. Kozlov and G. E. Zaikov

225

Estimation of Vapor Liquid Equilibrium of Binary
Systems Tert-Butanol+2-Ethyl-1-Hexanol and
N-Butanol+2-Ethyl-1-Hexanol Using
Artificial Neural Network
H. Ghanadzadeh and A. K. Haghi
Liquid-Liquid Equilibria of the MME (Methylcyclohexane
+ Methanol + Ethylbenzene ) System
H. Ghanadzadeh and A. K.Haghi
Sugar Carbamides
J. A. Djamanbaev, J. A. Abdurashitova
and G. E. Zaikov

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233


243
251


Contents
Chapter 20

Impact of Chain-End Structure, Basic Comonomer
Incorporation and Pendant Structure on the Stability
of Vinylidene Chloride Barrier Polymers
Bob A. Howell, Adeyinka O. Odelana
and Douglas E. Beyer

Index

vii

257

279

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PREFACE
If it’s green or wiggly, it’s Biology

If it’s stinky, it’s Chemistry
If it doesn’t work, it’s physics.
(Definitions of sciences on the back of Sasha Zaikova’s sweatshirt
High School, Perry, Ohio, U.S.A.)

“Inevitability (verity) is something that nobody knows;
the truth everybody knows but each has his or her own truth”
Proverb

The word truth is a multi-meaning word which can be applied both to science and life.
We will not raise social problems but we will go down to the science, particularly chemical
science (organic and physical chemistry). We choose chemical kinetics as a method of
research because chemical kinetics is a science about chemical processes, mechanisms of
reactions, and about possibilities of directing reactions. Parts of the articles in this volume
deal with chemical physics, biochemical physics, and physical organic chemistry. All of these
fields of science are interconnected with each other.
The authors and editors are all part of an international effort to bring these fields of
knowledge to readers around the world. All these efforts are collected at symposia to share
and exchange knowledge. Symposium is defined as a convivial meeting, usually following a
dinner, for drinking and intellectual conversation. It is derived from ancient Greek word
sympósion, which means drinking party, and where ancient philosophers gathered to discuss
ideas. It is well-known that the ancient Greek philosopher and scientist Plato loved to attend
symposiums very much and he even died during a symposium on his birthday, at the age of
81. These are just fun facts on the background of symposia and none of this concerns the
authors and editors of this volume. The papers of this volume focus on the different states of
modern chemistry (both reviews and original papers.) Editors and authors will be grateful to
the readers for valuable remarks that will be taken into account in further work and research.
In the U.S.A., in the times of the Wild West, there was a proverb, stating that “A good
word is appreciated, but a good word with a gun behind it is even better.” Interpreting and
applying this proverb to modern times and situations, one can say that new hypotheses and


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x

Y.G. Medvedevskikh, Artur Valente, Robert A. Howell, et al.

ideas are always appreciated, but new hypotheses and ideas with experimental data and other
proof behind them are better.
In the words of an engineer-mechanic of the Russian aviation, Vitalii K. Petukhov, said
“Try your best to do your best, because bad things will happen on their own”
We would like to accept this idea, because our dream is good volume for readers.
However, the last decision (the volume is good or bed) will be done by our readers.
Editors
Prof. Yurii G.Medvedevskikh
Branch of L.V.Pisarzhevskii Institute of Physical Chemistry
National Academy of Sciences
L’viv, Ukraine
Dr. Artur Valente,
Coimbra University
Coimbra, Portugal
Prof. Bob Howell
Central Michigan University
Mount Pleasant, Michigan, USA
Prof. Gennady Zaikov
N.M.Emanuel Institute of Biochemical Physics
Russian Academy of Sciences
Moscow, Russia


Chapter 1 - It was proposed a strict statistics of self–avoiding random walks in the d–
measured lattice and continuous space for intertwining chains in the concentrated solutions
and melts. On the basis of this statistics it was described the thermodynamics of conformation
and isothermal and adiabatic deformation of intertwining chains. It has been obtained the
equation of conformational state. It was shown, that in the field of chains overlap they are
stretched increasing its conformational volume. In this volume there are others chains with
the formation of m–ball. Free energy of a chain conformation does not depend upon the fact,
if the chains intertwined or they are isolated in the m–ball. Mixing entropy is responsible to
the chains interweaving in the m–ball. Dependencies of the conformational radius, free
energy and conformation pressure on respective concentration of polymeric chains have been
determined. Using the thermodynamics of intertwining polymeric chains of m–ball
conformational state and also the laws of isotropic media deformation into linear differential
form it were obtained the theoretical expressions for elasticity modules (namely, volumetric
volume, Young’s module and shift’s module) and for the main tensions appearing at the
equilibrium deformation of the m–ball. Poisson’s coefficient is a function only on the
Euclidean’s space and for the real 3–dimensional space is equal to 3/8. It was proposed a
simple model explaining the tensile strength of the m–ball by the chains intertwining effect
and, thereafter by the loss of the mixing entropy, but not by the chemical bonds breaking.

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Preface

xi

Calculations of the elastic properties, the main tensions and tensile strength of natural rubber
carried out without using the empirical adjusting parameters are in good agreement with the
experimental data.
Chapter 2 - It was proposed the analysis of osmotic pressure for diluted, semi–diluted and

concentrated polymeric solutions based on the taking into account a free energy of
macromolecules conformation as a component of their chemical potential. It was shown, that
only into diluted solutions a free energy of macromolecules conformation does not contribute
into osmotic pressure and it is described by Vant–Goff’s equation. In a case of semi–diluted
and concentrated solutions the contribution of the conformative component of chemical
potential of macromolecules into osmotic pressure is dominate. Obtained expressions for the
osmotic pressure in a cases of semi–diluted and concentrated solutions are more general than
proposed ones in the scaling method and self–consistent field method; generally they are in
good agreement with the experimental data and don’t contain the empirical constants. It was
discussed the especial role of the critical concentration c* of the polymeric chains
intertwining. It was shown, that in this point a free energy of the conformation and also
osmotic pressure were determined uniquely, whereas for their derivatives upon the
macromolecules concentrations the jump is observed. On the basis of these peculiarities the
concentration c* is the critical point of the second order phases transition for the polymeric
solutions. This in accordance with the de Clause assumes the Scaling’s ratios application near
c*, although does not establish the criteria for the indexes of corresponding power functions
estimation.
Chapter 3 - Approaches to the consideration of a coal swelling process, which were used
up to now and based on the theory of regular solutions, do not give the possibility to
generalize quantitatively the experimental data. Adequate relation between the physical–
chemical properties of the solvents and the degree of a coal swelling in them can be obtained
only with the use of linear multiparametric equations which take into account the effects of
the all processes proceeding in the system; besides, the basicity and a molar volume of the
liquids are determinative. Such approach is effective at the generalization of data concerning
to extraction of a coal.
Chapter 4 - On the basis of the diallylsilazanes, α,ω-dihydrideoligoorganosiloxanes and
1,4-bis(dimethylhydridesilyl)benzene, new polyfunctional siliconorganic polymers have been
synthesized. General regularities and feasible mechanism of the reaction for obtaining diallylsilazanes have been studied. Based on data of elemental, IR and NMR 1H spectral analysis,
the composition and structure of synthesized polymers have been established.
The kinetics of polyhydrosailylation reactions has been studied. Quantum-chemical

calculations of the model system and data of NMR 1H spectra of the real products of the polyaddition reaction have confirmed probability of passing polyhydrosilylation reaction according to the aforementioned two concurrent directions obtaining both α and β adducts. For the
evaluation of relative activity for selected monomers the algebraic-chemical approach has
been used.
Using Differential Scanning Calorimetric and Roentgen-phase analyses methods it has
been established that synthesized polymers are amorphous systems. Thermal (phase) transformation temperatures of synthesized polymers have been determined. Thermooxidation
stability of the synthesized polymers has been studied. There was shown that their
thermooxidation stability exceeded the analogical characteristic of polyorganocarbosiloxanes.

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xii

Y.G. Medvedevskikh, Artur Valente, Robert A. Howell, et al.

Using synthesized diallylsilazanes modification of the properties of some important industrial
polymer composites based on phenolformaldehide resins has been carried out. Preliminary
investigations showed that synthesized polymers in combination with phenolformaldehyde
resins were successfully used as binding-components for polymer/graphite and
polymer/carbon black electro-conducting composites.
Chapter 6 - There has been provided mathematical description of the processes of
thermonuclear destruction in deformed polypropylene melts; the aim was to use the criterion
of destruction estimation in modelling and optimising the processing of polypropylene into
products.
Chapter 7 - The application of methodology of spatial-energy interactions (P-parameter)
to main stages of photosynthesis is given. Their energy characteristics are calculated. The
values obtained correspond to the reference and experimental data.
Chapter 8 - Spatial-energy characteristics of many molecules and free radicals are
obtained. The possibilities of applying the P-parameter methodology to structural interactions
with free radicals and photosynthesis energetics evaluation are discussed. The satisfactory

compliance of calculations with experimental and reference data on main photosynthesis
stages is shown.
Chapter 10 - There has been investigated the effect of thermo-mechanical impact
conditions on destruction kinetics in polypropylene melts. The conditions served as a basis
for obtaining quantitative dependencies and mathematical expressions aimed at describing
destruction processes.
Chapter 11 - In work the synthesis technology of zinccontaining polymer - inorganic
composite on the basis of products of secondary raw material processing at joint precipitating
with carbamide and formaldehyde (ZnCFO) is described.
The structure and properties of ZnCFO are investigated by the differencial-thermal
analysis, electronic microscopy and IR-spectroscopy.
The ZnCFO action as vulcanization active component of elastomeric compositions on the
basis of rubbers of general and special assignment with various vulcanization systems is
investigated.
The comparative estimation of ZnCFO efficiency depending on type of vulcanization
system is given.
The ZnCFO influence on character of formed morphological structure of rubbers is
determined by the method of percalation analysis.
Chapter 12 - Spatial-energy criterion of structure stabilization was obtained. The
computation results for a hundred binary systems correspond to the experimental data. The
basic regularity of organic cyclic compound formation is given and its application for carbon
nanostructures is shown.
Chapter 13 - It was shown, that limiting conversion (in the given case - imidization)
degree is defined by purely structural parameter – macromolecular coil fraction, subjected
evolution (transformation) in chemical reaction course. This fraction can be correctly
estimated within the framework of fractal analysis. For this purpose were offered two
methods of macromolecular coil fractal dimension calculation, which gave coordinated
results.
Chapter 14 - It was shown, that the conception of reactive medium heterogeneity is
connected with free volume representations, that it was to be expected for diffusioncontrolled solid phase reactions. If free volume microvoids were not connected with one


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Preface

xiii

another, then medium is heterogeneous, and in case of formation of percolation network of
such microvoids – homogeneous. To obtain such definition is possible only within the
framework of the fractal free volume conception.
Chapter 15 - It was shown, that the reesterification reaction without catalyst can be
described by mean-field approximation, whereas introduction of catalyst
(tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results
to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is
considered as recombination reaction and treated within the framework of scaling approaches.
Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive
medium or its biased diffusion allows to decrease reaction duration approximately twofold.
Chapter 16 - It is shown, that there is principal difference between the description of
generally reagents diffusion and the diffusion defining chemical reaction course. The last
process is described within the framework of strange (anomalous) diffusion concept and is
controled by active (fractal) reaction duration. The exponent α, defining the value of active
duration in comparison with real time, is dependent on reagents structure.
Chapter 17 - Vapor-liquid equilibrium (VLE) data are important for designing and
modeling of process equipments. Since it is not always possible to carry out experiments at
all possible temperatures and pressures, generally thermodynamic models based on equations
on state are used for estimation of VLE. In this paper, an alternate tool, i.e. the artificial
neural network technique has been applied for estimation of VLE for the binary systems viz.
tert-butanol+2-ethyl-1-hexanol and n-butanol+2-ethyl-1-hexanol. The temperature range in
which these models are valid is 353.2-458.2K at atmospheric pressure. The average absolute

deviation for the temperature output was in range 2-3.3% and for the activity coefficient was
less than 0.009%. The results were then compared with experimental data.
Chapter 18 - The determination region of solubility of methanol with gasoline of high
aromatic content was investigated experimentally at temperature of 288.2 K. A type 1 liquidliquid phase diagram was obtained for this ternary system. These results were correlated
simultaneously by the UNIQUAC model. By application of this model and the experimental
data the values of the interaction parameters between each pair of components in the system
were determined. This revealed that the root mean square deviation (RMSD) between the
observed and calculated mole percents was 3.57% for methylcyclohexane + methanol +
ethylbenzene. The mutual solubility of methylcyclohexane and ethylbenzene was also
demostrated by the addition of methanol at 288.2 K.
Chapter 19 - The results of experimental researches on the synthesis of sugars derivatives
with glycosylamide and thioamide bonds have been presented in this work. The possibility of
using their in the preparative chemistry of sugars, some fields of medicine and agriculture has
been shown.

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In: Organic and Physical Chemistry Using Chemical Kinetics … ISBN: 978-1-60021-763-0
Editors: Y.G. Medvedevskikh, et al. pp. 1-21
© 2007 Nova Science Publishers, Inc.

Chapter 1

CONFORMATION AND DEFORMATION OF LINEAR
MACROMOLECULES IN CONCENTRATED SOLUTIONS
AND MELTS IN THE SELF–AVOIDING RANDOM

WALKS STATISTICS
Yu. G. Medvedevskikh*
Physical Chemistry of Combustible Minerals Department;
L. M. Lytvynenko Institute of Physical–Organic Chemistry
and Carbon Chemistry; National Academy of Sciences of Ukraine

ABSTRACT
It was proposed a strict statistics of self–avoiding random walks in the d–measured
lattice and continuous space for intertwining chains in the concentrated solutions and
melts. On the basis of this statistics it was described the thermodynamics of conformation
and isothermal and adiabatic deformation of intertwining chains. It has been obtained the
equation of conformational state. It was shown, that in the field of chains overlap they are
stretched increasing its conformational volume. In this volume there are others chains
with the formation of m–ball. Free energy of a chain conformation does not depend upon
the fact, if the chains intertwined or they are isolated in the m–ball. Mixing entropy is
responsible to the chains interweaving in the m–ball. Dependencies of the conformational
radius, free energy and conformation pressure on respective concentration of polymeric
chains have been determined. Using the thermodynamics of intertwining polymeric
chains of m–ball conformational state and also the laws of isotropic media deformation
into linear differential form it were obtained the theoretical expressions for elasticity
modules (namely, volumetric volume, Young’s module and shift’s module) and for the
main tensions appearing at the equilibrium deformation of the m–ball. Poisson’s
coefficient is a function only on the Euclidean’s space and for the real 3–dimensional
space is equal to 3/8. It was proposed a simple model explaining the tensile strength of
the m–ball by the chains intertwining effect and, thereafter by the loss of the mixing
entropy, but not by the chemical bonds breaking. Calculations of the elastic properties,
*

Yu. G. Medvedevskikh: 3a Naukova Str., 79053, Lviv, UKRAINE; e–mail:


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2

Yu. G. Medvedevskikh
the main tensions and tensile strength of natural rubber carried out without using the
empirical adjusting parameters are in good agreement with the experimental data.

Key words: intertwining chains, SARW statistics, conformation, polymer chain, random
walks, lattice, thermodynamics, modules of elasticity, forces, work..

1. INTRODUCTION
Self–avoiding random walks (SARW) statistics has been proposed [1] for single that is for
non–interacting between themselves ideal polymeric chains (free–articulated Kuhn’s chains
[2]) into ideal solvents, in which the all–possible configurations of the polymeric chain are
energetically equal. From this statistics follows, that under the absence of external forces the
conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar
radius Rf of which is described by known expression [3, 4]

R f = aN 3 /( d + 2 )

(1)

Here: a is statistical length of the chain’s link; N is number of the links in chain or its length;
d is the dimension of the Euclidean’s space.
Polymeric chains in the concentrated solutions and melts at molar–volumetric
concentration c of the chains more than critical one c* = (NARfd)-1 are intertwined. As a result,
from the author’s point of view [3] the chains are squeezed decreasing their conformational
volume. Accordingly to the Flory theorem [4] polymeric chains in the melts behave as the

single ones with the size R = aN1/2, which is the root–main quadratic radius in the random
walks (RW) Gaussian statistics.
SARW statistics leads to other result.

2. SARW STATISTICS FOR INTERTWINING CHAINS
IN D–DIMENSIONAL LATTICE SPACE
Let us introduce the d–dimensional lattice with the cell’s parameter equal to the statistical
length a of the chain’s link; let us notify, that Z is number of cells in a space and m chains are
represented in it; every chain has the length N. As same as earlier [1], we will disregard the
energetic effects considering the all–possible configurations of the chains as equivalent.
We appropriate the random chain and notify as ni the numbers of steps of the end of chain
random walk along i–directions of d–dimensional lattice. At this,

∑n

i

=N,

i = 1, d

(2)

i

The probability

ω ( n ) that at given ni the end of chain draws si = ni + − ni − efficient

steps is subordinated to Bernoulli’s distribution [1]


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Conformation and Deformation of Linear Macromolecules…

⎛1⎞
ω( N ) = ⎜ ⎟
⎝2⎠

N

∏ {n ! /[( n + s ) / 2 ]! [( n − s ) / 2 ]!}
i

i

i

i

3

(3)

i

Change of a sign si in eq. (3) doesn’t change the value

ω ( n ) ; that is why this probability


represents the probability of fact, that the RW trajectory per ni steps along i–directions of the
d–dimensional space will be finished in one of the 2d cells M(s), position data of which are
given by vectors s = (si), i = 1, d differing only by the signs of own components si.
Condition of the self–avoiding RW trajectories absence on the d–dimensional lattice
demands the circumstance at which more than one link of the chain can not be stood in every
cell. Links of the chain are inseparable; they cannot be divided one from another and located
into the cells in random order. Thereby, number of different methods of mN differing links
location per Z identical cells under condition that in every cell more than one link of the chain
cannot be stood is equal to Z! / (Z – mN)!.
By identify of the cells the antecedent probability of fact that the cell will be occupied by
presented link equal to 1/Z, and when will be not occupied – then (1 – 1/Z). Consequently,
probability ω ( z ) of mN differing links distribution per Z identical cells is determined by
Bernoulli’s distribution

Z!
⎛1⎞
ω( z ) =
⎜ ⎟
( Z − mN )! ⎝ Z ⎠

mN

⎛ 1⎞
⎜1 − ⎟
⎝ Z⎠

Z −mN

(4)


Distribution (3) describes the RW trajectory of one random chain whereas the expression
(4) assigns the links distribution of all m chains. That is why, the probability ω ( s ) of
common event consisting of the fact that the RW trajectory of random chain is also the SARW
trajectory and at given Z, n, N and ni will turned out by its own last step in one among 2d
equiprobable cells M(s) will be equal to

ω ( s ) = ( ω ( z ))1 / m ω ( n )

(5)

Using the Stirling’s formula under condition Z >> 1, N >> 1, ni >> 1 and factorizations ln(1–1/Z) ≈
–1/Z, ln(1–mN/Z) ≈ –mN/Z, ln(1±si/ni) ≈ ± si/ni–(si/ni)2/2 accordingly to condition si << ni, mN
<< Z and also assuming N(N–1) ≈ N2, we find the asymptotic (5) with accuracy to the
constant multiplier:

⎧ mN 2 1
⎫
2
− ∑ si / ni ⎬ ,
2 i
⎩ Z
⎭

ω ( s ) ≈ exp⎨−

m ≥1

(6)


As same as earlier [1], let us assume, that the fiducial cells M(s) generally appertain to
ellipsoid surface. Then we have [1]

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4

Yu. G. Medvedevskikh

Z = d d / 2 ∏| si |

(7)

i

Determination (7) means, that the d–dimensional space consisting of Z cells is disposable
for any random chain; this demands of their full mixing.
Combining the expressions (6) and (7) we will obtain

⎧

ω ( s ) = exp⎨− mN 2 / d d / 2 ∏| si | −
⎩

Function

i

⎫

1
2
si / ni ⎬
∑
2 i
⎭

(8)

ω ( s ) determines the probability that the RW trajectory of the random walk is

simultaneously also by SARW trajectory and by its own last step realizes the state M(s).
Hence, it is numerically equal to part of these SARW trajectories among general number (2d)N
of RW trajectories which realize the state M(s). Number L(s) of such SARW trajectories
determines the thermodynamical probability of the realization M(s):

L( s ) = ( 2d )N ω ( s )

(9)

By summing L(s) upon the all set of possible state of the chain’s end we find general
number L of SARW trajectory:

L = ( 2d ) N c( s )

(10)

where

⎧

⎫
1
2
c( s ) = ∑ exp⎨− mN 2 / d d / 2 ∏| si | − ∑ si / ni ⎬
2 i
s
i
⎩
⎭

(11)

Then function

w( s ) =

⎧
⎫
1
1
2
exp⎨− mN 2 / d d / 2 ∏| si | − ∑ si / ni ⎬
c( s )
2 i
i
⎩
⎭

(12)


normalized per unity and determines the end of chain distribution upon states M(s) of d–
dimensional lattice. It equal to ratio of number L(s) of SARW trajectories realizing the state
M(s) to general number L of SARW trajectories: w( s ) = L( s ) / L .
In turn, the ratio L/(2d)N equal to part of general number of SARW trajectories among
general number of RW trajectories in accordance with the adopted terms [3] is the fatigue
function g(N) of the SARW trajectories: g(N) = L/(2d)N = c(s).

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Conformation and Deformation of Linear Macromolecules…

5

3. SARW STATISTICS FOR INTERTWINING CHAINS
IN CONTINUOUS D–DIMENSIONAL SPACE
Let us introduce the variable of displacement xi, which is by semi–axis of conformational
ellipsoid; the state M(s) appertains to the surface of this ellipsoid [1]

xi = a | si | d 1/ 2
and parameter

(13)

σ i is a standard deviation of the Gaussian part of the distribution (12)

σ i 2 = a 2 ni d

(14)


In accordance with the expression (2) the following connection is imposed on the values

σi

∑σ

2
i

= a 2 Nd

(15)

i

Since si / ni = xi / σ i , d
2

2

2

d/2

∏| s |= a ∏ x
−d

i

i


i

ω( x ) =

the eq. (12) can be re–written as

i

⎧
1
1
2
2⎫
exp⎨− a d mN 2 / ∏ xi − ∑ xi / σ i ⎬
2 i
c( x )
i
⎩
⎭

⎧
1
2
2⎫
c( x ) = ∫ exp⎨− a d mN 2 / ∏ xi − ∑ xi / σ i ⎬dx
2 i
i
⎩
⎭


(16)

(17)

At this, c(x) is d–multiple integral upon all possible values xi, dx =

∏ dx . Since
i

i

c( x ) = a d d d / 2 c( s ) we have g(N) = c(x)/addd/2.
Integral c(x) can be taken with the adequate accuracy by saddle–point technique [1, 5].
Change of (13) introduces an essential difference between w( s ) and w( x ) : the last
determines the probability w( x )dx of fact that the SARW trajectory at given values m, N and

σ i will finished in the elementary volume dx = ∏ dxi lying on the surface of the ellipsoid
i

with the semi–axes xi, i = 1,d.

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6

Yu. G. Medvedevskikh

4. THERMODYNAMICS OF CONFORMATION AND DEFORMATION

OF INTERTWINING CHAINS
Maximum w( x ) at given m, N and

σ i determines the most expected or equilibrium

state of the polymeric chain. Semi–axes xi of equilibrium conformational ellipsoid we will
found from the condition ∂ ln w( x ) / ∂xi = 0 at xi = X i :
1 /( d + 2 )

⎛
⎞
X i = σ i ⎜⎜ a d mN 2 / ∏ σ i ⎟⎟
i
⎝
⎠

(18)

In the absence of external forces the all directions of the end of chain walking are
equiprobable accordingly to condition ni = N / d ; so

σ i 2 = σ 02 = a2 N

(19)

Substitution of (19) into (18) makes the semi–axes Xi of equilibrium ellipsoid the same
and equal to radius Rm of the conformational sphere; the same distribution density ω ( x )
corresponds to the surface of this conformational sphere:

Rm = aN 3 /( d +2 )m


1 /( d + 2 )

(20)

Expression (20) determines not only the conformational radius of one random chain, but
due to the chains intertwining effect also the conformational radius of all m chains. Thereby

Rmd is the conformational volume of m–ball disposable for every among the intertwining
chains. As we can see, m–ball is a fractal with two fractal indexes: first is 3/(d+2) and
determines the dependence Rm on the chain N, the second is 1/(d+2) and determines the
dependence on number of chain in m–ball.
We can see from the comparison of (20) and (1), that the conformational radius Rm of m–
ball and respectively of any random chain in it is more than the conformational radius Rf of
random chain: in m–ball the chains are stretched but are not twisted. The presence of other
chains diminishes the number of free cells of d–dimensional lattice accessible for SARW
trajectory of presented chains enforcing it to encroach more volume of the space.
In the presence of external forces acting along i–axes of the d–dimensional space,
σ i ≠ σ 0 and m–ball is deformated into the ellipsoid with semi–axes Xi accordingly to (18). It
is convenient to introduce the following variables as a measure of m–ball deformation

Λi = X i / Rm

(21)

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Conformation and Deformation of Linear Macromolecules…


7

which characterize the multiplicity of the linear deformation of m–ball along i–direction of a
space.
Next, let us determine the multiplicity Λv of volumetric deformation via expression

Λv = ∏ X i / Rmd = ∏ Λi
i

Λv

(22)

i

Due to (2) [1] at any deformations of the m–ball its conformational volume is decreased:
≤ 1 . The connection equation between Λi corresponds to connection equation (2):

∑Λ

i

i

2

= d / ∏ Λi

(23)


i

In continuous space the thermodynamical probability W ( x ) of the realization of state in
which the end of chain is located on the surface of the ellipsoid with the semi–axes Xi is equal
to

W ( x ) = Lω ( x )

(24)

As same as for the lattice space, general number L of SARW trajectories in continuous
space let us determine in the form (10), that is L ≈ ( 2d ) c( x ) . That is why
N

⎧
1
2
2⎫
W ( x ) ≈ ( 2d ) N exp⎨− a d mN 2 / ∏ xi − ∑ xi / σ i ⎬
2 i
i
⎩
⎭

(25)

Entropy S of presented conformational state is equal to S = k ln W ( x ) , free energy

F = −TS or F = −kT ln W ( x ) . From (25) follows F = F0 + F ( x ) where


F0 ≈ −kTN ln 2d ≈ −

d
kTN
2

(26)

⎧
1
2
2⎫
F ( x ) = kT ⎨a d mN 2 / ∏ xi + ∑ xi / σ i ⎬
2 i
i
⎩
⎭

(27)

Thereby, F0 represents by itself a free energy of random walks independent on the
conformational state of a chain; F(x) brings a positive contribution into F and the sense of this
consists in a fact that the terms F(x) and S(x) represent the limitations imposed on the
trajectories of random walk by request of the self–avoiding absence. These limitations form
the self–organization effect of the polymeric chain: the conformation of polymeric chain is
the statistical form of its self–organization.

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8

Yu. G. Medvedevskikh

Since F0 doesn’t depend on the conformational state of a chain we assume that the free
energy of a polymeric chain conformation is equal to F = F(x) accordingly to (27).
Expression for the free energy of equilibrium conformation of polymeric chain we will obtain by
substitution of the values xi = Xi in (27) in accordance with the (18):
2

⎛ d⎞ ⎛R ⎞
Fm = ⎜1 + ⎟kT ⎜⎜ m ⎟⎟ / Λv
⎝ 2 ⎠ ⎝ σ0 ⎠

(28)

For non–deformated m–ball we have

⎛ d⎞ ⎛R ⎞
F = ⎜1 + ⎟kT ⎜⎜ m ⎟⎟
⎝ 2 ⎠ ⎝ σ0 ⎠

Λv = 1 and

2

0
m

(29)


From this the expression follows for the deformation work ( A = ΔFdef in the system of
the mechanics signs) of m–ball into ellipsoid in calculation per one chain

⎛ d⎞ ⎛R ⎞
ΔFdef . = ⎜1 + ⎟kT ⎜⎜ m ⎟⎟
⎝ 2 ⎠ ⎝ σ0 ⎠
Since

2

⎛ 1
⎞
⎜⎜ − 1⎟⎟
⎝ Λv
⎠

(30)

Λv ≤ 1 , a work of the deformation is positive ΔFdef ≥ 0 , that is realized above the
0

polymeric chain. Let us compare a free energy Fm of the polymeric chain in non–deformated
m–ball with a free energy Ff of single deformated polymeric chain [1]

⎛ d⎞ ⎛R
F f = ⎜1 + ⎟kT ⎜⎜ f
⎝ 2 ⎠ ⎝ σ0
Here


2

⎞
⎟⎟ / λv
⎠

(31)

λv is a multiplicity of the volumetric deformation of Flory ball.

Let us assume that the chains in m–ball aren’t intertwined, every among them occupies
the isolated volume equal to Rmd/m. Then the multiplicity of the volumetric deformation of
Flory ball into m–ball will be equal to

λv = Rm d / mR f d = m −2 /( d +2 )

(32)

We will obtain for the conformation free energy of isolated chain into m–ball

⎛ d⎞ ⎛R
F f = ⎜1 + ⎟kT ⎜⎜ f
⎝ 2 ⎠ ⎝ σ0

2

⎞ 2 /( d +2 )
⎟⎟ m
⎠


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(33)


Conformation and Deformation of Linear Macromolecules…
that

is

equal

to

Fm0

accordingly

( Rm / σ 0 ) = ( R f / σ 0 ) m
2

2

2 /( d + 2 )

to

(29)

with


taking

into

9
account

that

.

Thereby, free energy of the conformation of single chain into m–ball for intertwining or
isolated one from another chains is the same. Free energy of the conformation is not the
factor, which facilitates or prohibits the chains intertwining.
In the absence of energetic interaction such factor is the entropy of mixing. It can be
estimated via the numbers of displacement methods of the all chains links into m–ball with
the exception of a displacement links in every chains: (mN)!/(N!)m. From this under the
Stirling’s approximation we will obtain the expression for the entropy of mixing ΔS c in
calculation per one chain,

ΔSc = kN ln m , and, respectively we will obtain for free energy

ΔFc of mixing
ΔFc = −kTN ln m
The value

(34)

ΔFc < 0 and can be sufficiently big per absolute value, for instance for melts,


in order to provide the chains intertwining of their mixing in m–ball.

6. EQUATION FOR THE CONFORMATIONAL STATE OF M–BALL
Let us determine the pressure P of a conformation via the ordinal thermodynamic ratio
(∂F / ∂V )T = − P as a connection measure between the free energy and the volume of
conformation. Taking into account the all chains into m–ball, we have F = mFm ,

V = Rmd Λv , that is why P = −m∂Fm / ∂Λv Rmd . By differing the eq. (28) we have
2

⎛ d⎞ ⎛R ⎞
P = ⎜1 + ⎟kT ⎜⎜ m ⎟⎟ m / Rmd Λ2v
⎝ 2 ⎠ ⎝ σ0 ⎠

(35)

By multiplying (35) on V = ( Rm Λv )
2

d

2

we will obtain the equation of the

conformational state of m–ball:

PV 2 = mkTβ


(36)
2

⎛ d ⎞⎛ R ⎞
β = ⎜1 + ⎟⎜⎜ m ⎟⎟ Rmd
⎝ 2 ⎠⎝ σ 0 ⎠

(37)

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