Ciencia de materiales para polímeros

1. COMPUTATIONAL MATERIALS SCIENCE
OF
POLYMERS

3. COMPUTATIONAL
MATERIALS SCIENCE
OF
POLYMERS

4. CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING

5. Published by
Cambridge International Science Publishing
7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK
http://www.cisp-publishing.com
First published January 2003
© A A Askadskii
© Cambridge International Science Publishing
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6. About the Author
Andrey Aleksandrovich Askadskii is a Professor of Chemistry at the In-stitute
of Organo-Element Compounds of the Russian Academy of Sciences.
He holds M.S. in Civil Engineering from the Moscow Civil Engineering Institute
(1959), M.S. in Chemistry from the Mendeleev Institute of Chemical Technology
(1962) and Ph.D. in Physics of Polymers (1968).
The main scientific interests of the author are: the development of a
physical approach to the quantitative evaluation of the physical properties
of linear and network polymers on the basis of their chemical structure;
development of computer programs for calculating the properties of poly-mers
and low-molecular liquids and also computer synthesis of polymers with
the required properties; experimental examination of the structure of properties
of heat-resistant aromatic polymers of different grades; development of new
methods of experimental and theoretical analysis of the relaxation proper-ties
of polymer materials; production of new types of polymers; production
and examination of electrically conducting polymer materials on the basis
of heat-resistant polymers and organo-element compounds; development of
gradient polymer materials with a variable modulus of elasticity within the
limits of the same material and retaining elastic (not viscoelastic) proper-ties
at any point of the gradient material.
Prof Askadskii is the author of more than 400 scientific studies and
20 books, six of which have been published abroad.

7. Contents
Preface
Introduction 3
Chapter I. Brief information on types of polymes and their chemical structure 9
Chapter II. Packing of macromolecules and polymers density 16
II.1. Increments method and basic physical assumption 16
Chapter III. Temperature coefficient of volumetric expansion 58
Chapter IV. Glass transition temperature of polymers 67
IV.I. Thermomechanical and other methods of evaluation of the glass
transition temperature of polymers 67
IV.2. Mechanism of glass transition 88
IV.3. Calculation of the glass transition temperature of linear polymers 108
IV.4. Influence of plasticization on the glass transition temperature of polymers 322
IV.5. Calculation of the glass transition 343
Chapter V. Temperature of transition into the viscous flow state for amorphous
polymers 385
V.1. Estimation of temperature of transition into the viscous flow state of
polymers 385
V.2. Dependence of Newtonian viscosity on molecular mass of polymer in a
wide range of its change 388
Chapter VI. Melting point of polymers 398
Chapter VII. Temperature of onset of intense thermal degradation of polymers 408
Chapter VIII. Optical and opto-mechanical properties of polymers 418
VIII.1. Refractive index 418
VIII. 2. Stress-optical coefficient 426
Chapter IX. Dielectric constant of polymers and organic solvents 445
Chapter X. Equilibrium rubber modulus for polymer networks 456
X.1. Calculation of the equilibrium modulus 456
X.2. Heteromodular and gradient-modulus polymers 466
Chapter XI. Description of relaxation processes in polymers 475
XI.1. Stress relaxation 475
XI. 2. Sorption and swelling processes 497
Chapter XII. Solubility of polymers 504
XII.1. Specific cohesive energy of organic liquids and polymers. Hildebrand
solubility parameter 504
XII.2. Solubility criterion 509
XII.3. Influence of molecular mass and degree of macromolecule orientation
on solubility 520
Chapter XIII. Surface properties of organic liquids and polymers 527
XIII.1. Surface tension of organic liquids 528
XIII.2. Surface tension of polymers 536
Chapter XIV. Miscibility of polymers 547
Chapter XV. Influence of the end groups on the properties of polymers 555
Chapter XVI. Thermophysical properties of polymers 562
XVI.1. Heat capacity 562
XVI.2. Thermal diffusivity and heat conductivity 564

8. Chapter XVII. Molecular design and computer synthesis of polymers with
predermined properties 567
Appendix 1. Examples of solution of direct problems of polymers synthesis 589
Appendix 2. Examples of solving the reverse problem of polymer synthesis 602
Appendix 3. The example of solving the complex problem – analysis of the
chemical structure of phenol formaldehyde resin 607
Appendix 4. Application of the approach to multicomponent copolymers 621
Appendix 5. Influence of strong intermolecular interaction occurring between
two dissimilar polymers on their miscibility 625
Appendix 6. On formation of super-molecular structure in amorphous polymers 645
1. Scheme of formation of the super-molecular structure 645
2. Calculation method of evaluation of dimensions of elements of super-molecular
structure of polymers
3. Phase state of polymers as a result of formation of the super-molecular
structure by one-cavity bond hyperboloids 653
References 669
Index 689

9. PREFACE
Published in the journal “Chemistry and Life”, No. 2, 1981 was the article by
me, titled by the editor as “Atom plus atom plus thousand atoms”. This article
discussed the possibility of calculating some physical properties of polymers on the
basis of the chemical structure of the repeat unit (it was then possible to calculate
properties of linear polymers only). In conclusion of the article, titled “A little
fantasy”, it was written: “Therefore, many properties of polymer can be predicted, if
nothing except the structural formula of the appropriate monomer is known. It is a
great progress: nowadays already, such calculations allow chemists to be drawn
away from heavy duty to synthesize hopeless monomers. Formerly, under empirical
selection of materials, many of such monomers had to be synthesized. Nevertheless,
calculations are to be made manually still. Moreover, when they are translated into
the machinery language, chalk and blackboard traditional for any chemical dispute
can be substituted by an electronic “pencil”. A chemist will draw a formula of the
suggested monomer on the screen by it, and the computer will answer immediately if
it is useful or not to synthesize it. Another opposite task seems to be much more
absorbing. If the computer is able to calculate properties by structural formulae,
apparently, it may be taught, vice versa, to calculate the formula of a suitable
monomer (or several formulae to choose) by any, even contradictory set of properties,
given to it. In this case, it will be able to substitute the chemist in his most problematic
part of work, one is able to succeed in on the basis of experience, intuition and luck.”
That was a fantasy, and it could be hardly imagined that these ideas would be realized
at any time in neat future. However, events were developing very fast, especially after
appearance of high-power personal computers. Before discussing stages of this great
work, methods of the quantitative estimation of polymer physical properties must be
presented in brief performed on the basis of their chemical structure. At the present
time, there are three main approaches to this estimation. One of them, developed by
Van Krevelen [214], is based on the idea of so-called ‘group contributions’, according
to which the simplest empirical expressions of the additive type are written down, the
present group, existing in different polymeric units, making one and the same
contribution to the calculated characteristic (for example, glass transition temperature,
melting, etc.). As the author states, this is just an empirical approach, which allows
the physical properties of many of linear polymers to be calculated with high
accuracy.
Another approach, being developed for a long time by the author of this
preface in company with Yu.I. Matveev [28, 128] is semi-empirical. According to it,
equations for calculation of the physical properties are deduced on the basis of ideas
of physics of solids, and calibration of the method is performed with the help of
physical characteristics of polymeric standards, the properties of which are studied
well. Consequently, parameters of equations possess a definite physical sense (energy
of dispersion interaction, energy of strong intermolecular interaction, including
hydrogen bonds, Van-der-Walls volume, etc.). Application of this approach makes
possible estimation with enough accuracy of many physical characteristics of
polymers (about 60 up to now). Therefore, the number of polymers of various
structures is unlimited.
The third approach developed by J. Bicerano [133] has appeared recently. It is
based on the so-called coherence indexes, reduced in practice to a search for various

10. 2
correlations of physical properties with many rules of obtaining coefficients of
correlation dependencies.
Discussed in the present monograph are principles of the approach, developed
by A.A. Askadskii and Yu.I. Matveev, special attention being paid particularly to
computer realization of the current calculation method for physical properties of
polymers. The first computer software has been composed by E.G. Galpern, I.V.
Stankevich and A.L. Chistyakov - investigators of quantum chemistry laboratory of
A.N. Nesmeyanov Institute of Organo-Element Compounds, RAS. Initially, computer
“synthesis” of polymers by this software was performed from so-called large
procurements representing residues of monomers, involved into the synthesis
reaction. In the second variant, computer synthesis was performed from smallest
procurements, from which the repeat unit of the polymer was constructed. This
broadens significantly capabilities of the software for solving both direct (calculation
of the polymer properties from its chemical structure) and reverse task (computer
‘synthesis’ of polymers with preliminarily programmed /assigned/ properties, the
ranges of which were set in the computer), because the amount of ‘synthesized’
olymers has increased sharply. Then principally new software was composed by A.F.
Klinskikh, in which chemical structure of the repeat unit was ‘constructed’from
atoms. Thus, the user needs just to depict chemical structure of the polymer on the
computer screen as chemist does it on the paper, and computer lays out all physical
properties of polymers, involved in the software (all about 60). This software also
provides for calculation of a sequence of properties of low-molecular weight organic
compounds, as well as, which is very important, properties of polymeric networks.
Solution of the reverse task is also provided. Of special importance is the possibility
to calculate properties of copolymers and their mixtures, to predict solubility and
compatibility of polymers, to construct dependencies of properties on temperature,
molecular mass, crystallinity degree, microtacticity (of special importance are
dependences of glass transition temperature and temperature of transition into the
viscous flow state on molecular mass).
It stands to reason that not all the problems are solved. Accuracy of the
calculation and various predictions of polymers behavior at dissolution and mixing
with each other must be increased, calculation schemes to estimate new properties of
polymers must be developed, and their computer realization must be performed, etc.
It is obvious that the present monograph possesses some drawbacks. The
authors will be thankful for any notes on the point of the book.

11. 3
INTRODUCTION
As mentioned above, the approach to estimation of the physical properties of
polymers, discussed in the monograph, is semi-empirical. When estimating the
thermal characteristics of polymers, such as glass transition temperature, melting
point, it is supposed that the repeat unit is composed of a set of anharmonic oscillators
representing atomic pairs, linked by intermolecular physical bonds. The critical
temperature of this set of anharmonic oscillators is that determines the above-mentioned
two transition temperatures. The thermal expansion coefficient is also
closely related to these characteristics. In the case of a characteristic as the
temperature of the onset of intensive thermal degradation, the polymeric unit is
considered as a set of anharmonic oscillators representing atomic pairs, linked by
chemical bonds. The critical temperature of such a set of oscillators characterizes the
temperature of the onset of intensive thermal degradation at the given rate of heating
(clearly at a different rate of heating, the temperature of the onset of intensive thermal
degradation will be different, i.e. kinetic effects play a significant role in this case). At
first glance, it may seem strange that thermal degradation is considered here not as a
kinetic, which is conventional, but as an original phase transition, at which, however,
the initial substance cannot be obtained from the products of thermal decomposition
by simple cooling down.
Equations for calculating other physical characteristics are based on physical
approaches, discussed in detail below, and we will not consider them in this part.
Common for all these equations is summarizing the sequence of atomic
constants, which characterize contributions to the energy of intermolecular
interaction, chemical bonds energy, Van-der-Waals volume, etc. Strictly speaking, the
present approach cannot be named additive in the common sense of the word, because
the calculated properties are not additive in relation to atoms and groups, which
compose the repeat unit of polymer.
Here additivity is applied to the characteristics which are really additive (Van-der-
Waals volume, molecular mass, intermolecular interaction energy, etc.). The
approach being described allows calculation of their properties of the unlimited
number of polymers and conduction of the computer synthesis of polymers with
assigned properties with the help of software created and described in the monograph
that is not possible using other existing programs.
As mentioned above, the approach discussed in the monograph is semi-empirical,
calibration of the method being based on the so-called polymeric standards,
the properties of which are studied in detail and common. Let us consider the essence
of calibration on an example of the equation calculating glass transition temperature
of a linear polymer, Tg:
Σ
Δ
i
Δ +
V
T ,
Σ Σ
=
j
j
i
i i
i
g a V b

12. 4
where ai are atomic constants; bj are constants bound to the energy of strong
intermolecular interaction (dipole-dipole, hydrogen bonds), occurred between
polymeric chains at the sacrifice of polar groups existing in them; ΣΔ
i
Vi is the Van-der-
Waals volume of the polymer repeat unit, summarized from Van-der-Waals
volumes of atoms participating in the composition of the unit.
Reduce the equation to the following view:
Σ Δ +Σ = ΣΔ
i
i
j
j g
i
i i V
T
a V b
1
.
Basing on this equation, the excessive system of linear equations is composed
as follows:

    

    

.........................................................................................................................

 


Σ
Δ + Δ + + Δ + α + β + + γ = Δ
Σ
Δ + Δ + + Δ + α +β + + γ = Δ

 

Δ + Δ + + Δ + α +β + + γ = Δ


 

 


 

 

Σ
.
1
... ...
;
1
... ...
;
1
... ...
1 ,
1 ,1 2 ,2 , 1 2
,1 2
1 2,1 2 2,2 2, 2 1 2 2 2
,1 1
1 1,1 2 1,2 1, 1 1 1 2 1
i
i m
g
m m n m n m m m k
i
i
g
n n k
i
i
g
n n k
V
T
a V a V a V b b b
V
T
a V a V a V b b b
V
T
a V a V a V b b b
Then the matrix of coefficients at the unknowns of this excessive system of
equations:

    


    

Δ Δ Δ
Δ Δ Δ
Δ Δ Δ
=
α β γ
α β γ
α β γ
and the column matrix of free terms of these equations

        


        




 




 
Δ
 

 
Δ
 

 
Δ
=
Σ
Σ
Σ
are composed.
Further on, a transposed matrix à is composed and multiplied by the initial
one – ÃA, as well as by the column matrix – ÃB. All this results in obtaining a

13. 5
canonic system of equations. This canonic system is solved, for example, by the
Gauss method. The whole procedure of calibration is performed by standard software.
Without considering features of such regressive analysis, let us note only that
polymers, selected for calibrating the method, must possess experimental values of
analyzed physical characteristics in broadest range, and the chemical structure of
polymeric standards must be sufficiently different. Usually, an excessive system
composed of 30–0 equations is to be solved, which corresponds to 30–40 polymers.
Next, the properties of other polymers are calculated from the coefficients obtained.
In this case, the energy of weak dispersion interaction, strong dipole–dipole
interactions and hydrogen bonds, their relative part and many other physical
parameters of the system are determined.
We are coming now to a brief description of the contents of individual
chapters of the monographs.
The first chapter discusses the data of modern classification of polymers and
their chemical structure. Of the outstanding importance, induced by the features of the
chemical structure and the application field, are interpolymers, dendric and staircase
(ladder) polymers.
The second chapter discusses the approach to computerized materials
technology of polymers on the atomic–olecular level, based on the method of
increments. The increments of various atoms and main groups of them are calculated.
The main physical ideas about structure of macromolecules of polymers and
parameters determining it are displayed. The method for calculating such an important
characteristic of the polymer structure, as the coefficient of molecular packing, is
given. A connection between the free volume of the polymer, the coefficient of
molecular packing and parameters of its porous structures is established. For
experimental determination of characteristics of the microporous structure of
polymers, the method of positron annihilation, the application of which indicated
structural changes in polymers in their relaxation, is used.
With consideration of weak dispersion and strong (dipole–dipole and
hydrogen bonds), the third chapter gives formulae for calculating the thermal
coefficient of the volume expansion in dependence on the chemical structure of the
polymer. In this case, the type of atoms in the polymeric chain and type of the
intermolecular interaction are estimated by a limited number of corresponding
increments, numerical values of which are determined.
The fourth chapter describes in detail the thermomechanical method of
determination of the glass transition temperature and fluidity of polymers, features of
interpreting thermomechanical curves for amorphous and crystalline polymers are
analyzed, the calculation method of determination of the mechanical segment from
the chemical structure of the polymer is displayed. Two main concepts of the
mechanism of vitrification processes of polymers, relaxation and intermolecular, are
discussed. The ‘atomistic approach’ which is more universal than the widespread so-called
‘group contributions method’ to calculation of polymer properties from their
chemical structure, is considered. This approach was used for deriving an analytical
expression to calculate the glass transition temperature of linear and network
polymers from their chemical structure. The influence of types of linear polymers
branching and the number of units between cross-link points, type and structure of
these points, existence and type of the network defects for network polymers on the
glass transition temperature of the polymers is analyzed.
Given in the fifth chapter is the method for calculating the fluidity temperature
of amorphous polymers and the temperature range of the rubbery state of polymers

14. 6
from their chemical structure, and conditions of appearance of the rubbery state in a
polymer depending on its molecular mass, as well, which is important for processing
of polymers.
The sixth chapter describes two approaches to calculating the melting point of
polymers from the chemical structure of the repeat unit. The first approach is based on
the experimental fact of closeness in parts of the empty volume in melting of a
crystalline polymer and in transition of an amorphous polymer of the same structure
from the glassy-like into the high-elastic state. The second approach is based on the
consideration of the repeat unit of a polymer as a selection of anharmonic oscillators.
Discussed in the seventh chapter is the most important characteristic of
thermal resistance of polymers — initial temperature of their intensive thermal
degradation. The formula to calculate this temperature based on the chemical structure
of the polymer was deduced, and necessity to take into account the resulting products
of thermal degradation which starts with the decay of end groups in polymer
macromolecules, are indicated.
In the eighth chapter, Lorenz–Lorentz equations are used for deriving
equations for calculation of the refractive index of polymers and copolymers from
their chemical structure. To obtain the stress-optical coefficient, empirical and semi-empirical
approaches are established, in which the contribution of each atom and the
type of intermolecular interaction are estimated by an appropriate increment. Using
the dependencies obtained for the stress-optical coefficient on the chemical structure
of the repeat unit of the polymer, the contribution of various atoms and polar groups
to the value of this coefficient is estimated, and a polymer with the properties unique
for the method of dynamic photo-elasticity is proposed.
The ninth chapter displays a scheme for calculating the dielectric constant of
polymers and organic liquids with respect to their chemical structure which is
important for both synthesis of polymers with the required dielectric constant and
prognosis of polymer solubility in organic liquids. Taking into account not only the
contribution of various polar groups to the dielectric constant of polymers and liquids,
but also different contributions of a polar group in the present class of liquids resulted
in the previously unobtainable agreement in the experimental and calculated values of
the dielectric constant for a broad spectrum of organic polymers and liquids.
Based on the notion of network polymers as an elastic and rotational–isomeric
subsystem and taking into account its structure as linear fragments and cross-linked
points, the tenth chapter indicates the deduction of formulae for calculating the
equilibrium rubbery modulus and molecular mass of a linear fragment between
neighboring cross-linked points. Further analysis of the resultant dependencies
allowed the formulation of conditions for obtaining a polymer with unique (unusual)
properties – different modulus and gradient polymers characterized by large changes
of the equilibrium rubbery modulus within the same article. Existence of these unique
properties is confirmed experimentally for synthesized network of polyisocyanurates.
The eleventh chapter describes the derivation of analytical expressions for
relaxation memory functions, necessary for determining the stress relaxation and
creep of the polymers. In this case, the production of entropy of a relaxing system is
represented by transition of relaxants (kinetic units of a polymer of different nature)
into non-relaxants by means of their interaction or diffusion, the mechanism of
interaction of relaxants in stress relaxation being found predominant. The apparatus
created for description of relaxation events in polymers is applied in description of
sorption and swelling processes. Thus, contrary to stress relaxation, the mechanism of
relaxants diffusion is predominant in sorption.

15. 7
The twelfth chapter is devoted to the problem of increasing the accuracy of
prediction of polymer solubility in organic liquids. It is shown that the predictive
ability of the solubility criterion, calculated with respect to the chemical structure of
the polymer and the solvent, sharply increases with consideration for the type of
supermolecular structure of the polymer and the degree of its polymerization.
Based on the chemical structure of the matter, the thirteenth chapter gives a
calculation method for the most important property of organic liquids and polymers,
i.e. surface tension. Contrary to the additive scheme for summation of parachors
which characterizes the contribution of separate atoms to the surface tension, the
approach developed allows estimation of the contribution of polar groups and specific
intermolecular interaction to the surface tension value and connection of it with the
solubility parameter and density of cohesion energy in substances.
Invoking the idea of solubility of a single homopolymer in another one, the
fourteenth chapter suggests a criterion for estimating the compatibility of polymers
basing on the data of the chemical structure of separate components. The analysis of
application of the criterion for compatible, partially compatible or incompatible
polymers indicates its high predictive ability.
On the example of the calculation of the Van-der-Waals volume, molar
refraction, heat capacity and other properties of a number of polymers, chapter fifteen
displays the role of the chemical structure of macromolecule end groups and
importance of their calculation in the study of regularities of changes in the polymer
properties on their molecular mass.
The sixteenth chapter indicates a method for calculating the molar heat
capacity with respect to the chemical structure of polymers. The method is based on a
supposition that the contribution of each atom to heat capacity is proportional to its
Van-der-Waals volume. It is noted that the heat capacity, thermal diffusivity and heat
conductivity of polymers depend not only on their chemical structure, but also on the
physical and phase states of the polymeric body.
The seventeenth chapter describes methodological ways of solving the direct
problem of computerized determination of the physical characteristics of polymers
and low-molecular liquids with respect to their chemical structure and the reverse one
— computer synthesis of polymers with the given set of properties. These problems
are solved by the methods of fragments and separate atoms. The corresponding
software which allows calculation of more than 50 chemical properties of linear and
network polymers and copolymers, and a number of the most important properties of
low molecular weight liquids, as well, is developed. Discussed is the method of
depicting diagrams of polymer properties compatibility, application of which may
significantly simplify solution of the direct and, especially, reverse problems of
computational materials sciences.
Appendices demonstrate abilities of the approach, described in the
monograph, to determine the properties of some natural polymers (the example of
solving the direct problem of polymers synthesis) with respect to their chemical
structure (Appendix 1); to search for chemical structures of polyetherketones (the
example of solving the reverse problem of polymer synthesis), the properties of which
must lie in a given range (Appendix 2); to solve a mixed problem of polymers
synthesis on the example of analyzing the chemical structure of phenoloformaldehyde
resin, when the direct problem — estimation of the properties of the ideal structures
of such resin with respect to their chemical formulae — and the reverse one —
searching for a combination of structures with which the chemical formula of
phenoloformaldehyde resin obtained provides experimentally observed values of its

16. 8
properties — are solved consecutively (Appendix 3); to analyze the structure and
properties of copolymers, composed of from three to five comonomers (Appendix 4);
and the influence of a strong intermolecular interaction appearing between two
heterogeneous polymers on their compatibility is analyzed (Appendices 5 and 6).

17. Chapter I. Brief information on types of polymers and their
chemical structure
The very large number of existing polymers may be subdivided into three
main classes forming the basis of the presently accepted classification. The first class
contains a large group of carbochain polymers whose macromolecules have a skeleton
composed of carbon atoms. Typical representatively of the polymers of this class are
polyethylene, polypropylene, polyisobutylene, poly(methyl methacrylate), poly(vinyl
alcohol) and many other. A fragment of a macromolecule of the first of them is of the
following structure
[–CH2–CH2–]n
The second class is represented by a similar large group of heterochain
polymers, the main chain of macromolecules of which contains heteroatoms, in
addition to carbon atoms (for example, oxygen, nitrogen, sulfur, etc.). Numerous
polyethers and polyesters, polyamides, polyurethanes, natural proteins, etc., as well as
a large group of elemento-organic polymers relate to this class of polymers. The
chemical structure of some representatives of this class of polymers is the following:
[–CH2–CH2–O–]n Poly(ethylene oxide)
(polyether);
Poly(ethylene terephthalate)
(polyester);
Polyamide;
Polydimethylsiloxane
(elemento-organic
polymer);
Polyphosphonitrile chloride
(inorganic polymer).
CH3
C l
The third class of polymers is composed of high-molecular compounds with a
conjugated system of bonds. It includes various polyacetylenes, polyphenylenes,
polyoxadiazoles and many other compounds. The examples of these polymers are:
[–CH=CH–]n Polyacetylene
Polyphenylene
Polyoxadiazole
(CH2)2 O C
O
C O
O n
NH (C H2)6 N H C (C H2)4
O
C
O n
S i O
CH3 n
N P
C l n
n
N N
C
C
O n

18. 10
An interesting group of chelate polymers possessing various elements in their
composition, able to form coordination bonds (usually, they are depicted by arrows),
also relates to this class. The elementary unit of these polymers is often complex, for
example:
H3C CH3
The most widely used type of material in the large group of polymeric
materials are still the materials based on the representatives of the first class of
polymers which are carbochain high-molecular compounds. The most valuable
materials could be produced from carbochain polymers, for example, synthetic
rubbers, plastics, fibers, films, etc. Historically, these polymers have been
implemented in practice first (production of phenoloformaldehyde resins, synthetic
rubber, organic glass, etc.). Many of carbochain polymers became subsequently the
classic objects for investigation and creation of a theory of the mechanical behaviour
of polymeric substances (for example, polyisobutylene, poly(methyl methacrylate),
poly-propylene, phenoloformaldehyde resin, etc.).
Subsequently, materials based on heterochain polymers – polyamide and
polyester fibers, films, varnishes, coatings and other materials and articles – became
widespread. This has given impetus to investigating the properties and formation of
notions, in particular, of anisotropic substances possessing extremely different
properties in different directions. A special place in the sequence of these polymers is
devoted to high-molecular elemento-organic compounds.
Finally, the representatives of the third class – polymers with conjugated
system of bonds – were used for the preparation of conducting materials.
Considering in general terms the chemical structure of polymers of different
classes, we have discussed the structural formula of the repeating unit in the
macromolecule. However, the existence of many such units in the macromolecule
immediately complicates the situation. Let us begin, for example, with an assumption
that each unit in the elementary act of macromolecule growth may be differently
attached to the neighbouring one; in this case, we are talking about the ‘head-to-head’,
‘tail-to-tail’ or ‘head-to-tail’ addition. Various variants of the unit addition to the
propagating macromolecule are possible for asymmetric monomers of the
type which possess R substituents on one of carbon atoms. Here, variants of ‘head-to-head’
... ...
and “head-to-tail”
H3C
O
P
O O
CH3
Zn
O
P
O
O
C H2 C H
R
CH2 CH CH CH2 CH2 CH CH CH2
R R R R
... ...
CH2 CH CH2 CH CH2 CH
R R R

19. 11
additions are possible.
Alternation of the types of addition is possible, i.e. units may be differently
attached to each other in a single macromolecule. Existence of a great number of units
in the polymeric chain and possibility of only several variants of their attachment
gives a huge number of isomers in relation to the whole macromolecule. To put it
differently, a polymer may contain (and indeed contains) not only the macromolecules
of the same chemical structure, but mixtures of a large number of macromolecules,
which, of course, makes the polymer to differ from low-molecular substances,
composed of identical molecules only.
We will not talk about a rapid increase of the number of possible isomers in
the sequence of substituted saturated hydrocarbons with the number of carbon atoms
(i.e. with propagation of the molecule); even at a small (compared with polymers)
number of them this number reaches a tremendous value. It is easy to imagine that
when the number of units becomes tens or hundreds of thousands, the number of
possible isomers becomes astronomically high [80].
Let us return to monosubstituted unsaturated hydrocarbons. When a polymeric
chain is formed during polymerization, the substituents R may dispose differently in
relation to the plane of single bonds. In one of possible cases, these substituents are
disposed irregularly in relation to the plane of single bonds; such polymers are called
irregular or atactic:
H
C
H
C
H
C
H
C
H
C
R
C
H
C
H
C
H
C
R
C
H
C
H
C
H
C
H
C
H
C
R
C
H
R
H
R
H
R
H
R
H
R
H
R
H
R
H
R
In other cases, synthesis may be performed in such a manner that substituents
would be disposed either by the same side of the plane of the main bonds
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
C
H
R
H
R
H
R
H
R
H
R
H
R
H
R
H
R
or by both sides, but with regular alternation of the substituents direction:
H
C
H
C
R
C
H
C
H
C
H
C
R
C
H
C
H
C
H
C
R
C
H
C
H
C
H
C
R
C
H
C
R
H
H
H
R
H
H
H
R
H
H
H
R
H
H
H
The polymers composed of the units with regular alternation of substituents
were called stereoregular. If the substituents are disposed on one side of the plane of
the main bonds, stereoregular polymers are called isotactic. If they are disposed on
both sides of the plane, the polymers are called syndiotactic.
The situation is more complicated with polymers synthesized from
disubstituted monomers. Already in the monomer, substituents may dispose on the
same (cis-isomer) or on both sides (trans-isomer) of the plane of the double bonds:
H
C C
R
H
R'
H
C C
R
R'
H

20. 12
Synthesis of macromolecules from cis-isomers leads to the formation of
erythro-diisotactic polymers
R
C
R'
C
R
C
R'
C
R
C
R'
C
R
C
R'
C
R
C
R'
C
H
H
H
H
H
H
H
H
H
H
and trans-isomers give treo-diisotactic polymers
R
C
R'
C
R
C
R'
C
R
C
R'
C
H
H
H
H
H
H
H
C
R'
C
H
C
R'
C
H
C
R'
C
H
C
R'
C
H
C
R'
C
H
C
R'
C
H
C
R'
C
H
C
R'
C
R
H
R
H
R
H
R
H
R
H
R
H
R
H
R
H
Needless to say, other more complex modifications are also possible, which
immediately cause a change of properties of polymeric materials.
The materials composed from stereoregular polymers are often easily
crystallized so that gives their physical structure and properties can be regulated.
Here we meet for the first time a modification of the properties of polymeric
materials, which is caused by practically any change in the chemical structure of
macromolecules and the physical structure of the polymeric substance. Physical
modification is often indicated by a change of the chemical structure, and sometimes
is completely defined by it.
One of the main methods of modification is the synthesis of copolymers, when
not a single but several monomers participate in the reaction. That is why the
macromolecule becomes composed from different units. These units may alternate
continuously:
–A–B–A–B–A–B–A–B–A–B– the alternating copolymer;
but, most often, they are arranged irregularly:
–A–A–B–A–B–B–A–A–A–B– the random copolymer.
The units may also be linked in separate blocks which are the linked to each other:
–A–A–A–A–A–B–B–B–B–B– the block-copolymer.
Obviously, each block may contain a different number of units. This is
immediately shown up in the properties of the future polymeric substance. In this
case, the copolymerization process becomes regulated. Running ahead, recall that
mechanical mixtures of polymers and copolymers of the same molar composition may
often possess rather different properties, but sometimes they are practically identical.
The considered schemes of addition of units during macromolecule growth
indicate the only case of copolymerization of two types of monomers. Even if many
combinations are realized in these simplest cases, their number grows immeasurably
when three or more monomers (or types of units) are used
All the above-discussed chains of polymers represent linear formations.
However, branched macromolecular chains could be easily synthesized. For this
purpose, it is even unnecessary to introduce multifunctional compounds into the chain
composition. Branching also occurs in polymerization of unsaturated hydrocarbons
with no functional groups. If no special steps are taken, the products of
polymerization of ethylene, propylene, isobutylene and other similar compounds will
always contain some amount of chains branched from the main chain. Concerning the
products of polycondensation (see the above discussion on polyesters and

21. 13
polyamides), introduction of a three-functional compound into the main chain always
leads to the formation of branched polymers:
... ...
A A A A' A A A A A A A
A
A
A
A
... A A A A' A A A A A A A
It is self-evident that the polymeric body based on the branched
macromolecules will differ in the structure and properties from a substance composed
of linear macromolecules. However, we must not hurry in concluding about the type
of physical structuring of the branched polymers. At first glance, it seems that the
presence of large branches will make obstacles to denser packing of the chains, as
well as to the crystallization process or regulation of macromolecules in general.
Indeed, this is sometimes the case. In other cases, the opposite situation is observed. It
depends upon the chemical structure of the main chain and its branches, which
determines the volume of units, interaction forces between them and neighbour
chains, etc.
Recently, special attention has been paid to the structure and properties of so-called
dendric polymers, the macromolecule of which is schematically depicted in
Figure 1 [98, 212]. Below, we will discuss in more detail the influence of the types of
branchings on the properties of the resulting polymers.
Figure 1. Schematic representation of dendric polymers
Branchings may be composed in different ways. They may contain the same
units, which compose the main chain. However, ‘grafted’ polymers have become
widely used; they are formed in grafting of previously obtained chains of a definite
structure to the main chain with an extremely different structure:
... ...
B
B
B
B
...

22. 14
Sometimes, such grafting is performed many times.
We can now easily pass from the branched to three-dimensional ‘cross-linked’
polymers. This requires just an increase of the concentration of multifunctional
compounds in the polymer chain. The chains could also be cross-linked by special
curing agents, i.e. by compounds containing active groups, capable of reaction with
functional groups of the main chain or the end groups. The classic example is the
curing of epoxy resins:
CH3
O C
CH3
O CH2
CH CH2
O
CH3
... O C
CH3
O CH2
CH CH2
O
NH2
R
NH2
+
CH3
O C
CH3
CH3
O C
CH3
O CH2 CH CH2
O CH2
OH
CH CH2
OH
NH
R
NH
...
...
Further on, the second hydrogen atom is substituted, and a network is formed.
According to the classification described in ref. [202], there exist several main
...
methods of obtaining network polymers:
1) Realization of a chemical reaction between two (or more) different functional
end groups, attached to a chain of low molecular mass. As a result, a dense network
with short chains between cross-link points is formed.
2) Chemical linking of high-molecular compounds by the end groups with the
help of a low-molecular cross-linking agent. Consequently, a network with long linear
fragments between the cross-linked points is formed.
3) Formation of a network by copolymerization of two- and polyfunctional
monomers. The example of such a network is the styrene–divinylbenzene system:
... ...
CH2 CH CH2 CH CH2
... ...
CH2 CH CH2 CH CH2
4) Vulcanization of polymeric chains by involving, in the reaction, functional
groups disposed along the main chain. The reaction is performed either by the
application of a low-molecular cross-linking agent or by means of radiation and other
types of influence on the functional groups.

23. 15
Other possible (and already realized in practice) ways of producing the
network systems should also be added.
5) Formation of networks with by means of a reaction of two (or more)
heterogeneous polymers by functional groups disposed along the chain of each
polymers (i.e. in the repeating units, but not at the ends).
6) Synthesis of polymeric networks with the help of the polycyclotrimerization
reaction. For this purpose, oligomers with end groups capable of forming cycles
during the reaction [56, 79, 101, 152] are formed. The example of such a reaction is
the trimerization of two-functional oligomers (or monomers) containing cyanate end
groups. Clearly, other ways of obtaining the polymeric networks are also possible.
Recently, a new type of polymer, called ‘interpolymers’ was produced [16,
215]. The interpolymer is a system composed of two (or more) macromolecules,
heterogeneous in the chemical structure, chemically bonded to each other through the
functional groups disposed in the repeating units of the each macromolecule. A
schematic representation of the interpolymer is displayed in Figure 2.
Figure 2. Schematic representation of interpolymer.
A specific example of this system is, for example, a product of interaction
between polystyrene and polytrichlorobutadiene:
... CH2 CH ... + ... CH2 CH CCl CCl2
...
AlCl3
... ...
CH2 CH CCl CCl
The formation of interpolymers gives new possibilities of modifying the structure and
properties of polymers.
Another type of ‘two-cord’ system is the ladder polymer, the example of
which is polyphenylsylsesquioxane [113]:
... ...
CH2 CH
... ...
Si O Si
O
... Si
...
O
O
O Si O

24. Chapter II. Packing of macromolecules and polymer
density
II.1. Increments method and basic physical assumptions
After discussing briefly the chemical structure of polymers, let us pass to the
volumetric representation of macromolecules, which is necessary for understanding
the features of structure formation in polymers. These considerations will be based on
the assumptions developed by A.I. Kitaigorodsky in organic crystal chemistry [75].
According to these assumptions, every atom is presented as a sphere with
intermolecular radius R. Values of these radii are determined from the data of X-ray
structural analysis of ideal crystals of organic substances. In this case, it is assumed
that valency-unbonded atoms, entering into an intermolecular (but not chemical)
interaction, contact each other along the borders of the spheres. This is schematically
represented in Figure 3. Then, if two identical atoms are in contact, the intermolecular
radius will be determined from the relation:
R = l/2, (II.1)
where l is the distance between mass centers of two identical valency-unbonded
atoms, which, however, are capable of intermolecular physical interaction.
Figure 3. Schematic representation of intermolecular (Van-der-Waals) interaction of two atoms
According to the same assumptions, chemical interaction between two atoms
always causes their compression, because the length of the chemical bond di is always
shorter than the sum of two intermolecular radii:
di R1 + R2. (II.2)
This is clear from Figure 4, which schematically depicts two chemically
bonded atoms. If the intermolecular radii Ri for all atoms participating in the repeat
unit, and all lengths of chemical bonds between these atoms are known, their own
(Van-der-Waals) volume of the repeat unit could be easily calculated, and a model of
this unit (or greater fragment of the macromolecule), in which the volume of each
atom is bordered by a sphere with intermolecular radius Ri, could be composed.

25. 17
Figure 4. Schematic representation of two chemically bonded atoms.
Figure 5. Model of polyethylene chain fragment.
Table 1 shows intermolecular radii of some widespread atoms, which compose
the majority of polymers.
Table 1
Van-der-Waals radii R of different atoms
Atom R, nm Atom R, nm
C 0.180 Si 0.210
H 0.117 Sn 0.210
O 0.136 As 0.200
N 0.157 S 0.180
F 0.150 P 0.190
Cl 0.178 Pb 0.220
Br 0.195 B 0.165
I 0.221 Ti 0.200
Table 2 displays bond lengths of various combinations of atoms, also
characteristic for most of existing polymers. If these values are known, the volume of
the repeat unit of any polymer may be calculated. To conduct this, the own volume of
each atom participating in the repeat unit should be preliminarily determined. It is
calculated from the formula
Δ = 3 π 3 −Σ π 2 −
(3 ),
1
Vi R hi R hi (II.3)
3
4
i
where ΔVi is the increment of the own (Van-der-Waals) volume of the present atom;
R is the intermolecular radius of this atom; hi is the height of the sphere segment, cut
off from the present atom by a neighbor one, chemically bonded to it. The value hi is
calculated from relation

26. 18
+ − h = R
− ,
(II.4)
2 2 2
R d R
2
i i
i
i d
where Ri is the intermolecular radius of a neighbor valency-bonded atom; di is the
length of the chemical bond (see Figure 4).
Table 2
Chemical bond length di for same pairs of atoms
Bond* di, nm Bond* di, nm Bond* di, nm
C–C 0.154 C–F 0.134 O–F 0.161
C–C 0.148 C–F 0.131 O=N 0.120
C=C 0.140 C–Cl 0.177 O=S 0.144
C=C 0.134 C–Cl 0.164 O=P 0.145
C=C 0.119 C–Br 0.194 N–P 0.165
C–H 0.108 C–Br 0.185 N–P 0.163
C–O 0.150 C–I 0.221 N–P 0.158
C–O 0.137 C–I 0.205 S–S 0.210
C–N 0.140 C–B 0.173 S–As 0.221
C–N 0.137 C–Sn 0.215 S=As 0.208
C=N 0.131 C–As 0.196 Si–Si 0.232
C=N 0.127 C–Pb 0.220 P–F 0.155
0.134 H–O 0.108 P–Cl 0.201
C ≡N 0.116 H–S 0.133 P–S 0.181
C–S 0.176 H–N 0.108 B–B 0.177
C–S 0.156 H–B 0.108 Sn–Cl 0.235
C–Si 0.188 O–S 0.176 As–Cl 0.216
C–Si 0.168 O–Si 0.164 As–As 0.242
* If the same pair of atoms is linked by a single bond, the longer bond corresponds to attachment of this
atom to an aliphatic carbon atom; the shorter bond corresponds to attachment of the same atom to an
aromatic carbon atom.
Increments of the volumes of various atoms and atomic groups are shown in
Table 3. Obviously, the volume of the given atom depends on its surrounding, i.e. on
the type of atoms chemically bonded to it. The greater the volume of the neighbor,
chemically bonded atom and the shorter the length of the chemical bond, the greater is
the compression of the given atom.
When increments of the volumes, ΔVi, of all the atoms entering into the repeat
unit of polymers are determined, the relative part of the occupied volume in the total
volume of the polymeric substance may be calculated. In the case of polymer,
calculations would be appropriate to conduct basing on molar volumes of the repeat
unit, because polymers are always polydispersional (i.e. they contain macromolecules
of various length), and also because at long lengths of the macromolecule the
influence of end groups may be neglected. Then, the own molar volume will equal
own = AΣΔ ,
V N Vi and the total molar volume Vtotal = M/ρ, ρ is density of the
i
polymeric substance; M is the molecular mass of the repeat unit; NA is the Avogadro
number. Numerous experiments and calculations show that in all cases the condition
Vown Vtotal is fulfilled. Hence, in the first approximation, the volume of the polymeric
substance could be divided into two parts: the own (Van-der-Waals) volume of atoms,
which they occupy in a solid, and the volume of spaces determined as the difference
of Vtotal and Vown. Of interest is determination of the part
C N

27. 19
Table 3
Van-der-Waals volumes of atoms

28. 20

29. 21

30. 22

31. 23

32. 24

33. 25

34. 26

35. 27

36. 28

37. 29
of the occupied volume or, according to the terminology used in organic crystal
chemistry, the molecular packing coefficient k:
Δ
N V
ρ
V
own
k i
= =
Σ
/
A
total
M
V
i
. (II.5)
Clearly, the value of k for the same polymer will depend on temperature and
the physical state of the polymer, because the value of ρ depends on them.
Calculations performed for many amorphous bulky polymers existing in the glassy
state have indicated that the first approximation of k gives its value constant and
practically independent of the chemical structure of the polymer [41]. Passing on to
polymers with a complicated chemical structure from those with a simple one causes
no significant change of the part of the occupied volume (e.g. the value of k).
Table 4 indicates the chemical structure and numerical values of coefficients
of the molecular packing of some glassy polymers. It also shows that first
approximations of the values of k for each of them are equal, indeed. To demonstrate
this experimental fact more clearly, Figure 6 displays the dependence of density ρ of
various polymers on the relation M NA ΣΔ
Vi . In Figure 6 it is clearly seen that all
i

38. 30
Table 4
Values of the coefficients of molecular packing for some glassy and semi-crystalline polymers
Structural formula of the repeat unit of polymer Van-der-Waals
volume of the
unit, cm3/mol
Packing
coefficient k
41.6 0.678
32.6 0.682
58.5 0.684
69.1 0.680
144.3 0.679
234.7 0.679
263.1 0.680
277.5 0.688
56.4 0.685
C H 3
H
C
C
CH2
N
CH3
C
C
CH2
O CH3
O
CH3
C
C
CH2
O C2H5
O
C O
O
CH3
C
CH3
O
O O
O O
CH2 CH
CH
CH2
C
O
C
O
C
O
C
O
(CH2)8 C
O
C
C
O
O
C
NH
NH
C
C
HN
C
C H 3
C H 2
C
O
O

39. 31
—CH2—CH=CH—CH2— 59.1 0.654
74.3 0.659
100 0.699
97.8 0.708
110.3 0.693
269.0 0.692
CH2
(CH2)5 NH C
O
CH3
O
—CF2—CF2— 43.9 0.753
72.4 0.663
—CH2—CHF— 33.8 0.700
54.9 0.666
—CH2—CCl2— 58.7 0.654
—CH2—CF2— 36.0 0.744
123.1 0.641
134.3 0.664
CH2 CH C
CH3
CH2 CH
CH
H2C CH2
HC CH2
N
CH2 C
O
C
C
O
N
C
C
O
N O
CH2 CH
O C CH3
O
CH2 CH
O
CH3
CH3
CH2 C
C
O
O CH
CH3
CH3
CH3
CH2 C
C
O
O C4H9

40. 32
168.3 0.651
120.0 0.607
85.9 0.696
163.0 0.687
88.8 0.705
111.6 0.669
115.5 0.657
65.6 0.638
89.3 0.650
40.0 0.681
CH3
CH2 C
C
O
O C6H13
CH3
Si
O
CH2 CH2
CF3
C
O
CH2 CH2 CH2 CH3
N
CH2 CH
N
CH2 CH
N
H2C C O
H2C CH2
CH2 CH
Cl
CH2 CH
CH3
CH2 CH S
CH3
CH2 CH
C
O
O C2H5
O CH2
C
O

41. 33
69.9 0.684
172.5 0.740
70.6 0.677
—CH2—O— 21.3 0.752
126.1 0.616
118.5 0.667
53.0 0.733
150.8 0.679
103.0 0.620
76.2 0.568
F
CH3
CH2
CH2
C2H5
CH3
—CH2—CH2—S— 46.4 0.680
144.4 0.692
227.7 0.693
O CH
CH3
CH2 C
O
C
O
C
O
NH NH
CH2 CH
C O
O
CH3
CH3
Si O
CH2 CH
C
O
O C4H9
C
Cl
CF2
CH2 C
C
O
O CH
CH2
CH2
CH2
Si O
C2H5
Si O
CH3
(CH2)2 O C
O
C O
O
O C
O
O

42. 34
154.1 0.696
157.0 0.721
—CH2—CH2— 30.2 0.682
46.3 0.666
99.6 0.665
262.1 0.726
Figure 6. Dependence of density ρ on ΣΔ
the values of ρ determined experimentally fit well the same linear dependence on the
relation of atoms mass on their volume. In accordance with Equation (II.5), the
tangent of this straight line represents the molecular packing coefficient which, in the
case of amorphous bulky systems, serves as an universal constant. If it is true, the
polymer density ρ may be calculated from the equation
kM
A
ρ , (II.6)
ΣΔ
=
N Vi
i
O C
O
O SO2
CH2 CH
CH3
CH2 CH
CH2 NH C O
O
(CH2)4 O C NH
O

43. 35
that yields directly from Equation (II.5) under the condition kavg = const. In the case of
amorphous bulky polymers, kavg = 0.681. For silicon-containing polymers, the average
coefficient of molecular packing is 0.603.
Hence, a change of the polymer chemical structure is unable to cause a
significant effect on the part of the occupied volume in amorphous polymeric
substance, and the value of density, ρ, itself depends on the relation of mass and the
Van-der-Walls volume of the repeat unit only.
Obviously, here we are dealing with true bulky substances of the amorphous
structure. In reality, a polymeric substance with any porosity may be formed, and the
coefficient k will have extremely different values. However, in this case, the notion of
the packing density, quantitatively estimated by the value of k, loses its usual meaning
and must be calculated for pore walls material only. We return to this problem below
when discuss parameters of the porous structure of polymers, determined by the
sorption method.
For copolymers, equation (II.6) has the form
( )

 


 
α α α
α α α


 


k M M M
Δ + +  

 
n i



Δ +  

 



 

Δ
+ + +
=
Σ Σ Σ
i n
i
i
i
i
n n
N V V V
ρ
...
...
2
2
1
A 1
avg 1 1 2 2 , (II.7)
where α1, α2, …, αn are molar parts of the components 1, 2, …, n; M1, M2, …, Mn are

molecular masses of the repeat units of the same components;
 

1

 

Δ Σi
Vi ,
2

 


 

Δ Σi
Vi ,
…,

ΣΔ 
are their Van-der-Waals volumes.
Vi  

i n
 

In the reduced form, expression (II.7) is:
k n
Σ
k M
=

k
k k
Σ Σ
N V
=
=
=

 

 
k i

Δ
=
k n
1
k i k
A
1
avg
α
α
ρ , (II.8)
where αk, Mk,

ΣΔ 
are the molar part, the molecular mass, and the Van-der-
Vi  

i k
 

Waals volume of the k-th component, respectively.
If we want to express the density of copolymer via densities ρ1, ρ2, …, ρn of
homopolymers based on the components 1, 2, …, n, expression (II.7) changes to the
following form:
= + + +
α α ...
α
M M M
n n
n
M M M
n
n
ρ
α
ρ
α
1
ρ
α
ρ
2
+ + ...
+
2
2
1
1
1 1 2 2 , (II.9)

44. 36
(in this case, it should be taken into account that α1 + α2 + … + αn = 1).
In the reduced form, the expression (II.9) is the following:
=
Σ
=
= =k n
α
1
ρ , (II.10)
Σ
M
k k
=
k
k
k n
k
k k
M
1
ρ
α
Expressions (II.7)–(II.10) may also be used for calculating the density of
miscible blends of polymers.
Let us now examine the temperature dependences of the molecular packing
coefficients of glassy polymers. Calculation of values of k at different temperatures
are performed by formulae yielding from the expression (II.5):
Δ
N V
A
1
k T i
i
[ ( )] g G g
( )
+ −
MV T T
=
Σ
α
, (T Tg); (II.11)
Δ
N V
A
1
k T i
i
[ ( )] g L g
( )
+ −
MV T T
=
Σ
α
, (T Tg); (II.12)
where Vg is the specific volume of the polymer at the glass transition temperature Tg;
αG and αL are the volume expansion coefficients of polymers below and above the
glass transition temperature, respectively.
Figure 7. Temperature dependences of the coefficients of molecular packing k for a series of polymers:
1 – poly(n-butyl methacrylate), 2 – poly(n-propyl methacrylate), 3 – poly(ethyl methacrylate), 4 –
polystyrene, 5 – poly(methyl methacrylate), 6 – polycarbonate based on bisphenol A.
Calculations by equations (II.11) and (II.12) indicate that temperature
dependences of the molecular packing coefficients are of the form depicted in Figure
7. A remarkable property of these temperature dependences in the real equality of the
molecular packing coefficient in the first approximation for all bulky polymers at any
temperature below the glass transition point. In the second, more accurate
approximation, the molecular packing coefficient is the same for every polymer at the
glass transition temperature. This value is kg ≈ 0.667.

45. Table 5
Coefficients of molecular packing k for a series of crystalline polymers
Name Type of elementary cell Chemical formula ρ, g/cm3 k
1 2 3 4 5
Polyethylene Rhombic
Pseudo-monoclinic
Triclinic
CH2CH2
1.000
1.014
0.965
1.013
0.736
0.746
0.710
0.745
Polypropylene:
- isotactic
- syndiotactic
Monoclinic
Monoclinic
0.936
0.910
0.693
0.674
1,2-poly(butadiene):
- isotactic
- syndiotactic
Rhombic
Rhombohedral
0.963
0.960
0.692
0.690
CH2 CH
CH3
CH2 CH
CH
CH2
1,4-trans-poly(butadiene) Pseudo-hexagonal CH2CH=CHCH2 1.020 0.733
1,4-cis-poly(butadiene) Monoclinic CH2CH=CHCH2 1.010 0.726
1,4-cis-polyisoprene Monoclinic 1.000 0.725
CH2 CH C CH2
CH3
Polychloroprene Rhombic 1.657 0.893
CH2 CH C CH2
Cl
Poly(ethylene terephthalate) Triclinic 1.455 0.776
Poly(hexamethylene
terephthalate)
O CH2
CH2 O C
O
C
O
Triclinic 1.131 0.652
O C
O
C
O
O (CH2)6
37

46. 38
1 2 3 4 5
Poly(ethylene isophthalate) Triclinic 1.358 0.724
O C
O
CH2
C
O
O CH2
Poly(ethylene adipate) Triclinic 1.274 0.782
Polyamide 6,6:
α-isomer
β-isomer
Triclinic
Triclinic
1.240
1.248
0.764
0.769
O (CH2
)2 O C (CH2
)4
O
C
O
C
O
(CH2)4 C HN
O
(CH2)6 NH
Polyamide 6,10 Triclinic 1.157 0.740
C
O
(CH2)8 C HN
O
(CH2)6 NH
Polyamide 6 Monoclinic 1.230 0.758
C
O
HN (CH2)5
Polyamide 11 Triclinic 1.192 0.789
C
O
HN (CH2)10
Poly-4-methylpentene-1 Tetragonal 0.813 0.598
CH2 CH
CH2
CH CH3
CH3
38

47. 39
1 2 3 4 5
Polyvinylchloride Rhombic
Monoclinic
1.440
1.455
0.680
0.687
Polytetrafluoroethylene Pseudo-hexagonal
Hexagonal
CH2 CH
Cl
–CF2–CF2– 2.400
2.360
0.794
0.781
Polyvinylfluoride Hexagonal 1.440 0.742
CH2 CH
F
Poly(vinyl alcohol) Monoclinic 1.350 0.770
CH2 CH
OH
Polyacrylonitrile Rhombic 1.110 0.677
Poly(methyl methacrylate)
isotactic
CH2 CH
C N
Pseudo-rhombic 1.230 0.719
C H 3
C H 2 C
C
O
O
C H 3
Polystyrene Rhombohedral 1.120 0.711
CH2 CH
Polyoxymethylene Hexagonal –CH2–O– 1.506 0.808
Polyethylene oxide Hexagonal –CH2–CH2–O– 1.205 0.723
39

48. 40
1 2 3 4 5
Polypropylene oxide Rhombic 1.102
1.154
0.663
CH2 CH O 0.694
CH3
40

49. 41
Taking into account that the specific volume at the glass transition temperature
Tg equals
N V
V i
k M
i
g
A
g
g 1
ΣΔ
= ρ = , (II.13)
where ρg is the polymer density at Tg; and substituting (13) into (11) and (12), we get
g
[ 1
( )] G g
( )
T T
k
k T
+ −
=
α
, (T Tg); (II.14)
g
k
[ 1
( )] L g
( )
T T
k T
+ −
=
α
, (T Tg); (II.15)
Equations (II.14) and (II.15) can be used for obtaining relations, which
describe temperature dependences of the density of polymers ρ in the glassy and
rubbery states. For this purpose, we substitute (II.14) and (II.15) into equation (II.6):
g
ρ , (T Tg); (II.16)
[ + ( − )] ΣΔ
=
T T N Vi
i
k M
T
G g A
1
( )
α
g
[ + ( − )] ΣΔ
=
T T N Vi
i
k M
k T
L g A
1
( )
α
, (T Tg); (II.17)
Because, as it is seen from the further considerations, values of expansion
coefficients αG and αL, as well as the glass transition temperature Tg, can be
calculated from the chemical structure of the repeating polymer unit, temperature
dependences of density ρ (T) can also be calculated from relations (II.16) and (II.17).
In conclusion, let us note that the constancy of the coefficient of molecular
packing k is true only for amorphous bulky substances composed of polymers. In the
case of crystalline polymeric substances, the situation is significantly changed. If the
coefficients of molecular packing for ideal polymeric crystals are calculated with the
help of the X-ray analysis data, one can assure himself that, in spite of amorphous
ones, the coefficients of molecular packing of crystalline polymers are extremely
different. The smallest values of k are typical of aliphatic systems with volumetric
side groups, for example, for poly-4-methylpentene-1 and poly-n-butyraldehyde. The
highest coefficients of packing are typical of 1,4-trans-β-polyisoprene and poly-chloroprene.
As an example, Table 5 shows the crystallographic values of densities and
molecular packing coefficients for a series of typical crystalline polymers. It is clear
that the values of k for them vary in a wide range. Hence, crystalline polymers display
a rather wide distribution curve of the coefficients of molecular packing (Figure 8).

50. 42
Figure 8. Curve of distribution of the coefficients of molecular packing k for crystalline polymers.
II.2. Relationship between free volume of polymers, coefficient of
molecular packing and porous structure
Before we start discussing the relationship between the above-mentioned
physical characteristics, the term of the ‘free volume’ must be discussed in brief.
There are three definitions of the free volume:
1) The free volume represents the difference between the true molar
volume of the substance, VM, and its Van-der-Waals molar volume ΣΔ
NA Vi :
i
Δ = − ΣΔ = − ΣΔ
V VM NA Vi M /ρ NA V . (II.18)
i
i
i
The value of ΔV obtained in this way is often called ‘the empty volume’.
Clearly, the empty volume depends on temperature, because the molar volume also
depends on it: VM = M/ρ. Substituting this relation into equations (II.16) and (II.17),
we obtain:
( )

 

 
−
+ −
1
G g
Δ ( )
= ΣΔ 1
g
A k
T T
V T N V
i
i
α
, (T Tg); (II.19)
( )

 

1
 
−
+ −
T T
L g
Δ ( )
= ΣΔ 1
g
V T N V
A k
i
i
α
, (T Tg); (II.20)
Relations (II.19) and (II.20) describe the temperature dependences of the
empty volume.
2) The free volume represents the difference between the volumes of the
substance at the absolute zero and at the assigned temperature; to put it differently, the
free volume represents an excessive volume occurring as a result of thermal
expansion of the substance. This definition of the free volume is most valuable.
Moreover, the present free volume is subdivided into the free volume of fluctuation
and the expansion volume.
3) The free volume represents the difference between the volume of
polymeric substance at the assigned temperature and the volume of the ideal crystal

51. 43
composed of a polymer of the same chemical structure. This definition of the free
volume is used extremely seldom.
Let us now pass to analysis of the relationship between the free volume of
polymers, the coefficient of molecular packing and the porous structure.
The porous structure mostly defines their properties. That is why the methods
of estimation of the porous structure of polymers and its connection with such
characteristics as the coefficient of molecular packing and the free volume of polymer
must be discussed in detail. The case is that the size of micropores depends on the
method of its estimation. Clearly, interpretation of their nature and the relationship of
the characteristics of the microporous structure with the properties of polymers
significantly depends on the method of their determination.
The properties of many bulky and film polymers significantly depend on the
density of packing of macromolecules, and for such systems as sorbents, ionites, etc.,
used in gel-chromatography and production of ion exchangers, the volume of pores is
very important, together with their size distribution, specific surface.
Let us present the definition, given in ref. [68]: “Pores are emptinesses or
cavities in solids usually connected with each other. They possess various and
different form and size, determined significantly by nature and the way of obtaining
absorbents”.
Usually, the characteristics of a microporous structure are judged by
experimental data on equilibrium adsorption, capillary condensation of vapor and
mercury pressing in (mercury porosimetry) [121]. Recently, the positron annihilation
method has been used [3, 48, 110, 123, 134, 140, 155, 164, 187, 211]. This method
helps in determining the characteristics of the microporous structure, when the size of
pores is commensurable with the molecule size. Such micropores are inaccessible for
sorbate molecules and especially for mercury when mercury porosimetry is used.
Polymers and materials prepared from them possess the feature (in contrast to
mineral sorbents) that they swell during sorption of vapors of organic liquids.
Consequently, their structure changes and usual methods of calculation give no
possibility of estimating the true porous structure of the initial material. It stands to
reason that vapors of organic liquids, in which polymer does not swell, can be used in
sorption experiments. Then the parameters of the porous structure of the initial
material can be determined, but these cases are quite rare [107].
Before passing to comparison of parameters of the porous structure with the
free volume of the polymer, it should be noted that parameters of the porous structure
for the same polymer could be significantly different due to conditions of its synthesis
and further processing. For example, a film or fibers may be obtained from various
solvents [81], as well as from a solvent–precipitant mixture [97], and will display a
different microporous structure and properties. The same can be said about materials
obtained by pressing and injection molding and with the help of hydrostatic extrusion
as well. Therewith, macropores may also be formed and their total volume may be
quite high. If special synthesis methods are used, materials based on polymer
networks may be obtained, which possess a large specific surface and extremely large
pore radii [115]. Clearly, such macropores are not defined by the packing density of
macromolecules. They may be formed by loose packing of formations larger than
macromolecules or may be caused by conduction of a chemical process of the
network formation under special conditions [167].
Several more general comments should be made. Besides macropores, as
mentioned above, micropores are present in a polymeric substance, the size of which
is commensurable with the size of sorbate molecules. Clearly, in this case, sorbate

52. 44
molecules cannot penetrate into these micropores (it is assumed that for sorbate
molecules to penetrate into pores, the volume of the latter must be several times
greater than that of penetrating molecules). Since sorbate molecules may be different,
i.e. may possess different sizes, parameters of the porous structure determined from
the sorption data will depend on types and sizes of molecules of sorbed substances.
That is why such terms as ‘porosity to nitrogen’, ‘porosity to benzene’, etc. have been
introduced. Of interest is that the sorption method of determination of the porous
structure of polymeric substances cannot be used in the case when a substance
contains quite large macropores. This is associated with the fact that under conditions
of polymolecular adsorption, when many molecular layers are formed on walls of
macropores, their fusion becomes difficult, i.e. capillary condensation is absent. Then,
the total volume of pores calculated by the amount of sorbate penetrated into the
polymeric substance will be smaller than the true volume of macropores.
Starting the analysis of relationship between the physical characteristics of the
polymeric substance and its microporous structure, let us introduce some definitions
and designations:
Ssp is the specific surface of micropores,
W0 is the total volume of pores,
W0
max is the maximal volume of pores accessible for sorbate molecules of any
size (per gram of the substance),
VF is the free volume (in the present case, the volume of expansion),
VE is the ‘empty volume’ (see above),
VT is the specific volume of the polymeric substance at given temperature,
VW is the Van-der-Waals volume (per gram of the substance),
Vid.cr. is the specific volume of the ideal crystal or bulky amorphous polymer (a
bulky amorphous polymer is the one in which no sorbate molecule can penetrate into
its pores).
Let us write down some relations connecting these characteristics:
VF = VT – V0; (II.21)
VE = VT – VW. (II.22)
Next, let connect these characteristics with the coefficient of molecular
packing k (see above):
k = VW/VT; 1 – k = VE/VT. (II.23)
As mentioned above, there are so-called non-porous sorbents (for example,
crystalline substances), into which no molecules of sorbate can penetrate without
swelling. Clearly, that for such substances W0
max = 0. At the same time, as seen from
the data in Table 5, coefficients of molecular packing of crystals fall within the range
from 0.64 to 0.89. Taking into account that the coefficient of molecular packing, by
definition, represents a part of the occupied (Van-der-Waals) volume, it can be said
that the part of empty (but inaccessible) volume is 1 – k = 0.11–0.36. This empty
volume is inaccessible for even small sorbate molecules to penetrate in; let mark it as
Vinacc.. Then the volume of the ideal crystal (or bulky amorphous polymer, Vblk) can be
written down as
Vid.cr. = VW + Vinacc.; Vblk = VW + Vinacc.. (II.24)
The volume of the real polymeric substance (which contains micropores
accessible for a sorbate) will be summed up from three parts:
VT = VW + Vinacc. + W0
max. (II.25)
Then
W0
max = VT – Vid.cr.; W0
max = VT – Vblk. (II.26)

53. 45
The coefficient of molecular packing in the bulky part of the polymer will be
determined from the relation
= . (II.27)
max
W
V
V W
T 0
k
−
In the case of estimation of the density of macromolecule packing for the real
polymeric substance containing micropores accessible for sorbate molecules, the
coefficient of molecular packing, k, should be calculated by the relation
W
V
= , (II.28)
V W
T 0
k
−
where W0 is the total volume of micropores (per gram of the substance), determined
on the basis of sorption measurements.
The value of W0
max that represents the difference between the specific volume
of the substance at the given temperature and volume of the true bulky substance is
conceptually identical to the porosity factor P = 1/ρs – 1/ρt, where ρs is the apparent
density; ρt is the true density. Therewith, ρs represents the density of the substance at
the current temperature, affected by the pores existing in it. It is best to measure the
apparent density of substances with the proper geometrical shape, because when using
no solvents ρs can be found by dividing the substance weight by its volume. If the
apparent density of substances with the improper shape is measured, the pycnometric
or dilatometric method can be used. The difficulty is in selection of a liquid that does
not wet the surface of the substance and does not penetrate deep into it. The true
density ρt represents density of the bulky part of the substance containing no pores. It
is best to measure the density of the ideal crystal, because it can be calculated on the
basis of crystalline lattice parameters. In the case of amorphous and partly crystalline
substances, the method of gradient tubes may be used applying liquids penetrating
well into pores. However, it should be taken into account that a mixture of two liquids
is used for creation of the density gradient in the tube, each of which may possess
different wettability and penetrability into pores. The picture is then distorted, and the
determined density is not true.
The relations shown above can be estimated unambiguously if a polymer
swells in the sorbate, used for estimation of the porous structure of the polymer. If the
experiment indicates that W0 is greater than W0
max, this indicates that the volume of
vapors absorbed by the polymer is greater than the volume of pores existing in it, i.e.
the polymer swells during sorption.
Let us now consider the experimental and calculated data on determination of
the parameters of the polymer structure and coefficients of their molecular packing.
These data are shown in Table 6. For ideal polyethylene crystallites, VE = Vinacc. and
W0
max = 0. The coefficient of molecular packing is quite high. For semi-crystalline
polyethylene, the empty volume, VE, is greater than in the case of the ideal crystal
and, therewith, a part of it is accessible for penetration of small sorbate molecules.
However, the total volume of pores determined by methanol sorption equals 0.01
cm3/g. The molecular packing coefficient for the bulky part of such polyethylene is
significantly lower than for the ideal crystal.
Polymers in the rubbery state (polyisobutylene, for example) also possess
comparatively low values of free volumes and are practically non-porous sorbents
(VE = Vinacc.).
Contrary to this, polymers produced by polycondensation or polymerization in
solution display immensely high values of W0
max. In this synthesis method, pores are
formed due to elimination of the solvent, distributed in the volume of the synthesized

54. 46
polymer. This is observed from the fact that the same polymers produced by
polymerization in the melt are practically non-porous, and values of VE for them are
very small, and W0
max = 0.
Table 6
Parameters of porous structure and coefficients of molecular packing of a series of polymers
Polymer
VE,
cm3/g
max,
cm3/g
W0
W0,
cm3/g
Vinacc.,
cm3/g
K
Polyethylene (100% crystallinity)
–CH2–CH2–
0.26 ~0 ~0 0.26 0.736
Polyethylene (crystallinity 100%)
–CH2–CH2–
0.35 0.08 0.01 0.27 0.675
Polyisobutylene
–CH2–C(CH3)2–
0.36 ~0 ~0 0.36 0.678
Polymethylidenphthalide
CH2 C
O
C
O
Polymerization in dimethylformamide solution
Polymerization in melt
1.28
0.22
1.06
~0

0.22
0.22
0.687
0.687
Polyarylate F-1
C O O
O
C
O
C
O
C
O
Polycondensation in chlorinated bisphenol solution
pressed at 360°C and under 312.5 MPa pressure
0.82
0.24
0.58
~0
0.31
~0
0.24
0.24
0.688
0.688
Pores formed during synthesis may be closed in polymer pressing under high
pressure, and the porous polymer then becomes non-porous. Therewith, in all cases,
W0 is smaller than W0
max that indicates the absence of swelling.
For all polymers, values of Vinacc. are close to these characteristics for the
density of crystallized samples. Of special attention is the fact that independently of
the production method, the molecular packing coefficient for amorphous and semi-crystalline
polymers in their bulky part is the same and close to the average value
kavg = 0.681, which was discussed above. For a crystalline sample, the value of k is
significantly higher.
There is one more interesting point to discuss, associated with molecular
packing, namely, the change of the system volume during polymerization, i.e. at
transition from monomer to polymer.
It is well known that transition from a monomeric liquid to a solid glassy
polymer is accompanied by a significant contraction, i.e. volume decrease [76]. The
specific volume of the polymer Vp is always smaller than that of monomer Vm, and
their difference ΔV = Vp – Vm 0. One of the reasons for contraction is substitution of
longer intermolecular bonds existing in liquid monomers by shorter chemical bonds

55. 47
formed between monomer molecules in the polymer. Therewith, the own Van-der-
Waals volumes of atoms decrease owing to their ‘compressing’ (see above).
Nevertheless, this is not the only reason of contraction. It follows from consideration
of the experimentally determined specific volumes that there is another reason for
contraction, which is more dense packing of polymeric chains compared with the
packing of monomeric molecules. This is indicated by the fact that the packing
coefficients of polymers are always greater than those of their monomers (kp km).
Let the total contraction, ΔVtotal, be presented as a sum of two values: ΔV1,
which is the contraction stipulated by substitution of intermolecular bonds by
chemical ones, and ΔV2, which is the contraction involved by more dense packing of
chains,
ΔVtotal = ΔV1 + ΔV2, (II.29)
and each of the summands estimated.
To do this, values of the specific volume of a polymer should be calculated on
the assumption that it displays the packing coefficient, the same as the monomer km,
i.e.


A p
m
N
p k
V
M
V
i
i 

 

Δ
′ = ⋅
Σ
, (II.30)
where
p

 


 

Δ Σi
Vi is the Van-der-Waals volume of atoms in the repeat unit of the
polymer; M is the molecular mass of the unit. Values of Vp′ for some polymers,
calculated in this way, are shown in Table 7. They are always greater than
experimentally measured values of specific volumes of the polymer, Vp.
The difference between Vp′ and Vm is
ΔV1 = Vp′ – Vm, (II.31)
and the remaining part of the contraction is calculated by the formula
ΔV2 = ΔVtotal – V1. (II.32)
Relative parts of contraction are determined from the relations:
α1 = ΔV1/ΔVtotal; (II.33)
α2 = ΔV2/ΔVtotal. (II.34)
The data shown in Table 7 indicate that in all the cases the smaller part of
contraction depends upon opening of double bonds, and the greater part — on dense
packing of polymer chains. Therewith, the chemical structure of a monomer and an
appropriate polymer significantly affects the values of α1 and α2.

56. 49
Table 7
Changes in volume of the system as a result of polymerization
Polymer (monomer) Vm, cm3/g Vn, cm3/g V′n, cm3/g Vtotal, cm3/g ΔV1, cm3/g ΔV2, cm3/g α1, % α2, %
1 2 3 4 5 6 7 8 9
1.068 0.855 0.968 0.213 0.080 0.133 37.6 62.4
1.102 0.890 1.031 0.212 0.071 0.141 33.5 66.5
1.109 0.928 1.045 0.181 0.064 0.117 35.4 64.6
1.046 0.815 0.951 0.231 0.095 0.136 41.1 58.9
1.082 0.873 1.000 0.209 0.082 0.127 39.2 60.8
CH3
CH2 C
C
O
O CH3
CH3
CH2 C
C
O
O C2H5
CH3
CH2 C
C
O
O C3H7
CH2 CH
C
O
O CH3
CH2 CH
C
O
O C2H5
48

57. 50
1 2 3 4 5 6 7 8 9
1.098 0.952 1.036 0.146 0.062 0.084 42.5 57.5
1.073 0.841 0.976 0.232 0.097 0.135 41.8 58.2
1.104 0.942 1.028 0.162 0.076 0.086 46.9 53.1
CH2 CH
C
O
O C4H9
CH2 CH
O C CH3
O
CH2 CH
49

58. 50
In the set of polyacrylates and polymethacrylates α2 grows first with the
volume of the side substituent and then decreases. Decrease of the intensity of the
effect of the dense packing of chains, apparently, depends upon steric hindrances.
Hence, it follows from the above-said that the notions of porosity and packing
density are inadequate. Porosity reflects almost always cavities greater than the
molecular size, i.e. quite large ones. As for the packing density of macromolecules
themselves, it may be judged by considering the non-porous part of the sample only.
As noted above, application of positron annihilation methods is preferable for
analyzing the microporous structure of polymers [3, 48, 110, 123, 134, 140, 155, 164,
187, 211]. With the help of these methods, qualitative and quantitative information
about the characteristics of submicropores (2–15 Å) in polymers may be obtained.
Let us discuss the results of studying annihilation of positrons in two
polymers, which are good models of the limiting characteristics of the packing density
of macromolecular chains. One of them is polyimide characterized by a highly
regular, quasi-crystalline structure, and the second is poly(1-trimethylsilyl-1-propyne)
(PTMSP) which, on the contrary, is characterized by a low coefficient of molecular
packing.
Consider structural changes in PTMSP, which appear during its long exposure
at room temperature after synthesis.
For comparison, we also display the data on annihilation of positrons for a
series of other model polymers. The chemical structures of all above-mentioned
systems are shown below.
Poly(1-trimethylsilyl-1-propyne)
CH3
C C
Si
CH3
H3C CH3
Polyisoprene
CH CH2
Polydimethylsiloxane
CH3
Polystyrene
n
Polytetraflouroethylene
[—CF2—CF2—]n
n
CH2 C
CH3
n
CH2 CH
n
Si
CH3
O

59. 51
Polyimide
O
C
C
O
N
O
C
C
N O
O n
Observation of the annihilation of positrons in PTMSP was performed with
the help of a method of detection of the lifetime spectra of positrons (measurements
were made by S.A. Tishin; data not published). Measurements were performed by a
thermostabilized spectrometer, which realizes the traditional fast–slow scheme of
detection, with a temporal photomultiplier selected and optimized due to an original
method [111].
Processing of experimental spectra was performed with the help of well-known
software ‘Resolution’ and ‘Positron FIT’.
Table 8 shows the results of separation of parameters of a long-living
component at three-component decomposition of positron lifetime spectra for
PTMSP, polyimide, polystyrene, polydimethylsiloxane and polytetrafluoroethylene.
Clearly, PTMSP possesses an anomalous long lifetime for an ortho-positronium atom,
to annihilation of which by a pick-off–decay the origin of a long-living component of
the lifetime spectrum in polymers is bound [3, 48, 110, 123, 134, 140, 155, 164, 187,
211]. Hitherto, the maximal lifetime of the long-living component, τD, was observed
in polydimethylsiloxane and teflon in solid polymers [123, 164]. Comparison with the
results of measurements in model polymers (see Table 8) indicates that neither the
presence of an unsaturated bond, nor the presence of a side group or silicon atom
separately is the explanation of so high τD for PTMSP.
Table 8
Parameters of the longest component of positron lifetime spectrum for a series of polymers and
rated values of radius R and volume V of micropores
Sample τD + 0.03, ns ID ± 0.25, % R0, Å R, Å V, Å3 E, eV
PTMSP 5.78 38.4 6.76 5.10 416.5 0.41
Polytetrafluoroethylene 4.27 21.6 6.05 4.39 265.8 0.51
Polydimethylsiloxane 3.23 41.3 5.45 3.79 170.9 0.63
Polyimide 2.77 38.1 5.14 3.48 132.1 0.71
Polystyrene (atactic) 2.05 40.5 4.56 2.90 76.9 0.90
Two suggestions about the reasons of anomalous long average lifetime of
positrons in PTMSP can be made.
First, molecular structure of the repeat unit allows a supposition that a high
concentration of bulky, low-mobile side groups creates a porous structure with the
pore size of about Van-der-Waals volume of –Si≡C3H9 side fragment.
Secondly, the size of pores may be associated with a long relaxation time of
synthesized PTMSP at room temperature. It may be suggested that the formation and
evolution of microcavities of a large size must depend on the motion of large
segments of macromolecules or even structural fragments with a long period of
regrouping.
The lifetime of an ortho-positronium atom regarding the pick-off–annihilation
allows estimation of the size of the microcavity in which it was localized before
annihilation [140]. The calculation results are also shown in Table 8.

60. 52
In line with the model [140], positronium is considered in a spherical pit
surrounded by a layer of electrons, ΔR thick. For wave functions in spherical
coordinates:
( ) ( )

 
−
= ⋅ ⋅
2 sin / in the pit;
0 outside the pit.
( )
0
1/ 2 1
R0 r R
r r (II.35)
The probability of positronium existence outside the limits of density will be:

 

 
R
= − +
2
sin
1
W R , (II.36)
2
0 0
( ) 1
R
R
R
where R = R0 – ΔR.
Suggesting that the rate of ortho-positronium annihilation inside the electron
layer equals 0.5 ns–1, the decomposition rate averaged over spins will be:
λD = 1/τD = 2W(R) (II.37)
with the constant ΔR = 1.66 Å, selected empirically for solids.
Let us consider the results of measurements of PTMSP films porous structure
because of their aging.
Long-term relaxation of PTMSP films was investigated with the help of
measuring positron lifetime spectra. As Table 9 and Figure 9 indicate displaying a
series of characteristics of time spectrum decomposition into three components and
the calculated radius of micropores R, and durability of samples aging, lifetime of the
long-living component decreases with growth of PTMSP exposure time at room
temperature. In practice, the intensity of the long-living component does not depend
on the relaxation time.
Table 9
Long-term relaxation of PTMSP from the data of measurement of the longest component
parameters of positron lifetime spectrum (τn is lifetime of intermediate component)
Aging time, days τD ± 0.03, ns RD ± 0.25, % τn ± 0.080, ns
13 5.78 38.40 0.687
17 5.68 37.53 0.607
24 5.72 38.09 0.678
83 5.40 38.08 0.507
210 5.09 37.91 0.453
Figure 9. Dependence of sizes R of the positron-sensitive microcavity on time of exposure tc at 25°C
for PTMSP

61. 53
The result observed is connected with slow structural relaxation but not the
‘aging’ (if by the ‘aging’ occurrence of the main chain fission is meant), because the
latter process is usually accompanied by changes in intensity ID (results of observing
long-term aging of polyethylene by the method of positron lifetime variation may be
displayed as an example, although ‘aging’ in polymers is a very specific process).
Taking into account the relation between τD and the radius of micropores in
polymers [140], it must be concluded that in long-term relaxation of PTMSP sizes of
pores decrease (see Figure 9) and, probably, the mobility of macromolecular chains
reduces due to free volume decrease.
As follows from the constancy of ID, the concentration of positronium traps is
independent of the exposure time in the studied time interval.
Let us now discuss the results of investigation of positron annihilation in
polyimide.
As the measurements have shown [48], annihilation of positrons in polyimide
is significantly different from the one usually observed in most polymers. The
annihilation spectrum in polymers is usually characterized by the presence of three or
four components with average lifetimes from 100 ps to 4 ns [54, 164, 187]. However,
the different structure of the spectrum is observed for polyimide. It displays a single,
short-term, component with τ0 = 0.388 ns (Figure 10). Time distribution is
approximated well by a single decay line, the tangent of which determines the average
lifetime.
Figure 10. Positron lifetime spectrum τ of the starting polyimide film (here N is the number of
readings in a channel)
The value of lifetime and the spectrum structure allow a supposition that
annihilation in polyimide proceeds from the positron state without forming a
positronium atom as it is typical of metals and semiconductors with high mobility of
electrons and a regular crystalline structure.
In this meaning, polyimide forms an electron structure unique for polymers,
characterized by high values and high homogeneity degree of the density function for
electrons.

62. 54
Figure 11. Lifetimes τ and intensities of components (%) in the spectra of the original sample (I) and
deformed samples of polyimide after recovery lasting for 1 (II) and 24 (III) hrs.
Table 10
Annihilation characteristics of polyimide film
Sample
Recovery
lasting, hr τ0, ps τ1, ps τ2, ps I2, %
Count rate,
k⋅10–9, s
Initial  385±5    
Deformed 1  294±30 440±17 59±5 0.60±0.15
Deformed 24  361±10 531±30 9±2 0.12±0.05
In relation to interaction with positrons, the microstructure of the initial
(undistorted) polyimide film possesses no defects. However, time spectra change after
deformation (Figure 11 and Table 10). Two components instead of a single one are
observed in the deformed sample: with shorter and longer lifetimes. After recovery
(resting) during 24 hours at room temperature, an increase of lifetimes of both
components and reduction of intensity of longer-term ones are observed. The
character of changes taking place allows a supposition that the submolecular structure
of polyimide is rebuilt during deformation; intermolecular bonds break, and
microdefect free volumes enough for positron localization – are formed. In this case,
the value of the long-term component τ2 must reflect changes in the average size, and
intensity I2 – concentration of these defects. Analogous changes in the spectra were
also observed in annealing defects in metals and semiconductors. These changes are
usually analyzed with the help of a positron entrapment model. This model is
qualitatively good in reflecting changes in the time spectra observed in polyimide
deformation. Reduction of the lifetime of the short component, bound to annihilation
in the undistorted part of the polymer, depends on the high rate of capture in the
deformed sample. After partial contraction during recovery, the concentration of
defects decreases and lifetime τ2 approaches the characteristic one of the original
polymer. Therewith, the intensity of the long-term component, I2, formed due to
positron annihilation on defects, decreases, too. Growth of the lifetime τ2 may be
explained by coagulation (consolidation of small defects into larger ones) during
recovery or fast relaxation of small pores and, consequently, by growth of the average
capture radius.
As indicated in estimations, the concentration of microdefects after partial
relaxation decreases more than 7-fold. Therewith, the free volume induced by
deformation decreases by a factor of 4 [48]. The values obtained indicate that two
processes proceed – fusion of microdefects and relaxation of the smallest ones,
though, apparently, the intensity of the latter process is higher.

63. 55
Hence the one-component spectrum is typical of the original polyimide film.
In deformed samples, at least two components are observed in time spectra, which are
bound to the positron annihilation from the free state and the one localized in
micropores, formed at stretching. The lifetime increases and the intensity of the defect
component decreases during relaxation.
The results obtained with the help of the model of positron capture describe
clearly the changes of time distributions observed and allow a supposition that the
structure of the free volume during relaxation changes not only as a result of fast
recombination of the smallest pores, but also because of their consolidation with the
formation of long-term large-size microcavities.
Basing on the analysis performed in ref. [48], the following model of positron
annihilation and relaxation mechanism bound to it are suggested: before deformation
all positrons, captured in small traps with the bond energy slightly higher that the heat
energy, annihilate; after deformation, rather long (compared with the positron
diffusion length) areas occur, in which the concentration of small traps (of the size
~10 nm) decreases significantly, loosened up areas with deep centers of positron
capture are formed simultaneously in which the lifetime of positrons is longer;
relaxation happens in the way that pores formed during deformation recombine and,
moreover, increase when consolidate.
Hence, measuring the lifetime of positrons, the data on changes in structure of
the free volume occurring after polymeric film deformation may be obtained.
However, interpretation of the information obtained requires a detailed study of the
nature of components of a complex time spectrum of annihilation typical for a non-equilibrium
state of polymer. No solution of this problem with the help of one of the
positron methods was obtained [3, 110, 156]. That is why a complex study of positron
annihilation was performed [49] in deformed polyimide with the help of measuring
the lifetime of positrons and angular correlation of annihilation radiation.
Two series of experiments are described in ref. [49]. In the first series, a
polyimide film was stretched by 20%. Then, the film was set free and relaxed freely.
Lifetime spectra for the freely relaxed film were measured every 1.5 hours.
Parameters of angular distribution were determined every hour during the day.
Table 11
Change of annihilation characteristics of polyimide film depending on duration of relaxation
after deforming by 20%
Lifetime Angul Relaxation lasting ar correlation
after deforming, h τavg±1,
ps
τ1±10,
ps
I2±1.5,
% FWMH±
0.05, mrad
Γ1±0.07,
mrad
Θρ±0.07,
mrad
Iρ±1.5,
%
0 365 201 74.3 10.44 10.49 7.14 28.2
1 360 176 73.6 10.77   
5 368 208 77.2 10.60   
24 362 205 73.0 10.48 10.64 7.14 34.7
240 364 200 74.1 10.43 10.72 6.95 32.3
Separated 368 220 76.3    
Note. τavg, τ1 and I2 are characteristics of positron lifetime spectra; FWMH is the full width on the
middle height of the full spectrum; Γ1 is FWMH of the first Gaussian; Θρ and Iρ are characteristics of
the parabolic component of the angular correlation spectrum.
In the second series of experiments, stress relaxation at deformation ε 0 = 20%
was studied. The characteristics of angular distributions were determined for films
with fixed ends. Measurements were performed with the help of a device that
performs deformation of samples directly in the measurement chamber. Stress

64. 56
relaxation curves (dependences of stress σ on time τ) and recovery curves
(dependences of deformation ε on time τ) were taken simultaneously.
The values of the positron lifetime obtained from spectra are shown in Table
11 and Figure 12. Similar to the above-described results of two-component analysis,
changes of annihilation characteristics, which then relaxed gradually to those typical
of the initial polyimide sample, were observed in the structure of the time spectrum,
approximated by three components, after deformation.
Figure 12. Positron lifetime spectrum as a function of relaxation time for freely relaxing polyimide
films (for designation see Table 11).
Three components were separated: the lifetime of the first short-term
components (170–220 ps) significantly depend on relaxation time; as displayed by
investigations [49], the lifetime of the second one (388±10 ps) is independent of or
weakly depends on the sample state. However, significant changes in the intensity of
this component are observed. The characteristics of the third component have not
changed during the experiment.
In the work cited, experiments on measuring the angular correlation were
performed (alongside the measurement of the positron lifetime). Making no detailed
analysis of the results of these measurements, note that in experiments with fixed ends
(under stress relaxation conditions) the free volume significantly increases after
deformation, and its further slow relaxation is displayed well, happened at the
sacrifice of a decrease of micropore concentration.
In most cases, changes of macro- and microparameters of the polyimide film
during stress relaxation and recovery after deformation were indicated by the method
of positron diagnostics. Non-monotonous changes in the characteristics of positron
lifetime spectra and angular distributions of annihilation photons during recovery
were observed. Two ranges of changes in positron-sensitive properties of polyimide,

65. 57
associated with ‘fast’ and ‘slow’ relaxation processes, were separated, and differences
in the type of relaxation of the polymer microporous structure depending upon the
condition of deformation and ‘rest’ were observed. The effects observed are stipulated
by formation of areas of the local ‘defrosting’ of molecular mobility.
All these experimental facts indicate that the microporous structure of the
polymer is rearranged during stress relaxation; this is expressed by the redistribution
of the sizes of micropores and their merging. Hence the method of positron
annihilation allows not only estimation of the microporous structure of polymers, but
also following its change under mechanical loading.