1. COMPUTATIONAL MATERIALS SCIENCEOFPOLYMERS

2. COMPUTATIONALMATERIALS SCIENCEOFPOLYMERS 3. CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING 4. Published byCambridge International Science Publishing7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UKhttp://www.cisp-publishing.comFirst published January 2003 A A Askadskii Cambridge International Science PublishingConditions of saleAll rights reserved. No part of this publication may be reproduced or transmittedin any form or by any means, electronic or mechanical, including photocopy,recording, or any information storage and retrieval system, without permissionin writing from the publisherBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 1 898326 6 22Production Irina StupakPrinted by Antony Rowe Ltd, Chippenham, Wiltshire, Great Britain 5. About the AuthorAndrey Aleksandrovich Askadskii is a Professor of Chemistry at the In-stituteof 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 aphysical approach to the quantitative evaluation of the physical propertiesof linear and network polymers on the basis of their chemical structure;development of computer programs for calculating the properties of poly-mersand low-molecular liquids and also computer synthesis of polymers withthe required properties; experimental examination of the structure of propertiesof heat-resistant aromatic polymers of different grades; development of newmethods of experimental and theoretical analysis of the relaxation proper-tiesof polymer materials; production of new types of polymers; productionand examination of electrically conducting polymer materials on the basisof heat-resistant polymers and organo-element compounds; development ofgradient polymer materials with a variable modulus of elasticity within thelimits of the same material and retaining elastic (not viscoelastic) proper-tiesat any point of the gradient material.Prof Askadskii is the author of more than 400 scientific studies and20 books, six of which have been published abroad. 6. ContentsPrefaceIntroduction 3Chapter I. Brief information on types of polymes and their chemical structure 9Chapter II. Packing of macromolecules and polymers density 16II.1. Increments method and basic physical assumption 16Chapter III. Temperature coefficient of volumetric expansion 58Chapter IV. Glass transition temperature of polymers 67IV.I. Thermomechanical and other methods of evaluation of the glasstransition temperature of polymers 67IV.2. Mechanism of glass transition 88IV.3. Calculation of the glass transition temperature of linear polymers 108IV.4. Influence of plasticization on the glass transition temperature of polymers 322IV.5. Calculation of the glass transition 343Chapter V. Temperature of transition into the viscous flow state for amorphouspolymers 385V.1. Estimation of temperature of transition into the viscous flow state ofpolymers 385V.2. Dependence of Newtonian viscosity on molecular mass of polymer in awide range of its change 388Chapter VI. Melting point of polymers 398Chapter VII. Temperature of onset of intense thermal degradation of polymers 408Chapter VIII. Optical and opto-mechanical properties of polymers 418VIII.1. Refractive index 418VIII. 2. Stress-optical coefficient 426Chapter IX. Dielectric constant of polymers and organic solvents 445Chapter X. Equilibrium rubber modulus for polymer networks 456X.1. Calculation of the equilibrium modulus 456X.2. Heteromodular and gradient-modulus polymers 466Chapter XI. Description of relaxation processes in polymers 475XI.1. Stress relaxation 475XI. 2. Sorption and swelling processes 497Chapter XII. Solubility of polymers 504XII.1. Specific cohesive energy of organic liquids and polymers. Hildebrandsolubility parameter 504XII.2. Solubility criterion 509XII.3. Influence of molecular mass and degree of macromolecule orientationon solubility 520Chapter XIII. Surface properties of organic liquids and polymers 527XIII.1. Surface tension of organic liquids 528XIII.2. Surface tension of polymers 536Chapter XIV. Miscibility of polymers 547Chapter XV. Influence of the end groups on the properties of polymers 555Chapter XVI. Thermophysical properties of polymers 562XVI.1. Heat capacity 562XVI.2. Thermal diffusivity and heat conductivity 564 7. Chapter XVII. Molecular design and computer synthesis of polymers withpredermined properties 567Appendix 1. Examples of solution of direct problems of polymers synthesis 589Appendix 2. Examples of solving the reverse problem of polymer synthesis 602Appendix 3. The example of solving the complex problem analysis of thechemical structure of phenol formaldehyde resin 607Appendix 4. Application of the approach to multicomponent copolymers 621Appendix 5. Influence of strong intermolecular interaction occurring betweentwo dissimilar polymers on their miscibility 625Appendix 6. On formation of super-molecular structure in amorphous polymers 6451. Scheme of formation of the super-molecular structure 6452. Calculation method of evaluation of dimensions of elements of super-molecularstructure of polymers3. Phase state of polymers as a result of formation of the super-molecularstructure by one-cavity bond hyperboloids 653References 669Index 689 8. PREFACEPublished in the journal Chemistry and Life, No. 2, 1981 was the article byme, titled by the editor as Atom plus atom plus thousand atoms. This articlediscussed the possibility of calculating some physical properties of polymers on thebasis of the chemical structure of the repeat unit (it was then possible to calculateproperties of linear polymers only). In conclusion of the article, titled A littlefantasy, it was written: Therefore, many properties of polymer can be predicted, ifnothing except the structural formula of the appropriate monomer is known. It is agreat progress: nowadays already, such calculations allow chemists to be drawnaway from heavy duty to synthesize hopeless monomers. Formerly, under empiricalselection of materials, many of such monomers had to be synthesized. Nevertheless,calculations are to be made manually still. Moreover, when they are translated intothe machinery language, chalk and blackboard traditional for any chemical disputecan be substituted by an electronic pencil. A chemist will draw a formula of thesuggested monomer on the screen by it, and the computer will answer immediately ifit is useful or not to synthesize it. Another opposite task seems to be much moreabsorbing. If the computer is able to calculate properties by structural formulae,apparently, it may be taught, vice versa, to calculate the formula of a suitablemonomer (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 problematicpart 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 realizedat any time in neat future. However, events were developing very fast, especially afterappearance of high-power personal computers. Before discussing stages of this greatwork, methods of the quantitative estimation of polymer physical properties must bepresented in brief performed on the basis of their chemical structure. At the presenttime, there are three main approaches to this estimation. One of them, developed byVan Krevelen [214], is based on the idea of so-called group contributions, accordingto which the simplest empirical expressions of the additive type are written down, thepresent group, existing in different polymeric units, making one and the samecontribution to the calculated characteristic (for example, glass transition temperature,melting, etc.). As the author states, this is just an empirical approach, which allowsthe physical properties of many of linear polymers to be calculated with highaccuracy.Another approach, being developed for a long time by the author of thispreface 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 ideasof physics of solids, and calibration of the method is performed with the help ofphysical characteristics of polymeric standards, the properties of which are studiedwell. Consequently, parameters of equations possess a definite physical sense (energyof dispersion interaction, energy of strong intermolecular interaction, includinghydrogen bonds, Van-der-Walls volume, etc.). Application of this approach makespossible estimation with enough accuracy of many physical characteristics ofpolymers (about 60 up to now). Therefore, the number of polymers of variousstructures is unlimited.The third approach developed by J. Bicerano [133] has appeared recently. It isbased on the so-called coherence indexes, reduced in practice to a search for various 9. 2correlations of physical properties with many rules of obtaining coefficients ofcorrelation dependencies.Discussed in the present monograph are principles of the approach, developedby A.A. Askadskii and Yu.I. Matveev, special attention being paid particularly tocomputer realization of the current calculation method for physical properties ofpolymers. The first computer software has been composed by E.G. Galpern, I.V.Stankevich and A.L. Chistyakov - investigators of quantum chemistry laboratory ofA.N. Nesmeyanov Institute of Organo-Element Compounds, RAS. Initially, computersynthesis of polymers by this software was performed from so-called largeprocurements representing residues of monomers, involved into the synthesisreaction. In the second variant, computer synthesis was performed from smallestprocurements, from which the repeat unit of the polymer was constructed. Thisbroadens significantly capabilities of the software for solving both direct (calculationof the polymer properties from its chemical structure) and reverse task (computersynthesis of polymers with preliminarily programmed /assigned/ properties, theranges of which were set in the computer), because the amount of synthesizedolymers has increased sharply. Then principally new software was composed by A.F.Klinskikh, in which chemical structure of the repeat unit was constructedfromatoms. Thus, the user needs just to depict chemical structure of the polymer on thecomputer screen as chemist does it on the paper, and computer lays out all physicalproperties of polymers, involved in the software (all about 60). This software alsoprovides for calculation of a sequence of properties of low-molecular weight organiccompounds, as well as, which is very important, properties of polymeric networks.Solution of the reverse task is also provided. Of special importance is the possibilityto calculate properties of copolymers and their mixtures, to predict solubility andcompatibility of polymers, to construct dependencies of properties on temperature,molecular mass, crystallinity degree, microtacticity (of special importance aredependences of glass transition temperature and temperature of transition into theviscous flow state on molecular mass).It stands to reason that not all the problems are solved. Accuracy of thecalculation and various predictions of polymers behavior at dissolution and mixingwith each other must be increased, calculation schemes to estimate new properties ofpolymers must be developed, and their computer realization must be performed, etc.It is obvious that the present monograph possesses some drawbacks. Theauthors will be thankful for any notes on the point of the book. 10. 3INTRODUCTIONAs mentioned above, the approach to estimation of the physical properties ofpolymers, discussed in the monograph, is semi-empirical. When estimating thethermal characteristics of polymers, such as glass transition temperature, meltingpoint, it is supposed that the repeat unit is composed of a set of anharmonic oscillatorsrepresenting atomic pairs, linked by intermolecular physical bonds. The criticaltemperature of this set of anharmonic oscillators is that determines the above-mentionedtwo transition temperatures. The thermal expansion coefficient is alsoclosely related to these characteristics. In the case of a characteristic as thetemperature of the onset of intensive thermal degradation, the polymeric unit isconsidered as a set of anharmonic oscillators representing atomic pairs, linked bychemical bonds. The critical temperature of such a set of oscillators characterizes thetemperature 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 thermaldegradation will be different, i.e. kinetic effects play a significant role in this case). Atfirst glance, it may seem strange that thermal degradation is considered here not as akinetic, which is conventional, but as an original phase transition, at which, however,the initial substance cannot be obtained from the products of thermal decompositionby simple cooling down.Equations for calculating other physical characteristics are based on physicalapproaches, discussed in detail below, and we will not consider them in this part.Common for all these equations is summarizing the sequence of atomicconstants, which characterize contributions to the energy of intermolecularinteraction, chemical bonds energy, Van-der-Waals volume, etc. Strictly speaking, thepresent approach cannot be named additive in the common sense of the word, becausethe calculated properties are not additive in relation to atoms and groups, whichcompose 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.). Theapproach being described allows calculation of their properties of the unlimitednumber of polymers and conduction of the computer synthesis of polymers withassigned properties with the help of software created and described in the monographthat 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 essenceof calibration on an example of the equation calculating glass transition temperatureof a linear polymer, Tg:i +VT , =jjii iig a V b 11. 4where ai are atomic constants; bj are constants bound to the energy of strongintermolecular interaction (dipole-dipole, hydrogen bonds), occurred betweenpolymeric chains at the sacrifice of polar groups existing in them; iVi is the Van-der-Waals volume of the polymer repeat unit, summarized from Van-der-Waalsvolumes of atoms participating in the composition of the unit.Reduce the equation to the following view: + = iijj gii i VTa V b1.Basing on this equation, the excessive system of linear equations is composedas follows: ......................................................................................................................... + + + + + + + = + + + + + + + = + + + + + + + = .1... ...;1... ...;1... ...1 ,1 ,1 2 ,2 , 1 2,1 21 2,1 2 2,2 2, 2 1 2 2 2,1 11 1,1 2 1,2 1, 1 1 1 2 1ii mgm m n m n m m m kiign n kiign n kVTa V a V a V b b bVTa V a V a V b b bVTa V a V a V b b bThen the matrix of coefficients at the unknowns of this excessive system ofequations: = and the column matrix of free terms of these equations = are composed.Further on, a transposed matrix is composed and multiplied by the initialone A, as well as by the column matrix B. All this results in obtaining a 12. 5canonic system of equations. This canonic system is solved, for example, by theGauss method. The whole procedure of calibration is performed by standard software.Without considering features of such regressive analysis, let us note only thatpolymers, selected for calibrating the method, must possess experimental values ofanalyzed physical characteristics in broadest range, and the chemical structure ofpolymeric standards must be sufficiently different. Usually, an excessive systemcomposed of 300 equations is to be solved, which corresponds to 3040 polymers.Next, the properties of other polymers are calculated from the coefficients obtained.In this case, the energy of weak dispersion interaction, strong dipoledipoleinteractions and hydrogen bonds, their relative part and many other physicalparameters of the system are determined.We are coming now to a brief description of the contents of individualchapters of the monographs.The first chapter discusses the data of modern classification of polymers andtheir chemical structure. Of the outstanding importance, induced by the features of thechemical structure and the application field, are interpolymers, dendric and staircase(ladder) polymers.The second chapter discusses the approach to computerized materialstechnology of polymers on the atomicolecular level, based on the method ofincrements. The increments of various atoms and main groups of them are calculated.The main physical ideas about structure of macromolecules of polymers andparameters determining it are displayed. The method for calculating such an importantcharacteristic of the polymer structure, as the coefficient of molecular packing, isgiven. A connection between the free volume of the polymer, the coefficient ofmolecular packing and parameters of its porous structures is established. Forexperimental determination of characteristics of the microporous structure ofpolymers, the method of positron annihilation, the application of which indicatedstructural changes in polymers in their relaxation, is used.With consideration of weak dispersion and strong (dipoledipole andhydrogen bonds), the third chapter gives formulae for calculating the thermalcoefficient of the volume expansion in dependence on the chemical structure of thepolymer. In this case, the type of atoms in the polymeric chain and type of theintermolecular interaction are estimated by a limited number of correspondingincrements, numerical values of which are determined.The fourth chapter describes in detail the thermomechanical method ofdetermination of the glass transition temperature and fluidity of polymers, features ofinterpreting thermomechanical curves for amorphous and crystalline polymers areanalyzed, the calculation method of determination of the mechanical segment fromthe chemical structure of the polymer is displayed. Two main concepts of themechanism of vitrification processes of polymers, relaxation and intermolecular, arediscussed. The atomistic approach which is more universal than the widespread so-calledgroup contributions method to calculation of polymer properties from theirchemical structure, is considered. This approach was used for deriving an analyticalexpression to calculate the glass transition temperature of linear and networkpolymers from their chemical structure. The influence of types of linear polymersbranching and the number of units between cross-link points, type and structure ofthese points, existence and type of the network defects for network polymers on theglass transition temperature of the polymers is analyzed.Given in the fifth chapter is the method for calculating the fluidity temperatureof amorphous polymers and the temperature range of the rubbery state of polymers 13. 6from their chemical structure, and conditions of appearance of the rubbery state in apolymer depending on its molecular mass, as well, which is important for processingof polymers.The sixth chapter describes two approaches to calculating the melting point ofpolymers from the chemical structure of the repeat unit. The first approach is based onthe experimental fact of closeness in parts of the empty volume in melting of acrystalline polymer and in transition of an amorphous polymer of the same structurefrom the glassy-like into the high-elastic state. The second approach is based on theconsideration of the repeat unit of a polymer as a selection of anharmonic oscillators.Discussed in the seventh chapter is the most important characteristic ofthermal resistance of polymers initial temperature of their intensive thermaldegradation. The formula to calculate this temperature based on the chemical structureof the polymer was deduced, and necessity to take into account the resulting productsof thermal degradation which starts with the decay of end groups in polymermacromolecules, are indicated.In the eighth chapter, LorenzLorentz equations are used for derivingequations for calculation of the refractive index of polymers and copolymers fromtheir chemical structure. To obtain the stress-optical coefficient, empirical and semi-empiricalapproaches are established, in which the contribution of each atom and thetype of intermolecular interaction are estimated by an appropriate increment. Usingthe dependencies obtained for the stress-optical coefficient on the chemical structureof the repeat unit of the polymer, the contribution of various atoms and polar groupsto the value of this coefficient is estimated, and a polymer with the properties uniquefor the method of dynamic photo-elasticity is proposed.The ninth chapter displays a scheme for calculating the dielectric constant ofpolymers and organic liquids with respect to their chemical structure which isimportant for both synthesis of polymers with the required dielectric constant andprognosis of polymer solubility in organic liquids. Taking into account not only thecontribution 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 resultedin the previously unobtainable agreement in the experimental and calculated values ofthe dielectric constant for a broad spectrum of organic polymers and liquids.Based on the notion of network polymers as an elastic and rotationalisomericsubsystem and taking into account its structure as linear fragments and cross-linkedpoints, the tenth chapter indicates the deduction of formulae for calculating theequilibrium rubbery modulus and molecular mass of a linear fragment betweenneighboring cross-linked points. Further analysis of the resultant dependenciesallowed the formulation of conditions for obtaining a polymer with unique (unusual)properties different modulus and gradient polymers characterized by large changesof the equilibrium rubbery modulus within the same article. Existence of these uniqueproperties is confirmed experimentally for synthesized network of polyisocyanurates.The eleventh chapter describes the derivation of analytical expressions forrelaxation memory functions, necessary for determining the stress relaxation andcreep of the polymers. In this case, the production of entropy of a relaxing system isrepresented by transition of relaxants (kinetic units of a polymer of different nature)into non-relaxants by means of their interaction or diffusion, the mechanism ofinteraction of relaxants in stress relaxation being found predominant. The apparatuscreated for description of relaxation events in polymers is applied in description ofsorption and swelling processes. Thus, contrary to stress relaxation, the mechanism ofrelaxants diffusion is predominant in sorption. 14. 7The twelfth chapter is devoted to the problem of increasing the accuracy ofprediction of polymer solubility in organic liquids. It is shown that the predictiveability of the solubility criterion, calculated with respect to the chemical structure ofthe polymer and the solvent, sharply increases with consideration for the type ofsupermolecular structure of the polymer and the degree of its polymerization.Based on the chemical structure of the matter, the thirteenth chapter gives acalculation method for the most important property of organic liquids and polymers,i.e. surface tension. Contrary to the additive scheme for summation of parachorswhich characterizes the contribution of separate atoms to the surface tension, theapproach developed allows estimation of the contribution of polar groups and specificintermolecular interaction to the surface tension value and connection of it with thesolubility parameter and density of cohesion energy in substances.Invoking the idea of solubility of a single homopolymer in another one, thefourteenth chapter suggests a criterion for estimating the compatibility of polymersbasing on the data of the chemical structure of separate components. The analysis ofapplication of the criterion for compatible, partially compatible or incompatiblepolymers indicates its high predictive ability.On the example of the calculation of the Van-der-Waals volume, molarrefraction, heat capacity and other properties of a number of polymers, chapter fifteendisplays the role of the chemical structure of macromolecule end groups andimportance of their calculation in the study of regularities of changes in the polymerproperties on their molecular mass.The sixteenth chapter indicates a method for calculating the molar heatcapacity with respect to the chemical structure of polymers. The method is based on asupposition that the contribution of each atom to heat capacity is proportional to itsVan-der-Waals volume. It is noted that the heat capacity, thermal diffusivity and heatconductivity of polymers depend not only on their chemical structure, but also on thephysical and phase states of the polymeric body.The seventeenth chapter describes methodological ways of solving the directproblem of computerized determination of the physical characteristics of polymersand low-molecular liquids with respect to their chemical structure and the reverse one computer synthesis of polymers with the given set of properties. These problemsare solved by the methods of fragments and separate atoms. The correspondingsoftware which allows calculation of more than 50 chemical properties of linear andnetwork polymers and copolymers, and a number of the most important properties oflow molecular weight liquids, as well, is developed. Discussed is the method ofdepicting diagrams of polymer properties compatibility, application of which maysignificantly simplify solution of the direct and, especially, reverse problems ofcomputational materials sciences.Appendices demonstrate abilities of the approach, described in themonograph, to determine the properties of some natural polymers (the example ofsolving the direct problem of polymers synthesis) with respect to their chemicalstructure (Appendix 1); to search for chemical structures of polyetherketones (theexample of solving the reverse problem of polymer synthesis), the properties of whichmust lie in a given range (Appendix 2); to solve a mixed problem of polymerssynthesis on the example of analyzing the chemical structure of phenoloformaldehyderesin, when the direct problem estimation of the properties of the ideal structuresof such resin with respect to their chemical formulae and the reverse one searching for a combination of structures with which the chemical formula ofphenoloformaldehyde resin obtained provides experimentally observed values of its 15. 8properties are solved consecutively (Appendix 3); to analyze the structure andproperties of copolymers, composed of from three to five comonomers (Appendix 4);and the influence of a strong intermolecular interaction appearing between twoheterogeneous polymers on their compatibility is analyzed (Appendices 5 and 6). 16. Chapter I. Brief information on types of polymers and theirchemical structureThe very large number of existing polymers may be subdivided into threemain classes forming the basis of the presently accepted classification. The first classcontains a large group of carbochain polymers whose macromolecules have a skeletoncomposed of carbon atoms. Typical representatively of the polymers of this class arepolyethylene, polypropylene, polyisobutylene, poly(methyl methacrylate), poly(vinylalcohol) and many other. A fragment of a macromolecule of the first of them is of thefollowing structure[CH2CH2]nThe second class is represented by a similar large group of heterochainpolymers, the main chain of macromolecules of which contains heteroatoms, inaddition to carbon atoms (for example, oxygen, nitrogen, sulfur, etc.). Numerouspolyethers and polyesters, polyamides, polyurethanes, natural proteins, etc., as well asa large group of elemento-organic polymers relate to this class of polymers. Thechemical structure of some representatives of this class of polymers is the following:[CH2CH2O]n Poly(ethylene oxide)(polyether);Poly(ethylene terephthalate)(polyester);Polyamide;Polydimethylsiloxane(elemento-organicpolymer);Polyphosphonitrile chloride(inorganic polymer).CH3C lThe third class of polymers is composed of high-molecular compounds with aconjugated system of bonds. It includes various polyacetylenes, polyphenylenes,polyoxadiazoles and many other compounds. The examples of these polymers are:[CH=CH]n PolyacetylenePolyphenylenePolyoxadiazole(CH2)2 O COC OO nNH (C H2)6 N H C (C H2)4OCO nS i OCH3 nN PC l nnN NCCO n 17. 10An interesting group of chelate polymers possessing various elements in theircomposition, 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, forexample:H3C CH3The most widely used type of material in the large group of polymericmaterials are still the materials based on the representatives of the first class ofpolymers which are carbochain high-molecular compounds. The most valuablematerials could be produced from carbochain polymers, for example, syntheticrubbers, plastics, fibers, films, etc. Historically, these polymers have beenimplemented in practice first (production of phenoloformaldehyde resins, syntheticrubber, organic glass, etc.). Many of carbochain polymers became subsequently theclassic objects for investigation and creation of a theory of the mechanical behaviourof polymeric substances (for example, polyisobutylene, poly(methyl methacrylate),poly-propylene, phenoloformaldehyde resin, etc.).Subsequently, materials based on heterochain polymers polyamide andpolyester fibers, films, varnishes, coatings and other materials and articles becamewidespread. This has given impetus to investigating the properties and formation ofnotions, in particular, of anisotropic substances possessing extremely differentproperties in different directions. A special place in the sequence of these polymers isdevoted to high-molecular elemento-organic compounds.Finally, the representatives of the third class polymers with conjugatedsystem of bonds were used for the preparation of conducting materials.Considering in general terms the chemical structure of polymers of differentclasses, we have discussed the structural formula of the repeating unit in themacromolecule. However, the existence of many such units in the macromoleculeimmediately complicates the situation. Let us begin, for example, with an assumptionthat each unit in the elementary act of macromolecule growth may be differentlyattached 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 thepropagating macromolecule are possible for asymmetric monomers of thetype which possess R substituents on one of carbon atoms. Here, variants of head-to-head... ...and head-to-tailH3COPO OCH3ZnOPOOC H2 C HRCH2 CH CH CH2 CH2 CH CH CH2R R R R... ...CH2 CH CH2 CH CH2 CHR R R 18. 11additions are possible.Alternation of the types of addition is possible, i.e. units may be differentlyattached to each other in a single macromolecule. Existence of a great number of unitsin the polymeric chain and possibility of only several variants of their attachmentgives a huge number of isomers in relation to the whole macromolecule. To put itdifferently, a polymer may contain (and indeed contains) not only the macromoleculesof 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 inthe 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 thatwhen the number of units becomes tens or hundreds of thousands, the number ofpossible isomers becomes astronomically high [80].Let us return to monosubstituted unsaturated hydrocarbons. When a polymericchain is formed during polymerization, the substituents R may dispose differently inrelation to the plane of single bonds. In one of possible cases, these substituents aredisposed irregularly in relation to the plane of single bonds; such polymers are calledirregular or atactic:HCHCHCHCHCRCHCHCHCRCHCHCHCHCHCRCHRHRHRHRHRHRHRHRIn other cases, synthesis may be performed in such a manner that substituentswould be disposed either by the same side of the plane of the main bondsHCHCHCHCHCHCHCHCHCHCHCHCHCHCHCHCHRHRHRHRHRHRHRHRor by both sides, but with regular alternation of the substituents direction:HCHCRCHCHCHCRCHCHCHCRCHCHCHCRCHCRHHHRHHHRHHHRHHHThe polymers composed of the units with regular alternation of substituentswere called stereoregular. If the substituents are disposed on one side of the plane ofthe main bonds, stereoregular polymers are called isotactic. If they are disposed onboth sides of the plane, the polymers are called syndiotactic.The situation is more complicated with polymers synthesized fromdisubstituted monomers. Already in the monomer, substituents may dispose on thesame (cis-isomer) or on both sides (trans-isomer) of the plane of the double bonds:HC CRHR'HC CRR'H 19. 12Synthesis of macromolecules from cis-isomers leads to the formation oferythro-diisotactic polymersRCR'CRCR'CRCR'CRCR'CRCR'CHHHHHHHHHHand trans-isomers give treo-diisotactic polymersRCR'CRCR'CRCR'CHHHHHHHCR'CHCR'CHCR'CHCR'CHCR'CHCR'CHCR'CHCR'CRHRHRHRHRHRHRHRHNeedless to say, other more complex modifications are also possible, whichimmediately cause a change of properties of polymeric materials.The materials composed from stereoregular polymers are often easilycrystallized so that gives their physical structure and properties can be regulated.Here we meet for the first time a modification of the properties of polymericmaterials, which is caused by practically any change in the chemical structure ofmacromolecules and the physical structure of the polymeric substance. Physicalmodification is often indicated by a change of the chemical structure, and sometimesis completely defined by it.One of the main methods of modification is the synthesis of copolymers, whennot a single but several monomers participate in the reaction. That is why themacromolecule becomes composed from different units. These units may alternatecontinuously:ABABABABAB the alternating copolymer;but, most often, they are arranged irregularly:AABABBAAAB the random copolymer.The units may also be linked in separate blocks which are the linked to each other:AAAAABBBBB the block-copolymer.Obviously, each block may contain a different number of units. This isimmediately shown up in the properties of the future polymeric substance. In thiscase, the copolymerization process becomes regulated. Running ahead, recall thatmechanical mixtures of polymers and copolymers of the same molar composition mayoften possess rather different properties, but sometimes they are practically identical.The considered schemes of addition of units during macromolecule growthindicate the only case of copolymerization of two types of monomers. Even if manycombinations are realized in these simplest cases, their number grows immeasurablywhen three or more monomers (or types of units) are usedAll the above-discussed chains of polymers represent linear formations.However, branched macromolecular chains could be easily synthesized. For thispurpose, it is even unnecessary to introduce multifunctional compounds into the chaincomposition. Branching also occurs in polymerization of unsaturated hydrocarbonswith no functional groups. If no special steps are taken, the products ofpolymerization of ethylene, propylene, isobutylene and other similar compounds willalways contain some amount of chains branched from the main chain. Concerning theproducts of polycondensation (see the above discussion on polyesters and 20. 13polyamides), introduction of a three-functional compound into the main chain alwaysleads to the formation of branched polymers:... ...A A A A' A A A A A A AAAAA... A A A A' A A A A A A AIt is self-evident that the polymeric body based on the branchedmacromolecules will differ in the structure and properties from a substance composedof linear macromolecules. However, we must not hurry in concluding about the typeof physical structuring of the branched polymers. At first glance, it seems that thepresence of large branches will make obstacles to denser packing of the chains, aswell 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. Itdepends upon the chemical structure of the main chain and its branches, whichdetermines the volume of units, interaction forces between them and neighbourchains, etc.Recently, special attention has been paid to the structure and properties of so-calleddendric polymers, the macromolecule of which is schematically depicted inFigure 1 [98, 212]. Below, we will discuss in more detail the influence of the types ofbranchings on the properties of the resulting polymers.Figure 1. Schematic representation of dendric polymersBranchings may be composed in different ways. They may contain the sameunits, which compose the main chain. However, grafted polymers have becomewidely used; they are formed in grafting of previously obtained chains of a definitestructure to the main chain with an extremely different structure:... ...BBBB... 21. 14Sometimes, such grafting is performed many times.We can now easily pass from the branched to three-dimensional cross-linkedpolymers. This requires just an increase of the concentration of multifunctionalcompounds in the polymer chain. The chains could also be cross-linked by specialcuring agents, i.e. by compounds containing active groups, capable of reaction withfunctional groups of the main chain or the end groups. The classic example is thecuring of epoxy resins:CH3O CCH3O CH2CH CH2OCH3... O CCH3O CH2CH CH2ONH2RNH2+CH3O CCH3CH3O CCH3O CH2 CH CH2O CH2OHCH CH2OHNHRNH......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 functionalend groups, attached to a chain of low molecular mass. As a result, a dense networkwith short chains between cross-link points is formed.2) Chemical linking of high-molecular compounds by the end groups with thehelp of a low-molecular cross-linking agent. Consequently, a network with long linearfragments between the cross-linked points is formed.3) Formation of a network by copolymerization of two- and polyfunctionalmonomers. The example of such a network is the styrenedivinylbenzene system:... ...CH2 CH CH2 CH CH2... ...CH2 CH CH2 CH CH24) Vulcanization of polymeric chains by involving, in the reaction, functionalgroups disposed along the main chain. The reaction is performed either by theapplication of a low-molecular cross-linking agent or by means of radiation and othertypes of influence on the functional groups. 22. 15Other possible (and already realized in practice) ways of producing thenetwork 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 eachpolymers (i.e. in the repeating units, but not at the ends).6) Synthesis of polymeric networks with the help of the polycyclotrimerizationreaction. For this purpose, oligomers with end groups capable of forming cyclesduring the reaction [56, 79, 101, 152] are formed. The example of such a reaction isthe trimerization of two-functional oligomers (or monomers) containing cyanate endgroups. 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 thefunctional groups disposed in the repeating units of the each macromolecule. Aschematic 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 interactionbetween polystyrene and polytrichlorobutadiene:... CH2 CH ... + ... CH2 CH CCl CCl2...AlCl3... ...CH2 CH CCl CClThe formation of interpolymers gives new possibilities of modifying the structure andproperties of polymers.Another type of two-cord system is the ladder polymer, the example ofwhich is polyphenylsylsesquioxane [113]:... ...CH2 CH... ...Si O SiO... Si...OOO Si O 23. Chapter II. Packing of macromolecules and polymerdensityII.1. Increments method and basic physical assumptionsAfter discussing briefly the chemical structure of polymers, let us pass to thevolumetric representation of macromolecules, which is necessary for understandingthe features of structure formation in polymers. These considerations will be based onthe assumptions developed by A.I. Kitaigorodsky in organic crystal chemistry [75].According to these assumptions, every atom is presented as a sphere withintermolecular radius R. Values of these radii are determined from the data of X-raystructural analysis of ideal crystals of organic substances. In this case, it is assumedthat valency-unbonded atoms, entering into an intermolecular (but not chemical)interaction, contact each other along the borders of the spheres. This is schematicallyrepresented in Figure 3. Then, if two identical atoms are in contact, the intermolecularradius will be determined from the relation:R = l/2, (II.1)where l is the distance between mass centers of two identical valency-unbondedatoms, which, however, are capable of intermolecular physical interaction.Figure 3. Schematic representation of intermolecular (Van-der-Waals) interaction of two atomsAccording to the same assumptions, chemical interaction between two atomsalways causes their compression, because the length of the chemical bond di is alwaysshorter than the sum of two intermolecular radii:diR1 + R2. (II.2)This is clear from Figure 4, which schematically depicts two chemicallybonded atoms. If the intermolecular radii Ri for all atoms participating in the repeatunit, 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 ofthis unit (or greater fragment of the macromolecule), in which the volume of eachatom is bordered by a sphere with intermolecular radius Ri, could be composed. 24. 17Figure 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 composethe majority of polymers.Table 1Van-der-Waals radii R of different atomsAtom R, nm Atom R, nmC 0.180 Si 0.210H 0.117 Sn 0.210O 0.136 As 0.200N 0.157 S 0.180F 0.150 P 0.190Cl 0.178 Pb 0.220Br 0.195 B 0.165I 0.221 Ti 0.200Table 2 displays bond lengths of various combinations of atoms, alsocharacteristic for most of existing polymers. If these values are known, the volume ofthe repeat unit of any polymer may be calculated. To conduct this, the own volume ofeach atom participating in the repeat unit should be preliminarily determined. It iscalculated from the formula = 3 3 2 (3 ),1Vi R hi R hi (II.3)34iwhere Vi 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, cutoff from the present atom by a neighbor one, chemically bonded to it. The value hi iscalculated from relation 25. 18+ h = R ,(II.4)2 2 2R d R2i iii dwhere Ri is the intermolecular radius of a neighbor valency-bonded atom; di is thelength of the chemical bond (see Figure 4).Table 2Chemical bond length di for same pairs of atomsBond* di, nm Bond* di, nm Bond* di, nmCC 0.154 CF 0.134 OF 0.161CC 0.148 CF 0.131 O=N 0.120C=C 0.140 CCl 0.177 O=S 0.144C=C 0.134 CCl 0.164 O=P 0.145C=C 0.119 CBr 0.194 NP 0.165CH 0.108 CBr 0.185 NP 0.163CO 0.150 CI 0.221 NP 0.158CO 0.137 CI 0.205 SS 0.210CN 0.140 CB 0.173 SAs 0.221CN 0.137 CSn 0.215 S=As 0.208C=N 0.131 CAs 0.196 SiSi 0.232C=N 0.127 CPb 0.220 PF 0.1550.134 HO 0.108 PCl 0.201C N 0.116 HS 0.133 PS 0.181CS 0.176 HN 0.108 BB 0.177CS 0.156 HB 0.108 SnCl 0.235CSi 0.188 OS 0.176 AsCl 0.216CSi 0.168 OSi 0.164 AsAs 0.242* If the same pair of atoms is linked by a single bond, the longer bond corresponds to attachment of thisatom to an aliphatic carbon atom; the shorter bond corresponds to attachment of the same atom to anaromatic carbon atom.Increments of the volumes of various atoms and atomic groups are shown inTable 3. Obviously, the volume of the given atom depends on its surrounding, i.e. onthe 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 isthe compression of the given atom.When increments of the volumes, Vi, of all the atoms entering into the repeatunit of polymers are determined, the relative part of the occupied volume in the totalvolume of the polymeric substance may be calculated. In the case of polymer,calculations would be appropriate to conduct basing on molar volumes of the repeatunit, because polymers are always polydispersional (i.e. they contain macromoleculesof various length), and also because at long lengths of the macromolecule theinfluence of end groups may be neglected. Then, the own molar volume will equalown = A ,V N Vi and the total molar volume Vtotal = M/, is density of theipolymeric substance; M is the molecular mass of the repeat unit; NA is the Avogadronumber. Numerous experiments and calculations show that in all cases the conditionVownVtotal is fulfilled. Hence, in the first approximation, the volume of the polymericsubstance 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 differenceof Vtotal and Vown. Of interest is determination of the partC N 26. 19Table 3Van-der-Waals volumes of atoms 27. 20 28. 21 29. 22 30. 23 31. 24 32. 25 33. 26 34. 27 35. 28 36. 29of the occupied volume or, according to the terminology used in organic crystalchemistry, the molecular packing coefficient k:N VVownk i= =/AtotalMVi. (II.5)Clearly, the value of k for the same polymer will depend on temperature andthe physical state of the polymer, because the value of depends on them.Calculations performed for many amorphous bulky polymers existing in the glassystate have indicated that the first approximation of k gives its value constant andpractically independent of the chemical structure of the polymer [41]. Passing on topolymers with a complicated chemical structure from those with a simple one causesno 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 coefficientsof the molecular packing of some glassy polymers. It also shows that firstapproximations of the values of k for each of them are equal, indeed. To demonstratethis experimental fact more clearly, Figure 6 displays the dependence of density ofvarious polymers on the relation M NA Vi . In Figure 6 it is clearly seen that alli 37. 30Table 4Values of the coefficients of molecular packing for some glassy and semi-crystalline polymersStructural formula of the repeat unit of polymer Van-der-Waalsvolume of theunit, cm3/molPackingcoefficient k41.6 0.67832.6 0.68258.5 0.68469.1 0.680144.3 0.679234.7 0.679263.1 0.680277.5 0.68856.4 0.685C H 3HCCCH2NCH3CCCH2O CH3OCH3CCCH2O C2H5OC OOCH3CCH3OO OO OCH2 CHCHCH2COCOCOCO(CH2)8 COCCOOCNHNHCCHNCC H 3C H 2COO 38. 31CH2CH=CHCH2 59.1 0.65474.3 0.659100 0.69997.8 0.708110.3 0.693269.0 0.692CH2(CH2)5 NH COCH3OCF2CF2 43.9 0.75372.4 0.663CH2CHF 33.8 0.70054.9 0.666CH2CCl2 58.7 0.654CH2CF2 36.0 0.744123.1 0.641134.3 0.664CH2 CH CCH3CH2 CHCHH2C CH2HC CH2NCH2 COCCONCCON OCH2 CHO C CH3OCH2 CHOCH3CH3CH2 CCOO CHCH3CH3CH3CH2 CCOO C4H9 39. 32168.3 0.651120.0 0.60785.9 0.696163.0 0.68788.8 0.705111.6 0.669115.5 0.65765.6 0.63889.3 0.65040.0 0.681CH3CH2 CCOO C6H13CH3SiOCH2 CH2CF3COCH2 CH2 CH2 CH3NCH2 CHNCH2 CHNH2C C OH2C CH2CH2 CHClCH2 CHCH3CH2 CH SCH3CH2 CHCOO C2H5O CH2CO 40. 3369.9 0.684172.5 0.74070.6 0.677CH2O 21.3 0.752126.1 0.616118.5 0.66753.0 0.733150.8 0.679103.0 0.62076.2 0.568FCH3CH2CH2C2H5CH3CH2CH2S 46.4 0.680144.4 0.692227.7 0.693O CHCH3CH2 COCOCONH NHCH2 CHC OOCH3CH3Si OCH2 CHCOO C4H9CClCF2CH2 CCOO CHCH2CH2CH2Si OC2H5Si OCH3(CH2)2 O COC OOO COO 41. 34154.1 0.696157.0 0.721CH2CH2 30.2 0.68246.3 0.66699.6 0.665262.1 0.726Figure 6. Dependence of density on the values of determined experimentally fit well the same linear dependence on therelation of atoms mass on their volume. In accordance with Equation (II.5), thetangent of this straight line represents the molecular packing coefficient which, in thecase of amorphous bulky systems, serves as an universal constant. If it is true, thepolymer density may be calculated from the equationkMA , (II.6)=N ViiO COO SO2CH2 CHCH3CH2 CHCH2 NH C OO(CH2)4 O C NHO 42. 35that yields directly from Equation (II.5) under the condition kavg = const. In the case ofamorphous bulky polymers, kavg = 0.681. For silicon-containing polymers, the averagecoefficient of molecular packing is 0.603.Hence, a change of the polymer chemical structure is unable to cause asignificant effect on the part of the occupied volume in amorphous polymericsubstance, and the value of density, , itself depends on the relation of mass and theVan-der-Walls volume of the repeat unit only.Obviously, here we are dealing with true bulky substances of the amorphousstructure. In reality, a polymeric substance with any porosity may be formed, and thecoefficient k will have extremely different values. However, in this case, the notion ofthe packing density, quantitatively estimated by the value of k, loses its usual meaningand must be calculated for pore walls material only. We return to this problem belowwhen discuss parameters of the porous structure of polymers, determined by thesorption method.For copolymers, equation (II.6) has the form( ) k M M M + + n i + + + += i niiiin nN V V V......221A 1avg 1 1 2 2 , (II.7)where 1, 2, , n are molar parts of the components 1, 2, , n; M1, M2, , Mn aremolecular masses of the repeat units of the same components; 1 iVi ,2 iVi ,, are their Van-der-Waals volumes.Vi i n In the reduced form, expression (II.7) is:k nk M=kk k N V=== k i=k n1k i kA1avg , (II.8)where k, Mk, 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 1, 2, , n ofhomopolymers based on the components 1, 2, , n, expression (II.7) changes to thefollowing form:= + + + ...M M Mn nnM M Mnn12+ + ...+22111 1 2 2 , (II.9) 43. 36(in this case, it should be taken into account that 1 + 2 + + n = 1).In the reduced form, the expression (II.9) is the following:=== =k n1 , (II.10)Mk k=kkk nkk kM1Expressions (II.7)(II.10) may also be used for calculating the density ofmiscible blends of polymers.Let us now examine the temperature dependences of the molecular packingcoefficients of glassy polymers. Calculation of values of k at different temperaturesare performed by formulae yielding from the expression (II.5):N VA1k T ii[ ( )] g G g( )+ MV T T=, (TTg); (II.11)N VA1k T ii[ ( )] g L g( )+ MV T T=, (TTg); (II.12)where Vg is the specific volume of the polymer at the glass transition temperature Tg;G and L are the volume expansion coefficients of polymers below and above theglass 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 temperaturedependences of the molecular packing coefficients are of the form depicted in Figure7. A remarkable property of these temperature dependences in the real equality of themolecular packing coefficient in the first approximation for all bulky polymers at anytemperature below the glass transition point. In the second, more accurateapproximation, the molecular packing coefficient is the same for every polymer at theglass transition temperature. This value is kg 0.667. 44. Table 5Coefficients of molecular packing k for a series of crystalline polymersName Type of elementary cell Chemical formula , g/cm3 k1 2 3 4 5Polyethylene RhombicPseudo-monoclinicTriclinicCH2CH21.0001.0140.9651.0130.7360.7460.7100.745Polypropylene:- isotactic- syndiotacticMonoclinicMonoclinic0.9360.9100.6930.6741,2-poly(butadiene):- isotactic- syndiotacticRhombicRhombohedral0.9630.9600.6920.690CH2 CHCH3CH2 CHCHCH21,4-trans-poly(butadiene) Pseudo-hexagonal CH2CH=CHCH2 1.020 0.7331,4-cis-poly(butadiene) Monoclinic CH2CH=CHCH2 1.010 0.7261,4-cis-polyisoprene Monoclinic 1.000 0.725CH2 CH C CH2CH3Polychloroprene Rhombic 1.657 0.893CH2 CH C CH2ClPoly(ethylene terephthalate) Triclinic 1.455 0.776Poly(hexamethyleneterephthalate)O CH2CH2 O COCOTriclinic 1.131 0.652O COCOO (CH2)637 45. 381 2 3 4 5Poly(ethylene isophthalate) Triclinic 1.358 0.724O COCH2COO CH2Poly(ethylene adipate) Triclinic 1.274 0.782Polyamide 6,6:-isomer-isomerTriclinicTriclinic1.2401.2480.7640.769O (CH2)2 O C (CH2)4OCOCO(CH2)4 C HNO(CH2)6 NHPolyamide 6,10 Triclinic 1.157 0.740CO(CH2)8 C HNO(CH2)6 NHPolyamide 6 Monoclinic 1.230 0.758COHN (CH2)5Polyamide 11 Triclinic 1.192 0.789COHN (CH2)10Poly-4-methylpentene-1 Tetragonal 0.813 0.598CH2 CHCH2CH CH3CH338 46. 391 2 3 4 5Polyvinylchloride RhombicMonoclinic1.4401.4550.6800.687Polytetrafluoroethylene Pseudo-hexagonalHexagonalCH2 CHClCF2CF2 2.4002.3600.7940.781Polyvinylfluoride Hexagonal 1.440 0.742CH2 CHFPoly(vinyl alcohol) Monoclinic 1.350 0.770CH2 CHOHPolyacrylonitrile Rhombic 1.110 0.677Poly(methyl methacrylate)isotacticCH2 CHC NPseudo-rhombic 1.230 0.719C H 3C H 2 CCOOC H 3Polystyrene Rhombohedral 1.120 0.711CH2 CHPolyoxymethylene Hexagonal CH2O 1.506 0.808Polyethylene oxide Hexagonal CH2CH2O 1.205 0.72339 47. 401 2 3 4 5Polypropylene oxide Rhombic 1.1021.1540.663CH2 CH O 0.694CH340 48. 41Taking into account that the specific volume at the glass transition temperatureTg equalsN VV ik MigAgg 1= = , (II.13)where g is the polymer density at Tg; and substituting (13) into (11) and (12), we getg[ 1( )] G g( )T Tkk T+ =, (TTg); (II.14)gk[ 1( )] L g( )T Tk T+ =, (TTg); (II.15)Equations (II.14) and (II.15) can be used for obtaining relations, whichdescribe temperature dependences of the density of polymers in the glassy andrubbery states. For this purpose, we substitute (II.14) and (II.15) into equation (II.6):g , (TTg); (II.16)[ + ( )] =T T N Viik MTG g A1( )g[ + ( )] =T T N Viik Mk TL g A1( ), (TTg); (II.17)Because, as it is seen from the further considerations, values of expansioncoefficients G and L, as well as the glass transition temperature Tg, can becalculated from the chemical structure of the repeating polymer unit, temperaturedependences of density (T) can also be calculated from relations (II.16) and (II.17).In conclusion, let us note that the constancy of the coefficient of molecularpacking k is true only for amorphous bulky substances composed of polymers. In thecase of crystalline polymeric substances, the situation is significantly changed. If thecoefficients of molecular packing for ideal polymeric crystals are calculated with thehelp of the X-ray analysis data, one can assure himself that, in spite of amorphousones, the coefficients of molecular packing of crystalline polymers are extremelydifferent. The smallest values of k are typical of aliphatic systems with volumetricside groups, for example, for poly-4-methylpentene-1 and poly-n-butyraldehyde. Thehighest coefficients of packing are typical of 1,4-trans--polyisoprene and poly-chloroprene.As an example, Table 5 shows the crystallographic values of densities andmolecular packing coefficients for a series of typical crystalline polymers. It is clearthat the values of k for them vary in a wide range. Hence, crystalline polymers displaya rather wide distribution curve of the coefficients of molecular packing (Figure 8). 49. 42Figure 8. Curve of distribution of the coefficients of molecular packing k for crystalline polymers.II.2. Relationship between free volume of polymers, coefficient ofmolecular packing and porous structureBefore we start discussing the relationship between the above-mentionedphysical 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 molarvolume of the substance, VM, and its Van-der-Waals molar volume NA Vi :i = = V VM NA Vi M / NA V . (II.18)iiiThe value of V obtained in this way is often called the empty volume.Clearly, the empty volume depends on temperature, because the molar volume alsodepends on it: VM = M/. Substituting this relation into equations (II.16) and (II.17),we obtain:( ) + 1G g ( )= 1gA kT TV T N Vii, (TTg); (II.19)( ) 1 + T TL g ( )= 1gV T N VA kii, (TTg); (II.20)Relations (II.19) and (II.20) describe the temperature dependences of theempty volume.2) The free volume represents the difference between the volumes of thesubstance at the absolute zero and at the assigned temperature; to put it differently, thefree volume represents an excessive volume occurring as a result of thermalexpansion of the substance. This definition of the free volume is most valuable.Moreover, the present free volume is subdivided into the free volume of fluctuationand the expansion volume.3) The free volume represents the difference between the volume ofpolymeric substance at the assigned temperature and the volume of the ideal crystal 50. 43composed of a polymer of the same chemical structure. This definition of the freevolume is used extremely seldom.Let us now pass to analysis of the relationship between the free volume ofpolymers, the coefficient of molecular packing and the porous structure.The porous structure mostly defines their properties. That is why the methodsof estimation of the porous structure of polymers and its connection with suchcharacteristics as the coefficient of molecular packing and the free volume of polymermust be discussed in detail. The case is that the size of micropores depends on themethod of its estimation. Clearly, interpretation of their nature and the relationship ofthe characteristics of the microporous structure with the properties of polymerssignificantly depends on the method of their determination.The properties of many bulky and film polymers significantly depend on thedensity 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 isvery important, together with their size distribution, specific surface.Let us present the definition, given in ref. [68]: Pores are emptinesses orcavities in solids usually connected with each other. They possess various anddifferent form and size, determined significantly by nature and the way of obtainingabsorbents.Usually, the characteristics of a microporous structure are judged byexperimental data on equilibrium adsorption, capillary condensation of vapor andmercury pressing in (mercury porosimetry) [121]. Recently, the positron annihilationmethod has been used [3, 48, 110, 123, 134, 140, 155, 164, 187, 211]. This methodhelps in determining the characteristics of the microporous structure, when the size ofpores is commensurable with the molecule size. Such micropores are inaccessible forsorbate molecules and especially for mercury when mercury porosimetry is used.Polymers and materials prepared from them possess the feature (in contrast tomineral sorbents) that they swell during sorption of vapors of organic liquids.Consequently, their structure changes and usual methods of calculation give nopossibility of estimating the true porous structure of the initial material. It stands toreason that vapors of organic liquids, in which polymer does not swell, can be used insorption experiments. Then the parameters of the porous structure of the initialmaterial can be determined, but these cases are quite rare [107].Before passing to comparison of parameters of the porous structure with thefree volume of the polymer, it should be noted that parameters of the porous structurefor the same polymer could be significantly different due to conditions of its synthesisand further processing. For example, a film or fibers may be obtained from varioussolvents [81], as well as from a solventprecipitant mixture [97], and will display adifferent microporous structure and properties. The same can be said about materialsobtained by pressing and injection molding and with the help of hydrostatic extrusionas well. Therewith, macropores may also be formed and their total volume may bequite high. If special synthesis methods are used, materials based on polymernetworks may be obtained, which possess a large specific surface and extremely largepore radii [115]. Clearly, such macropores are not defined by the packing density ofmacromolecules. They may be formed by loose packing of formations larger thanmacromolecules or may be caused by conduction of a chemical process of thenetwork formation under special conditions [167].Several more general comments should be made. Besides macropores, asmentioned above, micropores are present in a polymeric substance, the size of whichis commensurable with the size of sorbate molecules. Clearly, in this case, sorbate 51. 44molecules cannot penetrate into these micropores (it is assumed that for sorbatemolecules to penetrate into pores, the volume of the latter must be several timesgreater than that of penetrating molecules). Since sorbate molecules may be different,i.e. may possess different sizes, parameters of the porous structure determined fromthe 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 beenintroduced. Of interest is that the sorption method of determination of the porousstructure of polymeric substances cannot be used in the case when a substancecontains quite large macropores. This is associated with the fact that under conditionsof polymolecular adsorption, when many molecular layers are formed on walls ofmacropores, their fusion becomes difficult, i.e. capillary condensation is absent. Then,the total volume of pores calculated by the amount of sorbate penetrated into thepolymeric substance will be smaller than the true volume of macropores.Starting the analysis of relationship between the physical characteristics of thepolymeric substance and its microporous structure, let us introduce some definitionsand designations:Ssp is the specific surface of micropores,W0 is the total volume of pores,W0max is the maximal volume of pores accessible for sorbate molecules of anysize (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 (abulky amorphous polymer is the one in which no sorbate molecule can penetrate intoits 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 molecularpacking 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 withoutswelling. Clearly, that for such substances W0max = 0. At the same time, as seen fromthe data in Table 5, coefficients of molecular packing of crystals fall within the rangefrom 0.64 to 0.89. Taking into account that the coefficient of molecular packing, bydefinition, represents a part of the occupied (Van-der-Waals) volume, it can be saidthat the part of empty (but inaccessible) volume is 1 k = 0.110.36. This emptyvolume is inaccessible for even small sorbate molecules to penetrate in; let mark it asVinacc.. Then the volume of the ideal crystal (or bulky amorphous polymer, Vblk) can bewritten down asVid.cr. = VW + Vinacc.; Vblk = VW + Vinacc.. (II.24)The volume of the real polymeric substance (which contains microporesaccessible for a sorbate) will be summed up from three parts:VT = VW + Vinacc. + W0max. (II.25)ThenW0max = VT Vid.cr.; W0max = VT Vblk. (II.26) 52. 45The coefficient of molecular packing in the bulky part of the polymer will bedetermined from the relation= . (II.27)maxWVV WT 0kIn the case of estimation of the density of macromolecule packing for the realpolymeric substance containing micropores accessible for sorbate molecules, thecoefficient of molecular packing, k, should be calculated by the relationWV= , (II.28)V WT 0kwhere W0 is the total volume of micropores (per gram of the substance), determinedon the basis of sorption measurements.The value of W0max that represents the difference between the specific volumeof the substance at the given temperature and volume of the true bulky substance isconceptually identical to the porosity factor P = 1/s 1/t, where s is the apparentdensity; t is the true density. Therewith, s represents the density of the substance atthe current temperature, affected by the pores existing in it. It is best to measure theapparent density of substances with the proper geometrical shape, because when usingno solvents s can be found by dividing the substance weight by its volume. If theapparent density of substances with the improper shape is measured, the pycnometricor dilatometric method can be used. The difficulty is in selection of a liquid that doesnot wet the surface of the substance and does not penetrate deep into it. The truedensity t represents density of the bulky part of the substance containing no pores. Itis best to measure the density of the ideal crystal, because it can be calculated on thebasis of crystalline lattice parameters. In the case of amorphous and partly crystallinesubstances, the method of gradient tubes may be used applying liquids penetratingwell into pores. However, it should be taken into account that a mixture of two liquidsis used for creation of the density gradient in the tube, each of which may possessdifferent wettability and penetrability into pores. The picture is then distorted, and thedetermined density is not true.The relations shown above can be estimated unambiguously if a polymerswells in the sorbate, used for estimation of the porous structure of the polymer. If theexperiment indicates that W0 is greater than W0max, this indicates that the volume ofvapors 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 ofthe parameters of the polymer structure and coefficients of their molecular packing.These data are shown in Table 6. For ideal polyethylene crystallites, VE = Vinacc. andW0max = 0. The coefficient of molecular packing is quite high. For semi-crystallinepolyethylene, the empty volume, VE, is greater than in the case of the ideal crystaland, therewith, a part of it is accessible for penetration of small sorbate molecules.However, the total volume of pores determined by methanol sorption equals 0.01cm3/g. The molecular packing coefficient for the bulky part of such polyethylene issignificantly lower than for the ideal crystal.Polymers in the rubbery state (polyisobutylene, for example) also possesscomparatively low values of free volumes and are practically non-porous sorbents(VE = Vinacc.).Contrary to this, polymers produced by polycondensation or polymerization insolution display immensely high values of W0max. In this synthesis method, pores areformed due to elimination of the solvent, distributed in the volume of the synthesized 53. 46polymer. This is observed from the fact that the same polymers produced bypolymerization in the melt are practically non-porous, and values of VE for them arevery small, and W0max = 0.Table 6Parameters of porous structure and coefficients of molecular packing of a series of polymersPolymerVE,cm3/gmax,cm3/gW0W0,cm3/gVinacc.,cm3/gKPolyethylene (100% crystallinity)CH2CH20.26 ~0 ~0 0.26 0.736Polyethylene (crystallinity100%)CH2CH20.35 0.08 0.01 0.27 0.675PolyisobutyleneCH2C(CH3)20.36 ~0 ~0 0.36 0.678PolymethylidenphthalideCH2 COCOPolymerization in dimethylformamide solutionPolymerization in melt1.280.221.06~00.220.220.6870.687Polyarylate F-1C O OOCOCOCOPolycondensation in chlorinated bisphenol solutionpressed at 360C and under 312.5 MPa pressure0.820.240.58~00.31~00.240.240.6880.688Pores formed during synthesis may be closed in polymer pressing under highpressure, and the porous polymer then becomes non-porous. Therewith, in all cases,W0 is smaller than W0max that indicates the absence of swelling.For all polymers, values of Vinacc. are close to these characteristics for thedensity of crystallized samples. Of special attention is the fact that independently ofthe production method, the molecular packing coefficient for amorphous and semi-crystallinepolymers in their bulky part is the same and close to the average valuekavg = 0.681, which was discussed above. For a crystalline sample, the value of k issignificantly higher.There is one more interesting point to discuss, associated with molecularpacking, namely, the change of the system volume during polymerization, i.e. attransition from monomer to polymer.It is well known that transition from a monomeric liquid to a solid glassypolymer is accompanied by a significant contraction, i.e. volume decrease [76]. Thespecific volume of the polymer Vp is always smaller than that of monomer Vm, andtheir difference V = Vp Vm0. One of the reasons for contraction is substitution oflonger intermolecular bonds existing in liquid monomers by shorter chemical bonds 54. 47formed 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 considerationof the experimentally determined specific volumes that there is another reason forcontraction, which is more dense packing of polymeric chains compared with thepacking of monomeric molecules. This is indicated by the fact that the packingcoefficients of polymers are always greater than those of their monomers (kpkm).Let the total contraction, Vtotal, be presented as a sum of two values: V1,which is the contraction stipulated by substitution of intermolecular bonds bychemical ones, and V2, which is the contraction involved by more dense packing ofchains,Vtotal = V1 + V2, (II.29)and each of the summands estimated.To do this, values of the specific volume of a polymer should be calculated onthe assumption that it displays the packing coefficient, the same as the monomer km,i.e.A pmNp kVMVii = , (II.30)wherep iVi is the Van-der-Waals volume of atoms in the repeat unit of thepolymer; 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 thanexperimentally measured values of specific volumes of the polymer, Vp.The difference between Vp and Vm isV1 = Vp Vm, (II.31)and the remaining part of the contraction is calculated by the formulaV2 = Vtotal V1. (II.32)Relative parts of contraction are determined from the relations:1 = V1/Vtotal; (II.33)2 = V2/Vtotal. (II.34)The data shown in Table 7 indicate that in all the cases the smaller part ofcontraction depends upon opening of double bonds, and the greater part on densepacking of polymer chains. Therewith, the chemical structure of a monomer and anappropriate polymer significantly affects the values of 1 and 2. 55. 49Table 7Changes in volume of the system as a result of polymerizationPolymer (monomer) Vm, cm3/g Vn, cm3/g Vn, cm3/g Vtotal, cm3/g V1, cm3/g V2, cm3/g 1, % 2, %1 2 3 4 5 6 7 8 91.068 0.855 0.968 0.213 0.080 0.133 37.6 62.41.102 0.890 1.031 0.212 0.071 0.141 33.5 66.51.109 0.928 1.045 0.181 0.064 0.117 35.4 64.61.046 0.815 0.951 0.231 0.095 0.136 41.1 58.91.082 0.873 1.000 0.209 0.082 0.127 39.2 60.8CH3CH2 CCOO CH3CH3CH2 CCOO C2H5CH3CH2 CCOO C3H7CH2 CHCOO CH3CH2 CHCOO C2H548 56. 501 2 3 4 5 6 7 8 91.098 0.952 1.036 0.146 0.062 0.084 42.5 57.51.073 0.841 0.976 0.232 0.097 0.135 41.8 58.21.104 0.942 1.028 0.162 0.076 0.086 46.9 53.1CH2 CHCOO C4H9CH2 CHO C CH3OCH2 CH49 57. 50In the set of polyacrylates and polymethacrylates 2 grows first with thevolume of the side substituent and then decreases. Decrease of the intensity of theeffect of the dense packing of chains, apparently, depends upon steric hindrances.Hence, it follows from the above-said that the notions of porosity and packingdensity are inadequate. Porosity reflects almost always cavities greater than themolecular size, i.e. quite large ones. As for the packing density of macromoleculesthemselves, it may be judged by considering the non-porous part of the sample only.As noted above, application of positron annihilation methods is preferable foranalyzing the microporous structure of polymers [3, 48, 110, 123, 134, 140, 155, 164,187, 211]. With the help of these methods, qualitative and quantitative informationabout the characteristics of submicropores (215 ) in polymers may be obtained.Let us discuss the results of studying annihilation of positrons in twopolymers, which are good models of the limiting characteristics of the packing densityof macromolecular chains. One of them is polyimide characterized by a highlyregular, quasi-crystalline structure, and the second is poly(1-trimethylsilyl-1-propyne)(PTMSP) which, on the contrary, is characterized by a low coefficient of molecularpacking.Consider structural changes in PTMSP, which appear during its long exposureat room temperature after synthesis.For comparison, we also display the data on annihilation of positrons for aseries of other model polymers. The chemical structures of all above-mentionedsystems are shown below.Poly(1-trimethylsilyl-1-propyne)CH3C CSiCH3H3C CH3PolyisopreneCH CH2PolydimethylsiloxaneCH3PolystyrenenPolytetraflouroethylene[CF2CF2]nnCH2 CCH3nCH2 CHnSiCH3O 58. 51PolyimideOCCONOCCN OO nObservation of the annihilation of positrons in PTMSP was performed withthe help of a method of detection of the lifetime spectra of positrons (measurementswere made by S.A. Tishin; data not published). Measurements were performed by athermostabilized spectrometer, which realizes the traditional fastslow scheme ofdetection, with a temporal photomultiplier selected and optimized due to an originalmethod [111].Processing of experimental spectra was performed with the help of well-knownsoftware Resolution and Positron FIT.Table 8 shows the results of separation of parameters of a long-livingcomponent at three-component decomposition of positron lifetime spectra forPTMSP, polyimide, polystyrene, polydimethylsiloxane and polytetrafluoroethylene.Clearly, PTMSP possesses an anomalous long lifetime for an ortho-positronium atom,to annihilation of which by a pick-offdecay the origin of a long-living component ofthe 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, D, was observedin polydimethylsiloxane and teflon in solid polymers [123, 164]. Comparison with theresults of measurements in model polymers (see Table 8) indicates that neither thepresence of an unsaturated bond, nor the presence of a side group or silicon atomseparately is the explanation of so high D for PTMSP.Table 8Parameters of the longest component of positron lifetime spectrum for a series of polymers andrated values of radius R and volume V of microporesSample D + 0.03, ns ID 0.25, % R0, R, V, 3 E, eVPTMSP 5.78 38.4 6.76 5.10 416.5 0.41Polytetrafluoroethylene 4.27 21.6 6.05 4.39 265.8 0.51Polydimethylsiloxane 3.23 41.3 5.45 3.79 170.9 0.63Polyimide 2.77 38.1 5.14 3.48 132.1 0.71Polystyrene (atactic) 2.05 40.5 4.56 2.90 76.9 0.90Two suggestions about the reasons of anomalous long average lifetime ofpositrons in PTMSP can be made.First, molecular structure of the repeat unit allows a supposition that a highconcentration of bulky, low-mobile side groups creates a porous structure with thepore size of about Van-der-Waals volume of SiC3H9 side fragment.Secondly, the size of pores may be associated with a long relaxation time ofsynthesized PTMSP at room temperature. It may be suggested that the formation andevolution of microcavities of a large size must depend on the motion of largesegments of macromolecules or even structural fragments with a long period ofregrouping.The lifetime of an ortho-positronium atom regarding the pick-offannihilationallows estimation of the size of the microcavity in which it was localized beforeannihilation [140]. The calculation results are also shown in Table 8. 59. 52In line with the model [140], positronium is considered in a spherical pitsurrounded by a layer of electrons, R thick. For wave functions in sphericalcoordinates:( ) ( ) = 2 sin/ in the pit;0 outside the pit.( )01/ 2 1R0 r Rr r (II.35)The probability of positronium existence outside the limits of density will be: R= +2sin1W R , (II.36)20 0( ) 1RRRwhere R = R0 R.Suggesting that the rate of ortho-positronium annihilation inside the electronlayer equals 0.5 ns1, the decomposition rate averaged over spins will be:D = 1/D = 2W(R) (II.37)with the constant R = 1.66 , selected empirically for solids.Let us consider the results of measurements of PTMSP films porous structurebecause of their aging.Long-term relaxation of PTMSP films was investigated with the help ofmeasuring positron lifetime spectra. As Table 9 and Figure 9 indicate displaying aseries of characteristics of time spectrum decomposition into three components andthe calculated radius of micropores R, and durability of samples aging, lifetime of thelong-living component decreases with growth of PTMSP exposure time at roomtemperature. In practice, the intensity of the long-living component does not dependon the relaxation time.Table 9Long-term relaxation of PTMSP from the data of measurement of the longest componentparameters of positron lifetime spectrum (n is lifetime of intermediate component)Aging time, days D 0.03, ns RD 0.25, % n 0.080, ns13 5.78 38.40 0.68717 5.68 37.53 0.60724 5.72 38.09 0.67883 5.40 38.08 0.507210 5.09 37.91 0.453Figure 9. Dependence of sizes R of the positron-sensitive microcavity on time of exposure tc at 25Cfor PTMSP 60. 53The result observed is connected with slow structural relaxation but not theaging (if by the aging occurrence of the main chain fission is meant), because thelatter process is usually accompanied by changes in intensity ID (results of observinglong-term aging of polyethylene by the method of positron lifetime variation may bedisplayed as an example, although aging in polymers is a very specific process).Taking into account the relation between D and the radius of micropores inpolymers [140], it must be concluded that in long-term relaxation of PTMSP sizes ofpores decrease (see Figure 9) and, probably, the mobility of macromolecular chainsreduces due to free volume decrease.As follows from the constancy of ID, the concentration of positronium traps isindependent of the exposure time in the studied time interval.Let us now discuss the results of investigation of positron annihilation inpolyimide.As the measurements have shown [48], annihilation of positrons in polyimideis significantly different from the one usually observed in most polymers. Theannihilation spectrum in polymers is usually characterized by the presence of three orfour 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 0 = 0.388 ns (Figure 10). Time distribution isapproximated well by a single decay line, the tangent of which determines the averagelifetime.Figure 10. Positron lifetime spectrum of the starting polyimide film (here N is the number ofreadings in a channel)The value of lifetime and the spectrum structure allow a supposition thatannihilation in polyimide proceeds from the positron state without forming apositronium atom as it is typical of metals and semiconductors with high mobility ofelectrons 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 forelectrons. 61. 54Figure 11. Lifetimes and intensities of components (%) in the spectra of the original sample (I) anddeformed samples of polyimide after recovery lasting for 1 (II) and 24 (III) hrs.Table 10Annihilation characteristics of polyimide filmSampleRecoverylasting, hr 0, ps 1, ps 2, ps I2, %Count rate,k109, sInitial 3855 Deformed 1 29430 44017 595 0.600.15Deformed 24 36110 53130 92 0.120.05In relation to interaction with positrons, the microstructure of the initial(undistorted) polyimide film possesses no defects. However, time spectra change afterdeformation (Figure 11 and Table 10). Two components instead of a single one areobserved in the deformed sample: with shorter and longer lifetimes. After recovery(resting) during 24 hours at room temperature, an increase of lifetimes of bothcomponents and reduction of intensity of longer-term ones are observed. Thecharacter of changes taking place allows a supposition that the submolecular structureof polyimide is rebuilt during deformation; intermolecular bonds break, andmicrodefect free volumes enough for positron localization are formed. In this case,the value of the long-term component 2 must reflect changes in the average size, andintensity I2 concentration of these defects. Analogous changes in the spectra werealso observed in annealing defects in metals and semiconductors. These changes areusually analyzed with the help of a positron entrapment model. This model isqualitatively good in reflecting changes in the time spectra observed in polyimidedeformation. Reduction of the lifetime of the short component, bound to annihilationin the undistorted part of the polymer, depends on the high rate of capture in thedeformed sample. After partial contraction during recovery, the concentration ofdefects decreases and lifetime 2 approaches the characteristic one of the originalpolymer. Therewith, the intensity of the long-term component, I2, formed due topositron annihilation on defects, decreases, too. Growth of the lifetime 2 may beexplained by coagulation (consolidation of small defects into larger ones) duringrecovery or fast relaxation of small pores and, consequently, by growth of the averagecapture radius.As indicated in estimations, the concentration of microdefects after partialrelaxation decreases more than 7-fold. Therewith, the free volume induced bydeformation decreases by a factor of 4 [48]. The values obtained indicate that twoprocesses proceed fusion of microdefects and relaxation of the smallest ones,though, apparently, the intensity of the latter process is higher. 62. 55Hence the one-component spectrum is typical of the original polyimide film.In deformed samples, at least two components are observed in time spectra, which arebound to the positron annihilation from the free state and the one localized inmicropores, formed at stretching. The lifetime increases and the intensity of the defectcomponent decreases during relaxation.The results obtained with the help of the model of positron capture describeclearly the changes of time distributions observed and allow a supposition that thestructure of the free volume during relaxation changes not only as a result of fastrecombination of the smallest pores, but also because of their consolidation with theformation of long-term large-size microcavities.Basing on the analysis performed in ref. [48], the following model of positronannihilation and relaxation mechanism bound to it are suggested: before deformationall positrons, captured in small traps with the bond energy slightly higher that the heatenergy, annihilate; after deformation, rather long (compared with the positrondiffusion length) areas occur, in which the concentration of small traps (of the size~10 nm) decreases significantly, loosened up areas with deep centers of positroncapture 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 ofthe free volume occurring after polymeric film deformation may be obtained.However, interpretation of the information obtained requires a detailed study of thenature of components of a complex time spectrum of annihilation typical for a non-equilibriumstate of polymer. No solution of this problem with the help of one of thepositron methods was obtained [3, 110, 156]. That is why a complex study of positronannihilation was performed [49] in deformed polyimide with the help of measuringthe lifetime of positrons and angular correlation of annihilation radiation.Two series of experiments are described in ref. [49]. In the first series, apolyimide 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 11Change of annihilation characteristics of polyimide film depending on duration of relaxationafter deforming by 20%Lifetime Angul Relaxation lasting ar correlationafter deforming, h avg1,ps110,psI21.5,% FWMH0.05, mrad10.07,mrad0.07,mradI1.5,%0 365 201 74.3 10.44 10.49 7.14 28.21 360 176 73.6 10.77 5 368 208 77.2 10.60 24 362 205 73.0 10.48 10.64 7.14 34.7240 364 200 74.1 10.43 10.72 6.95 32.3Separated 368 220 76.3 Note. avg, 1 and I2 are characteristics of positron lifetime spectra; FWMH is the full width on themiddle height of the full spectrum; 1 is FWMH of the first Gaussian; and I are characteristics ofthe parabolic component of the angular correlation spectrum.In the second series of experiments, stress relaxation at deformation 0 = 20%was studied. The characteristics of angular distributions were determined for filmswith fixed ends. Measurements were performed with the help of a device thatperforms deformation of samples directly in the measurement chamber. Stress 63. 56relaxation curves (dependences of stress on time ) and recovery curves(dependences of deformation on time ) were taken simultaneously.The values of the positron lifetime obtained from spectra are shown in Table11 and Figure 12. Similar to the above-described results of two-component analysis,changes of annihilation characteristics, which then relaxed gradually to those typicalof 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 polyimidefilms (for designation see Table 11).Three components were separated: the lifetime of the first short-termcomponents (170220 ps) significantly depend on relaxation time; as displayed byinvestigations [49], the lifetime of the second one (38810 ps) is independent of orweakly depends on the sample state. However, significant changes in the intensity ofthis component are observed. The characteristics of the third component have notchanged during the experiment.In the work cited, experiments on measuring the angular correlation wereperformed (alongside the measurement of the positron lifetime). Making no detailedanalysis of the results of these measurements, note that in experiments with fixed ends(under stress relaxation conditions) the free volume significantly increases afterdeformation, and its further slo