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The effect of molecular chain flexibility on the glass temperature of linear polymers
THE EFFECT OF MOLECULAR CHAIN FLEX1RHJTY ON THE, GLASS TEMPERATURE OF LINEAR POLYMERS* V. P. PRIVALKO and Yr. S. LIPATOV High Polymor Chomistry Instituto, Ukrain.S.S.R. Aca.domy of Scioneos (Received 18 M a y 1970)
THE glass temperature T~ is one of the fundamental characteristics of polymers, although the theoretical aspects of its dependence on molecular parameters are not yet available. Considerable progress was made in understanding the mechanism of glass formation by polymer and of the factors affecting the position of the glass-liquid transition; it was achieved by the use of different variants of the vacancy theory (e.g. [1-4]), which postulates that the transition from the glasslike to the liquid state (highly elastic state) will take place at a temperature at which the amount of free space within a system will become larger than some critical value. A statistical theory of glass formation, developed rather recently [5, 6], states that the Tg is the temperature at which the cordigurational entropy of a liquid has a minimal value. (It was assumed in [5] that this entropy will approach zero at a second-order phase transition temperature of T0
* Vysokomol. soyod. A13: No. 12, 2733-2738, 1971. 3075
(1)
3076
V. P. P R I V A T X O a n d Y U . S. LIPATOV
in which a, b and c empirical constants. Processing the results in accordance with eqn. (1) in the case of vinyl-series polymers gave the following values of t h e constants: a h 8 2 5 , b--820, c----133°K. The analysis of eqn. (1) by using these values gave a maximum near T',----450°K, and this value practically coincided with a ~ - - - - 4 5 4 ° K [11], which is regarded as the maximal Tg of polymers with a fairly large sterie hindrance of the chain [11]. A similar processing for the second type of polymers (curve 2) unfortunately was not feasible because of the limited number of existing Tg- and a-values; nevertheless, one could see that the Tg increased more slowly here with increasing a, so that the appropriate T~' should be well above 500°K. T H E Tg- A N D G-VALUES OF S O M E LINEAR P O L Y M E R S
Polymer Polyisobutylene Atactic polypropylene Polymethacrylate Poly-n-butylmethacrylate Polyvinylacetate Polychlorotrifluorethylene Polyrnethylmethacrylate: isotactic atactic Polystyrene: isotactic atactic Atactic PVC Syndiotactic polyacrylonitrile Poly. 1-vinylnaphthalene Poly-2-vinylnaphthalene Polyvinylbiphenyl Polyoxytetramethylene Polyoxypropylene Polyoxyethylene Polyetyleneglycoladipate Polyoxystyreno Polyamido 6
T~, °C
a
--70 --35 9 22 29 45
1.8 1.87 2.05 1.98 2.12 2.03
45 104
2.14 2.4
87 89 82 96 159 151 161 --86 --75 --67 --57 40 65
2.34 2.22 2.32 2-37 2.59 2-59 2.81 1.68 1-62 1.63 1-68 1.85 1.87
The analysis of the tabulated results showed that the Tg-a function, when neglecting the terms of second order in eqn. (1), can be described with sufficient accuracy by the linear function: T~=A"(a--a") . (2) The respective diagram is Fig. lb and the calculation subsequently made of A " and. a" gave: 270°K and 1-0 (line 1), 630°K and 1.35 (line 2). This means that eqn. (2) is identical with the Simha-Boyer empirical function [4], Tgx=K,
(3)
Glass temperature of linear polymers
3077
in which z--differenee between the thermal expansion coefficients of the polymer in the glass-like and liquid state, or the thermal expansion coefficient in the liquid state, K b c o n s t a n t which is equivalent to J I " ~ K and (#--a")-*~x. I t can be qualitatively proved that correlation of the parameters of eqn. (2) with those in the Simha-Boyer function (eqn. (3)) is feasible in principle. Perepelkin [12] pointed out that the thermal expansion coefficient is inversely proportional to the elasticity modulus of the polymer. There is information to the effect [13] that a series of polymer melt properties, especially the high elasticity modulus [14], shows a semi-quantitative correlation with the conformational characteristics of the macromolecules. The inverse proportionality between (#--a") and z thus seems to have a true physical meaning.
b
o
O" 3
2.~
1"5
I
~50
a:
I
I
250
I
I
~0
I"
r, oK
450
1/T,--c, b: T=-~ functions; l--vinyl series polymers, 2---oxygen- or nitrogen-containing polymers.
The results obtained can therefore be taken as another proof of the principal equivalence [15, 16] of the theories based on the free volume concept [1-4] and on the statistical theory of the glass-like state [5, 6]. There is at the same time good reason to believe that eqn. (2) can be a better approximation than the empirical eqn. (3). It follows from the proportionality between A" and K, as well as between (#--a") -1 and z (eqns. (2) and (3)), that the Tg-~ function will probably be described for various polymer types by eqn. (3) at different values of constant K, while others [4] had said that K is a "universal" constant for polymers. The deviation, from universality observed in a number of cases, is probably explained by this divergence in opinions. Figure 15 shows t h a t the point at which line I intersects line 2 has the coordinates Tg=160°K and a=1-62, which are practically identical with the Tg=IS0-160°K [17, 18] and G=1.63 [19] for linear polyethylene. The latter can thus be regarded as a specific member in the "homologous" series 1 and 2. By taking into account that # = f (g) in an inverse-isomeric approximation [19], in which g = e x p (--As/]~T), one can therefore say that C=As/I~Tg ,
(4)
3078
V. P. PI~IVAT.lrOand Yu. S. LIPATOV
and should be constant for all the members within series 1 and 2. The numerical value of G will thus depend on the value of z18 of polyethylene, which is given in the literature as being from 540 [21] to 850 cal/mole [20]. The insertion into eqn. (4) of a value of zle=540 cal/mole gives C=1.69, which coincides with one of G = l . 7 [22] obtained by inserting the "universal" value for a free space (~0.025 [1-4]) in the theoretical Gibbs-Di!~arzio equation. Using z18----850 cal/ /mole gives C = 2.6 [23]. Furthermore, by eliminating Tg from equations (2) and (4), we get A~-~C. A " . R ( a - - a " ) . (5) An insertion of C-~2.6, A"----270°K, a----1.63 and a"----1 into eqn. (5) gives a A8=900 cal/mole ior polyethylene, which agrees with the recommended value [20]. This means that eqn. (2) does not contradict condition (4). It was interesting to find out the physical meaning of eqn. (5), which was transformed for this purpose into ~---- U-- U0,
(Sa)
in which U----C. A " . 1~. a, Uo= C. A " . / ~ . a". A comparison with the literature data made it clear that U = 2 . 3 kcal/mole in the case of polyethylene and was about equal to the height of the potential barrier to internal rotation in normal alkanes, which is of the order of 2-3 kcal/mole [20, 24]. (It is interesting to see that this potential barrier to internal rotation in n-alkanes, calculated from the semiempirical reaction potentials by Eyring and coworkers [25], was also 2.3 kcal/ /mole). It should be noted in this connection that the potential barrier is often presented in the shape of the sum of contributing electron exchange reaction energies in adjacent bonds (bond orientation) and of reaction energies between atoms not linked by valency bonds (steric effeot) [9, 24]. If U in eqn. (Sa) had the meaning of a potential barrier to rotation, then U0= U--zle will be obviously associated with the energy contributing to the barrier to bond orientation. According to eqn. (5a) UÜ will be constant for the particular polymer series, and this agrees with the theory [24, 26]. Please note that the use of C=2.6 in the evaluation of U and U 0 in accordance with eqn. (Sa) for series 2 will yield values greatly differing from the experimental [9, 20]. The explanation is (as shown on oxygen-containing polymers, for example) [27, 28] that the energy difference between trans- (t) and gauche- (g) isomers due to a rotation around C--O and C--C single bonds does not differ only in its absolute value, but also by the sign in front, so that Ae in eqn. (5a) must characterize the true energy difference between various conformations of monomer units [9]. For example, the energy difference between the (tttt) and ($tgg) conformations of polyoxytrimethylene totalled 200-400 cal/mole [28]. Similar results were got also for the polyoxytetramethylene [27]. Taking an average A8=300 cal/mole, and inserting into eqn. (4) a value of 187°K for Tg (see Table), we got C-----0.8 for polyoxytetramethylene, i.e. series 2 polymers. One
Glass temperature of linear polymers
3079
can then insert into eqn. (Sa) U : 1.7 and U0= 1.35 kcal/mole for the same polymer. These approximate values are of the same order as the experimental for the low molecular weight (mol.wt.), oxygen-containing compounds [9, 20]. I t was interesting to compare the A8 calculated from eqn. (5a) with the experimental value. Unfortunately there are very few reports dealing with this subject where polymers are concerned, and we therefore restricted the comparison to only two polymers from within series 1, i.e. polystyrene and polyvinylchloride (PVC). Using the values listed in the Table, eqn. (5) gave A ~ 1 . 7 1 and 1.85 kcal/mole respectively for these 2 polymers. This agreed reasonably well with values of 1-66~0.2 (polystyrene [12]) and 1-2 kcal/mole (PVC [30-32]), which had been determined from the NMR [29] and infrared spectra [30-32 I respectively. Finally, it should be noted t h a t the empirical functions produced by us can be used only with polymers showing relatively weak reactions of the dispersionary (polyethylene, polypropylene) or orientational type (PVC, polyacrylonitrile); it is obvious that the Tg of polymers with specific types of molecular interactions will chiefly depend on the concentrations of functional groups capable of such reactions [33, 34]. Furthermore, steric factor a will not always be an absolute measure of chain rigidity in the unit (block). This applies particularly to polymers containing aromatic rings in the main chain. The a of such polymers is usually small [35, 36], while the Tg can reach fairly large values [37]. This behaviour is linked with the differences between the statistical chains of the polymers examined in this study, and those having phenyl rings in the main chain, which have a symmetrical function of the energy potential of internal rotation around the C--O bonds [36]. I t appears t h a t other equations will be more suitable [38-40]. The authors express their thanks to A. A. Asl~adskii for valuable criticism. CONCLUSIONS
(1) On the basis of an analysis of literature data it is shown that the glass temperature (Tg) of polymers with molecular interactions of a dispersionary or orientational type will chiefly depend on the value of steric factor a, which reflects the thermodynamic chain flexibility. (2) The equations arrived at on the basis of the functions were found to agree with the energetical parameters of the respective low- and high-molecular weight substances. Translated by K. A. AImE~ REFERENCES
1, T. G. FOX and P. J. FLORY, J. Appl. Phys. 21: 581, 1950 2. D. TURNBULL and M. H. COHEN, J. Chem. Phys. 31: 1164, 1959; 34: 120, 1961 3. N. I~RAI and H. EYRING, J. Polymer Sei. 37: 51, 1959 4. R. SIMHA and R. F. BOYER, g. Chem. Phys. 37: 1003, 1962 5. J. H. GIBBS and E. A. DiMARZIO, J. Chem. Phys. 28: 373, 1958
3080
V. P. P ~ v ~ o
and Yu. S. L~ATov
6. J. H. GIBBS, In: Modern Aspects of the Vitreous State, vol. 1, p. 152, J. P. MacKenzie (Ed.), Butterworth, London, 1960 7. V. N. TSVETKOV, V. Ye. ESK1N and S. Ya. FRENKEL', Struktura makromolekul v ras. tvorakh (Macromolecular Structures in Solutions). Izd. "Nauka", 1964 8. P. J. FLORY, Principles of Polymer Chemistry, Cornell Univ. Press, Itlmca, New York, 1953 9. T. M. BIRSHTEIN and O. B. PTITSYN, Konformatsiya makromolekul (Macromolemdar Conforr~tions). Izd. "Nauka", 1964 10. V. P. PRIVALKO and Yu. S. LIPATOV, Vysokomol. soyed. Bl$: 102, 1970 (Not translated in Polymer Sci. U.S.S.R.) 11. A. F. LEWIS, J. Polymer Sci. Bl: 649, 1963 12. K. Ye. PEREPELKIN, Mekhanika polimerov, 174, 1967 13. V. R. ~LLEN and T. J. FOX, J. Chem. Phys. 41: 337, 344, 1964 14. V. O. KULICHIKHIN, Dissertation, 1969 15. J. MOACANIN and R. SIMHA, J. Chem. Phys. 45: 964, 1966 16. A. EISENBERG and S. SAITO, J. Chem. Phys. 45: 1673, 1966 17. M. J. BROADHURST, J. Res. l~at. Bur. Standards A67: 233, 1963 18. E. PASSAGLIA and H. K. KEVORKIA~, J. Appl. Polymer Sol. 7: 119, 1963 19. M. KURATA and W. H. STOCKMAYER, Fortsehr. Hoohpol. Forsehg. 3: 196, 1963 20. M. V. VOL'KENSHTEIN, Konfigurat~ionnaya statistika polimernykh tsepei (The Configurational Statistics of Polymer Chains). Izd. Akad. Nauk SSSR, 1959 21. A. L~[]~ERRI,C. NOEVE and P. FLORY, J. Am. Chem. See. 83: 1015, 1961 22. F. E. KARASZ and W. J. MACKNIGHT, Macromolecules 1: 537, 1968 23. A. A. MILLER, Macromolecules 2: 355, 1969 24. R. A. SCOTT and H. A. SCHERAGA, J. Chem. Phys. 42: 2209, 1965; 44: 3054, 1966 25. H. HErmIT, D. GRANT and H. EYRING, Molek. Phys. 3: 577, 1960 26. L. PAULING, Proe. Nat. Acad. Sei. U.S.A. 44: 211, 1958 27. J. E. MARK, J. Polymer Sci. B4: 825, 1966 28. J. E. MARK, J. Am. Chem. Soe. 88: 3708, 1966 29. F. A. BOVEY, F. P. HOOD, E. W. ANDERSON and L. C. SNYDER, J. Chem. Phys. 45: 3900, 1965 30. H. GERMAR, Kolloid-Z. und Z. f~r Polymere 193: 25, 1963 31. A. NAKAJIMA, H. HAMADA and S. HAYASHI, Makromol. Chemie 95: 40, 1966 32. Yu. V. GLAZKOVSKII and Yu. G. PAPULOV, Vysokomol. soyed. A10: 492, 1968 (Translated in Polymer Sci. U.S.S.R. 10: 3, 569, 1968) 33. S.N. ZHURKOV and B. Ya. LEVIN, Vestnik Leningrad. Gos. Univ., No. 3, 45, 1950 34. V. P. PRIVALK0, Yu. S. LIPATOV and Yu. Yu. KERCHA, Dokl. Akad. l~auk Ukrain. SSR, Seriya B, No. 3, 255, 1969 35. G. ~4LI.EN, J. MeAISH and C. STRAZIELLE, Europ. Polymer J. 5: 319, 1969 36. P. J. AKERS, G. ALLEN and M. J. BETHEL, Polymer 9: 575, 1968 37. A.A. ASKADSKII~ Vysokomol. soyed. A9: 418, 1967 (Translated in Polymer Sei. U.S.S.R. 9: 2, 471, 1967) 38. S. N. ZHURKOV, Dokl. Akad. Iqauk SSSR 49: 201, 1945; 47: 493, 1945 39. A. A. ASKADSKII~ Fiziko.khimiya poliarilatov (The Physical Chemistry of Polyarylares). Izd. "Khimiya", 1968 40. G. L. SLONIMSKII, A. A. ASKADSKII and A. I. KITAIGORODSgII, Vysokomol. soyed. A12: 494, 1970 (Translated in Polymer Sci. U.S.S.R. 12: 3, 556, 1970)
THE glass temperature T~ is one of the fundamental characteristics of polymers, although the theoretical aspects of its dependence on molecular parameters are not yet available. Considerable progress was made in understanding the mechanism of glass formation by polymer and of the factors affecting the position of the glass-liquid transition; it was achieved by the use of different variants of the vacancy theory (e.g. [1-4]), which postulates that the transition from the glasslike to the liquid state (highly elastic state) will take place at a temperature at which the amount of free space within a system will become larger than some critical value. A statistical theory of glass formation, developed rather recently [5, 6], states that the Tg is the temperature at which the cordigurational entropy of a liquid has a minimal value. (It was assumed in [5] that this entropy will approach zero at a second-order phase transition temperature of T0
* Vysokomol. soyod. A13: No. 12, 2733-2738, 1971. 3075
(1)
3076
V. P. P R I V A T X O a n d Y U . S. LIPATOV
in which a, b and c empirical constants. Processing the results in accordance with eqn. (1) in the case of vinyl-series polymers gave the following values of t h e constants: a h 8 2 5 , b--820, c----133°K. The analysis of eqn. (1) by using these values gave a maximum near T',----450°K, and this value practically coincided with a ~ - - - - 4 5 4 ° K [11], which is regarded as the maximal Tg of polymers with a fairly large sterie hindrance of the chain [11]. A similar processing for the second type of polymers (curve 2) unfortunately was not feasible because of the limited number of existing Tg- and a-values; nevertheless, one could see that the Tg increased more slowly here with increasing a, so that the appropriate T~' should be well above 500°K. T H E Tg- A N D G-VALUES OF S O M E LINEAR P O L Y M E R S
Polymer Polyisobutylene Atactic polypropylene Polymethacrylate Poly-n-butylmethacrylate Polyvinylacetate Polychlorotrifluorethylene Polyrnethylmethacrylate: isotactic atactic Polystyrene: isotactic atactic Atactic PVC Syndiotactic polyacrylonitrile Poly. 1-vinylnaphthalene Poly-2-vinylnaphthalene Polyvinylbiphenyl Polyoxytetramethylene Polyoxypropylene Polyoxyethylene Polyetyleneglycoladipate Polyoxystyreno Polyamido 6
T~, °C
a
--70 --35 9 22 29 45
1.8 1.87 2.05 1.98 2.12 2.03
45 104
2.14 2.4
87 89 82 96 159 151 161 --86 --75 --67 --57 40 65
2.34 2.22 2.32 2-37 2.59 2-59 2.81 1.68 1-62 1.63 1-68 1.85 1.87
The analysis of the tabulated results showed that the Tg-a function, when neglecting the terms of second order in eqn. (1), can be described with sufficient accuracy by the linear function: T~=A"(a--a") . (2) The respective diagram is Fig. lb and the calculation subsequently made of A " and. a" gave: 270°K and 1-0 (line 1), 630°K and 1.35 (line 2). This means that eqn. (2) is identical with the Simha-Boyer empirical function [4], Tgx=K,
(3)
Glass temperature of linear polymers
3077
in which z--differenee between the thermal expansion coefficients of the polymer in the glass-like and liquid state, or the thermal expansion coefficient in the liquid state, K b c o n s t a n t which is equivalent to J I " ~ K and (#--a")-*~x. I t can be qualitatively proved that correlation of the parameters of eqn. (2) with those in the Simha-Boyer function (eqn. (3)) is feasible in principle. Perepelkin [12] pointed out that the thermal expansion coefficient is inversely proportional to the elasticity modulus of the polymer. There is information to the effect [13] that a series of polymer melt properties, especially the high elasticity modulus [14], shows a semi-quantitative correlation with the conformational characteristics of the macromolecules. The inverse proportionality between (#--a") and z thus seems to have a true physical meaning.
b
o
O" 3
2.~
1"5
I
~50
a:
I
I
250
I
I
~0
I"
r, oK
450
1/T,--c, b: T=-~ functions; l--vinyl series polymers, 2---oxygen- or nitrogen-containing polymers.
The results obtained can therefore be taken as another proof of the principal equivalence [15, 16] of the theories based on the free volume concept [1-4] and on the statistical theory of the glass-like state [5, 6]. There is at the same time good reason to believe that eqn. (2) can be a better approximation than the empirical eqn. (3). It follows from the proportionality between A" and K, as well as between (#--a") -1 and z (eqns. (2) and (3)), that the Tg-~ function will probably be described for various polymer types by eqn. (3) at different values of constant K, while others [4] had said that K is a "universal" constant for polymers. The deviation, from universality observed in a number of cases, is probably explained by this divergence in opinions. Figure 15 shows t h a t the point at which line I intersects line 2 has the coordinates Tg=160°K and a=1-62, which are practically identical with the Tg=IS0-160°K [17, 18] and G=1.63 [19] for linear polyethylene. The latter can thus be regarded as a specific member in the "homologous" series 1 and 2. By taking into account that # = f (g) in an inverse-isomeric approximation [19], in which g = e x p (--As/]~T), one can therefore say that C=As/I~Tg ,
(4)
3078
V. P. PI~IVAT.lrOand Yu. S. LIPATOV
and should be constant for all the members within series 1 and 2. The numerical value of G will thus depend on the value of z18 of polyethylene, which is given in the literature as being from 540 [21] to 850 cal/mole [20]. The insertion into eqn. (4) of a value of zle=540 cal/mole gives C=1.69, which coincides with one of G = l . 7 [22] obtained by inserting the "universal" value for a free space (~0.025 [1-4]) in the theoretical Gibbs-Di!~arzio equation. Using z18----850 cal/ /mole gives C = 2.6 [23]. Furthermore, by eliminating Tg from equations (2) and (4), we get A~-~C. A " . R ( a - - a " ) . (5) An insertion of C-~2.6, A"----270°K, a----1.63 and a"----1 into eqn. (5) gives a A8=900 cal/mole ior polyethylene, which agrees with the recommended value [20]. This means that eqn. (2) does not contradict condition (4). It was interesting to find out the physical meaning of eqn. (5), which was transformed for this purpose into ~---- U-- U0,
(Sa)
in which U----C. A " . 1~. a, Uo= C. A " . / ~ . a". A comparison with the literature data made it clear that U = 2 . 3 kcal/mole in the case of polyethylene and was about equal to the height of the potential barrier to internal rotation in normal alkanes, which is of the order of 2-3 kcal/mole [20, 24]. (It is interesting to see that this potential barrier to internal rotation in n-alkanes, calculated from the semiempirical reaction potentials by Eyring and coworkers [25], was also 2.3 kcal/ /mole). It should be noted in this connection that the potential barrier is often presented in the shape of the sum of contributing electron exchange reaction energies in adjacent bonds (bond orientation) and of reaction energies between atoms not linked by valency bonds (steric effeot) [9, 24]. If U in eqn. (Sa) had the meaning of a potential barrier to rotation, then U0= U--zle will be obviously associated with the energy contributing to the barrier to bond orientation. According to eqn. (5a) UÜ will be constant for the particular polymer series, and this agrees with the theory [24, 26]. Please note that the use of C=2.6 in the evaluation of U and U 0 in accordance with eqn. (Sa) for series 2 will yield values greatly differing from the experimental [9, 20]. The explanation is (as shown on oxygen-containing polymers, for example) [27, 28] that the energy difference between trans- (t) and gauche- (g) isomers due to a rotation around C--O and C--C single bonds does not differ only in its absolute value, but also by the sign in front, so that Ae in eqn. (5a) must characterize the true energy difference between various conformations of monomer units [9]. For example, the energy difference between the (tttt) and ($tgg) conformations of polyoxytrimethylene totalled 200-400 cal/mole [28]. Similar results were got also for the polyoxytetramethylene [27]. Taking an average A8=300 cal/mole, and inserting into eqn. (4) a value of 187°K for Tg (see Table), we got C-----0.8 for polyoxytetramethylene, i.e. series 2 polymers. One
Glass temperature of linear polymers
3079
can then insert into eqn. (Sa) U : 1.7 and U0= 1.35 kcal/mole for the same polymer. These approximate values are of the same order as the experimental for the low molecular weight (mol.wt.), oxygen-containing compounds [9, 20]. I t was interesting to compare the A8 calculated from eqn. (5a) with the experimental value. Unfortunately there are very few reports dealing with this subject where polymers are concerned, and we therefore restricted the comparison to only two polymers from within series 1, i.e. polystyrene and polyvinylchloride (PVC). Using the values listed in the Table, eqn. (5) gave A ~ 1 . 7 1 and 1.85 kcal/mole respectively for these 2 polymers. This agreed reasonably well with values of 1-66~0.2 (polystyrene [12]) and 1-2 kcal/mole (PVC [30-32]), which had been determined from the NMR [29] and infrared spectra [30-32 I respectively. Finally, it should be noted t h a t the empirical functions produced by us can be used only with polymers showing relatively weak reactions of the dispersionary (polyethylene, polypropylene) or orientational type (PVC, polyacrylonitrile); it is obvious that the Tg of polymers with specific types of molecular interactions will chiefly depend on the concentrations of functional groups capable of such reactions [33, 34]. Furthermore, steric factor a will not always be an absolute measure of chain rigidity in the unit (block). This applies particularly to polymers containing aromatic rings in the main chain. The a of such polymers is usually small [35, 36], while the Tg can reach fairly large values [37]. This behaviour is linked with the differences between the statistical chains of the polymers examined in this study, and those having phenyl rings in the main chain, which have a symmetrical function of the energy potential of internal rotation around the C--O bonds [36]. I t appears t h a t other equations will be more suitable [38-40]. The authors express their thanks to A. A. Asl~adskii for valuable criticism. CONCLUSIONS
(1) On the basis of an analysis of literature data it is shown that the glass temperature (Tg) of polymers with molecular interactions of a dispersionary or orientational type will chiefly depend on the value of steric factor a, which reflects the thermodynamic chain flexibility. (2) The equations arrived at on the basis of the functions were found to agree with the energetical parameters of the respective low- and high-molecular weight substances. Translated by K. A. AImE~ REFERENCES
1, T. G. FOX and P. J. FLORY, J. Appl. Phys. 21: 581, 1950 2. D. TURNBULL and M. H. COHEN, J. Chem. Phys. 31: 1164, 1959; 34: 120, 1961 3. N. I~RAI and H. EYRING, J. Polymer Sei. 37: 51, 1959 4. R. SIMHA and R. F. BOYER, g. Chem. Phys. 37: 1003, 1962 5. J. H. GIBBS and E. A. DiMARZIO, J. Chem. Phys. 28: 373, 1958
3080
V. P. P ~ v ~ o
and Yu. S. L~ATov
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