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The relationship between volume and elasticity in polymer glasses
The relationship between volume and elasticity in polymer glasses R. N. HAWARD and J. R. MACCALLUM It is proposed that the adiabatic bulk modulus of a polymeric glass is determined by interchain forces which obey conventional intermolecular force relationships. Based on the Lennard-Jones expression for the intermolecular forces in liquids an equation is derived relating the bulk modulus and the volume of the polymer glass relative to its volume at the condition of zero expansion volume. The relationship deduced is tested on data available for poly(methyl methacrylate), polystyrene, polyethylene, and poly(vinyl chloride).
SOME YEARS ago it was suggested that the adiabatic bulk modulus of an organic glass (generally of the order of 2-5 × l0 4 bar*) was determined by the deformation of carbon-carbon bonds 1. More recently calculations have shown that the modulus associated with bond distortion is some two orders of magnitude higher than the Young's or adiabatic bulk modulus2, 3. Just prior to Tuckett's postulate Muller had measured the bulk modulus of hydrocarbon crystals both along and perpendicular to the axis of the molecular chain, obtaining values of 3.3 x l0 6 bar and 3 . 3 - 0.8 × l0 ~ bar respectively 4. Measurements of a similar nature have been made on oriented crystals of macromolecules. Thus a value of 2.4 x 106 bar has been obtained for the modulus due to bond extension for polyethylene ~ which compares well with the figure reported for the low molecular weight analogues 4. Robertson and Wilson examined a sample of polycarbonate and obtained a value of about 1.7 × l0 ~ bar for the elastic modulus along the polymer chains 6. They found that this longitudinal modulus showed little variation with temperature, whereas the temperature dependence of the transverse modulus was considerable. These authors concluded that the modulus for a sample of unoriented polycarbonate was almost totally due to the interchain modulus. In view of these observations it is now proposed that the adiabatic bulk modulus of an isotropic organic glass, in which the molecular chains are randomly oriented, is largely determined by the interchain modulus. This being so it is to be expected that the interchain forces follow conventional inter-molecular force relationships. The concept that the modulus is determined by intermolecular forces is not universally accepted and recently it has been suggested that bond rotations are involved 6. This view was supported by arguments based on the relation between volume and elasticity. In this paper an accepted model for describing intermolecular forces is used to relate volume and elasticity, and some experimental data examined in the light of the derived relationship. * 1 bar = 105Nm-z 189
R.
N.
HAWARD
AND
J. R.
MAcCALLUM
Intermolecular .forces The most commonly used expression which describes the intermolecular forces in a non-polar liquid is that of kennard-Jones 8. It can be written in the form A B U = - ~ / 3 - Vm/3 (1) in which U is the intermolecular potential for the system of molar volume V, A, B, n, and m are constants. (n and m usually have values of 9 and 6, or 12 and 6 respectively). Equation (1) can be rewritten thus u
=
g
--
m
(2) ,
in which K is a constant and V0 is the molar volume of the system for the condition that (dU/dV)y vo = 0. Vo can be considered as the molar volume when the expansion volume is zero 9. Since the adiabatic bulk modulus, B, e q u a l s - V(d2U/dV2)s, equation (2) can be used to derive the following relationship between B and V. Bo ( B--(n--m)l(n+
/Vo~ "+3 /V0~-m+3~ 3)_~-~]_ 3 _ ( m + 3 ) ~ ) 3 _t
(3)
B0 is the modulus corresponding to the volume V0. Recently, Haward has discussed the methods available for estimating V0 for liquids and polymers 9. The validity of equation (3) can be verified for various combinations ofn and m using a computed value for Vo, and measured values for B and V. Although the Lennard-Jones type potential was originally derived for liquids it should also be applicable to non-polar polymeric glasses in which thermal fluctuation of the molecular segments are minimal and inter-chain forces are of the van der Waals type. Some applications of equation (3) are described below.
Poly (methyl methacrylate) Equation (3) can be applied to data obtained by Asay, Lamberson and Guenther a°. Figure 1 shows a plot of B against V for a 12:6 potential. V0 was calculated by assuming Vo/V= 0-87 at Tg, with Vg, = 87.0 cm 3 tool -1, resulting in V0 = 75.7 cm z mo1-1 11. An alternative method for estimating Vo is the bond additive values derived by Biltz and by Sugden which yield values of 78.7 and 78-9 cm 3 mo1-1 respectively9. Using these values makes little difference to the linearity of the plot which, however, can be marginally improved by using a 9:6 potential. For the purposes of this paper, however, it is adequate to consider only the more generally used 12:6 potential. An alternative procedure can be employed for estimating B0. Ramon Rao compiled a set of atomic increments whereby the velocity of sound through a liquid can be calculated for a given density a2. The applicability of this technique has been confirmed for a series of hydrocarbons 13. Thus it is possible to estimate Bo for a polymeric glass using equations (4) and (5). S0 a/3 = k p0 190
(4)
VOLUME A N D ELASTICITY IN POLYMER GLASSES
So is the longitudinal velocity of sound, k is a constant which can be calculated using tabulated values for constituent atoms, and p0 is the density when the expansion volume is zero. Using conventional equations for So and for the relations between elastic constants, it follows that So" -
3B0 (1 ~) po (1 + (~)
(5)
where o is Poisson's ratio and is assumed to be 0.33 for a polymeric glass 14. The application of equation 4 and 5 to poly(methyl methacrylate) results in a value of 5.90 x l04 for B. F r o m Figure 1 Bo is calculated to be 14 ~ l04.
1000 MN/m2 O • {11
1.6 1"5
800MN/m20io(
1-4 ~-1.3 1.2
500MN/m2%
/
m1-1
-~ ~.0 ~3 ~0 09
F: 0-8
#
/0,®M./m 2 0'6
O MN/ma
0-5 0
|
2
I
1
i
t
3 /. 5 6 15 (V/Vo) 5 _ 9 {V/Vo) a
1
7
Ft:gure 1 A plot of B, the adiabatic bulk modulus, versus 15(V/V.) a 9(V/V~)):* for data on poly(methyl methacrylate) TM. The values were obtained at different temperatures, (' 25 °C, • 40c'C, E7 55°C, • 75r~C, and at the pressures indicated alongside the points The two quantities are of the right magnitude which suggests that equations 4 and 5 provide a useful means of estimating the approximate value of B0 for a given polymer glass.
Polystyrene Haward has proposed a modified van der Waals equation which makes it possible to estimate the isothermal compressibility at the volume Vo, and by applying this equation to data available for polystyrene he estimated a value o f 8.0 x 10 6 bar-a for this parameter 15. It is very reasonable to assume that the isothermal and adiabatic bulk moduli are equal under the conditions o f 191
R. N. HAWARD AND J. R. MACCALLUM zero expansion volume and therefore a value of 12.5 × 104 bar is obtained for Bo. This is very close to the measured B0 for poly(methyl methacrylate). Using this value for B0 and equation (3) it is possible to compare calculated and observed data on the bulk modulus. Gee has proposed a value of 3-0 × 104 bar for the experimentally determined isothermal bulk modulus at the Tg 16 and, assuming the difference between the isothermal and adiabatic moduli is small in the glassy state, this can be taken as B. It has been shown that at Tg Vo/V ~-- 0-8717, and substituting this value with the computed value of Bo in equation 3 results in a calculated B of 3-2 × 104 bar. Warfield et al measured B at room temperature to be 5.4 × 104 bar is, taking the density of polystyrene at this room temperature as 1.051 g cm-3,19 and using Breuer, Haward and Rehage's value for p017 we calculate B to be equal to 4.2 × 104 bar. In both examples the agreement between calculated and observed modulus is good. Using Rao's approach, i.e. equations (4) and (5), B0 is calculated to be 4.5 × 104 bar which, as for poly(methyl methacrylate), is between one half and one third of the measured value.
Polyethylene Using Doolittles value for P0, 1 g cm -3 20, and equation (4) and (5) a value of 4.5 × l04 bar is obtained for B0. Thus from equation (3) B is calculated to equal 0.8 × 104 bar for amorphous polyethylene with p = 0.85 g cm -3. Data for experimentally determined values of Young's modulus, determined from sound velocity measurements at a frequency of 2m cycle/s, which with = 0.33 is almost equal to B, are somewhat scattered, but a value of B 2 × 104 bar can be deducted for amorphous polyethylene21. The difference between observed and calculated values is most likely to be due to the uncertainty in evaluating B0 by Rao's method.
Poly(vinyl chloride) po calculated from Sugden's atomic increment values 22 is 1-0502g cm -3 and p at Tg was estimated from published data 19. Bo equal to 8.7 × 104 bar was calculated as previously, and thus, from equation (3), B is found to be 1.7 × 104 bar. The experimentally determined modulus with which this value can be compared is the isothermal bulk modulus which is equal to 3.3 × 104 bar at Tg 16. Assuming the adiabatic and isothermal bulk moduli are almost equal for a glass then the measured modulus is almost twice that calculated using B0 estimated by Rao's method. This difference is similar to that noted previously and indicates that Rao's method gives a useful, but rather low, guide to the approximate value of B0.
DISCUSSION The variation of bulk modulus with volume for poly(methyl methacrylate) can be satisfactorily accounted for using a Lennard-Jones type expression for the inter-molecular forces. Indeed the bulk moduli for the few polymers for 192
V O L U M E A N D ELASTICITY IN P O L Y M E R GLASSES
which data are available appear to depend on inter-molecular forces. Previous work has been concerned with crystalline polymers rather than glasses. It is of interest to note that the modulus for deformation of a C - C bond in an oriented crystallite is o f the order of the modulus for diamond. Graphite which may be considered as a macromolecule of a rather special type has been examined by Girifalco and Lad 23, who used the Lennard-Jones a p p p r o a c h to calculate the inter-planar modulus and obtained good agreement between this value (3.15 × 10~ bar) and that measured experimentally (3-37 × 105 bar). They concluded that the decrease in volume with increasing pressure was due to compression of inter-layer spacing. It would seem to be reasonable to conclude that the bulk modulus of a polymeric material, crystalline or glassy, especially when measured at high frequencies, is determined by inter-chain forces and is not dependent on the bonding along the macromolecule. However, where there is an appreciable dependence o f modulus on frequency 24 and, especially at long times under tension as in a creep experiment, local molecular adjustment including bond rotations will of course take place leading to a decrease in modulus at long times. The dependence of the thermal expansion of polymeric materials on inter rather than intra-chain forces has also been noted2L
University o f Manchester Institute ~?f Science and Technology, Sackville Street, Manchester 1, U K
R. N. Haward
Department o f Chemistry, The University, St. Andrews, Scotland
J. R. MacCallum (Received 7 September 1970)
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Tuckett, R. F. Chem. andlnd. 1943, 62, 430 Treloar, L. R. G. Polymer, Lond. 1960, 1, 95 Asahini, M., Enomoto, S. and Shimanouchi, T. J. Polymer Sci. 1962, 59, 93 Muller, A. Proc. Roy. Soc. 178A, 1941, 227 Sakurada, i., Ito, T. and Nakamae, K. Makromol. Chem. 1964, 75, 1 Robertson, R. E. and Wilson, C. M. Polymer Letters, 1965, 3, 427 Bowden, P. B. Polymer, Lond. 1968, 9, 449 Lennard-Jones, J. E. Proc. Roy. Soc. (A196) 1924, 463 Haward, R. N. Reviews Macromol. Chem. (C4) 1970, 45 Asay, J. R., Lamberson, D. L. and Guenther, A. H. J. Appl. Phys. 1969.40, 1768 Bondi, A. J. Polymer Sci. (A2) 1964, 3159 Rao Raman, M. J. Chem. Phys. 1941, 9, 682 Bakhshi, N. N. and Parthasawathy, S. J. Phys. Chem. 1953, 57, 453 Neilson, L. E., 'The Mechanical Properties of Polymers', Rheinhold, New York, 1962 Haward, R. N. J. Polymer Sci. (A2) 1969, 7, 219 Gee, G. Polymer, Lond. 1966, 7, 177 Breuer, J., Haward, R. N. and Rehage, G. Polymer Letters 1966, 4, 375 Barnet, F. R., Cuevas, J. E. and Warfield, R. W. J. Appl. Polymer Sci. 1968, 12, 1147 193
R. N. HAWARD AND J. R. MAcCALLUM 19 20 21 22 23 24
Lewis, O. C., 'Physical Constants of Linear Homopolymers', Springer, Berlin, 1968 Doolittle, A. K. J. Appl. Phys. 1951, 22, 1471 Schuyer, J. J. Polymer Sci 1959, 36, 475 Sugden, S. J. Chem. Soc. 1927, 1786 Girifalco, L. A. and Lad, R. A. J. Chem. Phys. 1956, 25, 693 Turner, S,, 'Testing of Polymers', (Ed. W. E. Brown), VoL 4., Interscience, 1969, pp 1, 73 25 Swan, P. R. J. Polymer Sci. 1962, 56, 403
194
SOME YEARS ago it was suggested that the adiabatic bulk modulus of an organic glass (generally of the order of 2-5 × l0 4 bar*) was determined by the deformation of carbon-carbon bonds 1. More recently calculations have shown that the modulus associated with bond distortion is some two orders of magnitude higher than the Young's or adiabatic bulk modulus2, 3. Just prior to Tuckett's postulate Muller had measured the bulk modulus of hydrocarbon crystals both along and perpendicular to the axis of the molecular chain, obtaining values of 3.3 x l0 6 bar and 3 . 3 - 0.8 × l0 ~ bar respectively 4. Measurements of a similar nature have been made on oriented crystals of macromolecules. Thus a value of 2.4 x 106 bar has been obtained for the modulus due to bond extension for polyethylene ~ which compares well with the figure reported for the low molecular weight analogues 4. Robertson and Wilson examined a sample of polycarbonate and obtained a value of about 1.7 × l0 ~ bar for the elastic modulus along the polymer chains 6. They found that this longitudinal modulus showed little variation with temperature, whereas the temperature dependence of the transverse modulus was considerable. These authors concluded that the modulus for a sample of unoriented polycarbonate was almost totally due to the interchain modulus. In view of these observations it is now proposed that the adiabatic bulk modulus of an isotropic organic glass, in which the molecular chains are randomly oriented, is largely determined by the interchain modulus. This being so it is to be expected that the interchain forces follow conventional inter-molecular force relationships. The concept that the modulus is determined by intermolecular forces is not universally accepted and recently it has been suggested that bond rotations are involved 6. This view was supported by arguments based on the relation between volume and elasticity. In this paper an accepted model for describing intermolecular forces is used to relate volume and elasticity, and some experimental data examined in the light of the derived relationship. * 1 bar = 105Nm-z 189
R.
N.
HAWARD
AND
J. R.
MAcCALLUM
Intermolecular .forces The most commonly used expression which describes the intermolecular forces in a non-polar liquid is that of kennard-Jones 8. It can be written in the form A B U = - ~ / 3 - Vm/3 (1) in which U is the intermolecular potential for the system of molar volume V, A, B, n, and m are constants. (n and m usually have values of 9 and 6, or 12 and 6 respectively). Equation (1) can be rewritten thus u
=
g
--
m
(2) ,
in which K is a constant and V0 is the molar volume of the system for the condition that (dU/dV)y vo = 0. Vo can be considered as the molar volume when the expansion volume is zero 9. Since the adiabatic bulk modulus, B, e q u a l s - V(d2U/dV2)s, equation (2) can be used to derive the following relationship between B and V. Bo ( B--(n--m)l(n+
/Vo~ "+3 /V0~-m+3~ 3)_~-~]_ 3 _ ( m + 3 ) ~ ) 3 _t
(3)
B0 is the modulus corresponding to the volume V0. Recently, Haward has discussed the methods available for estimating V0 for liquids and polymers 9. The validity of equation (3) can be verified for various combinations ofn and m using a computed value for Vo, and measured values for B and V. Although the Lennard-Jones type potential was originally derived for liquids it should also be applicable to non-polar polymeric glasses in which thermal fluctuation of the molecular segments are minimal and inter-chain forces are of the van der Waals type. Some applications of equation (3) are described below.
Poly (methyl methacrylate) Equation (3) can be applied to data obtained by Asay, Lamberson and Guenther a°. Figure 1 shows a plot of B against V for a 12:6 potential. V0 was calculated by assuming Vo/V= 0-87 at Tg, with Vg, = 87.0 cm 3 tool -1, resulting in V0 = 75.7 cm z mo1-1 11. An alternative method for estimating Vo is the bond additive values derived by Biltz and by Sugden which yield values of 78.7 and 78-9 cm 3 mo1-1 respectively9. Using these values makes little difference to the linearity of the plot which, however, can be marginally improved by using a 9:6 potential. For the purposes of this paper, however, it is adequate to consider only the more generally used 12:6 potential. An alternative procedure can be employed for estimating B0. Ramon Rao compiled a set of atomic increments whereby the velocity of sound through a liquid can be calculated for a given density a2. The applicability of this technique has been confirmed for a series of hydrocarbons 13. Thus it is possible to estimate Bo for a polymeric glass using equations (4) and (5). S0 a/3 = k p0 190
(4)
VOLUME A N D ELASTICITY IN POLYMER GLASSES
So is the longitudinal velocity of sound, k is a constant which can be calculated using tabulated values for constituent atoms, and p0 is the density when the expansion volume is zero. Using conventional equations for So and for the relations between elastic constants, it follows that So" -
3B0 (1 ~) po (1 + (~)
(5)
where o is Poisson's ratio and is assumed to be 0.33 for a polymeric glass 14. The application of equation 4 and 5 to poly(methyl methacrylate) results in a value of 5.90 x l04 for B. F r o m Figure 1 Bo is calculated to be 14 ~ l04.
1000 MN/m2 O • {11
1.6 1"5
800MN/m20io(
1-4 ~-1.3 1.2
500MN/m2%
/
m1-1
-~ ~.0 ~3 ~0 09
F: 0-8
#
/0,®M./m 2 0'6
O MN/ma
0-5 0
|
2
I
1
i
t
3 /. 5 6 15 (V/Vo) 5 _ 9 {V/Vo) a
1
7
Ft:gure 1 A plot of B, the adiabatic bulk modulus, versus 15(V/V.) a 9(V/V~)):* for data on poly(methyl methacrylate) TM. The values were obtained at different temperatures, (' 25 °C, • 40c'C, E7 55°C, • 75r~C, and at the pressures indicated alongside the points The two quantities are of the right magnitude which suggests that equations 4 and 5 provide a useful means of estimating the approximate value of B0 for a given polymer glass.
Polystyrene Haward has proposed a modified van der Waals equation which makes it possible to estimate the isothermal compressibility at the volume Vo, and by applying this equation to data available for polystyrene he estimated a value o f 8.0 x 10 6 bar-a for this parameter 15. It is very reasonable to assume that the isothermal and adiabatic bulk moduli are equal under the conditions o f 191
R. N. HAWARD AND J. R. MACCALLUM zero expansion volume and therefore a value of 12.5 × 104 bar is obtained for Bo. This is very close to the measured B0 for poly(methyl methacrylate). Using this value for B0 and equation (3) it is possible to compare calculated and observed data on the bulk modulus. Gee has proposed a value of 3-0 × 104 bar for the experimentally determined isothermal bulk modulus at the Tg 16 and, assuming the difference between the isothermal and adiabatic moduli is small in the glassy state, this can be taken as B. It has been shown that at Tg Vo/V ~-- 0-8717, and substituting this value with the computed value of Bo in equation 3 results in a calculated B of 3-2 × 104 bar. Warfield et al measured B at room temperature to be 5.4 × 104 bar is, taking the density of polystyrene at this room temperature as 1.051 g cm-3,19 and using Breuer, Haward and Rehage's value for p017 we calculate B to be equal to 4.2 × 104 bar. In both examples the agreement between calculated and observed modulus is good. Using Rao's approach, i.e. equations (4) and (5), B0 is calculated to be 4.5 × 104 bar which, as for poly(methyl methacrylate), is between one half and one third of the measured value.
Polyethylene Using Doolittles value for P0, 1 g cm -3 20, and equation (4) and (5) a value of 4.5 × l04 bar is obtained for B0. Thus from equation (3) B is calculated to equal 0.8 × 104 bar for amorphous polyethylene with p = 0.85 g cm -3. Data for experimentally determined values of Young's modulus, determined from sound velocity measurements at a frequency of 2m cycle/s, which with = 0.33 is almost equal to B, are somewhat scattered, but a value of B 2 × 104 bar can be deducted for amorphous polyethylene21. The difference between observed and calculated values is most likely to be due to the uncertainty in evaluating B0 by Rao's method.
Poly(vinyl chloride) po calculated from Sugden's atomic increment values 22 is 1-0502g cm -3 and p at Tg was estimated from published data 19. Bo equal to 8.7 × 104 bar was calculated as previously, and thus, from equation (3), B is found to be 1.7 × 104 bar. The experimentally determined modulus with which this value can be compared is the isothermal bulk modulus which is equal to 3.3 × 104 bar at Tg 16. Assuming the adiabatic and isothermal bulk moduli are almost equal for a glass then the measured modulus is almost twice that calculated using B0 estimated by Rao's method. This difference is similar to that noted previously and indicates that Rao's method gives a useful, but rather low, guide to the approximate value of B0.
DISCUSSION The variation of bulk modulus with volume for poly(methyl methacrylate) can be satisfactorily accounted for using a Lennard-Jones type expression for the inter-molecular forces. Indeed the bulk moduli for the few polymers for 192
V O L U M E A N D ELASTICITY IN P O L Y M E R GLASSES
which data are available appear to depend on inter-molecular forces. Previous work has been concerned with crystalline polymers rather than glasses. It is of interest to note that the modulus for deformation of a C - C bond in an oriented crystallite is o f the order of the modulus for diamond. Graphite which may be considered as a macromolecule of a rather special type has been examined by Girifalco and Lad 23, who used the Lennard-Jones a p p p r o a c h to calculate the inter-planar modulus and obtained good agreement between this value (3.15 × 10~ bar) and that measured experimentally (3-37 × 105 bar). They concluded that the decrease in volume with increasing pressure was due to compression of inter-layer spacing. It would seem to be reasonable to conclude that the bulk modulus of a polymeric material, crystalline or glassy, especially when measured at high frequencies, is determined by inter-chain forces and is not dependent on the bonding along the macromolecule. However, where there is an appreciable dependence o f modulus on frequency 24 and, especially at long times under tension as in a creep experiment, local molecular adjustment including bond rotations will of course take place leading to a decrease in modulus at long times. The dependence of the thermal expansion of polymeric materials on inter rather than intra-chain forces has also been noted2L
University o f Manchester Institute ~?f Science and Technology, Sackville Street, Manchester 1, U K
R. N. Haward
Department o f Chemistry, The University, St. Andrews, Scotland
J. R. MacCallum (Received 7 September 1970)
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Tuckett, R. F. Chem. andlnd. 1943, 62, 430 Treloar, L. R. G. Polymer, Lond. 1960, 1, 95 Asahini, M., Enomoto, S. and Shimanouchi, T. J. Polymer Sci. 1962, 59, 93 Muller, A. Proc. Roy. Soc. 178A, 1941, 227 Sakurada, i., Ito, T. and Nakamae, K. Makromol. Chem. 1964, 75, 1 Robertson, R. E. and Wilson, C. M. Polymer Letters, 1965, 3, 427 Bowden, P. B. Polymer, Lond. 1968, 9, 449 Lennard-Jones, J. E. Proc. Roy. Soc. (A196) 1924, 463 Haward, R. N. Reviews Macromol. Chem. (C4) 1970, 45 Asay, J. R., Lamberson, D. L. and Guenther, A. H. J. Appl. Phys. 1969.40, 1768 Bondi, A. J. Polymer Sci. (A2) 1964, 3159 Rao Raman, M. J. Chem. Phys. 1941, 9, 682 Bakhshi, N. N. and Parthasawathy, S. J. Phys. Chem. 1953, 57, 453 Neilson, L. E., 'The Mechanical Properties of Polymers', Rheinhold, New York, 1962 Haward, R. N. J. Polymer Sci. (A2) 1969, 7, 219 Gee, G. Polymer, Lond. 1966, 7, 177 Breuer, J., Haward, R. N. and Rehage, G. Polymer Letters 1966, 4, 375 Barnet, F. R., Cuevas, J. E. and Warfield, R. W. J. Appl. Polymer Sci. 1968, 12, 1147 193
R. N. HAWARD AND J. R. MAcCALLUM 19 20 21 22 23 24
Lewis, O. C., 'Physical Constants of Linear Homopolymers', Springer, Berlin, 1968 Doolittle, A. K. J. Appl. Phys. 1951, 22, 1471 Schuyer, J. J. Polymer Sci 1959, 36, 475 Sugden, S. J. Chem. Soc. 1927, 1786 Girifalco, L. A. and Lad, R. A. J. Chem. Phys. 1956, 25, 693 Turner, S,, 'Testing of Polymers', (Ed. W. E. Brown), VoL 4., Interscience, 1969, pp 1, 73 25 Swan, P. R. J. Polymer Sci. 1962, 56, 403
194