- Email: [email protected]
The thermoelasticity of natural rubber in torsion
The thermoelasticity of natural rubber in torsion P. H. BOYCE and L. R. G. TRELOAR Following a recent theoretical treatment of the thermoelasticity of rubber in torsion t, an apparatus has been devised to measure the stress/temperature coefficients of elastomers in torsion. The experiments carried out enable the temperature coefficient of the mean-square chain vector length r02 to be derived. For natural rubber a value for d In ro2/dT of 0"43 ~ 0"05 /. 1@3 deg 1 has been obtained; this compares favourably with values obtained by other workers from stresstemperature data for simple extension. No significant variation in this figure was obtained with variations of strain or crosslink density. UP TO the present time most of the work that has been done on the thermoelasticity of elastomers has been confined to the case of simple extension. On the basis of the Gaussian statistical theory, Flory" has derived the following relation for simple extension
.f = (vkT/lO (r~2/ro 2)
(a --
a
'-')
I)
where f i s the tensile force, v is the total number of chains in the specimen, I g = V 1,'3, where V is the volume in the final strained state, a is the extension ratio referred to the length /i, and k and T have their usual meaning. The mean-square vector length of network chains in the undistorted (isotropic) state of volume V is denoted by r& whilst r0 u is the mean-square chain vector length for the same set of chains in the absence of the constraints introduced by the crosslinks. A recent analysis by Treloar I for the case of the torsion of a rubber cylinder gives the corresponding relation M = 0r/2) (vkT/Vu) (ri2/ro2)¢ao 4
(2)
In equation (2) M is the torsional couple, ¢ is the torsion (expressed in radians per unit length of the strained axis), V,, is the volume in the unstrained (i.e. stress-free) state, a0 is the unstrained radius and the remaining symbols have their previous meaning. The stress/temperature coefficients at constant pressure can be found by differentiating equations (1) and (2), taking into account the changes in dimensions of the rubber with temperature. For simple extension the result obtained is3 [a In (f/T)/~,T]p,z == -- d In ro~/dT -- fi/(a 3 -- 1)
(3)
where fi, the coefficient of volume expansion of the rubber, is V t(i~V/i~T)~,. 21
P. H. BOYCE and L. R. G. TRELOAR
The corresponding relation in torsion 1 is* [c~ In (M/T)/~T]v,z,¢~ :
-- d In ro2/dT q- fi
(4)
which can be expressed in the alternative form
(3M/~T)v,~,¢~ = (M/T) (1 + fiT -- T d In ro2/d T)
(4a)
In simple extension it is sometimes found convenient to express the internal energy component of the tensile stress at constant volume as fev which is defined as (~E/Ol)T,V. It can be shown 3 that
fev/f : -- T[~3In (f/T)/~T] v,t = T d In roZ/d T
(3a)
Similarly in the case of torsion we may introduce the quantity Mev, representing the internal energy contribution to the couple at constant volume, given by
Mev/M : -- T[c3 In (M/T)/~T]vd,¢~ : T d In roZ/d T
(5)
The quantities Mev/M and fev/f are thus seen to be strictly analogous. By using equation (4a) in conjunction with equation (5) the relative contribution, Mev/M, of the internal energy to the couple can be derived from stress-temperature experiments at constant pressure, and hence the quantity d In roZ/d T may be obtained. The aim of this work was to obtain a value of this quantity from torsional experiments on natural rubber, and to compare the value obtained with existing data for the case of simple extension. EXPERIMENTAL
Materials Natural rubber vulcanizates were made with various degrees of crosslinking by curing pale crfipe with 2.7-3-1% by weight of dicumyl peroxide for 30-55 min at 140°C. Cylinders were moulded of approximately 185 m m length and 6.5 m m diameter. No treatment was given to extract any uncrosslinked material or excess vulcanizing reactants which might have been present. Each cylinder was gripped at either end in a half-inch long cylindrical split clamp made from brass. Reference marks for the measurement of the axial extension ratio were made in ink on the surface of the cylinder. The length between the brass clamps, the diameter of the cylinder and the length between the reference marks were all measured with a travelling microscope.
Apparatus The apparatus used for the measurement of the torsional stress (couple) as a function of temperature at constant pressure, length and twist, is shown in Figure 1. In the design of this apparatus close attention was paid to the * In equation (4) fl is defined as Vu-a(OVu/ST)p. Since ( V -- Vu)/Vu is very s m a l l ( ~ 1 0 - 4 ) , this does not differ significantly from fl in equation (3).
22
THE THERMOELASTICITY OF NATURAL RUBBER IN TORSION
|
i
I I
I k.
-r,
II
®
'i
r,
LI
Figure 1. Schematic front view of apparatus The numbered parts represent : (1) sample (2) upper clamp (3) resistive probe (Type 5110 G. Sangamo-Weston) (17) temperaupper rod (4) worm gear (5) ture-indieator controller (Type lower clamp (6) silvel-steel rod 6003-2, Fielden) (18) variable (7) torsion drum (8) tension a.c. transformer (Type spring (9) threads (10) transmisV5HMTF, Variac) (19) lowsion bar (11) counterbalancing pitch screw adjustment for tenbar (12) ball-bearing pulleys sion spring (20) clamps attached (13) weights (14) stops (15) to main frame (21) main frame. heating chamber (16) thermo23
P. H. BOYCE
and
L. R. G. T R E L O A R
necessity for accurate control of the axial and torsional strains to which the sample was subjected. In carrying out the experiments, a given amount of twist was inserted into the sample by rotating the upper clamp via the upper rod which was connected to the worm gear. The lower clamp, which was rigidly attached by a silversteel rod to the torsion drum, was prevented from rotating about its axis by adjustment of the spring by which the tension in the attached threads was controlled. This tension was transmitted via the transmission bar as shown
Figure 2. Schematic diagram of torsional system (The numbered parts are referred to in Figure 1) in Figure 2; by maintaining the vertical position of this bar constant, the angular position of the drum was automatically fixed. The arrangement of the threads around the drum and their connection to the transmission bar and counterbalancing bar offered a stable torsional device in which the couple could be calculated from the measured extension of the spring. The frictional component of the stresses in the ball-bearing pulleys was found to be negligibly small. The angle of twist of the sample was controlled to within ~ 0 . 5 degrees. The couple was measured to within 4-0.1 g cm in the range 10 - 200 g cm. The sample was also given a small axial extension sufficient to maintain a positive tension and to eliminate buckling in the twisted state. The extension was controlled by adjustment of the weights applied to the top of the torsion 24
THE THERMOELASTICITY OF NATURAL RUBBER IN TORSION
drum. A guide to the vertical alignment and position of the drum was provided by the stops. The length of the sample was controlled to within ! 0.05 cm.
The heating chamber was constructed from a Pyrex glass tube which was wound with standard electrical-resistance wire. The temperature of the sample was controlled by an electronic temperature-indicator controller which was connected to a variable transformer which supplied the appropriate voltage to the resistance wire. The temperature at the centre of the sample was controlled to within ~0.5°C, but a maximum temperature difference of 2~C was detected along the length of the sample. However, this was not sufficient to create appreciable errors in the resulting stress-temperature data.
Procedure The sample was allowed to relax at 100°C for 1 to 1½ h under the maximum torsional strain to be used in subsequent experiments. Stress-temperature data were then recorded in the range 20 - 60°C. Stress readings were taken all 5 or 10°C intervals after sufficient time had been allowed for the attainment of thermal equilibrium; from preliminary experiments this was found to be about 10 min when, between each interval, the sample was heated or cooled at the rate of 1-3°C per min. The decreasing/increasing temperature cycle was repeated until effective reversibility was obtained. The procedure was then repeated in a similar manner at different lower values of the strain. but excluding the initial relaxation at 100°C. RESULTS AND CONCLUSIONS
Stress relaxation The effects of stress relaxation are shown in Figure 3. The total reduction in stress during the preliminary, high temperature relaxation and the two subsequent temperature cycles was about 101~. The periods of time that are shown at 30 and 60°C illustrate the difference between the rates of stress relaxation at these two temperatures. The time taken to obtain the complete curve illustrated was 4 days, compared with l0 h for each subsequent curve for the same sample at a lower strain.
Analysis oj' stress-temperature data A typical set of stress-temperature plots for one particular sample at different strains is shown in Figure 4. In all cases a reversible, linear relationship was found to exist between couple and temperature. The slope of any one line was reproducible to within 2~i for successive temperature cycles. It should be noted that the broken line plot (at ~ba0 0.690) was obtained from a repeated run, after the curves at the lower strains had been recorded. The small displacement of this plot from the bold line curve signified that, although reversibility had been achieved within the time of one run, thermodynamic reversibility had not been achieved over the total time of the experiment. 25
and
P. H. BOYCE
L. R. G. T R E L O A R
140
ff S ~
130
SjSS
E u
pSSS ~ sSP~/ Xi
o) 120
,." /
e~
XSS
o
~X
Xt
/l ,sh .,~J
110
100 l
I
20
40
I
60 Temp. [°C]
1
I
80
100
Figure 3. Typical stress-relaxation plot for natural rubber in torsion O, decreasing temperature,
×, increasing temperature
However, the value of the stress-temperature coefficient had not changed significantly. The quantity (T/M)~M/~T [equation (4a)] was calculated from relative stress-temperature data for a temperature of 293°K. The resulting data are included in Table 1.
Effects o f strain and crosslink density upon MevlM In Table 1 the torsional strain is expressed in terms of the parameter ~ba0 corresponding to n turns of twist in the total length of the sample; a0 and ~b are defined as in equation (2). The crosslinking density is expressed in terms of G, the shear modulus, which was calculated from the reversible stress-temperature data at 20°C by use of the equation
G = 2M/rr~bao4 Both ~ba0 and A, the extension ratio, were referred to the original, unstrained dimensions at 20°C. Mev/M was calculated at 293°K from the (T/M) c~M/c~T values using equations (4a) and (5). The coefficient of volume expansion,/3, for natural rubber was taken as 6-6 x 10 -4 deg -1. 26
THE T H E R M O E L A S T I C I T Y OF N A T U R A L R U B B E R IN TORSION
90
u
70
f x
~
~
C
50 ...,...,--..~ d
,.__t-.---30 i
i
[
20
40 Temp.("C ]
60
Figure 4. Stress-temperature plots at different torsional strains C), decreasing temperature, x , increasing temperature Torsional strain parameter ~bao- (a) 0-888, (b) 0.690, (c) 0.492 (d) 0.393, (e) 0.297
27
P. H. BOYCE and L. R. G. TRELOAR
Table 1 Thermoelastic data for natmaI rubber in torsion
[kg c m -2] 2.1
(~ = 1'25)
turin'
[rad]
M \ e T / p,t,~
[ %]
9
0.888
1-06
13- 3
7 5
0.690 0.492
1.06 1-07
13'3 12.3
4 3
0.393 0.297
1.08 1.09
11. 3 10'3
3.0
7
0.683
1-04
(,~ = 1"19)
6 5
0'586 0.488
1"07 1.07
15"3 12" 3 12' 3
6
0"633
1"07
12' 3
(,~ = 1-15)
3-6
5.5 5
0.580 0.526
1.05 1.06
14"3 13-3
3.7
6.5
0.696
1.05
14'3
6
0.642
1.07
12. 3
5.5
0.589
1.10
9"3
(,~ = 1"13)
The average value of the quantity Mev/M, obtained from the figures in Table 1, is 12.6 ± 1.6~. The relative error in Mev/M is, of course, much larger than the relative error in (T/M)~M/OT (eight times as large); there is, however, no indication of any significant dependence of Mev/M on either the crosslink density or the amount of the torsional strain. From the mean value of Mev/M the resulting value of d In roZ/dT for natural rubber at 20°C is found to be 0.43 -- 0.05 × 10 -2 deg -1.
Comparison with simple extension data Although there is no difficulty in principle in carrying out comparable thermoelastic measurements in simple extension, the accuracy of the values of d In ro2/dT determined from such experiments is generally lower, owing to the greater sensitivity of the stress-temperature coefficient in extension to volume changes, discussed in an earlier paper 1. The presence of the term --fl/(a~ -- 1) in equation (3) implies a sensitivity to the absolute value of the strain, and hence to the precision with which the unstrained length can be determined, which is absent in the corresponding equation (4) for torsion. As a result the errors in the determination of the internal energy contribution to the tensile force fev/f (and hence d In ro2/dT) increase progressively as the strain decreases. It is presumably for this reason that published results show no agreement on the question of whether or not the internal energy contribution to the stress, fev/f, is dependent upon the strain; some authors 7,10,13,14,15 have reported an increase in this quantity as the strain becomes smaller, while others4,~,9,~v,is have found no significant change. In one case t6 a decrease has been reported. Table 2 gives some of the published data obtained from simple extension experiments; in cases where the figures varied according to the strain the values quoted refer to the region of higher strains, where 28
THE THERMOELASTICITY OF NATURAL RUBBER 1N TORSION t h e v a r i a t i o n s w e r e g e n e r a l l y r e d u c e d t o a m i n i m u m . It is s a t i s f a c t o r y t o n o t e t h a t t h e p r e s e n t r e s u l t f o r M e v / M in t o r s i o n , n a m e l y ,~~-,,/o, ~ " / is w i t h i n t h e r a n g e c o v e r e d b y t h e d a t a f o r t h e e q u i v a l e n t quantityJ~,~,/Ji
Table 2 Published values offe,,/f for natural rubber for thermoelastic measurements in simple extension.
/~,/f
Temperature
[%1
VC]
17
0 - 70
Ciferri 4
13
0 - 70
Ciferri, Hoeve and Flory s
20 20
45 30
Roe and Krigbaum 7 Allen, Bianchi and Price s
12.5
20 - 60
18 18
15
30 25
-
30
70
Authors
Method oJ determination
Yamamoto, Kusamizu and Fujita 9 Shen, McOuartie and Jackson t0 Barrie and Standen 1J Shen and Blatz 12
Constant pressure, equation (3) Calculated from experimental data of Wood and Roth ~ Constant pressure, equation (3) Constant volume - direct measurement of (~f/aT)v,z Constant pressure, modified Mooney form of equation (3) Con.~tant pressure, equation (3) Constant pressure-length v. temperature measurements at constant load, equation (3) Constant pressure, usingJ~.,.//'= 1 T(dln G/dT - fiT/3
It is c o n c l u d e d t h a t t h e t o r s i o n a l m e t h o d p r o v i d e s a c o n v e m e n t a n d a c c u r ate m e a n s o f d e t e r m i n i n g t h e q u a n t i t y d In roZ/dT w h i c h p o s s e s s e s d i s t i n c t experimental advantages over the more usual method of simple extension.
ACKNOWLEDGEMENTS T h e a u t h o r s w i s h to t h a n k M r G . D a l t o n o f t h i s D e p a r t m e n t the design of part of the apparatus.
f o r his h e l p in
Department o f Polymer and Fibre Science, University o f Manchester Institute o f Science and Technology
(Receil,ed I August 1969)
REFERENCES I 2 3 4 5 6 7 8
Treloar, L. R. G. PolymerLond., 1969, 10, 291 Flc.ry, P. J. J. Amer. Chem. Soe. 1956, 78, 5222 Flory, P. J. Trans Faraday Soc. 1961, 57, 829 Ciferri, A. Die Makromol. Chem. 1961,43, 152 Ciferri, A., Hoeve, C. A. J. and Flory, P. J. J. Amer. Chem. Soe. 1961,83, 1015 Wood, L. A. and Roth, F. L. J. appl. Physics 1944, 15, 781 Roe, R. J. and Krigbaum, W. R. J. Polymer Sci. 1962, 61, 167 Allen, G., Bianchi, U. and Price, C. Trans Faraday Soc. 1963, 59, 2493 29
P. H. BOYCE a n d L. R. G. TRELOAR 9 10 11 12 13 14 15 16 17 18
Yamamoto, K., Kusamizu, S. and Fujita, H. Die Makromol. Chem. 1966, 99, 212 Shen, M. C., McQuarrie, D. A. and Jackson, J. L. J. appl. Physics 1967, 38, 791 Barrie, J. A. and Standen, J. PolymerLond. 1967, 8, 97 Shen, M. C. and Blatz, P. J. J. appL Physics 1968, 39, 4937 Roe, R. J. and Krigbaum, W. R.J. Polym. Sci. (A) 1963, 1, 2049 Crespi, G. and Flisi, U. Die Makromol. Chem 1963, 60, 191 Opschoor, A. and Prins, W. J. Polym. Sci. (c) 1967, 16, 1095 Haly, A. R. J. Polym. Sci. (A) 1965, 3, 3331 Ciferri, A. Trans Faraday Soc. 1961,57, 846 Mark, J. E. and Flory, P. J.J. Amer. Chem. Soe. 1964, 86, 138
30
.f = (vkT/lO (r~2/ro 2)
(a --
a
'-')
I)
where f i s the tensile force, v is the total number of chains in the specimen, I g = V 1,'3, where V is the volume in the final strained state, a is the extension ratio referred to the length /i, and k and T have their usual meaning. The mean-square vector length of network chains in the undistorted (isotropic) state of volume V is denoted by r& whilst r0 u is the mean-square chain vector length for the same set of chains in the absence of the constraints introduced by the crosslinks. A recent analysis by Treloar I for the case of the torsion of a rubber cylinder gives the corresponding relation M = 0r/2) (vkT/Vu) (ri2/ro2)¢ao 4
(2)
In equation (2) M is the torsional couple, ¢ is the torsion (expressed in radians per unit length of the strained axis), V,, is the volume in the unstrained (i.e. stress-free) state, a0 is the unstrained radius and the remaining symbols have their previous meaning. The stress/temperature coefficients at constant pressure can be found by differentiating equations (1) and (2), taking into account the changes in dimensions of the rubber with temperature. For simple extension the result obtained is3 [a In (f/T)/~,T]p,z == -- d In ro~/dT -- fi/(a 3 -- 1)
(3)
where fi, the coefficient of volume expansion of the rubber, is V t(i~V/i~T)~,. 21
P. H. BOYCE and L. R. G. TRELOAR
The corresponding relation in torsion 1 is* [c~ In (M/T)/~T]v,z,¢~ :
-- d In ro2/dT q- fi
(4)
which can be expressed in the alternative form
(3M/~T)v,~,¢~ = (M/T) (1 + fiT -- T d In ro2/d T)
(4a)
In simple extension it is sometimes found convenient to express the internal energy component of the tensile stress at constant volume as fev which is defined as (~E/Ol)T,V. It can be shown 3 that
fev/f : -- T[~3In (f/T)/~T] v,t = T d In roZ/d T
(3a)
Similarly in the case of torsion we may introduce the quantity Mev, representing the internal energy contribution to the couple at constant volume, given by
Mev/M : -- T[c3 In (M/T)/~T]vd,¢~ : T d In roZ/d T
(5)
The quantities Mev/M and fev/f are thus seen to be strictly analogous. By using equation (4a) in conjunction with equation (5) the relative contribution, Mev/M, of the internal energy to the couple can be derived from stress-temperature experiments at constant pressure, and hence the quantity d In roZ/d T may be obtained. The aim of this work was to obtain a value of this quantity from torsional experiments on natural rubber, and to compare the value obtained with existing data for the case of simple extension. EXPERIMENTAL
Materials Natural rubber vulcanizates were made with various degrees of crosslinking by curing pale crfipe with 2.7-3-1% by weight of dicumyl peroxide for 30-55 min at 140°C. Cylinders were moulded of approximately 185 m m length and 6.5 m m diameter. No treatment was given to extract any uncrosslinked material or excess vulcanizing reactants which might have been present. Each cylinder was gripped at either end in a half-inch long cylindrical split clamp made from brass. Reference marks for the measurement of the axial extension ratio were made in ink on the surface of the cylinder. The length between the brass clamps, the diameter of the cylinder and the length between the reference marks were all measured with a travelling microscope.
Apparatus The apparatus used for the measurement of the torsional stress (couple) as a function of temperature at constant pressure, length and twist, is shown in Figure 1. In the design of this apparatus close attention was paid to the * In equation (4) fl is defined as Vu-a(OVu/ST)p. Since ( V -- Vu)/Vu is very s m a l l ( ~ 1 0 - 4 ) , this does not differ significantly from fl in equation (3).
22
THE THERMOELASTICITY OF NATURAL RUBBER IN TORSION
|
i
I I
I k.
-r,
II
®
'i
r,
LI
Figure 1. Schematic front view of apparatus The numbered parts represent : (1) sample (2) upper clamp (3) resistive probe (Type 5110 G. Sangamo-Weston) (17) temperaupper rod (4) worm gear (5) ture-indieator controller (Type lower clamp (6) silvel-steel rod 6003-2, Fielden) (18) variable (7) torsion drum (8) tension a.c. transformer (Type spring (9) threads (10) transmisV5HMTF, Variac) (19) lowsion bar (11) counterbalancing pitch screw adjustment for tenbar (12) ball-bearing pulleys sion spring (20) clamps attached (13) weights (14) stops (15) to main frame (21) main frame. heating chamber (16) thermo23
P. H. BOYCE
and
L. R. G. T R E L O A R
necessity for accurate control of the axial and torsional strains to which the sample was subjected. In carrying out the experiments, a given amount of twist was inserted into the sample by rotating the upper clamp via the upper rod which was connected to the worm gear. The lower clamp, which was rigidly attached by a silversteel rod to the torsion drum, was prevented from rotating about its axis by adjustment of the spring by which the tension in the attached threads was controlled. This tension was transmitted via the transmission bar as shown
Figure 2. Schematic diagram of torsional system (The numbered parts are referred to in Figure 1) in Figure 2; by maintaining the vertical position of this bar constant, the angular position of the drum was automatically fixed. The arrangement of the threads around the drum and their connection to the transmission bar and counterbalancing bar offered a stable torsional device in which the couple could be calculated from the measured extension of the spring. The frictional component of the stresses in the ball-bearing pulleys was found to be negligibly small. The angle of twist of the sample was controlled to within ~ 0 . 5 degrees. The couple was measured to within 4-0.1 g cm in the range 10 - 200 g cm. The sample was also given a small axial extension sufficient to maintain a positive tension and to eliminate buckling in the twisted state. The extension was controlled by adjustment of the weights applied to the top of the torsion 24
THE THERMOELASTICITY OF NATURAL RUBBER IN TORSION
drum. A guide to the vertical alignment and position of the drum was provided by the stops. The length of the sample was controlled to within ! 0.05 cm.
The heating chamber was constructed from a Pyrex glass tube which was wound with standard electrical-resistance wire. The temperature of the sample was controlled by an electronic temperature-indicator controller which was connected to a variable transformer which supplied the appropriate voltage to the resistance wire. The temperature at the centre of the sample was controlled to within ~0.5°C, but a maximum temperature difference of 2~C was detected along the length of the sample. However, this was not sufficient to create appreciable errors in the resulting stress-temperature data.
Procedure The sample was allowed to relax at 100°C for 1 to 1½ h under the maximum torsional strain to be used in subsequent experiments. Stress-temperature data were then recorded in the range 20 - 60°C. Stress readings were taken all 5 or 10°C intervals after sufficient time had been allowed for the attainment of thermal equilibrium; from preliminary experiments this was found to be about 10 min when, between each interval, the sample was heated or cooled at the rate of 1-3°C per min. The decreasing/increasing temperature cycle was repeated until effective reversibility was obtained. The procedure was then repeated in a similar manner at different lower values of the strain. but excluding the initial relaxation at 100°C. RESULTS AND CONCLUSIONS
Stress relaxation The effects of stress relaxation are shown in Figure 3. The total reduction in stress during the preliminary, high temperature relaxation and the two subsequent temperature cycles was about 101~. The periods of time that are shown at 30 and 60°C illustrate the difference between the rates of stress relaxation at these two temperatures. The time taken to obtain the complete curve illustrated was 4 days, compared with l0 h for each subsequent curve for the same sample at a lower strain.
Analysis oj' stress-temperature data A typical set of stress-temperature plots for one particular sample at different strains is shown in Figure 4. In all cases a reversible, linear relationship was found to exist between couple and temperature. The slope of any one line was reproducible to within 2~i for successive temperature cycles. It should be noted that the broken line plot (at ~ba0 0.690) was obtained from a repeated run, after the curves at the lower strains had been recorded. The small displacement of this plot from the bold line curve signified that, although reversibility had been achieved within the time of one run, thermodynamic reversibility had not been achieved over the total time of the experiment. 25
and
P. H. BOYCE
L. R. G. T R E L O A R
140
ff S ~
130
SjSS
E u
pSSS ~ sSP~/ Xi
o) 120
,." /
e~
XSS
o
~X
Xt
/l ,sh .,~J
110
100 l
I
20
40
I
60 Temp. [°C]
1
I
80
100
Figure 3. Typical stress-relaxation plot for natural rubber in torsion O, decreasing temperature,
×, increasing temperature
However, the value of the stress-temperature coefficient had not changed significantly. The quantity (T/M)~M/~T [equation (4a)] was calculated from relative stress-temperature data for a temperature of 293°K. The resulting data are included in Table 1.
Effects o f strain and crosslink density upon MevlM In Table 1 the torsional strain is expressed in terms of the parameter ~ba0 corresponding to n turns of twist in the total length of the sample; a0 and ~b are defined as in equation (2). The crosslinking density is expressed in terms of G, the shear modulus, which was calculated from the reversible stress-temperature data at 20°C by use of the equation
G = 2M/rr~bao4 Both ~ba0 and A, the extension ratio, were referred to the original, unstrained dimensions at 20°C. Mev/M was calculated at 293°K from the (T/M) c~M/c~T values using equations (4a) and (5). The coefficient of volume expansion,/3, for natural rubber was taken as 6-6 x 10 -4 deg -1. 26
THE T H E R M O E L A S T I C I T Y OF N A T U R A L R U B B E R IN TORSION
90
u
70
f x
~
~
C
50 ...,...,--..~ d
,.__t-.---30 i
i
[
20
40 Temp.("C ]
60
Figure 4. Stress-temperature plots at different torsional strains C), decreasing temperature, x , increasing temperature Torsional strain parameter ~bao- (a) 0-888, (b) 0.690, (c) 0.492 (d) 0.393, (e) 0.297
27
P. H. BOYCE and L. R. G. TRELOAR
Table 1 Thermoelastic data for natmaI rubber in torsion
[kg c m -2] 2.1
(~ = 1'25)
turin'
[rad]
M \ e T / p,t,~
[ %]
9
0.888
1-06
13- 3
7 5
0.690 0.492
1.06 1-07
13'3 12.3
4 3
0.393 0.297
1.08 1.09
11. 3 10'3
3.0
7
0.683
1-04
(,~ = 1"19)
6 5
0'586 0.488
1"07 1.07
15"3 12" 3 12' 3
6
0"633
1"07
12' 3
(,~ = 1-15)
3-6
5.5 5
0.580 0.526
1.05 1.06
14"3 13-3
3.7
6.5
0.696
1.05
14'3
6
0.642
1.07
12. 3
5.5
0.589
1.10
9"3
(,~ = 1"13)
The average value of the quantity Mev/M, obtained from the figures in Table 1, is 12.6 ± 1.6~. The relative error in Mev/M is, of course, much larger than the relative error in (T/M)~M/OT (eight times as large); there is, however, no indication of any significant dependence of Mev/M on either the crosslink density or the amount of the torsional strain. From the mean value of Mev/M the resulting value of d In roZ/dT for natural rubber at 20°C is found to be 0.43 -- 0.05 × 10 -2 deg -1.
Comparison with simple extension data Although there is no difficulty in principle in carrying out comparable thermoelastic measurements in simple extension, the accuracy of the values of d In ro2/dT determined from such experiments is generally lower, owing to the greater sensitivity of the stress-temperature coefficient in extension to volume changes, discussed in an earlier paper 1. The presence of the term --fl/(a~ -- 1) in equation (3) implies a sensitivity to the absolute value of the strain, and hence to the precision with which the unstrained length can be determined, which is absent in the corresponding equation (4) for torsion. As a result the errors in the determination of the internal energy contribution to the tensile force fev/f (and hence d In ro2/dT) increase progressively as the strain decreases. It is presumably for this reason that published results show no agreement on the question of whether or not the internal energy contribution to the stress, fev/f, is dependent upon the strain; some authors 7,10,13,14,15 have reported an increase in this quantity as the strain becomes smaller, while others4,~,9,~v,is have found no significant change. In one case t6 a decrease has been reported. Table 2 gives some of the published data obtained from simple extension experiments; in cases where the figures varied according to the strain the values quoted refer to the region of higher strains, where 28
THE THERMOELASTICITY OF NATURAL RUBBER 1N TORSION t h e v a r i a t i o n s w e r e g e n e r a l l y r e d u c e d t o a m i n i m u m . It is s a t i s f a c t o r y t o n o t e t h a t t h e p r e s e n t r e s u l t f o r M e v / M in t o r s i o n , n a m e l y ,~~-,,/o, ~ " / is w i t h i n t h e r a n g e c o v e r e d b y t h e d a t a f o r t h e e q u i v a l e n t quantityJ~,~,/Ji
Table 2 Published values offe,,/f for natural rubber for thermoelastic measurements in simple extension.
/~,/f
Temperature
[%1
VC]
17
0 - 70
Ciferri 4
13
0 - 70
Ciferri, Hoeve and Flory s
20 20
45 30
Roe and Krigbaum 7 Allen, Bianchi and Price s
12.5
20 - 60
18 18
15
30 25
-
30
70
Authors
Method oJ determination
Yamamoto, Kusamizu and Fujita 9 Shen, McOuartie and Jackson t0 Barrie and Standen 1J Shen and Blatz 12
Constant pressure, equation (3) Calculated from experimental data of Wood and Roth ~ Constant pressure, equation (3) Constant volume - direct measurement of (~f/aT)v,z Constant pressure, modified Mooney form of equation (3) Con.~tant pressure, equation (3) Constant pressure-length v. temperature measurements at constant load, equation (3) Constant pressure, usingJ~.,.//'= 1 T(dln G/dT - fiT/3
It is c o n c l u d e d t h a t t h e t o r s i o n a l m e t h o d p r o v i d e s a c o n v e m e n t a n d a c c u r ate m e a n s o f d e t e r m i n i n g t h e q u a n t i t y d In roZ/dT w h i c h p o s s e s s e s d i s t i n c t experimental advantages over the more usual method of simple extension.
ACKNOWLEDGEMENTS T h e a u t h o r s w i s h to t h a n k M r G . D a l t o n o f t h i s D e p a r t m e n t the design of part of the apparatus.
f o r his h e l p in
Department o f Polymer and Fibre Science, University o f Manchester Institute o f Science and Technology
(Receil,ed I August 1969)
REFERENCES I 2 3 4 5 6 7 8
Treloar, L. R. G. PolymerLond., 1969, 10, 291 Flc.ry, P. J. J. Amer. Chem. Soe. 1956, 78, 5222 Flory, P. J. Trans Faraday Soc. 1961, 57, 829 Ciferri, A. Die Makromol. Chem. 1961,43, 152 Ciferri, A., Hoeve, C. A. J. and Flory, P. J. J. Amer. Chem. Soe. 1961,83, 1015 Wood, L. A. and Roth, F. L. J. appl. Physics 1944, 15, 781 Roe, R. J. and Krigbaum, W. R. J. Polymer Sci. 1962, 61, 167 Allen, G., Bianchi, U. and Price, C. Trans Faraday Soc. 1963, 59, 2493 29
P. H. BOYCE a n d L. R. G. TRELOAR 9 10 11 12 13 14 15 16 17 18
Yamamoto, K., Kusamizu, S. and Fujita, H. Die Makromol. Chem. 1966, 99, 212 Shen, M. C., McQuarrie, D. A. and Jackson, J. L. J. appl. Physics 1967, 38, 791 Barrie, J. A. and Standen, J. PolymerLond. 1967, 8, 97 Shen, M. C. and Blatz, P. J. J. appL Physics 1968, 39, 4937 Roe, R. J. and Krigbaum, W. R.J. Polym. Sci. (A) 1963, 1, 2049 Crespi, G. and Flisi, U. Die Makromol. Chem 1963, 60, 191 Opschoor, A. and Prins, W. J. Polym. Sci. (c) 1967, 16, 1095 Haly, A. R. J. Polym. Sci. (A) 1965, 3, 3331 Ciferri, A. Trans Faraday Soc. 1961,57, 846 Mark, J. E. and Flory, P. J.J. Amer. Chem. Soe. 1964, 86, 138
30