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High-elastic deformation in the viscous flow of butyl rubber
1386
Cx. V. VINOGRADOVeta/.
8. B. I. LIOGON'KII, A. A. BERLIN, A. V. RAGIMOV and V. P. PARINI, I n t e r n a t i o n a l Symposium on Macromolecular Chemistry, Prague, 1965, Preprint 235 9. A. N. MACHYULIS and L. P. ZHILINSKAITE, Mekhanika polimerov, No. 1, 68, 1968 10. A.N. MACIIYULIS, L. P. ZHILINSKAITE and A. V. RAGIMOV, Materialy respublikanskoi nauehno-tekhnicheskoi konferentsii pc voprosom issledovaniya i primeneniya polimernykh materialov (Transactions of the Republican Scientific and Technical Conference on Problems in the Study and Use of Polymeric Materials). p. 70, 1965 11. Yu. V. POLEZHAYEV, Izv. Akad. Nauk SSSR, Mekhanika i mashinostroyenie, No. 5, 157, 1964
HIGH-ELASTIC DEFORMATION IN THE VISCOUS FLOW OF BUTYL RUBBER* G. V. VI~OQRXDOV, A. YA. IW_A~Kr~, I~I.P. ZAB~GI~X and V. F. SII~HSKrl A. V. Topchiev Institute of Petrochemical Synthesis, U.S.S.R. Academy of Sciences
(Received 22 January 1968)
THE ability to undergo large elastic (high-elastic) deformation is an inherent feature of the behaviour of polymeric systems in any physical state. Correlations in the manifestation of high:elastic properties have been studied extremely widely, but mainly for polymers in the high-elastic state. However, the ability to sustain large elastic deformations can be preserved by polymers in the glassy, crystalline and viscous-flow states. Despite the fact that it is widely recognized that in the flow of polymer solutions and melts high-elastic deformation is superimposed on plastic (irreversible) deformation and that this is often quoted to explain various phenomena, nevertheless there are practically no positive data on correlations in the development of high-elastic deformation in systems such as polymer melts, concentrated solutions and uncrosslinked rubbers. Among the few exceptions are some early papers by Leaderman and his collaborators [1, 2], and another not very useful paper (because of the vagueness of the conditions of deformation) [3]. The absence of direct experimental observation has given rise to many theoretical, semi-theoretical and simply empirical appraisals of the highyelasticity of flowing polymer systems, the validity, value and suitability of these appraisals being still an open question. The question of the applicability of some published formulae for calculation of high-elastic deformation will be considered to some extent in the section below devoted to discussion of the experimental results of the present work. * Vysokomol. soyed. A l l : No. 6, 1221-1228, 1969.
High-elastic deformation in viscous flow of butyl rubber
1387
This c o m m u n i c a t i o n is concerned w i t h direct m e a s u r e m e n t of high-elastic d e f o r m a t i o n in t h e flow of uncrosslinked, " r a w " b u t y l r u b b e r (BR). R e f e r e n c e [4] p r e s e n t s t h e results o f a detailed s t u d y o f t h e shear a n d n o r m a l stresses developing in flow of this p o l y m e r . C o m p a r i s o n o f d a t a on stresses a n d high-elastic d e f o r m a t i o n s w o u l d enable a c o m p l e t e view t o b e gained Of t h e rheological p r o p e r t i e s o f B R , w h i c h in m a n y respects is a t y p i c a l elastomer.
EXPERIMENTAL The material on which the measurement were made has been described in detail in reference [4]. Elastic shear deformation was measured with an apparatus of the cone-and-plate type, differing from the standard cone-and-plate viseometer in the following respects: /y
10
H
/5
0
f6 7
,
[
,
.
i
FIG. 1. Schematic diagram of the apparatus for measurement of high-elastic deformation in flow. (For denotation of the components see text). 1) the possibility of disconnecting the rotating member of the working cone and plate pair from the driving system, and of application of rapid breaking; 2) the possibility of providing either free rotation or a strictly stationary position of the second member of the cone and plate pair; 3) provision of a special operating control system.
1388
G . V . VI~OGRADOVe$ al.
The apparatus, which is illustrated schematically in Fig. 1, consists of the following basic components and units: a DSD-2 electric motor with an integral reducing gear 1, sm SD-09 electric motor 2, a semi-stepwise gear box with push-button control 3, a dog clutch ¢ an electromagnetic brake 5, the driven rotating p l a t e 6, the seoond working m e m b e r - - a demountable cone, firmly seated on a shaft 7, a bearing unit 8, adjustable in the axial direction, a casing 9, a stop bar 10, on which~is f i x e d a Small mirror 11, an eleetrom~g~et 12, a scale 13 for measuring the rotation of the mirror, a thermostatted h e a t i n g jacket 14, and an electrical control system (not shown in Fig. 1). At the beginning of an experiment casing 9 is rotated to the stop (the sample is not yet in the apparatus) and by means of the threaded collar 14 the bearing unit is clamped dpwn, to set the position of cone a n d plate contact, b u t with the desired axial clearance, The working unit is then opened up and the test specimen is placed between the cone and plate. The qasing is now rotated to the stop, whereby the cone .~-ld plate contact position is reset. The desired temperature is established b y means of the therniostatted heater jacket (which may be replaced by a liquid jacket connected to a thermostat bath, or by s n electric heater) and the system is left until thermal equilibrium is reached (usually about 1-1.5 hr). The apparatus is started up b y switching on the motor with the reducing gear, because the engaged position of the dog clutch is the switch-on position. The electric motors are mounted on the same shaft, and therefore to avoid damage the DSD-2 motor must be switched off before the SD-09 motor is switched on. The electrical circuit does not permit simultaneous switching on of both motors. After the plate has begtm to rotate the torque through the sample acts on the cone, which also begins to rotate, because before the start of the experiment the cone is not fixed. Under these conditions no deformation of the specimen occurs. This continues until the stop bar 10 comes to rest against the protruding core of the electromagnet 12. The moment of contact of the bar and core marks the beginning of deformation precisely, because at that point the cone is stopped and the plate continues to rotate, so that shear deformation of the specimen begins. I n addition a time relay, present for a given period of deformation, is switched on at the time of contact. After the set time the relay switches on the current in the coils of the electromagnet and the magnetic brake. The core of the electromagnet is thus retracted inside the coil and the stop bar and cone, which is firrhly a t t a c h e d to it, are allowed free rotation. Because the electromagnetic brake 5 is switched on at the same time the plate becomes detached from the driving shaft (the teeth o f ~ e d0g-cldtch become disengaged) and it stops in less t h a n 0" 1 see. Then movement of the stop bar and its attached mirror is caused only by elastic forces set up in the specimen and the rotation=of the bar corresponds to the elastic deformation retained in the specimen. The angle of rotation of the bar is measured by means of an optical system (lamp-mirror-scale). The size of the angle is a measure of the highelastic deformation. By varying the transmission ratio in the gear box (each step of which corresponds to a 5 : 1 reduction in speed), by changing over the electric motors (the rate of rotation of the DSD-2 motor is 2 rev/min and of the SD-09 motor 3000 rev/min) and by exchanging cones (the set contains cones with angles from 168 to 177 ° at the apex) it is possible to produce shear rates in the specimen from 10 -~ to ~ l0 S sec -1. Regnlation of the time relay permits setting up of deformation times from 1 see to 1 rain. If longer periods o f deformatiou axe required the time relay is switched out and the system is controlled manually. By means of the measuring system used the angle of rotation of the axis of the cone can be-ascertained with a precision of the order of 0.1 °, which is quite sufficient for the present. purpose because this corresponds (with a cone with an angle of ~174 °, which was the one mainly used) to approximately 3~o deformation of the m a t e r i a l . E s t i m a t i o n of the accuracy and reproducibility of the results showed that the total error does not exceed 10% of the deformation. I n order to obtain reliable values of elastic deformation
High-elastic deformation in viscous flow of b u t y l rubber
1389
i t is very i m p o r t a n t to follow the elastic recovery for a long time, sometimes for a m a t t e r of days. I n this connection heating up of the specimen can be of considerable help, because when heated it reverts much more rapidly to the unstressed state and the total measured elastic deformation is independent of the heating routine (see below). I n general, and as a rule, the duration of the complete spectrum of high-elastic deformation is greater the higher the viscosity of the material.
8 7
Q
_×
....
x
2 _ j ~ , - - - _- -_- × - - ~ - ~ - - ~
6 ~zx
5
x ! r~ 2
4J 3
0
I
Z
I
1
5
fO
f5
20
II
I
50 Time , min
I
100
I
150
II
I
300
I
900
FIG. 2. Kinetics of elastic recovery after cessation of deformation: / - - r e p e a t e d experiments at 22°; 2 - - e x p e r i m e n t at 22 °, elastic recovery measured at 80 ° (the point of attainment of equilibrium of 7e with the heater switched on is shown b y the arrow). Continuous line-value of 7e after equilibrium. The deformation of the material, 7, was calculated b y means of the formula ?=o)~-lt, where co is the angular velocity of the plate, t the period of deformation and 8 the angle between the cone and plate. The ratio co/e= 7"is the rate of deformation. The elastic deformation, ~e, was dalculated b y means of a similar formula, 7e=~/~, where ~ is the experimentally measured angle of rotation of the cone after cessation of deformation. Figure 2 shows a typical curve of the dependence of elastic recovery on the time of keeping the specimen after cessation of deformation. This graph includes d a t a from a few independent measurements, from which the reproducibility of the results can be judged. The measurements of elastic deformation shown in Fig. 2 relate to cessation of deformation during equilibrium flow conditions at total deformation of from 250 to 800 units. The effect of heating during elastic recovery is also seen from these results. Thus the measuring system permits determination of the total elastic deformation in a material when it is deformed at a constant rate of shear for different set times (corresponding to different total deformations) at different temperatures, and also study of the kinetics of elastic recovery. RESULTS AND DISCUSSION The build up of elastic deformation under a constant rate of shear (7=const) is v e r y r e m i n i s c e n t o f t h e d e v e l o p m e n t o f s h e a r a n d n o r m a l s t r e s s e s in t h i s m a t e r i a l [4], w h i c h is i n g e n e r a l f a i r l y t y p i c a l o f v i s c o u s - f l o w i n g p o l y m e r i c s y s t e m s in t h a t a s t h e t o t a l d e f o r m a t i o n i n c r e a s e s t h e h i g h - e l a s t i c c o m p o n e n t g r a d u a l l y i n c r e a s e s . T h e n i f t h e r a t e o f s h e a r is l o w 7e s t o p s a t s o m e e q u i l i b r i u m v a l u e . W h e n t h e r a t e o f s h e a r is h i g h e n o u g h t h e n a t u r e o f t h e 7e(7) r e l a t i o n s h i p c h a n g e s , i.e. 7~ p a s s e s t h r o u g h a m a x i m u m a n d t h e n f a l l s t o a n e q u i l i b r i u m v a l u e . R e d u c t i o n i n t e m p e r a t u r e h a s a n effect s i m i l a r t o t h a t o f
G. V. VI~OG~DOV e$ a~.
1390
increase in the rate of shear. Typical experimental results illustrating the different types of 7e(7) relationship are presented in l~ig. 3. I n reference [4] it was shown t h a t the a t t a i n m e n t of the equilibrium values of the shear stress, ~, a n d of the ~fference in the normal stresses, a, requires
7"5 !
5"0 2"5
/
I
0
I
50
I
fO0
I
150
i
I
I
200 300 ~00
FIG. 3. Typical curves of the dependence of y, on the total deformation, 9, at various tern. peratures: 1,2--25°; 3--60°; 4--100% 7=0.34 sec -1 (continuous lines) and 0.0655 sec -z (broken line). quite different times. Comparison of the v(7), a(7) a n d ?~(7) curves (Fig. 4) shows t h a t in this respect the 7e(7) relationship occupies an intermediate position between ~(7) a n d a(~). F o r example at a rate of shear of 0.34 see-Z a n d a tempera-
2e and ~/2v
frlS/OITt2
!
o--Ip--.o-
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It
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I
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I
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o
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60
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1~901~ 0
7 FzG. 4. Comparison of the functions a (y) (1), 9e (7) (2), ~ (7) (3) and y~ (7) (4) at 100° and ~= 0-34 sec -z. ture of 100 ° the equilibrium values of v are reached at T~8, of a at 7 ~ 6 0 and of ~e at 7 ~ 28. The same graph (Fig. 4) shows the dependence of a/2v on 7. According to some theories [5] the q u a n t i t y ~6=a/2~ corresponds to 7e- I n particular this correspondence always holds for equilibrium flow conditions in the linear re-
High-elastic deformation in viscous flow of butyl rubber
1391
gions of mechanical behaviour [6]. The experimental data in Fig. 4 show, however, t h a t there is no general correspondence between 7e and 7'e. The values of 7~ and 7'e differ very considerably in the transitional deformation region. Attention has previously been given to this [7] in a study of dilute polymer solutions. In the example given in Fig. 4 a marked difference between 7e and 7'e is maintained even under equilibrium flow conditions.
0"5
0
10
20
30
O0
5O
2,sec-' FIO. 5
Fro. 6
FIG. 5. Variation in the ratio of high-elastic and plastic deformations in the course of flow at temperatures of: 1 - 25°; 2--60°; 3--100 °. 7~0.34 see-1 (continuous lines) and 0.0655 sec-1 (broken line). " FIo. 6. Dependence of re (continuous lines) and 7 " e : a / 2 v (broken lines) on log ~ under equilibrium flowconditions at temperatures of: 1 , 1 ' - - 25°; 2,2" - - 60°; 3,3" - - 80°; 4 , 4 ' - - 100°; 5 , 5 ' -- 120°. I t should be recognized t h a t the relationship between the three characteristics of a polymer, namely high-elastic deformation, shear stress and normal stresses, can still not be established. This conclusion is valid not only for transitional deformation conditions, but also for equilibrium flow conditions, as will be shown below. In equilibrium flow the plastic (irreversible) component of the total deformation considerably exceeds the high-elastic component, which according to our measurements for :BR never exceeds a few units. However, under transitional conditions the ratio of the plastic to the high-elastic component can be quite different. I n answer to the question of the ratio of these components of the total deformation Fig. 5 shows experimental data on the dependence of t h e ratio 7 e ] 7 ~ - T e / T t on 7 for one rate of shear (0.34 sec -1) and three temperatures, and for one temperature (25 °) and two rates of shear. I t is seen from Fig. 5
G. V. VXNOGRADOVet al.
1392
that in the initial period of deformation which lasts a fairly long time (up to ~ 100% deformation), there is practically no flow and all the deformation is reversible. As the temperature is raised and the viscosity of the polymer is correspondingly lowered plastic deformation begins earlier and proceeds more rapidly than at lower temperatures; however in these eases the general relationship that up to ~ 1 0 0 % only high-elastic deformation is developed is preserved. /o9 C~,dyne/cm2
~'J~
!
I
l--1
I
0
I
zo#2, sec-'
1
FIG. 7. D e p e n d e n c e o f t h e h i g h - e l a s t i c i t y m o d u l u s on rate of shear at t e m p e r a t u r e s of: 1--25°; 2--60°; 3--80°; 4--100°; 6 - - 1 2 0 °.
I t is seen from Fig. 5 that eYen at 100 ° and a rate of Shear of 0.34 sec -1 at a total deformation of 400% half of it consists of the high-elastic component. Decrease in the rate of shear affects the ratio of the reversible and irreversible components ~in the same way as increase in temperature. Let us now consider in more detail the nature of the variation in high elastic deformation under equilibrium conditions in relation to temperature and rate of shear. First we shall consider only equilibrium flow conditions in the sense of attainment of time-invariant values of 7~, and also r and a on further prolonged deformation. The basic experimental data, obtained at rates of shear from 10 -2 to 10 sec -z and temperatures from 25 to 120 ° are represented b y continuous lines in Fig. 6. I t is obvious that the high-elastic deformation increases when the rate of shear is increased at constant temperature or when the temperature is reduced at constant rate of shear. The dependence of ~'e on 7, calculated from the data of reference [4] for the same region of shear rates and temperatures, is represented in this graph b y broken lines. As was mentioned above the quantity 7'~=a/2T should according to some theoretical ideas correspond to 7~. According to other theories (for example those of references [8] and [9]) correspondence between high-elastic deformations and stresses is found if ?e and 7~'=a/v are compared (obviously 7~'=27'e). Replacement of 7'e b y 7~' in Fig. 6 means shifting the broken lines upward b y a factor of two. Comparison of 7~ with 7'e and 7~ shows that in the studied range of temperatures and shear rates the experimental and theoretical values of the high-elastic deformation diverge, the divergence being greater the higher the rate of shear. Opinion in the literature is divided on the question of the correlation between 7~ and y: or y':. Thus
1393
High-elastic deformation in viscous flow of butyl rubber
in a study of dilute solutions of polyisobutylene, natural rubber and aluminium naphthenate [7] it was found that under equilibrium flow conditions 7~=7~. Philippoff and his collaborators [10, 11] assumed that 7e=7~. The experimental results of this work show that in fact the position with regard to the correlation
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3
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I
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FzG. 8. Dependence of high-elastic deformation on shear stress at temperatures of: 1--25°; 2--60°; 3--80°; 4--100°; 5--120 °. between stress and high-elastic deformation is more complex. It is seen from Fig. 6 that in the nonlinear region, fairly far from Newtonian flow conditions, 7~>~'>7'e. On the other hand it has been shown theoretically [6] that in the linear region 7e~7'e- Extrapolation of the data obtained here to ~->0 confirms the validity of this conclusion. It m a y thus be assumed that at very low rates of shear 7e=-a/2T, then as the rate is increased 7~ increases more rapidly than a/2T, and in a certain region of shear rates ?,e,,~a/~. Finally at still higher rates of shear 7~>a/~. The region of values of 7 to which one or other of these relationships applies is dependent on the material under test. All these problems require further refinement and experimental elucidation, b u t it may be affirmed with certainty that the theoretical formulae do not in the general case correctly describe the relationship between high-elastic deformation and stress. Therefore, the position with regard to the connection between stress and high-elastic deformation requires further refinement and study. The direct experimental measurements made here could be used to calculate the high-elastic modulus of the polymers in flow, G~, which is given b y the equation Ge-=~/?J e . The values of Ge at various rates of deformation and temperatures are shown in Fig. 7. The general trend of the Ge(7, T) relationships is seen fairly clearly from Fig. 7 and does not require further discussion.
1394
G. V. VI~0GRADOVe~~.
I t is very important to consider the high-elastic properties of flowing polymer systems in relation not to the rate of shear, as was done above in order to obtain a clear, general picture of the development of large elastic deformations, but ~',/0 -,Sdyne/cme
i
"
i
3OO
I
35O
I
4~0 T,°K
FIe. 9. Temperature dependence of shear stress at ?,=cons~, y~= l (1); 2 (2); 3 (3); 3.5 (4) units. to shear stress: Consideration of this is important primarily because stress is a factor determining both the viscosity properties of the polymer and the development of the normal stresses [4]. This follows from the possibility of construction of temperature-invariant characteristics of viscosity and the normal stresses, taking shear stress as the argument. Experimental data on the dependence of high-elastic deformation on shear stress in the studied temperature range are presented in Fig. 8. It is evident that there is a general tendency for the high-elastic deformation to increase as the shear stress is increased, but the 7e(~) relationship is rather complex. According to the entropic theory of high-elasticity, and similarly according to the i~ooney-Rivlin theory [12], one would expect a linear relationship between 7e and stress in simple shear. It is seen from Fig. 8 that this is in fact not so. Therefore, the so-called Hooke's law of shear, according to which in the general case ?e ~T, is not followed and the question of its applicability to individual special cases must be answered individually for each particular polymer and conditions of deformation. A still greater divergence between theoretical prediction and experimental results is found on examination of the temperature dependence of ~ when ?~~ const (Fig. 9). According to the entropie theory of high-elasticity the T(T) relationship should give a slightly rising curve. The complex nature of this relation-
High-elastic deformation in viscous flow of butyl rubber
1395
ship actually observed is due primarily to a structural transition occurring at about 60 °. Attention was first drawn to this transition in reference [13], where it was found t h a t there is a mechanical loss maximum in the region of 60-70 °. This was confirmed later by measurements made in our laboratory (Ye. A. Dzyura and L. I. Ivanova) on the sample of B R studied in the present work. The stress-temperature relationship also indicates the occurrence of a transition in BR at 60-70 ° [4]. I t is probably this transition, specific to BR, t h a t leads to the well-defined maximum near 60 ° in the dependence of T on T when 7e=const. However, the observed nonlinearity of the 7e(T) relationship shows t h a t high-elastic deformation in polymer systems in the viscous-flow state develops in a manner rather different from t h a t in crosslinked elastomers (vulcanized rubbers), quite apart from the occurrence of this transition. CONCLUSIONS
(1) Direct measurement has been made of high elastic deformation during flow of a typical highly viscous polymer--plasticized butyl rubber. These measurements were made in both the transitional deformation region and under conditions of equilibrium flow, at various rates of deformation and temperatures. (2) Comparison is made between the high-elastic deformation and the stresses arising during flow at a fixed rate of shear. I t was found t h a t the measured elastic deformations are not in agreement with the deformations calculated from the conventional theories involving the normal and shear stresses, and this divergence exists not only in the transitional deformation region. (3) In simple shear the modulus of high-elasticity increases with increase in the shear stress, i.e. "Hooke's law of shear" is not obeyed. (4) The observed types of dependence of high-elastic deformation on shear stress and temperature (the nonlinearity of the dependence on shear stress and the existence of an extremum in the temperature dependence) show t h a t the elasticity of uncrosslinked butyl rubber cannot be described within the framework of the concepts of the standard entropic theory of the high-elasticy of crosslinked rubbers. Translated by E. O. PHILLIPS REFERENCES
1. 2. 3. 4. 5. 6. 7. 8.
H. LEADERMAN, R. G. SMITH and R. W. JONES, J. Polymer Sci. 14: 47, 1954 H. LEADERMAN, R. G. SMITH and L. C. WILLIAMS, J. Polymer Sci. 36: 233, 1959 R. McCORD and B. MAXWELL, Modern Plastics 39: 116, 1961 G. V. VINOGRADOV, A. Ya. MALKIN and V. F. SHUMSKII, Vysokomol. soyed. A10: 2672, 1968; All: 663, 1969 A. S. LODGE, Elastic Liquids, Academic Press, London-New York, 1964 A. Ya. MALKIN, l~heol. Acta 7: 335, 1968 A. S. MOROZOV, Dissertation, 1967 N. I. MALININ, Kolloid. zh. 22: 201, 1960
N.I. ZHrR~ov et at.
1396
9. A. JOBLING and J. E. ROBERTS, Rheology (Ed. F. Eirich) 2: 503, New York, 1958 10. J. G. BRODNYAN, F. H. GASKINS, W. PHILIPPOFF and E. G. LENDRAT, Trans. Soc. Rheol. 2: 285, 1958 11. W. PHILIPPOFF, Trans. Soc. Rheol. l0 (1): 1, 1966 12. L. TRELOAR, Fizika uprugosti kauchuka (Physics of Rubber Elasticity). Foreign Literature Publishing House, 1953 13. S, GEHMAN, Rubber Chem. and Technol. 30: 1202, 1957
ESTIMATION OF THE THEORETICAL STRENGTH OF ORIENTATED POLYETHYLENE* N, I. ZHIRNOV,YE. G. KORYAK-DOROI~E:~KOand G. 1~I. BARTEI~EV V. I. Lenin Pedagogical Institute, Moscow
(Received 26 January 1968) 1KUCH experimental material has now accumulated on the strength properties of polyethylene (PE), which are dependent on temperature, the rate of stretching and other factors [1]. With regard to the theoretical strength of PE we know only of reference [2] in which a rough assessment was made of the strength of PE for stretching of the molecules along one of the C--C bonds, without account being taken of deformation of the valence angles. In reference [2] use was made of the Morse potential [3] in the form
U l(r)=De-2~(r-ro)-2De- ~(~-~o),
(1)
where ~ = ~ / 2Kr D
(2)
Here D is the energy of dissociation of the C--C bond, determined experimentally from the heat of combustion, r 0 the equilibrium internuclear distance and K~ a force constant characterizing the valency vibration of the C--C bond and determined from optical data. The stress applied to the molecule when it is stretched along the C--C bond is found by differentiating (1) with respect to r
a*----
dr
-~2aDe- ( -°)[1--e-~(r-r°)].
* Vysokomol. soyed. All: No. 6, 1229-1233, 1969.
(3)
Cx. V. VINOGRADOVeta/.
8. B. I. LIOGON'KII, A. A. BERLIN, A. V. RAGIMOV and V. P. PARINI, I n t e r n a t i o n a l Symposium on Macromolecular Chemistry, Prague, 1965, Preprint 235 9. A. N. MACHYULIS and L. P. ZHILINSKAITE, Mekhanika polimerov, No. 1, 68, 1968 10. A.N. MACIIYULIS, L. P. ZHILINSKAITE and A. V. RAGIMOV, Materialy respublikanskoi nauehno-tekhnicheskoi konferentsii pc voprosom issledovaniya i primeneniya polimernykh materialov (Transactions of the Republican Scientific and Technical Conference on Problems in the Study and Use of Polymeric Materials). p. 70, 1965 11. Yu. V. POLEZHAYEV, Izv. Akad. Nauk SSSR, Mekhanika i mashinostroyenie, No. 5, 157, 1964
HIGH-ELASTIC DEFORMATION IN THE VISCOUS FLOW OF BUTYL RUBBER* G. V. VI~OQRXDOV, A. YA. IW_A~Kr~, I~I.P. ZAB~GI~X and V. F. SII~HSKrl A. V. Topchiev Institute of Petrochemical Synthesis, U.S.S.R. Academy of Sciences
(Received 22 January 1968)
THE ability to undergo large elastic (high-elastic) deformation is an inherent feature of the behaviour of polymeric systems in any physical state. Correlations in the manifestation of high:elastic properties have been studied extremely widely, but mainly for polymers in the high-elastic state. However, the ability to sustain large elastic deformations can be preserved by polymers in the glassy, crystalline and viscous-flow states. Despite the fact that it is widely recognized that in the flow of polymer solutions and melts high-elastic deformation is superimposed on plastic (irreversible) deformation and that this is often quoted to explain various phenomena, nevertheless there are practically no positive data on correlations in the development of high-elastic deformation in systems such as polymer melts, concentrated solutions and uncrosslinked rubbers. Among the few exceptions are some early papers by Leaderman and his collaborators [1, 2], and another not very useful paper (because of the vagueness of the conditions of deformation) [3]. The absence of direct experimental observation has given rise to many theoretical, semi-theoretical and simply empirical appraisals of the highyelasticity of flowing polymer systems, the validity, value and suitability of these appraisals being still an open question. The question of the applicability of some published formulae for calculation of high-elastic deformation will be considered to some extent in the section below devoted to discussion of the experimental results of the present work. * Vysokomol. soyed. A l l : No. 6, 1221-1228, 1969.
High-elastic deformation in viscous flow of butyl rubber
1387
This c o m m u n i c a t i o n is concerned w i t h direct m e a s u r e m e n t of high-elastic d e f o r m a t i o n in t h e flow of uncrosslinked, " r a w " b u t y l r u b b e r (BR). R e f e r e n c e [4] p r e s e n t s t h e results o f a detailed s t u d y o f t h e shear a n d n o r m a l stresses developing in flow of this p o l y m e r . C o m p a r i s o n o f d a t a on stresses a n d high-elastic d e f o r m a t i o n s w o u l d enable a c o m p l e t e view t o b e gained Of t h e rheological p r o p e r t i e s o f B R , w h i c h in m a n y respects is a t y p i c a l elastomer.
EXPERIMENTAL The material on which the measurement were made has been described in detail in reference [4]. Elastic shear deformation was measured with an apparatus of the cone-and-plate type, differing from the standard cone-and-plate viseometer in the following respects: /y
10
H
/5
0
f6 7
,
[
,
.
i
FIG. 1. Schematic diagram of the apparatus for measurement of high-elastic deformation in flow. (For denotation of the components see text). 1) the possibility of disconnecting the rotating member of the working cone and plate pair from the driving system, and of application of rapid breaking; 2) the possibility of providing either free rotation or a strictly stationary position of the second member of the cone and plate pair; 3) provision of a special operating control system.
1388
G . V . VI~OGRADOVe$ al.
The apparatus, which is illustrated schematically in Fig. 1, consists of the following basic components and units: a DSD-2 electric motor with an integral reducing gear 1, sm SD-09 electric motor 2, a semi-stepwise gear box with push-button control 3, a dog clutch ¢ an electromagnetic brake 5, the driven rotating p l a t e 6, the seoond working m e m b e r - - a demountable cone, firmly seated on a shaft 7, a bearing unit 8, adjustable in the axial direction, a casing 9, a stop bar 10, on which~is f i x e d a Small mirror 11, an eleetrom~g~et 12, a scale 13 for measuring the rotation of the mirror, a thermostatted h e a t i n g jacket 14, and an electrical control system (not shown in Fig. 1). At the beginning of an experiment casing 9 is rotated to the stop (the sample is not yet in the apparatus) and by means of the threaded collar 14 the bearing unit is clamped dpwn, to set the position of cone a n d plate contact, b u t with the desired axial clearance, The working unit is then opened up and the test specimen is placed between the cone and plate. The qasing is now rotated to the stop, whereby the cone .~-ld plate contact position is reset. The desired temperature is established b y means of the therniostatted heater jacket (which may be replaced by a liquid jacket connected to a thermostat bath, or by s n electric heater) and the system is left until thermal equilibrium is reached (usually about 1-1.5 hr). The apparatus is started up b y switching on the motor with the reducing gear, because the engaged position of the dog clutch is the switch-on position. The electric motors are mounted on the same shaft, and therefore to avoid damage the DSD-2 motor must be switched off before the SD-09 motor is switched on. The electrical circuit does not permit simultaneous switching on of both motors. After the plate has begtm to rotate the torque through the sample acts on the cone, which also begins to rotate, because before the start of the experiment the cone is not fixed. Under these conditions no deformation of the specimen occurs. This continues until the stop bar 10 comes to rest against the protruding core of the electromagnet 12. The moment of contact of the bar and core marks the beginning of deformation precisely, because at that point the cone is stopped and the plate continues to rotate, so that shear deformation of the specimen begins. I n addition a time relay, present for a given period of deformation, is switched on at the time of contact. After the set time the relay switches on the current in the coils of the electromagnet and the magnetic brake. The core of the electromagnet is thus retracted inside the coil and the stop bar and cone, which is firrhly a t t a c h e d to it, are allowed free rotation. Because the electromagnetic brake 5 is switched on at the same time the plate becomes detached from the driving shaft (the teeth o f ~ e d0g-cldtch become disengaged) and it stops in less t h a n 0" 1 see. Then movement of the stop bar and its attached mirror is caused only by elastic forces set up in the specimen and the rotation=of the bar corresponds to the elastic deformation retained in the specimen. The angle of rotation of the bar is measured by means of an optical system (lamp-mirror-scale). The size of the angle is a measure of the highelastic deformation. By varying the transmission ratio in the gear box (each step of which corresponds to a 5 : 1 reduction in speed), by changing over the electric motors (the rate of rotation of the DSD-2 motor is 2 rev/min and of the SD-09 motor 3000 rev/min) and by exchanging cones (the set contains cones with angles from 168 to 177 ° at the apex) it is possible to produce shear rates in the specimen from 10 -~ to ~ l0 S sec -1. Regnlation of the time relay permits setting up of deformation times from 1 see to 1 rain. If longer periods o f deformatiou axe required the time relay is switched out and the system is controlled manually. By means of the measuring system used the angle of rotation of the axis of the cone can be-ascertained with a precision of the order of 0.1 °, which is quite sufficient for the present. purpose because this corresponds (with a cone with an angle of ~174 °, which was the one mainly used) to approximately 3~o deformation of the m a t e r i a l . E s t i m a t i o n of the accuracy and reproducibility of the results showed that the total error does not exceed 10% of the deformation. I n order to obtain reliable values of elastic deformation
High-elastic deformation in viscous flow of b u t y l rubber
1389
i t is very i m p o r t a n t to follow the elastic recovery for a long time, sometimes for a m a t t e r of days. I n this connection heating up of the specimen can be of considerable help, because when heated it reverts much more rapidly to the unstressed state and the total measured elastic deformation is independent of the heating routine (see below). I n general, and as a rule, the duration of the complete spectrum of high-elastic deformation is greater the higher the viscosity of the material.
8 7
Q
_×
....
x
2 _ j ~ , - - - _- -_- × - - ~ - ~ - - ~
6 ~zx
5
x ! r~ 2
4J 3
0
I
Z
I
1
5
fO
f5
20
II
I
50 Time , min
I
100
I
150
II
I
300
I
900
FIG. 2. Kinetics of elastic recovery after cessation of deformation: / - - r e p e a t e d experiments at 22°; 2 - - e x p e r i m e n t at 22 °, elastic recovery measured at 80 ° (the point of attainment of equilibrium of 7e with the heater switched on is shown b y the arrow). Continuous line-value of 7e after equilibrium. The deformation of the material, 7, was calculated b y means of the formula ?=o)~-lt, where co is the angular velocity of the plate, t the period of deformation and 8 the angle between the cone and plate. The ratio co/e= 7"is the rate of deformation. The elastic deformation, ~e, was dalculated b y means of a similar formula, 7e=~/~, where ~ is the experimentally measured angle of rotation of the cone after cessation of deformation. Figure 2 shows a typical curve of the dependence of elastic recovery on the time of keeping the specimen after cessation of deformation. This graph includes d a t a from a few independent measurements, from which the reproducibility of the results can be judged. The measurements of elastic deformation shown in Fig. 2 relate to cessation of deformation during equilibrium flow conditions at total deformation of from 250 to 800 units. The effect of heating during elastic recovery is also seen from these results. Thus the measuring system permits determination of the total elastic deformation in a material when it is deformed at a constant rate of shear for different set times (corresponding to different total deformations) at different temperatures, and also study of the kinetics of elastic recovery. RESULTS AND DISCUSSION The build up of elastic deformation under a constant rate of shear (7=const) is v e r y r e m i n i s c e n t o f t h e d e v e l o p m e n t o f s h e a r a n d n o r m a l s t r e s s e s in t h i s m a t e r i a l [4], w h i c h is i n g e n e r a l f a i r l y t y p i c a l o f v i s c o u s - f l o w i n g p o l y m e r i c s y s t e m s in t h a t a s t h e t o t a l d e f o r m a t i o n i n c r e a s e s t h e h i g h - e l a s t i c c o m p o n e n t g r a d u a l l y i n c r e a s e s . T h e n i f t h e r a t e o f s h e a r is l o w 7e s t o p s a t s o m e e q u i l i b r i u m v a l u e . W h e n t h e r a t e o f s h e a r is h i g h e n o u g h t h e n a t u r e o f t h e 7e(7) r e l a t i o n s h i p c h a n g e s , i.e. 7~ p a s s e s t h r o u g h a m a x i m u m a n d t h e n f a l l s t o a n e q u i l i b r i u m v a l u e . R e d u c t i o n i n t e m p e r a t u r e h a s a n effect s i m i l a r t o t h a t o f
G. V. VI~OG~DOV e$ a~.
1390
increase in the rate of shear. Typical experimental results illustrating the different types of 7e(7) relationship are presented in l~ig. 3. I n reference [4] it was shown t h a t the a t t a i n m e n t of the equilibrium values of the shear stress, ~, a n d of the ~fference in the normal stresses, a, requires
7"5 !
5"0 2"5
/
I
0
I
50
I
fO0
I
150
i
I
I
200 300 ~00
FIG. 3. Typical curves of the dependence of y, on the total deformation, 9, at various tern. peratures: 1,2--25°; 3--60°; 4--100% 7=0.34 sec -1 (continuous lines) and 0.0655 sec -z (broken line). quite different times. Comparison of the v(7), a(7) a n d ?~(7) curves (Fig. 4) shows t h a t in this respect the 7e(7) relationship occupies an intermediate position between ~(7) a n d a(~). F o r example at a rate of shear of 0.34 see-Z a n d a tempera-
2e and ~/2v
frlS/OITt2
!
o--Ip--.o-
3"0
2'O 3
It
Zl
w~
0
J
I
20
I
I
~
]
o
o-41- --o-
I
60
fO
1~901~ 0
7 FzG. 4. Comparison of the functions a (y) (1), 9e (7) (2), ~ (7) (3) and y~ (7) (4) at 100° and ~= 0-34 sec -z. ture of 100 ° the equilibrium values of v are reached at T~8, of a at 7 ~ 6 0 and of ~e at 7 ~ 28. The same graph (Fig. 4) shows the dependence of a/2v on 7. According to some theories [5] the q u a n t i t y ~6=a/2~ corresponds to 7e- I n particular this correspondence always holds for equilibrium flow conditions in the linear re-
High-elastic deformation in viscous flow of butyl rubber
1391
gions of mechanical behaviour [6]. The experimental data in Fig. 4 show, however, t h a t there is no general correspondence between 7e and 7'e. The values of 7~ and 7'e differ very considerably in the transitional deformation region. Attention has previously been given to this [7] in a study of dilute polymer solutions. In the example given in Fig. 4 a marked difference between 7e and 7'e is maintained even under equilibrium flow conditions.
0"5
0
10
20
30
O0
5O
2,sec-' FIO. 5
Fro. 6
FIG. 5. Variation in the ratio of high-elastic and plastic deformations in the course of flow at temperatures of: 1 - 25°; 2--60°; 3--100 °. 7~0.34 see-1 (continuous lines) and 0.0655 sec-1 (broken line). " FIo. 6. Dependence of re (continuous lines) and 7 " e : a / 2 v (broken lines) on log ~ under equilibrium flowconditions at temperatures of: 1 , 1 ' - - 25°; 2,2" - - 60°; 3,3" - - 80°; 4 , 4 ' - - 100°; 5 , 5 ' -- 120°. I t should be recognized t h a t the relationship between the three characteristics of a polymer, namely high-elastic deformation, shear stress and normal stresses, can still not be established. This conclusion is valid not only for transitional deformation conditions, but also for equilibrium flow conditions, as will be shown below. In equilibrium flow the plastic (irreversible) component of the total deformation considerably exceeds the high-elastic component, which according to our measurements for :BR never exceeds a few units. However, under transitional conditions the ratio of the plastic to the high-elastic component can be quite different. I n answer to the question of the ratio of these components of the total deformation Fig. 5 shows experimental data on the dependence of t h e ratio 7 e ] 7 ~ - T e / T t on 7 for one rate of shear (0.34 sec -1) and three temperatures, and for one temperature (25 °) and two rates of shear. I t is seen from Fig. 5
G. V. VXNOGRADOVet al.
1392
that in the initial period of deformation which lasts a fairly long time (up to ~ 100% deformation), there is practically no flow and all the deformation is reversible. As the temperature is raised and the viscosity of the polymer is correspondingly lowered plastic deformation begins earlier and proceeds more rapidly than at lower temperatures; however in these eases the general relationship that up to ~ 1 0 0 % only high-elastic deformation is developed is preserved. /o9 C~,dyne/cm2
~'J~
!
I
l--1
I
0
I
zo#2, sec-'
1
FIG. 7. D e p e n d e n c e o f t h e h i g h - e l a s t i c i t y m o d u l u s on rate of shear at t e m p e r a t u r e s of: 1--25°; 2--60°; 3--80°; 4--100°; 6 - - 1 2 0 °.
I t is seen from Fig. 5 that eYen at 100 ° and a rate of Shear of 0.34 sec -1 at a total deformation of 400% half of it consists of the high-elastic component. Decrease in the rate of shear affects the ratio of the reversible and irreversible components ~in the same way as increase in temperature. Let us now consider in more detail the nature of the variation in high elastic deformation under equilibrium conditions in relation to temperature and rate of shear. First we shall consider only equilibrium flow conditions in the sense of attainment of time-invariant values of 7~, and also r and a on further prolonged deformation. The basic experimental data, obtained at rates of shear from 10 -2 to 10 sec -z and temperatures from 25 to 120 ° are represented b y continuous lines in Fig. 6. I t is obvious that the high-elastic deformation increases when the rate of shear is increased at constant temperature or when the temperature is reduced at constant rate of shear. The dependence of ~'e on 7, calculated from the data of reference [4] for the same region of shear rates and temperatures, is represented in this graph b y broken lines. As was mentioned above the quantity 7'~=a/2T should according to some theoretical ideas correspond to 7~. According to other theories (for example those of references [8] and [9]) correspondence between high-elastic deformations and stresses is found if ?e and 7~'=a/v are compared (obviously 7~'=27'e). Replacement of 7'e b y 7~' in Fig. 6 means shifting the broken lines upward b y a factor of two. Comparison of 7~ with 7'e and 7~ shows that in the studied range of temperatures and shear rates the experimental and theoretical values of the high-elastic deformation diverge, the divergence being greater the higher the rate of shear. Opinion in the literature is divided on the question of the correlation between 7~ and y: or y':. Thus
1393
High-elastic deformation in viscous flow of butyl rubber
in a study of dilute solutions of polyisobutylene, natural rubber and aluminium naphthenate [7] it was found that under equilibrium flow conditions 7~=7~. Philippoff and his collaborators [10, 11] assumed that 7e=7~. The experimental results of this work show that in fact the position with regard to the correlation
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o! o2 03
3
2
!
~
1 1
2
I
3
I
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I
5"
t
6
I
7
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FzG. 8. Dependence of high-elastic deformation on shear stress at temperatures of: 1--25°; 2--60°; 3--80°; 4--100°; 5--120 °. between stress and high-elastic deformation is more complex. It is seen from Fig. 6 that in the nonlinear region, fairly far from Newtonian flow conditions, 7~>~'>7'e. On the other hand it has been shown theoretically [6] that in the linear region 7e~7'e- Extrapolation of the data obtained here to ~->0 confirms the validity of this conclusion. It m a y thus be assumed that at very low rates of shear 7e=-a/2T, then as the rate is increased 7~ increases more rapidly than a/2T, and in a certain region of shear rates ?,e,,~a/~. Finally at still higher rates of shear 7~>a/~. The region of values of 7 to which one or other of these relationships applies is dependent on the material under test. All these problems require further refinement and experimental elucidation, b u t it may be affirmed with certainty that the theoretical formulae do not in the general case correctly describe the relationship between high-elastic deformation and stress. Therefore, the position with regard to the connection between stress and high-elastic deformation requires further refinement and study. The direct experimental measurements made here could be used to calculate the high-elastic modulus of the polymers in flow, G~, which is given b y the equation Ge-=~/?J e . The values of Ge at various rates of deformation and temperatures are shown in Fig. 7. The general trend of the Ge(7, T) relationships is seen fairly clearly from Fig. 7 and does not require further discussion.
1394
G. V. VI~0GRADOVe~~.
I t is very important to consider the high-elastic properties of flowing polymer systems in relation not to the rate of shear, as was done above in order to obtain a clear, general picture of the development of large elastic deformations, but ~',/0 -,Sdyne/cme
i
"
i
3OO
I
35O
I
4~0 T,°K
FIe. 9. Temperature dependence of shear stress at ?,=cons~, y~= l (1); 2 (2); 3 (3); 3.5 (4) units. to shear stress: Consideration of this is important primarily because stress is a factor determining both the viscosity properties of the polymer and the development of the normal stresses [4]. This follows from the possibility of construction of temperature-invariant characteristics of viscosity and the normal stresses, taking shear stress as the argument. Experimental data on the dependence of high-elastic deformation on shear stress in the studied temperature range are presented in Fig. 8. It is evident that there is a general tendency for the high-elastic deformation to increase as the shear stress is increased, but the 7e(~) relationship is rather complex. According to the entropic theory of high-elasticity, and similarly according to the i~ooney-Rivlin theory [12], one would expect a linear relationship between 7e and stress in simple shear. It is seen from Fig. 8 that this is in fact not so. Therefore, the so-called Hooke's law of shear, according to which in the general case ?e ~T, is not followed and the question of its applicability to individual special cases must be answered individually for each particular polymer and conditions of deformation. A still greater divergence between theoretical prediction and experimental results is found on examination of the temperature dependence of ~ when ?~~ const (Fig. 9). According to the entropie theory of high-elasticity the T(T) relationship should give a slightly rising curve. The complex nature of this relation-
High-elastic deformation in viscous flow of butyl rubber
1395
ship actually observed is due primarily to a structural transition occurring at about 60 °. Attention was first drawn to this transition in reference [13], where it was found t h a t there is a mechanical loss maximum in the region of 60-70 °. This was confirmed later by measurements made in our laboratory (Ye. A. Dzyura and L. I. Ivanova) on the sample of B R studied in the present work. The stress-temperature relationship also indicates the occurrence of a transition in BR at 60-70 ° [4]. I t is probably this transition, specific to BR, t h a t leads to the well-defined maximum near 60 ° in the dependence of T on T when 7e=const. However, the observed nonlinearity of the 7e(T) relationship shows t h a t high-elastic deformation in polymer systems in the viscous-flow state develops in a manner rather different from t h a t in crosslinked elastomers (vulcanized rubbers), quite apart from the occurrence of this transition. CONCLUSIONS
(1) Direct measurement has been made of high elastic deformation during flow of a typical highly viscous polymer--plasticized butyl rubber. These measurements were made in both the transitional deformation region and under conditions of equilibrium flow, at various rates of deformation and temperatures. (2) Comparison is made between the high-elastic deformation and the stresses arising during flow at a fixed rate of shear. I t was found t h a t the measured elastic deformations are not in agreement with the deformations calculated from the conventional theories involving the normal and shear stresses, and this divergence exists not only in the transitional deformation region. (3) In simple shear the modulus of high-elasticity increases with increase in the shear stress, i.e. "Hooke's law of shear" is not obeyed. (4) The observed types of dependence of high-elastic deformation on shear stress and temperature (the nonlinearity of the dependence on shear stress and the existence of an extremum in the temperature dependence) show t h a t the elasticity of uncrosslinked butyl rubber cannot be described within the framework of the concepts of the standard entropic theory of the high-elasticy of crosslinked rubbers. Translated by E. O. PHILLIPS REFERENCES
1. 2. 3. 4. 5. 6. 7. 8.
H. LEADERMAN, R. G. SMITH and R. W. JONES, J. Polymer Sci. 14: 47, 1954 H. LEADERMAN, R. G. SMITH and L. C. WILLIAMS, J. Polymer Sci. 36: 233, 1959 R. McCORD and B. MAXWELL, Modern Plastics 39: 116, 1961 G. V. VINOGRADOV, A. Ya. MALKIN and V. F. SHUMSKII, Vysokomol. soyed. A10: 2672, 1968; All: 663, 1969 A. S. LODGE, Elastic Liquids, Academic Press, London-New York, 1964 A. Ya. MALKIN, l~heol. Acta 7: 335, 1968 A. S. MOROZOV, Dissertation, 1967 N. I. MALININ, Kolloid. zh. 22: 201, 1960
N.I. ZHrR~ov et at.
1396
9. A. JOBLING and J. E. ROBERTS, Rheology (Ed. F. Eirich) 2: 503, New York, 1958 10. J. G. BRODNYAN, F. H. GASKINS, W. PHILIPPOFF and E. G. LENDRAT, Trans. Soc. Rheol. 2: 285, 1958 11. W. PHILIPPOFF, Trans. Soc. Rheol. l0 (1): 1, 1966 12. L. TRELOAR, Fizika uprugosti kauchuka (Physics of Rubber Elasticity). Foreign Literature Publishing House, 1953 13. S, GEHMAN, Rubber Chem. and Technol. 30: 1202, 1957
ESTIMATION OF THE THEORETICAL STRENGTH OF ORIENTATED POLYETHYLENE* N, I. ZHIRNOV,YE. G. KORYAK-DOROI~E:~KOand G. 1~I. BARTEI~EV V. I. Lenin Pedagogical Institute, Moscow
(Received 26 January 1968) 1KUCH experimental material has now accumulated on the strength properties of polyethylene (PE), which are dependent on temperature, the rate of stretching and other factors [1]. With regard to the theoretical strength of PE we know only of reference [2] in which a rough assessment was made of the strength of PE for stretching of the molecules along one of the C--C bonds, without account being taken of deformation of the valence angles. In reference [2] use was made of the Morse potential [3] in the form
U l(r)=De-2~(r-ro)-2De- ~(~-~o),
(1)
where ~ = ~ / 2Kr D
(2)
Here D is the energy of dissociation of the C--C bond, determined experimentally from the heat of combustion, r 0 the equilibrium internuclear distance and K~ a force constant characterizing the valency vibration of the C--C bond and determined from optical data. The stress applied to the molecule when it is stretched along the C--C bond is found by differentiating (1) with respect to r
a*----
dr
-~2aDe- ( -°)[1--e-~(r-r°)].
* Vysokomol. soyed. All: No. 6, 1229-1233, 1969.
(3)