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Approximation of stress relaxation curves for polyoxadiazole and polyimide with the aid of new relaxation nuclei
1784
A . A . Asg.~sgu et at.
REFERENCES 1. Zh. A. ABILOV, Diss . . . . kand. khim. nank, In-t khim. nauk Akad. Nauk Kaz.SSR, Alma-Ata, 1982 2. K. V. MUSABEKOV, S. B. AIDAROVA and Dz. A. ABILOV, 5th Yugoslav. Symp. Surfaceactive Substance. p. 265, 1981 3. K. B. MUSABEKOV, R. Ye. LEGKUNETS and V. G. PALER, Kolloid. zh. 42: 1189, 1980 4. V. N. BERESNEV, L. R. KHACHATUROVA, N. A. FERMOR and N. I. SMIRNOV, Zh. prikl, khim. 38: 2655, 1965 5. V. N. IZMAILOVA and P. A. REBINDER, Strukturoobrazovaniye v belkovykh sistemakh (Structure Formation in Protein Systems). Moscow, 1974 6. V. S. PSHEZETSKH, G. A. MURTAZAEVA and V. A. KABANOV, Europ. Polymer J. 10: 581, 1974 7. G. MORAVETS, Makromolekuly v rastvore (Macromolecules in Solution). Moscow, 1987 8. A. KITAKARA and K. KON-NO, Mitsslloobrazovanyie, solubilizatsiya i microemulsii (Micelle Formation, Solubilization, and Micro,emulsions). Moscow, 1980 9. S. SAITO and H. HIRATA, Kolloid-Z B165: 162, 1959 10. T. I. YUI TSUN-SIN, V. G. PALMER and K. B. MUSABEKOV, Izv. Akad. Nauk KazSSR Set. khim., 2, 19, 1984 11. T. L YUI TSUN-SIN, V. G. PALMER and K. B. MUSABEKOV, Kolloid. zh. 49: 819, 1987
PolymerScicmceU.S.S.R.Vol. 30. No. 8, pp. 1784.-1791, 1988 Printed in Poland
0032-3950/88$10.00+.00 © 1989PergamonPress plc
APPROXIMATION OF STRESS RELAXATION CURVES FOR POLYOXADIAZOLE AND POLYIMIDE WITH THE AID OF NEW RELAXATION NUCLEI* A. A. ASKADSKH, R. B. BANYAVICHYUS, Z. S. VIKHAUSKAS, A. I. MARMA a n d A. L. BLYUMENFEL'D* A. N . Nesmeyanov Institute of Heteroorganic Compounds, U.S.S.R. Academy of Sciences
Antanas Snechkus Polytechnical Institute, Kannas Received 23 March 1987
The reMxational properties of two heat-resistant polymers, polyoxadiazole and polyimide are studied. In the linear region the stress relaxation curves are approximated with the aid of new relaxation nuclei, allowing for the change in entropy of the system during relaxation. Use of these nuclei provides a means not only of obtaining a good approximation of the relaxation curves, but also of assessing the physical nuclei parameters, which provides a basis for drawing conclusions as to the mechanism governing the relaxational process. * Vysokomol. soyed. A30: No. 8, 1684-1689, 1988.
Approximation of stress relaxation curves for polyoxadiazoleand polyimide
1785
ASKADSKU [1] proposed new relaxation nuclei, based on analysis of the changes in entropy of a system during stress relaxation. Two nuclei were proposed, i.e. T~(t) and T2(t), which were derived using the various mechanisms available for controlling the changes in entropy during relaxation as a starting point. The nucleus Tx(t) is obtained on the assumption that the interaction of relaxation oscillators, especially various types of microcavities of polymer units or their segments, etc., i.e. all kinetic units which can undergo interaction and restructuring, are the limiting stage of the process. For example, various microcavities in a material subjected to initial deformation can merge into a single microcavity and pass into an unrelaxing material, where the relaxation processes have already been completed. Consideration of the kinetics of such interaction in the form of an n-th order reaction, with allowance for the change in entropy of the system because of mingling of relaxation oscillators and non-relaxation oscillators, produces a relaxation unit of the following type Tl(t)=
So{
1
1 }
- k-a-m~ [fl(t)- ~o] In [fl(t)- %] + [1 "f~(t) + %] In [l -f~(t) + %]
In ()'5
'
(1) where f l ( t ) = 1/(1 +k'tiff) #, while f~(t) has physical significance only when it is I>0.5. The quantity ~o is constant, and determines the proportion of relaxation oscillators changing to non-relaxation oscillators over the small deformation time; this constant is ,-, I0-lo [1]. The other constants have the following meaning: S o is the initial entropy of the system, k8 is the Boltzmann constant, ml is the total number of kinetic units (relaxation oscillators and non-relaxation oscillators) in nucleus volume, k*=kcg-1, k is an interaction rate constant, Co is the initial concentration of relaxation oscillators (if the concentration is expressed by the number of kinetic nucleus, then Co=m1), and n is the reaction order, f l = ( n - 1 ) - L The second relaxation nucleus is obtained on the assumption that diffusion of the non-relaxation oscillators formed in the specimen material is the limiting stage. The relaxation nucleus then has the form [1]
T2(t) =
So I
- kB m-~
1
~f2(t)In f2(t) + [1 --f2(t)] In [l -f2(t)]
1 1
In ~3-5 '
(2)
where f2(t)=at ~, and indicates the number of sites occupied in a moment of time t by the kinetic nucleus during their disordered wandering to the lattice [2], a and ), parameters of the system. At the initial section of the stress relaxation curve, when k*t/fl<
So 1 ks ml (k*t + ~o) [In (k*t + ~o) - 1]
(3)
1786
A. A. AsK~ost.rl et aL
Under these conditions T2(t) is also simplified to
,o[
,
,]
T2(t) = - ks m2 aP(lnat ~ - 1) In 0.~ In order to use the functions Tl(t) and T2(t) for approximating the stress relaxation curves they must be inserted in the Boltzmann equation, which can be written in the form t
tr(t)=a o [ 2 - ~ T(~)dz]
(5)
0
(where tro is the initial stress). On inserting the functions Tl(t) and T2(t) in the above equation and converting it to the relaxation modulus, the equations obtained are t
Er( t) = Eo I I
So
tO t
(7) 0
where T*(r) and T~(r) are the variable parts of the Tl(t)and T2(t) nucleus, and 1
1
[ f l ( ~ ) - ct0] In [A(~) - ~0] + [1 -fl(~) + cto] In I-1-flCZ) + ~t0] In 0"5 and T~*(~) =
1
1
f2(z) lnf2(z)+ [1 --f2(~)] ha [1 --f2(z)] In 0"5
If (6) and (7) give the course of the stress relaxation curves correctly, in the coordit
nares a -
t
S T*(~)dr or a - S T2*(~)d~ then straight lines should be formed of slope 0
0
~roSo/k~ml o r ~So/kBm~. In order to use (6) and (7) it is essential to know the values of these integrals. The functions T*(z) and T~(r) can only be integrated in the general form by numerical methods. This procedure was used by Askadskii et aL [3] for various pairs of values of the parameters k* and fl in the case of the function T*(~), and a and ~, in the case t
t
of the function T~'(r). As a result, tabulated values of ~ T*(z)d~ and ~ T*(~)dr. were obtained, o o These values were used by the authors for calculating tke relaxation curves for polyoxadiozole (POD) and polyimide (PI).
Approximation of str e ss relaxation curves for polyoxadiazole and polyimide
1787
The calculation procedure is as follows: the tabulated values of the integrals f T*(r)dr and v
and
j T*(Q dr are recorded in a computer memory in the form of two maxima. Each block con-
tains values of one of these integrals at different upper limits of t, and each series of integrals at different valueg of t was found for the given ~ pair of parameters k*, fl or ~, 7. The parameters k*, fl, and 7 were selected with a specific interval so as to encompass all possible relaxation times from 0"5 to lO s min. Each stress relaxation curve, i.e. the experimental relation tr(t), was approximated by a linear equation in accordance with (6) or (7), using the least squares method: with the aid of the computer the values of the pair of parameters k*, p, and ct, y, for which the sum of the squares of the deviations of the experimental stresses from the theoretical values were a minimum, and the correlation coefficient was a maximum. The test results for blocks of POD and PI specimens are described by Vikhauskas et aL [4]. The stress relaxation curves were measured on specimens of size 3 x 3 x 4.5 mm under uniaxial compression conditions, with an initial specimen deformation rate of 1-5 mm/min. The compression curves for the specimens were determined previously, and from this the limit of the forced elasticity and the corresponding deformation value were found. These data were used as preliminary data for evaluating the deformation range over which the phenomenon of forced elasticity is observed. The experiments on stress relaxation were carried out at each selected temperature over a.wide range of initial deformations, encompassing the regions before and after the appearance of forced elasticity. Accordingly, at each temperature a series of stress relaxation curves is obtained both in the linear and clearly non-linear regions of mechanical behaviour. In the case of POD the test temperature range was 200-300°C, and for PI 20-320°C. The upper test temperature limit was approximated to the glass transition temperature of each of the polymers. To reveal the regions of linear and non-linear mechanical behaviour the stress relaxation curves were reconstructed so as to give the relaxation modulus Er=a(t)/eo as a function of log t. If the experimental curves in the given coordinates can be packed into a narrow bunch, this indicates linear mechanical behaviour. A significant deviation from this bunch indicates a transition to non-linear behaviour.
F i g u r e 1 shows the r e l a t i o n between Er a n d log t for P O D a n d P I at 20°C. T h e d a t a c o r r e s p o n d i n g to t h e linear region o f m e c h a n i c a l b e h a v i o u r was a v e r a g e d to give a single r e l a x a t i o n relation. I t can b e seen f r o m Fig. 1 t h a t a d e v i a t i o n f r o m the averaged r e l a t i o n in the d i r e c t i o n o f a decrease in E~ f o r P O D at 20°C is o b s e r v e d when Eo = 5.6 ~o. A t ao values o f 1.7 to 5"0 ~ linear m e c h a n i c a l b e h a v i o u r is observed. I n terms o f this c r i t e r i o n P O D surpasses o t h e r k n o w n p o l y m e r s for which the region o f n o n - l i n e a r b e h a v i o u r occurs a t significantly lower d e f o r m a t i o n s . I n t h e case o f P I d e v i a t i o n s f r o m l i n e a r m e c h a n i c a l b e h a v i o u r are a l r e a d y o b s e r v e d a t ~o = 4 ~ (Fig. lb), a n d t h e region o f linear m e c h a n i c a l b e h a v i o u r e n c o m p a s s e s values o f e0 o f 1.7 to 3 . 4 ~ , i.e. significantly less t h a n the c o r r e s p o n d i n g r a n g e f o r P O D . A t h i g h e r t e m p e r a t u r e s the d e f o r m a t i o n r a n g e in the region o f linear b e h a v i o u r is decreased, b u t n o t u n i f o r m l y . F i g u r e 2 shows the limiting d e f o r m a t i o n ~1, f o r which linear viscoelastic b e h a v i o u r is a l r e a d y observed. I n the case o f P O D et is significantly higher t h a n f o r PI. A small decrease in el with rise in t e m p e r a t u r e is a c o m m o n f e a t u r e o f all these p o l y m e r s . T h e r e is a t e m p e r a t u r e region for which ex is i n d e p e n d e n t o f t e m p e r a t u r e , after which, o n a p p r o a c h i n g the glass t r a n s i t i o n t e m p e r a t u r e , e~ is decreased,
A.A. ASr..AUSFdI¢t al.
1788
E~ OPa 2.7,
2.2,
1"75 O'E
2
8
qO
b
~x5 2.0
1.0 O.E
i
I 2
j S
~ 20
" T " - . . ~ t2 110 t',min
Fie. 1. Relaxation modulus Er as a function of log t at 20°C for POD (a) and PI (b) in the linear (1) and non-linear region (2-12). a: /-averaged unit; 8=5.56 (2), 6.67 (3), 7.23 (4), 7.78 (5), and 8.33~ (6) b: 1 -averaged unit; e--4.46 (2), 5.02 (3), 5.56 (49, 6.82 (5), 8.33 (6), 8-99 (7), 10.23 (8), 11.57 (9), 12.36 (10), 13.95 (11), and 15.34y0 (12). The results of approximating the stress relaxation curves in the linear region o f viscoelastic behavour will now be considered. The approximation was carried out for the plots o f E, against t, averaged over the linear region. Both nucleus Tl(t) and T2(t) were used for the approximation, and from (6) and (7) straight lines are obtained in the t
coordinates
t
E~- S T~(T)d~ and E~- ~ T~(z)dz. The results o f t h e approximation O
O
are given in Tables 1 and 2, where the values of the correlation coetficients r are given.
Approximation of stress relaxation curves for polyoxadiazole and polyimide
1789
ct %
-
\
I
l
I
100
200
300
\
\ \ \x
I
To
qO0
FIG. 2. Limiting deformation 8x, up to which linear viscoelastic behaviour is maintained, as a function of temperature for POD (1) and PI (2). The nucleus Tl(t) is the most useful for studying polymers; in using this nucleus the correlation coefficient is close to unity, and good agreement is obtained between the experimental and calculated values of Er (Fig. 3). Approximation by means of the nucleus T2(t) results in a significant decrease in the correlation coefficient. Since, as mentioned above, the Tl(t) nucleus describes the kinetics of interaction of the relaxation oscillators and their passage into the non-relaxing material, and the nucleus t2(t) the kinetics of diffusion of these kinetic units in the material, it can be concluded that TABLE 1. KINETIC PARAMETERS OF STRESS RELAXATION PROCESS FOR P O D , TWO RELAXATION NECLEUS
Tl(t)
AND
Nucleus Ta(t) T°
Eo, t mike GPa I ~
20 70 120 170 220 270 300
2"9144 I 1265 0"99761 0"4 2"7504 ] 1660 0"99311 0"8 2"6444 1670 0"99961 0"4 2"6049 1617 0'99661 0"6 2-52281 10,945 0"99601 0"2 2-7575 299,728 0"9986 0"2 1"8074 272 0"9934 0"3
r
fl
k*, min - 1 0-01 0"01 0"01 0"01 0"001 0.00001 0"01
n 3"50 2"25 3"50 2"67 6.00 6.00 4-33
AS DETERMINED BY USING
Tz(t)
Eo, G Pa 2"6158 2"5313 2"4394 2"3929 2"2933 2"2376 0"9532
Nucleus T2(t) m2kB -~o r a 428 631 579 601 613 175 52
0"9843 / 0-05 0'9627 [ 0"05 0"9888 [ 0"05 0"9753 [ 0"05 0"9946 t 0"401 0"9903 / 0-05 0"9762 / 0'05
r 0.50 0.50 0.50 0.50 0.50 0.50 0.50
in the given case the rate o f the relaxation process is restricted by the kinetics of relaxation oscillator interaction. Only the kinetic parameters of the Tl(t) will thus be analyzed. In the case of POD (Table 1) the initial elastic modulus Eo is somewhat decreased with rise in temperature, after which a temperature region is observed where Eo is almost independent of temperature; as the glass transition temperature is approached Eo is decreased. The value of k*, indicating the rate constant for the interaction of relaxation oscillators, is constant over a wide temperature range; in this temperature region, where
1790
A. A. ASKADSrdl et aL
in the ease of POD a significant decrease in the deformation range of linear viscoelasticity is observed, the rate constant k* falls rapidly, but the order of the reaction n and the number of relaxation oscillators m, (Table 1) increase rapidly. Increase in the reaction order n means that a large number of relaxation oscillators enter into active "collision", i.e. a large number of relaxation oscillators take part in the elementary act of interaction (for example the fusion of several microcavities in a row into one). Er,GPa
CL 2.q2"4 ~
l
+
2.0
2"3 T 140
I
,] 10
,1
I
180
220
I
i
22
3q
i
i
IL/O
180
10
i
Z20 ~ +
e2
FIG. 3. Relaxation modulus E, as a function of J T* (r) dr or
a 4 "~r*(~') z
T* ( 0 dr (2) for POD (a) O
and PI (b). Tests carried out at 70°e.
In the case of PI the picture is approximately the same. The initial modulus Eo is somewhat decreased with rise in temperature, and the reaction rate constants k* show little dependence on the temperature, while increase in the interaction rate constant for the relaxation oscillators is accompanied by an increase in reaction order. TABI.~ 2. KINETIC PARAMETERS OF THE STRESS RELAXATION PROCESS fOR P I , AS DETERMINED BY USING THE TWO RELAXATION NUCLEUS T l ( t )
AND
T2(t)
Nucleus Tl(t) T° 20 70 120 170 220 270 320
Eo, GPa
mtkn So
r
B
3-3259 3.1109 3.2992 2.9620 3.0221 2.9483 4"6264
694 608 83 10,375 727 493 33
0-9999 0.9998 0.9856 0.9974 0.9934 0.9973 0.9988
0-6 0.3 0-2 0.2 0-5 0.8 0.2
Anomalies relaxation decreased. associated
k* 9
•Fo•
min -l
GPa
0'01 0'01 0"10 0"001 0"01 0"01 0"10
2"70 4"30 6"00 6"00 3"00 2"25 6"00
2.6984 2.4561 2.5816 2.6770 2.4791 2.1603 2.1391
Nucleus T2(t) m2 ks i i I i
So 223 172 249 583 233 149 60
r
a
0.9846 0.05 0"9928 0"05 0.9473 !0"05 0"9925 0"403 0"9720 0"05 0"9724 0"05 0"9575 0"05
0.50 0.50 0.50 0.50 0.50 0.50 0.50
are observed a t t h e temperatures 170 and 320°C. A t 170°C the number of oscillators rnl is significantly increased, but the interaction rate constant is A t 3 2 0 ° C the initial modulus Eo increases anomalously, which is evidently with the special kinetic features of the relaxation curve, i.e. a sharp fall in
Rigid phase in styrene-butadiene-styrene thermoelastoplasts
1791
stress over the initial section (high value of the rate constant) and with an unjustified extrapolation of er to t-~0. On the whole, the correlation coefficient for PI is also very high, and the relaxation curves are described by eqn. (6) with a high degree of accuracy. Translated by N. STANDEN REFERENCES 1. A. A. ASKADSKII, Mekhanika kompozitnykh materialov (Mechanics of Composite Materials). p. 403, 1987 2. R. d. GAYLORD, B. JOSS, d. T. BENDLER and E. A. DI MARZIO, Brit. Polymer J. 17: 126, 1985 3. A. A. ASKADSKII, A. L. BLYUMENFELD, Ye. G. GAL'PERN and A. L. CHISKTYAKOV, Vysokomol. soyed. A30: 886, 1988 (Translated in Polymer Sci. S.S.S.R. 30: 4, 904, 1988) 4. Z. S. VIKHAUSKAS, R. B. PANYAVICHYUS, A. I. MARMA and A. A. ASKADSKII, Mekhanika kompozitnykh materialov (Mechanics of Composite Materials). 1090, 1982
PolymerScienceU.S.S.R.Vol.30, No. 8. pp. 1791-1796, 1988 Printed in Poland
0032-3950/88$10.00+.00 0) 1989PergamonPress plc
FRACTURE A N D RESTORATION OF THE RIGID P H A S E IN S T Y R E N E - B U T A D I E N E - S T Y R E N E THERMOELASTOPLASTS * N. P. BESSONOVA, 'Y'U. K. GODOVSKII, O. V. KOVRIGA, S. N. CHVALUN and V. S. SHIRETS L. Ya. Karpov Physicochemical Research Institute
(Received 23 March 1987) The fracture and restoration of the rigid phase in styrene-butadiene-styrenethermoplasts is studied for isotropic and "single crystal" DST-30 specimens, using strain calorimetry, small-angle X-ray scattering, and dynamometric methods. It is shown that stretching is accompanied by breaking of the polystyrene cylinders at least up to 800%, and that their length affects the mechanical properties only up to 150-200yo. After load removal the typical supermolecular lattice of the material is rapidly restored, but quantitative restoration of the lattice parameters and the mechanical properties of the material is restricted by the slow growing together of the fragments of destroyed cylinders, which is mostly governed by a diffusion mechanism.
THREE-block styrene-butadiene-styrene (SBS) copolymers are classical thermoplastic elastomers. Their unique properties (combination of high strength and one thousand per cent deformability) are determined by the presence in the block-copolymer of thermo* Vysokomol. soyed. A30: No. 8, 1690-1694, 1988.
A . A . Asg.~sgu et at.
REFERENCES 1. Zh. A. ABILOV, Diss . . . . kand. khim. nank, In-t khim. nauk Akad. Nauk Kaz.SSR, Alma-Ata, 1982 2. K. V. MUSABEKOV, S. B. AIDAROVA and Dz. A. ABILOV, 5th Yugoslav. Symp. Surfaceactive Substance. p. 265, 1981 3. K. B. MUSABEKOV, R. Ye. LEGKUNETS and V. G. PALER, Kolloid. zh. 42: 1189, 1980 4. V. N. BERESNEV, L. R. KHACHATUROVA, N. A. FERMOR and N. I. SMIRNOV, Zh. prikl, khim. 38: 2655, 1965 5. V. N. IZMAILOVA and P. A. REBINDER, Strukturoobrazovaniye v belkovykh sistemakh (Structure Formation in Protein Systems). Moscow, 1974 6. V. S. PSHEZETSKH, G. A. MURTAZAEVA and V. A. KABANOV, Europ. Polymer J. 10: 581, 1974 7. G. MORAVETS, Makromolekuly v rastvore (Macromolecules in Solution). Moscow, 1987 8. A. KITAKARA and K. KON-NO, Mitsslloobrazovanyie, solubilizatsiya i microemulsii (Micelle Formation, Solubilization, and Micro,emulsions). Moscow, 1980 9. S. SAITO and H. HIRATA, Kolloid-Z B165: 162, 1959 10. T. I. YUI TSUN-SIN, V. G. PALMER and K. B. MUSABEKOV, Izv. Akad. Nauk KazSSR Set. khim., 2, 19, 1984 11. T. L YUI TSUN-SIN, V. G. PALMER and K. B. MUSABEKOV, Kolloid. zh. 49: 819, 1987
PolymerScicmceU.S.S.R.Vol. 30. No. 8, pp. 1784.-1791, 1988 Printed in Poland
0032-3950/88$10.00+.00 © 1989PergamonPress plc
APPROXIMATION OF STRESS RELAXATION CURVES FOR POLYOXADIAZOLE AND POLYIMIDE WITH THE AID OF NEW RELAXATION NUCLEI* A. A. ASKADSKH, R. B. BANYAVICHYUS, Z. S. VIKHAUSKAS, A. I. MARMA a n d A. L. BLYUMENFEL'D* A. N . Nesmeyanov Institute of Heteroorganic Compounds, U.S.S.R. Academy of Sciences
Antanas Snechkus Polytechnical Institute, Kannas Received 23 March 1987
The reMxational properties of two heat-resistant polymers, polyoxadiazole and polyimide are studied. In the linear region the stress relaxation curves are approximated with the aid of new relaxation nuclei, allowing for the change in entropy of the system during relaxation. Use of these nuclei provides a means not only of obtaining a good approximation of the relaxation curves, but also of assessing the physical nuclei parameters, which provides a basis for drawing conclusions as to the mechanism governing the relaxational process. * Vysokomol. soyed. A30: No. 8, 1684-1689, 1988.
Approximation of stress relaxation curves for polyoxadiazoleand polyimide
1785
ASKADSKU [1] proposed new relaxation nuclei, based on analysis of the changes in entropy of a system during stress relaxation. Two nuclei were proposed, i.e. T~(t) and T2(t), which were derived using the various mechanisms available for controlling the changes in entropy during relaxation as a starting point. The nucleus Tx(t) is obtained on the assumption that the interaction of relaxation oscillators, especially various types of microcavities of polymer units or their segments, etc., i.e. all kinetic units which can undergo interaction and restructuring, are the limiting stage of the process. For example, various microcavities in a material subjected to initial deformation can merge into a single microcavity and pass into an unrelaxing material, where the relaxation processes have already been completed. Consideration of the kinetics of such interaction in the form of an n-th order reaction, with allowance for the change in entropy of the system because of mingling of relaxation oscillators and non-relaxation oscillators, produces a relaxation unit of the following type Tl(t)=
So{
1
1 }
- k-a-m~ [fl(t)- ~o] In [fl(t)- %] + [1 "f~(t) + %] In [l -f~(t) + %]
In ()'5
'
(1) where f l ( t ) = 1/(1 +k'tiff) #, while f~(t) has physical significance only when it is I>0.5. The quantity ~o is constant, and determines the proportion of relaxation oscillators changing to non-relaxation oscillators over the small deformation time; this constant is ,-, I0-lo [1]. The other constants have the following meaning: S o is the initial entropy of the system, k8 is the Boltzmann constant, ml is the total number of kinetic units (relaxation oscillators and non-relaxation oscillators) in nucleus volume, k*=kcg-1, k is an interaction rate constant, Co is the initial concentration of relaxation oscillators (if the concentration is expressed by the number of kinetic nucleus, then Co=m1), and n is the reaction order, f l = ( n - 1 ) - L The second relaxation nucleus is obtained on the assumption that diffusion of the non-relaxation oscillators formed in the specimen material is the limiting stage. The relaxation nucleus then has the form [1]
T2(t) =
So I
- kB m-~
1
~f2(t)In f2(t) + [1 --f2(t)] In [l -f2(t)]
1 1
In ~3-5 '
(2)
where f2(t)=at ~, and indicates the number of sites occupied in a moment of time t by the kinetic nucleus during their disordered wandering to the lattice [2], a and ), parameters of the system. At the initial section of the stress relaxation curve, when k*t/fl<
So 1 ks ml (k*t + ~o) [In (k*t + ~o) - 1]
(3)
1786
A. A. AsK~ost.rl et aL
Under these conditions T2(t) is also simplified to
,o[
,
,]
T2(t) = - ks m2 aP(lnat ~ - 1) In 0.~ In order to use the functions Tl(t) and T2(t) for approximating the stress relaxation curves they must be inserted in the Boltzmann equation, which can be written in the form t
tr(t)=a o [ 2 - ~ T(~)dz]
(5)
0
(where tro is the initial stress). On inserting the functions Tl(t) and T2(t) in the above equation and converting it to the relaxation modulus, the equations obtained are t
Er( t) = Eo I I
So
tO t
(7) 0
where T*(r) and T~(r) are the variable parts of the Tl(t)and T2(t) nucleus, and 1
1
[ f l ( ~ ) - ct0] In [A(~) - ~0] + [1 -fl(~) + cto] In I-1-flCZ) + ~t0] In 0"5 and T~*(~) =
1
1
f2(z) lnf2(z)+ [1 --f2(~)] ha [1 --f2(z)] In 0"5
If (6) and (7) give the course of the stress relaxation curves correctly, in the coordit
nares a -
t
S T*(~)dr or a - S T2*(~)d~ then straight lines should be formed of slope 0
0
~roSo/k~ml o r ~So/kBm~. In order to use (6) and (7) it is essential to know the values of these integrals. The functions T*(z) and T~(r) can only be integrated in the general form by numerical methods. This procedure was used by Askadskii et aL [3] for various pairs of values of the parameters k* and fl in the case of the function T*(~), and a and ~, in the case t
t
of the function T~'(r). As a result, tabulated values of ~ T*(z)d~ and ~ T*(~)dr. were obtained, o o These values were used by the authors for calculating tke relaxation curves for polyoxadiozole (POD) and polyimide (PI).
Approximation of str e ss relaxation curves for polyoxadiazole and polyimide
1787
The calculation procedure is as follows: the tabulated values of the integrals f T*(r)dr and v
and
j T*(Q dr are recorded in a computer memory in the form of two maxima. Each block con-
tains values of one of these integrals at different upper limits of t, and each series of integrals at different valueg of t was found for the given ~ pair of parameters k*, fl or ~, 7. The parameters k*, fl, and 7 were selected with a specific interval so as to encompass all possible relaxation times from 0"5 to lO s min. Each stress relaxation curve, i.e. the experimental relation tr(t), was approximated by a linear equation in accordance with (6) or (7), using the least squares method: with the aid of the computer the values of the pair of parameters k*, p, and ct, y, for which the sum of the squares of the deviations of the experimental stresses from the theoretical values were a minimum, and the correlation coefficient was a maximum. The test results for blocks of POD and PI specimens are described by Vikhauskas et aL [4]. The stress relaxation curves were measured on specimens of size 3 x 3 x 4.5 mm under uniaxial compression conditions, with an initial specimen deformation rate of 1-5 mm/min. The compression curves for the specimens were determined previously, and from this the limit of the forced elasticity and the corresponding deformation value were found. These data were used as preliminary data for evaluating the deformation range over which the phenomenon of forced elasticity is observed. The experiments on stress relaxation were carried out at each selected temperature over a.wide range of initial deformations, encompassing the regions before and after the appearance of forced elasticity. Accordingly, at each temperature a series of stress relaxation curves is obtained both in the linear and clearly non-linear regions of mechanical behaviour. In the case of POD the test temperature range was 200-300°C, and for PI 20-320°C. The upper test temperature limit was approximated to the glass transition temperature of each of the polymers. To reveal the regions of linear and non-linear mechanical behaviour the stress relaxation curves were reconstructed so as to give the relaxation modulus Er=a(t)/eo as a function of log t. If the experimental curves in the given coordinates can be packed into a narrow bunch, this indicates linear mechanical behaviour. A significant deviation from this bunch indicates a transition to non-linear behaviour.
F i g u r e 1 shows the r e l a t i o n between Er a n d log t for P O D a n d P I at 20°C. T h e d a t a c o r r e s p o n d i n g to t h e linear region o f m e c h a n i c a l b e h a v i o u r was a v e r a g e d to give a single r e l a x a t i o n relation. I t can b e seen f r o m Fig. 1 t h a t a d e v i a t i o n f r o m the averaged r e l a t i o n in the d i r e c t i o n o f a decrease in E~ f o r P O D at 20°C is o b s e r v e d when Eo = 5.6 ~o. A t ao values o f 1.7 to 5"0 ~ linear m e c h a n i c a l b e h a v i o u r is observed. I n terms o f this c r i t e r i o n P O D surpasses o t h e r k n o w n p o l y m e r s for which the region o f n o n - l i n e a r b e h a v i o u r occurs a t significantly lower d e f o r m a t i o n s . I n t h e case o f P I d e v i a t i o n s f r o m l i n e a r m e c h a n i c a l b e h a v i o u r are a l r e a d y o b s e r v e d a t ~o = 4 ~ (Fig. lb), a n d t h e region o f linear m e c h a n i c a l b e h a v i o u r e n c o m p a s s e s values o f e0 o f 1.7 to 3 . 4 ~ , i.e. significantly less t h a n the c o r r e s p o n d i n g r a n g e f o r P O D . A t h i g h e r t e m p e r a t u r e s the d e f o r m a t i o n r a n g e in the region o f linear b e h a v i o u r is decreased, b u t n o t u n i f o r m l y . F i g u r e 2 shows the limiting d e f o r m a t i o n ~1, f o r which linear viscoelastic b e h a v i o u r is a l r e a d y observed. I n the case o f P O D et is significantly higher t h a n f o r PI. A small decrease in el with rise in t e m p e r a t u r e is a c o m m o n f e a t u r e o f all these p o l y m e r s . T h e r e is a t e m p e r a t u r e region for which ex is i n d e p e n d e n t o f t e m p e r a t u r e , after which, o n a p p r o a c h i n g the glass t r a n s i t i o n t e m p e r a t u r e , e~ is decreased,
A.A. ASr..AUSFdI¢t al.
1788
E~ OPa 2.7,
2.2,
1"75 O'E
2
8
qO
b
~x5 2.0
1.0 O.E
i
I 2
j S
~ 20
" T " - . . ~ t2 110 t',min
Fie. 1. Relaxation modulus Er as a function of log t at 20°C for POD (a) and PI (b) in the linear (1) and non-linear region (2-12). a: /-averaged unit; 8=5.56 (2), 6.67 (3), 7.23 (4), 7.78 (5), and 8.33~ (6) b: 1 -averaged unit; e--4.46 (2), 5.02 (3), 5.56 (49, 6.82 (5), 8.33 (6), 8-99 (7), 10.23 (8), 11.57 (9), 12.36 (10), 13.95 (11), and 15.34y0 (12). The results of approximating the stress relaxation curves in the linear region o f viscoelastic behavour will now be considered. The approximation was carried out for the plots o f E, against t, averaged over the linear region. Both nucleus Tl(t) and T2(t) were used for the approximation, and from (6) and (7) straight lines are obtained in the t
coordinates
t
E~- S T~(T)d~ and E~- ~ T~(z)dz. The results o f t h e approximation O
O
are given in Tables 1 and 2, where the values of the correlation coetficients r are given.
Approximation of stress relaxation curves for polyoxadiazole and polyimide
1789
ct %
-
\
I
l
I
100
200
300
\
\ \ \x
I
To
qO0
FIG. 2. Limiting deformation 8x, up to which linear viscoelastic behaviour is maintained, as a function of temperature for POD (1) and PI (2). The nucleus Tl(t) is the most useful for studying polymers; in using this nucleus the correlation coefficient is close to unity, and good agreement is obtained between the experimental and calculated values of Er (Fig. 3). Approximation by means of the nucleus T2(t) results in a significant decrease in the correlation coefficient. Since, as mentioned above, the Tl(t) nucleus describes the kinetics of interaction of the relaxation oscillators and their passage into the non-relaxing material, and the nucleus t2(t) the kinetics of diffusion of these kinetic units in the material, it can be concluded that TABLE 1. KINETIC PARAMETERS OF STRESS RELAXATION PROCESS FOR P O D , TWO RELAXATION NECLEUS
Tl(t)
AND
Nucleus Ta(t) T°
Eo, t mike GPa I ~
20 70 120 170 220 270 300
2"9144 I 1265 0"99761 0"4 2"7504 ] 1660 0"99311 0"8 2"6444 1670 0"99961 0"4 2"6049 1617 0'99661 0"6 2-52281 10,945 0"99601 0"2 2-7575 299,728 0"9986 0"2 1"8074 272 0"9934 0"3
r
fl
k*, min - 1 0-01 0"01 0"01 0"01 0"001 0.00001 0"01
n 3"50 2"25 3"50 2"67 6.00 6.00 4-33
AS DETERMINED BY USING
Tz(t)
Eo, G Pa 2"6158 2"5313 2"4394 2"3929 2"2933 2"2376 0"9532
Nucleus T2(t) m2kB -~o r a 428 631 579 601 613 175 52
0"9843 / 0-05 0'9627 [ 0"05 0"9888 [ 0"05 0"9753 [ 0"05 0"9946 t 0"401 0"9903 / 0-05 0"9762 / 0'05
r 0.50 0.50 0.50 0.50 0.50 0.50 0.50
in the given case the rate o f the relaxation process is restricted by the kinetics of relaxation oscillator interaction. Only the kinetic parameters of the Tl(t) will thus be analyzed. In the case of POD (Table 1) the initial elastic modulus Eo is somewhat decreased with rise in temperature, after which a temperature region is observed where Eo is almost independent of temperature; as the glass transition temperature is approached Eo is decreased. The value of k*, indicating the rate constant for the interaction of relaxation oscillators, is constant over a wide temperature range; in this temperature region, where
1790
A. A. ASKADSrdl et aL
in the ease of POD a significant decrease in the deformation range of linear viscoelasticity is observed, the rate constant k* falls rapidly, but the order of the reaction n and the number of relaxation oscillators m, (Table 1) increase rapidly. Increase in the reaction order n means that a large number of relaxation oscillators enter into active "collision", i.e. a large number of relaxation oscillators take part in the elementary act of interaction (for example the fusion of several microcavities in a row into one). Er,GPa
CL 2.q2"4 ~
l
+
2.0
2"3 T 140
I
,] 10
,1
I
180
220
I
i
22
3q
i
i
IL/O
180
10
i
Z20 ~ +
e2
FIG. 3. Relaxation modulus E, as a function of J T* (r) dr or
a 4 "~r*(~') z
T* ( 0 dr (2) for POD (a) O
and PI (b). Tests carried out at 70°e.
In the case of PI the picture is approximately the same. The initial modulus Eo is somewhat decreased with rise in temperature, and the reaction rate constants k* show little dependence on the temperature, while increase in the interaction rate constant for the relaxation oscillators is accompanied by an increase in reaction order. TABI.~ 2. KINETIC PARAMETERS OF THE STRESS RELAXATION PROCESS fOR P I , AS DETERMINED BY USING THE TWO RELAXATION NUCLEUS T l ( t )
AND
T2(t)
Nucleus Tl(t) T° 20 70 120 170 220 270 320
Eo, GPa
mtkn So
r
B
3-3259 3.1109 3.2992 2.9620 3.0221 2.9483 4"6264
694 608 83 10,375 727 493 33
0-9999 0.9998 0.9856 0.9974 0.9934 0.9973 0.9988
0-6 0.3 0-2 0.2 0-5 0.8 0.2
Anomalies relaxation decreased. associated
k* 9
•Fo•
min -l
GPa
0'01 0'01 0"10 0"001 0"01 0"01 0"10
2"70 4"30 6"00 6"00 3"00 2"25 6"00
2.6984 2.4561 2.5816 2.6770 2.4791 2.1603 2.1391
Nucleus T2(t) m2 ks i i I i
So 223 172 249 583 233 149 60
r
a
0.9846 0.05 0"9928 0"05 0.9473 !0"05 0"9925 0"403 0"9720 0"05 0"9724 0"05 0"9575 0"05
0.50 0.50 0.50 0.50 0.50 0.50 0.50
are observed a t t h e temperatures 170 and 320°C. A t 170°C the number of oscillators rnl is significantly increased, but the interaction rate constant is A t 3 2 0 ° C the initial modulus Eo increases anomalously, which is evidently with the special kinetic features of the relaxation curve, i.e. a sharp fall in
Rigid phase in styrene-butadiene-styrene thermoelastoplasts
1791
stress over the initial section (high value of the rate constant) and with an unjustified extrapolation of er to t-~0. On the whole, the correlation coefficient for PI is also very high, and the relaxation curves are described by eqn. (6) with a high degree of accuracy. Translated by N. STANDEN REFERENCES 1. A. A. ASKADSKII, Mekhanika kompozitnykh materialov (Mechanics of Composite Materials). p. 403, 1987 2. R. d. GAYLORD, B. JOSS, d. T. BENDLER and E. A. DI MARZIO, Brit. Polymer J. 17: 126, 1985 3. A. A. ASKADSKII, A. L. BLYUMENFELD, Ye. G. GAL'PERN and A. L. CHISKTYAKOV, Vysokomol. soyed. A30: 886, 1988 (Translated in Polymer Sci. S.S.S.R. 30: 4, 904, 1988) 4. Z. S. VIKHAUSKAS, R. B. PANYAVICHYUS, A. I. MARMA and A. A. ASKADSKII, Mekhanika kompozitnykh materialov (Mechanics of Composite Materials). 1090, 1982
PolymerScienceU.S.S.R.Vol.30, No. 8. pp. 1791-1796, 1988 Printed in Poland
0032-3950/88$10.00+.00 0) 1989PergamonPress plc
FRACTURE A N D RESTORATION OF THE RIGID P H A S E IN S T Y R E N E - B U T A D I E N E - S T Y R E N E THERMOELASTOPLASTS * N. P. BESSONOVA, 'Y'U. K. GODOVSKII, O. V. KOVRIGA, S. N. CHVALUN and V. S. SHIRETS L. Ya. Karpov Physicochemical Research Institute
(Received 23 March 1987) The fracture and restoration of the rigid phase in styrene-butadiene-styrenethermoplasts is studied for isotropic and "single crystal" DST-30 specimens, using strain calorimetry, small-angle X-ray scattering, and dynamometric methods. It is shown that stretching is accompanied by breaking of the polystyrene cylinders at least up to 800%, and that their length affects the mechanical properties only up to 150-200yo. After load removal the typical supermolecular lattice of the material is rapidly restored, but quantitative restoration of the lattice parameters and the mechanical properties of the material is restricted by the slow growing together of the fragments of destroyed cylinders, which is mostly governed by a diffusion mechanism.
THREE-block styrene-butadiene-styrene (SBS) copolymers are classical thermoplastic elastomers. Their unique properties (combination of high strength and one thousand per cent deformability) are determined by the presence in the block-copolymer of thermo* Vysokomol. soyed. A30: No. 8, 1690-1694, 1988.