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Nuclear magnetic relaxation and the type of distribution of correlation times of segmental motion in rubber
1042
V. M. ~ o v
and V. D. FE~o~ov
ratios of ktr/kc, ktr/kt and kc/kt were found from gradients of curves and these appeared to be 25.4, 0.23 and 0.15, where [B-] is the concentration of the counterion, i.e. the concentrations HSO( and SO~- ions. Knowing the values of these constants (or their ratios), we may control the number of VFEG in the oligomers. Results of investigations are tabulated. Translated by E. S E ~ REFERENCES 1. G. A. KAZARYAN, Kh. A. KIRAKO8YAN, A. N. KARAPETYAN, V. A. SARKISYAN
and 8. R. ENTELIS, O mekhanizme l~olimerizatsii epikhlorogidrina sernoi kislotoi v nitrometane (Polymerization Mechanism of Epichlorohydrin in the Presence of Sulphuric acid in Nitromethane) Arm. khimich, zh., 1976 2. Kh. A. KHtAKOSYAN, G. A. KAZARYAN, V. A. 8ARKISYAN and 8. G. ENTELIS, Zavisimost' MVR i R T F ot mekhanizma kationnoi polimerizatsii (Dependence of M W D and D T F on the Mechanism of Cationic Polymerization). Arm. khimich, zh., 1977 3. P. PLESH, K a t i o n n a y a polimerizatsiya (Cationic Polymerization). Izd. "Mir", 1966
Polymer Science U.S.S.R. Vol. 23, No. 4, pp. 1042-1054, 1981 Printed in Poland
0032-3950/81/041042-18507.50/0 © 1982 Pergamon Press Ltd.
N U C L E A R MAGNETIC R E L A X A T I O N A N D T H E T Y P E OF D I S T R I B U T I O N OF CORRELATION TIMES OF S E G M E N T A L ' MOTION IN R U B B E R * V. M. CHER~OV a n d V. D . FEDOTOV S. M. K i r o v Chemico-Technological Institute, K a z a n
(Received 26 February 1980) A n analyms was carried out of the NMR pulse experiment, which involved the s t u d y of the form of reduction of longitudinal and transverse intensity of magnetization and t e m p e r a t u r e dependences of spin-spin (T,) and spin-lattice (~1 ~l~ld ~IP) relaxation times in rubbor (using polyisebutylene (PIB)), in order to explain the form and t y p e of formation of correlation time spectra of segmental motion. I t wa~ shown t h a t the correlation time spectrum of molecular motion in polyisobutylene is above i"s, which m a y be s~tisfactorily described b y the superposztion of two F u o s s - K i r k w o o d distribution functions of varying intensity, of which the centres are at a distance of several orders of magnitude. The high-frequency p a r t of the overall spectrum describes small-scale, segmental motion, while the low-frequency p a r t describes large-scale * Vysokomol. soyed. A23: No. 4, 932-942, 1981.
Correlation times of segmental motion in rubber
1043
motion related to network ttuotuations of linkages. It was established that the existen0e of this spectrum is determined by special structural features of the maoromolecular chain, while the heterogeneity of the sample expressed in structural features of the spin-system, reflects a second order effect--distribution of correlation time spectra.
ALTHOUGHin general nuclear magnetic relaxation in polymers follows theoretical predictions of magnetic relaxation according to Bloembergen, Purcell and Pound [1], developed for low molecular weight substances, it has a number of special features which reflect the specific chain structure of polymers. To describe the behaviour of nuclear relaxation in polymers, the Bloembergen, Purcell and Pound theory is normally used, which has been modified by the introduction of correlation time spectra. Various correlation time spectra were used to describe temperature dependences of spin-lattice relaxation times [2], line width [3, 4], the form of reduction of free induction [3, 5] and satisfactory agreement was obtained in m a n y fields. I n a previous study [6] to describe the behaviour of attenuated transverse magnetization in rubber, the correlation time spectrum was used which consisted of two i~ndividual values differing by 5-6 orders of magnitude. Even this simple spectrum model made it possible to give a qualitative description of the attenuation of transverse magnetization in a wide range of temperature, although the quantitative description is far from being always adequate, due to the need for introducing a continuous distribution of correlation times near each individual value. So far all studies concerned with the effect of correlation time spectra on nuclear relaxation dealt with individual parts of the experiment and not the entire pattern as a whole. Results of analysis of various experiments did not show agreement. This study is aimed at trying to obtain information about the type of formation and the form of correlation time spectra determining processes of segmental motion from a I ~ [ R pulse experiment examining both the form of reduction of various types of magnetization and temperature dependences of nuclear relaxation times. A typical rubber -- PIB was used for the exper,ment. We earrled out similar investigatlons of spin-lattice relaxation using c~s-1;4-polybutadlene and natural rubber, but in view of the fact that at temperature higher tha~ the melting point of these polymers results were slmdar from a qualitative point of view, in this study we restmct ourselves to analysing results obtained for only one experiment. Investigations were earned out using an NMR pulse relaxometer described previously [7] at a proton resonance frequency of 21.5 MHz m the temperature range of 170to 470°K. Further measurements of ~/'~were carried out using similar devices operating at frequencies of 10 and 40 MHz. Spin-lattice relaxation times were measured both m a laboratory (T1) and in rotary (Tip) systems of coordinates. T 1was determined using a 90~T--90°-pulse sequence and Tlp.--by the method of spin-locking in the magnetic field, ~/1 ranging from 0.1 to 17 Oe. Attenuation of transverse magnetization was analysed according to results previously published [6]. polymer samples used previously [6] were examined.
V. M. C ~ m ~ o v and V. D. I~DOTOV
1044
Analysis showed that the form of reduction of longitudinal magnetization is exponential in ~he majority of cases and may be described by a sole relaxation time (T1 or TI~), whereas the form of attenuation of transverse magnetization is more complex and depends on temperature [6]. For simplicity the value characterizing the rate of reduction of the attenuation of transverse magnetization will be determined by the time, during which magnetization decreases e number of times (T,e). ~igure 1 shows temperature dependences of relaxation times T~, Tip and T~, while Fig. 2 shows field dependences of times Tzp at several temperatures. TS , TI ~,7"z , m s e c
I00
I0 8 8 I0
-\
I'0
,3 ,2
0'I
.I 2-2
1 3.0
I 3.8
I q.6 /O~T, K -~
1~O. 1. Temperature dependences of spia.spm T~ (1, 1', 1") and spin-lattice Tx, Tt~, (2-10} relaxation in P I B : / - - e x p e r i m e n t a l dependence; 1', / " - - d e p e n d e n c e s calculated from thooretieal reductzons of the of attenuation of transverse magnetization shown in Fig. 6 ( l " - calculated when q==0); 2-10--dependencos l~lotted with field values of H~ and H0 corresponding to resonant frequencies of 12.8X 10' (2); 21.3× 108 (3); 42.6× 10 a (4); 55.3X 10 a (5); 72.3 × 10 = (6); 10 ~ (7); 2.15 X 10 v (8); 30"0 × 107 (9--results of a previous study [8]) and 4.0× 10~
Hz
(10).
Correlation times of segmental motion in rubber
104ff,
I t is k n o w n ' t h a t in the temperature dependences of T~ and T,p in P I B two minimum values are observed, one of which is above the glass temperature and is determined by segmental motion, while the other is observed below Tg and reflects local motions of methyl groups [8]. The first type of motion is only examined in this paper. As regards field (frequency) dependences of spin-lattice relaxation times, as shown by analysis, these are observed in the entire range of temperature; they are linear on a logarithm-logarithmic scale except for field dependences T~p at high temperatures and low H~ values, where deviations are observed from linearity (Fig. 2). At these temperatures the gradient in the temperature dependence of T~ decreases considerably.
+lp ~rn~ec
1"0
50
10
I
" q
30 2O
0"I
l'O
HI , Oe
lO
t
~-~G. 2. Field dependences of spin-latrine relaxation times Tlo m PIB. Points indicate erpemmentM values and continuous lines -- theoretleal vMues derived from eqn. (16) when /~=0-75; .6,----0.5; o'1:101° sec-2; 1--Zso=l.6× 10-2 sec; ~,°= cc and q~-~0; 2--Vlo=~5x ×10-*, ~,o----0-35×10-a and qS:8×10-2 see.; 3--Z,,o~0.18×10 -s, T,,-----0.18 ×10 -a and q=----5.2X10-5 see; 4--zj°----0.9×10-1°; T,o=0.25×10 -a and q-"=3.3×10 -5 see; T=80 (•); 120 (2); 160 (3) and 180° (4). As certain parameters of form and position of minima T 1 and Tip are required for the analysis of the experiment, they are tabulated. Apparent activation , energies Ea were determined from high temperature branches of minima T 1 and from high- (numerator) and low-temperature (denominator) branches of minima Tz. True activation energies Ea° were determined from temperature dependences of correlation frequencies Ve obtained from minimum conditions. I t should be noted t h a t these frequencies are satisfactorily situated on the correlation diagram plotted using results obtained by various methods. Since dependence on temperature conforms to the Williams-Lundell-Ferry rule, in determining Ea°, it was assumed t h a t this dependence is linear within the range of 10-20 ° .
V. M. C~rR~NOVand V. D. F~OTOV
1046
PA_~AMETER8 OF Ttt~ FOR~ A..N-D POSITION OF MINIMA T 1 AND
Form of relaxation
vc, ttz
~aa
~min K
kcal/rnole
exp.
theory
0.09
0.10
0"14
0-17
0"24
0'35
0"31
0-45
0"43
0-59
1(~ 21 29 38
14 30 42 56
w'
Tip
T~
2"6X 104
261
4"2 X 104
263
8"6X l0 t
267
11 × 104
268
14 × 104
270
1.6× 107 3.8x 107 4.9× l0 T 6.5 × 107
321 332 336 340
8.0 7.4 8:0 7.4 6.4 6.9 6.3 6-6 5.3 5.4 6.4 6.2 6.2 5.8
22::k2 22±2 22~2 22~:2 22 :t:2 14±1 14+1 14±1 14±1
.
0"36 0"35 0"36 0"34 0"29 0"31 0.29 0.3 0.24 0.25 0.46 0.44 0.44 0.41
Type and methods of calculating correlation time spectra. Before analysing experimental results let us examine possible causes of appearance of eorrela~on time spectra and theoretical methods of calculation. I t is known t h a t the formation of these spectra m a y be due to two causes. 1. The spectrum is formed as a result of the chain structure of the polymer molecule [9] and is due to the fact t h a t each relaxator (proton-proton vector) takes part simultaneously in several types of motion, for example, determined by the motion of various chain segments, as a result of which the correlation function is non-exponential. 2. The spectrum is formed as a result of the volumetric heterogeneity of the sample caused for example by density fluctuations, linkages, etc. In this case to consider the spectrum it is essential to assume t h a t each relaxator (proton-proton vector) does not interact with adjacent links and then its motion m a y be de.scribed by an exponential correlation function, i.e. the only correlation time. In well-kuown studies of nuclear relaxation in polymers [3, 5] the spectrum from the theory practically corresponds to this model. Theoretical calculations of correlation time spectra are carried out assuming t h a t fluctuations of dipole-dipole interaction as ~ result of moleculer motion of the amorphous polymer m a y be described by the total of independent Gaussian Markoff processes. This means that, as shown previously [10], the theory of nuclear relaxation developed by Kubo and Tomita [11] holds for the latter. WiShln the framework of this theory equations for the reduction of longitudinal relaxation A 1 and Alp and attenuation of transverse magnetization A S are described in the first case as follows
Correlation times of segmental motion in rubber
1047
ao
0
A~(t)=exp{-- f o(r)~z~'.f(--~)dr}, 't
(2)
o
and in the second case oo
0 oo
O
where G(z) is the distribution function of correlation times,
,(+)
....
,
l+e
T
~, 43
l[Tx=3a'[l-q-~o,)~+ 1+(-2~.-c,J
(5)
{a~ is the second moment of the solid lattice and coo and ~ox are the resonant frequencies of protons in fields H 0 and H1, respectively). With correlation time spectra formed as a result of volumetric heterogeneity, so-cMled rapid exchange is often observed, by which a situation is understood in which, while observing a reduction in magnetization, the correlation time of each relaxator extends to all z~ values available in the spectrum. It was shown [12], t h a t this situation m a y only arise if v~
t 0
a n d therefore, reductions in magnetization are simple exponents with.relaxation times dependent on spectrum p~rameters. I t should be noted t h a t a more general case m a y occur when the correlation time spectrum is arranged on both sides of a~ ~, I n this case reductions of attenua-
1048
V. M. C ~ o v
and V. D. F~.DOTOV
tio~ of transverse magnetization take a form which is the total of two components of Gaussian and exponential form. This situation has been examined in detail [12]. Let us compare eqns. (11 and (21 with eqns. (3) and (4). According to (1) and (2) the reduction of longitudinal magnetization is always described by a single exponent and the reduction of attenuation of transverse magnetization varies from the Gaussian to an almost exponential o~rve, whereas according to eqns. (3) and (4), a reduction in longitudinal magnetizatffon is always the total of exponents while the reduction of attenuation of transverse magnetization takes an even more complex form. To illustrate the foregoing, Fig. 2 shows reductions of magnetization calculated from formulae (1)-(4) assuming the existence of a fairly narrow log-rectangular spectrum. It can be seen that both cases can be easily distinguished. With wider spectra the differences increase. From a comparison of eqns. (1) and (2) with (7) it can be seen that the case of rapid exchange may only be distinguished from case ill in the form of the attenuation of transverse magnetization. It follows from the foregoing that analysing the form of reduction of different magnetization patterns the type of formation of correlation time spectra may, in principle, be determined. However, the solution of this fundamental problem is hindered by the fact that the form of reduction of magnetization is not only determined by special features of molecular motion, but also structural features of the spin-system. It was shown in previous studies [13, 14] that in some cases the form of the I~¢IR signal in polymers may be due to the fact that no single spin-system is formed in the sample but there are n independent sub-systems. Therefore, the signal is the total of signals and, generally speaking, reflects the heterogeneity of the sample and is the result of the distribution of certain characteristics of the spin-system in the volume. I f we follow the assumption put forward previously [13] that a spin system corresponds to each macromolecule, in the case of a spectrum formed as a result of the first cause, the formula for attenuation of transverse magnetization is written in the Form cO
A.(t)=
"
~-~
exp
rL-r-
,r.W
J
.
\~ /
(8)
0
where Pl is the relative importance of signals in each spin-system. Although it signifies distribution in the volume, eqn. (8) differs fundamentally from eqn. (4) as it means that the correlation time spectrum is determined by the structure of the macromolecular chain and volumetric distribution is determined by magnetic interactions between macromoleeules. It should be noted that eqn. (4) may be derived from eqn. (8) if we assume that each spin-system consists of one relaxator.
Correlation time6 of segmental mo~ion in rubber
1049
Analysis of the experimen$. In most experiments the form of reduction of longitudinal magnetization is exponential and the attenuation of transverse magnetization near the glass temperature varies from the Oaussian form to exponential without any indication of the form described by eqns. (4) and (7). The preliminary conclusion may hence be drawn that the most likely cause of the formation of a correlation time spectrum of segmental motion in P I B is the structural anisotropy of the macromolecule. This conclusion will be examined quantitatively. The experiment will be analysed quantitatively using results of a previous study [6]; this was improved by replacing two correlation times V and ~,, characterizing rapid (segmental related to glass transition) and slow (related to fluctuations of linkages) motion by two continuous spectra. Let qj(~) and (~s(v) be distribution functions of correlation times of these motions and T10and T~0be the most likely correlation times corresponding to these values. Let us introduce these spectra into eqn. (5) of a previous study [6] using a method corresponding to the first case. As a result the equation for attenuation of transverse magnetization takes the form cO
0
A similar equation for reductions of longitudinal magnetization takes the form t
A
1,
(m)
0
where q~ is the parameter introduced into eqn. (5) of a previous study [6] and q2a2, which denotes a gravimetric factor of two parts of the spectrum, characterizes residual dipole-dipole interaction which is the result of incomplete neutralization of the latter by rapid segmental motion due to its anisotropy. As it was shown previously [6] that the effect of slow motion in experiments of studying the attenuation of transverse magnetization only shows at fairly high temperatures, we will analyse our experiment by a simplified system, i.e. an analysis will first be made assuming the existence of only rapid motion, and as it becomes necessary (at high temperatures) we will involve a slow motion. We assume that the spectrum of rapid motion is described by the function of Fuoss-Kirkwood distribution [15], then according to a previous study [2], values for rates of spin-lattice relaxation are obtained from eqn. (1)
2
(2coo~I.)p~ "]
(Ookl+(O o f.)'p,
I
'P,J
(II)
50
V.M. C , u ~ o v and V. D. FzvoTov
1/T~= ~a, (2co~!.)p' 2o~z 1 -{- (2~o1~f.) Sp''
(12)
here p,. is the parameter of the width of the spectrum line ( 0 < p l < l ) . The ,,oss-Kirkwood function was selected because equations derived with the ~]p of this function for spin-lattice relaxation give a staisfactory description ' t h e experiment and are fairly simple and clear. I t follows from eqns. (11) and 2) that in the minima
T~m~,~Oo(/~,)-1; T~=~ojz(p,a,)-', A~
~oo
2oo
aoo
I
I
I
(13)
A/p.
0.3
I
0.I
II 20
J I 1,,. 60 I00 T i m e , psec
I
Fio. 3. Theoretical reductions of transverse (1, 2) and longitudinal magnetization (1', 29: 1,r--calcuh~ted from formulae (1) and (2), respectively; 2, 2'--from formulae (3) and (4). It was assumed that G(r) = I/log (~/5), where a = 10-*; b = 10-6 and ~rl~-~-10 ~0 SeC-=.
nd in the high and low temperature branches of minima T1, lp shows a dependace on oJa and wl determined b y the relation T l , l p OC ~LIO,I" 1~I
(14)
~rthermore, according tx) a previous s t u d y [2], apparent and true activation nergies are related to the parameter of spectrum width ~! b y a simple ratio
We use eqns. (13)-(15) to analyse the experiment. Analysis of dependences f T z and T,p on frequency o~0 and o~1 shows t h a t t h e y are step functions, in rhich the exponents < 1 when T > Tmm a n d > 1 when T < T rain, while exponents ~min f dependences of ~1,1p on o~o,1 are close to 1 (~0.95). Hence it follows that the
Correlatmn times of segmental motion in rubber
105!
Fuoss-Kirkwood distribution describes (at least qualitatively} field dependences and therefore parameter p may perhaps be determined from them. The Table shows that the E=/E~ratio is always >1~ which does not conflict with eqn. (15) and also enables parameter pf to be evaluated.
P O7
0"3
i 0
II
Ii
II
I 80
II
I[ ~141 160 T °
Fro. 4. Temperature dependences of parameter Bs, derived from field dependences of Tip (1), T1 (2) and from temperature dependences of TI (3) and TI~ (4). Figure 4 shows p! values determined at different temperatures from various experiments. I t can be seen that although absolute values of fll obtained b y various methods do not coincide, there are general tendencies in the temperature behaviour of Pl values. The value of p! increases in every case with an increase in temperature and tends to the same value of 0.75-0.8. The quantitative discrepancy of ~! values obtained from T I and Tip m a y be due both to the varying effect over a period of relaxation of the low-temperature mechanism and the possible inadequacy of describing the true spectrum of correlation times b y the Fuoss-Kirkwood function. Since, however, we cannot accurately distinguish between these two causes in this study, will assume t h a t the Fuoss-Kirkwood function gives a description within the experimental accuracy of the correlation time spectrum of rapid segmental motion in PIB. The accuracy of this conclusion confirms the approximate agreement between experimental values of T~ In and, T ~ In and theoretical ones (Table} calculated from eqn. (13) with ~f values taken from the Table (Ea[E~ratio). I t can be seen that most satisfactory agreement between theory and experiment is observed for relaxation times measured with lower fields. With an increase in field theoretical values become higher than experimental ones, which is due to an increase in the effect of the low-temperature mechanism of relaxation. In fact, with an increase in temperature, the frequencies of motion of two transitions converge and their mutual effect on magnetic relaxation increases. The increase in the Ea/E~ratio as the field of H 0 or H 1 decreases is evidently due to this effect. Let us now examine field dependences of Tip at high temperatures. It can be seen from Fig. 2 t h a t at temperatures > 100 ° these dependences are not step functions and therefore, cannot be described b y simple eqn. (14). It is known from previous studies [6, 16] that at these temperatures the effect of slow motion is clear and therefore to describe the behaviour of Tip, complete formula (10) has to be used. Assuming that the spectrum of slow motion is also described b y
1052
V.M. C m m ~ o v and V, D. FEDOTOV
the Fuoss-Kirkwood function with parameters ~so and fiB, from eqn. (10) we derive Pla=(1-q=) (2~1"c~.)p' ,/~,a~q2 (2co1~.) ~"
1/Tzp=
2(oz
l~-(2e)1~fo) 2~ -~ 2a~z 1@(2~1"C,o)~p"
(16)
Substituting in eqn. (16) parameters of rapid motion well known from a former analysis we select values of a2, vf0 and fll so that theoretical curves Tip (HI) most satisfactorily coincided with experimental curves. Results of calculations are shown in Fig. 2 which illustrate that with certain parameters satisfactory agreement m a y be obtained. I t should be noted t h a t a set of three parameters can only be selected clearly using results of attenuation of transverse magnetization. A rough evaluation of activation energies of slow motion gives a value of 3.5~=1.5 keal/mole. I t follows from the analysis carried out t h a t the form of correlation time spectrum of molecular motion in PIB above the glass temperature m a y be
G~ 1 /'0
o:
3
O °
5
O.3
x -
/
"c,c. .// 3
\\
'rso 6 FIG. 5
O
O O
\\~
20 9
log ~'/'c~o
O
SOo Time
FIG. 6
:Fzo. 5. Layout of the correlat]on time spectrum" for molecular motion in P I B above Tg presented b y the sum of two Fuoss-Kirkwood functions at 160°. FIG. 6. Experimental (points) and theoretical (continuous lines)curves of the attenuation of transverse magnetmation in P I B . Theoretzeal curves were calculated from formula (9) assuming that spectroscopic parameters correspond to data derived from an analysis of field a n d temperature dependences of spin-lattice relaxation tlmes; 1--tLme scale 20 #sec, 50°; 2--20/~sec, 17°; 3--70/~sec, 17°; 4--580/~sec, 80°; 5--1450 psec, 140°.
Correlationtimes of segmental motion in rubber
1053
presented by the superposition of two Fuoss-Kirkwood distribution functions, o of which t h e eentres are at a distance of several orders of magnitude from each other and the intensity of the low-frequency part is higher by four orders of magnitude than the high-frequency part. A layout of this spectrum is shown in Fig. 5. I t can be seen that spectrum obtained is similar from a qualitative point of view to the time spectrum of relaxations obtained from visco-elastie properties of PYB (e.g. in previous study [17]). To conclude the analysis let us examine if the spectrum obtained from the analysis of spin-lattice relaxation can give a quantitative description of the temperature behaviour of the form of attenuation of transverse magnetization. For this purpose we calculate the form of attenuation of transverse magnetization using formula (9), giving for each temperature the spectrum parameters obtained from a previous analysis. Results'of calculation are shown in Fig. 6 together with experimental results of attenuation of transverse magnetization taken from a previous study [6]. It may be seen that theoretical curves give satisfactory description of the experiment in practically the entire range of temperature. A narrow interval in the range of room temperature is an exception, in which experimental curves take a form close to the exponential, whereas theoretical curves appear to be "Gaussian" at all temperatures. According to eqn. (2), the form of attenuation of transverse magnetization with advanced motion may be converted to exponential form with G (~) described by the FuossKirkwood function only when f l ~ l , otherwise, the spectrum is described by another function, e.g. a Gaussian or rectangular function. This, however, conflicts with the entire analysis carried out. In order to emerge from this conflict it is necessary to consider, when analysing the form of attenuation of transverse magnetization, the structure of the spin-system. This can easily be done by joining eqns. (8) and (9). As a result we obtain QO
0
At low temperatures when the effect of slow motion is negligible, eqn. (17) becomes oo
t 0
It is seen from (18) that assuming some distribution according to G~ (v), a more effective transformation of the Gaussian form of attenuation of transverse magnetization to the exponential form or even to the super-exponential form may be obtained, which is observed in thermoplastic materials (see e.g. earlier studies [14, 18]).
1054
V.M. CB'EI~NOVand V. D. FEDOTOV
In the high temperature range (above 140 °) attenuation of transverse magnetization may be described by an equation consisting of two components, in which the rapidly decreasing component is described by eqn. (17) and the slowly decreasing one by eqn. (18). The latter reflects the non-exponential nature of the slowly decreasing component which has not been considered by the authors previously [6]. Analysis of an N-MR pulse experiment of rubber, carried out with PIB suggests t h a t the existence a n d f o r m o f correlation time s p e c t r u m o f segmental m o t i o n in p o l y m e r s is basically d e t e r m i n e d b y special s t r u c t u r a l features o f t h e p o l y m e r chain, while t h e h e t e r o g e n e i t y o f the sample expressed in s t r u c t u r a l features o f t h e spin s y s t e m reflects a second order effect---distribution o f correlation t i m e spectra. Finally, t h e a u t h o r s wish to t h a n k T. N. K h a z a n o v i e h a n d A. I. l ~ a k l a k o v
for discussing the study. Translated by E. S~.MERE REFERENCES
1. 2. 3. 4.
N. BLOEMBEROEN, B. M. PURCELL and R. W. POUND, Phys. Roy. 75: 679, 1948 T. M. CONNOR, Trans. Faraday Soc. 60: 1574, 1964 A. MIJAKE, J. Polymer Sci. 28: 476, 1958 V. P. GRIGOR'YEV and A. I. MAKLAKOY, Vysokomol. soyod. A15: 2576, 1973 (Translated m Polymer SoL U.S.S.R. 15: 11, 2922, 1973) 5. V. P. GRIGOR'YEV, A. I. MAKLAKOV, V. S. DERINOVSKII, Vysokomol. soyed. B16: 737, 1974 (Not translated in Polymer Scl. U.S.S.R.) 6. V. D.. FEDOTOV, V. M. CHERNOV and T. N. KHAZANOVICH, Vysokomol. soyod. A20: 919, 1978 (Translated m Polymer Sci. U.S.S.R. 20: 4, 1037, 1978) 7. G. M. KADIYEVSKII, V. M. CHERNOV, A. Sh. AGISHEV and V. D. FEDOTOV, Sb~ Nekotoryye voprosy fiizlkl zhidkosti (Some Problems of Physics of Liquids). Kazan', KGPI, 1974 8. V. P. SLIKTER, Syrup. Perekhody 1 relaksatmonnyye yavleniya v pohmorakh (Trsnsitlons and Relaxation Effects in Polymers) Izd. "Mar", 1968 9. Yu. Ya. GOTLIB, A. A. DARINSKII and I. M. NEYELOV, Vysokomol. soyed. A18: 1528, 1976 (Translated m Polymer SeL U.S.S.R. 18: 7, 1748, 1976) 10. N. N. KORST and T. N. KHAZANOVICH, Zh. eksp. i teeter, fizlki 45: 1523, 1963 11. R. KUBO and K. TOM1TA, J. Phys. See. Japan 9: 888, 1954 12. H. R. RESING, J. Chem. Phys. 43: 669, 1965; D. E. WOSSNER, J. Phys. Chem. 70: 1217, 1966 13. V. D. FEDOTOV and V. M. CHERNOV, Vysokomol. soyed. B21: 216, 1979 (Not translated m Polymer SeL U.S.S.R.) 14. D. DOKOCLOVA, B. SCHNEIDER and J. JAKES, J. Magn. Resonance 29: 79, 1978 15. B. R. FUOSS and ~I. ~. KIRKWOOD, J. Amor. Chem. Soc. 63: 385, 1941 16. V. D. FEDOTOV and V. M. CHERNOV, Dold. AN SSSR $$4: 891, 1975 17. I. UORD, Mekhmliehoskiye svoistva tverdykh polimerov (Mechanical Propergios of Solid Polymers). Izd. "Khimiya", 1975 18. V. D. SKIRDA, A. I. M~KI~KOY and Kh. SHNAIDER, Vysokomol. soyod. 20: 1442, 1978 (Not translated in Polymer Sci. U.S.S.R.)
V. M. ~ o v
and V. D. FE~o~ov
ratios of ktr/kc, ktr/kt and kc/kt were found from gradients of curves and these appeared to be 25.4, 0.23 and 0.15, where [B-] is the concentration of the counterion, i.e. the concentrations HSO( and SO~- ions. Knowing the values of these constants (or their ratios), we may control the number of VFEG in the oligomers. Results of investigations are tabulated. Translated by E. S E ~ REFERENCES 1. G. A. KAZARYAN, Kh. A. KIRAKO8YAN, A. N. KARAPETYAN, V. A. SARKISYAN
and 8. R. ENTELIS, O mekhanizme l~olimerizatsii epikhlorogidrina sernoi kislotoi v nitrometane (Polymerization Mechanism of Epichlorohydrin in the Presence of Sulphuric acid in Nitromethane) Arm. khimich, zh., 1976 2. Kh. A. KHtAKOSYAN, G. A. KAZARYAN, V. A. 8ARKISYAN and 8. G. ENTELIS, Zavisimost' MVR i R T F ot mekhanizma kationnoi polimerizatsii (Dependence of M W D and D T F on the Mechanism of Cationic Polymerization). Arm. khimich, zh., 1977 3. P. PLESH, K a t i o n n a y a polimerizatsiya (Cationic Polymerization). Izd. "Mir", 1966
Polymer Science U.S.S.R. Vol. 23, No. 4, pp. 1042-1054, 1981 Printed in Poland
0032-3950/81/041042-18507.50/0 © 1982 Pergamon Press Ltd.
N U C L E A R MAGNETIC R E L A X A T I O N A N D T H E T Y P E OF D I S T R I B U T I O N OF CORRELATION TIMES OF S E G M E N T A L ' MOTION IN R U B B E R * V. M. CHER~OV a n d V. D . FEDOTOV S. M. K i r o v Chemico-Technological Institute, K a z a n
(Received 26 February 1980) A n analyms was carried out of the NMR pulse experiment, which involved the s t u d y of the form of reduction of longitudinal and transverse intensity of magnetization and t e m p e r a t u r e dependences of spin-spin (T,) and spin-lattice (~1 ~l~ld ~IP) relaxation times in rubbor (using polyisebutylene (PIB)), in order to explain the form and t y p e of formation of correlation time spectra of segmental motion. I t wa~ shown t h a t the correlation time spectrum of molecular motion in polyisobutylene is above i"s, which m a y be s~tisfactorily described b y the superposztion of two F u o s s - K i r k w o o d distribution functions of varying intensity, of which the centres are at a distance of several orders of magnitude. The high-frequency p a r t of the overall spectrum describes small-scale, segmental motion, while the low-frequency p a r t describes large-scale * Vysokomol. soyed. A23: No. 4, 932-942, 1981.
Correlation times of segmental motion in rubber
1043
motion related to network ttuotuations of linkages. It was established that the existen0e of this spectrum is determined by special structural features of the maoromolecular chain, while the heterogeneity of the sample expressed in structural features of the spin-system, reflects a second order effect--distribution of correlation time spectra.
ALTHOUGHin general nuclear magnetic relaxation in polymers follows theoretical predictions of magnetic relaxation according to Bloembergen, Purcell and Pound [1], developed for low molecular weight substances, it has a number of special features which reflect the specific chain structure of polymers. To describe the behaviour of nuclear relaxation in polymers, the Bloembergen, Purcell and Pound theory is normally used, which has been modified by the introduction of correlation time spectra. Various correlation time spectra were used to describe temperature dependences of spin-lattice relaxation times [2], line width [3, 4], the form of reduction of free induction [3, 5] and satisfactory agreement was obtained in m a n y fields. I n a previous study [6] to describe the behaviour of attenuated transverse magnetization in rubber, the correlation time spectrum was used which consisted of two i~ndividual values differing by 5-6 orders of magnitude. Even this simple spectrum model made it possible to give a qualitative description of the attenuation of transverse magnetization in a wide range of temperature, although the quantitative description is far from being always adequate, due to the need for introducing a continuous distribution of correlation times near each individual value. So far all studies concerned with the effect of correlation time spectra on nuclear relaxation dealt with individual parts of the experiment and not the entire pattern as a whole. Results of analysis of various experiments did not show agreement. This study is aimed at trying to obtain information about the type of formation and the form of correlation time spectra determining processes of segmental motion from a I ~ [ R pulse experiment examining both the form of reduction of various types of magnetization and temperature dependences of nuclear relaxation times. A typical rubber -- PIB was used for the exper,ment. We earrled out similar investigatlons of spin-lattice relaxation using c~s-1;4-polybutadlene and natural rubber, but in view of the fact that at temperature higher tha~ the melting point of these polymers results were slmdar from a qualitative point of view, in this study we restmct ourselves to analysing results obtained for only one experiment. Investigations were earned out using an NMR pulse relaxometer described previously [7] at a proton resonance frequency of 21.5 MHz m the temperature range of 170to 470°K. Further measurements of ~/'~were carried out using similar devices operating at frequencies of 10 and 40 MHz. Spin-lattice relaxation times were measured both m a laboratory (T1) and in rotary (Tip) systems of coordinates. T 1was determined using a 90~T--90°-pulse sequence and Tlp.--by the method of spin-locking in the magnetic field, ~/1 ranging from 0.1 to 17 Oe. Attenuation of transverse magnetization was analysed according to results previously published [6]. polymer samples used previously [6] were examined.
V. M. C ~ m ~ o v and V. D. I~DOTOV
1044
Analysis showed that the form of reduction of longitudinal magnetization is exponential in ~he majority of cases and may be described by a sole relaxation time (T1 or TI~), whereas the form of attenuation of transverse magnetization is more complex and depends on temperature [6]. For simplicity the value characterizing the rate of reduction of the attenuation of transverse magnetization will be determined by the time, during which magnetization decreases e number of times (T,e). ~igure 1 shows temperature dependences of relaxation times T~, Tip and T~, while Fig. 2 shows field dependences of times Tzp at several temperatures. TS , TI ~,7"z , m s e c
I00
I0 8 8 I0
-\
I'0
,3 ,2
0'I
.I 2-2
1 3.0
I 3.8
I q.6 /O~T, K -~
1~O. 1. Temperature dependences of spia.spm T~ (1, 1', 1") and spin-lattice Tx, Tt~, (2-10} relaxation in P I B : / - - e x p e r i m e n t a l dependence; 1', / " - - d e p e n d e n c e s calculated from thooretieal reductzons of the of attenuation of transverse magnetization shown in Fig. 6 ( l " - calculated when q==0); 2-10--dependencos l~lotted with field values of H~ and H0 corresponding to resonant frequencies of 12.8X 10' (2); 21.3× 108 (3); 42.6× 10 a (4); 55.3X 10 a (5); 72.3 × 10 = (6); 10 ~ (7); 2.15 X 10 v (8); 30"0 × 107 (9--results of a previous study [8]) and 4.0× 10~
Hz
(10).
Correlation times of segmental motion in rubber
104ff,
I t is k n o w n ' t h a t in the temperature dependences of T~ and T,p in P I B two minimum values are observed, one of which is above the glass temperature and is determined by segmental motion, while the other is observed below Tg and reflects local motions of methyl groups [8]. The first type of motion is only examined in this paper. As regards field (frequency) dependences of spin-lattice relaxation times, as shown by analysis, these are observed in the entire range of temperature; they are linear on a logarithm-logarithmic scale except for field dependences T~p at high temperatures and low H~ values, where deviations are observed from linearity (Fig. 2). At these temperatures the gradient in the temperature dependence of T~ decreases considerably.
+lp ~rn~ec
1"0
50
10
I
" q
30 2O
0"I
l'O
HI , Oe
lO
t
~-~G. 2. Field dependences of spin-latrine relaxation times Tlo m PIB. Points indicate erpemmentM values and continuous lines -- theoretleal vMues derived from eqn. (16) when /~=0-75; .6,----0.5; o'1:101° sec-2; 1--Zso=l.6× 10-2 sec; ~,°= cc and q~-~0; 2--Vlo=~5x ×10-*, ~,o----0-35×10-a and qS:8×10-2 see.; 3--Z,,o~0.18×10 -s, T,,-----0.18 ×10 -a and q=----5.2X10-5 see; 4--zj°----0.9×10-1°; T,o=0.25×10 -a and q-"=3.3×10 -5 see; T=80 (•); 120 (2); 160 (3) and 180° (4). As certain parameters of form and position of minima T 1 and Tip are required for the analysis of the experiment, they are tabulated. Apparent activation , energies Ea were determined from high temperature branches of minima T 1 and from high- (numerator) and low-temperature (denominator) branches of minima Tz. True activation energies Ea° were determined from temperature dependences of correlation frequencies Ve obtained from minimum conditions. I t should be noted t h a t these frequencies are satisfactorily situated on the correlation diagram plotted using results obtained by various methods. Since dependence on temperature conforms to the Williams-Lundell-Ferry rule, in determining Ea°, it was assumed t h a t this dependence is linear within the range of 10-20 ° .
V. M. C~rR~NOVand V. D. F~OTOV
1046
PA_~AMETER8 OF Ttt~ FOR~ A..N-D POSITION OF MINIMA T 1 AND
Form of relaxation
vc, ttz
~aa
~min K
kcal/rnole
exp.
theory
0.09
0.10
0"14
0-17
0"24
0'35
0"31
0-45
0"43
0-59
1(~ 21 29 38
14 30 42 56
w'
Tip
T~
2"6X 104
261
4"2 X 104
263
8"6X l0 t
267
11 × 104
268
14 × 104
270
1.6× 107 3.8x 107 4.9× l0 T 6.5 × 107
321 332 336 340
8.0 7.4 8:0 7.4 6.4 6.9 6.3 6-6 5.3 5.4 6.4 6.2 6.2 5.8
22::k2 22±2 22~2 22~:2 22 :t:2 14±1 14+1 14±1 14±1
.
0"36 0"35 0"36 0"34 0"29 0"31 0.29 0.3 0.24 0.25 0.46 0.44 0.44 0.41
Type and methods of calculating correlation time spectra. Before analysing experimental results let us examine possible causes of appearance of eorrela~on time spectra and theoretical methods of calculation. I t is known t h a t the formation of these spectra m a y be due to two causes. 1. The spectrum is formed as a result of the chain structure of the polymer molecule [9] and is due to the fact t h a t each relaxator (proton-proton vector) takes part simultaneously in several types of motion, for example, determined by the motion of various chain segments, as a result of which the correlation function is non-exponential. 2. The spectrum is formed as a result of the volumetric heterogeneity of the sample caused for example by density fluctuations, linkages, etc. In this case to consider the spectrum it is essential to assume t h a t each relaxator (proton-proton vector) does not interact with adjacent links and then its motion m a y be de.scribed by an exponential correlation function, i.e. the only correlation time. In well-kuown studies of nuclear relaxation in polymers [3, 5] the spectrum from the theory practically corresponds to this model. Theoretical calculations of correlation time spectra are carried out assuming t h a t fluctuations of dipole-dipole interaction as ~ result of moleculer motion of the amorphous polymer m a y be described by the total of independent Gaussian Markoff processes. This means that, as shown previously [10], the theory of nuclear relaxation developed by Kubo and Tomita [11] holds for the latter. WiShln the framework of this theory equations for the reduction of longitudinal relaxation A 1 and Alp and attenuation of transverse magnetization A S are described in the first case as follows
Correlation times of segmental motion in rubber
1047
ao
0
A~(t)=exp{-- f o(r)~z~'.f(--~)dr}, 't
(2)
o
and in the second case oo
0 oo
O
where G(z) is the distribution function of correlation times,
,(+)
....
,
l+e
T
~, 43
l[Tx=3a'[l-q-~o,)~+ 1+(-2~.-c,J
(5)
{a~ is the second moment of the solid lattice and coo and ~ox are the resonant frequencies of protons in fields H 0 and H1, respectively). With correlation time spectra formed as a result of volumetric heterogeneity, so-cMled rapid exchange is often observed, by which a situation is understood in which, while observing a reduction in magnetization, the correlation time of each relaxator extends to all z~ values available in the spectrum. It was shown [12], t h a t this situation m a y only arise if v~
t 0
a n d therefore, reductions in magnetization are simple exponents with.relaxation times dependent on spectrum p~rameters. I t should be noted t h a t a more general case m a y occur when the correlation time spectrum is arranged on both sides of a~ ~, I n this case reductions of attenua-
1048
V. M. C ~ o v
and V. D. F~.DOTOV
tio~ of transverse magnetization take a form which is the total of two components of Gaussian and exponential form. This situation has been examined in detail [12]. Let us compare eqns. (11 and (21 with eqns. (3) and (4). According to (1) and (2) the reduction of longitudinal magnetization is always described by a single exponent and the reduction of attenuation of transverse magnetization varies from the Gaussian to an almost exponential o~rve, whereas according to eqns. (3) and (4), a reduction in longitudinal magnetizatffon is always the total of exponents while the reduction of attenuation of transverse magnetization takes an even more complex form. To illustrate the foregoing, Fig. 2 shows reductions of magnetization calculated from formulae (1)-(4) assuming the existence of a fairly narrow log-rectangular spectrum. It can be seen that both cases can be easily distinguished. With wider spectra the differences increase. From a comparison of eqns. (1) and (2) with (7) it can be seen that the case of rapid exchange may only be distinguished from case ill in the form of the attenuation of transverse magnetization. It follows from the foregoing that analysing the form of reduction of different magnetization patterns the type of formation of correlation time spectra may, in principle, be determined. However, the solution of this fundamental problem is hindered by the fact that the form of reduction of magnetization is not only determined by special features of molecular motion, but also structural features of the spin-system. It was shown in previous studies [13, 14] that in some cases the form of the I~¢IR signal in polymers may be due to the fact that no single spin-system is formed in the sample but there are n independent sub-systems. Therefore, the signal is the total of signals and, generally speaking, reflects the heterogeneity of the sample and is the result of the distribution of certain characteristics of the spin-system in the volume. I f we follow the assumption put forward previously [13] that a spin system corresponds to each macromolecule, in the case of a spectrum formed as a result of the first cause, the formula for attenuation of transverse magnetization is written in the Form cO
A.(t)=
"
~-~
exp
rL-r-
,r.W
J
.
\~ /
(8)
0
where Pl is the relative importance of signals in each spin-system. Although it signifies distribution in the volume, eqn. (8) differs fundamentally from eqn. (4) as it means that the correlation time spectrum is determined by the structure of the macromolecular chain and volumetric distribution is determined by magnetic interactions between macromoleeules. It should be noted that eqn. (4) may be derived from eqn. (8) if we assume that each spin-system consists of one relaxator.
Correlation time6 of segmental mo~ion in rubber
1049
Analysis of the experimen$. In most experiments the form of reduction of longitudinal magnetization is exponential and the attenuation of transverse magnetization near the glass temperature varies from the Oaussian form to exponential without any indication of the form described by eqns. (4) and (7). The preliminary conclusion may hence be drawn that the most likely cause of the formation of a correlation time spectrum of segmental motion in P I B is the structural anisotropy of the macromolecule. This conclusion will be examined quantitatively. The experiment will be analysed quantitatively using results of a previous study [6]; this was improved by replacing two correlation times V and ~,, characterizing rapid (segmental related to glass transition) and slow (related to fluctuations of linkages) motion by two continuous spectra. Let qj(~) and (~s(v) be distribution functions of correlation times of these motions and T10and T~0be the most likely correlation times corresponding to these values. Let us introduce these spectra into eqn. (5) of a previous study [6] using a method corresponding to the first case. As a result the equation for attenuation of transverse magnetization takes the form cO
0
A similar equation for reductions of longitudinal magnetization takes the form t
A
1,
(m)
0
where q~ is the parameter introduced into eqn. (5) of a previous study [6] and q2a2, which denotes a gravimetric factor of two parts of the spectrum, characterizes residual dipole-dipole interaction which is the result of incomplete neutralization of the latter by rapid segmental motion due to its anisotropy. As it was shown previously [6] that the effect of slow motion in experiments of studying the attenuation of transverse magnetization only shows at fairly high temperatures, we will analyse our experiment by a simplified system, i.e. an analysis will first be made assuming the existence of only rapid motion, and as it becomes necessary (at high temperatures) we will involve a slow motion. We assume that the spectrum of rapid motion is described by the function of Fuoss-Kirkwood distribution [15], then according to a previous study [2], values for rates of spin-lattice relaxation are obtained from eqn. (1)
2
(2coo~I.)p~ "]
(Ookl+(O o f.)'p,
I
'P,J
(II)
50
V.M. C , u ~ o v and V. D. FzvoTov
1/T~= ~a, (2co~!.)p' 2o~z 1 -{- (2~o1~f.) Sp''
(12)
here p,. is the parameter of the width of the spectrum line ( 0 < p l < l ) . The ,,oss-Kirkwood function was selected because equations derived with the ~]p of this function for spin-lattice relaxation give a staisfactory description ' t h e experiment and are fairly simple and clear. I t follows from eqns. (11) and 2) that in the minima
T~m~,~Oo(/~,)-1; T~=~ojz(p,a,)-', A~
~oo
2oo
aoo
I
I
I
(13)
A/p.
0.3
I
0.I
II 20
J I 1,,. 60 I00 T i m e , psec
I
Fio. 3. Theoretical reductions of transverse (1, 2) and longitudinal magnetization (1', 29: 1,r--calcuh~ted from formulae (1) and (2), respectively; 2, 2'--from formulae (3) and (4). It was assumed that G(r) = I/log (~/5), where a = 10-*; b = 10-6 and ~rl~-~-10 ~0 SeC-=.
nd in the high and low temperature branches of minima T1, lp shows a dependace on oJa and wl determined b y the relation T l , l p OC ~LIO,I" 1~I
(14)
~rthermore, according tx) a previous s t u d y [2], apparent and true activation nergies are related to the parameter of spectrum width ~! b y a simple ratio
We use eqns. (13)-(15) to analyse the experiment. Analysis of dependences f T z and T,p on frequency o~0 and o~1 shows t h a t t h e y are step functions, in rhich the exponents < 1 when T > Tmm a n d > 1 when T < T rain, while exponents ~min f dependences of ~1,1p on o~o,1 are close to 1 (~0.95). Hence it follows that the
Correlatmn times of segmental motion in rubber
105!
Fuoss-Kirkwood distribution describes (at least qualitatively} field dependences and therefore parameter p may perhaps be determined from them. The Table shows that the E=/E~ratio is always >1~ which does not conflict with eqn. (15) and also enables parameter pf to be evaluated.
P O7
0"3
i 0
II
Ii
II
I 80
II
I[ ~141 160 T °
Fro. 4. Temperature dependences of parameter Bs, derived from field dependences of Tip (1), T1 (2) and from temperature dependences of TI (3) and TI~ (4). Figure 4 shows p! values determined at different temperatures from various experiments. I t can be seen that although absolute values of fll obtained b y various methods do not coincide, there are general tendencies in the temperature behaviour of Pl values. The value of p! increases in every case with an increase in temperature and tends to the same value of 0.75-0.8. The quantitative discrepancy of ~! values obtained from T I and Tip m a y be due both to the varying effect over a period of relaxation of the low-temperature mechanism and the possible inadequacy of describing the true spectrum of correlation times b y the Fuoss-Kirkwood function. Since, however, we cannot accurately distinguish between these two causes in this study, will assume t h a t the Fuoss-Kirkwood function gives a description within the experimental accuracy of the correlation time spectrum of rapid segmental motion in PIB. The accuracy of this conclusion confirms the approximate agreement between experimental values of T~ In and, T ~ In and theoretical ones (Table} calculated from eqn. (13) with ~f values taken from the Table (Ea[E~ratio). I t can be seen that most satisfactory agreement between theory and experiment is observed for relaxation times measured with lower fields. With an increase in field theoretical values become higher than experimental ones, which is due to an increase in the effect of the low-temperature mechanism of relaxation. In fact, with an increase in temperature, the frequencies of motion of two transitions converge and their mutual effect on magnetic relaxation increases. The increase in the Ea/E~ratio as the field of H 0 or H 1 decreases is evidently due to this effect. Let us now examine field dependences of Tip at high temperatures. It can be seen from Fig. 2 t h a t at temperatures > 100 ° these dependences are not step functions and therefore, cannot be described b y simple eqn. (14). It is known from previous studies [6, 16] that at these temperatures the effect of slow motion is clear and therefore to describe the behaviour of Tip, complete formula (10) has to be used. Assuming that the spectrum of slow motion is also described b y
1052
V.M. C m m ~ o v and V, D. FEDOTOV
the Fuoss-Kirkwood function with parameters ~so and fiB, from eqn. (10) we derive Pla=(1-q=) (2~1"c~.)p' ,/~,a~q2 (2co1~.) ~"
1/Tzp=
2(oz
l~-(2e)1~fo) 2~ -~ 2a~z 1@(2~1"C,o)~p"
(16)
Substituting in eqn. (16) parameters of rapid motion well known from a former analysis we select values of a2, vf0 and fll so that theoretical curves Tip (HI) most satisfactorily coincided with experimental curves. Results of calculations are shown in Fig. 2 which illustrate that with certain parameters satisfactory agreement m a y be obtained. I t should be noted t h a t a set of three parameters can only be selected clearly using results of attenuation of transverse magnetization. A rough evaluation of activation energies of slow motion gives a value of 3.5~=1.5 keal/mole. I t follows from the analysis carried out t h a t the form of correlation time spectrum of molecular motion in PIB above the glass temperature m a y be
G~ 1 /'0
o:
3
O °
5
O.3
x -
/
"c,c. .// 3
\\
'rso 6 FIG. 5
O
O O
\\~
20 9
log ~'/'c~o
O
SOo Time
FIG. 6
:Fzo. 5. Layout of the correlat]on time spectrum" for molecular motion in P I B above Tg presented b y the sum of two Fuoss-Kirkwood functions at 160°. FIG. 6. Experimental (points) and theoretical (continuous lines)curves of the attenuation of transverse magnetmation in P I B . Theoretzeal curves were calculated from formula (9) assuming that spectroscopic parameters correspond to data derived from an analysis of field a n d temperature dependences of spin-lattice relaxation tlmes; 1--tLme scale 20 #sec, 50°; 2--20/~sec, 17°; 3--70/~sec, 17°; 4--580/~sec, 80°; 5--1450 psec, 140°.
Correlationtimes of segmental motion in rubber
1053
presented by the superposition of two Fuoss-Kirkwood distribution functions, o of which t h e eentres are at a distance of several orders of magnitude from each other and the intensity of the low-frequency part is higher by four orders of magnitude than the high-frequency part. A layout of this spectrum is shown in Fig. 5. I t can be seen that spectrum obtained is similar from a qualitative point of view to the time spectrum of relaxations obtained from visco-elastie properties of PYB (e.g. in previous study [17]). To conclude the analysis let us examine if the spectrum obtained from the analysis of spin-lattice relaxation can give a quantitative description of the temperature behaviour of the form of attenuation of transverse magnetization. For this purpose we calculate the form of attenuation of transverse magnetization using formula (9), giving for each temperature the spectrum parameters obtained from a previous analysis. Results'of calculation are shown in Fig. 6 together with experimental results of attenuation of transverse magnetization taken from a previous study [6]. It may be seen that theoretical curves give satisfactory description of the experiment in practically the entire range of temperature. A narrow interval in the range of room temperature is an exception, in which experimental curves take a form close to the exponential, whereas theoretical curves appear to be "Gaussian" at all temperatures. According to eqn. (2), the form of attenuation of transverse magnetization with advanced motion may be converted to exponential form with G (~) described by the FuossKirkwood function only when f l ~ l , otherwise, the spectrum is described by another function, e.g. a Gaussian or rectangular function. This, however, conflicts with the entire analysis carried out. In order to emerge from this conflict it is necessary to consider, when analysing the form of attenuation of transverse magnetization, the structure of the spin-system. This can easily be done by joining eqns. (8) and (9). As a result we obtain QO
0
At low temperatures when the effect of slow motion is negligible, eqn. (17) becomes oo
t 0
It is seen from (18) that assuming some distribution according to G~ (v), a more effective transformation of the Gaussian form of attenuation of transverse magnetization to the exponential form or even to the super-exponential form may be obtained, which is observed in thermoplastic materials (see e.g. earlier studies [14, 18]).
1054
V.M. CB'EI~NOVand V. D. FEDOTOV
In the high temperature range (above 140 °) attenuation of transverse magnetization may be described by an equation consisting of two components, in which the rapidly decreasing component is described by eqn. (17) and the slowly decreasing one by eqn. (18). The latter reflects the non-exponential nature of the slowly decreasing component which has not been considered by the authors previously [6]. Analysis of an N-MR pulse experiment of rubber, carried out with PIB suggests t h a t the existence a n d f o r m o f correlation time s p e c t r u m o f segmental m o t i o n in p o l y m e r s is basically d e t e r m i n e d b y special s t r u c t u r a l features o f t h e p o l y m e r chain, while t h e h e t e r o g e n e i t y o f the sample expressed in s t r u c t u r a l features o f t h e spin s y s t e m reflects a second order effect---distribution o f correlation t i m e spectra. Finally, t h e a u t h o r s wish to t h a n k T. N. K h a z a n o v i e h a n d A. I. l ~ a k l a k o v
for discussing the study. Translated by E. S~.MERE REFERENCES
1. 2. 3. 4.
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