Nuclear magnetic resonance and activation energy of segmental motion in polymers

NUCLEAR MAGNETIC RESONANCE AND ACTIVATION ENERGY OF SEGMENTAL MOTION IN POLYMERS * A. I. MAKLAKOV a n d G. G. PIMENOV V. I. Ul'yanov-Lenia State University, Kazan

(Received 28 April 1967)

THE motion of macromolecules and their components (various "pendants", segments, etc.) is usually described by the correlation time T, the temperature dependence of which is exponential in nature q--To exp (E/RT),

(1)

where 30 is the pre-exponential factor, E--the activation energy of the type of motion, R--gas constant, T--temperature. It is normally assumed that T0 and E are independent of T and the value of T can be determined from measurements of dielectric and mechanical losses, viscosity, etc. By studying segmental motion in polymers by nuclear magnetic resonance (N-MR) it is possible to determine correlation times (denoted as z~) firstly from the temperature dependence of the width 5 of the absorption line (or spin-spin relaxation time T2 making a certain assumption regarding the shape of the line) in the range of a marked variation according to formula (1) 2 2 To-- ~?d tan 2 dT--d d

(2)

where a is the gyromagnetic ratio of the resonant nucleus; 7, a constant of the order of unity [1], or ~0.16 [2]; dd and 5T line width with and without well-developed motion, respectively. Secondly, from the temperature dependence of the spin-lattice relaxation time T 1 according to the Kubo-Tomita equation [3]

1

A

4~

* Vysokomol. soyed. A10: No. 3, 662-670, 1968. 773

], J

(3)

774

A.I. MAKTJA~OVand G. Go.P ~ o v

where A is a constant determined by the structure of the substance; co, angular resonant frequency. It follows from (3) that when T 1 is minimum, the following condition is satisfied co%=0.6158. (4) Values of Tc obtained from equation (2) are of the order of 10-4-10 -5 s e c and from ratios (3) and (4) of the order of 10 -v sec. With the combination of equations (1) and (2) values of E and 30 can be derived, which correspond to this type of motion. Similar data can be obtained from measurements of T 1 and expressions (1) and (3) over the whole range of this motion, or in the low-temperature (o)%>>1) or high-temperature (~o%<<1) regions. By measuring T 1 in a wide frequency range the activation energy can be determined from the position of the minimum T 1 using equations (1) and (4). I t follows from (1) t h a t the dependence of log v on 1/T should be linear. For certain substances, both low- (e.g. [4]) and high-molecular weight [5], this is certainly the ease in a small temperature range (10-20°). However, in m a n y cases log ~ does not v a r y in proportion to 1/T 1. To "correct" this non-linearity, equation (1) is somewhat modified and becomes [6]: ~=To exp{

E

R(T--T~)

},

(5)

where T~ is the temperature at which relaxation processes take place at an infinitely low rate, i.e. z->oo. 'Thus, for values of v, determined from dielectric measurements, which differ from the correlation times determined from the 1WMR data by a constant factor of three, taking the example of glycerol, pro1 pylglycol and n-propanol it was shown [6] that dependences of log z on T--T~ are linear. Similar results were obtained for z in polyethyleneglycol [7]. Equation (5) is, no doubt, more reasonable from a physical point t h a n (1) because ~->oo is valid at temperatures which considerably exceed absolute zero. These attempts to "preserve" equation (1) with an activation energy independent of T are not always successful. Powles in one of his recent studies on molecular reorientation in liquids [8] indicates that the dependence of log z on I/T, contrary to widespread opinion, is not linear, i.e. a unique values of E, characterizing the liquid and independent of temperature, cannot be calculated. Much earlier Shishkin [9] studying electrical conductivity and viscosity of several amorphous substances, including phenol-formaldehyde resin, the above parameters of which, according to earlier assumptions, depend exponentially on temperature, obtained an empirical ratio for the dependence of E on T in the form:

E = R T ~ exp

(6)

Activation energy of segmental motion in polymers

775

where b, c are empirical constants characterizing the substance. From (6) it can be seen t h a t the activation energy increases continuously and exponentially with a reduction in the temperature of the substance. From an analysis of experimental data on the relaxation properties of m a n y substances, including polymers (NMR relaxation data, in fact, were not available), Williams, Landell and Ferry (WLF) derived another expression for E, which also depends on temperature [10] 4.14T 2 E = (T+51"6--Tg) 2 ,

(7)

where Tg is the glass transition temperature of the polymer. Investigating dielectric relaxation in polymethylacrylate, polyvinylacetate and polyvinylehloride, Mikhailov and Lobanov [11] revealed that with increased temperature the slope of the curve log ¢=f(1/T) decreases, i.e. E decreases qualitatively conforming to the W L F equation. In the authors' view this is due to a dimensional reduction in the kinetic unit of the macromolecule which takes part in the motion on increasing T. The dependence of activation energy on T determined from T 1 in polyethylene, polyoxymethylene, was indicated in a former paper [12]. The activation energy of polydimethylsiloxane, calculated from the temperature dependence of the NMR line width, increase from 2.6 to 8.7 kcal/mole on increasing T from 140 to 200°K [13]. Frequently, however, irrespective of the obvious non-linearity of the dependence of log zc on 1/T, the authors seek to introduce an activation energy value independent of lIT (e.g. [14]). Thus, the dependence of the activation energy of motion on temperature can, apparently, be considered as a general relation; and the cases of independence of E of T - - a s special cases which are correct within small temperature ranges. However, few data are available of E and r c obtained from the temperature dependence of NMR line width and are separate. For the systematic study of the behaviour of activation energy of segmental motion in polymers obtained from NMR data, we calculated the temperature dependence of r c from curves previously described ~=~0 (T) for a large number of polymers: natural rubber (NR) [5, 15], polyisobutylene (PIB) [5], atactie polypropylene [5], ataetic fractionated polystyrene (PS) of different molecular weights [16], isotactic and atactic unfraetionated PS [17, 18], polyvinylchloride, polyvinylaeetate and various polymethylmethacrylates (PMMA) [18, 19], polyethyleneterephthalate (PETP) and APA-3 aromatic polyamide [17], D-I, D-2, P-l, 1)-2 polyarylates. For simplicity ~c = ~ Y ~ = - - tan

2 2 2 g~--ga

A. I. Y_AV~AXOVand G. G. PrME~OV

776

was calculated instead of To. The curves showing dependences of log rc and log ~* on T are exactly the same, which makes it possible to determine the actir a t i o n energy from equation (1), according to the formula: E=2.303R

A (log z~) A (l/T)

v* was calculated in the range of marked variation of line width, due to initial segmental movement in the polymer. I t should be noted that it is sometimes

G

J=H?

(~, oe

2.2O 2

1'80

,.=L/1-1

=

1ook

IE,k=ll ol,

x(.xf4,

-tiY ll:

[00

o

~,o~

/)

0

!

1'0

2 ..,_//~

0~0 2"5

~8 ;7

' i/7",10a 2.8

2.e ~7 2.81/r,I~

1"8

2"0

2"21/7"/0a

Fio. 1. Dependence of log v* and $ on 1/T: a--for PS: /--amorphous isotactic, 2--crystalline isotactic, fractionated atactic with a molecular weight of 2× 10a, 3--7 x 104, 5"6 × 106 a n d 1"2 x 106; b--for amorphous (1) and crystalline (2) PETP, (3)--dependence of E o n lit for amorphous PETP; c--for polyarylates: 1--D-I, 2--amorphous D-2, 3--crystalline D-2. very difficult to determine accurately the range of marked variation of line width. Furthermore, when 6 varies slowly before suddenly contracting (which can be observed frequently) the value of 6r becomes somewhat uncertain. Figure 1 shows typical dependences of log vc* on inverse temperature for several polymers. Temperature dependences of 6 were obtained b y entirely different authors [17, 18, 19]. Figure la indicates that a linear relation between log T* and 1/T was not observed for any specimen. I t is true that for amorphous isotactic PS a very small temperature range can be found where linearity is shown. The slope of the tangent to the curve log v * = f ( 1 / T ) decreases with increase of temperature, i.e. the activation energy does not remain constant, b u t increases with temperature; this is rather strange if we consider that amorphous PS is subject to the W L F equation (equation 7).

Activation energy of segmental motion in polymers

777

As a matter of fact, if we consider that E depends on temperature and ~c,. but nevertheless satisfies equation (1), i.e. T~=r0 exp { E R ~ } , for vo independent of T, the gradient of the curve log ~-f(1/T) will be dd (log (I/T)v)

l°gelE(T)--T R dE(T)}.dT

If the dependence of the activation energy on T conforms to the WLF equation, the following will represent the second derivative: d2(logv) d (l/T) ~

8.28 (T~--51.6) log e R

T3(2T+To--51.6) (T~-51.6--Tg)*

Hence d 2 (log

d (1/T):)>O ,

(8)

as for all known polymers Tg>51.6°K and 2T~-Tg--51.6>0. Thus, with the increase of 1/T, the gradient of the carve increases, which contradicts the experimental curves plotted for various PS specimens. A similar non-linear dependence of log ~* on 1/Tis also observed for PETP, both in the amorphous and in the crystalline states (Fig. lb). It can be seen that with increase of temperature (>Tg) the value of E increases irrespective of the phase condition of the polymers. A more complex dependence of log ~* on inverse temperature is observed for D-2 polyarylates (Fig. lc). Although curves ei=~(1/T) differ for amorphous and crystalline D-2 specimens, curves log T*:f(1/T) agree, so that the linear part of this relation is in the middle of transition in the line width and for crystalline specimens at the inception of transition. The curve for D-1 is simpler and conforms to equation (8). Figure 2 gives the temperature dependence of J and E for various PMMA specimens. In this ease there are temperature sections where E is practically constant. Activation energy only increases markedly near T c for isotactic specimens which is roughly in qualitative agreement with the WLF equation. A somewhat different type of dependence of E on 1/T is observed for APA-3 aromatic polyamide (Fig. 3): with amorphous Specimens in the range of temperature corresponding to the inflexion of curve J : ~ (T) the maximum activation energy is observed, with crystalline polymers E increases with rise of temperature. We note that these anomalies in curves log 3" :f(1/T) are normally observed in polymers with fairly high glass-transition temperatures. In specimens of lower

~778

A. I. !WA~AKOV and G. G. PI~ENOV

Tg (much below 0 °) and consequently having more mobile chains in the solid •state, comparatively wide temperature ranges can be isolated where a linear relationship is observed between log ~e• and lIT. Thus, for P I B practically over the whole range of marked line contraction (220-270°K) linearity is retained (it should be noted that according to WLF data, the activation energy of the polymer varies in this range from 39 to 14 kcal/mole). The same can be said of NR, poly~propylene, polydimethylsilylene, but for these polymers there are ranges where

~,,08 5

r

E,kca/~mole~

goal ~

/I

",~---~/ x'

.3~

- -J_ , OJ-

[-¢ '

I

OI

I

1"7

l/T, tO~ FIG. 2

i

i

1"8

,

l/r*lOJ

1"0

FIG. 3

FIG. 2. Dependence of E a n d $ on 1/T for PMMA specimens: 1 - - a t a c t i c , molecular weight l0 s, 2--isotactic, molecular weight 1"2 × 10 e, 3--isotactic with a molecular weight of 3 × 105. FIG. 3. Dependence of E and $ on 1/T for APA-3 aromatic polyamide in amorphous (1) and crystalline (2) states.

linearity is disturbed. For a polymer such as polydimethylsiloxane which has very mobile chains, marked deviations from the linear are observed in the dependence of log T*=f(1/T) [13], so t h a t equation (8) does not apply. For polyvinylacetate and polyvinylchloride the activation energies determined from the temperature dependence of log ~ decrease with increase of T, which is in qualitative agreement with the WLF equation and subsequent dielectric data [11], though it is true t h a t these values are greatly reduced. I t was of interest to introduce a ratio which would make it possible to establish the possibility of introducing the activation energy independent of T in the temperature dependence of line width. In a general form, as follows from equations (1) and (2), this ratio is: T

log

tan

2

6 dT2- - 6~

For the actual definition of this ratio it is necessary to obtain ~ = ~ (T) analytically, which at present is impossible. Therefore the graphical dependence of log r ,e on

Activation energy of segmental motion in polymers

779

1/T, according to the curve ~ = ~ (I/T) is the simplest criterion for the possible introduction of an activation energy independent of temperature. An attempt to "rectify" the dependence of log r c* on 1/T by replacing formula (1) by (5) only gives a positive result if the temperature relationship of this curve satisfies equation (8), i.e. if the centres of curvature are situated above the curve itself (e.g. for D-1 polyarylate in Fig. lc). None of the other types of curve could be rectified by a similar procedure. I f we assume that equation (5) is really correct, we have: E--

R

d (log T)

R (T--T®)2 d (log ~)

Fulfilment of condition (8) requires t h a t the polymer conforms to the W L F equation. Therefore, substituting the values of log z from the expression obtained in [10] T--T d log av-=--c 1 T ~ % - - T ~ (9), where a T is the ratio of mechanical or dielectric relaxation time at temperature T to its value at temperature Tg; c 1, c2--WLF coefficients, corresponding to temperature Tu, we obtain: E. RClC2 ( T - - T ~ ) 2 log e (T-t-c2--Tu) ~

(10}

I f we assume that T~o-~T~--% ,

(11)

it appears that the activation energy is independent of T and equal to E - - RClC2 log e

(12)

Equation (11)is, in fact, valid for some substances if we bear in mind that the value of c2, according to literature data, m a y range from 20 to 120 ° for various polymers and low-molecular weight compounds. Miller [20] reported that T ~ , which is 25-50 ° below Tg, can be used as the initial point, when the viscosity of the polymer melt varies with temperature according to the Arrhenius curve for the same activation energy, if T changes to T--To~, i.e. equation (5) is used. The activation energy of glycerine with T g = 175°K [21] which is determined from dielectric measurements and spin-lattice relaxation time according to (5), when T ~ = 1 2 5 ° K , is 3.5-5.1 kcal/mole and independent of T [7], i.e. equations (11) and (12)hold good. A rough observance of these conditions and the linear dependence of log Tc* on l/T--Too are found with polyvinylchloride [18], natural rubber

780

A.I. MA~T.AXOVand G. G. PI~E~OV

[15], polyoxymethylene [7]. However, for isotactic PMMA [19] the above relationship can be "rectified" when T~=IS0°K, which is considerably lower than Tg=325°K. It may be concluded from the above that a non-linear dependence of log %* on 1/T is most general. Only in some cases (when E slightly depends on T) can small linear sections of this relation be isolated. However, even if this can be effected, it is difficult to relate the characteristics obtained to any parameters of the specimens. Thus, in some polymers crystallinity has absolutely no effect on activation energy (D-1 and D-2 polyarylates) although change in the phase state influences E. In other cases crystallinity has a marked effect on E (PETP, APA-3). Stereospecific characteristics in some cases considerably affect E (PMMA), in others, they have hardly any effect (PS). Nor does molecular weight have any definite effect on values of E (PS, PMMA). The temperature dependence of activation energy is different for various polymers. In some cases it may increase with higher temperature (various specimens of PS, PETP, APA-3, atactic polypropylene), in others it decreases (isotactic PMMA, D-1 polyarylate, polyvinylacetate, polyvinylchloride), in the third group it passes through a maximum (amorphous APA-3, P-2 polyarylate), in the fourth group it remains constant over a wide range of temperatures (PIB, NR, ]:)-2 polyarylate, atactic PMMA). If the constancy of E values or their reduction with higher temperature can somehow be explained, the passage of E through a maximum, or its increase with higher T are totally incomprehensible and have never been observed by other methods. In addition, the activation energies obtained appear to be markedly reduced, compared with the values calculated from dielectric and mechanical measurements and there is no correlation between E and other parameters of polymers. The explanation of experimental data by the dependence of log Tc* on 1/T should, apparently, be based on the correctness of the WLF equations (7) and (9), which for many polymers studied are satisfactorily fulfilled and have recently been again qualitatively confirmed (see [11]). The error of using existing theories of nuclear magnetic relaxation [3] to describe segmental movement in polymers is the most likely cause of anomalous temperature behaviour of E, reduced values and the absence of any relation .with other polymer characteristics. Equations (2) and (3) are, in fact, correct for spherical molecules participating in the Brownian movement. If this consideration can be extended to some extent to flexible molecular segments (NR, PIB, etc.) it is quite obvious that the segments of rigid-chain polymers, such as various polyarylates, aromatic polyamides, etc. cannot satisfy the assumptions made in existing nuclear relaxation theories. Additional discrepancies may arise as a result of the fact that intra- and intermolecnlar effects on NMR line width vary to different extents, according to temperature.. Furthermore, existing nuclear relaxation theories are based on the possibility

Activation energy of segmental motion in polymers

781

of describing motion b y correlation time alone. It is, however, well known [11] that polymer systems are characterized b y a wide distribution of T¢, i.e. this neglect of correlation frequencies m a y also influence the interpretation of results. I t is true that a former study [22] proves that the rectangular distribution of r o has practically no effect on activation energy. Activation energies and correlation time obtained from measurements and spin-lattice relaxation time should therefore be compared, as the effect of the presence and type of distribution spectrum of re on T1 has been thoroughly examined [7]. I t is indicated in this paper that the values of E determined from equations (1) and (3)from the low-temperature (core >>1) and high-temperature (cor~<
OF SEGMENTAL MOVEMENT IN N R

E, determined E, determined E, calculated Temperature of from T1 and equa- from 5 and equafrom (7), measurement, °K tions (1) and (4), tions (1) and (2), keal/mole kcal/mole kcal/mole 220 250 267 273 293

13.5 13.5 24 16.6 13

38 26 22 19 13

Comparison shows that, b y giving special attention to spectrum when calculating activation energies (data from T1) a satisfactory agreement is obtained with the W L F equation (7). This is one of the few eases when the E values obtained from dielectric, mechanical and NMR data coincide, which points to the possible use of existing theories of nuclear relaxation for describing segmental motion in the temperature range exceeding Tg b y 50-100 °, where only a few monomer

A. I. MAXLAXOVand G. G. PIMElCOV

782

units emerge as kinetic units. The constancy of activation energy of N R determined from 5 = 9 (T) in the temperature range of 210-250 °, where according to existing views [23] the effect of T on E should be even more marked, apparently indicates the limited use of this theory for polymers near Tg, when large chain sections emerge as kinetic units [11]. It can therefore be accepted that the activation energies sometimes obtained from N M R data as independent of temperature only prove the imperfection of the nuclear relaxation theory when applied to these polymers and not the invariable accuracy of the Arrhenius equation (1). In connection with Powles' [24] observations that r c determined from the temperature dependence of line width in the range where it suddenly decreases at the temperature, where 5=½ 5T is independent of the t y p e of spectrum showing the distribution of correlation times, we tried to determine the activation energy E 1 as follows. We assumed that, in the temperature range from the beginning of transition to 5 and up to the point where T 1 is minimum, E 1 remains unchanged. Values of ze were calculated for temperatures where 5 = ~ g T and T 1 is minimum, i.e. at points where the effect of distribution in ~c can be disregarded; E 1 was calculated from equation (1) (Table 2). The above values of E 1 do not agree (except for polymethylacrylate) either with activation energies calculated from temperature dependences of 5 and T 1, or with values obtained from dielectric and mechanical measurements made at frequencies of 103-105 c/s at T>Tg-$-50 °. The assumption of constant activation energy in this temperature range is therefore incorrect for these polymers. The reason is, apparently, the gradual dimensional reduction of the kinetic unit, which determines the motion of macrochains on increasing polymer temperature to Tg and above [11]. One can therefore agree with Schlichter's views [23] that the mechanisms of macroehain motion in the range where line width and the minimum of T 1 values markedly change are different. T A B L E 2. A C T I V A T I O ~ E N E R G I E S O B T A I N E D B Y V A R I O U S M E T H O D S

E from Polymer Atactie PMMA Polyvinylacetate Polymethylacrylate Crystalline PETP Polypropylene

I

E from

$----~(T),

T=~(T),

kcal/mole

kcal/mole

6

15. -

12 [25]

!

12 [25] 11 [ 2 6 ] 6.9 [26]

15

E from mechanical El, kcal/mole and dielectrio data, kcal/mole [23] 20 30 31 23 14

100 44-48 35 83 28

* The average is given for the temperature range of 323-- 353°K.

CONCLUSIONS

(1) A study was made of the inverse dependence of log % obtained from N M R line width on temperature for a large number of polymers; these dependences are non-linear in the majority of cases.

Activation energy of segmental motion in polymers

783

(2) It was shown that in several polymers the activation energy determined from NMR line width decreases with higher temperature, roughly conforming to the Williams, Landell and Ferry equation. (3) The anomalous values of activation energies obtained (increase and passage through the maximum of activation energies on increasing temperature) may be due to the inapplicability of existing nuclear magnetic relaxation theories to describing segmental movement in polymers, in particular rigid-chain polymers. Translated by E. SE.~ERE REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26.

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