Yamafuji, Theory of Dielectric Crystalline Dispersion in High Polymers

111

From the Department o/ Electronics, Faculty o/Engineering, Kyushu University, Fukuoka (Japan) *)

Theory of Dielectric Crystalline Dispersion in High Polymers By Kaoru Yama/uji With 1 figure (Received August 6, 1963)

Introduction The dielectric anomalous dispersions observed in a high polymer are successively called a, fl, y and so on from higher temperature and lower frequency sides. The dielectric a-dispersion observed in the highly crystallized sample of the polymer such as oxidized polyethylene or polypropylene is considered to come from the crystalline part. The dielectric a-dispersion of the amorphous polymer such as polyvinyl acetate, however, comes from the amorphous part and, therefore, its mechanism is different from that of the former one. In order to avoid such a confusion, the dielectric a-dispersion observed in the highly crystallized polymer sample is, in this paper, called the a~-dispersion where the suffix c means the crystalline part and dielectric dispersions coming from the amorphous part are called the as-dispersion, fl~-dispersion and so on where the suffix a means the amorphous part. As is well known, the dielectric as-dispersion is due to the dipole reorientation accompanied with segmental micro-Brownian motions of polymer chains in the amorphous part (1). The mechanism of the dielectric fladispersion also has been clarified recently (2, 3) and it is attributed to the dipole polarization accompanied with local motions of polymer chains in the glassy state of the amorphous part, where such motions are called the "local relaxation modes (4)". Recently, the property of the polymer crystallite seems to have been becoming one of the current subjects in the polymer physics. The behaviour of the dielectric at-dispersion is, however, not so clear in the present stage. Although the behaviours of the dielectric %dispersions in oxidized polyethylene, polypropylene and polytetrafluoroethylene were observed by several authors (5-9), these investigations seem to give only insufficient information upon the properties of the *) Now in the Department of Metallurgy, Carnegie Institute of Technology, Pittsburgh (USA.).

dielectric ac-dispersion such as the magnitude of the absorption, the activation enthalpy, the shape of the absorption curve and their dependence on the temperature or the degree of crystallinity. Such a situation might be attributed to the difficulty of the observation of the dielectric at-dispersion. At first, the dielectric at-dispersion can be observed in highly crystallized polymer samples. Most of the polymers with high crystallinity are nearly nonpolar polymers such as polyethylene. The observation of the dielectric absorption is, therefore, very difficult for these polymers. Next, the dielectric at-dispersion is observed at high temperatures where the ionic conduction is usually dominant. Since the property of the ionic conduction in the solid high polymer is not so clear, the elimination of the large ionic conduction from observed data seems to be extremely difficult. Theoretical investigation of the dielectric at-dispersion under these circumstances might be too early. However, the theoretical investigation of the dielectric at-dispersion under a reasonable assumption for its mechanism might be necessary for the development of this subject.

Mechanism of Dielectric at-Dispersion Now we shall consider the mechanism of the dielectric ae-dispcrsionl). The following two mechanisms could be considered for the dielectric ac-dispersion: (a) the MaxwellWagner type dispersion due to the co-existence of the crystalline and amorphous parts and (b) the dielectric dispersion attributed to the permanent dipole polarization accompanied with polymer chain motions in crystallites. If the dielectric ac-dispersion is attributed to the mechanism (a), it cannot be observed in a single crystal. On the other hand, the dielectric at-dispersion should be 1) The dielectric dispersion due to the rotational transition of polymer chains of polytetrafluoroethylene is not included in the dielectric ~c-dispersion. 8*

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KoUoid.Zeitschrifl und Zeitschrifl fi~r Polymere, Band 195 9 Heft 2

observed even in the single crystal if it is attributed to the mechanism (b). The choice between these mechanisms is, therefore, determined by the dielectric observation in single crystals. Such an observation seems to be possible with the aid of small electrodes and is now under planning in our group. Some information as to the behavior of polymer chains in crystallites can, however, be obtained from the nuclear magnetic data. For instance, the width of the broad component for polyethylene begins to decrease near 80 ~ (10), which suggests that the motions of polymer chains in crystallites become active above 80 ~ Recently, Peterlin et al. (11) gave a smart explanation for the problem of the chain folding in the polymer single crystal by taking account of longitudinal and torsional vibrations of polymer chains in the single crystal. These facts suggest that torsional and longitudinal vibrations occur actively at least in the temperature range where the dielectric ~c-dispersion is observed. Rempel et al. (12) also suggested from the nuclear magnetic data that the ~c-dispersion might be attributed to vibrational or rotational motions of polymer chains in crystallites for the mechanical case. The dielectric resonance absorptions due to thermal vibrations of polymer chains are usually observed at far infrared frequencies. In the incoherent vibrations of skeletal chains on the large scale, however, there are thermal vibrations with lowest frequency modes and such motions of polymer chains would be suffered by the strong internal friction. Since frictional terms become dominant compared with inertia terms under strong internal frictions, the dipole polarization accompanied with lowest vibrational modes would lose its resonance character and would be observed as the dispersion of the Debye type (5). In this paper, we shall adopt an assumption that the dielectric at-dispersion is due to the mechanism (b); that is, it is attributed to the dipole polarization accompanied with incoherent torsional and longitudinal vibrations of skeletal chains in crystallites. This mechanism is somewhat similar to the so called "local relaxation modes (4)" which bring about the dielectric fla-dispersion. The polymer chains in erystallites, however, stand in the more ordered state than those in the amorphous part. The activation entropy for the dielectric ~e-dispersion would, therefore, be smaller than that of the dielectric fla-dispersion. Moreover, the activation enthalpy for the motions of polymer chains in the crystal-

lite is larger than that in the amorphous part on account of, e. g., larger interchain interactions in the crystallite. According to the Eyring theory (13), the frequency at the maximum of the absorption curve depends upon the activation entropy and the activation enthalpy as is shown in eq. [4. 1]. The dielectric ~c-dispersion in the present mechanism should, therefore, be separated from the dielectric fla-dispersion.

Complex Dielectric Constant In this section, we shall derive the expression of the complex dielectric constant for the dielectric ~c-dispersion under the assumption that it is attributed to the dipole polarization accompanied with incoherent torsional and longitudinal motions of skeletal chains in crystallites. It seems, however, very difficult to deal with the dielectric dispersion of the polymer exactly. In this paper, we shall make the following simplifications. (a) The chemical bond length and the chemical bond angle are assumed to be kept respectively constant in thermal vibrations of skeletal chains with low frequency modes. (b) As the result of the simplification (a), only remain the skeletal vibrations of polymer chains which consist of torsional and longitudinal motions perpendicular and parallel to the molecular axis in the crystallite, respectively. I n this paper, we shall neglect the contribution from longitudinal motions, since the translation of the dipole does not contribute to the dielectric dispersion. (c) As for the potential energy for torsional motions of skeletal chains, we shall consider only nearest neighbor interactions for the intrachain interactions. The intrachain interactions among farther monomerie units than second neighbors and dipole-dipole interactions are neglected compared with nearest neighbor interactions. The incoherent interchain interaction is taken into account as only the averaged form. (d) Any choice of the boundary condition makes little difference to the result since the number of monomeric units n between successive loops in the crystallite is usually so large that 1Inis negligibly small. In this paper, we shall adopt the cyclic boundary condition, because the chain folds with the period n 3). Under these simplifications, the potential energy for torsional motions of a chain with 3) We shall, hereafter, use the word "chain" as the meaning of the chain part between successive loops in the crystallite.

Yamafuji, Theory of Dielectric Crystalline Dispersion in High Polymers

small amplitudes m a y be expressed as ~b

H = H0 + ~ . {Yl (~l -- g~l-1)2 -- YI' cos 2 ~/} l=l

- - E~ ~ #t sin O sin Ot~ {cos (q) - - ~l ~ -- ~Pl) l=1

-- COS ( ~ - - ~10)} ,

[3.1]

where ~vl is t h e r o t a t i o n a l angle of t h e dipole of t h e / t h m o n o m e r i e u n i t from its equilibrium direction ~10 in t h e plane perpendicular to t h e molecular axis in a crystallite, n is t h e n u m b e r of m o n o m e r i c units in a chain, (Oi~ q~z~ a n d (O, (/)) are respectively t h e directions of t h e l t h dipole in t h e equilibrium configuration a n d of t h e i n t e r n a l electric field E r w i t h respect to t h e u n i t cell coordinate in a cristallite where t h e molecular axis is t a k e n as t h e polar axis, Yl is t h e force c o n s t a n t of t h e dipole of t h e l t h m o n o m e r i c u n i t due to nearest n e i g h b o r i n t r a c h a i n interactions a n d Yl' is t h e force c o n s t a n t due to i n c o h e r e n t i n t e r c h a i n interactions. T h e e s t i m a t i o n for t h e values of 71 a n d 7l' a r e m a d e in A p p e n d i x 1 for a few typical polymers.

The equation of motion of the dipole of the /th monomeric unit is given by ~11

I0"t

~r

l=

1,2 .....

n,

[3.2]

113

In this collision process, only the velocity distribution becomes Maxwell's distribution with a relaxation time ~. The variation of the position occurs much more slowly compared with time intervals between successive collisions, and hence the position distribution is kept unMtered during the relaxation of the velocity distribution. This collision process will, therefore, be a good approximation also for the polymer crystMlite, though it was not derived for the polymer crystMlite. In order to solve the Boltzmann equation [3.4], we shall put /-:/0 + .q

[3.s]

F

[3.9]

and f 0 ( 1 + r ),

where

[3.10]

~0 --~ J"/0({~t~}, {~h}) d{r Then, the equation for g becomes

c~g

l

~" r. @

., @ ]

1= 1

where I is t h e m o m e n t of inertia of t h e m o n o m e r i c unit.

[3.11]

Now, we shall consider a distribution function f ({~vz}, {~), t) in the generalized phase

= / o r + - ~/o- E'~, 2~ p l S l n ~ s i n O l ~

space

The dipole polarization due to torsionM motions is given by

o f {~9l} = ~91 . . . .

~n

and

{9/}

=

@1 . . . . .

~0n. The function f is normalized as I I / ({~/}' {r

t) d{cfl } d{C~l} = 1

"

~ --q)l)(/l.

""

l=l

[3.3]

and can be obtained b y solving the BoItzmann equation:

av = ~-~

1

ffl s i n O sin Ol~ ; {cos ( ~ -- rFi~ -- ~l)

-- cos (~b -- ~oZo)}g d {~vl}d {ch}>a v ,

l=l

The equilibrium value o f / i n the absence of the applied electric field is given by r

1

n

2yl@l_q~l_O}2

2yl, COS2q~t] '

where N is t h e n u m b e r of dipoles per u n i t volume a n d < >av represents t h e average of crystallite coordinates with respect to t h e direction of t h e internal electric field E 7 .

The complex dielectric constant for the dielectric ~c-dispersion is given by

J

e(i~o)--e~

[3.5] where Co is t h e n o r m a l i z a t i o n constant, ~ is t h e Boltzmann c o n s t a n t a n d T is t h e absolute t e m p e r a t u r e .

To the thermal collisions of dipoles, we shall use Gross's collision process (14)which is given by

t

-

-

tl f__F({(vl},t)(2x~T)-nl2exp ~[-- ~

1

~'

1~1.=I

11

and

Yl' ==7',

av ~ ,

[3.13]

l -- 1 , 2 . . . . .

n

[3.14]

is made, the result is given by (i o~) - coo -. . . . 7 - -

where

-4n---

if the applied electric field E varies with an angular frequency o). The detailed calculation of the complex dielectric constant is made in Appendix 2. If the approximation of 7l=Y

[3.6]

[3.12]

~-mm av l, l' = 1

Zit-g'l F({~vl}, t) ~ f/({TI}, {q"/}, t) d{(/t}.

[3.7]

• Z --~ - - Z "

[3.15]

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Kolloid.Zeitschrift u n d Zeitschrift fiir Polymere, B a n d 195 9 Heft 2

In eq. [3.15], following abbreviations are used <&V>~v -~ {sin ~O sin Ol~ sin Ol,~ sin ( r - - ~l ~ sin (q~ -- ~v/,~ [3.16]

and Z~{I+ -

Y' + i c o r o +

(i~~

Y

\ t~

--

1+

] 1

Y + i m ~o + \ r

,

[3.17]

I 4y .

[3.is]

In order to calculate the summations with respect to l and l' in eq. [3.15], we should know the explicit form of av which depends upon the dipole configuration, values of dipole moments and the crystallite structure. At first we shall consider the usual pure polymer in which the dipole attaches to each monomeric unit, and make the approximation that /q--#

and

7g

Ol~ ~- - ~ ;

I=1,2

.....

n,

[3.19]

for simplicity. In the idealized helical dipole configuration: qPl~ ~ (Pc + 1 ~v0 [3.20] where % represents the configuration of the considering chain in the unit cell of the crystallite, < Su,>av in eq. [3.16] leads to 1

av = ~ - cos {% (l -- l')},

coo

=

e (i ~) - ~ -

N / t 2 exp (-- < ~2>/2) 3 { y ( 1 - - c o s % ) + y'}

l+i~ov0

Since the effect of defects can hardly be estimated, we shall deal with the limiting case of the random dipole configuration in the plane perpendicular to the molecular axis. In this case, ~,. m a y be approximated b y 1 av - - ~ exp (-- <~2>/2) 6 (l -- l'),

(i ~) - e~ =

~ N # ' exp (-- <92>/2) ( ~ )

3 {;~'(r' + 2 ~)}'/~

\co11!

{

~

[3.23]

1 + i~OVo + (~o-0)

after some calculations. In the dipole configuration ofeq. [3.20], the zigzag chain structure is also included as the limiting case of ~vo = 0Or~o =7tIn the actual crystallite of the pure polymer, dipoles stand in somewhat disordered configuration due to thermal agitations and the existence of defects besides the disordered configuration in the vicinity of loops. The effect of the disordered dipole configuration in the vicinity of loops is investigated in

'

[3.27]

where T1 ~

~~ T

~

r176. .

[3.28]

I 4 ( y ' + 2y) "

[3.29]

T2 ~ 2

T

and I wl~=4~y''

~

[3.26]

where 5 (1 - l') represents Kronecker's delta. Then, the complex dielectric constant leads to

3 {y (1 -- cos ~0) + Y'} ,'~ ico

ir 2 + (~-0)

X

[3.22]

for the random orientation of crystallites. With the aid of eq. [3.21], the complex dielectric constant becomes --

is,

x

1 {sin s O sin S (~ -- ~c)>av = ~ -

[3.24]

where <72> is the mean square of amplitudes of torsional vibrations. Comparing eq. [3.24] with eq. [3.21], we can see that the factor exp ( - < ~ * > / 2 ) i s multiplied to the complex dielectric constant in eq. [3.23] ; that

[3.21]

where we have used the fact

(i o~)

1

av ~-- - ~ exp (-- (~2}/2) cos {~v0(l -- l')),

where I T0=-- 4 y v , w ~

Appendix 2 and only gives a small correction term. In order to investigate the effect of thermal agitations, we shall make an intuitive approximation that the position of the dipole fluctuates with Gaussian distribution from the ideal dipole configuration of eq. [3.20]. Then, we can easily derive that < Szz,>~v is given by

c~

Next, we shall consider the oxidized polymer. In this case, dipoles attach sporadically to the chain and the dipole configuration will be nearly random. We could, therefore, use the approximation in eq. [3.26] to ~v. The summations with respect to 1 and l' in eq. [3.15] give rise to a factor q#S, where q is the percentage of oxidization in the crystallite. Then, the complex dielectric constant becomes s (i o~) -- e~ : q [!

3 {y' @' + 2 r)} '/~

[3.30]

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Yamafuji, Theory of Dielectric Crystalline Dispersion in High Polymers

Activation Energy

Now we shall estimate the activation energy for the dielectric a~-dispersion in the present mechanism. For the definition of the activation enthalpy (13), the angular frequency at the maximum of the absorption curve is given b y =_~_ AH where h is the Planck constant, AH is the activation enthalpy and AS is the activation entropy. We can, therefore, estimate theoretically from eq. [3.25] or eq. [3.30] in principle. The ~om for the complex dielectric constant of eq. [3.25] is 1/~0 given b y eq. [3.18]. Since the temperature dependence of r o is not so clear, we shall estimate the activation energy with F r 6 h l i c h ' s method (14). In the activated state of the chain, t h e / t h dipole would be deformed b y the torsional angle ~ with thermal agitations even the applied electric field is absent. The energy difference between the activated and the equilibrium states of the chain is given by n

A I I = ~ {? (~p/-- ~/_~)2 + ?, (1 -- cos 2 ~l)}

[4.2]

functions is, however, very tedious and leads to a complicated form. We shall, therefore, approximate eq. [4.4] by a linear differential equation of the second order: d~

2y' ~ : 0 .

dZ 2

[4.7]

y a2

The solution of eq. [4.7] with boundary conditions eqs. [4.5] and 4.6] can be obtained easily and the result is given b y

"~=

((q~m2>,1/2 cosh ,[[2Y' [\ 7aZ ,~1/2(Z

n~.

a)]/cosh

• ItS)

E4s]

Inserting eq. [4.8] into eq. [4.3], we obtain A H -- 2 (2 y y,)~/2 tanh [n { ?' ~/2]

[4.9]

V~-y/ l

In order to estimate the order of magnitude of AH, the information upon <~Vm2> is necessary. The square mean of the torsional angles of the monomeric units in the vicinity of loops due to thermal agitations is calculated in Appendix 3 and the result is given by <9~> _ n~T/8[y' (y' + 2y)]~/2.

[4.10]

Inserting eq. [4.10] into eq. [4.9], we obtain

/=1

with the aid of eq. [3.1] and under the approximation of eq. [3.14]. I f we approximate the summation b y the integral, eq. [4.2] becomes na

~?a [~-} + ?' (1 -- cos

~

[4.3]

0

where a is the length of the monomeric unit along the molecular axis of the crystallite. Since the activated state should be at least metastable, the variation of A H must be zero, which leads to E u l e r ' s differential equation ( A H ) = O: Ya2 d2~ -- y' sin 2 ~ = 0. [4.4] As the boundary condition for ~, we shall adopt the following two conditions: is symmetric with respect to Z = n a/2

[4.5]

and (Z : n a) : (<%,~>)t/2,

[4.6]

because the behavior of the chain in the crystallite would be symmetric with respect to the middle point between successive loops and because the torsional angle of the chain would be larger in the vicinity of the loop. Equation [4.4] can be solved exactly with the aids of J a c o b i ' s elliptic functions. The calculation of A l l b y the use of J a c o b i ' s elliptic

/

[4.11]

Discussions

In preceding sections, theoretical expressions for the complex dielectric constant and the activation energy were derived under several simplifications. In this section, we shall show some features of theoretical expressions derived in preceding sections and compare them with the observed data. M a g n i t u d e o] A b s o r p t i o n . The complex dielectric constant for the dielectric ~e-dispersion is given by eqs. [3.25] or [3.30]. In the usual pure polymer, the magnitude of the absorption (so - e~) is given b y eq. [3.25], that is, N #2 exp (-- <~v~>/2yl E~ \ . [5.1] In the limiting ease of the isotaetic dipole configuration for the zigzag chain polymer (% = 0), eq. [5.1] leads to s0--s~=

3 y'

.

[5.2]

In another limiting case of the syndiotactic dipole configuration for the zigzag chain polymer (% = ~r), eq. [5.1] leads to n N I'~ exp (--@v2>/2)(--~-). [5.3] eo--ec~ = 3 ( 2 y + 7')

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Kolloid-Zeitschrift und Zeitschriflfiir Polymere, Band 195. Heft 2

As is s h o w n in A p p e n d i x 1, t h e force cons t a n t ~ due to n e a r e s t n e i g h b o r i n t r a c h a i n i n t e r a c t i o n s is m u c h larger t h a n t h e force c o n s t a n t Y' due to i n t e r c h a i n i n t e r a c t i o n s ; e.g., y ~ 6 • 10-13erg, y' _ 3.5 x 10-~ erg in the vicinity of 100 ~ [5.4]

Since the o b s e r v e d d a t a for t h e dielectric %-dispersion in the p u r e zigzag chain p o l y m e r has n o t y e t been r e p o r t e d , we shall consider t h e case of oxidized p o l y e t h y l e n e . I n this case (e0 - e~) is g i v e n b y eq. [3.30]; t h a t is

in t h e case of p o l y e t h y l e n e . T h e m a g n i t u d e of t h e %-dispersion for t h e isotactic zigzag chain p o l y m e r is, t h e r e f o r e m u c h larger t h a n t h a t for t h e s y n d i o t a c t i e zigzag chain p o l y m e r , which can be e x p e c t e d b y inspection, too. I n t h e case of t h e helical chain p o l y m e r (0 < % < :~), t h e v a l u e of (e0 - e~) lies bet w e e n t h e values of eq. [5.2] a n d eq. [5.3], alt h o u g h the values of force c o n s t a n t s in helical chain p o l y m e r are different f r o m those in zigzag chain p o l y m e r s in general. N o w we shall consider t h e case of p o l y p r o p y l e n e , since p o l y p r o p y l e n e is o n l y t h e p o l y m e r in which t h e dielectric %-dispersion has b e e n observed. I n isotactic p o l y p r o p y l e n e , ~ 6 • 10-~3erg, y' _ 5 x 10-~4 erg in the vicinity of 100 ~ [5.5]

W i t h t h e aid of #3___ (1.2 • lO-aSesu)2, N = 4 • 1023A,,, <~3> ~ 0.07, E~/E ~ 2,

as is s h o w n in A p p e n d i x 1 a n d % = ~/3, N = 2 x 1022At (d = 0.92) , # ~ (0.4 • 10-is esu)~ , <~3> ~ 0.08,

[5.6]

where A~ is the degree of c r y s t a l l i n i t y a n d t h e v a l u e of <~> is e s t i m a t e d with t h e aids of eqs. [A. 3.6], [A. 3.9] a n d [5.5]. T h e r a t i o E~/E can be e s t i m a t e d b y t h e use of Onsager's expression (17) : E~_ 3~0 /~+ 2~ 3 [5.7] E 2e0+ e~ which leads to a b o u t 2 in t h e p o l y m e r w i t h small dipole m o m e n t s . Then, (e0 - e~r in eq. [5.1] leads to e0 - coo = 2 • 10-2 A c at about 100 ~ [5.8] T h e o b s e r v e d d a t a of t h e dielectric %-dispersion for p o l y p r o p y l e n e (9) are g i v e n as t h e t a n 0 vs. t e m p e r a t u r e curve, where t h e m a x i m u m of t a n 6 is a b o u t 2 • l0 -a n e a r 100 ~ S i n c e e ' = 2 ( 9 ) , e" = e' t a n 0:and (eo - e~) = 2 e"max for t h e Debye a b s o r p t i o n curve, t h e o b s e r v e d v a l u e of (e0 - e ~ ) will be a b o u t 0.8 • l0 -~. A l t h o u g h t h e theoretical v a l u e seems to be a little larger t h a n the observed value, t h e d i s c r e p a n c y m a y be attrib u t e d to t h e o v e r e s t i m a t i o n of t h e v a l u e of #2, Ac < 1 a n d t h e u n d e r e s t i m a t i o n of t h e o b s e r v e d v a l u e calculated b y a s s u m i n g t h e Debye a b s o r p t i o n curve.

e0 -- soo = q

~ N P~ exp (--<~2>/2)(~_~L) [5.9] 3 {y' (y' + 2 y)}:/2 " 9

[5.10]

a n d eq. [5.413), (e~ _ e~) becomes eo - e~ : 6 x 10-1 q A c at about 100 ~

[5.11]

T h e o b s e r v e d v a l u e of t a n ~ is 2 ~ 4 X 10 -3 n e a r 100 ~ (5, 6, 7), a n d hence t h e o b s e r v e d v a l u e of (e0 - e~) will be a b o u t 0.8 ~ 1.2 X 10 -2 . This v a l u e is u n d e r e s t i m a t e d b y t h e use of t h e Debye a b s o r p t i o n c u r v e since t h e a c t u a l a b s o r p t i o n c u r v e seems to be b r o a d e r t h a n Deby's. On t h e o t h e r h a n d , t h e v e r y small v a l u e of q a n d A c < 1 will m a k e t h e t h e o r e t i cal v a l u e smaller b y a b o u t the f a c t o r 10 ~. A l t h o u g h the t e m p e r a t u r e d e p e n d e n c e of (e0 - e~o) comes also f r o m the v a r i a t i o n s of 7, ~" a n d (Er/E) w i t h t e m p e r a t u r e , t h e m a i n c o n t r i b u t i o n m a y be a t t r i b u t e d to t h e t e m p e r a t u r e d e p e n d e n c e of t h e e x p o n e n t i a l factor in eq. [5.1] or [5.9]. As can be seen f r o m eqs. [A. 3.6] a n d [A. 3.9], <~v2> is p r o p o r t i o nal to n T. W e can, therefore, e x p e c t t h a t (e0 - e~) will v a r y as e x p ( - b~T) at higher t e m p e r a t u r e s , where b is t h e f u n c t i o n of y a n d 7'. The o b s e r v e d t a n ~ - t e m p e r a t u r e c u r v e also seems to show such a t e n d e n c y since t a n decreases w i t h increasing t h e o b s e r v e d f r e q u e n c y (9) a n d since t h e higher f r e q u e n c y corresponds to t h e higher t e m p e r a t u r e . F r o m the p r e s e n t m e c h a n i s m of t h e dielectric %dispersion, (e0 - e ~ ) will again decrease w i t h d e c r e a s i n g t e m p e r a t u r e because torsional m o t i o n s of skeletal chains in t h e c r y s t a l lite are e x p e c t e d to be " f r o z e n " at lower t e m p e r a t u r e s . No o b s e r v e d d a t u m has, h o w e v e r , y e t b e e n r e p o r t e d a b o u t the b e h a v i o r of (eo - e~) a t lower t e m p e r a t u r e s . Shape o/Absorption Curve. T h e expression of t h e c o m p l e x dielectric c o n s t a n t is g i v e n b y eq. [3.25] in the case of t h e usual p u r e p o l y mer. As is seen in eq. [3.18], r o has t h e s a m e order of m a g n i t u d e as a n g u l a r frequencies for t o r s i o n a l m o d e s of the chain a n d hence lies a t 3) As the value of the force constant for oxidized polyethylene, we can use the value of polyethylene in eq. [5.4] so far as the percentage of oxidization is small.

Yamafuji, Theory of Dielectric Crystalline Dispersion in High Polyesters

far infrared frequencies. The term (~o/o~0)2is, therefore, much smaller than unity in our observed range of {1 ~ 106 c/s), and can be neglected compared with other two terms since the maximum of the absorption curve is in the vicinity of o ~ 0 = l. Equation [3.25] leads, therefore, to: nearly the simple Debye absorption curve. In the expression of the complex dielectric constant in eq. [3.25], however, the alteration of the molecular configuration caused by defects in the crystallite is not considered. Since the contribution of this factor to the complex dielectric constant can hardly be estimated exactly, only the limiting case of the random dipole configuration in the plane perpendicular to the molecular axis is investigated and the result is given by eq. [3.27]. The absorption curve from eq. [3.27] is shown in fig. 1 for the case of rl/T 2 = 40, which is somewhat broader than the Debye absorption curve. The observed absorption curve is, therefore, expected to have the shape between these absorption curves. 0.5t

!

2

//

,~\

\

I

',

I

',

o't -5

// -,;

-3

, -2

-t

0

I

2

.~

4

/n w("ct'c2 )1/2 Fig. 1. The theoretical absorption curve from eq. [3.30] in which 7:1/v~ = 40. The dotted curve is the Debye absorption curve for comparison

In the present derivation of the complex dielectric constant, however, we have neglected the distribution of the value of the relaxation time v in the velocity space. Since several atomic groups are included in the polymer chain and the dipole configuration will not be in the completely ordered state due to defects in the actual polymer crystallite and so on, the distribution function of ~ will be broader than the delta function assumed in the present calculation. Then, the shape of the observed absorption curve may be broader than the present theoretical estimation. In the oxidized polymer, the shape of the dielectric ac-absorption curve is given b y eq. [3.30] in the first approximation. With the aid of eqs. [3.28] and [5.4], Vl/V2 becomes about 40. The absorption curve for vl/~2 = 40 is shown in fig. l. The absorption curve is

117

broader than the Debye absorption curve and the slope of the higher frequency side is a little gentler than that of the lower frequency side. The lower temperature side of the absorption curve corresponds to the higher frequency side, and hence this tendency suggests that the slope of the lower temperature side is a little gentler than that of the higher temperature side, if we assume that the effect of the structural alteration of the erystallite with temperature is relatively small. Although such a tendency of the absorption curve in eq. [3.30] seems to be found in the observed tan ~ vs. temperature curves (5, 6, 7), the observation by varying temperature might include the considerable effect of the structural alteration such as recrystallization. To say spitefully, even the existence of the dielectric ~c-dispersion might not be completely sure so long as the observations are made by varying the temperature at a fixed frequency. The observation by altering the frequency at a fixed temperature seems, therefore, desirable very much.

Activation Energy. The expression of the activation energy is given by eq. [4.11]. In the vicinity of 100 ~ where the dielectric %-dispersion is usually observed, the number of monomeric units n between successive loops in the crystallite changes rapidly with temperature. For instance, n increases from about 100 at about 80 ~ to about 200 at about 120 ~ for polyethylene (11). Since the activation energy in eq. [4.1 l] is proportional to n, it varies rapidly with temperature. In the case of oxidized polyethylene, we m a y use the values of eq. [5.4] for y and 7'. Then, the activation energy in eq. [4.1 l] leads to ~ l I ~- 20 kcal/mol (n ~ 100) -~ 40 kcal/mol (n = 200).

[5.~2]

Since observed values were estimated from the tan 5 vs. temperature curve, these data will not be correct if the activation energy varies rapidly with temperature. Actually, the observed value of the activation energy varies from 25 kcal/mol (7) to 60 kcal/mol (5) with various authors' observation in the case of oxidized polyethylene. Moreover, these observations might not be made in the single crystallite and, therefore, the value n might be larger than the value o f n = 100 ~ 200 in the single crystallite. Our theoretically estimated value of the activation energy for oxidized polyethylene in eq. [4.11] may be the correct order for the value in the single crystallite at about 100 ~

Kolloid.Zeitschrift und Zeitschrifl fiir Polymere, Band 195 9 Heft 2

118

In the case of polypropylene, the observed value of the activation energy is scattering in the very wide range, t h a t is 70 ~ 150 keal/ mol (8,18). Unfortunately, the value of n seems to have never been estimated for polypropylene. I f we assume t h a t n = 200 ~ 4004), the theoretical value leads to 40 ~ 80 kcal/mol with the aid of eq. [5.5] for ~ and y'. Since the value of 150 kcal/mol seems to be too large, the observation with altering the frequency at fixed temperatures is desirable. Frequency Region. The dielectric ~c-dispersion is observed at the audio frequency region. It is necessary to show t h a t the dielectric c%-dispersion in the present mechanism appears at the audio frequency region. As can be seen from eq. [3.25], the angular frequency at the maximum of the absorption curve is at 1/~0, because the term (eo/~0)~'can be neglected as mentioned in the discussion of the shape. In order to estimate ~0 in eq. [3.18], the informations about v and I are necessary. While the time during successive collisions between the Brownian particle and the molecules of the surrounding fluid is generally the order of 10-21 sec (19) in the liquid, the quantity in crystallites will be shorter than this value. On account of strong interactions in the polymer crystallite, the relaxation time ~ in the velocity space is not larger than time intervals between successive collisions. Accordingly, the order of~ will be not so far from l0 -21 sec. The moment of inertia for the torsional motion of the monomeric unit is approximately given by I --~ Mmonomer d 2 ,

[5.13]

where M is the mass and d is the length of the monomeric unit perpendicular to the molecular axis. Inserting these values into eq. [3.18], we obtain ],~ = o)m/2zt =

I0 -*2 X I0 -2.

10 -37

= 104 .

[5.14]

We can, therefore, conclude t h a t the dielectric ~c-dispersion in the present mechanism appears at the audio frequency region. As a conclusion, the theoretical expression for the dielectric c%-dispersion is derived in this paper with a mechanism t h a t the dielectric g~-dispersion is attributed to the permanent dipole polarization accompanied with incoherent torsional and longitudinal motions of skeletal chains in crystallites. The observ4) As c a n b e seen f r o m t h e c a l c u l a t i o n of t h e activ a t i o n e n e r g y , we s h o u l d t a k e t h e n u m b e r o f C - C b o n d s b e t w e e n successive loops as t h e v a l u e o f n.

ed results up to date seem, however, to give only insufficient information about the property of the dielectric %-dispersion. The more detailed experimental investigation is, therefore, desirable to make further progress in this subject. The author would like to express his thanks to Professor Fu~io Irie for valuable discussions with respect to theoretical derivations. He is also grateful to A. Professor Y6ichi Ishida of Applied Chemistry for helpful discussions with numerous problems through this work.

Appendix 1 In this section, we shall estimate force constants for small amplitude torsional motions of skeletal chains. Although the calculation of these force constants is very tedious, we shall estimate those here only approximately since the observed data to be compared are scattered in rather wide range. At first, we shall consider the nearest neighbor intrachain interaction. Suppose t h a t the skeleton of a polymer chain has the helical structure and let the bond length, bond angle and internal rotation angle be denoted by r/j, ~jk, and 2~jk~respectively, where i, j, k and 1 are any four consecutive numbers specifying skeletal atoms. These atoms lie on a helix which can be described by three coordinates, d~, ~lj and ~v~j,where d~ is the distance of the i th atom from the axis of the helix, ~/j is the translation along the axis on passing from the i t h atom to the ?" th and ~1 is the angle of rotation around the axis from the i th atom to the jth. The zigzag chain structure is included in the helical structure as the limiting case of ~/jo = 0 or q~jo = z, where q~jodenotes the value of ~/j in the equilibrium configuration. I f we use the potential energy of the UreyBradley type (20), the nearest neighbor intrachain potential energy m a y be given by 1 //intra = //~ntra

n

"4" -~ K 2 (~rii)~ i=1

1

H ~ (r 0 ~eiik) 2

+-2- i= t 1 + ~-F

n ~ (6qi_1, i+1) 2 i=1 n

+ K' 2 (r~

[A. 1.1]

i=1

where ~ represents the small deviation from the equilibrium value of each quantity, r 0 is

Yamnfuji, Theory of Dielectric Crystalline Dispersion in High Polymers

the value of r o. in the equilibrium configuration, qj_~, j+a is t h e d i s t a n c e b e t w e e n the (j - 1)th a n d the (j + 1)th skeletal a t o m s a n d K, H , F a n d K ' are r e s p e c t i v e force constants. Since 6rig, 09ijk, 6Atjlr a n d d q i _ 1 i + 1 are functions o f di, a O a n d ~o~), the force c o n s t a n t y defined b y eq. [3.1] is given b y -

1 ~2//intra 2 o (~o.) ~

n

Ilhm,r :-: ~'o - 7' ~" cos 2 ~/ /=!

[A. 1.6]

7' =: 7o exp (-- 2 (q_22),

[A. 1.7]

and

[A. 1.3]

for t h e helical chain with the s a m e cons t i t u e n t unit. I n t h e case of p o l y e t h y l e n e , % = 20 = n a n d cos % = -1/a. Then, y is given b y y = K ' ro~/sin ~ (~0/2) with the aid of eqs. [A. 1.1], [A. 1.2] a n d [A. 1.3]. I f we notice t h a t r 0 cos (~o/2) is the radius for the torsional r o t a t i o n a r o u n d the molecular axis, t h e height of the sinusoidal b a r r i e r V0 m a y be given b y V0 = K ' r o ~ cos(~0/2 ). T h e value of V0 is a b o u t 2750 cal/mol if we a p p r o x i m a t e l y use the value of V0 for ethane. Then, the value of y is e s t i m a t ed as _~ 6 • 10-~serg for polyethylene, [A. 1.4.] which is j u s t t h e s a m e as the v a l u e e s t i m a t e d b y Szigeti (21). A l t h o u g h p o l y p r o p y l e n e has two sorts of c o n s t i t u e n t unit, % = ~1~ = 2~1 = :z/3 a n d cos ~o = cos ~o = _ 1/a for t h e isotactie conf o r m a t i o n (16). Then, we can f o r t u n a t e l y use eq. [A. 1.3] in the first a p p r o x i m a t i o n . T h e similar calculation to t h e case of p o l y e t h y l e n e leads to a g a i n the s a m e v a l u e of 7 as eq. [A. 1.4]: y ~ 6 • lO ~aerg for polypropylene.

The force c o n s t a n t y' due to i n t e r c h a i n interactions is e s t i m a t e d b y P e t e r l i n et al. (I 1) for the case of p o l y e t h y l e n e . The results are given b y

[A. 1.2]

A l t h o u g h the e x a c t calculation of y f r o m eqs. [A. 1.1] a n d [A. 1.2] is v e r y tedious, the m a i n c o n t r i b u t i o n for t h e torsional force cons t a n t of the skeletal chain will come f r o m the i n t e r n a l r o t a t i o n of the skeletal b o n d since force c o n s t a n t s K , H a n d F are usually m u e h larger t h a n the force c o n s t a n t K ' a n d ru, 0~ j'k a n d q~_ 1, t+ 1 are a p p r o x i m a t e l y k e p t u n a l t e r e d in the torsional m o t i o n of t h e skeletal chain as was m e n t i o n e d on the simplification (a). U n d e r such an a p p r o x i m a t i o n , we h a v e o n l y to k n o w the relation b e t w e e n 520e I a n d d~o" which is g i v e n b y (22) sin ~o' @n - sins (o0/2) sin 2~0'5).0

119

where Y0 and y' are the a m p l i t u d e s of t h e potential e n e r g y a t rest a n d a t finite r o t a t i o n a l fluctuations of a d j a c e n t chains, respectively, a n d is the m e a n square of r o t a t i o n a l angles of dipoles a r o u n d the molecular axis. With the aid of 70 ~- 0.639 ~T in the vicinity of 100 ~

[A. 1.8]

given b y P e t e r l i n et M. ( l l ) a n d eqs. [A. 1.6], [A. 1.7], [A. 3.6] a n d [A. 3.9], we o b t a i n y' ~ 3.5 • 10-14erg

[A. 1.9]

for p o l y e t h y l e n e . Similar calculation for the case of the isotactic p o l y p r o p y l e n e w i t h the aid of the u n i t cell s t r u c t u r e g i v e n b y N a t t a (16) leads to 7 ' ~ 5 • 10 l~erg in the vicinity of 100~ [A. l.lO]

Appendix 2 T h e d e r i v a t i o n of eqs. [3.15] is as follows: Since we shall consider torsional m o t i o n s with small amplitudes, we h a v e only to solve eq. [3.11] in the a p p r o x i m a t i o n to the first orders of Fl's a n d E,, 5). I f we p u t g

:: (/o E,/~T) ~ {~I qsl + fit (it},

[A. 2.1]

t=l

a n d t a k i n g into a c c o u n t t h a t E~ varies with an a n g u l a r f r e q u e n c y o~, eq. [3.11] becomes

l--1

i

/=l

fll {2 7l (q~l -- q~l- 1) -- 2 Yl ~1 (~l + 1 -- cPl) 1=1

[A. 1.5]

A l t h o u g h such a simple calculation of y for p o l y p r o p y l e n e m a y not give a correct value, the sinusoidM p o t e n t i a l barrier Vo is y e t the order o f 3500 cal/mol in t h e m o r e rigid helical p o l y m e r as p o l y o x y m e t h y l e n e (23) a n d hence the order o f y in eq. [A. 1.5] will be reasonable for p o l y p r o p y l e n e .

+ 4 Yl' F1} =: ~ / q sin O sin 0t0 sin (O -- r I=1

ql [A. 2.2]

5) The result of the estimation in Appendix 3 shows that the mean amplitudes of the torsional motions of the skeletal chains due to thermal agitations are very small, for instance, (~} ~ 0.07 (radian) 2 in the case of polyethylene.

Kolloid-geitschrift und Zeitschrifl fiir Polymere, Band 195 9 Heft 2

120

With the aid of the cyclic boundary condition, we can easily see that ill {2

-

el-l)

-

+l (vt +l

Inserting eq. [A. 2.8] into eq. [A. 2.4], we obtain ~l -- ~ / ~ l ' sin O sin 0z, ~ sin (q~ -- ~vz,~

-

l'=l

/=1

n

[A. 2.3]

• ~. n -~ Q (k) {4 (y + ;/) -- 4 r cos (2 n kil0} k=l • exp {i 2 ~ k (1 -- l')/n}. [A. 2.10]

Inserting eq. [A. 2.3] into eq. [A. 2.2] and putting coefficients of ~v,'s and ~b~'s independently zero, we obtain

The dipole polarization

~v is given by eq. [3.12] which to the first order of qz's becomes

i w l a 1 -- 2 { ( y / + ~l+~ + 2 yt') fl/ -- Yl~ill+~


= ~

2 ~91 {()]/ 71- ~]l+l) ill - - Y l + I f l l + l

--

~]l i l l - - l }

l= ~

"

n -

7t ill- l} =

0,

~l + (ia~ + + ) ill = t q s i n O s i n Ol~

l, l' =1

Since skeletal chains are in the regular configuration except in the vicinity of loops, the values of force constants are nearly independent of I in the middle of the chain. We shall, therefore, make an approximation that 7t = ? and ~t' = ~' in the middle of the chain,

Since a relation (Fo/uT)

ul qDl = -- (1/i t~ I) ~ fit (~ Fo/~qDI) [A. 2.12] l=l

/=1

can be easily derived with the aid of eqs. [3.5], [3.10], [A. 2.3], [A. 2.8] and [A. 2.10], eq. [A. 2.11] leads to n

?t = ? -- Ay and 7{ = 7' in the vicinity of loops.

1

[A. 2.5] The fluctuations of yr in the vicinity of loops m a y be neglected compared with d ~ if we take account of 7 >~ $'. The sign of As m a y be positive because the value of y will be smaller in the vicinity of loops. With the aid of eqs. [A. 2.4] and [A. 2.5], the equation of fit becomes


1,1' = l

• exp {i 2 ~ k (1 -- l')/n},

[A. 2.13]

where ( S l l ' ) a v ~ (sin2 0 sin 0/o sin 0/,o sin (~b -- ~l 0)

• sin (@ -- tp/,~

.

[A. 2.14]

If we approximate the summation with respect to k b y the integral, we obtain

4 (~ -~- ~/')} fll - - 2 ~ (ill + l - - ill__ ] )

:2Ay(2ill--ill+~--ill_l).

1

k=l

[A. 2.15]

[A. 2.6]

For a while, we shall consider the case of A 7 = 0. Then, eq. [A. 2.6] can be solved by putting

where

•

n

ill= (1/n)l/=~u(k)exp(i2~kl/n),

ZI~-g'I

n -1 Q (k) exp {i 2 ~]c (1 -- l')/n} -- 2 y Z - ~ - Z '

-- i w I #t sin O sin 0t ~ sin (~b -- ~to)

[A. 2.7]

k=l

It 1 + - -"+~'i o J ~

o+

--

'ovl'

--1

[A. 2.16]

r ~ ~ 1/4 ~ .

[A. 2.17]

\too I J

and

3o ~ Wo2/~,

and the result is given by fit -- i eo I ~ / q , sin O sin Ol,~ sin (q) -- ~r ~ ~ n -~ Q (k) l'=l

k~l

• exp {i 2 ~ k (l -- l')/n},

[A. 2.8]

where Q (k) ~ [i~oI (i~o+ +)+4

o)

(~ -- qzt~ .

[A. 2.4]

{i ~o I (i o) + + ) +

~ #l sin O sin 01~

~TzT N E( v

[A. 2.9] (7+Y')--4y

cos (27~k/n)] -~.

In the right hand side of eq. [A. 2.16], the minus sign should be chosen from the condition that ]Z I< 1 within the contour of the integral. Inserting eq. [A. 3.15] into eq. [A. 3.13] and taking account of eq. [3.13], we obtain eq. [3.15] : e (i o)) -- e~ --

2~N

7

~

1

Z]l-t'[

~ ~-~l ~l' (St~')av Z_1 _ Z " l,l' = l

[3.15]

Yamafuji, Theory of Dielectric Crystalline Dispersion in High Polymers T h e c o r r e c t i o n for e (i o~) - e~ d u e t o A7 :~ 0 c a n be c a l c u l a t e d in t h e similar w a y a n d t h e r e s u l t is g i v e n b y

are g i v e n b y eq. [A. 3.3]. As is well k n o w n , (q~2} is given by (11) n

(~2) =

[e (i t~) -- e~]corrcction --

ZTZ-~--2 (Z-' -- Z) 2 Zl+l',

[A. 2.18]

w h e r e / i n is t h e n u m b e r o f m o n o m e r i c u n i t s b e l o n g i n g t o a loop. Since t h e c o r r e c t i o n t e r m is s m a l l e r t h a n t h e v a l u e o f eq. [3.15] b y t h e f a c t o r (~in~n) (d~,/~) ~ or m o r e , we c a n n e g l e c t t h e c o n t r i b u t i o n o f t h e d i s o r d e r e d dipole c o n f i g u r a t i o n in t h e v i c i n i t y o f loops in t h e first a p p r o x i m a t i o n . 3

I n this section, we shall d e r i v e a n expression o f (~m2}. T h e e q u a t i o n s o f m o t i o n o f dipoles for t o r s i o n a l v i b r a t i o n s w i t h small a m p l i t u d e s are g i v e n b y eq. [3.2] w h i c h b e c o m e s

I(~l+ 2 y ( 2 ~ - - ~ l + ~ - - ~ Z _ ~ ) + 4 y ' ~ v t : 0

[A. 3.1]

t o t h e first o r d e r s o f ~{s a n d u n d e r t h e a p p r o x i m a t i o n o f eq. [3.14]. Since e i g e n f u n c t i o n s o f this e q u a t i o n h a v e t h e f o r m n

~t = (lilt)'/* ~ u (k) exp (i 2 z k 1/n), k=l

[A. 3.2]

t h e e i g e n v a l u e s o f (o are g i v e n b y ~k ~ = (4/~) {~' + 2 r sin~ (k ~/n)}.

[A. 3.3]

Since ~m is t h e r o t a t i o n a l angle o f t h e dipole a t t h e loop, ~0m ~ ~n = ~n/2 + ~ (~t -- ~z-~) 9

[A. 3.4]

n -~- + 1

l=

N o t i c i n g ~0nl2 ~ 0 f r o m eq. [4.8], we o b t a i n -

+

-

-

ft

~=-~+ ~

[A. 3.7]

I n s e r t i n g eqs. [A. 3.3] a n d [A. 3.7] i n t o eq. [A. 3.6], we o b t a i n

n- #//tl,

~

l, l * = 1

Appendix

-n- wk -~ .

k=l

n

• (Sll'>av

121

~'l'=-~- + ~

[A. 3.5]

14:l'

I n t h e l i m i t i n g case o f ~' = 0, t h e c o n t r i b u t i o n o f t h e s e c o n d t e r m o f eq. [A. 3.5] t o (~} cancels e x a c t l y . Since y >~ 7', t h e c o n t r i b u t i o n of t h e s e c o n d t e r m is also small, and hence 7~

( ~ n ~} --~ -~ ( ~ } ,

[A. 3.6]

w h e r e (~2} is t h e s q u a r e m e a n o f t h e a m plitudes of thermal vibrations of the harm o n i c oscillators w h o s e a n g u l a r f r e q u e n c i e s

?/

(fra 2} - ~ ~ T

~' + 2 y sin 2 - -

.

[A. 3.8]

k=l

A p p r o x i m a t i n g t h e s u m m a t i o n in eq. [A. 3.8] b y t h e integral, we o b t a i n t h e final r e s u l t : <~m2> ~ -~-•

[yr (?' + 2 7)]-U 2 .

[A. 3.9]

Summary Theoretical investigation was attempted for the dielectric crystalline dispersion which is observed at high temperatures in highly crystallized polymer samples such as polypropylene and oxidized polyethylene. The dielectric dispersion of this kind had been confirmed experimentally to come from the crystalline part, but no theoretical investigation had been reported for this dielectric dispersion. The proposed mechanism was such that this dielectric dispersion is attributed to the dipole polarization due to torsional and longitudinal motions of skeletal chains in polymer crystallites. On the basis of such a mechanism, the expression of the complex dielectric constant was calculated by solving the Boltzmann equation for the motion of the skeletal chain. The magnitude of the absorption, its temperature dependence and the shape of the absorption curve were derived from the complex dielectric constant. The activation energy was also calculated by the use of Fr6hlich' s method. The agreement between the theoretical results and the observed data seems quantitatively good, which seems to show the appropriateness of the present mechanism for the dielectric dispersion of this sort. The observed data up to date are, however, given as the tan ~ vs. temperature curve, in which might be included in the effect of the structural alteration such as the recrystallization. The observation by varying the frequency at constant temperatures seems to be desirable for being compared with the present theory. Zusammen]assung Diese theoretische Untersuchung wurde fiir dielektrische Dispersion im Kristallinen, beobachtet bei hohen Temperaturen, in hochkristallisierten Polymerproben wie Propylen und oxydiertes Poly~thylen durchgefiihrt. Die dielektrische Dispersion dieser Art wurde als vom krist~llinen Tefl herriihrend experimentell gesichert, doch gibt es bisher noch keine theoretisehe Untersuehung dariiber. Als Meehanismus wurde vorgeschlagen, dal3 diese dielektrisehe Dispersion einer Dipolpolarisation gehSrig zu longitudinalen und tordierenden Bewegungen der Skelettketten in den polymeren Kristallen gehSrt. Auf Grundlage dieses Mechanismus wurde die komptexe Dielektrizit~tskonstante dureh LSsung der Boltzmann. Gleichung fiir die Bewegung der Skelettkette bereehnet. Die Gr6Be der Absorption, ihre Temperaturabh/~ngigkeit und die Gestalt der Absorptionskuve wurden aus

122

Kolloid-Zeitschrift und Zeitschrift fiiv Polymere, Band 195 9Heft 2

der komplexen Dielektrizit/itskonstante abgeleitet. Die Aktivierungsenergie wurde ebenfalls nach der Methode von FrShlich bereehnet. Die l~bereinstimmung zwischen theoretischen Ergebnissen und den Beobachtungen scheint quantitativ gut. ])as dfirfte zeigen, dal~ der Mechanismus der dielektrischen Dispersion tatsi~chlich yon dieser Art ist. Die bisherigen Beobachtungsdaten geben jedoch tg 6 als Funktion der Temperatur, so dab Sti~kturiinderungen wie z. B. solche durch Rekristallisation eingeschlossen sind. Untersuchungen bei Variation der Frequenz und konstanten Temperatuven w~ren zum Vergleich mit der vorgelegten Theorie erwiinscht. Re]erences l) An example of the theories of the dielectric ua-dispersion is given by Kirkwood, J. E. and R. M. Fuoss, J. Chem. Phys. 9, 329 (1941). 2) Yama/ufi, K., J. Phys. Soc. Japan 15, 2295 (1960). 3) Yama/u]i, K. and Y. Ishida, Kolloid-Z. u. Z. Polymere 183, 15 (1962). 4) Saito, N., K. Okano, S. Iwayanagi and T. Hideshima, Molecular Motion in Solid State Polymer in; Solid State Physics (edited by F. Seitz and D. Turnbull) (Vol. 14). 5) Oakes, W. G. and D. W. Robinson, J. Polymer Sci. 14, 505 (1954). 6) Mikhailov, G. K., Zhur. Fiz. 27, 2050 (1957).

7) Okamoto, S. and K. Takeuchi, J. phys. Soc. Japan 14, 378 (1959). 8) Kr~imer, H. and K. E. Hel/, Kolloid-Z. u. Z. Polymere 180, 114 (1962). 9) Krum, F., Kolloid-Z. 165, 77 (1959). 10) Slichter, W. P., J. Appl. Phys. 32, 2339 (1961). ll) Peterlin, A., J. Appl. Phys. 31, 1934 (1960); Peterlin, A. and E. W. Fischer, Z. Physik 159, 272 (1960); Peterlin, A., E. W. Fischer and Chr. Reinhold, J. Chem. Phys. 37, 1403 (1962). 12) Rempel, R. C., H. E. Weaver, R. H. Sands and R. L. Miller, J. Appl. Phys. 28, 1082 (1957). 13) Glasstone, S., K. J. Laidler and H. Eyring, The Theory of Rate Process (New York 1941). 14) Gross, E. P., Phys. Rev. 07, 395 (1955). 15) Fr6hlich, H., Proc. Phys. Soe. 54, 422 (1942). 16) Natta, G., J. Polymer Sci. 16, 143 (1955). 17) Onsager, L., J. Amer. Chem. Soc. 58, 1486(1936). 18) Ishida, Y. (private communications). 19) Chandrasekhar, S., Rev. Mod. Phys. 15, 1 (1943). 20) Urey, H. C. and C. A. Bradley, Phys. Rev. 38, 1969 (1931). 21) Szigeti, B., Trans. Faraday Soc. 48, 400 (1952). 22) Shimanouchi, T. and S. Mizushima, J. Chem. Phys. 23, 707 (1955). 23) Tadokoro, H. (private communications). Authors' address: Dr. Kaoru Yama]u]i, Department of Metallurgy, Carnegie Institute of Technology,Pittsburgh (USA}

From the Illrd Institute o/Physics, University, GSttingen

Dielectric Relaxation of PMMA as F u n c t i o n of Pressure, Temperature and Frequency By P. Heydemann With 8 figures in 9 details and 3 tables

(Received November 25, 1963)

I. Shifting factors and free volume O n l y r e c e n t l y t h e h y d r o s t a t i c pressure w a s i n t r o d u c e d as a t h i r d v a r i a b l e into t h e i n v e s t i g a t i o n of r e l a x a t i o n s in p o l y m e r s (1, 2, 3 4, 5, 6, 7). T w o c h a r a c t e r i s t i c q u a n t i t i e s derived f r o m such m e a s u r e m e n t s are t h e shifting f a c t o r s (dT/dP)~, a n d (d lgeo/dP)~. T h e att e m p t s to derive these shifting f a c t o r s f r o m other physical properties of the respective p o l y m e r s are manifold. I t was e v e n assum e d t h a t t h e shifting f a c t o r ( d T / d P ) ~ is a u n i v e r s a l c o n s t a n t for all p o l y m e r s . This is c e r t a i n l y n o t t h e case; b u t t h e differences b e t w e e n t h e shifting f a c t o r s of t h e glass t r a n s i t i o n of different p o l y m e r s are g e n e r a l l y small (table 1). A larger v a r i a t i o n is f o u n d in t h e shifting f a c t o r s r e l a t e d to s e c o n d a r y rel a x a t i o n r a n g e s a n d t h e d e t e r m i n a t i o n of t h e shifting f a c t o r s for these ranges will e v e n t u a l l y lead to interesting conclusions on t h e molecular r e a r r a n g e m e n t processes i n v o l v e d (5, 6).

S e c o n d a r y r e l a x a t i o n r a n g e s were o m i t t e d f r o m the studies of W i l l i a m s , L a n d e l a n d F e r r y (9) on t h e influence of free v o l u m e on r e l a x a t i o n s a n d t h e resulting m e t h o d s to o b t a i n r e d u c e d f r e q u e n c y or t e m p e r a t u r e curves. Because of the g r e a t significance of the shifting factors of s e c o n d a r y r e l a x a t i o n r a n g e s a n d in t h e absence of a b e t t e r theoretical c o n c e p t a s t r i c t l y empirical e x p l a n a tion of the shifting factors of m a i n a n d sec o n d a r y t r a n s i t i o n s m i g h t t h e r e f o r e be of interest. One e x p l a n a t i o n of the shifting f a c t o r s in t e r m s of free v o l u m e is b a s e d on t h e a s s u m p tion t h a t c o n s t a n t r e l a x a t i o n t i m e ( j u m p rate) requires c o n s t a n t free v o l u m e . T h e definition of t h e t e r m ,,free v o l u m e " still p r e s e n t s s o m e difficulties, hence F e r r y (10) calls it " a p a r a m e t e r w i t h a q u a l i t a t i v e m e a n i n g b u t no e x a c t o p e r a t i o n a l definition". F o r a m o r e detailed t r e a t m e n t of t h e free v o l u m e c o n c e p t