Intermolecular cooperativity in dielectric and viscoelastic relaxation

Joumal of Non-Crystalline Solids 131-133 (1991) 293-301 North-Holland

293

Intermolecular cooperativity in dielectric and viscoelastic relaxation S. Matsuoka and X. Quan AT&T Bell Laboratories, Murray Hill, NJ 07974, USA

A model for the glass transition which incorporates inter-molecular cooperativity is proposed. A domain of cooperativity is defined as a group of segments that must undergo relaxation simultaneously. As the density is increased by changes in temperature or pressure, the configurational entropy in the condensed state decreases much more rapidly than it does in isolated chains. The resulting equation is similar to the well-known Adam-Gibbs equation, although changes in the domain size, z, and the entropy are due primarily to intermolecular hindrance of the relaxation of individual conformers. A spectrum of relaxation times derived from the domain size distribution is similar to the KWW equation near the characteristic time-scale, but fits data better at high frequencies. The temperature dependence of the width of the spectrum is explained from the dependence of the temperature coefficients on the domain size.

I. Molecular model for intermolecular cooperativity W h e n a force is applied over a p o l y m e r chain through the surrounding media, the molecular segments will respond by readjusting themselves to relieve the stress. The m o m e n t a r i l y increased free energy will dissipate by a small a m o u n t every time a segment is reoriented to a new lower free energy state. The 'stress' referred to here can also be in the form of an electrical potential acting o n a polar segment. In polar polymers, the segmental dipoles react to a sudden imposition of an electric field b y reorienting f r o m the original r a n d o m and isotropic orientational state to a state of orientation in the direction of the field. The reorientation c a n n o t take place at once, however, because the rate at which the molecules can reorient is limited b y their ability to undergo configurational changes. The angular dependence of the potential energy for rotating the c a r b o n - c a r b o n b o n d s in b u t a n e has been investigated b y A b e and Flory [1]. The energy difference between the gauche and trans states is - 500 c a l / m o l , and the activation energy is - 3100-3500 c a l / m o l of bond. The relaxation time for the rotational m o t i o n of the C - C b o n d between two methylene units is about 10 -1° s at r o o m temperature. In polymers, the molecular seg-

ments u n d e r g o m o r e complex motions involving triads as suggested b y Helfand and co-workers [2], b u t the basic barrier is that of the C - C rotation. The barrier energy weakly increases with the size of the segments, as is discussed below. The smallest segmental unit of rotation (which we will call a conformer) is surrounded by other conformers in a solid or molten state. In order for a c o n f o r m e r to complete a rotational relaxation, its neighbors must also move in cooperation. The barrier for this process is intermolecular by nature. U n d e r such restrictions a c o n f o r m e r can change its c o n f o r m a t i o n a l state only if its close neighbors cooperate with it as if they are in a group of meshed gears. This is illustrated in fig. 1. A model incorporating meshed gears has been suggested b y A d a c h i [3], although his analysis is quite different f r o m ours. The effective d o m a i n size increases with the overall density. If equilibrium can be achieved at a low-temperature limit which we call TO, every c o n f o r m e r becomes meshed with all others, the entire b o d y of the polymer becomes one huge meshed domain, and the number of conformers in one d o m a i n is nearly infinite. This is the state of zero c o n f o r m a t i o n a l entropy at temperature TO >> 0 K. At the high-temperature limit, which we call T * , on the other hand, the conformers are sufficiently far apart that each can

0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation

294 p-x

t ~

f

/f

: :':, ,

",

x

~._.

._~-,,.,,

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I x

~..'.-.

"~

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perature-dependent, c I is also species-dependent, since some configurations are not possible in some polymers. For 1 mol (of conformers) in which there are N~ domains consisting of z conformers, the conformational entropy, So, for 1 mol of conformers is

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DOMAINS WITH z =G Fig. 1. Model for cooperative domains; all conformers are locked in the domain and can relax only if every one of them relax simultaneously such that the relaxation time of the domain is the z th power of the relaxation time for one of them alone.

relax independently from the neighbors, and only the intramolecular barrier must be overcome. The number of conformers in each domain is 1. Between these two extreme temperatures, the size of a domain is specified by the number z of conformers in the domain. For the example shown in fig. 1, z is shown to be 6. For illustration purposes only, we first assume that each conformer can take three different conformations. At the high-temperature limit T * , where no cooperativity is at work among the conformers, six independent conformers are able to exhibit ~ = 36 conformational states. In the state illustrated, where six conformers are meshed together, however, in order for one conformer to change its conformation, all others must change also. Thus it makes no difference which conformer the observer disturbs, the results are always that every conformer in the domain of cooperativity must change its conformation. Thus a domain can take only the same number of states as that one conformer can take, i.e., W~ = 3. We now generalize the above example by calling the number of states a conformer can take c~ instead of 3. Since the conformational probability depends on the populations of g a u c h e and trans conformations, the value of c~ is tern-

(1)

where k is the Boltzmann constant. Since z = N A / N z, where NA is Avogadro's number, z = N g k In c J N ~ k

In

C1 =

N A k In

Cl/S

(2)

c.

Thus the domain size, z, is inversely proportional to the conformational entropy. We now define s * as the conformational entropy of 1 mol of conformers in chains that are not mutually restricted by each other, i.e., where each conformer relaxes independently, s * = N A k In

(3)

C1 .

s * is the conformational entropy that corresponds to that calculated b y rotational isomeric state theory, and it, too, is temperature-dependent. Thus, we obtain the formula (4)

z = s * / S c.

At T * , Sc and s* are equal. Below T * , as the temperature is decreased, the conformational entropy, S~, (and enthalpy) will drop more rapidly than does s *, and will reach zero at T0, while s * depends on the equilibrium concentration of various conformations and will reach zero at 0 K. The comparison between the corresponding enthalpies is illustrated schematically in fig. 2. Curve A denotes the excess enthalpy, h*, without meshing of the conformers, and it is approximately proportional to the concentration of higher energy configurations calculated from rotational isomeric statistics. Curve B represents the conformational enthalpy, H~, that incorporates the meshing or cooperativity concept, which raises the zero-entropy temperature from 0 K to T0. If we assume that curves A and B have similar shape but differ only in the temperature scale, curve B can be obtained by compressing curve A through a change in the temperature scale from [0 ~ T * ] to [TO T * ], and we obtain the relationship Hc(T)/Hc(T*

h*(r)/h*(T*)

(T- To)/(T* -

)

=

T/T*

To)

,

(s)

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation

295

where >_

~,.,,.,,~ h~ ( T'~ ) = Hc(T~')

13-

A/~* = A/~

-r I-Z u.I

U.l

T*

o

In the equilibrium state, eq. (11) is completely equivalent to the Williams-Landel-Ferry (WLF) equation [6]

/-

/

T*-T

~[Hc(T }

)k

l°gL f

CI(T-

Tref)

c: + 7"-

(12)

I

0

TO

and the free volume equation by Doolittle [7]:

Tw T OK

Fig. 2. Excess enthalpy with (B) and without (A) cooperativity. Curve A is obtained from the concentration of the high-energy conformation(s) through calculation of the rotational isomeric states.

and since

Ho = h*

--

(6)

Ts* '

we obtain S~ T* s* - T * ~

T-To T

(7)

In order for the z meshed conformers in a domain to relax simultaneously together, the probability will be z th power of the probability for independent relaxation, and the apparent activation energy must be z times the i n t r a m o l e c u l a r activation energy for one conformer to relax. We define this energy barrier as A#. The relaxation time k at T is related to the relaxation time, ~*, at T* by the formula z

T T* but, from eq. (4), In

=--k-

TS c

(8)

'

T*

(9)

"

This equation is identical in form with the Adam-Gibbs's equation [4], although our Sc is largely intermolecular in nature and therefore is not limited to polymers. Substituting eq. (7) into eq. (9) will obtain in ~

A# T * - TO k

T*

1 T-

A# TO

kT*

'

(10)

which is the Vogel-Fulcher equation [5], ln?~

5#* k

1 T-T

A/z* o

k

1 T*-T

o'

(11)

k 1 ln~-~f = a f ( r - To)

1 o~f(Tref - To)'

(13)

based on an arbitrary reference temperature, TrefHowever, an important difference exists between the A d a m - G i b b s equation and the free volume equation. In the n o n - e q u i l i b r i u m glassy state, the former correctly predicts the temperature dependence of the loss peak frequency, while the latter predicts temperature independence [8-10]. Comparing eqs. (11) and (13), it is immediately clear that A # * / k should be numerically equal to 1 / a f . The quantity a r is the expansion coefficient of the Doolittle free volume fraction f = V f / V , where Vf is the free volume V - V0, and 110 is the 'occupied' volume that is obtained by extrapolating the equilibrium liquidus line to T0. The values of af are found experimentally to be nearly the same for many polymers, about 3 × 10 -4 K -1. This implies that A/~* should exhibit a 'universal' value of 3-3.5 kcal/mol of c o n f o r m e r . The experimental values of A/~* shown in table 1 were calculated from the shift of viscoelastic relaxation spectra with temperature in the equilibrium state, most showing values close to 3 or 4 kcal. In fact, A#* is approximately equal to the i n t r a - m o l e c u l a r activation energy for the C - C bond rotation in butane. (For polymers with heteroatom backbones such as siloxanes and ethers, A#* is smaller, as are the bond rotation activation energies.) This implies that A/~* does not depend on the polymer species. However, A/~ varies with the chemical structure, and the relationship between Art of a given polymer and the universal value A#* is described by the equation T* A~ = A/x* T* - TO"

(14)

296

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation

Table 1 Vogel-Fulcher energy. polymer

A/t* (kcal/mol)

polyisobutylene polycarbonate polyvinylacetate polystyrene poly-a-methylstyrene polymethylacrylate polymethylmethacrylate hevea rubber poly-1,4-butadiene poly-l,2-butadiene styrene-butadienecopolymer

4.2 3.6 3.4 3.2 3.1 3.8 11.7 4.2 3.1 2.1 8.3

polydimethylsiloxane polyoxymethylene Polypropyleneoxide

The E h above has been designated at the 'hole energy' by Kanig [11], since it is the energy required to create the free volume fraction fg = AVg/Vg at Tg. The enthalpy E h (in calories per mole of repeat units) calculated from the experimental data for over 70 polymers shows a clear tendency to be greater for polymers with a higher

2.1 1.8 1.8

Thus, the activation energy for the intra-molecular relaxation of one C - C bond is greater with conformers which exhibit a higher TO and thus a higher Tg. As we discuss below, the differences in the packing densities among polymers of different conformer sizes affect the excess cohesive energy. It can be shown from thermodynamics that the specific heat related to the configurational entropy with cooperativity is dSc

/ OS~

(15)

where ACe = G,liquid -- G,glass; hence, / 0S¢ ~

1

ACe

(16)

where the thermal expansion coefficient Aa = Cqiq,id -- aglass. For convenience, we shall take Tg to be the temperature at which the relaxation time has a chosen value, hg, even though we are aware that the glass transition temperature is not a thermodynamic temperature. This is similar to defining Tg to be the equilibrium temperature at which the shear viscosity has an arbitrary value, e.g., 1013 Poise. As is discussed below, Xg is about 2 h at

TO+ 50°c. Consider now that we increase the volume at Tg from V0 to Vg. The total increase in the enthalpy, Eh, is TgScg and AV~ ( ACp I

.

(17)

Equation (17) is, according to our model, on the basis of 1 mol of conformer, but the relationship between E h per gram and ACp per gram should be the same, since AVg/Vg and Aa do not depend on the chosen amount of mass. In 1955, Bunn [12] published his observation that the molar heat of fusion for conformers of various sizes and shapes can be scaled with their molecular weights, drawing an analogy to Trouton's rule that the ratio of the heat of evaporation to the temperature of vaporization is nearly constant for liquids. Bunn's tabulated data show an apparent proportionality between the heat of fusion and the molecular weight with a proportionality constant of - 5 0 c a l / g for conformers that are in the middle of a chain rather than at the end. (Chain ends are at a higher energy level.) However, Bunn finds the proportionality to be only an approximate relationship and, in order to be applicable to structures that are composed of a group of these conformers, a correction factor for the packing density must be taken into consideration. This correction can be illustrated in the following manner. First, if eq. (17) were written on the unit mass basis, e.g., per gram, and if we accept the empirical theory by Fox and Flory [17] that the free volume fraction takes a universal value at T~, then E h per gram should be proportional to ACp/Aa at Tg per gram. The data compiled by Wrasidlo [11] indicate that the latter quantity is essentially independent of Tg. Therefore, E h ought to be a constant. Kanig finds this to be incorrect, and devised a correction factor that is numerically the same as the correction factor by Bunn, although the physical reasoning is somewhat different. When segments are packed together, according to Bunn, the molecular volume is actually smaller than the sum of individual segments, and the cohesive energy is therefore greater than the sum.

S. Matsuoko, X. Quan / Intermolecular cooperativity in relaxation

Corrected values for the molecular volume and the cohesive energy can be made b y using E1 -- E2

E~

V2 -- V1

- 1.73--------~--1

(18)

for segmental species 1 and 2. This has been shown to hold for the substance in which intermolecular forces are of the van der Waals' type, whether or not there is a phase-change from liquid to sohd. We can apply this formula to obtain a correction for the intermolecular energy when the molar volume is increased from V1 to V2 by increasing the molecular weight from M 0 to M: E - E0

m - M 0 ..~ In M - In M 0.

(19)

Thus the hole energy per gram increases with In M, whereas from simple additivity, E h per gram would be independent of the conformer size. The conformational entropy Scg at Tg in eq. (17) can be substituted with the entropy without cooperativity, s~, using eq. (7) to obtain T*

To)s; r . _ o = Eh0 + C(ln M -

In M0) , (20)

where Eh0 and M 0 refer to a hypothetical polymer with TO= 0 K. F r o m the Vogel equation, T g - TO must be a constant for various conformers to satisfy the iso-~g condition. Thus eq. (20) states that among various species, the following relationship should hold: ( T * - TO)In M = C 3.

(21)

C3 has been evaluated as having a value of 1750 from a number of polymers, which translates to M 0 = 9.6 for a polymer with TO= 0 K. This hypothetical conformer does not need intermolecular cooperation to relax at any temperature, and therefore its TO is at 0 K. Also, for the methylene unit as a conformer with M = 14, we obtain 'Tg' = - 1 1 0 ° C. This value is nearer to the -/ transition temperature than to the fl transition temperature of polyethylene, although the latter has been considered as the glass transition of the amorphous

297

polyethylene. For To = - 75 o C (to obtain Tg = - 2 5 ° C ) , the conformer should be with a size which is equivalent of one and a half methylene units. In aromatic polymers, we find that a phenylene group alone cannot be counted as one conformer. A para-phenylene unit can rotate about its own longitudinal axis without disturbing the immediate neighbors on either side. Such a rotational mode does not contribute toward the relaxation of an applied stress through unkinking of the segments. The rotation of the same phenylene group about a bond once removed, however, would be accompanied by a change in the shape of the local structure. Thus, if an ether or a carbonyl group is in the adjacent position in the main chain, it is included as part of the conformer, i.e., a p h e n y l - O - or p h e n y l - C : O - g r o u p should be counted as one conformer, and this results in a value of C 3 of 1750 for m a n y aromatic polymers. If one follows this rule, a bis-phenol-A with - O in both sides should be considered as two conformers. The repeat unit of polycarbonate consists of three conformers. When phenyl groups are connected consecutively in the para position, the conformer size is equal to the entire sequence of para-phenylene groups. Such polymers can, in theory, attain TO= T * with a large number of uninterrupted phenyl units. Kevlar exhibits a similar feature since the phenyl group can be rotated without changing the conformation. For the cross-linked polymers such as epoxy novolac and bis-phenol-A type epoxies, TO values correspond to the state of cross-linking at which the solventlike effect of unreacted sites has just become negligible. At this point, they also exhibit a significant fl relaxation, signifying that the cross-linking is not so tight as to eliminate intramolecular relaxation. Equation (21) with C 3 = 1750 is plotted in fig. 3, together with points corresponding to various polymers. The conformer size is the average size and can be checked from the fl transition, as discussed below. Finally, we wish to summarize the useful features of this model for intermolecular cooperativity by showing the so-called 'transition m a p ' for polycarbonate and polyethylene terephthalate in fig. 4. A transition m a p is a plot of the frequency for the m a x i m u m rate of relaxation versus 1 / T K.

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation

298

The term including X* and TO in eq. (11) can be further reduced to a universal value by considering again the hypothetical polymer with a conformer size of M 0 which exhibits TO= 0 K and the relaxation time h~ at T * : in a

A#*

T

= kT

T-T

Ag* o

(11')

kT*"

All curves, including the /3 transition, seem to start at T * = 5 0 0 ° C , log a ' ~ - A # * / 2 . 3 k T * = - 1 1 . 4 (s). The limiting relaxation time a~ of 3 × 10 - I t s corresponds to the range of dielectric relaxation time for heptane extrapolated to T * [13]. The corresponding frequency log fo* is the upper limit that a relaxation process is observable. Similarly, T * = 500 ° C is the upper limit of temperature for To. From eq. (11'), a s -- 1.8 h is obtained for A g * = 3.5 kcal at T s = T 0 + 5 0 ° C . The slope of the curve multiplied by R will not obtain Agz, as z itself is a function of temperature. However, the apparent activation energy be-

10

~ L L O U I N

S

/ \

t.9 5

~

o,

L

0

PPE •

200

/s

PEI e,,,~.~

r.poxy Novolac

p opolyarylate, lysulfo~ PEEK

~ - - ~ p pltChalomide

P o

to +

100

PPS

1,2

PB

i,4 PB l

l

I

I

l

I

I

I

~

[

I

I

t00 200 SEGMENTAL MW/BOND

Fig. 3. Ts versus conformer size as determined by the rule as described in the text. The solid line is the plot of eq. (21). Tg= TO+50°C to make As =1.8 h, as discussed in the text.

2

~

\wC

Po,y~o,bo~o,,(PC) ~'-process

_

i , I i i i L INI 4 5 t O 0 0 / T (K)

i ~ ih i i 3

ni I i Ji

6

7

F i g . 4. T r a n s i t i o n m a p - f r e q u e n c y f o r t h e l o s s m a x i m u m v e r s u s 1/T K The curves denote the Adam-Gibbs equation for both a a n d fl p r o c e s s e s .

low the glass transition temperature yields the value of A#z at the temperature of freezing, Tf, s*

Ap,-~c

Kevlar

i i I at i i 4

\ "kA

? o_:,::.. \

o_

O

\

\

Tf

= A p , z = A/.t T f - ~

To'

(22)

from which the domain size, z (which depends on the temperature), can be estimated. In the glass transition temperature range, the apparent activation energy is - 4 0 k c a l / m o l . Since Ag = 4 - 6 kcal, z s is typically about 7 - 1 0 conformers. The/3 relaxation process typically is a constant activation energy process. The process can be considered as due to molecular segments trapped (such as gas molecules trapped in clathrates) but able to continue in liquid-like, although local, motions even after the rest is vitrified. The apparent activation energy is a few times greater than At**, which leads to the speculation that a chain is ' p i n n e d ' at various parts and intra molecular coordination is needed for this relaxation process. Actually, the number of bonds relaxing simultaneously in the intramolecular fl process cannot be larger than z¢, or else it would have been relaxed before the glass transition. For polycarbonate, the size of a unit of relaxation for the /3 process is between 2 and 3 conformers, whereas for polyethylene terephthalate (PET) this size is 4 conformers. These numbers correspond to the number of conformers per repeat unit for the a process. The molecular weight of the repeat unit divided by these numbers agrees

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation tl

We will discuss only stage 1. This stage is observed in the sohd-like state where the characteristic relaxation time, Xc, is in the order of hours. (A benchmark can be set at Xc of 1 h at Tg - 5 o C with a value of G(1 h) of 10 9 d y n / c m 2 . ) Our cooperative domain model has dealt up to this point with a single relaxation time associated with the characteristic domain size. However, the possibility of a variation in domain size must exist. At a given temperature, T, the cooperative domain with size z will exhibit a relaxation time, X, given by

9 o 8 0_.1 ~

7

J

Z

o

~ t I000C

\

\\

5

X

"~ 4 -...1 5 -4

h A# lnx--~ = ~-T(z - Zc), -2

0

:~

4

6

8

10

299

(23)

12

LOG TIME (hrs)

Fig. 5. The master relaxation modulus curve for amorphous polymer in the equilibrium state. The KWW equation is only for the stage 1, the short-time range involving the local segmental rearrangements.

with the molecular weight of the respective conformers used for calculation in eq. (22). Thus even though polycarbonate has a higher Tg than polyethylene terephthalate, its fl temperature is lower. This feature should be useful for designing a polymer molecule with a high Tg but a low T~ which has perhaps better low-temperature fracture properties. One point shown on the straight line labeled z = 1.2 in fig. 4 corresponds to the frequency and the temperature of the methyl group rotation which we think is a good example of the unhindered conformer, i.e., z is always 1. This line then is the limit of the shortest relaxation time for the glass transition of all kinds of polymers and olefins.

2. Distribution of relaxation times near the glass transition

A 'master' curve for the relaxation modulus of an amorphous polymer in the equilibrium state is shown in fig. 5. The wide range of the relaxation spectrum is divided into three stages: stage 1 is the comparatively fast-relaxing stage observed at low temperatures; stage 2 is the transition zone between the solid-like stage 1 and the much slower stage 3 observed in a typical melt.

where the subscript 'c' refers to the characteristic (maximum or 'cutoff') size. The ANz of domains, the size of which falls between z 1 and z 2, can be expressed by the normalized magnitudes of the modulus, G ( h : ) , at time t = h2 and G ( h l ) at time t=?h:

AN z cc

G(X:)- a(Xa) G(Xl ) = In G(X2) - In a( kl). (24)

The maximum size, z¢, is uniquely determined by the pressure and temperature. The energy per domain is proportional to the number of conformers in that domain, as we discussed above referring to Bunn's theory. The entropy per domain is, by definition, equal to the entropy of one conformer. From the equipartition principle, the total free energy for each ensemble of domains with a given size should be the same, and this is accomplished by making the product z N z constant. Since we have taken as our basis 1 mol of conformers, the total number of conformers residing in all of the various size domains is the Avogadro number, NA, and we obtain the relationship gc

zN~ = N A,

(25)

z=1

from which the following relationship should hold:

NA1

N~ = - - --. zc z

NJN

A

(26)

is the fractional number of domains of size

300

S. Matsuoka, X.. Quan / Intermolecular cooperativity in relaxation

z. The proportionality sign of eq. (24) can be replaced by the equality sign if Nz is taken to be such a fractional number. From eqs. (23) and (24), by substituting z 2 = zc - 1 and z 1 = z c - ~, ln G ( ~ 2 ) - l n G ( ~ l ) = l (

l zc

1

1 1 zc z c - 1 ×

zcl) _ 1 A/~ z~ k T

[1] ln(t/X~)

(27)

and we obtain for the relaxation modulus In G ( t ) = In G(0) + ~ k-T ln(t/X~)

"

(28)

The stage 1 relaxation process is frequently approximated by the well-known Kohlransch-Williams-Watts (KWW) equation [14]. This formula has as a convenient feature that its characteristic relaxation time corresponds to the dielectric loss maximum frequency as well as the time at which the relaxation modulus has decreased to 1 / e of the initial value. It turns out, however, that the KWW formula is very good when t or ~0-1 is near ~¢, but becomes poor at extremely short times such as 5 decades below, as shown with the dielectric loss data in fig. 6.

0.4 rr

o o

0.3

ffl

0 J

0.2

0 0C (..)

~,0.1

u.I

-6

-5

-4

-5

-2

-1

0

1

2

L 0 6 t (hrs) Fig. 6. Dielectric loss versus - l o g ~ hours (taken from polycarbonate data by J o h n s o n [15]). The dotted line is calculated from K W W equation using fl = 0.5 and log h c = 0.3, and the solid line is f r o m eqs. (28) and (29), with ),c = 0.6.

The relaxation spectrum H(ln ~) is approximately H ( l n X )--~

ddGIn( t )t ,=x "

(29)

The loss factor was calculated from this spectrum, and has been plotted in fig. 6. The fit is better than the K W W equation at the high-frequency end. We think that this means that all domains that are smaller than z¢ participate in the relaxation process, and they are responsible for the persistent tail of the stretched exponential function toward the high-frequency limit which even the KWW equation underpredicts. Both the K W W equation and our eq. (28) cut off fairly abruptly beyond t >__Xc, whereas the viscoelastic relaxation modulus continues on for many more decades, depicting the so-called external viscosity as against the internal viscosity with which we have been concerned here. The portion that is identified with the external viscosity is unique in polymers since its maximum relaxation time depends on the molecular weight, whereas the stage 1 we have discussed involves only the local segmental relaxation observed by the dielectric, volumetric and thermodynamic measurement. However, the values of the dielectric and mechanical ~'c for the stage 1 are nearly the same for many polymers, e.g., polyvinyl acetate, polystyrene and polycarbonate, when measured in the equilibrium state. If these 'stretched' exponential functions are in fact a reflection of the distribution of the size of cooperative domains, this may be the reason for the apparent success of the universal scheme created by Ngai and co-workers [16] that makes a connection between the KWW exponent, fl, and the value of the apparent activation energy. So far we have discussed only the equilibrium state. In the non-equilibrium state, the spectrum broadens at lower temperatures. This is because the relaxation time of the larger domains exhibit a greater coefficient than those of smaller domains. As we mentioned above the K W W parameter, fl, is equal to - d In G ( t ) / d I n t . Let us define the time, t o, such that

1/fl = In ~c - In t o.

(30)

S. Matsuoka, X. Quan / Intermolecular cooperativity in relaxation

t o depends on the thermal history but not on the temperature, as is shown below. When the temperature is decreased from Tg to T in the glassy state, let the characteristic relaxation time be changed from A~g to AcT. The new fi at T will be

1/fi = In X~T - In t o = In ~kcg-4- In t 0.

Zc r

Tg

=

l/fig + In Xc~ -- In Xcg.

equal to the viscoelastic kc, and their Vogel parameters A/x* and TO are also about equal. (5) The distribution of relaxation times broadens markedly in the non-equilibrium glassy state as the temperature is lowered because of the different activation energies for different domain sizes.

(31)

However In keg - In t o is 1/fio at Tg. Accordingly, the KWW parameter, /3, can be shown to depend on temperature according to

l/fi

301

(32)

3. Conclusions The features of the cooperative domain model and its implications are summarized as follows. (1) A model for the glass transition has been proposed which incorporates restrictions on individual segments relaxing independently of their inter-molecular neighbors. A domain of cooperativity is defined as a group of segments that must undergo relaxation simultaneously. The model has been used to explain several universal aspects of relaxation phenomena in polymers. (2) The origin of the distribution of relaxation times in 'solids' is the size distribution of the cooperatively relaxing domains. The A d a m - G i b b s formula is a natural consequence of the temperature dependence of the domain size. (3) The KWW equation is a simple and convenient formula which approximates the dielectric relaxation near Ac but not where t << ~c(4) For short time (stage 1) relaxation, the dielectric relaxation time, ~'c, is in general about

References

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