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Macromolecules 1992,25,

5765-5768 6766

Normalization of

the Temp erature Dependence

of

Segmental

Relaxation Times

C. M.

oland ' and

K.

L.Ngai

Naval Research Laboratory, Washington, D.C. 20375-5000

Received April

1,

1992; Revised M anuscript Received June

30

1992

ABSTRACT The validity of Tc-scaled Arrhenius plots of segmental relaxation times for glass-forming

liquids is assessed by comparing the results on polymers differing only in molecular weight. The differences

result n identical segmentalrelaxation dispersions which occur, however, at different times and temperatures.

It is verified herein that the Tc-normalization scheme is a se lf-consistent means to classify and distinguish

segmental relaxation behavior of polymers. Contrarily, changes in glass transition temperature effec ted

by

hydrostatic pressure alter the segmental dynamics in a manner such that the fragility of the polymer is

altered.

ntroduction

The time and tem perature dependencies of the rheo-

logical response of polymers are c entral issues in the study

of the viscoe lasticity of polymers. Over the range of most

experimental measurements, temperature dependencies

are non- Arrhenius; consequen tly, some normalization

scheme must be invoked

to

allow comparisons among

different fluids of the effect of temperature on the

measured relaxation times. Recen tly, a correlation was

demonstratedbetween he shape of the relaxation function

of several amorphous polymers and the magn itude of th e

time-temperature shift factors.' Specifically, a broader

segmental dispersion was associated with a stronger

temperature dependenceof the segmental elaxation ime.

Th is conclusion was reached by comparing he shift actors

as a function of inverse tempera ture normalized by TG,

the glass transitio n temperature. Although the glass

transition tem perature is an obvious normalizing param-

eter, this specific basis for comparing different m aterials

is not obviously the co rrect one.

Angel12t3 has proposed a classification based on the

density of available configu rations (corresponding

to

minima on a potential energy hypersurface ) for an

N-particle fluid. If the liquid gains accesstoa high density

of these energeticallypreferred con fii ati on al states when

temperature is raised through TG, he glass to liquid

transition will be accompanied by a large entropy increase;

consequently, he he at capacity and relaxation time change

markedly with temperature near TG. Such a liquid is

classified as fragile , an allusion to the loss of short- or

intermediate-range order accompanying the transition

from the glassy state. Strong liquids, characterized by

fewer potential energy minima in configuration space,

suffer little loss of local struc ture and thus exhib it smaller

incrementa n heat capacity a t

TG.

This herma lly induced

Structure degradation refers only to the loss of local order;

i t is unrelated

to

chemical stability.

consideration of the hea t capacity change

et

TG rovides

a me am for comparing the temperatu re depende ncies of

different glasa-forming

liquid^.^^^

In the Adam and Gibbs

theory of relaxation' the liquid is comprised of cooper-

atively rearrangingregions (CRR's), defined

as

he smallest

volume element that c n relax

to

a new configuration

independen tly of neighboring regions. Th e energy barrier

hindering transition of the CRR to a new configuration s

proportional

to

the size of th e CR R, with the average

transition rate for the C R R s given by

W T) A exp(-C/TS,) (1)

where

A

is a constant. Th e parameter

C

s proportional

to the product of Ap, the energy barrier per molecule, and

8 the configurational entropy of the sm allest CRR

C = A l S J 2)

where is the Boltzmann constant. The configurational

entropy of the macroscopic samp le, S,, can be expressed

in terms of the difference in heat capacity between the

liquid and glassy states s3

(3)

where

TK,

dentifiable with the K a u temperature?W

is the temperature a t whichS =0. Assuming a hyp erbolic

dependenc e of he at capacity on temp erature , it can be

shown that2a

4)

where a s a constant. Subs tituting into eq 1gives

W

=

A exp(DTK/T-

TK)

(5 )

with

D

= C / a 6)

Equation 5 corresponds

to

the empiricalVFTH quation@

with the average transition rate of th e C R R s identified

with the inverse relaxation time. Non -Arrhenius tem-

perature dep endency data from glass-forming iquids near

TG

an be represen ted well using this relation.

If

both the

energy barrier per molecule and the configurational

entropy of the smallest CRR are constant, then D is

constant, a t least

to

the extent the heat capacity can be

adequately d escribed by a hyp erbolic relation. It follows

from eq 5 th at two liquids having the sam e D and A (but

possibly differe nt TK s)willhave iden tical fragility plots ,

th at is, plots of log ( W ( T ) ) ersus TG /T,where TOcan be

operationally defined

to

be the temp erature at which the

transition rate attains a constant (arbitrary) value. Such

plots have been used

to

depict relaxation times and

viscosities of sm all-molecule glasses in order

to

illustrate

th e con trasting behavior of stron g versus fragile liquids.2*3

An

extension of the Adam and Gibbs model

to

include

This article not eubject to

U.S.

Copyright.

Published 1992

by

the American Chemical Society

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6766 Roland and

Ngai

dynamic correlations between the

CCR s

provides a

rationale for the observed correlations between time a nd

tempe rature dependen cies in small-molecule liquids.1

More recently this approach has been applied

to

amorphouspolymers,lJ1although the app ellations strong

and fragile may be misleading therein, since there is no

modification of struc ture a t the glass transition. For

polymers the term cooperativity plot is more accurate.

From comparisons of segme ntal relaxation times for

various polymers made on the basis of TG-scaled Arrhen ius

plots, correlations have been adduced .lJ1 These have been

used to draw inferences concerning fundamental rela-

tionships between ch ain structure , intermolecular inter-

actions, and segmental dynamics. Fragility plots have

also

been extended tomiscible polymer blends

to

demonstrate

the effect of local compositional variations on se gmen tal

relaxation.12-14

Frag ility plots embody the assu mp tions of constancy of

sc

and

Ap,

the correctness of which cannot be directly

demo nstrated. Th e need to take account of differences

in T among liquids is apparent, and

a

simple T/Tc

construct is plausible. However, it is not obvious th at

comparisons of different and decidedly non-Arrhenius

temperature dependen cies of different liquids are made

meaningful so simply.

To assess this app roach

to

the stu dy of chain dynamics,

it would be useful to demonstrate its validity with liquids

identical in chemical structure but differing in their

dynamics. This is conspicuously impossible for small-

molecule liquids; however, the chain-length depen dence

of a polymer's glass transition tem peratu re provides such

an opportunity. Chemically identical materials differing

only in molecular weight will necessarily have th e same

energy barrie r

to

conformational ransitions, and both sC

and AC, will be essentially equal. While the differences

in Tc confer differences in temperatu re dependenc ies,since

the degrees of intermo lecula r cooperat ivity of their seg-

mental relaxation are equivalent, the relaxation times

should superimpose when expressed in term s of th e

reduced variable T/

TG.

We emphasize

hat

TG-nOrmalized

Arrhenius plots may provide a meaningful basis for

comparing segmental relaxation of different polymers,

aside from the particu lar m odel (e.g., the theory of Adam

and Gibbs) from which th e appro ach originated. Accord-

ingly, herein we employ the VF TH expression only

to

interpolate experimental measu rements of segmental

relaxation. The adequacy of the fragility plot approach

is considered apart from the specific assumptions tha t

enable one to derive the TITGnormalization scheme.

The glass transition temperatu re of a given material is

strongly dependent on pressure, reflecting the m olecular

crowding nduced by pres . Isotherma l elaxation imea

and their temperature depend enceare therefore functions

of pressure. Th e changes in segmental dynam ics engen-

dered by presswe, however, are expectad to alter

Ap

and

AC,, thus undermining the basis of the fragility plot

approach. Speaking loosely, the strength of a liquid

should increase upon densification. It is interesting

to

explore the consequences of pressure variations

on

the

fragility characteristics of glass-forming liquids.

The presen t study was intended

to

sm th e utility of

fragility plots in accounting for changes in segmental

dynamics of amorp hous polymers induced by changes in

TG Th e latter were effected by variations in m olecular

weight and pressure.

Results

Polystyrene. Compliance measurements at temper-

atures encompassing the glass

to

liquid transition have

2

d O

D 2

0

r l

4

6

Macromolecules,

Vol. 25 N o. 21

I992

I

4r l

? I

i

2 . 4

2.6 2.8

3 0

EL

1 0 0 0 / T

I

I

I

I

I

a

r -

- 5 L L - U L L

0 . 3 6

0.96

1 . 0 0

1 . 0 2

TO T

Figure

1.

(a) Shift factors determined

from

compliance mea-

surementson polystyrenes of molecular weights equal to 3.4

109 .),1.6Xl r O),and1.2X

1o6(+).l6 (b)Segmentalrelaxation

times that become coincident when the temperatures

are scaled

by the respective TG S Table I) of each polymer.

Table

I

polymer

polystyrene

polystyrene

polystyrene

poly(propy1ene

oxide)

poly(propy1ene

oxide)

poly(phenylmethylsiioxane)

poly phenylmethylsiloxane)

M w

3.4

x

109

1.6X 10

1.2 x 106

1.0 x 109

9.5

106

2.5 X 109

8.3 x 106

T O

3530

36W

372O

202b

205b

235b

245b

1 Temperature at which

compliance

i n c r e w by a factor

of

10

from

the

glassy evel in

100 .

Temperature

at

which the segmental

relaxation time equals 1

8.

been reported for polystyrenes of various molecular

weight.lS The obtained

shift

factor s (the ratio of the

relaxation time,

T

measured a t a given temperature to the

relaxation ime a t an arbitrary reference temperature) are

displayed in Figure

1

for the three polystyrenes listed in

Table

I. It

is evident that the differences

in

molecular

weigh ts give rise tosubsta ntial differences in temp erature

dependencies for the segmental relaxation. The

VFTH

parameters, determined by fitting eq 5 with W taken to

be ~ - l ,ary with m olecular weight due to the variat ion of

TGwith m olecular weight. However, no molecular weight

dependence was found for the shape of the segmental

relaxation dispersion (reflecting the degree of intermo-

lecula r cooperativity16J7). Thus, hese data provide an

opportunity to evaluate the fragility approach

to

scaling

the results from polystyrenes of different TG.

From th e origina l compliance data,'6 a glass transitio n

temperature is def iied for each molecular weight as he

temperature at which the compliance increases 10-fold

from th e glassy compliance over

a

100-s

ime span. Using

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Macromoleculee, Vol.

26,

No. 1 1992

0

Segmental Relaxation Times 5767

6

a

a

a + o

00

ol

a + o

61

O

~ ~ ~ ~ ~ ~

0 . 8 5

0 90 0 95

oo

TO / T

Figure 3.

Shift

factors determined for amorphous

PPO MW

= 9.5 109 at

1

01,1000 +I, and 2000 0 ) tmospheres of

pressure.19 The TG

as

taken

tobe

the temperature

at

which the

relaxation time

equaled

1

s

at

that

pressure.

the TG S

o

determined , the shift factors of Figure l a are

replotted in Figure lb , with the temperatures normalized

by the respective

TG S

f each polystyrene. It is seen th at

t h i s

ecaling unifies the behavior, indicating

that

at least

for mechanical measurements on these polymers the

disparate characteristics of the molecular weight depend-

ent TG nd m olecular weight independent fragility can be

distinguished.

Poly propy1ene oxide).

Dielectric sp ectroscopy has

been carrie d out on both low1*and highlBmolecular weight

poly(propy1ene oxide) (PPO) in the vicinity of its glass

transition. Th e frequency, fp, of the maximum in the

dielectric ose defines a peak relaxation time, T~ = 2rfP)-l.

Th e variation with tempe rature of

T~

is found

to

be well

e

4 2

. 4

3 8

I I I I I I I I I I I I

0 . 8 8

0 94 1.00

TQ T

Figure

4.

(a) Shift factors determined for

PPMS

f molecular

weightequalto2.5X 108Wand8.3X 106 0).20b)Datareplotted

with the abscissavalues normalized by the

respective

Toof

each

polymer

(Table I).

described by the VTH F equation. The temperature at

which rp

= 1

s is determined and identified as

TG. A

l - s

time scale

is

employed since the dielectric data are available

only over a limited ran ge

of

frequencies.

TO

s arbitrary

as

far as dynamics

are

concerned, so that any convenient

operational definition suffices. Using the TGso deter-

mined, shift factors are calculated

as

shown in Figure 2a.

Since the glass transition temperatu res are different for

the two m olecular weights, nonparallelcurves are obtained.

Similarly

as

or PS above, we normalize the temperatures

by the respective

TG

of each species and replot t he d ata

(Figure 2b). Th e plots for the two molecular weights are

coincident, supporting the norma lization scheme.

The TGof a polymer can be altered by the ap plication

of hydrosta tic pressure. Dielectric oss measurements have

in fact been repo rted for th e higher molecular weightPPO

at high pressure.lS From this data we determine the

pressure dep endence of

TG

nd thereby construct fragility

curves (Figure3). However, it is seen th at t he changes in

temperature dependence brought about by changing

pressure cannot be accounted for simply by scaling the

temperatures by the TG- . Th e change in TGwith pressure

can be attributed to changes in the intramolecularly

correlated conform ational ra nsition rates, similar

to

the

effect of changing molecular weight. However, whereas

the increase in TGwith inc reasing molecular weight has

no effect on the normalized temperature dependence, he

TG ncrease brought abou t by higher pressure is accom-

panied by a redu ction in the fragility of the liquid.

Poly phenylmethylsiloxane). Dielectric and pho ton

correlation spectroscopy results have been obtained on

poly phenylmethylsi1oxane) PPMS)

f

molecular weight

=

8.3

X

106,20921

nd both techniques yielded equivalent

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6768

Roland and Ngai

Macromolecules, Vol.25

No. 21, 1992

I

r l

I

I I I I

1 1

0.92

0 95 0.96

1.01

To / T

Figure

6.

Shi ft factors determined for low molecular weights

PPMSatl -),450 - -),and900(-..)ba~hydrostaticpressure.~

TG as taken

to

be the temperature

at

which the relaxation time

equaled 1

s,

with temperature a nd pressure dependencies for the

latter obtained from eq

7.

temperature d ependencies for segmental relaxation. In

Figure 4a the shift factors for th e dielectric loss maximum

are displayed for this PPMS, along with shift factors

determined for a low molecular weight

=2.5

X

lo3)

PPMS.20 When a glass transitio n tem peratur e is defined

as he temperature at which the relaxation time equa ls

1

s,

the d ata are replotted in the TG-normalizedArrhenius

form (Figure 4b). Superpositioning of the d ata is again

observed.

The pressure dependence of the relaxation time was

determined for the low molecular weight PP M S to bem

= 7.94 X lo- exp((1659 0.813P)/(T 207)j (7)

where P is in unita of bar. Using eq

7,

the pressure

dependenceof the

7

vs TI

TG

elation was calcula ted,where

TG

s again the tem perature a t which the relaxation time

equals1

.

The

reeulta

(shown n

Figure

5) are in agreement

with the data in Figure

3

for PPO;

that

is, the fragility

curves diverge at different pressures, wi th increasing

pressure reducing the tem perature sensitivity when the

latter is normalized by TG.

Summary

The log r vs TG IT form has been tested herein and

verified to be a rational means to classify and distinguish

the segmentalrelaxatio n charac teristics of polymers. Th e

original derivation

of

this normalization relied on several

assumptions as discussed above; however, the corrobo-

ration of the fragility plot appro ach

Figures

b , 2b, nd

4b)

isnot taken asconfirmation

of

any connectionbetween

temperature dependenciesand the heat capacitydifference

of

the liquid and glassy

s t a t e s

In fact,

for

polymers no

such relationship is expected. Correlations have been

demonstrated previously between the degree of intermo-

lecular cooperativity of a polym eric liquid and the tem -

perature dependence of segmental relaxation.lJl The

present assessment of the T c - n o d e d caling procedure

is made possible because

of

the equivalence of this

cooperativity (and thus th e shape of segmental relaxation

dispersions)

for

polymers differing only in molecular

weight.

Acknowledgment. We thank D. J.Plazekfor the creep

compliance data on polystyrenes and

E.

Schlosser and

A.

Sch6nhals for providing prior to publication their da ta on

the dielectric relaxation of PPO . Thi s work was supporte d

by th e Office of N aval Research (toK.L.N. under Contra ct

NOQO1491WX24019).

References

and

Notes

(1)

Plazek, D. J.; Ngai,

K. L. Macromolecules 1991,24, 222.

(2)

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DC,

1985;

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