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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
nloaded from http://polymerphysics.net
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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)
Angell, C. A. In Relaxations in Complelc System s;Ngai, K.
.,
Wrig ht, G. B., Eds.;Government Printing
Office:
Washingon,
DC,
1985;
3;
roceedings
of
the Materials Research Society
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B a t o n ,
1990.
3)Angell, C. A. J Non-Cryst. Solids 1991,131-133, 3.
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J.
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Ferry, J.D.
Viscoelastic Properties ofPolymers;
Wiley: New
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W.;
Plazek,
J.J .
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Roland, C. M.; Ngai, K. L.
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14)Roland, C. M.; Ngai, K. L. J Rheol., in press.
15)
Plazek,
D. J.;
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V.
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Ngai, K. L.; Rendell, R. W. J .Non-Cryst.Solids 1991,131-133,
942.
(18)
Schonhals,
A.;
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