Transcript Relaxations in the Glassy State. Short

Relaxations in the
Glassy State.
Short-Range Dynamics
TUTURIAL 8
INTRODUCTION

Although molecular mobility is severely restricted below the
glass transition temperature, a cascade of subglass relaxation
phenomena may occur below Tg.

That reveal different modes of mobility.

These processes, called secondary relaxations, are labeled β, γ,
etc. , in order of decreasing temperatures.

The temperature at which a given group of segments
undergoes a relaxation usually depends on both: the polymer
structure, and the local molecular environment.

It is widely accepted that relaxations in the glassy state
basically imply intramolecular processes, and perhaps only the
β-relaxation (the highest temperature secondary relaxation) is
also related to intermolecular effects.

In fact, some authors consider the β-relaxation as the precursor
of the glass transitions, and in this sense it would be a
"universal feature" of the glass-forming materials, including low
molecular weight compounds.

As a general picture, it is considered that molecular units
associated with a dipole tend to rotate within a cage, cluster, or
mobile islands by the effect of the electric field.

The motion is restricted by limits of the cluster (island of
mobility) made up of other polymer atoms acting as the
boundaries of the cage.

The motion of the unit is described in terms of the coordinates
fixed with respect to the coordinates of the cavity, and an
intramolecular potential barrier expressed as a function of these
coordinates controls it.

Consequently, subglass relaxation processes can be envisaged
as a thermally activated motion between two potential wells
separated by a potential barrier.

The probability of location of molecules at each side of
the barrier is determined by the Boltzmann distribution.

The application of an external electric field alters the
equilibrium distribution by changing the relative depths
of the minima, thus causing a redistribution, the rate of
which is controlled by the activation energy barrier or
activation enthalpy
PHENOMENOLOGICAL CLASSIFICATION OF
SECONDARY RELAXATION PROCESSES

There are two competing theories concerning the origin of the
β-relaxation peak.

According to Johari and Goldstein, the appearance of the βpeak does not require any specific molecular motion.

The occurrence of the β-peak would be due to the so-called
"clusters" or "islands of mobility" present in glass-forming
materials.

Another authors interprets the presence of all
the observed secondary losses as the result of
some specific molecular motion.

It’s also possible to think that these theories
are not mutually exclusive; in fact, it is possible
to consider specific molecular motions taking
place in the above mentioned "islands of
mobility“.

Several cases of molecular motions causing dielectric
relaxations could be considered:
– 1 In polymers without prominent lateral chains, such as polyvinyl
chloride (PVC) or polycarbonates, local main chain motions can give rise
to secondary dielectric relaxations
– 2 Motions of side groups about the bonds linking them to the main
chain, as in the case of poly-n-alkyl methacrylates, are probably the best
studied examples of the polymers containing dipolar groups in their
lateral chains
– 3 Internal motions of the side chain groups without cooperation of the
main chain are typical of polymers containing flexible units or polar final
groups
– 4 Another possibility concerns the motions of small molecules, as water,
for example, embedded in a polymer matrix.
– 5 Secondary relaxations in semicrystalline polymers
– 6 Secondary relaxations in liquid crystalline polymers.

In any case, the relevant parameters concerning the secondary
relaxations are: the relaxation strength, , the frequency of
the maximum, and the shape and broadness of the relaxation
peak.

Unfortunately, in many cases, secondary relaxations mutually
overlap or appear as a shoulder of the prominent -relaxation

The determination of the relaxation strength and the maximum
of the peak may be subjected to some degree of uncertainty
owing to the errors involved in the deconvolution of the
overlapping peaks.
Local Main Chain Motions

Polymers such as polyvinyl chloride (PVC),
polycarbonates (PCs), and aromatic polyesters
derived from terephthalic acid (PET) and similar
polymers or isophthalic (PEIT) acid show
secondary peaks which in some cases can be
depressed by the effect of additives that
increase the modulus and decrease the
damping (antiplasticizers).
Journal of Non-Crystalline Solids 235-237 (1998) 623-627
Macromolecules, Vol. 19, No. 8, 1986
Effect of the crystallinity
in the relaxation
strength
Macromolecules, Vol. 19, No. 8,
1986
The higher the temperature, the narrower the distribution
Motions of Side Groups About
Their Link to the Main Backbone

In this category are included the β-relaxations of poly(alkyl
methacrylates) and poly(itaconates).

Experimental data show that the position of the β-relaxation in
polyalkyl methacrylates is insensitive to the length of the alkyl
group.

However, as the temperature of the -relaxation decreases by
the effect of adding successive methylene groups in the lateral
chain, the -relaxation tends to overlap with the β-relaxation,
giving rise to the β-peak

The intensity of the β-relaxation in PMMA
is higher than that of the -relaxation, an
uncommon fact in polymers.
Macromolecules, 29 (1), 247 -253, 1996
Motions within Side Groups

There are a number of examples corresponding to this
category, such as the polymers in which one or more hydrogen
atoms of alkyl groups of polyalkyl methacrylates,
polyalkylitaconates, etc., are substituted for halogen atoms.

This category also includes polymers containing flexible rings as
side groups.

These polymers may present ostensible β-relaxations
Polymer 1848 45 (2004) 1845–1855
(CH2)n-Cl
O
”
”
max=0,06
O
max=0,18
O
O
(CH2)n-Cl

γ
β
(a)Loss factor and (b) Electric loss
modulus as a function of the
temperature at a frequency of 1Hz
for PCHMA(), P4THPMA() and
PDMA(). Arrows show the
calorimetric glass transition
temperature (Tg), measured by
DSC

γ
β
CH3
CH3
CH3
O
O
O
O
O
O
O
O
O
PCHMA
P4THPMA
PDMA
Figure 4. Loss factor data (symbols) and their deconvolution in two FK functions (lines)
as function of temperature, at 10.3 Hz for P4THPMA
PDMA, 43.8 kJ/mol
PCHMA, 43.1 kJ/mol
P4THPMA, 48.0 kJ/mol
CH3
CH3
CH3
O
O
O
O
O
O
68.2 kJmol-1
O
O
O
PCHMA
P4THPMA
PDMA
Figure 3. Dependence of log fmax with the inverse of temperature in the range of α ( P4THPMA),  (
P4THPMA),  ( P4THPMA,  PCHMA,  PDMA) and  ( P4THPMA) relaxations.
-relaxation: parcial rotation of the
cyclohexyl ring as a whole
CH3
CH3
CH3
O
O
O
O
O
O
O
O
O
PCHMA
P4THPMA
PDMA
Figure 6. Potential energy (kJ mol-1) profile for rotation of O-C bond for (1: -180º to
+180º) for one-unit model compounds of () PCHMA, () P4THPMA and ()
PDMA.
γ-relaxation:
Chair to inverse-chair interconversion of the cyclohexyl ring
Chair  Boat  half-chair  chair
Energy Barrier (kJmol-1 )
P4THPMA
MMX
33.348
53.517
79.633
34.514
46.114
SE
-25.080
-5.016
21.318
-23.826
(48.0)

1.667
2.853
2.326
1.475
PDMA
MMX
31.655
53.333
80.164
35.133
43.769
SE
-46.816
-25.080
1.672
-43.054
(43.8)

1.495
1.489
2.387
3.272
PCHMA
MMX
41.495
63.9085
84.3315
41.9925
SE
-5.434
17.138
37.620
-5.016

1.762
1.749
1.752
1.754
43.03
(43.1)
Motions due to the Presence of Small
Molecules in the Polymer Matrix

Low molecular weight compounds not only act
as plasticizers depressing the glass transition
temperature of polymers but also interact with
the motions that cause subglass activity

A typical case is the effect of water on the
secondary relaxations of polymers containing
hydrophilic groups, such as hydroxylic or amide
groups
Secondary Relaxations in
Semicrystalline Polymers

The analysis of the dielectric spectroscopy of semicrystalline
polymers is more complicated than that of amorphous
polymers.

The reason is that, in addition to the difficulties concerning the
molecular assignation of the observed peaks, the phase in
which the relaxations occur must also be elucidated.

To start with, the existence of a double -process in many
semicrystalline polymers is well known.

The lower-temperature process, called the a-peak, is usually
related to the cooperative relaxation of the amorphous phase.

This fraction is sometimes called "intercrystalline", because it
refers to the fringing material existing between lamellar
structures.

Typical WLF behavior corresponding to an amorphous polymer
is expected for this peak.

At slightly higher temperatures, another narrower peak appears
that is related to some sort of mechanism in which crystalline
entities of the material are implied.

The dielectric spectrum of PE displays three characteristic
relaxational zones conventionally designated as γ, β , and  in
order of increasing temperatures

Semicrystalline halogen-polymers such as polyvinyl fluoride
(PVF), polytetrafluorethylene (PTFE), and related polymers also
show considerable sub-Tg activity.

The low-temperature relaxations in these polymers are
understood as "cooperative'', local mode, main chain motions.
3,5
Log f max
3,0
2,5
ln (fmax)= Ea/RT + k
2,0
0,0045
0,0050
1/T, K
-1
0,0055
Eyring equation
H and the activation energy Ea given by the Arrhenius equation
are related by Ea=H + RT.
The values of H and S can directly determined from ln (f/T)
versus 1/T plots.
Low values of the activation entropy suggests, that the processes
are a simple secondary relaxation.
High values of entropies indicate some degree of cooperativity.
Secondary relaxations near to the  relaxation sometimes shows
high entropy.
Eyring
Ln (f/T)
0,5
0,0
y = A+ Bx
-0,5
-1,0
0,0045
0,0050
1/T, K
-1
0,0055
Summary
Secondary relaxation in polymers revels
molecular mobility in the glassy state.
 Multiple causes could origin secondary
relaxation: motions of the main chain, motion
of side chain, motion of part of the side chain.
 To characterize secondary relaxation it is
necessary to know the evolution of the
relaxation strength, the relaxation map, and the
broadness of the peak.
