Transcript Dielectric Relaxation processes at temperatures above

Dielectric Relaxation processes
at temperatures above glass
transition.
Molecular chains dynamics
(1)
TUTORIAL 6
INTRODUCTION





Condensed matter has a structural fading memory reflected in the
velocity with which a perturbed system forgets the molecular
configuration that it had in the past.
In ordinary liquids, molecular reorganization occurs very rapidly
and structural memory at the molecular level is very short.
The relaxation time is roughly defined as the time necessary for
the system to forget the configuration it had prior to the
perturbation.
At the other extreme the solids are characterized for having a very
large structural memory at the molecular level, reflected in large
relaxation times.
From a strict point of view, the distinction between solids and
liquids need to be expressed in a no subjective way.

The solid or liquid character of the condensed matter is
expressed in terms of the Deborah number defined as
Time required for the
relaxation process
timescale of the experiment



For ideal liquids  → 0 and ND = 0, while for ideal solids
→∞ and ND → ∞
For the so-called viscoelastic systems  and exp are
comparable and their Deborah's number is of the order
of unity.
Polymers are the most important viscoleastic systems
PHENOMENOLOGICAL
DIELECTRIC
D(t)
RESPONSE IN THE TIME DOMAIN



Let us consider the response of an isotropic polar
system to a perturbation electric field defined as
The orientation of the dipoles by the effect of the field
is reflected in a continuous increase in the dielectric
displacement with time, until eventually a constant
value is reached.
The time dependence of the dielectric displacement can
be written as
r(ω=0) is the
relaxed dielectric
activity
Ψ(t) is a monotonous increasing function of time, the extreme
values of which are Ψ(0)=0 and Ψ(∞) = 1.
u (ω=∞) is the unrelaxed dielectric permittivity. It’s caused by the
distortion of the electronic cloud and the positions of the nuclei of the
atoms by the effect of the electric field.




The function Ψ(t) is of entropic nature.
It reflects the molecular motions that take
place in the system to accommodate the
orientation of the dipoles to the perturbation
field.
Owing to the entropic nature of the response at
t>0, the dielectric displacement does not
vanish when the perturbation field ceases.
As a result, D(t) not only depends on the actual
perturbation field but also on the electric
history undergone by the material in the past.

Under a linear behavior regime, the
response to perturbation fields is
governed by the Boltzmann superposition
principle
Electric perturbation
in past time
Electric Displacement at time t
Electric Permittivity, dependent of
the actual and past events.
Boltzmann superposition principle for
dielectric experiments in continuous form
DIELECTRIC RESPONSE IN THE
FREQUENCY DOMAIN


Do is the amplitude of the
response.
This equation indicates that the
dielectric displacement is a
complex quantity:
 component in phase with the
perturbation field: Do·cos
 90º out of phase: Do·sin
POLY VINYL ACETATE
=r-u
DIELECTRIC RELAXATION MODULUS IN
THE TIME AND FREQUENCY DOMAINS


The experimental determination of the
relaxed permittivity r may involve some
difficulties in cases where the conductivity
contribution to the dielectric loss overlaps
the dipolar one.
In this situation it is preferable to analyze
the dielectric results in terms of the
dielectric modulus M .



Let us consider the following electric history
If Do is very small, the linear dielectric
phenomenological theory predicts that
The dielectric relaxation modulus, M(t), is given by
Φ(t) is a monotonous decreasing function of time, the
extreme values of which are Φ(0)=1 and Φ(∞)=0.

For low dielectric displacements, the Boltzmann
superposition principle holds



In continuous form
If a sinusoidal dielectric displacement D(t)=Do·sinωt is
imposed on the material, the electric field can be
written as
E(t)=Eo·sin(ωt+)
=Eo·sin(ωt)·cos+Eo·cos(ωt)·sin
= Do(M’sin(ωt) + M" cos(ωt))

The electric field is a complex quantity with a
component in phase with the perturbation (Eocos) and
another 90º out of phase (Eosin).

The real component M’ and loss component M”
of the complex dielectric modulus are given by
Phase angle
From Kremer – Schönhals book
LOCAL AND COOPERATIVE
DYNAMICS:BASIC CONCEPTS



Relaxation response functions of liquids to weak
external perturbations may provide information on the
actual structural kinetics resulting from transitions in
the configurational space describing the system.
The normalized response function to an electric field
can be expressed in molecular terms by
N is the number of relaxing species and mi(t) is the
dipole moment of the species i at time t



The area of the normalized relaxation curve g(t) vs t,
at temperature T, defines the mean relaxation time (T)
The value of  in the liquid state increases rapidly with
decreasing temperature in the vicinity of the glass
transition temperature.
It can be fitted to a Vogel-Fulher-Tamman-Hesse
(VFTH) equation
Vogel temperature
D>10 STRONG glass
prefactor of the order
of picoseconds
strength parameter
D<10 WEAK glass






The importance of the dynamics is evident if a melt is cooled
at high enough cooling rate to avoid its crystallization.
The fast cooling leads the system in a super-cooled liquid,
and then, to a glass.
The glass transition temperature, at which the super-cooled
liquid forms a glass, is a dynamic property.
It is often defined as the temperature at which the
relaxation time is 200 s, a reasonable maximum relaxation
time for dynamic experiments, but obviously Tg,
increases/decreases as the timescale of the experiment
decreases/increases.
The relaxation time reflects the time involved in the
structural rearrangement (glass transition) of the molecules.
 changes by many orders of magnitude from the liquid to
the glass.
Molar enthalpy
supercooled
liquid
Cpliquid
glass
fast
Cpglass
slow
liquid
Hmelting
crystal Cpcrystal
Temperature
7/17/2015
18




The glass transition is a universal property of
condensed amorphous matter, independent of its
molecular weight.
Below Tg the dynamic response depends on the thermal
history, and aging phenomena may occur.
Owing to the latent heat of melting, a supercooled
liquid has substantially higher entropy than the crystal.
However, the liquid loses entropy faster than the
crystal below the melting temperature (Tm)


The difference between the entropies of the liquid and
the crystal becomes rather small just below the glass
transition.
Extrapolation of the entropy of the glass to low
temperatures, indicates the existence of a temperature
TK, called the Kauzmann temperature, at which the
crystal and the liquid attain the same entropy.



The most prominent theoretical approaches to the glass
transition are the Adams – Gibbs and the Free Volumes
theories.
The Adams-Gibbs theory is the first to consider the
glass transition as a cooperative process.
In the free volume approach Tv is considered to be the
temperature at which the free volume would be zero.



In the kinetic and fluctuation model the volume of the
cooperatively rearranging region is defined as the
smallest volume that can relax to a new configuration
independently.
The cooperativity continues well above Tg, and
increases with decreasing temperature.
Another approach to the study of the glass transition is
to determine the response of liquids deep in the liquid
state


The mode coupling theory explains the glass
transition in terms of a dynamic phase
transition occurring at a critical temperature T,
significantly above the glass transition
temperature.
A coupling scheme has been proposed by Ngai
and coworker, in which the first-time derivative
of the KWW equation is taken to be the master
equation in the time domain for application to
isothermal and non-isothermal experiments.
RESPONSES OF GLASS FORMERS ABOVE Tg
TO PERTURBATION FIELDS




The temperature dependence of the mean relaxation
time of supercooled liquids obeys Vogel equation.
This absorption, called the -relaxation process, may
display at high temperatures conductive contributions
on the low frequency side of ”(ω)
-relaxation displays a KWW stretched exponential
decay
The wider the -relaxation, the lower the value of the
stretch exponent(βKWW).
-Relaxation for Poly 2,4-Difluorbencyl methacrylate
10
-2
10
-4
10
-6
10
0,0024 0,0027 0,0030
-8
o
10
kww
0
0,5
0,4
0,3
1/T, K
-1



In the vicinity of Tg, the KWW dependence with
temperature is described by the VFTH equation.
The mean relaxation time associated with the
glass-rubber absorption is given by
According to the linear phenomenological
theory, the glass-liquid relaxation can be
obtained in the frequency domain by means of



The -relaxation in the frequency domain is described
by the empirical Havriliak-Negami equation
In this expression, HN is related to the maximum of the
-peak in the frequency domain, whereas HN and βHN
are fractional
the following
High fshape parameters fulfilling
Low f
conditions: βHN>0 and HNβHN<1.
HN, and βHN parameters are related to the limiting
values of the slopes in the low- and high-frequency
regions of the double logarithmic plot " vs ω, by
means of the expressions


Although m and n are uncorrelated parameters, for
polymeric materials 0<n<0.5
At frequency higher than the corresponding to the relaxation some low molecular glass formers shows an
excess loss or wing not accounted for by the empirical
expressions commonly used to describe the -process



For many systems in which the -relaxation is
described by the Davidson-Cole equation, the excess
wing contribution can be described by the scaling law
"~ωa with the exponent. a is lower than the value of
the exponent of the Davidson-Cole expression.
No accepted model exists to describe the macroscopic
origin of the excess wing.
Besides the -relaxation, many glass formers,
especially polymers, display at high frequencies one or
more relaxations (β, γ, , etc) that are believed to be
associated with local intramolecular relaxations in the
main chain or in side groups.



Another type of relaxation is the so-called JohariGoldstein β-relaxation, which seems to be a rather
universal property of glass formers.
This relaxation even appears in relatively simple
systems in which intramolecular motions are absent.
In some materials, the Johari-Goldstein β-relaxation
may overlap the excess wing of the -relaxation
process.
BROADBAND DIELECTRIC SPECTROSCOPY
OF SUPERCOOLED POLYMERS

Some polymers with
flexible side groups
are characterized for
displaying high
dielectric activity
even in the glassy
state.

β

poly(5-acryloxy
methyl-5-ethyl-1,3dioxacyclohexane)
(PAMED)

Below the glass transition temperature, the isotherms
present a wide β-relaxation whose maximum shifts to
higher frequency with increasing temperature
max/2



Usually, the β-peak in the frequency domain
is
Half width
symmetric and displays half-widths of 2-5 decades.
The normalized dielectric loss for this process is often
described by the empirical Fuoss-Kirkwood equation
The parameter m, accounts for the broadness of the βprocess in such a way that, the larger the value of m,
the narrower is the absorption


Also, the intensity of the β-process increases with
increasing temperature
The width of the β-relaxation is often explained in
terms of the distribution of both the activation energies
and the pre-exponential factor that results from the
variety of molecular environments that cause the
relaxation.


At the higher temperatures, the distance separating the
β- from the -peak decreases as a consequence of the
high activation energy of the -relaxation process.
A temperature is reached at which both relaxations
coalesce into a single peak, named β-relaxation, the
intensity of which seems to increase with increasing
temperature

The loss curves at T > Tg
can be fitted to a sum of
an - plus a β-relaxation



The relaxation strength can be evaluated as:
In the case of the β relaxation, the relaxation strength
for a FK equation became
The strength of the -relaxation decreases with
increasing temperature as a consequence of the
randomization of the dipolar orientation.


 reaches a maximum
value at temperatures
just slightly above Tg.
Then, undergoes a steep
decrease with increasing
temperature
Extrapolation to  = 0
(-onset) points to an
onset temperature of
97°C in the case analyzed
here



The strength of the β-relaxation displays a pattern by
which it increases steeply in the temperature range in
which the strength of the -relaxation steeply
decreases.
The total strength of the isotherms expressed as
=+β remains nearly constant in the whole
interval of temperatures.
The location of the -relaxation in the frequency
domain depends on the chemical structure but not on
molecular weight for high molecular weight polymers


This behavior suggests
some sort of
cooperativity on the
motions of the molecular
chains, otherwise this
relaxation should exhibit
a strong dependence on
chain length.
The relaxation time
associated with the relaxation shows a
stronger dependence on
temperature than the
secondary relaxation
processes.
Based on the analysis of the
type A scenario
splitting of the β-relaxation
 type D scenario
type C scenario
of condensed matter,
type
B scenario
 the separated
-onset can be characterized
by
 there are two different but touching
-relaxations,
with
including
polymers,
it the
has
is
similar
to
that
of
the
conventional
one,
but
here
minimum
cooperativity
for  that cannot
be

there
is
a
locally
coordinative
β-precursor
been
speculated
that for
five
a
sharp
crossover
between
them.
-relaxation
is above
and below
the splitting,
and
continued tocurve
a local,
noncooperative
process.
scenarios
are possible
the
cooperative -process at high
β is not the tangent to 
temperatures.
Conventional type
 the β-process continues
deep in the liquid state


γ
(a) Loss factor and (b) Electric loss
modulus as a function of the
temperature at a frequency of 1Hz
for PCHMA (),
β
P4THPMA () and
PDMA ().

γ
Arrows show the calorimetric glass
transition temperature (Tg),
measured by DSC
β
CH3
CH3
CH3
O
O
O
O
O
O
O
O
O
PCHMA
P4THPMA
PDMA
Summary


Polymer materials present “structural memory”.
The glass transition:
• universal property of condensed amorphous matter.
• it’s a dynamic phenomenon.
• has entropic nature.

The mean relaxation time for the -relaxation
show a Vogel dependency with the temperature
Summary



The D, parameter in the VFTH equation it’s
known as the strength parameter (If D>10,
strong glass former, If D<10, fragile glass
former)
Adams – Gibss theory: assume that the 
relaxation it’s a cooperative process.
Free volume theory: assume that Tv is the
temperature at which the free volume it’s zero.
Summary


Experimental data for
the  relaxation can be
fitted by mean of the
HN empirical equation.
 decreases when
increasing
temperature.
Summary



From the  we can infer about the number of entities
relaxing, and the mean square dipole moment.
The Arrhenius plot ( vs T-1), gives information about
the dynamic of the system.
From the shape parameters, we can infer information
about the distribution of the relaxation time.