TIME DOMAIN RESPONSES OF GLASSY POLYMERS
by
SHANKAR KOLLENGODU SUBRAMANIAN
A DISSERTATION
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
111111111111111111111111111111
Gregory B. McKenna
(Chairperson of the Committee)
111111111111111111111111111111
Sindee L. Simon
111111111111111111111111111111
Brandon Weeks
111111111111111111111111111111
Edward L. Quitevis
111111111111111111111111111111
Peggy Gordon Miller
Dean of the Graduate School
May, 2011
Copyright 2011 Shankar Kollengodu Subramanian
Texas Tech University, Shankar Kollengodu Subramanian, May 2011
ACKNOWLEDGEMENT
I would like to thank my advisor and chair of my committee Prof. Gregory
B.McKenna for his role in mentoring, guiding and encouraging me. He provided a
great environment to learn and ample space for me to think and grow. I would also
like to extend my gratitude to Prof Sindee Simon for her positive words and valuable
feedback on my dissertation. Further, I would like to extend my thanks to Prof
Edward Quitevis and Brandon Weeks for agreeing to be part of my committee and for
their insightful inputs.
I would like to appreciate the encouragement and camaraderie of all my
colleagues and coworkers at Texas Tech University. I would like to recognize Dr.
Prashanth Bardinarayanan for his words of encouragement and Dr. Paul O Connell,
Babji Srinivasan, Dr. Lameck Banda, Dr. Mataz Alcoutlabi, and Jing Zhao for helping
me with my experiments.
I would like to thank my mother, sisters, in-laws and my entire family for their
unconditional love, belief and support. Finally I would like to acknowledge my wife,
Harini for being my cheer leader.
I would like to dedicate this thesis to my late uncle Dr Natrajan, who has been one of
my inspirations for becoming a Chemical Engineer.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
TABLE OF CONTENTS
ACKNOWLEDGEMENTS……………………………………………………….......ii
ABSTRACT……………………………………………………………………...….viii
LIST OF TABLES………………………………………………………………….....x
LIST OF FIGURES…………………………………………………………………..xi
CHAPTER
1
INTRODUCTION…………………………………………………………….1
1.1
Glassy Phenomena……………………………………………..……...1
1.2
Dielectric responses in glassy polymers……………………………....2
1.3
Structural recovery and aging in glassy polymers…………………….6
1.3.1 Intrinsic isopiestics…………………………………………....6
1.3.2 Memory effect ………………………………………………...7
1.3.3 Asymmetry of approach………………………………………7
1.3.4 τeff paradox……………………………………………………8
1.3.5 Physical aging………………………………………………...9
2
1.4
Effect of plasticizer in glassy behavior……………………………….9
1.5
References…………………………………………………………...12
EXPERIMENTAL SYSTEM ……………………………………………….29
2.1
Time domain dielectric spectrometer……………………………..…29
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
2.2
Experimental setup for studying the aging and structural recovery
responses of glassy polymers subjected to plasticizer environment………..31
3
2.3
Linear Variable Differential Transformer calibration………………32
2.4
References…………………………………………………………..33
A DIELECTRIC STUDY OF POLY (VINYL ACETATE) USING A PULSEPROBE TECHNIQUE………………………………….…………………..39
3.1
Motivation…………………………………………………………..39
3.2
Introduction………………………………………………………....39
3.3
Sample preparation..………………………………………………...42
3.4
Method of analysis………………………..…………………………43
3.5
Results ………………………..……………………………………...45
3.5.1 Single step response………..………………………………..45
3.5.2 Time-frequency conversion……………...………………….48
3.5.3 Two step (pulse-probe) response………...………………….49
3.6
Discussion…………………………………………..……………….50
3.7
Conclusion…………...……………………………………………...52
3.8
Appendix……..……………………………………………………...53
3.8.1 Algorithm to obtain true compliance………………...……...53
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
3.8.2 Boltzmann superposition algorithm to delineate linear and
nonlinear behaviors………………….……………….……..57
3.9
4
References……………………………………………..…………...60
APPLICATION OF EMPIRICAL MODE DECOMPOSITION IN THE
FIELD OF POLYMER PHYSICS................................................................78
4.1
Motivation..........................................................................................78
4.2
Introduction.......................................................................................79
4.3
Methodology......................................................................................86
4.3.1 EMD based FFT filerting algorithm..........................................87
4.4
Results and discussion........................................................................88
4.4.1 Simulation studies..................................................................88
4.4.2 Filtering of the simulated data...............................................90
4.4.3 Discussion of results obtained from filtering methods..........96
4.4.4 Experimental case study I......................................................97
4.4.5 Experimental study II............................................................99
4.4.6 Practical benefits due to use of the proposed EMD based FFT
filtering algorithm............................................................................103
4.5
Conclusions.....................................................................................106
4.6
Appendix.........................................................................................107
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4.6.1 Mathematical details of various filtering techniques...........107
4.7
5
References........................................................................................110
AGING AND STRUCTURAL RECOVERY BEHAVIORS IN EPOXY
FILMS SUBJECTED TO CARBON DIOXIDE PLASTICIZATION JUMPS:
EVIDENCE FOR A NEW GLASSY STATE……………………………….……138
5.1
Motivation…………………………………………….…………...138
5.2
Introduction……………………………………..…………….…..138
5.3
Experimental……………….…………………………….………..140
5.4
Method of analysis………………………………..……………….141
5.5
Results and discussion……………………………………..……...142
5.5.1 Structural recovery experiments…………………..………142
5.5.2 Aging experiments………………………………………...143
5.6
Conclusions……………………………………………..…………146
5.7
References……………………………………..………………….147
6
SUMMARY AND CONCLUSIONS …………………………………….166
7
FUTURE WORK………………………………..………………………..170
7.1
Introduction ……………………………………………………..170
7.2
Time domain nonlinear dielectric……………………………......170
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7.3
Isobaric time domain dielectric measurements……………..……171
7.4
Physical aging and structural recovery of glassy polymers……...172
7.5
References……………………………………………..………...173
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
ABSTRACT
The properties of glassy polymers are generally studied in the vicinity of the glass
transition temperature using frequency and time domain responses. In this work, we
focus on the dielectric, the structural recovery, and aging responses in the time
domain. A proper understanding of these properties is essential for a better prediction
of the performances of polymers.
Time domain dielectric measurement provides a powerful means to study the
glassy responses in short times (as low as micro seconds) to long times (>200
seconds) in a single measurement device. We built a Time Domain Dielectric
Spectrometer (TDS) in our laboratory at Texas Tech to take advantage of this ability.
Successful working of the spectrometer was demonstrated by studying the dielectric
response of poly (vinyl acetate) in a pulse-probe experiment. We see memory effect
and this was in quantitative agreement with linear Boltzmann superposition for small
applied fields. However, evidence of breakdown of linearity was observed at larger
applied fields. This is the first demonstration in time domain dielectric measurement
of the ability to delineate between linear and nonlinear behaviors.
Noise in the TDS set up was a hindrance for performing more sensitive experiments.
The dielectric data obtained from TDS was non stationary in nature (with respect to
time) and corrupted with nonlinear noise. Popular methods like fast Fourier transform
or moving average are not entirely suitable to handle this type of noise. We introduce
a filtration tool called Empirical Mode Decomposition (EMD) to improve the data
analysis for these types of data corrupted with noise. An advantage of this method is
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
that it aids in the improvement of the experimental setup by giving useful information
on the noise. EMD based filtering was also applied to the data obtained from
structural recovery experiments which are the other time domain response studied in
this work.
Structural recovery and aging experiments of glassy polymers are very well
understood for temperature formed glasses compared to concentration formed glasses.
Previous work from our group has shown that concentration formed glasses
qualitatively mimic temperature formed glasses but were quantitatively different.
Further, our preliminary work on the structural recovery of an epoxy film subjected to
CO2 plasticizer jumps showed that the effective retardation time for concentration
formed glass and temperature formed glass (subjected to same final condition) do not
converge to the same point as equilibrium is approached. This result was unexpected;
as we had hypothesized that both concentration and temperature formed glasses come
to the same apparent equilibrium state. Hence, we further investigated this behavior
by studying the aging and structural recovery of epoxy film subjected to CO2
plasticizer jumps. We observe evidence for the existence of a new metastable glassy
state.
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LIST OF TABLES
3.1
Fit parameters for equation 4 at 308 K. Errors represent standard error of
estimate on the fit parameters……………………………………………………….. 64
3.2
A, B and T0 from VFT fits for the retardation time and the dc conductivity
term vs. temperature obtained by fitting the data in Figure 2a to equation 4. Errors
represent standard error of estimate on the VFT fit parameters……………...……... 64
4.1
Sum squared error between the MKWW model data and filtered data obtained
from various filtering approaches…………………………………………...………116
5.1
KWW parameters for the longest aging time (230400s) of T and P jump
experiments…………………………………………………………………….…...152
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LIST OF FIGURES
1.1
Specific volume vs Temperature plot for different cooling rates (q). Here, q1>
q2>q3 ………………………...………………………………………………………17
1.2
Classifications of dipoles in amorphous polymers………………...…………18
1.3
Schematic representation of two different sources of non-exponential
correlation decays……………………………….…………………………………...19
1.4
Schematic of the protocol for dielectric hole burning experiment……...……20
1.5
Time dependent dielectric permittivity of propylene carbonate at 157.4K......21
1.6
Empirical mode decomposition schematic to filter noise in time domain
responses………………………..……………………………………………………22
1.7
(a) Schematic of intrinsic isotherm experiment (b) Intrinsic isotherm
experiment performed by Kovacs on PVAc ………………………………………...23
1.8
(a) Schematic of memory effect experiment (b) Memory effect experiment of
PVAc performed by Kovacs………………………………..………………………..24
1.9
(a) Schematic for asymmetry of approach experiment (b) Asymmetry of
approach experiment of PVAc performed by Kovacs…………………………...…..25
1.10
Schematic of Struik’s protocol……………………………..………………..26
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1.11 Creep compliance of poly(vinyl chloride) quenched from above to below Tg at
different aging times after quenching…………...……………………………………27
1.12 Effective retardation time plot for temperature and concentration formed
glasses to the same final condition….………………………………………………..28
2.1
Schematic of time domain dielectric spectrometer built at Texas Tech taken
from reference ………………………………………………………….……………34
2.2
Sample setup schematic for the time domain dielectric spectrometer…..…...35
2.3
The creep apparatus that was built in our laboratory to perform experiments
under different CO2-pressure and temperature conditions…………………………..36
2.4
Pressure vessel used to perform the experiments under CO2 …………...…..37
2.5
LVDT calibration plot where the displacement of the core is measured as a
function of voltage……………………..……………………………………………38
3.1
(a) Isothermal measurement and (b) Master curve for PVAc with 308 K as
reference temperature ………………………………………………………..……..65
3.2
Equation 3.4 parameters ε1 and ε2 as a function of temperature…...………..66
3.3
Dielectric retardation time τ and dc conductivity τc term plotted as a function
of inverse temperature. The solid lines represent the VFT fits………………..……67
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
3.4
(a) Dielectric recoverable compliance and (b) master curve for PVAc with 308
K as reference temperature………..………………………………………….……68
3.5
Time-temperature shift factors as a function of temperature for PVAc.
Comparison of current dielectric results with literature reports for dielectric and
mechanical behaviors………………………………..…………………………….69
3.6
Dielectric storage compliance response for PVAc………………..………70
3.7
Dielectric loss compliance of PVAc ………………………..…………….71
3.8
Comparison of apparent dielectric compliance with true dielectric compliance
for an applied electric field of 5.4*105Vm-1 ………………………………….…..72
3.9
Dielectric compliance response in two step pulse-probe experiments with
varying time duration t1 of the first step ………………………………………….73
3.10 Dielectric compliance response in two step pulse-probe experiments with
varying electric field E1 of the first step………………………………….……….74
3.11 Dielectric strain response for the 1second jump of Figure 8. The solid line
represents the Boltzmann prediction……………………………………………...75
3.12 Single step responses for applied electric field of 44.6*105 Vm-1 and 89.2*105
Vm-1 …………………………………………………………………………...…76
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3.13 Two step dielectric compliance response (jump from 89.2*105 Vm-1 to
44.6*105 Vm-1) with linear Boltzmann prediction showing deviation from linear
behavior…………………………………………………………………………77
4.1
Data generated using the MKWW model (without noise)……………..117
4.2
Simulated noise added to the data generated from the MKWW model
(equation 4.4)…..………………………………………………………………118
4.3
Autocorrelation function of the white noise...........................................119
4.4
Simulated MKWW model data with noise n(t) added………………..120
4.5
MA filtered noisy model data – Window size 5................................121
4.6
MA filtered noisy model data – Window size 10..............................122
4.7
FFT filtered noisy model data...............................................................123
4.8
Schematic of DWT for the simulated data............................................124
4.9
Filtered data using Discrete Wavelet Transform – Five level
decomposition...............................................................................................125
4.10 Filtered data using Discrete Wavelet Transform – Six level
decomposition....................................................................................................126
4.11 IMFs and their corresponding magnitude at various frequencies obtained
from FFT for the simulated data with noise.................................................127
4.12
Filtered data using EMD based FFT approach......................................128
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
4.13 Squared error comparison plots for the filtered data obtained using FFT, DWT
and EMD based FFT Approach…………………………………………….…129
4.14 (a) Zero voltage time domain data, (b) Magnitude spectrum of the data at
various frequencies using FFT...........................................................................130
4.15
Experimental dielectric time domain data.........................................131
4.16 IMFs and their corresponding magnitude at various frequencies obtained for
the experimental dielectric data………………………..……………………...132
4.17
Comparison of experimental dielectric data and filtered data from EMD based
FFT approach (Inset shows a zoomed version)..................................................133
4.18
Comparison of volume recovery data after performing a down jump
experiment from 85oC to 75oC and 72oC without filtering............................134
4.19
Comparison of volume recovery data after performing a down jump
experiment from 85oC to 75oC and 72oC after filtering using EMD method.135
4.20
IMFs and their corresponding magnitude at various frequencies obtained for
the experimental volume recovery after a down jump in temperature to 75o.....136
4.21 MKWW generated data after removing the noise at 14Hz........................137
5.1
Schematic representation of specific volume as a function of temperature or
concentrations……………………..……………………………………………153
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5.2
A comparison of departure from equilibrium as a function of time for T jump
and P jump experiment subjected to same final condition of 72oC and 0 MPa……152
5.3
Effective retardation time as a function of departure from equilibrium for T
and P jump experiments of same final condition…………………………………..155
5.4
Creep compliance curves for different aging time plotted as a function of time
for T jump experiment………………………………………………………….......156
5.5
Time-aging time superposition of creep curves of the T jump
experiment…………………………………………………………………………..157
5.6
Creep compliance curves for different aging time plotted as a function of time
for P jump
experiment……………………………………………………………......................158
5.7
Time-aging time superposition of the creep curves of P jump experiment…159
5.8
Creep compliance curves for different aging time for T and P jump
experiments subjected to same final condition 0MPa and 69.3oC…………...……..160
5.9
Time aging time superposition curves for P jump experiment superposed to
the longest aging time of T jump experiment…………………………………….…161
5.10 Horizontal shift factor as a function of aging time. The concentration glasses
are shifted with respect to the longest aging time of temperature formed glass……162
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5.11 Retardation time obtained from KWW function for T and P jump creep curves
plotted against the aging………………………………………………...…………..163
5.12 Retardation time as function of aging time for humidity and T jump
experiments of same final condition taken from reference ………………………...164
5.13 Volume recovery of epoxy film showing the reversal of concentration formed
glass to temperature formed glass upon heating above its Tg followed by cooling to
room temperature and heating………………………………………….…………...165
7.1
Schematic of time domain dielectric spectrometer with modifications to
perform isobaric measurements of dielectric compliance………….……………….174
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
CHAPTER 1
INTRODUCTION
1.1 Glassy Phenomena
Glassy behavior of amorphous polymers is one of the most widely studied
topics in Applied Sciences and Engineering for the last half a century. When
amorphous polymers are cooled from a higher temperature, the thermodynamic
properties such as volume, enthalpy or entropy deviate from the equilibrium path. The
point at which the thermodynamic property (volume, entropy or enthalpy) depart from
equilibrium path is called the glass transition temperature (Tg) [1-3]. As shown in the
Figure 1.1, the glass transition temperature is rate dependent and hence can be looked
upon as a kinetic phenomenon [4, 5]. However, there are certain aspects of the glassy
behavior which support the idea of thermodynamic origin [5-7]. The question of
glassy behavior being a thermodynamic or kinetic phenomenon is not our area of
interest in this work. For the purpose of clarity, as stated by McKenna, we can
consider glassy behavior as a “kinetic phenomena with underlying thermodynamic
transition” [2].
Understanding the properties of amorphous (glassy) polymers in the vicinity
of, as well as below the glassy transition temperature is very important to predict their
long term performance and stability [8-10]. The glassy behaviors are generally
studied in both time and frequency domain. In this work, we primarily focus on
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
studying two time domain responses of glassy polymers, namely dielectric, and
structural recovery and aging responses.
1.2 Dielectric responses in glassy polymers
Dielectric spectroscopy is a powerful method to study the glassy behavior of
materials spanning 10 decades in both frequency and time response using a single
measurement device, and more than 18 decades using multiple devices [11]. They
work on the principle of measuring the polarization (dipole moment) of the material
when subjected to an applied field (current or voltage) [11-13]. Dipole is a pair of
electrical charges of “equal magnitude and opposite polarity” and dipole moment is
defined as the vector quantity obtained by the “product of the magnitude of one of the
poles and the distance separating the two poles” [14]. Polymers in general are weakly
polar when compared to small molecule glass formers. However, based on the
dipole’s arrangement they are classified as Type A, Type B and Type C, as shown in
the Figure 2 [15].
In Type A polymers, relatively strong dipole moments typically occur along
the backbone of the polymer chain. They are characterized by a slow relaxation mode
with strong molecular weight dependence [15]. In Type B polymers, the dipole
moments occur in the main chain, but perpendicular to the backbone chain. They are
characterized by a fast segmental relaxation with no molecular weight dependence
[15]. Type C polymers have the weakest dipole moments and they occur along the
side chains of the polymers. They also show no molecular weight dependence like
Type B polymers [15]. The material investigated in this work is poly (vinyl acetate)
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
(PVAc) which is a Type B dipole. PVAc is a model polymer used for dielectric
studies of segmental relaxation. Most of the work on PVAc has been performed in
frequency domain (linear and nonlinear studies) [16, 17] with limited work in time
domain responses [18]. However, it may be noted that the dielectric responses can be
transformed from time domain to frequency domain and vice versa using Laplace
transformation in the linear regime [11, 19]. The relationship between dielectric time
and frequency domain responses is given by equation1.1 and 1.2 [11, 18, 19].
= − "
(1.1)
= + "
(1.2)
Where ε(t) is the dielectric compliance, ε’(ω) is the dielectric storage compliance,
ε”(ω) is the dielectric loss compliance, M(t) is the dielectric modulus, M’(ω) is
dielectric storage modulus and M” (ω) is the dielectric loss modulus. Furthermore, the
dielectric compliance and modulus can also be transformed from one form to the
other using equations similar to that used for conversion between creep compliance
and shear modulus [11, 20].
In general, most of the dielectric measurements reported in the literature are
performed in frequency domain, despite time domain measurements being much
faster compared to frequency sweeps because of the difficulty is filtering noises in
time domain data [11, 16, 17, 21]. However, the potential of using time domain
dielectric spectrometry (TDS) to study the glassy responses in polymers is
tremendous. Probing dynamic heterogeneity is one of the potential areas of study
which could be tapped using TDS. As shown in Figure 1.3, the overall macroscopic
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
picture of a homogeneous and heterogeneous material is same, although
microscopically the each spatial domain in a heterogeneous material can be different
[22]. This behavior can be well understood by subjecting these materials to a high
sinusoidal frequency probe, wherein particular domain in the material relaxes slower
or faster compared to the macroscopic response of the material. This is known as
dynamic heterogeneity [22].
Hole burning experiment is an excellent technique to demonstrate the
dynamic heterogeneity in materials [23, 24]. Figure 1.4 is the protocol for the hole
burning experiment. It comprises of 4 experimental steps. In step 1, a positive
sinusoidal probe of high frequency is applied, followed by a suitable waiting time and
a small voltage step. In step 2, the same experiment is repeated with a negative pulse.
Then, the signals from step 1 and step 2 are summed to give the modified response.
Step 3 and step 4 are the positive and negative pulse respectively without the
sinusoidal probe. Summation of the signals from step 3 and 4 is called the unmodified
response. If there is no difference between the modified and unmodified response
then the material is homogeneous; else it is the evidence for dynamic heterogeneity
[23, 24]. Bohmer and coworkers, [23] and Richert and coworkers [24] have
demonstrated the hole burning technique to probe dynamic heterogeneity in small
molecule glass formers. Figure 1.5, is the result of a hole burning experiment of
propylene carbonate showing horizontal and vertical holes from Bohmer’s group [23].
Shi and McKenna, [25] and Qin and McKenna [26] have also probed dynamic
heterogeneity using mechanical hole burning experiments for small molecule glass
formers and polymers. It may be noted that the small molecule glass formers used in
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
these dielectric studies have a larger signal to noise ratio compared to polymers. This
is precisely the challenge we have currently in probing the dynamic heterogeneities in
glassy polymers.
We have demonstrated in our work on time domain dielectric responses
(Chapter 3) that using the simple Boltzmann superposition principle, linear and
nonlinear behaviors can be delineated [27]. It is a progress in the direction of tapping
the potential in TDS experiments. However, noise in the data was proving detrimental
for performing more sensitive experiments like the hole burning experiments with
polymers as the signal to noise ratio is very low to sense holes as discussed above.
Time domain dielectric data are time varying in nature and are corrupted with
nonlinear noise. Widely popular filtration techniques like Fast Fourier Transform
(FFT), and Moving Average (MA) filters are not appropriate for filtering these types
of data [28]. We found out that the Empirical Mode Decomposition (EMD) with FFT
based algorithm was an effective method to filter non stationary data corrupted with
nonlinear noise. EMD filter works on the principle of splitting the given signal into
various individual components in time domain called intrinsic mode functions (IMF),
using in-situ generated cubic splines as shown in Figure 1.6 [ 28, 29]. Then the
frequency information of each IMF component is obtained using FFT algorithm.
Based on the prior knowledge of these experiments, we can then determine whether
that IMF should be removed or retained. IMFs deemed as noise components are
omitted and the signal is reconstructed using other IMFs. The biggest advantage of
the EMD based approach is that we do not lose any information about the actual data
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
besides giving valuable information on the noise, which could be further used to
improve the instrumentation wherever possible [28, 29]. The EMD based approach is
covered more elaborately in chapter 4 of this thesis work.
With improved data analysis using EMD based filtration; there is a potential to
expand the horizon of time domain dielectric spectroscopy to probe the nonlinearity
and dynamic heterogeneity in glassy polymers. Below is another study in time domain
responses of glassy polymers, which has also been benefited from using EMD based
filtration.
1.3 Structural recovery and aging in glassy polymers
On cooling amorphous glassy polymers from above to below the glass
transition point, the sample moves into a non equilibrium state and hence will tend to
evolve back towards the equilibrium state. The study of thermodynamic properties
(volume, enthalpy or entropy) as the material evolves towards equilibrium is called
structural recovery [7] and the changes associated with the viscoelastic properties
such as mechanical, optical or dielectric during this process is called physical aging
[7]. Kovacs was the first to comprehensively demonstrate the structural recovery
phenomenon using three classic experiments namely the intrinsic isotherm, the
memory effect and the asymmetry of approach [7].
1.3.1 Intrinsic isotherm
Figure 1.7(a) is the schematic of the intrinsic isotherm experiment. When a
glassy material is cooled from above the glass transition temperature (Tg) to a certain
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
temperature Ta below the glass transition temperature and held there isothermally, the
thermodynamic property (volume, enthalpy or entropy) will slowly recover towards
equilibrium. The further the departure from Tg , the longer it takes to reach the
equilibrium. This could be well understood in terms of molecular mobility and free
volume. As we go deeper into the glassy regime, the free volume, and hence the
molecular mobility, decreases. Figure 1.7(b) is the intrinsic isotherm plot of PVAc,
performed by Kovacs in his classic structural recovery experiment, after quenching
from 40oC to different temperatures (Ta) [7].
1.3.2 Memory effect
Memory effect is a two step experiment as shown in Figure 1.8(a). Initially,
the material is cooled from above the glass transition temperature to Point 1 far below
Tg and partially aged for some time. After that, the material is then jumped to Point 2
(up jump), such that the departure to equilibrium is very close to zero and aged.
Instead of directly returning to equilibrium, the sample goes through a maximum
before reaching the equilibrium remembering the previous thermal history. It may be
noted that the farther the first jump (point 2) is, the bigger is the memory. The
memory effect is evidence that it needs more than one exponential function to capture
the spectrum of data. Figure 1.8(b) is the memory experiment on PVAc performed by
Kovacs [7].
1.3.3 Asymmetry of approach
Figure 1.9(a) is the schematic of the asymmetry of approach experiments. It
comprises of two experiments, namely, up jump and down jump experiments. In
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
down jump experiment, a jump in temperature is made from Point A closer to glassy
transition temperature, to Point B further away in the glassy regime. In up jump
experiments, a jump is made from Point C (well below Point B) to point B. It may be
noted that the magnitude of temperature jump for the up jump and down jump
experiments should be the same. On measuring the departure from equilibrium for
both cases, we can observe that the up jump and down jump responses are nonlinear.
The down jump experiment behaves like an auto retardation experiment such that
initially the recovery is faster. This is due to the higher mobility as a result of the
departure from equilibrium (δ) being greater than zero. In time, the recovery slows
down as the molecular mobility decreases. In the up jump experiments, the reverse
happens and it behaves like an auto catalytic experiment. Initially, the recovery is
slower as δ < 0. In time, the recovery accelerates due to the increase in molecular
mobility as δ approaches zero. Figure 1.9 (b) is the asymmetry of approach
experiment as performed by Kovacs on PVAc [7].
1.3.4 τ eff paradox
Kovacs’ initial work on the effective retardation time, which is the slope of
departure from equilibrium (δ) as a function of time plot for different magnitudes of
asymmetry of approach experiments, showed that the material doesn’t come to
equilibrium at the same time as the departure from equilibrium approaches zero. This
was considered a paradox because the material violates the fundamental law that the
equilibrium is path independent [7]. McKenna and his coworkers later reanalyzed
some of Kovacs’ original data, and concluded that the expansion gap and error in
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
measurements close to equilibrium could, to some extent, be the reason for the
paradox [30]. Simon and coworkers have shown in their volume recovery work, that
for a smaller δ, the paradox is resolved as the expansion gap vanishes [31]. Hence, the
expansion gap occurs only for large T-jumps, i.e., when the response is nonlinear.
1.3.5 Physical Aging
According to Struik’s protocol, the sample loading-unloading for the aging
experiment is performed in such a way that the loading-unloading time is one tenth of
the waiting time. This is done so that each loading-unloading step is independent of
the previous event [8]. As seen in the Figure 1.10, the loading-unloading time is
sequentially increased by a factor of 2. Figure 1.11 is the first demonstration of aging
experiment performed by Struik [2] on poly (vinyl chloride). He showed that the
creep curves at various aging times can be superimposed, similar to time temperature
superposition and is called time-aging time superposition [2].
1.4 Effect of plasticizers on glassy behavior
Plasticizers are small molecules like moisture or carbon dioxide, which cause
considerable changes in the properties of glassy polymers upon constant exposure
[32-34]. Understanding these behaviors can add to exploring new areas of application
as well as preventing unexpected material failures [35, 36]. Despite a considerable
amount of work on the effect of plasticizers on polymers in general, there are very
few studies in understanding the glassy behavior in terms of structural recovery and
aging [37-41]. In our group, there has been tremendous impetus in this aspect.
Previous work from our group has shown that concentration formed glasses using a
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
strongly polar plasticizer (H2O) and a weakly polar plasticizer (CO2) qualitatively
mimic temperature formed glasses but were quantitatively different [37-41]. The
concentration formed glasses had a larger departure from equilibrium than
temperature formed glasses. Further, our preliminary work on the structural recovery
of epoxy film subjected to CO2 plasticizer jumps, showed that the effective
retardation time for concentration formed glasses and temperature formed glasses
(subjected to same final condition) do not converge to the same point as equilibrium is
approached [see Figure 1.12]. This is similar to the τeff paradox observed by Kovacs.
This problem is discussed more elaborately in chapter 5.
This dissertation is organized as follows:
In chapter 1, the background of glassy phenomena, dielectric spectroscopy, its linear
and nonlinear behavior in time domain responses, correlation between frequency and
time domain responses, kinetic manifestation of glassy behaviors and impact of
plasticizers in aging and recovery responses are discussed.
In chapter 2, the experimental set up used for the time domain dielectric
spectrometer and the pressure vessel set up for aging recovery experiments are
discussed. Chapters 3, 4 and 5 are elaborated versions of publications in the peer
reviewed journals.
In chapter 3, we present an investigation of the dielectric behavior of
poly(vinyl acetate) (PVAc) using a two step pulse-probe technique. Time domain
dielectric experiments were performed in the vicinity of the glass transition
temperature. After establishing the linear response function in single step
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
experiments, two types of pulse-probe experiments were performed. In one, the time
duration t1 of the first step in the probe was varied. In the second case, the magnitude
of the field E1 applied to the sample for the first step was varied. We observe the
memory effect and the responses were analyzed in the context of a linear Boltzmann
rule. Evidence of deviations from linear superposition at the highest electric fields are
also presented.
Noise in the dielectric experimental data limited the application of the time
domain dilectric set up built at Texas Tech. In chapter 4, we propose an algorithm
for effective filtering of noise using an EMD based FFT approach for applicatons
in polymer physics. The advantages of the proposed approach are: (i) it uses the
precise frequency information provided by the FFT and therefore efficiently filters
a wide variety of noise and, (ii) the EMD approach can effectively obtain IMFs
from both non-stationary as well as nonlinear experimental data. The utility of the
proposed approach is illustrated using an analytical model and also through two
typical laboratory experiments, namely, the dilectric experiments and structural
recovery experiments in polymer physics, wherein the material response is nonstationary; standard filtering approaches are often inappropriate in such cases.
By taking advantage of the EMD based filtering technique, in chapter 5, we
investigate the structural recovery and physical aging of an epoxy film subjected to
carbon dioxide pressure jumps and compare the results with temperature jump
experiments, such that the final conditions are identical. This a continuation of work
done previously in our group where we have shown using strong and weakly polar
plasticizers, that they qualitatively mimic the behaviors of temperature jumps but
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
quantitatively they are different [8-10, 13] . In chapter 6, we summarize the
conclusion of this work and the future scope is discussed in chapter 7.
1.5 References
1.
Plazek DJ, Ngai KL. In: Mark JE, editor. Physical properties of polymers
handbook. NY: American Institute of Physics; 1996 (chapter 12).
2.
McKenna GB. In: Booth C, Price C, editors. Comprehensive Polymer Science,
Polymer Properties, Vol.2, Oxford: Pergamon Press, 1989 (chapter 2).
3.
McKenna GB, Simon SL. In: Cheng SZD, editor. Handbook of Thermal
Analysis and Calorimetry, Applications to Polymers and Plastics, Vol.3,
Elsevier Science, 2002.
4.
Kovacs AJ. Transition vitreuse dans les polymères amorphes. Etude
phénoménologique. Adv Polym Sci 1964; 3:394.
5.
Kovacs AJ. La contraction isotherme du volume des polymères amorphes.
J.Polym Sci. 1958; 30:131.
6.
Chang SS. Thermodynamic properties and glass transition of polystyrene.
J.Polym Sci, Poly. Sympo 1984; 71:59.
7.
Kovacs AJ. Transition vitreuse dans les polymères amorphes. Etude
phénoménologique. Fortschr. Hochpolym.-Forsch. 1963; 3:394.
8.
Struik LCE. Physical aging in polymer and other amorphous materials.
Elsevier: Amsterdam, 1978
9.
McKenna GB. On the physics required for the prediction of long term
performance of polymers and their composites. J. Res. NIST 1994; 99:169.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
10.
Alcoutlabi M, McKenna GB, Simon SL. Analysis of the development of
isotropic residual stresses in a bismaleimide/sprio orthocarbonate
thermosetting resin for composite materials. J Appl Polym Sci 2003; 88:227
11.
Kremer F, Schonhals A. Broadband Dielectric Spectroscopy. 1st ed. New
York: Springer-Verlog; 2003.
12.
Mopsik FI. Precision Time-Domain Dielectric Spectrometer. Rev. Sci. Inst.
1984; 55: 79.
13.
Smith JW. Electric dipole moments. London: Butterworths scientific; 1955.
14.
www.awnsers.com
15.
Watanabe H. Dielectric relaxation of type -A polymers in melts and solutions.
Macromol. Rapid Commun. 2001; 22:127.
16.
Mashimo S, Nozaki R, Yagihara S, Takeishi S. Dielectric relaxation of poly
(vinyl acetate). Journal of Chemical Physics. 1982; 77:6259.
17.
Rendell RW, Ngai KL, Mashimo S. Coupling model interpretation of
dielectric relaxation of poly (vinyl acetate) near Tg. Journal of Chemical
Physics. 1987; 87: 2359.
18.
Richert R, Wagner H. The dielectric modulus: relaxation versus retardation.
Solid State Ionics 1998; 105:167.
19.
Mopsik FI. The transformation of time-domain relaxation data into the
frequency domain. IEEE Trans. Elec. Insul 1985; 20:957.
20.
Ferry JD. Viscoelastic Properties of Polymers. 3rd ed. New York: John Wiley
and Sons; 1980.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
21.
Serghei A, Huth H, Schick C, Kremer F. Glass dynamics in thin polymer
layers having a free upper interface. Macromolecules 2008; 41:3636.
22.
Richert R. Homogeneous dispersion of dielectric responses in a simple glass. J
Non Cryst Sol 1994; 209.
23.
Bohmer R, Schiener B, Hemberger J, Chamberlin RV. Pulsed dielectric
spectroscopy of supercooled liquids. Z.Phys. B. 1995; 99:91-99.
24.
Duwuri K, Richert R. Dielectric hole burning in the high frequency wing of
supercooled glycerol. J Chem. Phys 2003, 118: 1356.
25.
Shi X, McKenna GB. Mechanical hole-burning spectroscopy. Demonstration
of hole-burning in the terminal relaxation regime. Phys. Rev. B 2006, 73:
014203-1.
26.
Qin Q, Shi X, McKenna GB. Mechanical holeburning spectroscopy in a SIS
tri-block copolymer. J. Polym. Sci Part B Polym. Phys 2007; 46:3277.
27.
Kollengodu-Subramanian S, McKenna GB. A dielectric study of poly (vinyl
acetate) using a pulse probe technique. Journal of Thermal analysis and
calorimetry 2010; 102:477.
28.
Kollengodu-Subramanian S, Srinivasan B, Rengaswamy R, Zhao J, McKenna
GB. Application of empirical mode decomposition in the field of polymer
physics. J. Polym Sci. Part B Polym phys, 2011; 49:277.
29.
Huang N.E.; et al. The empirical mode decomposition and the Hilbert
spectrum for nonlinear and non-stationary time series analysis. Proceedings of
Royal society of London 1998; 454:903.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
30.
McKenna GB, Vangel MG, Rukhin AL, Leigh SD, Lotz B, Straupe C. The τEffective paradox revisited: An extended analysis of Kovacs volume recovery
data on poly (vinyl acetate). Polymer 1999; 40:5183.
31.
Kolla S, Simon SL. Tau-effective paradox: New measurements towards a
resolution. Polymer 2005, 46:733.
32.
Knauss WG, Kenner VH. On the hygrothermomechanical characterization of
polyvinyl acetate. J. Appl. Phys. 1980; 51:5531.
33.
Wang WCh, Kramer EJ, Sachse WH. Effect of high pressure CO2 on the glass
transition temperature and mechanical properties of polystyrene. Journal of
polymer science Part B: Polymer Physics 1982; 20:1371.
34.
Chiou JS, Barlow JW, Paul DR. Plasticization of glassy polymers by CO2. J.
Appl. Polym. Sci. 1985, 30, 2633-2642.
35.
Van der Vegt NFA, Briels WJ, Wessling M, Strathman H. The sorption
induced glass transition in amorphous glassy polymers. J Chem Phys 1999;
110:11061.
36.
Cotugno S, Larobina D, Mensitieri G, Musto P, Ragotsa G. A novel
spectroscopic approach to investigate transport process in polymers: The case
of water-epoxy system. Polymer 2001; 42:6431.
37.
Alcoutlabi M, Vangosa Briatico F, McKenna GB. Effect of chemical activity
jumps on the viscoelastic behavior of an epoxy resin: physical aging response
in carbon dioxide pressure jumps. Journal of polymer science Part B: Polymer
Physics 2002, 40:2050.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
38.
Zheng Y, McKenna GB. Structural recovery in a model Epoxy: Comparison
of responses after temperature and humidity jumps. Macromolecules 2003,
36:2387.
39.
Zheng Y, Priestley RD, McKenna GB. Physical aging of an epoxy subsequent
to relative humidity jumps through the glass concentration. Journal of
polymer science Part B: Polymer Physics 2004; 42: 2107.
40.
Alcoutlabi M, Banda L, McKenna, GB. A comparison of concentrationglasses and temperature-hyperquenched glasses: CO2 formed versus
temperature formed glass. Polymers. 2004; 45:5629.
41.
Alcoutlabi M, Banda L, Kollengodu-Subramanian S, Zhao J, McKenna GB.
Environmental effects on the structural recovery responses of an epoxy resin
after carbon dioxide pressure-jumps: Intrinsic isopiestics, asymmetry of
approach and memory effect. Macromolecules 2010 (Under Review).
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.1: Specific volume vs Temperature plot for different cooling rates (q). Here,
q1> q2>q3
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.2: Classifications of dipoles in amorphous polymers [15]
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.3: Schematic representation of two different sources of non-exponential
correlation decays [24].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.4: Schematic of the protocol for dielectric hole burning experiment [23].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.5: Time dependent dielectric permittivity of propylene carbonate at 157.4K
[23].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.6: Empirical mode decomposition schematic to filter noise in time domain
responses [29]
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
δ x 1000
Figure 1.7 (a): Schematic of intrinsic isotherm experiment
5.0
o
4.5
T0 = 40 C
4.0
o
19.8 C
3.5
3.0
22.4
24.9
2.5
27.5
2.0
1.5
30
1.0
32.5
0.5
35
0.0
-3
-2
-1
0
1
10
10
10
10
10
2
10
t-ti (h)
Figure 1.7(b): Intrinsic isotherm experiment performed by Kovacs on PVAc digitized
by Zheng and McKenna [38]
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.8(a): Schematic of memory effect experiment
2.0
o
T 0=40 C
(1)
δ x 1000
1.5
1.0
(2)
(3)
(4)
0.5
0.0
-2
10
-1
10
10
0
1
10
2
10
10
3
t-ti (h)
Figure 1.8(b): Memory effect experiment of PVAc performed by Kovacs digitized by
Zheng and McKenna [38].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.9(a): Schematic for asymmetry of approach experiment
1.5
o
T0=40 C
1.0
δ x 1000
0.5
0.0
-0.5
o
T a=35 C
-1.0
-1.5
o
T0=30 C
-2.0
-2.5 -3
10
-2
10
-1
10
10
t-ti (h)
0
1
10
2
10
Figure 1.9(b): Asymmetry of approach experiment of PVAc performed by Kovacs
digitized by Zheng and McKenna [38].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.10: Schematic of Struik’s protocol
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 1.11: Creep compliance of poly(vinyl chloride) quenched from above to
below Tg at different aging times after quenching [8].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
-1
T-jump 85-62C at 0MPa
PCO2-jump 3.9 to 0MPa at 62C
-2
-log(τ)/s
-3
-4
-5
-6
-1
0
1
2
3
4
5
6
δ*1000
Figure 1.12: Effective retardation time plot for temperature and concentration formed
glasses to the same final condition [41].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
CHAPTER 2
EXPERIMENTAL SYSTEM
2.1 Time domain dielectric spectrometer
The dielectric responses of glassy polymers were studied using the time
domain dielectric spectrometer built in the Polymer and Condensed Material lab at
Texas Tech University. This setup was built as a part of this thesis work. The general
idea for this experimental system was based on the earlier works pioneered by Mopsik
[1] and Bohmer and his coworkers [2]. The working principle of the dielectric time
domain spectrometer is based on the ability to measure the capacitance of the sample
film as a function of time. It may be noted that when a polymer film is placed between
the two well polished metal plates, it acts as a capacitor. Using the capacitance of the
sample, we can calculate the dielectric compliance of the material as a function of
time.
The experimental set up comprises of a high voltage supply source, sample
setup and electrometer interfaced with PC using a DAQ board and controlled using a
LabView program [3]. The schematic of the system is shown in Figure 2.1 Trek
model 610 E is used as the high voltage supply source. The Trek system can be used
in two voltage ranges, namely, 0 to 1000V range and 0 to 10000V range. For the
current work, we have used the low voltage range as it also gives a better resolution.
Further, very high voltage on thin films also caused dielectric breakdown of the
material. Keithley 6514 model is used as the electrometer to measure voltage or
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
current response of the material under investigation. In this work, the electrometer is
used to measure the voltage response from the sample setup. The electrometer can
measure a maximum output voltage of 200V. The electrometer operates in 3 ranges
namely 0 to 2V, 0 to 20V and 0 to 200V range. The desired range is selected based on
the response of the output voltage. This type of measurement leads to resolution
problems which are addressed in chapter 4. A surge protector is used to prevent any
damage to the electrometer if the voltage exceeds 200V range. The measured output
voltage is then stored in the PC using an NI instrument DAQ board of 12 bit
resolution.
The schematic of the sample set is shown in Figure 2.2. It comprises of the
sample and integrating capacitor. The sample capacitor (SC) comprises of two well
polished flat stainless steel plates with thin polymer film placed between them. The
thickness of the SS steel plate is about 0.5 mm. The diameter of the lower and upper
plate is 3 cm and 2 cm respectively. The plates are held tight using a spring setup
shown in Figure 2. Mylar capacitor of capacitance 2.2 nF purchased from Digi-Key
was used as the integrating capacitor (IC). As shown in the schematic, the sample
capacitor and integrating capacitor are connected in series, such that one end of the IC
is connected to the SC and the other end is grounded. The high voltage is applied on
the SC and the response is measured between the junction of SC-IC and ground using
the electrometer. The entire sample set up is placed in an enclosed aluminum casing
and the temperature is applied using a cone heater and controlled to a range of ±0.1oC.
The total set up is placed in a temperature controlled box made from polycarbonate.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
The time domain dielectric spectrometer built at Texas Tech University can operate in
the temperature range of 25oC to 80oC.
The capacitance of the sample C(t) is measured using the equation 2.1.
×
= (2.1)
Where, Vin is the applied voltage from the high voltage supply, Vo is the
output voltage measured by the electrometer, and Ci is the capacitance of the
integrating capacitor.
The dielectric compliance ε(t) of the sample is then measured using equation
2.2.
=
×
×
=
∆
× ∆
(2.2)
Where, d is the thickness of the sample, A is the surface area of the plate, εo is
the dielectric permittivity in vacuum, ∆P is the polarization and ∆E is the applied
electric field.
2.2 Experimental setup for studying the aging and structural recovery responses
of glassy polymers subjected to plasticizer environment
The physical aging and structural recovery experiments after pressure jumps (P Jump)
were performed using the experimental setup built at Texas Tech University shown in
Figure 2.3 [4]. The set up comprises of a pressure vessel (Figure 2.4) with a capacity
to handle up to 8 MPa of pressure, the sample set up comprising of the sample holder,
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
linear variable differential transformer (LVDT) to measure the change in length (HR
100, Lucas Schaevitz inc;) and a motor to apply and retract the load, pressure sensors
(Omega Electrovalves, SV128) to control the pressurization and depressurization
rates, an oil bath filled with silicone oil and a heating coil to heat the pressure vessel
to the desired temperature with a stability of ±0.1oC. The entire system is controlled
using a DAQ board interfaced with the computer using a LabView program. It may
be noted that for the P jump experiments, the set up was first subjected to vacuum for
20 minutes, followed by pressurization to about 4MPa at a pressurization rate of
0.0016 MPa/s, maintained at 4 MPa for about an hour, and then depressurized to 0
MPa at the depressurization rate of 0.0016 MPa/s.
The temperature jump experiments were performed in an oven using the same
experimental setup instead of a pressure vessel. The reason for using the oven instead
of a pressure vessel for temperature jump (T jump) experiments is because of the very
low cooling rate in the pressure vessel. A complete detail on the experimental set up is
given in the reference [4].
2.3 Linear Variable Differential Transformer calibration
The LVDT used for the aging and recovery experiments for the current work
was performed using a higher resolution mode of the signal conditioner such that
maximum length change measured is about 1 mm. Figure 2.5 is the calibration plot
for the LVDT, where length change (the distance the core is moved inside the LVDT)
is plotted against the voltage response. From the calibration plot it was that calculated
that 0.0985mm length change corresponds to a 1V response.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
2.4 References
1. Mopsik FI. Precision Time-Domain Dielectric Spectrometer. Rev. Sci. Inst.
1984; 55: 79.
2. Schiener B, Bohmer R, Loidl A, Chamberlin RV. Non resonant spectral hole
burning in the slow dielectric response of super cooled liquids. Science. 1996;
274:752.
3. Kollengodu-Subramnain, S, McKenna, G.B. A dielectric study of poly(vinyl
acetate) using pulse probe technique. Journal of Thermal analysis and
calorimetry 2010, 102, 477.
4. Alcoutlabi, M.; Banda, L.; Kollengodu-Subramnain, S.; Zhao, J.; McKenna,
G.B. Environmental effects on the structural recovery responses of an epoxy
resin after carbon dioxide pressure-jumps: Intrinsic isopiestics, Asymmetry of
approach and memory effect. Macromolecules (2010 : Under review)
33
Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 2.1: Schematic of time domain dielectric spectrometer built at Texas Tech
taken from reference 3
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 2.2: Sample setup schematic for the time domain dielectric
spectrometer [3]
35
Texas Tech University, Shankar Kollengodu Subramanian, May 2011
A
D
B1
E
F1
G
H
Sample
C
F2
LVDT
CO2 vent
Weight
B2
Lift
Motor
I
Pressure vessel
CO2
Supply
Mixer
Oil bath
A
Pressure Controller
Temperature Controller
Signal Conditioner
Motor Controller
PC
A/D
A/D Board
Board
Air Drive
A) Regulator B1) Inlet automatic valve B2) Outlet automatic valve C) High pressure pump D) Filter E) Safety
valve F1) Inlet needle valve F2) Outlet needle valve G) Pressure sensor H) One way valve I) Three-way valve
Figure 2.3: The creep apparatus that was built in our laboratory to perform
experiments under different CO2-pressure and temperature conditions [4].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 2.4: The pressure vessel used to perform the experiments under CO2 pressure
[4].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 2.5: LVDT calibration plot where the displacement of the core is measured as
a function of voltage
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
CHAPTER 3
A DIELECTRIC STUDY OF POLY (VINYL ACETATE) USING A PULSEPROBE TECHNIQUE
3.1 Motivation
There is considerable literature available that describes our understanding of
the viscoelastic properties of polymers subjected to mechanical stresses or
deformations. What we refer to here as a pulse-probe technique is one method that is
commonly used to study the time dependent behavior of materials in histories, e.g.,
temperature-jump or step-deformations, that exhibit fading memory responses. In the
linear case the behavior is well understood in the context of Boltzmann superposition
ideas. However, there is only limited work available that investigates the dielectric
response of materials within this same context
3.2 Introduction
Dielectric spectroscopy is normally performed in the frequency domain and in
the linear response regime [1-3]. It is used as a tool to characterize materials and
often the results of dielectric response in the linear regime are compared with
mechanical and rheological measurements. The pulse-probe technique is one method
used to study time dependent responses in temperature jumps [4] and nonlinear
mechanical or rheological measurements [5-8]. In the present work, we present results
from time domain dielectric spectroscopy experiments in which we explore the limits
of linearity in poly (vinyl acetate) (PVAc). In particular, we used a pulse-probe
method. The way in which the experiments are carried out is similar to the single and
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
double step strain or stress experiments commonly used in nonlinear viscoelasticity
investigations. [5- 8].
Dielectric properties such as dielectric compliance and modulus are analogous
to mechanical properties such as shear compliance and modulus [9]. The equivalents
to mechanical stress and strain in a dielectric measurement are the dielectric stress
(applied electric field) and the dielectric strain (polarization). The equation relating
the time dependent dielectric compliance (ε(t)) to the applied field (∆E) and the
polarization (∆P) is [9]
ε (t ) =
DielectricStrain
∆P (t )
⇔
∆Eε 0
Dielectricstress
(3.1)
where ε0 is the dielectric permittivity of vacuum. Importantly, equation 3.1 is valid in
ideal situations in which the field is applied instantaneously, i.e., a step-pulse
measurement and does not decay with time. Here, we found that the time decay of ∆E
is small enough that errors introduced by treating the data as ideal constant dielectric
stress experiments are negligible.
In the present work we use time domain dielectric spectroscopy, a method
pioneered by Mopsik [10-12] to investigate the dielectric responses of small molecule
liquids and polymers. The method has been extended by Richert and Wagner [13]
through the development of time domain modulus spectroscopy to investigate
dynamic heterogeneity in small molecule glass formers and polymers. These works
were confined to the linear domain and the responses were well explained using
classic descriptions such as the Kohlrausch-Williams-Watts (KWW) [14, 15] and
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
modified KWW [16] functions to describe the time-domain responses and the VogelFulcher-Tammann [17-19] expression to describe the temperature dependence of the
relaxation or retardation times near to the glass transition temperature.
As noted above, linear dielectric spectroscopy is generally used in the
frequency domain rather than the time domain. Frequency domain measurements,
however, have also been used to study the nonlinear dielectric response of materials.
For example, Furukawa et al [20] studied the nonlinear dielectric response of PVAc
by obtaining the first and third harmonics in the frequency domain in samples
subjected to increasing electric field. This is similar to attempts by Davis and
Macosko [21] to use modified Boltzmann superposition [22, 23] to study the
nonlinear viscoelastic behavior of polymers subjected to large mechanical
deformations. A similar body of work has recently appeared from Wilhelm’s group in
which Fourier Transform Rheology is used to characterize the higher harmonics of the
extremely nonlinear rheolgical response of polymers and other complex fluids [24,
25]. In addition, mixed mode experiments have been carried out in which large
amplitude sine waves are followed by single step small probes or time domain
measurements to examine the nonlinear response of glass-forming liquids and
polymer melts and solutions. For example, Schiener et al [26] established the
dielectric hole burning method using experiments on supercooled propylene
carbonate, and Shi and McKenna [27] developed a mechanical hole burning
experiments using a polyethylene melt and a polystyrene solution as example systems.
Richert and coworkers [28, 29] have studied the nonlinear dielectric behavior of
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
supercooled liquids by subjecting them to several cycles of sinusoidal waves of high
amplitude followed by a wait time and a small step (time domain) probe.
Prior to the present work, there have been only limited reports in the literature
of using the pulse-probe technique to study the dielectric behavior of materials.
Bohmer et al [30] used the pulse-probe technique to study the dielectric responses of
supercooled liquids. They observed the memory effect similar to that observed in
thermal and mechanical measurements [4, 5]. An interesting early work from the
1890’s by Hopkinson reports a memory effect in simple glasses subjected to reversing
polarity is cited by Whitehead in a 1927 treatise [31]. There, the Boltzmann
superposition principle [22] was used to predict the same.
In the present investigation, we test the limits of Boltzmann superposition [22]
for the dielectric response of poly(vinyl acetate) (PVAc) in pulse-probe experiments.
For this, we have performed a single step time domain response and two step (pulseprobe) measurements having different amplitudes and durations in the vicinity of the
glass transition. This is the first of a series of work to be later extended to study the
nonlinear dielectric time domain response using the modified Boltzmann
superposition principle [21, 23] and the pulse-probe method as developed here.
3.3 Sample Preparation
Dielectric experiments were performed using a time domain dielectric
spectrometer built at Texas Tech University (Figure 2.1). The working of the
instrument is well described in the chapter 2.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Poly(vinyl acetate) of molecular weight 157,000 g mol-1 purchased from Scientific
Polymer Products, Inc. was used for the experiments. The glass transition temperature
Tg of this PVAc as previously measured by our group using DSC at a cooling rate of
10 K min-1 was reported to be 303.6 K [32]. The sample was made by placing the
pellets between thin brass sheets placed between two thick brass plates and then
pressed at 333 K in a platen press. After that, the sample is cut into a circular section
to fit the electrode plate, then held tight using a spring support and annealed at 338 K
(above the Tg) before performing the experiments. The loaded spring set up helps to
establish and maintain good contact between electrodes and polymer. The figure for
the sample support set up is given in Figure 2.2. PVAc films of thickness 185 ± 10
microns were used for the single step isothermal measurements used to examine the
time-temperature superposition behavior of the PVAc. The same film thickness was
used for two step pulse-probe experiments in which the first step duration t1 was
varied. Films of 112 ± 8 micron thickness were used for the experiments in which the
electric field E1 was varied. The sample was cut to size and placed into the dielectric
cell for measurement.
3.4 Methods of analysis
The dynamic response of many liquids can be described by the stretched
exponential or so-called KWW function shown in equation 3.2 [14-16, 33]. However,
because the shape of the dielectric compliance response shows a sigmoidal-like shape,
it can be necessary to apply a modified KWW function [16] as shown in equation 3.3
to capture the full response. Furthermore, when there is a long time process such as
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
the viscosity in viscoelastic creep measurements [34], or the conductivity in dielectric
creep it may be necessary to add an additional term as in equation 3.4.
ε (t ) = ε 1 * e
t
 
τ 
β
ε (t ) = ε 1 + ε 2 * (1 − e
ε (t ) = ε 1 + ε 2 * (1 − e
(3.2)
t
− 
τ 
β
t
− 
τ 
β
)
(3.3)
)+
t
(3.4)
τc
Where ε1, ε2, τ, τc, and β are fit parameters. Here τ is the retardation time and τc is the
conductivity term. We find that equation 3.3 describes the recoverable part of the
dielectric compliance and the full response of ε(t) is well described by equation 3.4.
This separation of equation 4 into a recoverable term and a conductivity term is used
subsequently.
To test the validity of Boltzmann superposition for the two step dielectric
response, we used the Boltzmann equation as rearranged by Riande et al [35] and
implemented numerically in our group [32]:
∆P (t ) = ε g E (t ) + ∫ E (t − t ' )
dε ( t )
dt '
dt '
(3.5)
Hence, ∆P is the predicted dielectric strain, E(t) is the applied dielectric stress for the
two step test in the experiment analyzed, εg is the dielectric compliance at zero time
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
and ε(t) is the modified KWW (equation 3) fit to the single step, recoverable, linear
dielectric compliance response.
3.5.0 Results
3.5.1 Single step response
Isothermal measurements of the dielectric response (compliance) for the
PVAc were performed in a temperature range of 303 K to 333 K and the data are
shown in Figure 3.1(a). The solid lines in Figure 3.1(a) are the fits to the data at each
temperature using the modified KWW function with conductivity contribution given
in equation 4. Similar studies on PVAc have been performed by various groups [1, 13,
36]. The reason for using equation 4 is to separate the effect of dc conductivity from
the dielectric compliance response and also to examine the time-temperature
superposability of the dielectric response separate from the dc conductivity. The fit
parameters for equation 4 for the 308 K reference temperature are given in Table 3.1.
The β parameter obtained at the reference temperature was kept fixed for fitting the
data at the remaining temperatures. We observed an increase in the ε1 parameter and
decrease in ε2 parameter with increasing temperature for the above used function as
shown in Figure 3.2. Both the conductivity term and retardation time decreased with
increasing temperature.
Figure 3.1(a) shows that, with increasing temperature, the curves shift to
shorter times. We attribute the steep rise after the secondary plateau to the dc
conductivity [9, 13]. Richert and Wagner used two KWW functions to capture the
entire spectrum of data in their work on dielectric modulus [13]. As noted above, we
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
have used a term similar to the viscosity term in linear viscoelasticity but for the
conductivity added to a modified KWW function to fit our data (see equation 4).
Figure 3.1(b) is the master curve for the data of Figure 3.1(a) with 308 K as the
reference temperature. The data at the longer times do not superimpose and we
hypothesize this to be due to the domination of dc conductivity over dielectric
response.
The retardation time τ and the conductivity term τc obtained from equation 4
for the isothermal measurements of PVAc shown in Figure 3.1(a) are plotted as a
function of inverse temperature in Figure 3.3. The data are fitted using the VFT
function given in equation 3.6 [37]. The fit parameters are given in Table 3.2. The fit
parameters for the dielectric retardation time and the dc conductivity term are
different which explains the spread of the dielectric response at longer times in Figure
3.2(b).
log τ = − A +
B
(T − T o )
(3.6)
Where A, B and To are fit parameters.
To confirm the above hypothesis, the recoverable dielectric compliance
(which is the difference between the dielectric compliance and the conductivity term)
was estimated using equation 3.7.


t
 ε (t ) − − ε 1 
τc

ε R (t ) = 
ε2
(3.7)
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Where εR(t) is the recoverable dielectric compliance
For each fit of equation 3.4 to the data of Figure 3.1(a) εR(t) was determined
and the recoverable dielectric compliance response functions are plotted in Figure
3.4(a). In fact, these are described by the modified KWW function presented in
equation 3.3. Figure 3.4(b) gives the master curve for the recoverable compliance
data with 308 K as the reference temperature. We also observed softening like
behavior of the secondary plateau with increasing temperature. This can be explained
by the increase in ε1 and decrease in ε2 parameters of equation 4 with temperature as
shown in Figure 3.2. In equation 3.4 the parameter ε1 is related to the glassy (short
time) response and ε2 is related to the transition towards the long time plateau
response. This kind of softening behavior with increasing temperature is also
observed for dielectric responses in the frequency domain in the literature [9]. By
shifting the curves vertically for the higher temperature data in addition to the
horizontal shift, we obtain a reasonable time temperature superposition. In Figure 3.5
we compare the horizontal shift factor data for the dielectric recoverable compliance
with the Plazek’s recoverable creep compliance data [38] and Richert’s (digitized)
dielectric modulus [13] data. The results are in good agreement with Richert’s
dielectric data. Somewhat surprisingly, the data of Figure 3.5 show that the dielectric
response seems to follow Plazek’s terminal dispersion data rather than the softening
dispersion data. A similar sort of behavior has also been observed by Zorn et al. in
their work on polybutadienes [12].
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
3.5.2 Time-frequency conversion
It is possible to calculate the frequency domain response namely the dielectric
loss compliance (ε”() and the dielectric storage compliance (ε’() from time
domain response and vice versa using Fast Fourier transformation. The relationship
between the dielectric compliance and its corresponding frequency components is
given by the equation 3.8.
= ′ − "
(3.8)
Figure 3.6 is the dielectric storage compliance plot calculated from the MKWW fit
(Equation 4) of isothermal measurements of dielectric compliance shown in Figure
3.2(a). Similar to what is observed in the compliance data, the secondary plateau
shows softening like behavior with increasing temperature. Figure 3.7 is the dielectric
loss compliance for the data given in Figure 3.1(a).
Dielectric compliance in general is a sum of contribution from dielectric relaxation,
electrode polarization and conductivity [9]. What is the contribution of electrode
polarization and conductivity to memory effect in the dielectric response is a
worthwhile question to ask here. In the dielectric loss compliance plot, the features of
dielectric relaxation, electrode polarization and conductivity appear in that order as
we move from a higher frequency to a lower frequency [9]. It can be clearly seen, for
the above temperature data, the conductivity as well as electrode polarization effect in
the given range is very small and we observe contribution mainly due to segmental
relaxation. Also, it might be worthwhile to note that the range of temperature
investigated is sufficient to the purpose of the study, which is to examine the
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Boltzmann superposition and the preliminary results of breakdown of Boltzmann
superposition is going through the glass transition range.
3.5.3 Two step (pulse-probe) response
Prior to performing a two step experiment, we checked the effects on the
dielectric compliance due to the small drift in the dielectric stress (∆E in equation 1)
as a function of time. We used the Boltzmann equation given in equation 5 to obtain
the true compliance data for the single step as implemented by our group previously
[32]. We observed that there is no significant difference between the apparent and true
compliance for the above material indicating that the effect of the time varying
dielectric stress is very weak. The comparison is shown in Figure 3.8. Hence, we
have used the data as obtained for the experiments without the additional use of the
full Boltzmann equation.
The dielectric compliance response of PVAc in the variable duration first step,
two step pulse-probe experiments is shown in Figure 3.9. The experiments were
performed at 302.8 K. The electric field-jump is made from 10.8*105 Vm-1 to 5.4*105
Vm-1 at a varying jump times t1 of 0.25s, 0.5s and 1s. In all the cases, we see a typical
non-monotonic memory response similar to what is observed in thermal [4] and
mechanical responses [5] to step-wise histories. In Figure 3.10, two step pulse- probe
responses after decreasing the field from E1 to 44.6*105 Vm-1 for different values of
E1 are shown. This is a ‘down-jump’ experiment. As the down-jump magnitude
increases the memory effect lasts for longer times.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
When the response is linear, we expect Boltzmann [22] superposition to hold. To
test this for the two step pulse-probe experiments, we applied equation 5, using the
single step response function determined above (recoverable part) and the
experimentally applied fields for each step, to calculate the expected second step
response. Equation 5 was solved using a MATLAB program based on a numerical
integration method used previously by our group [32]. Comparison of the dielectric
strain response from the experiment for the 1s jump depicted in Figure 3.9 with the
Boltzmann prediction is shown in Figure 3.11. The results are in good agreement
confirming that the dielectric response of the two step pulse-probe is in the linear
regime for the above experimental conditions.
On the other hand, for the largest down-jump investigated, that from 89.2 *105
Vm-1 to 44.6 *105 Vm-1, the outcome is different although the single step response at
this applied field is in good agreement with Boltzmann superposition (See Figure
3.12). As shown in Figure 3.13 the Boltzmann superposition prediction deviates from
the observed second step response showing evidence of nonlinear behavior. For
comparison, Richert and Weinstein observed nonlinear behaviors in small molecule
liquids at a high applied electric field [27, 28] in frequency domain spectroscopy.
3.6 Discussion
Time domain dielectric spectroscopy has been used to characterize the linear
response of PVAc. We found that the response was well fitted using equation 4,
which is basically the modified KWW function (equation 3) plus an additional term
for conductivity. This is equivalent to the use of a viscosity tem in creep compliance
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
determinations in linear viscoelasticity [34]. The conductivity term and the retardation
term show different temperature dependences which explains the spread of data at
longer times. Richert and Wagner [13] also observed this kind of a behavior in their
dielectric relaxation work on PVAc. By subtracting the conductivity term, we obtain
the dielectric recoverable compliance of the material which is analogous to the
recoverable creep compliance in linear viscoelasticity [34, 38].
The recoverable dielectric compliance plotted as a function of time in Figure 3.4a
clearly shows a softening-like behavior with increasing temperature similar to that for
frequency response data reported in the literature [9]. This can be attributed to the
changing in ε1 and ε2 parameters which are related to glassy and long time plateau
responses respectively. The recoverable dielectric compliance is found to follow timetemperature superposition with ε1 and ε2 changing with temperature. To the best of
our knowledge, other workers have not explicitly written the dielectric compliance in
an equivalent fashion (directly analogous to the creep compliance in mechanics;
equation 4). The temperature shift factors for the dielectric recovery are found to be
the same as those determined by Richert and Wagner [13] for dielectric modulus.
Surprisingly, the dielectric shift factors seem to follow the terminal shift factors
determined by Plazek [38] in shear experiments and differ from the mechanical
segmental shift factors. This is a surprise because the type B dipole in PVAc should
reflect molecular motions that are local rather than long chain motions related to the
terminal relaxations.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Two step pulse-probe experiments were performed to test the limits of linearity in
PVAc. We observed the classical memory response similar to that observed by others
for dielectric behavior [30, 31] and in volume recovery after temperature jumps [4]
and in two step mechanical experiments [5]. The experiments demonstrate that
memory depends on both the time duration of the first step and the magnitude of the
field jump. We observed that for small jumps, Boltzmann superposition is valid. For
larger jumps we find deviations from linearity as observed by the over prediction of
the memory response by Boltzmann superposition in spite of the fact that the single
step response at the same field magnitudes were in the linear regime as evidenced in
Figure 3.13. This suggests that the two step pulse-probe method may be a sensitive
approach in dielectric spectroscopy to delineate the linear to nonlinear transition in
behavior.
3.7 Conclusion
Time domain dielectric spectroscopic measurements were performed on a
PVAc polymer near to its glass transition temperature. Time temperature
superposition of the response was not strictly valid due to the existence of different
shift factors for the recoverable portion of the dielectric compliance and for the
conductivity contribution to the response. The shift factors agree with those for
dielectric modulus measurements [13] and terminal relaxation response in mechanical
measurements [38] reported in the literature. A modified KWW function with an
additional term for the dc conductivity was able to capture the entire experimental
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
regime with the retardation time τ for the recoverable compliance term and the
conductivity contribution τc showing different temperature dependence.
A two step pulse-probe technique was used to study the dielectric behavior of
PVAc in the context of the Boltzmann superposition. We observe a memory effect
similar to those observed in mechanical [5] and thermal responses [4]. The memory
effect observed was in quantitative agreement with linear Boltzmann superposition for
small applied fields. Evidence of nonlinearity is observed when the polymer was
subjected to higher electric field. However, the further probing of nonlinear behavior
as well as dynamic heterogeneity was found to be difficult because of the noise in the
data. Dielectric data are time varying in nature corrupted with nonlinear noise. This
problem is addressed in the next chapter.
3.8 Appendix
3.8.1 Algorithm to obtain true compliance (Adapted from the thesis of Stephen
Hutcheson, Texas Tech)
% y=Strain, Z=time, n=stress
% l=compliance obtained by experiment
% SR=Compliance obtained by Boltzmann
%n=zeros(1000);
k=0;
%y=zeros(1000);%y(strain)
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
%z=zeros(1000);
%n=zeros(1000);
for x=1:10000
z(x)=k+0.001;
y(x)=dstrain(x);%MKWWFit
n(x)=dstress(x);%Curve fit
k=z(x);
l(x)=y(x)/n(x);
end
semilogx(z,n)
semilogx(z,y)
semilogx(z,l)
dTLab=0.001;
dTLoop=0.001;
phi_g=1.0;
D1=zeros(1,10000);
C1=zeros(1,10000);
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
numbers=zeros(1,10000);
SR=zeros(1,10000);
for x=1:9999
T_Lab=z(x);
integral_tot=0;
June=0;
%Our D(z(x))
D=phi_g;
x1=x;
if (x1>=1)
for x2=1:x1
T_loop=x2*dTLoop;
T_loop_1=(x2-1)*dTLoop;
mat_time=T_Lab-T_loop_1;
c=int16(mat_time*1000);
torque_local=n(c);
integral_local=torque_local*numbers(x2);
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
integral_tot=integral_tot+integral_local;
end
Zero=5.0;
while Zero>0.0001
torque_local=n(dTLab*1000);
June=integral_tot+torque_local*D;
fd=June-y(int16(T_Lab*1000));
Zero=abs(fd)*10^15;
D1=D;
D=D-fd/torque_local;
end
else
D1=0;
end
numbers(x)=D1;
if (x>1)
SR(x)=numbers(x)+SR(x-1);
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
else
SR(x)=numbers(x);
end
end
semilogx(z,l,'+',z,SR)
3.8.2 Boltzmann superposition algorithm to delineate linear and nonlinear
behaviors
%MKWW Single step compliance response
P1=2.8;
P2=6.25;
P3=81.64019;
P4=0.38034;
k=0;
% Call the dielectric stress data of two step response
% tress1, Stress 2
for x=1:50000
z(x)=k+0.001;
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
k=z(x);
if (z(x) < 1.001)
y(x) = stress1(:,1);
J(x)= P1 + P2*(1-exp(-(z(x)/P3)^P4));
elseif (z(x) > 1.549)
y(x) = Stress2(:,1);
J(x)= P1 + P2*(1-exp(-(z(x)/P3)^P4));
end
end
semilogx(z,J,'+')
for x=1:49999
m(x) = x/1000;
DJ(x)=J(x+1)-J(x);
end
semilogx(m, DJ)
q=0;
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
%torque=0;
%torque=zeros(40000);
for x=1:40000
q(x)=x/1000;
torque(x)=0;
for j=1:(x-1)
if (x-j>0)
torque(x)=torque(x) + y(x-j)*DJ(j);
end
end
torque(x)=torque(x)+ 2.85*y(x);
end
for x=1: 40000
r(x)=J(x)*y(x)
end
semilogx(z,r,'o',q,torque)
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
3.9 References
1. Serghei A, Huth H, Schick C, Kremer F. Glass dynamics in thin polymer
layers having a free upper interface. Macromolecules. 2008; 41:3636.
2. Kovacs AJ. Transition vitreuse dans les polymeres amorphes. etude
phenomenoloqique. Fortschr. Hochpolym.-Forsch. 1963; 3:394.
3. McKenna GB, Zapas LJ. Non linear viscoelastic behavior of poly (methyl
methacrylate) in torsion. J. Rheology. 1979; 23:151.
4. Zapas LJ, Craft T. Correlation of large longitudinal deformations with
different strain histories. Res. Nat. Bur. Stand. 1965; 69A:541.
5. McKenna GB. Viscoelasticity. In: Encyclopedia of Polymer Science and
Technology, John Wiley and Sons; 2002.
6. Schapery RA. On the characterization of non linear viscoelastic materials.
Polym. Eng.Sci.1969; 9:295.
7. Kremer F, Schonhals A. Broadband Dielectric Spectroscopy. 1st ed. New
York: Springer-Verlog; 2003.
8. Mopsik FI. Precision Time-Domain Dielectric Spectrometer. Rev. Sci. Inst.
1984; 55: 79.
9. Mopsik FI. The transformation of time-domain relaxation data into the
frequency domain. IEEE Trans. Elec. Insul. 1985; 20:957.
10. Zorn R, Mopsik FI, McKenna GB, Willner L, Richter D. Dynamics of
polybutadienes with different microstructure. 2. Dielectric response and
comparisons with rheologigal behavior. Journal of Chemical Physics. 1997;
107 (9):3645.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
11. Richert R, Wagner H. The dielectric modulus: relaxation versus retardation.
Solid State Ionics. 1998; 105:167.
12. Kohlrausch F. Ueber die elastische Nachwirkung bei der Torsion. Pogg Ann
Phys Chem. 1863; 119:337.
13. Williams G, Watts DC. Non-symmetrical dielectric relaxation behavior
arising from a simple empirical decay function. Trans Faraday Soc. 1970;
66:80.
14. Alcoutlabi M, Francesco Briatico-Vangosa, McKenna GB. Effect of chemical
activity jumps on the viscoelastic behavior of an epoxy resin: Physical aging
response in carbon dioxide pressure jumps. Journal of Polymer Science Part
B: Polymer Physics. 2002; 40:2050.
15. Vogel H. The law of relation between the viscosity of liquids and the
temperature. Phys Z. 1921; 22:645.
16. Fulcher GS. Analysis of recent measurements of the viscosity of glasses. J
Am Ceram Soc. 1923; 8:339-355.
17. Tammann G, Hesse W. The dependence of viscosity upon the temperature of
supercooled liquids. Z Anorg Allg Chem. 1926; 156:245.
18. Furukawa T, Matsumoto K. Nonlinear dielectric relaxation spectra of
polyvinylacetate. Japanese Journal of Applied Physics Part 1. 1992; 31, 840.
19. Davis WM, Macosko CW. Nonlinear dynamic mechanical moduli for
polycarbonate and PMMA. Journal of Rheology. 1978; 22: 53..
20. Boltzmann L. Zur theorie der elastischen nachwirkung. Akad. Wiss. Wien.
Mathem.-Naturwiss. Kl. 1874; 70:275.
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21. Findley WN, Lai JS, Onaran K. Creep and relaxation of nonlinear
viscoelastic materials with an introduction to linear viscoelasticity. New
York: North-Holland Publication; 1976.
22. Wilhelm M, Maring D, Spiess HW. Fourier transform rheology. Rheol. Acta.
1998; 37:399.
23. Wilhem M, Reinheimer P, Ortseifer M. High sensitivity Fourier transform
rheology. Rheol. Acta. 1999; 38: 349.
24. Schiener B, Bohmer R, Loidl A, Chamberlin RV. Non resonant spectral hole
burning in the slow dielectric response of super cooled liquids. Science.
1996; 274:752.
25. Shi X, Mckenna GB. Mechanical hole burning spectroscopy: Demonstration
of hole burning in the terminal relaxation regime. Physical Review B. 2006;
73:0142303-1.
26. Richert R, Weinstein S. Nonlinear dielectric response and thermodynamic
heterogeneity in liquids. Phys. Rev.Lett. 2006; 97:095703 -1.
27. Richert R, Weinstein S. Nonlinear features in the dielectric behavior of
propylene glycol. Physical Review B. 2007; 75:064302-1.
28. Bohmer R, Schiener B, Hemberger J, Chamberlin RV. Pulsed dielectric
spectroscopy of supercooled liquids. Z.Phys. B. 1995; 99:91.
29. Whitehead JB. Lectures on Dielectric Theory and Insulation. 1st ed. New
York: McGraw Hill Book Company; 1927.
30. O'Connell PA, Hutcheson SA, McKenna GB. Creep behavior of ultra thin
polymer films. J. Poly Sci Part B. Poly Phys. 2008; 46:1952.
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31. Struik LCE. Physical aging in amorphous polymers and other materials.
Amsterdam: Elsevier; 1978.
32. Ferry JD. Viscoelastic Properties of Polymers. 3rd ed. New York: John
Wiley and Sons; 1980.
33. Riande E, Diaz-Calleja R , Prolongo M, Masegosa R. and Salom C . Polymer
Viscoelasticity: Stress and Strain in Practice. New York: CRC Press, Marcel
Dekker; 2000
34. Schlosser E, Schonhals A. Dielectric relaxation during physical aging.
Polymer. 1991; 32:2135-2140.
35. Shelby JE. Introduction to glass science and technology. 2nd ed. Cambridge:
The Royal Society of Chemistry; 2005.
36. Plazek DJ. The temperature dependence of the viscoelastic behavior of
poly(vinyl acetate). Polymer Journal. 1980; 12: 43-53
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Table 3.1 Fit parameters for equation 4 at 308 K. Errors represent standard error of
estimate on the fit parameters.
ε1
ε2
Β
τ/s
τc /s
3.21± 0.0096
5.34± 0.017
0.5±0.005
1.43±0.03
164±5.25
Table 3.2 A, B and T0 from VFT fits for the retardation time and the dc conductivity
term vs. temperature obtained by fitting the data in Figure 2a to equation 4. Errors
represent standard error of estimate on the VFT fit parameters.
Time constants
A
B /K
T0/K
log τ
13.5 ± 0.95
914 ±87
241± 4.8
log τc
6.54±0.85
437± 110
258± 9.8
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.1 (a): (a) Isothermal measurement for PVAc with 308 K as reference
temperature
Figure 3.1 (b): Master curve for PVAc with 308 K as reference temperature
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.2: Equation 3.4 parameters ε1 and ε2 as a function of temperature
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.3: Dielectric retardation time τ and dc conductivity τc term plotted as a
function of inverse temperature. The solid lines represent the VFT fits
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.4 (a): Dielectric recoverable compliance for PVAc with 308 K as reference
temperature
Figure 3.4 (b): Master curve for PVAc with 308 K as reference temperature.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.5: Time-temperature shift factors as a function of temperature for PVAc.
Comparison of current dielectric results with literature reports for dielectric [13] and
mechanical [37] behaviors.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.6: Dielectric storage compliance response for PVAc
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.7: Dielectric loss compliance of PVAc
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.8: Comparison of apparent dielectric compliance with true dielectric
compliance for an applied electric field of 5.4*105Vm-1
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.9: Dielectric compliance response in two step pulse-probe experiments with
varying time duration t1 of the first step.
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
Figure 3.10: Dielectric compliance response in two step pulse-probe experiments
with varying electric field E1 of the first step.
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Figure 3.11: Dielectric strain response for the 1second jump of Figure 8. The solid
line represents the Boltzmann prediction
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Figure 3.12: Single step responses for applied electric field of 44.6*105Vm-1 and
89.2*105 Vm-1
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Figure 3.13: Two step dielectric compliance response (jump from 89.2*105 Vm-1 to
44.6*105 Vm-1) with linear Boltzmann prediction showing deviation from linear
behavior
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CHAPTER 4
APPLICATION OF EMPIRICAL MODE DECOMPOSITION IN THE FIELD
OF POLYMER PHYSICS
4.1 Motivation
Noisy data has always been a problem to the experimental community.
Effective removal of noise from data is important for better understanding and
interpretation of experimental results. Over the years, several methods have
evolved for filtering the noise present in the data. Fast Fourier transform (FFT)
based filters are widely used since they provide precise information about the
frequency content of the experimental data which is used for filtering of noise.
However, FFT assumes that the experimental data is stationary. This means that:
(i) the deterministic part of the experimental data obtained from a system is at
steady state without any transients and has frequency components which do not
vary with respect to time and, (ii) noise corrupting the experimental data is wide
sense stationary, i.e., mean and variance of the noise does not statistically vary
with respect to time. Several approaches, e.g., Short Time Fourier Transform
(STFT) and Wavelet Transform (WT) based filters, have been developed to handle
transient data corrputed with nonstationary noise (mean and variance of noise
varies with respect to time) data. Both these approaches provide time and
frequency information about the data (time at which a particular frequency is
present in the signal). However, these filtering approaches have the following
drawbacks: (i) STFT requires identification of an optimal window length within
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which the data is stationary, which is difficult and, (ii) there are theoretical limits
on simultaneous time and frequency resolution. Hence, filtering of noise is
compromised. Recently, Empirical Mode Decomposition (EMD) has been used in
several applications to decompose a given nonstationary data segment into several
characteristic oscillatory components called intrinsic mode functions (IMFs).
Fourier transform of these IMFs identifies the frequency content in the signal,
which can be used for removal of noisy IMFs and reconstruction of the filtered
signal. In the present work, we propose an algorithm for effective filtering of
noise using an EMD based FFT approach for applicatons in polymer physics. The
advantages of the proposed approach are: (i) it uses the precise frequency
information provided by the FFT and therefore efficiently filters a wide variety of
noise and, (ii) the EMD approach can effectively obtain IMFs from both nonstationary as well as nonlinear experimental data. The utility of the proposed
approach is illustrated using an analytical model and also through two typical
laboratory experiments in polymer physics wherein the material response is nonstationary;standard filtering approaches are often inappropriate in such cases.
4.2 Introduction
Experimental data are invariably corrupted with some form of noise [1].
Except for the few cases where noise is within the experimental uncertainties, it is
necessary to eliminate this noise from the data [1,2]. Some common sources of
noise are fluctuations in: (i) power supply (ii) instrument performance, especially
when pushing the limits of a device, and (iii) ambient (temperature or other)
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conditions involved in the experiments [1, 2]. Over the last half-century, much
progress has been made in the area of denoising of signals, especially because of
the advent of widespread digital data acquisition. In addition, experimentalists are
invariably looking at ways to extend the boundaries of experimentation from
macro to micro, micro to nano level [1-5], thereby pushing the limits of their
instrumentation.
Ways to reduce noise have historically involved physical methods. For
example, shielded cables are used to eliminate or reduce the electrical noise
present in the system [1, 6, 7]. However, since noise comes from various sources
(corrupting the experimental data at various frequencies), it is impossible to
completely physically eliminate all sources of noise. Recent advances in the field
of digital signal processing (DSP) have addressed the denoising of signals by
using various filtering algorithms [2] in addition to physical methods. Moving
average and Fast Fourier transform (FFT) based filters are some of the most
commonly applied DSP techniques in experimental studies [1, 2, 5-7]. Moving
average (MA) filters are known to be optimal for removal of random noise present
in the signal. However, if the data is corrupted with noise at specific frequencies,
MA filters perform poorly by introducing biases [1]. Moving average filters act as
low pass filters with poor ability to separate/filter noise at individual frequencies
[8]. The drawbacks of using moving average filters are highlighted later in the
article by applying this technique for data generated using a modified KohlrauschWilliams-Watts (MKWW) model corrupted with noise at specific frequencies.
Fast Fourier transformation (FFT) is one of the most widely used methods to filter
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the noise present in experimental data as it provides the best frequency
information for a given time signal [9]. Unlike moving average filters, FFT based
filters are unbiased, which is one of the main reasons for their use in multiple
disciplines [1, 2]. In this approach, noise reduction is achieved by reducing the
gain at user specified frequency regions. The fundamental assumptions involved in
all FFT based filtering approaches are: (i) the noise is assumed to be
predominantly present in the user specified frequency bands and, (ii) the
experimental data is wide sense stationary (a weak form of stationarity condition)
[10, 11]. A signal x(t) generated by the process is said to be wide sense stationary
(WSS) if the following conditions are true: (i) mean of the signal x(t) generated
from the process denoted as E(x) (E is the expectation operator) is constant and,
(ii) the autocovariance function of the signal defined as E{x(t)x(t+τ)} = Rx(t,t+τ)
is independent of time and does not vary with respect to time [8]. The data
obtained from experiments is usually found to contain the following two
components: (i) a deterministic component from the system under study and, (ii) a
purely stochastic component arising due to the noise corrupting the process [12].
Fourier transformation (FT) assumes that the spectral content of the deterministic
component of the signal given by = ∑ ! "#$% . Here Xk is the Fourier
transform of the signal (which is complex) and ωk. Denotes the frequency of the
signal for all k [12]. Therefore, nonstationary signals (amplitude and/or frequency
characteristics changes with respect to time) cannot be analyzed or filtered using
Fourier transformation (since it violates the assumptions made in FT) [13].
Regarding the purely stochastic component of the data, FT assumes that the noise
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corrputing the data is wide sense stationary [14]. Fourier transform performs
poorly if the WSS assumption of the data is violated [8]. A detailed explanation
regarding the Fourier transform and its drawbacks for use in nonstationary signals
is provided in reference 10. For nonlinear signals, FFT provides fundamental
frequency along with harmonics (integer multiples of fundamental frequencies.)
[15]. However, harmonics provided by FFT are defined based on the stationary
nature of the signal. However, if the underlying signal is nonstationary, there is no
certainty that the spectral components provided by FFT really exist in the data
[16]. In short, traditional FFT based filtering approaches are not sutiable for
filtering of experimental non-stationary data corrupted with nonlinear noise
sources [10, 11].
Noise in non-stationary data can be handled using techniques like short
time Fourier transform (STFT), Wavelet or EMD based filters [10, 11, 13]. STFT
works on the principle of dividing the data into various stationary
segments/windows (mean of the signal remains constant in this segment) followed
by application of an FFT based filter for each individual segment [10]. In STFT,
optimal window length selection is important to obtain the required frequency
information from the data. Two major drawbacks associated with STFT are: (i) it
is hard to obtain an optimal window (segment length) for a given system and (ii) a
time-frequency resolution problem (for a detailed explanation refer to 9) occurs
due to the finite size of the stationary segments [10, 11, 13]. In summary, when
using STFT filters for experimental data, if the window size is small, it is not
possible to separate narrow frequency bands. This in turn leads to difficulty in
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filtering narrow band noise. On the other hand, it is also often not possible to find
large stationary segments in the experimental data of interest [11, 13].
Discrete wavelet transform (DWT) filters are widely used to overcome the
drawbacks associated with STFT filters [13] . While STFT uses a window function
with infinitely long (in time) sinusoidal orthonormal basis functions, wavelet
transform uses short lived (in time) mother wavelets as basis functions for signal
processing[13, 17]. This is considered as one of the main advantages of DWT over
STFT as this helps in better handling of both windowing and time-frequency
resolution issues related to processing of non-stationary signals [13]. DWT works
on the principal of splitting a signal into low and high frequency bands (levels) by
passing it through a series of band pass filters obeying the Nyquist criterion [13].
Filtering is then performed by removing the noise components in the user
specified frequency bands. The mathematical details involved in MA filters, FT
based and DWT filters are briefly discussed in the appendix section of the article
DWT based method is widely used for filtering wide band noise (has nonzero
magnitude over a large band of frequencies) [13]. Notice that like STFT, DWT is
also a time–frequency analysis algorithm for handling nonstationary signals. Also,
in DSP, there always exists a trade-off between time and frequency resolution of
the signal. This time-frequency tradeoff is referred to as the uncertainty principle
in the DSP community since it is analogous to the Heisenberg uncertainty
principle in quantum mechanics [10]. Therefore, DWT cannot be effectively used
for filtering signals corrupted with narrow band and nonlinear noise sources..
Hence, there is a requirement for a filtering algorithm that can take advantage of
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FFT as well as have the ability to handle non-stationary signals.
Recently, Empirical Mode Decomposition (EMD), a time domain
algorithm, has been developed for handling non-stationary and nonlinear signals
[11, 18]. It works on the principal of dividing the experimental data into various
intrinsic components called intrinsic mode functions (IMFs) which represent the
characteristic features of the data at various time scales [11, 18]. Fourier transform
of these characteristic IMFs are computed to obtain the frequency information
from the experimental data over the various time scales. Unlike STFT which
requires optimal window length selection for computing the frequency information
from the data, EMD based filtering approach provides all the frequency
information to the user without any user input. This frequency domain information
about the IMFs is then utilized to identify the noisy IMFs which are removed to
construct the filtered signal. The denoised signal is reconstructed by summing up
of the remaining IMFs. This EMD based DSP filtering approach is more
elaborately discussed in the Methodology section.
Noise in experimental data is also a well recognized problem in the
polymer field [6, 19-21]. A literature review over the last few decades clearly
highlights the development of numerous DSP based filtering and model based
approaches to treat experimental noise in this area [7, 22-25]. A discrete FFT
method with oversampling has been developed to both filter noise and investigate
nonlinear behavior in polymer rheology [7, 26-30]. Kremer and Schönhals [6]
proposed a filtering approach based on the sinusoidal response of the system. In
this approach, a sinusoidal input is provided to the system and the corresponding
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output from the system is obtained. The output could be a phase shifted sinusoidal
signal of the same frequency as the input along with harmonics (if the system is
nonlinear). A correlation analysis in frequency domain (obtained using FFT) is
performed between the input signal of known frequency and the output signal
(with phase shifts and/or harmonics). The result of correlation analysis is a noise
free sinusoidal output signal since it is well known that the correlation will be
maximum between the input and output signals at the phase shifted value when
there is no correlation between input and noise corrupting the system. This has
been cited as one of the major advantages of the frequency domain approach over
the time domain measurement in broad band dielectric spectroscopy (see page 39
of reference 6) as it is difficult to remove noise in time domain data. Kremer and
Schönhals have also recognized the difficulty of handling noise and nonlinearity in
time domain data. Further, it is to be noted that this frequency domain approach is
time consuming for lower frequencies [6]. In the present article we propose a
novel EMD based FFT filtering technique to remove the noise in time domain data
and that offers improved measurement of time domain phenomena.
Mopsik suggested the use of functional fitting techniques as an effective
way to reduce noise in time-domain dielectric spectroscopy [31]. However, these
are not filtering techniques and therefore can introduce artifacts in the data if the
fitting function is improper. Moreover, it is difficult to arrive at the exact function
which can capure all the features of experimental data and reduce the presence of
noise. In the present work we demonstrate the application of the empirical mode
decomposition (EMD) based FFT algorithm as an effective tool to filter the noise
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from time-domain non-stationary data in the field of polymer physics. We
illustrate the practical utility of the proposed methodology for filtering using the
following case studies: (i) an analytical form, specifically the modified
Kohlrausch-Williams-Watts (MKWW) [32, 33] model to generate data corrupted
with harmonic and random noise, (ii) time domain dielectric compliance data for a
poly(vinyl acetate polymer) in the glass transition regime, and (iii) volume
recovery data after a temperature jump in an epoxy glass-former.
4.3 Methodology
EMD is an adaptive data analysis technique used to decompose the given time
domain signal into various time domain components called intrinsic mode
functions (IMFs). The IMFs are representative of the characteristics of the signal
at various time scales [11, 18]. In other words, the EMD algorithm obtains various
IMFs by decomposing the given time domain signal into various shorter (high
frequency) time scale & and longer time scale (low frequency)
components '. Each of the IMFs obtained from the EMD algorithm satisfies
the following constraints: (i) the number of extrema and zero crossings must be
either equal or differ at most by one and, (ii) the mean value of the envelopes
defined by the local maxima and minima is zero. The EMD algorithm used to
obtain the IMFs is provided below as taken from references [34, 35]:
1. Identify all extrema of the given signal .
2. A cubic spline fit is used to obtain the envelope of minima (emin(t)) and
maxima of the signal (emax(t)).
3. Mean of the envelopes m(t) = (emin(t)+emax(t))/2 is computed.
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4. Extract the component ( = – '. If ( satisfies the conditions of
an IMF (mentioned earlier), then assign (=& , where
& represents the kth short time scale IMF. Otherwise, assign =
( and iterate steps from 1 to 4.
5. Obtain the residual ) = x – & . If ) contains a minimum of two
extrema, then set r(t) = x(t) and iterate steps 1 to 5. Otherwise set the trend
(zero frequency) component
'
*
= ).
Applying EMD algorithm to the signal gives
= '* + ∑+,- & (4.1)
where '* is the trend component, & is the kth intrinsic mode function
with k varying from 1 to the number of IMFs, N. Once IMFs are obtained from the
EMD algorithm, the next step is to identify and eliminate the IMFs corresponding
to noise components.
4.3.1 EMD based FFT filtering algorithm
The EMD based filtering procedure used in this work is outlined below:
1. Compute the magnitude at various frequencies using FFT of the IMFs
obtained from the EMD algorithm. The magnitude of a particular IMF
(x(t)) at various frequencies are computed using the following equation
|X(ω)|= |! "# |
2. Identify the IMFs corresponding to the noisy signal using their magnitudes
at the various frequencies obtained from step 1. Let the number of IMFs be
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N. The frequency domain information provided by the FFT of the IMFs are
used for selection of the noisy IMFs. Let the signal contain Y noisy IMFs
with ki (i = 1 to Y) representing the index of the noisy IMFs. The
magnitude of the IMFs are computed. Reconstruct the filtered signal
/ using the non noisy IMFs. The filtered signal is given by
/ = '* + ∑+0- & for all k ≠ ki
(4.2)
The proposed EMD based FFT filtering procedure is applied to various
simulation and experimental data and the results obtained are discussed in the
following section. We compare the FFT and DWT based filtering methods as well.
4.4 Results and Discussion
4.4.1 Simulation studies
To illustrate the proposed method and to make comparisons with other
filtering methods, we first use an analytical function to which noise is added:
specifically, we use a modified form of the Kolrausch-Williams-Watt (MKWW)
function [32, 33]. This model has been used to study the compliance and modulus
behaviors in mechanical measurements of polymers. The model equation is given
by:
1 = 2 + 31 − !
6 9
7
5 8
(4.3)
Here a, b, c and d are ‘material’ constants and the values used in the simulation
are 7, 2.5, 10 and 0.5 respectively. Y is the response function. Often, one is
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interested in determining the material parameters from noisy data. The data
obtained using he above equation is shown in Figure 4.1. Noise n(t) is added to the
MKWW function using the following expression
: = 0.2 ∗ sin2π14t + sin2π60t + 0.2 ∗ sin2π180t + 0.1 ∗ sin2π300t +
0.1 ∗ randt
(4.4)
Here n(t) is the noise generated which includes sinusoidal noise at a fundamental
frequency of 60Hz along with harmonic frequencies namely 180Hz and 300Hz. A
low frequency noise at 14 Hz is introduced along with a random white noise (with
zero mean and unit variance). The maximum variability in the signal due to noise
is ± 5% of the data simulated using the MKWW function. The data is sampled at a
sampling frequency of 1000 Hz which is the same as used in the experimental case
studies. Therefore, according to the Nyquist criterion [34], the maximum
frequency that can be present in the data is 500 Hz. The reasons for adding this
kind of noise to the simulated data are as follows: (i) the dielctric spectroscopy
data obtained from experiments (discussed in subsequently) is corrupted with a
60Hz sinusoidal power supply noise. However, due to truncation effects involved
in the electrometer used for the time domain spectroscopy, this noise gets
truncated such that there is a noise contribution to the signal with harmonic
frequencies in addition to the fundamental at 60 Hz, and, (ii) the data obtained
from volume recovery experiments (after performing a step change in temperature)
is corrupted with ambient noise which can be considered to be white. Figure 4.2
shows the noise generated using equation 4.4. To ensure that the ranodm noise
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added to the system is white, Autocorrelation function (Normalized
autocovariance function) of the random noise is computed and is shown in Figure
4.3. The autocovariance function of a signal x(t) is defined as E{x(t)x(t+τ)} =
Rx(t,t+τ), with τ denoting the lags which are used for computation of the
correlation between the signal at the current instant and previous instant. In
otherwords, autocovariance function provides the correlation between the signal at
current instant and its values in the past. For a pure white noise, this should be
maximum at at time t and zero at all instances. This is due to the fact that white
noise is correlated with itself at time t and does not have any correlation with its
past values. This can also be noticed from autocorrelation function shown in
Figure 4.3. Further, the confidence interval (confidence interval, J =
-
√+
with N
being the number of data points) for autocorrelation function is also provided in
Figure 4.3. Figure 4.4 shows the simulated data corrupted with the noise, obtained
by performing linear addition.
4.4.2 Filtering of the simulated data
In the next paragraphs, we consider four filtering methods for the just
described simulated data. We examine the results provided by MA filters (with
two different window lengths), FFT based filtering method, the DWT method,
and the proposed EMD based FFT filtering technique. Finally, we provide a
comparison of the results obtained from all these approaches.
Moving average filter
Moving average filters compute the average of the given data over a
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specified window length thereby reducing the noise present in the data. The results
vary with different window size and selection of optimal window length is
difficult. Further, as mentioned earlier, MA filters are known to perform poorly
when the data is corrupted with noise at specific frequencies (colored noise). MA
filter with window sizes 5 and 10 were used to filter the MKWW model generated
signal with noise. The results obtained using this approach are shown in Figure 4.5
and Figure 4.6. From the figures, it can be clearly seen that the MA filters (with
window sizes 5 and 10) perform poorly at lower time scales. This is due to: (i)
presence of less data at short time scales compared to the large time scales and, (ii)
the amount of frequency specific noise corrupting the system is high for the
shorter time scales. This is an example of how MA filters are not optimal for
filtering of nonstationary signals corrupted with colored noise [8].
FFT filter
The MKWW function (used to study compliance of systems) used for
simulation provides a transient signal. As mentioned in the introductory section of
the paper, transient signals have amplitudes and freqencies varying with respect to
time and hence they are nonstationary. Therefore, the current simulation data
violates the stationarity assumption made in FFT. However, the noise corrupting
this data set is stationary (mixture of sinusoidal signals and white noise with mean
and variance constant with respect to time). An FFT based low pass filter with
cut-off frequency as (60 Hz) is used to denoise the signal. This cut-off frequency
is generally chosen to remove the 60Hz sinusoidal electrical noise. The result
obtained after using the FFT filter is shown in Figure 4.7. From the Figure, it is
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clear that FFT filter performs poorly when the data is nonstationary. Though the
data looks stationary at the lower time scales, the number of data points is also less
there and therefore, beginning from the sharp slope change, the FFT filter is
unable to perform well. However, at longer time scales there is a large amount of
data and there are not sharp slope changes (since the amount of data is large,
differences between the previous and current data points in this region are less).
Further, this is coupled with the fact that for the longer times the signal magnitude
is higher while the noise magnitude remains the same. In other words, the signal to
noise ratio is larger at longer time scales. Hence FFT is able to provide reasonable
filtering at longer time scales. From, the figure, it can be seen clearly that the
FFT filter performed well when the mean of the data is changing slowly, i.e. at
long times. However, due to the non-stationarity of the short-time data, FFT
filtering is inefficient at this time scale and this leads to oscillations in the filtered
data.
DWT filters
A DWT based filter was applied to denoise the data. Discrete wavelet
transform (DWT) is a time-frequency analysis tool which is widely used to
analyze nonstationary data. Unlike Fourier transform which uses sine and cosine
functions, DWT uses short lived orthogonal wave functions to analyze the data.
The steps involved in DWT are as follows: (i) the signal x(t) = x(nT) (x(nT) is the
experimental data sampled at discrete time intervals with a sampling time T and n
= 1,2...N; N represents the maximum number of samples) is passed through a half
band low pass filter h(n) (n is used to indicate that the filter is discrete) and a half
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band high pass filter g(n) [13]. The half band low pass filter removes the
frequencies that are above half of the maximum frequency content in the signal to
give an approximation signal a(n) while the half band high pass filter removes the
frequencies that are below half of the maximum frequency content in the signal,
(ii) the signals obtained from these two filters are subsampled by a factor 2
(eliminating half the number of points based on the Nyquist criterion) [36]. These
two steps correspond to single level decomposition of signal using DWT. Now the
approximation signal s(n) is not decomposed any further, while the detail
coefficient is further decomposed and these two steps are repeated until a desired
level of decomposition is reached. After decomposition, detail components
(corresponding to high frequency ranges) are removed or an amplitude based
thresholding is performed (only frequencies at which amplitude is above a certain
threshold magnitude are retained). Then, signal reconstruction is performed using
the final approximation component and the retained detailed components. During
reconstruction of the filtered signal, upsampling is performed at every step to
obtain the final filtered signal [36]. A more detailed explanation on discrete
wavelet filters is provided in reference 13.
A five level decomposition of the data was performed using DWT. The
schematic of the five level decomposition of the simulated signal is shown in
Figure 4.8 (based on the Nyquist criterion, 500 Hz is the maximum frequency
content in the signal since the sampling frequency is 1000 Hz) . The denoised
signal is constructed using the coefficients a5 and d5 and the resulting denoised
signal is shown in Figure 4.9. From Figure 4.9, it is clear that DWT filter
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
performs poor and has as still an oscillatory component present in the data. This
suggests that the user has to perform more level of decomposition. However, there
is no fixed specific decomposition level and results vary: (i) depending upon the
number of decompositions performed and (ii) the number of approximation and
detail components used for constructing the filtered signal. If the underlying
original process data is unknown (usually this is the case in experiments),
selection of decomposition levels and coefficients for construction of filtered data
plays a key role in obtaining the filtered signal. Further, if the experimental data is
corrupted with noise at specific frequencies, it is difficult to select these
parameters. As an extreme case, one can include all the noise in the data (by
including all detail and approximation coefficients, equivalent to no filtering ) or
even remove the dynamics exhibited by the underlying process (excluding all the
coefficients except for those closer to zero frequencies, thus filteriing the dynamic
transient nature of the data produced by the process). In addition, this method does
not provide precise frequency information about the noise corrupting the process
output. Even if more levels of decomposition are performed, the user cannot
identify which components have to be utilized for construction of the filtered data
[13].
With the six level decomposition, as shown in Figure 4.10, the DWT
filtered data matches well with that of the original signal, using DWT. As
mentioned earlier, the results obtained from DWT vary depending upon the
number of levels of decomposition performed and number of coefficients used for
construction for filtered signal. From Table 4.1, it can be seen that the results
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
obtained from DWT vary depending upon the number of levels of decomposition.
Further, they also vary with the number of components used for reconstruction of
the data [6] . However, good filtering of the signal can be achieved if the number
of coefficients and decomposition level are selected properly, which is generally
difficult for narrow band noisy signals. In the following discussion, it will be
shown that the proposed EMD based FFT filtering approach provides a time
domain decomposition of the simulated signal which can be used to filter noise at
specific frequencies along with the broad band noise.
Proposed EMD based FFT filter approach
The proposed EMD based FFT filtering approach was also used to denoise
the MKWW function corrupted with noise The IMFs (components after time
domain decomposition of the signal) and their corresponding FFTs obtained after
the decomposition are shown in Figure 4.11. From the FFT of IMF 1, we can
clearly see sharp spikes in the second and fourth harmonics of 60 Hz and at 14 Hz
frequency. Thus IMF 1 provides information about the nonlinear noise corrupting
the signal at specified frequencies. During the filtering procedure this IMF can be
neglected since it represents the noise corrupting the signal. Similarly, FFT of the
second IMF shows the presence of white noise with higher amplitudes between 0
and 50 Hz. Thus, FFT of the IMFs helps in identifying the nature of the noise
corrupting the system. The data then obtained using the EMD based filtering
approach is shown in Figure 4.12 matches well with the original data sans noise.
The importance of this analysis is highlighted in the subsequently described
experimental case studies. It is to be noted that unlike DWT based filters the
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Texas Tech University, Shankar Kollengodu Subramanian, May 2011
proposed approach provides: (i) precise frequency information about noise which
can be used to improve the experimental setup (discussed as a separate subsection
in the end of experimental studies) and (ii) filtered results do not depend on the
number of decompositions, since the EMD algorithm itself has a termination
criterion.
4.4.3 Discussion of results obtained from filtering methods
In this section, we compare the goodness of the filtered data obtained from
various filtering approaches. Total sum squared error (TSE) is a measure which
indicates the effectivenss of the filtering technique to obtain better filtered data.
The filtering approach which provides the minimum TSE compared to other
filtering techniques performs the best filtering of the data. The total sum squared
error (TSE) between the original non-noisy data and filtered data is defined as
Q
LMN = ∑+
,-1 − 1/OP . The TSE values obtained from the various
approaches are shown in Table 1. The plot of squared error (e2(t) =( y(t) –
yfilt(t))2) from the various filtering approaches is shown in Figure 4.13. It can be
clearly seen that the proposed approach provides minimum squared error at all
times compared to other filtering approaches. Further, the TSE value for the
proposed approach is also minimum compared to other techniques which explains
the goodness of the proposed method over other methods.
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4.4.4 Experimental case study I
Time-domain dilectric experiment
Time-domain dielectric spectrometry is used to study the dielectric
response of polymers and other materials of interest [6]. The data obtained from
the dielectric response is time varying in nature and hence it is of interest to use
such data to demonstate the applicability of EMD based filters.
The data used for the present study was obtained from a single step
response performed in the time domain dielectric spectrometer (TDS) built at
Texas Tech. The TDS system consists of a high voltage supply using which a
known voltage is applied to a sample capacitor. The voltage output is measured
using an electrometer. The entire system is operated using a Labview program
interfaced with the computer using a DAQ board. Figure 2.1 is the schematic of
TDS system built at Texas Tech and a more detailed version of the experimental
set up is given in reference 37.
Poly vinyl (acetate) of molecular weight 198,000 g/mol and PDI of 2.73
with its glass transition temperature of 31oC obtained at the cooling rate of
10K/min is the material investigated in this experiment. The thickness of the film
used is about 120 microns. The specific experiment considered here was
performed at 29oC at a sampling frequency of 1000 Hz. The data obtained was
found to be corrupted with experimental noise.
To ascertain the noise in the system, a test was run on the same material at
0V (mean value =0V). The output data is shown in Figure 4.14 (a). It can be
clearly observed that the data is corrupted with sinusoidal noise. Figure 4.14 (b) is
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the FFT performed on the data shown in figure 4.14 (a). A distinct single peak at
60Hz can be observed in Figure 4.14 (b) indicating the presence of ac noise from
the laboratory electrical power supply. However, there is also a small amplitude at
a frequency of 180Hz due to minor distortions present in the observed sinusoidal
signal. It is well known that this type of single frequency ac noise can be removed
using the conventional FFT based filters [1]. However, the data collected as the
single step response after a 200V applied voltage input shown in Figure 4.15
indicates the non stationary nature of the signal. Further, from the inset in Figure
4.15, the presence of irregular pulses can be noticed. This is because of the
truncation effects arising due to the autoranging capacity of the electrometer
system (Keithley 6514) at various voltage levels in the experimental set up. These
truncations introduce nonlinear noise effects in the output voltage data.
To summarize, filtering of the data shown in Figure 4.15, poses the
following challenges: (i) nonstationary nature of the output signal of interest and
(ii) nonlinear nature of the noise corrupting the system output. Therefore, as
discussed in the simulation study using MKWW model, we use the EMD based
FFT approaches to filter the experimental data. The advantages of the information
provided by EMD based FFT filtering approach for filtering noisy data is
highlighted.
Proposed EMD based FFT filter approach
We now apply the EMD algorithm to the experimental voltage data set
shown in Figure 4.15. Following the algorithm presented earlier in section B, the
data is decomposed into several IMFs. Consequently, the filtering approach (using
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FFT of the IMFs) detailed in section B.1 is applied to obtain the filtered signal.
During filtering, the noisy IMFs are identified using the frequency domain
information provided by the FFT of the IMFs. The key noisy IMFs along with
their FFTs are shown in Figure 4.16. Notice, the presence of magnitude values at
frequencies of 60Hz (ω), 120Hz (2ω), 180 (3 ω) and so on from the FFT of the
noisy IMFs shown in Figure 4.16. These IMFs provide information that the noise
corrupting the experimental setup could be ac noise (60Hz signal) and their
harmonics. Unlike DWT, we can justify the removal of these particular IMFs
during reconstruction of filtered signal as detailed in section B.1. These keys
IMFs along with other IMFs representing the truncated ac noise are removed to
obtain the filtered dielectric time domain data (using equation 2). The ability of
the proposed methodology to remove the noise can be readily seen using the
comparison plot between the actual and filtered data shown in Figure 4.17
(magnified version is provided in the inset).
4.4.5 Experimental Study II
Volume recovery after a temperature jump
Structural recovery is the study of thermodynamic properties namely
volume, enthalpy or entropy evolving towards equilibrium, when the glassforming material is quenched from above its glass transition temperature (Tg) to a
known temperature below its Tg [38-40]. It is important to understand this
behavior as it is known to have a strong impact on the long term durability of
glassy materials, especially polymers [41, 42]. Structural recovery experiments
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require very sensitive measurements that are often close to the limits of the
transducing devices used to measure volume or length as a functions of time, and
temperature fluctuations play an important role in determining how well one can
estimate the equilibrium state [43, 44]. As a result, there is a need for a good
filtering method to enhance the signal to noise ratio. Here, we analyze the results
of a length-change dilatometric measurement from our own labs in which the
volume recovery of an epoxy glass is followed subsequent to a tempereature jump
from above to below, but still close to the glass transition temperature, i.e., the
volume changes are small.
The structural recovery test set up [45] shown in Figure 2.3 consists of a
temperature controlled chamber, the sample set up comprising of a support holding
the polymer film, and an LVDT system. The LVDT system comprises of the LVDT
(HR100, Lucas Schaevitz, Inc.), and a signal conditioner (ATA/1000 Schaevitz)
which is used for the length change measurement. The length change measurable
using this set up is in the range of 1mm with the resolution of 1 micron. The data
obtained from the LVDT system are collected using a DAQ board and stored in a PC.
A more detailed version of the experimental set up can be found in references [39,
45].
The epoxy film used in the specific experiment analyzed was of thickness
60 microns and length of 40 mm. The Tg of the epoxy film measured using the
DSC at the cooling rate of 10K/min is 79oC. The material is first heated to 85oC
and maintained at that temperature for an hour. This treatment is performed to
remove any previous thermal history effects. Then two sets of down-jump in
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temperature experiments are performed from 85oC to 75oC and 72oC, respectively,
with simultaneous measurement of the length change and the volume recovery is
calculated using the following equation:
R=
S
S
P
= % U − 1
PT
.
(4.5)
Here, V(t) and V(∞), are the instantaneous and equilibrium volume
respectively while l(t) and l(∞) denotes the instantaneous and equilibrium length
respectively. The above experiments were performed using the sampling frequency of
1 Hz. It may be noted that each data point is obtained using the LabView program is
an average of 150 points collected per second. This is due to the fact that the
experiments are run for duration of 2 to 5 days leading to large volume of data. The
length changes measured are in the range of 10 to 50 µm, which is at the low end of
the measurement range of the LVDT. Figure 4.18 shows the time evolution of the
volume recovery following the temperature jump from 85oC to 75oC and 72oC
respectively. As seen, the data obtained from this experiment was also found to be
corrupted with noise.
Similar to case study 1, this signal is also nonstationary. Further, due to the
non-linear nature of the noise corrupting the signal it is difficult to determine the
approach towards equilibrium. The possible noise sources for this data are:(i)
white noise due to thermal fluctuations in the experiment (ii) low frequency noise
induced due to the use of the LVDT at its lower limits to measure the change in
length, and (iii) unknown sources, e.g., due to vibrations in the lab. These types of
noise can also be successfully removed using EMD based filtering approach as
seen in Figure 4.19. The key IMFs and the corresponding FFT’s are shown in
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Figure 4.20. From Figure 4.20, it is observed that the contributing noise is white
noise and low frequency LVDT noise. Noises at lower frequencies might be due to
the use of the LVDT at the extreme lower end of the measurement system or
temperature variations. Due to the presence of a core, an inertial element, the
LVDT acts as low pass filter [46] and can introduce noise at low frequencies due
to smaller vibrations. We beleive that noise at low frequneices observed in the
FFT of IMFs is contributed by the LVDT as well as temperature variations while
the broad band noises are due to change in the ambient conditions.
It may be noted that when performing the down jump experiments from
above to below the glass transition, two factors contribute to volume/length
change namely the temperature change and the structural recovery. The
contribution of the changing temperature to the length change dominates for the
first 1000 seconds as it takes about that time to reach the derised down jump
temperature. After that, the length change observed is primarily due to structural
recovery with some contribution due to thermal fluctuations. The results shown in
Figure 4.18 were obtained after removing the data obtained during the first fifteen
minutes which contributes to the length change mainly due to temperature change.
Ideally, the length change goes from non equlibrium to equilibrium position
leading to a zero volume change at a finite time. However, due to drift in the data,
it is difficult to exactly zero in on the equilibrium position. Although, the DSP
techniques helps in the reduction of noise, it cannot fix systematic biases like
drifts occuring in the experiments. Hence, there remains a challenge in estimating
the exact equilibrium position as observed in Figure 4.20.
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4.4.6 Practical benefits due to use of the proposed EMD based FFT filtering
algorithm
The proposed EMD based approach provides a time domain decomposition
of the given signal (IMFs) along with the precise frequency information present
in each of the individual IMF. This can be used to identify the noise corrupting the
experimental setup and improve upon the same.
In simulation studies, the proposed approach provides correct information
about the simulated noise added to the MKWW model generated data. However,
as seen in Figure 4.12, at the shortest time scales there is a deviation of the filtered
data from the original data leading to a relatively higher squared error at this time
scale compared to other time scales. This can also be seen in Figure 4.13. Now,
the question of interest is, can we use the information about the noise provided by
the proposed approach to improve the experimental setup and obtain better filtered
data?. Let us assume that we have improved the experimental setup to eliminate
the source of the 14 Hz noise. Therefore, the modified KWW data contains noises
at 60Hz, 180 Hz and 300 Hz sinusoidal noise signals. The results obtained from
the EMD filtering approach for this dataset are shown in Figure 4.21. The sum
squared error value for this case is 0.0047 which is twice less than the earlier
simulation study. This improvment by a factor of 2 is achieived by removing a
single frequency noise component in the data. That is, by improving a single
sensor, the noise present in the output is reduced by a factor of 2. The information
provided by EMD based approach along with process knowledge can be used to
improve upon several sensors which could result in drastic reduction in the amount
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of noise in the system. Further, for the particular case study, the improvement
factor turns out to be 2. However, this factor could be more in specific
applications. Thus, the information about the noise content from proposed EMD
based approach can be used to improve the experimental setup to obtain better
experimental data.
In experimental case study I, frequency information obtained from the
IMFs of the time domain dielectric response data revealed the presence of
truncated ac noise (can be seen clearely from Figure 4.16). The truncation is due to
the use of an autoranging electrometer for different voltage ranges. Therefore, to
avoid the truncation effects and the ac noise the experimental would be improved
by: (i) using a high voltage power supply with inbuilt sophisticated 60 Hz
hardware filters and (ii) electrometers with higher resolution (to avoid the problem
with autoranging). Thus, the proposed approach helps in identification of noise
sources which can be used to improve the experimental setup. Further, if these
improvements are economically not attractive, as discussed in the simulation
studies using the MKWW model, it is better to use the EMD based FFT filtering
approach which practically requires zero additional input from the user except the
experimental data to do the filtering.
In the experimental case study II, the FFT of the individual IMFs revealed
the presence of low frequency noise. This is mainly due to the use of the LVDT at
its specifed lower limits. One possible improvement is to use a more sensitive
LVDT with much lower limits along with a better temperature controller unit. The
proposed filtering approach can also be used in several areas such as (i) Time
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domain dilatometry techniques (ii) time domain diffusive wave spectroscopy [47]
and (iii) all time varying signals corrutped with noises, obtained from various
experiments conducted in the field of polymer physics.
As discussed earlier in the introduction section, Kremer and Schönhals [6]
have argued that it is difficult to perform time domain dielectric spectroscopy
analysis due to the isssues invovled in filtering. Now, using the proposed filtering
approach filtering can be performed effectively so that time domain dielectric
spectroscopy experiments can be conducted with improved short time capability.
Here, by improved short time capability we refer to the following alternative
approach that instead of using sinusoidal waves as input to the system, arbitrary
waveforms containing multiple frequencies can be used as input. Thus, with the
design of single arbitray wave like Pseudo random binary signals (PRBS, which
contains all frequencies) as inputs, the corresponding noisy outputs have be
obtained. These outputs can be filtered effectively using the proposed method
effectively. Thus information at all frequencies can be obtained effectively with
use of signals apart from sinusoidal waves. Further, they claim that material
behaviors at low frequencies can be studied more readily in time domain as
opposed to frequency domain conditions since experiments with low frequency
sinusoidal signals take longer time In fact, design of input signals to obtain
maximum information from the object of interest is a growing area of interest in
the fields of process control and electrical engineering [48] and also in rheology
[49, 50] and thermal analysis [51]. The proposed EMD based filtering approach
provides an opportunity to broaden the benefits of time domain experiments by
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improving the short time data reliability by filtering. It may also be noted that each
frequency measurement requires multiple sine waves to capture the necessary
information whereas we could acheive much better results in a much less time
applying the EMD approach to the time domain data.
4.5 Conclusions
Application of Empirical Mode Decomposition (EMD) as an effective tool
to filter noise in time domain data in the field of polymer physics has been
proposed. The proposed EMD filtering algorithm provides: (i) information which
can be used to identify the noise corrupting the process and, (ii) precise frequency
content of the signal using FFT of the IMFs (Intrinsic Mode Functions). The utility
of this method as a tool to filter non-stationary data corrupted with nonlinear noise
is demonstarted successfully using the following case studies: (i) modified
Kohlrausch Williams Watts (MKWW) model generated data corrupted with
harmonic and random noise, (ii) time domain dielectric compliance data and, (iii)
volume recovery data after temperature jumps in polymer materials.
In the case study using the MKWW model, we have also illustrated the
effectiveness of the EMD based filtering approach by performing a comparison
study with MA filters, FFT and DWT based filtering methods. Further, the ability
of EMD based algorithm to separate data from noise, for dielertic time domain
data, without losing any features of the actual data, can increase the horizon of the
current experimental setup, to further probe into the nonlinear behavior and
dynamic heterogenity of glassy polymers using the sensitive holeburning
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technique.
We have demonstated that EMD based algoritm is also sensitive to filter noise in
the structural recovery experiments after temperature jumps, where the signal to
noise ratio is relatively low since measurements are made in the lower limits of the
LVDT. This method is applied in the next chapter to filter such noises from the
structural recovery experiments of temprature formed and concentration formed
glasses which further aid in confoming some suprising results.
4.6 Appendix
4.6.1 Mathematical Details of various filtering techniques
Moving Average (MA) filters or Boxcar averages
Moving average filters are the most commonly used filter in the field of
Digital Signal Processing (DSP) [8]. This is mainly due to the fact that it is easy to
use and can significantly reduce the random noises present in the data. MA filter
averages a number of points in the input signal (signal to be filtered) to produce a
single point corresponding to the filtered signal (output of the MA filter). In
otherwords, for a given noisy signal x, the output of the MA filter y is obtained
using the followoing formula
Z-
1
X V
+ YW
1 V
W =
",[
In the above equation, M is the number of points over which average is computed.
For example, in a 3 point MA filter, M is 3 MA filter can also be thought of as
performing a convolution of the input noisy signal with a simple rectangular
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kernel of size M to obtain the filtered signal. This moving average technique with
rectangular kernels are also known as Boxcar averages. Several variants of this
filter with different kernels are also widely used for filtering of noisy signals.
Moving average filters have little ability to separate one band of frequencies
from another [8]. Since MA filters operate by performing averages over a
particular rectangular window, it can introduce biases when the signal is
nonstationary.
Fourier Transform (FT) based filters
Fourier transform is applied to implement a well known class of filters called as
Finite impulse response (FIR) filters.
The equation governing the FIR filters is given by [52]
1: = X \V]W ∗ : − ]
In the above equation, y(n) is the filtered signal and x(n) is the input signal. At any
instant, the output of the filter is depends on the past input signal. The
corresponding frequency response of the system is given by
^_ = X \ V]W ∗ ! 5
Q`"
8/a
In the above equation, f denotes the frequency and j is the complex number. This
frequency response of the FIR filter is the same as the of the Fourier transform of
the filter coefficients. Therefore, the value of the filter coefficients of FIR filter is
obtained by taking inverse Fourier Transform of the desired frequency response..
For instance, if it is required to filter the 60 Hz frequency signal from the data, the
deisred frequency response corresponding to this requirement is constructed. Then,
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an inverse FT of the desired response is computed to obtain the filter coefficients.
Once the filter coefficients are obtained, the data is passed through the FIR filter
to obtain the filtered signal.
Discrete Wavelet Transform (DWT) based filters
DWT analyzes the given signal by passing it through a series of low and high pass
filters. In otherwords, the signal is decomposed into coarse and detail components
using the low pass and high pass filters respectively. Once the given signal is
decomposed at first stage as coarse and detailed components, the amount of
samples can be halved (which drastically reduces the computation burden) due the
Nyquist criterion after which next level of decomposition is performed. The
following equations describe the operation involved in DWT at every level [53]
1:bOcb = X V]W ∗ dV2: − ]W
1:Pef = X V]W ∗ ℎV2: − ]W
In the above equations, g(n) and h(n) denote the high and low pass frequency
filters respectively. The factor of 2 is present to indicate that subsampling is
performed in computation of DWT. Once the signal is decomposed to obtain
various frequencies along with the time information, the desired coarse and details
components are used for obtaining the filtered signal. Thus DWT can be thought
of as a bank of low and high pass filters providing both time and frequency
information to the user for filtering of the noisy signal.
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Table 4.1: Sum squared error between the MKWW model data and filtered data
obtained from various filtering approaches
Filtered data from various
approaches
MA filtered data
Window size 5
MA filtered data
Window size 10
FFT filtered data
DWT filtered data – 5 level
decomposition(with a5 )
DWT filtered data – 6 level
decomposition(with a5 )
EMD based FFT approach
Mean squared error (e2(t) = y(t) –
yfilt(t))2)
9.3259
2.2614
3.2240
0.3476
0.0172
0.0151
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Figure 4.1: Data generated using the MKWW model (without noise).
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Figure 4.2: Simulated noise added to the data generated from the MKWW model
(equation 4.4).
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Figure 4.3: Autocorrelation function of the white noise
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Figure 4.4: Simulated MKWW model data with noise n(t) added.
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Figure 4.5: MA filtered noisy model data – Window size 5
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Figure 4.6: MA filtered noisy model data – Window size 10.
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Figure 4.7: FFT filtered noisy model data.
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Figure 4.8: Schematic of DWT for the simulated data.
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Figure 4.9: Filtered data using Discrete Wavelet Transform – Five level decomposition
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Figure 4.10: Filtered data using Discrete Wavelet Transform – Six level decomposition
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Figure 4.11: IMFs and their corresponding magnitude at various frequencies obtained
from FFT for the simulated data with noise
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Figure 4.12: Filtered data using EMD based FFT approach
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Figure 4.13: Squared error comparison plots for the filtered data obtained using FFT,
DWT and EMD based FFT Approach
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Figure 4.14: (a) Zero voltage time domain data, (b) Magnitude spectrum of the data at
various frequencies using FFT
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Figure 4.15: Experimental dielectric time domain data.
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Figure 4.16: IMFs and their corresponding magnitude at various frequencies obtained for
the experimental dielectric data
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Figure 4.17: Comparison of experimental dielectric data and filtered data from EMD
based FFT approach (Inset shows a zoomed version)
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Figure 4.18: Comparison of volume recovery data after performing a down jump
experiment from 85oC to 75oC and 72oC without filtering
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Figure 4.19: Comparison of volume recovery data after performing a down jump
experiment from 85oC to 75oC and 72oC after filtering using EMD method
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Figure 4.20: IMFs and their corresponding magnitude at various frequencies obtained for
the experimental volume recovery after a down jump in temperature to 75oC
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Figure 4.21: MKWW generated data after removing the noise at 14Hz
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CHAPTER 5
AGING AND STRUCTURAL RECOVERY BEHAVIORS IN EPOXY FILMS
SUBJECTED TO CARBON DIOXIDE PLASTICIZATION JUMPS: EVIDENCE
FOR A NEW GLASSY STATE
5.1 Motivation
Structural recovery and physical aging of glassy polymers after temperature
jumps have been very well studied in literature [1-7]. On the contrary, there is only
limited work available on the aging and recovery behaviors of glassy polymers subjected
to plasticizer jumps [8-13]. Plasticizers are known for causing a reduction in the glass
transition temperature, which in turn alters the mechanical properties of the glassy
polymers. We have shown in our previous works, using strong and weakly polar
plasticizers, that they qualitatively mimic the behaviors of temperature jumps but
quantitatively are different [8-10, 13]. In this work, we further investigate this anomalous
behavior by studying the structural recovery and physical aging of an epoxy film
subjected to carbon dioxide pressure jumps and compare the results with temperature
jump experiments such that the final conditions are identical.
5.2 Introduction
Glassy polymers are used extensively in very many industrial applications. Most
of the applications use polymers in the vicinity or below its glass transition temperature
or concentration (Tg or Φg). As shown in Figure 5.1, when a material is cooled from a
higher temperature, the point at which the thermodynamic property such as volume,
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enthalpy or entropy deviates from the equilibrium path is called the glass transition
temperature (Tg) [1] and such a material is called temperature-formed glasses. The point
of deviation upon depressurization instead of cooling when the material is subjected to a
plasticizer environment is called the glass transition concentration (Φg) and the material
formed during this process is called concentration-formed glasses. Below the glass
transition point, the material is in non equilibrium state and hence will tend to evolve
towards the equilibrium state. The study of thermodynamic properties as the material
evolves towards equilibrium is called structural recovery [1] and the changes associated
in the viscoelastic properties such as mechanical, optical, or dielectric during this process
is called physical aging [1].
A proper understanding of aging and structural recovery behavior is essential to
predict the long term stability of glassy polymers [2, 14, 15]. Aging and structural
recovery of temperature-formed glasses have been widely studied and very well
documented in literature [1-7]. However, there is only limited work available in
understanding these behaviors of concentration-formed glasses [8-13, 16]. It is well
documented that the plasticizers are known to depress the glass transition temperature of
glassy polymers which can also create drastic impact on the properties of glassy polymers
[16-28]. Therefore, it is important to understand these behaviors of concentration-formed
glasses and their impact on material applications.
Concentration glasses formed using supercritical CO2 have been known to show
retrograde vitrification phenomenon [20, 25, 28]. Flemings and Koros [16] also observed
hysteresis effect while studying the CO2 sorption-desorption and volume dilation work on
polymers. The structural recovery work after humidity [9] and carbon dioxide jumps [13]
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from our group were found to mimic the behaviors of temperature jump experiments
qualitatively. However, quantitatively they were different. The concentration-formed
glasses showed a larger departure from equilibrium compared to temperature formed
glasses despite the excess volume [9, 13]. This anomalous behavior was also confirmed
with physical aging experiments for both humidity and CO2 jumps [8, 10].
In our prior work on structural recovery of epoxy film subjected to CO2
plasticizer jumps, we surprisingly observed that at 10oC below the glass transition
temperature, the effective retardation time for concentration-formed glasses and
temperature-formed glasses (Subjected to same final condition) do not converge to the
same point as equilibrium is approached [13]. This was kind of surprising. As a
continuation of that work, we report the aging and structural recovery results of an epoxy
film subjected to carbon dioxide plasticizer jumps (P-CO2) and compare the results with
temperature jump experiments to same final conditions.
5.3 Experimental
The material investigated in the current work is a thin epoxy film obtained by
curing diglycidyl ether of bisphenol A (DGEBA, DER332) with amine terminated poly
(propylene oxide) (T403, Huntsman). Epoxy resin DER 332 was first preheated at 55oC
for duration of one hour to ensure that the resin is free of any crystal. A stoichiometric
ratio of resin and curing agent is then mixed well in a container using a magnetic stirrer
for a period of one hour under vacuum at room temperature. The degassed mixture is
then neatly spread between two smooth brass plates and clamped. This is then placed
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inside an oven and cured at 2 atm pressure and 100oC for 1 day. The sample preparation
methodology is well described in references 9 and 13.
Two different samples were used to study the aging behavior and structural
recovery of glassy polymers respectively. The glass transition temperature of epoxy film,
used for structural recovery and aging studies measured at the cooling rate of 10o/min
using DSC, are 79oC and 75oC respectively. The dimension of the film used for structural
recovery is 40mm in length, 60 micron thickness and 10 mm in width and that for
physical experiments were 30 mm in length, 60 micron in thickness and 4mm in width.
The stress applied for the aging experiments performed was 0.9 MPa.
Both the physical aging and structural recovery experiments after pressure jump
(P Jump) experiments were performed using the experimental setup built at Texas Tech
University shown in Figure 2.3 [13]. A detailed explanation on the working of the
experimental set up is given in chapter 2.
5.4 Method of analysis
The departure from equilibrium (δ) for the structural recovery experiments were
calculated using the equation 5.1 [1]. Here, V(t) is the instantaneous volume, and V(∞) is
assumed to be the plateau volume.
R=
S
S
(5.1)
The effective retardation time (τ-1eff ) is obtained using the equation 5.2 [1].
- j
hi//
= − j (5.2)
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The creep compliance (D(t)) curves for aging experiments is calculated using equation
5.3 [8, 10].
k =
(5.3)
l
Here, ε(t) is the instantaneous strain and σo is the applied stress.
The creep curves obtained for T jump and P jump experiments were captured well using
the Kohlrauch William Watt function (KWW) given in equation 5.4 [29, 30].
% n
k = k- ! 5m8
(5.4)
Here, D1, τ (retardation time) and β (shape factor) are fit parameters.
5.5 Results and discussion
5.5.1 Structural recovery experiments
The comparison of the structural recovery experiments performed to same final
condition of 0 MPa and 72oC after T jump and P-CO2 jump respectively is shown in
Figure 5.2. The results reported in Figure 5.2, are the data obtained after performing
EMD based filtering to remove the noise in the raw data [31]. We observe that the
anomalous behavior, which is the departure from equilibrium, is much higher for P jump
experiments compared to T jump experiments. Further, the P jump experiments take a
longer time to reach the equilibrium compared to the T jump experiments. To further
understand this result, we studied the comparison in terms of effective retardation plot
(τ-1eff ) as shown in Figure 5.3. It may be noted that data were smoothened using the
origin software before calculating the effective retardation time for concentration and
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temperature formed glasses. Similar to results obtained previously in our group [13], we
observed that the results of temperature and concentration formed glasses do not
converge to the same value as equilibrium is approached. This result indicates that the
kinetics of the concentration formed glasses is different from temperature formed glasses
and the equilibrium is path dependant. To investigate this behavior further, we performed
the physical aging experiments on the epoxy film subjected to temperature and pressure
jumps.
5.5.2 Aging experiments
Aging results reported in this work, were performed following the Struik protocol
shown in Figure 1.9 [2]. According to Struik’s protocol, the sample loading-unloading for
the aging experiment is performed in such a way that the loading-unloading time is one
tenth of the waiting time. This is done so that each loading unloading step is independent
of the previous event [2]. As seen in the Figure 1.9, the loading-unloading time is
sequentially increased by a factor of 2. The creep compliance after a T- jump from 84oC
to 69.3oC for different aging times is shown in Figure 5.4. The time-aging time
superposition for the creep curves plotted in Figure 5.4 is shown in Figure 5.5. The solid
line in Figure 5.5 is the KWW fit for the creep curves of longest aging time.
Figure 5.6 shows the creep curves for different aging times after a PCO2- jump
from 4.2 to 0 MPa and 69.3oC, and Figure 5.7 is time-aging time superposition curves for
the P- jump experiments. Similar to the T- jump experiments, the KWW model captures
the entire spectrum of creep curves reasonably well. The parameters for the KWW model
obtained for both T- jump and PCO2 -jump experiments for the longest aging time of
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230400s is given in Table 1. It may be noted that the aging experiments for T- jumps and
PCO2 -jumps were performed to the same final condition of 0 MPa and 69.3oC.
Figure 5.8 shows the creep response from the T jump and PCO2 -jump
experiments of Figure 5.4 and 5.6 plotted together. Similar to the results in structural
recovery experiments, we observe that the concentration-formed glasses take longer time
to relax compared to the temperature-formed glasses, although the final conditions are
same for both these experiments. The creep curves of PCO2 -jump experiments collapsed
reasonably well to the longest aging time of T- jump experiments as shown in Figure 5.9,
although the shape parameters (β) are different as seen in Table 5.1. Figure 5.10 is the
horizontal shift factor plot for the time-aging time superposition curves of concentration
and temperature formed glasses. Similar to the departure from equilibrium experiments,
the horizontal shift factor of temperature and concentration formed glasses do not come
to equilibrium at the same aging time.
To further investigate the aging behavior of T- jump and PCO2- jump
experiments, the retardation time obtained using KWW function for the creep curves of
both these experiments were plotted as a function of aging time in Figure 5.11. The
relaxation time for different aging times was estimated using the relationship between the
shift factor and relaxation time given in equation 5.5.
2i = o
o
pqr
(5.5)
Where, ate is the time-aging time shift factor, τref is retardation time of the reference creep
curve and τ is the retardation time of required creep curve.
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It can be clearly observed that the retardation time as a function of aging time for
T -jump and PCO2 -jump experiments do not converge to the same point as the
equilibrium is approached. Moreover, there is close to a decade difference between the
retardation times of these two glasses as equilibrium is reached. A similar result was
reported in the earlier work of Alcoutlabi, Briatico-Vangossa, and McKenna [8], except
that the aging experiments were not performed long enough for the experiments to attain
equilibrium. Further, this result is surprising, as the retardation time results from Zheng
and McKenna [10], on humidity and T- jump experiments converge to the same point as
the equilibrium is reached as shown in Figure 5.12.
It may be noted that Berens and Hodge’s work [32-34] on the DSC study of enthalpy
recovery of concentration-formed glasses and hyper-quenched (temperature-formed)
observed a sub Tg endotherm for both these glasses. Based on this result, they concluded
that this is a common trait of aging phenomenon irrespective of type of glasses. In our
prior work, we have shown that their finding is only partially correct since it is not a
direct measurement method like volume recovery experiments [11]. The current results of
structural recovery and aging experiments confirm that the equilibrium reached by
temperature- and concentration-formed glasses due to carbon dioxide jumps is not the
same equilibrium state. Further, the kinetics of temperature- and concentration-formed
glasses are different.
A question which could be asked at this point is whether the properties altered
during the formation of concentration-formed glasses are reversible. Alcoutlabi, Banda
and McKenna [11], have shown that by heating the concentration-formed glasses after
complete depressurization ( partial equilibration for 24 hours), to above its glass
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transition temperature, followed by cooling to room temperature of 25oC, and heating
again above the glass transition temperature, the excess volume is shed and the material
returns to its normal state. This can be seen in Figure 5.13. This reversing of the
properties and kinetics of concentration formed glasses serves as evidence that the
equilibrium state attained during the PCO2 -jump experiments is not a true equilibrium
but a meta stable state.
5.6 Conclusion
Structural recovery experiments after T- jump and P- jump experiments to the
same final conditions were performed. We observe anomalous behavior similar to that
previously observed in our group for humidity and carbon dioxide jump experiments [810, 13]. In addition, we also observe that volume recovery of temperature- and
concentration-formed glasses do not reach the equilibrium at the same time. Further, the
fact that the effective retardation times as a function of departure from equilibrium do not
converge to same point as the departure from equilibrium approaches zero indicates two
things (1) The kinetics of concentration-formed glasses are different from that of the
temperature-formed glasses (2) The equilibrium state reached during the PCO2 -jump
experiments is not the same as the T- jump experiments.
To confirm this behavior, physical aging experiments were performed to the
same final conditions for T-jump and PCO2 -jump experiments. They also confirm the
anomalous behavior. Further, the retardation time plotted as a function of aging time
shows clearly that the equilibrium state obtained for concentration formed glasses using
carbon dioxide is different from the temperature formed glasses. The above result further
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enhances the finding of structural recovery experiments about the existence of different
equilibrium state for temperature- and concentration-formed glasses. Alcoutlabi, Banda,
and McKenna [11], have shown that the concentration-formed glasses return to normal
state upon heating above the glass transition temperature followed by cooling to room
temperature and heating again. This confirms that the equilibrium obtained during the
PCO2 -jump is an evidence for the existence of a new meta stable glassy state. This is
interesting because such a behavior was not observed for PRH -jump experiments. There
is no concrete explanation at this point for this underlying behavior and it needs to be
further investigated.
5.7 References
1. Kovacs AJ. Transition vitreuse dans les polymeres amorphes. etude
phenomenoloqique. Fortschritte der. Hochpolymeren-Forschung 1963; 3: 394.
2. Struik LCE. Physical aging in polymer and other amorphous materials. Elsevier:
Amsterdam, 1978
3. McKenna, GB.“Glass formation and glassy behavior,” in Comprehensive
Polymer Science. Vol.2, Polymer properties, ed. By C. Booth and C. Price,
Pergamon, Oxford 1989; 311-363.
4. Hutchinson JM. Physical aging of polymers. Progress in Polymer Science 1995;
20:703.
5. Hodge IM. Enthalpy relaxation and recovery in amorphous materials. J. NonCryst. Solids 1994; 169:211.
6. Simon SL, Soieseki JW, Plazek DJ. Volume and enthalpy recovery of
polystyrene. Polymer 2001; 42:2555.
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7. Bernazzani P, Simon SL. Volume Recovery of Polystyrene: Evolution of the
Characteristic Relaxation Time. Journal of Non-Crystalline Solids 2002; 307310C:470.
8. Alcoutlabi M, Vangosa Briatico F, McKenna GB. Effect of chemical activity
jumps on the viscoelastic behavior of an epoxy resin: physical aging response in
carbon dioxide pressure jumps. Journal of polymer science Part B: Polymer
Physics 2002; 40:2050.
9. Zheng Y, McKenna GB. Structural recovery in a model Epoxy: Comparison of
responses after temperature and humidity jumps. Macromolecules 2003; 36: 2387.
10. Zheng Y, Priestley RD, McKenna GB. Physical aging of an epoxy subsequent to
relative humidity jumps through the glass concentration. Journal of polymer
science Part B: Polymer Physics 2004; 42:2107.
11. Alcoutlabi M, Banda L, McKenna GB. A Comparison of Concentration-Glasses
and Temperature-Hyperquenched Glasses: CO2-Formed Glass vs. TemperatureFormed Glass. Polymer 2004; 45:5629.
12. McKenna GB. Glassy States: Concentration Glasses and Temperature Glasses
Compared. J. Non-Crystalline Solids 2007; 353:3820.
13. Alcoutlabi M, Banda L, Kollengodu-Subramanian S, Zhao J, McKenna GB.
Environmental effects on the structural recovery responses of an epoxy resin after
carbon dioxide pressure-Jumps: Intrinsic isopiestics, asymmetry of approach and
memory effect. Macromolecules 2010(Under Review).
14. McKenna GB. On the physics required for the prediction of long term
performance of polymers and their composites. J. Res. NIST 1994; 99:169.
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15. Alcoutlabi M, McKenna GB, Simon SL. Analysis of the development of isotropic
residual stresses in a bismaleimide/sprio Orthocarbonate thermosetting resin for
composite materials. Journal of Applied Polymer science 2003; 88: 227.
16. Fleming GK, Koros WJ. Dilation of polymers by sorption of carbon dioxide at
elevated pressures. Macromolecules 1986; 19: 2285.
17. Chan AH, Paul DR. Influence of history on gas sorption, thermal, mechanical
properties of glassy polycarbonate. J. Appl. Polym. Sci.1979; 24: 1539.
18. Koros WJ, Paul DR. Sorption and transport of CO2 above and below the glass
transition of poly(ethylene terephthalate). Polym. Eng. Sci. 1980; 20:14.
19. Knauss WG, Kenner VH. On the hygrothermomechanical characterization of
polyvinyl acetate. J. Appl. Phys. 1980; 51: 5531-5536.
20. Wang WCh, Kramer EJ, Sachse WH. Effect of high pressure CO2 on the glass
transition temperature and mechanical properties of polystyrene. Journal of
polymer science Part B: Polymer Physics 1982; 20: 1371.
21. Chiou JS, Barlow JW, Paul DR. Plasticization of glassy polymers by CO2. J.
Appl. Polym. Sci. 1985; 30:2633.
22. Chiou JS, Barlow JW, Paul DR. Polymer crystallization induced by sorption of
CO2 gas. J. Appl. Polym. Sci. 1985; 30:3911.
23. Wissinger RG, Paulaitis ME. Swelling and sorption in polymer-CO2 mixtures at
elevated pressures. Journal of polymer science Part B: Polymer Physics 1987; 25:
2497.
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24. Wissinger RG, Paulaitis ME. Glass transition in polymer/CO2 mixtures at
elevated pressures. Journal of polymer science Part B: Polymer Physics 1991; 29:
631.
25. Condo PD, Johnston KP. Retrograde virtrification of polymers with compressed
fluid diluents: Experimental confirmation. Macromolecules 1992; 25:6730.
26. Condo PD, Johnston KP. In situ measurement of glass transition temperature of
polymers with compressed fluid diluents. Journal of polymer science Part B:
Polymer Physics 1994; 32; 523.
27. Zhang C, Cappleman BP, Defibaugh-Chavez M, Weinkauf DH. Glassy polymersorption phenomena measured with quartz crystal microbalance technique.
Journal of polymer science Part B: Polymer Physics 2003; 41:2109.
28. Handa YP, Zhang Z. A new technique for measuring retrograde vitrification in
polymer-gas systems and for making ultramicrocellular foams from retrograde
phase. Journal of polymer science Part B: Polymer Physics 2000; 38:716.
29. Kohlrausch F. Ueber die elastische Nachwirkung bei der Torsion. Pogg Ann Phys
Chem. 1863; 119:337.
30. Williams G, Watts DC. Non-symmetrical dielectric relaxation behavior arising
from a simple empirical decay. Trans Faraday Soc. 1970; 66: 80.
31. Kollengodu-Subramanian S, Babji S, Zhao J, Rengaswamy R, McKenna G B.
Application of empirical mode decomposition in polymer physics. Journal of
polymer science. Part B. Polymer Physics 2011; 49:277.
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32. Berens AR, Hodge IM. Effects of annealing and prior history on the enthalpy
relaxation in glassy polymers. I Experimental study of polyvinyl chloride.
Macromolecules 1982; 756.
33. Hodge IM, Berens AR. Effects of annealing and prior history on the enthalpy
relaxation in glassy polymers. 2. Mathematical modeling. Macromolecules 1982;
762.
34. Hodge IM. Effects of annealing and prior history on the enthalpy relaxation in
glassy polymers. 4. Comparison of 5 polymers. Macromolecules 1983, 16:898.
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Table 5.1: KWW parameters for the longest aging time (230400s) of T and P jump
experiments
Parameter
T jump
P jump
D1/Pa
1.2 E-9
1.43E-9
β
0.28 ± 0.2
0.46 ± 0.2
τ/s
1820 ± 80
60534 ± 500
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Figure 5.1: Schematic representation of specific volume as a function of temperature or
concentrations
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Figure 5.2: A comparison of departure from equilibrium as a function of time for T
jump and P jump experiment subjected to same final condition of 72oC and 0 MPa.
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Figure 5.3: Effective retardation time as a function of departure from equilibrium for T
and P jump experiments of same final condition
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Figure 5.4: Creep compliance curves for different aging time plotted as a function of
time for T jump experiment
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Figure 5.5: Time-aging time superposition of creep curves of the T jump experiment
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Figure 5.6: Creep compliance curves for different aging time plotted as a function of
time for P jump experiment
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Figure 5.7: Time-aging time superposition of the creep curves of P jump experiment
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Figure 5.8: Creep compliance curves for different aging time for T and P jump
experiments subjected to same final condition 0MPa and 69.3oC
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Figure 5.9: Time aging time superposition curves for P jump experiment superposed to
the longest aging time of T jump experiment
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Figure 5.10: Horizontal shift factor as a function of aging time. The concentration
glasses are shifted with respect to the longest aging time of temperature formed glass
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Figure 5.11: Retardation time obtained from KWW function for T and P jump creep
curves plotted against the aging time
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Figure 5.12: Retardation time as function of aging time for humidity and T jump
experiments of same final condition taken from reference [10].
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Figure 5.13. Volume recovery of epoxy film showing the reversal of concentration
formed glass to temperature formed glass upon heating above its Tg followed by cooling
to room temperature and heating [11].
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CHAPTER 6
SUMMARY AND CONCLUSIONS
In this work, we have investigated the time domain responses of glassy polymers
in the vicinity of its glass transition temperature. As a part of this thesis work, Time
Domain Dielectric Spectrometer (TDS) was built in the Polymer and Condensed matter
group at Texas Tech. Its successful working is demonstrated by studying the dielectric
response of poly(vinyl acetate) subjected to pulse probe technique. After establishing the
linear response function in single step experiments, two types of pulse-probe experiments
were performed. In one, the time duration t1 of the first step in the probe was varied. In
the second case, the magnitude of the field E1 applied to the sample for the first step was
varied. The memory effect was observed similar to what is observed in mechanical or
thermal responses. This was in a quantitative agreement with linear Boltzmann
superposition for small applied fields. However, evidence of breakdown of linearity was
observed at a relatively larger applied field. This is an interesting result, as we were able
to demonstrate the ability to delineate between linear and nonlinear behaviors in time
domain, instead of the traditionally used high amplitude sinusoidal pulses in frequency
domain. Further, by using a modified KWW function with an additional term to capture
the dc conductivity, we successfully separated the conductivity effect from the dielectric
relaxation in the time domain itself.
Dielectric time domain data obtained from the TDS were nonstationary in nature
and were found to be corrupted with nonlinear noise. This hindered the application of the
current TDS setup for more sensitive dielectric applications, like probing the dynamic
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heterogeneity. To circumvent the noise problems in instrumentation, and in general to
address the filtration issues associated with nonstationary data corrupted with nonlinear
noise in the field of polymer physics, we used an Empirical Mode Decomposition (EMD)
algorithm to filter the noise.
EMD is an effective tool developed by the signal processing community to filter
non stationary data (time varying signal with no constant mean) corrupted with nonlinear
noises. They work on the principle of breaking the time domain data into various
components called intrinsic mode functions (IMF) such that the actual signal is the sum
of all IMFs. The frequency information of each IMF is then obtained using a fast Fourier
transform (FFT). Then based on the information obtained, the IMF’s related to the noises
are omitted and the remaining IMF’s are added to construct the filtered data.
The effectiveness of EMD filter for data analysis over the traditionally used MA,
FFT and DWT filters was demonstrated using a comparative study of modified KWW
generated model data corrupted with nonlinear and random noise. Further, dielectric data
obtained from TDS was also successfully filtered using the EMD based filtering
approach. Besides filtering the data, the EMD method also gives insightful information
about the noise, which could be used to improve the experimental set up.
Structural recovery and aging experiments of glassy polymers are also time
domain responses essential to predict the long term behavior of glassy polymers. These
behaviors are very well understood for temperature-formed glasses. On the contrary,
there is only limited work done in understanding these behaviors in the context of
concentration (Plasticizers)-formed glasses. Previous work from McKenna’s group has
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shown that concentration-formed glasses (P-RH, P-CO2) qualitatively mimic the
temperature-formed glasses but quantitatively they were different. The initial departure
from equilibrium for concentration-formed glasses after both humidity (strongly polar)
and carbon dioxide (weakly polar) jumps, were higher than the temperature-formed
glasses. Further, our preliminary work on the structural recovery of epoxy film subjected
to CO2 plasticizer jumps showed that the effective retardation time for concentrationformed glasses and temperature-formed glasses (subjected to same final condition) do not
converge to the same point as equilibrium is approached. Hence, we further investigated
this behavior by studying the aging and structural recovery of epoxy film subjected to
carbon dioxide plasticizer jumps and compared them with the temperature jump
experiments of same final condition.
It may be noted that the data obtained from the structural recovery experiments
after temperature and plasticizer (P-CO2) jumps were also found to be time varying in
nature corrupted with nonlinear noise. Moreover, this is a very sensitive experiment with
measurements made in the lower limits of LVDT which lead to relatively lower signal to
noise ratio. EMD based algorithm was successfully used to filter these noise in these
experiments. The data obtained after filtering the raw structural recovery data from the
temperature and pressure jump experiments to the same final condition validated the
anomalous observations made previously in our group. Further, due to better filtering, we
were able to clearly observe that the volume recovery for T jumps and P-CO2 jumps were
not coming to equilibrium at the same time. In addition, the effective retardation plot
showed that the concentration- and temperature-formed glasses do not converge to the
same value as the departure from equilibrium approaches zero, in agreement with
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preliminary results of our previous work. This behavior was further confirmed with a
comparative study of the aging experiments of temperature- and concentration-formed
glasses to the same final condition. It was found that the concentration-formed glasses
relax slower than the temperature-formed glasses. Further, the time-aging time
superposition curves of concentration-formed glasses to longest aging time curve of
temperature-formed glasses was good.
The retardation time of aging creep curves of temperature- and concentrationformed glasses was estimated using the KWW function. It was found that the retardation
time as equilibrium was approached, differed by almost a decade between T jump and P
jump experiments. Similar to the volume recovery results, the retardation time for PCO2 jump experiments were longer than T-jump experiments. This result indicates that the
equilibrium attained by the concentration- and temperature-formed glasses were not the
same. The existence of two equilibriums is a surprise, as our previous work with
humidity jumps did not show such a behavior. However, Alcoutlabi, Banda, and
McKenna have also shown that the equilibrium attained by the concentration formed
glasses is reversible. This result further suggests that this equilibrium obtained with
concentration-formed glasses is only a metastable.
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CHAPTER 7
FUTURE WORK
7.1 Introduction
Time domain responses of glassy polymers in the vicinity of glass transition have
been studied in this work using dielectric and mechanical measurements. The
understanding of these properties is essential to predict and modify the long term stability
and performance of glassy polymers. There is a tremendous potential in this area of
research.
7.2 Time domain nonlinear dielectric
In the time domain dielectric study of glassy polymer, we have demonstrated that
it is possible to delineate between linear and nonlinear behavior in glassy polymers using
linear Boltzmann superposition. This is a good start in this area which needs to be further
explored. The objective of future work would be to explore the possibility of using
nonlinear Boltzmann model to capture and explain this nonlinear behavior.
The modified Boltzmann equation similar to the one used in studying nonlinear
mechanical responses, given in equation 1 [1, 2, 3] can be applied to verify the same.
v
= ℎe s te s + ue ∆t − h vo ℎ- VshW&h
(7.1)
Where Jo is the glassy creep compliance, σ(τ) is the instantaneous stress and ε (t) is the
instantaneous stain.
The modified Boltzmann for nonlinear behavior is based on that assumption that
when you double your stress, the strain is doubled as a function of stress instead of just
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the stress as in linear case [1, 2, 3]. The effective working of this model for a single step
response in mechanical measurement of polycarbonate is well demonstrated by Luo and
his coworkers [1].
Experimental methodology
The nonlinear dielectric experiments can be performed using the time domain
dielectric spectrometer built at Texas Tech. The necessary protocols are discussed in the
experimental section in chapter 3. It may be noted that to study the nonlinear behaviors,
the experiments have to be performed at a higher applied electric field. This could be
achieved by reducing the thickness of the film, increasing the applied voltage or by using
the combination of both.
7.3 Isobaric time domain dielectric measurements
We have successfully demonstrated the working of time domain dielectric
spectrometer by performing isothermal measurements of PVAc in chapter 3. As per our
knowledge, there is no work done on understanding the dielectric behaviors of polymers
when they are subjected to plasticizer environment. Since we have the facility to perform
such experiments, it would be of fundamental interest to perform the same. This work
will be based on the hypothesis that, similar to mechanical measurements in polymers,
the dielectric measurements are sensitive to plasticizer concentration.
Experimental methodology
The schematic of the experimental set up for studying the dielectric responses of
glassy polymers subjected to plasticizer environment (Carbon dioxide) is shown in Figure
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7.1. The set up comprises of a pressure vessel with a maximum capacity of 6MPa inside
which the sample set up (See chapter 3, Figure) is placed. The working of the pressure
vessel is well explained in the experimental section in chapter 2. Dielectric compliance
can then be measured as a function of time at various pressures (Plasticizer
concentration) to verify the hypothesis. Further, there is also a possibility to probe
dynamic heterogeneity, by replacing carbon dioxide with a more polar plasticizer, which
then can be used to further perform hole burning experiments which are discussed in
Chapter 1.
7.4 Physical aging and structural recovery of glassy polymers
The other time domain study, namely, the aging and structural recovery of glassy
polymers subjected to carbon dioxide plasticizer jump, has indicated two things. One is
that similar to humidity jump experiments, P-CO2 jump experiments show anomalous
behavior compared to the temperature jump experiments of the same final condition. The
second thing is that we have observed the evidence for a new glassy state in P-CO2 jump
experiments. This is very interesting; as such a behavior was not observed with the
concentration-formed glasses subjected to humidity jumps and needs further
investigation.
An enthalpy recovery study after plasticizer and temperature jump would be
another method which could complement the current work. This experiment will not only
help in confirming the anomalous behavior irrespective of the measurement technique but
can also validate evidence of the new glassy state observed with the carbon dioxide
concentration glasses.
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7.5 References
1. Findley WN, Lai JS, Onaran K. Creep and relaxation of nonlinear viscoelastic
materials with an introduction to linear viscoelasticity. New York: North-Holland
Publication; 1976.
2. Schapery RA. On the characterization of nonlinear viscoelastic material. Polym.
Eng.Sci.1969; 9:295.
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Figure 7.1: Schematic of time domain dielectric spectrometer with modifications
to perform isobaric measurements of dielectric compliance
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