Time-temperature superposition

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Time-temperature superposition

• Time-temperature superposition is a tool to
  determine the material properties over broad
  range of times and temperatures? by shifting data.

Prior viewing Creep and stress relaxation Future
viewing Linear viscoelastic superposition Course
Name Polymeric Materials Level(UG/PG) PG
Author Manish Gupta, MTech student, IIT
Madras Mentor Dr. Abhijit P. Deshpande
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Learning objectives
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• After interacting with this Learning Object, the
  learner will be able to
• Appreciate the concept of time temperature
  superposition principle.
• Calculate shift factor.
• Predict material properties at extremely low and
  high time scales.

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Definitions and Keywords
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• Time-temperature superposition Time-temperature
  superposition is a tool to determine the material
  properties over broad range of times and
  temperatures? by shifting data.
• Shift factor It is the factor by which data need
  to be shifted.
• Storage modulus (G) It describes the elastic or
  energy storage behavior of the material.
• Relaxation modulus (E) It is defined as the
  ratio of stress (a function of time) to constant
  strain.

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Master Layout
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This animation consists of three parts Part 1
Importance of time-temperature superposition
(TTS) Part 2 Time-temperature superposition in
frequency domain Part 3 Time-temperature
superposition in time domain Part 4
Calculation of shift factor
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Slide 1
Part 1 Step 1
Importance of time-temperature superposition
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1.1
Conversion 70 Time of reaction 2 hrs
T 110o C
More than two hours
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Conversion 70 Time of reaction ???
Less than two hours
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Conversion 70 Time of reaction ???
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Slide 2
Part 1 Step 1
Importance of time-temperature superposition
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If the desired conversion is 80, then this
conversion can be obtained either by
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Analogously, polymer behavior obtained at a
particular combination of time (or frequency)
and temperature, can also be obtained at some
other combinations of time ( or frequency) and
temperature. And, the technique used for this
purpose is known as time-temperature
superposition.
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Part 1 step 1,2 and 3
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Action Description of the action Audio Narration
As shown in animation Picture should appear one after another as shown in the slides. Maroon call out ( in slide 1) should appear in sync with the sentence highlighted in red in audio narration para 3. Green call out ( in slide 1) should appear in sync with the sentence highlighted in pink in para 3. In slide 2, both yellow and pink item should appear in sync with sentence highlighted in green in audio narration para 5. In slide 2, green box should appear when narrator is narrating the sentence of para 7 highlighted in blue. To understand the principle of time temperature superposition, let me start with a chemical reaction as an analogy. Consider a chemical reaction between A and B which gives C as a product. Let us assume that this reaction is taking place at one hundred and ten degree Celsius for two hours, and conversion is seventy percent. If I ask you to achieve the same conversion, with the reaction temperature of sixty degree Celsius, then naturally you will have to wait for longer time. Time require to complete the reaction will definitely be more than two hours. Similarly, when reaction is carried out at temperature higher than one hundred and ten degree Celsius, time of reaction decreases. It means that the same conversion can be obtain at different combinations of temperature and time of reaction. If the desired conversion is eighty percent, then this conversion can be obtained, either by increasing the temperature or by increasing the time of reaction. Thus, conversion can be manipulated by adjusting temperature and time of reaction. Analogously, material property of a particular value obtained at a particular combination of temperature and time (or frequency), can also be obtained at some other combinations of temperature and time ( or frequency). And, the technique which is use for this purpose is known as time temperature superposition.
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Part 1 Step 1
Importance of time-temperature superposition
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1.1
What is time-temperature superposition ?
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Time-temperature superposition is a tool to
determine material properties over broad range of
times and temperatures by shifting data. Material
properties should be temperature dependent such
as creep compliance, relaxation modulus, loss and
storage moduli, viscosity etc. A material to
which this technique is applicable are said to be
thermorheologically simple. This terminology
was introduced by Schwarzl and Staverman.
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Part 1
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Time temperature superposition
G(t,T) vs. t G(at t,To) vs. at t
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Where, G is modulus t
is time T is temperature
To is reference temperature at
is horizontal shift factor

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Part 1 Step 1
Importance of time-temperature superposition
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1.1
What is the importance of time-temperature
superposition ?
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Any instrument, due to its mechanical limitations
can usually give data over a limited range of
time or frequency at a particular temperature,
and this is inadequate to determine viscoelastic
properties at very large time scale or at very
low frequency. Therefore, in order to probe the
viscoelastic properties of the material at
extreme time scales, time temperature
superposition principle is needed.
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Part 1 Step 1
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Importance of time-temperature superposition
1.1
Assumptions behind time-temperature superposition
principle
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1. The material does not undergo any chemical or
physical changes as a result of the temperature
change.
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2. There is no phase transition as a result of
change in temperature.
3. There is no heterogeneity in the sample.
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4. Applicable in linear viscoelastic regime only.
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Part 1 step 1,2 and 3
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Action Description of the action Audio Narration
As shown in animation Picture should appear one after another as shown in the slides. Now we can define time temperature superposition. Time-temperature superposition is a tool to determine material properties over broad range of times and temperatures by shifting data. Material properties should be temperature dependent such as creep compliance, relaxation modulus, loss and storage moduli, viscosity etc. A material to which this technique is applicable are said to be thermorheologically simple. This terminology was introduced by Schwarzl and Staverman. Before going into detail, we ought to know why we should go for time temperature superposition and what is the importance of time temperature superposition ? Time temperature superposition is needed because any instrument, due to its mechanical limitations can usually give data over a limited range of time or frequency at a particular temperature, and this is inadequate to determine viscoelastic properties at very large time scale or at very small time scale. Therefore, in order to probe the viscoelastic properties of the material at extreme time scales, time temperature superposition principle is needed. Before we move further, lets us discuss the assumptions involved in time temperature superposition. Firstly, The material does not undergo any chemical or physical changes as a result of the temperature change. Secondly, there is no phase transition as a result of change in temperature. Then, there should be no heterogeneity in the sample. Lastly, this technique is applicable only in linear viscoelastic regime.
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Master Layout
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This animation consists of three parts Part 1
Importance of time-temperature superposition
(TTS) Part 2 Time-temperature superposition in
frequency domain Part 3 Time-temperature
superposition in time domain Part 4
Calculation of shift factor
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Master curve
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Part 2
Time-temperature superposition in frequency domain
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Reference temperature is 140o C
Frequency sweep of polymer melt
110o C
120o C
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130o C
140o C
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150o C
160o C
170o C
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G- Storage modulus ? - Frequency
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Part 2
Time-temperature superposition in frequency domain
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Reference temperature is 140o C
Frequency sweep of polymer melt
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Shift violet curve to the right and green to the
left such that all curves collapses into a single
curve by means of horizontal shift.
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G- Storage modulus ? - Frequency
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Part 2
Time-temperature superposition in frequency domain
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Reference temperature is 140o C
Frequency sweep of polymer melt
at
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at
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G- storage modulus ? - frequency
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Part 2
Time-temperature superposition in frequency domain
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Generation of master curve at 140o C
Frequency sweep of polymer melt
at
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at
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G- storage modulus ? - frequency
at horizontal shift factor
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Part 1 Step 1
Time-temperature superposition in frequency domain
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Master curve at 140o C
Frequency sweep of polymer melt
at
at
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at
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at
at
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G- storage modulus ? - frequency
at horizontal shift factor
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Part 2
Importance of time-temperature superposition
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Master curve at 140o C
Frequency sweep of polymer melt
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Storage modulus at 170o C at higher frequency is
equivalent to storage modulus at 140o C at lower
frequency.
G- storage modulus ? - frequency
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Part 2
Time-temperature superposition in frequency domain
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1.1
The inefficiency of measuring the polymer
behavior at longer time scale is avoided by
utilizing the fact that, the polymer behavior at
higher temperature and smaller time scale will be
the same.
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Part 2
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Action Description of the action Audio Narration
As shown in animation Picture should appear one after another as shown in the slides. We will now consider two examples of time temperature superposition principle in both frequency and time domain. Let us , first discuss the this principle in frequency domain. Consider the plot of storage modulus as a function of frequency for a polymer melt system at different temperatures ranging from one hundred and ten degree celsius to one hundred and seventy degree celsius. You can see that at any frequency, modulus is increasing with decrease in temperature and over all behavior of each curve is the same. That is modulus is increasing with frequency at any temperature. Due to the sensitivity of the instrument, data at very low frequency cannot be obtained. That is if you are interested in probing sample behavior at longer time scale, then it is not possible to predict behavior with the help of instrument. In other words, data at longer time scale, at any temperature are not available. In order to observe the behavior of storage modulus at longer time scale or at lower frequency, time temperature superposition technique is used. Now let us see how this technique is applied. Suppose that you are interested in knowing behavior of polymer melt at one hundred and forty degree celsius at larger time scale or lower frequecy range. For this consider, the brown curve measured at one hundred and forty degree celsius. To observed the behavior of this polymer melt at larger time scale, curves higher than this one hundred and forty degree celsius should be shifted towards left and curves of lower temperature should be shifted towards right. Amount of shifting is determine by horizontal shift factor at . As shown in the figure. Similarly, all other curves of temperature higher than hundred and forty degree celsius should be shifted towards left and that of lower temperatures should be shifted towards right. Thus we obtained master curve for polymer melt at one hundred and forty degree celsius over broad range of time scales.
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Master Layout
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This animation consists of three parts Part 1
Importance of time-temperature superposition
(TTS) Part 2 Time-temperature superposition in
frequency domain Part 3 Time-temperature
superposition in time domain Part 4
Calculation of shift factor
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Master curve
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Part 3
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Time temperature superposition in time domain
Reference temperature is 25o C
Relaxation modulus data of polymer
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E - Relaxation modulus t - Time
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Part 3
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Time temperature superposition in time domain
Reference temperature is 25o C
Relaxation modulus data of polymer
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Shift the blue curve to the left and the yellow
one to the right such that all curves collapses
into a single curve by means of horizontal shift.
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E - Relaxation modulus t - Time
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Part 3
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Time temperature superposition in time domain
Relaxation modulus data of polymer
Reference temperature is 25o C
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E - Relaxation modulus t - Time
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Time temperature superposition in time domain
Relaxation modulus data of polymer
Generation of master curve at 25o C
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at
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E - Relaxation modulus t - Time
at horizontal shift factor
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Time temperature superposition in time domain
Relaxation modulus data of polymer
Master curve at 25o C
at
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at
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at
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at
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E - Relaxation modulus t - Time
at horizontal shift factor
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Time temperature superposition in time domain
Relaxation modulus data of polymer
Master curve at 25o C
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Relaxation modulus at 45o C at lower time scale
is equivalent to relaxation modulus at 25o C at
higher time scale.
E - Relaxation modulus t - Time
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Part 3
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Time-temperature superposition
1.1
If the relaxation behavior of the polymer at 25o
C were to obtained in the similar time scale,
then this behavior will be the same as the curve
produced by superimposing different curves
obtained at several temperatures.
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Part 3
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Action Description of the action Audio Narration
As shown in animation Picture should appear one after another as shown in the slides. Let us now discuss this superposition principle in time domain. For this, consider a log log plot of relaxation modulus versus time for a polymer at different temperatures. In this experiment, the material is subjected to rapid deformation and the stress on the material is monitored with time. Since stress is a function of time, therefore tensile relaxation modulus is also a function of time. To capture the behavior of stress, relaxation modulus is plotted against time and it is observed that modulus is decreasing with time at a given temperature. We can see that relaxation modulus is decreasing for a given temperature, but it decreases as temperature increases. This experiment has been carried for time scale of two decades. But, what if, we want to see the relaxation behavior for time scale over four decades ? If we use the same instrument, to capture the behavior of stress for several time decades then it may take few years to finish the experiment. Here, principle of time temperature superposition comes into play. For this we carry out the same experiment at several temperatures, and shift the data horizontally to generate a master curve showing the behavior at reference temperature that covers many decades of time. As a result, master curve is obtained at reference temperature of twenty five degree celsius as shown in the figure. Therefore, we can obtain long time relaxation behavior of polymer at desired reference temperature merely by horizontal shifting of data.
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Master Layout
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This animation consists of three parts Part 1
Importance of time-temperature superposition
(TTS) Part 2 Time-temperature superposition in
frequency domain Part 3 Time-temperature
superposition in time domain Part 4
Calculation of shift factor
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WLF equation
Arrhenius equation
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Part 4
Calculation of shift factor
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Horizontal shift factor can be calculated based
on Williams-Landel-Ferry (WLF) and Arrhenius
equations.
WLF equation
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C1 and C2 are constants T is temperature in
Kelvin To is reference temperature in Kelvin
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WLF equation is applicable for system having
glass transition temperature, Tg . If To Tg
then C1 17.44 and C2 51.6 K.
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Arrhenius equation
Ea is activation energy R is universal gas
constant
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Part 4
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Action Description of the action Audio Narration
As shown in animation As shown in animation We will learn now how to calculate shift factors. Shift factors are used for shifting raw data to generate master curve. By incorporating horizontal shift factor into raw data, new sets of data are produced, which when plotted shows behavior of the material property at several decades of time or frequency. For example, if you have a plot of modulus vs time at different temperatures, then by incorporation of shift factors, reduced modulus vs reduced time can be plotted. Horizontal shift factor can be calculated by using WLF equation and Arrhenius equation.
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Questionnaire
APPENDIX 1

• 1. Time temperature superposition is a tool used
  for determination of material property preferably
  at
• a) longer time scale
• b) Smaller time scale
• In a log-log plot of relaxation modulus vs. time,
  modulus obtained at 50o C is 100 Pa at time T1 .
  If you want to observed the same modulus at 25o
  C, then this will be observed at time scale (T2)
• a) T2 gt T1 b) T2 lt T1 c) T2
  T1
• 3. Arrhenius equation for calculating horizontal
  shift factor is valid near glass transition
  temperature
• a) True b) False
• A student wishes to apply the technique of time
  temperature superposition to determine the long
  time rheological behavior of polymer solution,
  but during the experiment it is found that
  solvent is evaporating. Would this render him
  violation of assumption involved in time
  temperature superposition principle ?
• a) Yes b) No
• Heterogeneity in the polymer sample should not be
  there for time temperature principle to be valid.
• a) True b) False

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Answers
APPENDIX 1

• 1. Time temperature superposition is a tool used
  for determination of material property preferably
  at
• a) longer time scale
• In a log-log plot of relaxation modulus vs. time,
  modulus obtained at 50o C is 100 Pa at time T1 .
  If you want to observed the same modulus at 25o
  C, then this will be observed at time scale (T2)
• a) T2 gt T1
• 3. Arrhenius equation for calculating horizontal
  shift factor is valid near glass transition
  temperature
• b) False
• A student wishes to apply the technique of time
  temperature superposition to determine the long
  time rheological behavior of polymer solution,
  but during the experiment it is found that
  solvent is evaporating. Would this render him
  violation of assumption involved in time
  temperature superposition principle ?
• a) Yes
• Heterogeneity in the polymer sample should not be
  there for time temperature principle to be valid.
• a) True

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APPENDIX 2
Links for further reading

• References
• Kumar, A. Gupta, R.K. Fundamentals of polymers,
  Tata-McGraw Hills international, pp 389-392
• http//en.wikipedia.org/wiki/Timetemperature_supe
  rposition

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Summary
APPENDIX 3

• Time-temperature superposition is a tool to
  determine material
• properties over broad range of times and
  temperatures by
• shifting data.
• The inefficiency of measuring the polymer
  behavior at longer
• time scale is avoided by utilizing the
  fact that, the polymer
• behavior at higher temperature and smaller
  time scale will be
• the same.
• Shift factor can be calculated by WLF and
  Arrhenius
• equations.