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 Title of the concept Author: Manish Gupta, MTech student, IIT Madras Mentor: Dr. Abhijit P. Deshpande

Learning objectives 1 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. 2 3 4 5

Definitions and Keywords 1 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. 2 3 4 5

Master Layout 1 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 2 1.1 A + B C Temperature = 110o C Conversion = 70% Time of reaction = 2 hrs Temperature Time of reaction or 3 4 5

3 Importance of time-temperature superposition 1 C 2 C 4 C 5 A + B Slide 1 Part 1: Step 1 Importance of time-temperature superposition 1 1.1 Conversion = 70% Time of reaction = 2 hrs T = 110o C 2 A + B C More than two hours 3 A + B C T = 60o C Conversion = 70% Time of reaction = ??? Less than two hours 4 Conversion = 70% Time of reaction = ??? A + B C T = 150o C 5

3 Importance of time-temperature superposition 1 C 2 4 5 A + B Slide 2 Part 1: Step 1 Importance of time-temperature superposition 1 1.1 A + B C Temperature = 110o C Conversion = 70% Time of reaction = 2 hrs 2 3 If the desired conversion is 80%, then this conversion can be obtained either by Temperature Time of reaction = 2 hrs or Temperature = 110o C Time of reaction 4 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. 5

3 1 2 4 5 Part 1: step 1,2 and 3: 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. 2 3 4 5

3 Importance of time-temperature superposition 1 2 4 5 Part 1: Step 1 Importance of time-temperature superposition 1 1.1 What is time-temperature superposition ? 2 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. 3 4 5

3 G(t,T) vs. t G(at t,To) vs. at t 1 Time temperature superposition 2 Part 1: 1 Time temperature superposition G(t,T) vs. t G(at t,To) vs. at t T TS 2 3 Where, G is modulus t is time T is temperature To is reference temperature at is horizontal shift factor 4 5

3 Importance of time-temperature superposition 1 2 4 5 Part 1: Step 1 Importance of time-temperature superposition 1 1.1 What is the importance of time-temperature superposition ? 2 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. 3 4 5

3 1 Importance of time-temperature superposition 2 4 5 Part 1: Step 1 1 Importance of time-temperature superposition 1.1 Assumptions behind time-temperature superposition principle: 2 1. The material does not undergo any chemical or physical changes as a result of the temperature change. 3 2. There is no phase transition as a result of change in temperature. 4 3. There is no heterogeneity in the sample. 4. Applicable in linear viscoelastic regime only. 5

3 1 2 4 5 Part 1: step 1,2 and 3: 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. 2 3 4 5

1 Master Layout 2 3 4 5 Log ω, Hz Log G’, dyne/cm2 Log at + Log ω, Hz 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 2 Master curve 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log at + Log ω, Hz at – horizontal shift factor 3 4 5

3 1 2 4 5 Time-temperature superposition in frequency domain Log ω, Hz Part 2 Time-temperature superposition in frequency domain 1 Frequency sweep of polymer melt Reference temperature is 140o C 110o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 120o C 2 130o C 140o C 3 150o C 160o C 170o C 4 G’- Storage modulus ω - Frequency 5

3 1 2 4 5 Time-temperature superposition in frequency domain Log ω, Hz Part 2 Time-temperature superposition in frequency domain 1 Frequency sweep of polymer melt Reference temperature is 140o C 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 2 3 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. 4 G’- Storage modulus ω - Frequency 5

3 1 2 4 5 Time-temperature superposition in frequency domain Log ω, Hz Part 2 Time-temperature superposition in frequency domain 1 Frequency sweep of polymer melt Reference temperature is 140o C 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 at 2 at 3 4 G’- storage modulus ω - frequency 5

3 1 2 4 5 Time-temperature superposition in frequency domain Log ω, Hz Part 2 Time-temperature superposition in frequency domain 1 Frequency sweep of polymer melt Generation of master curve at 140o C 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log at + Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 at 2 at 3 4 G’- storage modulus ω - frequency at – horizontal shift factor 5

3 1 2 4 5 Time-temperature superposition in frequency domain Log ω, Hz Part 1: Step 1 Time-temperature superposition in frequency domain 1 Master curve at 140o C Frequency sweep of polymer melt at 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log at + Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 at 2 at 3 at at 4 G’- storage modulus ω - frequency at – horizontal shift factor 5

3 Importance of time-temperature superposition 1 2 4 5 Log ω, Hz Part 2 Importance of time-temperature superposition 1 Master curve at 140o C Frequency sweep of polymer melt 150o C 160o C 170o C 110o C 120o C 130o C 140o C Log ω, Hz Log G’, dyne/cm2 -2 -1 6 5 4 3 -3 1 Log at + Log ω, Hz at – horizontal shift factor 2 3 4 Storage modulus at 170o C at higher frequency is equivalent to storage modulus at 140o C at lower frequency. G’- storage modulus ω - frequency 5

3 1 2 4 5 Time-temperature superposition in frequency domain Part 2 Time-temperature superposition in frequency domain 1 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. 2 3 4 5

3 1 2 4 5 Part 2 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. 2 3 4 5

1 Master Layout 2 3 4 5 Log t, hrs. Log E, dyne/cm2 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 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log at + Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C at – horizontal shift factor Master curve 3 4 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Part 3 1 Time temperature superposition in time domain Relaxation modulus data of polymer Reference temperature is 25o C 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C 3 4 E - Relaxation modulus t - Time 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Part 3 1 Time temperature superposition in time domain Relaxation modulus data of polymer Reference temperature is 25o C 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C 3 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. 4 E - Relaxation modulus t - Time 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Part 3 1 Time temperature superposition in time domain Relaxation modulus data of polymer Reference temperature is 25o C 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C 3 4 E - Relaxation modulus t - Time 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Relaxation modulus data of polymer Generation of master curve at 25o C 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log at + Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C at 3 at 4 E - Relaxation modulus t - Time at – horizontal shift factor 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Relaxation modulus data of polymer Master curve at 25o C at 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log at + Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C at 3 at at at 4 E - Relaxation modulus t - Time at – horizontal shift factor 5

1 Time temperature superposition in time domain 2 3 4 5 Log t, hrs. Relaxation modulus data of polymer Master curve at 25o C 2 1.1 Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 1.1 Log at + Log t, hrs. -2 10 8 6 4 -4 2 5 Log E, dyne/cm2 5o C 15o C 25o C 35o C 45o C 55o C at – horizontal shift factor 3 4 E - Relaxation modulus t - Time Relaxation modulus at 45o C at lower time scale is equivalent to relaxation modulus at 25o C at higher time scale. 5

3 1 Time-temperature superposition 2 4 5 Part 3 1 Time-temperature superposition 1.1 2 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. 3 4 5

3 1 2 4 5 Part 3 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. 2 3 4 5

1 Master Layout 2 3 4 5 WLF equation Arrhenius equation Log at = 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 2 WLF equation Arrhenius equation 3 Log at = -C1 (T-To ) [C2 + (T-To )] at = exp[(Ea /R)(1/T – 1/To )] 4 5

3 Calculation of shift factor 1 2 4 5 WLF equation Arrhenius equation Part 4 Calculation of shift factor 1 Horizontal shift factor can be calculated based on Williams-Landel-Ferry (WLF) and Arrhenius equations. WLF equation 2 C1 and C2 are constants T is temperature in Kelvin To is reference temperature in Kelvin Log at = -C1 (T-To ) [C2 + (T-To )] 3 WLF equation is applicable for system having glass transition temperature, Tg . If To ≈ Tg then C1 ≈17.44 and C2 ≈ 51.6 K. 4 Arrhenius equation Ea is activation energy R is universal gas constant at = exp[(Ea /R)(1/T – 1/To )] 5

3 1 2 4 5 Part 4: 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. 2 3 4 5

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 > T1 b) T2 < 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

APPENDIX 1 Answers 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 > 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

Links for further reading 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/Time–temperature_superposition

Summary Time-temperature superposition is a tool to determine material APPENDIX 3 Summary 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.