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The Glass Transition Temperature Depends on the Loading Rate

Introduction

Most engineers know that the following question is not that great: “What is the Young’s modulus of this specific medium-density polyethylene specimen”? One problem with this question is that the Young modulus of MDPE depends on the temperature. Another problem is that the Young modulus also depends on the applied loading rate. What is not as well-known is that the glass transition temperature (Tg) of a polymer is similarly not just a number, it will also depend on the loading rate! In this article I will use stress relaxation data to show just how much the glass transition temperature can change with the loading rate.

Definition of the Glass Transition Temperature

There are different ways to experimentally measure the glass transition temperature of a polymer. My favorite method is to perform a temperature sweep in a Dynamic Mechanical Analysis (DMA) test machine. This is a quick (and inexpensive) method that measures the storage and loss modulus of the material as a function of the temperature. The glass transition temperature (Tg) can be defined as the temperature at which the loss modulus has a max value (note that the exemplar image below is using a different definition).

Examplar DMA data for a polycarbonate [Atroiss, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons]

Results

To examine the glass transition temperature I will start with a perhaps unexpected data set that I studied in a previous article. The data consists of stress relaxation tests at 19 different temperatures.

I then combined each stress relaxation test file into one long file containing all data in 3 columns: (1) time, (2) engineering stress, (3) temperature. This file format is needed to use the Master Curve tool in MCalibration.

				
					0,0.00742640818667,-20.2
7.1682780134,0.00731628652409,-20.2
16.661246772,0.00720779778825,-20.2
29.2328093821,0.00710091776549,-20.2
45.8813616397,0.00695639595905,-20.2
67.9290816307,0.00680415739767,-20.2
97.1269331485,0.00665525052984,-20.2
135.793721987,0.00650898977211,-20.2
187.00024854,0.00636460445904,-20.2
254.81318156,0.00622342196536,-20.2
344.618020797,0.0060853712447,-20.2
463.546797229,0.00594577033622,-20.2
621.04447757,0.0058087986032,-20.2
829.619055205,0.00567498226545,-20.2
1105.83489751,0.00554424863267,-20.2
1471.62823821,0.0053985518687,-20.2
1956.0492873,0.00521414418824,-20.2
2597.56931154,0.00503603564011,-20.2
3447.13590715,0.00486401105395,-20.2
4572.219105,0.00469786260933,-20.2
0,0.00928831073754,-40.1
12.1088286772,0.00871988156513,-40.1
28.290538797,0.00818623931288,-40.1
49.9150699041,0.00786085373287,-40.1
78.813149411,0.00756286714625,-40.1
117.431281525,0.00727617653445,-40.1
...[more data]...
				
			

The Master Curve tool can then very quickly create the master stress relaxation curve as shown in the following video.

I then used MCalibration to calibrate a linear viscoelastic material model with time-temperature superposition to the master curve. Here I selected the Ansys Linear Elastic model with 15 Prony series terms and the WLF equation. The calibration results are shown in the figure below. The material model captures the master curve response very accurately!

I can now take the calibrated linear viscoelastic model and predict the storage and loss modulus as a function temperature and frequency. The following image shows the predicted response for frequencies between 0.001 rad/s and 1000 rad/s. The peak in loss modulus corresponds to the glass transition, so the glass transition temperature for this material is clearly dependent on the applied frequency. Specifically, Tg=-59°C at f=1000 rad/s, and Tg=-78°C at f=0.001 rad/s. 

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