Rheology: Short Note 4
The Question: How to understand and construct a master curve (using Excel)?
The Answer: There are number of literatures available (ppts, notes, papers, book chapters etc.) which speaks about master curve: a typical term used for polymeric materials. This note is not about the theoretical background on mastercurve. It is rather a simple note to understand the fundamentals about the construction of master curve along with the step by step process of constructing master curve using a simple excel sheet. So lets begin! Let us understand this by starting with viscoelastic materials. Viscoelastic, as the name suggests are materials which displays both elastic and viscous response. This response is a function of three important parameters: a. temperature, b. rate of loading (frequency) and c. amplitude of stress/strain. When the response of the material to any given loading condition depends on the first two factors only, i.e. temperature and frequency, it can be termed as a linear viscoelastic material. However, if the response will depend on all the three parameters, it will behave as a non-linear viscoelastic material. The behaviour of non-linear viscoelastic materials are more complex in the sense of its mechanical behaviour. This is the reason that asphalt binder/bitumen is assumed to behaving as a linear visco-elastic material (though it maynot be always true!). Hence, most of the theories underlying the response of asphalt binders considers that its rheological behaviour is a function of time (frequency) and temperature. With this background let us understand how time and temperature plays role in influencing the mechanical behaviour of asphalt binders and then try to explore the so called “Time-temperature superposition principal” (TTSP). Let us take an unconventional example: Take a chewing gum (mouth gum) , chew it for some time, take it out from your mouth and roll it into a small ball. This chewing gum is a form of polymer. Now try to stretch it slowly ( this ‘slowly’ indicates the rate of loading you are giving to the gum). You will see that it can be stretched to a long distance (rubbery/viscous). Now again roll it into a small ball as before. Try to grip it with your thumb and forefinger (of both the hands) and pull it fast (here ‘fast’ again indicates the rate of loading you are giving to the gum). You will notice that there is an instantaneous break (it doesnot stretch as before) (stiff!!). Hence rate of loading (slow/fast) plays an important role. Understanding the role of temperature is comparatively easier. Instead of the chewing gum, take a small sample of asphalt binder and roll it into a ball. Keep it inside a freezer for some time. Take it out: you will see that it is still in the shape of a ball and hard now (stiff!!). This ball you keep in a oven (may be at 60-70 deg. Cel.) for the same time. When taken out, you won’t see the ball anymore: it has liquefied (not literally). So this is how temperature influences its behaviour. Now you see, the behaviour’s corresponding to temperature and rate of loading can be superimposed. A fast rate
of loading (as in case of chewing gum) shows a similar behaviour as in case of lower temperature (binder kept in freezer). They both behaves like a stiff material. Whereas, a higher temperature phenomena is similar to that of slow loading rate. They both makes the material viscous (chewing gum could be stretched, binder started to flow).
Now that’s time-temperature superposition. Let us understand it in a better way using a complex modulusfrequency relationship. Below is a figure showing the variation of complex modulus with frequency at two different temperatures (40 and 60 °C) for an asphalt binder (VG 10 here). As said before, with increase in rate of loading (frequency), stiffness (complex modulus in figure below) increases. Also, with increase in temperature, stiffness decreases (so the curve of 60 °C is on the lower side). Interestingly, point 1 and 3 has the same value of modulus. Point 1 is at 40 °C while
point 3 belongs to 60°C (but at different frequencies!!). So, if the curve of 40 °C is shifted to the right, it will display the merge with the curve of 60 °C, increasing the domain of the x-axis (i.e a reduced frequency). This has been shown in the next figure. Mathematically, we are multiplying a factor (shift factor) to each value of ‘x’ of 40 °C, keeping the value of ‘y’ same. This leads to shift in the curve. Remember, the curve is a log-log plot.
Steps for its construction using Excel: Let us take an example: The steps will be illustrated using screen shots of excel sheet to make it clearer.
Step 1: Figure below is a screen shot of data inserted in excel from the measurements taken using dynamic shear rheometer. The binder is VG 30. Measurements (frequency sweep test) of complex modulus (in Pa) was made at eight different temperatures, 10-80 °C.
Step 2: Create a portion for shift factors: Keep the values to be 1 for all the temperatures now.
Step 3: Create columns for calculation reduced frequency. Here, for example P4 = N4 x B3, i.e Reduced frequency at a temperature T (P4) = Actual frequency (B3) x Shift Factor (N4). All the values in the figure below are same because, shift factors are kept 1. Do this for all temperatures.
Step 4: Suppose we want to create the master curve at a reference temperature of 50 °C. The concept is that you have to find the appropriate shift factors at different temperatures so that a superimposed curve is obtained. The shift factor at the reference temperature will be 1. For temperatures higher than the reference temperature, shift factor will be less than, 1, decreasing with increase in temperature (this means that you are trying to shift the curves on the left). For temperatures lower than the reference temperature, shift factor will be greater than, 1, increasing with decrease in temperature (this means that you are trying to shift the curves on the right). There are various methods in finding the appropriate shift factors Williams-landel- Ferry (WLF) and Arrhenius equations are the most common. But they have some boundary conditions, which is usually not satisfied for asphalt binders. It is recommended that one may use “manual” shifting procedure. Keep varying the shift factors till a smooth curve is obtained.
For the example above, the following shift factors were obtained:
The reduced frequencies so obtained are as follows:
The master curve so obtained can be seen below:
You may see now that the range of values in x-axis has been has been increased. It was from 0.159-15.9 Hz. Now it is from 0.0002-1000000 Hz.
The above were the simple steps for construction of master curve using excel. One may develop their own program for automated construction of master curve. One such method can be found in the link below. https://www.researchgate.net/publication/323003406_Equivalent_Slope_Method_for_Construction_of_Ma ster_Curves
Dr. Nikhil Saboo, Assistant Professor Indian Institute of Technology (BHU), Varanasi Phone: (+91) 7579048967 Email:
[email protected] 