Technical Report: Vibration Damping Materials and Their Performance Measurement (Part 9)

Polymeric materials exist in a state where crystalline and amorphous regions coexist, with each molecular chain randomly intertwined, and their temperature-frequency dependence depends on the viscoelasticity of the molecules. These material molecular chains have various motion modes, and each mode releases energy at specific temperatures and frequencies. Vibration damping performance is maximized in the transition region where this molecular motion becomes active. The local motion of the main chain of material molecules in the glass region changes to micro-Brownian motion of the main chain in the transition region, resulting in the greatest loss. Furthermore, the WLF (Williams, Landel, Ferry) equation, a temperature-frequency conversion formula, also holds true in this transition region. A fundamental concept here is that, based on the temperature characteristics of the elastic modulus of viscoelastic materials shown in Figure 1, similar elastic modulus characteristics can be expressed for frequency by swapping temperature and frequency (using the temperature-frequency conversion rule: Master Curve).

The complex elastic modulus (also known as longitudinal elasticity or transverse elasticity) of viscoelastic materials such as polymers and rubber used as vibration damping materials is a function of temperature T and frequency f. It is known that if a certain temperature T0 is set as the reference temperature, and the complex elastic modulus at temperature T1 is plotted on the vertical axis and frequency on the horizontal axis (logarithmic axis), and this is shifted horizontally, it will match the complex elastic modulus at the reference temperature T0. This means that increasing the temperature corresponds to decreasing the frequency (increasing the time), and decreasing the temperature corresponds to increasing the frequency (shortening the time). This ability to convert temperature changes into frequency changes is called the temperature-frequency conversion law. Since the vibration damping characteristics (dynamic characteristics) of vibration damping materials depend on both temperature and frequency, a three-dimensional representation is necessary to express the elastic modulus and loss coefficient in terms of both parameters. This is where the temperature-frequency conversion law becomes important. The temperature-frequency superposition rule holds well for viscoelastic materials (especially in the glass transition region), and by replacing temperature changes with frequency changes (converted frequency), the dynamic properties of damping materials can be expressed in two dimensions. This means finding a temperature-frequency superposition rule that simultaneously satisfies both the modulus of elasticity and the loss coefficient. In other words, it means finding a law that states that a change in temperature of T is equivalent to a change in frequency of f, and that can be applied simultaneously to both the modulus of elasticity and the loss coefficient over a wide temperature and frequency range. Research on this temperature-frequency superposition rule has been conducted in the field of rheology of polymer materials, and the most famous example is the WLF equation, which shows the temperature dependence of the shift factor αT obtained for many polymers.

The relationship between the shift factor αT and temperature T has been analyzed for various viscoelastic materials, and the Williams-Landel-Ferry (WLF) equation, as shown below, is generally used.

C1 and C2 are constants, chosen to match the relationship between the shift factor αT and [T − T0] found in the previous section.

When amorphous polymers (non-crystalline) exhibit relaxation phenomena, the movable space of the segments, i.e., the free volume, is thought to govern the mobility of the segments, i.e., the internal viscosity. Based on the concept of free volume, Doolittle's viscosity equation...

Taking the natural logarithm,

This shows the relationship between viscosity η and free volume from a molecular perspective. f is the free volume fraction, and B is approximately 1. In this case, Doolittle's equation is as follows:

Here, we compare the viscosity ηT at an arbitrary temperature T with the viscosity ηTg at the glass transition point Tg. The viscosity at these two temperatures is:

Here, f and fg are the free volume fractions at T and Tg. Assuming that the temperature dependence of the relaxation time τ is also of the Doolittle type, we get the following equation.

Next, the question arises as to how to express the temperature dependence of the free volume fraction f. In the relationship between specific volume and temperature, the thermal expansion coefficient of the glassy state does not differ much from that of the crystalline state, but it increases sharply above Tg. Assuming that the temperature dependence of f is roughly similar,

αf is the coefficient of the difference in free volume fraction above Tg.

Common logarithmic transformation

This equation has the same form as the WLF equation (Equation 1) mentioned earlier, and by comparing the coefficients, we can see that the coefficients C1 and C2 in the WLF equation are as follows.

 WLF style 

C1, C2: Constants
Tg: Reference temperature
αT: Shift Factor

The reduced frequency (fr) is expressed as the product of the shift factor and the measured frequency.

Figure 2 shows the relationship between the shift factor (αT: shift coefficient) and the WLF equation, and Figure 3 shows a graph of temperature versus shift factor (log αT) with C1 = 17.44 and C2 = 51.6 in the WLF equation, and Tg as the parameter.