Technical Report on Vibration-Damping Materials and Their Performance Measurement 9

Polymer materials exist in a state where each molecular chain is randomly intertwined, with a mixture of crystalline and amorphous regions, and their temperature and frequency dependence relies on the molecular viscoelasticity. The molecular chains of these materials have various modes of motion, each releasing energy at specific temperatures and frequencies. The damping performance is maximized in the transition region where this molecular motion becomes active. The local motion of the main chains of the material molecules in the glassy region transforms into micro-Brownian motion of the main chains in the transition region, resulting in the largest losses. Additionally, the W.L.F. (Williams, Landel, Ferry) equation, which is a temperature-frequency conversion formula, holds true in this transition region. Here, an important fundamental concept is that based on the temperature characteristics of the elastic modulus of viscoelastic materials shown in Figure 1, by interchanging temperature and frequency (using the temperature-frequency conversion rule: Master Curve), it is possible to represent similar elastic modulus characteristics with respect to frequency.

The complex elastic modulus (both longitudinal and transverse) of viscoelastic materials such as polymer materials and rubber used as vibration-damping materials is a function of temperature T and frequency f. By setting a certain temperature T ₀ as the reference temperature and plotting the complex elastic modulus at temperature T ₁ on the vertical axis and frequency on the horizontal axis (logarithmic scale), it is known that this aligns with the complex elastic modulus at the reference temperature T ₀ when shifted parallel in the horizontal direction. This means that increasing the temperature corresponds to decreasing the frequency (lengthening the time), and decreasing the temperature corresponds to increasing the frequency (shortening the time). The ability to convert changes in temperature to changes in frequency in this way is referred to as the temperature-frequency conversion principle. Since the vibration-damping characteristics (dynamic characteristics) of vibration-damping materials depend on both temperature and frequency, a three-dimensional representation is necessary when trying to express the elastic modulus and loss factor concerning these two parameters. The temperature-frequency conversion principle is crucial here. The temperature-frequency conversion principle is well established in viscoelastic materials (especially in the glass transition region), allowing the dynamic characteristics of vibration-damping materials to be expressed in two dimensions by replacing temperature changes with frequency changes (equivalent frequency). This implies finding a temperature-frequency conversion principle that simultaneously satisfies both characteristic values of the elastic modulus and loss factor. In other words, it will lead to discovering a law that states that a change in temperature by T is equivalent to a change in frequency by f, applicable simultaneously over a wide range of temperatures and frequencies for both the elastic modulus and loss factor. Research on this temperature-frequency conversion principle is conducted in the field of rheology of polymer materials, with the most famous being the WLF equation, which shows the temperature dependence of the shift factor α _T derived for many polymers.

The relationship between the shift factor α _T and temperature T has been analyzed for various viscoelastic materials, and the formulation organized as the Williams-Landel-Ferry (WLF) equation is generally used as follows.

C ₁ and C ₂ are constants, chosen to be consistent with the relationship between the shift factor α _T and [ T − T ₀] obtained in the previous section.

When amorphous polymers (not crystalline) exhibit relaxation phenomena, it is considered that the space available for segmental movement, namely the free volume, governs the mobility of the segments, that is, the internal viscosity. Based on the free volume concept, Doolittle's viscosity equation states that...

Taking the natural logarithm,

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

Here, we compare the viscosity η _T at any temperature T with the viscosity η _Tg at the glass transition temperature T _g. The viscosities at these two temperatures are,

Here, f and f _g are the free volume fractions at T and T _g. The temperature dependence of the relaxation time τ is also considered to be of the Doolittle type, as shown in the following equation.

The next issue is how to express the temperature dependence of the free volume fraction f. In the relationship between specific volume and temperature, the thermal expansion coefficient in the glass state does not differ much from that in the crystalline case, but it increases sharply above T _g. If the temperature dependence of f is generally similar to this,

α _f is the coefficient of the difference in free volume fraction at temperatures above T _g,

common logarithm transformation

This equation has the same form as the previous WLF equation (Equation 1), and by comparing the coefficient parts, it can be seen that the coefficients C ₁ and C ₂ in the WLF equation are as follows.

 WLF equation 

C ₁, C ₂: Constants (Constant)
T _g: Reference Temperature
α _T: Shift Factor (Shift Factor)

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

Figure 2 shows the relationship between the shift factor (α _T: moving coefficient) and the WLF equation, while Figure 3 presents the graph of temperature versus shift factor (log α _T) with C ₁ = 17.44, C ₂ = 51.6, and T _g as parameters.