Viscoelastic Properties as Functions of the Distribution of Activation Energies

1. Assuming that the elementary molecular deformation process conforms to the Maxwell model, and that the molecular elastic force G_i and viscous force η_i are functions (of unspecified forms) of the free energy of activation F*, the following expressions for the dynamic modulus G_d and dynamic viscosity (internal friction) η_d are obtained: $$G_d=\frac{1}{A}{\displaystyle {\int }_0^{\infty }}\frac{{\omega }^2{\tau }_i^2{G}_i{\mathrm{\phi (F}}^{*})}{{\omega }^2{\tau }_i^2\text{+1}}{\mathrm{dF}}^{*}$$ , and $${\eta }_d=\frac{1}{A}{\displaystyle {\int }_0^{\infty }}\frac{{\eta }_i{\mathrm{\phi (F}}^{*})}{{\omega }^2{\tau }_i^2\text{+1}}{\mathrm{dF}}^{*}$$ , where A=area of sample, τ_i=G_i/η_i, ω=vibration frequency, and φ(F*)dF*=the number of elementary processes having activation energies lying between F* and F*+dF*. 2. By employing an expression relating the relaxation time τ_i with F* for the elementary process, and adopting the so‐called ``box'' distribution of relaxation times, the following explicit form for the distribution of activation energies is deduced: $$\mathrm{\phi =}\mathrm{const}{\text{(1/kTF}}^{*}{\text{−1/F}}^{\text{*2}})$$ , where k=Boltzmann's constant and T=absolute temperature. When the box distribution, as represented by this explicit form for φ, is introduced into the foregoing expressions for G_d and η_d, the integrated results are found to predict temperature and frequency dependencies which are in gratifying agreement with experiment.