Grüneisen Numbers for Polymeric Solids

Recent considerations by Wada make it appropriate to extend a previous discussion of Grüneisen ``numbers'' for polymers and other molecular solids. Wada hypothesized that the proper Grüneisen constant for polymers is ${\gamma }_G{\mathrm{=\alpha V/\beta C}}_{\mathrm{vb}}$ , where C<sub>vb</sub> is the interchain contribution to the heat capacity at constant volume; α, β, and V are the volumetric thermal expansivity, the compressibility, and the volume. To the extent that a polymer can be treated as a vibrational lattice, the hypothesis appears to be consistent with ${\gamma }_G{\mathrm{=\Sigma \gamma }}_i{\mathrm{\epsilon (x}}_i{\mathrm{)/\Sigma \epsilon (x}}_i)$ , averaged over Einstein oscillator functions ε(x<sub>i</sub>) with x<sub>i</sub>=hv<sub>i</sub>/kT. At low temperatures, ε(x<sub>i</sub>) is much larger for the low‐frequency modes, so that they tend to determine γ<sub>G</sub> below the Debye −θ. Since γ<sub>i</sub>=−∂ lnν<sub>i</sub>/∂ lnV and since low ν<sub>i</sub> are likely to be more sensitive to changes in V, it is expected that γ<sub>G</sub> will be larger for molecular solids than for metallic, ionic, or covalent crystals. Earlier predictions and Wada's calculations agree that γ<sub>G</sub>≈4 might be typical for polymers and suggest that ∂γ/∂T>0. The correlation, Eα<sub>l</sub><sup>2</sup>≈15 N/m<sup>2</sup>°K<sup>2</sup>, between modulus E and linear expansivity α<sub>l</sub> led to the prediction, now verified, that there should be relations between the harmonic and anharmonic moduli. The anharmonic coefficients in the relation ΔV/V<sub>0</sub>=a<sub>1</sub>p+a<sub>2</sub>p<sup>2</sup>+a<sub>3</sub>p<sup>3</sup>+⋯are a<sub>2</sub>=C<sub>1</sub>a<sub>1</sub><sup>2</sup> and a<sub>3</sub>=C<sub>3</sub>a<sub>1</sub><sup>3</sup>, where for metals C<sub>2</sub>=−2.5±0.5, and for polymers C<sub>2</sub>=−4.0±0.1 and C<sub>3</sub>=8.8±0.2. A phenomenological theory based on a ``bundle of tubes'' model is developed which is in good agreement with data and according to which γ_G=−C₂ and ${\mathrm{d\gamma }}_G{\mathrm{/dT\sim \alpha C}}_2$ . The relation of γ_G to intermolecular potential functions also is discussed and some qualtitatively encouraging results are obtained.