Rubber Physics (Part 6)

The rubber band has a volume $V= A L$ and a mass $M = K N m$ with the number, $K$ , of polymer chains and the number, $N = L/l$ , of chain elements per polymer. The mass of an repeat unit is $m = 8 \cdot m_\text{H} + 5 \cdot m_\text{C} \approx 1.13 \cdot 10^{-25}\,\text{kg}$ . The density thus results $\rho = \frac{M}{V} = \frac{K m}{A l} \quad \Rightarrow \quad K = \frac{A l \rho}{m}$ The force on a single polymer chain is the fraction $f = f_\text{ext}/K$ of external force. The excitation energy results $\Delta E = 2 l f$ as shown in the last problem. Therefore, the energy ratio yields $\begin{aligned} \frac{\Delta E}{k_B T} &= \frac{2 l f_\text{ext}}{K k_B T} = \frac{2 m f_\text{ext}}{A \rho k_B T} \\ &\approx \frac{2 \cdot (1.13 \cdot 10^{-25}\,\text{kg}) \cdot (10 \,\text{N}) }{(3 \cdot 10^{-6} \,\text{m}^2) \cdot (950 \,\text{kg}/\text{m}^3) \cdot (1.38 \cdot 10^{-23} \,\text{J}/\text{K}) \cdot (293 \,\text{K})} \\ &\approx 0.2 \end{aligned}$ Thus, the thermal energy $k_B T$ and the excitation energy $\Delta E$ have a comparable order of magnitude, so that the flipping of individual chain links in the rubber can be excited by the thermal movement. Although the external force energetically favors parallel alignment of all the chain elements, the thermal motion causes the chain elements to flip randomly so that the polymer chains are only partially unraveled and have only a fraction of their actual length.