Polymer Extension

#Physics

$\displaystyle z=\frac{{\left\langle{L}\right\rangle}}{L_{\text{tot}}}$

• $\displaystyle {\left\langle{L}\right\rangle}$ is the expected polymer length
• $\displaystyle L_{\text{tot}}$ is the total length of the polymer when completely stretched out

$\displaystyle z_{\text{1D}}=\tanh(\beta fa)$

• Polymer can either move left or right
• $\displaystyle \beta$ is [coldness]
• $\displaystyle f$ is the force applied to the stretch
• $\displaystyle a$ is the length of one segment

$\displaystyle z_{\text{3D lattice}}=\frac{2\sinh(\beta fa)}{4+2\cosh(\beta fa)}$

• Polymer can move in $\displaystyle \pm e_{i}$ direction

$\displaystyle {\left\langle{L_{\text{3D}}}\right\rangle}=-\frac{\partial G }{\partial f}=Na\left[ \coth(\beta fa)-\frac{1}{\beta fa} \right]$

• Polymer can move in any direction in 3D