Determine Mesh Size through Monomer Mean-Square Displacement

A dynamic method to determine the main parameter of the tube theory through monomer mean-square displacement is discussed in this paper. The tube step length can be measured from the intersection of the slope- $\frac{1}{2}$ line and the slope- $\frac{1}{4}$ line in log-log plot, and the tube diameter can be obtained by recording the time at which $g_1$ data start to leave the slope- $\frac{1}{2}$ regime. According to recent simulation data, the ratio of the tube step length to the tube diameter was found to be about 2 for different entangled polymer systems. Since measuring the tube diameter does not require $g_1$ data to reach the slope- $\frac{1}{4}$ regime, this could be the best way to find the entanglement length from microscopic consideration.