The Randomly Joined Chain Model: Statistics of A Polymer Chain
Updated: Aug 13, 2021
The freely joined model is the simplest model to visualize the confirmation of the polymer chains. In this simple approach, no interactions between polymer segments are considered. The energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell–Boltzmann distribution.
The general form of the long-chain molecule is shown in Fig. 1, This general form is independent of the precise geometry of the chain, provided only that the number of bonds about which free (or relatively free) rotation can occur is sufficiently large.. This randomly joined chain model consists of a chain of n links of equal length /, in which the direction in space of any link is entirely random and bears no relation to the direction of any other link in the chain. Such a randomly jointed chain automatically excludes valence angle or other restrictions on the freedom of motion of neighboring links.
In order to define the statistical properties of the randomly jointed chain we consider one end A to be fixed at the origin of a Cartesian coordinate system Ox, Oy, Oz and allow the other end B to move in a random manner throughout the available space (Fig. 1). However, though the motion is random, all positions of B are not equally probable, and for any particular position P, having coordinates (x, y, z), there will be an associated probability that the end B shall be located within a small volume element dr in the vicinity of the point P, which for convenience may be taken as a rectangular block of volume dx dy dz. The general probability equation for finding the chain end B has been given by Kuhn (1934, 1936) and by Guth and Mark (1934). This formula gives the probability that the components of the vector r representing the end-to-end distance for the chain shall lie within the intervals x to x+dx, y to y +dy, and z to z+dz, respectively.
L. R. G. Treloar, “The elasticity of a network of long-chain molecules. I,” Trans. Faraday Soc., vol. 39, no. 0, pp. 36–41, Jan. 1943, doi: 10.1039/TF9433900036.