End-to-End Distance (Displacement Length) Calculation

[Sperling-173] [Flory-399] [Williams-234]

Flory defines:

displacement length- the distance between one end of the polymer chain and the other for a coiled polymer
contour length- the distance between one end of the polymer chain and the other when the polymer is stretched out. The contour length could be considered the maximum possible displacement length.

In the above figure both polymer chains have a degree of polymerization of 12.

There are two types of calculations:

The first is the random walk, which uses the correct bond length for the monomer, but ignores the angle between the one monomer line and the next, as shown in the figure below:

[Flory-414] and [Hiementz-52] show which Flory refers to as a 'freely jointed chain.'

and [Williams-230] shows which he refers to as a 'freely orienting chain model.'
- h (or r) is the end-to-end distance, or the displacement length.
- n (or sigma) is the number of monomers in the polymer chain
- l is the length of the monomer
The second builds in the correct bond angles. Notice that in the pictures above for displacement length and contour length that each angle is 120 degrees.

There are two angles to consider; in our two dimensional representation you can only see one.

If there were just then we could orient the first monomer in the plane of the screen, and the polymer would stay in the plane of the screen. This angle is defined by three atoms.

The angle is called a dihedral or torsional angle, and it gives the polymer the three dimensional character. This angle is defined by four atoms.

(Sperling-74) (Flory-418) (Hiementz-58)

The characteristic Ratio has to do with whether or not the polymer stretches out (because it is in a good solvent) or coils up. The length

and the angle

are constrained by the geometry of the monomer, but this is not so for the angle

. Of course, everything has to be constrained by something:

Mathematically, the variables determine whether or not a polymer is stretched out or tightly coiled up.
Physically, the solvent is the reason for the coil expansion of a polymer chain.
The angles depend on the solvent. Oh, it should make sense that if the growing chain was about to run into itself, that the phi variable would be influenced in such manner as to veer the growing polymer chain away from collision.

Hiementz p. 55- "To obtain isolated polymer chains, a solvent must be present. The solvent might be selectively excluded or imbibed by the coil, depending on the free energy of interaction, and thereby perturb the coil dimension."

Flory p. 424- "The configuration of the polymer molecule must depend also on its environment. In a good solvent, where the energy of interaction between a polymer element and a solvent molecule adjacent to it exceeds the mean of the energies of interaction between the polymer-polymer and solvent-solvent pairs, the molecule will tend to expand further so as to reduce the frequency of contacts between pairs of polymer elements. In a poor solvent, on the other hand, where the energy of interaction is unfavorable (endothermic), smaller configurations in which polymer-polymer contacts occur more frequently will be favored.

Last Update- September 2, 1995- wld