Symmetric Hindered Rotation

C = [(1 - cos theta)(1 + cos theta)^-1] [(1 - a)(1 + a)^-1]

where the zero of the torsion angle, phi, is assigned in the cis state. The average of the cosine of the torsion angle, denoted here by a, is obtained from the symmetric torsion potential energy function and temperature as the ratio of two integrals.

a = integral cos phi exp (-E/kT) d phi / integral exp (-E/kT) d phi

• If the torsion potential energy function favors torsion angles near the trans state, the value of C increases.
• If the torsion potential energy function favors torsion angles near the cis state, the value of C decreases.
• The freely rotating chain is recovered if all torsion angles are equally probable, i. e., if E is independent of phi, or if T is infinite.
• C indistinguishable from that of the freely rotating chain is obtained with any symmetric hindered potential energy function that yields a = 0.

Temperature coefficient

This model envisions a temperature dependence for C, because the average of the cosine of phi depends on kT. The temperature coefficient of the mean square unperturbed dimensions is usually reported as (d ln C / d T), which is equivalent to (d ln <r²>₀ / d T). The model described here has

d ln C / d T = - [2(1 - a)^-2] d a / d T

The order of magnitude is often 10^-3 deg^-1 (of either sign) for typical flexibile chains. For polyethylene, the temperature coefficient is -0.0011 deg^-1.

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