Molecular Chain

Consider a one-dimensional chain consisting of N molecules which exist in two configurations, α, β, with corresponding energies ε_α, ε_β and lengths a and b. The chain is subject to a tensile force f.

a) Write the partition function Z_N for the system.

b) Calculate the average length ⟨L⟩ as a function of f and the temperature τ.

c) Assume that ε_α > ε_β and a > b. Estimate the average length ⟨L⟩ in the absence of the tensile force f = 0 as a function of temperature. What are the high- and low-temperature limits, and what is the characteristic temperature at which the changeover between the two limits occurs?

d) Calculate the linear response function

Produce a general argument to show that χ > 0.

SOLUTION

a) Consider one link of the chain in its two configurations: α and β. The energy of the link is

(1)

The partition function for the entire chain is given by

(2)

b) The average length of the chain may be found from the partition function:

(3)

c) If f = 0, (3) becomes

(4)

If ε_{α, β} << τ, high temperature,

(5)

If ε_{α, β} << τ,

(6)

where we let δ = (ε_­α - ε_β)/τ. The changeover temperature is obviously ε_­α - ε_β.

d) From (3),

(7)

At small f, (7) becomes

(8)

Therefore,

(9)

as it should, since (for the specified direction of the tensile force f) it corresponds to a thermodynamic inequality for a system at equilibrium: