Flory's argument balances the entropic elastic free energy of stretching an ideal…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Variational and asymptotic methods

Soft Matter at the Thermal Scale, Ch The size and shape of a polymer chain

Problem

Flory's argument balances the entropic elastic free energy of stretching an ideal chain against the free-energy cost of segment-segment overlap. In

d

dimensions, for

N

Kuhn segments of length

b

confined to a region of size

R

, write

\frac{F(R)}{k_BT} \sim \frac{R^2}{Nb^2} + \frac{vN^2}{R^d},

where

v

is the excluded volume of a segment (of dimension

[\mathrm{length}]^d

d

dimensions). ((a)) Minimise

F(R)

with respect to

R

and show that

R\sim (vb^2)^{1/(d+2)}N^{3/(d+2)}

, so that the Flory exponent is

\nu=3/(d+2)

. ((b)) Evaluate

\nu

for

d=1

and

d=4

, and comment on what each limit means physically for a self-avoiding walk. ((c)) A polymer in a good solvent (

d=3

) has

N=800

Kuhn segments of length

b=0.60\,\mathrm{nm}

and excluded volume

v=0.50\,\mathrm{nm^3}

. Compute the Flory radius

R_F=b(v/b^3)^{1/5}N^{3/5}

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