A flexible polymer chain has Kuhn length and Kuhn segments (Kuhn-segment molar mass

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Polymer solutions and blends

Problem

A flexible polymer chain has Kuhn length

b = 1.0\,\mathrm{nm}

and

N = 5000

Kuhn segments (Kuhn-segment molar mass

M_0 = 120\,\mathrm{g\,mol^{-1}}

, so

M\approx 6\times10^5\,\mathrm{g\,mol^{-1}}

), dissolved in a good solvent (Flory exponent

\nu = 3/5

). The bulk polymer density is

\rho = 1.10\,\mathrm{g\,cm^{-3}} = 1100\,\mathrm{g\,L^{-1}}

. ((a)) Estimate the overlap volume fraction

\phi^\ast \approx N^{1-3\nu}

, and convert it to an overlap concentration

c^\ast

\mathrm{g\,L^{-1}}

using

c^\ast \approx \rho\phi^\ast

. ((b)) At a semidilute volume fraction

\phi = 0.050

, estimate the correlation length (mesh size)

\xi \approx b\phi^{-3/4}