A polybutadiene melt has plateau modulus

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 dynamics and viscoelasticity

Problem

G_N^0 = 1.2\,\mathrm{MPa}

, density

\rho = 900\,\mathrm{kg\,m^{-3}}

, at

T = 298\,\mathrm{K}

. A dilute solution of the same polymer, with

N = 5000

Kuhn segments of length

b = 0.90\,\mathrm{nm}

per chain, sits in a solvent of viscosity

\eta_s = 0.55\,\mathrm{mPa\,s}

. ((a)) Using the entanglement plateau relation

G_N^0 \approx \tfrac45\rho RT/M_e

(with

R

the gas constant: the permanent-network modulus

G=n_sk_BT

with strand density

n_s=\rho N_{\rm Av}/M_e

, times the quoted correction

4/5

), find the entanglement molar mass

M_e

. ((b)) Using the ideal-chain result

R_g^2 = Nb^2/6

for the dilute coil, find

R_g

, then estimate the Zimm relaxation time

\tau_{\rm Z}\approx \eta_sR_g^3/k_BT

and the diffusion coefficient

D=k_BT/(6\pi\eta_sR_g)

of the dilute coil.