A polymer melt in the linear-viscoelastic regime is well fitted by two Maxwell modes…

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

A polymer melt in the linear-viscoelastic regime is well fitted by two Maxwell modes acting in parallel, so that its relaxation modulus is

G(t) = G_1e^{-t/\tau_1} + G_2e^{-t/\tau_2}, \qquad t \gt 0,

with

G_1 = 0.30\,\mathrm{MPa}

\tau_1 = 0.010\,\mathrm{s}

G_2 = 0.050\,\mathrm{MPa}

\tau_2 = 1.0\,\mathrm{s}

. ((a)) Starting from

G^*(\omega) = i\omega\int_0^\infty G(t)e^{-i\omega t}\,\mathrm{d}t

, derive general expressions for the storage modulus

G'(\omega)

and the loss modulus

G''(\omega)

of a single Maxwell mode of modulus

G_0

and relaxation time

\tau

. ((b)) Hence write

G'(\omega)

and

G''(\omega)

for the two-mode melt above. ((c)) Evaluate

G'

and

G''

at an imposed angular frequency

\omega = 100\,\mathrm{rad\,s^{-1}}