A chain of Kuhn segments (length

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 chain of

N

Kuhn segments (length

b

, friction coefficient

\zeta

per segment) in an entangled melt is confined by its neighbours to a tube whose diameter

a

is set by the entanglement number

N_e

through the random-walk relation

a^2 = N_eb^2

(the

N_e

segments between entanglements perform a random walk, as any ideal chain does). The primitive path along the tube has contour length

L = (N/N_e)a

. The chain's centre of mass diffuses along this curvilinear coordinate with the Rouse diffusion coefficient of the whole chain,

D_{\rm c} = k_BT/(N\zeta)

, because the total friction on the chain is

N\zeta

. ((a)) Derive the scaling of the disengagement (terminal) time

\tau_d

with

N

, taking

\tau_d

to be the time for the chain to diffuse a curvilinear distance

L

along the tube,

\tau_d \sim L^2/D_{\rm c}

. ((b)) The plateau modulus

G_N^0

depends on

N_e

but not on

N

(for

N\gg N_e

). Using

\eta_0 \approx G_N^0\tau_d

, give the scaling of

\eta_0

with

N

. ((c)) The chain's real-space diffusion coefficient satisfies

D \sim R^2/\tau_d

, where

R^2 = Nb^2

is the mean-square end-to-end distance of the ideal chain. Give the scaling of

D

with

N

. ((d)) If

N

is tripled while

N_e

b

\zeta

and

T

are unchanged, by what factor do

\tau_d

\eta_0

and

D

change according to these scalings?