An unentangled melt of chains with Kuhn segments

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

An unentangled melt of chains with NN Kuhn segments, each of length bb and friction coefficient ζ\zeta, has the Rouse relaxation modulus G(t)=ρRTM∑p=1Ne−p2t/τR,τR=ζb2N26π2kBT,G(t) = \frac{\rho RT}{M}\sum_{p=1}^{N}e^{-p^2t/\tau_{\rm R}}, \qquad \tau_{\rm R} = \frac{\zeta b^2N^2}{6\pi^2k_BT}, where ρ\rho is the melt's mass density, MM the molar mass of one chain, RR the gas constant and TT the temperature. ((a)) Using η0=∫0∞G(t) dt\eta_0 = \int_0^\infty G(t)\,\mathrm{d}t and ∑p=1∞1/p2=π2/6\sum_{p=1}^{\infty}1/p^2 = \pi^2/6 (a good approximation for N≫1N\gg1, since the series converges quickly and no modes exist beyond p∼Np\sim N), show that η0=ρRTζb2N236MkBT.\eta_0 = \frac{\rho RT\zeta b^2N^2}{36Mk_BT}. ((b)) Writing RT=NAvkBTRT = N_{\rm Av}k_BT and M=NM0M = NM_0, where M0M_0 is the molar mass of one Kuhn segment, show that at fixed monomer mass density ρ\rho, η0\eta_0 is proportional to NN. ((c)) Two unentangled polyethylene melts have the same monomer mass density but N=50N=50 and N=200N=200 Kuhn segments per chain respectively. By what factor does η0\eta_0 differ between the two melts?

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