Relaxation time melts, molar mass

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

Consider a PDMS melt with molar mass A/ = 6 X 10 g mol . The relaxation time of a Kuhn monomer is tq — 10 s. The molar mass of a Kuhn monomer is Mo = 381g moC and the molar mass of an entanglement strand is jWg 12000g mol ... 

Reptation involves a one-dimensional (head-first) diffusion of the polymer in whidi lateral motion of the diain is restricted as though the chain were emfined to a tube whose effective dimensions reflect the mean entanglement spadng in the polymer melt It is widely appreciated that for reptating chains, the dqrendence of the centre-of-mass diffusion coefiBcient on molar mass (Af) is Dcm oc who eas for Rouse chains, D a oc M in addition, the melt viscosity (and terminal relaxation time) increase as for entangled chains [37]. 

We have already met such a molar mass dependence in Eqs. (6.119) and (6.120), when formulating the average viscoelastic relaxation time r of polymer melts. Roughly speaking, f gives the time required by a chain for a complete conformational reorganization. This also implies a full reorientation of the end-to-end distance vector of the chain. This is exactly this motion that shows up in the dielectric normal mode. 

Equation (8.96) provides a description of the normal mode, giving the relaxation strength as well as the relaxation time. Notice in particular that the observed molar mass dependence of Tnm for non-entangled melts as shown in Fig. 6.22,... 

Gpi denotes the plateau modulus and Tq, is the mean relaxation time of the Rouse mode part, agreeing with the mean relaxation time of a Rouse system of chains with Nr equal to F7r,c- Because the first term on the right-hand side of Eq. (8.117) is constant, for the molar mass dependence of the viscosity of entangled melts we obtain the expression... 

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