Explanation for the 3.4-power law for viscosity of polymeric liquids on the basis of the tube model

The viscosity η₀(M) of polymeric liquids of molecular weight M is calculated on the basis of the tube model formulated by Doi and Edwards (ref. 3). The contour length fluctuation of polymers along the tube, which was neglected in ref. 3, is now explicitly taken into account. The result is where M_c = 2M_e, and M_e is the molecular weight between the entanglement points. This result is numerically close to the empirical 3.4-power law, η₀(M) = η₀(M_c)(M/M_c)^3.4, for 10M_c ? M ? 100M_c but approaches the result in ref. 3 for very high molecular weight. We thus conclude that the 3.4-power law is actually an approximate expression for the real curve which slowly approaches the asymptotic form calculated in ref. 3.