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              Rheology of Polymeric Liquids                                                               247

              temporary couplings between neighboring chains) begin  is imposed on a polymer. The stress remaining in the spec-
              to dominate the resistance to flow. For concentrated solu-  imen at time t can be determined from a material property
              tions and molten polymers, the chain contours are exten-  referred to as the stress relaxation modulus G(t), which
              sively intermingled, so each chain is surrounded all along  for the Rouse model is given by
              its length by a mesh of neighboring chain contours. Re-
                                                                                       ∞
                                                                                 ρRT
              arrangement of macromolecular chains on larger scales        G(t) =        exp(−t/τ p ),    (45)
              is restricted because the chain cannot cross through its            M   p=1
              neighbors. The molecular weight between entanglement
                                                                where ρ is the density, R is the universal gas constant, T
              couplings M e is about one half of M c , i.e., M c ≈ 2M e .In
                                                                is the absolute temperature, M is the molecular weight,
              the case of polymer solutions, both K and M c in Eq. (44)
                                                                and τ p is given by
              change if a solvent is added to the polymer.
                One can interpret the critical molecular weight M c as a              ζb N  2
                                                                                        2
              material constant signifying the lower limit of molecular        τ p =  6π p k B T  ,       (46)
                                                                                       2 2
              weight for which non-Newtonian flow can be observed. It
                                                                in which ζ is the segmental friction coefficient, b is the
              would then be expected that the onset of non-Newtonian
                                                                Kuhn statistical length, N is the number of identical seg-
              behavior is strongly dependent on the molecular weight
                                                                ments, and k B is the Boltzmann constant. Note that the
              and the molecular weight distribution. Above M c , the on-
                                                                largest or terminal relaxation time τ 1 for the Rouse chain,
              set of non-Newtonian behavior occurs at lower shear rates
                                                                i.e., for p = 1 in Eq. (46), is given by
              as the molecular weight increases and as the molecular
                                                                                             2
              weight distribution broadens. A molecular interpretation                (Nζ)(Nb )
              of the viscoelastic behavior of polymeric liquids requires     τ r = τ 1 =  2    ,          (47)
                                                                                       6π k B T
              different concepts for the two regimes: (a) unentangled
                                                                                          2
              regime and (b) entangled regime.                  where the quantities Nζ and Nb describe the chains as a
                Rouse introduced a “bead-spring” model, in which it is  whole and are each proportional to the number of links in
              assumed that the long polymer molecule can be divided  the chain backbone. Hereafter τ r will be referred to as the
              into submolecules and that fluctuations of the end-to-end  Rouse relaxation time.
              length of a polymer molecule follow a Gaussian proba-  When G(t) is known, one can obtain expressions for
                                                                                                         0
              bility function. Then a polymer molecule is considered to  zero-shear viscosity η 0 , steady-state compliance J , dy-
                                                                                                        e

              be replaced by a chain of N identical segments joining  namic storage modulus G (ω), and dynamic loss modulus

              N + 1 identical beads with completely flexible spring at  G (ω) from
              each bead, as schematically shown in Fig. 14. The sem-                 ∞
              inal study of Rouse dealt with dilute polymer solutions,       η 0 =    G(t) dt,            (48)
                                                                                   0
              which was later extended to polymeric melts by Zimm.
              The Rouse theory being valid in the linear regime (i.e., in     0   1   ∞
                                                                             J =        tG(t) dt,         (49)
                                                                              e
              the Newtonian regime), later Zimm extended the Rouse                η 0 2  0
              theory to predict shear-thinning (non-Newtonian) viscos-                ∞

              ity of a polymer, as well as the effect of polydispersity on  G (ω) = ω  G(t) sin ωtdt,     (50)
              shear viscosity.                                                       0
                                                                                      ∞
                We will now discuss predictions of the linear viscoelas-
                                                                          G (ω) = ω    G(t) cos ωtdt,     (51)

              tic properties of unentangled polymer melts based on the               0
              Rouse theory. Consider the situation where a sudden strain
                                                                in which ω is angular frequency applied in oscillatory
                                                                shear flow. Substitution of Eq. (45) into Eqs. (48)–(51)
                                                                gives
                                                                                    2
                                                                             η 0 = (π KρRT/36)M,          (52)
                                                                              0
                                                                             J = 2M/5ρRT,                 (53)
                                                                              e
                                                                                             2 2
                                                                                        ∞
                                                                                  ρRT  
    ω τ p

                                                                           G (ω) =                ,       (54)
                                                                                   M      1 + ω τ
                                                                                               2 2
                                                                                       p=1       p
                                                                                        ∞
                                                                                  ρRT  
    ωτ p

              FIGURE 14 The bead-spring model for a linear polymer mole-  G (ω) =                 .       (55)
                                                                                               2 2
                                                                                   M      1 + ω τ
              cules.                                                                   p=1       p
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