Page 310 - Academic Press Encyclopedia of Physical Science and Technology 3rd Polymer
P. 310

P1: GQT/MBQ  P2: GPJ Final Pages
 Encyclopedia of Physical Science and Technology  EN014C-660  July 28, 2001  17:14






               248                                                                            Rheology of Polymeric Liquids


               Note that K in Eq. (52) is defined by
                                       2
                                     ζb N 2
                               K =          .            (56)
                                    π k B TM  2
                                     2
                 The Rouse model allows us to determine that stress σ
               contributed by the polymer chains by
                                      ∞

                                 σ =    σ p ,            (57)
                                     p=2
               where σ p denotes the stress contributed by the polymer
               chain at the pth mode (p = 1, 2,..., ∞), which can be
               evaluated from
                                    ∂

                              1 + τ p  σ p = G 0 δ,      (58)
                                   ∂t
               where ∂/∂t is the upper convected derivative, τ p is the
               relaxation times defined by Eq. (46), G 0 = ρRT/M, and
               δ is the Kronecker delta function.
                 In steady-state shear flow the Rouse model predicts

                                                                 FIGURE 15 Plots of log G versus log ω for a nearly monodis-
                                       ∞
                                  ρRT  
                         perse polystyrene with molecular weight of 9 × 10 at various tem-
                                                                                                    3
                             σ =         τ p ˙γ,         (59)
                                   M                             peratures: ( ) 120 C, ( ) 130 C, ( ) 140 C, and ( ) 150 C.
                                                                                              ◦
                                                                                                          ◦
                                                                                      ◦
                                                                              ◦
                                       p=1
                                        ∞
                                  2ρRT  
   2  2                 with molecular weight M less than M c . Figure 15 gives

                            N 1 =         τ ˙γ ,         (60)    plots of log G  versus log ω, and Fig. 16 gives plots of
                                            p
                                   M
                                        p=1                      log G versus log ω, measured at four different temper-

                            N 2 = 0.                     (61)    atures for a monodisperse polystyrene with M = 9000,
                                                                 which is much lower than the M c = 36,000 of polystyrene.
               Use of Eq. (46) in Eq. (59) gives the zero-shear viscosity,
                                                                 It can be seen in these figures that (a) at a constant value
                                     2
                                   ρb ζ N A N  2
                              η 0 =          ,           (62)
                                      36M
               where N A is Avogadro’s number. Since N ∝ M, it can be
               concluded from Eq. (62) that η 0 ∝ M and it is independent
               of shear rate ˙γ , i.e., the Rouse model cannot predict shear-
               dependent viscosity. Equation (62) can be rewritten as

                                     2
                                   ρb ζ 0 N A
                                     0
                             η 0 =      2   M,           (63)
                                     36M
                                        0
               where b 0 is the length of monomer, ζ 0 is the monomeric
               friction coefficient, and M 0 is the molecular weight of
               monomer. The ζ 0 is one of the most important properties
               of macromolecules, which can be calculated from Eq. (63)
               by measurement of η 0 . With the aid of Eq. (63), Eq. (46)
               can be rewritten as
                                     6η 0 M
                               τ p =        ,            (64)
                                     2 2
                                    π p ρRT
               and thus the terminal (Rouse) relaxation time becomes
                                       6η 0 M
                              τ r = τ 1 =  2  .          (65)
                                       π ρRT
                                                                 FIGURE 16 Plots of lot G versus log ω for a nearly monodis-

                 The Rouse model is useful to predict the linear vis-  perse polystyrene with molecular weight of 9 × 10 at various tem-
                                                                                                    3
               coelastic properties (Eqs. (52)–(55)) of polymer melts  peratures: ( ) 120 C, ( ) 130 C, ( ) 140 C, and ( ) 150 C.
                                                                                                          ◦
                                                                              ◦
                                                                                      ◦
                                                                                              ◦
   305   306   307   308   309   310   311   312   313   314   315