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              Rheology of Polymeric Liquids                                                               251

              It should be pointed out that Eq. (75) was derived with-  parameters with the molecular weight, molecular weight
              out invoking fluctuations of contour length (i.e., without  distribution, and molecular structure. The second reason
              considering the Rouse motion in a reptating chain). The  is that such an expression can be used, together with the
              main idea behind the derivation of Eq. (75) is that since  equation of continuity, to solve the equations of motion
              experimental data for η 0 is usually obtained from shear  that relate the rheological parameters to flow conditions
              measurement, stress effect must be included into the rep-  and die geometry.
              tation model; i.e., when a polymer is subjected to shear  In the past, much effort has been put into developing
              flow, a relaxation of polymer chains to reptate around the  rheological models for predicting the rheological prop-
              entangled junctions must be taken into consideration, in  erties of polymeric liquids. There are two approaches to
              addition to the reptation of the overall center-of-mass mo-  describing the rheological behavior of viscoelastic poly-
              tion. For very high-molecular-weight polymers, the ap-  meric liquids. One approach is to view the material as
              proach used to derive Eq. (75) yields             a continuum, and then to describe the response of this
                                                                continuum to stress or strain by a system of mathematical

                                        2
                              0.02916ρζ 0 b N A  3              statements having their origin in the theories of continuum
                                        0
                        η 0 =                 M .       (76)
                                    2
                                  M M  2                        mechanics. Another approach is to describe the rheolog-
                                    e  0
                                                                ical behavior of the material from molecular considera-
              It has been shown that the combination of Eqs. (75) and
                                                                tions. In either approach, one basically has to establish
              (76) predicts a gradual transition from a 3.4 power law to
                                                                a relationship (or relationships) on the basis of either a
              a 3.0 power law for 200M e < M < 2000M e , which com-
                                                                rigorous mathematical physical theory or empiricism, or
              pares very favorably with experiment.
                                                                both, which describes the deformation of a fluid in terms
                The tube model predicts, for steady-state shear flow,
                                                                of the components of the deformation history or the rate
              three material functions, η, N 1 , and N 2 , and they are ex-
                                                                of deformation and the components of the stress.
              pressed by
                                                                  As an important application of rheology, one can cite
                                                                processing of polymeric materials. The polymer process-

                                    ∞
                        η = (G 0 / ˙γ )  m(s)F 1 ( ˙γ s) ds,  (77)  ing industry constantly strives to improve its existing pro-
                                   0
                                                                cessing techniques and to develop new ones with the pur-
                                 ∞

                       N 1 = G 0 ˙γ  sm(s)F 1 ( ˙γ s) ds,  (78)  pose of finding optimum processing conditions for each
                                 0                              new material that comes on the market. Therefore, the
                                                                development of a method or methods for evaluating the
                                  ∞

                                    m(s)F 2 ( ˙γ s) ds,  (79)
                       N 2 =−G 0                                processibility of a new polymeric material and for im-
                                 0
                                                                proving existing processing conditions is an essential step
              where ˙γ denotes shear rate, m(s) is a memory function,
                                                                in improving the mechanical or other properties of the
              and F 1 ( ˙γ s) and F 2 ( ˙γ s) are complicated expressions not
                                                                final product. On the basis of the information presented
              shown here.
                                                                here, it can be concluded that proccesibility is very closely
                                                                related to the rheological properties of the polymeric ma-
              VI. CONCLUDING REMARKS                            terials in the molten state and that a good understanding
                                                                of any polymer processing operation requires knowledge
              In studying the rheology of a specific type of material,  of several branches of science and engineering, such as
              one needs to perform the following three basic steps: (1)  polymer chemistry, polymer physics, polymer rheology,
              define the flow field in terms of the velocity components  and mass and energy transports.
              and the coordinates that are most appropriate; (2) choose
              a rheological equation of state for the description of the  SEE ALSO THE FOLLOWING ARTICLES
              material under deformation, and (3) decide which of the
              experimentaltechniquesavailablearemostsuitableforde-  FLUID DYNAMICS (CHEMICAL ENGINEERING) • LIQUID
              termining the rheological properties of the material under  CRYSTALS • LIQUIDS,STRUCTURE ANDDYNAMICS • PLAS-
              consideration.                                    TICS ENGINEERING • POLYMER PROCESSING • POLYMERS,
                There are two reasons for seeking a precise mathemat-  MECHANICAL BEHAVIOR • RUBBER,SYNTHETIC • STERE-
              ical description of the rheological models, which relate  OCHEMISTRY
              the state of stress to the state of deformation. The first is
              that such an expression can be used to identify the signifi-  BIBLIOGRAPHY
              cant rheological parameters characteristic of the materials
              and to suggest the experimental procedure for measuring  Bird, R. B., Armstrong, R. C., and Hassager, O. (1987). “Dynamics of
              them. One would then like to correlate the rheological  Polymeric Liquid,” 2nd edition, Vol. 1, Wiley, New York.
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