The Rheology of Entangled Polymers

In nondilute polymer solutions and melts, the polymer coils interpenetrate each other enough that the molecular motions of one chain are greatly slowed by the interfering effects of other chains. These interferences are attributed to intermolecular entanglements. 

The effect of entanglements on the relaxation of polymer chains is illustrated in Fig. 3-22, which shows the storage modulus G for a series of polystyrene melts of differing 

It turns out that Me is about a fifth to a half as large as Me- The number of monomers between entanglements is denoted Ne = Me/Mo, where Mo is the molecular weight of a monomer. 

Note that p is independent of molecular weight, since for Gaussian chains M/ [R )q is a constant. At 140°C, the entanglement molecular weight Mg, plateau modulus G, and the tube diameter a (defined below in Section 3.7.1) are related to p by (Fetters et al. 1994) 

polyethylene PS, polystyrene PDMS, polydimethylsiloxane PIB, polyisobutylene PMMA, (atactic) polymethylmethacrylate l,4PBd, 1.4-polybutadiene 1,4PI, 1,4-polyisoprene. 

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Cates ME, McLeish TCB, Marmcd G (1993) The rheology of entangled polymers at very high shear rates. Europhys Lett 21(4) 451—456... 

Thurston GB, Peterlin A (1967) Influence of finite number of chain segments, hydrodynamic interaction, and internal viscosity on intrinsic birefringence and viscosity of polymer solutions in an oscillating laminar flow field. J Chem Phys 46(12) 4881 4884 Treloar LRG (1958) The physics of rubber elasticity. Oxford University Press, London Tsenoglou C (2001) Non-Newtonian rheology of entangled polymer solutions and melts. Macromolecules 34 2148-2155... 

As discussed in Chapters 10 and 11, rheology can be very sensitive to the microstructure of liquids. For example, the viscosity of entangled polymer melts depends on molecular weight to the 3.4 power, T]o Ml . Equilibrium creep compliance is very sensitive to molecular weight distribution. The yield stress and low frequency G are good indicators of the flocculation state of colloids. Extensional viscosity can be an important indicator of bread dough quality (Padmanabhan, 1993). 

Because of the exponential dependence of relaxation time on the potential, the relaxation of star polymers is extremely sensitive to the strength of the potential and therefore to the value of V and of Mg, which sets the value of Z. The correct value of v has been controversial a discussion of this and of non-quadratic corrections to Eq. 9.2 can be found in McLeish [14]. Recent fine-scale simulations using lattice models and real-space pearl necklace models of entangled polymers provide some justification for the quadratic potential and for the value v= 3/2 [15, 16]. As mentioned in Section 6.3, the relationship between and is also open to revision [17]. Hence, adjustments of either or v might be needed to obtain quantitative predictions of the rheology of star polymers. 

The DE equation was the first detailed molecular theory for the rheology of polymer melts and since its introduction has been the basis for almost all theories for the dynamics and rheology of entangled polymers. The predictions of the DE equation have been explored in detail in many publications (for a review, see [ 1,4,17,18]). Summing up these reviews, we can say that while the DE equation inspired all that followed and does capture some aspects of the rheology of melts, even for the simplest case of monodisperse linear polymers it is not a realistic, quantitative, theory except in a few special types of deformation. 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

The recognition of the two fundamental mechanisms of reptation and arm fluctuation for linear and branched entangled polymers respectively allows theoretical treatment of the hnear rheology and dynamics of more complex polymers. The essential tool is the renormahsation of the dynamics on a hierarchy of timescales, as for the case of star polymers. It is important to stress that experimental checks on well-controlled architectures of higher complexity are still very few due to the difficulty of synthesis, but the case of comb-polymers is an example where good data exists [7]. 

The effects of molecular entanglements on the rheology of polymers above the glass transition temperature are large and have been recognized for a long One of the most prominent effects is the appearance of a rubbery... 

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