Edwards theory

The EMT analysis indicated that the stress relaxes in proportion to the number of bonds removed. The initial linear decrease of E/Eq with is intuitively appealing and is the basis for many linear constitutive theories of polymers. An example is the Doi-Edwards theory of viscoelasticity of linear polymer melts [49] in which... 

In the Doi-Edwards theory the plateau modulus and the tube diameter are related according to Eq. (40). Inserting Eq. (40) into (52) we finally obtain... 

If the ideas of Marrucci [69] are correct and the non-monotonic predictions of the simple Doi-Edwards theory need to be modified in the case of polymer melts (for a recent development see [78]), then an explanation will be required for the apparent difference at high shear rates between melts and wormlike micelle solutions. There is also evidence that ordinary entangled polymer solutions do exhibit non-monotonic shear-stress behaviour [79]. As in the field of linear deformations, it may be that a study of the apparently more complex branched polymers in strong flows may shed light on their deceptively simple linear cous-... 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

M. Doi and S. F. Edwards. Theory of Polymer dynamics, Oxford University Press, (1986). 

According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a veuies as Thus the number of monomers between entanglements will scale as < ) . Accordingly, the reptation time x (relation 3-14) should be proportional to (]) as a first approximation, the zero-shear viscosity tio and the steady-state compliance J should respectively scale as [Pg.133]

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

Doi and Edwards (1978) and Kuzuu and Doi (1980) have solved the Smoluchowski equation (6-47)-(6-48) for simple shearing and elongational flows, and they obtained predictions of rheological behavior that are similar to those of the reptation theory for concentrated flexible polymers discussed in Section 3.7.5.1. Figure 6-19, for example, shows the shear-rate-dependence of the shear viscosity and first and second normal stress coefficients predicted by the Doi-Edwards theory for semidilute rods these results are similar to those predicted by the Doi-Edwards theory for entangled flexible molecules. At... 

The Doi-Edwards theory provides expressions for G%, r[, and D that contain two adjustable parameters the friction coefficient and the primitive path step length L. The friction coefficient can be obtained from the relationship between the viscosity and molecular weight in the Rouse theory [Eq. (11.36)] or from the relaxation spectrum discussed below. Moreover, the step length a can be determined from the plateau modulus G%. Actually, according to the Doi-Edwards theory... 

M Doi, SF Edwards. Theory of Polymer Dynamics. Oxford UK Clarendon Press, 1986. 

M Doi, SF Edwards. Theory of Liquids. New York Oxford Univ Press, 1986. PG de Gennes. Scaling Concepts in Polymer Physics. 2nd ed. Ithaca NY Cornell Univ Press, 1985. 

While several experimental results [22-25] give similar exponents for the length and concentration to those of the Doi-Edwards theory [11], the computations of Bitsanis et al. [26] show that both theories are valid in the appropriate concentration regimes. For dimensionless number densities in the range 5 < vZ <90 the rotational diffusivity obtained is consistent with Fixman s theory [19] and for vL 90 the Doi-Edwards theory [11] is found to match the computational results. 

Using the diffusivities predicted by the Doi-Edwards theory [11] and the homogeneous rate constant given by - 7.5 1/mol s, Agarwal and Khakhar... 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

Fig. 6. Comparisons of viscosity and recoverable compliance predictions by the Doi-Edwards theory with experimental observations. The predicted tio h too large, but its chmn length dependence is slightly weaker than observed. The predicted J is too small, but independent of chain length as observed. The dashed lines indicate predictions of the Rouse model...

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

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