Doi/Edwards limits

To describe the polymer stress in this equation, one can probe any of rheological constitutive models proposed for the long-chain branched polymers the partially extending convection (PEC) model of R. Larson, [117], the molecular stress function (MSP) theory of M. Wagner et al. [118,119], the modified extended pom-pom (mXPP) model of M.H. Verbeeten et al. [120], etc. Here, the PEC model has been chosen as it can be easily tuned to describe the overshoot position for a wide class of polymers by changing a value of the non-linear parameter Thus, = 1 in the case of linear polymer (Doi-Edwards limit), 0 < < 1 in the case of branched polymers, and = 0 in the... 

Table 1. Theoretical dependences on time (f), angular frequency (co), and molecular mass (M) predicted by the tube/reptation model for the mean squared segment displacement and the intrasegment spin-lattice relaxation time in the four Doi/Edwards limits. The factors Q, Cj/, Qff, and Cr are frequency and molecular mass independent constants ...

Segment diffusion in pores suggests itself as a typical model scenario representing the premisses of the tube/reptation model. NMR diffusometry is suitable to probe the time or length scales of the Doi/Edwards limits (II)de> (III)de and beyond. Since the wall adsorption effect in the PHEMA system is expected to be negligible, one can therefore expect that the reptation features of the anomalous segment diffusion regime, especially with respect to limit (III)de> are faithfully rendered by the experiments. 

Fig. 46. Theoretical mean squared segment displacement of a chain confined in a randomly coiled tube versus time according to the harmonic radial potential theory [70]. The tube diameter d is given in multiples of the Kuhn segment length b. The crossover tendency to free, unconfined Rouse chain dynamics with increasing tube diameter is obvious. The mean squared displacement is given in units the diffusion time t in units of the segmental fluctuation time Tj. The chain length was assumed to be N= 1,600 Kuhn segments. The three anomalous Doi/Edwards limits (see Table 1) are reproduced with finite tube diameters...

The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes. 

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... 

This result differs somewhat from the expression obtained using the Doi-Edwards model (Eq.40), and it gives a larger departure from neo-Hookean behavior for uniaxial extension (A q>endix II and Fig. 9). In the limit of small deformations the entire contribution to stress comes from the first term in Eq. 62. The entanglement contribution to the infinitesimal shear modulus is predsely the same as the Doi-Edwards expression for the plateau modulus (Eq. 37)... 

The limiting laws predicted by the Doi-Edwards model, i/o and a M , are thus... 

Even aside from the rather puzzling factor of 1/3 difference, the Curtiss-Bird result does not reduce to Doi-Edwards in the limit of linear viscoelasticity. For example, in Curtiss-Bird the dynamic viscoaty = G"(fi))l(o changes from % at tu = 0 to IBe... 

As the theory is based on the theoretical framework of the Doi-Edwards theory, it is referred to as the extended Doi-Edwards theory from the theoretical consideration at the same time, since the theory includes the intramolecular motions on top of the reptation process, physically it is referred to as the extended reptation theory. Considering that the Doi-Edwards theory covers the nonlinear region, while the extended reptation theory is limited to the linear region, the term extended reptation theory is used throughout this book instead of the extended Doi-Edwards theory. 

For almost two decades following the early 1960s there had been relatively limited research activities on the rheology of branched flexible homopolymers. However, in 1988 McLeish (1988) extended the concept of the Doi-Edwards tube model, which had been developed for linear flexible homopolymers (see Chapter 4), to describe the dynamics of branched flexible homopolymers. Since then, during the past several years, other investigators (Blackwell et al. 2000 Bourrigaud et al. 2003, Inkson et al. 1999 McLeish and Larson 1998 McLeish et al. 1999 Shie et al. 2003 Verbeeten et al. 2001) have actively engaged in further development of this theory. Such efforts have... 

The upper limit on the summation is r, the degree of polymerization, which is M/Mq. In a polydisperse system, this approach must be modified. Montfort etal. [48] account for the effects of polydispersity in two ways. First, they use the double reptation concept with the Doi-Edwards kernel function to account for constraint release, but they also let the relaxation times depend on the molecular weight distribution, a concept originally proposed by Graessley [49]. Specifically, they represent the terminal relaxation time in a polydisperse system as the harmonic average of the reptation time and a tube renewal time, Tp which depends on the molecular weight distribution. 

Equation 11.45 is the equation for the orientation tensor S. This equation is similar to that for the Doi-Edwards or DEMG theory see Eq. 11.8. Equation 11.46 is the stretch equation. It is also similar to its counterpart in the DEMG theory see Eq. 11.9. The main difference in the theory for the pom-pom is that the stretch X is limited to be equal to or less than q, the number of arms on each end of the backbone. If the stretch attains the value q, the arms start to be pulled into the backbone tube. The length of arm pulled into the backbone tube is defined by S, which is measured in units of numbers of entanglements. Thus, 1 - c = the fraction of each arm that has been pulled into the backbone tube. The evolution equation... 

Part I summarizes the main ideas of de Gennes, Doi and Edwards about tube models and reptation in entangled polymer systems. Attention has been limited to properties for which predictions can be made without invoking the independent alignment approximation macromolecular diffusion, linear viscoelasticity in the plateau and terminal regions, stress relaxation following a step strain from rest of arbitrary magnitude, and equilibrium elasticity in networks. 

A very elegant linear model of reptating macromolecules was proposed by Doi and Edwards [63], This model can be considered as a limiting case of the above model of Curtiss and Bird [47] at a oo. It was proposed that, in this case of limiting anisotropy, the macromolecule moves inside the tube of radius... 

Much research in the last few decades focused on the simulation of LCPs for various processes. Suitable rheological constitutive equations are required for this simulation. Leslie-Ericksen (LE) theory describes the flow behaviour and molecular orientation of many LCPs. LE model is limited to low shear rates and weak molecular distortions. But at high shear rate, the rate of molecular distortions is too fast. Doi and Edwards developed their model to describe the complex dynamics of macromolecules at high shear rate (Doi and Edwards 1978). Doi theory is applicable for lyotropic LCPs of small and moderate concentrations. Due to the complex nature of Doi theory, it is also challenging for simulation. Leonov s continuum theory of weak viscoelastic nematodynamics, developed on the basis of thermodynamics and constitutive relations, consider the nematic viscoelasticity, deformation of molecules as well as evolution of director. 

---