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Doi-Edwards tube

Note that the squared diameter of the Doi-Edwards tube relates to our intermediate length as follows... [Pg.125]

Two theories of viscoelasticity in reptating chain systems have appeared since the original Doi-Edwards publications, the theory of Marrucci and coworkers and the theory of Curtiss and Bird ° They differ in various ways from the Doi-Edwards tube model and from the model suggested in Part II. It is difficult and probably premature to provide detmled criticisms at the present time, but it is perhaps worthwhile to point out at least a few of the differences. [Pg.107]

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... [Pg.236]

The Doi-Edwards tube theory predicts the following damping function for tensile extension [24] ... [Pg.379]

Step strain experiments have been carried out using lubricated squeeze flow to determine the damping function for biaxial extension h e ). The Doi-Edwards tube model (without lA assumption) prediction of this function is as follows [24] ... [Pg.385]

Fig. 7. Illustration of a polymer chain (the tagged chain) confined in the fictitious tube of diameter d formed by the matrix. The contour line of the tube is called the primitive path having a random-walk conformation with a step length a=d. The four characteristic types of dynamic processes (dotted arrow lines) and their time constants Zs, Zg, Zr, and za defined in the frame of the Doi/Edwards tube/reptation model are indicated... Fig. 7. Illustration of a polymer chain (the tagged chain) confined in the fictitious tube of diameter d formed by the matrix. The contour line of the tube is called the primitive path having a random-walk conformation with a step length a=d. The four characteristic types of dynamic processes (dotted arrow lines) and their time constants Zs, Zg, Zr, and za defined in the frame of the Doi/Edwards tube/reptation model are indicated...
A further inconsistency refers to the treatment of shear stress relaxation. The Doi/Edwards tube cannot exist without long-living intermolecular correlations pertaining to a time scale of the order Nevertheless,... [Pg.36]

Figure 4 shows the transient evolution of the extensional viscosity at varying strain rates for PHA Sample I. The plots follow the start-up to reach a steady state plateau value. At short times, the curves fall on top of one another. At high strain rates, we observe a slight decrease in the steady state extensional viscosity for both PHA samples. This could indicate that the two PHA samples are strain-rate thinning, as predicted by the Doi-Edwards tube model [9]. The Trouton Ratio (Tr), defined as the ratio of the extensional viscosity to the zero-shear viscosity of the polymer, was also investigated for the two PHA samples. For both PHA samples, the Trouton Ratio maintained the Newtonian value of Tr=3 at low strain rates. [Pg.2150]

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... [Pg.53]

Figure 6.21 Comparison of Graessley and Doi-Edwards models for normalised viscosity versus normalised shear rate. Also shown is an estimate of the role of short time Rouse relaxation mechanisms within the tube... Figure 6.21 Comparison of Graessley and Doi-Edwards models for normalised viscosity versus normalised shear rate. Also shown is an estimate of the role of short time Rouse relaxation mechanisms within the tube...
The Doi-Edwards, reptation based model makes specific predictions for the relaxation dynamics of different portions of a polymer chain. Specifically, the relaxation of the chain ends is predicted to be substantially faster than the relaxation of the center. This is a result of the reptation dynamics, which have the ends first leaving the confines of the tube. Using polymer chains that were selectively deuterated either at the ends or at the middle, Ylitalo and coworkers [135] examined this problem and found that the Doi-Edwards model was able to successfully predict the observed behavior once the effects of orientational coupling was included. The same group further explored the phenomena of orientational coupling in papers that focused on its molecular weight [136] and temperature [137]... [Pg.198]

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). [Pg.128]

Fig. 3.12 (a) A pom-pom with three arms at each branch point (q = 3). At short times the polymer chains are confined to the Doi-Edwards tuhe. Sc is the dimensionless length of branch point retraction into the tube X is the stretch ratio where L is the curvilinear length of the crossbar and Lq is the curvilinear equilibrium length, (b) Relaxation process of a long-chain-branched molecule such as LDPE. At a given flow rate e the molecule contains an unrelaxed core of relaxation times t > g 1 connected to an outer fuzz of relaxed material of relaxation t < g 1, behaving as solvent. [Reprinted by permission from N. J. Inkson et al., J. Rheol., 43(4), 873 (1999).]... [Pg.128]

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... [Pg.129]

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. [Pg.58]

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. [Pg.59]

Each molecule is surrounded by the above distribution of chain lengths and the tube renewal time has to take into account the distribution of the attached constraint release times. For a monodisperse sample, Graessley [17] defines the constraint release time x from the Doi-Edwards relaxation function F(t) such as ... [Pg.124]

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]


See other pages where Doi-Edwards tube is mentioned: [Pg.199]    [Pg.195]    [Pg.296]    [Pg.327]    [Pg.497]    [Pg.366]    [Pg.741]    [Pg.211]    [Pg.236]    [Pg.134]    [Pg.199]    [Pg.195]    [Pg.296]    [Pg.327]    [Pg.497]    [Pg.366]    [Pg.741]    [Pg.211]    [Pg.236]    [Pg.134]    [Pg.201]    [Pg.93]    [Pg.268]    [Pg.269]    [Pg.208]    [Pg.242]    [Pg.243]    [Pg.200]    [Pg.127]    [Pg.228]    [Pg.196]    [Pg.127]    [Pg.127]    [Pg.350]    [Pg.107]    [Pg.111]   
See also in sourсe #XX -- [ Pg.199 ]




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