Star Dynamic dilution

Fig. 7. The effective free-energy potentials for retraction of the free end of arms in a mon-odisperse star polymer melt. The upper curve assumes no constraint-release, the lower two curves take the dynamic dilution approximation with the assumptions (Ball-...

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

Using the Rouse result, the left hand side of Eq. (30) is just l/2t. In the case of star polymers, using the approximate result for t(x) from Eq. (21) and the corresponding dynamic dilution result 0(x)=(l-x) the criterion becomes... 

So the criterion that the effective constraint-release must be fast enough to allow local pieces of umelaxed chain to explore any dilated tube fully confirms the assumption of dynamic dilution for nearly the whole range of relaxation timescales exhibited by star polymers. 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

Fig. 14. Data (points) for G (co) and G (co) for a range of compositions of a blend of two polyisoprene stars of molecular weights 28 and 144 kg mol The fractions of the bigger star are in order 0.0,0.2,0.5,0.8 and 1.0. Curves are theoretical predictions of the tube model with co-operative constraint release treated by dynamic dilution [56]. The choice of 2.0 rather than 7/3 for the dilution exponent p is compensated for by taking M = 5500 kg mol" ...

The correction to the coefficient of in the dynamic dilution (cubic) term in the potential (compare Eq. 22 for the pure star case) arises from the way the difference in arm molecular weights affects the fraction of unrelaxed arm at the same timescale. 

Exact form of the distribution function and following from it thermodynamical properties of linear pol5meric chain conformation are strictly determined in the SARW statistics for ideal diluted [28 and concentrated [29] solutions. Here this approach is spread on the regular polymeric stars in diluted and concentrated solutions with the description of their thermo-d5mamical and dynamical properties. 

Self-avoiding random walks statistics completely describes the thermodynamic and dynamic properties of the polymeric stars in diluted solutions as the function on a length and the number of rays in concentrated solutions additionally as the function on the concentration of polymer. 

Ball, R. G., and T. C. B. McLeish. 1989. Dynamic dilution and the viscosity of star polymer melts. Macromolecules 22 1911-1913. 

Comparing this result with Eq. 9.2, we find that dynamic dilution speeds up relaxation of a star arm by the exponential of an order unity prefactor times a large number Z3. Thus, the degree of acceleration of the relaxation can be truly enormous, i.e., factors of millions or billions. Ball and McLeish point out that inclusion of the dynamic-dilution effect is essential if truly quantitative, or even quahtative, predictions of the relaxation of star polymers are to be obtained. 

Finally, we remark that the idea of self-consistent dynamic dilution was applied first by Marrucci [20] to the case of monodisperse linear polymers, and was then adapted by BaU and McLeish [11] to monodisperse stars. We also note that theories combining reptation, primitive path fluctuations, and constraint release by dynamic dilution have been applied successfully by Milner and McLeish and coworkers to monodisperse linear polymers [21], monodisperse stars [13], bimodal star/star blends [22], and star/linear blends [23], as well as H-branched polymers [24], and combs [25]. The approach taken for all these cases is similar at early times after a small step strain, the star arms and the tips of linear molecules relax by primitive path fluctuations and dynamic dilution. At some later time, if there are linear chains that reach their reptation time, there is a rapid relaxation of these linear chains. This produces a dilation of the effective tubes that surround any remaining unrelaxed star arms by constraint-release Rouse motion (see Section 7.3). Finally, after dilation has finished, the primitive path fluctuations of remaining portions of star arms begin again, in the dilated tube. We refer to this set of theories for stars, linears, and mixtures thereof as the Milner-McLeish theory . The details of the Milner-McLeish theory are beyond the scope of this work, but the interested reader can learn more from the original articles as well as from McLeish and Milner [26], McLeish [14], Park and Larson [27], and by Watanabe [19]. 

The case of star/linear blends is a challenging one, because the description of constraint release that works best for pure star polymers is dynamic dilution, while for pure linear polymers, double reptation , or some variant of it, seems to be the better description. However, Milner, McLeish and coworkers [23] have developed a rather successful theory for the case of star/ linear blends. In the Milner-McLeish theory, at early times after a step strain both the star branches and the ends of the linear chains relax by primitive-path fluctuations combined with dynamic dilution, the latter causing the effective tube diameter to slowly increase with time. Then, at a time corresponding to the reptation time of the linear chains, the tube surrounding the unrelaxed star arms increases rather quickly, because of the sudden reptation of the linear chains. The increase in the tube diameter would be very abrupt, if it were not slowed by inclusion of the constraint release-Rouse processes, which leads to a square-root-in-time decay in the modulus (see Section 7.3). With this formulation, the Milner-McLeish theory yields very favorable predictions of polybutadiene data for star/linear blends see Fig. 9.13, where the parameters have the same values as were used for pure linears and pure stars. 

With respect to the arm relaxation time, we have to reconsider the role of dynamic dilution when both arms and backbones are present. For a melt of pure monodisperse stars, the Ball-McLeish theory for dynamic dilution predicts that the effective volume fraction of entangling chains decreases towards zero as the arms relax see Section 9.3.2. However, arms of... 

Notice that when the volume fraction of arms 0 approaches unity, Eq. 9.18 yields the expression for dynamically diluted star arms, while when 0 approaches zero, Eq. 9.18 reverts to the arm relaxation time in the absence of dynamic dilution. Eq. 9.18 should then be used in Eq. 9.13 to compute the backbone relaxation time. In Eq. 9.10 the number of entanglements Z should be taken to be the number of entanglements of the backbone with other backbones, because the arms have released all their constraints on the time scales over which the backbone is moving. Since the backbone volume fraction is 0, the number of backbone/backbone entanglements is 0(, where is the number of undiluted backbone entangle-... 

The symbols in Fig. 9.18 are experimental data of Daniels etal. [25]. The solid and dotted lines are predictions of the hierarchical model with monodisperse and polydisperse arms and backbone molecular weights, respectively. The parameters are given in the caption of Fig. 10.6 with a= 4/3 the parameter value = 1/12 is used in Eq. 9.9 for the branch-point mobility, as suggested by Daniels et al. [25]. Once the arms relax, the backbone is assumed to reptate in a tube dilated by the dynamic dilution due to relaxation of the star arms. 

Frischknecht, A. L., Milner, S. T. Self-diffusion with dynamic dilution in star polymer melts. 

Shanbhag, S., Larson, R. G., Takimoto, J., Doi, M. Deviations from dynamic dilution in the terminal relaxation of star polymers. Phys. Rev. Lett. (2001) 87, article no. 195502... 

The paper is organized in the following way In Section 2, the principles of quasi-elastic neutron scattering are introduced, and the method of NSE is shortly outlined. Section 3 deals with the polymer dynamics in dense environments, addressing in particular the influence and origin of entanglements. In Section 4, polymer networks are treated. Section 5 reports on the dynamics of linear homo- and block copolymers, of cyclic and star-shaped polymers in dilute and semi-dilute solutions, respectively. Finally, Section 6 summarizes the conclusions and gives an outlook. 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

The second example deals with dilute solutions of 12-arm stars, where the arm are built up by symmetrical diblocks of protonated and deuterated but otherwise identical monomers [149], Figure 48 displays the reduced relaxation rates r12/Q3- In addition the effective 1/e decay rates of the dynamic structure... 

In another series of experiments [149] the correlation between structure and dynamics was investigated on dilute solutions of 12-arm PS star systems (Mw = 14.9 104 g/mol) in d-tetrahydrofurane, where either only one or all 12 arms were protonated (labelled). 

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