Monodisperse Stars

3 Comparison of Milner-McLeish Theory to Linear Viscoelastic Data 9.3.3.1 Monodisperse Stars 

is required to give the good fits to data shown in Figs. 9.6 through 9.8 this value disagrees with that inferred from the experimental plateau modulus, suggesting that a = 1 is a less appropriate value for the dilution exponent. With determined, only is left as an adjustable parameter. A best-fit value in the case a= 4/3 leads to = 2.8 10 s, while a=l leads to 

Models such as the Milner-McLeish or dual constraint model appear to give good agreement with experimental star data for 1,4-polybutadiene, 1,4-polyisoprene, and polystyrene [8]. 

However, the most common commercial polymer for which long-chain branching is common is polyethylene, and it is therefore of great interest to see if similarly good agreement is obtained for star polyethylenes as well. Direct synthesis of polyethylenes cannot be carried out anionically, however, so ideal, model branched (or for that matter linear) polyethylenes cannot be made directly. 

Zero-shear viscosity versus molecular weight for nearly monodisperse linear hydrogenated 1,4-poly butadienes at 190 °CThe lines showthe fits ofthedatatotheMilner-McLeish model 

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Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

As we conjectured in the introduction, the fundamental role of topology in this approach to entangled polymer dynamics would indicate that changes to the topology of the molecules themselves would radically affect the dynamic response of the melts. In fact rheological data on monodisperse star-branched polymers, in which a number of anionically-polymerised arms are coupled by a multifunctional core molecule, pre-dated the first application of tube theory in the presence of branching [22]. Just the addition of one branch point per molecule has a remarkable effect, as may be seen by comparing the dissipative moduli of comparable linear and star polymer melts in Fig. 5. 

Figure 16 Complex shear modulus of two nearly monodisperse star-shaped polybutadiene samples at 25°C ( ) 3 branches, total M=164 000 ( ) 4 branches M= 45 000.

Figure 3.38 Reduced storage modulus G / versus reduced frequency mti for monodisperse star...

Problem 3.12 Suppose you have monodispersed star-branched melts of some polymer with molecular weights 20,000, 40,000, and 100,000. You measure the longest relaxation times of the two lower-molecular-weight samples and find Ti = 1 sec and 10 sec, respectively, for M =... 

Equation 98 is also applicable to monodisperse star polymers except in this case F(t) is given by Eq. 68, R(t) requires the eigenvalues appropriate to the star structure in Eq. 99, and Arj must be replaced by r calculated from Eq. 85. Values of r can be estimated by using the properties of for small m when Nb is large (see Appendix IV). 

An excellent summary of — M data by Fetters et al. provides estimates for these parameters in both 6 and good solvents for a number of linear polymersJ The parameters in Eq. (19) are not only influenced by the experimental conditions (solvent, temperature) but are also affected by the polymer s structure. For monodisperse stars, it has been found that increasing the number of arms decreases K g while u remains identical to that of the linear polymer (see Fig. 1). However, for randomly branched polymers, it has been found that u is closer to 0.5 and in some cases much lower. Such low values of this exponent might be considered as an indication of the branched polymer being in an unperturbed state however, this is not the case. An explanation as to why u is so small for randomly branched polymers has been found using fractal behavior, and an overview is given by Burchard. ... 

The dielectrically obtained (p (f) data and the CR time data were utilized in Equation (3.40) to evaluate a t), and the normalized modulus G(f)/G deduced from the partial-DTD picture was calculated by utilizing this a (t) in Equation (3.38). The modulus calculated for the PI/PI blends (Watanabe et al., 2004b) is shown with the solid curves in Figure 3.7. These curves are in agreement with the data (circles), demonstrating the validity of the partial-DTD picture. The validity was confirmed also for the monodisperse star/ Cayley-tree PI (see Watanabe et al., 2006,2008). 

Watanabe, H., T. Sawada, and Y. Matsumiya. 2006. Constraint release in star/star blends and partial tube dilation in monodisperse star systems. Macromolecules 39 2553-2561. 

Fig. 3.25. A comparison of complex viscosity and steady-state viscosity in reduced form for (a) nearly monodisperse linear polymers and (b) nearly monodisperse star polymers.

The variation of recoverable compliance with molecular weight also differs for linear and nonlinear polymers. In contrast to the behavior of nearly monodisperse linear polymers, for which becomes a constant 2/ G ) beyond about 5Mg, for stars simply continues to increase in direct proportion to Mb, which is exemplified by the comparison of data for linear polystyrene [71] and four-arm polystyrene stars [66] in Fig. 3.47. Experimentally, the behavior of for nearly monodisperse stars, irrespective of branch-point functionality, is described well by [52]... 

In an extremely important paper, Heuer, et al. used photomicrography to examine electrophoresis of synthetic DNA stars in linear polyacrylamide having an estimated molecular weight of 5-6 MDa(13). Polymer concentrations covered 0.5-10 g/1, the nominal overlap concentration (estimated in several ways) being near 1.6 g/1. Stars were created via the approach of Seeman(22) four short (44 bp) synthetic oligonucleotides, whose complementary sequences cause them to self-assemble into a four-arm star, and whose arms bind X-phage DNA ends, were synthesized, leading to monodisperse star polymers. 

Watanabe, H., Ykoshida, H., Kotaka T. Entanglement in blends of monodisperse star and linear polystyrenes. 1. Dilute blends. MacromoL (1988) 21, pp. 2175-2184... 

The simplest possible type of branched polymer is a monodisperse star. In some respects, monodisperse stars are actually easier to consider than linears, because for stars one can neglect reptation. This leaves only the relaxation mechanisms of primitive path fluctuations, constraint release, and high-frequency Rouse modes that need to be considered to describe the linear... 

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]. 

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... 

Such data of star-branched chains have been often compared with the prediction of bead-spring modds, namdy, Zimm-Kilb and Rouse-Ham modds that are based on the Zimm and Rouse modds for linear chains. For monodisperse star chains having / arms of molecular weight the rdaxation moduli deduced from the Zimm-Kilb and Rouse-Ham modds can be commonly expressed as ... 

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