Entangled star polymers

Arm retraction of entangled star polymers demonstrated by an octopus in a fishing net. The circles are permanent topological constraints. 

All the discussions of entangled polymer dynamics above were limited to linear chains. The molecular architecture of the chain (star vs. linear vs. 

The easiest way for the octopus to change the conformation of any of its arms without crossing the obstacles, represented by gray circles in Fig. 9.13, is by retracting that arm. Such arm retraction reduces the length La of its primitive path by forming loops. In Section 9.4.1, we demonstrated that such conformations with primitive path reduced by 

The number of Kuhn monomers in each arm of the star is and the effective spring constant of this harmonic potential is 7. Most of the time, the length of the confining tube of an arm is close to its equilibrium value 

Occasionally, there are large atypical fluctuations of the tube length (with [vl J.- [L) R) that are exponentially unlikely [Eg. (g.Slfl 

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Expressions for Viscosity and Recoverable Compliance of Entangled Star Polymers... 

Entangled star polymers relax by arm retractions with relaxation times and viscosities exponentially large in the number of entanglements per arm NJNg [Eqs (9.58) and (9.61)]. This leads to exponentially small diffusion coefficients [Eq. (9.62)] for entangled star polymers. 

Vega, D. A., Sebastian, J. M., Russel, W. B., and Register, R. A., Viscoelastic properties of entangled star polymer melts comparison of theory and experiment, Macromolecules, 35, 169-177 (2002). 

Nicol, E. Nicolai, T. Durand, D. (2001) Effect of Random End-Linking on the Viscoelastic Relaxation of Entangled Star Polymers, Macromolecules, Vol.34, 5205-5214. 

Tanaka, Y. (2009) Viscoelastic Behaviour for End-Linking of Entangled Star Polymer Application of the Calculation for Branched Polymer to End-Linked Polymer, Nihon Reoroji Gakkaishi, Vol.37,89-95. 

Star-shaped polymer molecules with long branches not only increase the viscosity in the molten state and the steady-state compliance, but the star polymers also decrease the rate of stress relaxation (and creep) compared to a linear polymer (169). The decrease in creep and relaxation rate of star-shaped molecules can be due to extra entanglements because of the many long branches, or the effect can be due to the suppression of reptation of the branches. Linear polymers can reptate, but the bulky center of the star and the different directions of the branch chains from the center make reptation difficult. 

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. 

The tube model gives a direct indication of why one might expect the strange observations on star melts described above. Because the branch points themselves in a high molecular weight star-polymer melt are extremely dilute, the physics of local entanglements is expected to be identical to the linear case each segment of polymer chain behaves as if it were in a tube of diameter a. However, in... 

Fig. 6. Proposed mechanism of entangled dynamics of a star polymer in a melt. Retractions as shown partially renew the tube, beginning with rapid retractions near the free end and much more rarely renewing deeper parts of the molecule...

In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go theta solvents suggest a mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

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... 

The recognition of the two fundamental mechanisms of reptation and arm fluctuation for linear and branched entangled polymers respectively allows theoretical treatment of the hnear rheology and dynamics of more complex polymers. The essential tool is the renormahsation of the dynamics on a hierarchy of timescales, as for the case of star polymers. It is important to stress that experimental checks on well-controlled architectures of higher complexity are still very few due to the difficulty of synthesis, but the case of comb-polymers is an example where good data exists [7]. 

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. 

J. Klein, D. Fletcher, and L. J. Fetters, Dynamics of Entangled Star-branched Polymers, Faraday Symp. Chem., 18, 159-171 (1983). 

Masuda et a/.[30] reported data collected for a series of polystyrene star polymers that seemingly conflict with the discovery made by Quack and Fetters [27]. They showed that the viscosity of polystyrene star polymers was dependent on the number of arms. Specifically, they showed that viscosity increased with the number of branches for a series of polystyrene stars with Mw, arm = 55 000 g/mol and the number of arms ranging from 7 to 39. However, the level of arm entanglement for the polystyrene stars was far lower than that of the polyisoprene stars studied by Quack and Fetters [27]. 

Clearly, in order for the viscosity of star polymers to be independent of the number of branches, a certain level of entanglement needs to be present. 

It is not clear why this transition should occur at such a higher level of arm entanglement for polystyrene stars than for other star polymers. This observation is in direct conflict with the standard assumption that through a proper scaling of plateau modulus (Go) and monomeric friction coefficient (0 that rheological behavior should be dependent only on molecular topology and be independent of molecular chemical structure. This standard assumption was demonstrated to hold fairly well for the linear viscoelastic response of well-entangled monodisperse linear polyisoprene, polybutadiene, and polystyrene melts by McLeish and Milner [24]. 

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