Branch-Point Motion

How can a branch point move The repertoire of polymer movements that we have considered up to now reptation, primitive path fluctuations, and Rouse motion within the tube do not allow for branch-point motion, at least not directly. Yet, clearly, the branch points do move, for if they did not, branched polymers, including stars, would have zero center-of-mass diffusivity. 

With this picture, the diffusion coefficient of the branch point is approximately = x /(2 t ), where x is the hopping distance that the branch point moves every time the arm disentangles itself [50, 51]. An estimate of the hopping distance is the tube diameter a, since the tube diameter is the distance over which the branch point is localized by the entanglements with its neighbors. Hence, we can estimate that 

What if the star has more than three arms In a star with q arms, a naive extension of the above picture would suggest that q - 2 of the arms must retract simultaneously to allow the chain to reptate along the path defined by the remaining two unretracted arms. This would imply that the time between hops would be of the order = 

Tpre exp[(q - 2) R Z3 ], which, for large q and large Z3, would be an enormously long time. However, the center of mass diffusivity of a star polymer, measured by Shull et al. [52], is only modestly dependent on the number of arms in the star, decreasing by a factor of around 40 as the number of arms (q) in a polystyrene star increases from 3 to 12. This modest dependence of star diffusivity on the number of branches shows that the above naive picture of branchpoint motion must be wrong. 

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Shanbhag, S. and Larson, R.G. (2004) A slip-link model of branch-point motion in entangled polymers. Macromolecules, 37 (21), 8160-8166. 

The theories considered thus far have been limited to linear and star polymers, which have no more than a single branch point. The simplest polymer with more than one branch point is an H polymer, depicted in Fig. 9.4. An H polymer contains two branch points the segment of polymer connecting the two branch points is called the backbone . What is especially significant about such polymers is that the backbone cannot relax its configuration unless the branch points move. This contrasts with star polymers, which can relax completely without the branch point moving. Theories for polymers with more than one branch point require consideration of branch-point motion. 

Figure 9.16 Sketch showing branch-point motion. The entanglements of the arms with neighboring invisible chains are shown as loops that represent "slip links" that confine each arm. The location of the branch point, on average, is taken to be the centroid of the three locations of the innermost entanglements of each of the three arms. When the arm shown as a bold line relaxes and re-entanglesythe slip links, including the inner-most slip link on this arm, move to a new position. Hence the centroid of the three innermost slip links, and therefore the average location of the branch point, shifts to the position marked with an X" as shown.

The above is a sketch of the hierarchical model for branched/linear mixtures. Details have been omitted, including a discussion of inclusion of early time fluctuations into the hierarchical model by Park and Larson [49], and a discussion of how branch-point motion affects primitive-path fluctuations, the latter of which is an open area of research activity. 

The success of the slip link model for symmetric and asymmetric star polymers inspires its application to more complex architectures, such as H polymers. The mechanisms of relaxation and branch point motion were established in studies of symmetric and asymmetric star polymers, as were the parameters (Mf,, and Tg) that allow the simulations to be compared... 

Slip-link and other simulation models, that treat entanglements as interactions between two molecules in an ensemble of chains, are producing important insights into the processes of constraint release and branch-point motion, and in some cases yield predictions that are more accurate than tube models. We can expect that considerable progress will be made using such models in the near future. 

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

Other complications may arise in dielectric relaxation spectra of polymers from chain branching, which may introduce a distinct relaxation process connected with molecular motion at a branch point, and from crosslinking, which greatly restricts certain kinds of molecular movement. 

For both linear and star polymers, the above-described theories assume the motion of a single molecule in a frozen system. In polymers melts, it has been shown, essentially from the study of binary blends, that a self-consistent treatment of the relaxation is required. This leads to the concepts of "constraint release" whereby a loss of segmental orientation is permitted by the motion of surrounding species. Retraction (for linear and star polymers) as well as reptation may induce constraint release [16,17,18]. In the homopol5mier case, the main effect is to decrease the relaxation times by roughly a factor of 1.5 (xb) or 2 (xq). In the case of star polymers, the factor v is also decreased [15]. These effects are extensively discussed in other chapters of this book especially for binary mixtures. In our work, we have assumed that their influence would be of second order compared to the relaxation processes themselves. However, they may contribute to an unexpected relaxation of parts of macromolecules which are assumed not to be reached by relaxation motions (central parts of linear chains or branch point in star polymers). 

This approach is very simple and powerful. It has been used in numerous studies (for references see [176, 177]) and generally captures the essentials of the adiabatic versus non-adiabatic branching. It is especially useful in circumstances where the nuclear motion is essentially classical (i.e. zero point motion and tunnelling can be ignored). 

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