Constraint release

2J Constraint Release The reptation theory assumes that the environment is frozen while the test chain moves. This assumption is questionable. The other 

Another important relaxation process in entangled melts is constraint release, depicted in Fig. 3-27. When an end of a surrounding chain moves past a test chain, an entanglement constraint restricting the motion of the test chain is released, and a portion of the latter is freed to reorient (Graessley 1982 Montfort et al. 1986 Pearson 1987 Viovy et al. 1991). Constraint release can only be completely neglected for the case of an isolated chain 

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Approximate Theory for Constraint Release in Star Polymer Melts. 216... 

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

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 approximate treatment described above accounts rather well for the linear rheology of star polymer melts. In fact it has been remarked that the case for the tube model draws its real strength from the results for star polymers rather than for linear chains, where the problems of constraint release and breathing modes are harder to account for (but see Sect. 3.2.4.). However, there are still some outstanding issues and questions ... 

Rubinstein has constructed on a reptation-fluctuation approach a detailed self-consistent theory of constraint release, allowing each loss of entanglement in one chain to permit a random jump in the tube of another [37]. When this is done the form of predicted relaxation functions are in good accord with experiments. However, in monodisperse linear melts it appears that the fluctuation contribution is more important than constraint release. 

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... 

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. 

The experiments on H-polymers confirm another aspect of the dynamic dilution theory for constraint release in branched polymers the range of relaxation times clearly attributable to the arms of the H-polymers is typically much... 

A strong test of this theory is presented by a blend of two dynamically different components (but of identical local chemistry) such that the volume fraction of both is large. Two cases of especial interest suggest themselves blends of linear with star polymers [42,55] and blends of star polymers with widely separated molecular weights [56]. Recent work on both these systems has shed further light on the nature of co-operative constraint release and the remarkable power of the theoretical tools we now have at hand. 

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" ...

Constraint release (CR). This takes place if a confining chain moves out of the way of a given chain and thus opens some freedom for lateral motion. This phenomenon is an intrinsic many-body effect and for monodisperse polymer melts becomes significant mainly in the creep regime. 

It is the virtue of the NSE experiments that they confirmed the CLF mechanism quantitatively on a molecular level in space and time. On the other hand, the constraint release process still remains to be probed on a molecular scale and it is expected that future NSE experiments on bimodal melts will clarify these processes on a molecular scale also. 

Newer rubber elasticity theories based on the tube model (35) consider special constraint release mechanisms which allow a physi-... 

M. H. Wagner, P. Rubio, and H. Bastian, The Molecular Stress Function Model for Polydisperse Polymer Melts with Dissipative Convective Constraint Release, J. Rheol., 45, 1387-1412 (2001). 

For 100 < P < 1000, the measured diffusion coefficients for N = P no longer follow the N 2 reptation prediction. In the same range of N values, D remains proportional to N 2 if P N, i.e. if the motion of the chains surrounding the test chain are frozen down during the diffusion time of the test chain. The comparison of the data obtained with N = P and with N P clearly puts into evidence the acceleration of the dynamics associated with the matrix chains, similarly to what has yet been observed with other polymers [11, 12, 42 to 44] or in solutions [10]. This acceleration, by a factor close to three, can be attributed to the constraint release mechanism [7, 8, 13], the effects of fluctuations of the test chain inside its tube [9] being a priori the same in the two situations P = N and P N. 

From the theoretical point of view, the first refinement of the reptation approach has been to introduce the collective dynamics of the chains in terms of the constraint release process[7, 8, 13]. Due to the motions of the surrounding chains, some constraints which constitute the tube may disappear during one reptation lime, and thus give more freedom to the test chain. Quantitative attempts have been made to take into account these additional... 

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