Verdier-Stockmayer chain

Verdier-Stockmayer model of unentangled chain dynamics... 

Mattice and coworkers performed their simulations of diblock copolymer micellization on a cubic lattice, typically of dimensions 44 x 44 x 44 and with a coordination number of c = 6. As in the case of Larson s model, only one energy parameter, f, is considered for all interactions, namely, the interaction between the tail and the solvophilic (head and solvent) beads. The base structure used for the copolymer is hiQtio. The chains are rearranged using reptation and Verdier-Stockmayer [53] type local motions, both of which are discussed in detail in Sec. III. C. Wijmans and Linse [50] also based their simulations on this model and surfactant structure. 

M. Rubinstein (Eastman Kodak Company) In the des Cloizeaux double reptation model which is similar to the Marrucci Viovy model, it is assumed that a release of constraint chain A imposes on chain B when chain A reptates away completely relaxes the stress in that region for both chains. This would imply that for a homopolymer binary blend of long and short chains would be completely relaxed after each of these K entanglements is released only once. But if an entanglement is released, another one is formed nearby. I believe that to completely relax this section one needs disentanglement events and that the Verdier-Stockmayer flip-bond model or the Rouse model is needed to describe the motion and relaxation of the primitive path due to the constraint release process, as was proposed by Prof, de Gennes, J. Klein, Daoud, G. de Bennes and Graessley and used recently by many other scientists. The fact that double reptation is an oversimplification of the constraint release process has been confirmed by experiments. 

The Verdier-Stockmayer (pure one-bead) and Kranbuehl-Verdier (pure two-bead) algorithms for the SAW or NRRW (but not the ORW) have peculiar conservation laws which inhibit the relaxation of the chain, thereby... 

Verdier,P.H., Stockmayer, W.H. Monte Carlo calculations on the dynamics of polymers in dilute solution. J. Chem. Phys. 36, 227-235 (1962). See also Verdier,P.H. Monte Carlo studies of lattice-model polymer chains. 1. Correlation functions in the statistical-bead model. J. Chem. Phys. 45,2118-2121 (1966). 

Dynamic Monte Carlo simulations were first used by Verdier and Stockmayer (5) for lattice polymers. An alternative dynamical Monte Carlo method has been developed by Ceperley, Kalos and Lebowitz (6) and applied to the study of single, three dimensional polymers. In addition to the dynamic Monte Carlo studies, molecular dynamics methods have been used. Ryckaert and Bellemans (7) and Weber (8) have studied liquid n-butane. Solvent effects have been probed by Bishop, Kalos and Frisch (9), Rapaport (10), and Rebertus, Berne and Chandler (11). Multichain systems have been simulated by Curro (12), De Vos and Bellemans (13), Wall et al (14), Okamoto (15), Kranbu ehl and Schardt (16), and Mandel (17). Curro s study was the only one without a lattice but no dynamic properties were calculated because the standard Metropolis method was employed. De Vos and Belleman, Okamoto, and Kranbuehl and Schardt studies included dynamics by using the technique of Verdier and Stockmayer. Wall et al and Mandel introduced a novel mechanism for speeding relaxation to equilibrium but no dynamical properties were studied. These investigations indicated that the chain contracted and the chain dynamic processes slowed down in the presence of other polymers. 

The MC method was first applied to polymer chain dynamics by Verdier and Stockmayer, using a bead model on a simple cubic lattice. Beads are moved from site to site on the lattice, in a way which satisfies certain criteria e.g. chain connectivity, excluded volume effects), and both the equilibrium average chain dimensions, and (by sampling to obtain the time correlation function) the relaxation behaviour of chains may be studied. One of the results is that excluded volume effects slowed the relaxation times significantly. Deutch and Boots have criticized the original model, attributing this unexpected result to unrealistic... 

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