Topological constraint

Having set the basis of Molecular Dynamics based topological constraint counting, we now review certain results obtained within this new framework. 

We first apply such methods to Ge-Se network glasses which are probably the most well documented systems in the field of rigidity transitions. We then use the same tools to investigate the constraint behavior in the liquid. 

Using this constraint, the configuration sum of a polymer with entanglement index m (with respect to another chain) can be calculated. This problem is the same as the Aharonov-Bohm effect encountered in quantum mechanics. Although the entropic force from the topological constraints has approximately been calculated, the dynamical consequences are still unexplored. 

The Alexander polynomial, encountered in the algebraic topology, can be used to distinguish different topological states particularly in computer simulations. Interesting quantities like the probability that two rings are entangled as a function of distance between their centers of mass can readily be computed. However, the modification on the Rouse relaxation time is unknown. 

It is to be noted that both invariants discussed above are not complete invariants and more work needs to be done. 

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Even when the secondary stmcture of a protein is known, there are a large number of ways that this stmcture can be packed together. Studies dealing with the identification of the topological constraints in the packing of heUces and sheets have revealed certain patterns, but as of this writing accurate prediction is not possible. 

Once the chains become larger and larger, the dynamics of the melt slows down dramatically, due to the topological constraints imposed by the chains on each other. For the chain diffusion one observes a transition... 

A similar anomalous behavior has been detected also in 3d polymer melts but only for rather short chains [41] for longer chains, several regimes occur because of the onset of entanglement (reptation ) effects. In two dimensions, of course, the topological constraints experienced by a chain from... 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

A polymer such as polyethylene is a long-chain molecule with repetitions of the same monomer. Due to topological constraints, the crystallization process of polymer chains is expected to be different from that of simple molecules as discussed so far [160]. 

The model describes the characteristic stress softening via the prestrain-dependent amplification factor X in Equation 22.22. It also considers the hysteresis behavior of reinforced mbbers, since the sum in Equation 22.23 has taken over the stretching directions with ds/dt > 0, only, implying that up and down cycles are described differently. An example showing a fit of various hysteresis cycles of silica-filled ethylene-propylene-diene monomer (EPDM) mbber in the medium-strain regime up to 50% is depicted in Figure 22.12. It must be noted that the topological constraint modulus Gg has... 

Kuhn H., Demidov V.V., Frank-Kame-NETSKii M.D. Rolling-circle amplification under topological constraints. Nucleic Acids Res. 2002 30 574-580... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

With respect to the intensity resolution relationship of NSE, PEB-2 [essentially PE with one ethyl branch every 50 main chain bonds the sample is obtained by saturating 1-4 polybutadiene, the residual 1-2 groups (7%) cause the ethyl branches Mw = 73200 g/mol Mw/Mn = 1.02] has two advantages compared to PEP (1) the Rouse rate W/4 of PEB-2 is more than two times faster than that of PEP at a given temperature [W/pEP (500 K) = 3.3 x 1013 A4s 1 W/pEB (509 K) = 7 x 1013A4s-1] (2) at the same time, the topological constraints are stronger. 

Closed buckyball polyhedra are known to be composed of cyclic pentameric and hexameric faces (see Fig. 5.29), with exactly twelve pentamers and variable numbers of hexamers (in,). Because each vertex is shared by three faces, the total number of vertices ( vertex) must satisfy the topological constraint... 

The most successful theoretical framework in which the accumulating data has been understood is the tube model of de Gennes, Doi and Edwards [2]. We visit the model in more detail in Sect. 2, but the fundamental assumption is simple to state the topological constraints by which contingent chains may not cross each other, which act in reality as complex many-body interactions, are assumed to be equivalent for each chain to a tube of width a surrounding and coarse-graining its own contour (Fig. 2). So, motions perpendicular to the tube contour are confined while those curvilinear to it are permitted. The theory then resembles a dynamic version of rubber elasticity with local dissipation, and with the additional assumption of the tube constraints. 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

However, this less-than-two-turn conformation clearly resulted from the system attempt to minimize the energy of loop bending within the topological constraints... 

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