Slithering snake

Clearly, both the pivot and the slithering snake algorithms are incapable of reproducing true chain dynamics at molecular basis, covering the time range of typical chain relaxation times. Therefore, in the following we focus on two alternative methods, broadly used at present to this end. 

Fig. 5.13. Relaxation time r3 plotted vs. temperature for the coarse-grained model of PE with N = 20, using the random hopping algorithm (upper set of data) or the slithering snake algorithm (lower set of data), respectively. The time r3 is of the same order as the Rouse relaxation time of the chains, and is defined in terms of a crossing criterion for the mean-square displacements [41], g3(t = r3) = g2(t = r3) [See Eqs. (5.2) and 5.3)]. From [32]...

On the other hand, one strength of the approach is the availability of algorithms (such as the slithering snake algorithm) by which undercooled polymer melts can be equilibrated at relatively low temperatures. This allows the static properties of the model to be established over a particularly wide parameter range. Furthermore, the lattice structure allows many questions to be answered in a well-defined, unique way, and conceptional problems of the approach can be identified and eliminated. Last but not least, the lattice structure allows the formulation of very efficient algorithms for many properties. 

The polymer literature yields a variety of specialized move types in particular for lattice homopolymers [110]. Sampling methods like the slithering snake and reptation algorithms (see ref. Ill and references therein) or the original configurational-bias/chain growth algorithms [112,113] were specifically... 

Fig. 49a. A representative configuration of block copolymers on the lattice (For clarity a square lattice is shown, while all work refers to a simple cubic lattice). Three symmetric diblock copolymers are shown, each of chain length N = 10. The two monomeric species are labeled A-type (full dots) and B-type (open dots). The vacancies are not shown explicitly, but are assumed to reside on each lattice site left unoccupied by either of the two species of monomer. A volume fraction of < >v = 0.2 is used, since experience with blends [107] has shown that such a system behaves like a very dense melt. The energy contributions eAA, eBB and eAB are shown, b Examples of typical slithering-snake [298,299] motion monomer situated at point labelled by 5 is removed, and one of sites 1,2,3 is randomly chosen for occupation. Note that unlike Refs. [298,299] also the junction point needs to be displaced accordingly, as shown in the figure. For the reverse process, monomer at 3 is removed and the sites 4,5,6 are considered for attachment (of course, a move to site 6 is rejected due to excluded volume constraints), c Interchange of A-Block and B-Block of a diblock copolymer chain. From Fried and Binder [325],...

By virtue of the slithering-snake algorithm it is (hitherto) possible to remove all non-equilibrium effects up to T 0.16 so that one can study the equilibrium dynamic properties of the model in the supercooled state. During supercooling the structural relaxation time increases over several orders of magnitude, and dynamic... 

The next important question now is how to calculate all observables of the previous section from the tube coordinates Vj(t). In the slithering snake model, we will not distinguish the chain from the tube and assume that N = Z and Ri = Vj. This means that R can be thought of as the centers of the blobs of size Ne = N/Z, and one understands that the results of such a model are not valid at timescales treptation motion is associated with the center-of-mass motion of the chain inside the tube, it is reasonable to assume that the jump time t scales linearly with the molecular weight, that is, t=Ztss. Thus, the parameters of the model areZ, a, and z s- Since the last two parameters are just imits of space and time (similar to b and zq in the Rouse model), Z is the only nontrivial parameter (analogous to N in the Rouse model). 

Figure 17 Slithering snake and reptation model results for chain length /V=32.

The expected property G(0) = 1 is rectified in the slithering snake model with the variable segment length, which is obtained if E( ) in eqn [57] is a random Gaussian vector with /s(t) = 0 and = 1. This leads to the results shown in... 

One disadvantage of the slithering snake model is that the time step cannot be controlled in one step the chain moves exactly one tube segment. This in particular leads to the unphysical oscillations in i,mid(t) at early time in Figure 17. In order to resolve the motion on smaller timescales (and more importantly to account for fluctuations of the chain inside the tube, see below), one has to distinguish between the tube and chain coordinates. Now we introduce the main set of variables of the tube model the 3D tube coordinates V),(t), fe = 0...Z as in the previous section plus the one-dimensional (ID) chain coordinates inside the tube Xj( ), i = 0...N. In total, we have 3(Z-r 1) -r (N-f 1) variables, and their equations of motion are coupled. The main idea of the tube theory is that the chain inside the tube moves independent of the tube coordinates, whereas the tube segments are deleted at the ends when the chain does not occupy them any more, and are created when the chain sticks out of the tube. In the pure reptation model, only the center-of-mass of X coordinates moves according to... 

Figure 18 Intermonomer distance function d(s) for slithering snake and tube models. The tube model puts beads on the line or in 3D as indicated in the legend and illustrated in the picture.

One very widely used and simple MC move that can easily be combined with local moves is the slithering snake (SS) algorithm (also sometimes called reptation moves ). " One first randomly selects one of the chain ends and removes this end monomer (plus the bond that connects it to its... 

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