Reptation moves

Figure 1. Illustration of the principle of operation of various elementary Monte Carlo moves from top to bottom single-site displacement moves, reptation moves, Continuum configurational-bias move, Extended continuum configurational-bias (ECCB) moves and also concerted-rotation moves (CONROT), and end-bridge moves.

In a reptation move, a segment of the molecule is cut off from one end of the chain and appended at the other end of the chain [20] (see Fig. 1). To satisfy microscopic reversibility, it is important to choose either end of the chain with an arbitrary but fixed probability if both ends are selected at random, the probabilities of making a head-to-tail or a tail-to-head transition are equal. 

Unfortunately, simple reptation moves cannot be used to simulate branched polymers or block copolymers, because the geometry of the molecule would be altered in the process. Furthermore, for a long molecule a significant amount of time is spent moving the chain ends back and forth, while leaving its center portions unchanged. For this reason, reptation moves are usually supplemented by other moves. 

The specific steps in performing a reptation move are as follows. First, one end of the chain, either the head or the tail, is chosen randomly to be the lead end of the reptation. A site is then chosen randomly from the five, potentially available, nearest neighbors of the lead end (the sixth site is occupied hy the bead adjacent to the end and is not considered). If this site is vacant (i.e., not occupied by a bead from another chain), then the end is moved to this new site and all other beads of the chain follow the end otherwise, no move is made. Either way, the reptation move is completed, the new configuration is temporarily recorded, and the Monte Carlo acceptance criterion described above is used to determine whether this move is to be kept or not. 

Figure 10 presents bond correlation functions, and end-to-end relaxation functions obtained from longer simulations. Both runs were made with 20% CCB end moves (Ncut = 15), 20% reptation moves, and 60% biased ECROTl moves with Atj) equal to 10°. The first run was started from a system configuration where the constituent chains have conformations close to the unperturbed ones the second run departed from a set of relatively collapsed chain conformations. It is demonstrated that, even with the present algorithm, it is still not possible to obtain fully-equilibrated melts for these long-chain systems. From the different relaxations of bond correlation functions and end-to-end distance vectors (Fig. 10 (top)), it is clear that the results obtained depend strongly on the initial conditions chosen. An examination of the fluctu-... 

Demonstration of bottlenecks in long nins with the C, Chaim system. The algcmthm was composed of 20% reptation moves — 20% CCB 60% Uased K3ROT1 with... 

A MC cycle is defined to consist of 10 attempted reptation moves per chain. Due to the high density, the fraction of accepted moves is only 0.031 for system Mq it decreases even more with increasing filler content, being as low as 0.014 for system Ms,50- On the other hand, all units are displaced in 12 cycles in the case of system Mq (i.e., no units are left having the same coordinates of any unit of the same chain before the 12 cycles), with more than 95% of the units displaced in 6 cycles. 

N = 60 and A N = 1000. Both local and nonlocal (ie, reptation) moves have been used in our Monte-Carlo simulations. Filled-in symbols are reproduced from Reference 35 ( , N = 100) and Reference 36 (B, Af = 64). The curves are drawn to guide the eye. Reprinted with permission from Ref 40. Copyright (1997) American Chemical Society. 

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... 

The slithering-snake (or reptation) move, which deletes a bond from one end of the walk and appends a new bond (in an arbitrary direction) at the other end (Fig. 2.4). 

The kink-end reptation move, which deletes a kink at one location along the walk and appends two new bonds (in arbitrary directions) at one of the ends of the walk (Fig. 2.6 ). 

Fig. 2.4 The slithering-snake (reptation) move. Head of the walk is indicated by a cross. Dashed lines indicate the proposed new step (resp. the just-abandoned old step).

Fig. 2.6 The kink-end reptation (— ) and end-kink reptation ( —) moves. In a kink has been...

Although reptation and tube renewal contribute equal magnitudes to viscoelastic relaxation and viscosity, reptation dominates diffusion. Consider a time when reptation moves the molecule s center of mass a distance of order of the radius of gyration, i.e., a squared distance agN. In there have also been about N tube renewal processes, each of which moves the center of mass Og/N. These add incoherently to produce a squared displacement NiuglN) a IN. The contribution of tube renewal to the diffusion constant is thus only 1/AT that of reptation. 

The main difference between Monte Carlo (MC) and Molecular Dynamics (MD) simulations is that we do not need to follow the physical trajectory of the system with MC, which, in turn, enables us to use unphysical moves to cover the relevant area of phase space more quickly. Such moves include chain breaking and reattachment,configurational bias, and reptation moves. ... 

---