Cluster length fluctuations

A discretized version of the reptation model is proposed. The tube is modeled by a one-dimensional lattice, and the polymer is modeled by a cluster of walkers, called reptons, on this lattice. Each repton represents a part of the chain. Reptons are allowed to hop between neighboring sites in such a way that the cluster always remains connected. The fluctuations in tube length correspond to cluster length fluctuations in the repton model and are not pre-averaged. In the experimentally accessible range of molecular weights M, the repton model predicts the diffusion coefficient and viscosity 0.1 ... 

Ishii and Murakami (1991) evaluated the CFB scaling relationships of Horio et al. (1989) using two cold CFB models. Solids flux, pressure drop, and optical probe measurements were used to measure a large number of hydrodynamic parameters to serve as the basis for the comparison. Fair to good similarity was obtained between the beds. Dependent hydrodynamic parameters such as the pressure drop and pressure fluctuation characteristics, cluster length and voidage, and the core diameter were compared between the two beds. The gas-to-solid density ratio was not varied between the beds. As seen in Table 7, the dimensionless solids flux decreased as the superficial velocity was increased because the solids flux was held constant. 

Figure II. N-dependent melting temperatures and coexistence temperature ranges A7. a—longtime averaged kinetic energy values at which rms bond length fluctuations S rise sharply (see Fig. 10). Dashed lines correspond to coexistence temperature ranges AT = T — 7) ) for clusters that display a bimodal form for short-time averaged temperatures. (Beck et al. )...

Between temperatures of 28 and 29 K the rms bond length fluctuations of the 13-particle system increase dramatically. Similar results have been obtained for all the other clusters N = 5, 6, and 7) for which S(T) is cal-culated. - The curves of S T) for these systems are similar to those occurring with first-order phase transitions of macroscopic systems.Lindemann s criterion states that melting occurs for such systems when rms fluctuations reach 10%.For the small clusters studied, the rise in this function occurs at values of S slightly below 10%—an effect that can be attributed to the large ratio of surface to core atoms. 

The rms bond length fluctuation data of Davis et al. are pictured in Fig. 4. 8(T) is similar to those previously obtained form Etters and Kaelberer and Jellinek et al. in that the apjjearance of a second, more diffusive form of the cluster is dramatically reflected by a sudden rise in this function occurring over a narrow range of temjjeratures. The increase in this function is observed at lower temjjeratures here than in the curve of Etters and Kaelberer. The... 

In this section the results obtained from AIMD-B-LYP simulations by energizing three isomers of Li9" clusters will be analyzed [39]. For this purpose three types of quantities will be used. One of them is the root-mean-square (RMS) bond-length fluctuation 8 [25], which is calculated at the end of trajectories. A sharp increase of the 8 value is known as the Lindemann criterion for bulk melting, while in the context of finite-size clusters it can be taken as an indication of transition from solid-like to liquid-like state. We find it particularly instructive to analyze the trajectories in terms of atomic equivalence indexes ,... 

In previous work on Van der Waals clusters, it was established that the behaviour of a parameter characterizing the relative root mean square bond length fluctuations,... 

Fig. 4. Averaged relative root mean square bond length fluctuations for the Ar-Ar and Ar-SF distances for certain SF -(Ax) clusters. The shaded band identifies the 8 values chosen to define the Lindemann criterion for melting.

For the calculation of the kinetic factor we usually performed about 5 trajectories. The length of the trajectory depends on whether the cluster size fluctuates around the critical size or not if not the simulation is stopped. From these simulations we calculated the attachment rate. The error estimates vary between a factor of one for the larger critical cluster sizes and a factor of two to three for the smaller cluster sizes. In the regime of smaller critical cluster sizes, the fluctuations in cluster size are almost on the order of the critical cluster size and it becomes therefore more difficult to get a good estimate. 

---