Lennard-Jones cluster

In linear elastic polymers, bonds that connect monomers are not stiff. Within a certain range, bonded monomers can adapt their distance to external perturbations without much energetic effort. One can imagine the bond as a rather floppy spring. For this reason, crystalline structures of elastic polymers with nonbonded monomers interacting with each other will exhibit similarities compared to atomic or Leimard-Jones clusters, provided the 

Many-particle systems governed by van der Waals forces are typically described by the Lennard-Jones (LJ) pair potential (1.34), 

It thus seems obvious that there are strong analogies in liquid-solid and solid-solid transitions of LJ clusters and classes of flexible polymers [52,53,161-165], but also between colloid and polymer systems (see, e.g., [166]). For this reason, we will now discuss some basic properties of generic icosahedral stmetures. 

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Northby J A 1987 Structure and binding of Lennard-Jones clusters 13< W < 147 J. Chem. Phys. 87 6166 Berry R S 1993 Potential surfaces and dynamics what clusters tell us Chem. Rev. 93 2379... 

Calvo, F. Neirotti, J.P. Freeman, D.L. Doll, J.D., Phase changes in 38-atom Lennard-Jones clusters. II. A parallel tempering study of equilibrium and dynamic properties in the molecular dynamics and microcanonical ensembles, J. Chem. Phys. 2000, 112, 10350-10357... 

J. Kostrowicki, L. Piela, B.J. Cherayil, H.A. Scheraga, Performance of the diffusion equation method in searches for optimum structures of clusters of Lennard-Jones clusters, J. Phys. Chem. 95 (1991), 4113. 

S. Schelstraete, H. Verschelde, Finding minimum energy configurations of Lennard-Jones clusters using an effective potential, J. Phys. Chem. AlOl (1997), 310. 

Nauchitel and Pertsin have studied the melting properties of 13-, 19-, and 55-particle Lennard-Jones clusters.Questioning the validity of results obtained from free-volume simulations of such systems, they have used hard-sphere boundaries to constrain their clusters to finite volumes. The results of Nauchitel and Pertsin are most interesting for the 55-particle cluster. For certain ranges of temperature and mean density, structural evidence for surface melting was obtained projections of the cluster s coordinates, and radial density distribution functions, like those given in Fig. 17, characterize the cluster as a 13-particle icosahedral core surrounded by a fluidlike shell. However, dynamic calculations like those described for other clusters in the previous section have yet to be obtained to determine how fluidlike these outer atoms really are. 

Table 1 Calculated number of (second line) inequivalent total-energy minima and (third line) inequivalent saddle points for Lennard-Jones clusters as a function of the number of atoms N in the cluster. From 8...

One-Component Lennard-Jones Clusters. Lennard-Jones clusters remain being the reference model system for testing new theoretical developments. For clusters consisting of only one type of atoms, the total energy is written as... 

In a later work, Doye and Wales53 studied the properties of a single Lennard-Jones cluster (N = 38) as a function of temperature. They found that at low temperatures, the structure is like that of a piece of the fee crystal structure, but at somewhat elevated temperatures it adopts a structure with significant similarity with an icosahedron, until it at even higher temperatures becomes liquid-like. 

Doye and Calvo54 included entropy effects for large Lennard-Jones clusters and used the results in estimating which structures will be dominating as a function of size and temperature of the cluster. Their main results are reproduced in Figure 9. Most interesting may be that the crystal structure is not found before N is well above 100 000. 

Frantz58 studied the temperature-dependent properties of Lennard-Jones clusters using a Monte Carlo method (Section 2.2). The two parameters e and a... 

Figure 9 (a) shows the energies of various structural types of Lennard-Jones clusters. 

In their study, Cheng et al.4S compared different Lennard-Jones clusters with 98 atoms using their proposed connectivity tables. The results for eight isomers are reproduced in Table 2. It is clear that all clusters have comparable total energies as well as number of nearest-neighbour pairs. Therefore, in order to clearly distinguish between the isomers, further descriptors are needed. Their connectivity tables are indeed one way of distinguishing, as the table clearly... 

Figure 10 From top to bottom AjE(N), A2E(N), and A12E(N) for Lennard-Jones clusters. Reproduced with permission of American Institute of Physics from 58...

Two-Component Lennard-Jones Clusters. - By generalizing the Lennard-... 

Figure 13 Temperature dependence of the average separation between the centers of mass of the A and B atoms in the A13B13 Lennard-Jones clusters for different values of the parameter A. Reproduced with permission of American Physical Society from 49...

In a much more recent study, Sieck et al.91 used molecular-dynamics calculations in studying some few, selected Srv clusters, i.e., for N = 25, 29, 35, 71, and 239, in connection with a parameterized density-functional method. The fact that they found several isomers for clusters of these sizes should not surprise, but is rather a confirmation of the finding for Lennard-Jones clusters... 

In their discussion of the eigenmode method, Tsai and Jordan8 illustrated the approach through two simple systems. One of those was Lennard-Jones clusters, and the other was small clusters of water molecules. For both they tried to identify as many local total-energy minima and saddle points as possible. The numbers for the Lennard-Jones clusters are reproduced in Table 1 and it is remarkable to see that the number of transition states exceeds by far the number of local total-energy minima. This was also the case for the clusters of water molecules. 

As mentioned above, Chaudhury et al.131 suggested to consider a different fitness function, Eqs. (80) and (81). As test system they, too, considered Lennard-Jones clusters and demonstrated the feasibility of their method. However, it shall be stressed that Lennard-Jones clusters are special except for scalings, only one type of interactions (including strength and range) is treated, and, moreover, it is trivial to calculate any derivatives of the total energy with respect to nuclear coordinates. 

Another improvement was proposed by Trygubenko and Wales135 who subsequently applied it to Lennard-Jones clusters with 7, 38, and 75 atoms. 

Figure 5. The K-entropy of the seven-particle Lennard-Jones cluster, a smooth, monotonic increasing function with no indication of any maximum. [Reprinted with permission from R. J. Hinde, R. S. Berry, and D. J. Wales, J. Chem. Phys. 96, 1376 (1992). Copyright 1992, American Institute of Physics.]...

The next two figures reveal how these overall cluster properties vary with the size of the cluster, even for quite small clusters. Figure 9 shows the same kind of distributions as Fig. 8, but for the four-particle Lennard-Jones cluster, and Figs. 10 and 11 do the same for the 5-particle cluster, a system with two kinds of saddles (but only one locally stable structure), so two sets of distributions are shown there. Only the distributions over the higher-energy saddle show any detectable differences from the distributions elsewhere on the surface. With still larger clusters, the distinctions between saddle regions and the other parts of the surface essentially disappear. 

Finally, in this particular investigation, we look at the same distributions for the six-particle Lennard-Jones cluster, the first that has two geometrically different locally stable structures. The lower-energy structure is a regular... 

Then, Fig. 14 shows the same kind of behavior for the four-particle Lennard-Jones cluster, as it passes through its saddle, on a single, typical trajectory [9]. 

Figure 15. Distributions of sample values of Liapunov exponents for the seven-particle Lennard-Jones cluster at energies corresponding to (a) 18.15 K, (b) 28.44 K, (c) 30.65 K, and (d) 36.71 K. The trajectories from top to bottom in each figure correspond to 256, 512, 1024, 2048, 4096, and 8192 steps. Only at the energy of part c does two-phase equilibrium occur. [Reprinted with permission from C. Amitrano and R. S. Berry, Phys. Rev. E 47, 3158 (1993). Copyright 1993, American Physical Society.]...

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