FRAGMENTATION OF A NON-ROTATING Nit9 CLUSTER: A MOLECULAR DYNAMICS STUDY

Collisionless fragmentation of a non-rotating Nil9 cluster is studied using constant-energy molecular dynamics computer simulations. The cluster is modelled by an embedded atom model (EAM) energy surface. Distribution of the channel-specific fragmentation probabilities, and global rate constants are computed and analyzed as functions of the internal energy of the cluster. The results are compared with those obtained using the RRK statistical approach, and also compared with the other multi-channel fragmentation work.


Collisionless
fragmentation of a non-rotating Nil9 cluster is studied using constant-energy molecular dynamics computer simulations.The cluster is modelled by an embedded atom model (EAM) energy surface.Distribution of the channel-specific fragmentation probabilities, and global rate constants are computed and analyzed as functions of the internal energy of the cluster.The results are compared with those obtained using the RRK statistical approach, and also compared with the other multi-channel fragmentation work.

Structural
and dynamical propenie of atomic and molecular clusters are size dependent Understanding the \ariation 01 stIuctures and properties as clusters grow will elucidate the transition from atoms and molecuies.on the one hand.and bulk matter on the other Therefore, they arc subject u 111\:re,1Illgh Il!Lll research ,lCtlvity theoretically and experimentally [1-4] Theoretical ,tudle ... ha e heen carried out for van der Waals [5-8], ionic [2] and metallic [9-! Uj C~)he"'I()n hlrthe' the fragmentation of energized clusters has been the subject of a number of \10 studIes.hov"ever, most of them considered Lennard-Jones [5-8] system Ejection of an atom.emerged in these studies as the only dissociation channel.In the study of fragmel tation \.)\ metal clusters (M)'h n= 12-J 4, based on many-body potential repol1ed by Lopez and .lelllJ1ct.. [10].evaporation of monomers appears to be the least probable channel In this present work, we report resuits of a MD study of the collisionless fragmentation of a i i19 cluster.
The Nil') cluster was described by an EAM and fragmentation of the cluster preferentially into monomers was predicted from the simulations.In fact, fragmentation into larger products, i > I, appears to be the least probable channels.however.there is an increasing trend for the fragmentation .intolarger products as the cluster energy increases We have calculated the survival and channel specific fragmentation probabilities.and diSSOCIation rate constants for the "super heated" Nil') cluster as functions of the clustel energy are obtai ned.
The semi-empirical potentials used in the simulations are usually litted III 11ll'd'-:::l'd properties of the bulk Some of these measured properties are indeed size depelldl'l1l Therefore the validity of the potentials will be questionable in the cluster studies Till' EAM of [11] employed in this study, properties of the bulk and Ni2 were used in fitting the embedding function and the pair interaction, So, one might expect that this potential could work better between the two limits i.e., range of clusters.This work is focused on the analysis of the dynamics of the fragmentation process In Sections 2 and 3 potential, and theoretical background and computational procedure are presented, respectively This work is concluded with the results and discussions Section.
The cluster is described by an n-body potential (EAM), and the cohesion energy of a metallic system is obtained by the sum of the interaction energies of each of the II atoms with a host modelling the effect of the remaining n-I atoms on the one in question The potential energy of the system is expressed as (2) Where 15, is the effective host electron density at the atom-i site, and obtained by linear superposition of the spherically symmetric atomic density tails p;' .The ;'1 (Pi) is the embedding energy function-energy required to embed atom-i into the background electron density at site i The ¢ij kJ) is a core-core.formed by the remaining charges, interaction between atoms i and.i separated by a distance 1;/.The EAM model of Voter and Chen [11] has a medium-range attractive contribution in the core-core interaction (whereas in the version of Foiles et at.[12] it is entirely repulsive), But the most important change is that properties of the Ni2 were used in fitting the embedding function and the pair interaction.The fit to the diatomic molecule lowers the binding energies of small Ni clusters, and results in better agreement with ah initio binding energies.
The coordinates and momenta of the atoms in the cluster are prepared starting from the zero total linear and angular momenta initial conditions.Their time evolution is generated by solving the classical equations of motion using the 4 th order predictorcorrector algorithm.The time step of /1t = 4x IO-16S is used in the simulation, and conservation of the total energy within 0.0 I% in the runs of 4.5x I0 6 steps is achieved.The quantities used to characterize the fragmentation of a cluster are, 1) channel specific fragmentation probability f1, where No is the total number of trajectories that lead to fragmentation of the cluster within a specified time interval-in the simulations we had No = 400.The N i is the number of trajectories that result in fragmentation into channel i, i.e., In our study the length of the maximum time interval is 1.8 ns-within this observation time all the trajectories considered in the simulations ended with fragmentation.
2) Timedependent survival probability P(tr No i=l where B is the step function, fOi is the survival time in which the cluster lives before it undergoes fragmentation along the trajectory i. 3) Global fragmentation rate constant k is detined by the equation, where N(t) is the number of surviving parent clusters at time t, N(t 0) No Assuming that k does not depend on time, we obtain survival probability as (8 The k is a function of the total energy of the cluster, and can be obtained by fitti ng /11 P(t) to a Iinear function.
StatIstical model RRK (Rice-Ramsperger-Kassel) is the simplest statisti~al approach 1\ \ '1'1imolecular processes [13-] 4]. which assumes that all the degrees of t,-eedol1l 0" till ~\ ", '.:'111 are vibrational and all are strongly coupled.It states that the system undergol'~, un! 'wlecular process when and if the energy localized in a specified vibrational degree, .j, ('cdom becomes equal or larger than a certain threshold value.Eo The rate constant kr" ot'lhe process is proportional to the probability of such a localization and is given by' \\ here E?: Eo is the total energy of the system, s is the number of its degrees of t'•eedonL and I' is a frequency proportionality constant (the value of v is calculated as in Ref [10]) We have computed the total probabilities and rate constants at different values of the total energy of the cluster in order to analyze the energy dependence of the fragmentation.
The initial conditions of the No number of the trajectories (coordinates and momenta) were generated and recorded every 10,000 propagation steps along the trajectory for a given total energy Therefore, phase-space coordinates are adequate to make these initial conditions independent from each other.We have selected 5 different total energies and generated of 400 initial conditions for each total energy.We had to consider energies considerably higher than the actual fi-agmentation energy of the cluster in order to obtain adequate statistics of the fragmentation process on the time scale of our simulations, approximately 2ns.The minimum energy geometry and the energetic of the Ni19 cluster is given in Fig. I.The fragmentation energies of the Nil9 are calculated from the minimum-energy structures of the parent and fragment clusters as (11) where E(i), J~(ll-i), and E(n) are the minimum energies of the parent, II, and fragment Ni l9 ---+ Ni r3 + Ni 6 , has lower fragmentation energy than the i=3-5 channels since Ni13 is a highly symmetric and stable cluster.The energy increases further and reaches to a maximum value al i=8.We would like to mention that the above analysis for dissociation channels is based on the total energy difference only.In actual fragmentation, the energy barriers may exist and play an important role.
The potential energy surface used in this work requires the lowest fragmentation energy for ejection of a monomer compared to the higher fragmentation channels.The next ones are the dissociation of a dimer and hexamer, respectively.In contrast, the potential energy surfaces based on the tight binding method, such as, Gupta [15] and Finnis-Sinclair [16 j, favors the ejection of a dimer, I.e., requires the least energy compared to the other fragmentation channels.
Figure 3 shows the distribution of the fragmentation channel probabilities for the II') parent cluster at five different total energies per atom.These probabilities are calculated from 400 trajectories for a given energy.The prominent feature of this distribution is that it peaks at channel i I-ejection of a single atom.At all energies (except energy 'a') fragmentation into channel i ~2 is observed, and their probabilities increase with increasing total energy of the cluster from "a" to "e" The channel specific probabilities for the channels i=3 and 4 are quite small, therefore, error is larger and the trend is not so clear The global fragmentation rate constants are obtained from linear fits of the logarithms of the time-dependent survival probabilities PO), Eq. (6) These probabilities.as calculated from the dynamical simulations, are shown in Fig 4 The rate of decay of the survival probabilities is indeed exponential and increases with the energy of the cluster In Ref [10] the clusters (M)n, n= 12-14, were mimicked by an n-body potential that is based on Gupta's expression [15] for the cohesive energy of bulk metals.The prominent features of their results were that distributions for these sizes of the clusters peaked at channel i 2. The evaporation of monomers was practically absent and the probabilities decrease monotonically with i for i > 2 For the ranges covered, the size and the energy of the clusters seem to have no major effect on the distributions.However, our channel  specific probabilities peaked at channel i I and higher fragmentation channeL i ~2.

probabilities are small
The main difference between these two simulation studies is the potential energy surfaces used.As a result, fragmentation of a cluster, as any dynamical process, depends on the topology of the potential-energy surface defining the forces acting on the atoms as well as on the total energy of the cluster.
The fragmentation energies are increasing with increasing channel number and alll1()~1 reaches to a threshold value in i=3-5 interval.
This also correlates with the monotonic decrease of the corresponding channel probabilities.However.fragmentation into i 2: :) channel s (despite the fact that the i=6 has lower fragmentation energy than i=3-4) was not observed.
As a result, there are some correlations between the fragmentation-channel probabilities, on the one hand, and the corresponding fragmentation energies, on the other.
The global fragmentation rate constants are shown in Fig 5 .The rate constants show dependence on the energy.Comparison of the results indicate that there is some correlation between the global decay rate constants obtained from the MD simulation and the statistical model RRK.However, energy dependence is stronger in RRK model than the MD (as the energy increases the difference between the two is increasing).

ACKNOWLEDGMENT
clusters i, and (ll-i) , respectively These fragmentation energies are shown in Fig 2. Fragmentation energy is increasing first with increasing channel number i, and then, remains more less constant from the i=3 to i=5.The channel i=6, which corresponds to

Fig. 3 .Fig. 5 .
Fig. 3. Channel specific fragmentation probabilities for the Ni l9 at different total energies per atom.The energies are defined with respect to the minimum of the potential well corresponding to its most stable geometry.