Chapter 2

Cluster Melting

A number of theoretical studies, largely by Berry and coworkers [12], showed in the 1980s, that finite clusters undergo an analogue of the solid-liquid melting phase transition. This change in the phase can be studied through a number of different indicators. All these attempt to compare properties of the system on either side of this finite-size first-order phase transition and we discuss the most commonly used among these in the next section. The Lindemann index which examines fluctuations in the interatomic distances is a measure that was originally [120] derived for bulk materials, but which also indexes the finite-size melting transition very well. The caloric curve [11, 97] and the specific heat are other similar indicators, adapted from analogous quantities used in the study of melting behaviour of bulk materials. These and other microscopic measures are discussed in the next section and their utility in the study of finite clusters is illustrated.

The study of the thermal stability and the melting transition of clusters is an active field of research as increasingly finer experiments have become pos-sible. It is possible to measure melting temperatures using the temperature dependence of fragmented mass spectra [172, 173]. A clear understanding of this phenomenon will help in the development of ideas concerning macro-scopic phase transitions as well.

17

-}~ o 200 400 600 800 1000

-:~ o 200 400 600 800 1000

-j&;M~ o 200 400 600 800 1000

Iterations number

Figure 2.1: The fluctuation parameter f as a function of time at different temperatures. The curves as obtained seems to indicate that the Ar7 cluster does not melt but fluctuates among its many isomers, even after the tempera-ture where the Lindemann index shows appreciable bond length fluctuations.

2.1 Indicators of the Melting transition The caloric curve is a plot of the temperature T versus the total energy E. The temperature T is defined in terms of the average kinetic energy per freedom,

2 T = (3N _ 6) kB (Ek )

(2.1)

where kB is the Boltzmann constant. The Lindemann parameter is essentially the fluctuation 8 in the interatomic separations, and this determines whether the cluster is in a "solid" or "liquid" phase. This quantity is defined as

(2.2)

At the cluster melting transition, the caloric curve mayor may not show a prominent bend but usually the Lindemann index undergoes a sharp increase, exceeding the value of 0.1 which is taken as an indication of melting [31]. This is a criterion derived from bulk phenomena, and thus may not necessarily be

18

-~~ ~ M~h ~~~~!~~M~M~"i ~ o 1000 2000 3000

-~hm~~~ o 1000 2000 3000

2~' 0,06 1.5

~ I·

0.5

OL-------~------~----~ o 1000 2000 3000 Iterations

Figure 2.2: The fluctuation graph at various temperatures for a generic 7 atom transition metal cluster. The three different curves reflects that the cluster has already melted at the reduced temp 0.06.

true for cluster systems [10, 149, 185]. Hence it becomes necessary to monitor other quantities which may help to index melting behaviour more accurately.

The specific heat capacity C in molecular dynamics (MD) simulations for an N atom cluster is defined as

~ = [N - N(l- 2 )(K)(K-1)t1 NkB 3N - 6

(2.3)

where ( ... ) denotes a time average. In Monte Carlo (MC) simulations the specific heat is evaluated as

(2.4)

with ( ... ) denoting configuration average. Another important indicator for understanding the melting behaviour is

the mean-square displacement curve [185].

(2.5)

19

':~. 2.26

2.22 0 1000 2000 3000

-:~C:::=:J o 1000 2000 3000 -:~E ,~I ,=-=;-] o 1000 2000 3000

Iterations number

Figure 2.3: The fluctuation parameter f at various temperatures for the (C6oh cluster. The cluster seems to be fluctuating vigorously around 400K and at 900K when the cluster has already melted as per the peak in the spe-cific heat capacity, the cluster fluctuates more vigorously among the various higher energy minima.

where nt is the number of different time origins tOj selected. The mean square displacement gives a higher slope when the cluster has melted than when it is in a solid state, nt being the number of different time origins. The diffusion coefficient is directly related to the mean squared displacement 'curve and is .. gIven as

(2.6)

The mean square displacement can also be used to track the trajectories of independent atoms and thereby one can gain information on the cluster dynamics in greater detail. This may be particularly useful in understanding surface effects and surface melting behaviour of clusters in mixed species as well. Cheng et ai., [31] studied the dynamical properties of argon clusters "layer by layer" to establish the relation between surface melting, surface atoms and core atoms: well below the bulk melting temperature, the cluster surface becomes soft and exhibits well-defined diffusion constants while the core remains rigid.'

lrj C"<) .......... () , t

J:: ~

....

0.8

0.6

0.4

0.2

o

2

1.5

1

0.5

0

4

3

.... 2

r-

o 200

0 200

-20K

400 600 800 1000

-30K

600 800 1000

-35K

o~----~----~------~----~----~ o 200 400 600 800 1000 Iterations number

Figure 2.4: The fluctuation parameter f for the Arl3 cluster, showing that the cluster actually melts. At temperatures around 34K the cluster switches back and forth among its many higher energy minima, liquid like states.

The fluctuation index f measures how the atomic structure of the cluster changes over a period of time at different temperatures.

f = !IIA - Aol12 (2.7) 2

A is the adjacency matrix corresponding to the structure present during the temporal evolution,

A .. _ {I, for Iri - rjl < rn ZJ - h . 0, ot erwise. (2.8)

The initial adjacency matrix Ao corresponds to the lowest energy config-uration, while r n are the average interatomic distances in the ground state

21

~J n : {; -5OOK

1 o 1000 2000 3000

3.---~----~--------~--------~ -7ooK

2 ...

OL-~~----~~~--~~----~--~

o 1000 2000 3000 3.---------~--------~--------_.

2 ...

1000 2000 3000

Figure 2.5: The fluctuation parameter f at various temperatures for the (C6o h3 cluster. The cluster still seems to be fluctuating even at 900K when one or more molecules may have sublimed.

configuration ofthe cluster. At low temperatures, the two adjacency matrices A and Ao usually have the same elements and hence f ~ o. At higher tem-peratures prior to melting, the cluster begins to fluctuate between its many higher energy minima and the ground state. After the cluster has melted, the nature of the curve changes appreciably and the cluster now moves among the many higher energy liquid-like minima [10, 11, 97].

The fluctuation parameter f behaves very differently in different cluster systems as can be seen in Figs. 2.1-2.5 where data is presented for 7 atom rare-gas, 13 atom rare-gas, metal and fullerene clusters respectively. The 7 atom rare gas cluster does not really melt, even though the Lindemann index crosses the value 0.1: as can be seen in Fig. 2.1, f keeps oscillating between o and 2 even at the highest temperature indicating that the cluster returns to its ground state geometry periodically. For the metal cluster, there is clear evidence of melting above the reduced temperature of '0.06, while for the 7 fullerene cluster, again there is melting, as also for the 13 atom Argon cluster. The 13 fullerene cluster probably sublimes.

22

Typically, three indicators are adequate to estimate the range of temper-ature over which the cluster can be said to melt [3]. Other indicators have also been studied. Lee et at., [114] examined the effect of melting on the potential energy distributions and they were consistently able to explain the non-monotonic variation of melting temperature with size. the dependence of melting, boiling and sublimation on the interatomic potentials. the existence of surface melted phase and the absence of a pre-melting peak in the heat capacity curves. They also identified a ne\\' type of premelting mechanism in the double icosahedral Pd19 clusters where one of the two internal atoms' escape to the surface at the premelting temperature.

In small clusters a remarkable phenomenon observed is phase co-existence, "'hereby the system fluctuates dynamically between its different phases. Co-existence requires that the timescales on which the fluctuation takes place to be smaller than the timescales in which the dynamics takes place on the potential energy surface. It appears that the change from solid phase to the liquid phase is mediated by intermediate phase changes, depending on whether the cluster has fully melted or not, characterized by increasing flu-idity. In finite systems the different indicators may not coincide, the system being far away from the thermodynamic limit and also the various order pa-rameters probe different aspects and regions of phase space. It is again widely acknowledged that the solid-liquid co-existence requires a gap between the quenched energy states. In mixed clusters such a co-existence can exist even in the absence of such gap. The timescales for fluctuating between the dif-ferent phases being effectively reduced. While different phases can coexist at the melting temperature a necessary condition for dynamical coexistence is that each of the phases has a long persistence time, long enough to establish equilibrium properties characteristic of those phases and properties of which can be experimentally be determined [15]. There is nothing to prevent the co-existence of more than two phases, so ensembles of clusters may exist in two or more phase-like forms, the distinction between 'phase' and 'compo-nent' having been lost for clusters and the probability of finding the cluster in an intermediate state being small [15, 16].

There can also be instances when there is a negative specific heat capac-ity in the microcanonical molecular dynamics simulations. This was earlier theoretically predicted using simple models by considering the distribution of local minima which characterize the potential energy landscape [19]. In this study it was found that for high values of the parameter involving the

23

ratios of the vibrational energy of the global minimum isomer and the local minima, the caloric curve showed an S-bend, with a negative heat capacity in the vicinity of the melting point [19]. Recently experimental results showed a negative heat capacity for a 147 atom sodium cluster [79, 173]. The negative heat capacity was explained by assuming that the cluster avoids the melting states and that the excess kinetic energy gained is converted into potential energy [173]. However ab-initio methods based on first principles have not hinted at the existence of negative heat capacity. This may be due to the short simulation time [1, 147].

To find out whether the cluster has actually melted or evaporated, one should rather analyze the structure, take snapshots and study the interatomic distances [30]. This can be used to identify and discriminate between melting and evaporation or sublimation. Normally during evaporation or sublimation one or more atoms undergo large displacements. From a purely analytical point of view though, a complete picture of melting requires estimation of the free energies G(P, T, N) in the solid and liquid phase; at the transition these should match. On the other hand, from a purely dynamical point of "iew at the melting transition, one may anticipate that the dynamics becomes unstable [130, 151]. One method of detecting such a change is to examine the eigenvalues of the Hessian matrix: when these are negative, the associated mode is unstable [23, 184]. Such behaviour can be studied through the computation of the instantaneous normal modes: this is described in the following section.

2.2 Instantaneous Normal Modes The concept of instantaneous normal modes (INMs), namely that it is pos-sible to define vibrational modes even for the liquid state, was initially de-veloped by Keyes and co-workers [102]. Although normal modes of vibration can only be properly defined for the solid state [207] it is possible to com-pute the normal modes for each atomic configuration in the liquid state and consider the distribution of these INM frequencies [184]. These ideas have found interesting applications in studies on clusters, liquids and supercooled liquids. It can be argued that diffusion phenomena are related to negative eigenvalues in the INM spectrum [76, 112, 118] and correspond to the an-harmonicities of the well. Simulation of supercooled liquids indicate that the

'24

---16

12

'8 8 8'

4

o ----------------------------4

0 4 8 12 16 20 24

20

16

12 -;--E 8

.!:!.. 8 4

o ---------------------------

4 8 12 16 20 24 T(oK)

Figure 2.6: The adiabatic curves from MC and lVID simulations. They do not show any mode softening (crossing of the zero axis) over the temperature range where other indicators may suggest melting of the Ar7 cluster.

frequency in diffusive directions tends to zero near the mode-coupling tem-perature [49, 148]. It has also been suggested [210] that barrier heights and hopping rates can be estimated from INM frequencies.

Here we study the behaviour of INM spectra in a variety of cluster sys-tems. These include the heterogeneous Ar9XelO cluster [93], the homogeneous Arlg, Ar, and Ar12 clusters as well as (C60 )?, (C60 h3 and a generic 7 atom transition metal cluster modeled by the Gupta potential [84].

To obtain INM frequencies, the potential is approximated at each instant in time as a quadratic function of coordinates,

where Rtis the configuration at time t, expanded around the reference con-figuration, Ro. The instantaneous force matrix is F, while D is the Hessian matrix, namely the second derivative of the many body potential energy

25

4~------~----~~--__ --~--~--, 3

~-.. 2 '5 1 ~

-1

-20L-------200~--~--~~--~--~~--~~~

4~------~--~--~~~ __ ~,_--__. 3

2

-1

-2~------~--~--~------~--~~ o 200 400 600 800 T(~

Figure 2.7: The lowest frequency modes of the (C6oh cluster softens as the cluster goes through the melting phase change.

surface, V(R), a2v

Djll,kll(Ro) = a a IRa' (2.10) rjll' rkll

Clearly this approximation will be valid only for short times, but within this limitation, the (3N) x (3N) matrix of eigenvectors Ua,jll(Ro), of D(Ro) defines a unique set of "harmonic" modes.

The eigenvalues of D give the squares of the normal mode frequencies,

Aa = mu-';' = L Ua,jll(Ro)Djll,kll(Ro)UL,a(Ro) a == 1, ... ,3N. (2.11) j Il,kll

INM spectra are obtained by accumulating the frequencies Wo. during a con-stant energy simulation, namely along a trajectory R t , and obtaining the

26

10

--... 5 E 0 '3

10

--... 5

5 '3 0

-5

200

-

0 200

400

400 T(oK)

600 800

600 800

Figure 2.8: The mode-softening for the (C6o h3 duster as obtained from the Monte-Carlo and molecular dynamics results. The softening of many modes hints at sublimation even at low temperatures.

distribution,

P(w)dw _ Probability that an INM frequency Wa has a value between wand u.; + dw. (2.12)

Quantities such as the Einstein frequency, the fraction of imaginary modes, etc. can be deduced from this distribution [184]. Note that since the squares of the frequencies are the eigenvalues of the Hessian, these can be negative. Such modes are unstable, having imaginary frequencies; traditionally, these are represented on the negative w axis. Three translational modes have frequency zero. These are usually not included in the analysis, as are the frequencies of the three purely rotational modes [184].

INMs are related to short time dynamics [108, 174, 183] and hence it is natural to consider these as being indicative of the melting process. The "universal" shape [33] for the density of states for all liquids (whether polar, nonpolar or atomic) is also typical of clusters. The \vide range in the melting temperature of different cluster types, for instance metallic ver-sus nonmetallic, makes comparisons less easy and correspondence cannot be hence properly ascertained. On the other hand, the mean field theory [112]

27

's

-1~~--~~--~--~~--~~--~~ o 0.Q1 0.Q2 0.03 0.04 0.05

1

's

-1~----~----~----~------~--~ o 0.01 0.02 0.03 0.04 0.05 T('K)

Figure 2.9: The adiabatic curves from Me and MD simulations for a generic 7 atom transition metal cluster.

suggests that since mass differences may not play a major role, both the homogeneous and heterogeneous clusters may show similar INM spectra at least to a first approximation. Mass differences affect the dynamics in a dif-ferent way in which the potential energy surface is explored and thereby alter the INM spectra. INM spectra [23, 184]' have been studied extensively in the past both within the classical dynamics [184] framework as well quantum mechanically [29].

In finite clusters the INM frequencies are well-separated and they do not mix owing to a consequence of a non-crossing rule, which applies to the Hessian matrix. The non-crossing rule due to [153] states that eigen-values

28

3

.... - 1 'e u ~ -1

-3

-5 ~----------~----------~----------~~ o 10 20 30

3

-3

-5 ~----~--~~--~----~----~----~--~ o 10 20 30 T(K)

Figure 2.10: MC and MD results for the adiabatic INM frequencies for the rare-gas cluster Ar12.

corresponding to eigen-vectors of the same symmetry of the Hessian cannot cross[l11] . This allows us to define adiabatic INMs [40, 142] and study them at the cluster melting transition region.

Specifically, during a MD (or MC) simulation, the eigenvalues of the Hessian matrix are evaluated at each step, and averages are computed over the different realizations of each eigenvalue. This is then converted into

29

an adiabatic normal mode frequency which can be studied as a function of temperature.

2.3 Applications Standard equilibrium molecular dynamics simulations of finite cluster sys-tems are performed in a free volume. Interactions between the constituent atoms or molecules are the potentials discussed in Chapter 1; between rare-gas atoms, this is taken to be the usual Lennard-Jones form with parame-ters chosen appropriately, while for metal atoms we use the Gupta potential [84, 185]. For the Gupta potential the following values, p = giro, q = 3/ro and A = 0.101035 are found to be appropriate for the transition metals [185]. U is related to the bulk cohesive energy Ebulk = 1.17674U which was kept at unity and ro was used as the unit of distance. For modeling (C60 )N clusters the Girifalco potential [77] was used. For the mixed clusters which we study here in Section 2.3.1 the Xe-Ar and Ar-Ar Lennard-Jones parameters were obtained by the Lorentz-Berthelot mixing rules [3], namely

where a and (3 indicate the atomic identities. The values used are given in Table 2.1.

II

II

Table 2.1: Parameters used in this study.

Parameter t/K a/A

Ar-Ar 120 3.40

Xe-Ar 177.6 3.65

Xe-Xe II

The total potential is thus a sum of two-body terms,

V(rij) = 4<a-P [ (a;:p) l2 _ (a;i~p) '] , (2.13)

the distance between atoms i (which is of type a) and j (of type (3) is rij =1 I4 - R j I, the R's being the coordinates of the various atoms.

30

The velocity Verlet algorithm was used to solve the equations of motion. For an N atom cluster, the internal temperature T is the (per mode) average kinetic energy Ek . All quantities are measured in reduced units. The time step in molecular dynamics simulations was fixed at 0.01 in reduced units and in Monte-Carlo simulations the total number of sweeps was 106 iterations after equilibrating for 105 iterations. The Boltzmann constant in reduced units was fixed at unity.

Melting of homogeneous or single component clusters differ from those which are mixed. A single impurity can cause the cluster to have dif-ferent thermodynamic behaviours and ground state geometries. The 13 atom cluster, both homogeneous [41, 69, 70, 97, 161, 187] and heteroge-neous [61, 80, 82, 122, 190] has been the subject of many studies. Investiga-tions carried out by Frantz [69] for the binary rare-gas cluster Ar13-nKrn for o ~ n ~ 13 suggest that homogeneous clusters exhibit "magic number" ef-fects for many of their properties due to their largely icosahedral ground state configurations. Because Ar is only about 11% smaller in radius than Kr, all 13-atom Ar-Kr clusters have icosahedral-like configurations. This results in categorization of various topological isomers which are similar to their homo-geneous counterparts and permutational isomers which are based on various re-arrangements of the component atoms within a topological form. Ne and Ar have different sizes ( Ne atom is about 19% smaller than Ar), result-ing in many non-icosahedral isomers with energies similar to the icosahedral like isomers. The size differences and the intermolecular potential differ-ences (cNe-Ne/cAr-Ar = 0.2982 and cAr-Ar/cKr-Kr = 0.7280) result in Ne-Ar cluster configurations dramatically different from their Ar-Kr counterparts [69].

Heat capacity curves for the 13-atom homogeneous clusters are charac-terized by a prominent peak in the solid-liquid "phase" transition [41, 69, 70, 123, 161, 187]. This owes to the fact that a large energy gap exists between the ground state icosahedral isomer and the higher energy non-icosahedral isomers [41, 69, 161, 187]. The heat capacity curves for mixed Ar-Kr clus-ters is very similar to the homogeneous counterpart except that a very small peak occurs at very low temperatures and some minor variations owing to whether the cluster is Ar centric or Kr centric [69]. This feature is reminis-cent of order-disorder transition known to occur in some bulk alloy materials [123]. For Ne-Ar clusters the heat capacity curves are qualitath·ely different. Another difference between the homogeneous and heterogeneous clusters is

31

that the latter generally suffer from inadequate mixing of the different com-ponents and hence show different characteristic thermodynamic behaviours. This also results in difficulty in optimization for finding the ground state con-figurations; see Chapters 3-5. At low temperatures the competition between interatomic forces and atomic sizes affects the dynamics causing the cluster's behaviour to change arising from different geometries available. Quenching carried out by Frantz [69] at low temperatures reveals that for the 13 atom Ar-Kr cluster, the low energy isomers were dominated by icosahedral like structures with the smaller Ar atom at the center. For the Ke-Ar, this was not the case especially in Ne12Ar and NellAr2 as both had :\fe core icosa-hedral like structures as their lowest energy isomers, but for all the other clusters the Ne core configurations were absent.

ArnXe13-n clusters have been studied by Nayak and Ramaswamy [150] and earlier Robertson and Brown [165] had compared the phase transition behaviour of mixed cluster to the homogeneous case. They found that the melting temperatures exhibits plateaus in the case of equal mixture of (Ar-Xeh. Temporal correlations studied by Nayak and Ramaswamy [150] in the Ar-Xe mixture for n = 1 and n = 12 revealed that the spectrum of potential energy fluctuations for individual atoms have a marked 1/ f character, de-pending upon whether the cluster was solid or liquid and whether the atom chosen was Ar or Xe.

Shown in Figs. 2.6-2.10 are the adiabatic INMs curves showing the mode-softening behaviour and consistent with other indicators. Both the MC and MD show similar behaviour. The Ar7 cluster does not melt and there is no mode softening as observed, whereas Ar12 does and the modes soften in this case. In contrast the other 7 atom cluster for the transition metal and the (C6o h cluster both melt and show mode softening. (C6o h3 shows sublimation effects as a number of modes soften at low temperature.

Interesting features such as mentioned above prompts us to study the behaviour of the INM spectra in the case of the 19 atom cluster (both mixed and homogeneous) at the melting phase transition. This larger cluster is convenient for study since the ground state geometry of the homogeneous entity has the double icosahedral motif. The somewhat larger size of the cluster also allows for a better understanding of surface effects.

32

2.3.1 Melting of the heterogeneous cluster Ar9XelO

Phase change phenomena in finite clusters mirror analogous bulk phase tran-sitions. The INM spectra of mixed clusters have been studied in some detail previously [69, 112] and effects that devolve from the relative sizes' of the con-stituent atoms are germane to this study. For instance, mixed Ar-Kr clusters have properties that are similar to the pure cases, while Ne-Ar clusters have a wide range of behaviour which is very distinct from that of the pure clusters [69]; furthermore the properties vary drastically with composition. For in-stance: Ne4Ar9 and Ne3ArlO are very similar, but their complements Ne9Ar4 and NelOAr3 differ greatly.

A number of studies have focussed on the behaviour of mixed cluster species [61, 80, 82, 106, 122, 123, 150, 167, 190] in order to examine the effect of the heterogeneity on the melting process induced in such clusters and other factors which influence the cluster solid -t liquid transition.

Mixed clusters (especially those with comparable numbers of different species) are glassy in the sense that there are a large number of degenerate or nearly degenerate configurations even in the solid state. This leads to some interesting cases such as making it more difficult to locate the global minimum as the number of associated configurations and isomers grow expo-nentially with the size of the cluster. Equilibrium properties may be prone to errors if there is inadequate mixing of the different components and this may lead to large systematic errors at low temperatures. : At the melting transition, there may be co-exisiting phases with approxirriately no energy gaps and this can happen even with a single impurity [130].

The ground state geometry of the 19 atom homogeneous rare-gas clus-ter is that of two interpenetrating icosahedra. The point group for this arrangement is DSh ; the high symmetry and the compact geometry endow this "magic" number cluster with a somewhat higher stability than the 18 or 20 atom cases. The mixed cluster, Ar9XelO, however, does not have the DSh symmetry, although the same double-ico arrangement is maintained in the ground state (see Fig. 2.11).

This feature plays a significant role in the nature of the normal modes of the cluster, which melts around 60K. Shown in Fig. 2.12. is the caloric curve for this system, along with the Lindemann index as a function of the internal temperature. In comparison, the pure Ar19 cluster (see Fig. 2.13) melts around at 30K [11].

33

Figure 2.11: Axial and transverse views of the ground state configurations for the Ar9XelO (left) and Ar19 (right) clusters. Xe atoms are coloured red while theAr atoms are white. The atoms are linked by bonds in order to show the ~ymmetry; both clusters share the double-icosahedral motif, though only the homogeneous cluster belongs to the DSh symmetry group.

The high level of symmetry for the pure Ar19 cluster, however, causes the normal modes (at OK) to have a high degree of degeneracy. Analysis of the vibrations of the 19 atom cluster show that the normal modes span the representations 5A~ + A~ + 6E~ + 5E~ + A~ +4A~ + 5 E~ +4 E~. From this it is clear that at low temperatures, in the cold solid phase, the 51 modes of vibration must be distributed in sets which are compatible with the above symmetry classes. (Accordingly we see that the levels at OK have significant degeneracies; see Fig. 2.16). The INM spectra at low tempera-tures takes the familiar form shown in (see Fig. 2.14). After melting, when there are significant imaginary normal mode. frequencies, the INM spectra becomes broadened (see Fig. 2.14). and picks up a large tail, both in the

34

100 1-~ ___ ~~_·_r ___ • __ .~1

80 r 1 ~ 60 r j ~ :: r ..... B) j o~· I -110 -100 -90 -SO -70

ElF.

I

0.3 ~ j I i

<Q 0.2\

I 0.1 ! :J oL ....... ······

0 20 40 60 80 100 TtK)

Figure 2.12: (a) The caloric curve and (b) the variation of the Lindemann parameter with temperature, for Ar9XelO'

~-----

30·

i2' 20 l ~ 10 I

.... ,.,.. ...... : ......... -::.: ... -~.-

o L --.------~. - a)

-75 -70 -65 -60 -55 ElF.

0.3 r-- . : .:: .... ~. 0.2

0.1 I'· o~ ___ ----~~~-=--=-~~ ____ ~_b_)~

o 10 20 30 TtK)

Figure 2.13: (a) The caloric curve and (b) the variation of the Lindemann parameter with temperature, for Ar19

high frequency regime, as well as in the range of negative w [184]. Ar9XelO behaves differently. Even at low temperatures, the large num-

ber of nearly degenerate levels for the mixed cluster gives rise to an inher-

35

Figure 2.14: INM spectra at (a) 10K, (b) 20K and (c) 32K for the homoge-neous cluster Ar19. The cluster melts at around 30K.

ently broader distribution (see Fig. 2.15). This is typical of glassy systems, where recently Parisi and coworkers have argued that random matrix the-ory should be applicable in order to determine the nature of the frequency distribution [38]. Melting still implies the broadening of the distribution, but this is mainly into the negative frequency domain rather than into the high-frequency region; see Fig. 2.15. Since for a cluster, the different modes are well-separated, it is possible to compute an average frequency for each mode as a function of the temperature. However, the noncrossing rule that applies to the Hessian matrix says that levels with the same symmetry do not cross. Here, since all the levels have no symmetry, they essentially have the same trivial symmetry and therefore do not mix [40, 142].

We define the adiabatic frequency of a mode as the time average of the mode frequency. The variation of these frequencies with temperature for Ar19 andArgXelO is shown in Fig. 2.16 and Fig. 2.17 respectively. It is clear that while these vary smoothly for the most part, melting is heralded by the softening of the lowest few vibrational modes whose frequencies decrease to zero and eventually become imaginary. As discussed in the previous Section this behaviour is typical of all clusters including Ar19 shown in Fig. 2.16. It is clear, however, that the high level of symmetry of Ar19 gives a different

36

0.04 ---

0.03-

0.02 -

a) 10k

0.01 '. n N o ~_--'-'--~-..L~ "._---20 0 20 40 60 80

0.04 .-, -----..., j

0.03 ~ 3 . ~ 0.02 L

t( 0.01 ~

,/""l"~,"~ .... ",

"

'x..

o ,--I _~_~~~ -20 o 20

0.04 ,,-----l

0.03 ~ r

0.02 r-

om ~ L

Oi __ _ -20 o

rc----,-.--

" '"~''''' ........ -.....

b) 40k

,-~-.;~----------'

40 60

60

c) 74k

80

, , 1

80

Figure 2.15: INM Spectra for Ar9XelO at (a) 10K, (b) 40K, (c) 14K. Contrast the spectra at 10K with Fig. 2.14 for Ar19' At such low temperatures the different glassy configurations of nearly the same energy gh-e rise to the jagged distribution.

degeneracy structure to the adiabatic frequencies as compared to the case of Ar9XelO'

For mixed as well as for homogeneous clusters, it is the 1m-vest few modes that appear to initiate the melting process. Eigenvector analysis of these can be illustrative ; in the pure cluster case, these have been seen to be highly symmetric modes which are like shear motions (Fig. 2.18). In the mixed case, the symmetry of the mode is more difficult to ascertain. The lowest mode for Ar9XelO is shown in Fig. 9, where it can be seen that the mobility rests largely with the heavier Xe atoms. Viewed in conjunction with Fig. 2.11 which gives the ground state structure of the cluster, it is clear that the' cluster melts first at the surface, after which the inner atoms begin to move independently.

37

-..

70 IF=========:''====-::-:==.-·:~='::~:::.::::-··,:,~ . . ~r .. ,.",

I I

'I: (,) 30 S

5 10 15 20 25 30 TtK)

35

Figure 2.16: Variation of the adiabatic frequencies with temperature for Ar19'

Figure 2.17: Adiabatic frequencies and their variation with temperature for Ar9XelO. Comparison with the Ar19 case (Fig. 2.16) shows that Ar9XelO is es-sentially a disordered glass-'-like system which lacks the degeneracy structure of Ar19'

2.4 Summary \Ve have reexamined the nature of melting in pure and mixed clusters from the point of view of instantaneolis normal modes. The main application has

38 .

0>=10.91 em-I

Figure 2.18: Lowest frequency adiabatic mode for the cluster Ar19. The displacements of the different atoms is indicated by the arrows (the length is proportional to the amplitude). Since the atoms are identical, the symmetric displacement of all atoms is clearly seen.

been to the study of a glassy 19 atom heterogeneous cluster; the underlying potential energy surface of such systems has a large number of nearly degen-erate minima, corresponding to configurations that are very close in energy. This has an effect on the nature of the INM frequencies which tend to have a wider distribution than the case of homogeneous clusters.

In general we have shown here that the melting process can be understood as a mode-softening process when the vibrational modes are examined in detail. As a function of the internal temperature, the lowest normal modes soften-the frequency decreases to zero, and eventually become imaginary. This corresponds to a mode becoming unstable, and this causes the cluster to melt. Analysis of the corresponding eigenvectors for the mixed cluster

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0>=5.61 cm- 1

Figure 2.19: As in (Fig. 2.18) for the mixed cluster Ar9XelO. The Xe atom displacements can be seen to be larger than the Ar atom displacements; thus this mode corresponds to a surface melting as discussed in the text.

studied in detail show that the atomic displacements are consistent with a surface melting picture, where the heavier Xe atoms first become more mobile and capable of large scale intra cluster motions.

The adiabatic mode softening is observed to be common to all the clusters studied. The study of such adiabatic modes thus throws considerable light on the detailed mechanics of the melting process. Similar analysis will be very useful in examining other phase change phenomena and may help in providing a novel and useful order-parameters for classifying phase transitions in finite systems.

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