Introduction

Introduction

Nanoclusters of atoms, molecules and ions play an important role. Their thermal, structural and dynamical properties are of interest in many fields such as crystal growth, gas phase nucleation, structure of amorphous materials, catalysis and atmosphere chemistry. Their study raises fundamental questions concerned with phase transitions and coexistence and the operational meaning of thermodynamic limit. Incidentally, we define a nanocluster as a microaggregate whose linear dimensions are of the order of 10^-⁹m.

Phase transitions and coexistence in nanoclusters of different sizes have attracted many investigations aiming at a comparative behaviour of such a small systems and the corresponding bulk systems, the underlying molecular mechanisms of melting, freezing, hysteresis, glass formation, annealing, and the development of theoretical models to correlate and predict simulation and experimental results.

A significant part of the studies has been carried out by computer simulation since the experimental methods to probe the behaviour of these systems are not of straightforward implementation and control.

Here, we present results of the melting, freezing, glass formation and annealing of unconstrained (zero external pressure) KCl nanoclusters by molecular dynamics, with the assistance of several computer animations.

Part of the calculations were performed by molecular dynamics at constant temperature. Recall that the temperature, T, is directly related to the particle velocities by the energy equipartition theorem:

where m_i and v_i are respectively the masses and velocities of the N particles in the model, k_B the Boltzmann constant, and <...> denotes a time average.

Shortly, to attain a pre-set temperature T, the velocities are adjusted so that the value T is maintained, on average, along the simulation. Suppose two different successive simulations at temperatures T₁ and T₂. If T₂ >> T₁ it means a fast heating of the system; T₂ ≈> T₁ means a slow heating. The inverse, of course, for fast cooling and slow cooling. This procedure simulates a thermostat, and from now on it is denoted control of temperature.
Imposing successive T values to the solid and liquid phases leads to jumps of the configurational energy and other properties at the transition points. Heating and cooling can produce hysteresis cycles, whose properties depend on the clusters size and type. Furthermore, it is shown that nanoclusters can also present glass-like transitions when the liquid is subjected to fast cooling. If the glasses are heated slowly they can turn into crystalline structures (annealing). Nonetheless, constraining the temperature does not allow for sustained phases coexistence as shall be explained ahead.

Other calculations were also carried out by means of successive small total energy fluxes. After each fux, given by scaling the velocities at the start of each simulation, the system is, however, put into isolation and let to relax until it reaches a final, not pre-set, temperature. Therefore, the total energy is kept constant, not the temperature. This procedure is able to unravel some important details of the transitions mechanisms, as phases coexistence, that otherwise remain hidden. It simulates a system in contact with hot or cold sources without controlling the temperature, but the total energy of the system. As such, from now on it is identifyed as conducted by the control of energy. Similary to the other procedure we can talk about fast or slow control of total energy.

Having defined the procedures, let´s see, shematically, the behaviours of bulk systems and nanoclusters concerning phase transitions and coexistence.

Figures 1 shows what is expected for bulk systems: (a) by the control of temperature and (b) by the control of energy. The horizontal line in (b) is the region of phases coexistence where the temperature remains naturally constant. In case (a), it suffices a tiny increase of temperature T_A to cause a transition of the system to the other phase at temperature T_B ≈ T_A. No phases coexistence is seen.

Figure 2 displays the existence of hysteresis. It is characterised by a overheating or undercooling generally due to the lack of very small impurities that can act as nuclei of the transition at the right temperature, or to high heating or cooling rates. Super-heated or super-cooled states are metastable and, sooner or later, jump to the stable situations.

Nanoclusters followed by a control of temperature have behaviours similar to the bulk systems. However, when they are followed by a control of energy the difference is remarkable as seen in Figure 3. For instance, when the solid reaches the onset of melting at T_A the size of the cluster is always less than the critical nucleus size and phase coexistence cannot be sustained, unless the system decreases its temperature freely until a critical nucleus is attained. Thus, the nature of controlling the energy, by small fluxes and subsquent isolation, allows the system to adjust the internal kinetic and potencial energies keeping, however, the total energy constant: melting decreases and freezing increases the temperature along the coexistence region. As the size of the cluster increases, the coexistent line turns asymptotically horizontal, that is, the bulk behaviour is approached.

It seems clear that the two processes, by controlling the total energy or the temperature, are not equivalent: one is able to unravel phases coexistence, the other is not. So, the total energy turns out as the natural variable at least for clusters.

(a) (b)

Figure 1. (a) control of temperature; (b) control of total

energy for bulk systems

(a) (b)

Figure 2. Same as Fig. 1, but with hysteresis

Phase Transitions and Coexistence in Nanoclusters of KCl