Molecular dynamics simulation of the melting of Na nanoclusters - - PowerPoint PPT Presentation

Molecular dynamics simulation of the melting of Na nanoclusters

Yoon Tiem Leong Talk given at Theoretical and Computational Group seminar, School of Physics, Universiti Sains Malaysia Friday, 29 March 2013

ABSTRACT

The paper by Aguado et al., J. Phys. Chem. B 2001, 105, 2386-2392, is briefly reviewed. Using orbital-free molecular dynamics (OFMD), Aguado have investigated the melting of Na55, Na92, Na143 clusters. Focus of the talk will be in what to look for and what quantities to monitor when one seek to investigate the melting of atomic clusters using molecular dynamics.

Cluster melting: Why so interesting

◮ ∼ 101 nm, N ∼ 102 − 103. ◮ Different than the first order phase transition of bulk melting. ◮ Melting point of Tm is not “sharp” as in bulk melting –

“melting-like” transition.

◮ Multiple melting stages – surface melting, homogeneous

melting.

◮ Tm scales with cluster size beyond certain size but not

extrapolated to bulk melting point.

◮ Tm oscillates with size. ◮ Difficult to study experimentally. ◮ MD complements the investigation. ◮ The mechanisms by which the melting-like transition proceeds

in these large clusters can be investigated via MD.

OFMD

◮ Orbital-free version of ab initio MD scale linearly with system

size (rather than ∼ N3).

◮ Contain approximated electron kinetic energy. ◮ Optimised for bulk and surface, but application to cluster is a

new attempt.

◮ Can account for the effects of detailed electronic contribution

• n the total energy on ions.

◮ Important for matelic clusters. ◮ Setback: less complete statistics can be obtained due to

expensive computational cost.

◮ Give more reliable identification of transitions.

Method

◮ Set box size (71 au and 64 au). ◮ Set 64 × 64 × 64 grid. ◮ Set energy cutoff 10 Ry. ◮ Set fictitious mass associated with the electron density

coefficients 1.0 × 1010 − 3.3 × 108 au.

◮ Set time step for Verlet algorithm ∆t = 0.73 fs.

Determination of low-temperature ground states

◮ Very difficult to determine global minimum for large clusters

as number of different local minima increases exponentially with N.

◮ Instead of finding them, Aguado et al. used structures

suggested from experiments - icosahedral.

◮ The GS structure are constructed using a mathematical

procedure known as icosahedral growth.

◮ “ · · · also used dynamical simulated annealing to generate

low-temperature isomers, but this procedure always led to amorphous structures for Na92, Na142 (less stable than the icosahedral ones) and to a nearly icosahedral structure for Na55”.

◮ Na92, Na142 icosahedral got point defects on outer surface,

while Na55 got none.

Icosahedron

Icosahedron Facts

Indicators for locating melting-like transition

◮ Specific heat per particle

CV = [N − M

• 1 −

2 3N−6

• EkintE −1

kin t]−1. ◮ Mean square displacement

r2(t) =

1 Nnt

ni

j=1

N

i=1

• Ri(toj + t) − Ri(toj)

2, where nt is the number of time origins, t0j, considered along a trajectory.

◮ Diffusion coefficient D = 1 6 d dt r2(t). ◮ Time evolution of the distance between each atom and the

center of mass of the cluster ri(t) = |Ri(t) − Rcm(t)|.

◮ Radial atomic density, averaged over an entire dynamical

trajectory ρ(r) = dNat(r)/dr

4πr2

, where dNat(r) is the number of atoms at distances from the center of mass between r + dr.

Internal cluster temperature and caloric curve

◮ Internal temperature as a function of the total energy,

T = 2Ek

3N−6. ◮ caloric curve: T vs. Total energy. ◮ “Several molecular dynamics simulation runs at different

constant energies were performed. The initial positions of the atoms for the first run were determined by slightly deforming the equilibrium low-temperature geometry of the isomer. The final configuration of each run served as the starting geometry for the next run at a different energy. The initial velocities for every new run were obtained by scaling the final velocities of the preceding run. The total simulation time was 20 ps for each run at constant energy”.

◮ Thermal phase transition shows up as (i) a change in the slope

• f caloric curve, and (ii) as spikes in specific heat per atom.

◮ Both are obtained independently. ◮ The height of the step in caloric curve during the change of

slope gives an estimate of the latent heat of fusion.

Caloric curve and specific heat for Na142

Na142

◮ Two-step melting, Ts = 240 K and Tm = 270 K. ◮ Experiment measured Ts − 280 K. ◮ Latent heat simulated qm ≈ 15 meV/atom, vs. 14 meV/atom

measured experimentally.

◮ Pre-melting peak not measured experimentally, since the two

peaks are so close, and the first peak is much lower than the second.

Na142

◮ Another MC simulations by Calvo and Spiegelmann, using a

semiempirical many-atom potential for Na139 also reported two-step melting with Ts ≈ 210 K and Tm ≈ 230 K.

◮ Calvo and Spiegelmann that these two temperatures become

closer as the cluster size increases.

◮ For more than 100 atoms, there is effectively just one peak in

the specific heat and a single-step melting.

◮ They also performed Tight-binding (TB) MD and found

different Tm and Ts from the semiempirical potentials

◮ TB (empirical potentials) overestimate (underestimate) the

experimental values, but the qualitative picture of melting in two close steps was the same.

Caloric curve and specific heat for Na92

Na92

◮ Two-step melting, Ts = 130 K and Tm = 240 K. Gap is more

prominent.

◮ Pre-melting peak is much weaker than the homogeneous

melting.

◮ Compares well with experiment. ◮ Calvo and Spiegelmann’s finding for Na93 using empirical

potential gives 100 K and 180 K; TB about 100 K higher.

Caloric curve and specific heat for Na55

Na55

◮ MD predict single step melting at 190 K, vs. experimental

value of 325 K.

◮ Suspect this anomalously higher effect is not produced by MD

because the OFMD does not include electronic shell-effects (thought to be responsible for the high melting point).

◮ Aguado suggest this needs to be further investigated.

ri(t)sta

◮ To zoom into the detail mechanism of the melting at the

atomic level, monitor ri(t)sta for each atom.

◮ ri(t)sta, short-time averages (sta) of the distances between

each atom and the center of mass of the cluster

◮ The cluster evolution during the trajectories was followed

visually using computer graphics.

ri(t)sta at low temperature

ri(t)sta at low temperature

◮ At low temperatures ri(t)sta is almost independent of step. ◮ Clearly display “shell” structure. ◮ Movies show clusters are solid (atoms merely vibrate about

equilibrium positions).

◮ Curve crossings are due to oscillatory motion and slight

structural relaxations rather than diffusive motion.

◮ Quasidegenerate groups of atoms that are characteristic of the

icosahedral symmetry can be distinguished (pattern of the grouping of lines traced out by the vibrating atoms can deduce the position).

The radial atomic density distributions with respect to the cluster center at 30 K for Na142

The radial atomic density distributions with respect to the cluster center at 30 K for Na142

◮ ρ(r) The atoms in the icosohedral isomer are distributed in

three well-separated shells

◮ The shell structure is still present at T = 130 K. ◮ Though the shell structure is still present at T = 160 K in the

Figure 7, movie reveals “isomerization transitions” (with no inter-shell diffusion).

◮ “Isomerizations”: motion of vacancies in the outer shell

surfaces, in such a way that different isomers are visited while the icosahedral structure is preserved.

◮ The onset of this motion is gradual. ◮ Does not show up in specific heat but do so in temperature

evolution of the diffusion coefficient (see later).

◮ The true surface melting stage does not develop until a

temperature of Ts ≈ 240 K is reached.

ri(t)sta at 361 K for Na142

ri(t)sta at 361 K for Na142

◮ Shell structure disappears. ◮ The cluster is liquid; all atoms diffuse throughout the cluster. ◮ The melting detail of Na92, as revealed by the radial atomic

density at various temperature, and ri(t)sta, is similar to that of Na142.

Na55

◮ A perfect two-shell icosahedron ◮ Surface atoms have no empty sites to hop to ◮ Diffusion within an atomic shell is almost as difficult as

diffusion across different shells.

◮ When the surface atoms have enough energy to exchange

positions with one another, they can just as easily migrate throughout the whole cluster, and melting proceeds in a single stage at 190 K.

◮ In effect, the melting will happen “in a sudden” and not

gradual.

◮ The one-step melting is associated with a large energy gap

between the ground-state icosahedral structure and the closest low-lying isomers.

Variation of diffusion coefficient with temperature for Na142

Variation of diffusion coefficient with temperature for Na142

◮ Features of D(T) for Na92 are very similar. ◮ At T ≤ 140 K, D ≈ 0 – only oscillatory motion of the atoms. ◮ For 140K < T < Ts, D increases, indicating atoms in the

cluster are not undergoing simple vibrational but “isomerisation” motion that preserves the icosahedral structure.

◮ Slope increases appreciably at Ts when surface melting occurs

but Tm is not noticeable when the cluster finally melts.

◮ Is sensitive to the surface melting stage but not the

homogeneous melting (masked by the former).

◮ Is a good indicator of homogeneous melting only if surface

melting stage is absent e.g., Na55.

Observations

◮ Large cluster: melting transition occurs over smaller

temperature range, at least near an icosahedral-shell closing.

◮ Pre-peak diminishes as cluster size increases (as surface effect

diminishes)

◮ Homogeneous melting less ambiguously defined for large

clusters.

◮ The above apply to caloric curve and specific heat. ◮ Microscopic indicators, i.e., D or ri(t)sta are sensitive to

microscopic reorganisation of atomic arrangement but not melting transition.

◮ Shape of radial atomic density distribution good indicator:

Pronounce structure at low T, smoothed out at intermediate temperature (when vacancy diffusion / surface melting begin to emerge), totally flatten above Tm.

Melting temperatures vs cluster size

Melting temperatures vs. cluster size

◮ Good agreement with experiments except for Na55 (due to

electronic-shell effects).

◮ Not seeing dynamic coexistence (DC) (shown up in back

bending of caloric curves), nor bimodal behaviour – possibly a consequence of time scale too short.

◮ DC – cluster fluctuate in time between solidlike and liquidlike

phase.

◮ It is difficult to see DC in large clusters. ◮ Importance of DC also decreases with increasing cluster size

due to entropic effect: difficult for liquid phase to “find its way back to solid phase catchment basin in the potential energy surface”.

Dynamic coexistence in aluminium cluster melting (Alavi, J.Phys. Chem. A 2006, 110, 1518 - 1523)

Sequel: Eur. Phys. J. D 15, 221227 (2001)

Sequel: PRL 94 (2005) 233401