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 Na 55 , Na 92 , Na 143 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 ◮ ∼ 10 1 nm, N ∼ 10 2 − 10 3 . ◮ Different than the first order phase transition of bulk melting. ◮ Melting point of T m is not “sharp” as in bulk melting – “melting-like” transition. ◮ Multiple melting stages – surface melting, homogeneous melting. ◮ T m scales with cluster size beyond certain size but not extrapolated to bulk melting point. ◮ T m 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 ∼ N 3 ). ◮ 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 on 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 × 10 10 − 3 . 3 × 10 8 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 Na 92 , Na 142 (less stable than the icosahedral ones) and to a nearly icosahedral structure for Na 55 ”. ◮ Na 92 , Na 142 icosahedral got point defects on outer surface, while Na 55 got none.

Icosahedron Icosahedron Facts

Indicators for locating melting-like transition ◮ Specific heat per particle � � � E kin � t � E − 1 2 kin � t ] − 1 . C V = [ N − M 1 − 3 N − 6 ◮ Mean square displacement � 2 , where n t is � n i � N � r 2 ( t ) � = 1 � R i ( t o j + t ) − R i ( t o j ) j =1 i =1 Nn t the number of time origins, t 0 j , considered along a trajectory. ◮ Diffusion coefficient D = 1 dt � r 2 ( t ) � . d 6 ◮ Time evolution of the distance between each atom and the center of mass of the cluster r i ( t ) = | R i ( t ) − R cm ( t ) | . ◮ Radial atomic density, averaged over an entire dynamical trajectory ρ ( r ) = dN at ( r ) / dr , where dN at ( r ) is the number of 4 π r 2 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 = 2 � E k � 3 N − 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 of 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 Na 142

Na 142 ◮ Two-step melting, Ts = 240 K and Tm = 270 K. ◮ Experiment measured T s − 280 K. ◮ Latent heat simulated q m ≈ 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.

Na 142 ◮ Another MC simulations by Calvo and Spiegelmann, using a semiempirical many-atom potential for Na 139 also reported two-step melting with T s ≈ 210 K and T m ≈ 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 T m and T s 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 Na 92

Na 92 ◮ 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 Na 93 using empirical potential gives 100 K and 180 K; TB about 100 K higher.

Caloric curve and specific heat for Na 55

Na 55 ◮ 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.

� r i ( t ) � sta ◮ To zoom into the detail mechanism of the melting at the atomic level, monitor � r i ( t ) � sta for each atom. ◮ � r i ( 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.

� r i ( t ) � sta at low temperature

� r i ( t ) � sta at low temperature ◮ At low temperatures � r i ( 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 Na 142

The radial atomic density distributions with respect to the cluster center at 30 K for Na 142 ◮ 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 T s ≈ 240 K is reached.

� r i ( t ) � sta at 361 K for Na 142

� r i ( t ) � sta at 361 K for Na 142 ◮ Shell structure disappears. ◮ The cluster is liquid; all atoms diffuse throughout the cluster. ◮ The melting detail of Na 92 , as revealed by the radial atomic density at various temperature, and � r i ( t ) � sta , is similar to that of Na 142 .

Na 55 ◮ 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 Na 142