Ogata Lab., Dept. Mech. Sci. and Bioeng., Grad. School Eng. Sci., Osaka Univ.

Diffusion modelling

Yun-Jiang Wang, Jun-Ping Du, Shuhei Shinzato, Lan-Hong Dai and Shigenobu Ogata, "A free energy landscape perspective on the nature of collective diffusion in amorphous solids" , Acta Materialia 157 (2018) 165-173.

The nature of collective diffusion in amorphous solids is in strong contrast with diffusion in crystals. However, the atomic-scale mechanism and kinetics of such collective diffusion remains elusive. Here the free energy landscape of collective diffusion triggered by single atom hopping in a prototypical Cu50Zr50 metallic glass is explored with well-tempered metadynamics which significantly expands the observation timescale of diffusion at atomic-scale. We clarify an experimentally suggested collective atomic diffusion mechanism in the deep glassy state. The collective nature is strongly temperature-dependent. It evolves from string-like motion with only several atoms to be large size collective diffusion at high temperature, which can promote the atomic transport upon glass transition temperature. We also clarify the apparent diffusivity is dominated by the highest free energy barrier of atomic diffusion among widely distributed free energy barriers due to the dynamic heterogeneity of metallic glass, which suggests the sequential nature of diffusion is a proper assumption to the metallic glasses with dynamic heterogeneity. The temperature and pressure dependence of diffusion free energy landscape are further quantified with activation entropy, (19.6 ± 2.5)kB, and activation volume, (7.9 ± 3.4) Å3, which agree quantitatively with experiments. Laboratory timescale simulations of atomic diffusion brings physical insights into the unique atomic motion mechanism in non-crystalline materials.

Shuhei Shinzato, Ei Kasuya, and Shigenobu Ogata, "Study of Interstitial Atom Diffusion in Nanowire using Duffusion Equation driven by Chemo-Mechanical Potential" , Journal of the Society of Materials Science, Japan, 67-2 (2018) 263-268.

Impurity atom distribution and its time evolution in cylindrical nanowire were studied using a diffusion equation based on chemo-mechanical potential, which considered coupling effects between chemical potential (chemical effect) and internal stress (mechanical effect) due to interstitial impurity atoms. We did consider diffusion blocking and nanowire yielding effects when we solved the diffusion equation. The yield criterion was described by comparing local volumetric strain caused by impurity atoms with critical volumetric strain at which the lattice instability appears. Obtained results used the diffusion equation with realistic materials parameters indicate a possibility of nanowire yielding due to interstitial atom diffusion and diffusion slowdown due to the yielding.

Hajime Kimizuka, Shigenobu Ogata, and Motoyuki Shiga , “Mechanism of fast lattice diffusion of hydrogen in palladium: Interplay of quantum fluctuations and lattice strain”, Physical Review B, 97-1 (2018) 014102-1-11.

Understanding the underlying mechanism of the nanostructure-mediated high diffusivity of H in Pd is of recent scientific interest and also crucial for industrial applications. Here, we present a decisive scenario explaining the emergence of the fast lattice-diffusion mode of interstitial H in face-centered cubic Pd, based on the quantum mechanical natures of both electrons and nuclei under finite strains. Ab initio path-integral molecular dynamics was applied to predict the temperature- and strain-dependent free energy profiles for H migration in Pd over a temperature range of 150–600 K and under hydrostatic tensile strains of 0.0%–2.4%; such strain conditions are likely to occur in real systems, especially around the elastic fields induced by nanostructured defects. The simulated results revealed that, for preferential H location at octahedral sites, as in unstrained Pd, the activation barrier for H migration ( Q ) was drastically increased with decreasing temperature owing to nuclear quantum effects. In contrast, as tetrahedral sites increased in stability with lattice expansion, nuclear quantum effects became less prominent and ceased impeding H migration. This implies that the nature of the diffusion mechanism gradually changes from quantum- to classical-like as the strain is increased. For H atoms in Pd at the hydrostatic strain of ∼ 2.4 % , we determined that the mechanism promoted fast lattice diffusion ( Q = 0.11 eV) of approximately 20 times the rate of conventional H diffusion ( Q = 0.23 eV) in unstrained Pd at a room temperature of 300 K.

Yushi Nakagami, Hajime Kimizuka, and Shigenobu Ogata , “Controlling Factors for the Formation of Guinier-Preston Zones in Al-Cu Alloys: An Atomistic Study”, Journal of the Japan Institute of Metals and Materials, 80-9 (2016) 575-584 [in Japanese].

Nanosized precipitates of solute atoms, which are often called "Guinier-Preston (GP) zones," are known to play a significant role in the precipitation-hardening effect on the Al alloys. In this study, the controlling factors for the formation of solute nanoclusters as in GP1 and GP2 zones, which are observed in the early and middle stages, respectively, of aging in the Al-Cu system, were investigated by extracting and examining the intra- and intercluster interaction energies of Cu in Al using density functional theory (DFT). DFT-calculated two- and three-body binding energies for Cu indicated that a Cu-Cu pair tends to bind at the first nearest-neighbor (1NN) positions, and a Cu-Cu-Cu triplet energetically prefers the arrangement of the triangular atomic cluster that contains two Cu-Cu pairs at the 1NN positions on the same {100} plane. Such a characteristic short-range ordering was suggested to be dominated by attractive three-body interactions due to the chemical (charge localization) effect, leading to planar clustering as found in the GP1 zones along the {100} planes. Intercluster interaction energies between two Cu planar clusters in Al were also calculated based on DFT. The results indicated that the energetically preferable configuration was the one in which two planar clusters are aligned at an intercluster distance of 2a (where a is the lattice constant of Al); it is noteworthy that this distance was the same to that in the basic structures of the GP2 zones observed in the experiments. A potential model for a dilute Al-Cu system was constructed on the basis of the extracted intra- and intercluster interactions, and then applied to atomistic Monte Carlo modeling for predicting the planar segregation of Cu atoms at finite temperatures. As a result, the formation of planar Cu clusters and the alignment of two planer clusters separated by 2a were successfully reproduced within a specific temperature range. This demonstrated that these interactions were important controlling factors for the formation of a characteristic pattern of solute clusters in the Al-Cu system.

Masakazu Tane, Hajime Kimizuka, and Tetsu Ichitsubo, “Two distinct crystallization processes in supercooled liquid”, Journal of Chemical Physics, 144 (2016) 194505-1-10.

Using molecular dynamics simulations we show that two distinct crystallization processes, depending on the temperature at which crystallization occurs, appear in a supercooled liquid. As a model for glass-forming materials, an Al2O3 model system, in which both the glass transition and crystallization from the supercooled liquid can be well reproduced, is employed. Simulations in the framework of an isothermal-isobaric ensemble indicate that the calculated time-temperature-transformation curve for the crystallization to gamma(defect spinel)-Al2O3 exhibited a typical nose shape, as experimentally observed in various glass materials. During annealing above the nose temperature, the structure of the supercooled liquid does not change before the crystallization, because of the high atomic mobility (material transport). Thus, the crystallization is governed by the abrupt crystal nucleation, which results in the formation of a stable crystal structure. In contrast, during annealing below the nose temperature, the structure of the supercooled liquid gradually changes before the crystallization, and the formed crystal structure is less stable than that formed above the nose temperature, because of the restricted material transport.

Marco Fronzi, Hajime Kimizuka, and Shigenobu Ogata, “Atomistic investigation of the vacancy-assisted diffusion mechanism in Mg ternary (Mg-RE-M) alloys”, Computational Materials Science, 98 (2015), 76-82.

To investigate the kinetics of the formation of solute cluster structures in some of the Mg ternary alloys, we perform a first-principles analysis of some fundamental quantities in doped Mg lattices. We calculate interaction energies between vacancy and solute atoms in both Mg hexagonal close packed (HCP) and face-centered cubic (FCC) crystal structures. In particular, we consider in this work Al, Gd, Y, and Zn solute atoms. Also, to understand the diffusion mechanism, we calculate vacancy-assisted diffusion for Mg and solute atoms in HCP and FCC lattices using the nudge elastic band method.

Hajime Kimizuka, Shu Kurokawa, Akihiro Yamaguchi, Akira Sakai, and Shigeobu Ogata, “Two-Dimensional Ordering of Solute Nanoclusters at a Close-Packed Stacking Fault: Modeling and Experimental Analysis”, Scientific Reports, 4 (2014), 7318-1-7318-9.

Predicting the equilibrium ordered structures at internal interfaces, especially in the case of nanometer-scale chemical heterogeneities, is an ongoing challenge in materials science. In this study, we established an ab-initio coarse-grained modeling technique for describing the phase-like behavior of a close-packed stacking-fault-type interface containing solute nanoclusters, which undergo a two-dimensional disorder-order transition, depending on the temperature and composition. Notably, this approach can predict the two-dimensional medium-range ordering in the nanocluster arrays realized in Mg-based alloys, in a manner consistent with scanning tunneling microscopy-based measurements. We predicted that the repulsively interacting solute-cluster system undergoes a continuous evolution into a highly ordered densely packed morphology while maintaining a high degree of six-fold orientational order, which is attributable mainly to an entropic effect. The uncovered interaction-dependent ordering properties may be useful for the design of nanostructured materials utilizing the self-organization of two-dimensional nanocluster arrays in the close-packed interfaces.

Shigenobu Ogata, “Diffusion Dynamics of Hydrogen and Carbon in Iron”, Bulletin of the Iron and Steel Institute of Japan, 19-11(2014), 745-750 [in Japanese].

Hajime Kimizuka and Shigenobu Ogata,"Predicting Atomic Arrangement of Solute Clusters in Dilute Mg Alloys", Materials Research Letters, 1-4 (2013) 213-219.

Theoretical prediction of the heterogeneous atomic structures in multicomponent alloys is one of the most challenging issues in materials science. Here we present a first-principles demonstration of this by constructing an on-lattice effective multibody potential model to describe the energetics of hexagonal close-packed Mg crystals containing stacking faults (SFs) with Al and Gd substitutions. Remarkably, we showed that our intuitive model can describe the segregation of solute atoms to SFs and reproduce the characteristic short-range chemical order in dilute Mg–Gd and Mg–Al–Gd systems, including long-period stacking ordered structures, in a manner consistent with recent scanning transmission electron microscopy measurements.

Yun-Jinag Wang, Guo-Jie Jason Gao, and Shigenobu Ogata, "Atomistic understanding of diffusion kinetics in nanocrystals from molecular dynamics simulations", Physical Review B, 88-11 (2013) 115413-1-7.

Understanding the grain size effect on diffusion in nanocrystals has been hampered by the difficulty of measuring diffusion directly in experiments. Here large-scale atomistic modeling is applied to understand the diffusion kinetics in nanocrystals. Enhanced short-circuit diffusivity is revealed to be controlled by the rule of mixtures for grain-boundary diffusion and lattice diffusion, which can be accurately described by the Maxwell-Garnett equation instead of the commonly thought Hart equation, and the thermodynamics of pure grain-boundary self-diffusion is not remarkably affected by varying grain size. Experimentally comparable Arrhenius parameters with atomic detail validate our results. We also propose a free-volume diffusion mechanism considering negative activation entropy and small activation volume. These help provide a fundamental understanding of how the activation parameters depend on size and the structure-property relationship of nanostructured materials from a physical viewpoint.

Hajime Kimizuka, Marco Fronzi, and Shigenobu Ogata, "Effect of alloying elements on in-plane ordering/disordering of solute clusters in Mg-based long-period stacking ordered structures: A first-principles analysis", Scripta Materialia, 69-8 (2013) 594-597.

Using density functional theory, we characterized the in-plane binding between L12-type solute clusters in Mg-M-RE (M = Al, Zn; RE = Y, Gd) long-period stacking ordered (LPSO) structures. The difference between the Al and Zn concentrations within the clusters determines whether the intercluster interaction is attractive or repulsive. Incomplete in-plane ordering observed experimentally in Mg-Zn-Y LPSO structures was suggested to be caused by the unlinked nature of the clusters owing to their significant inward contraction.

Yun-jiang Wang, Akio Ishii, and Shigenobu Ogata, "Entropic effect on creep in nanocrystalline metals", Acta Materialia, 61-10 (2013) 3866-3871.

We report a significant entropic effect on creep of nanocrystalline metal using molecular dynamics. Our simulations reveal that the activation entropy may contribute a multiplicative factor of many orders of magnitude to the steady-state creep rate. The relationship between activation entropy and enthalpy obeys an empirical Meyer-Neldel compensation rule. The activation volume is found to decrease with increasing temperature for dislocation nucleation creep, which agrees well with experimental results. The study opens up an avenue for quantitatively discussing the entropic effects on various thermally activated deformations in nanocrystals.

Akio Ishii, Ju Li, and Shigenobu Ogata, "”Conjugate Channeling” Effect in Dislocation Core Diffusion: Carbon Transport in Dislocated BCC Iron", PloS one, 8-4 (2013) e60586-1-7.

Dislocation pipe diffusion seems to be a well-established phenomenon. Here we demonstrate an unexpected effect, that the migration of interstitials such as carbon in iron may be accelerated not in the dislocation line direction ξ, but in a conjugate diffusion direction. This accelerated random walk arises from a simple crystallographic channeling effect. c is a function of the Burgers vector b, but not ξ, thus a dislocation loop possesses the same everywhere. Using molecular dynamics and accelerated dynamics simulations, we further show that such dislocation-core-coupled carbon diffusion in iron has temperature-dependent activation enthalpy like a fragile glass. The 71° mixed dislocation is the only case in which we see straightforward pipe diffusion that does not depend on dislocation mobility.

Narumasa Miyazaki, Masato Wakeda, and Shigenobu Ogata, "Temperature Dependence of Viscosity in Supercooled Liquid of Cu-Zr Bulk Metallic Glass by Molecular Dynamics", Journal of the Society of Materials Science, Japan, 62-3 (2013) 172-178. [in Japanese]

When a cooling rate is fast enough to prevent crystallization, molten metals are solidified into a disordered structure as known amorphous metals. During the cooling process, molten metals become supercooled liquid below the melting point, and dynamic factors such as viscosity and relaxation time rapidly increase, while static factors such as density show no significant change. In the present study, we investigated a temperature dependence of viscosity of Cu-Zr bulk metallic glass above the glass transition temperature Tg using both molecular dynamics (MD) technique and a recently developed energetic technique. A limitation of MD time scale prevents us to calculate viscosity at Tg, because a relaxation time of supercooled liquid becomes significantly long at lightly above Tg. On the other hand, a new method developed by Kushima et. al., analyzes transition state pathway trajectory from an energy viewpoint and provides viscosity of liquid state at wide temperature range from sufficient high temperature to Tg. We discuss a temperature dependence of viscosity of Cu-Zr bulk metallic glass both from atomistic and energetic viewpoints.

Takehiro Yoshikawa, Toshiyuki Takayanagi, Hajime Kimizuka, and Motoyuki Shiga, "Quantum-thermal crossover of hydrogen and tritium diffusion in alpha-iron", Journal of Physical Chemistry C, 116-43 (2012) 23113-23119.

The diffusion coefficients of hydrogen (H) and tritium (T) in α-Fe have been computed using two approximate quantum dynamical techniques, that is, centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD), in the temperature range of T = 100-1000 K using the embedded atom method (EAM) potential. It has been found that the RPMD and CMD methods give very similar results. From a further analysis based on quantum transition-state theory (centroid density QTST) combined with path integral molecular dynamics (PIMD), it has been clear that there is a crossover between thermal and quantum mechanisms at about T = 500 and 300 K for H and T diffusions, respectively. The importance of nuclear quantum effects at low temperatures has been illustrated in terms of the effective free-energy surface map.

Akio Ishii, Shigenobu Ogata, Hajime Kimizuka, and Ju Li, "Adaptive-boost molecular dynamics simulation of carbon diffusion in iron", Physical Review B, 85-6 (2012) 064303-1-7 .

We have developed an accelerated molecular dynamics (MD) method to model atomic-scale rare events. In this method, a smooth histogram of collective variables is first estimated by canonical ensemble molecular dynamics calculations, and then a temperature-dependent boost potential is iteratively constructed to accelerate the MD simulation. This method not only allows us to observe the rare events but also to evaluate the profile of free energy and trial frequency along the reaction coordinate. We employed this method to study carbon diffusion in bcc iron and evaluated carbon's temperature-dependent diffusivity. The obtained diffusivities agree well with the experimental measurements. Even at low temperature for which, to the best of our knowledge, no experimental data are available, the diffusivity can be evaluated accurately. Additionally, we study carbon diffusion inside the edge dislocation core in bcc iron, and demonstrate the applicability of the method to rare events on a rugged free-energy surface.

Akio Ishii, Hajime Kimizuka, and Shigenobu Ogata, "Multi-replica Molecular Dynamics Modeling", Computational Materials Science, 54-1 (2012) 240-248.

We propose a new unified modeling concept, multi-replica molecular dynamics (MRMD), which is a collective designation for molecular dynamics (MD) or particle dynamics (PD) that simultaneously follows the time evolution of two or more replica systems. Numerous methods in various fields (physics, chemistry, biology, materials science, astronomy, etc.) can be classified in MRMD, such as parallel replica methods, the nudged elastic band method, path integral methods, replica exchange methods, and the multi-walker metadynamics method. However, the relations between these MRMD methods have not been clarified, and a global understanding or unified methodological framework remains to be presented. Therefore, herein, we discuss the relations between various MRMD methods to provide a comprehensive understanding of these methods. Then, with this understanding, we describe a concise implementation of MRMD into conventional domain- and particle-decomposition parallel MD program codes, instead of the somewhat intuitive replica-parallel implementation. Finally, we demonstrate examples of different MRMD calculations performed using our parallel MRMD program “ParaMrMD.”

Yun-jiang Wang, Akio Ishii, and Shigenobu Ogata, "Grain Size Dependence of Creep in Nanocrystalline Copper by Molecular Dynamics", Materials Transactions, 53-1 (2012) 156-160.

The grain size dependence of creep is critical to understand the plastic deformation mechanism of nanoscale metals. Here we used molecular dynamics to study the stress-induced grain size exponent transition in creep of nanocrystalline copper. The grain size exponent was found to initially increase with increasing stress, then decrease after some critical stress. The derived grain size exponents are in agreement with experimental results for diffusional and grain boundary sliding creep at low stress. While, the founded decreasing grain size exponent with increasing stress for dislocation nucleation creep in nanocrystal is in contrast with conventional materials. We propose a constitutive equation for dislocation nucleation governed creep in nanocrystal to explain its grain size dependence transition with stress.

Yun-Jiang Wang, Akio Ishii, and Shigenobu Ogata, "Transition of creep mechanism in nanocrystalline metals", Physical Review B, 84-22 (2011) 224102-1-7.

Understanding creep mechanisms with atomistic details is of great importance to achieve the mechanical and thermodynamical stabilities of nanocrystalline (NC) metals over a wide temperature range. Here we report a molecular dynamics analysis of creep in NC copper dominated by competing deformation mechanisms. We found the dominating creep mechanism transits from grain boundary (GB) diffusion to GB sliding, and then dislocation nucleation with increasing stress. The derived stress exponent, small activation volume of 0.1−10b3, and grain size exponent all agree quantitatively with experimental values. We proposed a stress-temperature deformation map in NC metals accommodated by the competition among different stress-driven, thermally activated processes. The model is general to answer the question why deformation mechanism transits with stress in NC metals.

Hajime Kimizuka and Shigenobu Ogata, "Slow diffusion of hydrogen at a screw dislocation core in alpha-iron", Physical Review B, 84-2 (2011) 024116-1-6.

Here we demonstrate and characterize the H-diffusion behavior around a screw dislocation in body-centered cubic (bcc) α-Fe by performing path-integral molecular dynamics modeling and adopting an ab initio–based potential. Counterintuitively, our results indicate that the H diffusivity along the dislocation line is significantly lower than lattice diffusion. Thus, the “fast” pipe diffusion does not occur for H in α-Fe.

Hajime Kimizuka, Hideki Mori, and Shigenobu Ogata, "Effect of Temperature on Fast Hydrogen Diffusion in Iron: A Path-Integral Quantum Dynamics Approach", Physical Review B, 83-9 (2011) 094110-1-7.

Here we explicitly present the diffusion coefficients (D) and activation energies (Ea) of interstitial H in α-Fe over a temperature range of 100 to 1000 K. These values were predicted by applying path-integral molecular dynamics modeling based on first principles. The obtained D and Ea values exhibit clear non-Arrhenius temperature dependence and a transition from quantum to classical behavior at around 500 K. Our results show that the quantum effects not only significantly lower the diffusion barrier but also change the diffusion pathway even at room temperature; thus, fast diffusion becomes possible.

Hajime Kimizuka, Hideki Mori, Hiroki Ushida, and Shigenobu Ogata, "Evaluation of Hydrogen Diffusivity and Its Temperature Dependence in BCC Metals: A Path-Integral Centroid Molecular Dynamics Study", Journal of the Japan Institute of Metals, 73-8 (2009) 571-576. [in Japanese]

We have analyzed the diffusion behavior of interstitial hydrogen in bcc iron and niobium using path-integral centroid molecular dynamics (CMD) method, which can describe the real-time evolution of particles based on quantum statistical mechanics. In this study, the embedded-atom-method (EAM) potential model for the iron-hydrogen interaction is developed to reproduce the ab initio minimum energy path of hydrogen migration based on the density functional theory (DFT) data in the literature, while the description of niobium-hydrogen interaction is based on an empirical potential model. Time evolutions of mean-square displacements of hydrogen atoms in the two bulk metals are calculated at various temperatures, and then diffusion coefficients and activation energies of hydrogen migration are evaluated. Especially in the case of iron, the results are in good agreement with experimental measurements over a wide temperature range. In order to characterize the quantum effects on the hydrogen diffusion process, the CMD results are compared with those obtained from classical molecular dynamics method. The obtained results indicate that the quantum effects can play a significant role in hydrogen diffusivity over a wide temperature range in these bcc metals.

Masahiro Yamamoto, Kohei Kunizawa, Akinori Fujinami, Shigenobu Ogata, and Yoji Shibutani, "Formation of Atomistic Island in Al Film Growth by Kinetic Monte Carlo", Journal of Computational Science and Technology, 3-1 (2009) 148-158.

Kinetic Monte Carlo (KMC) method realizes the millisecond or second order atomistic thin film growth. Twenty five kinds of events which may occur on Al(111) surface were classified. An attempt frequency and an activation energy of each event were defined using vibration analyses and nudged elastic band (NEB) method by which the minimum energy path (MEP) can be reasonably predicted. Temperature and deposition rate dependences of Al(111) film growth were intensively investigated in the present paper. The higher temperature and the lower rate drive the layer-by-layer film structural change. Two types of islands (fcc and hcp) were seen by modeling without considering the events of diffusion of dimer and trimer, while only fcc islands remain with considering such events. Thus, we find that the primitive events of diffusion of dimer and trimer take important roles in determination of surface morphology.

Masahiro Yamamoto, Akinori Fujinami, Shigenobu Ogata, and Yoji Shibutani, "Hybridized Atomistic Modeling of Migration Observed on Thin Film Surface by Incident Particles", Journal of Computational Science and Technology, 1-1 (2008) 14-21.

Innovative thin film technology to realize the finer electric devices needs to understand the atomic level process of film growth and its relationship to the film characterization. In this paper, the long film growth phenomena for a few micro-second order with the short severe collisions by incident particles are analyzed by the proposed hybridized atomistic modeling. This method combined molecular dynamics (MD) with kinetic Monte Carlo (KMC) can directly treat two types of events of deposition and diffusion, which have quite different time scales. The solutions suggest that the large incident kinetic energy of deposited atoms compatible to the realistic physical vapor deposition (PVD) impels to fluctuate the equilibrium on Al (111) surface very drastically and affects the atomic level surface morphology. It is found that the faster incident atoms with 1.0×104 m/s can make the smoother surface than those with the velocity of 1.0×103 m/s. This is due to much activated atomic migration, which can be realized only by MD.

Masahiro Yamamoto, Kohei Kunizawa, Akinori Fujinami, Shigenobu Ogata, and Yoji Shibutani, "Formation of Atomistic Island in Al Film Growth by Kinetic Monte Carlo", Transactions of the JSME A, 73-728 (2007) 490-497. [in Japanese]

Kinetic Monte Carlo (KMC) method realizes the millisecond or second order atomistic thin film growth. Twenty five kinds of events which may occur on Al (111) surface were classified. An attempt frequency and an activation energy of each event were defined using vibration analyses and nudged elastic band (NEB) method by which the minimum energy path (MEP) can be reasonably predicted. Temperature and deposition rate dependences of Al(lll) film growth were intensively investigated in the present paper. The higher temperature and the lower rate drive the layer-by-layer film structural change. Two types of islands (fee and hep) were seen by modeling without considering the events of diffusion of dimer and trimer, while only fee islands remain with considering such events. Thus, we find that the primitive events of diffusion of dimer and trimer take important roles in determination of surface morphology.