Soft landing of metal clusters on graphite: a molecular dynamics study
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Soft landing of metal clusters on graphite: a molecular dynamics study
Alexey V. Verkhovtsev, ∗ Yury Erofeev,
1, 2 and Andrey V. Solov’yov † MBN Research Center, Altenh¨oferallee 3, 60438 Frankfurt am Main, Germany Department of Physics, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands
Structure and stability of 3 nm size Ag , Au and Ti clusters deposited on graphite undersoft landing conditions ( ∼ − − eV per atom) are studied by means of molecular dynamicssimulations. Parameters for the cluster–surface interaction are derived from complementary ab initio calculations. We found that the shape of clusters on the surface is governed by their elementalcomposition and depends also on the initial cluster structure and landing conditions. At depositionenergies below 0.1 eV/atom, the Ag cluster acquires an ellipsoidal shape, while Au and Ti clusters transform into oblate and prolate truncated spheroids, respectively, due to stronger adhesionto graphite. The clusters flatten over the surface and eventually disintegrate as the deposition energyincreases. Simulation results reveal that Ag and Au fragment at about 0 . − . . The contactangle, contact radius and height of the clusters as functions of deposition energy are determinedfrom fitting the positions of cluster surface atoms with a surface equation. The dependence of theseparameters on internal energy of the clusters is also analyzed. I. INTRODUCTION
The interaction of atomic clusters and nanoparticleswith surfaces has been a widely studied topic in clusterscience over the past several decades [1–3]. The dynam-ics of metal clusters and carbon fullerenes deposited ontodifferent surfaces (mainly, metal surfaces, graphite andsilicon oxide) was explored both experimentally [4–7] andcomputationally by means of molecular dynamics (MD)simulations [8–10]. The strong interest in the depositionof mass-selected clusters on surfaces has been motivatedby both fundamental research and technological applica-tions.From a fundamental physics viewpoint, an importantquestion is how structural, electronic, magnetic and op-tical properties of deposited clusters change with respectto free counterparts. A variety of phenomena emergealso when atomic clusters are brought in contact witha surface. Examples include fragmentation of clustersand implantation of cluster atoms into the substrate [11],penetration of energetic clusters through a substrate andsurface sputtering [12, 13], irradiation-induced structuralrearrangements of deposited clusters [14], cluster diffu-sion and aggregation into islands [15, 16], super-diffusionand L´evy flights [17], as well as the formation and frag-mentation of fractal nanostructures [18–20].From a technological viewpoint, an understanding ofthe cluster–surface interaction is crucial for the control-lable production of novel materials such as thin filmsand nanostructured surfaces [21], supported nanocata-lysts [22] as well as nanoscale components for electronicdevices [23]. If the process of cluster landing on a surfacesignificantly modifies their shape and morphology, the ∗ [email protected]; On leave from Ioffe Institute,Politekhnicheskaya 26, 194021 St. Petersburg, Russia † On leave from Ioffe Institute, Politekhnicheskaya 26, 194021 St.Petersburg, Russia major technical effort required to produce size-selectedcluster beams is largely in vain. Stability of size-selectedclusters on a surface is therefore a key goal of depositionstudies.Stability and electronic properties of small clusters de-posited on solid surfaces were also studied theoreticallywithin the framework of the liquid drop model [24–26].Analytical relations were derived for the deformation-dependent surface and curvature energies of small Na N and Ar N ( N < , Au and Ti , deposited ongraphite. To the best of our knowledge, a comparativeanalysis of the dynamics of clusters made of different el-ements and deposited at the same conditions has beenlacking in the previous MD papers, each of which hasfocused on one type of atomic clusters. Other than that,an important question in MD simulations of cluster de-position concerns the choice of an interaction potentialbetween the cluster and the substrate. Since the cluster–substrate interaction at soft-landing conditions is weak,the widely accepted approach is to describe this interac-tion using pairwise Lennard-Jones or Morse potentials.However, a literature review reveals that parameters ofthese potentials vary significantly across publications.Therefore, it is an open question how the parameters ofmetal–surface interatomic interactions would affect thestability of deposited clusters.In this paper, we focused on the deposition energyrange of 0 . − . ab initio calculations employing the second-order Møller-Plesset(MP2) perturbation theory. We found that the shape ofdeposited clusters depends strongly on the element type.The positions of cluster surface atoms obtained from MDsimulations are fitted with a surface equation. From thisfit, the contact angle, contact radius and height of theclusters are determined as functions of deposition energy.The dependence of these parameters on the initial clus-ter structure and internal energy of the clusters is alsoanalyzed. Pre-equilibration of clusters at elevated tem-peratures results in a significant decrease of the contactangle, although this trend is different for the silver, goldand titanium clusters. The shape of Ti is rather stableat cluster temperatures up to 900 K and deposition ener-gies up to 0.25 eV/atom, whereas the shape of thermallyexcited Ag and Au clusters changes significantlyeven at low deposition energies. II. COMPUTATIONAL METHODOLOGY
The simulations were performed using MBN Explorer[28] – a software package for advanced multiscale model-ing of complex molecular structure and dynamics. MBNStudio [29], a dedicated graphical user interface for MBNExplorer, was used to construct the systems, prepare allnecessary input files and analyze simulation outputs.
A. Metal clusters
As a first step, spherical clusters with the diameter of3 nm were cut from ideal silver, gold and titanium crys-tals. The resulting structures contained N Ag = N Au =887 and N Ti = 787 atoms. Energy minimization calcu-lations were conducted for the free clusters using the ve-locity quenching algorithm [30] with the time step of 1 fs.Interatomic interactions were described using the many-body Gupta potential [31] and the parameters were takenfrom Ref. [32].Energy-minimized structures were annealed by meansof MD simulations following the computational protocolfrom Ref. [33]. The annealing procedure enabled sam-pling of the configuration space of the clusters at elevatedtemperatures, at which the cluster surface has undergonea phase transition but the core remained, at least partly,in the solid phase. This protocol was validated [33] bythe comparison of annealed cluster geometries with thestructures obtained from scanning transmission electron FIG. 1. Geometries of Ag , Au and Ti clusters ob-tained after energy minimization (upper row) and the an-nealing process (bottom row). microscopy experiments. The clusters were heated upto 700 K for Au , 800 K for Ag and 900 K forTi . These values are about 70 −
100 K lower thanthe melting temperatures of the clusters which were de-termined through the analysis of caloric curves and root-mean-square displacement of all atoms. Each cluster washeated from 0 K up to the target temperature for 1 ns,then kept at this temperature for 2 ns and cooled downto 0 K over 1 ns, so that one annealing cycle was com-pleted in 4 ns. Three subsequent cycles were performedfor each cluster to get energetically favorable structures.Potential energy of each cluster decreased upon annealingby about 0 . − .
02 eV/atom with respect to the initialvalues obtained from energy minimization. The annealedcluster structures contained several grains with different(mainly, fcc and hcp) crystal lattices. Final structuresof the energy-minimized and annealed clusters are com-pared in Fig. 1.
B. Graphite substrate
The constructed graphite substrate contained 5 mono-layers with the size of 106 . × .
38 ˚A . These dimen-sions were chosen to replicate the system with periodicboundary conditions. The substrate size was more thanthree times larger than the cluster size and significantlyexceeded the range of interatomic interactions as de-scribed below. Thus, we ensured that there is no artificialinteraction across the simulation box boundaries. Priorsimulation of cluster deposition, the substrate was energyminimized using the velocity quenching algorithm. Themany-body Brenner potential [34] was used to describethe interaction between covalently-bonded carbon atomswithin each graphite layer whereas the Lennard-Jonespotential was employed to account for the van der Waalsinteraction between the layers. The Lennard-Jones po-tential is implemented in MBN Explorer in the following TABLE I. Parameters of the Lennard-Jones potential used inthis work to describe the C–C interaction between graphitelayers as well as between the metal (Ag, Au, Ti) and car-bon atoms. Other values (of the Lennard-Jones or the pair-wise Morse potentials) reported in literature are also listedfor comparison. ε (eV) r (˚A) Ref.C–C 0.00286 3.89 [35]Ag–C 0.020 3.49 this work (MP2)0.029 3.32 [36]0.030 3.37 [37]0.009 3.45 [40] (Morse)Au–C 0.044 3.49 this work (MP2)0.033 3.32 [36]0.013 3.36 [17]Ti–C 0.165 2.44 this work (MP2)Ni–C 0.023 3.20 [36]0.345 2.03 [39] (Morse)0.363 2.28 [40] (Morse) form: U ( r ) = ε (cid:20)(cid:16) r r (cid:17) − (cid:16) r r (cid:17) (cid:21) , (1)where ε is the depth of the potential energy well and r isthe equilibrium interatomic distance. Parameters for thecarbon–carbon interaction (see Table I) were taken fromRef. [35]. The Lennard-Jones potential was truncated ata cutoff distance of 10 ˚A that is about 3 times greaterthan the interplanar distance in graphite and an order ofmagnitude smaller than the size of the simulation box. C. Metal–carbon interaction
The cluster–substrate interaction was also described bythe Lennard-Jones potential using the cutoff distance of10 ˚A. Several MD studies of noble and transition-metalclusters (Ag, Au, Pt, Cu, Ni) interacting with graphiteand graphene were reported earlier, see e.g. [17, 36, 37].Parameters for the metal–carbon interaction were de-rived in these papers using empirical mixing rules: r C–M0 = 12 (cid:0) r C–C0 + r M–M0 (cid:1) , ε
C–M = √ ε C–C ε M–M , (2)where r M–M0 and ε M–M are the parameters of metal–metalinteractions [38]. Other studies have shown [39, 40] thatparameters for the pairwise interactions between metalatoms and sp carbon systems, such as fullerenes or car-bon nanotubes, differ significantly from those derived us-ing the mixing rules. Thus, a broad range of parametersfor various metal–carbon systems can be found in liter-ature, and the optimal choice of the parameters is notobvious.To elaborate on this issue, we performed ab initio cal-culations of potential energy scans for Ag, Au and Tiatoms interacting with a benzene molecule, which can be considered as a smallest structural unit of a graphitelayer. It is known (see, e.g., Refs. [41, 42] and refer-ences therein) that standard DFT methods do not ac-count properly for long-range dispersion interactions andrequire additional empirical corrections. The importanceof non-local correlation effects that govern dispersive in-teractions was particularly emphasized for the binding ofatoms and small clusters of silver and gold on graphite[43, 44].Since the dispersive interaction is naturally accountedfor in wave function-based ab initio methods, a series ofcalculations employing the second-order Møller-Plesset(MP2) perturbation theory were performed. The Gaus-sian 09 software package [45] using a LanL2DZ basis setwas employed. The metal atoms were placed in the hol-low position on top of the benzene molecule and displacedalong its main axis. The metal atom–benzene interac-tion energy was obtained from these scans and dividedby the number of carbon atoms to determine the inter-action energy per M–C pair of atoms. Since benzene isa highly symmetric molecule and the metal atoms weredisplaced along its main axis, all M–C bonds can be con-sidered as equivalent. The evaluated parameters of theLennard-Jones potential for Ag–C, Au–C and Ti–C aresummarized in Table I. Other values reported in litera-ture [17, 36, 37, 40] are also presented for completeness.Note that the parameters from Refs. [17, 36, 37] weredetermined using the mixing rules (Eq. (2)) while theparameters from Refs. [39, 40] were derived on the basisof DFT calculations. To the best of our knowledge, thereis no reference data available on the interaction betweenTi and C atoms. Therefore, in Table I we provide alsoparameters taken from literature [36, 39, 40] for the in-teraction between carbon and nickel, another open-shelltransition metal.Our MP2 calculations revealed that the metal–carboninteraction energy increases from silver to gold to tita-nium. The Au–C potential well depth is about two timeslarger than for the Ag–C interaction. This is in agree-ment with the results of earlier DFT calculations withdispersion corrections [43] which found that adsorptionenergy of a gold atom on graphite is about two timeshigher than for a silver atom. It was also found [43] thatno charge redistribution occurs between the Ag atom andgraphite, while the deposited Au atom receives a chargeof approximately 0.1 e from the carbon sheet. On thisbasis, it was concluded that the adsorption of silver ongraphite is purely of van der Waals type whereas smallhybridization in the density of states, i.e. a chemicalcontribution to the binding, occurs in the case of goldadsorbed on graphite. As it is indicated in Table I, theTi–C interaction is about eight times stronger than theAg–C interaction. A recent DFT-based study of the ti-tanium/graphite interface [46] reported the formation ofchemical bonding between interfacial Ti and C atoms.It was found also that an interfacial Ti atom acquires acharge of about 0 . e . Another DFT study of the adsorp-tion of a titanium slab on graphene demonstrated thata p − d hybridization occurs between atomic orbitals ofcarbon and titanium [47]. D. Deposition of clusters on graphite
For MD simulations of cluster deposition, a sim-ulation box of 106 . × . ×
110 ˚A was used.Each cluster was placed in the center of the simula-tion box approximately 40 ˚A above the topmost graphitelayer. The clusters were deposited with energies E dep =0 . , . , . , . , . , .
75 and 1.0 eV/atom at nor-mal incidence to the graphite surface. To determine frag-mentation threshold for the Ti cluster, higher deposi-tion energies from 2.0 to 5.0 eV/atom were also consid-ered.To simulate the deposition of clusters and their rear-rangement on the surface, 250-ps long MD simulationswere performed for the microcanonical (
N V E ) ensembleof particles. The simulation time was chosen such that,after hitting the surface, the clusters would relax for atleast 200 ps. Kinetic and potential energies of the sys-tem reached steady-state values within 20 −
40 ps afterthe collision and remained practically constant until theend of each simulation. Integration of equations of mo-tion was done using the velocity Verlet algorithm withthe time step of 1 fs. We ensured that a variation ofthe total energy of the system did not exceed 0.01% withthis time step. Two bottom graphite layers were fixedto avoid translational motion of the whole system afterthe collision. Test calculations were performed also forthicker graphite substrates containing 7 and 11 monolay-ers to check that the results obtained do not depend onsubstrate thickness.As described in Section III, disintegration of the clus-ters and scattering of cluster fragments over the wholegraphite sample were observed at deposition energies of0.75 eV/atom and above. To ensure that the simulationbox boundaries do not affect the results, a set of simula-tions was performed on a larger graphite substrate withthe size of 212 . × . . The total number of atomsin the systems thus varied from approx. 19,000 to 86,000. III. RESULTS AND DISCUSSION
We begin our analysis by considering deposition of thespherical Ag , Au and Ti clusters cut from idealbulk crystals. Then we discuss how alteration of the ini-tial cluster structure due to annealing affects their shapeupon deposition. Finally, we consider deposition of ther-mally excited clusters and compare the resulting shapeswith the case of deposition at zero temperature.For cluster sizes smaller than we considered in thisstudy, quantum effects such as even-odd oscillations incluster abundance spectra [48] and the appearance of“magic” numbers associated with electron shell closure[49], become more prominent and play a crucial role in determining the shape of clusters on a surface [25]. Sucheffects are particularly strong for the clusters containing
N < ∼
200 atoms but shrink with increasing the clustersize up to N ≈
800 [50].Figure 2 shows final snapshots of the spherical Ag ,Au and Ti clusters deposited on graphite at dif-ferent deposition energies E dep . The figure indicatesclearly that the cluster geometry depends on the elementtype and that it changes significantly with an increaseof deposition energy. At E dep = 0 .
001 eV/atom (up-per row) the Ag cluster retains its spherical shape,whereas the gold and titanium clusters are deformed dueto stronger adhesion to the surface. The Au clusterhas the shape of a truncated ellipsoid while Ti is acapped structure elongated in the direction normal to thesubstrate. At E dep = 0 .
05 eV/atom (second row) the sil-ver cluster acquires a slightly deformed quasi-ellipsoidalshape, while both gold and titanium clusters transforminto truncated ellipsoids. All three clusters deposited at E dep = 0 .
25 eV/atom (third row) transform into trun-cated ellipsoids with the gold cluster being the most flat-tened structure. Finally, at E dep = 0 .
75 eV/atom (bot-tom row) topology of the silver and gold clusters changesafter collision with the surface. The Ag cluster be-comes a hollow structure that remains stable over the250-ps long simulation. The Au cluster fragmentsinto a large pretzel-like structure and several smaller is-lands that are scattered over the surface. In contrast,Ti remains intact but gradually becomes more andmore flattened. Note that the simulations performed at E dep = 0 .
75 eV/atom were conducted on a large graphitesubstrate of 212 . × . .Coordinates of cluster atoms, extracted from each MDtrajectory, were used to parameterize the cluster shapeand to evaluate a contact angle with the substrate. Asfollows from the simulated trajectories, each depositedcluster is, to a good approximation, radially symmetricwith respect to its main axis. Thus, we introduced cylin-drical coordinates ρ and z , where ρ = p ( x − x CM ) + ( y − y CM ) , (3)with x CM and y CM being x - and y -projections of the cen-ter of mass of each cluster. The ρ -axis lies in the graphitesurface plane, whereas z -axis is perpendicular to the sur-face and z = 0 corresponds to the average position ofthe topmost graphite layer. Figure 3 shows by symbols( ρ, z ) projections of all atoms in the Ag , Au andTi clusters deposited at 0.01 eV/atom. Figure 3(a)shows that at low values of E dep the Ag cluster ac-quires the shape of a non-truncated ellipsoid which isslightly deformed from the side which is in contact withgraphite. The initially spherical Au cluster transformsinto a truncated oblate spheroid (Fig. 3(b)) while theTi cluster has a well pronounced prolate shape, i.e.its height is larger than the contact radius (Fig. 3(c)).Interestingly, atoms in the titanium cluster arrange intolayers oriented parallel to graphite planes, that is differ-ent from the atomic arrangement in the silver and gold FIG. 2. MD snapshots of the spherical Ag (left column), Au (middle column) and Ti (right column) clusters depositedon graphite at the deposition energies of 0.001, 0.05, 0.25 and 0.75 eV/atom (top to bottom rows). clusters. This can be explained by a stronger interactionbetween titanium and carbon atoms as compared to theAg–C and Au–C interactions.For each cluster we selected coordinates of atoms lo-cated on the surface and fitted the resulting profiles withthe following surface equation [51, 52]: ρ ( z ) = (cid:26) p a ( z − z ) + b ( z − z ) + c , z ≥ , z < , (4)where a , b , c and z are fitting parameters. This expres-sion enables a description of different cluster shapes witha single fitting function without any geometrical assump-tions on the cluster shape. From the least-squares fit of cluster profiles with Eq. (4) we determined the contactradius and height of each cluster as well as the contactangle with the substrate as functions of deposited energy.An inverse dependence z ( ρ ) is shown by solid red curvesin Fig. 3.The contact angle θ was evaluated by calculating thederivative of z ( ρ ) at the point ρ ′ which corresponds tothe average position of the bottom-most atomic layer ofthe clusters, z ′ = z ( ρ ′ ), see the dashed lines in Fig. 3. Anexpression for the contact angle is then given by [51] θ = arctan dzdρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ ′ ! = π (cid:18) dρdz (cid:12)(cid:12)(cid:12)(cid:12) z = z ′ (cid:19) . (5) FIG. 3. Radial profiles of the initially spherical Ag (a), Au (b) and Ti (c) clusters deposited at E dep = 0 .
01 eV/atom.Symbols (crosses) show the distribution of all atoms in each cluster at the end of a 250 ps-long simulation. Solid curves showthe best fit of the cluster profile with z ( ρ ) that is an inverse function of ρ ( z ), Eq. (4). Dashed lines indicate the averagedposition of the bottom-most atomic layer in each cluster.FIG. 4. Dependence of the contact angle (Eq. (5), top panel)and cluster deformation parameter (Eq. (7), bottom panel)on deposition energy for the initially spherical Ag , Au and Ti clusters. Solid curves show exponential fits usingEqs. (6) and (8). The contact angle for the optimized Ag , Au andTi clusters as a function of deposition energy is pre- sented in the upper panel of Fig. 4. The calculated valuesof θ were averaged over three distinct cluster geometriessampled from the last 100 ps of each simulation. It wasfound that the deposited clusters acquire their equilib-rium shapes shortly after reaching the surface (on thetimescales of few tens of picoseconds) and a variation of θ in the remaining part of simulations does not exceed10 degrees.The figure shows that the contact angle for the Ag cluster (black squares) evolves differently with an in-crease of E dep than for the gold and titanium clusters.The main distinction is that θ for Ag remains close to180 ◦ at deposition energies up to 0.1 eV/atom. A fur-ther increase of E dep up to 0.25 eV/atom leads to a rapiddecrease of the angle by about 70 degrees whereas it con-verges to a value of about 60 ◦ at E dep = 0 .
75 eV/atom.At higher deposition energies, the cluster fragments intoseveral small islands and therefore the contact angle wasnot determined. A similar trend was observed for Au (red circles). In this case, the contact angle drops rapidlyfrom approx. 130 ◦ at E dep = 0 .
001 eV/atom down to 20 ◦ at E dep = 0 . cluster (blue triangles), θ gradu-ally decreases from 100 ◦ to 45 ◦ in the deposition energyrange considered. This corresponds to the observationsshown in Fig. 2 that the contact area for Ti increasesgradually with E dep . Interestingly, the contact angle forall three clusters considered decreases exponentially with E dep , θ = θ + θ e − αE dep , (6)see solid curves in the upper panel of Fig. 4. The corre-sponding fitting parameters are summarized in Table II. FIG. 5. Contact angle θ as a function of deposition energy for the Ag , Au and Ti clusters, either optimized (opensymbols) or annealed and then equilibrated at a given temperature (closed symbols).FIG. 6. The cluster deformation parameter δ as a function of deposition energy for the Ag , Au and Ti clusters, eitheroptimized (open symbols) or annealed and then equilibrated at a given temperature (closed symbols).TABLE II. Fitting parameters describing an exponential de-crease of the contact angle (Eq. (6)) and an increase of thecluster deformation parameter (Eq. (8)) with E dep . Note thatonly the data points for E dep ≥ . cluster.Ag Au Ti θ (deg.) 48.87 17.11 43.47 θ (deg.) 219.93 110.30 48.52 α (eV − /atom) 5.13 9.58 2.83 δ -4.29 0.76 -1.04 δ β (eV − /atom) 0.85 10.74 1.12 Additional MD simulations were conducted at higherdeposition energies to determine the fragmentationthreshold for Ti . We found that the threshold de-position energy is between 2 and 3 eV per atom, whichis about four times larger than the corresponding frag-mentation thresholds for Ag and Au . At E dep =2 eV/atom the titanium cluster is flattened over the graphite surface but remains intact, whereas at E dep =3 eV/atom the cluster transforms into a flat pretzel-like structure (similar to the Au cluster deposited at0.75 eV/atom, see Fig. 2) with the maximal height of twoatomic layers. Deposition of Ti at 4 eV/atom resultsin the formation of small titanium islands that are scat-tered over the whole simulated substrate. Note that thefragmentation thresholds should depend not only on ele-mental composition of the clusters but also on their size.However, we leave a detailed analysis of this dependencefor further studies.To complement this analysis, the lower panel of Fig. 4shows the cluster deformation parameter δ , defined as aratio of cluster contact radius (radius of the bottom-mostatomic layer, see Fig. 3) to cluster height, δ = ρ ( z = z ′ ) z ( ρ = 0) , (7)for the initially spherical Ag , Au and Ti clustersdeposited at different energies. The dependence of δ on E dep also follows an exponential law, δ = δ + δ e βE dep , (8)see solid curves in the lower panel of Fig. 4. The cor-responding fitting parameters are also listed in Table II.The horizontal dashed line denotes the case when thecontact radius is equal to the height, which correspondsto a perfect semi-spheroidal shape. The figure shows thatthe shape of Au is close to a semi-spheroid at low depo-sition energies (up to 0.05 eV/atom) while highly oblatetruncated spheroids are formed in the course of deposi-tion at higher energies. A similar trend is observed for thesilver and titanium clusters but in this case the clustershape is close to a semi-spheroid at E dep = 0 .
25 eV/atom.Thus, by increasing the deposition energy the clustersevolve from an ellipsoid (in the case of silver) or a trun-cated prolate spheroid (in the case of titanium) to semi-spheroids to very flat structures with the contact radiusexceeding the cluster height by the factor of four.The above described analysis was carried out for thespherical clusters deposited at zero temperature. It isworth exploring how the shape of deposited clusterswould change upon altering the initial cluster structuredue to annealing. Results of this analysis are summa-rized in Figs. 5 and 6, which show how the contact angle θ and the deformation parameter δ evolve as functionsof E dep . Gray lines / open symbols correspond to thespherical cluster case described above in Fig. 4. Blacklines / filled squared represent the results for the an-nealed clusters (see Fig. 1) at initial temperature of 0 K.The figures show that alteration of the cluster structuredue to annealing have a moderate impact on the shape ofAg and Ti clusters in the deposition energy rangeconsidered, whereas annealing of Au results in a verydifferent geometry even at low deposition energy. At E dep = 0 .
01 eV/atom the annealed gold cluster wets thesurface stronger than its spherical counterpart and thecontact angle decreases from 130 ◦ down to 70 ◦ . However,the variation of θ with E dep is much smaller than for thespherical Au cluster so that at E dep = 0 .
25 eV/atomthe contact angle for the annealed Au is almost twotimes larger than for the spherical cluster. Deposition ofthe annealed cluster at higher energies results in its frag-mentation. Figure 6 indicates that the annealed noblemetal clusters are characterized by a larger deformationparameter as compared to the spherical clusters but an-nealing of the titanium Ti cluster has a minor impacton its shape after deposition.Finally, let us analyze how pre-equilibration of the clus-ters at different temperatures affects their stability andshape upon deposition. Figures 5 and 6 show the con-tact angle θ (Eq. (5)) and the deformation parameter δ (Eq. (7)) of the clusters after they were given an initialtemperature of 300 K, 600 K and 900 K. The latter valueis just above the melting point of Ag and Au butabout 100 degrees less than the melting temperature ofTi . Therefore, the silver and gold clusters equilibratedat 900 K are deposited as liquid droplets while the tita- FIG. 7. Radial distribution function for the Au cluster,pre-equilibrated at 300 K, at different time instances t afterthe collision. An RDF for the molten state was obtained froma simulation of melting of a free Au cluster. nium cluster has a molten surface but its core is still, atleast partly, in the solid phase. As one may expect, depo-sition of the clusters at elevated temperature leads to asignificant decrease of the contact angle and an increasedcontact area at low deposition energies. The contact an-gle for Ag saturates with an increase of E dep at thevalue of about 60 degrees, independent on the clusterinitial temperature. In contrast, the shape of Au de-pends strongly on the amount of internal energy storedin the cluster. While the profile of the θ ( E dep ) depen-dence does not change when the initial temperature isincreased from 0 K to 900 K, a decrease of the contactangle is evident.The contact angle for the thermalized silver cluster de-posited at energies below 0.1 eV/atom exceeds the cor-responding values for the gold and titanium clusters.This result agrees with experimental observations thatnanometer-size silver clusters deposited on graphite at E dep = 0 .
05 eV/atom [18–20] are highly mobile, whichleads to the formation of fractal-like silver nanostruc-tures.Further insights into the stability of clusters on the sur-face can be drawn from the analysis of radial distributionfunction (RDF). Figure 7 shows an exemplary RDF forthe Au cluster, pre-equilibrated at 300 K prior de-position, at different time instances t after the collision.Mechanical stress induced by the collision causes the for-mation of a liquid droplet within the first 10 ps afterthe collision (solid blue curve). Over the next 40 ps thedroplet re-crystallizes into a solid structure whose RDFresembles the initial one (dashed yellow curve). Thisstructure remains stable and practically does not changein the remaining part of the simulation.Further analysis reveals that the clusters undergostructural transformations induced by the collision. Fig-ure 8 compares RDFs for the annealed Ag , Au and FIG. 8. Radial distribution function for the Ag , Au and Ti clusters, pre-equilibrated at 300 K, deposited at energiesof 0.05 and 0.25 eV/atom. Arrows show the appearance of fine structures in the RDFs that are indicative for fcc → hcp (inthe case of silver) and hcp → fcc (in the case of titanium) structural transformations. Ti clusters deposited at 0.05 and 0.25 eV/atom. TheRDFs are calculated at the end of each 250-ps long tra-jectory. Lattice structure of the gold cluster deposited at0.25 eV/atom becomes compressed as compared to theinitial structure (see middle panel). This is evident fromthat the fact that all the peaks in the RDF are stronglysuppressed and uniformly shifted towards smaller inter-atomic distances. The silver cluster deposited at both0.05 and 0.25 eV/atom maintains the short-range order(see left panel). However, at E dep = 0 .
05 eV/atom thelong-range order is lost as the peaks in the interatomicdistance range of 6 − cluster deposited at 0.05 eV/atom can also be seen inFig. 2. In contrast, lattice structure of the silver clus-ter deposited at E dep = 0 .
25 eV/atom resembles, to alarge extent, the initial lattice structure. The only im-portant difference is the formation of a shoulder at theinteratomic distance of about 5.5 ˚A and a shift of anotherpeak centered at 7.1 ˚A to 6.9 ˚A (see green arrows). Thisis an indication of an increased ratio of hcp lattice pack-ing in the deposited cluster structure. RDFs for the Ti cluster (right panel) illustrate the opposite phenomenon,namely a shift of the peaks centered at 5.5 ˚A and 6.9 ˚A to-wards larger interatomic distances, that is indicative foran increased ratio of fcc lattice in the final cluster struc-ture.
IV. CONCLUSION
The deposition of metal clusters made of three differ-ent elements – silver, gold and titanium – on a graphitesubstrate was studied by means of molecular dynam- ics simulations using the MBN Explorer and MBN Stu-dio software packages. The clusters had the diameterof 3 nm and contained N Ag = N Au = 887 and N Ti =787 atoms. We focused on deposition energies in therange 0 . − . ab initio calculations employ-ing the second-order Møller-Plesset (MP2) perturbationtheory.We found that the shape and stability of depositedclusters depends strongly on the element type. Atlow deposition energies, the Ag cluster has a quasi-ellipsoid shape while the gold and titanium clusters re-arrange upon collision into truncated oblate and prolatespheroids. Both silver and gold clusters flatten over thesurface and eventually disintegrate as the deposition en-ergy increases up to 0 . − . ◦ down to 20 ◦ at E dep = 0 . ◦ to 45 ◦ in the rangeof deposition energies considered.We also analyzed how the initial structure (optimizedvs. annealed geometries) and internal energy of the clus-ters affects the shape and stability of the deposited clus-ters. Pre-equilibration of clusters at elevated temper-atures up to 900 K results in a significant decrease of0the contact angle, although this trend is different for thesilver, gold and titanium clusters. The shape of Ti is rather stable at cluster temperatures up to 900 Kand at deposition energies up to 0.25 eV/atom, whereasthe shape of thermally excited Ag and Au clusterschanges significantly even at low deposition energies. ACKNOWLEDGEMENTS
This work was supported in part by DeutscheForschungsgemeinschaft (Project no. 415716638) andby the European Union’s Horizon 2020 research and in-novation programme (the Radio-NP project within theH2020-MSCA-IF-2017 call, GA 794733 and the RADONproject within the H2020-MSCA-RISE-2019 call, GA872494). The possibility to perform calculations at theGoethe-HLR cluster of the Frankfurt Center for Scien-tific Computing and at the DeiC National HPC Center(University of Southern Denmark, Odense) is gratefullyacknowledged. [1] P. Jensen,
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