IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 46 (2013) 335302 (11pp) doi:10.1088/0022-3727/46/33/335302

Molecular dynamics study of neck growthin laser sintering of hollow silvernanoparticles with different heating ratesShan Jiang1, Yuwen Zhang2, Yong Gan2,3, Zhen Chen1 and Hao Peng2

1 Department of Civil & Environmental Engineering, University of Missouri, Columbia, MO 65211, USAState Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering

Mechanics, Dalian University of Technology, Dalian 116024, People’s Republic of China2 Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia,MO 65211, USA3 Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027,People’s Republic of China

E-mail: ChenZh@missouri.edu, zhangyu@missouri.edu and ganyong@zju.edu.cn

Received 17 April 2013, in final form 9 July 2013Published 29 July 2013Online at stacks.iop.org/JPhysD/46/335302

AbstractEngineered hollow nanoparticles have exhibited their potential in nanotechnologyapplications, but so far the investigation of the deformation mechanisms for these hollowparticles during the sintering process has rarely been reported. Hence, a comparative study ofboth solid and hollow spherical silver nanoparticles with different sizes under different heatingrates of laser sintering is conducted systematically in this paper, based on molecular dynamicssimulations. An interesting phenomenon is observed where the temperature for fast neckgrowth shows an inverse trend in all the hollow nanoparticle pairs at an ultrahigh heating rate,which is quite different from that known in the solid particle cases. This finding implies thatbesides the size and heating rate, the nanoparticle geometry could also play an important rolein the sintering process. At a low heating rate, the plastic deformation combined withstructural reconfigurations induced by the lattice sliding in the hollow shells is found to be animportant mechanism during the heating process. At an ultrahigh heating rate, the transitionfrom fcc crystal directly to disordered structure from both outside and inside surfaces becomesmore dominant than the structural reconfiguration with lattice defects, which is facilitated bythe introduction of the inner free surfaces in hollow nanoparticles. The entire hollow particlepairs thus show an obvious tendency to coalesce and melt at a lower temperature level thanwith a low heating rate.

(Some figures may appear in colour only in the online journal)

1. Introduction

Nanoparticles are of growing interest to academia and modernindustry, as fundamental building blocks with many potentialapplications in nanotechnology [1–4]. The use of laserradiation to selectively sinter nanoparticles has shown manyadvantages from the aspects of meeting performance and costrequirements [5, 6]. Since the late 1980s when it was initiallydeveloped, selective laser sintering (SLS) [7, 8] technology hasbeen gaining great improvements, and proved to be a verypromising tool for rapid manufacturing, which allows building

functional nanostructures from a wide range of commerciallyavailable powdered materials. However, strong size effecton the physical and chemical properties of the nanoparticles[9, 10] makes this research topic more complicated, due to highsurface-area-to-volume ratios and the possible phenomena ofmelting and evaporation under high laser intensity [11, 12].

Besides extensive experimental research [5, 6, 13, 14]on the sintering of nanoparticles, many theoretical andcomputational efforts have been made to model and evaluatethe sintering process over the past decades. When the particlesize is much smaller than the laser beam radius, the sintering

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J. Phys. D: Appl. Phys. 46 (2013) 335302 S Jiang et al

problem could be solved by a continuum-based model [15, 16]with the assumption that the powder bed is uniform andisotropic. However, a single particle level model [17, 18]should be considered if the particle size is comparable tothe laser beam size. Molecular dynamics (MD) method,as another suitable and effective approach, has been widelyemployed to study the fundamental physics. It could revealmore details such as neck growth [19], gyration radius [19, 20],densification and dislocation [21] from the molecular level ofthe particle sintering. By comparing the simulation-predictedneck growth with the empirical-theoretical formulations, thereliability of MD simulations has been proved for the sinteringof metallic nanoparticles [22]. Different elements, e.g. thenanoparticles of Au [19, 12, 23], Ag [14], Cu [21], Fe [24],Ni [20] and metal oxide [25] (among others), have beenconsidered in both previous experiments and simulations, andmost of these studied nanoparticles are spherical in shapes withsolid spheres.

With the development of experimental technology, moreand more functional nanoparticles with novel hollow structuresare synthesized [26, 27]. As mentioned by Yin et al [28],the ability to manipulate the morphology and structures ofporous solids in nanoscale would enable greater controlof the local chemical environment. They successfullydemonstrated the formation of hollow nanoparticles with amechanism analogous to void formation in the nanoscaleKirkendall effect. Schwartzberg et al [29] reported synthesis,characterization, and tunable optical properties of hollowgold nanoparticles, which have shown great potential forchemical and biological sensing applications. Chen et al [30]reported the preparation of ordered arrays of hollow silvernanoparticles by colloidal crystal templating. Anumol et al[31] also reported a general method for the synthesis of hollowstructures of various functional inorganics by partial sinteringof mesoporous nanoparticle aggregates, which is attractive formany applications, such as catalysis, drug delivery and bio-sensing.

Although much progress has been made on theexperiments of both solid and hollow nanoparticles, theexisting analytical and computational investigations ofnanoparticle sintering are mainly focused on solid spheres,and the study of hollow spheres is rarely reported, as can befound from the open literature. A basic question is thereforeraised: will there be any fundamental difference between thesolid and hollow nanoparticles in the sintering process? If so,how will the heating rates influence the properties such as neckgrowths, gyration radii and melting of the hollow nanoparticlesduring the laser sintering? These questions have not beenanswered yet. In this paper, the sintering of two hollowsilver (Ag) nanoparticles by laser heating is modelled usingMD simulations. Since the sizes of two particles are rarelyidentical in a real sintering process, a series of simulationswith different sizes and heating rates are preformed. For thepurpose of comparison, solid nanoparticles corresponding tohollow sphere cases are also considered. The effects of particlesize as well as the laser heating rates on the neck growth aresystematically investigated. Other key properties for both solidand hollow nanoparticles during the sintering process are alsodiscussed.

Figure 1. Sketches of (a) solid and (b) hollow nanoparticles. Thesectional views are taken from the planes cut from the middle. Theparticle sizes are determined by d1 and d2 and the initial distancebetween the two particles is 5 Å. The shell thickness for all thehollow particles in the current study is set as 3a.

2. Methodology

2.1. Atomic model preparation

The system geometries studied are depicted schematicallyin figure 1. The 3D atomic models of nanoparticles arecreated through a top-down method. First, a solid sphere(shown in figure 1(a)) is extracted from a bulk face-centredcubic (fcc) Ag crystal with the [1 0 0], [0 1 0] and [0 0 1]crystallographic directions, which coincide with the x-, y- andz-axes, respectively. Then, removing atoms from the centreof the nanoparticle leaves a hollow spherical space inside andthus forms a hollow nanoparticle. The shell thickness for allthe hollow nanoparticles studied in this paper is 3a, where a

is the lattice constant of Ag (a = 0.409 nm), as indicated infigure 1(b).

The two-nanoparticle-model is characterized by twoparameters d1 and d2, which are the diameters of the twospheres (or the outer diameters of the hollow spheres). To studythe particle size effects on the neck growth during sintering,d1 and d2 can be varied, as 10a, 16a and 24a, respectively.Thus, there are twelve combinations in total for the solid andhollow models, as listed in table 1. The centroids of bothparticles have the same x and y coordinates and the initialgap between two nanoparticles in the z-direction is 5 Å. Ineach simulation case, two solid/hollow nanoparticles in onepair are thermally equilibrated separately at 298 K for 1 ns,which is performed to reduce the initial stresses resulting fromthe introduction of free surfaces cut from the bulk. Thentwo equilibrated solid or hollow nanoparticles are assembled

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Table 1. Specification of the simulation models. The lattice lengthis a = 0.409 nm.

Simulation No d1 d2 Morphology Atoms contained

1 10a 10a Solid sphere 42462 10a 16a Solid sphere 10 7123 10a 24a Solid sphere 31 0204 16a 16a Solid sphere 17 1785 16a 24a Solid sphere 37 4866 24a 24a Solid sphere 57 7947 10a 10a Hollow sphere 39648 10a 16a Hollow sphere 84489 10a 24a Hollow sphere 18 66410 16a 16a Hollow sphere 12 93211 16a 24a Hollow sphere 23 14812 24a 24a Hollow sphere 33 364

to establish the initial MD system for sintering simulation,as shown with the configurations in figures 1(a) and (b),respectively. In addition, a set of simulations for equilibriumare also conducted at different temperatures (e.g. 10, 98, 198and 298 K) which is much lower than the melting point. Toavoid the deviation from the equilibrium and extra driving forceinduced by the deformation of particle, the potential energycurves and configurations of the particles are examined toensure that the equilibrium durations are long enough and theentire systems can approach the minimum energy state. Allthe configurations in these simulations show that the hollowspheres are able to remain the initial morphology with a cavityinterior after the equilibrium, and thus are ready for the sequentsintering simulation. In this study, we focus on the modelsprepared at the room temperature of 298 K.

2.2. MD simulations

In the MD simulations of sintering, non-periodical boundaryconditions are used along all the three dimensions. Theinteratomic interactions are modelled based on embedded-atom method (EAM) force field [32]. The total energy U fora system of Natoms is described as

U =N∑

i=1

Fi(ρi) +

1

2

N∑j �=i

�ij (rij )

, (1)

where Fi is the embedding function, ρi is the electron density atatom i, �ij is a pair interaction function, and rij is the distancebetween atom i and j . The detailed parameters of the EAMforce field for Ag atoms, which are specified in the study ofFoiles et al [33], are utilized in this study to describe betweenAg atoms.

After the initial MD model is established, thesimulation is carried out at room temperature (298 K) withisochoric-isothermal (NVT) ensemble using the Nose–Hooverthermostat [34, 35] and velocity-Verlet integrator [36]. Thisequilibration lasts for 1 ns to mimic the solid state sinteringat 298 K. Subsequently, the system temperature is linearlyincreased to model the laser sintering. It is shown that thetemperature of the metal nanoparticles due to laser heating isapproximately linear [37]. For simplification, the laser heating

of metal nanoparticles is simulated by linearly increasing thenanoparticle temperature. Three heating rates of 0.04, 0.2 and1 K ps−1 are considered for each simulation case as listed intable 1. The system temperature is computed and monitoredevery 10 steps during the simulations of the heating process. Ifthe difference between the computed and desired temperaturesis greater than 1 K, the atomic velocities are rescaled in orderto make sure that the system temperature is reset to the desiredvalue. For all simulations, a time step of 2 fs is used. No heatloss to the environment is considered.

2.3. Auxiliary computations and analysis

Common neighbour analysis (CNA) [38] was used to study thelocal crystal structure in the nanoparticles; this analysis canfacilitate the visualization of the evolution of the deformationconfigurations and has been widely used in previous research[39]. During the sintering simulation, at each desired time step,different colours with corresponding colour numbers will beassigned to each atom based on CNA. For instance, a seriesof numbers 1, 2, 3, 4 and 5 represents the colours of grey,blue, green, red and gold, respectively. These numbers areoutput along with the three coordinate values for the atoms,so that the crystalline structure evolution can be tracked byplotting the entire atomic configurations with a certain colourdistribution. In the results discussed below, fcc atoms areshown as grey, hexagonal close-packed (hcp) atoms blue andother twelvefold-coordinated atoms green, and the remainingatoms which are not in any typical ordered crystalline structureare classified as disordered atoms and shown as gold. A localcrystalline structure is classified as an intrinsic stacking faultif it contains two adjacent layers of atoms in the local hcpstructure, an extrinsic stacking fault if it contains two hcp layersseparated by a single fcc layer, and a twin boundary if it consistsof a single hcp layer. Disordered or non-structured atoms aretypically located at surfaces or dislocation cores [40, 41]. Apost-processing computer program was developed, which cancollect all the atoms in the same type of lattice structure basedon the same colour number and thus obtain the total number ofthe atoms in the specific lattice structure. The number densityshown in the figures of this paper is defined as the ratio ofthe number of atoms in a specific lattice structure to the totalatoms number in the system. For a clear illustration, all theatomic configuration profiles in this paper are extracted fromthe middle cross-section on the y − z plane with a thickness ofone unit cell length along the x-axis.

Gyration radius, as an important characteristic of asintering process, is defined as the root mean square distanceof atoms of two spheres measured from their mass centre:

R2g = 1

M

N∑i=1

mi(ri − rcm)2, (2)

where M is the total mass of the group, r is the position of eachatom, and rcm is the centre-of-mass position of the group, andthe subscript i runs over all atoms in the system.

The mean square displacement (MSD), as anotherimportant calculation, is a measure of the average distance

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Figure 2. Time histories of neck width and number density ofsurface and disordered atoms for (a) solid and (b) hollownanoparticle pair of 10a versus 16a. The insets are the atomicconfigurations at various time steps.

that an atom travels in the sintering process:

MSD = 1

N

N∑i=1

[ri(t) − ri(0)]2, (3)

where N is the total number of atoms, t is time, and r is theposition of each atom. The MSD is computed based on theinitial coordinates of atoms at the beginning of the subsequentlaser sintering.

3. Solid state sintering at room temperature

From the simulation results, it is found that the solidstate sintering of two bare nanoparticles can occur at roomtemperature without any laser irradiation, due to their ultrafinesizes and relatively large atomic potential gradients. Theevolution of the particle deformation during solid statesintering at room temperature is illustrated in figure 2, withthe snapshots of the atomic configurations obtained at differenttime steps corresponding to the points A, B, C and D on the neckwidth curves. The two nanoparticles are initially separatedwithout laser heating applied yet (points A in figure 2), and thesystem temperature is maintained at 298 K. At points B, it canbe clearly seen that the two nanoparticles move towards eachother and become connected with a rapid neck formation. Arelatively stable dumbbell-shaped particle is then formed in thelater stage (see points C and D). This phenomenon of solid statesintering at room temperature can be observed in all the casesfor both solid and hollow nanoparticles with different sizes.

The time histories of neck width and number densityof classified disordered and surface atoms for both solid

(simulation 2) and hollow (simulation 8) nanoparticles are alsoshown in figure 2. At the early stage of the solid sintering frompoints A to B, a rapid neck growth occurs with the emergence ofgrain boundary consisting of non-crystalline atoms, and thusresults in a sudden increase of the disordered-atom fraction(corresponding to the first peak on the number density curve).Then, during the thermal equilibration, particularly in the fastneck growth stage (e.g., from point B to C in figure 2(a),and point B to D in figure 2(b)), partial dislocation cores arenucleated from the neck region, and quickly lead to a slippagealong the {1 1 1} close-packed plane, forming the intrinsicstacking faults. Since partial dislocation cores also consistof non-crystalline atoms which are determined by CNA, theirnucleation results in another increase of the disordered-atomfraction (corresponding to the second peak on the numberdensity curve). After the first peak (temporary grain boundary)and second peak (partial dislocation cores) of the numberdensity curve, it can also be found that the surface-atomfraction finally decreases to a lower value than that in the initialstatus. This reflects the reduction of total free surface area andimplies that the decrease of free energy due to this reducedsurface area is the main driving force [25] for the solid statesintering at room temperature.

As shown in the configuration on points C and D infigure 2(a), a partial dislocation core is initially nucleatedfrom the neck region, then passes through the whole section ofthe smaller nanoparticle, leaving a stacking fault behind, andfinally ends on the other side of the free surface, forming stablestacking faults in the lattice during the later solid sinteringprocess. This process indicates that the lattice structuredeformation is mainly mediated by the partial dislocationactivity. Another different scenario is that spontaneousgeneration of dislocations near the neck still occurs, but formstransient stack faults that later can disappear, as shown inconfigurations on points C and D in figure 2(b). In allthe cases modelled in this paper, stable stacking faults wereformed in simulations 1, 2 and 7, in which relatively smallparticles were included. Transient stacking faults were foundin most simulations (simulations 3–6, 8, 9 and 11) with theincrease of the particle size. These results confirm that theplastic deformation via dislocation generation is an importantmechanism [21, 42], which contributes in a major way to thesintering of both solid and hollow nanoparticles. However,both stable and transient slippages of the lattice can be observedduring the solid state sintering at the room temperature, whichis not obviously shown in the previous work [19, 23].

Furthermore, either stable or transient stacking faults wereobserved to occur in the smaller one for each pair if thetwo sphere sizes are not identical (see the configurations infigure 2), indicating that the partial dislocations and lattice slipsare more prone to form in smaller particles. This finding couldbe explained by the mechanical rotation that occurs to somedegree in virtually all the simulations reported in this study,which is similar to the phenomenon in the previous work [42].Additional simulation of different initial orientations including〈1 1 0〉 and 〈1 1 1〉 combined with 〈1 0 0〉 were also performed,and it was found that the dislocations can be easily formedand slide on (1 1 1) planes in the 〈1 1 0〉 directions during

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Figure 3. The temperature dependence of the gyration radius andnumber density of surface and disordered atoms for the hollownanoparticle pair of 16aversus 16a. The insets are atomicconfigurations at different stages during the sintering with theheating rate of 0.04 K ps−1.

the solid sintering with an obvious mechanical rotation dueto the initial misorientation [21]. Similarly, for the particlepairs focused on in the current study, the mechanical rotationis found to be especially obvious in smaller nanoparticles,and the dislocations thus can be easily found in the smallerones because of the relative crystalline misorientation. Asillustrated in both figures 2(a) and (b) at the point B, thesmaller spheres rotate to a certain angle at the onset of neckformation. After the disordered atoms in the neck regionrecrystallize to fcc structure, it can be found that the ‘upper’smaller nanoparticles start to rotate back and forth around a‘balance position’, which is much more obvious for the hollowcases. That is why there are more ‘small oscillations’ on theneck width curve during the stable stage for the hollow particlesafter the neck is formed.

4. Laser sintering

The resulting dumbbell shape nanoparticles at 298 K are thensubjected to laser irradiation. Each particle pair is heatedlinearly from 298 K to 1098 K (or 1298 K for some largersize particle cases to obtain the melting points) and the laserheating lasts for 20 ns, 4 ns and 0.8 ns, corresponding to theheating rates of 0.04, 0.2 and 1 K ps−1, respectively. A typicalconfiguration evolution for the laser sintering of a hollownanoparticle pair, as well as the corresponding gyration radiuschanging, is illustrated in figure 3. The neck width and MSDevolutions for the solid and hollow particle pairs are plottedagainst temperature in figures 4–6.

4.1. Deformation evolution of the particle pair

Generally, for each nanoparticle pair with different sizes, nomatter whether it is hollow or solid, there are two turning pointson the axis of temperature (as indicated by the two vertical dashlines in figure 3 or the two arrows in the following figures 4and 5), marking the rapid drops of the gyration radii but the fast

growths of the neck widths. Thus, the deformation evolutionof the nanoparticle pair during the laser sintering process couldbe divided into three stages: relatively stable stage, transitionstage, and melting stage.

In the relatively stable stage, which is referred to asthe quasi-equilibrium state, the neck growth rate of the twonanoparticles is lower and relatively stable, in comparison withthat in the later transition stage (see figures 4 and 5). Thegyration radius curve also exhibits a relatively smooth feature,which also indicates that the shape change and deformationof the particles are not very severe in this early stage of lasersintering.

The end of the low neck growth rate stage is followedby the transition stage. As the temperature increases, thenanoparticles experience a transition from solid state topremelting state and quasi-liquid state. The neck growth ratebecomes faster than that in the previous stage and the gyrationradius starts to decreases to a lower value. The structuredeformation is much more severe, along with both plasticdeformation and surface melting. As a result, the numberdensity of the disordered atoms is continuously increasingand becomes dominant over the fcc atoms (>0.5) in the entiresystem. Specifically for the hollow spheres, atoms on the twoinner free surfaces as well as the outer free surface begin tomove very fast, and the inner hollow space becomes smallerand smaller during this stage (see the configurations at pointsC and D in figure 3). Driven by curvature induced surfacediffusion, the so-called ‘spherication’ process [19] begins andthe two particles keep moving towards each other to form intoone sphere. This transition stage is also accompanied by asmall portion of lattice structure changing from fcc to hcp(visually found in the configuration at point D), which is similarto the phenomenon reported for metallic nanorods by Wang andDellago [43] but has not been observed clearly in the previousstudy [19] on the laser sintering of nanoparticles. This latticestructural transition can be a very important mechanism relatedto the heating rate effect on the hollow nanoparticle sintering,as will be discussed later in this paper.

Lastly, as the temperature increases to the second turningpoint, the most rapid reduction of the gyration radius occurs,with the simultaneous ascending of the neck growth rateand MSD in figures 4–6. These sudden changes with steepslopes shown on the curves indicate the onset of the meltingstage with the coalescence of the two particles. For thehollow nanoparticles, the shell structures finally collapse inthis stage and the inner vacant spaces disappear, filled witha large number of amorphous atoms. As illustrated by theconfigurations at points E and F in figure 3, the melting tends tooccur and drive all the atoms from the two particles together toform one sphere. At point F around 1098 K, the fraction of thedisordered and surface atoms finally reaches 100%, denotingthat the entire transition from fcc crystal structure to amorphousis fully accomplished.

4.2. Size and heating rate effects

Figure 4 demonstrates the laser heating rate effect on theneck width growth of different-sized solid sphere pairs. By

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J. Phys. D: Appl. Phys. 46 (2013) 335302 S Jiang et al

Figure 4. Comparison of the neck width in the simulations 1–6 for the solid nanoparticle pairs at three different laser heating rates.

comparing the curves in figures 4(a)–(f ), it is found that ateach specific heating rate, melting or coalescence would startat a higher temperature when the particle diameter becomeslarger. This size effect at a constant heating rate also holdsfor the hollow nanoparticle case, as compared by each panelsin figure 5. In figure 6, the temperature at which the MSDincreases significantly indicates the occurrence of melting.From all the simulations for both hollow and solid pairswith different sizes, the results of MSD again confirm thearguments of the size effect discussed above. To determinethe melting temperature more accurately, the dependence ofthe potential energy on temperature during different heatingprocesses for each individual solid and hollow particle isevaluated (partially shown in figure 7). Similar to previouswork [44, 45], the melting transition for the nanoparticlescan be clearly identifiable by the sharp rise in energy overa relative small range of temperature change. Thus themelting temperatures are collected from these energy curvesand plotted in figure 8. The error bars indicate the amplitudesof the melting ranges, taken from the starting points and endpoints of the large upward jumps in the energy curves, and thesymbols their centres. It can be found that the size-dependentmelting temperature is valid for both solid and hollow particlesat the same heating rate. These results confirm the theoreticaldemonstration from Buffat and Borel [10] that the melt pointincreases with the increase of size, and are also in goodagreement with previous work [19, 23, 45].

Another finding for each solid nanoparticle case is that theneck width during the transition stage (the region between thetwo arrows in figure 4) is larger when the heating rate becomeslower, in spite of the neck width showing almost independentlyon the heating rate during the relatively stable stage (the regionbefore the first arrow). The corresponding MSD curves canalso show the same trend, that is, at the same temperature foreach solid particle pair (e.g. figure 6(a)), the MSD increaseswith the decrease of heating rates. This phenomenon can beexplained by the duration of the laser heating. At a lowerheating rate, it needs more time to raise the temperature from298 K to a certain value. Thus, the whole sintering processlasts longer and the atoms in the particles have longer timeto interact with each other. As a result, the sintering becomesmore thorough and the atoms would be much farther from theirinitial positions, leading to a larger neck width and MSD value.

A very interesting phenomenon is observed when thehollow nanoparticles are considered in the sintering process.Compared to those in figure 4, the results in figure 5 for all thehollow particle cases show an inverse trend of the temperaturefor fast neck growth at the end of the transition stage, asthe heating rate is increased from 0.2 K ps−1 to 1.0 K ps−1.This indicates that melting stage happens beforehand on thetemperature axis for the hollow spheres at an ultrahigh heatingrate, which is quite different from that in the solid particlecases. In figures 5 and 6(b), the second arrow in each panelsuggests the steep increases of the neck growth rate and MSD,

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Figure 5. Comparison of the neck width in the simulations 7–12 for the hollow nanoparticle pairs at three different laser heating rates.

which occur at a lower temperature for the heating rate of1.0 K ps−1. This implies that besides the size and heating rate,the geometrical structure of the nanoparticle could also playan important role in the sintering process.

This inverse trend can be found much more clearly infigure 9, which plots the melting temperature versus heatingrate for both solid and hollow particle pairs with different sizes.Similarly, as shown in figure 10, the temperature for the fastneck growth to start and end are also collected from the curvesat different heating rates in figures 4 and 5, and defined as thecoalescence temperature for the particle pairs in this study. Ascan be seen, the hollow particles show a very different tendencyfrom the solid ones. For each hollow particle pair, the meltingpoint and coalescence temperatures show an obvious inversetrend with the increase of heating rate at a high level, as plottedin figures 9(b) and 10(b).

4.3. Comparison of hollow and solid particles

To gain insight into the effect of hollow structure on the inversetrend of fast neck growth temperature with the increase ofheating rate to an ultrahigh level, a comparative study betweenthe deformation evolutions of the solid and hollow spheresis conducted in detail, as shown in figure 11. To betterunderstand the evolution of the inner free surface in hollowspheres, the variation of the cavity at different heating rates isalso calculated and plotted in figure 12.

In figures 11(a) and (b), the crystalline structure of atomsis identified to understand the lattice transition during thesintering at various heating rates. The statistic distributions ofthe surface and disordered atoms with respect to temperatureat three different heating rates actually reveal quite differentpatterns between the solid and hollow spheres. For the solidparticles in figure 12(a), the three number density curves of thesurface and disordered atoms almost coincide with each otheras the temperature increases, until the first melting temperatureof the lowest heating rate case is reached (see the curves fromthe initial point at 298 K to point C). In contrast, the numberdensity curves of hollow particles in figure 11(b) start to showmuch divergence right after point A at a lower temperature,until the entire crystal structure in all three cases transits toamorphous after point D with the fraction of disordered atomsequaling 100%. Specifically, at the same high heating rate of1.0 K ps−1, by comparing the configurations in figures 11(a)and (b), it can be found that introducing the inner free surfacesin hollow nanoparticles actually facilitates the surface diffusionand melting from both outside and inside, which makes theentire structure more unstable and easier to collapse. Theinitial larger surface-atom fraction in the hollow spheres alsoimplies the increased number of active atoms and the increasedfree surface-area-to-volume ratio, so that the surface tension-driven-atomic migration becomes more dominant than thatin the solid spheres during the sintering. Hence, the hollownanoparticles tend to coalesce into one sphere more thoroughly

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Figure 6. Temperature dependence of MSD for the (a) solid and(b) hollow nanoparticle pairs in simulations 1 and 7 at threedifferent laser heating rates.

Figure 7. Caloric curves for a single (a) solid and (b) hollowparticle with the diameter of 10a at different heating rates.

than the solid ones at the same temperature level, as comparingthe configurations at points C and D in figures 11(a) and (b).

Furthermore, for each specific hollow particle pair, theinverse trend of fast neck growth temperature at an ultrahigh

Figure 8. Melting temperature and melting ranges as a function ofparticle diameter for each single particle with the heating rate of0.2 K ps−1. The error bars indicate the amplitudes of the meltingranges and the symbols their centres.

Figure 9. Melting temperature and melting range as a function ofheating rate for the (a) solid and (b) hollow particle pairs withdifferent sizes.

heating rate could be explained by the different lattice structuretransition from that at a low heating rate. The different way ofstructure transition under different heating rates can be inferredby the curves in figure 11(b), in which the fraction of disordered

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Figure 10. Coalescence temperature as a function of heating rate forthe (a) solid and (b) hollow particle pairs with different sizes. Theerror bars indicate the amplitudes of the fast neck growthranges.

atoms for the 1.0 K ps−1 case is larger than those for the othertwo lower heating rate cases (see points B, C and D). It alsocan be demonstrated by the gyration radius versus temperaturecurves at different heating rates plotted in figure 11(c). Forthe two lower rates of 0.04 K ps−1 and 0.2 K ps−1, the dropsof gyration radius curves are step-like. In contrast, the rapidreduction of the gyration radius for the 1.0 K ps−1 happensat a lower temperature and appears much smoother, whichis consistent with the inverse trend of the neck-growth/MSDversus temperature curves in figures 5 and 6. The step-likefeature of the gyration radius evolution at lower heating ratesis mainly due to the structure reconfiguration as well as thelattice transformation, which can be explained by the slidingof the lattice structure [19] in the early neck growth stage, andthe transformation from fcc to hcp structure later during thepremelting stage.

From the atomic configurations of the hollow particles atthe lower heating rate of 0.04 K ps−1, as discussed above insection 4.1, in the transition stage with the plastic deformation(see point B in figure 3), a lot of intrinsic stacking faults(hcp atoms) are formed in the particles. These defects candisappear with the recrystallization of the lattice structure as thetemperature rises (see point C in figure 3). Later in the end ofthe transition stage when premelting occurs, the re-formationof the hcp atoms are observed at points D and E. However,the same hollow particle pair shows a different scenario ofstructure deformation at the highest heating rate of 1.0 K ps−1,

as illustrated by the atomic configurations in figure 11(b). Asdiscussed earlier, the higher heating rate means that there isa shorter duration to heat the system from 298 K to a desiredvalue, thus the duration for the interactions among the atoms islimited, which makes it harder to form regular planer defects.Instead of lattice sliding, the surface diffusion and melting fromboth inside and outside dominate the entire process of the lasersintering.

These arguments stated above can be further confirmedby a quantitative comparison of the hcp atoms density infigure 11(d) for the three different heating rates. For the0.04 K ps−1 and 0.2 K ps−1 cases, the number density of hcpatoms becomes much higher than that in the 1.0 K ps−1 caseafter ∼698 K. That is the reason why the correspondingfraction of disordered atoms in figure 11(b) becomes smaller,which postpones the thorough transition from crystal toamorphous state. On the other hand, the curves of normalizedcavity volume (the ratio of current cavity volume to initialcavity volume) versus temperature in figure 12 for the hollowspheres are also consistent with the tendency predicted byfigures 11(c) and (d). Similar to the decrease of gyration radiuscurves, the reduction of the cavity volume is more stepwiseand starts to occur at a much lower temperature below themelting point, at the lowest heating rate of 0.04 K ps−1. Thisresult confirms that structural reconfiguration induced by thelattice sliding in the hollow shells, is an important mechanismduring the low heating rate process. In contrast, at the highestheating rate of 1.0 K ps−1, the lattice structure transformationfrom fcc to hcp becomes much less dominant, and only avery small portion of hcp atoms is found in figure 11(d).Meanwhile, the sudden drop on the smoother cavity volumecurve around the melting stage in figure 12 implies the directcollapse of the hollow interior instead of the thorough structuraladjustment and reconfiguration. The transition from thefcc structure directly to disordered state is more likely tohappen at this ultrahigh heating rate, which makes hollowspheres prone to coalescence and melt earlier at a lowertemperature.

5. Conclusions

Based on MD simulations, a comparative study of both solidand hollow spherical silver nanoparticles with different sizesunder laser sintering is conducted systematically in this paper.Three heating rates of 0.04 K ps−1, 0.2 K ps−1 and 1 K ps−1 areconsidered. Major findings from the study are summarized asfollows:

• The solid state sintering of both solid and hollow nanopar-ticles can occur spontaneously at room temperature, andplastic deformation via partial dislocation activities andlattice slips are important mechanisms which contributein a major way to the sintering of both solid and hollownanoparticles. Either stable or transient stacking faultshave been observed to occur in the smaller one in a pairif the two sphere sizes are not identical, indicating thatthe partial dislocations and lattice slips are more prone toform in smaller particles. Mechanical rotation can also befound during the neck growth, which is more obvious inthe hollow nanoparticles.

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J. Phys. D: Appl. Phys. 46 (2013) 335302 S Jiang et al

Figure 11. The temperature dependence of the number density of surface and disordered atoms for (a) the solid nanoparticle pair of 16aversus 16a and (b) the hollow nanoparticle pair of 16a versus 16a at three different heating rates. The insets are the evolutions of atomicconfigurations during the sintering with the heating rate of 1.0 K ps−1. (c) The gyration radius versus temperature and (d) the numberdensity of hcp atoms versus temperature for the hollow nanoparticle pair of 16a versus 16a at three different heating rates.

• The deformation evolution of the nanoparticle pair duringlaser sintering process can be mainly divided into threestages: relatively stable stage, transition stage, andmelting stage. At each specific heating rate, it isfound that melting starts at a higher temperature whenthe particle diameter becomes larger for both solid andhollow nanoparticles. An interesting inverse trend ofthe temperature for fast neck growth has been observedat the end of the transition stage for all the hollowparticle cases, as the heating rate rises from 0.2 K ps−1

to ultrahigh 1.0 K ps−1, which is quite different from thesolid nanoparticle cases. This phenomenon implies thatin addition to the size and heating rate, the geometricalstructure of the nanoparticle could also play an importantrole in the sintering process.

• In hollow nanoparticles, more active atoms and larger freesurface-area-to-volume ratio make the surface-tension-driven atomic migration more dominant than that in thesolid spheres during the laser sintering. At a low heatingrate, the plastic deformation combined with structuralreconfigurations induced by the lattice sliding in thehollow shells is found to be an important mechanismduring the heating process. At an ultrahigh heating rate,though there is still a small portion of hcp atoms formed,transition from fcc crystal directly to disordered structurefrom both outside and inside becomes more dominant,facilitated by the introduction of the inner free surfacesin hollow shells. The entire hollow particle pairs thus

show an obvious tendency to coalesce and melt at a lowertemperature level than with a low heating rate.

The findings reported in this paper might be useful for thefuture investigation of different kinds of nanoparticle sinteringin which both solid and hollow structures are involved. In a reallaser sintering process, controlling heat flux is more commonand easier than controlling temperature. Further investigationscan be focused on the sintering process of particles heated bymeans of constant heating flux. Meanwhile, different metallicparticles besides the silver ones studied currently, should beconsidered in the future study to better understand the effectof interior structures during the sintering process.

Acknowledgments

This work was supported by the US Defense Threat ReductionAgency under grant number HDTRA1-10-1-0022, the USNational Science Foundation under grant number CBET-1066917, the National Natural Science Foundation of Chinaunder grant numbers 11102185 and 11232003, and theNational Key Basic Research Special Foundation of Chinaunder grant number 2010CB832704. SJ thanks Dr YouqingYang, Professor Thomas D Sewell and Professor DonaldL Thompson for some helpful discussions. The authorsare grateful to the anonymous reviewers for their insightfulcomments to improve the paper. The simulations wereperformed in part on the cluster ‘Lewis and Clark’ provided

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J. Phys. D: Appl. Phys. 46 (2013) 335302 S Jiang et al

Figure 12. Normalized cavity volume as a function of temperaturefor the hollow particle pairs of (a) 10aversus 10a and (b) 16a versus16a at different heating rates.

by the University of Missouri Bioinformatics Consortium(UMBC).

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