Size effect on the spontaneous coalescence of nanowires
SSize effect on the spontaneous coalescence of nanowires
Zhenyan Wu, X. Yang, and Zhao Wang ∗ Department of Physics, Guangxi University, Nanning 530004, P. R. China.
Abstract
This paper investigates the size effect on the coalescence process of contacting nanoparticles. Itis revealed by molecular dynamics that the nanometer-sized surface curvature coupled with theeffective melting temperature exhibits a strong influence on the atom diffusion at the interface,and is therefore critical to the coalescence time. This effect is particularly pronouncing for surfacecurvatures below 20 nm. A phenomenological model is derived from the melting-point reductionapproach to describe the kinetic process of nanowire coalescence and is validated against a va-riety of simulation datasets. The quantitative correlation between the sample size, the sinteringtemperature and the contact morphology evolution is demonstrated. a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r NTRODUCTIONS
The spontaneous coalescence is critical for the self-assembly of nanomaterials. It is knownto be strongly correlated with the surface atom diffusion, which manifests when the materialsize goes down below tens of nanometers. Experiments demonstrated significant surfacediffusion taking place even at room temperature at sub-10 nm lengthscale [1]. Similarbehaviors were reported by experiments of the cold-welding of gold nanowires (NWs) [2] andNPs [3, 4], and of self-assembly of nanoparticle (NP) aggregates [5]. Understanding the size-effect on the mass-diffusion process at nanoscale solid interfaces is crucial for developing self-assembly technologies of nanostructures, which hold promise for a wide range of applications[6].Recently, it is reported by simulations that the coalescence of NPs can start without thethermal activation [7], and that the NP size and the sintering temperature exhibit significanteffects on the densification of the sintered nanoparticals [8]. Lu et al. demonstrate that singlecrystalline gold NWs with diameters between 3 and 10 nm can be cold-welded together withinseconds by mechanical contacts alone under low applied pressures [2]. Su et al. report thatthe gold NPs can be self-assembled at silver NW junctions and nanogaps by rod-coating[9]. Sabelfeld and Kablukova develop a stochastic model of the growth of an ensemble ofGaN NWs to include the coalescence caused by bundling [10]. The surface chemistry andthe chemical nature of the material are found to strongly influence the process of the coldwelding of nano-objects [3].The coalescence process is driven by a natural need to minimize the surface chemicalpotential. Hence the mass diffusion at sub-10nm-curvature surface is mainly driven by adramatic increase in the surface energy [11]. The theory of macroscopic thermal groovingdescribes the evolving shape of particles or wires in coalescence by considering both evapora-tion and surface diffusion mechanisms [12]. Despite of successful applications in interpretinga number of experimental measurements, this model suffers from problems caused by itsassumptions ignoring the atomistic details of the system. Meanwhile, atomistic simulationshave intensively been used to study the sintering [13–15] and coalescence [16–21] processesin nanomaterial synthesize. Notably, the mechanisms of melting temperature variation andphase transformation have been quantified by Koparde and Cummings [22, 23]. However,little is known up to date, about the combined roles of the sample size and temperature in2he kinetic process of NP coalescence.To this end, here we simulate the spontaneous coalescence of two contacting NWs usingmolecular dynamics (MD) [24–28]. The essential role of the surface curvature coupled withthe thermal effect in the coalescence process is demonstrated. Moreover, the simulationresults are used as inputs for developing a phenomenological model. Unlike previous models,the present model takes the curvature-dependent surface diffusivity into account, and is ableto predict the coalescence time as a function of NW size and the temperature.
METHODS
We start by simulating the contact between two curved aluminum NW surfaces at zeroexternal load. This set-up mimics a NW cold-welding experiment [2], as shown in Fig.1(a).The simulations are performed in a two-dimensional plane-stress configuration, with a peri-odicity w ≈ .
62 nm in the direction perpendicular to the cross section plane. An embeddedatom method (EAM) is used to describe the potential energy ε of the interaction betweenthe Al atoms, ε ( r ) = [ V ( b − b ) ( b z b − b z b ) + δ ]Ψ( r − r c h ) , (1)where r c is the cutoff distance, z = r/r (cid:48) , and b , b , δ , V , h and r (cid:48) are fitting parameters,Ψ( x ) is a cutoff function. The parameterization of this EAM force field is provided in Ref.29.We use a Nos´ e -Hoover thermostat at a time step of 0 . −
13 ordersof magnitudes larger than that of classical MD [32].
RESULTS AND DISCUSSIONS
The ratio between the effective contact area and apparent one can be greatly enhancedby decreasing the surface curvature to nanometer-scale [28]. This size effect becomes most3
IG. 1: (a) Snapshots of the simulated contact between a pair of Al NWs (cross-section view along[100] crystallographic direction). The color scale corresponds to the von-Mises stress distribution.(b) Contact area ratio φ = A/A as a function of R . The squares represent the ground-state (0 Klimit) results, while the circles and triangles stand for those of MD for simulation time of 10 and20 ns, respectively. The contact area A is computed by defining an inter-particle spacing cutoff of0 .
286 nm as the equilibrium inter-atomic distance and A = 2 wR where w is the thickness. pronounced for tip radii below 10 nm, as shown in Fig.1(b). Similar to previously reportedexperimental [2, 33, 34] and computational [35] results, the reconstruction of the cubiclattice with very few defects is observed. The displacive plasticity is simply related tothe electrostatic nature of the inter-atomic force, as a sharp tip contains a larger fractionof surface atoms that are exposed to the atomistic attractive force of the adjoining surface,which decreases rapidly with increasing separation distance and tends to vanish after several4anometers. This is consistent with the experimentally observed size effect on the contactbetween NPs and NWs [36], and is strongly correlated with the inverse contact scaling inbiological adhesive systems [37, 38]. For instance, a ten-fold-increase is found for an R = 1nm contact at 600 K. FIG. 2: (a-d) Snapshots of the simulation on adhesion of two NWs ( R = 10 nm) with fixedboundaries at 900 K. The arrows represent the direction of surface atom flow. The surface atomsare labeled with different colors. Results are obtained at different temperatures. The “zero-K” results [squares in Fig.1(b)]are from molecular mechanics [39–41] which do not include the thermal effects, and thusonly represent the time-independent (so-called displacive) contribution to the contact area.Comparing the two sets of 600 K data [triangles and circles in Fig.1(b)] that were obtainedby MD with different simulation time, we see that the finite-temperature contribution tothe contact area is time-dependent. This time-dependency is strongly correlated with the5urface atom diffusion, as shown in Fig.2 for a rigid-boundary set-up. The colors contrastthe surface and in-body atoms, by which we clearly see that the increase in the contactarea is mainly contributed from immigrated surface atoms. We see that, starting from theinitial displacive contact [Fig.2(a)], the surface atoms of the two contacting bodies diffuseinto the neck region [Fig.2(b,c)] until the curvature radius tends to be uniform along thesurface [Fig.2(d)]. This is consistent with microscopy experiment observations [42, 43].
FIG. 3: (a-f) Snapshots of the coalescence of two NWs ( R = 10 nm) with free boundaries at900 K from cross-section view. The arrows represent the direction of surface atom flow. Thesurface atoms are labeled with different colors. Another set of simulation is performed for NWs with free boundaries. It can be seen inFig.3(a-d) that the neck is filled with diffused atoms with time, similar to that in the fixed-boundary case shown in Fig.2. However, the wire shape keeps evolving until the formationof a new wire, as shown in Fig.3(e-f). This observation is consistent with the field-emissionTEM results reported by Cheng et al. , which show the e-beam-induced coalescence betweenNWs and NPs takes place by the fast, massive atom transportation near their contact surface6egion [34, 44].To quantify the diffusion behavior of surface atoms, we consider a phenomenologicaldescription derivable from the concept of curvature-dependent surface potential energy [12],∆ φ = βA (cid:90) t ∆ Ddt, (2)where β is a positive constant and ∆ D is the difference in the diffusivity of the atoms at thefree surface and that of the atoms at the neck region. In contrast to the original model ofMullins [12], we consider D is no longer constant but changes with R . This introduces theconcept of curvature-dependent diffusivity of nano-crystals. This is based on the melting-point reduction approach [45, 46] that was used to study atom diffusion in sintering of silica-encapsulated Au [47] and Au-Ag NPs [48]. This allows describing the effective diffusivity ofsurface atoms as a function of the surface curvature radius and melting point shift, D = D exp (cid:20) − CT m, ∞ k B T (cid:16) − α R (cid:17)(cid:21) . (3)This equation is accompanied with the well-established Gibbs-Thomson equation [49, 50],by which the melting point is approximately a function of the surface curvature radius R ofsmall crystals, T m ( R ) = T m, ∞ (cid:16) − α R (cid:17) , (4)where T m, ∞ is the bulk thermodynamic melting point, k B is the Boltzmann constant, both D and C are temperature- and size-independent positive constants. The shape parameter α can be obtained by considering the effect of size and shape on the bulk melting temperature, α = 4 υ sl H f ρ s , (5)where υ sl is the solid-liquid interface energy, H f is the bulk enthalpy of fusion and ρ s is thesolid state density.Numerical analyses show that the time-dependent diffusivity difference ∆ D can be ap-proximated by an exponential-decay function of time t , as shown in the supplementarymaterial, ∆ D = D (cid:26) exp (cid:20) − CT m ( R ) k B T (cid:21) − exp (cid:20) − CT m, ∞ k B T (cid:21)(cid:27) exp( − γt ) , (6)7 IG. 4: Change in contact area ratio ∆ φ versus time for a pair of NWs at different temperatures(a), and for those of different sizes at a given temperature (b) with fixed boundaries. The symbolsstand for the simulation data and the curves represent the results obtained by Eq.7. where the effective melting temperature of the surface atoms at the neck region is assumedto be T m, ∞ , and the parameter γ denotes the decay rate of ∆ D . This yields∆ φ = βA (cid:90) t D (cid:26) exp (cid:20) − CT m ( R ) k B T (cid:21) − exp (cid:20) − CT m, ∞ k B T (cid:21)(cid:27) exp( − γt ) dt. (7)where the values of γ and β are obtained by fitting to simulation results. The parameter8alues used in this work are provided in the supplementary material.The simulated evolution of the contact morphology can be well predicted by Eq.7 fordifferent sizes and temperatures, as shown in Fig.4. It can be seen that the contact areareaches a limit with different adhesion velocities at different temperatures for a given wireradius [Fig.4(a)]. For instance, for R = 10nm, the time required for φ to saturate at 600 Kis over four orders of magnitude longer than that at 900 K. This observation is qualitativelyconsistent with the results reported by Cheng et al. [51, 52]. We also see that the smallcontact exhibits higher adhesion velocity than the large ones [Fig.4(b)]. The differencebetween the prediction by Eq.7 and MD becomes more significant when R decreases below5 nm. This may be due to the effect of surface roughness [53]. CONCLUSION
In conclusion, we have demonstrated by MD and an analytical model that the nanometer-sized surface curvature coupled with the effective melting temperature is critical to theNP coalescence. We develop a phenomenological model to predict quantitatively the NWmorphology evolution as a function of the NW size and the coalescence temperature, bytaking into account the curvature-dependent surface diffusivity. These results have strongimplications to our understanding of the mass diffusion at sub-10 nm lengthscale. We remarkthat classical MD with a typical time step of 1 . ∗ Electronic address: [email protected]
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