Size effect on the spontaneous coalescence of nanowires

TL;DR

This study explores how nanowire size influences coalescence, revealing that smaller surface curvatures significantly accelerate atom diffusion and coalescence, with a validated phenomenological model linking size, temperature, and morphology.

Contribution

It introduces a new size-dependent model for nanowire coalescence based on melting point reduction, validated by molecular dynamics simulations.

Findings

Surface curvature below 20 nm greatly accelerates coalescence.

A phenomenological model accurately predicts coalescence kinetics.

Size and temperature critically influence contact morphology evolution.

Abstract

This paper investigates the size effect on the coalescence process of contacting nanoparticles. It is revealed by molecular dynamics that the nanometer-sized surface curvature coupled with the effective 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 surface curvatures below 20 nm. A phenomenological model is derived from the melting-point reduction approach to describe the kinetic process of nanowire coalescence and is validated against a variety of simulation datasets. The quantitative correlation between the sample size, the sintering temperature and the contact morphology evolution is demonstrated.

Equations

$$\epsilon (r)=[\frac{V_0}{(b_2-b_1)}(\frac{b_2}{z^{b_1}}-\frac{b_1}{z^{b_2}})+\delta ]\Psi (\frac{r-r_c}{h}),$$

$$\Delta \varphi =\frac{\beta }{A_0}{\int }_0^t\Delta Ddt,$$

$$D=D_0\exp \left[-\frac{C{T}_{m,\infty }}{k_{\mathrm{B}}T}\left(1-\frac{\alpha }{2R}\right)\right].$$

$$T_m(R)=T_{m,\infty }\left(1-\frac{\alpha }{2R}\right),$$

$$\alpha =\frac{4{\upsilon }_{sl}}{H_f{\rho }_s},$$

$$\Delta D=D_0\left\{\exp \left[-\frac{C{T}_m(R_0)}{k_{\text{B}}T}\right]-\exp \left[-\frac{C{T}_{m,\infty }}{k_{\text{B}}T}\right]\right\}\exp (-\gamma t),$$

$$\Delta \varphi =\frac{\beta }{A_0}{\int }_0^t{D}_0\left\{\exp \left[-\frac{C{T}_m(R_0)}{k_{\text{B}}T}\right]-\exp \left[-\frac{C{T}_{m,\infty }}{k_{\text{B}}T}\right]\right\}\exp (-\gamma t)dt.$$

Full text

I Introductions

The spontaneous coalescence is critical for the self-assembly of nanomaterials. It is known to be strongly correlated with the surface atom diffusion, which manifests when the material size goes down below tens of nanometers. Experiments demonstrated significant surface diffusion taking place even at room temperature at sub-$10\;\mathrm{nm}$ lengthscale Akande et al. (2010). Similar behaviors were reported by experiments of the cold-welding of gold nanowires (NWs) Lu et al. (2010) and NPs Wagle and Baker (2015); Sun et al. (2014), and of self-assembly of nanoparticle (NP) aggregates Klajn et al. (2007). 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 Liu et al. (2013).

Recently, it is reported by simulations that the coalescence of NPs can start without the thermal activation Li et al. (2016), and that the NP size and the sintering temperature exhibit significant effects on the densification of the sintered nanoparticals Yang et al. (2018). Lu et al. demonstrate that single crystalline gold NWs with diameters between 3 and 10 nm can be cold-welded together within seconds by mechanical contacts alone under low applied pressures Lu et al. (2010). Su et al. report that the gold NPs can be self-assembled at silver NW junctions and nanogaps by rod-coating Su et al. (2019). Sabelfeld and Kablukova develop a stochastic model of the growth of an ensemble of GaN NWs to include the coalescence caused by bundling Sabelfeld and Kablukova (2018). The surface chemistry and the chemical nature of the material are found to strongly influence the process of the cold welding of nano-objects Wagle and Baker (2015).

The coalescence process is driven by a natural need to minimize the surface chemical potential. Hence the mass diffusion at sub-10nm-curvature surface is mainly driven by a dramatic increase in the surface energy Jose-Yacaman et al. (2005). The theory of macroscopic thermal grooving describes the evolving shape of particles or wires in coalescence by considering both evaporation and surface diffusion mechanisms Mullins (1957). Despite of successful applications in interpreting a number of experimental measurements, this model suffers from problems caused by its assumptions ignoring the atomistic details of the system. Meanwhile, atomistic simulations have intensively been used to study the sintering Cheng and Ngan (2013); Buesser et al. (2011); Koparde and Cummings (2005) and coalescence Wang et al. (2016); Lim et al. (2009); Hawa and Zachariah (2005); Li et al. (2016); Guevara-Chapa and Mejia-Rosales (2014); Grammatikopoulos et al. (2014) processes in nanomaterial synthesize. Notably, the mechanisms of melting temperature variation and phase transformation have been quantified by Koparde and Cummings Koparde and Cummings (2008a, b). However, little is known up to date, about the combined roles of the sample size and temperature in the kinetic process of NP coalescence.

To this end, here we simulate the spontaneous coalescence of two contacting NWs using molecular dynamics (MD) Wang (2018); Wang and Devel (2011); Qi et al. (2018); Wang and Philippe (2009); Guo et al. (2015). The essential role of the surface curvature coupled with the thermal effect in the coalescence process is demonstrated. Moreover, the simulation results 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 able to predict the coalescence time as a function of NW size and the temperature.

II Methods

We start by simulating the contact between two curved aluminum NW surfaces at zero external load. This set-up mimics a NW cold-welding experiment Lu et al. (2010), as shown in Fig.1(a). The simulations are performed in a two-dimensional plane-stress configuration, with a periodicity $w\approx 1.62\,$nm in the direction perpendicular to the cross section plane. An embedded atom method (EAM) is used to describe the potential energy $\varepsilon$ of the interaction between the Al atoms,

[TABLE]

where $r_{c}$ is the cutoff distance, $z=r/r^{{}^{\prime}}$, and $b_{1}$, $b_{2}$, $\delta$, $V_{0}$, $h$ and $r^{{}^{\prime}}$ are fitting parameters, $\Psi(x)$ is a cutoff function. The parameterization of this EAM force field is provided in Ref.Zope and Mishin, 2003. We use a Nos$\acute{e}$-Hoover thermostat at a time step of $0.5\;\mathrm{fs}$ to simulate the shape evolution of the contact Yang et al. (2015). The system temperature is controlled to be relatively high for letting the system reach an equilibrium state in the time scale accessible to MD, i.e. in the order of nanoseconds, since the atomistic diffusion can be strongly accelerated at temperatures close to the bulk melting point Guo et al. (2015); He et al. (2017). Note, that surface diffusion is observable even at room temperature in experiments Lu et al. (2010), since the experimental time scale is typically $10-13$ orders of magnitudes larger than that of classical MD Li et al. (2011).

III Results and Discussions

The ratio between the effective contact area and apparent one can be greatly enhanced by decreasing the surface curvature to nanometer-scale Guo et al. (2015). This size effect becomes most pronounced for tip radii below $10\;\mathrm{nm}$, as shown in Fig.1(b). Similar to previously reported experimental Lu et al. (2010); Bay (1983); Cheng et al. (2019) and computational Pereira and da Silva (2011) results, the reconstruction of the cubic lattice with very few defects is observed. The displacive plasticity is simply related to the electrostatic nature of the inter-atomic force, as a sharp tip contains a larger fraction of 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 several nanometers. This is consistent with the experimentally observed size effect on the contact between NPs and NWs Tang et al. (2002), and is strongly correlated with the inverse contact scaling in biological adhesive systems Autumn et al. (2000); Lee et al. (2007). For instance, a ten-fold-increase is found for an $R_{0}=1$ nm contact at $600\;\mathrm{K}$.

Results are obtained at different temperatures. The “zero-K” results [squares in Fig.1(b)] are from molecular mechanics Wang (2009); Wang and Devel (2007); Wang et al. (2007) which do not include the thermal effects, and thus only represent the time-independent (so-called displacive) contribution to the contact area. Comparing the two sets of $600\;\mathrm{K}$ data [triangles and circles in Fig.1(b)] that were obtained by MD with different simulation time, we see that the finite-temperature contribution to the contact area is time-dependent. This time-dependency is strongly correlated with the surface atom diffusion, as shown in Fig.2 for a rigid-boundary set-up. The colors contrast the surface and in-body atoms, by which we clearly see that the increase in the contact area is mainly contributed from immigrated surface atoms. We see that, starting from the initial displacive contact [Fig.2(a)], the surface atoms of the two contacting bodies diffuse into the neck region [Fig.2(b,c)] until the curvature radius tends to be uniform along the surface [Fig.2(d)]. This is consistent with microscopy experiment observations Laza et al. (2013); Honey et al. (2015).

Another set of simulation is performed for NWs with free boundaries. It can be seen in Fig.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 formation of a new wire, as shown in Fig.3(e-f). This observation is consistent with the field-emission TEM results reported by Cheng et al., which show the e-beam-induced coalescence between NWs and NPs takes place by the fast, massive atom transportation near their contact surface region Cheng et al. (2018, 2019).

To quantify the diffusion behavior of surface atoms, we consider a phenomenological description derivable from the concept of curvature-dependent surface potential energy Mullins (1957),

[TABLE]

where $\beta$ is a positive constant and $\Delta D$ is the difference in the diffusivity of the atoms at the free surface and that of the atoms at the neck region. In contrast to the original model of Mullins Mullins (1957), we consider $D$ is no longer constant but changes with $R$. This introduces the concept of curvature-dependent diffusivity of nano-crystals. This is based on the melting-point reduction approach Jiang et al. (2004); Guisbiers and Buchaillot (2008) that was used to study atom diffusion in sintering of silica-encapsulated Au Dick et al. (2002) and Au-Ag NPs Shibata et al. (2002). This allows describing the effective diffusivity of surface atoms as a function of the surface curvature radius and melting point shift,

[TABLE]

This equation is accompanied with the well-established Gibbs-Thomson equation Buffat and Borel (1976); Zhang et al. (2000), by which the melting point is approximately a function of the surface curvature radius $R$ of small crystals,

[TABLE]

where $T_{m,\infty}$ is the bulk thermodynamic melting point, $k_{\rm B}$ is the Boltzmann constant, both $D_{0}$ and $C$ are temperature- and size-independent positive constants. The shape parameter $\alpha$ can be obtained by considering the effect of size and shape on the bulk melting temperature,

[TABLE]

where $\upsilon_{sl}$ is the solid-liquid interface energy, $H_{f}$ is the bulk enthalpy of fusion and $\rho_{s}$ is the solid state density.

Numerical analyses show that the time-dependent diffusivity difference $\Delta D$ can be approximated by an exponential-decay function of time $t$, as shown in the supplementary material,

[TABLE]

where the effective melting temperature of the surface atoms at the neck region is assumed to be $T_{m,\infty}$, and the parameter $\gamma$ denotes the decay rate of $\Delta D$. This yields

[TABLE]

where the values of $\gamma$ and $\beta$ are obtained by fitting to simulation results. The parameter values used in this work are provided in the supplementary material.

The simulated evolution of the contact morphology can be well predicted by Eq.7 for different sizes and temperatures, as shown in Fig.4. It can be seen that the contact area reaches a limit with different adhesion velocities at different temperatures for a given wire radius [Fig.4(a)]. For instance, for $R_{0}=10$nm, the time required for $\phi$ to saturate at $600\;\mathrm{K}$ is over four orders of magnitude longer than that at $900\;\mathrm{K}$. This observation is qualitatively consistent with the results reported by Cheng et al. Cheng and Robbins (2010); Cheng et al. (2010). We also see that the small contact exhibits higher adhesion velocity than the large ones [Fig.4(b)]. The difference between the prediction by Eq.7 and MD becomes more significant when $R_{0}$ decreases below $5\;\mathrm{nm}$. This may be due to the effect of surface roughness Luan and Robbins (2005).

IV 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 the NP coalescence. We develop a phenomenological model to predict quantitatively the NW morphology evolution as a function of the NW size and the coalescence temperature, by taking into account the curvature-dependent surface diffusivity. These results have strong implications to our understanding of the mass diffusion at sub-$10\;\mathrm{nm}$ lengthscale. We remark that classical MD with a typical time step of $1.0\;\mathrm{fs}$ could be limiting to adequately capture real material processes taking place on much longer time scale. Also the simplistic way of modeling the surface diffusion used in this work may not realistically represent the effect of local environment, such as surface passivation (e.g. oxygen, hydrogen, liquid solutions and other adsorbates) in real experiments. One can however envision implementing the Monte Carlo Henkelman and Jonsson (2001) or diffusive MD Li et al. (2011) simulations in light of the above-introduced kinetic model. The combination of atomistic simulations and continuum contact theories should be further applicable to a wider range of surface types, and is expected to provide useful guidelines for experimentalists working on welding and self-assembly of nano-objects, that are otherwise mostly limited by a time- and effort-consuming trial-and-error procedure.