Mechanical Behaviors of Copper Nanoparticle Superlattices: Role of Lattice Structure

Abstract

Nanoparticle superlattices, periodic assemblies of nanoscale building blocks, offer opportunities to tailor mechanical behavior through controlled lattice geometry and interparticle interactions. Here, classical molecular dynamics simulations were performed to investigate the compressive responses of copper nanoparticle superlattices with face-centered cubic (FCC), hexagonal close-packed (HCP), body-centered cubic (BCC), and simple cubic (SC) arrangements, as well as disordered assemblies. The flow stresses span 0.5–1.5 GPa. Among the studied configurations, the FCC and HCP superlattices exhibit the highest strengths (~1.5 GPa), followed by the disordered assembly (~1.0 GPa) and the SC structure (~0.8 GPa), while the BCC superlattice exhibits the lowest strength (~0.5 GPa), characterized by pronounced stress drops and recoveries resulting from interfacial sliding. Atomic-scale analyses reveal that plastic deformation is governed by two coupled geometric factors: (i) the number of interparticle contact patches, controlling the density of dislocation sources, and (ii) their orientation relative to the loading axis, which dictates stress transmission and slip activation. A combined parameter integrating particle coordination number and contact orientation is proposed to rationalize the structure-dependent strength, providing mechanistic insight into the deformation physics of metallic nanoparticle assemblies.

1. Introduction

Nanoparticle superlattices, periodic assemblies of nanoparticles, are promising materials for structural, optical, electrical, and energy-related applications [1,2,3,4]. Beyond their functional roles, the mechanical properties of these assemblies are of fundamental im-portance, since stability, strength, and deformation resistance ultimately dictate their re-liability and performance in practical systems [5,6]. When nanoparticles are organized into long-range ordered structures, the resulting assemblies display emergent behaviors that cannot be predicted from isolated particles. These responses are controlled not only by particle properties, but also by interparticle contacts, lattice symmetry, and packing geometry [7,8]. Understanding such structure–property relationships is essential for linking nanoscale mechanics to macroscale performance and for designing high-strength, multifunctional nanomaterials [1,9].

The mechanics of individual metallic nanoparticles provide the foundation for understanding superlattice behavior. Metallic nanoparticles often exhibit mechanical responses distinct from bulk metals, including size-dependent elasticity [10,11,12], pseudoelastic/orientation-dependent effects [13,14,15,16], and material strengths approaching theoretical limits [17,18]. For instance, silver nanoparticles have been reported to deform in a liquid-like fashion [16], while gold nanoparticles can display reversible dislocation-mediated plasticity [17]. Copper nanoparticles, in particular, show exceptionally high hardness values and unique atomic-scale plasticity mechanisms [19], attributed to their high surface-to-volume ratios and dislocation-controlled deformation [20,21]. These findings highlight that nanoscale mechanics are governed by mechanisms not present in bulk metals, raising the key question of how such behaviors manifest in nanoparticle assemblies.

When assembled into ordered superlattices, nanoparticles undergo cooperative deformation processes such as sliding, rotation, densification, and dislocation activity [22]. These collective mechanical behaviors are strongly dependent on lattice symmetry, packing density, and ligand interactions [23]. For example, molecular dynamics (MD) simulations have shown that nanoparticle superlattices exhibit asymmetric deformation and pressure-induced collapse [22,23,24], and ligand interactions can generate chiral responses or enhance stiffness [25,26]. While these studies establish that superlattice mechanics are highly structure-dependent, most prior work has focused on noble-metal systems, leaving the mechanical behavior of copper superlattices largely unexplored.

Copper nanoparticle superlattices are especially compelling for both scientific and practical reasons. At the single-particle level, copper nanoparticles already exhibit distinctive strengthening mechanisms [19,21], making them ideal system for studying how such effects evolve in ordered assemblies. In addition, copper is abundant, inexpensive, and indispensable in electronics, catalysis, and structural applications [27,28]. Moreover, various synthesis routes for copper nanoparticle superlattices have been developed, such as inkjet printing [29], digestive ripening and aging [30], thermal evaporation [31], and mechanical sinter [32], with recent studies highlighting their potential in electronic devices and power packaging [29,32]. Despite these advances, the mechanical properties of copper superlattices, particularly the role of lattice structure in governing strength and plasticity, remain poorly understood.

In this study, we address this gap by investigating the mechanical behaviors of copper nanoparticle superlattices through MD simulations of uniaxial compression. Guided by experimental observations, we consider four representative lattice structures, including face-centered cubic (FCC), hexagonal-closed cubic (HCP), body-centered cubic (BCC), and simple cubic (SC) superlattices [33,34,35]. For comparison, disordered assemblies of copper nanoparticles are also considered [32]. By contrasting different lattice symmetries, we reveal how packing geometry governs both macroscopic mechanical responses and atomic-scale deformation mechanisms. These insights provide some preliminary structure–property relationships for copper nanoparticle assemblies, bridging individual particle mechanics with assembly-level behavior and informing the design of high-performance nanostructured materials.

2. Material and Methodology

2.1. Construction of Copper Nanoparticle Superlattice and Assembly

In the present study, copper was selected as the model material. Both ordered nanoparticle superlattices and disordered assemblies were considered. To focus on the influence of superlattice structure and to reduce the complexity and computational cost, monodisperse copper nanoparticles with a fixed radius of 5.0 nm were used.

To construct a sample of ordered nanoparticle superlattice, an individual copper nanoparticle was first prepared via cutting a single-crystal bulk copper model with a spherical surface, which served as the basic building block. Totally, four types of superlattice structures were considered, including FCC, HCP, BCC, and SC superlattices. Each superlattice was then scaled according to the nanoparticle size. In the following stage, duplicates of the prepared nanoparticle were assigned random orientations, and were placed at each superlattice site to construct the intended superlattice structure.

To construct a disordered nanoparticle assembly, granular simulations were first conducted to generate some random patterns of nanoparticle assemblies. In such simulations, nanoparticles were modeled as elastic spheres that gathered in a simulation box with periodic boundary conditions. The contact between spheres was described by the Hertzian model [36]. For the details of granular simulations, please refer to reference [37]. When the shape of the simulation box was changed reversibly, different granular configurations could be obtained. In the next step, each sphere in the simulation box was replaced by one copper nanoparticle with a random rotation to construct a series of disordered assemblies of copper nanoparticles.

2.2. Setup of Atomic Simulation

To investigate the mechanical properties, uniaxial compressions were conducted. The loading directions were chosen along the normal to the close-packed plane for each superlattice: [111] for FCC, [0001] for HCP, [110] for BCC, and [001] for SC. For the disordered assembly, loading direction was along one of the three mutually orthogonal directions of the simulation box. Figure 1 shows the atomic models of superlattices and disordered assembly. Depending on the structure of different samples, the simulation box size ranges from approximately ~29 × 29 × 29 nm³ to ~34 × 29 × 32 nm³. The number of copper nanoparticles per sample varies between 27 and 48, corresponding to a total of roughly 1.20 to 2.14 million copper atoms.

With the preparation of nanoparticle superlattices and disordered assemblies, atomic simulations were conducted by using the open-source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), version 28-Mar-2023 [38]. The atomic interactions among copper atoms were described by using the embedded atom method (EAM), with a potential parameter set developed by Mishin et al. [39]. The EAM with this potential parameter set has been widely used to investigate the mechanical properties and deformation behaviors of different types of copper nanostructures, exhibiting excellent accuracy and stability [40,41]. Periodic boundary conditions were applied in all three directions for the cubic/rectangular atomic regions as shown in Figure 1. Evolution of the atomic systems was described within the framework of isothermal–isobaric ensemble (NPT), which was quantified by such quantities as the number of particles, pressure, and temperature of the system throughout the simulation. Temperature of the system was controlled at 300.0 K by using a Nosé–Hoover thermostat [42,43]. The integration time step was set to 0.002 ps.

Before applying compressive load, energy minimization was performed through a conjugate gradient method, in order to obtain an energy-favorable initial state. In the following stage, the atomic systems were further dynamically equilibrated at 300.0 K for about 100.0 ps, which was sufficient for the structural relaxation of the superlattices. After the equilibration, uniaxial compression tests were conducted with a strain rate of 1.0 × 10⁸ s⁻¹, consistent with common strain rate applied in MD simulations [11,12,13,14,17,18,19,20,21,22]. During loading stage, the stress tensor at each atom site was calculated based on the formula of the virial stress [44]. The compressive stress was represented by the component of the pressure tensor of the entire system in loading direction. To track the nucleation and evolution of the atomic defects, the local atomic structure was identified based on a method of common neighbor analysis (CNA) [45,46]. Using the CNA parameters, atoms in a perfect crystalline lattice could be clearly distinguished from those associated with defects such as surface, interface, stacking faults, twins, and grain boundaries. To further differentiate from the structures of superlattices, lowercase labels, such as fcc, bcc, and hcp, are used to identify atoms belonging to different atomic crystal lattices. All the atomic structures were visualized by using the OVITO software of version 3.10 [47].

3. Results and Discussions

3.1. Compressive Loading Responses

Figure 2a displays the compressive stress–strain curves of different superlattices and the disordered assembly under uniaxial compression. Depending on the structure, various characteristics could be observed. For FCC and HCP superlattices, the loading curves nearly overlaps. In the initial stage, compression stress increases linearly up to a strain of ~0.07. In following stage, stress saturates with small fluctuations. Although strain hardening behavior exists, the up-limit could not surpass 1.5 GPa. The stress–strain curve of the disordered assembly is close to that of the SC superlattice but slightly higher. Moreover, the range of the linearly increasing stage narrows to 0 to ~0.05, and the slopes are also reduced. Strain-hardening behaviors lasts till the end of the compression with up-limit smaller than 1.0 GPa. Compared to other cases, a drastic different compressive behavior is exhibited in the BCC superlattice. In the initial stage, the stress steadily increases to a local maximum at a strain of 0.08. In the following stage, evident stress-drop and stress rebuild-up appear on the loading curve, e.g., at strain ~0.10 and ~0.15.

Figure 2b compares the flow stress of different superlattices and the disordered assembly. The flow stress is defined as the average compressive stress in the strain range from 0.10 to 0.20. It is noted that FCC and HCP superlattices exhibit the highest flow stress, ~1.5 GPa. Disordered assembly shows an evidently lower flow stress, ~1.0 GPa, but still larger than that of the SC superlattices, ~0.8 GPa. BCC superlattice has the lowest flow stress, ~0.5 GP. Figure 2b also shows the variation in coordination number of different superlattices and disordered assembly. For the disordered assembly, the coordination number is calculated based on the Voronoi diagram around each nanoparticle, and is equal to the average face number of Voronoi cell. Notably, the coordination number of superlattice can be used to predict the overall variation trend of the flow stress. In FCC and HCP superlattices, nanoparticles have the largest packing density and thus the highest coordination number, resulting in the highest flow stresses. However, some exceptions to this trend have been observed. For example, the coordination number in BCC superlattice is higher than the SC superlattice owing to a larger packing density, while BCC superlattice exhibits the lower flow stress than the SC superlattice. The phenomenon indicates the strength of superlattices cannot be solely determined by the coordination number of nanoparticles.

3.2. Atomic Deformation Mechanisms in Nanoparticle Superlattices

By examining the atomic deformation characteristics, it is found that the plastic responses of ordered superlattices and disordered assembly are mainly governed by two factors: (i) interparticle contact patch number per particle, and (ii) orientation of the contact patches, i.e., normal direction of contact patch with respect to loading axis. The first factor can be quantified by the particle coordination number of structures, whereas the second one can be described by a defined fabric tensor spanned by the orientation of contact patches. In this section, the effects of the contact patch number are first investigated, followed by the effects of their orientations. Finally, one parameter combining the coordination number and the fabric tensor is proposed to qualitatively assess the strength of nanoparticle superlattices.

3.2.1. Effects of Contact Patch Number

In the ordered superlattices and disordered assembly, the transmission of forces among particles occurs primarily through the discrete contact patches where neighboring nanoparticles physically touch. Since both the FCC and HCP superlattice are the most close-packed particle assemblies, the packing density, the interparticle contact orientation, and contact patch number per particle are identical, mechanical responses are similar. To keep concise, only the FCC superlattice is adopted to demonstrate the mechanical responses for these two superlattices. Figure 3 depicts the distribution of the contact patches in ordered superlattices and disordered assembly. The contact patch number per particle depends on the superlattice structure. Except for the disordered assembly, the contact patches in superlattices are regular. In the FCC superlattice, each nanoparticle has 12 contact patches, the highest among the considered structures, while nanoparticles in the disordered assembly have an average of 9 contact patches. The BCC and SC superlattices have eight and six contact patches per particle, respectively. The first two structures (FCC and disordered) are taken as examples to analyze the influence of contact patch number.

Figure 4 depicts atomic models of superlattices and disordered assembly after structural relaxations. Owing to the interfacial adhesion, copper atoms around the interparticle contact patches slightly deviate from perfect lattice, and initial dislocation embryos nucleate as marked. It is noted that, in ordered superlattices, the initial dislocation patterns are regular within individual nanoparticles, in contrast to the disordered assembly, where larger stacking faults may be formed in nanoparticles with a specific position. Although the initial dislocations exhibit position-dependent sizes, they are distributed nearly uniformly throughout the entire simulation box for all cases. The contact patches serve as dislocation sources during further deformation. A larger number of interparticle contact patches result in more dislocation sources.

As the compression proceeds, atomic stress concentrates at contact patches, as shown in Figure 5a. In the subsequent stages, while existing dislocation embryos continue to grow up, additional dislocations nucleate from the contact patches, as shown in Figure 5b,c. After one dislocation nucleates, it glides through the central region of nanoparticles, and finally arrives at the surface on the other side, leaving an intrinsic stacking behind. Similar dislocation-mediated plasticity owing to interparticle contact has also been observed in the sintering of dual-particle systems [48]. During the gliding, dislocations on intersecting or parallel slip planes interact with each other, forming dislocation structures such as the Lomer–Cottrell junctions and twin dislocations, which were observed in uniaxial compression of individual copper nanoparticle [21]. However, multiple contact patches functioning as dislocation sources increases the reaction probability. As the atomic nucleation process repeats, dislocation multiplications are promoted, and dislocation debris and stacking faults pervade within nanoparticles. As a result, the higher dislocation density leads to a higher flow stress, as shown in Figure 2a.

To quantitatively characterize the plastic deformation in the particle superlattices, Figure 6 shows the increments of two types of the primary defective atoms, i.e., the hcp atoms, composing the stacking faults and twin boundaries, and the non-crystal atoms, composing the dislocation cores and surfaces/interfaces.

It is noted that, in Figure 6a, the hcp atom fraction in the SC superlattice exhibit a decrease after a strain rate of 0.1. This results from the nucleation the trailing dislocations following the leading ones, which restores the previously slipped atoms from hcp to perfect fcc atoms and reduce the hcp fraction. Since the packing density, number of contact patch per particle, and their orientations are identical for FCC and HCP superlattices, the dislocation-mediated plastic responses are similar, yielding comparable fractions of defective atoms at a given strain.

Generally, a superlattice with a higher coordination number results in more contact patches among nanoparticles. On the one hand, the higher number of contact patches leads to producing more dislocations, stacking faults, and twin boundaries. In Figure 6a, the fraction of hcp atoms in superlattice with higher contact patch number does exhibit a higher value. On the other hand, a greater number of contact patches also indicates a larger interparticle contact area during plastic deformation, which create more non-crystal atoms besides the dislocation cores. Figure 6b confirms this overall trend of the defective non-crystal atoms.

However, in addition to the characteristics of flow stress, the evolution of defective atoms also exhibits some exceptions. For example, although the BCC superlattice has a higher contact patch number per particle than the SC superlattice, the increment in defective hcp atoms is at a lower level. In contrast, the fraction of defective non-crystal atoms of BCC superlattice evidently surpasses that of the SC superlattice. These phenomena cannot be explained solely by the number of contact patches, and the orientation of contact patches should be considered.

3.2.2. Effects of Contact Patch Orientations

By examining the atomic structure and deformation characters, it is found that the orientations of the contact patches, i.e., their normal directions with respect to loading axis play important roles for the different mechanical responses between SC and BCC superlattice.

When viewed along the [001] loading direction, nanoparticles in SC superlattice are perfectly aligned in center-to-center contact, as shown in Figure 3d and Figure 4d. Consistent with the coordination number, each nanoparticle has six contact patches connecting to its neighbors, of which two are normal to the loading direction and four are parallel. Owing to differences in orientation, the compressive load is primarily transmitted through the normal contact patches. While in BCC superlattice, each nanoparticle has eight contact patches connecting to its neighbors. When viewed along the [110] loading direction, four contact patches are parallel to the loading direction, and four contact patches are inclined. Interparticle forces are mainly transmitted through the four inclined patches owing to the orientations.

During loading, the contact patches normal to the loading direction sustain higher load than those inclined to it, as no resolved tangential force induced interfacial sliding. Consequently, stress concentrates more around the normal contact patches in the SC superlattice than around the inclined contact patches in the BCC superlattice. Higher stress concentration causes dislocation embryos to grow more evidently. As shown in Figure 7a, nanoparticles in SC superlattice have already nucleated multiple dislocations at a strain of 0.05. As the strain increases to 0.10, dislocation nucleation becomes nearly uniform within the nanoparticles, leading to a plateau in the fraction of defective stacking-fault (hcp) atoms. Additionally, as the contact patches grow in size, more atoms around the contact interface transform into a non-crystalline structure.

In contrast, at a strain of 0.05 (Figure 7b), the dislocation embryos in the BCC superlattice grow only slightly, and fewer stacking faults are observed. Only at a larger strain, e.g., 0.18, do massive dislocations and stacking faults begin to nucleate. These behaviors are associated with the orientations of the contact patches. In the BCC superlattice, the force-bearing contact patches are inclined to the loading direction. As shown in the zoomed-in local view, a resolved shear force exists along the contact interface. When the load exceeds a critical threshold, the contact surfaces begin to slip relative to each other. The weak resistance to interparticle sliding causes a drop in the stress–strain curves (Figure 2a). Under compression, distance between nanoparticles at the same [110] plane increases, while the distance between nanoparticles in different [110] planes decreases. This local structure arrangement helps densify the superlattice structure. When the contact patches reach new stable configurations and can bear higher force, the load build-up leads to an increase in the slope of the stress–strain curve (Figure 2a). Dislocation nucleation prevails within nanoparticles until a highly stressed state is reached, and fraction of hcp atoms starts to increases rapidly. Due to interfacial sliding at the contact patches, the contact size and the number of interfacial dislocations increases more evidently than in the SC superlattice, resulting in a higher fraction of non-crystal atoms compared with the SC superlattice (Figure 6b).

To gain deeper insight into deformation, Figure 8 shows the evolution of dislocation density with increasing strain. It evidently shows that the overall variation in the dislocation density (Figure 8a) closely follows that of the fraction of non-crystal atoms (Figure 7b). The increase in dislocation density leads to a higher fraction of non-crystalline atoms associated with dislocation core structures. In the BCC superlattice, the increase in interface size has negligible effects on the stacking faults within nanoparticles, but contributes significantly to the interfacial dislocation density. As shown in Figure 8b–e, the final dislocation morphology is governed primarily by the distribution and number of the contact patches.

3.3. A Parameter to Explain the Structure-Dependent Flow Stress

The above section shows how the strength and atomic deformation are influenced by the number of contact patches per particle and their orientation. In this section, considering these two factors, one parameter is proposed to predict the strength of the nanoparticle superlattices and assembly.

For a nanoparticle superlattice or assembly, fabric tensor is a mathematical tool used to describe the geometric arrangement and the orientation of contact patches [49], which quantifies the anisotropy of how particles are packed and how forces or contacts are distributed. The global second-order fabric tensor F is defined as

$$F={\displaystyle \frac{1}{w}}{\displaystyle \sum _{i=1}^N{w}_i\cdot n_i\otimes n_i}$$

(1)

where N is the total number of interparticle contacts, w_i is a weight factor, defined as a function of the angle, θ, between the normal of contact patch and the loading direction, i.e., w_i = cos²(θ). w is the total sum of w_i. n_i is a unit vector along the normal of the i-th contact patch (from the center of one nanoparticle to the other). Symbol ‘$\otimes$’ is the dyadic product of a tensor. The contact tensor encodes the information about the preferred directions of particle contacts. Given a unit direction vector d, the directional fabric component is

$$F_{\mathrm{d}}=d^T\cdot F\cdot d$$

(2)

which is a scalar that quantifies how strongly contacts are aligned with direction d. Therefore, F_d can be used to characterize how the normal of contact is aligned with a specific loading direction. If total number of interparticle contact is fixed, a larger F_d indicate a higher resistance to external load resulting from orientation of contact patches.

Based on the analysis in Section 3.2, it is found that (i) the coordination number of a superlattice or disordered assembly determine the overall variation trend of contact flow stress, and (ii) the orientation of contact patches noticeably affects strength of loosely packed BCC and SC superlattices. By considering these two factors, a parameter φ is proposed to characterize the strength of superlattice, which is expressed as

$$\phi =C_N\cdot F_{\mathrm{d}}=C_N\cdot d^T\cdot F\cdot d$$

(3)

where C_N is the coordination number of a superlattice, and F_d is defined in Equation (2). Generally, structures with higher materials density could resist a higher external load; it governs the overall trend of the contact flow stress. The directional fabric component is a reflection of the geometrical structure of superlattice and assembly, which takes greater effect on a loosely packed superlattice and assembly. In Figure 9, the variation in the mean flow stress and the parameter φ is compared. As shown, the proposed parameter φ can qualitatively predict the relative contact flow stresses of nanoparticle superlattices and disordered assembly.

4. Conclusions

In summary, classical molecular dynamics simulations were employed to investigate the compressive behavior of copper nanoparticle superlattices with FCC, HCP, BCC, and SC arrangements, as well as disordered assemblies. The FCC and HCP superlattices exhibit the highest strengths, while the BCC superlattice shows the weakest response due to extensive interfacial sliding. Atomic-scale analyses reveal that plastic deformation is governed by two key geometric factors: the number of interparticle contact patches, which controls dislocation nucleation density, and the orientation of these contacts relative to the loading axis, which dictates stress transmission and interfacial sliding. A combined structural parameter integrating coordination number and contact orientation effectively captures the structure-dependent strength of nanoparticle assemblies. These findings establish clear structure–property relationships, providing mechanistic insight into the dislocation-mediated deformation of metallic nanoparticle superlattices and guiding the design of mechanically robust nanostructured materials.