In order to investigate the synergistic effect of different deformation modes (dislocations and local atomistic rearrangements) on the strength of mixed crystalline and amorphous materials, tensile and compressive deformation analyses of a two-dimensional binary system with various microstructures is performed through molecular dynamics simulations. The binary system is composed of atoms with two different atomic radii. By varying the mixing ratio and the interaction force between different atoms, 66 binary system models with various structures are represented, and each model is classified into three categories, crystalline, amorphous, and mixed crystalline/amorphous, through structural analysis. Deformation analysis shows that the strength of the mixed crystalline/amorphous models tends to be weaker than that of the crystal and amorphous models. The is because the edge of the force chain in the amorphous phase appears at the crystalline/amorphous interface, where dislocation release from the interface to the crystalline phase occurs easily.

This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 71 (2022) 135–142.

Fig. 10 Force chain development around the amorphous/crystalline interface shown in Fig. 9 under compressive deformation.

Solid materials deform plastically due to relative changes in the atomic configuration. Therefore, the introduction of obstacles to an activated plastic deformation mode increases the strength of the material. Increasing the deformation resistance of the plastic deformation mode with deformation, i.e., work hardening, can reduce deformation localization and improve ductility. The intrinsic plastic deformation modes are strongly influenced by the physical properties and microstructure of the solid material: in metallic materials with crystalline structures, the dominant plastic deformation mode is dislocation motion, whereas in metallic glass materials with amorphous structures,^1,2) the dominant deformation mode is local atomistic rearrangement, which produces shear deformation in a local region called the shear transformation zone.³⁾ In the case of pure single crystals, the Peierls potential, which is the resistance to dislocation motion, is generally small; hence, the dislocations preexisting in the grain can move easily, resulting in low yield strength. However, the dislocation density increases as the deformation progresses, and the interaction between dislocations results in a high work-hardening rate, which is characterized by high ductility although the strength is not so high. In contrast, amorphous materials such as metallic glasses have high strength but low ductility. This can be interpreted as the fact that high stress is required for the nucleation of local atomistic rearrangement, while work softening occurs owing to the increase in atomic free volume in the shear deformation zone where the local atomistic rearrangement has propagated.⁴⁾ Therefore, it is difficult to achieve both high strength and high ductility with only a single plastic deformation mode corresponding to a certain single-phase microstructure of a solid material.

To achieve excellent mechanical properties that cannot be realized using a single-phase material, the mixing of materials with different mechanical properties has been studied. Numerous synergistic effects of the different mechanical behaviors in composite structures have been reported to contribute to the realization of both high strength and high ductility, such as the strain and stress distribution between the matrix and second phase in duplex stainless steels,^5–8) the suppression of the plastic instability of the brittle layer by the ductile layer in multilayered laminated materials,^9–11) the backstress hardening and formation of unique dislocation structures that do not occur in homogeneous structures in gradient materials,^12–16) and the connectivity of the shell network in core–shell harmonic structure materials.¹⁷⁾ Therefore, the coexistence of crystalline and amorphous structures with completely different elementary deformation processes has the potential to produce materials with excellent mechanical properties. For example, laminates of crystalline and amorphous materials have been reported to have both high strength and high ductility.^18–20)

However, the deformation mechanism of materials with a mixture of crystalline and amorphous phases is still not fully understood because the mechanism of plastic deformation propagation differs for each phase. In polycrystalline materials, the accumulation of dislocations at grain boundaries causes large stress concentrations in the neighboring grains, facilitating the propagation of plastic deformation from grain to grain at low external loads. On the other hand, in metallic glasses, some local atomistic rearrangements are concentrated in a band region and grow into a macroscopic shear band to relieve loading stress. Therefore, clarifying the relationship between the deformation mechanism and strength of mixed structures in which these different plastic deformation modes operate simultaneously is essential for the development of next-generation structural materials.

In this study, the deformation mechanism of mixed crystalline/amorphous models is analyzed using molecular dynamics simulations to understand the mechanical properties produced by the synergistic effects of the multiple plastic deformation modes arising from the structure of the mixed crystalline/amorphous material. For this purpose, binary material models with various structures are fabricated by changing the mixing ratio of virtual atoms with two different atomic radii and the interaction forces between the atoms. By analyzing the deformation, we elucidate the relationship between the atomic structure, mechanical properties, and deformation mechanism. In particular, we focus on the strength and deformation mechanisms of mixed crystalline/amorphous models. In addition, we clarify the details of plastic deformation propagation between two different structures through compressive deformation analysis of layered models consisting of crystalline and amorphous materials. The concept of force chains, which play an important role in the deformation and flow of powders and colloidal systems,^21,22) is introduced, their existence is verified, and their effect on the strength of mixed crystalline/amorphous materials is investigated.

In this study, two-dimensional analysis is performed to facilitate the comparison of the deformation mechanisms arising from different structures. To represent virtual material models with crystalline and amorphous structures, a binary system comprising hypothetical S and L atoms with atomic radii of 0.07 nm and 0.13 nm, respectively, is considered. The atomic radii are based on the following relation, which can simultaneously satisfy a pentagon and decagon without translational symmetry (Fig. 1(a)), to represent the amorphous structure:²³⁾   

Fig. 1

Modeling of the binary system which can represent a crystalline/amorphous mixed state. (a) Atomic size relationship between small and large atoms. (b) Shapes of interatomic potentials for the binary system.

The shifted force potential, which is an extension of the Morse potential, is applied as the atomic interaction in the binary system model in this study. The Morse potential is given by the following:²⁴⁾   

To represent the binary system, the interactions between atoms S and S, L and L, and S and L expressed by eq. (3) are set as shown in Fig. 1(b). The cutoff distance (r_c) and α are common values, set as 0.5475 nm and 14.57364 1/nm, respectively. The potential parameters of the S and S, L and L, and S and L atoms are D_SS = 8.16 × 10⁻²⁰ J and r₀,_SS = 0.1744 nm, D_LL = 8.16 × 10⁻²⁰ J and r_0,LL = 0.2633 nm, and D_SL = KD_SS and r_0,SL = 0.2308 nm, respectively. Here, we set K = 0.75, 1, 1.25, 1.5, 1.75, and 2, and create several binary system models with different bonding forces between different atoms. These potential parameters are set such that the lattice constants of the S and L atoms are 0.07 nm and 0.13 nm, respectively, and eq. (1) is satisfied.

The analysis models consist of a total of 10,000 S and L atoms. Eleven values for the ratio of S atoms, f_S, are considered: 1%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 99%. For each f_S, there are six values for D_SL, which is a parameter representing the binding energy of the S and L atoms, as described in Section 2.1. Therefore, a total of 66 binary system models are created according to the following procedure. First, a random atomic configuration is created and then melted by heating the system from 10–2500 K over 40 ps under periodic boundary conditions, after which relaxation calculations are performed for 20 ps. Then, the configuration is cooled to 10 K over 40 ps, and relaxation calculations are performed for 5 ps at 10 K to obtain the various binary structures arising for the given f_S and D_SL values. The dimensions of the analytical model are controlled using the Parrinello–Rahman method²⁶⁾ such that the vertical stresses become zero. For this post-heat-treatment analysis model, tensile or compressive deformation analysis is performed at a constant temperature of 10 K and strain rate of 2 × 10⁸ 1/s.

It is necessary to identify the microstructures obtained through heat treatment with respect to the various binary system models described in the previous section. Here, we use the atomic configuration of the first neighbors of each atom to evaluate the local regularity of the atom. Focusing on atom j in the analysis model, the local atomic structure formed by the neighboring atoms centered on atom j can be classified as regular or irregular. To express the degree of irregularity, D_j, which is the six-fold symmetric order variable for atom j, is used.²⁷⁾ In a two-dimensional triangular lattice, if the first nearest-neighbor atoms of atom j are arranged in a hexagonal shape, atom j will have a regular structure; in this case, D_j is almost zero. If the first neighboring atoms deviate from a hexagonal configuration, D_j will have a large value. The threshold value of D_j to determine the local atomic structure is discussed in Section 3.2. Using D_j and the ratio of atoms with a regular structure, f_c, the microstructures of the two-dimensional models are classified into three phases: crystalline, amorphous, and mixed crystalline/amorphous.

In this section, we show that the binary system models obtained by heat treatment (described in Section 2.2) have various structures ranging from crystalline to amorphous, depending on the atomic ratio of S atoms (f_S) and the interatomic bonding force (D_SL). The atomic configurations after cooling are depicted in Fig. 2. The large gray atoms in the figure are L atoms, whereas the small white atoms are S atoms. Calculating the area occupied by each type of atom using the atomic radii of S and L atoms shows that the area occupied by S atoms in the entire system is approximately 50% when f_S ≈ 78%; i.e., the matrix phase switches when f_S reaches 70%–80%.

Fig. 2

Atomic configurations of the binary systems with different mixing ratio f_S and biding energy of small and large atoms D_SL/D_SS (= D_SL/D_LL). Light and dark gray circles represent small and large atoms, respectively. The identification of structures enclosed in squares is shown in Fig. 3.

We first focus on the structure with f_S = 20%, where the L atoms comprise the matrix phase. It can be confirmed that when D_SL/D_SS > 1, the individual S atoms are solid-soluble in the matrix phase, whereas when D_SL/D_SS < 1, a group of S atoms are precipitated. This occurs because the binding energy between the different types of atoms is lower than that between the same type of atoms (D_SL/D_SS < 1), and it is more energetically advantageous to increase the number of bonds between the same type of atoms than between different types of atoms.

We next focus on f_S = 70%, where the area ratios of the L and S atoms are almost the same. When D_SL/D_SS > 1.5, a structure with alternating atomic layers composed of the same type of atoms can be observed. This structure can be understood to be generated to increase the number of bonds between different types of atoms as much as possible. On the other hand, when D_SL/D_SS < 1, the two phases are separated to minimize the total length of heterophase boundaries, which increases the number of bonds between the different types of atoms.

Direct observation of the atomic configurations after cooling confirms that the material model fabrication method used in this study can produce binary system models with various microstructures depending on the atomic ratio of S atoms (f_S) and the binding energy of S and L atoms (D_SL).

Figure 2 shows the various mixed states of S and L atoms; however, it is not possible to determine the structure of each configuration by visual inspection alone. Therefore, using disorder variable D_j, the microstructure of each binary system model is classified into three types: crystalline phase, amorphous phase, and mixed crystalline/amorphous phase. Figures 3(a), (b), and (c) show the three binary models enclosed in squares in Fig. 2. Figures 3(d), (e), and (f) show the atomic configurations colored according to disorder variable D_j, and Figs. 3(g), (h), and (i) show the distributions of D_j. The phase shown in Fig. 3(a) has an irregular structure based on its atomic configuration. In addition, Fig. 3(g) shows that the D_j of the microstructure for this phase is distributed with a peak at D_j = 2. The phase in Fig. 3(c) has a crystalline structure based on its atomic configuration, and the distribution of D_j exhibits a monotonically decreasing trend from zero (Fig. 3(i)). In the phase shown in Fig. 3(b), the distribution of D_j exhibits two peaks (Fig. 3(h)). The microstructure of this phase includes a mixture of crystalline and amorphous phases, as observed from the atomic configuration colored according to D_j shown in Fig. 3(e). Based on these results, we set D_j = 1.5 as the threshold for the regularity of the atoms in this study, i.e., atoms with D_j < 1.5 have a regular structure with a hexagonal configuration, whereas atoms with D_j > 1.5 have an irregular structure.

Fig. 3

Atomic structure analysis to identify a phase state of the binary systems by the disorder valuable D_j. (a), (d), (g) f_S = 50%, D_SL/D_SS = 2, (b), (e), (h) f_S = 20%, D_SL/D_SS = 1.5, and (c), (f), (i) f_S = 50%, D_SL/D_SS = 0.75.

Figure 4 shows the ratio of atoms determined to have a regular structure, f_c, with D_j < 1.5. The continuous distribution of f_c in the figure is obtained by averaging and interpolating the values of f_c for each condition. It is confirmed that f_c has large values in the region where D_SL is small and f_S is below 20% or above 90%; f_c has small values in the region where D_SL is large and f_S is between 40% and 80%. In this study, models with f_c > 0.35 are classified as crystalline phases, whereas those with f_c < 0.1 are classified as amorphous phases; models with f_c between the 0.1 and 0.35 are classified as mixed crystalline/amorphous phases. The blue, yellow, and red circles in Fig. 4 represent crystalline, mixed, and amorphous phases, respectively. It can be observed that a mixed phase is present at the boundary between the crystalline and amorphous phases.

Fig. 4

Classification of the binary system into crystalline, mixed, and amorphous phases by the fraction of crystalline atom f_c identified by the disorder variable D_j.

Figure 5 shows the structural classifications visualized by coloring the atomic structures in Fig. 2 according to D_j. Although each atomic structure is a representation of part of the analytical model, it can be seen that the individual structures colored according to D_j correspond well with the three phases classified according to f_c. Thus, it is possible to create binary system models of various microstructures by changing the ratio of S atoms (f_S) and the binding energy between different atoms (D_SL). In addition, based on the ratio of crystalline atoms, f_S, we can classify the configurations into three types: a crystalline phase (the polycrystalline structure when f_S is below 20% or above 90% or the microstructures with separated crystalline phases of S and L atoms for the other values of f_S), an amorphous phase, and a crystalline/amorphous phase. In the next section, uniaxial tensile and compressive deformation analyses are performed on these binary system models to investigate the effect of the mixture of crystalline and amorphous materials on the strength.

Fig. 5

Classification of the binary systems into crystalline, amorphous, and mixed states based on the disorder variable D_j.

Uniaxial tensile and uniaxial compressive deformation analyses are performed on each binary system model. Figure 6 shows the stress–strain curves for a typical crystalline phase (f_S = 50%, D_SL/D_SS = 0.75), mixed phase (f_S = 20%, D_SL/D_SS = 1.25), and amorphous phase (f_S = 50%, D_SL/D_SS = 1.5) under tensile deformation. To investigate the relationship between the phases and the strength, the average stress in the strain interval of 0.2–0.45 is considered as the flow stress, σ_f, which is representative of the strength of each binary system model.

Fig. 6

Stress–strain curves for the binary system identified as crystalline (f_S = 50%, D_SL/D_SS = 0.75), mixed phases (f_S = 20%, D_SL/D_SS = 1.25), and (f_S = 50%, D_SL/D_SS = 1.5).

Figure 7 depicts the flow stress, σ_f, under tensile and compressive deformation as a function of f_S and D_SL. As the Young’s modulus, E, varies for each binary system model, the flow stress, σ_f, is normalized by the Young’s modulus. Here, E is calculated by linearly fitting the stress–strain curves for tensile and compressive deformation in the strain interval of 0–0.01 using the least-squares method. The continuous distribution of the flow stress in the figure is obtained by averaging and interpolating the flow stresses, σ_f, of each model. The blue, yellow, and red circles in Fig. 7 indicate binary system models in the crystalline, mixed, and amorphous phase, respectively. Under both tensile and compressive deformation, σ_f/E of the crystalline and amorphous phases is high, whereas that of the mixed crystalline/amorphous phase at the boundary is low. This confirms that the strengths of the binary system models with a mixture of crystalline and amorphous phases tend to be weaker than that of single-phase crystalline and single-phase amorphous models. A similar trend has previously been reported in a model using the Lennard–Jones potential,²⁸⁾ suggesting that the results of this study are a common property of two-dimensional binary system models.

Fig. 7

Flow stress map for the binary system related to the mixing rate f_S and biding energy of small and large atoms D_SL/D_SS under (a) tensile and (b) compressive deformations. Red, yellow, and blue circles represent the amorphous, mixed, and crystalline phases, respectively. The flow stress σ_f is normalized by the Young’s modulus E for each model. The deformation mechanisms for the states indicated by the symbols α, β, and γ are shown in Fig. 8.

Considering a simple mixture law for the strength, where σ_c is the strength of the crystalline phase and σ_a is the strength of the amorphous phase, the strength of the mixed crystalline/amorphous phase can be expressed as σ_cf_c + σ_a(1 − f_c); i.e., the mixed crystalline/amorphous phase should have a strength between σ_c and σ_a. However, the strength of the mixed crystalline/amorphous phase models is lower than σ_c and σ_a; hence, the simple mixture law does not hold. This suggests that a different deformation mechanism may be occurring in the mixed phase compared to the deformation that occurs in the crystalline and amorphous phases. Therefore, we investigate the deformation mechanism of each phase in the next section.

Figure 8 shows the atomic strain, ε_vM,²⁹⁾ under compressive deformation of the amorphous phase (α), mixed phase (β), and crystalline phase (γ) circled in black in Fig. 7. Atoms with higher atomic strains, ε_vM, have a different configuration of neighboring atoms compared to the initial state, and the region where these atoms exist corresponds to the location where plastic deformation occurs. Phases α, β, and γ have the same binding energy between different atoms, D_SL, and differ only in the fraction of S atoms, f_S. In the amorphous model shown in Fig. 8(a), areas with large atomic strain occur in the amorphous region in the maximum shear stress direction (45° diagonally), forming a band-like pattern, as shown in the enlarged figure in the square; this pattern resembles a shear band. This pattern is due to the cooperative occurrence of local atomistic rearrangement, which is the main deformation mechanism of metallic glasses. In the crystal phase model shown in Fig. 8(c), large atomic strains are observed in only two layers along the atomic planes of the crystal region, as shown in the enlarged image in the square, indicating the occurrence of dislocation motion. In the mixed phase, β, between the amorphous, α, and crystalline, γ, phases shown in Fig. 8(b), the atomic strain patterns observed in the α and γ phases appear to exist simultaneously. For a more detailed view, an enlarged image of the section in the square is presented, where the gray and white atoms belong to the amorphous and crystalline phases, respectively, and the red squares indicate atoms with an atomic strain ε_vM > 0.15. The results show both sliding deformation due to dislocation and local atomistic rearrangement, as well as the linkage of the two deformation modes. Although there are heterophase interfaces between the crystalline and amorphous phases, there is no significant atomic strain in these regions, indicating that a phenomenon wherein the interface itself produces plastic strain, such as interface slip, does not occur. Thus, the deformation mechanism of mixed phase β is a combination of the deformation mechanisms occurring in the amorphous phase, α, and crystalline phase, γ. A possible reason for the weakening of the strength of the mixed phase is the effect of the heterophase interface between the crystalline and amorphous phases in the elementary deformation processes. Therefore, in the next section, we focus on the plasticity phenomenon produced by the heterophase interface.

Fig. 8

Atomic strain distributions to investigate the deformation mechanism of each phase shown in Fig. 7. (a) Amorphous phase: α (f_S = 60%, D_SL/D_SS = 1.5), (b) mixed phase: β (f_S = 30%, D_SL/D_SS = 1.5), and (c) crystalline phase: γ (f_S = 10%, D_SL/D_SS = 1.5). Characteristic plastic deformation modes of local atomistic rearrangement in shear transformation zone for amorphous structure and dislocation for crystal structure are shown in the enlarged pictures.

To investigate the cause of the weakening of the crystalline/amorphous mixed phase, we create a simple layered model with crystalline and amorphous phases, as shown in Fig. 9, and perform compressive deformation analysis. For the amorphous phase, we use a model with f_S = 50% and D_SL/D_SS = 2. For the crystalline phase, we use a single phase of L atoms. Periodic boundary conditions are applied in all directions. The analytical conditions for deformation are the same as those applied in Section 2.2.

Fig. 9

The lamellar model of crystalline and amorphous phases to investigate the effect of the force chain in the amorphous phase on the dislocation emission from the interface under compressive deformation. The crystalline region consists of the large atoms, while the amorphous region consists of the large and small atoms with f_S = 50% and D_SL/D_SS = 2. The detailed force chain in the squire box is shown in Fig. 10.

Figure 10 shows the atomic structure of the square region in Fig. 9, which includes the heterointerface; the upper part is the amorphous phase, and the lower part is the crystalline phase. The compressive forces between atoms are shown in red. Figure 10(a) indicates that there is no compressive force exceeding −1 nN in the crystalline region, and the force field is homogeneous. In contrast, in the amorphous region, large compressive forces are observed in only a few atomic bonds, indicating a nonuniform force field. Furthermore, the atomic bonds with large compressive force are connected in the direction of the load and may function as a column, generating resistance to external compressive loads. Such a load-bearing structure formed by the connection of local forces is called a force chain; this type of structure has attracted attention as a unique mechanical field of irregular structures, mainly in the analysis of the deformation and flow characteristics of powders and colloidal systems.^21,22) Although force chains are observed in systems composed of direct contact interactions, such as powders, their existence has also recently been reported in atomic systems composed of non-direct contact interactions.³⁰⁾

Fig. 10

Force chain development around the amorphous/crystalline interface shown in Fig. 9 under compressive deformation.

The role of force chains in compressive deformation can be clarified by observing the atomic-scale dynamics. As shown in Fig. 10, the force chain repeatedly collapses and forms again during the process of compressive deformation. As shown in Fig. 10(b), when the end of the force chain in the amorphous phase reaches the crystalline phase, dislocations are released from the heterointerface at the end toward the crystalline phase, as shown in Figs. 10(c) and (d); i.e., the force chain developed in the amorphous phase may promote dislocation release from the heterointerface. Although further analysis of the relationship between the force chain and dislocation release is necessary, a heterogeneous distribution of the compressive load is considered to promote dislocation release. The atoms at the end of the force chain in the heterointerface are subjected to strong compressive loading, thus forming strong stress concentrations in their vicinity and leading to the formation and release of dislocations. This phenomenon can be one reason for the lower strength of the mixed crystalline/amorphous phase compared to that of the crystalline and amorphous single phases.

Dislocation emission from grain boundaries between crystalline phases is generally strongly influenced by interfacial dislocations. If the dislocation to be released can utilize the Burgers vector of the interface dislocation, dislocation release from the interface can be easily achieved.^31–33) This implies that the design of interface structures is important. On the other hand, the force chain generated within the amorphous region strongly influences dislocation emission from the crystalline–amorphous interface, as observed in this study. Because force chains are force linkages between atoms, their magnitude changes with external loading, and slight changes in atomic structure will result in branching, collapse, and recombination of force chains. The length of a force chain from branching point to branching point and the density of the force chain may change due to the species of elements constituting the amorphous phase, their composition ratio, and the free volume distribution, which may affect the magnitude of stress concentration near the interface. This may be a crucial factor in the strength design of materials with a mixture of crystalline and amorphous phases, which needs to be investigated in the future.

As two-dimensional analysis is applied in this study, dislocations appear as zero-dimensional point-like structures, whereas force chains appear as one-dimensional linear structures. Therefore, the force field of the force chain acts on the entire dislocation line for the dislocation release from the interface, as shown in Fig. 10. However, in an actual three-dimensional system, both the dislocations and the force chain have linear morphologies, and when dislocations are released from the interface by the force field of the force chain, they exhibit orthogonal positioning. This implies that the force field of the force chain acts only on a part of the dislocation line. Therefore, the role of the force chain observed in the amorphous phase in this study may appear more pronounced than in an actual three-dimensional system. The effect of the force chain on the plasticity through the interface in a three-dimensional structure requires further investigation.

To investigate the synergistic effect of different deformation modes (dislocations and local atomistic rearrangements) on the strength of mixed crystalline and amorphous materials, we performed tensile and compressive deformation analyses of two-dimensional binary models with various microstructures through molecular dynamics simulations. Binary system models were constructed using atoms with two different atomic radii. By varying the mixing ratio and interaction force between different atoms, 66 binary models with various structures were produced, and each model was classified as crystalline, amorphous, or mixed crystalline/amorphous based on structural analysis. Deformation analysis showed that the strength of the mixed crystalline/amorphous model tended to be weaker than that of the crystalline and amorphous models. This was due to the development of a force chain in the amorphous phase, as observed in powder systems, and easy dislocation emission to the crystalline phase when the force chain reaches the crystalline/amorphous interface.

This study was supported by JSPS Grants-in-Aid for Scientific Research JP16K17764, JP17K06049, JP18H05453, JP20K03783, and JST CREST (JPMJCR1994).

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