# Hierarchical structures of amorphous solids characterized by persistent homology

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Edited by Giorgio Parisi, University of Rome, Rome, Italy, and approved April 22, 2016 (received for review October 22, 2015)

## Significance

Persistent homology is an emerging mathematical concept for characterizing shapes of data. In particular, it provides a tool called the persistence diagram that extracts multiscale topological features such as rings and cavities embedded in atomic configurations. This article presents a unified method using persistence diagrams for studying the geometry of atomic configurations in amorphous solids. The method highlights hierarchical structures that conventional techniques could not have treated appropriately.

## Abstract

This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.

- amorphous solid
- hierarchical structure
- persistent homology
- persistence diagram
- topological data analysis

The atomic configurations of amorphous solids are difficult to characterize. Because they have no periodicity as found in crystalline solids, only local structures have been analyzed in detail. Although short-range order (SRO) defined by the nearest neighbor is thoroughly studied, it is not sufficient to fully understand the atomic structures of amorphous solids. Therefore, medium-range order (MRO) has been discussed to properly characterize amorphous solids (1⇓–3). Many experimental and simulation studies (4⇓⇓–7) have suggested signatures of MRO such as a first sharp diffraction peak (FSDP) in the structure factor of the continuous random network structure, and a split second peak in the radial distribution function of the random packing structure. However, in contrast to SRO, the geometric interpretation of MRO and the hierarchical structures among different ranges are not yet clear.

Among the available methods, the distributions of bond angle and dihedral angle are often used to identify the geometry beyond the scale of SRO. They cannot, however, provide a complete description of MRO because they only deal with the atomic configuration up to the third nearest neighbors. Alternatively, ring statistics are also applied as a conventional combinatorial topological method (2, 8, 9). However, this method is applicable only for the continuous random network or crystalline structures, and furthermore it cannot describe length scale. Therefore, methodologies that precisely characterize hierarchical structures beyond SRO and are applicable to a wide variety of amorphous solids are highly desired.

In recent years, topological data analysis (10, 11) has rapidly grown and has provided several tools for studying multiscale data arising in physical and biological fields (11⇓⇓⇓⇓–16). A particularly important tool in the topological data analysis is persistence diagram (PD), a visualization of persistent homology as a 2D histogram (e.g., see Fig. 2). The input to the PD is given by an atomic configuration with scale parameters, and the output consists of various multiscale information about topological features such as rings and cavities embedded in the atomic configuration. Here, the atomic configurations are generated by molecular dynamics simulations in this article. Importantly, in contrast to other topological tools, PDs not only count topological features but also provide the scales of these features. Hence, PDs can be used to classify topological features by their scales and clarify geometric relationships among them; this is presumably the most desired function for deeper analysis of amorphous structures.

This article proposes a method using PDs for various amorphous solids in a unified framework. The method is applied to atomic configurations and enables one to study hierarchical geometry embedded in amorphous structures that cannot be treated by conventional methods. We first applied the method to silica glass as an example of the continuous random network structure and obtained the following results. (*i*) We found three characteristic curves in the PD of silica glass. These curves classify the SRO rings in the *ii*) The PD reproduced the wavelength of the FSDP and clarified a real space origin of the FSDP. (*iii*) Each curve in the PD represents a geometric constraint on the ring shapes and, as an example, an MRO constraint on rings consisting of three oxygen atoms was explicitly derived as a surface in a parameter space of the triangles. Moreover, we verified that these curves are preserved under strain, indicating that the PD properly encodes the material property of elastic response. Next, as examples of the random packing structure, the Lennard-Jones (LJ) system and Cu-Zr metallic glass were studied by the PDs, and we clarified the following. (*iv*) These amorphous solids were also characterized well by the distributions of curves and islands in the PDs. (*v*) In the LJ system, the global connectivity of dense packing regions was revealed by dualizing octahedral arrangements. (*vi*) In Cu-Zr alloys, we found that the pair-distribution function defined by the octahedral region in the PD shows the split second peak. Furthermore, a relationship between the hierarchical ring structure and high glass-forming ability was discovered in Cu-Zr alloys.

## PDs of Atomic Configuration

The input to PDs is a pair *i*th atom, respectively. To characterize the multiscale properties in *Q*, we introduce a parameter *α*, which controls resolution, and generate a family of atomic balls *α* and detect rings and cavities at each *α*, where

Let *α*. To be more precise, a ring [respectively (resp.) cavity] here means a generator of the homology

In this article, our basic strategy is that we transform a complicated atomic configuration into PDs and try to identify meaningful shape information from specific distributions such as curves or islands in the PDs. Namely, we reconstruct characteristic atomic subsets from each distribution. To this aim, we compute the optimal cycle for each point

For a mathematically rigorous introduction of these concepts see *Supporting Information* or refs. 10 and 11. In this article, the PDs are computed by CGAL (20) and PHAT (21).

## PDs for Continuous Random Network Structure

Fig. 2 shows the PDs *R* are set to be *Supporting Information*.

We discovered that the PDs in Fig. 2 distinguish these three states. The liquid, amorphous, and crystalline states are characterized by planar (2-dim), curvilinear (1-dim), and island (0-dim) regions of the distributions, respectively. Here, the 0 and 2 dimensionality of the PDs result from the periodic and random atomic configurations of the crystalline and liquid states, respectively. Furthermore, we emphasize that the presence of the curves in

As shown in Fig. 2, *Supporting Information*). Through further analysis of the persistent homology using optimal cycles, we found the following three geometric characterizations. (*i*) The rings on *α*, each ring on *ii*) The rings on *iii*) The rings on

## Decomposition of FSDP

The FSDP observed in the structure factor *q* values of the FSDP fairly well. Moreover, we classified the MRO rings as a real space origin of the FSDP.

We first note that the death scales of the rings on *α* is the parameter controlling the radius *i*th atomic ball

From the aforementioned argument, we define a distribution*δ* is the Dirac delta function, which is used to count the contribution of each MRO ring in *q*-space.

Fig. 3 shows the plots of *q* values of the FSDP and the peak of *q* values and, hence, the rings in *q* values, respectively. It should be emphasized that the

## Curves and Shape Constraints

The presence of curves

For example, the shape of a ring on *Right*), is determined by specifying the first and second minimum edge lengths *θ* between them, and hence is realized in a 3D parameter space. Then, the constraint for

## Response Under Strain

The presence of the curves also indicates variations in the shapes of the rings. That is, by following each curve along its tangential direction and studying its rings, we can observe the deformation of the rings. It is reasonable to suppose that these variations are due to thermal fluctuations, and hence the deformations of the ring configurations along the curves are probably softer than those in the normal directions. Consequently, the response under strain is expected to follow the same shape constraints.

To verify this mechanical response of PDs, we performed simulations of isotropic compression for our amorphous state and computed the PD of the strained state. The strain is set to be

Fig. 5 shows that the contours of the strained state shift along the original curves

We have revealed that PDs encode information about elastic response, similar to how the radial distribution function encodes volume compressibility (23). It should also be emphasized that the curves or islands appear in the PDs of only the solids. This evidence suggests that these isolated distributions are related to the rigidity of the materials. This hypothesis follows from the fact that isolations represent geometric constrains reflecting mechanical responses. Future numerical and theoretical studies to unravel this relationship would be of great value.

## PDs for Random Packing Structures

We next study the geometry of amorphous states close to random packing structures. In this case, we found that both *R* are set to be zero, because changing *r* only causes translations of the PDs for the single component system. The details of the simulation are explained in *Supporting Information*.

Similar to the case of silica, the crystalline structure is characterized by the island distributions in the PDs (top panels in Fig. 6). They correspond to the regular triangles in

In random packings, it is known that the atomic configuration can be divided into dense packing regions built from tetrahedra and the complement that patches those regions together (25). In particular, the network structure of the dense packing regions characterizes the global connectivity beyond MRO, which has not yet been investigated in detail. Note that *Left*). Here, we set

Fig. 7, *Right* shows

As an example of multicomponent systems, we also studied metallic glasses composed of Cu and Zr (26), in particular, focusing on *Top*). These values are obtained by the same procedure as for the silica. Even in the multicomponent system, the PDs basically show similar behaviors to those of the LJ system. Specifically, the island distribution corresponding to octahedra appears in

In the random packing structure, the split second peak of the radial distribution function has been supposed to be a signature of MRO (27). The shaded region of the radial distribution function in Fig. 9, *Bottom* shows the split second peak of *Top*) in the same length scale. Here, *B* is chosen to be a region around the octahedral distribution. Meanwhile the pair-distance distribution of generators other than *B* shows a slight change there (pink line in Fig. 9). This means that the generators around the octahedral distribution play a significant role for the split second peak. Therefore, similar to the FSDP in the silica, this result demonstrates that the PDs classify the length scale of MRO from other scales.

We also studied the PDs of only the Zr component. The PD *Middle Left*) represents the existence of a hierarchical MRO structure similar to that of the silica in Fig. 2, whereas the PD *Bottom Left*) does not show any hierarchical curves. An example of the hierarchical rings in

We here remark that

## Conclusion

We have presented that PD is a powerful tool for geometric characterizations of various amorphous solids in the short, medium, and even further ranges. In this work, we have addressed two different types of amorphous systems: continuous random network and random packing structures. Both types of amorphous systems are characterized well by the existence of the curve and island distributions in the PDs. These specific distributions characterize the shapes of rings and cavities in multiranges, and the analysis using optimal cycles explicitly captures hierarchical structures of these shapes. We have shown that these shape characteristics successfully reproduce the FSDP for the continuous random network and the split second peak for the random packing and provide further geometric insights to them. Furthermore, the global connectivity of dense packing regions in the LJ system is revealed by the iterative application of the PD. For the binary random packing of the Cu-Zr metallic glass, we have also shown that the presence of the hierarchical MRO rings in the single component suggests the relationship with the glass-forming ability.

The methodology presented here can be applied to a wide variety of disordered systems and enables one to survey the geometric features and constraints in seemingly random configurations. Furthermore, because we investigated the mechanical response of the PDs, studying dynamical properties of materials using the PD method would be of great importance to understand the relationship between hierarchical structures and mechanical properties. We believe that further developments and applications of topological data analysis will accelerate the understanding of amorphous solids.

## Computational Homology

Here we summarize the computational homological tools used in this article. For further mathematical and computational details, please refer to ref. 10.

Given a pair *Q*, where *α*, we assign a ball *i*th atom. Then, the alpha shape *Q* such that an edge

One of the important properties of the alpha shape is that the atomic ball model *α* controls the resolution from fine (

Fig. S2, *Left* shows a schematic illustration of atomic ball models

The persistence diagram is a 2D histogram recording the birth and death scales of the topological features in a one-parameter family *c* be the yellow ring in Fig. S2. We observe that *c* first appears at *b* and *d*, respectively, of the ring *c* are determined by *Right* shows the persistence diagram of the left. From the definition of the persistence diagram, the birth scale indicates the maximum distance between adjacent atoms in the ring, whereas the death scale indicates the size of the ring. It should also be mentioned that persistence diagrams

## MD Simulation for the Silica System

The atomic configurations of silica in a liquid

Starting from the beta-cristobalite structure, the crystalline configuration

The liquid configuration

The amorphous configuration

The set *i*th atom (

The glass transition temperature *Left*) and *Right*), and the curve

## MD Simulation for the LJ System and the Cu-Zr Alloy

The LJ system is composed of 4,000 monatomic particles interacting via LJ potential *ε*, *σ*, and mass equal to unity. The time step is set to be 0.005 in LJ units.

Starting from the FCC configuration, *Left* the enthalpy normalized by the number of particles is described as a function of *T*. As is observed, there is no glass transition in the LJ system. Therefore, we use an inherent structure for the liquid state as *Left*).

For the Cu-Zr alloy system, MD simulations have been performed using the embedded-atom method potential (34). The masses of Cu and Zr atoms were set to be 63.54 and 91.22 g/mol, respectively. The number of particles was 16,000, in which there were 8,000 of both copper and zirconium atoms for

In this case, we can observe the glass transition (Fig. S5, *Right*) during a cooling simulation with the quench rate

All MD simulations in this article were performed using LAMMPS (35).

## Acknowledgments

We thank Mingwei Chen, Hajime Tanaka, Masakazu Matsumoto, and Daniel Miles Packwood for valuable discussions and comments. This work was sponsored by World Premier International Research Center Initiative, Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. This work was also supported by Japan Science and Technology Agency (JST) CREST Mathematics Grant 15656429 (to Y.H.), Structural Materials for Innovation Strategic Innovation Promotion Program D72 (Y.H., A.H., and Y.N.), MEXT Coop with Math Program (K.M.), Japan Society for the Promotion of Science (JSPS) Grant 26310205 (to Y.N.), and JSPS Grant 15K13530 and JST PRESTO (to T.N.).

## Footnotes

↵

^{1}Y.H. and T.N. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: hiraoka{at}wpi-aimr.tohoku.ac.jp.

Author contributions: Y.H., T.N., A.H., K.M., and Y.N. designed research; Y.H., T.N., A.H., E.G.E., K.M., and Y.N. performed research; Y.H. contributed new reagents/analytic tools; Y.H., T.N., and E.G.E. analyzed data; and Y.H., T.N., A.H., and E.G.E. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1520877113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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