^{1}, Jens Eggers

^{2}, C. Patrick Royall

^{3,a)}, Stephen R. Williams

^{4}and Hajime Tanaka

^{5,b)}

### Abstract

We study the relationship between local structural ordering and dynamical heterogeneities in a model glass-forming liquid, the Wahnström mixture. A novel cluster-based approach is used to detect local energy minimum polyhedral clusters and local crystalline environments. A structure-specific time correlation function is then devised to determine their temporal stability. For our system, the lifetime correlation function for icosahedral clusters decays far slower than for those of similarly sized but topologically distinct clusters. Upon cooling, the icosahedra form domains of increasing size and their lifetime increases with the size of the domains. Furthermore, these long-lived domains lower the mobility of neighboring particles. These structured domains show correlations with the slow regions of the dynamical heterogeneities that form on cooling towards the glass transition. Although icosahedral clusters with a particular composition and arrangement of large and small particles are structural elements of the crystal, we find that most icosahedral clusters lack such order in composition and arrangement and thus local crystalline ordering makes only a limited contribution to this process. Finally, we characterize the spatial correlation of the domains of icosahedra by two structural correlation lengths and compare them with the four-point dynamic correlation length. All the length scales increase upon cooling, but in different ways.

We thank M. Leocmach for his kind help in bond orientational order analysis and its comparison with TCC analysis. A.M. is funded by Engineering and Physical Sciences Research Council (EPSRC(GB)) Grant No. EP/E501214/1. C.P.R. thanks the Royal Society for funding. H.T. acknowledges support from a grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan and the Aihara Project, the FIRST program from JSPS, initiated by CSTP. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol.

I. INTRODUCTION

II. METHODOLOGY

A. Model and simulation details

B. The topological cluster classification method

III. RESULTS AND DISCUSSION

A. Overall dynamics

B. Topological cluster classification analysis

C. Network of icosahedra

D. Relationship of icosahedral domains to crystalline order

E. Comparison of structural and dynamic length scales

IV. SUMMARY

### Key Topics

- Crystal structure
- 57.0
- Cluster dynamics
- 16.0
- Liquid crystals
- 12.0
- Correlation functions
- 10.0
- Liquid crystal structure
- 8.0

## Figures

The self-intermediate scattering function .

The self-intermediate scattering function .

The 13A icosahedral cluster and clusters that have icosahedral-like structure. Clusters 11E and 12D occur at α-interlocking icosahedral sites, ^{ 13 } the rest are contained within a single icosahedron. Clusters in dotted rectangles are ground state clusters of *n* particles for the Wahnström mixture. ^{ 37 } The colors of the particles and the bonds highlight the detection methods in the TCC algorithm. Grey particles are part of 3, 4, or 5 membered shortest path rings. These rings are shown by the white, blue, pink, and green colored bonds. Yellow particles are spindle particles for the shortest path rings which together form the basic clusters. Red particles are additional particles bonded to smaller clusters forming a new cluster. The detection routines for clusters are described in Ref. ^{ 38 } .

The 13A icosahedral cluster and clusters that have icosahedral-like structure. Clusters 11E and 12D occur at α-interlocking icosahedral sites, ^{ 13 } the rest are contained within a single icosahedron. Clusters in dotted rectangles are ground state clusters of *n* particles for the Wahnström mixture. ^{ 37 } The colors of the particles and the bonds highlight the detection methods in the TCC algorithm. Grey particles are part of 3, 4, or 5 membered shortest path rings. These rings are shown by the white, blue, pink, and green colored bonds. Yellow particles are spindle particles for the shortest path rings which together form the basic clusters. Red particles are additional particles bonded to smaller clusters forming a new cluster. The detection routines for clusters are described in Ref. ^{ 38 } .

Clusters with topology distinct from icosahedral order. Clusters highlighted with dotted rectangles are ground state clusters of *n* particles for the Wahnström mixture. ^{ 37 } Colors are the same as in Fig. 2 . Detection routines for 10W and 11W clusters are described in Sec. II .

(a) Non-Arrhenius increase in structural relaxation time versus the inverse of temperature. The data are fitted with an Arrhenius exponential at high temperatures and VFT at low temperatures. The onset temperature of the super-Arrhenius behavior is *T** = 1.48 (vertical dashed line in both (a) and (b)). (b) The fraction of particles detected within icosahedral 13A clusters of all lifetimes *N* _{13A}(τ_{ℓ} ⩾ 0)/*N*. Note the fast increase in *N* _{13A}(τ_{ℓ} ⩾ 0)/*N* starts around *T**.

(a) Non-Arrhenius increase in structural relaxation time versus the inverse of temperature. The data are fitted with an Arrhenius exponential at high temperatures and VFT at low temperatures. The onset temperature of the super-Arrhenius behavior is *T** = 1.48 (vertical dashed line in both (a) and (b)). (b) The fraction of particles detected within icosahedral 13A clusters of all lifetimes *N* _{13A}(τ_{ℓ} ⩾ 0)/*N*. Note the fast increase in *N* _{13A}(τ_{ℓ} ⩾ 0)/*N* starts around *T**.

(a) Cluster lifetime correlation functions P(τ_{ℓ} > *t*) for icosahedral (denoted 13A—solid line), octahedral (6A—dashed line), and HCP_{13} (dotted line) at *T* = 0.604. All clusters with structure distinct from icosahedral order fall in the grey shaded region between P(τ_{ℓ} > *t*) for 6A and HCP_{13}. (b) The fraction of particles detected within icosahedral clusters with lifetime τ_{ℓ} > *t* (*T* = 0.604). (c) The mean-square displacement ⟨δ*r* ^{2}(*t*)⟩ of all particles (black line) and particles starting within icosahedral clusters with lifetime τ_{ℓ} > *t* as specified in the legend (colored lines—*T* = 0.604).

(a) Cluster lifetime correlation functions P(τ_{ℓ} > *t*) for icosahedral (denoted 13A—solid line), octahedral (6A—dashed line), and HCP_{13} (dotted line) at *T* = 0.604. All clusters with structure distinct from icosahedral order fall in the grey shaded region between P(τ_{ℓ} > *t*) for 6A and HCP_{13}. (b) The fraction of particles detected within icosahedral clusters with lifetime τ_{ℓ} > *t* (*T* = 0.604). (c) The mean-square displacement ⟨δ*r* ^{2}(*t*)⟩ of all particles (black line) and particles starting within icosahedral clusters with lifetime τ_{ℓ} > *t* as specified in the legend (colored lines—*T* = 0.604).

(a)–(c) Domains of icosahedral clusters form on cooling from high to low temperature (slices through 3D simulation box). Particles in icosahedral clusters are shown full size in green, other particles are blue dots. (d) The radius of gyration *R* _{ g } of the domains versus the number of particles in the domain *m* for *T* = 0.620. *R* _{ g } is well fitted by *m* ^{0.47} indicating the domains have a fractal dimension *d* _{ f } ≃ 2. (e) The mean lifetime of icosahedral clusters versus the domain size *m*. (f) Icosahedral domains retard the motion of neighboring particles. The MSD ⟨δ*r* ^{2}(τ_{ h })⟩ as a function of distance from icosahedral domains *d* (solid line). The dotted line is the MSD over τ_{ h } of all particles not in icosahedra, independent of *d*. Note that the particles located around *d* = 1 amount to 40% of the system at this temperature.

(a)–(c) Domains of icosahedral clusters form on cooling from high to low temperature (slices through 3D simulation box). Particles in icosahedral clusters are shown full size in green, other particles are blue dots. (d) The radius of gyration *R* _{ g } of the domains versus the number of particles in the domain *m* for *T* = 0.620. *R* _{ g } is well fitted by *m* ^{0.47} indicating the domains have a fractal dimension *d* _{ f } ≃ 2. (e) The mean lifetime of icosahedral clusters versus the domain size *m*. (f) Icosahedral domains retard the motion of neighboring particles. The MSD ⟨δ*r* ^{2}(τ_{ h })⟩ as a function of distance from icosahedral domains *d* (solid line). The dotted line is the MSD over τ_{ h } of all particles not in icosahedra, independent of *d*. Note that the particles located around *d* = 1 amount to 40% of the system at this temperature.

(a) The relationship between icosahedra and spatial heterogeneities in the dynamics at *T* = 0.604 (2D slice of simulation box). Mobility (⟨δ*r* ^{2}(τ_{ h })⟩) is depicted by the size of the particle. Large particles have ⟨δ*r* ^{2}(τ_{ h })⟩ < 0.043, small particles otherwise. Particles always within icosahedral clusters with lifetime are colored green, and others are grey. (b) Growing structural ξ_{ Rg } (squares) and ξ_{ S13A} (crosses), and dynamical ξ_{4} (circles) correlation lengths.

(a) The relationship between icosahedra and spatial heterogeneities in the dynamics at *T* = 0.604 (2D slice of simulation box). Mobility (⟨δ*r* ^{2}(τ_{ h })⟩) is depicted by the size of the particle. Large particles have ⟨δ*r* ^{2}(τ_{ h })⟩ < 0.043, small particles otherwise. Particles always within icosahedral clusters with lifetime are colored green, and others are grey. (b) Growing structural ξ_{ Rg } (squares) and ξ_{ S13A} (crosses), and dynamical ξ_{4} (circles) correlation lengths.

(a) Relationship between icosahedral 13A domains and the crystalline “Frank-Kasper,” and icosahedral clusters for *T* = 0.604. The quantity *N* _{c}(τ_{ℓ} > *t*)/*N* is the fraction of particles detected within clusters lifetime τ_{ℓ} > *t*. The dashed line is for Frank-Kasper clusters, and the dotted line is for icosahedral clusters with *B*-particles in the center, and arrangements of *A*- and *B*-species in the shell of the icosahedron compatible with the bulk crystal phase. ^{ 17 } Both of these clusters exist in the MgZn_{2} structure of the Wahnström crystal. The solid line is for all icosahedra, irrespective of the arrangement of *A*- and *B*-species. (b) Snapshot at *T* = 0.604 showing the overlap between icosahedral domains and Frank-Kasper clusters. Green particles are the icosahedral domains, red particles are those in both Frank-Kasper and icosahedral clusters, and black are Frank-Kasper particles *not* in icosahedral domains. The small blue particles are not members of icosahedral or Frank-Kasper clusters.

(a) Relationship between icosahedral 13A domains and the crystalline “Frank-Kasper,” and icosahedral clusters for *T* = 0.604. The quantity *N* _{c}(τ_{ℓ} > *t*)/*N* is the fraction of particles detected within clusters lifetime τ_{ℓ} > *t*. The dashed line is for Frank-Kasper clusters, and the dotted line is for icosahedral clusters with *B*-particles in the center, and arrangements of *A*- and *B*-species in the shell of the icosahedron compatible with the bulk crystal phase. ^{ 17 } Both of these clusters exist in the MgZn_{2} structure of the Wahnström crystal. The solid line is for all icosahedra, irrespective of the arrangement of *A*- and *B*-species. (b) Snapshot at *T* = 0.604 showing the overlap between icosahedral domains and Frank-Kasper clusters. Green particles are the icosahedral domains, red particles are those in both Frank-Kasper and icosahedral clusters, and black are Frank-Kasper particles *not* in icosahedral domains. The small blue particles are not members of icosahedral or Frank-Kasper clusters.

(a) The four-point dynamic susceptibility χ_{4}(*t*). The maximum of χ_{4}(*t*) occurs at time τ_{ h }. (b) The four-point dynamic structure factor *S* _{4}(*k*, τ_{ h }). The solid lines are fits to the data with the Ornstein-Zernike equation (Eq. (4) ). (c) The structure factor of the icosahedral particles *S* _{13A }(*k*). The solid lines are fits to the data with the OZ function.

(a) The four-point dynamic susceptibility χ_{4}(*t*). The maximum of χ_{4}(*t*) occurs at time τ_{ h }. (b) The four-point dynamic structure factor *S* _{4}(*k*, τ_{ h }). The solid lines are fits to the data with the Ornstein-Zernike equation (Eq. (4) ). (c) The structure factor of the icosahedral particles *S* _{13A }(*k*). The solid lines are fits to the data with the OZ function.

## Tables

The ground state clusters of the Wahnström mixture in TCC notation for *n* = 5–13 particles containing *n* _{ A } *A*-species particles. The energy *E* is the binding energy of the cluster. Clusters highlighted in **bold face** are the lowest energy states for a given value of *n*. The final column is the cluster detected by the Topological Cluster Classification algorithm from the ground state configuration. Note that for the Wahnström mixture the ground state clusters with *n* _{ A } = 0 and *n* _{ A } = *n* are identical for each *n* as the *AA* and *BB* Lennard-Jones interactions are the same. The *n* _{ A } = 0 clusters have therefore been omitted from the table for brevity.

The ground state clusters of the Wahnström mixture in TCC notation for *n* = 5–13 particles containing *n* _{ A } *A*-species particles. The energy *E* is the binding energy of the cluster. Clusters highlighted in **bold face** are the lowest energy states for a given value of *n*. The final column is the cluster detected by the Topological Cluster Classification algorithm from the ground state configuration. Note that for the Wahnström mixture the ground state clusters with *n* _{ A } = 0 and *n* _{ A } = *n* are identical for each *n* as the *AA* and *BB* Lennard-Jones interactions are the same. The *n* _{ A } = 0 clusters have therefore been omitted from the table for brevity.

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