A great success of solid state physics comes from the characterization of crystal structures in the reciprocal (wave vector) space. The power of structural characterization in Fourier space originates from the breakdown of translational and rotational symmetries. However, unlike crystals, liquids and amorphous solids possess continuous translational and rotational symmetries on a macroscopic scale, which makes Fourier space analysis much less effective. Lately, several studies have revealed local breakdown of translational and rotational symmetries even for liquids and glasses. Here, we review several mathematical methods used to characterize local structural features of apparently disordered liquids and glasses in real space. We distinguish two types of local ordering in liquids and glasses: energy-driven and entropy-driven. The former, which is favoured energetically by symmetry-selective directional bonding, is responsible for anomalous behaviours commonly observed in water-type liquids such as water, silicon, germanium and silica. The latter, which is often favoured entropically, shows connections with the heterogeneous, slow dynamics found in hard-sphere-like glass-forming liquids. We also discuss the relationship between such local ordering and crystalline structures and its impact on glass-forming ability.
Liquids and glasses possess neither long-range translational nor orientational order but still tend to form local structural order, which is not easily accessed by conventional scattering (wave vector) analysis.
Local structural analysis in real space — based on modern mathematical descriptions of geometrical structures and topologies — is a powerful tool to characterize the structure and elucidate its link to the physical properties of liquids.
Local structural ordering lowers the free energy locally and can be driven energetically and entropically.
Energetically driven ordering — which is commonly seen in liquids with specific directional bonding, such as water and silica — usually leads to the formation of spatially localized structures with translational order.
Entropically driven ordering — which is often seen in liquids interacting with isotropic potentials, such as hard spheres — usually leads to the formation of spatially extended structures with orientational order.
The understanding of local structural ordering in liquids brings new physical insights into the role of many-body correlations in many mysterious phenomena of fundamental importance in liquids such as the anomalies of water, liquid–liquid transition, glass transition and crystallization.
The knowledge of local orders in liquids provides invaluable information on the interpretation of the physical quantities experimentally accessed by scattering and spectroscopic measurements.
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This study was partially supported by Grants-in-Aid for Specially Promoted Research (grant no. JP25000002) and Scientific Research (A) (18H03675) from the Japan Society for the Promotion of Science (JSPS). J.R. acknowledges support from the European Research Council Grant DLV-759187 and the Royal Society University Research Fellowship.
- Structure factor
The two-point correlation function of the Fourier component of the microscopic density.
- Radial distribution function
The two-point correlation function of variation of density as a function of distance from a reference particle.
- Directional bondings
The bondings between atoms with preferred symmetry and directions.
- Second coordination shell
The spherical shell formed by neighbours at a distance between the first and second minima of the radial distribution function from the central molecule, ion or atom.
- Rosenfeld relation
The scaling relation between diffusion constant and excess entropy presented by Yaakov Rosenfeld in 1977.
In the description of the melting of 2D crystals, disclinations are elemental topological defects (particles with five or seven neighbours).
Topological defects that consist of pairs of disclinations.
- Kauzmann temperature
The temperature at which the extrapolated entropy of the glass becomes smaller than that of the crystal and therefore an ideal glass transition is hypothesized; originated by Walter Kauzmann in 1948.
Refers to the relation that a quantity scales proportional to the system size.
Refers to the relation that a quantity scales slower than the system size.
- Isoconfigurational ensemble
An ensemble of trajectories that are run from an identical configuration of particles with random initial momenta sampled from the equilibrium Boltzmann distribution.
- Dynamic propensity
The mean displacement of individual particles averaged over the isoconfigurational ensemble at the structural relaxation time, describing the tendency of particles’ movement.