# Revealing the three-dimensional structure of liquids using four-point correlation functions

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Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved May 4, 2020 (received for review March 25, 2020)

## Significance

The structure of disordered systems like liquids, gels, granular materials, etc. is considered to be isotropic, and hence very few studies exist that have investigated the three-dimensional structure of such systems. Here, we introduce a method that allows characterization of this structure. Considering the examples of a hard sphere-like liquid and of silica, an open network liquid, we show that the local three-dimensional arrangement of the particles is highly anisotropic up to distances of several particle diameters and shows a simple symmetry.

## Abstract

Disordered systems like liquids, gels, glasses, or granular materials are not only ubiquitous in daily life and in industrial applications, but they are also crucial for the mechanical stability of cells or the transport of chemical and biological agents in living organisms. Despite the importance of these systems, their microscopic structure is understood only on a rudimentary level, thus in stark contrast to the case of gases and crystals. Since scattering experiments and analytical calculations usually give only structural information that is spherically averaged, the three-dimensional (3D) structure of disordered systems is basically unknown. Here, we introduce a simple method that allows probing of the 3D structure of such systems. Using computer simulations, we find that hard sphere-like liquids have on intermediate and large scales a simple structural order given by alternating layers with icosahedral and dodecahedral symmetries, while open network liquids like silica have a structural order with tetrahedral symmetry. These results show that liquids have a highly nontrivial 3D structure and that this structural information is encoded in nonstandard correlation functions.

The microscopic structure of many-particle systems is usually determined from scattering experiments that give access to the static structure factor

Microscopy on colloidal systems and computer simulations have shown that for hard sphere-like systems, the local structure can be surprisingly varied, in particular, if the liquid is constituted by more than one type of particle. The geometry of these locally favored structures depends on packing fraction and is rather sensitive to parameters like the composition of the system, the size of the particles, or the interaction energies (7, 11). As a consequence, it has so far not been possible to come up with a universal description of the structure on the local scale, and it is unlikely that such a universal description exists.

In view of this difficulty, it is not surprising that very little effort has been made so far to investigate the structure of disordered systems on length scales beyond the first few nearest neighbors (11, 16). A further reason for this omission is the fact that the characterization of the structure on larger scales seems to be a daunting task, since already the classification of the local structure is highly complex. Other studies have therefore focused on the possible existence of orientational order that extents to larger distances (10, 18, 19), but no such order was found beyond the scale of a few particle diameters (19). However, whether or not disordered systems have indeed a structural order that extends beyond a few particle diameters is an important question since it is, e.g., related to the formation of the critical nucleus for crystallization or the possible growth of a static length scale that is often invoked for rationalizing the slow dynamics in glass-forming systems (3, 10, 20⇓⇓⇓–24). In the present work, we use an approach to reveal that liquids do have a highly nontrivial 3D structure that is surprisingly simple at length scales beyond the first few neighbors.

In order to show the generality of our results, we will consider two systems that have a very different local structure: a binary mixture of Lennard–Jones particles (BLJM), with 80% A particles and 20% B particles (25), and silica (*Materials and Methods*). The former liquid has a close-packed local structure that is similar to the one of a hard sphere system, while the latter is a paradigm for an open network liquid with local tetrahedral symmetry (3).

We study the equilibrium properties of the BLJM in a temperature range in which the system changes from a very fluid state to a moderately viscous one, i.e., *A*): take any three A particles that touch each other, i.e., they form a triangle with sides that are less than the location of the first minimum in the radial distribution function *SI Appendix*, Fig. S1*A*). We define the position of particle 1 as the origin, the direction from particles 1 to 2 as the *z* axis, and the plane containing the three particles as the *z*–*x* plane. (For the case of

For the BLJM, Fig. 1*D* shows the 3D normalized distribution *B*). Here, we denote by *SI Appendix*, Fig. S1). If temperature is decreased to *E*), the angular signal can be easily detected up to *F*), even at

Furthermore, one notes that *C*). For *C*), and hence the local dips formed by particles in the first minimum will be occupied by the particles in the subsequent shell. As shown below, this “duality mechanism” works even at large distances, thus leading to a nontrivial angular correlation in which, as a function of r, density distributions with icosahedral symmetry alternates with distributions with dodecahedral symmetry. Fig. 1 *E* and *F* show that with decreasing temperature the intensity of the signal at intermediate and large distances increases, indicating an enhanced order at low T.

Also for the case of the network liquid

To analyze these findings in a quantitative manner, we use the standard procedure to decompose the signal on the sphere into spherical harmonics *Materials and Methods*, and to consider the square root of the angular power spectrum *SI Appendix*, Fig. S2*A*), independent of r, a result that is reasonable in view of the icosahedral and dodecahedral symmetries that we find in the density distribution. For *SI Appendix*, Fig. S2*B*) since this mode captures well the tetrahedral symmetry of the density field.

In Fig. 3, we show the *r*-dependence of *F* shows that at distance *A*, one sees that at this r, the absolute value of *A* as well. We see that for the BLJM, *SI Appendix* for a precise definition of *T*-dependence [*SI Appendix*, Fig. S5 ].) For distances smaller than two- to three-particle diameters, there is no direct correlation between *SI Appendix*, Fig. S3).

Most remarkable is the observation that for the case of the BLJM, the height of the local maxima in *A*). This observation can be rationalized by the fact that a dodecahedron has 20 vertices [i.e., regions in which *SI Appendix*, Fig. S4), demonstrating that these findings hold also for liquids that are not supercooled.

In contrast to the BLJM, we find that for silica (Fig. 3*B*), the locations of the maxima in *SI Appendix*, Fig. S1*D* for the Si–O partial radial correlation function.) This shows that for liquids that have an open network structure, the distances at which one finds the highest orientational symmetry is not associated with a dense packing of particles, in contrast to hard sphere-like systems. Finally, we note that for both systems, the decay of

Fig. 3*B* shows that also for silica,

Since we have found that the distribution of the particles around a central particle is anisotropic, it is of interest to consider also the radial distribution functions in which one probes the correlations in a specific direction with respect to the local coordinate system shown in Fig. 1*A*. This type of information can be obtained for colloidal systems from confocal microscopy experiments and, more indirectly, from scattering experiments (8). *Insets* in Fig. 4 *A* and *C* show the directions we considered for the two type of liquids: for the BLJM, the directions corresponding to the vertices of the icosahedra/dodecahedra and the directions given by the midpoints between these two type of vertices; for silica, the directions of the vertices of the tetrahedra, the points given by the midpoints of the faces of the tetrahedra, and the directions given by the midpoints between the two former directions.

In Fig. 4*A*, we show for the BLJM, the radial distribution functions for these special directions, and one recognizes that the amplitude of the signal depends indeed strongly on the direction considered. For the directions of the icosahedra and of the dodecahedra,

Furthermore, Fig. 4*A* shows that the amplitudes of the oscillations in *B* shows these distribution functions on a logarithmic scale. (For the sake of clarity, only the maxima and minima of the functions are shown.) One notices that the slope of the curves for

For the case of silica, the connection between the extrema in *C*, *Inset*) is not straightforward. One finds that the peaks in *C*). In fact, the extrema of

The radial distribution function for the “neutral” direction N has a signal that is in phase with *D* shows on logarithmic scale the maxima of *C*. One recognizes that all of them decay in the same exponential manner with a slope that is independent of the direction.

To give a comprehensive view of the particle arrangement in three dimensions, we present in Fig. 5 the density distribution of the two systems. The colored regions correspond to the zones in which the particle density is high, and, by construction, they cover 35% of the sphere. For the BLJM at intermediate and large distances, one recognizes clearly the presence of high density zones with icosahedral symmetry (bluish color) interlocked with zones with dodecahedral symmetry (reddish color). The directions in which the blue and red regions touch each other correspond to the neutral direction N defined above and in which the particle correlation is weak. For silica, one finds instead interlocked tetrahedra at all distances (Fig. 5*B*). Again, the neutral direction corresponds to the one in which the blue and yellow regions touch.

In conclusion, we have demonstrated that liquids have nontrivial structural symmetries at beyond short range that have gone unnoticed so far. This result has been obtained by using a method that takes into account the 3D angular dependence of the structure and which can be readily applied to any system for which the particle coordinates are accessible, such as colloidal and granular systems, or materials in which some of the particles have been marked by fluorescence techniques (9, 28⇓⇓–31). Since we find that the nature of the orientational order in three dimensions depends on the system considered, the method allows to make a more precise classification of the structure of liquids, an aspect that should trigger the improvement of experimental techniques that probe this structural order.

## Materials and Methods

### System and Simulations.

The BLJM we study is a 80:20 mixture of Lennard–Jones particles (types A and B) with interactions given by *SI Appendix*, Fig. S1). Also, all of our results show a completely smooth dependence on temperature, and hence it is unlikely that they are affected by the presence of crystalline order.

For the simulation of silica, we use a recently optimized interaction potential proposed by Sundararaman et al. and which has been show to be able to describe reliably the properties of real silica (33). Although this potential is based only on pair interactions, it has been found to be able to describe better the structural and mechanical properties than other potentials for silica, including potentials with three-body interactions (33, 34). A cubic simulation box containing 120,000 atoms was used, which corresponds at room temperature and zero pressure to a box size of about 120 Å. The simulation was carried out in the isothermal–isobaric ensemble at 3,000 K for

### Angular Power Spectrum.

The coefficient

### Data Availability.

The data discussed in the paper have been deposited in Zenodo (https://zenodo.org/record/3783469#.XsfaDxZS_qN).

## Acknowledgments

We thank D. Coslovich, G. Monaco, M. Ozawa, and K. Schweizer for discussions. Part of this work was supported by China Scholarship Council Grant 201606050112 and Agence Nationale de la Recherche (ANR) Grant ANR-15-CE30-0003-02.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: walter.kob{at}umontpellier.fr.

Author contributions: Z.Z. and W.K. designed research; Z.Z. and W.K. performed research; Z.Z. analyzed data; and Z.Z. and W.K. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: The data discussed in the paper have been deposited in Zenodo (https://zenodo.org/record/3783469#.XsfaDxZS_qN).

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2005638117/-/DCSupplemental.

Published under the PNAS license.

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