Liquids and glasses have strongly disordered atomic structures which are difficult to characterize with precision. Explaining the origin of such structures is an even more daunting task, which has been a subject of discussion for a long time (Egelstaff, 1967; Croxton, 1974; Hansen and McDonald, 1976; Debenedetti and Stillinger, 2001; March and Tosi, 2002; Parisi and Zamponi, 2010). In the absence of symmetry, the most popular approach is the bottom-up approach, starting with local structural units, such as the nearest neighbors of an atom (short-range order—SRO) and building up the global structure with preferred local structures (Miracle, 2004; Sheng, et al., 2006; Robinson et al., 2019). The preferred structures, such as icosahedral clusters, are often geometrically frustrated as a building unit (Sadoc, 1981; Nelson, 1983; Sethna, 1983; Tarjus, et al., 2005), resulting in a disordered global structure. However, in this approach the frustration condition depends on details of the cluster geometry, and the universal principle of structure formation remains deeply hidden.

We propose a holistic concept to depict this frustration through global and local features of potential energy landscape, by going beyond the idea of geometrical frustration. On top of the bottom-up approach we add a top-down approach, in which we start with a high-density gas state and apply interatomic potentials to all atoms at once. We show that a high-density gas state is unstable against the global density wave state, once the interatomic potential is introduced in reciprocal space. The local force to form atomic units and the global force to form density waves are in conflict against each other. The real structure results from the competition and compromise between these two frustrated driving forces. The medium-range order (MRO) in the oscillation in density correlations beyond the nearest neighbors reflects this competition. The competition depends on temperature. At high temperatures the local forces are dominant, whereas as temperature is reduced the importance of global force increases. This change in their roles explains strong temperature dependence in various properties of liquid.

The atomic structure of liquid and glass is usually described in terms of the pair-distribution function (PDF), g(r), which shows the distribution of distances between two atoms. The PDF is obtained by Fourier-transforming the structure function, S(Q), where Q is the momentum transfer in scattering, determined by x-ray or neutron diffraction (Warren, 1969). Figure 1 shows an example of the PDF, that of Argon liquid model with the Lennard-Jones (LJ) potential (Rowley, et al., 1975), with 34,461 atoms in the cubic supercell of the edge size, L = 112.01 Å. The first peak of the PDF describes the distribution of the distances to the nearest neighbor atoms from a central atom and describes the radial SRO. The oscillations in the PDF beyond the first peak describes the MRO, and they decay with the form, G ( r ) = 4 π r ρ 0 [ g ( r ) − 1 ] = G 0 ( r ) exp ( − r / ξ s ) (1) where ρ ₀ is the macroscopic number density of atoms and ξ _s is the structural coherence length, as shown in Figure 2.

The form of Eq. 1 was first suggested by Ornstein and Zernike. (1914), but it can be derived by a more general argument (Ryu and Egami, 2021). Also, whereas the OZ theory connects the SRO directly to the MRO and regards the MRO as a consequence of the SRO, we found that the nature of the MRO is substantially distinct from that of the SRO. For instance, the variation of the MRO with temperature shows a clear change at the glass transition temperature, T _g , but the SRO is continuous through T _g (Ryu and Egami, 2021). The first peak of the PDF is relatively narrow (<1 Å) and describes the atom-atom distances to the nearest neighbors, which number in the order of ten. But the MRO peaks are wider (∼1 Å), and cover hundreds of atomic distances. Therefore, they describe the correlation between the center atom and aggregates of atoms, or density fluctuations (Egami, 2020). In other words, the SRO describes the point-to-point correlation, whereas the MRO describes the point-to-set correlation (Berthier and Kob, 2012).

For metallic liquids the structural coherence length, ξ _s, which characterizes the MRO, obeys the Curie-Weiss law (Ryu, et al., 2019), ξ s ( T ) = a C T − T I G (2) where a is the nearest neighbor distance, and T _IG (<0) is the ideal glass temperature. At T = T _IG , ξ _s diverges and G (r, T _IG ) = G ₀(r). Therefore, G ₀(r), Figure 3, describes the structure of an ideally coherent glass state (Ryu, et al., 2019), which has long-range positional correlation without periodicity. Its Fourier-transform, the structure function S ₀(Q) shown in Figure 4, has a Bragg-like sharp first peak at Q ₁, of which height depends on the model size. Small sharp peaks are noise due to the finite model size. Because the structure is isotropic, in three-dimensions the first peak forms a Bragg sphere.

The ideally coherent glass state exists only in extrapolation, because T _IG is negative and the structure freezes at T _g . However, the Curie-Weiss temperature dependence of the structural coherence length, ξ _s, suggests that there is a driving force for a liquid towards this state. We suggest that the origin of such a force is the attractive interatomic potential. Let us consider a simple monoatomic liquid of atoms interacting with a two-body spherical potential, ϕ(r). We express the local atomic density, ρ( r ), in terms of the density waves, ρ ( r ) = ∫ ρ ( q ) exp ( i q ⋅ r ) d q (3) In order to make the density real, we assume, ρ ( − q ) = ρ * ( q ) . The ρ( q ) is a complex number, ρ ( q ) = | ρ ( q ) | exp ( i δ ( q ) ) (4) and the phase factor, δ( q ), has a nearly random distribution to avoid wave pile-up which would put more than one atom at a place. The total potential energy is given by U = ∫ | ρ ( q ) | 2 ϕ ( q ) d q = ρ 0 2 ∫ S ( q ) ϕ ( q ) d q (5) where ϕ( q ) is the Fourier-transform of ϕ( r ), ϕ ( q ) = ∫ ϕ ( r ) exp ( − i q ⋅ r ) d r (6)

For a spherical potential, ϕ( r ) = ϕ(r), r = | r | , and ϕ( q ) = ϕ(q), q = | q | . The ϕ(q) is dominated by the strong repulsive part of ϕ(r) at small r. However, the strongly repulsive part of the potential with the energy much larger than k _B T is irrelevant, because atoms never come so close to each other. Therefore, we define a “pseudopotential,” ϕ _pp(r), in which the strongly repulsive part of ϕ(r) is removed and   ϕ p p ( r ) = ϕ ( r c ) is assumed for r < r _c , where r _c is the cut-off distance. This is equivalent to disregarding the inaccessible high energy states from the potential energy landscape. It is also similar to the concept of the pseudopotential in the quantum scattering theory: A strong attractive potential simply adds a multiple of 2π to the scattering phase shift, and taking just the principal part of the phase shift corresponds to replacing it with the pseudopotential (Wu and Ohmura, 1962). For electrons removing the redundant phase shift is equivalent to reinforcing orthogonality to the core states (Phillips and Kleinman, 1959). For a repulsive potential we remove the strongly repulsive part. We separate ϕ(r) into two parts; ϕ _pp(r) and the repulsive part, ϕ _R (r), ϕ ( r ) = ϕ R ( r ) + ϕ p p ( r ) (7) then, the potential energy is, U = U R + U p p (8) U R = 1 V ∬ ϕ R ( r ) ρ ( r ′ ) ρ ( r + r ′ ) d r d r ′ (9) U p p = 1 V ∬ ϕ p p ( r ) ρ ( r ′ ) ρ ( r + r ′ ) d r d r ′ (10) where V is volume. The ρ( r ) of the ground state is determined by minimizing U. However, for the range r < r _c where ϕ _R (r) is non-zero, there is no pair of atoms with such small separations because of the strong repulsion. Therefore, U R = 0 ,       U = U p p , (11) and the structure is determined by minimizing U _pp with, ϕ p p ( q ) = ∫ ϕ p p ( r ) exp ( − i q ⋅ r ) d r (12)

The pseudopotentials of the Lennard-Jones potential for Argon with various cut-off values and their Fourier-transforms are shown in Figure 5. Interestingly ϕ _pp(q) has a minimum at q ₁, which is not too far from the position of the maximum in S(Q) at T _g , Q ₁ = 2.06 Å^-1. The dependence of q ₁ on ϕ ( r c ) is shown in Figure 6. The value of Q ₁ decreases with increasing temperature, down to 1.93 Å⁻¹ at T = 100 K. A minimum in ϕ _pp(q) at q ₁ implies that a density wave with q ₁ would be energetically preferred. Because both S(Q) and ϕ(q) are dominated by the region in q around Q ₁ (q ₁), a density wave which minimize U must be that with q close to q ₁. Thus, in the zero-th order the structure is dominated by the global density waves with q ₁. In other words, a high-density gas state is unstable against the global density wave state with q ₁.

However, the density wave with q ₁ has a fairly long wavelength, which results in a wide first peak of the PDF, as wide as the higher order PDF peaks. This is in conflict with the real-space requirement that the first peak of g(r) should be narrow around the bottom of ϕ(r) at r = a to reduce the potential energy. Thus, the requirement to reduce the potential energy in q space by density waves and the same requirement in real space by better SRO are orthogonal, resulting in frustration. This is a more general energetic statement of the geometrical frustration discussed by many in more specific forms, such as formation of icosahedral clusters (Sadoc, 1981; Nelson, 1983; Sethna, 1983). This conflict also explains the distinct natures of the MRO and SRO (Ryu and Egami, 2021). The final structure results from the compromise between the two, requiring additional density waves with high q values.

The conventional approach to elucidate the structure of liquid is the bottom-up approach, in which the interatomic potential is applied to a small number of atoms to form a good local cluster, and the structure is extended by adding more clusters (Miracle, 2004; Sheng, et al., 2006; Robinson, et al., 2019). A major problem of this approach is that by starting with good clusters the boundaries between good clusters become strongly strained. This heterogeneous strain concentration increases the total strain energy. Actually, in an extended structure the peripheral atoms in one cluster are also central atoms in other clusters, thus the clusters are overlapping. Focusing just on “good” clusters is biased and dangerous. An orthogonal top-down approach is to consider all atoms at once, and to apply the potential simultaneously to all atoms. This can be done more effectively in reciprocal space. By removing the irrelevant strongly repulsive part of the interatomic potential to form the pseudopotential, the Fourier-transform of the potential, ϕ _pp (q), was found to have a minimum at q ₁, close to the position of the maximum in S(Q) at Q ₁. This means that there is a driving force to form a density wave at q ₁. A small set of coherent density waves can form the basis for a crystal structure (Alexander and McTague, 1978), but a large set of density waves with random phase factors can form a state with long-range correlation without periodicity, the structurally coherent ideal glass state. The nature of such a state will be discussed elsewhere. Quasicrystal was the first of such state with long-range correlation without periodicity, which has a periodic crystalline lattice in six-dimensions (Levine and Steinhardt, 1984). The ideal glass forms a crystal in infinite dimensions.

A model of the coherent ideal glass state can be constructed by using G ₀(r) as a guide. It was found the model has very diverse local structures (Ryu et al., 2019). For instance, the fraction of the icosahedral local environment is only 0.7%. Apparently, in order to maintain long-range density correlation the local structure is sacrificed. This is another proof of the conflict between the force to produce long-range density wave state and the force to create short-range atomic order. The final structure is determined through the compromise between these two conflicting forces. The balance between the two is reflected to the coherence length; the stronger the SRO is, for instance because of covalency, the shorter the ξ _s is. The extent of this balance can be expressed as the ideality of the structure (Ryu, et al., 2020). Furthermore, temperature also affects this balance. Local density fluctuations due to the atomic-level pressure fluctuations reduces the coherence of the density wave. This argument quantitatively explains the Curie-Weiss law, Eq. 2 (Egami and Ryu, 2021).

The idea of geometric frustration is based on the hard-sphere (HS) packing argument (Nelson, 1983). In the DRP-HS model dynamics is controlled by jamming, because no potential energy is involved. To keep the density constant external pressure is applied, which plays the role of the attractive potential. It is interesting to note, however, if we use the pseudopotential, ϕ _HS(q) has a minimum at q _1,HS as shown in Figure 7, with D _HS q _1,HS = 5.76, where D _HS = 2R _HS and R _HS is the hard-sphere radius. This value is not far from the value of D _HS Q _1,HS = 6.9 for the hard-sphere model, where Q _1,HS is the Q ₁ for hard-spheres (Lange, et al., 2009). It could also be related to the long-range oscillation found recently (Rissone et al., 2021), but its value, D _HS Ql = 7.5, is considerably higher. If the pseudopotential concept applies even for the hard-sphere mode, it means that the mere action of exclusion on a gas under pressure may induce a density wave state. The height of the pseudopotential is irrelevant as long as it is substantially larger than k _B T. The ϕ _HS (q) has a long oscillating tail, and it influences the structure through the high Q part of S(Q). However, the strong minimum of ϕ _HS (q) has a dominant effect on the first peak of S(Q), producing the density wave and the MRO even in the hard-sphere model.

Thus, the idea of the minimum of ϕ(q) producing the density wave and the MRO appears to work widely, independent of details of the interatomic potential, possibly including even the hard-sphere model. As will be described elsewhere the dynamics of the density wave, through the amplitudons and phasons, is directly related to atomic dynamics. Thus, the stiffness of the density wave controls the cooperative atomic dynamics, affecting many physical properties of liquid. To what extent the same argument applies to covalently bonded network glass needs to be examined.

To characterize the atomic structure of liquid and glass, it is customary to start with focusing on the local structural units made of several atoms, and to consider adding them up to form the global structure. This bottom-up approach is so prevalent that its wisdom has not even been questioned. However, glasses are not formed in such a manner. In the process of glass formation through the glass transition all atoms equally contribute to glass formation during cooling through cooperative processes. To capture this many-body aspect of global glass formation, we propose a new holistic concept, made of dual bottom-up and top-down approaches. In the top-down approach we consider an assembly of many atoms in a high-density gas state and apply interatomic potentials to all atoms at once. The interatomic potential in reciprocal space causes collective density wave instability, which tries to create the ideal glass state. The final structure is determined through the competition and compromise of frustration between these two, local and global, driving forces. The balance between the two driving forces can be quantified in terms of the coherence length of the MRO, ξ _s, and the ideality of the liquid structure (Ryu et al., 2020). This approach explains the distinct natures of short-range order and medium-range order, and strong temperature dependence of various properties of liquid.