Ordering on different length scales in liquid and amorphous materials

The ordering on different real-space length scales is considered for a variety of glass-forming materials, ranging from densely packed amorphous metals and hard-sphere glassy colloids, to simple tetrahedral systems that include amorphous silicon and patchy colloids, to decorated tetrahedral systems that include amorphous ice and network-forming glasses with the AX2 stoichiometry (A = Si, Ge or Zn; X = O, S, Se or Cl). The ordering manifests itself as distinct peaks in the total structure factor S(k), where k denotes the magnitude of the scattering vector, with positions ki (i = 1, 2 or 3) that scale with the nearest-neighbour distance. Different length scales emerge with complexity of the bonding scheme. A peak at k3 is a generic feature associated with nearest-neighbour contacts, and is therefore present in S(k) for all of the materials. A second longer-length scale emerges as a peak at k2 < k3 if the bonding scheme assumes a directional character, leading to the formation of tetrahedral motifs in amorphous silicon and patchy colloids, or to Se-Se-Se chain segments in glassy selenium. A third still-longer-length scale appears for AX2 glasses as a first sharp diffraction peak at k1 < k2, where the scaled peak position depends on the character of the local network of A atoms. The geometrical origin of the peaks in S(k) and corresponding partial structure factors is considered, and equations are given for predicting the peak positions. The change in system fragility with the emergence of ordering on different length scales is discussed, along with the effect of pressure.