Can a periodic boundary model reproduce the longer-range density fluctuations in a real amorphous material?☆

Abstract Whereas the conventional definition of the static structure factor, S ( Q ), means that, for any sample or structural model, its value at zero Q , S (0), is identically equal to zero, the structures of ideally-disordered materials, such as single-phase liquids and amorphous solids, incorporate long-range density fluctuations that are characterised by a non-zero limiting value ( S 0 ) of S ( Q  ≠ 0) as Q  → 0. An analysis of these density fluctuations in terms of their Fourier components leads to the definition of an ideally-disordered material as one that exhibits a continuous, isotropic distribution of Fourier wavelengths, A ( Λ ), that decays asymptotically to zero at Λ  = ∞. On the other hand, a similar analysis for a periodic boundary model reveals that the form of the intermediate-range order at higher inter-atomic distances, r , and that of the long-range density fluctuations are fundamentally different from those of a real amorphous material. The severely limited number of (especially the longer) allowed Fourier wavelengths, Λ , coupled with their strictly defined orientations within the unit cell of a periodic boundary model, means that such a model is inherently crystalline, and that no amount of orientational (polycrystalline) averaging can overcome this problem. The various methods of deriving S ( Q ) for both periodic-boundary and cluster models are discussed, and it is shown that, since a periodic boundary model is not ideally-disordered, a polycrystalline average does not yield a consistent value for S 0 , but one that is dependent on its exact method of calculation. It is therefore concluded that, to investigate the longer-range density fluctuations in amorphous materials, it is essential to employ a cluster model, rather than one generated with a periodic boundary.