Abstract
We study the coarsening of three-dimensional antiphase domain structures via Monte Carlo simulations. Linear lattice dimensions of $N=1024$ enable us to reach a scaling regime covering about 2.5 orders of magnitude of linear scale. With short-range interactions on cubic lattices at temperatures of $0.75\phantom{\rule{0.16em}{0ex}}{T}_{\text{c}}$, the resulting antiphase domain structures are isotropic, which allows us to describe the real-space correlation functions by a common function scaled by a time-dependent parameter. We compare abstract Potts models and realistic models of atomic order in compounds and show that those with same ground-state degeneracy $q$ lead to equivalent antiphase domain structures, while the scaling functions for different $q$ show slight but significant deviations. Finally, we quantitatively discuss notions of real-space scale (specific interface area) and reciprocal-space scale (superstructure peak width) and thus give numerically exact values for the corresponding parameter $K$ of the Scherrer equation.
Highlights
Ordering transitions in compounds, where the lattice of a priori equivalent sites decays into sublattices with preferred occupations by different elements, can have profound effects on structural as well as functional materials properties [1,2,3]
In addition we will show that, as long as interactions are short range and give rise to approximately isotropic antiphase domains (APDs) interface energies, the obtained scaling functions do not depend on the specific microscopic model but are universal with a universality class determined by the ground-state degeneracy of the ordered state, which opens the possibility to quantitatively relate the length scales determined by realspace microscopy to those determined by the broadening of diffraction peaks
As we have shown that the scaling functions are universal for given q, the values we give for the full width at half maximum (FWHM) and integral breadth of the peak shape in Table I allow us to relate scales determined by microscopy and diffraction quite obviating the need for a detailed phenomenological modeling in terms of distribution functions [62,63]
Summary
Ordering transitions in compounds, where the lattice of a priori equivalent sites decays into sublattices with preferred occupations by different elements, can have profound effects on structural as well as functional materials properties [1,2,3]. The assumption of infinitely sharp domain interfaces leads to the correlation function behaving like g(r) ≈ 1 − αr at small distances, with α being proportional to the specific interface area σ [20] This small-scale linear decay with a discontinuous derivative at the origin when seen as a function of three-dimensional real space corresponds to the so-called Porod’s law S(k) ∝ k−4 for the scaled structure factor at large k [21], which can conveniently be probed via the small-angle scattering in phase-separation processes [22]. In addition we will show that, as long as interactions are short range and give rise to approximately isotropic APD interface energies, the obtained scaling functions do not depend on the specific microscopic model but are universal with a universality class determined by the ground-state degeneracy of the ordered state, which opens the possibility to quantitatively relate the length scales determined by realspace microscopy to those determined by the broadening of diffraction peaks
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