A Fractal Simulation Phase Growth Analysis of Spinodal Decomposition

Spinodal decomposition is a diffusion-controlled phase-separation process that produces complex, interconnected microstructures in multi-component systems. In this study, we investigate the emergence of scale-invariant morphological complexity during spinodal decomposition through phase-field simulations governed by the Cahn–Hilliard equation. Two- and three-dimensional simulations are performed, and the evolving concentration fields are quantitatively characterized using box-counting analysis to determine their effective fractal dimension. Spinodal patterns emerge from continuous concentration fields with diffuse interfaces, in contrast to classical geometric fractals characterized by sharp, self-similar boundaries (e.g., the Koch curve). As such, the fractal dimension reported here measures the multi-scale spatial occupancy and textural complexity of the evolving morphology rather than the Hausdorff dimension of well-defined geometric interfaces. The effective fractal dimension increases systematically with time: from values typical of sparse, filamentary structures in the early stage (~1.3 in 2D) to significantly higher values (~1.7–1.8 in 2D slices; ~2.8 in 3D volume) during late-stage coarsening, reflecting enhanced connectivity and near space-filling behavior. The numerical sensitivity to grid resolution and interface smoothing is carefully assessed, which confirms stable scaling behavior with appropriate refinement. The contrast between spinodal morphology and deterministic geometric fractals in this work clarifies the fundamental difference between exact geometric self-similarity and statistically self-similar patterns driven by thermodynamic free-energy minimization. The results provide robust, quantitative descriptors of interfacial complexity and domain connectivity. This offers practical metrics for the design and optimization of nanostructured materials in applications such as energy storage, catalysis, and microelectronics.