A fractal model for characterizing multi-scaling particle diffusion behaviors in alkali-activated materials system

This paper proposes a fractal derivative model with a non-linear distributed-order (DOFM) to describe particle diffusion with multi-scaling behaviors in alkali-activated materials. The distributed derivative order is a power law function of the scaling factor, which generalizes the linear uniform case. The mean squared displacement in terms of the DOFM is derived as a non-linear form with the dilogarithm function that can describe multi-scaling diffusion behaviors. The Brownian motion running with a non-linear clock can clearly interpret the proposed DOFM from the perspective of particle motion. The DOFM is tested by using the experimental data of particles with different curing ages in alkali-activated materials. It is found that the diffusion coefficient and the scaling factor are power law dependent of curing age. Compared with the power law fractal derivative model, the proposed DOFM provides an efficient tool to describe the multi-scaling diffusion behaviors of the moving par-ticles in alkali-activated materials.

Automated summary

This paper proposes a fractal derivative model with a non-linear distributed-order (DOFM) to describe the multi-scaling behaviors of particle diffusion in alkali-activated materials. The DOFM is tested using experimental data and is found to effectively describe the multi-scaling diffusion behaviors of particles in the materials.