Scaling relations in the diffusive infiltration in fractals.

Area of Science:

• Physics
• Complex Systems
• Fluid Dynamics

Background:

• Previous work showed fluid infiltration in Sierpinski carpets (SCs) exhibits anomalous diffusion (exponent n < 1/2).
• The infiltration exponent (n) differed from single-particle diffusion exponents (ν) in SCs.
• This discrepancy highlighted a need to understand the relationship between macroscopic infiltration and microscopic diffusion in fractal geometries.

Purpose of the Study:

• To establish a scaling relationship between the fluid infiltration exponent (n) and fractal dimensions of the bulk (D_F) and border (D_B).
• To validate this relationship through numerical simulations in various fractal geometries.
• To investigate log-periodic oscillations in single-particle diffusion and diffusion front roughness.

Main Methods:

• Developed a scaling approach to relate infiltration and diffusion exponents: n = ν(D_F - D_B).
• Performed numerical simulations of single-particle random walks (RWs) and diffusive infiltration models.
• Utilized Sierpinski carpets (SCs) and generalized Menger sponges (MSs), including one with a fractal external border.

Main Results:

• Confirmed the scaling relation n = ν(D_F - D_B) in SCs and MSs.
• Observed log-periodic oscillations in the mean-square displacement of RWs and diffusion front roughness, linked to discrete scale invariance.
• Demonstrated that the exponent relation explains the quadratic dependence of n on D_F in SCs.

Conclusions:

• The study provides a unified scaling theory for anomalous diffusion in fluid infiltration within fractal porous media.
• Log-periodic oscillations in diffusion processes are a signature of discrete scale invariance in fractals.
• The findings enhance understanding of fluid transport in complex geological and synthetic materials.