Reactive capacitance of flat patches of arbitrary shape

Area of Science:

• Physics
• Physical Chemistry
• Mathematical Modeling

Background:

• Understanding particle trapping and diffusion is crucial in various scientific fields.
• Reactive capacitance quantifies particle flux onto a reactive surface, impacting reaction dynamics.

Purpose of the Study:

• To investigate the reactive capacitance of a flat, partially reactive patch of arbitrary shape in 3D diffusion.
• To develop bounds and approximations for reactive capacitance, separating reactivity and shape effects.

Main Methods:

• Spectral expansion of reactive capacitance using a Steklov eigenvalue problem.
• Derivation of monotonicity bounds and probabilistic interpretations.
• Development of an efficient numerical tool for arbitrary patch shapes.

Main Results:

• Established bounds on reactive capacitance, showing monotonicity with reactivity and patch shape.
• Validated a simple, explicit approximation for reactive capacitance based on surface area and electrostatic capacitance.
• Demonstrated the utility of the approximation for diffusion-controlled reactions in complex domains.

Conclusions:

• The developed approximation simplifies the calculation of reactive capacitance.
• This work provides insights into diffusion-controlled reactions and their applications in statistical physics and physical chemistry.
• The numerical tools and theoretical framework are applicable to general domains with multiple reaction sites.