Trapping and Escape in a Turbid Medium

TL;DR

This paper analyzes how diffusing molecules are absorbed or escape in a spherical beaker with many small traps, deriving universal asymptotic behaviors based on trap density and size.

Contribution

It provides a theoretical framework for calculating escape and absorption probabilities in a beaker with many small traps, revealing universal asymptotic behaviors.

Findings

Escape fraction E scales as λ^{-1/2} for large λ.

Absorption fraction T approaches 1 as λ approaches 0.

Results are extended to various three-dimensional shapes.

Abstract

We investigate the absorption of diffusing molecules in a fluid-filled spherical beaker that contains many small reactive traps. The molecules are absorbed either by hitting a trap or by escaping via the beaker walls. In the physical situation where the number $N$ of traps is large and their radii $a$ are small compared to the beaker radius $R$, the fraction of molecules $E$ that escape to the beaker wall and the complementary fraction $T$ that eventually are absorbed by the traps depend only on the dimensionless parameter combination $\lambda = Na/R$. We compute $E$ and $T$ as a function of $\lambda$ for a spherical beaker and for beakers of other three-dimensional shapes. The asymptotic behavior is found to be universal: $1- E\sim \lambda$ for $\lambda\to 0$ and $E\sim\lambda^{-1/2}$ for $\lambda\to\infty$.