Poisson–Boltzmann Equation for Microfluidic Transport Phenomena with Statistical Thermodynamics Approach

Abstract

With the advent of microfluidics and lab-on-chip systems, DNA and protein separation technologies are being developed for biology, diagnostics, and health purposes. Fully realizing these applications requires developing numerical models for sample transport. In this paper, a thorough investigation of electrokinetics and microfluidics transport phenomena reviews the background of the Poisson–Boltzmann equation. A detailed derivation of the equation is presented, which is not available in the microfluidic literature at one place, with the view of providing a more consolidated and comprehensive understanding of it. This equation is then applied to find the electric potential and charge density distributions in the electric double layer (EDL). The present study provides a detailed derivation of the Boltzmann distribution, and principles of probability are used to identify the most-probable ion distribution. This distribution is subject to constraints of constant number of particles and total energy of the system; Lagrangian multipliers are used to solve the resulting constrained optimization problem. Classical thermodynamics is shown to be consistent with the distribution of ions: the Boltzmann distribution. Then, based on Coulomb’s law, the derivation of Poisson’s equation, and its special form of Laplace’s equation, the electric potential distribution in the EDL and in the bulk flow is derived and presented. By applying classical thermodynamics and integrating the Boltzmann distribution and Poisson equation together, the Poisson–Boltzmann equation is achieved.