Nonlinear Waves in Capillary Electrophoresis

Electrophoretic separation of a mixture of chemical species is a fundamental technique of great usefulness in biology, health care, and forensics. In capillary electrophoresis, the sample migrates in a microcapillary in the presence of a background electrolyte. When the ionic concentration of the sample is sufficiently high, the signal is known to exhibit features reminiscent of nonlinear waves including sharp concentration "shocks." In this paper, we consider a simplified model consisting of a single sample ion and a background electrolyte consisting of a single coion and a counterion in the absence of any processes that might change the ionization states of the constituents. If the ionic diffusivities are assumed to be the same for all constituents the concentration of sample ion is shown to obey a one dimensional advection diffusion equation with a concentration dependent advection velocity. If the analyte concentration is sufficiently low in a suitable nondimensional sense, Burgers' equation is recovered, and thus the time dependent problem is exactly solvable with arbitrary initial conditions. In the case of small diffusivity, either a leading edge or trailing edge shock is formed depending on the electrophoretic mobility of the sample ion relative to the background ions. Analytical formulas are presented for the shape, width, and migration velocity of the sample peak and it is shown that axial dispersion at long times may be characterized by an effective diffusivity that is exactly calculated. These results are consistent with known observations from physical and numerical simulation experiments.