Single-component shock layer analysis in elution chromatography

One of the fundamental results of the theory of nonlinear chromatography is that a propagation velocity is associated with each concentration. This velocity is related to the slope of the isotherm at the corresponding concentration. It follows that if a continuous concentration gradient is injected into a column, the gradient profile will not propagate in a mere translation but will progressively change shape. In the most common case of a convex upward isotherm (e.g., Langmuir), a linear gradient will become curved upward, the high concentrations migrating faster than the low ones. However, high concentrations cannot pass low ones, so the concentrations pile up, a concentration shock forms, and its height increases. In practice, axial dispersion and the mass-transfer resistances combine and prevent the formation of a true shock. A shock layer, a region where the concentration gradient is very steep, is formed. This shock layer migrates at the same velocity as the ideal shock would. Many characteristics of concentration shocks and shock layers have been determined previously, but not the time that it takes for a continuous gradient to turn into a shock layer, the circumstances of the birth of the shock layer, and of its growth. Yet, this is important to know to understand certain aspects of gradient elution. We have derived simple equations relating the circumstances of the birth of shocks to the phase equilibrium isotherm and to the column characteristics. The results of experimental measurements made with a high-efficiency analytical column are in excellent agreement with these theoretical predictions.