Viscous fingering in packed chromatographic columns: Linear stability analysis

When a fluid is displaced by a less viscous one in a porous medium, a hydrodynamic instability appears leading to the formation of some kind of fingers of the upstream fluid invading the downstream one, hence the name “viscous fingering” (VF) given to this instability. In a LC column, such an instability is likely to appear at that of the two interfaces between the sample and the eluent which exhibits an unfavorable viscosity contrast. It leads to distorted peak shapes and contributes to peak broadening. This phenomenon has been observed for long in SEC and more recently in RPLC on elution peak shapes as well as with various methods of in-column visualization. A simplistic LC column model is described to explain the origin of the VF instability and its characteristics. The general principles for analyzing hydrodynamic instabilities are described and the results of the linear stability analysis performed by Tan and Homsy [C.T. Tan, G.M. Homsy, Phys. Fluids 29 (1986) 3549 [1]], at the onset of the VF phenomenon for a step interface between two fluids are here applied to typical operating conditions encountered in analytical LC. The most probable growth rate and wavelength (linked to the finger width) of the instability are given in terms of particle size and solute diffusion coefficient, with particular emphasis on the role of the carrier velocity. Previously published qualitative observations about VF in chromatography are examined and interpreted at the light of this theory. The role of the column geometry on the development of the instability, the possible sources of noise or fluctuations triggering the instability, and the various experimental situations in which a significant viscosity contrast is encountered in LC are discussed.