Exact solution of the hydrodynamic focusing driven by hydrostatic pressure

Microfluidic nanoprecipitation makes use of hydrodynamic focusing (HF) to accurately control the diffusive mixing of reactants. Both stability and precise handling of flow streams are essential for this application. However, flow stability is hardly attained when fluids are supplied by syringe pumps, due to the unavoidable fluctuations associated to the driving mechanical system. The alternative use of hydrostatic pressure is constantly increasing in microfluidic laboratories, though precise mathematical descriptions have not been reported so far. This paper presents a quantitative model for the HF driven by gravity in slit microchannels. The model analytically predicts the focusing width of the sample stream from the relative heights of the sample and sheath reservoirs. Fluids with different densities and viscosities are considered, which impact on the pressure provided by the hydrostatic columns, as well as on the flow pattern of HF. Theoretical predictions were successfully validated against experimental data. Flow-focusing experiments were carried out in hybrid PMMA/OCA chips with slit microchannels, using fluids with different physicochemical properties. Finally, a color reaction induced by pH-shift was implemented as a practical example. Different levels of diffusive mixing and reaction were attained along the focused stream by varying the relative heights of the fluid columns, precisely as predicted by calculations. The model thus provides a rational basis for the design of HF experiments using hydrostatics.