Determination of the Navier slip coefficient of microchannels exploiting the streaming potential

Abstract

For most microchannels made of hydrophobic materials such as polymers, velocity slip occurs at the wall, affecting volumetric flow rate of electroosmotic flow Q(eof) and streaming potential (∂ϕ(str)/∂z). Since most techniques exploit Q(eof) or (∂ϕ(str)/∂z) to determine the zeta potential, ζ, it is very difficult to measure ζ of hydrophobic walls, if the slip coefficient b is not found a priori. Until now, Q(eof) and (∂ϕ(str)/∂z) are known to depend on ζ and b in a same functional form, which makes it impossible to estimate ζ or b separately using measurements of Q(eof) and (∂ϕ(str)/∂z). However, exploiting the analytic formula for Q(eof) and (∂ϕ(str)/∂z) derived in the present work, it is found that the effect of ζ and that of b on Q(eof) and (∂ϕ(str)/∂z) can be separated from each other by varying the bulk ionic concentration. Thus, the slip coefficient as well as the zeta potential of hydrophobic microchannels can be found with reasonable accuracy by means of a nonlinear curve fitting method using measured data of Q(eof) and (∂ϕ(str)/∂z) at various bulk ionic concentrations. The present method allows an accurate estimation of slip coefficient of hydrophobic microchannels, which is quite simple and cheap compared with methods employing microparticle velocimetry.

Electrophoresis of particles with Navier velocity slip

“…Usually zeta potential depends on bulk ionic concentration. Plots of zeta potential versus the natural logarithm of ionic concentration are approximately linear at a given pH for organic or inorganic buffers . Thus, an appropriate functional form relating the dimensionless zeta potential and bulk ionic concentration may be:where C is a small number representing a slight deviation from the linear relationship between and .…”

“…. This method of estimation is a typical nonlinear curve‐fitting problem that can be solved using the Levenberg‐Marquardt method . Equation may be rewritten symbolically as:where denotes the value of the left hand side of Eq.…”

“…valid as closely as possible. To solve this nonlinear curve‐fitting problem using the Levenberg‐Marquardt method, we pose the following χ 2 merit function :where is the measurement error of the i th data point, presumed to be known and N is the number of data points. When a becomes the exact value, , χ 2 is at a minimum and .…”

Electrokinetics over hydrophobic surfaces

“…A remedy to this was suggested as obtaining (ς+ l s g) from a pressure‐driven flow that includes electrokinetic interactions and obtaining l s from measurement of flow rate using the same setup but without electrokinetic effects . However, appreciating the low sensitivity of this arrangement, variation of the bulk ionic concentration was proposed as a way to remove the degeneracy . Park and Sohn proposed a method for measuring the high zeta‐potential of macroscopic surfaces with Navier‐slip.…”

“…Park and Sohn proposed a method for measuring the high zeta‐potential of macroscopic surfaces with Navier‐slip. The mathematical formulation adopted by them was similar to Park .…”