e., Equation 1), and to upwards curvature for z 1 >1. For simplicity, we shall consider in this letter only the case z 1 = 1. Note also that z e may depend on position and time via the n(x,t) dependence. Impurity trapping probabilities as a function
of z eand n The role played by z e in our model will be in fact twofold. First, it affects how large the distance is within which if the impurity approximates the inner wall then the latter attracts the former so much as to consider it as a collision. This attraction distance may be seen as an effective radius, ρ e , of the impurity (see Figure 1), so if the distance from the center of the impurity to the center of the channel is larger than r e −ρ e , learn more the impurity will actually touch the click here wall (dressed with already trapped impurities). Let us discuss the ρ e (z e ) dependence. We consider first the simplest case of an unscreened electrostatic interaction, in which the potential energy of an impurity at a distance ρ e from the wall is . Its kinetic energy associated to the thermal agitation is . By equating both and also taking into account the finite bare size of impurities, we obtain as a reasonable approximation, where is a constant inversely proportional to temperature. More interesting is the case in which ions in the carrying fluid partly screen
out the electrostatic interaction. The precise algebraic distance dependence of
the screened electrostatic energy may be different for each specific channel, fluid, and impurity, but we adopt here the common Debye-Hückel approximation in which this energy at a distance ρ e from the surface is taken as where λ D is the so-called Debye length. In aqueous liquids, λ D is a function of the ionic strength, and for concreteness, we will consider it to be dominated by the background electrolytes in the fluid rather than by the impurities to be filtered out (this seems to be the case at least of the measurements in [5, 6]), so λ D is essentially independent on Astemizole the concentration of the impurities to be trapped by the channel walls. By equating now the screened potential energy at ρ e to the thermal kinetic energy, we get (2) In the right-hand side of this equation, for convenience, we have expressed the thermal kinetic energy in units of the unscreened potential in the clean channel at a distance ρ 0 from the surface, so ρ 1 is an nondimensional coefficient proportional to T. We have also taken into account the finite bare size of the impurities by using ρ e −ρ 0 instead of ρ e in the potential energy term. From the above equation, ρ e can be obtained with the help of the principal Lambert W function as follows: (3) Although W(x) can be easily evaluated by modern computers, it is worthwhile to mention its asymptotes W(x)≃x for x ≪ 1 and for .