Electrokinetic Phenomena Deen 15.4-15.5 ψ
ψ0
ound anions b ositively chargedsurface p Diffuseions y A layer of adsorbed counter-ions (open Zeta Potential (ζ). The electrostatic potential circles) centered at the inner Helmholtz corresponding to a mobile charge – free Stern plane (IHP) and a mobile layer of layer inside a diffuse layer of mobile point charges counter-ions and co-ions centered at centered at a distance d from the surface. The the outer Helmhotz plne (OHP) “slip plane” (y=d) at which ψ =ζ is the position at which The position of slip plane is often defined as y = 0. Electrokinetic Phenomena
Volumetric charge density (C/m3):
ρe = e∑ ziρi i The non-zero volumetric charge density in a diffuse double layer creates the possibility of an electric body force in the liquid when an external electric field is applied. Transport phenomena that arise from the resulting coupling between electric fields and fluid motion are called electrokinetic phenomena. Gouy-Chapman-Stern Model
ψ
ψ0
ρe = 0
ound anions 2 b ∇ Ψ = −ρ /εε ∇Ψ = C e 0 ositively chargedsurface p Diffuseions y The region, 0
Navier-Stokes Equation including body force by E field neglecting gravity and assuming no imposed pressure gradients: 2 µ∇ v = −ρeE viscous force Coulomb force charges drag fluid along 2 Combine with Poisson’s Equation: − ρ = εε ∇ Ψ e 0 d 2v εε E d 2ψ x = 0 dy2 µ dy2 Electro-osmotic flow near a charged planar surface
∇Ψ
vx ( y)
x
Ψ = ζ d 2v εε E d 2ψ x = 0 dy2 µ dy2 Boundary conditions assuming plates are infinitely far apart from each other: dv dψ x (∞) = 0, (∞) = 0 dy dy
vx (0) = 0, ψ (0) = ζ Electro-osmotic flow near a charged planar surface
∇Ψ
vx ( y)
x
Ψ = ζ
εε E v ( y) = − 0 x [ζ −ψ ( y)] x µ There is a linear relationship between velocity and electrostatic potential. Therefore, the two profiles have the same shape. What should be the scale of the velocity boundary layer??? Electro-osmotic flow near a planar surface Relate to Debye length
∇Ψ
vx ( y)
x
Ψ = ζ
εε E v ( y) = − 0 x [ζ −ψ ( y)] x µ Combine with Debye-Huckel Approximation:
εε0Exζ −κy vx ( y) = − [1− e ] 1/κ is the Debye length µ 2 2 1/ 2 κ = (∑ρ∞ie zi /εε0kT ) Electro-osmotic flow near a planar surface Beyond a few Debye lengths, velocity is constant
∇Ψ
v ( y) εε E ζ x v (∞) = − 0 x x µ x
Ψ = ζ
εε E ζ −κ v ( y) = − 0 x [1− e y ] x µ εε E ζ 1 εε E ζ v ( y > 4 /κ ) = − 0 x 1− ~ − 0 x = v (∞) x µ e>4 µ x Electro-osmotic flow near a planar surface Slip Velocity
∇Ψ
v ( y) εε E ζ x v (∞) = − 0 x x µ x
Ψ = ζ
εε E ζ v ( y > 4 /κ ) ~ − 0 x x µ The double layer, described by the Debye length, is often small (10s of nanometers), so velocity BL is very thin. Therefore, there would appear to be slip at the wall and plug flow! εε E ζ v (∞) = − 0 x So x µ is often called the “slip velocity”. Slip Velocity - Example
10V 10V ∆h 1 cm 1 cm
εε E ζ v (∞) = − 0 x x µ
E =10V/cm and ζ=-25mV The flow can be reversed in the center solvent water 25 °C by applying pressure gradient (NS eqn.) εε E ζ (78.5)(8.854x10−12 CV −1m−1)(−25x10−3V )(10V /.01m) v (∞) = − 0 x = x µ 0.85X10−3 Nsm−2 −5 vx (∞) = 20X10 m / s = 20µm / s This is a significant flow velocity for small scale systems – e.g. microfluidics Lipid Bilayer Vesicles in an Electric Field
The field strength is much higher at the lower vesicle part, facing the glass, than at the top part. Such asymmetric field distribution leads to special membrane flow patterns, consisting of concentric closed trajectories organized in four symmetric quadrants, each extending from the bottom to the top of the vesicle
Membrane flow patterns in multicomponent giant vesicles induced by alternating electric fields, Rumiana Dimova et. al, Soft Matter, 2008, 4, 2168–2171 Electrophoresis Velocity (U) of a sphere
1 / κ
1 / κa << 1
εε E ζ U ~ − 0 ∞ particle µ The sphere moves at a velocity –U if the fluid velocity is zero (e.g. no pressure gradient) This result by Smoluchowski applies to any particle shape if the double layer is sufficiently thin. Therefore, ζparticle can be measured.