10.3 Relations between q′′, φ₀, and ζ [zeta potential top]

If the Gouy-Chapman model of the electrical double layer is employed, the Poisson-Boltzmann equation is assumed to apply for all regions up to the solid wall, the no-slip condition is assumed, and fluid properties are assumed constant, the relations between q′′, φ₀, and ζ are straightforward. The relation between q′′ and φ₀ is determined by the electrostatic boundary condition

(10.22)

combined with the Poisson-Boltzmann relation for the potential distribution. The result can conveniently be written in general using the Grahameequation, obtained by transforming the Poisson-Boltzmann and solving it for q′′² at the wall:

(10.23)

Where the sign function sgn(x) ≡ x∕|x|. The Grahame equation can be written for symmetric electrolytes as

(10.24)

and both forms simplify in the Debye-Hückel limit to q′′_wall = . If the no-slip condition applies and fluid properties are uniform and equal to the bulk value, then ζ = φ₀.

Despite the material presented so far, prediction of the surface potential or charge density of microchannel substrates has largely eluded researchers. The arguments above give the impression that prediction of q′′, φ₀, and ζ is rather straightforward—the Poisson-Boltzmann diffuse double layer theory presented in Chapter 9 is well-established as a correct description of electrical double layers if the surface potential or charge density is specified and ion concentration is small. Further, the spatially-uniform fluid property analysis from Chapter 6 is a straightforward and intuitive description of the fluid flow induced in an electrical double layer. Unfortunately, if direct experimental measurements of surface charge (e.g., charge titration, described later in this chapter) are combined with the analysis in Chapter 6, the predictions, while qualitatively sound, overpredict electrokinetic velocities and are of relatively poor quantitative accuracy.

Physical models that go beyond the Poisson-Boltzmann relation, uniform fluid properties, and the no-slip condition have been introduced to explain the links between the values of q′′, φ₀, and ζ observed experimentally. In particular, these models describe departures from the assumptions that fluid properties are uniform or that all electrolytes are indifferent to the surface. This section explains these physical models. While Chapter 9 discusses the diffuse components of the electrical double layer, this section focuses on adsorbed and condensed regions.

10.3.1 Extended interface models: modifications to φ₀

The fluid velocity in an electrokinetic system is related to the surface potential and, in turn, the surface charge density. However, experimentally measured surface charge densities predict electroosmotic velocities much higher than observed. Extended interface models seek to reconcile this mismatch by postulating an additional structure to the interface that leads to lower velocities.

The most ubiquitous model is the Stern layer model, which postulates that counterions near the surface are so tightly bound that the fluid cannot move with respect to the interface. This layer is termed the Stern layer. This model defines an inner Helmholtz plane that corresponds to the interface between the wall and the Stern layer, and an outer Helmholtz plane that corresponds to the interface between the Stern layer and the diffuse electrical double layer. The Stern model typically defines the fluid boundary condition as = 0 at the outer Helmholtz plane, while solving the chemical equilibrium equations at the inner Helmholtz plane. In this sense, the effect of a Stern layer on the fluid mechanical prediction is simply to reduce the “effective” surface charge, thus reconciling the mismatch between charge titration data and electrokinetic data. Similar ideas are used to explain the suppression of ζ by many polymers.

10.3.2 Fluid inhomogeneity models: relation between φ₀ and ζ

Various models have been used to describe fluid properties in electrical double layers, with the key properties being the fluid electrical permittivity and viscosity. Some of these models seek to replace extended interface models such as the Stern layer model, while some are designed to be part of an extended interface description.

We start by reconsidering the integral analysis from Chapter 6. Consider a surface with potential φ₀ and an extrinsic electric field applied transverse to the surface. We write the Navier-Stokes equations (with nonuniform viscosity) for a thin region near the wall, and include the electrostatic body force term ρ_Eext:

As before, we assume that _ext is caused by an external power supply and is uniform within the electrical double layer, with a value equal to _ext,wall. The intrinsic electric field - is spatially nonuniform. For steady isobaric flow strictly along a wall aligned in the x-direction, with velocity and potential gradients only in the y-direction, Equation 10.25 reduces to a simple conservation of x-momentum equation:

(10.26)

and, inserting the Poisson equation with nonuniform permittivity:

(10.27)

using Equation 10.27 to substitute for ρ_E, and retaining only y-gradients, we find

(10.28)

This can be integrated to obtain

(10.29)

The no-slip condition and boundedness requires that C₁ and C₂ be zero, and rearrangement leads to

Equation 10.30 thus describes the bulk velocity observed in the general case where ε and η vary in the double layer. Equation 10.30 reduces to u_bulk = -ε_bulkφ₀E_ext,wall∕η_bulk (Equation 6.14) if ε and η are assumed to retain their bulk values throughout the double layer. Thus the bulk electroosmotic velocity can be determined by numerical integration if the permittivity and viscosity can be predicted as a function of the local potential.

Unfortunately, water is a difficult molecule to model in detail, given finite computational resources and a limited ability to make measurements with the spatiotemporal resolution required to define all relevant water properties. Because of this, the properties of water at interfaces and in the presence of high electric fields are under current debate. Appendix H gives some perspective on this, in terms of the variety of models used to predict water behavior.

Several models have been proposed to describe water properties under extreme conditions relevant to flow in electrical double layers and near walls, including effects of electric fields and high ion densities. The viscoelectric model postulates that the viscosity of water is anisotropic in the presence of an electric field. In particular, since the large intrinsic electric fields in the double layer are aligned normal to the surface while the flow induced in a double layer is tangent to the surface, the most relevant component of viscosity is the component normal to the intrinsic electric field. The viscoelectric model postulates that the viscosity normal to the electric field is given by

(10.31)

where k_ve is the viscoelectric coefficient, postulated to be on the order of 1×10^-15 m²∕V². This model thus predicts that a large electric field (on the order of k_ve^-(1∕2)) increases the viscosity of fluid moving normal to the electric field.

If the Debye-Hückel approximation is used, double layers are assumed thin, permittivity is assumed uniform, and Equation 10.31 is used to define the viscosity, the integral for the electroosmotic velocity at a surface can be evaluated analytically, leading to

(10.32)

Equation 10.32 shows that the electrokinetic potential ζ is equal to the surface potential φ₀, except for a correction factor that is a function of φ₀∕λ_D, ranging from unity at low surface potential to zero as the surface potential approaches infinity. This relation provides a quick qualitative estimate of the effects one might expect from a viscoelectric description of the fluid properties in an electrical double layer; however, for the ion concentrations typically employed in microfluidic experiments, the correction factor deviates from unity only in a range of surface potential for which the Debye-Hückel approximation is invalid, and the integral analysis of flow in the electrical double layer must be performed numerically.

The viscoelectric model seeks to include electric field effects on viscosity, but disregards the role of ion concentration and retains a dilute solution model for the purposes of modeling the viscosity. Few experimental or numerical results substantiate this theory, but it can explain the electrokinetic potentials observed with dilute electrolytes and modest interfacial potential. The most compelling concern with the viscoelectric models asserted to date is that the magnitude of the viscoelectric coefficient postulated for water (1×10^-15 m²∕V²) implies that water’s viscosity changes significantly at electric fields on the order of 3×10⁷ V∕m, at which the water is weakly oriented—pE∕kT is approximately 0.03 for water at E = 3×10⁷ V∕m, and thus the ensemble-averaged dipole moment of water is on the order of 0.03 at these voltages. The physical foundation for such a large viscosity change based on such minor orientation is yet to be justified.

Alternately, models have been proposed inspired by the jamming transition of granular materials, which seek to address complications arising from concentrated electrolytes and large interfacial potentials. The jamming transition in granular materials corresponds to the transition from liquid-like behavior to solid-like behavior as a function of the energy of the system and the particle density. Some everyday examples of these transitions can be seen in the behavior of coffee grounds, sand, and snow. Coffee grounds behave much like a liquid when loosely packed, and can be poured from a bag as if they were liquid. However, if one purchases a vacuum-packed bag of coffee grounds, the bag behaves as a solid. Similarly, a pile of sand dumped onto ground flows like a liquid at first (when the gravitational energy is high), until settles into a stationary cone shape corresponding to the point where the gravitational energy is no longer enough to jostle the sand grains out of their jammed state. Avalanches correspond to a transition from jammed to flowing state occurring because of a sudden addition of energy to the system.

At high ion concentrations, the viscosity of electrolyte solutions can be described using the relation

(10.33)

where c_i is the concentration of the dominant counterion and c_i,max is the sterically-limited concentration of that counterion. Equation 10.33 is inspired by rheological characterization of dense suspensions, and the parameters α and β are typically on the order of unity in rheological characterizations. Physically, this relation implies that, if ions are tightly packed by electrostatic forces near the point where the electron orbitals of the ions are the only repulsion keeping them from collapsing further, those tightly-packed ions behave rheologically like a vacuum-packed bag of coffee grounds. Given the absence of data on the rheology of aqueous solutions with highly-packed ions, little can be said for what α and β should be in this formulation, and c_i,max is furthermore known only approximately. The significance of this model is that it predicts that viscosity approaches infinity as the electrolyte reaches some maximum concentration.

The permittivity of water in the double layer is another area of interest, both in terms of its electric field dependence and its dependence on ion concentration. The permittivity of water drops precipitously near E = 1×10⁹ V∕m, and drops as small ion concentration increases. See Section 5.1.4 for more information on water permittivity.

10.3.3 Slip and multiphase interface models: hydrophobic surfaces

As discussed in Chapter 1, the tangential stress boundary condition at an interface is usually implemented using the no-slip condition. This no-slip condition was central to the integral analysis presented in Chapter 6, and is relevant for hydrophilic surfaces. However, hydrophobic surfaces also exhibit electrokinetic phenomena, in which case the integral analysis from Chapter 6 must be reexamined. If the velocity boundary condition is replaced with the Navier slip condition and a slip length b, a Debye-Hückel analysis leads to the result

(10.34)

while a Poisson-Boltzmann solution with a symmetric electrolyte leads to

(10.35)

The similitude between velocity gradients and intrinsic electric field in the double layer means that these relations have a functional form that is recapitulated when we discuss the capacitance of an electrical double layer in Section 16.2.1.

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