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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

10.2 Chemical and physical origins of interfacial charge [zeta potential top]

The simplest origin of interfacial charge occurs simply if we place two electrodes in contact with a fluid and apply a potential across the two electrodes to create surface charge. In this case, a potential is applied across the system, and the potential is dropped in part across electrical double layers at the electrodes and in part across the bulk solution. However, the conductivity of the surface prohibits the steady-state transverse extrinsic field one would use to drive steady-state electroosmosis, and thus the double layer at a conductor is irrelevant for equilibrium descriptions of steady fluid flow. We thus delay a description of the double layer at a conductor until Chapter 16.

On insulating surfaces at equilibrium, interfacial charge typically comes from (a) adhesion of charged solutes at the interface or (b) chemical reaction at the surface. This interfacial charge can be described with equilibrium relations. In particular, the electrochemical potential defines equilibrium in a convenient fashion, and that leads to discussion of potential-determining ions, which help frame the parameters that must be examined when interfacial potential is measured or analyzed.

10.2.1 Electrochemical potentials

The molar chemical potential (or partial molar Gibbs free energy) of a species gi [Jmol] is defined in the ideal solution limitas:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where gi is the chemical potentialat concentrationci. Here the term species implies both chemical and physical state—thus species in solid, liquid, aqueous, or surface-bound states are all treated differently. See also the description of electrochemical potential in Appendix B.

In systems that involve both electrical potential drops and chemical reaction, equilibrium is defined in the ideal solution limit using the electrochemicalpotential gi:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The system is at equilibrium when the electrochemical potentials of all species are uniform—if the electrochemical potential were nonuniform, there would be a driving force to initiate a chemical change. The electrochemical potential relates chemical properties to electrical potential, and thus relates chemistry to the electrostatic boundary condition used to solve our differential equations (e.g., the Poisson-Boltzmann equation) to obtain parameters (e.g., electroosmotic mobility) that directly impact microscale fluid mechanics.

The electrochemical potential formulation in the ideal solution limit is consistent with our earlier derivation of the Poisson-Boltzmann equation. The Boltzmann prediction for chemical species in a double layer (Equation 9.5) can be derived from Equation 10.3 by setting gi to be a constant (See Exercise 11.2). The electrochemical potential description has the added benefit that it also can be used to treat binding, chemical reaction, and other mechanisms that are the source of interfacial charge. We can write the electrochemical potential in the mean-field limit but without making the ideal solution approximation:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where ai and ai are the activities (i.e., effective concentrations) of species i at the current and reference concentrations. Equation 10.4 applies for solutions at all concentrations, while Equation 10.3 applies only at low concentrations. Note that the use of the mean-field electrical potential here ignores the structuring present at high concentration or in the monolayer next to a wall; at high concentration or within an atomic radii or two of a surface, the potential of mean force(see Appendix, Section H.2.1) must be used rather than the mean-field electrical potential.

10.2.2 Potential-determining ions

Describing the equilibrium chemistry at a liquid-solid interface, in general, involves writing an equality for electrochemical potential in the bulk and at the surface:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.5)

for every species i. Equation 10.5 thus implies that differences in the reference state chemical potential of a species gi and the electrical potential φ control the equilibrium activity ai and thus the concentration ci.

In determining the charge state at an insulating interface, Equation 10.5 shows that, if the reference state chemical potential is different for the species bound to a surface as compared to the species in solution, the species either preferentially adsorbs or desorbs from the surface. If the species in question carries an electric charge, this process controls the charge density at the surface. Similarly, if a chemical reaction at the surface leads to a change in the charge state of the surface, the reaction equilibrium dictates the charge state at the surface.

Consider, as an example, a glass microchannel in contact with an aqueous solution of sodium chloride and barium chloride. Electrochemical potential arguments can be used to describe the equilibrium of the sodium, barium, chloride, hydronium, and hydroxyl ions in this system. Sodium and chloride are generally (but not always) treated as indifferent electrolytes when in contact with a glass surface, meaning that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.6)

for both sodium and chloride ions. Thus the indifferent electrolyteapproximation assumes that the interaction between sodium and chloride ions and a glass surface is strictly a bulk electrostatic interaction, without chemical affinity. When this is the case, the equilibrium statement for the sodium and chloride ions is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.7)

which, in the ideal solution limit is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.8)

Equation 10.8 can then be rearranged to give the Boltzmann distribution (Equation 9.5), if ci is defined as ci, and the constant is set to zero.

In contrast, barium at a glass surface is an example of a specifically-adsorbing ion, implyingthat
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.9)

which means that barium has a chemical affinity for a glass surface that is different from its affinity for the water molecules that serve as the solvent. In this case, the Boltzmann statistics of Equation 9.5 can describe the barium concentration in the bulk but not at the wall—the concentration at the wall is then dictated by the energy of adsorption, often described using a reaction constant:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.10)

In fact, all species in general have a different reference state chemical potential at an interface as compared to the bulk—this phenomena is seen in many aspects of fluid mechanics; for example, the variation of Gibbs free energy at an interface was discussed as the foundation of surface tension in Chapter 1. Thus, all species act as surfactants to some extent. Indifferent electrolytes are those for which this energy variation and the attendant effects is small enough to be ignored.

In contrast to the sodium and barium cations, the protons (or hydronium ions) in solution can react chemically with a glass surface. The surface of silicates in contact with water has a large concentration of silanol (SiOH) groups, and these chemical groups undergo acid dissociation as described in Appendix B:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.11)

where Ka is the acid dissociation constant, often described using the pKa, defined as pKa = -logKa. Reaction 10.11 thus links the H+ concentration at the wall to the relative surface density of SiOH and SiO-. Since the concentrations of OH- and H+ are linked by water dissociation, the SiO-, H+, and OH- concentrations are all linked by chemical equilibrium, as described by the Henderson-Hasselbach equation. In the ideal solution limit, the pKa is evaluated using species concentrations and is a single value. For real systems at high concentrations, the pKa must be evaluated using species activities, and the pKa varies statistically based on the details of the spatial distribution of sites.

If the surface charge density is governed by an acid dissociation equation as described by Equation 10.11, the surface charge density q′′ is predicted in the ideal solution limit as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.12)

where C is the summed density of SiO- and SiOH sites [m-2], e is the elementary charge, and pKa is the acid dissociation constant of the site. For typical surfaces, the surface sites exhibit a range of pKa’s, and the surface charge density is more accurately exhibited with the relation
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.13)

where α is a value between 0 and 1 which describes the spread of pKa’s—1 denotes a single pKa and lower values correspond to a spread of pKa’s. Values between 0.3-0.7 are common.

Most systems are controlled by a small set of ions, called charge-determining ions or potential-determining ions. In the example above, thecharge at the surface is dictated primarily by protonation and deprotonation of silanol groups, and thus the charge is determined by H+ and OH- ions. Thus H+ and OH- are the primary charge-determining ions for glass, and the electrokinetic potential for a glass surface is a function of pH. Indifferent electrolytes can be added to an electrolyte solution without changing the charge density at the surface—indifferent electrolytes serve only to change the Debye length (which changes the surface potential) and, if present at high enough concentrations, to change fluid properties (which changes the electrokinetic potential). Specific adsorbers or species that react with the surface change the charge density at the surface directly. The process of defining charge-determining ions tends to greatly simplify the process of defining the chemical system that determines the double layer potential drop. As examples, Figure 10.1 shows the role of the charge-determining ions on a glass surface, and Figure 10.2 shows the role of the charge-determining ions on a silver iodide surface.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 10.1: Charge-determining ions on a glass surface.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 10.2: Charge-determining ions on a silver iodide surface.


10.2.3 Nernstian and non-Nernstian surfaces

A general equilibrium relation for an ion in a solution in contact with a wall is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.14)

where ai is the activity, and ai is the activity at a reference condition. In the ideal solution limit, ai = ci. This equation specifies the equilibrium condition but is rarely used in this form, because we rarely know gi or ai at the wall for the ions in question. However, this equation can be manipulated into a form that is useful if some experimental parameters are known. Consider the state for which φ at the wall is zero:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.15)

where the abbreviation pzc stands for point of zero charge, and denotes the solution concentrations at which the interfacial potential is zero. Note that pzc does not refer to a spatial location. Now subtract the equation at zero interfacial potential (Equation 10.15) from the general relation (Equation 10.14), and solve for φ0 to get
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.16)

This expression relates the surface potential to the activities of the ions in the bulk and on the surface, both at the point of zero charge and at the given experimental condition.

Equation 10.16, in itself, does not identify ai(pzc) and does not give ai at the surface at any condition. However, this equation frames the parameters we want to know (φ0), in terms of parameters we can directly change (ai,bulk, by controlling the concentration of the ions in solution), parameters we can easily measure (ai,bulk(pzc), by changing ion concentrations until the surface charge is zero), and, unfortunately, one typically unknown parameter (ai,wall).

Nernstian surfaces lead to a simplification of Equation 10.16 because, for Nernstian surfaces, the one problematic and unknown parameter is constant across experiments. A Nernstian surface is a surface for which ai,wall is independent of ai,bulk, meaning that we can add ions to the bulk solution without changing the surface ion activity, i.e., the effective surface ion concentration. This implies that changes in the surface ion concentration, while they are important when we calculate the surface charge for electrokinetic purposes, do not affect the role of those ions in surface chemical reaction. An analogous assumption, used when deriving acid dissociation reactions such as the Henderson-Hasselbach equation, is that the concentration of water is unchanged by acid dissociation. This assumption is made for acid dissociation because [H2O] is enormous as compared to [H+], and changes in [H+] have little effect on [H2O]. This assumption is sound for surfaces when the concentration of ions at the surface is large as compared to the concentration of ions in the solution, and thus typical Nernstian surfaces are weakly soluble ionic crystals such as AgI. Their surface activity is independent of the bulk activity because so many ions cover the surface that the small number of ions that might adsorb or desorb from the surface owing to changes in the bulk activity are only a small perturbation to the effective surface ion concentration. In this case, ai,wall = ai,wall(pzc) and thus
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.17)

or, rewriting in terms of base 10 logarithms of the activities,
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.18)

where log denotes a base 10 logarithm. In this case, φ0 can be predicted as a function of ai as long as an experimental observation of ai(pzc) is made. For example, the pzc for AgI at room temperature is at pAg = -log[Ag+] = 5.5. We can thus write the above equation for AgI in the ideal solution limit at room temperature as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(10.19)

Equation 10.18 is knownas the Nernst equation. It sets an upper limit for how strongly a surface can be charged owing to electrochemical equilibrium. The quantity RT ln10∕ziF is approximately 57 mV at room temperature for monovalent ions, and thus the maximum interfacial potential that a surface can achieve is approximately 57 mV per decade of concentration change from its point of zero charge.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Systems that are well described by Equation 10.18 (because of the independence of the wall ion activity to bulk ion concentration) are called Nernstian systems, while systems that are not well described by Equation 10.18 (because wall ion activity is a function of wall ion concentration and thus a function of bulk ion concentration) are called non-Nernstian systems. Nernstian surfaces (e.g., AgCl, AgBr, AgI, CaPO4) are relatively straightforward to analyze, because the only parameter of the surface that must be known is the bulk concentration of potential-determining ions at the point of zero charge, and the surface potential at a different concentration can be predicted from this parameter if the solution concentrations are known. This is particularly convenient, since the potential at the surface is closely related to the bulk electroosmotic or electrophoretic velocity observed in microsystems with thin double layers. For Nernstian surfaces, the Nernst equation dictates the surface potential and, for thin double layers, the bulk fluid velocity—the surface charge need not be calculated. If it needs to be calculated, the surface charge can be predicted by combining the surface potential with an electrical double layer model such as those discussed in Chapter 9.

Unfortunately, most of the surfaces present in microfabricated systems (e.g., oxides and polymers) are not Nernstian, because the adhesion of ions to the wall or reaction of the ions with the wall has a significant change on the effective concentration or activity of surface species. We thus cannot skip calculation of the surface charge density if our desired final result is the velocity of the fluid or a particle. For non-Nernstian surfaces, we typically must use chemical equilibrium to predict the surface charge density at the interface, and use an electrical double layer model such as those discussed in Chapter 9 to relate this surface charge density to a surface potential or directly to the fluid flow. Further, because of the effect of reaction or adsorption on the surface activity, we observe surface potentials on non-Nernstian surfaces that vary less per decade of concentration change of the potential-determining ions than that of Nernstian surfaces. The complexity of this process and the uncertainty with regards to the Gibbs energy of surface reactions or adsorption require that the surface potentials on non-Nernstian surfaces be measured directly. We describe these observations in some detail later in the chapter.

The non-Nernstian nature of most microdevice substrates can also be understood in terms of the density of charge sites. The charge site density for glass at high pH is usually reported as approximately 6Å2. Such a site density is depicted pictorially in Figure 10.3, and this schematic indicates that the density of sites is relatively low and the potential at the surface is, in fact, nonuniform within the plane of the surface. Care must thus be used when 1D models are used to describe the electrical double layer.

Related work on zeta potential from our research group can be found here.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 10.3: Schematic of charge sites at a 6Å2 surface.


[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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