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[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]
[InducedCharge 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 steadystate transverse extrinsic field one would use to
drive steadystate 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 potentialdetermining 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 g_{i} [J∕mol] is defined in the ideal
solution limitas:
where g_{i}^{∘} is the chemical potentialat concentrationc_{i}^{∘}. Here the term species implies both chemical and physical
state—thus species in solid, liquid, aqueous, or surfacebound 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 g_{i}:
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 PoissonBoltzmann
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 PoissonBoltzmann equation. The Boltzmann prediction for chemical species in a
double layer (Equation 9.5) can be derived from Equation 10.3 by setting g_{i}^{∘} 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 meanfield limit but without making the ideal solution
approximation:
where a_{i} and a_{i}^{∘} 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 meanfield 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 meanfield electrical
potential.
10.2.2 Potentialdetermining ions
Describing the equilibrium chemistry at a liquidsolid interface, in general, involves writing an equality for
electrochemical potential in the bulk and at the surface:
 (10.5) 
for every species i. Equation 10.5 thus implies that differences in the reference state chemical potential of a
species g_{i} and the electrical potential φ control the equilibrium activity a_{i} and thus the concentration
c_{i}.
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
 (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
 (10.7) 
which, in the ideal solution limit is
 (10.8) 
Equation 10.8 can then be rearranged to give the Boltzmann distribution (Equation 9.5), if c_{i}^{∘} is defined as
c_{i,∞} and the constant is set to zero.
In contrast, barium at a glass surface is an example of a specificallyadsorbing ion, implyingthat
 (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:
 (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:
 (10.11) 
where K_{a} is the acid dissociation constant, often described using the pK_{a}, defined as pK_{a} = logK_{a}.
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 HendersonHasselbach equation. In the ideal solution limit, the
pK_{a} is evaluated using species concentrations and is a single value. For real systems at high concentrations, the
pK_{a} must be evaluated using species activities, and the pK_{a} 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
 (10.12) 
where C is the summed density of SiO^{} and SiOH sites [m^{2}], e is the elementary charge, and pK_{a} is the acid
dissociation constant of the site. For typical surfaces, the surface sites exhibit a range of pK_{a}’s, and the surface
charge density is more accurately exhibited with the relation
 (10.13) 
where α is a value between 0 and 1 which describes the spread of pK_{a}’s—1 denotes a single pK_{a} and lower values
correspond to a spread of pK_{a}’s. Values between 0.30.7 are common.
Most systems are controlled by a small set of ions, called chargedetermining ions or potentialdetermining 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 chargedetermining 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 chargedetermining 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 chargedetermining ions on a glass
surface, and Figure 10.2 shows the role of the chargedetermining ions on a silver iodide surface.
10.2.3 Nernstian and nonNernstian surfaces
A general equilibrium relation for an ion in a solution in contact with a wall is given by
 (10.14) 
where a_{i} is the activity, and a_{i}^{∘} is the activity at a reference condition. In the ideal solution limit, a_{i} = c_{i}. This
equation specifies the equilibrium condition but is rarely used in this form, because we rarely know g_{i}^{∘} or
a_{i} 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:
 (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 φ_{0} to
get
 (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 a_{i}(pzc) and does not give a_{i} at the surface at any condition.
However, this equation frames the parameters we want to know (φ_{0}), in terms of parameters we can directly change
(a_{i,bulk}, by controlling the concentration of the ions in solution), parameters we can easily measure (a_{i,bulk}(pzc), by
changing ion concentrations until the surface charge is zero), and, unfortunately, one typically unknown parameter
(a_{i,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
a_{i,wall} is independent of a_{i,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 HendersonHasselbach equation, is that the concentration of water is unchanged
by acid dissociation. This assumption is made for acid dissociation because [H_{2}O] is enormous as
compared to [H^{+}], and changes in [H^{+}] have little effect on [H_{2}O]. 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, a_{i,wall} = a_{i,wall}(pzc) and
thus
 (10.17) 
or, rewriting in terms of base 10 logarithms of the activities,
 (10.18) 
where log denotes a base 10 logarithm. In this case, φ_{0} can be predicted as a function of a_{i} as long as an
experimental observation of a_{i}(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
 (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∕z_{i}F 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.
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 nonNernstian systems. Nernstian surfaces (e.g., AgCl, AgBr, AgI, CaPO_{4}) are relatively
straightforward to analyze, because the only parameter of the surface that must be known is the bulk
concentration of potentialdetermining 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 nonNernstian 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 nonNernstian surfaces that vary less per decade of concentration change of the
potentialdetermining 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
nonNernstian surfaces be measured directly. We describe these observations in some detail later in the
chapter.
The nonNernstian 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 6∕Å^{2}. 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.
[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]
[InducedCharge 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.
Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not
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