Donations keep this resource free! Give here:
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.
[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]
Much can be learned about doublelayer dynamics by simply modeling the bulk fluid as a resistor
and the electrical double layers as capacitors. In so doing, we discretize the spatial variations in the
system (which are in reality continuous) into simplified and isolated components, just as we did in
Chapter 3. Bulk fluid can be modeled as a resistor, since it conducts current but does not store charge.
Double layers can be modeled as capacitors because (a) they primarily store charge and (b) they conduct
current only when the double layer is changing. The capacitor analogy for double layers works best
for surfaces that have intrinsic voltages owing to their surface chemistry but transmit no current, or
for electrodes that do not lead to electrode reactions (Faradaic reactions) becausethe voltages applied
are low or the electrode material is poor at catalyzing Faradaic reactions. If Faradaic reactions are
occurring, modeling becomes more complex. We restrict our discussion here to surfaces that do not conduct
current, though Chapter 5 describes voltagecurrent relationships for conducting electrodes with Faradaic
reactions.
RC circuit theory indicates that the characteristic time for response of a circuit is given by τ = RC; this is the
basis for low and highpass RC circuits, as well as many much more complicated systems. specifically, for current
passed between two electrodes through an electrolyte solution, if the applied voltage is a sinusoid at
ω = 1∕RC, half of the voltage is dropped across the electrical double layers while half is dropped across
the bulk solution. For ω ≪ 1∕RC, all voltage is dropped across the electrical double layer, and for
ω ≫ 1∕RC, all voltage is dropped across the bulk fluid. This means that for ω ≪ 1∕RC, the applied voltage
creates charge that can induce electroosmosis, while for ω ≫ 1∕RC the applied voltage does not have
enough time to localize charge at the interface and thus the electroosmotic effects are minimal. If we
create an electrical double layer by applying voltage at an electrode, the temporal relation between
φ_{0} and the applied voltage V is dictated by an RC circuit model. If we then wish to predict fluid flow
generated by these voltages, we must combine an RC model to predict φ_{0} combined with fluid flow
equations to describe fluid flow. To predict the dynamics of electrical double layers, then, we need
only model the resistance of the fluid in the system (for simple geometries, R = L∕σA) as well as the
capacitance.
16.2.1 The double layer as a capacitor
Given the overall structure described above, we need to predict the electrical circuit properties of the electrical
double layer. To so do, we review the definition of a capacitor and evaluate the differential capacitance of an
electrical double layer.
Chapter 3 discussed the response of a capacitor, but not its fundamental physical attributes. Acapacitor
is a fundamental electrical circuit element that stores charge in response to a voltage applied across
it:
where ΔV is the (positive) voltage drop across the capacitor and q is the (positive) difference in charge between
the capacitor ends. The capacitance is positive by definition. Ideal capacitors have a linear chargevoltage response,
so the totalcapacitance is the same as thedifferential capacitance C = for all ΔV . Most physics and circuits
texts define the capacitance using the total capacitance; however, the differential capacitance is the property that
governs the AC performance of a circuit or the dynamics of double layer charging, so the differential capacitance is
what is used here.
A parallelplate capacitor is a gap of distance d separating two conductors of area A, where d^{2} is small in
comparison to A. The capacitance of a parallelplate capacitor is
The double layer as a capacitor
We now wish to model an electrical double layer as a capacitor, because the diffuse electrical double
layer stores charge in response to a potential drop. If we assume that the charged surface does not pass
any current, the electrical double layer is purely capacitive. Since double layers are typically thin, the
assumptions for parallelplate capacitors are good, and we typically refer to capacitances per unit area
C′′:
where the charge per unit area in the electrical double layer is defined as q′′_{dl} = q_{dl}∕A. If the double layer is thin, the
charge per unit area in the electrical double layer is balanced by an equal and opposite charge on the
surface:
 (16.5) 
For a surface connected to an infinite reservoir of fluid, the wall is a voltage source, the bulk reservoir is the ground,
and the double layer is the capacitor in between.
Double layer charge
The total charge in a double layercan be determined in one of two ways, each described below. In either case, for a
symmetric electrolyte at low wall potentials(DebyeHückel approximation), we can show that the net
charge in the double layer is given by q′′_{dl} = εφ_{0}∕λ_{D}, and the capacitance per unit area is given by
C′′ = ε∕λ_{D}. Thus a DebyeHückel double layer acts as a parallelplate capacitor with plate spacing λ_{D}.
Potential gradient at the wall
Recall that thePoisson equation gives
 (16.12) 
If we consider the wall as a thin layer with nonzero charge density and integrate over this layer, then take the limit as
this layer thickness approaches zero, we can show that
 (16.13) 
where implies a derivative normal to the wall. This relation sets a boundary condition that relates the gradient of
potential to the bound surface charge at the wall (and thus to the total net diffuse charge in the double layer). This
relation is suitable only if the electrical double layer is thin.
Integrated charge density
We can also determine thenet charge density in the double layer by integrating directly:
 (16.14) 
where n is the coordinate normal to the wall. This relation is general and applies for double layers that cannot be
assumed thin.
Differential double layer capacitance: model dependence
Recallthat, if we make theDebyeHückel approximation, we can show that the linearized PoissonBoltzmann
equation predicts that the net charge (per unit area) in the double layer is given by
The capacitance per unit area is given by the ratio of the difference of the stored charge between the ends to the
magnitude of the potential drop. For a wall in isolation, the second surface of the capacitor is assumed to be at
infinity. The capacitance per unit area is thus given by
 (16.16) 
Where the negative sign comes from the fact that if φ_{0} is positive, the end that carries the positive charge is at
infinity, and thus the differential charge is q′′_{dl}. Alternately, if φ_{0} is negative, the differential charge is q′′_{dl} but the
potential drop is φ_{0}. From Equation 16.16, the capacitance per unit area for a thin, DebyeHückel double layer in
equilibrium is C′′ = ε∕λ_{D}. If we do not make the DebyeHückel assumption, we can show (Exercise 16.4) that the
PoissonBoltzmann equations predict that, for asymmetric electrolyte, thenet charge in the double layer is given
by
 (16.17) 
and that the differential capacitance per unit area dq′′∕dφ_{0} is given by
In both cases, the nonlinearity is seen through a hyperbolic correction factor in brackets, which approaches unity as
φ_{0}^{*}→ 0. The PoissonBoltzmann model improves on the DebyeHückel model in that, in addition to correctly
predicting the capacitance for small voltages, it correctly describes the differential capacitance for surface
potentials that are on the order of RT ∕F ; however, this model predicts that lim_{φ0→∞}C′′ = ∞, which is not
observed in experiments. The inaccuracy of the PoissonBoltzmann model stems from its pointcharge
approximation, which leads to unrealistically large ion concentration at large wall voltages. Thus the
PoissonBoltzmann model erroneously predicts that the charge stored by the electrical double layer goes up while
the effective double layer thickness is unchanged. Clearly, a correction is required to handle large
voltages.
Several approaches have been implemented to correct the PoissonBoltzmann result at large voltages. We
discuss the Stern and modified PoissonBoltzmann approaches here. The Stern model postulates that the interface
consists both of (1) the diffuse double layer described by GouyChapman theory and (2) a thin layer of condensed
ions. By “condensed”, we mean that these ions do not move normal to the surface. This thin layer (the Stern layer) is
given a thickness ofλ_{S} and a permittivity of ε_{S}, leading to a capacitance per unit area of C′′ = ε_{S}∕λ_{S}.
Using the series relation for a capacitor, and modeling the Stern layer as a linear capacitor, whose
differential capacitance is not a function of potential drop, the total double layer capacitance is given
by
Figure 16.2 shows the condensed layer used in the Stern electrical double layer model.
From Equation 16.19, we see that the Stern layer model puts an upper limit on the capacitance of the system
(the limit as φ_{0}^{*}→∞ is C′′ = ε_{S}∕λ_{S}), which matches experiments better. ε_{S} is usually modeled using a value
ranging from 6ε_{0}–30ε_{0} and λ_{S} is modeled using a value ranging from 1–10 Å.
An alternate approach is to incorporate amodified PoissonBoltzmann model (see Chapter 9) with steric
hindrance, which similarly limits the capacitance. Given finite ion size, the region near the wall becomes
saturated at high φ_{0}^{*}. When this happens, the thickness of the dense ion layer expands and the layer
becomes less able to store charge. Both of these results correspond to lower differential capacitance.
Quantitatively, the modified PoissonBoltzmann model predicts (again for a symmetric electrolyte)
that
 (16.20) 
As we did in Chapter 9, we define (for symmetric electrolytes) apacking parameter ξ = 2c_{∞}N_{A}λ_{HS}^{3}, which is the
volume fraction of ions in the bulk. The modified PoissonBoltzmann model also predicts that the differential
capacitance per unit area dq′′∕dφ_{0} is given by
The effect of steric hindrance is to make the differential capacitance (a) much lower at high wall voltages
than that predicted by the PoissonBoltzmann relation and (b) a nonmonotonic function of the wall
voltage.
Note that, even within the applicability of equivalent circuit analysis, both techniques presented have problems.
The Stern model discretizes the electrical double layer and its properties in a nonphysical way, and
the parameters that go into the model are poorly known and generally treated as phenomenological.
The modified PoissonBoltzmann model removes discretization, but also depends on a poorlyknown
parameter (the ion hardsphere radius). The modified PoissonBoltzmann model cannot account for specific
ion adsorption to walls (while the Stern model can) and ignores the variation of solution permittivity
with large electric field (which is handled, albeit discretely, with the Stern layer permittivity). A more
thorough equivalent circuit model requires that the equations that predict ion distribution be solved in
more detail, to provide better equivalent circuits to add to the model. Beyond that, full treatment of the
governing equations for the whole system further improves modeling of the dynamics of the double
layer.
Despite the limitations of the capacitor models, they nonetheless frame approximately what the capacitance of
the electrical double layer is and thus predict approximately what the potential drop is across the double layer. From
Chapter 6, we know that defining the potential drop across the double layer specifies the effective electroosmotic
slip at the interface; this boundary condition combined with the fluid flow equations predicts the fluid flow from
voltages applied at conducting surfaces.
__________________________________________________________________________________________________________________________________________________________ EXAMPLE PROBLEM 16.2: Consider two electrodes separated by a distance 2l and containing a symmetric
electrolyte with all ions having mobilities and diffusivities of equal magnitude. Suppose that two different
voltages are applied to the two electrodes. How long does it take for the double layers at the electrodes to
form?
Treat the bulk fluid as a resistor. Show that the resistance R = (where σ is the bulk conductivity
and A is the crosssectional area of the electrolyte linking the two electrodes) can also be written as
.
Treat each double layer in the DebyeHückel limit as a Helmholtz capacitor with thickness equal to
λ_{D} (C = ). Model the system as a capacitor, resistor, and capacitor in series, and show that RC for this model
system is equal to .
Given this result, what is the characteristic time for the double layer to equilibrate if you have two
microelectrodes separated by 40 μm in a 1 mM NaCl solution? Approximate D for these ions as
1.5×10^{9} m^{2}∕s.
Solution:
For the capacitor, we have:
 (16.22) 
and for two capacitors in series:
 (16.23) 
For the resistor, we have:
 (16.24) 
To define σ in terms of λ_{D}, we write:
 (16.25) 
leading to
 (16.26) 
Substituting in for σ, we find
 (16.27) 
leading to
.
__________________________________________________________________________________________________
[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
constitute an endorsement by the Kirby Research Group or Cornell University.
Donations keep this resource free! Give here:
