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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]

16.2 Equivalentcircuits [AC electrokinetics top]

Much can be learned about double-layer 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 voltage-current relationships for conducting electrodes with Faradaic reactions.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 16.1: Schematic of circuit between two parallel plate electrodes.


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 high-pass 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 φ0 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 φ0 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 charge-voltage response, so the totalcapacitance microfluidics textbook nanofluidics textbook Brian Kirby Cornell is the same as thedifferential capacitance C = microfluidics textbook nanofluidics textbook Brian Kirby Cornellfor 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 parallel-plate capacitor is a gap of distance d separating two conductors of area A, where d2 is small in comparison to A. The capacitance of a parallel-plate capacitor is

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 parallel-plate capacitors are good, and we typically refer to capacitances per unit area C′′:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where the charge per unit area in the electrical double layer is defined as q′′dl = qdl∕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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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(Debye-Hückel approximation), we can show that the net charge in the double layer is given by q′′dl = -εφ0∕λD, and the capacitance per unit area is given by C′′ = ε∕λD. Thus a Debye-Hückel double layer acts as a parallel-plate capacitor with plate spacing λD.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Potential gradient at the wall Recall that thePoisson equation gives
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.13)

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 theDebye-Hückel approximation, we can show that the linearized Poisson-Boltzmann equation predicts that the net charge (per unit area) in the double layer is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.16)

Where the negative sign comes from the fact that if φ0 is positive, the end that carries the positive charge is at infinity, and thus the differential charge is -q′′dl. Alternately, if φ0 is negative, the differential charge is q′′dl but the potential drop is -φ0. From Equation 16.16, the capacitance per unit area for a thin, Debye-Hückel double layer in equilibrium is C′′ = ε∕λD. If we do not make the Debye-Hückel assumption, we can show (Exercise 16.4) that the Poisson-Boltzmann equations predict that, for asymmetric electrolyte, thenet charge in the double layer is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.17)

and that the differential capacitance per unit area dq′′∕dφ0 is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

In both cases, the nonlinearity is seen through a hyperbolic correction factor in brackets, which approaches unity as φ0*0. The Poisson-Boltzmann model improves on the Debye-Hückel 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 limφ0→∞C′′ = , which is not observed in experiments. The inaccuracy of the Poisson-Boltzmann model stems from its point-charge approximation, which leads to unrealistically large ion concentration at large wall voltages. Thus the Poisson-Boltzmann 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 Poisson-Boltzmann result at large voltages. We discuss the Stern and modified Poisson-Boltzmann approaches here. The Stern model postulates that the interface consists both of (1) the diffuse double layer described by Gouy-Chapman 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 ofλS and a permittivity of εS, leading to a capacitance per unit area of C′′ = εS∕λS. 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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Figure 16.2 shows the condensed layer used in the Stern electrical double layer model.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 16.2: Stern 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 φ0*→∞ is C′′ = εS∕λS), which matches experiments better. εS is usually modeled using a value ranging from 6ε0–30ε0 and λS is modeled using a value ranging from 1–10 Å.

An alternate approach is to incorporate amodified Poisson-Boltzmann 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 φ0*. 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 Poisson-Boltzmann model predicts (again for a symmetric electrolyte) that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.20)

As we did in Chapter 9, we define (for symmetric electrolytes) apacking parameter ξ = 2cNAλHS3, which is the volume fraction of ions in the bulk. The modified Poisson-Boltzmann model also predicts that the differential capacitance per unit area -dq′′∕dφ0 is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The effect of steric hindrance is to make the differential capacitance (a) much lower at high wall voltages than that predicted by the Poisson-Boltzmann relation and (b) a non-monotonic 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 non-physical way, and the parameters that go into the model are poorly known and generally treated as phenomenological. The modified Poisson-Boltzmann model removes discretization, but also depends on a poorly-known parameter (the ion hard-sphere radius). The modified Poisson-Boltzmann 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 = microfluidics textbook nanofluidics textbook Brian Kirby Cornell (where σ is the bulk conductivity and A is the cross-sectional area of the electrolyte linking the two electrodes) can also be written as microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

Treat each double layer in the Debye-Hückel limit as a Helmholtz capacitor with thickness equal to λD (C = microfluidics textbook nanofluidics textbook Brian Kirby Cornell). Model the system as a capacitor, resistor, and capacitor in series, and show that RC for this model system is equal to microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

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 m2s.

Solution: For the capacitor, we have:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.22)

and for two capacitors in series:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.23)

For the resistor, we have:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.24)

To define σ in terms of λD, we write:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.25)

leading to
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.26)

Substituting in for σ, we find
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(16.27)

leading to

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

.

__________________________________________________________________________________________________

[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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