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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]

9.1 The Gouy-Chapman electrical double layer [electrical double layer top]

VIDEO: Intro to 1D electrical double layer analysis.

VIDEO: 1D Poisson-Boltzmann equation.

At equilibrium, solid surfaces have a netsurface charge density q′′ [C/m2] because of ionization and adsorption processes. The immobile charge on the surface is balanced at equilibrium by a mobile, diffuse volumetric charge density [C/m3]. Equivalently, ions with like charge to the wall (coions) arerepelled from the region near the wall, while ions with opposite charge to the wall(counterions) are attracted to the region near the wall. We describe the distribution of ions and electrical potential using equilibrium between electrostatic forces and Brownianthermal motion. By describing these distributions, we refine the integral result from Chapter 6 (i.e., the bulk electroosmotic velocity) to give the distribution of velocity near the wall of a microdevice.

This model, the Gouy-Chapman model (a schematic of which is shown in Figure 9.1), accounts for the diffuse nature of the counterion distribution, uses bulk fluid properties throughout, treats the ions and solvent as ideal, and takes the interface properties as given. We build on this later in the chapter by describing modifications to this theory.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.1: The electrical double layer consists of a region near an interface in which the net charge density is nonzero. As compared to the bulk solution, the counterions (ions with charge opposite the wall) are present at higher concentration, while the coions (ions with charge of same sign as the wall) are present at lower concentration.


We start by deriving the ion and potential distributions in the Gouy-Chapman electrical double layer.

9.1.1 Boltzmann statistics for ideal solutions of ions

Statistical thermodynamicspredicts that the likelihood of a system at temperature T being in a specific state with average energy per ion e1 is proportional to
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.1)

Here, kB is theBoltzmann constant (1.38×10-23JK). For an ion, the “system” is the ion, and the “state” is its location. For an ideal solution, in which ions are treated as noninteracting point charges suspended in a continuum, mean-field solvent, Equation 9.1 means that the likelihood of an ion being in a specific location is a function of the electrostatic potential energy at that point.

The same relation can be written on a per mole basis, as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.2)

where ê1 = e1NA is the energy per mole. R is the universal gas constant [JmolK], given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.3)

where NA isAvogadro’s number (6.022×1023). We typically use molar values in this text.

The distribution of ions is thus governed by the potential energy of the ions:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.4)

where z is the ion valence, and F isFaraday’s constant, given by F = eNA = 95465 Cmol, e is the magnitude of the charge of an electron (e = 1.6×10-19C), and ϕ denotes the local electrical potential. The relations above apply for an indifferent ion, which is an ion that interacts with a wall only through its electrostatic interaction, with no chemicalinteraction.

9.1.2 Ion distributions and potential: Boltzmann relation

Consider first the bulk solution far from any wall. Aqueous solutions contain dissolved ionic species, which we denote with subscript i. Each ion has a bulk concentration ci and avalence or charge number zi. For example, if we have a 1 mM solution of NaCl, cNa+ = 1mM and cCl- = 1mM and zNa+ = 1 and zCl- = -1. We use the subscript “bulk” or to define properties in the bulk, far from walls. Further, we define a double layer potential φ = ϕ-ϕbulk; thus φ is zero in the bulk. The value of φ thus specifies how the electrical potential at a point differs from that in the bulk far from walls. From the arguments ofBoltzmann statistics, we can write in general that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.5)

In the bulk, φ = 0, and Equation 9.5 reduces to ci = ci,. We can then write an expression for the localnet charge density ρE as a function of the local potential:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.6)

or, inserting Equation 9.5,

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The ideal solution Boltzmann statistics result (Equation 9.7) can be combined with thePoisson equation to write a governing equation for the potential (or the ion distributions) in the double layer. We typically write and solve the equation for potential, and use that solution and Equation 9.7 to calculate species concentrations.

9.1.3 Ion distributions and potential: Poisson-Boltzmann equation

The Poisson equation (recall Section 5.1.5), which links potential to the local net charge density, is written for uniform ε as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.8)

Equations 9.7 and 9.8 can be combined to derive thePoisson-Boltzmann equation, which describes double layers if the solvent can be modeled as a mean field and the ions can be approximated aspoint charges. When we combine these relations, we find

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Equation 9.9 is the general formulation of the Poisson-Boltzmann equation. This equation can be simplified by normalization, as detailed in the Appendix, Section E.2.3. To do this, we normalize the concentrations by the ionic strength in the bulk, and normalize the potentials by the thermal voltage:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.10)

where the thermal voltage RT ∕F is a measure of the voltage (approximately 25 mV at room temperature) that induces a potential energy on an elementary charge on the order of the thermal energy. We normalize the lengthsby the Debye length λD, given by:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The Debye length plays an important role in all of our discussions of the electrical double layer. The Debye length, which is a property of the electrolyte solution, gives a rough measure of the characteristic length over which the overpotential at a wall decays into the bulk. The Debye length is calculated with the ionic strength of the bulk, and is a parameter of the bulk fluid. The nondimensionalized form of the Poisson-Boltzmann equation is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


9.1.4 Simplified forms of the nonlinear Poisson-Boltzmann equation

The nonlinear Poisson-Boltzmann equation is difficult to solve analytically owing to the summation of the charge density terms as well as their strongly nonlinear character. General implementation requires numerical solution. However, we must develop a physical intuition for these solutions, and simplified forms of the equations provide straightforward analytical solutions that illustrate key physical concepts.

Simplified forms of the nonlinear Poisson-Boltzmann equation: 1-D

If the curvature of the wall is low (i.e., if the local radius of curvature of the wall is large as compared to λD), we can simplify Equation 9.12 by considering a one-dimensional form. Assuming an infinite wall aligned perpendicular to the y-axis, we obtain

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Simplified forms of the nonlinear Poisson-Boltzmann equation: 1D, symmetric electrolyte

If the solution is composed of one symmetric electrolyte, we can simplify Equation 9.14 by noting that c1 = c2 = c and defining |z1| = |z2| = z (see Exercise 9.2), resulting in:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Simplified forms of the nonlinear Poisson-Boltzmann equation: 1D linear Poisson-Boltzmann equation

If ziφ* is small as compared to unity, the exponential term in Equation 9.12 or 9.14 can be replaced with a first-order Taylor series expansion by setting exp(x) = 1+x. Starting with Equation 9.14, we find
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.16)

which, from the definition of the ionic strength and the normalization of the species concentrations (Equation E.19), becomes:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

See Exercise 9.3. This approximation is referred to as the Debye-Hückel approximation, and Equation 9.17 is the 1D form of the linearized Poisson-Boltzmann equation (the Poisson-Boltzmann equation is sometimes termed the nonlinear Poisson-Boltzmann equation to make it clear that the full nonlinear term has been retained). The linearized Poisson-Boltzmann equation is valid for all regions in which φ* is small. For cases where φ0* is small as compared to unity, this relation is valid throughout the flowfield, and this linearization simplifies the analysis. Because of this, we often analyze systems in the Debye-Hückel limit in this text to provide qualitative insight. A discussion of the accuracy of the Debye-Hückel approximation for various double layer parameters is presented in Section 9.4.

9.1.5 Solutions of the Poisson-Boltzmann equation

Solutions of the Poisson-Boltzmann equation are most easily studied starting from the simplified versions (which are amenable to analytical solution) and ending with the more complete solutions (which require numerical solution). We must always check our results to ensure that they are consistent with the approximations used to obtain the results. Two examples, which illustrate regions in which the linearized solution is alternately accurate or inaccurate, are shown in Figures 9.2– 9.3.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.2: Potential, velocity, species, ionic strength, and charge density distributions in a double layer with surface potential of -12 mV in the dilute solution limit (no steric hindrance). Here, the wall voltage is below the thermal voltage, and the species concentrations deviate only slightly from that predicted using theDebye-Hückel approximation. The local ionic strength, which is equal to the bulk in the Debye-Hückel limit, increases slightly near the wall.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.3: Potential, velocity, species, ionic strength, and charge density distributions in a double layer with surface potential of -100 mV in the dilute solution limit (no steric hindrance). The strong nonlinearity of the species distribution is apparent—populations of the sodium counterion increase by a factor of 50 or so.


Linearized 1D Poisson-Boltzmann solution—semi-infinite domain

The semi-infinite domain approximation can be used when double layers are thin, i.e., when λD is small as compared to the depth or width of the channels under examination. In this case, the bulk fluid can effectively be treated as being at infinity. The linearized Poisson-Boltzmann equation (Equation 9.17) then leads to an exponential solution. Defining the origin at the wall and applying the boundary conditions φ* = 0 at y* = and φ* = φ0* at y* = 0, we find
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.18)

or, returning to dimensional form,

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

This is one of the solutions shown in Figure 9.4. Since we have solved for φ*, we can also write the solution for the concentration of the species

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

If the charge density q′′ at the wall is known rather than the wall potential, we can show (Exercise 9.17) that, in the Debye-Hückel limit,

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where q′′* = q′′microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

Overall, the linearized solution, despite failing quantitatively for large wall potentials, gives a good qualitative picture of the electrical double layer surrounding most insulating walls.

Linearized 1D Poisson-Boltzmann solution—two parallel plates

The semi-infinite domain solution is applicable for all cases for which the double layer is thin (λD d, where d is a characteristic dimension of the channel). In that case, the solution near each wall is independent of any other walls. If double layers are not thin, the solution must account for the presence of the other walls. This can be done straightforwardly only for certain geometries.

Given two parallel plates separated by a distance 2d with the origin (y = 0) set at the midpoint, the hyperbolic cosine solutions of the linearized Poisson-Boltzmann equation (Equation 9.17) satisfy the boundary conditions. Applying the boundary conditions φ* = φ0* at y* = -d* and y* = d* and invoking symmetry at y* = 0, we find (see Exercise 9.5):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.22)

where d* = d∕λD. Returning to dimensional form, we get

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

This solution highlights the effect of proximity of surfaces,and is discussed further in Chapter 15. If the surface charge density is known rather than the potential, we apply the boundary conditions q′′* = q0′′* at y* = -d* and y* = d*. Invoking symmetry at y* = 0, we find (see Exercise 9.6):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(9.24)

where d* = d∕λD. Returning to dimensional form, we get

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

1D, symmetric electrolyte Poisson-Boltzmann solution—semi-infinite domain

The 1D, symmetric electrolyte Poisson-Boltzmann equation (Equation 9.15) leads to a solution with a sharper-than-exponential dependence. Applying the boundary conditions φ* = 0 at y* = and φ* = φ0* at y* = 0, we find (see Exercise 9.7):

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Equation 9.26 is reminiscent of the linearized solution (Equation 9.18) except for the hyperbolic tangent, and the two are equal if tanhmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell = 1. This solution is quantitatively useful and is applicable since many electrolyte systems are symmetric or well approximated by symmetric electrolytes.

Several differences are observed in the nonlinear solution as compared to the linearized solution: (a) the potential decays from the wall more sharply than exponentially—this difference is large when φ* is large as compared to unity; (b) λD still describes the characteristic decay length in the farfield, where the local φ* becomes small, but does not in general describe the characteristic decay length near the wall; and (c) the gradients in coions and counterions near the wall are larger than for the linearized solution. Figure 9.5 illustrates on a logarithmic axis how the solution to the Poisson-Boltzmann equation departs from purely exponential behavior at high surface potential.

General 1D Poisson-Boltzmann solution

For a general electrolyte, the Poisson-Boltzmann equation (Equation 9.9 or 9.12) has no analytical solution, and must be solved numerically.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.4: Potential distributions in the electrical double layer for the semi-infinite, symmetric electrolyte case, modeled both with the nonlinear and linearized Poisson-Boltzmann relations and using z = 1. Note that, as φ0* →∞, the potential distribution approaches the linearized solution for 0 = 4, as derived in Exercise 9.10. Note that the Poisson and Boltzmann relations can become inaccurate as φ* becomes large.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 9.5: Departure from exponential behavior of the solution to the Poisson-Boltzmann equation as the surface potential increases. Note that pure exponential behavior (linear behavior on these semilog axes) is observed far from the wall in all cases. These solutions are for z = 1. Note that the Poisson and Boltzmann relations can become inaccurate as φ* becomes large.


[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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