VIDEO: Intro to 1D electrical double layer analysis.

VIDEO: 1D Poisson-Boltzmann equation.

At equilibrium, solid surfaces have a netsurface charge density q′′ [C/m^{2}] because of ionization and adsorption
processes. The immobile charge on the surface is balanced at equilibrium by a mobile, diffuse volumetric charge
density [C/m^{3}]. 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.

Figure 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 e_{1} is proportional to

(9.1)

Here, k_{B} is theBoltzmann constant (1.38×10^{-23}J∕K). 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

(9.2)

where ê_{1}= e_{1}N_{A} is the energy per mole. R is the universal gas constant [J∕molK], given by

(9.3)

where N_{A} isAvogadro’s number (6.022×10^{23}). We typically use molar values in this text.

The distribution of ions is thus governed by the potential energy of the ions:

(9.4)

where z is the ion valence, and F isFaraday’s constant, given by F = eN_{A}= 95465C∕mol, e is the magnitude of the
charge of an electron (e = 1.6×10^{-19}C), 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 c_{i} and avalence or charge number z_{i}. For example, if we
have a 1 mM solution of NaCl, c_{Na+}= 1mM and c_{Cl-}= 1mM and z_{Na+}= 1 and z_{Cl-}= -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-ϕ_{bulk};
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

(9.5)

In the bulk, φ = 0, and Equation 9.5 reduces to c_{i}= c_{i,∞}. We can then write an expression for the localnet charge
density ρ_{E} as a function of the local potential:

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

(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

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:

(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 λ_{D}, given by:

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

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 λ_{D}), we
can simplify Equation 9.12 by considering a one-dimensional form. Assuming an infinite wall aligned
perpendicular to the y-axis, we obtain

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 c_{1}= c_{2}= c_{∞}
and defining |z_{1}| = |z_{2}| = z (see Exercise 9.2), resulting in:

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

If z_{i}φ^{*} 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

(9.16)

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

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 φ_{0}^{*} 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.

Figure 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.

Figure 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.

The semi-infinite domain approximation can be used when double layers are thin, i.e., when λ_{D} 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^{*}= φ_{0}^{*} at
y^{*}= 0, we find

(9.18)

or, returning to dimensional form,

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

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,

where q^{′′*}= q′′∕.

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.

The semi-infinite domain solution is applicable for all cases for which the double layer is thin (λ_{D}≪ 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^{*}= φ_{0}^{*} at y^{*}= -d^{*} and y^{*}=d^{*} and invoking symmetry at y^{*}= 0, we find (see
Exercise 9.5):

(9.22)

where d^{*}= d∕λ_{D}. Returning to dimensional form, we get

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^{′′*}= q_{0}^{′′*} at y^{*}= -d^{*} and
y^{*}=d^{*}. Invoking symmetry at y^{*}= 0, we find (see Exercise 9.6):

(9.24)

where d^{*}= d∕λ_{D}. Returning to dimensional form, we get

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^{*}= φ_{0}^{*} at y^{*}= 0,
we find (see Exercise 9.7):

Equation 9.26 is reminiscent of the linearized solution (Equation 9.18) except for the hyperbolic tangent, and the
two are equal if tanh= 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) λ_{D} 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.

Figure 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 φ_{0}^{*}→∞, the potential distribution approaches the linearized solution for zφ_{0}= 4, as derived in
Exercise 9.10. Note that the Poisson and Boltzmann relations can become inaccurate as φ^{*} becomes large.

Figure 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.