Charge & Potential at the Interface¶

Charge is simple to model in air/vacuum, however when we start to introduce solvation, things can get kind of messy. One reason for this is that we need to consider the dielectric constant of the solvent, since it can greatly interfere (decrease) the interaction strength of the ions in solution.

Example 1 - $\ce{NaCl}$ dissolution

Using the coulombic approximation we can apply the following logic:

\[ V_{Coulombic}=\frac{q_1q_2}{4\pi D\varepsilon_0d} \]

If we calculate the attractive energy for $\ce{Na+}$ and $\ce{Cl-}$ in air ($D=1$), we get an energy of $-2.31\times10^{-19}\:J$

In water ($D=78.54$), this comes out as $-2.94\times10^{-21}\:J$.

The thermal energy associated with this system is $4.11\times10^{-21}\:J$, which is greater than the attractive enrgry of the ions in water, but not in air.

Realistically, in solution, with ions (including $\ce{H3O+/OH-}$, which are inevitable), the surface will be balanced with different counter ions and will have different properties form the bulk.

Helmholtz Model¶

Helmholtz posed a model where there is simplistically a charged surface with an equal number of counter charges to balance it out. This is rather innacurate

Guoy-Chapman Model¶

The Guoy-Chapman model describes a charged surface, balanced by a diffuse layer of counterions, with an gradient concentration.

The Stern Model¶

Stern pretty much mixed the two models and removed any point charge assumptions. He assumed that there is a linear decrease for the first layer of counterions, followed by an exponential decrease through the diffuse layer

The inner layer of adsorbed counterions is called the “Stern Layer”.

Quantifiction¶

It is incredibly difficult to quantify the elecrical double layer so we will follow the Guoy-Chapman model and make some asusmptions

The surface is flat with infinite and uniform charge
Ions are point charges with a Boltzmann based distribution in the diffuse layer
The solvent dielectric won’t vary through the double layer
We’ll use a symmetric electrolyte (both ions in the electrolyte have the same charge)

Poisson Equation¶

For a planar interface, the potential is related to the charge as such: $$ \frac{d^2\Psi(x)}{dx2}=-\frac{\rho(x)}{\varepsilon_0D} $$

This means we just need to find $\rho(x)$ (the charge density) in $C\cdot m^{-3}$

Calculating $\rho(x)$¶

The net charge density $\rho(x)$ should be equal to the ion population at a given point in solution, measured in $molecules\cdot m^{-3}$

Given that the potential is expected to follow Boltzmann statistics, we can model this with a Boltzmann distribution: $$ \rho_i(x)=\rho_i(bulk)\exp\bigg[-\frac{Z_iq\Psi(x)}{kT}\bigg] $$ (The signs on these charges are important)

Example 2

Calculate the $\ce{Na+}$ ion population at distance $x$ from the surface if the bulk concentration is $0.0150\:M$ and $\Psi(x) =-28\:mV\text{ at }25^\circ C$

\[ \begin{align}\rho_i(x)&=\rho_i(bulk)\exp\bigg[-\frac{Z_iq\Psi(x)}{kT}\bigg]\\ &=(0.0150)(6.022\times10^{23})(1000)\exp\bigg[-\frac{+1(1.602\times10^{-19})(-0.028)}{(1.381\times10^{-23})(298)}\bigg]\\ &=2.69\times10^{25}\: molecules\\ &=0.0446\:M \end{align} \]

If the potential is positive, then counter-ions will be in higher proportion compared to in the bulk. if the potential is negative, this is flipped and the concentration of co-ions is greater than in the bulk.

Net Charge Density¶

When accounting for multiple (symmetric) ions, we need to add another term:

\[ \rho_i(x)=-Zq\rho_i(bulk)\bigg[\exp\bigg(-\frac{Z_iq\Psi(x)}{kT}\bigg)-\exp\bigg(\frac{Z_iq\Psi(x)}{kT}\bigg)\bigg] \]

The previous equation can be simplified and substituted into the Poisson equation:

\[ \frac{d^2\Psi}{dx^2}=\frac{-2Zq\rho(bulk)}{\varepsilon_0D}\sinh\bigg(\frac{Zq\Psi(x)}{kT}\bigg) \]

Poisson-Boltzmann Equation¶

When combining this back into the Poisson equation, we get a complex series of equations as follows:

\[ \Psi(x)=\frac{2kT}{Zq}\ln\bigg(\frac{1+\gamma\exp(-X)}{1-\gamma\exp(-X)}\bigg) \]

Where:

\[ \gamma=\frac{\exp\bigg(\frac{Zq\Psi_0}{2kT}\bigg)-1}{\exp\bigg(\frac{Zq\Psi_0}{2kT}\bigg)+1} \]

and:

\[ X=\kappa x \]

$X$ is a dimensionless, scaled distance from the planar surface, the inverse scaling parameter $\kappa^{-1}$ has dimensions of length ands is given by:

\[ \kappa^{-1}=\bigg(\frac{\varepsilon_0DkT}{2q^2Z^2\rho(bulk)}\bigg)^{\frac{1}{2}} \]

Example 3

Calculate the Debye length of a $1.00\times10^{-4}\:M$ solution of NaCl in water ($D = 78.54$) and ethanol ($D = 25.3$)

$k=1.381\times10^{-23}\:JK_{-1}$
$q=1.602\times10^{-19}\:C$
$N_A=6.022\times10^{23}$
$T=298\:K$
$\varepsilon_0=8.854\times10^{-12}\:C\cdot(V\cdot m)^{-1}$
$\rho(bulk)=molecules\cdot m^{-3}$

\[ \begin{align}\kappa^{-1}&=\bigg(\frac{\varepsilon_0DkT}{2q^2Z^2\rho(bulk)}\bigg)^{\frac{1}{2}}\\&=\bigg(\frac{(8.854\times10^{-12})(78.54)(1.381\times10^{-23})(298)}{2(1.602\times10^{-19})(1)^2(1\times10^{-4})(6.022\times10^{23})(1000)}\bigg)\\ &=3.04\times10^{-8}\:m\\ &=30.4\:nm \end{align} \]

Debye-Hückel Approximation¶

This can be simplified further if we assume that $\Psi<25\:mV$ to:

\[ \frac{d^2\Psi(x)}{dx^2}\approxeq\kappa^2\Psi \]

and

\[ \Psi(x)\approxeq\Psi_0\exp(-\kappa x) \]

We can see from the last equation that $\kappa^{-1}$ is a measure of how the potential decays with distance from the surface

We can simplify EVEN FURTHER in water at $298\:K$ in a $1:1$ electrolyte:

\[ \kappa^-1=\frac{0.304}{\sqrt{C}} \]

For non symemtrical electrolytes

\[ \kappa^2 = \sum_i\frac{(Z_ie)^2\rho_i(bulk)}{D\varepsilon_0kT} \]