3: Electrified Interface

when an electric field is applied to a suspension, in most cases there will be a net displacement of the particles in the direction parallel to the field

the presence of charge of any sign on the particle surface requires, for an electrically neutral system, a charge of opposite sign and the same magnitude on the liquid phase side

thermal agitation in the liquid phase distributes ions (which would otherwise attach to the particle surface) toward the bulk of the solution

the constancy of the chemical potential ($μ^α = μ^β = …$)

an additional electrical work is involved in the transference of matter between phases due to differing electrostatic potential of each phase

$μ^α := μ^α_0 + z_i F ψ^α$, where z is the charge of ion i, F is the Faraday constant, $ψ^α$ is the electrostatic potential of phase α

$\nabla^2ψ=\frac{d^2ψ}{dx^2}=–\frac{ρ_q}{ε}$, where ψ is the electrostatic potential at any point in space with fixed charge density $ρ_q$ (i.e., electrical charge per unit volume) and ε is the dielectric coefficient (or constant) of the medium

in the absence of external forces, the motion of the charges in solution is random so that we can consider an average distribution; when the system is in thermal equilibrium (or close to it) the ions must follow the Boltzmann distribution

$c_i(r) = c^\infin e^\frac{–z_i e ψ(r)}{kT}$, where c(r) is the molar concentration of ion i at the point r in space, ψ(r) is the potential at the same point, $c^\infin$ is the concentration of the same ion in the solution bulk, z is the charge of that ion, e is the elementary charge, (exp e is the Euler's number), k is the Boltzmann constant and T the absolute temperature

$ρ_q(r)=\Sigma z_i F c_i(r)$

$\nabla^2 \psi(r) = -\frac{F}{ε} \Sigma z_i c^\infin_i e^\frac{–z_i e ψ_i(r)}{kT} = -\frac{F}{ε} \Sigma z_i c^\infin_i e^\frac{–z_i F ψ_i(r)}{RT}$, where the rightmost expression is fully written in molar units, R being the molar gas constant; the summation extends over all the ionic species present in the solution

the solution is considered to be a continuous medium of constant ε, and the ions are considered point charges

$e^{-\frac{z_i F \psi}{RT}}≈1-\frac{z_i F \psi}{RT}$

valid for $\frac{z_i F \psi}{RT}<1$ or ψ < 25 mV

$\psi (r)=\frac{ze}{4 \pi εr} \frac{e^{κ(r_0-r)}}{1+κr_0}$

the region where potential and concentration values differ from those at the bulk phase but decay toward it asymptotically (when around a particle or single ion, respectively)

$κ^{-1}$ where $κ=\sqrt{\frac{2F^2I}{εRT}}$

$I=\frac{1}{2} \Sigma z^2_i c^\infin_i$, an important property of electrolyte solutions

determines (near the central ion): κ, concentration profiles

in the case of spherical particles, where the Debye–Hückle theory can be in principle applied, the radius is greater, which would make the surface potential lower, but the total charge is several times that of a single ion, leading to a relatively high surface potential

a treatment of the PB equation in which the diffuse layer is in contact with a plane, infinite surface having a nonzero surface charge density $\sigma_s$

the two oppositely charged regions at both sides of the interface, which can also be regarded as a capacitor

there are charges of opposite sign and equal magnitude at both sides of the interface

in the GC model, there is no separation in the double layer, resulting in the capacitance (per unit area) going to infinity (consider $C=\frac{ε}{d}$ [per unit area])

$C=\left( \frac{\partial \sigma_s}{\partial \Delta \psi} \right)_{T,P,\mu_i} =\left( \frac{\partial \sigma_s}{\partial \psi_0} \right)_{T,P,\mu_i}$, where $\Delta \psi$ is the potential difference across the interface; in the right-hand side, it has been considered that $\psi=0$ in the solution bulk

the distance of closest approach to the surface for an ion, located at $x_2$ so that the properties of the diffuse layer may be found using the framework of the GC theory by replacing $\psi_0$ with $\psi_2$ and $x$ with $x-x_2$

• the space between the Stern plane (“2” plane) and the surface
• filled with solvent
• behaves as a dielectric

from an electrical point of view, the model interprets the double layer as two capacitors in series, i.e., $\frac{1}{C}=\frac{1}{C_i}+\frac {1}{C_d}$, where $C_i$ is the inner layer or Stern capacitance, and $C_d$ is the diffuse layer capacitance given by the GC theory

the minimum capacitance is identified as the point where the surface charge becomes null; here, the diffuse layer capacitance is determining the total surface capacitance (because in a series connection, the lower capacitance dominates the result), and as $C_d$ shows a minimum when both $\psi_2$ and $\sigma_d$ vanish, the potential (measured against a reference) of the electrode at this point should correspond to zero surface charge

- determined by the potential on conducting electrodes under external control
- on nonconducting and/or isolated surfaces, it is determined by the medium conditions, usually the pH

the Gouy–Chapman–Stern theory predicts well the interfacial capacitance of the interface between mercury electrodes and some electrolytes such as NaF or NaClO4, but fails for others such as KCl or KI; this is due to the fact that some ions, especially polarizable anions such as Br- and I-, can reach the surface closer than the Stern plane; thus *two* planes of closest approach are defined

the plane of closest approach located at $x_1$, the distance between the surface and the center of ions with which it is in direct contact, and with potential $\psi_1$

the plane of closest approach located at $x_2$, the distance between the surface and the center of ions that are separated by a layer of water molecules, and with potential $\psi_2$

these descriptions are based on a metallic electrode in laboratory conditions under potential control, a situation that is not, in principle, representative of what can be found in a complex natural system as a soil; these results are presented because liquid mercury is a surface as devoid of structure as possible so that the contributions to the charge from the solution side were focused upon; nevertheless, the diffuse layer, OHP and IHP are all present in the interaction between colloidal particles and solution

$R=k_BN_A=1.381(10^{-23}) \text{ J/K}*6.022(10^{23}) \text{ entities/mol}=8.314 \text{ J/(K-mol)}$

$F=\frac{e}{n}=(e)(N_A)=1.602(10^{-19}) \text{ C}*6.022(10^{23}) \text{ entities/mol}=96.5(10^3) \text{ C/mol}$