Double Layer Theory

solution causes a perturbation of the solution composition near the charged

surface.149

For example, a surface that presents a positive charge in a given set of conditions would induce a higher concentration of anions near the surface, and a depletion of cations (Figure 1.8a,b). The region near the surface in which the composition of the electrolyte is altered from bulk solution is termed the diffuse, or electrical double layer (EDL). The composition and size of the EDL are highly sensitive to the nature of the surface (e.g. conducting or insulating), the magnitude of

the charge on the surface, and the concentration of electrolyte in the bath solution.150

Much of the foundational work in the study of charged interfaces was motivated by the relationship between applied potential and surface charge at electrode

surfaces.151–153

While the applications in this thesis typically consider the charge presented at either insulators such as glass, or in more complex systems such as living cells, the basic principles described herein are transferable between the different experimental setups.

Figure 1.8. Structure of the electrical double layer (EDL). The general situation at the interface of a charged surface and an electroneutral solution is the perturbation of ionic concentrations from bulk concentration near the surface (a,b). For a positively- charged surface, there is an increase in anion concentration (a) and a decrease in cation concentration (b) adjacent to the surface. The Gouy-Chapman (GC) model of the EDL (c) does not consider the finite size of ions and is thus unsuitable at high electrolyte concentrations. Modifications of the GC model by Stern and Grahame account for phenomena such as specifically-adsorbed ions and a solvent layer at the surface (e). Plots of the electric potential that arise from the charge at the surface (d) differ between models depending on the treatment of the ions closest to the interface.

The first person to realise that the introduction of a charged surface into an electrolytic solution would cause a rearrangement of ions near that surface was Helmholtz, and he demonstrated the first notion of the EDL as a nanoscale dielectric. His initial model assumed a constant differential capacitance, meaning that the

charge stored by the EDL was linearly dependent on the applied potential, a notion

disproved independently by Gouy and Chapman in the early 20th

Century. The Gouy-Chapman (GC) model of the EDL introduced the idea of a diffuse layer of ions close to the interface in order to balance out the charge on the surface (Figure 1.8c). This diffuse region of counterions is at its highest concentration directly adjacent to the surface, while the concentration is lower at greater separation due to the reduction in strength of electrostatic forces. This leads to an exponential drop in the electric potential away from the surface (orange line, Figure 1.8d), the profile of which is dependent on many factors including the magnitude of the surface potential and the concentration of electrolyte species. For example, if the concentration of electrolyte is very low, the characteristic length of the EDL will be relatively long as the volume required to provide the counterions needed to balance a charge at the surface is larger than in a high concentration solution. In the GC model, the

relationship between the surface charge density (!!) and the surface potential (!!) is

given by the following:154

!! = (8!"#!!!×10!)! !sinh !_!!"!!! (1.1)

where ! is the molar gas constant, ! the temperature, ! the dielectric constant of the

solvent (78.54 for water at 298 K), !! the permittivity of free space, ! the molar

concentration of electrolyte, ! the charge magnitude of the !:! electrolyte used, and

! the Faraday constant. This model of the EDL is best applied in systems with low

ionic strength and at low potentials.149

At low ionic strength, the use of the dielectric constant of water is a reasonable approximation (for an aqueous solution), but this becomes far less accurate as the concentration is increased. At low surface potentials, the model predicts reasonable electrolyte concentrations near the interface, but the omission of the finite size of the ions in solution means that it can be unrealistic at high surface potentials, when the number of ions required to balance the charge at the surface is large. Despite these limitations, the GC model can be used to extract the Debye parameter (!), the inverse of which is described as the

!= !!_!"#!!×!"_! !

!

!

!

(1.2)

where ! is the ionic strength of the electrolyte solution. As described above, ! is proportional to the square root of the ionic strength, and thus the thickness of the EDL will decrease as the ionic strength increases. Table 1 shows some values of

double layer thickness, !!!_{, for several different 1:1 electrolyte concentrations at}

298 K.

Table 1.1. Relationship between electrolyte concentration and double layer thickness

for a 1:1 electrolyte at 298 K.150 ! (mM) !!! (nm) 0.1 30.4 1 9.62 10 3.04 100 0.96 1000 0.3

To accommodate for the breakdown of the GC model at high electrolyte concentrations and surface charges, Stern proposed a modification that incorporated many of the ideas of Helmholtz. He realised that ions have a finite size, that an ion could not approach closer than an ionic radius, and that they could thus not be treated as point charges. He proposed a compact layer directly adjacent to the electrode, now known as the Stern layer, comprised of solvated ions at the point of closest approach (Figure 1.8e). The plane passing through the middle of those solvated ions is termed the outer Helmholtz plane (OHP). As the thickness of the Stern layer is pre-determined by the electrolyte and solvent used, the exponential decay in electric potential seen in the GC model does not occur, and the drop in potential from the surface to the OHP is instead linear (blue line, Figure 1.8d). The diffuse portion of the EDL outside of the Stern layer is still treated with the GC model, and the electric potential at the OHP is significantly lower than at the surface. Grahame suggested the inclusion of a second plane (now known as the inner Helmholtz plane, IHP) to accommodate ions that had been specifically adsorbed to

the surface, as they would be able to approach closer than those that were fully solvated (Figure 1.8e). As the studies herein typically use low concentrations of

electrolyte (≤ 0.1 M), and low surface charge magnitudes (≤ 100 mC m-2

), it was decided that the GC model was sufficiently accurate when constructing FEM simulations of the experimental systems. Full details of surface charge simulations are given in Chapters 3 and 4.