Electroanalytical Chemistry. Gary A. Mabbott

of the surrounding solution, whereas the diffuse charge model implied that it all was susceptible to thermal motion. In 1924, Otto Stern proposed a new model that was a synthesis of the earlier two [8]. Stern proposed that part of the compensating charge on the solution side is held tightly in the IHP plus the OHP and the remaining fraction of charge is contained in a diffuse zone of freely moving counter ions with a concentration that decreases with distance from the electrode surface. The IHP and OHP are sometimes collectively called the Stern layer. This layer is considered a stagnant zone of solution that clings to the surface. Figure 1.8 shows the arrangement of ions at a negatively charged electrode according to the Stern model. The charge balance equation for the double layer becomes

      (1.20)

Figure 1.8 Stern model of the electrical double layer. Charges and solvent at the OHP cling to the solid surface. Not all of the charge on the surface of the electrode is compensated by the excess of cations at the outer Helmholtz plane (OHP). A fraction of the counter charge is represented in a diffuse region just beyond the OHP. (Only the charges for the excess cations – the number of cations greater than the local number of anions – are pictured for clarity.)

Notice how the electrical potential changes as a function of distance from the electrode surface. The Stern model has several implications. One is that at higher electrolyte concentrations, the diffuse region of excess counter ions becomes more compact. The distance from the surface to the outer edge of the diffuse region is known as the Debye length, λ_D. The Debye length is inversely proportional to the square root of the electrolyte concentration. A good benchmark to keep in mind is that the Debye length is about 100 Å (10 nm) for a sodium chloride solution of 0.01 M [9]. Most electroanalytical measurements are made at electrolyte concentrations of 0.01 M or higher. This profile for the potential has an important significance for voltammetry experiments where molecules must be transported to the electrode surface in order to exchange electrons with the surface. In most cases, the molecules can approach no closer than the OHP. Consequently, the electric potential that they experience is that of the OHP rather than the true potential of the electrode surface. As will be discussed in Chapter 5, this has important implications for the rate of electron transfer and the magnitude of the corresponding current in voltammetry experiments.

It seems appropriate to pause here and note an application of electrochemical principles to phenomena outside of the field of electroanalysis. Naturally occurring phase boundaries involving electrified surfaces often occur in a variety of environments. Here is just one example in which the structure of the electrical double layer is especially relevant. River waters frequently carry tiny soil particles that remain suspended in the water because of the negative charges on the mineral surface (see Figure 1.9). These surface charges arise from the crystal structure in which some Al³⁺ and Si⁴⁺ ions are replaced by lower valence cations, such as Mg²⁺. Cations from the surrounding solution form an electrical double layer with the surface of each clay platelet. In the river, where the electrolyte concentration is low, these charged particles have electrostatic fields that extend far enough into solution to keep neighboring particles from approaching each other closely. However, whenever a river flows into the sea, the ionic strength of the mixture increases suddenly (the sea is about 0.5 M in NaCl) decreasing the distance that the electrostatic field reaches from the surface of a particle. Under these conditions, collisions bring the particles close enough for attractive interactions, such as van der Waals forces and hydrogen bonding, to overcome the repulsion of the charges. The particles cling to each other and can settle out faster as a result.

Figure 1.9 The electrical double layer plays an important role in the suspension or sedimentation of tiny particles, such as clay platelets. (a) Montmorillonite clay mineral structure with lysine bridging plates. (b) The surfaces of clay platelets are negatively charged as a result of lower valence cations replacing Al³⁺ and Si⁴⁺ ions in the crystal lattice. (c) At low ionic concentrations, the diffuse charge region of the sides extends several nanometers out into solution effectively pushing neighboring particles apart. (d) At high electrolyte levels, the field from the diffuse part of the electric double layer compacts allowing clay particles to approach more closely [12]. (e) Neutral polymers, such as naturally occurring polysaccharides, can adsorb at multiple points to neighboring particles leading to aggregation. Aggregation processes are known as coagulation and flocculation.

      Source: Adapted with permission from Zhu et al. [17]. Copyright 2019, Elsevier.

      Seawater and soil particles are complicated mixtures and multiple mechanisms for binding particles together have been described. In some cases, calcium and other di‐ or tri‐valent cations can bridge between adjacent particles [10]. Another mechanism has been exploited in water treatment and industrial applications of clay materials. Water soluble, neutral polymers, such as polysaccharide chains, are added to clay suspensions in industrial processes to bridge between particles by hydrogen bonding to oxygen atoms in the clay surface [11]. Polymers that attach at several points but still loop out into the solution appear to work best. Presumably, the loops in the chains extend far enough to reach across the electric double layer of neighboring particles. The compression of the electrical double layer is a key part of the mechanism in both industrial and natural processes. As these particles agglomerate at the mouth of rivers, they settle out of solution‐carrying nutrients, and sometimes pollutants, into the sediments. This process has very important implications for the ecology of estuaries and the biological productivity of marine environments [12].

      1.4.2 The Relationship Between Double Layer Charge and the Potential at the Electrode Interface

One can gain a lot of insight about electrochemical processes from approximating the behavior of an electrified interface with that of a capacitor. For example, it is interesting to think about how many charges are involved in creating a voltage across the boundary between two phases. An estimate can be made by modeling the electrical double layer as a simple capacitor where the solid metal constitutes one plate of the capacitor and the solution at the OHP (plus the diffuse region) serves as the second plate (Figure 1.10). (A similar argument can be made for other types of phase boundaries, such as between two ion‐containing liquids.)

Figure 1.10 The electrical double layer can be modeled as a capacitor where the charge Q is the charge on one plate.

Consider how much charge would be required to create an interface potential difference of 1.0 V. For the sake of discussion, consider a metal surface that has a net positive charge. The magnitude of charge, Q, that a capacitor accumulates on each side of the interface for a given voltage, V, separating the two plates is given by Eq. (1.21) [5].

(1.21)

      where the proportionality