Electric double layer potential distribution

Biological macromolecules are often handled through microfluidic systems, in which these molecules can transport and react. A common driving force behind such microfluidic transport processes is the electrokinetic force, which originates as a consequence of interaction between the electric double-layer potential distribution and the applied electric field. This entry discusses some of the important features of the biochemical reactions in such microfiuidic systems. 

Solving the Poisson-Boltzmann equation with proper boundary conditions will determine the local electrical double layer potential field y/ and hence, via Eq.(3), the local net charge density distribution. 

FIGURE 2-1 Helmholtz model of the electrical double layer, (a) Distribution of counterions in the vicinity of the charged surface. (b) Variation of electrical potential with distance from the charged surface. 

When two conducting phases come into contact with each other, a redistribution of charge occurs as a result of any electron energy level difference between the phases. If the two phases are metals, electrons flow from one metal to the other until the electron levels equiUbrate. When an electrode, ie, electronic conductor, is immersed in an electrolyte, ie, ionic conductor, an electrical double layer forms at the electrode—solution interface resulting from the unequal tendency for distribution of electrical charges in the two phases. Because overall electrical neutrality must be maintained, this separation of charge between the electrode and solution gives rise to a potential difference between the two phases, equal to that needed to ensure equiUbrium. 

On the electrode side of the double layer the excess charges are concentrated in the plane of the surface of the electronic conductor. On the electrolyte side of the double layer the charge distribution is quite complex. The potential drop occurs over several atomic dimensions and depends on the specific reactivity and atomic stmcture of the electrode surface and the electrolyte composition. The electrical double layer strongly influences the rate and pathway of electrode reactions. The reader is referred to several excellent discussions of the electrical double layer at the electrode—solution interface (26-28). 

Previous considerations have shown that the interface between two conducting phases is characterised by an unequal distribution of electrical charge which gives rise to an electrical double layer and to an electrical potential diflFerence. This can be illustrated by considering the transport of charge (metal ions or electrons) that occurs immediately an isolated metal is immersed in a solution of its cations ... 

Figure 23. Electric potential distribution in electric double layer. HL, Helmholtz layer DL, diffuse layer.

When an electrode is in contact with an electrolyte, the interphase as a whole is electroneutral. However, electric double layers (EDLs) with a characteristic potential distribution are formed in the interphase because of a nonuniform distribution of the charged particles. 

From the equilibrium requirement that the chemical potential involving all ionic species be uniform throughout the phase boundary, the distribution of ions within the electrical double layer can be expressed by the Boltzmann equation ... 

Fig. 4.1 Structure of the electric double layer and electric potential distribution at (A) a metal-electrolyte solution interface, (B) a semiconductor-electrolyte solution interface and (C) an interface of two immiscible electrolyte solutions (ITIES) in the absence of specific adsorption. The region between the electrode and the outer Helmholtz plane (OHP, at the distance jc2 from the electrode) contains a layer of oriented solvent molecules while in the Verwey and Niessen model of ITIES (C) this layer is absent...

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The physical sense of the distribution potential can be demonstrated on the example of the distribution equilibrium of the salt of a hydrophilic cation and a hydrophobic anion between water (wt) and an organic solvent that is immiscible with water (org). After attaining distribution equilibrium the concentrations of the anion and the cation in each of the two phases are the same because of the electroneutraUty condition. However, at the phase boundary an electrical double layer is formed as a result of the greater tendency of the anions to pass from the aqueous phase into the organic phase, and of the cations to move in the opposite direction. This can be characterized quantita tively by quantities—and — AGJJ. ", for which... 

We commence with the adsorption of nonionic surfactants, which does not require the consideration of the effect of the electrical double layer on adsorption. The equilibrium distribution of the surfactant molecules and the solvent between the bulk solution (b) and at the surface (s) is determined by the respective chemical potentials. The chemical potential /zf of each component i in the surface layer can be expressed in terms of partial molar fraction, xf, partial molar area a>i, and surface tension y by the Butler equation as [14]... 

In Eq. 30, Uioo and Fi are the activity in solution and the surface excess of the zth component, respectively. The activity is related to the concentration in solution Cioo and the activity coefficient / by Uioo =fCioo. The activity coefficient is a function of the solution ionic strength I [39]. The surface excess Fi includes the adsorption Fi in the Stern layer and the contribution, f lCiix) - Cioo] dx, from the diffuse part of the electrical double layer. The Boltzmann distribution gives Ci(x) = Cioo exp - Zj0(x), where z, is the ion valence and 0(x) is the dimensionless potential (measured from the Stern layer) obtained by dividing the actual potential, fix), by the thermal potential, k Tje = 25.7 mV at 25 °C). Similarly, the ionic activity in solution and at the Stern layer is inter-related as Uioo = af exp(z0s)> where tps is the scaled surface potential. Given that the sum of /jz, is equal to zero due to the electrical... 

Fig. 3. The structure of electrical double layer at a semiconductor-electrolyte interface (a) and the distribution of the potential (b) and charge (c) at the interface. The electrode is charged negatively. is the space-charge region thickness, La is the Helmholtz layer thickness, Qlc and Qtl are the charge of the semiconductor and ionic plates of the double layer, respectively (for further notations see the text).

The purpose of this chapter is to introduce the basic ideas concerning electrical double layers and to develop equations for the distribution of charges and potentials in the double layers. We also develop expressions for the potential energies and forces that result from the overlap of double layers of different surfaces and the implication of these to colloid stability. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

Finally, if the thickness of the electrical double layer (diffuse layer) in the droplet is comparable with r, the r dependence of kqbs will be dependent on the TBA+ concentration in the droplet since the spatial distribution of the inner electric potential of the droplet varies with [TBA+TPB ], However, since results analogous with those in Figure 14a ([TBA+TPB ] = 10 mM) have been obtained even at [TBA+TPB"] = 5mM (Aodiffuse layer effect does not contribute to the r effect on kobs at r > 1 /an. 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

The net result is shown schematically in Figure 16. Instead of the array of surface charges leading to a well defined surface charge density and surface potential there is now a distribution of charges in space which contribute to the electrical double layer surrounding the particle. At low electrolyte concentrations the latter will extend into the space beyond the polyelectrolyte... 

An important consequence of (2) is the extension of the Boltzmann distributions to the non-equilibrium case provided (I> is replaced by the relative potential Tp = — ip. Thus, unlike the streaming potential w which only appears at non-equilibrium conditions, the excess Tp plays the role of a potential purely related to electrical double layer effects. Using the above change of variables one can rephrase (1) in terms of cb and Tp as follows... 

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