Electrical spherical double layer

Chapters 11 and 12 in the present edition focus exclusively on the theories of electrical double layers and forces due to double-layer interactions (Chapter 11) and electrokinetic phenomena (Chapter 12). Chapter 11 includes expressions for interacting spherical double layers, and both chapters provide additional examples of applications of the concepts covered. 

Chan, B.K.C. Chan, D.Y.C. Electrical doublelayer interaction between spherical colloidal particles an exact solution. J. Colloid Interface Sci. 1983, 92 (1), 281-283 Palkar, S.A. Lenhoff, A.M. Energetic and entropic contributions to the interaction of unequal spherical double layers. J. Colloid Interface Sci. 1994, 165 (1), 177-194 Qian, Y. Bowen, W.R. Accuracy assessment of numerical solutions of the nonlinear Poisson-Boltzmann equation for charged colloidal particles. J. Colloid Interface Sci. 1998, 201 (1), 7-12 Carnie, S.L. Chan, D.Y.C. Stankovich, J. Computation of forces between spherical colloidal particles Nonlinear Poisson-Boltzmann theory. J. Colloid Interface Sci. 1994, 165 (1), 116-128 Stankovich, J. Carnie, S.L. Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate nonlinear Poisson-Boltzmann theory. Langmuir 1996,12 (6), 1453-61. 

To determine the distribution of electric potential, as modified by surface conductance, we again take the electric field to be spherically symmetric and to satisfy the Laplace equation. A thin spherical double layer shell is considered to surround the particle, and the conductivity of this shell is taken to have the mean value cr(. In reality the conductivity in the thin double layer varies continuously. Outside of the double layer shell the bulk conductivity is that of the electrolyte. This electrostatics problem is a straightforward one in which, from the Laplace equation, the solution for the potential is... 

For small particles, however, we see that the factor 1/r causes the electric potential to fall off more rapidly than the purely exponential expression found for the flat double layer. If indeed, the particle is small in comparison to the thickness of the double layer, then the charge in the surrounding ionic atmosphere, going from the particle surface to the bulk of the solution, must be distributed among spherical shells of increasing volume. This explains the characteristic difference between spherical and flat double layers, and shows that the equations for the spherical double layer are especially important if xa 1. 

In a similar way we may understand another remarkable point ro be derived from M ii 11 e r s tables. For a flat double layer we found that the form of the electric potential curve is radically changed by an increase in the ionic charge. M ii 11 e r s data however, show, that for a spherical particle (with jco 1) the valency of the ions in the solution has only a minor influence upon the decline of the electric potential in the diffuse layer. Hence, also in this respect the Debyc-Hiickel theory is a much better approximation for the spherical double layer field than for the flat double layer, once we wish to apply this theory to cases where the potential is no longer small. 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, The Electrical Double Layer Around a Spherical Particle, MIT Press, Cambridge, MA, 1961. 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

An illustration of the effect of micelle/nanoparticle volume fraction on contact line motion is found in [57]. They used 0.1 M NaCl solution to reduce the electrical double layer thickness surrounding the NaDS micelle. At a given number concentration of micelles, decreasing the size of each micelle decreases the volume fraction greatly, since the volume of each spherical micelle varies as the third power of the radius. Thus, the addition of electrolyte effectively reduced the micellar volume fraction in the aqueous medium. The authors found that the oil droplet that would otherwise become completely detached from the solid surface, came back to reattach itself to the solid when electrolyte was present. They rationalized this finding as being caused by the inability of the weakened structural disjoining forces to counteract the attraction of the oil drop to the solid surface. 

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),... 

Before we proceed to the Gouy-Chapman theory of electrical double layers, it is worthwhile to note that relations similar to Equations (45) and (47) can also be derived for double layers surrounding spherical particles. The equation for surface charge density takes the form... 

Show that the one-dimensional Poisson equation for planar electrical double layers discussed in the text has the following analog in the case of spherically symmetric double layers ... 

FIG. 12.1 Streamlines (which also represent the electric field) around spherical particles of radius Rs. The dashed lines are displaced from the surface of the spheres by the double-layer thickness k. In (a) kRs is small in (b) kRs is large. 

If there are sufficiently strong repulsive interactions, such as from Ihe electric double-layer lorce. then the gas bubbles at the lop of u froth collect together without bursting. Furthermore, their interfaces approach as closely as these repulsive forces allow typically on the order of 100 nm. Thus bubbles on top of a froth can pack together very closely and still allow most uf the liquid to escape downward under the influence of gravity while maintaining their spherical shape. Given sufficient liquid, such a foam can resemble the random close-packed structure formed by hard spheres. 

Figure D3.5.8 (A) Three possible models for systems (spherical particle) with an electrical double layer. (B) Corresponding electrical potential as a function of the separation distance.

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 calculation of the interaction energy, VR, which results from the overlapping of the diffuse parts of the electric double layers around two spherical particles (as described by Gouy-Chapman theory) is complex. No exact analytical expression can be given and recourse must be had to numerical solutions or to various approximations. 

The history of PB theory can be traced back to the Gouy-Chapmann theory and Debye-Huchel theory in the early of 1900s (e.g., see Camie and Torrie, 1984). These two theories represent special simplified forms of the PB theory Gouy-Chapmann theory is a one-dimensional simplification for electric double-layer, while the Debye-Huchel theory is a special solution for spherical symmetric system. The PB equation can be derived based on the Poisson equation with a self-consistent mean electric potential tj/ and a Boltzmann distribution for the ions... 

In the simplest example of colloid stability, suspension partides would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The repulsive energy VR for spherical particles, or rigid droplets, is given approximately as ... 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

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