Colloidal particles Permittivity

When working with colloidal dispersions, we mostly deal with aqueous solutions, in which colloidal particles are immersed. From this point of view, all colloidal material can be separated into the two categories—dielectrics, with d <5C ew (fid and w are dielectric constants of a dielectric and water, respectively), and conductors, with c > w (sc is the dielectric permittivity of a conductor). 

Electrophoresis — Movement of charged particles (e.g., ions, colloidal particles, dispersions of suspended solid particles, emulsions of suspended immiscible liquid droplets) in an electric field. The speed depends on the size of the particle, as well as the -> viscosity, -> dielectric permittivity, and the -> ionic strength of the solution, and it is directly proportional to the applied electric field. In analytical as well as in synthetic chemistry electrophoresis has been employed to separate species based on different speeds attained in an experimental setup. In a typical setup the sample is put onto a mobile phase (dilute electrolyte solution) filled, e.g., into a capillary or soaked into a paper strip. At the ends of the strip connectors to an electrical power supply (providing voltages up to several hundred volts) are placed. Depending on their polarity and mobility the charged particles move to one of the electrodes, according to the attained speed they are sorted and separated. (See also - Tiselius, - electrophoretic effect, - zetapotential). 

A dielectric sphere of dielectric coefficient e embedded in an infinite dielectric of permittivity 82 is an important case from many points of view. The idea of a cavity formed in a dielectric is routinely used in the classical theories of the dielectric constant [67-69], Such cavities are used in the studies of solvation of molecules in the framework of PCM [1-7] although the shape of the cavities mimic that of the molecule and are usually not spherical. Dielectric spheres are important in models of colloid particles, electrorheological fluids, and macromolecules just to mention a few. Of course, the ICC method is not restricted to a spherical sample, but, for this study, the main advantage of this geometry lies just in its spherical symmetry. This is one of the simplest examples where the dielectric boundary is curved and an analytic solution is available for this geometry in the form of Legendre polynomials [60], In the previous subsection, we showed an example where the SC approximation is important while the boundaries are not curved. As mentioned before, using the SC approximation is especially important if we consider curved dielectric boundaries. The dielectric sphere is an excellent example to demonstrate the importance of curvature corrections . 

Impedance is the ratio of the voltage across a system to the current passing through the system. It measures the dielectric properties (permittivity and conductivity) of the system. The dielectric behavior of colloidal particles in suspension is generally described by Maxwell s mixture theory [26]. This relates the complex permittivity of the suspension to the complex permittivity of the particle, the suspending medium and the volume fraction. Based-on Maxwell s mixture theory, shelled-models have been widely used to model the dielectric properties of particles in suspension [35-40]. A single shelled spherical model is shown in Fig. la. 

In the high relative permittivity, r, of water most polymer colloid particles carry an electric charge. Frequently, depending on the synthetic method used to prepare the particles this arises from ionizable groups on the surface, such as... 

Schwartz (1962) proposed a model with a tightly bound layer of adsorbed counterions on the sphere surface to explain the high permittivity increment found (hundreds of 8q) in a suspension of colloidal particles. Diffusion, and not migration, may govern ionic motion in a double layer. Diffusion processes are not necessarily exponential, but as an... 

Dielectric dispersion. It is the change with the frequency of an applied alternating current (AC) field of the dielectric permittivity of a suspension of colloidal particles. The phenomenon... 

The Theoretical Basis of Latex Stabiiization. The colloidal stabihty of each class of latex is primarily dependent on the effectiveness of the sindactant. In the high permittivity of water, most polymer colloid particles carry an electric... 

Note R = radius of solid particles, e and /j, = respective permittivity and permeability the index 2 refers to the colloidal particle the index 1 denotes the swelling agent. 

Cj is the permittivity (dielectric constant) 78.6 for water at 25 °C. e, is the permittivity of free space, k is the Boltzmann constant and T is the absolute temperature, n is the number of ions per unit volume of each type present in bulk solution and Zj is the valency of the ions, e is the electronic charge. When charged colloidal particles in a... 

There are various expressions for describing the electrostatic (typically) repulsive forces between colloidal particles but all of them indicate that the potential energy decreases exponentially with the distance and increases with increasing Debye length, relative permittivity and surface potential (see Table 10.5). Thus the key properties appearing in the various mathematical expressions are ... 

Stable colloids are achieved if the Debye length (double layer thickness) is very high (i.e. low salt content, low valency ions), if the colloid particles are in a medium with high relative permittivity, if they have low (or even negative) Hamaker constants and high values of the surface or zeta potential. Control of the ionic concentration and surface charge are crucial. 

The above theory describing relaxations due to polarization of counterion cloud and processes of adsorption of coimterions onto a central colloidal particle can be referred to as a "macromolecule" theory." The permittivity increment from the coimterion relaxation theory was offered as [11] ... 

In Equation 19.12, Cq = 8.854 x j-i qi -1 jg jjjg dielectric constant in vacuum, e is the relative dielectric permittivity of the solvent (e = 78.5 for water at room temperature 298 K), and are the electrokinetic zeta potential defined at the shear plane (see Figure 19.3), r is the dynamic viscosity of the solvent (q = 8.91 x 10 kgm" s for water at room temperature 298 K), and E is the externally applied electric field. The first equation in Equation 19.12 represents the fluid motion in a stationary channel under the action of an externally appUed electric field. The motion is called electro-osmosis and the velocity is v. The second equation in Equation 19.12 gives the velocity v, of charged suspended colloidal particle (or a dissolved molecule) driven by the same electric field. This phenomenon is called electrophoresis. The EDL thickness 1/k depends on the concentration of background electrolyte [18,19,25,26]. 

For many colloids, the particle permittivity is much less than that of water, and in this case the surface average tangential electric field is given by (see Equation 4.18 below for small X,) ... 

Models of colloidal electrohydrodynamics relate the electrophoretic mobility /t to the zeta potential , the particle radius / , the composition of the solution via the Debye length /k, and the solvent viscosity and permittivity, // and f. Dimensional analysis shows that these variables must be related by... 

Exact numerical solutions of the full colloidal electrohydrodynamic problem have appeared in recent years. For computational convenience, the numerical schemes treat as an independent variable the user varies until the predicted and measured values of n agree. The earliest numerical solutions [132] were hampered by convergence difficulties at relatively low values of . O Brien and White [133] resolved these numerical problems, and their solution is widely used today. Some recent publications [134,135] document the evolution of analytical and numerical solutions of the colloidal electrohydrodynamics problem and report numerical solutions for particle mobility, suspension conductivity, and suspension dielectric permittivity for both constant and oscillatory applied electric fields. 

We must solve the problem of finding the steady velocity acquired by a colloidal sphere of radius a and total surface charge Q upon application of an electric field that, far from the particle, equals E. The particle is considered nonconducting and with an electric permittivity much smaller than that of the dispersion medium. Eor the moment, we will also assume that the electrolyte concentration is very low and that a is small enough for the following inequality to hold between the double-layer thickness (Equation (3.5)) and the radius ... 

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