Dielectric spherical particle

Figure 4.26. Electrophoretic mobility as a function of the potential at the slip plane, after O Brien and White (1978). Smooth, dielectric spherical particles (1-1) electrolyte. Conduction behind the slip plane neglected has its classical meaning as the potential of a discrete slip plane - - approximation (4.6.44] for high Ka. (Redrawn from R.W. O Brien, R.J. Hunter, loc. cit. (In the original the factor 3/2 on the ordinate is missing.).)...

Example 3.1.4 Calculate the magrritude of the force due to radiation pressure on a lossless dielectric spherical particle of radius 0.5145 pm and density Pp = Ig/cm subjected to a cw argon laser light of power 1 watt at a wavelength A = 0.5145 pm. Calculate also the instantaneous acceleration experienced by the particle. 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

The simplest shape for the cavity is a sphere or possibly an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ moleculai shaped cavities, generated for example by interlocking spheres located on each nuclei. Taking the atomic radius as a suitable factor (typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such a surface may have small pockets where no solvent molecules can enter, and a more appropriate descriptor may be defined as the surface traced out by a spherical particle of a given radius rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and illustrated in Figm e 16.7. 

This result implies that AA should be a quadratic function of the ionic charge. This is exactly what is predicted by the Bom model, in which the ion is a spherical particle of radius a and the solvent is represented as a dielectric continuum characterized by a dielectric constant e [1]... 

The sum rule (4.81) for extinction was first obtained by Purcell (1969) in a paper which we belive has not received the attention it deserves. Our path to this sum rule is different from that of Purcell s but we obtain essentially the same results. Purcell did not restrict himself to spherical particles but considered the more general case of spheroids. Regardless of the shape of the particle, however, it is plausible on physical grounds that integrated extinction should be proportional to the volume of an arbitrary particle, where the proportionality factor depends on its shape and static dielectric function. 

Up to this point we have considered only the conditions for resonances in the cross sections of small spherical particles of various kinds we have said nothing quantitative about their strengths and the frequencies at which they might occur other than brief introductory remarks about ionic crystals in the infrared and metals in the ultraviolet. To determine if a resonance is realizable, where it occurs, and its strength, we need to know how the dielectric function varies with frequency. Therefore, in the following sections we shall examine some of the preceding resonance conditions in the light of simple, but realistic, dielectric functions. 

As an example of extinction by spherical particles in the surface plasmon region, Fig. 12.3 shows calculated results for aluminum spheres using optical constants from the Drude model taking into account the variation of the mean free path with radius by means of (12.23). Figure 9.11 and the attendant discussion have shown that the free-electron model accurately represents the bulk dielectric function of aluminum in the ultraviolet. In contrast with the Qext plot for SiC (Fig. 12.1), we now plot volume-normalized extinction. Because this measure of extinction is independent of radius in the small size... 

The shapes of the absorption band cease to be independent of size for particles smaller than about 26 A, which suggests that the bulk dielectric function is inapplicable. Indeed, the broadening and lowering of the absorption peak can be explained by invoking a reduced mean free path for conduction electrons (Section 12.1). Thus, the major features of surface modes in small metallic particles are exhibited by this experimental system of nearly spherical particles well isolated from one another. But when calculations and measurements with no arbitrary normalization are compared, some disagreement remains. Measurements of Doremus on the 100-A aqueous gold sol, which agree with those of Turkevich et al., are compared with his calculations in Fig. 12.18 the two sets of calculations are for optical constants obtained... 

It is not difficult to show that the emissivity of small spherical particles, composed of both insulating and metallic crystalline solids, is expected to vary as 1/A2 in the far infrared. For example, if the low-frequency limit of the dielectric function for a single Lorentz oscillator (9.16) is combined with (5.11), the resulting emissivity is... 

The LS theory was applied to the localization of a Brownian particle in a three-dimensional optical trap [89] a transparent dielectric spherical silica particle of diameter 0.6 pm suspended in a liquid [88]. The particle moves at random within the potential well created with a gradient three-dimensional optical trap—a technique widely used in biophysical studies. The potential was modulated by a biharmonic force. By changing the phase shift between the two harmonics it was possible to localize the particle in one of the wells in very good quantitative agreement with the predictions based on the LS. 

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R you cannot take Eq. (6.1), insert the total surface charge as Qi and Q2, use the dielectric permittivity of water and expect to get a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

Quantitative measurements of electrokinetic phenomena permit the calculation of the zeta potential by use of the appropriate equations. However, in the deduction of the equations approximations are made this is because in the interfacial region physical properties such as concentration, viscosity, conductivity, and dielectric constant differ from their values in bulk solution, which is not taken into account. Corrections to compensate for these approximations have been introduced, as well as consideration of non-spherical particles and particles of dimensions comparable to the diffuse layer thickness. This should be consulted in the specialized literature. 

HLC have suggested that the solvent dipoles near the colloidal particles are preferentially aligned. This effect is well known in theories of the electrical double layer. One simple way of accounting for this effect is through the use of a Stern layer of low dielectric constant near the colloidal particles. It is difficult to calculate this correction for spherical particles. As a result, HLC considered a hard sphere fluid between two hard walls and with a region of low dielectric constant near the walls. They found that Eq. (62) should be generalized to... 

Boundary effects on the electrophoretic mobility of spherical particles have been studied extensively over the past two decades. Keh and Anderson [8] applied a method of reflections to investigate the boundary effects on electrophoresis of a spherical dielectric particle. Considered cases include particle motions normal to a conducting wall, parallel to a dielectric plane, along the centerline in a slit (two parallel nonconducting plates), and along the axis of a long cylindrical pore. The double layer is assumed to be infinitely thin... 

This may be clear from Fig. 56. Consider a conducting, solid, spherical particle of radius a, carrying a positive charge q, immersed in a liquid of dielectric constant D. The potential of the sphere is q/Da. Next consider the contribution to the potential difference between the sphere and the liquid made by a spherical shell in the liquid, of radius r and thickness dr the charge dq on this will be opposite in sign to that on the sphere, and the contribution to the difference in potential between the surface of the solid and the liquid will be dqjDr. The total difference in potential between the surface of the solid and the liquid beyond the outer limit of the double layer will be the sum of the contributions from the sphere and all the shells, i.e. 

Here e = er + i e, is the complex dielectric function, that describes in electromagnetic theory the reaction of matter on an external electric field and a the polarizability1 of a homogeneous spherical particle per unit volume. It is assumed that the particle is embedded in a material with real part em of its dielectric function and vanishing imaginary part. For particles embedded in vacuum (= ISM) we have em = 1. The complex dielectric function has to be determined by laboratory measurements for the materials of interest. 

In their investigation of dilute solutions of simple ions, Debye and Hiickel regarded the jth small ion as a spherical particle with radius a such that the small ion has an electric charge zf at its spherical center and the dielectric constant e inside the sphere. In this case, the total electrostatic energy of the system is obtained as... 

Line-width broadening may also be caused by other fast relaxation mechanisms in addition to a small particle size. For example, it is well known that, for spherical particles, radiation losses become more pronounced with increasing radius. In some metals, these relaxation mechanisms are so strong that a well-defined plasmon resonance is not observed, as in Fe, Pd, and Pt. Nanosized particles are interesting because the optical resonance can be designed in. For example, in a nanoshell consisting of a dielectric core surrounded by a metallic outer layer, the relative dimensions of these components can be varied. This, in turn, varies the optical resonance, possibly over several-hundred nanometers in wavelength. 

FIGURE 10.10 (a) Plot of the attraction and repulsion forces acting on two spherical particles as a function of particle separation for constant dielectric constant, surface potential, and double layer thickness resulting from 1 1 salt (b) Plot of the sum... 

So far, only a limited number of full dielectric relaxation spectra for well defined systems are available. Apart from the technical problems involved in the measurements (sec. 4.5e) there is the colloidal problem of synthesizing sufficiently concentrated sols with homodisperse spherical particles, preferably having different radii but fixed surface properties. Latices are popular objects because the particles are easily made homodisperse and spherical. Nevertheless they are somewhat suspect because there may be hairs on the surface, drastically affecting lateral hydrodynamic motion close to the surface. Moreover, changing the radius requires new syntheses and it is difficult to guarantee exact reproducibility of the surface structure. Inorganic particles do not have these drawbacks but it is not so ea to synthesize these as perfect spheres. 

Figure 4.38. Dielectric relaxation and conductivity spectrum for haematite sols with spherical particles a = 257 nm. in KCl solution (O) pH = 4.9,

Maxwell-Wagner Dispersion.—Macroscopic heterogeneities of the complex permittivity must always result in an apparent overall dielectric dispersion, even if the above-discussed orientation effect cannot occur. We may investigate this for the case of spherical particles of radius a and complex permittivity eg which are suspended in a medium with complex permittivity eg. It follows from electrostatic theory that the presence of one such sphere is equivalent to a dipole moment given by... 

In Eq. (1), So is the permittivity of free space (8.8 X 10 F/m), E is the maximum electric field before breakdown of air occurs (3 x lO V/m), a is the particle radius in m, and Sr is the relative dielectric constant of a powder particle. This equation shows that the maximum charge that can be acquired is proportional to the electric field and the square of the particle radius. It is also weakly dependent on the relative dielectric constant of the particle. The charge-to-mass ratio takes into account the mass of the particle and is determined by dividing Eq. (1) by the mass of a spherical particle, (4/3) na p, where p is the density of the... 

Here e is the electron charge, e is the dielectric constant of the medium, a is the distance of the maximum approach of a small ion and a spherical particle, often taken equal to (b-H 2.5A), X is the Debye-Hiickel parameter, 3.57 is the constant coefficient for aqueous systems at 25°C on condition that distances are measured in angstroms), pkjnt is the intrinsic ionization constant, Z = —on is the average value of the total charge of a particle (it is equal numerically to the number of protons having formed on dissociation), n is the number of dissociating groups in a particle. 

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