Charged sphere

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

In an aqueous system with large particles weU-separated by a distance, s, (D and 5 > t ) the electrostatic repulsion energy between two identical charged spheres may be approximated (1) ... 

Many simple sehemes have been put forward for these repulsion integrals, which are usually written. They are taken to depend on the type of atoms that basis funetion x, and Xj are centred on, and on the distance between the atomic centres. Pariser and Parr made use of the uniformly charged sphere representation illustrated in Figure 8.1. 

The Electrical A nalogue of Magnetic Cooling. Three Processes bg Which Ions Are Introduced into Solution.. 1 Polar Dielectric in an Electrostatic Field. The Concepts of Faraday and Maxwell. The Electrostatic Energy in the Fields of Ions. The. Charging of a Condenser. The Amount of Free Energy Lost, by a Dielectric. The Behavior of Solvents in an Electrostatic Field. A Dielectric in the Field of a Charged Sphere. Two Types of Process Contrasted. 

When we deal with any spherical atomic ion in a vacuum, we may regard it as a charge sphere of radius a bearing a charge + or —q. We shall find that the correct expression for the total energy in the field is obtained by integrating (3) over all space outside the sphere. The electrical capacity of any spherical conductor is equal to its radius. The work to place a charge + or — q on this sphere is... 

After discussing in Chapter 1 a charged sphere in a dielectric, we saw in Sec. 14 that, when any pair of ions is added to a solvent, there will be a change of entropy in the co-sphere of each ion. If, for example, we knew the value of this change for the ion pair (Ag+ + Cl ) and likewise for the ion pair (Ag+ + I ), any difference between the two quantities could at once be ascribed to the difference between the co-spheres of the chloride ion Cl and the iodode ion I-, since the contribution from the Ag+ ion and its co-sphere would be the same in the two cases. 

In discussing the loss of entropy in an electric field, we may consider a charged sphere immersed either in an alcohol or in a dioxane-water mixture. In (19) in Sec. 8 we obtained an expression for the total amount... 

Let us now consider the same charged sphere immersed in various liquids with widely different values of n. By diluting water with dioxane at constant temperature, we can reduce n from 3.3 X 1022 toward zero. Clearly when n, the number of dipoles per unit volume, approaches zero, the total entropy lost per unit volume must approach zero. From this point of view the expression (170) is seen to have a somewhat paradoxical appearance, since e, which, according to Table 32, is roughly proportional to n, occurs in the denominator. This means that, as the number of dipoles per cubic centimeter decreases, the total amount of entropy lost progressively increases. The reason for this is that, when... 

The Coulomb interaction of the (point) nucleus with the potential Vo, which is also part of the monopole interaction, was neglected in (4.5) because it yields only an offset of the total energy. The subscript u in is introduced to distinguish the radius of the uniformly charged sphere from the usual mean square radius which can be obtained from scattering experiments. 

The radius rs is sometimes called the Wigner-Seitz radius and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rs and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rs is proportional to l/rs, or, equivalently, to p( r,)17 3. Hence, we arrive at the following approximate expression for Ex (Cx is a numerical constant),... 

The Na+ and Cl- ions can be considered as negatively and positively charged spheres that attract each other. Since positive (+) and negative (-) charges form an electric field in all directions, the electrostatic force of attraction (ionic bond) is not just in one direction. In the NaCl crystal, each Na+ ion is surrounded by six Cl- ions and each Cl- ion is surrounded by six Na+ ions (Figure 2). Because of this, the structure of NaCl is not a molecule but it is in the form of an ionic crystal in which many ions are found together. 

Thompson developed an atomic model, the raisin pudding model, which described the atom as being a diffuse positively charged sphere with electrons scattered throughout. 

The theory proposed by Debye and Huckel dominated the study of aqueous electrolytes from around 1920 to near the end of the 1950 s. The Debye-Huckel theory was based on a model of electrolyte solutions in which the ions were treated as point charges (later as charged spheres), and the solvent was considered to be a homogeneous dielectric. Deviations from ideal behaviors were assumed to be due only to the long range electrostatic forces between ions. Refinements to include ion-ion pairing and ion... 

Fig. 8. Attractive force between equal-sized, equally and oppositely charged spheres at various particle clearances.

Equation (50) may also be used to calculate the attractive force between equal sized, equally and oppositely charged spheres that are perfect insulators. For this condition, the value of ksA and of ksg is taken as unity if polarization possibilities are neglected, and if the charge is initially uniformly distributed. Any polarization will tend toward an approach to the conductor condition (which basically represents a condition of infinite polarization). Figure 8 presents a plot of Eq. (50). Thus, if we assume that the maximum possible field intensity is 200 V/micron, the attractive force between equally but... 

The atomic model in 1903. Thomson viewed the atom as a positively charged sphere emhedded with sufficient numbers of electrons to balance (neutralize) the total charge. 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

The Huggins coefficient kn is of order unity for neutral chains and for polyelectrolyte chains at high salt concentrations. In low salt concentrations, the value of kn is expected to be an order of magnitude larger, due to the strong Coulomb repulsion between two polyelectrolyte chains, as seen in the case of colloidal solutions of charged spheres. While it is in principle possible to calculate the leading virial coefficients in Eq. (332) for different salt concentrations, the essential feature of the concentration dependence of t can be approximated by... 

Ishikawa and coworkers [15,24] have shown that G-spinors, with orbitals spanned in Gaussian-type functions (GIF) chosen according to (14), satisfy kinetic balance for finite c values if the nucleus is modeled as a uniformly-charged sphere. 

This divergent behaviour at the origin can be avoided by considering instead of a point-like nucleus a uniform charged sphere of radius i [ 10,11,12], Then the density is forced to drop to zero at the center of the nucleus, which makes it normalizable, and the energy is finite. However, this quantity as well as p near the nucleus are highly overstimated, and for example the relativistic correction to the energy... 

Self-assembly of highly charged colloidal spheres can, under the correct conditions, lead to 3D crystalline structures. The highly charged spheres used are either polystyrene beads or silica spheres, which are laid down to give the ordered structures by evaporation from a solvent, by sedimentation or by electrostatic repulsion (Figure 5.34). The structures created with these materials do not show full photonic band gap, due to their comparitively low relative permittivity, although the voids can be in-filled with other materials to modify the relative permittivity. 

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