Polarizability of particles

EXAMPLE 5.1 Polarizability of Particles. Criticize or defend the following proposition on the basis of the units of a/4ire0 The larger the volume of a particle is, the easier it is to induce polarity in that particle. 

Polarizability of particles Molecular weights from Rayleigh ratios Use of Zimm plots... 

Dielectrophoresis is the motion of polarizable particles that are suspended in an ionic solution and subjected to a spatially non-uniform electric field. Polarizability of particle relative to the suspending medium determines the basic direction of DEP force (positive/negative DEP), which also strongly depends on the frequency of the applied electric field. In the case of electric field with constant phase, time-averaged DEP force can be represented as [8]... 

This energy is summated from the change in the eneigy of interaction of dipoles A, B, and A.B with the medium and the change in the polarization eneigy of particles by the medium (aA and ae are the polarizabilities of particles A and B)... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

In addition to the interactions discussed above, which all depend in part on the ioniz-ability, or at least polarizability, of the surface and the adsorbates, hydrophobic parts of ligands may bind to corresponding parts of surfaces. Thus, if a metal ion is complexed or irreversibly bonded to a hydrophobic molecule, the metal may be incorporated into the bulk or surface of a particle via hydrophobic interaction between the molecule and the solid phase. Such interactions may be quantitatively significant in systems with high concentrations of dissolved and particulate organic matter. 

The particles of the preceding treatment are here replaced by small elements of volume of the solution. The excess polarizability of one of these volume elements due to the deviation of its concentration from the average may be written... 

Mixed solvents are generally unsatisfactory for use in the determination of polymer molecular weights owing to the likelihood of selective absorption of one of the solvent components by the polymer coil. The excess of polarizabilit f of the polymer particle (polymer plus occluded solvent) is not then equal to the difference between the polarizabilities of the polymer and the solvent mixture. For this reason the refractive increment dn/dc which would be required for calculation of K, or of i7, cannot be assumed to equal the observed change in refractive index of the medium as a whole when polymer is added to it, unless the refractive indexes of the solvent components happen to be the same. The size Vmay, however, be measured in a mixed solvent, since only the dissymmetry ratio is required for this purpose. 

Of course, encapsulation can also be governed by other factors besides the charge control. Thus, the anion-radical of fullerene Cgg exhibits better inclusion ability with cyclodextrins than neutral Cgg (Liu et al. 2007). Polarizability of the anion-radical is much larger than that of neutral fullerene. This factor enhances dispersion forces and assists encapsulation of the charged particle by cyclodextrin. 

First item in a series (for example, the first carbon linked to a carboxyl group). 2. Abbreviation for alpha particle. 3. Symbol for angle of optical rotation. 4. Symbol for degree of dissociation. 5. Symbol for electric polarizability of a molecule. 6. Often with a subscript number (i) the coefficient of [A] in the numerator of the generalized rate expression. 7. Symbol for is proportional to. 8. Symbol for Napierian absorption coefficient. 9. In brackets, symbol for specific optical rotation. 

A parameter used to characterize ER fluids is the Mason number, which describes the ratio of viscous to electrical forces, and is given by equation 14, where 8 is the solvent dielectric constant T 0, th solvent viscosity 7, the strain or shear rate P, the effective polarizability of the particles and E, the electric field (117). 

It might seem at first glance that arriving at the dipole moment p of an ellipsoidal particle via the asymptotic form of the potential < p is a needlessly complicated procedure and that p is simply t>P, where v is the particle volume. However, this correspondence breaks down for a void, in which P, = 0, but which nonetheless has a nonzero dipole moment. Because the medium is, in general, polarizable, uP, is not equal to p even for a material particle except when it is in free space. In many applications of light scattering and absorption by small particles—in planetary atmospheres and interstellar space, for example—this condition is indeed satisfied. Laboratory experiments, however, are frequently carried out with particles suspended in some kind of medium such as water. It is for this reason that we have taken some care to ensure that the expressions for the polarizability of an ellipsoidal particle are completely general. 

We noted in the preceding section that the polarizability of an ellipsoid is anisotropic the dipole moment induced by an applied uniform field is not, in general, parallel to that field. This anisotropy originates in the shape anisotropy of the ellipsoid. However, ellipsoids are not the only particles with an anisotropic polarizability in fact, all the expressions above for cross sections are valid regardless of the origin of the anisotropy provided that there exists a coordinate system in which the polarizability tensor is diagonal. 

It is interesting to compare and contrast an isotropic ellipsoid and an anisotropic sphere the polarizability of both particles is a tensor, the principal values of which are... 

Solution The ratio a/4ire0 has the units (C2 m N -1)/(C2 m-2 N-1) = m3, which are units of volume. In fact, polarizabilities of actual molecules are on the order of 10 29 m3 molecule-1 = 0.01 nm3 molecule-1, which is the magnitude of molecular volumes. For individual atoms, we expect the polarizability to increase with atomic volume since the outermost electrons are less tightly restrained by the nucleus in such cases. Extension of this principle to covalently bonded species must be done cautiously, however, since the bonding affects the overall picture. The accuracy of the proposition, then, depends on the nature of the particle under consideration. Even when the principle stated in the proposition is not literally true, it offers a convenient mnemonic for the definition of polarizability. ... 

Now let us consider the effect of such a field on a particle with no permanent dipole of its own. The field will induce a dipole in the second molecule the magnitude of the induced dipole moment n2 is proportional to the field with the polarizability of the second molecule a2 the proportionality constant (see Equation (5.11)) ... 

In this formula V is the volume of one gram mol of a substance containing N particles per gram mol. If the substance consists of molecules, then a — Eoq, where oc is the polarizability of the ith atom in the molecule. In gases where is nearly unity, the formula can be further simplified by putting n% 2 = 3 which gives... 

The refractive index, nD, defined as the ratio of light speed at the sodium D-line in a vacuum to that in the medium, is used in obtaining the polarizability, a, of solvent molecules. The relationship between a and nD is given by a = (3Vm/ 4 rNA)( D-l)/(wD + 2), where NA is the Avogadro constant and Vm is the molar volume.4 Solvent molecules with high a-values tend to interact easily with one another or with other polarizable solute particles by dispersion forces.5 ... 

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