Quadrupolar forces

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

It is well-known that the interaction energy falls off more rapidly the higher the order of the multipole. Thus, for the interaction of an n-pole with an m-pole, the potential energy varies with distance as E o l/(r ). The reason for the faster decrease is that the array of charges seems to blend into neutrality more rapidly with distance the higher the number of individual charges contributing to the multipole. Consequently, quadrupolar forces die off faster than dipolar forces. 

The forced electric dipole mechanism was treated in detail for the first time by Judd (1962) through the powerful technique of irreducible tensor operators. Two years later it was proposed by Jorgensen and Judd (1964) that an additional mechanism of 4/-4/ transitions, originally referred to as the pseudo-quadrupolar mechanism due to inhomogeneities of the dielectric constant, could be as operative as, or, for some transitions, even more relevant than, the forced electric dipole one. 

Before this equation can be applied to describe the movement of ions in a quadrupolar field the parameters a, q, and must be defined. This is possible using classical mechanics. The force in the x-direction, Fx, experienced by an ion of mass m and charge e may be expressed as... 

The ion trap works on a similar principle to the linear quadrupole. Oscillating RF and DC voltages are applied to the electrodes. These voltages create a quadrupolar field in the ion trap, and ions can be retained in stable trajectories in this field. The force acting upon the ions in the trap is directly proportional to the distance of the ions from the center of the ion trap. Therefore the quadrupolar field acts to store the ions as a packet at the center of the trap. For an ion to be retained in the ion trap it needs to have a stable trajectory in both the axial (z) and radial (r) directions. Whether the ion is stable in one or both of these directions is given by the solutions to the Mathieu equation previously used to describe ion motion through the linear quadrupole with the reduced Mathieu parameters ... 

Chemisorption systems are sometimes used for removing trace concentrations of contaminants, but the difficulty of regeneration makes such systems unsuitable for most process applications so most adsorption processes depend on physical adsorption. The forces of physical adsorption are weaker than the forces of chemisorption so the heats of physical adsorption are lower and the adsorbent is more easily regenerated. Several different types of force are involved. For nonpolar systems the major contribution is generally from dispersion-repulsion (van der Waals) forces, which are a fundamental property of all matter. When the surface is polar, depending on the nature of the sorbate molecule, there may also be important contributions from polarization, dipole, and quadmpole interactions. Selective adsorption of a polar species such as water or a quadrupolar species such as CO2 from a mixture with other nonpolar species can therefore be accomplished by using a polar adsorbent. Indeed, adjustment of surface polarity is one of the main ways of tailoring adsorbent selectivity. 

One reason for this at first sight unexpected result is the fact that probably 70... 90% of the solute/solvent interaction term is caused by London dispersion forces, which are more or less equal for the cis and trans isomers. Another important reason is that one has to take into account higher electric moments the trans isomer has a quadrupole moment, and the cis isomer also has moments of a higher order than two. Calculations of solute/solvent interactions of both diastereomers using a reaction field model led to the conclusion that the quadrupolar contribution of the trans isomer is comparable to the dipolar contribution of the cis isomer. It has been pointed out that the neglect of solute/solvent interactions implying higher electric moments than the dipole moment can lead to completely false conclusions [202],... 

Fig. 69 a Formation of supramolecular polymers between an AFM-tip and a SAM via quadrupolar hydrogen bonds, b Force-extension curves of the aggregates. Reprinted with permission from [237]... 

The quadrupolar splittings depend mainly on the electric field gradient at the nucleus, which is caused by the spatial distribution of the electrons around it, via electric Coulomb forces. Both electrons (valence contribution f/yai) and neighboring anions and cations (lattice contribution U x) in the vicinity of the nucleus contribute to this electric field gradient U, which can be expressed as follows ... 

Unlike water, C02 has no dipole moment. Therefore, hydrophilic molecules are practically insoluble even in supercritical C02. Van der Waals forces, arising principally from quadrupolar interactions are weaker even than in hydrocarbons. In principle, addition of surfactants would improve the solubilisation properties of compressed C02 and could also result in self-assembled microstructures. However, most commercially available surfactants are insoluble in supercritical C02 [48]. 

Thus, the magnitude of physisorption forces is most strongly affected by the size, polarity, polarizability, and quadrupolarity of sorbate atoms or molecules, as well as the electric field strength and the local field gradient of the solid surface. 

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