Dispersion interactions, retardation

The theoretical treatment of the hydrophobic effect is limited to pure aqueous systems. To describe chromatographic separations in RPC Horvath and Melander developed the solvophobic theory [47]. In this theory, no special assumptions are made about the properties of solute and solvent, and besides hydrophobic interaction electrostatic and other specific interactions are included. The theory has been valuable to describe the retention of nonpolar [48], polar [49], and ionizable [50] solutes in RPC. The modulation of selectivity via secondary equilibria (variation of pH, ion pair formation [51]) can also be described. On the other hand, it is not a problem to find examples of dispersive interactions in literature, e.g., separation of carotinoids with a long chain (C30) RP gives a higher selectivity compared to standard RP C18 cyclohexanols are preferentially retarded on cyclohexyl-bonded phases compared to phases with linear-bonded alkyl groups. 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

To summarize, the retarded interactions are important only for small wave vectors, of the order of that of the photons. For larger wave vectors the retarded interactions are uncoupled, in the sense that they do not contribute to the local field which describes the interaction between dipoles. This property allows us to understand why in global effects (cohesion energy, dispersion, etc.) retarded interactions make very small contributions, although for small K, the retarded interactions may show very strong effects (such as the quasi-metallic reflection of certain dyes,1 s or of the second singlet of the anthracene crystal). In particular, in all phenomena that involve interactions between excitons and free radiation, the retarded effects are by no means essential. 

The description of transport is closely connected to the terms convection, diffusion dispersion, and retardation as well as decomposition. First, it is assumed that there are no interactions between the species dissolved in water and the solid phase, through which the water is flowing. Moreover, it is assumed that water is the only fluid phase. The multiphase flow water-air, water-organic phase (e g. oil or DNAPL) or water-gas is not considered here. 

A final interesting observation is the existence of a frequency scale, 3x10 see in Eq. (2-39). This is the frequency at which the electronic cloud around an atom fluctuates it is therefore the rate at which the spontaneous dipoles fluctuate. Since the electromagnetic field created by these dipoles propagates at the speed of light c = 3 x lO cm/sec, only a finite distance c/v 100 nm is traversed before the dipole has shifted. Since the dispersion interaction is only operative when these dipoles are correlated with each other, and this correlation is dismpted by the time lag between the fluctuation and the effect it produces a distance r away, the dispersion interaction actually falls off more steeply than r when molecules or surfaces become widely separated. This effect is called the retardation of the van der Waals force. The effective Hamaker constant is therefore distance dependent at separations greater than 5-10 nm or so. 

The asymptotic behavior of the dispersion interaction at large intermolecular separations does not obey Equation 5.166 instead, oc lf due to the electromagnetic retardation effect established by Casimir and Polder. Various expressions have been proposed to account for this effect in the Hamaker constant. ... 

J.K. Jenkins, A. Salam, T. Thirunamachandran, Retarded Dispersion Interaction Energies Between Chiral Molecules. Phys. Rev. A 50 (1994) 4767. 

For most substances, the retardation effect in the dispersion interaction energy becomes significant at distances greater than 100 A. The important point is that the Hamaker constant A defined in Eq. (167) is no longer constant but depends on the separation of interacting bodies. 

Further studies of the van der Waals interaction among molecules having permanent dipole moments have been carried out using quantum mechanical methods by Dzyaloshinskii et Their approach includes the retardation effect as well as a polarization effect due to molecular dipoles. Then, the total dispersion interaction force (F) acting on a unit area of each of the slabs is given by... 

As mentioned in an earlier section, the dispersion interactions exhibit a quantum-mechanical retardation effect at large (on the atomic or molecular scale) distances. Such effects are brought out explicitly by the Lifshitz theory, so that, for example, long range interactions become proportional to r instead of r, where r is the distance of separation, as is observed experimentally. Luckily, however, the effects are seldom of concern in practical systems since their magnitude is extremely small relative to other factors. 

The non-retarded Hamaker constant Ai32,o reflects the contribution of both static as well as dispersion interactions. At large surface distances h, the latter vanish and the Hamaker function tends towards its static part Ai32,s ... 

Let us first deal with the dispersion (London) interaction. This interaction is of a non-polar nature, in a non-polar liquid such as carbon tetrachloride, London s dispersion interaction is the only force present between two molecules. These non-polar molecules do not possess any permanent dipole moment. The interaction is a resultant of Instantaneous dipoles formed between the nuclei and electrons at zero-point motion of the molecule. Dispersion forces are weak. When two non-polar molecules of the same type approach each other closely enou for their electronic orbitals to overlap, the weak attraction changes to repulsion. Thus, non-polar molecules exist in a state of random distribution to give a disordered array. Another non-polar molecule (whether a solute or a solvent) will mix in all proportions since neither kind of the molecule has any attraction between them. From the foregoing, it is easy to understand that a non-pK)lar solute molecule will interact more with the phase which is non-polar this solute molecule will move fast if the non-polar phase is the mobile phase or will be retarded more and move slowly if the non-p>olar phase is the stationary phase. 

The microscopic method, credited to Hamaker, came first and is based on pair-wise summation of the individual dispersion interaction between molecules. Casmir and Polder later supplemented this approach by including the correction for electromagnetic retardation. The molecular interaction potential used is typically represented by the following expression ... 

There is one circumstance in which the retardation could be significant, and that is in dispersion interactions, which are generally fairly weak. Here it is possible to have one-center densities in the exchange integrals, and the distances can be large. For the He dimer, for example, the dependence of the dispersion energy on the intemuclear separation R changes from l/R to l/R due to retardation (Jamieson et al. 1995). 

The interaction of two macroscopic bodies can sometimes be obtained by summing the dispersion energy between all pairs of molecules or unit cells in the two bodies. There is no electrostatic or induction contribution when the material is uncharged and isotropic. If the separation of the units is large compared to the reduced wavelength associated with the strong electronic transitions, the dispersion interaction is retarded and therefore weakened it varies as R rather than [34]. If the dispersion energy between the units is... 

For distances larger than 5-10 nm, the finite speed of light leads to a reduction of the London dispersion interaction. This effect is called retardation. Retarded van der Waals forces exhibit a distance dependence with a power law exponent that is larger by 1 than for the unretarded case. 

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