Kirkwood factors

A simple example of how molecular electronic structure can influence condensed phase liquid crystalline properties exists for molecules containing strongly dipolar units. These tend to exhibit dipolar associations in condensed phases which influence many thermodynamic properties [29]. Local structural correlations are usually measured using the Kirkwood factor g defined as... 

Table 16-1 shows results for the dielectric constant e(0), Kirkwood -factor gK, and the static dipole cross-correlation parameter g° = ( M(0) 2) /(Np ) — 1 where M(f) = IFit) is the system s collective dipole at time t, for a selected set of thermodynamic states. The experimental values for e(0) are shown within parentheses. The overall trend of these quantities with density and temperature is consistent with the expectation of a higher degree of dipolar correlation at higher densities and/or lower temperatures. At liquid-like densities (states 10-12), where polarizability effects are known to be important, the simulated model underestimates e(0), a feature common to most non-polarizable water models. Given the error bars and differences in thermodynamic states, our estimates for e(0) for states 10-12 are... 

In any case, either with periodic boundary conditions including Ewald sums or reaction field, the central quantity to calculate is the fluctuation of the total dipole of the sample, measured by the Kirkwood factor ... 

Kirkwood factor is then combined with the polarity index y = 4npfl j9kin different ways, according to the boundary conditions implemented, to get dielectric permittivity. 

The collective structure of aqueous IL solutions was studied by means of MD simulations [101]. Various concentrations of [C4mim][BF4] and TIP3P water were simulated at the very same size of the simulation box. For the analysis, the ternary system cation/anion/water was subdivided into binary networks. The local structure of each of these six networks and the mutual orientation of the network constituting partners were studied. Furthermore, the collective structure of the whole samples was characterized by the contribution of each species to the static dielectric constant e(co= 0) and to the Kirkwood factor [101]. 

G. S. Rushbrooke, MoL Phys., 37, 761 (1979). See also the work of Gray and Gubbins, MoL Phys., 30, 1481 (1975), for expansions of the Kirkwood -factor for simple models that include quadrupolar terms and Murad et al., Chem. Phys. Lett., 65, 187 (1979) for... 

Figure 9.8 Difference from experiment of simulated Kirkwood -factor, as a function of (solid symbols). Symbol types are given in Fig. 9.7. The linear regression given in Eq. 9.5 for of models with only inplane chaise (dotted line) and with out-of-plane charge (dashed line), using appropriate values of AS. Since is multiplied by /Tq for the model so that...

This effect originates from the anisotropic dipole-dipole correlations not accounted for by the Maier-Meier theory operating with a single particle distribution function. When, with decreasing temperature, the smectic density wave p(z) develops (even at the short-range scale) the longitudinal dipole moments prefer to form antiparallel pairs and the apparent molecular dipole moment becomes smaller. This would reduce positive s. Theoretically, dipole-dipole correlations may be taken into account by introducing the so-called Kirkwood factors. 

When molecules are distributed on a cubic lattice or form an ideal gas, this contribution is zero. In other systems this contribution is introduced as the Kirkwood factor( )... 

For flexible chains the cross-correlation terms decrease rapidly as k - k l is increased. Model calculations for polyethers (35,36) show that the cross-correlation terms may be positive or negative, giving a Kirkwood factor greater or less than unity (see refs. 5 and 36). 

The reader should note that, in case of centrosymmetric structures the term z=0, and the Kirkwood factor is 1. In this case Ac provides no information about the average orientational order. However, that does not occur for our samples, because poliols ar linear chains and very unlikely they can form centrosymmetric stmctures via hydrogen intermolecular bondings... 

---