Yukawa

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

This r dependence is also known as a Yukawa potential. This type of potential has been used to describe the behaviour of latex suspensions at low ionic strength. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

Robbins M O, Kremer K and Grest G S 1988 Phase diagram and dynamics of Yukawa systems J. Chem. Phys. 88 3286-312... 

To meet the point that the amount of resonance interaction in the transition state will be dependent upon the nature of the electrophile, Yukawa and Tsuno have put forward a modified equation with three parameters. The physical interpretation of such an equation is interesting, but it is not surprising that it correlates experimental data better than does the equation with two parameters. ... 

One approach is to correct for the added resonance interaction. This is done in a modification of the Hammett equation known as the Yukawa-Tsuno equation. ... 

The Yukawa-Tsuno relationship expanded to include both the [Pg.210]

Another example of enhanced sensitivity to substituent effects in the gas phase can be seen in a comparison of the gas-phase basicity for a series of substituted acetophenones and methyl benzoates. It was foimd that scnsitivtiy of the free energy to substituent changes was about four times that in solution, as measured by the comparison of A( for each substituent. The gas-phase data for both series were correlated by the Yukawa-Tsuno equation. For both series, the p value was about 12. However, the parameter r" ", which reflects the contribution of extra resonance effects, was greater in the acetophenone series than in the methyl benzoate series. This can be attributed to the substantial resonance stabilization provided by the methoxy group in the esters, which diminishes the extent of conjugation with the substituents. 

The relative stabilities of 1-phenylvinyl cations can be measured by determining the gas-phase basicity of the corresponding alkynes. The table below gives some data on free energy of protonation for substituted phenylethynes and 1-phenylpropynes. These give rise to the corresponding Yukawa-Tsuno relationships. 

It is observed that solvolyses of the tertiary benzylic / nitrobenzoates A, B and C are correlated by Yukawa-Tsuno equation with reduced resonance components as indicated by a lower coefficient of the resonance parameter r. Offer an explanation. 

The Yukawa-Tsuno r values have been measured for the solvolysis reactions fonning benzyl cations and several a-substituted derivatives, 6-3IG charges and bond orders have been calculated for the presumed cationic intermediates. Analyze the data for relationships between r and the structural parameters. (Hint. Plot r versus the bond orders and the charges at C-1, C-2, C-3, and C-4.)... 

In the Yukawa potential, A is an inverse range parameter. The value A = 1.8 is appropriate for the inert gases. Each of the above potentials has a hard core. Real molecules are hard but not infinitely so. A slightly softer core is more desirable. The Lennard-Jones potential... 

Since the results for thermodynamics from the Yukawa potential, with A = 1.8, are similar to the results of the LJ potential, it is quite possible that the DHH closure may be applicable to the Yukawa potential. With A = 1.5, the thermodynamics of the square well fluid are also similar. Here, too, the DHH closure may be useful. However, the DHH closure has not been applied to either of these potentials. 

The Yukawa potential is of interest in another connection. According to the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, colloidal... 

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

The (T and scales of substituent effects result from changes in the standard reaction that defines the cr scale. An alternative approach to dealing with substituents that possess more than one mechanism of electronic interaction with the reaction site is to make use of more than one substituent constant. Yukawa and Tsuno ... 

H. Yukawa (Kyoto) prediction of the existence of mesons on the basis of theoretical work on nuclear forces. 

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