Yukawa attraction

The simplest possible situation is a one-component system of hard spheres with an added attractive interaction. For attractive Yukawa spheres the pair potential W(h) for h 0 is a simple exponential  

In this equation we introduced the inverse (relative) range k= l/q. 

Earlier [9] we simplified the expressions for as given by the first-order mean spherical approximation by Tang et al. [20, 21], which is based upon pair-wise additivity of the interaction. The result is  

Inserting Equation 7.6 into Equation 7.1 gives (q, e, q), and Equation 7.3 then provides p(q, e, q) and pv r, e, q). Explicit analytical forms for these latter quantities were presented in a recent publication [8j. The pressure in the system is given 

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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... 

J. G. Malherbe and S. Amokrane (1999) Asymmetric mixture of hard particles with Yukawa attraction between unhke ones a cluster algorithm simulation... 

To bring this subsection to a close some comments on the QHS system including attractive forces are in order. The introduction of attractive tails can modify greatly the structural and thermodynamic behavior of the underlying bare QHS system. This has been shown by the present author and L. Bailey [108] with the use of pair-wise Yukawa attractive tails and PIMC simulations. By doing so, the onset of critical behavior in the quantum hard-sphere Yukawa (QHSY) system was identified under conditions in which the QHS system remains in the normal fluid phase. The Hamiltonian for the QHSY system can be cast as... 

In this chapter we have shown that the critical endpoint (cep), which corresponds to the lowest (relative) attraction range q where a stable colloidal liquid exists, does not depend on the details of the pair potential. It is situated at = 1/3 and an interaction strength e p = 2 kTfor four different systems hard spheres with a Yukawa attraction ( Yuk ), a Lennard-Jones type (LJ) fluid (both onethree-component colloid/polymer/solvent systems with either a fixed depletion thickness ( fix ) or a depletion thickness that varies with the polymer concentration ( var ). In the latter case the depletion thickness decreases from (roughly) the coil radius R in dilute solutions to the blob size in semi-dilute solutions. 

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]. 

M.H.J. Hagen and D. Frenkel Determination of Phase Diagrams for the Hard-Core Attractive Yukawa System. J. Chem. Phys. 101, 4093 (1994). 

We will briefly discuss the molecular dynamics results obtained for two systems—protein-like and random-block copolymer melts— described by a Yukawa-type potential with (i) attractive A-A interactions (saa < 0, bb = sab = 0) and with (ii) short-range repulsive interactions between unlike units (sab > 0, aa = bb = 0). The mixtures contain a large number of different components, i.e., different chemical sequences. Each system is in a randomly mixing state at the athermal condition (eap = 0). As the attractive (repulsive) interactions increase, i.e., the temperature decreases, the systems relax to new equilibrium morphologies. 

FIGURE 3.10 Force-distance curves. The curvilinear line is the prediction of the coulombic attraction theory. The straight line has been drawn through simulated error bars in the previous line to illustrate the approximately linear In P vs. x (or In P vs. r) plots expected from the coulombic attraction theory. The upper dotted line is the corresponding Yukawa prediction. If we scale the plot such that the 1996 data of Crocker and Grier [14] lies along the upper dotted line, the data for the same sample in their 1994 paper [13] is as represented by the lower dotted line. 

In 1935, the Japanese physicist Hideki Yukawa proposed that a force between protons that is stronger than the electrostatic repulsion can exist between protons. Later research showed a similar attraction between two neutrons and between a proton and a neutron. This force is called the strong force and is exerted by nucleons only when they are very close to each other. All the protons and neutrons of a stable nucleus are held together by this strong force. 

The Yukawa-Tsuno equation (also referred to as Linear Aromatic Substituent Reactivity relationship, LASR) modifies the Hammett equation, taking into account the exaltation of the resonance effects of electron-releasing and electron-attracting substituents on the reaction centre [Yukawa et al., 1972a Yukawa et ai, 1972b]. 

Analogously, for electron-attracting substituents, the Yukawa-Tsuno equation is ... 

Several molecular simulation studies have been published in the past decade regarding the solid-phase coexistence behavior of purely repulsive [187-195] and attractive [169,170,196,197] Yukawa models binary mixtures have also been examined approximately [198]. 

A similar approach can be used to construct a molecular model for microemulsions [55-57]. All three components—oil, water, and amphiphile—are taken to be hard spheres. The interactions, W r), between oil molecules and between water molecules are taken to be isotropic and attractive, and the interactive between oil and water, to be isotropic and repulsive. A Lennard-Jones or Yukawa form has been used for W(r). The interaction between a surfactant molecule located at ri and water molecule at rj on the other hand, is anisotropic, with the dependence... 

We now present some mathematical properties of two-body Hamiltonians. In one or two dimensions, any attractive potential supports at least one bound state (some local attraction is sufficient) [29]. In three dimensions, this is not true any more. For instance, a Yukawa potential V = -B exp(-ar)/r, with a > 0, does not provide any bound state if B is too small. For a = 0, on the other hand, one gets the Coulomb potential, which has an infinite number of bound states, with E<0, as well as a continuum spectrum of positive energies. In simple models of quarkonium, we are dealing with confining potentials, which support an infinite number of bound states. 

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