Yukawa particles

Phase Behavior of Hard-Core Yukawa Particles. 170... 

Subsequently, we employ the Gibbs-Duhem method as first proposed by Kofke [31,32] to determine the phase coexistence lines in the (rj, kct) plane for a fixed Pe. Using these methods, we study the phase behavior of hard-core Yukawa particles, whose interactions are described by the pair potential given by Equation 8.8. The phase diagrams are calculated for fixed contact values Pe and they are given in the (ri, l/xa) representation. We calculate the phase diagram for four contact... 

FIGURE 8.3 Phase diagram of hard-core Yukawa particles with pe = 20 presented in the (q, 1/ko) plane. The symbols and lines are the same as in Fignre 8.2. Note the difference in the q scale compared to Fignres 8.2 throngh 8.5. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68 021407. With permission.)... 

In conclusion, we have shown that the phase diagram of charged colloids, where the interactions are given within the DLVO theory by hard-core repnlsive Yukawa pair potential, can be obtained for any snfficiently high contact valne Pe by mapping the well-known phase boundaries of the point Yukawa particles onto those of the hard-core repulsive Yukawa system and bearing in mind that the stable bcc region is bounded by a bcc-fcc coexistence at ti 0.5. 

Hynninen AP and Dijkstra M. 2003. Phase diagrams of hard-core repulsive Yukawa particles. Physical Review E 68 021407. 

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. 

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]. 

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... 

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... 

In 1934 the Japanese physicist Hideki Yukawa postulated the existence of yet another force particle, which he called the meson. In 1932 Yukawa began his academic career with an appointment at Osaka Imperial University, which had been founded the previous year. The discovery of the neutron and the publication of Fermi s theory started him thinking about the nature of the force that bound protons and neutrons together in an atomic nucleus. He realized that, though... 

Yukawa proceeded by writing down a mathematical formula for the force. It wasn t especially difficult to do this. He looked for the simplest mathematical form that was consistent with experimental facts. He knew that, if necessary, refinements could be added later. Then, applying the principles of quantum mechanics, he deduced that, if the force did have that form, there had to exist a previously unobserved particle that had a mass approximately 200 times greater than that of the electron. 

Yukawa published a paper on his theory in 1934. Three years later Anderson discovered a new particle in cosmic rays. Its mass was in line with Yukawa s predictions, and at first physicists believed that Yukawa s theory had been confirmed. In reality, it hadn t been. It turned out that the new particle was wot Yukawa s meson. It had about the right mass, but its other properties were inconsistent with his theory. 

The English physicist Cedi Powell discovered Yukawa s meson in 1947. Powell found evidence of its existence in photographic plates that had been exposed to cosmic rays in the Bolivian Andes. The particle was found to be a little heavier than the muon, and it interacted strongly with nuclei, as Yukawa s particle was expected to do. Unlike the muon, which always carried a negative charge, the new particle could have either a positive or a negative charge, or it could be electrically neutral. 

A considerable amount of evidence indicates that nuclear forces are charge-independent, i.e, the neutron-neutron, neutron-proton, and proton-proton forces are identical. The meson theory of nuclear forces, originated by Yukawa, postulates the atomic nucleus being held together by an exchange force in which particles, now called mesons, are exchanged between individual nucleons within the nucleus. 

The picture, however, had changed rapidly after the end of the Second World War. The experiment of Conversi, Pancini, and Piccioni59 had shown that this particle had an interaction with nuclei much weaker than that expected for the Yukawa mediator. At the beginning of October of the same year 1947, Lattes, Occhialini, and Powell60 in Bristol had discovered in cosmic rays a new particle, that they called ir-meson. It is unstable and decays, with a mean life of 10-8 sec, into a neutrino and the particle of Anderson and Neddermeyer that was called p-meson or muon. 

J. P. Vigier, H. Yukawa, and E. Katayama, An approach to the unified theory of elementary particles, Prog. Theor. Phys. 29, 468 (1963). 

Figure 11. The difference between the free-energy densities of fee and bcc phases of particles interacting through a Yukawa potential, as a function of temperature, determined through the FG methods discussed in Section V.C. The error bars reflect the difference between the upper and lower bounds provided by FG switches between the phases (along the Bain path [85]) in the two directions.

The introduction of the van der Waals potential in combination with a Yukawa potential produces a curve in which the primary minimum is always deeper than the secondary minimum. This must be so because the primary minimum state is that for which the particles have coalesced and the valency of the nth plate Zn has dropped to zero since Z —> 0 as Xmn —> 2a, —> 0 as —> 2a, and the van der Waals force... 

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