Hard-and Soft-Sphere Systems

This determines the glass transition line in the softness-concentration state space [i.e., in the ( x, 1) plane)]. By observing this glass transition phase diagram, the value of p, whose glass transition volume fraction (J) coincides with the experimentally reported value of 0.644, it was concluded that 

FIG U R E 1.5 Comparison of the SCGLE collective correlator,/(fc, t) = F(k, t)/S(k), of the hand-sphere system at the position of the main peak of S k) (solid lines) with the experimental results of van Megen and Underwood [24] (symbols) corresponding to the experimental volume fractions ) = 0.494,0.528,0.535,0.574, 0.581, and 0.587 (from bottom to top). The SCGLE theory predicts )j = 0.563, and hence, the comparison is made at the same values of the separation parameter e = (( ) — )g)/ )g. (From Ramirez-Gonzalez, P. and Medina-Noyola, M. 2009. J. Phys. Condens. Matter. 21 075101. With permission.) 

FIGURE 1.6 Comparison of the SCGLE collective correlator/(k, t) = F(k, t)/S(k) of the soft-sphere system in Equation 1.29 with p = 14 at the position of the main peak of S(k) in the vicinity of the glass transition, 

FIGURE 1.7 (a) Theoretical fit (solid line) of the static structure factor, and (b) SCGLE theoretical predictions (solid line) for the nonergodicity parameter/(A ) of the mono-disperse charged sphere system of reference [33], modeled by the pair potential of Equation 1.36 at ( ) = 0.27, z = 3.1587, and K = 11.66. The symbols correspond to the experimental data. The inset in (b) enlarges the region where experimental data for/(A) are available. (From Yeomans-Reyna, L. et al. 2007. Phys. Rev. E 76 041504. With permission.) 

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It is possible but not easy to imagine conditions in which two phases of the same laboratory substance could have identical entropies and also maintain the identity over a range of temperatures it is not possible, however, in the case of classical hard and soft sphere systems since, at constant pressure, equal entropy in these cases implies equal volume, hence the same phase. Since, at constant pressure, there is only one point in temperature— the fusion point—where the free energies of the fluid and ciystal phases of the same substance can be equal, a cannot exist for hard spheres. This raises the question of whether there are other occurrences that might terminate the supercooled fluid state above 7 . Two have been suggested. 

With these possibilities in mind, we plot the courses in temperature of the fluid- and solid-phase entropies for hard- and soft-sphere systems, and compare them with the MD results in Figs. 21 and 22. It should be noted that in classical mechanical models the entropy is usually defined relative... 

Before comparing the glass transition observed in simulations with that observed in the laboratory, it is necessary to review briefly the temperature and density dependence of transport properties. In some of the model systems studied (specifically, hard and soft spheres) there is only one system variable, and temperature- or density-dependent representations of the properties are a matter of choice only. With other systems and of course with all laboratory systems, the two types of plot display independent aspects of the system s behavior. 

Between the polarized nematic and soHd phases a density region seems to exist where dipolar hard and soft spheres form ferroelectric columns with square ordering in the plane perpendicular to the coliunns [102,134], The stabihty and density range of this phase needs further investigation. Such a columnar phase has also been identifled in a system of extended dipoles where the coliunns order on a hexagonal lattice [ 146]. 

To further highlight the differences in the crystalUne and liquid states Table 9.1 lists the entropies of melting calculated in both two- and three-dimensions from the present work (taken at the respective coexistence curve maxima) and from previous work focussed on relatively simple models hard and soft spheres, a Lennard-Jones potential, and the one-component plasma [102]. The absolute values reported here are larger (and consistent with known values for the conformal systems Si and Ge [103]). However, the ratio of the three- and two-dimensional system values appears consistent throughout, reflecting the more ordered nature of the liquid state when confined to two dimensions. 

Colloidal suspensions can be classified as soft sphere systems because the repulsive intoactions occur at some characteristic distance from the particle surface. For electrostatic and stoic stabilization, this distance is the Debye length (1/ K) and the thickness of the adsorbed polymer layer, respectively. For stoically stabilized suspensions, the adsorbed polymer layer leads to an increase in the hydrodynamic radius of the particle. When the adsorbed layer is densely packed, the principles described above for hard sphere systems are applicable, provided that the volume fraction of particles/is replaced by an effective volume fraction /gy given by... 

Polymer colloids are important model systems for investigating fundamental aspects of colloid science. Traditionally rigid particles that have rather simple, for example, hard sphere-like or Yukawa, interaaion potentials have been employed. Such systems have been reviewed in the past, see, for example, a review on hard spheres by Pusey et al In this chapter, we focus on recent developments depletion interaction and soft spheres. Concerning the latter topic, two different types of materials are discussed star polymers and microgels. 

The simplest models of molecular dynamics simulations (hard or soft spheres) under stationary conditions have the property of ergodicity and mixing. A statistical mechanical system is ergodic if... 

The hard sphere (HS) interaction is an excellent approximation for sterically stabilized colloids. However, there are other interactions present in colloidal systems that may replace or extend the pure HS interaction. As an example let us consider soft spheres given by an inverse power law (0 = The energy scale Vq and the length scale cr can be com-... 

An important conclusion of this discussion is the fact that at very high <)> thermodynamic stability is re-established. Restabilisation is not a kinetic effect, as suggested by Feigin and Napper (10, 11), but is a consequence of lower free energy of the dispersion as compared to the floe. This conclusion is supported by experimental evidence for soft spheres (3, 5, 23). We should add, however, that for hard spheres is so high that experimental verification is difficult for most polymer-solvent systems due to the high viscosity of the solution. 

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

We would like to mention that RHNC is able to reproduce qualitatively accurate bridge functions inside the core, especially when a reference system (RS) of soft spheres is utilized rather than the conventional hard-sphere fluid. However, the prescription used to determine the RS requires a priori knowledge of both the RS and the system under investigation. This feature restricts the applicability of RHNC to sytems for which the EOS is available. The optimization of the RS proposed by Lado does not have such a drawback, but requires intensive computation since the optimization has to be performed at each state point. Furthermore, it was shown [85] that for the LJ potential, there exists a region around the critical point where the RHNC [54] has no solution. 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

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