Models soft sphere

The noble gases are mostly unreactive. In some instances, they act mostly as a place holder to fill a cavity. For dynamical studies of the bulk gas phase or liquid-phase noble gases, hard-sphere or soft-sphere models work rather well. 

Fig. 1. Models of contact force (Soft sphere model)...

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. 

Similar simulations have been carried out by other research groups. Ouyang and Li (1999) developed a slightly different version of this model. Helland et al. (1999) recently developed a DPM in which hard-sphere collisions are assumed, but where a time-driven scheme (typically found in the soft-sphere model) is used to locate the collisional particle pair. Effect of the gas turbulence has also... 

It is also interesting to note that soft-sphere models have also been applied to other applications such as gas-particle heat transfer by Li and Mason (2000) and coal combustion by Zhou et al. (2003). Clearly, these methods open a new way to study difficult problems in fluidized bed reactors. 

Comparison between Hard- and Soft-Sphere Models... 

Comparison between Hard- and Soft-Sphere Models. The Symbols Indicate Good (+ +), Normal ( + ), and Not Suitable (—)... 

Fig. 11. Graphical representation of the linear spring-dashpot soft-sphere model. From Hoo-mans, Ph.D. thesis, University of Twente (2000).

To perform simulations of relatively large systems for relatively long times, it is essential to optimize the computational strategy of discrete particle simulations. Obviously, the larger the time step 5t, the more efficient the simulation method. For the soft-sphere model, the maximum value for 5t is dictated by the duration of a contact. Since there are two different spring-dashpot systems in our current model, it is essential to assume that tcontact>n — tcontacUU so that... 

The conclusion is that the soft-sphere model can be used as an alternative for the hard-sphere model, as far as the calculation of the excess compressibility is concerned. 

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.

If the rays possess ideal flexibility to allow application of Gaussian statistics, the resultant structure will resemble a soft sphere. This was the reason why the present author introduced the soft sphere model [38]. This model reduces to dendrimers in the narrow sense when no spacer chains between the branching units are present. 

In this lab, the students determine the compression factor, (9) Z = PV/nRT, for Argon using the hard sphere model, the soft sphere model, and the Lennard-Jones model and compare those results to the compression factor calculated using the van der Waals equation of state and experimental data obtained from the NIST (70) web site. Figure 3 shows representative results from these experiments. The numerical accuracy of the Virtual Substance program is reflected by the mapping of the Lennard-Jones simulation data exactly onto the NIST data as seen in Figure 3. 

Fig. 24. a The three-functional regularly branched chain model with Gaussian behavior of the subchains, called the soft sphere model s). b The Berry-plot of the reciprocal particlescattering factor of the soft sphere model. Compare also Figs. 19, 25 and 27... 

Finally we mention the Kratky plot which also may help to detect branching. Here ((S2) q2) Pz (q2) is plotted against q. Figures 26 and 27 show the Kratky plots for regular star-molecules and for the soft sphere model. Linear randomly coiled chains result in... 

Fig. 25. Guinier plot of the soft sphere model. The numbers denote the number of branching shells the filled and open circles are lightscattering results from polyvinyl acetate (PVAc) microgels in methanol at 20 °C at A0 = 546 nm and 436 nm, respectively. The dot-dash line corresponds to the Rayleigh-Gans behavior of hard spheres, i.e. no Mie scattering93)...

Fig. 37. Dependence of the reduced and normalized first cumulant F/Dq2 = Dap,/D on u for the soft sphere model of different numbers of branching shells93 ...

Fig. 39. Measurement of the apparent diffusion coefficient Dapp = 17q2 for two concentrations of a PVAc microgel in methanol 88 189. The full lines are theoretical curves for a soft sphere model with 7 branching shells93 ...

To our knowledge there have been no reported measurements of equilibrium defect concentrations in soft-sphere models. Similarly, relatively few measurements have been reported of defect free energies in models for real systems. Those that exist rely on integration methods to connect the defective solid to the perfect solid. In ab initio studies the computational cost of this procedure can be high, although results have recently started to appear, most notably for vacancies and interstitial defects in silicon. For a review see Ref. 109. 

Example 4.2 Soft-Sphere Model DEM Treatment of Two Disks Deformed by Two Rigid Walls To demonstrate the basic and simple physical model used in DEM, we turn to a pair of two disks, X and Y, compressed between two rigid walls, as shown in Fig. E4.2. 

Figure 15. The d l phase diagram for Pd nanociystals thiolized with different alkane thiols. The mean diameter, 4, was obtained from the TEM measurements on as-prepared sols. The length of the thiol, I, is estimated by assuming an all-nuns conformation of the alkane chain. The thiol is indicated by the number of carbon atoms, C . The bright area in the middle encompasses systems which form close-paced organizations of nanocrystals. The surrounding darker area includes disordered or low-order arrangements of nanocrystals The area enclosed by the dashed line is derived from calculations from the soft sphere model (reproduced with permission from ref. [40]).

The local elastic constant G is assumed to be controlled by the rubber phase around fillers, i.e., it is primary attributed to bound rubber. The elastic constant Q is controlled by van der Waals forces between fillers. The amplitude gb is the failure strain amplitude for breaking the contact between the constructing particles. Krau [36] derived gb within a soft sphere model as... 

Fig. 3.58. Pair correlation functions from Monte Carlo simulation for Charged soft sphere model for LiCI liquid at 883 Kand 28.3 cm mol" .

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