Binary mixtures simulation

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

The breaking up of azeotropic mixtures. The behaviour of constant boiling point mixtures simulates that of a pure compound, because the composition of the liquid phase is identical with that of the vapour phase. The composition, however, depends upon the pressure at which the distillation is conducted and also rarely corresponds to stoichiometric proportions. The methods adopted in practice will of necessity depend upon the nature of the components of the binary azeotropic mixture, and include —... 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

In binary mixtures of water, surfactants, or lipids the most common structure is the gyroid one, G, existing usually on the phase diagram between the hexagonal and lamellar mesophases. This structure has been observed in a very large number of surfactant systems [13-16,24—27] and in the computer simulations of surfactant systems [28], The G phase is found at rather high surfactant concentrations, usually much above 50% by weight. 

Figure 8.2 Phase separation in binary mixtures of model spherical particles at a planar interface generated by Brownian dynamics simulation. The three 2-D images refer to systems in which (A) light particles form irreversible bonds, (B) light particles form reversible bonds, and (C) neither dark nor light particles form bonds, but they repel each other. Picture D shows a 3-D representation. Reproduced from Pugnaloni et al. (2003b) with permission.

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

P. Suppan, Time-resolved luminescence spectra of dipolar excited molecules in liquid and solid mixtures - dynamics of dielectric enrichment and microscopic motions, Faraday Discuss., (1988) 173-84 L. R. Martins, A. Tamashiro, D. Laria and M. S. Skaf, Solvation dynamics of coumarin 153 in dimethylsulfoxide-water mixtures Molecular dynamics simulations, J. Chem. Phys., 118 (2003) 5955-63 B. M. Luther, J. R. Kimmel and N. E. Levinger, Dynamics of polar solvation in acetonitrile-benzene binary mixtures Role of dipolar and quadrupolar contributions to solvation, J. Chem. Phys., 116 (2002) 3370-77. 

Lennard-Jones binary mixture of particles is a prototypical model that describes glass-forming liquids [52,53,158,162-165]. The temperature and the density dependence of diffusivity D(T, p) have been obtained by computer simulations for the Lennard-Jones binary mixture in the supercooled state. To relate fragility of binary Lennard-Jones mixture to thermodynamic properties necessitates determination of the configurational entropy SC(T, p) as well as the vibration entropy Sv,h(T, p) at a given temperature and density. 

Mayur and Jackson (1971) simulated the effect of holdup in a three-plate column for a binary mixture, having about 13% of the initial charge distributed as plate holdup and no condenser holdup. They found that for both constant reflux and optimal reflux operation, the batch time was about 15-20% higher for the holdup case compared to the negligible holdup case. Rose (1985) drew similar conclusion about column holdup but mentioned that the adverse effects of column holdup depends entirely on the system, on the performance required (amount of product, purity), and on the amount of holdup. Logsdon (1990) found that column holdup had a small but positive effect on their column operation. 

A liquid binary mixture with B0 = 10 kmol (Hc) and xB0 = <0.6, 0.4> (xj) molefraction is subject to inverted batch distillation shown in Figure 4.12. The relative volatility of the mixture over the operating temperature range is assumed constant with a value of (a-) 2. The number of plates is, N= 10. The vapour boilup rate is, V = 10.0 kmol/hr. The total plate holdup is 0.3 kmol and the reboiler holdup is 0.1 kmol. The total batch time of operation is 4 hr with two time intervals. The first interval is of duration 1 hr and the column is operated with a reboil ratio of 0.8. The second interval is of duration 3 hrs when the column is operated with a reboil ratio of 0.9. The column operation is simulated with the type III model (section 4.3.2.1). 

Table 7.1 summarises the results of repetitive simulation following the strategy described in section 7.2. Table 7.1 shows that a column with 70 plates will give the best design and operation for the binary mixture being processed in the column in a campaign mode for 8500 hrs a year. However, it was not clear in Al-Tuwaim and Luyben (1991) whether the actual optimum value of N lies in between 60-70 or 70-80 plates. 

As discussed in the previous section, the work of Mayur et al. (1970) and Christensen and Jorgensen (1987) on the optimal recycle policy was restricted to binary mixtures. The benefits of recycling were measured in terms of a reduction in batch time although increase in productivity could be a possible alternative. Luyben (1988) considered this productivity measure (as defined as "capacity" which includes both batch time and a constant charging and cleaning time) in a simulation of multicomponent batch distillation with recycle. Luyben (1988), however, showed the effect of different parameters (no of plates, relative volatilities, etc.) on the productivity and did not actually consider the effect of off-cuts recycle on the productivity. 

The proposed modeling approach has been validated for distillation of non-reactive mixtures. For this purpose, the use is made of the total reflux distillation data for the binary mixture chlorobenzene/ethylbenzene (CB/EB) and ternary mixture methanol/acetonitrile/water (MEOH/ACN/WATER) obtained by Pelkonen (1997) as well as for the ternary mixture methanol/ethanol/water (MeOH/EtOH/WATER) measured by Mori et al. (2006). The experiments of Pelkonen (1997) were carried out in a column of 100 mm diameter, equipped with Montz-Pak A3-500 structured packing. The measured concentrations, temperature and flow rates at the condenser outlet are used as input values for simulations. 

The area available to a disk (or to a sphere) to be adsorbed on a surface during a random sequential adsorption (RSA) of a binary mixture of disks (spheres) is in general estimated via Monte-Carlo simulations, which are vety time consuming. A simple analytical expression is proposed, which provides good estimates of the jamming points for the RSA of a binary mixture of disks or spheres. The predictions of the simple formula have been compared with the Monte-Carlo simulations existing in the literature [8.10],... 

A simple analytical expression is proposed for the area available to a disk on a surface for a Random Sequential Adsorption (RSA) of binary mixtures of disks. The expression was obtained by combining the low-order terms of the density expansion of the available area with the asymptotic behaviour of the surface coverage near the jamming point. Comparison with Monte Carlo simulations shows that this approach provides a fair estimation of the jamming coverage for both kinds of disks. 

The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12], However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much Less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [ 19,20], the available area vanishes only for the unphysical total coverage 9 = 9 +0p = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming . Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21], The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22], However, there are no such calculations for the RSA model. 

The jamming point can also be evaluated without using the information provided by the Monte Carlo simulations [30]. While this approach is not useful for monodisperse disks, for which a much better interpolating expression can be obtained using the accurate values for 6c and K predicted by Monte Carlo simulations, it can he employed to estimate the available area and the jamming coverage for binary mixtures of disks, for which in general the values of 6c and K are not known. 

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