Lorentz-Berthelot rules

The choice of appropriate potential parameters to use in the study of a certain mixture can be a significant problem. Traditionally the Lorentz Berthelot rules. 

The LJ parameters, elk and a, have been determined from the critical constants, Tc and pc, by adopting the recommendation of Nicolas et al. [11] kTJe = 1.35 andpco3/e = 0.142. However, different values for the potential depth of benzene, 22, have been determined so as to fit the vapor pressure at temperatures from 307.2 K to 553.2 K. The LJ parameters used in this work are summarized in Table 1, where the parameters for graphitic carbon atom are taken from those suggested by Steele [10]. We used the modified Lorentz -Berthelot rule for the cross parameters, that is, the arithmetic mean for o and the geometric mean for e by introducing the binary parameter ktj defined as Eq. (4). 

The energetic inhomogeneity of the surface along the x and y directions is not taken into account, but this is not expected to affect the results significantly at 308 and 333 K [39]. The cross interaction potential parameters between different sites were calculated according to the Lorentz-Berthelot rules Oap = aa + and eafi= ( The potential energy t/ due to the walls inside the slit pore model for each atom of the CO2 molecule is given by the expression C/ = + Uw(H-r where H is the distance between the carbon centers across... 

The water-water intermolecular interaction is described by the TIP4P potential. The ethane molecule consists of two interaction sites, each of which interacts with each other via Lennard-Jones (LJ) potential. The reference of ethane molecule is spherical and is of LJ type interaction with size and energy parameters of 4.18 A and 1.72 kj/mol. The LJ parameters for methyl group of ethane are 3.78 A and 0.866 kj/mol. For the water-guest interaction, the Lorentz-Berthelot rule is assumed. The interaction potentials for all pairs of molecules are truncated smoothly at... 

Finally, /o and Cio corresponding to the 12-6 and m-n LJ parameters were found by fitting to eqn. (2). The Lorentz-Berthelot rules were adopted for the description of the cross-interactions, where epo and Coo have the values corresponding to the SPC/E model, 0.1554 kcal/mol and 3.1655A respectively ... 

The interactions between pairs of gas molecules were estimated using the LJ potential =As alr) - Glr), where r is the site-site distance). Two sets of LJ parameters for gas molecules (i.e., and cr) were chosen. One was slkB = 149 K (fe is the Boltzmann constant) and G = 0.378 nm, and the other was dkB 231 K and a = 0.405 nm. The former corresponded to the parameter set for a spherically-approximated CH4 molecule, and the latter for a Xe molecule. The interactions between H2O and gas molecules were also estimated using the LJ potential, and the s and a values for the H20-gas LJ potential were determined using the Lorentz-Berthelot rules. Long-range interactions were smoothly truncated at an intermolecular distance of 1 nm using a switching function. 

The parameters in simple potential models for interactions between unlike molecules A and B are often deduced from the corresponding parameters for the A-A and B-B interactions using combination rules . For example, the a and s parameters are often estimated from the Lorentz-Berthelot rules ... 

Equations 13 and 14 follow from the Lorentz-Berthelot rules, an arithmetic mean for unlike-molecule size parameters and a geometric mean for unlike-molecule energy parameters, with deviations allowed for either rule. 

Equations (2) and (3) are directly equivalent to the Lorentz-Berthelot rules in their more familiar form using e and a, the well depth and collision diameter associated with the intermolecular potential (r) ... 

In complete contrast the measurements by Sigmund et al. of the second and third virial coefficients of CF4 + SF showed that the Lorentz-Berthelot rules predict the experimentally determined interaction parameters for the unlike interactions within experimental error. Mixtures of spherically symmetric fluorocarbons thus closely resemble similar hydrocarbon mixtures in this respect. Lange and Stein reported measurements of the second and third virial coefficients for CF3H + CF4 mixtures. A distinct weakness in the unlike interactions was noted although no detailed calculations were made. 

For any kinds of mixtures, in addition to LJ parameters for each component, combining rule (or mixing rule) for unlike interaction should be prepared. Even for simple liquid mixtures, conventional Lorentz-Berthelot rule is not good answer. 

Given the failure of the Lorentz-Berthelot rules, in order to increase the predictability of mixture models (be it equations of state or computer simulations) different combining rules have been proposed and investigated. Schnabel et have recently provided a quantitative assessment of eleven combining rules in terms of their performance in describing binary mixtures classified by the molecular shape and polarity of the components. Of particular interest is a... 

The fluids are modelled by a truncated and shifted Lennard-Jones (LI) pair potential [7], The potential parameters and eqiutions of state of the fluids are given in Table 1 with the parameters for carbon cross-interaction parametos are obtained by the Lorentz-Berthelot rules. The cut-off radius for simulations in the solid is 2.Sa, whilst that used in generating the single pore isotherms for determining the PSDs is Sag, inline uitfa standard procedure. 33. Simulation details... 

This consideration is underlined by the analysis of isothermal slices through the miscibility gap in the plane of variables pressure versus molar concentration of this system (Fig. 17). It is seen that the MC results that include the quadrupolar interactions q = 0.47) and respect the Lorentz-Berthelot rule (( = 1) fit to the... 

When the nonbond interactions of a system that contains multiple particle types and multiple molecules are modeled using a Lennard-Jones type nonbond potential, it is necessary to be able to define the values of a and e that apply to the interaction between particles of type I and /. The parameters for these cross interactions are generally found using one of the two following mixing rules. One common mixing rule is the Lorentz-Berthelot rule where the value of o// is found from the arithmetic mean of the two pure values and the value of ej/ is the geometric mean of the two pure values ... 

Oa = 0.5(oa-I-Ob)). Figure 1.3 shows a projection of the phase diagram and the critical line onto the pressure-temperature plane. Even though the model is very simple and lacks atomistic details it can describe adequately the phase behavior of the mixture without additional fitting parameters. If, however, the quadrupolar moment of CO2 is not taken into account, a modification of the Lorentz-Berthelot rule is required < 1). This emphasizes that atomistic detail is not always required to describe the phase behavior correctly. However, it is advisable to include physically relevant quantities like the quadrupolar moment of CO2. 

To illustrate these techniques, we consider the coarse-grained model for hexadecane and carbon dioxide, which we have discussed in Section 1.2. The correction of the Lorentz-Berthelot rule is set to = 0.9 and the temperature is fixed atllcBr/E = 0.92. (In this case carbon dioxide was modeled as a simple Lennard-Jones bead without quadrupolar moment.) The short- and long-ranged interactions at the polymer-solid contact are described by a 9-3-potential of the form ... 

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