Combining rule

Equations of state are used in engineering to predict the thermodynamic properties in particular the phase behaviour of pure substances and mixtures. However, since there is neither an exact statistical-mechanical solution relating the properties of dense fluids to their intermolecular potentials, nor detailed information available on intermolecular potential functions, all equations of state are, at least partially, empirical in nature. The equations of state in common use within both industry and academia are described elsewhere in this book and can be arbitrarily classified as follows (1), cubic equations derived from the observation of van der Waals that are described in Chapter 4 (2), those based on the virial equation discussed in Chapter 3 (3), equations based on general results obtained from statistical mechanics and computer simulations mentioned in Chapter 8 and (4), those obtained by selecting, based on statistical means, terms that best represent the available measurements obtained from a broad range of experiments as outlined in Chapter 12. The methods used for mixtures are also alluded to in these chapters and in Chapter 6. 

The development of an equation of state typically commences with the representation of the thermodynamic properties of pure fluids and the functions are then extended to provide estimates of the properties of mixtures. The purpose of this chapter is to provide the methods used to permit this calculation 

Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Peters International Union of Pure and Applied Chemistry 2010 Published by the Royal Society of Chemistry, www.rsc.org 

The methods most frequently used to predict the properties of mixtures for over 100 years have inevitably undergone only minor additions and corrections to, it is claimed, improve the representation of experimental data for specific categories of substances. It is, however, possible that completely different alternatives to these traditional approaches are required, particularly for a method to be both predictive and applicable over a wide range of fluids and conditions. Such methods might arise from future research and methods based on statistical mechanics and quantum-mechanical calculations are ultimately sought rather than empiricism. 

One of the exact results from statistical mechanics, discussed in Chapter 3, is the virial equation of state expressed in terms of the definition of the 

---

For the 6-12Atom VDWForm at entry set to RfitarEpsilon, the combination rules used are ... 

For the buffered 14-7 functional form more elaborate combination rules are employed ... 

Halgren T A 1992. Representation of van der Waals (vdW) Interactions in Molecular Mechanics Force Fields Potential Form, Combination Rules, and vdW Parameters. Journal of the American Chemical Society 114 7827-7843. 

The force fields in HyperChem that use the above 6-12 potential allow six ways of specifying the constants Ay and B.. three by single atom type and three by pairs of atom types. Single atom type means that there are constants for individual atom types, i, that are combined by a combining rule that results in a parameters for a specific pair, ij, of atom types. Pairs of atom types means that parameter files contain explicit parameters for a pair of atom types... 

The form that single atom type constants take is selected by setting the Registry/chem.ini parameter set entry 6-12AtomVDWFormat to one of RStarEpsilon, SigmaEpsilon or SlaterKirkwood. This specifies the combination rules that are used for the file pointed to by the 6-12AtomVDW entry in the Registry or the chem.ini file for the same parameter set. 

The functional form for van der Waals interactions in AMBER is identical with that shown in equation (13) on page 175. The coefficients A. and B.. are computed from the parameters in the file pointed to by the 6-12AtomVDW entry for the parameter set in the Registry or the chem. ini file, usually called nbd.txt(dbf), and optionally with the file pointed to by the 6-12PairVDW entry for the parameter set, usually called npr.txt(dbf). The standard AMBER parameter sets use equations (15) and (16) for the combination rules by setting the 6-12AtomVDWFormat entry to RStarEpsilon. The 1 van der Waals interactions are usually scaled in AMBER to half their nominal value (a scale factor of 0.5 in the Force Field Options dialog box). 

The OPLS atom types are a superset of the AMBER united atom types and the bonding parameters are just those of AMBER, supplemented where needed by the OPLS developers. The bond stretch, angle bending, dihedral angle and improper dihedral angle terms are identical to those of AMBER. Unlike AMBER, different combination rules are used for the van der Waals parameters, no hydrogen bonding term is used and no lone pairs are used. 

Newer, published CHARMM parameter sets override some of the combination rule generated parameters for some atom type pairs. These parameters are found in the file pointed to by the 6-12PairVDW entry for the parameter set, usually called npr.txt(dbf). The values of Ay and By for these are computed using equations (22) and (23) on page 178 by setting the 6-12PairVDWFormat entry to RStarEpsilon. 

The most recendy developed model is called UNIQUAC (21). Comparisons of measured VLE and predicted values from the Van Laar, Wilson, NRTL, and UNIQUAC models, as well as an older model, are available (3,22). Thousands of comparisons have been made, and Reference 3, which covers the Dortmund Data Base, available for purchase and use with standard computers, should be consulted by anyone considering the measurement or prediction of VLE. The predictive VLE models can be accommodated to multicomponent systems through the use of certain combining rules. These rules require the determination of parameters for all possible binary pairs in the multicomponent mixture. It is possible to use more than one model in determining binary pair data for a given mixture (23). 

Calculations performed using MC BOSS [66] with the CHARMM combination rules. Partial atomic charges (C = —0.27 and H = 0.09) were identical for all three simulations. 

Combine the modified probabilities to give the overall error probabilities for the task. The combination rules for obtaining the overall error probabilities follow the same addition and multiplication processes as for standard event trees (see last section). 

For interactions between different atom pairs, we should ideally try to deduce experimental parameters for the interactions of these atoms. What is done instead is to use combination rules, which allow us to relate the parameters for unlike atom pairs A- -B to those for the two like pairs A- A and B- -B. If we use symbols i and j to label the two atoms, then there are a number of different such... 

The quantities e and a are the force constants of the Lennard-Jones interaction solute molecule K with an element of the wall of the hydroquinone cage (cf. Eq. 30). It is assumed that for this interaction the frequently-used combining rules for the interaction between two unlike particles hold,... 

In this respect the values of A /kT = ze/kT used by Barrer and Stuart for the two kinds of cavities in the hydrates of the inert gases show a serious discrepancy (cf. Table V of ref. 4). Their procedure leads to Af/Ax CzL 1.9, whereas for a L-J-D field one must have A% /Ax = zt/zx = 1.2, independent of the combining rules. 

In conclusion, it is suggested that a spin combination rule may be an important criterion in determining whether or not reactants may follow an adiabatic, potential curve corresponding to a low lying state of an intermediate. This, in turn, may determine whether or not there will be strong attraction or weak, or even a barrier preventing fast reaction at low energy. 

The Trebb/e-Bishnoi EoS is a cubic equation that may utilize up to four binary interaction parameters, k=[ka, kb, kc, k[Pg.228]

An issue as interesting as it is contentious is that of electrolyte inhibition of bubble coalescence. Recently, a number of studies have reported the ion-specific nature of electrolyte inhibition of bubble coalescence, albeit in static (non-acoustic) fields [43 -9]. Some electrolytes appear to be highly efficacious whereas others almost completely ineffectual in inhibiting coalescence and ion combination rules have been devised to predict the behavior of various ion pairs. Various explanations have been proposed, most implying a gas-liquid interfacial mechanism. Christenson and Yaminsky [44] have reported a correlation between the inverse Marangoni factor, (dy/ dc]) 2, and coalescence inhibition ability for several different... 

The OPLS model is an example of pair potential where non-bonded interactions are represented through Coulomb and Lennard-Jones terms interacting between sites centred on nuclei (equation (51). Within this model, each atomic nucleus has an interaction site, except CH groups that are treated as united atoms centered on the carbon. It is important to note that no special functions were found to be needed to describe hydrogen bonding and there are no additional interaction sites for lone pairs. Another important point is that standard combining rules are used for the Lennard-Jones interactions such that An = (Ai As )1/2 and Cu = (C Cy)1/2. The A and C parameters may also be expressed in terms of Lennard-Jones o s and e s as A = 4ei Oi and C ... 

Two of them, 6 and p, can be deduced from the properties of pure A and B by applying the theorem of corresponding states which is implied in assumption (1). The two other parameters, 0 and a, were eliminated by assuming the combination rules ... 

---