Modern Quasi-Chemical Theory

In this section, we will explore the foundations of quasi-chemical theory, starting from its original model to the more advanced and modern formulations, such as COSMO-SAC. We will also delve into semi-empirical approximations of the theory, including well-known models like Wilson, NRTL, UNIQUAC, and others. These models have been pivotal in advancing our understanding of solution thermodynamics and play a key role in various applications, including phase equilibria and mixture property prediction.

Original quasi-chemical model

The quasi-chemical model was pioneered by Guggenheim¹ in 1940s as a thermodynamic model to describe the non-random arrangement of molecules in a mixture, particularly in liquids. In its original version, it is based on a lattice picture of the liquid state assuming interactions only between nearest neighbour compounds.

In this framework, interactions can be interpreted as chemical reactions at equilibrium. For a binary mixture of compounds A and B, the following reaction holds:

$$AA + BB \rightleftharpoons 2 AB$$

When either compound A or compound B is pure, they can only interact with other identical molecules, forming AA or BB pairs. However, in mixtures, there exists an equilibrium between AA, BB, and AB complexes.

Using the notation introduced by Soares and Staudt², there should be Boltzmann factor $\Psi_{AB}$ in terms of the pair formation (or interaction) energy $u_{AB} = u_{BA}$, given by: $$\Psi_{AB} = \exp\left(-\frac{u_{AB}}{kT}\right)$$

Then, the probabylity of finding AB pairs would be given by: $$\theta_{AB}^2 = \theta_{AA} \theta_{BB} \frac{\Psi_{AB}^2}{\Psi_{AA} \Psi_{BB}}$$

Note: The above equation is equivalent to the quasi-chemical treatment, Eq.(4.09.1) of Guggenheim¹. If the mixture where to be completely random: $\theta_{AB} = \Theta_{A} \Theta_{B}$, where $\Theta_{A}$ is the surface area fraction of compound A in mixture.

Extensions in terms of groups or small segments

In its original and simplest form, the quasi-chemical method is expressed in molecular terms, which imposes several limitations on the model. For instance, the model is unable to describe positive excess entropies (found for instance in the acetone/n-heptane system). The simple preference for cross-compound interactions invariably leads to an increase in order within the mixture ³.

In a more sophisticated form (assuming the presence of functional groups with different interaction energies), quasi-chemical models become quite flexible. However, expressing the problem in terms of groups results in an expanded set of equations that need to be solved. Notably, Panayiotou⁴ have already presented a formulation in terms of surface area fractions in the 1980s and shortly after Larsen⁵ investigated the numerical solution of the resulting system of equations.

The surface fragmentation can be further refined to small surface segments, as depicted below²:

This refinement results in sets of equations equivalent to those used in COSMO-RS ⁶ or COSMO-SAC ⁷ models. Many studies have recognized COSMO-based models as equivalent to the quasi-chemical treatment ⁸⁶⁹¹⁰².

Elliott ¹¹ extends this concept by characterizing COSMO-RS/SAC models as Small Segment Quasi-Chemical Theory (SS-QCT), considering them essentially equivalent, despite the absence of an explicit lattice reference.

We prefer to refer to this category of models simply as modern quasi-chemical models.

Semi-empirical approximations

When no additional approximations are assumed, quasi-chemical models lack explicit expressions for activity coefficients, requiring the solution of a set of non-linear equations⁵. Many semi-empirical developments - like Wilson¹², NRTL¹³, and UNIQUAC¹⁴ - originate from quasi-chemical theory, but their simplifications allow for explicit expressions for the activity coefficients⁵. These models became widely used due to their simplicity and flexibility³.

UNIFAC (UNIQUAC Functional-group Activity Coefficients) original¹⁵ and modified versions like the UNIFAC (Do)¹⁶ are still usually very good options when one needs to predict activity coefficients.