A general framework for modelling association

Abstract

A general formulation of association theory is presented which, unlike Wertheim's association theory as proposed in the Statistical Associating Fluid Theory, allows association strengths to vary with different sizes of oligomer. It is shown that association theory is governed by a few simple relationships which apply in all cases. The formulation is compatible with any equation of state. Potential applications are suggested.

Introduction

The concept of association is not new; it is often said that the idea was first proposed in 1908 by Dolezalek [1]. According to kinetic theory, fluids consist of a large number of particles in random thermal motion, and the simplest interpretation is that the particles correspond to the fluid's molecules. However, molecules are known to associate, typically through hydrogen bonding, to form oligomers. Since the oligomers are relatively long lasting, association theory identifies the fluid particles with these oligomers rather than the molecules, leading to the prediction that the pressure exerted by an associating fluid will be lower than otherwise expected.

Over the years, association theory has been incorporated into many thermodynamic models in various ways. To cite a few examples: an early work was that of Kretschmer and Wiebe [2] who proposed a continuous linear association scheme to describe the thermodynamics of alcohol molecules in solution. Others have developed similar ideas [[3], [4], [5]]. It is well known that the vapour phase of carboxylic acids dimerise strongly, and that an adequate thermodynamic description of them must include some treatment of the monomer-dimer equilibrium; see for example [6]. Wenzel adopted the idea of computing the equilibrium of specific oligomers in solution to model a number of polar solutions [7]. Heidemann and Prausnitz demonstrated that it was possible to obtain a closed form of the association of a mixture containing an associating substance with a continuous linear association scheme using an equation of state [8]. Anderko [9] added an additional association term to the Soave-Redlich-Kwong (SRK) equation [10] to extend the SRK equation to model solutions containing an associating substance. Ikonomou and Donohue proposed the Associated Perturbed Anisotropic Chain Theory [11] as a general model for associating solutions. For an extensive review of the application of the concept of association in a variety of thermodynamic model, see Economou and Donohue [12].

The Statistical Associating Fluid Theory (SAFT) proposed in 1990 by Chapman et al. [13] has been particularly influential. SAFT is a complex equation of state designed to have realistic physical terms based on statistic knowledge. In common with most equations of state, however, it has difficulty representing associating mixtures, but SAFT overcomes the problem by having an additional association term based on Wertheim's association theory [14,15]. It has been shown that SAFT is able to represent the thermodynamics of strongly associating mixtures with considerable accuracy. Inspired by the success of SAFT, Kontogeorgis and co-workers proposed the Cubic Plus Association (CPA) equation [16,17], which is based on the SRK equation to which a Wertheim association term is added. The CPA equation is much simpler mathematically than SAFT, and reduces to the SRK equation when no association is present. Like SAFT it can model associating mixtures.

The Wertheim association term as defined in SAFT is a first order treatment of Wertheim's theory; it has an attractive simplicity which results from the assumption that association is always chain-like and bonds are pair-wise additive, i.e. the association strengths of any type of bond are constant (at a given temperature) and independent of the size of the oligomers in which they occur. Unfortunately, these approximations are probably not particularly good. Take water for example: traditionally it was assumed that water forms a three-dimensional hydrogen-bonded network in which tetragonal structures are common. However, a different concept was proposed by Benson [18] who used thermodynamic arguments to suggest that water must have around 1.75 hydrogen bonds per molecule, compared with 2 for ice, while at the same time water molecules must be mobile. Benson felt this could only occur if water molecules were largely present as closed oligomers; he proposed an equilibrium between cyclic tetramers and cubic-shaped octomers. The idea would only have been regarded as speculative were it not that Benson was able to predict the heat capacity of water between 0 and 100 °C to within ±2% using only thermodynamic data for ice and water vapour. There is no fundamental reason for water oligomers not to be cyclic, and some molecular simulation studies support the idea that cyclic structures will form; see for example [19].

Alcohols are another example of highly associated compounds which have also been studied by Benson [20]. Using thermodynamic data, he concluded that, as with water, they tend to form cyclic oligomers with the cyclic tetramer probably the predominant species. Others have come to the same conclusion: using molecular orbital simulation, Curtiss [21] concluded that the most stable structure of the trimer and tetramer of methanol is cyclic; from measurements of alcohol dipole moments in solutions of varying concentration, Brink et al. [22] concluded that the cyclic tetramer is one of the predominant species of liquid alcohols. More recent molecular simulation studies [23] have supported the idea of cyclic oligomers forming in liquid alcohols and also ‘lasso’ structures where further association occurs around a cyclic core.

The implication of the finding that oligomers may form cyclic or other closed structures is that the association strength cannot be constant; as an oligomer grows, the thermodynamics of the addition of an extra molecule will depend on how many new bonds will form, whether the bonds are strained, the degree of steric hindrance and the cooperative effect which occurs when two or more hydrogen bonds occur. The Wertheim association term is therefore a considerable approximation. The success of the CPA equation with its highly empirical physical terms makes the obvious point that, in an associating system, the association term is the dominant term and the formulation of the physical terms is of lesser importance. It implies that the key to improving the description of associating systems is to improve the association term.

Since the Wertheim association term in SAFT is a first order expression, there is clearly scope to explore other less restrictive formulations. In order to describe experimental data more accurately, many researchers have investigated the implications of not assuming pair-wise additivity of bonds using a variety of formalisms. (See Hendriks et al. [24] for a demonstration that lattice theory, chemical theory and perturbation theory are equivalent at least to first order.) A common idea is bond cooperativity which implies that, compared with the dimer, larger oligomers should show greater association strength. For example, Sear and Jackson [25] introduced a three-body term into a perturbation treatment and showed that it had a significant impact on predicted levels of dimerisation, on coexistence curves and critical points. Missopolinou and Panayiotou [26] presented a lattice-fluid model in which dimerisation strength was different from that of all larger oligomers. A similar idea was introduced into a modified form of the Wertheim association term by Marshall and Chapman [27] which gave excellent agreement with molecular simulations. The effect of intramolecular association and formation of cyclic oligomers was also considered by Missopolinou and Panayiotou. Modelling cyclic association was investigated by Sear and Jackson [28]. Ghonasgi et al. [29,30] have investigated the impact of intramolecular bonding on association. The works above show that it is possible to extend formulations of association to incorporate a wider range of phenomena at the expense of varying increases of complexity.

The approach presented in this paper is to show that, starting from simple concepts, association theory can be formulated in a very general way which does not make assumptions about the values of the association strength, and the model is always governed by the same compact relationships whatever association scheme is assumed. The theory is set out in a way that is compatible with any equation of state.