THERMODYNAMIC PERTURBATION THEORY FOR ASSOCIATING MOLECULES

BENNETT D. MARSHALL1 and WALTER G. CHAPMAN2

1ExxonMobil Research and Engineering, Spring, TX, USA

2Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX, USA

CONTENTS

• I. Introduction
• II. A Brief Introduction to Cluster Expansions
• III. Single Association Site: Bond Renormalization
• IV. Single Association Site: Two-Density Approach
  • A. The Monovalent Case
  • B. The Divalent Case
• V. Multiple Association Sites: Multi-Density Approach
• VI. The Two-Site AB Case
  • A. Steric Hindrance in Chain Formation
  • B. Ring Formation
  • C. Bond Cooperativity
• VII. Spherically Symmetric and Directional Association Sites
• VIII. Density Functional Theory
• IX. Concluding Remarks
• Acknowledgments
• References

I. INTRODUCTION

Since the time of van der Waals, scientists have sought to describe the macroscopic behavior of fluids in terms of the microscopic interactions of the constituent molecules. By the early 1980s, accurate theories based on statistical mechanics had primarily been developed for near-spherical molecules. Successes of the 1960s and 1970s particularly by Chandler, Weeks, and Andersen [1] and by Barker and Henderson [2] produced perturbation theories for the properties of Lennard-Jones (LJ) fluids. Site-site theories such as reference interaction site model (RISM) [3] were developed, in part, to provide reference fluid structure to extend these perturbation theories to polyatomic molecules. However, for certain classes of fluids, the accurate description of the fluid phase in terms of the microscopic interactions has proven much more challenging. Hydrogen bonding interactions are strong, short-ranged, highly directional interactions that lie somewhere between a dipole/dipole attraction and a covalent bond. The short range and directionality of hydrogen bonds result in the phenomena of bond saturation, giving a limited valence of the hydrogen bonding attractions.

The same properties of the hydrogen bond, which complicate the theoretical description of these fluids, also give rise to a number of macroscopic physical properties that are unique to fluids that exhibit hydrogen bonding. Hydrogen bonding is responsible for the remarkable properties of water [4], folding of proteins [5] and is commonly exploited in the self-assembly [6] of advanced materials. More recently patchy colloids, a new class of materials that shares many qualities with hydrogen bonding fluids, have been developed. Patchy colloids are colloids with some number of attractive surface patches giving rise to association like anisotropic inter-colloid potentials [7]. For the purposes of this review, patchy colloids and hydrogen bonding fluids are treated on equal footing and will simply be termed "associating fluids."

The first models used to describe hydrogen bonding fluids were developed using a chemical approach, where each associated cluster is treated as a distinct species created from the reaction of monomers and smaller associated clusters [8, 9]. The "reactions" are governed by equilibrium constants that must be obtained empirically. This type of approach has been incorporated into various equations of state including a van-der-Waals-type equation of state [10], the perturbed anisotropic chain theory equation of state (APACT) [11], and the Sanchez-Lacombe [12] equation of state.

Alternatively, lattice theories may be employed to model hydrogen bonding fluids. These approaches generally follow the method of Veytsman [13] who showed how the free energy contribution due to hydrogen bonding could be calculated in the mean field by enumerating the number of hydrogen bonding states on a lattice. Veytsman's approach was incorporated into the Sanchez-Lacombe equation of state by Panayiotou and Sanchez [14] who factored the partition function into a hydrogen bonding contribution and a non-hydrogen bonding contribution. The lattice approach has also been applied to hydrogen bond cooperativity [15] and intramolecular [16] hydrogen bonds.

Both the chemical and lattice theory approaches to hydrogen bonding yield semi-empirical equations of state, which are useful for several hydrogen bonding systems [8]. The drawback of these approaches is a result of their simplistic development. As discussed earlier, it is desired to describe the macroscopic behavior of fluids through knowledge of the microscopic intermolecular interactions and distributions. This cannot be accomplished using a lattice or chemical theory. To accomplish this goal, we must incorporate molecular details of the associating fluid from the outset.

The starting place for any molecular theory of association is the definition of the pair potential energy ?(12) between molecules (or colloids). Molecules are treated as rigid bodies with no internal degrees of freedom. In total, six degrees of freedom describe any single molecule: three translational coordinates represented by the vector and three orientation angles represented by O1. These six degrees of freedom are represented as . It is assumed that the intermolecular potential can be separated as

where ?as(12) contains the association portion of the potential and ?ref(12) is the reference system potential, which contains all other contributions of the pair potential including a harsh short-ranged repulsive contribution.

Considering molecules (or colloids) that have a set of association sites , where association sites are represented by capital letters, the association potential is decomposed into individual site-site contributions

The potential ?AB(12) represents the association interaction between site A on molecule 1 and site B on molecule 2. One of the challenges in developing theoretical models for associating fluids stems from the short-ranged and directional nature of the association potential ?AB, which results in the phenomena of bond saturation. For instance, considering molecules which consist of a hard spherical core of diameter d

and a single association site A (see Fig. 1), bond saturation arises as follows. When spheres 1 and 2 are positioned and oriented correctly such that an association bond is formed, the hard cores of these two spheres may, depending on the size and range of the association site, prevent sphere 3 from approaching and sharing in the association interaction. That is, if and , then , meaning that each association site is singly bondable (has a valence of 1). In hydrogen bonding it is usually the case that each association bond site is singly bondable, although there are exceptions. For the case of patchy colloids, the patch size can be controlled to yield a defined valence controlling the type of self-assembled structures that form.

Figure 1. Illustration of bond saturation for hard spheres with a single monovalent association site.

Conical square well (CSW) association sites are commonly used as a primitive model for the association potential ?AB. First introduced by Bol [17] and later reintroduced by Chapman et al. [18, 19], CSWs consider association as a square well interaction which depends on the position and orientation of each molecule. Kern and Frenkel [20] later realized that this potential could describe the interaction between patchy colloids. For these CSWs the association potential is given by

where rc is the maximum separation between two colloids for which association can occur, ?A1 is the angle between and the orientation vector passing through the center of the patch on colloid 1, and ?c is the critical angle beyond which association cannot occur. Equation (4) states that if the spheres are close enough , and both are oriented correctly and , then an association bond is formed and the energy of the system is decreased by eAB. Figure 2 gives an illustration of two single-site spheres interacting with this potential. The size of the patch is dictated by the critical angle ?c that defines the solid angle to be . The patch size determines the maximum number of other spheres to which the patch can bond. Specifically considering a hard sphere reference fluid with association occurring at hard sphere contact , it is possible for a patch to associate at most once for , twice for , thrice for , and four times for [21]. The advantage of the CSW model is that it separates the radial and angular dependence of the potential and allows for analytic calculations in the model while allowing for quick calculation of association in a simulation since only two dot products are needed to determine that the molecular orientation criteria is satisfied for association.

Figure 2. Association parameters for conical association sites.

In the following sections we review some of the existing theories to model associating fluids with potentials of the form of Eqs. (1)-(2). We focus mainly on the...