Solvent Design Using a Quantum Mechanical Continuum Solvation Model T. J. Sheldon, M. Folic ´ , and C. S. Adjiman* Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, London SW7 2AZ, U.K. The design of solvents for solutes typically found in the pharmaceutical and agrochemical industries is considered. These solutes are usually aromatic, with several heteroatoms, and they therefore exhibit complex interactions with solvents. As a result, detailed models are often needed to predict the behavior of solute/ solvent systems. The use of the SM5.42 continuum model of solvation 1-4 in solvent design is investigated. This model is based on a quantum mechanical representation of the solute. An optimization-based molecular design problem is formulated with the simple objective of minimizing the free energy of solvation. The solvent properties needed to calculate the free energy of solvation are obtained using group contribution methods. The resulting problem is a nonconvex mixed-integer nonlinear program with mixed-integer algebraic constraints. The outer-approximation algorithm is implemented to solve this optimization problem, using a combination of analytical and numerical gradients. Several case studies are solved, based on HF/6-31G* quantum calculations. Monofunctional and bifunctional aromatic compounds are used as test solutes. The design of the solvent is based on combinations of up to 41 different atom groups. In all cases, the algorithm identifies the best solvent in a small number of iterations. The required CPU time is up to 9 times smaller than that needed to evaluate all possible solvent structures. 1. Introduction Solvents are frequently used in the process industry and are especially common in the pharmaceutical and agrochemical industries 5 where they are used to carry out a variety of tasks such as reaction, transport, and separation. In many of these operations, the affinity between solvent and solute plays an important role, determining the feasibility and/or performance of the processing task. The infinite dilution activity coefficient for a solute (1) in a solvent (2), γ 12 ∞ , is often used as a measure of this affinity and is therefore a useful tool in solvent selection. 6 Several approaches have been proposed for the prediction of γ 12 ∞ . The UNIFAC group contribution method, based on the concept of a mixture of groups, can be used in its original form 7 or one of its modified forms. 8,9 One issue which often arises with complex solutes is that all group interaction parameters may not be available. An approach has recently been proposed to estimate missing parameters when few experimental mea- surements are available. 6,10 Another issue which can prevent the reliable prediction of γ 12 ∞ for complex solutes is the presence of multiple functional groups and of aromatic rings, which can lead to significant proximity effects. The development of a second-order version of UNIFAC, KT-UNIFAC, 11 extends the applicability of the approach by including connectivity information and correcting for proximity effects. Nevertheless, the full connectivity of the solute, which is always known during solvent design, is not used as an input for the predictions, and this loss of information may result in poor predictions for complex systems. Another class of methods that has recently been developed is based on computational chemistry. The COSMO-RS ap- proach 12 uses separate quantum mechanical calculations for solute and solvent. These calculations yield molecular surface descriptors for the individual molecules. These are then used in a simple procedure to predict the phase behavior of the solvent/solute mixture. This approach allows solvent screening by using a library of molecular descriptors for a wide range of solvents. It has also been shown 13-15 that the free energies of solvation calculated using a quantum mechanical continuum model of solvation 16,17 can be related to infinite dilution activity coefficients. The free energy of solvation, defined as the difference between the free energy of a solute in the gas phase and its free energy in solution, is, thus, a useful tool for solvent selection. One of the main benefits of continuum models of solvation is that the solute molecule is represented in detail and to a high degree of accuracy. The connectivity of the solute is an input to the model, and proximity effects are, therefore, well accounted for. As a result, complex organic molecules can readily be studied. The solvent, which is typically a small molecule, is represented implicitly via a parameterization based on a few bulk properties. Despite the wide applicability of such an approach, the computational cost of each free energy of solvation calculation is high and it is not yet clear how these techniques can be incorporated in the larger, process-centered, solvent design problem, where several criteria are typically taken into account. 18-26 A first step in this direction has been taken with the work of Lehmann and Maranas, 27 who have considered the use of quantum mechanics (QM) to design refrigerants and solvents. The use of computationally demanding property prediction techniques for molecular design has also been discussed by Harper and Gani, 28 who proposed a multilevel generate-and-test approach in which only molecules which pass inexpensive property tests are then screened with more expen- sive tests. In recent years, optimization-based molecular design tech- niques have emerged as a promising framework for solvent design (see ref 29 for an overview), in which it seems possible to handle expensive computations. One characteristic of interest in this context is that convergence of the optimization algorithm to the optimal solvent(s) typically occurs without testing all possible solvents in the design space. Another characteristic is that these approaches allow several design criteria to be taken into account simultaneously, in the form of constraints on the * To whom correspondence should be addressed. E-mail: c.adjiman@imperial.ac.uk. Tel.: +44 (0)20 7594 6638. Fax: +44 (0)- 20 794 6606. 1128 Ind. Eng. Chem. Res. 2006, 45, 1128-1140 10.1021/ie050416r CCC: $33.50 © 2006 American Chemical Society Published on Web 01/06/2006