Thermophysical properties of fluids: From simple models to applications Ivo NEZBEDA

1. Thermophysical properties of fluids: From simple models to applications Ivo NEZBEDA E. Hala Lab. of Thermodynamics, Acad. Sci., 165 02 Prague, Czech Rep. Dept. of Physics, J. E. Purkyne University, 900 46 Usti n. Lab., Czech Rep. COLLABORATORS: J. Kolafa M. Lisal M. Predota L. Vlcek SUPPORT: Grant Agency of the Czech Republic Grant Agency of the Academy of Sciences

2. ULTIMATE GOAL OF THE PROJECT: Using a molecular-based theory, to develop workable (and reliable) expressions for the thermodynamic properties of fluids With availability of fast and powerful computers, molecular simulations have become the major tool to study properties of condensed matter. Yet there are instances, both academic and practical, for which close analytic formulae are indispensable. METHOD: For realistic (complex) intermolecular potential models the only route towards analytic expressions is via a perturbation expansion.

3. PERTURBATION EXPANSION – general considerations Given an intermolecular pair potential u, the perturbation expansion method proceeds as follows: (1) u is first decomposed into a reference part, uref, and a perturbation part, upert: u = uref+ upert The decomposition is not unique and is dictated by both physical and mathematical considerations. This is the crucial step of the method that determines convergence (physical considerations) and feasibility (mathematical considerations) of the expansion. (2) The properties of the reference system must be estimated accurately and relatively simply so that the evaluation of the perturbation terms is feasible. (3) Finally, property X of the original system is then estimated as X = Xref + X where X denotes the contribution that has its origin in the perturbation potential upert.

4. STEP 1:Separation of the total u into a reference part and a perturbation part, u = uref + upert THIS PROBLEM SEEMS TO HAVE BEEN SOLVED DURING THE LAST DECADE AND THE RESULTS MAY BE SUMMARIZED AS FOLLOWS:  Regardless of temperature and density, the effect of the long-range forces on the spatial arrangement of the molecules is very small. Specifically: (1) The structure of both polar and associating realistic fluids and their short- range counterparts, described by the set of the site-site correlation functions, is very similar (nearly identical). (2) The thermodynamic properties of realistic fluids are very well estimated by those of suitable short-range models; (3) The long-range forces affect only details of the orientational correlations and hence, to a certain extent, also pressure. However, integral quantities, such as e.g. the dielectric constant, remain unaffected.  THE REFERENCE MODEL IS A SHORT-RANGE FLUID: uref = ushort-range model

5. STEP 2:Estimate the properties of the short-range reference accurately (and relatively simply) in a CLOSED form PARTIAL GOAL:ACCOMPLISH STEP 2 HOW?? HINT: Recall theories of simple fluids: uLJ = usoft spheres + Δu(decomposition into ‘ref’ and ‘pert’ parts) XLJ = Xsoft spheres + ΔX XHARD SPHERES + ΔX SOLUTION: Find a simple model (called primitive model) that (i) approximates reasonably well the short-range reference, and (ii) is amenable to theoretical treatment

6. SUBSTEPS OF STEP 2: • construct a primitive model • apply (develop) theory to get its properties Re SUBSTEP (1): Early (intuitive/empirical) attempts Ben-Naim, 1971; M-B model of water (2D) Dahl, Andersen, 1983; double SW model of water Bol, 1982; 4-site model of water Smith, Nezbeda, 1984; 2-site model of associated fluids Nezbeda, et al., 1987, 1991, 1997; models of water, methanol, ammonia Kolafa, Nezbeda, 1995; hard tetrahedron model of water Nezbeda, Slovak, 1997; extended primitive models of water PROBLEM: These models capture QUALITATIVELY the main features of real associating fluids, BUT they are not linked to any realistic interaction potential model.

7. GOAL 1: Given a short-range REALISTIC (parent) site-site potential model, develop a methodology to construct from ‘FIRST PRINCIPLES’ a simple (primitive) model which reproduces the structural properties of the parent model. IDEA: Use the geometry (arrangement of the interaction sites) of the parent model, and mimic short-range repulsions by a HARD-SPHERE interaction, , Example: carbon dioxide  and short-range attractions by a SQUARE-WELL interaction. PROBLEM: We need to specify the parameters of interaction 1. HARD CORES (size of the molecule) 2. STRENGTH AND RANGE OF ATTRACTION

8. 1. HOW (to set hard cores): ??? FACTS: Because of strong cooperativity, site-site interactions cannot be treated independently. HINT: Recall successful perturbation theories of molecular fluids (e.g. RAM) that use sphericalized effective site-site potentials and which are known to produce quite accurate site-site correlation functions. SOLUTION: Use the reference molecular fluid defined by the average site-site Boltzmann factors, and apply then the hybrid Barker-Henderson theory (i.e. WCA+HB) to get effective HARD CORES (diameters dij):

9. EXAMPLES:  SPC water OPLS methanol  carbon dioxide 

10. 2. HOW (to set the strength and range of attractive interaction): ??? HINT: Make use of (i) various constraints, e.g. that no hydrogen site can form no more than one hydrogen bond. This is purely geometrical problem. For instance, for OPLS methanol we get for the upper limit of the range, λ, the relation: The upper limit is used for all models. (ii) the known facts on dimer, e.g. that for carbon dioxide the stable configuration is T-shaped.

11. SELECTED RESULTS (OPLS methanol): filled circles: OPLS methanol solid line: primitive model Average bonding angles θ and φ: θ φ prim. model 147 114 OPLS model 156 113

12. APPLICATIONS (of primitive models): 1. As a reference in perturbed equations for the thermodynamic properties of REAL fluids. Example: equation of state for water [Nezbeda & Weingerl, 2001] Projects under way: equations of state for METHANOL, ETHANOL, AMMONIA, CARBON DIOXIDE 2. Used in molecular simulations to understand basic mechanism governing the behavior of fluids. Examples: (i) Hydration of inerts and lower alkanes; entropy/enthalpy driven changes [Predota & Nezbeda, 1999, 2002; Vlcek & Nezbeda, 2002] (ii) Solvation of the interaction sites of water [Predota, Ben-Naim & Nezbeda, 2003] (iii) Preferential solvation in mixed (e.g. water-methanol) solvents

13. Re SUBSTEP (2): Theory of primitive models • METHOD: Thermodynamic perturbation theory • PROBLEMS: • First-order theory is only fairly accurate • Oxygen sites may form simultaneously up to two H-bonds (violation of the • steric incompatibility conditions) • GOAL 2: Develop 2nd order theory and implement it for double-bonding sites • RESULT [Vlcek L., Nezbeda I.,Mol. Phys. 2003, in press] • Contributions of three classes of graphs contributing to the second-order of the thermodynamic • perturbation theory have been evaluated. It has been shown that the contributions of linear • chains bring only a marginal improvement over the first-order theory. The most significant • contribution comes from the graph accounting for double bonding of the oxygen site. • Neglecting the linear chain diagrams and retaining only this graph, general analytic • expressions for the thermodynamic properties have derived and it has shown that the theory • within this approximation is in agreement with simulation data.