MOLECULAR MODELING OF MATTER : FROM REALISTIC HAMILTONIANS TO SIMPLE MODELS AND THEIR APPLICATIONS Ivo NEZBEDA E. Hala Lab. of Thermodynamics, Acad. Sci., Prague, Czech Rep. and Dept. of Theoret. Physics, Charles University, Prague, Czech Rep. COLLABORATORS: J. Kolafa, M. Lisal, M. Predota, L. Vlcek; Acad. Sci., Prague A. A. Chialvo; Oak Ridge Natl. Lab., Oak Ridge P. T. Cummings; Vanderbilt Univ., Nashville M. Kettler; Univ. of Leipzig, Leipzig SUPPORT: Grant Agency of the Czech Republic Grant Agency of the Academy of Sciences

Roughly speaking, there are TWO WAYS OF MOLECULAR MODELING OF MATTER: SPECULATIVE: MATHEMATICAL: to model INTUITIVELY certain features (e.g. to capture the essence of intermolecular interactions) or a specific property (e.g. free energy models) starting from a certain mathematical description of the problem (e.g. from a realistic Hamiltonian), well-defined approximations are introduced in order to facilitate finding a solution. Although both methods may end up with the same model, there is a substantial difference between them.

WAYS OF MODELING…contd. SPECULATIVE: MATHEMATICAL: EXAMPLE 1: van der Waals EOS Molecules are not immaterial points but OBJECTS with their own impenetrable volume  molecules may be viewed as HARD SPHERES  EXAMPLE 1: Theory of simple fluids The simple fluid is defined by a REALISTIC potential, u (r). Too complex for theory. 1. The structure is determined by short-range REPULSIVE interactions.  u(r)=u(r) rep + Δu(r) and use a perturbation expansion 2. To solve the problem for the reference, u(r) rep, is still too difficult  properties of the u(r) rep fluid are mapped onto those of hard spheres: X rep  X hard sphere 

WAYS OF MODELING…contd. SPECULATIVE: MATHEMATICAL: PROBLEMS: (1) How to determine Δz? (2) How to refine/improve the approach? (3) How this is related to reality? Hard sphere model is a part of the well-defined scheme The correction term is well-defined  There are evident ways to improve performance of the method

HOW TO REACH THE GOAL? Start from the best realistic potential models and use a perturbation expansion. SIMPLE (THEORETICALLY FOOTED) MODELS ARE AN INDISPENSABLE PART OF THIS SCHEME. ULTIMATE GOAL OF THE PROJECT: Using a molecular-based theory, to develop workable (and reliable) expressions for the thermodynamic properties of fluids

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, u ref, and a perturbation part, u pert : u = u ref + u pert 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 = X ref +  X where  X denotes the contribution that has its origin in the perturbation potential u pert.

STEP 1: Separation of the total u into a reference part and a perturbation part, u = u ref + u pert 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 of PURE fluids 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.  THE REFERENCE MODEL IS A SHORT-RANGE FLUID: u ref = u short-range model

STEP 2: Estimate the properties of the short-range reference accurately (and relatively simply) in a CLOSED form HOW TO ACCOMPLISH THIS STEP ? HINT: Recall theories of simple fluids: u LJ = u soft spheres + Δ u (decomposition into ‘ref’ and ‘pert’ parts) X LJ = X soft spheres + ΔX X HARD SPHERES + ΔX SOLUTION: Find a simple model (called primitive model) that (i) approximates reasonably well the STRUCTURE of the short-range reference, and (ii) is amenable to theoretical treatment

Re SUBSTEP (1): Early (intuitive/empirical) attempts (for associating fluids) 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. SUBSTEPS OF STEP 2: (1)construct a primitive model (2)apply (develop) theory to get its properties

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: PROBLEM: We need to specify the parameters of interaction 1. HARD CORES (size of the molecule) 2. STRENGTH AND RANGE OF ATTRACTION

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 d ij ): 1-1. 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.

EXAMPLES: SPC water OPLS methanol carbon dioxide   

1-2. 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.

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

AMMONIA circles ….. reality lines ……. prim. model

ETHANOL

SPC/E WATER circles ….. reality lines ……. prim. model

POLAR FLUIDS Although they do not form hydrogen bonds, the same methodology can be applied also to them.

ACETONE site-site correlation functions

ACETONE site-site correlation functions

APPLICATIONS (of primitive models) : 1. As a reference in perturbed equations for the thermodynamic properties of REAL fluids [STEP 3 of the above scheme: X = X ref +  X ]. 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) Mixtures: water-alcohols, water-carbon dioxide,… (iv) Preferential solvation in mixed (e.g. water-methanol) solvents (v) Aqueous solutions of polymers (with EXPLICIT solvent) (vi) Clustering/Condensation (i.e. nucleation) (vii) Water at interface

Excess thermo-properties WATER-METHANOL at ambient conditions circles….. exptl. data squares… prim. model

Excess thermo-properties CO_2 - WATER at supercritical CO_2 conditions circles….. exptl. data squares… prim. model

THANK YOU for your attention