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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

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slide1

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

slide2

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.

slide3

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.

slide4

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

slide5

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

slide6

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.

slide7

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

slide8

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):

slide9

EXAMPLES:

SPC water

OPLS methanol

carbon dioxide

slide10

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.

slide11

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

slide12

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

slide13

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.