Thermophysical properties of fluids: From simple models to applications

Ivo NEZBEDAE. 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. KolafaM. LisalM. PredotaL. Vlcek

SUPPORT:Grant Agency of the Czech RepublicGrant Agency of the Academy of Sciences

General bla, bla

PERTURBATION THEORY – general considerations (1) Given an intermolecular pair potential u, the perturbation expansion method proceeds as follows: u is first decomposed into a reference, uref, and a perturbation, upert, part,

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.

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

STEP 2: Estimate the properties of the reference accurately (and relatively simply) in a CLOSED form

HOW?? HINT: Recall theories of simple fluids:

uLJ = usoft spheres + Δu

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

GOAL: 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,

ji

ijijrangeshort ruu,

)()2,1( ,

Example:3-site modelof methanol

PROBLEM:PROBLEM: How to set the parameters of interaction

and mimic short-range repulsion by a HARD-SPHERE interaction, and short-range attraction by a SQUARE-WELL interaction.

Early (intuitive/empirical) attempts

Ben-Naim, 19xx; M-B model of water (2D)Dahl, Andersen, 19xx; double SW model of waterBol, 19xx; 4-site model of waterSmith, Nezbeda, 198x; 2-site model of associated fluidsNezbeda, et al., 198x, 199x, 199x; models of water,

methanol, ammoniaKolafa, Nezbeda, 199x; hard tetrahedron model of water

SOLUTION: Use the reference molecular fluid defined by the average site-site Boltzmann factors,

and apply then the hybrid Barker-Henderson theory to get effective HARD CORES(diameters dij):

constr

ijijij

ijBij

ij

ddure

reTkru

)2()1(]exp[)(

)(ln)(

RAM

r

uij/kT

repulsive part: uijrep

attractive part: uijatr

full angle averaged potential: uijRAM

rmin

0

rrudr

repijij dexp1

min

0

HOW: ???FACTS: Because of strong cooperativity, site-site interactions cannot be treated

independently.HINT: Recall successful perturbation theories of molecular fluids that use sphericalized effective site-site potentials and which are known to produce quite accurate site-site correlation functions.