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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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From simple models to applications
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.
Grant Agency of the Czech Republic
Grant Agency of the Academy of Sciences
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.
For realistic (complex) intermolecular potential models the only route towards
analytic expressions is via a perturbation expansion.
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
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
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
(and relatively simply) in a CLOSED form
PARTIAL GOAL:ACCOMPLISH STEP 2
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
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,
Kolafa, Nezbeda, 1995; hard tetrahedron model of water
Nezbeda, Slovak, 1997; extended primitive models of water
These models capture QUALITATIVELY the main features of real
associating fluids, BUT
they are not linked to any realistic interaction potential model.
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,
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
FACTS: Because of strong cooperativity, site-site interactions cannot be treated
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.
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):
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.
filled circles: OPLS methanol
solid line: primitive model
Average bonding angles θ and φ:
prim. model 147 114
OPLS model 156 113
1. As a reference in perturbed equations for the thermodynamic properties of
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.
(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 &
(iii) Preferential solvation in mixed (e.g. water-methanol) solvents