Thermophysical properties of fluids:

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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 SUPPORT Grant Agency of the
Czech Republic Grant Agency of the Academy of
Sciences
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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.
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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.
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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
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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 ?u (decomposition into ref
and pert parts) XLJ Xsoft
spheres ?X
XHARD SPHERES ?X
SOLUTION Find a simple model (called primitive
model) that (i) approximates
reasonably well the short-range reference, and
(ii) is amenable to theoretical
treatment
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• 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.
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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
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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. WCAHB) to get
effective HARD CORES (diameters dij)
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EXAMPLES
?
SPC water
OPLS methanol
?
carbon dioxide
?
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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, ?, 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.
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SELECTED RESULTS (OPLS methanol)
filled circles OPLS methanol solid line
primitive model
Average bonding angles ? and f ?
f prim. model 147 114 OPLS
model 156 113
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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
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• 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.