Twodimensional model of water: Theory and computer simulations

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Twodimensional model of waterTheory and
computer simulations
Toma Urbic Fakulteta za kemijo in kemijsko
tehnologijo Univerza v Ljubljani Mentor Prof.
Dr. Branko Bortnik
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SUMMARY

• Some facts about water
• Simple model of water
• Integral equation theory
• Thermodynamic perturbation theory
• Wertheims theory
• Results
• Conclusion

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WATER

• is the most important substance on Earth
• molecular formula H2O
• H-O distances around 0.1 nm
• H-O-H angle around 104.5

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WATER

• has some properties that distinguish it from
  simpler liquids
• the most important anomalies of pure water are
• high melting point (Tm0 C)
• high boiling point (Tb100 C)
• temperature of maximum density in the liquid
  phase (4C)
• large and almost constant heat capacity in liquid
  phase
• unusually low compresibility

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WATER
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HYDROGEN BOND

• The anomalies are related to the ability of water
  molecules to form tetrahedrally coordinated
  hydrogen bonds.
• Directional bond (the O-H hydrogen atom being
  donated to the O-atom acceptor atom on another
  H2O molecule)
• The bond strength depend on its length and angle
  (exponential decay with distance)

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Simple model of water

• Many models have been developed to capture water
  properties
• One difficulty is that simulations that include
  a high degree of atomic realism are often
  computationally limited and cannot reliably
  predict the most subtle properties like heat
  capacity
• Hence there is a need for simplified models
• The simplest is the MB model, originally
  proposed by Ben--Naim in 1971

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MB model
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MB model
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MB model
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Integral equation theory

• The total interaction of one particular
  configuration is assumed to be given as a sum of
  the pair-wise interaction.

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Integral equation theory

• g2(r1,r2) is defined by the conditional
  probability of finding a particle in volume dr1
  at r1, and another particle in dr2 at r2,
  regardless of the positions of all other
  particles. In an isotropic fluid g2(r1,r2)
  g2(r1-r2) g(r).
• The total correlation function
• h(r) g(r) 1 measures an excess above the
  ideal gas value which is 1.

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Configurational integral Z(r1,.,rN) and the pair
distribution function g(r)
For ßU 0, we obtain the ideal contribution
ZVN.
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Ornstein-Zernike integral eq. defines c(r)
For ? ? 0 the two particles are only directly
correlated and h (r) c(r) the second term is
due to the presence of all other particles in the
system!
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The probability of finding a particle in dr is
changed due to the presence of particle 1 and is
given by ?h(r)dr. The 3rd particle makes a
contribution c(r-r)?h(r)dr.
?h(r)dr
1
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The closure condition
The Ornstein-Zernike equation in form

• is merely a definition and another relation
  between h(r) and c(r) (closure) is needed to
  solve this equation.
• For ?? 0 and r ? 8 an exact result is derived
  showing that c(r) - ßu(r). This is the
  simplest possible continuation.

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HNC closures

• Diagrammatic expansion yields an exact equation

• HNC approximation follows for the bridge
  graphs B(r) 0 only numerical solution is
  possible.

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Perturbation theory

• Structure of simple fluids is largely determined
  by geometric factors associated with the packing
  of molecular hard cores.
• The properties of a given fluid can be related to
  those of a hard-core reference system and the
  atractive part of the potential being treated as
  a perturbation.
• The choice of the hard-core fluid as reference
  system is an obvious one, since the thermodynamic
  and structural properties are well known.

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Hamiltonian of the system is written as
H0 is the Hammiltonian of hard-core fluid, while
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Perturbation theory example

• Lennard-Jones fluid (Baker-Henderson)

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Wertheimova teorija

•  Modern liquid theories are based on Mayer
  density series
•  For systems with weak interactions ( )
  we got ideal gas
• equation with this series.
•  
• In case of strong interaction between pairs (
  ) the correct
• eqution for ideal gas of dimers is
  .

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• To get the right equation of state we need to
  calculate infinite series.
• Wertheim developed the formalism which solved
  this problem of infinite series. With this theory
  we can study asociative fluids.
• Theory split potential in two parts on strong
  asociative part and nonasociative part. In our
  case HB potential is asociative part and LJ
  interaction non-asociative.
• Because of spliting of the potential the density
  of the molecules is splited into two parts into
  density of asociated and density of nonasociated
  molecules.
•  

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Results
Comparison for pair-correlation functions for MB
molecules between IET and MC simulations at
presure p0.19. Presented are the following
temperatures (a) T0.36, (b) T0.28, (c)
T0.24, (c) T0.21, (d) T0.18 and (e)
T0.16. MC results are presented with symbols
and IET results with solid line.
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Results
Temperature dependence of termodynamic properties
of MB molecules (a) the molar volume, (b) the
thermal expansion coefficient, (c) heat capacity
at constant pressure and (c) the
isothermal Compressibility s obtained by the
Monte Carlo simulation (symbols), the
thermodynamic perturbation theory (continuous
line) and integral equation theory (dashed line).
1.0M
0.425M
0.2M
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Conclusion

• Two statistical--mechanical theories developed
  for associated fluids by Wertheim were applied
  to the Mercedes--Benz model of water
• Both theories qualitatively reproduce the
  temperature trends of key thermodynamic
  properties
• Both theories improve at high temperatures,
  where the orientational averaging approximation
  is least problematic.
• The direction for improvement is to solve the
  orientationally dependent Ornstein--Zernike
  equation.