TIP4P model of water

2007-07-04T14:17:15Z

83.149.19.6:

The '''TIP4P''' is a rigid planar model of [[water]], having a similar geometry to the Bernal and Fowler ([[BF]]) model.
==Parameters==
{| style="width:75%; height:100px" border="1"
|-
| r (OH) (

Mean spherical approximation

2007-07-04T10:59:47Z

83.149.19.6:

The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) [[Closure relations | closure relation]] is given by

:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>

In the '''Blum and Høye''' mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by

:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) 1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i \sigma_j)/2</math>

and

:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>

where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

:<math>g(r) = \frac{c(r) \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>

where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the
[[Weeks-Chandler-Anderson perturbation theory | Weeks-Chandler-Anderson division]]
of the [[Lennard-Jones model | Lennard-Jones]] potential.
By introducing the definition (Eq. 10 Ref. 4)

:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>

one can arrive at (Eq. 11 in Ref. 4)

:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1 s)-s</math>

The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.

==Thermodynamic consistency==

See Ref. 5.

==References==
#[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251 - 258 (1966)]
#[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, '''19''' pp. 317-324 (1978)]
#[http://dx.doi.org/10.1007/BF01013935 Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal of Statistical Physics, '''22''' pp. 661-672 (1980)]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]
#[http://dx.doi.org/10.1063/1.2712181 Andrés Santos "Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials" Journal of Chemical Physics '''126''' 116101 (2007)]
[[Category:Integral equations]]

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TIP4P model of water

Mean spherical approximation