http://www.sklogwiki.org/SklogWiki/api.php?action=feedcontributions&user=83.149.19.6&feedformat=atom SklogWiki - User contributions [en] 2020-07-04T12:17:41Z User contributions MediaWiki 1.30.0 http://www.sklogwiki.org/SklogWiki/index.php?title=TIP4P_model_of_water&diff=3219 TIP4P model of water 2007-07-04T14:17:15Z <p>83.149.19.6: </p> <hr /> <div>The '''TIP4P''' is a rigid planar model of [[water]], having a similar geometry to the Bernal and Fowler ([[BF]]) model.<br /> ==Parameters==<br /> {| style=&quot;width:75%; height:100px&quot; border=&quot;1&quot;<br /> |- <br /> | r (OH) (</div> 83.149.19.6 http://www.sklogwiki.org/SklogWiki/index.php?title=Mean_spherical_approximation&diff=3212 Mean spherical approximation 2007-07-04T10:59:47Z <p>83.149.19.6: </p> <hr /> <div>The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) [[Closure relations | closure relation]] is given by<br /> <br /> <br /> :&lt;math&gt;c(r) = -\beta \omega(r), ~~~~ r&gt;\sigma.&lt;/math&gt;<br /> <br /> <br /> In the '''Blum and Høye''' mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by<br /> <br /> <br /> :&lt;math&gt;{\rm g}_{ij}(r) \equiv h_{ij}(r) 1=0 ~~~~~~~~ r &lt; \sigma_{ij} = (\sigma_i \sigma_j)/2&lt;/math&gt;<br /> <br /> <br /> and<br /> <br /> :&lt;math&gt;c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} &lt; r&lt;/math&gt;<br /> <br /> where &lt;math&gt;h_{ij}(r)&lt;/math&gt; and &lt;math&gt;c_{ij}(r)&lt;/math&gt; are the total and the direct correlation functions for two spherical<br /> molecules of ''i'' and ''j'' species, &lt;math&gt;\sigma_i&lt;/math&gt; is the diameter of '''i'' species of molecule.<br /> Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as<br /> <br /> <br /> :&lt;math&gt;g(r) = \frac{c(r) \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}&lt;/math&gt;<br /> <br /> <br /> where &lt;math&gt;\Phi_1&lt;/math&gt; and &lt;math&gt;\Phi_2&lt;/math&gt; comes from the <br /> [[Weeks-Chandler-Anderson perturbation theory | Weeks-Chandler-Anderson division]] <br /> of the [[Lennard-Jones model | Lennard-Jones]] potential.<br /> By introducing the definition (Eq. 10 Ref. 4) <br /> <br /> <br /> :&lt;math&gt;\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)&lt;/math&gt;<br /> <br /> <br /> one can arrive at (Eq. 11 in Ref. 4)<br /> <br /> <br /> :&lt;math&gt;B(r) \approx B^{\rm MSA}(s) = \ln (1 s)-s&lt;/math&gt;<br /> <br /> <br /> The [[Percus Yevick]] approximation may be recovered from the above equation by setting &lt;math&gt;\Phi_2=0&lt;/math&gt;.<br /> <br /> ==Thermodynamic consistency==<br /> <br /> See Ref. 5.<br /> <br /> ==References==<br /> #[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus &quot;Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids&quot;, Physical Review '''144''' pp. 251 - 258 (1966)]<br /> #[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye &quot;Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture&quot;, Journal of Statistical Physics, '''19''' pp. 317-324 (1978)]<br /> #[http://dx.doi.org/10.1007/BF01013935 Lesser Blum &quot;Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure&quot; Journal of Statistical Physics, '''22''' pp. 661-672 (1980)]<br /> #[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet &quot;Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function&quot;, Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]<br /> #[http://dx.doi.org/10.1063/1.2712181 Andrés Santos &quot;Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials&quot; Journal of Chemical Physics '''126''' 116101 (2007)]<br /> [[Category:Integral equations]]</div> 83.149.19.6