SklogWiki - User contributions [en]

Computation of phase equilibria

2010-08-06T17:11:24Z

161.111.20.32: /* Gibbs ensemble Monte Carlo for one component systems */

Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>:
* equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math>
* equal [[pressure]]s: <math> p_{\alpha} = p_{\beta} </math>
* equal [[chemical potential]]s: <math> \mu_{\alpha} = \mu_{\beta} </math>
The computation of phase equilibria using computer simulation can follow a number of different strategies. Here we will focus mainly
on [[first-order transitions]] in fluid phases, usually [[Gas-liquid phase transitions |liquid-vapour]] equilibria.
== Independent simulations for each phase at fixed temperature in the [[canonical ensemble]] ==
Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique.
Assuming that one has some knowledge on the [[phase diagrams | phase diagram]] of the system, one can try the following recipe:
# Fix a temperature and a number of particles
#Perform a limited number of simulations in the low density region (where the gas phase density is expected to be)
# Perform a limited number of simulations in the moderate to high density region (where the liquid phase should appear)
# In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the [[Widom test-particle method]])
===== A quick 'first guess' method =====
Using the previously obtained results the following, somewhat unsophisticated, procedure can be used to obtain a first inspection of the possible phase equilibrium:
#Fit the simulation results for each branch by using appropriate functional forms: <math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math>
#Use the fits to build, for each phase, a table with three entries: <math> \rho, p, \mu </math>, then plot for both tables <math> \mu </math> as a function of <math> p </math> and check to see if the two lines intersect.
#The crossing point provides (to within statistical uncertainty, the errors due to [[finite size effects]], etc.) the coexistence conditions.
===== Improving the 'first guess' method =====
It can be useful to take into account classical thermodynamics to improve the previous analysis. This is
because is is not unusual have large uncertainties in the results for the properties.
The basic idea is to use [[thermodynamic consistency]] requirements to improve the analysis.
== Methodology in the [[Isothermal-isobaric ensemble|NpT]] ensemble ==
=== Low temperature: <math> \left. T << T_c \right. </math> ===
For temperatures well below the [[critical points | critical point]], provided that the calculation of the chemical potential
of the liquid phase using [[Widom test-particle method]] gives precise results, the following strategy can be used to obtain a 'quick' result:
#Perform an <math> NpT </math> simulation of the liquid phase at zero pressure, i.e. <math> p \simeq 0 </math>
#Arrive at an initial estimate, <math> \mu^{(1)} </math> for the coexistence value of the chemical potential by computing, in the liquid phase: <math> \left. \mu^{(1)} = \mu_l (N,T,p=0) \right. </math>
#Make a first estimate of the coexistence pressure, <math> p^{(1)} </math>, by computing, either via simulation or via the [[Virial coefficients of model systems |virial coefficients]] of the gas phase, the pressure at which the gas phase fulfils: <math> \left. \mu_g(N,T, p^{(1)} ) = \mu^{(1)} \right. </math>
#Refine the results, if required, by performing a simulation of the liquid phase at <math> \left. p^{(1)} \right. </math>, or use estimates of <math> \left( \partial V / \partial p \right)_{N,T,p=0} </math> (from the initial simulation) and the gas equation of state data to correct the initial estimates of pressure and chemical potential at coexistence.
Note that this method works only if the liquid phase remains metastable at zero pressure.
=== Weak first order transitions ===
There are situations where other strategies based on [[Isothermal-isobaric ensemble|NpT]] ensemble simulations can be used.
Similar approaches have also been applied in the [[Monte Carlo in the grand-canonical ensemble|Grand Canonical]] ensemble.
In practice, the ''free energy barrier'' between the two phases has to be low enough to allow the simulated system
to cross from one phase to the other when it is being simulated. Somehow, the situation is now the opposite to that
described in Section 2.1.
Taking into the account the classical partition function in the [[isothermal-isobaric ensemble|NpT]] ensemble it can be written:

: <math> \left. P(V|N,p,T) \propto \exp \left[ - \frac{ p V}{k_B T} \right] \exp \left[ - \frac{A(N,V,T)}{k_B T} \right] \right. </math> ,

where <math> \left. P(V|N,p,T) \right. </math> stands for the probability of the volume <math> \left. V \right. </math> at given
conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a
case the following scenario is expected for <math> \left. P(V|N,p,T) \right. </math>:
*<math> \left. P(V|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with <math> \left. P(V_v|N,p_{eq},T) = P(V_l|N,p_{eq},T) = P_{v/l} \right. </math>

*The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as:

:<math> \left. P(V|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>,

where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface,
and <math> \left. {\mathcal A} \right. </math> is the
surface area, which depends on the thermodynamic
variables <math> \left. (N,V,T) \right. </math> and the geometry of the [[Periodic boundary conditions |simulation box]].
For small values of the surface tension, small system sizes and good simulation algorithms it could be possible for
pressures close to <math> p_{eq} </math> to sample in a simulation the whole region of densities between
vapour and liquid densities. If such is the case the phase equilibria conditions can be computed
by reweighing techniques applied on the volume histograms of the simulation.
==== Simple reweighing of the volume probability distribution ====
Suppose that a precise <math> \left. NpT \right. </math> simulation has been carried out at pressure <math> \left. p_0 \right. </math>. From that simulation a volume probability distribution, <math> \left. P_0(V|N,p_0,T) \right. </math>
has been computed. It is possible to use this function to estimate the distributions of volume for pressure values
close to <math> \left. p_0 \right. </math>;

: <math> \left. P(V|N,p,T) \propto \exp \left[ - \frac{ (p-p_0) V }{k_B T } \right] P_0(V|N,p_0,T) \right. </math>

Using this reweighing procedure we can estimate the value of <math> \left. p_{eq} \right. </math>; that is: the value
of pressure for which the volume distribution presents two peaks of equal height.
The analysis of the form of the distributions at the equilibrium conditions for different system sizes (i.e.
in the current case for different values of <math> \left. N \right. </math>) can be useful to distinguish between
continuous and first order [[phase transitions]].
=== See also ===
[[Surface tension]]
== Van der Waals loops in the [[canonical ensemble|canonical ]] ensemble ==
It is possible to compute the liquid-vapour equilibrium without explicit calculations of the chemical potential (or the pressure)
by performing a number of simulations sampling appropriately the vapour, liquid, and intermediate regions.
As an example, consider a simple fluid at a given subcritical temperature (<math> \left. T < T_c \right. </math>). We can perform a number of simulations for a given number of particles, <math> \left. N \right. </math> and different densities:

: <math> \rho_i = i \times \Delta \rho; \; \; \; \; i=1, 2, \cdots, m </math>

In these simulations, we can compute the pressure (or the chemical potential) and fit the result to an appropriate equation.
With such an ''equation of state'' the phase equilibria can be estimated.
If two phase equilibria exists, a ''loop'' in the representation of <math> \left. p = p (\rho) \right. </math> (or <math> \left. \mu = \mu (\rho) \right. </math>)
should appear.
* Computing <math> \left. \mu(\rho) \right. </math> from the equation of state given as <math> \left. p(\rho) \right. </math>:
For fixed temperature and number of particles:

: <math> \left. d A = - p d V \right. </math>
: <math> \left. d A = \frac{ p N }{\rho^2} d \rho ; \; \; \; \; d a = \frac{ p }{\rho^2} d \rho \right. </math>

where <math> \left. A \right. </math> is the [[Helmholtz energy function]] and
<math> \left. a = A/N \right. </math>, integrating:

:<math> \left. a(\rho) = a(\rho_0) + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \rho' \right. </math>

On the other hand

:<math> \left. A = - p V + \mu N ; \; \; \; a = - p/ \rho + \mu \right. </math>

therefore:

: <math> \left. \mu(\rho) - p(\rho)/\rho = \mu(\rho_0) - p(\rho_0)/\rho_0 + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \rho' \right. </math>
: <math> \left. \mu(\rho) = \mu(\rho_0) + p(\rho)/\rho - p(\rho_0)/\rho_0 + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \rho' \right. </math>

A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math>
from <math> \left. \mu(\rho) \right. </math>.
Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point.
==== Practical details ====
Some precautions should be taken if this procedure is used:
* The precision of the simulation results in the two phase region will be poor (so, large simulations are required to have a good estimation of the equation of state)
* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)
== Direct simulation of the two phase system==
An example using the [[Lennard-Jones model]],
*[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]
and its application to [[water]]
*[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]
== Gibbs ensemble Monte Carlo for one component systems==
The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See Ref. 1).
The simulation is, therefore, carried out at constant volume, temperature and number of particles.
The whole system is divided into two non-interacting parts, each one has its own simulation
box with its own [[periodic boundary conditions]].
This separation of the two phases into different boxes is in order to suppress any influence due to [[interface | interfacial]] effects.
The two subsystems can interchange volume and particles. The rules for these interchanges are
built up so as to guarantee conditions of both chemical and mechanical equilibrium between
the two phases.
If the overall conditions are of phase separation, it is expected that two phases will appear in
different simulation boxes.
====External links====
*[http://kea.princeton.edu/jerring/gibbs/ Gibbs ensemble Monte Carlo code] on the [http://kea.princeton.edu/ Panagiotopoulos Group Homepage]
[[category: Monte Carlo]]
[[category: Computer simulation techniques]]

== Mixtures ==
=== Symmetric mixtures ===
Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models | Ising model]] can be
viewed as mixture of two different chemical species which de-mix at low
temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams.
== See also==
*[[Gibbs-Duhem integration]]
==References==
#[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]
[[category: computer simulation techniques]]

Carnahan-Starling equation of state

2009-11-19T14:32:43Z

161.111.20.32: /* The 'Percus-Yevick' derivation */

The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).

== See also ==

*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Carnahan-Starling equation of state

2009-11-19T14:29:21Z

161.111.20.32: /* Thermodynamic expressions */changing betas

The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Carnahan-Starling equation of state

2009-11-19T14:26:04Z

161.111.20.32: /* Thermodynamic expressions */

The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{\beta p^{CS}}{\rho} = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[Helmholtz energy function]]:

:<math> \beta A_{ex}^{CS} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

Configurational [[chemical potential]]:

:<math>\beta \overline{\mu }^{CS} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{kT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Researchers and research groups

2009-11-11T17:36:46Z

161.111.20.32: /* United Kingdom */ hyperlink corrected M A Bates

==Argentina==
*[http://www.iflysib.unlp.edu.ar/index.html Instituto de Física de Líquidos y Sistemas Biológicos] IFLYSIB, La Plata
*[http://linux0.unsl.edu.ar/~fisica/superfi.htm Laboratorio de Ciencias de Superficies y Medios Porosos] Universidad Nacional de San Luis
*[http://www.tandar.cnea.gov.ar/~pastorin/ Claudio Pastorino] Comisión Nacional de Energía Atómica
*[http://cabtes55.cnea.gov.ar/personales/jagla/homepage.htm Eduardo A. Jagla] Centro Atómico Bariloche, Comisión Nacional de Energía Atómica

==Australia==
*[http://rsc.anu.edu.au/~evans/index.php Denis Evans] Australian National University
*[http://chem.sci.gu.edu.au/~s384166/d_bernhardt.html Debra J. Bernhardt (née Searles)] Griffith University
*[http://www.unisanet.unisa.edu.au/staff/PhilAttard/ Phil Attard] University of South Australia
*[http://www.phys.unsw.edu.au/%7Egary/statmech.html Statistical Mechanics and Dynamical Systems Group] University of New South Wales
*[http://www.nanochemistry.curtin.edu.au/research/theoretical.cfm Nanochemistry Research Institute: Theoretical and Computational] Curtin University of Technology
*[http://www.it.swin.edu.au/centres/cms/ Centre for Molecular Simulation] Swinburne University of Technology
*[http://www.rmit.edu.au/browse;ID=xono5nykwdrm The Condensed Matter Theory Group] RMIT (Royal Melbourne Institute of Technology) University

==Austria==
*[http://tph.tuwien.ac.at/smt/index.html Soft Matter Theory] Vienna University of Technology
==Belgium==
*[http://www.ulb.ac.be/sciences/polphy/ Laboratoire de Physique des Polymères] Université Libre de Bruxelles
*[http://www.grasp.ulg.ac.be/ Group for Research and Applications in Statistical Physics (GRASP)] Université de Liège
*[http://www.ulb.ac.be/rech/inventaire/unites/ULB658.html Laboratoire de Physique de la Matière molle (LPMM)] Université Libre de Bruxelles

==Brazil==
*[http://www.cbpf.br/GrupPesq/StatisticalPhys/ Group of Statistical Physics] Centro Brasileiro de Pesquisas Físicas
*[http://www.if.ufrgs.br/fcomplex/ Complex Fluids] Universidade Federal do Rio Grande do Sul

==Canada==
*[http://www.physics.uoguelph.ca/~des/ Research on Liquid Crystals and Complex Fluids] University of Guelph
*[http://mse.mcmaster.ca/faculty/johari/ Gyan Johari] McMaster University
*[http://www.pmc.umontreal.ca/~mousseau/site_an/index.php?n=Main.Welcome Normand Mousseau] Université de Montréal
*[http://www.apmaths.uwo.ca/~mkarttu/ The Karttunen group] University of Western Ontario in London
*[http://www.theory.chem.uwo.ca/drupal5/ Styliani Constas] University of Western Ontario in London
*[http://www.chem.queensu.ca/people/faculty/cann/ Natalie M. Cann] Queen's University, Kingston, Ontario
*[http://www.chem.utoronto.ca/staff/JMS/schofield_j.html Jeremy Schofield] University of Toronto
*[http://www.chem.utoronto.ca/~swhittin/ Stu Whittington - Chemical Physics Theory Group] University of Toronto
*[http://physics.sfu.ca/people/profiles/plischke Michael Plischke] Simon Fraser University, Burnaby, British Columbia
*[http://www.sfu.ca/~boal/ Dave Boal] Simon Fraser University, Burnaby, British Columbia
*[http://homepages.ucalgary.ca/~pkusalik/ Kusalik Research Group] University of Calgary
*[http://poole.stfx.ca/ Peter H. Poole] St. Francis Xavier University

==China==
*[http://www.zhuzit.edu.cn/en/edu_res/units.asp Dr. Zhou Shiqi, Modern Statistic Mechanics Research Institute] Hunan University of Technology
==Czech Republic==
*[http://www.natur.cuni.cz/~pmc/namecard.php?id=47 Tomáš Boublík] Univerzita Karlova v Praze
*[http://www.icpf.cas.cz/theory/IvoNez.html Ivo Nezbeda] Akademie věd České republiky
*[http://www.vscht.cz/fch/en/research/theory.html Department of Physical Chemistry: Theory] (Dir.: Anatol Malijevsky) Prague Institute of Chemical Technology
==Denmark==
*[http://www.mip.sdu.dk/~jperram/ John Perram] University of Southern Denmark
*[http://glass.ruc.dk/ "Glass and Time"] DNRF Centre for Viscous Liquid Dynamics, Roskilde University, Denmark

==France==
*[http://www.lcp.u-psud.fr/ Laboratoire de Chimie Physique] CNRS/Université Paris-Sud
*[http://www.lps.ens.fr/ Laboratoire de physique statistique] Ecole Normale Superieure
*[http://www.lptl.jussieu.fr/Welcome.html Laboratoire de Physique Théorique de la Matière Condensée] (Dir. Bertrand Guillot) Université Pierre et Marie Curie/CNRS
*[http://www.msc.univ-paris7.fr/site/index.html Matière et Systèmes Complexes] (Dir.: Jean-Marc di Meglio) Université Paris 7 - Denis Diderot
*[http://w3.lcvn.univ-montp2.fr/~kob/ Walter Kob] Universite Montpellier II
*[http://www.th.u-psud.fr/rubrique.php3?id_rubrique=8 Groupe de physique statistique] Laboratoire de Physique Théorique d'Orsay, CNRS et de l'Université Paris-Sud 11
*[http://www.lpm.u-nancy.fr/activite_physique_statistique/index.php?lang=en_GB Groupe de Physique Statistique ], Institut Jean Lamour, Nancy Université

==Germany==
*[http://www.cond-mat.physik.uni-mainz.de/ Condensed Matter Theory Group KOMET 331] (Dir: Kurt Binder) Johannes Gutenberg-University, Mainz
*[http://www.icp.uni-stuttgart.de/ Institute for Computational Physics] (Dir: Christian Holm) Stuttgart University
*[http://www.mpip-mainz.mpg.de/theory/ Polymer Theory Group Prof. Dr. Kurt Kremer] Max Planck Institute for Polymer Research, Mainz
*[http://www.theorie.physik.uni-goettingen.de/forschung/mm/index.en.html Marcus Müller's research group] Georg-August-Universität Göttingen
*[http://www.physik.uni-bielefeld.de/theory/cm/ Condensed Matter Theory Group] Universität Bielefeld
*[http://www.staff.uni-marburg.de/~germano/ Computer Simulation Group] Philipps-Universität Marburg
*[http://constanze.materials.uni-wuppertal.de/ Dr. R. Hentschke] Universität Wuppertal
*[http://van-der-waals.pc.uni-koeln.de/persons/kraskaE.html Dr. Thomas Kraska] Universität zu Köln
*[http://www.uni-koeln.de/math-nat-fak/phchem/deiters/index.html Statistische Thermodynamik] (Dir. Prof. Dr. Deiters) Universität zu Köln
*[http://www2.tu-berlin.de/%7Einsi/ag_schoen/people/profile/klapp/welcome.html Emmy-Noether research group: Complex fluids in external fields] (Dir. Dr. Sabine H. L. Klapp) Stranski-Laboratorium für Physikalische und Theoretische Chemie
*[http://theorie.physik.uni-konstanz.de/lsfuchs/ Soft Matter Theory Group] Universität Konstanz
*[http://www.theo.chemie.tu-darmstadt.de/front/joomla/ Theoretical Physical Chemistry Group] (Dir.: Dr. Florian Müller-Plathe) Technische Universität Darmstadt
*[http://ganter.chemie.uni-dortmund.de/index.shtml The Simulation and Theory Group] (Dir.: Prof. Dr. Alfons Geiger) Universität Dortmund
*[http://thwww.chemietechnik.uni-dortmund.de/en/textonly/content/staff/head/Sadowski.html Prof. Dr. Gabriele Sadowski] Universität Dortmund
*[http://agknapp.chemie.fu-berlin.de/agknapp/ Macromolecular Modelling Group] (Dir.: Ernst-Walter Knapp) Freie Universität Berlin

==Greece==
*[http://mmml.chem.demokritos.gr/ Molecular Thermodynamics and Modeling of Materials Laboratory] National Research Center for Physical Sciences "Demokritos"
*[http://www.matersci.upatras.gr/SoftMat/ Soft Matter Theory and Simulations Group] University of Patras

==Hungary==
*[http://www.chem.elte.hu/departments/elmkem/baranyai/index.htm András Baranyai] Eötvös Loránd University
*[http://www.chem.elte.hu/departments/kolloid/personnel/jp/ Pál Jedlovszky] Eötvös Loránd University
*[http://www.chem.elte.hu/departments/elmkem/toth/ Gergely Tóth] Eötvös University

==India==
*[http://www.iitk.ac.in/phy/New01/phy_CMT.html Condensed Matter Theory Group] Indian Institute of Technology (Kanpur)
*[http://home.iitk.ac.in/~amalen/ Prof Amalendu Chandra] Indian Institute of Technology (Kanpur)
*[http://www.physics.iitm.ac.in/%7Elabs/cfl/index.html Complex Fluids Laboratory] Indian Institute of Technology (Madras)
*[http://web.iitd.ac.in/~charusita/ Professor Charusita Chakravarty] Indian Institute of Technology (Delhi)
*[http://www.bhu.ac.in/science/faculty/Department_of_Physics_Dr_S_Singh.htm Dr. Shri Singh] Banaras Hindu University

==Iran==
*[http://sina.sharif.edu/~chinfo/parsafarh.html Gholamabbas Parsafar] Sharif University of Technology
==Israel==
*[http://chemistry.huji.ac.il/cgi-bin/chemistry/show_page.pl?L=E&Id=11 Prof. Arieh Ben-naim] Hebrew University of Jerusalem
*[http://www.fh.huji.ac.il/~viki/ Professor Victoria Buch] Hebrew University, Jerusalem

==Italy==
*[http://www2.fci.unibo.it/~bebo/z/index.html Claudio Zannoni home page] Università di Bologna
*[http://abaddon.phys.uniroma1.it/ GCI Computational Physics Group] (Dir. Giovanni Ciccotti) Universita’ di Roma La Sapienza
*[http://glass.phys.uniroma1.it/sciortino/index.html Francesco Sciortino] Universita’ di Roma La Sapienza
*[http://www.ictp.it/pages/research/cmsp.html CMSP - Condensed Matter and Statistical Physics at ICTP] Trieste
*[http://www.pv.infn.it/~romano/ Silvano Romano] Pavia
*[http://www.unive.it/nqcontent.cfm?a_id=36652&persona=000244 Domenico Gazzillo] Università Ca' Foscari Venezia
*[http://www.sissa.it/sbp/web_2008/index.html Statistical and Biological Chemistry Sector] Scuola Internazionale Superiore di Studi Avanzati (SISSA)

==Japan==
*[http://theochem.chem.okayama-u.ac.jp/?lang=en Theoretical Chemistry Group] Okayama University
*[http://www.ifs.tohoku.ac.jp/tokuyama-lab/ Michio Tokuyama Laboratory] Tohoku University

==Mexico==
*[http://abaco.izt.uam.mx/espanol/investigacion/liquidos/index.html Física de Líquidos] Universidad Autónoma Metropolitana
*[http://abaco.izt.uam.mx/espanol/profesores/estadistica.html Mecánica Estadística] Universidad Autónoma Metropolitana
*[http://www.ifug.ugto.mx/Investigacion/MecanicaEstadistica.php Mecánica Estadística] Universidad de Guanajuato
*[http://www.iquimica.unam.mx/pizio.html Orest Pizio] Universidad Nacional Autónoma de México
*[http://xml.cie.unam.mx/xml/tc/ft/mlh/ Mariano López de Haro] Universidad Nacional Autónoma de México
*[http://quimica.izt.uam.mx/Areas/QuimCuant/datosJRA.htm Jose Alejandre] Universidad Autónoma Metropolitana

==Netherlands==
*[http://www1.phys.uu.nl/scm/default.htm Soft Condensed Matter Group] Utrecht University
*[http://fcc.chem.uu.nl/peopleindex/henk/henk.htm Henk N.W. Lekkerkerker] Debye Research Institute, Utrecht University
*[http://www.science.uva.nl/~bolhuis/ Simulation of complex fluids] University of Amsterdam
*[http://www.rug.nl/gbb/research/researchgroups/molecularDynamics/ Marrink's MD group] University of Groningen

==Norway==
*[http://home.phys.ntnu.no/instdef/personale/hjemmesider/johan.hoye/index.html Johan Skule Høye] Norwegian University of Science and Technology (NTNU)
==Poland==
*[http://ichfit.ch.pwr.wroc.pl/?q=node/10 Molecular Modelling and Quantum Chemistry] Wrocław University of Technology
*[http://poczta.umcs.lublin.pl/zmpfch/index_en.htm Department for the Modelling of Physico-Chemical Processes] Maria Curie-Skłodowska University
*[http://ichf.edu.pl/person/ciach.html Professor Alina Ciach] Instytut Chemii Fizycznej, Polskiej Akademii Nauk
*[http://th-www.if.uj.edu.pl/zfs/ Statistical Physics Division] (Dir. Prof. Lech Longa) Uniwersytet Jagielloński
*[http://www.ifmpan.poznan.pl/zp10/zp10_www.htm Nonlinear Dynamics and Computer Simulations] (Dir. Prof. Dr. Habil. K. W. Wojciechowski) Institute of Molecular Physics, Polish Academy of Sciences

==Portugal==
*[http://cftc.cii.fc.ul.pt/index.php Centro de Física Teórica e Computacional] Universidade de Lisboa
==Russia==
*[http://theor.jinr.ru/~kuzemsky/ Alexander L. Kuzemsky] Bogoliubov Laboratory of Theoretical Physics
*[http://www.ihed.ras.ru/norman Атомистическое моделирование и теория конденсированного состояния и неидеальной плазмы] Институт теплофизики экстремальных состояний

==Spain==
*[http://www.qft.iqfr.csic.es/ Theoretical Physical Chemistry Group] (Dir. Enrique Lomba García) Instituto de Química-Física "Rocasolano" (IQFR), CSIC
*[http://www.icmm.csic.es/Teoria/ Condensed Matter Theory] Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC
*[http://emoles.quim.ucm.es/ Carlos Vega Statistical Thermodynamics of Molecular Fluids Group] Universidad Complutense de Madrid
*[http://seneca.fis.ucm.es/ Group of Statistical Mechanics (GISC)] Universidad Complutense de Madrid
*[http://www.uhu.es/filico/home.html Physics of Complex Liquids Group] (Dir.Dr. Enrique de Miguel Agustino) University of Huelva
*[http://gisc.uc3m.es/~cuesta/Science/scientific.html Scientific page of José A. Cuesta] Universidad Carlos III de Madrid
*[http://valbuena.fis.ucm.es/gisc/ Grupo Interdisciplinar de Sistemas Complejos] (GISC)
*[http://www.icmm.csic.es/mossnoho/ MOSSNOHO] Madrid
*[http://www.uam.es/departamentos/ciencias/fisicateoricamateria/propia/fluidos.html Investigación en física estadística de líquidos complejos y biofísica] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/dduque/ Daniel Duque] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/gnavascu/ Guillermo Navascués] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/evelasco/ Enrique Velasco] Universidad Autónoma de Madrid
*[http://www.uned.es/dpto-fisicoquimica/personas/lorna.htm Lorna Bailey Chapman] Universidad Nacional de Educación a Distancia (UNED)
*[http://www.uned.es/dpto-fisicoquimica/personas/luis.htm Luis M. Sesé Sánchez] Universidad Nacional de Educación a Distancia (UNED)
*[http://oboe.quim.ucm.es/jfg.html Juan J. Freire] Universidad Nacional de Educación a Distancia (UNED)
*[http://www.fisfun.uned.es/~pep/ Pep Español] Universidad Nacional de Educación a Distancia (UNED)
*[http://oboe.quim.ucm.es/ Simulation of Chain Molecules] Universidad Complutense de Madrid
*[http://www.upo.es/depa/webdex/quimfis/slagara/slagara.htm Santiago Lago Aranda] Universidad Pablo De Olavide
*[http://www.upo.es/depa/webdex/quimfis/miembros/Web_Sofia/Sofia_archivos/Group.htm Sofía Calero Materials Computational Group] Universidad Pablo De Olavide
*[http://www.grupo.us.es/gmecest/ Group of Statistical Mechanics] University of Seville
*[http://www.icmab.es/molsim/ Lourdes F. Vega Molecular Simulation Group] Instituto de Ciencia de Materiales de Barcelona (ICMAB), CSIC
*[http://grupos.unican.es/GTFE/ Grupo de Termodinámica y Física Estadística] Universidad de Cantabria
*[http://complex.ffn.ub.es/ Physics of Complex Systems Group] Universitat de Barcelona
*[http://simcon.upc.edu/ Computer Simulation in Condensed Matter Group SIMCON] Universitat Politècnica de Catalunya
*[http://www-fen.upc.es/cscmcs/index.html Complex Systems. Computer Simulation of Materials and Biological Systems] Universitat Politècnica de Catalunya
*[http://web.ffn.ub.es/node/6&id=1061 Grupo de Física Estadística] Universitat de Barcelona
*[http://www.unex.es/eweb/fisteor/index_eng.html Statistical Physics Group at the University of Extremadura (SPHINX)] University of Extremadura
*[http://www.ual.es/GruposInv/FQM-230/componentes/jcaballe.htm José Baldomero Caballero Moraleda] Universidad de Almería
*[http://ergodic.ugr.es/ Statistical Physics Group] University of Granada
*[http://www.etseq.urv.es/COMPLEX/index_cs.htm Complex Systems] (Dir.: Allan Mackie) Universitat Rovira i Virgili

==Sweden==
*[http://folding.bmc.uu.se/ David van der Spoel] Uppsala University
*[http://www.fos.su.se/page.php?pid=155&id=407 Alexander Lyubartsev] Stockholms universitet

==Switzerland==
*[http://www.igc.ethz.ch/ The van Gunsteren group's home page] ETH Hönggerberg, HCI
*[http://www.chemie.unibas.ch/~huber/index.html Prof. Dr. Hanspeter Huber] University of Basel
*[http://www.rgp.ethz.ch/ Professor Michele Parrinello's Research Group] ETH Zurich (Swiss Federal Institute of Technology Zurich)
*[https://www.cecam.org/ CECAM] Centre Européen de Calcul Atomique et Moléculaire, Lausanne, Switzerland
==Ukraine==
*[http://ph.icmp.lviv.ua/~lyuda/ Department for Statistical Theory of Condensed Systems] Ukrainian National Academy of Sciences

==United Kingdom==
*[http://www.md-net.org.uk/ MD network]
*[http://www.dur.ac.uk/mark.wilson/ The Wilson Group] Durham University
*[http://cmt.dur.ac.uk/ Condensed Matter Theory] (Dir.: Professor Richard Abram) University of Durham
*[http://www.shu.ac.uk/research/meri/mmg/ Materials Modelling group] Sheffield Hallam University
*[http://www2.warwick.ac.uk/fac/sci/physics/theory/research/simulation/ Molecular Simulation Group] (Dir.: Dr. M. Allen) University of Warwick
*[http://www2.warwick.ac.uk/fac/sci/chemistry/research/molsaw/ MOLecular Simulations At Warwick] University of Warwick
*[http://www.ceas.manchester.ac.uk/research/groups/multiscale/ Multi-scale and Multi-phase Systems] University of Manchester
*[http://www.sci-eng.mmu.ac.uk/facstaffdetails/mneal/default.htm Maureen P. Neal] Manchester Metropolitan University
*[http://www3.imperial.ac.uk/ceMMT/ Molecular modelling and thermodynamics] Imperial College London
*[http://www.ch.ic.ac.uk/quirke/ Computational Physical Chemistry Group] (Dir.: Nick Quirke) Imperial College London
*[http://www.ch.ic.ac.uk/bresme/ Dr. Fernando Bresme Group] Imperial College London
*[http://www.csec.ed.ac.uk/main.html Center for Science at Extreme Conditions] University of Edinburgh
*[http://www.ph.ed.ac.uk/cmatter/ Condensed Matter Group] University of Edinburgh
*[http://www.homepages.ed.ac.uk/pjc01/ Philip J. Camp] University of Edinburgh
*[http://www.chem.ucl.ac.uk/people/catlow/ Richard Catlow FRS] University College London
*[http://www.chem.ucl.ac.uk/people/coveney/ Professor Peter V. Coveney] University College London
*[http://titus.phy.qub.ac.uk/ Atomistic Simulation Centre] Queen's University Belfast
*[http://www.ucl.ac.uk/msl Thomas Young Centre at UCL] University College London
*[http://www-theor.ch.cam.ac.uk/people/jphgroup/ Research group of Professor Hansen] Cambridge University
*[http://www.ch.cam.ac.uk/staff/df.html Professor Daan Frenkel] University of Cambridge
*[http://www-theor.ch.cam.ac.uk/people/sprikgroup/ Sprik Group] University of Cambridge
*[http://www-wales.ch.cam.ac.uk/ Wales group home page] University of Cambridge
*[http://www.bath.ac.uk/physics/research/theory/ Condensed Matter Theory] University of Bath
*[http://staff.bath.ac.uk/chsscp/ Computational Solid State Chemistry Group] University of Bath
*[http://www.irc.leeds.ac.uk/~phy6pdo/ Peter D. Olmsted] University of Leeds
*[http://wheatley.chem.nottingham.ac.uk/ Dr. Richard Wheatley] University of Nottingham
*[http://www.strings.ph.qmul.ac.uk/~cmsmg/ Condensed Matter and Statistical Mechanics Group] (Dir.: Dr. Bob Jones) Queen Mary, University of London
*[http://www-thphys.physics.ox.ac.uk/user/ArdLouis/ Ard Louis research group] University of Oxford
*[http://physchem.ox.ac.uk/%7Edoye/index.html Jonathan Doye's Research Group] University of Oxford
*[http://www-thphys.physics.ox.ac.uk/user/JuliaYeomans/ Julia Yeomans' research group] University of Oxford
*[http://sbcb.bioch.ox.ac.uk/ Structural Bioinformatics and Computational Biochemistry] (Dir.: Prof. Mark S. P. Sansom) University of Oxford
*[http://www-thphys.physics.ox.ac.uk/user/SoftBio/ Theory of Soft and Biological Matter] University of Oxford
*[http://www.strath.ac.uk/chemeng/research/groupdetails/drmartinsweatman-seniorlecturer/ Dr. Martin Sweatman] University of Strathclyde
*[http://www.chm.bris.ac.uk/pt/jeroen/jsvdhome.html Jeroen van Duijneveldt's research group] University of Bristol
*[http://www.umi.surrey.ac.uk/research/scm Soft Condensed Matter Physics Group] University of Surrey
*[http://www.chm.bris.ac.uk/pt/allan/Research/ Computational Materials Chemistry] (Dir.: Professor Neil L. Allan) University of Bristol
*[http://www.stp.dias.ie/~dorlas/ Professor Teunis C. Dorlas] Dublin Institute for Advanced Studies
*[http://www.york.ac.uk/chemistry/staff/academic/a-c/mbates/ Dr Martin Bates] University of York
*[http://www.physics.leeds.ac.uk/pages/JRHenderson J. R. Henderson] University of Leeds
*[http://www.physics.leeds.ac.uk/pages/TCBMcLeish T. C. B. McLeish] University of Leeds
*[http://www.mth.kcl.ac.uk/~tcoolen/nnds/nnds.html Disordered Systems Group] King's College, University of London
*[http://www.shef.ac.uk/materials/staff/kptravis.html Dr Karl P. Travis] University of Sheffield
*[http://www.ma.hw.ac.uk/~oliver/ Professor Oliver Penrose] Heriot-Watt University
*[http://www.ccp5.ac.uk/ Collaborative Computational Project 5 - The Computer Simulation of Condensed Phases]
*[http://www.uwethiele.de/ Uwe Thiele] Loughborough University

== United States of America ==
*[http://faculty.ucmerced.edu/lhirst/index.html The Hirst Group] University of California, Merced
*[http://www.ecs.umass.edu/che/NSF_WWW/ Nanoscale Interdisciplinary Research Team (NIRT)] University of Massachusetts Amherst and University of Delaware
*[http://www.sas.upenn.edu/chem/groups/klein/klein.html The Klein Group] University of Pennsylvania
*[http://chumba.che.ncsu.edu/ Keith E. Gubbins' Research Group] North Carolina State University
*[http://turbo.che.ncsu.edu/index.html Carol K. Hall's Research Group] North Carolina State University
*[http://www.physics.ncsu.edu/people/faculty_lado.html Fred Lado] North Carolina State University
*[http://dl9s6.chem.unc.edu/ Polymer Theory Group] (Dir.: Michael Rubinstein) University of North Carolina at Chapel Hill
*[http://www.che.vanderbilt.edu/cummings1.htm Peter T. Cummings] Vanderbilt University and Oak Ridge National Laboratory
*[http://people.vanderbilt.edu/~c.mccabe/ McCabe Group] Vanderbilt University
*[http://www.cbe.buffalo.edu/kofke.htm David A. Kofke] University at Buffalo
*[http://www.chemical.buffalo.edu/ Errington Research Group] University at Buffalo
*[http://www.sunysb.edu/chemistry/faculty/gstell.htm George Stell] Stony Brook University
*[http://inka.mssm.edu/~mezei/ Mihaly Mezei] Mount Sinai School of Medicine, New York
*[http://pablonet.princeton.edu/ Pablo Gaston Debenedetti Group] Princeton University
*[http://cherrypit.princeton.edu/index.html Complex Materials Theory Group] (Dir.: Salvatore Torquato) Princeton University
*[http://www.princeton.edu/che/people/faculty/panagiotopoulos/group/ Panagiotopoulos Group Homepage] Princeton University
*[http://www.princeton.edu/~cargroup/ The Car Group] (Dir. Dr. Roberto Car) Princeton University
*[http://polymer.bu.edu/hes/ H. Eugene Stanley] Boston University
*[http://physics.bu.edu/people/show/161 Nicolas Giovambattista] Boston University
*[http://spider.pas.rochester.edu/mainFrame/people/pages/Shapir_Yonathan.html Yonathan Shapir] University of Rochester
*[http://bly.colorado.edu/index.html Liquid Crystal Physics Group] (Dir.: Noel Clark) University of Colorado at Boulder
*[http://www.columbia.edu/cu/chemistry/groups/berne/ Berne Group] Columbia University in the City of New York
*[http://people.chem.byu.edu/doug Douglas J. Henderson] Brigham Young University
*[http://www.chem.umn.edu/groups/siepmann/index.html Siepmann Group] University of Minnesota
*[http://www.engr.wisc.edu/groups/mtsm/ Molecular Thermodynamics and Statistical Mechanics Research Group] (Dir.: Juan J. de Pablo) University of Wisconsin-Madison
*[http://ising.phys.cwru.edu/ Soft Condensed Matter Theory Group of Professor Philip Taylor] Case Western Reserve University
*[http://liq-xtal.cwru.edu/ Case Liquid Crystal and Complex Fluids Group] (Dir.: Charles Rosenblatt) Case Western Reserve University
*[http://www.cchem.berkeley.edu/jmpgrp/index.htm John M. Prausnitz] University of California, Berkeley
*[http://cheme.berkeley.edu/people/faculty/smit/smit.html Berend Smit] University of California Berkeley
*[http://gold.cchem.berkeley.edu:8080/index.html The Chandler Group] University of California, Berkeley
*[http://europa.chem.uga.edu/ Allinger's Molecular Mechanics Research Lab] University of Georgia
*[http://zarbi.chem.yale.edu/ William L. Jorgensen Research Group] Yale University
*[http://www.inl.gov/cams/ Center for Advanced Modeling and Simulation] Idaho National Laboratory
*[http://www.mwdeem.rice.edu/djearl/index.html David J. Earl group] University of Pittsburgh
*[http://www.pitt.edu/~jordan/index.html Ken Jordan Theoretical and Computational Chemistry] University of Pittsburgh
*[http://www.wag.caltech.edu/ Materials and Process Simulation Center] (Dir.: Dr. William A. Goddard III) California Institute of Technology
*[http://dasher.wustl.edu/ Jay Ponder Lab] Washington University School of Medicine
*[http://www.chemistry.wustl.edu/~gelb/ Lev David Gelb Research Group] Washington University in St. Louis
*[http://www.glue.umd.edu/~xpectnil/ Michael E. Fisher] University of Maryland
*[http://www.glue.umd.edu/~jdw/ John D. Weeks] University of Maryland
*[http://www.chem.wisc.edu/~yethiraj/ The Yethiraj group] University of Wisconsin
*[http://www.ksu.edu/chem/people/faculty/smith.html Dr. Paul E. Smith] Kansas State University
*[http://www.chem.unl.edu/faculty/eachfaculty/zeng.shtml Xiao Cheng Zeng] University of Nebraska-Lincoln
*[http://www.chm.colostate.edu/bl/ Branka M. Ladanyi] Colorado State University
*[http://www.engr.ucr.edu/~jwu/ Jianzhong Wu] University of California, Riverside
*[http://tigger.uic.edu/~mansoori/TRL_html Thermodynamics Research Laboratory] (Dir.: Dr. G. Ali Mansoori) University of Illinois at Chicago
*[http://www.chem.cornell.edu/faculty/index.asp?fac=45 Professor Benjamin Widom] Cornell University
*[https://engineering.purdue.edu/ChE/Directory/Faculty/Corti.html David S. Corti] Purdue University
*[http://boltzmann.rockefeller.edu/ E. G. D. Cohen Laboratory] The Rockefeller University
*[http://www.phys.washington.edu/users/thouless/cmt.html Condensed Matter Theory group] University of Washington
*[http://www.math.rutgers.edu/~lebowitz/ Joel L. Lebowitz] Rutgers University
*[http://www.physics.rutgers.edu/cmt/group-cmt.html Theoretical Condensed Matter Physics] Rutgers University
*[http://www.science.duq.edu/faculty/talbot.html Julian Talbot] Duquesne University
*[http://cbme.ou.edu/faculty/lee.htm Lloyd L. Lee] University of Oklahoma
*[http://www.public.asu.edu/~caangell/ C. Austen Angell] Arizona State University
*[http://www.ruf.rice.edu/~saft/ Walter G. Chapman] Rice University
*[http://www.dartmouth.edu/~chem/faculty/JEGL.html Prof. Jane E. G. Lipson] Dartmouth College
*[http://thglab.lbl.gov/ Teresa Head-Gordon's Lab] Lawrence Berkeley National Laboratory
*[http://www.chm.tcu.edu/faculty/huckaby/ Dale A. Huckaby] Texas Christian University
*[http://www.engin.umich.edu/dept/che/research/glotzer/index.html Glotzer group] University of Michigan
*[http://www.nd.edu/~gezelter/Main/ Gezelter Lab] University of Notre Dame
*[http://www.cbms.utah.edu/ Voth Group] University of Utah
*[http://williamhoover.info/ Herr Professor Doctor William Graham Hoover] UC Davis, University of California
*[http://www.egr.msu.edu/~priezjev/ Nikolai Priezjev] Michigan State University
*[http://chemistry.uchicago.edu/fac/rice.shtml Prof. Stuart A. Rice] University of Chicago

[[category: Miscellaneous]]

MOSSNOHO Workshop 2009

2009-06-03T13:58:48Z

161.111.20.32:

[[Image:log_mossnoho1.gif|300px|frameless|right]]
La reunión del Programa de Investigación [http://www.icmm.csic.es/mossnoho/ MOSSNOHO/MODELICO] 2009 se celebrará el día 16 de junio en la [http://www.ucm.es/info/ccquim/ Facultad de Ciencias Químicas] de la [http://www.ucm.es/ Universidad Complutense de Madrid]. Por favor, utilizad esta Wiki para completar/modificar los títulos de las charlas.

Aquí esta el programa '''preliminar'''. Las charlas son de 20 minutos más 10 minutos para preguntas.
==Horario de las charlas==
{| style="width:90%; height:85px" border="3"
|-
|Horario || Ponente || Titulo de la charla || Institución || Departamento/Grupo de investigación
|-
| 10:00-10:30 || [http://cab.inta.es/03_01_01_curriculum.php?lab_id=&laboratorio=&per_id=30&idioma=ESP Susanna Cuevas Manrubia] || ''Evolución y adaptación de poblaciones moleculares y virales de RNA'' || [http://cab.inta.es/ Centro de Astrobiología] || Grupo de Sistemas Evolutivos
|-
| 10:30-11:00 || José Maria Guil || ''Microcalorimetria de adsorción'' || [http://www.iqfr.csic.es/ CSIC Instituto de Química-Física "Rocasolano"] || [http://www.iqfr.csic.es/electroquimica/Departamento.htm Química-Física de Interfases y Electroquímica]
|-
| 11:00-11:30 || [http://www.icmm.csic.es/Teoria/basjav.htm José Antonio Verges] || ''Estadísticas cuánticas del transporte electrónico a través de sistemas desordenados'' || [http://www.icmm.csic.es/ CSIC Instituto de Ciencia de Materiales de Madrid] || [http://www.icmm.csic.es/Teoria/ Condensed Matter Theory]
|-
| 11:30-12:00 || '''Café''' || || ||
|-
| 12:00-12:30 || [http://cacharro.quim.ucm.es/ José Luis F. Abascal] || ''Propiedades del agua subenfriada'' || [http://www.ucm.es/ Universidad Complutense de Madrid] || [http://www.ucm.es/info/quifi/ Departamento de Química Física I]
|-
| 12:30-13:00 || [http://www.dmae.upm.es/bartolo.html Bartolome Luque] || || [http://www2.upm.es/institucional Universidad Politécnica de Madrid] || [http://www.dmae.upm.es/ Departamento Matemática Aplicada y Estadística]
|-
| 13:00-13:30 || Alfonso Paez || || [http://www.uam.es/ Universidad Autónoma de Madrid] || [http://www.uam.es/departamentos/ciencias/fisicateoricamateria/propia/fluidos.html Investigación en física estadística de líquidos complejos y biofísica]
|-
| 13:30-14:30 || '''Comida''' || || ||
|-
| 14:30-15:00 || [http://www.qft.iqfr.csic.es/personal/alberto/ Alberto Gallardo] || ''Modelado y simulación de adsorción en arcillas con pilares'' || [http://www.iqfr.csic.es/ CSIC Instituto de Química-Física "Rocasolano" ] || [http://www.qft.iqfr.csic.es/ Grupo de Química Física Teórica]
|-
| 15:00-15:30 || [http://oboe.quim.ucm.es/jfg.html Juan Freire] || || [http://www.uned.es/ Universidad Nacional de Educación a Distancia (UNED)] || [http://www.uned.es/dpto-fisicoquimica/index.htm Departamento de Ciencias y Técnicas Fisicoquímicas]
|-
| 15:30-16:00 || [http://emoles.quim.ucm.es/carlos.html Carlos Vega] || ''La importancia de los efectos cuánticos en el agua'' || [http://www.ucm.es/ Universidad Complutense de Madrid] || [http://emoles.quim.ucm.es/ Termodinámica Estadística de Fluidos Moleculares]
|-
| 16:00-16:30 || '''Café''' || || ||
|-
| 16:30-17:00 || [http://www.uam.es/personal_pdi/ciencias/rdelgado/ Rafael Delagado Buscalioni] || || [http://www.uam.es/ Universidad Autónoma de Madrid] || [http://www.uam.es/departamentos/ciencias/fisicateoricamateria/propia/ Departamento de Física Teórica de la Materia Condensada]
|-
| 17:00-17:30 || José A. Capitan || ''Cuasiecosistemas y la catástrofe de extinción'' || [http://www.uc3m.es/portal/page/portal/inicio Universidad Carlos III de Madrid] ||
|-
| 17:30-18:00 || [http://www.iem.csic.es/fmacro/fismacr_doc/fismacr_bio/tiberio.htm Tiberio Ezquerra] || || [http://www.iem.csic.es/ CSIC Instituto de Estructura de la Materia] ||

|}
==Como llegar==
La [http://www.ucm.es/info/ccquim/ Facultad de Ciencias Químicas] se encuentra situada en el Campus de Moncloa de la UCM. [http://www.ucm.es/info/ccquim/pags.php?p=5 Mapa].
El workshop tendrá lugar en el salón de actos situado en el sotano del la biblioteca:
[[Image:UCM_salon_biblio.png|400px|frameless|center]]

[[Image:logo_sumaeducmadrid-org.jpg|300px|frameless|center]]