Pressure

2012-06-01T17:30:13Z

Kevin.dale.parrish: /* Virial pressure */

'''Pressure''' (<math>p</math>) is the force per unit area applied on a surface, in a direction perpendicular to that surface, i.e. the scalar part of the [[stress]] tensor under equilibrium/hydrosatic conditions.
==Thermodynamics==
In thermodynamics the pressure is given by

:<math>p = - \left.\frac{\partial A}{\partial V} \right\vert_{T,N} = k_BT \left.\frac{\partial \ln Q}{\partial V} \right\vert_{T,N}</math>

where <math>A</math> is the [[Helmholtz energy function]], <math>V</math> is the volume, <math>k_B</math> is the
[[Boltzmann constant]], <math>T</math> is the [[temperature]] and <math>Q (N,V,T)</math>
is the [[Canonical ensemble | canonical ensemble partition function]].

==Units==

The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m2, or 1 J/m3. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 105 Pa by definition. This is very close to the standard atmosphere (atm), approximately equal to typical air pressure at earth mean sea level:
atm, standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa = 1.01325 bar
==Stress==
The '''stress''' is given by

:<math>{\mathbf F} = \sigma_{ij} {\mathbf A}</math>

where <math>{\mathbf F}</math> is the force,
<math>{\mathbf A}</math> is the area, and <math>\sigma_{ij}</math> is the stress tensor, given by

:<math>\sigma_{ij} \equiv \left[{\begin{matrix}
\sigma _x & \tau _{xy} & \tau _{xz} \\
\tau _{yx} & \sigma _y & \tau _{yz} \\
\tau _{zx} & \tau _{zy} & \sigma _z \\
\end{matrix}}\right]</math>

where where <math>\ \sigma_{x}</math>, <math>\ \sigma_{y}</math>, and <math>\ \sigma_{z}</math> are normal stresses, and <math>\ \tau_{xy}</math>, <math>\ \tau_{xz}</math>, <math>\ \tau_{yx}</math>, <math>\ \tau_{yz}</math>, <math>\ \tau_{zx}</math>, and <math>\ \tau_{zy}</math> are shear stresess.
==Virial pressure==
The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since [[Newtons laws#Newton's second law of motion |forces]] are evaluated and readily available. For pair interactions, one has:

:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]].
In this equation one can recognize an [[Equation of State: Ideal Gas |ideal gas]] contribution, and a second term due to the [[Virial theorem |virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress | stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

If the interaction is central, the force is given by
:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>
where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential |intermolecular potential]] <math>\Phi(r)</math>:

:<math>-\partial \Phi(r)/\partial r.</math>

For example, for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>

Notice that most [[Realistic models |realistic potentials]] are attractive at long ranges; hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model | hard spheres]], are always positive.

==Pressure equation==
For particles acting through two-body central forces alone one may use the [[Thermodynamic relations | thermodynamic relation]]

:<math>p = -\left. \frac{\partial A}{\partial V}\right\vert_T </math>

Using this relation, along with the [[Helmholtz energy function]] and the [[partition function | canonical partition function]], one
arrives at the so-called
'''pressure equation''' (also known as the '''virial equation'''):
:<math>p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r</math>

where <math>\beta := 1/k_BT</math>,
<math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]].
==See also==
*[[Test volume method]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1063/1.3245303 Aidan P. Thompson, Steven J. Plimpton, and William Mattson "General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions", Journal of Chemical Physics '''131''' 154107 (2009)]
*[http://dx.doi.org/10.1063/1.3316134 G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics '''132''' 074902 (2010)]
*[http://dx.doi.org/10.1063/1.3582905 Nikhil Chandra Admal and E. B. Tadmor "Stress and heat flux for arbitrary multibody potentials: A unified framework", Journal of Chemical Physics '''134''' 184106 (2011)]
*[http://dx.doi.org/10.1063/1.3626410 Takenobu Nakamura, Wataru Shinoda, and Tamio Ikeshoji "Novel numerical method for calculating the pressure tensor in spherical coordinates for molecular systems", Journal of Chemical Physics '''135''' 094106 (2011)]
*[http://dx.doi.org/10.1063/1.3692733 Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics '''136''' 104109 (2012)]

[[category: statistical mechanics]]
[[category: classical thermodynamics]]
[[category: classical mechanics]]

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Pressure