Pressure
Pressure (\(p\)) 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.
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[edit] Thermodynamics
In thermodynamics the pressure is given by
\[p = - \left.\frac{\partial A}{\partial V} \right\vert_{T,N} = k_BT \left.\frac{\partial \ln Q}{\partial V} \right\vert_{T,N}\]
where \(A\) is the Helmholtz energy function, \(V\) is the volume, \(k_B\) is the Boltzmann constant, \(T\) is the temperature and \(Q (N,V,T)\) is the canonical ensemble partition function.
[edit] 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
[edit] Stress
The stress is given by
\[{\mathbf F} = \sigma_{ij} {\mathbf A}\]
where \({\mathbf F}\) is the force, \({\mathbf A}\) is the area, and \(\sigma_{ij}\) is the stress tensor, given by
\[\sigma_{ij} \equiv \left[{\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\right]\]
where where \(\ \sigma_{x}\), \(\ \sigma_{y}\), and \(\ \sigma_{z}\) are normal stresses, and \(\ \tau_{xy}\), \(\ \tau_{xz}\), \(\ \tau_{yx}\), \(\ \tau_{yz}\), \(\ \tau_{zx}\), and \(\ \tau_{zy}\) are shear stresess.
[edit] 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 forces are evaluated and readily available. For pair interactions, one has:
\[ p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, \]
where \(p\) is the pressure, \(T\) is the temperature, \(V\) is the volume and \(k_B\) is the Boltzmann constant. In this equation one can recognize an ideal gas contribution, and a second term due to the virial. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in Monte Carlo; \(d\) is the dimension of the system (3 in the "real" world). \( {\mathbf f}_{ij} \) is the force on particle \(i\) exerted by particle \(j\), and \({\mathbf r}_{ij}\) is the vector going from \(i\) to \(j\): \({\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i\).
This relationship is readily obtained by writing the partition function in "reduced coordinates", i.e. \(x^*=x/L\), etc, then considering a "blow-up" of the system by changing the value of \(L\). This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the 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 \[ {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , \] where \(f(r)\) the force corresponding to the intermolecular potential \(\Phi(r)\):
\[-\partial \Phi(r)/\partial r.\]
For example, for the Lennard-Jones potential, \(f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r\). Hence, the expression reduces to \[ p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. \]
Notice that most 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 spheres, are always positive.
[edit] Pressure equation
For particles acting through two-body central forces alone one may use the thermodynamic relation
\[p = -\left. \frac{\partial A}{\partial V}\right\vert_T \]
Using this relation, along with the Helmholtz energy function and the canonical partition function, one arrives at the so-called pressure equation (also known as the virial equation): \[p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r\]
where \(\beta := 1/k_BT\), \(\Phi(r)\) is a central potential and \({\rm g}(r)\) is the pair distribution function.
[edit] See also
[edit] References
Related reading
- 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)
- G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics 132 074902 (2010)
- 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)
- 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)
- Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics 136 104109 (2012)
