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.

Thermodynamics

In thermodynamics the pressure is given by

$p=-{\frac{\partial A}{\partial V}|}_{T,N}=k_{B}T{\frac{\partial \ln Q}{\partial V}|}_{T,N}$

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.

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

$\mathbf{F}={\sigma }_{ij}\mathbf{A}$

where $\mathbf{F}$ is the force, $\mathbf{A}$ is the area, and ${\sigma }_{ij}$ is the stress tensor, given by

${\sigma }_{ij}\equiv [\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_y& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_z\\ \end{array}]$

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.

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:

$p=\frac{k_{B}TN}{V}-\frac{1}{dV}\overline{{\sum }_{i<j}{\mathbf{f}}_{ij}{\mathbf{r}}_{ij}},$

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 $\mathrm{\Phi }(r)$:

$-\partial \mathrm{\Phi }(r)/\partial r.$

For example, for the Lennard-Jones potential, $f(r)=24ϵ(2(\sigma /r{)}^{12}-(\sigma /r{)}^6)/r$. Hence, the expression reduces to

$p=\frac{k_{B}TN}{V}+\frac{1}{dV}\overline{{\sum }_{i<j}f(r_{ij})r_{ij}}.$

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.

Pressure equation

For particles acting through two-body central forces alone one may use the thermodynamic relation

$p=-{\frac{\partial A}{\partial V}|}_T$

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):

$p^{*}=\frac{\beta p}{\rho }=\frac{pV}{N{k}_{B}T}=1-\beta \frac{2}{3}\pi \rho \underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{\mathrm{d}\mathrm{\Phi }(r)}{\mathrm{d}r}r)\mathrm{g}(r)r^2\mathrm{d}r$

where $\beta :=1/k_{B}T$, $\mathrm{\Phi }(r)$ is a central potential and $\mathrm{g}(r)$ is the pair distribution function.

See also

• Test volume method

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)