# Pressure

(Redirected from Virial pressure)

Pressure (${\displaystyle p}$) 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/hydrostatic conditions.

## Thermodynamics

In thermodynamics the pressure is given by

${\displaystyle p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T,N}=k_{B}T\left.{\frac {\partial \ln Q}{\partial V}}\right\vert _{T,N}}$

where ${\displaystyle A}$ is the Helmholtz energy function, ${\displaystyle V}$ is the volume, ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle T}$ is the temperature and ${\displaystyle Q(N,V,T)}$ 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

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

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

${\displaystyle \sigma _{ij}\equiv \left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right]}$

where where ${\displaystyle \ \sigma _{x}}$, ${\displaystyle \ \sigma _{y}}$, and ${\displaystyle \ \sigma _{z}}$ are normal stresses, and ${\displaystyle \ \tau _{xy}}$, ${\displaystyle \ \tau _{xz}}$, ${\displaystyle \ \tau _{yx}}$, ${\displaystyle \ \tau _{yz}}$, ${\displaystyle \ \tau _{zx}}$, and ${\displaystyle \ \tau _{zy}}$ 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 (Eq. 2 in [1]):

${\displaystyle p={\frac {k_{B}TN}{V}}+{\frac {1}{Vd}}{\overline {\sum _{i

where ${\displaystyle p}$ is the pressure, ${\displaystyle T}$ is the temperature, ${\displaystyle V}$ is the volume and ${\displaystyle k_{B}}$ 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; ${\displaystyle d}$ is the dimension of the system (3 in the "real" world). ${\displaystyle {\mathbf {f} }_{ij}}$ is the force on particle ${\displaystyle i}$ exerted by particle ${\displaystyle j}$, and ${\displaystyle {\mathbf {r} }_{ij}}$ is the vector going from ${\displaystyle i}$ to ${\displaystyle j}$: ${\displaystyle {\mathbf {r} }_{ij}={\mathbf {r} }_{j}-{\mathbf {r} }_{i}}$.

This relationship is readily obtained by writing the partition function in "reduced coordinates", i.e. ${\displaystyle x^{*}=x/L}$, etc, then considering a "blow-up" of the system by changing the value of ${\displaystyle L}$. 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

${\displaystyle {\mathbf {f} }_{ij}=-{\frac {{\mathbf {r} }_{ij}}{r_{ij}}}f(r_{ij}),}$

where ${\displaystyle f(r)}$ the force corresponding to the intermolecular potential ${\displaystyle \Phi (r)}$:

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

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

${\displaystyle p={\frac {k_{B}TN}{V}}+{\frac {1}{Vd}}{\overline {\sum _{i

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

${\displaystyle p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T}}$

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):

${\displaystyle p^{*}={\frac {\beta p}{\rho }}={\frac {pV}{Nk_{B}T}}=1-\beta {\frac {2}{3}}\pi \rho \int _{0}^{\infty }\left({\frac {{\rm {d}}\Phi (r)}{{\rm {d}}r}}~r\right)~{\rm {g}}(r)r^{2}~{\rm {d}}r}$

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