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Pressure

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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[edit]

In thermodynamics the pressure is given by

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 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[edit]

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[edit]

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.

Virial pressure[edit]

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]):

p={\frac  {k_{B}TN}{V}}+{\frac  {1}{Vd}}\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}TN}{V}}+{\frac  {1}{Vd}}\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[edit]

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

See also[edit]

References[edit]

  1. Enrique de Miguel and George Jackson "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations", Journal of Chemical Physics 125 164109 (2006)

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