Virial stress

Virial stress is a measure of mechanical stress on an atomic scale. It is given by

\tau _{ij}={\frac {1}{\Omega }}\sum _{k\in \Omega }\left(-m^{(k)}(u_{i}^{(k)}-{\bar {u}}_{i})(u_{j}^{(k)}-{\bar {u}}_{j})+{\frac {1}{2}}\sum _{\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}\right)

where

$k$ and $\ell$ are atoms in the domain,
$\Omega$ is the volume of the domain,
$m^{(k)}$ is the mass of atom k,
$u_{i}^{(k)}$ is the i^th component of the velocity of atom k,
${\bar {u}}_{j}$ is the j^th component of the average velocity of atoms in the volume,
$x_{i}^{(k)}$ is the i^th component of the position of atom k, and
$f_{i}^{(k\ell )}$ is the i^th component of the force applied on atom $k$ by atom $\ell$ .

At zero kelvin, all velocities are zero so we have

\tau _{ij}={\frac {1}{2\Omega }}\sum _{k,\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}

This can be thought of as follows. The τ₁₁ component of stress is the force in the x₁-direction divided by the area of a plane perpendicular to that direction. Consider two adjacent volumes separated by such a plane. The 11-component of stress on that interface is the sum of all pairwise forces between atoms on the two sides.

In an isotropic system, at equilibrium the atomic pressure is usually defined as

P_{at}=-{\frac {1}{3}}Tr(\tau )

It's worth noting that some articles and textbook ^[1] use a slightly different version of the equation

\tau _{ij}={\frac {1}{\Omega }}\sum _{k\in \Omega }\left(-m^{(k)}(u_{i}^{(k)}-{\bar {u}}_{i})(u_{j}^{(k)}-{\bar {u}}_{j})-{\frac {1}{2}}\sum _{\ell \in \Omega }x_{i}^{(k\ell )}f_{j}^{(k\ell )}\right)

where $x_{i}^{(k\ell )}$ is the i^th component of the vector oriented from the $\ell$ ^th atoms to the k^th calculated via the difference

$x_{i}^{k\ell }=x_{i}^{(k)}-x_{i}^{(\ell )}$

Both equation being strictly equivalent, the definition of the vector can still lead to confusion.

References

↑ Allen, MP; Tildesley, DJ (1991). Clarendon Press, ed. Computer Simulations of Liquids. Oxford. pp. 46–50.

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Virial stress

See also

References

External links