Virial stress

Virial stress is a measure of mechanical stress on an atomic scale for homogeneous systems. The expression of the (local) virial stress can be derived as the functional derivative of the free energy of a molecular system with respect to the deformation tensor^[1].

Volume averaged Definition

The instantaneous volume averaged virial stress 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.

The volume averaged virial stress is then the ensemble average of the instantaneous volume averaged virial stress.

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

{\mathcal {P}}_{at}=-{\frac {1}{3}}Tr(\tau ).

The pressure then is the ensemble average of the instantaneous pressure^[2]

P_{at}=\langle {\mathcal {P}}_{at}\rangle .

This pressure is the average pressure in the volume $\Omega$ .

Equivalent Definition

It's worth noting that some articles and textbook^[2] use a slightly different but equivalent 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.

Inhomogeneous Systems

If the system is not homogeneous in a given volume the above (volume averaged) pressure is not a good measure for the pressure. In inhomogeneous systems the pressure depends on the position and orientation of the surface on which the pressure acts. Therefore in inhomogeneous systems a definition of a local pressure is needed^[3]. As a general example for a system with inhomogeneous pressure you can think of the pressure in the atmosphere of the earth which varies with height.

Instantaneous local virial stress

The (local) instantaneous virial stress is given by^[1]:

$\tau _{ab}({\vec {r}})=-\sum _{i=1}^{N}\delta ({\vec {r}}-{\vec {r}}^{(i)})\left(m^{(i)}u_{a}^{(i)}u_{b}^{(i)}+{\frac {1}{2}}\sum _{j=1,j\neq i}^{N}({\vec {r}}^{(i)}-{\vec {r}}^{(j)})_{a}{\vec {f}}_{b}^{(ij)}\right),$

References

1 2 Morante, S., G. C. Rossi, and M. Testa. "The stress tensor of a molecular system: An exercise in statistical mechanics." The Journal of chemical physics 125.3 (2006): 034101, http://aip.scitation.org/doi/abs/10.1063/1.2214719.
1 2 Allen, MP; Tildesley, DJ (1991). Clarendon Press, ed. Computer Simulations of Liquids. Oxford. pp. 46–50.
↑ Numerical Simulations of a Smectic Lamellar Phase of Amphiphilic Molecules, p. 40, https://books.google.de/books?id=rPpegGthzO4C&lpg=PA40&dq=local%20pressure%20tensor&hl=de&pg=PA40#v=onepage&q=local%20pressure%20tensor&f=false

External links

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[morante06-1] 1 2 Morante, S., G. C. Rossi, and M. Testa. "The stress tensor of a molecular system: An exercise in statistical mechanics." The Journal of chemical physics 125.3 (2006): 034101, http://aip.scitation.org/doi/abs/10.1063/1.2214719.

[allen91-2] 1 2 Allen, MP; Tildesley, DJ (1991). Clarendon Press, ed. Computer Simulations of Liquids. Oxford. pp. 46–50.

[3] Numerical Simulations of a Smectic Lamellar Phase of Amphiphilic Molecules, p. 40, https://books.google.de/books?id=rPpegGthzO4C&lpg=PA40&dq=local%20pressure%20tensor&hl=de&pg=PA40#v=onepage&q=local%20pressure%20tensor&f=false