Virial stress

Virial stress is a measure of mechanical stress on an atomic scale for homogeneous systems. The name is derived from the Latin word vis, meaning force: "Virial is then derived from Latin as well, stemming from the word virias (plural of vis) meaning forces."^[1] 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.^[2]

Volume averaged Definition edit

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 a three dimensional, isotropic system, at equilibrium the "instantaneous" atomic pressure is usually defined as the average over the diagonals of the negative stress tensor:

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

The pressure then is the ensemble average of the instantaneous pressure^[3]

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

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

Equivalent Definition edit

It's worth noting that some articles and textbook^[3] 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.

Derivation edit

The virial pressure can be derived, using the virial theorem and splitting forces between particles and the container^[4] or, alternatively, via direct application of the defining equation $P=-{\frac {\partial F(N,V,T)}{\partial V}}$ and using scaled coordinates in the calculation.

Inhomogeneous Systems edit

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.^[5] 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 edit

The (local) instantaneous virial stress is given by:^[2]

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

Measuring the virial pressure in molecular simulations edit

The virial pressure can be measured via the formulas above or using volume rescaling trial moves.^[6]

References edit

^ Wasalwar, Yash (May 23, 2023). "Spotlight: the Virial Theorem". Medium. Archived from the original on February 3, 2024.
^ ^a ^b 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.
^ ^a ^b Allen, MP; Tildesley, DJ (1991). Clarendon Press (ed.). Computer Simulations of Liquids. Oxford. pp. 46–50.{{cite book}}: CS1 maint: location missing publisher (link)
^ Navet, M.; Jamin, E.; Feix, M. R. (1980-02-01). "" Virial " pressure of the classical one-component plasma". Journal de Physique Lettres. 41 (3): 69–73. doi:10.1051/jphyslet:0198000410306900. ISSN 0302-072X. S2CID 122678419.
^ Loison, Claire (2005). Numerical Simulations of a Smectic Lamellar Phase of Amphiphilic Molecules. Cuvillier Verlag. ISBN 978-3-86537-655-8.
^ Miguel, Enrique de; Jackson, George (2006-10-30). "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations". The Journal of Chemical Physics. 125 (16): 164109. Bibcode:2006JChPh.125p4109D. doi:10.1063/1.2363381. hdl:10272/9584. ISSN 0021-9606. PMID 17092065.

External links edit

Physical Interpretation of the volume averaged Virial Stress
Allen, M.P.; Tildesley, DJ. Computer Simulation of Liquids (PDF). Archived from the original (PDF) on 2016-12-21.
2017 edition (second): Allen, Michael Patrick; Tildesley, Dominic J. (2017). Computer simulation of liquids (PDF) (2nd ed.). Oxford: Oxford University Press. ISBN 9780198803201. Archived (PDF) from the original on November 26, 2022.
Python and Fortran code examples for Computer Simulation of Liquids
Zhou, Min (8 September 2003). "A new look at the atomic level virial stress: on continuum-molecular system equivalence" (PDF). Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 459 (2037): 2347–2392. doi:10.1098/rspa.2003.1127. ISSN 1364-5021. Archived (PDF) from the original on February 3, 2024.

[1] Wasalwar, Yash (May 23, 2023). "Spotlight: the Virial Theorem". Medium. Archived from the original on February 3, 2024.

[morante06-2] 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.

[allen91-3] Allen, MP; Tildesley, DJ (1991). Clarendon Press (ed.). Computer Simulations of Liquids. Oxford. pp. 46–50.{{cite book}}: CS1 maint: location missing publisher (link)

[4] Navet, M.; Jamin, E.; Feix, M. R. (1980-02-01). "" Virial " pressure of the classical one-component plasma". Journal de Physique Lettres. 41 (3): 69–73. doi:10.1051/jphyslet:0198000410306900. ISSN 0302-072X. S2CID 122678419.

[5] Loison, Claire (2005). Numerical Simulations of a Smectic Lamellar Phase of Amphiphilic Molecules. Cuvillier Verlag. ISBN 978-3-86537-655-8.

[6] Miguel, Enrique de; Jackson, George (2006-10-30). "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations". The Journal of Chemical Physics. 125 (16): 164109. Bibcode:2006JChPh.125p4109D. doi:10.1063/1.2363381. hdl:10272/9584. ISSN 0021-9606. PMID 17092065.

[1]

[2]

[3]

[4]

[5]

[6]