# Viscosity from DFT (VASP) using the Green-Kubo relation

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• Polyamorph
In summary: OK, so which equation should I use to input the stress tensor components into the Green-Kubo equation?The equation you should use is:##\eta = \frac{V}{3k_{\rm{B}}T}\int_{0}^{\infty} dt \left< P_{xy}(t)P_{xy}(0)\right>##
Polyamorph
Hello! In this paper https://pdfs.semanticscholar.org/e8a2/02f25555cd8c4f947bbbdff5a61a0ea0efd2.pdf the authors use VASP to determine MgSiO3 viscosity using the Green-Kubo relation

## \eta = \frac{V}{3k_{\rm{B}}T}\int_{0} \left<\sum_\limits{i<j}\sigma_{ij}(t+t_{0}).\sigma_{ij}(t_{0})\right>dt## where ##\sigma_{ij}## (i and j = x, y, z) is the stress tensor, t is time and t0 is the time origin. But I've seen other papers use:

## \eta = \frac{V}{3k_{\rm{B}}T}\int_{0}^{\infty} dt \left< P_{xy}(t)P_{xy}(0)\right>##, where ##P_{xy}## is the off-diagonal component of the stress tensor ##P_{αβ}## ( α and β are Cartesian components).

OK, so clearly these are essentially exactly the same equation but the second uses only the xy component whereas the first seems to suggest a summation? So which is correct.

Also, VASP outputs the stress tensor components as XX YY ZZ XY YZ ZX. So which of these should I use to input into the Green-Kubo equation? And are there missing components? what about yx, zx, yx?

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OK, I've worked out that due to symmetry, XY=YX, YZ=ZY, ZX=XZ, so the tensor reduces from 9-components to 6. So that explains my second question (note typo when I wrote "what about yx, zx, yx?" should have said "what about yx, zy, xz?"

But I still don't understand why the first equation is suggesting a sum of the off-diagonal components whereas the second equation suggests only using one of the off-diagonal components ##P_{xy}##.

Polyamorph said:
But I've seen other papers
Can you give an example of these other papers? Were they looking at a 2D case? Without the proper context, it's really difficult to tell what' s going on.

Polyamorph said:
Sure, I got the second equation from this paper: http://www.homepages.ucl.ac.uk/~ucfbdxa/pubblicazioni/PRL05161.pdf
I think this is just a schematic equation. Further into that paper (page 2), there's a paragraph starting with "There are five independent components of the traceless stress tensor..." The paper goes on to say that they find the shear viscosity by averaging the autocorrelation functions of these five components and integrating them from 0 to ##t##, taking the limit as ##t \to \infty##. (beginning of the paragraph that starts with "In Fig. 2") I haven't done the nitty gritty math, but it looks like this makes the two definitions in your OP equivalent.

jim mcnamara
Yes, looks like you're right. Thanks for spotting that.

Polyamorph said:
OK, I've worked out that due to symmetry, XY=YX, YZ=ZY, ZX=XZ, so the tensor reduces from 9-components to 6. So that explains my second question (note typo when I wrote "what about yx, zx, yx?" should have said "what about yx, zy, xz?"

But I still don't understand why the first equation is suggesting a sum of the off-diagonal components whereas the second equation suggests only using one of the off-diagonal components ##P_{xy}##.

As liquids are isotropic, viscosity is a 4th order tensor which can be parametrized with only two parameters, ##\eta## and ##\zeta##, ##\eta## is the same whether determined with xy or xz or whathever component of the deviator of the viscosity gradient or of their sums. On the other hand, the correlation of the Trace of the ##\sigma_{ii}## will probably yield the longitudinal viscosity ##\zeta##.

DrDu said:
As liquids are isotropic, viscosity is a 4th order tensor which can be parametrized with only two parameters, ##\eta## and ##\zeta##, ##\eta## is the same whether determined with xy or xz or whathever component of the deviator of the viscosity gradient or of their sums. On the other hand, the correlation of the Trace of the ##\sigma_{ii}## will probably yield the longitudinal viscosity ##\zeta##.
I think you mean the dilatational viscosity (proportionality constant between the trace of the stress tensor and the trace of the rate of deformation tensor).

## 1. What is viscosity and how is it measured?

Viscosity is a measure of a fluid's resistance to flow. It is typically measured by the force required to move a layer of fluid over another layer. In the context of computational chemistry, viscosity can be calculated using the Green-Kubo relation, which relates the time correlation function of stress to the viscosity of the fluid.

## 2. What is DFT and how is it related to viscosity calculation?

DFT, or Density Functional Theory, is a computational method used to calculate the electronic structure of a material. In the context of viscosity calculation, DFT is used to calculate the stress tensor of the fluid, which is then used in the Green-Kubo relation to obtain the viscosity.

## 3. How does VASP contribute to the calculation of viscosity from DFT?

VASP, or Vienna Ab initio Simulation Package, is a software package commonly used to perform DFT calculations. It provides the necessary tools and algorithms to accurately calculate the electronic structure and stress tensor of the fluid, which are essential for the Green-Kubo relation to be applied.

## 4. What are the limitations of using DFT and VASP in viscosity calculations?

DFT and VASP are powerful tools for calculating viscosity, but they have limitations. DFT relies on certain approximations and assumptions, which can affect the accuracy of the results. Additionally, the size and complexity of the system being studied can also impact the accuracy of the viscosity calculation.

## 5. How can the results of viscosity calculations from DFT be validated?

There are several ways to validate the results of viscosity calculations from DFT. One method is to compare the calculated value to experimental data. Additionally, the results can also be compared to other theoretical methods or simulations. It is also important to ensure that the system is properly equilibrated and that the simulation time is long enough for accurate results.

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