# #16 Bulk Modulus that Describes the Mechanical Property of a Solid

Viewed : 840 times,  2019-09-03 14:59:01
To find the most stable cell parameter, we should find the most stable point in the aspect of energy. In general, in DFT, the procedure can be done by repeatedly performing SCF by changing the cell parameters and the atomic positions to find the most stable energy. The 'Variable-Cell Relaxation (vc-relax),' which is covered at the last Weekly Tip #15, automatically performs this series of steps.
vc-relax can be manually executed by performing SCF several times.
At this time, you can calculate the bulk modulus by applying the data obtained from several executions of SCF.

1. Bulk Modulus Calculation

Assuming that a material is isotropically compressed by external pressure (ΔP), its volume will change from V to V+ΔV. At a small volume change, ΔP is proportional to the volume strain ΔV/V, and the proportional constant (K) is called the bulk modulus.[1]

s bulk modulus depends on the material density, it is a measure of the hardness of a solid and is used to decide the mechanical property of the material.[1],[2] It can be obtained by fitting data by following Birch’s formula or the Murnaghan equation of state.[3]

B0' (B0P), B0, V0, E0, R4 can be variables. At the time, the volume V0 and the total energy E0 fitting are sensitive variables during the fitting. Therefore, it is recommended to set a variable depending on the data with the minimum energy. A program that can fit data such as the ev.x code in Quantum Espresso, Gnuplot, and origin using a function can calculate the bulk modulus by entering the abovementioned formula. First, create the data with the scf calculation for diverse volumes. This weekly tip uses the following data for the bulk modulus fitting of the calcium bulk unit cell.

cell parameter (Å) Volume (Å3) Energy (Ry)
5.15 136.590875 -299.743606
5.2 140.608000 -299.747529
5.25 144.703125 -299.750230
5.3 148.877000 -299.751902
5.35 153.130375 -299.752747
5.4 157.464000 -299.752752
5.45 161.878625 -299.751882
5.5 166.375000 -299.750149
5.55 170.953875 -299.747565
5.6 175.616000 -299.744154
5.65 180.362125 -299.739960
5.7 185.193000 -299.735040
5.75 190.109375 -299.729446
5.8 195.112000 -299.723231
5.85 200.201625 -299.716439

👉 Check the calculation results

Furthermore, this weekly tip fits data by using Gnuplot, which is a strong plotting utility that allows the previously mentioned data to be used in diverse operating systems. See the following link for script examples to fit in Gnuplot. Gnuplot Example

2. Bulk modulus Results

When you perform the fitting in Gnuplot, proper fitting can be realized through iteration. The iteration process is displayed in the terminal SSH screen, and the final result is saved in the fit.log file. The fit.log file displays how the data is fit and shows the standard error of the parameter fitting result. The newly calculated B0 value after the fitting is the bulk modulus as follows. The unit is eV/Å3. If it is multiplied by 160.22, it can be converted into GPa.

 Final set of parameters       Asymptotic Standard Error ================     ==================== B0P             = 1.22513         +/- 0.2021       (16.5%) B0              = 0.0994759     +/- 0.0004809    (0.4835%) V0              = 155.233          +/- 0.05381      (0.03466%) E0              = 0.0221405     +/- 0.0002391    (1.08%) R4              = -29.0013         +/- 11.46        (39.52%) correlation matrix of the fit parameters: ............ ......B0P      B0        V0       E0       R4 B0P            1.000 B0             -0.005  1.000 V0             -0.889  0.252  1.000 E0             -0.250 -0.696  0.107  1.000 R4              0.915 -0.383 -0.867  0.017  1.000

With the fitting result, you can obtain the equation of state (EOS) graph as follows:

According to the fitting result, the bulk modulus of the calcium bulk is 15.94 GPa, and the lattice parameter has a minimum value (minimum energy) at 5.37 Å. In addition, the energy gap between the point and a single atom can be used as cohesive energy. We’ve learned the calculation of the bulk modulus that can find the mechanical property of a solid by applying the data with the scf calculation, which can be easily performed through Materials Square. The mechanical property is accurate without additional corrections in DFT and is very useful in finding the property of a material. You can also compute the shear modulus and Young’s modulus by applying this.

[1] Doi, M. (2013). Soft matter physics. Oxford University Press.
[2] Faccio, R., Denis, P. A., Pardo, H., Goyenola, C., & Mombrú, A. W. (2009). Mechanical properties of graphene nanoribbons. Journal of Physics: Condensed Matter, 21(28), 285304.