Rep:MgO egn14

2017-01-23T17:01:51Z

Egn14: /* Introduction */

== Introduction ==

The aim of this computational experiment is to simulate the thermal expansion of a magnesium oxide (MgO) crystal lattice using different models, and in the process calculate the coefficient for thermal expansion.

The system under investigation is MgO in the solid state. In this state, MgO has crystalline structure where there is long-range order. This means that if the relative positions of an atom and its neighbours are known at a particular point, it is then possible to pin-point the positions of these atoms throughout the crystal by virtue of the periodic structure. Hence, solid MgO can be represented by a unit cell - a basic building block that is repeated periodically to generate the entire crystal lattice. The conventional unit cell of MgO is the 'NaCl unit cell' (Image 1.) which can be viewed as a simple face-centred cubic (FCC) cell where the Mg atoms occupy the octahedral holes of the oxygen's sub-lattice and the oxygen atoms occupy the octahedral holes of the magnesium sub-lattice. Another way of viewing this unit cell is as two interpenetrating FCC cells of Mg and O displaced from each by half of the body-diagonal. A less common representation of the MgO lattice is by using its rhombohedron primitive cell (Image 2.). In either case, the crystal lattice is held together by strong ionic interactions between the oppositely charged Mg 2+ and O 2- ions.

== Results and Discussion ==

=== Lattice Vibrations - Phonon Computation ===

An appropriate grid size of the MgO crystal had to be determined prior to performing computations in the quasi-harmonic approximation. The grid size is represented by shrinking factors along each direction of the crystal. This was done by examining phonon Density of States (DOS) graphs as a function of shrinking factor. The number of k-points included in the DOS computation varies as a function of shrinking factor.

For the shrinking factor of 1, i.e a lattice of grid size 1x1x1, one k-point which was 0.5 multiplied by the lattice vector in each direction of the crystal was included. This k-point (0.5, 0.5, 0.5) corresponds to the symmetry point L.

The density of states graphs were plotted for the shrinking factors 1, 2, 3, 4, 8, 16, 32 and 64. It was observed that larger shrinking factors gave smoother DOS curves of higher resolution. This is due to the fact that a larger shrinking factor corresponds to a smaller Brillouin zone. This means that a greater number of k-points is used in the computations and more phonon frequencies are included.

To determine an appropriate grid size for a reasonable DOS approximation, a sufficiently well-resolved DOS curve had to be observed. From observing the DOS curves in table 1, a shrinking factor of 16 was determined to be the minimum grid size which produced a reasonable DOS curve. The 16x16x16 DOS curve was determined to be reasonable by comparing its appearance to those of smaller and larger shrinking factors. There was a significant change in the DOS curve appearance going from the 8x8x8 grid to the 16x16x16 grid but only a minor improvement in resolution going to the larger grid sizes of 32x32x32 and 64x64x64. Therefore, if computational power or time was extremely limited, performing computations using a shrinking factor of 16 would suffice.

However, the optimal grid size for the proceeding computations in the quasi-harmonic model was determined to be that with a shrinking factor of 32. This is because the shrinking factor of 32 gave a more detailed DOS curve than the shrinking factor of 16, and both calculations took roughly the same amount of time to complete. A shrinking factor of 64 was not chosen because the minor improvement in resolution in the DOS curve was greatly offset by the significantly longer computational time and was determined to be inordinately computationally costly.

{| class="wikitable"
! Shrinking factor
! 1
! 2
! 3
! 4

|-
| DOS Curve
| [[File:DOS 1x1x1.png|350px]]
| [[File:DOS2x2x2.png|350px]]
| [[File:DOS3x3x3.png|350px]]
| [[File:DOS4x4x4.png|350px]]
|-
| Shrinking Factor
| 8
| 16
| 32
| 64
|-
| DOS Curve
| [[File:DOS8x8x8.png|350px]]
| [[File:DOS16x16x16.png|350px]]
| [[File:DOS32x32x32.png|350px]]
| [[File:DOS64x64x64.png|350px]]
|-

|}

[[File:Dispersiondiagramegn14.png|thumb|400px|Image 1. Phonon dispersion of MgO.|right]]

The dispersion curve is an alternative representation of the lattice vibrational states and their energies. The variation of the energies of vibrations with respect to different k-points is illustrated in a dispersion curve. Symmetry points are k-points of extra importance and are highlighted in the dispersion curve. Information such as the energies and number of vibrational states at different k-points within the crystal can be extracted from the dispersion curve. This is in contrast to the DOS curves which show the proportion of
vibrational states at a given energy interval based on the number of k-points provided which is governed by the grid size.

Based on the optimal grid size with shrinking factor of 32 for the MgO lattice, several assumptions on the optimal grid sizes for other species can be made by considering their lattice sizes relative to that of MgO. Namely, lattices with similar lattice parameter (a) and inverse lattice parameter (b) magnitudes would be expected to share a common optimal grid size with the MgO lattice. The lattice parameter (a) of MgO is 4.2 Å<ref>http://www.crystec.de/daten/mgo.pdf</ref>. Firstly, for a similar oxide such as CaO (a = 4.7 - 4.8 Å<ref>1.II-VI and I-VII Compounds; Semimagnetic Compounds, 1999, 1-3.</ref> with a primitive cell of similar dimensions to MgO, the inverse lattice parameter would be similar, hence the Brillouin zone would be similar in size, which means that the same number of k-points would be needed to generate a DOS curve of sufficient resolution. This means that the same shrinking factor of 32 would be adequate for CaO.
Zeolites generally have larger structures with larger primitive cells. For instance, Faujasite has a lattice parameter around 24.6Å<ref> D. N. Stamires, Clays and Clay Minerals, 1973, '''21''', 379-389</ref> which is significantly larger than MgO. This means that the cells in reciprocal space of Zeolites are significantly smaller than MgO. Hence, a smaller shrinking factor than 32 which corresponds to fewer sampled k-points would likely be adequate for a well-resolved DOS curve.
A smaller number of k-points from a smaller shrinking factor than 32 would also suffice for a regular metal lattice like Li. This is due to the higher DOS or narrower band widths characteristic of regular metal lattices. This narrower band width can be attributed to the cushioning of the repelling positive cations undergoing vibrational motion by the sea of electrons surrounding the cations. As a consequence, there is minimal fluctuation in the vibrational energy levels.

=== Quasi-Harmonic Approximation ===
==== Computing the Helmholtz Free Energy ====

To further justify the choice of grid size with shrinking factor 32, calculations of the Helmholtz free energies as a function of grid size was performed. As the grid sizes increased, the Helmholtz free energy converged to a greater extent towards the value of the infinite grid. This is evident in the decreasing degree of fluctuation with grid size. i.e a smaller change in Helmholtz free energy values was observed for the larger grid sizes. There was no change in Helmholtz free energy value going from shrinking factor 32 to 64, which indicates complete convergence. A shrinking factor of 2 results in a free energy value accurate to 1 meV and 0.5 meV, and a shrinking factor of 4 gives a free energy value accurate to 0.1 meV.

{| class="wikitable"
!Shrinking Factor
!Helmholtz Free Energy (eV)
!Change in Energy
|-
|1
|<nowiki>-40.9303</nowiki>
|
|-
|2
|<nowiki>-40.9266</nowiki>
|3.69x10-3
|-
|4
|<nowiki>-40.9264</nowiki>
|1.59x10-4
|-
|8
|<nowiki>-40.9264</nowiki>
|2.80x10-5
|-
|16
|<nowiki>-40.9264</nowiki>
|4.00x10-6
|-
|32
|<nowiki>-40.9264</nowiki>
|1.00x10-6
|-
|64
|<nowiki>-40.9264</nowiki>
|0
|}

[[File:Helmholtz_Gridsize.png|thumb|400px|Image 2. Convergence of Helmholtz Free Energy Values with increasing grid sizes.|none]]

==== Thermal Expansion of MgO ====

[[File:HelmholtzTemp egn14.png|thumb|Image 3. Plot of Helmholtz free energy as a function of temperature.|480x480px|none]][[File:LatParamTemp egn14.png|thumb|Image 4. Plot of lattice parameter as a function of temperature.|480x480px|right]]

From the plots in images 4 and 5, it can be seen that there is an increase in lattice parameter and hence increase in cell volume with increasing temperature. i.e Thermal expansion of the MgO lattice occurs.
It follows from this that the Helmholtz free energy becomes more negative with increasing temperature as can be seen in the plot of Image 3. This is due to the positive change in entropy (deltaS) associated with thermal expansion as the system becomes less configurationally constrained and hence more disordered. The -TdeltaS contribution to the Helmholtz free energy is thus negative and becomes progressively more negative with increasing temperature.

The coefficient of thermal expansion is defined as:
<math>
\alpha_V = \frac{1}{V_0}\,\left(\frac{\partial V}{\partial T}\right)_p
</math>
, where V0 represents the initial lattice volume.

Using V0 = 18.8364 Å and the gradient from the plot of cell volume against temperature at constant pressure, the coefficient of thermal expansion αV was found to be 2.654x10-5 K-1. The experimental values for a similar temperature range of 300 to 1000 K found in literature was 3.99x10-5 K-1.<ref> O.L. Anderson and K. Zou, J Phys Chem Ref Data, 1990, '''19''', 71</ref> This has the same order of magnitude as the computed coefficient of thermal expansion and both values were in agreement.

[[File:LatVolTemp egn14.png|thumb|Image 5. Plot of cell volume as a function of temperature.|505x505px|none]]

=== Molecular Dynamics ===