= The Free Energy and Thermal Expansion of MgO =

== Introduction ==
Using DLVisualize installed on the Chemistry department's computers, MgO can be graphically modelled allowing the behaviour of MgO to be studied when heated. MgO's free energy and thermal expansion coefficient can be calculated through both quasi-harmonic approximation and molcular dynamics calculations.

In a diatomic molecule with an exactly harmonic potential the equilibrium bond length remains constant; however the quasi-harmonic approximation can be applied to materials in their solid state and takes into account that the lattice constant can increase from fluctuations in bond lengths(due to increased internal energy) in the structure; the harmonic approximation can then be applied to the larger lattice constant values, giving more accurate calculations.

Thermal expansion arises from an larger vibrations along harmonic oscillator due to the increased internal energy in the system, and thus the lattice constant and the equilibrium bond length increases, hence an expansion in 3 dimensions occurs. The thermal expansion coefficient can be determined using the equation below:
<center><math>{\alpha} = \frac{\mathrm{1}}{\mathrm{V}_0} \frac({\mathrm{\partial V}}{\mathrm{\partial}T}) \ \ (1)</math></center>

== Method ==
[[File: Loading_MgO.png|thumb|Fig a) MgO visualised by DLVis]]DLVisualize was launched from RedHat Linux, and MgO was graphically modelled '''(fig. a).'''

The internal energy per unit cell of MgO crystal was determined using a single point calculation, where the force field between Mg(+2e) and O (-2e) can be represented through a Buckingham potential. An output file from the GULP execution provides a break down of the total energy.

The phonon modes were computed for 50 points along the conventional path in k-space and graphed as a dispersion curve '''(Fig b)'''. The phonon mode at around 117 was animated '''(Fig C?)''' to aid in the understanding of plotting the density of states graph.

Initially the DOS (density of states) was calculated for a single point in k-space (1x1x1) which consisted of four peaks, the grid size was then increased so that more k-points could be sampled and a smooth curve for the DOS could be graphed '''(Fig D).'''

The suitable k-space grid was determined by increasing the grid size of the k-points, and identifying where the free energy converges at an appropriate value. The free energy of MgO was computed with respect to the quasi-harmonic approximation, for increasing grid sizes at 300 K and 0 GPa starting from an initial 1x1x1 up to a 32x32x32 grid, whereby the free energy converges.

With the grid size suitable at 16x16x16, the free energy was minimised and lattice constant was calculated for every 100 K between 0 K and 1000 K by optimising the structure of MgO at these temperatures.

MgO with 32 Mg-O units was loaded into DLVis and molecular dynamic simulations were ran.The number of particles in the simulation and the external pressure were kept constant. The molecular dynamic simulation was used to determine a final equilibrium volume for several different temperatures.

== Results ==
[[File:Jtl14Disp_vs_DOS.png|thumb|Density of states (left) aligned with phonon dispersion (right)|390x390px]]

'''Q5''' The peaks from the DOS plotted for a single k-point aligns with the high symmetry point L.'''(Fig )'''

Q3) Show that Helmholts free energy converges (and every other property as more K points are sampled) '''excel'''

The optimal grid size for DOS computations was determined to be 16x16x16, as smaller grid sizes did not provide a smooth curve, and larger grids took longer to compute.

Running GULP calculations on similar structures to MgO (eg. CaO) would indicate that it would also be appropriate to use a similar sized grid. Structures which are much larger than MgO (eg. Zeolites) would require a smaller grid size, as they have a larger lattice parameter and thus a smaller reciprical space to be sampled. On the other hand a metal (eg. Lithium) has a much large reciprical space, and requires more k-points for adequate convergence.

[[File:jtl14helmholtz_vs_temperature.png|thumb|Graph of free energy against temperature]]
The free energy was plotted against temperature '''(FIG)''' and it can be seen that when the temperature is increased the Helmholtz free energy can be minimised, '''(refer to equation in intro 'A=U-TS') '''this becomes more prominent at higher temperatures.

As the temperature nears the melting point of MgO, the phonon modes poorly represent the actual motions of the ions as the solid structure of the lattice behaves less like a solid and more like a liquid, where it cannot be modelled by the quasi-harmonic appoximation
[[File:jtl14lattice_constant_vs_temperature.png|thumb|Graph of lattice constant against temperature]]
The lattice constant was also plotted against temperature to examine how the volume of the cell changes with respect to temperature. The graph was fitted with a second-degree polynomial and shows that the volume per unit cell increases with increasing temperature at a relatively constant rate,

''Q8)?? check vs energy stated on page one of wiki larger grid should give calc accurate to smaller increments''

Using '''equation (thermal expansion coefficient) '''the thermal expansion coefficient can be calculated.

The thermal ''volume'' expansion coefficient of MgO was calculated with the aid of a volume/temperature '''graph''', the fitted second-degree polynomial was differentiated at 300 K and with the use of '''equation, '''was found''' '''to be 1.6x10-5 K-1 at 300 K. Literature values were often reported as thermal ''linear ''expansion which ranged from (9-12)x10-6 K-1 '''LIT;'''which is equivalent to a thermal ''volume ''coefficient of (1.8-2.4)x10-5 K-1, around double the calculated value.

[[File:jtl14MD_LD.png|thumb|Graph of normalised cell volume against temperature. LD = quasi-harmonic approximation,
MD = molecular dynamics
]]
A graph of normalised cell volume per formula unit vs temperature was plotted for both the quasi-harmonic approximation and the molecular dynamics '''(FIG)'''. It can be seen that points from the quasi-harmonic approximation follow closely a second-degree polynomial, whereas the molecular dynamics plot follows a linear trend. Molecular dynamics gives the thermal volume expansion coefficient of MgO to be 3x10-5 K-1, which matches the literature values very well. This is because MD can simulate random motion and allows for the system to develop and equilibriate following Newton's equations of motions.

As the temperature increases, the difference between the predicted thermal expansion coefficients reduces and the value computed within quasi-harmonic approximation tends to that predicted by molecular dynamics. As the quasi-harmonic approximation is fitted with a second-degree polynomial, a higher temperature will also lead to a higher predicted thermal expansion coefficient which becomes more accurate as anharmonic effects are taken into account. '''<nowiki>http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.8250&rep=rep1&type=pdf#page=33</nowiki>'''

== Conclusion ==
For this limited temperature range the MD prediction is accurate, however as the thermal expansion coefficient is a function of temperature, temperatures above 1000K are unlikely to follow the trend shown. accuracy is not very high kpoints