19990614: lattice constant from GULP, but NO thermal expansion

GOAL

It looks like the lattice constant influences very heavily the quality of the MSD plots obtained from a given simulation (compare 19981124 and 19981126). The lattice constant used in 19981126 (which was the same used in 19981113-19981117-19981119), which gives a good MSD plot, was taken from the paper of Brinkman [2], which deals with YSZ (and not ceria-zirconia). I'd like to find an acceptable method for selecting the lattice constant for the simulations. It seems that taking thermal expansion into account yelds lattice constants too large (see 19981124). So a new attempt: take GULP optimized lattice constant without correction for thermal expansion. By the way the lattice constants obtained from GULP agree very well with values obtainable from the Vegard law.

Results

GULP input

The system to be simulated is: $Zr_{54}Ce^{IV}_{38}Ce^{III}_{16}O_{208}$. This corresponds to the following occupancies:

$$\begin{array}{llcl} Zr&54/108&=&0.5\\ Ce^{IV}&38/108&=&0.35... ...{III}&16/108&=&0.14814815\\ O&208/216&=&0.96296296 \end{array}$$

I run GULP with the above occupancies, polarizable oxygen and rigid cations (same conditions as the DL_POLY run) and get the follwoing value for the optimized lattice constant: $5.337004\;\mbox{\textit{\AA}}$.

Vegard's law

The lattice constant of the fluorite structure is related to the sum of the cationic ($r_C$) and anionic ($r_A$) radii by the following relation:

$$\begin{eqnarray*} a&=&\frac{4}{\sqrt{3}}(r_C+r_A) \end{eqnarray*}$$

(the above follows immediately from the fact that the oxygen ion is at $1/4a,1/4a,1/4a$, i.e. along the diagonal of the cubic unit cell. Thus, assuming that the cation and the anion are in contact, one can say:

$$\begin{eqnarray*} (r_C+r_A)^2&=&\left(\left(\frac{1}{4}a\right)^2+\left(\frac{1}{4}a\right)^2\right)+\left(\frac{1}{4}a\right)^2 \end{eqnarray*}$$

)

According to Vegard's law (see for instance p.367 of reference [4]), the cationic radius for the $Zr_{54}Ce^{IV}_{38}Ce^{III}_{16}O_{208}$ system should be an average of the values for the three cation components:

$$\begin{eqnarray*} r_C&=&0.5r_{Zr}+0.35185185r_{Ce^{IV}}+0.14814815r_{Ce^{III}} \end{eqnarray*}$$

This can be substituted in the expression for the lattice constant above, taking the following values for the ionic radii [5]:

$$\begin{array}{ll} r_{Zr^{IV}} & 0.84\;\mbox{\textit{\AA}}\\ ... ...{\AA}}\\ r_{O^{2-}} & 1.38\;\mbox{\textit{\AA}}\\ \end{array}$$

The value so obtained is: $a=5.33514415\;\mbox{\textit{\AA}}$, in close agreement with GULP.

(In the Vegard's law calculation I've not considered the fractional occupancy of the oxygen sites: I think this should be more or less correct)

MSD results

Got rather nice MSD's. Have to keep this as a starting point for future developments.

QUESTION:

why are the MSD's for oxygen cores and oxygen shells shifted?

From the slope of the oxygen MSD's I get the following diffusion coefficient, using the relation:

$$\begin{eqnarray*} MSD&=&6Dt+B \end{eqnarray*}$$

$$\begin{array}{lll} &\mbox{slope}\ (\mbox{\textit{\AA}}^2/ps)... ...{\mathit{shell}} & 0.0166864825 & 2.78108041667 \\ \end{array}$$

(MSD data fitted in the interval $[20.0:160.0]\;ps$)

The diffusion coefficients for $O_{\mathit{core}}$ and $O_{\mathit{shell}}$ are identical and match quite well that in 19981127.