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Material for a paper on phase relative stabilities in the $Ce_xZr_{(1-x)}O_2$ system

Gabriele Balducci

last update: August 18, 1999


Contents


19990507: SHELL vs GULP and DL_POLY

GOAL

Getting familiarity with SHELL: comparison SHELL vs GULP on a static calculation

Input files

GULP input file

single
title
ceria energy: single point calc
end
cell
   5.429078 5.429078 5.429078    90.000000  90.000000  90.000000

pressure 0.0 GPa
fractional   
############
# Cores
############
Ce4  core  0.000000   0.000000   0.000000  -3.700000 1.0
O    core  0.250000   0.250000   0.250000   0.07700  1.0000  
############
# Shells
############
Ce4  shel  0.000000   0.000000   0.000000   7.700000 1.0
O    shel  0.250000   0.250000   0.250000  -2.07700  1.0000
space
225

#########################
# Short range potentials
#########################
buck
Ce4 shel O  shel    1986.830000 0.351070   20.40000  0.000 10.000
#  buck
#  Ce3 shel O  shel    1731.61808  0.36372    14.43256  0.000 10.000
#  buck
#  Zr  shel O  shel     985.869000 0.376000    0.00000  0.000 10.000
buck
O   shel O  shel   22764.300000 0.149000   27.89000  0.000 10.000
#########################
# Spring parameters
#########################
cuts 0.2
spring
Ce4    291.750000
spring
O      27.29

SHELL input file

NOTE: the ``INVERSE'' part of the Buckingham potential is taken ``as is'': this means that the minus sign must be included in the input (see below and the ``SHELL user's guide'')

TITLE ceria structure optimization
LATTICE CP 5.429078
PARTICLES
Ce-shel    0.0      7.7
Ce-core  140.12    -3.7
O-shel     0.0     -2.077
O-core    16.0      0.077

BASIS
Ce-core  0.0    0.0     0.0   
Ce-core  0.5    0.5     0.0   
Ce-core  0.5    0.0     0.5   
Ce-core  0.0    0.5     0.5   
O-core   0.25   0.25    0.25  
O-core   0.75   0.25    0.25  
O-core   0.75   0.75    0.25  
O-core   0.25   0.75    0.25  
O-core   0.25   0.25    0.75  
O-core   0.75   0.25    0.75  
O-core   0.75   0.75    0.75  
O-core   0.25   0.75    0.75  
#                     
Ce-shel  0.0    0.0     0.0   
Ce-shel  0.5    0.5     0.0   
Ce-shel  0.5    0.0     0.5   
Ce-shel  0.0    0.5     0.5   
O-shel   0.25   0.25    0.25  
O-shel   0.75   0.25    0.25  
O-shel   0.75   0.75    0.25  
O-shel   0.25   0.75    0.25  
O-shel   0.25   0.25    0.75  
O-shel   0.75   0.25    0.75  
O-shel   0.75   0.75    0.75  
O-shel   0.25   0.75    0.75  


POTENTIALS       
    EXPO     Ce-shel  O-shel       1986.83   0.351070    0.0   10.0
    INVERSE  Ce-shel  O-shel        -20.4    6.0         0.0   10.0
    EXPO     O-shel   O-shel      22764.300  0.149000    0.0   10.0
    INVERSE  O-shel   O-shel        -27.89   6.0         0.0   10.0
    SPRING   Ce-shel  Ce-core       291.75               0.0    0.2
    SPRING   O-shel   O-core         27.29               0.0    0.2
CUTSHELL 0.3
OPTION
MESH 0
TEMP 0.0
PRESS 0.0
RUN THERMO
END

DL_POLY input files

CONTROL:
pure ceria, to check the SHELL code
zero
pres                      0.001013 kbar  (0.001013 kbar=1.0 atm)
steps                  1
equilibration steps    0
timestep                 0.002 ps
multiple timestep           1
scale                       1
cutoff     8.00    # maximum allowed value from genlat
delr       1.0
rdf sampling every   1 steps
print rdf
eps        1.0
ewald precision 1.0e-5
ensemble nve
stats 1
trajectory nstraj 1 istraj 1 keytrj 0
stack  1
job time  10700
close time  300
finish

FIELD:
CeO_2
units eV
molecular types 2
CeriumFour
nummols 108
atoms 1
Ce4     140.120     4.0
finish
Oxygen
nummols 216
atoms 2
O          14.40000    0.077     1
O_s        1.600000   -2.077     1
shell 1
    1    2   27.29
finish
vdw    2
Ce4     O_s     buck 1986.830000 0.351070    20.40000
O_s     O_s     buck 22764.300   0.149000    27.89000
CLOSE

Results

A good accord is found.

\begin{tabular}{ld} \multicolumn{1}{l}{code}& \multicolumn{1}{c}{energy ({$kJ/mo... ... \texttt{GULP}&-40644.6225\\ \texttt{DL\_POLY}&-40641.75534040\\ \end{tabular}


19990519: convergence test on MESH

GOAL

Convergence test for relevant quantities upon increasing the MESH value

Results

Pure ceria was taken as a test system.

Five runs with increasing MESH values were performed, each consisting of an OPTIMISE CELL optimization at $300\;K$ and $0.00010133\;GPa=1.0\;atm$

Gibbs free energy and lattice constant seem to converge properly.

\begin{center}\vbox{\input{19990519-01.pslatex} }\end{center}

\begin{center}\vbox{\input{19990519-02.pslatex} }\end{center}


19990520: setting up the ceria-zirconia solid solution

GOAL

Calculation of a series of hybrid potentials to simulate the truly disordered ceria-zirconia solid solution

Results

We simulate the ceria-zirconia solid solution by creating an hybrid cation with properties scaled according to the composition.

If we call:

$M$ the hybrid cation
$x$ zzirconia fraction in the solid solution
$M_X$ molar mass of species $X$
$Y_X$ shell charge of species $X$
$V_{X-O}(r)$ Buckingham potential for the interaction of $X$ with oxygen
$S_{X}(r)$ spring potential between core and shell of $X$

then we have:

\begin{eqnarray*} M_M&=&xM_{Zr}+(1-x)M_{Ce}\\ Y_M&=&xY_{Zr}+(1-x)Y_{Ce}\\ V_{M... ...r}+(1-x)k_{Ce}\right)r^2\\ &=&\frac{1}{2}\left(k_M\right)r^2\\ \end{eqnarray*}



( We write $+C/r^6$, so that $C$'s are negative, as SHELL requires that)

The values to be combined are:

\begin{tabular}{ldld} \multicolumn{2}{c}{cerium}& \multicolumn{2}{c}{zirconium}\... ... 0.00000 \\ {$k_{Ce}$} & 291.750000 & {$k_{Zr}$} & 169.617000 \\ \end{tabular}

These are the values of the hybrid properties as a function of the zirconia fraction:

\begin{tabular}{D{.}{.}{3}dddd} \hline \multicolumn{1}{c}{{$x$}} &\multicolumn{1... ...1100 & 1.9850 & 2.0150 & \\ 1.0 & 91.2200 & 1.3500 & 2.6500 & \\ \end{tabular}

\begin{tabular}{D{.}{.}{3}dddd} \hline \multicolumn{1}{c}{{$x$}} &\multicolumn{1... ...0.351070 \\ 1.0 & 985.86900000 & 0.3760 & 0.0000 & 0.351070 \\ % \end{tabular}

\begin{tabular}{D{.}{.}{3}dddd} \hline \multicolumn{1}{c}{{$x$}} &\multicolumn{1... ....04000000 & 181.83030 & & \\ 1.0 & 0.00000000 & 169.61700 & & \\ \end{tabular}

\begin{center}\vbox{\input{19990520-01.pslatex} }\end{center}

\begin{center}\vbox{\input{19990520-02.pslatex} }\end{center}

The following compares SHELL vs GULP for a static single point energy calculation on the complete range of compositions:

$ZrO_2\;\%$ GULP SHELL  
10 -40778.0111 -40777.89813 $kJ/(\mathit{mol\ of}\ M_4O_8)$
20 -40917.8574 -40917.74343 $kJ/(\mathit{mol\ of}\ M_4O_8)$
30 -41065.2598 -41065.14482 $kJ/(\mathit{mol\ of}\ M_4O_8)$
40 -41219.8563 -41219.74031 $kJ/(\mathit{mol\ of}\ M_4O_8)$
50 -41382.9012 -41382.78406 $kJ/(\mathit{mol\ of}\ M_4O_8)$
60 -41555.2593 -41555.14099 $kJ/(\mathit{mol\ of}\ M_4O_8)$
70 -41738.1029 -41737.98320 $kJ/(\mathit{mol\ of}\ M_4O_8)$
80 -41932.2041 -41932.08300 $kJ/(\mathit{mol\ of}\ M_4O_8)$
90 -42139.1513 -42139.02862 $kJ/(\mathit{mol\ of}\ M_4O_8)$

Same as above, but in a graphical form:

\begin{center}\vbox{\input{19990520-03.pslatex} }\end{center}


19990814: setting up monoclinic zirconia and comparison SHELL vs GULP

GOAL

SHELL input set up for monoclinic zirconia and comparison of results with GULP

Results

Structural determinations of monoclinic zirconia (baddeleyte) can be found in references smith:65, mccullough:59 and adam:59.

The following are the structural parameters from ref. smith:65:


\begin{displaymath} \begin{array}{lll} \mbox{space group}&P2_1/c&\\ a&5.145&\mb... ...}&0.7549&fractional\\ O_{2,z}&0.4789&fractional\\ \end{array}\end{displaymath}

The following compares SHELL and GULP results for a static cell only optimisation:

\begin{tabular}{D{.}{.}{0}D{.}{.}{8}D{.}{.}{8}D{.}{.}{8}} &\multicolumn{1}{D{.}{... ...0841730 & 5.390877 \\ \beta& 99.230 & 97.67381585 & 97.676752 \\ \end{tabular}

The following compares SHELL and GULP results for a static, symmetry constrained (cell+coordinates) optimisation:

\begin{tabular}{D{.}{.}{0}D{.}{.}{8}D{.}{.}{8}D{.}{.}{8}} &\multicolumn{1}{D{.}{... ...30517 &0.718212000 \\ O_{2,z}&0.4789&0.392597215 &0.395742000 \\ \end{tabular}

Bibliography

1
Smith, D.; Newkirk, H. Acta Cryst. 1965, 18, 983-91.

2
McCullough, J.; Trueblood, K. Acta Cryst. 1959, 12, 507-11.

3
Adam, J.; Rogers, M. Acta Cryst. 1959, 12, 951.

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