2000-03-03: correction for partial occupation of oxygen site

GOAL

In the $Ce_{0.8}M_{0.2}O_{1.9}$ systems, a correction must be applied for the partial occupation of the oxygen site.

Mean field analysis

In the $Ce_{0.8}M_{0.2}O_{1.9}$ system, oxygen sites are only partially occupied. This means that GULP assigns a scaled charge to the oxygen ions: in other words, each anion site is assumed to contain a $O^{-1.9}$ ion, instead of the $O^{-2}$ oxide species. To put it another way, the following correspondence is established:

REAL SYSTEM	MEAN FIELD ( GULP)
Anionic sites are occupied by $O^{-2}$ ions only in part: a number of them are vacant (this number is related to the fraction of the $+3$ dopant)	Anionic sites are all occupied by fictitious $O^{-1.9}$ ions

From the mean field viewpoint, the water incorporation reaction should be written in the following way:

$$H_2O_{(g)}+V_O^{+1.9}+O_O^{-0.1}=2OH_O^{+0.9} E_{H_2O}$$ (1)

where $V_O^{+1.9}$ is an oxygen site vacancy, whose charge is $+1.9\;e$ and not $+2\;e$, since GULP treats each oxygen as having a charge of $-1.9\;e$ because of partial occupancy; $O_O^{-0.1}$ means ``a real $O^{2-}$ ion (because we need that to make up an $OH$ group) at an anionic site'': since the anionic site has charge $+1.9\;e$, an $O^{2-}$ oxide at that site will have a charge of $-0.1\;e$; finally, the actual charge on the $OH_O$ defect will be $+0.9\;e$ and not $+1\;e$ for the same reasons.

What we actually can evaluate with GULP is the energy change for the following processes:

$$O^{2-}_{(g)}+O_O^\times=O_O^{-0.1}+O_{(g)}^{-1.9} E_{O_O^{-0.1}}$$ (2)

$$O_O^\times=V_O^{+1.9}+O_{(g)}^{-1.9} E_{V_O^{+1.9}}$$ (3)

$$OH^-_{(g)}+O_O^\times=OH_O^{+0.9}+O_{(g)}^{-1.9} E_{OH_O^{+0.9}}$$ (4)

The evaluation of the above energy terms is coded in GULP as follows ( O1 is a ``genuine'' $O^{2-}$, while O2 and H2 are the oxyegn and hydrogen atoms of the $OH$ group):

energy	eqn.	GULP code
$E_{O_O^{-0.1}}$	(2)	IMPURITY O1 0.25 0.25 0.25
$E_{V_O^{+1.9}}$	(3)	VACANCY 0.25 0.25 0.25
$E_{OH_O^{+0.9}}$	(4)	IMPURITY O2 CORE 0.25 0.25 0.25
		INTERSTITIAL H2 CORE 0.3735 0.3735 0.25

Equation 1 can be decomposed in terms of equations 2-4 and of the usual gas phase reaction 5

$$H_2O_{(g)}+O^{2-}_{(g)}=2OH^-_{(g)} E_{PT}$$ (5)

We have:

$$\begin{eqnarray*} % latex2html id marker 2379(\ref{2000-03-03:eq:100})&=&(\ref... ...})-(\ref{2000-03-03:eq:500})+2\times(\ref{2000-03-03:eq:700})\\ \end{eqnarray*}$$

or:

$$\begin{eqnarray*} H_2O_{(g)}+O^{2-}_{(g)}=2OH^-_{(g)}&&E_{PT}\\ V_O^{+1.9}+O_{(... ..._{H_2O}=E_{PT}-E_{V_O^{+1.9}}-E_{O_O^{-0.1}}+2E_{OH_O^{+0.9}}\\ \end{eqnarray*}$$

Relation with the ``usual'' decomposition scheme

Equation 1 is usually decomposed into the following three steps:

$$H_2O_{(g)}+O^{2-}_{(g)}=2OH^-_{(g)} E_{PT}$$

$$2OH^-_{(g)}+2O_O^{-0.1}=2OH_O^{+0.9}+2O^{2-}_{(g)} E_1$$ (6)

$$V_O^{+1.9}+O^{2-}_{(g)}=O_O^{-0.1} E_2$$ (7)

$$\rule{60mm}{0.5pt} \rule{38mm}{0.5pt}$$

$$H_2O_{(g)}+V_O^{+1.9}+O_O^{-0.1}=2OH_O^{+0.9} E_{H_2O}=E_{PT}+E_1+E_2$$

The first step is just equation 5. The second and third steps consist of the following:

step 6:

substitution of a ``real'' $O^{-2}$ ion sitting at an anionic site by an $OH$ group

step 7:

occupation of an anionic vacancy by a ``real'' $O^{-2}$ ion

$E_1$ and $E_2$ are not directly evaluated with GULP: the energies that we can calculate directly with GULP are only those for processes 2-4.

However, steps 6 and 7 can be expressed as proper combinations of processes 2-4.

For step 6 we have:

$$\begin{eqnarray*} % latex2html id marker 2472(\ref{2000-03-03:eq:900})&=&2\times(\ref{2000-03-03:eq:700})-2\times(\ref{2000-03-03:eq:500})\\ \end{eqnarray*}$$

or:

$$\begin{eqnarray*} 2OH^-_{(g)}+2O_O^\times=2OH_O^{+0.9}+2O_{(g)}^{-1.9}&&2E_{OH_O... ..._O^{+0.9}+2O^{2-}_{(g)}&&E_1=2E_{OH_O^{+0.9}}-2E_{O_O^{-0.1}}\\ \end{eqnarray*}$$

For step 7 we have:

$$\begin{eqnarray*} % latex2html id marker 2498(\ref{2000-03-03:eq:1000})&=&-(\ref{2000-03-03:eq:600})+(\ref{2000-03-03:eq:500})\\ \end{eqnarray*}$$

or:

$$\begin{eqnarray*} V_O^{+1.9}+O_{(g)}^{-1.9}=O_O^\times&&-E_{V_O^{+1.9}}\\ O^{2-... ...1.9}+O^{2-}_{(g)}=O_O^{-0.1}&&E_2=-E_{V_O^{+1.9}}+E_{O_O^{-0.1}} \end{eqnarray*}$$

Using the values so obtained for $E_1$ and $E_2$, the final expression for $E_{H_2O}$ is obtained:

$$\begin{eqnarray*} E_{H_2O}&=&E_{PT}+E_1+E_2\\ &=&E_{PT}+2E_{OH_O^{+0.9}}-2E_{O_... ...}\\ &=&E_{PT}+2E_{OH_O^{+0.9}}-E_{O_O^{-0.1}}-E_{V_O^{+1.9}}\\ \end{eqnarray*}$$

Results

Taking the $E_{OH_O^{+0.9}}$ values from 2000-02-23 and the $E_{V_O^{+1.9}}$ values from 2000-02-02, we have:

As expected, the mean field correction is zero for pure ceria. For the doped systems, its effect is that of increasing the $E_{H_2O}$ values, making the difference with pure ceria much smaller. This sounds reasonable. Water incorporation is still more unfavorable for pure than the doped cerias. The problem is that the trend within the mixed systems is still opposite with the experimental one.

Representative input file

opti conp defect regi_before
#conp defect single
dump every 1 nd-O-impurity.dump
maxcyc  200
cutb 3.0 # what does this mean? Ronny uses it.
title
Neodymium solid solution
N.Sakai et al. Solid State Ionics 125(1999)325-331
end
cell
   5.490000 5.490000 5.490000  90.000000  90.000000  90.000000

size        9.82697125     20.0
centre      0.25           0.25    0.25
#vacancy     0.25           0.25    0.25
impurity   O1     0.25           0.25    0.25
fractional   
############
# Cores
############
Ce4  core  0.000000   0.000000   0.000000  -3.700000 0.8
Nd   core  0.000000   0.000000   0.000000   3.00     0.2
O1   core  0.250000   0.250000   0.250000   0.07700  0.95
############
# Shells
############
Ce4  shel  0.000000   0.000000   0.000000   7.700000 0.8
O1   shel  0.250000   0.250000   0.250000  -2.07700  0.95
space
225

#########################
# Shell-core charges
#########################
species   
Ce3   shel    7.700000
Ce3   core   -4.700000   
O2    core   -1.4263
H2    core    0.4263         
#########################
# Short range potentials
#########################
buck 
Ce4 shel O1 shel    1986.830000 0.351070   20.40000  0.0 15.000
Ce3 shel O1 shel    1731.61808  0.36372    14.43256  0.0 15.000
Nd  core O1 shel    1379.9      0.36010     0.00000  0.0 15.000
O1  shel O1 shel   22764.300000 0.149000   27.89000  0.0 15.000
O2  core O1 shel   22764.300000 0.149000   27.89000  0.0 15.000
H2  core O1 shel     311.970    0.35        0.0      0.0 15.000
Ce4 shel O2 core    1986.830000 0.351070   20.40000  0.0 15.000
Nd  core O2 shel    1379.9      0.36010     0.00000  0.0 15.000
morse
H2  core O2 core       7.05250  2.19860     0.9485   1.0 0.0 2.0
#########################
# Spring parameters
#########################
spring 
Ce4    291.750000
Ce3    291.750000
O1      27.29