2000-08-10: activation energy for oxygen migration

GOAL

Evaluation of the activation energy for oxygen migration in $Ce_{(1-x)}D_xO_{2-\frac{x}{2}}, D=Y,Gd,La,Mn$

Results

Mean field analysis

Let's re-work the mean field analysis of the $Ce_{(1-x)}D_xO_{2-\frac{x}{v-1}}$ system, including the activation energy for oxygen migration.

The present discussion is basically equivalent to that at 2000-06-21 (section 2), the only differences being:

notation:

which I find clearer

system composition:

at 2000-06-21 (section 2) we were still dealing with the three cation system $Ce_{(1-x)}Zr_{(x-y)}D_{y}O_{2-\frac{y}{v-1}},\;(v=3,2,\;y<x)$. Since the work has evolved towards the study of two cation systems of type $Ce_{(1-x)}D_xO_{2-\frac{x}{v-1}}$, we restrict the discussion accordingly, in order to avoid unnecessary complications

oxygen migration:

we include the mean field analysis of the activation energy for oxygen migration

$D$ is a tri- or di-valent dopant ($v=3$ or $v=2$, respectively). In order to compensate for the lower valence of $D$, a number of oxygen vacancies must be distributed in the (supposed) fluorite structure of the mixed oxide.

If we denote with $N$ the total number of cations in the crystal, then the ``real'' situation is described as follows:

$$\begin{eqnarray*} \mbox{formula}&=&Ce_{(1-x)}D_xO_{2-\frac{x}{v-1}}\\ \mbox{n. ... ...ac{2N-\frac{x}{v-1}N}{2N}\\ &=&1-\frac{x}{2\left(v-1\right)}\\ \end{eqnarray*}$$

The mean field description of the above system is based on fractional occupancies: instead of considering the cationic sites as randomly occupied by distinct $Ce^{4+}$ and $D^{v+}$ species, we assume that each cationic site is occupied by both a $Ce^{4+}$ and a $D^{v+}$ cation, in a proportion dictated by the chemical composition. In a similar way, we do not consider the anionic sites to be only partially occupied, but we take each anionic site to be occupied by both an oxide ion and a vacancy, in a proportion which is again related to the chemical composition.

This is equivalent to assume that the crystal is composed by only two fictitious chemical species: a cation $C$ which results by the ``linear combination'' of a pure $Ce^{4+}$ and a pure $D^{v+}$ species and an anion $A$, given by the linear combination of a pure oxide species and a, let's say, $nil$ species.

$$\begin{eqnarray*} C&=&\left(1-x\right)Ce^{4+}+xD^{v+}\\ A&=&\left(1-\frac{x}{2\... ...ight)O^{2-}+\left(\frac{x}{2\left(v-1\right)}\right)\mathit{nil} \end{eqnarray*}$$

The crystal does not contain different cations and oxygen vacancies any more; rather, all crystal sites of a given type (cationic or anionic) are identical and contain the same (hybrid) species. It follows that the ``mean field'' formula of the mixed oxide is to be written as $CA_2$, with $C$ and $A$ defined above.

The properties of the ``mean field species'' are averages derived from those of pure component species. In particular, the electric charges $Q_C$ and $Q_A$ are given by:

$$\begin{eqnarray*} Q_C&=&\overbrace{\left(1-x\right)4}^{Ce^{4+}}+\overbrace{xv}^{... ...\left(v-1\right)}\right)0}^{\mathit{nil}}\\ &=&-2+\frac{x}{v-1} \end{eqnarray*}$$

The charge of a substitutional point defect is given by the difference between the charge of the substitutional and the charge of the crystal species being substituted. It follows that in the mean field treatment, putting, say, a pure $Ce^{4+}$ species at a cationic site of the lattice will give raise to a charged defect, whose charge is: $4-\left(4-x\left(4-v\right)\right)=x\left(4-v\right)$. The most interesting (for us) point defects are reported below:

$C_C^\times$

this isn't a defect at all. It means a ``mean field'' cation $C$ at its regular site

$A_A^\times$

ditto for the ``mean field'' anion

$Ce_C^{\left[4-\left(4-x\left(4-v\right)\right)\right]}=Ce_C^{x\left(4-v\right)}$

this means a pure $Ce^{4+}$ cation at a cationic site. Note that this is seen as a charged point defect and will have its formation energy, which can be calculated with GULP using the defect description:
`` impurity Ce4 0.0 0.0 0.0''

$Ce_C^{\left[3-\left(4-x\left(4-v\right)\right)\right]}=Ce_C^{x\left(4-v\right)-1}$

this means a pure $Ce^{3+}$ cation at a cationic site. Differently from the ceria-zirconia system, the charge is less negative than $-1$ because of the low valent dopant

$D_C^{\left[v-\left(4-x\left(4-v\right)\right)\right]}=D_C^{x\left(4-v\right)-\left(4-v\right)}$

this means a pure $D^{v+}$ dopant cation at a cationic site. For the same above reasons, the charge is less negative than $-\left(4-v\right)$

$V_A^{\left[0-\left(-2+\frac{x}{v-1}\right)\right]}=V_A^{2-\frac{x}{v-1}}$

this means an $A$ vacancy. The charge is less positive than $+2$ because the charge of a ``mean field'' anion is less negative than $-2$

$O_A^{\left[-2-\left(-2+\frac{x}{v-1}\right)\right]}=O_A^{-\frac{x}{v-1}}$

this means a pure oxide ion at an anionic site. Again, the defect has a negative charge due to the fact that a ``mean field'' anion is less negative than $-2$

We now turn our attention to the $Ce^{4+}/Ce^{3+}$ reduction in a $Ce_{(1-x)}D_xO_{2-\frac{x}{v-1}}$ mixed oxide.

The usual formulation of the reaction would be:

$$\begin{eqnarray*} 2Ce_{Ce}^\times+O_O^\times&=&2Ce_{Ce}^\prime+V_O^{\cdot\cdot}+\frac{1}{2}O_{2\;(g)} \end{eqnarray*}$$

whose meaning is: two $Ce^{4+}$ cations at their lattice sites become two reduced $Ce^{3+}$ species at the same sites; at the same time, one oxide ion is released as gaseous oxygen, leaving a corresponding vacancy at an anionic site.

The equivalent formulation in the mean field frame is a bit different, for we have to reformulate the pure species as charged defects, as discussed above:

$$\begin{eqnarray*} 2Ce_C^{x\left(4-v\right)}+O_A^{-\frac{x}{v-1}}&=&2Ce_C^{x\left(4-v\right)-1}+V_A^{2-\frac{x}{v-1}}+\frac{1}{2}O_{2\;(g)} \end{eqnarray*}$$

This reaction can be decomposed, as usual, into steps whose energy change can be either directly calculated with GULP or is known from the literature:

$$\begin{eqnarray*} 2Ce_C^{x\left(4-v\right)}+2C^{4-x\left(4-v\right)}_{(g)}=2Ce^{... ...t(4-v\right)-1}+V_A^{2-\frac{x}{v-1}}+\frac{1}{2}O_{2\;(g)}&&\\ \end{eqnarray*}$$

Summing up the various contributions, the $Ce^{4+}/Ce^{3+}$ reduction energy becomes:

$$\begin{eqnarray*} E_{Ce^{4+}/Ce^{3+}}&=&-2E_{Ce_C^{x\left(4-v\right)}}-2I_{4,Ce}... ...1}}-E_{O_A^{-\frac{x}{v-1}}}+E_{V_A^{2-\frac{x}{v-1}}}-83.39\;eV \end{eqnarray*}$$

As usual, the constant terms which add up to $-83.39\;eV$ are taken from Table 5 of Sayle et al. (1994). The other energy contributions are evaluated with GULP:

energy term	GULP code
$E_{Ce_C^{x\left(4-v\right)}}$	IMPURITY Ce4 0.00 0.00 0.00
$E_{Ce_C^{x\left(4-v\right)-1}}$	IMPURITY Ce3 0.00 0.00 0.00
$E_{O_A^{-\frac{x}{v-1}}}$	IMPURITY O 0.25 0.25 0.25
$E_{V_A^{2-\frac{x}{v-1}}}$	VACANCY O 0.25 0.25 0.25

We now consider the migration of oxide ions. The activation energy for oxygen migration is evaluated as the difference between the energy of the ``activated state'' and the energy of the corresponding reference state. The reference state is an oxide ion at an anionic site of the lattice having an available vacant anionic site in nearest neighbor position. The ``activated state'' is taken to consist of the migrating oxide ion along the migration path towards the vacant site, in a position which maximizes the energy of the system. The process can be depicted as follows:

$$\begin{eqnarray*} \begin{xy} (0,0);<1em,0em>: p+/d5em/*+{O_A^{-\frac{x}{v-1}}},c... ...'fromHere''; p+/r5em/;p+/u5em/*+{\mbox{activated state}} \end{xy}\end{eqnarray*}$$

Using our mean field notation, the process can be described by the following equation:

$$\begin{eqnarray*} \left(O_A^{-\frac{x}{v-1}}V_A^{2-\frac{x}{v-1}}\right)^{2-2\fr... ..._i^{\prime\prime}V_A^{2-\frac{x}{v-1}}\right)^{2-2\frac{x}{v-1}} \end{eqnarray*}$$

The defect cluster on the LHS indicates a pure oxide ion at an anionic site with a neighboring vacancy (see picture above); the defect cluster on the RHS stands for a pure oxide ion at an interstitial site (the exact position is the one which maximizes the energy) with two neighboring vacancies.

As said before, an estimate of the activation energy for oxygen migration is given by the energy change for this reaction.

The decomposition of the reaction into steps whose energy change can be calculated directly with GULP is as follows:

$$\begin{eqnarray*} O_{(g)}^{2-}+2A_A^\times=\left(V_A^{2-\frac{x}{v-1}}O_i^{\prim... ...c{x}{v-1}}&&E_{\mathit{mig}}=E_{\mathit{trans}}-E_{\mathit{ref}} \end{eqnarray*}$$

The two component energies are evaluated with GULP using the following directives:

$E_{\mathit{ref}}$

IMPURITY  O  0.25 0.25 0.25
VACANCY      0.75 0.25 0.25

$E_{\mathit{trans}}$

bulk_nooptimize trans defect
...
VACANCY          0.25 0.25 0.25
VACANCY          0.75 0.25 0.25
INTERSTITIAL O   0.45 0.23 0.27    
...

$Ce_{\left(1-x\right)}Mn_xO_{2-\frac{x}{2}}$

For all compositions the position of the migrating oxide in the ``activated state'' found by GULP was exactly midway between the two vacant sites.

Taking the cluster described in the previous mean field analysis as reference state one gets negative activation energies:

This is because the energy profile for an interstitial oxide moving along the line between two nearest-neighbor anionic sites has a maximum midway (as expected and found by the `` trans'' calculations), but also two symmetrical minima near the anionic sites. The following is a plot of the energy profile for the $10\%Mn$ composition (obtained with `` INTERSTITIAL O $\cdots$ FIX''):

In the light of these findings, the activation energy for oxygen migration should be evaluated according to the following procedure:

1. for each composition, evaluate the energy profile for the migrating oxygen like the one shown above
2. evaluate the activation energy as the difference between the maximum and minimum energy of the profile

$Ce_{\left(1-x\right)}Y_xO_{2-\frac{x}{2}}$

For the reasons explained in the ceria-$Mn_2O_3$ subsection, I have not evaluated the $E_{\mathit{ref}}$ term.

$Ce_{\left(1-x\right)}Gd_xO_{2-\frac{x}{2}}$

For the reasons explained in the ceria-$Mn_2O_3$ subsection, I have not evaluated the $E_{\mathit{ref}}$ term.

$Ce_{\left(1-x\right)}La_xO_{2-\frac{x}{2}}$

For the reasons explained in the ceria-$Mn_2O_3$ subsection, I have not evaluated the $E_{\mathit{ref}}$ term.

Representative input files

Trans calcs

The trans run has to be done with an already optimized bulk structure. Hence there are two steps:

Bulk optimization

opti conp
dump every 1 la-bulk-60.dump

maxcyc opt      200
maxcyc fit      200
title
Activation energy for oxygen migration in Ce_{1-x}La_xO_{2-\frac{x}{2}}
end

cell
   5.329267   5.329267   5.329267  90.000000  90.000000  90.000000

fractional    5
   Ce4  core  0.0000  0.0000  0.0000  -3.7000  0.4000  0.0000 
    La  core  0.0000  0.0000  0.0000   3.0000  0.6000  0.0000 
     O  core  0.2500  0.2500  0.2500   0.0770  0.8500  0.0000 
   Ce4  shel  0.0000  0.0000  0.0000   7.7000  0.4000  0.0000 
     O  shel  0.2500  0.2500  0.2500  -2.0770  0.8500  0.0000  
                                                     
space
225


size   10.45067684    21.94642136
centre          0.50   0.25   0.25
interstitial O  0.45   0.23   0.27  
vacancy         0.25   0.25   0.25
vacancy         0.75   0.25   0.25


buck
Ce4   shel O     shel  1986.8300     0.351070  20.400      0.000 15.000
La    core O     shel  1439.7        0.36510    0.0        0.0   15.0
O     shel O     shel 22764.300      0.149000  27.890      0.000 15.000
spring
Ce4    291.75000
O      27.290000

Actual trans calc

# 
# Keywords:
# 
bulk_nooptimize trans defect
# 
# Options:
# 
title
Activation energy for oxygen migration in Ce_{1-x}La_xO_{2-\frac{x}{2}}         
end
cell
   5.586480   5.586480   5.586480  90.000000  90.000000  90.000000
fractional    5
Ce4   core 0.0000000 0.0000000 0.0000000 -3.7000000 0.40000 0.00000             
La    core 0.0000000 0.0000000 0.0000000 3.00000000 0.60000 0.00000             
O     core 0.2500000 0.2500000 0.2500000 0.07700000 0.85000 0.00000             
Ce4   shel 0.0000000 0.0000000 0.0000000 7.70000000 0.40000 0.00000             
O     shel 0.2500000 0.2500000 0.2500000 -2.0769999 0.85000 0.00000             
space
225
centre  0.5000 0.2500 0.2500
size  10.4507  21.9464
vacancy  0.250000 0.250000 0.250000
vacancy  0.750000 0.250000 0.250000
interstitial O       0.450000 0.230000 0.270000    
buck     
O     shel Ce4   shel  1986.8300     0.351070 20.400      0.000 15.000
buck     
La    core O     shel  1439.7000     0.365100 .00000E+00  0.000 15.000
buck     
O     shel O     shel  22764.300     0.149000 27.890      0.000 15.000
spring
Ce4    291.75000    
spring
O      27.290000    
maxcyc opt      200
maxcyc fit      200
dump every   1 la-trans-60.dump

NOTE: the input for the trans calc can be generated automatically from the dump file of the bulk optimization in a shell script. E.g.:

...
gulp < la-bulk-60 > la-bulk-60.out
wait
sed -e's/^opti conp  *$/bulk_nooptimize trans defect/' \
   -e'/^dump every/s/la-bulk/la-trans/' la-bulk-60.dump > la-trans-60
gulp < la-trans-60 > la-trans-60.out
wait
...