Computing and analyzing the migration map in β-alumina, Na₂O(Al₂O₃)₁₁ 


Algorithm:

1. Go to the Na₂O(Al₂O₃)₁₁ record. Compute the adjacency matrix.
   Now run Dirichlet, to ignore Na atoms you should El - Na in both lines in the Continuous tab of Dirichlet options. After running reply Yes to the request:
   
   
   
   After computing voids and channels duplicate the record in the electrolytes# database and remove from the adjacency matrix all voids (ZA) that cannot contain Na atoms: their radius (Rsd) is less than the Na radius in oxygen environment, 1.54 Å (see Blatov V.A. (2004) Cryst. Rev. 10, 249-318). From 23 voids (ZA1-ZA23) only six remain (ZA1-ZA6).
   
   
   
   
   
   Remove all channels inaccessible for Na atoms, they have radius less than 2.4 - 2.4⋅0.1 ≈ 2.15 Å. Look for the value of Rad for all ZA-ZA for each ZA. Here below we show the change for ZA1 where the channels (bonds ZA1-ZA1, ZA1-ZA2, ZA1-ZA3) are < 2.15 Å. Do the same for all ZA, checking all ZA-ZA Rad.
   
   
   
   
   
   Finally, remove all the voids (ZA) having only one adjacent channel (one ZA-ZA) and all the voids that have no adjacent channels (no ZA-ZA) because they do not provide the ion migration. The final adjacency matrix contains only four voids. Save the change.
   
   
   
   
   
   
2. Alternatively to the procedure above (1), the removal from the adjacency matrix of all voids and channels that cannot contain Na atoms can be done automatically using the Modify adjacency matrix procedure as follows:
   
   Duplicate the initial record Na₂O(Al₂O₃)₁₁ in the electrolytes# database.
   Use the option Modify adjacency matrix (Compound/Auto Determine/Modify Adjacency matrix):
   
   
   
   In the open window select as atoms all the voids (Z). For Rsd apply the values 0:1.54 Å. In column Bonds to change select all types of bonds without bond Atom-Void. And Apply parameters to - Change bonds. As a result all voids less 1.54 Å are deprived of all contact.
   
   
   
   
   
   Now we change to no bond all channels inaccessible for Na atoms, changing the Z-Z bonds to “none” for the ones that have radius (Rad) less than 2.15 Å, select Z both for Atom A and Atom B
   
   
   
   
   
   Remove all voids adjacent to one or no channel by option Simplify ADM (Compound/Auto Determine/Simplify Adjacency matrix). The final result is the same as we did with (1):
   
   
   
   
   
   
3. Draw the migration map in IsoCryst. Select and draw only voids (ZA nodes). What is the dimensionality of the channel system? What is the orientation?
   
   
   
   
4. Open an ADS window. Check the Common/Dimen. Calc. option and uncheck all other options. Run ADS and specify all atoms as central. You will get the following output:
   
   

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Structural group No 1
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Structure consists of 3D framework with Al11NaO17
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Structural group No 2 -------------------------
Structure consists of plane layers ( 0 0 1) with ZA

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Be sure that the ADS conclusion about dimensionality of the channel system coincides with the IsoCryst analysis.

Exercise: analyze the dimensionality of the migration map in Li5(GaO4). Assume Rsd = 1.38 Å for Li⁺ cations.

Answer: The system of voids contains 13 types of voids. All voids are larger than 1.38 Å.



When analyzing the channels we convince that they have Rad are larger than 1.8 Å (R(Li-O) – 10% R(Li-.O)).



A void/channel will be available for Li-cations if it is determined only by oxygen atoms (Solid State Ionics 179 (2008) 2248–2254). We then must check for all voids that the environment does not have the other cations too close. In this case we see that many voids are located too close to Ga. We remove voids using the criteria of solid angles, specifically all bonds to cations with SA large than 10%. After this procedure do not forget remove 0-coordinated voids ZA from adjacency matrix. We get a one-dimensional system of conduction channels formed by three types of voids.







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