Model Building An Introduction to Atomistic Simulation

Transcription

1 Materials and Modelling MPhil COURSE MP3: MONTE CARLO AND MOLECULAR DYNAMICS COMPUTING CLASS 1 Model Building An Introduction to Atomistic Simulation Wednesday 22 nd November Aims and objectives The aim of this class is to introduce the concept of molecular mechanics for atomistic simulation. We will look at what is required to build a simple model of a periodic ionic solid and use it to calculate some defect energies as a means of illustrating some of the issues that must be considered when performing simulations. This information will then be used in the second class when we consider molecular dynamics. By the end of this class you will be able to :- 1. Generate the unit cell of a crystal given its cell dimensions, space group and atomic coordinates. 2. Appreciate what is required to define the interaction between ions. 3. Calculate defect energies and understand the problems finite size effects introduce. 2 Practical 2.1 Getting started All the software we will be using in this session is available free within the UK academic community. The packages are by no means unique in their capabilities and other software packages are available (including commercial software). However, whatever software you use in the future, the methodology described below will provide a useful routine to follow when generating a computer model of a material. The code we will be using to generate our models and at the end of the session to do some simple calculation is METADISE 1. To visualise our models we will be using VIEWER-LITE, which is a simplified version of the commercial package Materials Studio. 1 Minimum Energy Techniques Applied to Dislocation, Interface and Surface Energies 1

2 2.2 Model building Building a simple crystal with two different atomic species To define the structure of a three-dimensional solid requires the input of three main sets of information - the unit cell, the fractional coordinates and types of the atoms, and finally the space group symmetry. If all this sounds a little confusing, don t worry too much, everything should become clear as you work through this practical session. If you have any questions though, please ask a demonstrator for help that is what we are here for. The following will generate a NaCl-type lattice. If you haven t done so already, log into one of the Window s PCs and Double-Click on the My Computer icon on your desktop. Then double click on the ion representing the network drive labelled i: (This is the MPhil file server and saving any files you create during the class here means that they will be available on any of the PCs in the workroom and not just the one you are currently logged into). Create a new folder by choosing File >> New >> New Folder from the menu and name it class1. Double click on its icon to change into the folder and create a new file called mgo.txt by choosing File >> New >> Text Document from the menu. Double click on the new text file you have just created and add a magnesium atom at fractional coordinate (0,0,0) and an oxygen atom at fractional coordinate (0.5,0.5,0.5) by typing the following. fractional Mg O ends A keyword the data assocaiated with it follows on the next lines The position of the Anion The position of the Cation Tells the program we have stopped giving it a block of data It is also possible to insert atoms in Cartesian coordinates, as we will see later in the class, however in simple bulk structures it is easier to use fractional coordinates. 2

3 Next add the information about the unit cell, MgO is cubic and has a cell parameter of 4.21Å. This goes above the fractional key word. Finally we need to add information about the space group, Fm-3m in this case. Once you have done that, the input file should look like this: cell Space full fm-3m 225 fractional Mg O ends The values of a, b, c, α, β, and γ for our unit cell FM-3M is the space group that represents a simple cubic lattice The additional information in the space line tells the program not to try and simplify the cell using symmetry and gives the reference number for the space group. Don t worry if the idea of space groups is new to you. All you need to know is that they are a way of defining the symmetry and position of the atoms in a unit cell. Finally, we need to add some keywords to tell the program what type of calculation we want to do and to define the force field we want to use. At this stage we only want to generate the crystal structure, so all you need to add to the bottom of your file is: Nopotential Check Start Stop No force field (or potential parameters) will be defined Just read in the input we give it then write it back out again We have finished giving input to the program so it can do any calcaultion we have asked for Gives us control of the computer back when the calculation has finished At this point you might want to check with a demonstrator you have set the file up correctly. 3

4 Save the file and rename it MgO.met. Assuming your computer is set up correctly all you need to do is double click on the file you have created and METADISE will run and generate your crystal structure. The program will only take a second or so to run and will produce several output files. The code*.out file is the main output file from METADISE and sum*.out is a summary of this file. Both can be opened in word pad or notepad. The remaining files display the crystal structure in formats that can be read by various visualisation packages and by METADISE itself (in the case of the.res files). The.xyz.car and.cif are all in a format that can be read by VIEWER-LITE. Double click on the aft*.car file and VIEWER-LITE should open displaying the MgO crystal you have just created. If you are not sure what the structure should look like, ask a demonstater to check your results. Experiment with using the viewer to manipulate the cell. For example you might try to change the display type or grow the crystal. Once you are happy with using the viewer to manipulate a crystal, close the window down and try to make a ceria (CeO 2 ) cell, which has the fluorite structure, by editing the mgo.met file you created above to reflect the following crystal cell information: Unit cell a = b = c = 5.51Å α = β = γ = 90, space group FM3-M (225) Fractional coordinates: Ce , O Once you have made the changes, save the file as ceo2.met and run METADISE again. If you have done everything correctly you should get another aft*.car file containing a ceria structure (as well as the other files METADISE produces). Again if you are not sure what the structure should look like, ask a demonstater to check your results for you. This procedure is always the starting point for building a model of a periodic solid. However, to ease the process there are numerous databases of experimentally determined crystal structures available online, for example. 4

5 2.2.2 Building a Simple Molecular Crystals It should also be noted at this point that we can also generate the structure of molecular crystals using METADISE, although this is not what the code was designed for and thus the process is less intuitive and there are far better codes available, designed more for this purpose. As an example however the the file urea.met on the course website will generate a crystal cell or urea (something we will look at in more detail in the next lab class). 2.3 Choosing a Potential Model Being able to generate the crystal structure of our system is only the first stage of the battle. We also need to choose a suitable potential model to describe the interactions in our system and calculate key properties such as energies, elastic constants and electrostatic constants. This will be the topic of the remainder of the class. We are now going to select the potential parameters required to model Ceria. At this point you might want to refer back to the section in your lecture notes on force fields and potential models. Just as when we were considering crystal structures there are various online data bases of potential parameters available, one of the best is maintained by the Royal Institution in London and can be found at 5

6 Open the input file you used to create the ceria structure above and change the keyword nopotential to potential and on the next line add the word ends. The details of the potential model will go between these two lines. First add the long-range Columbic interactions. This simply requires details of the atomic species in the cell and their charge. The keyword Buckingham is used to indicate how the short-range interactions will be defined. In our model we will assume the Ce Ce interaction is totally Columbic (a common assumption in these models) so we only need to add details of the Ce O and O O interactions. Once you have done this the potential section of your input file should look like this. potential keyword to indicate the potential model will follow species keyword to indicate a list of ionic species and charges will follow Ce 4.0 Ce 4+ O -2.0 O 2 ends The end of the list of ionic species Buckingham keyword to indicate a list of potential parameters will follow Ce O O O ends The end of the list of potential parameters ends The end of the potential model definition You will no doubt remember ( ) from the lecture course that the buckingham potential takes rij the form Φ ij = A ij exp ρ ij C ij which means we need to add the values of A, rho and r 6 C for the two potentials defined above. 6

7 Open the link to A periodic table should appear. Click on Ce (found at the left-hand end of the lanthanide series). There are several models containing Ce 4+ and O 2 listed and each will have its strengths and weaknesses. We are going to use the Grimes99 library. Double click on its link and copy and paste the parameters for O O and Ce O you find listed (Don t worry about references to cores and shells we aren t including these in our simple rigid-ion model). The five numbers on each line are the values of A in ev, ρ in Å, C in evå 6 and the range over which the interaction is to act in Å. Once you have added these parameters to your potential model, the final step is change the check keyword so it reads conp to request that METADISE performs an energy minimisation calculation to constant pressure. Once METADISE has completed open the code*.out file and scroll to the bottom of the file. If the calculation was successful you will find the phrase valid minimisation to constant pressure a few screens up from the bottom. DON T WORRY IF THE PHRASE error! must set mode, add either dislocation, bulk, surface, film, slab or nano THIS DOSE T REFER TO A PROBLEM WITH YOUR CALCULATION IGNORE IT! Below this you will find the lattice vectors of the relaxed system, the lattice energy and various bulk properties. These serve as a useful check on how well our chosen model reproduces the true behaviour of the system we are studying. The fin*.res file contains the coordinates of the relaxed system in METADISE input file format, useful as the starting point for additional calculations. Of course you can also view the relaxed structure in VIEWER-LITE. The aft*.car file contains the structure after minimisation and the bef*.car the initial structure we defined in the input file. 7

8 Question Look at the relaxed dimensions of the system (you can get this information from the code*.out file or the fin*.out file). By how much have the system dimensions changed during the simulation? What reasons might there be for any change you see? Based on this, comment on the validity of this simple model for studying bulk ceria. 2.4 Calculating Defect Energies Having built our model we can now actually do some simulations. In the next class we will look at how we perform Molecular dynamics simulations but these calculations, even with the advance of computing power, are slow. Energy minimization provides a faster alternative, however its disadvantage is that it is a static technique that totally disregards temperature. Ceria is an important component of commercial 3-way catalytic exhaust systems and thus an understanding of the stability and formation mechanism of defects, particularly oxygen vacancies is important and is something that can be studied routinely using energy minisation. Open the fin*.res file containing the relaxed structure of ceria. Notice that the system is now expressed in cartesian coordinates. To create an oxygen vacancy add the following block of commands to the file above the potential keyword: defect bulk1 O core vacancy Ends keyword used to add defects to the cell type of defect (this should be followed by the defect s coordinates) end of declaration of defects You will also need to add the coordinates of one of the oxygen atoms on the same line as O core vacancy, the easiest way to do this is to copy and paste them from the BASIS section above. Also change the conp keyword to conv, as we don t want the crystal structure to change during the minimisation. 8

9 Save the file as something new and run METADISE and check it completes OK. To calculate the energy required to create an oxygen vacancy, subtract the energy of this defective cell from the energy of the same cell without the defect. If you are unsure of what to do or need help ask one of the demonstrators. Repeat the process described above to create a Ce vacancy and calculate the energy to create an isolated Ce vacancy. The Shottky defect energy is the energy required to remove a complete formula unit from the crystal, as isolated defects, and recombine them an infinite distance from the crystal: (na x B y (n 1)A x B y + A x B y ) and is one of the most common intrinsic defect mechanisms found in simple ionic crystals. Using the lattice energy of CeO 2 and the energies to create Oxygen and Ceria vacancies you just calculated calculate the Shottky defect energy. HINT: You can calculate the energy of a reaction using a Gibbs energy cycle. Applying this to the general equation above would give the Shottky defect energy as: E shottky = (xe Avaca + ye Bvaca ) + E latt where E Avaca and E Bvaca are the formation energy of a A and B vacancy respectively and E latt is the lattice energy of ONE formula unit of A x B y. It is also usual to quote E shottky as an energy per defect so your answer should be divided by (x + y). If this makes no sense ask for help Question In the paper our potential model was taken from Grimes quotes the Shottky energy as being 3.53 ev/defect. How does this compare to the number you have just calculated? Can you account for the difference and thus suggest a way of achieving better agreement? 9

10 By adding the keyword grow x y z where x, y and z are integers above the start keyword you can increase the size of the simulation cell and thus reduce the defect concentration. Use this keyword to calculate how the energy to create an Oxygen vacancy, a Ce vacancy and hence the Shottky defect energy varies with increasing simulation cell size. Plot your results and determine the optimum size for the simulation cell (The smallest cell which gives results close to the converged value). Comment on these results. 3 Summary In this class we have looked at the information required to build a simple model of an ionic crystal and have used this model to compute some simple defect energies. You should now be able to: 1. Generate the unit cell of a crystal given its cell dimensions, space group and atomic coordinates. 2. Appreciate what is required to define the interaction between ions. 3. Calculate defect energies and understand the problems finite size effects introduce. In the next class we will use some of the models we have built in conjunction with the molecular dynamics program DL POLY 2. 10