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Interpretation of deformation electron density maps (for Al alloys)

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  1. Oct 11, 2012 #1
    I am posting this from a friend's account since I've been unable to register for a while.

    Brace yourselves for this is going to be a long post.

    -----------------------
    TLDR:

    I am trying to figure out the reason for AlSi's lower than expected from atomic misfit solid solution hardening (compared to AlZn, AlMg and AlCu). After about two weeks of digging I couldn't come up with any other explanation than Si's different electronic structure/being a metalloid. This article (http://www.springerlink.com/content/80408u1202556086/?MUD=MP ; for coloured images http://crm-eac.imr.ac.cn/pdf/全文链接44-高磊.pdf) reinforced my assumptions. Fast forward several months, and I'm researching (together with Dr Banach) this subject at the Institute of Low Temperature and Structure Research of Polish Academy of Sciences during an internship. There I've found another interesting article (Science Magazine: Sign In) that hinted where to look, i.e. the tetrahedral interstices. Now we're done with most of the calculations for those alloys, particularly electron density maps, but can't seem to find any apparent tendency among those alloys (unlike the first article).

    Does anyone know how to analyze deformation density maps (in metal alloys)? Is there any literature that could be helpful? I haven't found any, unfortunately. Maybe some researcher here is well versed in this sort of thing?
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    Let me first introduce myself. I'm a student at the Mechanical Faculty of Wroclaw University of Technology (Poland) with a keen interest in materials science, metal alloys in particular, but also smart materials and metamaterials. During one of the Engineering Material Design lectures the professor asked us, without hinting anything, why solid solution hardening of Al alloys with Si is weaker than one would expect from the atomic misfit of the two elements, and showed us the table below (turned out I overlooked one word - "% składnika" = "% of ingredient", i.e. solute):

    tab1_2_kopia.jpg

    Now, I'm not the kind of person who's satisfied with cop-out answers (which pretty much all other students turned out to be) and the question intrigued me, so I analyze the problem. After two weeks of digging and verifying various hypotheses, I concluded that there's no correlation between electronegativity of the solutes or the difference in electronegativity of Al and the solutes and solution hardening. Same goes for the the type of equilibrium crystal lattice of the solutes, i.e. elements crystalizing in BCC were not better at hardening Al's FCC or any such thing.

    I noted that YS strengthening (at%) is proportional to the misfit factor, while TS strengthening (at%) is not, as well as the fact that TS increase is always bigger than that of YS. Because of that, I concluded that most likely the only factor in YS strengthening is the atomic misfit (because it's unlikely other strengthening mechanisms were correlated with the atom size), while TS strengthening is influenced by more factors.

    Knowing that Si is a metalloid, this made me believe that maybe the atom size of Si in Al alloys is different from that assumed by the table above (regular atom radius). I hypothesized that maybe the bonding between Al and Si is more covalent than in case of the remaining solutes, and thus it's more adequate to compare their covalent radii instead. In that case, the atomic misfit would be roughly 8.3%, placing Si right below Zn, making solution hardening indeed proportional to the misfit factor:

    tab1_3_kopia.jpg

    Al-Si bond's covalence would result in lowered free electron density e/a. The lower the e/a, the weaker is the Stacking Fault Energy decrease. If the SFE decrease is too weak, then cross slip of a screw dislocation is not counteracted enough, leading to easier deformation compared to higher e/a.

    Furthermore, based on Quantitative prediction of solute strengthening in aluminium alloys (Nature Materials), I learned that Si (in Al alloys) exhibits unusually low solute-dislocation interaction and volume deformation, at least compared to the likes of Mg or Cu. The latter would further reinforce the hypothesis of different than expected atom size of Si in Al's FCC structure. The interaction energy gradient for Si is much weaker and smoother, which could mean that Si atoms are interacting with Al matrix stronger with dislocations.

    Solute-dislocation interaction energies and volume deformations of discussed solutes/alloys:

    nature.jpg

    I concluded that in order to understand, why the way Si interacts with the dislocations/Al matrix so different compared to other solutes discussed here, I'd have to investigate its electronic structure. Based on this article (http://www.springerlink.com/content/80408u1202556086/) I decided that ELF (electron localization function) might be the right tool, as in this article the authors explained the opposite phenomenon - Y and Gd strengthening Mg more than expected due to increased bonding covalence. I gathered that the electronic structure (electron density or such) could potentially explain Si's comparativly weak solution hardening. I asked several researchers for help, but none of them could really help me. However, Dr Banach from the Institute of Low Temperature and Structure Research of Polish Academy of Sciences offered me an internship where we could try to solve this problem together.

    Together with Dr Banach, I have calculated electron densities and deformation electron densities of pure Al and four other Al alloys (AlSi, AlZn, AlCu and AlMg), using 2x2x2 supercells (basically a unit cell copied x2 in each ortogonal direction, resulting in a structure made of 32 atoms, 31 Al and 1 solute) and substituted the central supercell atom - Si/Zn/Cu/Mg instead of Al. (We've also done some calculations of 3x3x3 structures but haven't got round to analyzing any of them.) For calculations we've used Density Functional Theory programme WIEN2k, and for visualization - VESTA (although this was the first time Dr Banach used it, normally he used to work with XCrysDen but VESTA boasts much more impressive functionality).

    Our prime achievement, though, was repeating the results from Dr Nakashima's article (http://www.sciencemag.org/content/331/6024/1583.full.pdf, both Fig. 3 and Fig. 4., although we're still struggling to properly visualize the latter. We're on our way of getting the equivalents of Fig. 4. for all the Al alloys I'm discussing here, however, due to being unfamiliar with VESTA, we're unable to extract the deformation density values in each direction. Once we manage to do that, we're going to compare those different alloys and see whether Si's weaker hardening is a result of lower Young modulus in certain (all?) directions. All in all, this article was a true revelation for us. It gave us a clear direction (analyzing tetrhedral interstices instead of the whole structure).

    We have calculated not only deformation densities for each alloy, but also the diference bewteen AlX (X = Cu, Si, Mg or Zn) and pure Al in terms of total electron density and each structure's respective deformation density (in order to see how different those alloys are from pure Al in terms of electron gradient). The problem is we can't find the right scale for the gradient maps to contain all the details and nuances, because the difference between core and non-core areas is several orders of magnitude, ad the difference between positive and negative peaks of the non-core area is of at least one order of magnitude.

    Here's the deformation electron density map for pure Aluminium (the core regions are slightly different due to differences in the method of obtaining deformation density data, ours was based purely on WIEN2k):

    Al_new.png

    The next step will be integrating the charge density from the interstices in order to compare their strength and see whether AlSi structure is weaker in that regard. The problem is, we still haven't figured out how to do that.

    It would be interesting to see how much weaker a metallic bond like that in Al is compared to a covalent bond, which according to this lecture from Yale (www.cosmolearning.com/video-lectures/seeing-bonds-by-electron-difference-density-6652/)/) is equivaent to 0.1 eV (single bond).

    Here are the deformation electron density maps for each alloy. The color scale is different from used by Nakashima et al. In my notes I wrote down that the max value (red) is +0.006 and min value (blue) is -0.006, although I am not sure what the unit used by VESTA is. WIEN2k used Ry/bohr^3 for electron density by defualt, but VESTA has the option of converting it to eV/A^3. I'm gonna have to check that out next week, but it's not relevant to my questions, fortunately. The second picture of each alloy tries to imitate Nakashima's scale, although ignoring negative values.


    For comparison, I recommend opening each alloys' map in a separate tab.

    Al:

    image.png
    Al_new.png


    AlCu:

    Al_Cu_2_nm.png
    Al_Cu_2_nm_new.png


    AlMg:

    Al_Mg_2_nm.png
    Al_Mg_2_nm_new.png


    AlSi:

    Al_Si_2_nm.png
    Al_Si_2_nm_new.png


    AlZn:

    Al_Zn_2_nm.png
    Al_Zn_2_nm_new.png


    So my questions are:

    Does anyone know how to properly interpret deformation density maps (or AlX - pure Al difference maps I haven't posted yet)? How can we integrate electron density in VESTA or some other software? The densities do change depending on the alloy, but do they tell us anything meaningful like with MgY and MgGd (Chinese article)? What are the aspects I should pay attention to? - electron gradient morphology? Peaks? Integrated charge of the intersticial areas?
     
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