First-principles quantum-mechanical calculations are computationally expensive; thus, even on today's fastest computers, the system size which can be simulated by first-principles approaches are limited by computational resources, and thus some "scaling up" of length scales is necessary to treat thermodynamic alloy problems. One method for this scaling up which is currently utilized in NREL's Solid State Theory Group is in mapping accurate first-principles data onto simpler energy functionals which can then be used on a much larger length scale. The method is known as LEGO, or "Linear Expansion in Geometric Objects". Examples of this type of approach are in precipitation hardening in Al-Cu and Al-Zn alloys: Like most pure metals, aluminum is relatively weak, and therefore need to be strengthened via alloying additions. In precipitation hardening, common in Al-alloys, a small amount of a solute element is added to Al at high temperatures, and then the alloy is quenched down in temperature past the solubility limit of the alloying element. Thus, the solute element begins to precipitate out of the Al matrix, and these precipitates act to pin dislocations, and hence improve mechanical strength. However, without understanding the structure and stability of the precipitated phase, alloy designers cannot fully understand the strengthening mechanism. NREL researchers have predicted the thermodynamic stability and atomic-scale structure of Cu precipitates embedded in an Al matrix. The precipitation is determined by a combination of strain and interfacial energies, and an example of calculated strain energies for Cu embedded in Al is shown in Fig. 1. By using both strain and interfacial in thermodynamic Monte Carlo simulations, one can predict the complete atomic-scale structure of precipitates in Al. An exampleof this kind of hybrid "first-principles/scale up" approach is shown in Fig. 2, which shows the atomic-scale structure of an ordered Cu precipitate in a dilute Al-Cu alloy.
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