In 2010, the Gaussian approximation potential (GAP)1 was introduced as an approach to create IPs with ab initio accuracy, using kernel regression and invariant many-body representations of the atomic neighborhood. Since their introduction, they have been effective at modeling potential energy surfaces2,3,4 and reactivity5 of molecules6 and solids7,8, defects9, dislocations10, and grain boundary systems11. Recently BartÃk et al.12 showed that a GAP model using a smooth overlap of atomic position (SOAP) kernel can be systematically improved to reproduce even complex quantum-mechanical effects13. SOAP-GAP has thus become a standard by which to judge the effectiveness of numerical approximations to ab initio data. There are a number of other machine-learned potentials that also perform well and have overlapping applications with SOAP-GAP, although applications of several of these methods to alloys are still nascent14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31. While recent work32 compares the performance of multiple methods for alloy systems, it does not address dynamical quantities such as phonon dispersion or temperature-composition dependence in phase diagrams. Note that although the GAP framework can be used with arbitrary kernels, for simplicity we will use the GAP abbreviation to mean SOAP-GAP exclusively in the rest of this paper.
The moment tensor potential (MTP)33 is another approach to learning quantum-mechanical potential energy surfaces. Due to the efficiency of its polynomial basis of interatomic distances and angles, MTP is significantly faster than GAP and has already been shown to be capable of reaching equivalent accuracy for modeling chemical reactions34, single-element systems35,36, single-phase binary systems37, or ground states of multicomponent systems25. In this work, we demonstrate that both GAP and MTP are capable of fitting the potential energy function of a binary metallic system, the Ag-Pd alloy system, with DFT accuracy across the full space of configuration and composition for solid and liquid systems. In addition to reproducing energies, forces, and stress tensor components with near-DFT accuracy, we show that these potentials can also approximate phononic band structure quite well and can be used to model compositional phase diagrams. These new capabilities of quantum-accurate IPs for alloys would pave the way to accelerated materials discovery and optimization.
Phonon band structures directly describe phase stability at moderate temperatures via the quasi-harmonic approximation. We first show that both SOAP-GAP and MTP potentials can accurately reproduce DFT-calculated phonon band structures for alloy configurations that are not in the training set. As a demonstration of speed and transferability, we use the MTP potential to calculate melting lines and transition temperatures for the Ag-Pd phase diagram using the nested sampling (NS) method39,40,41. We then compare the performance of GAP and MTP across a low energy transition pathway between two stable configurations to demonstrate the importance of regularization and active learning.
CE has been a go-to tool for computing energies across configuration space for alloys. Because of its speed and applicability over the full range of compositions, it is useful for performing ground-state searches, and even for temperature-dependent phase mapping in certain systems. However, it cannot address dynamic processes that involve structural perturbations, which often limits its usefulness.
This work demonstrates that machine-learned IPs are nearly as good as CE for on-lattice computation of energies81. Nevertheless, for on-lattice systems where the atomic displacements are small, CE is still the best choice. Ref. 38 discusses metrics to determine when CE performs well as a function of atomic displacement. But unlike CE, the machine-learned IPs can compute forces, virials, and hessians across the compositional space as well. These additional derivatives of the potential energy surface are sufficiently accurate to approximate dynamic properties like phonon dispersion curves, as well as map out the temperature-composition phase diagram for an alloy. Software for creating datasets and fitting potentials is readily available and easy to use. These potentials, therefore, offer a viable alternative to CE models, and arguably represent the future direction of first-principles computational alloy design.
In addition to temperature and pressure, other thermodynamic properties may be graphed in phase diagrams. Examples of such thermodynamic properties include specific volume, specific enthalpy, or specific entropy. For example, single-component graphs of temperature vs. specific entropy (T vs. s) for water/steam or for a refrigerant are commonly used to illustrate thermodynamic cycles such as a Carnot cycle, Rankine cycle, or vapor-compression refrigeration cycle.
One type of phase diagram plots temperature against the relative concentrations of two substances in a binary mixture called a binary phase diagram, as shown at right. Such a mixture can be either a solid solution, eutectic or peritectic, among others. These two types of mixtures result in very different graphs. Another type of binary phase diagram is a boiling-point diagram for a mixture of two components, i. e. chemical compounds. For two particular volatile components at a certain pressure such as atmospheric pressure, a boiling-point diagram shows what vapor (gas) compositions are in equilibrium with given liquid compositions depending on temperature. In a typical binary boiling-point diagram, temperature is plotted on a vertical axis and mixture composition on a horizontal axis.
The temperature scale is plotted on the axis perpendicular to the composition triangle. Thus, the space model of a ternary phase diagram is a right-triangular prism. The prism sides represent corresponding binary systems A-B, B-C, A-C.
The classification of miscible and immiscible systems of binary alloys plays a critical role in the design of multicomponent alloys. By mining data from hundreds of experimental phase diagrams, and thousands of thermodynamic data sets from experiments and high-throughput first-principles (HTFP) calculations, we have obtained a comprehensive classification of alloying behavior for 813 binary alloy systems consisting of transition and lanthanide metals. Among several physics-based descriptors, the slightly modified Pettifor chemical scale provides a unique two-dimensional map that divides the miscible and immiscible systems into distinctly clustered regions. Based on an artificial neural network algorithm and elemental similarity, the miscibility of the unknown systems is further predicted and a complete miscibility map is thus obtained. Impressively, the classification by the miscibility map yields a robust validation on the capability of the well-known Miedema's theory (95% agreement) and shows good agreement with the HTFP method (90% agreement). Our results demonstrate that a state-of-the-art physics-guided data mining can provide an efficient pathway for knowledge discovery in the next generation of materials design.
Nano-particle binary alloy phase diagrams have been evaluated from the information on the Gibbs energy and the surface tension of the bulk size on the basis of the regular solution model. As the size of the particle decreases, the liquidphase region is enlarged in those binary phase diagrams, in other words, the liquidus temperature decreases. Effect of the size of the particle on the phase equilibria is remarkable when the excess Gibbs energy is positive and its absolute value is large in solid and liquid phases.
Phase diagrams are a great tool for the Brazing Engineer. While its is certainly true that most brazing applications involve systems more complicated than a binary alloy represented by the common phase diagram, nevertheless, the binary phase diagram is an invaluable tool both for answering questions about why a particular braze alloy and substrate interact the way they do and it can also help to predict what to expect from a novel application. While they are extremely useful, like any power tool, they can be difficult to use and must be fully understood to be of most use.
So how do you read a phase diagram? This phase diagram Phase Diagram shows a typical binary system that happens to contain a eutectic. This is a fairly common characteristic of bimetallic alloys, the copper-silver system for instance has a phase diagram very similar to the one in the link. First lets consider the information that is displayed.
There is a lot more information that you can glean about the characteristics of the alloy formed between the two elements from this phase diagram and also information about how this alloy might perform in relationship to a braze application but that will be the subject of a future post or two.
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