Periodic Reporting for period 3 - Materials 4.0 (Advancing materials design by high-accuracy finite-temperature first principles calculations accelerated by machine learning potentials)
Okres sprawozdawczy: 2024-01-01 do 2025-06-30
The main objective of “Materials 4.0” is to develop and apply simulation techniques that facilitate the computation of highly accurate phase diagrams, which constitute a fundamental tool in materials design. The term “Materials 4.0” is inspired by the concept of “Industry 4.0”, which denotes a new era of industrial processes that are connected by means of data exchange. Likewise, “Materials 4.0” is to herald a new era in material design, in which quantum mechanical simulations accelerated by machine-learning techniques allow for a significantly improved prediction of thermodynamic and kinetic materials properties.
For some time now, these so-called ab initio methods—i.e. methods that are based on quantum-mechanical principles and that require no experimental input—have been used within materials science. Until now, however, the methods and applications have been severely limited, since most calculations had to assume unrealistic conditions, in particular very low temperatures near absolute zero (-273 °C). This shortcoming is reflected in present day’s ab initio databases that contain many millions of entrees of materials properties, however, almost all of them restricted to zero kelvin. This restriction will be overcome in the “Materials 4.0” project and accurate ab initio high-temperature materials properties will be obtained. High-temperature properties are clearly relevant in high-temperature applications, such as for example in airplane turbines. Yet, even if materials are used at room temperature, their high-temperature properties are often indispensable during their synthesis, in particular in the form of the above-mentioned phase diagrams. Within “Materials 4.0”, a combination of ab initio methods, statistical physics concepts, and machine-learning techniques will be utilized to develop a novel holistic simulation framework. This framework will enable an efficient yet highly accurate computation of materials properties up to the melting point; materials properties that will serve as the basic input to phase-diagram predictions. The framework will be applied to a wide range of materials, and a high-quality database of materials properties will be established. “Materials 4.0” will thus pave the way to future material design.
For some time now, these so-called ab initio methods—i.e. methods that are based on quantum-mechanical principles and that require no experimental input—have been used within materials science. Until now, however, the methods and applications have been severely limited, since most calculations had to assume unrealistic conditions, in particular very low temperatures near absolute zero (-273 °C). This shortcoming is reflected in present day’s ab initio databases that contain many millions of entrees of materials properties, however, almost all of them restricted to zero kelvin. This restriction will be overcome in the “Materials 4.0” project and accurate ab initio high-temperature materials properties will be obtained. High-temperature properties are clearly relevant in high-temperature applications, such as for example in airplane turbines. Yet, even if materials are used at room temperature, their high-temperature properties are often indispensable during their synthesis, in particular in the form of the above-mentioned phase diagrams. Within “Materials 4.0”, a combination of ab initio methods, statistical physics concepts, and machine-learning techniques will be utilized to develop a novel holistic simulation framework. This framework will enable an efficient yet highly accurate computation of materials properties up to the melting point; materials properties that will serve as the basic input to phase-diagram predictions. The framework will be applied to a wide range of materials, and a high-quality database of materials properties will be established. “Materials 4.0” will thus pave the way to future material design.
As originally envisaged, a novel holistic simulation framework for computing accurate high- temperature properties could be developed and implemented during the first half of “Materials 4.0”. The framework—termed “direct upsampling”—combines ab initio methods, statistical physics concepts, and machine-learning potentials in a unique manner, and facilitates thereby an efficient, yet very accurate prediction of high-temperature thermodynamic properties. The direct upsampling framework has already been applied to various material systems, from simpler single-element systems where experimental data is available for benchmarking to more complex alloys where the simulations touch unknown grounds and mark the only available information.
Examples are given in the figures. The first figure shows a 3x3 matrix of plots for the thermodynamic properties of the bcc refractory elements tantalum (Ta; first column), molybdenum (Mo; second column), and tungsten (W; third column). All these elements have high melting temperatures (Tm) of around 3000 K as denoted in the figure. The first row displays the heat capacity, the second row the expansion coefficient and third one the bulk modulus, all of which are key benchmark quantities since they very sensitively probe the accuracy of the high-temperature calculations (specifically of the free energy). The solid blue lines obtained with the novel direct upsampling methodology are compared to experimental and CALPHAD values from literature (symbols). CALPHAD is an empirical approach to calculate phase diagrams based on experimental input.
An excellent agreement can be seen between theory and experiment over the whole temperature range. This is a remarkable result clearly verifying the performance of the introduced methodology even under extreme conditions. It should be stressed that the experimental data can have uncertainties as highlighted for the expansion coefficient of Ta. At temperatures higher than 60% of the melting point, there are two largely diverging experimental datasets (pluses versus circles), and only one (if any) can be the correct one. The agreement of our calculations with the lower experimental dataset gives confidence that this is the one to trust.
In the second figure, the heat capacity of Mo is repeated, however, with additional curves included. The gray dotted line marks the result obtained with the state-of-the-art low- temperature approximation (quasiharmonic approximation) available prior to direct upsampling. It is obvious that this result severely underestimates the true experimental curve. Utilizing the novel direct upsampling methodology, we can see that anharmonic vibrations (red) and electronic excitations including coupling to vibrations (blue) contribute significantly at high temperatures, bringing the final result close to experimental data. In general, all relevant high-temperature excitations need to be included to obtain an accurate prediction of thermodynamic properties. This is achieved by the direct upsampling methodology.
In the third figure, an example is shown where no experimental data is available. The computations thus serve as a prediction in an unknown territory. The system is a so-calledhigh-entropy alloy, a comparatively new type of alloy class in which several elements at high concentrations are mixed together (here TaVCrW, all at 25 atomic %). High entropy alloys have shown superior material properties such as high-temperature strength and stability and good corrosion resistance. On the other hand, they pose critical challenges to modelling due to the chemical complexity. The direct upsampling methodology could be successfully applied to such a high-entropy alloy. In the three plots in the third figure, the thermal expansion, the heat capacity, and the bulk modulus are shown. The shaded regions highlight the contribution of the different excitation mechanisms. Clearly, at high temperatures the excitations need to be considered, as is possible with the direct upsampling methodology.
Examples are given in the figures. The first figure shows a 3x3 matrix of plots for the thermodynamic properties of the bcc refractory elements tantalum (Ta; first column), molybdenum (Mo; second column), and tungsten (W; third column). All these elements have high melting temperatures (Tm) of around 3000 K as denoted in the figure. The first row displays the heat capacity, the second row the expansion coefficient and third one the bulk modulus, all of which are key benchmark quantities since they very sensitively probe the accuracy of the high-temperature calculations (specifically of the free energy). The solid blue lines obtained with the novel direct upsampling methodology are compared to experimental and CALPHAD values from literature (symbols). CALPHAD is an empirical approach to calculate phase diagrams based on experimental input.
An excellent agreement can be seen between theory and experiment over the whole temperature range. This is a remarkable result clearly verifying the performance of the introduced methodology even under extreme conditions. It should be stressed that the experimental data can have uncertainties as highlighted for the expansion coefficient of Ta. At temperatures higher than 60% of the melting point, there are two largely diverging experimental datasets (pluses versus circles), and only one (if any) can be the correct one. The agreement of our calculations with the lower experimental dataset gives confidence that this is the one to trust.
In the second figure, the heat capacity of Mo is repeated, however, with additional curves included. The gray dotted line marks the result obtained with the state-of-the-art low- temperature approximation (quasiharmonic approximation) available prior to direct upsampling. It is obvious that this result severely underestimates the true experimental curve. Utilizing the novel direct upsampling methodology, we can see that anharmonic vibrations (red) and electronic excitations including coupling to vibrations (blue) contribute significantly at high temperatures, bringing the final result close to experimental data. In general, all relevant high-temperature excitations need to be included to obtain an accurate prediction of thermodynamic properties. This is achieved by the direct upsampling methodology.
In the third figure, an example is shown where no experimental data is available. The computations thus serve as a prediction in an unknown territory. The system is a so-calledhigh-entropy alloy, a comparatively new type of alloy class in which several elements at high concentrations are mixed together (here TaVCrW, all at 25 atomic %). High entropy alloys have shown superior material properties such as high-temperature strength and stability and good corrosion resistance. On the other hand, they pose critical challenges to modelling due to the chemical complexity. The direct upsampling methodology could be successfully applied to such a high-entropy alloy. In the three plots in the third figure, the thermal expansion, the heat capacity, and the bulk modulus are shown. The shaded regions highlight the contribution of the different excitation mechanisms. Clearly, at high temperatures the excitations need to be considered, as is possible with the direct upsampling methodology.
The examples discussed in the previous section have clearly revealed that substantial progress beyond the state-of-the-art has been achieved within “Materials 4.0”. The direct upsampling methodology facilitates the inclusion of the relevant thermal excitation mechanisms in the whole temperature range, up to the melting point. It does so efficiently, yet at the level of ab initio (density functional theory) accuracy.
In the forthcoming “Materials 4.0” period, we will continue to apply the direct upsampling methodology to other materials. One interesting class of materials that we will consider are group IV elements in the periodic table, i.e. Ti, Zr, and Hf. These elements exhibit polymorphism which means that as temperature increases their crystallographic structure changes, i.e. they undergo a phase transition. The solid phases occurring close to the melting point have a very complicated low temperature behavior (dynamical instability) and are very difficult to treat computationally. We will extend the direct upsampling methodology to these types of materials. Another important direction will be the computation of a complete phase diagram with the direct upsampling methodology. We will also investigate the extension of the methodology to kinetic processes, specifically to vacancy migration.
In the forthcoming “Materials 4.0” period, we will continue to apply the direct upsampling methodology to other materials. One interesting class of materials that we will consider are group IV elements in the periodic table, i.e. Ti, Zr, and Hf. These elements exhibit polymorphism which means that as temperature increases their crystallographic structure changes, i.e. they undergo a phase transition. The solid phases occurring close to the melting point have a very complicated low temperature behavior (dynamical instability) and are very difficult to treat computationally. We will extend the direct upsampling methodology to these types of materials. Another important direction will be the computation of a complete phase diagram with the direct upsampling methodology. We will also investigate the extension of the methodology to kinetic processes, specifically to vacancy migration.