dataesr/scientific-paragraphs-categorization huggingface dataset:
e. , when the system has low density of phase boundaries), and (iii) ternary intermetallic phases are not present. Because the locations of the phase boundaries and possibility of formation of ternary phases are not known when evaluating novel composition space, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require additional experimental verification. We estimate the level of the thermodynamic description which would be still sufficient to correctly predict thermodynamic properties of quaternary alloy systems. The main factors affecting the accuracy of the thermodynamic predictions in quaternary alloys are identified by comparing the results of CALPHAD calculations where quaternary phase space is extrapolated from binary descriptions to those resulting from ternary system descriptions.
Introduction Computational approaches are now a vital aspect of materials science, which enable the prediction of unknown data, such as materials properties and phase stability, and the more rapid exploration of design space. CALPHAD (Calculation of Phase Diagrams) is an efficient computational thermodynamic technique which permits the prediction of the phase equilibria and thermodynamic properties of multicomponent systems from those of the respective binary and ternary subsystems. CALPHAD practices also include the simulation of solidification, mapping of onset driving forces and nucleation barriers for precipitation  [1] , modeling of phase transformations  [2] , and estimation of stacking fault energy  [3] , for example. The development of multicomponent thermodynamic databases caused a growing interest in the calculations of phase diagrams for high entropy alloys (HEAs) and complex concentrated alloys (CCAs)  [3] [4] [5] [6] [7]  allowing the exploration of the composition space to be accelerated  [8] [9] [10].
In this context, it is of paramount importance to analyze the reliability of the predictions made using commercial dedicated HEA databases. The reliability of the predictions depends upon the quality of the assessment of the Gibbs energy parameters of the phases stored in the databases. The CALPHAD approach faces new challenges when applied to the development of multi-principal element alloys (HEAs and CCAs) due to large extrapolations from the thermodynamic models which may lead to inaccurate evaluations of the Gibbs energies. Many CALPHAD databases have been developed for traditional alloys so the full thermodynamic assessment of the Gibbs energies is usually given only for compositions enriched with one main element. In contrast, HEAs and CCAs lie in the central regions of compositional space, far from the known boundaries delimited by the binary and ternary subsystems.
Consequently, the calculations in these vast composition spaces rely on the availability and quality of the descriptions of the lower order constituent systems and require large extrapolations from these lower order systems by assigning higher order interaction parameters, which are refined using available experimental data. Reliability and accuracy of the predictions increases when using more complete, experimentally verified, thermodynamic descriptions specially designed for CCA compositions (e. g. , TCHEA and PanHEA special HEA databases developed by Thermo-Calc Software AB and CompuTherm LLC, respectively). A complete thermodynamic description (full assessment) of binary and ternary systems is generally sufficient to correctly describe thermodynamic properties of higher order (quaternary, quinary, etc.
) systems because the probability of the occurrence of quaternary or higher order intermetallic phases is low and decreases rapidly with an increase in the number of components. Therefore, complete thermodynamic description of binary and ternary systems is considered to be sufficient for the correct prediction of thermodynamic properties of higher order systems by extrapolation from the lower order systems. Unfortunately, even the most developed thermodynamic databases lack in the complete thermodynamic description (full assessment) of all binary and, especially, ternary systems based on the elementary components present in the database. For example, the TCHEA3 database includes 26 elements that can form 325 binary and 2600 ternary systems, but only 294 binary and 136 ternary systems are assessed in the full range of composition and temperature.
Therefore, complete thermodynamic description is currently unavailable for a huge number of quaternary (or higher order) alloy systems, which may have a potential practical interest. For those reasons, it is very important to know what the minimum level of the thermodynamic assessment of binary and ternary systems is to be sufficient to qualitatively predict equilibrium phases and phase transformations in higher order CCAs. The goal of this paper is to evaluate the quality of prediction of phases and phase compositions in a quaternary system at different levels of the assessment (thermodynamic description) of the respective ternary systems and a complete description of the respective binary systems. To achieve this goal, we used the current capabilities of a commercially specialized CALPHAD database and identified quaternary alloys systems that have a complete description of their binary and ternary sub-systems.
We then calculated ternary and quaternary phase diagrams for these quaternary systems and considered them as "reference diagrams", assuming that they are 100% accurate relative to known experimental data due to a complete thermodynamic description. A parallel coordinate plots method, which is briefly described in the following section, was used to display three-dimensional data related to the extent of phase regions in the quaternary compositional space. Finally, we artificially reduced the levels of the thermodynamic description of the ternary systems, calculated the ternary and quaternary phase diagrams using this incomplete description, and compared the calculations with the reference diagrams. This analysis allowed us to identify conditions when the databases with incomplete thermodynamic descriptions can still be used to reliably predict thermodynamic properties of quaternary systems.
Methods Senkov et al. [8, 9]  defined credibility criteria for CALPHAD calculations based on the fraction of fully thermodynamically assessed binary systems (FAB) and the fraction of fully assessed ternary systems (FAT) included in the database. By definition, a fully assessed system is one that has a complete thermodynamic description (of the respective elements, binary systems and ternary systems) within a given thermodynamic database allowing CALPHAD software to calculate thermodynamic properties for this system, which agree with the experimentally available data within the whole composition and temperature range. Using these theoretical credibility criteria, Wertz et al. [11]  have recently demonstrated the importance of (a) the number of assessed constituent binary systems and (b) the compositional distance from them, on the accuracy of the predictions for ternary systems.
Their study was carried out by comparing equilibrium calculations of the same ternary systems using eight different databases developed by CompuTherm LLC (Madison, WI, USA). Each of the studied ternary systems had complete thermodynamic descriptions (FAB = 1, FAT = 1) in at least one database, while other databases calculated the ternary space by extrapolation from binary descriptions with different levels of FAB and FAT = 0 (Figure  1a ). Entropy 2018, 20, x 3 of 9 systems (FAT) included in the database. By definition, a fully assessed system is one that has a complete thermodynamic description (of the respective elements, binary systems and ternary systems) within a given thermodynamic database allowing CALPHAD software to calculate thermodynamic properties for this system, which agree with the experimentally available data within the whole composition and temperature range.
Using these theoretical credibility criteria, Wertz et al. [11]  have recently demonstrated the importance of (a) the number of assessed constituent binary systems and (b) the compositional distance from them, on the accuracy of the predictions for ternary systems. Their study was carried out by comparing equilibrium calculations of the same ternary systems using eight different databases developed by CompuTherm LLC (Madison, WI, USA). Each of the studied ternary systems had complete thermodynamic descriptions (FAB = 1, FAT = 1) in at least one database, while other databases calculated the ternary space by extrapolation from binary descriptions with different levels of FAB and FAT = 0 (Figure  1a ). Here we expand a similar approach to higher order systems to evaluate the ability to predict quaternary phase diagrams.
A quaternary system has 6 constituent binaries and 4 ternaries, so the possible FAB levels are 0, 1/6, 1/3, 1/2, 2/3, 5/6 or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, 4, 5 or 6 of the constituent binary systems, and the FAT values are 0, 1/4, 1/2, 3/4, or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, or 4 of the constituent ternary system (Figure  1b ). TCHEA3 is an encrypted database which does not permit access to the values of the Gibbs free energy parameters of the phases. However, it is possible to fix the value of any parameter using the commands available in the Gibbs Energy System (GES) module implemented in Thermo-Calc® Here we expand a similar approach to higher order systems to evaluate the ability to predict quaternary phase diagrams.
A quaternary system has 6 constituent binaries and 4 ternaries, so the possible FAB levels are 0, 1/6, 1/3, 1/2, 2/3, 5/6 or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, 4, 5 or 6 of the constituent binary systems, and the FAT values are 0, 1/4, 1/2, 3/4, or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, or 4 of the constituent ternary system (Figure  1b ). TCHEA3 is an encrypted database which does not permit access to the values of the Gibbs free energy parameters of the phases. However, it is possible to fix the value of any parameter using the commands available in the Gibbs Energy System (GES) module implemented in Thermo-Calc® software. Using this command for all possible relevant parameters allows a comparison of the predictions for the reference systems, referred to as FAT=1 for Co-Cr-Fe-Ni and Co-Cu-Fe-Ni, and FAT = 3/4 for Al-Co-Ni-Ti (the Co-Ni-Ti ternary subcomponent is only partially assessed in TCHEA3), with the results obtained when (1) setting to zero all the ternary interaction parameters for all the relevant phases (solid solutions and a binary intermetallic having ternary solubility) but retaining ternary intermetallics and (2) additionally suspending all the ternary intermetallics.
The scenarios (1) and (  2 ) are referred to as FAT = 0 since they are equivalent to a database in which the ternary systems are not assessed for the three selected quaternary systems (the description of the quaternary relies only on the descriptions of the 6 constituent binaries). We have selected one comparison metric which is the extent of the compositional range of each phase throughout the 3D composition space. These data were retrieved from the phase point coordinates calculated at a fixed temperature for 12,341 equally spaced system points (2. 5 at. % compositional steps) throughout the quaternary diagram. It is worth mentioning that our methodology uses the same commercial database for calculation of thermodynamic properties of a quaternary alloy at different levels of assessment of the respective ternary systems, which is achieved by artificial removal of the ternary interaction parameters.
At the same time, the binary interaction parameters in the database containing the description of ternary systems can slightly differ from the published binary assessment to better reproduce the ternary phase diagram. However, these modifications are made in such a way that the calculated thermodynamic properties of the respective binary systems still agree with the accepted experimental data. Therefore, setting all ternary terms to zero (FAT=0) can still be treated as a condition with FAB = 1. As previously discussed by Miracle and Senkov  [12] , and Wertz et al. [11] , visualizing the extension and topology of binary and ternary terminal solid solution phase fields into a higher order dimensional composition space remains difficult because the properties of high dimensional objects diverge our two or three-dimensional intuition.
Therefore, a higher credibility database is still preferable, while the calculations using lower The different sensitivities to FAT of the calculated phase diagrams for Co-Cr-Fe-Ni, Co-Cu-Fe-Ni and Al-Co-Ni-Ti systems result from the different thermodynamic properties and topologies of these systems. In all the cases, the number of solid solution phases and solid-solution phase identifications match perfectly between FAT = 1 (or 3  4 ) and FAT = 0 calculations. The main differences in the calculations occur at or near the boundaries between different phase fields. Therefore, higher density of these boundaries should result in lower accuracy of the FAT = 0 calculations. This observation is in agreement with earlier results by Wertz et al. [11]. The presence of ternary phases in the ternary systems, which the evaluated higher order system contains, will result in incorrect FAT = 0 calculations, as the extrapolations from the binary interactions cannot predict formation of ternary phases.
Unfortunately, the locations of the phase boundaries and possible presence of ternary phases are not known when evaluating novel composition space. Therefore, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require experimental verification. The Co-Cr-Fe, Co-Cr-Ni, Co-Fe-Ni and Cr-Fe-Ni behave like near-ideal mixtures of the constituent elements so the Co-Cr-Fe-Ni quaternary does too, which results in the stabilization of the disordered solid solutions and the absence of ternary intermetallic phases. In such cases, neglecting ternary interaction parameters when considering FAT = 0 predictions gives approximatively the same results as for FAT = 1 because their contributions to the Gibbs free energy are also negligible.
This sensitivity depends upon the neighboring regions. In particular, the assemblage around the fcc and L1 2 regions consists of various binary and ternary phase fields involving B2, Ni 3 Ti and not containing ternary intermetallics, so the extent of the fcc region is not constrained when ternary intermetallics are removed. In consequence, using an incomplete database (FAT = 0) does not substantially affect the composition and boundaries of the fcc and L1 2 phases. On the other hand, bcc and B2 regions are surrounded by several ternary intermetallics (γ and H_L2 1 ), so neglecting them (FAT = 0) shifts significantly the bcc and B2 composition and boundaries, thus providing significant disagreement between FAT = 3/4 and FAT = 0 calculations (Figure  3 ).
Conclusions The ability of a commercial CALPHAD database for high entropy alloys to predict quaternary phase diagrams has been evaluated through comparisons of predictions made from full binary system descriptions with the results from full ternary system descriptions. This point is of prime interest because even the most comprehensive databases (such as TCHEA3 from Thermo-Calc) include a limited number (about 5%) of fully assessed ternary systems. This entails that quaternary and higher order predictions are systematically obtained from incomplete thermodynamic descriptions and rely almost entirely on extrapolations from binary subsystem descriptions. The analysis has led to the following conclusions. (1) The quality of CALPHAD calculations for 4-component alloys mainly depends on the ability to correctly describe (a) all the phases present in binary and ternary sub-systems, including novel ternary phases, and on (b) positions (composition and temperature) of boundaries between different phase-field regions.
(2) Direct extrapolation from binaries to quaternary (i. e. , without ternary descriptions) systems may be acceptable only when binary miscibility gaps are not present, binary intermetallic phases are not present or present in a few quantities (i. e. , when the system has low density of phase boundaries), and ternary intermetallic phases are not present. (3) When ternary phases are present but not included in the thermodynamic descriptions, the prediction accuracy of the FAT = 0 calculations will be strongly affected only for neighboring phase regions. Considering that quaternary and higher order interactions are negligibly weak, and the formation of novel quaternary intermetallics very unlikely, the present conclusions for quaternary alloys can be extended to higher order systems. Because the locations of the phase boundaries and possibility of formation of ternary phases are not known when evaluating novel composition space, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require additional experimental verification.
The commercial database TCHEA3 from Thermo-Calc Software AB (Stockholm, Sweden) used in the present work includes 26 elements, 294 fully assessed binary and 136 fully-assessed ternary descriptions, providing few quaternary systems with FAB = 1 and FAT = 1 (e. g. , Co-Cr-Fe-Ni and Co-Cu-Fe-Ni), and FAB = 1 and FAT = 3/4 (e. g. , Al-Co-Ni-Ti, Al-Cu-Fe-Ni, Al-Cu-Fe-Ti, Al-Cu-Ni-Ti, Al-Mn-Ni-Ti, Al-Fe-Ni-Ti, Al-Ni-Si-Ti). Among them, we have selected three phase diagram templates for this study: • Co-Cr-Fe-Ni (FAB = 1, FAT = 1), which consists of 3 disordered solid solutions (fcc, bcc, hcp) and one binary intermetallic having ternary solubility (σ); • Co-Cu-Fe-Ni (FAB = 1, FAT = 1), which has only disordered solid solutions (fcc and bcc phases), with miscibility gap for fcc; • Al-Co-Ni-Ti (FAB = 1, FAT = 3/4), which includes ordered and disordered solid solutions (fcc/L12 and bcc/B2) and several binary and ternary intermetallics.
Data sources Climate data for model simulations include monthly temperature and cloudiness from the Climate Research Unit (CRU)'s time series (TS) 3. 1 data  (Mitchell and Jones, 2005)  and monthly gridded precipitation from the Global Precipitation Climatology Centre (GPCC, version 5)  (Becker et al. , 2013). Monthly climate data are interpolated by the model internally to daily values by linear interpolation for temperature and cloudiness and by an internal weather generator  (Gerten et al. , 2004)  for precipitation using the number of wet days as described by  New et al. (2000). Global annual values for atmospheric CO 2 concentration are used from the Mauna Loa station (NOAA/ESRL; www. esrl. noaa. gov/gmd/ ccgg/trends/). Thermal and hydraulic characteristics of the soils are derived from the Harmonized World Soil Database (version 1.
Soil and litter decomposition is controlled by soil moisture and soil temperature. All soil processes are computed on a daily basis. For further details, see  Schaphoff et al. (2013). Carbon and water dynamics are linked so that the effects of changing temperatures, water availability and CO 2 concentrations are accounted for  (Gerten et al. , 2004 (Gerten et al. , , 2007)). Physiological and structural plant traits of each PFT determine its water requirements and consumption. Competition between PFTs due to differences in their performance under given climate conditions can lead to changes in vegetation composition. Changes in the PFT distribution in turn affect the productivity of individual PFTs in subsequent time steps, leading to changes in carbon and water fluxes. These fluxes are also impacted by the dynamics of the input data (weather data, soil), accounting for long-term climate trends, interannual climate variability and the impact of extreme events.
Sustainable potentials In order to derive biomass use potentials for livestock grazing at the global scale, simulation results with option G D and Taylor Diagram   2008 ) for combinations of grassland management options. "G D max" denotes values displayed in Fig. 13b , "G D best" in Fig. 13c  and "all best" in Fig. 13d. Perfect match to the estimates would mean that the simulated data points (colored dots) are at the location of the black box on the x axis, having a standard deviation of 118 g C m -2 a -1 (distance to origin) and a correlation of 1 (angle) as well as a centered RMSD of 0 (gray contour lines). livestock densities between 0 and 2 LSU ha -1 were analyzed (see Sect. 2. 6. 2). First, the livestock density value LSU harv was determined for each grid cell for which the maximum harvest was simulated (Fig.
For Germany, phenological data are collected by the German Weather Service  (Kaspar et al. , 2014)  also for haymaking dates which give the beginning of June for the first cut (average day of year 157 ± 12 days). Usually, the spring cut is triggered by a threshold amount of biomass buildup before the flowering. This threshold is given, e. g. , as 1700 kg DM ha -1 (= 77 g C m -2 ) in recommendations for pastures in temperate regions  (IKC, 1993 ), but it is hardly possible to derive a common rule for a global application from literature values. Also, information on conditions for a second mowing is scarce, with estimates such as leaf biomass accumulation of 3000 kg DM ha -1 (= 135 g C m -2 ) or day of year 222 (± 20 days), in the German phenological database. For the reduction of frost damage  (IKC, 1993) , it is recommended to mow not too late in fall (e.
, 2008)  reveals that the best agreement with observed grass yields can be achieved assuming heterogeneous livestock densities. Managed grasslands are still heavily under-researched in terms of global distributions of grazing livestock and wild herbivores and the implications of overgrazing in boreal and polar regions. With the model extension presented in this paper, the DGVM LPJmL can also contribute to the assessment of the ecological "hoofprint" of livestock. Here, simulations of potential grass yields and the effects on soil carbon stocks may help to frame guidelines for sustainable grassland management and to better understand the implications of livestock production and climate mitigation targets. Code and data availability. The code of the LPJmL model is available upon request for review purposes and for collaboration projects.
Acknowledgements. Susanne Rolinski  and  Christoph Müller  acknowledge financial support from the  MACMIT  project ( 01LN1317A ) and the  Kulunda  project ( 01LL0905L ) funded through the  German Federal Ministry of Education and Research (BMBF)  and support by  Hermann Lotze-Campen  and the  Landuse Working Group at PIK. Isabelle Weindl  and  Benjamin Leon Bodirsky  received funding from the  European Union 's  Horizon 2020 research and innovation program  under grant agreement no. 652615  ( SUSTAg FACCE-JPI ). Eltje R. Boons-Prins ,  Johnny A. te Roller  and  Peter A. Leffelaar  acknowledge the financial support of the organization " Statutory Research in Nature and Environment " in Wageningen. We enjoyed discussions and help from  Bernhard Schauberger ,  Dieter Gerten ,  Yvonne Jans ,  Femke Lutz ,  Sara Minoli  and  Stephen Wirth.