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Results for “Density functional calculations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 451 records · Page 25
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Modeling the thermodynamics of the FeTi hydrogenation under para-equilibrium: An ab-initio and experimental study

We report FeTi-based hydrides have recently re-attracted attention as stationary hydrogen storage materials due to favorable reversibility, good sorption kinetics and relatively low costs compared to alternative intermetallic hydrides. Employing the OpenCalphad software, the thermodynamics of the (FeTi) 1-x H x (0 ≤ x ≤ 1) system were assessed as a key basis for modeling hydrogenation of FeTi-based alloys. New thermodynamic data were acquired from our experimental pressure-composition-isotherm (PCI) curves, as well as first-principles calculations utilizing density functional theory (DFT). The thermodynamic phase models were carefully selected based on critical analysis of literature information and ab-initio investigations. Key thermodynamic properties such as dissociation pressure, formation enthalpies and phase diagrams were calculated in good agreement to our performed experiments and literature-reported data. This work provides an initial perspective, which can be extended to account for higher-order thermodynamic assessments and subsequently enables the design of novel FeTi-based hydrides. In addition, the assessed thermodynamic data can serve as key inputs for kinetic models and hydride microstructure simulations.

08 HYDROGEN↗
DOE OSTI2026

A Robust Methodology to Elucidate Kinetics of Room Temperature Electrochemical Propane Adsorption on Platinum

Electrocatalytic activation of alkanes can further decarbonize chemical manufacturing by leveraging affordable renewable electricity and readily available shale gas reserves in the United States. Earlier works have identified the unique role of Pt in adsorbing and activating alkanes, like propane, at room temperature in acidic, aqueous electrolytes, revealing spontaneous formation of deeply dehydrogenated propane-derived surface species with an intact C 3 - backbone. Although an adsorption mechanism was hypothesized, it has not been explicitly investigated to date, preventing the quantification of kinetic rate parameters. A robust methodology to investigate and benchmark propane adsorption kinetics on Pt is critical for the rational design of electrocatalysts that exhibit higher selectivity toward desired partially oxidized products. Herein, we analyze an oxidative current transience that appears during the adsorption of propane on Pt in aqueous electrochemical conditions and develop a methodology that elucidates the adsorption mechanism and enables quantification of rate parameters such as order dependences and apparent activation barriers. This method yields an expected first-order dependence with respect to propane concentration at low coverage and reveals a second-order dependence with respect to the concentration of surface active sites. Additionally, the apparent activation barrier for propane adsorption was calculated using an Arrhenius analysis of the current transience under temperature control. The experimentally measured activation barrier of 35 kJ mol –1 is in excellent agreement with the theoretical barrier calculated by density functional theory (DFT). The kinetic analysis was extended, via the use of transition state theory, to extract entropy and enthalpy of activation, yielding consistent results with the proposed two-step adsorption mechanism and DFT calculations. These results demonstrate reliable quantification of kinetic parameters for electrocatalytic activation of C–H bonds in alkanes that can be employed for rational catalyst development for a versatile range of electrocatalytic conditions.

alkane activation↗
DOE OSTI2020

Electronic Structure of Atomically Dispersed Supported Iridium Catalyst Controls Iridium Aggregation

Supported iridium complexes, Ir(C 2 H 4 ) 2 /support, were characterized by X-ray absorption spectroscopy during a temperature ramp to 120 °C in flowing H 2 . Iridium in complexes bonded to weak and moderate electron-donor supports, SiO 2 and γ-Al 2 O 3 , underwent aggregation, forming nanoparticles and clusters, respectively. When the support was a strong electron-donor (MgO), iridium remained site-isolated. Density functional theory calculations confirm the dependence of iridium–support bond strength on the support’s electron-donor character. Coating the SiO 2 -supported complexes with 1-n-ethyl-3-methyl-imidazolium acetate enhanced electron density on the iridium, hindering its aggregation. Finally, these results demonstrate opportunities for stabilizing atomically dispersed supported noble metals under reducing conditions by choice of support/ionic liquid sheath combinations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Probing radical–molecule interactions with a second generation energy decomposition analysis of DFT calculations using absolutely localized molecular orbitals

Intermolecular interactions between radicals and closed-shell molecules are ubiquitous in chemical processes, ranging from the benchtop to the atmosphere and extraterrestrial space. While energy decomposition analysis (EDA) schemes for closed-shell molecules can be generalized for studying radical–molecule interactions, they face challenges arising from the unique characteristics of the electronic structure of open-shell species. In this work, we introduce additional steps that are necessary for the proper treatment of radical–molecule interactions to our previously developed unrestricted Absolutely Localized Molecular Orbital (uALMO)-EDA based on density functional theory calculations. A “polarize-then-depolarize” (PtD) scheme is used to remove arbitrariness in the definition of the frozen wavefunction, rendering the ALMO-EDA results independent of the orientation of the unpaired electron obtained from isolated fragment calculations. The contribution of radical rehybridization to polarization energies is evaluated. It is also valuable to monitor the wavefunction stability of each intermediate state, as well as their associated spin density profiles, to ensure the EDA results correspond to a desired electronic state. The examples suggest that this second generation uALMO-EDA is a useful tool for furthering one's understanding of both energetic and property changes associated with radical–molecule interactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

A density-wave-like transition in the polycrystalline V 3 Sb 2 sample with bilayer kagome lattice

Recently, transition-metal-based kagome metals have aroused much research interest as a novel platform to explore exotic topological quantum phenomena. Here we report on the synthesis, structure, and physical properties of a bilayer kagome lattice compound V 3 Sb 2 . The polycrystalline V 3 Sb 2 samples were synthesized by conventional solid-state-reaction method in a sealed quartz tube at temperatures below 850 °C. Measurements of magnetic susceptibility and resistivity revealed consistently a density-wave-like transition at T dw ≈ 160 K with a large thermal hysteresis, even though some sample-dependent behaviors were observed presumably due to the different preparation conditions. Upon cooling through T dw , no strong anomaly in lattice parameters and no indication of symmetry lowering were detected in powder x-ray diffraction measurements. This transition can be suppressed completely by applying hydrostatic pressures of about 1.8 GPa, around which no sign of superconductivity was observed down to 1.5 K. Specific-heat measurements revealed a relatively large Sommerfeld coefficient γ = 18.5 mJ∙mol –1 ∙K –2 , confirming the metallic ground state with moderate electronic correlations. Density functional theory calculations indicate that V 3 Sb 2 shows a non-trivial topological crystalline property. Furthermore, our study makes V 3 Sb 2 a new candidate of metallic kagome compound to study the interplay between density-wave-order, nontrivial band topology, and possible superconductivity.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Computationally Predicted High-Throughput Free-Energy Phase Diagrams for the Discovery of Solid-State Hydrogen Storage Reactions

The design of multinary solid-state material systems that undergo reversible phase changes via changes in temperature and pressure provides a potential means of safely storing hydrogen. However, fully mapping the stabilities of known or newly targeted compounds relative to competing phases at reaction conditions has previously required many stringent experiments or computationally demanding calculations of each compound’s change in Gibbs energy with respect to temperature, G(T). Here, we have extended the approach of constructing chemical potential phase diagrams based on ΔG f (T) to enable the analysis of phase stability at non-zero temperatures. We first performed density functional theory calculations to compute the formation enthalpies of binary, ternary, and quaternary compounds within several compositional spaces of current interest for solid-state hydrogen storage. Temperature effects on solid compound stability were then accounted for using our recently introduced machine learned descriptor for the temperature-dependent contribution G δ (T) to the Gibbs energy G(T). From these Gibbs energies, we evaluated each compound’s stability relative to competing compounds over a wide range of conditions and show using chemical potential and composition phase diagrams that the predicted stable phases and H2 release reactions are consistent with experimental observations. This demonstrates that our approach rapidly computes the thermochemistry of hydrogen release reactions for compounds at sufficiently high accuracy relative to experiment to provide a powerful framework for analyzing hydrogen storage materials. This framework based on G(T) enables the accelerated discovery of active materials for a variety of technologies that rely on solid-state reactions involving these materials.

08 HYDROGEN↗
DOE OSTI2021

2D Copper Tetrahydroxyquinone Conductive Metal–Organic Framework for Selective CO 2 Electrocatalysis at Low Overpotentials

Abstract Metal–organic frameworks (MOFs) are promising materials for electrocatalysis; however, lack of electrical conductivity in the majority of existing MOFs limits their effective utilization in the field. Herein, an excellent catalytic activity of a 2D copper (Cu)‐based conductive MOF, copper tetrahydroxyquinone (CuTHQ), is reported for aqueous CO 2 reduction reaction (CO 2 RR) at low overpotentials. It is revealed that CuTHQ nanoflakes (NFs) with an average lateral size of 140 nm exhibit a negligible overpotential of 16 mV for the activation of this reaction, a high current density of ≈173 mA cm −2 at −0.45 V versus RHE, an average Faradaic efficiency (F.E.) of ≈91% toward CO production, and a remarkable turnover frequency as high as ≈20.82 s −1 . In the low overpotential range, the obtained CO formation current density is more than 35 and 25 times higher compared to state‐of‐the‐art MOF and MOF‐derived catalysts, respectively. The operando Cu K‐edge X‐ray absorption near edge spectroscopy and density functional theory calculations reveal the existence of reduced Cu (Cu + ) during CO 2 RR which reversibly returns to Cu 2+ after the reaction. The outstanding CO 2 catalytic functionality of conductive MOFs (c‐MOFs) can open a way toward high‐energy‐density electrochemical systems.

Majidi, Leily↗
DOE OSTI2022

Fluorination Enables Simultaneous Improvements of a Dialkoxybenzene-Based Redoxmer for Nonaqueous Redox Flow Batteries

Redoxmers or redox-active organic materials, arc one critical component for nonaqueous redox flow batteries (RFBs), which hold high promise in enabling the time domain of the grid. While tuning redox potentials of redoxmers is a very effective way to enhance energy densities of NRFBs, those improvements often accompany accelerated kinetics of the charged species, undermining stability and cycling performance. In this paper, a strategy for designing redoxmers with simultaneous improvements in redox potential and stability is proposed. Specifically, the redoxmer 1,4-di-tert-butyl-2,5-bis(2,2,2-trifluoroethoxy)benzene (ANL-C46) is developed by incorporating fluorinated substitutions into the dialkoxybenzene-based platform. Compared to the non-fluorinated analogue, ANL-C46 demonstrates not only an increased (similar to 0.41 V) redox potential but also much enhanced stability (1.6 times) and cyclability (4 times) evidenced by electron paramagnetic resonance kinetic study, H-cell and flow cell cycling. In fact, the cycling performance of ANL-C46 is among the best of high potential (>1.0 V vs Ag/Ag + ) redoxmers ever reported. Density functional theory calculations suggest that while the introduced fluorine substitutions elevate the redox potentials, they also help to depress the decomposition reactions of the charged redoxmers, affording excellent stability. The findings represent an interesting strategy for simultaneously improving energy density and stability, which could further prompt the development of high-performance redoxmers.

25 ENERGY STORAGE↗
DOE OSTI2023

Ferroelectric switching pathways and domain structure of SrBi 2 ⁢(Ta,Nb) 2 ⁢O 9 from first principles

Several families of layered perovskite oxide ferroelectrics exhibit a coupling between polarization and structural order parameters, such as octahedral rotation distortions. This coupling provides opportunities for novel electric field-based manipulation of material properties, and it also stabilizes complex domain patterns and domain-wall vortices. Among layered perovskites with such coupled orders, the Aurivillius-phase oxides SrBi 2 ⁢B 2 ⁢O 9 (B = Ta,Nb) are well-known for their excellent room-temperature ferroelectric performance. This work combines group theoretic analysis with density functional theory calculations to examine the ferroelectric switching processes of SrBi2⁢B2⁢O9. Low-energy two-step ferroelectric switching paths are identified, with polarization reversal facilitated by structural order-parameter rotations. Analysis of the domain structure reveals how the relative energetics of the coupled order parameters translates into a network of several distinct domain-wall types linked by domain-wall vortex structures. Comparisons are made between the ferroelectric switching and domain structure of SrBi 2 ⁢B 2 ⁢O 9 and those of the layered n = 2 Ruddlesden-Popper hybrid improper ferroelectrics. The results provide insight into how ferroelectric properties may be optimized by engineering the complex crystal structures of Aurivillius-phase oxides.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Image-enhanced bipolaron formation at organic semiconductor/electrode interfaces

We explore the image charge interaction for organic semiconductor bipolarons near a conducting interface and find that the cross term between one of the constituent charges and the image of its neighbor stabilizes the bipolaron by up to ~0.3 eV, dramatically increasing the concentration of this species near the interface. Using density functional theory calculations for the common hole transport molecule N,N'-bis(3-methylphenyl)- N,N'-diphenylbenzidine, we validate a simple point charge description of this effect and incorporate it within an interface energy level alignment model to predict the density of polarons and bipolarons near the interface. We find that the image effect greatly enhances bipolaron formation in the first few monolayers, leading to the expectation that bipolarons account for more than 1% of the total interface charge in many cases of practical interest. Furthermore, this result reinforces the notion that bipolarons are robust near the contacts of many organic semiconductor devices and thus helps to rationalize their involvement in the phenomenon of unipolar organic magnetoresistance.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Magnetic states of atomic vacancies in graphite probed by scanning tunneling microscopy

Intrinsic defects in graphitic materials, like vacancies and edges, have been expected to possess magnetic states from the many-body interaction of localized electrons. However, charge screening from graphite bulk carriers significantly reduces the localization effect and hinders the observation of those magnetic states. Here, we use an ultra-low-temperature scanning tunneling microscope with a high magnetic field to observe the magnetic states of atomic vacancies in graphite generated by ion sputtering. Scanning tunneling spectroscopy reveals localized states at the vacancies, which exhibit splitting at a certain magnetic field whose separation increases with the field strength. The transition is well described by the “Anderson model,” which describes the emergence of localized magnetic states inside the metallic reservoir through electron–electron interaction. The interaction strength is estimated to be between 1 meV and 3 meV, which is supported by the density functional theory calculation. The observation provides an important foundation for application of intrinsic defects to carbon-based spintronic devices.

36 MATERIALS SCIENCE↗
DOE OSTI2026

MnRhBi3: A Cleavable Antiferromagnetic Metal

This dataset contains DFT input and output files supporting the theoretical modeling in the associated publication (Chem. Mater. 2024, 36, 11306-11316). The calculations characterize MnRhBi3, an orthorhombic (Cmmm) van der Waals-layered intermetallic compound that cleaves easily between neighboring Bi layers. The dataset is organized into three calculation types: (i) Bulk: Structural relaxations of the periodic MnRhBi3 crystal in antiferromagnetic (AFM) and ferromagnetic (FM) configurations, using the vdW-DF-optB86b functional. These provide the equilibrium lattice constants, magnetic energy differences (AFM is 0.5 meV/f.u. lower than FM), and magnetic moments (4.4 µB/Mn, 0.17 µB/Rh, 0.18 µB/Bi) reported in Table 1 of the main text. (ii) Slab: Same magnetic configurations computed with an 18 Ang vacuum layer introduced between Bi layers, used to calculate the cleavage energy Ec = 0.56 J/m2 (AFM) and 0.57 J/m2 (FM), establishing MnRhBi3 as a van der Waals-layered material comparable to graphite, MoS2, and CrI3. (iii) ELF: Single-point calculation on the relaxed bulk AFM geometry with LELF=.TRUE., producing the ELFCAR file used to generate electron localization function isosurfaces and contour maps (Fig. 2, main text) showing Bi lone pairs directed into the van der Waals gaps. All folders contain CONTCAR, INCAR, KPOINTS, OUTCAR, and POSCAR. The ELF/ folder additionally contains ELFCAR. Calculations were performed using VASP 6.3.2 with PBE + vdW-DF-optB86b, PAW potentials, and an energy cutoff of 800 eV.

36 MATERIALS SCIENCE↗
NASA NTRS2021

Machine Learning the COSMO Model for Predicting Thermodynamics of Electrolyte Mixtures

Bottom-up design of electrolyte mixtures for battery systems requires predicting macro thermodynamic properties from molecular constituents. For instance, molten salt electrolyte batteries require conditions far above room temperature to operate. Therefore, discovering mixtures with increasingly lower eutectic melting points is desirable. A model that can approximate chemical activity is a valuable tool to search through the vast compositional design space. Machine learning can predict properties of materials such as vibrational free energies, electronic energy gaps, and thermal conductivities. Moreover, they can learn physical models such as interatomic potentials. The COSMO-SAC model uses theory and empirical parameterization to predict liquid-vapor and liquid-solid properties using first-principles calculations. However, obtaining activity coefficients required for parameterizing the COSMO-SAC model is costly and limited to a select chemical space. In this work, we explored if machine learning methods could improve the COSMO-SAC model and bridge density functional theory calculations to liquid phase thermodynamic properties. Our data-driven approach uses existing databases for sigma-profiles of organic solvents and reconciles their methodological differences via ensemble averaging. First, an optimal machine learning model is constructed for each dataset. Our machine learning algorithms use the sigma-profile as an input feature to predict binary mixtures' activity coefficients using multi-output regression. Each dataset uses different choices of functionals, methods, and basis sets. Therefore, our ensemble model attempts to predict corrected activity coefficients given the combination of all the model outputs. The activity coefficients used for training are generated using the COSMO-SAC model. This approach enables the extraction of meaningful information from the existing datasets to improve the COSMO-SAC model for obtaining thermodynamic properties of electrolyte mixtures. With the liquid phase activities, we can identify electrolyte mixtures that meet desired phase equilibria conditions.

Thermodynamics↗
DOE OSTI2021

Ferroelectricity in boron-substituted aluminum nitride thin films

This manuscript demonstrates ferroelectricity in B-substituted AlN thin films and a complementary set of first-principles calculations to understand their structure-property relationships. Al 1–x B x N films are grown by dual-cathode reactive magnetron sputtering on (110)W/(001)Al 2 O 3 substrates at 300°C at compositions spanning x = 0 to x = 0.20. X-ray diffraction studies indicate a decrease in both the c and a lattice parameters with increasing B concentration, resulting in a decrease in unit cell volume and a constant c/a axial ratio of 1.60 over this composition range. Films with 0.02 ≤ x ≤ 0.15 display ferroelectric switching with remanent polarizations exceeding 125 μCcm –2 while maintaining band gap energies of > 5.2eV. The large band gap allows low frequency hysteresis measurement (200 Hz) with modest leakage contributions. At B concentrations of x > 0.15, c-axis orientation deteriorates and ferroelectric behavior is degraded. Furthermore, density-functional theory calculations corroborate the structural observations and provide predictions for the wurtzite u parameter, polarization reversal magnitudes, and composition-dependent coercive fields.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Catalytic decomposition of methane into hydrogen and high-value carbons: combined experimental and DFT computational study

Thermocatalytic decomposition (TCD) of methane can produce hydrogen and valuable nanocarbon co-products with low to near-zero CO2 emission. In this study, a series of Pd promoted Ni catalysts, prepared with various Ni/Pd ratios on a CNT support, were evaluated for methane TCD performance. Characterization and calculations using density functional theory (DFT) were carried out to elucidate the activity–structure relationship and growth mechanism of carbon nanomaterials. It was found that the methane conversion and stability of the catalysts were highly dependent on the Ni/Pd ratio and reaction temperature. DFT calculations revealed that the diffusion of carbon in the metal sublayer required for CNT growth was more favorable in the Ni–Pd alloy lattice suggesting that the buildup of carbon in the metal alloy sublayer facilitated the formation of CNTs and CNFs. A cyclic reaction–regeneration process for self-sustained TCD was experimentally demonstrated. In each cycle, a portion of the separated CNT product was used to re-synthesize the Ni–Pd/CNT catalyst for use in the next reaction cycle. After five cycles of operation, the CH4 conversion, morphology and crystallinity of the carbon product remained unchanged.

Wang, I-Wen↗
DOE OSTI2025

Prediction of Solute Segregation at Metal/Oxide Interfaces Using Machine Learning Approaches

The atomic structure and chemistry at metal/oxide interfaces play a crucial role in determining their properties. However, studying semi-coherent metal/oxide interfaces that include misfit dislocations through density functional theory (DFT) is often computationally expensive due to the large number of atoms involved, ranging from hundreds to thousands. In this study, we explore solute segregation behavior at the Fe/Y 2 O 3 interface—an important model interface for cladding applications in nuclear fission reactors—by combining DFT calculations with a machine learning (ML) approach. ML models are trained using DFT-calculated segregation energies (𝐸 𝑆𝑒𝑔 ) to identify the key chemical and geometric factors influencing solute segregation at metal/oxide interfaces, revealing the competition between these features in determining 𝐸 𝑆𝑒𝑔 . Moreover, the segregation behavior at a specific Fe/Y 2 O 3 interface is predicted with high accuracy using ML models trained on data from this interface. Furthermore, it is found that the ML models could also predict solute segregation at a different Fe/Y 2 O 3 interface with a new orientation relationship (OR), at a computational cost of less than 1/45 of that required for similar DFT calculations.

36 - MATERIALS SCIENCE↗
DOE OSTI2024

Electronic structure and lattice dynamics of 1⁢𝑇−VSe 2 : Origin of the three-dimensional charge density wave

To characterize in detail the charge density wave (CDW) transition of 1⁢𝑇−VSe 2 , its electronic structure and lattice dynamics are comprehensively studied by means of x-ray diffraction, muon spectroscopy, angle resolved photoemission (ARPES), diffuse and inelastic x-ray scattering, and state-of-the-art first-principles density functional theory calculations. Resonant elastic x-ray scattering does not show any resonant enhancement at either V or Se, indicating that the CDW peak at the 𝐾 edges describes a purely structural modulation of the electronic ordering. ARPES experiments identify (i) a pseudogap at 𝑇 > 𝑇⁢ CDW , which leads to a depletion of the density of states in the ML-M'L' plane at 𝑇 < 𝑇⁢ CDW , and (ii) anomalies in the electronic dispersion reflecting a sizable impact of phonons on it. A diffuse scattering precursor, characteristic of soft phonons, is observed at room temperature (RT) and leads to the full collapse of the low-energy phonon (𝜔 1 ) with propagation vector (0.25 0 −0.3) r.l.u. Here, we show that the frequency and linewidth of this mode are anisotropic in momentum space, reflecting the momentum dependence of the electron-phonon interaction (EPI), hence demonstrating that the origin of the CDW is, to a much larger extent, due to the momentum dependent EPI with a small contribution from nesting. The pressure dependence of the 𝜔 1 soft mode remains nearly constant up to 13 GPa at RT, with only a modest softening before the transition to the high-pressure monoclinic C2/m phase. The wide set of experimental data is well captured by our state-of-the art first-principles anharmonic calculations with the inclusion of van der Waals corrections in the exchange-correlation functional. The comprehensive description of the electronic and dynamical properties of VSe 2 reported here adds important pieces of information to the understanding of the electronic modulations in the family of transition-metal dichalcogenides.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

CO 2 -mediated oxidative dehydrogenation of propane enabled by Pt-based bimetallic catalysts

The greenhouse gas CO 2 is a promising soft oxidant for the oxidative dehydrogenation of light alkanes. However, the occurrence of side reactions including cracking, hydrogenolysis, and reforming results in lower olefin yields compared with direct dehydrogenation. Here, we report that Pt-M (M = Sn/In/Zn) bimetallic catalysts on non-redox-active silica support can break the equilibrium limit of direct propane dehydrogenation using CO 2 as a co-reactant to consume the hydrogen formed in propane dehydrogenation. Unlike the commonly postulated direct CO 2 -assisted dehydrogenation mechanism, we confirm that CO 2 -oxidative dehydrogenation of propane (ODHP) proceeds in two tandem steps on these bimetallic catalysts, i.e., propane dehydrogenation and reverse water-gas shift, with the latter being the rate-determining step. In situ X-ray absorption studies and density functional theory calculations suggest that the Pt m M n -MO x (e.g., Pt 3 Sn-SnO x ) interfaces are likely active sites.

03 NATURAL GAS↗