Free Energy Expansion Research Articles

Expansion of Landau theory with Magnetic Susceptibility and Heat Capacity

Abstract. Landau theory employs free energy to depict the work potential of molecules and utilizes order parameters to signify the degree of molecular organization. By applying a Taylor expansion of the free energy with respect to the order parameter in the vicinity of a phase transition, the theory elucidates how molecular arrangements can influence the system's energy. The universality of Landau theory stems from the order parameter within the framework of free energy expansion. The application of critical exponents exemplifies the universality of Landau theory, showcasing the properties of materials that share the same dimensionality and similar correlation lengths. This article aims to derive formulas and discuss practical applications of Landau theory, such as analyzing the heat capacity and magnetic susceptibility of materials across a range of temperatures.

Effect of arm-length polydispersity on the phase behaviour of V-shaped molecules

Phase behaviour of a melt of polydisperse V-shaped particles is considered within the Landau theory of phase transitions. It is assumed that particles are composed of two semi-flexible segments, one of which has a constant length, while the other has an arbitrary length distribution; the segments are joined at an external angle. The contributions up to the sixth order into the Landau–de Gennes free energy expansion in terms of the alignment tensor are presented in a diagrammatic form. The phase diagrams have been obtained for a special case characterised by rigid segments and Schulz–Zimm distribution for the length of the segment. It is shown that an increase in the polydispersity index of the segment leads to an increase of the isotropic–nematic transition temperature, substantial increase of the stability region of a uniform biaxial nematic ( N B ) phase, and a decrease in the area occupied by a uniform oblate nematic phase.

To the theory of phase transition of a binary solution into an inhomogeneous phase

ABSTRACT In the framework of the theoretical model of the phase transition of binary solutions into spatially inhomogeneous states proposed earlier by the authors [Poluektov Y, Soroka AA. Transition of a binary solution into an inhomogeneous phase. Phase Transition. 2022;95:267–280], which takes into account nonlinear effects, the role of the cubic in concentration term in the expansion of free energy was studied. It is shown that taking into account the cubic term contributes to the stabilization of a homogeneous state. The existence of two inhomogeneous phases in an isotropic medium, considered in [Poluektov Y, Soroka AA. Transition of a binary solution into an inhomogeneous phase. Phase Transition. 2022;95:267–280], proves to be possible only at half the concentration of the solution. The contribution of inhomogeneity effects to thermodynamic quantities is calculated. Phase transitions from a homogeneous state and between inhomogeneous phases are second-order phase transitions.

Structural mode coupling in perovskite oxides using hypothesis-driven active learning

Finding the ground-state structure with minimum energy is paramount to designing any material. In ABO3-type perovskite oxides with Pnma symmetry, the lowest energy phase is driven by an inherent trilinear coupling between the two primary order parameters such as rotation and tilt with antiferroelectric displacement of the A-site cations as established via hybrid improper ferroelectric mechanism. Conventionally, finding the relevant mode coupling driving phase transition requires performing first-principles calculations which is computationally time-consuming as well as expensive. It involves following an intuitive iterative hit and trial method of (a) adding two or multiple mode vectors, followed by (b) evaluating which combination would lead to the ground-state energy. In this study, we show how a hypothesis-driven active learning framework can identify suitable mode couplings within the Landau free energy expansion with minimal information on amplitudes of modes for a series of double perovskite oxides with A-site layered, columnar and rocksalt ordering. This scheme is expected to be applicable universally for understanding atomistic mechanisms derived from various structural mode couplings behind functionalities, for e.g. polarization, magnetization and metal–insulator transitions.

Pressure dependence of elastic constants using Landau free energy and a numerical modeling analysis of W-InN

ABSTRACT In this work, we study the pressure dependence of elastic constants of wurtzite indium nitride semiconductor based on Landau free energy expansion. Elastic constants are obtained as functions pressure up to second order. We have also evaluated the bulk modulus (B) and anisotropies of compressibility and shear for wurtzite indium nitride as functions of pressure. Agreement with previously obtained results based on first-principles calculations is found to be satisfactory.

Symmetry Breaking and Ascending in the Magnetic Kagome Metal FeGe

Spontaneous symmetry breaking—the phenomenon in which an infinitesimal perturbation can cause the system to break the underlying symmetry—is a cornerstone concept in the understanding of interacting solid-state systems. In a typical series of temperature-driven phase transitions, higher-temperature phases are more symmetric due to the stabilizing effect of entropy that becomes dominant as the temperature is increased. However, the opposite is rare but possible when there are multiple degrees of freedom in the system. Here, we present such an example of a symmetry-ascending phenomenon upon cooling in a magnetic kagome metal FeGe by utilizing neutron Larmor diffraction and Raman spectroscopy. FeGe has a kagome lattice structure with simple A-type antiferromagnetic order below Néel temperature TN≈400 K and a charge density wave (CDW) transition at TCDW≈110 K, followed by a spin-canting transition at around 60 K. In the paramagnetic state at 460 K, we confirm that the crystal structure is indeed a hexagonal kagome lattice. On cooling to around TN, the crystal structure changes from hexagonal to monoclinic with in-plane lattice distortions on the order of 10−4 and the associated splitting of the double-degenerate phonon mode of the pristine kagome lattice. Upon further cooling to TCDW, the kagome lattice shows a small negative thermal expansion, and the crystal structure gradually becomes more symmetric upon further cooling. A tendency of increasing the crystalline symmetry upon cooling is unusual; it originates from an extremely weak structural instability that coexists and competes with the CDW and magnetic orders. These observations are against the expectations for a simple model with a single order parameter and hence can only be explained by a Landau free energy expansion that takes into account multiple lattice, charge, and spin degrees of freedom. Thus, the determination of the crystalline lattice symmetry as well as the unusual spin-lattice coupling is a first step towards understanding the rich electronic and magnetic properties of the system, and it sheds new light on intertwined orders where the lattice degree of freedom is no longer dominant. Published by the American Physical Society 2024

Response of Mg2X (X = Si, Ge and Sn) compounds to extreme uniaxial compression: first-principles calculations

Abstract In this study, we perform first-principles calculations using density functional theory to examine the structural, electronic, thermodynamic, and thermoelectric properties of the Mg2X (X = Si, Ge and Sn) compounds under uniaxial compression within the generalized gradient and modified Becke–Johnson approximations. It is found that the band gap of Mg2Si, Mg2Ge and Mg2Sn decreases with applied uniaxial pressure and changes its direction from Γ-X to Γ-K. The results of phonon frequencies indicate that the studied compounds are dynamically stable at zero and higher uniaxial strains. Furthermore, the uniaxial compression and temperature dependence of the Gibbs free energy, heat capacity and thermal expansion coefficient are investigated in the frame of the quasi-harmonic approximation. The semiclassical-Boltzmann method is used to study the Seebeck coefficient, electrical conductivity, thermal conductivity and figure of merit ZT as a function of both temperature and uniaxial pressure. It is shown that the Seebeck coefficient decreases with increasing pressure whereas thermal conductivity increases, which leads to the lowering in the value of ZT and thus to a worse thermoelectric performance of the studied materials.

Holographic thermal observables and M2-branes

We use holography in conjunction with recent results from supersymmetric localization to compute certain thermal observables for 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs arising on the worldvolume of N M2-branes. We obtain results for the thermal free energy density on S1 × ℝ2, the Casimir energy on T2 × ℝ, and the three leading coefficients in the large temperature limit of the free energy on S1 × S2 valid to subleading order in the large N limit. As a byproduct of our holographic analysis we also present a conjecture for the structure of the large temperature expansion of the thermal free energy of general 3d CFTs on S1 × S2.

Structural, electronic, optical, mechanical and thermal properties of TM5Ge3C (TM= Cr, Mo and W): A theoretical prediction

Structural, electronic, optical, mechanical and thermal properties of hexagonal TM5Ge3C (TM = Cr, Mo, W) were explored using first-principles calculation. The results demonstrated the thermodynamic and dynamic stability of TM5Ge3C (TM = Cr, Mo and W). Based on the electronic band structure, density of states (DOS), and charge density mapping (CDM), the electronic ground state was determined. The metallic properties of TM5Ge3C (TM = Cr, Mo and W) were evaluated from the DOS and optical properties. According to Poisson's ratio, G/B, C13-C44 and C12-C66, it was determined that TM5Ge3C (TM = Cr, Mo and W) is ductile. The elastic anisotropy of TM5Ge3C (TM = Cr, Mo and W) was systematically analysed based on the elastic anisotropy index, three-dimensional spatial distributions and planar contours, which show that the elastic anisotropy of TM5Ge3C (TM = Cr, Mo and W) is in the order of W5Ge3C > Mo5Ge3C > Cr5Ge3C. In addition, the sound velocity, Debye temperature and thermal conductivity of TM5Ge3C (TM = Cr, Mo and W) were obtained from the elastic constants and moduli. Finally, the temperature dependence of the Helmholtz free energy, internal energy, entropy, specific heat and thermal expansion coefficient of TM5Ge3C (TM = Cr, Mo and W) were studied in detail. The thermal properties of TM5Ge3C (TM = Cr, Mo and W) were studied systematically, and it was found that the high operating temperature of these compounds makes them suitable for applications such as aero-engine turbine blades.

Impact of Sb degrees of freedom on the charge density wave phase diagram of the kagome metal CsV3Sb5

Elucidating the microscopic mechanisms responsible for the charge density wave (CDW) instability of the AV$_3$Sb$_5$ (A=Cs, K, Rb) family of kagome metals is critical for understanding their unique properties, including superconductivity. In these compounds, distinct CDW phases with wave-vectors at the $M$ and $L$ points are energetically favorable, opening the possibility of tuning the type of CDW order by appropriate external parameters. Here, we shed light on the CDW landscape of CsV$_3$Sb$_5$ via a combination of first-principles calculations and phenomenology, which consists of extracting the coefficients of the CDW Landau free-energy expansion from density functional theory. We find that while the main structural distortions of the kagome lattice in the staggered tri-hexagonal CDW phase are along the nearest-neighbor V-V bonds, distortions associated with the Sb ions play a defining role in the energy gain in this and all other CDW states. Moreover, the coupling between ionic displacements from different unit cells is small, thus explaining the existence of multiple CDW instabilities with different modulations along the c-axis. We also investigate how pressure and temperature impact the CDW phase of CsV$_3$Sb$_5$. Increasing pressure does not change the staggered tri-hexagonal CDW ground state, even though the $M$-point CDW instability disappears before the $L$-point one, a behavior that we attribute to the large nonlinear coupling between the order parameters. Upon changing the temperature, we find a narrow regime in which another transition can take place, toward a tri-hexagonal Star-of-David CDW phase. We discuss the implications of our results by comparing them with experiments on this compound.

Gaussian fluctuations and free energy expansion for Coulomb gases at any temperature

On obtient des résultats de concentration pour les fluctuations du gaz de Coulomb en toute dimension et dans un large régime de température, incluant des températures très petites et très grandes qui peuvent dépendre du nombre de points. On obtient un Théorème Central Limite (TCL) complet pour les fluctuations des statistiques linéaires en dimension 2, valable pour la première fois jusqu’aux échelles microscopiques et pour des températures pouvant tentre vers 0 ou l’infini quand le nombre de points diverge. On montre qu’un TCL semblable peut aussi être obtenu en toute dimension sous une condition d’absence de transition de phase, dès lors qu’on peut obtenir une erreur suffisamment petite dans le développement de l’énergie libre – un tel développement est prouvé en toute dimension, mais l’erreur n’est pas suffisamment petite pour conclure. Ces TCL sont valables dès que la fonction-test vit à une échelle supérieure à l’échelle minimale ρβ dépendant de la température, introduite dans le précédent travail (Ann. Probab. 49 (2021) 46–121).

Proximity of superconducting LaCoSi to a ferromagnetic quantum critical point.

The physical characteristics of the 4 K superconductor LaCoSi are studied using first-principles density functional theory inside the local density approximation (LDA) framework. We discover that LDA predicts a ferromagnetic ground state for LaCoSi, which is in disagreement with the experiments. Even though LDA rarely overestimates the local magnetic moment associated with the magnetic ion in itinerant systems, such occurrences highlight the significance of spin fluctuations in the system. In this view, the Ginzburg-Landau free energy expansion and its variation as a function of pressure are used to calculate the amplitude of the zero-point fluctuations. Based on our calculations, we contend that the superconductivity associated with LaCoSi is closely related to a ferromagnetic quantum critical point, making it an intriguing candidate for the category of quantum materials.

Effect of Disparity in Self Dispersion Interactions on Phase Behaviors of Molten A-b-B Diblock Copolymers.

Phase behaviors of molten A-b-B diblock copolymers with disparity in self dispersion interactions are revisited here. A free energy functional is obtained for the corresponding Gaussian copolymers under the influence of effective interactions originating in the localized excess equation of state. The Landau free energy expansion is then formulated as a series in powers of A and B density fluctuations up to 4th order. An alternative and equivalent Landau energy is also provided through the transformation of the order parameters to the fluctuations in block density difference and free volume fraction. The effective Flory χ is elicited from its quadratic term as the sum of the conventional enthalpic χH and the entropic χS that is related to energetic asymmetry mediated by copolymer bulk modulus. It is shown that the cubic term is balanced with Gaussian cubic vertex coefficients in corporation with energetics to yield a critical point at a composition rich in a component with stronger self interactions. The full phase diagrams with classical mesophases are given for the copolymers exhibiting ordering upon cooling and also for others revealing ordering reversely upon heating. These contrasting temperature responses, along with the skewness of phase boundaries, are discussed in relation to χH and χS. The pressure dependence of their ordering transitions is either barotropic or baroplastic; or anomalously exhibits anomalously both at different stages. These actions are all explained by the opposite responses of χH and χS to pressure.

The Thermal Stability of Asymmetric Separated Configurations inside Alloy Nanoparticles: Atomic-Scale Modeling of Pd-Ir Nanophase Diagrams.

Compared to alloy bulk phase diagrams, the experimental determination of phase diagrams for alloy nanoparticles (NPs), which are useful in various nanotechnological applications, involves significant technical difficulties, making theoretical modeling a feasible alternative. Yet, being quite challenging, modeling of separation nanophase diagrams is scarce in the literature. The task of predicting comprehensive nanophase diagrams for Pd-Ir face-centered cubic-based three cuboctahedra is facilitated in this study by combining the computationally efficient statistical-mechanical Free-energy Concentration Expansion Method, which includes short-range order (SRO) with coordination-dependent bond-energy variations as part of the input and with rotationally symmetric site grouping for extra efficiency. This nanosystem has been chosen mainly because of the very small atomic mismatch that simplifies the modeling, e.g., in the assessment of vibrational entropy contributions based in this work on fitting to the Pd-Ir experimental bulk critical temperature. This entropic effect, together with SRO, leads to significant destabilization of low-T Quasi-Janus (QJ) asymmetric configurations of the NP core, which transform to symmetric partially mixed nanophases. First-order and second-order intracore transitions are predicted for dilute and intermediate-range compositions, respectively. Caloric curves computed for the former case yield the NP-size dependent transition latent heat, and in the latter case critical temperatures exhibit a specific scaling behavior. The computed separation diagrams and intracore solubility diagrams reflect enhanced elemental mixing in smaller QJ nanophases. In addition to these diagrams, the revealed near-surface compositional variations are likely to be pertinent to the utilization of Pd-Ir NPs, e.g., in catalysis.

First-Principles Calculations of Elastic and Thermodynamic Properties for Multi-component Co-based Superalloys

First-principles calculations were performed to investigate the elastic and thermodynamic properties for multi-component Co-based superalloy systems and explored the effect of alloying on stabilizing the γ′ phase. First, the comparisons were carried out for the γ′ phase in Co3(Al,TM) (TM being transition metals) and Ni3Al systems between the present computational results using the EMTO-CPA method and other available DFT calculations as well as experimental data. The lattice parameters, elastic constants, and Debye temperatures are consistent with experimental results and other calculations. The predicted thermodynamic properties, e.g., the Gibbs free energy, excess entropy, and linear thermal expansion coefficient, agree well with CALPHAD results, experimental results, and other available first-principles calculations. A combination of EMTO-CPA method and Debye–Grüneisen model is utilized in this work to ensure that the alloying effect on the stability of the γ′ phase in a multi-component Co-based system is captured efficiently. This could open the path for designing novel multi-component Co-based alloys based on first-principles calculation. To demonstrate this, predictions for the properties of multicomponent systems were undertaken. Our results show that Ni aids in the stabilization of the (CoNi)3(Al, Mo, Nb) phase.Graphical

T − E phase diagrams and electrocaloric effect in PNN-PT solid solutions

Using the sixth-order free energy expansion of the Landay thermodynamic potential, T − E phase diagrams and the electrocaloric effect (ECE) in solid solutions (1-x)Pb(Ni1∕3Nb2∕3)O3-xPbTiO3 (PNN-PT) were studied under an electric field along the crystallographic directions [001], [101] and [111]. The composition of the samples, temperature range, as well as magnitude and direction of the electric field significantly affect the sequence of phase transitions, stability of various crystalline phases and electrocaloric properties. Direct measurements of the intensive ECE in the 0.8PNN-0.2PT compound were carried out by the method of adiabatic calorimetry at electric field strength E = 6 kV/cm. The large electrocaloric coefficient ΔT∕ΔE = 10 μK ⋅ cm/kV indicates that this compound and other PNN-PT solutions are promising for application as solid-state refrigerants.

Insights into Cation Ordering of Double Perovskite Oxides from Machine Learning and Causal Relations

This work investigates the origins of cation ordering of double perovskites using first-principles theory computations combined with machine learning (ML) and causal relations. We have considered various oxidation states of A, A', B, and B' from the family of transition metal ions to construct a diverse compositional space. A conventional framework employing traditional ML classification algorithms such as Random Forest (RF) coupled with appropriate features including geometry-driven and key structural modes leads to highly accurate prediction (~98%) of A-site cation ordering. We have evaluated the accuracy of ML models by entailing analyses of decision paths, assignments of probabilistic confidence bound, and finally introducing a direct non-Gaussian acyclic structural equation model to investigate causality. Our study suggests that the structural modes are the most important features for classifying layered, columnar and rock-salt ordering. For clear layered ordering, the charge difference between the A and A' is the most important feature which in turn depends on the B, B' charge separation. Based on the outputs from ML models, we have designed functional forms with these features to derive energy differences forming clear layered ordering. The trilinear coupling between tilt, rotation, and A-site antiferroelectric displacement in Landau free-energy expansion becomes the necessary condition behind the formation of A-site cation ordering.

On the stacking fault forming probability and stacking fault energy in carbon-doped 17 at% Mn steels via transmission electron microscopy and atom probe tomography

Assessing the stacking fault forming probability (Psf) and stacking fault energy (SFE) in medium- or high-Mn base structural materials can anticipate and elucidate the microstructural evolution before and after deformation. Typically, these two parameters have been determined from theoretical calculations and empirical results. However, the estimation of SFE values in Fe–Mn–C ternary systems is a longstanding debate due to the complicated nature of carbon: that is, whether the carbon doping indeed plays an important role in the formation of stacking faults; and how the amount of carbon atoms exist at grain boundaries or at internal grains with respect to the nominal carbon doping contents. Herein, the use of atom probe tomography and transmission electron microscopy (TEM) unveils the influence of carbon-doping contents on the structural properties of dual-phase Fe–17Mn–xC (x = 0–1.56 at%) steels, such as carbon segregation free energy at grain boundaries, carbon concentration in grain interior, interplanar d-spacings, and mean width of intrinsic stacking faults, which are essential for SFE estimation. We next determined the Psf values by two different methods, viz., reciprocal-space electron diffraction measurements and stacking fault width measurements in real-space TEM images. Then, SFEs in the Fe–17Mn–xC systems were calculated on the basis of the generally-known SFE equations. We found that the high amount of carbon doping gives rise to the increased SFE from 8.6 to 13.5 mJ/m2 with non-linear variation. This SFE trend varies inversely with the mean width of localized stacking faults, which pass through both other stacking faults and pre-existing ε-martensite plates without much difficulty at their intersecting zones. The high amount of carbon doping acts twofold, through increasing the segregation free energy (due to more carbon at grain boundaries) and large lattice expansion (due to increased soluble carbon at internal grains). The experimental data obtained here strengthens the composition-dependent SFE maps for predicting the deformation structure and mechanical response of other carbon-doped high-Mn alloy compositions.

Evidence for Realignment of the Charge Density Wave State in ErTe3 and TmTe3 under Uniaxial Stress via Elastocaloric and Elastoresistivity Measurements

We report the evolution of the charge density wave (CDW) states in the quasi-two-dimensional rare-earth tritellurides (RTe3 for R=Er, Tm) under the influence of in-plane uniaxial stress. Measurements of the elastocaloric effect, resistivity, and elastoresistivity allow us to demonstrate the importance of in-plane antisymmetric strain on the CDW and to establish a phase diagram. We show that modest tensile stress parallel to the in-plane a axis can reversibly switch the direction of the ordering wave vector between the two in-plane directions, and present a free-energy expansion which reproduces the general structure of the observed phenomena. This work opens a new avenue in the study of RTe3 in its own right, and more generally establishes RTe3 as a promising model system for the study of strain-CDW interactions in a quasi-two-dimensional square lattice.5 MoreReceived 7 January 2022Revised 13 March 2022Accepted 15 April 2022DOI:https://doi.org/10.1103/PhysRevX.12.021046Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCharge density wavesStrainTechniquesCalorimetryResistivity measurementsCondensed Matter, Materials & Applied Physics

Formation of diblock copolymer nanoparticles: Theoretical aspects

We explore the shape and internal structure of diblock copolymer (di-BCP) nanoparticles (NPs) by using the Ginzburg-Landau free-energy expansion and the dynamic self-consistent field theory (DSCFT). The self-assembly of di-BCP lamellae confined in emulsion droplets can form either ellipsoidal or onion-like NPs. The corresponding inner structure is a lamellar phase that is either perpendicular to the ellipsoid long axis (L⊥) or forms a multi-layer concentric shell (C∥), respectively. We focus on the effects of the interaction parameters between the A and B monomers χ, and polymer/solvent χPS, as well as the NP size on the nanoparticle shape and internal morphology. The aspect ratio (lAR), defined as the length ratio between the long and short axes, is used to characterize the overall NP shape. Our results show that when the solvent is neutral towards the two blocks, the ratio lAR increases as χ increases, indicating that the NP becomes more elongated. Likewise, decreasing χPS or increasing the NP size also results in a more elongated NP. Furthermore, as the solvent preference towards one of the A or B blocks increases, the NP undergoes a shape transition from a striped ellipsoid (L⊥) to onion-like sphere (C∥). Our results are in good agreement with previous experiments, and some of the predictions could be tested in future experiments.