Phase Transition Point Research Articles

Frequency stabilization of adiabatic temperature change in Fe50Rh50 alloy in a cyclic magnetic field of 1.2 T

We present the results of direct measurements of the adiabatic temperature change (ΔTad) for the Fe50Rh50 alloy in a cyclic magnetic field (CMF) of 1.2 T. It is shown that increasing the frequency of the CMF from 1 to 30 Hz is accompanied by a shift of the position of temperature dependence ΔTad(T) maximum, Tmax, toward low temperatures. With an increase in the CMF frequency from 1 to 5 Hz, the ΔTmax value decreases by ∼12%. A further increase in frequency leads to stabilization of the effect. In the vicinity of the antiferromagnetic-ferromagnetic phase transition point TC = 370 K, ΔTad exhibits unconventional frequency behavior: while at T well above TC, the value of ΔTad monotonously decreases as frequency increases, at T = 370.4 K; an interval of frequency-independent ΔTad up to 10 Hz is observed, and at 368 K < T < TC, the maximum of ΔTad(f) dependence is found in the interval 1 < f < 10 Hz. Such behavior in the future can be applied in magnetic cooling technology due to large values of ΔTad and the frequency stability of the effect in alternating fields. The specific cooling power reaches giant values of ∼22 W/g at 20 Hz, which is comparable to the values under the same conditions for Gd −21.6 W/g. The unconventional behavior of ΔTad in the CMF is discussed in the context of the role of secondary phase localization, which leads to an enhanced internal local magnetic field and dynamic effects of ΔTad.

Particle shape descriptors as a tool for classification of grain breakage

This study explores particle breakage of a calcareous sand, utilising dynamic image analysis and direct shear tests at varying normal stress levels to analyse the sand crushing mechanism. Tests were arrested at four phase transition points encountered during shear. Particle breakage was traced along a proposed scale based on the literature. Distinct behaviours in shape descriptors for particles of different diameters were observed under changing stress conditions. Agglomerated particle-level behaviour indicates that applied stress levels lead to different soil responses; at low stress, grains roll or slide, whereas at high stresses, grain movement is restricted. Particle breakage was evident even at minimal stress, with increasing grain size reduction as shear strain and stress levels heightened. This study introduces a parameter loading intensity (LI), amalgamating the effects of force chains and loading duration to model grain crushing. Relationships among the shape-angularity group indicator (SAGI), convexity (Cx) and sphericity (S) were also established, enabling calculation of these descriptors, after any loading intensity is applied, based on the initial values. Finally, a methodology for forecasting changes in S and Cx, under any given LI for materials with a known SAGI is presented.

Indirect to direct band gap engineering of cubic bromide perovskite AlMgBr3 under pressure: First-principles calculations for enhanced optoelectronic applications

Through first principles, the structural, electronic, optical, mechanical, and thermodynamic properties of AlMgBr3 at different pressures are investigated for the first time. At 0 GPa, cubic AlMgBr3 is an indirect band gap insulator (Eg = 3.958 eV). Interestingly, its band gap type changes from an indirect band gap to a direct band gap at 5 GPa, making it a more favorable material for optoelectronic device applications. In addition, as the pressure increases, the conduction band electrons move downward, resulting in a reduction of the band gap, and the band gap transition from the ultraviolet to visible region. The optical properties show that AlMgBr3 has a high dielectric constant, refractive index, and conductivity, which is favorable for its application in solar cells, waveguides and other optoelectronic devices. The phase transition point of AlMgBr3 was determined to be 39.6 GPa based on the mechanical stability conditions, and the mechanical properties revealed the ductility, anisotropy and machinability of AlMgBr3. By using the quasi-harmonic Debye model, the effect of temperature and pressure on the bulk modulus, coefficient of thermal expansion, isobaric heat capacity, Debye temperature, and entropy were also calculated. It is worth mentioning that the electronic, optical, mechanical and thermodynamic properties of AlMgBr3 have been significantly enhanced under pressure.

Polariton-polariton interaction induced by the hopping effect between two monolayers

Investigating the quantum many-body interaction in microcavity semiconductor systems is a significant objective in current research. Hence, in this study, Green’s function was employed to investigate the hopping effect between two monolayers denoted as L and R. The interaction between excitons in L monolayer and R monolayer, characterized by spin-like properties, leads to the generation of Josephson currents and emergence of two quasiparticle branches, known as LP+ and LP− polaritons (i.e., two low polariton states). And compared with other papers, such as reference 5, the research shows that presence of hopping interaction and many-body interaction between monolayers causes significant enhancement phenomenon of residue Z, which will further reduce the effective mass. This is more conducive to Bose–Einstein condensates (BEC). Furthermore, we investigated the Parity-Time symmetry (PT symmetry) of the system. The PT broken phase transition point will appear when the condition of PT symmetry is satisfied.

Relativistic hydrodynamics with phase transition

Assessing the applicability of hydrodynamic expansions close to phase transition points is crucial from either theoretical or phenomenological points of view. We explore this within the gauge/gravity duality, using the Einstein–Klein–Gordon model, a bottom-up string theory construction. This model incorporates a parameter, B4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_4$$\\end{document}, that simulates different types of phase transitions in the strongly coupled field theory existing at the boundary. We thoroughly examine the thermodynamics and dynamics of time-dependent, linearized perturbations in the spin-2, spin-1, and spin-0 sectors. Our findings suggest that ‘hydrodynamic series breakdown near transition points” is valid exclusively for second-order phase transitions, not for crossovers or first-order phase transitions. Additionally, we observe that the high-temperature and low-temperature limits of the radius of convergence for the hydrodynamic series (qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_c$$\\end{document}) are equal. We also discover that the relationship (Max|qc2|)spin-2<(Max|qc2|)spin-0<(Max|qc2|)spin-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ ext {Max}|q^2_c|)_{\ ext {spin-2}}< (\ ext {Max}|q^2_c|)_{\ ext {spin-0}} < (\ ext {Max}|q^2_c|)_{\ ext {spin-1}}$$\\end{document} is consistent for different spin sectors, regardless of the phase transition type. At the chaos point, we observe the emergence of pole-skipping behavior for both gravity and scalar perturbations at ωn=-2πTni\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _n = -2\\pi T n i$$\\end{document}. Lastly, comparing the chaos momentum with qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_c$$\\end{document}, we find that qps2<qc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^2_{ps} < q^2_c$$\\end{document}, except for extremely high temperatures.

Spinodal slowing down and scaling in a holographic model

The dynamics of first-order phase transitions in strongly coupled systems are relevant in a variety of systems, from heavy ion collisions to the early universe. Holographic theories can be used to model these systems, with fluctuations usually suppressed. In this case the system can come close to a spinodal point where theory and experiments indicate that the behaviour should be similar to a critical point of a second-order phase transition. We study this question using a simple holographic model and confirm that there is critical slowing down and scaling behaviour close to the spinodal point, with precise quantitative estimates. In addition, we determine the start of the scaling regime for the breakdown of quasistatic evolution when the temperature of a thermal bath is slowly decreased across the transition. We also extend the analysis to the dynamics of second-order phase transitions and strong crossovers.

Electrically charged black holes in gravity with a background Kalb–Ramond field

We derive the exact solutions for electrically charged black holes both in the absence and presence of a cosmological constant in the gravitational theory with Lorentz violation induced by a background Kalb–Ramond (KR) field. The corresponding thermodynamic properties are investigated. It is found that the standard first law of thermodynamics and the Smarr formula remain valid for the charged KR black holes. Nevertheless, the Lorentz-violating effect influences their ranges of local thermodynamic stability and the first- and second-order phase transition points. Furthermore, to examine the impact of Lorentz violation on the motion of test particles in the spacetime, we analyze the shadow and the innermost stable circular orbit (ISCO) of these black holes. Our results reveal that both the shadow and ISCO radii exhibit a high sensitivity to the Lorentz-violating parameter ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, with a decrease observed as ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} increases.

Spontaneous piezoeletric effect as evidence of ferroelastoelectricity in guanidine zinc sulfate crystal

The crystal of [C(NH2)3]2Zn(SO4)2 guanidine zinc sulfate was grown and its structure, dilatometric, dielectric, elastic and piezoelectric properties were studied in a broad temperature range, covering the phase transition point. The crystal undergoes a continuous phase transition at 178 K from the room temperature tetragonal phase with a space group I4¯2d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{I}\\overline{4}2\ ext{d }$$\\end{document} to the tetragonal low temperature phase with a space group I4¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{I}\\overline{4 }$$\\end{document}. The structural X-ray studies allowed proposing molecular mechanism associated with the rearrangement in the configuration of N–H⋯O hydrogen bonds and reorientation of guanidine cations in the structure, leading to a change in the symmetry of the low temperature phase. Results of thermal expansion and dielectric studies are typical of a structural nonferroelectric continuous transition. Also measurement of piezoelectric and elastic properties revealed small anomalies at 178 K. Below the transition temperature, a new piezoelectric component, that is a ferroelastoelectric macroscopic order parameter, was found.

Thermodynamic stability versus chaos bound violation in D-dimensional RN black holes: Angular momentum effects and phase transitions

We compute the Lyapunov exponents for test particles orbiting in unstable circular trajectories around D-dimensional Reissner-Nordström (RN) black holes, scrutinizing instances of the chaos bound violation. Notably, we discover that an increase in particle angular momentum exacerbates the breach of the chaos bound. Our research centrally investigates the correlation between black hole thermodynamic phase transitions and the breaking of the chaos limit. Findings suggest that the chaos bound can only be transgressed within thermodynamically stable phases of black holes. Specifically, in the four-dimensional scenario, the critical point of the thermodynamic phase transition aligns with the threshold condition that delineates the onset of chaos bound violation. These outcomes underscore a deep-rooted link between the thermodynamic stability of black holes and the constraints imposed by the chaos bound on particle dynamics.

In-depth investigation of phase transition phenomena in network models derived from lattice models

Lattice models exhibit significant potential in investigating phase transitions, yet they encounter numerous computational challenges. To address these issues, this study introduces a Monte Carlo-based approach that transforms lattice models into a network model with intricate inter-node correlations. This framework enables a profound analysis of Ising, JQ, and XY models. By decomposing the network into a maximum entropy component and a conservative component, under the constraint of detailed balance, this work derives an estimation formula for the temperature-dependent magnetic induction in Ising models. Notably, the critical exponent β in the Ising model aligns well with the established results, and the predicted phase transition point in the three-dimensional Ising model exhibits a mere 0.7% deviation from numerical simulations.

Mechanical properties and corrosion resistance improvement of TiAlSnMo alloys via Mo addition

In this paper, Ti-5Al-2.5Sn-xMo (TASxM, x = 0,5,10,15 wt%) alloys were designed and prepared. Phase compositon and microstructure analysis results indicate that the addition of Mo reduces the phase transition point of TASxM alloys, stabilizes more β phase to room temperature, and refines the α grains. The strength of the TASxM alloys exhibits a positive correlation with the increasing Mo content within 10 wt%, and the highest ultimate tensile strength (UTS) can be obtained in TAS10M (1260 MPa), which can be primarily attributed to the synergistic effects of phase transition, solid solution strengthening, and fine grain strengthening. Interestingly, the UTS of TAS15M unexpectedly decreases to 724 MPa, which is strongly influenced by the altered slip system numbers resulting from the β/α phase ratio. Even more intriguing is the observation of stress-induced martensitic transformation in the deformed TAS15M alloy, often resulting in a substantial influence on elongation. The corrosion resistance of the TASxM alloy has also been investigated through electrochemical experiments. The findings reveal that the corrosion resistance of the TASxM alloy is enhanced with the elevation of Mo content, peaking at TAS15M. This improved resistance is attributed to the augmented β phase content and the formation of protective MoO2 and MoO3 oxide films on the surface of alloy.

Revealing failure law of single-track traffic-loaded concrete airport pavement from a thermodynamic perspective

ABSTRACT With the growth of global air traffic and the popularity of large aircraft, the design and life prediction of airport pavements have received increasing attention from scholars. However, concrete airport pavement evaluations based on cracking patterns are often uncertain. A thermodynamic-based failure analysis of the FAA Phase 4 Construction Cycle 8 (CC8) strength and fatigue traffic test is conducted in this work to investigate the failure laws of airport concrete pavements. First, this work equates concrete pavement failure to a phase transition. It converts strain data from full-scale traffic tests into state variables. Based on the Wilson theory definition of phase transition, network-free renormalisation is applied to reveal the concrete slab stressing state evolution and phase transition features. The clustering algorithm that utilises the attractor effect of phase transition points determines characteristic points. In summary, this work applies network-free renormalisation and clustering algorithms to reveal robust concrete pavement failure points, which can provide a reference for airport pavement design, assessment, and failure warning. Furthermore, this work further validates the stressing state theory and network-free renormalisation under high-speed time-varying conditions.

Relaxation dynamics in the alternating XY chain following a quantum quench

We investigate the relaxation dynamics of the fermion two-point correlation function Cmn(t)=〈ψ(t)∣cm†cn∣ψ(t)〉 in the XY chain with alternating nearest-neighbor hopping interaction after a quench. We find that the deviation δ C mn (t) = C mn (t) − C mn (∞) decays with time following the power law behavior t −μ , where the exponent μ depends on whether the quench is to the commensurate phase (μ = 1) or incommensurate phase ( μ=12 ). This decay of δ C mn (t) arises from the transient behavior of the double-excited quasiparticle occupations and the transitions between different excitation spectra. Furthermore, we find that the steady value C mn (∞) only involves the average fermion occupation numbers (i.e. the average excited single particle) over the time-evolved state, which are different from the ground state expectation values. We also observe nonanalytic singularities in the steady value C mn (∞) for the quench to the critical points of the quantum phase transitions (QPTs), suggesting its potential use as a signature of QPTs.

Topological AdS black holes surrounded by Chaplygin dark fluid: From stability to geometrothermodynamic analysis

Implementing the concept of Dark Fluid with a Chaplygin-like equation of state within General Relativity, we construct a new higher-dimensional, static, and spherically symmetric anti-de Sitter (AdS) black hole solution. Energy conditions are explored alongside curvature singularity tools. The inspection at the level of the phase structure and P−v critical behavior is carried out in the context of the extended phase space, where the cosmological constant appears as pressure. Our findings disclose non-trivial similarities between the small/large phase transition of AdS black holes surrounded by Chaplygin dark fluid and van der Waals systems’ liquid/gas phase transition. This analysis offers insights into the physical interpretation of the P−v diagram and identifies critical exponents that reveal the scaling behavior of thermodynamic quantities close to criticality in a universal manner. We finally deepen our understanding of the thermodynamic properties and microstructure of AdS black holes by leveraging the geometrothermodynamic formalism. Specifically, we employ tools, including Weinhold, Ruppeiner, Hendi–Panahiyan–Eslam–Momennia (HPEM), and Quevedo classes I and II. We show that each class of metrics predicts either the physical limitation point and/or the phase-transition critical points, with HPEM and Quevedo formulations providing richer information about the phase transitions. Altogether, this study contributes to advancing our knowledge of the role of Chaplygin gas in General Relativity and thoroughly examining the thermodynamic phase structure of high-dimensional AdS black holes under extreme conditions.

Ultrasensitive Ratiometric Fluorescent Nanothermometer with Reverse Signal Changes for Intracellular Temperature Mapping.

Sensing temperature at the subcellular level is pivotal for gaining essential thermal insights into diverse biological processes. However, achieving sensitive and accurate sensing of the intracellular temperature remains a challenge. Herein, we develop a ratiometric organic fluorescent nanothermometer with reverse signal changes for the ultrasensitive mapping of intracellular temperature. The nanothermometer is fabricated from a binary mixture of saturated fatty acids with a noneutectic composition, a red-emissive aggregation-caused quenching luminogen, and a green-emissive aggregation-induced emission luminogen using a modified nanoprecipitation method. Different from the eutectic mixture with a single phase-transition point, the noneutectic mixture possesses two solid-liquid phase transitions, which not only allows for reversible regulation of the aggregation states of the encapsulated luminogens but also effectively broadens the temperature sensing range (25-48 °C) across the physiological temperature range. Remarkably, the nanothermometer exhibits reverse and sensitive signal changes, demonstrating maximum relative thermal sensitivities of up to 63.66% °C-1 in aqueous systems and 44.01% °C-1 in the intracellular environment, respectively. Taking advantage of these outstanding thermometric performances, the nanothermometer is further employed to intracellularly monitor minute temperature variations upon chemical stimulation. This study provides a powerful tool for the exploration of dynamic cellular thermal activities, holding great promise in unveiling intricate physiological processes.

Colossal Barocaloric Effect near Ambient Temperature in 1-Dodecanol under a Low Pressure.

Solid-state refrigeration based on the barocaloric effect is an effective alternative to traditional vapor compression refrigeration. Here 1-dodecanol has been studied due to its large latent heat at a solid-liquid phase transition point around room temperature. The transition temperature will vary with the applied hydrostatic pressure, exhibiting with a sensitivity of 0.14 K MPa-1, which indicates its potential for refrigeration. A pressure of 40 MPa can result in a large isothermal entropy change of 520 J kg-1 K-1 (equivalent to that obtained in vapor compression refrigeration) at 297 K. A large adiabatic temperature change of >20 K in 1-dodecanol was acquired by direct measurement. A wide temperature window of ∼50 K (288-337 K) can be obtained in 1-dodecanol, which demonstrates broad application prospects. These discoveries offer promising prospects for barocaloric cooling and high-efficiency refrigeration technologies relying on solid-liquid phase transitions.

Comparative Study of the Mechanical, Electronic Structure, and Optical Properties of Cubic Lithium‐Based Perovskite LiMgX3 (X = Cl, Br, I) under Pressure Effects: First‐Principles Calculations

The mechanical, electronic structure, and optical properties of lithium‐based perovskite LiMgX3 (X = Cl, Br, I) are investigated for the first time at 0–20 GPa using density‐functional theory. The Born stability criteria reveal that the phase transition points of LiMgCl3, LiMgBr3, and LiMgI3 are 20.7, 20.9, and 23.4 GPa, respectively. At 0 GPa, studies on the electronic properties using the Heyd‐Scuseria‐Ernzerhof (HSE06) functional show that LiMgCl3 and LiMgBr3 are indirect bandgap insulators with values of 5.336 and 4.113 eV, whereas LiMgI3 is an indirect bandgap semiconductor with a value of 2.055 eV. In addition, the bandgap calculated using both the PBEsol and HSE06 functionals decreases with increasing pressure, and the bandgap trends with pressure are consistent. Both functionals are also used to study the optical properties of LiMgX3 compounds, which show that they have potential for use in vacuum ultraviolet and photovoltaic applications. The mechanical and optical characteristics of the materials are significantly enhanced under pressure.

ZEROS OF PARTITION FUNCTION FOR Z5-SYMMETRIC MODEL ON SQUARE LATTICE-PARTICULAR CASE

We study a statistical mechanics model of multiple-phase transitions called the ZQ-symmetric model on the square lattice for five possible spin directions (Q = 5). This study aims to investigate the existence of phase transition(s) and discuss the distribution of partition function zeros on the complex-temperature plane. Some cases of energy list χ are considered where they are written in an arbitrary arrangement of integer numbers (usually in decreasing value due to the angle of separation). The partition functions are computed using a transfer matrix approach, and their zeros are found numerically using the NewtonRaphson method. A Monte-Carlo Metropolis simulation is then performed to study the critical point of phase transition. The results are compared to the zeros distribution of the partition functions for the considered cases. Studying this generalised model is vital to mimic the system in the real world and to help experimental works use a more sustainable approach.

Dynamical analysis of the low-temperature phase diagram of a cuprate superconductor

For a cuprate superconductor, specifically the yttrium barium copper oxide system, we analyze the low-temperature phase diagram consisting of the superconducting, pseudogap, and normal phases by using a phonon-mediated attraction and Coulomb repulsion mechanism. Through this analysis, we show this mechanism in the cuprate is quite important for its key dynamics. For the pairing mechanism, a variational method is developed and applied, and a generalized mean field theory is used to calculate the condensate energy and the energy gap of the quasiparticle. We obtain a self-consistent system of equations for these quantities. The superconducting and pseudogap phases, which involve pair-condensates, are assumed to be equilibrium systems of the condensate and normal subphases. The condition for the phase transition point is given by the equal total electronic energy gains. Assuming that the difference in the pairing range of the pair condensate is small between just above and below the transition temperature, computations are performed to solve the self-consistent system of equations. We find solutions compatible with experimental data. Plausible values of the energy gap in the superconducting phase and in the pseudogap phase are obtained. The present analysis shows that the phonon-mediated attraction and Coulomb repulsion mechanism itself is compatible with the experimental data. The importance of the screening effect through one-phonon exchange in the pairing mechanism and of the structure of the valence band is discussed.

Noise-induced phase transition in the Vicsek model through eigen microstate methodology

Abstract This paper presents a comprehensive framework for analyzing phase transitions in collective models like the Vicsek model under various noise types. The Vicsek model, focuses on understanding the collective behaviors of social animals, is known for its discontinuous phase transitions under vector noise. However, its behavior under scalar noise remains less conclusive. Renowned for its efficacy in the analysis of complex systems under both equilibrium and non-equilibrium state, the eigen microstate method is employed here for a quantitative examination of the phase transitions in the Vicsek model under both vector and scalar noise. The study finds that the Vicsek model exhibits discontinuous phase transitions regardless of noise type. Furthermore, the dichotomy method is utilized to identify the critical points for these phase transitions. A significant finding is the observed increase in the critical point for discontinuous phase transitions with the escalation of population density.