Weiss Molecular Field Theory Research Articles

New evaluation of the thermodynamics stability for bcc-Fe

The thermodynamic properties for bcc-Fe were predicted by combination of the first-principles calculations, the quasiharmonic approximation, the CALPHAD method and the Weiss molecular field theory. The hybrid method considers the effects of the lattice vibration, electron, intrinsic magnetism and external magnetic fields on the thermodynamic properties at finite temperature. Combined with experimental data, the calculated heat capacity without external magnetic fields was used to verify the validity of the hybrid method. Close to the Fermi level the high electronic density of states leads to a significant electronic contribution to free energy. Near the Curie temperature lattice vibrations dominant the Gibbs free energy. The order of the other three excitation contributions to Gibbs free energy from high to low is: intrinsic magnetism > electron > external magnetic fields. The investigation suggests that all the excitation contributions to Gibbs free energy are not negligible which provides a correct direction for tuning the thermodynamic properties for Fe-based alloy.

Insights into the assessment of the magnetic-field-induced precipitation behavior of alloy carbides M7C3 in steels

A combination of first-principles calculations, Weiss molecular field theory and density functional perturbation theory is combined to systematically evaluate the magnetic-field-induced thermodynamic properties of the alloy carbide M7C3 in steel. Special emphasis is focused on the self-magnetic and the external magnetic-field-induced magnetic contribution determined by extending the magnetic heat capacity. The self-magnetism origins from the 3d orbit contribution of the Fe atoms which constituted the carbon-containing trigonal prism. The magnetic-field-induced Curie temperature which triggers the boundary of the long and short-range magnetic order closely related to the Cr content and the magnetic field strengths. The higher external magnetic-field-induced Gibbs energy change provides a potential precipitation strategy in advance for the alloy carbides M7C3 which is well in agreement with the experimental results. Our insights further will extend the mechanistic understanding of magnetic-field-induced phase transformation in ferromagnetic Fe-based alloys, emphasizing the role of external field in minimizing energy and guiding new alloy design.

Thermodynamic mechanism of the magnetic-field-induced carbon content variation in bainitic ferrite during super bainite transformation

The experimental results demonstrate the decrease in carbon content of bainitic ferrite under a 12 T high magnetic field. A hybrid approach combining electron probe microanalyzer (EPMA), first-principles calculations and Weiss molecular-field theory was used to evaluate the relationship between the carbon content and magnetism in bainitic ferrite. The calculated results demonstrate that the origins of the magnetic structural characteristics are controlled by interactions amongst the various sites of iron atoms and Fe-C bond distances. The negative magnetic Gibbs free energy changes increased the transformation driving force between the ferrite (α) and austenite (γ) phases, leading to an increased paraequilibrium carbon concentration at the α/γ interface, leading to an increase in the carbon diffusion in the austenite near the interface. As a consequence, the carbon atoms in bainitic ferrite decrease which is consistent with the experimental results.

Thermodynamic mechanism of the high magnetic-field-induced morphology of ferrite

The stability of the ferrite morphology under a high magnetic field has been investigated according to the combination of the extended Weiss molecular field theory and the principle of minimum energy. The results showed that the ferromagnetic phase region decreases significantly with the increase of total angular momentum, whereas the paramagnetic phase region has the opposite tendency. The magnetic-field-induced morphology is noted to be closely related to the angular momentum, magnetic moment and interfacial energy. Based on the principle of minimum energy, the magnetic-field-induced aspect ratio (2.48:1) of the ferrite grains is in good agreement with the previous experimental data.

Thermodynamic Analysis for the Magnetic-Field-Induced Precipitation Behaviours in Steels

Alloy carbide M23C6 plays a significant role in the creep strength of reduced activation steels. Experiments have proven that a magnetic field accelerates the precipitation of M23C6 at intermediate temperature. A scheme that combines first-principle calculations, Weiss molecular field theory and equilibrium software MTDATA is proposed to investigate the thermodynamic features of magnetic-field-induced precipitation. The calculated results reveal that the origin of the magnetic moment is the NaCl-like crystal structure. The magnetic field enhances the exchange coupling and stabilizes the ferromagnetic phase region. The external field influences the Curie temperature, thereby changing the magnitude and position of the maximum magnetic heat capacity, magnetic entropy and enthalpy. The strong magnetic field improves the stability of M23C6, and the theoretical results agree well with the previous experiment. The study provides a theoretical basis for the magnetic-field-induced precipitation behaviours in steels.

Microscopic origin of electric-field-induced modulation of Curie temperature in cobalt

The Curie temperature TC is one of the most fundamental physical properties of ferromagnetic materials and can be described by the Weiss molecular field theory with the exchange interaction of neighboring atoms. Here, we demonstrate the electrical control of exchange coupling in cobalt films through direct magnetization measurements. We find that the reduction in magnetization with temperature, which is caused by thermal spin wave excitation and scales with Bloch’s law, clearly depends on the applied electric field. Furthermore, we confirm that the correlation between the electric-field-induced modulation of TC and that of exchange coupling follows the Weiss molecular field theory.

Refrigeration Performance and Entropy Generation Analysis for Reciprocating Magnetic Refrigerator with Gd Plates.

In the current work, a novel 2D numerical model of stationary grids was developed for reciprocating magnetic refrigerators, with Gd plates, in which the magneto-caloric properties, derived from the Weiss molecular field theory, were adopted for the built-in energy source of the magneto-caloric effect. The numerical simulation was conducted under the conditions of different structural and operational parameters, and the effects of the relative fluid displacement (φ) on the specific refrigeration capacity (qref) and the Coefficient of Performance (COP) were obtained. Besides the variations of entropy, the generation rate and number were studied and the contours of the local entropy generation rate are presented for discussion. From the current work, it is found that with an increase in φ, both the qref and COP followed the convex variation trend, while the entropy generation number (Ns) varied concavely. As for the current cases, the maximal qref and COP were equal to 151.2 kW/m3 and 9.11, respectively, while the lowest Ns was the value of 2.4 × 10−4 K−1. However, the optimal φ for the largest qref and COP, and for the lowest Ns, were inconsistent, thus, some compromises need be made in the optimization of magnetic refrigerators.

Magnetic dipole interactions in crystals

The influence of magnetic dipole interactions (MDIs) on the magnetic properties of local-moment spin systems is investigated. The eigenvalues and eigenvectors of the MDI tensor are determined for a wide variety of spin lattices and magnetic wavevectors in the magnetically ordered state with collinear moment alignments. The predicted easy axes in the ordered state are compared with experimental results for antiferromagnetic (AFM) compounds. The theory is generalized to include noncollinear AFM structures and compared with experiments. Many properties caused or influenced by MDIs in the ordered and paramagnetic states are calculated within Weiss molecular field theory.

Magnetic-field-induced magnetism and thermal stability of carbides Fe6−xMoxC in molybdenum-containing steels

A hybrid method combining first-principles calculations and Weiss molecular field theory with thermodynamic data has been implemented to explore the origin of magnetic-field-induced precipitation behaviors for alloy carbides. The paramagnetic Mo atom disturbed the order of magnetic moment and resulted in a decrease in the Curie temperature for alloy carbide Fe6−xMoxC. The temperature dependence of magnetic moment and saturation magnetization of Fe atoms at different Wyckoff positions, as well as the saturation or induced magnetization of Fe6−xMoxC, decreased with increasing temperature. The higher Fe content and external magnetic field greatly increased the magnetization of alloy carbides. Two kinds of stella quadrangula lattices were employed to account for the total magnetism which was derived from the contribution of different Wyckoff sites of Fe atoms and Fe–C distances. The calculated total free energy taking into account magnetic field, temperature and composition was sufficient to provide quantitative agreement with experiment. The investigation of the effects of external field on the carbide precipitation behaviors led to a better understanding of the magnetic-field-induced phase transformation mechanism in heat resistant steels.

Helical antiferromagnetic ordering inLu1−xScxMnSi

Polycrystalline samples of Lu{1-x}Sc{x}MnSi (x= 0, 0.25, 0.50) are studied using powder x-ray diffraction (XRD), heat capacity Cp, magnetization M, magnetic susceptibility chi, and electrical resistivity rho measurements versus temperature T and magnetic field H. This system crystallizes in the primitive orthorhombic TiNiSi-type structure (space group Pnma) as previously reported. The rho(T) data indicate metallic behavior. The Cp(T), chi(T), and rho(T) measurements consistently indicate long-range antiferromagnetic (AF) transitions with AF ordering temperatures TN = 246, 215 and 188 K for x = 0, 0.025 and 0.50, respectively. A second transition is observed at somewhat lower T for each sample from the chi(T) and rho(T) measurements, which we speculate are due to spin reorientation transitions; these second transitions are completely suppressed in H = 5.5 T. The Cp data below 10 K for each composition indicate an enhanced Sommerfeld electronic heat capacity coefficient for the series in the range gamma = 24 to 29 mJ/mol K^2. The chi(T) measurements up to 1000 K were fitted by local-moment Curie-Weiss behaviors which indicate a low Mn spin S ~ 1. The chi data below TN are analyzed using the Weiss molecular field theory for a planar noncollinear helical AF structure with a composition-dependent pitch, following the previous neutron diffraction work of Venturini et al. [J. Alloys Compd. 256, 65 (1997)]. Within this model, the fits indicate a turn angle between Mn moments along the helix axis of ~100 degrees or ~145 degrees, either of which indicate dominant AF interactions between the Mn spins in the Lu{1-x}Sc{x}MnSi series of compounds.

The use of the first and of the second order phase magnetic transition alloys for an AMR refrigerator at room temperature: A numerical analysis of the energy performances

The magnetic refrigeration is a new highly efficient and environmentally protective technology, which could be an actual solution for the typical problems of the vapour compression systems. The core of a magnetic refrigerator system is the Active Magnetic Regenerator (AMR). It is a special kind of thermal regenerator made of magnetic material which works both as a refrigerating medium and as a heat regenerating medium. The performance of an AMR system strongly depends on the magnetocaloric effect of the magnetic material used to build the regenerator.In the present paper, a model to simulate the thermal behaviour of an AMR has been introduced for predicting the performance of an AMR refrigerator system. Different magnetic materials have been considered as refrigerant: pure gadolinium, second order phase magnetic transition binary rare earth alloys (SOMT) and first order phase magnetic transition alloys (FOMT). The SOMT are GdxDy1−x and GdxTb1−x whereas the FOMT are Gd5(SixGe1−x)4 and MnAs1−xSbx. The magnetocaloric behaviour of gadolinium can be correctly predicted by the Weiss molecular field theory. This approach can be generalized for binary alloys with a second order phase transition. The behaviour of the MnAs1−xSbx alloys is described by means of a model based on the phenomenological approach of Bean Rodbell. Interpolation of empirical data is utilized for the evaluation of the magnetocaloric effect of Gd5(SixGe1−x)4 alloys.With this model, the refrigeration capacity, the power consumption and consequently the Coefficient of Performance of the cycle can be predicted. The results of the simulation clearly show that Gd5(SixGe1−x)4 is the best magnetic material with a COP that is always greater then that of a traditional vapour compression plant in the same operating conditions (from a minimum of +40% to a maximum of +62%).

Study on the phase transitions of ferroelectric systems by Weiss's molecular field theory with an external field

The descriptions of the paramagnetic-ferromagnetic and paraelectric-ferroelectric phase transitions (PTs) by Weiss's molecular field theory (WMFT) are quite successful, and the WMFT is also a theoretical basis of initial analysis of the PTs of structural disorder and complex compositional systems because of its mean-field characteristic. However, there is not any study on the PT behaviors of the WMFT with external field, and only the case of two orientational states of molecules for ferroelectric systems has been investigated by the WMFT. Although the descriptions of the above two kinds of PTs by the WMFT are quite similar, the exact ones and the corresponding results are different more or less due to the difference in microscopic unit between the magnetization and polarization In this work, the exact descriptions of the ferroelectric systems with arbitrary orientational states of molecules by the WMFT are provided, including the temperature dependences of the spontaneous polarization, the internal energy, the specific heat and the static polarizability, and then the evolutions of the PTs with an external field are studied. The obtained results are as follows. 1) Without the external field, the PTs of the systems are of the second order, and the transition temperatures and spontaneous polarizations decrease, which are different from those of the ferromagnetic systems, but the internal energy, the specific heat and the polarizability increase with the increase of the orientational states. 2) The external field drives the second order PT to a diffusive one, and diffusive temperature range becomes wider as the field is increases. The results mentioned above would benefit the deep studies of ferroelectric PT, especially the diffusive one.

Study on the variation of static dielectric constant with temperature and the corresponding orientational correlation in polar liquids by using Weiss's molecular field theory

There is no widely accepted microscopic theoretical model of the static dielectric constant of liquids so far. This is mainly because the orientational correlation between molecules in liquids, belonging to the strong correlation systems, is still not clear. In this paper, the variations of the static dielectric constants of 4 kinds of polar liquids, i.e., water, methanol, ethanol and 1-propanol, specifically the Curie-Weiss constant, Curie temperature and Weiss molecular field factor with temperature are studied according to the Weiss's molecular field theory (WMFT), and it is concluded that 1) ferroelectric correlations (FCs) and anti-ferroelectric correlations (AFCs) between molecules coexist in the liquids, and FC is stronger than AFC, as well as FC becomes weak and/or AFC stronger with temperature decreasing; 2) WMFT of homogeneous structure cannot quantitatively describe the abnormally large values of εs of the liquids at low enough T. It can be expected that the coexistence of FC and weak AFC must lead to the spatial distribution of the correlation orders, and consequently, we propose an WMFT of coarse grain approximation of the spatial distribution correlation orders, and use the WMFT to explain why the static dielectric constant varies rapidly with temperature. The above results can benefit the understanding of the liquid physics, including the glass transition mechanism.

Effect of magnetic field on interfacial energy and precipitation behavior of carbides in reduced activation steels

This work investigated the influence of high temperature and high magnetic field on the carbide precipitation behavior in reduced activation steels. As-quenched steels are tempered at 923 K for 3 h with and without a 10 T magnetic field. The applied field can effectively prevent the directional growth of rod-shaped M 23C 6 carbides along martensite packet boundaries. The aspect ratio of M 23C 6 carbides decreased from 2.3 to 1.2 due to an increase of the carbide/ferrite interfacial energy under the high magnetic field. Applications of the Weiss molecular field theory to calculate the difference in interfacial energy caused by the high magnetic field, and of the Langer–Schwartz theory to model metal carbide (MC) precipitation behavior under the magnetic field were described. Results indicated that the density of MC decreased by nearly an order of magnitude and its mean size increased by 40% owing to an increase of 0.03 J/m 2 of the carbide/ferrite interfacial energy.

Thermodynamic analysis using experimental magnetization data of the austenite/ferrite phase transformation in Fe–xNi alloys (x = 0, 2, 4 wt%) in a strong magnetic field

The effect of a strong magnetic field on the transformation temperature from ferrite to austenite and from austenite to ferrite is examined in Fe–xNi alloys (x = 0, 2, 4 wt%). Alloy magnetization measurements are used to calculate the magnetic contribution to the driving force and to account thermodynamically for the field dependence of the transformation temperatures. The predicted shift of the transformation temperatures is compared with experimental dilatometry measurements of transformation temperatures, up to 16 T. The use of experimental magnetization measurements is found to be an accurate alternative to the Weiss molecular field theory for the prediction of the magnetization of ferrite close to its Curie point.

Cluster mean-field approach including correlation effects between clusters

Mean-field approaches provide a qualitative understanding of the behavior of interacting many-body systems. Weiss's molecular-field theory, in particular, has been successfully applied to the studies of various phenomena. However, since local fluctuations from the average are ignored, this simple mean-field theory does not work for systems whose distinctive behavior is caused by fluctuation effects. In the present work, we introduce a new cluster mean-field approach, which we refer to as the "correlated cluster mean-field approach", and the applications to classical and quantum spin systems are demonstrated. We divide the lattice sites into clusters consisting of several sites and include the effect of cluster-cluster correlations into the effective fields acting on the spins. In doing so, the effect of spin fluctuations can be taken into account, and our method gives qualitatively and even semiquantitatively correct results for both classical and quantum systems.

Magnetization of the canted antiferromagnetic CoCO 3 in Abragam–Pryce approximation

Weiss molecular field theory was used to calculate the magnetization of the canted antiferromagnetic CoCO 3 ( T N = 1 8.1 K ). Wave functions of magnetic doublets near Co 2+ ground state in Abragam–Pryce approximation were determined. One of the crystal field variables, free Co 2+ ion isotropic exchange interaction inside, and between magnetic sublatticies, and rotation angle ϕ , characterizing nonequivalence ion Co 2+ positions, were used as parameters. From comparison with the experimental data exchange interaction anisotropy and g-factors g ⊥ , g ∥ were obtained. At low temperatures T < 4 0 K the coincidence of calculated and experimental results are good and g-factor values are almost the same as have been obtained from EPR data in Co 2 + ( 1 % ) + CdCO 3 single crystals. At high temperatures in the paramagnetic region, experimental data differs from calculated ones by more than two times. It is shown that this discrepancy cannot be described within the frames of used approximations.

Diffuse ferroelectric phase transitions in Pb-substitutedPbFe1∕2Nb1∕2O3

We investigate effects of cation (K, Ca, Sr, Ba, La, Bi, Tl, Ag) substitution at Pb-site in PbFe1/2∕Nb1/2O3 (PFN) on its diffuse ferroelectric phase transition (DPT) through measurements and modeling of the temperature dependence of its dielectric constant. We find that the chemical trends in experimentally determined rates of change of transition temperature and diffuseness with substituent concentration are governed by two parameters, nominal and Born effective charges calculated within first-principles density functional theory. We introduce “ferroactivity” of an atom using these charges and use it in a generalized Weiss molecular field theory that accounts for the disorder to explain chemical trends in the DPT found experimentally, suggesting guidelines to tune properties of ferroelectrics.

Influence of magnetic field on the kinetics of proeutectoid ferrite transformation in iron alloys

The kinetics of proeutectoid ferrite transformation in Fe-C base alloys in a strong magnetic field (7.5 T) were studied. The transformation kinetics were accelerated at temperatures not only below but also significantly above the Curie temperature. The free energy and equilibrium ferrite/austenite phase boundaries in applied magnetic fields were calculated using the reported experimental magnetic susceptibility and Weiss molecular field theory. The persistence of magnetic field effects above the Curie temperature can be attributed to the rapidly diminishing difference in relative stability between ferrite and austenite toward the Ae 3 temperature of iron.

Temperature evolution of the internal magnetic hyperfine field of Metglas: amorphous and crystallized phases

We present differential scanning calorimetry (DSC), X-ray diffractometry, in situ Mössbauer spectroscopy (MS), and transmission electron microscopy (TEM) studies in Metglas ribbons subjected to different heat treatments. The temperature evolution of the hyperfine field H hf( T) and the Curie temperature ( T c) of the amorphous phase are determined. The magnetic field originally present in the amorphous phase has a ‘normal’ behavior, in the sense that it can be described by the Weiss molecular field theory. The total angular momentum of the iron atoms turns out to be 5/2 and this implies Fe 3+ in which the electronic spins are uncoupled. When the samples are maintained near T c (≈673 K), three new magnetic phases are detected in the Mössbauer spectra, indicating an onset of a crystallization process well below the first crystallization temperature ( T X1), as determined by DSC (≈820 K). The magnetic behavior of these phases is also described by universal Brillouin curves with J 1= 3 2 , J 2= 5 2 and J 3= 5 2 . A pre-identification of the crystalline phases is made through the Mössbauer parameters of the three observed magnetic phases, and the final characterization by X-ray and electron diffraction. Two of the three phases were identified as α-FeSi and Fe 2B. The third phase could be the cubic phase of Fe 3Si or a solid solution of FeSi. The size of the nanocrystals formed in a sample heated at 673 K during different time intervals were determined by TEM. Even though the number of the nanocrystals increases with time, their average size remains the same.