Invariant Points Research Articles

(Solid + liquid) phase equilibria in the quaternary systems LiCl–Li2SO4–MgCl2–MgSO4–H2O and KCl–K2SO4–MgCl2–MgSO4–H2O at 288.15 K

The experimental solubility of LiCl–Li2SO4–MgCl2–MgSO4–H2O and KCl–K2SO4–MgCl2–MgSO4–H2O systems was measured at 288.15 K. The corresponding stable phase diagrams and water content diagrams were plotted by experimental results. Double salt was formed in both two systems. There are six phase fields, nine univariant solution curves and four invariant points in the phase diagram of LiCl–Li2SO4–MgCl2–MgSO4–H2O system. And the diagram of another system has seven invariant points and eight crystallization regions. By using multiple linear regression method, parameter ΨLi,Cl,SO4 and the solubility equilibrium constants of double salt KCl·MgSO4·3H2O were obtained by solubility data fitting at 288.15 K. The equilibrium solubility of salts in above systems at the same temperature was predicted with Pitzer's electrolyte solution model and mathematical calculation software. The results indicate that the predicted data are in correspondence with experiments.

Enhanced electrocaloric effect in the Sm and Hf co-doped BaTiO3 ceramics

In this work, the electrocaloric (EC) effect was studied in the (100-x)BaHf0.2Ti0.8O3-xBa0.94Sm0.04TiO3 (BHT-xBST) ceramics fabricated by the conventional solid-reaction method. By tuning composition to their diffuse phase transition (DPT), invariant critical point (ICP) and first-order phase transition (FPT), the large electrocaloric response with a temperature change of ΔT = 0.46 °C (at 64 °C) under 30 kV/cm and an electrocaloric strength of ΔT/ΔE = 0.18 K mm/kV under 12 kV/cm were achieved in BHT-70BST. In addition, BHT-50BST shows a broad EC working temperature window from 28 °C to 68 °C because of its diffusion behavior near the ferroelectric-to-paraelectric phase transition and multiphase coexistence. The maximum ΔT of BHT-50BST is larger than that of BHT-40BST with only diffusion character but no multiphase coexistence. This work may provide a design strategy to enhance EC performance with a broad working temperature region in lead-free rare-earth doped BaTiO3-based ceramics by combining the diffusion behavior, multiphase coexistence and heterovalent substitution.

Lie symmetries and singularity analysis for generalized shallow-water equations

Abstract We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm (CH) equation and the generalized Benjamin-Bono-Mahoney (BBM) equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the CH equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.

Solubility determination and thermodynamic modeling of solid−liquid equilibria in the LiBO2−Li2B4O7−H2O system at 298.15 K and 323.15 K

The solubilities and the refractive indices in the ternary system LiBO2−Li2B4O7−H2O at 298.15 K and 323.15 K were determined with the isothermal dissolution method. The phase diagrams and refractive index diagrams were drawn with the experimental values. The phase diagrams at 298.15 K and 323.15 K consist of one invariant point, two univariant curves and two crystallization regions corresponding to Li2B4O7·3H2O, LiBO2·8H2O (at 298.15 K) and LiBO2·2H2O (at 323.15 K). Based on the Pitzer model, the mixing ion-interaction parameters (θB(OH)4−,B4O5(OH)42− and ΨLi+,B(OH)4−,B4O5(OH)42−) at 298.15 K and 323.15 K were fitted with the solubility data. A comparison shows that the calculated solubilities in the ternary system at 298.15 and 323.15 K agree well with the experimental data. The phase diagram and Pitzer model for the ternary system were then used to conduct isothermal evaporation simulation. Different salt will be obtained if the original solution point in different unsaturated regions.

Dynamic analysis of the nonlinear energy sink with local and global potentials: geometrically nonlinear damping

In this paper, the considered two-DOF system consists of a linear oscillator (LO) under external harmonic excitation and an attached lightweight nonlinear energy sink (NES) with local potential and geometrically nonlinear damping. With the application of complex-averaging method, the steady-state dynamical behavior of the system is investigated by the slow invariant manifold, folding singularities and equilibrium points. Different scenarios of strongly modulated responses are presented based on the geometry of SIM, and the numerical simulation results are in consistent with the analytical prediction. The incremental harmonic balance method is applied to detect the frequency response curves of the system around the fundamental resonance, and the accuracy of the theoretical analysis is fully verified by the numerical results obtained by direct integration of equations of motion of the system. It is demonstrated that the increase in external forcing amplitude, global nonlinear stiffness and local nonlinear stiffness can drive the frequency response curves move toward the right and widen the frequency bandwidth of the coexistence of multiple steady-state response regimes, while the increase in nonlinear damping the reverse. The numerical simulation results also show that the addition of geometrically nonlinear damping and local potential in the proposed NES can drastically enhance the capacity of the nonlinear vibration absorber to suppress the shock-induced response of the LO, and the proposed NES is effective for a comparatively broad range of applied impulsive energies, particularly for the high impulsive energies.

Phase Equilibria for the Reciprocal Aqueous Quaternary System Li+, Rb+//Cl−, Borate–H2O at 323.2 K

The isothermal dissolution equilibrium method was applied for the solid–liquid equilibria experiments of the reciprocal aqueous quaternary system Li+, Rb+//Cl−, borate–H2O at 323.2 K, and the relevant diagrams (phase diagram, water content diagram, density/refractive index against composition diagram) were plotted. It was found that there are two commensurate type quaternary invariant points with three-salt cosaturated, five univariant curves, and four crystallization fields corresponding to four single salts, rubidium chloride (RbCl), lithium chloride monohydrate (LiCl·H2O), lithium tetraborate trihydrate (Li2B4O7·3H2O), and rubidium pentaborate tetrahydrate (RbB5O8·4H2O), respectively. LiCl·H2O has a salting out effect on other coexisting salts and the size of the LiCl·H2O crystallization field is the smallest. Li2B4O7·3H2O has the largest crystallization field, which demonstrates that Li2B4O7·3H2O can be more easily separated from the solutions.

Solid–Liquid Phase Equilibria in the Quinary System Na+, K+, Mg2+//Cl−, $${\text{NO}}_{3}^{ - }$$–H2O and Its Subsystems at 258 K

The phase equilibrium of the quaternary system Na+, K+, Mg2+//Cl−–H2O and the quinary system Na+, K+, Mg2+//Cl−, $${\text{NO}}_{3}^{ - }$$ –H2O were investigated at 258 K using the isothermal dissolution equilibrium method. The phase diagrams were obtained based on the measured data. Double salt KCl·MgCl2·6H2O was found in the quaternary and the quinary systems at 258 K. In the quaternary system Na+, K+, Mg2+//Cl−–H2O, there are two invariant points, five univariant curves, and four crystallization fields. The quinary system saturated with NaCl·2H2O contains four invariant points, nine univariant curves, and six crystallization fields corresponding to KCl, NaNO3, KNO3, Mg(NO3)2·6H2O, KCl·MgCl2·6H2O and MgCl2·8H2O. Mg(NO3)2·6H2O and MgCl2·8H2O that have higher concentrations and stronger salting-out effect on other salts. Therefore, low-temperature pretreatment can offer an alternative treatment of brines in accordance with the necessity for current brine treatment processes to reduce the presence of double salts and be crucial for purer products to be separated.

Performance Analysis of SIFT/SURF Algorithms in Neural Networks for Optimized Feature Detection

This paper is an experiment on the implementation of scale-invariant feature transform (SIFT) and speeded up robust features (SURF) algorithms into multi-dimensional neural networks. We are attempting to perform a comparative performance evaluation by using different scale factors of the SIFT algorithm in multi-layered neural networks. This method will help us to understand the best way of implementing the above algorithms in neural networks and from a given sample, extracting distinctive invariant features and finding points of interests. Hence performing a large data set computation would be made much easier because of the neural network implementation. The conventional method of performing SIFT has computational limitations and we aim to achieve best possible way of performing the feature detection when using SIFT and neural network combined, hence transcending computational limitations that SIFT previously had. This approach to recognition of features can robustly find results much faster on bigger dataset and at the same time have the benefits of SIFT algorithm.

Stable Tetrahedron NaCl – KCl – PbCl2 – PbWO4 in the quaternary reciprocal system Na, K, Pb // Cl, WO4

The phase diagram of the stable tetrahedron NaCl – KCl – PbCl2 – PbWO4 of the quaternary reciprocal system Na, K, Pb // Cl, WO4 was first studied using methods of differential thermal analysis. Its phase diagram was triangulated and stable triangulating internal sections of NaCl – PbWO4 – KCl.2PbCl2, NaCl – PbWO4–2KCl. PbCl2 and obtained tetrahedra NaCl – PbCl2 – PbWO4 – KCl.2PbCl2, NaCl – KCl – PbWP4, and NaCl – KCl – PbWP4. There are coordinates of three quadruple invariant points revealed.

Measurements and theoretical simulations of phase equilibria in the quaternary systems NaBr + KBr + SrBr2 + H2O, NaBr + MgBr2 + SrBr2 + H2O and their subsystems at 273.15 K

Since various mineral elements (such as sodium, potassium, strontium, bromine) are rich in underground brine in Sichuan Basin of China, to utilize the bromine resources rationally, the phase equilibria of the quaternary systems NaBr + KBr + SrBr2 + H2O, NaBr + MgBr2 + SrBr2 + H2O and their subsystems at 273.15 K were studied with the means of isothermal dissolution equilibrium method. The research systems are three ternary systems (NaBr + SrBr2 + H2O, KBr + SrBr2 + H2O and MgBr2 + SrBr2 + H2O) and two quaternary systems (NaBr + KBr + SrBr2 + H2O and NaBr + MgBr2 + SrBr2 + H2O). The corresponding solubility phase diagrams of the research systems listed above were plotted based on the measured experimental data.The experimental results indicate that there is no formation of solid solution or double salts in five systems. Five phase diagrams all belong to hydrate type I and contain one invariant point. In three ternary systems, there are two isothermal solubility curves and two crystallization fields, and the equilibrium solid phases are NaBr•2H2O and SrBr2•6H2O in the ternary system NaBr + SrBr2 + H2O, KBr and SrBr2•6H2O in the ternary system KBr + SrBr2 + H2O, and MgBr2•6H2O and SrBr2•6H2O in the ternary system MgBr2 + SrBr2 + H2O, respectively. There are three isothermal solubility curves and three crystallization fields in two quaternary systems. The equilibrium solid phases are NaBr•2H2O, KBr, and SrBr2•6H2O in the quaternary system NaBr + KBr + SrBr2 + H2O. In the quaternary system NaBr + MgBr2 + SrBr2 + H2O, the salts NaBr•2H2O, MgBr2•6H2O and SrBr2•6H2O coexist at the invariant point.The unreported mixing ion-interaction parameters of Pitzer equations θMg, Sr, ΨMg, Sr, Br, ΨNa, Sr, Br, and ΨK, Sr, Br are fitted with the solubility data at 273.15 K. The Pitzer ionic interaction model and the required parameters were employed to predict the solubilities of salts in five systems. The calculated results show that the predicted values are in accordance with the measured data, which verify the reliability of the selected parameters.

Solid-Liquid Phase Equilibria in the Quaternary System (NaCl–KCl–SrCl2–H2O) at 288.15 K

Phase equilibria of the quaternary system (NaCl–KCl–SrCl2–H2O) at 288 K are studied by the isothermal dissolution equilibrium method. The liquid phase was subjected to chemical analysis, and the corresponding solid phase was identified via X-ray diffraction. The phase diagram was plotted according to the experimental data. In the system of (NaCl–KCl–SrCl2–H2O), there are one invariant point, three univariant curves and three crystallization regions. The invariant point corresponding to NaCl + KCl + SrCl2 · 6H2O + Liquid, three crystallization regions corresponding to the NaCl, KCl, SrCl2 · 6H2O. Double salt or solid solution was not formed between the three original components NaCl, KCl and SrCl2 · 6H2O that without dehydration. The crystallization field of KCl is the largest, it is also found that strontium chloride has the salting-out effect on NaCl and KCl in this quaternary system.

Study and Performance Evaluation Binary Robust Invariant Scalable Keypoints (BRISK) for Underwater Image Stitching

In this paper, performance evaluation of Binary Robust Invariant Scalable Key points (BRISK) for underwater image stitching is presented. Underwater image stitching is quite challenging because usually images produced by underwater cameras have few features and noise such as motion blur and underexposure. In this paper, we proposed a robust algorithm for the underwater image stitching which consists of several stages: taking an image frame, getting a key point using BRISK, feature matching using Random Sample Consensus (RANSAC), homography estimation, and perspective warping. The experimental result shows that the proposed algorithm can be implemented for underwater image and achieve better matching results even the less detected keypoints.

Determination of Low-Melting Alloys in the Ternary Systems NaCl–NaI–Na2WO4 and KCl–KI–K2WO4

The ternary systems with a common cation, NaCl–NaI–Na2WO4 and KCl–KI–K2WO4, were studied by differential thermal analysis. The composition point of the congruently melting compound NaCl ⋅ Na2WO4 formed in the NaCl–Na2WO4 system divides the NaI–NaCl–Na2WO4 system into two triangles, in each of which there is a ternary eutectic. The characteristics of invariant points were determined, and phase equilibria in the studied systems were described.

Solid-Liquid Phase Equilibria of the Ternary System (CsNO3 + NH4NO3 + H2O) at (298.15 and 348.15) K and 101.325 kPa

In this paper, the solubilities and the densities of the ternary system (CsNO3 + NH4NO3 + H2O) at 298.15 and 348.15 K and 101.325 kPa were measured using the isothermal dissolution equilibrium method, and the phase diagrams and density versus composition diagrams at these two temperatures were established on the basis of experimental data for the first time. In the ternary system at both temperatures, there are all one invariant point, two solubility isotherm curves, and two crystallization zones corresponding to cesium nitrate (CsNO3) and ammonium nitrate (NH4NO3). The phase diagram at two temperatures belongs to the simple-type of salt–water system, neither double salt nor solid solution were formed. With increasing of the concentration of ammonium nitrate in solution at the two temperatures, the density of the solution is increased gradually, firstly to reach the maximum value at the invariant point, and then decreased gradually. Moreover, on the basis of empirical equations of density of the electrolytes, the calculated values of densities at the two temperatures agree well with the experimental data.

Phase complex of the system Na,Ca||SO4,CO3,HCO3-H2O at 100 ºC

The article discusses the results of determining the possible phase equilibria in geometric images of a five-component reciprocal water-salt system of sulfates, carbonates, sodium bicarbonates and calcium at 100 ºC with subsequent construction of its phase complex diagram. The laws that determine the structure of the phase complex diagram of this system are needed to be obtained for the production of scientific data used both as a reference material and also to create the optimal conditions for the recycling of liquid waste industrial production of aluminum-containing sulfate carbonate and bicarbonate salts of sodium and calcium. It was established that the system under study at 100ºC is characterized by the presence of 31 divariant double saturation fields, 25 monovariant trisaturation curves and 14 invariant points.

Solid–Liquid Phase Equilibria of the Quaternary System MgCl2–PbCl2–ZnCl2–H2O at 323 K

To provide theoretical data for the hydrometallurgical process of lead–zinc ores and the separation process of secondary lead recovery, the quaternary system MgCl2–PbCl2–ZnCl2–H2O at 323 K was investigated using the isothermal dissolution equilibrium method. The isothermal solubility data were obtained, based on which the corresponding dry salt phase diagram and the water content diagram were constructed. The double salts MgZnCl4·6H2O and 3MgCl2·2PbCl2·18H2O were found in the equilibrium solid phases as its ternary subsystems. The dry salt phase diagram contains three invariant points, seven isothermal solubility curves, and five crystallization fields (the crystal salts are PbCl2, 3MgCl2·2PbCl2·18H2O, ZnCl2, MgZnCl4·6H2O, and MgCl2·6H2O(Bis), respectively), and the crystallization region of PbCl2 is the largest. At the same time, the invariant data of this quaternary system and its ternary subsystems at different temperatures were compared and discussed.

“Not- A ”, representation symmetry-protected topological, and Potts phases in an S3 -invariant chain

We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in the region of a U(1)-invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Also nearby is a phase with "representation" symmetry-protected topological (RSPT) order. Its dual exhibits an unusual "not-A" order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact RSPT ground state is similar to that of Affleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-A phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. A field-theory analysis shows that all transitions are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.

Electro-caloric effects in the BaTiO3-based solid solution ceramics

Electro-caloric effect (ECE) was investigated in BaTiO3 (BT)-based solid solution ceramics, Ba(Zr,Ti)O3 (BZT), Ba(Sn,Ti)O3 (BST) and (Ba,Ca)(Zr,Ti)O3 (BCZT) with the composition near an invariant critical point (ICP). The samples were fabricated by the solid-state reaction method and the ECE was obtained by an indirect measurement. The 12BZ–88BT, 9BS–91BT, and 32BCT–68BZT samples showed the best polarization–electric field (P–E) hysteresis characteristics at room temperature and displayed dielectric peaks at 46, 58, and 66 °C, which are the Curie temperatures. With increasing temperature, the P–E loops changed from typical ferroelectric square shapes to paraelectric slanted shapes in the BT-based solid solution ceramics. The adiabatic temperature change due to the ECE (ΔTECE) showed the maximum values of 0.46 °C at 80 °C, 0.5 °C at 65 °C, and 0.47 °C at 75 °C, respectively, in the 12BZ–88BT, 9BS–91BT, and 32BCT–68BZT samples. The BT-based solid solution ceramics showed smaller maximum ΔTECE, but broader ΔTECE peaks at nearer room temperature than the BT ceramic. The enhancement of the ECE due to the multi-phase coexistence was not observed in BT-based solid solutions with the compositions near an invariant critical point (ICP) at which several phases coexist.

Solid–Liquid Phase Equilibria of the Quaternary System (Li2B4O7 + Na2B4O7 + K2B4O7 + H2O) at 323.15 K and Its Application in Industry

The solubilities and refractive indices of the quaternary system (Li2B4O7 + Na2B4O7 + K2B4O7 + H2O) were attained at 323.15 K by the isothermal dissolution equilibrium method. The dry-salt phase diagram, water content diagram and refractive index diagram were plotted. The phase diagram contains three univariant curves, three crystallization fields (Li2B4O7·3H2O, Na2B4O7·10H2O, K2B4O7·4H2O), and one invariant point. The system is a simple quaternary solid–liquid equilibrium type without double salt or solid solution. A comparison of the phase diagram of the quaternary system at 288.15 and 323.15 K shows that the crystallization regions of Li2B4O7·3H2O becomes larger with the increasing of temperature. The refractive indices calculated with semiempirical formula are in good agreement with the experimental results. The brine separation process was demonstrated with the phase diagrams at 288.15 and 323.15 K. All Li2B4O7·3H2O, most of Na2B4O7·10H2O, and K2B4O7·4H2O can be separated from the solution with the phase diagrams at 288.15 and 323.15 K.

Solid Liquid Phase Equilibria in the Ternary Systems KCl–ZnCl2–H2O and MgCl2–ZnCl2–H2O at 278 K

This paper presents the solid–liquid equilibria phase diagrams of two ternary systems KCl–ZnCl2–H2O and MgCl2–ZnCl2–H2O at 278 K using the method of isothermal dissolution equilibrium. Integrated with the determined equilibrated liquid phase and humid residue composition, the phase diagrams of the two systems were generated. At the same time, the invariant points, the univariant solubility curves and the solid crystalline phase regions are displayed and discussed. The results of the experiment show that the two ternary systems are classified as complex owing to the formation of double salt. The phase diagram of the ternary system KCl–ZnCl2–H2O at 278 K is comprised of two invariant points, three univariant solubility curves and three solid crystalline phase regions corresponding to KCl, ZnCl2·2H2O and 2KCl·ZnCl2·2H2O. The phase diagram of the ternary system MgCl2–ZnCl2–H2O at 278 K includes two invariant points, three univariant solubility curves and three solid crystalline phase regions corresponding to MgCl2·6H2O, MgCl2·ZnCl2·6H2O and ZnCl2·2H2O. The densities of ternary mixtures at 278 K were measured. The possibility of Laliberte model to predict densities of investigated ternary saturated solution was checked. The results show that this model is allowed to offer fundamental description for densities of ternary mixtures and calculated results are in good agreement with the experimental values.