Critical Point Tc Research Articles

On the classical critical behavior of the specific heat capacity of a model of structural phase transitions with a long-range interaction

The critical specific heat capacity c of a d-dimensional model describing structural phase transitions in an anharmonic crystal with a long-range interaction (decreasing at large distances r as r −d−σ , 0 < σ ≤ 2) is studied near the classical critical point Tc . At temperatures T > Tc and for dimensions σ < d < 2σ (σ and 2σ are the lower and the upper critical dimensions, respectively) the critical specific heat capacity is obtained in the form c ≈ 1 − Dεαs , where D > 0 and αs < 0 depend only on the ratio d/σ, and ε = T/Tc −1 is a measure of the deviation from the critical point. For three fixed values of the ratio d/σ the dependence c ≈ c(ε) is graphically presented. It is shown that at all temperatures T ≤ Tc the specific heat capacity retains its maximum value, c max = 1. The critical exponent αs , obtained here, coincides with that of the known mean spherical model, while c max is different for the two models.

Thermodynamics of the Ising Model Encoded in Restricted Boltzmann Machines.

The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.

Thermoresponsive ionic liquid for electrochemical low-grade heat harvesting

Thermally regenerative electrochemical cycle (TREC) is a promising technology for low-grade heat harvesting by employing the thermogalvanic effect of the electrodes. Whereas the electrolytes applied in TREC systems have a negligible response to temperature variation. In this study, a thermoresponsive ionic liquid (TRIL) is added to an electrolyte to endow it with temperature-driven phase change behavior, and the electrolyte is then utilized in a copper hexacyanoferrate-based TREC system for ultralow-grade heat harvesting. The TREC system is operated between 10 and 30 °C across the phase change critical point (Tc), so that the solvation states of the ions varied during the charging and discharging process, and a high energy density of 1.30 J g−1 and high energy conversion efficiency of 1.32% (20.0% for the Carnot efficiency) are achieved. The energy efficiency is 10 times that achieved by the conventional non-TRIL system under the same conditions. Moreover, the Tc of the TRIL can be tuned according to the species and concentrations of the electrolyte salt, which enhances the feasibility and resilience of the TRIL-containing TREC system. This study provides a novel perspective for electrolyte design in electrochemical cells, promoting the applicability of electrochemical cells in high-performance ultralow-grade thermal energy harvesting systems.

Penentuan Daya Tampung Sungai Mayang Terhadap Beban Pencemaran Menggunakan Persamaan Streeter-Phelps

The Mayang River in Jember Regency has a length of approximately 145.50 km which is the longest river and can irrigate an area of 5,860 hectares. The use of the Mayang River by the community can produce waste, both domestic waste, agricultural waste, and livestock waste. Waste that enters directly into river bodies can cause a decrease in river water quality and can create a pollution load. Therefore, the measurement of the load capacity of the Mayang River was conducted. Mayang River water samples were taken using the grab sampling method, while the river's capacity was determined using the Streeter Phelps method. The results showed that the average value of the Mayang River pollution load was 219,483 kg/day. Some of the parameters used to determine the carrying capacity of the river are the daily deoxygenation rate of 0.137 mg/L and the dailyy reaeration rate of 1.097 mg/L. It has a critical point (tc) of 0.186, a critical distance (Xc) of 4.753 km, and a critical deficit (Dc) of 0.281 mg/L. Based on these parameters, it is concluded that the Mayang River can accommodate a pollution load of 1.5 kg/day and can carry out proper purification. Keywords : grab sampling method, Streeter Phelps, the deoxygenation rate, the reaeration rate

Heider balance of a chain of actors as dependent on the interaction range and a thermal noise

The Heider balance is investigated in a chain of actors, with periodic boundary conditions and the neighborhood of range r, with r as a parameter. Two model dynamics are applied: a deterministic cellular automaton [Malarz et al., Physica D 411 (2020) 132556] and the heat-bath algorithm, with the density of unbalanced–balanced triads in the role of energy. The outcome is a spectrum of energy in stationary and blinking states and a balanced–unbalanced network transition driven by thermal noise. The critical point Tc increases with the range r and it does not depend on the system size.

From the vapor-liquid equilibrium to the supercritical condition. Molecular dynamics modeling of 1,3-butadiene

Molecular interactions encode macroscopic phenomena. Molecular dynamics simulations (MD) based on classical force fields can provide structural and energetic information about phase equilibrium at different conditions as well as about systems above the critical point. In this study, vapor/liquid interface of 1,3-butadiene is investigated using classical MD and Two Phase Molecular Dynamics (2ϕMD) techniques in order to test force field parameters as well as to analyze structural and energetic details from ambient conditions to the supercritical phase. A new approach is proposed based on Voronoi-tessellation to categorize vapor, liquid and ‘phase transferring’ molecules.The force field (OPLS-AA with 1.14*CM1A partial atomic charges) combined with the 2ϕMD method is used to reproduce experimental data (density and surface tension) of 1,3-butadiene well for all investigated phases. Density (ρ), potential energy (Epot), surface tension (γ) and specific surface area (SSA) based methods have also been tested to calculate the critical parameters of the molecular system. It is found that the density based approach is the best alternative for retrieving experimental data for the critical point (Tc,exp = 425.0 K, ρc,exp = 245 kg/m3, Pc,exp = 43.2 bar and Tc,ρ = 430.25 ± 2.50 K, ρc,ρ = 234.25 ± 2.66 kg/m3, Pc,ρ = 39.09 ± 0.73 bar), while from the SSA and potential energy the critical temperature can be reproduced within 1% of deviation (Tc,SSA = 426.5, Tc,Epot = 426.0 K).

Direction for High-Performance Supercritical CO2 Centrifugal Compressor Design for Dry Cooled Supercritical CO2 Brayton Cycle

To overcome the degradation of the cycle efficiency of a supercritical carbon dioxide (S-CO2) Brayton cycle with dry cooling, this study proposes an improved design of an S-CO2 centrifugal compressor. The conventional air centrifugal compressor can achieve higher efficiency as backsweep angle increases. However, the structural issue restricts the maximum allowable angle (−50~−56°). In this study, an S-CO2 centrifugal compressor performance was examined while changing the backward sweep angle at impeller exit to study if the previous optimum backsweep angle for an air centrifugal compressor is still valid when the fluid has changed. It is shown through an analysis that an S-CO2 centrifugal compressor can achieve the highest efficiency at −70° backsweep angle, which is greater than the typical design value. The S-CO2 centrifugal compressor is less restricted from a structural integrity issue because it has low relative Mach number regardless of the low sound speed near critical point (Tc = 304.11 K, Pc = 7377 kPa). It is also shown in the paper that the variation of compressibility factor does not impact on its total to total efficiency since its Mach number is still lower than unity. Finally, it is also shown that a backward sweep impeller can achieve higher pressure ratio and operate stably in wider range as the mass flow rate is decreased. As further works, the suggested concept will be validated by the structural analysis and the compressor performance test.

Universality class of explosive percolation in Barab\xe1si-Albert networks

In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree k = m, for both product rule (PR) and sum rule (SR) of the Achlioptas process. For m = 1 we find that the critical point tc = 1 which is the maximum possible value of the relative link density t; Hence we cannot have access to the other phase like percolation in one dimension. However, for m > 1 we find that tc decreases with increasing m and the critical exponents ν, α, β and γ for m > 1 are found to be independent not only of the value of m but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality α + 2β + γ ≥ 2 but always close to equality. PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd.

Measuring the dynamical critical exponent of an ordering alloy using x-ray photon correlation spectroscopy

The dynamics of the order fluctuations in the AuAgZn2 close to the critical point (Tc≃609 K) was observed by coherent x-ray scattering. With the high beam intensity of the ID10 ESRF beamline and with the new pixel detector, the dynamics was measured with a few tens of millisecond resolution. The intensity connected to the diffuse scattering corresponding to fluctuations was unambiguously distinguished from the surface pretransitional ordering occurring in this system close to Tc. The variations of the fluctuation time with temperature and wave vectors were measured in this system belonging to the universality class of Ising second order transition with a nonconserved order parameter. The direct observation of the critical slowing down in the vicinity of the second-order transition led to an estimate of the dynamic exponent z≃1.96(11), in rough agreement with theory (model “A” of Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)]).

Controlled pinewood fractionation with supercritical ethanol: A prerequisite toward pinewood conversion into chemicals and biofuels

Abstract The objective of this work was to investigate the ability of supercritical (SC) ethanol conditions to attack preferentially the lignin fraction against the carbohydrate fraction and their effects on the product distribution among gases, light products, bio-oils, and chars. In this study, the conversion of each pinewood component was determined by the analysis of solid residues to quantify cellulose, hemicellulose, lignin, and char contents. It is shown that, by tuning the temperature, hemicellulose and lignin are already transformed in subcritical ethanol conditions, lignin being more reactive than hemicellulose. In contrast, native wood cellulose is recalcitrant to liquefaction in SC ethanol near the critical point (Tc = 241 °C and Pc = 61 bar), but 20% of native wood cellulose is converted in SC ethanol at 280 °C. Besides, the severity of the conditions, in terms of temperature and treatment time, does not significantly influence the yields of gases, light products, and bio-oils but strongly enhances char formation. Interestingly, the increase in SC ethanol density does not change the conversion of biomass components but has a marked effect on bio-oil yield and prevents char formation. The optimum fractionation conditions to convert the lignin component, while keeping unattacked the cellulose fraction with a minimum formation of char, are dense SC ethanol, at 250 °C for 1 h, in batch conditions. However, although lignin is more reactive than hemicellulose under these conditions, these fractions are converted, in a parallel way, to around 50% and 60%, respectively.

Computer simulation of adsorption properties of polymer on surface under external driving force

Monte Carlo simulation is performed to study the adsorption properties of polymers on an attractive surface. Annealing method is adopted to simulate the adsorption characteristics and conformational changes of polymer chains driven by an external driving force F. In simulations using cooperative motion algorithm, the ensembles of monomers located at lattice sites are connected by non-breakable bonds. When the external force is F=0, the finite-size scale method can be used to determine the critical adsorption temperature (Tc) of the polymer chain on the attractive surface, but when the external force is F&gt;0, the dependence of the average number of surface contacts M&gt; on the chain length N is unrelated to temperature T. Therefore, Tc cannot be obtained by the finite-size scale method. However, the pseudo-critical adsorption temperature Tc can be estimated by a function of the average number of surface contacts M&gt; and the temperature T for the chain length N=200. And then Tc decreases with external force F increasing. The phase diagram is obtained for the polymer chain between the desorbed state and the adsorbed state under temperature T and external driving force F. Furthermore, the influence of the external driving force on the conformation of the polymer chain is analyzed by the mean square radius of gyration of polymer chains. The critical adsorption point Tc can be checked roughly by the minimum location of the mean square radius of gyration or by the variation of its components in the Y and Z direction perpendicular to the external force. With the increase of the external force F for adsorbed polymer, the temperature T can determine whether polymer is changed from the adsorption state to the desorption state and where the force is located at the transformation. There are two different cases, that is, the polymer can be desorbed at the temperature Tc* TTc and the polymer cannot be desorbed at T Tc*. In this paper, we discuss these two cases for the adsorption of polymer on the attractive surface:weak and strong adsorption. In the first case, the adsorption is strongly influenced by the external driving force. By contrast, in the strong adsorption, the adsorption is weakly influenced by the external force. Our results unravel the dependence of adsorption of polymer on external driving force, which is also consistent with the phase diagram of adsorption and desorption of polymer chains.

The critical parameters and congruent vapor-liquid diagram of ten metallic alkali and alkaline earth fluids and one H-bond organic (methanol)

We report the first reliable prediction of the critical point (CP) (Tc,Pc,ρc) and two separate branches of the coexistence curve (CXC) for 10 alkali and alkaline earth fluids metals as well as for methanol. It is based on the recently proposed methodology of the congruent vapor-liquid (CVL) – diagram following from the developed earlier model of the fluctuational thermodynamics (FT). Both CXC-branches are predicted by the quite different (i.e. asymmetric and non-“symmetrized” artificially) FT-correlations. They are completely conformed in the asymptotic CP-region not only to the non-classical (with the exponent β≈1/3) projection of CXC on the (T,ρ)-plane but also to the well-known classical Clausius-Clapeyron approximation of the vapor-pressure curve in the (P,T)-plane. The respective interrelations between two phase-dependent factors of thermodynamic similarity introduced, originally, by Riedel (Ri) and Trouton (Tr) in the classical principle of corresponding states (PCS) are estimated without any fitting. We show that neither any analytic expansion (it fails near CP) nor the known non-analytic Wegner's expansion (it diverges far from CP) are necessary to predict with the reasonable accuracy the entire range of CXC including its CP. The only input data for this aim are the standard low-temperature measurements of one-phase liquid performed at atmospheric pressure. The reasons of violation of the classical PCS for the metallic and ionic fluids have been analyzed. Despite the widespread belief on the contrary, FT-methodology predicts the striking similarity of all studied fluid metals revealed by their CVL-diagrams.

Liquid bridging of cylindrical colloids in near-critical solvents.

Within mean field theory, we investigate the bridging transition between a pair of parallel cylindrical colloids immersed in a binary liquid mixture as a solvent that is close to its critical consolute point Tc. We determine the universal scaling functions of the effective potential and of the force between the colloids. For a solvent that is at the critical concentration and close to Tc, we find that the critical Casimir force is the dominant interaction at close separations. This agrees very well with the corresponding Derjaguin approximation for the effective interaction between the two cylinders, while capillary forces originating from the extension of the liquid bridge turn out to be more important at large separations. In addition, we are able to infer from the wetting characteristics of the individual colloids the first-order transition of the liquid bridge connecting two colloidal particles to the ruptured state. While specific to cylindrical colloids, the results presented here also provide an outline for identifying critical Casimir forces acting on bridged colloidal particles as such and for analyzing the bridging transition between them.

Simulation of Temperature Dependences of Emission Characteristics of Nano-Layer Injection Lasers Based on Symmetric Heterostructures

The exact problem of electromagnetic wave propagation in a multilayer nanostructure with complex values of dielectric permittivity is solved. The contribution to the refractive index of an additive to the dielectric constant connected with the injected carriers is taken into account. Within the framework of this problem, it is shown that an anomalous temperature dependence of the threshold current of injection lasers based on nanoheterostructures is related to the anti-waveguide action of injected carriers. A quantum-well heterostructure based on InGaAs/AlGaAs/GaAs nanosystems used for fabrication of 0.94-1.14 μm lasers is considered. The applied techniques and approaches are also acceptable for the optimization of multilayer nanostructures based on other solid solutions. As an optical model of the active region of injection lasers based on nanostructures, a planar multilayer dielectric waveguide with complex values of dielectric permittivity in the layers is considered. It is shown that with decreasing thickness of the active region of injection lasers, the dependence of the mode gain on the local gain is essentially sublinear. The reason for this is the antiwaveguide action of electrons. The results of calculations of the temperature dependence of the threshold current of injection lasers indicate the presence of a critical point Tc, at which a sharp decrease in the characteristic temperature occurs. The performed calculations and optimization of the temperature dependence of emission characteristics of injection lasers based on nanostructures show that the anomalous behavior of the temperature dependence of the threshold current is also associated with the weakening of the waveguide properties of theirs active region.

Critical behavior and microscopic structure of charged AdS black holes via an alternative phase space

It has been argued that charged Anti-de Sitter (AdS) black holes have similar thermodynamic behavior as the Van der Waals fluid system, provided one treats the cosmological constant as a thermodynamic variable (pressure) in an extended phase space. In this paper, we disclose the deep connection between charged AdS black holes and Van der Waals fluid system from an alternative point of view. We consider the mass of an AdS black hole as a function of square of the charge Q2 instead of the standard Q, i.e. M=M(S,Q2,P). We first justify such a change of view mathematically and then ask if a phase transition can occur as a function of Q2 for fixed P. Therefore, we write the equation of state as Q2=Q2(T,Ψ) where Ψ (conjugate of Q2) is the inverse of the specific volume, Ψ=1/v. This allows us to complete the analogy of charged AdS black holes with Van der Waals fluid system and derive the phase transition as well as critical exponents of the system. We identify a thermodynamic instability in this new picture with real analogy to Van der Waals fluid with physically relevant Maxwell construction. We therefore study the critical behavior of isotherms in Q2–Ψ diagram and deduce all the critical exponents of the system and determine that the system exhibits a small–large black hole phase transition at the critical point (Tc,Qc2,Ψc). This alternative view is important as one can imagine such a change for a given single black hole i.e. acquiring charge which induces the phase transition. Finally, we disclose the microscopic properties of charged AdS black holes by using thermodynamic geometry. Interestingly, we find that scalar curvature has a gap between small and large black holes, and this gap becomes exceedingly large as one moves away from the critical point along the transition line. Therefore, we are able to attribute the sudden enlargement of the black hole to the strong repulsive nature of the internal constituents at the phase transition.

Critical phenomena and estimation of the spontaneous magnetization by the analysis of the magnetic entropy change for the Nd0.6Sr0.3Ca0.1Mn0.975Fe0.025O3 manganite

In this actual work, we have already studied the universal critical behavior in perovskite-manganite compound Nd0.6Sr0.3Ca0.1Mn0.975Fe0.025O3. An approve of a second order ferromagnetic-paramagnetic phase transition through the magnetic measurements using Banerjee's criteria is completely clarified. The critical exponents values near the critical point TC are determined through several techniques such as modified Arrott plot (MAP), Kouvel Fisher (KF) method and critical isotherm (CI). The critical exponents values obtained agree toughly with the mean field model (β = 0.478 ± 0.01 with TC = 232 K ± 0.9 K and γ = 1.039 ± 0.02 with TC = 230 K ± 0.94 K). We have confirmed the obtained critical exponents with the single scaling equation of: M(H,ε)=εβf±(H/ε(β+γ)) With ε = (T−TC)/TC is the reduced temperature.We have verified the agreement between two different methodologies of determining the spontaneous magnetization (MS) in Nd0.6Sr0.3Ca0.1Mn0.975Fe0.025O3 manganite. Used the spontaneous magnetization (MS) obtained through the magnetic entropy change (ΔSM) vs M2; we have fitted MS(T) curves to obtain an excellent agreement between the theoretical and experimental value of β. This result leads to confirm the validity of the magnetic entropy change approach in order to estimate the spontaneous magnetization MS in a ferromagnetic system.

Critical behavior in polycrystalline La0.7Sr0.3CoO3 from bulk magnetization study

The critical behavior of hole-doped perovskite cobaltite La0.7Sr0.3CoO3 has been investigated around the second-order paramagnetic-ferromagnetic (PM-FM) phase transition based on the static magnetization. Reliable critical exponents (β = 0.272 ± 0.001, γ = 1.291 ± 0.004, and δ = 5.54 ± 0.02 with critical point TC = 227.2 ± 0.2 K) have been determined by using different techniques, including the Modified Arrott plot, the Kouvel–Fisher method, and critical isotherm analysis. The obtained critical exponents not only obey the Widom relation δ = 1+γ/β, but also can collapse the magnetization data M(H,T) into two curves below and above TC following a single scaling equation M(H,ε) = εβf±(H/εβ+γ), which implies the reliability and accuracy of the exponents. Temperature variation of the effective exponents resemble with those for disordered ferromagnets. The exponents analysis related to the magnetocaloric effect is studied. The field dependence of the relative cooling power (RCP) is found to obey RCP∝H1+1/δ with the exponent δ. The asymptotic critical exponents (βeff and γeff) are close to those predicted by 3D-Ising model with the exchange interaction J(r) decaying as r−4.97 in the system, which reflects the existence of short-range FM coupling.

Information-entropic signature of the critical point

We investigate the critical behavior of continuous (second-order) phase transitions in the context of (2+1)-dimensional Ginzburg–Landau models with a double-well effective potential. In particular, we show that the recently-proposed configurational entropy (CE)—a measure of the spatial complexity of the order parameter in momentum space based on its Fourier-mode decomposition—can be used to identify the critical point. We compute the CE for different temperatures and show that large spatial fluctuations near the critical point (Tc)—characterized by a divergent correlation length—lead to a sharp decrease in the associated configurational entropy. We further show that the CE density goes from a scale-free to an approximate scaling behavior |k|−5/3 as the critical point is approached. We reproduce the behavior of the CE at criticality with a percolating many-bubble model.

Achievement of local thermodynamic equilibrium for ns laser-induced plasmas on aluminium sample at different wavelengths

The Collisional-Radiative model CoRaM-Al is elaborated and implemented in a 0D numerical approach in the purpose of describing the formation of the plasma resulting from the interaction between a τ = 4 ns Nd:YAG laser pulse and an aluminium sample in vacuum. The influence of the four harmonics at 266, 355, 532 and 1064 nm on the behavior of the nascent plasma is studied. In each case, the fluence is set equal to the threshold above which a phase explosion takes place (fluence of 7.7, 7.4, 6.8, 5.1 J cm−2 in order of increasing wavelength). The model takes into account free electrons and excited states of Al, Al+, Al2+ and Al3+. Both groups of particles are characterized by their translation temperature in thermal non-equilibrium. Besides, each population density is assumed to be in chemical non-equilibrium and to behave freely through the involved seven elementary processes (electron impact induced excitation and ionization, elastic collisions, multiphoton ionization, inverse laser Bremsstrahlung, direct thermal Bremsstrahlung and spontaneous emission). Atoms passing from the sample to the gas are described by considering classical vaporization phenomena (governed by the Hertz-Knudsen law) so that the surface temperature is limited to values less than the critical point (Tc = 6700 K). The relative role of the elementary processes is discussed and the time-evolution of the excitation of the species is analyzed for the four considered wavelengths. This study allows to determine the different excitation temperatures as well as their evolution in time. Thus the conditions required for the achievement of the Local Thermodynamic Equilibrium can be precisely described.

Critical adsorption of an end-grafted diblock copolymer on a flat surface

An end-grafted flexible diblock copolymer exists a critical adsorption point Tc from a desorbed state to an adsorbed state. The finite-size scaling law is adopted to determine Tc and the crossover exponent φ. The non-adsorbed block B weakens the adsorption of the adsorbed block A. Then the effective influence of the B block is reduced with increase in the length of the A block but it is independent of the length at Tc.