Free Energy Model Research Articles

DIFFERENT STAGES OF LIQUID FILM GROWTH IN A MICROCHANNEL: TWO-PHASE LATTICE BOLTZMANN STUDY

Abstract A free energy model is used to describe the droplet formation and break-up process in a T-junction bio-microchannel. Droplets are created as a result of interaction of two immiscible liquids. Different stages for the droplet formation process are analyzed which are: a) growing in the x and y directions, b) growing in the x direction and c) detachment process. The effects of capillary number and flow rate ratio on the droplet formation stages are also studied. The influences of Capillary number, flow rate ratio, viscosity ratio and geometrical parameters on droplet break up, droplet size and detachment time are systematically studied. By increasing the flow rate ratio; the duration of droplet formation and the length of the x-growth stage are decreased for small capillary numbers. For larger capillary numbers; the droplet penetrates toward the downstream; therefore, the length of the x-growth stage is increased. The start of detachment process in the microchannel is also reported, which is related to narrowing of the neck of the liquid film. The results show that the detachment time is increased by decreasing the Capillary number. For Ca>0.02, the detachment time is independent of the flow rate ratios. Moreover; the effects of viscosity ratios on detachment time are not significant in comparison to the effects of capillary number. For Ca<0.04, the size of the droplet is independent of the viscosity ratio, but after the critical Capillary number (i.e., Ca=0.04), the size of the droplet is varied by the viscosity ratio. The time between two consecutive drops is also decreased by increasing the Capillary number. Moreover, this time is decreased by increasing the flow rate ratio until Ca=0.04. After this Capillary number, the flow rate ratios have no significant effect on the time between two consecutive droplets. An exhaustive validation study is performed including (a) the Laplace equation in the stationary droplet; (b) a contact angle test; (c) Taylor deformation test in shear flow and (d) comparison of the droplet length as a function of flow rate ratio between the present work and other studies.

Parameter estimation in reactive systems subject to sufficient criteria for thermodynamic stability

For process simulation involving chemical reactions, thermodynamically accurate model parameters for chemical equilibrium are indispensable. We demonstrate that regression subject to the well-known chemical equilibrium constant entails violation of the sufficient conditions for thermodynamic stability. As a consequence, erroneous values of the equilibrium conversion are obtained upon use of the respective parameter values in, e.g., process simulation. In order to alleviate the problem, we propose a methodology for estimating parameters subject to necessary and sufficient criteria for thermodynamic stability. In particular, we propose an extension of Mitsos et al. (2009) to parameter estimation in reactive systems; an upper-level model-experiment mismatch is minimized subject to several lower-level minimization problems (LLP), which ensure the fulfilment of the reactive tangent plane criterion (RTPC) of Smith et al. (1993), along with other desired behavior of the Gibbs free energy. A fictitious single-phase dimerization reaction employing Margules and NRTL as Gibbs free energy models illustrates the thermodynamic guarantees provided by the proposed formulation, when the conventional ones fail. Our formulation conceptually allows any type of phase equilibrium, e.g., VLLE, as well as any type of activity coefficient model. The subproblems of the bilevel algorithm pose challenges to state-of-the-art commercial solvers.

Origin of large electrostrain in Sn4+ doped Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 ceramics

A relatively large bipolar strain of 0.23% was recently reported in 3% Sn4+ doped (BaTi0.8Zr0.2)O3-x (Ba0.7Ca0.3)TiO3 (BZT-xBCT) ceramics under a field of 20 kV/cm. This strain is 53% higher than that in BZT-50BCT (0.15% at 20 kV/cm). To systematically study the mechanism and origin of the large electrostrain enhancement in Sn4+ doped BZT-53.3BCT, we develop a parameterized Landau free energy model and perform phase field simulations. Our results indicate that the softening of the elastic modulus C′=C11−C12 accompanied by the reduction of anisotropy energy due to the 3% Sn4+ doping is primarily responsible for the large electrostrain enhancement.

Coupling two order parameters in a quantum gas.

Controlling matter to simultaneously support coupled properties is of fundamental and technological importance1 (for example, in multiferroics2-5 or high-temperature superconductors6-9). However, determining the microscopic mechanisms responsible for the simultaneous presence of different orders is difficult, making it hard to predict material phenomenology10,11 or modify properties12-16. Here, using a quantum gas to engineer an adjustable interaction at the microscopic level, we demonstrate scenarios of competition, coexistence and mutual enhancement of two orders. For the enhancement scenario, the presence of one order lowers the critical point of the other. Our system is realized by a Bose-Einstein condensate that can undergo self-organization phase transitions in two optical resonators17, resulting in two distinct crystalline density orders. We characterize the coupling between these orders by measuring the composite order parameter and the elementary excitations and explain our results with a mean-field free-energy model derived from a microscopic Hamiltonian. Our system is ideally suited to explore quantum tricritical points18 and can be extended to study the interplay of spin and density orders19 as a function of temperature20.

Predicting Model and Algorithm in RNA Folding Structure Including Pseudoknots

The prediction of RNA structure with pseudoknots is a nondeterministic polynomial-time hard (NP-hard) problem; according to minimum free energy models and computational methods, we investigate the RNA-pseudoknotted structure. Our paper presents an efficient algorithm for predicting RNA structure with pseudoknots, and the algorithm takes O([Formula: see text]) time and O([Formula: see text]) space, the experimental tests in Rfam10.1 and PseudoBase indicate that the algorithm is more effective and precise. The predicting accuracy, the time complexity and space complexity outperform existing algorithms, such as Maximum Weight Matching (MWM) algorithm, PKNOTS algorithm and Inner Limiting Layer (ILM) algorithm, and the algorithm can predict arbitrary pseudoknots. And there exists a [Formula: see text] ([Formula: see text]) polynomial time approximation scheme in searching maximum number of stackings, and we give the proof of the approximation scheme in RNA-pseudoknotted structure. We have improved several types of pseudoknots considered in RNA folding structure, and analyze their possible transitions between types of pseudoknots.

Free-energy model for nanoparticle self-assembly by liquid crystal sorting.

We modeled the experimentally observed self-assembly of nanoparticles (NPs) into shells with diameters up to 10 μm, via segregation from growing nematic domains. Using field-based Monte Carlo simulations, we found the equilibrium configurations of the system by minimizing a free-energy functional that includes effects of excluded-volume interactions among NPs, orientational elasticity, and the isotropic-nematic phase-transition energy. We developed a Gaussian-profile approximation for the liquid crystal (LC) order-parameter field that provides accurate analytical values for the free energy of LC droplets and the associated microshells. This analytical model reveals a first-order transition between equilibrium states with and without microshells, governed mainly by the competition of excluded-volume and phase-transition energies. By contrast, the LC elasticity effects are much smaller and mostly confined to setting the size of the activation barrier for the transition. In conclusion, field-based thermodynamic methods provide a theoretical framework for the self-assembly of NP shells in liquid crystal hosts and suggest that field-based kinetic methods could be useful to simulate and model the time evolution of NP self-assembly coupled to phase separation.

Ternary Free-Energy Entropic Lattice Boltzmann Model with a High Density Ratio.

A thermodynamically consistent free energy model for fluid flows comprised of one gas and two liquid components is presented and implemented using the entropic lattice Boltzmann scheme. The model allows a high density ratio, up to the order of O(10^{3}), between the liquid and gas phases, and a broad range of surface tension ratios, covering partial wetting states where Neumann triangles are formed, and full wetting states where complete encapsulation of one of the fluid components is observed. We further demonstrate that we can capture the bouncing, adhesive, and insertive regimes for the binary collisions between immiscible droplets suspended in air. Our approach opens up a vast range of multiphase flow applications involving one gas and several liquid components.

Entropic multi-relaxation free-energy lattice Boltzmann model for two-phase flows

The entropic multi-relaxation lattice Boltzmann method is extended to two-phase systems following the free-energy approach. Gain in stability is achieved by incorporating the force term due to Korteweg's stress into the redefined entropic stabilizer, which allows simulation of higher Weber and Reynolds numbers with an efficient and explicit algorithm. Results for head-on droplet collisions and droplet impact on super-hydrophobic substrates are matching experimental data accurately. Furthermore, it is demonstrated that the entropic stabilization leads to smaller spurious currents without affecting the interface thickness. The present findings demonstrate the universality of the simple and explicit entropic lattice Boltzmann models and provide a viable and robust alternative to existing methods.

Change in the entropy during a first-order phase transition induced by a magnetic field in an isotropic non-Heisenberg ferromagnet

We describe a first-order phase transition in an isotropic non-Heisenberg ferromagnet induced by a magnetic field with a fourth-order spin exchange interaction at temperatures above the Curie point. The field behavior of the magnetization is analyzed and the temperature dependences of the critical fields of the stability of the paramagnetic and ferromagnetic phases as well as the critical field of the phase transition between them are determined. It is shown that this first-order phase transition has a finite amplitude of the magnetization jump and can occur in small magnetic fields. An H–T phase diagram is presented. It is found that the amplitude of the phase jump during the magnetic field-induced transition from the paramagnetic to the ferromagnetic phase can exceed the magnitude of the entropy change of an isotropic Heisenberg ferromagnet by two orders of magnitude when it is magnetized at the Curie point. It is shown that the expansion of the free energy model up to the 6th power in magnetization can only qualitatively describe the entropy behavior in the phase transition between the paramagnetic and ferromagnetic phases.

Validation and application of a kinetic model for biomass gasification simulation and optimization in updraft gasifiers

Biomass gasification has attracted great interest recently for its great potential as a thermal degradation method that converts biomass or carbonaceous solids into combustible gases with a usable heating value. However, accurate simulation of biomass gasification faces significant challenges as it has a complicated dependence on reaction kinetics, reactor geometry, processing methodology and operating condition. In this paper, a reaction model (RXN model) based on comprehensive biomass gasification kinetics is introduced and validated to predict the syngas and tar compositions. By comparing the simulating predictions with data from two updraft gasification experiment findings available in the literature, it is demonstrated that the RXN model is able to provide a more accurate description for updraft biomass gasification than the minimizing Gibbs free energy model (MGFE model), which can substitute for the widely used yet not accurate MGFE model. Parametric optimization studies were conducted to investigate the impacts of equivalence ratio (ER), gasifier temperature, biomass feed types on syngas and unreacted tar compositions. The results of this work provide vital information for large-scale gasifier design, operating decisions, and optimization.

Nanoscale multiphase phase field approach for stress- and temperature-induced martensitic phase transformations with interfacial stresses at finite strains

A thermodynamically consistent, novel multiphase phase field approach for stress- and temperature-induced martensitic phase transformations at finite strains and with interfacial stresses has been developed. The model considers a single order parameter to describe the austenite↔martensitic transformations, and another N order parameters describing N variants and constrained to a plane in an N-dimensional order parameter space. In the free energy model coexistence of three or more phases at a single material point (multiphase junction), and deviation of each variant-variant transformation path from a straight line have been penalized. Some shortcomings of the existing models are resolved. Three different kinematic models (KMs) for the transformation deformation gradient tensors are assumed: (i) In KM-I the transformation deformation gradient tensor is a linear function of the Bain tensors for the variants. (ii) In KM-II the natural logarithms of the transformation deformation gradient is taken as a linear combination of the natural logarithm of the Bain tensors multiplied with the interpolation functions. (iii) In KM-III it is derived using the twinning equation from the crystallographic theory. The instability criteria for all the phase transformations have been derived for all the kinematic models, and their comparative study is presented. A large strain finite element procedure has been developed and used for studying the evolution of some complex microstructures in nanoscale samples under various loading conditions. Also, the stresses within variant-variant boundaries, the sample size effect, effect of penalizing the triple junctions, and twinned microstructures have been studied. The present approach can be extended for studying grain growth, solidifications, para↔ferro electric transformations, and diffusive phase transformations.

Stretch-Induced Interdigitation of a Phospholipid/Cholesterol Bilayer.

The interdigitated gel (LβI) phase is one of the membrane phases of phospholipid molecules in which the hydrophobic tails of the phospholipid molecules penetrate the opposite leaflet of the bilayer. Recent molecular dynamics (MD) simulations have shown that interdigitation can take place as a phase transition from the liquid-ordered (Lo) phase to the LβI phase under stretching. However, there is still no conclusive experimental evidence for this process, so its existence remains controversial. In this study, to explain the transition from energy balance, we propose a free-energy model. The model consists of three energy components: the elastic deformation energy, surface energy at the bilayer-water interface, and interphase boundary energy. To determine the parameters of the model, we perform MD simulations of a stretched 1,2-dipalmitoyl- sn-glycero-3-phosphocholine/cholesterol bilayer. The phase diagrams from our model are in good agreement with those obtained from MD simulations. The energy balance among the components in the stretched bilayer quantitatively explains the stretch-induced transition. In the model, increasing the system size to that used in experiments shows that interdigitation is favorable for rigid bilayers under stretching or in alcohol solutions. These results suggest that the stretch-induced interdigitation might be observed in microscopic experiments.

Benchmark First-Principles Calculations of Adsorbate Free Energies

Adsorbate free energies are fundamental quantities in the microkinetic modeling of catalytic reactions. In first-principles modeling, finite-temperature free energies are generally obtained by combining density functional theory energies with standard approximate models, such as the harmonic oscillator, the hindered translator, or the two-dimensional ideal gas. In this work, we calculate accurate free energies directly from first-principles potential energy surfaces combined with exact quantum mechanical solutions for the translational energy states to benchmark the reliability of common approximations. Through a series of case studies of monatomic adsorbates on metal surfaces, we show that no one free energy model performs satisfactorily in all cases. Moreover, even combinations of different approximations sometimes deviate significantly from the free energies calculated by our first-principles approach. Using observations from these case studies, we discuss how a full quantum mechanical approach can be extended to calculate accurate free energies for arbitrary adsorbate potential energy surfaces at computational cost similar to standard models.

Effect of NiO substitution on the structural and dielectric behaviour of NaNbO3

The structural and dielectric properties of NiO substituted NaNbO3 ceramics are reported. The orthorhombic (Pmna) crystal structure of NaNbO3 transforms to a lower symmetry monoclinic phase (Pbma) after the dilute dispersion of NiO. X-ray photoelectron spectroscopy reveals pentavalent “Nb,” monovalent “Na,” and divalent “Ni” states along with the signatures of non-local screening effects. The antiferroelectric to paraelectric transition (TAFE) accompanied by a structural change from the orthorhombic to the tetragonal phase shifts by 55 °C toward the low-temperature side, whereas the morphotropic phase boundary (TO-M) moves toward a higher temperature by 28 °C for nominal substitutions (x≤0.10). The generalized Lyddane-Sachs-Teller expression (ε0−S′ε∞)=(ωlωt)2 and thermodynamic free energy models are employed to explain the anomalous behaviour of the temperature dependence of relative dielectric permittivity (εr(T)) across TAFE and TO-M. The frequency dependence of ac-conductivity σac(ω) follows the Jonscher power law (σac = σ(0) + Aωs), suggesting the dominance of the phonon-assisted hopping mechanism, whereas the frequency independent term (σ(0)) was explained by Funke's Jump-Relaxation Model.

Formation of three-dimensional colloidal crystals in a nematic liquid crystal.

We investigate the possible structures of three-dimensional colloidal crystals formed when these spherical particles are dispersed in a liquid crystal. The case of a strong homeotropic boundary condition is considered here. Their corresponding defect structures in the space-filler nematic liquid crystal are induced by the presence of the spherical surface of the colloids and produce an attraction between colloidal particles. Here, a standard Landau-de Gennes free energy model for a spatially inhomogeneous liquid crystal is numerically minimized to yield an optimal configuration of both spherical colloids and the orientational field. The stable and metastable structures obtained in this work are described and analyzed according to the type of periodic liquid-crystal defect lines that couple the colloidal spheres together. A large range of the spherical size is covered in this study, corresponding to a 5CB-liquid-crystal comparison for assembling micron- to nano-sized colloidal spheres. Multiple configurations are found for each given particle size and the most stable state is determined by a comparison of the free energies. From large to small colloidal particles, a sequence of structures, which range from quasi-one-dimensional (columnar), to quasi-two-dimensional (planar), and to truly three-dimensional, are found to exist.

Faceted and dendritic morphology change in alloy solidification

We present methods and results for the simulation of faceted and dendritic crystal growth. Using a thermodynamically realistic isothermal alloy model for AlSi we demonstrate, in confirmation of experimental observations, a change in morphology from perfectly faceted hexagons at smaller undercooling to dendritic growth at larger undercoolings. We also demonstrate that there exists a cut off temperature which separates the two distinct morphologies, and indeed hybrid morphologies. These results suggest that the mechanism for morphology variation observed experimentally primarily lies in anisotropic surface free energy modelling, which we adopt in preference to kinetic anisotropy.

On the co-operative formation of local ground states and frustration fields in the structural glass As2Se3

A microscopic free energy model with an order parameter was introduced earlier that included a local frustration gauge field with U1 symmetry. Ad hoc values of this field were introduced to fit the NQR distribution of a powdered sample of As2Se3 glass and predict the glass and Kauzmann temperatures. Here we derive initial values of the local frustration force amplitudes from an associated molecular free energy. We show that these amplitudes as calculated solely as gradients of the free energy are too large, and that a force term derived from energy leaving the glass and entering the bath must be included to give the correct magnitude. The modification to the bath also acts as a U1 gauge transformation of the frustration field, and insures that the “freezing” process is irreversible, non-equilibrium, and approximately steady state.

Thermophysical properties of paramagnetic Fe from first principles

A computationally efficient, yet general, free-energy modeling scheme is developed based on first-principles calculations. Finite-temperature disorder associated with the fast (electronic and magnetic) degrees of freedom is directly included in the electronic structure calculations, whereas the vibrational free energy is evaluated by a proposed model that uses elastic constants to calculate average sound velocity of the quasiharmonic Debye model. The proposed scheme is tested by calculating the lattice parameter, heat capacity, and single-crystal elastic constants of $\ensuremath{\alpha}$-, $\ensuremath{\gamma}$-, and $\ensuremath{\delta}$-iron as functions of temperature in the range 1000--1800 K. The calculations accurately reproduce the well-established experimental data on thermal expansion and heat capacity of $\ensuremath{\gamma}$- and $\ensuremath{\delta}$-iron. Electronic and magnetic excitations are shown to account for about 20% of the heat capacity for the two phases. Nonphonon contributions to thermal expansion are 12% and 10% for $\ensuremath{\alpha}$- and $\ensuremath{\delta}$-Fe and about 30% for $\ensuremath{\gamma}$-Fe. The elastic properties predicted by the model are in good agreement with those obtained in previous theoretical treatments of paramagnetic phases of iron, as well as with the bulk moduli derived from isothermal compressibility measurements [N. Tsujino et al., Earth Planet. Sci. Lett. 375, 244 (2013)]. Less agreement is found between theoretically calculated and experimentally derived single-crystal elastic constants of $\ensuremath{\gamma}$- and $\ensuremath{\delta}$-iron.

A posteriori error estimators for the Frank-Oseen model of liquid crystals

This paper derives a posteriori error estimators for the nonlinear first-order optimality conditions associated with the Frank-Oseen elastic free-energy model of nematic and cholesteric liquid crystals, where the required unit-length constraint is imposed via either a Lagrange multiplier or penalty method. Furthermore, theory establishing the reliability of the proposed error estimator for the penalty method is presented, yielding a concrete upper bound on the approximation error of discrete solutions. The error estimators herein are composed of readily computable quantities on each element of a finite-element mesh, allowing the formulation of an efficient adaptive mesh refinement strategy. Four elastic equilibrium problems are considered to examine the performance of the error estimators and corresponding adaptive mesh refinements against that of a simple uniform refinement scheme. The adapted grids successfully provide significant reductions in computational work while producing solutions that are highly competitive with those of uniform mesh in terms of constraint conformance and computed free energies.

Quantum-mechanical analysis of effect of alloying elements on \u03b5-martensite start temperature of steels

With regard to the transformation mechanism of austenitic high manganese steel, the prediction of the ε-martensite start temperature is a critical consideration in alloy design. Evaluation of the ε-martensite start temperature makes it possible to predict the microstructure and to understand the phase transformation occurring during deformation. Here we use the quantum mechanical calculation of random alloys to understand the physics for ε-martensitic transformation in steels. We could find the linear relationship between the measured ε-martensite start temperatures and the crystal structure stability for various compositions. We also could estimate the effect of several alloying elements. It is expected that the effect of decreasing the temperatures for the same amount of alloying elements addition will be larger moving farther from Group VIII. By creating a free-energy model that reflects the temperature effect, we were able to calculate the average driving force required for the ε-martensitic transformations.