Free Internal Energy Research Articles

Evaluation of the thermodynamics of a four level system using canonical density matrix method

Abstract We consider a four-level system with two subsystems coupled by weak interaction. The system is in thermal equilibrium. The thermodynamics of the system, namely internal energy, free energy, entropy and heat capacity, are evaluated using the canonical density matrix by two methods. First by Kronecker product method and later by treating the subsystems separately and then adding the evaluated thermodynamic properties of each subsystem. It is discovered that both methods yield the same result, the results obey the laws of thermodynamics and are the same as earlier obtained results. The results also show that each level of the subsystems introduces a new degree of freedom and increases the entropy of the entire system. We also found that the four-level system predicts a linear relationship between heat capacity and temperature at very low temperatures just as in metals. Our numerical results show the same trend.

WANG–LANDAU SIMULATION ON THERMODYNAMIC AND MAGNETIC PROPERTIES OF HONEYCOMB LATTICE

Wang–Landau algorithm of Monte Carlo simulation is performed to understand the thermodynamic and magnetic properties of antiferromagnetic Ising model on honeycomb lattice. The internal energy, specific heat, free energy and entropy are calculated to present the thermodynamic behavior. For magnetic property, the magnetization and magnetic susceptibility are discussed at different temperature upon different magnetic field. The antiferromagnetic order is confirmed to be the ground state of the system, and it can be destroyed by a large magnetic field.

Scaling, bounds, and inequalities for the noninteracting density functionals at finite temperature

Finite-temperature density functional theory requires representations for the internal energy, entropy, and free energy as functionals of the local density field. A central formal difficulty for an orbital-free representation is the construction of the corresponding functionals for noninteracting particles in an arbitrary external potential. That problem is posed here in the context of the equilibrium statistical mechanics of an inhomogeneous system. The density functionals are defined and shown to be equal to the extremal state for a functional of the reduced one-particle statistical operators. Convexity of the latter functionals implies a class of general inequalities. First, it is shown that the familiar von Weizs\"acker lower bound for zero temperature functionals applies at finite temperature as well. An upper bound is obtained in terms of a single-particle statistical operator corresponding to the Thomas-Fermi approximation. Next, the behavior of the density functionals under coordinate scaling is obtained. The inequalities are exploited to obtain a class of upper and lower bounds at constant temperature, and a complementary class at constant density. The utility of such constraints and their relationship to corresponding results at zero temperature are discussed.

Ab initio investigation of structural, electronic, mechanical, and thermodynamic properties of AlSc2 intermetallic compound under pressure

The dependences of the structural, electronic, mechanical, and thermodynamic properties of AlSc2 compound on pressure were investigated for the first time by means of the first-principles method based on the density functional theory with generalized gradient approximation and local density approximation. It was found that the pressure has significant effects on the equilibrium volume, mechanical properties, electronic properties and the heat capacity. Our calculated structural data are in good agreement with the previous experimental and theoretical data results. The calculated elastic constants indicate that the AlSc2 compound is mechanically stable in the pressure range of 0–50 GPa. The Zener ratio A and linear compressibility coefficients kc/ka are used to estimate anisotropic elasticity. The isotropic bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν of polycrystalline AlSc2 compound were determined using the Voigt-Reuss-Hill averaging scheme. The B/G, Poisson’s ratio and microhardness parameter are calculated. The band structure and density of states are also discussed. The Debye temperature can be obtained from the elastic constants and the sound velocities and the pressure has little influence. The temperature-dependent behavior of thermodynamical properties (internal energy, entropy, free energy, and heat capacity) at 0, 20, and 40 GPa have been obtained from phonon density of states.

Combined ab Initio/DFT and Monte Carlo Calculation of Relative Standard Chemical Potentials in Solution

A method has been proposed for pure theoretical estimation of relative standard chemical potentials (1 mol/dm(3) standard state) and related K(c) equilibrium constants for tautomers/conformers dissolved in some solvents. The relative internal free energy could be obtained by means of in-solution ab initio/DFT methods. Using the free energy perturbation method for the transformation of the involved species, the solvation contribution to the relative standard chemical potential can be determined by considering 1 M solution models. Comparison of the ΔG(solv)/MC value calculated for the 1 M solution with those obtained for the system at other concentrations helps explore the ratio of the activity coefficients in nonstandard states. The method has been applied for the study of the tautomeric pair of formaldoxime and nitrosomethane with large structural differences. It was pointed out that the ΔG(solv)/MC values for the 1 and 0.11 M solutions differ by up to 0.2 kcal/mol, when atomic charges are derived on the basis of the in-solution IEF-PCM/B3LYP/aug-cc-pvtz molecular electrostatic potential. On the basis of calculated free energy differences, the ratio of the CH(3)NO and CH(2)NOH activity coefficients in 0.11 molar aqueous and dichloromethane solutions were predicted as of 1.4 and 0.8, respectively. The 0.11 M model is hypothetical by assuming that only one of the tautomers exists in the solution box. As an extension of the method, a procedure has been outlined where contributions to the change of the solvation free energy by terms related to relative activity coefficients might be assessed at physically relevant concentrations for the equilibrated tautomers.

Structural, electronic, elastic and thermodynamic properties of AlSi2RE (RE = La, Ce, Pr and Nd) from first-principle calculations

It is of academic interest to study the ternary intermetallic compounds of the Al–Si–RE system for the development of both structural and functional materials. In this work, the structural, electronic, elastic and thermodynamic properties of the AlSi2RE (RE=La, Ce, Pr and Nd) compounds was investigated using first-principle calculations based on density functional theory. The calculated structural parameters of AlSi2RE compounds are consistent with the experimental data. Due to the fact that there is strong Coulomb correlation among the partially filled 4f electron for RE atoms, we present a combination of the GGA and the LSDA+U approaches to investigate the electronic structures of Al3RE compounds in order to obtain the appropriate results. The elastic constants were determined from a linear fit of the calculated stress–strain function according to Hooke’s law. The bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν of polycrystalline AlSi2RE compounds were determined using the Voigt–Reuss–Hill (VRH) averaging scheme. The Debye temperature of AlSi2RE compounds can be obtained from elastic constants. The temperature dependence of the internal energy, free energy, entropy and heat capacity for AlSi2RE compounds were also calculated by using the quasi-harmonic approximation.

Stochastic thermodynamics of single enzymes and molecular motors

For a single enzyme or molecular motor operating in an aqueous solution of non-equilibrated solute concentrations, a thermodynamic description is developed on the level of an individual trajectory of transitions between states. The concept of internal energy, intrinsic entropy and free energy for states follows from a microscopic description using one assumption on time scale separation. A first-law energy balance then allows the unique identification of the heat dissipated in one transition. Consistency with the second law on the ensemble level enforces both stochastic entropy as third contribution to the entropy change involved in one transition and the local detailed balance condition for the ratio between forward and backward rates for any transition. These results follow without assuming weak coupling between the enzyme and the solutes, ideal solution behavior or mass action law kinetics. The present approach highlights both the crucial role of the intrinsic entropy of each state and the physically questionable role of chemiostats for deriving the first law for molecular motors subject to an external force under realistic conditions.

Quantitative study of fluctuation effects by fast lattice Monte Carlo simulations. I. Compressible homopolymer melts

Using fast lattice Monte Carlo (FLMC) simulations both in a canonical ensemble and with Wang-Landau–Transition-Matrix sampling, we have studied a model system of compressible homopolymer melts (or equivalently, homopolymers in an implicit, good solvent). Direct comparisons of the simulation results with those from the corresponding lattice self-consistent field (LSCF) and Gaussian fluctuation (LGF) theories, all of which are based on the same Hamiltonian (thus without any parameter-fitting among them), unambiguously and quantitatively reveal the fluctuations and correlations in the system. At finite chain number density C and N/κ > 0 (where N is the number of segments on a chain and κ the Helfand compressibility), the LSCF theory underestimates the internal energy and free energy but overestimates the entropy per chain, and does not capture the chain swelling due to excluded-volume interactions. At large C, the LSCF predictions of internal energy, free energy and entropy per chain, as well as the mean-square chain end-to-end distance and radius of gyration, are all approached by FLMC results at a rate of 1/C. LGF theory predicts this 1/C behavior at all C, independent of the system dimensionality. For our model system, both theories become exact only in the limit of C → ∞, where the excluded-volume interactions are fully screened (thus no chain correlations) and the system is in the ground state (with no fluctuations).

ROLE OF DISTRIBUTION FUNCTION ON CHARACTERISTIC PROPERTIES OF SUPERNOVA MATTER

Supernova matter consisting of protons, neutrons and electrons with proton fraction yp = 0.3 are studied within finite temperature Brueckner–Goldstone approach with effective two-body Sussex interaction for various values of densities and at temperatures T = 5, 7 and 10 MeV. It is found that at a given density, temperature and proton fraction, the entropy production, internal energy per nucleon, free energy per nucleon and pressure generated by protons and electrons are not equal. Entropy produced in the supernova matter is larger than that of corresponding asymmetric nuclear matter. The rise in temperature with densities in this matter under adiabatic condition is relatively suppressed with respect to corresponding asymmetric nuclear matter. Contribution to internal energy and free energy due to electron components is more pronounced than those of nuclear components. But the contribution to entropy and pressure due to nuclear components is larger than those of electron components. It is observed that for the matter with proton fraction yp = 0.1, the internal energy, free energy and pressure generated due to protons are density-independent whereas, for supernova matter, these quantities are density-dependent. Distribution function, fraction of particles and mean field are the key factors to explain the characteristic properties of the constituent particles of supernova matter.

Thermodynamic functions of classical systems

The internal energy, pressure, free energy, and entropy of classical systems have been calculated using the exact basic thermodynamic relationship between the energy and the pressure in combination with the virial theorem for classical systems.

Specific thermodynamical potentials on surfaces under relativistic conditions

Expressions of specific thermodynamical potentials on surfaces ( J / cm 2 ) were obtained (for internal energy, enthalpy, free energy, and free enthalpy) under relativistic conditions.

Finite element method based on a minimization theorem to obtain unique magnetization distribution

A new approach to obtain unique magnetization distribution M and the corresponding internal magnetic field H inside magnetic materials subject to an applied field is introduced. The proposed finite element method obtains M directly by finding the minimum of a given mathematical function WM(M) that can be interpreted as the sum of the internal free energy of polarization, and internal and external magnetic energies. According to a theorem in magnetostatics, the magnetization state that satisfies the constitutive equation minimizes WM(M) and is unique. It is shown that the numerical solution satisfies both the condition for a minimum state and the constitutive equation. Therefore there is theoretical confidence that the proposed method yields the unique distributions M and H.

Micellization Behavior of an Amphiphilic Statistical Copolymer in Water-Methanol Mixtures

Sedimentation equilibrium measurements were made on an amphiphilic statistical copolymer p(AMPS/C6) made of sodium 2-(acrylamido)-2-methylpropane-sulfonate (AMPS) and n-hexyl methacrylate (C6), dissolved in water-methanol mixtures with 0.2M LiClO4 and different methanol contents over a wide polymer concentration range. The experimental results were analyzed in terms of a micellar solution theory for amphiphilic polyelectrolytes to estimate the aggregation number m of the micelle and the internal free energy change Δm/m (per polymer chain) due to micellization. Both m and Δm/mdecreased with increasing the methanol content in the solvent. In addition, dynamic light scattering for the same copolymer system indicated that p(AMPS/C6) formed a uni-core star-like micelle in water-methanol mixture with the methanol mole fractions of 0 and 0.06.

Internal energy variational principles and governing equations in electroelastic analysis

The first thermodynamic law contains a universal thermodynamic variational principle. The complete internal energy variational principle in the electroelastic analysis is not discussed in previous papers. In this paper this principle will be discussed. From this principle the simple complete governing equations can be deduced, and the Maxwell stress can be naturally derived from this variational principle. It is shown that the Maxwell stress has slightly different forms determined by using internal energy or electric Gibbs free energy variational principle, but substantially they are the same. In the second-order precision the Maxwell stress is uniquely determined, and its expression has the same form for all deformable and rigid dielectrics. The electroelastic analyses in the dielectric should be studied together with its environment, because the electric field exists in all materials except the ideal conductor. The complete governing equations under finite deformation in the initial configuration are also discussed.

A new front-tracking method to model anisotropic grain and phase boundary motion in rocks

Microstructures of rocks play an important role in determining rheological properties and help to reveal the processes that lead to their formation. Some of these processes change the microstructure significantly and may thus have the opposite effect in obliterating any fabrics indicative of the previous history of the rocks. One of these processes is grain boundary migration (GBM). During static recrystallisation, GBM may produce a foam texture that completely overprints a pre-existing grain boundary network and GBM actively influences the rheology of a rock, via its influence on grain size and lattice defect concentration. In this paper we present a new front-tracking method to simulate GBM. Generally, any movement of boundaries is driven by the minimisation of the internal free energy of a system. The new method moves boundaries along the energy gradient towards a lower total energy state of the system. The calculation of the energy gradient is not necessarily limited to (an)isotropic boundary energies but may also include metamorphism, melting, reaction energies, surface energies and elastic stresses, etc. Two examples are included in this paper where we simulate grain growth with isotropic boundary energy functions and grain growth with isotropic and anisotropic boundary energy functions in a system with a melt present.

Density of states of a two-dimensionalXYmodel from the Wang-Landau algorithm

Using the Wang-Landau algorithm combined with the analytic method, the density of states of a two-dimensional XY model on square lattices of sizes 16x16, 24x24, and 32x32 is accurately calculated. Thermodynamic quantities, such as internal energy, free energy, entropy, and specific heat are obtained from the resulted density of states by numerical integration. It is found that both the density of states and the extensive thermodynamic quantities obey the scaling law. From the entropy curve, symptoms of phase transition are observed. A general method of calculation of the density of states of continuous models by simulation combined with analytical method is proposed.

Towards a unification of hierarchical reference theory and self-consistent Ornstein-Zernike approximation: Analysis of exactly solvable mean-spherical and generalized mean-spherical models

The hierarchical reference theory (HRT) and the self-consistent Ornstein-Zernike approximation (SCOZA) are two liquid state theories that both furnish a largely satisfactory description of the critical region as well as the phase coexistence and equation of state in general. Furthermore, there are a number of similarities that suggest the possibility of a unification of both theories. Earlier in this respect we have studied consistency between the internal energy and free energy routes. As a next step toward this goal we here consider consistency with the compressibility route too, but we restrict explicit evaluations to a model whose exact solution is known showing that a unification works in that case. The model in question is the mean spherical model (MSM) which we here extend to a generalized MSM. For this case, we show that the correct solutions can be recovered from suitable boundary conditions through either SCOZA or HRT alone as well as by the combined theory. Furthermore, the relation between the HRT-SCOZA equations and those of SCOZA and HRT becomes transparent.

Interactions and dynamics of mesogenic molecules in the vicinity of the isotropic to nematic phase transition

A role of the molecular structure and polarity in the dielectric behavior of nematogenic compounds in the vicinity of the isotropic to nematic phase transition is discussed. Temperature dependences of the static permittivity and its derivative, in relation to the basic thermodynamic quantities (internal energy, entropy and Helmholtz free energy), as well as an anomaly in the rotational diffusion of the mesogenic molecules and corresponding activation energy, are analyzed.

Nonlinear Dynamic Response of Piezoelectric Plates Considering Damage Effects

Based on the Talreja’s tensor valued internal state variables damage model and the Helmhotlz free energy of piezoelectric material, the constitutive relations of the piezoelectric plates with damage are derived. Then, the nonlinear dynamic equations of the piezoelectric plates considering damage are established. By using the finite difference method and the Newmark scheme, these equations are solved and the effects of damage and electric loads on the nonlinear dynamic response of piezoelectric plates are discussed.

Transient State Kinetics of Transcription Elongation by T7 RNA Polymerase

The single subunit DNA-dependent RNA polymerase (RNAP) from bacteriophage T7 catalyzes both promoter-dependent transcription initiation and promoter-independent elongation. Using a promoter-free substrate, we have dissected the kinetic pathway of single nucleotide incorporation during elongation. We show that T7 RNAP undergoes a slow conformational change (0.01-0.03 s(-1)) to form an elongation competent complex with the promoter-free substrate (dissociation constant (Kd) of 96 nM). The complex binds to a correct NTP (Kd of 80 microM) and incorporates the nucleoside monophosphate (NMP) into RNA primer very efficiently (220 s(-1) at 25 degrees C). An overall free energy change (-5.5 kcal/mol) and internal free energy change (-3.7 kcal/mol) of single NMP incorporation was calculated from the measured equilibrium constants. In the presence of inorganic pyrophosphate (PPi), the elongation complex catalyzes the reverse pyrophosphorolysis reaction at a maximum rate of 0.8 s(-1) with PPi Kd of 1.2 mM. Several experiments were designed to investigate the rate-limiting step in the pathway of single nucleotide addition. Acid-quench and pulse-chase kinetics indicated that an isomerization step before chemistry is rate-limiting. The very similar rate constants of sequential incorporation of two nucleotides indicated that the steps after chemistry are fast. Based on available data, we propose that the preinsertion to insertion isomerization of NTP observed in the crystallographic studies of T7 RNAP is a likely candidate for the rate-limiting step. The studies here provide a kinetic framework to investigate structure-function and fidelity of RNA synthesis and to further explore the role of the conformational change in nucleotide selection during RNA synthesis.