Derivative Of Free Energy Research Articles

Solubility of KF and NaCl in water by molecular simulation

The solubility of two ionic salts, namely, KF and NaCl, in water has been calculated by Monte Carlo molecular simulation. Water has been modeled with the extended simple point charge model (SPC/E), ions with the Tosi-Fumi model and the interaction between water and ions with the Smith-Dang model. The chemical potential of the solute in the solution has been computed as the derivative of the total free energy with respect to the number of solute particles. The chemical potential of the solute in the solid phase has been calculated by thermodynamic integration to an Einstein crystal. The solubility of the salt has been calculated as the concentration at which the chemical potential of the salt in the solution becomes identical to that of the pure solid. The methodology used in this work has been tested by reproducing the results for the solubility of KF determined previously by Ferrario et al. [J. Chem. Phys. 117, 4947 (2002)]. For KF, it was found that the solubility of the model is only in qualitative agreement with experiment. The variation of the solubility with temperature for KF has also been studied. For NaCl, the potential model used predicts a solubility in good agreement with the experimental value. The same is true for the hydration chemical potential at infinite dilution. Given the practical importance of solutions of NaCl in water the model used in this work, whereas simple, can be of interest for future studies.

On kinetics of hysteresis

An initiation criterion and a kinetic relation describing hysteresis in shape memory alloys consistent with non-equilibrium thermodynamics are proposed. The initiation of phase transition begins always at zero driving force, defined as the negative derivative of free energy with respect to the volume fraction of high strain phase. The kinetic relation is obtained from the dissipation potential which is proportional to the magnitude of the volume fraction rate times the newly formed volume fraction. The proposed theory turns out to be rate-independent, but history dependent, and can describe all features of hysteresis loops observed in experiments.

Using cluster dynamics to model electrical resistivity measurements in precipitating AlSc alloys

Electrical resistivity evolution during precipitation in AlSc alloys is modeled using cluster dynamics. This mesoscopic modeling has already been shown to correctly predict the time evolution of the precipitate size distribution. In this work, we show that it leads too to resistivity predictions in quantitative agreement with experimental data. We only assume that all clusters contribute to the resistivity and that each cluster contribution is proportional to its area. One interesting result is that the resistivity excess observed during coarsening mainly arises from large clusters and not really from the solid solution. As a consequence, one cannot assume that resistivity asymptotic behavior obeys a simple power law as predicted by LSW theory for the solid solution supersaturation. This forbids any derivation of the precipitate interface free energy or of the solute diffusion coefficient from resistivity experimental data in a phase-separating system like AlSc supersaturated alloys.

Pure shear deformation of physical and chemical gels of poly(vinyl alcohol)

Pure shear deformation reveals the significant differences in elastic properties of the poly(vinyl alcohol) (PVA) gels with almost identical initial modulus, but with different types of crosslinks, physical crosslinks formed by microcrystallites and chemical crosslinks made of covalent bonds. The ratio of the two principal stresses steeply increases with elongation in the physical gels, while that remains almost constant independently of stretching in the chemical gels. The marked growth of the stress ratio with elongation in the physical gels leads to the negative values of the derivative of the elastic free energy ( W 2) with respect to the second invariant of the deformation tensor in the whole range of deformation, which is firstly observed for elastomeric materials. By contrast, the chemical gels exhibit the positive values of W 2 like most chemically crosslinked rubbers. Among the existing theories of rubber elasticity, the classical non-Gaussian three-chain model considering the effect of finite chain-length is qualitatively successful in accounting for the steep increase of the stress ratio and the negative values of W 2 in the physical gels, although it fails to reproduce the large difference in the stress–strain behavior among uniaxial, pure shear and equi-biaxial deformations. These features of the physical gels are expected to stem from the structural characteristics such as fewer amounts of slippery-trapped entanglement along network strands compared to the chemical PVA gels.

Toward Efficient Chemical Potential Calculations by Expanded Ensemble Simulations; to Make the Free Energy Pathway Fairly Level

A scheme is suggested of how to construct good bias potentials ("balancing factors") to be used in expanded ensemble (EE) calculations of chemical potentials of solutions. A combination of two strategies are used: (i) to use a pathway for particle insertions that avoids large variations in free energy and (ii) to use calculated free energy derivatives to construct a bias potential that makes the pathway fairly level. Only a few very short simulations are needed to accomplish the latter, and then, a full EE simulation is done to obtain the chemical potential. By practical calculations of the chemical potential of benzene, cyclohexane, and benzylamine in water, it is shown that this method is at least equally efficient to the recent adaptive EE (AEE) method by Aberg et al. (J. Chem. Phys. 2004, 120, 3370). Furthermore, the new method provides an alternative strategy that complements existing EE methods.

Microscopic Derivation of Free Energy Under Electric Field in Ferroelectric and Ferroelectric Heterostructures Containing Free Carriers

We derive Free energy and the macroscopic charge distribution in ferroelectric and ferroelectric heterostructures from a microscopic quantum mechanical model, and attempt to bridge the macroscopic and the atomic/quantum mechanical pictures. We postulate that ferroelectric should be defined in a heterostructure, especially when it is nanometer-scale. In addition, the separation of the polarization energy into the pure electrostatic and the material reconfiguration energy is proposed.

Ordering of geometrically frustrated classical and quantum triangular Ising magnets

A systematic study of both classical and quantum geometric frustrated Ising models with a competing ordering mechanism is reported in this paper. The ordering comes in the classical case from a coupling of 2D layers and in the quantum model from the quantum dynamics induced by a transverse field. By mapping the Ising models on a triangular lattice to elastic lattices of non-crossing strings, we derive an exact relation between the spin variables and the displacement field of the strings. Using this map both for the classical (2+1)D stacked model and the quantum frustrated 2D system, we obtain a microscopic derivation of an effective Hamiltonian which was proposed before on phenomenological grounds within a Landau-Ginzburg-Wilson approach. In contrast to the latter approach, our derivation provides the coupling constants and hence the entire transverse field--versus--temperature phase diagram can be deduced, including the universality classes of both the quantum and the finite--temperature transitions. The structure of the ordered phase is obtained from a detailed entropy argument. We compare our predictions to recent simulations of the quantum system and find good agreement. We also analyze the connections to a dimer model on the hexagonal lattice and its height profile representation, providing a simple derivation of the continuum free energy and a physical explanation for the universality of the stiffness of the height profile for anisotropic couplings.

First-Order Phase Transition and Bayesian Image Processing by Loopy Belief Propagation

The framework is presented of Bayesian image restoration for multi-valued images based on the Q-state Potts model. Hyperparameters in the probabilistic model are determined so as to maximize the marginal likelihood. A practical algorithm is described for multi-valued image restoration based on loopy belief propagation. We conclude that the maximization of marginal likelihood can provide good results even if the ap rioriprobabilistic model exhibits first-order phase transition. In Bayesian image restoration, we have to determine certain hyperparameters, and in practice it is common for the hyperparameters to be determined so as to maximize a marginal likelihood. 1) In some probabilistic models treated in statistical mechanics, some types of phase transition occur at a critical point. First-order and second-order phase transitions are familiar types of phase transition. In the case of first-order phase transition, the first derivative of the free energy is discontinuous at the critical point; if the probabilistic model exhibits first-order phase transition, the free energy is not differentiable at a critical value of the hyperparameter. The marginal likelihood can be expressed in terms of the free energies of the ap riori and a posteriori probability distributions. This means that the marginal likelihood is not differentiable at a critical value of the hyperparameter present in the ap riori probability distribution. In the context of the expectation-maximization algorithm, which is a powerful method for maximizing the marginal likelihood, the marginal likelihood is differentiable at any value of the hyperparameter. In the present paper, we investigate Bayesian image restoration with hyperparameters estimated by maximizing the marginal likelihood when we adopt a Q-state Potts model 2) as the ap rioriprobability distribution. The free energy of the Q-state Potts model for Q≥3 is not differentiable at a certain value of the hyperparameter when we analyze the model by means of advanced mean-field approximations. The maximization of the marginal likelihood is achieved by using loopy belief propaga

Phonon thermodynamics versus electron-phonon models

Applying the path integral formalism we study the equilibrium thermodynamics of the phonon field both in the Holstein and in the Su-Schrieffer-Heeger models. The anharmonic cumulant series, dependent on the peculiar source currents of the e-ph models, have been computed versus temperature in the case of a low energy oscillator. The cutoff in the series expansion has been determined, in the low $T$ limit, using the constraint of the third law of thermodynamics. In the Holstein model, the free energy derivatives do not show any contribution ascribable to the e-ph anharmonic effect. We find signatures of large e-ph anharmonicities in the Su-Schrieffer-Heeger model mainly visible in the temperature dependent peak displayed by the phonon heat capacity.

EQUILIBRIUM CURRENTS IN A QUANTUM DOUBLE RING SYSTEM: A NON-TRIVIAL ROLE OF SYSTEM-RESERVOIR COUPLING

Amperes law states that the magnetic moment of a ring is given by current times the area enclosed. Also from equilibrium statistical mechanics, it is known that magnetic moment is the derivative of free energy with respect to magnetic field. In this work, we analyze a quantum double ring system interacting with a reservoir. A simple S-Matrix model is used for system-reservoir coupling. We see complete agreement between the aforesaid two definitions when coupling between system and reservoir is weak, increasing the strength of coupling parameter, however, leads to disagreement between the two, thereby signifying the important role played by the coupling parameter in mesoscopic systems.

Overcoming free energy barriers using unconstrained molecular dynamics simulations.

Association of unconstrained molecular dynamics (MD) and the formalisms of thermodynamic integration and average force [Darve and Pohorille, J. Chem. Phys. 115, 9169 (2001)] have been employed to determine potentials of mean force. When implemented in a general MD code, the additional computational effort, compared to other standard, unconstrained simulations, is marginal. The force acting along a chosen reaction coordinate xi is estimated from the individual forces exerted on the chemical system and accumulated as the simulation progresses. The estimated free energy derivative computed for small intervals of xi is canceled by an adaptive bias to overcome the barriers of the free energy landscape. Evolution of the system along the reaction coordinate is, thus, limited by its sole self-diffusion properties. The illustrative examples of the reversible unfolding of deca-L-alanine, the association of acetate and guanidinium ions in water, the dimerization of methane in water, and its transfer across the water liquid-vapor interface are examined to probe the efficiency of the method.

Free energy simulations: use of reverse cumulative averaging to determine the equilibrated region and the time required for convergence.

A method is proposed for improving the accuracy and efficiency of free energy simulations. The essential idea is that the convergence of the relevant measure (e.g., the free energy derivative in thermodynamic integration) is monitored in the reverse direction starting from the last frame of the trajectory, instead of the usual approach, which begins with the first frame and goes in the forward direction. This simple change in the use of the simulation data makes it straightforward to eliminate the contamination of the averages by contributions from the equilibrating region. A statistical criterion is introduced for distinguishing the equilibrated (production) region from the equilibrating region. The proposed method, called reverse cumulative averaging, is illustrated by its application to the well-studied case of the alchemical free energy simulation of ethane to methanol.

Kinetics of SN1 Reactions in Binary Liquid Mixtures near the Critical Point of Solution

We have measured the rates of the SN1 hydrolysis reactions of 2-chloro-2-methylbutane in isobutyric acid + water and 2-bromo-2-methylpropane in trietylamine + water near their respective consolute points. In the former mixture, two phases coexist below the critical solution temperature, while in the latter, two phases coexist above the critical solution temperature. Because each hydrolysis reaction produced a strong electrolyte among its products, we could follow the progress of the reaction by making measurements of the conductance. To interpret these measurements properly, however, we had to take into account the following phenomena: (1) the effect of the composition on the critical temperature, (2) the effect of the liquid−liquid-phase transition when it occurred during the course of the run, and (3) the competition between the effects of temperature and the effects of concentration on the response of the conductance probe. With these phenomena accounted for, however, we found in the case of both reactions that the specific rate of hydrolysis slowed at temperatures above the critical solution temperature and accelerated at temperatures below it. From the point of view of thermodynamics, the causative factor in slowing down is the derivative of the reaction Gibbs free energy with respect to the extent of reaction, which goes to zero at the critical point. In contrast, speeding up has no ready theoretical explanation. The experimental results imply, nevertheless, that speeding up is not confined to the two-phase side of the phase diagram.

Landau theory of domain walls for one-dimensional asymmetric potentials

Investigation of a great number of physical systems shows that a Landau free energy of the form F(φ)=Hφ+(A/2)φ2+(B/3)φ3+(C/4)φ4 describes a first-order phase transition in an internal or external field H. To study the formation of static domain walls in these systems we include a spatial gradient (Ginzburg) term of the scalar order parameter φ. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation exactly for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the two different phases which are created in the system. In addition, we obtain soliton lattice (domain wall array) solutions, calculate their formation energy and the asymptotic interaction between the solitons. By introducing certain transformations, we show that the solutions obtained here can be used to study domain walls in other physical systems such as described by asymmetric double Morse potentials. Finally, we apply our results to the specific cases of liquid crystals and the jam phenomena in traffic flows.

Cubic curves from matrix models and generalized Konishi anomalies

We study the matrix model/gauge theory connection for three different N=1 models: U(N) x U(N) with matter in bifundamental representations, U(N) with matter in the symmetric representation, and U(N) with matter in the antisymmetric representation. Using Ward identities, we explicitly show that the loop equations of the matrix models lead to cubic algebraic curves. We then establish the equivalence of the matrix model and gauge theory descriptions in two ways. First, we derive generalized Konishi anomaly equations in the gauge theories, showing that they are identical to the matrix-model equations. Second, we use a perturbative superspace analysis to establish the relation between the gauge theories and the matrix models. We find that the gauge coupling matrix for U(N) with matter in the symmetric or antisymmetric representations is_not_ given by the second derivative of the matrix-model free energy. However, the matrix-model prescription can be modified to give the gauge coupling matrix.

Free Energy Perturbation Calculations with Combined QM/MM Potentials Complications, Simplifications, and Applications to Redox Potential Calculations

The present work deals with a formal discussion on complications associated with using combined quantum mechanical and molecular mechanical (QM/MM) potentials in free energy perturbation simulations. Because quantum mechanical potentials are not trivially separable and because of the difficulty associated with QM calculations for “fractional” electronic Hamiltonians, conventional computational strategies cannot be straightforwardly applied to free energy simulations with QM/MM potentials. Here, we propose a unique coupling scheme, termed the “dual-topology-single-coordinate” scheme, in which the two chemical states are forced to adopt the same set of Cartesian coordinates during the perturbation simulation; the method is exact because the free energy difference is independent of the coupling path, although a formal proof is also given. The scheme combines the merits of the conventional dual- and single-topology approaches: both coupling potential and free energy derivative are straightforward to compute ...

Direct Derivation of the Interfacial Free Energy of Charged Colloids from (N,V,T) Monte Carlo Simulations

(N,V,T) Monte Carlo simulations are used to simulate the equilibrium distribution of the neutralizing monovalent counterions condensed in the vicinity of two charged colloids of various shapes (spheres, disks, and cylinders). The corresponding configurational entropy is used, in addition to the total electrostatic energy, to estimate the free energy of a pair of colloids at different distances and relative orientations (in the case of nonspherical colloids). By calculating the balance between energetic and entropic contributions to the total free energy of charged colloids, this study illustrates the large influence of the entropy on the net repulsion between colloids carrying electric charges of the same sign and neutralized by monovalent counterions. This procedure leads also to an effective pair potential useful for describing dispersions of charged colloids in the one-component plasma framework, implicitly including the contribution of the labile counterions. The treatment is based on a pairwise additivity of the effective potential. The validity of this hypothesis is carefully examined for the case of three charged cylinders.

Energetics and Self-Assembly of Amphipathic Peptide Pores in Lipid Membranes

We present a theoretical study of the energetics, equilibrium size, and size distribution of membrane pores composed of electrically charged amphipathic peptides. The peptides are modeled as cylinders (mimicking α-helices) carrying different amounts of charge, with the charge being uniformly distributed over a hydrophilic face, defined by the angle subtended by polar amino acid residues. The free energy of a pore of a given radius, R, and a given number of peptides, s, is expressed as a sum of the peptides’ electrostatic charging energy (calculated using Poisson-Boltzmann theory), and the lipid-perturbation energy associated with the formation of a membrane rim (which we model as being semitoroidal) in the gap between neighboring peptides. A simple phenomenological model is used to calculate the membrane perturbation energy. The balance between the opposing forces (namely, the radial free energy derivatives) associated with the electrostatic free energy that favors large R, and the membrane perturbation term that favors small R, dictates the equilibrium properties of the pore. Systematic calculations are reported for circular pores composed of various numbers of peptides, carrying different amounts of charge (1–6 elementary, positive charges) and characterized by different polar angles. We find that the optimal R's, for all (except, possibly, very weakly) charged peptides conform to the “toroidal” pore model, whereby a membrane rim larger than ∼1 nm intervenes between neighboring peptides. Only weakly charged peptides are likely to form “barrel-stave” pores where the peptides essentially touch one another. Treating pore formation as a two-dimensional self-assembly phenomenon, a simple statistical thermodynamic model is formulated and used to calculate pore size distributions. We find that the average pore size and size polydispersity increase with peptide charge and with the amphipathic polar angle. We also argue that the transition of peptides from the adsorbed to the inserted (membrane pore) state is cooperative and thus occurs rather abruptly upon a change in ambient conditions.

Kinetics of phase separation in iron-based ternary alloys. II. Numerical simulation of phase separation in Fe–Cr–X(X=Mo, Cu) ternary alloys

Numerical simulations were performed using a model based on the Cahn–Hilliard equation in order to investigate asymptotic behavior of a minor element associated with phase decomposition of the major element in Fe–Cr–Mo and Fe–Cr–Cu ternary alloys. Bifurcation of peaks of Mo along peak tops of Cr concentration occurs in an Fe-40at.%Cr–5at.%Mo alloy at temperatures between 750 and 850 K. Bifurcation of peaks of Cr is also shown in an Fe–40at.%Mo–5at.%Cr alloy. The asymptotic behavior of Mo or Cr along a trajectory of a peak top of Cr or Mo concentration depends on the sign of the second derivative of the chemical free energy with respect to the concentrations of Cr and Mo. In an Fe–40at.%Cr–5at.%Cu alloy,the amplitude of Cu along peak tops of Cr concentration decreases with time, Simulated asymptotic behavior of Mo and Cr in Fe–Cr–Mo ternary alloys and Cu in an Fe–Cr–Cu alloy is in good agreement with that predicted by the theory proposed by the present authors.

CALCULATING ACCURATE REDOX POTENTIALS IN ENZYMES WITH A COMBINED QM/MM FREE ENERGY PERTURBATION APPROACH

An approach for computing accurate redox potentials in enzymes is developed based on the free energy perturbation technique in a QM/MM framework. With an appropriate choice of the QM level and QM/MM coupling scheme, the intermolecular interaction between the redox center and the protein environment can be adequately described; the speed of QM/MM methods also allows a sufficient configurational sampling for the convergence of free energy derivatives. Following the implementation into the simulation package CHARMM, the method was tested with an application to the first reduction potential of FAD in cholesterol oxidase (Chox). In addition to an accurate QM level and adequate conformational samplings, the effect of long-range electrostatic interactions due to the bulk solvent was also found to be essential. Using a semi-empirical density functional theory (SCC-DFTB) as the QM level, and a multi-stage charge-scaling scheme based on Poisson–Boltzmann calculations for the solvation effect, satisfactory agreements with experimental measurements were obtained. The study of Chox also indicates that large errors in the calculated redox potential might arise if changes in the conformational properties of the protein during the redox process are not taken into account, such as in energy minimization type of studies based on only the X-ray structure of the enzyme in one redox state.