Second Derivatives Of Energy Research Articles

Ion diffusion retarded by diverging chemical susceptibility

For first-order phase transitions, the second derivatives of Gibbs free energy (specific heat and compressibility) diverge at the transition point, resulting in an effect known as super-elasticity along the pressure axis, or super-thermicity along the temperature axis. Here we report a chemical analogy of these singularity effects along the atomic doping axis, where the second derivative of Gibbs free energy (chemical susceptibility) diverges at the transition point, leading to an anomalously high energy barrier for dopant diffusion in co-existing phases, an effect we coin as super-susceptibility. The effect is realized in hydrogen diffusion in vanadium dioxide (VO2) with a metal-insulator transition (MIT). We show that hydrogen faces three times higher energy barrier and over one order of magnitude lower diffusivity when it diffuses across a metal-insulator domain wall in VO2. The additional energy barrier is attributed to a volumetric energy penalty that the diffusers need to pay for the reduction of latent heat. The super-susceptibility and resultant retarded atomic diffusion are expected to exist universally in all phase transformations where the transformation temperature is coupled to chemical composition, and inspires new ways to engineer dopant diffusion in phase-coexisting material systems.

Hydrodynamics and instabilities of relativistic superfluids at finite superflow

We study the linear response of relativistic superfluids with a non-zero superfluid velocity. For sufficiently large superflow, an instability develops via the crossing of a pole of the retarded Green’s functions to the upper half complex frequency plane. We show that this is caused by a local thermodynamic instability, i.e. when an eigenvalue of the static susceptibility matrix (the second derivatives of the free energy) diverges and changes sign. The onset of the instability occurs when ∂ζ(nsζ) = 0, with ζ the norm of the superfluid velocity and ns the superfluid density. The Landau instability for non-relativistic superfluids such as Helium 4 also coincides with the non-relativistic version of this criterion. We then turn to gauge/gravity duality and show that this thermodynamic instability criterion applies equally well to strongly-coupled superfluids. In passing, we compute holographically a number of transport coefficients parametrizing deviations out-of-equilibrium in the hydrodynamic regime and demonstrate that the gapless quasinormal modes of the dual planar black hole match those predicted by superfluid hydrodynamics.

Consistent Analytical Second Derivatives of the Kohn-Sham DFT Energy in the Framework of the Conductor-Like Screening Model through Gaussian Charge Distributions.

The use of implicit solvation models such as the conductor-like screening model (COSMO) in quantum chemical calculations is very common, as both a rough estimate of solvation effects as well as a general tool for stabilizing ionic molecular structures. In order to generate a smooth potential energy surface as well as consistent gradients, it is necessary to apply the Gaussian charge model (GCM) for the COSMO charges. This work introduces an efficient implementation for consistent analytical second derivatives of the electronic energy with COSMO-GCM in the framework of the Kohn-Sham density functional theory. This is used to investigate the infrared spectroscopy of amino acids in aqueous solution, where the impact of pH on the molecular structure and vibrational spectra is examined. Furthermore, the structure and stability of selected all-metal aromatic cluster ions are assessed.

Performance of machine learning algorithms in spectroscopic ellipsometry data analysis of ZnTiO3 nanocomposite

In this research, the optical properties of the PVP: ZnTiO3 nanocomposite are studied using the spectroscopic ellipsometry technique. The preparation procedure of the ZnTiO3 nanocomposite is explained in detail. The absorbance/transmittance, surface morphology, structural information, chemical identification, and surface topography of the ZnTiO3 nanocomposite is studied using UV–Vis spectroscopy, field-emission scanning electron microscopy, Raman spectroscopy, Fourier transform infra-red, and atomic force microscopy, respectively. The ellipsometry method is used to obtain the spectra of the real and imaginary parts of the dielectric function and refractive index in the photon energy range of 0.59–4.59 eV. Moreover, using two machine learning algorithms, namely artificial neural network and support vector regression methods, the ellipsometric parameters ψ and Δ are analyzed and compared with non-linear regression. The error and accuracy of each three methods, as well as the time required for their execution, are calculated to compare their suitability in the ellipsometric data analysis. Also, the absorption coefficient was used to determine the band gap energy of the ZnTiO3 nanocomposite, which is found to be 3.83 eV. The second-energy derivative of the dielectric function is utilized to identify six critical point energies of the prepared sample. Finally, the spectral-dependent optical loss function and optical conductivity of the ZnTiO3 nanocomposite are investigated.

On the nature of screening in charge-regulated macroion solutions.

We present a derivation of the screening length for a solution containing a charge-regulated macroion, e.g. protein, with its counterions. We show that it can be obtained directly from the second derivatives of the total free energy by taking recourse to the "uncertainty relation" of the Legendre transform, which connects the Hessians or the local curvatures of the free energy as a function of density and its Legendre transform, i.e., osmotic pressure, as a function of chemical potentials. Based on the Fowler-Guggenheim-Frumkin model of charge regulation, we then analyze the "screening resonance" and the "overscreening" of the screening properties of the charge-regulated macroion solution.

Structural Fluctuation, Relaxation, and Folding of Protein: An Approach Based on the Combined Generalized Langevin and RISM/3D-RISM Theories

In 2012, Kim and Hirata derived two generalized Langevin equations (GLEs) for a biomolecule in water, one for the structural fluctuation of the biomolecule and the other for the density fluctuation of water, by projecting all the mechanical variables in phase space onto the two dynamic variables: the structural fluctuation defined by the displacement of atoms from their equilibrium positions, and the solvent density fluctuation. The equation has an expression similar to the classical Langevin equation (CLE) for a harmonic oscillator, possessing terms corresponding to the restoring force proportional to the structural fluctuation, as well as the frictional and random forces. However, there is a distinct difference between the two expressions that touches on the essential physics of the structural fluctuation, that is, the force constant, or Hessian, in the restoring force. In the CLE, this is given by the second derivative of the potential energy among atoms in a protein. So, the quadratic nature or the harmonicity is only valid at the minimum of the potential surface. On the contrary, the linearity of the restoring force in the GLE originates from the projection of the water’s degrees of freedom onto the protein’s degrees of freedom. Taking this into consideration, Kim and Hirata proposed an ansatz for the Hessian matrix. The ansatz is used to equate the Hessian matrix with the second derivative of the free-energy surface or the potential of the mean force of a protein in water, defined by the sum of the potential energy among atoms in a protein and the solvation free energy. Since the free energy can be calculated from the molecular mechanics and the RISM/3D-RISM theory, one can perform an analysis similar to the normal mode analysis (NMA) just by diagonalizing the Hessian matrix of the free energy. This method is referred to as the Generalized Langevin Mode Analysis (GLMA). This theory may be realized to explore a variety of biophysical processes, including protein folding, spectroscopy, and chemical reactions. The present article is devoted to reviewing the development of this theory, and to providing perspective in exploring life phenomena.

Numerical 3D‐bifurcation analysis of star‐shaped crack patterns using the energy method

AbstractThe present research deals with a three‐dimensional (3D) Finite Element Method (FEM) bifurcation analysis based on the global mechanical potential that can be used to find the parameters at which a crack pattern changes. In our case we want to analyze at which point the star‐shaped shrinkage cracks in an aqueous colloidal suspension filled in a glass cylinder change from four to three or two cracks growing. The driving force for the crack growth is shrinkage caused by diffusion controlled drying. The 3D crack front geometry is described efficiently by using a Fourier series approach. Based on steady‐state crack growth, the Fourier coefficients are determined in a first step using an optimization algorithm. As a result, the time dependent crack growth can be determined. In a second step, the bifurcation point is determined by an eigenvalue analysis of the second order derivatives of the potential energy of the system. If the lowest eigenvalue reaches zero the fundamental solution becomes unstable and a transition will occur. Our analysis shows that the transition from four to two cracks is preferred over the transition from four to three cracks.

Thermodynamics of Highly Interacting Blend PCHMA/dPS by TOF-SANS.

We investigate the thermodynamics of a highly interacting blend of poly(cyclohexyl methacrylate)/deuterated poly(styrene) (PCHMA/dPS) with small-angle neutron scattering (SANS). This system is experimentally challenging due to the proximity of the blend phase boundary (>200 °C) and degradation temperatures. To achieve the large wavenumber q-range and flux required for kinetic experiments, we employ a SANS diffractometer in time-of-flight (TOF) mode at a reactor source and ancillary microscopy, calorimetry, and thermal gravimetric analysis. Isothermal SANS data are well described by random-phase approximation (RPA), yielding the second derivative of the free energy of mixing (G″), the effective interaction (χ̅) parameter, and extrapolated spinodal temperatures. Instead of the Cahn-Hilliard-Cook (CHC) framework, temperature (T)-jump experiments within the one-phase region are found to be well described by the RPA at all temperatures away from the glass transition temperature, providing effectively near-equilibrium results. We employ CHC theory to estimate the blend mobility and G″(T) conditions where such an approximation holds. TOF-SANS is then used to precisely resolve G″(T) and χ̅(T) during T-jumps in intervals of a few seconds and overall timescales of a few minutes. PCHMA/dPS emerges as a highly interacting partially miscible blend, with a steep dependence of G″(T) [mol/cm3] = -0.00228 + 1.1821/T [K], which we benchmark against previously reported highly interacting lower critical solution temperature (LCST) polymer blends.

Analytical harmonic vibrational frequencies with VV10-containing density functionals: Theory, efficient implementation, and benchmark assessments.

VV10 is a powerful nonlocal density functional for long-range correlation that is used to include dispersion effects in many modern density functionals, such as the meta-generalized gradient approximation (mGGA), B97M-V, the hybrid GGA, ωB97X-V, and the hybrid mGGA, ωB97M-V. While energies and analytical gradients for VV10 are already widely available, this study reports the first derivation and efficient implementation of the analytical second derivatives of the VV10 energy. The additional compute cost of the VV10 contributions to analytical frequencies is shown to be small in all but the smallest basis sets for recommended grid sizes. This study also reports the assessment of VV10-containing functionals for predicting harmonic frequencies using the analytical second derivative code. The contribution of VV10 to simulating harmonic frequencies is shown to be small for small molecules but important for systems where weak interactions are important, such as water clusters. In the latter cases, B97M-V, ωB97M-V, and ωB97X-V perform very well. The convergence of frequencies with respect to the grid size and atomic orbital basis set size is studied, and recommendations are reported. Finally, scaling factors to allow comparison of scaled harmonic frequencies with experimental fundamental frequencies and to predict zero-point vibrational energy are presented for some recently developed functionals (including r2SCAN, B97M-V, ωB97X-V, M06-SX, and ωB97M-V).

New features of the TFmix code: Thermodynamic properties of electrons in mixtures

New features of the TFmix code: Thermodynamic properties of electrons in mixtures

Theory of multidimensional quantum capacitance and its application to spin and charge discrimination in quantum dot arrays

Quantum states of a few-particle system capacitively coupled to a metal gate can be discriminated by measuring the quantum capacitance, which can be identified with the second derivative of the system energy with respect to the gate voltage. This approach is here generalized to the multivoltage case, through the introduction of the quantum capacitance matrix. The matrix formalism allows us to determine the dependence of the quantum capacitance on the direction of the voltage oscillations in the parameter space and to identify the optimal combination of gate voltages. As a representative example, this approach is applied to the case of a quantum dot array, described in terms of a Hubbard model. Here, we first identify the potentially relevant regions in the multidimensional voltage space with the boundaries between charge stability regions, determined within a semiclassical approach. Then, we quantitatively characterize such boundaries by means of the quantum capacitance matrix. Altogether, this provides a procedure for optimizing the discrimination between states with different particle numbers and/or total spins.

Anharmonic thermo-elasticity of tungsten from accelerated Bayesian adaptive biasing force calculations with data-driven force fields

The elastic properties of tungsten, a ubiquitous material in future energy systems, are investigated up to its melting temperature by means of a data-driven approach. The proposed workflow combines machine learning of the force field and enhanced sampling of the crystalline structure. While the machine learning force field achieves the accuracy of ab initio calculations, its implementation in sampling methods is often limited due to its high computational cost, which is commonly a few orders of magnitude larger than that of traditional potentials. To overcome this limitation, we propose a fast and robust Bayesian sampling scheme aiming at estimating the fully anharmonic free energy of crystalline solids with the help of an improved adaptive biasing force method. This method performs a thermodynamic integration from a harmonic reference system, wherein zero frequencies associated with the periodic boundaries are screened off. The proposed sampling method drastically improves the speed of convergence and overall accuracy. We demonstrate the efficiency of the improved method by calculating the second order derivatives of the free energy, such as the elastic constants, which are performed almost 100 times faster than with the standard methods. The proposed method enables the prediction of the elastic properties of tungsten in the range of temperatures that cannot be investigated experimentally, from 2100 K to the melting temperature. The accuracy and numerical efficiency of the proposed strategy open up many avenues for the reliable prediction of finite-temperature properties of materials, such as the relative stability of structural defects and elastic constants.

Dynamic performance of triple-column bridge pier under barge collision

Growing incidents of pier cracks or even failure of the whole structure induced by ship-pier collision have attracted social attention to the safety of water-crossing bridges. In this study, finite element (FE) models are successfully established to gain insight into the collision between a modified barge and a three-column bridge pier in the non-navigable zone. Firstly, numerical models of the bridge pier and barge are satisfactorily verified against experimental results, through which the dynamic performance of the bridge under typical collision scenarios is investigated with respect to the kinetic energy, internal energy, and impact force. A new performance index, i.e., internal energy power slope (IEPS), calculated from the second derivative of the structural internal energy, is then proposed to prescribe the damage states of the impacted column. The usefulness and applicability of the IEPS are further explored through structural reliability analysis and a parametric study. Moreover, the efficacy of two protective jackets made in alloy honeycomb (AH) and aluminum foam (AF) for enhancing the structural anti-impact capacity is investigated by comparing the associated values of IEPS. It is observed that the proposed IEPS has a strong correlation with the structural damage state, and thus can be favourably used as a new performance metric for analyzing barge-pier collisions. The results also evidence that adopting UHPC (ultra-high performance concrete), densifying stirrup spacing, and adding protective jackets are effective measures to reduce the IEPS and hence improve the structural anti-impact capacity.

Achieving highest Young's modulus in Al-Li by tracing the size and bonding evolution of Li-rich precipitates

For decades, it has been well accepted that every 1 wt.% Li addition to Al will reduce Al alloy's density by 3% and increase its Young's modulus by 6%. However, the fundamental mechanism of modulus improvements stays controversial though all studies agreed that the contribution of such a substantial boosting comes from Li-rich clusters either in solid solution or precipitations. In this study, we experimentally produce nano-sized Li-rich clusters by non-equilibrium solidification using centrifugal casting and trace their evolutions as a function of subsequent heat treatments. High-resolution transmission electron microscopy (HRTEM) reveals a further decrease in the lattice constants of Li-rich regions from the as-cast (0.406 nm), solid solution (0.405 nm) to the aged state (0.401 nm), while Young's modulus of the Al-Li alloy reaches 89.16 GPa. Small-angle neutron scattering (SANS) experiments and first-principle calculations based on density functional theory have shown both the bond strength around precipitates and the size of those Li-rich region dominate Young's modulus. At the beginning, it is volumetric compression due to Li addition that increases modulus, tightening the Al-Al potential curves. In the end, it is the Al-Al and Al-Li valence bonds in Al3Li at large size and high-volume fraction which increase its second derivative of internal energy and thus Young's modulus.

Isothermal and adiabatic elastic constants from virial fluctuations.

We derive expressions for classical isothermal and adiabatic elastic constants for periodic systems with the boundary contributions included explicitly. The potential-dependent part of these expressions is written in terms of potential energies of atomic groups that make up the total potential energy. It is shown that in the thermodynamic limit, the Born term, which depends on the second derivatives of potential energy, can be expressed exactly in terms of equilibrium averages that involve two types of atomic-group virials. As a result, the new form of the Born term involves only first derivatives of either atomic-group or total potential energies. The derived elastic constant expressions involving the two forms of the Born terms are tested and compared using molecular-dynamics simulations of crystalline argon and silicon. For both materials, the elastic constants obtained using the two forms of the Born term are in good agreement. In particular, the new form of the Born term converges to the same value as the original Born term but at a slower rate. The results for silicon also agree well with the results from the previous molecular-dynamics studies.

The second derivative of the electronic energy with respect to the compression scaling factor in the XP-PCM model: Theory and applications to compression response functions of atoms.

We present the analytical theory for the second derivative of the electronic energy with respect to the scaling factor of the compression cavity within the eXtreme pressure polarizable continuum model (XP-PCM) for the study of compressed atomic and molecular systems. The theory has been exploited to study compression response functions describing how the atomic/molecular properties are effected by an external pressure. The response functions considered include the atomic compressibility and the pressure coefficients of the ionization energy (IE) and electron affinity (EA). The theory has been validated by numerical application to compressed neon, argon, and krypton atoms.

Anomalous thermodynamics in a mixed spin-1/2 and spin-1 hexagonal nanowire system

The mixed spin-1/2 Ising model and spin-1 Blume–Capel model in an hexagonal nanowire structure under the presence of crystal field is considered. The free energy is obtained through the transfer matrix technique, which is solved numerically. Our main result lies in the presence of pseudo-transition in low temperature region near the ferrimagnetic/ferromagnetic boundary, due to the influence of a crystal field. The evidence of a pseudo-transition is observed in several quantities. Free energy first derivative quantities like entropy and internal energy show an abrupt but continuous jump, whereas quantities associated with second derivatives of the free energy like the specific heat exhibit a strong sharp peak, quite similar to a second order phase transition. We also investigate magnetization patterns and do not find evidence of spontaneous magnetization. Nevertheless, assuming a small magnetic field, we can induce a magnetization which resembles a spontaneous magnetization at a pseudo-critical temperature.

On-the-Fly Ring-Polymer Molecular Dynamics Calculations of the Dissociative Photodetachment Process of the Oxalate Anion.

The dissociative photodetachment dynamics of the oxalate anion, C2O4H− + hν → CO2 + HOCO + e−, were theoretically studied using the on-the-fly path-integral and ring-polymer molecular dynamics methods, which can account for nuclear quantum effects at the density-functional theory level in order to compare with the recent experimental study using photoelectron–photofragment coincidence spectroscopy. To reduce computational time, the force acting on each bead of ring-polymer was approximately calculated from the first and second derivatives of the potential energy at the centroid position of the nuclei beads. We find that the calculated photoelectron spectrum qualitatively reproduces the experimental spectrum and that nuclear quantum effects are playing a role in determining spectral widths. The calculated coincidence spectrum is found to reasonably reproduce the experimental spectrum, indicating that a relatively large energy is partitioned into the relative kinetic energy between the CO2 and HOCO fragments. This is because photodetachment of the parent anion leads to Franck–Condon transition to the repulsive region of the neutral potential energy surface. We also find that the dissociation dynamics are slightly different between the two isomers of the C2O4H− anion with closed- and open-form structures.

Analytical second derivatives of the free energy in solution by the reference interaction site model self-consistent field explicitly including constrained spatial electron density distribution

The application of analytical derivative methods to solution systems is important because several chemical reactions occur in solution. The reference interaction site model (RISM) is one of the solvation theories used to study solution systems and has shown good performance, especially in the polar solvent systems. Although the analytical first derivative based on the RISM coupled with quantum methods (RISM-SCF) has already been derived, the analytical second derivative has not been proposed yet. Therefore, in this study, the analytical second derivative was derived using RISM-SCF explicitly including constrained spatial electron density distribution (RISM-SCF-cSED). The performance of this method was validated with the Hessian calculations of formaldehyde and para-nitroaniline in solution, and the results demonstrated that the method accurately calculated frequency values at a small computational cost.

High-order series expansion of non-Hermitian quantum spin models

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-symmetry. To this end we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region.