Mechanical Equation Of State Research Articles

From the Fokker–Planck equation to a contact Hamiltonian system

Abstract The Fokker–Planck equation is one of the fundamental equations in nonequilibrium statistical mechanics, and this equation is known to be derived from the Wasserstein gradient flow equation with a free energy. This gradient flow equation describes relaxation processes and is formulated on a Riemannian manifold. Meanwhile contact Hamiltonian systems are also known to describe relaxation processes. Hence a relation between these two equations is expected to be clarified, which gives a solid foundation in geometric statistical mechanics. In this paper a class of contact Hamiltonian systems is derived from a class of the Fokker–Planck equations on Riemannian manifolds. In the course of the derivation, the Fokker–Planck equation is shown to be written as a diffusion equation with a weighted Laplacian without any approximation, which enables to employ a theory of eigenvalue problems.

The Nanoscale Wetting Parameter and Its Role in Interfacial Phenomena: Phase Transitions in Nanopores.

Through analysis of the statistical mechanical equations for a thin adsorbed film (gas, liquid, or solid) on a solid substrate or confined within a pore, it is possible to express the equilibrium thermodynamic properties of the film as a function of just two dimensionless parameters: a nanoscale wetting parameter, αw, and pore width, H*. The wetting parameter, αw, is defined in terms of molecular parameters for the adsorbed film and substrate and so is applicable at the nanoscale and for films of any phase. The main assumptions in the treatment are that (a) the substrate structure is not significantly affected by the adsorbed layer and (b) the diameter of the adsorbate molecules is not very small compared to the spacing of atoms in the solid substrate. We show that different surface geometries of the substrate (e.g., slit, cylindrical, and spherical pores) and various models of wall heterogeneity can be accounted for through a well-defined correction to the wetting parameter; no new dimensionless variables are introduced. Experimental measurements are reported for contact angles for various liquids on several planar substrates and are shown to be closely correlated with the nanoscale wetting parameter. We apply this approach to phase separation in nanopores of various geometries. Molecular simulation results for the phase diagram in confinement, obtained by the flat histogram Monte Carlo method, are reported and are shown to be closely similar to experimental results for capillary condensation, melting, and the triple point. The value of the wetting parameter, αw, is shown to determine the qualitative behavior (e.g., increase vs decrease in the melting temperature, capillary condensation vs evaporation), whereas the pore width determines the magnitude of the confinement effect. The triple point temperature and pressure for the confined phase are always lower than those for the bulk phase for all cases studied.

Multiscale modelling of fracture in graphene sheets

Most of the continuum scale processes, such as fracture, plasticity, etc., trace their origin to atomistic scale phenomena. To gain deeper insights into these processes, one needs to understand the behaviour of materials through the lens of multiscale methods. In this manuscript, we study the problem of fracture crack propagation in graphene sheets through a sequential multiscaling technique. The continuum-mechanical smoothed particle hydrodynamics (SPH) is coupled with the atomistic scale molecular dynamics (MD) simulations through proper constitutive modelling — the non-linear material properties and the mechanical equation of state which serve as the inputs to the SPH model are evaluated directly from the MD simulations. Such handshaking ensures that the continuum-scale SPH model is able to faithfully reproduce the atomistic scale stress–strain behaviour until failure. Using a pre-notched continuum scale graphene sheet, we show that the mode-I stress intensity factor obtained from our SPH model agrees well with the published literature. We subsequently study crack propagation in pre-notched graphene sheets, where the influence of the orientation of the notch is evaluated. Lastly, we take the case of a continuum-scale graphene sheet having randomly oriented cracks and identify the changes in the stress–strain behaviour vis-à-vis a pristine graphene sheet.

Experimental Verification of the Mechanical State Equation Adequacy for Description of the Creep of Some Polymer Materials under Non-stationary Modes of Static and Cyclic Loading in Plain Stress

The equation of mechanical condition for the description of deformation of polymer and polymer composite materials under complex modes of long static and cyclic loading in condition of the plain stress state is developed. The experimental data show its efficiency. The means of accounting the influence of natural aging and the damage accumulation data of a material are suggested. The opportunity of using of a simplified variants of the equation is shown.

The thermodynamics of the van der Waals dimer and trimer formation in gaseous carbon dioxide and methane and their heterodimers with dinitrogen

The equilibrium thermochemical properties of the dimers and trimers of the greenhouse gases carbon dioxide and methane, as well as their heterodimers with dinitrogen, have been calculated from the CO2 and CH4 equations of state (EOSs) and statistical mechanics. The standard dimer and trimer equilibrium constants, along with the respective ΔHoandΔSo values are reported for CO2 and CH4 between 220 and 260, and 95 and 135 K, respectively, and for the heterodimers between 100 and 250 K. Statistical mechanical calculations based on experimental and computational data of the dimers provide qualitative agreement with the EOS results. The equilibrium thermochemical properties of the CH4 dimer and the CO2–N2 and CH4–N2 heterodimers are calculated using a Morse potential to represent pairwise interactions in an “atom–diatom” model that accounts for both bound and metastable rovibrational states.

Dynamic overlap concentration scale of active colloids.

By introducing the notion of a dynamic overlap concentration scale, we identify additional universal features of the mechanical properties of active colloids. We codify these features by recognizing that the characteristic length scale of an active particle's trajectory, the run length, introduces a concentration scale ϕ^{*}. Large-scale simulations of repulsive active Brownian particles (ABPs) confirm that this run-length dependent concentration, the trajectory-space analog of the overlap concentration in polymer solutions, delineates distinct concentration regimes in which interparticle collisions alter particle trajectories. Using ϕ^{*} and concentration scales associated with colloidal jamming, the mechanical equation of state for ABPs collapses onto a set of principal curves that contain several overlooked features. The inclusion of these features qualitatively alters previous predictions of the behavior for active colloids, as we demonstrate by computing the spinodal for a suspension of purely repulsive ABPs. Our findings suggest that dynamic overlap concentration scales should help unravel the behavior of active and driven systems.

Hypervolume scalarization for shape optimization to improve reliability and cost of ceramic components

In engineering applications one often has to trade-off among several objectives as, for example, the mechanical stability of a component, its efficiency, its weight and its cost. We consider a biobjective shape optimization problem maximizing the mechanical stability of a ceramic component under tensile load while minimizing its volume. Stability is thereby modeled using a Weibull-type formulation of the probability of failure under external loads. The PDE formulation of the mechanical state equation is discretized by a finite element method on a regular grid. To solve the discretized biobjective shape optimization problem we suggest a hypervolume scalarization, with which also unsupported efficient solutions can be determined without adding constraints to the problem formulation. FurthIn this section, general properties of the hypervolumeermore, maximizing the dominated hypervolume supports the decision maker in identifying compromise solutions. We investigate the relation of the hypervolume scalarization to the weighted sum scalarization and to direct multiobjective descent methods. Since gradient information can be efficiently obtained by solving the adjoint equation, the scalarized problem can be solved by a gradient ascent algorithm. We evaluate our approach on a 2 D test case representing a straight joint under tensile load.

Expansion of Wave Rays and Fronts in Media with Inhomogeneous Structure

In the article on the principles of Fermet, Huygens obtaine the differential equations in the form of Hamilton, which describe the ray trajectories and wave fronts in inhomogeneous media. Established that the vector of Poynting-Umov's determining the direction of energy propagation in inhomogeneous medium is coincident with the vector tangent to the ray. In the second part of the article established that the equations of the theory of rays' propagation in inhomogeneous media have the form of equations of nonlinear dynamics and describe the emergence of deterministic chaos in the geometry of rays for a wide variety of types of heterogeneous structures. In this case, the rays behave randomly and their description you must go to the description based on the theory of random functions and fields. In the third part of the paper is considered a model which is equivalent to the random medium and the calculation of the coordinates of the ray (the mathematical expectation and correlation functions). Understanding of these characteristics gives information about the behavior of the trajectories of the rays for these models of media. The description of the behavior of rays on the basis of the equations of statistical mechanics is discussed in the article for functions of Markov's type.

The pressure in interfaces having cylindrical geometry.

While much work has been reported on the statistical mechanics and molecular simulation of interfaces of planar and spherical geometries, very little has been published on the interfaces of cylindrical geometry. The cylindrical geometry is important for the study of cylindrical micelles and particularly for nano-phases confined within cylindrical pores since the most well-defined porous materials (e.g., carbon and silicon nanotubes, SBA-15 and KIT-6 silicas) that are presently available are of this geometry. In this work, we derive the statistical mechanical equations for the pressure tensor for an interfacial region of cylindrical geometry via the virial route and for the condition of mechanical (hydrostatic) equilibrium. We also report the equation for the surface tension via the mechanical route. Monte Carlo and molecular dynamics simulation results are obtained for two example systems involving a fluid nano-phase of Lennard-Jones argon: a gas-liquid interface of cylindrical geometry and a confined nano-phase within a cylindrical carbon pore. All three diagonal elements of the pressure tensor are reported in each case, the component normal to the interface, PN = Pρρ , and the two tangential components PTϕ = Pϕϕ and PTz = Pzz , where (ρ, z, ϕ) are the usual cylindrical polar coordinates. For the cylindrical pore, the tangential pressures, Pϕϕ and Pzz , show strong compression in the adsorbed layers, as has been found in slit-shaped and spherical pores.

Quantum Machine Learning in Chemical Compound Space.

Rather than numerically solving the computationally demanding equations of quantum or statistical mechanics, machine learning methods can infer approximate solutions, interpolating previously acquired property data sets of molecules and materials. The case is made for quantum machine learning: An inductive molecular modeling approach which can be applied to quantum chemistry problems.

A complete equation of state for non-ideal condensed phase explosives

The objective of this work is to improve the robustness and accuracy of numerical simulations of both ideal and non-ideal explosives by introducing temperature dependence in mechanical equations of state for reactants and products. To this end, we modify existing mechanical equations of state to appropriately approximate the temperature in the reaction zone. Mechanical equations of state of the Mie-Grüneisen form are developed with extensions, which allow the temperature to be evaluated appropriately and the temperature equilibrium condition to be applied robustly. Furthermore, the snow plow model is used to capture the effect of porosity on the reactant equation of state. We apply the methodology to predict the velocity of compliantly confined detonation waves. Once reaction rates are calibrated for unconfined detonation velocities, simulations of confined rate sticks and slabs are performed, and the experimental detonation velocities are matched without further parameter alteration, demonstrating the predictive capability of our simulations. We apply the same methodology to both ideal (PBX9502, a high explosive with principal ingredient TATB) and non-ideal (EM120D, an ANE or ammonium nitrate based emulsion) explosives.

Thermodynamic properties of lubricant/refrigerant mixtures using statistical mechanics and artificial intelligence

In this research, the volumetric properties of sixteen lubricant/refrigerant mixtures are predicted using the developed statistical mechanical equation of state at a broad range of temperatures, pressures and mole fractions. The equation of state have been examined using corresponding states correlation based on just one input parameter (density at room temperature) as scaling constants. Besides, the artificial neural network (ANN) based on back propagation training with 19 neurons in hidden layer was tested to predict the behavior of binary mixtures of lubricant/refrigerant. The AADs% of a collection of 3961 data points for all binary mixtures using the EOS and the ANN at various temperatures and mole fractions are 0.92% and 0.34%, respectively. Furthermore, the excess molar volume of all binary mixtures calculated from obtained densities of ANN, and the results shown these properties have good harmony with literature.

Communication: Kirkwood-Buff integrals in the thermodynamic limit from small-sized molecular dynamics simulations.

We present an accurate and efficient method to obtain Kirkwood-Buff (KB) integrals in the thermodynamic limit from small-sized molecular dynamics simulations. By introducing finite size effects into integral equations of statistical mechanics, we derive an analytical expression connecting the KB integrals of the bulk system with the fluctuations of the number of molecules in the corresponding closed system. We validate the method by calculating the activity coefficients of aqueous urea mixtures and the KB integrals of Lennard-Jones fluids. Moreover, our results demonstrate how to identify simulation conditions under which computer simulations reach the thermodynamic limit.

Structure, thermodynamic and surface properties of liquid metals investigated by Square well potential

The auto correlation function, S(k) (where k = 4π/λsinθ) and its Fourier component g(r) are important quantities characterizing the structure of a liquid. Experimentally these quantities have been determined using neutron or X-ray scattering intensities. Wertheim's solution of the fundamental statistical mechanical equation given by Percus and Yevick for hard spheres mixture is invoked with square well attractive part as a perturbation tail to evaluate the direct correlation function C(k) in momentum space through which structure factor, S(k) of liquid metals were derived. Radial distribution function, g(r) is obtained by Fourier analysis of computed S(k) from which the coordination numbers of ten liquid metals were computed. Surface tension of atomic liquids in terms of diffusion coefficient has been determined under square well interaction. The computed values of structural dependent coordination number, diffusion coefficient and surface tension of these liquid metals are compared with the available experimental results and a good agreement is found between the computed and experimental results

Statistical mechanics and artificial intelligence to model the thermodynamic properties of pure and mixture of ionic liquids

In this paper, the volumetric properties of pure and mixture of ionic liquids are predicted using the developed statistical mechanical equation of state in different temperatures, pressures and mole fractions. The temperature dependent parameters of the equation of state have been calculated using corresponding state correlation based on only the density at 298.15K as scaling constants. The obtained mean of deviations of modified equation of state for density of all pure ionic liquids for 1662 data points was 0.25%. In addition, the performance of the artificial neural network (ANN) with principle component analysis (PCA) based on back propagation training with 28 neurons in hidden layer for predicting of behavior of binary mixtures of ionic liquids was investigated. The AADs of a collection of 568 data points for all binary systems using the EOS and the ANN at various temperatures and mole fractions are 1.03% and 0.68%, respectively. Moreover, the excess molar volume of all binary mixtures is predicted using obtained densities of EOS and ANN, and the results show that these properties have good agreement with literature.

Canonical ensemble in non-extensive statistical mechanics

The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, using the maximization of entropy. In this paper we investigate the canonical ensemble in the non-extensive statistical mechanics using a more traditional way, by considering a small system interacting with a large reservoir via short-range forces. The reservoir is characterized by generalized entropy instead of the Boltzmann–Gibbs entropy. Assuming equal probabilities for all available microstates we derive the equations of the non-extensive statistical mechanics. Such a procedure can provide deeper insight into applicability of the non-extensive statistics.

Investigation of structure, thermodynamic and surface properties of liquid metals using square well potential

In the present paper surface tension, Debye temperature, coordination numbers along with microscopic correlations of ten liquid metals are determined using square-well model of correlation functions. Wertheim’s solution of the fundamental statistical mechanical equation given by Percus and Yevick for hard spheres is invoked with a square well attractive part as a perturbation tail to get exact solution of the direct correlation function, C(k) in momentum space and the analytical expressions are obtained for structure factor, S(k). These expressions are then used to predict static structure factors for ten liquid metals, leading to fair agreement with experimental data. Radial distribution function g(r) is obtained by Fourier analysis of computed S(k), from which the coordination numbers and the nearest neighbor distances of liquid metals are evaluated. Computed coordination numbers and surface properties of liquid metals using such a simple technique are in good agreement with experimental results.

Pressure-volume-temperature and excess molar volume prediction of amorphous and crystallizable polymer blends by equation of state

In this work the statistical mechanical equation of state was developed for volumetric properties of crystalline and amorphous polymer blends. The Ihm–Song–Mason equations of state (ISMEOS) based on temperature and density at melting point (Tm and ρm) as scaling constants were developed for crystalline polymers such as poly(propylene glycol)+poly(ethylene glycol)-200 (PPG+PEG-200), poly(ethylene glycol) methyl ether-300 (PEGME-350)+PEG-200 and PEGME-350+PEG-600. Furthermore, for amorphous polymer blends containing poly(2,6-dimethyl-1,4-phenylene oxide) (PPO)+polystyrene (PS) and PS+poly(vinylmethylether) (PVME), the density and surface tension at glass transition (ρg and γg) were used for estimation of second Virial coefficient. The calculation of second Virial coefficients (B2), effective van der Waals co-volume (b) and correction factor (α) was required for judgment about applicability of this model. The obtained results by ISMEOS for crystalline and amorphous polymer blends were in good agreement with the experimental data with absolute average deviations of 0.84% and 1.04%, respectively.

Emergence of patterns in random processes. II. Stochastic structure in random events.

Random events can present what appears to be a pattern in the length of peak-to-peak sequences in time series and other point processes. Previously, we showed that this was the case in both individual and independently distributed processes as well as for Brownian walks. In addition, we introduced the use of the discrete form of the Langevin equation of statistical mechanics as a device for connecting the two limiting sets of behaviors, which we then compared with a variety of observations from the physical and social sciences. Here, we establish a probabilistic framework via the Smoluchowski equation for exploring the Langevin equation and its expected peak-to-peak sequence lengths, and we introduce a concept we call "stochastic structure in random events," or SSRE. We extend the Brownian model to include antipersistent processes via autoregressive (AR) models. We relate the latter to describe the behavior of Old Faithful Geyser in Yellowstone National Park, and we devise a further test for the validity of the Langevin and AR models. Given our analytic results, we show how the Langevin equation can be adapted to describe population cycles of three to four years observed among many mammalian species in biology.

An extended expression for the radial distribution function of HFD-like fluid for investigation of many-body and quantum effects in some transport properties

The previous expression of Abbaspour and Keyvanloo (2013) for radial distribution function (RDF) of the HFD-like fluid has been developed as a continuous function of reduced interparticle distance, temperature, and density. This expression reproduces the RDF data in the range 1.1≤T⁎≤4.4 and 0.43≤ρ⁎≤0.82 with fairly good accuracy. We have also calculated shear viscosity, self-diffusion coefficient, and thermal conductivity of two-body HFD-like potential using the statistical mechanical equations for liquid neon. Quantum corrections have been also considered using the Feynman–Hibbs (FH) approach for the HFD-like system. To take many-body forces into account, the simple three-body potentials of Wang and Sadus (2006) and Hauschild and Prausnitz (1993) have been used with the two-body HFD-like potential without requiring an expensive three-body calculation. The results show that the quantum and three-body effects improve the prediction of the transport properties of the HFD-like fluid. We have also concluded that our (quantum and many-body) corrected HFD-like potential is superior than a recently quantum LJ model.