Green-Kubo Method Research Articles

Thermal conductance bottleneck of a three dimensional graphene-CNT hybrid structure: a molecular dynamics simulation.

Three dimensional (3D) graphene-CNT hybrid structures (GCNTs) are promising materials for applications including capacitors and gas storage and separation devices, however until now their thermal conductance mechanism has scarcely been studied. These hybrid nanomaterials are particularly suitable as next-generation thermal interface materials due to the excellent thermal properties of carbon nanotubes and single atomic layer graphene. In this paper, the out-of-plane thermal conductivities of GCNTs, graphene nanomesh (GNM), and graphene sheets are investigated using molecular dynamics (MD) simulations which apply the Green-Kubo method. Distinct from GNMs and graphene sheets, the GCNTs exhibit a relatively high out-of-plane thermal conductivity, stemming from the CNTs' ability to accelerate the energy flow. However, the GCNT out-of-plane thermal conductivity is still far lower than that of pristine graphene due to extreme phonon localizations, which are concentrated on the graphene-CNT junction regions as evidenced by the participation ratio, phonon vibrational density of states, and overlap energy. This study provides microscopic insight into the GCNT heat transfer mechanism and offers design guidelines for application of GCNTs in thermal management devices.

Transport coefficients of the Lennard-Jones fluid close to the freezing line.

Molecular dynamics simulations have been carried out along four Lennard-Jones (LJ) fluid isomorphs close to the freezing line, covering a temperature, T, in the range of 0.8-350 and a number density, ρ, in the range of 1.1-3.0 in LJ units. Analysis of the transport coefficients is via the Green-Kubo time correlation function method. The radial distribution function, percolation threshold connectivity distance, self-diffusion coefficient, and shear viscosity are shown to be invariant along an isomorph to a very good approximation when scaled with Rosenfeld's macroscopic units, although there are some small departures for T ≃ 1 and lower temperatures. The thermal conductivity is shown for the first time also to be isomorph invariant. In contrast, the Einstein and moment-based frequencies, and especially the bulk viscosity, ηb, show poor isomorphic collapse at low T but not surprisingly tend to an "inverse power" potential limiting value in the high T limit. In the case of the bulk viscosity, the significant departures from invariance arise from oscillations in the pressure autocorrelation function at intermediate times, which scale for inverse power potential systems but not for the LJ case, at least in part, as the pressure and bulk elastic moduli are not isomorph invariant.

Accessing thermal conductivity of complex compounds by machine learning interatomic potentials

While lattice thermal conductivity is an important parameter for many technological applications, its calculation is a time-consuming task, especially for compounds with a complex crystal structure. In this paper, we solve this problem using machine learning interatomic potentials. These potentials trained on the density functional theory results and provide an accurate description of lattice dynamics. Additionally, active learning was applied to significantly reduce the number of expensive quantum-mechanical calculations required for training and increases reliability of the potential. The ${\mathrm{CoSb}}_{3}$ skutterudite was considered as an example, and the solution of the Boltzmann transport equation for phonons was compared with the Green-Kubo method. We demonstrated that accurate and reliable potentials can be obtained by performing just a few hundred quantum-mechanical calculations. The potentials reproduce not only the vibrational spectrum, but also the lattice thermal conductivity, as calculated by various methods.

Contributions of force field interaction forms to Green-Kubo viscosity integral in n-alkane case

Decades of molecular simulation history proved that the Green-Kubo method for shear viscosity converges without any problems in atomic and simple molecular liquids, unlike liquids with high values of viscosity. In the case of highly viscous liquids, the time decomposition method was developed in 2015 by Maginn and co-authors [Y. Zhang, A. Otani, and E. J. Maginn, J. Chem. Theory Comput. 11, 3537-3546 (2015)] which allows us to improve the convergence of the Green-Kubo integral. In this paper, the contributions of intramolecular and intermolecular force field parts to the viscosity integral are discovered to gain the understanding of the Green-Kubo method. The n-alkanes from n-ethane to n-pentane at 330 K in the optimized potentials for liquid simulations-all atom force field are used as reference models. The dependencies of these contributions and decay times of the corresponding correlation functions on the chain length are observed. The nonequilibrium simulations are carried out to verify the Green-Kubo results. The obtained values of viscosity are compared with experimental data.

Heat Flux for Many-Body Interactions: Corrections to LAMMPS

The virial stress tensor-based instantaneous heat flux, which is used by LAMMPS, is only valid for the small subset of simulations that contain only pairwise interactions. For systems that contain many-body interactions using 3- or 4-body potentials, a more complete derivation is required. We have created a software patch to LAMMPS that implements the correct heat flux calculation approach for 3- and 4-body potentials, based on the derivation by Torii et al. ( J. Chem. Phys. 2008 , 128 , 044504 ) Using two example systems, the error in the uncorrected code for many-body potential heat flux is shown to be significant and reaches nearly 100% of the many-body potential heat flux for the systems we studied; hence, the error of the total heat flux calculation is proportional to the fraction of the total heat flux transferred through the many-body potentials. This error may have consequences for thermal conductivities calculated using the Green-Kubo method or any NEMD method that uses the instantaneous heat flux. We recommend that all researchers using LAMMPS for heat flux calculations where significant heat is transferred via the many-body potentials adopt the corrected code.

A new correlation for viscosity of model water-carbon nanotube nanofluids: Molecular dynamics simulation

In the present study, the viscosity of a model water-based nanofluid including an armchair-type (6,6) single-wall carbon nanotube is examined via Molecular Dynamics Simulation. The flexible four-site water model and the Tersoff empirical interatomic potential model are employed as an intermolecular force between liquid water molecules and as an interatomic potential function of carbon nanotube, respectively. Nevertheless, the interactions among water‑carbon nanotube atoms are modeled by the Lennard-Jones potential. The viscosity of nanofluid is sought using the Green–Kubo method at temperatures of 273 K, 283 K, 293 K, 303 K, and 313 K with volume fractions of 0.418%, 0.557% and 0.6%. The results are assessed with the related experimental data. The Radial Distribution Function is also calculated for its importance and usage to link the microscopic and macroscopic data. In conclusion, a relationship is obtained for the viscosity of water‑carbon nanotube nanofluids,μ=BζmTAζn+ζCT, in which volume fraction, temperature and viscosity are linked to each other. The recommended correlation is revealed to be acceptable and accurate for volume fractions in the range of 0.25 % ≤ ϕ ≤ 0.65%.

Temperature-dependent phonon spectrum of transition metal dichalcogenides calculated from the spectral energy density: Lattice thermal conductivity as an application

Predicting the mechanical and thermal properties of quasi-two-dimensional (2D) transition metal dichalcogenides (TMDs) is an essential task necessary for their implementation in device applications. Although rigorous density-functional-theory--based calculations are able to predict mechanical and electronic properties, mostly they are limited to zero temperature. Classical molecular dynamics facilitates the investigation of temperature-dependent properties, but its performance highly depends on the potential used for defining interactions between the atoms. In this study, we calculated temperature-dependent phonon properties of single-layer TMDs, namely, ${\mathrm{MoS}}_{2}$, ${\mathrm{MoSe}}_{2}$, ${\mathrm{WS}}_{2}$, and ${\mathrm{WSe}}_{2}$, by utilizing Stillinger-Weber--type potentials with optimized sets of parameters with respect to the first-principles results. The phonon lifetimes and contribution of each phonon mode in thermal conductivities in these monolayer crystals are systematically investigated by means of the spectral-energy-density method based on molecular dynamics simulations. The obtained results from this approach are in good agreement with previously available results from the Green-Kubo method. Moreover, detailed analysis of lattice thermal conductivity, including temperature-dependent mode decomposition through the entire Brillouin zone, shed more light on the thermal properties of these 2D crystals. The LA and TA acoustic branches contribute most to the lattice thermal conductivity, while ZA mode contribution is less because of the quadratic dispersion around the Brillouin zone center, particularly in ${\mathrm{MoSe}}_{2}$ due to the phonon anharmonicity, evident from the redshift, especially in optical modes, by increasing temperature. For all the considered 2D crystals, the phonon lifetime values are compelled by transition metal atoms, whereas the group velocity spectrum is dictated by chalcogen atoms. Overall, the lattice thermal conductivity is linearly proportional with inverse temperature.

Shear stress relaxation and diffusion in simple liquids by molecular dynamics simulations: Analytic expressions and paths to viscosity.

The results are reported of an equilibrium molecular dynamics simulation study of the shear viscosity, η, and self-diffusion coefficient, D, of the Lennard-Jones liquid using the Green-Kubo (GK) method. Semiempirical analytic expressions for both GK time correlation functions were fitted to the simulation data and used to derive analytic expressions for the time dependent diffusion coefficient and shear viscosity, and also the correlation function frequency transforms. In the case of the shear viscosity for a state point near the triple point, a sech function was found to fit the correlation function significantly better than a gaussian in the ballistic short time regime. A reformulation of the shear GK formula in terms of a time series of time integrals ("viscuits") and contributions to the viscosity from components based on the initial stress ("visclets") enable the GK expressions to be recast in terms of probability distributions which could be used in coarse grained stochastic models of nanoscale flow. The visclet treatment shows that stress relaxation is statistically independent of the initial stress for equilibrium and metastable liquids, suggesting that shear stress relaxation in liquids is diffusion controlled. By contrast, the velocity autocorrelation function is sensitive to the initial velocity. Weak oscillations and a plateau at intermediate times originate to a greater extent from the high velocity tail of the Maxwell-Boltzmann velocity distribution. Simple approximate analytic expressions for the mean square displacement and the self Van Hove correlation function are also derived.

Development of interatomic potentials for the complex binary compound Sb2Te3 and the prediction of thermal conductivity

Molecular dynamics (MD) simulation has emerged as a powerful predictive tool to study mechanical and thermal properties of materials, without the requirement for any fitting parameter inputs such as phonon relaxation time. However, the lack of suitable interatomic potentials for many complex materials greatly prohibits effective use of MD simulations to investigate properties of bulk materials and nanostructures. In this paper, we use the method of fitting to an ab initio energy surface to develop interatomic potential parameters for the complex binary material antimony telluride, which has important applications in thermoelectric energy generation. Density-functional theory is used to calculate the ground-state electronic structure of the ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ crystal, following which the total energies of a series of artificially distorted lattice configurations are calculated to create the energy surface. A Morse potential functional form is fitted to the energy surface and experimental data, and the parameters are used to calculate the bulk crystal properties and phonon spectra using lattice dynamics. Our parameters are able to reproduce the lattice structure, elastic constants, and acoustic phonon dispersion in good agreement with experimental data. MD simulations are performed using the fitted potential to calculate the thermal conductivity of bulk ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ using the Green-Kubo method. The predicted thermal conductivity shows a $1/T$ variation in both in-plane and cross-plane directions with the results in the range of experimental measurements. Frequency domain normal mode analysis is used to calculate the modal phonon relaxation times and the accumulation of thermal conductivity with respect to phonon mean free paths. The results show that phonons with mean free paths between 3 and 100 nm contribute to 80% of the total cross-plane thermal conductivity.

Molecular dynamics modelling of the thermal conductivity of off-stoichiometric UO2±x and (UyPu1−y)O2±x using equilibrium molecular dynamics

Thermal conductivities of off-stoichiometric single crystal mixed oxide nuclear fuels are estimated by equilibrium molecular dynamics simulations using the Green-Kubo method. The charge neutrality in non-stoichiometric systems is imposed using a charge rescaling method. The atomic charges are modified to account for charge removed or added depending on the defect concentration. The thermal conductivity for stoichiometric, hyperstoichiometric and hypostoichiometric nuclear fuels are estimated. A reduction in thermal conductivity is observed when the (U/Pu):O ratio deviates from 2.0, and the decrease is more pronounced in the hypostoichometric material. However, the U/Pu ratio had little effect on the thermal conductivity. The results are consistent with experimental and theoretically-determined values obtained elsewhere.

Homogeneous nonequilibrium molecular dynamics method for heat transport and spectral decomposition with many-body potentials

The standard equilibrium Green-Kubo and nonequilibrium molecular dynamics (MD) methods for computing thermal transport coefficients in solids typically require relatively long simulation times and large system sizes. To this end, we revisit here the homogeneous nonequilibrium MD method by Evans [Phys. Lett. A \textbf{91}, 457 (1982)] and generalize it to many-body potentials that are required for more realistic materials modeling. We also propose a method for obtaining spectral conductivity and phonon mean free path from the simulation data. This spectral decomposition method does not require lattice dynamics calculations and can find important applications in spatially complex structures. We benchmark the method by calculating thermal conductivities of three-dimensional silicon, two-dimensional graphene, and a quasi-one-dimensional carbon nanotube and show that the method is about one to two orders of magnitude more efficient than the Green-Kubo method. We apply the spectral decomposition method to examine the long-standing dispute over thermal conductivity convergence vs divergence in carbon nanotubes.

Phonon heat transport properties of graphene based on molecular dynamics simulations and lattice dynamics

To choose an ideal method to study the phonon properties of graphene, the results of thermal conductivity (TC) of graphene computed using the equilibrium molecular dynamics (EMD), reverse nonequilibrium molecular dynamics (RNEMD) and direct nonequilibrium molecular dynamics (DNEMD) with Tersoff potential are compared, and we find that all of them are very close to each other. While two of them have been compared in the past, there is a lack of comparison of the three methods. Eventually, we choose the Green–Kubo method to study the temperature dependence of TC in graphene and find that the [Formula: see text] diverges with the system temperature T as [Formula: see text]T[Formula: see text] with [Formula: see text] and [Formula: see text] for the direction of armchair and zigzag, respectively, which is in reasonable agreement with the one in recent theoretical and experimental researches. To gain further insight into the TC, the phonon dispersion and the phonon density of states (PDOS), which depend on evaluating the eigenvalues and the eigenvectors of dynamical matrix, are calculated for graphene with dimensions of 30 × 30 unit cell by a combination of EMD simulations and lattice dynamics calculations.

Transport property prediction and inhomogeneity analysis of supercritical n-Dodecane by molecular dynamics simulation

Predicting the transport properties of fuels in high-pressure conditions such as those for supercritical combustion has been a major challenge. In this study, molecular dynamics method of optimised potentials for liquid simulations force field is used to model the diesel fuel surrogate, i.e. n-Dodecane molecule. The Green-Kubo and Einstein methods are used to predict basic transport properties with conditions ranging from the high-pressure nozzle condition (150 MPa, 363 K) to the supercritical chamber environment (6 MPa, 900 K). The computed values agree well with experimental results in conditions with temperature and pressure far away from the critical point. But large discrepancies are observed in the neighborhood of the critical region due to density fluctuation and the critical divergence. The bulk viscosity of n-Dodecane is calculated and found to be hundreds of times greater than shear viscosity in supercritical conditions which cannot be ignored in computational fluid dynamics simulations. The inhomogeneity of supercritical fluids associated with the Widom and Frenkel lines is studied by mapping the loci of the extrema and “ridges” for compressibility coefficient, heat capacity, density fluctuation and velocity autocorrelation function on the P-T phase diagram. In addition, a new criterion is developed to distinguish inhomogeneity and structural transition of n-Dodecane into the “rigid-like”, “liquid-like” and “gas-like” fluids, by evaluating the decay of radial distribution function peak heights quantitatively.

The Effect of the Liquid Layer Around the Spherical and Cylindrical Nanoparticles in Enhancing Thermal Conductivity of Nanofluids

In this work, the equilibrium molecular dynamics (MD) simulation combined with the Green–Kubo method is employed to calculate the thermal conductivity and investigate the impact of the liquid layer around the solid nanoparticle (NP) in enhancing thermal conductivity of nanofluid (argon–copper), which contains the liquid argon as a base fluid surrounding the spherical or cylindrical NPs of copper. First, the thermal conductivity is calculated at temperatures 85, 85.5, 86, and 86.5 K and for different volume fractions ranging from 4.33% to 11.35%. Second, the number ΔN of argon atoms is counted in the liquid layer formed at the solid–liquid interface with the thickness of Δr = 0.3 nm around the NP. Finally, the number density n of argon atoms in this layer formed is calculated in all cases. Also, the results for spherical and cylindrical NPs are compared with one another. It is observed that the thermal conductivity of the nanofluid increased with the increasing volume fraction and the number ΔN. Likewise, the thermal conductivity of nanofluid containing spherical NPs is higher than that of nanofluid containing cylindrical NPs. Furthermore, the number density n of argon atoms near the surface of spherical NPs is higher than that of argon atoms attached in the curved surface of cylindrical NPs. As a result, the liquid layer around the solid NP has been considered one of the mechanisms responsible contributing to the thermal conductivity enhancement in nanofluids.

Simulation of dental resin monomer mixtures

In this work, molecular dynamics simulations of Bis-GMA/TEGDMA resin systems were carried out to investigate the rheological and dynamic properties of dental materials. The effect of the resin diluent monomer TEGDMA content rd on the viscosity and diffusion behaviors of the resin monomer mixtures was studied as well. The viscosity of the resin monomer mixture was calculated using the Green-Kubo method. It was shown that the mixture viscosity decreases apparently with rd, showing good agreements with experiments. Moreover, the diffusion coefficient of the Bis-GMA/TEGDMA monomer mixtures was evaluated, and it rises from 0.273 × 10−11 m/s to 9.710 × 10−11 m/s as rd increases from 0 to 100 wt%. In particular, as rd is ˂ 50 wt%, the diffusion dynamics of the resin monomer system was significantly suppressed due to the rigid benzene rings and hydroxyl groups in Bis-GMA. On the other hand, the diffusion coefficient of the materials increases rapidly with rd at rd ˃ 50 wt%. The findings in this work are believed to improve the atomic/molecular scale understanding of the rheological and dynamic performances of Bis-GMA/TEGDMA dental materials.

Improvements and limitations of Mie λ-6 potential for prediction of saturated and compressed liquid viscosity

Over the past decade, the Mie λ-6 (generalized Lennard-Jones) potential has grown in popularity due to its improved accuracy for predicting vapor-liquid coexistence densities and pressure compared to the traditional Lennard-Jones 12-6 potential. This manuscript explores the hypothesis that greater accuracy in characterizing the coexistence properties may lead to greater accuracy for viscosity predictions. Four united-atom force fields are considered in detail: the Transferable Potentials for Phase Equilibria (TraPPE-UA and the recently developed TraPPE-2) model of Siepmann and coworkers, the Transferable Anisotropic Mie (TAMie) model of Gross and coworkers, the fourth generation anisotropic-united-atom (AUA4) model of Ungerer and coworkers, and the model of Potoff and coworkers.Equilibrium molecular dynamics simulations are analyzed using the Green-Kubo method for viscosity characterization. Simulations are performed for linear alkanes with two to twenty-two carbons and branched alkanes with four to eight carbons. Simulation conditions follow the saturated liquid from reduced temperatures of 0.5–0.85 and along the 293 K isotherm in the dense liquid region.In general, the more accurate force fields for coexistence properties do indeed predict viscosity more accurately. For saturated liquids, both Mie-based potential models (Potoff and TAMie) provide roughly 10% accuracy for linear alkanes, while deviations are between 20 and 50% for TraPPE-UA. For branched alkanes, the performance is slightly diminished, but Potoff still provides roughly 15–20% accuracy, while the TAMie force field results in deviations of 20–40%, and TraPPE-UA has deviations of approximately 25–60%. The AUA4 deviations are 10–20% for ethane and 30–60% for 2,2-dimethylpropane, the only compounds tested with the AUA4 force field. The TraPPE-2 deviations for ethane are similar to those using the original TraPPE force field, namely, between 10 and 20%. The percent deviations for each compound and force field tend to increase with decreasing temperature, with the exception of the Potoff deviations for propane, which are nearly constant to the triple point temperature.For compressed liquids, the Mie-based potential models perform better once again than the Lennard-Jones-based force fields, but tend to over estimate the viscosity at very high densities. As the Potoff and TAMie models also tend to over estimate the pressure at high densities, a fortuitous cancellation of errors leads to predictions of viscosity with respect to pressure that are accurate to within about 10%. The comparison with experimental viscosity data is limited to pressures below 200 MPa for most normal and branched alkanes. However, accurate predictions are obtained for propane near 1000 MPa with the Potoff force field.

A reliable approach for calculating thermophysical properties of liquid using molecular dynamics simulations

This paper presents a new data processing method for calculating more reliable thermophysical properties of liquid using Green-Kubo Method based molecular dynamics (MD) simulation. In this study, MD simulations of water are first performed using three common water models. A new approach for analysing the simulation data is then developed to obtain statistically meaningful thermophysical properties such as thermal conductivity and viscosity. It is demonstrated that for a given desired standard deviation for a specific simulation, a suitable simulation box size can be determined. The proposed approach can assure the repeatability and reliability of the calculated thermophysical properties of liquid.

Understanding the structural, dynamic and thermodynamic properties of 5-Nonylsalicylaldoxime: Molecular dynamics and experimental studies

5-Nonylsalicylaldoxime (NSAO) is a well-known and strong extractant reagent, which is mainly used in the solvent extraction of copper and other metal ions. In this study, for the first time, the dynamic and thermodynamic properties and structure of 5-Nonylsalicylaldoxime were investigated by molecular dynamics (MD) simulation using the CHARMM force field. The results were validated against the experimental data of 5-Nonylsalicylaldoxime, which was synthesised by these authors. The self-diffusion coefficient was calculated based on both the Green-Kubo and Einstein methods that were in excellent agreement with each other. To understand the structure of NSAO, the FTIR spectroscopy of this reagent was computed by MD simulation and compared with the FTIR spectrum of the synthesised one. There was a good agreement between the calculated and experimental spectra. Moreover, based on the radial distribution function (RDF) and coordination numbers, it was shown that the formation probability of non-complexing isomer (Z-isomer) for NSAO molecules was greater than that of the complexing isomer (E-isomer).

The validity of the potential model in predicting the structural, dynamical, thermodynamic properties of the unary and binary mixture of water-alcohol: Methanol-water case

Two intermolecular potential models of methanol (TraPPE-UA and OPLS-AA) have been used in order to examine their validity in reproducing the selected structural, dynamical, and thermodynamic properties in the unary and binary systems. These two models are combined with two water models (SPC/E and TIP4P). The temperature dependence of density, surface tension, diffusion and structural properties for the unary system has been computed over specific range of temperatures (200-300K). The very good performance of the TraPPE-UA potential model in predicting surface tension, diffusion, structure, and density of the unary system led us to examine its accuracy and performance in its aqueous solution. In the binary system the same properties were examined, using different mole fractions of methanol. The TraPPE-UA model combined with TIP4P-water shows a very good agreement with the experimental results for density and surface tension properties; whereas the OPLS-AA combined with SPCE-water shows a very agreement with experimental results regarding the diffusion coefficients. Two different approaches have been used in calculating the diffusion coefficient in the mixture, namely the Einstein equation (EE) and Green-Kubo (GK) method. Our results show the advantageous of applying GK over EE in reproducing the experimental results and in saving computer time.

Viscous field-aligned water exhibits cubic-ice-like structural motifs.

Strong electric fields are known to greatly accelerate the freezing of water in molecular dynamics simulations, and have also been shown to affect the thermodynamics of the phase transition. In this work, a mechanistic explanation for field-induced crystallization of water is presented. Due to the coupling between the rotational and the translational degrees of freedom of individual water molecules, an applied field can directly drive the formation of cubic-ice like local motifs in water. Analysis of the angular distributions of water molecules in TIP4P-2005 water at field strengths between 0.0 and 0.32 V Å-1 demonstrates the existence of such motifs in the field-aligned liquid phase that is observed prior to the onset of the freezing transition. The dynamic properties of this field-aligned liquid phase are also studied, and its viscosity is shown to be within a factor of two of that of regular liquid water using the Green-Kubo method as well as mean squared displacements. The choice between the NPT and the NVT ensembles is shown to have a strong impact on the evolution of molecular dynamics trajectories at field strengths close to the threshold for the freezing transition, and the importance of properly accounting for the electric field terms in the pressure virial is emphasized.