Green-Kubo Method Research Articles

An investigation into application of the Green-Kubo method in molecular simulation to help understand the mechanisms of thermal conductivity of alcohols.

An investigation into application of the Green-Kubo method in molecular simulation to help understand the mechanisms of thermal conductivity of alcohols.

Molecular Fin Effect from Heterogeneous Self-Assembled Monolayer Enhances Thermal Conductance across Hard-Soft Interfaces.

Thermal transport across hard-soft interfaces is critical to many modern applications, such as composite materials, thermal management in microelectronics, solar-thermal phase transition, and nanoparticle-assisted hyperthermia therapeutics. In this study, we use equilibrium molecular dynamics (EMD) simulations combined with the Green-Kubo method to study how molecularly heterogeneous structures of the self-assembled monolayer (SAM) affect the thermal transport across the interfaces between the SAM-functionalized gold and organic liquids (hexylamine, propylamine and hexane). We focus on a practically synthesizable heterogeneous SAM featuring alternating short and long molecular chains. Such a structure is found to improve the thermal conductance across the hard-soft interface by 46-68% compared to a homogeneous nonpolar SAM. Through a series of further simulations and analyses, it is found that the root reason for this enhancement is the penetration of the liquid molecules into the spaces between the long SAM molecule chains, which increase the effective contact area. Such an effect is similar to the fins used in macroscopic heat exchanger. This "molecular fin" structure from the heterogeneous SAM studied in this work provides a new general route for enhancing thermal transport across hard-soft material interfaces.

Atomistic insights into the thermodynamics, structure, and dynamics of ionic liquid 1-hexyl-3-methylimidazolium hexafluorophosphate via molecular dynamics study

Molecular dynamics (MD) simulations were performed to compute the properties of the ionic liquid 1-hexyl-3-methylimidazolium hexafluorophosphate, [hmim][PF6], at three different temperatures (298, 323, and 348K) and atmospheric pressure. Thermodynamic, structural, and dynamical properties, such as average density, isobaric thermal expansion coefficient, isothermal compressibility factor, radial distribution function (RDF), spatial distribution function (SDF), mean square displacement (MSD) and velocity autocorrelation function (VACF) were computed and interpreted. The ionic self-diffusion coefficients were calculated using both the Einstein and Green–Kubo formulas and it was found that the cation diffuses faster than the anion and contributes more in the electric current. The ionic self-diffusion coefficients obtained from the Green–Kubo method agreed reasonably well with the experimental data in compared to the Einstein method. Also, the self-diffusion coefficients obtained from the two methods were used to calculate the ionic conductivity from the Nernst–Einstein equation. The effects of the system size and simulation run-time on determining diffusivity using the Einstein method were also investigated. The RDFs show that anions interact preferentially with the cation ring especially in direction of the acidic hydrogen atom of the imidazolium ring. This interaction was identified as hydrogen bonding interaction and observed using SDF.

First-principles Green-Kubo method for thermal conductivity calculations

We present a first-principles approach to calculate the phonon thermal conductivity based on the Green-Kubo formalism. In this approach, the density functional theory energy is distributed to each atom, and a two-step method in the molecular dynamics is introduced to avoid the atomic position $\mathbf{R}$ wrapping problem in a periodic system when the heat current is calculated. We show that this first-principles Green-Kubo approach is particularly suitable for disordered systems like amorphous and liquid, where the thermal conductivities are small due to strong phonon scattering but difficult to be calculated using anharmonic interaction energy. We have applied our method to liquid Ar, liquid Si, and amorphous Si. The calculated thermal conductivities agree well with previous theoretical and experimental results. We have also compared our method to previous works combining first-principles simulations with the Green-Kubo formalism.

A molecular dynamics study of nuclear quantum effect on diffusivity of hydrogen molecule.

In this paper, the nuclear quantum effect of the hydrogen molecule on its diffusivity was analyzed using the molecular dynamics (MD) method. The centroid MD (CMD) method was applied to reproduce the time evolution of the molecules. The diffusion coefficient of hydrogen was calculated using the Green-Kubo method over a wide temperature region, and the temperature dependence of the quantum effect of the hydrogen molecule on its diffusivity was addressed. The calculated results were compared with classical MD results based on the principle of corresponding state (PCS). It was confirmed that the difference in the diffusion coefficient calculated in the CMD and classical MD methods was small, and the PCS appears to be satisfied on the temperature dependence of the diffusion coefficient, even though the quantum effect of the hydrogen molecules was taken into account. It was clarified that this result did not suggest that the quantum effect on the diffusivity of the hydrogen molecule was small but that the two changes in the intermolecular interaction of hydrogen due to the quantum effect offset each other. Moreover, it was found that this tendency was related to the temperature dependence of the ratio of the kinetic energy of the quantum fluctuational motion to the classical kinetic energy.

Isovector dipole resonance and shear viscosity in low energy heavy-ion collisions

The ratio of shear viscosity over entropy density in low energy heavy-ion collision has been calculated by using the Green-Kubo method in the framework of an extended quantum molecular dynamics model. After the system almost reaching a local equilibration for a head-on $^{40}$Ca+$^{100}$Mo collision, thermodynamic and transport properties are extracted. Meanwhile, iso-vector giant dipole resonance (IVGDR) of the collision system is also studied. By the Gaussian fits to the IVGDR photon spectra, the peak energies of IVGDR are extracted at different incident energies. The result shows that the IVGDR peak energy has a positive correlation with the ratio of shear viscosity over entropy density. This is a quantum effect and indicates a difference between nuclear matter and classical fluid.

Effect of lattice relaxation on thermal conductivity of fcc-based structures: an efficient procedure of molecular dynamics simulation

This work studied the computational details of the Green–Kubo method with molecular dynamics (MD) simulation for thermal conductivity prediction. In MD thermal conductivity calculation, little consensus has been made about the inclusion of zero-pressure volume relaxation in the isobaric–isothermal (NpT) ensemble, which determines the simulation lattice parameters. Simulations of fcc-based structures with different lattice parameters were performed to calculate lattice thermal conductivities and phonon density of states, and the results were compared to experimental reports and ab initio results to conclude that NpT volume relaxation is crucial for accurate prediction of thermal conductivity. In addition, the relation between thermal conductivity and interatomic potential cutoff distance was also analysed in the context of lattice relaxation. The results suggested that calculated thermal conductivity is strictly dependent on the lattice parameter and essentially independent of the cutoff distance. It was also shown that reducing the cutoff distance can greatly accelerate the thermal conductivity calculation, even without sacrificing the accuracy of thermal conductivity.

Comparison of the Green–Kubo and homogeneous non-equilibrium molecular dynamics methods for calculating thermal conductivity

Different molecular dynamics methods like the direct method, the Green–Kubo (GK) method and homogeneous non-equilibrium molecular dynamics (HNEMD) method have been widely used to calculate lattice thermal conductivity (). While the first two methods have been used and compared quite extensively, there is a lack of comparison of these methods with the HNEMD method. Focusing on the underlying computational parameters, we present a detailed comparison of the GK and HNEMD methods for both bulk and vacancy Si using the Stillinger–Weber potential. For the bulk calculations, we find both methods to perform well and yield within acceptable uncertainties. In case of the vacancy calculations, HNEMD method has a slight advantage over the GK method as it becomes computationally cheaper for lower values. This study could promote the application of HNEMD method in calculations involving other lattice defects like nanovoids, dislocations, interfaces.

Overestimation of Viscosity by the Green-Kubo Method in a Dusty Plasma Experiment

The Green-Kubo (GK) method is widely used in simulations of strongly coupled plasmas to obtain the viscosity coefficient. However, the method's applicability, which is often taken for granted, has not been tested experimentally. We report an experimental test using a two-dimensional strongly coupled dusty plasma. We find that the GK viscosity is ≈60% larger than the result of a benchmark hydrodynamic method, obtained in the same experiment with the same conditions except for the presence of shear.

AbInitio Green-Kubo Approach for the Thermal Conductivity of Solids.

We herein present a first-principles formulation of the Green-Kubo method that allows the accurate assessment of the phonon thermal conductivity of solid semiconductors and insulators in equilibrium abinitio molecular dynamics calculations. Using the virial for the nuclei, we propose a unique abinitio definition of the heat flux. Accurate size and time convergence are achieved within moderate computational effort by a robust, asymptotically exact extrapolation scheme. We demonstrate the capabilities of the technique by investigating the thermal conductivity of extreme high and low heat conducting materials, namely, Si (diamond structure) and tetragonal ZrO_{2}.

Method to manage integration error in the Green-Kubo method.

The Green-Kubo method is a commonly used approach for predicting transport properties in a system from equilibrium molecular dynamics simulations. The approach is founded on the fluctuation dissipation theorem and relates the property of interest to the lifetime of fluctuations in its thermodynamic driving potential. For heat transport, the lattice thermal conductivity is related to the integral of the autocorrelation of the instantaneous heat flux. A principal source of error in these calculations is that the autocorrelation function requires a long averaging time to reduce remnant noise. Integrating the noise in the tail of the autocorrelation function becomes conflated with physically important slow relaxation processes. In this paper we present a method to quantify the uncertainty on transport properties computed using the Green-Kubo formulation based on recognizing that the integrated noise is a random walk, with a growing envelope of uncertainty. By characterizing the noise we can choose integration conditions to best trade off systematic truncation error with unbiased integration noise, to minimize uncertainty for a given allocation of computational resources.

Isotope effect on quantum thermal transport in a polyethylene chain

both the theoretical and the experimental aspects. Bulk polyethylene is regarded as a thermal insulator because its thermal conductivity is typically on the order of 0.35 W·m-1·K-1. However, recent studies demonstrate that a polyethylene chain has an extremely high thermal conductivity and the reported thermal conductivity of ultra-drawn polyethylene nanofibers is as high as 104 W·m-1·K-1, about 300 times higher than that of bulk polyethylene. In order to cast off this dilemma, several simulation methods are used to detect the unusually high thermal conductivity of a polyethylene chain. Molecular dynamics (MD) simulation results are highly sensitive to the choice of empirical potential or simulation method. Even using the same potential (AIREBO potential), the obtained thermal conductivity of a polyethylene chain is different. By combining the Green-Kubo method with a modal decomposition approach, equilibrium molecular dynamics (EMD) indicates that the thermal conductivity is able to exceed 100 W·m-1·K-1 while the polyethylene chain is longer than 40 nm at room temperature. Compared with the simulation result obtained by equilibrium molecular dynamics, the simulation result provided by using the non-equilibrium molecular dynamics (NEMD) method is only 57 W m·m-1·K-1 for a 160-nm-long polyethylene chain at room temperature. We use the first-principles method to calculate the force constant tensor, and the characteristics of quantum thermal transport in a polyethylene chain can be revealed. In our algorithm, several shortcomings of molecular dynamics, i.e., different potential functions or simulation methods may lead to obviously different results for the same quantum thermal transport system, are overcome. Based on the density functional theory (DFT), the central insertion scheme (CIS) combined with nonequilibrium Green's function (NEGF) is used to evaluate the isotope effect on quantum thermal transport in a polyethylene chain, which includes 432 atoms in scattering region and has a length of 18.533 nm. It is found that the upper limit of thermal conductivity of a 100-nm-long pure 12C polyethylene chain reaches a high value of 314.1 W·m-1·K-1 at room temperature. Moreover, for the case of a pure polyethylene chain of 12C, with other conditions unchanged, the reduction of average thermal conductance caused by 14C impurity is more remarkable than that by 13C. The most outstanding isotope effect on quantum thermal transport can be detected in the polyethylene chain. When the doping concentration of 14C in 12C is 50% at room temperature, the average thermal conductance will be reduced by 51%. It is of great significance for studying the mechanism of isotope effect on thermal transport in polyethylene.

Effect of temperature and diameter of narrow single-walled carbon nanotubes on the viscosity of nanofluid: A molecular dynamics study

Equilibrium molecular dynamics simulations by all-atom model have been employed to investigate the adsorption properties and temperature dependence of the shear viscosity of ethanol molecules confined in narrow single-walled carbon nanotubes (SWCNTs). Since the properties of the narrow tubes (diameters < 7 Å) are strongly influenced by their curvature, the narrow SWCNTs with diameters ranging from 0.54 to 1.08 nm and the length of 2.5 nm in ethanol reservoir at 273.0, 298.0 and 323K and 1 bar have been studied. The results indicate that endo-adsorption occurs in nanotubes with the diameter larger than 0.81 nm and as a result ethanol nano-wires can be formed inside nanotubes.Calculations of shear viscosity of the nanofluid using Green–Kubo method reveal that the relative viscosity of the confined ethanol decreases with increasing temperature.All the CNT@ethanol fluids have lower viscosity than the base liquid ethanol at all temperatures and the viscosity increases with the increase of CNT radius and endo-adsorption.Chirality of the CNTs has minor effects on shear viscosity and endo-adsorption and the main factor is CNTs diameter.The results of the present work provide deep understanding of the viscosity changes with temperature at the nanofluids and can describe the mechanisms for base fluid property modifications caused by adding nanoparticles. These results are applicable in design and preparation of diesohols and fuel-grade ethanol.

Self-consistent molecular dynamics calculation of diffusion in higher n-alkanes.

Diffusion is one of the key subjects of molecular modeling and simulation studies. However, there is an unresolved lack of consistency between Einstein-Smoluchowski (E-S) and Green-Kubo (G-K) methods for diffusion coefficient calculations in systems of complex molecules. In this paper, we analyze this problem for the case of liquid n-triacontane. The non-conventional long-time tails of the velocity autocorrelation function (VACF) are found for this system. Temperature dependence of the VACF tail decay exponent is defined. The proper inclusion of the long-time tail contributions to the diffusion coefficient calculation results in the consistency between G-K and E-S methods. Having considered the major factors influencing the precision of the diffusion rate calculations in comparison with experimental data (system size effects and force field parameters), we point to hydrogen nuclear quantum effects as, presumably, the last obstacle to fully consistent n-alkane description.

Thermal Conductivity of Monolayer MoSe2 and MoS2

Two-dimensional (2D) MoSe2 and MoS2 monolayers, two prototype transition metal dichalcogenides (TMDCs) materials, have attracted growing interest as promising 2D semiconductors. In this work, thermal conductivity (κ) of the monolayer MoSe2 is computed using large-scale classical nonequilibrium molecular dynamics (NEMD) simulations for the first time. The predicted κ of monolayer MoSe2 with infinite length (or MoSe2 2D sheets) are 43.88 ± 1.33 and 41.63 ± 0.66 W/(m·K) in armchair and zigzag directions, respectively. These simulation results are further confirmed by independent simulations using the Green–Kubo method (GKM), which yield computed κ of 44.38 ± 2.08 and 44.63 ± 2.50 W/(m·K), respectively. For 2D MoS2 sheet, the computed κ based on the NEMD method are 101.43 ± 1.13 and 110.30 ± 2.07 W/(m·K), respectively, in armchair and zigzag directions, whereas those based on the GKM are 102.32 ± 6.05 and 108.74 ± 6.68 W/(m·K), respectively. The predicted κ values of MoS2 monolayer are 2 times larger than those of MoSe2 monolayer. Both types of 2D monolayers exhibit isotropic properties in thermal conduction. Effects of system dimensions, heat flux, and temperature on κ are investigated comprehensively. The predicted κ value increases monotonically with the system length but decreases with temperature.

Rheology of liquid n-triacontane: Molecular dynamics simulation

Molecular dynamics is applied to calculate diffusion coefficients of n-triacontane C30H62 using Einstein-Smoluchowski and Green-Kubo relations. The displacement 〈Δr2〉(t) has a subdiffusive part 〈Δr2〉 ∼ t α, caused by molecular crowding at low temperatures. Longtime asymptotes of 〈v(0)v(t)〉 are collated with the hydrodynamic tail t-3/2 demonstrated for atomic liquids. The influence of these asymptotes on the compliance of Einstein-Smoluchowski and Green-Kubo methods is analyzed. The effects of the force field parameters on the diffusion process are treated. The results are compared with experimental data.

Effects of guest atomic species on the lattice thermal conductivity of type-I silicon clathrate studied via classical molecular dynamics

The effects of guest atomic species in Si clathrates on the lattice thermal conductivity were studied using classical molecular dynamics calculations. The interaction between a host atom and a guest atom was described by the Morse potential function while that between host atoms was described by the Tersoff potential. The parameters of the potentials were newly determined for this study such that the potential curves obtained from first-principles calculations for the insertion of a guest atom into a Si cage were successfully reproduced. The lattice thermal conductivities were calculated by using the Green-Kubo method. The experimental lattice thermal conductivity of Ba8Ga16Si30 can be successfully reproduced using the method. As a result, the lattice thermal conductivities of type-I Si clathrates, M8Si46 (M = Na, Mg, K, Ca Rb, Sr, Cs, or Ba), were obtained. It is found that the lattice thermal conductivities of M8Si46, where M is IIA elements (i.e., M = Mg, Ca, Sr, or Ba) tend to be lower than those of M8Si46, where M is IA elements (i.e., M = Na, K, Rb, or Cs). Those of mM8Si46, where m was artificially modified atomic weight were also obtained. The obtained lattice thermal conductivity can be regarded as a function of a characteristic frequency, fc. That indicates minimum values around fc=2-4 THz, which corresponds to the center of the frequencies of the transverse acoustic phonon modes associated with Si cages.

Thermal Conductivity of 3D Boron-Based Covalent Organic Frameworks from Molecular Dynamics Simulations

Covalent organic frameworks (COFs) have been widely investigated for use in gas storage and separation, while their thermal properties have been scarcely studied. In the study reported in this paper, the thermal conductivities of 3D boron-based COFs were investigated for the first time using molecular dynamics simulations (MD) employing the Green–Kubo method. The predicted thermal conductivities of COF-102, COF-103, COF-105, and COF-108 were on the order of 0.1 W/(m·K) at 300 K. The thermal conductivity decreased by up to 47% with the increase in temperature from 200 to 500 K. This resulting low thermal conductivity was due to the short mean free path of the phonon in the COFs, which was deduced to be 2.7–9.2 nm. The low-frequency phonon modes below 50 THz contributed mostly to heat conduction. By analyzing the phonon vibrational density of states and overlap energy between per two bonded atoms, it was revealed that the connection between phenylene rings in COF-102 and COF-103 weakens the phonon coupling ...

Estimates of crystalline LiF thermal conductivity at high temperature and pressure by a Green-Kubo method

Given the unique optical properties of LiF, it is often used as an observation window in high-temperature and pressure experiments; and, hence, estimates of its transmission properties are necessary to interpret observations. Since direct measurements of the thermal conductivity of LiF at the appropriate conditions are difficult, we resort to molecular simulation methods. Using the Belonoshko et al. (2000) empirical potential validated against ab initio phonon density of states, we estimate the thermal conductivity of LiF at high temperatures (1000-4000K) and pressures (100-400 GPa) with the Green-Kubo method. We also compare these estimates to those derived directly from ab initio data. To ascertain the correct phase of LiF at these extreme conditions we calculate the (relative) phase stability of the B1 and B2 structures using a quasiharmonic ab initio model of the free energy. We also estimate the thermal conductivity of LiF in an uniaxial loading state that emulates initial stages of compression in high-stress ramp loading experiments and show the degree of anisotropy induced in the conductivity due to deformation.

Fluid/gravity correspondence: A nonconformal realization in compactified D4 branes

We develop the framework of boundary derivative expansion (BDE) formalism of fluid/gravity correspondence in a compactified D4-brane system, which is a nonconformal background used in top-down holographic QCD models. Such models contain the D4-D6 model and the Sakai-Sugimoto (SS) model, with the background of the compactified black D4 branes under the near-horizon limit. By using the dimensional reduction technique, we derive a 5D Einstein gravity minimally coupled with three scalar fields from the 10D D4-brane background. Following the BDE formalism of fluid/gravity correspondence in the conformal background, we directly derive all the first order transport coefficients for nonconformal gluonic matter. The results of the ratio of the bulk to shear viscosity and the sound speed agree with those obtained from the Green-Kubo method. This agreement guarantees the validity of the BDE formalism of fluid/gravity duality in the nonconformal D-brane background, which can be used to calculate the second order transport coefficients in nonconformal background.