Fick Diffusion Coefficients in Ternary Liquid Systems from Equilibrium Molecular Dynamics Simulations

Mutual Diffusivities of Mixtures of Carbon Dioxide and Hydrogen and Their Solubilities in Brine: Insight from Molecular Simulations

H2-CO2 mixtures find wide-ranging applications, including their growing significance as synthetic fuels in the transportation industry, relevance in capture technologies for carbon capture and storage, occurrence in subsurface storage of hydrogen, and hydrogenation of carbon dioxide to form hydrocarbons and alcohols. Here, we focus on the thermodynamic properties of H2-CO2 mixtures pertinent to underground hydrogen storage in depleted gas reservoirs. Molecular dynamics simulations are used to compute mutual (Fick) diffusivities for a wide range of pressures (5 to 50 MPa), temperatures (323.15 to 423.15 K), and mixture compositions (hydrogen mole fraction from 0 to 1). At 5 MPa, the computed mutual diffusivities agree within 5% with the kinetic theory of Chapman and Enskog at 423.15 K, albeit exhibiting deviations of up to 25% between 323.15 and 373.15 K. Even at 50 MPa, kinetic theory predictions match computed diffusivities within 15% for mixtures comprising over 80% H2 due to the ideal-gas-like behavior. In mixtures with higher concentrations of CO2, the Moggridge correlation emerges as a dependable substitute for the kinetic theory. Specifically, when the CO2 content reaches 50%, the Moggridge correlation achieves predictions within 10% of the computed Fick diffusivities. Phase equilibria of ternary mixtures involving CO2-H2-NaCl were explored using Gibbs Ensemble (GE) simulations with the Continuous Fractional Component Monte Carlo (CFCMC) technique. The computed solubilities of CO2 and H2 in NaCl brine increased with the fugacity of the respective component but decreased with NaCl concentration (salting out effect). While the solubility of CO2 in NaCl brine decreased in the ternary system compared to the binary CO2-NaCl brine system, the solubility of H2 in NaCl brine increased less in the ternary system compared to the binary H2-NaCl brine system. The cooperative effect of H2-CO2 enhances the H2 solubility while suppressing the CO2 solubility. The water content in the gas phase was found to be intermediate between H2-NaCl brine and CO2-NaCl brine systems. Our findings have implications for hydrogen storage and chemical technologies dealing with CO2-H2 mixtures, particularly where experimental data are lacking, emphasizing the need for reliable thermodynamic data on H2-CO2 mixtures.

Calculating Thermodynamic Factors for Diffusion Using the Continuous Fractional Component Monte Carlo Method

Thermodynamic factors for diffusion connect the Fick and Maxwell-Stefan diffusion coefficients used to quantify mass transfer. Activity coefficient models or equations of state can be fitted to experimental or simulation data, from which thermodynamic factors can be obtained by differentiation. The accuracy of thermodynamic factors determined using indirect routes is dictated by the specific choice of an activity coefficient model or an equation of state. The Permuted Widom's Test Particle Insertion (PWTPI) method developed by Balaji et al. enables direct determination of thermodynamic factors in binary and multicomponent systems. For highly dense systems, for example, typical liquids, it is well known that molecular test insertion methods fail. In this article, we use the Continuous Fractional Component Monte Carlo (CFCMC) method to directly calculate thermodynamic factors by adopting the PWTPI method. The CFCMC method uses fractional molecules whose interactions with their surrounding molecules are modulated by a coupling parameter. Even in highly dense systems, the CFCMC method efficiently handles molecule insertions and removals, overcoming the limitations of the PWTPI method. We show excellent agreement between the results of the PWTPI and CFCMC methods for the calculation of thermodynamic factors in binary systems of Lennard-Jones molecules and ternary systems of Weeks-Chandler-Andersen molecules. The CFCMC method applied to calculate the thermodynamic factors of realistic molecular systems consisting of binary mixtures of carbon dioxide and hydrogen agrees well with the NIST REFPROP database. Our study highlights the effectiveness of the CFCMC method in determining thermodynamic factors for diffusion, even in densely packed systems, using relatively small numbers of molecules.

Ab initio determination on diffusion coefficient and viscosity of FeNi fluid under Earth's core condition

The Earth's outer core is mainly composed of Fe and Ni. The geodynamo of the Earth's core are closely correlated with the transport properties of the fluid in the Earth's core. We selected the typical FeNi fluid, and systemically calculated its diffusion coefficient and viscosity under Earth's core condition by quantum molecular dynamics simulation. The diffusion coefficients are almost constant along the core adiabatic curve. The self-diffusion coefficients of Ni along the core adiabatic curve range from 2.47 × 10^-9 to 3.37 × 10^-9 m²s^-1. The diffusion coefficient increases with temperature increase, while viscosity decrease with temperature increase. The calculations on the transport properties suggest that the Ni impurities have a negligible effect on the diffusion coefficient and viscosity of Earth's core.

Kirkwood-Buff integrals: From fluctuations in finite volumes to the thermodynamic limit

The Kirkwood-Buff theory is a cornerstone of the statistical mechanics of liquids and solutions. It relates volume integrals over the radial distribution function, so-called Kirkwood-Buff integrals (KBIs), to particle number fluctuations and thereby to various macroscopic thermodynamic quantities such as the isothermal compressibility and partial molar volumes. Recently, the field has seen a strong revival with breakthroughs in the numerical computation of KBIs and applications to complex systems such as bio-molecules. One of the main emergent results is the possibility to use the finite volume KBIs as a tool to access finite volume thermodynamic quantities. The purpose of this Perspective is to shed new light on the latest developments and discuss future avenues.

How sensitive are physical properties of choline chloride-urea mixtures to composition changes: Molecular dynamics simulations and Kirkwood-Buff theory

Deep eutectic solvents (DESs) have emerged as a cheaper and greener alternative to conventional organic solvents. Choline chloride (ChCl) mixed with urea at a molar ratio of 1:2 is one of the most common DESs for a wide range of applications such as electrochemistry, material science, and biochemistry. In this study, molecular dynamics simulations are performed to study the effect of urea content on the thermodynamic and transport properties of ChCl and urea mixtures. With increased mole fraction of urea, the number of hydrogen bonds (HBs) between cation-anion and ion-urea decreases, while the number of HBs between urea-urea increases. Radial distribution functions (RDFs) for ChCl-urea and ChCl-ChCl pairs shows a significant decrease as the mole fraction of urea increases. Using the computed RDFs, Kirkwood-Buff Integrals (KBIs) are computed. KBIs show that interactions of urea-urea become stronger, while interactions of urea-ChCl and ChCl-ChCl pairs become slightly weaker with increasing mole fraction of urea. All thermodynamic factors are found larger than one, indicating a non-ideal mixture. Our results also show that self- and collective diffusivities increase, while viscosities decrease with increasing urea content. This is mainly due to the weaker interactions between ions and urea, resulting in enhanced mobilities. Ionic conductivities exhibit a non-monotonic behavior. Up to a mole fraction of 0.5, the ionic conductivities increase with increasing urea content and then reach a plateau.

Fick Diffusion Coefficient in Binary Mixtures of [HMIM][NTf2] and Carbon Dioxide by Dynamic Light Scattering and Molecular Dynamics Simulations

Dynamic light scattering (DLS) experiments and equilibrium molecular dynamics (EMD) simulations were performed in the saturated liquid phase of the binary mixture of 1-hexyl-3-methylimidazolium bis(trifluormethylsulfonyl)imide ([HMIM][NTf₂]) and carbon dioxide (CO₂) to access the Fick diffusion coefficient (D₁₁). The investigations were performed within or close to saturation conditions at temperatures between (298.15 and 348.15) K and CO₂ mole fractions (x_CO2) up to 0.81. The DLS experiments were combined with polarization-difference Raman spectroscopy (PDRS) to simultaneously access the composition of the liquid phase. For the first time in an electrolyte-based system, D₁₁ was directly calculated from EMD simulations by accessing the Maxwell-Stefan (MS) diffusion coefficient and the thermodynamic factor. Agreement within combined uncertainties was found between D₁₁ from DLS and EMD simulations for CO₂ mole fractions up to 0.5. In general, an increasing D₁₁ with increasing x_CO2 could be observed, with a local maximum present at a CO₂ mole fraction of about 0.75. The local maximum could be explained by an increasing MS diffusion coefficient with increasing x_CO2 over the entire studied composition range and a decreasing thermodynamic factor at x_CO2 above 0.7. Finally, PDRS and EMD simulations were combined to investigate the influence of the fluid structure on the diffusive process.

Machine Learning-Based Upscaling of Finite-Size Molecular Dynamics Diffusion Simulations for Binary Fluids

Molecular diffusion coefficients calculated using molecular dynamics (MD) simulations suffer from finite-size (i.e., finite box size and finite particle number) effects. Results from finite-sized MD simulations can be upscaled to infinite simulation size by applying a correction factor. For self-diffusion of single-component fluids, this correction has been well-studied by many researchers including Yeh and Hummer (YH); for binary fluid mixtures, a modified YH correction was recently proposed for correcting MD-predicted Maxwell-Stephan (MS) diffusion rates. Here we use both empirical and machine learning methods to identify improvements to the finite-size correction factors for both self-diffusion and MS diffusion of binary Lennard-Jones (LJ) fluid mixtures. Using artificial neural networks (ANNs), the error in the corrected LJ fluid diffusion is reduced by an order of magnitude versus existing YH corrections, and the ANN models perform well for mixtures with large dissimilarities in size and interaction energies where the YH correction proves insufficient.

Enhanced diffusion and the eigenvalue band structure of Brownian motion in tilted periodic potentials

We consider enhanced diffusion for Brownian motion on a tilted periodic potential. Expressing the effective diffusion in terms of the eigenvalue band structure, we establish a connection between band gaps in the eigenspectrum and enhanced diffusion. We explain this connection for a simple cosine potential with a linear force and then generalize to more complicated potentials including one-dimensional potentials with multiple frequency components and nonseparable multidimensional potentials. We find that potentials with multiple band gaps in the eigenspectrum can lead to multiple maxima or broadening of the force-diffusion curve. These features are likely to be observable in experiments.

Frame-invariant Fick diffusion matrices of multicomponent fluid mixtures

Extension of a description of mass diffusion in binary fluids based on Fick's law to multicomponent fluids requires introduction of diffusion matrices. A problem is that Fick diffusion matrices commonly adopted for multicomponent fluids depend on the velocity frame of reference. In this paper we show how one can define Fick diffusion matrices for multicomponent fluids that are frame invariant.

Generalized Form for Finite-Size Corrections in Mutual Diffusion Coefficients of Multicomponent Mixtures Obtained from Equilibrium Molecular Dynamics Simulation

The system-size dependence of computed mutual diffusion coefficients of multicomponent mixtures is investigated, and a generalized correction term is derived. The generalized finite-size correction term was validated for the ternary molecular mixture chloroform/acetone/methanol as well as 28 ternary LJ systems. It is shown that only the diagonal elements of the Fick matrix show system-size dependency. The finite-size effects of these elements can be corrected by adding the term derived by Yeh and Hummer (J. Phys. Chem. B2004, 108, 15873–15879). By performing an eigenvalue analysis of the finite-size effects of the matrix of Fick diffusivities we show that the eigenvector matrix of Fick diffusivities does not depend on the size of the simulation box. Only eigenvalues, which describe the speed of diffusion, depend on the size of the system. An analytic relation for finite-size effects of the matrix of Maxwell–Stefan diffusivities was developed. All Maxwell–Stefan diffusivities depend on the system size, and the required correction depends on the matrix of thermodynamic factors.

Mutual and Thermal Diffusivities as well as Fluid-Phase Equilibria of Mixtures of 1-Hexanol and Carbon Dioxide

This work contributes to an improved understanding of the fluid-phase behavior and diffusion processes in mixtures of 1-hexanol and carbon dioxide (CO₂) at temperatures around the upper critical end point (UCEP) of the system. Raman spectroscopy and dynamic light scattering were used to determine the composition at saturation conditions as well as Fick and thermal diffusivities. An acceleration of the Fick diffusive process up to CO₂ mole fractions of about 0.2 was found, followed by a strong slowing-down approaching vapor-liquid-liquid equilibrium or critical conditions. The acceleration of the Fick diffusive process vanished at temperatures much higher than the UCEP. Experimental Fick diffusivity data were compared with predictions from equilibrium molecular dynamics simulations and excess Gibbs energy calculations using interaction parameters from the literature. Both theoretical methods were not able to predict that the thermodynamic factor is equal to zero at the spinodal composition, stressing the need for new methodologies under such conditions. Thus, new sets of temperature-dependent interaction parameters were developed for the nonrandom two-liquid model, which improve the prediction of the Fick diffusion coefficient considerably. The link between the Fick diffusion coefficient and the nonrandomness of the liquid phases is also discussed.

Multicomponent mutual diffusion in the warm, dense matter regime

We present the formulation, simulations, and results for multicomponent mutual diffusion coefficients in the warm, dense matter regime. While binary mixtures have received considerable attention for mass transport, far fewer studies have addressed ternary and more complex systems. We therefore explicitly examine ternary systems utilizing the Maxwell-Stefan formulation that relates diffusion to gradients in the chemical potential. Onsager coefficients then connect the macroscopic diffusion to microscopic particle motions, evinced in trajectories characterized by positions and velocities, through various autocorrelation functions (ACFs). These trajectories are generated by molecular dynamics (MD) simulations either through the Born-Oppenheimer approximation, which treats the ions classically and the electrons quantum-mechanically by an orbital-free density-functional theory, or through a classical MD approach with Yukawa pair-potentials, whose effective ionizations and electron screening length derive from quantal considerations. We employ the reference-mean form of the ACFs and determine the center-of-mass coefficients through a simple reference-frame-dependent similarity transformation. The Onsager terms in turn determine the mutual diffusion coefficients. We examine a representative sample of ternary mixtures as a function of density and temperature from those with only light elements (D-Li-C, D-Li-Al) to those with highly asymmetric mass components (D-Li-Cu, D-Li-Ag, H-C-Ag). We also follow trends in the diffusion as a function of number concentration and evaluated the efficacy of various approximations such as the Darken approximation.

The Investigation of Protein Diffusion via H-Cell Microfluidics

In this study, we developed a microfluidics method, using a so-called H-cell microfluidics device, for the determination of protein diffusion coefficients at different concentrations, pHs, ionic strengths, and solvent viscosities. Protein transfer takes place in the H-cell channels between two laminarly flowing streams with each containing a different initial protein concentration. The protein diffusion coefficients are calculated based on the measured protein mass transfer, the channel dimensions, and the contact time between the two streams. The diffusion rates of lysozyme, cytochrome c, myoglobin, ovalbumin, bovine serum albumin, and etanercept were investigated. The accuracy of the presented methodology was demonstrated by comparing the measured diffusion coefficients with literature values measured under similar solvent conditions using other techniques. At low pH and ionic strength, the measured lysozyme diffusion coefficient increased with the protein concentration gradient, suggesting stronger and more frequent intermolecular interactions. At comparable concentration gradients, the measured lysozyme diffusion coefficient decreased drastically as a function of increasing ionic strength (from zero onwards) and increasing medium viscosity. Additionally, a particle tracing numerical simulation was performed to achieve a better understanding of the macromolecular displacement in the H-cell microchannels. It was found that particle transfer between the two channels tends to speed up at low ionic strength and high concentration gradient. This confirms the corresponding experimental observation of protein diffusion measured via the H-cell microfluidics.

Interplay of structure and diffusion in ternary liquid mixtures of benzene + acetone + varying alcohols

The Fick diffusion coefficient matrix of ternary mixtures containing benzene + acetone + three different alcohols, i.e., methanol, ethanol, and 2-propanol, is studied by molecular dynamics simulation and Taylor dispersion experiments. Aiming to identify common features of these mixtures, it is found that one of the main diffusion coefficients and the smaller eigenvalue do not depend on the type of alcohol along the studied composition path. Two mechanisms that are responsible for this invariant behavior are discussed in detail, i.e., the interplay between kinetic and thermodynamic contributions to Fick diffusion coefficients and the presence of microscopic heterogeneities caused by hydrogen bonding. Experimental work alone cannot explain these mechanisms, while present simulations on the molecular level indicate structural changes and uniform intermolecular interactions between benzene and acetone molecules in the three ternary mixtures. The main diffusion coefficients of these ternary mixtures exhibit similarities with their binary subsystems. Analyses of radial distribution functions and hydrogen bonding statistics quantitatively evidence alcohol self-association and cluster formation, as well as component segregation. Furthermore, the excess volume of the mixtures is analyzed in the light of intermolecular interactions, further demonstrating the benefits of the simultaneous use of experiment and simulation. The proposed framework for studying diffusion coefficients of a set of ternary mixtures, where only one component varies, opens the way for further investigations and a better understanding of multicomponent diffusion. The presented numerical results may also give an impulse to the development of predictive approaches for multicomponent diffusion.

Influence of Liquid Structure on Fickian Diffusion in Binary Mixtures of n-Hexane and Carbon Dioxide Probed by Dynamic Light Scattering, Raman Spectroscopy, and Molecular Dynamics Simulations

This study contributes to a fundamental understanding of how the liquid structure in a model system consisting of weakly associative n-hexane ( n-C₆H₁₄) and carbon dioxide (CO₂) influences the Fickian diffusion process. For this, the benefits of light scattering experiments and molecular dynamics (MD) simulations at macroscopic thermodynamic equilibrium were combined synergistically. Our reference Fickian diffusivities measured by dynamic light scattering (DLS) revealed an unusual trend with increasing CO₂ mole fractions up to about 70 mol %, which agrees with our simulation results. The molecular impacts on the Fickian diffusion were analyzed by MD simulations, where kinetic contributions related to the Maxwell-Stefan (MS) diffusivity and structural contributions quantified by the thermodynamic factor were studied separately. Both the MS diffusivity and the thermodynamic factor indicate the deceleration of Fickian diffusion compared to an ideal mixture behavior. Computed radial distribution functions as well as a significant blue-shift of the CH stretching modes of n-C₆H₁₄ identified by Raman spectroscopy show that the slowing down of the diffusion is caused by a structural organization in the binary mixtures over a broad concentration range in the form of self-associated n-C₆H₁₄ and CO₂ domains. These networks start to form close to the infinite dilution limits and seem to have their largest extent at a solute-solvent transition point at about 70 mol % CO₂. The current results not only improve the general understanding of mass diffusion in liquids but also serve to develop sound prediction models for Fick diffusivities.

Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO2)x-1(Y2O3)x Electrolyte

The calculation of the entropy production rate within an operational high temperature solid oxide fuel cell (SOFC) is necessary to design and improve heating and cooling strategies. However, due to a lack of information, most of the studies are limited to empirical relations, which are not in line with the more general approach given by non-equilibrium thermodynamics (NET). The SOFC 1D-model presented in this study is based on non-equilibrium thermodynamics and we parameterize it with experimental data and data from molecular dynamics (MD). The validation of the model shows that it can effectively describe the behavior of a SOFC at 1300 K. Moreover, we show that the highest entropy production is present in the electrolyte and the catalyst layers, and that the Peltier heat transfer is considerable for the calculation of the heat flux in the electrolyte and cannot be neglected. To our knowledge, this is the first validated model of a SOFC based on non-equilibrium thermodynamics and this study can be extended to analyze SOFCs with other solid oxide electrolytes, with perovskites electrolytes or even other electrochemical systems like solid oxide electrolysis cells (SOECs).

Finite-Size Effects of Binary Mutual Diffusion Coefficients from Molecular Dynamics

Molecular dynamics simulations were performed for the prediction of the finite-size effects of Maxwell-Stefan diffusion coefficients of molecular mixtures and a wide variety of binary Lennard-Jones systems. A strong dependency of computed diffusivities on the system size was observed. Computed diffusivities were found to increase with the number of molecules. We propose a correction for the extrapolation of Maxwell-Stefan diffusion coefficients to the thermodynamic limit, based on the study by Yeh and Hummer ( J. Phys. Chem. B , 2004 , 108 , 15873 - 15879 ). The proposed correction is a function of the viscosity of the system, the size of the simulation box, and the thermodynamic factor, which is a measure for the nonideality of the mixture. Verification is carried out for more than 200 distinct binary Lennard-Jones systems, as well as 9 binary systems of methanol, water, ethanol, acetone, methylamine, and carbon tetrachloride. Significant deviations between finite-size Maxwell-Stefan diffusivities and the corresponding diffusivities at the thermodynamic limit were found for mixtures close to demixing. In these cases, the finite-size correction can be even larger than the simulated (finite-size) Maxwell-Stefan diffusivity. Our results show that considering these finite-size effects is crucial and that the suggested correction allows for reliable computations.

Thermal, Mutual, and Self-Diffusivities of Binary Liquid Mixtures Consisting of Gases Dissolved in n-Alkanes at Infinite Dilution

In the present study, dynamic light scattering (DLS) experiments and molecular dynamics (MD) simulations were used for the investigation of the molecular diffusion in binary mixtures of liquids with dissolved gases at macroscopic thermodynamic equilibrium. Model systems based on the n-alkane n-hexane or n-decane with dissolved hydrogen, helium, nitrogen, or carbon monoxide were studied at temperatures between 303 and 423 K and at gas mole fractions below 0.06. With DLS, the relaxation behavior of microscopic equilibrium fluctuations in concentration and temperature is analyzed to determine simultaneously mutual and thermal diffusivity in an absolute way. The present measurements document that even for mole gas fractions of 0.007 and Lewis numbers close to 1, reliable mutual diffusivities with an average expanded uncertainty ( k = 2) of 13% can be obtained. By use of suitable molecular models for the mixture components, the self-diffusion coefficient of the gases was determined by MD simulations with an averaged expanded uncertainty ( k = 2) of 7%. The DLS experiments showed that the thermal diffusivity of the studied systems is not affected by the dissolved gas and agrees with the reference data for the pure n-alkanes. In agreement with theory, mutual diffusivities and self-diffusivities were found to be equal mostly within combined uncertainties at conditions approaching infinite dilution of the gas. Our DLS and MD results, representing the first available data for the present systems, reveal distinctly larger mass diffusivities for mixtures containing hydrogen or helium compared to mixtures containing nitrogen or carbon monoxide. On the basis of the broad range of mass diffusivities of the studied gas-liquid systems covering about 2 orders of magnitude from about 10^-9 to 10^-7 m²·s^-1, effects of the solvent and solute properties on the temperature-dependent mass diffusivities are discussed. This contributed to the development of a simple semiempirical correlation for the mass diffusivity of the studied gases dissolved in n-alkanes of varying chain length at infinite dilution as a function of temperature. The generalized expression requiring only information on the kinematic viscosity and molar mass of the pure solvent as well as the molar mass and acentric factor of the solute represents the database from this work and further literature with an absolute average deviation of about 11%.

Mutual diffusion governed by kinetics and thermodynamics in the partially miscible mixture methanol + cyclohexane

To gain an understanding of the transport and thermodynamic behavior of the highly non-ideal mixture methanol + cyclohexane, three complementary approaches, i.e. experiment, molecular simulation and predictive equations, are employed. The temperature and composition dependence of different diffusion coefficients is studied around the miscibility gap at ambient pressure. On the one hand Fick diffusion coefficients are measured experimentally by interferometric probing and on the other hand Maxwell-Stefan diffusion coefficients and intradiffusion coefficients are sampled by equilibrium molecular dynamics simulation at five temperatures below the upper critical temperature of ∼319 K. The spinodal curve is determined from extrapolation of the experimental Fick diffusion coefficient data and compared to predictions from excess Gibbs energy models. It is found that these models are not capable to correctly describe the activity coefficients over the whole composition range of the studied mixture. Thus, different parameter sets for a modified Wilson model are used for calculations of the thermodynamic factor, which is needed to transform Maxwell-Stefan into Fick diffusion coefficients and vice versa. Further, predictive equations for the Maxwell-Stefan diffusion coefficient, which are based on intradiffusion coefficients, are compared to simulation results. Using different approaches provides a clearer understanding of the relations between kinetic and thermodynamic properties contributing to the diffusion behavior of partially miscible mixtures.

Diffusion in Multicomponent Liquids: From Microscopic to Macroscopic Scales

In spite of considerable research on the nature of aqueous alcohol mixtures that are characterized by microscopic inhomogeneity or incomplete mixing at the molecular level, transport properties have received little attention. We report the results of a study on diffusion in the ternary mixture of water with two alcohols, that is, water + methanol + ethanol, which is investigated on microscopic and macroscopic scales by means of molecular simulation and Taylor dispersion experiments. A novel protocol is developed for the comparison of mutual diffusion coefficients sampled by two fundamentally different approaches, which allows for their critical analysis. Because of complex intermolecular interactions, given by the presence of hydrogen bonding, the analysis of transport processes in this mixture is challenging for not only on the microscopic scale for simulation techniques but also on the macroscopic scale due to unfavorable optical properties. Binary limits of the Fick diffusion matrix are used for validation of the experimental ternary mixture results together with the verification of the validity of the phenomenological Onsager reciprocal relations. The Maxwell-Stefan diffusion coefficients and the thermodynamic factor are sampled by molecular simulation consistently on the basis of given force field models. The protocol for the comparison of the results from both approaches is also challenging because Fick diffusion coefficients of ternary mixtures depend on the frame of reference. Accordingly, the measured coefficients are transformed from the volume-averaged to the molar-averaged frame of reference, and it is demonstrated that both approaches provide not only similar qualitative behavior along two concentration paths but also strong quantitative agreement. This coordinated work using different approaches to study diffusion in multicomponent mixtures is expected to be a significant step forward for the accurate assessment of cross-diffusion.

The reaction enthalpy of hydrogen dissociation calculated with the Small System Method from simulation of molecular fluctuations

We show how we can find the enthalpy of a chemical reaction under non-ideal conditions using the Small System Method to sample molecular dynamics simulation data for fluctuating variables. This method, created with Hill's thermodynamic analysis, is used to find properties in the thermodynamic limit, such as thermodynamic correction factors, partial enthalpies, volumes, heat capacities and compressibility. The values in the thermodynamic limit at (T,V, μj) are then easily transformed into other ensembles, (T,V,Nj) and (T,P,Nj), where the last ensemble gives the partial molar properties which are of interest to chemists. The dissociation of hydrogen from molecules to atoms was used as a convenient model system. Molecular dynamics simulations were performed with three densities; ρ = 0.0052 g cm(-3) (gas), ρ = 0.0191 g cm(-3) (compressed gas) and ρ = 0.0695 g cm(-3) (liquid), and temperatures in the range; T = 3640-20,800 K. The enthalpy of reaction was observed to follow a quadratic trend as a function of temperature for all densities. The enthalpy of reaction was observed to only have a small pressure dependence. With a reference point close to an ideal state (T = 3640 K and ρ = 0.0052 g cm(-3)), we were able to calculate the thermodynamic equilibrium constant, and thus the deviation from ideal conditions for the lowest density. We found the thermodynamic equilibrium constant to increase with increasing temperature, and to have a negligible pressure dependence. Taking the enthalpy variation into account in the calculation of the thermodynamic equilibrium constant, we found the ratio of activity coefficients to be in the order of 0.7-1.0 for the lowest density, indicating repulsive forces between H and H2. This study shows that the compressed gas- and liquid density values at higher temperatures are far from those calculated under ideal conditions. It is important to have a method that can give access to partial molar properties, independent of the ideality of the reacting mixture. Our results show how this can be achieved with the use of the Small System Method.

Partial molar enthalpies and reaction enthalpies from equilibrium molecular dynamics simulation

We present a new molecular simulation technique for determining partial molar enthalpies in mixtures of gases and liquids from single simulations, without relying on particle insertions, deletions, or identity changes. The method can also be applied to systems with chemical reactions. We demonstrate our method for binary mixtures of Weeks-Chandler-Anderson particles by comparing with conventional simulation techniques, as well as for a simple model that mimics a chemical reaction. The method considers small subsystems inside a large reservoir (i.e., the simulation box), and uses the construction of Hill to compute properties in the thermodynamic limit from small-scale fluctuations. Results obtained with the new method are in excellent agreement with those from previous methods. Especially for modeling chemical reactions, our method can be a valuable tool for determining reaction enthalpies directly from a single MD simulation.

Calculation of the chemical potential and the activity coefficient of two layers of CO2 adsorbed on a graphite surface

Thermodynamics of two layers of CO₂ on a graphite surface obtained directly from the simulations and the Small System Method.

Atomistic molecular dynamics simulations of CO₂ diffusivity in H₂O for a wide range of temperatures and pressures

Molecular dynamics simulations were employed for the calculation of diffusion coefficients of CO2 in H2O. Various combinations of existing force fields for H2O (SPC, SPC/E, and TIP4P/2005) and CO2 (EPM2 and TraPPE) were tested over a wide range of temperatures (283.15 K < T < 623.15 K) and pressures (0.1 MPa < P < 100.0 MPa). All force-field combinations qualitatively reproduce the trends of the experimental data; however, two specific combinations were found to be more accurate. In particular, at atmospheric pressure, the TIP4P/2005-EPM2 combination was found to perform better for temperatures lower than 323.15 K, while the SPC/E-TraPPE combination was found to perform better at higher temperatures. The pressure dependence of the diffusion coefficient of CO2 in H2O at constant temperature is shown to be negligible at temperatures lower than 473.15 K, in good agreement with experiments. As temperature increases, the pressure effect becomes substantial. The phenomenon is driven primarily by the higher compressibility of liquid H2O at near-critical conditions. Finally, a simple power-law-type phenomenological equation is proposed to correlate the simulation values; the proposed correlation should be useful for engineering calculations.

Multicomponent diffusion in molten LiCl-KCl: dynamical correlations and divergent Maxwell-Stefan diffusivities

Multicomponent diffusional mechanisms in the ternary LiCl-KCl system are elucidated using the Green-Kubo formalism and equilibrium molecular dynamics simulations. The Maxwell-Stefan (MS) diffusion matrix is evaluated from the Onsager dynamical matrix that contains the diffusion flux correlation functions. From the temporal behavior of the correlation functions, we observe that the Li-Li and Li-Cl ion pairs have a pronounced cage dynamics that remains noticeably strong even at high temperatures. Even though the Onsager coefficients, which are the time integrals of the diffusion flux correlation functions, portray a relatively smooth variation across various compositions and temperatures, we observe a sign change and a divergent-like behavior for the MS diffusivity of the K-Li ion pair at a temperature of ~1100 K for the eutectic composition, and at a KCl mole fraction of ~0.49 at 1043 K. Negative MS diffusivities, while unusual, are however shown to satisfy the nonnegative entropic constraints.

Kirkwood-Buff Integrals for Finite Volumes

Exact expressions for finite-volume Kirkwood-Buff (KB) integrals are derived for hyperspheres in one, two, and three dimensions. These integrals scale linearly with inverse system size. From this, accurate estimates of KB integrals for infinite systems are obtained, and it is shown that they converge much better than the traditional expressions. We show that this approach is very suitable for the computation of KB integrals from molecular dynamics simulations, as we obtain KB integrals for open systems by simulating closed systems.