Maxwell-Stefan Diffusion Coefficients Research Articles

Theoretical Study of Maxwell-Stefan Binary Diffusion Coefficients of n-alkanes-N2 at High Densities

ABSTRACT In this study, the Maxwell-Stefan binary diffusion coefficients ( D ij ) of C1-C6 n-alkanes-N2 are calculated using the Green-Kubo formula. The ratio of pD ij at different pressures (from 1 to 1000 bar) to that at 1 atm (reference condition) is studied. It is found that the ratio pD ij /(pD ij ) ref first decreases and then increases with increasing density, finally reaching 1.3 for n-C6H14-N2 at 1000 bar, 600 K. The gradual increase in molecular attraction at the beginning of the density increase results in a slower pressure increase compared to that of an ideal gas, contributing to the decreasing trend of pD ij /(pD ij ) ref . As the density continues to increase, the enhanced molecular repulsion and the molecular volume exclusion effect accelerate the pressure increase beyond that of an ideal gas, leading to an increasing trend in pD ij /(pD ij ) ref . Finally, the results obtained by the Green-Kubo method are compared with those obtained from empirical methods, and an equation for the refinement of D ij is formulated.

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.

Predicting the Water Sorption in ASDs

Water decreases the stability of amorphous solid dispersions (ASDs) and water sorption is, therefore, unwanted during ASD storage. This work suggests a methodology to predict the water-sorption isotherms and the water-sorption kinetics in amorphous pharmaceutical formulations like ASDs. We verified the validity of the proposed methodology by measuring and predicting the water-sorption curves in ASD films of polyvinylpyrrolidone-based polymers and of indomethacin. This way, the extent and the rate of water sorption in ASDs were predicted for drug loads of 0.2 and 0.5 as well as in the humidity range from 0 to 0.9 RH at 25 °C. The water-sorption isotherms and the water-sorption kinetics in the ASDs were predicted only based on the water-sorption isotherms and water-sorption kinetics in the neat polymer on the one hand and in the neat active pharmaceutical ingredient (API) on the other hand. The accurate prediction of water-sorption isotherms was ensured by combining the Perturbed-Chain Statistical Association Theory (PC-SAFT) with the Non-Equilibrium Thermodynamics of Glassy Polymers (NET-GP) approach. Water-sorption kinetics were predicted using Maxwell–Stefan diffusion coefficients of water in the ASDs.

Water Sorption in Glassy Polyvinylpyrrolidone-Based Polymers.

Polyvinylpyrrolidone (PVP)-based polymers are excellent stabilizers for food supplements and pharmaceutical ingredients. However, they are highly hygroscopic. This study measured and modeled the water-sorption isotherms and water-sorption kinetics in thin PVP and PVP-co-vinyl acetate (PVPVA) films. The water sorption was measured at 25 °C from 0 to 0.9 RH, which comprised glassy and rubbery states of the polymer-water system. The sorption behavior of glassy polymers differs from that in the rubbery state. The perturbed-chain statistical associating fluid theory (PC-SAFT) accurately describes the water-sorption isotherms for rubbery polymers, whereas it was combined with the non-equilibrium thermodynamics of glassy polymers (NET-GP) approach to describe the water-sorption in the glassy polymers. Combined NET-GP and PC-SAFT modeling showed excellent agreement with the experimental data. Furthermore, the transitions between the PC-SAFT modeling with and without NET-GP were in reasonable agreement with the glass transition of the polymer-water systems. Furthermore, we obtained Fickian water diffusion coefficients in PVP and in PVPVA from the measured water-sorption kinetics over a broad range of humidities. Maxwell-Stefan and Fickian water diffusion coefficients yielded a non-monotonous water concentration dependency that could be described using the free-volume theory combined with PC-SAFT and NET-GP for calculating the free volume.

Fick and Maxwell-Stefan Diffusion Coefficients in the Ternary Liquid Mixture of Acetone-Methanol-Chloroform

Fick diffusion coefficients are calculated by means of molecular simulation for liquid mixtures containing acetone, methanol and chloroform at 1 atm and 298 K for different compositions. For this means, Maxwell-Stefan (MS) diffusion coefficients were calculated using physical properties of the components and thermodynamic factors, Γ, using three different models of Wilson, NRTL and UNIQUAC. Because of the lack of experimental data for Fick diffusivities for ternary system, the validity of the model was tested by comparing Fick diffusivities obtained for binary subsystems with experimental data for acetone- chloroform system. The results were in good agreement. So the Fick coefficients, Maxwell-Stefan diffusion coefficient and the thermodynamic factors were predicted for the ternary mixture as well as its binary subsystems by molecular simulation in a consistent manner. The presented ternary diffusion data should facilitate the development of aggregated predictive models for diffusion coefficients of polar and hydrogen-bonding systems and allows for an efficient and consistent prediction of multicomponent Fick diffusion coefficients from molecular models.

Hydrodynamic density functional theory for mixtures from a variational principle and its application to droplet coalescence.

Dynamic density functional theory (DDFT) allows the description of microscopic dynamical processes on the molecular scale extending classical DFT to non-equilibrium situations. Since DDFT and DFT use the same Helmholtz energy functionals, both predict the same density profiles in thermodynamic equilibrium. We propose a molecular DDFT model, in this work also referred to as hydrodynamic DFT, for mixtures based on a variational principle that accounts for viscous forces as well as diffusive molecular transport via the generalized Maxwell-Stefan diffusion. Our work identifies a suitable expression for driving forces for molecular diffusion of inhomogeneous systems. These driving forces contain a contribution due to the interfacial tension. The hydrodynamic DFT model simplifies to the isothermal multicomponent Navier-Stokes equation in continuum situations when Helmholtz energies can be used instead of Helmholtz energy functionals, closing the gap between micro- and macroscopic scales. We show that the hydrodynamic DFT model, although not formulated in conservative form, globally satisfies the first and second law of thermodynamics. Shear viscosities and Maxwell-Stefan diffusion coefficients are predicted using an entropy scaling approach. As an example, we apply the hydrodynamic DFT model with a Helmholtz energy density functional based on the perturbed-chain statistical associating fluid theory equation of state to droplet and bubble coalescence in one dimension and analyze the influence of additional components on coalescence phenomena.

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 (xCO2) 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 xCO2 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 xCO2 over the entire studied composition range and a decreasing thermodynamic factor at xCO2 above 0.7. Finally, PDRS and EMD simulations were combined to investigate the influence of the fluid structure on the diffusive process.

A predictive model for self-, Maxwell-Stefan, and Fick diffusion coefficients of binary supercritical water mixtures

Self-, Maxwell-Stefan, and Fick diffusion coefficients of binary supercritical water mixtures (species 1: H2O, species 2: H2, CH4, CO, O2, CO2) at 673-973 K and 250 atm were calculated by molecular dynamic simulations using mean-squared displacement, Onsager coefficient, and thermodynamic factor. The mole fraction of species 2 is from 1% to up to 30%. The results show that self- and Maxwell-Stefan diffusion coefficients are largely determined by temperature and mole fraction. Despite the small fluctuations, Fick diffusion coefficient can be treated as a constant at certain temperature of a binary mixture regardless of mole fraction. We developed a general predictive model for different types of diffusion coefficients of binary supercritical water mixtures. Shared modification factors for equations are defined in this model. Overall, the average deviation for our predictive model is 4.15% for all types of diffusion coefficients

Comparison and validation of the lattice thermal conductivity formulas used in equilibrium molecular dynamics simulations for binary systems

The Green-Kubo relations have been widely utilized in equilibrium molecular dynamics (MD) simulations to evaluate the lattice thermal conductivity (TC) of condensed matter. In previous studies, however, three different formulas have been used to calculate the TC. In the present study, focusing on binary systems, we investigate differences among the three TC formulas to evaluate the appropriateness of each formula and estimate possible errors. First, by using theoretical means, the differences are explicitly expressed in terms of material properties such as the Maxwell-Stefan (MS) diffusion coefficient, the partial specific enthalpy, and the reduced heat of transport. Subsequently, MD simulations are conducted to quantify the differences over wide temperature ranges in Li2O and TiO2 model systems including crystal, amorphous and liquid phases. The results show that although the three TC formulas give virtually identical results at low temperatures, one TC formula may cause significant errors at high temperatures when the MS diffusion coefficient reaches approximately 10−6–10−7 cm2/s. These large diffusion coefficients usually occur in liquid phases, often occur in amorphous and super-ionic crystal phases, and may occur in defective systems. Finally, a simple equation to roughly estimate the error in this TC formula is derived.

On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of general collision kernels and parameters of the kinetic model.

Fick diffusion coefficients of binary fluid mixtures consisting of methane, carbon dioxide, and propane via molecular dynamics simulations based on simplified pair-specific ab initio-derived force fields

For the calculation of thermophysical properties of fluids with molecular dynamics (MD) simulations, effective force fields (FFs) which are optimized against experimental vapor-liquid equilibria are often used. Examples for this FF category are TraPPE, OPLS, and AMBER. An alternative are simplified pair-specific, ab initio-based FFs (AI-FFs), which are derived from quantum-chemical calculations in the limit of zero-density in combination with the kinetic theory of gases, and, thus, feature a predictive character. In the present study, we show with the help of selected binary fluid mixtures that the predictive power of MD simulations in calculating Fick diffusion coefficients can be improved using simplified pair-specific AI-FFs based on corresponding FFs for the pure substances. To evaluate the performance of the new AI-FFs in comparison with the TraPPE FFs, binary mixtures consisting of methane, carbon dioxide, and propane were investigated from the superheated vapor to the gas state and supercritical region up to the compressed liquid state. For the determination of the Fick diffusion coefficient at pressures between (0.1 and 12) MPa, temperatures between (293 and 355) K, and mole fractions between 0.05 and 0.95, separate simulations for the analysis of the Maxwell–Stefan diffusion coefficient and the thermodynamic factor were performed considering system size effects. With the exception of the compressed liquid state and regions in vicinity of the two-phase boundary, where the TraPPE FFs are generally superior, the AI-FFs show improved predictions for the Fick diffusion coefficient with average expanded statistical uncertainties of 12%. This could be demonstrated by comparison of the simulation results with the theoretical ab initio calculations and the available experimental data, resulting in average absolute deviations of 7% and 13% for the AI-FFs and TraPPE FFs.

Maxwell–Stefan diffusion coefficient model derived from entropy generation minimization principle for binary liquid mixtures

A new model for the Maxwell–Stefan (MS) diffusion coefficient was developed in this study. The entropy generation minimization principle was used to derive the expression of the MS diffusion coefficient, which has a similar form as Eyring’s model. Molar excess internal energy based on two-liquid theory was derived to correct the error induced by the linear mixing rule. The local surface area fraction was used to replace the bulk fraction by considering the intermolecular friction force exerting on the molecular surfaces. The new model was applied to calculate the Fick diffusion coefficients for four binary systems, which were measured experimentally by digital holographic interferometry. Comparison of the experimental and calculation results indicated that the proposed model has a good performance for calculating the diffusion coefficients for the mentioned binary systems.

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell–Stefan type

In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of the collisional kernels and parameters of the kinetic model.

Correction: Maxwell-Stefan diffusion coefficient estimation for ternary systems: an ideal ternary alcohol system.

Correction for 'Maxwell-Stefan diffusion coefficient estimation for ternary systems: an ideal ternary alcohol system' by Tariq Allie-Ebrahim et al., Phys. Chem. Chem. Phys., 2017, 19, 16071-16077.

Diffusion Coefficients of a Highly Nonideal Ternary Liquid Mixture: Cyclohexane–Toluene–Methanol

To better understand diffusion phenomena in highly nonideal ternary liquid mixtures, cyclohexane–toluene–methanol is studied by equilibrium molecular dynamics (EMD) simulation. Intradiffusion and Maxwell–Stefan (MS) diffusion coefficients, being strictly kinetic properties, are predicted by EMD over the entire composition range at ambient conditions. The thermodynamic contribution to the Fick diffusion coefficients is studied with an excess Gibbs energy model. Predictive results from the combination of these two approaches are in convincing agreement with experimental Fick diffusion coefficient data. Different aspects determining the composition dependence of diffusion coefficients, such as their behavior at the binary limits, hydrogen bonding, and stability criteria, are discussed. While the intradiffusion coefficients exhibit only a weak composition dependence, the MS diffusion coefficients are strongly affected by the nonideality of the present mixture. Fick diffusion coefficients reveal pronounced dif...

Fluctuating Hydrodynamics and Debye-Hückel-Onsager Theory for Electrolytes

Abstract We apply fluctuating hydrodynamics to strong electrolyte mixtures to compute the concentration corrections for chemical potential, diffusivity, and conductivity. We show these corrections to be in agreement with the limiting laws of Debye, Huckel, and Onsager. We compute explicit corrections for a symmetric ternary mixture and find that the co-ion Maxwell-Stefan diffusion coefficients can be negative, in agreement with experimental findings.

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.

Predicting the Kinetics of Ice Recrystallization in Aqueous Sugar Solutions.

The quality of stored frozen products such as foods and biomaterials generally degrades in time due to the growth of large ice crystals by recrystallization. While there is ample experimental evidence that recrystallization within such products (or model systems thereof) is often dominated by diffusion-limited Ostwald ripening, the application of Ostwald-ripening theories to predict measured recrystallization rates has only met with limited success. For a model system of polycrystalline ice within an aqueous solution of sugars, we here show recrystallization rates can be predicted on the basis of Ostwald ripening theory, provided (1) the theory accounts for the fact the solution can be nonideal, nondilute and of different density than the crystals, (2) the effect of ice-phase volume fraction on the diffusional flux of water between crystals is accurately described, and (3) all relevant material properties (involving binary Fick diffusion coefficients, the thermodynamic factor of the solution, and the surface energy of ice) are carefully estimated. To enable calculation of material properties, we derive an alternative formulation of Ostwald ripening in terms of the Maxwell–Stefan instead of the Fick approach to diffusion. First, this leads to a cancellation of the thermodynamic factor (a measure for the nonideality of a solution), which is a notoriously difficult property to obtain. Second, we show that Maxwell–Stefan diffusion coefficients can to a reasonable approximation be related to self-diffusion coefficients, which are relatively easy to measure or predict in comparison to Fick diffusion coefficients. Our approach is validated for a binary system of water and sucrose, for which we show predicted recrystallization rates of ice compare well to experimental results, with relative deviations of at most a factor of 2.

A compositional model based on SAFT-VR and Maxwell-Stefan equations for pervaporative separation of aroma compounds from aqueous solutions

In this study, the SAFT-VR equation of state combined with the Maxwell-Stefan theory was employed to develop a mass transfer model for the pervaporation process. For this purpose, the SAFT-VR equation of state was used for thermodynamic modeling of the sorption step as one of the main steps of the mass transfer in the pervaporation process and the Maxwell-Stefan theory was applied to describe the transport of permeating compounds through the membrane. The Maxwell-Stefan diffusion coefficients were estimated based on the free volume theory. The model was then validated using the experimental data obtained from the sorption and pervaporative separation of volatile aroma compounds like 3-methyl butanal, isoamyl acetate and n-hexanol from their binary aqueous solutions with the polydimethylsiloxane (PDMS) membrane. The proposed model was successfully able to predict the influence of feed aroma concentration and operating temperature on the aroma and water partial fluxes and aroma separation factors. The predicted partial and total fluxes of 3-methyl butanal, isoamyl acetate and n-hexanol aqueous solutions showed good agreement with the measured fluxes at various feed aroma concentrations and operating temperatures. Therefore, the proposed mass transfer model based on the SAFT-VR and Maxwell-Stefan equations satisfactorily described the mass transport in the hydrophobic pervaporation process without the need of any adjustable parameters dependent on the sorption and pervaporation experiments.

Predicting mass fluxes in the pervaporation process using Maxwell-Stefan diffusion coefficients

In the past decades, it has been proven that pervaporation is an effective and energy-efficient membrane process for the separation of liquids that are difficult to separate in classical processes. The demand for new process applications has increased the need for mass transfer simulation methods, considering the interactions between the system components and the influence of process parameters on the permeation fluxes and at the same time requiring as few experiments as possible.The aim of the study was to find out whether the calculation of mass fluxes of multicomponent fluids based on the system of generalized Maxwell-Stefan equations (GMSE), using Maxwell-Stefan (M-S) diffusion coefficients, performed according to the method recently proposed by Kubaczka [J. Membr. Sci. 470, 2014, 389–398] could meet these expectations. For this purpose, computational experiments were conducted where the predicted molar fluxes, concerning pervaporative dehydration of ethanol and isopropanol with a poly(vinyl alcohol) (PVA) membrane, were compared with the experimental results. Free volume parameters, necessary for determining the self-diffusion coefficients, were estimated according to Vrentas and Vrentas’ method [Eur. Polym. J. 34, 1998, 797–803] using the PRODE database correlations for viscosity and density of the components of the analyzed systems.Moreover, the study examined the importance and influence of the controversial parameter ξip, being the ratio of the molar volume of the solvent jumping unit to the molar volume of the polymer jumping unit, on the achieved permeation fluxes. The correlations derived from the experimental data necessary for their correlation are also given.The equilibrium concentrations in PVA were calculated by using the modified UNIFAC-FV method.The results obtained from the computational experiments have confirmed that the proposed method provides accurate predictions of mass fluxes in pervaporation systems.