Thermal, Mutual, and Self-Diffusivities of Binary Liquid Mixtures

Subscriber access provided by - Access paid by the | UCSB Libraries

Article

Thermal, Mutual and Self-Diffusivities in Binary Liquid Mixtures Consisting of n-Alkanes with Dissolved Gases at Infinite Dilution Cédric Giraudet, Tobias Klein, Guanjia Zhao, Michael Heinrich Rausch, Thomas M. Koller, and Andreas Paul Fröba J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b00733 • Publication Date (Web): 27 Feb 2018 Downloaded from http://pubs.acs.org on March 13, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

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

Cédric Giraudet,*,a Tobias Klein,a Guanjia Zhao,a,b Michael H. Rausch,a Thomas M. Koller,a and Andreas P. Fröbaa

a

Institute of Advanced Optical Technologies – Thermophysical Properties (AOT-TP),

Department of Chemical and Biological Engineering (CBI) and Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander-University ErlangenNürnberg (FAU), Paul-Gordan-Straβe 6, 91052 Erlangen, Germany

b

Thermal Engineering, College of Electrical and Power Engineering, Taiyuan University of Technology, Shanxi Taiyuan, CN 030024, China

__________________________ * Author to whom correspondence should be addressed. Tel. +49-9131-85-23022, fax +499131-85-25851, email [email protected].

ACS Paragon Plus Environment

1

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 42

Abstract 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-alkanes 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. Using 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. Based on the broad range of mass diffusivities of the studied gas-liquid systems covering about two orders of magnitude from about 10-9 to 107

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 semi-empirical 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


2

Page 3 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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%.


3


Page 4 of 42

INTRODUCTION At present, there exists an increasing interest in liquids containing dissolved gases in chemical and energy technology.1–3 Examples refers to the separation of greenhouse gases from flue gas components using proper working fluids4,5 and the synthesis of high-valued hydrocarbons from hydrogen and carbon monoxide in the Fischer-Tropsch process.2 The transport property which strongly affects the design and optimization of the aforementioned processes is the mutual diffusivity in corresponding mixtures. Therefore, exact knowledge of the mutual diffusivity at a well-defined thermodynamic state, together with the specification of the associated uncertainty, is required. In addition to its relevance in process design, the mutual diffusivity is an important transport property in connection with the development of fluid models. Until now, no rigorous theory can describe the diffusive mass transport in fluids. Some auspicious theoretical6–8 and semi-empirical9–12 approaches for the prediction of the mutual diffusivity of binary mixtures of liquids with dissolved gases have been developed. In most of the correlations, the mutual diffusivity is expressed as a function of the temperature, the dynamic viscosity of the solvent, as well as the molar masses and/or molar volumes of the gaseous solute and the liquid solvent.10 Nevertheless, in the case of, e.g., small solutes such as hydrogen and helium and/or solvents with relatively large viscosities, the models show poor agreement with experimental results.9,10,13 The lack of appropriate models for the mutual diffusivity in liquid systems is caused by the scarce experimental data situation in literature. The metrological challenge in studying binary liquid mixtures is further intensified in the case of liquid-gas systems. Here, due to the gas pressure that has to be applied inside the measurement volume, the control and analysis of the system composition is challenging. Even at large pressures, the concentration of many gases dissolved in liquids is very low, weakening the measured signals and increasing the


4

Page 5 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


measurement uncertainties. During the relaxation of sufficiently large macroscopic concentration gradients subjected to the system and analyzed by transient methods, the occurrence of not only diffusive mass transport but also of additional perturbing flow contributions caused by viscosity, pressure, or temperature gradients is likely possible for systems consisting of liquids and dissolved gases. Such effects, if not accounted for in the data evaluation, can falsify the measured results. Methods which can overcome the limitations of transient methods are based on the analysis of mixtures at a steady-state using, e.g., light scattering techniques.14–17 Here, for two-phase systems, the application of dynamic light scattering (DLS) at macroscopic thermodynamic equilibrium is of particular interest. This technique analyzes short-ranged statistical fluctuations which originate from the random thermal movement of molecules and are related to the corresponding macroscopic properties. During the past twenty years, we have demonstrated that the analysis of the dynamics of the scattered light rising from fluctuations at fluid interfaces or in the bulk of fluids allows for the accurate determination of various thermophysical properties in an absolute way.17,18 Recently, our studies on mixtures of gases dissolved in liquids such as hydrocarbons19,20 or ionic liquids21,22 could document the capability of the method for an accurate determination of the mutual diffusivity with typical expanded uncertainties (k = 2) ranging between 5% and 30% depending on the thermodynamic state. Large uncertainties are associated to the vicinity of a Lewis number Le = a/D11 equal to 1. In our previous investigations on systems consisting of liquids with dissolved gases, the mean expanded uncertainty was of less than 10%. Given the large experimental effort and the increasing computational power, molecular dynamics (MD) simulation represents a useful method for the prediction of mass diffusivities and various other thermophysical properties of fluid systems.23–25 A further benefit of MD simulations is that relations between the chemical structures and the macroscopic properties


5


Page 6 of 42

can be investigated.26,27 Nevertheless, the capability of MD simulations in predicting thermophysical properties reliably depends on how accurately the underlying molecular model, often called force field (FF), can describe the inter- and intramolecular interactions over a broad range of thermodynamic states. The use of different molecular models can result in strongly varying values for the computed properties, especially for dynamic properties such as viscosity and mass diffusivities. In the present study, our aim is to combine DLS experiments and equilibrium MD simulations for the investigation of molecular diffusion in binary mixtures of the solvents nhexane (n-C6H14) or n-decane (n-C10H22) with dissolved gaseous solutes hydrogen (H2), helium (He), nitrogen (N2), and carbon monoxide (CO) at infinite dilution of the gas. By DLS from the bulk of a binary mixture, one single Fick mutual diffusivity is determined. At macroscopic thermal equilibrium where the Soret effect28 is absent, according to Fick’s first law of diffusion, this collective property of the gaseous solute (species 1) and the liquid solvent (species 2) relates the molar flux of one species to a driving force, i.e. the gradient in the molar concentration of this component. While the symbol D12 is often used in literature for the mutual diffusivity to highlight the inter-diffusive process between unlike species, it is not appropriate in connection with multicomponent mixtures.6,29–31 Therefore, to be consistent with the generalized formulation of Fick’s law of diffusion for multicomponent mixtures,23,32 in the present manuscript we denote the Fick mutual diffusivity of a binary mixture by the symbol D11. The latter property can be obtained from independent calculations of the Maxwell-Stefan (MS) mutual diffusivity Ð12 and the thermodynamic factor Γ11.23 In the equilibrium MD simulations performed here, the self-diffusivity of the gas in the liquid, D1, is calculated. It describes the random motion of the dissolved gas molecules in the mixture in the absence of a driving force for diffusion. Only in the limit of an infinite dilution of the gas in the liquid solvent, the different diffusivities D11, Ð12, and D1 are identical. In this


6

Page 7 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


case, the self-diffusivity D1, which is much easier accessible in MD simulations than the Fick and MS diffusivity, can be directly compared with experimental D11 data for small gas mole fractions. The paper is structured in the following way. After an introduction to the DLS method, the materials and sample preparation as well as the experimental setup and procedure are described. Thereafter, the development of an empirical model for the estimation of the concentration of the gases dissolved in the liquids as well as measurement examples and the data evaluation procedure related to the DLS experiments are presented. In the subsequent chapter on the MD simulations, the main information on the selection of the molecular force field, the computational details, and the data evaluation is provided. The section on the results and discussion is divided into three subsections. First, for the eight different gas-liquid combinations studied in the limit of an infinite dilution of the dissolved gases, the mutual and thermal diffusivities obtained by DLS and the gas self-diffusivities obtained by MD simulations are presented and compared with each other. Thereafter, the mass diffusivity results are discussed in connection with thermodynamic and molecular influences as well as available semi-empirical correlations. Finally, a new simple semi-empirical correlation for the mutual diffusivity of the studied gases dissolved in n-alkanes of varying chain length in the limit of an infinite dilution is suggested.

EXPERIMENTAL SECTION Rayleigh Scattering from the Bulk of Binary Fluid Mixtures. Binary fluid mixtures at macroscopic thermodynamic equilibrium are locally subjected to microscopic statistical fluctuations in the thermodynamic variables temperature, pressure, and concentration at a given time. In the case that the size of these fluctuations is much larger than the mean free path of the molecules, the fluid behaves like a continuum and the response of the statistical


7


Page 8 of 42

fluctuations is governed by the hydrodynamic equations.33,34 Information on these fluctuation-dissipation processes can be obtained from dynamic light scattering (DLS). Here, by irradiating the bulk of the fluid sample with coherent laser light and analyzing the temporal behavior of the scattered light intensity rising from refractive index fluctuations at a defined scattering angle, various thermophysical properties can be determined in an absolute way, see refs 17, 18 and 35. In the following, only the main information on DLS relevant for the present study is given. For binary fluid mixtures as studied here, the frequency-unshifted Rayleigh component of the spectrum of the scattered light is related to the relaxation behavior of microscopic fluctuations in temperature and concentration. With DLS, the mean relaxation times of both hydrodynamic modes are analyzed in the time domain by calculating the pseudo-cross correlation function of the scattered light intensity as a function of the delay time τ. For heterodyne conditions, where the scattered light is superimposed with coherent reference light of much higher intensity, the normalized intensity pseudo-cross correlation function from now on called correlation function (CF) - is

g(2) (τ ) = b0 + bt exp(−τ / τ C,t ) + bc exp(−τ / τ C,c ) .

(1)

In eq 1, τC,t and τC,c represent the decay times of the concentration and temperature fluctuations in the binary mixture. While b0 represents a background contribution, the constants bt and bc are mainly related to the intensities of the scattered light due to fluctuations in temperature and concentration, as well as to the intensity of the reference light. The decay times τC,t and τC,c are connected with the thermal diffusivity a and mutual diffusivity D11 according to τC,t = (a q2)‒1 and τC,c = (D11 q2)‒1. Assuming quasi-elastic scattering,

the

wavenumber,

i.e.

the

modulus

of

the

scattering

vector,

q = (4πnfluid·λ0‒1)·sin(Θs/2), is defined by the refractive index of fluid nfluid, the laser ACS Paragon Plus Environment

8

Page 9 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


wavelength in vacuo λ0, and the scattering angle Θs. For small scattering angles, the smallangle approximation and the Snell-Descartes’ law can reliably be applied to simplify the expression for the wavenumber to q = (2π·λ0‒1)·sin(Θi), where Θi is the easily accessible incident angle. In all present measurements, small scattering angles, i.e. small incident angles (Θi ≤ 6.5°), were adjusted. Materials and Sample Preparation. All gases used in this study were provided by Linde AG. The purities for H2, He, CO, and N2 are specified to be larger than 99.996, 99.9999, 99.997, and 99.9999 vol %, respectively. The purities of the sample for n-C6H14 applicable for spectroscopy and provided by Merck KGaA as well as for n-C10H22 provided by Alfa Aesar GmbH & Co. KG are both larger than 99.0 mass %. All chemicals were used without further purification. The experimental setup including the optical arrangement, the manifold system, the sample cell, and the temperature control system is the same as that employed in our previous studies.19–22 Here, only the information on the sample preparation and the measurement conditions necessary for the current study is reported. All studied gas-liquid samples were conditioned inside a hydrogen-resistant sample cell with a total inner volume of 40 mL providing four optical accesses. To minimize the interaction of the fluid sample with the cell material and, thus, the formation of complexes disturbing DLS experiments, the entire wall of the sample cell was coated first with a nickel layer and then with a gold layer. The control and measurement of the temperature of the sample cell was achieved by two Pt 100 Ω resistance probes with an absolute expanded uncertainty (k = 2) of 15 mK. While the temperature is controlled with one temperature probe placed in the wall of the cell close to the resistance heating, the second probe, which is also located inside the cell material but close to the fluid, measures the sample temperatures T reported within this study. An average temperature stability of 0.4 mK within a single measurement and 1.5 mK within a set of


9


Page 10 of 42

measurements contributing to one thermodynamic state could be achieved. The total pressure p inside the sample cell was measured using a oscillating quartz crystal pressure sensor (DPM1, DH-Budenberg) with a resolution of 10-4 MPa and an expanded uncertainty (k = 2) of 6×10-4 MPa. The average pressure stability was 4×10-3 MPa within a single measurement and 5×10-2 MPa within a set of measurements. The latter value resulting in a maximum change in the concentration of the dissolved gases of smaller than 0.07 mol %. Approximately 30 mL of the liquid n-alkane solvents were filled into the sample cell connected to the manifold system. To ensure high sample purity, vacuum was applied to the sample inside the cell and to the manifold system at about 293 K using an oil-sealed vacuum pump. During this step, the minimum pressure corresponds to the vapor pressure of the pure n-alkanes at room temperature. From the manifold system, the gases were injected in the sample cell by adjusting initial pressures which satisfy the conditions of an infinite dilution, i.e. mole fractions of the gases in the liquids below 6 mol %. Due to the relatively low solubilities of the studied gases, the pressure in the cell decreased only weakly as long as diffusive and convective mass transport of the gas into the liquid occurred. DLS measurements were performed at equilibrium conditions between 6 and 12 h after the first gas injection and 2 h after a temperature variation. These time steps were sufficient to ensure steady-state conditions verified by the stable system pressures between 0.5 and 6.1 MPa and temperatures between 303 and 423 K. Furthermore, based on the static shadowgraph of an expanded beam passing through the sample consisting of the liquid and the dissolved gas, macroscopic advection-free measurements could be guaranteed for all systems. Only for mixtures containing CO, DLS measurements were limited to a temperature of 373 K due to the formation of complexes at larger temperatures. These complexes are supposed to originate from a reaction between CO and nickel. The presence of the latter


10

Page 11 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


component is related to inevitably occurring small scratches on the gold coating above the nickel layer of the inner wall of the sample cell. Estimation of Mixture Composition. For the specification of the thermodynamic state related the diffusivity data of the studied mixtures, besides the temperature and pressure also the composition is necessary. Furthermore, the detailed information on the composition is important from an engineering perspective, for example, in connection with the FischerTropsch process.2 Here, gas solubilities or vapor-liquid equilibria strongly affect the product distributions in a reactor.36,37 It is clear that the detailed knowledge on the composition is more important in concentrated systems where concentration differences as driving forces for mutual diffusion are relatively large compared to those in diluted systems studied in the present work. In the latter case, the mutual diffusivity can be considered to be independent of composition. Nevertheless, our aim is to develop a simple empirical model for the quantitative estimation of the mole fraction xgas of the gases dissolved in the liquid n-alkanes at infinite dilution. For the model development, experimental solubility data for the eight systems investigated in the present study were collected from literature. However, solubility data and corresponding uncertainties could not be found for all binary mixtures over the entire range of temperatures and pressures studied by DLS. For example, investigations on He-based systems are limited to a pressure of 0.1 MPa and temperatures between 283 and 318 K.38 Therefore, we have developed an empirical model for the concentration of the gases H2, He, N2 and CO dissolved in the two n-alkane systems as a function of temperature and pressure. The proposed model is based on Henry’s law and the integration of the relationship between solubility and heat of solution.39 Details on the model for the estimation of the solubility of the gases H2, He, CO, and N2 dissolved in n-C6H14 and n-C10H22 are given in the Supporting Information. For the totally 524 data points which were all considered with the same


11


Page 12 of 42

statistical weight, the AAD between the experimental and modeled solubilities of the binary mixtures of H2, He, N2, and CO dissolved in n-C6H14 and n-C10H22 is 2.4%. Experimental Procedure. After reaching macroscopic thermodynamic equilibrium, at least six individual measurements at different small incident angles between about 3.5 and 6.5° were performed in the liquid phase at each thermodynamic state. In our DLS setup, the scattering angle is adjusted by fixing the detection direction and determining the incident angle Θi with an expanded uncertainty (k = 2) of ±0.01°, see ref 19. During each experiment at a given incident angle, the CF was calculated by two different digital correlators (ALV GmbH) simultaneously. The digital correlators are a linear-tau correlator (LTC) featuring equally spaced channels and a multiple-tau correlator (MTC) exhibiting a logarithmic time structure. Further details on the used digital correlators are given in ref 31. Since for the LTC the number of correlation channels depends on the sampling time, in the present study only sampling times of at least 250 ns were used. This allows to access all 2048 correlation channels for calculating the CFs with the LTC. In addition, the total lag time of the LTC was set to at least ten times the maximum relaxation time present in the CFs. This is a prerequisite to well describe disturbances rising from vibrations, particles, incoherent external stray light, or convection.19 Finally, the totally at least twelve independent CFs recorded by the LTC and MTC were analyzed to determine the averaged diffusivity data at a defined thermodynamic state. During one set of measurement at a thermodynamic state lasting approximately 1.25 hours, the pressure and the temperature of the sample cell were continuously recorded. The average temperature and pressure values were then used to estimate the composition of the liquid phase. The incident laser power was reduced to a minimum for preventing any laser heating effect on the studied samples which may induce temperature gradients and additional disturbing signals in the CFs due to convection. Further information on our study of the


12

Page 13 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


influence of the incident laser power on the DLS experiments is given in the Supporting Information. Data Analysis. Because of the inevitable presence of incoherent stray light, vibrations or particles, long time disturbances are always present in the CFs. Therefore, to access the correct values for the mean lifetimes of temperature (τC,t) and concentration (τC,c) fluctuations, an additional term accounting for these disturbances was added into the theoretical model given in eq 1. In the present study, a linear term was sufficient to obtain residual plots free of any systematic. The physical origin of the disturbances present in the CFs was not investigated systematically because it is not of special interest in this study. In the following, reduced CFs are considered, where it is assumed that the disturbance term can be subtracted from the original CF and does not influence the final results. All fitting procedures applied to the CFs were performed by nonlinear regression based on a Levenberg-Marquardt algorithm in which the squared sum of residuals is minimized. In a dense fluid phase far from the demixing point, the faster hydrodynamic mode is attributed to temperature fluctuations and the slower one to concentration fluctuations. The upper panel in Figure 1A exemplarily illustrates an experimental CF calculated by the LTC for a binary CO + n-C10H22 mixture at T = 372.65 K, p = 0.80 MPa, xCO ≈ 0.8 mol %, and Θi = 5.98. Here, the hydrodynamic modes related to temperature as well as concentration fluctuations and indicated by the dashed red as well as the dotted green lines, have distinctly different characteristic times as well as amplitudes and can thus be separated. In the present case, the Lewis number, Le = τC,c/τC,t, is equal to 5.4. In general, when Le ≥ 1.6 or Le ≤ 0.6, which was valid for 75% of the measurements, the CF can be well fitted by eq 1. This fit model is drawn by the solid blue line in the upper panel of Figure 1A, while the associated absolute residuals of the measured data from the fit are shown in the lower panel of the figure. Absolute


13


Page 14 of 42

residuals are the difference between the CF calculated by the correlator and the CF estimated by non-linear regression. While the theoretical model given by eq 1 is, in principle, valid for each thermodynamic state, in the vicinity of a Lewis number equal to one, both hydrodynamic modes are not distinguishable from each other due to the convergence of both decay times. In addition, the dissociation of both modes for Le→1 is further impeded in the case of mixtures of liquids with dissolved gases where also the associated amplitudes of the modes become similar, i.e. bc/bt ≈ 1. This can most probably be explained by a divergence of the osmotic compressibility and/or a convergence of the refractive index derivatives of both modes according to the expression of the Rayleigh ratio for a binary mixture.40 From a statistical point of view, the determination of two exponential decays with similar decay times by a non-linear regression analysis is also function of the SNR. A systematic variation of the measurement times between 36 and 1200 s during an individual DLS experiment where Le ≈ 1 showed that even if the SNR is increasing from 10 to 700 with increasing the measurement time, only one reliable effective exponential decay could be extracted from the CFs. In all cases where 1.6 ≤ Le ≤ 0.6, applying a fit model containing two exponentials provides results for τC,c and

τC,t which strongly vary depending on the initial fit parameters, and uncertainties in the decay times and the amplitudes of both modes of more than 80%. The inability of the non-linear regression to dissociate both hydrodynamic modes when Le is close to 1 was also verified using the Nelder-Mead non-linear regression algorithm. Thus, for the aforementioned range of Le numbers, only one effective hydrodynamic mode with the effective decay time τC,eff related to an effective diffusivity Deff was determined by non-linear regression based on eq 1 which is reduced to a single exponential decay. Such an exemplary CF is shown in Figure 1B. It was obtained by the LTC for a binary He + n-C6H14 mixture at T = 397.34 K, p = 1.99 MPa, xHe ≈ 3.6 mol %, and Θi = 4.79. It can be seen that the residuals in the lower


14

Page 15 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


panel of Figure 1B are free of systematic, validating the single exponential decay approach performed in the upper panel of the figure. These effective diffusivities, are not considered in the following discussion and data comparison. Nevertheless, according to our previous studies, these effective diffusivities differ from the mutual diffusivities by less than 30%.

Figure 1. Exemplary normalized CFs obtained by the LTC (upper panel) and their residual plots (lower panel): (A) CO + n-C10H22 mixture at T = 372.65 K, p = 0.80 MPa, and Le = 5.4. CF fitted according to eq 1 (solid blue line) containing a sum of two exponential decays in the form of the concentration mode (dotted green line) and the thermal mode (dashed red line). (B) He + n-C6H14 mixture at T = 397.34 K, p = 1.99 MPa, and Le ≈ 1. CF fitted by one exponential decay (solid blue line) representing an effective mode.


15


Page 16 of 42

MOLECULAR DYNAMICS (MD) SIMULATION In addition to the experimental determination of the mutual diffusivity, equilibrium MD simulations were performed. Based on an accurate molecular model of a selected statistical ensemble of the studied system and the solution of the equations of motion, various equilibrium and transport properties can be deduced at virtually every desired state. Details on the principles of MD simulation can be found in ref 41. In this section, only the relevant information in connection with the present investigations is summarized. To describe the interactions in the studied binary model mixtures of liquids with dissolved gases realistically, first different commonly applied force fields (FFs) for the pure n-alkanes n-C6H14 and n-C10H22 were tested regarding their performance in predicting the density and dynamic viscosity η. The study of density and viscosity of the pure n-alkanes provides first indications on the reliability of the FF models for the prediction of equilibrium as well as dynamic properties of binary mixtures consisting of n-alkanes with dissolved gases at infinite dilution. In fact, in the infinite dilution regime, the self-diffusivity of the gas is mainly governed by the solvent properties, cross-interactions between unlike species play a minor role. Since dynamic viscosity and self-diffusivity are dynamic properties which show in a good approximation a reciprocal behavior, overestimations in the dynamic viscosity of the liquid solvent can be directly associated with underpredictions in the self-diffusivity of the gas in the liquid and vice versa. Taking this into account, the model performing best for the pure liquids was then applied to their binary mixtures with dissolved gases to predict the gas self-diffusivity at infinite dilution of the gas.

Force Field and Computational Details. The non-polarizable FFs used in the present study are characterized by pair-additive potential energy functions consisting of intramolecular and intermolecular interactions. Details on the functional form of the FFs for the various


16

Page 17 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


components and the corresponding parameters can be found in the respective references detailed in the following. For the liquid n-alkanes, we have focused on investigating two well-established molecular models in the form of the Transferable Potentials for Phase Equilibria (TraPPE) united-atom (UA) FF42 and the modified version of the Optimized Potentials for Liquid Simulations (OPLS) all-atom (AA) FF,43 commonly known as L-OPLS FF.44 The latter model was optimized for simulating dynamic properties such as the viscosity for a broad range of n-alkanes with varying carbon number. For the gas molecules, common FFs from literature were employed. Lennard-Jones (LJ) spheres as proposed by Hirschfelder et al.45 were used to model He and CO. For the latter gas, the simplified single-site model has been shown to provide reliable mass diffusivities in connection with liquid n-alkanes of varying carbon number.46,19 For N2, a two-center FFs with a flexible bond47 was employed. In the case of H2, two commonly used FFs were considered. While the model from Hirschfelder et al.45 treats H2 as a single Lennard-Jones sphere, the more sophisticated model proposed by Darkrim et al.48 applied two charged real atoms and a charged virtual LJ site. Standard Lorentz-Berthelot combinations rules for the Lennard-Jones parameters of different atom types were used. This means that an arithmetic average was employed for the size parameters, while a geometric average was applied to the energy parameters. These mixing rules were adapted for both the pure liquids and their binary mixtures with dissolved gases. No additional binary interaction parameter in the combination rule for the energy parameter commonly used for optimization purposes was used. In accordance with the frameworks of the OPLS FF43 and AMBER FF,49 intramolecular 1-4 interactions for the electrostatic Coulomb and the Lennard potential were scaled by a factor of 5/6 and 1/2, respectively. All simulations were carried out using the GROMACS package, version 5.1.2, in single precision.50 Computational details on the used thermostat, barostat, cutoff values, long-range interactions, and constraint as well as integration algorithm are provided in the Supporting


17


Page 18 of 42

Information. Therein, also relevant information on the number of molecules, the energy as well as equilibration procedure, and the ensembles used for the production runs for the calculation of the density and viscosity of the pure liquids and of the self-diffusivity of the gases in the liquids at a mole fraction of the dissolved gas of xgas = 0.99 mol % are given. For all pure and binary systems, temperatures between 303.15 and 423.15 K were investigated.

Data Evaluation. To get a good statistical representation of the thermophysical properties calculated by own developed post-processing codes, five independent runs using different initial configurations were carried out for all systems and thermodynamic states. The statistical uncertainties of the properties were calculated from the double standard deviation of the results obtained from the individual runs. For the density, expanded uncertainties (k = 2) of 0.08% for the T-dependent densities of the two n-alkanes could be obtained using both sets of FF types. The Green-Kubo formalism51,52 analyzing the pressure autocorrelation function of the off-diagonal elements of the stress tensor was used to calculate the dynamic viscosity of pure n-C6H14 and n-C10H22 from the plateau of the autocorrelation function sampled each 10 fs. Typical expanded uncertainties (k = 2) for η of 5% for n-C6H14 and 8% for n-C10H22 were accessed. The self-diffusion coefficient D1 of the gases (species 1) dissolved in the n-alkanes (species 2) was calculated from the linear part of the mean-square displacement (MSD) of the molecules of type 1 as a function of the delay time τ in steps of each 1 ps according to the Einstein equation.53 To specify the time range in which the slope of the MSD1 can be extracted reliably from the linear formulism based on eq 2, the coefficient

β = d(log(MSD1))/d(log(τ)) was calculated. Details on the evaluation of the MSD plots according to the β value and the calculation of the self-diffusion coefficients of the dissolved gases from the linear part of the Einstein equation53 are given in the Supporting Information. To correct for the system size dependency of the gas self-diffusion coefficient, all results


18

Page 19 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


computed in this study were obtained by extrapolation to infinite box sizes, as it was proposed by Yeh and Hummer.54 For pure n-C6H14 and n-C10H22, the results calculated for density and viscosity at temperatures between 303.15 and 423.15 K in steps of 30 K and a pressure of 2 MPa using the TraPPE and L-OPLS FFs were compared with reference data from literature at the same or similar thermodynamic states. For density of both alkanes, the AAD between the predicted values and the values from the Refprop database55 was lower using the TraPPE FF with a value of 0.3% than using the L-OPLS FF with 4%. This is reasonable since the TraPPE FF has been fitted against vapor-liquid equilibrium data,42 while the L-OPLS model has been fitted against dihedral potentials from ab initio calculations, as well as data for liquid densities and heat of vaporizations in order to also obtain improved results for the dynamic viscosity.44 The latter fact could be validated in our study where a distinctly better prediction of the dynamic viscosity of the pure liquids could be found using the L-OPLS FF compared to the TraPPE FF. Here, the AAD value between the data from the MD simulations and those from the Refprop database55 is 12% in the case of the L-OPLS FF, while it is 28% in the case of the TraPPE FF systematically underestimating the viscosity. Thus, for the representation of the n-alkanes in their binary mixtures with the dissolved gases, the L-OPLS FF was selected as underlying model for both n-alkanes due to its better representation of the dynamic behavior.

RESULTS AND DISCUSSION First, the diffusivity results for all eight studied binary mixtures obtained by DLS and MD simulations at macroscopic thermodynamic equilibrium and infinite dilution of the gas are presented. Then, the self-diffusivities calculated by MD simulations are compared to the mutual diffusivities measured by DLS. Besides the comparison between self- and mutual


19


Page 20 of 42

diffusivities, the results are discussed in connection with thermodynamic as well as structural influences and empirical correlations.

Summary of Diffusivity Data. For each thermodynamic state investigated by DLS, the reported values for D11 and a are based on a statistical analysis of at least 12 independent measurements calculated by the LTC and MTC for different scattering angles. For data representation, instead of a standard arithmetic average of the individual measurements, a weighting scheme was applied because it considers the variation of the SNR between individual measurements and, thus, provides more reliable data for the diffusivities and their uncertainty. In the present study, the statistical weight wλ for each individual value of the transport coefficient λ (= a or D11) is defined by the inverse of the propagated uncertainties related

to

the

corresponding

working

equation

for

λ.

This

means

that

wa = (∆a/a)‒1 = (∆τC,t/τC,t + 2∆Θi/tan(Θi))‒1 in connection with the thermal diffusivity and wD11 = (∆D11/ D11)‒1 = (∆τC,c/τC,c + 2∆Θi/tan(Θi))‒1 in connection with the mutual diffusivity. A similar weighting scheme was also used to calculate the standard deviation of the individual diffusivity results for each thermodynamic state. It was found that the diffusivity values obtained from the weighted and arithmetic averages are in agreement within their combined uncertainties. For the corresponding uncertainties of the a and D11 data, the standard deviations of the individual diffusivity data considering their weights are mostly larger than those not considering their weights, and are hence used for all measurements reported here. The mutual diffusivities D11 and thermal diffusivities a obtained by DLS for binary mixtures of n-C6H14 or n-C10H22 containing dissolved H2, He, N2, and CO as well as their relative expanded uncertainties (k = 2) are summarized in Table 1 at the studied temperatures T, pressures p, and estimated mole fractions of the gases xgas. The data marked by an asterisk


20

Page 21 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


in Table 1 are effective diffusivities Deff where thermal and mutual diffusivity show the same value. Table 1. Thermal Diffusivity a and Mutual Diffusivity D12 as well as their Relative Uncertainties (k = 2) Measured by DLS for Binary Mixtures of n-C6H14 or n-C10H22 with Dissolved H2, He, CO, and N2 at Different Temperatures, Pressures, and Estimated Mole Fractions of Dissolved Gases. T (K)

p (MPa)

xgas (mol %)

a (10 m2⋅s-1) -9

∆a/a (%)

D11 (10 m2⋅s-1) -9

∆D11/D11 (%)

H2 + n-C6H14 303.04

2.55

1.9 ± 0.2

77.7

2.8

22.5

10.2

302.96

3.78

2.8 ± 0.2

77.2

2.8

22.0

5.2

322.85

2.79

2.1 ± 0.2

73.1

4.2

27.8

9.8

347.64

2.75

2.2 ± 0.2

67.3

5.4

34.1

10.6

372.44

3.00

2.6 ± 0.2

60.8

11.4

39.3

20.2

397.28

2.08

1.8 ± 0.1

53.2

422.08

2.61

2.4 ± 0.2

(*)

47.4

2.8

(*)

53.2

(*)

2.8

(*)

18.8

64.3

57

H2 + n-C10H22 303.01

1.55

1.03 ± 0.07

79.6

1.8

14.8

16.6

322.80

1.55

1.11 ± 0.08

76.1

2.0

18.2

9.6

347.58

1.35

1.08 ± 0.08

70.6

2.0

22.8

12.4

372.37

2.43

2.1 ± 0.2

66.5

3.2

28.7

7.0

397.11

3.11

3.1 ± 0.2

62.6

5.4

36.0

13.6

422.04

2.34

2.6 ± 0.2

53.7

(*)

8.4

(*)


53.7

(*)

8.4

(*)

21


Page 22 of 42

He + n-C6H14 303.02

6.08

2.18 ± 0.03

78.2

1.8

24.1

4.8

322.88

5.43

3.28 ± 0.04

73.3

4.0

29.1

9.4

347.71

5.68

5.98 ± 0.08

68.4

9.8

35.9

17.4

372.55

3.79

5.91 ± 0.08

54.5

33.8

39.0

50.6

397.34

1.99

3.62 ± 0.04

54.5

422.24

1.43

2.11 ± 0.03

52.2

(*)

(*)

2.2 6.4

(*)

(*)

54.5 52.2

(*)

(*)

2.2 6.4

(*)

(*)

He + n-C10H22 303.01

3.44

0.88 ± 0.03

80.6

3.2

17.2

17.6

322.87

2.76

0.83 ± 0.02

76.3

2.2

20.7

12.1

347.87

2.43

0.86 ± 0.03

72.8

4.4

30.2

13.8

372.53

2.38

0.97 ± 0.03

68.4

7.4

37.3

24.0

397.38

1.89

0.87 ± 0.03

61.4

56

53.3

34.2

422.24

5.57

2.87 ± 0.08

50.2

(*)

5.4

(*)

50.2

(*)

5.4

(*)

CO + n-C6H14 302.97

1.48

2.31 ± 0.09

77.1

6.0

10.4

4.2

322.81

1.47

2.26 ± 0.08

73.0

8.8

13.3

16.2

347.48

1.41

2.17 ± 0.08

68.6

5.2

17.5

9.0

372.21

1.41

2.14 ± 0.08

61.4

5.2

20.6

8.4

4.8

6.0

18.0

CO + n-C10H22 303.00

0.68

1.00 ± 0.01

78.6


22

Page 23 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


322.82

0.50

0.74 ± 0.01

74.6

4.1

7.5

16.3

347.75

0.68

1.03 ± 0.01

71.0

2.6

9.5

10.8

372.65

0.80

1.26 ± 0.02

66.3

4.8

12.2

7.0

N2 + n-C6H14 303.05

1.46

1.92 ± 0.09

78.7

4.4

10.2

4.0

322.94

1.50

1.88 ± 0.09

73.4

5.4

13.0

9.2

322.93

2.1

1.88 ± 0.09

72.0

9.6

12.8

11.0

347.82

1.52

1.86 ± 0.09

-

-

17.2

8.6

372.70

1.60

1.93 ± 0.09

-

-

21.2

10.4

397.68

1.51

1.67 ± 0.08

-

-

26.0

22.0

N2 + n-C10H22 303.05

1.83

1.7 ± 0.2

79.3

2.6

5.38

2.2

323.00

1.72

1.6 ±0.2

75.7

2.2

7.2

8.4

347.85

1.48

1.4 ± 0.1

73.0

0.8

9.3

6.4

372.77

1.46

1.5 ± 0.1

66.5

2.4

12.8

4.6

397.70

1.24

1.3 ± 0.1

62.2

7.8

14.5

11.0

The superscript (*) denotes effective diffusivities Deff The thermal diffusivity of binary mixtures of n-C6H14 or n-C10H22 with dissolved He, H2, N2 and CO in the limit of an infinite dilution of the gas with an average expanded uncertainty (k = 2) of 6.8% decreases with increasing temperature and with decreasing chain length of the n-alkane solvent. Furthermore, the a data are not significantly affected by the type and amount of the gas. This fact is verified by comparison of our experimental temperature-


23


Page 24 of 42

dependent results for the thermal diffusivity a of the various binary mixtures with the corresponding values aref for the pure n-alkanes from Watanabe and Seong,56 see Figure 2. The relative deviations of all measured values from the reference data are within combined uncertainties, resulting in an AAD value of 1.7%.

Figure 2. Comparison between the temperature-dependent thermal diffusivities of the binary mixtures of n-C6H14 (upper panel) or n-C10H22 (lower panel) with dissolved gases obtained by DLS (symbols) and the corresponding thermal diffusivities of the respective pure n-alkanes from Watanabe and Seong56 (continuous black line). For clarity purposes, error bars of the experimental data are not plotted. The experimental mutual diffusivity data D11 obtained by DLS are represented by filled symbols in Figure 3. The associated error bars displayed in the graphs illustrate the expanded


24

Page 25 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


uncertainties (k = 2), which range from 5 to 35% over the whole temperature range and are in average 13%. The latter value is slightly larger than the typical uncertainties obtained in our former studies.19–22,31,57 This can be explained by the large number of state points close to Le = 1 where the separation of the hydrodynamic modes related to temperature and concentration remains difficult. Nevertheless, it could be documented that even for the studied small gas mole fractions ranging between 0.7 mol % and 6.0 mol %, uncertainties in the experimental D11 data of typically smaller than 10% can be obtained when Le ≥ 2. According to our literature survey, to the best of knowledge, our experimental mutual diffusivity results are the first data for the investigated binary mixtures of short-chained nalkanes with dissolved gases. The self-diffusivity data of the gases D1 obtained by MD simulation are reported in Table 3 and represented by open symbols in Figure 3. In the MD simulations performed at or close to the temperature and pressure conditions as studied by DLS, the gas mole gas fraction was fixed to 0.99 mol % for all binary mixtures. While this composition is not entirely representative for the experimental thermodynamic state, an exemplary study on the variation of the mole fraction of N2 dissolved in n-C10H22 between (0.9 and 5.0) mol % revealed no effect of composition on the gas self-diffusivity data within combined uncertainties. The average expanded statistical uncertainty (k = 2) for the gas self-diffusion coefficients related to all eight systems is 7% which is relatively low under consideration of the small concentration of the dissolved gases in the infinite diffusion limit. For the simulations with H2, the two different FFs proposed by Darkrim et al.48 and by Hirschfelder et al.45 were used. To the best of our knowledge, until now, no comparison of the calculated self-diffusivity data for H2 in its binary mixtures with liquid n-alkanes using these two FFs has been performed. Ferrando and Ungerer58 have compared the ability of the two models to predict the compressibility of pure H2 and phase equilibria data of


25


Page 26 of 42

H2 + hydrocarbons mixtures. They found that the FF proposed by Hirschfelder et al.45 allows for a better estimation of the compressibility, while phase equilibria data could be better predicted by the model proposed by Darkrim et al.48 Makrodimitri et al.46 and Koller et al.22 used the H2 model proposed by Hirschfelder et al.45 to estimate dynamic properties of H2 dissolved in heavy hydrocarbons and ionic liquids because it leads to less computation efforts. For consistency, in the present study we performed a comparison between the two models for H2 in its two binary mixtures with n-C6H14 or n-C10H22 over the entire temperature range. Within combined uncertainties, both FFs provide matching self-diffusivity data. In Table 2, only the results obtained by the model proposed by Darkrim et al.48 showing 1.8% lower uncertainties for D1 than those obtained with the model of Hirschfelder et al.45 are reported. Table 2. Self-Diffusivities (k = 2) of H2, He, CO, and N2 Dissolved in n-C6H14 or n-C10H22 and their Relative Uncertainties (k = 2) Simulated by MD at Different Temperatures, Pressures and a Gas Mole Fraction xgas = 0.01. T (K)

p (MPa)

D1 (10 m2⋅s-1) -9

∆D1/ D1 (%)

p (MPa)

H2 + n-C6H14

D1 (10 m2⋅s-1) -9

∆D1/ D1 (%)

H2 + n-C10H22

303.15

4.20

20.3

3.4

1.29

10.8

5.5

333.15

4.15

29.9

7.1

1.00

17.1

11.3

363.15

3.17

42.2

6.1

1.13

25.2

9.9

393.15

3.90

59.2

2.0

0.84

35.7

2.7

423.15

3.90

84.5

1.2

0.71

49.2

2.8

He + n-C6H14 303.15

3.77

26.1

He + n-C10H22 3.5

3.91


15.1

4.0

26

Page 27 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


333.15

3.52

35.4

6.6

3.04

22.2

11.7

363.15

3.43

48.5

6.6

2.90

30.9

9.4

393.15

3.45

63.6

8.2

2.58

39.1

9.6

423.15

3.42

87.1

7.2

2.13

52.0

12.5

CO + n-C6H14

CO + n-C10H22

303.15

1.69

9.7

2.9

1.49

4.3

5.9

333.15

1.64

13.4

6.1

1.66

6.9

9.0

363.15

1.43

19.4

6.1

1.54

10.0

5.6

393.15

1.39

26.5

5.1

1.60

13.8

11.6

423.15

1.42

38.0

7.2

1.31

19.0

7.1

N2 + n-C6H14

N2 + n-C10H22

303.15

1.20

9.8

10.8

0.81

4.6

3.1

333.15

1.12

14.1

4.2

0.84

7.0

9.8

363.15

1.07

19.7

5.2

0.57

10.6

7.0

393.15

0.93

27.6

4.2

0.55

14.2

3.5

423.15

0.90

40.5

8.0

0.56

19.6

12.1

Comparison Between Mutual and Self-Diffusivities. All measured mutual diffusivities D11 and calculated gas self-diffusivities D1 obtained in this study are shown as a function of temperature in Figure 3. Here, diffusivities of mixtures based on the solvents n-C6H14 and nC10H22 are reported on the left and right side of the figure, respectively. While the upper part of Figure 3 shows the absolute diffusivity results, relative deviations of the experimental from the simulation data are illustrated in the lower panel of Figure 3 as it will be discussed later


27


Page 28 of 42

on. The error bars plotted in Figure 3 illustrate the expanded uncertainties (k = 2) of the DLS and MD data as given in Tables 1 and 2. The typical Arrhenius-like behavior for the temperature dependency of D11 and D1 can be observed for all binary systems. Due to the restricted temperature range accessible by DLS for mixtures with CO, the variation of D11 with temperature is almost linear. In order to compare mutual diffusivity and self-diffusivity data at infinite dilution, we have fitted the temperature dependency of the simulated self-diffusivities by an Arrheniustype equation.22,31 The latter model represents the self-diffusivities with a mean AAD of 2.4%. The relative deviations of the experimental D11 data from the fits of the calculated D1 data are shown in the lower panel in Figure 3. Here, the dashed lines correspond to the expanded uncertainties (k = 2) of the fit correlations related to the systems with the same nalkane solvent. For this, the averaged expanded uncertainty of all corresponding simulated D1 data and the doubled AAD values between the simulated and correlated data are summed up. For comparable thermodynamic states, our experimental mutual diffusivities and simulated gas self-diffusivities of the studied binary mixtures of n-C6H14 and n-C10H22 with dissolved H2, He, CO, and N2 at infinite dilution show very good agreement mostly within combined uncertainties, see Figure 3. Thus, the matching of the two properties valid for binary systems at small gas mole fractions below 6.0 mol % could be proven here. An overview of the statistical relative deviations between the experimental D11 and correlated D1 data in the form of values for the AAD, the bias, and the maximum deviation ∆max is given in Table 3 for each binary mixture. In average, the AADs are 8.3% and 18.2% for the n-C6H14- and n-C10H22based systems, respectively.


28

Page 29 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. Comparison between experimental mutual diffusivities D11 (filled symbols) and simulated self-diffusivities of the gases D1 (open symbols) for binary mixtures of n-C6H14 or n-C10H22 with dissolved gases for different temperatures T. In the upper panels, the absolute diffusivity values are shown, while the lower panels illustrate the relative deviations of the mutual diffusivities from the Arrhenius-type correlations of the self-diffusivities for each binary mixture. In the left and right column, the corresponding results for mixtures based on n-C6H14 and n-C10H22 are given. Table 3. Statistical Relative Deviations between the Experimental Mutual Diffusivities D11 and the Arrhenius-Type Correlations of the Simulated Gas Self-Diffusivities D1 for the Investigated Binary Mixtures of n-Alkanes with Dissolved Gases. CO

He

N2

H2

5.8

7.0

n-C6H14 AAD (%)

6.9

13.6


29


Page 30 of 42

bias (%)

6.0

-13.6

-0.7

-1.6

∆max (%)

9.2

25.6

12.3

5.9

n-C10H22 AAD (%)

30.7

15.3

10.2

16.5

bias (%)

24.6

15.3

-7.1

-11.7

∆max (%)

38.7

20.9

15.6

30.3

Regarding the variation of the gases on the diffusivity results, Table 3 and Figure 3 indicate the best agreement between experiment and simulation within combined uncertainties for systems containing N2. While also the results for the small gases H2 and He are represented well by the MD simulations mostly within combined uncertainties supporting the choice of the used gas models, the largest deviations are found for the systems containing CO, with a maximum relative deviation of up to 39% in the case of the solvent n-C10H22. Here, the simplified single-site LJ model of Hirschfelder et al.45 cannot reliably capture the dynamics of the dissolved diatomic CO molecules. A similar behavior was also found in our study on the mutual diffusivity of CO in n-octacosane (n-C28H58).19 From the lower panels in Figure 3, it can also be seen that the increase of D11 with increasing temperature is slightly smaller than that of D1 for all gas-liquid combinations, except of the system He + n-C10H22. In addition, the effect of the solvent on the diffusivity data is that the deviations of the mutual diffusivities from the self-diffusivities are by trend shifted to more positive values from mixtures based on n-C6H14 to mixtures based on nC10H22. In particular for n-C10H22-based systems below 373 K, most of the mutual diffusivity data are outside combined uncertainties with respect to the generally lower values for the correlated self-diffusivities of the gases. This discrepancy can mostly be explained by the employed L-OPLS FF for the pure n-alkane, where we found at low temperatures overly


30

Page 31 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


large dynamic viscosities calculated by our MD simulations in comparison with the reference data given in the Refprop database.55 For increasing temperatures, the deviations between simulated and reference dynamic viscosities decreased, resulting in a better agreement of the diffusivity data. A similar but weaker tendency was observed for mixtures consisting of nC6H14. Therefore, it can be concluded that the more distinct discrepancies observed at low temperatures originate from the lack of the L-OPLS FF in reproducing the intermolecular and structural effects here well. Our findings are in agreement with the original work from Siu et al.44 developing the L-OPLS FF, who obtained a better agreement between their simulated dynamic viscosities at 298.15 K and experimental literature data in the case of shorter nalkanes such as n-C6H14 compared to longer n-alkanes such as n-C10H22. Structure-Property Relationships. Taking into account the infinite number of possible combination of liquids with dissolved gases to be investigated by experiments and/or simulations, correlations for the mass diffusivity based on conveniently accessible properties are of considerable interest for engineering purposes. In this connection, numerous empirical and semi-empirical correlations allowing for a prediction of D11 or D1 of gases in liquids typically within about 20% exist in literature, most of which are summarized in refs 9 and 11. Often employed correlations are, for example, the models developed by Wilke and Chang,59Arnold,60 and Scheibel.61 Based on these and other models, it can be concluded that the commonly used properties affecting the mass diffusivity are the molar mass of solute and/or solvent, the dynamic viscosity of the solvent, and the molar liquid volume of the solvent and/or the solute, for the latter commonly taken at its normal boiling point. In a first step, general relationships between our determined mass diffusivities and the properties of the solute and solute are discussed in the following. Regarding the influence of the n-alkane solvent on the mass diffusivities for a given gas and temperature, decreasing D11 and D1 values with increasing carbon chain length can be


31


Page 32 of 42

found. This observation is in accordance with the increasing viscosities as well as molar masses for longer-chained n-alkane solvents. With respect to the effect of the gaseous solutes on their mass diffusion process in binary mixtures with the studied n-alkanes, two groups having similar diffusivities at comparable temperatures for a given solvent can be observed according to Figure 3. The diffusivities for the systems with CO and N2 match within combined uncertainties, which can be related to their identical molar masses. In the same context, the mass diffusivities for the systems with the H2 and He solutes of significantly lower molar masses exceed those related to the heavier CO and N2 solutes by at least a factor of two in the case of both solvents. Nevertheless, for each thermodynamic state studied, the diffusivities of the mixtures with the heavier He are larger than the ones for the mixtures with H2, yet within combined uncertainties. This trend is in contraction with the reciprocal dependency between mass diffusivity and molar mass of the solute, which is generally expressed in available correlations.9 The same discrepant behavior with respect to H2 and He is also be observed if the molar volume of the solute at its normal boiling point is used instead of the molar mass. The apparent failure of structureproperty relationships for the mass diffusion of the small gases H2 and He has also been observed in literature for at least mixtures containing H2 dissolved in further n-alkane systems,19,20,46 ionic liquids,22 and water.9 Development of a Semi-Empirical Correlation. In consideration of the trends found for our experimental and simulated data, we intend to develop a new semi-empirical correlation for the mass diffusivity D11 of binary mixtures of the studied gases H2, He, CO, and N2 dissolved in n-alkanes of varying chain length at the limit of infinite dilution. This correlation should require only conveniently accessible physical properties of the solvent and solute. One correlation approach is to describe the temperature dependency of the mass diffusivity data related to a certain gas in the form of an Arrhenius-like behavior as a function of the


32

Page 33 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


reduced temperature Tr = T/Tc of the n-alkane solvents. Such solute-specific correlations containing only two constants and requiring only information on the critical temperature Tc of the liquids have shown to provide reliable estimations over a broad range of n-alkanes from n-C6H14 up to n-hexanonacontane (n-C96H194) and will be investigated in more detail in the future. In this study, a simple correlation accounting for the influence of the solvent and solute on the mutual diffusivity is of interest. The suggested variables are the temperature T, the molar masses of the solute M1 as well as of the solvent M2, and the kinematic viscosity of the solvent ν2. To account for the larger diffusivities of the heavier He compared to the lighter H2, the acentric factor ω1 of the dissolved gas was considered as additional parameter, which is a measure for the non-sphericity of the molecule. For the establishment of the correlation scheme, 175 temperature-dependent mass diffusivities were considered. The database includes all experimental D11 and simulated D1 data obtained in the present study with the exception of the measured Deff data as well as further experimental D11 data from the literature related to binary mixtures containing H2 or CO dissolved in n-alkanes. Besides our DLS results on mixtures with the solvents n-dodecane (n-C12H26)20 and n-C28H58,19 we also incorporated experimental results obtained by Taylor dispersion on mixtures with the solvents n-heptane (n-C7H16),62 n-C12H26,62 n-hexadecane (nC16H34),62 n-eicosane (n-C20H42),63 and n-C28H58.64 Our D11 data for mixtures containing ntetracontane (n-C40H82)20 could not be used due to the lack of corresponding kinematic viscosities. For n-C6H14, n-C7H16, n-C10H22, and n-C12H26, ν data from the Refprop database55 at the T and p conditions corresponding to the D11 and D1 data were used. Regarding n-C16H34 and n-C20H42, density and dynamic viscosity data measured by Matthews et al.65 and by Rodden et al.63 were employed to deduce the kinematic viscosity. The kinematic viscosities


33


Page 34 of 42

of n-C28H58 were calculated based on the T-dependent dynamic viscosities and densities reported by Koller et al.66 The suggested semi-empirical correlation for the mutual diffusivity of binary mixtures of the gases H2, He, N2, and CO dissolved in n-alkanes at infinite dilution is a modified version of the model of Wilke and Chang59 and has the form D11,calc = 1.274 ⋅ 10−8

T ⋅ M 20.5 . ν 20.9 [ M 1 ⋅ (0.45 + ω1 )]0.25

(2)

In eq 2, T is the temperature in K, M1 and M2 are the molar masses of solute and solvent, respectively, in g⋅mol-1, ν2 is kinematic viscosity of the solvent in m2⋅s-1, ω1 is the acentric factor of the solute, and D11,calc is the calculated mutual diffusivity in m2⋅s-1. A comparison of the 175 experimental and simulated mass diffusivity data with the results calculated from eq 2 and those predicted according to the Wilke-Chang model59 is shown in Figure 4.

Figure 4. Comparison of the 175 temperature-dependent mass diffusivities D11 and D1 for binary mixtures of H2, He, N2, and CO dissolved in the n-alkanes n-C6H14, n-C7H16, n-C10H22, n-C12H26, n-C16H34, n-C20H42, and n-C28H58 in the limit of infinite dilution obtained from this work by DLS and MD simulations as well as from experimental literature data19,20,62-64 with the corresponding values D11,calc calculated according to the model of Wilke and Chang59 and ACS Paragon Plus Environment

34

Page 35 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the new model based on eq 2. For clarity purposes, error bars of the experimental and simulated data are not plotted.

The relative AAD of the experimental and simulated mass diffusivities from the calculated ones reduces from 101.2% in the case of the Wilke-Chang model59 to 10.7% in the case of our new model in eq 2. Considering the average uncertainty of the used database of 10.5%, the quality of the proposed correlation scheme is underlined. In particular the data for mixtures containing the light gases H2 or He, which are systematically underestimated by the Wilke-Chang model59 as it can be seen in the left part of Figure 4, are represented well by the new correlation mainly due to the consideration of the acentric factor of the dissolved gases. In addition, the adaptation of the kinematic viscosity instead of the dynamic viscosity used in the Wilke-Chang model59 is not only advantageous with respect to a better description of the mass diffusivity data, but also due to the fact that no information on the density is required. A transfer of the developed model based on eq 2 to systems containing H2O or CO2 dissolved in n-C12H26 or n-C28H58 results in an relative AAD of the testing data for the 28 experimental mutual diffusivities19,20,62,64 from the predicted results of 9.8%. For a further improved representation of mass diffusivity data in liquids with dissolved gases, we believe that in the developed model further physico-chemical characteristics of the solutes and solvents such as their polarities and/or molecular structures need to be considered. This can only be performed if a comprehensive and reliable experimental database is given, which cannot be confirmed at present. In the future, we propose to investigate further binary mixtures of liquids with dissolved gases of varying solute polarities as well as sizes and solvent viscosities as a function of composition in order to develop a reliable semi-empirical correlation method for D11.


35


Page 36 of 42

CONCLUSIONS Thermal and mutual diffusivities as well as self-diffusivities in binary mixtures consisting of the n-alkanes n-C6H14 and n-C10H22 with dissolved H2, He, CO, and N2 were determined at macroscopic thermodynamic equilibrium between 303 and 423 K in the limit of an infinite dilution of the gas. It could be demonstrated that even for gas concentrations smaller than 6 mol % as well as for gases with very small scattering cross sections, DLS measurements allowed for the simultaneous determination of thermal and mutual diffusivities with typical expanded uncertainties (k = 2) of 13%. Only in the vicinity of the Lewis number close to 1, a dissociation of the two superimposed hydrodynamic modes is not possible. On the basis of appropriate molecular force fields, corresponding MD simulations of the self-diffusivity of the various gases in their mixtures with the liquids at comparable thermodynamic states could reproduce the mutual diffusivities mostly within combined uncertainties, resulting in a mean AAD between MD and DLS results of about 13%. In addition, the measured thermal diffusivities of the studied binary mixtures are not influenced by the type as well as amount of gas and deviate from the literature values for the pure n-alkanes by less than 5%. While the matching mass diffusivities for the systems containing CO and N2 follow the expected trend, the larger values for the systems with He compared to those with H2 cannot be explained with the help of available empirical models. Based on the data obtained in this study and further literature data, a new semi-empirical model for the prediction of the mass diffusivity of H2, He, CO, and N2 dissolved in n-alkanes of varying chain length at infinite dilution was developed. This generalized model represents the collected database with an absolute average deviation of 11% and can, thus, serve as a starting point for a further improved prediction scheme covering arbitrary binary combinations of liquids and dissolved gases.

AUTHOR INFORMATION


36

Page 37 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Corresponding Author *E-mail: [email protected]. Tel.: +49-9131-85-23022.

SUPPORTING INFORMATION Details on the model for the estimation of the solubility of the gases H2, He, CO, and N2 dissolved in n-C6H14 or n- C10H22, on the influence of the incident laser power on the DLS experiments, and on the computational procedure as well as the data evaluation of the selfdiffusivity of the dissolved gases by MD simulations. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENTS This work was financially supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) within the project FR 1709/11-1 and by funding the Erlangen Graduate School in Advanced Optical Technologies (SAOT) within the German Excellence Initiative. The authors also gratefully acknowledge the compute resources and support provided by the Erlangen Regional Computing Center (RRZE). The authors would also like to acknowledge Korbinian Batz for his support in the collection and modeling of solubility data.

REFERENCES (1) (2) (3) (4)

(5)

Hasib-ur-Rahman, M.; Siaj, M.; Larachi, F. Ionic Liquids for CO2 CaptureDevelopment and Progress. Chem. Eng. Process. Process Intensif. 2010, 49, 313–322. Greener Fischer-Tropsch Processes; Maitlis, P. M., de Klerk, A., Eds.; Wiley-VCH.; Weinheim, Germany, 2013. Li, Y.-N.; Ma, R.; He, L.-N.; Diao, Z.-F. Homogeneous Hydrogenation of Carbon Dioxide to Methanol. Catal. Sci. Technol. 2014, 4, 1498–1512. Figueroa, J. D.; Fout, T.; Plasynski, S.; McIlvried, H.; Srivastava, R. D. Advances in CO2 Capture Technology - The U.S. Department of Energy’s Carbon Sequestration Program. Int. J. Greenh. Gas Control 2008, 2, 9–20. Rochelle, G. T. Carbon Capture and Sequestration. Science 2009, 325, 1652–1654.


37


(6)

(7)

(8)

(9) (10) (11)

(12)

(13)

(14)

(15) (16) (17)

(18)

(19)

(20)

(21)

Page 38 of 42

Liu, X.; Schnell, S. K.; Simon, J.; Bedeaux, D.; Kjelstrup, S. Fick Diffusion Coefficients of Liquid Mixtures Directly Obtained From Equilibrium Molecular Dynamics. J. Phys. Chem. B 2011, 115, 12921–12929. Liu, X.; Vlugt, T. J. H.; Bardow, A. Maxwell–Stefan Diffusivities in Liquid Mixtures: Using Molecular Dynamics for Testing Model Predictions. Fluid Phase Equilib. 2011, 301, 110–117. Hellmann, R.; Bich, E.; Vogel, E.; Vesovic, V. Intermolecular Potential Energy Surface and Thermophysical Properties of the CH4–N2 System. J. Chem. Phys. 2014, 141, 224301. Himmelblau, D. M. Diffusion of Dissolved Gases in Liquids. Chem. Rev. 1964, 64, 527–550. Umesl, N. O.; Danner, R. P. Predicting Diffusion Coefficients in Nonpolar Solvents. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 662–665. Condemarin, R.; Scovazzo, P. Gas Permeabilities, Solubilities, Diffusivities, and Diffusivity Correlations for Ammonium-Based Room Temperature Ionic Liquids with Comparison to Imidazolium and Phosphonium RTIL Data. Chem. Eng. J. 2009, 147, 51–57. Pizarro, C.; Suárez-Iglesias, O.; Medina, I.; Bueno, J. L. Binary Diffusion Coefficients for 2,3-Dimethylaniline, 2,6-Dimethylaniline, 2-Methylanisole, 4-Methylanisole and 3-Nitrotoluene in Supercritical Carbon Dioxide. J. Supercrit. Fluids 2009, 48, 1–8. Morgan, D.; Ferguson, L.; Scovazzo, P. Diffusivities of Gases in Room-Temperature Ionic Liquids: Data and Correlations Obtained Using a Lag-Time Technique. Ind. Eng. Chem. Res. 2005, 44, 4815–4823. Giraudet, C.; Bataller, H.; Croccolo, F. High-Pressure Mass Transport Properties Measured by Dynamic Near-Field Scattering of Non-Equilibrium Fluctuations. Eur. Phys. J. E 2014, 37, 107. Magatti, D.; Alaimo, M. D.; Potenza, M. A. C.; Ferri, F. Dynamic Heterodyne Near Field Scattering. Appl. Phys. Lett. 2008, 92, 2–4. Fröba, A. P.; Leipertz, A. Diffusion Measurements in Fluids by Dynamic Light Scattering. Diffus. Fundam. 2005, 225, 1–63. Fröba, A. P. Dynamic Light Scattering (DLS) for the Characterization of Working Fluids in Chemical and Energy Engineering. Habil. Thesis, Friedrich-AlexanderUniversität, Erlangen, Germany2009. Fröba, A. P.; Will, S.; Nagasaka, Y.; Winkelmann, J.; Wiegand, S.; Köhler, W. Optical Methods. In Experimental Thermodynamics Volume IX: Advances in Transport Properties of Fluids; Assael, M. J., Goodwin, A. R., Vesovic, V., Wakeham, W. A., Eds.; The Royal Society of Chemistry, 2014; pp 19–74. Heller, A.; Koller, T. M.; Rausch, M. H.; Fleys, M. S. H.; Bos, A. N. R.; van der Laan, G. P.; Makrodimitri, Z. A.; Economou, I. G.; Fröba, A. P. Simultaneous Determination of Thermal and Mutual Diffusivity of Binary Mixtures of n-Octacosane with Carbon Monoxide, Hydrogen, and Water by Dynamic Light Scattering. J. Phys. Chem. B 2014, 118, 3981–3990. Heller, A.; Fleys, M. S. H.; Chen, J.; Van Der Laan, G. P.; Rausch, M. H.; Fröba, A. P.; Fröba, A. P. Thermal and Mutual Diffusivity of Binary Mixtures of n-Dodecane and n-Tetracontane with Carbon Monoxide, Hydrogen, and Water from Dynamic Light Scattering (DLS). J. Chem. Eng. Data 2016, 61, 1333–1340. Rausch, M. H.; Heller, A.; Herbst, J.; Koller, T. M.; Bahlmann, M.; Schulz, P. S.; Wasserscheid, P.; Fröba, A. P. Mutual and Thermal Diffusivity of Binary Mixtures of the Ionic Liquids [BMIM][C(CN)3] and [BMIM][B(CN)4] with Dissolved CO2 by Dynamic Light Scattering. J. Phys. Chem. B 2014, 118, 4636–4646.


38

Page 39 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22)

(23)

(24)

(25)

(26)

(27)

(28) (29)

(30)

(31)

(32)

(33) (34) (35) (36) (37) (38)

(39)

Koller, T. M.; Heller, A.; Rausch, M. H.; Wasserscheid, P.; Economou, I. G.; Fröba, A. P. Mutual and Self-Diffusivities in Binary Mixtures of [EMIM][B(CN)4] with Dissolved Gases by Using Dynamic Light Scattering and Molecular Dynamics Simulations. J. Phys. Chem. B 2015, 119, 8583–8592. Liu, X.; Martín-Calvo, A.; McGarrity, E.; Schnell, S. K.; Calero, S.; Simon, J.-M.; Bedeaux, D.; Kjelstrup, S.; Bardow, A.; Vlugt, T. J. H. Fick Diffusion Coefficients in Ternary Liquid Systems from Equilibrium Molecular Dynamics Simulations. Ind. Eng. Chem. Res. 2012, 51, 10247–10258. Koller, T. M.; Rausch, M. H.; Ramos, J.; Schulz, P. S.; Wasserscheid, P.; Economou, I. G.; Fröba, A. P. Thermophysical Properties of the Ionic Liquids [EMIM][B(CN)4] and [HMIM][B(CN)4]. J. Phys. Chem. B 2013, 117, 8512−8523. Janzen, T.; Zhang, S.; Mialdun, A.; Guevara-Carrion, G.; Vrabec, J.; He, M.; Shevtsova, V. Mutual Diffusion Governed by Kinetics and Thermodynamics in the Partially Miscible Mixture Methanol + Cyclohexane. Phys. Chem. Chem. Phys. 2017, 19, 31856–31873. Galliéro, G.; Duguay, B.; Caltagirone, J.-P.; Montel, F. Thermal Diffusion Sensitivity to the Molecular Parameters of a Binary Equimolar Mixture, a Non-Equilibrium Molecular Dynamics Approach. Fluid Phase Equilib. 2003, 208, 171–188. Galliéro, G.; Boned, C. Molecular Dynamics Study of the Repulsive Form Influence of the Interaction Potential on Structural, Thermodynamic, Interfacial, and Transport Properties. J. Chem. Phys. 2008, 129, 74506. Köhler, W.; Morozov, K. I. The Soret Effect in Liquid Mixtures – A Review. J. NonEquilibrium Thermodyn. 2016, 41, 151–197. Mialdun, A.; Sechenyh, V.; Legros, J. C.; Ortiz De Zárate, J. M.; Shevtsova, V. Investigation of Fickian Diffusion in the Ternary Mixture of 1,2,3,4Tetrahydronaphthalene, Isobutylbenzene, and Dodecane. J. Chem. Phys. 2013, 139, 1– 16. Ortiz de Zárate, J. M.; Giraudet, C.; Bataller, H.; Croccolo, F. Non-Equilibrium Fluctuations Induced by the Soret Effect in a Ternary Mixture. Eur. Phys. J. E 2014, 37, 77. Heller, A.; Giraudet, C.; Makrodimitri, Z. A.; Fleys, M. S. H.; Chen, J.; van der Laan, G. P.; Economou, I. G.; Rausch, M. H.; Fröba, A. P. Diffusivities of Ternary Mixtures of n-Alkanes with Dissolved Gases by Dynamic Light Scattering. J. Phys. Chem. B 2016, 120, 10808–10823. Liu, X.; Schnell, S. K.; Simon, J.-M.; Krüger, P.; Bedeaux, D.; Kjelstrup, S.; Bardow, A.; Vlugt, T. J. H. Diffusion Coefficients from Molecular Dynamics Simulations in Binary and Ternary Mixtures. Int. J. Thermophys. 2013, 34, 1169–1196. Molecular Hydrodynamics; Boon, J. P., Yip, S., Eds.; Courier Corporation, 1980. Hydrodynamic Fluctuations in Fluids and Fluid Mixtures; Ortiz de Zárate, J. M., Sengers, J. V., Eds.; Elsevier, 2006. Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics; Berne, B. J., Pecora, R., Eds.; Dover publ.; 2000. Marano, J. J.; Holder, G. D. Characterization of Fischer-Tropsch Liquids for VaporLiquid Equilibria Calculations. Fluid Phase Equilib. 1997, 138, 1–21. Masuku, C. M.; Hildebrandt, D.; Glasser, D. The Role of Vapour-Liquid Equilibrium in Fischer-Tropsch Product Distribution. Chem. Eng. Sci. 2011, 66, 6254–6263. Lawrence Clever, H.; Battino, R.; Saylor, J. H.; Gross, P. M. The Solubility of Helium, Neon, Argon and Krypton in Some Hydrocarbon Solvents. J. Phys. Chem. 1945, 61, 1078–1082. Tsonopoulos, C. Thermodynamic Analysis of the Mutual Solubilities of Normal


39


(40) (41) (42) (43) (44) (45) (46)

(47) (48)

(49)

(50)

(51)

(52)

(53) (54)

(55)

(56)

(57)

(58)

Page 40 of 42

Alkanes and Water. Fluid Phase Equilib. 1999, 156, 21–33. Segrè, P. N.; Sengers, J. V. Nonequilibrium Fluctuations in Liquid Mixtures under the Influence of Gravity. Physica A 1993, 198, 46–77. Computer Simulation of Liquids; Allen, M. P., Tildesley, D. J., Eds.; Oxford university press, 1987. Martin, M. G.; Siepmann, J. I. Transferable Potentials for Phase Equilibria. 3. ExplicitHydrogen Description of Normal Alkanes. J. Phys. Chem. B 1998, 102, 2569–2577. Jorgensen, W. L.; Madura, J. D.; Sweson, C. J. Optimized Intermolecular Potential Functions for Liquid Hydrocarbons. J. Am. Chem. Soc. 1984, 106, 6638–6646. Siu, S. W. I.; Pluhackova, K.; Böckmann, R. A. Optimization of the OPLS-AA Force Field for Long Hydrocarbons. J. Chem. Theory Comput. 2012, 8, 1459–1470. Molecular Theory of Gases and Liquids; Hirschfelder, J. O., Bird, R. B., Curtiss, C. F., Eds.; Wiley-VCH: New York, 1954. Makrodimitri, Z. A.; Unruh, D. J. M.; Economou, I. G. Molecular Simulation of Diffusion of Hydrogen, Carbon Monoxide, and Water in Heavy n-Alkanes. J. Phys. Chem. B 2011, 115, 1429–1439. Rivera, J. L.; Alejandre, J.; Nath, S. K.; De Pablo, J. J. Thermodynamic and Transport Properties of Nitrogen and Butane Mixtures. Mol. Phys. 2000, 98, 43–55. Darkrim, F.; Vermesse, J.; Malbrunot, P.; Levesque, D. Monte Carlo Simulations of Nitrogen and Hydrogen Physisorption at High Pressures and Room Temperature. Comparison with Experiments. J. Chem. Phys. 1999, 110, 4020–4027. Cornell, W. D.; Cieplak, P.; Bayly, C. I.; Gould, I. R.; Merz, K. M.; Ferguson, D. M.; Spellmeyer, D. C.; Fox, T.; Caldwell, J. W.; Kollman, P. A. A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules. J. Am. Chem. Soc. 1995, 117, 5179–5197. Berendsen, H. J. C.; van der Spoel, D.; van Drunen, R. GROMACS: A MessagePassing Parallel Molecular Dynamics Implementation. Comput. Phys. Commun. 1995, 91, 43–56. Green, M. S. Markoff Random Processes and the Statistical Mechanics of Time‐Dependent Phenomena. II. Irreversible Processes in Fluids. J. Chem. Phys. 1954, 22, 398–413. Kubo, R. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Japan 1957, 12, 570–586. Understanding Molecular Simulation: from Algorithms to Applications; Frenkel, D., Smit, B., Eds.; Elsevier, 2002. Yeh, I. C.; Hummer, G. System-Size Dependence of Diffusion Coefficients and Viscosities from Molecular Dynamics Simulations with Periodic Boundary Conditions. J. Phys. Chem. B 2004, 108, 15873–15879. Lemmon, E. W.; Huber, M. L.; McLinden, M. O. REFPROP, Standard Reference Data Program. National Institute of Standards and Technology: Gaithersburg, MD, USA, 2013. Watanabe, H.; Seong, D. J. The Thermal Conductivity and Thermal Diffusivity of Liquid n-Alkanes: CnH2n+2 (n=5 to 10) and Toluene. Int. J. Thermophys. 2002, 23, 337–356. Rausch, M. H.; Hopf, L.; Heller, A.; Leipertz, A.; Fröba, A. P. . Binary Diffusion Coefficients for Mixtures of Ionic Liquids [EMIM][N(CN)2], [EMIM][NTf2], and [HMIM][NTf2]with Acetone and Ethanol by Dynamic Light Scattering (DLS). J. Phys. Chem. B 2013, 117, 2429–2437. Ferrando, N.; Ungerer, P. Hydrogen/hydrocarbon Phase Equilibrium Modelling with a


40

Page 41 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(59) (60) (61) (62)

(63) (64)

(65)

(66)

Cubic Equation of State and a Monte Carlo Method. Fluid Phase Equilib. 2007, 254, 211–223. Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solutions. Aiche J. 1955, 1, 264–270. Arnold, J. H. Studies in Diffusion. II. A Kinetic Theory of Diffusion in Liquid Systems. J. Am. Chem. Soc. 1930, 52, 3937–3955. Scheibel, E. G. Correspondence. Liquid Diffusivities. Viscosity of Gases. Ind. Eng. Chem. 1954, 46, 2007–2008. Matthews, M. A.; Rodden, J. B.; Akgerman, A. High-Temperature Diffusion of Hydrogen, Carbon Monoxide, and Carbon Dioxide in Liquid n-Heptane, n-Dodecane, and n-Hexadecane. J. Chem. Eng. Data 1987, 32, 319–322. Rodden, J. B.; Erkey, C.; Akgerman, A. High-Temperature Diffusion, Viscosity, and Density Measurements in n-Eicosane. J. Chem. Eng. Data 1988, 33, 344–347. Rodden, J. B.; Erkey, C.; Akgerman, A. Mutual Diffusion Coefficients for Several Dilute Solutes in n-Octacosane and the Solvent Density at 371-534 K. J. Chem. Eng. Data 1988, 33, 450–453. Matthews, M. A.; Rodden, J. B.; Akgerman, I. A. High-Temperature Diffusion, Viscosity, and Density Measurements in n-Hexadecane. J. Chem. Eng. Data 1987, 32, 317–319. Koller, T. M.; Klein, T.; Giraudet, C.; Chen, J.; Kalantar, A.; Van Der Laan, G. P.; Rausch, M. H.; Fröba, A. P. Liquid Viscosity and Surface Tension of n-Dodecane, nOctacosane, Their Mixtures, and a Wax between 323 and 573 K by Surface Light Scattering. J. Chem. Eng. Data 2017, 62, 3319–3333.


41


Page 42 of 42

TOC Graphic


42

Thermal, Mutual, and Self-Diffusivities of Binary Liquid Mixtures

Recommend Documents