Diffusivities of Ternary Mixtures of n

Article pubs.acs.org/JPCB

Diffusivities of Ternary Mixtures of n‑Alkanes with Dissolved Gases by Dynamic Light Scattering Andreas Heller,† Cédric Giraudet,† Zoi A. Makrodimitri,‡ Matthieu S. H. Fleys,§ Jiaqi Chen,§ Gerard P. van der Laan,§ Ioannis G. Economou,‡,∥ Michael H. Rausch,† and Andreas P. Fröba*,† †

Department of Chemical and Biological Engineering (CBI) and Erlangen Graduate School in Advanced Optical Technologies (SAOT), University of Erlangen-Nuremberg, Paul-Gordan-Straße 6, 91052 Erlangen, Germany ‡ National Centre for Scientific Research “Demokritos”, Institute of Nanoscience and Nanotechnology, Molecular Thermodynamics and Modelling of Materials Laboratory, GR-15310 Aghia Paraskevi Attikis, Greece § Shell Global Solutions International B.V., Grasweg 31, 1031 HW Amsterdam, The Netherlands ∥ Texas A&M University at Qatar, Chemical Engineering Program, Education City, PO Box 23874, Doha, Qatar S Supporting Information *

ABSTRACT: Theoretical approaches suggest that dynamic light scattering (DLS) signals from low-molecular-weight ternary mixtures are governed by fluctuations in temperature as well as two individual contributions from fluctuations in concentration that are related to the eigenvalues of the Fick diffusion matrix. Until now, this could not be proven experimentally in a conclusive way. In the present study, a detailed analysis of DLS signals in ternary mixtures consisting of n-dodecane (n-C12H26) and n-octacosane (nC28H58) with dissolved hydrogen (H2), carbon monoxide (CO), or water (H2O) as well as of n-C12H26 or n-C28H58 with dissolved H2 and CO is given for temperatures up to 523 K and pressures up to 4.1 MPa. Thermal diffusivities of pure n-C12H26 and n-C28H58 as well as thermal and mutual diffusivities of their binary mixtures being the basis for the ternary mixtures with dissolved gas were studied for comparison purposes. For the investigated ternary mixtures, three individual signals could be distinguished in the time-resolved analysis of scattered light intensity by using photon correlation spectroscopy (PCS). For the first time, it could be evidenced that these signals are clearly associated with hydrodynamic modes. In most cases, the fastest mode observable for ternary mixtures is associated with the thermal diffusivity. The two further modes obviously related to the molecular mass transport are observable on different time scales and comparable to the modes associated with the concentration fluctuations in the respective binary mixtures. Comparison of the experimental data with results from molecular dynamics simulations revealed very good agreement.

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INTRODUCTION Transport properties of multicomponent mixtures are not only of technological but also of fundamental scientific interest. For example, questions in connection with the diffusive mass transport in such systems are still open. Mass diffusivities in ternary fluid mixtures can be studied by a large variety of experimental measurement techniques. Most recently, such systems were investigated with the Taylor dispersion (TD) technique,1,2 with the diaphragm cell,3,4 or by Raman spectroscopy.5 All these techniques make use of macroscopic gradients that must be large enough to give rise to a measurable effect but small enough to cause only very little perturbation in the system under investigation. In contrast, dynamic light scattering (DLS) analyzes microscopic fluctuations in temperature or species concentration in macroscopic thermodynamic equilibrium. DLS, however, is restricted to the hydrodynamic regime, where the linearized Navier−Stokes equations are valid. For the determination of mass diffusivities, the solute concentration has to be large enough to obtain detectable DLS signals attributable to the molecular diffusion processes. © 2016 American Chemical Society

On the other hand, DLS allows for the determination of transport and other thermophysical properties in an absolute way over a wide range of thermodynamic states.6,7 In the literature, reports on the application of DLS for studying multicomponent mixtures are very rare. Only Ivanov et al.8,9 and Das et al.10 report homodyne DLS experiments for low-molecular-weight ternary fluid mixtures with a special focus on mass diffusion and without providing thermal diffusivity data. Ivanov et al.8,9 investigated mixtures consisting of glycerol, acetone, and water in the near-critical region, while Das et al.10 report on measurements for aqueous solutions of decylsulfobetaine with dodecylsulfobetaine. Both groups compared their DLS results with data obtained from TD.10,11 Ivanov et al.11 only found one eigenvalue of the Fick diffusion matrix from DLS measurements to coincide with their TD measurements. A second DLS signal that they attributed to mass diffusion yields Received: August 11, 2016 Revised: September 26, 2016 Published: September 27, 2016 10808

DOI: 10.1021/acs.jpcb.6b08117 J. Phys. Chem. B 2016, 120, 10808−10823

Article

The Journal of Physical Chemistry B ∂δx1 = D11∇2 δx1 + D12∇2 δx 2 ∂t

an eigenvalue that differs by a factor of more than 1000 from their TD results. Das et al.10 report good agreement between their DLS and TD results. However, they did not prove that their homodyne DLS signals are associated with hydrodynamic modes. Hence, the question of what information can be gained from DLS signals from low-molecular-weight multicomponent mixtures has not been answered conclusively so far. Theoretical derivations of scattering spectra originating from ternary mixtures consisting of polymers dissolved in binary fluid mixtures are available in the literature.12,13 Such systems were also studied experimentally by using DLS where mutual and translational diffusivities could be determined.14−16 More recently, Bardow17 developed a simple theory that describes the signals observable in the spectrum for ternary mixtures. He concluded that two hydrodynamic modes associated with mass diffusion contribute to the signal, but one of these modes is strongly enhanced for situations typically encountered in DLS experiments. He suggests that signals from ternary mixtures are governed by temperature fluctuations related to thermal diffusivity and two individual contributions from concentration fluctuations that are connected with the eigenvalues of the Fickian diffusion matrix. Bardow’s derivation was confirmed by Ortiz de Zárate et al.18 using fluctuating hydrodynamics. Both groups used the large Lewis number approximation, which is also employed in this study. In the present study, we demonstrate for the first time that the thermal diffusivity as well as mass diffusivities in binary and ternary mixtures relevant for the Fischer−Tropsch synthesis19,20 can be obtained by using DLS by a systematic evaluation of the recorded correlation functions. The investigated mixtures consist of different combinations of the components n-dodecane (n-C12H26), n-octacosane (n-C28H58), hydrogen (H2), carbon monoxide (CO), and water (H2O). From our ongoing study of binary mixtures consisting of nC12H26 or n-C28H58 with dissolved H2, CO, or H2O, we already gained valuable knowledge on the thermal and mutual diffusivities of binary subsystems of the ternary mixtures investigated here.21,22 To complete the systematic investigation of all possible binary subsystems by using DLS, thermal and mutual diffusivities of binary mixtures consisting of n-C12H26 and n-C28H58 are also reported. All these investigations should point out the information accessible by using DLS for lowmolecular-weight ternary fluid mixtures.

(1)

and ∂δx 2 = D21∇2 δx1 + D22∇2 δx 2 ∂t

(2)

as well as the differential equation for the temperature field ∂δT = a∇2 δT ∂t

(3)

In eqs 1 and 2, δx1 and δx2 denote fluctuations in the mole fractions of species 1 and 2 in the presence of a third species as a function of time and space. Dij represents the Fickian diffusion coefficients in a molar-averaged velocity reference frame. In eq 3, δT are fluctuations in temperature and a is the thermal diffusivity. If thermal diffusion coefficients DT tend to zero, which holds for systems at infinite dilution, neglecting the coupling between thermal and concentration fluctuations can be justified.5,23−25 In the case in which thermal and mass diffusivities differ by more than 2 to 3 orders of magnitude, a coupling can be neglected as well.23,24 These assumptions hold for all mixtures investigated in the present study. eqs 1 and 2 are commonly written in matrix form ∂ ⎛ δx1 ⎞ ⎡ D11 D12 ⎤ 2 ⎛ δx1 ⎞ ⎜⎜ ⎟⎟ = ⎢ ⎥∇ ⎜⎜ ⎟⎟ = [D]∇2 [δx] ∂t ⎝ δx 2 ⎠ ⎣ D21 D22 ⎦ ⎝ δx 2 ⎠

(4)

where [D] denotes the Fickian diffusion matrix and [δx] is a vector containing fluctuations in mole fractions of species 1 and 2. To derive the spectrum of scattered light, eqs 1 and 2 need to be decoupled. Bardow utilized the eigenvalue-decomposition method and derived the spectrum of the fluctuations to be17 S(q , ω) =

∑ i = 1,2

Ai

2Dî q2 2aq2 + B ω 2 + (Dî q2)2 ω 2 + (aq2)2

(5)

Here, ω denotes the temporal angular frequency, Ai and B are prefactors that depend on various parameters,17 and D̂ i are eigenvalues of the 2 × 2 Fickian diffusion matrix [D] according to

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̂ = 0.5[D11 + D22 ± D1/2

METHOD When coherent light irradiates a transparent fluid sample in macroscopic thermodynamic equilibrium, light scattered from the sample can be observed in all directions. The underlying scattering process is governed by spontaneous microscopic fluctuations in temperature, pressure, and species concentration in fluid mixtures. These fluctuations cause a spectral broadening of the scattered light. In the following, a coupling between temperature and concentration fluctuations as well as the existence of pressure fluctuations are neglected, which simplifies the theoretical treatment of light scattering experiments considerably. For binary fluid mixtures, the derivation of the scattering spectrum results in a superposition of two Lorentzian lines.23,24 The widths of these two lines are related to the thermal diffusivity a and the mutual diffusivity D12. In the case of ternary fluid mixtures, the spectrum of scattered light can be derived from Fick’s second laws5,24

(D11 − D22)2 + 4D12D21 ] (6)

In eq 5, q represents the modulus of the wave vector of the fluctuations studied, which is equivalent to the modulus of the scattering vector defined by q = |kI⃗ − kS⃗ |. Here, kI⃗ and kS⃗ are the wave vectors of the incident and scattered electric fields. The modulus of the scattering vector q can be expressed by q = (4πn/λ0) sin ΘS, where n is the refractive index of the fluid, λ0 the laser wavelength in vacuo, and ΘS is the scattering angle. For small scattering angles and by using Snell’s law, the modulus of the scattering vector can also be reformulated to be q ≅ (2π/λ0) sin Θi. Here, Θi represents the incident angle, and no information on the refractive index of the fluid is needed, see, e.g., ref 7. In DLS experiments, the time-dependent correlation function (CF) of the scattered light intensity is calculated. For a ternary fluid mixture, it takes the form 10809


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The Journal of Physical Chemistry B G(2)(τ ) = (ILO + It +

∑

manifold system to provide the gas solutes, the sample cell, and the temperature control system, is the same as used in our former studies.21 With the temperature control of the sample cell, a temperature stability of better than ±5 mK could be achieved. The sample cell temperature was measured with calibrated Pt-100 Ω resistance probes with an absolute expanded uncertainty (k = 2) of 15 mK. For the measurement of the pressure inside the cell, a high pressure transducer (KS Sensortechnik GmbH) with a specified uncertainty of 0.05% was used. The used n-alkanes n-C12H26 (Merck GmbH) and nC28H58 (Alfa Aesar GmbH & Co. KG) had purities of >99 mass % and >98 mass % according to the manufacturers’ specifications. As in our previous studies, the n-C28H58 sample was filtered at about 340 K with a syringe filter with a pore size of 200 nm to remove particulate contaminations. The n-C12H26 sample could be used without further purification. CO and H2 were used as provided by Linde AG with purities of 99.997 vol % and 99.9999 vol %. Deionized water was generated by an ionexchanger. The two binary n-alkane mixtures consisting of 30 mol % of n-C12H26 and 70 mol % of n-C28H58 (0.3n-C12H26/0.7n-C28H58) as well as of 70 mol % of n-C12H26 and 30 mol % of n-C28H58 (0.7n-C12H26/0.3n-C28H58) were investigated close to saturation pressure in the temperature range from (373 to 523) K. Filtered n-C28H58 was liquefied in a beaker at about 340 K before an appropriate amount was filled into the sample cell that was preheated to about 370 K. n-C12H26 was directly filled from a beaker into the sample cell without heating. To achieve the desired compositions of the n-alkane mixtures, both liquid compounds were weighed before and after filling with a balance (Sartorius BP110) with an expanded uncertainty (k = 2) of 1 mg, taking into account the complete weighing procedure. After the filling procedure, the cell was evacuated with an oil-sealed vacuum pump to 50 Pa. Only a small vapor phase volume was allowed in the cell to minimize changes in the liquid composition caused by different evaporated amounts of the components. The inevitable variation in composition over the investigated temperature range was estimated based on vapor pressure, volume of the vapor phase, vapor and liquid densities, and mixing rules to be less than 0.05 mol %. For the investigations of ternary mixtures, binary n-alkane mixtures were provided as described above and H2 or CO was added from the manifold system. Before the manifold was filled with gas, it was flushed with nitrogen several times and evacuated with the oil-sealed vacuum pump. Ternary systems of the two n-alkanes with H2O were generated by providing the binary n-alkane mixture and adding between (2 and 5) mL of deionized water, such that a water-rich liquid phase remained visible. After heating the sample cell to a temperature of about 423 K, this liquid water phase could not be observed visually any more. Nevertheless, the measured pressures being in agreement with the vapor pressure of water at the respective temperatures showed that the aqueous solutions were investigated at saturation conditions. This information is required for the determination of the concentration of water in the n-alkane mixture. Ternary mixtures consisting of H2 and CO dissolved in nC12H26 or n-C28H58 were produced by filling the individual gases consecutively into the manifold system at room temperature. First, 2.02 MPa of CO was filled into the manifold. Then, H2 was added, leading to a total pressure of 6.90 MPa in the manifold. The provided gas mixture composition in the manifold system was calculated based on

Ic, i)2

= 1,2 i 

background + 2ILOIt exp( −τ /τC,t) +

∑

2ILOIc, i exp( −τ /τC,c, i)

 i = 1,2

heterodyne term + It2 exp( −2τ /τC,t) +

∑

Ic,2i exp(− 2τ /τC,c, i)

i = 1,2  

1. homodyne term + 2ItIc,1 exp( −τ /τC,t − τ /τC,c,1) + 2ItIc,2 exp( −τ /τC,t − τ /τC,c,2) + 2Ic,1Ic,2 exp( −τ /τC,c,1 − τ /τC,c,2)   2. homodyne term

(7)

Here, reference light with intensity ILO, e.g., stray light from the windows of the sample cell, coherently superimposed with the scattered light from the sample is considered. It and Ic,i denote scattered light intensities caused by fluctuations in the temperature and concentration of component i. The relaxation times τC,t and τC,c,i characteristic for the mean decay behavior of fluctuations in temperature and species concentration are related to the thermal diffusivity a and the eigenvalues of the Fickian diffusion matrix D̂ 1 and D̂ 2 by 1 1 1 a = 2 , D1̂ = 2 , and D̂2 = 2 q τC,t q τC,c,1 q τC,c,2 (8) respectively. Due to the large number of parameters in the CF, it is very difficult to obtain decay times τC,t and τC,c,i and other experimental constants. The problem is simplified if heterodyne conditions can be arranged experimentally, i.e., It ≪ ILO and Ic,i ≪ ILO. Then, the normalized CF for ternary fluid mixtures is reduced to a superposition of three exponentials reflecting the mean lifetimes of temperature and concentration fluctuations according to g(2)(τ ) = b0 + bt exp( −τ /τC,t) + bc,1 exp(−τ /τC,c,1) + bc,2 exp( −τ /τC,c,2)

(9)

The constants b0 , bt, bc,1, and bc,2 include both the corresponding terms from eq 7 and effects caused by the imperfect signal collection due to incoherent background and the finite detector area. For binary fluid mixtures, the CF as stated in eq 9 reduces to a superposition of two exponentials according to g(2)(τ ) = b0 + bt exp( −τ /τC,t) + bc exp(−τ /τC,c)

(10)

with characteristic decay times τC,t and τC,c that are connected with thermal and mutual diffusivity, a and D12,binary. Whether all decay times in binary or ternary fluid mixtures are simultaneously accessible within one CF depends on the time scales at which the individual fluctuations relax.

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EXPERIMENTAL SECTION Experimental Setup and Sample Preparation. The experimental setup, including the optical arrangement, the 10810


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The Journal of Physical Chemistry B density data26 at 293 K assuming ideal mixing behavior. The mixture consisting of 30 mol % of CO and 70 mol % of H2 was then added to the evacuated sample cell filled with pure nC12H26 or n-C28H58. These systems were investigated in a temperature range between (298 and 422) K and at pressures from (2.3 to 3.4) MPa. Due to formation of particles in the sample at elevated temperatures, further DLS measurements were impossible. The origin of the formed particles has been described in detail in our previous study21 and could be mainly attributed to the interaction of CO with the inner sample cell walls. Further efforts were made to study ternary mixtures consisting of one n-alkane as well as H2O and H2 or CO. Here, the initial n-alkane−H2O mixture was either provided by injecting liquid water into the n-alkane as described above or by providing water vapor with pressures up to about 0.5 MPa from a separately heated supply vessel. After equilibration of the binary mixture, the further gas component was added at initial pressures up to 5 MPa. In all cases, formation of particles hindered a reliable examination of these ternary mixtures by DLS. Experimental Procedure. For each binary and ternary system, after reaching equilibrium, two to eight measurements at a defined thermodynamic state were performed at different adjustments of the optical setup. Incident angles were varied between (−9 to +9)° relative to the detection direction. During the experiment, the pseudocross CF is calculated simultaneously by a linear tau (LT) digital correlator featuring equally spaced channels and a multiple tau (MT) digital correlator (ALV GmbH) that exhibits a quasi-logarithmic time structure. The spacing between correlator channels using the LT correlator can be adjusted by means of the sample time. Together with the amount of correlator channels, the selected sample time defines the total time range in which experimental CFs are recorded. The MT correlator provides a fixed time structure featuring 264 correlator channels to record CFs in a total time range from (25 × 10−9 to 805) s. For an individual measurement, the measurement time was between (10 to 200) min depending on light scattering intensities. For each DLS measurement, pressure and temperature were measured as a function of time. Based on these time-dependent pressure and temperature data, mean values were evaluated and used for the calculation of mixture compositions. Calculation of Mixture Compositions. Solubility data for the investigated gases dissolved in binary n-alkane mixtures consisting of n-C12H26 and n-C28H58 are not available in the literature. Thus, mixture compositions of studied ternary fluid mixtures were calculated based on estimated mole fractions of dissolved gases in the pure individual n-alkanes.27−31 The available solubility data for the binary mixtures consisting of H2 or CO dissolved in n-C12H26 or n-C28H58 were fitted with a surface polynomial as a function of pressure and temperature. From this polynomial, the mole fraction of gases dissolved in the n-alkanes could be estimated at our experimental conditions. For the estimation of the amount of dissolved H2O in n-C28H58, the pseudo-Henry constant provided by Breman et al.27 was utilized. Here, it was assumed that the partial pressure of water dominates the pressure in the binary mixture. Hence, the vapor pressure of pure water at the investigated temperatures26 was used for composition calculations. The amount of dissolved H2O in n-C12H26 is estimated based on data provided by Maczynski et al.,32 who summarized liquid−liquid equilibrium data for various n-alkanes with chain

lengths from n = 5 to 11. These data together with the correlation from Economou et al.28 serve as a basis to estimate the amount of dissolved H2O in n-C12H26. In this case, the available solubility data had to be extrapolated with respect to the carbon number to n = 12. The mole fractions of the dissolved gas (xternary Gas ), n-C12H26 ternary (xC12H26), and n-C28H58 (xternary ) in the studied ternary mixtures C28H58 consisting of two n-alkanes with dissolved H2, CO, or H2O were calculated from the mole fractions of corresponding binary subsystems. Here, it is assumed that the two binary subsystems mix ideally and the amount of gas dissolved in the 12H26 28H58 ternary mixture adds up according to nCGas + nCGas , where C12H26 C28H58 nGas and nGas represent the number of moles of dissolved gas in n-C12H26 and n-C28H58 in the corresponding binary subsystem. From this, the mole fractions of the three components are calculated according to ternary = xGas

X12xC12H26 + X13xC28H58 X12xC12H26 + X13xC28H58 + 1

= xCternary 12 H 26

1 X12 + X3(X13 + 1) + 1

= xCternary 28H58

1 X13 + X 2(X12 + 1) + 1

(11)

with the abbreviations X12 =

xGC12H26

,

1 − xGC12H26 xC12H26 X2 = , 1 − xC12H26

X13 =

xGC28H58

, 1 − xGC28H58 xC28H58 and X3 = 1 − xC28H58

(12)

Here, the indices 1, 2, and 3 indicate the connection to the dissolved gas, n-C12H26, and n-C28H58, respectively. xCG12H26 and xCG28H58 correspond to the mole fractions of gas dissolved in nC12H26 and n-C28H58 in the binary n-alkane-gas subsystems and were calculated at the experimental conditions at which the corresponding ternary mixtures were investigated. xC12H26 and xC28H58 are the mole fractions of the individual n-alkanes in the investigated binary subsystem consisting of n-C12H26 and nC28H58. As already discussed in our previous study, the cell exhibits leakage of about (20 to 100) kPa h−1 for different gases due to inevitable diffusion through the window sealings.22 A worst case analysis taking into account the recorded pressure data for each experimental run at constant temperature as well as estimated uncertainties related to the determination of the underlying solubility data for the binary subsystems was performed. The resulting uncertainty in the calculated compositions of the ternary mixtures based on binary n-alkane mixtures calculated from eqs 11 and 12 is estimated to be 0.03 for systems containing H2 or CO, and 0.06 for systems with H2O. The mixture composition of the investigated ternary mixtures consisting of the two gases H2 and CO dissolved in nC12H26 or n-C28H58 was estimated using solubility data.29 To calculate the mole fraction of dissolved H2 and CO, ideal mixing behavior was assumed and the vapor pressure of nC12H26 and n-C28H58 was neglected. From this calculation procedure, an uncertainty of 2 mol % is estimated. All calculated mixture compositions of the studied ternary fluid mixtures are summarized in Table S1 in the Supporting Information. 10811


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Figure 1. Correlation function for ternary systems: (a) H2 dissolved in a binary 0.3n-C12H26/0.7n-C28H58 mixture recorded at T = 422.73 K and a calculated composition of xH2 = 0.05, xC12H26 = 0.29, and xC28H58 = 0.66. (b) H2 and CO dissolved in n-C12H26 at T = 298.11 K with xH2 = 0.01, xCO = 0.01, and xC12H26 = 0.98.

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DATA EVALUATION AND PROOF OF HYDRODYNAMIC MODES For the analysis of experimental CFs obtained for binary fluid mixtures, the data evaluation is described in detail in context with our previous study.21 A nonlinear regression based on a Levenberg−Marquardt (LM) algorithm is applied to calculate characteristic relaxation times τC,t and τC,c as well as experimental constants b0, bt, and bc as given in eq 10. For the analysis of decay times associated with thermal and mass diffusivities in ternary mixtures, it is assumed that three exponentials are present in the experimental CF as described in context with eq 9. These exponentials can be superimposed with disturbing signals caused by vibrations, the presence of particles, incoherent external stray light, or convection in the sample. The disturbing signals need to be taken into account by the fit model to receive reliable results for the decay times of the exponentials.21 In the fit model

in eq 13. The total time range was selected between (1 and 1536) μs, and disturbances could be described well with a firstorder polynomial. The deviation of the experimental data from the theoretical model is illustrated in the lower part of Figure 1a and shows no systematic behavior, which indicates an appropriate selection of the fit model. Estimated uncertainties on a 95% confidence level (k = 2) in the calculated decay times τC,t, τC,c,1, and τC,c,2 were obtained from the fit applying the LM algorithm. Figure 1b shows a CF recorded with the LT correlator for the ternary mixture consisting of CO and H2 dissolved in nC12H26 at 298.11 K and 3.51 MPa. Here, a superposition of three exponentials can be resolved by the LM algorithm-based analysis using a fit model according to eq 9, i.e. without any terms accounting for disturbances. In the following, the fast relaxation mode is assumed to be associated with temperature fluctuations. From their mean decay time τC,t, the thermal diffusivity a can be calculated according to eq 8. The two slower relaxation modes with characteristic mean decay times τC,c,1 and τC,c,2 are assumed to result from concentration fluctuations and will be referred to as fast and slow mass diffusion modes. From these modes, the eigenvalues of the Fickian diffusion matrix D̂ 1 and D̂ 2 can be calculated, cf. eq 8. The capability of the LM algorithm to distinguish between all three exponential modes depends on the amplitudes and the decay times of the individual modes. In our previous study, we specified a range of 0.5 < τC,c,1·τ−1 C,t < 2 for decay time ratios that do not allow for distinguishing two exponentials by the LM algorithm. In case the noise level in experimental CFs is small and the amplitudes of both exponentials are nearly identical, the described limitations are reduced to the range 0.7 < τC,c,1·τ−1 C,t < 1.5. In the present study, distinguishing between the slow mass diffusion mode and the other modes was always possible.

2

g(2)(τ ) =

∑ b0,kτ k + bt exp(−τ /τC,t) k=0

+

∑ i = 1,2

bc, i exp( −τ /τC,c, i) (13)

the disturbances are described by a polynomial of zeroth, first, or second order. Higher-order polynomials are not considered in the present study because of the increasing degree of freedom in the fit model. The upper part in Figure 1a exemplarily illustrates an experimental CF calculated by the MT correlator for a ternary system consisting of H2 dissolved in a (0.3n-C12H26/0.7nC28H58) mixture at 422.73 K and 2.76 MPa. In this example, the LM algorithm was applied to calculate relaxation times and the other experimental constants according to the model given 10812


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Figure 2. Comparison of different results from MT and LT correlators as well as different evaluation strategies for the same system as given in Figure 1a: (a) Evaluation of a MT CF recorded at Θi = 7.999° for τC,c,2 in the time range between (179.2 and 1536) μs. (b) Evaluation of the same CF as in (a) for τC,t and τC,c,1 in the time range between (1 and 204.8) μs. (c) Evaluation of a LT CF simultaneously recorded with the CF in (a). (d) Evaluation of a LT CF at Θi = −2.999° for τC,t and τC,c,1.

Further distinguishing between the fast mass diffusion mode and the mode related to temperature fluctuations was only hindered for the systems where H2 was dissolved in binary nalkane mixtures. Nevertheless, the analysis of the fast mass diffusion mode in these cases was possible by employing the decay time τC,t or the thermal diffusivity data available from the investigation of the corresponding binary n-alkane subsystem. The decay time τC,t calculated in this way was used as a constant to access relaxation times connected with the fast mass diffusion process. This is justified because the thermal diffusivities of the ternary mixtures do not differ significantly from those of the corresponding binary n-alkane subsystems, cf. Results and Discussion. A detailed discussion regarding the reliability and achievable uncertainties of this data evaluation strategy is presented in our previous study for binary mixtures. 22 Results obtained in this way are marked correspondingly in Table S1 of the Supporting Information. In the present study, an additional data evaluation strategy by using the LM algorithm was applied, where individual exponentials in experimental CFs were treated separately within the different time scales on which they decay. According to eq 8, relaxation times scale with the inverse square of the modulus of the scattering vector. Hence, faster relaxation of the DLS signals for all modes is given for larger scattering angles. The signal strength at these scattering angles is rather low, which leads to long measurement durations and bad signal statistics. On the other hand, disturbing signals influence experimental CFs very weakly, as they are commonly found on a time scale of several milliseconds. Utilizing small incident angles of about 3° allows for shorter measurement durations and leads to better signal statistics. Small scattering angles, however, result in slowly relaxing DLS signals which are more

strongly affected by disturbances. Hence, in particular the slow mass diffusion mode can be superimposed by disturbing signals in these CFs, such that it cannot be properly evaluated any more. Faster relaxing modes, however, are still accessible from such measurements at small scattering angles. To study to the reliability of respective results, the CF shown in Figure 1a was also evaluated by considering different time ranges in which the individual modes were found. Figure 2a illustrates the slow mass diffusion mode that is evaluated in a time range between (179.2 and 1536) μs and that is described with a single exponential fit model including a linear term to account for disturbances. As a starting point for the evaluation of τC,c,2, the correlation channel related to 179.2 μs was chosen on the basis of the results obtained from the evaluation approach described in context with Figure 1a. By such selection, it must be ensured that the contribution of the fast relaxing modes to the experimental CF is negligible. In the given example, a relaxation time τC,c,1 of 16 μs was obtained according to the data evaluation including three superimposed exponentials. On this basis, it could be estimated that, at the selected correlation channel of 179.2 μs, the fast mass diffusion mode contributes only about 0.005% of its initial amplitude to the considered CF illustrated in Figure 2a. Splitting up the short-time range to treat the thermal diffusion mode and the fast mass diffusion mode separately is not reasonable, because the analysis according to Figure 1a showed that τC,t and τC,c,1 differ only by a factor of 3. Hence, these two modes relax in a similar time range from (1 to 204.8) μs that was chosen for their evaluation employing a fit model including two exponentials. In this case, the slow mass diffusion mode is not described sufficiently, but significantly contributes to the time-dependent behavior of the CF. In the fit model, the 10813


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Figure 3. Data evaluation using the CONTIN algorithm: (a) H2 dissolved in a binary 0.3n-C12H26/0.7n-C28H58 mixture recorded at T = 422.73 K and a calculated composition of xH2 = 0.05, xC12H26 = 0.29, and xC28H58 = 0.66. (b) H2 and CO dissolved in n-C12H26 at T = 298.11 K with xH2 = 0.01, xCO = 0.01, and xC12H26 = 0.98.

a fit model which does not rely on a fixed number of expected exponential modes should be employed to validate the above assumptions. A suitable model for such a task is given by

contribution from the slow mode was treated as a disturbing signal, which could be represented well by additionally introducing a second-order polynomial for the given example. Figure 2b shows the result of the described evaluation. The CF illustrated in Figure 2c was recorded with the LT correlator simultaneously with the CF shown in Figure 2a and is evaluated in a time range between (0.6 and 153) μs. A fit model consisting of two superimposed exponentials and an additional linear term was utilized to calculate relaxation times. The relative uncertainties in the relaxation times obtained from the CF illustrated in Figure 2c are smaller compared to those calculated from the CF in Figure 2b because of the about four times larger number of data points available from the relevant correlator channels. Nevertheless, all relaxation times calculated from the CFs in Figures 2b and 2c are within their combined uncertainties (k = 2). To achieve better signal statistics, an additional measurement was performed using an incident angle of 3°, where the corresponding CF is illustrated in Figure 2d. This CF is described with a fit model consisting of two superimposed exponentials and a second-order polynomial. The relative uncertainties in the relaxation times found for the CF recorded at an incident angle of 3° are about three times smaller than those obtained from the CF calculated at the incident angle of 8°. Further evaluation showed that the resulting diffusivity data agree within their combined uncertainties. Moreover, the measurement duration of 30 min for the CF recorded at the small scattering angle is significantly shorter than that of 130 min for the CF obtained for the large scattering angle. In the data evaluation approaches described above, the LM algorithm minimizes the root-mean square deviation of experimental data from a predefined theoretical model. Thus, this approach cannot be considered to be impartial. This is why

g(2)(τ ) =

∫τ

τC,max

C,min

exp( −τ /τC)G(τC)dτC

(14)

where the decay time distribution G(τC) needs to be found. In eq 14, τC,min and τC,max are the minimum and maximum relaxation times allowed for the data evaluation. In the corresponding regression, the adjustable parameters must be found using appropriate inversion algorithms. In this study, the correlator software provided by ALV GmbH that operates the CONTIN algorithm was used.33,34 The CONTIN algorithm seeks for exponentials present in experimental CFs, where additional disturbing signals originating from the experimental conditions cannot be considered. For a detailed description of the CONTIN algorithm and inversion algorithms in general, the reader is referred to specialized literature.34,35 The CFs illustrated in Figure 1a and 1b were evaluated with the CONTIN algorithm, where the minimum and maximum decay times τC,min and τC,max were set to (1 and 1000) μs. The open symbols in the upper part of Figure 3 illustrate the decay time distribution obtained from the CONTIN algorithm implemented in the ALV correlator software. In the lower part of the figure, deviations of the experimental CF from the theoretical model similar to eq 9 employing all four decay times identified by the CONTIN algorithm are illustrated. A systematic behavior of these deviations can be observed in the time range between about (300 and 1,500) μs. This indicates that the CF is not properly described by the fit model in this time range. 10814


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Figure 4. Proportionality of the inverse of the relaxation times τC,t, τC,c,1, and τC,c,2 on the squared modulus of the scattering vector for ternary −1 −1 −1 −1 −1 systems. (a) 0.3n-C12H26/0.7n-C28H58/H2: ●, τ−1 C,t; ◓, τC,c,1; ◒, τC,c,2; (b) n-C12H26/H2/CO: ■, τC,t; ⬒, τC,c,1; ⬓, τC,c,2.

performed as presented in Figure 4a. At incident angles of ±3° (q2 ≈ 382 × 109 m−2) and ±3.5° (q2 ≈ 462 × 109 m−2), only the thermal and fast mass diffusion mode could be evaluated as described above. For incident angles between ±3° and ±8° (q2 ≈ 2701 × 109 m−2), the dependence of the inverse mean lifetimes of the fluctuations as a function of the squared modulus of the scattering vector is described very well by a straight line through the origin. The same statement holds for the n-C12H26/H2/CO mixture at 298.11 K illustrated in Figure 4b. As mentioned above, the fast relaxation mode is in general associated with thermal fluctuations reflecting the thermal diffusivity a. The slower relaxation modes reflect concentration fluctuations. According to theory, the latter modes are associated with the eigenvalues D̂ 1 and D̂ 2 of the Fickian diffusion matrix. The correct allocation of the individual hydrodynamic modes is supported by the temperaturedependent behavior of the corresponding calculated diffusivities. Figure 5 exemplarily shows the temperature dependence of the mean values of all three calculated diffusivities for a 0.3nC12H26/0.7n-C28H58/H2 ternary mixture. The calculated thermal diffusivity of the ternary mixture decreases with increasing temperature, which is in agreement with our

Uncertainties in the mean peak positions obtained from the CONTIN algorithm cannot be provided on a reliable basis. However, the first three of the four obtained mean peak positions and the mean decay times determined with the LM algorithm-based evaluation procedures are in excellent agreement. Thus, it could be proven that these three exponentials are present in the experimental CFs calculated for the investigated ternary mixtures. The fourth exponential mode indicated by the CONTIN algorithm at a mean position of about 1000 μs can be attributed to disturbances present in the long time range of the CF, which cannot be represented by the fit model given in eq 14. Because disturbances in the CFs cannot be considered within the CONTIN algorithm, all CFs in the present study were analyzed using the approaches based on the LM algorithm. The agreement of their results within uncertainties in the cases in which both approaches are applicable, which is not always given, indicates that there is no substantive reason why one of these approaches should be preferable. Thus, the reported diffusivities in this study represent mean values calculated from relaxation times obtained from CFs which were recorded with two different correlators for different optical adjustments and evaluated with all applicable evaluation strategies using the LM algorithm. The reported uncertainties in the calculated mean diffusivities for a defined temperature represent double standard deviations implying a 95% confidence level. Error bars in the figures correspond to these uncertainties, where the absence of error bars indicates that the estimated uncertainties range within the symbol size. The expanded uncertainties (k = 2) of the diffusivities obtained from the evaluation of the individual CFs were also determined by propagation calculations, where agreement with the reported diffusivities within combined uncertainties could be found. For ternary fluid mixtures, the above analysis of experimental CFs demonstrates that they can be represented well by three exponentials. Yet, proof is required whether these exponentials are associated with hydrodynamic modes. For that, it has to be shown that the inverse of the relaxation time τ−1 C is directly proportional to the squared modulus of the scattering vector q2. Figure 4 illustrates this proportionality for the ternary mixtures 0.3n-C12H26/0.7n-C28H58/H2 and n-C12H26/H2/CO. For the 0.3n-C12H26/0.7n-C28H58/H2 mixture, 18 individual measurements at various scattering or incident angles were

Figure 5. Calculated diffusivities for a ternary system consisting of (0.3n-C12H26/0.7n-C28H58) and dissolved H2 as a function of temperature: ●, thermal diffusivity a; ◓, fast mass diffusivity D̂ 1; ◒, slow mass diffusivity D̂ 2. 10815


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Figure 6. Thermal diffusivities (a) for n-C12H26, n-C28H58, and their binary mixtures 0.3n-C12H26/0.7n-C28H58 and 0.7n-C12H26/0.3n-C28H58 obtained with DLS, and mutual diffusivities (b) for the binary mixtures 0.3n-C12H26/0.7n-C28H58 and 0.7n-C12H26/0.3n-C28H58 from DLS and MD: ▲, pure n-C12H26;22 ⧫, pure n-C28H58;21 ■, 0.3n-C12H26/0.7n-C28H58 (DLS); ●, 0.7n-C12H26/0.3n-C28H58 (DLS); 1/4-filled square, 0.3n-C12H26/ 0.7n-C28H58 (MD); 1/4-filled circle, 0.7n-C12H26/0.3n-C28H58 (MD); pentagon, n-C12H26 in n-C28H58 at infinite dilution (TD);38 , fit according to eq 15 for pure n-C28H58, ---, fit according to eq 15 (a) or eq 16 (b) for 0.3n-C12H26/0.7n-C28H58; -·-, fit according to eq 15 (a) or eq 16 (b) for 0.7nC12H26/0.3n-C28H58; ···, fit according to eq 15 for pure n-C12H26; -··-, fit according to eq 16 for n-C12H26 in n-C28H58 at infinite dilution (TD).38

previous investigations.21 Diffusivities calculated from the two slow relaxation modes increase with increasing temperature. This behavior is characteristic for mass diffusivities. Expanded uncertainties (k = 2) are estimated from up to 8 single measurements performed at different scattering angles for each defined temperature. For 523 K, where only two measurements were performed, the mean maximum error resulting from an error propagation analysis is stated. For this calculation, expanded (k = 2) uncertainties in the incident angle of 0.01° and in the relaxation time resulting from the regression are considered. For a, the relative uncertainty ranges between (3 and 6)%. For D̂ 1 and D̂ 2, uncertainties between (11 and 26)% and (1 and 9)% could be found. The uncertainty in D̂ 1 is considerably larger compared to the uncertainties in a or D̂ 2. This is because in the experimental CFs, the exponential associated with D̂ 1 exhibits a lower amplitude.

a basis to calculate all components of the Fickian diffusion matrix for comparison with the experimental results. In the MD simulations, the TraPPE force field36 for the n-alkanes as well as a model of Hirschfelder et al.37 for H2 were used. Within the SAFT calculations, the temperature-independent binary interaction parameters kij = 0.3151 and kij = 0.4413 were used for the cross-interaction dispersion energies between n-C12H26 and H2 and between n-C28H58 and H2. For mixtures containing CO, only self-diffusivities of the dissolved gas are compared with experimental data. Furthermore, the results obtained for the ternary mixtures are compared with results for the respective binary subsystems. For that, thermal and mutual diffusivities of binary mixtures consisting of n-C12H26 or n-C28H58 with CO, H2, or H2O from our previous studies19,20 are used. Numerical thermal and mass diffusivity data of all binary and ternary fluid mixtures measured in the present study are summarized in Table S1 in the Supporting Information. Binary Mixtures of n-Alkanes. For the analysis of DLS results from ternary mixtures, thermal and mass diffusivities of binary subsystems are helpful. Binary subsystems consisting of n-C12H26 or n-C28H58 with dissolved CO, H2, or H2O were already investigated in our previous studies.21,22 The binary subsystems 0.3n-C12H26/0.7n-C28H58 and 0.7n-C12H26/0.3nC28H58 were studied in the present work, where a and D12,binary could be determined simultaneously from the experimental CFs. The corresponding results are illustrated in Figure 6. Each datum represents the average value of measurements for one defined temperature at different angles of incidence. Calculated uncertainties in a for both binary mixtures range between (1.7 and 6.0)% (k = 2) over the complete temperature range. In Figure 6a, thermal diffusivities for pure n-C12H26 and nC28H58 from our previous studies are also included.21,22 For all systems, thermal diffusivities decrease with increasing temper-

■

RESULTS AND DISCUSSION Several ternary systems were investigated in the present study. One group of ternary systems is based on the binary n-alkane subsystems 0.3n-C12H26/0.7n-C28H58 and 0.7n-C12H26/0.3nC28H58, in which H2 (systems 0.3n-C12H26/0.7n-C28H58/H2 and 0.7n-C12H26/0.3n-C28H58/H2), CO (systems 0.3n-C12H26/ 0.7n-C28H58/CO and 0.7n-C12H26/0.3n-C28H58/CO), or H2O (systems 0.3n-C12H26/0.7n-C28H58/H2O and 0.7n-C12H26/ 0.3n-C28H58/H2O) was dissolved. The second group consists of n-C12H26 or n-C28H58 with dissolved H2 and CO (systems nC12H26/H2/CO and n-C28H58/H2/CO). Furthermore, the two binary n-alkane mixtures 0.3n-C12H26/0.7n-C28H58 and 0.7nC12H26/0.3n-C28H58 were investigated for comparison purposes. For ternary mixtures containing H2, molecular dynamics (MD) simulations and the SAFT equation of state were used as 10816


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Figure 7. Comparison of experimental thermal diffusivity (a) and experimental as well as simulated D̂ 2 data (b) of ternary fluid mixtures with corresponding results for the binary n-alkane subsystems calculated according to eq 15 (a) or eq 16 (b): ◨, 0.3n-C12H26/0.7n-C28H58 with CO (DLS); ◑, 0.7n-C12H26/0.3n-C28H58 with CO (DLS); ◧, 0.3n-C12H26/0.7n-C28H58 with H2 (DLS); ◐, 0.7n-C12H26/0.3n-C28H58 with H2 (DLS); □, 0.3n-C12H26/0.7n-C28H58 with H2O (DLS); ○, 0.7n-C12H26/0.3n-C28H58 with H2O (DLS); 1/4-filled square, 0.3n-C12H26/0.7n-C28H58 with H2 (MD + SAFT); 1/4-filled circle, 0.7n-C12H26/0.3n-C28H58 with H2 (MD + SAFT). →1 dilution. Self-diffusivities at infinite dilution DCxC28H58 are 12H26,self xC12H26→1 38 adopted from literature, while DC28H58,self is approximated by the procedure outlined in the Supporting Information in →1 →1 x x context with eq S1 and Figure S1. DCC12H26 and DCC28H58 were 12H26,self 28H58,self calculated by MD simulations. The Fick diffusivities calculated according to eqs 17 and 18 differ less than 2% from the temperature-dependent correlation of our experimental DLS data. The same holds in comparison with the concentrationdependent correlations according to eq S1, as shown in Figure S1 in the Supporting Information. This shows the good quality of the employed MD data as well as of the described approach to estimate Fick diffusivities. This is important because the prediction of self-diffusivities according to eq 18 was also used to model self-diffusivities in ternary mixtures. The selfdiffusivities of the pure compounds calculated from MD, modeled self-diffusivities at a given composition based on eq 18, and resulting Fick diffusivities calculated according to eq 17 are summarized in Table S2 in the Supporting Information. Ternary Mixtures of n-Alkanes with Dissolved H2, CO, or H2O. In Figure 7a, the experimental thermal diffusivities aexp,ternary of all ternary mixtures consisting of the two n-alkanes with dissolved H2, CO, or H2O are plotted versus the thermal diffusivity acalc,binary of the corresponding binary n-alkane subsystems calculated according to eq 15. For the complete temperature range, the thermal diffusivities for all ternary mixtures could be determined with an average estimated uncertainty of about 5% (k = 2). Thermal diffusivities of all ternary mixtures agree with the corresponding data of binary nalkane subsystems within combined measurement uncertainty. Thus, it can be concluded that the thermal diffusivities of these ternary mixtures and their binary n-alkane subsystems are identical. In Figure 7b, the slow mass diffusivities or smaller eigenvalues of the diffusion matrix D̂ 2 measured for the ternary mixtures are compared with mutual diffusivities D12,binary,calc of the binary n-alkane subsystems calculated according to eq 16. For all ternary mixtures, D̂ 2 could be measured with an average uncertainty of about 7% over the complete temperature range. Figure 7 illustrates the good agreement between the experimental D̂ 2 values and D12,binary,calc. This indicates that the presence of dissolved gases does not affect the slow diffusion mode in the ternary mixture significantly.

ature and are fitted as a function of temperature with a linear equation according to a(T ) = a0 + a1T

(15)

For the regression, each datum was weighted with its inverse relative expanded uncertainty (k = 2). For a given temperature, a increases with increasing concentration of n-C28H58. The D12,binary data presented in Figure 6b increase with increasing temperature as expected. These data were fitted as a function of temperature with an Arrhenius-type equation according to D(T ) = D0 exp(D1/T )

(16)

Also in this regression, each datum was weighted according to its inverse expanded relative uncertainty (k = 2), which ranges from (1.0 to 6.4)% over the complete temperature range for both binary mixtures. D12,binary data of n-C12H26 in n-C28H58 at infinite dilution measured with the TD technique38 are included in Figure 6b for comparison. Within the investigated temperature range, uncertainties in D12 measured with the TD technique are reported to range from less than (1 to 4)%, where good agreement with our DLS data can be found. Furthermore, calculated D12,binary,MD data based on selfdiffusivity data of the pure components n-C12H26 and n-C28H58 from MD simulations at (443, 473, 500, and 523) K are included in Figure 6b. To calculate Maxwell−Stefan (MS) diffusivities Đbinary,MD, which are equal to Fick diffusivities in the case of ideal mixtures or at infinite dilution, the Darken equation D12,binary,MD ≅ Đ binary,MD = xC12H26DC28H58,self + xC28H58DC12H26,self

(17)

was used. The required self-diffusivities of the individual nalkanes at a given concentration were modeled according to 1 Di,self

n

=

xj xj → 1 j = 1 Di ,self

∑

(18) 39

as proposed by Liu et al., where Di,self corresponds to the selfdiffusivity of the ith compound at a given composition xi. In eq xj→1 18, Di,self represents the self-diffusivity of species i at infinite 10817


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Table 1. MS Diffusivities Đij Estimated from the Multicomponent Darken-like Equation (eq 20) and Components of the Thermodynamic Factor Matrix [Γ] Calculated from Fugacity Coefficients ϕi of the Individual Components in the Ternary Fluid Mixture

To further investigate this phenomenon, all components of the diffusion matrix [D] as well as the modal matrix [P], ⎡ D̂2 − D22 ⎤ ⎢ ⎥ 1 D21 ⎥ ⎢ P=⎢ ⎥ ⎢ D1̂ − D11 ⎥ 1 ⎢ D ⎥ ⎣ ⎦ 12

(19)

should be calculated. From DLS, however, only eigenvalues of [D] are accessible according to theory. Hence, the components of [D] were calculated on the basis of MD simulations and the SAFT equation of state for the two example ternary mixtures containing H2. In a first step, MS diffusivities Đij at desired mixture compositions in the ternary mixture are estimated with the multicomponent Darken-like equation proposed by Liu et al.39 according to Đij =

T

Đ12

Đ13

Đ23

K

10−9 m2 s−1

10−9 m2 s−1

10−9 m2 s−1

443 473 500 523 443 473 500 523

Γ11

Γ12

Γ21

Γ22

n-C12H26 + n-C28H58 + H2 (27 mol % + 63 mol % + 10 mol %) 27.0 29.1 2.4 1.05 −0.15 0.06 30.7 32.4 3.1 1.04 −0.15 0.06 38.2 35.7 3.9 1.04 −0.14 0.06 43.0 41.5 4.7 1.04 −0.12 0.05 n-C12H26 + n-C28H58 + H2 (63 mol % + 27 mol % + 10 mol %) 34.0 31.5 2.5 1.02 −0.03 −0.03 37.2 34.6 3.3 1.01 −0.02 0.01 42.4 35.3 4.1 1.01 −0.01 0.01 43.7 37.4 4.8 1.00 0.00 0.02

0.80 0.81 0.81 0.82 0.99 0.99 0.98 0.97

Di ,self Dj ,self

1 = Dmix

Dmix n

∑ i=1

Based on the estimated MS diffusivities and the thermodynamic factor matrix [Γ], [D] in the molar-averaged velocity reference frame is obtained. The transformation to the molar volume-averaged velocity reference frame is accomplished with transformation matrices according to Taylor and Krishna40 that are given in the Supporting Information, eqs S5 and S6. The required information on the partial molar volume of the individual compounds in the ternary mixture was also obtained from the SAFT equation of state. All components of [D] in the molar volume-averaged velocity reference frame calculated from a combination of the SAFT equation of state and MD simulation are summarized in Table 2.

(20)

xi Di ,self

(21)

Here, Di,self and Dj,self are the self-diffusivities of the ith and jth components in the ternary mixture at a given composition, and xi is the mole fraction of the corresponding species. For the estimation of Đij in the ternary mixture, the self-diffusivities of H2 at desired compositions from our own MD simulations were used. The self-diffusivities of n-C12H26 and n-C28H58 were calculated according to eq 18. For this, the self-diffusivities of the n-alkanes at infinite dilution in H2 had to be estimated. In a conservative estimation, these self-diffusivities are between that of H2 in the ternary mixture and the fastest self-diffusivity of the n-alkanes present in the mixture. For the two limiting cases, calculations showed that the larger and smaller eigenvalues of the diffusion matrices differ by less than (1 and 8)%. Finally, both solutions were averaged and the uncertainty in the calculated eigenvalues was estimated to be at least 10%. In eqs 20 and 18, D1,self, D2,self, and D3,self correspond to the selfdiffusivities of H2, n-C12H26, and n-C28H58, respectively, in the present case. For the calculations of Fickian diffusion coefficients from the MS diffusion coefficients, which are given in more detail in the Supporting Information in context with eqs S2−S4, indices were selected correspondingly for the individual components. The estimated MS diffusivities and the components of the thermodynamic factor matrix [Γ], which is required for the transformation to Fickian diffusion coefficients and calculated according to Γij = δij + xi

Table 2. Theoretically Calculated Coefficients Dij of the Fickian Diffusion Matrix in the Molar Volume-Averaged Velocity Reference Frame and Their Eigenvalues D̂ i

T ,p,Σ

D11

D12

D21

D22

D̂ 1

D̂ 2

K

10−9 2 −1

10−9 2 −1

10−9 2 −1

10−9 2 −1

10−9 2 −1

10−9 m2 s−1

443 473 500 523 443 473 500 523

m s

m s

n-C12H26 + n-C28H58 + H2 30.3 1.8 33.9 2.1 38.6 2.3 44.5 2.7 n-C12H26 + n-C28H58 + H2 39.3 3.5 43.1 3.9 47.2 4.3 49.3 4.6

m s

m s

m s

(27 mol % + 63 mol % + 10 mol %) 2.0 2.3 30.5 2.2 2.6 3.1 34.1 2.9 3.2 3.9 38.8 3.7 4.0 4.6 44.8 4.4 (63 mol % + 27 mol % + 10 mol %) 0.2 2.7 39.3 2.7 1.6 3.6 43.3 3.4 3.1 4.5 47.6 4.2 5.0 5.4 49.9 4.9

Figure 7b indicates that for both ternary mixtures containing H2, the theoretically calculated smaller eigenvalues, D̂ 2, tend to agree with D12,binary,calc for the binary n-alkane subsystems. Furthermore, Figure 8 shows that the theoretical D̂ 2 data differ by less than 8% from the DLS data over the complete temperature range. This supports the theoretical approach that suggests that DLS provides information on the eigenvalues of the diffusion matrix for ternary mixtures.5 With the calculated components of [D], it can additionally be shown that the modal matrix [P] approximately equals the unity matrix. From this and from the similar values of D̂ 1 and D11 as well as of D̂ 2 and D22, it can be concluded that the investigated ternary mixtures

∂ ln ϕi ∂xj

T

(22)

are summarized in Table 1. In eq 22, δij represents the Kronecker delta, and xi is the mole fraction of species i. The partial derivative of the logarithm of the fugacity coefficient ϕi of species i with respect to the mole fraction of species j, xj, is calculated at constant temperature, pressure, and species concentration other than j, as indicated by the subscripts T, p, and Σ, respectively. The required fugacity coefficients were calculated from the SAFT equation of state. 10818


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Figure 8. Theoretically calculated diagonal elements and eigenvalues of the Fickian diffusion matrix in comparison with experimental results obtained from DLS for 0.3n-C12H26/0.7n-C28H58/H2 (a) and 0.7n-C12H26/0.3n-C28H58/H2 (b): ■, D̂ 1, DLS; ●, D̂ 2, DLS; □, D̂ 1, MD; ○, D̂ 2, MD; △, D11, MD; pentagon, D22, MD; , fit according to eq 16 for D̂ 1, DLS; ---, fit according to eq 16 for D̂ 2, DLS.

Figure 9. Fast mass diffusivities D̂ 1 of ternary mixtures of 0.3n-C12H26/0.7n-C28H58 or 0.7n-C12H26/0.3n-C28H58 with dissolved H2, CO, and H2O as well as mutual diffusivity D12,binary of the corresponding binary n-alkane-gas subsystems: ●, D12,binary for n-C12H26 with H2 (DLS);22 ○, D12,binary for nC28H58 with H2 (DLS);21 ◓, D̂ 1 for 0.3n-C12H26/0.7n-C28H58 with H2; ◒, D̂ 1 for 0.7n-C12H26/0.3n-C28H58 with H2; ■, D12,binary for n-C12H26 with CO (DLS);22 □, D12,binary for n-C28H58 with CO (DLS);21 ⬒, D̂ 1 for 0.3n-C12H26/0.7n-C28H58 with CO; ⬓, D̂ 1 for 0.7n-C12H26/0.3n-C28H58 with CO; ⧫, D12,binary for n-C12H26 with H2O (DLS);22 ◊, D12,binary for n-C28H58 with H2O (DLS);21 top-half-filled diamond, D̂ 1 for 0.3n-C12H26/0.7nC28H58 with H2O; bottom-half-filled diamond, D̂ 1 for 0.7n-C12H26/0.3n-C28H58 with H2O; ---, correlation for D12,binary according to eq 16; − −, guide for the eye for ternary mixtures containing the 0.3n-C12H26/0.7n-C28H58 subsystem; , guide for the eye for ternary mixtures containing the 0.7nC12H26/0.3n-C28H58 subsystem.

measured D̂ 1 values. These are caused by the crossing of the slow mass diffusion mode with that related to temperature fluctuations. Corresponding findings were obtained for the 0.7n-C12H26/0.3n-C28H58/H2 mixture; cf. Figure 8b. In Figure 9, fast mass diffusivities D̂ 1 of all investigated ternary fluid mixtures based on the binary n-alkane mixtures 0.3n-C12H26/0.7n-C28H58 and 0.7n-C12H26/0.3n-C28H58 with dissolved H2, CO, and H2O are presented as a function of temperature and compared with the mutual diffusivities of the corresponding binary n-alkane-gas subsystems.21,22 All individ-

containing H2 can be regarded as ideal mixtures.40 Furthermore, this result indicates that the diffusive flux of the dissolved gas is governed only by the concentration gradient of the dissolved gas in the mixture. For the 0.3n-C12H26/0.7n-C28H58/H2 mixture, differences of less than (12 and 25)% between the greater theoretically calculated eigenvalue D̂ 1 and the fast mass diffusivity information from DLS can be observed as illustrated in Figure 8a. The somewhat higher deviations in comparison to the slow mass diffusivity may be related to the higher uncertainties in the 10819


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Figure 10. Mass (top) and thermal diffusivity (bottom) of ternary mixtures n-C12H26/H2/CO (left) and n-C28H58/H2/CO (right) and their binary and pure subsystems: ⬒, D̂ 1, present study (DLS); top-half-filled diamond, D̂ 2, present study (DLS); □, D12,binary, n-C12H26 with H2 (TD);42 ◊, D12,binary, n-C12H26 with CO (TD);42 ■, D12,binary n-C28H58 with H2 (DLS);21 ◆, D12,binary n-C28H58 with CO (DLS);21 ◐, a, n-C12H26/H2/CO (left) and n-C28H58/H2/CO (right), present study (DLS); ●, a, pure n-C28H58 (DLS);21 ···, a, pure n-C12H26 (Refprop);26 − −, correlation of D12,binary data from TD42 according to eq 16; , correlation of D12,binary or a data from DLS according to eq 16 or 15

concentration fluctuations cannot be observed for 0.07 mol % of H2 dissolved in n-C12H26.22 For the ternary n-C12H26/nC28H58/H2 mixtures investigated in the present study at about (498 and 523) K, Le is estimated to be smaller than 1.5 and 1, respectively. Here, the recorded CFs showed some irregularities in the short-time range. These irregularities, however, could not be related to coupling effects as described by Anisimov et al.25 and could be easily handled in the data evaluation by neglecting the corresponding correlator data for the fitting procedure. The obtained diffusivity data agree with the general temperaturedependent behavior of the investigated systems. For all other ternary mixtures and thermodynamic states, no such irregularities could be observed in the CFs. In the latter cases, the Lewis number was in the range between 2 and 77 for the ternary mixtures containing CO or H2O. For mixtures with dissolved H2, Le ranges between 72 and 1.6 for temperatures between (372.9 and 448.3) K. In summary, the irregularities which may be related to coupling effects were only observed in DLS measurements where the exponential modes associated with the thermal diffusivity and the fast mass diffusivity could not be separated in corresponding CFs. Hence, it seems that in all other cases where a mode separation was possible, also the approximation of large Lewis numbers is applicable. For further data comparison, a prediction of an effective binary diffusivity D1* of the diluted gas in the binary subsystem according to41

ual data sets were fitted with an Arrhenius-type equation, where the corresponding fits are included in Figure 9 as guides for the eye. The D̂ 1 data measured for the 0.3n-C12H26/0.7n-C28H58/H2 mixture agree within the measurement uncertainty with the D12 data obtained from the binary mixture consisting of n-C28H58 and dissolved H2. Although the theoretically calculated diffusivities suggest an ideal mixture behavior, cf. Table 2, experimental observations indicate a nonideal behavior. It was expected that D̂ 1 is inversely related to the viscosity of the binary n-alkane subsystem. However, D̂ 1 data for the 0.7nC12H26/0.3n-C28H58/H2 mixture exhibit greater values compared with the 0.3n-C12H26/0.7n-C28H58/H2 mixture. For both ternary systems containing CO or H2O, measured D̂ 1 data inversely correlate with the viscosity of the binary nalkane subsystems; that is, D̂ 1 decreases with increasing nC28H58 content. For both ternary mixtures containing CO, estimated uncertainties in D̂ 1 range from (6 to 13)% over the temperature range from (372 to 498) K. These uncertainties are distinctly smaller than those for the respective systems with dissolved H2 or H2O. For the latter, uncertainties range between (7 and 26)% and between (11 and 35)% over the complete temperature range investigated. Although the mole fractions of CO, H2, or H2O in the binary n-alkane mixtures are similar, systems containing CO exhibit stronger DLS signals compared with other solutes. Measured a and D̂ 1 data are approaching each other at about 498 K for the 0.7n-C12H26/0.3n-C28H58/H2 mixture; cf. Table S1 in the Supporting Information. In such case, the Lewis number Le = aD̂ −1 1/2 approaches unity and may also take up smaller values; that is, the assumption of large Lewis numbers is no longer valid. For the binary subsystem n-C12H26 with dissolved H2, such a mode crossing could be observed in a previous study, where possible coupling effects between thermal and concentration fluctuations were not observed in recorded CFs.22 Based on the work of Anisimov et al.,25 we concluded that a mode coupling between temperature and

* + x3D13 * D1* = x 2D12

(23)

was performed. This approach assumes ideal mixture behavior and can be considered as a simple tool for engineering practice. * and D13 * in For ternary mixtures of two n-alkanes and a gas, D12 eq 23 are the mutual diffusivities of binary subsystems consisting of n-C12H26 and n-C28H58 with dissolved CO, H2, or H2O. x2 and x3 correspond to the mole fractions of n-C12H26 and n-C28H58 in the ternary mixtures. The maximum deviation of D1* from DLS data for D̂ 1 is about 16% for the 0.7n-C12H26/ 10820


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The Journal of Physical Chemistry B 0.3n-C28H58/H2 mixture. Thus, the predicted D*1 data are in good agreement with our experimental results for D̂ 1, which indicates ideal mixture behavior. For the 0.3n-C12H26/0.7nC28H58/H2 mixtures, larger deviations of more than 35% and outside combined uncertainties could be found, which may be attributed to the nonideal behavior of the mixture as discussed earlier. The predicted D1* data for both ternary mixtures containing CO differ less than 16% from DLS data for D̂ 1 over the complete temperature range, suggesting ideal mixture behavior. With a maximum deviation of about 25%, the prediction of D̂ 1 for the two mixtures with H2O is in good agreement with our DLS data. Ternary Mixtures of an n-Alkane and Dissolved Gases. It was already reported in the Experimental Section that for the ternary mixtures containing one n-alkane as well as H2O as one of the solutes, formation of particles and/or ongoing reactions were the reason for strong perturbations. In only one case, a noisy CF containing information on the molecular mass transport could be obtained for the n-C28H58/H2O/CO mixture. Here, the disturbances hindered a separation of the two modes related to mass diffusion, which should decay on very similar time scales according to the results for the underlying binary subsystems.22 In the upper part of Figure 10, the diffusivities D̂ 1 and D̂ 2 for ternary mixtures consisting of n-C12H26 (left) or n-C28H58 (right) with dissolved H2 and CO are summarized. In the lower part of Figure 10, thermal diffusivities of the corresponding ternary mixtures are compared with data for the pure n-alkanes. Experimental difficulties connected with chemical reactions limited the accessible temperature range for these ternary mixtures. The n-C12H26/H2/CO mixture was investigated at (298.11 and 322.82) K, where formation of particles in the sample prevented further DLS measurements at higher temperatures. Similar observations were made for the nC28H58/H2/CO mixtures that were investigated at (372.32, 396.99, and 421.43) K. In the following, our D̂ 1 and D̂ 2 data for the ternary mixture n-C12H26/H2/CO are compared with mutual diffusivities measured with the TD technique.42 Further comparison is conducted with mutual diffusivities previously measured by DLS in binary subsystems consisting of n-C12H26 and CO or H2 as well as of n-C28H58 and CO or H2.21 For both ternary mixtures, the measured fast and slow mass diffusivities D̂ 1 and D̂ 2 coincide with the mutual diffusivities of the binary subsystems containing H2 and CO. In the upper part of Figure 10, the solid lines represent a correlation of the binary diffusivity data according to eq 16. For the n-C12H26/H2/CO mixture, only two CFs were recorded at 322.82 K, resulting in a mean estimated uncertainty of about 26%. At this temperature, data for D̂ 1 show distinctly larger values compared with mutual diffusivities for n-C12H26 with H2 measured by TD. However, within experimental uncertainty, it can be stated that D̂ 1 and D̂ 2 obtained for the ternary mixture are identical with mutual diffusivities measured by DLS for the corresponding binary subsystems. A possible physical interpretation based on this observation is that diffusive processes of the individual gases in the ternary mixture do not influence each other. Thus, the off-diagonal elements in the diffusion matrix D12 and D21 are close to zero. Within combined uncertainties, thermal diffusivities a for the ternary mixture n-C28H58/H2/CO are in agreement with the data measured for pure n-C28H58.21 For the n-C12H26/H2/CO system, thermal diffusivity data could be compared with data

from the Refprop database.26 Here, good agreement between our data and the Refprop database can be found within estimated uncertainty.

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CONCLUSION

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ASSOCIATED CONTENT

Up to now, it was unclear what kind of DLS signals can be expected from ternary mixtures. In this study, we presented experimental DLS data for ternary fluid mixtures consisting of n-C12H26, n-C28H58, and the solutes CO, H2, or H2O. For the first time, we were able to separate three individual contributions or modes present in CFs recorded from DLS experiments. It could be proven experimentally that each individual contribution represents a hydrodynamic mode. The fastest decaying mode could be associated with temperature fluctuations. The two slower decaying modes could be shown to originate from concentration fluctuations. In agreement with our previous studies, we showed that thermal fluctuations in the liquid phase are not influenced by the dissolved gases. This is valid for binary and as well as for ternary mixtures. Within experimental uncertainty, measured thermal diffusivities for pure n-alkanes match with those from respective binary nalkane gas mixtures. The same holds for binary n-alkane mixtures compared with ternary mixtures additionally containing one dissolved component. From MD simulation and SAFT equations of state, all components of the Fickian diffusion matrix were calculated for ternary mixtures consisting of the two n-alkanes and dissolved H2. From the comparison between DLS and calculated results, a first indication that the eigenvalues of the Fickian diffusion matrix are accessible by DLS measurements could be obtained. Although this still needs further experimental proof, our presented results strongly support the recently published derivation of scattering spectra resulting from low-molecular-weight ternary fluid mixtures. In all investigated ternary mixtures including the two n-alkanes, the slow mass diffusivities coincide with mutual diffusivities of the binary n-alkane subsystems. A comparison between the fast mass diffusivities for the same ternary mixtures with mutual diffusivities of binary subsystems consisting of one n-alkane and the dissolved gas showed that this diffusive process scales inversely to the n-alkane mixture viscosity. For the ternary mixtures consisting of one n-alkane, H2, and CO, the measured fast and slow mass diffusivities agree with the mutual diffusivities of the binary n-alkane−gas subsystems.

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b08117. Diffusivity data for the studied binary and ternary →1 mixtures. Approximation for DCxC12H26 . Comparison 28H58 ,self between experimental and predicted mutual diffusivities for n-C12H26/n-C28H58 mixtures. Self-diffusivities of pure n-C12H26 and n-C28H58 from MD simulations, their modeled self-diffusivities at xC12H26 = 0.3 and xC12H26 = 0.7, and estimated Fick diffusivities for their binary mixtures. Calculations for the comparison of MD with DLS results for ternary mixtures. (PDF) 10821


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(17) Bardow, A. On the Interpretation of Ternary Diffusion Measurements in Low-Molecular Weight Fluids by Dynamic Light Scattering. Fluid Phase Equilib. 2007, 251, 121−127. (18) Ortiz de Zárate, J. M.; Hita, J. L.; Sengers, J. V. Fluctuating Hydrodynamics and Concentration Fluctuations in Ternary Mixtures. C. R. Mec. 2013, 341, 399−404. (19) Klerk, A.; Li, Y. W.; Zennaro, R. Greener Fischer−Tropsch Processes for Fuels and Feedstocks; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, 2013. (20) van der Laan, G. P.; Beenackers, A.; Krishna, R. Multicomponent Reaction Engineering Model for Fe-Catalyzed Fischer− Tropsch Synthesis in Commercial Scale Slurry Bubble Column Reactors. Chem. Eng. Sci. 1999, 54, 5013−5019. (21) 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. (22) Heller, A.; Fleys, M. S. H.; Chen, J.; van der Laan, G. P.; Rausch, M. H.; 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. (23) Berne, B. J.; Pecora, R. Dynamic Light Scattering; WileyInterscience: New York, 1976. (24) Mountain, R. D.; Deutch, J. M. Light Scattering from Binary Solutions. J. Chem. Phys. 1969, 50, 1103−1108. (25) Anisimov, M. A.; Agayan, V. A.; Povodyrev, A. A.; Sengers, J. V.; Gorodetskii, E. E. Two-Exponential Decay of Dynamic Light Scattering in Near-Critical Fluid Mixtures. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 57, 1946−1961. (26) Lemmon, E. W.; McLinden, M. O.; Huber, M. L. NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties - REFPROP, Version 9.0 (Standard Reference Data Program); National Institute of Standards and Technology: Gaithersburg, MD, 2010. (27) Breman, B. B.; Beenackers, A.; Rietjens, E. W. J.; Stege, R. J. H. Gas-Liquid Solubilities of Carbon-Monoxide, Carbon-Dioxide, Hydrogen, Water, 1-Alcohols (1 ≤ n ≤ 6), and n-Paraffins (2 ≤ n ≤ 6) in Hexadecane, Octacosane, 1-Hexadecanol, Phenanthrene, and Tetraethylene Glycol at Pressures up to 5.5 MPa and Temperatures from 293 To 553 K. J. Chem. Eng. Data 1994, 39, 647−666. (28) Economou, I. G.; Heidman, J. L.; Tsonopoulos, C.; Wilson, G. M. Mutual Solubilities of Hydrocarbons and Water: III. 1-Hexene, 1Octene, C10-C12 Hydrocarbons. AIChE J. 1997, 43, 535−546. (29) Gao, W.; Robinson, R. L.; Gasem, K. A. M. High-Pressure Solubilities of Hydrogen, Nitrogen, and Carbon Monoxide in Dodecane from 344 to 410 K at Pressures to 13.2 MPa. J. Chem. Eng. Data 1999, 44, 130−132. (30) Park, J.; Robinson, R. L.; Gasem, K. A. M. Solubilities of Hydrogen in Heavy Normal-Paraffins at Temperatures from 323.2 to 423.2 K and Pressures to 17.4 MPa. J. Chem. Eng. Data 1995, 40, 241− 244. (31) Srivatsan, S.; Yi, X. H.; Robinson, R. L.; Gasem, K. A. M. Solubilities of Carbon-Monoxide in Heavy Normal-Paraffins at Temperatures from 311 K To 423 K and Pressures to 10.2 MPa. J. Chem. Eng. Data 1995, 40, 237−240. (32) Maczynski, A.; Wisniewska-Goclowska, B.; Goral, M. Recommended Liquid-Liquid Equilibrium Data. Part 1. Binary Alkane-Water Systems. J. Phys. Chem. Ref. Data 2004, 35, 549−577. (33) Provencher, S. W. A Constrained Regularization Method for Inverting Data Represented by Linear Algebraic or Integral-Equations. Comput. Phys. Commun. 1982, 27, 213−227. (34) Provencher, S. W. Contin - a General-Purpose Constrained Regularization Program for Inverting Noisy Linear Algebraic and Integral-Equations. Comput. Phys. Commun. 1982, 27, 229−242.

AUTHOR INFORMATION

Corresponding Author

*Tel. +49-9131-85-29789, Fax +49-9131-85-25851, E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) by funding the Erlangen Graduate School in Advanced Optical Technologies (SAOT) within the German Excellence Initiative. Financial support from Shell Global Solutions International BV through a contracted research agreement is gratefully acknowledged.

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