Simultaneous Determination of Thermal and Mutual Diffusivity of

Article pubs.acs.org/JPCB

Simultaneous Determination of Thermal and Mutual Diffusivity of Binary Mixtures of n‑Octacosane with Carbon Monoxide, Hydrogen, and Water by Dynamic Light Scattering Andreas Heller,† Thomas M. Koller,† Michael H. Rausch,†,‡ Matthieu S. H. Fleys,§ A. N. René Bos,§ Gerard P. van der Laan,§ Zoi A. Makrodimitri,∥ Ioannis G. Economou,∥,⊥ and Andreas P. Fröba*,†,‡ †

Erlangen Graduate School in Advanced Optical Technologies (SAOT), University of Erlangen-Nuremberg, Paul-Gordan-Straße 6, D-91052 Erlangen, Bavaria, Germany ‡ Department of Chemical and Biological Engineering, Institute of Engineering Thermodynamics, University of Erlangen-Nuremberg, Am Weichselgarten 8, D-91058 Erlangen, Bavaria, Germany § Shell Global Solutions International B.V., Grasweg 31, 1031 HW Amsterdam, The Netherlands ∥ National Centre for Scientific Research “Demokritos”, Institute of Advanced Materials, Physicochemical Processes, Nanotechnology and Microsystems, Molecular Thermodynamics and Modelling of Materials Laboratory, GR-15310 Aghia Paraskevi Attikis, Greece ⊥ Chemical Engineering Program, Texas A&M University at Qatar, Education City, PO Box 23874, Doha, Qatar ABSTRACT: It is demonstrated that thermal and mutual diffusivities of binary mixtures of n-octacosane (n-C28H58) with carbon monoxide (CO), hydrogen (H2), and water (H2O) are simultaneously accessible by dynamic light scattering (DLS). As the light-scattering signals originating from thermal and concentration fluctuations appear in similar time scales, different data evaluation strategies were tested to achieve minimum uncertainties in the resulting transport properties. To test the agreement of the respective theoretical model with the DLS signals in the regression, an improved multifit procedure is introduced. With the selected data evaluation strategy, uncertainties of 4 to 15% and 4 to 30% in the thermal and mutual diffusivities, respectively, could be obtained for the binary mixtures. The mutual diffusivities for the mixtures measured at temperatures ranging from 398 to 523 K and pressures of 5 to 30 bar at saturation conditions are in good agreement with molecular dynamics simulations and data from the literature.

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INTRODUCTION The mutual diffusivity of liquids containing dissolved gases is of technological interest in different industrial fields of chemical and energy engineering.1−3 The characterization of such systems over a wide region of thermodynamic states represents a metrological challenge. In the literature, many conventional measurement techniques capable of determining mutual diffusivities of liquids containing dissolved gases can be found, all of them having their own restrictions. For example, the Taylor dispersion (TD) can be used to determine the mutual diffusivity by studying the concentration distribution of a solute dispersed in a flowing solvent stream. To the best of our knowledge, up to now the TD has been applied only to systems at infinite dilution.4−6 Furthermore, the transient gravimetric7,8 and open tube methods9−11 can be representatively mentioned. They make use of macroscopic gradients in concentration which 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. Dynamic light scattering (DLS), which is used in the present study, is restricted to the hydrodynamic regime where the linearized Navier−Stokes equations are valid. For the determination of mutual diffusivities in binary mixtures, the solute concentration has to be large enough to obtain © 2014 American Chemical Society

detectable DLS signals that can be attributed to the molecular diffusion process. On the other hand, DLS allows the determination of transport and other thermophysical properties in macroscopic thermodynamic equilibrium in an absolute way over a wide range of thermodynamic states.12,13 DLS was recently applied for the determination of mutual diffusivities of mixtures of ionic liquids with molecular solvents.14,15 In the present study, the DLS technique was utilized for the first time to investigate mutual and thermal diffusivities of liquids containing dissolved gases. Both properties are related to the line width of the Rayleigh component of the spectrum of scattered light which arises from temperature and concentration fluctuations. From the calculated time-dependent correlation function (CF) of the intensity of scattered light, the mutual and thermal diffusivity can be evaluated.16,17 For mixtures exhibiting mutual and thermal diffusivities on the same order of magnitude, the corresponding hydrodynamic modes appear on the same time scale in the CF. Investigations on a methane−ethane mixture in the nearReceived: January 10, 2014 Revised: February 14, 2014 Published: March 20, 2014 3981

dx.doi.org/10.1021/jp500300y | J. Phys. Chem. B 2014, 118, 3981−3990

The Journal of Physical Chemistry B

Article

fluctuations observed, the mutual diffusivity D12 can be determined by

critical region showed that a simultaneous determination of mutual and thermal diffusivity is possible in the case when both transport properties expose similar values.18 In the present study, it is demonstrated that the mutual and thermal diffusivity for liquids containing dissolved gases are also simultaneously accessible by DLS. As a model system, noctacosane (n-C28H58) containing carbon monoxide (CO), hydrogen (H2), or water (H2O) was chosen. Information on the mutual diffusivity of such systems is required for a fundamental understanding of the Fischer−Tropsch synthesis.1 Earlier studies4,19−21 have shown that the mutual diffusivity of n-C28H58 mixtures with dissolved CO, H2, or H2O is on the same order of magnitude as the thermal diffusivity of n-C28H58, which should allow for a simultaneous determination of both transport properties by DLS. In this case, the key to precisely accessing mutual and thermal diffusivity from the recorded CFs containing superimposed signals is the applied data evaluation strategy. A new data evaluation procedure for the chosen systems probed at process-relevant temperatures ranging from 398 to 523 K and pressures from 5 to 30 bar as well as the corresponding data are presented and discussed.

D12 =

4πn q = |k ⃗i − ks⃗ | ≅ 2k i sin(Θs /2) = sin(Θs /2) λ0

(4)

which is defined by the difference of the wave vectors of incident and scattered light, ki⃗ and ks⃗ , respectively. Assuming elastic scattering (ki ≅ ks), the modulus of the scattering vector is given in terms of the fluid refractive index n, the laser wavelength in vacuo λ0, and the scattering angle Θs. For small scattering angles, an approximation for the modulus of the scattering vector can be deduced, q ≅ (2π/λ0) sin Θi , where Θi denotes the incident angle. In this case, no information on the refractive index of the fluid is needed. Because of the large number of parameters in the correlation function, eq 1, it is very difficult to obtain the decay times τC,t and τC,c. The problem is simplified if heterodyne conditions can be arranged, i.e., It ≪ ILO and Ic ≪ ILO. The normalized correlation function is reduced to a sum of two exponentials reflecting the mean lifetime of temperature and concentration fluctuations according to

METHOD The principles of DLS for the determination of transport and other thermophysical properties are described elsewhere in detail.12,16,17 Here, only the essential features relevant for the present study are presented. The decay of the two hydrodynamic modes present in binary fluid mixtures follows the same laws that are valid for macroscopic systems. Thus, the decay of temperature and concentration fluctuations is governed by the thermal diffusivity and the mutual diffusivity, respectively. Information about these equilibration processes can be derived by the temporal analysis of the scattered light intensity by DLS. The mean decay times of the two hydrodynamic modes are calculated from the time-dependent CF of the intensity of scattered light. Taking into account that in a realistic experimental situation reference light, e.g., stray light from the windows of the sample cell, is superimposed coherently on the scattered light from the sample, the intensity correlation function G(2)(τ) takes the form

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

(5)

The experimental constants b0 , bt , and bc include both the corresponding terms from eq 1 and effects caused by the imperfect signal collection due to incoherent background and the finite detector area. If thermal and mutual diffusivity of an investigated system are on the same order of magnitude, the two corresponding hydrodynamic modes present in binary fluid mixtures are observable on the same time scale. Whether it is possible to resolve both signals of the correlation function given by eq 5 is restricted mainly by the ratio of the scattering intensities It/ Ic , which depends on the relative difference of the refractive indices of the two components and their concentration. In general, the simultaneous determination of both signals in binary mixtures is possible if the refractive indices of the two components are comparable. In the case of a large difference between the refractive indices, the scattered light intensity arising from concentration fluctuations dominates the correlation function. If the refractive indices of the pure components of the mixtures match, no information on the concentration fluctuations can be found in the scattered light.

Here, ILO is the intensity of the reference light or the socalled local oscillator and It and Ic denote the scattering light intensities caused by temperature and concentration fluctuations, respectively. In contrast to the homodyne term and the cross term, which are due to the scattered light from the sample alone, the heterodyne term is the result of the interference of the scattered and the reference light. This term is characterized by decay times of fluctuations in temperature and concentration, τC,t and τC,c. The decay time τC,t , which is equivalent to the mean lifetime of the temperature fluctuations observed, is related to the thermal diffusivity a by 1 τC,tq2

(3)

In eqs 2 and 3, q is the modulus of the scattering vector

■

a=

1 τC,cq2

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EXPERIMENTAL SECTION A scheme of the optical setup used in this study is illustrated in Figure 1. A frequency-doubled Nd:YVO4 laser source operated at 532 nm and 500 mW output power is focused into the sample cell (SC) by mirrors (M) and a lens (L) with a focal length of f = 2 m. The intensity of the main beam indicated by the solid line can be adjusted by a combination of a half-wave plate (λ/2 retardation) and a polarization beam splitter (PBS). With the setup, scattering angles ranging from −9° to +9° can be realized, where the approximation for q can be applied. The incident angle Θi is adjusted by the mirror M6 and measured with a rotational table using the autocollimation technique with an uncertainty of ±0.01°.

(2)

Equivalently, from the characteristic decay time τC,c , which is equivalent to the mean lifetime of the concentration 3982



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Figure 1. Optical setup.

Figure 2. Manifold for dosing CO or H2 into the sample cell.

Two stops (P1, P2) with a distance of about 1 m are installed behind the sample cell defining the detection direction. To achieve heterodyne conditions, a local oscillator indicated by the dashed line is superimposed with the scattered light. Its intensity is adjusted by a gray filter (GF). The resulting signal is detected by two photomultiplier tubes (PMTs). During the experiment, the pseudocross correlation function is calculated by a single-tau correlator featuring 255 equally spaced channels and a multitau correlator simultaneously. The sample cell with a total inner volume of 40 mL is made of hydrogen-resistant steel and discloses four optical accesses, which are sealed by an O-ring system. First investigations showed that the sample was contaminated with particles stemming from the corrosion of the cell material. To render the surface of the sample cell inert, it was galvanically gold plated. However, because of strong thermal and mechanical stress induced by the measurement and cleaning procedure, the gold layer became unstable after some time. A more chemically stable nickel coating countered the effects of mechanical stress. At elevated temperatures, however, the nickel reacted with CO to form carbonyl complexes. Taking these aspects into account, the measurements for pure n-C28H58 and its mixtures with CO were carried out with the uncoated and gold-coated sample cell. Investigations for the mixtures with H2 and H2O were carried out with the nickel-coated sample cell. A maximum sample temperature of 573 K can be achieved by resistance heating with a temperature stability of better than 3 mK. Two calibrated Pt 100 Ω resistance probes are used to measure the cell temperature with an absolute uncertainty (k = 2) of less than 10 mK. The temperature control loop is realized with a temperature probe placed in the wall of the cell close to the resistance heating. To measure the temperature of the fluid, the second probe is placed inside the cell material, located close to the fluid. The sample cell can be closed by two bellow-type valves. It is connected to a manifold allowing evacuation, flushing, and dosing of the sample gases (Figure 2). An additional resistance heating system is installed at the tubing directly connected to the sample cell to eliminate convection. The corresponding temperature control loop is probed by a further calibrated temperature sensor with an absolute uncertainty of less than 10 mK. Here, temperature stability better than 0.1 K was achieved.

The manifold including the sample cell is installed within a fume hood. CO or H2 is loaded into the gas buffer after the system was flushed with nitrogen (N2) and evacuated with an oil-sealed vacuum pump (0.5 mbar) several times. The manifold and the measurement cell were designed to handle pressures up to 80 bar. The pressure in the sample cell is recorded by a pressure transducer with an uncertainty of less than 0.05%. The investigated n-C28H58 was purchased from Alfa Aesar GmbH & Co. KG with a purity of 99% by mass. At ambient conditions, the sample is solid. The liquefied sample shows particle-like impurities presumably resulting from the manufacturing process. To obtain particle-free samples as necessary for DLS experiments, the n-C28H58 was filtered with a syringe filter with 200 nm pore size at about 353 K. CO and H2 were provided by Linde AG with purities of 99.997 vol % and 99.9999 vol %. About 30 mL of liquid filtered alkane were filled into the sample cell at a temperature of 353 K. At this temperature, the filled sample cell is connected to the manifold. Because of the low vapor pressure of n-C28H58 at 353 K, the remaining air in the sample cell can be removed by an oil-sealed vacuum pump (0.5 mbar) before it is filled with CO or H2. After equilibration of the two-phase system for at least 24 h, first measurements were carried out. After that time, no significant changes in the DLS signals could be observed, indicating a steady state of the investigated system. For the preparation of the mixtures with H2O, 2−5 mL of distilled water were added to the sample cell containing liquid n-C28H58. After the cell was assembled and connected to the manifold, the remaining air in the cell was removed by an oilsealed vacuum pump. The investigations for the mixtures with CO and H2 were carried out in the liquid phase in the presence of a gas phase above. For the mixture with H2O, a second liquid phase mainly consisting of H2O could be observed directly after filling at 353 K. This water phase could not be observed after heating the sample cell to temperatures higher than 448 K. Nevertheless, the system can be specified regarding its composition, assuming the recorded pressure corresponds to the saturated pressure of water. To specify the concentration of the dissolved solute in the n-alkane, the composition of each mixture was estimated from solubility data,22−24 the recorded temperature, and the measured saturation pressure. Solubility data for mixtures with CO and H2 are given from 348.2 to 423.2 K and had to be extrapolated to temperatures up to 523 K. This extrapolation 3983



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Figure 3. Normalized CF for the mixture with CO as obtained from the experiment (a), residual plot (b), and MFP results (c, d).

recorded. On the basis of this time-dependent pressure data, mean values were calculated and used for the calculation of the mixture composition.

was performed by describing the solubility data with a surface polynomial with pressure and temperature dependence to estimate the composition at our experimental conditions. The error in each calculated mole fraction is estimated to be smaller than 0.02. The composition of the mixture with H2O was calculated based on pseudo-Henry constants reported by Breman et al.24 in the temperature range from 428.8 to 519.0 K. According to the authors, the pseudo-Henry constant deviates by less than 10% from the real Henry constant; this results in an estimated error in the calculated mole fraction of less than 0.02. The calculated compositions for the CO and H2 mixtures based on solubility data reported by Srivatsan et al.22 and Park et al.23 were compared with composition data derived from pseudo-Henry constants reported by Breman et al.24 Between both estimates of the mixture composition, a maximum deviation of 0.04 in the mole fraction could be observed over the whole temperature range. During the measurements, the bellow-type valves which connect the manifold with the measurement cell are closed to ensure steady-state conditions. Unpreventable diffusion of gas through the soft sealings of the optical accesses caused ongoing pressure loss. For the mixture with H2 at a temperature of 473.04 K and an absolute pressure of about 32 bar, a maximum pressure drop of 1 bar h−1 was observed. According to solubility data, however, such a pressure loss results in a change of the H2 mole fraction of less than 0.006 per hour. At lower temperatures, this pressure drop reduces to less than 0.2 bar h−1. The observed pressure drop for the mixtures with CO or H2O was less than 0.2 bar h−1 at all investigated temperatures. This described pressure loss causes convective effects in the sample which can be one reason for disturbances observable in the DLS signals. However, these disturbing signals are present in a different time range than the signals related to molecular or thermal diffusion processes. For each defined temperature, six measurements at different scattering angles ranging from −6° to +6° were performed. For each single measurement lasting from 10 to 120 min, depending on the light scattering intensities, the pressure was

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DATA EVALUATION In general, the mutual diffusivity is smaller than the thermal diffusivity, so it can be assumed that the more slowly decaying exponential mode in the recorded CF is related to the mutual diffusivity, and hence is denoted with τC,c. On the basis of this assumption, the faster decaying mode can be attributed to the thermal diffusivity and is denoted with τC,t. To access the decay times attributable to thermal and mutual diffusivity for the different mixtures, the recorded CFs can be described by a theoretical model according to eq 5. Because of vibrations, particles, incoherent external stray light, or convection in the sample, the CFs are superimposed with additional disturbing signals (see the example given in Figure 3a). Here, a normalized CF obtained from the experiment for the mixture of n-C28H58 with CO at a temperature of 498.15 K and a pressure of 38.5 bar is shown. When a nonlinear regression is performed to find the experimental constants and decay times, the theoretical model has to take these disturbances into account. In general, the disturbance term can be expressed by a polynomial up to the third order. In the present study, a quadratic term was sufficient to describe the disturbing signals well, resulting in the fit model

This can be shown by performing a multifit procedure (MFP), where all nonlinear regressions are performed on the basis of a Levenberg−Marquardt algorithm. The MFP serves as a tool to qualify a good description of the CF. On the basis of an initial regression taking into account the complete time init range of the recorded CF, the decay times τinit C,t and τC,c are determined. Both decay times and their uncertainties (k = 2) which result from the nonlinear regression based on eq 6 are 3984



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Figure 4. (a) Reduced normalized CF: ---, exponential mode related to mutual diffusivity; -·-, exponential mode related to thermal diffusivity; residual plot (b); and MFP results (c, d).

fit variation shows systematic deviations in case the chosen disturbance term does not describe the CF sufficiently in the long-time range. If both MFP runs verify the applicability of the selected fit model, the decay times obtained from the initial regression are considered to be the correct ones and are used for further data processing. The difference between the results obtained by a fit model using, e.g., a polynomial of second order or a quadratic extension, can be found in the uncertainty of the calculated decay time. Because of the higher degree of freedom, the polynomial extension yields an uncertainty larger than that of the quadratic extension. The decay times calculated considering the complete time range based on the two fit models, however, agree within their combined uncertainties. A comparison of the variation plots obtained from the MFP for both fit models also indicates no significant difference. On the basis of the MFP results and the comparison of several fit models using different disturbance terms, a quadratic extension term according to eq 6 was found to describe the disturbances in all the CFs recorded within the present study best. The physical origin of the disturbances present in the CFs was not investigated systematically because it is not of special interest in this study. Hence, reduced CFs will be treated in the following. Here, it is assumed that the disturbance is welldescribed by the disturbance term, can be subtracted from the original CF, and does not influence the final results. In Figure 4a, the reduced CF from the measurement example given in Figure 3a is shown, where the two superimposed exponential functions are indicated separately by the dashed and the chain lines. It is obvious that not only the determined decay times but also the variation plots calculated for the reduced CF are in exact accordance with the nonreduced CF (see Figures 3 and 4). The uncertainty in the decay times calculated from the reduced CF over the complete time range is smaller than the uncertainty calculated from the nonreduced CF because the degree of freedom in the fit model applied to the reduced CF is smaller.

given in Figure 3a. Figure 3b depicts the deviation of the correlator data from the initial fit in percent, which does not show any systematic behavior in this case. After the initial regression, the MFP is applied. In each of the two required MFP runs, only one exponential mode is variable while the second mode is treated as a constant using the parameters found in the initial regression. The disturbance term is treated to be constant as calculated from the initial regression in both MFP runs. In the example shown in Figure 3, the MFP run focuses on the more slowly decaying mode, where the faster mode is treated as a constant. During each MFP run, the time range considered for the regression is varied. For the start-offit variation, the data from the first correlator channels are omitted step by step. The last nonlinear regression for the start-of-fit variation is performed for the time range from 0.5τinit C,c,t up to the last correlator datum. This variation results in several decay times obtained for the different time ranges considered. In Figure 3c, a start-of-fit variation plot is illustrated, showing the deviation of each single decay time from the average of all decay times obtained in the start-of-fit variation in percent. The x-axis represents the considered start time divided by the individual decay time calculated for the corresponding time range. This variation procedure is also applied to the end of the data set, which is called end-of-fit variation. Here, the considered time range starts at the beginning of the data set while its end varies from 3.5τinit C,c,t to 6.5τinit C,c,t. This variation results in an end-of-fit variation plot where the x-axis represents the end of the considered time range divided by the corresponding decay time (Figure 3d). Each single calculated decay time exhibits an uncertainty resulting from the regression which is represented by the error bars in the fit-variation plots. In the example in Figure 3c,d, the deviations of the single decay times from their mean in the start- and end-of-fit variation do not exceed 1.5%. Because this deviation is within the calculated uncertainty (k = 2) of each single decay time, it can be assumed that the observed CF is well-described by the theoretical model including the selected disturbance term. Especially the end-of3985



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The calculated uncertainties in both decay times τC,c and τC,t strongly depend on the ratios χ = τC,c/τC,t and β = bc /bt. The decay time ratio χ indicates how strongly both decay times are separated in time, whereas β describes the amplitude ratio between both exponential modes in the recorded CF. A limiting case in conjunction to the amplitude ratio β could not be found. However, a ratio of less than 0.1 drastically increases the uncertainty in the decay time exhibiting the smaller amplitude. In the present study, amplitude ratios less than 0.1 were found only for the mixture with H2O at temperatures of 398.77 and 423.49 K, as shown in the upper panel of Figure 5.

Figure 6. Reduced example CFs: ---, exponential mode related to mutual diffusivity; -·-, exponential mode related to thermal diffusivity, recorded for the mixture with H2 at temperatures of 448.34 K (a), 473.04 K (b), and 498.15 K (c) including evaluated (a, b) and estimated decay time ratios (c).

Figure 5. Amplitude (upper panel) and decay time (lower panel) ratios found for the investigated systems: ■, mixture with CO, evaluated; ●, mixture with H2, evaluated; ▲, mixture with H2O, evaluated; ○, mixture with H2, estimated.

the decay time connected with the mutual diffusivity, the decay time which describes the thermal fluctuations was calculated based on thermal diffusivity data of the pure nC28H58 measured by DLS and included into the fit model for the CF of the mixture as a constant. This can be justified because the thermal diffusivity of the mixtures does not change significantly compared with the that of pure n-C28H58, see Results and Discussion. The uncertainty in the decay time of the more slowly decaying mode using the evaluation strategy given in Figure 6c is underestimated as the uncertainty in the decay time calculated from the thermal diffusivity of the pure n-C28H58 was not considered. The thermal diffusivity of pure n-C28H58 was measured before the experiments for the mixtures were carried out. It can be assumed that the thermal diffusivity of the mixture is dominated by n-C28H58 because of the very small gas concentration in the mixture on a mass basis. This assumption and the measurements for pure n-C28H58 confirm the initial hypothesis that the faster relaxing mode in the recorded CFs of the mixtures denoted by τC,t can be attributed to the decay time of the hydrodynamic mode governed by thermal fluctuations. The concentration fluctuations characterized by the mean decay behavior τC,c have therefore to be associated with the more slowly decaying mode. The allocation of the

The empirically determined limiting values for the evaluation of two superimposed exponential functions are decay time ratios larger than about 0.5 but smaller than 2. The nonlinear regression does not recognize two exponential modes if χ is within this range; thus, a mode separation is not possible. Hence, in this range it has to be concluded that the decay times related to fluctuations in temperature and concentration match and imply larger uncertainties. The mixtures with CO and H2O did not exhibit decay time ratios smaller than 2 or larger than 0.5 over the whole temperature range (see the lower panel of Figure 5). Estimated decay time ratios of less than 2 but more than 0.5 were found only for the mixture with H2 at temperatures of 498.15 and 523.15 K, which are indicated in Figure 5 by the open symbols. Here, the procedure described in the following was applied for an estimation of the second present mode. Reduced example CFs recorded for the mixture with H2 at temperatures from 448.34 to 498.15 K are illustrated in Figure 6. The decay times of the two modes approach each other with increasing temperature. At a temperature of 498.15 K, a mode separation is not possible anymore. However, to access 3986



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Table 1. Thermal and Mutual Diffusivity Data and Their Uncertainties at the Investigated Temperatures, Mean Pressures, and Estimated Compositions T (K)

p (MPa)

xCO,H2,H2O (mol %)

372.83 397.79 422.75 448.21 472.93 498.16 518.15

0.15 0.16 0.20 0.18 0.23 0.23 0.26

− − − − − − −

446.96* 448.28 471.82* 473.28 498.15* 498.15 518.15

3.52 2.03 4.00 2.27 3.85 3.06 3.54

0.08 0.05 0.09 0.05 0.08 0.06 0.06

372.90 372.94* 397.78 397.83* 422.72 422.72* 448.34 473.04 498.16 523.15

3.06 1.77 3.20 3.33 2.67 3.12 3.94 3.59 3.12 1.58

0.04 0.03 0.05 0.05 0.05 0.05 0.07 0.07 0.07 0.06

398.77 423.49 448.27* 449.13 473.31* 473.15 498.15 498.15* 523.15

0.20 0.36 0.29 0.63 0.47 0.91 0.77 0.48 0.71

− 0.15 0.08 0.18 0.10 0.20 0.13 0.08 0.09

a (10−9 m2 s−1) n-C28H58 79.8 75.6 71.3 68.7 63.4 59.8 58.2 n-C28H58 + CO 65.2 65.4 60.8 63.0 56.8 58.5 57.5 n-C28H58 + H2 82.3 80.3 75.8 78.7 69.7 69.2 66.8 70.1 − − n-C28H58 + H2O 72.4 65.4 66.5 66.1 60.8 63.6 59.4 56.3 55.9

Δa/a

D12 (10−9 m2 s−1)

ΔD12/D12

0.04 0.04 0.04 0.04 0.07 0.08 0.04

− − − − − − −

− − − − − − −

0.10 0.07 0.08 0.07 0.15 0.11 0.11

9.17 9.79 11.2 12.1 12.8 13.9 16.4

0.21 0.12 0.12 0.11 0.30 0.21 0.17

0.11 0.05 0.05 0.09 0.06 0.10 0.11 0.58 − −

15.0 13.2 18.5 19.6 21.2 21.1 26.9 33.4 37.0 47.4

0.27 0.15 0.11 0.14 0.28 0.26 0.25 0.42 0.29 0.19

0.04 0.04 0.01 0.03 0.07 0.05 0.05 0.04 0.03

7.40 8.43 9.99 11.8 14.3 14.6 18.4 17.1 21.6

0.27 0.16 0.48 0.07 0.14 0.16 0.29 0.19 0.14

*

Data not illustrated in Figures 7 and 8.

solubility limit. For CO and H2, the mole fraction is equal to 0.22. For n-C28H58, a united atom (UA) representation was used. In particular, the transferable potential for phase equilibria (TraPPE)26 was employed. TraPPE has shown to be very accurate for the thermodynamic properties of nalkanes in pure state and in mixtures over a wide range of conditions, including the critical point and the range of temperature and pressure values of importance to the current project. Details on the force field functional form and parameters for the various terms can be found in the literature.19 Standard Lorentz−Berthelot combining rules were used to describe nonbonded Lennard-Jones interactions between sites of different types

two superimposed exponentially decaying modes to their physical origin is additionally confirmed by the temperature dependence of both decay times. The decay time of the faster decaying mode increases with increasing temperature, while the decay time of the more slowly decaying mode decreases. This means that in connection with the inverse proportionality of the decay times to the corresponding transport properties, eqs 2 and 3, the fact that the thermal diffusivity decreases and the mutual diffusivity increases with increasing temperature helps to identify which mode corresponds to which kind of fluctuations. Maxwell−Stefan Diffusion Calculations. Long NVT molecular dynamics (MD) simulations at the experimental density of n-C28H5825 and on the order of 30 ns were performed for the estimation of the Maxwell−Stefan (MS) diffusion coefficient DMS 12 of n-C28H58 mixtures with CO, H2, or H2O. Various temperatures were examined where the simulated systems consisted of 30 n-C28H58 molecules. The MS diffusivities for H2O correspond to a low H2O mole fraction of 0.03 in order to be sure that we are below the

εij =

εiiεjj and σij =

σii + σjj 2

(8)

In these simulations, an increased time step of 2 fs was used and the equations of motion were integrated using the velocity Verlet algorithm.27 In all cases, to maintain the 3987



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temperature fixed at its prescribed value, the Berendsen thermostat28 was used with a coupling constant equal to 0.1 ps. Simulations were performed in machines based on Intel Xeon CPU 2.8 GHz processor. To decrease the uncertainty in DMS 12 calculations, four initially different structures of each mixture were examined. The Maxwell−Stefan diffusion coefficient is a kinetic factor and can be determined from the mean-square displacement of the center of mass of particles of species 1 (here the solute)29,30 MS D12 =

⎞2 1 ⎛ m1 d ⎜ x1 + x 2⎟ lim 6Nx1x 2 ⎝ m2 ⎠ t →∞ dt

⎡ N1 ⎢∑ rk(0) − ⎢⎣ k = 1

N1

⎤2

k=1

⎥⎦

∑ rk(t )⎥

(9)

where N is the total number of particles; x1 = N1/N and x2 = N2/N are the mole fractions of species 1 and 2, respectively; N1 and N2 are the number of particles of type 1 and 2, respectively; and m1 and m2 are the masses of the two different types of particles. rk(t) is the position of particle k of species 1 at time t.

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RESULTS AND DISCUSSION The measured thermal and mutual diffusivity data and their uncertainties for pure n-C28H58 and its mixtures with CO, H2, or H2O over a wide temperature range at the corresponding mean pressures and estimated compositions are listed in Table 1. Each diffusivity value stated in the table represents a mean thermal or mutual diffusivity calculated from 12 CFs, which were recorded by two correlators at six different adjustments of the optical setup. The expanded experimental uncertainty (k = 2) of each datum is based on the standard deviation calculated from the 12 individual measurements. The data marked by an asterisk in Table 1 are not illustrated in Figures 7 and 8. The illustrated data are chosen because of comparable compositions of the respective mixtures.

Figure 8. Comparison of mutual diffusivities: ■, mixture with CO (xCO = 0.05), DLS; ●, mixture with H2 (xH2 = 0.06), DLS; ▲, mixture with H2O (xH2O = 0.15), DLS; () fit from DLS data; ◨, mixture with CO, MD; ◑, mixture with H2, MD; ◮, mixture with H2O, MD; □, mixture with CO, Rodden et al.;4 ○, mixture with H2, Rodden et al.4

a(T ) = [− 5.826 × 10−8 + 1.237 × 10−9(T /K) − 3.280 × 10−12(T /K)2 + 2.558 × 10−15(T /K)3 ]m 2 s−1 (10)

In the regression, each thermal diffusivity datum was weighted inversely to its calculated uncertainty, which is between 4 and 8% over the whole temperature range. Compared with that of the pure n-C28H58, the uncertainty in the thermal diffusivity of the mixtures is larger because the thermal diffusivity had to be extracted from two superimposed exponentially decaying functions in the recorded CFs. Uncertainties in the thermal diffusivity for the mixtures range from 4 to 15% over the temperature range where a mode separation was possible. An uncertainty of more than 50% resulted for the mixture with H2 at 473.04 K, as a clear mode separation became difficult because of a decay time ratio of about two. In comparison, uncertainties of less than 1% in the thermal diffusivity can be achieved by the DLS technique for pure fluids or for binary mixtures where the thermal and mutual diffusivity differ by more than one order of magnitude.32,14 Furthermore, the large uncertainties found here result from the challenging sample handling described in the Experimental Section. Regarding the data evaluation for the mutual diffusivity of the mixture with H2 at temperatures of 498.15 and 523.15 K, a general trend for the deviation of the thermal diffusivity of the pure n-C28H58 from its mixtures is of special interest. Figure 7 shows that the thermal diffusivity data for pure nC28H58 and the mixtures agree within combined uncertainties. Consequently, it is justified to utilize the data of pure nC28H58 to evaluate the CFs obtained for the mixtures regarding the mutual diffusivity. The literature data shown in Figure 7 indicated by the open symbols were calculated

Figure 7. Comparison of thermal diffusivity of pure n-C28H58 and its mixtures: () eq 10; ▼, pure, this study; ▽, pure, calculated based on literature data;4,20,21 ■, mixture with CO, this study; ●, mixture with H2, this study; ▲, mixture with H2O, this study.

The thermal diffusivity of pure n-C28H58 and its mixtures with CO, H2, or H2O decreases with increasing temperature. This behavior is in agreement with results for pure shorterchain n-alkanes.31 A significant dependence of the thermal diffusivity on the composition of the mixtures in the probed composition range could not be detected. Our data for pure n-C28H58 could be fitted by a third-order polynomial found by applying a nonlinear regression 3988



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from the thermal conductivity,20 density,4 and specific heat capacity at constant pressure21 and exhibit an estimated uncertainty on the order of 10%. This uncertainty represents the maximum error which was calculated based on the reported uncertainties in density of 0.1% and the specific heat capacity of 4%. An uncertainty in the thermal conductivity was not stated in the literature and was estimated to be 3%. Within combined uncertainties, the thermal diffusivities calculated from literature data are also in good agreement with our data, the first thermal diffusivity data directly measured for n-C28H58. Our experimental mutual diffusivity data are shown in the upper panel of Figure 8. The symbols in the figure represent the mean mutual diffusivities obtained by DLS in the present study. The error bars illustrate the uncertainties (k = 2) in the mutual diffusivity, which are estimated to range from 4 to 30% over the whole temperature range. The mutual diffusivity data in dependence on temperature for the mixtures with CO, H2, and H2O show an Arrhenius-like behavior. In a corresponding regression, each data point was weighted inversely to its uncertainty. In the lower panel of Figure 8, the deviations of the data obtained by MD simulations and literature data measured with the Taylor dispersion (TD) technique4 from the Arrhenius-fit of our data are illustrated. The DMS 12 values obtained from MD simulations and their uncertainties are listed in Table 2.

impossible; thus, the data evaluation strategy based on the knowledge of the thermal diffusivity of the pure n-C28H58 was used. In this case, the uncertainty in the mutual diffusivity is estimated to be smaller than 30%. The mutual diffusivities investigated by Rodden et al.4 by TD at infinite dilution differ by less than 15% from the data measured in this study for the mixture with H2 over the whole temperature range. For the mixture with CO, a deviation greater than 25% can be observed only at a temperature of 371 K. This can be attributed to the regression applied to our data which could be performed only with a starting temperature of 448 K and obviously cannot sufficiently describe the temperature dependence of the mutual diffusivity in the low-temperature region. The mutual diffusivities for the mixtures with CO and H2 calculated by MD simulations deviate by less than 22% and 20% from the DLS data, respectively. Larger deviations between experimental and simulated data are observed for the mixture with H2O at temperatures below 473 K. Here, one can conclude that a more realistic force field is needed to account for the H2O−nalkane interactions, possibly by accounting explicitly for the polarizability effects. The simulations for the mixtures with CO and H2 were carried out at a mole fraction of 0.22 and for the mixture with H2O at a mole fraction of 0.03. As these data are in good agreement with our data, where the composition of all mixtures is estimated to be less than 0.1, a potential concentration dependence of the mutual diffusivity in the investigated composition range cannot be resolved. This result is in agreement with the work of Makrodimitri et al.,19 where mutual diffusivities of the mixtures n-C28H58 with CO and H2 were simulated over a wide concentration range.

Table 2. Mutual Diffusivity Data and Their Uncertainties Obtained from MD Simulations T (K) 373 443 473 495 523

−9 DMS m2 s−1) 12 (10

xCO = 0.22 3.78 7.96 10.5 11.0 15.4 xH2 = 0.22

MS ΔDMS 12 /D12

0.14 0.15 0.01 0.07 0.03

373 443 473 495 523

14.4 23.5 30.3 32.2 35.0 xH2O = 0.03

0.07 0.10 0.13 0.10 0.03

373 443 473 495 523

8.03 17.1 19.4 21.1 23.0

0.03 0.14 0.13 0.05 0.09

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CONCLUSIONS

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AUTHOR INFORMATION

It was demonstrated that thermal and mutual diffusivities are simultaneously accessible by DLS without applying macroscopic gradients for binary mixtures of n-C28H58 with CO, H2, and H2O. Both transport properties could be determined over a wide range of thermodynamic states with satisfactory uncertainties. In the temperature range where the superimposed signals attributable to thermal or mutual diffusivity approached each other, the uncertainty in both properties increased dramatically. With an improved data evaluation procedure, however, even in these cases reliable data could be obtained. Although the DLS experiment allows measurements at different compositions, no dependence of the thermal or mutual diffusivity on the dissolved gas concentration could be evidenced within the measurement uncertainties. This result is in agreement with results from MD simulations of DMS 12 for the same mixtures. For pure n-C28H58, the first directly measured thermal diffusivity data were obtained at saturation conditions with an uncertainty of 4 to 8% (k = 2) for temperatures from 372 to 518 K.

Uncertainties in the experimental mutual diffusivity of less than 12% could be achieved when the decay time ratio is larger than two and the amplitude ratio is close to one as observed for the mixture with CO at temperatures of about 473 K. The mixture with H2O at temperatures larger than 450 K exhibited amplitude ratios smaller than 0.2. In these cases, the uncertainty in the mutual diffusivity reaches 50% as the thermal fluctuations dominate the CF. For the mixture with H2 at 473.04 K, the limiting condition for the data evaluation strategy was reached. A decay time ratio equal to two resulted in an uncertainty in the mutual diffusivity greater than 40%. At higher temperatures of 498.15 and 523.15 K, a decay time ratio less than 2 but greater than 0.5 made a mode separation

Corresponding Author

*Tel.: +49-9131-85-29789. Fax: +49-9131-85-29901. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 3989



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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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Simultaneous Determination of Thermal and Mutual Diffusivity of

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