The Soret effect in ternary mixtures of water+ethanol+triethylene glycol of equal mass fractions: Ground and microgravity experiments

Measurements of the Soret and thermodiffusion coefficients of a symmetric ternary mixture with equal mass fractions of water, ethanol, and triethylene glycol have been performed by two-color optical beam deflection (2-OBD) and the thermogravitational column technique (TGC) in the laboratory and under microgravity conditions in the Selectable Optical Diagnostics Instrument (SODI) aboard the International Space Station. The results from all three experimental techniques agree within the experimental error bars, which result mainly from the inversion of the contrast factor matrices. TGC shows by far the lowest, 2-OBD the highest error amplification. The microgravity measurements are in between. The agreement with the microgravity results shows that thermosolutal convection could be well controlled in the 2-OBD experiments by a proper orientation of the temperature gradient. Despite the different condition numbers, the results are invariant under the choice of the independent compositions. Based on the orientation of the confidence ellipsoid in the ternary composition diagram, not all coefficients are equally affected by experimental errors. Although there are appreciable uncertainties for water and ethanol, the Soret and the thermodiffusion coefficients of triethylene glycol could be obtained with a good accuracy due to the favorable orientation of the confidence ellipsoid. We have found that water behaves thermophobic, corresponding to a positive Soret coefficient, whereas both ethanol and triethylene glycol are thermophilic with negative Soret coefficients. This resembles the behaviour of the binary system ethanol/water above the ethanol concentration of the sign change.


Introduction
The number of unknown parameters in diffusion and thermodiffusion processes in liquid mixtures increases in leading order quadratically with the number of components. Because of mass or volume conservation, one diffusion and one thermodiffusion coefficient are sufficient for the characterization of a binary mixture. These two coefficients are frequently obtained from optical experiments, where the transient refractive index change in response to a thermal perturbation is detected by, e.g., deflection of a laser beam [1][2][3]. In a ternary mixture the number of independent coefficients amounts to six: four for diffusion and two Contribution to the Topical Issue "Thermal Non-Equilibrium Phenomena in Soft Matter", edited by Fernando Bresme, Velisa Vesovic, Fabrizio Croccolo, Henri Bataller. a e-mail: werner.koehler@uni-bayreuth.de for thermodiffusion. This strong increase in the number of degrees of freedom complicates both the theoretical description and, in particular, the experimental approach to the problem. It can be shown that only four out of six independent parameters can be extracted from a conventional single-color optical thermodiffusion experiment on a ternary mixture. In order to resolve all unknowns, an independent measurement with, e.g., a second detection wavelength is necessary, thereby relying on the different refractive index dispersions of the constituents [4][5][6]. Early data on Soret coefficients of ternary mixtures were only partly consistent [7,5] and the need for systematic thermodiffusion measurements on ternary mixtures was clearly recognized [6]. A difficult problem in laboratory experiments is the exclusion of convective perturbations. The occurrence of instabilities in such triplediffusive ternary mixtures is quite complex [8], and the solutal Rayleigh numbers are neither known nor predictable without prior knowledge of the Soret coefficients. While strong convection is easily recognized in typical optical experiments, very slow convective currents are particularly dangerous and difficult to identify [9]. They can lead to significantly reduced amplitudes in, e.g., optical beam deflection (OBD) experiments and, thus, to wrong Soret coefficients. The typical shape of the transients may, however, hardly change and not show the characteristic oscillations observed in the case of strong convection.
Phenomenologically, the time evolution of the two independent mass fractions c 1 and c 2 in a ternary quiescent mixture is described by the extended diffusion equation that accounts for both Fickian diffusion and thermodiffusion: Here, c = (c 1 , c 2 ) T and D T = (D T,1 , D T,2 ) T are column vectors that contain the two independent concentrations and the two thermodiffusion coefficients, respectively. D is the 2 × 2 diffusion matrix. The two Soret coefficients are defined as Equations (1) and (2) contain six unknown coefficients and provide the basis for the interpretation of nonisothermal diffusion experiments on ternary mixtures [26,13].
In this contribution, we will discuss both ground and microgravity measurements of a ternary mixture consisting of equal weight fractions of water, ethanol, and triethylene glycol (H 2 O/EtOH/TEG). This system was flown in sample cell two of the DCMIX3 mission in 2016 [10]. The microgravity experiments were performed by means of the Selectable Optical Diagnostics Instrument (SODI) onboard the International Space Station (ISS). This instrument comprises a Soret cell with two-color interferometric detection (optical digital interferometry, 2-ODI). Measurements on ground were performed by means of twocolor optical beam deflection (2-OBD) and by means of a thermogravitational column (TGC) equipped with combined refractive index and density measurement, which employs the coupling between diffusion and convection for the species separation.

Experiment
Sample. The technical details of the DCMIX3 campaign have been summarized in a first mission report [10]. Both the samples for the space and the 2-OBD experiments were prepared from high-purity liquids: ethanol absolute 99.5%, extra dry, AcroSeal (CAS: 64-17-5) from Acros Organics (lot: 1545382, product code: 39769-0025); triethylene glycol 99% (CAS: 112-27-6), Acros Organics (lot: A0364772, product code: 139590025); de-ionized and filtrated water (resistivity 18.5 MΩcm, 0.22 μm PAK-filter) retrieved from a Millipore Milli-Q filtration station. The lot numbers refer to the chemicals used in the microgravity experiments. Since bubbles are detrimental and may lead to a complete loss of the sample on the ISS, any residual gases have been removed from the liquids by freeze-pump-thaw degassing. The sample discussed here was contained in cell number two of the DCMIX3 cell array (table 1 in ref. [10]). It has a composition of 0.333 mass fraction of each of the components water (H 2 O), ethanol (EtOH) and triethylene glycol (TEG). The naming convention throughout this paper will be c 1 (H 2 O) and c 2 (EtOH) for the two independent concentrations and c 3 = 1 − c 1 − c 2 (TEG) for the dependent concentration. SODI microgravity experiments. The SODI instrument was employed for the microgravity experiments. It comprises a one-and a two-color Mach-Zehnder interferometer that are permanently stored aboard the ISS. The cell array contains five individually addressable ternary samples for the movable two-color and one binary reference sample for the fixed one-color interferometer. It was built by QinetiQ Space (Antwerp, Belgium) and filled with the samples on ground. After its arrival on the ISS, the SODI instrument was assembled within the Microgravity Science Glovebox (MSG) and control was handed over to the Spanish User Support and Operations Center (E-USOC) in Madrid. The two-color interferometer employed for the ternary mixture in cell 2 is equipped with lasers of λ 1 = 670 nm (moving red, MR) and λ 2 = 935 nm (moving near-infrared, MN) wavelength. The Soret cell has a quadratic cross section of 10 × 10 mm 2 and a height of 5 mm along the direction of the temperature gradient. The design and operation of SODI have already been described in a number of previous publications [10,25,27].
The experiment with the symmetric ternary mixture in cell two of the cell array reported here was conducted at a mean temperature of T 0 = 298 K (25 • C) and a temperature difference of ΔT = 5 K between the hot and the cold plate. A total of four runs (run numbers 2, 7, 12, 17) with identical temperature and timing parameters were performed. Every run was preceded by 20 h of unmonitored and 1 h of monitored thermalization. This isothermal phase was followed by 20 h, during which the temperature gradient was switched on. The image acquisition frequency was 0.05 Hz during the initial temperature gradient build-up and later stepwise reduced to 0.005 Hz during the three so-called Soret phases. The timing and the individual phases of the experiment are described in detail in table 3 of ref. [10]. Every sampled "data point" comprises one image stack for the MR and one for the MN laser. An image stack contains five images with a phase shift of π/2 between consecutive images, which is accomplished by current-detuning of the laser. All details of the experimental procedure are defined in the ESA document DCMIX3 Experiment Scientific Requirements (ESR).
As reported in ref. [10], some of the DCMIX3 experiments suffered from laser instabilities, but none of the four runs discussed here were affected. There are various measures that can be used to identify optical stability issues. One such quantity is the Michelson contrast C M averaged over the contrasts C i M of all image stacks acquired during an entire run: N x and N y refer to the 20% × 60% central region of the cropped and rotated images as defined in the ESR document. The coordinates (x, y) identify the individual pixels. I max (x, y) and I min (x, y) are calculated for every pixel from the five images of every stack. The Michelson contrasts of the four runs are listed in table 1. The ideal phase stepping angle from image to image is π/2. Its actual value can be obtained from the Carre estimation [28] tan (ω 0 /2) = 3( Here, the I k are the intensities of a given pixel in four successive images of a stack. The estimated phase step is averaged over the entire region of interest for signal to noise improvement. Invalid interferometric data with negative arguments of the square root are discarded. Since only four consecutive images are required, the phase step can be estimated both from images 1 → 4 and from images 2 → 5 of an image stack. Both numbers are also listed in table 1. Finally, the last row in the table shows the temperature fluctuations of the two plates, which did not exceed a couple of millikelvins. All data in table 1 are within the proper parameter ranges and do not indicate any serious problem. Only the FR laser developed significant phase stepping issues during run 17. Since this affects only the binary reference cell, it is of no concern for the here discussed measurements.

2-OBD laboratory experiments.
The 2-OBD measurements were performed in the laboratory with the instrument described in refs. [5,12] following a suggestion of Haugen and Firoozabadi [4]. The core of the setup is an optical Soret cell with a geometric path length of l = 10 mm and a vertical spacing of h = 1.43 mm between the hot and the cold plate. The two detection lasers are operating at wavelengths of 405 nm and 635 nm, respectively. The temperature difference is typically on the order of 1 K, corresponding to gradients of 10 3 K/m, comparable to the ones in the microgravity experiments.
TGC laboratory experiments. The thermodiffusion coefficients have also been measured directly under ground conditions by means of the TGC technique [22]. With this technique, a horizontal temperature gradient, that is applied between both vertical walls of the column, couples to convection and creates a vertical separation of the components in the sample. In this work, the so-called long thermogravitational column (LTC) has been used because of its amplification of the component separation and increased sensitivity, which is particularly favorable in the case of ternary mixtures. The dimensions of this column are L z = 980 mm, L y = 50 mm, and L x = 1.02±0.005 mm, and the volume required for each experiment is approximately 60 ml.

SODI microgravity experiments
Once the data have been collected, there are two separate and to a large extent independent tasks that need to be accomplished: the signals need to be fitted by a suitable analytical model function and they need to be transformed from the refractive index to the composition space. Typically, these two steps are combined by, e.g., writing the fit equations in terms of the diffusion and thermodiffusion coefficients as free parameters. Mialdun et al. have discussed different evaluation schemes that are more or less along these lines [15]. In our following discussion we will make a clear separation between these two tasks by first fitting the measured transients by a suitable model function in the refractive index space. In a second step, the diffusive transport coefficients will then be computed from the obtained fit parameters [29]. This logically separates the parameterization of the data from the transformation to the composition space and allows for a reevaluation with improved contrast factors without the need to re-analyze the original raw data. As will be shown below, such a separation also allows for a much better identification of characteristic error sources. It also appears quite natural, since the SODI data and the contrast factors are determined independently in separate experiments -the former in microgravity and hardly repeatable, the latter in the laboratory and easily repeatable.

Analysis in the refractive index space
There are various approaches for the evaluation of the image stacks and/or the phase images. Some of these methods have been compared during a recent benchmark campaign for a DCMIX1 sample [11]. They either fit the entire refractive index profile after computation of the unwrapped phase image [17], determine the phase difference between the two plates [24,30], or evaluate the phase field after filtering in the Fourier domain [23]. We present here an alternative approach, which is inspired by the similarity between optical digital interferometry (ODI), as used in the SODI instrument, and optical beam deflection (OBD). Since both methods employ a Soret cell, they share the identical solution of the coupled heat and diffusion equations.
In an OBD experiment only the refractive index gradient in the center of the cell is probed by deflection of a laser beam that travels along the optical symmetry axis [3,5,13,29]. For small gradients, the beam deflection δz is proportional to the refractive index gradient: Here, A is an instrument factor that depends on the geometry of the setup and on the laser wavelength. Due to the finite beam diameter, the refractive index gradient needs to be averaged over the Gaussian beam profile in the OBD experiment [3]. In an ODI experiment, dn/dz can directly be calculated from the 2d-phase field without averaging.
In the temporal phase stepping approach, the 2d-phase field is computed on a per-pixel basis for every image stack. In a first step, the images are aligned, rotated and cropped. Then, the so-called wrapped phase image Φ(x, y) is computed from the five phase shifted images I 1 , . . . , I 5 of a stack by a modified Hariharan algorithm [17,31,32]: Because of the four-quadrant arctan 2 -function, Φ(x, y) is wrapped into the interval (−π, π]. After phase unwrapping [10] and subtraction of a reference image, typically recorded during the initial isothermal state, the unwrapped phase image is obtained ( fig. 1). Finally, a phase change δΦ can directly be related to a refractive index change δn λ at wavelength λ by δn λ = δΦλ/(2πl), with l being the geometric path length of the cell. After averaging over a central rectangular region that is not affected by distortions of the temperature field near the lateral walls, the refractive index gradient dn λ /dz in the midplane is obtained for every image stack by fitting a polynomial to the averaged refractive index changes along the z-direction. The final result of this procedure are OBD-like time series for the two detection wavelengths that can be evaluated in the same way as 2-OBD experiments [5,13,29].
The information content of optical two-color thermodiffusion experiments on ternary mixtures has been studied in detail in ref. [29]. The beam deflection signal for the laser wavelength λ i , after normalization to the amplitude A th,i of the respective thermal contribution without Soret effect, can be written as Here, M ij are the entries of a 2 × 2 amplitude matrix. The somewhat lengthy expression f OBD,j (t) (eq. (8) in ref. [29]) depends on the eigenvalueD j of the diffusion matrix.  Although six independent parameters are contained in the measured two-color time series, it is very difficult to separate the two diffusion eigenvalues if they are too close. As a result, only five independent and stable quantities can be extracted from a model fit to the refractive index signals. If the six independent fit parameters are chosen as the two diffusion eigenvaluesD j and the four amplitudes M ij , the stable quantities in the refractive index space are measured at the two detection wavelengths λ i , plus a mean diffusion coefficientD [29]. Both a i and b i /q 2 are merely shortcuts for the respective right-hand sides of the two equations. The notation b i /q 2 originally stems from TD-FRS experiments, where q is the wave vector of the holographic interference grating [29]. In the context of the here discussed 2-OBD experiments, q alone has no particular meaning and can be set to q = 1 for convenience. Table 2 lists the fit results for the four microgravity measurements together with the thermal amplitudes A th,i . Figure 2 shows for run 2, as an example, the raw OBDlike signals extracted from the SODI phase images, the two normalized signals and the fits of eq. (7). The fits describe the measured data very well for long times, which is important for the determination of the amplitudes and the Soret coefficients. At very short times there is some overshooting, whose origin is not yet fully understood and which is not reproduced by the fit.

Transformation to the concentration space
The contrast factors are required to transform the measurements from the 2d-space of the recorded signals to the space of the two independent concentrations c 1 and c 2 . In the case of the optical experiments, the measurements are performed by detecting refractive index changes at two different wavelengths λ 1 and λ 2 . These refractive index changes δn = (δn 1 , δn 2 ) T result from the concentration changes δc = (δc 1 , δc 2 ) T according to The entries of the 2 × 2 contrast factor matrix are defined by (N c ) ij = (∂n i /∂c j ) p,T,c k =j . In the case of the TGC experiment, the refractive index at the sodium D line (589 nm) and the density ρ are used instead: (N c ) 1j = (∂n D /∂c j ) p,T,c k =j and (N c ) 2j = (∂ρ/∂c j ) p,T,c k =j . All contrast factors are summarized in table 3. The optical contrast factors at the SODI-wavelengths have been calculated on the basis of the refractive index parameterizations given by Sechenyh et al. [33] for 670 nm and 925 nm. The IR-wavelength does not exactly match the corresponding SODI wavelength of 935 nm but, due to the low dispersion in the IR, this small difference can be neglected.
The Soret and thermodiffusion coefficients of the two independent components one (H 2 O) and two (EtOH) can be computed from eqs. (8) and (9) according to [13,29] Here, a = (a 1 , a 2 ) T and b/q 2 = (b 1 /q 2 , b 2 /q 2 ) T . The solutal contrast factor matrix (N c ) ij = (∂n i /∂c j ) p,T,c k =j has already been defined before and the numbers are listed in table 3. The thermal contrast factors at the SODI wavelengths, (N T ) ij = (∂n i /∂T ) p,c1,c2 δ ij , could not be determined interferometrically due to the lack of a proper infrared interferometer wavelength. As an alternative, they have been Table 3. Contrast factors for the SODI, the 2-OBD and the TGC experiments. SODI-wavelengths λ1 = 670 nm and λ2 = 935 nm. 2-OBD-wavelengths λ 1 = 405 nm and λ2 = 635 nm. TGC-wavelength λ1 = 589 nm. T = 298 K. Note that the thermal contrast factors for SODI are in parentheses, since they have not been corrected by tomographic reconstruction [17] (see text). calculated from the temperature difference and the thermal amplitudes A th,i of the SODI signals: Here, n px is the number of vertical image pixels between the hot and the cold plate. Note that the thermal amplitudes in table 2 are in units of phase gradients measured in rad/pixel. Hence, A th,i n px is the phase difference between the hot and the cold plate (assuming a linear temperature profile across the cell). The resulting values are included in table 3. They are in parentheses to indicate that they have not been corrected by tomographic reconstruction [17], which would also take the distortion of the temperature field near the back and front window along the beam propagation direction into account. We have neglected this aspect for the moment, since it would only change all Soret and thermodiffusion coefficients by not more than approximately ten percent [17]. Finally, the Soret and thermodiffusion coefficients computed from the mean values in table 2 and the contrast factors in table 3 are listed in the SODI-column in table 8. The coefficients of the dependent third component (TEG) have been calculated from the mass conservation condition

2-OBD laboratory experiments
The 2-OBD measurements performed in the laboratory were carried out as described in refs. [5,13]. The transient beam deflection signals were treated essentially in the same way as the time series for the central phase gradient extracted from the SODI experiments. The transients were fitted in the refractive index space with eq. (7), followed by a transformation to the concentration space for the computation of the Soret and thermodiffusion coefficients according to eqs. (11) and (12). One major difference between the 2-OBD and the SODI measurements are the employed detection wavelengths, namely 405 nm and 635 nm in the laboratory versus 670 nm and 935 nm in space. Some care is required to avoid convection. The question, whether a particular orientation of the temperature gradient leads to a stable stratification or to a convective instability, can generally not be answered a priori without knowledge of all Soret coefficients. In particular in ternary systems with their two independent diffusion eigenvalues, an initially stable heated-from-above configuration can eventually be destabilized by one or both thermodiffusion modes. Figure 3 shows, as an example, the deflection of the red laser in a 2-OBD measurement for two different orientations of the temperature gradient. The positive value of ΔT = +1 K corresponds to the heated-fromabove configuration and yields a convection-free transient. For the opposite orientation of the temperature gradient, ΔT = −1 K, the system is still below the critical thermal Rayleigh number, but the density stratification Table 4. Evaluation of the 2-OBD laboratory measurements in the refractive index space. Fit values A th,i , ai, and bi/q 2 measured at the two 2-OBD wavelengths λi with λ1 = 405 nm and λ 2 = 635 nm. Tmean = 298 K. Note that the thermal amplitudes A th,i in tables 2 and 4 possess different units, which are owed to the different ways the signals are recorded. The interpretation of A th,i in the case of the SODI experiment is given in eq. (13), whereas in table 4 it is directly the refractive index gradient. These different units for the amplitudes are of no concern, since they cancel out after normalization of the signals for evaluation according to eq. (7).

+1 K +2 K Mean
Because of the poor condition number of N c for the 2-OBD experiment, we have measured the contrast factors at the 2-OBD-wavelengths with particular care. For this purpose, 19 samples with compositions up to approximately 0.2 weight fractions in all directions around the symmetric composition were prepared. The refractive index of each sample was measured at wavelengths of 405, 437, 488, 532, 589, 633, and 684 nm using commercial Abbe refractometers (Anton Paar, Abbemat WR-MW, and Abbemat WR, modified to 405 nm). The mixtures were prepared in vials and continuously stirred during sample withdrawal to counteract concentration changes near the surface caused by ethanol evaporation. In order to reduce experimental noise, the wavelength dependence of n was fitted for every sample by a Cauchy equation n(λ) = A+Bλ −2 +Cλ −4 , from which the refractive indices at 405 and 635 nm were obtained. These values where then fitted by a 2d-Taylor expansion around the symmetric composition in order to compute the partial derivatives in the direction of the two independent concentrations. Since the partial derivatives of the third and fourth order expansion agree within approximately 0.2 percent, the average of both values is listed in table 3.
The thermal contrast factors (∂n i /∂T ) p,c1,c2 for the 2-OBD-wavelengths have been measured interferometrically as described in ref. [34] with the proper correction for the cuvette window from ref. [3]. The thermal contrast factors for the SODI measurements will be discussed later on.

TGC laboratory experiments
In the TGC experiments, the thermodiffusion coefficients are obtained from the vertical separation in the thermogravitational column according to [22] Here, L x and L z are the dimensions of the column, α is the thermal expansion coefficient, ν the kinematic viscosity, and g the acceleration of gravity. The vertical concentration profiles c i (z) are determined from the density ρ and the refractive index n D of samples extracted from the column after the stationary state has been reached. In order to define the transformation to the concentration space, ρ and n D were measured for a large number (25) of mixtures around the composition of interest by means of an Anton Paar DMA 5000 densimeter and an Anton Paar RXA 156 refractometer at the Na-D line (589 nm). Other than in the cases of SODI and 2-OBD, TEG and EtOH were chosen as independent concentrations, and the two measured quantities are parameterized by (17) Figure 4 shows the 2d-fits to both quantities. The fit coefficients are listed in table 5.
Once the fit coefficients are determined, the concentrations can be computed according to For a direct comparison with the other techniques, the contrast factors transformed to the independent concentrations H 2 O and EtOH are also included in table 3. In total, three TGC measurements were performed at a mean temperature of 298 K and with a temperature difference of 8 K between the two plates. After the mixture had reached the stationary state, five samples were extracted at different heights of the column and their compositions were determined from their densities and refractive indices according to eqs. (18)- (20). The concentration variations along the height of the column are plotted in fig. 5.
Finally, the three thermodiffusion coefficients are obtained from the slopes dc i /dz (table 6) according to eq. (15). The averages over the three runs are included   in table 8 together with the Soret coefficients calculated from eq. (2) and the diffusion matrix from ref. [35]. The thermophysical properties entering eq. (15) are summarized in table 7.

Discussion
The investigation of ternary systems is rather complex and the accuracy that is routinely achieved in experiments on binary mixtures can, as a rule, not be reached in the case of ternaries. As outlined before, the two Soret cell techniques 2-OBD and SODI are capable to determine both the thermodiffusion and the Soret coefficients. TGC yields only the thermodiffusion coefficients, and the Soret coefficients have been computed according to eq. (2) with the help of the diffusion matrix from ref. [35]. The Soret and thermodiffusion coefficients obtained by the two different laboratory techniques and in microgravity are summarized in table 8. The error bars in the table represent only the dominant contribution due to the inversion of the contrast factor matrix. They are estimated by a Monte Carlo-simulation based on one percent Gaussian noise on the SODI and 0.5 percent on the 2-OBD amplitude fit parameters a i . In the absence of convection and other systematic error sources, one of the most crucial steps is the transformation of the primary measured signals to the concentration space. This step requires the inversion of the contrast factor matrix N c , whose condition number gives a rough estimate for the error amplification incurred during this step. As can be seen from table 3, TGC shows by far the most favorable and 2-OBD the least favorable condition number. From this perspective, TGC appears very well suited for the ternary mixture investigated here. The technique is, however, limited to mixtures with positive separation ratios, i.e., mixtures with a solutal density increase on the cold side. This is only the case for the DCMIX3 samples in cells two and three, but not for the other cells. The accessibility by TGC has been a central argument for the focus on cell two in this work. The small error bars of the TGC data reflect the very good condition number of the corresponding contrast factor matrix. Additional error sources have intentionally not been taken into account, and the true but unknown systematic errors may well exceed the indicated numbers. Table 8. Summary of the Soret and thermodiffusion coefficients measured in microgravity (SODI) and by the two ground-based techniques 2-OBD and TGC. In case of TGC, the Soret coefficients have been computed from the thermodiffusion coefficients with the help of the diffusion matrix from ref. [35]. Soret coefficients in units of (10 −3 K −1 ), thermodiffusion coefficients in (10 −13 m 2 (sK) −1  The agreement between the three techniques is acceptable to good for all six coefficients. In particular the signs always agree between the different techniques and also between all Soret and corresponding thermodiffusion coefficients. The positive sign for H 2 O and the negative signs for EtOH and TEG correspond to a thermophobic behaviour of water and a thermophilic one of the other two components. The positive Soret coefficient of H 2 O and the negative one of EtOH are in agreement with the behaviour of the corresponding binary mixture, where ethanol is thermophobic at low concentrations but becomes thermophilic above 0.29 ethanol mass fractions [3,36]. How the locus of the sign change extends into the ternary composition space and evolves as a function of TEG concentration is not known, but some insight can be expected from the upcoming data evaluation of the remaining four DCMIX3 cells.
We have chosen H 2 O and EtOH as the independent concentrations, but this choice is arbitrary. A transformation of the contrast factors to another pair of independent concentrations c * i is accomplished by If (H 2 O/TEG) are selected as concentration base instead of (H 2 O/EtOH), i.e., c * 1 = c 1 and c * 2 = c 3 , the condition number of N c reduces from 1132 to 470 for 2-OBD and from 283 to 119 for SODI. Of course, this arbitrary choice of the independent concentrations cannot lead to more accurate transport coefficients, since their computation is still based on the identical experimental data set. The improved condition of the matrix can, however, shed some light on the accuracy of the three Soret and thermodiffusion coefficients. Figure 6 shows the SODI-results for the Soret coefficients measured for the four runs in table 2. Also shown are the confidence regions resulting from one percent Gaussian noise added to the mean amplitudes a i . These elongated ellipsoidal regions are a direct consequence of the transformation from the refractive index to the concentration space. The minor and the major axis of the ellipses are determined by the right-singular vectors u and v of N c , which are shown in fig. 6 for the independent concentrations (H 2 O/EtOH) and (H 2 O/TEG). The large aspect ratio of the ellipses indicates strongly correlated errors of the Soret coefficients with a large uncertainty in the direction of the major but only a very small one in the direction of the minor axis.
For (H 2 O/EtOH) as independent concentrations and N c as given in table 3, the right-singular vector in the direction of the major axis is v = (0.63238, −0.77466) T . With (S 0 T,1 , S 0 T,2 ) T = (1.64, −1.38) T × 10 −3 K −1 from table 8 and with eq. (14), the following relations must hold between the three Soret coefficients S T,i to fulfill the condition that they lie on the straight lines defined by the major axes of the two ellipsoids in fig. 6: Inserting the numbers finally yields The lower condition number for the independent concentrations (H 2 O/TEG) is reflected in the smaller slope of the major axis of the corresponding ellipsoid. While the uncertainty for S T,1 (H 2 O) remains the same as in the case of (H 2 O/EtOH) as independent concentrations, the histogram projected onto the y-axis in fig. 6 is much narrower for TEG than for EtOH. It should be noted that exactly the same result is obtained, if (H 2 O/TEG) instead of (H 2 O/EtOH) are taken as independent concentrations. In any case, the Soret coefficient of the dependent component is computed from eq. (14).
The irrelevance of a particular choice of the dependent and independent concentrations is further emphasized by the ternary diagram shown in fig. 7 for the data from fig. 6. Such a ternary diagram can always be constructed, if the sum of the three variables is constant, in our case zero. Since all three Soret coefficients are represented simultaneously, there is only one unique data set instead of two in the xy-plot in fig. 6. The colored projection intervals onto the three S T -axes correspond to ±3σ of the simulated distribution. The favorable narrow projection onto the S T (TEG)-axis becomes immediately clear from the orientation of the major axis of the confidence region.
The TGC data points in figs. 6 and 7 lie almost perfectly on the straight lines defined by the SODI confidence regions.
Also included in fig. 7 are the two 2-OBD measurements from table 4 together with the confidence region constructed in the same way as for the SODI data, however with an assumed noise amplitude of only 0.5 percent. Due to the higher condition number of the contrast factor matrix, it still extends over a wider region than in the case of SODI. The two 2-OBD-measurements are almost symmetric to and their mean value almost coincides with the TGC result. Given the poor condition number and the scatter between both measurements, this excellent agreement may, however, merely be a lucky coincidence. Very convincingly, the SODI and the 2-OBD confidence regions overlap and extend along the same direction.

Summary and conclusion
By employing two ground based techniques and one microgravity experiment, it has been possible to obtain reliable and consistent information about the Soret and thermodiffusion coefficients of a symmetric ternary mixture. Although the small error bars known from binary mixtures cannot be achieved for ternaries, all experiments reasonably agree and, most important, there is no ambiguity in the signs. All techniques yield a thermophobic behaviour of H 2 O and a thermophilic one of the other two components EtOH and TEG. The thermophobicity of water is in agreement with the sign of the Soret coefficient of water at the same weight fraction in pure ethanol [3,36].
For the here investigated mixture with a positive separation ratio, the TGC method excels with a very low condition number of the contrast factor matrix, which avoids problems during the transformation of the measured signals to the composition space. In the case of an ill-conditioned contrast factor matrix, the symmetric circle of uncertainty is transformed into a stretched ellipse, whose major and minor axes are defined by the rightsingular vectors of N c . The accuracy of the Soret and thermodiffusion coefficients is high in the direction of the minor axis but significantly lower in the direction of the major axis.
Although a change of the independent concentrations from (H 2 O/EtOH) to (H 2 O/TEG) leads to a contrast factor matrix with a lower condition number, no direct advantage can be drawn. After fixing the third Soret coefficient, the one of the dependent concentration, by the requirement of eq. (14), identical results are obtained for all choices of independent concentrations.
A low condition number is very advantageous for precise measurements, but a large one does not automatically exclude the extraction of precise data. How the uncertainty caused by the transformation to the composition space affects the individual Soret and thermodiffusion coefficients depends on the orientation of the major axis of the elliptical confidence region. As can be seen in fig. 7, its projection onto one axis can be much narrower than onto the other ones. As a consequence, the corresponding Soret coefficients can still be obtained with a high accuracy. For the here investigated mixture, the uncertainty for TEG is approximately by a factor of five smaller than for H 2 O and EtOH. Even in case of arbitrary orientation, two equations can be extracted that pin the three unknown coefficients to two straight lines (eqs. (22), (23)), leaving only one unresolved degree of freedom. In this case, the condition number eventually determines how well the positions along these lines can further be narrowed down.

Author contribution statement
TT analyzed the DCMIX3 microgravity experiments and performed and analyzed 2-OBD measurements. DS analyzed DCMIX3 microgravity data. MS and FS performed contrast factor measurements. EL and MMB-A performed and analyzed the TGC measurements. WK analyzed the data and wrote the manuscript. All authors discussed the content of the paper.
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