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Simultaneous Analysis of Equilibrium Fluctuations at the Surface and in the Bulk of a Binary Liquid Mixture by Dynamic Light Scattering Thomas M. Koller, Tobias Klein, Jiaqi Chen, Ahmad Kalantar, Gerard P. van der Laan, Michael Heinrich Rausch, and Andreas Paul Fröba J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b09770 • Publication Date (Web): 10 Nov 2017 Downloaded from http://pubs.acs.org on November 16, 2017

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The Journal of Physical Chemistry

Simultaneous Analysis of Equilibrium Fluctuations at the Surface and in the Bulk of a Binary Liquid Mixture by Dynamic Light Scattering

Thomas M. Koller,*,a Tobias Klein,a Jiaqi Chen,b Ahmad Kalantar,b Gerard P. van der Laan,b Michael H. Rausch,a and Andreas P. Fröbaa

a

Institute of Advanced Optical Technologies ‒ Thermophysical Properties, Department of Chemical and Biological Engineering (CBI) and Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander-University Erlangen-Nürnberg (FAU), Paul-Gordan-Straße 6, 91052 Erlangen, Germany

b

Shell Global Solutions International B.V., Grasweg 31, 1031 HW Amsterdam, The Netherlands

Corresponding Author * Author to whom correspondence should be addressed. Tel. +49-9131-85-23279, fax +49-9131-8525851, email [email protected].

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ABSTRACT For the first time, we demonstrate that it is possible to simultaneously analyze microscopic fluctuations at the surface and in the bulk of a binary liquid mixture by dynamic light scattering in macroscopic thermodynamic equilibrium. For a model system containing n-octacosane and ethanol, three individual signals distinguishable in the time-resolved analysis of the scattered light intensity appear on different time scales. One oscillatory signal from surface fluctuations at the vapor-liquid interface in the short-time range and two exponential Rayleigh signals from fluctuations in temperature and concentration in the bulk of fluid in the long-time range could be associated with hydrodynamic modes. This microscopic information allows for a simultaneous determination of the macroscopic properties interfacial tension, kinematic viscosity, thermal diffusivity, and mutual diffusivity within a single experimental run. The presented approach represents a worthwhile strategy, for example in the context of sensor development for an effective multi-property determination of fluid systems.

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INTRODUCTION Dynamic light scattering (DLS) analyzes microscopic statistical fluctuations originating from the random thermal movement of molecules in macroscopic thermodynamic equilibrium. The interaction of coherent laser light with these fluctuations causes a modulation of the scattered light which contains information on the dynamics of the molecules and thus on the macroscopic transport properties.1 The basic justification for this relation is Onsager's regression hypothesis2,3 which states that on a statistical average, microscopic fluctuations are governed by the very same macroscopic transport equations. For light scattering from the bulk of fluids, the scattering process is governed by microscopic fluctuations in pressure, temperature, and species concentration in mixtures. The relaxation of temperature and concentration fluctuations, which are of interest in this study and contribute to the central unshifted Rayleigh component of the spectrum of the scattered light, is related to thermal diffusivity and mutual diffusivity, respectively.1 Our studies on mixtures of, e.g., methane and ethane,4 ionic liquids and cosolvents,5,6 or liquids with dissolved gases7-11 document the capability of the method for an accurate determination of thermal and/or mutual diffusivity. By the application of DLS to fluid interfaces, also called surface light scattering (SLS), fluctuations on the liquid surface or, in a more general formulation, at phase boundaries are probed. The relaxation behavior of these surface waves which are quantized in so-called ”ripplons” can be used to measure viscosity and surface or interfacial tension.12 Our investigations on toluene13 contributing to the development of a low-viscosity standard14 and systems based on, e.g., refrigerants,15-17 n-alkanes,18 or ionic liquids19-21 showed that SLS is a useful tool in thermophysical property research for different fluids covering a broad viscosity and surface tension range. Until now, to the best of our knowledge no investigation has been carried out to verify if it is possible to simultaneously analyze surface and bulk fluctuations by DLS within a single experimental setup or even within a single experimental run in a quantitative way. Decisive for resolving the corresponding scattering process is the adjustment of the scattering geometry and the scattering volume. In SLS experiments, laser light is directed onto the vapor-liquid interface, and 3 ACS Paragon Plus Environment


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scattered light is observed close to the transmitted or reflected beam. For standard DLS measurements from the bulk of fluids, laser light irradiates a fluid phase without passing any phase boundary of the fluid. Here, the scattered light is detected at a defined scattering angle which represents the angle between the transmitted and the scattered light. While in the latter case a detection of light scattered from surface fluctuations is not possible, light scattering at liquid surfaces or interfaces is always accompanied by scattering from the bulk of fluid. The intensity of the light scattered from the bulk of fluid is typically weaker than that from the surface, and the relaxation time of fluctuations in the bulk is often several orders of magnitude larger than that of surface fluctuations. In the present study, we demonstrate that it is possible to simultaneously study surface and bulk fluctuations of a binary liquid mixture from the analysis of light scattered at the vapor-liquid interface region. For this purpose, the signals related to the equilibrium fluctuations at the surface and in the bulk of fluid are separated into different characteristic time ranges. The three recorded signals could be proven to be hydrodynamic modes. Thus, it could be evidenced that the four macroscopic properties interfacial tension, viscosity, thermal diffusivity, and mutual diffusivity can be accessed simultaneously within a single experimental run in an accurate way.

LIGHT SCATTERING FROM BULK FLUIDS AND SURFACE WAVES In the following, only brief information on the principles of the DLS technique is given. For more details, the reader is referred to the specialized literature for light scattering from bulk fluids1,22 and from surface waves.12,23 The scattering geometry normally applied for SLS experiments is shown in Figure 1a. Scattered light is observed near refraction and perpendicular to the surface plane. From the angle of incidence ΘE resulting in a specific scattering angle ΘS, the r r r r scattering vector q ripplon = k I′ − k S′ of the observed surface vibration mode is determined. Here, k I′ r r r and k S′ denote the projections of the wave vectors of the refracted ( k I ) and the scattered ( kS ) light into the surface plane, respectively. For elastic scattering (i.e. k I ≅ kS ) and the case that the 4 ACS Paragon Plus Environment

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scattering angle ΘS is equal to the angle of refraction, the modulus of the scattering vector is given by qripplon = (2π λ0‒1) sinΘE, where λ0 is the laser wavelength in vacuo. The relaxation behavior of a selected surface wave with a defined wavelength Λ or modulus of the wave vector qripplon = 2π Λ‒1 depends on the thermophysical properties of the fluid. In the case of large viscosity and/or small interfacial tension, the surface waves show an overdamped behavior and do not propagate. In the case of small viscosity and/or large interfacial tension, the surface waves show an oscillatory behavior and propagate. In this study, the latter case has always been observed and is solely considered in the following.

Figure 1. (a) Scattering geometry used for SLS. (b) Schematic representation of the scattering volume providing in the case of the illumination of the vapor-liquid interface an additional scattered light contribution induced by Rayleigh and Brillouin scattering processes from the liquid and/or vapor phase of the fluid.

Here, scattered light is analyzed by a post-detection filtering scheme using photon correlation spectroscopy. In this type of detection, the time-dependent correlation function of the scattered light intensity is measured at a point in the far field. For heterodyne conditions, where the scattered light is superimposed with coherent reference light of much higher intensity, the normalized intensity 5 ACS Paragon Plus Environment


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correlation function for the analysis of surface fluctuations with an oscillatory behavior takes the form24

g ( 2 ) (τ ) = a R + bR cos(ωR τ − φ ) exp(− τ τ C,R ) .

(1)

The experimental constants aR and bR in eq 1 are essentially determined by the total number of counts registered, the ratio of scattered light to reference light intensity, and the coherence properties of the optical system. While the phase term φ mainly accounts for the deviations of the spectrum from the Lorentzian form, the correlation time τC,R and the frequency ωR are identical with the mean lifetime or the inverse of the damping constant ΓR (= τC,R‒1) of the surface waves and their frequency of propagation, respectively.12 For an accurate determination of liquid kinematic viscosity ν' or liquid dynamic viscosity η' (η' = ν' ρ') and surface or interfacial tension σ by SLS, the dispersion relation D(η', η'', ρ', ρ'', σ, Γ, ωq, q) = 0 for hydrodynamic surface fluctuations at the interface between contacting liquid (') and vapor ('') phases must be considered in its complete form.24 For this, measured data for the dynamics of surface waves, i.e., the frequency ωq and damping Γ at a defined wave number qripplon, are combined with data for the dynamic viscosity of the vapor phase η'' and density data for both phases, ρ' and ρ''. Owing to the finite size of illumination and detection optics, the scattered light originating from the fluctuating surface cannot be observed without any superposition of scattering contributions from the bulk of fluid, i.e., its vapor and/or liquid phase. This is shown in Figure 1b where the corresponding scattering volume for light scattering at the surface region is given. Besides the light scattered from the surface itself, also light scattered in the vapor and liquid phase of the fluid is registered in detection direction. In theory, contributions from the bulk of fluid related to the Rayleigh and the Brillouin scattering process may be superimposed on the surface light scattering process. However, Brillouin signals related to periodic pressure fluctuations show typically very small characteristic lifetimes25 and cannot be detected without using a frequency-shifted local 6 ACS Paragon Plus Environment

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oscillator. Furthermore, in the case of low vapor densities, Rayleigh contributions from the vapor phase are much weaker than those from the liquid phase and can be neglected in this study. Thus, for binary fluid mixtures as studied here, any Rayleigh signal may be connected with the liquid phase and originate from microscopic fluctuations in temperature and concentration. The mean relaxation times of both hydrodynamic modes can also be analyzed via the correlation function of the scattered light intensity. For a heterodyne detection scheme, the normalized intensity correlation function consists of two exponentials,4

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

(2)

where the decay times τC,t and τC,c represent the mean lifetimes of the temperature and concentration fluctuations in the binary mixture. The experimental constants b0, bt, and bc are mainly determined by the intensities of the scattered and the reference light as well as by effects that are caused by imperfect signal collection.8 In the hydrodynamic regime where the mean free path length of the molecules is much smaller than the inverse value of the modulus of the scattering vector,1 the decay times τC,t and τC,t are directly related to the thermal diffusivity a and the mutual diffusivity D12 according to τC,t = (a q2)‒1 and τC,c = (D12 q2)‒1. For elastic scattering and small scattering angles r r ΘS < 10°, the modulus of the scattering vector in the bulk of fluid, q bulk = k I′ − k S′ , as given in Figure 1a, can be calculated in good approximation with the help of the angle of incidence ΘE by qbulk = (2π λ0‒1) sin(ΘE).8 For most mixtures where the relative differences in the refractive index of the pure substances are (5 to 10)%, the signal contributions to the correlation function related to temperature and concentration fluctuations have a similar magnitude (bt ≈ bc) and can thus be resolved simultaneously. If the relative refractive index difference is below 1%, usually the signal associated with temperature fluctuations predominate the one associated with concentration fluctuations (bt > bc).23 Whether Rayleigh scattering signals appear in the correlation function besides a SLS signal depends on the thermophysical properties of the fluid and the relative signal strength. The latter is 7 ACS Paragon Plus Environment


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mainly determined by the extension and position of the scattering volume, the wave number of the studied fluctuations, and the thermodynamic state of the fluid. For conditions far away from the critical point as valid in the present investigations, only a weak signal can be found for the Rayleigh scattering process. However, the scattering intensity for SLS strongly decreases with increasing wave number, while it is constant for the Rayleigh scattering from the bulk of fluid.23

EXPERIMENTAL SECTION In this work, a binary mixture of n-octacosane (n-C28H58) and ethanol (C2H5OH or EtOH) with a mass fraction of ethanol of wEtOH = 0.030 was investigated as a model system for a temperature of T = 473.15 K close to saturation conditions. Details on the materials and sample preparation are given in the Supporting Information. Therein, also the used experimental setup which has already been employed in our former investigations by SLS13,21,26-28 and the measurement cell18 are specified in some detail. Scattered light is analyzed at relatively high moduli of the scattering vector q = qripplon = qbulk for fluctuations at the liquid surface and in the bulk of the binary liquid mixture, which is why instrumental broadening effects are negligible.13 For a simultaneous detection of surface and bulk fluctuations, the scattering volume has to include both the interface as well as the bulk of fluid. To compensate the expected higher scattering intensity from surface fluctuations compared to that from bulk fluctuations, the scattering volume is shifted closer to the bulk of the liquid than to the surface. Since the different scattering signals can appear over a broad time range, a multi-tau correlator featuring a broad time range between a few nanoseconds up to many seconds was used to calculate the normalized intensity correlation function g(2)(τ). While the application of smaller incident angles is favorable with respect to the signal quality and intensity, larger angles shift the relaxation times of the signals to shorter times and closer to each other. At the same time, a heterodyne detection scheme has to be guaranteed for both signal contributions in order to apply the corresponding working equations, eqs 1 and 2. This means that in practice, the optimization of the


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optical alignment for the simultaneous analysis of surface and bulk fluctuations under heterodyne conditions is associated with a large experimental effort.

MEASUREMENT EXAMPLES AND DATA EVALUATION Figure 2 shows a correlation function obtained from scattering on the surface of the model binary mixture as a function of the lag time τ, which is plotted on a logarithmic scale. The adjusted incident angle was ΘE = 6°, and the total measurement time was about 60 min. According to Figure 2, three signals related to the decay of three kinds of fluctuations can be found in different time domains. In the first part of the correlation function at short times up to a few microseconds, a damped oscillation is visible. This signal can clearly be related to the oscillatory decay of surface fluctuations for low-viscosity fluids. At larger times, only contributions from Rayleigh light scattering in the bulk of fluid can be found, which arise from the relaxation of temperature and concentration fluctuations. The Rayleigh scattering contributions are also present at short times and superimposed on the signal from surface fluctuations. An indication for the presence of two superimposed Rayleigh signals related to thermal and mutual diffusivity is the change in the slope at about 30 µs in Figure 2.

Figure 2. Measurement example for a normalized intensity correlation function for a binary mixture of n-C28H58 and C2H5OH (wEtOH = 0.03) at a temperature of T = 473.15 K and an incident angle of

ΘE = 6.0° measured by a multi-tau correlator. Different signals are caused by the relaxation of surface, temperature, and concentration fluctuations. 9 ACS Paragon Plus Environment


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The individual time ranges given in Figure 2, which are related to the SLS signal and to the Rayleigh light scattering signals from the liquid bulk of fluid, were fitted separately by two fits. All fitting procedures were performed by nonlinear regression based on a Levenberg-Marquardt algorithm in which the squared sum of residuals is minimized. For the SLS signal in the short time range between (0.1 and 4) µs, a fit model on the basis of eq 1 was used, where a logarithmic weighing scheme was applied to the correlator data. Compared to eq 1, an additional linear term in the long-time range of the surface signal was added accounting for the beginning of the signals from temperature and concentration fluctuations. To ensure heterodyne conditions, the amplitude of the signals should be lower than 1.5% which is given for the SLS signal. Based on the fit, values and expanded uncertainties on a confidence level of more than 95% (coverage factor k = 2) of

τC,R = (0.485 ± 0.013) µs and ωR = (5.523 ± 0.076) MHz are obtained for the characteristic quantities of the ripplon signal. For the description of the Rayleigh light scattering signals, a single exponential was used as a first fit model. Distinct systematic deviations in the residual plot suggested that only two modes related to the decay of temperature and concentration fluctuations can represent the correlator data well. The existence of a mode related to concentration fluctuations is supported by the comparably large difference in the refractive indices of pure n-C28H58 (n ≈ 1.43 at T ≈ 343 K)29 and pure C2H5OH (n ≈ 1.36 at T ≈ 298 K).30 Despite the relatively low concentration of C2H5OH in the liquid phase, it is still sufficient to give rise to a Rayleigh light scattering signal associated with concentration fluctuations. Thus, a fit model in the form of the sum of two exponentials according to eq 2 was used for the time range between (4.4 and 512) µs, where the correlator data are weighted with a logarithmic weighing scheme. A disturbing signal which appears in the long-time range at about 200 µs and may arise from, e.g., vibrations and particle scattering, could be described well by an additional quadratic term in the fit model and is already subtracted from the correlation function shown in Figure 2 to improve legibility. Within the entire fit range, no systematic 10 ACS Paragon Plus Environment

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deviations between the measured data and the fit can be observed. The deduced decay times of

τC,t = (10.75 ± 0.25) µs and τC,c = (120.8 ± 8.1) µs can be associated with the thermal diffusivity a and the mutual diffusivity D12, respectively. Taking into account our previous studies on binary mixtures of n-C28H58 with dissolved gases,7,8 the faster mode must be related to the thermal diffusivity and the slower one to the mutual diffusivity for the present mixture. The larger uncertainty of τC,c is mainly related to the lower signal contrast of about 0.1% compared to about 0.4% for the signal associated with τC,t. In the following, our strategy is to show that the results obtained from the simultaneous analysis of surface and bulk fluctuations agree with those which can be accessed from separate, consecutive light scattering measurements focusing on the surface or the liquid phase. Compared to the simultaneous measurement in the interface region, these consecutive measurements are expected to be more easily feasible since either the SLS signal or the Rayleigh scattering signal from the bulk of fluid is registered without any superposition from the respective other signal. It should be mentioned that also the consecutive analysis of both surface and bulk fluctuations within the same experimental setup without further manipulation of the sample represents a new development. This approach is also utilized to prove by comparison between experiment and theory that the observable signals recorded by the simultaneous and consecutive measurements are in agreement with the classical hydrodynamic theory. For this purpose, a variation of the modulus of the scattering vector was performed in order to validate the working equations for the SLS and Rayleigh scattering signals. The proof of concept regarding the simultaneous analysis of surface and bulk fluctuations within one experimental run will finally be given by checking the agreement of the values for the four thermophysical properties determined from the simultaneously and consecutively analyzed signals. In the following, single measurement examples for the SLS signal and the Rayleigh scattering signal recorded from consecutive measurements are shown and discussed. Here, a single-tau 11 ACS Paragon Plus Environment


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correlator was used where all 255 correlator channels have a fixed spacing. The sample time was adjusted to access maximum information on the time range where the different signals can be found. For the investigation of fluctuations at the liquid surface and in the bulk of fluid of the studied binary mixture, the correlation functions recorded with the single-tau correlator are illustrated in Figure 3.

Figure 3. Normalized intensity correlation functions for a binary mixture of n-C28H58 and C2H5OH (wEtOH = 0.03) at 473.15 K measured by the single-tau correlator and obtained (a) at the liquid surface for ΘE = 3.0° and (b) from the liquid bulk of fluid for ΘE = 7.0°.

For the study of the dynamics of surface fluctuations, the system was aligned in a way that the incident beam and the detection direction intersect in the middle of the sample cell at the liquid surface, c.f. Figure 1a. For ΘE = 3.0° and a measurement time of 10 mins, the surface signal in the form of a damped oscillation given in Figure 3a is present within a short time range of about 10 µs. Based on a fit according to eq 1 where all correlator data were considered with the same statistical weight, the quantities τC,R and ωR can be determined with expanded relative uncertainties (k = 2) of 12 ACS Paragon Plus Environment

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about 1.5% and 0.5%, respectively. Owing to the relatively strong SLS signal, the latter uncertainties are about a factor of 2 lower than those obtained from the simultaneous analysis of surface and bulk fluctuations given in Figure 2. Within the entire fit range, no systematic deviations or further signals from the liquid phase in the long-time range of the correlation function could be found for the SLS signal. A signal contrast of about 1.4% is sufficiently small to ensure a heterodyne detection scheme. By moving the scattering volume from the surface into the liquid phase by shifting the incident beam correspondingly, it was also possible to analyze only light scattered from the bulk of the fluid. A Rayleigh signal obtained in this way within a measurement time of about 30 mins is shown in Figure 3b. In comparison with the SLS signal, the correlation function obtained for ΘE = 7.0° covers a much larger time range of about 470 µs and contains signals related to the relaxation of temperature and concentration fluctuations. The correlation function was evaluated based on eq 2 to obtain the decay times of temperature and concentration fluctuations, τC,t and τC,c, with corresponding expanded relative uncertainties (k = 2) of about 7% and 12%, respectively. The larger uncertainities for τC,t and τC,c obtained from the current measurement example and from all further separate measurements compared to those obtained from the simultaneous measurement are partially related to the shorter measurement times of the consecutive measurements between about (30 and 50) min, but mainly to the lower applied intensity of the incident laser light causing weaker intensities of the scattered light. Disturbing effects in the long-time range of the correlation function could be well described by an additional quadratic term in comparison with the theoretical model according to eq 2. For visualization purposes, this term was subtracted from the correlation function shown in Figure 3b. The signal contrasts of about 1.1% and 0.4% for the signals related to the temperature and concentration fluctuations, respectively, confirm the validity of a heterodyne detection scheme.



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The results for the characteristic dynamic properties related to the surface and bulk signals, i.e.,

ωR, τC,R, τC,t, and τC,c, obtained from the consecutive measurements carried out over a wide range of scattering angles from both sides of incidence are shown in Figure 4 as a function of the squared modulus of the scattering vector q. In Figure 4a, the measured values for the damping ΓR (= τC,R‒1) and the frequency ωR of the surface fluctuations are given for eight different values for q, which corresponds to eight different incident angles between (2.5 and 5)°. Here, the evaluation has always been performed on the basis of a full theoretical solution of the dispersion relation.13 The measured results for τC,R and ωR are compared to the theoretical values, for which input data for liquid and vapor density, liquid and vapor viscosity, as well as interfacial tension are required. The data for liquid and vapor density as well as vapor viscosity are adopted from literature and are discussed in the Supporting Information. For the liquid kinematic viscosity and interfacial tension, the measured mean values averaged over all individual measurements at different q values were used because no corresponding reference data are available. For all wave vectors investigated in this study, agreement can be found between the measured values for ΓR and ωR and the theoretical ones predicted by an exact solution of the dispersion equation.13 Hence, the oscillatory signals analyzed in our experiments are well described by classical hydrodynamic theory.31 Figure 4a also shows that first-order approximations13,23 of the dispersion relation for τC,R and ωR, which are given as dashed lines and are often adopted in the literature,32,33 cannot be used for an accurate description of the dynamics of surface fluctuations.


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Figure 4. Dispersion relation for the dynamics of fluctuations at the surface and in the bulk of the fluid for a binary mixture of n-C28H58 and C2H5OH (wEtOH = 0.03) at a temperature of 473.15 K measured by a single-tau correlator. Damping and frequency of surface fluctuations (a) as well as damping of fluctuations in temperature and in concentration in the liquid bulk of the fluid (b) as a function of the squared modulus of the scattering vector.

Regarding the measurements in the bulk of the fluid, Figure 4b shows the damping, i.e. the inverse of the experimentally determined decay times for temperature and concentration fluctuations as a function of the squared modulus of the scattering vector in the upper and lower part, where incident angles of (4, 5, 6, and 7)° were studied. Furthermore, a theoretical description is included where the slopes of the straight lines represent the thermal and mutual diffusivity values which were averaged from the eight single measurements. In both cases, the measured data agree within their expanded uncertainties with the theory. The linear trend of the experimental data is evidence that the two Rayleigh signals are also related to hydrodynamic modes.



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RESULTS AND DISCUSSION To prove that the correlation function shown in Figure 2 indeed gives access to the four different thermophysical properties from one single measurement, the determined properties are compared to the average values obtained from the consecutive measurements at the liquid surface and in the liquid bulk of the fluid at different scattering angles. Also for the evaluation of the correlation function given in Figure 2, an exact numerical solution of the dispersion equation13 was carried out to determine the liquid kinematic viscosity ν' and interfacial tension σ from the signal contributions associated with the surface fluctuations. For all individual measurements performed in this study, the expanded uncertainties (k = 2) were calculated from error propagation schemes13 taking into consideration the uncertainties of the measured quantities as well as of the reference data adopted for data evaluation. The data determined from the SLS signal contribution in the correlation function shown in Figure 2 are ν' = (1.245 ± 0.021) × 10‒6 m2·s‒1 and σ = (17.13 ± 0.47) mN m‒1. In comparison, the values averaged from the 16 measurements at eight different incident angles with focus on the surface fluctuations according to Figure 4a are ν' = (1.257 ± 0.034) × 10‒6 m2 s‒1 and

σ = (17.28 ± 0.45) mN m‒1. In the latter case, the specification of the expanded uncertainties (k = 2) is based on the average of the individual uncertainties calculated for each data point by error propagation. These stated uncertainties are larger than the double standard deviation of the individual results for ν' and σ contributing to the reported mean values. The relative absolute deviations between the data sets from the two different experimental approaches of 0.9% for both liquid kinematic viscosity and interfacial tension are within combined uncertainties. For the light scattering measurements focusing at the bulk of the fluid, eight individual measurements at different values for the modulus of the scattering vector, c.f. Figure 4b, result in average values of a = (6.29 ± 0.25) × 10‒8 m2 s‒1 and D12 = (6.06 ± 0.98) × 10‒9 m2 s‒1. From the light scattering signal contributions in the correlation function in Figure 2 related to temperature and concentration

fluctuations,

values

of

a = (6.10 ± 0.14) × 10‒8 m2 s‒1


and

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D12 = (5.43 ± 0.36) × 10‒9 m2 s‒1 could be deduced. Also here, agreement between the two data sets for both transport coefficients a and D12 with relative absolute deviations of 2.9% and 10.4%, respectively, is given within combined uncertainties which were determined as described above.

CONCLUSIONS In this work, we have shown for the first time that microscopic equilibrium fluctuations at the surface and in the bulk of a binary liquid mixture can be resolved within a single DLS experiment in thermodynamic equilibrium. By the analysis of light scattered in the region of the vapor-liquid interface, three different signals appearing on distinctly different time ranges covering about three orders of magnitude could be found. One is related to an oscillatory behavior of surface fluctuations, while two further signals are associated with hydrodynamic fluctuations in the bulk of the fluid. Based on this microscopic information, it could be demonstrated that in principle, the four macroscopic properties interfacial tension, kinematic viscosity, thermal diffusivity, and mutual diffusivity can be determined simultaneously within one experimental run. Thus, the main benefit of the simultaneous measurement of multiple thermophysical properties is their reliable determination in a single experimental setup probing the very same sample at a well-defined thermodynamic state. To date, however, the additionally introduced approach of measuring the four thermophysical property data consecutively by varying the optical alignment while the sample remains unaffected bears some practical advantages. Nevertheless, the determination of the four fluid properties within a single measurement represents a worthwhile strategy, for example in the context of sensor development. It is clear that the approach still possesses considerable potential for further improvement which has to be addressed before the technique can be established as an accurate and routine method in thermophysical property research. In this connection, optimizations in the illumination and detection strategies by, e.g., irradiating a larger portion of interface region are considered to be promising ways of handling the difficult simultaneous detection of surface and bulk signals of strongly varying signal strengths. By this, measurement uncertainties of the 17 ACS Paragon Plus Environment


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accessible thermophysical properties in the low percentage level at short measurement times are aimed to be achieved. Within a light scattering setup particularly optimized for such a measurement task, the procedure suggested in this work can be further developed for various fluid systems covering a broad range of values for the accessible transport and equilibrium properties.

ASSOCIATED CONTENT Supporting Information. Information on the materials, the sample preparation, as well as the experimental setup and conditions is given. In addition, the sources for the reference data for the liquid density, vapor density, and vapor dynamic viscosity used for the evaluation of the liquid kinematic viscosity and interfacial tension by surface light scattering are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Tel.: +49-9131-85-23279.

ACKNOWLEDGEMENTS 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 B.V. through a contracted research agreement is gratefully acknowledged.

REFERENCES (1) Leipertz, A.; Fröba, A. P. Diffusion Measurements in Fluids by Dynamic Light Scattering. In Diffusion in Condensed Matter - Methods, Materials, Models; Heitjans, P., Kärger, J., Eds.; Springer: Berlin, Germany, 2005; pp 579-618. 18 ACS Paragon Plus Environment

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(2) Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev. 1931, 37, 405-426. (3) Onsager, L. Reciprocal Relations in Irreversible processes. II. Phys. Rev. 1931, 38, 2265-2279. (4) Fröba, A. P.; Will, S.; Leipertz, A. Diffusion Modes of an Equimolar Methane-Ethane Mixture from Dynamic Light Scattering. Int. J. Thermophys. 2000, 21, 603-620. (5) Rausch, M. H.; Hopf, L.; Heller, A.; Leipertz, A.; Fröba, A. P. Binary Diffusion Coefficients for Mixtures of Ionic Liquids [EMIM][N(CN)2], [EMIM][NTf2], and [HMIM][NTf2] with Acetone and Ethanol by Dynamic Light Scattering (DLS). J. Phys. Chem. B 2013, 117, 2429-2437. (6) Rausch, M. H.; Lehmann, J.; Leipertz, A.; Fröba, A. P. Mutual Diffusion in Binary Mixtures of Ionic Liquids and Molecular Liquids by Dynamic Light Scattering (DLS). Phys. Chem. Chem. Phys. 2011, 13, 9525-9533. (7) 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. (8) 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. (9) Koller, T. M.; Heller, A.; Rausch, M. H.; Wasserscheid, P.; Economou, I. G.; Froba, A. P. Mutual and Self-Diffusivities in Binary Mixtures of [EMIM][B(CN)4] with Dissolved Gases by Using Dynamic Light Scattering and Molecular Dynamics Simulations. J. Phys. Chem. B 2015, 119, 8583-8592. (10) Rausch, M. H.; Heller, A.; Herbst, J.; Koller, T. M.; Bahlmann, M.; Schulz, P. S.; Wasserscheid, P.; Fröba, A. P. Mutual and Thermal Diffusivity of Binary Mixtures of the Ionic Liquids [BMIM][C(CN)3] and [BMIM][B(CN)4] with Dissolved CO2 by Dynamic Light Scattering. J. Phys. Chem. B 2014, 118, 4636-4646. 19 ACS Paragon Plus Environment


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(11) Heller, A.; Giraudet, C.; Makrodimitri, Z. A.; Fleys, M. S. H.; Chen, J.; van der Laan, G. P.; Economou, I. G.; Rausch, M. H.; Fröba, A. P. Diffusivities of Ternary Mixtures of n-Alkanes with Dissolved Gases by Dynamic Light Scattering. J. Phys. Chem. B 2016, 120, 10808-10823. (12) Fröba, A. P.; Will, S. Light Scattering by Surface Waves ‒ Surface Light Scattering. In Experimental Thermodynamics, Volume IX: Advances in Transport Properties of Fluids; Assael, M. J., Goodwin, A. R. H., Wakeham, W. A., Eds.; Royal Society of Chemistry: Cambridge, United Kingdom, 2014; pp 22-35. (13) Fröba, A. P.; Leipertz, A. Accurate Determination of Viscosity and Surface Tension Using Surface Light Scattering (SLS): Toluene under Saturation Conditions between 260 and 380 K. Int. J. Thermophys. 2003, 24, 895-921. (14) Santos, F. J. V.; Nieto de Castro, C. A.; Dymond, J. H.; Dalaouti, N. K.; Assael, M. J.; Nagashima, A. Standard Reference Data for the Viscosity of Toluene. J. Phys. Chem. Ref. Data 2006, 35, 1-8. (15) Fröba, A. P.; Botero, C.; Leipertz, A. Thermal Diffusivity, Sound Speed, Viscosity, and Surface Tension of R227ea (1,1,1,2,3,3,3-Heptafluoropropane). Int. J. Thermophys. 2006, 27, 16091625. (16) Fröba, A. P.; Kremer, H.; Leipertz, A.; Flohr, F.; Meurer, C. Thermophysical Properties of a Refrigerant Mixture of R365mfc (1,1,1,3,3-Pentafluorobutane) and Galden® HT 55 (Perfluoropolyether). Int. J. Thermophys. 2007, 28, 449-480. (17) Fröba, A. P.; Leipertz, A. Thermophysical Properties of the Refrigerant Mixtures R410A and R407C from Dynamic Light Scattering (DLS). Int. J. Thermophys. 2003, 24, 1185-1206. (18) Koller, T. M.; Klein, T.; Giraudet, C.; Chen, J.; Kalantar, A.; van der Laan, G. P.; Rausch, M. H.; Fröba, A. P. Liquid Viscosity and Surface Tension of n-Dodecane, n-Octacosane, Their Mixtures, and a Wax between 323 and 573 K by Surface Light Scattering (SLS). J. Chem. Eng. Data 2017, 62, 3319-3333.


Page 20 of 23

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(19) Fröba, A. P.; Kremer, H.; Leipertz, A. Density, Refractive Index, Interfacial Tension, and Viscosity of Ionic Liquids [EMIM][EtSO4], [EMIM][NTf2],[EMIM][N(CN)2], and [OMA][NTf2] in Dependence on Temperature at Atmospheric Pressure. J. Phys. Chem. B 2008, 112, 12420-12430. (20) Hasse, B.; Lehmann, J.; Assenbaum, D.; Wasserscheid, P.; Leipertz, A.; Fröba, A. P. Viscosity, Interfacial Tension, Density, and Refractive Index of Ionic Liquids [EMIM][MeSO3], [EMIM][MeOHPO2], [EMIM][OcSO4], and [BBIM][NTf2] in Dependence on Temperature at Atmospheric Pressure. J. Chem. Eng. Data 2009, 54, 2576-2583. (21) Koller, T. M.; Ramos, J.; Schulz, P. S.; Economou, I. G.; Rausch, M. H.; Fröba, A. P. Thermophysical Properties of Homologous Tetracyanoborate-Based Ionic Liquids Using Experiments and Molecular Dynamics Simulations. J. Phys. Chem. B 2017, 121, 4145-4157. (22) Berne, B. J.; Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics; Robert E. Krieger: Malabar, 1990. (23) Fröba, A. P. Simultane Bestimmung von Viskosität und Oberflächenspannung transparenter Fluide mittels Oberflächenlichtstreuung, Dr.-Ing. Thesis, Friedrich-Alexander-University ErlangenNuremberg, Erlangen, Germany, 2002. (24) Langevin, D. Light Scattering by Liquid Surfaces and Complementary Techniques; Marcel Dekker: New York, 1992. (25) Kraft, K.; Leipertz, A. Sound Velocity Measurements by the Use of Dyamic Light Scattering: Data Reduction by the Application of a Fourier Transformation. Appl. Opt. 1993, 32, 3886-3893. (26) Fröba, A. P.; Leipertz, A. Viscosity and Surface Tension of Saturated Toluene from Surface Light Scattering (SLS). Int. J. Thermophys. 2001, 22, 41-59. (27) Fröba, A. P.; Leipertz, A. Viscosity of Diisodecyl Phthalate by Surface Light Scattering (SLS). J. Chem. Eng. Data 2007, 52, 1803-1810. (28) Rausch, M. H.; Kretschmer, L.; Will, S.; Leipertz, A.; Fröba, A. P. Density, Surface Tension, and Kinematic Viscosity of Hydrofluoroethers HFE-7000, HFE-7100, HFE-7200, HFE-7300, and HFE-7500. J. Chem. Eng. Data 2015, 60, 3759-3765. 21 ACS Paragon Plus Environment


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(29) ChemicalBook. http://www.chemicalbook.com/ProductMSDSDetailCB2339962_EN.htm/ (accessed July 12, 2017). (30) Aralaguppi, M. I.; Jadar, C. V.; Aminabhavi, T. M. Density, Viscosity, Refractive Index, and Speed of Sound in Binary Mixtures of Acrylonitrile with Methanol, Ethanol, Propan-1-ol, Butan-1ol, Pentan-1-ol, Hexan-1-ol, Heptan-1-ol, and Butan-2-ol. J. Chem. Eng. Data 1999, 44, 216-221. (31) Lucassen-Reynders, E. H.; Lucassen, J. Properties of Capillary Waves. Advan. Colloid Interface Sci. 1969, 2, 347-395. (32) Yue, H.; Liu, Z. Surface Tension of Binary Mixtures of 2,2,2-Trimethylpentane + 1-Alkanols from 298.15 to 323.15 K. J. Chem. Eng. Data 2016, 61, 1270-1279. (33) Tin, P.; Adin Mann, J.; Meyer, W. V.; Taylor, T. W. Fiber-Optics Surface-Light-Scattering Spectrometer. Appl. Opt. 1997, 36, 7601-7604.


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