1 Introduction

Fluid systems consisting of liquids with dissolved gases with more than two different substances play an important role in many fields of chemical and biological engineering and energy technology. Applications where the working fluid consists of such a multicomponent system with dissolved gases include synthesis processes, for instance, the production of alcohols from carbon monoxide (CO) or carbon dioxide (CO2) and hydrogen (H2) in hydrogenation processes [1] or the production of methanol or dimethyl ether from synthesis gas using liquid-phase catalysis [2, 3]. In the latter example, mixtures will contain not only the desired products, but also intermediate reaction products, side products, as well as the reactants in the form of dissolved gases, and can be characterized as a multicomponent mixture with liquids and dissolved gases. For designing related equipment as well as modeling and optimization of involved processes, reliable knowledge on the thermophysical properties of such working fluids is required at process-relevant conditions. In this work, the liquid dynamic viscosity ηL and interfacial tension σ are of particular interest due to their requirement for the design and modeling of processes and related apparatuses in chemical and energy engineering. In addition, accurate knowledge on ηL and σ also helps to develop and validate prediction models which can be applied to novel systems. However, reliable data for ηL and σ of liquids with dissolved gases of representative fluid systems at process relevant conditions with respect to temperature T, pressure p, and composition, specifically for systems having more than two species, is still limited.

One method, which enables the absolute determination of ηL and σ at macroscopic thermodynamic equilibrium is the optical technique surface light scattering (SLS). SLS probes the dynamics of thermal fluctuations at phase boundaries in a contactless manner via an analyzes of the temporal behavior of the light scattered at such boundaries. Using hydrodynamic theory to describe the dynamics of such surface fluctuations, this technique has been used to determine viscosity and surface or interfacial tension of various fluid classes, including reference fluids [4, 5], liquid mixtures [6,7,8] as well as liquids with dissolved gases [9,10,11], with typical relative expanded experimental uncertainties (coverage factor k = 2) of ur = 0.02 for both ηL and σ. Complementary to experiments, a powerful tool used in thermophysical property research is molecular dynamics (MD) simulations. Based on an accurate description of the inter- and intramolecular interactions, MD simulations predict trajectories of molecules in a statistical ensemble by solving Newton’s equations of motion. From the resulting information, various thermophysical properties, including ηL and σ, can be determined. Since MD simulations provide detailed information on the molecular and atomistic trajectories, an insight into the fluid structure is possible, which is helpful to develop an understanding of structure–property relationships. Another advantage of MD simulations is their ability to operate at thermodynamic states which are only hardly accessible by experiments. However, the accuracy of MD simulations relies on the underlying molecular force fields (FFs) describing the intra- and intermolecular interactions between atoms or molecules. To develop and validate the FFs used in MD simulations for the prediction of thermophysical properties, reliable experimental data with low uncertainties are necessary.

The present work is part of an ongoing project at the institute of Advanced Optical Technologies – Thermophysical Properties (AOT-TP). With a special focus on ηL and σ, the main aim of this project is to contribute to a reliable database of thermophysical properties of systems consisting of liquids with dissolved gases and to develop a fundamental understanding of structure–property relationships. For this purpose, long linear, branched, and cyclic hydrocarbons and related alcohols as well as their mixtures with and without dissolved gases are systematically investigated up to process-relevant conditions. The reliable data from SLS serve as the basis to validate and develop FFs used in equilibrium MD (EMD) simulations. The EMD simulation results are used to develop structure–property relationships, e.g., the influence of surface enrichment on the interfacial tension [9, 12, 13]. Within the overall project, which aims to get a fundamental understanding on how the molecular characteristics of the solvent and solute influence the thermophysical properties of the mixture, the first publication in the series reported thermophysical properties of 12 different long linear and branched alkanes and alcohols with a carbon number between 12 and 40 up to T = 573 K [14]. Within this context, based on the results from our own experiments, a modification to the optimized potentials for liquid simulations FF for long hydrocarbons (L-OPLS) [15, 16] could be proposed for a better representation of fluid behavior up to 573 K for all 12 pure liquids. This modification incorporates a T-dependency to the description of dispersive and repulsive interactions defined by the Lennard–Jones (LJ) potential and will be identified as L-OPLS-T FF [14]. Then, SLS and EMD simulations were extended to investigations of liquid–liquid and gas–liquid binary systems. ηL and σ of three binary liquid mixtures of n-hexadecane with n-octacosane, 1-hexadecanol, or 2,2,4,4,6,8,8-heptamethylnonane (HMN) over the complete composition range and up to T = 573 K were studied [13]. Furthermore, ηL and σ of binary gas–liquid systems were investigated up to T = 573 K and a solute mole fraction xsolute = 0.20. First, ηL and σ of binary gas–liquid mixtures based on six different solutes in the solvent n-hexadecane were investigated to probe the influence of solute characteristics on mixture properties [9]. The gases H2, helium (He), methane (CH4), H2O, CO, and CO2 were selected due to their difference in size, weight, sphericity, and polarity. Investigations were then further extended to include an analysis of how solvent characteristics influence ηL and σ of binary mixtures consisting of liquids with dissolved gases. Systematic investigations were performed for binary systems based on the same six gases dissolved in solvents, featuring difference in molecular size, weight, branching, hydroxylation, and chain length [11]. These solvents are 1-hexadecanol, 2-butyl-1-octanol, HMN, 2,6,10,15,19,23-hexamethyltetracosane (squalane), n-octacosane, n-triacontane, and the liquid organic hydrogen carrier reference systems diphenylmethane (DPM) and its hydrogenated analogue dicyclohexylmethane (DCM).

In the present work, ηL and σ of ternary gas–liquid mixtures are determined via SLS and EMD simulations with the L-OPLS-T FF [14,15,16] up to T = 573 K and solute mole fractions of up to 0.20. To investigate the influence of constituent species with different molecular characteristics on the thermophysical properties of a ternary system, one solvent with two different dissolved gases or two solvents with one dissolved gas species are investigated. The ternary mixtures of interest for the former category are n-hexadecane with dissolved N2 and CO2 at different mole fractions of \(x_{{{\text{N}}_{{2}} }}\) and \(x_{{{\text{CO}}_{{2}} }}\). For the latter category, mixtures of the linear alkanes n-octacosane and n-hexadecane with dissolved CO2 are studied. To investigate the influence of hydroxylation and hydrogen bonding on the studied properties of ternary systems, mixtures of n-octacosane and 1-hexadecanol with either dissolved CO2 up to \(x_{{{\text{CO}}_{{2}} }}\) = 0.20 or dissolved H2O up to \(x_{{{\text{H}}_{{2}}\text{O}}}\) = 0.09 are investigated with EMD simulations.

In the following, first, the theory, experimental procedure, and data evaluation for the SLS experiments are presented. Afterwards, details necessary for the EMD simulations including the simulation parameters, applied FFs, and data evaluation methods used in this work are outlined. Finally, the results for ηL and σ of the ternary mixtures are presented and the influence of the dissolved gas are discussed by comparing the properties of the ternary mixtures to those of the binary subsystems or the pure solvents investigated in our previous studies.

2 Experimental Section

2.1 Materials and Sample Preparation

The names, sources, purities in mass fraction w or volume fraction φ as specified by suppliers, CAS numbers, and molar masses M of all substances used in this work are provided in Table 1. The solvents n-hexadecane (n-C16H34) and n-octacosane (n-C28H58) were filtered with a polytetrafluoroethylene filter with a pore size of 220 nm to remove particle-like impurities. Filtering was performed under ambient condition for n-hexadecane and approximately 20 K above its melting point for n-octacosane. The filtered alkanes were then degassed at around (313 and 353) K at p below 10 Pa for approximately 3 h and stored under an argon atmosphere for further usage. For each investigation, the cleaned sample cell is first filled with approximately 40 mL of the liquid sample. The cell is then flushed with He to create an inert atmosphere at 0.1 MPa before introducing the investigated gas or gases.

Table 1 Specifications of the chemical supplier, molar mass M, purification method, and purity of the investigated solvents and solutes
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For the first ternary mixture of n-hexadecane + CO2 + N2, this 40 mL sample is entirely n-hexadecane and is filled into the sample cell under ambient conditions. The first investigated gas which is then introduced to the sample cell is CO2 due to its high solubility. The cell temperature is adjusted to 298.15 K and CO2 is introduced up to pressures to estimate (10 or 20)% dissolved CO2 according to the solubility data of Breman et al. [17]. Once the system has equilibrated, the second gas N2 is introduced with an added pressure to approximate (2.5 and 5.0)% dissolved N2 at T = 298.15 K according to the solubility data of Tremper and Prausnitz [18] and Lin et al. [19]. This totals four different ternary mixtures. The fill pressures of each gas are provided in the footnote of Table 2. It is assumed that the solubilities of the two gases in n-hexadecane are independent of each other. For each solute, the mole fraction in the liquid phase xi is estimated according to xi = pi / Hij, where pi is the partial pressure of the solute in the vapor phase and Hij is the Henry’s law constant acquired from refs 17–19. Once filled, the cell is sealed, and the measurement series is performed under isochoric conditions.

Table 2 Liquid dynamic viscosity ηL and interfacial tension σ of the ternary mixtures of n-hexadecane and n-octacosane, with dissolved CO2 or of n-hexadecane with dissolved N2 and CO2 obtained by SLS under saturation conditions as well as the estimated compositions of each component in the liquid phase x1, x2, and x3, the liquid density ρL, vapor density ρV, and the vapor viscosity ηV for each investigated temperature T and pressure p
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The second investigated system, containing n-hexadecane + n-octacosane + CO2, consists of six unique ternary mixtures. The 40 mL which is filled into the sample cell is a binary liquid mixture consisting of n-hexadecane and n-octacosane with mole fraction n-hexadecane of 0.25, 0.50, or 0.75. The binary liquid mixture is prepared with the same procedure as described in ref 13 and the uncertainty of the mixture compositions is estimated to be 0.01. The mixture is heated to approximately 373 K and then filled into the sample cell which is preheated to about T = 373 K. The cell is flushed with Argon before and after filling. The investigated gas, CO2, is then introduced at T = 373.15 K and pressures to approximate (10 or 20)% dissolved CO2 at this T. The solubility of CO2 in the binary alkane mixture is approximated from a mole fraction-weighted Henry constant considering solubility data of the two binary mixtures of n-hexadecane + CO2 and n-octacosane + CO2 from Breman et al. [17] and Huang et al. [20]. These six mixtures are not investigated under isochoric conditions. At each temperature, p may be adjusted to maintain the (10 or 20)% dissolved CO2. Recorded experimental pressures for all mixtures at each temperature are provided in Table 2.

2.2 Surface Light Scattering: Liquid Viscosity and Interfacial Tension

The liquid dynamic viscosity ηL and interfacial tension σ are simultaneously determined in an absolute way at macroscopic thermodynamic equilibrium for all investigated mixtures via the contactless, optical technique surface light scattering (SLS). The reader is referred to the literature [21,22,23,24] as well as to our previous studies for a detailed description of the theory and application of this technique [6, 7, 9, 11, 13, 14, 25]. In the present work, the probed surface fluctuations exhibit an oscillatory behavior, identical in nature to that of our previous studies of binary mixtures containing n-hexadecane with n-octacosane [13] or with a dissolved gas [9].

All ten ternary mixtures in this work were investigated on the experimental SLS setup detailed in our previous publication for the binary mixtures containing He, N2, H2O, or CO2 [9, 11]. The combined expanded uncertainty (k = 2) for T accounting for the difference of the two T probes is estimated to be U(T) = 0.02 K at 298.15 K and U(T) = 0.8 K at 573.15 K and can be extrapolated linearly in between. The overall relative uncertainty for the reported pressure, which includes the pressure stability during a measurement series, is ur(p) = 0.005 (k = 2).

The determination of ηL and σ is possible by solving the complete form of the dispersion equation D(ηL, ηV, ρL, ρV, σ, Γ, ωq, q) for surface waves at the boundary between a liquid phase (subscript L) and a vapor phase (subscript V). Here, ρ and q refer to the density and the modulus of the wave vector, while Γ and ωq describe the damping and the frequency of the surface wave of a specific q value. In this work, four to six measurements are performed at each thermodynamic state, where different q values corresponding to different external angles of incidence relative to the normal of the vapor–liquid interface ΘE between (2.5 and 3.2)° were adjusted. Literature data on ρL could not be found for the present mixtures. However, ρL is available for the binary mixtures n-hexadecane + n-octacosane at xn-hexadecane = (0.25, 0.50, and 0.75) [13] and n-hexadecane + CO2 at \(x_{{{\text{CO}}_{{2}} }}\) = (0.069, 0.59, and 0.73) [26] up to 473 K, and for n-hexadecane + N2 up to 475 K at atmospheric pressure [27]. Here, the estimated amount of dissolved N2 is less than 0.02% across the complete temperature range. Ashcroft et al. [28] measured n-hexadecane + CO2 and n-hexadecane + N2 at T = 298 K and atmospheric p and found CO2 to have a seven times stronger impact on ρL than N2. Their estimated mole fraction of dissolved gas was 1.34 × 10−3 and 1.39 × 10−2 for N2 and CO2 [28]. For the present work, the impact of dissolved N2 on ρL in the n-hexadecane + CO2 + N2 mixtures is neglected. For input into the dispersion equation, ρL is calculated with the estimated \(x_{{{\text{CO}}_{{2}} }}\) provided in Table 2. In the ternary mixture n-hexadecane + n-octacosane + CO2, ρL for the binary liquid mixture n-hexadecane + n-octacosane are used with deviations between n-hexadecane + CO2 and pure n-hexadecane applied to account for the influence of dissolved CO2. In our previous publication, it is mentioned that a dissolved gas should have a relatively small impact on the mixture ρL when the molecular weight between components is large [11]. Therefore, the impact of CO2 in n-octacosane is assumed negligible compared to that observed for n-hexadecane + CO2. Should ρL data for these ternary mixtures become available, the values for ηL and σ reported in this work can be corrected using the expressions provided in the work of Klein et al. [9].

The vapor phase properties ηV and ρV were calculated via the Lucas model [29] and Dalton’s law using the aforementioned solubility data. The composition of the vapor phase was calculated with Raoult’s law, assuming that the partial pressure pi in the vapor phase can be calculated by multiplying the liquid phase mole fraction xi with the vapor pressure of the pure component under saturation conditions. The partial pressures in the vapor phase for the gases are calculated with the REFPROP software version 10.0 [30] from the recorded experimental pressures assuming isochoric conditions starting from the respective fill pressures for CO2 and N2 of (0.773 and 1.86, 0.837 and 4.22, 1.56 and 2.09, and 1.53 and 3.97) MPa. Estimated relative expanded uncertainties (k = 2) for ηV and ρV are assumed to be ur = (0.10 and 0.05) [29].

Reported uncertainties in ηL and σ were calculated via an error propagation scheme [4, 6] considering uncertainties introduced by the measured variables and adopted reference data or predictions. Total expanded relative measurement uncertainties (k = 2), averaged over all investigated systems and thermodynamic states, were determined to be ur(ηL) = 0.021 and ur(σ) = 0.019.

3 EMD Simulations

Using EMD simulations, multiple thermophysical properties are accessible by modeling the dynamics of molecules in a statistical ensemble under equilibrium conditions. Theses simulations are, however, highly sensitive to the inter- and intramolecular interactions described by the underlying force field (FF) using atomistic pair-wise potentials. The dynamics of the atoms and molecules can then be calculated using classical Newtonian equations of motion. For a detailed description of the underlying theory of EMD simulations, the reader is referred to the respective literature [31,32,33].

In this work, EMD simulations were performed using the GROMACS software package, version 5.1.2 [34] at T = (323.15, 348.15, 373.15, 423.15, 473.15 and 573.15) K. The bonds between the lighter hydrogen atoms and the heavier carbon or oxygen atoms in the solvent molecules were constrained using the LINCS algorithm [35] and the simulation timestep was 2 fs. Dispersive and electrostatic interactions are calculated within a cutoff radius of 1.6 nm. Long-range electrostatic interactions beyond the cutoff radius are modeled via the particle-mesh Ewald (PME) algorithm [36]. For simulations in the slightly compressed liquid phase, standard dispersion corrections for energy and pressure are considered beyond the cutoff radius. For simulations in vapor–liquid equilibrium (VLE), long-range LJ interactions are considered using the LJ-PME algorithm [37].

For systems of two solvents mixed with one gas species, simulations were initialized by randomly inserting the solvent molecules into a cubic simulation box with periodic boundary conditions in all directions. The total number of solvent molecules of both species consist of approximately 30,000 atoms. This amount was selected to maintain comparable simulation effort between all simulations. Approximate atomic proximity is then estimated with a steepest-decent energy minimization of the binary mixture. After this, solute molecules are added to match the calculated solute mole fractions in the liquid phase according to the aforementioned solubility data. A further energy minimization for the ternary mixture is performed. Then, the desired T and p are realized in a 1 ns simulation run in the canonical ensemble (NVT) followed by a 10 ns simulation run in the isothermal-isobaric ensemble (NpT). To avoid a possible phase separation, the reference p in the latter equilibration run is set 0.2 MPa above the calculated saturation p. The liquid density ρL is determined from this same simulation run in the NpT ensemble, neglecting the first 0.5 ns to account for the equilibration of T and p. ηL is calculated from a subsequent simulation in the NVT ensemble. Here, simulation runtimes are approximately (40 to 60) ns long. The calculation of ηL from these runs is based on the Green–Kubo method [38, 39] and the reader is referred to our previous publications for further detail [10, 14].

To calculate σ, the cubic simulation boxes are then extended in the z-direction to create two vapor–liquid interfaces and introduce VLE. For typical simulation boxes with a box length of (6.0 to 7.4) nm, previous studies have shown that an extension by a factor of three is sufficient to ensure that the long solvent molecules in the vapor phase only interact with one of the two interfaces at any given time [14]. To calculate the number of solute molecules for the vapor phase, the vapor densities of the solute at the required pi were obtained from the REFPROP database [30]. Afterwards, VLE simulations of about (15 to 20) ns in the NVT ensemble were performed to calculate σ from the diagonal elements of the pressure tensor, i.e. pxx, pyy, and pzz [33], neglecting the first 1 ns from calculations to account for equilibration.

4 Results and Discussion

The results from SLS experiments for ηL and σ are summarized in Table 2 along with their expanded experimental uncertainties, the liquid-phase mole fraction, and the input parameters for solving the dispersion relation at each indicated state point. Headers for the six mixtures consisting of n-hexadecane, n-octacosane, and CO2 indicate the average mixture compositions. Due to the large variation in the liquid-phase compositions for the mixtures consisting of n-hexadecane, N2, and CO2, the desired composition is given in the headers. The data for ηL and σ calculated from EMD simulations, together with their thermodynamic states and statistical uncertainties, are reported in Table 3.

Table 3 Liquid density ρL, liquid dynamic viscosity ηL, and interfacial tension σ with their expanded statistical uncertainties (k = 2) of the ternary mixtures of n-hexadecane and n-octacosane with dissolved CO2, of n-hexadecane with dissolved N2 and CO2, and of 1-hexadecanol and n-octacosane with dissolved CO2 or H2O obtained by EMD at or close to saturation conditions for each investigated temperature T and pressure p
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The data for ηL and σ of the ternary mixtures are shown and discussed in the following sections. First, both SLS and EMD results for the mixtures n-hexadecane + n-octacosane + CO2 and the n-hexadecane + N2 + CO2 are directly compared to check the ability of the EMD simulations to predict the properties of the ternary mixtures. Initially in the form of a parity plot in Fig. 1 and later shown with previously reported correlations for the pure component n-hexadecane or the mixture’s binary subsystem.

Fig. 1
figure 1

Comparison of the liquid dynamic viscosity ηL (left) and interfacial tension σ (right) of the ternary mixtures consisting of n-hexadecane, n-octacosane, and CO2 as well as n-hexadecane, N2, and CO2 studied in this work by EMD and SLS. Expanded uncertainties (k = 2) are shown exemplary for the results for mixtures containing approximately 20% dissolved CO2. The error bar is not shown if it is smaller than the symbol size

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The results in Fig. 1 show that EMD simulations tend to overpredict ηL and σ of the ternary mixtures in comparison to the experimental results. The average absolute relative deviation of ηL and σ between EMD and SLS results is (34 and 12)% for all ten mixtures where a direct comparison could be made. This agrees with all previous investigations within this series, where the L-OPLS-T FF was applied for the prediction of ηL and σ of the pure solvents [14, 40], binary liquid mixtures [13], as well as binary mixtures consisting of a liquid with a dissolved gas [9, 11, 12]. The results also show that by applying the T-dependent correction, the overprediction of both properties is T independent. Therefore, the influence of the dissolved gas can still be investigated using EMD simulations by comparing the results for the ternary mixtures to the results for the pure solvents or binary subsystems which were also investigated by EMD simulations. Congruently, the reported SLS experimental results are compared to our previously reported SLS results for the corresponding pure solvents or binary mixtures [13, 14]. Since no data are available in the literature for ηL and σ of the ternary mixtures investigated in this work, a literature review was not possible.

4.1 Viscosity

The results for ηL of the ternary mixtures consisting of n-octacosane, n-hexadecane, and CO2 from SLS experiments (left) and EMD simulations (right) are shown in the upper part of Fig. 2. In addition, the results for ηL of the binary solvent mixtures, i.e. binary mixtures of n-octacosane and n-hexadecane, investigated by SLS and EMD simulations [13] and represented by corresponding T-dependent correlations are shown in Fig. 2. While the T-dependent correlations for ηL of the binary solvent mixtures investigated by SLS can be found in our previous publication [13] the corresponding correlations for ηL from EMD simulations were obtained by fitting ηL data reported in the same publication with respect to T using a Vogel-type equation according to

$$\eta_{{\text{L,calc}}} \left( T \right)/{\text{mPa}} \cdot {\text{s}} = \exp \left( {\sum\limits_{i = 0}^{n} {\frac{{\eta_{{{\text{L}},i}} }}{{T^{i} }}} } \right).$$
(1)
Fig. 2
figure 2

(Top) Liquid dynamic viscosity ηL of the ternary mixtures of n-hexadecane, n-octacosane, and CO2 by SLS (left) and EMD simulations (right) as a function of T and solute mole fraction (open and closed symbols). For comparison, the correlations for ηL of the binary mixtures of n-hexadecane + n-octacosane from SLS and EMD simulations [13] are shown (lines). (Bottom) Relative deviations of ηL of the ternary mixtures from ηL of the binary mixtures of n-hexadecane and n-octacosane. The dash-dotted lines mark the average expanded uncertainty (k = 2) of the data for the binary mixtures. Error bars are shown only exemplarily for the mixtures containing 20% CO2 for clarity. The mole fractions of CO2 and n-hexadecane given in the legend are approximate values. The real composition at each T can be taken from Tables 2 and 3

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Here, all simulated data had the same statistical weight in the fitting and ηL,i were the mixture-specific fit coefficients. To study the influence of the dissolved CO2 in more detail, the relative deviation between ηL of the ternary mixtures and that of the binary solvent mixtures is given in the lower part of Fig. 2, which will be discussed in the following.

The results in Fig. 2 obtained by SLS show that ηL tends to decrease when CO2 is dissolved in the solvent mixtures. This reduction increases with increasing amount of dissolved CO2. The influence of the dissolved CO2 here is more pronounced at low T with a maximum deviation of 18% at T = 373 K and reduces with increasing T. This T-dependent trend for ηL was already observed for binary mixtures of CO2 dissolved in linear and branched alkanes and alcohols [9, 11]. The results for the three different binary solvent mixtures at a given \(x_{{{\text{CO}}_{{2}} }}\), however, are mostly within combined uncertainty which suggests that the composition of the solvent mixture seems to have a negligible influence on ηL of the ternary mixtures. This is surprising given the fact that dissolving CO2 leads to a relative reduction of 21% when dissolved in n-hexadecane [9] and 10% when dissolved in n-octacosane [11] at T = 373 K. An investigation of the influence of dissolved CO2 on ηL of the binary solvent mixtures by EMD simulations is difficult due to the large relative expanded statistical uncertainties (k = 2) averaging about ur(ηL) = 0.15. The results for ηL of a given ternary mixture always fall within combined uncertainties with that of the respective binary solvent mixture. For the ternary mixture with \(x_{{{\text{CO}}_{{2}} }}\) = 0.20, however, the EMD simulations seem to predict the qualitative influence of the dissolved CO2 correctly with larger relative deviations from the binary solvent mixture of about − 10% at T ≤ 473 K and smaller deviations of up to − 6% at T = 573.15 K. For the ternary mixture with \(x_{{{\text{CO}}_{{2}} }}\) = 0.10, the results for ηL scatter around the values of the binary solvent mixtures with relative deviations between (− 6 and 12)%. This indicates that the L-OPLS-T FF might not be able to accurately predict the amount of free volume in the fluid mixture, which was already discussed in our previous publications [9, 11, 14].

The results for ηL for the ternary mixtures consisting of n-hexadecane, CO2, and N2 from SLS experiments (left) and EMD simulations (right) are shown in the upper part of Fig. 3. Additionally, corresponding T-dependent correlations of the results for ηL of pure n-hexadecane investigated by SLS [25] and EMD simulations [14] are shown in Fig. 3. The correlation for data from SLS can be found in our previous publication [25]. For EMD simulations, the results for ηL of pure n-hexadecane were correlated with respect to T using Eq. 1. In the present work, the influence of the dissolved gases is studied by calculating the relative deviation between ηL of the ternary mixtures and that of pure n-hexadecane. This is shown in the lower part of Fig. 3.

Fig. 3
figure 3

(Top) Liquid dynamic viscosity ηL of the ternary mixtures of n-hexadecane, N2, and CO2 by SLS (left) and EMD simulations (right) as a function of T. Symbol fill (open or closed) and shape (diamond or triangle) refer to mole fraction N2 and CO2. For comparison, the correlations for ηL of pure n-hexadecane from SLS [25] and EMD simulations [14] are shown (solid lines). (Bottom) Relative deviations of ηL of the ternary mixtures from that of pure n-hexadecane. The dash-dotted lines mark the average expanded uncertainty (k = 2) of the data for pure n-hexadecane. Error bars are shown only exemplarily for the mixtures containing \(x_{{{\text{CO}}_{{2}} }}\) = 0.20 for clarity. The mole fractions of the dissolved gas given in the legend are approximate values. The real composition at each T can be taken from Tables 2 and 3

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The results in Fig. 3 show that ηL for the mixtures with \(x_{{{\text{N}}_{{2}} }}\) = (0.025 and 0.050) at a given \(x_{{{\text{CO}}_{{2}} }}\) are always within 3% of each other. The influence of dissolved CO2, on the other hand, is much more pronounced, especially at low T. At T = 298 K, for example, the relative reduction in ηL in comparison to pure n-hexadecane is approximately 20% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.10 and 30% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.20. These values agree well with the finding for the binary mixture of CO2 dissolved in n-hexadecane, where a relative reduction in comparison to pure n-hexadecane was found in ηL to be approximately 19% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.10 and 30% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.20 [9]. Also, the T-dependent trend of the deviations between ηL of the ternary mixtures and that of pure n-hexadecane, which shows a reducing influence of the dissolved gases with increasing T, agrees well with the observations made for the binary subsystem consisting of CO2 dissolved in n-hexadecane [9]. This suggests that the influence of the dissolved N2 is much smaller than the influence of the dissolved CO2 and can be neglected in the case of the studied ternary mixture. This is also in agreement with our previous investigation of the subsystems consisting of N2 dissolved in n-hexadecane, where ηL of the binary mixtures was found to be mostly within combined uncertainties with that of pure n-hexadecane [9].

The results for ηL for the ternary mixtures from EMD simulations, which are shown on the right side of Fig. 3, have a large relative expanded statistical uncertainty (k = 2) of ur(ηL) = 0.15. Similar to the case for the ternary mixture of n-hexadecane, n-octacosane, and CO2, only a qualitative evaluation of the influence of the dissolved gas is possible, since ηL of all ternary mixtures are within combined uncertainties of pure n-hexadecane. For the mixtures with \(x_{{{\text{CO}}_{{2}} }}\) = 0.20, the EMD simulations predict a reduction in ηL with respect to pure n-hexadecane with relative deviations up to − 31% at the lowest investigated T = 323 K. Additionally, the influence of the dissolved gases decreases with increasing T, which is qualitatively in agreement with the SLS results. For \(x_{{{\text{CO}}_{{2}} }}\) = 0.10, the predicted reduction in ηL of about − (10 to15)% at low T and the convergence of ηL with increasing T to that of pure n-hexadecane seem to agree with the SLS results. For a quantitative analysis of the influence of the dissolved gases, however, the statistical uncertainties of the EMD simulations must be reduced, which would require either more independent simulation runs or longer simulation runtimes and, therefore, higher computational costs.

Finally, the ternary mixtures consisting of an equimolar mixture of n-octacosane and 1-hexadecanol and the solutes CO2 or H2O were studied using EMD simulations to investigate whether the presence of a polar solvent, in this case the primary alcohol 1-hexadecanol, has an influence on the thermophysical properties of mixtures. The results for ηL from EMD simulations are shown in the upper part of Fig. 4. Additionally, also ηL of the pure solvents n-octacosane and 1-hexadecanol [14] are shown in the form of a T-dependent correlation according to Eq. 1 as well as ηL for the equimolar mixture based on the EMD results for the pure solvents and the mixing rule \(\ln (\eta_{{\text{L,mix}}} \left( T \right)) = \sum\limits_{i} {w_{i} \cdot \ln (\eta_{{\text{L,}}i} \left( T \right))}\). To study the influence of the solutes in more detail, the relative deviation between ηL of the ternary mixtures and that of the binary solvent mixture according to the aforementioned mixing rule is given in the lower part of Fig. 4.

Fig. 4
figure 4

(Top) Liquid dynamic viscosity ηL of the ternary mixtures consisting of n-octacosane and 1-hexadecanol with dissolved CO2 or H2O investigated by EMD simulations as a function of T and solute mole fraction (open and closed symbols). For comparison, ηL of the pure solvents n-octacosane and 1-hexadecanol investigated by EMD simulations [14] are shown (dashed and dotted lines). Additionally, ηL,mix calculated via a mass-based logarithmic mixing rule described in the work of Lenahan et al. [13] for an equimolar binary mixture of n-octacosane and 1-hexadecanol is shown (solid line). (Bottom) Relative deviations of ηL of the ternary mixtures from ηL,mix. The dash-dotted lines mark the average expanded statistical uncertainty (k = 2) of the pure solvents

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For the solvent H2O, the EMD simulations of the ternary mixtures predict smaller ηL which are within combined uncertainties with ηL,mix and have an average deviation from ηL,mix of − 11%. These results can be compared to the previous work of Klein et al. [11] who reported results for ηL and σ of the binary subsystems consisting of H2O dissolved in the solvents n-octacosane or 1-hexadecanol obtained from SLS and EMD simulations. Both methods showed agreement between ηL of the binary mixture and that of the pure solvent. While the results in the present work agree with these findings for the n-octacosane + H2O mixture, the SLS results of Klein et al., compared to their EMD results, have shown up to 37% larger values for the mixtures consisting of H2O in 1-hexadecanol [11]. This was related to the inability of the applied FFs to accurately describe the relatively strong hydrogen bonding between H2O and 1-hexadecanol. Since the same FFs are used in this work, the same outcome is expected for the present results for ηL of the ternary mixtures. However, no experimental data are available for the ternary mixtures shown here, so a failure of the EMD simulations to predict larger ηL of the ternary mixtures in comparison to ηL,mix cannot be validated. Regardless, since the applied SPC/E FF [41] for the H2O molecules does not consider a T-dependency in its FF parameters, it is strongly suggested that the hydrogen bonds are also not accurately represented in the ternary mixture, Also for the ternary mixtures containing 1-hexadecanol, n-octacosane, and CO2, the results for ηL of the ternary mixture are within combined uncertainty with ηL,mix due to the large statistical uncertainty in determining ηL by EMD simulations. Here, however, we can find the typical T-dependent trend for ηL caused by dissolved CO2. This is a reduction of ηL which is more pronounced at low T. The relative reduction, which is between − (30 and 17)% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.20, is larger than the one seen by SLS and EMD simulations for the binary mixture of CO2 dissolved in n-octacosane and closer to the ones observed by EMD simulations for CO2 dissolved in the primary alcohol 1-dodecanol [12]. This can be explained by stronger interactions between CO2 and the polar alcohol in comparison to the non-polar n-alkane and was also found when comparing binary mixtures of CO2 dissolved in n-dodecane with mixtures of CO2 dissolved in 1-dodecanol by EMD simulations. Here, a stronger reduction of − 44% for the 1-dodecanol-based mixture was observed in comparison to − 22% for the n-dodecane-based mixture at T = 323.15 K and \(x_{{{\text{CO}}_{{2}} }}\) = 0.20 [12].

4.2 Interfacial Tension

The results for σ of the ternary mixtures consisting of n-octacosane, n-hexadecane, and CO2 from SLS experiments (left) and EMD simulations (right) are shown in the upper part of Fig. 5. Additionally, the results for σ of the binary solvent mixtures investigated by SLS and EMD simulations [13] and represented by corresponding T-dependent correlations are shown in Fig. 5. While the T-dependent correlations for σ of the binary solvent mixtures investigated by SLS can be found in our previous publication [13], the corresponding correlations for σ from EMD simulations were obtained by fitting σ reported in the aforementioned publication with respect to T using a second-order polynomial function, which considers all simulated data with the same statistical weight. As was done for ηL, the influence of dissolved CO2 is investigated by calculating the relative deviation between σ of the ternary mixtures and that of the binary solvent mixtures, which is given in the lower part of Fig. 5.

Fig. 5
figure 5

(Top) Interfacial tension σ of the ternary mixtures of n-hexadecane, n-octacosane, and CO2 by SLS (left) and EMD simulations (right) as a function of T and solute mole fraction (open and closed symbols). For comparison, the correlations for σ of the binary mixtures of n-hexadecane and n-octacosane from SLS [ref] and EMD simulations [ref] are shown (lines). (Bottom) Relative deviations of σ of the ternary mixtures from σ of the binary mixtures of n-hexadecane and n-octacosane. The dash-dotted lines mark the average expanded uncertainty (k = 2) of the data for the binary mixtures. Error bars are shown only exemplarily for the mixtures containing 20% CO2 for clarity. The mole fractions of CO2 and n-hexadecane given in the legend are approximate values. The real composition at each T can be taken from Tables 2 and 3

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The results in Fig. 5 obtained by SLS show a reduction in σ when CO2 is dissolved in the solvent mixtures in all cases. This reduction increases with increasing \(x_{{{\text{CO}}_{{2}} }}\). The influence of the dissolved CO2 is more pronounced at large T with a maximum deviation of 22% at T = 573 K and \(x_{{{\text{CO}}_{{2}} }}\) = 0.20. Similar to ηL, the composition of the binary solvent mixture seems to have a negligible influence on σ of the ternary mixture, which is expressed by a very similar relative reduction in σ for the three different solvent compositions at a given T and \(x_{{{\text{CO}}_{{2}} }}\). The T-dependent trend of the reduction in σ due to the dissolved CO2 agrees well with the one observed for the binary subsystems of CO2 dissolved in n-hexadecane [9] or n-octacosane [11].

In contrast to ηL, where the influence of the dissolved CO2 could only be investigated qualitatively due to the large uncertainties, the smaller statistical uncertainty related to the determination of σ allows a better investigation of the influence of the dissolved gas by EMD simulations. By comparing the relative deviation between σ of the ternary mixture and the binary solvent mixtures from SLS and EMD simulations, we can observe that EMD simulations are able to predict the reduction in σ by dissolving CO2 correctly over the entire T range. The simulations can predict the T-dependent trend of this reduction, its absolute value, as well as the influence of \(x_{{{\text{CO}}_{{2}} }}\). This shows that EMD simulations are still able to correctly predict the relative influence of the dissolved gas on σ, even though they overpredict the absolute values for σ of the pure solvents and the mixtures in comparison to the experimental results from SLS [9, 11, 13, 14].

The results for σ for the ternary mixtures consisting of n-hexadecane, CO2, and N2 from SLS experiments (left) and EMD simulations (right) are shown in the upper part of Fig. 6. Additionally, the results for σ of pure n-hexadecane investigated by SLS [25] and EMD simulations [14] and represented by corresponding T-dependent correlations are shown here. The T-dependent correlation for σ of pure n-hexadecane investigated by SLS can be found in our previous publication [25]. The results for σ of pure n-hexadecane investigated by EMD simulations [14] were correlated with respect to T using a second-order polynomial function. The influence of the dissolved gases was studied by calculating the relative deviation between σ of the ternary mixtures and that of pure n-hexadecane. This is given in the lower part of Fig. 6.

Fig. 6
figure 6

(Top) Interfacial tension σ of the ternary mixtures of n-hexadecane, N2, and CO2 by SLS (left) and EMD simulations (right) as a function of T. Symbol fill (open or closed) and shape (diamond or triangle) refer to mole fraction N2 and CO2. For comparison, the correlations for σ of pure n-hexadecane from SLS [25] and EMD simulations [14] are shown (solid lines). (Bottom) Relative deviations between σ of the ternary mixtures from that of pure n-hexadecane. The dash-dotted lines mark the average expanded uncertainty (k = 2) of the data for pure n-hexadecane. Error bars are shown only exemplarily for the mixtures containing \(x_{{{\text{CO}}_{{2}} }}\) = 0.20 for clarity. The mole fractions of the dissolved gas given in the legend are approximate values. The real composition at each T can be taken from Tables 2 and 3

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The results in Fig. 6 show a reduction in σ caused by the dissolved gas mixture over the entire T range for all four investigated ternary mixtures. Deviations from σ of pure n-hexadecane are in the range of – (9.4 to 27)%. An increase of \(x_{{{\text{CO}}_{{2}} }}\) and \(x_{{{\text{N}}_{{2}} }}\) leads to a decrease in σ and, therefore, an increased reduction. When compared to the binary sub-systems consisting of n-hexadecane with dissolved CO2 or N2 [9], it seems that the influence of the different solutes add up in the ternary mixture. For example, the relative reduction in σ from that of pure n-hexadecane for the ternary mixture with \(x_{{{\text{CO}}_{{2}} }}\) = 0.10 and \(x_{{{\text{N}}_{{2}} }}\) = 0.05 is − 20%. For the binary sub-systems this value is − (10 and 11)% for \(x_{{{\text{CO}}_{{2}} }}\) = 0.10 and \(x_{{{\text{N}}_{{2}} }}\) = 0.05 [9]. The T-dependent trend of the reduction decreases with increasing T up to T = 473 K. At larger T, the reduction increases, resembling that observed for the binary sub-systems. These observations suggest that the influence of multiple different solutes on σ can be approximated using the assumption that the influence of the solute can be transferred from binary mixtures to multicomponent mixtures provided no strong interactions between the solute molecules are possible. Whether such an assumption also holds true for systems where strong interactions between the components are possible, e.g. polar solutes in an alcohol, cannot be answered here and must be further investigated.

The results for σ for the ternary mixtures from EMD simulations, which are shown on the right side of Fig. 6, predict a similar influence of the solutes. Due to the larger uncertainties related to σ of the ternary mixtures from EMD simulations in comparison to those from SLS, σ of the four different ternary systems are always within uncertainties. Nevertheless, the relative reduction in σ in comparison to the pure n-hexadecane predicted by EMD simulations are of the same magnitude and show a similar T-dependent trend to those from SLS. This suggests that, also for such ternary mixtures, EMD simulations are able to accurately predict the influence of multiple dissolved gases on σ.

Finally, the results for σ of the ternary mixtures consisting of an equimolar mixture of n-octacosane and 1-hexadecanol and the solutes CO2 or H2O, which were studied only by EMD simulations, are shown in the upper part of Fig. 7. Also here, σ of the pure solvents n-octacosane and 1-hexadecanol [14] as well as σ for the equimolar mixture based on the EMD results for the pure solvents [14] and the mixing rule \(\sigma_{{{\text{mix}}}} \left( T \right) = \sum\limits_{i} {w_{i} \cdot \sigma_{i} \left( T \right)}\) are shown in the form of a T-dependent correlation using a second-order polynomial function. To study the influence of the solutes in more detail, the relative deviation between σ of the ternary mixtures and that of the binary solvent mixture according to the aforementioned mixing rule is given in the lower part of Fig. 7.

Fig. 7
figure 7

(Top) Interfacial tension σ of the ternary mixtures consisting of n-octacosane and 1-hexadecanol with dissolved CO2 (blue) or H2O (orange) investigated by EMD simulations as a function of T and solute mole fraction (open and closed symbols). For comparison, σ of the pure solvents n-octacosane and 1-hexadecanol investigated by EMD simulations [14] are shown (dashed and dotted lines). Additionally, σmix calculated via a mass-based linear mixing rule described by Lenahan et al. [13] for an equimolar mixture of n-octacosane and 1-hexadecanol is shown (solid line). (Bottom) Relative deviations between σ of the ternary mixtures and σmix. The dash-dotted lines mark the average expanded uncertainty (k = 2) of the pure solvents. Error bars are not shown when symbol size exceeds bar length (Color figure online)

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For the solvent H2O, the EMD simulations predict smaller values within combined uncertainties for σ of the ternary mixtures in comparison to σmix at T ≤ 423.15 K. This reduction in σ increases with increasing T with a maximum deviation of − 10% for \(x_{{{\text{H}}_{{2}}\text{O}}}\) = 0.090 and T = 573.15 K. The influence of H2O is very small, so σ of the ternary mixtures with \(x_{{{\text{H}}_{{2}}\text{O}}}\) = (0.045 and 0.090) are within combined uncertainties over the entire T range. Since SLS experiments have shown σ of the binary mixture consisting of 1-hexadecanol and H2O to be larger than that of the pure solvent [11], it is suggested that EMD simulations fail to predict that also σ of the ternary mixture is larger than that of the binary solvent mixture. The reason for this failure is most probably due to a failure of the FF combination to accurately predict the hydrogen bonds at large T. This was already discussed in connection with ηL in this work as well as in our previous publications [11, 12].

Also for the ternary mixtures containing 1-hexadecanol, n-octacosane, and CO2, the results for σ predict a reduction in σ by dissolving CO2 with respect to the binary solvent mixture with relative deviations of – (7.1 and 16)% for \(x_{{{\text{CO}}_{{2}} }}\) = (0.10 and 0.20) at T = 373.15 K. With increasing T, the relative deviations increase and are – (16 and 26)% for \(x_{{{\text{CO}}_{{2}} }}\) = (0.10 and 0.20) at T = 573.15 K. The relative deviations and the T-dependent trend are in agreement with the ones found for the binary mixture of n-octacosane and CO2 [11]. In order to estimate the influence of the polar alcohol, the results for the binary mixture of 1-dodecanol and CO2 from our previous publication is used [12]. Here, results are compared to those of a mixture based on the non-polar n-octacosane with dissolved CO2. A stronger reduction of σ by approximately 5% for a given \(x_{{{\text{CO}}_{{2}} }}\) over the entire T-range was found for the alcohol-based mixture. Given the statistical uncertainty of σ, it seems that the presence of the polar 1-hexadecanol has only a negligible impact. Therefore, the influence of CO2 in the ternary mixture can be estimated based on the binary sub-system consisting of n-octacosane and CO2. This is not surprising, since such long primary alcohols tend to orient themselves with their alkane tail towards the vapor–liquid interface [10, 13, 42].

5 Conclusions

SLS and EMD simulations were used to characterize ternary gas–liquid mixtures through the determination of ηL and σ in the T range between (298 and 573) K and solute mole fractions up to about 0.20. The mixtures consisting of n-hexadecane, n-octacosane, and CO2 as well as n-hexadecane, N2, and CO2 were investigated with both SLS and EMD simulations, while the ternary mixtures consisting of 1-hexadecanol, n-octacosane, and CO2 as well as 1-hexadecanol, n-octacosane, and H2O are investigated by EMD simulations only. The average relative expanded uncertainties for SLS experiments and EMD simulations are ur(ηL) = (0.021 and 0.15) and ur(σ) = (0.019 and 0.061), considering all investigated systems. By comparing the thermophysical properties of the ternary mixtures with the ones of the pure solvents as well as binary subsystems, i.e. binary liquid mixture or binary mixtures consisting of a liquid with a dissolved gas, the influence of the dissolved gas on ηL and σ could be analyzed. Due to the large uncertainties for ηL from EMD simulations, the results for ternary mixtures are often within combined uncertainties with the ones for the pure solvents or binary subsystems, which allowed only a qualitative comparison. The results for the ternary mixture consisting of n-hexadecane, n-octacosane, and CO2, which include a variation of the composition of the binary subsystem, i.e. the solvent mixture, have shown the reduction of both properties due to dissolving CO2 to be independent of the solvent composition. This suggests that the influence of the dissolved gases can be estimated using the results of the binary subsystems, i.e. CO2 dissolved in n-hexadecane or n-octacosane, and treating the binary solvent subsystem as a pseudo-pure liquid. Also for the ternary mixture consisting of n-hexadecane, N2, and CO2, a reduction in both properties due to the dissolved gases was determined using both SLS and EMD simulations. Here, this reduction can be estimated by adding up the reduction determined for the binary subsystem, i.e. n-hexadecane with dissolved N2 or CO2. This means that for such ternary mixtures the influence of the dissolved gases can also be estimated only with the help of the binary subsystems.