Fluid Theory Research Articles

Experimental measurement and modeling of the solubility of fluorinated compounds derived from dichlone in supercritical carbon dioxide

Dichlone, also known as 2,3-dichloronaphthalene-1,4-dione, is a solid organic substance employed in the field of agriculture for its fungicidal properties and as a retardant for vegetable decomposition. The bioactive properties of dichlone can be enhanced by modifying its structure, specifically through the synthesis of new derivatives achieved by replacing the functional groups within its molecular structure. Two new solid dichlone derivatives were synthesized in this work, namely 2-chloro-3-((4-fluorobenzyl)amino)naphthalene-1,4-dione (dCl-2B-F) and 2-chloro-3-((4-fluorophenethyl)amino)naphthalene-1,4-dione (dCl-3 P-F) and measured their solubility in supercritical carbon dioxide at (313, 323, and 333) K and pressures between (9. to 32) MPa. The results indicated that solubility ranged between 30.5 and 47.9 µmol of solute/mol of CO2 for dCl-2B-F, and from 2.2 to 243.5 µmol of solute/mol of CO2 for dCl-3 P-F. The solubility data of dichlone and its synthesized derivatives (dCl-2B-F, dCl-3 P-F, 2-chloro-3-((4-chlorobenzyl)amino)naphthalene-1,4-dione (dCl-2B-Cl), 2-chloro-3-((4-chlorophenethyl)amino)naphthalene-1,4-dione (dCl-3 P-Cl), 2-(benzylamino)-3-chloronaphthalene-1,4-dione (dCl-2B) and 2-chloro-3-(phenethylamino)naphthalene-1,4-dione (dCl-3 P)) was compared using the density-based correlation of Chrastil and the Statistical Associating Fluid Theory of Variable Range Mie-potential (SAFT-VR Mie) equation of state (EoS), to better comprehend the effects of the structural differences on the solubility. As a result, for the Chrastil model, a root mean square deviation (rmsd) of 3% was obtained for dCl-2B-F and 16% for dCl-3 P-F, whereas for the SAFT-VR Mie equation, it averaged 24% for dCl-2B-F and 28% for dCl-3 P-F. It was found that the solubility of the homologous compounds, differing only in one methylene group, increased with solute size (-2B derivatives were less soluble in CO2 than the −3 P ones), contrary to the expected trend, which could be attributed to the increased probability of ring-to-ring interactions as the chain length connecting the rings decreases. This demonstrates that geometric factors, along with the pressure and temperature, affect the behavior of the solubility and these should be accurately represented in the predictive models.

Hydrophobic interactions described using hetero-segmented PC-SAFT: 1. Alcohol/water mixtures

The hydrophobic effect plays a central role in aqueous systems containing molecules with hydrophobic moieties. Despite the relevance of this effect for chemical processes or pharmaceutical applications, modeling remains challenging even using molecular-based models such as different versions of the Statistical Associating Fluid Theory. This work shows the feasibility of hetero-segmented PC-SAFT for this purpose. This model was used as a group contribution method to build molecules from different functional groups, namely from CH2, CH3, and CH2OH groups. Saturated-liquid densities and vapor pressures of n-alkanes and n-alcohols as well as binary mixtures of these molecules were accurately described by this approach. The description of aqueous mixtures containing n-alkanes and n-alcohols was improved compared to state-of-the-art modeling by explicitly accounting for intermolecular interactions resulting in the hydrophobic effect. The so-obtained framework describes the two liquid−liquid equilibrium phases in mixtures of n-alkanes or n-alcohols with water equally well using a single set of transferable parameters. The model was also validated for vapor–liquid equilibria, solid–liquid equilibria, infinite-dilution properties, as well as octanol/water partition coefficients and showed quantitative agreement with experimental data.

Bottom Multi-Parameter Bayesian Inversion Based on an Acoustic Backscattering Model

The geoacoustic and physical properties of the bottom, as well as spatial distribution, are crucial factors in analyzing the underwater acoustic field structure and establishing a geoacoustic model. Acoustic inversion has been widely used as an economical and effective method to obtain multi-parameters of the bottom. Compared with traditional inversion methods based on acoustic propagation models, acoustic backscattering models are more suitable for multi-parameter inversion, because they contain more bottom information. In this study, a Bayesian inversion method based on an acoustic backscattering model is proposed to obtain bottom multi-parameters, including geoacoustic parameters (the sound speed and loss parameter), partial physical parameters of the sediment, and statistical parameters of the seafloor roughness and sediment heterogeneity. The bottom was viewed as a kind of fluid medium. A high-frequency backscattering model based on fluid theory was adopted as the forward model to fit the scattering strength between the model prediction and the measured data. The Bayesian inversion method was used to obtain the posterior probability density (PPD) of the inversion parameters. Parameter estimation, uncertainty, and correlation were acquired by calculating the maximum a posterior (MAP), the mean values, the one-dimensional marginal distributions of the PPD, and the covariance matrix. Finally, the high-frequency bottom backscattering strength from the Quinault Range site was employed for inversion tests. The estimated values and uncertainties of various bottom parameters are presented and compared with the directly measured bottom parameters. The comparison results demonstrate that the method proposed herein can be used to estimate the sediment/water sound speed ratio, the sediment/water density ratio, and the spectral exponent of the roughness spectrum effectively and reliably.

Predicting Solvation Free Energies from the Minnesota Solvation Database Using Classical Density Functional Theory Based on the PC-SAFT Equation of State.

We critically assess the capabilities of classical density functional theory (DFT) based on the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state to predict the solvation free energies of small molecules in various hydrocarbon solvents. We compare DFT results with experimental data from the Minnesota solvation database and utilize statistical methods to analyze the accuracy of our approach, as well as its weaknesses. The mean absolute error of the solvation free energies is 3.7 kJ mol-1 for n-alkane solvents, ranging from pentane to hexadecane, with 473 solute-solvent systems. For solvents consisting of cyclic hydrocarbons (cyclohexane, benzene, toluene, and ethylbenzene) with 245 solute-solvent systems, we report a slightly larger mean absolute error of 4.2 kJ mol-1. We identify three possible sources of errors: (i) the neglect of solute-solvent and solvent-solvent Coulomb interactions, which limits the applicability of PC-SAFT DFT to nonpolar and weakly polar molecules; (ii) the solute's Lennard-Jones parameters supplied by the general AMBER force field, which are not parametrized toward solvation free energies; and (iii) the application of the Lorentz-Berthelot combining rules to the dispersive interactions between a segment of the PC-SAFT solvent and a Lennard-Jones interaction site of the solute. The approach is more accurate than standard implementations of phenomenological models in common chemistry software packages, which exhibit mean absolute errors larger than 9.12 kJ mol-1, even though newer phenomenological models achieve a mean absolute error of about 2 kJ mol-1. PC-SAFT DFT is more computationally efficient than state of the art explicit molecular simulations in combination with free energy perturbation methods. It is predictive with respect to solvation free energies, i.e., the input for the model is the (element-specific) molecular force field, the solute configuration from molecular dynamics simulations, and the (substance-specific) PC-SAFT parameters. The PC-SAFT parametrization uses pure-component data and does not require experimental solvation free energies. The PC-SAFT equation of state, without applying a DFT formalism, can also be used to calculate solvation free energies, provided that the PC-SAFT parameters for the solute are available. A large number of substances was recently parametrized by members of our group (Esper, T.; Bauer, G.; Rehner, P.; Gross, J. Ind. Eng. Chem. Res. 2023, 62), which enables a comparison to the DFT approach for 103 substances. Accurate results are obtained from the PC-SAFT equation of state with an MAE below 2.51 kJ mol-1. The DFT approach does not require PC-SAFT parameters for the solute and can be applied to all solutes that can be represented by the molecular force field.

Density and viscosity measurement of R513A and a modified residual entropy scaling model for predicting the viscosity of HFC/HFO refrigerants

Liquid density and viscosity of alternative refrigerants R513A were measured with a vibrating-wire viscosimeter in the temperature of 253 K to 363 K when pressure changes from 1 MPa to 12 MPa. The combined expended uncertainties with a confidence level of 0.95 (k = 2) of density and viscosity are 0.2 % and 2 %, respectively. In addition, a viscosity model of HFCs (fluorocarbons)/, HFOs (hydro-fluoroolefins) and their mixtures was provided by coupling the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) equation of state and a modified residual entropy scaling (RES) method. Obvious quasi-linear relation between the proposed entropy variable and the modified residual viscosity has been confirmed. The overall average absolute percentage deviations between the calculated results with the proposed RES model and the literature experimental data of pure and mixture refrigerants are 2.63 % and 2.05, respectively.

Viscosities of inhomogeneous systems from generalized entropy scaling

This study extends entropy scaling to inhomogeneous fluids by using the classical density functional theory together with a new viscosity reference that takes into account the influence of solid–fluid interactions on the fluid viscosity. The density functional theory uses a Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory; the local residual entropy per particle is determined from the temperature derivative of the Helmholtz energy functional in combination with an appropriate weighted density profile. The weighted density calculation requires a single transferable parameter, which is adjusted to a reference molecular dynamics simulation. In particular, local viscosity values for fluids under nanoconfinement near solid–fluid interfaces are predicted using the same entropy scaling parameters as for homogeneous fluids. We validate the model by comparing viscosity and velocity profiles with results from non-equilibrium molecular dynamics simulations of a Couette flow in a slit pore. Good agreement is found between the entropy scaling model and the non-equilibrium molecular dynamics results for both the viscosity and velocity profiles of the Lennard–Jones truncated and shifted fluid. The proposed model extrapolates well to systems with different temperatures, fluid densities, and shear forces as well as to systems with different wetting behaviors. These results demonstrate that entropy scaling can be generalized to inhomogeneous fluids using an appropriate combination of residual entropy profile and viscosity reference.

Computer simulations of soft matter. On 60-th anniversary of Jaroslav Ilnytskyi

This special issue Condensed Matter Physics, entitled "Computer Simulations of Soft Matter" is dedicated to Jaroslav Ilnytskyi on the occasion of his 60-th birthday. Prof. J. Ilnytskyi is one of scientific leaders of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine (NASU), who iswell known for his contributions to the theory and computer simulation studies of polymer fluids, liquid crystals, macromolecular and colloid solutions. Versatility of Jaroslav works is due to his excellent background commonly resulting in original ideas, his feeling and ability to choose and apply appropriate methods for the solution of various problems. Finally, Jaroslav works are distinguished by the profound analyses of either theoretical or simulation results with respect to experimental observations frequently complemented by the discussion of possible applications.

Predicting the density and viscosity of deep eutectic solvents at atmospheric and elevated pressures

Deep eutectic solvents (DESs) are promising and sustainable substitutes for organic solvents in various applications. The knowledge of their density and viscosity at different temperatures and pressures is crucial for the corresponding process design. However, most literature primarily focuses on predicting these properties at atmospheric pressure. In this study, we employed the perturbed chain polar statistical associating fluid theory (PCP-SAFT) model and entropy scaling method to predict DESs’ density and viscosity at atmospheric and elevated pressures. 2831 density data points of 38 DESs and 1018 viscosity data points of 31 DESs at different temperatures and pressures were used to evaluate the accuracy of the applied methods. The pure component PCP-SAFT parameters of choline chloride (ChCl) were regressed using density data of ChCl-based DES at various temperatures and pressures. It was found that the PCP-SAFT model can provide satisfactory density predictions for DESs at atmospheric pressure, even without using the system-specific binary interaction parameter (kij=0). However, accurate viscosity prediction at atmospheric pressure for DESs necessitated the inclusion of the binary interaction parameter (kij=aijT+bij) in the PCP-SAFT model. The system-specific coefficients aij andbij, derived from experimental data at atmospheric pressure, were used to predict the density or viscosity of DES at elevated pressures. Accurate predictions for density were observed across the entire pressure range, whereas effective viscosity predictions were confined to a specific pressure range.

Three-Dimensional Chiral Morphogenesis of Active Fluids.

Chirality is an essential nature of biological systems. However, it remains obscure how the handedness at the microscale is translated into chiral morphogenesis at the tissue level. Here, we investigate three-dimensional (3D) tissue morphogenesis using an active fluid theory invoking chirality. We show that the coordination of achiral and chiral stresses, arising from microscopic interactions and energy input of individual cells, can engender the self-organization of 3D papillary and helical structures. The achiral active stress drives the nucleation of asterlike topological defects, which initiate 3D out-of-plane budding, followed by rodlike elongation. The chiral active stress excites vortexlike topological defects, which favor the tip spheroidization and twisting of the elongated rod. These results unravel the chiral morphogenesis observed in our experiments of 3D organoids generated by human embryonic stem cells.

Viscosity measurement and correlation for dense fluid mixture of carbon dioxide and ethanol at 313–343 K and 15–30 MPa

The viscosity ηm of dense fluid mixtures of carbon dioxide and ethanol was determined from the pressure drop between the inlet and outlet of a packed bed using the Ergun equation at temperatures from 313.2 K to 343.2 K and pressures from 15 MPa to 30 MPa over the entire range of ethanol composition. The ηm values for the fluid mixtures decreased with increasing temperature and increased with increasing pressure under isocratic conditions. In addition, the values measured ηm with Ergun eq. were correlated with the equation combining the Eyring theory and the perturbed chain statistical associating fluid theory equation of state (PC-SAFT EoS) and with that based on the rough hard-sphere (RHS) model proposed by Dymond and Assael. The accuracies of the Eyring + PC-SAFT EoS and the RHS model were 8.7% and 12%, respectively, in terms of average relative deviation for 80 conditions over the entire range of ethanol composition.

Dissolution Mechanisms of Amorphous Solid Dispersions: Application of Ternary Phase Diagrams To Explain Release Behavior.

The use of amorphous solid dispersions (ASDs) in commercial drug products has increased in recent years due to the large number of poorly soluble drugs in the pharmaceutical pipeline. However, the release behavior of ASDs is complex and remains not well understood. Often, the drug release from ASDs is rapid and complete at lower drug loadings (DLs) but becomes slow and incomplete at higher DLs. The DL where release becomes hindered is termed the limit of congruency (LoC). Currently, there are no approaches to predict the LoC. However, recent findings show that one potential cause leading to the LoC is a change in phase morphology after water-induced phase separation at the ASD/solution interface. In this study, the phase behavior of ASDs in contact with aqueous solutions was described thermodynamically by constructing experimental and computational ternary phase diagrams, and these were used to predict morphology changes and ultimately the LoC. Experimental ternary phase diagrams were obtained by equilibrating ASD/water mixtures over time. Computational ternary phase diagrams were obtained by Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT). The morphology of the hydrophobic phase was studied with fluorescence confocal microscopy. It was demonstrated that critical point (plait point) composition approximately corresponded to the ASD DL, where the hydrophobic phase, formed during phase separation, became interconnected and hindered ASD release. This work provides mechanistic insights into the ASD release behavior and highlights the potential of in silico ASD design using phase diagrams.

Thermodynamic analysis of a dual-pressure evaporation high-temperature heat pump with low GWP zeotropic mixtures for steam generation

High-temperature heat pumps (HTHPs) have gained attention due to their economic and environmental benefits in utilizing industrial waste heat for steam generation. This study focuses on improving the HTHPs’ performance by employing two-stage separation and dual-pressure evaporation technology in the auto-cascade heat pump cycle. Additionally, binary zeotropic refrigerant mixtures with low-GWP are selected using the polar perturbed-chain statistical associated fluid theory (PC-SAFT) equation of state. The findings indicate that the R1234ze(E)&R1233zd(E) refrigerant mixture outperforms other potential alternatives, exhibiting a thermodynamic effectiveness 0.85%–1.86% higher than the benchmark mixture, R134a&R245fa. The improved cycle demonstrates significant enhancements, achieving a 45.17% increase in heat source utilization efficiency and a 24.48% improvement in COP compared to the basic auto-cascade cycle. Effective temperature matching between refrigerant mixtures and heat sources/sinks is guaranteed in the improved cycle. Moreover, a parameter analysis reveals that increasing the subcooling degree of the cascaded heat exchanger and the separation dryness fraction at separator 2 enables improvements in both COP and heat source utilization efficiency. These findings provide valuable insights into the design of steam generation heat pumps and the selection of refrigerant mixtures.

Thermophysical properties of ethyl levulinate and n-alkanol systems

Some thermophysical properties such as densities, speeds of sound, refractive indices, and heat flows of the binary systems ethyl levulinate and n-alkanols (methanol, ethanol, 1-propanol, and 1-butanol) are reported in the temperature range T = (283.15–313.15) K and at p = 100 kPa. From these experimental data, the excess properties were calculated and correlated with the composition using a modified Redlich–Kister equation. The excess molar volumes were negative for short-chain alcohols and positive for 1-propanol and 1-butanol. The excess refractive indices exhibited the opposite behaviour. The excess isentropic compressibilities were negative and the excess molar enthalpies, were positive. The perturbed chain statistical associating fluid theory equation of state (PC-SAFT EoS) was successfully applied to correlate the densities of the mixtures, and good qualitative results in the computation of the excess molar enthalpy were obtained. Furthermore, Schaaff’s collision factor theory (SCFT) and the Laplace mixing rule were coupled with PC-SAFT to predict the speed of sound and the refractive index, respectively.

Learning fast, accurate, and stable closures of a kinetic theory of an active fluid

Important classes of active matter systems can be modeled using kinetic theories. However, kinetic theories can be high dimensional and challenging to simulate. Reduced-order representations based on tracking only low-order moments of the kinetic model serve as an efficient alternative, but typically require closure assumptions to model unrepresented higher-order moments. In this study, we present a learning framework based on neural networks that exploits rotational symmetries in the closure terms to learn accurate closure models directly from kinetic simulations. The data-driven closures demonstrate excellent a-priori predictions comparable to the state-of-the-art Bingham closure. We provide a systematic comparison between different neural network architectures and demonstrate that nonlocal effects can be safely ignored to model the closure terms. We develop an active learning strategy that enables accurate prediction of the closure terms across the entire parameter space using a single neural network without the need for retraining. We also propose a data-efficient training procedure based on time-stepping constraints and a differentiable pseudo-spectral solver, which enables the learning of stable closures suitable for a-posteriori inference. The coarse-grained simulations equipped with data-driven closure models faithfully reproduce the mean velocity statistics, scalar order parameters, and velocity power spectra observed in simulations of the kinetic theory. Our differentiable framework also facilitates the estimation of parameters in coarse-grained descriptions conditioned on data.

Solubility of digitoxin in supercritical CO2: Experimental study and modeling

In this communication, the solubility of digitoxin drug in supercritical CO2 was studied at different operating conditions (311 < T (K) < 343, 120 < P (bar) < 300). The results revealed digitoxin drug solubility (in mole fraction) was between 0.095 × 10−5 to 1.12 × 10−5. In the case of thermodynamic solubility modeling, cubic and non-cubic equation of states i.e. SAFT (statistical associating fluid theory), SRK (Soave-Redlich-Kwong) and sPC-SAFT (simplified perturbed chain SAFT) EoSs and six density-based correlations (Chrastil, Kumar-Johnston (KJ), Mendez-Santiago-Teja (MST), Garlapati and Madras (GM), Bartle et al. and Sung-Shim models) were considered. All used equations indicated reasonable behavior with appropriate accuracy for the solubility of the digitoxin drug. Meanwhile, sPC-SAFT EoS and Kumar-Johnston correlation with AARD% set to 8.96 % and 6.25 %, respectively exhibited greater accuracy in fitting the solubility data. Moreover, total, solvation and vaporization enthalpies of the digitoxin/supercritical carbon dioxide binary mixture were calculated based on KJ, Chrastil and Bartle et al. models.

CO2 solubility and absorption rate analysis of a mixed poly(oxymethylene) dimethyl ethers (PODE3–5) solution

Efficient and economical carbon capture is critical to counteract the diverse problems of global warming. In this work, the solubility of carbon dioxide (CO2) in different poly(oxymethylene) dimethyl ethers (PODE) solutions was experimentally investigated at temperatures from 278.15 to 308.15 K and pressures up to 1.2 MPa. The Henry's law constants of CO2 in the PODE system were determined and compared with other types of physical absorbents, which revealed PODE3–5 has great CO2 absorption ability in terms of molar and mass fraction. Meanwhile, the initial absorption rate of the physical absorption process of PODE3–5 was calculated and compared with commercially available absorbents using a gas absorption rate relationship. It was shown that the absorption rate of PODE3–5 in the CO2 absorption process is faster than that of other commercial absorbents. The CO2 solubility data of PODE3–5 were also correlated using the Jou-Mather model and the perturbed-chain statistical association fluid theory (PC-SAFT) equation of state model, and the recycling performance of the PODE3–5 absorbent was tested.

Cosmology of Tsallis and Kaniadakis holographic dark energy in Saez–Ballester theory and consideration of viscous van der Waals fluid

In the framework of Saez–Ballester (SB) theory, the current study proposes a reconstruction scheme for Tsallis and Kaniadakis holographic dark energy, particular cases of a more generalised Nojiri–Odintsov holographic dark energy (Nojiri and Odintsov, 2006; The European Physical Journal C, 2017, 77, pp.1-8.), with Hubble horizon and future event horizons serving as the universe’s respective trapping horizons. The Hubble parameter was rebuilt by placing a constraint on the metric potential inside the conservation equation. As a result, the holographic dark energy of Tsallis and Kaniadakis are recreated, and the cosmology and thermodynamics are examined. Lastly, an interacting van der Waals (VDW) coupled fluid model was examined in the context of bulk viscosity, and the behaviour of the equation of state (EoS) parameter was examined.

Hydrodynamics of quantum vortices on a closed surface

We develop a neutral vortex fluid theory on closed surfaces with zero genus. The theory describes collective dynamics of many well-separated quantum vortices in a superfluid confined on a closed surface. Comparing to the case on a plane, the covariant vortex fluid equation on a curved surface contains an additional term proportional to Gaussian curvature multiplying the circulation quantum. This term describes the coupling between topological defects and curvature in the macroscopic level. For a sphere, the simplest nontrivial stationary vortex flow is obtained analytically and this flow is analogous to the celebrated zonal Rossby-Haurwitz wave in classical fluids on a nonrotating sphere. For this flow the difference between the coarse-grained vortex velocity field and the fluid velocity field generated by vortices is solely driven by curvature and vanishes in the corresponding vortex flow on a plane when the radius of the sphere goes to infinity. Published by the American Physical Society 2024

Reconstruction of aether scalar tensor theory for variouscosmological scenarios* *Supported by the National Natural Science Foundation of China (11975072, 11875102, 11835009), National SKA Program of China (2022SKA0110200, 2022SKA0110203), Shaanxi Provincial Education Department (21JK0556), and Shaanxi Provincial Department of Science and Technology (2023-JC-QN-0077)

In this paper, we present several explicit reconstructions for the aether scalar tensor (AeST) theory derived from the background of the Friedmann-Lemaître-Robertson-Walker cosmological evolution. It is shown that the Einstein-Hilbert Lagrangian with a positive cosmological constant is the only Lagrangian capable of accurately replicating the exact expansion history of the Λ cold dark matter (ΛCDM) universe filled solely with dust-like matter. However, the ΛCDM-era can be produced within the framework of the AeST theory for some other fluids, including a perfect fluid with , multifluids, and nonisentropic perfect fluids. Moreover, we demonstrate that the ΛCDM-era can be replicated with no real matter field for the AeST theory. The cosmic evolution resulting from both the power-law and de-Sitter solutions can also be obtained.

Application of PC‐SAFT EoS and SCFT for the modeling of thermophysical properties comprising deep eutectic solvent and alcohols

AbstractZinc (II) chloride based deep eutectic solvent (DES) were formed by mixing zinc (II) chloride with phosphoric acid. Fourier‐transform infrared spectroscopy was used to identify any possible shifts when the two compounds were assorted, and differential scanning calorimetry (SDT Q600 V20.9 Build 2D) was utilized to evaluate the formation of deep eutectic solvent. Densities, , and speed of sound, u, of [ZnCl2][phosphoric acid] 1:2.5 with methanol, ethanol, or propanol have been measured at T = (293.15, 303.15, 308.15, and 313.15) K and at atmospheric pressure. Excess molar volumes ( ), isentropic compressibilities ( ), deviation in isentropic compressibility ( ), and intermolecular free length ( ) were calculated from the densities, and speed of sound, respectively. To fit the excess molar volumes and deviation in isentropic compressibilities, the Redlich–Kister smoothing polynomial was used. Also, we have used perturbed chain statistical associating fluid theory equation of state (PC‐SAFT EoS) for modeling the densities of the binary mixtures, and Schaaff's collision factor theory (SCFT) and Nomoto's relation (NR) were used for modeling the speed of sounds for the binary mixtures.