Ternary Mixture Model Research Articles

A ternary mixture model with dynamic boundary conditions.

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

Critical micelle concentration and partition coefficient of mixed micelles: Analysis of ternary systems based on Markov chain model and simple mixture model

When two or more surfactants are mixed, the mixed system exhibits improved performance compared with a single surfactant system in solution. The Markov chain model can analyze the critical micelle concentration (CMC) of mixed micelles, yielding results similar to those of a simple mixture model, which is typically referred to as the “regular solution theory”. In this study, two hypotheses were tested: (1) the Markov chain model for ternary systems can be simplified by approximating the association constant of surfactants i and j as Kij = Kji and (2) the quasi-simple ternary mixture model, that is, a simple mixture model analogous to the Markov chain model, helps interpret the interaction parameter of the simple mixture model that can describe the partition coefficient of the binary mixed micelle. Equations were derived for (1) the Markov chain model for ternary systems by assuming Kij = Kji and (2) the interaction parameter of the simple mixture model that can describe the partition coefficient of the binary mixed micelle. The models were compared with the experiment data, and the derived equations described the data of the CMC and partition coefficient well.

Molecular dynamic simulation on the density of titanium dioxide and silver water-based nanofluids using ternary mixture model

This study utilizes molecular dynamics (MD) simulations to scrutinize the influence of type and volume fraction of nanoparticle on the density of nanofluids, i.e. silver (as a hydrophilic case) and titanium dioxide (as a hydrophobic case) water-based nanofluids. Ionic structure of TiO2 distinguishes the solid–liquid interactions with those in silver-water nanofluid. The thickness of nanolayer is estimated from density profile formed around the nanoparticle and is included in the density calculation. It is found that the type of interatomic potential between the nanoparticle and surrounding water can highly affect the thickness of nanolayer. The hydrophobicity/hydrophilicity of nanoparticle is of great importance in nanolayer density and its thickness. It is observed that the difference between the traditional binary mixture and the ternary mixture formula for calculating the density of nanofluid increases by increasing the volume fraction of nanoparticle. This indicates that the effect of nanolayer gets more considerable at high volume fractions.

Source Identification and Quantification of Seepage Water in a Coastal Mine, in China

The Sanshandao gold mine, which is the largest coastal mine in China, is under threat from seawater intrusion and water inrush. The objective of this study is to determine the water end-members (seawater, freshwater, and brine) of the seepage water in the mine and quantify the proportion of end-members. Non-conservative ions and ion exchange were identified by using hydrogeochemical analysis. Then, the principal component analysis (PCA) was used to identify the end-members of mine water. Three end-members were identified, so a ternary mixture model was applied to compute the mixing ratios. The potential water flow channels and the prevailing supply patterns were inferred by combining the results of mixing ratios with the tectonic and engineering geological conditions. The results indicate that the proportion of seawater in mine water is about 57%, the freshwater is about 16% and the brine is about 27% for the entire mine area, the prevailing supply pattern of seawater was lateral recharge, the water samples which were located in −510 m sublevel or in the northeast of prospecting line 2260 had high proportions of seawater, the freshwater supplied the groundwater mainly through the secondary fractures developed area in a vertical recharge and the influence depth was about −500 m, and F3 was the largest tensile-shear fault in the study area and it was both a watercourse for seawater and fresh water.

Lead Pollution and Isotope Tracing of Surface Sediments in the Huainan Panji Coal Mining Subsidence Area, Anhui, China.

In this study, the provenance of anthropogenic lead, a major pollutant of surface sediments, was determined in Huainan Panji coal mining area. The lead concentrations and the pollution degree were investigated by the enrichment factor. Tessier five-step morphology extraction method was used to discuss its potential hazard. By comparing the lead isotope ratios of surface sediments and surrounding potential polluted end-members, the lead isotope ternary mixture model was appropriated to explore the sources and relative contribution fractions. The results showed that: (1) The lead concentrations ranged from 31.44 to 64.07mg/kg which was mild-moderate pollution. (2) The chemical forms of lead were residue state > iron-manganese oxidation state > exchangeable state > organic state > carbonate state. (3) The anthropogenic lead in surface sediments originated from soil, vehicle exhaust, coal gangue. And the relative contribution fractions were 51.70%, 30.90%, and 17.40%.

Experimental investigation and modeling of the separation of ternary mixtures by hydrophilic pervaporation

ABSTRACTIn the course of the present study, the pervaporative dehydration of two ternary mixtures: ethanol/n-butanol/water and ethanol/ethyl acetate/water were investigated through hydrophilic polymer composite PERVAP 1210 membrane. The effects of temperature and the concentrations of water and organic components on permeation were studied through pervaporation experiments. The influence of Hansen solubility parameters of the components and mixtures on their permeation was also examined. It has been found that changes of organic ratios in the feed solution affect organic permeation due to mutual interactions and flux coupling. This interaction was built in a suggested semi-empirical mass-transport model for ternary mixtures with optional simplifications. The modeled partial fluxes are in good accordance with the measured values.

Triphase Separation of a Ternary Symmetric Highly Viscous Mixture.

We discuss numerical results of diffusion-driven separation into three phases of a symmetric, three-component highly viscous liquid mixture after an instantaneous quench from the one-phase region into an unstable location within the tie triangle of its phase diagram. Our theoretical approach follows a diffuse-interface model of partially miscible ternary liquid mixtures that incorporates the one-parameter Margules correlation as a submodel for the enthalpic (so-called excess) component of the Gibbs energy of mixing, while its nonlocal part is represented based on a square-gradient (Cahn–Hilliard-type) modeling assumption. The governing equations for this phase-field ternary mixture model are simulated in 3D, showing the segregation kinetics in terms of basic segregation statistics, such as the integral scale of the pair-correlation function and the separation depth for each component. Based on the temporal evolution of the integral scales, phase separation takes place via the simultaneous growth of three phases up until a symmetry-breaking event after which one component continues to separate quickly, while phase separation for the other two seems to be delayed. However, inspection of the separation depths reveals that there can be no symmetry among the three components at any instant in time during a triphase segregation process.

Critical behaviour of microemulsions: Monte Carlo simulations of the Widom model

The critical behaviour of a lattice model for ternary mixtures of hydrophilic, hydrophobic and amphiphilic (surfactant) molecules is investigated by means of Monte Carlo simulations. A lattice model formulated by Widom as a simple approach of these mixtures is adopted taking advantage of its equivalence to a spin-12 Ising model, which adds next nearest neighbors interactions on a simple cubic lattice. In this work we focus on the critical behaviour of the order-disorder phase transition of this model. A simulation strategy combining a standard Metropolis sampling and multiple histogram reweighting technique is used. The location of the phase boundary is well determined, being consistent with the 3D Ising critical temperature. In addition, a suitable finite size scaling analysis is performed to obtain critical amplitudes of magnitudes as magnetization, susceptibility and interfacial tension. The critical behaviour results compatible with the 3D Ising universality class, as expected. Likewise, critical amplitudes ratios give a reasonable agreement with the universal values obtained from different systems and methods.

Dissolution or Growth of a Liquid Drop via Phase-Field Ternary Mixture Model Based on the Non-Random, Two-Liquid Equation.

We simulate the diffusion-driven dissolution or growth of a single-component liquid drop embedded in a continuous phase of a binary liquid. Our theoretical approach follows a diffuse-interface model of partially miscible ternary liquid mixtures that incorporates the non-random, two-liquid (NRTL) equation as a submodel for the enthalpic (so-called excess) component of the Gibbs energy of mixing, while its nonlocal part is represented based on a square-gradient (Cahn-Hilliard-type modeling) assumption. The governing equations for this phase-field ternary mixture model are simulated in 2D, showing that, for a single-component drop embedded in a continuous phase of a binary liquid (which is highly miscible with either one component of the continuous phase but is essentially immiscible with the other), the size of the drop can either shrink to zero or reach a stationary value, depending on whether the global composition of the mixture is within the one-phase region or the unstable range of the phase diagram.

Multivariate statistical and lead isotopic analyses approach to identify heavy metal sources in topsoil from the industrial zone of Beijing Capital Iron and Steel Factory.

Heavy metals are considered toxic to humans and ecosystems. In the present study, heavy metal concentration in soil was investigated using the single pollution index (PIi), the integrated Nemerow pollution index (PIN), and the geoaccumulation index (Igeo) to determine metal accumulation and its pollution status at the abandoned site of the Capital Iron and Steel Factory in Beijing and its surrounding area. Multivariate statistical (principal component analysis and correlation analysis), geostatistical analysis (ArcGIS tool), combined with stable Pb isotopic ratios, were applied to explore the characteristics of heavy metal pollution and the possible sources of pollutants. The results indicated that heavy metal elements show different degrees of accumulation in the study area, the observed trend of the enrichment factors, and the geoaccumulation index was Hg>Cd>Zn>Cr>Pb>Cu≈As>Ni. Hg, Cd, Zn, and Cr were the dominant elements that influenced soil quality in the study area. The Nemerow index method indicated that all of the heavy metals caused serious pollution except Ni. Multivariate statistical analysis indicated that Cd, Zn, Cu, and Pb show obvious correlation and have higher loads on the same principal component, suggesting that they had the same sources, which are related to industrial activities and vehicle emissions. The spatial distribution maps based on ordinary kriging showed that high concentrations of heavy metals were located in the local factory area and in the southeast-northwest part of the study region, corresponding with the predominant wind directions. Analyses of lead isotopes confirmed that Pb in the study soils is predominantly derived from three Pb sources: dust generated during steel production, coal combustion, and the natural background. Moreover, the ternary mixture model based on lead isotope analysis indicates that lead in the study soils originates mainly from anthropogenic sources, which contribute much more than the natural sources. Our study could not only reveal the overall situation of heavy metal contamination, but also identify the specific pollution sources.

Evolving machine learning models to predict hydrogen sulfide solubility in the presence of various ionic liquids

Substituting conventional solvents for gas sweetening with ionic liquids (ILs) is an interesting way to specify superior design from energy consumption in regeneration and reduction solvent loss. In this study, based on the critical temperature (Tc), critical pressure (Pc), and molecular weight (Mw) of pure ionic liquids, a feed forward Multi-Layer Perceptron Artificial Neural Network (MLP-ANN), an Adaptive Neuro-Fuzzy Inference System (ANFIS) and a Radial Basis Function Artificial Neural Network (RBF-ANN) were developed to predict solubility of Hydrogen Sulfide in the presence of various ILs over wide ranges of temperature, pressure and concentration. To develop the aforementioned methods, 664 experimental data points collected from the literatures were employed. Moreover, to investigate the Hydrogen Sulfide solubility in ternary mixture containing Carbon Dioxide, Hydrogen Sulfide and ILs, MLP-ANN model was proposed. To propose MLP-ANN method for estimating H2S solubility in ternary mixture, 89 experimental data points collected from the previous published works were employed. To examine the ability of the methods suggested in this study different statistical criteria including R-Squared (R2), Mean Squared Error (MSE), Standard Deviation (STD) and Mean Absolute Relative Error (MARE) were used. The values of R2 and MSE achieved for the MLP-ANN model are 0.9951 and 0.000117 respectively. Furthermore, the values of R2 and MSE for both ANFIS and RBF-ANN methods obtained 0.901, 0.002268 and 0.9679, 0.000787 respectively. In addition, R2 and MSE of the MLP-ANN model for ternary mixtures are 0.9955 and 0.000082 correspondingly. Therefore, the ability and acceptable performance of using the MLP-ANN as an accurate model for estimating Hydrogen Sulfide solubility in ILs was showed versus other computational intelligence models.

Electrochemical Measurement and Modeling of Corrosion Inhibition of Ternary Mixtures of Homologous Surfactants

In the present study, the corrosion inhibition efficiency (IE) of different mixtures of three homologous cationic surfactants, benzalkonium chlorides (ADBAC), were investigated using electrochemical measurements. There is excellent agreement between IE measured from EIS and LPR of the discussed mixture of ADBAC surfactants: C12Cl/C14Cl/C16Cl=0.70/0.25/0.05. The calculated adsorption free energy indicates that the adsorption mechanism of mixtures 0.70/0.25/0.05 is a mixed form of physisorption and chemisorption. A new critical micelle concentration (CMC) prediction model for ternary mixtures of homologous surfactants, on the basis of previous work, is developed over a wide concentration range of salt from 0.2% to 5.0% NaCl. The developed corrosion IE prediction model based on CMC prediction is an effective method that can be used for corrosion IE prediction of various mixed surfactant systems at various salt concentrations.

Prediction of the dependence of the line tension on the composition of linactants and the temperature in phase separated membranes.

We calculate the line tension between domains in phase separated, ternary membranes that comprise line active molecules (linactants) that tend to increase the compatibility of the two phase separating species. The predicted line tension, which depends explicitly on the linactant composition and temperature, is shown to decrease significantly as the fraction of linactants in the membrane increases toward a Lifshitz point, above which the membrane phase separates into a modulated phase. We predict regimes of zero line tension at temperatures close to the mixing transition and clarify the two different ways in which the line tension can be reduced: (1) The linactants uniformly distribute in the system and reduce the compositional mismatch between the two bulk domains. (2) The linactants accumulate at the interface with a preferred orientation. Both of these mechanisms have been observed in recent experiments and simulations. The second one is unique to line active molecules, and our work shows that it is increasingly important at large fraction of linactants and is necessary for the emergence of a regime of zero line tension. The methodology is based on the ternary mixture model proposed by Palmieri and Safran [Palmieri, B.; Safran, S. A. Langmuir 2013, 29, 5246], and the line tension is calculated via variationally derived, self-consistent profiles for the local variation of composition and linactant orientation in the interface region.

Hybrid lipids increase nanoscale fluctuation lifetimes in mixed membranes

A recently proposed ternary mixture model is used to predict fluctuation domain lifetimes in the one phase region. The membrane is made of saturated, unsaturated, and hybrid lipids that have one saturated and one unsaturated hydrocarbon chain. The hybrid lipid is a natural linactant which can reduce the packing incompatibility between saturated and unsaturated lipids. The fluctuation lifetimes are predicted as a function of the hybrid lipid fraction and the fluctuation domain size. These lifetimes can be increased by up to three orders of magnitude compared to the case of no hybrids. With hybrid, small length scale fluctuations have sizable amplitudes even close to the critical temperature and, hence, benefit from enhanced critical slowing down. The increase in lifetime is particularly important for nanometer scale fluctuation domains where the hybrid orientation and the other lipids composition are highly coupled.

Hybrid Lipids Increase the Probability of Fluctuating Nanodomains in Mixed Membranes

A ternary mixture model is proposed to describe composition fluctuations in mixed membranes composed of saturated, unsaturated, and hybrid lipids (with one saturated and one unsaturated hydrocarbon chain). The hybrids are line-active and can reduce the packing incompatibility between the saturated and unsaturated lipids. We introduce a lattice model that extends previous studies by taking into account the dependence of the interactions of the hybrid lipids on their orientations in a simple way. A methodology to recast the free energy of the lattice model in terms of a continuous, isotropic field theory is proposed and used to analyze composition fluctuations in the one-phase region (above the critical temperature). The effect of hybrid lipids on fluctuation domains rich in saturated/unsaturated lipids is predicted. The correlation length of such fluctuations decreases significantly with increasing amounts of hybrids; this implies that nanoscale fluctuation domains are more probable compared to the case with no hybrids. Smaller correlated fluctuation domains arise even when the temperature is close to a critical point, where very large correlation lengths are normally expected. This decrease in the correlation length is largest as the hybrid composition tends toward a crossover value above which stripelike fluctuations are predicted. This crossover value defines the Lifshitz line. The characteristic wavelength of the stripelike fluctuations is large close to the Lifshitz point but decreases toward a molecular size in a membrane that contains only hybrids. Micrometer size, stripelike domains have recently been observed experimentally in giant unilamelar vesicles (GUVs) made of saturated, unsaturated, and hybrid lipids. These results suggest that the line activity of hybrid lipids in such mixtures may be significant only at large hybrid fractions; in that regime, the interface between domains can be diffuse and several hybrid molecules with correlated orientations can separate saturated and unsaturated lipid regions.

Can the joint effect of ternary mixtures be predicted from binary mixture toxicity results?

The joint effect of the majority of chemical mixtures can be predicted using the reference model of Concentration Addition (CA). It becomes a challenge, however, when the mixtures include chemicals that synergise or antagonise the effect of each other. In this study we examine if the deviation from CA of seven ternary mixtures of interacting chemicals can be predicted from knowledge of the binary mixture responses involved. We hypothesise that the strongest interactions will take place in the binary mixtures and that the size of the ternary mixture response can be predicted from knowledge of the binary interactions. The hypotheses were tested using a stepwise modelling approach of incorporating the information held in binary mixtures into a ternary mixture model, and comparing the model predictions with observed ternary mixture toxicity data derived from studies of interacting chemical mixtures on the floating plant Lemna minor and the bacteria Vibrio fischeri. The results showed that for both the antagonistic and the synergistic ternary mixtures the ternary model predictions were superior to the conventional CA reference model and provided robust estimations of the size of the experimentally derived ternary mixture toxicity effects.

Mean Field Based Coarse-Grained Simulations of Ternary Mixtures

Traditional methods of bilayer simulation at an atomistic level fail to capture length and time scales of biological interest. For example, the formation of lipid rafts in ternary mixtures is not observable with Molecular Dynamics (MD) simulations. To address this fundamental problem we have developed a coarse-grained model to simulate lipid bilayers based on self-consistent mean-field theory (SCMFT) that uses MD trajectories to extract simulation parameters, and a library of chain states that is used for statistical mechanical calculations. We have applied this model to two ternary mixtures: DOPC, SM and Cholesterol and POPC, SM and cholesterol. The thermodynamic behavior of these two systems is of interest because they are known from experiment to exhibit coexisting regions of lipid order and disorder, related to lipid rafts, at certain temperatures and concentrations. In this poster we describe predictions of the SCMFT model for both ternary mixtures, over a range of mixture concentrations, and over microsecond time scales. The existence of the single double bond in POPC influences packing among neighboring lipids differently from two double bonds of DOPC, and this has an effect on the system-wide level of lateral organization. The two systems are compared with experiments conducted with these mixtures.

Theoretical analysis and comparison with experimentation of the molecular and thermal diffusion coefficients for a ternary hydrocarbon mixture

The thermal diusion phenomena, also known as the Soret eect, is important for the study of compositional variation in hydrocarbon reservoirs. Several theoretical models have been developed over the years to predict the thermodiusion coefficient of an organic mixture. Firoozabadi et al. [1] developed an analytical relationship between the thermal, molecular, and pressure diusion coefficients based on the approach of Dougherty and Drickamer [2]. The distinctive feature of Firoozabadi's model is its ability to be used not only for binary mixtures but also for multicomponent mixtures. The verification of Firoozabadi's model by numerical simulations was limited to binary mixtures only because of a lack of experimental data on thermodiusion coefficients in multicomponent hydrocarbon mixtures. However, Platten et al. [3] were able to measure the thermodiusion coefficients of a ternary hydrocarbon mixture experimentally by utilizing the thermogravitational column technique, hence making it possible to verify the theoretical model. This paper will compare the density, thermodiusion, and molecular diusion coefficients obtained with the Fontainebleau benchmark values for binary mixtures of n-dodecane, isobutylbenzene, and tetrahydronaphtalene. The accuracy of the numerical results from the Firoozabadi's model for ternary hydrocarbon mixtures is investigated. Results reveal a positive agreement between experimental and numerical data. © Copyright 2005 Walter de Gruyter.

Reduced Two-Phase Model for Analysis of the Anode of a DMFC

An isothermal two-phase ternary mixture model that takes into account conservation of momentum, mass, and species in the anode of a direct methanol fuel cell (DMFC) is presented and analyzed. The slenderness of the anode allows a considerable reduction of the mathematical formulation, without sacrificing the essential physics. The reduced model is then verified and validated against data obtained from an experimental DMFC outfitted with a transparent end plate. Good agreement is achieved. The effect of mass-transfer resistances in the flow field and porous backing are highlighted. The presence of a gas phase is shown to improve the mass transfer of methanol at higher temperatures (>30°C). It is also found that at a temperature of around 30°C, a one-phase model predicts the same current density distribution as a more sophisticated two-phase model. Analysis of the results from the two-phase model, in combination with the experiments, results in a suggestion for an optimal flow field for the liquid-fed DMFC. © 2004 The Electrochemical Society. All rights reserved.

A microemulsion model on sc, bcc and fcc lattices: Ground state properties

The microemulsion model proposed by Ciach, Høye, and Stell [J. Phys. A 21, L777 (1988)] is a lattice gas model for ternary mixtures of water, oil, and amphiphile, which explicitly takes into account orientational degrees of freedom of amphiphilic molecules. Usually only a discrete set of orientations (towards nearest-neighbor sites) is permitted, and the model is lattice dependent. An investigation about the ground state phase diagram on the body-centered cubic (bcc) and face-centered cubic (fcc) lattices, compared with previously known results on the simple cubic (sc) lattice, is performed. Different structured phases are found, resembling experimentally observed phases and not always considered in the literature. The role of the lattice structure is discussed.