Excess volume of mixing and equation of state theories

Equation-of state theories of Flory and of Sanchez and Lacombe describe both enthalpy and volume of mixing of binary systems using single component properties and only one binary parameter X₁₂. We have evaluated this parameter from literature enthalpy data for numerous mixtures of two aromatic hydrocarbons, of alkanes with aromatic compounds, and of alkanes with carbonyl compounds. We have used this X₁₂ for calculation of excess volumes and compared the results with our previously measured experimental data. The agreement was fair for mixtures of two nonpolar components. Nevertheless, mixtures containing either cyclohexane or benzene displayed anomalies that could be traced to special packing of molecules in these compounds when pure. For mixtures of carbonyl compounds with alkanes, the theories predicted the qualitative trends correctly, but the quantitative agreement was rather poor. These results tend to support a model in which the enthalpy(cohesive energy) is inversely proportional to volume (as in the theories considered) only for dispersive interaction. When polar-polar interactions are involved, the dependence of excess volume on the excess enthalpy is much weaker.     

Bubble pressures and saturated liquid molar volumes of trichlorofluoromethane—chlorodifluoromethane mixtures. Representation of refrigerant-mixtures vapor—liquid equilibrium data by a modified form of the Peng—Robinson equation of state

Experimental vapor—liquid equilibrium data and saturated liquid molar volumes of chlorodifluoromethane—trichlorofluoromethane binary mixtures have been obtained at four temperatures (298.15, 323.15, 348.15 and 373.15 K) using apparatus described previously.The experimental vapor—liquid equilibria are represented well by a modified form of the Peng—Robinson equation of state with one interaction parameter, but the mean deviation between the calculated and experimental densities is 5%.Vapor—liquid data for binary refrigerant mixtures from the literature are treated using the modified form of the Peng—Robinson equation of state with one adjusted interaction parameter in the mixing rule for a. The representation is fair and is not improved by introducing an additional parameter in the mixing rule for b.     

Modeling of phase equilibria with CPA using the homomorph approach

For association models, like CPA and SAFT, a classical approach is often used for estimating pure-compound and mixture parameters. According to this approach, the pure-compound parameters are estimated from vapor pressure and liquid density data. Then, the binary interaction parameters, k_ij, are estimated from binary systems; one binary interaction parameter per system. No additional mixing rules are needed for cross-associating systems, but combining rules are required, e.g. the Elliott rule or the so-called CR-1 rule. There is a very large class of mixtures, e.g. water or glycols with aromatic hydrocarbons, chloroform-acetone, esters-water, CO₂-water, etc., which are classified as “solvating” or “induced associating”. The classical approach cannot be used and the cross-association interactions are difficult to be estimated a priori since usually no appropriate experimental data exist, while the aforementioned combining rules cannot capture the physical meaning of such interactions (as at least one of the compounds is non-self-associating). Consequently, very often one or more of the interaction parameters are optimized to experimental mixture data. For example, in the case of the CPA EoS, two interaction parameters are often used for solvating systems; one for the physical part (k_ij) and one for the association part (β^cross). This limits the predictive capabilities and possibilities of generalization of the model. In this work we present an approach to reduce the number of adjustable parameters in CPA for solvating systems. The so-called homomorph approach will be used, according to which the k_ij parameter can be obtained from a corresponding system (homomorph) which has similar physical interactions as the solvating system studied. This leaves only one adjustable parameter for solvating mixtures, the cross-association volume (β^cross). It is shown that the homomorph approach can be used with success for mixtures of water and glycols with aromatic hydrocarbons as well as for mixtures of acid gases (CO₂, H₂S) with alcohols and water. The homomorph approach is less satisfactory for mixtures with fluorocarbons as well as for aqueous mixtures with ethers and esters. In these cases, CPA can correlate liquid-liquid equilibria for solvating systems using two adjustable parameters. The capabilities and limitations of the homomorph approach are discussed.     

Liquid–liquid phase separation in multicomponent polymer solutions. IX. Concentration-dependent pair interaction parameter from critical miscibility data on the system polystyrene–cyclohexane

Critical miscibility data obtained from measurements of phase-volume ratios have been used to calculate the concentration dependence of the pair interaction parameter for the system polystyrene–cyclohexane. The measured temperature and concentration ranges are 11–30°C and 4–18% polymer by weight, respectively. With the Gibbs free energy of mixing expressed in polymer segment mole fractions, x^*, the pair interaction parameter is g(x^*, T) = 0.4961 + 71.92/T + 0.2312x^* + 0.0750x^*2. In a polymer volume fraction formulation the parameter is g(φ, T) = 0.4099 + 90.65/T + 0.2064 φ + 0.0518 φ², which approximates to χ(φ, T) = 0.2035 + 90.65/T + 0.3092 φ + 0.1554 φ². Comparison of the temperature and concentration dependence with that obtained by other authors shows very good agreement, even when extensive extrapolations in temperature and concentration are applied. The present function is believed to be the most accurate. Solutions of mixtures of two narrow-distribution polystyrenes in cyclohexane show separation into three liquid phases under the exact conditions predicted by theoretical calculation with the present pair-interaction function.     

A local composition extension of the van der Waals mixing rule for PR cubic equation of state

New mixing rules (VWLC-I and II) capable of connecting van der Waals (VDW) to CEOS/A^E mixing rule models were developed. These models are able to incorporate the same multi-component mixture parameters obtained for the van der Waals and CEOS/A^E models simultaneously. The VWLC mixing rules directly incorporate local compositions into the cubic equations of state (CEOS). The energy parameters required for the local compositions are calculated from the CEOS parameters. The Peng–Robinson (PR) CEOS was used for this study. Binary interactions parameters were obtained by adjusting the vapor pressure of the binary mixture for several low and high-pressure systems. The predictive capabilities of the VWLC mixing rules were tested by vapor–liquid equilibria calculations for low and high-pressure multicomponent systems. The results were compared with the predictions of the VDW mixing rule and a Huron–Vidal (HV) kind of CEOS/A^E-NRTL mixing rule. The VWLC mixing rules are consistent models giving good results in a broad range of pressures and temperatures in binary and multicomponent mixtures. They compare favorably with the CEOS/A^E-NRTL mixing rule for low-pressure systems. In high-pressure ternary systems VWLC-I and II give good predictions, much better, in fact, than the CEOS/A^E-NRTL mixing rule.     

On the resistance to the interpenetration between macromolecular coils of different segment densities

Using viscometry techniques on polymer fractions, we determine the critical concentrationc ^* (separating the dilute and semi dilute solutions). The same measurements have been conducted with mixtures of these fractions (mixtures 1:1 by weight of fractions differing in molecular mass and chemical nature, or fractions differing only in molecular mass). The determined values of critical concentrationc ^* of the mixtures are higher than the values calculated based on the critical concentrations of the corresponding fractions. This deviation from the additivity rule is attributed to the resistance in the interpenetration (delay to the attainment of the homogeneous state) between macromolecular coils of different chemical nature or of the same chemical nature but of different molecular mass. Higher values of the reduced viscosities of the mixture of the fractions, compared to the values calculated using the reduced viscosities of the corresponding fractions, are observed above the critical concentrationc ^*. In this concentration region the interaction parameter between two different polymers is calculated.     

Interaction in Binary Mixtures of Gemini Surfactant G12-6-12 and CTAB by NMR

The interaction between N, N′-bis（dimethyldodecyl）-1,6-hexanediammoniumdibromide （G12-6-12） and cetyltrimethylammonium bromide （CTAB） in D20 aqueous medium has been investigated by NMR at 298 K. The G12-6-12 and CTAB are about 0.773 and measured critical micelle concentration （cmc） of 0.668 mmol/L, respectively. The cmc^＊ （cmc of mixture） values are less than CMC^＊ （cmc of ideally mixed solution） in the mixed system, and the interaction parameter βM〈0 at different molar fractions α of G12-6-12 in the mixed systems, but just when α≤0.3, cmc^＊ values are much smaller than CMC^＊, and βM satisfies the relation of ｜βM｜〉｜ln（cmc1/cmc2）｜ （cmcl： cmc of pure G12-6-12 and cmc2： cmc Of pure CTAB）. The results indicate that there exists synergism between G12-6-12 and CTAB, and they can form mixed micelles, which is further proven by 2D NOESY and self-diffusion coefficient D experiments. There are intermolecular cross peaks between G12-6-12 and CTAB in 2D NOESY, and the radius of micelles in mixed solution is bigger than that in G12-6-12 pure solution in D experiments, indicating there are mixed micelles. However, when α〉0.3, we find that cmc^＊≈CMC^＊, βM≈0, obviously, the two surfactants are almost ideal mixing fitting the pseudo-phase separation model and regular solution theory.     

New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures

Huron, M.-J. and Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271.Good correlations of vapour-liquid equilibria can be achieved by applying the same two-parameter cubic equation of state to both phases. The results primarily depend on the method used for calculating parameters and, for mixtures, on the mixing rule. True parameters are the covolume b and the energy parameter a/b. For this latter one, deviations from a linear weighting rule are closely connected to the excess free energy at infinite pressure. Thus any mixing rule gives a model for the excess free energy, or any accepted models for this property can be used as mixing rules.From the above, an empirical polynomial mixing rule is used for data smoothing and evaluation, while for practical work a local composition model is used. The mixing rule thus obtained can be reduced to the classical quadratic rule for some easily predicted values of the interaction energies. For highly polar systems, it includes three adjustable parameters. Using literature data, the new mixing rule is applied, in the low and high pressure range, to binary mixtures with one or two polar compounds, giving good data correlation and sometimes avoiding false liquid-liquid immiscibility.     

Mixing rules for hardsphere chain mixtures and their extension to heteronuclear hardsphere polyatomic fluids and mixtures

Mixing rules are very important for the calculation of fluid properties using different equations of state. In order to find the theoretical lead of the mixing rule for the size parameter, a mixing rule [1] for hardsphere mixtures has been proposed on the basis of Carnahan-Starling equation and Boublik-Mansoori equation. As its extension, mixing rules for hardsphere chain mixtures are proposed in this work. A mixing rule for the segment number (or chain length) is derived on the limitation of the equality of segment diameters, from the first order thermodynamic perturbation theories (TPT1) for pure chain fluids and for chain mixtures. Meanwhile, the mixing rule for the segment diameter is the same as the mixing rule for hardsphere mixtures on the limitation of monomer mixtures. The two mixing rules are checked together over wide ranges of conditions for hardsphere chain mixtures and compared with the first order thermodynamic perturbation theory (TPT1) and also with simulation data available in literature. An another interesting usage of new mixing rules is to describe the heteronuclear hardsphere polyatomic pure fluids, which consist of hardspheres with different segment diameters as in methane and ethane in which carbon and hydrogen atoms are looked as bonded spheres, and heteronuclear hardsphere polyatomic mixtures. The comparison with simulation data shows the validity of the mixing rules.     

Application of the Perturbed-Chain SAFT equation of state to polar systems

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to pure polar substances as well as to vapor–liquid and liquid–liquid equilibria of binary mixtures containing polar low-molecular substances and polar co-polymers. For these components, the polar version of the PC-SAFT model requires four pure-component parameters as well as the functional-group dipole moment. For each binary system, only one temperature-independent binary interaction k_ij is needed. Simple mixing and combining rules are adopted for mixtures with more than one polar component without using an additional binary interaction parameter. The ability of the model to accurately describe azeotropic and non-azeotropic vapor–liquid equilibria at low and at high pressures, as well as liquid–liquid equilibria is demonstrated for various systems containing polar components. Solvent systems like acetone–alkane mixtures and co-polymer systems like poly(ethylene-co-vinyl acetate)/solvent are discussed. The results for the low-molecular systems also show the predictive capabilities of the extended PC-SAFT model.     

Solubility and Solution Thermodynamics of Some Sulfonamides in 1-Propanol + Water Mixtures

The solubilities of sulfadiazine (SD), sulfamerazine (SMR) and sulfamethazine (SMT) in some 1-propanol + water co-solvent mixtures were measured at five temperatures from 293.15 to 313.15 K over the polarity range provided by the aqueous solvent mixtures. The mole fraction solubility of all these sulfonamides was maximal in the 0.80 mass fraction of 1-propanol solvent mixture (δ _solv = 28.3 MPa^1/2) and minimal in water (δ = 47.8 MPa^1/2) at all temperatures studied. The apparent thermodynamic functions Gibbs energy, enthalpy, and entropy of solution were obtained from these solubility data by using the van’t Hoff and Gibbs equations. Apparent thermodynamic quantities of mixing were also calculated by using the ideal solubilities reported in the literature. Nonlinear enthalpy–entropy relationships were observed for these drugs in the plots of enthalpy versus Gibbs energy of mixing. The plot of ?_mix H° versus ?_mix G° shows different trends according to the slopes obtained when the mixture compositions change. Accordingly, the mechanism for the solution process of SD and SMT in water-rich mixtures is enthalpy driven, whereas it is entropy driven for SMR. In a different way, in 1-propanol-rich mixtures the mechanism is enthalpy driven for SD and SMR and entropy driven for SMT. Ultimately, in almost all of the intermediate compositions, the mechanism is enthalpy driven. Nevertheless, the molecular events involved in the solution processes remain unclear.     

Effect of interphase interaction within zinc-filled composite coating on the potential of the cathodic protection of steel

The potential of cathodic protection of steel with composite coatings based on polystyrene (PS) filled by highly dispersed powders of zinc (Zn) of different forms and dispersivities is investigated. The potential for all PS/Zn systems with a low content of filler is shown to be less than for pure metal; as some critical value (the percolation threshold) is reached, the negative values of potential increase abruptly and become higher in absolute value than the potential of pure metal. The threshold zinc contents are found to depend on the particle shape and rise from 9 to 30 vol% upon transitioning from spear-shaped to spherical particles. The electrochemical protection properties of a composite coating are correlate with the enthalpy of mixing of filled composites. The enthalpy of mixing in regions of low filling is shown to be negative, indicating strong interphase interaction; that in regions of high filling is positive. The positive enthalpy of mixing corresponds to compositions that generate the potential of cathodic protection. It is found that an increase in the concentration range of positive values of a composition’s enthalpy of mixing occurs symbatically with a decrease in the threshold concentration of metallic zinc within the composition. We conclude that cathodic protection by zinc-filled polymeric composites is due to weak interphase interaction that results in the aggregation of particles of metallic zinc within a polymeric matrix and the appearance of an infinite cluster. In the case of PS/Zn compositions, it is shown that the infinite cluster appears at enthalpies of mixing greater than 0.6 J/g of the composition.     

Molecular weight dependence of phase behavior of PEO/P(EO‐b‐DMS) blends: Application of Sanchez‐Lacombe lattice fluid theory

Molecular weight dependence of phase separation behavior of the Poly (ethylene oxide) (PEO)/Poly(ethylene oxide‐block‐dimethylsiloxane) (P(EO‐b‐DMS)) blends was investigated by both experimental and theoretical methods. The cloud point curves of PEO/P(EO‐b‐DMS) blends were obtained by turbidity method. Based on Sanchez‐Lacombe lattice fluid theory (SLLFT), the adjustable parameter, (quantifying the interaction energy between different components), was evaluated by fitting the experimental data in phase diagrams. To calculate the spinodals, binodals, and the volume changes of mixing for these blends, three modified combining rules of the scaling parameters for the block copolymer were introduced. The calculated binodals with those modified combining rules show good agreement with the experimental data. Furthermore, the calculated volume changes during mixing decrease with increasing molecular weight of PEO, and the relationship between the volume changes and temperature is quite different for the mixtures with different molecular weight of PEO. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 452–459, 2008     

Solution and solvation thermochemistry of copper(II) and nickel(II) nitrates in water and acetonitrile mixtures at 298 K

The calorimetric enthalpies of mixing of aqueous solutions of Cu(NO₃)₂ and Ni(NO₃)₂ in mixtures of water with acetonitrile (AN) at 298 K are measured over the entire range of the compositions. The ionic enthalpies of transfer from pure water to water mixed with AN are calculated. The behavioral features of d cations, as distinct from simple ions, are recognized. The contributions from the universal, chemical, and electrostatic interactions between the ion and solvent to the enthalpy of ion transfer are calculated. The structural term of the enthalpy of ion transfer that reflects the energy (enthalpy) changes in the solution induced by the chemical interaction of an ion with the solvent is obtained; the regions of d-cation resolvation in the solution are recognized as a result.     

Vapor–liquid equilibria predictions of hydrogen–hydrocarbon mixtures with the Huron–Vidal mixing rule

An excess Gibbs-equation of state (G^E-EoS) framework based on the Huron–Vidal mixing rule, has been applied to study vapor–liquid equilibria (VLE) of hydrogen–hydrocarbon mixtures. The mixing rule couples the Peng–Robinson–Stryjek–Vera (PRSV) EoS with a local composition solution model. The solution model is based on one-fluid theory treatment and assigns a single energy parameter to each binary pair. This energy parameter relates to the preference of the molecules for like to unlike interactions. The allocation of a system's number of interactions to the individual species in a binary mixture, incorporates the use of size parameters which gain significance only in the liquid phase. In a two parameter form, the framework has been used for the simultaneous data reduction of a large number of binary and several ternary hydrogen–hydrocarbon mixtures. These systems were taken over an extended range of pressures and temperatures. Results from the data reduction are reported in both tabular and graphical forms. Correlations for the model parameters have been identified with the acentric factor of the hydrocarbon in hydrogen–hydrocarbon binary mixtures. In a fully predictive mode, the model has shown to describe well VLE of binary hydrogen–linear alkane systems. Comparisons of these results with calculations from the Peng–Robinson (PR) EoS and the classical mixing rule (vdW) are included.     

Effects of Cs and Na ions on the interfacial properties of dodecyl sulfate solutions

The competitive binding of counterions to anionic dodecyl sulfate ions in aqueous solutions of cesium dodecyl sulfate (CsDS) and sodium dodecyl sulfate (SDS) mixtures, which significantly influences the critical micelle concentration (cmc) and surface (or interfacial) tension of surfactant solutions, was investigated. The cmc and degree of counterion binding were obtained through electrical conductivity measurements. The curve of cmc versus the mole fraction of CsDS in the surfactant mixture was simulated by Rubingh's equations, which enabled us to estimate the interaction parameter in micelles (W _R) based on the regular solution approximation. The curve-fitting exhibited a slightly negative value (W _R=−0.1), indicating that the mixing (SDS+CsDS) enhances micelle formation owing to a greater interaction between surfactant molecules and counterions than in pure systems (SDS). On going from SDS, SDS:CsDS(75:25), SDS:CsDS(50:50), SDS:CsDS(25:75) to CsDS, interfacial tension at the hexadecane/surfactant-solution interface showed a negative deviation from the mixing rule (interaction parameter in adsorbed film W _A=−0.38), indicating the replacement of Na⁺ bound to anionic dodecyl sulfate by Cs⁺ ions owing to the stronger interaction between the Cs⁺ and the dodecyl sulfate ions. Droplet sizes of emulsion formed with hexadecane and aqueous dodecyl sulfate solutions were investigated using the light scattering spectrophotometer. The higher binding capacity of Cs⁺, having a smaller hydrated ionic size than Na⁺, also resulted in a negative deviation in emulsion droplet size in mixed systems. Received: 10 May 2000/Accepted: 11 August 2000     

Thermochemistry of the binary system nitrocellulose+2,4-dinitrotoluene

Melting enthalpy and mixing enthalpy of binary system 2,4-dinitrotoluene and nitrocellulose were determined by DSC method. The maximum value of mixing enthalpy was H _max ^M=1.38 kJ mol⁻¹ for molar fraction x _w24DNT = 0.501. The Flory-Huggins parameter (c) was estimated. The solubility curves and glass transition temperatures were predicted and compared with the experimental results. The measurements were performed for the samples with different times of storage at room temperature. The analysis of melting peaks for the mixture leads to the conclusion that for the long periods of storage the melting of 2,4-dinitrotoluene takes place in the confined spaces (pores) and unconfined space (bulk). The crystallization and melting is observed during the short time of storage in mixtures with low nitrocellulose content and in the case of mixtures with a large amount of NC the glass transition is additionally observed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A new mixing rule—modified conventional mixing rule

The asymmetry of the corrective cohesion parameter a^c against the mole fraction is found out to be a controlling factor of the vapour—liquid equilibrium(VLE) calculations.A new mixing rule which satisfies the asymmetry of a^c and has two adjustable parameters is proposed for systems containing strongly polar substances. Furthermore, it still keeps the functional form of the conventional mixing rule.The parameters of the proposed mixing rule for the MVDW equation of state are correlated by the system temperature and well applied to the ternary systems containing strongly polar substances.