Prediction of global vapor-liquid equilibria for mixtures containing polar and associating components with improved renormalization group theory

Our recently improved renormalization group (RG) theory is further reformulated within the context of density functional theory. To improve the theory for polar and associating fluids, an explicit and complete expression of the theory is derived in which the density fluctuation is expanded up to the third-order term instead of the original second-order term. A new predictive equation of state based on the first-order mean spherical approximation statistical associating fluid theory (FMSA-SAFT) and the newly improved RG theory is proposed for systems containing polar and associating fluids. The calculated results for both pure fluids and mixtures are in good agreement with experimental data both inside and outside the critical region. This work demonstrates that the RG theory incorporated with the solution of FMSA is a promising route for accurately describing the global phase behavior of complex fluids and mixtures.     

Theoretical study of Sutherland fluids with long-range, short-range, and highly short-range potential parameters

On the basis of the first-order mean spherical approximation (FMSA) theory the behavior of Sutherland fluids with a number of parameters (gamma=3.1-36) is investigated. The investigation includes its modification by the simplified exponent approximation, renormalization group (RG) transformation, and density functional theory (DFT). For long-range parameters, the original FMSA is found sufficiently good to describe the global phase behavior, including inside the critical region. For short-range parameters, the modified FMSA by the simplified exponent approximation outside the critical region and RG transform inside the critical region are applied. For extremely short-range forces, the success is achieved by its combination with the DFT. This work gives a general sense about the capability of a theory for different ranges of potential, as well as for different thermodynamic states.     

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.     

Time evolution of density fluctuation in supercritical region. I. Non-hydrogen-bonded fluids studied by dynamic light scattering

The time evolution of the density fluctuation of molecules inhomogeneously dispersing in a mesoscopic volume is investigated by dynamic light scattering in several fluids in supercritical states. This study is the first time-domain investigation to compare the dynamics of density fluctuation among several fluids. The samples used are non-hydrogen-bonded fluids in the supercritical states: CHF(3), C(2)H(4), CO(2), and xenon. These four molecules have different properties but are of similar size. Under these conditions, the relationship between dynamic and static density inhomogeneities is studied by measuring the time correlation function of the density fluctuation. In all cases, this function is characterized by a single exponential function, decaying within a few microseconds. While the correlation times in the four fluids show noncoincidence, those values agree well with each other when scaled to a dimensionless parameter. From the results of this scaling based on the Kawasaki theory and Landau-Placzek theory, the relation between dynamics and static structures is analyzed, and the following four insights are obtained: (i) viscosity is the main contributor to the time evolution of density fluctuation; (ii) the principle of corresponding state is observed by the use of time-domain data; (iii) the Kawasaki theory and the Landau-Placzek theory are confirmed to be applicable to polar, nonpolar, and nondipolar fluids that have no hydrogen bonding, at temperatures relatively far from critical temperature; and (iv) the density fluctuation correlation length and the value of density fluctuation are estimated from the time-domain data and agree with the values from other experiments and calculations.     

Time evolution of density fluctuation in the supercritical region. 2. Comparison of hydrogen- and non-hydrogen-bonded fluids

The time evolution of the density fluctuation of molecules is investigated by dynamic light scattering in six neat fluids in supercritical states. This study is the first to compare the dynamics of density inhomogeneity between hydrogen- and non-hydrogen-bonded fluids. Supercritical methanol and ethanol are used as hydrogen-bonded fluids, whereas four non-hydrogen-bonded fluids were used: CHF(3), C(2)H(4), CO(2), and Xe. We measure the time correlation function of the density fluctuation of each fluid at the same reduced temperatures and densities and investigate the relationship between the dynamic and static density inhomogeneities of those supercritical fluids. In all cases, the profile of the time correlation function of the density fluctuation is characterized by a single-exponential function, whose decay is responsible for the dynamics characterized by hydrodynamic conditions. We obtain correlation times from the time correlation function and discuss dynamic and static inhomogeneity using the Kawasaki theory and the Landau-Placzek theory. While the correlation times in the six fluids show noncoincidence, those values agree well with each other except for the supercritical alcohols when scaled to a dimensionless parameter. Although the principle of corresponding state is observed in the non-hydrogen-bonded fluids, both the supercritical methanol and ethanol deviate from that principle. This deviation is attributed to the presence of hydrogen bonding among alcohol molecules at high temperature and low density. The average cluster size of each fluid is estimated under the same thermodynamic conditions, and it is shown that the clusters of supercritical alcohols are on average 1.5-1.7 times larger than those of the four non-hydrogen-bonded fluids. Moreover, the thermal diffusivity of each neat fluid is obtained over wide ranges of density and temperature.     

Studies on fluids and their mixtures using the Born-Green-Yvon integral equation technique

We have developed the Born-Green-Yvon (BGY) integral equation theory for investigating the equilibrium properties of fluids and their mixtures both on the lattice and in the continuum. Using the continuum theory we have studied hard sphere fluids over a range in density having chain lengths between one and fifty sites. We have also investigated the collapse transition of a square well chain and a square well ring, each having up to four hundred sites, and have predicted the theta temperature for these systems. Turning to the case of a dilute (hard-sphere) solution we have been able to show the effect of solvation on a hard sphere chain, and captured the dependence of this effect on the ratio of hard sphere diameters of the solvent and chain segments. In all the continuum studies we have found good to excellent agreement with simulation results. We have also derived a lattice BGY theory which, while less sophisticated than the continuum version, has the advantage of producing simple closed-form expressions for thermodynamic properties of interest. This theory is capable of exhibiting the full range of miscibility behaviour observed experimentally, including upper and lower critical solution temperatures and closed-loop phase diagrams. We find that the theory does an excellent job of fitting to different kinds of experimental data and, making use of the parameters derived from fits to pure component data alone, we have been able to predict properties ranging from pure fluid vapour pressures and critical temperatures to changes in the volume and enthalpy on mixing as well as coexistence curves for solutions.     

Effective potential approach to bulk thermodynamic properties and surface tension of molecular fluids I. Single component n-alkane systems

The aim of this work is to develop spherically symmetric effective potentials allowing bulk thermodynamic properties and surface tension of molecular fluids to be predicted semiempirically by the use of statistical mechanical methods. Application is made to the straight chain alkane fluids from methane to decane. An effective Lennard-Jones potential is generated with temperature-dependent parameters fitted to the critical temperature and pressure and to Pitzer's acentric factor. Insertion of this potential into the generalised van der Waals (GvdW) density functional theory yields bulk properties in good agreement with experiments. The surface tension is overestimated for the longer alkane chains. In order to account for the surface tension, an independently adjustable attractive range of interaction is required and obtained through the use of square-well potentials chosen so as to leave the bulk thermodynamics unaltered while the attractive range is fitted to the surface tension at a single temperature. The GvdW theory, which includes binding energy, entropic and profile shape contributions, then generates surface tension estimates that are of good accuracy over the full range of available experimental data. It appears that, given a sufficiently flexible form, effective potentials combined with simple statistical mechanical theory can reproduce both bulk and non-uniform fluid data of great variety in an insighful and practically useful way.     

Mechanism analysis of interface characteristics of sequential Co-injection self-reinforced parts

The mechanism analysis of the interface characteristics of sequential co-injection self-reinforced parts starts from the following three aspects. Firstly, the formation process of the bonding interface and the factors affecting the bonding strength were investigated in theory. A theoretical model of interface bonding degree, which explains the bonding degree of the interface by the penetration depth of polymer chains through the initial solid-melt interface, was proposed, and then combines the average critical penetration depth to analyze the influence of different molding parameters on the interface bonding degree. Then, the interface bonding strength of the prepared polypropylene sequential co-injection self-reinforced parts under different molding parameters (including melt temperature, mold temperature and injection speed, etc.) was studied experimentally, and the reliability of the model was verified. Finally, the interrelationship function between effective penetration degree of molecular chain diffusion and interface bonding degree on both sides of the interface is established, which provides a new theoretical basis for interfacial mechanism analysis of sequential co-injection self-reinforced parts. The results show that the interface bonding degree between experiment and theory has a good consistency, and the deviation between them is within 10%, which proves that the theoretical model is reliable. In addition, the theoretical critical penetration depth and the theoretical maximum degree of bonding sequential co-injection polypropylene products are 1.399 nm and 0.857, respectively. The performance test results of different molding parameters show that the temperature change of reinforced melt is the key factor affecting the interfacial fusion of sequential co-injection self-reinforced parts. The high temperature melt is beneficial to the diffusion and intertwining of molecular chains on both sides of the interface and to the improvement of the degree of interfacial bonding.     

Interactions in the B(+)-RG complexes and comparison with Be(+)-RG (RG = He-Rn): evidence for chemical bonding

Potential energy curves for the interaction of B(+) ((1)S) with RG ((1)S), RG = He-Rn, have been calculated at the CCSD(T) level of theory employing quadruple-ζ and quintuple-ζ quality basis sets. The interaction energies from these curves were subsequently point-by-point extrapolated to the basis set limit. Rovibrational energy levels have been calculated for each extrapolated curve, from which spectroscopic parameters are determined. These are compared to previously determined experimental and theoretical values. The potentials have also been employed to calculate the transport coefficients for B(+) traveling through a bath of RG atoms. We also investigate the interactions between B(+) and the rare gases via contour plots, natural population analysis (NPA), and molecular orbital diagrams. In addition, we consider the atoms-in-molecules (AIM) parameters. The interactions here are compared and contrasted with those for Li(+)-He and Be(+)-RG; it is concluded that there is significant and increasing dative covalent bonding for the Be(+)-RG and B(+)-RG complexes for RG = Ar-Rn, while the other species are predominantly physically bound.     

纯流体在临界区内外的表面张力研究

An equation of state(EOS)applicable for both the uniform and non-uniform fluids was established by using thedensity-gradient expansion,in which the influence parameter к[p(r),T] was obtained by the use of direct correlationfunction.The density functional theory(DFT)provides a framework under which both the phase equilibria and in-terfacial properties can be investigated within a single set of molecular parameters.The phase equilibria inside thecritical region can be improved by the renormalization group theory(RGT).However,the correction of interracialproperties by DFT and RGT is computationally difficult.In the present work,the density gradient theory(DGT)inwhich к[p(r),T] is treated as a constant is used to combine with the RGT for interfacial properties inside the criticalregion.     

Surface tension and scaling of critical nuclei in diatomic and triatomic fluids

Density functional theory has been used to investigate surface tension and scaling of critical clusters in fluids consisting of diatomic and rigid triatomic molecules. The atomic sites are hard spheres with attractive interactions obtained from the tail part of the Lennard-Jones potential. Asymmetry in attractive interactions between the atomic sites has been introduced to cause molecular orientation and oscillatory density profiles at liquid-vapor interfaces. The radial dependence of cluster surface tension in fluids showing modest orientation in unimolecular layer at the interface or no orientation at all resembles the surface tension behavior of clusters in simple monoatomic fluids, although the surface tension maximum becomes more pronounced with increasing chain length of the molecule. Surface tension of clusters having multiple oscillatory layers at the interface shows a prominent maximum at small cluster sizes; however, the surface tension of large clusters is lower than the planar value. The scaling relation for the number of molecules in the critical cluster and the nucleation barrier height developed by McGraw and Laaksonen [Phys. Rev. Lett. 76, 2754 (1996)] are well obeyed for fluids with little structure at liquid-vapor interface. However, fluids having enhanced interfacial structure show some deviation from the particle number scaling, and the barrier height scaling breaks up seriously.     

Using diatomics-in-molecules in calculating semiempirical band structures

The diatomics-in-molecules approach usually limited to small-sized molecules is applied to infinitely extended systems, thereby relating the empirical potentials used as DIM input data to the well-known Slater-Koster parameters of tight-binding theory. Based on this parameters the band structure and the density of states of the rare-gas (RG) valence bands are calculated. Our results are in very good agreement with experimental data like the width of the bands and the vacuum-energy. Moreover, the characteristic features of the calculated density of states reflect the spectroscopically determined peak positions satisfactorily. We discuss further applications of this work.     

Crossover Peng-Robinson equation of state with introduction of a new expression for the crossover function

In this research, we use the original Peng-Robinson (PR) equation of state (EOS) for pure fluids and develop a crossover cubic equation of state which incorporates the scaling laws asymptotically close to the critical point and it is transformed into the original cubic equation of state far away from the critical point. The modified EOS is transformed to ideal gas EOS in the limit of zero density. A new formulation for the crossover function is introduced in this work. The new crossover function ensures more accurate change from the singular behavior of fluids inside the regular classical behavior outside the critical region. The crossover PR (CPR) EOS is applied to describe thermodynamic properties of pure fluids (normal alkanes from methane to n-hexane, carbon dioxide, hydrogen sulfide and R125). It is shown that over wide ranges of state, the CPR EOS yields the thermodynamic properties of fluids with much more accuracy than the original PR EOS. The CPR EOS is then used for mixtures by introducing mixing rules for the pure component parameters. Higher accuracy is observed in comparison with the classical PR EOS in the mixture critical region.     

Structure of liquid solutions near the critical point

Specific features of the structure of the critical state of binary liquid solutions leading to an anomalous behavior of the Rayleigh line due to a dramatic increase in concentration and density fluctuations are considered. It is shown that an experimental treatment must deal with two fluctuation regions near the critical point of solvent vaporization. In the first region, one can achieve a sufficient degree of accuracy by using theories like selfconsistent field theory. In the second region, which is closer to the critical point than the first region, scaling theory of secondorder phase transitions may be applied. It is found that the anomalous behavior of the Rayleigh line associated with kinetic coefficients is determined by the equilibrium thermodynamic properties and by the radius of fluctuation correlation (rinc). A general theory is developed for calculating thermodynamic potentials, especially the chemical potential and its concentration derivative in the fluctuation region. The results of these calculations are compared with the experimental data briefly described in the paper. Translated fromZhumal Strukturnoi Khimii, Vol. 39, No. 4, pp. 655–668, July–August, 1998.     

Analysis to obtain precise density fluctuation of supercritical fluids by small-angle X-ray scattering

A procedure of analysis for small-angle X-ray scattering (SAXS) data has been established to obtain density fluctuation of supercritical fluids near the critical point. It is indispensable for the certain analysis to utilize both of high-quality SAXS data measured under stable thermodynamic condition and accurate P–ρ–T data in supercritical region. As a standard example, SAXS measurements have been performed for supercritical CO₂, which is a suitable sample satisfying the condition for both experiment and analysis. The measurements were carried out along four isothermal conditions at reduced temperature of T_r = T/T_c = 1.020, 1.022, 1.043 and 1.064. Comparing the experimental density fluctuation with calculated one from the most reliable equation of state, the differences are within 8% at most.     

Density functional theory for semiflexible and cyclic polyatomic fluids

The effects of bond angle and chain stiffness on the structures of semiflexible polyatomic fluids are investigated by incorporating the bending potential into a density functional theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 2368 (2002)] that combines a modified fundamental measure theory for the excluded-volume effects and the first-order thermodynamics perturbation theory for the chain connectivity. The refined density functional theory faithfully reproduces the density profiles and conformational properties of a variety of triatomic fluids near a hard wall in which extensive Monte Carlo simulation data are available. In particular, the theory is able to capture the structures of rigid cyclic trimers where all segments are identical. The variation of local density profiles with respect to the chain length of confined polyatomic fluids is also explored. For quadratomic fluids confined in slit pores, the density profile of the middle segments exhibits novel double peaks that are absent in a fully flexible chain model. In addition, the density functional theory is applied to predicting the conformational properties and adsorption behavior of heterogeneous triatomic fluids of type "ABB" mimicking surfactant molecules. The competition between surface adsorption and self-association of trimers consisting of surface active and self-binding "A" segments and neutral "B" segment is explored.     

Efficient prediction of thermodynamic properties of quadrupolar fluids from simulation of a coarse-grained model: the case of carbon dioxide

Monte Carlo simulations are presented for a coarse-grained model of real quadrupolar fluids. Molecules are represented by particles interacting with Lennard-Jones forces plus the thermally averaged quadrupole-quadrupole interaction. The properties discussed include the vapor-liquid coexistence curve, the vapor pressure along coexistence, and the surface tension. The full isotherms are also accessible over a wide range of temperatures and densities. It is shown that the critical parameters (critical temperature, density, and pressure) depend almost linearly on a quadrupolar parameter q=Q(*4)T*, where Q* is the reduced quadrupole moment of the molecule and T* the reduced temperature. The model can be applied to a variety of small quadrupolar molecules. We focus on carbon dioxide as a test case, but consider nitrogen and benzene, too. Experimental critical temperature, density, and quadrupolar moment are sufficient to fix the parameters of the model. The resulting agreement with experiments is excellent and marks a significant improvement over approaches which neglect quadrupolar effects. The same coarse-grained model was also applied in the framework of perturbation theory in the mean spherical approximation. As expected, the latter deviates from the Monte Carlo results in the critical region, but is reasonably accurate at lower temperatures.     

Thermodynamic properties of Lennard-Jones chain molecules: renormalization-group corrections to a modified statistical associating fluid theory

A modified version of the statistical associating fluid theory (SAFT), the so-called soft-SAFT equation of state (EOS), has been extended by a crossover treatment to take into account the long density fluctuations encountered when the critical region is approached. The procedure, based on White's work from the renormalization group theory [Fluid Phase Equilibria 75, 53 (1992); L. W. Salvino and J. A. White, J. Chem. Phys. 96, 4559 (1992)], is implemented in terms of recursion relations where the density fluctuations are successively incorporated. The crossover soft-SAFT equation provides the correct nonclassical critical exponents when approaching the critical point, and reduces to the original soft-SAFT equation far from the critical region. The accuracy of the global equation is tested by direct comparison with molecular simulation results of Lennard-Jones chains, obtaining very good agreement and clear improvements compared to the original soft-SAFT EOS. Excellent agreement with vapor-liquid equilibrium experimental data inside and outside the critical region for the n-alkane series is also obtained. We provide a set of transferable molecular parameters for this family, unique for the whole range of thermodynamic properties.     

Dependence of the critical temperature on molecular parameters

At the critical temperature the surface tension between coexisting liquid and vapor phases must be zero, and the repulsive contributions associated with cavity formation must exactly counterbalance those from interactions of a molecule in the cavity and the bulk. An expression for the critical temperature of pure fluids in terms of the parameters of scaled particle theory (SPT) has been obtained, and the calculated critical temperatures are compared with experimental data for a range of pure fluids. These include noble and diatomic gases, short and medium length hydrocarbons, aromatic compounds, halogenated compounds, oxygen-containing compounds, and water. Considering the simplicity of this approach, a remarkably good correlation between calculated and experimental values is found for most of these fluids.