Theoretical investigation of the phase behaviour of model ternary mixtures containing n-alkanes,perfluoro-n-alkanes,and perfluoroalkylalkane diblock surfactants

We have used the hetero-SAFT-VR approach developed by McCabe and collaborators [Mol. Phys. 104, 571 (2006)] to investigate the phase equilibria of a number of binary and ternary mixtures of n-alkanes, perfluoro-n-alkanes, and perfluoroalkylalkane diblock surfactants. We focused our work on the understanding of the microscopic conditions that control the phase behaviour of these mixtures, with a particular emphasis of the effect on the liquid–liquid separation and the stabilisation of n-alkane + perfluoro-n-alkane mixtures when a diblock surfactant is added. We used very simple molecular models for n-alkanes, and perfluoro-n-alkanes that describe the molecules as chains with tangentially bonded segments with molecular parameters taken from the literature. In the particular case of semifluorinated alkanes or SFA surfactants, we used an hetero-segmented diblock chain model where the parameters for the alkyl and perfluoroalkyl segments taken from the corresponding linear alkanes and perfluoroalkanes, as shown in our previous work [J. Phys. Chem. B 111, 2856 (2007)]. Our goal was to identify the main effects on the phase behaviour when different perfluoroalkylalkane surfactants are added to mixtures of n-alkanes and perfluoro-n-alkanes. We selected the n-heptane + perfluoromethane binary mixture, and studied the changes on the phase behaviour when a symmetric (same number of alkyl and perfluoroalkyl chemical groups) or an asymmetric (different number of alkyl and perfluoroalkyl chemical groups) diblock surfactants is added to the binary mixture. We have obtained the phase diagrams of a wide range of binary and ternary mixtures at different thermodynamic conditions. We have found a variety of interesting behaviours as we modify the alkyl or/and the perfluoroalkyl chain-length of the diblock surfactants: the usual changes in the vapour–liquid phase separation, changes in the type of phase diagrams (typically from type I to type V phase behaviour according to the Scott and Konynenburg classification), azeotropy, and Bancroft points. We noted that the main effect of adding a symmetric or an asymmetric surfactant to the n-heptane + perfluoromethane mixture is to stabilise the system, i.e. to decrease the two-phase (liquid–liquid) immiscibility region of the ternary diagram as the surfactant concentration is increased. This effect becomes larger as the chain length of the surfactant is increased, which is consistent with a higher number of alkyl–alkyl and perfluoroalkyl–perfluoroalkyl favourable interactions in the mixture.     

On the thermodynamics of diblock chain fluids from simulation and heteronuclear statistical associating fluid theory for potentials of variable range

The SAFT-VR equation of state is extended to treat heteronuclear chain fluids, focusing, in particular, on symmetric and asymmetric diblock chains. The chain molecules studied are composed of segments of different size and/or energy of interaction. Both symmetric and asymmetric systems are considered. The theoretical predictions are compared with isothermal–isobaric and Gibbs ensemble Monte Carlo simulation data. Excellent agreement is obtained between the hetero-SAFT-VR predictions and the simulation data, validating the use of the SAFT-VR approach for heteronuclear chains in more realistic models of polymers and small molecules composed of different functional groups.     

Coarsening and self-organization in dilute diblock copolymer melts and mixtures

This paper explores the evolution of a sharp interface model for phase separation of copolymers in the limit of low volume fraction. Particles both exchange material as in usual Ostwald ripening, and migrate because of an effectively repulsive nonlocal energetic term. Coarsening via mass diffusion only occurs while particle radii are small, and they eventually approach a finite equilibrium size. Migration, on the other hand, is responsible for producing self-organized patterns.We construct approximations based upon an ansatz of spherical particles similar to the classical LSW theory to derive finite dimensional dynamics for particle positions and radii. For large systems, kinetic-type equations which describe the evolution of a probability density are constructed. For systems larger than the screening length, we obtain an analog of the homogenization result of Niethammer & Otto [B. Niethammer, F. Otto, Ostwald ripening: The screening length revisited, Calc. Var. Partial Differential Equations 13-1 (2001) 33-68]. A separation of timescales between particle growth and migration allows for a variational characterization of spatially inhomogeneous quasi-equilibrium states.     

Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn–Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L¹/m, where h is the space grid size, L¹ is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.     

Theory of lateral and flip-flop phase separations in bilayer biomembranes and surfactants

In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A and B. Under a variation of a suitable parameter, such as temperature or difference of lengths of hydrophobic chains, these systems undergo a phase separation from a homogeneous liquid-phase to two distinct liquid-phases. Two physical situations can be distinguished: (1) The amphiphiles A and B prefer to jump from a monolayer to the other (flip-flop transition), (2) the mixture phase separates on each monolayer, and there is no jump from one sheet towards the second one (lateral transition). To investigate the associated critical phase behavior, we first introduce a field theory, constructed with two order parameters (or fields) φ and ψ, which are nothing else but the composition fluctuations relative to the monolayers. Beside the usual terms proportional to φ², ψ², φ⁴ and ψ⁴, the free energy contains an extra one, −Cφψ, which describes the lowest order coupling between the two monolayers. The coupling constant C is positive for the lateral phase separation, and negative for the vertical one. We show that its sign results from a competition between the chemical segregation of amphiphiles and the curvature asymmetry. With the help of this free energy, we first identify the liquid-phases, and show the existence of a critical point, T_c, of which the location depends naturally on the value of the coupling constant C. In particular, for those bilayer biomembranes or surfactants made of amphiphiles of the same chemical nature but with different lengths, and at fixed temperature, we show the existence of a critical line in the -plane, along which the bilayer undergoes a phase separation. Here, and account for the curvature gap and the length difference, respectively. Second, we determine the behavior of the composition fluctuations, φ and ψ, and the total one, Φ=φ+ψ, upon temperature, T, and chemical potential difference, , in the critical region. Third, we determine the critical behavior of the partial compressibilities, κ_φφ, κ_ψψ and κ_φψ, and the overall one, . Finally, we remark that the flip-flop phase separation shows some analogy with the classical para-ferrimagnetic transition of coupled paramagnetic materials of Curie-Weiss type.     

超临界二氧化碳和模型共聚物的相平衡和成核性质研究

在Weeks-Chandler-Andersen （WCA）微扰理论的基础上,建立了一个状态方程,研究了温度和压力以及模型共聚物分子链长对体系相平衡和临界胶束浓度的影响. 关键词： 超临界二氧化碳 模型共聚物 相平衡 临界胶束浓度     

Morphology, photocatalytic and antibacterial activities of radial spherical ZnO nanorods controlled with a diblock copolymer

Radial spherical ZnO nanorods were synthesized directly from an aqueous zinc acetate dihydrate solution in the presence of the poly(ethylene oxide)-b-poly(propylene oxide) copolymer at a mole ratio of Zn²⁺:OH⁻ = 1:10. The diameter of the hexagonal facet and the length of each rod decreased with an increase of the copolymer concentrations. The blue-shift in the optical band gap was caused by an increase of the compressed lattice. The efficiency of photocatalytic degradation of methylene blue in aqueous solution increased with an increase of their surface areas. However, the decrease of their sizes did not improve their antibacterial activities.     

Molecular simulation of phase equilibria for mixtures of polar and non-polar components

Grand canonical histogram-reweighting Monte Carlo simulations were used to obtain the phase behaviour of several binary mixtures. The main goal of this work was to test the predictive capabilities of recently developed intermolecular potential models that accurately reproduce the phase behaviour of pure components. These united-atom potentials utilize the exponential-6 functional form for repulsive and dispersion interactions and fixed point charges for electrostatic interactions. The mixtures studied were n-pentane—methane, ethane—CO₂, propane—CO₂, n-pentane-CO₂, H₂O-ethane, CH₃OH-n-hexane and CH₃OH-CO₂. The conventional Lorentz-Berthelot combining rules, as well as a set of combining rules due to Kong (1973, J. chem. Phys., 59, 2464) were used to obtain unlike-pair potential parameters. The Lorentz—Berthelot rules generally result in more attractive unlike-pair interactions than the Kong rules. For the n-alkane—CO₂ systems, predicted phase diagrams are in excellent agreement with experiment when the Kong combining rules are used. For mixtures with CH₃OH and H₂O, the Lorentz—Berthelot rules yield better agreement with experiment than the Kong rules, but statistically significant differences remain. Our results suggest that relatively simple intermolecular potential models can be used to predict the phase behaviour of broad classes of binary systems. For mixtures with large differences in polar character of the components, however, present models do not predict the phase behaviour in quantitative agreement with experiment. New models that include higher order interactions such as polarizability may be suitable for this purpose, a hypothesis that will need to be tested in the future.     

Experimental investigation and thermodynamic calculation of the Bi-Ga-Sn phase equilibria

Binary thermodynamic data, successfully used for phase diagram calculations of binary systems Bi-Ga, Bi-Sn, and Ga-Sn, were used for prediction of phase equilibria in ternary Bi-Ga-Sn system. The thermodynamic functions, such as enthalpy of formation and activity, were calculated using the Redlich-Kister-Muggianu model and compared with experimental data reported in the literature. The liquidus surface, invariant equilibria and three vertical sections with molar ratio Ga:Sn=1, Bi:Sn=1 and Bi:Ga=1 of the Bi-Ga-Sn ternary system were calculated by the CALPHAD method. Alloys, situated along three calculated vertical sections, were investigated by Differential Scanning Calorimetry (DSC). The experimentally determined phase transition temperatures were compared with calculation results and good mutual agreement was noticed.     

Theoretical study on tailoring symmetric and asymmetric thin films of diblock copolymers

We used density functional theory (DFT) to investigate the formation of symmetric and asymmetric thin film of diblock copolymer melts by tuning the size of the slit confinement. In this work, the DFT contains a modified fundamental measure theory for the excluded volume effect and the first-order thermodynamic perturbation theory for the chain connectivity as well as the mean-field approximation for van der Waals attraction. For the symmetric A₈B₈ linear copolymers, it is observed that with the increase of the width of the slit, morphologies of copolymer in the slits undergo an evolution of “non-layered structure → ABA → ABAB → BABAB → disordered structure”, while the morphologies of asymmetric copolymer with the increase of the width of the slit exhibit a process of “ABA → ABAB → ABABA → ABABAB lamellar structure” in all the cases studied. It suggests that the ratio of two blocks of a copolymer plays an important role on the structure of copolymer film. By adjusting the ratio of two blocks, some copolymer films with novel morphologies, including asymmetric ABAB lamellar structure, can be tailored. Furthermore, it is found that the bonding orientation distribution introduced into the DFT can act as a criterion to identify the disordered and ordered states of copolymers.     

Spectral method for exploring patterns of diblock copolymers

A numerical method in Fourier-space is developed to solve the polymeric self-consistent field equations. The method does not require a priori symmetric information. More significantly, periodic structure can be adjusted automatically during the iteration process. In this article, we apply our method to AB linear diblock copolymer melt, thus reproduce all known stable phases, and reveal some meta-stable phases. It is worthy to point out that we also give an efficient strategy to estimating initial values for diblock copolymer system. Finally, by comparing with Matsen–Schick’s method, we show some advantages of our method.     

Employment, Production and Consumption model: Patterns of phase transitions

We have simulated the model of Employment, Production and Consumption (EPC) using Monte Carlo. The EPC model is an agent based model that mimics very basic rules of industrial economy. From the perspective of physics, the nature of the interactions in the EPC model represents multi-agent interactions where the relations among agents follow the key laws for circulation of capital and money. Monte Carlo simulations of the stochastic model reveal phase transition in the model economy. The two phases are the phase with full unemployment and the phase with nearly full employment. The economy switches between these two states suddenly as a reaction to a slight variation in the exogenous parameter, thus the system exhibits strong non-linear behavior as a response to the change of the exogenous parameters.     

Thermodynamic properties and phase equilibria of charged hard sphere chain model for polyelectrolyte solutions

A recent thermodynamic perturbation theory for flexible polyelectrolyte solutions is extended to study thermodynamic properties and phase equilibria for polyelectrolyte solutions with different polyion, chain lengths. Osmotic pressure, activity coefficient of an individual ion (polyion or counterion) and average activity coefficient are calculated. Except for the activity coefficient of the polyion, the chain length dependence of the properties is small. Also presented are vapour-liquid equilibrium coexisting curves, vapour pressures and critical points. As the polyion chain length approaches infinity the critical properties (temperature, pressure and density) become constants.     

Analysis of phase transition points for a two-color agent-based model

The agent-based model treated in the present study describes dynamics of two types of population in a gravity-like potential field. In previous studies, the model was known to exhibit various spatiotemporal patterns on two-dimensioanl lattice systems. However, the patterns were classified depending purely on eye observations, and the underlying dynamics of these patterns were not fully explored. It remained a question to be answered if these eye observation-based classifications could be confirmed by any analytical means. To pursue the question, we first suggest several analytic quantities, such as convergence time steps and reaction speed, to replace the eye observations. As a result, we show that a phase diagram can be reasonably drawn on the contour diagram of the time steps. In addition, we find a power-law scaling in the reaction speed, confirming that a phase transition really is involved there. Next, as a main part of the present study, we apply analytical methods to calculate two important phase transition points from the system. The results from the analytical approach agreed well with the numerically obtained phase transition points from the agent-based model. In general, the paper serves as an example study of estimating global phenomena of complex systems in terms of local parameters of the system.     

Experimental study and thermodynamic calculation of Au–Bi–Sb system phase equilibria

Phase equilibria in the Au–Bi–Sb ternary system have been studied experimentally and calculated by the CALPHAD method. Three calculated isopleths with molar ratios Au:Bi=1, Bi:Sb=1 and Au:Sb=1 were compared with the DTA results from this work. The liquidus projection has been calculated. Two ternary invariant reactions were noted. Calculated phase diagram of isothermal section at 573 K was compared with the results of SEM/EDX analysis.     

Conversion from phase map to coordinate: Comparison among spatial carrier, Fourier transform, and phase shifting methods

For full-field phase measurement methods, many algorithms have been developed to extract a phase map from fringe image(s). Both phase wrapping and unwrapping algorithms have been extensively investigated by many researchers, but few papers can be found on how to calculate the coordinates of surface points from a phase map. This paper focuses on algorithms that show how a phase map can be used to calculate coordinates. Details are given for single image methods such as Fourier transform, spatial carrier methods, and multiple image methods like traditional phase-shifting methods. Algorithms that can be used to convert a phase map to coordinates and some issues related to these conversion algorithms are discussed. An artifact is measured using these phase measurement methods. The results show that using the correct algorithm to convert a phase map to coordinates is a key to obtaining accurate measurement results.     

Induction of B1 phase in binary mixtures of rod- and bent-shaped mesomorphic compounds

The miscibility of a rod-shaped (R) and a bent-shaped (BC) compound forming an induced B₁ phase is reported. B₁ phase is not present in both R and BC in the entire phase sequence. R exhibits partially bilayered smectic A_d phase with cholesteric and TGB while BC shows a B₂ phase. It is interesting to show that the B₂ phase is totally diminished in low composition (weight%) of BC, B₁ phase is induced at lower temperature range (~90-135 °C). A dielectric study is also incorporated to confirm the induction of B₁ phase in the binary mixture.     

Comparison of phase behavior between water soluble and insoluble surfactants at the air-water interface

The surface phase behavior of 2-hydroxyethyl myristate (2-HEM) has been studied in Langmuir monolayers by measuring surface pressure (π)-area (A) isotherms with a film balance and observing monolayer morphology with a Brewster angle microscope (BAM). These results are compared with the phase behavior of 2-hydroxyethyl laurate (2-HEL) in Gibbs monolayers studied by measuring π-time (t) curves and observing monolayer morphology. The π-A isotherms of 2-HEM show a first-order phase transition from a liquid expanded (LE) phase to a liquid condensed (LC) phase in the temperature range between 5 and 35 °C whereas the π-t curves of 2-HEL represent a similar phase transition in the temperature range between 2 and 25 °C. The critical surface pressure, π_c necessary for the phase transitions increases with increasing temperature in both the cases. The LC domains formed in 2-HEM show circular shapes, which are independent of the temperature. In contrast, the circular domains having stripe texture formed at lower temperatures show a shape transition to fingering domains with uniform brightness at 15 °C. The amphiphile, 2-HEM having 13-carbon chain has higher line tension than 2-HEL that has 11-carbon chain as tail. Thus, for 2-HEM, this high line tension always dominates over other factors giving rise to circular domains at the all studied temperatures.