Colloidal aggregation in polymer blends

We consider here a low-density assembly of colloidal particles immersed in a critical polymer mixture of two chemically incompatible polymers. We assume that, close to the critical point of the free mixture, the colloids prefer to be surrounded by one polymer (critical adsorption). As result, one is assisted to a reversible colloidal aggregation in the nonpreferred phase, due the existence of a long-range attractive Casimir force between particles. This aggregation is a phase transition driving the colloidal system from dilute to dense phases, as the usual gas-liquid transition. We are interested in a quantitative investigation of the phase diagram of the immersed colloids. We suppose that the positions of particles are disordered, and the disorder is quenched and follows a Gaussian distribution. To apprehend the problem, use is made of the standard phi(4) theory, where the field phi represents the composition fluctuation (order parameter), combined with the standard cumulant method. First, we derive the expression of the effective free energy of colloids and show that this is of Flory-Huggins type. Second, we find that the interaction parameter u between colloids is simply a linear combination of the isotherm compressibility and specific heat of the free mixture. Third, with the help of the derived effective free energy, we determine the complete shape of the phase diagram (binodal and spinodal) in the (Psi,u) plane, with Psi as the volume fraction of immersed colloids. The continuous "gas-liquid" transition occurs at some critical point K of coordinates (Psi(c) = 0.5,u(c) = 2). Finally, we emphasize that the present work is a natural extension of that, relative to simple liquid mixtures incorporating colloids.     

Phase behavior of attractive and repulsive ramp fluids: integral equation and computer simulation studies

Using computer simulations and a thermodynamically self-consistent integral equation we investigate the phase behavior and thermodynamic anomalies of a fluid composed of spherical particles interacting via a two-scale ramp potential (a hard core plus a repulsive and an attractive ramp) and the corresponding purely repulsive model. Both simulation and integral equation results predict a liquid-liquid demixing when attractive forces are present, in addition to a gas-liquid transition. Furthermore, a fluid-solid transition emerges in the neighborhood of the liquid-liquid transition region, leading to a phase diagram with a somewhat complicated topology. This solidification at moderate densities is also present in the repulsive ramp fluid, but in this case inhibits the fluid-fluid separation.     

Kinetic appearance of first-order gas-liquid expanded and liquid expanded-liquid condensed phase transitions below the triple point

Phase diagram of Gibbs monolayers of mixtures containing n-hexadecyl phosphate (n-HDP) and L-arginine (L-arg) at a molar ratio of 1:2 has been constructed by measuring surface-pressure-time (pi-t) isotherms with film balance and by observing monolayer morphology with Brewster angle microscopy (BAM). This phase diagram shows a triple point for gas (G), liquid expanded (LE), and liquid condensed (LC) phases at around 6.7 degrees C. Above this triple point, a first-order G-LE phase transition occurring at 0 surface pressure is followed by another first-order LE-LC phase transition taking place at a certain higher surface pressure that depends upon temperature. The BAM observation supports these results. Below the triple point, the pi-t measurements show only one first-order phase transition that should be G-LC. All of these findings are in agreement with the general phase diagram of the spread monolayers. However, the BAM observation at a temperature below the triple point shows that the thermodynamically allowed G-LC phase transition is, in fact, a combination of the G-LE and LE-LC phase transitions. The latter two-phase transitions are separated by time and not by the surface pressure, indicating that the G-LC phase transition is kinetically separated into these two-phase transitions. The position of the LE phase below the triple point in the phase diagram is along the phase boundary between the G and LC phases.     

Phase diagrams of binary mixtures of oppositely charged colloids

Phase diagrams of binary mixtures of oppositely charged colloids are calculated theoretically. The proposed mean-field-like formalism interpolates between the limits of a hard-sphere system at high temperatures and the colloidal crystals which minimize Madelung-like energy sums at low temperatures. Comparison with computer simulations of an equimolar mixture of oppositely charged, equally sized spheres indicate semiquantitative accuracy of the proposed formalism. We calculate global phase diagrams of binary mixtures of equally sized spheres with opposite charges and equal charge magnitude in terms of temperature, pressure, and composition. The influence of the screening of the Coulomb interaction upon the topology of the phase diagram is discussed. Insight into the topology of the global phase diagram as a function of the system parameters leads to predictions on the preparation conditions for specific binary colloidal crystals.     

The phase behavior of two-dimensional symmetrical mixtures in a weak external field of square symmetry

Using Monte Carlo simulation methods in the grand canonical and semigrand canonical ensembles, we study the phase behavior of two-dimensional symmetrical binary mixtures of Lennard-Jones particles subjected to a weakly corrugated external field of a square symmetry. It is shown that the both vapor-liquid condensation and demixing transition in the liquid phase are not appreciably affected by a weakly corrugated external field. On the other hand, even a weakly corrugated external field considerably influences the structure of solid phases and the liquid-solid transition. In particular, the solid phases are found to exhibit uniaxially ordered distorted hexagonal structure. The triple point temperature increases with the corrugation of the external field, while the triple point density becomes lower when the surface corrugation increases. The changes in the location of the triple point are shown to lead to the changes of the phase diagram topology. It is also demonstrated that the solid phase undergoes a demixing transition, which is also very slightly affected by the weakly corrugated external potential. The demixing transition in the solid phase is shown to belong to the universality class of the Ising model.     

Folding kinetics of a polymer

We present the results of computer simulations giving a kinetic insight into the liquid-to-solid transition of a homopolymer chain with short-range interactions. By calculating the absolute rates in each direction of the transition, using molecular dynamics employing the forward flux sampling scheme, we provide the phase diagram based on purely kinetic data, and compare it with the results from Monte Carlo simulations. Additionally, we present and discuss a remarkably simple and general relation between the polymer topology and the folding pathway, and show that the eigenvalue spectrum of a matrix defined by non-bonded contacts (the Laplacian matrix) provides an insight into the nonequilibrium ensembles of these trajectories. In particular, the Laplacian matrix seems to identify a large fraction of configurations on the folding pathway at the free energy maximum that have a very low probability of reaching the crystallized state. This implies that the eigenvalues of this matrix may be suitable additional reaction coordinates to describe the folding transition of chain molecules.     

Microphase segregation in bridging polymeric brushes: Regular and singular phase diagrams

A mean-field theory of deformation-induced microphase segregation in bridging polymeric brushes anchored to two parallel surfaces is presented. Models with isotropic and orientation-dependent liquid-crystalline interactions between segments are considered. For the first model, the problem is similar to that of classical liquid-vapor phase separation, and the phase diagram in the P-T plane has a line of first-order transitions terminating at the critical point. We show that the critical pressure is negative implying that a free brush tethered only to one surface always exists at supercritical conditions and hence cannot undergo the collapse phase transition. In the second model, the free energy density depends on two coupled order parameters, one related to segment density and the other to the orientational order, which strongly modifies the phase behavior. Depending on the grafting density the system is described by a phase diagram of a regular or a singular type. In the regular phase diagram the first-order transition line terminates at the critical point. In a singular diagram, the first-order transition line extends to infinity; the critical point corresponds to infinite pressure so that the system undergoes the phase transition at arbitrary external pressures. Regular phase diagrams correspond to dense grafting, and singular ones to sparse grafting. The change from a regular phase behavior to another occurs at a certain marginal value of the grafting density. On approaching this value the critical point on the regular diagram moves to infinity, logarithmically with the deviation from the critical grafting density. We relate the analytical properties of the free energy density as a function of the segment concentration to the type of the phase diagram and the shape of the coexistence curve in the temperature- concentration plane.     

Phase Behavior of Solutions of Polymers with Multiply Aggregating Groups

We have theoretically investigated phase diagrams of solutions of aggregating polymers. In our model all polymer chains in the solution carry a certain number of aggregating groups (“stickers”), capable of the formation of thermoreversible aggregates, each assumed to consist of exactly m stickers. The treatment of aggregation corresponds to the Flory‐like theory of gelation. We have studied the dependence of the phase diagram on the parameters of aggregation, such as its strength (fraction of monomeric units carrying stickers and aggregation constant) and cooperativity (aggregation number). In the considered system of homopolymer chains the stickers have been found to always promote phase separation. The value of aggregation number m is crucial for the topology of the phase diagram: for m ≥ 5 the triple point may appear in the phase diagram. Conditions on the properties of the system ensuring the appearance of this feature have been obtained. The interrelation between gelation and phase separation for different values of parameters is elucidated.     

Phase and orientational ordering of low molecular weight rod molecules in a quenched liquid crystalline polymer matrix with mobile side chains

We study the phase diagram and orientational ordering of guest liquid crystalline (LC) rods immersed in a quenched host made of a liquid crystalline polymer (LCP) matrix with mobile side chains. The LCP matrix lies below the glass transition of the polymer backbone. The side chains are mobile and can align to the guest rod molecules in a plane normal to the local LCP chain contour. A field theoretic formulation for this system is proposed and the effects of the LCP matrix on LC ordering are determined numerically. We obtain simple analytical equations for the nematic/isotropic phase diagram boundaries. Our calculation show a nematic-nematic (N/N) first order transition from a guest stabilized to a guest-host stabilized region and the possibility of a reentrant transition from a guest stabilized nematic region to a host only stabilized regime separated by an isotropic phase. A detailed study of thermodynamic variables and interactions on orientational ordering and phases is carried out and the relevance of our predictions to experiments and computer simulations is presented.     

Orientational ordering in hard rectangles: The role of three-body correlations

We investigate the effect of three-body correlations on the phase behavior of hard rectangle two-dimensional fluids. The third virial coefficient B3 is incorporated via an equation of state that recovers scaled particle theory for parallel hard rectangles. This coefficient, a functional of the orientational distribution function, is calculated by Monte Carlo integration, using an accurate parametrized distribution function, for various particle aspect ratios in the range of 1-25. A bifurcation analysis of the free energy calculated from the obtained equation of state is applied to find the isotropic (I)-uniaxial nematic (N(u)) and isotropic-tetratic nematic (N(t)) spinodals and to study the order of these phase transitions. We find that the relative stability of the N(t) phase with respect to the isotropic phase is enhanced by the introduction of B3. Finally, we have calculated the complete phase diagram using a variational procedure and compared the results with those obtained from scaled particle theory and with Monte Carlo simulations carried out for hard rectangles with various aspect ratios. The predictions of our proposed equation of state as regards the transition densities between the isotropic and orientationally ordered phases for small aspect ratios are in fair agreement with simulations. Also, the critical aspect ratio below which the N(t) phase becomes stable is predicted to increase due to three-body correlations, although the corresponding value is underestimated with respect to simulation.     

Phenomenology of Polymorphism: IV. The Trimorphism of Ferrocene and the Overall Metastability of Its Triclinic Phase

Structural and thermodynamic data reported in the literature for the three crystalline phases of ferrocene are used to build a topological representation of its (p, T) phase diagram. Two phases (orthorhombic and monoclinic) exhibit stable phase regions whose temperature ranges increase with increasing pressure. The triclinic phase is found to be metastable at any temperature and pressure. In fact, in the (p, T) diagram, it is not associated with any state of lowest energy although it transforms into the monoclinic phase according to a transition which is reversible. This transition occurs in the phase region where the orthorhombic phase is the stable one.     

An algorithm for semi-empirical design of nucleation rate surface

During the last half of century, Classical Nucleation Theory (CNT) has been developed and there have been advances in the molecular theory of nucleation. Most of these efforts have been directed towards small molecule system modeling using intermolecular potentials. Summarizing the nucleation theory, it can be concluded that the current theory is far from complete. Agreement is generally not obtained between experimental and theoretical results. In practical applications, parametric theories can be used for the systems of interest. However, experimental measurements are still the best source of information on nucleation. Experiments are labor intensive and costly, and thus, it is useful to extend the value of limited experimental measurements to a broader range of nucleation conditions. The available nucleation parameters represent only small regions of possible nucleation conditions over the range from the critical temperature to absolute zero. Thus, it is useful to develop better tools to use the data to estimate semi-empirical nucleation rate surfaces. Following our published approach, the nucleation rate surface for any system can be constructed over its phase diagram. This concept involves using the phase equilibrium diagram to establish lines of zero nucleation rates. Nucleation rate surfaces arise from equilibrium lines and their extensions that are representing unstable equilibria. Only limited experimental data is available for use in normalizing the slopes of the linearized nucleation rate surfaces. The nucleation rate surface is described in terms of steady-state nucleation rates. To design the surfaces of nucleation rates, several assumptions are presented. In the present study, an algorithm for the semi-empirical design of nucleation rate surfaces is introduced. The topology of the nucleation rate surface for a unary system using the example of water vapor nucleation is created semi-empirically. The nucleation of two concurrent (stable and unstable) phase states of critical embryos is considered in the context of multi-surface nucleation rates. Only one phase transition (melting) in the condensed state of water is considered for simplicity. The nucleation rate surface is constructed numerically using the available experimental results for vapor nucleation and phase diagram for water. The nucleation rate for water vapor is developed for the full temperature interval, i.e. from critical point to absolute zero. The results help to suggest a new direction for experimental nucleation research.     

Several routes to the glassy states in the one component soft core system: revisited by molecular dynamics

Molecular dynamics simulations have been performed to study the glass transition for the soft core system with a pair potential φ(n)(r) = ε(σ∕r)(n) of n = 12. Using the compressibility factor, PV/Nk(B)T=P?(ρ*), its phase diagram can be represented as a function of a reduced density, ρ? = ρ(ε∕k(B)T)(3∕n), where ρ = Nσ(3)∕V. In the present work, NVE relaxations to the glassy or crystalline states starting from the unstable states in the phase diagram have been revisited in details and compared with other processes. Relaxation processes can be characterized by the time dependence of the dynamical compressibility factor (PV/Nk(B)T)(t)?(≡g(ρ(t)*)) on the phase diagram. In some cases, g(ρ(t)*) reached a crystal branch in the phase diagram; however, metastable states are found in many cases. With connecting points for the metastable states in the phase diagram, we can define a glass branch where the dynamics of particles are almost frozen. The structures observed there have common properties characterized as glasses. Although overlaps of glass forming process and nanocrystallization process are observed in some cases, these behaviors are distinguishable to each other by the characteristics of structures. There are several routes to the glass branch and we suggest that all of them are the glass transition.     

Phase behavior of the lattice restricted primitive model with nearest neighbor exclusion

The global phase behavior of the lattice restricted primitive model with nearest neighbor exclusion has been studied by grand canonical Monte Carlo simulations. The phase diagram is dominated by a fluid (or charge-disordered solid) to charge-ordered solid transition that terminates at the maximum density rho*(max)= sqrt 2 and reduced temperature T* approximately equal to 0.29. At that point, there is a first-order phase transition between two phases of the same density, one charge-ordered, and the other charge-disordered. The liquid-vapor transition for the model is metastable, lying entirely within the fluid-solid phase envelope.     

Structural Features of Solid Phases and Topology of Phase Diagrams

The paper considers model phase diagrams of binary and ternary systems involving transformations (transitions) of phase equilibria. The relationship between the type of structural solidstate transformation and the type of phase diagram is shown. Topological series of phase diagrams of systems with continuous and limited types of solid solutions are considered, including phase diagrams with polymorphic transitions between intermediate solid phases of variable composition.     

Type-IV phase behavior in fluids with an internal degree of freedom

We have identified a fourth archetype of phase diagram in binary symmetrical mixtures, which is encountered when the ratio of the interaction between the unlike and the like particles is sufficiently small. This type of phase diagram is characterized by the fact that the lambda line (i.e., the line of the second-order demixing transition) intersects the first-order liquid-vapor curve at densities smaller than the liquid-vapor critical density.     

Lateral phase separation in cholesterol/diheptadecanoylphosphatidylcholine binary bilayer membrane

We investigated the phase behavior of cholesterol/diheptadecanoylphosphatidylcholine (C17:0-PC) binary bilayer membrane as a function of the cholesterol composition (X(ch)) by fluorescence spectroscopy using 6-propionyl-2-(dimethylamino)naphthalene (Prodan) and differential scanning calorimetry (DSC). The fluorescence spectra showed that the wavelength at the maximum intensity (lambda(max)) changed depending on the bilayer state: ca. 440 nm for the lamellar gel ( [Formula: see text] or L(beta)) and the liquid ordered (L(o)) phases and ca. 490 nm for the liquid-crystalline (L(alpha)) phase. The transition temperatures were determined from the temperature dependence of lambda(max) and endothermic peaks of the DSC thermograms. Both measurements showed that the pre- and main transition disappear around X(ch)=0.05 and 0.30, respectively. The constructed temperature-X(ch) phase diagram resembled a typical phase diagram for a eutectic binary mixture containing a peritectic point. The presence of a peritectic point at X(ch)=0.15 suggested that a complex of cholesterol and C17:0-PC is stoichiometrically formed in the gel phase. Consideration based on the hexagonal lattice model revealed that the compositions of 0.05 and 0.15 correspond to the bilayer states where cholesterol molecules are regularly distributed in different ways. The former is nearly equal to the composition for the membrane occupied entirely with Units (1:18), composed of a cholesterol and 18 surrounding C17:0-PC molecules within the next-next nearest neighbor sites. The latter is represented by a Unit (1:6), including a cholesterol and 6 surrounding C17:0-PC molecules. Further, the disappearance of the main transition at X(ch)=0.30 indicates that the pure L(o) phase can exist in X(ch)>0.30. The eutectic behavior observed in the phase diagram was explainable in terms of phase separation between two different types of regions with different types of regular distributions of cholesterol.     

Phase Diagram of an Ethylene Glycol–Hexamethylphosphorotriamide System

The phase diagram of an ethylene glycol (EG)–hexamethylphosphorotriamide (HMPT) system is studied over two wide temperature intervals (+25°С…?90°С…+40°С) and (?150°С…+40°С) by means of differential scanning calorimetry using INTERTECH DSC Q100 and METTLER TA4000 DSC instruments (Switzerland) in the DSC30 mode with variable cooling/heating rates. Substantial overcooling of the liquid phase, a glass transition, and different types of interaction are observed in the system. No thermal effects are observed in intermediate range of concentrations during the slow cooling/heating processes, and the system remains liquid until the glass transition. The presence of such a metastable phase is attributed to a sharp rise in the viscosity of the system due to different kinds of interaction between the components. HMPT: 2EG and HMPT: EG compounds with crystallization temperatures of +5 and ?0.5°С, respectively, are observed upon rapid cooling and slow heating. Changes in enthalpy are calculated for all of the observed thermal effects. The distinction from the phase diagram of H₂O–HMFT (literary data) is explained by the difference in the interactions between system components and by the structural differences between EG and H₂O.