Phase coexistence in polydisperse athermal polymer-colloidal mixture

A theoretical scheme developed earlier [Y. V. Kalyuzhnyi et al., Chem. Phys. Lett. 443, 243 (2007)] is used to calculate the full phase diagram of polydisperse athermal polymer-colloidal mixture with polydispersity in both colloidal and polymeric components. In the limiting case of bidisperse polymer-colloidal mixture, theoretical results are compared against computer simulation results. We present the cloud and shadow curves, critical binodals, and distribution functions of the coexisting phases and discuss the effects of polydispersity on their behavior. According to our analysis polydispersity extends the region of the phase instability, shifting the critical point to the lower values of the pressure and density. For the high values of the pressure polydispersity causes strong fractionation effects, with the large size colloidal particles preferring the low-density shadow phase and long chain length polymeric particles preferring the high-density shadow phase.     

Examining the effect of chain length polydispersity on the phase behavior of polymer solutions with the statistical associating fluid theory (Wertheim TPT1) using discrete and continuous distributions

Polymers are naturally polydisperse. Polydispersity may have a large effect on the phase behavior of polymer solutions, in particular, on the liquid-liquid phase equilibria. In this paper, we determine the cloud and shadow curves bounded by lower critical solution temperatures for a number of polymer+solvent systems where the polymer is polydisperse in terms of molecular weight (chain length). The moment method [P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem. Phys. 116, 265 (2001)] is applied with the SAFT approach to determine cloud and shadow curves with continuous Schulz-Flory distributions. It is seen that chain length polydispersity always enhances the extent of liquid-liquid phase equilibria. The predicted cloud curves obtained for continuous distributions are very similar to those obtained for simple ternary mixtures with the same polydispersity index, while the corresponding shadow curves can be very different depending on the composition of the parent distribution. The ternary phase behavior can be used to provide an understanding of the shape of the cloud and shadow curves. Regions of phase equilibria between three liquid phases are found for ternary systems when the chain length distribution is very asymmetrical; such regions are not observed for Schulz-Flory distributions even in the case of a large degree of polydispersity.     

Liquid-gas coexistence and critical point shifts in size-disperse fluids

Specialized Monte Carlo simulations and the moment free energy (MFE) method are employed to study liquid-gas phase equilibria in size-disperse fluids. The investigation is made subject to the constraint of fixed polydispersity, i.e., the form of the "parent" density distribution rho(0)(sigma) of the particle diameters sigma, is prescribed. This is the experimentally realistic scenario for, e.g., colloidal dispersions. The simulations are used to obtain the cloud and shadow curve properties of a Lennard-Jones fluid having diameters distributed according to a Schulz form with a large (delta approximately 40%) degree of polydispersity. Good qualitative accord is found with the results from a MFE method study of a corresponding van der Waals model that incorporates size dispersity both in the hard core reference and the attractive parts of the free energy. The results show that polydispersity engenders considerable broadening of the coexistence region between the cloud curves. The principal effect of fractionation in this region is a common overall scaling of the particle sizes and typical interparticle distances, and we discuss why this effect is rather specific to systems with Schulz diameter distributions. Next, by studying a family of such systems with distributions of various widths, we estimate the dependence of the critical point parameters on delta. In contrast to a previous theoretical prediction, size dispersity is found to raise the critical temperature above its monodisperse value. Unusually for a polydisperse system, the critical point is found to lie at or very close to the extremum of the coexistence region in all cases. We outline an argument showing that such behavior will occur whenever polydispersity affects only the range, rather than the strength of the interparticle interactions.     

Phase coexistence in a polydisperse charged hard-sphere fluid: polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.     

Isotropic–nematic phase separation and demixing in mixtures of spherical nanoparticles with length‐polydisperse nanorods

An extension of Onsager theory is developed to simulate isotropic–nematic phase separation in a mixture of spheres with length‐polydisperse system of rods. This work is motivated by recent experimental data on nanorod liquid crystals. Prior theoretical investigations indicate that both polydispersity and the presence of spheres should increase the biphasic–nematic phase transition, that is, the nematic cloud point. Results indicate that the phase diagrams undergo drastic changes depending upon both particle geometry and rod length polydispersity. The key geometric factor is the ratio between the sphere diameter and the rod diameter. In general, length fractionation is enhanced by the addition of spheres, which may be experimentally advantageous for separating short nanorods from a polydisperse population. Simulation results also indicate that the nematic cloud and shadow curves may cross one another because of the scarcity of spheres in the shadow phase. In general, these results do indicate that the nematic cloud point increases as a function of sphere loading; however, in certain areas of phase space, this relationship is nonmonotonic such that the nematic cloud point may actually decrease with the addition of spheres. This work has application to a wide range of nanoparticle systems, including mixtures of spherical nanoparticles with nanorods or nanotubes. Additionally, a number of nonspherical particles and structures may behave as spheres, including crumpled graphene and tightly coiled polymers. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Phase behavior of weakly polydisperse sticky hard spheres: perturbation theory for the Percus-Yevick solution

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys. 49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys. 22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.     

Phase behavior of polydisperse spheres: simulation strategies and an application to the freezing transition

The statistical mechanics of phase transitions in dense systems of polydisperse particles presents distinctive challenges to computer simulation and analytical theory alike. The core difficulty, namely, dealing correctly with particle size fractionation between coexisting phases, is set out in the context of a critique of previous simulation work on such systems. Specialized Monte Carlo simulation techniques and moment free energy method calculations, capable of treating fractionation exactly, are then described and deployed to study the fluid-solid transition of an assembly of repulsive spherical particles described by a top-hat "parent" distribution of particle sizes. The cloud curve delineating the solid-fluid coexistence region is mapped as a function of the degree of polydispersity δ, and the properties of the incipient "shadow" phases are presented. The coexistence region is found to shift to higher densities as δ increases, but does not exhibit the sharp narrowing predicted by many theories and some simulations.     

Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: high temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere fluid mixtures. It is demonstrated that in the frames of the HTA the model belongs to the class of "truncatable free energy models," i.e., the models with thermodynamical properties (Helmholtz free energy, chemical potential, and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and size distribution functions of the coexisting phases of several different models of polydisperse fluids. In particular, we consider polydisperse one-Yukawa hard-sphere mixture with factorizable Yukawa coefficients and polydisperse Lennard-Jones (LJ) mixture with interaction energy parameter and/or size polydispersity. To validate the accuracy of the HTA we compare theoretical results with previously published results of more advanced mean spherical approximation (MSA) for the one-Yukawa model and with the Monte Carlo (MC) computer simulation results of [Wilding et al. J. Chem. Phys. 121, 6887 (2004); Phys. Rev. Lett. 95, 155701 (2005)] for the LJ model. We find that overall predictions of the HTA are in reasonable agreement with predictions of the MSA and MC, with the accuracy range from semiquantitative (for the phase diagram) to quantitative (for the size distribution functions).     

Effect of polydispersity and soft interactions on the nematic versus smectic phase stability in platelet suspensions

We theoretically discuss, using density-functional theory, the phase stability of nematic and smectic ordering in a suspension of platelets of the same thickness but with a high polydispersity in diameter, and study the influence of polydispersity on this stability. The platelets are assumed to interact like hard objects, but additional soft attractive and repulsive interactions, meant to represent the effect of depletion interactions due to the addition of nonabsorbing polymer, or of screened Coulomb interactions between charged platelets in an aqueous solvent, respectively, are also considered. The aspect (diameter-to-thickness) ratio is taken to be very high, in order to model solutions of mineral platelets recently explored experimentally. In this regime a high degree of orientational ordering occurs; therefore, the model platelets can be taken as completely parallel and are amenable to analysis via a fundamental-measure theory. Our focus is on the nematic versus smectic phase interplay, since a high degree of polydispersity in diameter suppresses the formation of the columnar phase. When interactions are purely hard, the theory predicts a continuous nematic-to-smectic transition, regardless of the degree of diameter polydispersity. However, polydispersity enhances the stability of the smectic phase against the nematic phase. Predictions for the case where an additional soft interaction is added are obtained using mean-field perturbation theory. In the case of the one-component fluid, the transition remains continuous for repulsive forces, and the smectic phase becomes more stable as the range of the interaction is decreased. The opposite behavior with respect to the range is observed for attractive forces, and in fact the transition becomes of first order below a tricritical point. Also, for attractive interactions, nematic demixing appears, with an associated critical point. When platelet polydispersity is introduced the tricritical temperature shifts to very high values.     

Effect of quenched size polydispersity on the ordering transitions of hard polyhedral particles

Monodisperse polyhedral nanocrystals with O(h) (octahedral) symmetry self-assemble into various mesophases and crystal structures at intermediate and high concentrations. In this work, the effect of quenched size polydispersity on phase and jamming behavior has been studied via molecular simulations for three representative O(h) polyhedral shapes; namely, cubes, cuboctahedrons, and truncated octahedrons. Polydispersity is set by the standard deviation "δ" of an underlying Gaussian distribution of particle sizes, and is "quenched" in that it is fixed in a given uniphase sample. Quenched polydisperse states are relevant to: (i) equilibrium behavior for small enough δ when phase segregation does not occur, and (ii) actual experimental behavior for arbitrary δ when dense states are reached at a rate faster than the relaxation of slow diffusion-driven fractionation modes. Space-filling polyhedrons (cubes and truncated octahedrons) are found to be more robust with respect to the nucleation of orientational and translational order at high polydispersities compared to the non-space-filling cuboctahedron, with the former shapes exhibiting an onset of jamming behavior at a critical polydispersity δ(t) that is about twice larger than that for the latter (δ(t) ≈ 0.08). Further, the orientational ordering in cubes is found to be highly resilient to polydispersity, leading to the formation of a dense, orientationally aligned, and translationally jammed state. Overall, increasing size polydispersity enhances the range of pressures where the mesophases occur.     

Effect of particle size distribution on the collection efficiency of Brownian particles

The main purpose of the present paper is to investigate the effect of the normal Gaussian size distribution on the deposition of Brownian particles onto a spherical collector, by applying the Brownian dynamic simulation method and the Kuwabara flow field model with different types of DLVO interaction energy curves and the shadow effect. The simulation results show that the collection efficiency of Brownian particles always increases with a wider particle size distribution region. The same increased tendencies are also observed for the case of increasing Reynolds number and for the case of increasing the particle size to the collector size ratio. When compared to the available experimental data, the present simulation method fits well with the experimental data when the specific deposit per collector is not large.     

Refined Continuous Thermodynamic Treatment for the Liquid-Liquid Equilibrium of Copolymer Solutions

Based on an improved calculation of activity coefficients, continuous thermodynamics using a generalized divariate Stockmayer distribution is applied to the liquid-liquid equilibrium of random copolymer solutions. The effects of the chemical polydispersity on the cloud-point curve, the shadow curve, the spinodal, and the critical point are discussed. The theory can account for the occurrence of three-phase equilibria as well as for the phase separation in pure copolymers.     

Cloud-Point Curves and Coexistence Curves of Several Polydisperse Polystyrenes in Cyclohexane

Cloud-point curves, critical points, and coexistence curves with feed concentrations close to the critical concentration were measured in three systems involving cyclohexane + different polydisperse polystyrenes. The shape of the coexistence curves is analyzed by using a scaling expression. In two systems the critical exponent β possesses values somewhat larger than in actual binary systems (where β ≈ 1/3), whereas in the third system a somewhat smaller value is found. By using a three-parameter Gibbs free energy relation, cloud-point curves and coexistence curves are calculated from the critical point data and from the slope of the cloud-point curve at this point. To account for polydispersity, the method of continuous thermodynamics is applied. The cloud-point curves are well described, but the prediction of the coexistence curves is bad due to the mean-field character of the Gibbs free energy relation resulting in β = 1/2. Hence, the often used practice of fitting the parameters of a mean-field Gibbs free energy relation to the critical point and to some cloud points and then to calculate the coexistence data is to be considered with great care.     

Effect of quenched size polydispersity on the fluid-solid transition in charged colloidal suspensions

We study the effect of quenched size polydispersity on the phase behavior of charged colloidal suspensions using free-energy calculations in Monte Carlo simulations. The colloids are assumed to interact with a hard-core repulsive Yukawa (screened-Coulomb) interaction with constant surface potential, so that the particles are polydisperse both in size and charge. In addition, we take the size distribution to be fixed in both the fluid and crystal phase (no size fractionation is allowed). We study the fluid-solid transition for various screening lengths and surface potentials, finding that upon increasing the size polydispersity the freezing transition shifts toward higher packing fractions and the density discontinuity between the two coexisting phases diminishes. Our results provide support for a terminal polydispersity above which the freezing transition disappears.     

Effects of Polydispersity on the Phase Behavior of Additive Hard Spheres in Solution

The ability to separate enzymes, nucleic acids, cells, and viruses is an important asset in life sciences. This can be realised by using their spontaneous asymmetric partitioning over two macromolecular aqueous phases in equilibrium with one another. Such phases can already form while mixing two different types of macromolecules in water. We investigate the effect of polydispersity of the macromolecules on the two-phase formation. We study theoretically the phase behavior of a model polydisperse system: an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We calculate the theoretical liquid–liquid phase separation boundary (the binodal), the critical point, and the spinodal. We vary the distribution of the polydisperse component in terms of skewness, modality, polydispersity, and number of sub-components. We compare the phase behavior of the polydisperse mixtures with their concomittant monodisperse mixtures. We find that the largest species in the larger (polydisperse) component causes the largest shift in the position of the phase boundary, critical point, and spinodal compared to the binary monodisperse binary mixtures. The polydisperse component also shows fractionation. The smaller species of the polydisperse component favor the phase enriched in the smaller component. This phase also has a higher-volume fraction compared to the monodisperse mixture.     

Preparation and characterization of particles with small differences in polydispersity

Colloidal particles are widely used both in fundamental research and in materials science. One important parameter influencing the physical properties of colloidal materials is the particle size distribution (polydispersity) of the colloidal particles. Recent work on colloidal crystallization has demonstrated that even subtle changes in polydispersity can have significant effects. In this study we present centrifugation techniques for subtly manipulating the width and the shape of the particle size distribution, for polydispersities less than 10%. We use scanning electron microscopy as well as dynamic and static light scattering to characterize the particle size distributions. We compare the results and highlight the difficulties associated with the determination of accurate particle size distributions.     

Solid-liquid phase transition of tin particles observed by UHV high resolution transmission electron microscopy: pseudo-crystalline phase

Solid-liquid transition of fine tin particles having diameter of 2–10 nm is studied in-situ by high-resolution transmission electron microscopy under a ultra-high vacuum condition. Melting temperature is confirmed to decrease with the decrease of particle diameter. The particles less than the critical size, 2r _c?5 nm, are found to have a specific phase between the solid and the liquid phase. The particle in this “pseudo-crystalline” phase contains crystalline embryos in it. Particles larger than the critical size have sharp liquid-solid transition, which completed within the time resolution of our microscope observation, 33 ms upon heating or cooling process. Large solid particles have Wulff's polyhedron, while particles around the critical diameter have rather spherical shape. Structural anomaly at the critical size occurs all over the outer most surface layer slightly below the melting temperature. Origin of the “pseudo-crystalline” phase and surface pre-melting phenomena are discussed.     

Particle Penetration into Polydisperse Polymer Brushes: A Theoretical Analysis

Based on theoretical analysis, the effect of polydispersity on particle penetration into polydisperse polymer brushes is investigated. Three different polydispersities representing sharp, moderate, and extremely wide chain length distributions are chosen, since the corresponding explicit expressions of brush density at these polydispersities are available. To simplify the discussion, this study is restricted to spherical particles of small size which ensure that the particle insertion only causes local conformational perturbations. By analyzing the particle distribution, it is found that polydispersity always facilitates particle penetration. This prediction is confirmed by analyzing the surface fluctuations of the brushes. Interestingly, uniform scaling relations are observed for particles penetrating into monodisperse and moderately polydisperse brushes. The uniformity predicted by monodisperse and moderately polydisperse brushes originates from the same asymptotic behavior of their densities approaching the brush edge. This indicates that polydispersity brings significant influence only at high polydispersities.     

Water in nanopores. I. Coexistence curves from Gibbs ensemble Monte Carlo simulations

Coexistence curves of water in cylindrical and slitlike nanopores of different size and water-substrate interaction strength were simulated in the Gibbs ensemble. The two-phase coexistence regions cover a wide range of pore filling level and temperature, including ambient temperature. Five different kinds of two-phase coexistence are observed. A single liquid-vapor coexistence is observed in hydrophobic and moderately hydrophilic pores. Surface transitions split from the main liquid-vapor coexistence region, when the water-substrate interaction becomes comparable or stronger than the water-water pair interaction. In this case prewetting, one and two layering transitions were observed. The critical temperature of the first layering transition decreases with strengthening water-substrate interaction towards the critical temperature expected for two-dimensional systems and is not sensitive to the variation of pore size and shape. Liquid-vapor phase transition in a pore with a wall which is already covered with two water layers is most typical for hydrophilic pores. The critical temperature of this transition is very sensitive to the pore size, in contrast to the liquid-vapor critical temperature in hydrophobic pores. The observed rich phase behavior of water in pores evidences that the knowledge of coexistence curves is of crucial importance for the analysis of experimental results and a prerequiste of meaningful simulations.