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.     

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.     

Isotropic-nematic phase transition of nonaqueous suspensions of natural clay rods

A novel model system for studying the behavior of hard colloidal rods is presented, consisting of sterically stabilized particles of natural sepiolite clay. Electron microscopy and scattering results confirmed that the organophilic clay particles were individual, rigid rods when dispersed in organic solvents. With a length-to-diameter ratio of approximately 27, the particles showed nematic ordering for volume fractions phi > 0.06. Polarizing microscopy revealed that the phase separation process involved nucleation, growth, and coalescence of nematic domains. The phase volumes and particle concentrations in the coexisting phases were determined. The dependence of these quantities on the total concentration of the suspension agrees well with Onsager's [Ann. N. Y. Acad. Sci. 51, 627 (1949)] isotropic-nematic phase transition theory extended to bidisperse and polydisperse rod systems, and with previous experimental results for rigid rodlike particles. Particle size distributions were obtained by analyzing transmission electron microscopy images. A significant fractionation with respect to rod length (but not diameter) was observed in the coexisting isotropic and nematic phases. The relative polydispersity of both daughter phases was distinctly smaller than that of the parent suspension. The phase behavior of these daughter fractions agrees well with the predictions for hard spherocylinders of corresponding aspect ratios. An isotropic-nematic-nematic phase equilibrium was seen to develop in phase separated samples after 1 month standing and is ascribed to the effect of polydispersity and possibly gravity. The second nematic phase appearing is dominated by very long rods.     

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.     

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.     

Nematic–isotropic phase behaviors of polydisperse polymer/liquid‐crystal systems: Chain‐length‐dependent interaction parameter

Phase behaviors of polydisperse polystyrene (PS)/nematic liquid‐crystal systems [P‐ethoxy ‐ benzylidene ‐ p ‐ n‐butylaniline (EBBA)] are investigated with a thermo‐optical analysis technique. We also develop a thermodynamic framework to describe the phase behaviors of polydisperse PS/EBBA systems. The proposed model is based on a modified double‐lattice model to describe isotropic mixing and Maier–Saupe theory for anisotropic ordering. To correlate the polymer chain length and energy parameters in a nematic–isotropic biphasic region and to apply the primary interaction parameter in an isotropic–isotropic phase‐transition behaviors of polydisperse PS/EBBA systems. The proposed model shows remarkable agreement with experimental data for the model systems in comparison with an existing model. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1031–1039, 2006     

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.     

Towards Predicting Partitioning of Enzymes between Macromolecular Phases: Effects of Polydispersity on the Phase Behavior of Nonadditive Hard Spheres in Solution

The ability to separate enzymes, or cells or viruses, from a mixture is important and can be realized by the incorporation of the mixture into a macromolecular solution. This incorporation may lead to a spontaneous phase separation, with one phase containing the majority of one of the species of interest. Inspired by this phenomenon, we studied the theoretical phase behavior of a model system composed of an asymmetric binary mixture of hard spheres, of which the smaller component was monodisperse and the larger component was polydisperse. The interactions were modeled in terms of the second virial coefficient and could be additive hard sphere (HS) or nonadditive hard sphere (NAHS) interactions. The polydisperse component was subdivided into two subcomponents and had an average size ten or three times the size of the monodisperse component. We gave the set of equations that defined the phase diagram for mixtures with more than two components in a solvent. We calculated the theoretical liquid–liquid phase separation boundary for the two-phase separation (the binodal) and three-phase separation, the plait point, and the spinodal. We varied the distribution of the polydisperse component in skewness and polydispersity, and we varied the nonadditivity between the subcomponents as well as between the main components. We compared the phase behavior of the polydisperse mixtures with binary monodisperse mixtures for the same average size and binary monodisperse mixtures for the same effective interaction. We found that when the compatibility between the polydisperse subcomponents decreased, the three-phase separation became possible. The shape and position of the phase boundary was dependent on the nonadditivity between the subcomponents as well as their size distribution. We conclude that it is the phase enriched in the polydisperse component that demixes into an additional phase when the incompatibility between the subcomponents increases.     

Structure formation and fractionation in systems of polydisperse attractive rods

Fractionation effects and the formation of structured domains are investigated in polydisperse systems of attractive spherocylinders with the help of Monte Carlo simulations. For sufficiently high attractive interaction strength and pressure, the large rods in the system aggregate and form a highly ordered hexatic monolayer that coexists with an isotropic fluid of smaller rods. Fractionation diminishes with decreasing interaction strength but is still observed for hard rod systems, in which the large rods form a nematic droplet rather than a monolayer. Results for polydisperse systems are accompanied by phase diagrams for monodisperse systems of attractive spherocylinders. Here, the phase behavior is shown as a function of rod length and pressure.     

Polymer solubility in mixed solvents

Modeling results indicate that polymer chains in mixtures of a good with a bad solvent exhibit preferential adsorption of the good solvent. That phenomenon is found to be strongly dependent on molecular weight and it increases with a decrease in chain length. These results have important consequences on polymer solubility. Thus, a low molecular weight chain in a solvent mixture behaves as if it were dissolved in the pure good solvent component, whereas the solubility of a longer chain is controlled by the average mixture composition. As a result, quenching a polydisperse system below the cloud point may induce molecular weight segregation between the two phases: the longer chains, which precipitate out first, tend to populate the polymer rich phase whereas the shorter chains, having greater solubility, remain in the solvent phase. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2782–2787, 1999     

Chemical potential of a hard sphere fluid adsorbed in model disordered polydisperse matrices

We consider a model for adsorption of a simple fluid in disordered polydisperse adsorbents. The fluid consists of hard sphere particles. On the other hand, the adsorbents of this study are modeled as a collection of hard spheres with their diameter obeying a certain distribution function. Our focus is in the evaluation of the chemical potential of the fluid immersed in such a polydisperse material. It permits us to obtain porosity and pore size distribution for the adsorbent, as well as a set of adsorption isotherms. The latter have been calculated theoretically and by grand canonical Monte Carlo simulations. We observe that the width of assumed polydispersity distribution affects all the properties of the system. Nevertheless, the effect of matrix packing is dominant in determining adsorption for this class of models. We are convinced that the matrix structures generated via more sophisticated algorithms would exhibit stronger effects of polydispersity on the entire set of properties of adsorbed simple fluids.     

Depletion interaction mediated by polydisperse rods

The interaction between a colloidal hard sphere of radius R and a wall or between two spheres in a dilute suspension of infinitely thin rods of length L is calculated numerically. The method allows the study of depletion potentials for any value of LR and, consequently, the influence of rod length polydispersity can be investigated. It was observed that both the depth and the range of the potential increase drastically if the relative standard deviation sigma of the length distribution is larger than 0.25, while the potential is virtually indistinguishable from that caused by monodisperse rods, if sigma < or similar to 0.1.     

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.     

Hollow spheres of aromatic polyamide prepared by reaction‐induced phase separation

Hollow spheres of aromatic polyamide are obtained by the reaction‐induced phase separation during polymerization of 5‐hydroxyisophthalic acid and 1,4‐phenylene diamine in an aromatic solvent at a concentration of 1–2% at 320 °C without stirring. The hollow sphere has a dimple hole and the diameters of the hollow spheres are 3–4 μm. The droplets are initially generated via liquid–liquid phase separation and then rigid cross‐linked network structure formed the rigid skin layer on the surface of the droplets. The solidification of the droplets occurred owing to the further polymerization in them with maintaining the morphology to form the hollow spheres. The hollow spheres exhibit outstanding thermal stability. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Diffusion of spheres in isotropic and nematic networks of rods: electrostatic interactions and hydrodynamic screening

Translational diffusion of a small charged tracer sphere in isotropic and nematic suspensions of long and thin charged rods is investigated as a function of ionic strength and rod concentration. A theory for the diffusive properties of a small sphere is developed, where both (screened) hydrodynamic interactions and charge interactions between the tracer sphere and the rod network are analyzed. Hydrodynamic interactions are formulated in terms of the hydrodynamic screening length. As yet, there are no independent theoretical predictions for the hydrodynamic screening length for rod networks. Experimental tracer-diffusion data are presented for various ionic strengths as a function of the rod concentration, both in the isotropic and nematic states. Orientational order parameters are measured for the same ionic strengths as a function of the rod concentration. The hydrodynamic screening length is determined from these experimental data and scaling relations obtained from the above mentioned theory. For the isotropic networks, a master curve is found for the hydrodynamic screening length as a function of the rod concentration. For the nematic networks the screening length turns out to be a very sensitive function of the orientational order parameter.     

Velocity‐correlation analysis in polymer–solvent mixtures

The subject of this article is the combined interpretation of intradiffusion and mutual‐diffusion data for polymer–solvent mixtures in terms of integrals over velocity self‐correlation functions and velocity cross‐correlation functions. The combination of mutual‐diffusion, intradiffusion, and activity data allows the evaluation of velocity‐correlation coefficients (VCCs) and distinct‐diffusion coefficients in systems containing one monodisperse solute. This study is the first attempt to extend these approaches to polymers that are polydisperse solutes. Because of the polydispersity, this correlation analysis may become critical for polymers. Its application to polydisperse samples requires the reduction of intradiffusion and mutual‐diffusion coefficients to the same average. After such a reduction, the VCCs and distinct‐diffusion coefficients are evaluated for a homologous series of poly(ethylene glycol)s (PEGs). Attractive PEG–PEG interactions depend on the chain length and concentration of PEG. In this analysis, network formation in PEG–water systems appears to be a smooth process. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 40: 43–51, 2002     

Phase separation behavior of aqueous solutions of a thermoresponsive polymer

Cloud‐point and binodal curves of the LCST type were obtained for aqueous solutions of a thermoresponsive polymer, poly [2‐(2‐ethoxy)ethoxyethyl vinyl ether], poly(EOEOVE). The cloud‐point curve obtained was very flat except in a dilute region, that is the cloud‐point temperature was insensitive to the polymer concentration, resembling the cloud‐point curve for aqueous solutions of poly(N‐isopropylacrylamide). On the other hand, the binodal curve obtained was parabolic, and located within the two‐phase region of the cloud‐point curve. Accompanied with the phase separation, a sharp endothermic peak was observed in a region including the cloud‐point and binodal temperatures. The reciprocal of the osmotic compressibility ?Π/?c obtained by sedimentation equilibrium indicated that water changes from a good to poor solvent for poly(EOEOVE) with increasing temperature. Analyzing the ?Π/?c data by a thermodynamic perturbation theory, we determined the interchain interaction parameters, the hard‐core diameter d and the depth ε of the square‐well potential. Theoretical binodal and endothermic curves calculated by the perturbation theory using the estimated interaction parameters reproduced experimental ones semiquantitatively, but the theoretical binodal disagreed with the experimental flat cloud‐point curve. The disagreement at high concentrations was in the opposite direction to that expected from the sample polydispersity in the molecular weight. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2937–2949, 2005     

Monte carlo simulations of polydisperse polymers grafted on spherical surfaces

This article presents effects of polydispersity in polymers grafted on spherical surfaces on grafted polymer chain conformations, grafted layer thickness, and free‐end monomer distribution within the grafted layer. At brush‐like grafting densities, as polydispersity index (PDI) increases, the scaling exponent of radius of gyration of grafted chains approaches that of a single chain grafted on the same nanoparticle, because polydispersity alleviates monomer crowding within the brush. At high PDI, the chains shorter than the number average chain length, N_n, have more compressed conformations, and the chains longer than N_n overall stretch less than in the monodisperse case. As seen in polydisperse flat brushes at high grafting densities, the grafted layer thickness on spherical nanoparticle increases with PDI. Polydispersity eliminates the region near the surface devoid of free‐end monomers seen in monodisperse cases, and it reduces the width of free‐end monomer distribution and shifts the free‐end monomer distribution close to the surface. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Phase behavior and particle size cutoff effects in polydisperse fluids

We report a joint simulation and theoretical study of the liquid-vapor phase behavior of a fluid in which polydispersity in the particle size couples to the strength of the interparticle interactions. Attention is focused on the case in which the particle diameters are distributed according to a fixed Schulz form with degree of polydispersity delta = 14%. The coexistence properties of this model are studied using grand canonical ensemble Monte Carlo simulations and moment free energy calculations. We obtain the cloud and shadow curves as well as the daughter phase density distributions and fractional volumes along selected isothermal dilution lines. In contrast to the case of size-independent interaction [N. B. Wilding et al., J. Chem. Phys. 121, 6887 (2004)], the cloud and shadow curves are found to be well separated, with the critical point lying significantly below the cloud curve maximum. For densities below the critical value, we observe that the phase behavior is highly sensitive to the choice of upper cutoff on the particle size distribution. We elucidate the origins of this effect in terms of extremely pronounced fractionation effects and discuss the likely appearance of new phases in the limit of very large values of the cutoff.     

Design of liquid-crystalline aqueous suspensions of rutile nanorods: evidence of anisotropic photocatalytic properties

TiO2 rutile nanorods of average length L = 160 +/- 40 nm and average diameter D = 15 +/- 5 nm have been synthesized through a seed-mediated growth process by TiCl4 hydrolysis in concentrated acidic solution. These nanorods were dispersed in water to yield stable (aggregation-free) colloidal aqueous suspensions. At volume fractions phi > 3%, the suspensions spontaneously display a phase separation into an isotropic liquid phase and a liquid-crystalline phase identified as nematic by X-ray scattering. At phi > 12%, the suspensions form a nematic single phase, with large order parameter, S = 0.75 +/- 0.05. Very well aligned rutile films on glass substrate were produced by spin-coating, and their photocatalytic properties were examined by monitoring the decomposition of methylene blue under UV light. We found that UV-light polarized along the quadratic axis of the rutile nanorods was most efficient for this photocatalytic reaction.