Polydisperse Polymer Solutions Research Articles

Theory of nematic-isotropic phase transitions in solutions of rodlike aggregates

ABSTRACT A mean-field theory is introduced to describe phase behaviours for solutions of rodlike aggregates. By combining the Onsager model for rodlike aggregates and the Flory-Huggins model for polydisperse polymer solutions, we derive the free energy for the solutions of the rodlike aggregates. We calculate the mean aggregation number and orientational order parameter of rodlike aggregates. The mean aggregation number jumps at the first-order nematic–isotropic transition (NIT). We also derive the Landau-de Gennes expansion of the free energy and calculate the isotropic-nematic coexistence curves on the temperature-concentration plane. The NIT curves are qualitatively consistent with the experimental results of rodlike aggregates.

Cell-Sized Confinements Alter Molecular Diffusion in Concentrated Polymer Solutions Due to Length-Dependent Wetting of Polymers.

Living cells are characterized by the micrometric confinement of various macromolecules at high concentrations. Using droplets containing binary polymer blends as artificial cells, we previously showed that cell-sized confinement causes phase separation of the binary polymer solutions because of the length-dependent wetting of the polymers. Here, we demonstrate that the confinement-induced heterogeneity of polymers also emerges in single-component polymer solutions. The resulting structural heterogeneity also leads to a slower transport of small molecules at the center of cell-sized droplets than that in bulk solutions. Coarse-grained molecular simulations support this confinement-induced heterogeneous distribution by polymer length and demonstrate that the effective wetting of the shorter chains at the droplet surface originates from the length-dependent conformational entropy. Our results suggest that cell-sized confinement functions as a structural regulator for polydisperse polymer solutions that specifically manipulates the diffusion of molecules, particularly those with sizes close to the correlation length of the polymer chains.

Many-body effects in a binary nano-particle mixture dispersed in ideal polymer solutions.

A new mean-field theory is developed to treat a binary mixture of nanoparticles imbedded in a polydisperse polymer solution. The theory is based on a many-body polymer-mediated potential of mean force (PMF) between the particles and remains accurate even in the protein regime, where the particles' diameters cannot necessarily be considered large compared to the polymer radius of gyration. As implemented here, the theory is strictly valid for dilute to semi-dilute polymer solutions near the theta temperature (the so-called theta regime) or when the range of the PMF is strongly affected by the polymer size. For non-adsorbing particles, this is the same regime where the celebrated Asakura-Oosawa (AO) model is often used. Unlike the traditional AO model, however, our approach includes polymer flexibility and is accurate in the protein regime. We use the theory to calculate phase diagrams for a binary mixture of unequal-sized particles, both adsorbing and non-adsorbing. To test the theory, we carry out comparisons with simulations and obtained good quantitative agreement, which gives support to its accuracy. On the other hand, the oft-used approach assuming pairwise-additive potentials of mean force produce quantitatively (and sometime qualitatively) different phase diagrams.

Quantifying Size Exclusion by Diffusion NMR: A Versatile Method to Measure Pore Access and Pore Size.

Size-exclusion quantification NMR spectroscopy (SEQ-NMR) is introduced for measuring equilibrium distribution coefficients, Keq, in porous media. The porous medium is equilibrated with a polydisperse polymer solution. The original bulk polymer solution and the polymer solution after equilibration but in the absence of the porous medium are analyzed by NMR diffusion experiments. The joint evaluation of the two diffusion attenuation curves under suitable constraints provides the extent by which polymer fractions of particular size were depleted from the solution by pore access. This procedure yields Keq versus polymer probe size, the selectivity curve that in turn can provide the pore size and its distribution. Simulations probe the performance of the method that is demonstrated experimentally in chromatographic media using dextran polymers. SEQ-NMR and inverse size-exclusion chromatography (ISEC) yield selectivity curves that virtually coincide. Crucial advantages with SEQ-NMR, such as versatility with regard to both the polymer used and porous system explored, high speed, potential for automation, and small required sample volume, are discussed.

Continuous Thermodynamics of Polydisperse Polymer/Solvent Systems: Polystyrene/Cyclohexane and Polystyrene/Methyl Acetate Mixtures

Various types of liquid–liquid equilibrium of polydisperse polymer solutions, including upper critical solution temperature (UCST), both UCST and lower critical solution temperature (LCST), and closed loop miscibility, are investigated. Determination of the cloud-point curves for polydisperse polystyrene (PS)/cyclohexane and PS/methyl acetate systems is conducted by thermal optical analysis. In addition, the molecular weight distribution of polydisperse PS is determined by a gel permeation chromatography technique. To describe the phase equilibria of polydisperse polymer solutions, a continuous thermodynamic approach is incorporated into the incompressible lattice model. This appropriately describes the characteristic phase behaviors of polydisperse polymer solutions, and the calculated results show good agreement with the experimental data.

Effects of polydispersity on liquid–liquid equilibrium of polymeric fluids

The effects of polymer polydispersity on the liquid–liquid equilibrium (LLE) of polymer solutions were investigated for the simple case of polydisperse polymer solutions: mixtures composed of a solvent and two monodisperse polymers of different molecular weights. Various quasi-binary cloud point curves including the upper critical solution temperature (UCST), lower critical solution temperature (LCST), both UCST and LCST, and closed loop miscibility were determined from thermal optical analysis (TOA). To describe the phase equilibrium of polydisperse polymer solutions, we extended the chain length contribution term in the modified double lattice chain length dependent model (extended MDL-CL). Phase equilibrium calculations were conducted for several ternary systems including hypothetical hourglass type LLE. The extended MDL-CL model appropriately described the effect of polymer polydispersity on the LLE of polymer solutions and the calculated results were in satisfactory agreement with the experimental data.

Dispersity and spinnability: Why highly polydisperse polymer solutions are desirable for electrospinning

We develop new criteria that describe the minimum concentration limits controlling the spinnability of dilute and semi-dilute flexible polymer solutions with high molecular weight and varying polydispersity. By asserting that the finite and bounded extensional viscosity of the solution is the key material property determining the stability of a filament during spinning, we propose a new scaling relating the minimum necessary concentration of a polymer cspin to its molecular weight M and the quality of the solvent (through the excluded volume exponent ν) of the form cspin∼M−(ν+1). This new scaling differs from the classical interpretation of the coil overlap concentration c∗ or entanglement concentration ce as the minimum concentration required to increase the viscosity of the spinning dope, and rationalizes the surprising spinnability of high molecular weight polymers at concentrations much lower than ce. Furthermore, we introduce the concept of an extensibility average molecular weight ML as the appropriate average for the description of polydisperse solutions undergoing an extension-dominated spinning process. In particular it is shown that this extensibility average measure, and thus the solution spinnability, is primarily determined by the extensibility of the highest molecular weight fractions. For highly polydisperse systems this leads to an effective lowering of the minimum required concentration for successful fibre spinning (in comparison to narrowly distributed polymer solutions of similar weight average molecular weights). These predictions are validated with experimental observations of the electrospinnablity of mono- and polydisperse poly(methyl methacrylate) (PMMA) solutions as well as a model bimodal blend, and through comparison to published literature data on the minimum spinnable polymer concentration for a variety of flexible long chain polymers over a range of molecular weights.

Molecular Competition between Large and Small Polyethylene-Glycols (PEGs) Partitioning into OMPC Porin Channels

Molecular crowding by polyethylene glycols (PEGs) has been employed in a wide range of inquiries into molecular structure and dynamics. Investigating the partitioning of crowded polymers into biological nanopores reveals properties also seen in, e.g., DNA ejection and compaction into viral capsids, as well as in facilitated molecule transport. Capitalizing on our previous work on the partitioning of binary PEG mixtures into the benchmark Alpha-Hemolysin nanopore, we have now studied partitioning of mono- and polydisperse polymer solutions into comparably sized, yet differently structured, diffusion channels. We find that in the presence of large non-partitioning PEG 3400, the small, otherwise equipartitioning PEG 200 partitions disproportionately. Specifically, with 15% PEG 3400, an added 15% PEG 200 reduces channel conductance by 65% (compared with channel conductance in polymer-free solutions), while 15% PEG 200 alone reduces channel conductance by only 45%. We employed open-channel noise analysis to investigate channel diffusion properties of individual polymers, and to compare them with their properties in bulk water solution. In this way we demonstrate and quantify crowding-assisted transport through nanopores in mixed-polymer systems.

Polymers Pushing Polymers: Polymer Mixtures in Thermodynamic Equilibrium with a Pore

Studies on the ejection of viral genomes from capsids use osmotic stress to push a DNA molecule into the viral capsid by the action of PEG in solution. Identities of the polymer forced into the nano-cavity (DNA) and the polymer pushing it (PEG) differ. We analyze a variation on the problem with a simple model of a binary polymer mixture and find that polymer size dependent partitioning into a pore also has a complex dependence on the composition of the polymer mixture and the pore-penetration penalty. We analyze a polydisperse polymer solution of big and small polymer chains (bPEG and sPEG, respectively) of which only the sPEG is allowed to pay an energy penalty and enter the pore. We introduce an ansatz for the free energy of the polymer mixture that is consistent with our previous phenomenological fit to the equation of state of bulk uncharged polymers. We then analyze the osmotic pressure of this solution in equilibrium with a pore for various amounts of sPEG. Calculating the corresponding partition coefficient allows us to assess the pushing forces exerted by the external polymer solution via its osmotic pressure and ascertain that some polymers can push others to enter the pore in a kind of osmotic tug-of-war. We find that pore penetration is governed by two different penetration free energy scales. One scale represents the osmotic pushing strength of the external polymer mixture. The other scale comprises the interplay between the pore energy penalty and the translational mixing entropy (van't Hoft) and configurational (des Cloizeaux) fluctuations of the external solution. This principle should have applications in many areas of nano-science dealing with partitioning of polymer chains into small enclosures.

The effect of depletion layer on diffusion of nanoparticles in solutions of flexible and polydisperse polymers

We introduce a model of diffusion of nanoparticles in solutions of flexible, polydisperse polymers. The model takes into account the effect of depletion layer with soft boundaries. The presence of depletion layer leads to nonlinear dependence of the mean square displacement (MSD) on time. Our model may be an alternative choice for the study of those experimental systems where the crossover between subdiffusion and normal diffusion is observed. Its advantage is mathematical simplicity: it allows easy identification of the crossover times and distances, which are here associated with the depletion layer thickness. The soft boundaries of the depletion layer, generated by the flexible and polydisperse polymers, are here approximated by two shells enclosed one in another, which may be interpreted as approximations of polymer density profiles around the probe. We show a very good agreement of the model with dynamic light scattering (DLS) measurements of diffusion of nanoparticles in solutions of polyethylene glycol (PEG).

Non-linear dynamics of semi-dilute polydisperse polymer solutions in microfluidics: A study of a benchmark flow problem

The complex flow behaviour of semi-dilute (15 < c/ c * < 22.5) polydisperse polyethylene oxide (PEO) aqueous solutions flowing through a planar microfluidic geometry with an 8:1:8 contraction–expansion is systematically studied. The molecular weight and distribution of the PEO samples are analysed by Gel Permeation Chromatography (GPC). Full rheometric characterizations using various techniques including piezoelectric axial vibrator (PAV) measurements at frequencies as high as 6700 Hz are carried out for one semi-dilute PEO solution. Complex flows over a wide range of elasticity numbers (20 ⩽ El ⩽ 120), Weissenberg numbers (7 ⩽ Wi ⩽ 121) and Reynolds numbers (0.08 ⩽ Re ⩽ 4.5) are characterized using micro-particle image velocimetry (μ-PIV) and pressure drop measurements. The evolution of vortex formation and dynamics has been visualized through a step-flow-rate experiment. The effect of El on vortex stability has been studied. Various flow dynamics regimes have been quantified and are presented in a Wi–Re diagram. The experimental results reveal that the elastic behaviour of polymer solutions is very sensitive to high molecular weight polymer in the polydisperse polymer samples, and the contraction ratio and the aspect ratio of flow geometry are the important design parameters in controlling the non-linear dynamics of semi-dilute polymer solutions in microfluidics.

Non-linear dynamics of semi-dilute polydisperse polymer solutions in microfluidics: effects of flow geometry

The non-linear dynamics of a semi-dilute (c/c* = 15) polydisperse polyethylene oxide (PEO) solution in microfluidics are studied experimentally using benchmark contraction–expansion flow geometries with three contraction–expansion ratios (4:1:4, 8:1:8 and 16:1:16) and two narrow channel lengths (L c/D h = 53 and 5.3, where L c is the length of the narrow channel and D h is its hydraulic diameter). Complex flows over a range of elasticity numbers (El), Weissenberg numbers (Wi) and Reynolds numbers (Re) are characterized using micro-particle image velocimetry ( $\upmu$ -PIV) and pressure drop measurements. The evolution of vortex formation and dynamics has been visualized through a step-flow-rate experiment. Various flow dynamics regimes have been quantified and are presented in a Wi–Re diagram. The experimental results reveal that the contraction ratio can result in qualitatively different vortex dynamics of semi-dilute polymer solutions in microfluidics, whereas the length of the narrow channel merely affects the dynamics at a quantitative level. A single elasticity number, if defined by the size of the narrow channel, is not sufficient to account for the effects of contraction ratio on the non-linear vortex dynamics.

Characteristic Shear Rate for Nonlinear Viscoelastic Behavior in a Polydisperse Polymer Solution

The characteristic shear rate for nonlinear viscoelastic behavior in a polydisperse polymer system was investigated by the study of the stress overshoot behavior of polybutadiene solutions. The frequency at the crossing point of the dynamic moduli () was determined by oscillation frequency sweeps. For the stress overshoot observed in the shear flow at a fixed rate, the dependency of the stress drop, , defined as the difference between the stress at the peak and the stress at the steady state, on the shear rate can be well described by a fitting function: , and a critical shear rate () can be obtained using this fitting function. The correlation between and for the solutions can be expressed as and the proportionality factor is dependent on the degree of chain interpenetration. When the shear rate was normalized by , the dependency of the stress drop on this normalized shear rate for the solutions with different concentrations can be superposed well. This observation indicates that can be viewed as a characteristic shear rate for nonlinear viscoelastic behavior in a polydisperse polymer solution.

Application of Continuous Polydisperse Molecular Thermodynamics for Modeling Asphaltene Precipitation in Crude Oil Systems

In this work, a new algorithm for polydisperse asphaltene modeling through application of continuous molecular thermodynamics is introduced. The Scott−Magat theory of polydisperse polymer solutions is applied at equilibrium condition and a necessary and sufficient condition is defined to minimize the Gibbs free energy relation. Three commonly used distribution functions were examined for characterization of asphaltenes, and experimental data reported in the literature is used to adjust parameters of the distribution functions. Among the three distribution functions, the fractal molecular weight distribution function which gives accurate results is selected. A new exponential binary interaction coefficient between asphaltene and crude oil is introduced. The solubility parameter, volume fraction of asphaltene, and the amount of asphaltenes precipitated are calculated through minimization of Gibbs free energy and phase equilibrium condition. It has been shown that the calculated results are in good agreement with experimental data.

Minimal model of relaxation in an associating fluid: Viscoelastic and dielectric relaxations in equilibrium polymer solutions

Cluster formation and disintegration greatly complicate the description of relaxation processes in complex fluids. We systematically contrast the viscoelastic and dielectric properties for models of equilibrium polymers whose thermodynamic properties have previously been established. In particular, the monomer-mediated model allows chain growth to proceed only by monomer addition, while the scission-recombination model enables all particles to associate democratically, so that chain scission and fusion occur at the interior segments as well as at chain ends. The minimal models neglect hydrodynamic and entanglement interactions and are designed to explore systematically the competition between chemical reaction and internal chain relaxation and how this coupling modifies the dynamics from that of a polydisperse solution of Rouse chains with fixed lengths (i.e., "frozen" chains). As expected, the stress relaxation is nearly single exponential when the assembly-disassembly reaction is fast on the time scale of structural chain rearrangements, while multiexponential or nearly stretched exponential relaxation is obtained when this reaction rate is slow compared to the broad relaxation spectrum of almost unperturbed, nearly "dead" chains of intrinsically polydisperse equilibrium polymer solutions. More generally, a complicated intermediate behavior emerges from the interplay between the chemical kinetic events and internal chain motions.

Miscibility of styrene/unsaturated polyester quasibinary systems: Unsaturated polyester chemical composition and molecular weight influence

AbstractUnsaturated polyester prepolymers were synthesized with different chemical compositions and molecular weights. Cloud‐point curves (CPC) were measured in St‐UP quasibinary solutions, showing UCST behavior in all cases. The miscibility of the first UP samples series in St comonomer was enhanced when AA chemical comonomer concentration in UP prepolymer increased. In the second series, UP prepolymer miscibility increased with the molecular weight up to a maximum and, after that, the miscibility decreased. A thermodynamic analysis of experimental CPCs was performed using the Flory‐Huggins (F‐H) theory for polydisperse polymer solutions. A simple relationship between the interaction parameter and the temperature inverse could fit the measured CPCs in wide concentrations and molecular weight ranges. In the temperature interval where this fit took place, the positive enthalpic contribution to the interaction parameters determined the miscibility dependence with temperature in both UP sample series. The St‐UP miscibility behavior was also correlated with UPs structural chemical parameters as: (a) the final HO and HOOC high polar groups concentration, (b) the chain backbone polar adipate and phthalate groups concentration, and (c) the UP size dependent mixing entropy. All these parameters are molecular weight dependent. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 6064–6073, 2006

Polydispersity Effects in Dilute Polymer Solutions in the Good-Solvent Regime

We study the behavior of dilute polydisperse polymer solutions in the good-solvent regime. We compute the second virial coefficient for polymers of different degree of polymerization and for general polydispersity distributions. We also determine the effective center-of-mass pair potential for polymers of different length. These results, combined with the integral-equation formalism and the hypernetted-chain closure, allow us to determine the osmotic pressure and the intermolecular structure function in the dilute regime for general polydispersity distributions.

Prediction of multiple overshoots in shear stress during fast flows of bidisperse polymer melts

We present a differential constitutive model of stress relaxation in polydisperse linear polymer melts and solutions that contains contributions from reptation, contour-length fluctuations, and chain stretching. The predictions of the model during fast start-up and steady shear flows of polymer melts are in accord with experimental observations. Moreover, in accordance with reported experimental literature (Osaki et al. in J Polym Sci B Polym Phys 38:2043–2050, 2000), the model predicts, for a range of shear rates, two overshoots in shear stress during start-up of steady shear flows of bidisperse polymer melts having components with widely separated molar masses. Two overshoots result only when the stretch or Rouse relaxation time of the higher molar mass component is longer than the terminal relaxation time of the lower molar mass component. The “first overshoot” is the first to appear with increasing shear rate and occurs as a result of the stretching of longer chains. Transient stretching of the short chains is responsible for the early time second overshoot. The model predictions in steady and transitional extensional flows are also remarkable for both monodisperse and bidisperse polymer solutions. The computationally efficient differential model can be used to predict rheology of commercial polydisperse polymer melts and solutions.

Effect of molecular weight and its distribution on the spinodal for polydisperse polymer solutions

The Flory–Huggins interaction parameter χ(ri) is considered as dependent on the chain length of a polymer. Therefore, a modified free energy expression of Flory–Huggins theory is obtained for the polydisperse polymer solutions. Based on this modified free energy expression and the thermodynamics of Gibbs, the expression of spinodal for polydisperse polymer solutions is obtained. For a given χ(ri) according to de Gennes, the spinodals are calculated for polydisperse polymer solutions at different molecular weights and their distributions. It is found that all the interested variables rn, rw, rz and molecular weight distribution have an effect on the spinodal for polydisperse polymer solutions, where the effect of changing rw is much greater than that of changing rn, rz and molecular weight distribution.

Phase behavior applications of SAFT based equations of state—from associating fluids to polydisperse, polar copolymers

The statistical associating fluid theory (SAFT) equation of state has been shown to accurately predict the effects of molecular weight, intermolecular association, and compressibility on the phase behavior of mixtures containing solvents, monomers, and polymers. In this paper, we briefly review the physical basis for the SAFT approach and present recent extensions of the equation of state. An advantage of theoretically based equations of state is that the theory can be systematically improved to address weaknesses or to extend the theory to more complex fluid systems. Recent developments beyond SAFT allow us to predict the effects of multiple polar groups on the phase behavior of polar copolymers, model the effects of intramolecular association, and efficiently calculate phase equilibrium for polydisperse polymer solutions. These points are illustrated through comparisons with phase equilibria data for monomers, solvents, and copolymer systems.