Statistical thermodynamics of polydisperse polymer blends with strong interaction

A statistical thermodynamics theory of polydisperse polymer mixtures with strong interaction between dissimilar components based on a lattice fluid model is formulated. Expressions for the free energy, equation of state, phase stability and spinodal for a polydisperse, binary polymer mixture with strong interaction are derived.     

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.     

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

The Flory–Huggins interaction parameter χ(r_i) 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(r_i) 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 r_n, r_w, r_z and molecular weight distribution have an effect on the spinodal for polydisperse polymer solutions, where the effect of changing r_w is much greater than that of changing r_n, r_z and molecular weight distribution.     

Statistical thermodynamics in the framework of the lattice fluid model, 1. Polydisperse polymers of special distribution

The Gibbs free energies and equations of state of polymers with special molar mass distributions, e.g., Flory distribution, uniform distribution and Schulz distribution, are derived based on a lattice fluid model. The influence of the polydispersity (or the chain length) on the close-packed mass density, the close-packed volume of a mer and the mer-mer interaction energy or the scaling temperature is discussed. The diagrams of the Gibbs free energies as a function of temperature and chain length are simulated with a computer. The results suggest that a polydisperse polymer is thermodynamically more stable than the corresponding monodisperse polymer and that the thermodynamical properties of a polydisperse polymer are identical with those of the corresponding monodisperse polymer when the average degree of polymerization is sufficiently high.     

Spinodal decomposition in asymmetric polymer mixtures

We present a preliminary numerical study of spinodal decomposition in an asymmetric polymer mixture, i.e., of polymer with different chain lengths, in three dimensions with full Flory-Huggins-de Gennes free energy, numerically integrating the time evolution equations for the conserved order parameter. For the sake of comparison, we also present a numerical study of the symmetric polymer mixture. The results indicate that the scaled structure factor for the asymmetric polymer mixture is much broader than that of a symmetric polymer mixture. It is interesting that the growth exponents are not symmetric around the critical quench, i.e., growth exponents on the two sides of the critical composition are different. In addition to that, the magnitudes of the pair correlation functions of asymmetric mixtures are very small for x larger than the characteristic domain size r_g and the oscillations seen in the symmetric mixture are almost absent. We have attributed this finding to the rough interfaces and broader domain size distribution in the phase separated asymmetric polymer mixtures. Therefore, the simulation reveals that the asymmetry plays an important role for the spinodal decomposition dynamics of polymer mixtures.     

Effect of polydispersity on the cloud-point curves of polymer mixtures

A method is presented for the calculation of cloud-point curves of polymer–polymer mixtures when the polymers involved are polydisperse. The method is based on the Flory–Huggins free energy of mixing with a concentration-independent χ parameter. Numerical results are given for cases in which the molecular weight distributions are represented by the Schulz–Flory type. When the two polymers have similar average molecular weights and polydispersities, the cloud-point curves become flatter as the polydispersity increases. When the two polymers have similar average molecular weights but differ in polydispersity, the cloud-point curves become more skewed as the difference in the polydispersity increases. The results point out that, if the polydispersity effect is not properly accounted for, the value of χ deduced from experimental cloud points is liable to be in error, especially with regard to its temperature coefficient and its concentration dependence.     

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.     

APplication of Continuous Thermodynamics to the Stability of Polymer Systems. II

A determinant criterion for the critical state in solutions and mixtures of polydisperse polymers is established within the general framework of Gibbs theory. The treatment continues an earlier paper by considering more general Gibbs free energy relations: The function replacing the x-term in the classic Flory-Huggins equation is permitted to depend on a finite number of moments of the polymer distribution(s) so as to embrace most Gibbs free energy relations of practical use. The new criterion leads to a very large reduction of computer time and of needed storage capacity compared to the traditional Gibbs determinant criterion. Some relations known from the literature are shown to be special cases of the established new criterion.     

Sorption and pervaporation properties of sulfonyl‐containing polyimide membrane to aromatic/non‐aromatic hydrocarbon mixtures

Sorption of single‐component vapors of benzene (Bz), n‐hexane (Hx), and cyclohexane (Cx), and of binary liquid mixtures of Bz/Hx and Bz/Cx in a polyimide from 3,3′,4,4′‐diphenylsulfone‐tetracarboxylic dianhydride (DSDA) and 2,8(6)‐dimethyl‐3,7‐diaminobenzothiophene‐5,5‐dioxide (DDBT) were investigated in detail at 333 K. Sorption and desorption of vapors followed the non‐Fickian kinetics and the sorption isotherms were concave to the vapor activity. For the binary liquids, the sorption isotherms of the Bz component were concave to the Bz composition in feed, whereas those of Hx and Cx were convex because of competitive sorption. As a result, the solubility selectivity was much larger than the sorption ratio of pure liquids. The concentration‐averaged diffusion coefficients of Bz (D̄_Bz) and Hx (D̄_Hx) were evaluated using the sorption and pervaporation data of the polyimide membrane toward the binary mixtures. A kind of coupling effect of the coexisting component on D̄ was observed. That is, D̄ of penetrant with smaller molecular size (Hx and Bz for Bz/Hx and Bz/Cx systems, respectively) was reduced by the presence of penetrant with larger molecular size (Bz and Cx, respectively) and vice versa. D̄_Bz was similar to D̄_Hx, but much larger than D̄_Cx. The difference in PV behavior between Bz/Hx and Bz/Cx systems for glassy polymer membranes was understood based on the aforementioned features of sorption and diffusion. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2954–2964, 2000     

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.     

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.     

Polymer Polydispersity Effect on Depletion Interaction between Colloidal Particles

The effect of polymer polydispersity on the polymer‐induced interaction between colloidal particles due to non‐adsorbing ideal chains is investigated. An analytical theory is developed for the polymer‐segment density between two plates and in the space surrounding two spheres by extending a recently proposed superposition approximation to include polymer polydispersity. Monte Carlo computer simulations were made to test the validity of the analytical theory. The polymer densities predicted by the superposition approximation are in reasonable agreement with simulation results for the polydisperse case. The simulations show that depletion leads to a size fractionation of the polymers. It is shown that size polydispersity has a small effect on the interaction between two parallel plates but a more significant effect on the interaction between two spheres. The range of the potential increases and the contact potential drops with increasing polydispersity.

Polymer‐segment density as a function of y for three values of x, as indicated, in the space surrounding two colloidal spheres with radius R = R_g0 and h = 0.48R_g0. Symbols are the MC results: polydisperse polymer (○; z = 1) and monodisperse polymer (•) samples. Curves are the predictions of the product‐function approximation for monodisperse polymer (solid lines) and polydisperse polymer (z = 1, dashed lines).     

Modelling free-radical copolymerization kinetics—evaluation of the pseudo-kinetic rate constant method, 2. Molecular weight calculations for copolymers with long chain branching

The full moment equations and equations using pseudo-kinetic rate constants for binary copolymerization with chain transfer to polymer in the context of the terminal model have been developed and solved numerically for a batch reactor operating over a wide range of conditions. Calculated number- and weight-average molecular weights (M̄_n and M̄_w) were compared with those found using the pseudo-kinetic rate constant method (PKRCM). The results show that the weight-average molecular weights calculated using PKRCM are in agreement with those found using the method of full moments for binary copolymerization when polymeric radical fractions φ₁^˙ and φ₂^˙ of type 1 and 2 (radical centers are on monomer types 1 and 2 for a binary copolymerization) are calculated accounting for chain transfer to small molecules and polymer reactions in addition to propagation reactions. Errors in calculating M̄_w using PKRCM are not always negligible when polymer radical fractions are calculated neglecting chain transfer to small molecules and polymer. In this case, the relative error in M̄_w by PKRCM increases with increase in monomer conversion, extent of copolymer compositional drift and chain transfer to polymer rates. The errors in calculating M̄_w, however, vanish over the entire monomer conversion range for all polymerization conditions when chain transfer reactions are properly taken into account. It is theoretically proven that the pseudo-kinetic rate constant for chain transfer to polymer is valid for copolymerizations. One can therefore conclude that the pseudo-kinetic rate constant method is a valid method for molecular weight modelling for binary and multicomponent polymerizations.     

Compositionally symmetric diblock copolymer blends of moderate polydispersity

Recent experimental evidence and theoretical predictions indicate that binary blends of relatively monodisperse diblock copolymers remain miscible if the molecular weight disparity of the constituent copolymers is not too great. In this work, we examine the effect of moderate copolymer polydispersity on both the microstructural characteristics and phase behavior of blends prepared from four compositionally symmetric poly(styrene-b-isoprene) (SI) diblock copolymers ranging in polydispersity (M̄_w/M̄_n) from 1.02 to 1.30. Blend periodicities, measured by small-angle X-ray scattering, compare favorably with predictions from a strong segregation theory proposed for lamellar diblock copolymer blends composed of monomolecular copolymers. Transmission electron microscopy, employed to ascertain the real-space morphological characteristics of these blends, reveals that a lamellar → cylindrical transition occurs in macrophase-separated blends. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2653–2658, 1997     

Effect of tetraalkylammonium salts on the tacticity of poly(methyl methacrylate), 2.. Initiation by potassium tert-butoxide

The anionic polymerization of methyl methacrylate was carried out in the presence of potassium tert-butoxide (t-BuOK)/quaternary ammonium salts (QAS) in toluene and tetrahydrofuran at −60°C. It was found that in toluene some QAS additives substantially increase the syndiotacticity of poly(methyl methacrylate). Two types of QAS were distinguished, quite different in their action. The addition of QAS with one or two longchain alkyl groups (>C₁₂), does not change significantly the mode of the monomer addition, whereas the polymerization in the presence of tetraalkylammonium salts with four equal substituents and dimethyldidodecylammonium bromide yields predominantly a syndiotactic polymer with high conversion and comparatively low polydispersity (M̄_w/M̄_w = 1.3−1.5). In some cases QAS additives are more effective modifiers than cryptand [2.2.2].     

Calculation of the stability and of the phase equilibrium in the system polystyrene+cyclohexane+carbon dioxide based on equations of state

In order to calculate spinodals for polymer systems with an equation of state (EOS), we developed a stability theory using continuous thermodynamics. Here, the mixture considered consists of a polydisperse polymer and two monodisperse components as for example a solvent and a gas. We derived the spinodal equation on the base of the segment-molar Helmholtz energy of the mixture. As a result, a determinant similar to that of the monodisperse case is obtained, but the polydisperse polymer is identified by its weight average of the molecular weight. Furthermore, our paper provides the equations for the cloud-point curve derived with the aid of continuous thermodynamics. The final equations are applied to the system polystyrene+cyclohexane+carbon dioxide using the EOS of Sako, Wu and Prausnitz (SWP-EOS). For parameter fit and to prove the accuracy of the treatment, experimental data of the high-pressure equilibrium of the binary subsystems and of the ternary system were taken from literature.     

Polydispersity treated by continuous thermodynamics

Continuous thermodynamics is a suitable concept for performing phase equilibrium calculations of polydisperse systems such as polymer solutions or polymer mixtures. In contrast to the traditional pseudo-component method the continuous molar mass distribution density function obtained by fractionation or gel permeation chromatography is used directly for thermodynamic calculations. This paper describes the application of continuous thermodynamics to calculate cloud-point curves, shadow curves and coexistence curves for solutions and mixtures of homopolymers and random copolymers. Particular attention is devoted to special problems such as fractionation or the transition from polymers to oligomers requiring the application of sophisticated Gibbs energy functions. In all cases the use of continuous thermodynamics leads to a drastic reduction of the required computer time. For Schulz-Flory distributions the occurring integrals often may be calculated analytically leading to simple closed end formulas.     

Liquid crystalline,highly luminescent poly(2,5-dialkoxy-p-phenyleneethynylene)

A high-molecular-weight poly(2,5-dialkoxy-p-phenyleneethynylene) derivative has been prepared by the Heck reaction of 1,4-bis(2-ethylhexyloxy)-2,5-diiodobenzene and 1,4-diethynyl-2,5-dioctyloxybenzene. The highly luminescent polymer exhibits excellent solubility and can readily be processed into high-optical-quality films. The weight-average molecular weight M̄_w was 240000 g · mol⁻¹, with a polydispersity index of 2.9. Thermal analysis revealed a glass transition around 90°C, and an onset of chemical crosslinking at 130°C. The high M̄_w and the remarkable solubility enabled the preparation of liquid crystalline solutions of the new PPE.     

Reverse atom transfer radical polymerization of methyl acrylate using AIBN as the initiator under heterogeneous conditions

Reverse atom transfer radical polymerization of methyl acrylate in the presence of a conventional radical initiator (2,2′-azoisobutyronitrile, AIBN) in bulk was successfully implemented via a new polymerization procedure. The system first reacts at 65–70°C for ten hours, then polymerizes at 100°C. Various mole ratios of AIBN to Cu^IICl₂ were used in this work, all of which result in a well-controlled radical polymerization with high initiation efficiency and narrow molecular weight distribution, i.e., the polydispersity is as low as M̄_w/M̄_n = 1.36.