Concentration fluctuation in binary polymer blends: chi parameter, spinodal and Ginzburg criterion

A theory for concentration fluctuations in binary polymer blends is developed using field-theoretic techniques. The theory provides a simple, unified framework for addressing a number of important issues. First, consideration of the fluctuation and correlation effects on different length scales leads to a clarification of three different chi parameters and their interrelationship. By incorporating interaction (modeled by the bare χb) and packing effects up to the polymer size, an effective chie emerges as the natural parameter for characterizing the molecular compatibility of the two polymer species. The measured quantity in small-angle neutron scattering (SANS) experiments is an apparent chia that includes long wavelength critical and spinodal fluctuations, and is related to χe through a self-consistent equation. χa exhibits the typical upward parabolic composition dependence observed in experiments and computer simulations. Second, a unified Ginzburg criterion involving both the composition and temperature (or temperaturelike variable) is derived that is applicable to both the critical and the off-critical spinodal regimes. The common characterization of the Ginzburg criterion in terms of a range of temperature (or temperaturelike variable) alone is generally inadequate. The molecular weight scaling proposed by de Gennes and Binder in the respective critical and off-critical spinodal regimes are recovered as special cases in the limit of large molecular weights. For typical molecular weights used in experiments the Ginzburg region is larger than commonly believed. Finally, the nature of the thermodynamic spinodal is examined. It is shown that a true off-critical thermodynamic spinodal does not exist in spatial dimensions less than 4. In its place, a pseudo-spinodal can be defined where the susceptibility reaches a finite maximum. The pseudo-spinodal precedes the mean-field spinodal but approaches the latter in the limit of infinite molecular weights. The pseudo-spinodal correlates strongly with the free energy barrier for nucleation becoming order kT. Thus it provides a kinetic limit for the physically accessible metastable state, beyond which phase separation may exhibit features characteristic of spinodal decomposition. The calculated location of the pseudo-spinodal for two samples used in a recent experiment of Balsara and co-workers agrees with the onset of spinodal-decomposition-like nucleation observed in the experiment

Effects of Ion Solvation on the Miscibility of Binary Polymer Blends

We study the effects of adding salt ions on the miscibility of a binary blend of polymers having different dielectric constants. The competition between the preference of the ions to be solvated by the component of the higher dielectric constant and the entropic tendency for the ions to be distributed uniformly results in nontrivial effects on the miscibility. We first study the thermodynamics of the polymer blend−ion mixture using a simple Born model in a uniform dielectric medium of the average composition of the polymer blend. We then study the effect of local enrichment of the higher dielectric constant polymer near the ion. We find that when the dielectric constants of the polymers are both low, adding salt decreases the miscibility, while when the dielectric constants of the polymers are both high, the addition of salt enhances the miscibility. When the blend consists of a high dielectric constant polymer and a low dielectric constant polymer, miscibility is decreased if the low dielectric constant component is the majority and is increased if the high dielectric constant component is the majority. The effect becomes significant at ion concentrations corresponding to an order of one ion per polymer chain. The quantitative change in the effective χ parameter depends on the functional form of the composition dependence of the dielectric constant of the mixture. We also illustrate the difference between fixed ion concentration and fixed chemical potential of the ions

Theory of Side-Chain Liquid Crystal Polymers: Bulk Behavior and Chain Conformation

We study the thermodynamics and chain conformation of side-chain liquid crystal polymers (SCLCPs) in the bulk using the self-consistent-field approach and a new model to account for the coupling between the orientation of the side-chain liquid-crystal (LC) groups and that of the backbone segments. The new model accounts for both a global coupling between the polymer backbone and the nematic field and a local coupling between the polymer backbone and its attached LC group. Here, the terms global and local refer to the chemical (backbone) distance between the groups. A phenomenological parameter is introduced to represent the coupling strength and nature of the attachment, i.e., end-on vs side-on. The nematic field is shown to control the chain conformation through both the global and the local coupling effects. For the side-on SCLCPs, these two coupling effects act cooperatively so that the chain conformation is always prolate. For the end-on SCLCPs, these two effects act competitively. The chain conformation can be either oblate or prolate in this case, and depends on the relative strengths of these two couplings. On the other hand, the chain conformation also affects the nematic field, primarily through the global coupling. The prolate conformation enhances the nematic field and increases the phase transition temperature, whereas the oblate conformation frustrates the nematic field and decreases the transition temperature. The nematic order parameter is found to be determined mainly by the reduced temperature, and is not sensitive to the coupling effects. Furthermore, we show that the grafting density of the LC side groups has a significant effect on the chain conformation due to the orientational competition between the LC attached and unattached segments. For the end-on SCLCPs with lower graft density, the conformation of the chain backbone can be oblate at higher temperatures and prolate at lower temperatures, in agreement with the re-entrant nematic phase observed in experiments

Effects of ion solvation on phase equilibrium and interfacial tension of liquid mixtures

We study the bulk thermodynamics and interfacial properties of electrolyte solution mixtures by accounting for electrostatic interaction, ion solvation, and inhomogeneity in the dielectric medium in the mean-field framework. Difference in the solvation energy between the cations and anions is shown to give rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The ion solvation affects the phase equilibrium of the solvent mixture, depending on the dielectric constants of the solvents, reflecting the competition between the solvation energy and translation entropy of the ions. Miscibility is decreased if both solvents have low dielectric constants and is enhanced if both solvents have high dielectric constant. At the mean-field level, the ion distribution near the interface is determined by two competing effects: accumulation in the electrostatic double layer and depletion in a diffuse interface. The interfacial tension shows a nonmonotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations and decreases approximately as the square root of the salt concentration for dilute solutions, reaching a minimum near 1 mM. We also find that, for a fixed cation type, the interfacial tension decreases as the size of anion increases. These results offer qualitative explanations within one unified framework for the long-known concentration and ion size effects on the interfacial tension of electrolyte solutions

On the theoretical description of weakly charged surfaces

It is widely accepted that the Poisson-Boltzmann (PB) theory provides a valid description for charged surfaces in the so-called weak coupling limit. Here, we show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. In the presence of dielectric discontinuity, there is no limiting condition for which the PB theory is valid

Inhomogeneous screening near a dielectric interface

Screening is one of the most important concepts in the study of charged systems. Near a dielectric interface, the ion distribution in a salt solution can be highly nonuniform. Here, we develop a theory that self-consistently treats the inhomogeneous screening effects. At higher concentrations when the bulk Debye screening length is comparable to the Bjerrum length, the double layer structure and interfacial properties are significantly affected by the inhomogeneous screening. In particular, the depletion zone is considerably wider than that predicted by the bulk screening approximation or the WKB approximation. For asymmetric salts, the inhomogeneous screening leads to enhanced charge separation and surface potential.Comment: 5 figure

Salt-doped block copolymers: ion distribution, domain spacing and effective χ parameter

We develop a self-consistent field theory for salt-doped diblock copolymers, such as polyethylene oxide (PEO)–polystyrene with added lithium salts. We account for the inhomogeneous distribution of Li+ ions bound to the ion-dissolving block, the preferential solvation energy of anions in the different block domains, the translational entropy of anions, the ion-pair equilibrium between polymer-bound Li+ and anion, and changes in the χ parameter due to the bound ions. We show that the preferential solvation energy of anions provides a large driving force for microphase separation. Our theory is able to explain many features observed in experiments, particularly the systematic dependence in the effective χ-parameter on the radius of the anions, the observed linear dependence in the effective χ on salt concentration, and increase in the domain spacing of the lamellar phase due to the addition of lithium salts. We also examine the relationship between two definitions of the effective χ parameter, one based on the domain spacing of the ordered phase and the other based on the structure factor in the disordered phase. We argue that the latter is a more fundamental measure of the effective interaction between the two blocks. We show that the ion distribution and the electrostatic potential profile depend strongly on the dielectric contrast between the two blocks and on the ability of the Li+ to redistribute along the backbone of the ion-dissolving block

Kinetic Pathways of Order-Disorder and Order-Order Transitions in Weakly Segregated Microstructured Systems

The kinetics of hexagonal to disordered and hexagonal to body-centered-cubic phase transitions in weakly segregated, microstructured systems (e.g., diblock copolymers) is studied using a time-dependent Ginzburg-Landau (TDGL) approach. Both computer simulation of the TDGL equation and analysis of a simplified two-mode model reveal nontrivial pathways during the transition