A self‐consistent‐field approach to surfaces of compressible polymer blends

Entropic surface segregation from athermal polymer blends: Polymer flexibility vs bulkiness

We examine athermal binary blends composed of conformationally asymmetric polymers of equal molecular volume next to a surface of width ξ. The self-consistent field theory (SCFT) of Gaussian chains predicts that the more compact polymer with the shorter average end-to-end length, R0, is entropically favored at the surface. Here, we extend the SCFT to worm-like chains with small persistence lengths, ℓ p, relative to their contour lengths, ℓ c, for which [Formula: see text]. In the limit of ℓ p ≪ ξ, we recover the Gaussian-chain prediction where the segregation depends only on the product ℓ p ℓ c, but for realistic polymer/air surfaces with ξ ∼ ℓ p, the segregation depends separately on the two quantities. Although the surface continues to favor flexible polymers with smaller ℓ p and bulky polymers with shorter ℓ c, the effect of bulkiness is more pronounced. This imbalance can, under specific conditions, lead to anomalous surface segregation of the more extended polymer. For this to happen, the polymer must be bulkier and stiffer, with a stiffness that is sufficient to produce a larger R0 yet not so rigid as to reverse the surface affinity that favors bulky polymers.

Nematic ordering of worm-like polymers near an interface

The phase behavior of semi-flexible polymers is integral to various contexts, from materials science to biophysics, many of which utilize or require specific confinement geometries as well as the orientational behavior of the polymers. Inspired by collagen assembly, we study the orientational ordering of semi-flexible polymers, modeled as Maier-Saupe worm-like chains, using self-consistent field theory. We first examine the bulk behavior of these polymers, locating the isotropic-nematic transition and delineating the limit of stability of the isotropic and nematic phases. This effort explains how nematic ordering emerges from the isotropic phase and offers insight into how different (e.g., mono-domain vs multi-domain) nematic phases form. We then clarify the influence of planar confinement on the nematic ordering of the polymers. We find that while the presence of a single confining wall does not shift the location of nematic transition, it aligns the polymers in parallel to the wall; above the onset of nematic transition, this preference tends to propagate into the bulk phase. Introducing a second, perpendicular, wall leads to a nematic phase that is parallel to both walls, allowing the ordering direction to be uniquely set by the geometry of the experimental setup. The advantage of wall-confinement is that it can be used to propagate mono-domain nematic phases into the bulk phase.

Effect of chain stiffness on the entropic segregation of chain ends to the surface of a polymer melt

Entropic segregation of chain ends to the surface of a monodisperse polymer melt and its effect on surface tension are examined using self-consistent field theory (SCFT). In order to assess the dependence on chain stiffness, the SCFT is solved for worm-like chains. Our focus is still on relatively flexible polymers, where the persistence length of the polymer, ℓ _p , is comparable to the width of the surface profile, ξ, but still much smaller than the total contour length of the polymer, ℓ _c . Even this small degree of rigidity causes a substantial increase in the level of segregation, relative to that of totally flexible Gaussian chains. Nevertheless, the long-range depletion that balances the surface excess still exhibits the same universal shape derived for Gaussian chains. Furthermore, the excess continues to reduce the surface tension by one unit of k _B T per chain end, which results in the usual N ^-1 reduction in surface tension observed by experiments. This enhanced segregation will also extend to polydisperse melts, causing the molecular-weight distribution at the surface to shift towards smaller N _n relative to the bulk. This provides a partial explanation for recent quantitative differences between experiments and SCFT calculations for flexible polymers.

Entropic segregation of short polymers to the surface of a polydisperse melt

Chain ends are known to have an entropic preference for the surface of a polymer melt, which in turn is expected to cause the short chains of a polydisperse melt to segregate to the surface. Here, we examine this entropic segregation for a bidisperse melt of short and long polymers, using self-consistent field theory (SCFT). The individual polymers are modeled by discrete monomers connected by freely-jointed bonds of statistical length a , and the field is adjusted so as to produce a specified surface profile of width [Formula: see text]. Semi-analytical expressions for the excess concentration of short polymers, [Formula: see text], the integrated excess, [Formula: see text] , and the entropic effect on the surface tension, [Formula: see text], are derived and tested against the numerical SCFT. The expressions exhibit universal dependences on the molecular-weight distribution with model-dependent coefficients. In general, the coefficients have to be evaluated numerically, but they can be approximated analytically once [Formula: see text]. We illustrate how this can be used to derive a simple expression for the interfacial tension between immiscible A- and B-type polydisperse homopolymers.

Segregation of chain ends to the surface of a polymer melt: Effect of surface profile versus chain discreteness

Silberberg has argued that the surface of a polymer melt behaves like a reflecting boundary on the random-walk statistics of the polymers. Although this is approximately true, independent studies have shown that violations occur due to the finite width of the surface profile and to the discreteness of the polymer molecule, resulting in an excess of chain ends at the surface and a reduction in surface tension inversely proportional to the chain length, N . Using self-consistent field theory (SCFT), we compare the magnitude of these two effects by examining a melt of discrete polymers modeled as N monomers connected by Hookean springs of average length, a , next to a polymer surface of width [Formula: see text]. The effects of the surface width and the chain discreteness are found to be comparable for realistic profiles of [Formula: see text] ∼ a. A semi-analytical approximation is developed to help explain the behavior. The relative excess of ends at the surface is dependent on the details of the model, but in general it decreases for shorter polymers. The excess is balanced by a long-range depletion that has a universal shape independent of the molecular details. Furthermore, the approximation predicts that the reduction in surface energy equals one unit of kBT for every extra chain end at the surface.

Superposition in Flory-Huggins χ and Interfacial Tension for Compressible Polymer Blends

We investigate theoretically the response of interfacial tension γ for compressible polymer blends to thermodynamic variables. Helfand's self-consistent field theory is first extended to be combined with an off-lattice equation-of-state model to describe compressibility. Typical incompatible blends reveal that the effects of temperature and pressure (T-P) on γ are superposed into a single curve by a dimensionless pressure variable through the superposition in Flory-Huggins χ and density. In the case of polymer blends with strong compressibility difference, γ shows an anomaly upon pressurization with no T-P superposition, even though χ still follows the superposition.

Compressible or incompressible blend of interacting monodisperse linear polymers near a surface

We consider a lattice model of a mixture of repulsive, attractive, or neutral monodisperse linear polymers of two species, A and B, with a third monomeric species C, which may be taken to represent free volume. The mixture is confined between two hard, parallel plates of variable separation whose interactions with A and C may be attractive, repulsive, or neutral, and may be different from each other. The interactions with A and C are all that are required to completely specify the effect of each surface on all three components. We numerically study various density profiles as we move away from the surface, by using the recursive method of Gujrati and Chhajer [J. Chem. Phys. 106, 5599 (1997)] that has already been previously applied to study polydisperse solutions and blends next to surfaces. The resulting density profiles show the oscillations that are seen in Monte Carlo simulations and the enrichment of the smaller species at a neutral surface. The method is computationally ultrafast and can be carried out on a personal computer (PC), even in the incompressible case, when Monte Carlo simulations are not feasible. The calculations of density profiles usually take less than 20 min on a PC.

Predicting nonpolymeric materials structure with real-space self-consistent field theory

Polymer self-consistent field theory of the Edwards-Helfand kind is the state-of-the-art method for predicting the morphologies of block copolymer materials. The methodology of block copolymer self-consistent field theory is transported to classical density functional theory such that a wide range of self-consistent field theory tools can be applied to completely nonpolymeric materials, such as liquid crystal, molecular, or colloidal systems. This allows for the prediction of structure in nonpolymeric condensed matter systems without any prior knowledge of the possible phases, using calculations that take a fraction of the time needed for simulations. The approach is applied to a simple interaction site density functional theory representing adsorbed nitrogen molecules, and plastic crystal as well as herringbone phases are found in the phase diagram.

Approaching criticality in polymer-polymer systems

The interfacial width of polyolefins blends has been probed as a function of distance away from the critical point by using neutron reflectivity. For strongly immiscible polymer pairs, the width of the interface increases slowly when the degree of immiscibility is decreased and the interfacial width varies with the interaction parameter chi of the polymers. Closer to the critical point the dependence on the degree of miscibility becomes stronger and the way in which the interfacial width diverges, as criticality is approached, is related to both the chain length and chi. The self-consistent field theory numerical calculations, with the additional contribution due to capillary waves, provides a good description of the width of the interface between two polymer bulk phases in particular at intermediate values of the degree of immiscibility.

Comparison of random-walk density functional theory to simulation for bead-spring homopolymer melts

Density profiles for a homopolymer melt near a surface are calculated using a random-walk polymeric density functional theory, and compared to results from molecular dynamics simulations. All interactions are of a Lennard-Jones form, for both monomer-monomer interactions and surface-monomer interactions, rather than the hard core interactions which have been most investigated in the literature. For repulsive systems, the theory somewhat overpredicts the density oscillations near a surface. Nevertheless, near quantitative agreement with simulation can be obtained with an empirical scaling of the direct correlation function. Use of the random phase approximation to treat attractive interactions between polymer chains gives reasonable agreement with simulation of dense liquids near neutral and attractive surfaces.