On the origins of spontaneous spherical symmetry-breaking in open-shell atoms through polymer self-consistent field theory.

An alternative approach to density functional theory based on self-consistent field theory for ring polymers is applied to neutral atoms hydrogen to neon in their ground-states. The spontaneous emergence of an atomic shell structure and spherical symmetry-breaking of the total electron density are predicted by the model using the ideas of polymer excluded-volume between pairs of electrons to enforce the Pauli-exclusion principle and an exact electron self-interaction correction. The Pauli potential is approximated by neglecting inter-atomic correlations along with other types of correlations, and comparisons to Hartree-Fock theory are made, which also ignores correlations. The model shows excellent agreement with Hartree-Fock theory to within the standards of orbital-free density functional theory for the atomic binding energies and density profiles of the first six elements, providing exact matches for the elements hydrogen and helium. The predicted shell structure starts to deviate significantly past the element neon, and spherical symmetry-breaking is first predicted to occur at carbon instead of boron. The self-consistent field theory energy functional that describes the model is decomposed into thermodynamic components to trace the origin of spherical symmetry-breaking. It is found to arise from the electron density approaching closer to the nucleus in non-spherical distributions, which lowers the energy despite resulting in frustration between the quantum kinetic energy, electron-electron interaction, and the Pauli exclusion interaction. The symmetry-breaking effect is found to have a minimal impact on the binding energies, which suggests that the spherical-averaging approximation used in previous work is physically reasonable when investigating atomic systems. The pair density contour plots display behavior similar to polymer macro-phase separation, where individual electron pairs occupy single lobe structures that together form a dumbbell shape analogous to the 2p orbital shape. It is further shown that the predicted densities satisfy known constraints and produce the same total electronic density profile that is predicted by other formulations of quantum mechanics.

Molecular Bonding in an Orbital-Free-Related Density Functional Theory.

A density functional theory based on polymer self-consistent field theory is applied to systems of two atoms in order to show that this approach is capable of predicted molecular bonding. Periodic table elements from hydrogen up to neon are examined and homonuclear diatomic molecules are found to form for H2, N2, O2, and F2, in agreement with known results. The heteronuclear molecules CO and HF, which are known to exist under ambient conditions, are also found to be stable. Bond lengths for most of these molecules agree with experimental results to within less than 8%, with the exception of O2 and F2 which deviate more significantly. The bonding energy for H2 is given and is within 16% of the known value, but fundamental vibrational frequencies do not agree well with experiment. The main approximations of the theory are very simple and include a Fermi-Amaldi correction to the electron-electron interaction to account for self-interactions and a basic expression for the Pauli potential to account for the exclusion principle. The self-consistent equations are solved in terms of basis functions that encode the cylindrical symmetry of diatomic molecules. Since orbitals are not used, the approach is related to orbital-free density functional theory.

An alternative derivation of orbital-free density functional theory.

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory and to reduce to classical density functional theory, each under appropriate conditions. The obtained noninteracting kinetic energy functional is, in principle, exact but suffers from the usual orbital-free approximation of the Pauli exclusion principle in addition to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the timelike parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing pointlike quantum particles instead as nonlocal, polymerlike, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation. Thus, the present formalism may be considered as a type of "pilot-wave," realist, perspective on density functional theory.

Application of a constant hole volume Sanchez-Lacombe equation of state to mixtures relevant to polymeric foaming.

A variant of the Sanchez-Lacombe equation of state is applied to several polymers, blowing agents, and saturated mixtures of interest to the polymer foaming industry. These are low-density polyethylene-carbon dioxide and polylactide-carbon dioxide saturated mixtures as well as polystyrene-carbon dioxide-dimethyl ether and polystyrene-carbon dioxide-nitrogen ternary saturated mixtures. Good agreement is achieved between theoretically predicted and experimentally determined solubilities, both for binary and ternary mixtures. Acceptable agreement with swelling ratios is found with no free parameters. Up-to-date pure component Sanchez-Lacombe characteristic parameters are provided for carbon dioxide, dimethyl ether, low-density polyethylene, nitrogen, polylactide, linear and branched polypropylene, and polystyrene. Pure fluid low-density polyethylene and nitrogen parameters exhibit more moderate success while still providing acceptable quantitative estimations. Mixture estimations are found to have more moderate success where pure components are not as well represented. The Sanchez-Lacombe equation of state is found to correctly predict the anomalous reversal of solubility temperature dependence for low critical point fluids through the observation of this behaviour in polystyrene nitrogen mixtures.

Communication: Molecular-level insights into asymmetric triblock copolymers: network and phase development.

Molecularly asymmetric triblock copolymers progressively grown from a parent diblock copolymer can be used to elucidate the phase and property transformation from diblock to network-forming triblock copolymer. In this study, we use several theoretical formalisms and simulation methods to examine the molecular-level characteristics accompanying this transformation, and show that reported macroscopic-level transitions correspond to the onset of an equilibrium network. Midblock conformational fractions and copolymer morphologies are provided as functions of copolymer composition and temperature.

Tilt grain boundaries in a diblock copolymer ordered nanocomposite lamellar phase.

A hybrid self-consistent field theory/density functional theory method is applied to predict tilt (kink) grain boundary structures between lamellar domains of a symmetric diblock copolymer with added spherical nanoparticles. Structures consistent with experimental observations are found and theoretical evidence is provided in support of a hypothesis regarding the positioning of nanoparticles. Some particle distributions are predicted for situations not yet examined by experiment.

Reduction of polymer surface tension by crystallized polymer nanoparticles.

Self-consistent field theory is applied to investigate the effects of crystallized polymer nanoparticles on polymer surface tension. It is predicted that the nanoparticles locate preferentially at the polymer surface and significantly reduce the surface tension, in agreement with experiment. In addition to the reduction of surface tension, the width of the polymer surface is found to narrow. The reduced width and surface tension are due to the smaller spatial extent of the nanoparticles compared to the polymer. This allows the interface to become less diffuse and so reduces the energies of interaction at the surface, which lowers the surface tension. The solubility of the surrounding solvent phase into the polymer melt is mostly unchanged, a very slight decrease being detectable. The solubility is constant because away from the interface, the system is homogeneous and the replacement of polymer with nanoparticles has little effect.

Ordering and reverse ordering mechanisms of triblock copolymers in the presence of solvent.

Self-consistent field theory is used to study the self-assembly of a triblock copolymer melt. Two different external factors (temperature and solvent) are shown to affect the self-assembly. Either one or two-step self-assembly can be found as a function of temperature in the case of a neat triblock melt, or as a function of increasing solvent content (for non-selective solvents) in the case of a triblock-solvent mixture. For selective solvents, it is shown that increasing the solvent content leads to more complicated self-assembly mechanisms, including a reversed transition where order is found to increase instead of decreasing as expected, and re-entrant behavior where order is found to increase at first, and then decrease to a previous state of disorder.

Nanoparticle-regulated phase behavior of ordered block copolymers.

Although block copolymer motifs have received considerable attention as supramolecular templates for inorganic nanoparticles, experimental observations of a nanostructured diblock copolymer containing inorganic nanoparticles-supported by theoretical trends predicted from a hybrid self-consistent field/density functional theory-confirm that nanoparticle size and selectivity can likewise stabilize the copolymer nanostructure by increasing its order-disorder transition temperature.

Bidirectional mapping between self-consistent field theory and molecular dynamics: application to immiscible homopolymer blends.

A bidirectional mapping scheme that bridges particle-based and field-based descriptions for polymers is presented. Initial application is made to immiscible homopolymer blends. The forward mapping (upscaling) approach is based on the use of molecular dynamics simulations to calculate interfacial density profiles for polymer molecular weights that can be readily relaxed using standard simulation methods. These profiles are used to determine the optimal, effective interaction parameter that appears in the one-parameter self-consistent field theory treatment employed in the present work. Reverse mapping from a field representation to a particle-based description is accomplished by the application of a density-biased Monte Carlo method that generates representative chain configurations in the blend using statistical weights derived from fields obtained from self-consistent field theory.

Predicting the phases of a two-dimensional hard-rod system with real-space self-consistent field theory.

Polymer self-consistent field theory numerical tools are applied to a two-dimensional hard-rod colloidal system. Rods are represented through an interaction site model density functional theory that is derived and expressed from a self-consistent field theory perspective. A weighted density approximation is used within the density functional theory, and the phase space is sampled without bias for any particular morphology. A completely ordered crystal phase is found as well as a liquid crystal state.

Self-assembly of a binary mixture of particles and diblock copolymers.

Using theoretical models, we undertake the first investigation into the synergy and rich phase behavior that emerges when binary particle mixtures are blended with microphase-separating copolymers. We isolate an example of spontaneous hierarchical self-assembly in such hybrid materials, where the system exhibits both nanoscopic ordering of the particles and macroscopic phase transformation in the copolymer matrix. Furthermore, the self-assembly is driven by entropic effects involving all the different components. The results reveal that entropy can be exploited to create highly ordered nanocomposites with potentially unique electronic and photonic properties.

Entropically driven formation of hierarchically ordered nanocomposites.

Using theoretical models, we undertake the first investigation into the rich behavior that emerges when binary particle mixtures are blended with microphase-separating copolymers. We isolate an example of coupled self-assembly in such materials, where the system undergoes a nanoscale ordering of the particles along with a phase transformation in the copolymer matrix. Furthermore, the self-assembly is driven by entropic effects involving all the different components. The results reveal that entropy can be exploited to create highly ordered nanocomposites with potentially unique electronic and photonic properties.

Binary hard sphere mixtures in block copolymer melts.

We perform a self-consistent-field/density-functional-theory hybrid analysis for a system of diblock copolymers mixed with polydisperse, hard, spherical particles of various chemical species. We apply this theory to study the equilibrium morphologies of two different binary sphere/diblock melts. First, we examine the case where the particles have two different sizes, but both types are preferentially wetted by one of the copolymer blocks. We find that the single-particle distributions for the two species do not track one another and that the particles show a degree of entropically generated separation based on size, due to confinement within the diblock matrix. Second, we study the case where the particles are all the same size, but are of two different chemical species. We find that, as expected, the particle distributions reveal a degree of enthalpically driven separation, due to the spheres' preferential affinities for different blocks of the copolymer.