Solution Of Self-consistent Field Equations Research Articles

Ball-of-Yarn Conformation of a Linear Gradient Copolymer in a Homopolymer Melt

The conformation of a linear gradient copolymer chain in a homopolymer melt is investigated using theory and numerical solutions of self-consistent field equations. In the limit of large comonomer immiscibility and chain length, it is found that the copolymer collapses into a globule with monomers self-assembled into a “ball-of-yarn” conformation. The spatial heterogeneity of monomers within the globule is in striking contrast to the “tadpole” conformation of a collapsed symmetric diblock copolymer and the disordered globular state of a collapsed homopolymer or random copolymer. By simple free energy calculations, we find that the same thermodynamic factors which drive a melt of linear gradient copolymers to self-assemble into lamellar microphases in the strong-segregation regime act to drive a single copolymer to self-assemble its own monomers into the yarn ball conformation when in a homopolymer melt with which it is immiscible. Moreover, by considering self-assembly of monomers within the globule of a ...

Self-Assembly of Lamellar Microphases in Linear Gradient Copolymer Melts

We investigate lamellar microphases in linear gradient copolymer melts by a combination of techniques, including numerical solutions of self-consistent field equations, scaling theory, and analysis of the strong-segregation limit. In particular, we construct a Flory theory to predict the scaling of the equilibrium lamellar width Leq as a function of comonomer incompatibility measured by χN. The Flory theory balances chain stretching with increased monomer repulsive interactions resulting from conformational fluctuations about a state of uniformly stretched chains, that obtains in the strong-segregation limit. We find Leq/Rg ∼ (χN)1/6, which agrees well with numerical results. Remarkably, this is the same result as for symmetric diblock copolymers, although for quite different physical reasons.

Genuine converging solution of self-consistent field equations for extended many-electron systems

Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schrödinger equation for singleparticle orbitals. Due to nonlinearity and complexity this set of equations, one believes in the iterative method for the solution that should consist in consecutive improvement of the potential and the electron density until the self-consistency is attained. Though this approach exists for a long time there are two grave problems accompanying its implementation to infinitely extended systems.The first of them is related with the Poisson equation and lies in possible incompatibility of the boundary conditions for the potential with the electron density distribution renewed by means of the Schrödinger equation. Rigorously speaking, it means the iteration process cannot be continued. The resolution of this difficulty is proposed for both infinite conducting systems in jellium approximation and periodic structures. It provides the existence of self-consistent solution for the potential at every iteration step due to realization of a screening effect.The second problem results from the existence of continuous spectrum of Hamiltonian eigenvalues for unbounded systems. It needs to have a definition of Hilbert space basis with eigenfunctions of continuous spectrum as elements, which would be convenient in numerical applications. It is proposed to insert a limiting transition into the definition of scalar product specifying the Hilbert space. It provides self-adjointness of Hamiltonian and, respectively, the orthogonality of eigenfunctions corresponding to the different eigenvalues. In addition, it allows to normalize them effectively to delta-function and to prove the orthogonality of the 'right' and 'left' eigenfunctions belonging to twofold degenerate eigenvalues.

Kinetics of formation of physically end-adsorbed polystyrene layers from the melt

Abstract We have used neutron reflectivity and helium (3) nuclear reaction analysis to measure adsorbed amounts and concentration profiles of carboxy-terminated polystyrene chains in the melt, which are physically end-adsorbed onto the native oxide layer on silicon substrates. The excess of adsorbed chains at the substrate increases with annealing time, but more slowly than predicted by a model. We have measured the adsorption isotherm for this system. By comparison with numerical solutions of self-consistent field equations we deduce a sticking energy of 8.6 kT per molecule.

Solution of self-consistent field equations by the recursion method

We show that the equations of the Coherent Potential Approximation (C.P.A.) can be solved in real space using the recursion method. The Green's function, from which the density of states is calculated, is then represented by a continued fraction, and the method provides a quick and precise solution. A first application is presented

Local solutions of self-consistent field equations for coupled oscillators and their symmetry

Exact solutions of self-consistent field (SCF) equations are obtained for a class of coupled oscillators (CO). For some CO models the solutions localized in various domains of configurational space are found. These local solutions (LS) correspond to the critical points (CP) of SCF energy functionals of various types, e.g. local minima, saddle points and others which are Thom's catastrophes. The symmetry properties of LS are discussed. Broken symmetry LS and a new class of LS with symmetry different from the symmetry of the CO Hamiltonian are also found.