Renormalized one-loop theory of correlations in disordered diblock copolymers

A renormalized one-loop (ROL) theory developed in previous work [P. Grzywacz, J. Qin, and D. C. Morse, Phys. Rev E. 76, 061802 (2007)] is used to calculate corrections to the random phase approximation (RPA) for the structure factor S(q) in disordered diblock copolymer melts. Predictions are given for the peak intensity S(q?), peak position q?, and single-chain statistics for symmetric and asymmetric copolymers as functions of χ(e)N, where χ(e) is an effective Flory-Huggins interaction parameter and N is the degree of polymerization. The ROL and Fredrickson-Helfand (FH) theories are found to yield asymptotically equivalent results for the dependence of the peak intensity S(q?) upon χ(e)N for symmetric diblock copolymers in the limit of strong scattering, or large χ(e)N, but to yield qualitatively different predictions for symmetric copolymers far from the ODT and for asymmetric copolymers. The ROL theory predicts a suppression of S(q?) and a decrease of q? for large values of χ(e)N, relative to the RPA predictions, but an enhancement of S(q?) and an increase in q? for small χ(e)N. The decrease in q? near the ODT is shown to be unrelated to any change in single-chain statistics, and to be a result of inter-molecular correlations. Conversely, the predicted increase in q? at small values of χ(e)N is a direct result of non-Gaussian single-chain statistics.     

The structure and phase transitions in polymer blends,diblock copolymers and liquid crystalline polymers: The Landau-Ginzburg approach

The polymer systems are discussed in the framework of the Landau-Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi-block copolymer melts, random copolymer melts, ring polymers, rigid-flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean-field model, self consistent one-loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid-flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.     

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: an ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.     

Phase behavior of diblock copolymers; pressure and temperature dependence studied by small-angle neutron scattering

The structure factor of a poly(ethylene-propylene)poly(dimethylsiloxane) diblock copolymer has been measured by SANS as a function of temperature and pressure. In contradiction to the random phase approximation the conformational compressibility exhibits a pronounced maximum at the order-disorder phase transition. The phase boundary shows an unusual shape: with increasing pressure it first decreases and then increases. Its origin is an increase in, respectively, the entropic and the enthalpic part of the Flory-Huggins interaction parameter. The Ginzburg parameter describing the limit of the mean-field approximation is not influenced by pressure.     

First-principles calculations of zero-field splitting parameters

In this work, an implementation of an approach to calculate the zero-field splitting (ZFS) constants in the framework of ab initio methods such as complete active space self-consistent field, multireference configuration interaction, or spectroscopy oriented configuration interaction is reported. The spin-orbit coupling (SOC) contribution to ZFSs is computed using an accurate multicenter mean-field approximation for the Breit-Pauli Hamiltonian. The SOC parts of ZFS constants are obtained directly after diagonalization of the SOC operator in the basis of a preselected number of roots of the spin-free Hamiltonian. This corresponds to an infinite order treatment of the SOC in terms of perturbation theory. The spin-spin (SS) part is presently estimated in a mean-field fashion and appears to yield results close to the more complete treatments available in the literature. Test calculations for the first- and second-row atoms as well as first-row transition metal atoms and a set of diatomic molecules show accurate results for the SOC part of ZFSs. SS contributions have been found to be relatively small but not negligible (exceeding 1 cm(-1) for oxygen molecule). At least for the systems studied in this work, it is demonstrated that the presented method provides much more accurate estimations for the SOC part of ZFS constants than the emerging density functional theory approaches.     

Instabilities in block copolymer films induced by compressible solvents

We combine a simple lattice-gas model for fluid mixtures along with polymer mean-field theory for block copolymer melts to study the stability of thin films of diblock copolymers in the presence of compressible fluid solvents. Using a free energy analysis, the stable and unstable thicknesses of a copolymer thin film are obtained for given solvent conditions. Our results suggest that the interplay between confinement, the compressibility of the solvent, and its selectivity to polymer component can lead to significant changes on the ordering and stability of the diblock copolymer thin films. Our results are in qualitative agreement with recent experimental results.     

Neutron and light scattering studies of polymer blends and block copolymers

The structure of compatible polymer mixtures and its temperature dependence can be well described in many cases by the mean-field approximation. Deviations with regard to the temperature dependence of the correlation length of concentration fluctuations can be explained by taking into account the spatial structure of the interaction potential. Concentration fluctuations and the microphase separation in diblock copolymers are in good agreement with theories based on the random phase approximation. In the two-phase region the temperature dependence of the scattered X-ray or neutron intensity is governed by the difference in the thermal expansion coefficients of the two components of the diblock copolymer. Photon correlation spectroscopy (PCS) yields valuable information on the concentration dependence of the mutual diffusion coefficient D_AB.     

Developing a coarse-grained force field for the diblock copolymer poly(styrene-b-butadiene) from atomistic simulation

We have developed a coarse-grained force field for the poly(styrene-b-butadiene) diblock copolymer. We describe the computational methods and discuss how they were applied to develop a coarse-grained force field for this diblock copolymer from the atomistic simulation. The new force field contains three different bonds, four angles, five dihedral angles, and three nonbonded terms. We successfully tested this coarse-grained model against the chain properties, including static and dynamic properties, derived from the atomistic simulation; the results suggest that the coarse-grained force field is an effective model.     

Effects of interaction range and compressibility on the microphase separation of diblock copolymers: Mean-field analysis

Using the random-phase approximation and self-consistent field calculations, we have investigated the effects of finite interaction range and compressibility on the order-disorder transition (ODT) and the lamellar structure of symmetric diblock copolymers. While the compressibility does not affect the ODT, both the values of chiN and bulk lamellar period at the ODT increase with increasing interaction range. On the other hand, both the free-energy density and bulk period of the lamellae increase with either increasing interaction range or decreasing compressibility. Even with a finite compressibility, the mean-field ODT is still a second-order phase transition. The scaling exponent of bulk lamellar period with chiN, however, decreases with increasing compressibility. Our mean-field analysis provides a well understood reference for the study of fluctuation effects in diblock copolymers with finite interaction range and compressibility.     

Polymer brush in mixed solvent. The role of flory interaction parameter χAB of the solvent components

The properties of polymer brush in binary solvent are considered on the base of the mean-field theory. The role of the interaction parameter χ_AB of the two components on the brush parameters was specially considered. The mixture of two precipitants was shown to be a good solvent for the polymer brush provided that their interaction parameter χ_AB is sufficiently great.     

Fluctuation effects and the stability of the Fddd network phase in diblock copolymer melts

We examine the effect of composition fluctuations on the stability of the orthorhombic Fddd network phase in the diblock copolymer melt phase diagram within the self-consistent Hartree approximation to the Landau-Brazovskii theory. For weak fluctuations, the Fddd structure is an equilibrium phase; however, stronger fluctuations render this phase metastable. These results suggest a reinterpretation of a recent experiment beyond mean-field theory. Fluctuations may also explain why an equilibrium Fddd phase is not generally observed in analogous self-assembling systems.     

Scaling behavior of quantum nanosystems: emergence of quasi-particles, collective modes, and mixed exchange symmetry states

Examples of quantum nanosystems are graphene nanoribbons, molecular wires, and superconducting nanoparticles. The objective of the multiscale theory presented here is to provide a new perspective on the coupling of processes across scales in space and time underlying the dynamics of these systems. The long range objective for this multiscale approach is to serve as an efficient computational algorithm. Long space-time dynamics is derived using a perturbation expansion in the ratio ? of the nearest-neighbor distance to a nanometer-scale characteristic length and a theorem on the equivalence of long time-averages and expectation values. This dynamics is shown to satisfy a coarse-grained wave equation (CGWE) which takes a Schro?dinger-like form with modified masses and interactions. The scaling of space and time is determined by the orders of magnitude of various contributions to the N-body potential. If the spatial scale of the coarse-graining is too large, the CGWE would imply an unbounded growth of gradients; if it is too short, the system's size would display uncontrolled growth inappropriate for the bound states of interest, i.e., collective motion or migration within a stable nanoassembly. The balance of these two extremes removes arbitrariness in the choice of the scaling of space-time. Since the long-scale dynamics of each Fermion involves its interaction with many others, we hypothesize that the solutions of the CGWE have mean-field character to good approximation, i.e., can be factorized into single-particle functions. This leads to a coarse-grained mean-field approximation that is distinct in character from traditional Hartree-Fock theory. A variational principle is used to derive equations for the single-particle functions. This theme is developed and used to derive an equation for low-lying disturbances from the ground state corresponding to long wavelength density disturbances or long-scale migration. An algorithm for the efficient simulation of quantum nanosystems is suggested.     

Liquid-crystalline ordering in two-dimensional systems with discrete symmetry

The mean-field theories of liquid-crystalline (nematic) ordering developed for three-dimensional systems are applied to describe two-dimensional systems of both geometrically anisotropic and anisotropically interacting particles. Systems with discrete symmetry (lattice models) for which long-range order is possible are considered on the base of the Landau free-energy expansion. It is shown that the Hamiltonian describing the energy of intermolecular interactions may be written in a common form for lyotropic and thermotropic systems. The mean-field theory gives a continuous phase transition (second-order phase transition) for a square lattice, whereas for a triangular lattice it gives a phase transition with latent heat (first-order phase transition) like for three-dimensional systems. These results are compared with results of the exact theories (two-dimensional Ising and Potts models). It is concluded that for realistic two-dimensional models the orientational in-plane ordering is not sharper than a second-order phase transition.     

Transferability of interatomic correlation energies

A simple procedure is presented to estimate the interatomic correlation energy of a bond A-B on the basis of intrabond correlations of the homopolar parents A-A and B-B and the bond polarity of A-B. We use a local approach supplemented by a semiempirical mean-field Hamiltonian within the bond orbital approximation. The capability of the interpolation scheme is demonstrated for several molecules formed by main-group atoms.     

A new approach for efficient simulation of Coulomb interactions in ionic fluids

We propose a simplified version of local molecular field (LMF) theory to treat Coulomb interactions in simulations of ionic fluids. LMF theory relies on splitting the Coulomb potential into a short-ranged part that combines with other short-ranged core interactions and is simulated explicitly. The averaged effects of the remaining long-ranged part are taken into account through a self-consistently determined effective external field. The theory contains an adjustable length parameter sigma that specifies the cutoff distance for the short-ranged interaction. This can be chosen to minimize the errors resulting from the mean-field treatment of the complementary long-ranged part. Here we suggest that in many cases an accurate approximation to the effective field can be obtained directly from the equilibrium charge density given by the Debye theory of screening, thus eliminating the need for a self-consistent treatment. In the limit sigma-->0, this assumption reduces to the classical Debye approximation. We examine the numerical performance of this approximation for a simple model of a symmetric ionic mixture. Our results for thermodynamic and structural properties of uniform ionic mixtures agree well with similar results of Ewald simulations of the full ionic system. In addition, we have used the simplified theory in a grand-canonical simulation of a nonuniform ionic mixture where an ion has been fixed at the origin. Simulations using short-ranged truncations of the Coulomb interactions alone do not satisfy the exact condition of complete screening of the fixed ion, but this condition is recovered when the effective field is taken into account. We argue that this simplified approach can also be used in the simulations of more complex nonuniform systems.     

Extrapolated intermediate Hamiltonian coupled-cluster approach: theory and pilot application to electron affinities of alkali atoms

The intermediate Hamiltonian (IH) coupled-cluster method makes possible the use of very large model spaces in coupled-cluster calculations without running into intruder states. This is achieved at the cost of approximating some of the IH matrix elements, which are not taken at their rigorous effective Hamiltonian (EH) value. The extrapolated intermediate Hamiltonian (XIH) approach proposed here uses a parametrized IH and extrapolates it to the full EH, with model spaces larger by several orders of magnitude than those possible in EH coupled-cluster methods. The flexibility and resistance to intruders of the IH approach are thus combined with the accuracy of full EH. Various extrapolation schemes are described. A pilot application to the electron affinities (EAs) of alkali atoms is presented, where converged EH results are obtained by XIH for model spaces of approximately 20,000 determinants; direct EH calculations converge only for a one-dimensional model space. Including quantum electrodynamic effects, the average XIH error for the EAs is 0.6 meV and the largest error is 1.6 meV. A new reference estimate for the EA of Fr is proposed at 486+/-2 meV.     

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.     

Dynamical mean-field theory from a quantum chemical perspective

We investigate the dynamical mean-field theory (DMFT) from a quantum chemical perspective. Dynamical mean-field theory offers a formalism to extend quantum chemical methods for finite systems to infinite periodic problems within a local correlation approximation. In addition, quantum chemical techniques can be used to construct new ab initio Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-field theory to connect to quantum chemical language. Next, we describe an implementation of dynamical mean-field theory where we start from an ab initio Hartree-Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the configuration interaction hierarchy in DMFT as an approximate solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions.