Nanoparticle dynamics: a multiscale analysis of the Liouville equation

A theory of nanoparticle dynamics based on scaling arguments and the Liouville equation is presented. We start with a delineation of the scales characterizing the behavior of the nanoparticle/host fluid system. Asymptotic expansions, multiple time and space scale techniques, the resulting coarse-grained dynamics of the probability densities of the Fokker-Planck-Chandrasekhar (FPC) type for the nanoparticle(s), and the hydrodynamic models of the host medium are obtained. Collections of nanoparticles are considered so that problems such as viral self-assembly and the transition from a particle suspension to a solid porous matrix can be addressed via a deductive approach that starts with the Liouville equation and a calibrated atomic force field, and yields a generalized FPC equation. Extensions allowing for the investigation of the rotation and deformation of the nanoparticles are considered in the context of the space-warping formalism. Thermodynamic forces and dissipative effects are accounted for. The notion of configuration-dependent drag coefficients and their implications for coagulation and consolidation are shown to be natural consequences of the analysis. All results are obtained via formal asymptotic expansions in mass, size, and other physical and kinetic parameter ratios.     

Multiscale theory of collective and quasiparticle modes in quantum nanosystems

A quantum nanosystem (such as a quantum dot, nanowire, superconducting nanoparticle, or superfluid nanodroplet) involves widely separated characteristic lengths. These lengths range from the average nearest-neighbor distance between the constituent fermions or bosons, or the lattice spacing for a conducting metal, to the overall size of the quantum nanosystem (QN). This suggests the wave function has related distinct dependencies on the positions of the constituent fermions and bosons. We show how the separation of scales can be used to generate a multiscale perturbation scheme for solving the wave equation. Results for electrons or other fermions show that, to lowest order, the wave function factorizes into an antisymmetric (fermion) part and a symmetric (bosonlike) part. The former manifests the short-range/exclusion-principle behavior, while the latter corresponds to collective behaviors, such as plasmons, which have a boson character. When the constituents are bosons, multiscale analysis shows that, to lowest order, the wave function can also factorize into short- and long-scale parts. However, to ensure that the product wave function has overall symmetric particle label exchange behavior, there could, in principle, be states of the boson nanosystem where both the short- and long-scale factors are either boson- or fermionlike; the latter "dual fermion" states are, due to their exclusion-principle-like character, of high energy (i.e., single particle states cannot be multiply occupied). The multiscale perturbation analysis is used to argue for the existence of a coarse-grained wave equation for bosonlike collective behaviors. Quasiparticles, with effective mass and interactions, emerge naturally as consequences of the long-scale dynamics of the constituent particles. The multiscale framework holds promise for facilitating QN computer simulations and novel approximation schemes.     

Hierarchical Order Parameters for Macromolecular Assembly Simulations I: Construction and Dynamical Properties of Order Parameters

Macromolecular assemblies often display a hierarchical organization of macromolecules or their sub-assemblies. To model this, we have formulated a space warping method that enables capturing overall macromolecular structure and dynamics via a set of coarse-grained order parameters (OPs). This article is the first of two describing the construction and computational implementation of an additional class of OPs that has built into them the hierarchical architecture of macromolecular assemblies. To accomplish this, first, the system is divided into subsystems, each of which is described via a representative set of OPs. Then, a global set of variables is constructed from these subsystem-centered OPs to capture overall system organization. Dynamical properties of the resulting OPs are compared to those of our previous nonhierarchical ones, and implied conceptual and computational advantages are discussed for a 100ns, 2 million atom solvated Human Papillomavirus-like particle simulation. In the second article, the hierarchical OPs are shown to enable a multiscale analysis that starts with the N-atom Liouville equation and yields rigorous Langevin equations of stochastic OP dynamics. The latter is demonstrated via a force-field based simulation algorithm that probes key structural transition pathways, simultaneously accounting for all-atom details and overall structure.     

Order parameters for macromolecules: application to multiscale simulation

Order parameters (OPs) characterizing the nanoscale features of macromolecules are presented. They are generated in a general fashion so that they do not need to be redesigned with each new application. They evolve on time scales much longer than 10(-14) s typical for individual atomic collisions/vibrations. The list of OPs can be automatically increased, and completeness can be determined via a correlation analysis. They serve as the basis of a multiscale analysis that starts with the N-atom Liouville equation and yields rigorous Smoluchowski/Langevin equations of stochastic OP dynamics. Such OPs and the multiscale analysis imply computational algorithms that we demonstrate in an application to ribonucleic acid structural dynamics for 50 ns.     

Polymer-mediated spatial organization of nanoparticles in dense melts: Transferability and an effective one-component approach

We study two problems in the framework of the integral equation theory of polymer-mediated spatial organization of nanoparticles in dense melts motivated by multiscale simulation and many body physics issues. How nonspherical nanoparticle shape modifies polymer-induced interactions under dilute nanoparticle conditions is investigated over a range of primary particle sizes and interfacial cohesion strengths. Nonuniversal consequences of nonspherical shape are found for the pair-correlation function on local scales and some qualitative differences on larger scales due primarily to intraparticle connectivity constraints. For a large enough nanoparticle site diameter, the potentials of mean force (PMF) for all shapes studied (sphere, rod, disk, compact tetrahedral cluster) exhibit linear scaling with the size ratio of nanoparticle to polymer monomer site diameter and quite good "transferability." The ability of a simple effective one-component approach, based on the dilute nanoparticle PMF as an effective pair-decomposable potential, to describe interparticle structure at nonzero volume fractions is also studied. Although not generally quantitatively accurate due to neglect of many body correlation effects, especially at high nanoparticle loadings and near contact separations, the simple approach captures rather well many aspects of the real space structure. The errors incurred depend systematically on whether interfacial cohesion strength results in contact aggregation, steric stabilization, or bridging. For the filler collective static structure factor, many body effects are weakest for local cage scale correlations and grow significantly at smaller wavevectors under depletion or bridging conditions.     

Modeling real dynamics in the coarse-grained representation of condensed phase systems

This work presents a systematic multiscale methodology to provide a more faithful representation of real dynamics in coarse-grained molecular simulation models. The theoretical formalism is based on the recently developed multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B. 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] and relies on the generalized Langevin equation approach and its simpler Langevin equation limit. The friction coefficients are determined in multiscale fashion from the underlying all-atom molecular dynamics simulations using force-velocity and velocity-velocity correlation functions for the coarse-grained sites. The diffusion properties in the resulting CG Brownian dynamics simulations are shown to be quite accurate. The time dependence of the velocity autocorrelation function is also well-reproduced relative to the all-atom model if sufficient resolution of the CG sites is implemented.     

Dynamics of charge transfer states on metal surfaces: the competition between reactivity and quenching

The dynamics of excited states of adsorbates on surfaces caused by charge transfer is studied. Both negative and positive charge transfer processes are possible. In particular we are interested in positive charge transfer from a metal surface to molecular or atomic oxygen adsorbed on the surface. Once the negatively charged oxygen on the surface loses an electron it becomes chemically activated. The ability of this species to react depends on the quenching time or back transfer. The analysis of these processes is based on a set of diabatic potential energy surfaces each representing a different charged oxygen species. The dynamics is followed by solving the multichannel time-dependent Schr?dinger equation or Liouville von Neumann equation. Due to the nonadiabatic character of these reactions large isotope effects are predicted.     

Multiscale geometric modeling of macromolecules II: Lagrangian representation

Geometric modeling of biomolecules plays an essential role in the conceptualization of biolmolecular structure, function, dynamics, and transport. Qualitatively, geometric modeling offers a basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. Quantitatively, geometric modeling bridges the gap between molecular information, such as that from X‐ray, NMR, and cryo‐electron microscopy, and theoretical/mathematical models, such as molecular dynamics, the Poisson–Boltzmann equation, and the Nernst–Planck equation. In this work, we present a family of variational multiscale geometric models for macromolecular systems. Our models are able to combine multiresolution geometric modeling with multiscale electrostatic modeling in a unified variational framework. We discuss a suite of techniques for molecular surface generation, molecular surface meshing, molecular volumetric meshing, and the estimation of Hadwiger's functionals. Emphasis is given to the multiresolution representations of biomolecules and the associated multiscale electrostatic analyses as well as multiresolution curvature characterizations. The resulting fine resolution representations of a biomolecular system enable the detailed analysis of solvent–solute interaction, and ion channel dynamics, whereas our coarse resolution representations highlight the compatibility of protein‐ligand bindings and possibility of protein–protein interactions. © 2013 Wiley Periodicals, Inc.     

Numerical coupled Liouville approach: Application to second hyperpolarizability of molecular aggregate

In our previous study [Int. J. Quant. Chem., to appear], we have developed a novel numerical calculation scheme for a dynamics of quantum network for linear molecular aggregates under intense time‐dependent electric fields. In this approach, each molecule is assumed to be an electric dipole arranged linearly with an angle from the longitudinal axis, and the molecular interactions are taken into account by adding the radiations from these dipoles to the external electric fields. The effects of the radiations from all the dipoles involve the intermolecular distance, the speed of light, retarded polarization, and its first‐ and second‐order time derivatives at the position of each dipole. The quantum dynamics is performed by solving coupled Liouville equations composed of the Liouville equation for each dipole. In the present study, we develop a calculation approach of nonperturbative second hyperpolarizability γ in our novel approach and examine the γ of dimer models composed of two‐state molecules under the one‐photon near resonant intense laser fields. Similar phase transition‐like behavior in the field‐intensity dependence of the γ is observed. We also investigate the second hyperpolarizability spectra in the three‐photon resonant region for dimers composed of three‐state molecules, which mimic the electronic states of allyl cation. Contrary to the one‐photon resonant case, phase transition‐like behavior is not observed in the intensity dependence of γ in the three‐photon resonant region. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 295–306, 1999     

Growth of giant two-dimensional crystal of protein molecules from a three-phase contact line

A novel method to fabricate a two-dimensional (2D) crystal of protein molecules has been developed. The method enables us to control both the position of nucleation and the direction of the crystal growth. The crystal obtained using a protein molecule, ferritin, was found to be composed of a number of densely packed single crystal domains with an unprecedentedly large size of approximately 100 microm(2). This method also reveals characteristic behavior of the spatiotemporal evolution of the crystal; for example, "fusion" of the crystal domains, which is never observed in an ordinary crystal composed of atoms or ions, was demonstrated. Our approach could have potential in fabricating extraordinarily large and highly ordered nanoparticle arrays of organic or inorganic materials.     

A reliable new three-dimensional potential energy surface for H(2)-Kr

An improved three-dimensional potential energy surface for the H(2)-Kr system is determined from a direct fit of new infrared spectroscopic data for H(2)-Kr and D(2)-Kr to a potential energy function form based on the exchange-Coulomb model for the intermolecular interaction energy. These fits require repetitive, highly accurate simulations of the observed spectra, and both the strength of the potential energy anisotropy and the accuracy of the new data make the "secular equation perturbation theory" method used in previous analyses of H(2)-(rare gas) spectra inadequate for the present work. To address this problem, an extended version of the "iterative secular equation" method was developed which implements direct Hellmann-Feynman theorem calculation of the partial derivatives of eigenvalues with respect to parameters of the Hamiltonian which are required for the fits.     

Gelation and yielding behavior of polymer–nanoparticle hydrogels

Polymer–nanoparticle hydrogels are a unique class of self-assembled, shear-thinning, yield-stress fluids that have demonstrated potential utility in many impactful applications. Here, we present a thorough analysis of the gelation and yielding behavior of these materials with respect to the polymer and nanoparticle component stoichiometry. Through comprehensive rheological and diffusion studies, we reveal insights into the structural dynamics of the polymer nanoparticle network that identify that stoichiometry plays a key role in gelation and yielding, ultimately enabling the development of hydrogel formulations with unique shear-thinning and yield-stress behaviors. Access to these materials opens new doors for interesting applications in a variety of fields including tissue engineering, drug delivery, and controlled solution viscosity.     

Multiscale Lagrangian fluid dynamics simulation for polymeric fluid

We have developed a simulation technique of multiscale Lagrangian fluid dynamics to tackle hierarchical problems relating to historical dependency of polymeric fluid. We investigate flow dynamics of dilute polymeric fluid by using the multiscale simulation approach incorporating Lagrangian particle fluid dynamics technique (the modified smoothed particle hydrodynamics) with stochastic coarse‐grained polymer simulators (the dumbbell model). We have confirmed that our approach is well in agreement with the macroscopic results obtained by a constitutive equation corresponding to the dumbbell model, and observed that microscopic thermal fluctuation appears in macroscopic fluid dynamics as dispersion phenomena. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 886–893, 2010     

Adaptive integration of molecular dynamics

This article presents a particle method framework for simulating molecular dynamics. For time integration, the implicit trapezoidal rule is employed, where an explicit predictor enables large time steps. Error estimators for both the temporal and spatial discretization are advocated, and facilitate a fully adaptive propagation. The framework is developed and exemplified in the context of the classical Liouville equation, where Gaussian phase-space packets are used as particles. Simplified variants are discussed briefly. The concept is illustrated by numerical examples for one-dimensional dynamics in double well potential.     

Hierarchical superstructure of alkylamine-coated ZnS nanoparticle assemblies

We describe methodology for producing highly uniform, ordered and reproducible superstructures of surfactant-coated ZnS nanorod and nanowire assemblies, and propose a predictive multiscale "packing model" for superstructure formation based on electron microscopy and powder X-ray diffraction data on the superstructure, as well as on individual components of the nanostructured system. The studied nanoparticles showed a hierarchical structure starting from the individual faceted ZnS inorganic cores, onto which the crystalline surfactant molecules are adsorbed, to the superstructure of the nanoparticle arrays. Our results point out the critical role of the surfactant headgroup and polarity in nanoparticle assembly, and demonstrate the relationship between the molecular structure of the surfactant and the resulting superstructure of the nanoparticle assemblies.     

Semiclassical Liouville method for the simulation of electronic transitions: single ensemble formulation

In this paper, we describe a single ensemble implementation of the semiclassical Liouville method for simulating quantum processes using classical trajectories. In this approach, one ensemble of trajectories supports the evolution of all semiclassical density matrix elements, rather than employing a distinct ensemble for each. The ensemble evolves classically under a single reference Hamiltonian, which is chosen based on physical grounds; for electronic relaxation of an initially excited state, the initially populated upper surface Hamiltonian is the natural choice. Classical trajectories evolving on the reference potential then represent the time-dependent upper state population density and also the electronic coherence and the ground state density created by electronic transition. The error made in the classical motion of the trajectories for these latter distributions is compensated for by incorporating the difference between the correct and reference Liouville propagators into the calculation of the coefficients of the individual trajectories. This approach gives very accurate results for a number of model problems and cases describing ultrafast electronic relaxation dynamics.