Simulation of micro-behaviors including nucleation,growth, and aggregation in particle system

A new method for the solution of population balance equations (PBE) describing the micro-processes such as nucleation, growth, aggregation of particle swarms in a multiphase system is proposed. The method is based on the fixed pivot moment and allows arbitrary number of moments to be tracked simultaneously. By expressing PBEs for both batch and continuous operations in a general form, and using weighted residual method to derive the moment equations, different moments can be tracked directly. The numerical density function is assumed to be a summation of several weighted Dirac Delta functions, and the integral and derivative terms in PBEs are transformed to a summation in order to reduce computational cost. Simulations of a batch nucleation-growth process and a continuous aggregation-growth process have demonstrated good agreement with the corresponding analytical solutions, with relative errors less than 10⁻⁸%. Simulation of a combined nucleation-growth-aggregation process, which does not have an analytical solution, is also included, so as to reproduce the micro-behaviors of such a complex system, demonstrating the feasibility and reliability of this method. Supported by the National Basic Research Program of China (Grant No. 2004CB720208), National Natural Science Foundation of China (Grant Nos. 40675011 and 10872159) and Key Laboratory of Mechanics on Disaster and Environment in Western China (Grant No. 200707)     

Extended cell average technique for the solution of coagulation equation

Sectional (zero order) methods constitute a very important class of methods for the solution of the population balance equation offering distinct advantages compared to their competitors, namely, higher order and moment methods. For the last ten years a particular sectional method, the so-called fixed pivot technique has been the most extensively used in the scientific community for the solution of the coagulation equation because it offers arbitrary grid choice and conservation of two moments of the particle size distribution. Very recently, a new method (called cell average technique) has been developed which gives more accurate results than the fixed point technique. In the present work, the extension of this new method in order to conserve three moments is attempted. A stable algorithm for the solution of the coagulation equation is developed. Although the new method allows improved computation of moments of practical interest, this is not always the case with respect to complete particle size distribution.     

On the Importance of Mixing for the Production of Nanoparticles

The precipitation process of nanoparticles of barium‐sulphate in a confined impinging jet reactor (CIJR) is here modelled by means of a precipitation model based on computational fluid dynamics (CFD). The micromixing was described with the interchange by exchange with the mean model coupled with the direct quadrature method of moments algorithm (DQMOM‐IEM), and the population balance equation (PBE), which describes the nucleation, growth and aggregation steps, was solved with the approach of the quadrature method of moments (QMOM). In this work preliminary results for the validation of the model are presented.     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

离散相系统群体平衡模型的求解算法

准确预测离散相系统中微观粒子的尺度分布演变对系统动态流动行为的准确确定起关键性作用. 粒子的尺度分布演变以及引起尺度分布变化的离散相微观行为(聚并、破碎、长大等)由群体平衡模型来描述. 该模型是关于数值密度函数的非线性双曲型方程, 数值求解为主要手段. 本文对群体平衡方程的直接离散方法、Monte Carlo、矩方法从实现难易程度、计算机资源消耗、计算精度三方面进行了详细阐述, 并着重介绍了几种性能优越的矩方法—— 矩积分方法(QMOM)、矩直接积分方法(DQMOM)、可调节矩积分方法(M-QMOM)、自适应矩直接积分方法(ADQMOM)、定点矩积分方法(FPQMOM)、粒子游动算法(MPEM)和局部定点矩积分方法(LFPQMOM). 最后根据算法的优缺点及其当前发展状况对不同算法的未来发展做了预测.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples

This article is the first of two articles on the adaptive multilevel finite element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well‐known test problems. The PBE is first discretized with piece‐wise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh‐based finite difference or box‐method algorithms, including multilevel methods, require substantially more time to reach a target PBE solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000     

Determining partial differential cross sections for low-energy electron photodetachment involving conical intersections using the solution of a Lippmann-Schwinger equation constructed with standard electronic structure techniques

A method for obtaining partial differential cross sections for low energy electron photodetachment in which the electronic states of the residual molecule are strongly coupled by conical intersections is reported. The method is based on the iterative solution to a Lippmann-Schwinger equation, using a zeroth order Hamiltonian consisting of the bound nonadiabatically coupled residual molecule and a free electron. The solution to the Lippmann-Schwinger equation involves only standard electronic structure techniques and a standard three-dimensional free particle Green's function quadrature for which fast techniques exist. The transition dipole moment for electron photodetachment, is a sum of matrix elements each involving one nonorthogonal orbital obtained from the solution to the Lippmann-Schwinger equation. An expression for the electron photodetachment transition dipole matrix element in terms of Dyson orbitals, which does not make the usual orthogonality assumptions, is derived.     

Quadrature method of moments for aggregation-breakage processes

Investigation of particulate systems often requires the solution of a population balance, which is a continuity statement written in terms of the number density function. In turn, the number density function is defined in terms of an internal coordinate (e.g., particle length, particle volume) and it generates integral and derivative terms. Different methods exist for numerically solving the population balance equation. For many processes of industrial significance, due to the strong coupling between particle interactions and fluid dynamics, the population balance must be solved as part of a computational fluid dynamics (CFD) simulation. Such an approach requires the addition of a large number of scalars and the associated transport equations. This increases the CPU time required for the simulation, and thus it is clear that it is very important to use as few scalars as possible. In this work the quadrature method of moments (QMOM) is used. The QMOM has already been validated for crystal growth and aggregation; here the method is extended to include breakage. QMOM performance is tested for 10 different cases in which the competition between aggregation and breakage leads to asymptotic solutions.     

Honeycomb-patterned films of polystyrene/poly(ethylene glycol): preparation,surface aggregation and protein adsorption

Highly ordered honeycomb-patterned polystyrene (PS)/poly(ethylene glycol) (PEG) films were prepared by a water-assisted method using an improved setup, which facilitated the formation of films with higher regularity, better reproducibility, and larger area of honeycomb structures. Surface aggregation of hydrophilic PEG and adsorption of bovine serum albumin (BSA) on the honeycomb-patterned films were investigated. Field emission scanning electron microscopy (FESEM) and atomic force microscopy (AFM) were used to observe the surface morphologies of the films before and after being rinsed with water. As confirmed by the FESEM images and the AFM phase images, PEG was enriched in the pores and could be gradually removed by water. The adsorption of fluorescence-labeled BSA on the films was studied in visual form using laser scanning confocal microscopy. Results clearly demonstrated that the protein-resistant PEG was selectively enriched in the pores. This water-assisted method may be a latent tool to prepare honeycomb-patterned biofunctional surfaces. Supported by the National Natural Science Foundation of China (Grant No. 50803053), the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 50625309), the National Postdoctoral Science Foundation of China (Grant Nos. 20070421172 & 20081466) and the National Undergraduate Innovative Test Program     

Density functional theory study of 1:1 glycine-water complexes in the gas phase and in solution

We present a systematic study of 1:1 glycine-water complexes involving all possible glycine conformers. The complex geometries are fully optimized for the first time both in the gas phase and in solution using three DFT methods (B3LYP, PBE1PBE, X3LYP) and the MP2 method. We calculate the G3 energies and use them as the reference data to gauge hydrogen bond strength in the gas phase. The solvent effects are treated via the integral equation formalism-polarizable continuum model (IEF-PCM). Altogether, we locate fifty-two unique nonionized (N) structures and six zwitterionic (Z) structures in the gas phase, and fifty-five N structures and thirteen Z structures in solution. Both correlation and solvation are shown to be important in geometry determination. We found that in the gas phase, a water molecule binds more strongly to the carboxylic acid group of glycine than to its amine group, whereas in solution phase the reverse is true. The most stable Z structure is isoenergetic with the most stable N structure.     

Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations

We extendthe application of moment methods to multivariate suspended particle population problems-those for which size alone is insufficient to specify the state of a particle in the population. Specifically, a bivariate extension of the quadrature method of moments (QMOM) (R. McGraw, Aerosol Sci. Technol. 27, 255 (1997)) is presented for efficiently modeling the dynamics of a population of inorganic nanoparticles undergoing simultaneous coagulation and particle sintering. Continuum regime calculations are presented for the Koch-Friedlander-Tandon-Rosner model, which includes coagulation by Brownian diffusion (evaluated for particle fractal dimensions, D(f), in the range 1.8-3) and simultaneous sintering of the resulting aggregates (P. Tandon and D. E. Rosner, J. Colloid Interface Sci. 213, 273 (1999)). For evaluation purposes, and to demonstrate the computational efficiency of the bivariate QMOM, benchmark calculations are carried out using a high-resolution discrete method to evolve the particle distribution function n(nu, a) for short to intermediate times (where nu and a are particle volume and surface area, respectively). Time evolution of a selected set of 36 low-order mixed moments is obtained by integration of the full bivariate distribution and compared with the corresponding moments obtained directly using two different extensions of the QMOM. With the more extensive treatment, errors of less than 1% are obtained over substantial aerosol evolution, while requiring only a few minutes (rather than days) of CPU time. Longer time QMOM simulations lend support to the earlier finding of a self-preserving limit for the dimensionless joint (nu, a) particle distribution function under simultaneous coagulation and sintering (Tandon and Rosner, 1999; D. E. Rosner and S. Yu, AIChE J., 47 (2001)). We demonstrate that, even in the bivariate case, it is possible to use the QMOM to rapidly model the approach to asymptotic behavior, allowing an immediate assessment of when previously established asymptotic results can be applied to dynamical situations of current/future interest. Copyright 2001 Academic Press.     

Influence of heteroelement on dipole and quadrupole moments of a series of three-membered rings containing a second,third, fourth,or fifth-row atom: a theoretical investigation

The influence of heteroelements on the molecular dipole and traceless quadrupole moments of a series of twenty-two three-membered rings (1–22) was theoretically estimated employing levels of theory such as MP2, CCSD, and PBE1PBE in combination with standard Pople’s basis set. To an accurate evaluation of these properties, additional calculations on the optimized geometries were performed using the correlation-consistent cc-pVDZ and aug-cc-pVDZ basis sets on the three mentioned methods. In particular, the dipole and quadrupole moments from MP2 and CCSD approaches are comparable to each other for the studied molecules, while PBE1PBE calculations were significantly deviated compared to MP2 and CCSD levels. High level of theory and large basis sets seemed to be needed to obtain reliable electrical properties in the heterocycles containing heavy atoms. Results demonstrated that the dipole and quadrupole moments are strongly determined by the nature of the heteroatom into ring skeleton, and its magnitude depends on the polarity of C-heteroelement bond. Dipole moment of these molecules 1–22 showed a clear increase with the increase of electronegativity and the atomic size of heteroatom into ring skeleton. Then, high relative dipole moment was found for three-membered rings containing II, IIIA, VIA, and VIIA elements, which is associated to the high polarization of the C-heteroatom bond. A similar behavior was observed for the quadrupole moments of these three-membered rings.

     

A microchip to analyze single crystal growth and size-controllability

A microfluidic device to control single crystallization on the micron scale has been developed. The salt solution was stored in the nano-volume gaps between the arrays of protrudent circular plots in the microchip. The mixed organic solvent was injected into the chip as the counter diffusion phase for crystallization forming. This device provides a liquid-liquid interface through which only one phase flows while the other stays at the fixed plot. Therefore, it is possible to control the position of crystallization on the fixed plot. We can control the size and the uniformity of single crystals from 5 to 50 μm in length by adjusting the relative factors, such as interface lifetime, breeds of the mix-organic solvents and injecting velocities. The longer interface lifetime and lower organic solvent injecting velocities can bring up larger and more asymmetric crystals, which nearly shows the same trend compared with the macroscopic crystallization. Finally, the effect of the surfactant on the crystallization in the micro-device was studied. By adding the surfactant into the liquid-liquid interface, smaller sizes of crystals can be obtained without changing the crystal configuration. Supported by the National Natural Science Foundation of China (Grant No. 20775042) and the National Basic Research Program of China (Grant No. 2007CB714507)     

A dynamical Lie algebraic mehtod for quantum reactive scattering

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Numerical solutions for a coupled non-linear oscillator

A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge–Kutta methods of order two and four).     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation II. Refinement at solvent‐accessible surfaces in biomolecular systems

We apply the adaptive multilevel finite element techniques (Holst, Baker, and Wang 21 ) to the nonlinear Poisson–Boltzmann equation (PBE) in the context of biomolecules. Fast and accurate numerical solution of the PBE in this setting is usually difficult to accomplish due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domains, and rapid (exponential) nonlinearity. However, these adaptive techniques have shown substantial improvement in solution time over conventional uniform‐mesh finite difference methods. One important aspect of the adaptive multilevel finite element method is the robust a posteriori error estimators necessary to drive the adaptive refinement routines. This article discusses the choice of solvent accessibility for a posteriori error estimation of PBE solutions and the implementation of such routines in the “Adaptive Poisson–Boltzmann Solver” (APBS) software package based on the “Manifold Code” (MC) libraries. Results are shown for the application of this method to several biomolecular systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1343–1352, 2000     

A new theory for symmetry orbital and tensor (II)

The symmetry orbital-symmetry orbital tensor methtd is applied to the evaluation of molecular integrals (one-electron) and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation sheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries. Projcct supported by the National Natural Science Foundation of China (Grant No. 29473119).     

Preparation of electrolyte membranes for micro tubular solid oxide fuel cells

Yttria-stabilized zirconia (YSZ) micro tubular electrolyte membranes for solid oxide fuel cells (SOFCs) were prepared via the combined wet phase inversion and sintering technique. The as-derived YSZ mi- cro tubes consist of a thin dense skin layer and a thick porous layer that can serve as the electrode of fuel cells. The dense and the porous electrolyte layers have the thickness of 3-5 μm and 70-90 μm, respectively, while the inner surface porosity of the porous layer is higher than 28.1%. The two layers are perfectly integrated together to preclude the crack or flake of electrolyte film from the electrode. The presented method possesses distinct advantages such as technological simplicity, low cost and high reliability, and thus provides a new route for the preparation of micro tubular SOFCs.     

Numerical interpretation of molecular surface field in dielectric modeling of solvation

Continuum solvent models, particularly those based on the Poisson‐Boltzmann equation (PBE), are widely used in the studies of biomolecular structures and functions. Existing PBE developments have been mainly focused on how to obtain more accurate and/or more efficient numerical potentials and energies. However to adopt the PBE models for molecular dynamics simulations, a difficulty is how to interpret dielectric boundary forces accurately and efficiently for robust dynamics simulations. This study documents the implementation and analysis of a range of standard fitting schemes, including both one‐sided and two‐sided methods with both first‐order and second‐order Taylor expansions, to calculate molecular surface electric fields to facilitate the numerical calculation of dielectric boundary forces. These efforts prompted us to develop an efficient approximated one‐dimensional method, which is to fit the surface field one dimension at a time, for biomolecular applications without much compromise in accuracy. We also developed a surface‐to‐atom force partition scheme given a level set representation of analytical molecular surfaces to facilitate their applications to molecular simulations. Testing of these fitting methods in the dielectric boundary force calculations shows that the second‐order methods, including the one‐dimensional method, consistently perform among the best in the molecular test cases. Finally, the timing analysis shows the approximated one‐dimensional method is far more efficient than standard second‐order methods in the PBE force calculations. © 2017 Wiley Periodicals, Inc.