Evaluation of the Chain Length Distribution in Free‐Radical Polymerization, 1. Bulk Polymerization

A method for the direct computation of the chain length distribution in a bulk polymerization is developed, based on the discretization procedure introduced by Kumar and Ramkrishna (Chem. Eng. Sci. 1996 , 51, 1311) in the context of particle size distribution. The overall distribution of chain lengths is partitioned into a finite number of classes which are supposed to be concentrated at some appropriate pivotal chain lengths. Several of the involved reactions lead to the formation of chain whose length differs from the pivotal values. Rules have been introduced in order to share chains between two contiguous classes, which have been designed so as to preserve two well‐defined properties of the distribution, such as, for example, two of its moments. The method has been applied to a polymerization system including propagation, bimolecular terminations and two different chain branching mechanisms: chain transfer to polymer and crosslinking. In addition, complex systems such as one with chain length‐dependent kinetic constants or a two‐dimensional distribution of chain length and number of branches have been considered.     

Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from approximately 8 to approximately 21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets (on the vapor side) and bubbles (on the liquid side) of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume (or length scale), which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems (i.e., larger than about 15 molecular diameters) at low temperature (lower than about 0.85 times the critical temperature). This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to (finite system size) bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.     

受高斯白噪声影响的有限化学反应体系的随机热力学——外噪声对化学反应体系影响的有效热力学量度

通过分析噪声对跃迁概率的不同影响,借助Novikov定理及MSR理论,建立了受多源白噪声影响的有限化学反应体系的有效主方程及有效熵平衡方程,导出非平衡定态时这类噪声体系熵产生的一般表达式,揭示涨落熵产生的统计内涵及噪声贡献,并针对非宏观量级的外噪声,借助扰动按分布参数分离法及有效主方程的Kramers-Moyal展开,进一步对简单加合性噪声建立了非平衡定态宏观稳定性判据的随机模拟,论证了噪声对化学反应体系定态稳定性的弱化作用.     

A three-dimensional finite element approach towards molecular SCF computations

Fully three-dimensional, SCF ground-state computations for the Hartree equation are carried out by a finite element approach that completely avoids forming or storing the Fock matrix. A combination of strategies is used to reduce storage and computational requirements by orders of magnitude over the traditional finite element approach, which makes three-dimensional molecular orbital computations feasible. Results using the three-dimensional formulation and computer program are shown for one-electron systems: He⁺ and H₂⁺, and for two-electron systems: He and H₂. The best results are within about 30–100 micro-Hartrees of the exact values of the total energies for the ground states of these systems, indicating that our three-dimensional approach has been correctly implemented in the computer code.     

Time-convolutionless master equation for mesoscopic electron-phonon systems

The time-convolutionless master equation for the electronic populations is derived for a generic electron-phonon Hamiltonian. The equation can be used in the regimes where the golden rule approach is not applicable. The equation is applied to study the electronic relaxation in several models with the finite number of normal modes. For such mesoscopic systems the relaxation behavior differs substantially from the simple exponential relaxation. In particular, the equation shows the appearance of the recurrence phenomena on a time scale determined by the slowest mode of the system. The formal results are quite general and can be used for a wide range of physical systems. Numerical results are presented for a two level system coupled to Ohmic and super-Ohmic baths, as well as for a model of charge-transfer dynamics between semiconducting organic polymers.     

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.     

A quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution based on the framework of mixed quantum-classical molecular dynamics

We present a quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution in the framework of mixed quantum-classical molecular dynamics, where the reactant and product states are explicitly defined by dividing the double-well potential into the reactant and product wells. The equation can describe quantum reaction processes such as tunneling and thermal excitation and relaxation assisted by the solvent. Fluctuations of the zero-point energy level, the height of the barrier, and the curvature of the well are all included in the equation. Here, the equation was combined with the surface hopping technique in order to describe the motion of the classical solvent. Applying the present method to model systems, we show two numerical examples in order to demonstrate the potential power of the present method. The first example is a proton transfer by tunneling where the high-energy product state was stabilized very rapidly by solvation. The second example shows a thermal activation mechanism, i.e., the initial vibrational excitation in the reactant well followed by the reacting transition above the barrier and the final vibrational relaxation in the product well.     

A New General Equation of the Polarographic Waves

A new general equation of the polarographic waves that improves the accuracy of the analysis has been obtained. The theoretical approach has been developed for Current Sampled DC Polarography or TAST Polarography, and can be used to study reversible, irreversible and quasi‐reversible waves. In order to facilitate the use of the new equation proposed, we have developed a computer program, TOP v1.0. This version works on Windows 32 bits systems (95/98/NT/Xp).     

The Poisson-Boltzmann model for tRNA: Assessment of the calculation set-up and ionic concentration cutoff

Using tRNA molecule as an example, we evaluate the applicability of the Poisson-Boltzmann model to highly charged systems such as nucleic acids. Particularly, we describe the effect of explicit crystallographic divalent ions and water molecules, ionic strength of the solvent, and the linear approximation to the Poisson-Boltzmann equation on the electrostatic potential and electrostatic free energy. We calculate and compare typical similarity indices and measures, such as Hodgkin index and root mean square deviation. Finally, we introduce a modification to the nonlinear Poisson-Boltzmann equation, which accounts in a simple way for the finite size of mobile ions, by applying a cutoff in the concentration formula for ionic distribution at regions of high electrostatic potentials. We test the influence of this ionic concentration cutoff on the electrostatic properties of tRNA.     

Finite element simulation of the amperometric response of recessed and protruding microband electrodes in flow channels

The convective mass transfer for recessed and protruding microband electrodes, two geometries found in practical devices, is determined by the finite element method. The problem is solved by a two-step method, first solving the Navier-Stokes equation to compute the real hydrodynamic flow, then solving the convective diffusion equation to calculate the electrochemical response of the systems. In this work, we focus on the chronoamperometric response of recessed and protruding microband electrodes, emphasizing the role of the edge effects and convection streamlines, in particular the role of stagnant recirculating eddies due to these particular geometries.     

Empirical Multicomponent Equilibrium and Film-Pore Model for the Sorption of Copper,Cadmium and Zinc onto Bone Char

The adsorption of three metal ions onto bone char has been studied in both equilibrium and kinetic systems. An empirical Langmuir-type equation has been proposed to correlate the experimental equilibrium data for multicomponent systems. The sorption equilibrium of three metal ions, namely, cadmium (II) ion, zinc (II) ion and copper (II) ion in the three binary and one ternary systems is well correlated by the Langmuir-type equation. For the batch kinetic studies, a multicomponent film-pore diffusion model was developed by incorporating this empirical Langmuir-type equation into a single component film-pore diffusion model and was used to correlate the multicomponent batch kinetic data. The multicomponent film-pore diffusion model shows some deviation from the experimental data for the sorption of cadmium ions in Cd-Cu, Cd-Zn and Cd-Cu-Zn systems. However, overall this model gives a good correlation of the experimental data for three binary and one ternary systems.     

Calculation of binding energy differences for receptor–ligand systems using the Poisson-Boltzmann method

An approach using the finite difference solution of the Poisson-Boltzmann equation to estimate binding free energy changes for two receptor–ligand systems, arabinose binding protein and sulfate binding protein, is presented. The eight calculated binding free energy changes agree with experiment, showing a correlation coefficient of 0.92 and energy deviations of 1 kcal/mol or less. More importantly, the decomposition of solvation and assembly energies in this approach provides an understanding of binding mechanisms and therefore could suggest directions to alter binding affinities. The method is demonstrated to be useful in analyzing experimental binding structures and predicting binding effects of mutants or modified ligands for macromolecular systems, in which the electrostatic forces dominate the overall interaction and the structural perturbations upon modifications are small. © 1995 by John Wiley & Sons, Inc.     

Calculation of wetting angles in crude oil/water/quartz systems

A method for the determination of water-advancing wetting angles has been developed and tested. The method allows measurements in black oils, as opposed to traditional techniques which substitute transparent model oils prior to measurements. The method is based on the Laplace equation and axisymmetric drop shape analysis. The main source of error is the determination of the drop volume. Results in transparent systems are comparable to results using other techniques. Wetting angles are determined for water in two different crude oil systems, using quartz as the substrate. The quartz surfaces are water wet over large pH ranges, but it is possible to accurately identify pH intervals where the surfaces are intermediate or oil wet.     

Theory of inhomogeneous weakly charged polyelectrolytes

A theory of inhomogeneous multicomponent systems containing weakly charged polyelectrolytes is developed. The theory treats the polymer conformation and the electrostatics simultaneously using a functional integral representation of the partition function. A mean‐field approximation to the theory leads to two sets of coupled mean‐field equations: a Poisson‐Boltzmann type equation describing the electrostatic potential, and a set of self‐consistent field equations describing the equilibrium densities. Asymptotic forms of the theory at weak and strong segregation limits are derived. The theory can be used to study the interfacial properties, microphase structures, and adsorptions of a variety of weakly charged polyelectrolyte systems. As a simple example, the interface between the polymer‐rich and polymer‐poor phases of a polyelectrolyte solution is studied.     

Modeling and simulation of stationary catalytic processes

Generalized models for steady state catalytic processes are presented in matrix form. Multistep reaction rate control is assumed. Numerical algorithms for solving of the created linear and nonlinear equation systems are developed and tested. Four examples are considered: an Eley–Rideal-mechanism, a Langmuir–Hinshelwood mechanism, a dual route, dual site mechanism, and a monomolecular decomposition with steady state multiplicity. The overall reaction rates are simulated as a function of the reactant concentrations. A maximum reaction rate is obtained in the case of a Langmuir–Hinshelwood mechanism (example 2), the location of the rate maximum in the concentration domain is shifted towards the concentration of the reactant with the lowest adsorption constants. An Eley–Rideal mechanism (example 1) has always monotonously increasing rate curves. In the case of steady state multiplicity (example 4) all steady states could be simulated with the proposed algorithm. The computation of reaction rate surfaces is important in investigating the behavior of complicated catalytic systems (e.g., systems with multistep rate control and/or steady state multiplicity), in planning of experiments and in chemical reactor simulation.     

Elastic properties and network structure in silicone gum vulcanizates

A model of a rubber network formed from finite, linear polymer chains is treated, using expressions from accepted elastic theory as well as relationships that have been less generally used or only implied. The resultant empirical mathematical expressions conform very closely to the predictions of the simple kinetic theory of rubber elasticity and are mathematically consistent, in that the major properties and measurements are interrelated, and apparent inconsistencies in the literature are reconciled. As a consequence, an empirical equation is developed for the relationship between elastic properties and network structure in silicone gum vulcanizates. This equation is then applied to the estimation of peroxide crosslinking efficiencies.     

A method,based on statistical moments,to evaluate the kinetic parameters involved in unstable enzyme systems

The evaluation of individual rate constants involved in any reaction mechanism of an enzymatic systems first requires experimental monitoring of the time course of the concentration or product rate creation or of any enzyme species. The experimental progress curves obtained must then be fitted to the corresponding theoretical symbolic equation. Nevertheless, in some cases, e.g. when the equation involves two or more exponential terms, this fit is not easy and sometimes impossible. Simplification of the equation is usually required by assuming, for example, that the system has reached the steady-state, assuming an initial steady-state of a segment in the scheme of the reaction mechanism or assuming rapid equilibrium in one or more of the reversible steps, if there are any. But, obviously, simplified equations produce either fewer individual rate constants or global constants consisting of algebraic associations of individual rate constants or individual rate constants or global constants that might considerably differ from the real ones due to the approaches made. In this contribution, we suggest an alternative procedure for evaluating the rate constants of enzyme reactions corresponding to enzyme systems where one or more of the species involved is unstable or where one or more of the enzyme species is irreversibly inhibited, or both. The procedure is based on the numerical determination of statistical moments from experimental time progress curves. The fitting of these experimentally obtained moments to the corresponding theoretical expressions allows us, in most cases, to evaluate of all of the rate constants involved, with only a small error. To verify the goodness of the suggested procedure, it was applied to an unstable enzyme system which had previously been analysed with other methods. Finally, it is indicated how this procedure could also be extrapolated for application to any stable or unstable enzyme system.