Time-dependent Diffusion Coefficient and Conventional Diffusion Constant of Nanoparticles in Polymer Melts by Mode-coupling Theory (cited: 2)

Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa-tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site modelwith Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode-coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.     

Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized Fick-Jacobs equation

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.     

Generalized Kubo relations and conditions for anomalous diffusion: physical insights from a mathematical theorem

The paper describes an approach to anomalous diffusion within the framework of the generalized Langevin equation. Using a Tauberian theorem for Laplace transforms due to Hardy, Littlewood, and Karamata, generalized Kubo relations for the relevant transport coefficients are derived from the asymptotic form of the mean square displacement. In a second step conditions for anomalous diffusion are derived for the asymptotic forms of the velocity autocorrelation function and the associated memory function. Both spatially unconfined and confined diffusion processes are considered. The results are illustrated with semi-analytical examples.     

Interdiffusion of solvent into glassy polymer films: a molecular dynamics study

Large scale molecular dynamics and grand canonical Monte Carlo simulation techniques are used to study the behavior of the interdiffusion of a solvent into an entangled polymer matrix as the state of the polymer changes from a melt to a glass. The weight gain by the polymer increases with time t as t(1/2) in agreement with Fickian diffusion for all cases studied, although the diffusivity is found to be strongly concentration dependent especially as one approaches the glass transition temperature of the polymer. The diffusivity as a function of solvent concentration determined using the one-dimensional Fick's model of the diffusion equation is compared to the diffusivity calculated using the Darken equation from simulations of equilibrated solvent-polymer solutions. The diffusivity calculated using these two different approaches are in good agreement. The behavior of the diffusivity strongly depends on the state of the polymer and is related to the shape of the solvent concentration profile.     

Transfer of electronic excitation energy in partially disordered polymer chain

A new form of the generalized master equation for electronic excitation transfer in a partially disordered polymer chain was derived. Within this equation, in order to differentiate the transfer in ordered parts of the partially disordered polymer chain, we decomposed the memory function of electronic excitation into several parts. These parts are memory functions associated with various types of subdynamics of electronic excitation. The partially disordered polymer chain where the disorder is caused by conformational defects is discussed.     

Kinetics of the Photodimerization of Anthracene in an Inhomogeneous Polymer Matrix

The photodimerization of anthracene in polyethylene was investigated. It was shown that the kinetics of photodimerization are not described in terms of nonstationary diffusion. A method is proposed for determining the probability of the existence a volume of inhomogeneous polymer layer with a specific diffusion coefficient of the impurity anthracene molecule. An equation describing the kinetics of photodimerization is derived, and the limiting values of the diffusion coefficient (minimum and maximum) are obtained. A probability distribution for the existence of regions of polymer layer with diffusion coefficient between D and D + dD is also obtained.     

Free energy of an inhomogeneous polymer–polymer–solvent system. II

A complete expression for the enthalpy of mixing of inhomogeneous polymer–polymer–solvent systems applicable for small as well as large concentration fluctuations has been developed. This is used to express the free energy of inhomogeneous polymer–polymer–solvent systems in an extended form of the Landau-Ginzburg functional. The gradient energy parameters obtained here are consistent with the published results. The free energy functional has been applied to develop a generalized continuity equation for spinodal decomposition in polymer–polymer systems. A linearized version of this continuity equation has been used to study the effect of the gradient terms on the dominant wavelength during spinodal decomposition.     

Generalized diffusion equation of interacting brownian particles

Macroscopic behavior of a system of brownian particles interacting with each other through potential forces is described by a generalized diffusion equation (GDE) for the density of particles. The diffusion coefficient in the GDE is given by the generalized Stokes—Einstein relation and generally depends on the density. In the presence of long-range interactions, the GDE becomes non-local in space. When a Coulomb interaction exists, the GDE corresponds to an improvement of the Poisson—Boltzmann equation.     

Modelling of water transport in ionic hydrophilic polymers

Water and ion transport in thin sheets of initially dry, ionic, hydrophilic crosslinked polymers was modelled throughout the dynamic swelling process. The water transport was expressed in terms of a non-Fickian equation with a diffusion term containing a Fujitatype concentration-dependent diffusion coefficient coupled with a pseudoconvective term arising from the reasonable assumption that the stress in ionic polymers is proportional to the total number of ionized pendant groups in the polymer. Ion transport was expressed in terms of generalized Fickian equations with water concentration-dependent diffusion coefficients. These equations were solved with appropriate boundary conditions to establish the water uptake as a function of time, pH and ionic strength in a citrate-phosphate-borate buffer solution. A new dimensionless number, the Stress Swelling number, A, was defined to quantify the relative importance of stress in the overall swelling process. Water uptake was a strong function of A. © 1994 John Wiley & Sons, Inc.     

Adsorption kinetics in the polyethylenimine–cellulose fiber system

The rate of adsorption of fractionated polyethylenimine (PEI) from water onto regenerated cellulose fibers was studied as a function of the polymer diffusion coefficient. Differences in polymer molecular weight, salt concentration, and pH were employed to vary the diffusion coefficient which was measured independently by a free-diffusion technique. The sorption rate was measured at the same conditions and found to increase with decreasing molecular weight, increasing polymer concentration, decreasing salt concentration, and increasing pH. A simplified rate equation based on diffusion control with Langmuirian adsorption in stirred solution was developed by utilizing the concept of a Nernst diffusional film. The equation was successful in predicting the relationship between adsorption rate and diffusion coefficient for most cases studied. It was found, however, that a very large barrier to mass transfer retards the adsorption rate. For the system studied it was concluded that this barrier is a result of diffusion into and subsequent adsorption onto the internal porous structure of the cellulose.     

Evaluation of unperturbed and thermodynamic parameters of polymer solutions from viscometric data

The existing approximate theories of polymer solutions are generalized by a simple equation. It is combined with the Kirkwood- Riseman perturbation treatment of viscosity together with the Kuhn model of random coil dimensions to produce two expressions for the unperturbed and thermodynamic parameters of polymers in solvents. A test of these equations on a large variety of polymer systems yields results which are comparable with the experimental values.     

Relaxation of chemical potential and a generalized diffusion equation

The effect of slow structural relaxation in a solvent of high viscosity on the chemical potential driving the diffusion of penetrant molecules is described by a generalized diffusion equation with a memory term. The linearized version of this equation is solved for some special cases, and the correlation function of concentration fluctuations in thermodynamic equilibrium is calculated. As a result of the memory term, for very slow relaxation two different stages of the diffusion process can be distinguished.     

Diffusion of Nanoparticles in Semidilute Polymer Solutions: A Multiparticle Collision Dynamics Study

The diffusion of nanoparticles immersed in semidilute polymer solutions is investigated by a hybrid mesoscopic multiparticle collision dynamics method. Effects of polymer concentration and hydrodynamic interactions among polymer monomers are focused. Extensive simulations show that the dependence of diffusion coefficient D on the polymer concentration c agrees with Phillies equation D-exp (-αc^δ) with a scaling exponent δ≈0.97 which coincides with the experimental one in literature. For increasing nanoparticle size, the scaling prefactor α increases monotonically while the scaling exponent always keeps fixed. Moreover, we also study the diffusion of nanoparticle without hydrodynamic interactions and find that mobility of the nanoparticle slows down, and the scaling exponent is obviously different from the one in experiments, implying that hydrodynamic interactions play a crucial role in the diffusion of a nanoparticle in semidilute polymer solutions.     

聚合物单晶生长的三维相场模拟研究

利用元胞自动机方法与相场模型的结合建立新型三维模拟相场模型.同时,为模拟真实的、三维的高分子结晶的过程,采用元胞自动机方法离散方程,且元胞几何形状的选取符合真实聚合物晶格扩散方式的物理规律,以及新建立的相场模型套用间规聚丙烯的实验参数.利用该模型模拟了多种三维立方体或者薄层的晶体形貌及其相互之间的演化过程,包括正方形、长方形、菱形、六边形、多层单晶等.通过模拟结果与真实形貌作对比来证明所建立的相场模型真实可靠性.     

Model of processing‐induced microstructure formation in polymeric materials

A general, mechanistic, kinetic model is presented to predict polymer microstructure formation during processing. Applications of the model are presented for three specific cases. The model represents polymer molecules as Kramers chains which may or may not have nucleated. Three forces (hydrodynamic, Brownian, and intermolecular) that act on polymer molecules during processing were considered, which resulted in the presentation of the model as a diffusion equation. The input parameters account for the rheological and thermal history of the polymer melt, the specific type of polymer molecule, and the initial morphology. The solution of the diffusion equation yields a probability distribution function from which the transient and equilibrium morphology can be determined. The three specific cases were chosen to illustrate the versatility of the model and include: the extensional flow‐induced growth of extended chain crystals; the orientation of stiff molecules in solution undergoing shear flow well above the crystallization temperature; and the formation of folded chain vs. extended chain crystals in an extensional flow. Data are available for the first two cases and agree favorably with the model predictions. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2571–2585, 1999     

Electronic energy transfer in an impurity band

In this paper we explore the dynamics of triplet electronic energy transfer in an impurity band of a substitutionally-disordered material. The general characteristics of the solutions of the master equation in the strong scattering regime have been studied, leading to a generalized diffusion equation that contains a memory term, which results in a time-dependent diffusion coefficient. Approximate results for the average density of excitation were derived within the framework of the pair-approximation, resulting in explicit expressions for the time-resolved spectral diffusion, the mean-square displacement, as well as the time-dependent diffusion coefficient for electronic transfer processes induced by exchange interactions.     

水、甲醇和乙醇溶剂在PVA膜材料中无限稀释扩散系数测定

用气相色谱法研究小分子溶剂与聚合物材料之间的相互作用是一种快速、准确和方便的方法,该方法可以测量多种小分子溶剂在聚合物中的溶解和扩散行为.采用该方法测定了水、甲醇、乙醇在固定液PVA中的保留时间和半峰宽,运用van Deemter模型进行数据处理,得到了上述几种小分子在PVA膜材料中的无限稀释扩散系数,获得了有意义的结果.     

Calculation of diffusion impedance in the terms of locally nonequilibrium diffusion

An equation for diffusion impedance is derived in the terms of the lattice diffusion model based on the assumption as to the local nonequilibrium distribution of diffusing particles across the sites of different types. This equation is valid at low lattice occupation. Unlike the multiple trapping model, all site types are interpreted symmetrically. In the boundary condition, it is assumed that there is a unique relationship between the electric potential variation, on one side of the interface and variation of the generalized particle activity on the other side.     

极性和非极性溶剂在聚乙烯中的无限稀释扩散系数的测定

 气相法研究小分子溶剂与高分子聚合物材料之间的相互作用是一个快速、准确、方便的方法 ,可以测量多种小分子溶剂在聚合物中的溶解、扩散参数。通过气相法测定了 5种小分子溶剂 (正己烷、正庚烷、正癸烷、乙醇和水 )在固定液聚乙烯中的保留时间和半峰宽 ,运用vanDeemter模型进行数据处理 ,得到上述 5种小分子在聚乙烯中的无限稀释扩散系数 ,获得了十分有意义的结果。