Emulsion polymerization. III. Theoretical prediction of the effects of slow termination rate within latex particles

The Smith-Ewart theory predicts that there is an interval during an isothermal homopolymerization when the conversion varies linearly with time. This prediction rests on the assumptions that, during this interval II, the particle number is constant, the monomer concentration in the particles is constant, and the termination rate within the particles is instantaneous, so that the average number of radicals per particle Q is half. In this paper this latter assumption is abandoned. If the termination rate is slow, two or more radicals can coexist in a particle. The termination rate within a particle becomes a function of the particle size because of the decreased probability that two radicals meet for termination in a given time when the volume in which these radicals are located increases. It follows that with increasing conversion the termination rate decreases. Stockmayer's calculations based on this model neglected the variation of particle volume with time, and it was assumed that a steady state of radical concentration in particles exists. In the present calculations these restrictive assumptions were not used. Stockmayer calculated only how Q should vary with conversion. In the present paper several experimentally verifiable consequences of the model are shown. The new calculations show that the interval II conversion-time curve can be represented by the formula At² + Bt, where B is the Smith-Ewart rate and is proportional to the particle number and the parameter A is independent of the particle number and depends mainly on initiation and termination rates. From A and B and propagation and termination rate constants can be calculated. With the aid of parameters A and B the conversion dependence of molecular weight and of Q can also be predicted for interval II. In the theoretical calculations the distribution of radicals among particles is established. It is shown that for a given value of Q this distribution is unique, independent of the experimental conditions leading to this Q. This distribution was derived solely from kinetic considerations and is analogous to the statistical Poisson distribution. With increasing Q, i.e., with increasing conversion, this distribution broadens. Since each particle grows proportionally to the number of radicals in it, particles must grow at greatly varying rates if there is broad distribution of radicals among them. It follows that the particle size distribution has to broaden with increasing conversion, contrary to predictions based upon the Smith-Ewart model. At present it is not yet possible to predict quantitatively the shape of the conversion-time curve in interval III, the interval following the disappearance of monomer droplets. The reason for this is that the functional dependence of the termination rate constant upon monomer concentration in the particles is not known. However, once the conversion-time curve is experimentally determined, it is possible to calculate from it the interval III values of Q and of molecular weight.     

Modeling and analysis of high‐impact poly(propylene) production processes, 1. Effect of chemical poisoning on particle size distribution and gel formation

This work presents a simple model for a two‐stage process of high impact poly(propylene) (HIPP) production. The model predicts the bivariate distribution of particle size and polymer composition. It takes into account the effect of chemical poisoning on gel particle formation. The result shows that poisoning the solid catalyst is not an effective method for gel reduction. A better approach is to saturate the polymer particles with a co‐catalyst in reactor 1 and poison the co‐catalyst in reactor 2. It is also shown that the residence time distribution (RTD) of reactor 1 has a strong effect on the gel particle formation. A continuous reactor with narrow RTD is advantageous for gel reduction. The model provides some guidance for the analysis and design of the HIPP production process.     

The role of polymer adsorption kinetics in flocculation

The adsorption of polymeric flocculants on the surfaces of suspended solid particles is a non-equilibrium process. It is postulated that the process is controlled by an irreversible attachment between polymer molecules and solid particles. The frequency of such collisions determines the kinetics of adsorption and the distribution of adsorbed polymer on individual particle surfaces. A simplified model is presented in which polymer–particle collision frequencies determine the distribution of adsorbed polymer, and therefore, the adhesion efficiency of the particle–particle collisions that lead to flocculation. The implications of the model with regard to the effects of process variables, such as polymer molecular weight, particle size distribution, solids concentration and mixing conditions, are discussed at length. The critical importance of initial dispersion stability on polymer dosage requirements and overall process performance is demonstrated. The model provides considerable insight into the mechanisms involved in the use of progressive polymer addition to control adsorption and enhance flocculation efficiency.     

Axial mass transport and distribution of particles in D.C. arc plasma

A model is proposed to establish the axial distribution of added substances in an arc plasma. Particles with a higher ionization degree in the plasma are retained by the cathode. This causes a decrease in the axial transport velocity of particles newly arriving in the vicinity of the cathode. Consequently, this decrease of transport velocity causes an increase in the density of particles and their radiation density. Such an assumption is confirmed by measurement of the axial transport velocities. The theoretical consideration here is based on the works of Boumans and Krinberg and Smirnova, and takes into account the stated phenomena of decrease of axial transport velocities near the cathode. Using the results of the experimentally determined axial radiation density distribution, the axial distribution of particle transport velocities was calculated. The proposed model to establish the axial distribution of added substances contributes to the explanation of cathode layer enrichment of radiation density.     

A Critical Assessment of Particle Temperature Distributions During Plasma Spraying: Numerical Studies for YSZ

A three-dimensional computational model is used to simulate the in-flight particle melting behavior during plasma spraying process. The stochastic model is used for the particle size distribution. The particles surface temperature distributions at various spray distances have been presented. The results show that the surface temperature distribution varies with the spray distance. Single peak to double peaks and back to single peak has been observed in the simulations and also in the experiment. The effects of particle size and its distribution and plasma composition on the pattern shift have been investigated. Understanding the pattern shift may enable the design of a good control indicator to determine the particle melting status.     

Heterogeneous polymerization with polyaspartate macromonomer having vinyl pendant groups. II. Control of particle diameter and diameter distribution in dispersion copolymerization

To control particle diameter and particle diameter distribution in dispersion copolymerization of styrene and sodium polyaspartate macromonomer containing vinylbenzyl pendant groups, effects of some polymerization parameters, water contents, initiator concentration, styrene monomer concentration, reaction temperature, and type of initiator on the particle diameter and the diameter distribution were investigated. Variation of the water contents from 20 to 80 vol % controls the resultant particle diameter from 0.066 to 0.47 μm. The diameter increased with increasing initiator concentration. This tendency is similar to dispersion polymerization system using a nonpolymerizable stabilizer. Particle diameter distribution broadened with increasing styrene monomer concentration. This trend was attributed to the increase of a period of particle formation. This result indicated that the period of particle formation affected the resultant particle diameter distribution. Particle diameter distribution was successfully improved (CV = 9.1 from 23.6%) by shortening of decomposition time of initiator. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2281–2288, 2009     

Size-controlled flow synthesis of gold nanoparticles using a segmented flow microfluidic platform

Segmented flow is often used in the synthesis of nanomaterials to achieve narrow particle size distribution. The narrowness of the distribution is commonly attributed to the reduced dispersion associated with segmented flows. On the basis of the analysis of flow fields and the resulting particle size distribution, we demonstrate that it is the slip velocity between the two fluids and internal mixing in the continuous-phase slugs that govern the nature of the particle size distribution. The reduction in the axial dispersion has less impact on particle growth and hence on the particle size distribution. Synthesis of gold nanoparticles from HAuCl(4) with rapid reduction by NaBH(4) serves as a model system. Rapid reduction yields gold nuclei, which grow by agglomeration, and it is controlled by the interaction of the nuclei with local flow. Thus, the difference in the physical properties of the two phases and the inlet flow rates ultimately control the particle growth. Hence, a careful choice of continuous and dispersed phases is necessary to control the nanoparticle size and size distribution.     

Effect of reactor residence time distribution on the size distribution of polymer particles made with heterogeneous Ziegler-Natta and supported metallocene catalysts. A generic mathematical model

A generic mathematical model for analyzing the effect of ideal and non-ideal reactor residence time distributions on the size distribution of polymer particles produced with heterogeneous Ziegler-Natta and supported metallocene catalysts was developed. It was shown that the residence time distribution in polymerization reactors can have a significant effect on the size distribution of polymer particles and this can lead to imperfect replication of the catalyst particle size distribution.     

L.D.A. Simultaneous Measurements of Local Density and Velocity Distribution of Particles in Plasma Fluidized Bed at Atmospheric Pressure

Laser Doppler anemometry (L.D.A.) is an efficient and nonintrusive technique. Today, improved in its configuration, the L.D.A. has been applied even in flowing plasmas. ^(1,2) In-flight simultaneous measurements were performed for local density and velocity of particle distribution. The measurements provide an insight into thermal and mass transfer, chemical reactivity, and the distribution of residence times of particles in a plasma fluidized bed. The difficulties of L.D.A. in a plasma fludized bed such as high emission intensity of the plasma torch, high temperature, high particle density, and large distribution of particle granulometry were overcomed in the present investigation. The aims achieved were the characterization of the plasma fluidized bed distribution together with accurate measurements of local particle density and velocity as measured by L.D.A.     

Simulation of the latex particle morphology

A model is presented for the simulation of the structuration of polymer particles under conditions in which the new polymer chains are compatible with the polymer previously formed. The model involves the calculation of the monomer concentration gradients within the particles due to discrepancies in thermodynamic interactions between the monomer and the different polymers present in each part of the polymer particle. In addition, the distribution of free radicals in the latex particle is taken into account. This distribution results from the anchoring of the hydrophilic end-group of the growing polymer chain on the surface of the particle. The model was applied to the simulation of the polymerization of vinyl acetate on a butyl acrylate–vinyl acetate copolymer seed. It was found that the development of the particle morphology was mainly due to the profile of concentration of radicals in the particle. On the other hand, the effect of the monomer–polymer thermodynamic interactions on the particle morphology was found to be negligible. However, it has to be pointed out that this is because, for the system studied, the interaction parameters of vinyl acetate with polyvinyl acetate and polybutyl acrylate are nearly identical.     

Electrical phenomena of soft particles. A soft step function model

Simple analytic expressions are derived for the electrophoretic mobility of a soft particle consisting of the hard particle core covered with an ion-penetrable surface layer of polyelectrolyte for the case where the electric potential is low. The effect of the distribution of the polymer segments is taken into account by modeling the surface layer as a soft step function with the inhomogeneous distribution width δ. It is shown that the electrophoretic mobility becomes lower than that for the hard step function model and that the maximum deviation of the soft step function model from the hard step function model, which is a function of λδ (where 1/λ is the softness parameter) and κ/λ (where κ is the Debye-Hückel parameter), is 2.7% at λδ = 0.1, 5.1% at λδ = 0.2, and 11% at λδ = 0.5. In the limit of very high electrolyte concentrations, the obtained mobility expression tends to the result derived from the conventional hard step function model. In addition, an analytic expression for the interaction energy between two similar soft plates is derived on the basis of the present soft step function model. The magnitude of the interaction energy is shown to decrease by a factor 1/(1 + κδ)(2). Approximate analytic expressions for the interaction energies between two similar soft spheres and between two similar soft cylinders are also derived with the help of Derjaguin's approximation.     

Role of adsorption in formation of inorganic nanoparticles: Kinetic model

A simple kinetic model is proposed for the formation of inorganic nanoparticles in the presence of additives of readily adsorbing organic compounds. Additives and monomers may occupy the same sites on the surface of a growing particle. The maximum sizes and size distribution of formed particles are estimated under the assumption that the surface curvature of a growing particle has equivalent effects on the rate constants of the growth and adsorption. Equations are derived that relate the polydispersity indices for particle mass and radius distributions to the variances of particle radius distribution. The conditions are determined for the formation of virtually monodisperse nanoparticles.     

The Effect of Residence Time Distribution on the Slurry‐Phase Catalytic Ethylene Polymerization: An Experimental and Computational Study

This work is focused on the development and validation of a model accounting for the impact of the reactor residence time distribution in well‐stirred slurry‐phase catalytic polymerization of ethylene. Particle growth and morphology are described through the Multigrain model, adopting a two‐site model for the catalyst and a conventional kinetic scheme. Particle size distribution and polymer properties (average molecular weights and polydispersity) are computed as a function of particle size through a segregated model, assuming that neither breakage nor aggregation occur. Reactors are modeled by means of fundamental mass conservation equations. The model is applied to a system constituted by a series of two ideal continuous stirred tank reactors, where the synthesis of polyethylene with bimodal molecular weight distribution is performed, employing the initial catalyst size distribution as the only adjustable parameter. The model provides insights at the single particle scale for each specific size, thus highlighting the inhomogeneity which arises from the synergic effects of chemical kinetics and residence time distributions in both reactors. The satisfactory agreement between model results and experimental data, in terms of particle size distribution and average molecular weights, confirmed the suitability of the model and underlying assumptions.     

Emulsion polymerization of styrene in a single continuous stirred-tank reactor

The effects of mean residence time, initiation rate, and emulsifier concentration on particle formation, particle growth, and polymerization rate are examined for the emulsion polymerization of styrene in a completely mixed continuous stirred-tank reactor. Experimental measurements of number of particles, particle size distribution, polymerization rate, and molecular weights are compared with theoretical predictions. A theoretical model which incorporates Stockmayer's modification of the Smith-Ewart theory into the particle growth equation allows reasonably accurate prediction of polymerization rate, particle formation rate, and particle size distribution. Agreement between experimental measurements of number-average and weight-average molecular weights and a theory based on Smith-Ewart case 2 kinetics is also reasonable.     

Accurate particle size distribution determination by nanoparticle tracking analysis based on 2-D Brownian dynamics simulation

A physical model is presented to simulate the average step length distribution during nanoparticle tracking analysis experiments as a function of the particle size distribution and the distribution of the number of steps within the tracks. Considering only tracks of at least five steps, numerical simulation could be replaced by a normal distribution approximation. Based on this model, simulation of a step length distribution allows obtaining a much more reliable estimation of the particle size distribution, thereby reducing the artificial broadening of the distribution, as is typically observed by direct conversion of step length to particle size data. As this fitting procedure also allowed including data from particles that were followed for a relatively low number of steps, the measurement time could be reduced for particles that are known to be monodisperse. Whereas the inversion is less sensitive towards the particle size distribution width, still similar values were obtained for both the average diameter and standard deviation of a polystyrene latex sample irrespective of the track length, provided that the latter included at least five steps.     

Molecular weight distribution in emulsion polymerization. I. The homopolymer case

A model for evaluating the instantaneous degree of polymerization distribution of homopolymers produced in emulsion, based on the mathematics of the Markov chains, is developed. The model accounts for any number of active chains per particle, as well as for the two fundamental mechanisms of chain termination: mono- and bi-molecular, both by combination and by disproportionation. The core of the model is the so called subprocessmain process treatment, which allows us to correctly evaluate the degree of polymerization of the chains growing in the polymer particles, by distinguishing between the events experienced by the polymer chain which imply a change of its degree of polymerization (subject transitions) and those which imply only a change in the particle state (environment transitions). This is obtained by properly defining the one-step transition probability matrix of the relevant Markov process. Once this is done, the evaluation of the distribution of the degrees of polymerization reduces to a few simple operations among matrices. Explicit expressions for the instantaneous probability density functions and the relative cumulative distributions are obtained. The application of such relationships is facilitated by the numerical procedures reported in the Appendices. The results of the model developed in this work are in agreement with those of earlier models in the range of parameter values of practical interest. In the limit of very low molecular weights, only the model developed in this work provides the correct answer. Moreover, a much more significant result is its applicability to the case of emulsion copolymerization, as it is shown in Part II.     

超声衰减谱测量电池浆料的粒度分布

电池浆料中颗粒状活性物质的粒度大小和分散均匀性对电池的内阻、 电压、 局部表面电流和总极化程度等性能有直接影响, 实现对其的在线实时测量对电池的质量控制具有重要意义. 基于电池浆料的高固含量、 高黏度和低透光性的特点, 本文利用超声衰减谱的方式测量了其粒度分布(PSD). 应用于电池浆料的粒度分布测量的最大难点是其利用超声衰减谱法预测粒度分布的模型需要难以获得的分散相和连续相的物性参数. 本文采用主成分分析(PCA)结合误差反向传播(BP)神经网络建立预测模型解决了超声衰减谱法的难点, 并引入遗传算法(GA)优化BP神经网络的初始阈值和权值. 通过以LiCoO₂为活性物质的电池浆料进行了验证, 结果表明, PCA-GA-BP神经网络能够有效对不同固含量电池浆料的粒度分布进行预测, 预测值与真实值的峰形重合度高, 峰高偏差小, 两者的均方误差为0.1358, 拟合度(R²)为0.9816, 说明超声衰减谱法可作为测量电池浆料粒度分布的重要方式.     

纳米氧化锌粒子分散性对其吸收光谱的影响

在异丙醇中用氢氧化钠分别与醋酸锌及溴化锌反应制备了纳米氧化锌粒子. 分别用高分辨率电子显微镜及原位紫外吸收光谱测定了粒子大小及分布. 实验结果表明, 粒子的增大服从LSW (Lifshitz-Slyozov-Wagner)模型, 即粒子体积随老化时间线性增大；但粒子的分布不符合LSW模型, 这与他人的研究结果不一致. 用计算机数值模拟确定了纳米氧化锌分布函数对其紫外吸收光谱的影响, 发现在紫外吸收边附近光谱发生弯曲, 且随粒子分布标准方差(SD)的增大, 弯曲更显著, 引起紫外吸收光谱红移, 这将导致用吸收边计算氧化锌粒子大小时产生正误差. 就单分散(SD/γ<5%, γ是粒子的平均半径)纳米氧化锌而言, 这种正误差仅为2%, 但当粒子分布的SD/γ达到15%时, 所产生的正误差可高达15.1%.     

Polymerization in nonuniform latex particles. II. Kinetics of two-phase emulsion polymerization

A new theory, based on the concept of nonuniform distribution of free radicals in polymerizing latex particles, has been developed for the kinetics of two-phase emulsion polymerization reactions. This theory also takes into account the diffusion controlled termination and propagation reactions to describe the gel effect and limiting conversion. The kinetic model permits prediction of the distribution of free radicals in the two polymer phases and rate of polymerization as a function of reaction conditions. Experimental data for polystyrene/polymethyl methacrylate and polymethyl methacrylate/polystyrene (postformed polymer/preformed polymer) in the literature have been used to assess the proposed idea of nonuniform distribution of free radicals in the latex particle.