Molecular weight distribution in emulsion polymerization. II. The copolymer case

A model for evaluating instantaneous degree of polymerization distribution and the chain composition distribution of copolymers produced in emulsion is developed. The approach adopted is based on the mathematics of Markov processes and represents an extension of the one developed for homopolymers in Part I. As in the homopolymer case, the main aspect of the theoretical treatment is the definition of the proper one step transition probability matrix through the so called subprocess-main process procedure. The model accounts for monomolecular and bimolecular termination (both by combination and disproportionation) and, in principle, it can be applied to any number of reacting monomer species as well as to any number of active chains per particle. However, only the 0–1–2 and 0–1–2–3 emulsion copolymerization systems are discussed in detail. In the case of the chain composition distribution, the model allows the calculation of its moments only, through the method of the Generating Function associated with the probability density function. The expression obtained for the instantaneous probability density functions, as well as for the corresponding cumulative distributions, are all in explicit form and involve only algebraic operations among matrices. Efficient numerical procedure for their application are reported in the Appendix. Illustrative calculations are reported for a 0–1–2–3 copolymerization system, simulating the copolymer styrene–methylmethacrylate. The effect of the various termination mechanisms on the distribution of degrees of polymerization and on the first two moments of the chain composition distribution is discussed in detail. Finally, the three dimensional overall distribution function of both chain length and composition is shown under the assumption of Gaussian type chain composition distribution.     

非线性色谱的非平衡热力学分离理论 Ⅱ. 非线性-传质动力学过程的0-1模型

在制备色谱的优化设计和控制过程中,若试图把基于偏微分方程(PDE)-Eulerian描述的Wilson色谱理论框架和基于离散时间状态的优化控制方法(如Markov决策过程(MDP)和模型预测控制(MPC)等)衔接在一起时,就会出现明显的障碍。本文提出基于Lagrangian-Eulerian描述(L-ED)的非线性传质色谱(NTC)的0-1模型来克服这些障碍。该模型把一个溶质微元单元划分为在流动相中并以其线速度移动的流动相溶质微元(SCm)和在固定相中其移动速度为0的固定相溶质微元(SCs)。引入由溶质微元的序号集合、溶质微元的位置矢量、固定相溶质浓度矢量和流动相溶质浓度矢量组成的热力学状态矢量Sk，并用其来描述色谱过程的局域热力学路径(LTP)和宏观热力学路径(MTP)。在非线性-理想-传质色谱的理论分析和数值实验中,0-1模型的数值解表现出很好的一致性、稳定性、守恒性及精确性等。该模型能很好地与控制论中的Markov决策过程或其他基于离散时间状态的优化控制方法相衔接。     

Synthesis and crystal structure of 1D polyoxometalate-based composite compound, [{Gd(NMP)6}(PMo12O40)]n (NMP=N-methyl-2-pyrrolidone)

A novel compound, [{Gd(NMP)₆}(PMo₁₂O₄₀)]_n, has been synthesized and characterized by IR, and UV spectroscopy, and single crystal X-ray structural analysis. It forms an unprecedented one-dimensional zigzag chain built from alternating polyanions and cationic units through Mo-O_t-Gd-O_t-Mo links in the crystal. In the compound, Gd³⁺ is eight-coordinated with a bicapped trigonal prism geometry environment of oxygen atoms. The results of the single crystal X-ray diffraction analyses and IR are agreement and both show the metal cation units are coordinately bonded to the Keggin cluster. The UV spectrum of the title compound suggests that the compound is entirely dissociated in dilute solution.     

Lower bounds on entropy for polymer chains on a square and a cubic lattice

Rigorous lower bounds on the entropy per particle as a function of the fractiong of thegauche bonds of a system of semiflexible polymer chains is obtained in the thermodynamic limit. Only square and cubic lattices are considered. For the case of a single chain havingl monomers, the bound is obtained for all gg=2/3. For the case of p>1 chains, each havingl monomers, wherel is a multiple of 4, the bound is obtained for all gg=13/90. In both cases, it is shown that the entropy is alwaysnonzero for all 0<gm(l), whereg _m(l) =(l-2)/l. Thiscontradicts the prediction from the Flory-Huggins approximations that the entropy is zero for allgg₀, whereg ₀ is some finite nonzero number. It is also pointed out that it isnot impossible to pack a lattice with disordered configurations of rodlike chains with finite entropy, again contradicting an assertion by Flory that it is impossible to do so. Finally, it is concluded that onecannot trust the Flory-Huggins approximations at least at low temperatures. The study also casts doubts on the validity of the Gibbs-DiMarzio theory of glass transitions in polymeric systems.     

Doeblin's big limit theorem

Doeblin regarded his paper(²) as his hardest work. The big limit is that ofP(ⁿ)(x, E) asn tends to infinity, in a measurable non-topologized space. An exposition of part one of this paper was published in Ref. 1. This is the exposition of part two, which contains some reparation as well as clarification.Doeblin's great contributions to probability theory were remembered at a conference 50 Years after Doeblin: Developments in the Theory of Markov Chains, Markov Processes and Sums of Random Variables held at Blaubeuren. Germany November 2–7, 1991. Professor Chung contributed this article to further celebrate this occasion. The editor thanks him.     

One-dimensional random field Ising model and discrete stochastic mappings

Previous results relating the one-dimensional random field Ising model to a discrete stochastic mapping are generalized to a two-valued correlated random (Markovian) field and to the case of zero temperature. The fractal dimension of the support of the invariant measure is calculated in a simple approximation and its dependence on the physical parameters is discussed.Contribution to the symposium Statistical Mechanics of Phase Transitions—Mathematical and Physical Aspects, Trebo, CSSR, September 1–6, 1986.     

Stationary states for a mechanical system with stochastic boundary conditions

We consider a system of Newtonian particles, with a long-range repulsive pair potential, moving in a cavity whose surface temperature is spatially varying. When a particle hits the surface, it is thermalized at the temperature of the collision point. We prove that this system has a unique stationary ensemble, to which any initial distribution converges for large times. We show that this stationary ensemble depends continuously on the surface temperature profile.     

Nuclearity controlled cyanide-bridged bimetallic CrIII-MnII compounds: synthesis, crystal structures, magnetic properties and theoretical calculations

The preparation, X-ray crystallography and magnetic investigation of the compounds PPh4[Cr(bipy)(CN)4].2 CH3CN.H2O (1) (mononuclear), [[Cr(bipy)(CN)4]2Mn-(H2O)4].4H2O (2) (trinuclear), [[Cr(bipy)(CN)4]2Mn(H2O)2] (3) (chain) and [[Cr(bipy)(CN)4]2Mn(H2O)].H2O.CH3CN (4) (double chain) [bipy=2,2'-bipyridine; PPh4 (+)=tetraphenylphosphonium] are described herein. The [Cr(bipy)(CN)4]- unit act either as a monodentate (2) or bis-monodentate (3) ligand toward the manganese atom through one (2) or two (3) of its four cyanide groups. The manganese atom is six-coordinate with two (2) or four (3) cyanide nitrogens and four (2) or two (3) water molecules building a distorted octahedral environment. In 4, two chains of 3 are pillared through interchain Mn-N-C-Cr links which replace one of the two trans-coordinated water molecules at the manganese atom to afford a double chain structure where bis- and tris-monodenate coordination modes of [Cr(bipy)(CN)4]- coexist. The magnetic properties of 1-4 were investigated in the temperature range 1.9-300 K. A Curie law behaviour for a magnetically isolated spin quartet is observed for 1. A significant antiferromagnetic interaction between CrIII and MnII through the single cyanide bridge [J=-6.2 cm(-1), the Hamiltonian being defined as H=-J(SCr1.SMn+SCr2.SMn] occurs in 2 leading to a low-lying spin doublet which is fully populated at T <5 K. A metamagnetic behaviour is observed for 3 and 4 [the values of the critical field Hc being ca. 3000 (3) and 1500 Oe (4)] which is associated to the occurrence of weak interchain antiferromagnetic interactions between ferrimagnetic Cr2III MnII chains. The analysis of the exchange pathways in 2-4 through DFT type calculations together with the magnetic bevaviour simulation using the quantum Monte Carlo methodology provided a good understanding of their magnetic properties.     

Fast and Accurate Approximation of the Full Conditional for Gamma Shape Parameters

The gamma distribution arises frequently in Bayesian models, but there is not an easy-to-use conjugate prior for the shape parameter of a gamma. This inconvenience is usually dealt with by using either Metropolis–Hastings moves, rejection sampling methods, or numerical integration. However, in models with a large number of shape parameters, these existing methods are slower or more complicated than one would like, making them burdensome in practice. It turns out that the full conditional distribution of the gamma shape parameter is well approximated by a gamma distribution, even for small sample sizes, when the prior on the shape parameter is also a gamma distribution. This article introduces a quick and easy algorithm for finding a gamma distribution that approximates the full conditional distribution of the shape parameter. We empirically demonstrate the speed and accuracy of the approximation across a wide range of conditions. If exactness is required, the approximation can be used as a proposal distribution for Metropolis–Hastings. Supplementary material for this article is available online.