Radial jet in rotating outer flow

The rotating radial jet formed by fluid introduced radially outwards into a rotating environment is studied by the momentum integral method and by perturbation for weak swirl, assuming incompressible laminar flow. For non-swirling outer fluid, the results slightly extend previously published results. For swirling outer fluid, the jet is characterized by a finite radial penetration distance. Its development over that distance is divided into three regions: an initial adjustment, a central pseudo-similarity region, and a final decay.     

The modified stochastic game

We present a new tool for the study of multiplayer stochastic games, namely the modified game, which is a normal-form game that depends on the discount factor, the initial state, and for every player a partition of the set of states and a vector that assigns a real number to each element of the partition. We study properties of the modified game, like its equilibria, min–max value, and max–min value. We then show how this tool can be used to prove the existence of a uniform equilibrium in a certain class of multiplayer stochastic games.     

Moderately branched ultra‐high molecular weight polyethylene by using N,N′‐nickel catalysts adorned with sterically hindered dibenzocycloheptyl groups

Five examples of unsymmetrical 1,2‐bis (arylimino) acenaphthene ( L1 – L5 ), each containing one N‐2,4‐bis (dibenzocycloheptyl)‐6‐methylphenyl group and one sterically and electronically variable N‐aryl group, have been used to prepare the N,N′‐nickel (II) halide complexes, [1‐[2,4‐{(C₁₅H₁₃}₂–6‐MeC₆H₂N]‐2‐(ArN)C₂C₁₀H₆]NiX₂ (X = Br: Ar = 2,6‐Me₂C₆H₃ Ni1 , 2,6‐Et₂C₆H₃ Ni2 , 2,6‐i‐Pr₂C₆H₃ Ni3 , 2,4,6‐Me₃C₆H₂ Ni4 , 2,6‐Et₂–4‐MeC₆H₂ Ni5 ) and (X = Cl: Ar = 2,6‐Me₂C₆H₃ Ni6 , 2,6‐Et₂C₆H₃ Ni7 , 2,6‐i‐Pr₂C₆H₃ Ni8 , 2,4,6‐Me₃C₆H₂ Ni9 , 2,6‐Et₂–4‐MeC₆H₂ Ni10 ), in high yield. The molecular structures Ni3 and Ni7 highlight the extensive steric protection imparted by the ortho‐dibenzocycloheptyl group and the distorted tetrahedral geometry conferred to the nickel center. On activation with either Et₂AlCl or MAO, Ni1 – Ni10 exhibited very high activities for ethylene polymerization with the least bulky Ni1 the most active (up to 1.06  ×  10⁷ g PE mol^?1(Ni) h^?1 with MAO). Notably, these sterically bulky catalysts have a propensity towards generating very high molecular weight polyethylene with moderate levels of branching and narrow dispersities with the most hindered Ni3 and Ni8 affording ultra‐high molecular weight material (up to 1.5  ×  10⁶ g mol^?1). Indeed, both the activity and molecular weights of the resulting polyethylene are among the highest to be reported for this class of unsymmetrical 1,2‐bis (imino)acenaphthene‐nickel catalyst.     

cis‐Dicarbonyl‐cis‐dichlorido‐trans‐bis(triphenylphosphine‐κP)iridium(III) tetrafluoridoborate,formed by reaction of (cycloocta‐1,5‐diene)bis(triphenylphosphine)iridium tetrafluoridoborate with carbonyl fluoride in dichloromethane solution

The reaction between carbonyl fluoride and [Ir(COD)(PPh₃)₂]BF₄ (COD is cycloocta‐1,5‐diene) in dichloromethane solution affords the novel title iridium salt, [IrCl₂(C₁₈H₁₅P)₂(CO)₂]BF₄. The cation lies across a twofold rotation axis in the space group P2₁2₁2 and its structure confirms the presence in a cis relationship of two metal‐bound chlorides, while the phosphine ligands occupy a trans pair of sites. The anion also lies across a twofold rotation axis, and the F atoms are disordered over two sets of sites.     

Alkylaluminum activator effects on polyethylene branching using a N,N′‐nickel precatalyst appended with bulky 4,4′‐dimethoxybenzhydryl groups

Ten unsymmetrical N,N'‐bis (imino) acenaphthene‐nickel (II) halide complexes, [1‐[2,6‐{(4‐MeOC₆H₄)₂CH}₂–4‐MeC₆H₂N]‐2‐(ArN)C₂C₁₀H₆]NiX₂, each appended with one N‐2,6‐bis(4,4'‐dimethoxybenzhydryl)‐4‐methylphenyl group, have been synthesized and characterized. The molecular structures of Ni1 , Ni3 , Ni5 and Ni6 highlight the variation in steric protection afforded by the inequivalent N‐aryl groups; a distorted tetrahedral geometry is conferred about each nickel center. On activation with diethylaluminum chloride (Et₂AlCl) or methylaluminoxane (MAO), all complexes showed high activity at 30°C for the polymerization of ethylene with the least bulky bromide precatalysts ( Ni1 and Ni4 ), generally the most productive, forming polyethylenes with narrow dispersities [M_w/M_n: < 3.4 (Et₂AlCl), < 4.1 (MAO)] and various levels of branching. Significantly, this level of branching can be influenced by the type of co‐catalyst employed, with Et₂AlCl having a predilection towards polymers displaying significantly higher branching contents than with MAO [T_m: 33.0–82.5°C (Et₂AlCl) vs. 117.9–119.4°C (MAO)]. On the other hand, the molecular weights of the materials obtained with each co‐catalyst were high and, in some cases, entering the ultra‐high molecular weight range [M_w range: 6.8–12.2 × 10⁵ g mol^?1 (Et₂AlCl), 7.2–10.9 × 10⁵ g mol^?1 (MAO)]. Furthermore, good tensile strength (ε_b up to 553.5%) and elastic recovery (up to 84%) have been displayed by selected more branched polymers highlighting their elastomeric properties.     

Exceptionally high molecular weight linear polyethylene by using N,N,N′‐Co catalysts appended with a N′‐2,6‐bis{di(4‐fluorophenyl)methyl}‐4‐nitrophenyl group

The bis(arylimino)pyridines, 2‐[CMeN{2,6‐{(4‐FC₆H₄)₂CH}₂–4‐NO₂}]‐6‐(CMeNAr)C₅H₃N (Ar = 2,6‐Me₂C₆H₃ L1 , 2,6‐Et₂C₆H₃ L2 , 2,6‐i‐Pr₂C₆H₃ L3 , 2,4,6‐Me₃C₆H₂ L4 , 2,6‐Et₂–4‐MeC₆H₂ L5 ), each containing one N′‐2,6‐bis{di(4‐fluorophenyl)methyl}‐4‐nitrophenyl group, have been synthesized by two successive condensation reactions from 2,6‐diacetylpyridine. Their subsequent treatment with anhydrous cobalt (II) chloride gave the corresponding N,N,N′‐CoCl₂ chelates, Co1 – Co5 , in excellent yield. All five complexes have been characterized by ¹H/¹⁹F NMR and IR spectroscopy as well as by elemental analysis. In addition, the molecular structures of Co1 and Co3 have been determined and help to emphasize the differences in steric properties imposed by the inequivalent N‐aryl groups; distorted square pyramidal geometries are adopted by each complex. Upon activation with either methylaluminoxane (MAO) or modified methylaluminoxane (MMAO), precatalyts Co1 – Co5 collectively exhibited very high activities for ethylene polymerization with 2,6‐dimethyl‐substituted Co1 the most active (up to 1.1 × 10⁷ g (PE) mol^?1 (Co) h^?1); the MAO systems were generally more productive. Linear polyethylenes of exceptionally high molecular weight (M_w up to 1.3 × 10⁶ g mol^?1) were obtained in all cases with the range in dispersities exhibited using MAO as co‐catalyst noticeably narrower than with MMAO [M_w/M_n: 3.55–4.77 ( Co1 – Co5 /MAO) vs. 2.85–12.85 ( Co1 – Co5 /MMAO)]. Significantly, the molecular weights of the polymers generated using this class of cobalt catalyst are higher than any literature values reported to date using related N,N,N‐bis (arylimino)pyridine‐cobalt catalysts.     

Stochastic games with two non-absorbing states

In the present paper we consider recursive games that satisfy an absorbing property defined by Vieille. We give two sufficient conditions for existence of an equilibrium payoff in such games, and prove that if the game has at most two non-absorbing states, then at least one of the conditions is satisfied. Using a reduction of Vieille, we conclude that every stochastic game which has at most two non-absorbing states admits an equilibrium payoff. This paper is part of the Ph.D. thesis of the author completed under the supervision of Prof. Abraham Neyman at the Hebrew University of Jerusalem. I would like to thank Prof. Neyman for many discussions and ideas and for the continuous help he offered. I also thank Nicolas Vieille for his comments on earlier versions of the paper.     

Flexible N,N,N-chelates as supports for iron and cobalt chloride complexes; synthesis, structures, DFT calculations and ethylene oligomerisation studies

The aryl-substituted N-picolylethylenediamine and diethylenetriamine ligands, (ArNHCH(2)CH(2))[(2-C(5)H(4)N)CH(2)]NH and (ArNHCH(2)CH(2))(2)NH (Ar = 2,6-Me(2)C(6)H(3), 2,4,6-Me(3)C(6)H(2)), have been prepared by employing palladium-catalysed N-C(aryl) coupling reactions of the corresponding primary amines with aryl bromide. Treatment of MCl(2) with (ArNHCH(2)CH(2))[(2-C(5)H(4)N)CH(2)]NH affords [[(ArNHCH(2)CH(2))((2-C(5)H(4)N)CH(2))NH]CoCl(2)](Ar = 2,6-Me(2)C(6)H(3) 1a; 2,4,6-Me(3)C(6)H(2)) 1b and [[(ArNHCH(2)CH(2))((2-C(5)H(4)N)CH(2))NH]FeCl(2)](n)(n= 1, Ar = 2,6-Me(2)C(6)H(3) 2a; n= 2, 2,4,6-Me(3)C(6)H(2) 2b) in high yield. The X-ray structures of 1a and 1b are isostructural and reveal the metal centres to adopt distorted trigonal bipyramidal geometries with the N,N,N-chelates adopting fac-structures. A facial coordination mode of the ligand is also observed in bimetallic 2b, however, in 2a the N,N,N-chelate adopts a mer-configuration with the metal centre adopting a geometry best described as square pyramidal. Solution studies indicate that mer-fac isomerisation is a facile process for these systems at room temperature. Quantum mechanical calculations (DFT) have been performed on 1a and 2a, in which the ligands employed are identical, and show the fac- to be marginally more stable than the mer-configuration for cobalt (1a) while for iron (2a) the converse is evident. Reaction of (ArNHCH(2)CH(2))(2)NH with CoCl(2) gave the five-coordinate complexes [[(ArNHCH(2)CH(2))(2)NH]CoCl(2)](Ar = 2,6-Me(2)C(6)H(3) 3a, 2,4,6-Me(3)C(6)H(2) 3b), in which the ligand adopts a mer-configuration; no reaction occurred with FeCl(2). All complexes 1-3 act as modest ethylene oligomerisation catalysts on activation with excess methylaluminoxane (MAO); the iron systems giving linear alpha-olefins while the cobalt systems give mixtures of linear and branched products.     

A Luminescent One-Dimensional Copper(I) Polymer

The synthesis and structure of a one-dimensional polymer of formula [Cu₃(dpphp)₂(MeCN)_1.7(CH₂Cl₂)_0.3](ClO₄) _n is reported (where dpphp=6-diphenylphosphino-2-pyridonate). There are three distinct copper(I) sites within the polymer. Two of these sites are four-coordinate, bound to one P- and two O-atoms and a solvate molecule; the third site is bound only to two N-atoms. The Cu*Cu distances within the polymer are around 2.8 Å. The compound is luminescent in the solid-state.