Solid-state NMR spectra and long intradimer bonds in the pi-[TCNE]22- dianion

The principal (13)C chemical-shift values for the pi-[TCNE](2)(2-) dimer anion within an array of counterions have been measured to understand better the electronic structure of these atypical chemical species in several related TCNE-based structures. The structure of pi-[TCNE](2)(2-) is unusual as it contains two very long C-C bond lengths (ca. 2.9 Angstroms) between the two monomeric units and has been found to exist as a singlet state, suggestive of a (1)A(1g) (b(2u)(2)b(1g)(0)) electronic configuration. A systematic study of several oxidation states of [TCNE](n) (n = 0, 1-, 2-) was conducted to determine how the NMR chemical-shift tensor values change as a function of electronic structure and to understand the interactions that lead to spin-pairing of the monomer units. The density functional theory (DFT) calculated nuclear shielding tensors are correlated with the experimentally determined principal chemical-shift values. Such theoretical methods provide information on the tensor magnitudes and orientations of their principal tensor components with respect to the molecular frame. Both theoretical and experimental ethylenic chemical-shielding tensors reveal high sensitivity in the component, delta(perpendicular), lying in the monomer molecular plane and perpendicular to the pi-electron plane. This largest shift dependence on charge density is observed to be about -111 ppm/e(-) for delta(perpendicular). The component in the molecular plane but parallel to the central C=C bond, delta(parallel), exhibits a sensitivity of approximately -43 ppm/e(-). However, the out-of-plane component delta'(perpendicular) shows a minimal dependence of -2.6 ppm/e(-) on the oxidation state (n) of [TCNE](n). These relative values support the claim that it is changes within the ethylenic pi-electrons and not the sigma-electrons that best account for the dramatic variations in bonding and shift tensors in this series of compounds. Concerning the intraion bonding, relatively weak Wiberg bond orders between the two monomeric components of the dimer correlate with the long bonds linking the two [TCNE(*)](-) monomers. The chemical-shift tensors for the cyano group, compared to the ethylene shifts, exhibit a reduced sensitivity on the TCNE oxidation state. The experimental principal chemical-shift components agree (within typical errors) with the calculated quantum mechanical shieldings used to correlate the bonding. The embedded ion model (EIM) was used to investigate the typically large electrostatic lattice potential in these ionic materials. Chemical-shielding principal values calculated with the EIM model differ from experiment by +/-3.82 ppm on average, whereas in the absence of an electrostatic field model, the experimental and theoretical results agree by +/-4.42 ppm, which is only a modest increase in error considering the overall ionic magnitudes associated with the tensor variations. Apparently, the effects of the sizable long-range electrostatic fields cancel when the shifts are computed because of lattice symmetry.     

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e^|k|/k_d{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e^{-C(k/k_d) ln(|k|/k_d)}{{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C _* = 1/ ln 2. The same behavior with a universal constant C _* is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.     

Approach to Equilibrium for a Class of Random Quantum Models of Infinite Range

We consider random generalizations of a quantum model of infinite range introduced by Emch and Radin. The generalizations allow a neat extension from the class l ₁ of absolutely summable lattice potentials to the optimal class l ₂ of square summable potentials first considered by Khanin and Sinai and generalised by van Enter and van Hemmen. The approach to equilibrium in the case of a Gaussian distribution is proved to be faster than for a Bernoulli distribution for both short-range and long-range lattice potentials. While exponential decay to equilibrium is excluded in the nonrandom l ₁ case, it is proved to occur for both short and long range potentials for Gaussian distributions, and for potentials of class l ₂ in the Bernoulli case. Open problems are discussed.     

Time evolution of a fractal distribution: Particle concentrations in free-surface turbulence

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal structure, a rare manifestation of a strange attractor observable in real space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions D_q(t) (q=1 to 6) of the floating particles starting with the uniform distribution D_q(0)=2 for Taylor Microscale Reynolds number Re_λ?160. Focus is on the time evolution of the correlation dimension D₂(t) as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.     

Upper bound on the slope of steady water waves with small adverse vorticity

We consider the angle of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. There is an upper bound of 31.15° in the irrotational case [1] and an upper bound of 45° in the case of favorable vorticity [13]. On the other hand, if the vorticity is adverse, the profile can become vertical. We prove here that if the adverse vorticity is sufficiently small, then the angle still has an upper bound which is slightly larger than 45°.     

Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Hölder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.     

A Note on the Morozov Principle via Lagrange Duality

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov’s regularization parameter. The numerical relevance of our approach is established by means of an illustrative example from nonparametric instrumental regression, a standard problem in statistics.     

Pressure beneath a Stokes wave

We investigate the pressure within an irrotational fluid in a periodic, steady, two‐dimensional gravity wave above a flat bed. We prove that the pressure in the fluid strictly decreases horizontally away from the crest line. Furthermore, the pressure strictly increases with depth. We also investigate the effect of an underlying current on the paths of the particles. © 2009 Wiley Periodicals, Inc.     

Potential-modulated, attenuated total reflectance spectroscopy of poly(3,4-ethylenedioxythiophene) and poly(3,4-ethylenedioxythiophene methanol) copolymer films on indium-tin oxide

We report the first application of a potential-modulated spectroelectrochemical ATR (PM-ATR) instrument utilizing multiple internal reflections at an optically transparent electrode to study the charge-transfer kinetics and electrochromic response of adsorbed films. A sinusoidally modulated potential waveform was applied to an indium-tin oxide (ITO) electrode while simultaneously monitoring the optical reflectivity of thin (2-6 equivalent monolayers) copolymer films of poly(3,4-ethylenedioxythiophene) (PEDOT) and poly(3,4-ethylenedioxythiophene methanol) (PEDTM), previously characterized in our laboratory. At high modulation frequencies the measured response of the polymer film is selective toward the fastest electrochromic processes in the film, presumably those occurring within the first adsorbed monolayer. Quantitative determination of the electrochromic switching rate, derived from the frequency response of the attenuated reflectivity, shows a linear decrease in the rate, from 11 x 10(3) s(-1) to 3 x 10(3) s(-1), with increasing proportions of PEDTM in the copolymer, suggesting that interactions between the methanol substituent on EDTM and the ITO surface slow the switching process by limiting the rate of conformational change in the polymer film.     

Dependence of the zero shear-rate viscosity and the viscosity function of linear high-density polyethylenes on the mass-average molar mass and polydispersity

Linear high-density polyethylenes with molar masses M _w between 240 and 1,000,000 g/mol, obtained by metallocene catalysts, were characterized in shear using oscillatory and creep tests. The polydispersities of the molar mass distributions (MMDs) lay between 1 and 16. The resulting zero shear-rate viscosities η₀ covered a range from 2.5×10⁻³ to around 10⁸ Pas. Above a critical molar mass of M _c≈2,900 g/mol, the experimental results can be described by the relation η₀ ∼ M _w^3.6, independently of the MMD. The oscillatory data were fitted with a Carreau–Yasuda equation. The resulting parameters were correlated to molecular structure. The parameter a, being a quantity for the width of the transition between the Newtonian and the non-Newtonian regime, showed a dependence on the molar mass M _w but not on M _w/M _n. The parameter λ of the Carreau-Yasuda equation was found to be the reciprocal crossover frequency for all samples with a log-Gaussian MMD. λ depends on the molar mass M _w and also on M _w/M _n.