Janus Pyrrolopyrrole Aza-dipyrrin: Hydrogen-Bonded Assemblies and Slow Magnetic Relaxation of the Cobalt(II) Complex in the Solid State

A novel pyrrolopyrrole azadipyrrin ( Janus-PPAD ) with Janus duality was synthesized by a Schiff base–forming reaction of diketopyrrolopyrrole. The orthogonal interactions of the hydrogen-bonding ketopyrrole and metal-coordinating azadipyrrin moieties in Janus-PPAD enabled the metal ions to be arranged at regular intervals: zinc(II) and cobalt(II) coordination provided metal-coordinated Janus-PPAD dimers, which can subsequently form hydrogen-bonded one-dimensional arrays both in solution and in the solid state. The supramolecular assembly of the zinc(II) complex in solution was investigated by ¹H NMR spectroscopy based on the isodesmic model, in which a binding constant for the elongation of assemblies is constant. Owing to the tetrahedral coordination, in the solid state, the cobalt(II) complex exhibited a slow magnetic relaxation due to the negative D value of −27.1 cm⁻¹ with an effective relaxation energy barrier U_eff of 38.0 cm⁻¹. The effect of magnetic dilution on the relaxation behavior is discussed. The relaxation mechanism at low temperature was analyzed by considering spin lattice interactions and quantum tunneling effects. The easy-axis magnetic anisotropy was confirmed, and the relevant wave functions were obtained by ab initio CASSCF calculations.     

Exact Averaging of Stochastic Equations for Transport in Random Velocity Field

We present new examples of exactly averaged multi-dimensional equation of transport of a conservative solute in a time-dependent random flow velocity field. The functional approach and a technique for decoupling the correlations are used. In general, the averaged equation is non-local. We study the special cases where the averaged equation can be localized and reduced to a differential equation of finite-order, where the problem of evolution of the initial plume (Cauchy problem) can be solved exactly. We present in detail the results of the analyses of two cases of exactly averaged problems for Gaussian and telegraph random velocity with an identical exponential correlation function, which are informative and convenient models for continuous and discontinuous random functions. The problems in which the field has sources of solute and boundaries are also examined. We study the behavior of different initial plumes for all times (evolutions and convergence) and show the manner in which they approach the same asymptotic limit for two stochastic distributions of flow-velocity. A comparison between exact solutions and solutions derived by the method of perturbation is also discussed.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.