Synthesis And Characterization Of 1-(2H-tetrazol-5-YL)-1,4,7,10-tetraazacyclododecane And Its Zn(II), Ni(II), AND Cu(II) complexes

As part of an ongoing effort to develop new metal complexes of tetraazamacrocycles with novel properties in coordination or functionalization we report here the synthesis of a new derivative of 1,4,7,10-tetraazacyclododecane (cyclen) with a tetraazole moiety directly bound to the azamacrocycle. The new ligand was obtained by reaction with cyanogen bromide giving the cyanamide, followed by a [2+3] cycloaddition with NaN3 to yield the tetraazole. The ligand and its Zn(II), Ni(II), and Cu(II) complexes were fully characterized by analytical methods. X-ray structure analysis of the Ni(II) compound shows the formation of a stable dimer by coordination of each of the two tetraazole substituents to the neighboring metal cation. Potentiometric titrations of the metal complexes indicate a possible conversion of the monomer to the dimeric structure in solution and show the pKa of the NH-atom on the tetraazole substituent to be between 4.03 and 5.3 depending on the metal ion coordinated by cyclen

A Max–Min Control Problem Arising in Gradient Elution Chromatography

Gradient elution chromatography is an industrial process used to separate and purify multi-component chemical mixtures. In this article, we consider an optimal control problem in which manipulative variables in the chromatographic process need to be determined to maximize separation efficiency. This problem has two nonstandard characteristics: (i) the objective function is nonsmooth, and (ii) each state variable is defined over a different time horizon. The final time for each state variable, the so-called retention time, is not fixed and actually depends on the control variables. To solve this optimal control problem, we first introduce a set of auxiliary decision variables to govern the ordering of the retention times. Then, we approximate the control by a piecewise-constant function and apply a novel time-scaling transformation to map the retention times and control switching times to fixed points in a new time horizon. The retention times and control switching times become decision variables in the new time horizon. On this basis, the optimal control problem is reduced to an approximate nonlinear optimization problem that can be solved using a recently developed exact penalty method. Numerical results show that our approach is both accurate and efficient