The regularized 13 moment (R13) equations are a macroscopic model for the description of rarefied gas flows in the transition regime. The equations have been shown to give meaningful results for Knudsen numbers up to about 0.5. Here, their range of applicability is extended by deriving and testing boundary conditions for evaporating and condensing interfaces. The macroscopic interface conditions are derived from the microscopic interface conditions of kinetic theory. Tests include evaporation into a half-space and evaporation/condensation of a vapor between two liquid surfaces of different temperatures. Comparison indicates that overall the R13 equations agree better with microscopic solutions than classical hydrodynamics

Copper(II) and copper(III) complexes were prepared with two novel ligands, N,N′‐bis(2‐(1‐hydroxyimino‐2‐methyl‐1‐phenyl)propyl)dimethylmalondiamide (H4mal55) and N,N′ bis(‐(1‐hydroxyiminoethyl)phenyl)dimethylmalondiamide (H4mal66), both of which contain two amide and two oxime functionalities as potential ligand donor groups. The two copper(II) complexes (NEt4)[Cu(Hmal55)] (1) and (NEt4)[Cu(Hmal66)] (2) can be reversibly oxidized in acetonitrile at a redox potential of –0.120 and –0.075 V vs. the Fc/Fc+ redox couple, respectively. While the quantitative electrolysis of 1 results in the preparation of the oxidized complex [Cu(Hmal55)] (3), which is sufficiently stable to be isolated, isolation of the oxidation product of 2 was not attempted because of its long‐term instability. The properties of the complexes were investigated by means of various spectroscopic methods (UV‐vis, ESR, NMR, and IR spectroscopy) and by X‐ray structure analysis. The structure determinations and the spectroscopic investigations of the complexes reveal a square‐planar CuN4coordination environment for each complex in the solid state and in acetonitrile solution. In both the oxidized and reduced oxidation states of the complexes, the coordinated ligands remain triply deprotonated with a hydrogen atom bridging both oxime oxygen atoms. The ligands can therefore be regarded as pseudo‐macrocyclic. The characterization of the oxidation products clearly identifies the electron‐transfer reaction as being metal‐centered. For the first time, the structure of a copper(III) complex with a ligand containing oximes as donor groups was determined. The redox potentials of the copper complexes are compared to related CuIII/CuII redox couples.

Due to failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as DSMC to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow r egime. H ere, e vaporation b oundary c onditions f or t he R 13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be d etermined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier-Stokes-Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement to DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

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