Effects of internal hydrogen bonds between amide groups: protonation of alicyclic diamides

The proton affinity (PA) of cyclopentane carboxamide 1, cyclohexane carboxamide 2 and their secondary and tertiary amide derivatives S1, S2, T1 and T2, was determined by the thermokinetic method and the kinetic method [PA(1) = 888 +/- 5 kJ mol(1); PA(2) = 892 +/- 5 kJ mol(1); PA(S1) = 920 +/- 6 kJ mol(1); PA(S2) = 920 +/- 6 kJ mol(1); PA(T1) = 938 +/- 6 kJ mol(1); PA(T2) = 938 +/- 6 kJ mol(1)]. Special entropy effects are not observed. Additionally, the effects of protonation have been studied using an advanced kinetic method for all isomers 37 of cyclopentane dicarboxamides and cyclohexane dicarboxamides (with the exception of cis-cyclopentane-1,2-dicarboxamide) and their bis-tertiary derivatives T3T7 by estimating the PA and the apparent entropy of protonation Delta(DeltaS(app)). Finally, the study was extended to bicyclo[2.2.1]hepta-2,5-diene-2,3-dicarboxamide 8 and its bis-tertiary derivative T8, to all stereoisomers of bicyclo[2.2.1]heptane-2,3-dicarboxamide 9, their secondary and tertiary amide derivatives S9 and T9, and to endoendobicyclo[2.2.1]heptane-2,5-dicarboxamide 10 and the corresponding secondary and tertiary derivatives S10 and T10. Compared with 1 and 2, all alicyclic diamides exhibit a significant increase of the PA (DeltaPA) and special entropy effects on protonation. For alicyclic diamides, which can not accommodate a conformation appropriate for building a proton bridge, the values of DeltaPA and Delta(DeltaS(app)) are small to moderate. This is explained by ion / dipole interactions between the protonated and neutral amide group which stabilize the protonated species but hinder the free rotation of the amide groups. If any of the conformations of the alicyclic diamide allows formation of a proton bridge, DeltaPA and Delta(DeltaS(app)) increase considerably. A spectacular case is cis-cyclohexane-1,4-dicarboxamide 7c which is the most basic monocyclic diamide, although generation of the proton bridge requires the unfavorable boat conformation with both amide substituents at a flagpole position. A pre-orientation of the two amide groups in such a 1,4-position in 10 results in a particularly large PA of < 1000 kJ mol(1). The observation of comparable values for Delta(DeltaS(app)) for linear and monocyclic diamides indicates that a major part of the entropy effects originates from freezing the free rotation of the amide groups by formation of the proton bridge. This is corroborated by observing corresponding effects during the protonation of dicarboxamides containing the rigid bicyclo[2.2.1]heptane carbon skeleton, where the only internal movements of the molecules corresponds to rotation of the amide substituents.     

Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework,a volume distribution of holes

This paper treats the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. (For example, in the three-dimensional case, the obstacles have a size of ³ and are located at the nodes of a regular mesh of size .) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).     

Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II: Non-critical sizes of the holes for a volume distribution and a surface distribution of holes

This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size ², located at the nodes of a regular plane mesh of period .) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.