Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ ? u + ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ ? u + ā. A calibrated sub-action u gives the possibility to construct, for any point x ε Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.We prove that separating sub-actions are generic among Hölder sub-actions. We also prove that, under certain conditions on Ω(A), any calibrated sub-action is of the form u(x) = u(x _i ) + h _A (x _i , x) for some x _i ∈ Ω(A), where h _A (x, y) denotes the Peierls barrier of A. We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type \((\hat \Sigma , \hat \sigma )\).     

Theoretical investigation on 1H and 13C NMR chemical shifts of small alkanes and chloroalkanes

Using density functional theory (DFT) with the B3LYP, PBE, and PBE0 exchange-correlation functionals as well as the Moller-Plesset second-order perturbation theory (MP2) combined with a series of rather extended basis sets, 1H and 13C chemical shifts of small alkanes and chloroalkanes (with different numbers of chlorine atoms on specific positions) have been simulated and compared to experimental data. For the 1H chemical shifts, theory tends to reproduce experiment within the limits of the experimental errors. In the case of 13C chemical shift, the differences between theory and experiment increase monotonically with the number of chlorine atoms and exhibit a deviation from additivity. This behavior is related to the saturation of the experimental 13C chemical shifts with the number of chlorine atoms, whereas the evolution is mostly linear at both DFT and MP2 levels of approximation. This difference has been traced back to the relativistic spin-orbit coupling effects, which are exalted as a result of the enhancement of the s character of the C atom when increasing the number of linked Cl atoms. Thus, it was demonstrated that not only electron correlation but also relativistic effects have to be considered for estimating the 13C chemical shifts when several Cl atoms are directly attached to the C atom. Linear (theory/experiment) regressions have then been performed for the different types of C atoms, i.e., bearing one, two, and three Cl atoms, with excellent correlation coefficients. The linear correlation relationships so obtained can then serve to predict and facilitate the interpretation of the nuclear magnetic resonance spectra of more complex compounds. Furthermore, by investigating the basis set effects, the correlation between the chemical shifts calculated using the 6-311 + G(2d,p) basis set and the more extended 6-311 + G(2df,p) and aug-cc-pvtz basis sets is excellent, demonstrating that the choice of the 6-311 + G(2d,p) basis set for calculating the 1H and 13C chemical shifts is relevant.     

Chemo-regioselectivity in heterogeneous catalysis: competitive routes for C = O and C = C hydrogenations from a theoretical approach

The usual empirical rule stating that the C=C bond is more reactive than the C=O group for catalytic hydrogenations of unsaturated aldehydes is invalidated from the present study. Density functional theory calculations of all the competitive hydrogenation routes of acrolein on Pt(111) reveals conversely that the attack at the C=O bond is systematically favored. The explanation of such catalytic behavior is the existence of metastable precursor states for the O-H bond formation showing that the attack at the oxygen atom follows a new preferential mechanism where the C=O moiety is not directly bonded with the Pt surface atoms, hence yielding an intermediate pathway between Langmuir-Hinshelwood and Rideal-Eley general types of mechanisms. When the whole catalytic cycle is considered, our results reconcile with experimental studies devoted to hydrogenation of acrolein on Pt, since the desorption step of the partially hydrogenated product (unsaturated alcohol versus saturated aldehyde) plays a key role for the selectivity.     

Conformationally gated photoinduced processes within photosensitizer-acceptor dyads based on osmium(II) complexes with triarylpyridinio-functionalized terpyridyl ligands: insights from theoretical analysis

A theoretical analysis, based on density functional theory, has been carried out to study photoinduced processes within a recently experimentally characterized (Lainé, P. P.; Bedioui, F.; Loiseau, F.; Chiorboli, C.; Campagna, S. J. Am. Chem. Soc. 2006, http://dx.doi.org/10.1021/ja058357w.) series of Os(II) bis-tpy complexes (tpy = 2,2':6'2' '-terpyridine) functionalized by 2,4,6-triarylpyridinium groups, TP+. These dyad systems, designed to produce a charge-separated state (CSS) upon light excitation, are made up of a photosensitizer unit (P, the metal complex) and a tunable acceptor unit (A, the TP+). A full ab initio characterization of the electronic and structural properties of the lowest-lying triplet excited states, as well as of the CSS, allowed for a complete rationalization of the photoinduced processes taking place within the dyads. Among salient insights, theory allowed (i) substantiation of the inner P structural planarization as the relaxation mode of the MLCT states, (ii) confirmation of the existence of a ligand-centered triplet excited state (3LC) shown to essentially involve the nitro substituent of A (TP+-NO2) and lying very close in energy to the P-centered 3MLCT state, and (iii) a demonstration that the energy of the 3LC level is independent of intercomponent tilt angle (theta1). On this basis, the occurrence of a reversible electronic energy transfer between the 3MLCT and the 3LC states could be substantiated and shown to depend on the intramolecular conformation represented by theta1, which actually governs their electronic coupling (essentially via the degree of intercomponent conjugation). These computational issues were checked against experimental data already available and the results of a specifically undertaken photophysical study. Finally, CSS formation has been confirmed by studying the spin density patterns of reduced nitro-derivatized dyads. Taken together, these findings accurately account for the different excited-state behaviors of the dyads as a function of the level of structural restriction of their intercomponent conformation (and related amplitude for torsional fluctuations), thus providing theoretical evidence of conformationally gated photoinduced electron- and energy-transfer processes.     

Experimental and theoretical studies of magnetic exchange in silole-bridged diradicals

Five bis(tert-butylnitroxide) diradicals connected by a silole (7 a-d) or a thiophene (12) ring as a coupler were studied. Compound 12 crystallizes in the orthorhombic space group Pna2(1) with a = 20.752(5), b = 5.826(5), and c = 34.309(5) A. X-ray crystal structure determination, electronic spectroscopy, variable-temperature EPR spectroscopy, SQUID measurements and DFT computations (UB3LYP/6-31+G*) were used to study the molecular conformations and electronic spin coupling in this series of molecules. Whereas compounds 7 b, 7 c, and 7 d are quite stable both in solution and in the solid state, 7 a and 12 undergo a partial electronic rearrangement to both a diamagnetic quinonoidal form and a monoradical species owing to the fact that they correspond to the open form of a pi-conjugated Kekulé structure. In the solid state, magnetic measurements indicate that the diradicals are all antiferromagnetically coupled, as expected from their topology. These interactions are best reproduced by means of a "Bleaney-Bowers" model that gives values of J = -142.0 cm(-1) for 7 a, -1.8 cm(-1) for 7 b, -1.3 cm(-1) for 7 c, -4.2 cm(-1) for 7 d, and -248.0 cm(-1) for 12. The temperature dependence of the EPR half-field transition in frozen dichloromethane solutions is consistent with singlet ground states and thermally accessible triplet states for diradicals 7 b, 7 c, and 7 d with DeltaE(T-S) values of 3.48, 2.09, and 8 cm(-1), respectively. No evidence of a populated triplet state was found for diradicals 7 a and 12. Similarities between the DeltaE(T-S) and J values (DeltaE(T-S) = -2 J) clearly show the intramolecular origin of the observed antiferromagnetic interaction. Analyses of the data with a "Karplus-Conroy"-type equation enabled us to establish that the silole ring, as a whole, allows a more efficient magnetic coupling of the two nitroxide radicals attached to its 2,5-positions than the thiophene ring. This superiority probably originates from the nonaromaticity of the silole which thus permits a better magnetic interaction through it. DFT calculations also support the experimental results, indicating that the magnetic exchange pathway preferentially involves the carbon pi system of the silole.     

Identity SN2 reactions X- + CH3X --> XCH3 + X- (X=F, Cl, Br, and I) in vacuum and in aqueous solution: a valence bond study

The recently developed (L. Song, W. Wu, Q. Zhang, S. Shaik, J. Phys. Chem. A 2004, 108, 6017) valence bond method coupled with a polarized continuum model (VBPCM) has been applied to the identity SN2 reaction of halides in the gas phase and in aqueous solution. The barriers computed at the level of the breathing orbital VB method (P. C. Hiberty, J. P. Flament, E. Noizet, Chem. Phys. Lett. 1992, 189, 259), BOVB and VBPCM//BOVB, are comparable to CCSD(T) and CCSD(T)//PCM results and to experimentally derived barriers in solution (W. J. Albery, M. M. Kreevoy, Adv. Phys. Org. Chem. 1978, 16, 85). The reactivity parameters needed to apply the valence bond state correlation diagram (VBSCD) method (S. Shaik, J. Am. Chem. Soc. 1984, 106, 1227), were also determined by VB calculations. It has been shown that the reactivity parameters along with their semiempirical derivations provide a satisfactory qualitative and quantitative account of the barriers.     

Development of a lattice-sum method emulating nonperiodic boundary conditions for the treatment of electrostatic interactions in molecular simulations: a continuum-electrostatics study

Artifacts induced by the application of periodic boundary conditions and lattice-sum methods in explicit-solvent simulations of (bio-)molecular systems are nowadays a major concern in the computer-simulation community. The present article reports a first step toward the design of a modified lattice-sum algorithm emulating nonperiodic boundary conditions, and therefore exempt of such periodicity-induced artifacts. This result is achieved here in the (more simple) context of continuum electrostatics. It is shown that an appropriate modification of the periodic Poisson equation and of its boundary conditions leads to a continuum-electrostatics scheme, which, although applied under periodic boundary conditions, exactly mimics the nonperiodic situation. The possible extension of this scheme to explicit-solvent simulations is outlined and its practical implementation will be described in more details in a forthcoming article.