On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Confined dynamics, forms and transitions in colloidal systems: from clay to DNA

Colloidal suspensions are a classic example of confining systems developing large specific surfaces, presenting a rich variety of shapes and exhibiting complex organization on a length scale ranging from 1 nm to several micrometers. Two distinct confined dynamics are generally considered in such systems: (1) the embedded fluid dynamics entrapped in the pore network with two main contributions, surface interaction and long-range connectivity, and (2) the dynamics of the host matrix, associated with a time evolution of the interfacial geometry. This last contribution is particularly important during dynamic and structural transitions of colloidal suspensions such as jamming, glass transition, phase separations and flocculation. It is generally believed that the characteristic time scale needed to describe colloidal movement and interfacial geometrical reorganization is much slower than the dynamics of the embedded fluid (except in the trivial situation where the fluid molecule is irreversibly adsorbed to a colloidal surface). Thus, few connections are made between these two distinct dynamics. In this presentation, we show how the slow and confined water dynamics at proximity of a colloidal surface provides an original way to probe colloidal shape and colloidal orientation dynamics. Two topics are presented. First of all, water field-cycling NMR relaxometry is used to probe the glass transition and the strong rotational slowing down of a colloidal system made of plate-like particles, a synthetic clay (laponite). Second, we analyze the case of long colloidal thin rods (either mineral or biologic such as DNA cylinders) dispersed in very diluted suspensions. At large distance and/or long time, these particles appear as a portion of a line. We discuss how the embedded fluid dynamics can be sensitive to this morphological crossover and may provide information about the particle shape. Some comparisons with recent experiments are presented.     

Universal nuclear spin relaxation and long-range order in nematics strongly confined in mass fractal silica gels

We show how the low-frequency dependence of the proton spin-lattice relaxation time T1(nu) of octylcyanobiphenyl liquid crystals confined in high-density silica gels evidences a long-range order nematic phase in spite of the strong confinement and random disorder of the gels. The universal value and frequency dependence observed, T1(nu) proportional, variant nu(2/3), is interpreted within a relaxation model due to director fluctuations in nematic liquid crystals confined to mass fractal porous media. The model provides a relation T1(nu) proportional, variant nu(2-d/2), giving a reliable value of the structural fractal dimension d(f)=2.67 for all the host silica gels.     

Multiscale dynamics of 1H and 19F in confined ionogels for lithium batteries

We present the first results of the nuclear magnetic relaxation dispersion (NMRD) of the confined proton-bearing cation (BMI) and fluorine-bearing anion (TFSI) pair of ionic liquids (Li⁺-ionogels) confined within a silica-like mesoporous matrices designed for lithium batteries. These results are in favour of a very-correlated dynamical motion of the anion–cation pair within the solid and disordered silica matrix.     

Water dynamics in ionomer membranes by field-cycling NMR relaxometry

The dynamic behavior of water within two types of ionomer membranes, Nafion and sulfonated polyimide, has been investigated by field-cycling nuclear magnetic relaxation. This technique, applied to materials prepared at different hydration levels, allows the proton motion on a time scale of microseconds to be probed. The NMR longitudinal relaxation rate R(1) measured over three decades of Larmor angular frequencies omega is particularly sensitive to the host-water interactions and thus well-suited to study fluid dynamics in restricted geometries. In the polyimide membranes, we have observed a strong dispersion of R(1)(omega) following closely a 1/square root omega law in a low-frequency range (correlation times from 0.1 to 10 micros). This is indicative of a strong interaction of water with "interfacial" hydrophilic groups of the polymeric matrix (wetting situation). Variations of the relaxation rates with water uptake reveal a two-step hydration process: solvation and formation of disconnected aqueous clusters near polar groups, followed by the formation of a continuous hydrogen bond network. On the contrary, in the Nafion we observed weak variations of R(1)(omega) at low frequencies. This is typical of a nonwetting behavior. At early hydration stages, R(1)(omega) evolves logarithmically, suggesting a confined bidimensional diffusion of protons in the microsecond time range. Such an evolution is lost at higher swelling where a plateau related to three-dimensional diffusion is observed.     

NMR relaxivity of coated and non-coated size-sorted maghemite nanoparticles

ABSTRACT

The effect of polymer coating on MNR relaxometry of maghemite nanoparticles has been studied. The samples were carefully sorted by size in order to reach narrow size distribution (<0.2) with size ranging from 4.5 to 12.5?nm. Relaxation dispersion profile as well as studies at a fixed Larmor frequency, were recorded for numerous either uncoated or polymer coated samples. The NMR relaxivities r₁ and r₂ increase with nanoparticle diameter. We have analysed the role of polydispersity for nanoparticles with the same mean size on the dispersion curves. We have compared the role of coating on nanoparticles NMR relaxivity between bare and poly(sodium acrylate-co-maleate) coated nanoparticles. We have investigated the influence of nanoparticle size on the T₁ and T₂ activation energy Ea. While Ea decreases with nanoparticle diameter when determined from T₁, it increases from T₂ determination. The influence is more important for small particles (<9?nm) than for big particles (>9?nm). Moreover, the PAAMA coating changes the energy Ea obtained from T₂: Ea becomes independent of the nanoparticle diameter. These results highlight the need of a complete characterisation of the role of the coating on the relaxation of magnetic particles.     

Slow dynamics in colloidal glasses and porous media as probed by NMR relaxometry: assessment of solvent levy statistics in the strong adsorption regime

Mesoscopic media such as porous materials or colloidal pastes develop large specific surface area which strongly influence the dynamics of the embedded fluid. This fluid confinement can be used either to probe the interfacial geometry (frozen porous media) or the particle dynamics (paste and colloidal glass). In the strong adsorption regime, it was recently proposed that the effective surface diffusion on flat surface is anomalous and exhibits long time pathology (Lévy walks). This phenomena is directly related to the time and space properties of loop trajectories appearing in the bulk between a desorption and a readsorption step. The Lévy statistics extends the time domain of the embedded fluid dynamics toward the low frequency regime. An interesting way to probe such a slow interfacial process is to use field cycling NMR relaxometry. In the first part of this paper, we propose a simple theoretical model of NMR dispersion which only involves elementary time steps of the solvent dynamics near an interface (loops, trains, tails in relation with the confining geometry). In the second part, field cycling NMR relaxometry is used to probe the slow solvent dynamics in two type of interfacial systems: (i) a colloidal glass made of thin and flat particles (ii) two fully saturated porous media, the Vycor glass and MCM48 respectively. Experimental results are critically compared to closed-form analytical expressions and numerical simulations.     

Mean First-Passage Time of Surface-Mediated Diffusion in Spherical Domains

We present an exact calculation of the mean first-passage time to a target on the surface of a 2D or 3D spherical domain, for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion. The presented approach is based on an integral equation which can be solved analytically. Numerically validated approximation schemes, which provide more tractable expressions of the mean first-passage time are also proposed. In the framework of this minimal model of surface-mediated reactions, we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface.     

Brownian dynamics of water confined in AOT reverse micelles: A field-cycling deuteron NMR relaxometry study

Reversed micelles and water in oil micro-emulsions can be used to solubilize biopolymers and genetic materials allowing analyzing their properties in a confined geometry. Nuclear Magnetic Resonance Dispersion (NMRD) provides a powerful and a noninvasive experimental technique to probe the long-term dynamics of these confined systems. However, the first step is to analyze and understand the slow dynamics of water inside these micro-reactors without any guest molecule. This is the aim of this presentation. Experimental results have been obtained for deuteron ²H NMRD of water confined in reverse micelles of bis (2-ethylhexyl) sodium sulfosuccinate (AOT) dispersed in isooctane C₈H₁₈. The water content is expressed as the molar ratio W₀ = [Water]/[AOT]. The radius of the spherical reversed micelles, R_m, increases almost linearly with W₀. In our case, W₀ is chosen in the range 20 ≤ W₀ ≤ 50 (35 ≤ R_m ≤ 80 Å). The frequency dependence for the spin-lattice relaxation rate R₁(ω) exhibits two regimes, for all W₀ values: a plateau at low frequency, proportional to 1/R_m, followed by the beginning of an algebraic decay. These experimental observations are discussed and compared to a numerical simulation of the intermittent Brownian diffusion of a water molecule inside a rotating reverse micelle. The possibility to probe some properties of the confinement, such as the localisation time on the sulfonated palisade and/or the water self-diffusion inside the water pool is emphasised.