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Here we present an approach to functionalize silanized single-walled carbon nanotubes (SWNTs) through copper-free click chemistry for the assembly of inorganic and biological nanohybrids. The nanotube functionalization route involves silanization and strain-promoted azide–alkyne cycloaddition reactions (SPACC). This was characterized by X-ray photoelectron spectroscopy, scanning electron microscopy, transmission electron microscopy, Raman spectroscopy and Fourier transform infra-red spectroscopy. Silane–azide-functionalized SWNTs were immobilized from solution onto patterned substrates through dielectrophoresis (DEP). We demonstrate the general applicability of our strategy for the functionalization of SWNTs with metal nanoparticles (gold nanoparticles), fluorescent dyes (Alexa Fluor 647) and biomolecules (aptamers). In this regard, dopamine-binding aptamers were conjugated to the functionalized SWNTs to perform real-time detection of dopamine at different concentrations. Additionally, the chemical route is shown to selectively functionalize individual nanotubes grown on the surface of silicon substrates, contributing towards future nano electronic device applications.

The structural parameters of the rare-earth diacetate halide trihydrates, RE(OAc)2Hal·3H2O with RE = Ce − (Pm) − Lu and Hal = Cl, Br, have been determined by low temperature, high-resolution SCXRD in order to examine the effect of lanthanide contraction on the coordination geometry in this series of isomorphous compounds consisting of cationic, acetate-bridged, non-linear, one-dimensional coordination polymers of composition [RE(H2O)3(OAc)2]+ and laterally hydrogen bonded halide ions, Hal−. Although the shrinkage of the unit cell volume follows lanthanide contraction very well over the complete range of investigated RE elements, many other parameters (i.e., lattice constants, angles and distances in the RE··· RE alignment, RE-O bond lengths, etc.) exhibit a more complex response on lanthanide contraction often expressed by sigmoid curves that can be ascribed to a continuous transition from CN9 (RE = Ce) to CN8 (RE = Lu) as one acetate group loses the chelate function, an effect accompanied by significant structural changes of the carboxylate group. Therefore, data are best analyzed by use of two subsets represented by the two different structure types of Ce and Lu, the structural features of which change with decreasing/increasing the size of RE3+, up to the borderline between both subsets.

The existing range of the centrosymmetric, triclinic RE(OAc)3 · 2AcOH structure type has been extended for RE = Eu and Gd while the structure data of the Nd- and Sm-compounds have been revised and corrected, respectively, using low temperature (T = 100 K), well resolved (2θmax = 56°), highly redundant SCXRD data in order to evaluate the structural evolution within this class of acetic acid solvates by statistical methods. Within the nine-fold mono-capped square-antiprismatic coordination spheres of the RE3+ ions, RE-O distances decrease as a result of lanthanide contraction; some with different rates depending on the coordination modes (2.11/2.21) of the acetate ions. The experimental data show that the internal structural parameters of the acetate ions also correlate with their coordination modes. Both acetic acid molecules act as hydrogen bond donors but only one as monodentate ligand. The geometries of the hydrogen bonds reveal that they are strongly influenced by the size of the rare earth atom. The non-linear, one-dimensional coordination polymer propagates with unequal RE···RE distances along the a-axis. Rods of the coordination polymer are arranged in layers congruently stacked above each other with the hydrogen bonded acetic acid molecules as filler in between. In most cases, data fitting is best described in terms of a quadratic rather than a linear regression analysis.

We interrelate the real interpolation spaces associated with the couples (X,Y),(X+Y,Y),(X,X∩Y)(X,Y), (X+Y,Y), (X,X\\cap Y), and (X+Y,X∩Y)(X+Y,X\\cap Y), proving among others the identities (X+Y,X)θ,p∩(X+Y,Y)θ,pamp;=(X+Y,X∩Y)θ,p,(X+Y,X)θ,p∩(X+Y,Y)1−θ,pamp;=(X,Y)θ,p,(X,X∩Y)θ,p+(Y,X∩Y)θ,pamp;=(X+Y,X∩Y)θ,p,(X,X∩Y)θ,p+(Y,X∩Y)1−θ,pamp;=(X,Y)θ,p\\begin{align*} (X+Y,X)_{\\theta ,p} \\cap (X+Y, Y)_{\\theta ,p} & = (X+Y, X \\cap Y)_{\\theta ,p}, (X+Y,X)_{\\theta ,p} \\cap (X+Y, Y)_{1-\\theta ,p} & = (X,Y)_{\\theta ,p}, (X,X\\cap Y)_{\\theta ,p} + (Y,X\\cap Y)_{\\theta ,p} & = (X+Y, X\\cap Y)_{\\theta ,p}, (X,X\\cap Y)_{\\theta ,p} + (Y,X\\cap Y)_{1-\\theta ,p} & = (X,Y)_{\\theta ,p} \\end{align*} for all p∈[1,∞],θ∈[0,1]p \\in [1,\\infty ], \\theta \\in [0,1].

Let f = f ( z , t ) be a function holomorphic in z ∈ O ⊆ C d for fixed t ∈ Ω and measurable in t for fixed z and such that z ↦ f ( z , · ) is bounded with values in E : = L p ( Ω ) , 1 ≤ p ≤ ∞ . It is proved (among other things) that ⟨ t ↦ φ ( f ( · , t ) ) , μ ⟩ = φ ( z ↦ ⟨ f ( z , · ) , μ ⟩ ) whenever μ ∈ E ′ and φ is a bp-continuous linear functional on H ∞ ( O ) .

High sensitizer and activator concentrations have been increasingly examined to improve the performance of multi-color emissive upconversion (UC) nanocrystals (UCNC) like NaYF 4 :Yb,Er and first strategies were reported to reduce concentration quenching in highly doped UCNC. UC luminescence (UCL) is, however, controlled not only by dopant concentration, yet by an interplay of different parameters including size, crystal and shell quality, and excitation power density ( P ). Thus, identifying optimum dopant concentrations requires systematic studies of UCNC designed to minimize additional quenching pathways and quantitative spectroscopy. Here, we quantify the dopant concentration dependence of the UCL quantum yield ( Φ UC ) of solid NaYF 4 :Yb,Er/NaYF 4 :Lu upconversion core/shell nanocrystals of varying Yb 3+ and Er 3+ concentrations (Yb 3+ series: 20%–98% Yb 3+ ; 2% Er 3+ ; Er 3+ series: 60% Yb 3+ ; 2%–40% Er 3+ ). To circumvent other luminescence quenching processes, an elaborate synthesis yielding OH-free UCNC with record Φ UC of ∼9% and ∼25 nm core particles with a thick surface shell were used. High Yb 3+ concentrations barely reduce Φ UC from ∼9% (20% Yb 3+ ) to ∼7% (98% Yb 3+ ) for an Er 3+ concentration of 2%, thereby allowing to strongly increase the particle absorption cross section and UCNC brightness. Although an increased Er 3+ concentration reduces Φ UC from ∼7% (2% Er 3+ ) to 1% (40%) for 60% Yb 3+ . Nevertheless, at very high P (> 1 MW/cm 2 ) used for microscopic studies, highly Er 3+ -doped UCNC display a high brightness because of reduced saturation. These findings underline the importance of synthesis control and will pave the road to many fundamental studies of UC materials.

We developed a procedure to prepare luminescent LiYF 4 :Yb/LiYF 4 and LiYF 4 :Yb,Er/LiYF 4 core/shell nanocrystals with a size of approximately 40 nm revealing luminescence decay times of the dopant ions that approach those of high-quality laser crystals of LiYF 4 :Yb (Yb:YLF) and LiYF 4 :Yb,Er (Yb,Er:YLF) with identical doping concentrations. As the luminescence decay times of Yb 3+ and Er 3+ are known to be very sensitive to the presence of quenchers, the long decay times of the core/shell nanocrystals indicate a very low number of defects in the core particles and at the core/shell interfaces. This improvement in the performance was achieved by introducing two important modifications in the commonly used oleic acid based synthesis. First, the shell was prepared via a newly developed method characterized by a very low nucleation rate for particles of pure LiYF 4 shell material. Second, anhydrous acetates were used as precursors and additional drying steps were applied to reduce the incorporation of OH − in the crystal lattice, known to quench the emission of Yb 3+ ions. Excitation power density ( P )-dependent absolute measurements of the upconversion luminescence quantum yield ( Φ UC ) of LiYF 4 :Yb,Er/LiYF 4 core/shell particles reveal a maximum value of 1.25% at P of 180 Wcm −2 . Although lower than the values reported for NaYF 4 :18%Yb,2%Er core/shell nanocrystals with comparable sizes, these Φ UC values are the highest reported so far for LiYF 4 :18%Yb,2%Er/LiYF 4 nanocrystals without additional dopants. Further improvements may nevertheless be possible by optimizing the dopant concentrations in the LiYF 4 nanocrystals.

Despite considerable advances in synthesizing high-quality core/shell upconversion (UC) nanocrystals (NC; UCNC) and UCNC photophysics, the application of near-infrared (NIR)-excitable lanthanide-doped UCNC in the life and material sciences is still hampered by the relatively low upconversion luminescence (UCL) of UCNC of small size or thin protecting shell. To obtain deeper insights into energy transfer and surface quenching processes involving Yb 3+ and Er 3+ ions, we examined energy loss processes in differently sized solid core NaYF 4 nanocrystals doped with either Yb 3+ (YbNC; 20% Yb 3+ ) or Er 3+ (ErNC; 2% Er 3+ ) and co-doped with Yb 3+ and Er 3+ (YbErNC; 20% Yb 3+ and 2% Er 3+ ) without a surface protection shell and coated with a thin and a thick NaYF 4 shell in comparison to single and co-doped bulk materials. Luminescence studies at 375 nm excitation demonstrate back-energy transfer (BET) from the 4 G 11/2 state of Er 3+ to the 2 F 5/2 state of Yb 3+ , through which the red Er 3+ 4 F 9/2 state is efficiently populated. Excitation power density ( P )-dependent steady state and time-resolved photoluminescence measurements at different excitation and emission wavelengths enable to separate surface-related and volume-related effects for two-photonic and three-photonic processes involved in UCL and indicate a different influence of surface passivation on the green and red Er 3+ emission. The intensity and lifetime of the latter respond particularly to an increase in volume of the active UCNC core. We provide a three-dimensional random walk model to describe these effects that can be used in the future to predict the UCL behavior of UCNC.

We give a simplified proof of the complex inversion formula for semigroups and, more generally, solution families for scalar-type Volterra equations, including the stronger versions on unconditional martingale differences (UMD) spaces. Our approach is based on (elementary) Fourier analysis.

We prove pointwise convexity (Jensen-type) inequalities of the form where F is a convex function defined on a convex subset of some Banach space X and T is the X -valued extension of a positive operator on some function space. Examples include the pointwise Holder inequality T ( fg ) = ( Tf p ) 1/ p ( Tf q ) 1/ q for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a \"real\" proof of the Riesz-Thorin theorem for positive operators.