Synthesis and properties of 1,3,5-benzene periodic mesoporous organosilica (PMO): novel aromatic PMO with three point attachments and unique thermal transformations

A new aromatic periodic mesoporous organosilica material containing benzene functional groups that are symmetrically integrated with three silicon atoms in an organosilica mesoporous framework is reported. The material has a high surface area, well-ordered mesoporous structure and thermally stable framework aromatic groups. The functional aromatic moieties were observed to undergo sequential thermal transformation from a three to two and then to a one point attachment within the framework upon continuous thermolysis under air before eventually being converted to periodic mesoporous silica devoid of aromatic groups at high temperatures and longer pyrolysis times. The mesoporosity of the material was characterized by powder X-ray diffraction (PXRD), transmission electron microscopy (TEM), and nitrogen porosimetry, whereas the presence and transformation of the aromatic groups in the walls of the materials were characterized by solid-state NMR spectroscopy, mass spectrometry, and thermogravimetric analysis. The attachment of a benzene ring symmetrically onto three siloxanes of the framework was used advantageously as a cross-linker to enhance the thermal stability of the organic group. Some of these properties are investigated in comparison with other aromatic PMOs that have only two point attachments and an amorphous phenylsilica gel that has only one point attachment. The successful synthesis of the first aromatic PMO with its organic group attached within the framework through more than two points is an important step toward the synthesis of PMOs having organic groups with more complex and multiple attachments within the framework.     

Metamorphosis of ordered mesopores to micropores: periodic silica with unprecedented loading of pendant reactive organic groups transforms to periodic microporous silica with tailorable pore size

Ordered porous silicas with unprecedented loadings of pendant vinyl groups have been synthesized via co-condensation of tetraethyl orthosilicate (TEOS) and triethoxyvinylsilane (TEVS) under basic conditions in the presence of cetyltrimethylammonium surfactant. The resulting organosilicate-surfactant composites exhibited at least one low-angle X-ray diffraction (XRD) peak up to the TEVS:TEOS molar ratio of 7:3 (70% TEVS loading) in the synthesis gel. The surfactant was removed from these composites without any structural collapse. Nitrogen adsorption provided strong evidence of the presence of uniformly sized pores and the lack of phase separation up to TEVS:TEOS ratios as high as 13:7 (65% TEVS loading), whereas (29)Si MAS NMR and high-resolution thermogravimetry showed essentially quantitative incorporation of the organosilane. Thus, a hitherto unachieved loading level for pendant groups, considered by many to be impossible to achieve for stable organosilicas because of the expected framework connectivity constraints, has been obtained. The resulting vinyl-functionalized silicas exhibited gradually decreasing pore diameter (from 2.8 to 1.7 nm for TEVS loadings of 25-65%) and pore volume as the loading of pendant groups increased, but the specific surface area was relatively constant. Because of the reactivity of vinyl groups, ordered silicas with very high loadings of these groups are expected to be robust starting materials for the synthesis of other organic-functionalized ordered microporous materials. Herein, we demonstrate that these starting materials can also be transformed via calcination into ordered microporous silicas with pore diameters tailorable from 2.5 to as little as 1.4 nm simply by using an appropriate loading of the vinyl-functionalized precursor. This ease of the micropore size adjustment and the attained degree of structural ordering (as judged from XRD) have not been reported before. The novel ordered microporous materials reported herein are promising as adsorbents and catalyst supports.     

Maximum principles for a class of nonlinear second order elliptic differential equations

In this paper we investigate maximum principles for functionals defined on solutions to special partial diff'erential equations of elliptic type, extending results by Payne and Philippin. We apply such maximum principles to investigate one overdetermined problem.     

A Blinking Mesoporous TiO2−x Composed of Nanosized Anatase with Unusually Long‐Lived Trapped Charge Carriers

A mesoporous TiO_2?x material comprised of small, crystalline, vacancy‐rich anatase nanoparticles (NPs) shows unique optical, thermal, and electronic properties. It is synthesized using polymer‐derived mesoporous carbon (PDMC) as a template. The PDMC pores serve as physical barriers during the condensation and pyrolysis of a titania precursor, preventing the titania NPs from growing beyond 10 nm in size. Unlike most titania nanomaterials, during pyrolysis the NPs undergo no transition from the anatase to rutile phase and they become catalytically active reduced TiO_2?x. When exposed to a slow electron beam, the NPs exhibit a charge/discharge behavior, lighting up and fading away for an average period of 15 s for an extended period of time. The NPs also show a 50 nm red‐shift in their UV/Vis absorption and long‐lived charge carriers (electrons and holes) at room temperature in the dark, even long after UV irradiation. The NPs as photocatalysts show a good activity for CO₂ reduction.     

Cobalt‐Embedded Nitrogen‐Rich Carbon Nanotubes Efficiently Catalyze Hydrogen Evolution Reaction at All pH Values

Despite being technically possible, splitting water to generate hydrogen is still practically unfeasible due mainly to the lack of sustainable and efficient catalysts for the half reactions involved. Herein we report the synthesis of cobalt‐embedded nitrogen‐rich carbon nanotubes (NRCNTs) that 1) can efficiently electrocatalyze the hydrogen evolution reaction (HER) with activities close to that of Pt and 2) function well under acidic, neutral or basic media alike, allowing them to be coupled with the best available oxygen‐evolving catalysts—which also play crucial roles in the overall water‐splitting reaction. The materials are synthesized by a simple, easily scalable synthetic route involving thermal treatment of Co²⁺‐embedded graphitic carbon nitride derived from inexpensive starting materials (dicyandiamide and CoCl₂). The materials’ efficient catalytic activity is mainly attributed to their nitrogen dopants and concomitant structural defects.     

Optimization of Active Sites via Crystal Phase,Composition, and Morphology for Efficient Low-Iridium Oxygen Evolution Catalysts

Reducing the amount of iridium in oxygen evolution electrocatalysts without compromising their catalytic performances is one of the major requirements in proton-exchange-membrane water electrolyzers. Herein, with the help of theoretical studies, we show that anatase-type TiO₂-IrO₂ solid solutions possess more active iridium catalytic sites for the oxygen evolution reaction (OER) than IrO₂, the benchmark OER catalyst. Note that the same is not observed for their rutile-type counterparts. However, owing to their thermodynamic metastability, anatase-type TiO₂-IrO₂ solid solutions are generally hard to synthesize. Our theoretical studies demonstrate that such catalytically active anatase-type solid-solution phases can be created in situ on the surfaces of readily available SrTiO₃-SrIrO₃ solid solutions during electrocatalysis in acidic solution as the solution can etch away Sr atoms. We experimentally show this with porous SrTiO₃-SrIrO₃ solid-solution nanotubes synthesized by a facile synthetic route that contain 56 % less iridium than IrO₂ yet show an order of magnitude higher apparent catalytic activity for OER in acidic solution.     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

Ramanujan’s Master Theorem

S. Ramanujan introduced a technique, known as Ramanujan??s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan??s Master Theorem is discussed.     

Arithmetical properties of a sequence arising from an arctangent sum

The sequence {xn} defined by xn=(n+xn−1)/(1−nxn−1), with x₁=1, appeared in the context of some arctangent sums. We establish the fact that xn≠0 for n?4 and conjecture that xn is not an integer for n?5. This conjecture is given a combinatorial interpretation in terms of Stirling numbers via the elementary symmetric functions. The problem features linkage with a well-known conjecture on the existence of infinitely many primes of the form n²+1, as well as our conjecture that (1+¹²)(1+²²)?(1+n²) is not a square for n>3. We present an algorithm that verifies the latter for n?¹⁰³²⁰⁰.