Multivariate Composition Distribution in Free‐Radical Multicomponent Polymerization, 1

Statistical multicomponent polymerization is a typical example of a Markovian process for which the generating function approach can be applied. Up to the present, generating functions have been used mainly to obtain analytical solutions. However, recent advances of computer software capable of handling symbolic calculations can throw new light on the old mathematical technique. After formulating the equations representing the instantaneous composition distribution of polymers for a given chain length, r, the illustrative numerical calculations are conducted by using the symbolic calculator. For a multicomponent polymerization consisting of more than two components, the second component distribution is dependent on the composition of the first component (F₁), which is represented by the conditional probability given r and F₁, . It is found that is well approximated by the Gaussian distribution with the variance following the relationship, , as in the case of the first component distribution , where A and B are the constants. With the knowledge of chain length distribution, it is now possible to conduct the full analysis of multivariate distribution of chain length and compositions for multicomponent free‐radical polymerization.

Bivariate distribution of composition F₁ and F₂ for chain length r = 100 in a three‐component system.     

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

Let ${\mathbb{B}}_H$ be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ₂. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in $\mathbb{C}$ and are sharp in the case with values in ℓ₂.     

Size exclusion chromatography of branched polymers: Star and comb polymers

Monte Carlo simulations were conducted to estimate the elution curve of size exclusion chromatography (SEC). The present simulation can be applied to various types of branched polymers, as long as the kinetic mechanism of nonlinear polymer formation is given. We considered two types of detector systems, (1) a detector that measures the polymer concentration in the elution volume to determine the calibrated molecular weights, such as by using the differential refractive index detector (RI), and (2) a detector that determines the weight‐average molecular weight of polymers within the elution volume directly, such as a light scattering photometer (LS). For polydisperse star polymers, both detector systems tend to give a reasonable estimate of the true molecular weight distribution (MWD). On the other hand, for comb‐branched polymers, the RI detector underestimates the molecular weight of branched polymers significantly. The LS detector system improves the measured MWD, but still is not exact. The present simulation technique promises to establish various types of complicated reaction mechanisms for nonlinear polymer formation by using the SEC data quantitatively. In addition, the present technique could be used to reinvestigate a large amount of SEC data obtained up to the present to estimate the true MWD.     

The pummerer and the thio-claisen–type rearrangements of naphtho[1,8-bc]-1,5-dithiocin monooxide and N-p-tosylsulfilimine

Naphtho[1,8-bc]-1,5-dithiocin N-p-tosylsulfilimine ( 8 ) and monosulfoxide ( 9 ) were prepared. On treatment with conc. H₂SO₄, both the sulfilimine ( 8 ) and sulfoxide ( 9 ) gave the dithia dication which was converted to the sulfoxide by hydrolysis. The H–D exchange reaction of ( 8 ) took place highly regioselectively to afford the monodeuterated ( 8–D ) at the a-position of the N-tosyl group. The Pummerer rearrangement reaction of monooxide ( 9 ) with acetic anhydride gave the a-acetoxy derivative by the dication ( 10b ), while a new thio-Claisen rearrangement of sulfilimine ( 8 ) t-BuOK in CH₂Cl₂ gave 2-allyl-naphtho[1,8-bc]-1,5-dithiole. © John Wiley & Sons, Inc.     

Superconductivity in an Orbital-Reoriented SnAs Square Lattice: A Case Study of Li0.6Sn2As2 and NaSnAs

Searching for functional square lattices in layered superconductor systems offers an explicit clue to modify the electron behavior and find exotic properties. The trigonal SnAs₃ structural units in SnAs-based systems are relatively conformable to distortion, which provides the possibility to achieve structurally topological transformation and higher superconducting transition temperatures. In the present work, the functional As square lattice was realized and activated in Li_0.6Sn₂As₂ and NaSnAs through a topotactic structural transformation of trigonal SnAs₃ to square SnAs₄ under pressure, resulting in a record-high T_c among all synthesized SnAs-based compounds. Meanwhile, the conductive channel transfers from the out-of-plane p_z orbital to the in-plane p_x+p_y orbitals, facilitating electron hopping within the square 2D lattice and boosting the superconductivity. The reorientation of p-orbital following a directed local structure transformation provides an effective strategy to modify layered superconducting systems.     

Discovery of Two-dimensional Hexagonal MBene HfBO and Exploration on its Potential for Lithium-Ion Storage

The practical applications of two-dimensional (2D) transition-metal borides (MBenes) have been severely hindered by the lack of accessible MBenes because of the difficulties in the selective etching of traditional ternary MAB phases with orthorhombic symmetry (ort-MAB). Here, we discover a family of ternary hexagonal MAB (h-MAB) phases and 2D hexagonal MBenes (h-MBenes) by ab initio predictions and experiments. Calculations suggest that the ternary h-MAB phases are more suitable precursors for MBenes than the ort-MAB phases. Based on the prediction, we report the experimental synthesis of h-MBene HfBO by selective removal of In from h-MAB Hf₂InB₂. The synthesized 2D HfBO delivered a specific capacity of 420 mAh g⁻¹ as an anode material in lithium-ion batteries, demonstrating the potential for energy-storage applications. The discovery of this h-MBene HfBO added a new member to the growing family of 2D materials and provided opportunities for a wide range of novel applications.     

Monte Carlo simulation of size exclusion chromatography for branched polymers formed through free‐radical polymerization with chain transfer to polymer

The elution curves of size exclusion chromatography (SEC) for branched polymers formed through free‐radical polymerization that involves chain transfer to polymer were theoretically investigated by using a Monte Carlo method. We considered two types of measured molecular weight distribution (MWD), (1) the calibrated MWD relative to standard linear polymers, and (2) the MWD obtained by using a light scattering photometer (LS) in which the weight‐average molecular weight of polymers within the elution volume is determined directly. It was found that the calibrated MWD clearly underestimates the high molecular weight tail, and the measured distributions are narrower than the true MWD. On the other hand, the present simulation results showed that the LS method gives reasonable estimates of the true MWDs. The mean square radius of gyration of the polymer molecules having the same molecular weight was also investigated. The radii of gyration showed clear deviation from the Zimm‐Stockmayer equation^[1] because of the non‐random nature of branched structure and the difference in the primary chain length distribution.