Scaling by Binormalization

We present an iterative algorithm (BIN) for scaling all the rows and columns of a real symmetric matrix to unit 2-norm. We study the theoretical convergence properties and its relation to optimal conditioning. Numerical experiments show that BIN requires 2–4 matrix–vector multiplications to obtain an adequate scaling, and in many cases significantly reduces the condition number, more than other scaling algorithms. We present generalizations to complex, nonsymmetric and rectangular matrices.     

Dynamic light scattering and cryogenic transmission electron microscopy investigations on metallo-supramolecular aqueous micelles: evidence of secondary aggregation

Metallo-supramolecular diblock copolymers consisting of a polystyrene (PS) block connected to a poly(ethylene oxide) (PEO) block by a bis(terpyridine)ruthenium complex (PS₂₀-[Ru]-PEO _y ) were used to prepare aqueous micelles. The length of the PS block was kept constant, while two PEOs of different molecular weight were used. The resulting hydrated micelles and aggregates were characterized by a combination of cryogenic transmission electron microscopy (cryo-TEM) and dynamic light scattering measurements. The results were compared to those obtained for a covalent counterpart (PS₂₂-b-PEO₇₀). Cryogenic transmission electron microscopy allowed visualization of the PS core of the micelles. Moreover, the aggregates result from clustering of individual micelles.     

Supramolecular encapsulation of redox-active monomers to enable free-radical polymerisation

Extended polymeric structures based on redox-active species are of great interest in emerging technologies related to energy conversion and storage. However, redox-active monomers tend to inhibit radical polymerisation processes and hence, increase polydispersity and reduce the average molecular weight of the resultant polymers. Here, we demonstrate that styrenic viologens, which do not undergo radical polymerisation effectively on their own, can be readily copolymerised in the presence of cucurbit[n]uril (CB[n]) macrocycles. The presented strategy relies on pre-encapsulation of the viologen monomers within the molecular cavities of the CB[n] macrocycle. Upon polymerisation, the molecular weight of the resultant polymer was found to be an order of magnitude higher and the polydispersity reduced 5-fold. The mechanism responsible for this enhancement was unveiled through comprehensive spectroscopic and electrochemical studies. A combination of solubilisation/stabilisation of reduced viologen species as well as protection of the parent viologens against reduction gives rise to the higher molar masses and reduced polydispersities. The presented study highlights the potential of CB[n]-based host–guest chemistry to control both the redox behavior of monomers as well as the kinetics of their radical polymerisation, which will open up new opportunities across myriad fields.

Extended polymeric structures based on redox-active species are of great interest in emerging technologies related to energy conversion and storage.

Polyviologens are redox-active polymers based on N-substituted bipyridinium derivatives which have emerged as promising materials for energy conversion and storage.^1–5 Their physicochemical properties can be adjusted through copolymerisation of the redox-active viologen monomers.^6–8 The resultant materials are stable, water soluble and exhibit fast electron transfer kinetics. Polyviologens have been commonly fabricated through step-growth polymerisation in linear and dendritic architectures,^9–13 as supramolecular polymers,^14–16 networks,^6,17,18 and covalent organic frameworks.^19,20 Alternatively, anionic/cationic or metathesis-based polymerisations are used to avoid interference of radical-stabilising monomers with the radical initiators, however, these techniques are highly water- and/or oxygen-sensitive.^21,22 When free-radical polymerisation (FRP) is conducted in the presence of viologen species, its reduction can cause a depletion of active radicals and thus disruption of the polymerisation process. Despite varying solvents, comonomers and initiator loadings, the direct FRP of viologen-containing monomers remains therefore limited to molar masses of 30 kDa.^23–25 Accessing higher molar masses has been possible via post-polymerisation modification,^26–28 which has impacted the electrochemical properties of the resultant materials.^29,30 Alternative strategies to access higher molar masses of redox-active polymers and control their polymerisation are highly desirable.Incorporation of cucurbit[n]uril (CB[n]) macrocycles have lead to a variety of functional materials through host–guest chemistry.^31–34 Moreover, the redox chemistry of viologens can be modulated through complexation with CB[n].^35–38 Specifically, CB[n] (n = 7, 8) can tune the redox potential of pristine viologens and efficiently sequester monoreduced viologen radical cations, avoiding precipitation in aqueous environments. Further to this, we recently demonstrated that the viologen radical cation is stabilised by −20 kcal mol⁻¹ when encapsulated in CB[7].³⁹Consequently, we envisioned that incorporating CB[n]s as additives prior to polymerisation could (i) overcome current limits in accessible molar masses, (ii) increase control over FRP of viologen-based monomers through encapsulation and (iii) enable separation of radical species avoiding aggregation.Here, we demonstrate a new approach to control FRP of redox-active monomers leading to high molar masses and decreased dispersity of the resultant polymers. In absence of CB[n], co-polymerisation of the N-styryl-N′-phenyl viologen monomer 1²⁺ and N,N-dimethylacrylamide (DMAAm) only occurs at high initiator loadings (>0.5 mol%, Fig. 1a), leading to low molecular weights and high polydispersity. Using our synthetic approach, 1²⁺ is efficiently copolymerised with DMAAm in the presence of CB[n] (n = 7, 8) macrocycles resulting in control of the polymer molar mass across a broad range, 4–500 kDa (Fig. 1b). Finally, CB[n] are successfully removed from the polymer via competitive host–guest binding and dialysis. Spectroscopic and electrochemical studies revealed that solubilisation/stabilisation of the reduced species and/or shielding of the redox-active monomers from electron transfer processes was responsible for this enhancement.Open in a separate windowFig. 1Schematic representation of the investigated polymerisation. (a) Conventional free radical polymerisation either completely fails to copolymerise redox-active monomers (low initiator loading) or delivers copolymers with limited molar masses and high dispersities (high initiator loading). (b) CB[n]-mediated protection suppresses interference of viologen monomers with radicals formed through the initiation process facilitating copolymerisation. The molar mass of the resulting copolymers is readily tunable via the amount of present CB[n] macrocycles and the CB[n] is post-synthetically removed via competitive binding to yield the final copolymer with desired molar mass. Cl⁻ counter-ions are omitted for clarity.Recent studies on symmetric aryl viologens demonstrated 2 : 2 binding modes with CB[8] and high binding constants (up to K_a ∼ 10¹¹ M⁻²).^40,41 Incorporation of polymerisable vinyl moieties, in combination with the relatively static structure of their CB[n] host–guest complexes, was postulated to allow polymerisation without unfavorable side reactions. The asymmetric N-styryl-N′-phenyl viologen monomer 1²⁺ prepared for this study (Fig. S1a and S2–S13^†) displays a linear geometry and was predicted to bind CB[n] (n = 7, 8) in a 2 : 1 and 2 : 2 binding fashion (Fig. S1b^†).^40,42 Binding modes between CB[n] (n = 7, 8) and 1²⁺ were investigated through titration experiments (¹H NMR and ITC) which confirmed the formation of 1·(CB[7])₂ and (1)₂·(CB[8])₂ (see Fig. S25 and S26^†). ¹H NMR titration of CB[7] with 1²⁺ demonstrates encapsulation of both aryl moieties (including the vinyl group) through upfield chemical shifts of the respective signals (Fig. 2a). Similar upfield shifts were observed for CB[8] (Fig. 2c). Different para-aryl substituents (vinyl vs. hydrogen) resulted in either head-to-tail or head-to-head (1)₂·(CB[8])₂ dimers (Fig. S1b and S26^†), a previously reported phenomenon.⁴³ Nonetheless, the reversible nature of the complex renders the vinyl group temporarily available for copolymerisation. In the presence of CB[8], 1²⁺ yields polymer molar masses of up to 500 kDa as its complexation is more robust. ITC data confirmed binding stoichiometry, with binding constants of K_a = 2.64 × 10⁶ M⁻¹ for 1·(CB[7])₂ and K_a = 9.02 × 10¹⁰ M⁻² for (1)₂·(CB[8])₂ (Table S2, Fig. S29a and b^†).Open in a separate windowFig. 2Supramolecular complexation of 1²⁺ and CB[n]. ¹H NMR spectra of 1²⁺ at (a) χ_CB[7] = 2, (b) χ_CB[n] = 0 and (c) χ_CB[8] = 1 in D₂O. Cl⁻ counter-ions are omitted for clarity.The free radical copolymerisation of 1²⁺ and DMAAm ([M] = 2 M), in the absence of CB[n], was based on optimised DMAAm homopolymerisations (Fig. S14 and S15^†) and full conversion was confirmed by ¹H NMR spectroscopy (Table S1 and Fig. S16^†). 1²⁺ was maintained at 1 mol% relative to DMAAm and by varying the radical initiator concentration molar masses of up to 30 kDa with broad dispersities (Đ = 11.4) were obtained (Fig. S17^†). Lower initiator concentrations (<0.25 mol%) limited polymerisation (M_n = 3.7 kDa) and size exclusion chromatography elution peaks exhibited extensive tailing, suggesting that 1²⁺ engages in radical transfer processes.To verify our hypothesis that CB[n] macrocycles can modulate the redox behavior of 1²⁺, FRP of 1²⁺ and DMAAm was conducted with varying amounts of CB[n] (n = 7, 8) (Fig. 3, S18 and S20^†). Full conversion of all monomers including their successful incorporation into the polymer was verified via¹H NMR spectroscopy and SEC (Fig. S18 and S21–S23^†). Using CB[7], the molar mass of the copolymers was tunable between M_n = 3.7–160 kDa (Fig. 3b and S21a^†). Importantly, in the presence of CB[8], a broad range of molar masses M_n = 3.7–500 kDa were accessible for 0 < χ_CB[8] < 1.2 (Fig. S20 and S21b^†). Increasing the CB[n] (n = 7, 8) concentration caused dispersity values to converge to Đ = 2.2 (χ_CB[8] = 1.2, χ is the ratio of CB[n] to the redox-active monomer, Fig. S20^†). The copolymers were purified by addition of adamantylamine (competitive binder) prior to dialysis to deliver CB[n]-free redox-active copolymers (Fig. S23^†).Open in a separate windowFig. 3(a) In situ copolymerisation of DMAAm with 1²⁺ and CB[7]. (b) Molar mass and dispersity vs. amount of CB[7] in the system. Fitted curve is drawn to guide the eye. Cl⁻ counter-ions are omitted for clarity.The range of molar masses obtainable through addition of CB[n] (n = 7, 8) correlated with the measured K_a (Fig. 3b and S20^†). Binding of 1²⁺ to CB[8] was stronger and therefore lower concentrations of CB[8] were required to shift the binding equilibrium and mitigate disruption of the polymerisation. Dispersity values reached a maximum at χ_CB[7] = 0.6 or χ_CB[8] = 0.3, suggesting 1⁺˙ is only partially encapsulated. Consequently, higher CB[n] concentrations can enable FRP with lower initiator concentrations (0.10 mol%, Fig. S19^†), which demonstrates the major role of complexation to modulate electron accepting properties of 1²⁺.The redox-active monomer 1²⁺ can engage with propagating primary radicals (P_m˙) to either be incorporated into the growing polymer chain (P_m–1²⁺˙) or to abstract an electron deactivating it (P_m). This deactivation likely occurs through oxidative termination producing 1⁺˙ (energetic sink), inactive oligo- and/or polymer chains (P_m) and a proton H⁺, causing retardation of the overall polymerisation. Oxidative terminations have been previously observed in aqueous polymerisations of methyl methacrylate, styrenes and acrylonitriles that make use of redox initiator systems.^44–47 Another example by Das et al. investigated the use of methylene blue as a retarder, with the primary radical being transferred to a methylene blue electron acceptor via oxidative termination, altogether supporting the outlined mechanism of our system (extended discussion see ESI, Section 1.4^†).⁴⁸The process of retardation can, however, be successfully suppressed, when monomer 1²⁺ is encapsulated within CB[n] macrocycles. Herein the formation of 1·(CB[7])₂ or (1)₂·(CB[8])₂ results in shielding of the redox-active component of 1²⁺ from other radicals within the system, hampering other electron transfer reactions. This inhibits termination and results in extended polymerisation processes leading to higher molar mass polymers through mitigation of radical transfer reactions. Moreover, suppressing the formation of 1⁺˙ through supramolecular encapsulation minimises both π and σ dimerisation of the emerging viologen radical species,³⁹ preventing any further reactions that could impact the molar mass or polydispersity of the resulting polymers.Cyclic voltammetry (CV) and UV-Vis titration experiments were conducted to provide insight into the impact of CB[n] on the redox behavior and control over FRP of 1²⁺. Excess of CB[n] (n = 7, 8) towards 1²⁺ resulted in a complete suppression of electron transfer processes (Fig. S31 and S32^†). Initially, 1²⁺ shows a quasi-reversible reduction wave at −0.44 V forming 1⁺˙ (Fig. 4a). Increasing χ_CB[7], this reduction peak decreases and shifts towards more negative potentials (−0.51 V, χ_CB[7] = 1) accompanied by the formation of 1²⁺·(CB[7])₁. A second cathodic peak emerges at −0.75 V due to the increased formation of 1²⁺·(CB[7])₂. At χ_CB[7] = 2, this peak shifts to −0.80 V, where it reaches maximum intensity, once 1²⁺·(CB[7])₂ is the dominating species in solution. When 2 < χ_CB[7] < 4, the intensity of the reduction peak decreases and the complexation equilibrium is shifted towards the bound state, complete suppression of the reduction peak occurs at χ_CB[7] = 4. Similarly, the oxidation wave intensity is reduced by 95% at χ_CB[7] = 4 causing suppression of potential oxidative radical transfer processes (Fig. 4c).Open in a separate windowFig. 4Mechanism of the CB[n]-mediated (n = 7, 8) strategy for the controlled copolymerisation of redox-active monomer 1²⁺. (a) Cyclic voltammogram with varying amounts of CB[7]. (b) UV-Vis titration of 1²⁺ with varying amounts of CB[7]. (c) Intensity decay of the oxidation peak at −0.27 V and change in absorption maximum of 1⁺˙ at 590 nm vs. χ_CB[7]. (d) Electron transfer processes of 1²⁺ to generate 1⁺˙ and 1⁰. (e) Reduction of 1²⁺ resulting in precipitation of 1⁰. (f) Stabilisation of 1⁺˙ through encapsulation with CB[7]. (g) Protection of 1²⁺ from redox processes through CB[7]-mediated encapsulation.The concentration of 1⁺˙ can be monitored using UV-Vis (Fig. 4b and S34^†).⁴⁹ Absorbance at 590 nm (λ_max) vs. χ_CB[7] was plotted and the concentration of 1⁺˙ increases, reaching a maximum at χ_CB[7] = 4 (Fig. 4c). When χ_CB[7] > 4, a decrease in concentration of 1⁺˙ was observed. We postulate the following mechanism: at χ_CB[7] = 0, 1²⁺ is reduced to produce high concentrations of 1⁺˙ that partially disproportionates to form 1⁰, which precipitates (Fig. 4e and S34^†). When 0 < χ_CB[7] < 4, increasing amounts of green 1⁺˙ are stabilised through encapsulation within CB[7] suppressing disproportionation (Fig. 4c (cuvette pictures), Fig. 4f). For χ_CB[7] > 4, 1²⁺ is protected from reduction through encapsulation (Fig. 4g).To further demonstrate applicability of this strategy, we chose another viologen-based monomer 2²⁺ for copolymerisation (Fig. 5a). As opposed to 1²⁺, CB binds predominantly to the styryl moiety of 2²⁺ (Fig. S27 and S28^†).⁵⁰ ITC data showed that 2²⁺ binds CB[7] in a 1 : 1 fashion with a binding affinity of K_a = 2.32 × 10⁶ M⁻¹ (Fig. S30 and Table S2^†). Monomer 2²⁺ was also analysed via CV and showed three reversible reduction waves at −0.91 V, −0.61 V (viologen) and 0.40 V (styrene). Similar to 1²⁺, excess CB[7] selectively protects the molecule from redox processes, while the vinyl moiety remains accessible (Fig. 5a, S33c and d^†). For CB[8], only partial suppression of electron transfer processes was observed (Fig. S33e and f^†). We therefore chose CB[7] as an additive to increase control over FRP of 2²⁺ (Fig. 5b). Copolymerisation of 2²⁺ (1 mol%) and DMAAm ([M] = 2 M) at χ_CB[7] = 0 resulted in M_n = 28 kDa. When χ_CB[7] = 0.1, 0.2 or 0.3, M_n increased gradually from 124 to 230 and 313 kDa, respectively, demonstrating the potential of this strategy for FRP of redox-active monomers. Higher percentages of CB[7] led to copolymers with presumably higher molar masses causing a drastic decrease in solubility that prevented further analysis. Investigations on a broader spectrum of such copolymers, including those with higher contents of 2²⁺ are currently ongoing.Open in a separate windowFig. 5(a) Cyclic voltammogram of viologen-containing monomer 2²⁺ and its complexation with CB[n] (n = 7, 8) at a concentration of 1 mM using a scan rate of 10 mV s⁻¹ in 0.1 mM NaCl solution. (b) Molar mass and dispersity of 2²⁺-containing copolymers vs. equivalents of CB[7]. Cl⁻ counter-ions are omitted for clarity.In conclusion, we report a supramolecular strategy to induce control over the free radical polymerisation of redox-active building blocks, unlocking high molar masses and reducing polydispersity of the resulting polymers. Through the use of CB[n] macrocycles (n = 7, 8) for the copolymerisation of styrenic viologen 1²⁺, a broad range of molar masses between 3.7–500 kDa becomes accessible. Our mechanistic investigations elucidated that the redox behavior of monomer 1²⁺ is dominated by either CB[n]-mediated stabilisation of monoradical cationic species or protection of the encapsulated pyridinium species from reduction. In the stabilisation regime (χ_CB[7] < 4), 1²⁺ is reduced to form the radical cation 1⁺˙, which is subsequently stabilised through CB[7] encapsulation. Upon reaching a critical concentration of CB[7] (χ_CB[7] > 4), the system enters a protection-dominated regime, where reduction of 1²⁺ is suppressed and the concentration of 1⁺˙ diminishes. The resulting copolymers can be purified by use of a competitive binder to remove CB[n] macrocycles from the product. This strategy was successfully translated to a structurally different redox-active monomer that suffered similar limitations. We believe that the reported strategy of copolymerisation of redox-active monomers will open new avenues in the synthesis of functional materials for energy conversion and storage as well as for applications in electrochromic devices and (nano)electronics.     

Aerobic Iron‐Based Cross‐Dehydrogenative Coupling Enables Efficient Diversity‐Oriented Synthesis of Coumestrol‐Based Selective Estrogen Receptor Modulators

An iron‐based cross‐dehydrogenative coupling (CDC) approach was applied for the diversity‐oriented synthesis of coumestrol‐based selective estrogen receptor modulators (SERMs), representing the first application of CDC chemistry in natural product synthesis. The first stage of the two‐step synthesis of coumestrol involved a modified aerobic oxidative cross‐coupling between ethyl 2‐(2,4‐dimethoxybenzoyl)acetate and 3‐methoxyphenol, with FeCl₃ (10 mol %) as the catalyst. The benzofuran coupling product was then subjected to sequential deprotection and lactonization steps, affording the natural product in 59 % overall yield. Based on this new methodology other coumestrol analogues were prepared, and their effects on the proliferation of the estrogen receptor (ER)‐dependent MCF‐7 and of the ER‐independent MDA‐MB‐231 breast cancer cells were tested. As a result, new types of estrogen receptor ligands having an acetamide group instead of the 9‐hydroxyl group of coumestrol were discovered. Both 9‐acetamido‐coumestrol and 8‐acetamidocoumestrol were found more active than the natural product against estrogen‐dependent MCF‐7 breast cancer cells, with IC₅₀ values of 30 and 9 nM , respectively.     

Lensless diffractive imaging using tabletop coherent high-harmonic soft-X-ray beams

We present the first experimental demonstration of lensless diffractive imaging using coherent soft x rays generated by a tabletop soft-x-ray source. A 29 nm high harmonic beam illuminates an object, and the subsequent diffraction is collected on an x-ray CCD camera. High dynamic range diffraction patterns are obtained by taking multiple exposures while blocking small-angle diffraction using beam blocks of varying size. These patterns reconstruct to images with 214 nm resolution. This work demonstrates a practical tabletop lensless microscope that promises to find applications in materials science, nanoscience, and biology.     

Antimicrobial silica particles loaded with quaternary ammonium polyethyleneimine network

Silica particles functionalized with quaternary ammonium groups were prepared by interpenetrating polyethylenimine (PEI) into silica particles and crosslinking with diiodopentane, followed by octyliodide alkylation and methyliodide quaternarization (S‐QA‐PEI). The synthesized S‐QA‐PEI particles were identified with a slight particle size increase of 2–3 µm. Different ratios of PEI:silica particles were prepared and analyzed. While silica particles are negatively charged, ?16.7 ± 5.11 mV, the prepared S‐QA‐PEI particles are positively charged, +50–60 mV. These particles were embedded in poly(ethylene vinyl acetate) and poly(ethylene methacrylic acid) coatings which exhibited strong antibacterial activity. Copyright © 2014 John Wiley & Sons, Ltd.     

Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons

We present an experimental study on wave propagation in highly nonlocal optically nonlinear media, for which far-away boundary conditions significantly affect the evolution of localized beams. As an example, we set the boundary conditions to be anisotropic and demonstrate the first experimental observation of coherent elliptic solitons. Furthermore, exploiting the natural ability of such nonlinearities to eliminate azimuthal instabilities, we perform the first observation of stable vortex-ring solitons. These features of highly nonlocal nonlinearities affected by far-away boundary conditions open new directions in nonlinear science by facilitating remote control over soliton propagation.     

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics. Partially supported by Minerva Grant No. 8441 and by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272. Partially supported by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.     

Combinatorial sums,identities and trace identities of the 2 × 2 matrices

An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions c_n(F₂) and the trace codimensions t_n(F₂) of F₂, the 2 × 2 matrices, are asymptotically equal: $\text{c}{}_{\mathrm{n}}\text{(F}{}_2\text{) ? t}{}_{\mathrm{n}}\text{(F}{}_2\text{)}$.