How do we understand mathematical practices in non-mathematical fields? Reflections inspired by cases from 12th and 13th century China

Recent studies of the history of mathematics have shed light on the diversity of mathematical practices in the ancient world. In this article, I address this issue from two perspectives. First, I analyze different mathematical instruments used in different domains in 12^th and 13^th century China. Second, by analyzing how the text is related to the problems written down in the same period, I argue that there existed two categories of mathematical problems. As a result, I suggest that problems, textual procedures, and material operations should be considered together when regarding various mathematical practices in mathematical and non-mathematical fields.     

Formation of porous polymer morphology by microsyneresis during divinylbenzene polymerization

This article describes the investigation of the importance of various reaction conditions on microsyneretic pore formation during polymerization of divinylbenzene (DVB) under so‐called “solvothermal” conditions. To induce microsyneretic pore formation, the most important parameter is an unusually high dilution of monomers with a “good” porogen solvating the polymer chains. High dilution and solvation of the growing poly(DVB) chains promote the prolongation of the polymer chains rather than their interconnection by crosslinking. Consequently, when the polymer gel density reaches the point where syneresis starts, the polymer network is geometrically too extensive to be broken up into precipitating entities and, instead, porogen droplets are formed within the continuous polymer gel. The pore geometry created by microsyneresis offers high surface area in wide mesopores and hence, high capacity for supporting functional groups or reactions with much better accessibility than narrow pores between polymer microspheres produced by macrosyneresis in conventional styrenic polymer supports. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 774–781     

Synthesis of partially fluorinated poly(arylene ether sulfone) multiblock copolymers bearing perfluorosulfonic functions

Partially fluorinated poly(arylene ether sulfone) multiblock copolymers bearing perfluorosulfonic functions (ps‐PES‐FPES), with ionic exchange capacity (IEC) ranging between 0.9 and 1.5 meq H⁺/g, are synthesized by regioselective bromination of partially fluorinated poly(arylene ether sulfone) multiblock copolymers (PES‐FPES), followed by Ullman coupling reaction with lithium 1,1,2,2‐tetrafluoro‐2‐(1,1,2,2‐tetrafluoro‐2‐iodoethoxy)ethanesulfonate. The PES‐FPES are prepared by aromatic nucleophilic substitution reaction by an original approach, that is, “one pot two reactions synthesis.” The chemical structures of polymers are analyzed by ¹H and ¹⁹F NMR spectroscopy. The resulted ionomers present two distinct glass transitions and α relaxations revealing phase separation between the hydrophilic and the hydrophobic domains. The phase separation is observed at much lower block lengths of ps‐PES‐FPES as compared with the literature. AFM and SANS observations supported the phase separation, the hydrophilic domains are well dispersed but the connectivity to each other depends on the ps‐PES block lengths. The thermomechanical behavior, the water up‐take, and the conductivity of the ps‐PES‐FPES membranes are compared with those of Nafion 117^® and randomly functionalized polysulfone (ps‐PES). Conductivities close or higher to those of Nafion 117^® are obtained. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 1941–1956     

Effect of additive length and chemistry on the morphology of blends of conjugated thiophenes and fullerene derivative acceptor molecules

Small molecule additives have been shown to increase the device efficiency of conjugated polymer (donor) and fullerene derivative (acceptor) based organic solar cells by modifying the morphology of the device active layer. In this paper we conduct a systematic study of how additives affect the donor‐acceptor morphology using molecular dynamics simulations of blends of thiophene‐based oligomers, mimicking poly(3‐dodecylthiophene) (P3DDT) or poly(2,2′:5′,2”‐3,3”‐didocyl‐terthiophene) (PTTT), and fullerene derivatives with additives of varying length and chemical functionalization, mimicking experimentally used additives like methyl ester additives, diiodooctane, and alkanedithiols. We find that functionalization of additives with end groups that are attracted to acceptor molecules are necessary to induce increased donor‐acceptor macrophase separation. In blends where acceptors intercalate between oligomer alkyl side chains, functionalized additives decrease acceptor intercalation. Functionalized additives with shorter alkyl segments increase acceptor macrophase separation more than additives with same chemical functionalization but longer alkyl segments. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1046–1057     

Charge transport properties of MDMO PPV thin films cast in different solvents

Charge transport properties in thin films of Poly(2‐methoxy‐5‐(3′,7′‐dimethyloctyloxy)‐1,4‐phenylenevinylene) (MDMO PPV) cast using either chloroform (CF), toluene (TOL), or chlorobenzene (CB) as solvent were investigated. Hole mobility (μ) in these thin films measured using time‐of‐flight transient photoconductivity showed an increasing trend with respect to the solvent used in the same order, that is, μ_CF (2.4 × 10^?7 cm²/Vs) < μ_TOL (6.9 × 10^?7 cm²/Vs) < μ_CB (2.3 × 10^?6 cm²/Vs). Observed variations in mobilities were attributed to different morphologies of MDMO PPV chains in thin films cast using the aforesaid solvents. Nature of the interchain interactions and aggregate formation were obtained using photoluminescence (PL), Raman spectroscopy, and AFM studies. Ratio of PL peak intensities of 0–0 and 0–1 transitions, which is a direct measure of interchain interaction, was the highest in CB and lowest in CF. Variation in the relative intensities of out‐of‐plane wagging of vinylene group (～963 cm^?1 mode) in Raman spectra suggested different extent of coiling of polymer chains in these thin films. From these observations, it was elicited that aggregate size and interchain interactions are highest in CB and least in CF. AFM‐based topographic images of thin films further supported these variations in the size of aggregates. Variation in the aggregate sizes and interchain interactions explained the corresponding variation in the mobility. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1431–1439     

Morphology and microphase separation of star copolymers

Star copolymers have attracted significant interest due to their different characteristics compared with diblock copolymers, including higher critical micelle concentration, lower viscosity, unique spatial shape, or morphologies. Development of synthetic skills such as anionic polymerization and controlled radical polymerization have made it possible to make diverse architectures of polymers. Depending on the molecular architecture of the copolymer, numerous morphologies are possible, for instance, Archimedean tiling patterns and cylindrical microdomains at symmetric volume fraction for miktoarm star copolymers as well as asymmetric lamellar microdomains for star‐shaped copolymers, which have not been reported for linear block copolymers. In this review, we focus on morphologies and microphase separations of miktoarm (A_mB_n and ABC miktoarm) star copolymers and star‐shaped [(A‐b‐B)_n] copolymers with nonlinear architecture. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1–21     

Effects of top confinement and diluents on morphology in crystallization of poly(l‐lactic acid) interacting with poly(ethylene oxide)

Effects of top confinement and diluent poly(ethylene oxide) (PEO) on poly(l ‐lactic acid) (PLLA) crystal morphology have been investigated. When crystallized at 120 °C, uncovered neat PLLA sample exhibits higher growth rate ringless spherulites; while the covered sample exhibits lower growth rate ring‐banded spherulites. As PEO is introduced into PLLA, the morphology also undergoes significant changes. For the same T_c,PLLA = 120 °C, the PEO/PLLA blend with PEO composition greater than 25% exhibits ring‐banded patterns even in uncovered sample. However, in much greater PEO composition (>80 wt %), uncovered samples exhibit ring bands diverging into dendritic patterns, while top covered samples tend to maintain the spiral ring‐band patterns. Both PEO inclusion in PLLA and top cover on films impose growth kinetic alterations. Additionally, the top glass cover tends to prevent the lower surface tension PLLA to be accumulated on the surface, resulting in the formation of ring‐band pattern. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1160–1170     

Chemorheology investigation of a glassy epoxy thermoset on tensile plastic flow and fracture morphology

Reproducible and uncharacteristic tensile stress–strain behavior of cured glassy epoxy‐amine networks produces distinctive fracture surfaces. Test specimens exhibiting plastic flow result in mirror‐like fracture surfaces, whereas samples that fail during yield or strain softening regions possess nominal mirror‐mist‐hackle topography. Atomic force microscopy and scanning electron microscopy reveal branched nodule morphologies in the 50‐nm size scale that may be responsible for the unusual tensile properties. Current hypothesis is that plastic flow of the glassy thermoset occurs through the existence and deformation of these nodular nanostructures. The thermal cure profile of the epoxy‐amine thermoset affects the size and formation of the nodular nanostructure. Eliminating vitrification during thermoset polymerization forms a more continuous phase, reduction in size of the nodules, and eliminates the capacity of the material to yield in plastic flow. This maximizes nanostructure connectivity of the glassy epoxy‐amine thermoset and reduces strain to failure significantly. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1333–1344.     

Behavior of poly(ε-caprolactone)-based diblock copolymers in a bulk state and in solution

Abstract

A series of amphiphilic MOPEO-b-PCL copolymers (DBCs), based on biocompatible methoxypoly(ethylene oxide) with M_n=4.5?kDa and poly(ε-caprolactone) of a variable chain length, was synthesized by an anionic ring-opening block copolymerization. The structural investigations, performed by DSC and WAXS methods, revealed the microphase separation in DBC bulk structure and the existence of separate amorphous regions and microcrystalline domains of MOPEO and PCL blocks. Spectrophotometry and SLS were used to study the self-assembling of DBC macromolecules in selective dioxane/aqueous solution and to determine the main micellization parameters (CMC and -ΔG°). The DBC micelles morphology and their specific aggregation in mixed solvent were shown.     

Thermoset modified with polyethersulfone: Characterization and control of the morphology

Thermoset (TS) epoxy resins can be toughened with a thermoplastic (TP) for high-performance applications. The final structure morphology has to be controlled to achieve high mechanical properties and high impact resistance. Four polyethersulfone-modified epoxy resins are considered. They consist of different epoxy monomer structure (TGAP, triglycidyl-p-aminophenol and TGDDM, tetraglycidyl diaminodiphenylmethane) and a fixed amount of thermoplastic, and they are cured with two different amounts of curing agent. A reaction-induced phase separation occurs for all formulations generating morphologies, different in shapes and scales. The aim is to control the final morphology and in particular its dominant length scale. This morphology depends on the phase separation process, from the initiation to its final stage. The initiation relies on the relative miscibility of the components and on the stoichiometry between epoxy and curing agent. The kinetics depends on the viscosity of the systems. The different morphologies are characterized by electron microscopy or neutron scattering. Dynamic mechanical analysis allows confirming the presence of a phase separation even when it is not observable by electron microscopy. Vermicular morphologies with few hundreds nanometer width are obtained for the systems containing the TGAP as epoxy monomer. Systems formulated with TGDDM presents morphologies on much smaller scale of order a few tens of nanometers. We interpret the different sizes of the morphologies as a consequence of a larger viscosity for the TGDDM systems as compared to the TGAP ones rather than by a latter initiation of phase separation.