Incorporating glycidyl methacrylate into block copolymers using poly(methacrylate‐ran‐styrene) macroinitiators synthesized by nitroxide‐mediated polymerization

Methyl methacrylate/styrene (MMA/S), ethyl methacrylate/styrene (EMA/S) and butyl methacrylate/styrene (BMA/S) feeds (>90 mol % methacrylate) were copolymerized in 50 wt % p‐xylene at 90 °C with 10 mol % of additional SG1‐free nitroxide mediator relative to unimolecular initiator (BlocBuilder^®) to yield methacrylate rich copolymers with polydispersities _w/ _n = 1.23–1.46. k_pK values (k_p = propagation rate constant, K = equilibrium constant) for MMA/S copolymerizations were comparable with previous literature, whereas EMA/S and BMA/S copolymerizations were characterized by slightly higher k_pK's. Chain extensions with styrene at 110 °C initiated by the methacrylate‐rich macroinitiators (number average molecular weight _n = 12.9–33.5 kg mol^?1) resulted in slightly broader molecular weight distributions with _w/ _n = 1.24–1.86 and were often bimodal. Chain extensions with glycidyl methacrylate/styrene/methacrylate (GMA/S/XMA where XMA = MMA, EMA or BMA) mixtures at 90 °C using the same macroinitiators resulted frequently in bimodal molecular weight distributions with many inactive macroinitiators and higher _w/ _n = 2.01–2.48. P(XMA/S) macroinitiators ( _n = 4.9–8.9 kg mol^?1), polymerized to low conversion and purified to remove “dead” chains, initiated chain extensions with GMA/MMA/S and GMA/EMA/S giving products with _w/ _n ～ 1.5 and much fewer unreacted macroinitiators (<5%), whereas the GMA/BMA/S chain extension was characterized by slightly more unreacted macroinitiators (～20%). © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2574–2588, 2009     

Amphiphilic Poly(4‐acryloylmorpholine)/Poly[2‐(N‐carbazolyl)ethyl acrylate] Random and Block Copolymers Synthesized by NMP

A series of random copolymers and block copolymers containing water‐soluble 4AM and fluorescent VAK are synthesized by NMP. The homopolymerizations of 4AM and VAK and 4AM/VAK random copolymerization are performed in 50 wt% DMF using 10 mol% SG1, resulting in a linear increase in versus conversion, and final polymers with narrow molecular weight distributions ( < 1.4). Reactivity ratios r_VAK = 0.64 ± 0.52 and r_4AM = 0.86 ± 0.66 are obtained for the 4AM/VAK random copolymerization. In addition, a poly(4AM) macroinitiator is used to initiate a surfactant‐free suspension polymerization of VAK. After 2.5 h, the resulting amphiphilic block copolymer has = 12.6 kg · mol^?1, = 1.48, molar composition F_VAK = 0.38 with latex particle sizes between 270 and 475 nm.

     

Direct Cyclodextrin‐Mediated Ring Opening Polymerization of ϵ‐Caprolactone in the Presence of Yttrium Trisphenolate Catalyst

Unmodified β‐cyclodextrin has been directly used to initiate ring‐opening polymerization of ϵ‐caprolactone in the presence of yttrium trisphenolate. Well‐defined cyclodextrin (CD)‐centered star‐shaped poly(ϵ‐caprolactone)s have been successfully synthesized containing definite average numbers of arms (N_arm = 4–6) and narrow polydispersity indexes (below 1.10). The number‐average molecular weight ( ) and average molecular weight per arm ( ) are controlled by the feeding molar ratio of monomer to initiator. The prepared star‐PCL with of 2.7 × 10³ is in fully amorphous and that with of 13.3 × 10³ is crystallized. In addition, the obtained poly(e‐caprolactone) (PCL) stars with various molecular weights have different solubilities in methanol and tetrahydrofuran, which can be applied for further modifications.     

Macrocyclic and Polymeric Oxaziridine‐Derivatives

Macrocyclic and polymeric imines 5,5′ and 6,6′ are obtained in excellent yields by template‐free polycondensation of 1,6‐bis(4‐formylbenzoyloxy)hexane (1) with commercially available 4,4′‐methylene‐bis(cyclohexylamine) (2) and with bis(2‐amino‐2‐methylprop‐1‐yl)adipate dihydrochloride (4), respectively. The degree of macrocyclization during imine synthesis strongly depends on the diamine. Matrix‐assisted laser desorption–ionization time‐of‐flight (MALDI‐TOF) mass spectrometry analysis and gel permeation chromatography (GPC) measurements show that (2) leads to more macrocyclic adducts than (4). The subsequent meta‐chloroperoxybenzoic acid oxidation of polyimines 5,5′ and 6,6′ ( = 1650–11 200 g mol⁻¹, = 3800–27 350 g mol⁻¹) yields the corresponding polyoxaziridines 7,7′ and 8,8′ consisting of macrocyclic and linear polymeric structures ( = 1750–8050 g mol⁻¹, = 3250–15 800 g mol⁻¹). The synthesized polyoxaziridines are relatively stable and storable at room temperature.     

A Helical Poly(macromonomer) Consisting of a Polyacetylene Main Chain and Polystyrene Side Chains

A novel helical poly(macromonomer) [poly(M‐PS): absolute = 82 800–252 000, determined by GPC/RALLS] with a polyacetylene main chain and polystyrene (PS) side chains was synthesized by the polymerization of acetylene‐terminated M‐PS [ = 2 000, / = 1.20, = 18] with an Rh catalyst. M‐PS was prepared by ATRP of styrene using the acetylene‐containing initiator 2‐bromo‐2‐methylpropionic acid (S)‐1‐methylpropargyl ester ( l ). In solutions, poly(M‐PS) exhibited an intense CD signal at 345–355 nm, indicating that it possessed a predominantly one‐handed helical conformation. Poly(M‐PS) had a stable helical conformation irrespective of solvents and temperature.

     

Synthesis of Cyclopentadienyl Capped Polyethylene and Subsequent Block Copolymer Formation Via Hetero Diels‐Alder (HDA) Chemistry

In the current contribution it is demonstrated – for the first time – that poly(ethylene) ( = 1 400 as well as 2 800 g · mol⁻¹, PDI = 1.2) can be readily equipped with highly reactive cyclopentadienyl (Cp) end groups. The Cp terminal poly(ethylene) can subsequently be reacted in an efficient hetero Diels‐Alder (HDA) reaction with macromolecules (poly(isobornyl acrylate) ( = 4 600 g · mol⁻¹, PDI = 1.10) and poly(styrene) ( = 6 300 g · mol⁻¹, PDI = 1.13) featuring strongly electron withdrawing thiocarbonyl thio end groups, prepared via reversible addition fragmentation chain transfer (RAFT) polymerization employing benzylpyridin‐2‐yldithioformate (BPDF) as transfer agent. The resulting block copolymers have been analyzed via high‐temperature size exclusion chromatography (SEC) as well as nuclear magnetic resonance (NMR) spectroscopy. The current system allows for the removal of the excess of the non‐poly(ethylene) containing segment via filtration of the poly(ethylene)‐containing block copolymer. However, the reaction temperatures need to be judiciously selected. Characterization of the generated block copolymers at elevated temperatures can lead – depending on the block copolymer type – to the occurrence of retro Diels‐Alder processes. The present study thus demonstrates that RAFT‐HDA ligation can be effectively employed for the generation of block copolymers containing poly(ethylene) segments.

     

Naked and Self‐Clickable Propargylic‐Decorated Single‐Chain Nanoparticle Precursors via Redox‐Initiated RAFT Polymerization

Protection of acetylenic monomers is a common practice to avoid parasitic side reactions during polymerization. Herein, we report that redox‐initiated RAFT polymerization allows the direct, room temperature synthesis of a variety of single‐chain nanoparticle precursors (displaying narrow molecular weight dispersity, / = 1.12 –1.37 up to = 100 kDa) containing well‐defined amounts of naked, unprotected acetylenic functional groups available for rapid and quantitative intrachain cross‐linking via metal‐catalyzed carbon–carbon coupling (i.e., C–C “click” chemistry). To illustrate the useful “self‐clickable” character of the new unprotected acetylenic precursors, single‐chain nanoparticles have been prepared for the first time in a facile and highly efficient manner by copper‐catalyzed alkyne homocoupling (i.e., Glaser–Hay coupling) at room temperature under normal air atmosphere.     

Determination of thermodynamic properties of macroporous glycidyl methacrylate‐based copolymers by inverse gas chromatography

Macroporous crosslinked poly(glycidyl methacrylate‐co‐ethylene glycol dimethacrylate) (PGME) was synthesized by suspension copolymerization and modified by ring‐opening reaction of the pendant epoxy groups with ethylene diamine (EDA). Inverse gas chromatography (IGC) at infinite dilution was applied to determine the thermodynamic interactions of PGME and modified copolymer, PGME‐en. The specific surface areas of the initial and modified copolymer samples were determined by the BET method, from low‐temperature nitrogen adsorption isotherms. The specific retention volumes, V, of 10 organic compounds of different chemical nature and polarity (nonpolar, donor, or acceptor) were determined in the temperature range 333–413 K. The weight fraction activity coefficients of test sorbates, , and Flory–Huggins interaction parameters, , were calculated and discussed in terms of interactions of sorbates with PGME and PGME‐en. Also, the partial molar free energy, , partial molar heat of mixing, , sorption molar free energy, ΔG, sorption enthalpy ΔH, and sorption entropy, ΔS, were calculated. Glass transitions in PGME and PGME‐en, determined from IGC data, were observed in the temperature range 373–393 K and 363–373 K, respectively. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2524–2533, 2005     

A Fully Substituted 3‐Silolene Functions as Promising Building Block for Hyperbranched Poly(Silylenevinylene)

A 3‐silolene derivative, 2,2,5,5‐tetrakis(dimethylsilyl)‐1,1‐dimethyl‐3,4‐diphenyl‐3‐silolene (TDMSHS), is first synthesized and characterized by X‐ray diffraction crystallography and spectroscopic methods. Hydrosilylation polymerization of TDMSHS with 1,1‐dimethyl‐2,5‐bis(4‐ethynylphenyl)‐3,4‐diphenylsilole in the presence of Karstedt's catalyst generates a stereoregular silole‐containing hyperbranched poly(silylenevinylene) (hb‐SPSV) with a high molecular weight ( = 146 000, / = 1.5) in high yield (≈95%). hb‐SPSV exhibits excellent thermal stability and strong fluorescence, and the emission of its aggregates in aqueous mixture can be quenched efficiently by picric acid with large quenching constants K_SV up to 414400 M ⁻¹.     

Controlled Synthesis of a Chitosan‐Based Graft Copolymer Having Polysarcosine Side Chains Using the NCA Method with a Carboxylic Acid Additive

Summary: A novel chitosan derivative with polysarcosine side chains, i.e., chitosan‐graft‐polysarcosine [chitosan‐graft‐poly(N‐methylglycine)], was synthesized by ring‐opening polymerization of sarcosine N‐carboxyanhydride (NCA) with chitosan as a macroinitiator in the presence of carboxylic acids in dimethyl sulfoxide at 27 °C. Degree of substitution ( ) for polysarcosine side chains introduced to chitosan was controlled successfully by the feed amount of the additive nicotinic acid ( = 0.21–0.71). Independent of control, degree of polymerization ( ) for polysarcosine side chains was controlled by adjusting feed ratios of NCA monomer to chitosan ( = 14–43). Kinetic analysis of the propagation of sarcosine NCA was conducted by measuring CO₂ evolution. Apparent k_p values decreased with increased feed amounts of nicotinic acid, supporting the theory that propagation of NCA in the presence of nicotinic acid proceeds via equilibrium between active amine and dormant ammonium species.

Propagation mechanism of carboxylic acid‐mediated polymerization of sarcosine N‐carboxyanhydride.     

Palladium(II) and platinum(II) complexes of (1R,2R)‐(−)‐1,2‐diaminocyclohexane (DACH) with various carboxylato ligands and their cytotoxicity evaluation

Several palladium(II) and platinum(II) complexes analogous to oxaliplatin, bearing the enantiomerically pure (1R,2R)‐(?)‐1,2‐diaminocyclohexane (DACH) ligand, of the general formula {MX₂[(1R,2R)‐DACH]}, where M = Pd or Pt, X (COO)₂, CH₂(COO)₂, , , {1,1′‐C₅H₈(CH₂COO)₂}, [1,1′‐C₆H₁₀(CH₂COO)₂], [1,1′‐(COO)₂ferrocene], , , , MeCOO and Me₃CCOO, were synthesized. All the complexes prepared were characterized physicochemically and spectroscopically. Some selected complexes were screened in vitro against several tumor cell lines and the results were compared with reference standard drug, oxaliplatin. Copyright © 2009 John Wiley & Sons, Ltd.     

Multilayer Assemblies Consisting of Tri‐Vanadium‐Substituted Heteropolyanions and Its Electrocatalytic Properties

We describe the controlled fabrication of ultrathin multilayer films consisting of tri‐vanadium‐ substituted heteropolytungstate anions (denoted as P₂W₁₅V₃) and a cationic polymer of quaternized poly (4‐vinylpyridine) partially complexed with osmium bis(2,2′‐bipyridine) (denoted as QPVP‐Os) on the 4‐aminobenzoic acid (4‐ABA) modified glassy carbon electrode (GCE) surface based on layer‐by‐layer assembly. Cyclic voltammetry and UV‐vis absorption spectrometry have been used to easily monitor the thickness and uniformity of thus‐formed multilayer films. The V‐centered redox reaction of P₂W₁₅V₃ in the multilayer films can effectively catalyze the reduction of BrO and NO . The resulting P₂W₁₅V₃/QPVP‐Os multilayer film modified electrode behaves as a much promising electrochemical sensor because of the low overpotential for the catalytic reduction of BrO and NO , and the catalytic oxidation of ascorbic acid.     

Towards Clathrates: Frozen States of Hydration of tert‐Butylamine

The dilution of tert‐butylamine (tBA) with water and subsequent cooling leads to a large series of different crystalline hydrates by an in situ IR laser melting‐zone procedure. The crystal structures were determined for tBA?n H₂O, with n=0, , 1, 7 , 7 , 9 , 11, and 17. For the two lower hydrates (n= , 1), one‐ and two‐dimensional hydrogen‐bonded networks are formed, respectively. The higher hydrates (n>1) exhibit a clathrate‐like three‐dimensional water framework with the tBA molecules as part of, or sitting inside, the cages. In all cases, tBA is hydrogen‐bonded to the H₂O framework. In the intermediate range (1相似文献   

New Strategy Targeting Well‐Defined Polymethylene‐block‐Polystyrene Copolymers: The Combination of Living Polymerization of Ylides and Atom Transfer Radical Polymerization

Well‐defined polymethylene‐block‐polystyrene (PM‐b‐PS) diblock copolymers were synthesized via a combination of living polymerization of ylides and atom transfer radical polymerization (ATRP) of styrene. A series of hydroxyl‐terminated polymethylenes (PM‐OHs) with different molecular weight and narrow molecular weight distribution were prepared using living polymerization of ylides following efficient oxidation in a quantitive functionality. Then, the macroinitiators (PM‐MIs ( = 1 900–15 000; PDI = 1.12–1.23)) transformed from PM‐OHs in ≈ 100% conversion initiated ATRPs of styrene to construct PM‐b‐PS copolymers. The GPC traces indicated the successful extension of PS segment ( of PM‐b‐PS = 5 000–41 800; PDI = 1.08–1.23). Such copolymers were characterized by ¹H NMR and DSC.

     

Temperature and shear dependencies of rheology of poly(acrylonitrile‐co‐itaconic acid) dope in DMF

Poly(acrylonitrile‐co‐itaconic acid) (poly(AN‐co‐IA)) precursor required for carbon fiber production is made into a dope and spun into fibers using a suitable spinning technique. The viscosity of the resin dope is decided by the polymer concentration, polymer molecular weight, temperature, and shear force. The shear rheology of concentrated poly(AN‐co‐IA) polymer solutions in N,N‐dimethylformamide (DMF), in the range of 1 × 10⁵–1 × 10⁶ g mol^?1, has been investigated in the shear rate (γ′) range of 1 × 10¹–5 × 10⁴ min^?1. The zero shear viscosity (η₀) has been evaluated at different temperatures. The temperature dependence of zero shear viscosity conformed to the Arrhenius–Frenkel–Eyring model. The free energy of activation of viscous flow (ΔG_V) values were in the range 5–32 kJ mol^?1 and this value increased with increase in polymer concentration and molecular weight. A master equation for the ΔG_V value of the polymer solution of any and concentration (c) is suggested. The power law fitted well for the shear dependency of viscosity of these polymer solutions. The pseudoplasticity index (n) diminished with increase in polymer concentration and molecular weight. An empirical relation between viscosity (η) and was found to exist at constant shear rate, concentration and temperature. For each , the equation relating n, c, and T was established. Copyright © 2008 John Wiley & Sons, Ltd.     

Lamellar Copper(I) Thiocyanate‐Based Coordination Polymers containing the Alkali Cation ligating Thiacrown Ether 1,10‐Dithia‐18‐crown‐6

The lamellar coordination polymer [(CuSCN)₂(μ‐1,10DT18C6)] (1,10DT18C6 = 1,10‐dithia‐18‐crown‐6), in which staircase‐like CuSCN double chains are bridged by thiacrown ether ligands, may be prepared in two triclinic modifications 1 a and 1 b by reaction of CuSCN with 1,10DT18C6 in respectively benzonitrile or water. Performing the reaction in acetonitrile in the presence of an equimolar quantity of KSCN leads, in contrast, to formation of the K⁺ ligating 2‐dimensional thiocyanatocuprate(I) net [{Cu₂(SCN)₃}^–] of 2 , half of whose Cu(I) atoms are connected by 1,10DT18C6 macrocycles. The potassium cations in [{K(CH₃CN)}{Cu₂(SCN)₃(μ‐1,10DT18C6)}] ( 2 ) are coordinated by all six potential donor atoms of a single thiacrown ether in addition to a thiocyanate S and an acetonitrile N atom. Under similar conditions, reaction of CuI, NaSCN and 1,10DT18C6 affords [{Na(CH₃CN)₂}{Cu₄I₄(SCN)(μ‐1,10DT18C6)}] ( 3 ), which contains distorted Cu₄I₄ cubes as characteristic molecular building units. These are bridged by thiocyanate and thiacrown ether ligands into corrugated Na⁺ ligating sheets. In the presence of divalent Ba²⁺ cations, charge compensation requirements lead to formation of discrete [Cu(SCN)₃(1,10DT18C6‐κS)]^2– anions in [Ba{Cu(SCN)₃(1,10DT18C6‐κS)}] ( 4 ).     

Lewis acid‐base properties of cellulose acetate butyrate by inverse gas chromatography

The dispersive component of the surface‐free energy, , of cellulose acetate butyrate (CAB) has been determined using the net retention volume, V_N, of n‐alkanes (C₅? C₈) probes in the temperature range 323.15–393.15 K. The values decrease nonlinearly with increase in temperature, and the temperature coefficients of are ? 0.32 (mJ/m²K) and ? 0.10 (mJ/m²K) in the range 323.15–353.15 K and 353.15–393.15 K, respectively. This variation in has been attributed to the structural changes that take place on the surface of CAB at ～353.15 K. The specific components of the enthalpy of adsorption, , and entropy of adsorption, , calculated using V_N of polar solutes are negative. The values are used to evaluate Lewis acidity constant, K_a, and Lewis basicity constant, K_b, for the CAB surface. The K_a and K_b values are found to be 0.126 and 1.109, respectively, which suggest that the surface is predominantly basic. The K_a and K_b results indicate for the necessary surface modifications of CAB which act as biodegradable adsorbent material. Copyright © 2010 John Wiley & Sons, Ltd.     

Fe(CN)$\rm{ {_{6}^{4-}}}$‐Doped‐Glutaraldehyde‐Cross‐Linked Poly‐L‐Lysine Film Electrode. Part 1: Electrochemical Characterization and Its Electrocatalytic Activity Towards Oxidation of Ascorbic Acid

The present work describes preparation, characterization, and electrocatalytic behavior of a hexacyanoferrate‐doped‐glutaraldehyde‐cross‐linked poly‐L ‐lysine (PLL‐GA‐Fe(CN) film modified glassy carbon electrode. The modified electrode has been successfully prepared by electrostatically binding negatively charged Fe(CN) mediator into cross‐linked poly‐L ‐lysine cationic film. The dependence of the peak current of the modified electrode in pure supporting electrolyte (pH 6.8 phosphate buffer solution; PBS) shows that the charge transport in the film is fast and relatively unimpeded at lower scan rates. Cyclic voltammetry and rotating disk electrode (RDE) techniques are used to investigate the electrocatalytic activity of modified electrode towards oxidation of ascorbic acid. The rate constant (k), of catalytic reaction between electrogenerated Fe(CN) ions and ascorbic acid, obtained from RDE analysis was found to be 5.53×10⁵ cm³ mol^?1 s^?1. Finally, the PLL‐GA‐Fe(CN) film electrodes are successfully used for the individual estimation of ascorbic acid in the concentration range of physiological interest.     

Structure and Bonding of the Hexameric Platinum(II) Dichloride,Pt6Cl12 (β‐PtCl2)

The crystal structure of Pt₆Cl₁₂ (β‐PtCl₂) was redetermined ( a_h = 13.126Å, c_h = 8.666Å, Z = 3; a_rh = 8.110Å, α = 108.04°; 367 hkl, R = 0.032). As has been shown earlier, the structure is in principle a hierarchical variant of the cubic structure type of tungsten (bcc), which atoms are replaced by the hexameric Pt₆Cl₁₂ molecules. Due to the 60° rotation of the cuboctahedral clusters about one of the trigonal axes, the symmetry is reduced from to ( ). The molecule Pt₆Cl₁₂ shows the (trigonally elongated) structure of the classic M₆X₁₂ cluster compounds with (distorted) square‐planar PtCl₄ fragments, however without metal‐metal bonds. The Pt atoms are shifted outside the Cl₁₂ cuboctahedron by Δ = +0.046Å ( (Pt—Cl) = 2.315Å; (Pt—Pt) = 3.339Å). The scalar relativistic DFT calculations results in the full symmetry for the optimized structure of the isolated molecule with d(Pt—Cl) = 2.381Å, d(Pt—Pt) = 3.468Å and Δ = +0.072Å. The electron distribution of the Pt‐Pt antibonding HOMO exhibits an outwards‐directed asymmetry perpendicular to the PtCl₄ fragments, that plays the decisive role for the cluster packing in the crystal. A comparative study of the Electron Localization Function with the hypothetical trans‐(Nb₂Zr₄)Cl₁₂ molecule shows the distinct differences between Pt₆Cl₁₂ and clusters with metal‐metal bonding. Due to the characteristic electronic structure, the crystal structure of Pt₆Cl₁₂ in space group is an optimal one, which results from comparison with rhombohedral Zr₆I₁₂ and a cubic bcc arrangement.     

Cationic Graft Copolymers as Carriers for Delivery of Antisense‐Oligonucleotides

Self‐assembling systems based on ionic complexes of DNA fragments (36 base pairs), bcl‐2 antisense oligonucleotides (octadecamer), oligophosphates (25 phosphate groups) or acrylic oligomers (18 groups of phosphonic acid) with poly(L ‐lysine) (PLL) ( = 130 000 and 88 000) grafted with short poly[N‐(2‐hydroxypropyl)methacrylamide] (PHPMA) chains ( = 4 300 or 8 600) were studied by static and dynamic light scattering methods as systems suitable for gene therapy applications. The graft copolymers (GPLLs) with shorter PHPMA grafts ( = 4 300) provide polyelectrolyte complexes (PECs) with smaller and R_H than the corresponding GPLLs with longer grafts ( = 8 600) and the same content of PLL. The lowest aggregation number of 2 was observed for PECs prepared from the GPLL with short grafts and 40 wt.‐% of PLL. The complexes of oligonucleotides and DNA fragments with GPLLs showed quite similar behavior to that with oligophosphates and acrylic oligomer. The complexes prepared from GPLLs containing 40 wt.‐% of PLL and at excess of oligophosphate were stable for at least 48 h under physiological conditions (0.15 M NaCl) and in bovine serum albumin solutions (1 mg · mL^?1). Additionally, polyanion exchange reactions of the PECs in contact with poly(styrenesulfonate) and DNA were studied in 0.15 M NaCl solutions. The oligophosphates in complexes were at least partially substituted with high‐molecular‐weight polyanions. The structure of the initial PECs dominated the PEC structure after the exchange reaction.

The dependence of the molecular weight (a) and the hydrodynamic radius R_H (b) of complexes of the oligophosphate (OPP) and four graft copolymers (GPLLi, i = 0–3) on the mixing ratio X.