Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

MICELLIZATION STUDIES OF DODECYL BENZENESULFONIC ACID AND ITS INTERACTION WITH POLYVINYLPYRROLIDONE

Dodecyl benzenesulfonic acid (DBSA) surfactant was used in the present study to find the effect of concentration on its electrical conductance in solution from 293-323K above and below the critical micelle concentration (CMC). The micellization parameters i.e. degree of counter ion binding (β), aggregation number (n) and number of counter ion micelle(m) were measured. The interaction of DBSA with polyvinylpyrrolidone (PVP) was also studied at 293K throughconductance and surface tension measure ments. A number of important parameters i.e. critical aggregation concentration (CAC), Gibb‘s free energy (△G) and binding ratio (R) were determined and the effect of NaCl on the CAC and polymer saturation point (PSP) was also investigated.     

Organocatalytic Asymmetric Mannich Cyclization of Hydroxylactams with Acetals: Total Syntheses of (−)‐Epilupinine, (−)‐Tashiromine,and (−)‐Trachelanthamidine

An asymmetric, organocatalytic, one‐pot Mannich cyclization between a hydroxylactam and acetal is described to provide fused, bicyclic alkaloids bearing a bridgehead N atom. Both aliphatic and aromatic substrates were used in this transformation to furnish chiral pyrrolizidinone, indolizidinone, and quinolizidinone derivatives in up to 89 % yield and 97 % ee. The total syntheses of (?)‐epilupinine, (?)‐tashiromine, and (?)‐trachelanthamidine also achieved to demonstrate the generality of the process.     

Organotin(IV) Carboxylates of Substituted α‐Cinnamic Acid,RnSn(OCOC(R2)=CHR1)4 – n: Synthesis,Spectroscopic Characterization and Biological Evaluation

Di‐ and triorganotin(IV) carboxylates, R_nSn(OCOC(R²)=CHR¹)_4–n (n = 2 and 3; R = Me, Et, n‐Bu, Ph; R¹ = 3‐CH₃O‐4‐OHC₆H₃, R² = C₆H₅) were prepared by reacting the corresponding organotin(IV) chloride with the silver salt of the (E)‐3‐(4‐hydroxy‐3‐methoxyphenyl)‐2‐phenylpropenoic acid. The title compounds were investigated and characterized by elemental analysis, infrared (FT‐IR), multinuclear (¹H, ¹³C, ¹¹⁹Sn) NMR, and mass spectrometry, and possible structures were proposed. The complexes and ligand acid ( HL ) have been evaluated in vitro against various bacteria and fungi. The results noticed during the biocidal activity screenings proved their in vitro biological potential. They were also tested for cytotoxicity.     

Switching in Metal–Organic Frameworks

In recent years, metal–organic frameworks (MOFs) have become an area of intense research interest because of their adjustable pores and nearly limitless structural diversity deriving from the design of different organic linkers and metal structural building units (SBUs). Among the recent great challenges for scientists include switchable MOFs and their corresponding applications. Switchable MOFs are a type of smart material that undergo distinct, reversible, chemical changes in their structure upon exposure to external stimuli, yielding interesting technological applicability. Although the process of switching shares similarities with flexibility, very limited studies have been devoted specifically to switching, while a fairly large amount of research and a number of Reviews have covered flexibility in MOFs. This Review focuses on the properties and general design of switchable MOFs. The switching activity has been delineated based on the cause of the switching: light, spin crossover (SCO), redox, temperature, and wettability.     

Bifurcations of a creeping air–water flow in a conical container

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height \(H_{\mathrm{w}}\) and cone half-angle \(\beta \) vary, numerous flow metamorphoses occur. They are investigated for \(\beta =30^{\circ }, 45^{\circ }\), and \(60^{\circ }\). For small \(H_{\mathrm{w}}\), the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as \(H_{\mathrm{w}}\) exceeds a threshold depending on \(\beta \). For all \(\beta \), the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.     

Ultrasound-assisted addition of alcohols to N-acyliminium ions mediated by In(OTf)3 and synthesis of 1,2,3-triazoles

An easy and mild approach using ultrasound-assisted reaction addition of alcohols to N-acyliminium ion mediated by Lewis acid, In(OTf)₃, allowed the synthesis of ether pyrrolidinones; next, the products were converted to 1,2,3-triazoles using click chemistry reaction conditions. The products in both reactions were afforded in moderate to good yields.     

Polyol‐enhanced dispersive liquid–liquid microextraction coupled with gas chromatography and nitrogen phosphorous detection for the determination of organophosphorus pesticides from aqueous samples,fruit juices,and vegetables

Polyol‐enhanced dispersive liquid–liquid microextraction has been proposed for the extraction and preconcentration of some organophosphorus pesticides from different samples. In the present study, a high volume of an aqueous phase containing a polyol (sorbitol) is prepared and then a disperser solvent along with an extraction solvent is rapidly injected into it. Sorbitol showed the best results and it was more effective on the extraction recoveries of the analytes than inorganic salts such as sodium chloride, potassium chloride, and sodium sulfate. Under the optimum extraction conditions, the method showed low limits of detection and quantification within the ranges of 12–56 and 44–162 pg/mL, respectively. Enrichment factors and extraction recoveries were in the ranges of 2799–3033 and 84–92%, respectively. The method precision was evaluated at a concentration of 10 ng/mL of each analyte, and relative standard deviations were found to be less than 5.9% for intraday (n = 6) and less than 7.8% for interday (n = 4). Finally, some aqueous samples were successfully analyzed using the proposed method and four analytes (diazinon, dimethoate, chlorpyrifos, and phosalone) were determined, some of them at ng/mL level.     

Electromembrane extraction and anodic stripping voltammetric determination of mercury(II) using a glassy carbon electrode modified with gold nanoparticles

This study presents a method for the selective determination of Hg(II) using electromembrane extraction (EME), followed by square wave anodic stripping voltammetry (SWASV), using a gold nanoparticle-modified glassy carbon electrode, (AuNP/GCE). By applying an electrical potential of typically 60 V for 12 min through a thin supported liquid membrane (1-octanol), Hg(II) ions are extracted from a donor phase (i.e., the sample solution) to an acidic acceptor solution (15 μL) placed in the lumen of a hollow fiber. The influences of experimental parameters during EME were optimized using face-centered central composite design. The calibration plot, established at a working voltage of 0.55 V (vs. Ag/AgCl), extends from 0.2 to 10 μg^.L^?1 of Hg(II). The limit of detection, at a signal to noise ratio of 3, is 0.01 μg^.L^?1 and the relative standard deviations (for 5 replicate determinations at 3 concentration levels) are between 7.5 and 8.7 %. The method was successfully applied to the determination of Hg(II) in spiked real water samples to give recoveries ranging from 89 to 97 %. The results were validated by cold vapor atomic absorption spectroscopy.

Open image in new window

Graphical abstract Hg(II) ions were extracted from a donor phase into an acidic acceptor phase (15 μL) placed in the lumen of a hollow fiber using electromembrane extraction. The acceptor phase was then analyzed using anodic stripping voltammetry.