On the fundamentals of extended thermodynamics (ET) of a one-dimensional rarefied gas

Extended thermodynamics (ET) of degreer for a one-dimensional rarefied gas based, by definition, on a finite set A^r={a⁰, a²,..., a^r} of the firstr–1(3r) direct internal moments of the one-point distribution functionf is carefully investigated. With the aid of the second axiom of thermodynamics, the new representation forf, depending in a local and nonlinear way onA ^r , is explicitly derived. It is demonstrated that in ET of degreer an infinite sequence {b^r+1, b^r+2,...} ofhigher order Hermite coefficients, which normally drops out of Grad's proposition forf fashioned by mathematical apparatus such as the Hermite polynomials, cannot be considered negligible in the case when nonlinear constitutive functions are established. Using Ma's kinetic equation corresponding to a one-dimensional rarefied gas as well as the generalized representation forf, collision productions in the nonconservative moment equations are then calculated for a special choice of the rate of collisions between particles.     

Multiple SLE and the complex Burgers equation

In this note we ask whether one can take the limit of multiple SLE as the number of slits goes to infinity. In the special case of n slits that connect n points of the boundary to one fixed point, one can take the limit of the Loewner equation that describes the growth of those slits in a simultaneous way. In this case, the limit is a deterministic Loewner equation whose vector field is determined by a complex Burgers equation.     

New extension of dispersionless Harry Dym hierarchy

The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.     

Graphs to chemical structures 3. General theorems with the use of different sets of sphericity indices for combinatorial enumeration of nonrigid stereoisomers

The extended sphericity indices of k-cycles, which were defined in Part 2 of this series (S. Fujita, Theor Chem Acc, Online: http://www.springerlink.com/index/10.1007/s00214-004-0606-z) according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle, are further extended to prove more general theorems for enumerating nonrigid stereoisomers with rotatable ligands. One of the extended points is the use of different sets of sphericity indices to treat one or more orbits contained in skeletons and ligands. Another is to take account of chirality in proligands and sub-proligands, the latter of which are introduced to consider further inner structures of ligands. Two theorems for enumerating nonrigid stereoisomers are proved by adopting two schemes of their derivation, i.e., the scheme ``positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand ⇐ sub-proligands)' and the scheme ``(positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand)) ⇐ sub-proligands'. The theorems are applied to the stereoisomerism of trihydroxyglutaric acids. Thereby, it is demonstrated where Pólya's theorem and other previous methods are deficient, when applied to the enumeration of stereoisomers.     

Osmotic and Activity Coefficients of the {<Emphasis Type="Italic">x</Emphasis>ZnCl<Subscript>2</Subscript>+(1−<Emphasis Type="Italic">x</Emphasis>)ZnSO<Subscript>4</Subscript>}(aq) System at 298.15 K

Isopiestic vapor pressure measurements were made for {xZnCl₂+(1−x)ZnSO₄}(aq) solutions with ZnCl₂ molality fractions of x=(0,0.3062,0.5730,0.7969, and 1) at the temperature 298.15 K, using KCl(aq) as the reference standard. These measurements cover the water activity range 0.901–0.919≤a _w≤0.978. The experimental osmotic coefficients were used to evaluate the parameters of an extended ion-interaction (Pitzer) model for these mixed electrolyte solutions. A similar analysis was made of the available activity data for ZnCl₂(aq) at 298.15 K, while assuming the presence of equilibrium amounts of ZnCl⁺(aq) ion-pairs, to derive the ion-interaction parameters for the hypothetical pure binary electrolytes (Zn²⁺,2Cl⁻) and (ZnCl⁺,Cl⁻). These parameters are required for the analysis of the mixture results. Although significant concentrations of higher-order zinc chloride complexes may also be present in these solutions, it was possible to represent the osmotic coefficients accurately by explicitly including only the predominant complex ZnCl⁺(aq) and the completely dissociated ions. The ionic activity coefficients and osmotic coefficients were calculated over the investigated molality range using the evaluated extended Pitzer model parameters.     

Sequential peptide chemical ligation by the thioester method and extended chemical ligation

The sequential chemical ligation of peptide thioesters by a combination of the thioester method and extended chemical ligation using a photoremovable auxiliary, 2-mercapto-1-(2-nitrophenyl)ethyl group, is described. The thiazolidine ring was used as a protecting group for the N-terminal 1,2-aminoethanethiol moiety of the auxiliary in the middle peptide thioester. After the first thioester coupling, the thiazolidine ring was opened by treatment with O-methylhydroxylamine. Second coupling by extended chemical ligation followed by UV irradiation gave the target polypeptide.     

A quantum-mechanical study of the chain-length dependent stability of the extended and 310-helix conformations in dehydroalanine oligopeptides

Summary A quantum-chemical study of the chain-length dependent stability of the extended, 2₇-ribbon and 3₁₀-helix conformations in dehydroalanine (Ala) oligopeptides has been performed by using both semiempirical AM1 and ab initio 4–31G methodologies. The validity of both methods in the study of the conformational properties of Ala oligopeptides was tested first on the dipeptide. The results of this test showed that 4–31G and AM1 calculations are in good agreement with 6–31G^* calculations and experimental data. In order to monitor the conformational conversions, Ala oligopeptides comprising two to six residues were constructed. Molecular geometries were fully optimized using AM1, and the final conformations were verified to be minima by analysis of the corresponding second-derivative matrices. Conformational studies revealed that the 3₁₀-helix is stabilized with respect to the 2₇-ribbon when the number of residues is three or four, at the AM1 and ab initio 4–31G level respectively, while the extended form is the most stable in all the calculations performed. On the other hand, if a linear behaviour is assumed for longer chains, our calculations show a trend that would predict a conversion from extended form to 3₁₀-helix in oligopeptides with around six (ab initio 4–31G) or eight (AM1) Ala residues. In order to explain these conformational changes, the cooperative effects for the different conformers were investigated. Large cooperative energy effects were found for the 3₁₀-helix conformation.     

Theoretical analysis of the electronic structure and reactivity of the CoP3 core in [(triphos)CoP3]

Summary The electronic structure and reactivity of the compound [(triphos)-CoP₃] [triphos=1, 1, 1-tris(diphenylphosphinomethyl)ethane] have been investigated using the semiempirical, extended Hückel (EH) approach, and density functional theory (DFT). The calculations have been performed on the model complex [(PH₃)₃CoP₃]. The orientation of the P₃ ring with respect to the (triphos)Co unit and electrophilic addition reactions have been investigated within the local density approximation (LDA). The staggered and eclipsed conformations of the P₃ group have been found to have comparable energies and the molecule is stabilized by a strong interaction within thee atomic orbitals (C_3v symmetry) mainly involving the 3p _z orbitals of phosphorus and 3d _xz and 3d _yz hybrid metal orbitals. Using H⁺ as the electrophilic reagent four preferential sites of attack have been probed. The optimized structure of the most stable arrangement of the adduct corresponds to that experimentally observed in the monopositive cation [(triphos)CoP₃H]⁺, which was obtained by protonation of the neutral species. The arrangements of two other favourable sites for the attack correspond to the geometries observed in the derivatives obtained by the electrophilic additions of CH ₃ ⁺ and, respectively, HgCH ₃ ⁺ . The geometry of the model complex [(PH₃)₃Co(P₃CH₃)]⁺ has been optimized using DFT-LDA and compared to that of [(triphos)Co(P₃CH₃)]⁺.     

An elastic-plastic hybrid adhesion model for contacting rough surfaces in the presence of molecularly thin lubricant

The study of adhesion has received considerable attention in recent years, partly due to advances in the design and fabrication of micro/nano devices. Many adhesion investigations are centered on single-spherical-contact models, which include the classic Johnson-Kendall-Roberts (JKR), improved Derjaguin-Muller-Toporov (IDMT), and Maugis-Dugdale (MD) models. Based on the IDMT single-asperity model, adhesive rough surface contact models have also been developed, which are valid for elastic and elastic-plastic contact conditions. A limitation of the IDMT-based models is that they are only valid for application cases with low adhesion parameter values. In this research, a contacting rough surface adhesion model was developed by combining an extended Maugis-Dugdale (EMD) model (which is only valid for elastic contacts) with an IDMT-based elastic-plastic adhesion model. The proposed model, termed the elastic-plastic hybrid adhesion model, is valid for the entire adhesion parameter range and also for elastic-plastic contacts. The proposed model gives results similar to the EMD rough surface model when the contact is primarily elastic. Moreover, the proposed model was compared to an IDMT-based model (ISBL model) and both gave similar results for contacts with low adhesion parameter values. With high adhesion parameter values, the ISBL model fails, whereas the proposed model correctly predicts higher adhesion. Last, based on the stiffness of the external force, the instability for adhesive rough surfaces in contact was also discussed, and it was postulated that a high peak value of the external force stiffness directly relates to the unstable contact process.     

Rotational Phenomena on Substituent Phenyl Rings of (η5-Cyclopentadienyl)(η4-Tetraphenylcyclobutadiene)Cobalt

We used extended Hückel calculations, variable-temperature, homonuclear long-range shift-correlated 2-D ¹H NMR and dynamic NOE measurements to investigate rotational phenomena of substituent phenyl rings on (η⁵-cyclopentadienyl)(η⁴tetraphenylcyclobutadiene)cobalt (1). Two closely related compounds, (η⁵-cyclopentadienyl)(η⁴-1,3-diphenylcyclobutadiene)cobalt (A) and (η⁵-cyclopentadienyl)(η⁴-1,2-diphenylcyclobutadiene)cobalt (A′), were prepared. Energy minima appeared at conformations of which the dihedral angles between phenyl and cyclobutadiene rings are about 30° for I and 0° for A according to extended Hückel calculations. In 1, ortho protons of phenyl rings belong to one set of multiplet in ¹H NMR; meta and para protons belong to the other. It was supported by a long-range coupling 2-D ¹H NMR and NOE experiments. A sharp line due to phenyl rings was observed in the low-temperature ¹H NMR spectrum of A, which indicates that the five protons are magnetically equivalent at that temperature.