PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.     

Micelle formation of a cationic surfactant in the presence of 1,n-alkanediol and the miscibility of alcohols in micelles

Micelle formation of dodecyltrimethylammonium bromide (DTAB) was examined in the presence of α,ω-alkanediols applying conductivity measurements. Octanediol and hexanediol promoted the formation of mixed micelles of DTAB and the alcohol, but butanediol interfered with micellization. Analysis of the critical micelle concentration (cmc) based on the lattice model for mixed solution with the Bragg–Williams approximation indicated an unfavorable interaction between alcohol and water and a favorable interaction between the alcohol and surfactant, with the exception of butanediol. The exchange energy between alcohol and water was 0.5kT higher for alkanediol (C_2n(OH)₂) than for the corresponding regular alcohol (C_nOH), which is believed to have resulted from the smaller mixing entropy for the alkanediol than for the corresponding regular alcohol. It was inferred from the analysis that the cmc increase for C₄(OH)₂ was caused by favorable interaction with water but unfavorable interaction with the micellar surfactant.     

Determination of critical micellar concentrations of two monoketo derivatives of cholic acid

Critical micellear concentrations (CMC) were determined for two novel promoters of membrane permeability—7-monoketocholic acid (7-MKC) and 12-monoketocholic acid (12-MKC), using two non-invasive (¹H NMR relaxation experiment and conductometry) and two invasive (spectral shift and partition coefficient of the probe molecule) methods. Studies by the former methods suggest the different aggregation abilities of the investigated bile acid derivatives. In an aqueous solution, 7-MKC has a somewhat lower CMC value (43 mM) than 12-MKC (50 mM). Further, it was found that, in addition to primary micelles, 7-MKC forms also secondary micelles. In the experiments with probe (hydrophobic) molecules, the aggregation properties of investigated bile acids did not differ in water, whereas the presence of urea altered the aggregation of 7-MKC.Based on the CMC value, 7-MKC is more hydrophobic than 12-MKC. The apparent hydrophobicity of 7-MKC is a consequence of the formation of secondary micelles, shifting the monomer equilibrium to the direction of primary micelles, which is manifested as a decrease in the CMC value.     

Full separation of oligomers in block copolymers of ethylene oxide and propylene oxide

In this study di‐ and triblock copolymers of ethylene oxide (EO) and propylene oxide (PO) are analyzed by using CAP for one block and adsorption for the other block. This gives a complete picture of EO‐ and PO‐based block copolymer with respect to the oligomers of both blocks. A full resolution of the individual oligomers can be achieved for both blocks up to an average molecular weight of 1000–1500 of each block.     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.     

Indecomposability graph and critical vertices of an indecomposable graph

Given a directed graph G=(V,A), the induced subgraph of G by a subset X of V is denoted by G[X]. A subset X of V is an interval of G provided that for a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For instance, 0?, V and {x}, x∈V, are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial, otherwise it is decomposable. Given an indecomposable directed graph G=(V,A), a vertex x of G is critical if G[V?{x}] is decomposable. An indecomposable directed graph is critical when all its vertices are critical. With each indecomposable directed graph G=(V,A) is associated its indecomposability directed graph defined on V by: given x≠y∈V, (x,y) is an arc of if G[V?{x,y}] is indecomposable. All the results follow from the study of the connected components of the indecomposability directed graph. First, we prove: if G is an indecomposable directed graph, which admits at least two non critical vertices, then there is x∈V such that G[V?{x}] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs G which have a unique non critical vertex x and such that G[V?{x}] is critical. Third, we propose a new approach to characterize the critical directed graphs.     

Re-evaluation of the thermodynamic activity quantities in aqueous alkali metal nitrate solutions at T = 298.15 K

The Hückel equation used in this study to correlate the experimental activities of dilute alkali metal nitrate solutions up to a molality of about 1.5 mol · kg⁻¹ contains two parameters being dependent on the electrolyte: B [that is related closely to the ion-size parameter (a∗) in the Debye–Hückel equation] and b₁ (this parameter is the coefficient of the linear term with respect to the molality and this coefficient is related to hydration numbers of the ions of the electrolyte). In more concentrated solutions up to a molality of 7 mol · kg⁻¹, an extended Hückel equation was used, and it contains additionally a quadratic term with respect to the molality and the coefficient of this term is parameter b₂. All parameter values for the Hückel equations of LiNO₃, NaNO₃, and KNO₃ were determined from the isopiestic data measured by Robinson for solutions of these salts against KCl solutions [J. Am. Chem. Soc. 57 (1935) 1165]. In these estimations, the Hückel parameters determined recently for KCl solutions [J. Chem. Eng. Data 54 (2009) 208] were used. The Hückel parameters for RbNO₃ and CsNO₃ were determined from the reported osmotic coefficients of Robinson [J. Am. Chem. Soc. 59 (1937) 84]. The resulting parameter values were tested with the vapour pressure and isopiestic data existing in the literature for alkali metal nitrate solutions. These data support well the recommended Hückel parameters up to a molality of 7.0 mol · kg⁻¹ for LiNO₃ and NaNO₃, up to 4.5 mol · kg⁻¹ for RbNO₃, up to 3.5 mol · kg⁻¹ for KNO₃, and up to 1.4 mol · kg⁻¹ for CsNO₃ solutions. Reliable activity and osmotic coefficients of alkali metal nitrate solutions can, therefore, be calculated by using the new Hückel equations, and they have been tabulated at rounded molalities. The activity and osmotic coefficients obtained from these equations were compared to the values suggested by Robinson and Stokes [Electrolyte Solutions, second ed., Butterworths Scientific Publications, London, 1959], to those calculated by using the Pitzer equations with the parameter values of Pitzer and Mayorga [J. Phys. Chem. 77 (1973) 2300], and to those calculated by using the extended Hückel equation of Hamer and Wu [J. Phys. Chem. Ref. Data 1 (1972) 1047].     

Bubble formation in a quiescent pool of gold nanoparticle suspension

This paper begins with an extensive review of the formation of gas bubbles, with a particular focus on the dynamics of triple lines, in a pure liquid and progresses into an experimental study of bubble formation on a micrometer-sized nozzle immersed in a quiescent pool of aqueous gold nanofluid. Unlike previous studies of triple line dynamics in a nanofluid under evaporation or boiling conditions, which are mainly caused by the solid surface modification due to particle sedimentation, this work focuses on the roles of nanoparticles suspended in the liquid phase. The experiments are conducted under a wide range of flow rates and nanoparticle concentrations, and many interesting phenomena are revealed. It is observed that nanofluids prevent the spreading of the triple line during bubble formation, i.e. the triple line is pinned somewhere around the middle of the tube wall during the rapid bubble formation stage whereas it spreads to the outer edge of the tube for pure water. A unique ‘stick-slip’ movement of the triple line is also observed for bubbles forming in nanofluids. At a given bubble volume, the radius of the contact line is found to be smaller for higher particle concentrations, but a reverse trend is found for the dynamic bubble contact angle. With the increase of particle concentration, the bubble frequency is raised and the bubble departure volume is decreased. The bubble shape is found to be in a good agreement with the prediction from Young-Laplace equation for given flow rates. The influence of nanoparticles on other detailed characteristics related to bubble growth inside, including the variation of bubble volume expansion rate, the radius of the curvature at the apex, the bubble height and bubble volume, is revealed. It is suggested that the variation of surface tensions and the resultant force balance at the triple line might be responsible for the modified dynamics of the triple line.     

Nucleation kinetics of selenium (+4) precipitation from an acidic copper sulphate solution

The removal of selenium from copper sulphate solution prior to the electrowinning of copper is desirable in order to minimise contamination of the copper cathodes by selenium and other impurities. The selenium removal is effected by a precipitation process that takes place under high supersaturation conditions, which favour nucleation over any other particle formation processes. There is currently no fundamental information on the nucleation kinetics of this important process. In this study, the nucleation kinetics of selenium (+4) precipitation from an acidic copper sulphate solution was determined using the classical nucleation theory (CNT). Experiments were carried out by varying the levels of supersaturation from 8.66×10¹⁵ to 4.33×10¹⁷ at a temperature of 95 °C under atmospheric pressure. The nucleation rates for four different levels of supersaturation, the nucleation work and the nucleus size were determined. The kinetic constant A was found to be 3.92×10²⁷ m⁻³ s⁻¹, which shows that the nucleation process takes place through a homogeneous mechanism. The associated thermodynamic parameter (B) was determined to be 8.98×10⁰⁴.