Study on the retention behaviour of low-molar-mass polystyrenes and polyesters in reversed-phase liquid chromatography by evaluation of thermodynamic parameters

Polymers can be characterized under sorption conditions, to obtain information on molar mass and chemical composition. In order to get a better understanding of their retention behaviour under such conditions, the evaluation of thermodynamic parameters obtained from van't Hoff analyses on low-molar-mass polystyrenes (PS) and polyesters (PE) in various THF–water mixtures on a C₁₈ column is described in this study. Linear van't Hoff behaviour was observed in almost all cases. Negative values for both ΔH and ΔS were found for both PS and PE oligomers, which increase with increasing %THF. For ΔS this is explained from multi-site attachment effects. For PS, the non-linear relations between ΔH and ΔS, and degree of polymerization (p) could be properly described by the Stockmayer–Fixman equation. Although less clear, similar trends were found for PE. For PS, evidence for penetration effects of oligomer chains into the bonded chains was obtained. Martin plots for both PS and PE were shown to be non-linear in all investigated eluent compositions. The extent of non-linearity is suggested to depend on the conformation of a polymer in solution. No distinct enthalpy–entropy-compensation temperature (EECT) independent of p was found for PS, thus confirming the findings of an earlier study in which no exact molar mass independence was found under critical conditions. Further evaluation of EECT for PS oligomers revealed a retention mechanism independent of the binary eluent composition. This indicates that conclusions from this study can also be used for a qualitative understanding of sorption mechanisms in the gradient elution mode. Finally, for PS it was shown that ΔG equals zero under critical conditions, thus confirming theoretical predictions.     

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.     

On the eigenvalue distribution of a class of preconditioning methods

Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC ^–1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.     

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.     

A general approach to analyse preconditioners for two‐by‐two block matrices

Two‐by‐two block matrices arise in various applications, such as in domain decomposition methods or when solving boundary value problems discretised by finite elements from the separation of the node set of the mesh into ‘fine’ and ‘coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimisation problems. A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two‐by‐two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element‐by‐element approximations and/or geometric or algebraic multigrid/multilevel methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Electrophoretic transport of latex particles in lipid nanotubes

Lipid vesicles can be connected by membrane nanotubes to build networks with promising bioanalytical properties. Here we characterize electrophoretic transport in such membrane tubes, with a particular eye to how their soft-material nature influences the intratube migration. In the absence of field, the tube radius is 110 +/- 26 nm, and it remains in this range during electrophoresis even though the applied electric field causes a slight decrease in the tube radius (approximately 6-11%). The electrophoretic velocity of the membrane wall (labeled with quantum dots) varies linearly with the field strength. Intratube migration is studied with latex spheres of radii 15, 50, 100, and 250 nm. The largest particle size does not enter the tube at fields strengths lower than 1250 V/m because the energy cost for expanding the tube around the particles is too high. The smaller particles migrate with essentially the same velocity as the membrane at low fields. Above 250 V/cm, the 15 nm particles exhibit an upward deviation from linear behavior and in fact migrate faster than in free solution whereas the 100 nm particles deviate downward. We propose that these nonlinear effects arise because of lipid adsorption to the particles (dominating for 15 nm particles) and a pistonlike compression of the solvent in front of the particles (dominating for 100 nm). As expected from such complexities, existing theories for a sphere migrating in a rigid-wall cylinder cannot explain our velocity results in lipid nanotubes.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

Dependence of Pressure,Combustion and Frequency Characteristics on Valved Pulsejet Combustor Geometries

An experimental analysis of valved pulsejets based on the Curtis-Dyna design and the concomitant results are discussed in the current paper. By altering the combustor length, the tail pipe length and by adding a flare at the aft-end, twelve different pulsejet configurations are tested. An axially-distributed array of piezoelectric pressure sensors and ion probes reveal the pressure and combustion dynamics inside these devices. Evidence is attained to support the claim that valved Curtis-Dyna pulsejets of the tested configurations behave like a Helmholtz resonator. Each cycle of a pulsejet is composed of temporally and spatially restrained combustion events. Altering the geometry induces an amplitude modulated low frequency instability inside the pulsejet that is characterized by sinusoidally-varying peak cycle pressures. The operating frequency, peak pressures and combustion activity of the pulsejets are characterized to reveal that reliable pulsejet operation requires proper amount of coupling — defined by low time lags — between the pressure peaks and combustion events.     

Chromatographic behaviour of low molar-mass polyesters in normal-phase high-performance liquid chromatography

Summary The normal-phase chromatographic retention behaviour of polyesters on bare silica and on a polymer-based polyamine (PA) column has been studied with a variely of binary mobile phases under isocratic conditions. The dependence of experimental retention data on the degree of polymerization (p) and on mobile phase composition (φ) was characterized by to an approach developed by Jandera et al. The bulky repeating unit and the relatively highly polar end groups of the polyesters both had a large influence on retention behaviour. The two effects in combination explain the molar-mass-independent retention observed experimentally at a particular mobile phase composition for all the mobile phase—stationary phase combinations investigated. These conditions were found to be independent of the type of end group. End group separation on a silica column improves when the polarity of the less polar solvent is increased. End group separation is better on the PA column because of a greater difference between the adsorption energy of the alcohol and acid end groups. Better prediction of retention data on the PA column was achieved by use of an approach which assumes two different types of adsorption site. Results enabled further understanding of retention behaviour in normalphase gradient polymer-elution chromatography (NPGPEC) and explained both the dependence of the order of elution onp and differences between the end-group selectivity of different systems.