Electrochemical digital simulations with an exponentially expanding grid: General expressions for higher order approximations to spatial derivatives: The special case of four-point formulas and their application to multipulse techniques in planar and any size spherical electrodes

This paper deals with the consideration, in digital simulation, of a grid in which there is a geometrical relationship between the distances between neighbouring pairs of points, leading to an exponentially expanding sequence of intervals. We show that the expressions for derivatives for higher order spatial discretisations take a very simple form, with coefficients which are dependent on the number of points taken to the left and right of the point considered, but independent of the absolute position of the point in the grid. This enables convenient and efficient manipulation and incorporation in computer programs of these derivatives in the context of finite difference approach. Moreover, the explicit expressions obtained lead to very interesting particular situations. The most interesting is the optimal behaviour of the four-point spatial discretisation for very high expansion factors, which, in combination with fully implicit time-integration schemes, leads to very fast and accurate calculations both for planar and any size spherical electrodes, including microelectrodes. Applications of these coefficients in the simulation of some multipulse and square wave experiments are also presented.     

Relation of glass transition temperature to the hydrogen bonding degree and energy in poly(N-vinyl pyrrolidone) blends with hydroxyl-containing plasticizers: 3. Analysis of two glass transition temperatures featured for PVP solutions in liquid poly(ethylene glycol)

The phase behaviour of poly(N-vinyl pyrrolidone)-poly(ethylene glycol) (PVP-PEG) blends has been examined in the entire composition range using Temperature Modulated Differential Scanning Calorimetry (TM-DSC) and conventional DSC techniques. Despite the unlimited solubility of PVP in oligomers of ethylene glycol, the PVP-PEG system under consideration demonstrates two distinct and mutually consistent glass transition temperatures (T_g) within a certain concentration region. The dissolution of PVP in oligomeric PEG has been shown earlier (by FTIR spectroscopy) to be due to hydrogen bonding between carbonyl groups in PVP repeat units and complementary hydroxyl end-groups of PEG chains. Forming two H-bonds through both terminal OH-groups, PEG acts as a reversible crosslinker of PVP macromolecules. To characterise the hydrogen bonded complex formation between PVP (M_w=10⁶) and PEG (M_w=400) we employed an approach described in the first two papers of this series that is based on the modified Fox equation. We evaluated the fraction of crosslinked PVP units and PEG chains participating to the complex formation, the H-bonded network density, the equilibrium constant of complex formation, etc. Based on the established molecular details of self-organisation in PVP-PEG solutions, we propose a three-stage mechanism of PVP-PEG H-bonded complex formation/breakdown with increase of PEG content. The two observed T_gs are assigned to a coexisting PVP-PEG network (formed via multiple hydrogen bonding between a PEG and PVP) and a homogeneous PVP-PEG blend (involving a single hydrogen bond formation only). Based on the strong influence of coexisting regions on each other and the absence of signs of phase separation (evidenced by Optical Wedge Microinterferometry) we conclude that the PVP-PEG blend is fully miscible on a molecular scale.