Phosphate-doped borosilicate enamel coating used to protect reinforcing steel from corrosion

Phosphate-doped sodium borosilicate glasses were developed for enamel coatings to provide corrosion protection for reinforcing steel in concrete. Phosphates have low solubilities in borosilicate glasses, generally less than about 4 mol%, and are incorporated mainly as isolated PO43- and P2O74- anions, along with some borophosphate bonds. With the addition of P2O5, the silicate network is repolymerized and tetrahedral borates are converted to trigonal borates, because the phosphate anions scavenge sodium ions from the borosilicate network. The addition of Al2O3 significantly increases the solubility of P2O5 due to the formation of aluminophosphate species that increases the connectivity of phosphate sites to the borosilicate glass network. The addition of P2O5 has an insignificant effect on thermal properties like Tg, Ts and CTE, and chemical durability in alkaline environments can be improved, particularly for glasses co-doped with alumina. When the solubility limit is reached, a phase-separated microstructure forms, degrading the chemical durability of the glass because of the dissolution of a less durable alkali borate phase. The dissolution rates of glasses in a simulated cement pore water solution are orders of magnitude slower than in Ca-free solutions with similar pH. In Ca-saturated solutions, a passivating layer of calcium silicate hydrate (C-S-H) forms on the glass surface. Phosphate-rich glasses react with the simulated cement pore water to form crystalline hydroxyapatite (HAp) on the glass surface. Electrochemical tests reveal that the HAp precipitates that form on the surfaces of intentionally-damaged phosphate-doped enamel coatings on reinforcing steel suppress corrosion. The enhanced corrosion protection is retained in chloride-containing pore water, where the development of hydroxyl-chlorapatites immobilizes the free chloride ions in solution --Abstract, page iv

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes

Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn-Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524

A Semi-Implicit Scheme for Stationary Statistical Properties of the Infinite Prandtl Number Model

We propose a semisecret in time semi-implicit numerical scheme for the infinite Prandtl model for convection. Besides the usual finite time convergence, this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite Prandtl number model at vanishing time stop. One of the key characteristics of the scheme is that it preserves the dissipativity of the infinite Prandtl number model uniformly in terms of the time stop. So far as wo know, this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite dimensional dissipative complex systems. © 2008 Society for Industrial and Applied Mathematics

A Uniformly Dissipative Scheme for Stationary Statistical Properties of the Infinite Prandtl Number Model

The purpose of this short communication is to announce that a class of numerical schemes, uniformly dissipative approximations, which uniformly preserve the dissipativity of the continuous infinite dimensional dissipative complex (chaotic) systems possess desirable properties in terms of approximating stationary statistics properties. in particular, the stationary statistical properties of these uniformly dissipative schemes converge to those of the continuous system at vanishing mesh size. the idea is illustrated on the infinite Prandtl number model for convection and semi-discretization in time, although the general strategy works for a broad class of dissipative complex systems and fully discretized approximations. as far as we know, this is the first result on rigorous validation of numerical schemes for approximating stationary statistical properties of general infinite dimensional dissipative complex systems. © 2008 Elsevier Ltd. All rights reserved

Index of Dimensional Projection : an Index Supporting Search for Spatial Objects by Region�

In image databases and other spatial data retrieval systems, the techniques for storing and indexing data objects require different kinds of search and query from those in traditional databases and data retrieval systems. In order to handle spatial data more efficiently, a new index structure supporting search for spatial objects by region, the Index by Dimensional Projection is proposed in this thesis. By this method, the number of pages accessed for searching a point region has a logarithmic relationship with the number of objects in data space and the number of comparisons required for searching an entry within a disk page has logarithmic relationship with the number of entries in the disk page.Computer Scienc