Stochastic PAC models for vacancy motions with trapping and detrapping

In order to explain PAC data for tetragonal zirconia at temperatures between 900 and 1300oC, we have developed a four-state stochastic model. The model simulates vacancies which trap and detrap at a PAC probe nucleus. While trapped, the vacancies hop around the probe in equivalent sites. The four states in this Winkler–Gerdau stochastic theory are three trapped states with equivalent electric field gradients (EFGs) of different orientations and a detrapped state with a weaker EFG whose axis of symmetry is oriented along the diagonal between the three trapped EFGs. There are three hopping rates in this model: w, the rate a trapped vacancy hops around the probe, w_D, the detrapping rate, and w_t, the trapping rate. We report results of calculations for values of these hopping rates implied by our tetragonal zirconia data, and we report heuristic fitting functions which summarize the computer results and can be used to fit data efficiently for a wide range of parameters. This revised version was published online in August 2006 with corrections to the Cover Date.     

Molecular origin of differences in hole and electron mobility in amorphous Alq3--a multiscale simulation study

In order to determine the molecular origin of the difference in electron and hole mobilities of amorphous thin films of Alq(3) (meridional Alq(3) (tris(8-hydroxyquinoline) aluminium)) we performed multiscale simulations covering quantum mechanics, molecular mechanics and lattice models. The study includes realistic disordered morphologies, polarized site energies to describe diagonal disorder, quantum chemically calculated transfer integrals for the off-diagonal disorder, inner sphere reorganization energies and an approximative scheme for outer sphere reorganization energies. Intermolecular transfer rates were calculated via Marcus-theory and mobilities were simulated via kinetic Monte Carlo simulations and by a Master Equation approach. The difference in electron and hole mobility originates from the different localization of charge density in the radical anion (more delocalized) compared to the radical cation (more confined). This results in higher diagonal disorder for holes and less favourable overlap properties for the hole transfer integrals leading to an overall higher electron mobility.     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

Dynamics and thermodynamics of tetragonal zirconia determined by 111In/Cd TDPAC

For several years this research group has been using ¹¹¹In/Cd TDPAC to study oxygen vacancies in pure and lightly doped zirconia powders at temperatures up to 1400°C. This paper includes a brief survey of important results from those studies and some recent results. In particular, our new measurement showing negligible relaxation due to vacancy hopping above 600°C provides a major breakthrough in our ability to analyze PAC data and to determine accurately hopping and trapping rates and enthalpies for vacancies in tetragonal zirconia. This revised version was published online in August 2006 with corrections to the Cover Date.     

A smoothness preserving fictitious domain method for elliptic boundary-value problems

^** Email: mommer{at}math.uu.nl We introduce a new fictitious domain method for the solutionof second-order elliptic boundary-value problems with Dirichletor Neumann boundary conditions on domains with C² boundary.The main advantage of this method is that it extends the solutionssmoothly, which leads to better performance by achieving higheraccuracy with fewer degrees of freedom. The method is basedon a least-squares interpretation of the fundamental requirementsthat the solution produced by a fictitious domain method shouldsatisfy. Careful choice of discretization techniques, togetherwith a special solution strategy, leads then to smooth solutionsof the resulting underdetermined problem. Numerical experimentsare provided which illustrate the performance and flexibilityof the approach.     

Differentiable evaluation of objective functions in sampling design with variance-covariance matrices

In this short note we consider the differentiable evaluation of the objective function of the sampling design optimization problem based on the inverse of the Fisher information matrix, and where the integer design variables have been converted into real variables using a relaxation technique. To ensure differentiability and cover the full range of the variables, and thus improve the convergence behavior of derivative-based optimization algorithms, we propose applying a Cholesky decomposition on the Fisher information matrix, but using a special higher precision floating point arithmetic to ensure stability. While each evalu-ation of the functional becomes slower, the algorithm is much simpler, amenable to be used with automatic differentiation directly, and can be shown to be very stable. For many practical situations, this is a valuable trade-off. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new fictitious domain method in shape optimization

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from Mommer (IMA J. Numer. Anal. 26:503–524, 2006) to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method can be applied to three dimensional shape optimization problems.     

Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.     

Tailored Thiol‐Functional Polyamides: Synthesis and Functionalization

In this article, a synthetic concept for the preparation of polyamides with functional side groups is described. First, the synthesis of a bis(thiolactone) monomer is shown in a concise three‐step route from itaconic acid and DL‐homocysteine thiolactone. The reactivity of the resulting bis(thiolactone) toward hexyl amine is examined. Next, the bis(thiolactone) is reacted as A,A‐type monomer with different B,B‐type comonomers (1,12‐diaminododecane and 1,3‐bis(aminopropyl)tetramethyldisiloxane). Ring opening of the thiolactones by the diamines leads to polyamides with pendant thiol groups. Using two diamines in different ratios, the properties of the resulting polyamides are tuned (thermal properties are determined) and different molecular weights are acquired. Subsequently, the thiol groups are reacted with methyl acrylate via Michael addition to functionalize the polyamides. Functionalization of thiol‐functional polyamides using poly(ethylene glycol) monomethyl ether (mPEG) acrylates ( = 480 and 1700 g mol⁻¹) results in water‐soluble amphiphilic poly­amides with molecular weights higher than 10 000 g mol⁻¹.