0-7803-9062-8/05/$20.00 C 2005 IEEE 6th. Int. Conf. on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2005 —1— Parametric Model Order Reduction for Scanning Electrochemical Microscopy: Fast Simulation of Cyclic Voltammogram L. H. Feng 1 , D. Koziol 2 , E. B. Rudnyi 2 , and J. G. Korvink 2 1 ASIC & State Key Laboratory, Microelectronics Department, Fudan University, China. lhfeng@fudan.edu.cn 2 IMTEK-Institute for Microsystem Technology, University of Freiburg, Freiburg, Germany Abstract We propose the use of parametric model reduction for fast simulation of cyclic voltammograms. The model for a cyclic voltammogram is treated as a system with a parameter (applied voltage) to be preserved during model reduction. The voltage is preserved in the symbolic form during model reduction and we can accurately simulate the cyclic voltammograms with a reduced system by spending much less time and memory as compared with direct simulation based on the original large-scale model. 1. Introduction Model order reduction allows us to find an accurate low-dimensional approximation for a high-dimensional system of ordinary differential equations (ODEs) obtained after the discretization in space by the finite element method [1][2]. A formal and computationally efficient procedure based on the Arnoldi algorithm takes as input the system of ODEs in the form Ed v x / dt + K v x = bu (1) with a state vector v x and convert it to a similar system ) Ed v z / dt + ) K v z = ) b u (2) that, however, has much smaller dimension of the state vector v z . In many engineering applications, the matrix K depends on parameters that should be preserved in the symbolic form. For example, these are film coefficients [3] in simulation of a thermal problem and the flow velocity in simulation of an anemometer [4]. In order to treat these important cases, conventional model reduction should be extended to parametric model reduction. In our paper we present an application of parametric model reduction to scanning electrochemical microscopy (SECM) [5]. In the next section, we construct a mathematical model for SECM that after discretization in space produces a high-dimensional ODE system. Then we introduce the parametric model reduction technique in section 3 and discuss how it can be applied to the SECM model in section 4. In section 5, the simulation results are presented. Finally, we give conclusions. 2. Case study We consider a cylindrical electrode operating in the feedback mode of SECM as shown in Figure 1. The computation domain under a 2D-axisymmetrical approximation includes the electrolyte under the electrode. We assume that the concentration does not depend on the rotation angle. A single chemical reaction takes place on the electrode: Red e Ox f b k k ⇔ + − (3) According to the theory of SECM [6], the species transport in the electrolyte is described by diffusion only. The diffusion partial differential equations are given by Fick’s second law as follows 1 2 1 1 / c D dt dc ∇ ⋅ = (4) 2 2 2 2 / c D dt dc ∇ ⋅ = (5) The Buttler-Volmer equation has been used to describe the reaction rate on electrode surfaces for the chemical reaction (3) d d Ox Ox c k c k j Re Re ⋅ − ⋅ = (6) The reaction constants for the forward reaction and the backward reaction are given as follows k f = k Ox = k 0 e αzF( E− E 0 ) RT       (7) k b = k Re d = k 0 e −(1−α )zF ( E−E 0 ) RT       (8) where 0 k is the heterogeneous standard rate constant, and α = 0.5 is an empirical transmission factor for a heterogeneous reaction. F is the Faraday-constant, R is the gas constant, T is the temperature and n is the number of exchanged electrons per reaction. This allows us to write the boundary conditions at the electrode as follows: j n c = ⋅ ∇ v 1 and j n c − = ⋅ ∇ v 2 (9) The control volume method has been used for the spatial discretization of the equations above. The resulting system of ordinary differential equations is as follows Ed v c / dt + K{ U (t )} v c = b , v c (0) = v c 0 ≠ 0 (10) where E and K{U(t)} are system matrices, n R c ∈ v is the vector of unknown concentrations. n R means that there are n elements in the vector c v and n is usually referred to as the dimension of the system (10). The vector b is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk of the electrolyte. There are two important differences between Eq (10) and (1). First, the matrix K depends on the voltage that in turn depends on time in the simulation of a cyclic voltammogram. This feature must be preserved in the reduced model. Second, the initial condition of the system is always nonzero in our case, i.e. 0 0 ≠ c v .