Implementation of a diffusion convection surface evolution model in WallDYN

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Abstract

In thermonuclear fusion experiments with multiple plasma facing materials the formation of mixed materials is inevitable. The formation of these mixed material layers is a dynamic process driven the tight interaction between transport in the plasma scrape off layer and erosion/(re-) deposition at the surface. To track this global material erosion/deposition balance and the resulting formation of mixed material layers the WallDYN code has been developed which couples surface processes and plasma transport. The current surface model in WallDYN cannot fully handle the growth of layers nor does it include diffusion. However at elevated temperatures diffusion is a key process in the formation of mixed materials. To remedy this shortcoming a new surface model has been developed which, for the first time, describes both layer growth/recession and diffusion in a single continuous diffusion/convection equation. The paper will detail the derivation of the new surface model and compare it to TRIDYN calculations.

Introduction

One of the most important questions for future long pulse thermonuclear fusion experiments is the redistribution of the first wall material in time due to the combination of erosion, plasma-transport and (re-) deposition. The net balance between erosion and deposition determines the life time of the first wall and co-deposition with fuel species dominates fuel retention in machines, particularly in those with a low-Z first wall elements (e.g. Be in ITER) [1]. Additionally the surface composition that evolves in time is different from the initial material configuration since mixed material layers are formed by the continuous erosion/(re-) deposition. These mixed layers typically have quite different properties than the initial pure elements and can potentially hamper machine operation.

The global erosion/deposition balance and the formation of mixed layers are highly dynamic processes. They result from the tight coupling between transport of eroded impurities in the scrape off layer (SOL) plasma and the erosion/(re-) deposition processes at the first wall surface. However most current SOL codes operate under the assumption of a wall with static composition and most wall codes assume a static particle influx spectrum. Both of these assumptions do not hold in reality. Therefore the WallDYN code [2] has been developed which couples surface processes and plasma transport to track the global material erosion/deposition balance. In [2] it was shown that long range transport of material in a tokamak is a multi step process: Impurities are transported from one position to another by series of erosion/deposition/re-erosion and re-deposition steps. Therefore to describe the global erosion/deposition balance one must track these steps by following the evolution of the surface composition in time. In WallDYN this is done by describing the changes in the impurity influx at a given poloidal position due to erosion of impurities at all other poloidal positions as an algebraic equation system. The change in the surface composition resulting form this impurity influx is modeled by a simple ordinary differential equation (ODE) [3]. This approach allows one to couple plasma transport and surface dynamics in a single differential algebraic equation (DAE) system. This DAE system can be solved using linear multistep methods [4] which allows to truly (i.e. not iteratively, all processes occur simultaneously) couple different physical processes. As was also already pointed out in [2] this has numerous advantages over iteratively coupling SOL codes to Monte Carlo (MC) surface codes or even to Molecular Dynamics (MDs) and Density Functional Theory codes: Iterative coupling always occurs on different time scales and the error propagation during iterative coupling can be considerable. Also the continuous description does not suffer from sampling artifacts like MC codes (e.g. TRIM) do for incident particle spectra with very small flux fractions.

It should however also be noted that the continuous descriptions of physical processes have the disadvantage that the required rate determining parameters like sputter or reflection yields, with their potentially complex composition dependence, must be included in the models in a parameterized way. As was shown in [2] this is possible and allows to include the experimental data from 2¨0 years of plasma wall interaction research and the output of sophisticated codes in WallDYN. Therefore WallDYN is not meant to replace MD, MC or DFT methods but is a tool to properly include their output in global material transport simulations.

The current surface model in use in WallDYN only tracks the surface composition evolution but it is well known that mixed material formation, in particular at elevated temperatures, is strongly influenced by diffusion. Further the current surface model does not fully handle layer growth/recession which is important to properly model co-deposition with fuel species. Following the WallDYN concept a continuous description of layer growth/recession and diffusion is required to be able to describe all processes in a DAE system. The current approaches to model thickness changes in a surface due to erosion/deposition typically involve discrete histograms of layers whose thickness and number are changed to model layer growth and recession. However this discrete description cannot in a clean way (i.e. non-iteratively) be coupled to diffusion which by nature is a continuous process described by Fick’s second law. Therefore a new surface model was developed which for the first time describes both layer growth and recession together with diffusion in a single, continuous partial differential equation.

The paper will describe in detail the derivation of the PDE for the new surface model and how it is integrated into the WallDYN DAE system approach. Then the model will be compared to TRIDYN [5] which is the current standard to describe surface composition changes due to sputtering.

Section snippets

Model description

The change in thickness of a sample during erosion and implantation of elemental species is due to a relaxation of the total local number density ρTOT. When material is net deposited (implanted) ρTOT increases by ΔρTOT whereas when material is net eroded ρTOT decreases by ΔρTOT. As a response to this change in ρTOT the system relaxes either by reducing the thickness by δx (for ΔρTOT<0) or increasing the thickness by δx (for ΔρTOT>0). A local change in thickness by δx at position x within the

TRIDYN comparison

Eq. (8) is not specific to erosion deposition modeling but is generally applicable to 1D systems subject to thickness changes due to the density relaxations. To make it applicable to model erosion/deposition due to plasma impact an expression for Φi(x,t), the change in number density of species i at position x, has to be specified. For the first tests of the model a comparison with the TRIDYN Monte Carlo code was chosen. To that end Eq. (9) was used for Φi(x,t).Φi(x,t)=ΓiIn(1-Ri(Ei))ξiDep(x)-j=

Conclusions

The WallDYN approach allows to model complex coupling between plasma transport and surface PWI processes leading to a global erosion/deposition balance and the formation of mixed materials. The fundamental WallDYN paradigm is to truly couple the processes on the same time scale (i.e. not iteratively). This makes it necessary to describe all processes in a continuous form (PDEs, ODEs and algebraic equation systems) such that they can be combined in a large DAE system which can then be solved

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