微流道中無電鍍問題之數學建模與數值分析

Abstract

微流道中的無電鍍製程是一項在工業中具有發展潛力的技術。從物理角度來看，這是一個多重物理耦合問題：需要考慮流體力學、質傳、化學反應、相變態等等，特別是在微小尺度下需要考慮更多細膩的物理行為。本論文主要從數學建模與數值分析的角度研究無電鍍問題，並將它們寫成三個章節：第一章將簡介無電鍍製程並回顧其相關文獻；第二章將討論無電鍍問題在單相流中的數學模型以及數值分析；第三章則是連無電鍍生成氣泡的狀況也一起討論，並探討如何模擬以及一些數值分析。 • 在第二章中，我們不考慮無電鍍反應中氣泡產生的效應，因此我們只考慮單相不可壓縮流與質傳的耦合問題，此外我們利用蒸發逼近的技巧來模擬物質上鍍造成的小幅度邊界移動。因此簡化的模型方程為Navier-Stokes 方程以及擴散散對流方程的耦合並考慮非線性且非典型的邊界條件。我們將證明其中濃度方程解之存在性與唯一性並利用數值分析證明我們提出的數值方法與演算法之可行性。 • 氣體產生與氣相流體的移動問題在第三章中會被探討。由於氣泡會在整個物理空間中隨機地產生，因此我們採用平均形式的二相流方程以及擴散對流方程與通量邊界條件耦合作為模型方程，此外上鍍造成的邊界移動也有考慮。在數值方法方面，時間離散採用一階特徵線法；空間離散採用有限算法，且我們證明了數值方法的適定性。數值驗證方面我們將比較一、二維問題並與微流道實驗進行比較。 • 在附錄中，我們研究更加簡化的二相流模型。在這個模型中，我們忽略對流項的效應。我們考慮三條濃度方程：其中兩條代表參與無電鍍反應的化學物質傳輸以及一條代表溶解於水中的氣體傳輸，並與一條描述液體體積分率的常微分方程耦合。為了描述表面反應，我們考慮滿足電荷平衡的通量邊界條件。我們證明了此模型方程的解之存在性與唯一性，此證明將兩種化學物質推廣至N種化學物質也適用。

Electroless plating process in microchannel is a rising technology in industry. From a physical point of view, it is a multiphysics problem including fluid dynamics, mass transfer, chemical reaction, phase change, etc.. Especially in micrometer scale, more subtle physical phenomena are of interest. In this thesis, electroless plating problem is mostly taken care by mathematical modeling and numerical analysis. There are three chapters in this thesis: A quick review and introduction of electroless plating process are given in Chapter 1. Analysis of an electroless plating problem in a single phase liquid flow is presented Chapter 2. The numerical simulation on the electrolss plating plating problem with gas generation is discussed in Chapter 3. • In Chapter 2, the gas generation due to the electroless plating is neglected. Instead, single phase incompressible flow coupled with mass transfer is considered. The small boundary motion owing to the deposited chemical species is modeled by a transpiration approximation. With this simplification, the mathematical model, consists of a Navier- Stokes flow and an equation for the concentration of the plating chemical coupled by non-standard and nonlinear boundary conditions. Existence and uniqueness are proven for the concentration equation. Numerical analysis is carried out and justifies the proposed numerical schemes and nonlinear algorithm. • In Chapter 3, the gas generation and motion of gaseous phase are taken into account. Since the bubbles are generated randomly and everywhere, a volume averaged two phase flow model is applied. This simplification is coupled with convection-diffusion equations subject to flux boundary conditions satisfying electron balance. A first-order phase volume conservative method and finite element method are carried out for numerical simulation and the well-posedness of numerical scheme is proved. Numerical studies in one and two-dimensional cases with comparison to experiment are performed to justify the proposed model. • In Appendix B, a further simplified model for chemical species transport in two phase flow is considered. In this case, the convection terms are neglected so that the volume fraction of liquid phase depends only on the concentration of dissolving gas in the electrolyte. Three concentration equations for two chemical species transport and dissolving gas coupling with an ODE for volume fraction of liquid phase are considered. The flux boundary condition on the reacting surface with electron balance is taken care. The existence and uniqueness are proven for the coupling equations. It is shown that the two species case can be generalized to N-species case.