由於計算機的運算速度以及數值方法的功能日益增強，利用計算流體力學(Computational fluid dynamics; CFD)來完成系統之模擬分析以輔助實驗之不足與降低風險，並提昇產品品質與節省研發所耗之成本，已成為未來之趨勢。
本論文以二維模擬於雷射基座加熱生長(Laser-heated pedestal growth; LHPG) 系統之微浮動熔區氣液與固液介面形狀的研究，數值方法係使用有限體積差分方法(control-volume finite difference method )之非正交體適網格系統(non-orthogonal body-fitting grid system)，使得介面形狀能夠被有效率與精準測定。在穩定長晶情況下，各種長晶速度的熔區在CO2雷射加熱下，其所允許最高與最低加熱功\\\率的熔區長度和形狀能夠被詳細檢測出來，並且重力對微浮動熔區的效應也被模擬分析，它顯示水平生長方向LHPG 的可能性，此水平系統將有利於長晶纖的生長操作。
以上熔區形狀的實驗與模擬分析比對後，對於LHPG的熔區之熱傳(heat transfer)與流體流動(fluid flow)作了分析，在長度尺度約為一公分的系統中，完整的描述了包含原棒與熔區以及晶纖所組成的全域溫度與軸向溫度梯度。當比較原棒直徑尺度約為幾公分大之塊材的長晶系統時，較小的尺度熔區內，自然對流(natural convection)強度將下降約六個數量級。因而是熱傳導決定了熔區內的溫度分布而不是熱對流效應。對流呈現由熱毛細對流(thermocapillary convection)與質傳對流(mass-transfer convection)主控，而且熱毛細對流強度最後演化也變得比質傳對流大，主控了熔區內流場分布。當熱毛細對流係數大到約為10-4~10-3時，熔區的形狀呈現對稱與雙渦流分佈，因縮徑對熔區內流場所造成的影響是非常小的。經由以上的分析，其結果將可更進一步提供爾後濃度分析與水平生長製程之基礎。

Recently the computational speed and the functions of the numerical methods are advancing rapidly. It is the future trend that using the computational fluid dynamics (CFD) to perform simulation for making up the experimental deficiency, reducing the risk, improving the quality of the product, and saving the cost of research and development.
A two-dimensional simulation was employed to study the melt/air and melt/solid interface shapes of the miniature molten zone formed in the laser-heated pedestal growth (LHPG) system. Using non-orthogonal body-fitting grid system with control-volume finite difference method, the interface shape can be determined both efficiently and accurately. During stable growth, the dependence of the molten-zone length and shape on the heating CO2 laser is examined in detail under both the maximum and the minimum allowed powers with various growth speeds. The effect of gravity for the miniature molten zone is also simulated, which reveals the possibility for a horizontally oriented LHPG system. Such a horizontal system is good for the growth of long crystal fibers.
After comparing with the shape of the molten zone in terms of the experiment and the analysis of the simulation shown as above. Heat transfer and fluid flow in the LHPG system are analyzed near the deformed interfaces. The global thermal distributions of the crystal fiber, the melt, and the source rod are described by temperature and its axial gradient within length of ~10 mm. As compared with the growth of bulk crystal of several centimeters in dimension, natural convection drops six orders in magnitude due to smaller melt volume; therefore, conduction rather than convection determines the temperature distribution in the molten zone. Moreover, thermocapillary convection rather than mass-transfer convection becomes dominant. The symmetry and mass flow rate of double eddy pattern are significantly influenced by the molten-zone shape due to the diameter reduction and the large surface-tension-temperature coefficient in the order of 10-4~10-3. According to the analysis shown as above, the results could be further extended for the analysis of the concentration profile and study of horizontal growth.

中文摘要……………………………………………………………………….. i
Abstract ……………………………………………………………………….. ii
Table of Content. ……………………………………………………………... iv
List of Tables…………………………………….…………………………….. vi
List of Figures…………………………………………………………….…… vii

Chapter 1 Introduction……………………………………………………….. 1
1.1 History of crystal growth…………………………………………………. 1
1.2 Micro-floating zone by the LHPG method………………………….……. 4
1.3 Numerical simulation of the floating zone method…………….………… 4
1.4 Thesis structure……………………………………………………….…... 5
Chapter 2 Crystal growth theory and experiment…………………….……. 7
2.1 Crystal growth theory……………………………..………………….…... 7
2.1.1 Fundamentals……………………………………………….……. 7
2.1.2 Nucleation……………………………………………….……….. 8
2.1.3 Growth…………………………………………………………… 13
2.1.4 Solute distribution during freezing……………………….……… 14
2.2 LHPG micro-floating zone system……………………………………….. 18
2.3 Physical properties of yttrium aluminum garnet melt…………………... 23
2.3.1 Density…………………………………………………………… 24
2.3.2 Surface tension…………………………………………………... 25
2.3.3 Contact angle…………………………………………………….. 26
2.3.4 Viscosity…………………………………………………………. 27
Chapter 3 Conservation laws of fluid dynamics……………………………. 28
3.1 Governing equation of the low-doped incompressible fluid under three-dimensional space……………………………………………….….
28
3.1.1 Continuity equation…………………………….…………..…….. 28
3.1.2 Equation of motion…...……………………………….................. 31
3.1.3 Equation of energy…………………………………………….…. 38
3.1.4 Equation of mass-transfer………………………………………... 42
3.2 Governing equation of the stream function under two-dimensional coordinates………………………………………………………….……..
48
3.2.1 Stream function..………………………………............................ 48
3.2.2 Two-dimensional stream function in cylindrical coordinates……. 50
Chapter 4 Boundary conditions and two-dimensional numerical model of the LHPG system……………………………………………….….
52
4.1 Physical model of the LHPG system………………................................... 53
4.2 Boundary conditions………….………………………………….…..…… 54
4.3 Coordinate transformation and numerical model…..................................... 57
4.4 Shape of the free surface…….…………..................................................... 59
4.5 Numerical method..……………………...................................................... 60
Chapter 5 Comparison between experiments and simulations…………….. 62
5.1 Laser intensity profile..…………………………………..…………..…... 62
5.2 Shape of molten zone……………………………………………..……… 64
5.3 Slope of the melt-zone length divided by input power……………….….. 67
Chapter 6 Discussions on the free surface, temperature distribution, and flow field……………………………………………………………
69
6.1 Free surface shape………………………………………….……………... 69
6.1.1 Curvature of the free surface……………………………………... 70
6.1.2 Gravity effect on micro-floating zone…………….……………… 72
6.2 Temperature distribution and flow field..…................................................. 73
6.2.1 Global isotherms and streamlines………………………………... 73
6.2.2 Diameter reduction effect on isotherms and streamlines…….…... 77
6.3 Convection effects on isotherms and streamlines……………..….………. 78
Chapter 7 Conclusions and future work.......................................................... 83
7.1 Conclusions….…………….…..…………………..……………………… 83
7.2 Future work……..…………..…………………………………………….. 84

References ……..………………………………………..……………………. 89
Biography ………..……………………………………..…………………….. 100
Publication List……………………………………………..……………..….. 101

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[7.4]P. Y. Chen, C. L. Chang, K. Y. Huang, C. W. Lan, W. H. Cheng, and S. L. Huang, “Experiment and simulation on interface shapes of yttrium-aluminium-garnet miniature molten zone formed using laser-heated pedestal growth method for single crystal fibres,” to be published in J. Appl. Cryst. 42, (2009).
[7.5]K. Y. Huang, K. Y. Hsu, & S. L. Huang, “Analysis of ultra-broadband amplified spontaneous emissions generated by Cr4+:YAG single and glass-clad crystal fibers,” IEEE J. Lightwave Technol. 26, 1632 (2008).
[7.6]K. Y. Huang, K. Y. Hsu, D. Y.Jheng, W. J. Zhuo, P. Y. Chen, P. S. Yeh, & S. L. Huang, “Low-loss propagation in Cr4+:YAG double-clad crystal fiber fabricated by sapphire tube assisted CDLHPG technique,” Opt. Exp. 16, 12264 (2008).
[7.7]C. W. Lan, C. J. Chen, “Dynamic three-dimensional simulation of facet formation and segregation in Bridgman crystal growth,” J. Cryst. Growth 303, 287 (2007).
[7.8]J. C. Chen, K. Y. Huang, C. Nan Tsai, Y. S. Lin, C. C. Lai, G. Y. Liu, F. J. Kao, S. L. Huang, C. Y. Lo, Y. S. Lin, and P. Shen, “Composition dependence of the micro-spectroscopy of Cr ions in double-clad Cr:YAG crystal fiber,” J. of Appl. Phys. 99, 093113 (2006).
[7.9]J. C. Chen, C. Y. Lo, K. Y. Huang, F. J. Kao, S. Y. Tu, and S. L. Huang, “Fluorescence mapping of oxidation states of Cr ions in YAG crystal fibers,” J. Cryst. Growth 274, 522 (2005).