在軟凝態材料(soft condensed matter materials)中，分子在空間中的排列顯著影響其材料性質，如材料的強度和黏度。在眾多實驗工具中，小角度中子散射(small- angle neutron scattering, SANS)技術一直被用於探究軟凝態材料在分子尺度下的結構，而透過實球諧函數(real spherical harmonics, RSH)展開法分析不同樣品橫截面的 二維(2D) SANS光譜來重建三維(3D)異向性結構是一個方便的做法，但是當軟凝態材料受外場作用時，分子空間排列的原始對稱性被打破，導致需要更多的樣品橫截面2D SANS光譜參與分析和更複雜的計算才能完成重建三維結構。
為了解決上述的挑戰，我們提出一種可行的分析方法，利用對2D SANS光譜使用座標轉換的技巧，我們可以簡單的使用RSH展開法從實驗測量的2D SANS光譜中提取出3D扭曲結構，並透過高斯鏈(Gaussian chain)聚合物在仿射剪切變形(affine shear deformation)及仿射單軸拉伸變形(affine uniaxial tension deformation)下的研究進一步驗證了此方法的有效性，其中在不同變形條件下，數值與解析計算的結果之間有非常低的誤差。因此，我們的方法使SANS實驗能夠定量分析微觀結構的改變。此外，只要可以建構線性轉換矩陣，我們就可以利用此方法應用於經歷嚴重變形的軟凝態材料，其中不限制對稱性的種類。最後，提取的資訊為我們提供全面且深入的瞭解在嚴重扭曲下，材料原子核之間的結構變化。

In soft condensed matter materials, the spatial arrangement of molecular constituents significantly impacts their mechanical properties. Among existing experimental tools, small-angle neutron scattering (SANS) technique has been widely utilized for investigating materials structure at molecular levels. Conventionally, the real spherical harmonics (RSH) expansion serves as a convenient method to reconstruct three-dimensional (3D) anisotropic structures by analyzing the RSH components of two-dimensional (2D) SANS scattering spectra across different material cross-sections. However, when stress is introduced to materials, the original symmetry of the molecular spatial arrangements is broken, complicating the implementation of this method.
To address the aforementioned challenge, we introduce a feasible analytical approach that utilizes the coordinate and rotation transformation of 2D SANS spectra on different sample planes. With the aid of this method, the 3D distorted structure can be extracted from the experimentally measured 2D SANS spectra using RSH expansion. The validity of this approach was further verified by the case study of Gaussian chain polymer undergoing affine shear deformation, where the consistent agreement between the numerical results and the analytical ground truth under different deformation conditions has been observed. Therefore, our proposed approach enables quantitative determination of the microscopic distortion from SANS experiments. As a result, the extracted information provides a comprehensive and profound understanding of how molecular arrangements influence the material properties under substantial deformation.

論文審定書i
誌謝ii
摘要iii
Abstractiv
目錄v
圖目錄vii
表目錄ix
第一章 緒論1
第二章 背景知識與文獻回顧3
2.1小角度中子散射3
2.2實球諧函數展開法7
2.3 仿射變形9
2.4 二次型12
2.5 Gaussian chain13
2.6 擬合橢圓方程式13
第三章 實驗步驟15
3.1仿射變形後散射光譜的變化15
3.1.1 簡單剪切15
3.1.2 單軸拉伸17
3.1.2.1 拉伸軸沿z軸17
3.1.2.2 拉伸軸沿非z軸(方向已知)19
3.1.2.3 拉伸軸方向未知21
3.2 利用二次型定量分析應變23
3.2.1 簡單剪切23
3.2.2 單軸拉伸25
3.2.2.1 拉伸軸沿z軸25
3.2.2.2 拉伸軸方向未知27
3.3從2D散射光譜計算應變28
3.3.1簡單剪切31
3.3.2單軸拉伸33
3.3.2.1拉伸軸沿z軸33
3.3.2.2拉伸軸方向未知33
3.4 利用Debye function模擬散射光譜38
3.4.1簡單剪切38
3.4.2單軸拉伸40
3.4.2.1 拉伸軸沿z軸40
3.4.2.2 拉伸軸方向未知41
3.5 RSH展開等向性2D散射光譜42
第四章 結果與討論44
4.1 簡單剪切變形44
4.1.1 藉由光譜分佈計算剪應變的數值44
4.1.2 利用解析結果將系統回復至等向性狀態50
4.1.3 利用數值積分定量分析異向性散射光譜52
4.2 單軸拉伸變形(拉伸軸沿z軸)54
4.2.1 藉由光譜分佈計算拉伸應變的數值54
4.2.2 利用解析結果將系統回復至等向性狀態57
4.2.3 利用數值積分定量分析異向性散射光譜58
4.3 單軸拉伸變形(拉伸軸方向未知)60
4.3.1 藉由光譜分佈計算拉伸應變的數值60
4.3.2 利用解析結果將系統回復至等向性狀態64
4.3.3 利用數值積分定量分析異向性散射光譜65
第五章 結論68
參考文獻69

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