臺灣博碩士論文加值系統

原子力顯微鏡在多模態激發情形下之探針反應，對於量測奈米尺寸形貌和表面性質是非常有效率的，基於修正耦合應力理論為基礎，推導受到雙模激發、凡德瓦力及尺寸效應參數影響之原子力顯微鏡探針的數學模型，研究得到的結果如下：(1)尺寸效應參數越大、相互作用距離越小，頻移越大。(2)採用越小相互作用距離，頻移越明顯，雖變化微量的距離，就會產生明顯的頻移，可提高量測精度。(3)雙模共振的反應頻譜與單模共振有顯著的不同。(4)具有不同尺寸效應參數的兩個系統的自然頻率關係。
另以修正偶應力理論為基礎，推導原子力顯微鏡非均質探針的數學模型，得到一種解析方法滿足整數倍頻率關係之原子力顯微鏡探針的設計，得到兩個相似性條件中，即使有尺寸效應影響，仍可保持整數倍頻率關係，並發現自然頻率、等效彈性常數和幾何參數之間的關係，特定頻率和等效彈性常數可以經由關係式調整探針參數來獲得。研究得到有關特定頻率和等效彈性常數的探針尺寸設計方法，如下：(1)如探針厚度固定且不同探針的寬度斜率相同，則它們的無因次頻率整數倍關係是相同的，且該現象與尺寸效應無關。(2)根據推導之公式，可得到探針的原始厚度和長度，進而求得特定頻率。(3)根據推導之公式和原始厚度和長度，可得到原始寬度，進而求得特定的等效彈性常數。

We presented a mathematical model of AFM probe subjected to multimode excitation based on the modified couple stress theory. We proposed the semianalytical solution of the system. We investigated the response spectrum and transient behavior of AFM probe subjected to multimode excitations. It is very helpful to predict the surface properties and nanotopography based on the response of multimodes excitation. The effects of the interacting distance, size parameter, and root excitation on the frequency shift and response spectrum are investigated. We discovered the resonant frequency relation of the two systems with different size parameters and expressed in a formula. The natural frequencies determined by the semianalytical method and those predicted via the formula are extremely consistent.
We derived the mathematical model of an AFM nonuniform probe in the modified couple stress theory here. The analytical method is suggested to design the configuration of the AFM probe satisfactory the integer-multiples relation among frequencies. The changful cross-section of the probe is adjusted to adaptation the specific requirement. One invents two resemblance conditions such that the integer-multiples relation is retentive in spite of the size-dependency parameter. In addition, the relations among the geometry parameters, the effective spring constant and the natural frequencies are discovered. The spring constant and specific frequencies can be received by adjusting the probe parameters via the relations.

中文摘要 i
ABSTRACTii
誌謝 iv
表目錄 vii
圖目錄 viii
符號說明 ix
一、緒論 1
1.1前言 1
1.2原子力顯微鏡簡介 6
1.3凡德瓦力簡介 11
1.4研究動機及目的 13
二、基本理論與研究方法 28
2.1以修正耦合應力理論為基礎受到多模態激發的動態系統 28
2.1.1統御方程式和邊界條件 28
2.1.2解析法 31
2.1.3瞬態反應的數值結果及討論 40
2.2預測自然頻率 53
2.2.1具有尺寸效應的系統 53
2.2.2無尺寸效應的傳統系統 54
2.2.3自然頻率與尺寸效應參數之間的關係 56
三、簡諧探針設計方法 59
3.1以修正耦合應力理論為基礎的數學模型 59
3.2解析法 60
3.2.1分離變數法 60
3.2.2頻率方程 61
3.2.3使用修正Frobenius方法求精確的基本解 61
3.2.4原子力顯微鏡簡諧探針和尺寸效應 65
3.3共振頻率與尺寸效應參數之間的關係 67
3.3.1修正耦合應力理論中的統御方程式和邊界條件 67
3.3.2傳統理論中的統御方程式和邊界條件 69
3.3.3共振頻率與尺寸相關參數之間的關係 69
3.3.4整數倍關係的數值驗證 70
四、結論與建議 80
4.1結論 80
4.2未來研究建議 81
參考文獻 82
附錄A：無探針尖端質量系統之模態 87
附錄B：有探針尖端質量系統之頻率方程式 92
附錄C：標準基本解多項式 93
附錄D：Frobenius 解法 100

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