在本篇論文中，我們推導出Lifshitz理論在不同幾何結構下的凡得瓦自由能方程式，我們嘗試在兩平行介質平板各插入一定密度的細圓柱陣列，並假設當密度趨近於無窮小時，這兩摻雜的平行介質平板可以近似為兩平行細圓柱對。碳奈米管為類圓柱狀結構，我們將利用DFT計算出來不同極化方向的碳奈米管介電函數，代入我們的Lifshitz理論中，計算出(8,4)、(10,0) 碳奈米管陣列間的Hamaker常數，與(8,4)、(10,0)、(16,0)碳奈米管對的Hamaker常數。

In this thesis, we use the Lifshitz theory to figure out the van der Waals free energy function of various geometrical configurations. We try to add parallel thin rods into two parallel planer bodies across isotropic media, and then we control the density of rods, if density approaches infinitely small, these two “doped” parallel planer bodies will be looked like a pair of two parallel rods. Then we compute the vdW interaction by using the results of DFT theory. Here we figure out the dielectric functions of (8,0), (10,0), (16,0) CNT with polarizations at different orientation, and then we get the Hamaker constant of (8,4), (10,0) CNT arrays and also Hamaker constant of (8,4), (10,0), (16,0) CNT pairs.

Chapter 1導論..................................................1
	1-0 研究動機................................................1
	1-1 凡得瓦力的三種型式......................................1
	1-2 Hamaker理論.............................................3
	1-3 Casimir效應.............................................4
	1-4 論文架構................................................5

Chapter 2 Lifshitz 理論........................................6
	2-0 Lifshitz理論簡介.......................................6
2-1 Hamaker常數方程式......................................7
2-2空間中的自由能..........................................8
2-3 電磁波行進之色散關係...................................10
2-4 允許表面模式下的自由能.................................13

Chapter 3圓柱間之凡得瓦力.....................................18
	3-0 簡介...................................................18
3-1 旋轉系統...............................................18
3-2 旋轉細圓柱對系統.......................................22
Chapter 4碳奈米管之間的凡得瓦力計算...........................30
	4-0 簡介...................................................30
4-1 介電函數...............................................30
4-2 碳奈米管陣列之凡得瓦力.................................37
4-3 碳奈米管之間之凡得瓦力.................................38

Chapter 5結論.................................................40

附錄..........................................................41
參考文獻......................................................48

表目錄
表1-1 凡得瓦力的三種類型， 代表的是吸引力， 代表的是斥力。		............................................................2
表2-1 不同幾何結構下的Hamaker自由能方程式							............................................................7
表4-1 純水的介電函數 參數											...........................................................31
表4-2 兩8埃碳奈米管陣列之間的Hamaker常數 (zJ)					...........................................................38
表4-3 奈米管對之Hamaker常數							...........................................................38
表4-4 常用的Hamaker常數							...........................................................39

 
圖目錄
圖1-1成對相加近似與凡得瓦力..................................3
圖1-2 Casimir效應裝置........................................4
圖2-1 Lifshitz理論...........................................6
圖2-2 兩平行無窮長介質板L、R挾著一個厚度為 的介質m，各自的介質常數分別以 、 、 表示。..............................9
圖2-3	對虛數頻率軸做柯西積分。...............................14
圖3-1	兩平行無窮長介質板L、R挾著一個厚度為 的介質m，各自的介質常數分別以 、 、 表示。.......................19
圖3-2	在複合材料A與B當中，管子們的主軸與X軸平行，而且在管子與管子之前填滿著介電常數為 的介質，與兩塊複合材料之中的介質相同。..................................................23
圖3-2b	管子的直徑大小為2a，軸向與徑向的介電常數分別為 和 。...................................................23
圖3-3	在yz平面上看，兩複合材料L、R相距 ，圖中箭頭虛線表示在平板L與平板R裡取一小段截面 中的平行圓柱 、 、 有作用力產生。..................................................27
圖4-1純水的介電函數 ......................................32
圖4-2碳水化合物奈米管的介電函數 ..........................32
圖4-3 (8,4)碳奈米管在不同極化時的介電函數虛部 .............34
圖4-4 (10,0)碳奈米管在不同極化時的介電函數虛部 ............34
圖4-5 (16,0)碳奈米管在不同極化時的介電函數虛部 ............35
圖4-6 (8,4)碳奈米管的介電函數 .............................35
圖4-7 (10,0)碳奈米管的介電函數 ............................36
圖4-8 (16,0)碳奈米管的介電函數 ............................36圖4-9 在 面積裡共有n=9根碳管................................37
圖4-10不同N的截面；左圖(圖4-10a)，N=0.006；右圖(圖4-10b) N=0.01.................................................37
圖4-11 角度對碳奈米管對之Hamaker常數的影響...................39

[1] Iijima S, Nature, 354 56, 1991.
[2] van der Waals, J. D. Thesis, Leiden, 1873. 
[3] Keesom, W. H. Prov. R. Acad. Sci. Amsterdam 1915, 18, 636.
[4] Keesom, W. H. Prov. R. Acad. Sci. Amsterdam 1920, 23, 939.
[5] Keesom, W. H. Phys. Z. 1921, 22, 129.
[6] Keesom, W. H. Phys. Z. 1921, 22, 643.
[7] Debye, P, Phys. Z. 1920, 21, 178.
[8] Debye, P, Phys. Z. 1921, 22, 302.
[9] London, F. Z. Phys. 1930, 63, 245.
[10] Overbeek, J. Th. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevire; Amsterdam, 1952; Vol. Ⅰ, p 245.
[11] Fowkes, F. M. In Physicochemical Aspects of Polymer Surfaces; Mittel, K. L., Ed.; Plenum; New York, 1983; Vol. 2, p 583.
[12] Hamaker, H. C.“The London – van der Waals attraction between spherical particles. “ Physica 4(10), 1058–1072.(1937)
[13] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
[14] Lifshitz, E.M. Zh. Eksp. Theo. Fiz. 1955, 29, 94.
[15] V. Adrian Parsegian, van der Waals Forces – A Handbook for Biologists, Chemists, Engineers, and Physicists, Table Pr.1.
[16] M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, Cambridge, 1958)
[17] V. Adrian Parsegian, Nonretarded van der Waals Interaction berween Anisotropic Long Thin Rods at all Angles, J. Chem. Phys., 56,9 (1972).
[18] Lord Rayleigh (J.W. Strutt), Phil. Mag. 34, 481 (1892).
[19] I. Runge, Z.Tech. Phys. 6, 61(1925), quoted in Merideth and Tobias, Advan. Electrochem. Electrochem. Eng. 2, 15(1962)
[20] J. Mahanty and B.W. Ninham, Dispersion forces, p 67.
[21] V. Adrian Parsegian, van der Waals Forces – A Handbook for Biologists, Chemists, Engineers, and Physicists, Table L2.1
[22] J. Chun and J.L. Li, Anisotropic absorption of molecular assemblies on crystalline surfaces, J. Phys. Chem. B 2006, 110, 16624-16632.
[23] G.L. Zhao, D. Bagayoko, Optical properties of aligned carbon nanotube mats for photonic applications, J. Phys. 99, 114311(2006).
[24] Robert J. Hunter, Foudations of Colloid Science Vol. 1, p. 203.
[25]Jacob Israelachvili, “Intermolecular & Surface Forces,”p.186.
[26] J. L. Lee, “Use the dielectric functions in theory of dispersion forces,” Phys. Rev. B 71, 235412 (2005).