原子力學顯微鏡(AFM)可用來量測形貌。由AFM量測的表面高度z(x)，可得到表面的均方標準高度誤差、均方斜率以及均方曲率，再將其轉換成為粗糙表面的統計粗糙度參數，例如粗糙峰密度，粗糙峰標準高度誤差，以及平均表面高度(Ra)與平均粗糙峰高度(ys)之間距。此外，為了決定粗糙表面的碎形粗糙度參數，使用2D variation method來獲得粗糙表面的碎形維度(D)。且scaling constant (Cp)可從表面能量頻譜獲得。因為碎形維度和scaling constant已知，可得topothesy (G)。圓球探針頭和粗糙表面之接觸可以模擬成為一光滑平面和粗糙表面之接觸。此粗糙表面假設有高斯分布的圓球粗糙峰聚集。而表面粗糙度特性可藉由統計或碎形參數來描述。分子動力學(MD)模擬用來觀察原子尺度下，光滑平面和圓球粗糙峰之乾式接觸（滑動）。圓球粗糙峰由8682個銅原子組成。上方光滑表面由8464顆碳原子組成。使用絕熱狀態，且開始溫度為300K。本研究也已經完成微觀模擬程序，用來獲得具圓球探針頭之探針以及具等效圓球粗糙峰半徑之粗糙表面之間的乾接觸（滑動）之接觸力和摩擦力。因為接觸力和摩擦力與干涉量之關係已知，可計算出每個粗糙峰之接觸力期望值和摩擦力期望值。根據粗糙表面統計粗糙度參數以及每個粗糙峰的接觸力和摩擦力期望值，總接觸力和總摩擦力可由個別粗糙峰的加總獲得。粗糙表面的碎形粗糙度參數，碎形維度、topothesy以及scaling constant已知，粗糙表面的等效曲率半徑以及等效干涉量可以決定。根據等效曲率半徑、等效干涉量，以及光滑平面和圓球粗糙峰之間接觸力、摩擦力與干涉量之關係，可以得到圓球探針頭和粗糙表面乾式接觸（滑動）之接觸力、摩擦力。分子動力學模擬顯示，光滑平面與圓球粗糙峰之間的乾式滑動摩擦力表現出原子尺寸的stick-slip行為，其顯示出近週期性的變動，週期接近0.36nm，而且發現摩擦係數是隨速度及接觸面積而增加的函數。在相同負載速度之下，增加干涉量使得接觸力、摩擦力與摩擦係數呈現上升的趨勢。在相同干涉量下，隨著滑動速度的加快，摩擦力和摩擦係數都現上升的趨勢。這說明摩擦力與界面間接觸面積以及滑動速度有關。由統計理論、碎形理論以及實驗所獲得的鑽石探針頭和銅粗糙表面之間的乾式接觸（滑動）之接觸力與摩擦力結果顯示，碎形理論所計算之接觸力與摩擦力比統計理論來得接近實驗結果。與統計理論不同的是，碎形理論以接觸面積上整體表面形貌之特性，配合分子動力學模擬或JKR理論，作為計算接觸力與摩擦力之依據。因此，碎形理論計算結果較接近實驗結果，而統計理論明顯低估接觸力與摩擦。此外，在較大干涉量下，接觸區域之變形機制改變，使得JKR彈性變形理論明顯高估接觸力與摩擦力，而分子動力學模擬能改進這樣的缺點。

　　The measurement of the topography was made with an atomic force microscope (AFM). It can determine the mean of surface heights (Ra), the surface standard height deviation, and the statistical or fractal roughness parameters of the rough surface. Translating the mean square standard height deviation, mean square slop and mean square curvature of the surface determined by the surface height z(x) measured by AFM, into the statistical roughness parameters of rough surface, such as the summit density, the summit standard height deviation, and the difference in mean of surface height (Ra) and mean of asperity heights (ys). In order to determine the fractal roughness parameters of rough surface, a two-dimensional variation method was used to obtain the fractal dimension (D) of the rough surface. And the scaling constant (Cp) can be obtained from the power spectrum of the rough surface. Because the value of the fractal dimension and scaling constant are known, we can obtain the topothesy (G) from the relationship among topothesy, fractal dimension and scaling constant. The contact between a spherical tip and a rough surface can be modeled by a smooth plane in contact with a rough surface. The rough surfaces are modeled by a collection of spherical asperities with Gaussian height distribution and the surface roughness characters can be described by the statistical parameters or the fractal parameters. Molecular dynamics (MD) simulation was carried out to investigate the dry contact (sliding) between a smooth plane and a spherical asperity and the stick-slip on an atomic scale by using Morse potential for C-Cu. The spherical asperity comprised of 8682 copper atoms. The upper smooth plane was composed of 8464 carbon atoms. The atoms of bottom layer of the spherical asperity were fixed at their original positions. Adiabatic thermal conditions were imposed, and the starting temperature was 300K. In dry contact (sliding) between the smooth plane and the spherical asperity, the dependence of contact force and friction force on interference determined by the molecular dynamics simulation, Hertz or JKR contact mechanics theories. We also have carried out a program of microscopic simulations to gain the contact force and friction force in dry contact (sliding) between the probe with spherical tip and the rough surface with equivalent spherical asperities. Because the dependence of contact force and friction force on interference in dry contact (sliding) between the smooth plane and the spherical asperity are known, the expected contact force and friction force of each asperity can be calculated. According to the statistical parameters of the rough surface and the expected contact force and expected friction force of each asperity, the total contact force and friction force are obtained by summing the individual asperity contributions. Moreover, the fractal parameters, the fractal dimension, topothesy and scaling constant, of the rough surface are known, the effective radius of curvature and interference of the rough surface can be determined. According to the effective radius of curvature and the interference and the dependence of contact force and friction force on interference between the smooth plane and the spherical asperity, the contact force and friction force in dry contact (sliding) between the spherical tip and the rough surface are obtained. The molecular simulation results show that the dry sliding friction between the smooth plane and the spherical asperity reveal atomic scale stick-slip behavior which display a quasiperiodic variation with a period of approximately 0.36nm , and found the coefficient of friction is increasing function of velocity and contact area. The results of contact force and friction force in dry contact (sliding) between the diamond tip and copper rough surface obtained by statistical theory, fractal theory and experiment also showed.

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