蘭吉芬方法在分子動力學模擬中，常用來提供一個等溫的平板邊界，以模擬液、氣態分子在固態壁面的運動情形。在本文，我們則利用此方法來計算一塊厚度為奈米級的白金平板，分析上下板溫度不同時，其內部的溫度分佈是否符合傅立葉定律。從模擬結果得知，在白金平板兩側溫度相同，其厚度為2~3奈米時，白金平板內的溫度分佈十分均勻，與統計熱力學所得到的理論值最多只有10%的差距。同樣的模擬方法用在兩側溫度不相同時，在16奈米以下，白金平板內部的溫度分佈是呈現等溫狀態，在兩側的原子層則有大幅度的溫度變動。但是在100奈米以上，其白金平板內部的溫度分佈是呈現梯度分佈，從高溫慢慢降至低溫。從模擬結果可以發現，當白金平板厚度小於16奈米，蘭吉芬方法可以作為一良好的溫度控制系統，傅利葉定律的溫度分佈情形並不適用。當白金平板厚度大於100奈米，其內部溫度分佈便符合巨觀的傅利葉定律。

The Langevin model frequently provides a numerical method to simulate the dynamic of liquid or gaseous molecules over an isothermal plate. In the present study, we utilize the Langevin model to investigate the validity of Fourier’s law for the temperature distribution of a platinum plate with a thickness is of nanometers. When temperatures on the top and bottom of plate are identical, the results show that the temperature distribution inside the plate is entirely uniform with a maximum uncertainty of 10% for the thickness of 2 ~ 3 nm. This approach is also applied to the conditions of two sides of the plate at different temperatures. When the thickness of the plate is smaller than 16 nm, the temperature distribution in the most part of plate is isothermal. The temperature jump is observed on both edges of the plate. Obviously, the Fourier’s law is not valid for the thickness below 16 nm. The results also reveal that the Langevin model is a good temperature controlling tool. As the plate thickness is increased to 100 nm, a constant gradient of temperature distribution inside the plate is obtained; thereby the Fourier’s law is applicable.

[1] Alder B. J. and Wainwright T. E. （1957）. “Phase transition for a hard sphere system,” J. Chem. Phys., vol.27, p1208-1209.
[2] Rahman A. （1964）. “Correlations in the motion of atoms in liquid argon,” Phys. Rev., vol.136, p405-411.
[3] Andersen H. C.（1980）. “Molecular dynamics at constant pressure and/or temperature,” J. Chem. Phys., vol.72, p2384-2393.
[4] Berendsen J. K. C., Postma J. P. M., van Gunsteren W. F., DiNola A. and Haak J. R.（1984）. “Molecular dynamics with coupling to an external bath,” J. Chem. Phys., vol.81, p3684-3690.
[5] Hoover W. G.（1985）. “Canonical dynamics：equilibrium phase-space distribution,” Phys. Rev. A., vol.31, p1695-1697.
[6] Parrinello M. and Rahman A.（1982）. “Strain fluctuations elastic constants,” J. Chem. Phys., Vol.76, p2662-2666.
[7] Martha C. Mitchell, James D. Autry and Tina M. Nenoff（2001）. “Molecular dynamics simulations of binary mixtures of methane and hydrogen in zeolite A and a novel zinc phosphate,” Molecular Phys. , vol.99, No.22, p1831-1837.
[8]Song Hi Lee.（2003） ”Molecular Dynamics simulations for Transport Coefficient of Liquid Argon” Chem. Soc., vol 24, No.2.
[9]A.J.H.McGaughey and M.Kaviany（2004）.“Thermal conductivity decoposition and analysis using molecular dynamics simulations Part I. Lennard-Jones argon,” J. Heat and Mass Transfer, vol 47, p1783-1798.
[10] Takashi Tokumasu and Kenjiro Kamijo （2004）.“Molecular dynamics study for the thermal conductivity of diatomic liquid,” Superlattices and Microstructures, vol.35, p217-225.
[11] Song Hi Lee, Gyeong Keun Moon, and Sang Gu Choi（1991）. “Non-equilibrium Molecular Dynamics Simulations of Thermal Transport Coefficients of Liquid Water,” Chem. Soc. Vol. 12, No.3, p315-322.
[12] Dmitry Bedrov and Grant D. Smith（2000）. “Thermal conductivity of molecular fluids from molecular dynamics simulations: Application of a new imposed-flux method,” J. of Chem. Phy. Vol.113, NO.18, p.8080-8084.
[13] G. A. Fernandez, J. Vrabec, H. Hasse （2004）. “A molecular simulation syudy of shear and bulk viscosity and thermal conductivity of simple real fluids,” Fluid Phase Equilibria, vol.221, p157-163.
[14] D. K. Dysthe, A. H. Fuchs and B. Rousseau（1998）. “Prediction of Fluid Mixture Transport Properties by Molecular Dynamic,” J. of Thermophsics, vol.19, p437-448.
[15]G. A. Fernandez, J. Vrabec, and H. Hasse（2004）. “Self Diffusion and Binary Maxwell-Stefan Diffusion in Simple Fluids with the Green-Kubo Method,” vol. 25, No.1, p175-186.
[16] S. A. Adelman and J. D. Doll（1974）. “Generalized Langevin equation approach for atom/solid-surface scatterig：Collinear atom/harmonic chain model,” J. Chem. Phys., vol.61, p4242-4245.
[17] J. C. Tully（1980）. “Dynamics of gas-surface interactions：3D generalized Langevin model applied to fcc and bcc surfaces,” J. Chem. Phys., vol.73, p1975-1985.
[18] Spohr E.（1989）. “Computer simulation of the water/platinum interface,” J. Chem. Phys., vol.93, p6171-6180.
[19] I. –C. Yeh and M. L. Berkowitz（1998）. “Structure and dynamics of water at water/Pt interface as seen by molecular dynamics computer simulation,” J. Elec. Chem., vol.450, p313-325.
[20] Shigeo Maruyama and Tatsuto Kimura（1999）. “A study on thermal resistance over a solid-liquid interface by the molecular dynamics method,” Thermal Science & Engineering, vol.7, p.63-68.
[21] J. Blomer and A. E. Beylich（1999）. “Molecular dynamics simulation of energy accommodation of internal and translational degrees of freedom at gas-surface interfaces,” Surface Science, vol.423, p127-133.
[22] Maruyama S.（2000）. “Molecular Dynamics method for microscale heat transfer,” vol. 2, Chap. 6, p.189-226, Advances in Numerical Heat Transfer, W. J. Minkowycz and E. M. Sparrow(Eds), Taylor & Francis, New York, 2000.
[23] 楊宗翰, “以分子動力學模擬液態水之薄膜蒸發與奈米液滴在恆溫白金表面上的物理過程” 國立清華大學工程與系統科學所碩士論文, 2004.
[24] 王耀塵, “利用分子動力學評估奈米液體的熱力學性質” 國立清華大學動力機械工程研究所碩士論文, 2004.
[25] D. C. RAPAPORT ”The Art of Molecular Dynamics simulation”