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为探讨分形基底结构对生长表面标度行为的影响, 本文采用Kinetic Monte Carlo(KMC)方法模拟了刻蚀模型(etching model)在谢尔宾斯基箭头和蟹状分形基底上刻蚀表面的动力学行为. 研究表明,在两种分形基底上的刻蚀模型都表现出很好的动力学标度行为, 并且满足Family-Vicsek标度规律. 虽然谢尔宾斯基箭头和蟹状分形基底的分形维数相同, 但模拟得到的标度指数却不同, 并且粗糙度指数与动力学指数z也不满足在欧几里得基底上成立的标度关系+z=2. 由此可以看出, 标度指数不仅与基底的分形维数有关, 而且和分形基底的具体结构有关.In order to investigate the effect of the structure of fractal substrates on dynamic scaling behavior of the surfaces, the etching model growing on the Sierpinski arrowhead and crab fractal substrates is simulated by means of Kinetic Monte Carlo (KMC). It is found that the etching model evolving on two kinds of fractal substrates can exhibit dynamic scaling behavior, and can still be described by the Family-Vicsek scaling relation. Although the Sierpinski arrowhead and crab fractal substrates have the same fractal dimension, the obvious different values of roughness exponent and dynamic exponent z, however, are obtained on these two substrates, and they neither of them satisfy the scaling relation +z=2, which is satisfied in the usual Euclid space. It can be seen from the results obtained here that the scaling exponents of the etching model growing on fractal substrate are determined by not only the fractal dimension but also the fractal structure.
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Keywords:
- etching model /
- fractal substrate /
- dynamic scaling behavior
[1] Barabási A L, Stanley H E 1995 Fractal Concepts in Surface Growth (Cambridge: Cambridge University Press)
[2] Family F, Vicsek T 1991 Dynamics of Fractal Surfaces (Singapore: World Scientific Press)
[3] Tang G, Ma B K 2002 Acta Phys. Sin. 51 994 (in Chinese) [唐刚, 马本堃 2002 51 994]
[4] Hao D P, Tang G, Xia H, Chen H, Zhang L M, Xun Z P 2007 Acta Phys. Sin. 56 2018 (in Chinese) [郝大鹏, 唐刚, 夏辉, 陈华, 张雷明, 寻之朋 2007 56 2018]
[5] Xun Z P, Tang G, Han K, Hao D P, Xia H, Zhou W, Yang X Q, Wen R J, Chen Y L 2010 Chin. Phys. B 19 070516
[6] Tang G, Hao D P, Xia H, Han K, Xun Z P 2010 Chin. Phys. B 19 100508
[7] Family F, Vicsek T 1985 J. Phys. A 18 L75
[8] Meakin P 1998 Fractals, scaling and growth far from equilibrium (Cambridge: Cambridge University Press)
[9] Edwards S F, Wilkinson D R 1982 Proc. R. Soc. London A 381 17
[10] Kardar M, Parisi G, Zhang Y C 1986 Phys. Rev. Lett. 56 889
[11] Meakin P, Ramanlal P, Sander L M, Ball R C 1986 Phys. Rev. A 34 5091
[12] Jullien R, Botet R 1985 Phys. Rev. Lett. 54 2055
[13] Kim J M, Kosterlitz J M 1989 Phys. Rev. Lett. 62 2289
[14] Krug J 1997 Adv. Phys. 46 139
[15] Bab M A, Fabricius G, Albano E V 2008 Europhys. Lett. 81 10003
[16] Lee K E, Sung J Y, Cha M-Y, Maeng S E, Bang Y S, Lee J W 2009 Phys. Lett. A 373 4260
[17] Weber S, Klafter J, Blumen A 2010 Phys. Rev. E 82 051129
[18] Lee S B, Kim J M 2009 Phys. Rev. E 80 021101
[19] Kim D H, Kim J M 2010 J. Stat. Mech. p08008
[20] Horowitz C M , Romá F, Albano E V 2008 Phys. Rev. E 78 061118
[21] Tang G, Xun Z P,Wen R J, Han K, Xia H, Hao D P, ZhouW, Yang X Q, Chen Y L 2010 Physica A 389 4552
[22] Lee S B, Jeong H C, Kim J M 2008 J. Stat. Mech. p12013
[23] Huynh H N, Chew L Y, Pruessner G 2010 Phys. Rev. E 82 042103
[24] Mello B A 2001 Phys. Rev. E 63 041113
[25] Lee C, Lee S B 2010 Physica A 389 5053
[26] Aar?ao R F D A 2004 Phys. Rev. E 69 021610
[27] Paiva T, Aar?ao R F D A 2007 Surface Science 601 419 020511-6
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[1] Barabási A L, Stanley H E 1995 Fractal Concepts in Surface Growth (Cambridge: Cambridge University Press)
[2] Family F, Vicsek T 1991 Dynamics of Fractal Surfaces (Singapore: World Scientific Press)
[3] Tang G, Ma B K 2002 Acta Phys. Sin. 51 994 (in Chinese) [唐刚, 马本堃 2002 51 994]
[4] Hao D P, Tang G, Xia H, Chen H, Zhang L M, Xun Z P 2007 Acta Phys. Sin. 56 2018 (in Chinese) [郝大鹏, 唐刚, 夏辉, 陈华, 张雷明, 寻之朋 2007 56 2018]
[5] Xun Z P, Tang G, Han K, Hao D P, Xia H, Zhou W, Yang X Q, Wen R J, Chen Y L 2010 Chin. Phys. B 19 070516
[6] Tang G, Hao D P, Xia H, Han K, Xun Z P 2010 Chin. Phys. B 19 100508
[7] Family F, Vicsek T 1985 J. Phys. A 18 L75
[8] Meakin P 1998 Fractals, scaling and growth far from equilibrium (Cambridge: Cambridge University Press)
[9] Edwards S F, Wilkinson D R 1982 Proc. R. Soc. London A 381 17
[10] Kardar M, Parisi G, Zhang Y C 1986 Phys. Rev. Lett. 56 889
[11] Meakin P, Ramanlal P, Sander L M, Ball R C 1986 Phys. Rev. A 34 5091
[12] Jullien R, Botet R 1985 Phys. Rev. Lett. 54 2055
[13] Kim J M, Kosterlitz J M 1989 Phys. Rev. Lett. 62 2289
[14] Krug J 1997 Adv. Phys. 46 139
[15] Bab M A, Fabricius G, Albano E V 2008 Europhys. Lett. 81 10003
[16] Lee K E, Sung J Y, Cha M-Y, Maeng S E, Bang Y S, Lee J W 2009 Phys. Lett. A 373 4260
[17] Weber S, Klafter J, Blumen A 2010 Phys. Rev. E 82 051129
[18] Lee S B, Kim J M 2009 Phys. Rev. E 80 021101
[19] Kim D H, Kim J M 2010 J. Stat. Mech. p08008
[20] Horowitz C M , Romá F, Albano E V 2008 Phys. Rev. E 78 061118
[21] Tang G, Xun Z P,Wen R J, Han K, Xia H, Hao D P, ZhouW, Yang X Q, Chen Y L 2010 Physica A 389 4552
[22] Lee S B, Jeong H C, Kim J M 2008 J. Stat. Mech. p12013
[23] Huynh H N, Chew L Y, Pruessner G 2010 Phys. Rev. E 82 042103
[24] Mello B A 2001 Phys. Rev. E 63 041113
[25] Lee C, Lee S B 2010 Physica A 389 5053
[26] Aar?ao R F D A 2004 Phys. Rev. E 69 021610
[27] Paiva T, Aar?ao R F D A 2007 Surface Science 601 419 020511-6
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