於兩相流研究中，藉由模擬液珠於不同構造與材質之表面上的行為，可以提供材料科學與微流道設計作為資料庫。於此篇論文中，我們利用Zheng等人所建立的三維晶格波茲曼兩相流模型配合Briant等人發展出的部分沉浸邊界條件，模擬液珠置於平面與微結構上的接觸角度，並且藉由與Yoshimitsu等人所發表的實驗數值相互比較，驗證此兩相流模型的可靠度。於微結構平面模擬中，由於邊界條件複雜，因此我們實施三種不同的邊界處理方式，利用實驗數值跟理論數值驗證何者為最佳。另一方面，為了改善計算效率，我們採用簡化擴散項的Allen-Cahn方程式，並結合Zheng等人與Takada等人所提出的兩相流模組，藉由表面濕潤性控制模擬，審視此方法之可行性。

In two-phase flow field, the present researches of the droplet behavior simulation on the different structured-surface and different material provide the helpful information for the material research and micro-fluidic channel design. In this thesis, the three dimensional lattice Boltzmann model based on Zheng et al., which is for high density ratio, with partial wetting boundary condition by Briant et al. is adopted. This method is used to simulate a droplet on a partial wetting surface with given contact angle, and the results are compared with the experiment by Yoshimitsu et al. to verify the accuracy. Three different strategies which focus on dealing with the complicated boundary conditions on edges and corners are implemented, and the most appropriate approach is found out by contrasting the apparent contact angles with the experimental values. On the other hand, in order to improve the computational efficiency, Allen-Cahn equation is also considered and implemented due to simpler diffusion term. The 3D lattice Boltzmann model based on Zheng et al. and Takada et al. cooperating with Allen-Cahn equation is implemented and scrutinized for applicability.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Multiphase and multicomponent fluid systems . . . . . . . . . 3
1.1.3 Partial wetting boundary . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Lattice Boltzmann multiphase model . . . . . . . . . . . . . . 4
1.2.2 Allen-Cahn equation . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Wettability control on the solid surface . . . . . . . . . . . . . 7
1.2.4 Wetting behaviors on micro-structured surfaces . . . . . . . . 8
1.3 Motivation and objective . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Theory and governing equations 11
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 BGK and the low-Mach-number approximation . . . . . . . . . . . . 12
2.2.1 BGK approximation . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 The low-Mach-number approximation . . . . . . . . . . . . . . 14
2.3 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . 15
2.3.1 Discretization of phase space . . . . . . . . . . . . . . . . . . . 15
2.3.2 Dicretization of time . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 The free-energy model . . . . . . . . . . . . . . . . . . . . . . . . . . 17
i2.5 Lattice Boltzmann models for multiphase flows . . . . . . . . . . . . . 19
2.5.1 The model proposed by Zheng et al. . . . . . . . . . . . . . . 19
2.5.2 The revised Allen-Cahn model based on the models by Takada
et al. and Zheng et al. . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Wetting theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Numerical algorithm 24
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Boundary conditions for the computational domain . . . . . . . . . . 25
3.2.1 Introduction of the boundary conditions in Zheng’s model with
Cahn-Hilliard equation . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Introduction of the boundary conditions in the model based on
Takada et al. and Zheng et al. with revised Allen-Cahn equation 27
3.3 Wetting boundary implementations on the gradient and the Laplacian
of order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Inner grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Wall grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.3 Edge grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.4 Corner grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Similarities between experiment & simulation . . . . . . . . . . . . . 36
3.5 Periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Parallel algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Numerical results 39
4.1 The simulation of droplet on pillar-structured surface . . . . . . . . . 39
4.1.1 Introduction to the background and essential parameters in
the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.2 Influence of partial wetting boundary treatment . . . . . . . . 40
4.1.3 Comparison of the results . . . . . . . . . . . . . . . . . . . . 44
ii4.2 Wettability control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 The results based on Cahn-Hilliard equation model . . . . . . 46
4.2.2 The results based on revised Allen-Cahn equation model . . . 46
5 Conclusions 55

[1] U. Frisch, B. Hasslacher, and Y. Pomeau, “Lattice-gas automata for the Navier-Stokes equation,” Phys. Rev. Lett. 56, 1505, (1986).
[2] S. Wolfram, “Cellular automaton fluids 1: Basic theory,” J. Stat. Phys. 45, 471, (1986).
[3] F. J. Higuera, S. Sussi, and R. Benzi, “3-dimensional flows in complex geometries with the lattice Boltzmann method,” Europhys. Lett. 9, 345, (1989).
[4] F. J. Higuera, and J. Jem´enez, “Boltzmann approach to lattice gas simulations,” Europhys. Lett. 9, 663, (1989).
[5] P. L. Bhatnagar, E. P. Gross, and M. Grook, “A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems,” Phys. Rev. E 94, 511, (1954).
[6] S. Harris, “An introduction to the throry of the Boltzmann equation,” Holt, Rinehart and Winston, New York, (1971).
[7] U. Frisch, D. d’Humi`eres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. P. Rivet, “Lattice gas hydrodynamics in two and three dimensions,” Complex Syst. 1, 649, (1987).
[8] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, “Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics,” Phys. Fluids. 6, 1285, (1994).
[9] S. Osher and R. P. Fedkiw, “Level set method: An overview and some recent results,” J. Comput. Phys. 169, 463, (2001).
[10] R. Scardovelli and S. Zaleski, “Direct numerical simulation of free-surface and interfical flow,” Annu.Rev.Fluid Mech. 31, 567, (1999).
[11] T. Y. Hou, J. S. Lowengrub, M. J. Shelley, “Boundary integral methods for multicomponent fluids and multiphase materials,” J. Comput.Phys. 169, 302, (2001).
[12] M. R. Swift, W. R. Osborn, J. M. Yeomans, “Lattice Boltzmann simulation of nonideal fluids,” Phys. Rev. Lett. 75(5), 830-833, (1995).
[13] M. R. Swift, W. R. Osborn, J. M. Yeomans, “Lattice Boltzmann simulations of liquid-gas and binary-fluid systems,” Phys. Rev. E,54, 5041-5052, (1996).
[14] A. J. Briant, P. Papatzacos, J. M. Yeomans, “Lattice Boltzmann simulations of contact line motion in a liquid-gas system, ” Phil. Trans. Roy. Soc. A. 360, 485, (2002).
[15] A. J. Briant, Wagner, A. J. Wagner, J. M. Yeomans, “Lattice Boltzmann simulations of contact line motion: I. Liquid-gas systems, ” Phys. Rev. E 69, 031602, (2004).
[16] A. J. Briant and J. M. Yeomans, “Lattice Boltzmann simulations of contact line motion: II. Binary fluids,” Phys. Rev. E 69, 031603, (2004).
[17] A. Dupuis and J. M. Yeomans, “Lattice Boltzmann modelling of droplets on chemically heterogeneous surfaces,” Future Gener. Comput. Syst. 20, 993-1001, (2004).
[18] H. W. Zheng, C. Shu, Y. T. Chew, “A lattice Boltzmann model for multiphase flows with large density ratio,” JCP, Phys. 218, 353-371, (2006).
[19] X. Shan and H. Chen, “Lattice Boltzmann model for simulating flows with multiple phases and components,” Phys. Rev. E. 47, 1815-1819, (1993).
[20] X. Shan and H. Chen, “Simulation of Nonideal Gases and Liquid-GasPhase Transitions by the Lattice Boltzmann Equation,” Phys. Rev. E. 49, 2941-2948, (1994).
[21] X. Shan, and G. D. Doolen, “Multicomponent Lattice-Boltzmann Model With Interparticle Interaction,” J. Stat. Phys. 81, 379-393, (1995).
[22] J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system.-Interfacial energy,” J. Chem. Phys. 28(2), (1958).
[23] T. Inamuro, T. Ogata, S. Tajima, N. Konishi, “A lattice Boltzmann method for incompressible two-phase flows with large density differences,” J. Comput. Phys. 198, 628, (2004).
[24] T. Lee and C. L. Lin, “A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio,” J. Comput. Phys. 206, 16-47, (2005).
[25] X. He, S. Chen, R. Zhang, “A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability,” J. Comput. Phys. 152(2), 642-663, (1999).
[26] D. Jamet, O. Lebaigue, N. Coutris, J. M. Delhaye, “The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change,” J. Comput. Phys. 169, 624-651, (2001).
[27] S. M. Allen and J. W. Cahn, “A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,” Acta. Metall. 27, 1085-1095, (1979).
[28] Y. Sun, C. Beckermann,“Sharp interface tracking using the phase-field equation,” J. Comput. Phys. 220, 626-653, (2007).
[29] P. H. Chiu and Y. T. Lin,“A conservative phase field method for solving incompressible two-phase flows,” J. Comput. Phys. 230, 185-204, (2011).
[30] N. Takada, J. Matsumoto and S. Matsumoto, “Phase-field model-based simulation of motion of a two-phase fluid on solid surface,” Journal of Computational Science and Technology 7, 322-337, (2013).
[31] J. J. Huang, C. Shu, Y. T. Chew, H. W. Zheng, “Numerical study of 2d multiphase flows over grooved surface by lattice Boltzmann method,” Int. J. Mod. Phys C. 18(4), 492-500, (2007).
[32] Y. Y. Yan and Y. Q. Zu, “lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio,” J. Comput. Phys., 227, 763-775, (2007).
[33] C. H. Shih, C. L. Wu, L. C. Chang, C. A. Lin, “Lattice Boltzmann simulations of incompressible liquid-gas system on partial wetting surface,” Phil. Trans. R. Soc. A. 369, 2510-2518, (2011).
[34] R. N. Wenzel, “Surface roughness and contact angle,” J. Phys. Collid Chem. 53, 1466, (1949).
[35] A. B. D. Cassie and S. Baxter, “Wettability of porous surfaces,” Trans. Faraday Soc. 40, 546, (1944).
[36] C. H. Choi, C. J. Kim, “Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface,” Phys. Rev. Lett. 96, 066001, (2006).
[37] C. M. Cheng and C. H. Liu, “An electrolysis-bubble- actuated micropump based on the roughness gradient design of hydrophobic surface,” J. Microelectromech. Syst. 16, 1095-1105, (2007).
[38] M. Y. Lin, C. S. Yu, Y. C. Hu, S. C. Cheng, H. T. Hu, “Droplet-based microtexture biochip system for triglycerides and methanol measurement,” 27th Annual International Conference of the Engineering in Medicine and Biology Society, 2005. IEEE-EMBS 2005. 530-533, (2005).
[39] A. Dupuis and J. M. Yeomens, “Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions,” Langmuir. 21, 2624-2629, (2005).
[40] N. Moradi, F. Varnik, I. Steinbach, “Roughness-gradient-induced spontaneous motion of droplets on hydrophobic surfaces: A lattice Boltzmann study,” Europhysics Letters. 89, 26006, (2010).
[41] J. J. Huang, C. Shu, Y. T. Chew, “Lattice Boltzmann study of droplet motion inside a grooved channel,” Phys. Fluids. 21, 022103, (2009).
[42] Y. Q. Zu and Y. Y. Yan, “Lattice Boltzmann method for modelling droplets on chemically heterogeneous and microstructured surfaces with large liquid-gas density ratio,” Journal of Applied Mathematics. 76, 743-760, (2011).
[43] Y. T. Ju, C. A. Lin,“Simulation of droplet on micro-structured surface by lattice Boltzmann method.” (2012).
[44] F. C. Kou, C. A. Lin,“Simulation of structure effect on droplet using lattice
Boltzmann method.” (2013).
[45] D. Oner and T. J. McCarthy, “Ultrahydrophobic surfaces. Effects of topography length scales on wettability,” Langmuir. 16, 7777-7782, (2000).
[46] C. Sun, X. W. Zhao, Y. H. Han, Z. Z. Gu, “Control of water droplet motion by alteration of roughness gradient on silicon wafer by laser surface treatment,” Thin Solid Films. 516, 4059V4063, (2008).
[47] Tamas I. Gombosi, “Gas kinetic theorym,” Cambridge University Press, (1994).
[48] X. He, and L. S. Luo, “Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation,” Phys. Rev. E 56, 6811-6817, (1997).
[49] D. A. Wolf-Gladrow, “Lattice-gas cellular automata and lattice Boltzmann models - an introduction,” Springer, Lecture Notes in Mathematics, p.159, (2000).
[50] J. S. Rowlinson and B. Widom, “Molecular Theory of Capillarity, Clarendon,” Oxford, (1989).
[51] V. M. Kendon, M. E. Cates, I. Pagonabarraga, J.C. Desplat, P. Bladon, “Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study,” J. Fluid Mech. 440, 147-203, (2001).
[52] H. W. Zheng, C. Shu, Y. T. Chew, “Lattice Boltzmann interface capturing method for imcompressible,” Phys. Rev. E.72, 056705, (2005).
[53] D. Jacqmin, “Calculation of two-phase Navier-Stokes flows using phase-field modeling,” J. Comput. Phys. 155, 96-127, (1999).
[54] Z. Guo, C. Zhen, B. Shi, “Discrete lattice effects on the forcing term in the lattice Boltzmann method,” Phys. Rev. E 65(4), 046308, (2002).
[55] C. Huber, A. Parmigiani, B. Chopard, M. Manga and O. Bachmann,“Lattice Boltzmann model for melting with natural convection,” Int. J. Heat Fluid Flow 29, 1469V1480, (2008).
[56] J. W. Cahn, “Critical point wetting,” J. Chem. Phys. 66, 3667-3672, (1977).
[57] C. F. Ho, C. Chang, K. H. Lin, C. A. Lin, “Consistent Boundary Conditions for 2D and 3D Lattice Boltzmann Simulations,” CMES-Comp. Model Eng. 44, 137-155, (2009).
[58] M. Cheng, J. S. Hua, J. Lou, “Simulation of bubble-bubble interaction using a lattice Boltzmann method,” Comput. Fluids 39, 260-270, (2010).
[59] J. Zhang, “Lattice Boltzmann method for microfluidics: models and applications,” Microfluid. Nanofluid. 10, 1-28, (2011).
[60] Z. Yoshimitsu, A. Nakajima, T. Watanabe, K. Hashimoto, “Effects of surface structure on the hydrophobicity and sliding behavior of water droplets,” Langmuir 18, 5818V5822, (2002).
[61] Bo He, Neelesh A. Patankar, and Junghoon Lee, “Multiple Equilibrium Droplet Shapes and Design Criterion for Rough Hydrophobic Surfaces,” Langmuir 19, 4999 (2003).
[62] T. Zhang, B. Shi, Z. Guo, Z. Chai and J. Lu,“General bounce-back scheme for concentration boundary condition in the lattice-Boltzmann method,” Phys. Rev. E 85, 016701, (2012).
[63] H. Yoshida and M. Nagaoka,“Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation,” J. Comput. Phys. 229, 7774-7795, (2010).
[64] A. J. C. Ladd,“Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation,” J. Fluid Mech. 271, 285-309, (1994).
[65] N. Takada, M. Misawa, A. Tomiyama, S. Hosokawa, “Simulation of bubble motion under gravity by lattice Boltzmann method,” J. Nucl. Sci. Technol. 38(5), 330, (2001).