臺灣博碩士論文加值系統

在本研究中，透過分子動力學、即時動態穿透式電子顯微鏡壓縮技術與奈米壓痕實驗，探討鎳鈦銅合金奈米柱在壓縮試驗下的機械性能分析，並針對材料的塑性變形與力學性能進行分析。文中探討了鎳鈦銅合金奈米柱在不同銅含量百分比、晶粒尺寸、環境溫度的影響，並藉由聚焦離子術切割出不同直徑與結晶與非晶相的鎳鈦銅合金奈米柱進行壓縮試驗。模擬結果表明，根據銅的摻雜比例不同以及晶粒尺寸的不同，對於合金奈米柱的楊氏模數與最大應力等機械性能有直接的影響。摻雜銅含量10 %的鎳鈦奈米柱，相較於其他摻雜比例的奈米柱，有著較好的機械強度。隨著銅含量的提高(超過10 %)，則會對奈米柱產生軟化的結果。在鎳鈦合金與鈦銅合金的模擬中具有較高的非晶相結構，在較低的溫度下有較高的楊氏模數。在晶粒尺寸為40、50、60與70 Å下，具有較為明顯的變形產生，而隨著晶粒尺寸的增加，其機械強度會符合Hall-Petch理論。在即時動態壓縮實驗中，相同壓縮條件下，摻雜20 %銅的奈米柱產生的壓縮位移量高於含銅量10 %，且機械強度也隨之下降。於結晶與非晶奈米柱壓縮實驗結果中得知，非晶奈米柱相較於結晶奈米柱具有較高的機械強度。最後，透過理論探討與分子動力學模擬分析驗證，鎳鈦銅合金奈米柱的機械力學特性與結構的變化，為奈米柱的機械特性建立重要性能與結構相關連的機制。

In this article, the study investigations of the mechanical characteristics of Ni-Ti-Cu alloys nanopillars using in-situ TEM compression, nanoindentation, and molecular dynamics (MD) simulations. The effects of different percentages of Cu content, various grain sizes, and different nanopillar diameter are evaluated. The results show that the mechanical properties such as Young's modulus and maximum stress are strongly dependent on the Cu composition and grain size. The Cu-doped Ni-Ti nanopillars with the Cu content of 10 % is harder to compress than other ration one. The NiTi alloys and TiCu alloys have more amorphous structures. The lower temperature, the higher Young's modulus. The Cu-doped Ni-Ti nanopillars is deformed more strongly in the specimens with the grain sizes of 40, 50, 60, and 70 Å. Moreover, the strength presents an agreement with the Hall-Petch relationship as the grain size increases. In experiment, the displacement of the specimens with 20 % Cu-doped is more significant than the displacement of the specimens with 10 % Cu-doped nanopillars in the same compressive conditions. The amorphous pillar is harder to compress than a nanocrystalline one. Finally, through theoretical exploration and simulation analysis, the mechanical properties of micropillar should establish the important properties and the mechanism of structures and theory.

摘 要 IV
ABSTRACT V
誌　　謝 VII
Contents VIII
List of figure captions X
Symbol table XIII
Chapter 1 Introduction 1
1.1 Introduction 1
1.2 Shape memory alloy materials 4
1.3 Motivation 6
Chapter 2 Literature review 7
2.1 Molecular Dynamics 7
2. 1. 1 Basic theory of Molecular Dynamics 7
2. 1. 2 Intermolecular force and potential energy function 8
2. 1. 3 Initial condition setting 11
2. 1. 5 Calculation skills of Molecular dynamics 16
2. 1. 6 Boundary condition 20
2. 1. 7 Software and parallel computing methods 22
2. 1. 8 Molecular dynamics simulation process 24
2.2 In-situ TEM compression 26
2.3 Nickel - titanium – copper alloy 28
Chapter 3. Molecular dynamics simulation method and parameter setting 30
Chapter 4. Experiment and detection methods 34
4. 1 Preparation of the material 35
4. 2 Measurement devices 37
Chepter 5 Results and discussion 49
5.1 Simulation 49
5. 1. 1 Mechanical properties variation of Ni-Ti-Cu nanopillars at different proportion 49
5. 1. 2 Mechanical properties variation of Ni-Ti-Cu nanopillars at different grain sizes 56
5. 1. 3 Mechanical properties variation of Ni-Ti-Cu nanopillars at different lengths 60
5. 2 Experiment 64
5. 2. 1 Mechanical properties of the Cu-doped Ni-Ti nanopillars under the in-situ TEM compression 64
5. 2. 2 Mechanical properties of Ni-Ti-Cu alloy in　nano　crystalline and amorphous 73
5. 2. 3 Morphology and microstructure of Ni-Ti-Cu substrate structure 80
Chapter 6 Conclusion 83
Acknowledgements 85
References 86
Curriculum vitae 96
Publication list 97

[1]Schodek, D.L., Ferreira, P. and Ashby, M.F., 2009. Nanomaterials, nanotechnologies and design: an introduction for engineers and architects. Butterworth-Heinemann.
[2]Binnig, G. and Rohrer, H., 1983. Scanning tunneling microscopy. Surface Science, 126(1-3), pp.236-244.
[3]Binnig, G., Quate, C.F. and Gerber, C., 1986. Atomic force microscope. Physical Review Letters, 56(9), p.930.
[4]Iijima, S., 1991. Helical microtubules of graphitic carbon. Nature, 354(6348), pp.56-58.
[5]Ayappa, K.G., Malani, A., Kalyan, P. and Thakkar, F., 2007. Molecular simulations: probing systems from the nanoscale to mesoscale. Journal of the Indian Institute of Science, 87(1), p.35.
[6]Salvetr, P., Kubatík, T.F. and Novák, P., 2016. Preparation of Ni-Ti shape memory alloy by spark plasma sintering method. Manufacturing Technology, 16, pp.804-808.
[7]Chang, L.C. and Read, T.A., 1951. Plastic deformation and diffusionless phase changes in metals—The gold-cadmium beta phase. Journal of Metals, 3(1), pp.47-52.
[8]Buehler, W.J., Gilfrich, J.V. and Wiley, R.C., 1963. Effect of low‐temperature phase changes on the mechanical properties of alloys near composition TiNi. Journal of Applied Physics, 34(5), pp.1475-1477.
[9]Park, H.S., 2006. Stress-induced martensitic phase transformation in intermetallic nickel aluminum nanowires. Nano Letters, 6(5), pp.958-962.
[10]Park, H.S. and Laohom, V., 2007. Surface composition effects on martensitic phase transformations in nickel aluminium nanowires. Philosophical Magazine, 87(14-15), pp.2159-2168.
[11]Alavi, A., Mirabbaszadeh, K., Nayebi, P. and Zaminpayma, E., 2010. Molecular dynamics simulation of mechanical properties of Ni–Al nanowires. Computational Materials Science, 50(1), pp.10-14.
[12]Sutrakar, V.K. and Mahapatra, D.R., 2010. Asymmetry in structural and thermo-mechanical behavior of intermetallic NiAl nanowire under tensile/compressive loading: A molecular dynamics study. Intermetallics, 18(8), pp.1565-1571.
[13]Hsu, K.C., Chen, J.Y., Fang, T.H. and Lin, M.H., 2018. Size-dependent strength and interface-dominated deformation mechanisms of Cu/Zr multilayer nanofilms. Results in Physics, 11, pp.684-689.
[14]Zhang, Y., Jiang, S. and Wang, M., 2020. Atomistic investigation on superelasticity of NiTi shape memory alloy with complex microstructures based on molecular dynamics simulation. International Journal of Plasticity, 125, pp.27-51.
[15]Jani, J.M., Leary, M., Subic, A. and Gibson, M.A., 2014. A review of shape memory alloy research, applications and opportunities. Materials & Design (1980-2015), 56, pp.1078-1113.
[16]Haile, J.M., 1992. Molecular dynamics simulation: elementary methods. John Wiley & Sons, Inc.
[17]Irving, J.H. and Kirkwood, J.G., 1950. The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. The Journal of Chemical Physics, 18(6), pp.817-829.
[18]Barshilia, H.C. and Rajam, K.S., 2002. Characterization of Cu/Ni multilayer coatings by nanoindentation and atomic force microscopy. Surface and Coatings Technology, 155(2-3), pp.195-202.
[19]Hoover, W.G., 1985. Canonical dynamics: Equilibrium phase-space distributions. Physical Review A, 31(3), p.1695.
[20]Smith, R. and Jakas. M., 2005. Atomic and ion collisions in solids and at surfaces: theory, simulation and applications. Cambridge University Press.
[21]Nowak, U., 2001. Thermally activated reversal in magnetic nanostructures. In Annual Reviews of Computational Physics IX, pp. 105-151.
[22]Jones, J.E., 1924. On the determination of molecular fields—I. From the variation of the viscosity of a gas with temperature. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 106(738), pp.441-462.
[23]Rosato, V., Guillope, M. and Legrand, B., 1989. Thermodynamically and structural properties of fcc transition metals using a simple tight-binding model. Philosophical Magazine A, 59(2), pp.321-336.
[24]Hsu, Q.C., Wu, C.D. and Fang, T.H., 2005. Studies on nanoimprint process parameters of copper by molecular dynamics analysis. Computational Materials Science, 34(4), pp.314-322.
[25]Hoover, W.G., 1985. Canonical dynamics: Equilibrium phase-space distributions. Physical Review A, 31(3), p.1695.
[26]Stone, A.J. and Wales, D.J., 1986. Theoretical studies of icosahedral C60 and some related species. Chemical Physics Letters, 128(5-6), pp.501-503.
[27]Lutsko, J.F., 1989. Generalized expressions for the calculation of elastic constants by computer simulation. Journal of Applied Physics, 65(8), pp.2991-2997.
[28]Allen, M.P. and Tildesley, D.J., 2017. Computer Simulation of Liquids. Oxford university press.
[29]Haile, J.M., Johnston, I., Mallinckrodt, A.J. and McKay, S., 1993. Molecular dynamics simulation: elementary methods. Computers in Physics, 7(6), pp.625-625.
[30]Frenkel, D., Smit, B., Tobochnik, J., McKay, S.R. and Christian, W., 1997. Understanding molecular simulation. Computers in Physics, 11(4), pp.351-354.
[31]Rapaport, D.C. and Rapaport, D.C.R., 2004. The art of molecular dynamics simulation. Cambridge university press.
[32]Plimpton, S., 1993. Fast parallel algorithms for short-range molecular dynamics (No. SAND-91-1144). Sandia National Labs., Albuquerque, NM (United States).
[33]Greer, J.R. and De Hosson, J.T.M., 2011. Plasticity in small-sized metallic systems: Intrinsic versus extrinsic size effect. Progress in Materials Science, 56(6), pp.654-724.
[34]Uchic, M. D., Dimiduk, D. M., Florando, J. N., and Nix, W. D., 2004, “Sample Dimensions Influence Strength and Crystal Plasticity,” Science, 305(5686), pp. 986–989.
[35]Banhart, F., 2008. In-situ electron microscopy at high resolution. World Scientific.
[36]Kiener, D. and Minor, A.M., 2011. Source-controlled yield and hardening of Cu (1 0 0) studied by in situ transmission electron microscopy. Acta Materialia, 59(4), pp.1328-1337.
[37]McDowell, M.T., Ryu, I., Lee, S.W., Wang, C., Nix, W.D. and Cui, Y., 2012. Studying the kinetics of crystalline silicon nanoparticle lithiation with in situ transmission electron microscopy. Advanced Materials, 24(45), pp.6034-6041.
[38]Haque, M.A. and Saif, M.T.A., 2004. Deformation mechanisms in free-standing nanoscale thin films: A quantitative in situ transmission electron microscope study. Proceedings of the National Academy of Sciences, 101(17), pp.6335-6340.
[39]Jiang, L. and Chawla, N., 2010. Mechanical properties of Cu6Sn5 intermetallic by micropillar compression testing. Scripta Materialia, 63(5), pp.480-483.
[40]Han, X., Liu, P., Sun, D. and Wang, Q., 2019. Quantifying the role of interface atomic structure in the compressive response of Ti 2 AlN/TiAl composite using MD simulations. Journal of Materials Science, 54(7), pp.5536-5550.
[41]Fang, T.H., Hsiao, Y.J. and Kang, S.H., 2015. Mechanical characteristics of copper indium gallium diselenide compound nanopillars using in situ transmission electron microscopy compression. Scripta Materialia, 108, pp.130-135.
[42]Kang, S.H. and Fang, T.H., 2014. Size effect on compression properties of GaN nanocones examined using in situ transmission electron microscopy. Journal of Alloys and Compounds, 597, pp.72-78.
[43]Kang, S.H., Fang, T.H., Chen, T.H. and Kuo, C.H., 2013. Size effect on nanomechanical properties of ZnO cones using in situ transmission electron microscopy. Current Applied Physics, 13(8), pp.1689-1696.
[44]Fukamachi, M. and Kajiwara, S., 1980. Lattice Imaging Study of Boundaries in Martensite in Shape Memory Alloys. Japanese Journal of Applied Physics, 19(8), p.479.
[45]Gosavi, S., Gosavi, S. and Alla, R., 2013. Titanium: A Miracle Metal in Dentistry. Trends in Biomaterials & Artificial Organs, 27(1), PP. 42-46.
[46]Saburi, T., 1998. Ti-Ni shape memory alloys. Shape Memory Materials, pp.49-96.
[47]Moberly, W.J. and Melton, K.N., 1990. Ni--Ti--Cu Shape Memory Alloys. Butterworth-Heinemann, Engineering Aspects of Shape Memory Alloys (UK), 1990, pp.46-57.
[48]Sharma, N. and Kumar, K., 2018. Mechanical characteristics and bioactivity of porous Ni50− x Ti50Cux (x= 0, 5 and 10) prepared by P/M. Materials Science and Technology, 34(8), pp.934-944.
[49]Gil, F.J., Solano, E., Pena, J., Engel, E., Mendoza, A. and Planell, J.A., 2004. Microstructural, mechanical and cytotoxicity evaluation of different NiTi and NiTiCu shape memory alloys. Journal of Materials Science: Materials in Medicine, 15(11), pp.1181-1185.
[50]Otsuka, K. and Ren, X., 2005. Physical metallurgy of Ti–Ni-based shape memory alloys. Progress in Materials Science, 50(5), pp.511-678.
[51]Dimiduk, D.M., Nadgorny, E.M., Woodward, C., Uchic, M.D. and Shade, P.A., 2010. An experimental investigation of intermittent flow and strain burst scaling behavior in LiF crystals during microcompression testing. Philosophical Magazine, 90(27-28), pp.3621-3649.
[52]De Araújo, C.J., Da Silva, N.J., Da Silva, M.M. and Gonzalez, C.H., 2011. A comparative study of Ni–Ti and Ni–Ti–Cu shape memory alloy processed by plasma melting and injection molding. Materials & Design, 32(10), pp.4925-4930.
[53]Doan, D.Q., Fang, T.H., Tran, A.S. and Chen, T.H., 2019. Residual stress and elastic recovery of imprinted Cu-Zr metallic glass films using molecular dynamic simulation. Computational Materials Science, 170, p.109162.
[54]Saikrishna, C.N., Ramaiah, K.V. and Bhaumik, S.K., 2006. Effects of thermo-mechanical cycling on the strain response of Ni–Ti–Cu shape memory alloy wire actuator. Materials Science and Engineering: A, 428(1-2), pp.217-224.
[55]Li, J., Fang, Q., Liu, B., Liu, Y. and Liu, Y., 2016. Atomic-scale analysis of nanoindentation behavior of high-entropy alloy. Journal of Micromechanics and Molecular Physics, 1(01), p.1650001.
[56]Fang, Q., Yi, M., Li, J., Liu, B. and Huang, Z., 2018. Deformation behaviors of Cu29Zr32Ti15Al5Ni19 high entropy bulk metallic glass during nanoindentation. Applied Surface Science, 443, pp.122-130.
[57]Fabregat-Sanjuan, A., Gispert-Guirado, F., Ferrando, F. and De la Flor, S., 2018. Identifying the effects of heat treatment temperatures on the Ti50Ni45Cu5 alloy using dynamic mechanical analysis combined with microstructural analysis. Materials Science and Engineering: A, 712, pp.281-291.
[58]Daw, M.S. and Baskes, M.I., 1984. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Physical Review B, 29(12), p.6443.
[59]Pham, A.V., Fang, T.H., Tran, A.S. and Chen, T.H., 2020. Effect of annealing and deposition of Cu atoms on Ni trench to interface formation and growth mechanisms of Cu coating. Superlattices and Microstructures, 139, p.106402.
[60]Stukowski, A., 2012. Structure identification methods for atomistic simulations of crystalline materials. Modelling and Simulation in Materials Science and Engineering, 20(4), p.045021.
[61]Dovchinvanchig, M., Zhao, C.W., Zhao, S.L., Meng, X.K., Jin, Y.J. and Xing, Y.M., 2014. Effect of Nd addition on the microstructure and martensitic transformation of Ni-Ti shape memory alloys. Advances in Materials Science and Engineering, 2014, p.489701.
[62]Gorji, M.R. and Sanjabi, S., 2012. Corrosion behavior of ion implanted NiTi shape memory alloy thin films. Materials Letters, 73, pp.179-182.
[63]Elahinia, M.H., Hashemi, M., Tabesh, M. and Bhaduri, S.B., 2012. Manufacturing and processing of NiTi implants: A review. Progress in Materials Science, 57(5), pp.911-946.
[64]Tosun, G., Orhan, N. and Özler, L., 2012. Investigation of combustion channel in fabrication of porous NiTi alloy implants by SHS. Materials Letters, 66(1), pp.138-140.
[65]Lin, K.N. and Wu, S.K., 2010. Multi-stage transformation in annealed Ni-rich Ti49Ni41Cu10 shape memory alloy. Intermetallics, 18(1), pp.87-91.
[66]Etaati, A. and Dehghani, K., 2011. Microstructural evolution of NiTi47. 7Cu6. 3 alloy during hot deformation. Journal of Materials Science & Technology, 27(10), pp.951-960.
[67]Chen, T.H. and Wu, J.H., 2016. Mechanical behavior and fracture properties of NiAl intermetallic alloy with different copper contents. Applied Sciences, 6(3), p.70.
[68]Dimiduk, D.M., Woodward, C., LeSar, R. and Uchic, M.D., 2006. Scale-free intermittent flow in crystal plasticity. Science, 312(5777), pp.1188-1190.
[69]KOZIEŁ, T., Pajor, K., Grzegorz, C.I.O.S. and Piotr, B.A.Ł.A., 2019. Effect of strain rate and crystalline inclusions on mechanical properties of bulk glassy and partially crystallized Zr52. 5Cu17. 9Ni14. 6Al10Ti5 alloy. Transactions of Nonferrous Metals Society of China, 29(5), pp.1036-1045.
[70]Murty, B.S., Ranganathan, S. and Rao, M.M., 1992. Solid state amorphization in binary Ti-Ni, Ti-Cu and ternary Ti-Ni-Cu system by mechanical alloying. Materials Science and Engineering: A, 149(2), pp.231-240.
[71]Wang, T.H., Fang, T.H. and Lin, Y.C., 2007. A numerical study of factors affecting the characterization of nanoindentation on silicon. Materials Science and Engineering: A, 447(1-2), pp.244-253.
[72]Fang, T.H. and Kang, S.H., 2008. Effect of indium dopant on surface and mechanical characteristics of ZnO: In nanostructured films. Journal of Physics D: Applied Physics, 41(24), p.245303.
[73]Fang, T.H. and Chang, W.J., 2006. Nanomechanical characterization of amorphous hydrogenated carbon thin films. Applied Surface Science, 252(18), pp.6243-6248.
[74]Fang, T.H., Wang, T.H. and Kang, S.H., 2009. Nanomechanical and surface behavior of polydimethylsiloxane-filled nanoporous anodic alumina. Journal of Materials Science, 44(6), pp.1588-1593.
[75]Smith, R.L. and Sandly, G.E., 1922. An accurate method of determining the hardness of metals, with particular reference to those of a high degree of hardness. Proceedings of the Institution of Mechanical Engineers, 102(1), pp.623-641.
[76]Goryczka, T. and Van Humbeeck, J., 2008. NiTiCu shape memory alloy produced by powder technology. Journal of Alloys and Compounds, 456(1-2), pp.194-200.
[77]Frost, M., Sevcik, M., Kaderavek, L., Sittner, P. and Sedlak, P., 2020. Reconstruction of phase distributions in NiTi helical spring: comparison of diffraction/scattering computed tomography and computational modeling. Smart Materials and Structures, 29(7), p.075036.
[78]Sehitoglu, H., Karaman, I., Zhang, X., Viswanath, A., Chumlyakov, Y. and Maier, H.J., 2001. Strain–temperature behavior of NiTiCu shape memory single crystals. Acta Materialia, 49(17), pp.3621-3634.
[79]Kang, S., Kim, H. and Seo, J., 2010. A reliable multidomain model for speech act classification. Pattern Recognition Letters, 31(1), pp.71-74.
[80]Tran, A.S. and Fang, T.H., 2020. Size effect and interfacial strength in nano-laminated Cu/CuxTa100-x composites using molecular dynamics. Computational Materials Science, 184, p.109890.
[81]Lai, T.Y., Lin, K.P., Liang, S.W. and Fang, T.H., 2019. Mechanical properties and mechanism of NiTi pillars using in-situ compression and indentation. Materials Research Express, 6(4), p.045036.
[82]Povoden-Karadeniz, A., Cirstea, D.C. and Kozeschnik, E., 2016. Prediction of precipitate evolution and martensite transformation in Ti-Ni-Cu shape memory alloys by computational thermodynamics. Materials Science and Engineering, 123(1), p.012038.
[83]Zhang, H., He, Y., Yang, F., Liu, H. and Jin, Z., 2013. Thermodynamic assessment of Cu–Ni–Ti ternary system assisted with key measurements. Thermochemical Acta, 574, pp.121-132.
[84]Brantley, W.A. and Eliades, T., 2001. Orthodontic materials: scientific and clinical aspects. American Journal of Orthodontics and Dentofacial Orthopedics, 119(6), pp.672-673.
[85]Sharma, N., Kumar, K., Raj, T. and Kumar, V., 2019. Porosity exploration of SMA by Taguchi, regression analysis and genetic programming. Journal of Intelligent Manufacturing, 30(1), pp.139-146.
[86]Thomasová, M., Seiner, H., Sedlák, P., Frost, M., Ševčík, M., Szurman, I., Kocich, R., Drahokoupil, J., Šittner, P. and Landa, M., 2017. Evolution of macroscopic elastic moduli of martensitic polycrystalline NiTi and NiTiCu shape memory alloys with pseudoplastic straining. Acta Materialia, 123, pp.146-156.
[87]Liang, S.W., Fang, T.H., Lai, T.Y., Lin, K.P. and Fan, Y.C., 2019. Mechanical characteristics of NiTiCu alloys from experiments and molecular dynamics simulations. Journal of Non-Crystalline Solids, 525, p.119676.
[88]Šittner, P., Heller, L., Pilch, J., Curfs, C., Alonso, T. and Favier, D., 2014. Young’s modulus of austenite and martensite phases in superelastic NiTi wires. Journal of Materials Engineering and Performance, 23(7), pp.2303-2314.
[89]Fazeli, S., Vahedpour, M. and Sadrnezhaad, S.K., 2018. Effect of copper content on tensile mechanical properties of ternary NiTiCu alloy nanowire: Molecular dynamics simulation. Materials Today: Proceedings, 5(1), pp.1552-1555.
[90]Rodrigues, P.F., Fernandes, F.B., Magalhães, R., Camacho, E., Lopes, A., Paula, A.S., Basu, R. and Schell, N., 2020. Thermo-mechanical characterization of NiTi orthodontic archwires with graded actuating forces. Journal of the Mechanical Behavior of Biomedical Materials, p.103747.
[91]Povoden-Karadeniz, E., Cirstea, D.C., Lang, P., Wojcik, T. and Kozeschnik, E., 2013. Thermodynamics of Ti–Ni shape memory alloys. Calphad, 41, pp.128-139.
[92]Shamsolhodaei, A., Sun, Q., Wang, X., Panton, B., Di, H. and Zhou, Y.N., 2020. Effect of Laser Positioning on the Microstructure and Properties of NiTi-Copper Dissimilar Laser Welds. Journal of Materials Engineering and Performance, 29(2), pp.849-857.
[93]Abbott, S., 2015. Adhesion science: principles and practice. DEStech Publications, Inc.
[94]Wang, C., Zhu, J., Lu, Y., Guo, Y. and Liu, X., 2014. Thermodynamic description of the Cu-Ni-Si system. Journal of Phase Equilibria and Diffusion, 35(1), pp.93-104.