# Colloidal Crystallization in 2D for Short-Ranged Attractions: A Descriptive Overview

## Abstract

**:**

## 1. Introduction

## 2. Simulations

## 3. Description of the Subprocesses

#### 3.1. The Metastable Fluid-Fluid Phase Separation

#### 3.2. Crystalline Nucleation

#### 3.3. Ostwald Ripening

- (i)
- At any instant of the ripening process, there exists a critical radius ${r}_{c}$. Grains with a larger radius grow and smaller grains shrink. During the ripening process ${r}_{c}$ increases with time.
- (ii)
- In the long time limit, the grain size distribution becomes self-similar, when the sizes are scaled with ${r}_{c}$.
- (iii)
- The critical radius ${r}_{c}$ is equal to the number-average ${r}_{N}$ of the limiting self-similar size distribution.
- (iv)
- The ripening rate, defined as $d{r}_{N}^{3}/dt$, is constant and given by

#### 3.4. Grain Boundary Formation and Dynamics

#### 3.5. Grain Coarsening by Dislocation Disappearance after Grain Boundary Healing

^{o}is now situated on the opposite side as the case before.

#### 3.6. Some Theoretical Developments of Grain Coarsening

## 4. Discussion and Conclusions

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## References

- Pieranski, P. Colloidal crystals. Contemp. Phys.
**1983**, 24, 25–73. [Google Scholar] [CrossRef] - Gasser, U. Crystallization in three- and two-dimensional colloidal suspensions. J. Phys. Condens. Matter
**2009**, 21, 203101. [Google Scholar] [CrossRef] [PubMed] - Schall, P.; Cohen, I.; Weitz, D.A.; Spaepen, F. Visualization of dislocation dynamics in colloidal crystals. Science
**2004**, 305, 1944–1948. [Google Scholar] [CrossRef] [PubMed] - Schall, P.; Cohen, I.; Weitz, D.A.; Spaepen, F. Visualizing dislocation nucleation by indenting colloidal crystals. Nature
**2006**, 440, 319–323. [Google Scholar] [CrossRef] [PubMed] - Suresh, S. Colloid model for atoms. Nat. Mater.
**2006**, 5, 253–254. [Google Scholar] [CrossRef] [PubMed] - Schall, P. Laser difraction microscopy. Rep. Prog. Phys.
**2009**, 72, 076601. [Google Scholar] [CrossRef] - Wang, Z.; Wang, F.; Peng, Y.; Zheng, Z.; Han, Y. Imaging the homogeneous nucleation during the melting of superheated colloidal crystals. Science
**2012**, 338, 87–90. [Google Scholar] [CrossRef] [PubMed] - See for example Crystallization of Nucleic Acids and Proteins: A Practical Approach, 2nd ed.; Ducruix, A.; Giegé, R. (Eds.) Oxford University Press: Oxford, UK, 2000.
- Pieranski, P. Two-dimensional interfacial colloidal crystals. Phys. Rev. Lett.
**1980**, 45, 569–572. [Google Scholar] [CrossRef] - Grimes, C.C.; Adams, G. Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons. Phys. Rev. Lett.
**1979**, 42. [Google Scholar] [CrossRef] - Onoda, G.Y. Direct observation of two-dimensional, dynamical clustering and ordering with colloids. Phys. Rev. Lett.
**1985**, 55, 226–229. [Google Scholar] [CrossRef] [PubMed] - Saito, Y. Monte Carlo studies of two-dimensional melting: Dislocation vector systems. Phys. Rev. B
**1982**, 26, 6239–6253. [Google Scholar] [CrossRef] - Chui, S.T. Grain-boundary theory of melting in two dimensions. Phys. Rev. Lett.
**1982**, 48, 933–935. [Google Scholar] [CrossRef] - Tang, Y.; Armstrong, A.J.; Mockler, R.C.; O’Sullivan, W.J. Free-expansion melting of a colloidal monolayer. Phys. Rev. Lett.
**1989**, 62, 2401–2404. [Google Scholar] [CrossRef] [PubMed] - Lansac, Y; Glaser, M.A.; Clark, N.A. Discrete elastic model for two-dimensional melting. Phys. Rev. E
**2006**, 73, 041501. [Google Scholar] [CrossRef] [PubMed] - Murray, C.A.; van Winkle, D.H. Experimental observation of two-stage melting in a classical two-dimensional screened coulomb system. Phys. Rev. Lett.
**1987**, 58, 1200–1203. [Google Scholar] [CrossRef] [PubMed] - Kusner, R.E.; Mann, J.A.; Kerins, J.; Dahm, A.J. Two-stage melting of a two-dimensional colloidal lattice with dipole interactions. Phys. Rev. Lett.
**1994**, 73, 3113–3116. [Google Scholar] [CrossRef] [PubMed] - Bladon, P.; Frenkel, D. Dislocation unbinding in dense two-dimensional crystals. Phys. Rev. Lett.
**1995**, 74, 2519–2522. [Google Scholar] [CrossRef] [PubMed] - Zahn, K.; Lenke, R.; Maret, G. Two-stage melting of paramagnetic colloidal crystals in two dimensions. Phys. Rev. Lett.
**1999**, 82, 2721–2724. [Google Scholar] [CrossRef] - Von Grünberg, H.H.; Keim, P.; Zahn, K.; Maret, G. Elastic behavior of a two-dimensional crystal near melting. Phys. Rev. Lett.
**2004**, 93, 255703. [Google Scholar] [CrossRef] [PubMed] - Keim, P.; Maret, G.; von Grünberg, H.H. Frank’s constant in the hexatic phase. Phys. Rev. E
**2007**, 75, 031402. [Google Scholar] [CrossRef] [PubMed] - Marcus, A.H.; Rice, S.A. Observations of first-order liquid-to-hexatic and hexatic-to-solid phase transitions in a confined colloidal suspension. Phys. Rev. Lett.
**1996**, 77, 2577–2580. [Google Scholar] [CrossRef] [PubMed] - Lin, B.J.; Chen, L.J. Phase transitions in two-dimensional colloidal particles at oil/water interfaces. J. Chem. Phys.
**2007**, 126, 034706. [Google Scholar] [CrossRef] [PubMed] - Bernard, E.P.; Krauth, W. Two-step melting in two dimensions: First-order liquid-hexatic transition. Phys. Rev. Lett.
**2011**, 107, 155704. [Google Scholar] [CrossRef] [PubMed] - Engel, M.; Anderson, J.A.; Glotzer, S.C.; Isobe, M.; Bernard, E.P.; Krauth, W. Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods. Phys. Rev. E
**2013**, 87, 042134. [Google Scholar] [CrossRef] [PubMed] - Kapfer, S.C.; Krauth, W. Two-dimensional melting: From liquid-hexatic coexistence to continuous transitions. Phys. Rev. Lett.
**2015**, 114, 035702. [Google Scholar] [CrossRef] [PubMed] - Kosterlitz, J.M.; Thouless, D.J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys.
**1973**, 6, 1181–1203. [Google Scholar] [CrossRef] - Nelson, D.R.; Halperin, B.I. Dislocation-mediated melting in two dimensions. Phys. Rev. B
**1979**, 19, 2457–2484. [Google Scholar] [CrossRef] - Young, A.P. Melting and the vector coulomb gas in two dimensions. Phys. Rev. B
**1979**, 19, 1855–1866. [Google Scholar] [CrossRef] - Dillman, P.; Maret, G.; Keim, P. Polycrystalline solidification in a quenched 2D colloidal system. J. Phys. Condens. Matter
**2008**, 20, 404216. [Google Scholar] [CrossRef] - Lekkerkerker, H.N.W.; Poon, W.C.K.; Pusey, P.N.; Stroobants, A.; Warren, P.B. Phase behaviour of colloid + polymer mixtures. Europhys. Lett.
**1992**, 20, 559–564. [Google Scholar] [CrossRef] - Hagen, M.H.J.; Frenkel, D. Determination of phase diagrams for the hard-core attractive Yukawa system. J. Chem. Phys.
**1994**, 101, 4093–4097. [Google Scholar] [CrossRef] - Ilett, S.M.; Orrock, A.; Poon, W.C.K.; Pusey, P.N. Phase behavior of a model colloid-polymer mixture. Phys. Rev. E
**1995**, 51, 1344–1352. [Google Scholar] [CrossRef] - Asherie, N.; Lomakin, A.; Benedek, G.B. Phase diagram of colloidal solutions. Phys. Rev. Lett.
**1996**, 77, 4832–4835. [Google Scholar] [CrossRef] [PubMed] - Hobbie, E.K. Metastability and depletion-driven aggregation. Phys. Rev. Lett.
**1998**, 81, 3996–3999. [Google Scholar] [CrossRef] - Zhang, T.H.; Liu, X.Y. How does a transient amorphous precursor template crystallization. J. Am. Chem. Soc.
**2007**, 129, 13520–13526. [Google Scholar] [CrossRef] [PubMed] - Savage, J.R.; Dinsmore, A.D. Experimental evidence for two-step nucleation in colloidal crystallization. Phys. Rev. Lett.
**2009**, 102, 198302. [Google Scholar] [CrossRef] [PubMed] - Berland, C.R.; Thurston, G.M.; Kondo, M.; Broide, M.L.; Pande, J.; Ogun, O.; Benedek, G.B. Solid-liquid phase boundaries of lens protein solutions. Proc. Natl. Acad. Sci. USA
**1992**, 89, 1214–1218. [Google Scholar] [CrossRef] [PubMed] - Ten Wolde, P.R.; Frenkel, D. Enhancement of protein crystal nucleation by critical density fluctuations. Science
**1997**, 277, 1975–1978. [Google Scholar] [CrossRef] [PubMed] - Talanquer, V.; Oxtoby, D.W. Crystal nucleation in the presence of a metastable critical point. J. Chem. Phys.
**1998**, 109, 223–227. [Google Scholar] [CrossRef] - Galkin, O.; Vekilov, P.G. Control of protein crystal nucleation around the metastable liquid-liquid phase boundary. Proc. Natl. Acad. Sci. USA
**2000**, 97, 6277–6281. [Google Scholar] [CrossRef] [PubMed] - Lomakin, A.; Asherie, N.; Benedek, G.B. Liquid-solid transition in nuclei of protein crystals. Proc. Natl. Acad. Sci. USA
**2003**, 100, 10254–10257. [Google Scholar] [CrossRef] [PubMed] - Mao, Y.; Cates, M.E.; Lekkerkerker, H.N.W. Depletion force in colloidal systems. Phys. A
**1995**, 222, 10–24. [Google Scholar] [CrossRef] - Rosenbaum, D.; Zamora, P.C.; Zukoski, C.F. Phase behavior of small attractive colloidal particles. Phys. Rev. Lett.
**1996**, 76, 150–153. [Google Scholar] [CrossRef] [PubMed] - Kelton, K.F. Crystal nucleation in liquids and glasses. In Solid State Physics; Ehrenreich, H., Turnbull, D., Eds.; Academic Press: San Diego, CA, USA, 1991; Volume 45, pp. 75–177. [Google Scholar]
- Gasser, U.; Weeks, E.R.; Schofield, A.; Pusey, P.N.; Weitz, D.A. Real-space imaging of nucleation and growth in colloidal crystallization. Science
**2001**, 292, 258–262. [Google Scholar] [CrossRef] [PubMed] - Rubin-Zuzic, M.; Morfill, G.E.; Ivlev, A.; Pompl, R.; Klumov, B.A.; Bunk, W.; Thomas, H.M.; Rothermel, H.; Havnes, O.; Fouquet, A. Kinetic Develpment of crystallization fronts in complex plasmas. Nat. Phys.
**2006**, 2, 181–185. [Google Scholar] [CrossRef] - Yau, S.T.; Vekilov, P.G. Quasi-planar nucleus structure in apoferritin crystallization. Nature
**2000**, 406, 494–497. [Google Scholar] [PubMed] - Pan, A.C.; Chandler, D. Dynamics of nucleation in the Ising model. J. Phys. Chem. B
**2004**, 108, 19681–19686. [Google Scholar] [CrossRef] - Ostwald, W. Studien uber die Bildung und Umwandlung fester Korper. Z. Phys. Chem.
**1897**, 22, 289–330. [Google Scholar] - Penn, R.L.; Banfield, J.F. Morphology development and crystal growth in nanocrystalline aggregates under hydrotermal conditions: Insights from titania. Geochim. Cosmochim. Acta
**1999**, 63, 1549–1557. [Google Scholar] [CrossRef] - Madras, G.; McCoy, B.J. Growth and ripening kinetics of crystalline polymorphs. Cryst. Growth Des.
**2003**, 3, 981–990. [Google Scholar] [CrossRef] - Huang, F.; Zhang, H.; Banfield, J.F. Two-stage crystal-growth kinetics observed during hydrotermal coarsening of nanocrystalline ZnS. Nano Lett.
**2003**, 3, 373–378. [Google Scholar] [CrossRef] - Streets, A.M.; Quake, S.R. Ostwald ripening of clusters during protein crystallization. Phys. Rev. Lett.
**2010**, 104, 178102. [Google Scholar] [CrossRef] [PubMed] - Iacopini, S.; Palberg, T.; Schöpe, H.J. Ripening-dominated crystallization in polydisperse hard-sphere-like colloids. Phys. Rev. E
**2009**, 79, 010601. [Google Scholar] [CrossRef] [PubMed] - Stavans, J. The evolution of cellular structures. Rep. Prog. Phys.
**1993**, 56, 733–789. [Google Scholar] [CrossRef] - Gokhale, S.; Nagamanasa, K.H.; Ganapathy, R.; Sood, A.K. Grain growth and grain boundary dynamics in colloidal polycrystals. Soft Matter
**2013**, 9, 6634–6644. [Google Scholar] [CrossRef] - Edwards, T.D.; Yang, Y.; Beltran-Villegas, D.J.; Bevan, M.A. Colloidal crystal grain boundary formation and motion. Sci. Rep.
**2014**, 4. [Google Scholar] [CrossRef] [PubMed] - Nagamanasa, K.H.; Gokhale, S.; Ganapathy, R.; Sood, A.K. Confined glassy dynamics at grain boundaries in colloidal crystals. Proc. Natl. Acad. Sci. USA
**2011**, 108, 11323–11326. [Google Scholar] [CrossRef] [PubMed] - Skinner, T.O.E.; Aarts, D.G.A.L.; Dullens, R.P.A. Supercooled dynamics of grain boundary particles in two-dimensional colloidal crystals. J. Chem. Phys.
**2011**, 135, 124711. [Google Scholar] [CrossRef] [PubMed] - Trautt, Z.T.; Upmanyu, M.; Karma, A. Interface mobility from interface random walk. Science
**2006**, 314, 632–635. [Google Scholar] [CrossRef] [PubMed] - Skinner, T.O.E.; Aarts, D.G.A.L.; Dullens, R.P.A. Grain-boundary fluctuations in two-dimensional colloidal crystals. Phys. Rev. Lett.
**2010**, 105, 168301. [Google Scholar] [CrossRef] [PubMed] - Porter, D.A.; Easterling, K.E. Phase Transformations in Metals and Alloys, 2nd ed.; CRC Press: Boca Raton, FL, USA, 1992. [Google Scholar]
- Kikuchi, K.; Yoshida, M.; Maekawa, T.; Watanabe, H. Metropolis Monte Carlo method as a numerical technique to solve the Fokker-Planck equation. Chem. Phys. Lett.
**1991**, 185, 335–338. [Google Scholar] [CrossRef] - Kikuchi, K.; Yoshida, M.; Maekawa, T.; Watanabe, H. Metropolis Monte Carlo method for Brownian dynamics simulation generalized to include hydrodynamics interactions. Chem. Phys. Lett.
**1992**, 196, 57–61. [Google Scholar] [CrossRef] - Yoshida, M.; Kikuchi, K. Metropolis Monte Carlo Brownian dynamics simulation of the ion atmosphere polarization around a rodlike polyion. J. Phys. Chem.
**1994**, 98, 10303–10306. [Google Scholar] [CrossRef] - Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] - Honeycutt, J.D.; Andersen, H.C. The effect of periodic boundary conditions on homogeneous nucleation observed in computer simulations. Chem. Phys. Lett.
**1984**, 108, 535–538. [Google Scholar] [CrossRef] - Honeycutt, J.D.; Andersen, H.C. Small system size artifacts in the molecular dynamics simulation of homogeneous crystal nucleation in supercooled atomic liquids. J. Phys. Chem.
**1986**, 90, 1585–1589. [Google Scholar] [CrossRef] - González, A.E.; Ixtlilco-Cortés, L. Fractal structure of the crystalline-nuclei boundaries in 2D colloidal crystallization: Computer simulations. Phys. Lett. A
**2012**, 376, 1375–1379. [Google Scholar] [CrossRef] - Halperin, B.I.; Nelson, D.R. Theory of two-dimensional melting. Phys. Rev. Lett.
**1978**, 41, 121–124. [Google Scholar] [CrossRef] - Fraser, D.P.; Zuckermann, M.J.; Mouritsen, O.G. Simulation technique for hard-disk models in two dimensions. Phys. Rev. A
**1990**, 42, 3186–3195. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jaster, A. Computer simulation of the two-dimensional melting transition using hard disks. Phys. Rev. E
**1999**, 59, 2594–2602. [Google Scholar] [CrossRef] - Terao, T.; Nakayama, T. Crystallization in a quasi-two-dimensional colloidal system at an air-water interface. Phys. Rev. E
**1999**, 60, 7157–7162. [Google Scholar] [CrossRef] - Huerta, A.; Naumis, G.G.; Wasan, D.T.; Henderson, D.; Trokhymchuk, A. Attraction driven disorder in a hard-core colloidal monolayer. J. Chem. Phys.
**2004**, 120, 1506–1510. [Google Scholar] [CrossRef] [PubMed] - Dillman, P.; Maret, G.; Keim, P. Two-dimensional colloidal systems in time-dependent magnetic fields. Eur. Phys. J. Spec. Top.
**2013**, 222, 2941–2959. [Google Scholar] [CrossRef] - Lutsko, J.F.; Nicolis, G. Theoretical evidence for a dense fluid precursor to crystallization. Phys. Rev. Lett.
**2006**, 96, 046102. [Google Scholar] [CrossRef] [PubMed] - Mandelbrot, B.B. The Fractal Geometry of Nature; W. H. Freeman & Co.: San Francisco, FL, USA, 1988. [Google Scholar]
- González, A.E.; Martínez-López, F.; Moncho-Jordá, A.; Hidalgo-Álvarez, R. Two-dimensional colloidal aggregation: Concentration effects. J. Colloid Interface Sci.
**2002**, 246, 227–234. [Google Scholar] [CrossRef] [PubMed] - González, A.E.; Martínez-López, F.; Moncho-Jordá, A.; Hidalgo-Álvarez, R. Concentration effects on two- and three-dimensional colloidal aggregation. Phys. A
**2002**, 314, 235–245. [Google Scholar] [CrossRef] - Lifshitz, I.M.; Slyozov, V.V. The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids
**1961**, 19, 35–50. [Google Scholar] [CrossRef] - Wagner, C. Theorie de alterung von niederschlägen durch umlösen. Ber. Bunsen-Ges. Phys. Chem.
**1961**, 65, 581–591. [Google Scholar] - Söhnel, O.; Garside, J. Precipitation; Butterworth-Heinemann: Oxford, UK, 1992. [Google Scholar]
- Ng, J.D.; Lorber, B.; Witz, J.; Theóbald-Dietrich, A.; Kern, D.; Giegé, R. The crystallization of biological macromolecules from precipitates: Evidence for Ostwald ripening. J. Cryst. Growth
**1996**, 168, 50–62. [Google Scholar] [CrossRef] - Ståhl, M.; Åslund, B.; Rasmuson, Å.C. Aging of reaction-crystallized benzoic acid. Ind. Eng. Chem. Res.
**2004**, 43, 6694–6702. [Google Scholar] [CrossRef] - Finsy, R. On the critical radius in Ostwald ripening. Langmuir
**2004**, 20, 2975–2976. [Google Scholar] [CrossRef] [PubMed] - Qing-bo, W.; Finsy, R.; Hai-bo, X.; Xi, L. On the critical radius in generalized Ostwald ripening. J. Zhejiang Univ. Sci. B
**2005**, 6, 705–707. [Google Scholar] - Job, G.; Herrmann, F. Chemical potential—A quantity in search of recognition. Eur. J. Phys.
**2006**, 27, 353–371. [Google Scholar] [CrossRef] - Brailsford, A.D.; Wynblatt, P. The dependence of Ostwald ripening kinetics on particle volume fraction. Acta Metall.
**1979**, 27, 489–497. [Google Scholar] [CrossRef] - Voorhees, P.W.; Glicksman, M.E. Solution to the multi-particle diffusion problem with applications to Ostwald ripening—I. Theory. Acta Metall.
**1984**, 32, 2001–2011. [Google Scholar] [CrossRef] - Marqusee, J.A.; Ross, J. Theory of Ostwald ripening: Competitive growth and its dependence on volume fraction. J. Chem. Phys.
**1984**, 80, 536–543. [Google Scholar] [CrossRef] - Tokuyama, M.; Kawasaki, K. Statistical-mechanical theory of coarsening of spherical droplets. Phys. A
**1984**, 123, 386–411. [Google Scholar] [CrossRef] - Enomoto, Y.; Tokuyama, M.; Kawasaki, K. Finite volume fraction effects on Ostwald ripening. Acta Metall.
**1986**, 34, 2119–2128. [Google Scholar] [CrossRef] - Yao, J.H.; Elder, K.R.; Guo, H.; Grant, M. Theory and simulation of Ostwald ripening. Phys. Rev. B
**1993**, 47. [Google Scholar] [CrossRef] - Baldan, A. Progress in Ostwald ripening theories and their applications to nickel-based superalloys. J. Mater. Sci.
**2002**, 37, 2171–2202. [Google Scholar] [CrossRef] - Rosehain, W.; Ewen, D. The intercrystalline cohesion of metals. J. Inst. Met.
**1913**, 10, 119–149. [Google Scholar] - Zhang, H.; Srolovitz, D.J.; Douglas, J.F.; Warren, J.A. Grain boundaries exhibit the dynamics of glass-forming liquids. Proc. Natl. Acad. Sci. USA
**2009**, 106, 7735–7740. [Google Scholar] [CrossRef] [PubMed] - Janssens, K.G.F.; Olmsted, D.; Holm, E.A.; Foiles, S.M.; Plimpton, S.J.; Derlet, P.M. Computing the mobility of grain boundaries. Nat. Mater.
**2006**, 5, 124–127. [Google Scholar] [CrossRef] [PubMed] - Hoyt, J.J.; Asta, M.; Karma, A. Atomistic and continuum modeling of dendritic solidification. Mater. Sci. Eng. R
**2003**, 41, 121–163. [Google Scholar] [CrossRef] - Hernández-Guzmán, J.H.; Weeks, E.R. The equilibrium intrinsic crystal-liquid interface of colloids. Proc. Natl. Acad. Sci. USA
**2009**, 106, 15198–15202. [Google Scholar] [CrossRef] [PubMed] - Sides, S.W.; Grest, G.S.; Lacasse, M.D. Capillary waves at liquid-vapor interfaces: A molecular dynamics simulation. Phys. Rev. E
**1999**, 60, 6708–6713. [Google Scholar] [CrossRef] - Aarts, D.G.A.; Schmidt, M.; Lekkerkerker, H.N.W. Direct visual observation of thermal capillary waves. Science
**2004**, 304, 847–850. [Google Scholar] [CrossRef] [PubMed] - Fisher, M.P.A.; Fisher, D.S.; Weeks, J.D. Agreement of capillary-wave theory with exact results for the interface profile of the two-dimensional Ising model. Phys. Rev. Lett.
**1982**, 48. [Google Scholar] [CrossRef] - Hapke, T.; Patzold, G.; Heermann, D.W. Surface tension of amorphous polymer films. J. Chem. Phys.
**1998**, 109, 10075–10081. [Google Scholar] [CrossRef] - Li, J.C.M. Possibility of subgrain rotation during recrystallization. J. Appl. Phys.
**1962**, 33, 2958–2965. [Google Scholar] [CrossRef] - Harris, K.E.; Singh, V.V.; King, A.H. Grain rotation in thin films of gold. Acta Mater.
**1998**, 46, 2623–2633. [Google Scholar] [CrossRef] - Nabarro, F.R.N. Theory of Crystal Dislocations; Oxford Univ. Press: London, UK, 1967. [Google Scholar]
- Weertman, J.; Weertman, J.R. Elementary Dislocation Theory; Oxford Univ. Press: Oxford, UK, 1992. [Google Scholar]
- Hirth, J.P.; Lothe, J. Theory of Dislocations, 2nd ed.; Krieger Publishing Co.: Malabar, FL, USA, 1992. [Google Scholar]
- Hull, D.; Bacon, D.J. Introduction to Dislocations, 5th ed.; Butterworth-Heinemann: Oxford, UK, 2011. [Google Scholar]
- Anderson, M.P.; Srolovitz, D.J.; Grest, G.S.; Sahni, P.S. Computer simulation of grain growth—I. Kinetics. Acta Metall.
**1984**, 32, 783–791. [Google Scholar] [CrossRef] - Srolovitz, D.J.; Anderson, M.P.; Sahni, P.S.; Grest, G.S. Computer simulation of grain growth—II. Grain size distribution, topology, and local dynamics. Acta Metall.
**1984**, 32, 793–802. [Google Scholar] [CrossRef] - Rollet, A.D.; Srolovitz, D.J.; Anderson, M.P. Simulation and theory of abnormal grain growth—Anisotropic grain boundary energies and mobilities. Acta Metall.
**1989**, 37, 1227–1240. [Google Scholar] [CrossRef] - Landau, D.P. Monte Carlo studies of critical and multicritical phenomena. In Applications of the Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Springer: Berlin, Germany, 1984; pp. 93–124. [Google Scholar]
- Kawasaki, K.; Nagai, T.; Nakashima, K. Vertex models for two-dimensional grain growth. Philos. Mag. B
**1989**, 60, 399–421. [Google Scholar] [CrossRef] - Weygand, D.; Bréchet, Y.; Lépinoux, J. A vertex dynamics simulation of grain growth in two dimensions. Philos. Mag. B
**1998**, 78, 329–352. [Google Scholar] [CrossRef] - Raabe, D. Cellular automata in materials science with particular reference to recrystallization simulation. Annu. Rev. Mater. Res.
**2002**, 32, 53–76. [Google Scholar] [CrossRef] - Doherty, R.D.; Hughes, D.A.; Humphreys, F.J.; Jonas, J.J.; Juul Jensen, D.; Kassner, M.E.; King, W.E.; McNelley, T.R.; McQueen, H.J.; Rollet, A.D. Current issues in recrystallization: A review. Mater. Sci. Eng. A
**1997**, 238, 219–274. [Google Scholar] [CrossRef] - Raabe, D.; Becker, R.C. Coupling of a crystal plasticity finite-element model with a probabilistic cellular automaton for simulating primary static recrystallization in aluminum. Model.Simul. Mater. Sci. Eng.
**2000**, 8, 445–462. [Google Scholar] [CrossRef] - Raabe, D. Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data. Comp. Mater. Sci.
**2000**, 19, 13–26. [Google Scholar] [CrossRef] - Chen, L.-Q. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res.
**2002**, 32, 113–140. [Google Scholar] [CrossRef] - Kobayashi, R.; Warren, J.A.; Carter, W.C. Vector-valued phase field model for crystallization and grain boundary formation. Phys. D
**1998**, 119, 415–423. [Google Scholar] [CrossRef] - Kobayashi, R.; Warren, J.A.; Carter, W.C. A continuum model of grain boundaries. Phys. D
**2000**, 140, 141–150. [Google Scholar] [CrossRef] - Warren, J.A.; Kobayashi, R.; Lobkovsky, A.E.; Carter, W.C. Extending phase field models of solidification to polycrystalline materials. Acta Mater.
**2003**, 51, 6035–6058. [Google Scholar] [CrossRef] - Krill, C.E., III; Chen, L.-Q. Computer simulation of 3-D grain growth using a phase field model. Acta Mater.
**2002**, 50, 3057–3073. [Google Scholar] [CrossRef] - Kobayashi, R.; Warren, J.A. Modeling the formation and dynamics of polycrystals in 3D. Phys. A
**2005**, 356, 127–132. [Google Scholar] [CrossRef] - Kim, S.G.; Kim, D.I.; Kim, W.T.; Park, Y.B. Computer simulations of two-dimensional and three-dimensional ideal grain growth. Phys. Rev. E
**2006**, 74, 061605. [Google Scholar] [CrossRef] [PubMed] - Bjerre, M.; Tarp, J.M.; Angheluta, L.; Mathiesen, J. Rotation-induced grain growth and stagnation in phase-field crystal models. Phys. Rev. E
**2013**, 88, 020401. [Google Scholar] [CrossRef] [PubMed] - Gránásy, L.; Pusztai, T.; Warren, J.A. Modelling polycrystalline solidification using phse field theory. J. Phys. Condens. Matter
**2004**, 16, R1205–R1235. [Google Scholar] [CrossRef] - Singer-Loginova, I.; Singer, H.M. The phase field technique for modeling multiphase materials. Rep. Prog. Phys.
**2008**, 71, 106501. [Google Scholar] [CrossRef] - Hansen, J.P.; McDonald, I.R. Theory of Simple Liquids; Academic Press: London, UK, 1986. [Google Scholar]
- Becker, R.; Döring, W. Kinetische behandlung der keimbildung in übersättigten Dämpfen. Ann. Phys.
**1935**, 24, 719–752. [Google Scholar] [CrossRef] - Turnbull, D.; Fisher, J.C. Rate of nucleation in condensed systems. J. Chem. Phys.
**1949**, 17, 71–73. [Google Scholar] [CrossRef] - Binder, K.; Stauffer, D. Statistical theory of nucleation, condensation and coagulation. Adv. Phys.
**1976**, 25, 343–396. [Google Scholar] [CrossRef] - Pusey, P.N.; van Megen, W. Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. Nature
**1986**, 320, 340–342. [Google Scholar] [CrossRef] - Zhu, J.; Li, M.; Rogers, R.; Meyer, W.; Ottewill, R.H.; STS-73 Space Shuttle Crew; Russel, W.B.; Chaikin, P.M. Crystallization of hard-sphere colloids in microgravity. Nature
**1987**, 387, 883–885. [Google Scholar] - Auer, S.; Frenkel, D. Prediction of absolute crystal-nucleation rate in hard sphere colloids. Nature
**2001**, 409, 1020–1023. [Google Scholar] [CrossRef] [PubMed] - Anderson, V.J.; Lekkerkerker, H.N. Insights into phase transition kinetics from colloid science. Nature
**2002**, 416, 811–815. [Google Scholar] [CrossRef] [PubMed] - Cacciuto, A.; Auer, S.; Frenkel, D. Onset of heterogeneous crystal nucleation in colloidal suspensions. Nature
**2004**, 428, 404–406. [Google Scholar] [CrossRef] [PubMed] - Auer, S.; Frenkel, D. Numerical simulations of crystal nucleation in colloids. Adv. Polym. Sci.
**2005**, 173, 149–207. [Google Scholar] - Schilling, T.; Schöpe, H.J.; Oettel, M.; Opletal, G.; Snook, I. Precursor-mediatedi crystallization process in suspensions of hard spheres. Phys. Rev. Lett.
**2010**, 105, 025701. [Google Scholar] [CrossRef] [PubMed] - Deutschländer, S.; Dillmann, P.; Maret, G.; Keim, P. Kibble-Zurek mechanism in colloidal monolayers. Proc. Natl. Acad. Sci. USA
**2015**, 112, 6925–6930. [Google Scholar] [CrossRef] [PubMed] - Swygenhoven, H.V. Grain boundaries and dislocations. Science
**2002**, 296, 66–67. [Google Scholar] [CrossRef] [PubMed] - Cherkaoui, M.; Capolungo, L. Atomistic and Continuun Modeling of Nanocrystalline Materials: Deformation Mechanisms and Scale Transition; Springer: Berlin, Germany, 2009. [Google Scholar]

**Figure 1.**The free energy barrier of crystal nuclei. Generally, nuclei of subcritical size ($r<{r}_{c}$) shrink and disappear, while nuclei that reach a postcritical size ($r>{r}_{c}$) can grow larger, decreasing in this way their energy.

**Figure 2.**The potential of interaction between two particles. r is the distance between centers, normalized by the hard core diameter [70]. Reprinted from Physics Letters A, 376, González, A.E. and Ixtlilco-Cortés, L., Fractal structure of the crystalline-nuclei boundaries in 2D colloidal crystallization: Computer simulations, 1375–1379, Copyright (2012), with permission from Elsevier.

**Figure 3.**The phase diagram for systems of particles attracting with a short-ranged potential. There is a metastable fluid-fluid (or liquid-gas) coexistence, indicated by the dashed curve, which lies inside the final liquid-solid (or fluid-crystal) coexistence region. The vertical axis (T) is temperature while the horizontal axis (ϕ) is the volume fraction (or area fraction in 2D).

**Figure 4.**The simulation box where, on panel (

**a**), are shown all the particles after one Monte Carlo sweep while, on panel (

**b**), the MC time has advanced to 4001.

**Figure 5.**The global bond orientational order parameter ${\psi}_{6}$ (see the text) as a function of the Monte Carlo (MC) time t, for t = 4001, 8001, 12001, etc.

**Figure 6.**A section of the simulation box for the MC time of t = 5866 where, on panel (

**a**), are shown all the particles in that section while, on panel (

**b**), only the particles belonging to crystallites with more than 10 particles [70]. Reprinted from Physics Letters A, 376, González, A.E. and Ixtlilco-Cortés, L., Fractal structure of the crystalline-nuclei boundaries in 2D colloidal crystallization: Computer simulations, 1375–1379, Copyright (2012), with permission from Elsevier.

**Figure 7.**(

**a**) A section of the simulation box, where in (a) are shown all the particles in that section; while in (

**b**) only the particles belonging to crystallites with more than 10 particles. In this case we obtain a single crystallite with 75 particles, which is in the range of the critical crystallite size [70]. Reprinted from Physics Letters A, 376, González, A.E. and Ixtlilco-Cortés, L., Fractal structure of the crystalline-nuclei boundaries in 2D colloidal crystallization: Computer simulations, 1375–1379, Copyright (2012), with permission from Elsevier.

**Figure 8.**An example of (

**a**) two small crystallites with a chainy structure and (

**b**) a more rounded and compact (in the interior) crystal.

**Figure 9.**In panel (

**a**) we see two nearby crystals, at the MC time of 21,001, “interacting” through a cloud of particles; while in panel (

**b**), for the MC time of 35,001, we can see that the smaller of the the two crystals has almost disappeared, leaving a cloud of particles as the only vestige of its existence. In subsequent times this cloud would also vanish.

**Figure 10.**The aspect of a grain boundary formed after two grains started to touch during their growth, for the MC time 11,001.

**Figure 11.**In panels (

**a**)–(

**d**) are shown the whole simulation box at the MC times (

**a**) t = 14,001; (

**b**) t = 35,001; (

**c**) t = 56,001 and (

**d**) t = 98,001. The straight lines labeled A through D define the average position of the grain boundaries (GBs) at the MC time 14,001 for the lines A through C, while the line D is defined for the MC time 35,001. Once such lines are defined they stay fixed for the whole simulation time.

**Figure 12.**In panel (

**a**) we see two crystalline grains above the critical size, shown at the time t = 6101, that start to touch in their growth. Note how the two symmetry axes of both crystals differ in their orientation; In panel (

**b**), for t = 6801, we observe that the formed grain boundary tries to crystallize on its left side, a process known as healing, although the two symmetry axes do not exactly coincide. Nonetheless, as the misalingment is not very pronounced, this is a low-angle grain boundary (LAGB).

**Figure 13.**In panel (

**a**) we observe the same two crystal grains as in Figure 12, where now their grain boundary has healed on both sides, at the MC time of 8001; In panel (

**b**), for the same MC time, a Burgers circuit has been added, traversed counterclockwise (according to one of the conventions in the literature), that consists of three sides of length of 12 lattice spacings each, along the three axes of symetry of the crystal. The Burgers vector shown joins the initial to the final point of such circuit.

**Figure 14.**A triangular lattice containing a dislocation, consisting of two nearest-neighbor disclinations of coordination number 5 and 7. The two extra half-rows as well as the Burgers vector are shown highlighted.

**Figure 15.**(

**a**) In this figure we can appreciate, for the time t = 8501, the way in which the crystal tries to eliminate the dislocation. It opens up a little channel towards the boundary with the surrounding fluid, that will be filled up with particles, first with particles inside the channel and then with particles from the surrounding fluid; (

**b**) For the time t = 10,001, the filling up of the channel with more particles continues.

**Figure 16.**At the time t = 10,501 the dislocation has been eliminated, which could be said that went out from the upper-left boundary of the new, bigger crystal grain. Note how the three axes of symmetry of the crystal are well defined now. At the same time, the formation of a new grain boundary at the upper-right corner of the figure can be observed.

**Figure 17.**The aspect of a grain boundary at the times (

**a**) t = 9001; (

**b**) t = 28,001; (

**c**,

**d**) t = 68,001. In (c) the angle between the symmetry axes of the two joining crystals is shown.

**Figure 18.**The way in which the crystal gets rid of the complex dislocations, at the times (

**a**) t = 72,501; (

**b**) t = 80,001; (

**c**) t = 80,501 and (

**d**) t = 89,001. Note the different scales of the figures on the vertical axis. In (d) the angle between the two joining crystals is shown.

**Figure 19.**The monitoring of the last complex dislocation inside the crystal grain, at the times (

**a**) t = 95,001; (

**b**) t = 99,001; (

**c**) t = 101,001 and (

**d**) t = 101,501. Note the different scales of the figures on the vertical axis.

**Figure 20.**The Delaunay triangulation of one of the CDs (the third from the left) shown in Figure 17c,d. It consists on seven 5-7 “topological dipoles” and two 5-8-5 “topological cuadrupoles”, where the “topological charge” of a particle refers to the difference between its coordination number minus six.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

González, A.E.
Colloidal Crystallization in 2D for Short-Ranged Attractions: A Descriptive Overview. *Crystals* **2016**, *6*, 46.
https://doi.org/10.3390/cryst6040046

**AMA Style**

González AE.
Colloidal Crystallization in 2D for Short-Ranged Attractions: A Descriptive Overview. *Crystals*. 2016; 6(4):46.
https://doi.org/10.3390/cryst6040046

**Chicago/Turabian Style**

González, Agustín E.
2016. "Colloidal Crystallization in 2D for Short-Ranged Attractions: A Descriptive Overview" *Crystals* 6, no. 4: 46.
https://doi.org/10.3390/cryst6040046