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Mathematical Challenges in a New Phase of Materials Science pp 1–26Cite as

Generation of Defects and Disorder from Deeply Quenching a Liquid to Form a Solid

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 166))

Abstract

We show how deeply quenching a liquid to temperatures where it is linearly unstable and the crystal is the equilibrium phase often produces crystalline structures with defects and disorder. As the solid phase advances into the liquid phase, the modulations in the density distribution created behind the advancing solidification front do not necessarily have a wavelength that is the same as the equilibrium crystal lattice spacing. This is because in a deep enough quench the front propagation is governed by linear processes, but the crystal lattice spacing is determined by nonlinear terms. The wavelength mismatch can result in significant disorder behind the front that may or may not persist in the latter stage dynamics. We support these observations by presenting results from dynamical density functional theory calculations for simple one- and two-component two-dimensional systems of soft core particles.

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Notes

  1. 1.

    The result in Eq. (2) also applies to non-uniform liquids, i.e. to liquids initially at equilibrium with a density profile ρ old (r) in an external potential V old (r), disturbed by an infinitesimal change to the external potential, \(V _{old}(\mathbf{r}) \rightarrow V _{new}(\mathbf{r})\). The resulting change in the density profile \(\delta \rho (\mathbf{r}) \equiv \rho _{new}(\mathbf{r}) -\rho _{old}(\mathbf{r})\) is then also given by Eq. (2), where \(\delta V _{ext}(\mathbf{r}) \equiv V _{new}(\mathbf{r}) - V _{old}(\mathbf{r})\).

  2. 2.

    The early time profile displayed in the top right panel in Fig. 12 of Ref. [7] shows the corresponding plot for Φ = 0. 25 instead of Φ = 0. 5 as labelled there. This error is corrected here in Fig. 10.

  3. 3.

    The specific criterion for deciding to which subset a given density peak belongs is as follows: After performing the Delauney triangulation on the set of peaks, we consider each triangle. The corner angles are \(\theta _{1}\), \(\theta _{2}\) and \(\theta _{3}\). The triangle is defined as equilateral if \(\vert \theta _{i} -\theta _{j}\vert <5^{\circ }\) for all pairs i, j = 1, 2, 3. The vertices of these triangles are coloured black. Triangles are defined as right-angled if for the largest angle \(\theta _{1}\) we have \(\vert \theta _{1} - 90^{\circ }\vert <5^{\circ }\) and for the other two angles \(\vert \theta _{2} -\theta _{3}\vert <5^{\circ }\). The vertices of these triangles are coloured red. All remaining vertices which fall into neither of these categories are displayed as open circles.

References

  1. Debenedetti, P.G.: Metastable Liquids: Concepts and Principles. University Press, Princeton (1996)

    Google Scholar 

  2. Oxtoby, D.W.: Homogeneous nucleation: theory and experiment. J. Phys.: Condens. Matter 4, 7627 (1992)

    Google Scholar 

  3. Oxtoby, D.W.: Nucleation of first-order phase transitions. Acc. Chem. Res. 31, 91 (1998)

    Article  Google Scholar 

  4. Sear, R.P.: Nucleation: theory and applications to protein solutions and colloidal suspensions. J. Phys.: Condens. Matter 19, 033101 (2007)

    Google Scholar 

  5. Trudu, F., Donadio, D., Parrinello, M.: Freezing of a Lennard-Jones fluid: from nucleation to spinodal regime. Phys. Rev. Lett. 97, 105701 (2006)

    Article  Google Scholar 

  6. Klein, W., Leyvraz, F.: Crystalline nucleation in deeply quenched liquids. Phys. Rev. Lett. 57, 2845 (1986)

    Article  Google Scholar 

  7. Archer, A.J., Walters, M.C., Thiele, U., Knobloch, E.: Solidification in soft-core fluids: disordered solids from fast solidification fronts. Phys. Rev. E 90, 042404 (2014)

    Article  Google Scholar 

  8. Archer, A.J., Robbins, M.J., Thiele, U., Knobloch, E.: Solidification fronts in supercooled liquids: How rapid fronts can lead to disordered glassy solids. Phys. Rev. E 86, 031603 (2012)

    Article  Google Scholar 

  9. van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29 (2003)

    Article  MATH  Google Scholar 

  10. Marconi, U.M.B., Tarazona, P.: Dynamic density functional theory of fluids. J. Chem. Phys. 110, 8032 (1999)

    Article  Google Scholar 

  11. Marconi, U.M.B., Tarazona, P.: Dynamic density functional theory of fluids. J. Phys.: Condens. Matter 12, A413 (2000)

    Google Scholar 

  12. Archer, A.J., Evans, R.: Dynamical density functional theory and its application to spinodal decomposition. J. Chem. Phys. 121, 4246 (2004)

    Article  Google Scholar 

  13. Archer, A.J., Rauscher, M.: Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic? J. Phys. A: Math. Gen. 37, 9325 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids, 4th edn. Academic, London (2013)

    MATH  Google Scholar 

  15. Liu, F., Goldenfeld, N.: Dynamics of phase-separation in block copolymer melts. Phys. Rev. A 39, 4805 (1989)

    Article  Google Scholar 

  16. Galenko, P.K., Elder, K.R.: Marginal stability analysis of the phase field crystal model in one spatial dimension. Phys. Rev. B 83, 064113 (2011)

    Article  Google Scholar 

  17. Chen, L.Y., Goldenfeld, N., Oono, Y., Paquette, G.: Selection, stability and renormalization. Physica A 204, 113 (1994)

    Article  Google Scholar 

  18. van Teeffelen, S., Backofen, R., Voigt, A., Löwen, H.: Derivation of the phase-field-crystal model for colloidal solidification. Phys. Rev. E 79, 051404 (2009)

    Article  Google Scholar 

  19. Emmerich, H., Löwen, H., Wittkowski, R., Gruhn, T., Tóth, G.I., Tegze, G., Gránásy, L.: Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Adv. Phys. 61, 665 (2012)

    Article  Google Scholar 

  20. Evans, R.: Nature of the liquid-vapor interface and other topics in the statistical-mechanics of nonuniform, classical fluids. Adv. Phys. 28, 143 (1979)

    Article  Google Scholar 

  21. Evans, R.: Fundamentals of Inhomogeneous Fluids. Dekker, New York (1992)

    Google Scholar 

  22. Lutsko, J.F.: Recent developments in classical density functional theory. Adv. Chem. Phys. 144, 1 (2010)

    Google Scholar 

  23. Dee, G., Langer, J.S.: Propagating pattern selection. Phys. Rev. Lett. 50, 383 (1983)

    Article  Google Scholar 

  24. Ben-Jacob, E., Brand, H., Dee, G., Kramer, L., Langer, J.S.: Pattern propagation in nonlinear dissipative systems. Physica D 14, 348 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Likos, C.N.: Effective interactions in soft condensed matter physics. Phys. Rep. 348, 267 (2001)

    Article  Google Scholar 

  26. Dautenhahn, J., Hall, C.K.: Monte Carlo simulation of off-lattice polymer chains: effective pair potentials in dilute solution. Macromolecules 27, 5399 (1994)

    Article  Google Scholar 

  27. Likos, C.N., Löwen, H., Watzlawek, M., Abbas, B., Jucknischke, O., Allgaier, J., Richter, D.: Star polymers viewed as ultrasoft colloidal particles. Phys. Rev. Lett. 80, 4450 (1998)

    Article  Google Scholar 

  28. Louis, A.A., Bolhuis, P.G., Hansen, J.P., Meijer, E.J.: Can polymer coils be modeled as “soft colloids”? Phys. Rev. Lett. 85, 2522 (2000)

    Article  Google Scholar 

  29. Bolhuis, P.G., Louis, A.A., Hansen, J.P., Meijer, E.J.: Accurate effective pair potentials for polymer solutions. J. Chem. Phys. 114, 4296 (2001)

    Article  Google Scholar 

  30. Jusufi, A., Dzubiella, J., Likos, C.N., von Ferber, C., Löwen, H.: Effective interactions between star polymers and colloidal particles. J. Phys.: Condens. Matter 13, 6177 (2001)

    Google Scholar 

  31. Dzubiella, J., Jusufi, A., Likos, C.N., von Ferber, C., Löwen, H., Stellbrink, J., Allgaier, J., Richter, D., Schofield, A.B., Smith, P.A., Poon, W.C.K., Pusey, P.N.: Phase separation in star-polymer-colloid mixtures. Phys. Rev. E 64, 010401(R) (2001)

    Google Scholar 

  32. Louis, A.A., Bolhuis, P.G., Finken, R., Krakoviack, V., Meijer, E.J., Hansen, J.P.: Coarse-graining polymers as soft colloids. Physica A 306, 251 (2002)

    Article  MATH  Google Scholar 

  33. Likos, C.N., Harreis, H.M.: Star polymers: from conformations to interactions to phase diagrams. Condens. Matter Phys. 5, 173 (2002)

    Article  Google Scholar 

  34. Götze, I.O., Harreis, H.M., Likos, C.N.: Tunable effective interactions between dendritic macromolecules. J. Chem. Phys. 120, 7761 (2004)

    Article  Google Scholar 

  35. Mladek, B.M., Fernaud, M.J., Kahl, G., Neumann, M.: On the thermodynamic properties of the generalized Gaussian core model. Condens. Matter Phys. 8, 135 (2005)

    Article  Google Scholar 

  36. Likos, C.N.: Soft matter with soft particles. Soft Matter 2, 478 (2006)

    Article  Google Scholar 

  37. Lenz, D.A., Blaak, R., Likos, C.N., Mladek, B.M.: Microscopically resolved simulations prove the existence of soft cluster crystals. Phys. Rev. Lett. 109, 228301 (2012)

    Article  Google Scholar 

  38. Archer, A.J., Evans, R.: Binary Gaussian core model: fluid-fluid phase separation and interfacial properties. Phys. Rev. E 64, 041501 (2001)

    Article  Google Scholar 

  39. Archer, A.J., Evans, R.: J. Phys.: Wetting in the binary Gaussian core model. Condens. Matter 14, 1131 (2002)

    Google Scholar 

  40. Archer, A.J., Evans, R., Roth, R.: Microscopic theory of solvent-mediated long-range forces: influence of wetting. Europhys. Lett. 59, 526 (2002)

    Article  Google Scholar 

  41. Archer, A.J., Likos, C.N., Evans, R.: Binary star-polymer solutions: bulk and interfacial properties. J. Phys.: Condens. Matter 14, 12031 (2002)

    Google Scholar 

  42. Archer, A.J., Likos, C.N., Evans, R.: J. Phys.: Soft-core binary fluid exhibiting a lambda-line and freezing to a highly delocalized crystal. Condens. Matter 16, L297 (2004)

    Google Scholar 

  43. Götze, I.O., Archer, A.J., Likos, C.N.: Structure, phase behavior, and inhomogeneous fluid properties of binary dendrimer mixtures. J. Chem. Phys. 124, 084901 (2006)

    Article  Google Scholar 

  44. Mladek, B.M., Gottwald, D., Kahl, G., Neumann, M., Likos, C.N.: Formation of polymorphic cluster phases for a class of models of purely repulsive soft spheres. Phys. Rev. Lett. 96, 045701 (2006)

    Article  Google Scholar 

  45. Mladek, B.M., Gottwald, D., Kahl, G., Neumann, M., Likos, C.N.: Clustering in the absence of attractions: density functional theory and computer simulations. J. Phys. Chem. B 111, 12799 (2007)

    Article  Google Scholar 

  46. Moreno, A.J., Likos, C.N.: Diffusion and relaxation dynamics in cluster crystals. Phys. Rev. Lett. 99, 107801 (2007)

    Article  Google Scholar 

  47. Likos, C.N., Mladek, B.M., Gottwald, D., Kahl, G.: Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density-functional theory. J. Chem. Phys. 126, 224502 (2007)

    Article  Google Scholar 

  48. Mladek, B.M., Charbonneau, P., Likos, C.N., Frenkel, D., Kahl, G.: Multiple occupancy crystals formed by purely repulsive soft particles. J. Phys.: Condens. Matter 20, 494245 (2008)

    Google Scholar 

  49. Likos, C.N., Mladek, B.M., Moreno, A.J., Gottwald, D., Kahl, G.: Cluster-forming systems of ultrasoft repulsive particles: statics and dynamics. Comput. Phys. Commun. 179, 71 (2008)

    Article  Google Scholar 

  50. Overduin, S.D., Likos, C.N.: Clustering in nondemixing mixtures of repulsive particles. J. Chem. Phys. 131, 034902 (2009)

    Article  Google Scholar 

  51. Overduin, S.D., Likos, C.N.: Phase behaviour in binary mixtures of ultrasoft repulsive particles. Europhys. Lett. 85, 26003 (2009)

    Article  Google Scholar 

  52. van Teeffelen, S., Moreno, A.J., Likos, C.N.: Cluster crystals in confinement. Soft Matter 5, 1024 (2009)

    Article  Google Scholar 

  53. Camargo, M., Moreno, A.J., Likos, C.N.: Dynamics in binary cluster crystals. J. Stat. Mech. Theor. Exp. 2010, P10015 (2010)

    Google Scholar 

  54. Camargo, M., Likos, C.N.: Interfacial and wetting behaviour of phase-separating ultrasoft mixtures. Mol. Phys. 109, 1121 (2011)

    Article  Google Scholar 

  55. Nikoubashman, A., Kahl, G., Likos, C.N.: Flow quantization and nonequilibrium nucleation of soft crystals. Soft Matter 8, 4121 (2012)

    Article  Google Scholar 

  56. Carta, M., Pini, D., Parola, A., Reatto, L.: A density-functional theory study of microphase formation in binary Gaussian mixtures. J. Phys.: Condens. Matter 24, 284106 (2012)

    Google Scholar 

  57. Pini, D.: A density-functional theory investigation of cluster formation in an effective-potential model of dendrimers. Trans. R. Norw. Soc. Sci. Lett. 3, 99 (2014)

    Google Scholar 

  58. Archer, A.J., Rucklidge, A.M., Knobloch, E.: Quasicrystalline order and a crystal-liquid state in a soft-core fluid. Phys. Rev. Lett. 111, 165501 (2013)

    Article  Google Scholar 

  59. Prestipino, S., Saija, F.: Hexatic phase and cluster crystals of two-dimensional GEM4 spheres. J. Chem. Phys. 141, 184502 (2014)

    Article  Google Scholar 

  60. Hari, A., Nepomnyashchy, A.A.: Nonpotential effects in dynamics of fronts between convection patterns. Phys. Rev. E 61, 4835 (2000)

    Article  MathSciNet  Google Scholar 

  61. Couairon, A., Chomaz, J.M.: Pattern selection in the presence of a cross flow. Physica D 108, 236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  62. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer-Verlag, Heidelberg (2008)

    Book  MATH  Google Scholar 

  63. Robbins, M.J., Archer, A.J., Thiele, U., Knobloch, E.: Modelling fluids and crystals using a two-component modified phase field crystal model. Phys. Rev. E 85, 061408 (2012)

    Article  Google Scholar 

  64. Archer, A.J.: Dynamical density functional theory: binary phase-separating colloidal fluid in a cavity. J. Phys.: Condens. Matter 17, 1405 (2005)

    Google Scholar 

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Acknowledgements

A.J.A. and U.T. thank the Center of Nonlinear Science (CeNoS) of the University of Münster for recent support of their collaboration. M.C.W. is supported by an EPSRC studentship. The work of E.K. was supported in part by the National Science Foundation under Grant No. DMS-1211953.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    A. J. Archer & M. C. Walters

  2. Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, D-48149, Münster, Germany

    U. Thiele

  3. Center of Nonlinear Science (CeNoS), Westfälische Wilhelms Universität Münster, Corrensstr. 2, 48149, Münster, Germany

    U. Thiele

  4. Department of Physics, University of California at Berkeley, Berkeley, CA, 94720, USA

    E. Knobloch

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  1. A. J. Archer
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Correspondence to A. J. Archer .

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Editors and Affiliations

  1. WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi, Japan

    Yasumasa Nishiura

  2. WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi, Japan

    Motoko Kotani

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Archer, A.J., Walters, M.C., Thiele, U., Knobloch, E. (2016). Generation of Defects and Disorder from Deeply Quenching a Liquid to Form a Solid. In: Nishiura, Y., Kotani, M. (eds) Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics & Statistics, vol 166. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56104-0_1

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