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Crystallization in two dimensions

Crystallization in two dimensions. Workshop on Crystallization and Melting in Two-Dimensions MTA-SZFKI, Budapest, Hungary, May 18, 2010 by Hartmut Löwen ( Heinrich-Heine-Universität Düsseldorf) . Outline :. Introduction: crystallization and melting in two dimensions

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Crystallization in two dimensions

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  1. Crystallization in two dimensions Workshop on Crystallization and Melting in Two-Dimensions MTA-SZFKI, Budapest, Hungary, May 18, 2010 by Hartmut Löwen (Heinrich-Heine-Universität Düsseldorf) Outline: • Introduction: crystallization and melting in two dimensions • Dynamical density functional theory • Glass transitions in magnetic colloids • 3) Crystallization in 2d binary mixtures • 4) Conclusions

  2. Classical many body system in strict two dimensions Ground state of a repulsive potential - hard disks, - repulsive dipoles - plasma , is a triangular lattice Temperature T=0: Long-range translational order, periodic density field potential energy minimization

  3. Temperature T>0: long-range translational order does not exist in 2d under certain general conditions Mermin, Wagner PRL 17, 1133 (1968) spin systems Mermin PRE 176, 250 (1968) more general mathematical proof by Fröhlich, Pfister, Communications in Mathematical Physics 81, 277 (1981) (not for plasma and hard disks) but long-ranged bond orientational order exists!

  4. Communications in Mathematical Physics 81, 277 (1981)

  5. More quantitative: correlation functions translational order, pair correlation fucnction j band-orientational order i fixed reference axis fluid: solid:

  6. The debate about two-dimensional melting/crystallization coexistence 1) first order fluid solid 1. order 2) Kosterlitz-Thouless-Nelson-Halperin-Young (KTNHY) scenario thermal unbinding of defects disclinations dislocations fluid hexatic solid 2. order 2. order intermediate hexatic phase

  7. Experimental realizationofclassicaltwo-dimensional systems • Colloids at an air-water interface • (or between two parallel glass plates) • b) Granulates on a vibrating horizontal table • c) Dusty (complex) plasma sheets

  8. a) 2dcolloidal dispersions (Keim, Maret, Zahn et al.) tilt angle • spherical colloids confined to water/air interface • superparamagnetic due to Fe2O3 doping • external magnetic field •  induced dipole moments •  tunable interparticle potential surface normal repulsive no interaction attractive

  9. Particle configurations for different fields (Maret, Keim, Eisermann 2004) f B perp. to surface, liquid in-plane B i k B perp. to surface, crystal KTNHY scenario confirmed binary mixtures also realizable

  10. b) granulates on a vibrating table one-component hard disks: consistent with KTNHY (Shattuck et al, 2006) binary mixtures M.B. Hay, R.K. Workman, S. Manne, Phys. Rev. E 67, 012401 (2003)

  11. G.K. Kaufman, S.W. Thomas III, M. Reches, B.F. Shaw, J. Feng, G.M. Whitesides Soft Matter5, 1188 (2009)

  12. c) dusty complex plasmas R.A. Quinn, J. Goree, Phys. Rev. E 64, 051404 (2001)

  13. Donko, Hartmann: theoretical work on 2d Yukawa consistent with KTNHY

  14. 2) Dynamical density functional theory Equilibrium Density Functional Theory (DFT) Basic variational principle: There exists a unique grand-canonical free energy-density-functional , which gets minimalfor the equilibrium density and then coincides with the real grandcanoncial free energy. → is also valid for systems which are inhomogeneous on a microscopic scale. In principle, all fluctuations are included in an external potential which breaks all symmetries. For interacting systems, in 3d (2d), is not known.

  15. exceptions: i) soft potentials in the high density limit, ideal gas (how density limit) ii) 1d: hard rod fluid, exact Percus functional strategy: 1) chose an approximation 2) parametrize the density field with variational parameters gas, liquid: solid: with lattice vectors of bcc or fcc or ... crystals, spacing sets , vacancies? variational parameter Gaussian approximation for the solid density orbital is an excellent approximation 3) minimize with respect to all variational parameters → bulk phase diagram EPL 22, 245 (1993)

  16. b) approximations for the density functional + defines the excess free energy functional A) Ramakrishnan-Youssuf (RY) 1979 (for hard spheres) results in a first order solid-fluid transition

  17. dynamical density functional theory (DDFT) Starting point: Smoluchowski equation (exact) for Brownian dynamics (colloids) integrate out (Archer and Evans, JCP 2004) adiabatic approximation: such that time-dependent one particle density field is the same (in excellent agreement with BD computer simulations)

  18. Dynamics of crystal growth at externally imposed nucleation clusters Idea: impose a cluster of fixed colloidal particles (e.g. by optical tweezer) Does this cluster act as a nucleation seed for further crystal growth? cf: homogeneous nucleation: the cluster occurs by thermal fluctuations, here we prescribe them How does nucleation depend on cluster size and shape? (S. van Teeffelen, C.N. Likos, H. Löwen, PRL, 100,108302 (2008))

  19. equilibrium functional by Ramakrishnan-Yussouff (2d) hexatic phase?? (S. van Teeffelen et al, EPL 75, 583 (2006); J. Phys.: Condensed Matter, 20, 404217 (2008)) (magnetic colloids with dipole moments) coupling parameter equilibrium freezing for connection to phase field crystal models (L. Granasy et al) by gradient expansion (van Teeffelen, Backofen, Voigt, HL, Phys. Rev. E 79, 051404 (2009)

  20. procedure

  21. imposed nucleation seed cut-out of a rhombic crystal with N=19 particles

  22. nucleation + growth

  23. no nucleation

  24. „island“ for heterogeneous nucleation in Brownian dynamics computer simulation strongly asymmetric in A symmetric in

  25. 3) Crystallization in 2d binary mixtures • binary spherical colloids confined to water/air interface • superparamagnetic due to Fe2O3 doping • external magnetic field •  induced dipole moments •  tunable interparticle potential

  26. phase diagram at zero temperature composition A. Lahcen, R. Messina and HL, EPL80 48001 (2007)

  27. Some important phases X=2/3 X=0 X=1/2 X=1 X=1/3 X=1/2

  28. Experimental snapshots at and (F. Ebert, P. Keim, G. Maret, EPJE 26, 161 (2008)

  29. composition Found in experiment

  30. Ultra-fast temperature quench can be realized by increasing the magnetic field

  31. Brownian Dynamics computer simulation in agreement with experiments “patches“ of crystallites is this a glass?? dynamical heterogeneities L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. Löwen, Phys. Rev. Lett. 102, 238301 (2009)

  32. non-monotonic behaviour (in time) for 2-2- structure

  33. 4) Ground state of 2d oppositely charged mixtures 3d, textbook knowledge: NaCl, CsCl, ZnS structures are stable Lattice sum minimization (penalty method for hard spheres) L. Assoud, R. Messina, H.L., EPL 89, 36001 (2010)

  34. 5) Conclusions • 2d melting/crystallization is still interesting • mixtures • tetratic phase? • (2+ ) confinement (e.g. between plates • with finite spacing)

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