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Decay of Metastable States in Spin-Crossover Solids

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Decay of Metastable States in Spin-Crossover Solids

  1. 1. Decay of Metastable States in Spin-Crossover Solids Ranjit Chacko 1,2 Harvey Gould 1,2 W. Klein 2 1. Clark University 2. Boston University
  2. 2. h temperatures because it where g = gHS / gLS denotes the degeneracy ratio between the ition between the LS and HS and LS states.ges of the pressure, mag- The other important characteristic is the intermolecular Spin-Crossover Solids 7 When interactions be- interaction. For the cooperative property in the SC transition, raction changes smoothly the interactions become RHerative phenomena.8 The Vintra(R) RLs sharper with increasingthe interaction exceeds adiscontinuous. In order toperties of SC compounds,stable nature of such mo-the spin-crossover transi-acteristics of the system. he intramolecule Hamil- an energy difference be- and different degenera- the HS state and the LS D RL pin state at the ith site by Ror HS. The intramolecule RHy FIG. 1. Color online Schematic picture of the energy structuresi . 1 of a molecule. The left right minimum corresponds to the LS HS state. In the inset, schematic pictures of a lattice of LS molecules of the degeneracy as a left , and the distortion caused by a HS molecule in a lattice of LS • Molecules have low spin(LS) and high spin(HS) states.n use an effective Hamil- molecules right , are illustrated. • HS state has larger radius, higher energy, and higher degeneracy. 014105-1 ©2008 The American Physical Society • Effective long range interaction because of lattice distortion.
  3. 3. Wajnflasz-Pick Model 1H = −J σi σj + (D − kb T ln g)σi <i,j> 2 i
  4. 4. Wajnflasz-Pick Model NN Ising interaction 1H = −J σi σj + (D − kb T ln g)σi <i,j> 2 i
  5. 5. Wajnflasz-Pick Model NN Ising interaction effective field 1H = −J σi σj + (D − kb T ln g)σi <i,j> 2 i
  6. 6. Wajnflasz-Pick Model NN Ising interaction effective field 1 H = −J σi σj + (D − kb T ln g)σi <i,j> 2 ienergy difference between HS and LS states D>0
  7. 7. Wajnflasz-Pick Model NN Ising interaction effective field 1 H = −J σi σj + (D − kb T ln g)σi <i,j> 2 ienergy difference between HS and LS states g is the ratio of degeneracy of HS D>0 to LS states
  8. 8. the ratio of the radii to be RHS =RLS ˆ 1:1. H nnn expresses on the details of the c that the data given here Modeling Elastic Interactions elastic interaction for next-nearest-neighbor pairs (hhi; jii). In this study, we set the ratio of the spring constants, k1 =k2 , we perform simulation to be 10 [35]. MCSs at several point For the simulation, we adopt the NPT-MC method [36] First, we study how t Konishi et al., PRL, 2008 for the isothermal-isobaric ensemble with the number of the HS fraction fHS of molecules N, the pressure of the system P, and the k1 …ˆ 10k2 †. In Fig. 2 w Miyashita et al, Phys. Rev. B, 2008 temperature T. The thermodynamic potential for the of k1 with g…ˆ gHS = isothermal-isobaric ensemble is the enthalpy, H ˆ k1 ˆ 10, 20, 30,40, an• Replace NN Ising interaction with elastic interaction. of E ‡ PV, where E is the energy and V is the volume 10, the transition is g• Model exhibits mean fieldstates of the system are specified by 4N ‡ the system. The behavior. transition becomes sha • mean field 1 variables (n1 ; ; nN ; r1 ; ; rN ; V). In the NPT-MC critical exponents method, we have the following detailed balance condition 1 • spinodal nucleation HS molecule 0.8 LS molecule HS fraction 0.6 0.4 Rl Ri . . 0.2 Spring constant k1 Spring constant k2 0 0 0.2 Rj . . Rk FIG. 2 (color online). fraction fHS …T† with g FIG. 1 (color online). Schematic illustration of the present (red squares), 20 (green model. HS/LS molecule consists of Fe atom (red/blue circle) (blue triangles), and 50 (
  9. 9. Classical vs. Spinodal Nucleation Classical nucleation Spinodal Nucleation Geometry Compact Ramified Small difference from Density Same as stable phase metastable phaseGrowth Mode Grows at surface Fills in
  10. 10. Long Range Ising SC Model• Model ordering behavior of SC solids with Ising interaction as in WP model.• But, use weak long range interaction between spins to model the effective long range interaction due to elastic forces in SC solids.
  11. 11. Modeling Long-Range Interactions 1H = −J σi σj + (D − kb T ln g)σi 2 i [i,j] F 2 1 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 m -1 -2 -3
  12. 12. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  13. 13. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  14. 14. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  15. 15. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  16. 16. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  17. 17. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  18. 18. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  19. 19. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  20. 20. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  21. 21. Ising Model and Percolation• Mapping Ising model to percolation model gives us a geometric way of thinking about critical behavior.• Coniglio-Klein bond probability gives same critical point and same critical exponents as Ising model.• Clusters are statistically independent.• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang. p = 1 − e−2βJ A. Coniglio and W. Klein, J. Phys. A 1980
  22. 22. Percolation at the Spinodal Unger, Klein, Phys. Rev. B, 1994 p = 1 − e−2βJ(1−ρs )• For Ising model with long range interactions spinodal line can also be mapped to percolation transition.• Bond probability now depends on density of stable phase spins.
  23. 23. Entropic Barrier to Nucleation Klein et al., Phys. Rev. E 2007• Typically many clusters of connected spins will span one correlation volume.• Coalescence of clusters occurs at nucleation.• Reduction in number of independent clusters means loss of entropy.
  24. 24. Percolation in Infinite Range Ising Model• Bonds can be placed between any two aligned spins.• Equivalent to Erdos-Renyi graph theory.• In Erdos-Renyi graphs a “giant component” appears at a critical value of the probability.• This critical value corresponds to the Unger-Klein probability at the spinodal density.• Coalescence of clusters begins before nucleation.
  25. 25. Spinodal Nucleation in SC Solids• Investigate alternative derivation of bond probability through branching process argument.• More detailed study of spinodal nucleation in elastic model. • identify precursors to nucleation?• Investigate compressible Ising model as model for phase transition kinetics in SC solids.
  • YogendraSingh241

    Nov. 14, 2019

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