Tethered method to approach the equilibrium fluid-solid coexistence of hard spheres Beatriz Seoane Bartolomé in collaboration with L. A. Fernández, V. Martín-Mayor and P. Verrocchio Dipartimento di Fisica La Sapienza Università di Roma Physics of glassy and granular materials Kyoto, 18 of July of 2013 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 1 / 14
HS Crystallization Computational Problem Computational problem Approaching equilibrium in granular systems requires in general a huge amount of time even for very small system sizes, N . References V. Martin-Mayor, B. Seoane and D. Yllanes : J. Stat. Phys. 144 , 554 (2011). L. A. Fernandez, V. Martin-Mayor, B. Seoane and P. Verrocchio : Phys. Rev. Lett. 108 , 165701 (2012). B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 2 / 14
HS Crystallization Computational Problem Computational problem Approaching equilibrium in granular systems requires in general a huge amount of time even for very small system sizes, N . 1 We present a general Monte Carlo method (Tethered method) to approach equilibrium in reasonable times. References V. Martin-Mayor, B. Seoane and D. Yllanes : J. Stat. Phys. 144 , 554 (2011). L. A. Fernandez, V. Martin-Mayor, B. Seoane and P. Verrocchio : Phys. Rev. Lett. 108 , 165701 (2012). B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 2 / 14
HS Crystallization Computational Problem Computational problem Approaching equilibrium in granular systems requires in general a huge amount of time even for very small system sizes, N . 1 We present a general Monte Carlo method (Tethered method) to approach equilibrium in reasonable times. 2 Our method boosts the traditional umbrella sampling making practical the study of constrained free energies to several order parameters. References V. Martin-Mayor, B. Seoane and D. Yllanes : J. Stat. Phys. 144 , 554 (2011). L. A. Fernandez, V. Martin-Mayor, B. Seoane and P. Verrocchio : Phys. Rev. Lett. 108 , 165701 (2012). B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 2 / 14
HS Crystallization Computational Problem Computational problem Approaching equilibrium in granular systems requires in general a huge amount of time even for very small system sizes, N . 1 We present a general Monte Carlo method (Tethered method) to approach equilibrium in reasonable times. 2 Our method boosts the traditional umbrella sampling making practical the study of constrained free energies to several order parameters. 3 We apply the method to a well understood problem, but still extremely hard from the numerical point of view: the crystallization of hard spheres References V. Martin-Mayor, B. Seoane and D. Yllanes : J. Stat. Phys. 144 , 554 (2011). L. A. Fernandez, V. Martin-Mayor, B. Seoane and P. Verrocchio : Phys. Rev. Lett. 108 , 165701 (2012). B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 2 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) Several metastable phases B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) Several metastable phases Equilibrium ↔ all phases must be sampled B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) Several metastable phases Equilibrium ↔ all phases must be sampled Interfaces ∼ L 2 : appear with probability e − βγ L 2 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) Several metastable phases Equilibrium ↔ all phases must be sampled Interfaces ∼ L 2 : appear with probability e − βγ L 2 Frequency of jumps decreases exponentially with N 2 / 3 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Difficulty Glass transition ⇒ intrinsic slow dynamics. First order transition ⇒ exponential divergence of times with N . Exponential Dynamic Slowing Down (EDSD) Several metastable phases Equilibrium ↔ all phases must be sampled Interfaces ∼ L 2 : appear with probability e − βγ L 2 Frequency of jumps decreases exponentially with N 2 / 3 SOLUTION: AVOID THESE METASTABILITIES B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 3 / 14
HS Crystallization Computational Problem Hard Spheres 20.0 Seek the simplest system that HS suffers crystallization: HS. fcc 15.0 p ( k B T / σ 3 ) p co 10.0 fluid 5.0 η f η m 0.0 0.2 0.3 0.5 0.6 0.7 0.4 T. η Zykova-Timan, J. Horbach y K. Binder: J. Chem. Phys 133, 014705 (2010). . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 4 / 14
HS Crystallization Computational Problem Hard Spheres 20.0 Seek the simplest system that HS suffers crystallization: HS. fcc 15.0 The fluid-FCC was found in p ( k B T / σ 3 ) p co simulations in 1957, but no 10.0 equilibrium work is able to thermalize more than 500 fluid 5.0 particles. (Errington 2004) η f η m 0.0 0.2 0.3 0.5 0.6 0.7 0.4 T. η Zykova-Timan, J. Horbach y K. Binder: J. Chem. Phys 133, 014705 (2010). . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 4 / 14
HS Crystallization Computational Problem Hard Spheres Seek the simplest system that suffers crystallization: HS. 1.1 hot start The fluid-FCC was found in fcc start 1.08 simulations in 1957, but no 1.06 equilibrium work is able to 1.04 v 1.02 thermalize more than 500 1 particles. (Errington 2004) 0.98 All previous methods, are unable 0.96 50 100 150 200 250 300 350 400 450 500 to synthesize the FCC from a t random configuration. . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 4 / 14
HS Crystallization Computational Problem Hard Spheres Seek the simplest system that suffers crystallization: HS. 1.5 hot start 1.45 fcc start The fluid-FCC was found in 1.4 1.35 simulations in 1957, but no 1.3 1.25 equilibrium work is able to v 1.2 thermalize more than 500 1.15 1.1 particles. (Errington 2004) 1.05 1 All previous methods, are unable 0.95 0 100 200 300 400 500 600 700 800 900 1000 to synthesize the FCC from a t random configuration. Need observables that label univocally one of the branches: bond order parameters . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 4 / 14
HS Crystallization Order Parameters Crystal order parameter Q 6 (rotational symmetry) Q 6 has been widely used in crystallization studies (ten Wolde et al., 1995) . � 1 / 2 � l 4 π Perfect lattices values � | Q lm | 2 Q l ≡ 2 l + 1 Q 6 m = − l fluid FCC BCC N b ( i ) � N i = 1 q lm ( i ) � Y lm (ˆ Q lm ≡ , q lm ( i ) ≡ r ij ) 0 0.574 0.510 � N i = 1 N b ( i ) j = 1 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 5 / 14
HS Crystallization Order Parameters Crystal order parameter Q 6 (rotational symmetry) Q 6 has been widely used in crystallization studies (ten Wolde et al., 1995) . Compute the bonds joining neighboring particles (if r ij < 1 . 5 σ , with r ij = r i − r j ). � 1 / 2 � l 4 π Perfect lattices values � | Q lm | 2 Q l ≡ 2 l + 1 Q 6 m = − l fluid FCC BCC N b ( i ) � N i = 1 q lm ( i ) � Y lm (ˆ Q lm ≡ , q lm ( i ) ≡ r ij ) 0 0.574 0.510 � N i = 1 N b ( i ) j = 1 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 5 / 14
HS Crystallization Order Parameters Crystal order parameter Q 6 (rotational symmetry) Q 6 has been widely used in crystallization studies (ten Wolde et al., 1995) . Compute the bonds joining neighboring particles (if r ij < 1 . 5 σ , with r ij = r i − r j ). In a crystal bonds sum coherently (Steinhartdt et al. 1983) . � 1 / 2 � l 4 π Perfect lattices values � | Q lm | 2 Q l ≡ 2 l + 1 Q 6 m = − l fluid FCC BCC N b ( i ) � N i = 1 q lm ( i ) � Y lm (ˆ Q lm ≡ , q lm ( i ) ≡ r ij ) 0 0.574 0.510 � N i = 1 N b ( i ) j = 1 B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 5 / 14
HS Crystallization Order Parameters Crystal order parameter Q 6 (rotational symmetry) Q 6 has been widely used in crystallization studies (ten Wolde et al., 1995) . Compute the bonds joining neighboring particles (if r ij < 1 . 5 σ , with r ij = r i − r j ). In a crystal bonds sum coherently (Steinhartdt et al. 1983) . B. Seoane (Roma La Sapienza) HS crystallization 18 / 07 / 2013 5 / 14
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