Tethered method to approach the equilibrium fluid-solid coexistence - - PowerPoint PPT Presentation

▶
tethered method to approach the equilibrium fluid solid
SMART_READER_LITE
LIVE PREVIEW

Tethered method to approach the equilibrium fluid-solid coexistence - - PowerPoint PPT Presentation

Nov 30, 2022 11 likes •464 views

Tethered method to approach the equilibrium fluid-solid coexistence of hard spheres Beatriz Seoane Bartolom in collaboration with L. A. Fernndez, V. Martn-Mayor and P. Verrocchio Dipartimento di Fisica La Sapienza Universit di Roma

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

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

slide-4
SLIDE 4

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

slide-5
SLIDE 5

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

slide-6
SLIDE 6

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

slide-7
SLIDE 7

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

slide-8
SLIDE 8

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

slide-9
SLIDE 9

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

slide-10
SLIDE 10

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 ∼ L2: appear with probability e−βγL2

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 3 / 14

slide-11
SLIDE 11

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 ∼ L2: appear with probability e−βγL2 Frequency of jumps decreases exponentially with N2/3

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 3 / 14

slide-12
SLIDE 12

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 ∼ L2: appear with probability e−βγL2 Frequency of jumps decreases exponentially with N2/3 SOLUTION: AVOID THESE METASTABILITIES

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 3 / 14

slide-13
SLIDE 13

HS Crystallization Computational Problem

Hard Spheres

Seek the simplest system that suffers crystallization: HS.

0.2 0.3 0.4 0.5 0.6 0.7 η 0.0 5.0 10.0 15.0 20.0 p (kBT/σ3) fluid fcc HS ηm ηf pco

T. 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

slide-14
SLIDE 14

HS Crystallization Computational Problem

Hard Spheres

Seek the simplest system that suffers crystallization: HS. The fluid-FCC was found in simulations in 1957, but no equilibrium work is able to thermalize more than 500

  • particles. (Errington 2004)

0.2 0.3 0.4 0.5 0.6 0.7 η 0.0 5.0 10.0 15.0 20.0 p (kBT/σ3) fluid fcc HS ηm ηf pco

T. 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

slide-15
SLIDE 15

HS Crystallization Computational Problem

Hard Spheres

Seek the simplest system that suffers crystallization: HS. The fluid-FCC was found in simulations in 1957, but no equilibrium work is able to thermalize more than 500

  • particles. (Errington 2004)

All previous methods, are unable to synthesize the FCC from a random configuration.

0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 50 100 150 200 250 300 350 400 450 500 v t hot start fcc start

.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 4 / 14

slide-16
SLIDE 16

HS Crystallization Computational Problem

Hard Spheres

Seek the simplest system that suffers crystallization: HS. The fluid-FCC was found in simulations in 1957, but no equilibrium work is able to thermalize more than 500

  • particles. (Errington 2004)

All previous methods, are unable to synthesize the FCC from a random configuration.

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 100 200 300 400 500 600 700 800 900 1000 v t hot start fcc start

Need observables that label univocally one of the branches: bond order parameters .

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 4 / 14

slide-17
SLIDE 17

HS Crystallization Order Parameters

Crystal order parameter Q6 (rotational symmetry)

Q6 has been widely used in crystallization studies (ten Wolde et al., 1995). Ql ≡

  • 4π

2l + 1

l

  • m=−l

|Qlm|2 1/2 Qlm ≡ N

i=1 qlm(i)

N

i=1 Nb(i)

, qlm(i) ≡

Nb(i)

  • j=1

Ylm(ˆ r ij)

Perfect lattices values

Q6 fluid FCC BCC 0.574 0.510

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 5 / 14

slide-18
SLIDE 18

HS Crystallization Order Parameters

Crystal order parameter Q6 (rotational symmetry)

Q6 has been widely used in crystallization studies (ten Wolde et al., 1995). Compute the bonds joining neighboring particles (if rij < 1.5σ, with r ij = r i − r j). Ql ≡

  • 4π

2l + 1

l

  • m=−l

|Qlm|2 1/2 Qlm ≡ N

i=1 qlm(i)

N

i=1 Nb(i)

, qlm(i) ≡

Nb(i)

  • j=1

Ylm(ˆ r ij)

Perfect lattices values

Q6 fluid FCC BCC 0.574 0.510

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 5 / 14

slide-19
SLIDE 19

HS Crystallization Order Parameters

Crystal order parameter Q6 (rotational symmetry)

Q6 has been widely used in crystallization studies (ten Wolde et al., 1995). Compute the bonds joining neighboring particles (if rij < 1.5σ, with r ij = r i − r j). In a crystal bonds sum coherently (Steinhartdt et al. 1983). Ql ≡

  • 4π

2l + 1

l

  • m=−l

|Qlm|2 1/2 Qlm ≡ N

i=1 qlm(i)

N

i=1 Nb(i)

, qlm(i) ≡

Nb(i)

  • j=1

Ylm(ˆ r ij)

Perfect lattices values

Q6 fluid FCC BCC 0.574 0.510

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 5 / 14

slide-20
SLIDE 20

HS Crystallization Order Parameters

Crystal order parameter Q6 (rotational symmetry)

Q6 has been widely used in crystallization studies (ten Wolde et al., 1995). Compute the bonds joining neighboring particles (if rij < 1.5σ, 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

slide-21
SLIDE 21

HS Crystallization Order Parameters

Breaking rotational symmetry. . .

Second order parameter: C

Consider a second order parameter with cubic symmetry. (S.

Angioletti-Uberti et al., 2010). Replace spherical harmonics by

cα(ˆ r) = x4y4(1 − z4) + x4z4(1 − y4) + y4z4(1 − x4). C = 2288 79 N

i=1

Nb(i)

j=1 cα(ˆ

r ij) N

i=1 Nb(i)

− 64 79 Perfect lattice values: FCC: 1, BCC: -0.26, Fluid: 0.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 6 / 14

slide-22
SLIDE 22

HS Crystallization Order Parameters

Breaking rotational symmetry. . .

Second order parameter: C

Consider a second order parameter with cubic symmetry. (S.

Angioletti-Uberti et al., 2010). Replace spherical harmonics by

cα(ˆ r) = x4y4(1 − z4) + x4z4(1 − y4) + y4z4(1 − x4). C = 2288 79 N

i=1

Nb(i)

j=1 cα(ˆ

r ij) N

i=1 Nb(i)

− 64 79 Perfect lattice values: FCC: 1, BCC: -0.26, Fluid: 0. We need to fix simultaneously C and Q6 to avoid metastabilities.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 6 / 14

slide-23
SLIDE 23

HS Crystallization Tethered method for HS

Tethered ensemble ˆ Q6 ˆ CNpT

Order Parameters: Q6(R), C(R)

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 7 / 14

slide-24
SLIDE 24

HS Crystallization Tethered method for HS

Tethered ensemble ˆ Q6 ˆ CNpT

Order Parameters: Q6(R), C(R) In the NpT ensemble, the probability of Q6(R) = Q6 and C(R) = C for

p1(Q6, C, p) ∝ ∞ dV e−βpV

  • dR e−βU(R) δ (Q6 − o(R)) δ (C − o(R))
  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 7 / 14

slide-25
SLIDE 25

HS Crystallization Tethered method for HS

Tethered ensemble ˆ Q6 ˆ CNpT

Order Parameters: Q6(R), C(R) In the NpT ensemble, the probability of Q6(R) = Q6 and C(R) = C for

p1(Q6, C, p) ∝ ∞ dV e−βpV

  • dR e−βU(R) δ (Q6 − o(R)) δ (C − o(R))

MC simulation: softer constraint Q6(R) ≈ Q6 and C(R) ≈ C.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 7 / 14

slide-26
SLIDE 26

HS Crystallization Tethered method for HS

Tethered ensemble ˆ Q6 ˆ CNpT

Order Parameters: Q6(R), C(R) In the NpT ensemble, the probability of Q6(R) = Q6 and C(R) = C for

p1(Q6, C, p) ∝ ∞ dV e−βpV

  • dR e−βU(R) δ (Q6 − o(R)) δ (C − o(R))

MC simulation: softer constraint Q6(R) ≈ Q6 and C(R) ≈ C. New variables ˆ Q6 = Q6 + αN

i=1 ηi/αN (same for C), where ηi are

Gaussians variables (µ = 0, σ2 = 1).

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 7 / 14

slide-27
SLIDE 27

HS Crystallization Tethered method for HS

Tethered ensemble ˆ Q6 ˆ CNpT

Order Parameters: Q6(R), C(R) In the NpT ensemble, the probability of Q6(R) = Q6 and C(R) = C for

p1(Q6, C, p) ∝ ∞ dV e−βpV

  • dR e−βU(R) δ (Q6 − o(R)) δ (C − o(R))

MC simulation: softer constraint Q6(R) ≈ Q6 and C(R) ≈ C. New variables ˆ Q6 = Q6 + αN

i=1 ηi/αN (same for C), where ηi are

Gaussians variables (µ = 0, σ2 = 1). The probability of finding ( ˆ Q6, ˆ C) is the convolution of both

  • probabilities. The ηis can be integrated out leading to

p( ˆ Q6, ˆ C, p) ∝ e−βpV e−βU(R)e− αN

2 [ ˆ

Q6−Q(R)]

2

e− αN

2 [ˆ

C−C(R)]

2

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 7 / 14

slide-28
SLIDE 28

HS Crystallization Tethered method for HS

Tethered ensemble (II)

The tethered mean values of an observable A(R) at fixed ˆ Q6, ˆ C A ˆ

Q6,ˆ C ≡

  • dV
  • dR A(R)ωN(R, p; ˆ

Q6, ˆ C)

  • dV
  • dR ωN(R, p; ˆ

Q6, ˆ C) , are obtained with the Metropolis algorithm using this modified weight ωN(R, p; ˆ Q6, ˆ C) =

  • αN

2π e−βpV e−βU(R)e− αN

2

  • ( ˆ

Q6−Q6(R))

2+(ˆ

C−C(R))

2

.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 8 / 14

slide-29
SLIDE 29

HS Crystallization Tethered method for HS

Tethered ensemble (II)

The tethered mean values of an observable A(R) at fixed ˆ Q6, ˆ C A ˆ

Q6,ˆ C ≡

  • dV
  • dR A(R)ωN(R, p; ˆ

Q6, ˆ C)

  • dV
  • dR ωN(R, p; ˆ

Q6, ˆ C) , are obtained with the Metropolis algorithm using this modified weight ωN(R, p; ˆ Q6, ˆ C) =

  • αN

2π e−βpV e−βU(R)e− αN

2

  • ( ˆ

Q6−Q6(R))

2+(ˆ

C−C(R))

2

. We can define a Helmholtz effective potential

e−NΩN( ˆ

Q6, ˆ C,p) =

βp N!Λ3N

  • αN

2π

  • dV
  • dR e−βpV e−βU(R) e

− αN

2

  • ( ˆ

Q6−Q(R))

2+( ˆ

C−C(R))

2

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 8 / 14

slide-30
SLIDE 30

HS Crystallization Tethered method for HS

Tethered ensemble (II)

The tethered mean values of an observable A(R) at fixed ˆ Q6, ˆ C A ˆ

Q6,ˆ C ≡

  • dV
  • dR A(R)ωN(R, p; ˆ

Q6, ˆ C)

  • dV
  • dR ωN(R, p; ˆ

Q6, ˆ C) , are obtained with the Metropolis algorithm using this modified weight ωN(R, p; ˆ Q6, ˆ C) =

  • αN

2π e−βpV e−βU(R)e− αN

2

  • ( ˆ

Q6−Q6(R))

2+(ˆ

C−C(R))

2

. We can define a Helmholtz effective potential

e−NΩN( ˆ

Q6, ˆ C,p) =

βp N!Λ3N

  • αN

2π

  • dV
  • dR e−βpV e−βU(R) e

− αN

2

  • ( ˆ

Q6−Q(R))

2+( ˆ

C−C(R))

2

Saddle point approx. [where ( ˆ Q∗

6(p), ˆ

C ∗(p)) fulfills ∇ΩN = 0] tells us gN(p, T) = ΩN( ˆ Q∗

6, ˆ

C ∗, p) + O(1/N)

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 8 / 14

slide-31
SLIDE 31

HS Crystallization Tethered method for HS

Tethered ensemble (III)

Gradients

∇ΩN( ˆ Q6, ˆ C, p) =

  • ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ Q6

, ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ C

  • =
  • α
  • ˆ

Q6 − Q6

  • ˆ

Q6,ˆ C,p,

  • α
  • ˆ

C − C

  • ˆ

Q6,ˆ C,p

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 9 / 14

slide-32
SLIDE 32

HS Crystallization Tethered method for HS

Tethered ensemble (III)

Gradients

∇ΩN( ˆ Q6, ˆ C, p) =

  • ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ Q6

, ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ C

  • =
  • α
  • ˆ

Q6 − Q6

  • ˆ

Q6,ˆ C,p,

  • α
  • ˆ

C − C

  • ˆ

Q6,ˆ C,p

  • ΩN is recovered using a line integral ∆ΩN =
  • C ∇ΩN · dl

gfluid

N

(pco) = gFCC

N

(pco) ⇔ ∆ΩN(pco) = 0 ∆ΩN(p) = ΩN ˆ QFCC

6

(p), ˆ C FCC(p), p

  • − ΩN

ˆ Qfluido

6

(p), ˆ C fluido(p), p

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 9 / 14

slide-33
SLIDE 33

HS Crystallization Tethered method for HS

Tethered ensemble (III)

Gradients

∇ΩN( ˆ Q6, ˆ C, p) =

  • ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ Q6

, ∂ΩN( ˆ

Q6,ˆ C) ∂ ˆ C

  • =
  • α
  • ˆ

Q6 − Q6

  • ˆ

Q6,ˆ C,p,

  • α
  • ˆ

C − C

  • ˆ

Q6,ˆ C,p

  • ΩN is recovered using a line integral ∆ΩN =
  • C ∇ΩN · dl

gfluid

N

(pco) = gFCC

N

(pco) ⇔ ∆ΩN(pco) = 0 ∆ΩN(p) = ΩN ˆ QFCC

6

(p), ˆ C FCC(p), p

  • − ΩN

ˆ Qfluido

6

(p), ˆ C fluido(p), p

  • Reweighting method

O ˆ

Q6,ˆ C,p+δp =

O e−βδpV ˆ

Q6,ˆ C,p

e−βδpV ˆ

Q6,ˆ C,p

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 9 / 14

slide-34
SLIDE 34

HS Crystallization Tethered method for HS

Tethered method vs. umbrella sampling

This formulation of the tethered weight ωN(R, p; ˆ Q6, ˆ C) is the umbrella sampling’s one ⇒ simulations are equal (!)

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 10 / 14

slide-35
SLIDE 35

HS Crystallization Tethered method for HS

Tethered method vs. umbrella sampling

This formulation of the tethered weight ωN(R, p; ˆ Q6, ˆ C) is the umbrella sampling’s one ⇒ simulations are equal (!)

The real difference appear in the analysis

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 10 / 14

slide-36
SLIDE 36

HS Crystallization Tethered method for HS

Tethered method vs. umbrella sampling

This formulation of the tethered weight ωN(R, p; ˆ Q6, ˆ C) is the umbrella sampling’s one ⇒ simulations are equal (!)

The real difference appear in the analysis

Umbrella sampling reconstructs P(Q, C) (and the thermodynamic potential) using multi-histogram reweighting methods. Problem: measuring P ˆ

Q6,ˆ C(Q, C) is very imprecise (bidimensional histogram).

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 10 / 14

slide-37
SLIDE 37

HS Crystallization Tethered method for HS

Tethered method vs. umbrella sampling

This formulation of the tethered weight ωN(R, p; ˆ Q6, ˆ C) is the umbrella sampling’s one ⇒ simulations are equal (!)

The real difference appear in the analysis

Umbrella sampling reconstructs P(Q, C) (and the thermodynamic potential) using multi-histogram reweighting methods. Problem: measuring P ˆ

Q6,ˆ C(Q, C) is very imprecise (bidimensional histogram).

Tethered obtains ΩN by means of a line integration a MC time average

  • f the conjugated field ∇ΩN( ˆ

Q6, ˆ C, p). The number of constrains does not modify the efficiency.

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 10 / 14

slide-38
SLIDE 38

HS Crystallization Tethered method for HS

Results

  • 0.2

0.2 0.6 0.2 0.4 C Q6 ˆ ˆ fluid fcc bcc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 C Q6 ˆ ˆ fluid fcc 10-1∇ΩN

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 11 / 14

slide-39
SLIDE 39

HS Crystallization Tethered method for HS

Results

  • 0.2

0.2 0.6 0.2 0.4 C Q6 ˆ ˆ fluid fcc bcc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 C Q6 ˆ ˆ fluid fcc 10-1∇ΩN

∇SΩN, projection ∇ΩN on the line

  • 0.30

0.00 0.30 0.60 ∇SΩN 0.98 1 1.02 1.04 1.06 0.2 0.4 0.6 0.8 1 v S N=108 N=256 N=500 N=864 N=1372 N=2048 N=2916 N=4000

  • 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 11 / 14

slide-40
SLIDE 40

HS Crystallization Tethered method for HS

Results

  • 0.2

0.2 0.6 0.2 0.4 C Q6 ˆ ˆ fluid fcc bcc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 C Q6 ˆ ˆ fluid fcc 10-1∇ΩN

Integrating

  • 0.04
  • 0.02

0.00 0.02 0.04 10.6 10.8 11 11.2 11.4 11.6 ∆Ω p(kBT/σ3) N=108 N=256 N=500 N=864 N=1372 N=2048 N=2916 N→∞

  • 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 11 / 14

slide-41
SLIDE 41

HS Crystallization Coexistence Pressure

Coexistence Pressure

11.2 11.3 11.4 11.5 11.6 0.001 0.002 0.003 0.004 pco 1/N data Errington Wilding Vega cuadratic fit

Our estimation in (kBT/σ3) units

pN=∞

co

= 11.5727(10) Previous eq. estimation (Wilding & Bruce 2000) pN=∞

co

= 11.50(9) best nonequilibrium pN=∞

co

= 11.576(6)

(N = 1.6 × 105, Zykova-Timan et al., 2010)

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 12 / 14

slide-42
SLIDE 42

HS Crystallization Geometric transitions and interfacial free energy

Geometric transitions and interfacial free energy

slab

S=0.4

bubble

S=0.8

cylinder

S=0.6785

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 13 / 14

slide-43
SLIDE 43

HS Crystallization Geometric transitions and interfacial free energy

Geometric transitions and interfacial free energy

  • 0.30

0.00 0.30 0.60 ∇SΩN 0.98 1 1.02 1.04 1.06 0.2 0.4 0.6 0.8 1 v S N=108 N=256 N=500 N=864 N=1372 N=2048 N=2916 N=4000

γ(N)

{100} = kBT N (Ωs∗ − ΩFCC) /(2 Nv2/3 S∗ )

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 13 / 14

slide-44
SLIDE 44

HS Crystallization Geometric transitions and interfacial free energy

Geometric transitions and interfacial free energy

Our estimation γ{100} = 0.636(11) Mu et al. 2005 γ{100} = 0.64(2) Cacciuto et al. 2003 γ{100} = 0.619(3) Davidchack et al. 2010 γ{100} = 0.5820(19) Härtel et al. 2012 γ{100} = 0.639(11)

0.56 0.58 0.6 0.62 0.64 11.4 11.45 11.5 11.55 11.6 11.65 γ100 p N=2048 N=2916 N=4000

  • 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 13 / 14

slide-45
SLIDE 45

HS Crystallization Conclusions

Conclusions

We have proposed a new method to control crystallization. Tethered weight is formally equal than umbrella sampling but allow us to recover the potential by means of a thermodynamic potential. Results presented improve previous estimations by several orders of precision with rather short simulations. We can track the geometric transitions, giving us a tool to control the interfaces and thus obtain the interfacial free energy. THANK YOU VERY MUCH

  • B. Seoane (Roma La Sapienza)

HS crystallization 18/07/2013 14 / 14