Computer Simulation of Glasses: Jumps and Self-Organized Criticality

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1 Computer Simulation of Glasses: Jumps and Self-Organized Criticality Katharina Vollmayr-Lee Bucknell University November 2, 2007 Thanks: E. A. Baker, A. Zippelius, K. Binder, and J. Horbach

2 V glass crystal Introduction liquid Glass: system falls out of equilibrium T m T Crystal Glass Liquid Structure: discordered

3 V glass crystal Introduction liquid Glass: system falls out of equilibrium T m T Crystal Glass Liquid Structure: discordered Dynamics: frozen in

4 Introduction 12 log η [Poise] T /T 1.0 g [C.A. Angell and W. Sichina, Ann. NY Acad.Sci. 279, 53 (1976) Dynamics: slowing down of many decades

5 Model Binary Lennard-Jones System V αβ (r) = 4 ɛ αβ ( (σαβ r ) 12 ( σαβ r ) 6 ) A B σ AA = 1.0 σ AB = 0.8 σ BB = 0.88 ɛ AA = 1.0 ɛ AB = 1.5 ɛ BB = 0.5 [W. Kob and H.C. Andersen, PRL 73, 1376 (1994)] 800 A and 200 B

6 Numerical Solution: Euler Step Initialize: x(t 0 ), v(t 0 ), a(t 0 ) x(t + t), v(t + t), a(t + t) x(t 0 +2 t), v(t 0 +2 t), a(t 0 +2 t) = Iteration Step: etc. x(t+ t)=x(t)+v(t) t v(t+ t)=v(t)+ a(t) t a(t)=f(t)/m= (du/dx)(t)

7 Molecular Dynamics Simulation Initialize: x (t ), v (t,) a (t ) i 0 i 0 i 0 particles i=1,...,n three dimensions x (t + t), v (t + t), a (t + t) i 0 i 0 i 0 x (t +2 t), v (t +2 t), a (t +2 t) i 0 etc. = Iteration Step: i 0 i 0 x i(t+ t)=x i(t)+v i (t) t+a i(t)( t) /2 v (t+ t)=v (t) +(a (t)+a (t+ t)) t/2 i i a (t)=f (t)/m = U(t)/m i i i i i i i 2

8 Simulations Dynamics V liquid below glass transition: T = T c = glass crystal here T

9 Cage-Picture Mean-Squared Displacement: r 2 (t) = 1 N N i=1 (r i (t) r i (0)) 2 r data of T. Gleim & W. Kob T=0.446 A particles t ballistic motion cage jump diffusion here r 2 t 2 r 2 t

10 Definition: Jump Occurrence r i Single Particle Trajectory σ r R R > 20 σ r t

11 Definition: Jump Occurrence r i σ r Single Particle Trajectory R R > 20 σ r Definition: Jump Type Irreversible Jump r i r i Reversible Jump t t t

12 Outline Jump Statistics Correlated Single Particle Jumps History Dependence Summary

13 Number of Jumping Particles number of jumping particles irrev. A irrev. B rev. A rev. B T = increasing with increasing T = both A & B particles jump = irrev. & reversible jumps at all temperatures T

14 Fraction of Irreversibly Jumping Particles fraction of irrev. jumpers = number of irrev. jump. part. number of jump. part.

15 Fraction of Irreversibly Jumping Particles fraction of irrev. jumpers = number of irrev. jump. part. number of jump. part. fraction of irrev. jumpers A B = fraction of irrev. jumpers increases with increasing T T

16 Fraction of Irreversibly Jumping Particles fraction of irrev. jumpers = number of irrev. jump. part. number of jump. part. fraction of irrev. jumpers A B = fraction of irrev. jumpers increases with increasing T interpretation: door closing T

17 Jump Size r i σ r R avg t

18 Jump Size r i σ r R avg t R avg irrev. A irrev. B rev. A rev. B = increasing with increasing T = (smaller) B-particles jump farther = irreversible jumps farther T

19 Time Scale r i t d t b t

20 Time Scale r i t d t b t t t b irrev. t b rev. t d irrev. t d rev. = t b t d A & B = t b independent of temperature (whole simulation 10 5 ) T Time Scales Extra

21 Summary: Jump Statistics At larger temperature relaxation: not via t b (indep. of T ) via larger jumpsizes via more jumping particles

22 Outline Jump Statistics Correlated Single Particle Jumps Simultaneously Jumping Particles Temporally Extended Cluster History Dependence Summary

23 Simultaneously Jumping Particles Definition: Correlated in Time & Space x t

24 Simultaneously Jumping Particles Definition: Correlated in Time & Space x Cluster: nearest neighbor connections (via g(r)) t

25 Simultaneously Jumping Particles Cluster Size x = number of particles in cluster Cluster: nearest neighbor connections s= t

26 Cluster Size Distribution of Simultaneously Jumping Particles 10 0 T=0.43 P(s) = ln P = a τ ln s = P (s) s τ τ = 1.89 ± 0.03 = critical behavior simultaneously jump.part s

27 Critical Behavior Example: Liquid Gas ρ liquid gas Other Examples: Magnet (Ising Model) Synchronization Percolation T critical point (Phase Transition) At Critical Point: powerlaws α f(x) = x α α α f( λ x) = λ x = λ f(x) rescale x axis scale invariance rescale y axis looks same from any distance lack of specific length scale large fluctuations

28 Cluster Size Distribution of Simultaneously Jumping Particles 10 0 T=0.43 P(s) = ln P = a τ ln s = P (s) s τ τ = 1.89 ± 0.03 = critical behavior = Percolation? simultaneously jump.part s

29 Cluster Size Distribution of Simultaneously Jumping Particles P(s) simultaneously jump.part s T=0.15 T=0.25 T=0.35 T=0.38 T=0.41 T=0.43 = P (s) s τ percolation? NO because = power law for all temperatures = self-organized criticality τ(t )

30 Self Organized Criticality powerlaw not only at critical point but independent of details of external paramters [P. Bak, C. Tang, and K. Wiesenfeld, PRL 59, 381 (1987)] Examples: sandpile avalanches forest fire solar flares financial market earth quakes

31 Outline Jump Statistics Correlated Single Particle Jumps Simultaneously Jumping Particles Temporally Extended Cluster History Dependence Summary

32 Temporally Extended Cluster x Definition: cluster of events (r i, t i ) connected if: r < r cutoff and t < t cutoff s= t

33 Cluster Size Distribution of Temporally Extended Clusters P(s) extended cluster s T=0.15 T=0.25 T=0.35 T=0.38 T=0.41 T=0.43 = P (s) s τ = for all temperatures (self-organized crit.)

34 Shape of Clusters z = number of nearest neighbors within cluster s = number of particles (cluster size) z = average of z over particles 1,... s s=7 s=7 z =3.4 z =1.7

35 Shape of Clusters z = number of nearest neighbors within cluster s = number of particles (cluster size) z = average of z over particles 1,... s 10 = string-like clusters z 5 simultaneous extended cluster sphere string s

36 Shape of Clusters z = number of nearest neighbors within cluster s = number of particles (cluster size) z = average of z over particles 1,... s 10 = string-like clusters z 5 simultaneous extended cluster sphere string s same color = same time

37 Outline Jump Statistics Correlated Single Particle Jumps Simultaneously Jumping Particles Temporally Extended Cluster History Dependence Summary

38 History Dependence simulation run t win= t

39 History Dependence simulation run t win= t win = 5 t Power Law (Simultan. Jump. Part.) P(s) simultaneously jump. part s t win = 4 t win = 3 t win = 2 t win = 1 = aging independent τ(t )

40 Outline Jump Statistics Correlated Single Particle Jumps Simultaneously Jumping Particles Temporally Extended Cluster History Dependence Summary

41 Summary: Jump Statistics reversible and irreversible jumps: reversible jump irreversible jump At larger temperature relaxation: via more jumping particles via larger jumpsizes not via t b (indep. of T ) History Production Runs

42 Summary: Jump Statistics reversible and irreversible jumps: reversible jump irreversible jump At larger temperature relaxation: via more jumping particles history dependent via larger jumpsizes history independent not via t b (indep. of T ) history independent

43 Summary: Correlated Single Particle Jumps simultaneously jump. part. & extended clusters single particle jumps are correlated spatially and temporally Distribution of Cluster Size: P (s) s τ indep. of cluster definition and waiting time for all temp. self-organized criticality (critical behavior gets frozen in) string-like clusters

44 Future/Present SiO 2 (R. A. Bjorkquist & J. A. Roman & J. Horbach) granular media (T. Aspelmeier & A. Zippelius ) Acknowledgments A. Zippelius, K. Binder, E. A. Baker, J. Horbach Support from Institute of Theoretical Physics, University Göttingen, SFB 262 and DFG Grant No. Zi 209/6-1

45 Time Scales one MD step: 0.02 time units, Ar: s 0.02 = 6fs one oscillation: 100 MD steps, 0.6 ps time a jump takes: 200 MD steps, 1.2 ps time resolution (time bin): MD steps, 240 ps time betw. successive jumps t b : MD steps, 9 ns whole simulation run: MD steps, 30 ns Time Scales Cooperative Processes: N t,bcl

46 History of Production Runs T well equilibrated 10 initial configurations for each T etc NVT NVT NVE T init = not aged T init = 0.5 not aged T init = aged t

47 History Dependence T well equilibrated 10 initial configurations for each T etc. NVT NVE NVT T init = not aged T init = 0.5 not aged T init = aged t no. jumping part. / no. part A & B T init = not aged T init = aged T init = 0.5 not aged irreversible & reversible T Number of Jump. Part. = history dependent

48 History Dependence r i σ r R avg t 1 A & B T init = not aged T init = aged T init = 0.5 not aged irreversible jumps Jump Size = history independent R avg 0.5 reversible jumps T

49 History Dependence r i t b t T init = not aged T init = aged T init = 0.5 not aged reversible jumps Time Between Jumps A & B = history independent t b Summary: Jump Statistics irreversible jumps T

50 Exponent τ(t ) t win = 1 t win = 3 t win = τ P (s) T τ simult. extended cl.i extended cl.ii P (s) twin T T

51 Most Cooperative Processes x s bcl = largest cluster size s = 4 bcl s = 7 bcl t s bcl simultaneous extended cluster irreversible & reversible = highly correlated single particle jumps many particles T

52 x Most Cooperative Processes N t,bcl = no. of time bins of largest cluster N t,bcl = 3 t N t,bcl irreversible reversible irrev. & rev T = highly correlated single particle jumps many particles many time bins (maximum = 125) Time Scales Extra

53 History Dependence simulation run t win= t s bcl = largest cluster size s bcl t twin =1 t twin =2 t twin =3 t twin =4 t twin =5 simultaneously jumping particles = aging dependent 1st t-window: highly cooperative 2nd - 5th t-window: same, cooperative T s bcl extended cluster

54 History Dependence simulation run t win= t N t,bcl = no. of t-bins of largest cluster N t,bcl t twin =1 t twin =2 t twin =3 t twin =4 t twin =5 extended cluster = less aging dependent = highly cooperative T

55 History Dependence simulation run t win= t s bcl t twin =1 t twin =2 t twin =3 t twin =4 t twin =5 extended cluster = aging dependent 1st t-window: highly cooperative 2nd - 5th t-window: same, cooperative T s bcl simult. jump.

56 Normalized Jump Size Distribution 0.2 irrev. A & B 0.15 P Delta R/sigma

57 Jump Size Distribution 2 irrev.& rev. A&B 1.5 P Delta R

58 Jump Size Distribution irrev.&rev. A&B P Delta R

59 Distribution of t b 4e-05 corr irrev A&B 3e-05 P 2e-05 1e e+05 Delta t_b

60 Distribution of t b corr irrev.&rev. A&B P 1e-05 1e e+05 Delta t_b

61 History Dependence simulation run t win= t s bcl t twin =1 t twin =2 t twin =3 t twin =4 t twin =5 extended cluster = aging dependent 1st t-window: highly cooperative 2nd - 5th t-window: same, cooperative T s bcl simult. jump.

62 Time Scales one MD step: 0.02 time units, Ar: s 0.02 = 6fs one oscillation: 100 MD steps, 0.6 ps time a jump takes: 200 MD steps, 1.2 ps time resolution (time bin): MD steps, 240 ps time betw. successive jumps t b : MD steps, 9 ns whole simulation run: MD steps, 30 ns Time Scales Cooperative Processes: N t,bcl

Microscopic Picture of Aging in SiO 2 : A Computer Simulation

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