Long-range correlations in glasses and glassy fluids, and their connection to glasses elasticity

Transcription

1 Long-range correlations in glasses and glassy fluids, and their connection to glasses elasticity Grzegorz Szamel Department of Chemistry Colorado State University Ft. Collins, CO 80523, USA Workshop on Critical Phenomena in Random and Complex Systems Thanks to: Elijah Flenner &

2 Outline 1 Long-range density correlations in glasses and glassy fluids Long-range density correlations in crystalline solids Dynamic glass transition Long-range density correlations in glasses Computer simulation results 2 Long-range displacement correlations and (visco) elasticity Stress auto-correlation function, and all that Correlations of particle displacements Correlations of transverse displacements and the shear modulus 3 Summary

3 Spontaneously broken translational symmetry elasticity and long-range density correlations D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, And Correlation Functions P.W. Anderson, Basic Notions of Condensed Matter Physics

4 Long-range density correlations in crystalline solids Broken translat. symmetry long-range correlations In crystalline solids translational symmetry is broken n( r) - density field n 0 - spatially averaged density n( r) = n 0 + G n G ei G r G - reciprocal lattice vectors n G - Bragg-peak amplitudes (order parameters) Rigid translation: an equivalent but different state A rigid translation of a crystal by a constant vector a produces an equivalent but different state of the crystal. This does not cost any energy/does not require any force. Under such translation the density field changes: n( r) n( r a) n G n G ei G a for G 0 Rigid translations zero free energy cost excitations (Goldstone modes) The existence of such zero-free energy excitations is the reflection of a broken translational symmetry.

5 Long-range density correlations in crystalline solids Long-range density correlations Density fluctuation for a wavevector close to G: n( G + q) = i e i( G+ q) r i δn( G + q) = n( G + q) n( G + q) Bogoliubov inequality A 2 B 2 AB 2 A = V 1/2 δn ( G + q) & B = V 1/2ˆ n g( q) where g( q) = m v i e i q ri 1 V ) 2 δn( G + q) 2 1 (k B T) 2 n G (ˆ n 2 G q ˆ q 2 σ ( q) ˆ n 2 lim q 0 1 V i ˆ n - an arbitrary unit vector σ ( k) - microscopic stress tensor Small wavevector divergence long-range correlations in direct space. GS & M. Ernst, PRB 48, 112 (1993); H. Wagner, Z. Phys. 195, 273 (1966)

6 Dynamic glass transition Dynamic glass transition: dynamics & statics Dynamic approach At the dynamic glass transition the relaxation time diverges and the time-dependent density correlation function does not decay: δn( k; t)δn( k) = ns(k)f (k) > 0 lim t δn( k; t) = i e i k r i(t) i e i k r i(t) - density fluctuation S(k) - static structure factor f (k) - non-ergodicity parameter Static (replica) approach (Franz and Parisi, PRL 79, 2486 (1997)) N particles r 1,..., r N tethered to a quenched configuration r 0 1,..., r 0 N : attractive potential = ɛ i,j w( r i r 0 j ). At the dynamic transition non-trivial correlations survive in the ɛ 0 limit.

7 Dynamic glass transition Replicas Averaging over a distribution of quenched configurations = s replicas of the system & s 0 (or m = s + 1 & m 1). quenched conf. Dynamic glass transition non-trivial inter-replica correlations: e i k r iα i k r jβ = ns(k)f (k) lim ɛ 0 i,j α, β - replica indices S(k) - static structure factor f (k) - non-ergod. parameter

8 Long-range density correlations in glasses Symmetry transformation hidden in replica approach Glass can be moved as a rigid body The system can be tethered to a rigidly shifted quenched configuration: attractive potential = ɛ i,j w( r i r 0 j a ). As before: at the dynamic transition nontrivial correlations in the ɛ 0 limit. Physically, nothing changes: we get a glass that is shifted rigidly by a. However: (some) replica off-diagonal correlation functions change. For α > 0 : h α0 ( r 1, r 2 ) h α0 ( r 1 a, r 2 ) All other pair correlations are unchanged (note: this breaks replica symmetry). Rigid translations zero energy cost excitations (Goldstone modes) The existence of such zero-free energy excitations is the reflection of a randomly broken translational symmetry.

9 Long-range density correlations in glasses Long-range density correlations Fourier transform of the joint microscopic density in replicas α and 0: n α0 ( k; q) = i,j e i k r iα i( k+ q) r j0 (analogue of n( G + q) for crystals) Bogoliubov inequality A 2 B 2 AB 2 A = V 1/2 s 1 δn α0( k; q) where δn α0 ( k; q) = n α0 ( k; q) n α0 ( k; q) α>0 B = V 1/2 s 1 α>0 ˆ n g α ( q) where g α ( q) = i m v iα e i q riα ˆ n - an arbitrary unit vector 1 δn 10 ( k; q) 2 δn 10 ( k; q)δn 20 ( k; q) V 2 1 (k B T) 2 (ns(k)f (k)) (ˆ n 2 k) q 2 1 lim q 0 V Σ( q) f (k) - non-ergod. parameter Σ( q) = ˆ q σ 1 ( q) ˆ n 2 (ˆ q σ 1 ( q) ˆ n)(ˆ k σ 2 ( q) ˆ n)

10 Long-range density correlations in glasses Long-range density correlations: crystals vs. glasses Crystals 1 V ) 2 δn( G + q) 2 1 (k B T) 2 n G (ˆ n 2 G q ˆ q 2 σ ( q) ˆ n 2 lim q 0 1 V Small wavevector divergence long-range correlations in direct space. Glasses GS & M. Ernst, PRB 48, 112 (1993); H. Wagner, Z. Phys. 195, 273 (1966) 1 δn 10 ( k; q) 2 δn 10 ( k; q)δn 20 ( k; q) V 2 1 (k B T) 2 (ns(k)f (k)) (ˆ n 2 k) q 2 1 lim q 0 V Σ( q) f (k) - non-ergod. parameter Σ( q) = ˆ q σ 1 ( q) ˆ n 2 (ˆ q σ 1 ( q) ˆ n)(ˆ k σ 2 ( q) ˆ n) Small wavevector divergence long-range correlations in direct space. GS & E. Flenner, PRL 107, (2011)

11 Long-range density correlations in glasses Simplified version of the divergent correlation function 1 δn 10( k; q) 2 δn 10( k; q)δn 20( k, q) N 1 e i k r i1 i( k+ q) r j0 e i k r l1 +i( k+ q) r m0 N i,j i,j l,m e i k r i1 i( k+ q) r j0 e i k r l2 +i( k+ q) r m0 l,m Self (diagonal) part only 1 e i k r i1 i( k+ q) r i0 e i k r l1+i( k+ q) r l0 N i l

12 Long-range density correlations in glasses Replacing replicas with t limit Interpreting replica off-diagonal correlation function dynamically 1 e i k r i1 i( k+ q) r i0 e i k r l1+i( k+ q) r l0 N i l 1 lim e i k r i(t) i( k+ q) r i(0) t N i l e i k r l(t)+i( k+ q) r l(0) Connection to the four-point structure factor 1 e i k r i(t) i( k+ q) r i(0) e i k r l(t)+i( k+ q) r l(0) = N i l 1 e i k ( r i(t) r i(0)) i q r i(0) e i k ( r l(t) r l(0))+i q r l(0) S N 4( q, k; F t) i l This function is a version of a four-point structure factor used to investigate dynamic correlations in glassy fluids!

13 Long-range density correlations in glasses Four-point functions used to investigate dynamic corrs. Four-point structure factor: correlations of slow particles S 4 (q; t) = 1 w(δ r i (t))e i q ri(0) w(δ r j (t))e i q rj(0) N i j δ r i (t) = r i (t) r i (0), w( r j (t)) = θ(a r j (t) ) But one could select slow particles in a different way S4 cos ( q, k; t) = 1 cos( k δ r i (t))e i q ri(0) cos( k δ r j (t))e i q rj(0) N i j or one could choose S4( q, k; F t) = 1 e i k δ r i(t) e i q ri(0) N i j e i k δ r j(t) e i q rj(0) BTW, the self-intermediate scattering function F s (k; t), F s (k; t) = 1 N i ei k δ r i(t) 1 N i cos( k δ r i (t)).

14 Long-range density correlations in glasses Different four-point functions are sensitive to different aspects of dynamic correlations S cos 4 ( q, k; t) S cos 4 ( q, k; t) = 1 N cos( k δ r i (t))e i q ri(0) i j cos( k δ r j (t))e i q rj(0) S cos 4 ( q, k; t) quantifies dynamic heterogeneity in glasses and glassy fluids. S F 4 ( q, k; t) S4( q, k; F t) = 1 e i k δ r i(t) e i q ri(0) N i j e i k δ r j(t) e i q rj(0) S F 4 ( q, k; t) quantifies (visco)elastic fluctuations in glasses and glassy fluids (it is also sensitive to dynamic heterogeneity).

15 Computer simulation results Simulation details 50:50 mixture of harmonic spheres V(r) = { ( ) 2 ɛ 2 1 r σ αβ if r σ αβ 0 otherwise ɛ = 10 4, σ 11 = 1.0, σ 12 = 1.2, σ 22 = 1.4 number density n = N/V = volume fraction = π ( 0.5Nσ Nσ3 22) /(6V) = Very large systems: N = 100, 000 and N = 800, 000 Small wavevectors accessible: q min = for the N = 100, 000 system Temperature range: fluid: 5 T 20; T mct = 5.2, T onset 13 glass: T = 3

16 Computer simulation results Self-intermediate scattering function F s (k; t) F s (q;t) T=3 T=5 T=6 T=7 T=8 T=9 T=10 T=15 T= t T = 5 - the lowest temperature at which we can equilibrate the fluid; τ α (T = 5) =

17 Computer simulation results Long-range correlations in glasses S 4 F (q,k;t) t=30 t=960 t=7680 t=61500 q S4( q, k; F t)= 1 e i k ( r i(t) r i(0)) i q r i(0) N i l q T=3 (glass) e i k ( r l(t) r l(0))+i q r l(0) At finite t, lim q 0 SF 4 ( q, k; t) is finite and can be calculated independently. lim q 0 SF 4 ( q, k; t = 30) - showed as the thin horizontal line. As t, lim q 0 SF 4 ( q, k; t) (in the glass)., k =6.1, q k

18 Computer simulation results Long-range correlations in glasses 10 2 q 2 S 4 F (q,k;t) t=30 t=960 t=7680 t=61500 T=3 (glass) S4( q, k; F t)= 1 e i k ( r i(t) r i(0)) i q r i(0) N i l 10-1 q e i k ( r l(t) r l(0))+i q r l(0), k =6.1, q k

19 Computer simulation results Transient viscoelastic fluctuations in glassy fluids 10 2 t = τ α = 92 t = τ α = 931 T=12 T= S F (q,k;t) 10 2 t = τ α = t = T=5 T= S F (q,k;t) S4( q, k; F t) = 1 e i k ( r i(t) r i(0)) i q r i(0) N i l q q e i k ( r l(t) r l(0))+i q r l(0) k = 6.1 closed: k q open: k q

20 Computer simulation results Time-dependence of transient fluctuations q k q k S 4 F (q,k;t) τ α 2τ α 4τ α 6τ α S 4 F (q,k;t) ξ 4 F (t/τα ) q q Time dependence of S F 4 ( q, k; t) t/τ α close: k q, open: k q. T=15 (above the onset temperature)

21 Computer simulation results Transient viscoelastic correlations vs. dyn. heterogen. S cos (q,k;τ α ), S F (q,k;τ α ) 10 2 T=25 T= T=8 T= S cos (q,k;τ α ), S F (q,k;,τ α ) q q circles: S4 F ; squares: Scos 4, closed symbols: k q, open symbols: k q.

22 Computer simulation results Transient viscoelastic density correlations (S F 4 ( k; τ α )) vs. dynamic heterogeneity (S cos 4 ( k; τ α )) S 4 cos (q,k;τα ), S 4 F (q,k;τα ) 10 2 T= q S 4 cos (q,k;τα ), S 4 F (q,k;τα ) T = q circles: S4 F ; squares: Scos 4 ; k q. Very different small q limits very different dynamic correlation lengths.

23 Computer simulation results Viscoelastic length ξ F 4 (τ α) vs. dynamic heterogeneity length ξ cos 4 (τ α) ξ 4 cos (τα ), ξ 4 F (τα ) τ α circles: ξ4 F ; squares: ξcos 4, closed symbols: k q, open symbols: k q.

24 Computer simulation results Dynamic fluctuations in glassy fluids vs. in glasses S 4 F (q,k;τα ) T=3 T=6 T=8 T= q S 4 cos (q,k;τα ) 10 2 T=3 T=6 T=8 T= q black - glassy fluids; red: glass, at t = τ α (T = 5); k q. Transient viscoelastic correlations become long-range in glasses; dynamic heterogeneity correlations decrease in glasses.

25 Computer simulation results Dynamic correlation lengths in glassy fluids vs. in glasses ξ 4 F (τα ) ξ 4 cos (τα ) T T black - glassy fluids; red: glass, at t = τ α (T = 5); closed: k q, open: k q. Note: in the glass, as t, ξ F 4 (t) whereas ξcos 4 (t) is almost t-independ. Transient viscoelastic correlations become long-range in glasses; dynamic heterogeneity correlations decrease in glasses.

26 Computer simulation results Decrease of dynamic heterogeneity at the dynamic glass transition was observed in experiments ξ 4 cos (τα ) T Weeks et al., Science 287, 627 (2000) Also seen in: Ballesta et al., Nature Physics 4, 550 (2008)

27 Long-range displacement correlations and (visco) elasticity Connection between long-range dynamic correlations and viscoelastic response Glass transition: divergence of viscosity and emergence of elasticity Are correlations of particle dynamics related to the increase of viscosity and the emergence of elasticity? Correlations of particle displacements Correlations in the glass & the signature of elasticity Correlations in the fluid & signatures of viscoelasticity Elijah Flenner & GS, arxiv:

28 Long-range displacement correlations and (visco) elasticity Stress auto-correlation function, and all that Stress tensor auto-correlation function Long-time part well fitted by a stretched exponential. T=20 T=15 T=10 T=9 T=8 T=7 T=6 T= σxy(t)σxy(0) /(kbtv) σxy(t)σxy(0) /(kbtv) t t G β initial value of hσxy (t)σxy (0)i /(VkB T) = G0 stretched exponential = GP exp (t/τσ )β Gp 0.07 At the lowest temps GP and β are T-independ T 20

29 Long-range displacement correlations and (visco) elasticity Stress auto-correlation function, and all that Stress tensor auto-correlation function σ xy (t)σ xy (0) /(k B TV) T=20 T=15 T=10 T=9 T=8 T=7 T=6 T=5 T=3 (glass) t In the glass, σ xy (t)σ xy (0) does not decay. 1 shear modulus µ = lim t Vk B T σ xy(t)σ xy (0)

30 Long-range displacement correlations and (visco) elasticity Stress auto-correlation function, and all that Standard formula for the shear modulus D. R. Squire, A. C. Holt, and W. G. Hoover, Physica 42, 388 (1969): where B αβ = 1 2 [ µ = V 1 B xy (k B TV) 1 (σ xy ) 2 σ xy 2] }{{}}{{} Born term fluctuation term n m n (r α nm) 2 r 2 nm [ (rnm) β 2 d2 V nm (r nm ) drnm 2 + (rnm 2 (rnm) β 2 ) 1 ] dv nm (r nm ), r nm dr nm and r 2 nm = (r x nm) 2 + (r y nm) 2 + (r z nm) 2 with r α nm = r α n r α m Can be calculated for a fluid, a glass or a crystalline solid. In a fluid, it gives 0 (within error bars), in a solid it gives a nonzero result. Gives a zero shear modulus unless there are long range density correlations. Both stress tensor auto-correlation function and the above formula are very difficult (computationally expensive) to calculate in a simulation.

31 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of particle displacements Self part of density correlations S4( q, k; F t) = 1 e i k δ r i(t) e i q ri(0) N i j e i k δ r j(t) e i q rj(0) δ r i (t) = r i (t) r i (0) Correlations of transverse displacements S4 (q; t) = 1 δ r i (t)e i q ri(0) δ r j (t)e i q rj(0) 2N i j δ r i (t) q = 0 δ r i (t) = r i (t) r i (0)

32 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of transverse displacements Finite times, small wavevector limit lim q 0 S 4 (q; t) = k BT m t2 due to the momentum conservation Note that lim q 0 S 4 depends on microscopic dynamics! 1 q 0 2N (q; t) lim 1 2N i δ r i (t)e i q ri(0) j δ r j (t)e i q rj(0) i δ r i (t) j δ r j (t) 0 in a P total = 0 ensemble Finite times, large wavevector limit lim q S 4 (q; t) = 1 3N δ ri (t) 2 as t increases, it becomes i proportional to t in the fluid and it saturates in the glass

33 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of transverse displacements in a glass, in a viscous fluid, and in a moderately viscous fluid δr 2 (t) 10-1 T=20 T=5 T= t T = 20 - a moderately viscous fluid (above T onset ) T = 5 - the most most deeply supercooled fluid we could equilibrate T = 3 - a glass (quenched from T = 5 and well aged); the slight upturn of δ r 2 (t) at the longest times suggests that it is still aging.

34 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of transverse displacements in a glass S 4 (q;t) t=61,500 t=7,680 t=960 t=120 t=30 T=3 (glass) q t δr 2 (t) T=20 T=5 T= q horizontal lines indicate predicted q 0 limits At small wavevectors S 4 (q; t) saturates, q 2 behavior becomes apparent.

35 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of transverse displacements in a very viscous fluid S 4 (q;t) t=61,500 t=7,680 t=960 t=120 t=30 T=5 (fluid) q t δr 2 (t) T=20 T=5 T= q horizontal lines indicate predicted q 0 limits At small wavevectors S 4 (q; t) exhibits transient q 2 behavior.

36 Long-range displacement correlations and (visco) elasticity Correlations of particle displacements Correlations of transverse displacements in a moderately viscous fluid t=1024 t=256 t=64 t=16 t=8 S 4 (q;t) δr 2 (t) T=20 T=5 T= T=20 (fluid) q horizontal lines indicate predicted q 0 limits t

37 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlations of transverse displacements and the stress tensor auto-correlation function Adapting arguments presented in the Supplementary Material to C.L. Klix, F. Ebert, F. Weysser, M. Fuchs, G. Maret, and P. Keim, PRL 109, (2012) one can argue that if particles displacements are bounded (as they are in a glass) then: lim lim q 0 t S 4 2nk B T 1 = lim (q; t)q2 t Vk B T σxy (t)σ xy (0) = µ

38 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlations of transverse displacements in a glass 10-1 T=3 (glass) 2nk B T [q 2 S 4 (q;t)] t=61,500 t=7,680 t=120 t=30 δr 2 (t) T=20 T=5 T= t q Saturation of 2nk B T[S 4 (q; t)q2 ] 1 at long times allows us to get an estimate of the shear modulus.

39 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Shear modulus obtained from correlations of transverse displacements agrees well with that calculated from the stress tensor auto-correlation function and from the standard formula. σ xy (t)σ xy (0) /(k B TV) correlations of particle displacements: µ = ± long-time limit of the stress tensor correlations: µ = ± standard formula: µ = ± t The particle displacements route allows to use simulations two orders of magnitude shorter than the other two routes.

40 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlations of transverse displacements in a very viscous fluid 10-1 T=5 (fluid) 2nk B T [q 2 S 4 (q;t)] t=61,500 t=7,680 t=960 t=120 t=30 δr 2 (t) T=20 T=5 T= t q For times in the mean-square-displacement plateau region 2nk B T[S 4 (q; t)q2 ] 1 is equal to the plateau value of the stress tensor auto-correlation function.

41 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlations of transverse displacements: scaling Rescaling S4 (q; t), in the fluid and in the glass S 4 (q;t)/χ4 (t) t=4 t=8 t=16 t=64 t=256 t=1024 T=20 (fluid) 1/(1+(qξ 4 ) 2 ) Scaling hypothesis S 4 qξ 4 (t) S 4 (q;t)/χ4 (t) /(1+(qξ 4 ) 2 ) χ 4 (t) lim q 0 S 4 (q; t) = k BT m t2 T=3 (glass) qξ 4 (t) t = 32 t = 64 t = 3000 t = 6000 t = t = t = t = (q; t) χ 4 (t) = f (qξ4 (t)) f (x) x 2 for x 1 and f (x) x 2+η for x 1 η 0.23 ± 0.07 in the fluid and η = 0 in the glass

42 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlation length of transverse displacements A combination of the scaling form of S4 (q; t): S 4 (q; t) = χ 4 (t)f (qξ 4 (t)) & f (x) x 2+η for x 1 and the formula for the shear modulus: µ = lim q 0 lim t the glass η = 0 and ξ 4 (t) t µ/(2nm). ξ 4 (t) T=20 T=15 T=12 T=10 T=8 T=7 T=6 T=5 T=3 t [µ/(2ρm)] 1/2 t 1/2 [g(β)η/(2ρm)] 1/ t S 4 2nk B T implies that in (q; t)q2 Glass: ξ 4 (t) t µ/(2nm) Supercooled fluid: on intermediate time scales, ξ 4 (t) t µ/(2nm), where µ is the shear modulus of the glass at the longest times ξ 4 (t) t1/2.

43 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Growth of correlation length in the fluid Empirical observation: for the supercooled fluid, the transition between ξ4 (t) t and ξ 4 (t) t1/2 occurs when 1 Vk σ BT xy(t)σ xy (0) is approximately 0.22 G P (where G P is the amplitude of the stretched exponential part) σ xy (t)σ xy (0) /(k B TV) t

44 Long-range displacement correlations and (visco) elasticity Correlations of transverse displacements and the shear modulus Correlation length of transverse displacements (2) A combination of the last empirical observation with the dominance of the contribution of the stress tensor auto-correlation function to the viscosity suggests that the amplitude of t 1/2 dependence is ηg(β)/(2nm) where η is the fluid s viscosity and g(β) is a known function of the stretching exponent β Glass: ξ4 (t) t µ/(2nm) ξ 4 (t) T=20 T=15 T=12 T=10 T=8 T=7 T=6 T=5 T=3 t [µ/(2ρm)] 1/2 t 1/2 [g(β)η/(2ρm)] 1/ t Supercooled fluid: on intermediate time scales, ξ 4 (t) t µ/(2nm), where µ is the shear modulus of the glass. Supercooled fluid: asymptotically ξ 4 (t) t1/2 ηg(β)/(2nm).

45 Summary Summary Dynamic glass transition implies the existence of long-range density correlations. Long-range density correlations can be seen in computer simulations of glasses. Pronounced remnants of long-range density correlations can be seen in glassy fluids. Shear modulus can be obtained from the small wavevector correlations of particle displacements. Solid and fluid: different growth of the correlation length of particle displacements with time. Correlations of particle displacements directly reflect (transient) elasticity. Note: correlations of particle displacements depend on the microscopic dynamics (Newtonian vs. Brownian).