Theory of Disordered Condensed-Matter Systems

Transcription

1 Introduction: Types of disorder in condensed-matter systems Theory of Disordered Condensed-Matter Systems Walter Schirmacher University of Mainz, Germany Summer School on Soft Matters and Biophysics, SJTU Shanghai July 3-6, 2017 July 5, 2017 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 1 / ,

2 Introduction: Types of disorder in condensed-matter systems Mainz in Germany Mainz Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 2 / ,

3 Introduction: Types of disorder in condensed-matter systems Mainz in Germany Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 3 / ,

4 Introduction: Types of disorder in condensed-matter systems City of Mainz Mainz Cathedral and the Rhein river Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 4 / ,

5 Introduction: Types of disorder in condensed-matter systems University Mainz School of Physics Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 5 / ,

6 Introduction: Types of disorder in condensed-matter systems Outline 1 Disorder and the Glass transition Introduction: Types of disorder in condensed-matter systems Crystalline order and topological disorder Impurity disorder in Crystals Annealed and quenched disorder General phenomenology Overview over the present glass-transition theories Linearized hydrodynamics and mode-coupling theory Heterogeneous viscoelasticity: A model for dynamic heterogeneity Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 6 / ,

7 Diffusion in quenched-disordered systems Electronic hopping transport in disordered semiconductors Outline 1 Disorder and the Glass transition 2 Diffusion in quenched-disordered systems Electronic hopping transport in disordered semiconductors Phonon-assisted tunneling Mott optimization Master equation and Percolation construction Random walk on a lattice and regular diffusion diffusion in a disordered system: spatially fluctuating diffusivity Perturbation theory Analogy with a mass-spring vibrational problem disordered diffusivity and AC conductivity Coherent-potential approximation (CPA) Comparison of CPA calculations with measured AC conductivity data Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 2 / ,

8 Vibrational anomalies in glasses Outline 1 Disorder and the Glass transition 2 Diffusion in quenched-disordered systems 3 Vibrational anomalies in glasses Elasticity theory and Debye s wave spectra Observation of Vibrational anomalies in glasses Representation of vibrational spectra in terms of complex moduli Experimentally and simulationally observed anomalies Experimental and simulational evidence for heterogeheous elasticity Heterogeneous elasticity theory Self-consistent Born approximation (SCBA) Coherent-potential approximation (CPA) Raman Scattering in Glasses Boson peak, anharmonicity and marginal stability Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU6, Shanghai 2017 July 2 / ,

9 Vibrational anomalies in glasses Outline 1 Disorder and the Glass transition 2 Diffusion in quenched-disordered systems 3 Vibrational anomalies in glasses 4 Anderson localization Electrons in disordered materials Scattering and diffusion of electrons Interference effects Scaling theory of Anderson localization Classical waves with disorder Sound waves with disorder Evaluation of the stochastic field theory: Nonlinear σ model Saddle Point: Self-consistent Born approximation (SCBA) Beyond the Saddle Point: wave diffusivity and localization Experimental evidence for localized sound waves in 2d Localization of light Transverse localization of light Two conflicting theories and experimentum crucis Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU6, Shanghai 2017 July 2 / ,

10 Background literature: N. W. Ashcroft and N. D. Mermin Solid State Physics Holt, Rinehard and Winston, New York, 1976 E. N. Economou Green's Functions in Quantum Physics Springer, Heidelberg, 1990 S. R. Elliott The Physics of Amorphous Materials Longman, New York, 1984 K. Binder and W. Kob, Glassy Materials and Disordered Solids World Scientific, New Jersey, 20 W. Schirmacher, Theory of Liquids and Other Disordered Media Springer, Heidelberg, 2015

11 Original and Review papers: W. Schirmacher, G. Diezemann, and C. Ganter, Harmonic Vibrational Excitations in Disordered Solids and the Boson Peak2 Phys. Rev. Lett. 81, 136 (1998) W. Schirmacher, Thermal Conductivity of Glassy Materials and the Boson Peak Europhys. Lett. 73, 892 (26) C. Ganter and W. Schirmacher, Rayleigh Scattering, Long-Time Tails and the Harmonic Spectrum of topologically disordered systems Phys. Rev. B 82, (2010) W. Schirmacher, The Boson Peak Phys. Stat. Sol. (b) 250, 937 (2013) S. Köhler, G. Ruocco and W. Schirmacher, Coherent-Potential Approximation for Diffusion and Wave Propagation in topologically disordered systems Phys. Rev. B 88, (2013) W. Schirmacher, T. Scopigno and G. Ruocco, Theory of Vibrational Anomalies in Glasses J. Noncryst. Sol. 407, 133 (2014) W. Schirmacher, B. Abaie, A. Mafi, G. Ruocco, M. Leonetti, What is the Right Theory for Anderson Localization of Light? arxiv: (2017) Please download all these papers (and more) from

12 Lecture Notes in Physics 887 Walter Schirmacher Theory of Liquids and Other Disordered Media A Short Introduction

13 Introduction: Types of disorder in condensed-matter systems Outline 1 Disorder and the Glass transition Introduction: Types of disorder in condensed-matter systems Crystalline order and topological disorder Impurity disorder in Crystals Annealed and quenched disorder General phenomenology Overview over the present glass-transition theories Linearized hydrodynamics and mode-coupling theory Heterogeneous viscoelasticity: A model for dynamic heterogeneity Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 6 / ,

14 Introduction: Types of disorder in condensed-matter systems Crystalline order and topological disorder Crystalline order Topological disorder Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 7 / ,

15 Introduction: Types of disorder in condensed-matter systems Impurity disorder in Crystals Substitutional disorder Vacancies and interstitials Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 8 / ,

16 Introduction: Types of disorder in condensed-matter systems Annealed and quenched disorder Annealed disorder: The atoms or molecules are mobile and are in thermal equilibrium Liquid Quenched disorder: The atoms or molecules are frozen-in and are out of thermal equilibrium Glass Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 9 / ,

17 If a liquid is cooled or compressed and crystallization is avoided the fluidity goes to zero and an amorphous solid (glass) develops, which has a finite shear rigidity. The structure remains essentially the same Walter Schirmacher (University of Mainz, Germany[.5cm] Summer School on Soft Matters and Biophysics, SJTU Shanghai July 3-6, Theory of Disordered Condensed-M. Systems July 5, / 41

18 Maxwell s viscoelastic theory J. C. Maxwell, Philos. Trans. Roy. Soc Stress-Strain relation for a solid and a liquid: solid: σ xz = Gǫ xz liquid: σ xz = ηv xz strain ǫij = 1 2 ( iu j + j u i ) strain rate vij = 1 2 ( iv j + j v i ) = d dt ǫ ij Steady state: Maxwell s relaxation time η d dt ǫ xy +Gǫ xy = 0 ǫ xy (t) e t/τ τ = η/g strain strain rate For times smaller than τ the material acts like a solid For times larger than τ the material acts like a liquid Walter Schirmacher If τ is(university muchof larger Mainz, Germany[.5cm] than Theory of adisordered Glassblower s Summer Condensed-M. School Soft characteristic Systems Matters and Biophysics, time July SJTU 5, (or 2017 Shanghai larger July / ,

19 Maxwell s viscoelastic theory Steady state: η d dt ǫ xy +Gǫ xy = 0 ǫ xy (t) e t/τ Maxwell s relaxation time τ = η/g For times smaller than τ the material acts like a solid For times larger than τ the material acts like a liquid If τ is much larger than a Glassblower s characteristic time (or larger than the duration of a PhD) then the liquid has transformed into a glass! Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 12 July / ,

20 Viscosity/relaxation time as a function of inverse temperature Viscosity Relaxation time logη/pa s "strong" T g 1 logτ/s 10 2 "strong" T g 1 "fragile" "fragile" / T /T In this representation a straight line means an Arrhenius law η τ e E A/k B T A non-arrhenius temperature curve is often parametrized according to the Vogel-Fulcher-Tammann equation η τ e B/[T T 0] Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 13 July / ,

21 Angell plot log [viscosity (poise)] T /T g log [relaxation time (s)] Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 14 July / ,

22 Entropy, specific heat and heating/cooling rate S C T g liquid T g liquid glass different cooling/heating rates T glass different heating rates different cooling rates T The heating or cooling rate sets the time scale of the freezing, which controls Whether or not crystallization is avoided Whether partial equilibration is possible during the freezing Therefore the transition temperature T g depends on the freezing rate Because the relaxation during freezing becomes history dependent a cooling-heating hysteresis is observed in calorimetric experiments Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 15 July / ,

23 Kauzmann s Paradox S glass T K Tg liquid crystal T The slower the relaxation the lower is T g, leaving a decreasing entropy S(T) with decreasing T. For slow enough cooling one might achieve a situation at which the entropy S glass of the glass becomes equal to that of the crystal S cryst. Clearly S glass cannot be smaller than S cryst. Therefore an ideal glass transition was postulated by Kauzmann, whith transition temperature T K < T g. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 16 July / ,

24 Kauzmann s Paradox S T K Tg liquid glass crystal T Most important counter argument: Kauzmann s argument depends on a linear extrapolation of the temperature dependence of the entropy. There is no justification for this! Nevertheless the search for a theory of a Kauzmann-type transition is one of the present roadmaps for a theory of the glass transition. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 17 July / ,

25 Overview over the present glass-transition theories Kinetic facilitation theoriy [Chandler, Garrahan] Spin-glass like theories [Kirkpatrick, Thirumalai PRl 1978, Phys. Rev. A 1989, Franz, Parisi, PRL 1997] Dynamic Heterogeneity [Ediger, Ann. Rev. Phys. Chem. 20] In simulations it appears that the time scale of the motions fluctuate strongly in space. For this regime a Coherent-Potential description has been put forward recently [WS, Ruocco,Mazzone] next section! Mode-Coupling theory [Götze 1983, Leutheusser 1983] Microscopic theory of structural arrest High predictive power in the liquid regime Predicts a sharp transition towards a non-ergodic state for a critical temperature T c > T g. Breaks down below Tc, because activated processes, which re-establish ergodicity and which cause the dynamic heterogeneities, are not included. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 18 July / ,

26 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 19 July / ,

27 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 20 July / ,

28 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 21 July / ,

29 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 22 July / ,

30 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 23 July / ,

31 Kinetic facilitation Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 24 July / ,

32 Kinetic facilitation A hierarchy of possible hopping transition is established Regions with lower densities allow more transitions But most importantly all motions depend on previous motions By this the history dependence of the dynamics is included The glass transition is just a freezing out of the kinetics Drawback: The model is just verified by simulations, there is no real theory Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 25 July / ,

33 Spin-glass like theories A spin glass model is defined by a Spin Hamiltonian H = ij J ij S i Sj (vector) or H = ij J ijs i S j (Ising), S = ±1 The exchange couplings are quenched-disordered random quantities In the mean-field version all Ising spins are coupled to each other The mean-field theory predicts a thermal glass transition Generalizations (Pott s models: more than 2 spins) lead to a succession of two transitions: A Kauzmann-like transition and a Mode-coupling-like transition Drawback: The theory (and all its generalizations) starts with quenched disorder. But the appearance of the quenched disorder should be the object of the theory! Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 26 July / ,

34 Linearized hydrodynamics and mode-coupling theory If we denote ρ(r, t) the number density, j the corresponding current density, g(r,t) the momentum density, τ ij the components of the tensor of the momentum current density, we get from particle number conservation, momentum conservation, the continuity equations t ρ(r,t)+ j(r,t) = 0 t g j(r,t)+ i i τ ij (r,t) = 0 In addition one has the constitutive relations between ρ,j,g, and τ: g(r,t) = mj(r,t) ] ρ 0 τ im = p(r,t)δ im η S [ i j m + m j i 2 3 jδ im η B jδ im Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 27 July / ,

35 Linearized hydrodynamics and mode-coupling theory the continuity equations t ρ(r,t)+ j(r,t) = 0 t g j(r,t)+ i i τ ij (r,t) = 0 In addition one has the constitutive relations between ρ,j,g, and τ: g(r,t) = mj(r,t) ρ 0 τ im = p(r,t)δ }{{ im η } S [ i j m + m j i 2 ] 3 jδ im η B jδ im pressure term }{{} viscous damping p(r,t) = mc 2 ρ(r,t) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 28 July / ,

36 Separation into longitudinal and transverse currents g = g l +g t = m(j l +j t ) g l = 0 = j l g t = 0 = j t By this procedure the hydrodynamic equations decouple as follows [ m t + η ] S 2 j t (r,t) = 0 ρ 0 t ρ(r,t)+ j l(r,t) = 0 m t + 1 ρ 0 ( 4 3 η S +η B ) }{{} longitudilal viscosity η l 2 j l(r,t)+c 2 ρ(r,t) = 0 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 29 July / ,

37 Spatial and temporal Fourier transform In Fourier space the linar hydrodynamic equations are [ iω +q 2 η ] S j t (q,ω) = 0 mρ 0 ( [ω 2 q 2 c 2 iω )] η l ρ(q,ω) = 0 mρ 0 c 2 = mρ 0 K Bulk modulus The transverse equation is a diffusion equation with diffusivity D t = η S /mρ 0 = kinematic viscosity The longitudinal equation is a damped-wave equation with damping constant Γ = η l /mρ 0 (Sound attenuation coefficient) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 30 July / ,

38 Inelastic scattering and density correlation functions Scattering cross-section: d 2 σ dωdω S(q,ω) = 1 e iωt S(q,t) }{{} 2π }{{} dynamical structure factor density-density-correlation function S(q,t) =< ρ(q,t) ρ(q,0) > From linearized hydrodynamics: ρ 0 k B T/K S(q,ω) Re iω + iω Ω2 ; Ω2 = q 2 /ρ 0 mk = q 2 c 2 +Γ This is just the solution of a damped-harmonic oscillator equation S(q,t)+ΓṠ(q,τ)+Ω2 S(q,t) = 0 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 31 July / ,

39 Generalized hydrodynamics for the extended q range S(q) S(q,ω) Re Ω 2 q iω+ iω +M(q,ω) ; Ω 2 q =q2 k B T/mS(q) which is the solution of a generalized damped-oscillator equation with a memory function M(q, t) t S(q,t)+ dτm q (t τ)ṡ(q,τ)+ω 2 q S(q,t) = 0 0 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 32 July / ,

40 Closure: Mode-coupling theory Self-consistent Mode-Coupling equations t S(q,t)+ dτm q (t τ)ṡ(q,τ)+ω 2 q S(q,t) = 0 0 Memory function M q (t) = q 1,q 2 V{S(q)}S(q 1,t)S(q 2,t) derived within the Mori-Zwanzig projection formalism The equation describe the feedback of the surrounding particles ( cage effect ) Götze et al. J. Phys. C 1983 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 33 July / ,

41 Mode-Coupling Theory for S(q, ω) Input: Static structure factor S(q) (that s all! NO ADJUSTABLE PARAMETER!) Output: Scattering law S(q,ω) Memory function Mq (t) Numerical code: especially adapted to cover many orders of magnitude Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 34 July / ,

42 The idealized glass transition and MCT Idealized glassy state = non-ergodic state { S(q,t) t 0 ergodic f(q) > 0 non-ergodic Property of MCT: If density or inverse temperature (which change S(q)) exceed a critical value ρ c,t c the system becomes non-ergodic Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 35 July / ,

43 MCT calculation for hard spheres M. Voigtmann, Doctoral thesis, = F(q,t)/S(q) MCT calculation experiment colloidal particles increasing density Experiment by T. Eckert, E. Bartsch, Faraday Discuss. 123, 51 (23) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 36 July / ,

44 Power-law time dependences 1 φ (t) t a t b "beta relaxation" alpha relaxation two scaling laws log t The β and α scales are moving away from each other as the ideal glass transition is approached Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 37 July / ,

45 Power-law frequency dependences χ "(ω) = ωf.t[ φ(t)] a ω ω b alpha relaxation "beta relaxation" two scaling laws log These features have been observed in many glass-forming liquids ω Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 38 July / ,

46 Critical law for the viscosity log [viscosity (poise)] T /T g T g /T c log [relaxation time (s)] Idealized-glass-transition law η(t) [T c T] γ γ = 1 2a + 1 2b MCT does neither address the non-diverging viscosity below T c nor the dynamical heterogeneities in this regime. In this regime one can apply a different approach, which I shall present right now. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 39 July / ,

47 Heterogeneous visco-elastic theory WS, G. Ruocco, V. Mazzone, PRL 2015 Recall Maxwell s ansatz for viscoelasticity: solid: σ xz = Gǫ xz strain liquid: σ xz = ηv xz strain rate ǫ ij = 1 2 ( iu j + j u i ) v ij = 1 2 ( iv j + j v i ) Superposition of solid and liquid strain rates: v xz,eff = v xz + d dt ǫ xz 1 η eff (ω) = iω G eff (ω) = 1 η + iω = 1 ( ) 1 G G τ + iω

48 Maxwells visco-elastic theory Consequence for the mechanical loss modulus G (ω): G (ω) = Im { G eff (ω) } ωτ = G 1 + (ωτ) G (ω) ω 1 ω -1 "Debye spectrum"

49 Mechanical relaxation spectra of metallic glasses Asymmetric relaxation Secondary (β)relaxation Wang et al. JCP 28 Samwer et al. Materials Today 2013

50 tend to be more symmetric (if it were not for the beta wing) Typical Dielectric loss spectra of glasses Poly-Carbonate (T g = 420 K) P. Lunkenheimer et al. J. Noncryst. Sol , 336 (22) 10 5 α relaxation β wing Hz 0 " boson peak " mev In comparison with the mechanic loss spectra the dielectric ones

51 Dynamic and elastic heterogeneities in simulations of liquids/glasses dynamical heterogenity (Berthier, Physics, 20) Elastic heterogeneity (Leonforte et al. PRB 25

52 dynamical heterogenity (Berthier, Physics, 20) Elastic heterogeneity (Leonforte et al. PRB 25 Most mobile cluster (Glotzer, JNCS 20) strongest non-affine cluster (Leonforte et al. PRB 25

53 Heterogeneous visco-elastic model W. Schirmacher, G. Ruocco, V. Mazzone, PRL 5, (2015) Linearized Navier-Stokes equations with space and frequency-dependent viscosity iωρ m v i (r,ω) = K iω 2 v i + 2 j j η eff (r,ω) v ij (r,ω) v ij = 1 2 ( iv j + j v i ) v ij = v ij 1 3 δ ijtr{v ij } 1 η eff (r,ω) = iω G eff (r,ω) = 1 η(r) + iω G(r) η(r) = η 0 e [E(r) TS(r)]/k BT η(r) = η 0 e E(r) [ β {}} eff { 1 k B T α] S(r)/k B = αe(r) (Meyer-Neldel compensation) E(r) = V A G(r)

54 Linearized Navier-Stokes equations with space and frequency-dependent viscosity iωρ m v i (r,ω) = K iω 2 v i + 2 j j η eff (r,ω) v ij (r,ω) v ij = 1 2 ( iv j + j v i ) v ij = v ij 1 3 δ ijtr{v ij } 1 η eff (r,ω) = iω G eff (r,ω) = 1 η(r) + iω G(r) η(r) = η 0 e [E(r) TS(r)]/k BT η(r) = η 0 e E(r) [ β {}} eff { 1 k B T α] S(r)/k B = αe(r) (Meyer-Neldel compensation) E(r) = V A G(r) E and G are related: shoving model (J. Dyre et al. 1996)

55 1 η eff (r,ω) = iω G eff (r,ω) = 1 η(r) + iω G(r) η(r) = η 0 e [E(r) TS(r)]/k BT η(r) = η 0 e E(r) [ β {}} eff { 1 k B T α] S(r)/k B = αe(r) (Meyer-Neldel compensation) E(r) = V A G(r) E and G are related: shoving model (J. Dyre et al. 1996) Coarse-grained Navier-Stokes equations with macroscopic frequency-dependent viscosity iωρ m v i (r,ω) = K iω 2 v i + 2η(ω) j j v ij (r,ω)

56 η(r) = η 0 e E(r) [ β {}} eff { 1 k B T α] (Meyer-Neldel compensation) E(r) = V A G(r) E and G are related: shoving model (J. Dyre et al. 1996) Coarse-grained Navier-Stokes equations with macroscopic frequency-dependent viscosity iωρ m v i (r,ω) = K iω 2 v i + 2η(ω) j j v ij (r,ω) The disorder leads to an additional frequency dependence The solution η eff (r,ω) η(ω) is obtained via the coherent-potential approximation (CPA) using a Gaussian statistics of E(r).

57 Angell Plot of viscosities we are here T g /T log [viscosity (poise)]

58 CPA for the spatially fluctuating viscosity W. Schirmacher et al., PRL 5, (2015) continuous effective medium obeying the coarse-grained Navier-Stokes equation for η(ω) η(e) eff local viscosity Self-consistent CPA condition: η(ω) η eff (E,r,ω) 1 2ν 3 [η(ω) η eff(e,r,ω)]/η(ω) E = 0 The transition matrix due to the perturbation induced by replacing the macroscopic by the microscopic viscosity should be on the average zero. Derived as saddle-point of replica field theory S. Köhler et al., PRB 88, (2013)

59 Relaxation data in supercooled liquid metals and viscosity CPA Wang et al. JCP 128, (28) CPA calculation with Gaussian energy distribution P(E) = P 0 e 1 2 (E E 0) 2 /σ 2 θ(e)

60 Dielectric and mechanic loss functions in CPA { } ǫ 1 (ω) Im 1 + fg(ω) Gemant-DiMarzio-Bishop equation, Niss et al. JCP 123, (25) ε"(ω) G"(ω) G (ω) f * = f * = f * = f * = f * = f * = ω.η(0)

61 tend to be more symmetric (if it were not for the beta wing) Typical Dielectric loss spectra of glasses Poly-Carbonate (T g = 420 K) P. Lunkenheimer et al. J. Noncryst. Sol , 336 (22) 10 5 α relaxation β wing Hz 0 " boson peak " mev In comparison with the mechanic loss spectra the dielectric ones

62 Model for beta relaxation 1.5 P(E) E P(E) = xp G (E,E α,σ α )+(1 x)p G (E,E β,σ β ) P G (E,E i σ i ) = P 0 e 1 2 (E E i) 2 /σ 2 i x = 0.2 E α = 1 E β = 0.1 σ α = 0.25 σ β = 0.1 The Gaussians are cut off at E=0 (as in the previous calculations).

63 eta relaxation in metallic glasses compared with the two-gaussian CPA G"(ω,T) T Samwer et al. Materials Today 2013 At temperatures below the maximum / the traditional averaging applies, and the curves reflect approximately the distribution of activation energies. This is not true above the maximum. G (ω,t) T/ T α The different curves correspond to different frequencies

64 Take-home messages Glasses exhibit quenched (frozen-in) disorder. The structure of a liquid (and a glass) is described by the pair distribution function g(r) and its Fourier transform, the structure factor S(k). There is not yet a consensus in the theoretical community, whether the glass transition is a phase transition or a pure kinetic phenomenon. The kinetics of the glassy freezing above the transition is well described by Mode-coupling theory, which is based on generalized linear hydrodynamics. Mode-coupling theory does not describe dynamic heterogeneities near the transition. These can be well described by a heterogeneous viscoelasticity theory, solved in CPA Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 40 July / ,