A Toy Model. Viscosity

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1 A Toy Model for Sponsoring Viscosity Jean BELLISSARD Westfälische Wilhelms-Universität, Münster Department of Mathematics SFB 878, Münster, Germany Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics

2 Main References C. A. Angell Formation of Glasses from Liquids and Biopolymers, Science, 267, No (Mar. 31, 1995), pp T. Egami, Elementary Excitation and Energy Landscape in Simple Liquids, Mod. Phys. Lett. B, 28, (2014), :1-19. J. Bellissard Delone Sets and Material Science: a Program, in Mathematics of Aperiodic Order, J. Kellendonk, D. Lenz, J. Savinien, Eds., Progress in Mathematics series, 309, Birkhäuser, pp , (2015).

3 Content Warning This talk is reporting on a work in progress. 1. Motivation 2. Anankeons and Phonons 3. Constructing the Model 4. Computing Maxwell s Time 5. Conclusion

4 I - Motivation

5 Motivation C. A. Angell, Formation of Glasses from Liquids and Biopolymers, Science, 267, No (Mar. 31, 1995), pp

6 Motivation J. H. Magill, D. J. Plazek, Physical Properties of Aromatic Hydrocarbons. II. Solidification Behavior of 1,3,5-Tri-α-Naphthylbenzene, J. Chem. Phys., 45, (1967),

7 Motivation Change of slope in Arrhenius behavior

8 Motivation Comparison between the Maxwell relaxation time τ M and the relaxation time τ LC for local configurations. T. Iwashita, D. M. Nicholson, T. Egami, Phys. Rev. Lett., 110, :5, (2013).

9 Motivation Specific Heat for Fragile Glasses M. H. Cohen, D. Turnbull, J. Chem. Phys., 31, 1164 (1959).

10 Motivation Specific Heat for Fragile Glasses: contributions of phonons & anankeons

11 Motivation At temperature T > T co larger than the crossover temperature, only the anankeons contribute to the specific heat. If T g < T < T co phonons and anankeon interact. Question: How?

12 II - Anankeons and Phonons

13 Atomic Configurations The set L of position of atomic nuclei is a Delone set, namely The minimum distance between atoms is 2r 0 > 0. The maximum diameter of a hole without atoms is 2r 1 <.

14 Atomic Configurations

15 Atomic Configurations

16 Atomic Configurations

17 Atomic Configurations

18 Atomic Configurations

19 Atomic Configurations

20 Atomic Configurations

21 Voronoi Cells Let L be Delone. If x L its Voronoi cell is defined by V(x) = {y R d ; y x < y x x L, x x} V(x) is open. Its closure T(x) = V(x) is called the Voronoi tile of x

22 Voronoi Cells Let L be Delone. If x L its Voronoi cell is defined by V(x) = {y R d ; y x < y x x L, x x} V(x) is open. Its closure T(x) = V(x) is called the Voronoi tile of x

23 Voronoi Cells Let L be Delone. If x L its Voronoi cell is defined by V(x) = {y R d ; y x < y x x L, x x} V(x) is open. Its closure T(x) = V(x) is called the Voronoi tile of x Proposition: If L is Delone, the Voronoi tile of any x L is a convex polytope

24 Voronoi Cells Let L be Delone. If x L its Voronoi cell is defined by V(x) = {y R d ; y x < y x x L, x x} V(x) is open. Its closure T(x) = V(x) is called the Voronoi tile of x Proposition: If L is Delone, the Voronoi tile of any x L is a convex polytope containing the closed ball B(x; r 0 ) and contained in the ball B(x; r 1 )

25 The Delone Graph Proposition: the Voronoi tiles of a Delone set touch face-to-face

26 The Delone Graph Proposition: the Voronoi tiles of a Delone set touch face-to-face Two atoms are nearest neighbors if their Voronoi tiles touch along a face of maximal dimension.

27 The Delone Graph Proposition: the Voronoi tiles of a Delone set touch face-to-face Two atoms are nearest neighbors if their Voronoi tiles touch along a face of maximal dimension. An edge is a pair of nearest neighbors. E denotes the set of edges.

28 The Delone Graph Proposition: the Voronoi tiles of a Delone set touch face-to-face Two atoms are nearest neighbors if their Voronoi tiles touch along a face of maximal dimension. An edge is a pair of nearest neighbors. E denotes the set of edges. The family G = (L, E) is the Delone graph.

29 The Delone Graph taken from J. D. Bernal, Nature, 183, , (1959)

30 The Delone Graph Modulo graph isomorphism, the Delone graph encodes the local topology

31 Atomic Movements small, deterministic, conserves local topology large, unpredictable, quick jump, change the local topology

32 III - Constructing the Model

33 Harmonic Motion The vibration of an atom around its equilibrium position is assumed to be harmonic. To simplify further, the oscillator will be supposed to be onedimensional. The frequency ω of the harmonic oscillator is defined by the curvature of the potential energy near the equilibrium position. Let q denotes the position of the harmonic particle relative to the equilibrium position. Then X will denote the phase space vector X = ωq q = [ u v ]

34 Harmonic Motion Equation of motion m d2 q dt 2 + kq = 0, ω = k m. Equivalently dx dt = ωjx, J = [ ] Or else X(t) = e ωtj X(0) = [ cos ωt sin ωt sin ωt cos ωt ] [ u(0) v(0) ]

35 Anankeon Interaction At random times < τ n 1 < τ n < τ n+1 <, the atom jumps quickly from one potential well to another one. Each jump will be considered as instantaneous. In the new potential well, the curvature is different, hence, after the time τ n, the new frequency is ω n. After the jump, the new relative phase space position is changed by a vector ξ n, namely X(τ n + 0) = X(τ n 0) + ξ n

36 Randomness: Assumptions The random times are Poissonian, namely the variables τ n+1 τ n are i.i.d, with exponential distribution and average E{τ n+1 τ n } = τ n+1 τ n = τ LC The frequencies ω n are also random and i.i.d, such that E(ω n ) = ω, E{(ω n ω) 2 } = σ 2 The phase-space initial positions ξ n are also random and i.i.d, with Gibbs distribution Prob{ξ n Λ} = e βm ξ 2 /2 d 2 ξ 2πk B T/m, β = 1 k B T Λ

37 Correlation Function The goal is to compute the stress-stress correlation function. It is sufficient to compute C f (t) = E{ f (X(t)) f (X(0))} for any complex valued function f defined on the phase space and vanishing at infinity. The viscosity is given by the Green-Kubo formula for a suitable f η = V k B T 0 C f (t) dt

38 Correlation Function Then, the goal is to show that C f (t) t e t/τ M Hence η τ M, which allows to interpret τ M as the Maxwell relaxation time.

39 Correlation Function The dissipative evolution operator P t acting on the set of functions f is defined by P t f (x) = E{ f (X(t)) X(0) = x} Then C f (t) = R 2 f (x) P t f (x) d 2 x

40 IV - Computing Maxwell s Time

41 Laplace Transform The Laplace transform of C(t) is defined by LC(ζ) = 0 e tζ C(t) dt The function C admits the asymptotic C(t) t e t/τ M if and only if LC(ζ) is holomorphic w.r.t. ζ in the domain Rζ > 1/τ M In practice, 1/τ M is the singularity nearest to the origin in the complex plane.

42 Dual Actions Phase Space Rotation: If f is a function, and x = (u, v) a point in the phase space, then f ( e ωt J x ) = ( e ωtj f ) (x), J = v u u v Hence J is the phase-space angular momentum. Phase Space Translation: Similarly, if ξ = (ξ 1, ξ 2 ) is a phasespace vector, f (x + ξ) = ( e ξ f ) (x), ξ = ξ 1 u + ξ 2 v Hence ξ is the phase-space momentum along the vector ξ.

43 Stochastic Evolution Let X(t) the stochastic value of the phase-space position at time t, with initial condition X(0) = x. If n anankeons occurred during this time then τ n t < τ n+1, with τ 0 = 0, so that f (X(t)) = n ( e (τ j τ j 1 )ω j 1 J e ξ ) j e (t τ n)ω n J f (x) j=1 In this expression the the τ j τ j 1 s, the ω j s and the ξ j s are random, independent and identically distributed.

44 Stochastic Evolution Averaging and taking the Laplace transform leads to LP ζ f (x) = 0 e tζ P t f (x) dt, ζ C Equivalently LP ζ f (x) = E n=0 τn+1 τ n e tζ n ( e (τ j τ j 1 )ω j 1 J e ) ξ j e (t τ n)ω n J f (x) dt j=1 Remark: τ 0 = 0, thus t = t τ n +(τ n τ n 1 )+ +(τ 1 τ 0 ). Hence ω j 1 J can be replaced by ω j 1 J + ζ and the exponential pre-factor e tζ disappears.

45 Stochastic Evolution First evaluate the integral over time between τ n and τ n+1, by setting s = t τ n τn+1 τn 0 e sζ e sω nj ds = 1 e(τ n+1 τ n )(ζ+ω n J) ζ + ω n J Second, average over the τ j τ j 1 s, using the formula (here A is an operator) E τ { e (τ j τ j 1 )A } = 0 e s/τ LC sa ds τ LC = τ LC A

46 Stochastic Evolution Reminder: if two random variables X, Y are stochastically independent then E{XY} = E{X} E{Y}. The average over the τ j τ j 1 s gives LP ζ f (x) = τ LC E n=0 n ( ) τ LC (ζ + ω j 1 J) eξ j j= τ LC (ζ + ω n J) f (x) This gives a new averaging over a product of independent variables.

47 Stochastic Evolution Averaging over the ξ j s can be done using (Gibbs average is a Gaussian integral) E ξ { e ξ j } = e k BT /2m, = = 2 u + 2 v Averaging over the ω j s leads to defining the following operator 1 A(ζ) = E 1 + τ LC (ζ + ω j 1 J) Remark: A(ζ) does not depend on j since all the ω j s have same distribution.

48 Stochastic Evolution Inserting into the expression of the Laplace transform leads to Questions: LP ζ f (x) = τ LC n=0 { A(ζ) e k B T /2m } n A(ζ) f (x) = τ LC 1 1 A(ζ) e k BT /2m A(ζ) f (x) How do we evaluate this function? How can we compute the domain of analyticity in ζ?

49 Angular Momentum Trick: the operator A(ζ) is a function of the phase-space angular momentum J. The polar coordinates in the 2D-phase space are given by It follows that { u = r cos θ v = r sin θ J = θ { r 2 = u 2 + v 2 tan θ = v/u

50 Angular Momentum Consequently the eigenvalues of J are given by ıl with l = 0, ±1, ±2,, namely l Z. The eigenfunctions have the form g l (r, θ) = ĝ l (r) e ılθ Projecting a function f onto the eigenspace of eigenvalue ıl is given by 2π Π l f (r, θ) = e ılθ f (r, θ) e ılθ dθ 2π Hence the spectral decomposition gives J = ı l Π l 0 l Z

51 Angular Momentum Any function of J can be written as F(J) = F(ıl) Π l l Z It leads to A(ζ) = a l (ζ) Π l l Z with a l (ζ) a complex number given by 1 a l (ζ) = E ω 1 + τ LC (ζ + ılω j )

52 Angular Momentum New Trick: the operator commutes with J, more precisely, the polar decomposition gives = p 2 r + J2 r 2 p 2 r = 2 r + 1 r r This gives 1 LP ζ = τ LC l Z 1 a l (ζ)e k BT(p 2 r +l2 /r 2 ) a l(ζ) Π l

53 Analyticity Remark that if ω > 0 the function (1 + τ LC (ζ + ılω)) 1 admits a pole at ζ = 1 τ LC ılω It follows that a l (ζ) is analytic in R(ζ) > 1/τ LC The operator p 2 r + l2 /r 2 is positive, and its spectrum is the entire positive real line. Hence, LP ζ is analytic in the domain for which there is no l Z nor any p 0 such that a l (ζ) = e p2 1. It means the set of ζ must satisfy a l (ζ) [1, )

54 Analyticity The domain of analyticity of a l (ζ) is given by Rζ > 1/τ LC

55 Analyticity The correlation function for X(t) involves only the angular momentum l = ±1. Assume that the distribution of ω j is uniform with average ω and variance σ. Then the ω j s are uniformly distributed in the interval [ω 3 σ, ω + 3 σ] and 1 a ±1 = 2ı 3τ LC σ ln 1 + τ LC(ζ + ıω + ı 3σ) 1 + τ LC (ζ + ıω ı 3σ) If ζ + ıω is not real, then the imaginary part of the r.h.s. is non zero, so that a ±1 [1, ).

56 Analyticity If ξ = ζ + ıω is real then 1 a ±1 = 2ı 3τ LC σ ln 1 + τ LC(ζ + ıω + ı 3σ) 1 + τ LC (ζ + ıω ı 3σ) = θ 3τLCσ where tan θ = 3τLCσ 1 + τ LC ξ θ < π 2

57 Maxwell Relaxation Time After some algebra, this gives τ M = τ LC (1 τ LC 3τLC σ tan 3τ LC σ if 3τLCσ π/2 ) 1 > τlc otherwise Using a uniform distribution of oscillator frequencies is a reasonable approximation. If τ LC σ decreases to zero as T 0, then This gives a crossover temperature T co above which the anankeon dominates and τ LC = τ M for T T co If T < T co, the phonons resist and τ M /τ LC > 1.

58 Maxwell Relaxation Time The variance σ of the random frequencies depend upon the modification of the local landscape by anankeons. However, each anankeon involves several atoms, at least d + 2 atoms in dimension d. So the landscape is modified in a region that might be large compared with the mean atomic distance. For this reason, σ is expected to be proportional to a Gibbs factor, namely to follow also an Arrhenius law σ = σ e W v/k B T σ = lim T σ(t) Question: What is the meaning of W v?

59 Maxwell Relaxation Time Similarly the local configuration time τ LC is also given by a similar expression, thanks to Kramers formula, where now W is the potential energy barrier to be crossed when an anankeon occurs τ LC = τ e W/k BT τ = lim T τ(t) If the hypothesis made on σ is correct, then K(T) = τ LC (T)σ(T) = K e (W v W)/k B T K = σ τ The condition K(T) T 0 0 requires W v > W

60 Maxwell Relaxation Time Hence, if T g < T < T co τ M τ LC = 1 1 3K(T) tan ( 3K(T)) T 0 e2(w v W)/k B T K 2 Similarly the crossover temperature is reached whenever 3τ LC σ = π/2, which gives (W v W) = k BT co 10.2

61 V - Conclusion

62 To Summarize The liquid phase of fragile glasses is dominated by anankeons at least above the crossover temperature T > T co. If T g < T < T co, the phonon-anankeon interaction becomes essential to explain the difference between the Maxwell and the local configuration times. Hence the change of the Arrhenius behavior of the viscosity is explained through a dynamical effect. As the temperature decreases, phonons become more coherent and limit the dissipative effect of the anankeons. The crossover temperature is related to the difference W s W between the activation energies associated with the phonon frequency fluctuation and the anankeon potential barrier.

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