Interfaces with short-range correlated disorder: what we learn from the Directed Polymer

Transcription

1 Interfaces with short-range correlated disorder: what we learn from the Directed Polymer Elisabeth Agoritsas (1), Thierry Giamarchi (2) Vincent Démery (3), Alberto Rosso (4) Reinaldo García-García (3), Damien Vamderbroucq (3) Lev Truskinovsky (3,5) (1) PSM, LIPHY, Grenoble (2) DQMP, Genève (3) ESPCI, Paris (4) LPTMS, Orsay (5) LMS, Polytechnique Lyon October 1st, 2015 Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

2 Introduction Motivations 1D Interfaces Interfaces in magnetic films Growth in liquid crystals Large range of physical scales Wide spectrum of phenomena from Metaxas et al. from Takeuchi & Sano APL (2009) PRL (2010) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

3 Introduction Motivations 1D Interfaces Growth in liquid crystals Large range of physical scales Wide spectrum of phenomena from Metaxas et al. from Takeuchi & Sano APL (2009) PRL (2010) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

4 Introduction Modelisation Disordered elastic systems Elasticity: tends to flatten the interface H el = c dz ( u(z) ) 2 [Short-range] 2 H el = c ( u(z) u(z dzdz ) ) 2 2π (z z ) 2 [Long-range] z u(z) Disorder: tends to bend it = dz V ( u(z), z ) H dis V Competition btw order and disorder Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

5 Introduction Modelisation Disordered elastic systems Elasticity: tends to flatten the interface H el = c dz ( u(z) ) 2 [Short-range] 2 H el = c ( u(z) u(z dzdz ) ) 2 2π (z z ) 2 [Long-range] z u(z) Disorder: tends to bend it = dz V ( u(z), z ) H dis V Force: induces motion of the interface Competition btw order and disorder Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

6 Introduction Depinning Depinning zero temperature threshold force f c Ú ÐÓ ØÝ v v (f f c ) β f c ÓÖ f Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

7 Introduction Depinning Depinning finite temperature Ú ÐÓ ØÝ v thermal rounding creep regime T > 0 T = 0 v (f f c ) β Ö Ô Ö Ñ f c ÓÖ f Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

8 Introduction Depinning Uncorrelated disorder: V(z, x)v(z, x ) = D δ(z z)δ(x x) Correlated disorder on a lengthscale ξ: V(z, x)v(z, x ) = D δ(z z)r ξ (x x) R Ξ x scaling as R ξ (x) = 1 ξ R ξ=1(x/ξ) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

9 Introduction Depinning Uncorrelated disorder: V(z, x)v(z, x ) = D δ(z z)δ(x x) Correlated disorder on a lengthscale ξ: V(z, x)v(z, x ) = D δ(z z)r ξ (x x) Can ξ play a role at lengthscales ξ? R Ξ x scaling as R ξ (x) = 1 ξ R ξ=1(x/ξ) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

10 Introduction Outline Study 1D models with correlated disorder (ξ > 0) 1 Static properties & creep regime (T > 0 and T 0) short-range elasticity Identification of lengthscales [with Elisabeth Agoritsas, Thierry Giamarchi] Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

11 Introduction Outline Study 1D models with correlated disorder (ξ > 0) 1 Static properties & creep regime (T > 0 and T 0) short-range elasticity Identification of lengthscales [with Elisabeth Agoritsas, Thierry Giamarchi] 2 Creep law (T 0) short-range elasticity Non-equilibrium velocity & equilibrium free-energy [with Reinaldo García-García, Elisabeth Agoritsas, Damien Vandembroucq, Lev Truskinovsky] Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

12 Introduction Outline Study 1D models with correlated disorder (ξ > 0) 1 Static properties & creep regime (T > 0 and T 0) short-range elasticity Identification of lengthscales [with Elisabeth Agoritsas, Thierry Giamarchi] 2 Creep law (T 0) short-range elasticity Non-equilibrium velocity & equilibrium free-energy [with Reinaldo García-García, Elisabeth Agoritsas, Damien Vandembroucq, Lev Truskinovsky] 3 Depinning force f c long-range elasticity Role of disorder correlator [with Vincent Démery, Alberto Rosso] (T = 0) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

13 Short-range elasticity Directed Polymer language 1D Interface in the Directed Polymer (DP) language [Step n 1] z (z,u (z) ) u (z+r) t y(t) No bubbles No overhangs Interface lengthscale r DP time t u z (r) u (z) r t (0, 0) y (t, y(t)) x working at fixed time t integration of fluctuations at scales smaller than t Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

14 Short-range elasticity Directed Polymer language 1D Interface in the Directed Polymer (DP) language [Step n 1] z (z,u (z) ) u (z+r) t y(t) No bubbles No overhangs Interface lengthscale r DP time t u z (r) u (z) r t (0, 0) y (t, y(t)) x working at fixed time t integration of fluctuations at scales smaller than t lengthscale time duration Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

15 Short-range elasticity Directed Polymer language Disordered elastic systems Elasticity: tends to flatten the interface H el [y(t ), t] = c 2 t Disorder: tends to bend it H dis V [y(t ), t] = 0 t 0 dt [ t y(t ) ] 2 dt V ( t, y(t ) ) [short-range elasticity] Competition btw order and disorder Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

16 Short-range elasticity Directed Polymer language Disordered elastic systems Elasticity: tends to flatten the interface H el [y(t ), t] = c 2 t Disorder: tends to bend it H dis V [y(t ), t] = 0 t 0 dt [ t y(t ) ] 2 dt V ( t, y(t ) ) [short-range elasticity] Competition btw order and disorder Ingredients up to now: elastic constant c disorder potential V(t, y) trajectory weight e H V /T {}}{ temperature T Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

17 Short-range elasticity Questions & known results Questions Nature of fluctuations of y(t) V(t, y) 0: diffusive (y t 1/2 ), Edwards-Wilkinson (EW) V(t, y) 0: super-diffusive (y t 2/3 ), Kardar-Parisi-Zhang (KPZ) This holds at large times. What about intermediate times? Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

18 Short-range elasticity Questions & known results Questions Nature of fluctuations of y(t) V(t, y) 0: diffusive (y t 1/2 ), Edwards-Wilkinson (EW) V(t, y) 0: super-diffusive (y t 2/3 ), Kardar-Parisi-Zhang (KPZ) This holds at large times. What about intermediate times? Role of (experimentally ineluctable) disorder correlations? zero mean, Gaussian, V(t, y)v(t, y ) = D δ(t t)r ξ (y y) R Ξ y y scaling as R ξ (y) = 1 ξ R ξ=1(y/ξ) [standard uncorrelated case: ξ = 0] Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

19 Short-range elasticity Questions & known results Questions Nature of fluctuations of y(t) V(t, y) 0: diffusive (y t 1/2 ), Edwards-Wilkinson (EW) V(t, y) 0: super-diffusive (y t 2/3 ), Kardar-Parisi-Zhang (KPZ) This holds at large times. What about intermediate times? Role of (experimentally ineluctable) disorder correlations? zero mean, Gaussian, V(t, y)v(t, y ) = D δ(t t)r ξ (y y) R Ξ y y scaling as R ξ (y) = 1 ξ R ξ=1(y/ξ) [standard uncorrelated case: ξ = 0] Summary of ingredients: elastic constant c temperature T disorder corr. amplitude length D ξ Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

20 Short-range elasticity Questions & known results Free-energy fluctuations [Step n 2&3] Partition function Z V vs. Free-energy F V y(t)=y Z V (t, y) = Dy(t )e 1 T H V[y(t ),t] Z V (t, y) = exp { 1 T F V(t, y) } y(0)=0 Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

21 Short-range elasticity Questions & known results Free-energy fluctuations [Step n 2&3] Partition function Z V vs. Free-energy F V y(t)=y Z V (t, y) = Dy(t )e 1 T H V[y(t ),t] Z V (t, y) = exp { 1 T F V(t, y) } y(0)=0 Statistical Tilt Symmetry F V (t, y) = c y2 2t + T 2πTt log } 2 {{ c } thermal contribution F V 0 Tilted KPZ equation for F V (t, y) + F V (t, y) } {{ } disorder contribution t F V + y t yf V = T 2c 2 y F V 1 2c[ y F V ] 2 + V(t, y) Non-linear, additive noise, F V (0, y) 0: simple initial cond. (STS) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

22 Short-range elasticity Questions & known results Known = 0 [ > 0] Central tool: 2-point correlation function R(t, y 2 y 1 ) = y F V (t, y 1 ) y F V (t, y 2 ) Infinite- time limit (steady state) F(t =, y) distributed as a Brownian Motion i.e. : Prob [ F(t =, y) ] Gaussian, of correlator: R(t =, y) = D ξ=0 δ(y) with Dξ=0 = cd T Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

23 Short-range elasticity Questions & known results Known = 0 [ > 0] Central tool: 2-point correlation function R(t, y 2 y 1 ) = y F V (t, y 1 ) y F V (t, y 2 ) Infinite- time limit (steady state) F(t =, y) distributed as a Brownian Motion i.e. : Prob [ F(t =, y) ] Gaussian, of correlator: R(t =, y) = D ξ=0 δ(y) with Dξ=0 = cd T Roughness function B(t) B(t) = y(t) 2 = [variance of end-point fluct.] dy y 2 Z V (t,y) dy ZV (t,y) B(t) [ Dξ=0 /c 2] 2/3 t 4/3 as t Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

24 Results DP toymodel Effective > 0 & numerical results ξ > 0 not obtained from perturbation of ξ = 0 Distribution of free-energy scales closely to the ξ = 0 case Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

25 Results DP toymodel Effective > 0 & numerical results ξ > 0 not obtained from perturbation of ξ = 0 Distribution of free-energy scales closely to the ξ = 0 case 2-point correlation function of amplitude D R sat (y) = lim t R(t, y) DR ξ (y) T = R(t, y) 2.0 R sat (y) 2.0 R(t, y) t 2 [0.1, 25] 0.5 t 2 [25, 40] y y y -0.5 Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

26 Results DP toymodel High- and low-temperature regimes R sat (0) / e D 1 (T, ) /T c ( ) 1.0 [T T c ] Characteristic temperature T c = (ξcd) 1/3 1/T [T T c ] T (Advanced) scaling analysis T T c one optimal trajectory D = cd T c T T c many trajectories cd D = T Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

27 Results DP toymodel High- and low-temperature regimes R sat (0) / e D 1 (T, ) /T c ( ) 1.0 [T T c ] Characteristic temperature T c = (ξcd) 1/3 1/T [T T c ] (Advanced) scaling analysis [ Note: again B(t) = T ξ plays a role at all scales {}}{ [ D/c 2 ] 2/3 t 4/3 ] T T c one optimal trajectory D = cd T c T T c many trajectories cd D = T Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

28 Results Dynamics Lengthscales & dynamics Geometry of interface Directed Polym. free-energy fluctuat. T T c : ξ plays a role at all lengthscales [T c = (ξcd) 1/3 ] focus on the free-energy 2-point correlator amplitude D understanding of time - (i.e. length) multiscaling Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

29 Results Dynamics Lengthscales & dynamics Geometry of interface Directed Polym. free-energy fluctuat. T T c : ξ plays a role at all lengthscales [T c = (ξcd) 1/3 ] focus on the free-energy 2-point correlator amplitude D understanding of time - (i.e. length) multiscaling B th (t) ζ = 2 3 KPZ B(t) B dis (t) T = t ζ = 1 2 diffusive Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

30 Results Dynamics Lengthscales & dynamics Geometry of interface Directed Polym. free-energy fluctuat. T T c : ξ plays a role at all lengthscales [T c = (ξcd) 1/3 ] focus on the free-energy 2-point correlator amplitude D understanding of time - (i.e. length) multiscaling B th (t) ζ = 2 3 KPZ ζ = 1 2 diffusive intermediate B(t) B dis (t) T =0.35 t Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

31 Results Dynamics Lengthscales & dynamics Geometry of interface Directed Polym. free-energy fluctuat. T T c : ξ plays a role at all lengthscales [T c = (ξcd) 1/3 ] focus on the free-energy 2-point correlator amplitude D understanding of time - (i.e. length) multiscaling Non-Gaussian steady state: D cd T t R(t, y) = T c 2 y R(t, y) 1 c R 3(t, y) DR ξ (y) (Itō s lemma) R 3 (t, y) = { a 3-point correlation function of F } = 0 if Gaussian st.st. Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

32 Results Dynamics Lengthscales & dynamics Geometry of interface Directed Polym. free-energy fluctuat. T T c : ξ plays a role at all lengthscales [T c = (ξcd) 1/3 ] focus on the free-energy 2-point correlator amplitude D understanding of time - (i.e. length) multiscaling Non-Gaussian steady state: D cd T t R(t, y) = T c 2 y R(t, y) 1 c R 3(t, y) DR ξ (y) (Itō s lemma) R 3 (t, y) = { a 3-point correlation function of F } = 0 if Gaussian st.st. Interpretation in other incarnations of the KPZ class growth interfaces with F(t, y) = height at (real) time t experimental probe of the importance of ξ through replicæ: 1D quantum bosons with softened delta attractive interaction Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

33 Results Dynamics Creep law [in progress] [with R García-García, E Agoritsas, D Vandembroucq, L Truskinovsky] Creep law: non-linear response to small force depends on c, D, T, ξ { velocity exp [ {}}{ critical force] } 1/4 force Very well verified numerically and experimentally Usual energetic arguments: self-inconsistent (ζ = ) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

34 Results Dynamics Creep law [in progress] [with R García-García, E Agoritsas, D Vandembroucq, L Truskinovsky] Creep law: non-linear response to small force depends on c, D, T, ξ { velocity exp [ {}}{ critical force] } 1/4 force Very well verified numerically and experimentally Usual energetic arguments: self-inconsistent (ζ = ) Effective model: Equilibrium drifted polymer in a tilted potential generalized STS for its free-energy F [linear response works for F!] quasi-static picture: focus on extremities Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

35 Results Dynamics Effective picture velocity 1 mean first passage time. Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

36 Results Dynamics Effective potential seen by the polymer extremities Saddle-point analyis (after well-chosen scaling) creep law, finite-size corrections Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

37 Results Dynamics Depinning transition [with V Démery, A Rosso] [following V Démery, L Ponson, A Rosso EPL (2014)] Equation of evolution [long-range elasticity] t u(z, t) = f el [u(, t)](z) σ u V ( z, u(z, t)) + f Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

38 Results Dynamics Depinning transition [with V Démery, A Rosso] [following V Démery, L Ponson, A Rosso EPL (2014)] Equation of evolution [long-range elasticity] t u(z, t) = f el [u(, t)](z) σ u V ( z, u(z, t)) + f Method: [T = 0] add a confining potential moving at constant velocity perform a 1 st order perturbation in disorder obtain force(velocity) = send velocity to zero to get f c send confining potential to zero Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

39 Results Dynamics Depinning transition [with V Démery, A Rosso] [following V Démery, L Ponson, A Rosso EPL (2014)] Equation of evolution [long-range elasticity] t u(z, t) = f el [u(, t)](z) σ u V ( z, u(z, t)) + f Method: [T = 0] add a confining potential moving at constant velocity perform a 1 st order perturbation in disorder obtain force(velocity) = send velocity to zero to get f c send confining potential to zero Disorder correlations: [for the disorder-induced random force] z V(z, x) z V(z, x ) = u (z z) x (x x) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

40 Results Dynamics Critical force Result: f c = σ2 x (0) 4πc dk u k u u (k u ) (σ 0) Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

41 Results Dynamics Critical force Result: f c = σ2 x (0) 4πc dk u k u u (k u ) (σ 0) Fc = fc/σ model A model B model C prediction Σ = σ x (0) 4πc ku u (k u )dk u Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20

42 Results Dynamics Thank you for your attention! References: Elisabeth Agoritsas, VL & Thierry Giamarchi:. Phys. Rev. E (2013). Phys. Rev. E (2013). Physica B (2012) Vincent Démery, VL & Alberto Rosso:. J. Stat. Mech. P03009 (2014) Reinaldo García-García, Elisabeth Agoritsas, VL, Damien Vandembroucq & Lev Truskinovsky:. soon on Arxiv Vivien Lecomte (LPMA - Paris) Interfaces & short-range correl. disorder 01/10/ / 20