Fractional Derivatives and Fractional Calculus in Rheology, Meeting #3

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1 Fractional Derivatives and Fractional Calculus in Rheology, Meeting #3 Metzler and Klafter. (22). From stretched exponential to inverse power-law: fractional dynamics, Cole Cole relaxation processes, and beyond. Thomas J. Ober June 22, 2 Part of the summer 2 Reading Group: Fractional Derivatives and Fractional Calculus in Rheology Non-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley

2 Lexicon Maxwell-Debye relaxation Kohlrausch-Williams-Watts (stretched exponential) function Nutting law Lorentzian (Cauchy distribution) Mittag-Leffler relaxation Probability density function Markoffian diffusion-relaxation Fokker-Planck equation Riemann-Liouville operator Einstein-Stokes equation 2

3 Maxwell-Debye Stress Relaxation Relaxation process characterized by a single timescale. SIMPLE! φ( t) = e t τ d dt φ t = τ φ t ; φ = For a Maxwell material, the memory kernal is G φ t. = G φ( t s) σ t For a step strain rate, we recover the familiar result: t γ ( t)ds σ ( t) G = e t τ Maxwell Debye σ ( t) G γ t φ t s 3

4 Non-Maxwell-Debye Relaxations Often material relax by a stretched exponential, called a Kohlrausch- Williams-Watts (KWW) relaxation. = e t τ α < α < φ t Alternatively, relaxations may obey asymptotic power laws, called the Nutting law. φ( t) = δ δ > lim t + t τ φ( t) = ( t τ ) δ φ(t)!!2!3!4!5 KWW α =.25 α =.5 α =.75 α =! t τ φ(t)!!2!3 Nutting α =.25 α =.5 α =.75 α =!2! t 2 τ 4

5 Experimentally Observed Stretched Exponential Non-linear viscoelastic relaxation processes observed in micellar systems. φ( t) = e ( t τ )α Varying salt concentration for CPyCl:NaSal system facilitates transition from reptative to combined reptation/breaking behavior. CPyCl/NaSal Rehage & Hoffmann, Molecular Physics. 74(5): (99). Rauscher, Rehage, Hoffmann. Progr. Colloid Polymer Sci. 84 (99). 5

6 Stretched Exponential can be Predicted From Doi-Edwards theory, stress relaxation function (fraction of chain remaining in tube) µ ( t) = 8 π 2 p=odd p 2 ( e tp 2 T d ) Number density of chains N ( L) = 2c e L L c 2 Td = L 2 /Dcπ 2 τrep L = chain length D c = chain diffusion coefficient τrep = reptation time c = breakage rate constant c2 = recombination rate constant L = mean chain length Integrate of number density and stress relaxation function µ ( t) = L Le L L µ ( L,t)dL N ( L) µ ( L,t)dL Steepest decent analysis yields µ ( t) e ( t τ rep ) 4 A stretched exponential with α = /4 is recovered for micelles in the case of τrep << τbreak Cates, M. E. Macromolecules. 2(9): (987) 6

7 Maxwell-Debye Oscillatory Response Impose a time periodic deformation on Maxwell material. = e iωt d φ ( t) χ ω Complex susceptibility χ ( ω ) = + iωτ = iωτ + ωτ & G G G G!!2!3!4!2! 2 ωt G G G G 2 Maxwell Moduli = τ e iωt e t τ dt Power Spectrum χ ( ω ) 2 = Lorentzian (Cauchy Distribution) associated with a resonant behavior. G '( ω ) = G G = G '' ω + ωτ ( ωτ ) ωτ 2 ωτ 2 + ωτ Lorentz Cauchy

8 Real(χ(ω))!!2!3!4 Non-Maxwell-Debye Oscillatory Response Previously developed oscillatory response may be found to be too restrictive base on experimental results. Empirical phenomenological fit χ ( ω ) = α =.25 α =.5 α =.75 α = χ (ω) α + iωτ!2! 2 ωt Imag(χ(ω)) < α < χ(ω) = χ (ω) iχ (ω)!!2 χ (ω) α =.25 α =.5 α =.75 α =!2! 2 ωt Cole-Cole Objective of this paper: Develop a dynamic framework to obtain these complicated relaxation functions. χ α =.25 α =.5 α =.75 α = χ 8

9 Generalize Diffusion Equation Classical Markoffian diffusion equation: Diffusion process in which only instantaneous quantity, Markov P(x,t), influences rate of chance of that quantity. P(x,t) = probability density P 2 function (PDF) (e.g. = K 2 P ( x,t ) concentration) t x Generalize Markoffian to fractional Fokker-Planck equation include memory effects. Governs evolution of PDF with external driving potential. P V ( x ) 2 = + K 2 P ( x,t ) t x mγ x V(x) = external potential m = mass γ = friction coefficient K = diffusion coefficient P 2 α V ( x ) = Dt + Kα 2 P ( x,t ) t x x mγ α Metzler R. et al. Europhysics Letters. 46(4): (999) Fokker Planck Fractional EisteinStokes Equation Kα = k BT mγ α 9

10 Riemann-Liouville Operator Fractional Fokker-Planck Equation P t = D t α Reimann-Liouville Operator Convolution of PDF with a power law memory kernal D α t P( x,t) = x Γ α V ( x) 2 + K α mγ α x 2 t dt ' P( x,t ') ( t t ') α P x,t Riemann Liouville Operator has the property that D α t P( x,t)e pt dt = p α P ( x, p)

11 Solving the Fractional Fokker-Planck Equation Separate variables P( x,t) = T ( t)ϕ ( x) Spatial φ(x) is the same as for classical F-P equation, but can also be ignored provided material is spatially uniform. dt n ( t) = λ n D α t T n ( t) Tn() =, τ -α λn dt Use of identity and substitution of variables yields T n ( p) = p + pτ ( α ) Inverse Laplace transform yields Mittag-Leffler function Eα T n ( t) = E ( α ( t τ ) α ) ( z) E α ( z) n = Γ + αn n= Mittag-Leffler

12 Key Results of the Fractional Fokker-Planck Equation Under no driving force, V(x), mean square displacement is x 2 t ( t) α = 2K α Γ + α Mittag-Leffler function interpolates between initial KWW and terminal inverse power-law, with index α. E α ( ( t τ ) α ) = (( t τ ) α ) n E Γ + αn α t τ n= Mittag-Leffler function is very similar to the Taylor series expansion of an exponential function, and is identical to it for α =. e x = + x! + x2 2! + x3 3! +... = x n n= n! Γ( n) = ( n )! ( α ) t α Γ( + α ), t τ ( Γ( α )( t τ ) α ), t τ Nutting Law KWW M-L 2

13 Additional Results of Mittag-Leffler Function Fractional Fokker-Planck describes a diffusing particle occasionally trapped for a time β. Waiting times for trapping events characterized by PDF: Ψ(β) ~ Aα β --α Characteristic waiting time, ζtrap, diverges ζ trap = Ψ( β)βdβ No finite ζtrap yields ultimate power-law behavior. Complex susceptibility χ '( ω ) = χ ''( ω ) = χ(ω) = χ (ω) iχ (ω) + 2 ωτ + ( ωτ ) α cos( πα 2) α cos( πα 2) + ( ωτ ) 2α ( ωτ ) α sin( πα 2) α cos( πα 2) + ( ωτ ) 2α + 2 ωτ 3

14 Concluding Notes Fractional dynamics describe diffusion in systems marked by multiple trapping events. Mittag-Leffler functions retain some degree of the time-history of diffusion and interpolate between exponential functions and power law behavior. Brownian Diffusion Lévy Flights Whereas both trajectories are statistically selfsimilar, the Lévy walk trajectory possesses a fractal dimension, characterising the island structure of clusters of smaller steps, connected by a long step. Metzler, R. and J. Klafter. Physics Reports. 399: -77. (2) 4

15 Questions... 5

16 Solving Mittag-Leffler Governing equation dt n ( t) dt Laplace transform governing equation I dt ( t) n dt = λ ni D t From identities p Tn Rearrange to obtain T n ( p) = p + pτ = λ n D α t T n ( t) Tn() =, τ -α λn ( t) { α T } n ( p) = λ n p α Tn ( p) where τ -α λn ( α ) identity D α t P( x,t)e pt dt = p α P ( x, p) 6

17 Mittag-Leffler Plots for α =.5 Tn E ( ( t τ ) α ) = e t τ erfc t τ Linear-Linear ( 2 ) t τ KWW Nutting M-L Semilog x KWW: Nutting: Tn!!2!3 e ( t τ ) 2 lim t Log-Log E 2 Γ 2 KWW Nutting M-L ( t τ ) 2!4!3!2! t τ Semilog y Tn KWW Nutting M-L!4!3!2! t τ Tn!!2!3 KWW Nutting M-L t τ 7

18 Scale Invariance Scale invariance is a feature of objects or laws that do not change if length scales (or [other] scales) are multiplied by a common factor. Koch Curve Koch Curve Polymer Chains Ar tist holding a picture of himself holding a picture of himself, holding a picture of himself, holding... N chains of length b rescale to N/λ chains of length λ /2 b R 2 = Nb 2 goes to R 2 = N/λ(λ /2 b) 2, and is unchanged. Rescaling each image is analogous to multiplying the features of the image by a single constant. Doi, E. Introduction to Polymer Physics. Oxford

19 Scale Invariance and Power Laws Power law functions are scale invariant, because a change in their argument is equivalent to multiplying the function by a constant. f(x) = x n where n is a constant therefore f(sx) = (sx) n = Cx n where s, C are constants 2 f (x) =x n n =.5 2 f (x) =x n n =.5 f (sx) 5 5 s = s =2 s =3 f (sx) s = s =2 s = x!! 2 x 9