Fractional Derivatives and Fractional Calculus in Rheology, Meeting #3

Thomas J. OberJune 22, 2010

Part of the summer 2010 Reading Group: Fractional Derivatives and Fractional Calculus in RheologyNon-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley

Metzler and Klafter. (2002). “From stretched exponential to inverse power-law: fractional dynamics, Cole–Cole relaxation processes, and beyond.”

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

Maxwell-Debye Stress Relaxation

• Relaxation process characterized by a single timescale. SIMPLE!

• For a Maxwell material, the memory kernal is .

• For a step strain rate, we recover the familiar result:

3

ddtφ t( ) = −

1τφ t( ); φ 0( ) = 1φ t( ) = e− t τ

G0φ t( )

σ t( ) = G0φ t − s( )0

t

∫ γ t( )ds

σ t( )G0

= 1− e− t τ

σ t( )G0

γ t( )φ t − s( )

Maxwell1831-1879

Debye1884-1966

Non-Maxwell-Debye Relaxations

• Often material relax by a stretched exponential, called a Kohlrausch-Williams-Watts (KWW) relaxation.

• Alternatively, relaxations may obey asymptotic power laws, called the Nutting law.

4

φ t( ) = e− t τ( )α 0 < α < 1

φ t( ) = 11+ t τ( )δ δ > 0 lim

t→∞φ t( ) = t τ( )−δ

10!1 100 10110!5

10!4

10!3

10!2

10!1

100

t

!

"(t

)

# = 0.25# = 0.5# = 0.75# = 1

10!2 10!1 100 101 10210!3

10!2

10!1

100

t

!

"(t

)

# = 0.25# = 0.5# = 0.75# = 1

KWW Nutting

Experimentally Observed Stretched Exponential

• Non-linear viscoelastic relaxation processes observed in micellar systems.

• Varying salt concentration for CPyCl:NaSal system facilitates transition from reptative to combined reptation/breaking behavior.

5Rehage & Hoffmann, Molecular Physics. 74(5): 933-973 (1991).Rauscher, Rehage, Hoffmann. Progr. Colloid Polymer Sci. 84 (1991).

φ t( ) = e− t τ( )α

CPyCl/NaSal

Stretched Exponential can be Predicted

• From Doi-Edwards theory, stress relaxation function (fraction of chain remaining in tube)

• Number density of chains

• Integrate of number density and stress relaxation function

• Steepest decent analysis yields

6Cates, M. E. Macromolecules. 20(9):2289-2296. (1987)

Td = L2/Dcπ2 ≅τrepµ t( ) = 8π 2 p−2e − tp2 Td( )

p=odd∑

L = chain lengthDc = chain diffusion coefficientτrep = reptation time

N L( ) = 2c1c2

e−L L

c1 = breakage rate constantc2 = recombination rate constant = mean chain lengthL

µ t( ) e− t τ rep( )1 4

µ t( ) = 1L

Le−L L

0

∞

∫ µ L,t( )dL N L( )0

∞

∫ µ L,t( )dL

A stretched exponential with α = 1/4 is recovered for micelles in the case of τrep << τbreak

Maxwell-Debye Oscillatory Response

• Impose a time periodic deformation on Maxwell material.

• Complex susceptibility

7

χ ω( ) = e− iω td −φ t( )( )0

∞

∫ =1τ

e− iω te− t τ0

∞

∫ dt

10!2 10!1 100 101 10210!4

10!3

10!2

10!1

100

101

!t

G!

G0&

G!!

G0

G!

G0G!!

G0

Lorentzian (Cauchy Distribution) associated with a resonant behavior.

Power Spectrum

Cauchy1789-1857

Lorentz1853-1928

G ' ω( )G0

=ωτ( )2

1+ ωτ( )2

G '' ω( )G0

=ωτ

1+ ωτ( )2

Maxwell Moduli

χ ω( ) 2 = 11+ ωτ( )2χ ω( ) = 1

1+ iωτ=1− iωτ1+ ωτ( )2

Non-Maxwell-Debye Oscillatory Response

• Previously developed oscillatory response may be found to be too restrictive base on experimental results.

• Empirical phenomenological fit

8

•Objective of this paper:

• Develop a dynamic framework to obtain these complicated relaxation functions.

0 < α < 1 χ(ω) = χ’(ω) – iχ’’(ω)

10!2 10!1 100 101 10210!2

10!1

100

!t

Imag

("(!

))

# = 0.25# = 0.5# = 0.75# = 1

10!2 10!1 100 101 10210!4

10!3

10!2

10!1

100

!t

Rea

l("

(!))

# = 0.25# = 0.5# = 0.75# = 1

Cole-Coleχ’(ω) χ’’(ω)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

!!

!!!

" = 0.25" = 0.5" = 0.75" = 1

χ ω( ) = 11+ iωτ( )α

Generalize Diffusion Equation

• Classical Markoffian diffusion equation:

• Diffusion process in which only instantaneous quantity, P(x,t), influences rate of chance of that quantity.

• Generalize Markoffian to fractional Fokker-Planck equation include memory effects.

• Governs evolution of PDF with external driving potential.

9

Markov1856-1922

∂P∂t

= K∂2

∂x2P x,t( )

P(x,t) = probability density function (PDF) (e.g.

concentration) Fokker1887-1972

Planck1858-1947

∂P∂t

=∂∂xV x( )mγ

+ K∂2

∂x2⎛⎝⎜

⎞⎠⎟P x,t( )

V(x) = external potentialm = massγ = friction coefficientK = diffusion coefficient

∂P∂t

= 0Dt1−α ∂

∂xV x( )mγ α

+ Kα∂2

∂x2⎛⎝⎜

⎞⎠⎟P x,t( ) Kα =

kBTmγ α

Fractional Eistein-Stokes Equation

Metzler R. et al. Europhysics Letters. 46(4):431-436. (1999)

Riemann-Liouville Operator

• Fractional Fokker-Planck Equation

• Reimann-Liouville Operator

• Convolution of PDF with a power law memory kernal

• Operator has the property that

10

0Dt−αP x,t( ) = 1

Γ α( ) dt '0

t

∫P x,t '( )t − t '( )1−α

Riemann1826-1866

Liouville1809-1882

∂P∂t

= 0Dt1−α ∂

∂xV x( )mγ α

+ Kα∂2

∂x2⎛⎝⎜

⎞⎠⎟P x,t( )

0Dt−αP x,t( )e− ptdt

0

∞

∫ = p−α P x, p( )

Solving the Fractional Fokker-Planck Equation

• Separate variables

• Spatial φ(x) is the same as for classical F-P equation, but can also be ignored provided material is spatially uniform.

• Use of identity and substitution of variables yields

• Inverse Laplace transform yields Mittag-Leffler function Eα

11

P x,t( ) = T t( )ϕ x( )

dTn t( )dt

= −λn 0Dt1−αTn t( )

Tn p( ) = 1p 1+ pτ( )−α( )

Tn(0) = 1, τ-α ≡ λn

Tn t( ) = Eα − t τ( )α( ) Eα z( ) = −z( )nΓ 1+αn( )n=0

∞

∑

Mittag-Leffler1846-1927

Key Results of the Fractional Fokker-Planck Equation

• Under no driving force, V(x), mean square displacement is

• Mittag-Leffler function interpolates between initial KWW and terminal inverse power-law, with index α.

12

x2 t( ) = 2Kαtα

Γ 1+α( )

Eα − t τ( )α( ) = t τ( )α( )nΓ 1+αn( )n=0

∞

∑

Eα − t τ( )α( ) 1− tα Γ 1+α( ), t τ

Γ 1−α( ) t τ( )α( )−1 , t τ

⎧⎨⎪

⎩⎪Eq. (12) can be solved by the method of sepa-

ration of variables. By help of the separationansatz P !x; t" # T !t"u!x" [16], the fractionalFokker–Planck Eq. (12) can be decoupled. Thespatial eigenequation is the same as for the clas-sical Fokker–Planck equation; however, for thetemporal eigenequation for the eigenvalue kn, thefractional relaxation equation [12,21],

dTn!t"dt

# $kn 0D1$at Tn!t"; !16"

obtains. In the limit a # 1, the fractional operatorreduces to the identity operator, and Eq. (16) isbut the standard relaxation equation (2). Eq. (16)can be solved via Laplace transformation, makinguse of the property (14), or through a power seriesansatz. In Laplace space, Eq. (16) is algebraic, andits solution is

~TTn!p" # p$1=!1% !ps"$a" !17"

with Tn!0" # 1 and s$a & kn. Via inverse Laplacetransformation, the solution

Tn!t" # Ea!$!t=s"a" !18"

is recovered. Here, Ea denotes the Mittag–Le!erfunction [22] whose series expansion reads

Ea!$z" #X

1

n#0

!$z"n

C!1% an" : !19"

For 0 < a6 1, the Mittag–Le!er function is pos-itive and strictly monotonically decreasing [22].The series expansion (19) shows the proximitybetween the Mittag–Le!er function and the ex-ponential function, and it is obvious that in thelimit a # 1, it reduces to the the exponentialfunction. For a # 1=2, the Mittag–Le!er functioncan be expressed in terms of the complementaryerror function, E1=2!$!t=s"1=2" # et=s erfc!!t=s"1=2".In Fig. 1, we compare the Mittag–Le!er relax-ation for a # 1=2 with the KWW relaxation ofindex 1/2 and the asymptotic inverse power-law!t=s"$1=2. In general, Ea!$!t=s"a" exhibits the lim-iting behaviours

Ea!$!t=s"a" ' 1$ ta=C!1% a"; t ( s;!C!1$ a"!t=s"a"$1; t ) s:

!

!20"

Thus, the Mittag–Le!er function interpolates be-tween an initial stretched exponential (KWW)pattern and a terminal inverse power-law decay,both of index a. This interpolating behaviour iswell known from rheological modelling, see, forinstance, Ref. [23]. We note that by Tauberiantheorems, the power-law asymptote !t=s"$a fort ) s corresponds to the p ( s$1 behaviour of theLaplace transform (17).

The Mittag–Le!er pattern dominates the moderelaxation, and the relaxation of moments in sys-tems with a restoring force of the fractional Fok-ker–Planck equation (12). The multiple trappingmodel underlying Eq. (12) states that a di"usingparticle can get occasionally trapped at a givenspace point, and only be released after a givenwaiting time t. After release, the particle di"usesuntil it gets trapped again, and so forth. Thereby,the duration of individual waiting periods, t, is arandom variable distributed according to thewaiting time pdf w!t" ' Aat

$1$a [18]. The slow re-laxation manifested in the Mittag–Le!er decay(18) is thus directly related to the waiting timeswhose characteristic scale T #

R10

w!t"tdt di-verges, and therefore also the Mittag–Le!er pat-tern is scale-free, i.e., it possesses a diverging T.

Due to the dynamic origin of the scale-freememory which gives rise to the fractional equa-tions, the Mittag–Le!er relaxation might also beconnected to the crossing of an activation barrierin the generalisation of the Kramers model if the

Fig. 1. Mittag–Le!er relaxation in log10–log10 representation.The full line represents the Mittag–Le!er function for index 1/2. The dashed lines depict the initial stretched exponential be-haviour and the final inverse power-law pattern, demonstratingthe interpolating character between both limiting forms.

84 R. Metzler, J. Klafter / Journal of Non-Crystalline Solids 305 (2002) 81–87

ex = 1+ x1!+x2

2!+x3

3!+ ... = xn

n!n=0

∞

∑Γ n( ) = n −1( )!

• Mittag-Leffler function is very similar to the Taylor series expansion of an exponential function, and is identical to it for α = 1.

KWW

Nutting Law

M-L

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:

• Characteristic waiting time, ζtrap, diverges

• No finite ζtrap yields ultimate power-law behavior.

• Complex susceptibility

13

Ψ(β) ~ Aα β-1-α

ζ trap = Ψ β( )βdβ → ∞0

∞

∫

χ ' ω( ) = 1+ ωτ( )α cos πα 2( )1+ 2 ωτ( )α cos πα 2( ) + ωτ( )2α

χ '' ω( ) = ωτ( )α sin πα 2( )1+ 2 ωτ( )α cos πα 2( ) + ωτ( )2α

χ(ω) = χ’(ω) – iχ’’(ω)

low di!usivity limit applies, as was demonstratedin Ref. [24].

The complex susceptibility corresponding to theMittag–Le"er pattern (18) is obtained from Eq.(17) by virtue of relation (6), the result being ex-actly the Cole–Cole function (9), as obtained ear-lier by Weron and Kotulski [5] in a similar context.

As mentioned above, the Mittag–Le"er func-tion is a strictly monotonically decreasing function[22], so that all poles of the complex susceptibilityv!x" must necessarily lie in the upper half of thecomplex plane. Therefore, the real part

v0!x" # 1$ !xs"a cos!pa=2"1$ 2!xs"a cos!pa=2" $ !xs"2a

!21"

and the imaginary part

v00!x" # !xs"a sin!pa=2"1$ 2!xs"a cos!pa=2" $ !xs"2a

!22"

are connected through the Kramers–Kronig rela-tion [25]

v00!x" # % 1

pPZ 1

%1

v0!m"m% x

dm; !23"

where PR

denotes the Cauchy principal value.In Fig. 2, we use the Cole–Cole plot which re-

lates the behaviour of the real and imaginary partsof the complex susceptibility (9) for di!erent val-ues of the fractional order a. Accordingly, thesemicircle which corresponds to the exponential

relaxation pattern is increasingly compressed ver-tically for decreasing values of a. However, theproperty that the parametric plot increases strictlymonotonically up to 1/2 and then decays strictlymonotonically is preserved.

For not too low frequencies, both the real part(21) and the imaginary part (22) scale likev0!x" & v00!x" & x%a, and their ratio is giventhrough

v00!x"v0!x" & tan

pa2

! "

; !24"

i.e., it is independent of x, in contrast to the Debyerelaxation where the same ratio is &xs. Relation(24) corresponds to the universality advocated byJonscher [26], and it is a direct consequence of theKramers–Kronig relation (23).

The Cole–Cole behaviour (9) and the corre-sponding Mittag–Le"er relaxation function arethe characteristic response patterns which stemfrom anomalous di!usion related to the waitingtime pdf w!t" & Aat

%1%a within the continuous timerandom walk scheme. In the system under con-sideration, there might be combinations of di!u-sion processes which have a di!erent value for thisinternal time scale, and they might also be a!ecteddi!erently by the disorder in the system, i.e., ex-hibit a di!erent a. The combined relaxation pro-cess might therefore be composed of two or moreindividual Mittag–Le"er patterns, giving rise tomore elaborate relaxation forms.

A more general a priori form used to fit ex-perimental data is the HN pattern [6]

v!x" # 1

'1$ !ixs"a(c ; !25"

which has become a widely used formula to de-scribe experimental data. Usually, one restricts theparameters to a > 0 and ac6 1. A special case ofthe HN pattern (25) for c # 1 is therefore theCole–Cole function (9). The HN parameters a andc determine the slopes of v00!x" in the double-logarithmic plot, these being a on the low fre-quency side and %ac on the high frequency side ofthe peak. It can be shown that a combination oftwo fractional processes with di!erent internaltime scales can reproduce this behaviour but hasthe advantage that all related functions are known

Fig. 2. Cole–Cole plot of the real and imaginary parts (21) and(22) of the susceptibility v!x" associated with the Mittag–Le"errelaxation, in comparison to the semicircular shape corre-sponding to the exponential relaxation pattern, a # 1 (- - -). Thevalues for a are, from the bottom curve, 1/4, 1/2, 3/4.

R. Metzler, J. Klafter / Journal of Non-Crystalline Solids 305 (2002) 81–87 85

Fig. 11. Comparison of the trajectories of a Brownian or subdi!usive random walk (left) and a LeH vy walk with index!"1.5 (right). Whereas both trajectories are statistically self-similar, the LeH vy walk trajectory possesses a fractaldimension, characterising the island structure of clusters of smaller steps, connected by a long step. Both walks are drawnfor the same number of steps (approx. 7000).

The solution of the FDE (58) in (x, t) space can again be obtained analytically by making use ofthe Fox functions, the result being [195,217]

=(x, t)" 1!"x"

H!"!#"#! "x"

(K!t)!$! "(1, 1/!), (1, 1/2)

(1, 1), (1, 1/2) # . (61)

This is a closed-form representation of a LeH vy stable law, see Appendix C for details. For lim!!#,

the classical Gaussian solution is recovered, by standard theorems of the Fox functions. Asexpected, one can infer from Eq. (61) the power-law asymptotics [47,217]

=(x, t)& K!t"x"!"! , !(2 , (62)

typical for LeH vy distributions. Due to this property, the mean squared displacement diverges:

#x#(t)$PR (63)

R. Metzler, J. Klafter / Physics Reports 339 (2000) 1}77 27

Lévy Flights

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.

14Metzler, R. and J. Klafter. Physics Reports. 399: 1-77. (2000)

“Whereas both trajectories are statistically self-similar, the Lévy walk trajectory possesses a fractal dimension, characterising the island structure of clusters of smaller steps, connected by a long step.”

Brownian Diffusion

15

Questions...

Solving Mittag-Leffler

• Governing equation

• Laplace transform governing equation

• From identities

• Rearrange to obtain

16

dTn t( )dt

= −λn 0Dt1−αTn t( ) Tn(0) = 1, τ-α ≡ λn

0Dt−αP x,t( )e− ptdt

0

∞

∫ = p−α P x, p( )

ℑdTn t( )dt

⎧⎨⎩

⎫⎬⎭= −λnℑ 0Dt

1−αTn t( ){ }

p Tn p( ) −1 = −λn p

1−α Tn p( ) where τ-α ≡ λn

Tn p( ) = 1p 1+ pτ( )−α( )

identity

0 5 10 15 20 2510!3

10!2

10!1

100

t!

Tn

KWWNuttingM-L

10!4 10!3 10!2 10!1 100 101 102 103 1040

0.2

0.4

0.6

0.8

1

t!

Tn

KWWNuttingM-L

Mittag-Leffler Plots for α = 0.5

17

E1 2 − t τ( )α( ) = e t τ( )erfc t τ( )1 2( )

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

t!

Tn

KWWNuttingM-L

10!4 10!3 10!2 10!1 100 101 102 103 10410!3

10!2

10!1

100

t!

Tn

KWWNuttingM-L

Linear-Linear

Log-Log

Semilog x Semilog y

limt→∞

E1 2 → Γ 1 2( ) t τ( )1 2⎡⎣

⎤⎦−1

KWW:

Nutting:

e− t τ( )1 2

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.”

18

Koch Curve Koch Curve

Ar t i s t ho ld ing a picture of himself holding a picture of himself, holding a picture of himself, holding...

Polymer Chains

• Rescaling each image is analogous to multiplying the features of the image by a single constant. http://en.wikipedia.org/wiki/Scale_invariance

Doi, E. “Introduction to Polymer Physics.” Oxford. 1997.

N chains of length b

N/λ chains of length λ1/2b

rescale to

〈R2〉= Nb2 goes to〈R2〉= N/λ(λ1/2b)2,

and is unchanged.

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.

19

f(x) = xn where n is a constant

f(sx) = (sx)n = Cxn where s, C are constants

therefore

0 20 40 60 80 1000

5

10

15

20f (x) = xn n = 0.5

x

f(s

x)

s = 1s = 2s = 3

10!1 100 101 10210!1

100

101

102f (x) = xn n = 0.5

x

f(s

x)

s = 1s = 2s = 3