Fractal Kinetics Bruyères-le-Châtel

38
David L. Griscom impactGlass research international Mexico City Paris Tokyo Washington Commissariat à l’Energie Atomique, Bruyères-le-Châtel, France 1 December 2005

Transcript of Fractal Kinetics Bruyères-le-Châtel

Page 1: Fractal Kinetics Bruyères-le-Châtel

David L. Griscom

impactGlass research international Mexico City Paris Tokyo Washington

Commissariat à l’Energie Atomique, Bruyères-le-Châtel, France

1 December 2005

Page 2: Fractal Kinetics Bruyères-le-Châtel

An

Ideal Wedding of a Mathematical Formalism

often written off as “just another method of curve fitting”

with a Remarkable Body of Data

which has defied simple mathematical description,

thus severely limiting its utility for its intended purposes

Page 3: Fractal Kinetics Bruyères-le-Châtel

Acknowledgements

The Experimental data for the Ge-doped-silica fibers were

recorded at the Naval Research Laboratory by E.J. Friebele

with important assistance from M.E. Gingerich, M. Putnum,

G.M. Williams, and W.D. Mack.

A full account of this work has been published:

D.L. Griscom, Phys. Rev. B64, 174201 (2001)

Page 4: Fractal Kinetics Bruyères-le-Châtel

Classical Kinetic Solutions for Radiolytic Defect Creation with

Thermal Decay: Dependencies on Dose Rate

103

104

105

106

107

108

0.01

0.1

1

10

100

1000

Classical Kinetic Solutions:

Red Curves: 2nd

-Order; Small Circles: 1st-Order

340 rad/s

17 rad/s

0.45 rad/s

Experimental Data:

17 rad/s

0.45 rad/s

340 rad/s

Ind

uced

Lo

ss (

dB

/km

)

Dose (rad)

Slope = 1.0

Page 5: Fractal Kinetics Bruyères-le-Châtel

Radiation-Induced Absorption (1.3 m) in Ge-Doped-Silica-Core Optical Fibers:

Failure of Classical Kinetics to Fit Data as Functions of Dose Rate

103

104

105

106

107

108

0.01

0.1

1

10

100

1000

Classical Kinetic Solutions:

Red Curves: 2nd

-Order; Small Circles: 1st-Order

340 rad/s

17 rad/s

0.45 rad/s

Experimental Data:

17 rad/s

0.45 rad/s

340 rad/s

Ind

uced

Lo

ss (

dB

/km

)

Dose (rad)

Slope = 1.0

It is Impossible to Fit

These Data with These Solutions!

Page 6: Fractal Kinetics Bruyères-le-Châtel

•Gottfried von Leibnitz (1695): “Thus it follows that d½x will be equal to xdx:x,

… from which one day useful consequences will be drawn.”

What is Fractal (Fractional) Kinetics?

•I.M. Sokovov, J. Klafter, A. Blumen, Physics Today, November, 2002, p. 48:

“Equations built on fractional derivatives describe the anomalously slow diffusion

observed in systems with a broad distribution of relaxation times.”

•R. Kopelman, Science 241, 1620 (1988).

•Science 297, 1268 (2002): News article on “Tsallis entropy”.

(q 1)

Page 7: Fractal Kinetics Bruyères-le-Châtel

Fractal Kinetics in Brief

Fractal spaces differ from Euclidian spaces by having fractal dimensions df

such that

df < d,

where d is the dimension of the Euclidian space in which the fractal is embedded.

Each fractal also possesses a spectral dimension ds (< df < d), defined by the

probability P of a random walker returning to its point of origin after a time t:

P(t) t-ds/2.

The present work introduces a parameter, ds/2. Thus, for many amorphous

materials, values of 2/3 might be expected ...

– which

serves as a prototype for many amorphous materials.

It is known that ds 4/3 for the entire class of random fractals embedded in

Euclidian spaces of dimensions d 2 , including the percolation cluster

Page 8: Fractal Kinetics Bruyères-le-Châtel

Supercomputer Simulations of Fractal Kinetics Raoul Kopelman, Science 241, 1620 (1988)

A + B AB

Sierpinski “gasket”: df =1.585, ds = 1.365 Percolation Cluster: df =1.896, ds = 1.333

Page 9: Fractal Kinetics Bruyères-le-Châtel

First-order growth kinetics with thermally activated decay.

The classical rate equation for this situation can be written

dN(t)/dt = KDN* - RN,

and its solution is given by

N(t) = Nsat{1 - exp[-Rt]},

where K and R are constants, D is the dose rate, N* is a

number of unit value and dimensions of number density

(e.g., cm-3), and

Nsat = (KD/R)N*.

Rate Equations for Defect Creation under Irradiation

Page 10: Fractal Kinetics Bruyères-le-Châtel

Result of Change in Dimensionless Variable kt (kt)

First-order growth kinetics with thermally activated decay.

The fractal rate equation for this situation can be written

dN((kt))/d(kt) = (KD/R) N* - N

0 < <1 k = R

with solution

N((kt)) = Nsat{1 - exp[-(kt)]},

where Nsat = (KD/R) N*. •

Page 11: Fractal Kinetics Bruyères-le-Châtel

Second-order growth kinetics with thermally activated decay.

The classical rate equation for this situation can be written

dN(t)/dt = KDN* - RN2/N*,

and its solution is given by

N(t) = Nsattanh(kt),

where Nsat = (KD/R)1/2N* and k = (KDR)1/2.

Rate Equations for Defect Creation under Irradiation

• •

Page 12: Fractal Kinetics Bruyères-le-Châtel

Result of Change in Dimensionless Variable kt (kt)

Second-order growth kinetics with thermally activated decay.

The fractal rate equation for this situation can be written

dN((kt))/d(kt) = (KD/R) /2N* - (R/KD) /2N2/N*

0 < <1 k = (KDR)1/2

with solution

N((kt)) = Nsattanh[(kt)],

where Nsat = (KD/R) /2N*. •

• •

Three Fitting

Parameters

Page 13: Fractal Kinetics Bruyères-le-Châtel

Experimental Curves Fitted by Fractal First-Order Solutions Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

1000 10000 100000 10000000.1

1

10

100

0.0009 rad/s

=0.82

17 rad/s

=0.79

0.45 rad/s

=0.94

340 rad/s

=0.71

In

du

ce

d L

oss (

dB

/km

)

Dose (rad)

(Reactor Irradiation)

γ Irradiation

Page 14: Fractal Kinetics Bruyères-le-Châtel

1E-3 0.01 0.1 1 10 100

10

100

(c)

Slope =

Slope = /2

Satu

ration L

oss (

dB

/km

)

Dose Rate (rad/s)

1E-3 0.01 0.1 1 10 10010

-8

10-7

10-6

10-5

10-4

10-3

Linear!!!

(b)Classical 1st-Order Kinetics

Classical 2nd-Order

Kinetics

Rate

Coeff

icie

nt (s

-1)

1E-3 0.01 0.1 1 10 100

0.7

0.8

0.9(a)

Stretched 2nd Order Kinetics

Stretched 1st-Order Kinetics

Pow

er-

Law

Exponent

Fractal-Kinetic Fitting Parameters: Both Kinetic Orders Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Slope = 1

Classical 1st Order

Classical 2nd

Order Fractal 1st & 2nd Order (Slope = 1) Empirical!!!

Fractal 1st & 2nd Order

(Slope Variable)

Annoying Cusp

Annoying

k

Nsat

Dose Rate (rad/s)

Classical 1st Order

Slope = ½

Classical 2nd Order

Page 15: Fractal Kinetics Bruyères-le-Châtel

1000 10000 100000 10000000.1

1

10

100

0.0009 rad/s

=1.0

17 rad/s

=0.70

0.45 rad/s

=1.00

340 rad/s

=0.62

Ind

uce

d L

oss (

dB

/km

)

Dose (rad)

Experimental Curves Fitted by Fractal Solutions (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

“Population B” Included in Fits

Population B (included in all four curve fits)

Dose-Rate-Independent

N.B. The dominant “Population A” comprises

all defects with thermally activated decays

Page 16: Fractal Kinetics Bruyères-le-Châtel

Fractal-Kinetic Fitting Parameters (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Parameters for

dose-rate-

independent

“Population B”

included in fits

for all four dose

rates

1E-3 0.01 0.1 1 10 1001

10

100

(c)

Slope =

Slope = /2

Satu

ration L

oss (

dB

/km

)

Dose Rate (rad/s)

1E-3 0.01 0.1 1 10 10010

-9

10-8

10-7

10-6

10-5

10-4

10-3

Linear

(b)

Rate

Coeff

icie

nt (s

-1)

1E-3 0.01 0.1 1 10 1000.6

0.7

0.8

0.9

1.0

(a)

Pow

er-

Law

Exponent

No More

Annoying Cusp

Here!

Approximately

Straight Line Here

Intended Result of

Introducing

“Population B”:

Rate Coefficient

Is More Perfectly

Linear than Before !!!

k

Nsat

Dose Rate (rad/s)

Slope=/2

Empirical!!!

Happy Colateral

Consequences:

Page 17: Fractal Kinetics Bruyères-le-Châtel

Fractal-Kinetic Fitting Parameters (Second Order) Multi-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Proposed

“Canonical Form”

for

Classical Fractal

Phase Transition

1E-3 0.01 0.1 1 10 100

1

10

100

Slope = 1/2

(c)

Slope = /2

Sa

tura

tio

n L

oss (

dB

/km

)

Dose Rate (rad/s)

1E-3 0.01 0.1 1 10 100

10-7

10-6

10-5

10-4

10-3

Slope = 1/2

(b)

Slope = 1

Ra

te C

oeff

icie

nt

(s-1)

1E-3 0.01 0.1 1 10 1000.6

0.7

0.8

0.9

1.0

(a)

Classical Fractal

Po

we

r-La

w E

xp

on

en

t

Here “Population B”

was contrived to give

classical behavior

below the point where

= 1

k

Nsat

Slope=/2

Slope=1

Slope=1/2

Slope=1/2

Classical Fractal

Dose Rate (rad/s)

Page 18: Fractal Kinetics Bruyères-le-Châtel

1000 10000 100000 10000000.1

1

10

100

C ( = 0.66)B

0.0009 rad/s

=1.0

17 rad/s

=0.61

0.45 rad/s

=0.94

340 rad/s

=0.46

In

du

ce

d L

oss (

dB

/km

)

Dose (rad)

Experimental Curves Fitted by Fractal Solutions (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Dose-Rate-Independent “Populations B and C” Included in Fits

(Reactor Irradiation)

Populations B and C

Page 19: Fractal Kinetics Bruyères-le-Châtel

1E-3 0.01 0.1 1 10 1000.1

1

10

Slope = 1/2

Slope = /2(c)

Satu

ration L

oss (

dB

/km

)

Dose Rate (rad/s)

1E-3 0.01 0.1 1 10 1001E-8

1E-7

1E-6

1E-5

1E-4

1E-3

Slope = 1/2

Slope = 1(b)

Rate

Coeff

icie

nt (1

/s)

1E-3 0.01 0.1 1 10 100

0.5

0.6

0.7

0.8

0.9

1.0

Single-Population Fits

Fits Including Effects

of Populations B & C(a)

Exponent

Fractal-Kinetic Fitting Parameters (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

However… All fits are constrained

by the (questionable)

assumption that the

dosimetry for reactor-

irradiation is equivalent

to that for γ irradiation

vis-à-vis the induced

optical absorption. γ-Rays

Reactor

Irradiation Dose Rate (rad/s)

k

Nsat

Slope=1/2

Slope=1/2

Inclusion of Populations

B & C does not alter the

fundamental result:

There still seems to be a

classical fractal

transition.

Page 20: Fractal Kinetics Bruyères-le-Châtel

Experimental Curves Fitted by Fractal Solutions (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Dose-Rate-Independent “Populations B and C” Included in Fits

1000 10000 100000 10000000.1

1

10

100

C

B

0.011 rad/s

=1.017 rad/s

=0.66

0.45 rad/s

=0.85340 rad/s

=0.52

In

duced

Lo

ss (

dB

/km

)

Dose (rad)

( Irradiation)

Page 21: Fractal Kinetics Bruyères-le-Châtel

Fractal-Kinetic Fitting Parameters (Second Order) Single-Mode Ge-Doped-Silica-Core Fibers, =1300 nm

Now All Fits

Pertain to

γ-Irradiated

Fibers Only.

Classical Fractal Transition

0.01 0.1 1 10 100

1

10

Slope = /2

(c)

Satu

ration L

oss (

dB

/km

)

Dose Rate (rad/s)

0.01 0.1 1 10 10010

-8

10-7

10-6

10-5

10-4

10-3

Slope = 1

(b)

Rate

Coeffic

ient (1

/s)

0.01 0.1 1 10 100

0.5

0.6

0.7

0.8

0.9

1.0

Fits Including Influences

of Populations B and C

Single-Population Fits(a)

Exponent

No Data for Reactor-

Irradiated Fibers Are

Included. Empirical Result

of Fractal Kinetics!

Slope=1

Slope=/2

Dose Rate (rad/s)

k

Nsat

But Caution: May Now Be

Asymptotic to 1.0

as the Dose Rate

Approaches Zero.

? ?

Page 22: Fractal Kinetics Bruyères-le-Châtel

Fractal Kinetics of Defect Creation in Ge-Doped-Silica Glasses:

What Have We Learned by Simulation of the Growth Curves?

==========================================================

Parameters

__________________________________________________

First-Order Solution Second-Order Solution ==========================================================

Specified by k = R k = (KDR)½ New Formalisms

Nsat = (KD/R) Nsat = (KD/R) /2 ______________________________________________________________

Empirically R D R D1/2

Inferred in This Work

K D½ K D1/2

==========================================================

Note:

In classical cases

(=1), both K and

R are constants.

In fractal cases

(0<<1), both K

and R are dose-

rate dependent.

• •

• •

• •

Empirically

Page 23: Fractal Kinetics Bruyères-le-Châtel

Post-Irradiation Thermal Decay Curves and Fractal-Kinetic Fits

for γ-Irradiated Ge-Doped-Silica Core Fibers

1 10 100 1000 10000 100000 10000000

10

20

30

40

50

60

70

80

90

SM Fiber Data

MM Fiber Data

Naive Fractal Second-Order

Prediction from Growth-Curve Fit

(=0.62)

Fractal Second-Order

Best Fit (=0.51)

Fractal 1st-Order

Best Fit

(=0.44)

Fractal Secnd-Order

Best Fit (=0.54)

Naive Fractal

First-Order

Prediction from

Growth-Curve Fit

(=0.71)

Ind

uce

d L

oss (

dB

/km

)

Time (s)

Non-Decaying Component

Fractal Second-Order

Best Fit (=0.54)

Fractal Second-Order

Best Fit (=0.51)

(Equal to

Cumulative

Populations

B and C

Used in

Fitting the

Growth

Curves!)

Page 24: Fractal Kinetics Bruyères-le-Châtel

101

102

103

104

105

106

Time (s)

= 0.66

Fractal 2nd

-Order

Solution

Fractal

2nd

-Order

Solution

(Kohlrausch

Function)

102

103

104

105

106

107

1

10

100

= 0.66

Fractal

1st-Order

Solution

Fractal

2nd

-Order

Solution

Ind

uce

d L

oss (

dB

/km

)

Dose (rad)

Idealized Fractal Kinetics of Radiation-Induced Defect

Formation and Decay in Amorphous Insulators

During Irradiation After Cessation of Radiation

1st

Page 25: Fractal Kinetics Bruyères-le-Châtel

101

102

103

104

105

106

Time (s)

= 0.66

Fractal 2nd

-Order

Solution

Fractal

2nd

-Order

Solution

(Kohlrausch

Function)

102

103

104

105

106

107

1

10

100

= 0.66

Fractal

1st-Order

Solution

Fractal

2nd

-Order

Solution

Ind

uce

d L

oss (

dB

/km

)

Dose (rad)

During Irradiation After Cessation of Radiation

1st

Slope

Slope -

Idealized Fractal Kinetics of Radiation-Induced Defect

Formation and Decay in Amorphous Insulators

Page 26: Fractal Kinetics Bruyères-le-Châtel

100

101

102

103

104

105

106

Factor of 4

No-Adjustable-Parameters

Prediction Based on

Fitted Growth Curve

Time (s)

101

102

103

104

105

106

107

1

10

Data

Fitted Decaying Part

Fitted Non-Decaying

Parts

In

du

ce

d L

oss (

dB

/km

)

Dose (rad)

Fractal Kinetics of Radiation-Induced Defect

Formation and Decay in Amorphous Insulators: The Reality

C

B

B+C

Page 27: Fractal Kinetics Bruyères-le-Châtel

100

101

102

103

104

105

106

Factor of 4

No-Adjustable-Parameters

Prediction Based on

Fitted Growth Curve

Time (s)

101

102

103

104

105

106

107

1

10

Data

Fitted Decaying Part

Fitted Non-Decaying

Parts

In

du

ce

d L

oss (

dB

/km

)

Dose (rad)

C

B

B+C

N.B. These data

prove the existence

of (non-decaying)

dose-rate independent

components.

Fractal Kinetics of Radiation-Induced Defect

Formation and Decay in Amorphous Insulators: The Reality

Page 28: Fractal Kinetics Bruyères-le-Châtel

Fractal kinetics

of optical bands

in pure silica

glass…

0

5000

10000

15000

20000

250006,470 s

102 rad/s 15.3 rad/s

33

120

240

480

960 s 0

Ind

uce

d A

bsorp

tio

n (

dB

/km

)

400 500 600 700 800 900 1000 1100 1200 1300 1400 15000

500

1000

1500

2000

2500

Wavelength (nm)

N.B. These

bands appear

to arise from

self- trapped

holes.

Note absorption in all

three communications

windows.

Page 29: Fractal Kinetics Bruyères-le-Châtel

Growth and Disappearance of “660- and 760-nm” Bands:

Optical Spectroscopy D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175

High-purity, low-OH,

low-Cl, pure-silica-core

fiber (KS-4V) under γ

irradiation for 240 s at

1.0 Gy/s

NBOHCs

760 nm

N.B. These results are

remarkably similar to

those for a low-OH, low-Cl

F-doped silica-core fiber

measured simultaneously.

660 nm

(Bands near 660, 760, and 900 nm are due to self-trapped holes.)

Page 30: Fractal Kinetics Bruyères-le-Châtel

t-1

It appears that

the material is

“reconfigured”

by long-term,

low-dose-rate

irradiation in

such a way that

color centers

(STHs) are no

longer formed,

even when the

irradiation

continues

Loss at 760

nm during

γ irradiation

in the dark

at 1 Gy/s,

T=27 oC

Experimenter-Introduced

“Mid-Course” Transients

Growth and Disappearance of “660- and 760-nm” Bands:

Overview of Kinetics D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175

KS-4V, 760 nm

Page 31: Fractal Kinetics Bruyères-le-Châtel

t-1

It appears that

the material is

“reconfigured”

by long-term,

low-dose-rate

irradiation in

such a way that

color centers

(STHs) are no

longer formed,

even when the

irradiation

continues

– or

is repeated at a

later time.

Growth and Disappearance of “660- and 760-nm” Bands:

Overview of Kinetics D.L. Griscom, Appl. Phys. Lett. 71 (1997) 175

Same Fiber Re-Irradiated

KS-4V, 760 nm

Page 32: Fractal Kinetics Bruyères-le-Châtel

Growth and Disappearance of “660- and 760-nm” Bands:

Dose-Rate Dependence D.L. Griscom, Phys. Rev. B64 (2001) 174201

100

1000

Stretched 2nd Order, =0.96

Stretched 2nd Order: =0.90

Kohlrausch: =0.95

Stretched 2nd Order: =0.53

Kohlrausch: =0.60102 rad/s

15.3 rad/s

102 10

3 10

4 10

5 10

6 10

7

In

duced L

oss (

dB

/km

)

Time (s)

…Dependent

Only on Time

(Not Dose Rate)

at Long Times!

Radiation

Bleaching

Large Initial

Dose-Rate

Dependence

Two Lengths of

Virgin Fiber,

Irradiated

Separately

KS-4V, 900 nm

Page 33: Fractal Kinetics Bruyères-le-Châtel

Growth and Disappearance of “660- and 760-nm” Bands:

Optical Bleaching During Irradiation D.L. Griscom, Phys. Rev. B64 (2001) 174201

100 10001000

10000

(a) Light On

F-Doped

KS-4V

Ind

uce

d L

oss (

dB

/km

)

Time (s)

100 1000

= 670 nm

(b) Light Off

535 rad/s

25 rad/s

Time (s)

is independent of dose rate

in case of KS-4V core fiber.

depends strongly on dose

rate but is independent of

the type of silica in the core. F-doped is slightly different.

Page 34: Fractal Kinetics Bruyères-le-Châtel

Growth and Disappearance of “660- and 760-nm” Bands:

Isothermal Fading (Radiation Interupted), Regrowth D.L. Griscom, Phys. Rev. B64 (2001) 174201

100 1000 10000

1000

10000

(a)

Stretched 2nd Order, =0.71

Kohlrausch, =0.52

In

du

ce

d L

oss (

dB

/km

)

Time after Irradiation (s)

100 1000

(b)

F-Doped-Silica-Core Fiber,

Dose Rate = 102 rad/s

760 nm

Best Fits:

Stretched 2nd Order: =0.45

Kohlrausch: =0.53

660 nm

Best Fits:

Stretched 2nd Order: =0.60

Kohlrausch: =0.69

Irradiation Time (s)

Data Points for

t=0 were Used

in Fitting These

Data.

Fitted Values of

Are Independent

of Wavelength.

Fitted Values of

Are Strongly

Dependent on

Wavelength.

Fading Regrowth

Page 35: Fractal Kinetics Bruyères-le-Châtel

Fractal-Kinetic Fitting Parameters (Both Orders) Multi-Mode Low-OH, Low-Cl Pure-Silica-Core Fibers During Irradiation

Data due to

Nagasawa et al.

(1984) pertain to

a silicone-clad

pure-silica core

fiber.

Gaussian

resolutions were

performed to

extract intensities

of the 660- and

760-nm bands

separately.

My data for F-

doped-silica-clad

pure-silica-core

fiber with an Al

jacket.

Measurements

were made at

fixed wavelengths

of 670 and 900

nm (no Gaussian

resolutions)

The same fiber

was subjected to

the 3 different

dose rates in

progression

beginning with

the lowest. 10 100 1000

1000

10000

Slope=1/2

Polymer-Clad

Silica-Core

Fiber KS-4V Silica-Core

Fiber, Aluminum

Jacketed

Slope=/2

Slope=

(c)

Satu

ration L

oss (

dB

/km

)

Dose Rate (rad/s)

10 100 1000

10-4

10-3

10-2

11/9/00 16:18:58

Weighted Contributions

of Overlapping Bands

660-nm Band

Only

760-nm Band

Only

Slope=0.78(b)

Rate

Coeff

icie

nt (1

/s)

10 100 1000

0.5

0.6

0.7

0.8

0.9

1.0

=670 nm

Initial

Response

Recovery from

Optical Bleaching

Initial Response, =900 nm(a)

Exponent

Slope=0.78

k

Nsat

Page 36: Fractal Kinetics Bruyères-le-Châtel
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ÇA TERMINE MA DEUX-HEURS-LONG PRESENTATION. EST-CE QUE IL Y A DES QUESTIONS ?

C’EST EXPRÉS QUE VOTRE PRESENTATION ÊTRE INCOMPERHENSIBLE? OU EST-CE QUE VOUS AVEZ UN ESPÈS DE INCAPACITÉ «POWER POINT» ?

EST-CE QUE IL Y A DES QUESTIONS SUR LE CONTENU ?

IL FUT DE CONTENU ?

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