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NONLINEAR DOUBLE DAY, Sevilla, May 17-18, 2004 Macroscopic effects of anharmonic excitations Discrete breathers for understanding low temperature reconstructive tranformations JFR Archilla, J Cuevas, JM Trillo and F Palmero Nonlinear Physics

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NONLINEAR DOUBLE DAY, Sevilla, May 17-18, 2004 Macroscopic effects of anharmonic excitations

Discrete breathers for understanding low temperature reconstructive tranformations JFR Archilla, J Cuevas, JM Trillo and F Palmero Nonlinear Physics Group (GFNL). University of Sevilla, Spain. http://www.us.es/gfnl

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Known facts

LTRT can be described by:

Breaking of the Arrhenius law
An increase of the reaction speed
A diminution of the activation energy

No explanation has been provided for LTRT Mackay and Aubry [Nonlinearity, 1994] suggested the breaking of Arrhenius law as a consequence of discrete breathers LTRT take place in the presence of a cation layer

2 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Discrete breathers hypothesis

Adventurous but worth trying Objectives:

Calculate 2D breathers in the cation layer for mica muscovite
Obtention of their energies
Are those energies enough to provide the increase of the reaction rate?

Problems:

Choose a vibration mode
Construct the model
Obtain the parameter values

3 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Mode: vibration of the K+ perpendicular to the cation layer

Geometry of the K+ layer

a1 a2 (n,n) (n,n+1) (n,n−1) (n−1,n−1) (n−1,n) (n+1,n−1) (n+1,n) 4 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Mathematical model

Hamiltonian H =

2m ˙ u2

n + V (u

n) + 1

2k

n′∈NN

(u

n − u n′)2

Harmonic coupling
k=10±1 N/m
D. R. Lide Ed., Handbook of Chemistry and Physics (CRC press 2003-2004)

On-site potential V

Assignment of far infrared (30-230 cm−1) bands through dichroic experiments,

[Diaz et al, Clays and clay Miner.71, 701]

Linear frequency ωo=143cm−1
Nonlinearity of the potential unknown

5 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Infrared spectrum at LADIR-CNRS

200 250 300 350 400 450 500 2000 4000 6000 8000 10000 12000 14000

cm−1

Arb. units

Muscovite far infrared transmission spectrum

6 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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On site potentials

Fitting of the on-site potentials to the observed IR bands
Method: Crystal structures and proton dynamics in potassium and cesium hy-

drogen bistrifluoroacetate salts with strong symmetric hydrogen bonds A Cousson, JFR Archilla, J Tomkinson and F Fillaux, submitted physics/0404083.

−3 −2 −1 1 2 3 −200 200 400 600 800 1000 1200

x (Amstrongs) E (cm−1) 7 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Choice of the on-site potential

−3 −2 −1 1 2 3 −200 200 400 600 800 1000 1200

x (Amstrongs) E (cm−1)

V (x) = D(1 − exp(−b2x2)) + γx6
D=453cm−1, b2=36 ˚

A−2, γ=49884cm−1˚ A−6

Consistent with the space available to the K+: 2×1.45˚

A

8 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Breather profiles

Energy density profiles for breathers

5 10 15 5 10 15 0.5 1

nx ny Energy density (KJ/mol)

5 10 15 5 10 15 2 4 6

nx ny Energy density (KJ/mol)

νb = 164.15 cm−1 E=22.82 KJ/mol νb = 142.38 cm−1 E=28.75 KJ/mol

9 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Breather energies

Energy with respect to the frequency

130 140 150 160 170 20 25 30 35 40

Breather frequency (cm−1) Energy (KJ/mol)

10 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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Decrease of the activation energy

Arrhenius law: Reaction rate: k = A exp(− Ea RT ) Breather energy: ∆Ea New reaction rate: k′ = A exp(−Ea − ∆Ea RT ) Increase of the reaction rate k′ k = exp(∆Ea RT ) Several hundreds of times faster

11 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004

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CONCLUSION

Breathers in the K+ layer have enough energy to provide the observed increase of the reaction speed

PROBLEMS

Quantum breathers? Is the band assignment correct? Mechanism for the transmission of energy? Discrete breathers for understanding reconstructive mineral processes at low tem- peratures, J Cuevas, JFR Archilla, MD Alba, M Naranjo and JM Trillo. Submitted, arXiv:nlin.PS/0404030.

12 JFR Archilla, J Cuevas, JM Trillo and F Palmero, Nonlinear Double Day, Sevilla, 2004