Vibrations in jammed ibrations in jammed solids: Beyond linear - - PowerPoint PPT Presentation

SLIDE 1

Vibrations in jammed ibrations in jammed solids: Beyond linear solids: Beyond linear response esponse

Thibault Ber Thibault Bertrand trand1

1

Carl F. Schreck1 Corey S. O’Hern1 Mark D. Shattuck1,2

Physics of Glassy and Granular Materials YITP 2013

1 Yale University 2 City College of the City University of New York

SLIDE 2

Nonlinear Effects in Granular Solids

Non-linear effects in real granular packings:

• Breaking existing and forming new contacts
• Non linear interactions (Hertzian)
• Sliding and rolling friction
• Energy dissipation

Isolate the effects of fluctuations in the network of contacts! Nonlinear vibrational properties of granular solids – Vibration dampening, solitary modes, dispersion, deviations from elasticity theory See Carl Schreck’s poster for details on Hertzian interactions

SLIDE 3

Absence of Linear Response

M↵, = ✓ ∂2V ∂r↵∂r ◆

~ r= ~ r0

,

Dynamical Matrix:

Diagonalize the dynamical matrix to access eigenfrequencies:

ˆ ei, i ∈ {1, . . . , 2N} λi = mω2

i

δ ∼ ∆φ1/2

V (rij) = ✏ 2 ✓ 1 − rij ij ◆2 Θ ✓ 1 − rij ij ◆

✏

2✏

SLIDE 4

Absence of Linear Response

1 2kδ2 = T δ = r 2T k

σeff = σ − δ φeff = φ ✓ 1 − δ σ ◆2

φ = φJ ⇣ 1 − q

2T kσ2

⌘2

Need to increase the volume fraction to rejam the system at a given T: Apparent diameter of a particle: Temperature allow particle to explore its surrounding on a distance δ: δ ¡ σ

SLIDE 5

Absence of Linear Response







 

?

φ = φJ ⇣ 1 − q

2T kσ2

⌘2

SLIDE 6

Generating Jammed Packings

SLIDE 7

Beyond the Harmonic Approximation…

N = 20

• Molecular Dynamics

Simulation

• Constant energy
• Linear Spring Repulsion
• Frictionless
• No dissipation
• At t=0, add temperature

SLIDE 8

Non-harmonicity in Disordered Solids

Schreck, ¡Bertrand, ¡Sha.uck, ¡O’Hern, ¡Phys. ¡Rev. ¡Le+. ¡107 ¡(2011) ¡078301 ¡

N ¡= ¡12 ¡ Δφ ¡= ¡10-­‑5 ¡ mode ¡= ¡6 ¡ Pr Protocol:

• tocol:
• Perturb along eigenmode by δ
• Let the system evolve at constant energy
• Study the FT of the particle motion

First ¡contact ¡breaks! ¡

SLIDE 9

Beyond the Harmonic Approximation…

Under har harmonic appr monic approximation:

• ximation:

M = kBTC−1

V = 1 N hvvTi V = kBTI

M = VC−1

Solution 1: Solution 1: probing the correlation of particles displacements via Solution 2: Solution 2: looking for vibrational frequencies emerging in the Fourier Transform of the velocity autocorrelation function via

˜ d(ω) = F[d(t)]

d(t) = PN

i=1hvi(t).vi(0)i0

PN

i=1hvi(0).vi(0)i0

SLIDE 10

Assessing the Vibrational Frequencies







 

0.5 1 1.5 2 10

−6

10

−4

10

−2

10

ω D(ω)

0.5 1 1.5 2 10

−6

10

−4

10

−2

10

ω D(ω)

(a) (b) (b) (a)

SLIDE 11

Assessing the Vibrational Frequencies

0.5 1 1.5 2 10

−6

10

−4

10

−2

10

ω D(ω)

Non trivial evolution of the covariance matrix prediction and Fourier transform of Velocity autocorrelation function w/ T

SLIDE 12

Temperature Dependence of the Frequencies







 

x ¡

ωk(T) = ωd

k +

ω∗

k − ωd k

⇣ 1 + lc(∆φ)/ √ T ⌘ν

SLIDE 13







 

Temperature Dependence of the Frequencies

x ¡

SLIDE 14

Testing Resonance in the Modes

• Drive one particle
• Record average kinetic

energy per particle in steady state

N = 10 ∆φ = 10−8

hKppi

SLIDE 15



 



 



 



 



 



 



 



 

 

 

 

 

 

 

     

   

Rearrangement probability

Packing did not did not rearrange earrange, relate to the same inherent structure Packing rearranged earranged, relate to a different inherent structure 100 snapshot over the course of the simulation

SLIDE 16



 



 



 



 



 



 



 



 

 

 

 

 

 

 

     



  

 





 



 



 



 



 



 



 



 

 

 

 

 

 

 

     



 



 



 



 





     

  

 



Introducing a new Phase Diagram

ziso = dN − d + 1

ICS = Iso-coordinated Solid HCS = Hypo-coordinated Solid HPL = Hard Particle Liquid DL = Dense Liquid

SLIDE 17



 



 



 



 



 



 



 



 

 

 

 

 

 

 

     



  

 



Density of States

          





  

           

 

            

  

x ¡ x ¡ x ¡ ICS ¡ HPL ¡ HCS ¡

SLIDE 18

Conclusions & Future directions

• No linear response for a wide range
• f parameters
• Need of a new description for the

vibrational dynamics of jammed packings

• Transition from resonant to non-

resonant modes

• Investigating effect of friction,

particle shape and order “Vibrations in jammed solids: Beyond linear response”, T.Bertrand, C.F.Schreck, C.S.O’Hern and M.D.Shattuck, submitted to PRL (arXiv:1307.0440)

SLIDE 19

Acknowledgements

Grants: DTRA Grant No. 1-10-1-0021 NSF MRSEC DMR-1119826

Thank you!

• Corey O’Hern
• Mark Shattuck
• Carl Schreck
• The O’Hern Group
• Yale High Performance Computing