The effect of elastic vibrations on collisions of fine powder with - - PowerPoint PPT Presentation

The effect of elastic vibrations

• n collisions of fine powder with walls

Ryo Murakami and Hisao Hayakawa Yukawa Institute for Theoretical Physics Kyoto University Physics of Granular Flows Yukawa Institute for Theoretical Physics, Kyoto, June 23-July 6, 2013

Contents

Introduction Recent experiments and simulations of collisions of granular particles Model The elastic wave equation and the wall potential Results The energy stored in the vibration is transformed into translational energy. Discussion Perturbation theory Conclusion

Introduction

Collisions of Granular Particles

Restitution Coefficient Vibration : store and release Nonlinear function

Collisions of Granular Particles

Restitution Coefficient Vibration : store and release Nonlinear function steel, 6mm Glass 40 cm

• F. Müller, M. Heckel, A. Sack and T. Pöschel,
• Phys. Rev. Lett. 110, 254301 (2013).

Experiment

Super Rebounds

！

Super Rebounds Breaking the second law?

• H. Kuninaka and H. Hayakawa,
• Phys. Rev. E 79, 031309 (2009).

Molecular Dynamics

Two Approaches

Continuum Model Molecular Dynamics Focus on Computational Cost System Many-Particle Continuum Microscopic structures Depending on the size Independent of the size Macroscopic motions larger than 100 nm

• H. Kuninaka and H. Hayakawa,

PRE 86, 051302 (2012)

Previous Studies

Our Study Previous Studies Attraction Dimension 2D 3D × ◯

• F. Gerl and A. Zippelius,
• Phys. Rev. E 59, 2361 (1999).
• H. Hayakawa and H. Kuninaka,
• Chem. Eng. Sci. 57, 239 (2002).

Fast ~ Sound velocity / 10 Ultra-slow ~ Thermal velocity Impact Velocity Viscosity × ◯ Collision with Wall Wall, ball

Model

Elastic Wave Equation

Elastic Wave Equation Divergence term Rotation term

: Vertical sound velocity : Horizontal sound velocity

Elastic Wave Equation

Elastic Wave Equation Stress Free Solutions Spheroidal modes Divergence term Rotation term Breathing Quadrupole Dipole

: Vertical sound velocity : Horizontal sound velocity : Principal quantum number : Azimuthal quantum number : Magnetic quantum number

Elastic Wave Equation

Elastic Wave Equation Stress Free Solutions Spheroidal modes Neglected No contribution in head-on collisions Torsional modes Divergence term Rotation term Breathing Quadrupole Dipole

: Vertical sound velocity : Horizontal sound velocity : Principal quantum number : Azimuthal quantum number : Magnetic quantum number

Wall Potential

Modified Lennard-Jones

• A. Awasthi et al., Phys. Rev. B 76, 115437 (2007).
• P. M. Agrawal et al., Surf. Sci. 515, 21 (2002).

Wall Potential Equation of motion

: Mass density : Mass : Number density : Cohesive parameter

: borrow from copper

Wall Potential

Modified Lennard-Jones

• A. Awasthi et al., Phys. Rev. B 76, 115437 (2007).
• P. M. Agrawal et al., Surf. Sci. 515, 21 (2002).

Wall Potential Equation of motion

: Mass density : Mass : Number density : Cohesive parameter

: borrow from copper Center of mass

Initial Conditions

Center of Mass Distribution of vibrational modes Fix : at the position V = 0 Control : 0.0001 ~ 0.1 sound velocity Using normal random number

Results

Restitution Coefficient vs Impact Velocity

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.0001 0.001 0.01 0.1 e v0 / c(t) R = 1 µm R = 100 nm

: No reduce Cohesive Parameter Temperature

Restitution Coefficient vs Impact Velocity

0.98 0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1e-06 1e-05 0.0001 0.001 0.01 0.1 e v0 / c(t) R = 1 µm R = 100 nm

Cohesive Parameter Temperature

Restitution Coefficient vs Impact Velocity

0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1e-06 1e-05 0.0001 0.001 0.01 e v0 / c(t) R = 1 µm

Cohesive Parameter Temperature

Restitution Coefficient vs Impact Velocity

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 0.0001 0.001 0.01 e v0 / c(t) R = 100 nm

Cohesive Parameter Temperature

Excitation Energy

: Excitation energy of each mode : Initial kinetic energy

• 0.001

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 500 1000 1500 ∆Hnlm / H0 i = (n, l, m) R = 100 nm, v0 = 0.0001c(t)

Cohesive Parameter Temperature

Excitation Energy

: Excitation energy of each mode : Initial kinetic energy Quadrupole mode

• 0.001

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 500 1000 1500 ∆Hnlm / H0 i = (n, l, m) R = 100 nm, v0 = 0.0001c(t)

Cohesive Parameter Temperature

Restitution Coefficient vs Initial Phase

R = 100 nm, v0 = 0.0001c(t) 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03

• π
• π/2

π/2 π e α020

Cohesive Parameter Temperature : Amplitude : Phase

Discussion

Perturbation Theory

: Mass : Radius : Initial velocity Unit Expansion 0th order 1th order 2th order

Perturbation vs Simulation

5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05

• 300
• 200
• 100

100 200 300 ∆H020 / H0

: Simulation Perturbation

Phase Dependence

:

Averaging out except Energy conservation : Excitation energy : Initial kinetic energy

Two Ball Collisions

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.05 0.1 0.15 force displacement Two-ball ball-wall

Conclusion

Simulation Super rebounds are found when the radius, the temperature and the velocity are 100 nm, 300 K and 10-4 sound velocity, respectively. The quadrupole mode is the most excited in this condition. Sinusoidal structure is found in the restitution coefficient as a function

• f the initial phase of the quadrupole mode.

Perturbation theory The perturbation theory is good agree with our simulation when the initial velocity is lower than 10-5 sound velocity. The sinusoidal structure of the restitution coefficient is derived using this theory.

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.1 0.2 0.3 0.4 ∆H000 / H0 v0 g = 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.1 0.2 0.3 0.4 ∆H020 / H0 v0 g = 0