Chaotic properties of nonequilibrium liquids under planar elongational flow

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Chaotic properties of nonequilibrium liquids
under planar elongational flow

• Federico Frascoli
• PhD candidate
• Centre for Molecular Simulation
• Swinburne University of Technology
• Melbourne

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Main aspects of the model

• A liquid system of atoms subject to PEF is
  simulated via the SLLOD algorithm of NEMD. This
  translates boundary driven effects into
  external mechanical fields.

• Heat is extracted from the system with the use of
  synthetic thermostats (Gauss/NH), so that a
  steady state is reached.

• The atoms interact via a WCA potential a
  truncated and shifted version of the
  Lennard-Jones potential

• Lees-Edwards type of pbcs are necessary to remap
  atoms outside the simulation cell and to preserve
  homogeneity.

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Lees-Edwards pbcs
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Planar elongational flow

• What is planar elongational flow?

Initial unit lattice
Rotated unit lattice
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Kraynik-Reinelt (KR) conditions
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Lyapunov exponents

• Lyapunov exponents measure the mean exponential
  rate of expansion and contraction of initially
  nearby phase-space trajectories.

G1
DG(t2)
DG(t1)
t1
t2
G2
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Conjugate pairing rule (CPR)

• In a system with N degrees of freedom, if we
  order the exponents according to their value and
  form pairs coupling the highest with the lowest,
  the second highest with the second lowest and so
  on, the CPR is satisfied when each sum of pairs
  has the same value.

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Conjugate Pairing rule for PEF

• PEF systems of 8 and 32 particles under an
  isokinetic Gaussian thermostat are considered.
  Results for Isoenergetic dynamics are analogous.

• The differences between the sum of maximum and
  minimum exponents (lmax lmin) and all the other
  sums in the spectrum are used to show if the CPR
  holds.

• A comparison with PSF is given at the same energy
  dissipation rate note that PEF has adiabatic
  Hamiltonian equations, PSF has adiabatic non
  Hamiltonian equations.

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Differences for PEF at rates 0.5, 1.0.
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Differences for PSF at rates 1.0, 2.0.
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• For PEF, the Avg. deviation for 8 particles is
  1.4 and 2.1 respect to the sum of max and min
  exponents.
• For PEF, the Avg. deviation for 32 particles is
  0.4 and 0.3. This is a clear indication of a
  size dependence.

• The Max deviation for PEF is 1 for 32 particles.
  
• The Max deviation for equivalent shear flow
  systems is instead 10.

• CPR is violated by systems under PSF and holds
  for systems under PEF, with small size-related
  effects for very small particle numbers due to
  the fluctuations in the thermostat.

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Constant T and constant P

• Experimental results for PEF fluids (polymer
  melts, colloids, etc.) are predominantly at
  constant P and T, not at constant T and V.

• The Nose-Hoover (NH) mechanism is the preferred
  choice to constrain the average pressure of a
  NESS system.

• The KR conditions and the peculiar evolution of
  the unit cell under PEF require some care when
  the rescaling of cell volume takes place.

• Some methods of (isotropic) rescaling have been
  discussed, in relation to the different ways to
  code the evolution of the unit cell.

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Conjugate Pairing rule for NpT

• The NH mechanism introduces two extra degrees of
  freedom in the system a piston-like coupling
  constant and the system volume V. Two Lyapunov
  exponents are associated to these variables.

• If we use the avg. pressure from NVT runs as a
  target pressure for NpT runs, Lyapunov spectra of
  (q, p) basically coincide. This is expected, as
  we are simulating the same state point.

• The two exponents associated to the NH mechanism
  sum up to zero. Except at high rates of PSF, this
  sum is independent of the type and magnitude of
  the external field applied, and of the value of
  the target pressure.

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Values of NH exponents for N8
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A string phase exists for PSF!
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An open issue

• Results for viscosity of PEF do not show dramatic
  thinning as in PSF at high rates.

where are strings?
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• Enhanced ordering in the fluid is an artefact
  caused by the wrong assumption of a linear
  velocity profile any thermostat interprets the
  deviations from linearity as excess heat to be
  extracted from the system.

• Preliminary results seem to point at the non
  existence of linear strings for PEF.

• May another strings profile (hyperbolic?) exist
  due to the different nature of streamlines of
  PEF, so that viscosity does not drop as in PSF?

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Acknowledgements

• A/Prof. D. J. Bernhardt (nee Searles),Nanoscale
  Science and Technology Centre,School of
  Biomolecular and Physical Sciences,Griffith
  University, QLD.

• Prof. B. D. Todd,
• Centre for molecular simulation
• Swinburne University of Technology, VIC.