11

Chaotic Chaotic properties of nonequilibrium properties of nonequilibrium liquids under planar elongational flowliquids under planar elongational flow

Federico FrascoliFederico FrascoliPhD candidatePhD candidate

Centre for Molecular SimulationCentre for Molecular Simulation

Swinburne University of Swinburne University of TechnologyTechnology

Melbourne Melbourne

22

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

Main aspects of the modelMain aspects of the model

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

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

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

33

Lees-Edwards pbcsLees-Edwards pbcs

xu t xu t

44

PlanarPlanar elongational flow elongational flow

• What is planar elongational flow?What is planar elongational flow?

0 x

y

y x

x

y

0

Initial unit Initial unit latticelattice

Rotated unit Rotated unit latticelatticex

y

y x

55

Kraynik-Reinelt (KR) conditionsKraynik-Reinelt (KR) conditions

0

66

Lyapunov exponentsLyapunov exponents

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

1

2

t1

t2

(t1) (t2)

77

Conjugate pairing rule (CPR)Conjugate pairing rule (CPR)

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

88

Conjugate Pairing rule for PEFConjugate Pairing rule for PEF

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

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

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

99

0.0 0.2 0.4 0.6 0.8 1.0-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

8 particles 32 particles

PEF IK at

m

ax +

m

in)

-

i +

N-i+

1)

scaled pair-number

Differences for PEF at rates 0.5, 1.0.Differences for PEF at rates 0.5, 1.0.

0.0 0.2 0.4 0.6 0.8 1.0-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

PEF IK at

m

ax +

m

in)

-

i +

N-i+

1)

scaled pair-number

8 particles 32 particles

1010

Differences for PSF at rates 1.0, 2.0.Differences for PSF at rates 1.0, 2.0.

0.0 0.2 0.4 0.6 0.8 1.0-0.035

-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

m

ax +

m

in)

-

i +

N-i+

1)

scaled pair-number

8 particles 32 particles

PSF IK at

0.0 0.2 0.4 0.6 0.8 1.0

-0.12-0.10-0.08-0.06-0.04-0.020.000.020.040.060.080.100.120.140.160.18

scaled pair-number

8 particles 32 particles

PSF IK at

m

ax +

m

in)

-

i +

N-i+

1)

1111

•The Max deviation for PEF is The Max deviation for PEF is 1%1% for 32 particles. for 32 particles. •The Max deviation for equivalent The Max deviation for equivalent shearshear flow flow

systems is instead systems is instead 10%10%..

•For PEF, the Avg. deviation for For PEF, the Avg. deviation for 8 particles8 particles is is 1.4%1.4% and and 2.1%2.1% respect to the sum of max and min respect to the sum of max and min exponents. exponents.

•For PEF, the Avg. deviation for For PEF, the Avg. deviation for 32 particles32 particles is is 0.4%0.4% and and 0.3%0.3%.. This is a clear indication of a size This is a clear indication of a size dependence.dependence.

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

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Constant Constant TT and constant and constant PP

•Experimental results for PEF fluids (polymer Experimental results for PEF fluids (polymer melts, colloids, etc.) are melts, colloids, etc.) are predominantlypredominantly at constant at constant PP and and TT, not at constant , not at constant TT and and VV..

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

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

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

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Conjugate Pairing rule for Conjugate Pairing rule for NpTNpT

•If we use the avg. pressure from If we use the avg. pressure from NVTNVT runs as a runs as a target pressure for target pressure for NpT NpT runs, Lyapunov spectra of runs, Lyapunov spectra of ((qq,, p p) basically ) basically coincidecoincide. This is expected, as we are . This is expected, as we are simulating the simulating the samesame state point. state point.

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

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

1414

Values of “Values of “NHNH” exponents for ” exponents for NN=8=8

1515

A “string phase” exists for PSF!A “string phase” exists for PSF!

-8 -4 0 4 8-8

-4

0

4

8

Z

Y

1616

An open issueAn open issue

1E-3 0.01 0.1 1 100.0

0.5

1.0

1.5

2.0

2.5

3.0

NpT NVT

where are where are strings?strings?

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

1717

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

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

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

1818

AcknowledgementsAcknowledgements

• Prof.Prof. B. D. ToddB. D. Todd, ,

Centre for molecular simulationCentre for molecular simulation

Swinburne University of Technology, VIC.Swinburne University of Technology, VIC.

• A/Prof.A/Prof. D. J. Bernhardt (nee Searles)D. J. Bernhardt (nee Searles),,

Nanoscale Science and Technology Centre,Nanoscale Science and Technology Centre,

School of Biomolecular and Physical Sciences,School of Biomolecular and Physical Sciences,

Griffith University, QLD.Griffith University, QLD.