1 Chaotic properties of nonequilibrium liquids under planar elongational flow Federico Frascoli PhD candidate Centre for Molecular Simulation Swinburne University of Technology Melbourne

2 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. 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. Main aspects of the model Heat is extracted from the system with the use of synthetic thermostats (Gauss/NH), so that a steady state is reached. 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 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. Lees-Edwards type of pbcs are necessary to remap atoms outside the simulation cell and to preserve homogeneity.

3 Lees-Edwards pbcs

4 Planar elongational flow What is planar elongational flow? What is planar elongational flow? Initial unit lattice Rotated unit lattice

5 Kraynik-Reinelt (KR) conditions

6 Lyapunov exponents Lyapunov exponents measure the mean exponential rate of expansion and contraction of initially nearby phase-space trajectories. Lyapunov exponents measure the mean exponential rate of expansion and contraction of initially nearby phase-space trajectories. 11 22 t1t1 t2t2  (t 1 )  (t 2 )

7 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. 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.

8 Conjugate Pairing rule for PEF The differences between the sum of maximum and minimum exponents ( max + min ) and all the other sums in the spectrum are used to show if the CPR holds. The differences between the sum of maximum and minimum exponents ( max + min ) and all the other sums in the spectrum are used to show if the CPR holds. PEF systems of 8 and 32 particles under an isokinetic Gaussian thermostat are considered. Results for Isoenergetic dynamics are analogous. PEF systems of 8 and 32 particles under an isokinetic Gaussian thermostat are considered. Results for Isoenergetic dynamics are analogous. 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. 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.

9 Differences for PEF at rates 0.5, 1.0.

10 Differences for PSF at rates 1.0, 2.0.

11 The Max deviation for PEF is 1% for 32 particles. The Max deviation for PEF is 1% for 32 particles. The Max deviation for equivalent shear flow systems is instead 10%. The Max deviation for equivalent shear flow systems is instead 10%. 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 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. For PEF, the Avg. deviation for 32 particles is 0.4% and 0.3%. This is a clear indication of a size dependence. 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. 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.

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

13 Conjugate Pairing rule for NpT 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. 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 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. 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. 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. 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.

14 Values of “NH” exponents for N=8

15 A “string phase” exists for PSF!

16 An open issue where are strings? Results for viscosity of PEF do not show dramatic “thinning” as in PSF at high rates. Results for viscosity of PEF do not show dramatic “thinning” as in PSF at high rates.

17 Preliminary results seem to point at the non existence of linear strings for PEF. Preliminary results seem to point at the non existence of linear strings for PEF. 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. 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. May another strings profile (hyperbolic?) exist due to the different nature of streamlines of PEF, so that viscosity does not drop as in PSF? May another strings profile (hyperbolic?) exist due to the different nature of streamlines of PEF, so that viscosity does not drop as in PSF?

18 Acknowledgements Prof. B. D. Todd, Prof. B. D. Todd, Centre for molecular simulation Swinburne University of Technology, VIC. A/Prof. D. J. Bernhardt (nee Searles), Nanoscale Science and Technology Centre, School of Biomolecular and Physical Sciences, Griffith University, QLD. A/Prof. D. J. Bernhardt (nee Searles), Nanoscale Science and Technology Centre, School of Biomolecular and Physical Sciences, Griffith University, QLD.