The Role of Friction and Shear stress in the Jamming Transition Antonio Coniglio Università di Napoli “Federico II” Lorentz Center Leiden 6-10 July 2009 M.Pica Ciamarra, Univ Naples, Italy M. Nicodemi, Univ Warwick, UK R. Pastore Univ Naples, Italy D. Grebenkov Ecole Polytechnique Orsay, France Collaborators:

JAMMING PHASE DIAGRAM Liu, Nagel Nature 2000 ShearstressTemperature1/Density O’Hern, Silbert,Liu,Nagel PRE 2003

Zhang et al, Nature 2009 Jamming in the  plane  The J point is found to influence the behavior at T0

Study the Jamming phase diagram at T= 0 in the density-shear stress plane: OUTLINE a) Effect of the applied shear stress on the Jamming transition for a system of soft frictionless particles. b) Extend the study to soft frictional particles. Conclusions

Schematic behaviour at T=0,  J point) Z – Z iso  (  -  J   P=P 0 (  -  J )  G=G 0 (  -  J )   (  J -  ) -x viscosity P. Olsson, S. Teitl, PRL 99, (2007) C. O’Hern et al., PRE 68, (2003); Number of contacts Pressure Elastic modulus Nd = NZ/2  Z= Z iso = 2d = 6 Mechanical equilibrium of N frictionless grains Number of constraints: N d Number of equations: NZ/2 Z P G

Grebenkov, Pica Ciamarra, Nicodemi, A.C,, PRL 100, ,  JAMMING AT T=0 AND APPLIED SHEAR STRESS Plus viscous force Linear spring-dashpot model 8d z x 16 d y F MD simulations Control parameters: Shear Stress :  Volume Fraction: Φ Pica Ciamarra A.C. 2009, Pica Ciamarra, Pastore, Nicodemi A.C 2009 See also Heussinger and Barrat PRL2009 (2D at fixed shear strain)

MD simulation The system is prepared using the Lubachevsky-Stillinger procedure: Particles are placed into the system with small radii, which are rapidly increased to their final value. The system is allowed to relax until the kinetic energy vanishes. The maximum volume fraction at which the system is able to relax in an unjammed state (zero pressure) for 0 shear stress is found to be :

(Hysteresis) Viscosity Elastic modulus Viscosity and  Elastic modulus at small and large shear stress  small  large  MD simulations Pica Ciamarra, A.C. 2009

Jamming Phase Diagram (No Friction) Pica Ciamarra, A.C Two critical lines ending at J point: 1)Jamming line where the viscosity diverges 2) Unjamming line where the elastic modulus goes to zero. The two lines coincide in the limit of large fluid viscosity

Jamming phase diagram in the limit of large fluid viscosity

Hysteretic behaviour reproduced by a simple model Model: motion of a particle in an energy landscape with many minima W(x) subject to a driving force f and to a viscous force. For semplicity Pica Ciamarra, A.C. 2009

Results from the model V plays the role of f plays the role of 

MD simulation let us fix  very small

Structural signature at the jamming transition Different colours correspond to different values of  Sharp discontinuity only at  0 Results from MD simulations for small values of applied stress , where no hysteresis is present number of contacts P pressure

If particles i and j touch (do not touch) Dynamic signature of the jamming transition Results from MD simulations for small values of applied stress , Where no hysteresis is present K = (0, 2  d, 0)

Jamming at T=0 and finite shear stress (zero friction) Summary The applied shear stress s induces hysteretic effect, resulting in two lines where the viscosity diverges and the elastic modulus goes to zero. For small values of s, where the hysteris is negligible, the jamming transition, exhibits a continuous transition in one time quantities, (contact number Z, Pressure P ), while the asymptotic value of two time correlation functions jumps discontinuously. This glass-like transition converge to the jamming point in the limit of s = 0. The relaxation time and the viscosity diverge with a power law, therefore it is easier to locate the transition, compared to the case where the temperature is the control parameter.

Hernan Makse Talk Song,Wang, Makse Nature  Jamming for frictional particles Phase diagram of jamming Jammed states lie only within the yellow triangle. The range of  where jammed states are allowed increases with friction Silke Henkes Talk Somfai et al PRE 2007 Shundiak, van Hecke, van Saarloos PRE 2007

Grebenkov, Pica Ciamarra, Nicodemi, A.C. PRL 100, , Pica Ciamarra, Pastore, Nicodemi, A.C LzLz Linear spring-dashpot model Coulomb friction coefficient   8d z x 16 d y F Control parameters: Shear Stress :  Volume Fraction: Φ Coulomb friction:  MD simulations The values of the control parameters are chosen such that no hysteresis is present JAMMING FOR FRICTIONAL PARTICLES UNDER SHEAR

Jamming Phase Diagram Three lines separating four regions. MD simulations

Velocity of the plate in the four regions      

flow Flow and Jam

Shear velocity v shear and contact number Z fluctuates but are highly correlated Mechanism for Flow and Jam When the contact number Z reaches a value Zc close to 4, the system jams and shear velocity is zero Two different initial configurations (bleu and red) for the same fixed value of f=0.625 and s=0.002 Z Pastore et al 2009

Data collapse close to the jamming line Data colapse close to the jamming value Z jam for three different values of Φ Fixed value of F

The flow and jamming region is found to shrink when the number of particles gets larger, due to the decrease of fluctuations of the contact number Z. However the flow and jamming region is present even when the number of particles N is of the order of

Flow and Jam region shrinks with the system size

 jam slip and jam

friction  

 =0.1 Contact Number Pressure Elastic modulus Structural properties Red symbols  correspond to the Jammed phase.

Jamming phase diagram  at  for frictional particles   

Jamming at T=0, finite shear stress (finite friction) The presence of friction leads to a complex phase diagram with four distinct regions, separated by three lines. In the limit of zero friction the three lines end at the J point The behaviour of the structurural properties change drastically in the presence of friction. SUMMARY