Team Project 3Analysis of Granular SystemsAdvisor – Dr. Paul V. QuinnT.A. – Justin HotchkissDaniel Farnoly, Julie Wu, Hansun Hsiung, Tiffany Leung, William Nicoll, David Kelley, Benjamin Golub, Sonya Nikolaidis,Adam Shpigel, Jeremy Pfund
MORE INFO • These particles display interesting behavior • Solid-like and fluid-like nature • This has led to much scientific study • Use of Computer Simulations
Theory The fun stuff
Relating Density and Height • Under gravity, pressure varies with vertical height • Pressure function for hard spheres • Used to find density profile
Two dimensions: • Three dimensions:
Condensation • Solid layers form at critical temperature • Critical temperature found by:
Packing Density • Square packing for solid layers in one dimension • Density of solid layers ρ=π/4 → Φ=1
Phase Transition of Granular Systems • Possibility of first-order transition • Implications Ehrenfest Classification of Phase Transitions First-Order Second Order => undefined at a certain point. => undefined at a certain point.
Phase Transition of Granular Systems • The total energy of the granular system • Using the First and Second Laws of Thermodynamics,
Phase Transition of Granular Systems • If undefined, then undefined. • T < TC => • T > TC =>
Event Driven Simulations • Hard-spheres in a temperature reservoir • Simulated using FORTRAN • “Event Driven” momentum calculations at varying time steps • Did multiple runs for varying μ
More on Event-Driven Simulation • Program calculates • KEtot • Position • density • <z> • <z> v. T plotted • Using T=KE • Using T fit from density profile
Molecular Dynamics Simulation • Temperature reservoir can simulate vibrating plate • Vibrational Strength
Soft-Sphere Model • Fixed value of Δt • Leonard Jones Potential:
The Kink • Obtain average center of mass, <z> • Density equation is fit to data to obtain T • Plot <z> v. T for different μ
Visual Representation • Designed using Visual Basic • Displays particles and their movements in time • Observer can clearly visualize the system
Program Visual Representation of Simulation
How can we get kinki pictures? • Computational approach: measuring the kinetic energy using ½ m < v2> (This takes forever. We could have been doing it since the chalcolithic era.) • Fitting the density profiles – get T from ζ(Φ)
Theoretical Way • Boltzman: T = 2.5 E -7 • Enskog: T= 2.5 E -7 This is the 2D temperature!
More proof of a kink This makes me so damn excited.
CONCLUSIONS Why was he so damn excited?
First Order Transition Goes from solid-expansion parabolic to liquid-expansion linear function.
Works Like 2D density profile -Boltzman -Hard-sphere
Only box size changes. 1D and 2D can be compared because the mechanics are the same.
Modeling Granular Systems Can use a known, simple theory to model more complex systems.
New Tasks for a New Millennium • Phase transitions in two and three dimensional systems. • Does a three dimensional density profile fit a one dimensional system? • Explore the vibrating wall system in more detail.
Future applications of granular systems… • Studying earthquakes
More applications... • Mixing and processing substances in industry (pharmaceuticals)
Even Breakfast Cereals The packing of breakfast cereals requires the efficient packing of granular units
SURACE!!!!!! AAAHHH I think I’ll call him Mini-Surace (double points)