$\alpha_2$ in Monodisperse Systems

November 7, 2014

Wendell Smith

Results for the Physics Paper

  • $\alpha_2 \le 1.6$ for equilibrated systems
    • monodisperse and possibly bidisperse
  • This maximimum occurs at the glassy transition
  • Its higher for unequilibrated systems

Recent Data

$(N=100)$

  • Long simulations that are more definitively equilibrated
    • Run for 4–7 days
  • MSD has a classic shape
    • Almost linear for small $\phi$
    • Plateau region for larger $\phi$

Fourth Power

$\left< x^4 \right>$

  • Ratio of $\left< x^4 \right>$ to MSD determines $\alpha_2$:
    $$\alpha_2 = \frac{\left< x^4 \right>}{3 \left< x^2 \right>^2 }- 1$$
  • $\left< x^4 \right>$ looks similar to $\left< x^2 \right>$

Fourth Power

Without the time-component

  • Filled area is between
    $3\left< x^2 \right>$ and $\left< x^4 \right>$
  • Area corresponds to $\alpha_2$
  • $\alpha_2 = \frac{\left< x^4 \right>}{3 \left< x^2 \right>^2 }- 1$

$\alpha_2$

  • Goes up to $1.6$, but no higher
  • At higher densities, this is increasingly difficult to measure
  • As density increases, we expect $\alpha_2$ to remain under $1.6$

Relaxation Time

the duration required for a particle to move one diameter

  • Should scale exponentially with $\Delta \phi$
  • At higher densities, this is "washed out" by center-of-mass motion

Next Steps

Short-Term

  1. Change $\Delta \phi$ incrementally to see smooth transitions
  2. Track crystallization
  3. Look at bidisperse systems
  4. Fix ISF to not include center-of-mass motion