Free-energy landscape for cage breaking of three hard disks

We investigate cage breaking in dense hard-disk systems using a model of three Brownian disks confined within a circular corral. This system has a six-dimensional configuration space, but can be equivalently thought to explore a symmetric one-dimensional free-energy landscape containing two energy minima separated by an energy barrier. The exact free-energy landscape can be calculated as a function of system size by a direct enumeration of states. Results of simulations show the average time between cage breaking events follows an Arrhenius scaling when the energy barrier is large. We also discuss some of the consequences of using a one-dimensional representation to understand dynamics through a multidimensional space, such as diffusion acquiring spatial dependence and discontinuities in spatial derivatives of free energy.

Figure 1

(a) Three distinguishable hard disks are confined within a circular corral of fixed size. The variables $h$ and $\theta$ are macrostate variables, to be defined later. Dashed lines are lines of constant $h$ and $\theta$. (b) A 2D slice through the 6D configuration space with constant $({\vec{r}}_1,{\vec{r}}_2)$. The light gray (yellow) region is accessible to the center of disk 3, whereas the dark gray region is energetically forbidden. Upper and lower dashed lines represent possible system macrostates at fixed $h$, where the length of the line determines the entropy of the state.

Figure 2

Trajectories through $h$ space for (from top to bottom) $\epsilon =0.10,0.20,\mathrm{and}0.61$. Dotted lines pass through $h=0$ and dashed lines pass through $h=\pm 2$. The length of time between crossing events is clearly a sensitive function of $\epsilon$. Small fluctuations about $h=0$ (such as those in the boxed regions for $\epsilon =0.20$) are not considered as true cage-breaking events.

Figure 3

Mean-square displacement in $h$ space for systems with (from bottom to top, dark gray to light gray) $\epsilon =0.045,0.06,0.10,0.15,0.25,0.40,0.61,1.0,2.0,\mathrm{and}4.0$. The dashed line has a slope of 1.

Figure 4

(a) Short-time diffusion coefficient. (b) Short-time diffusion coefficient scaled by squared system size. (c) Average time between cage-breaking events. Error bars are calculated from the statistical uncertainty based on the number of events. Where not visible, error bars are smaller than the symbol. (d) Height of the free-energy barrier as a function of $\epsilon$. Circles are the results from simulations and the solid line is from the calculations described in Sec. 5. The dashed line grows as $\ln \left[{\epsilon }^{-7/2}\right]$ and shows scaling behavior as $\epsilon \to 0$.

Figure 5

Comparison between theoretical energy landscapes (solid lines) and results from simulations (symbols). Symbols denote the following: circles, $\epsilon =0.06$; squares, $\epsilon =0.25$; and triangles, $\epsilon =1.00$.

Figure 6

Symbols denote the measured relaxation time as a function of theoretical free-energy barrier height. The solid line is $378\text{exp}\left(\beta F_B\right)$, with $\beta =0.98\pm 0.04$, and comes from a weighted fit to data with $F_B\gtrsim 7.0$. For these larger barrier heights, transition times grow consistently with Arrhenius theory to within statistical error.

Figure 7

First derivatives of $F\left(h\right)$ for $\epsilon =$ 0.10 (solid line), 0.31 (dotted line), and 0.62 (dashed line). Discontinuities exist at $h=\pm 2$. Additionally, a discontinuity exists in the second derivative at $h=\pm \sqrt{3}$, as indicated by the arrow.

Figure 8

Spatially dependent diffusion coefficients for (from bottom to top) $\epsilon =0.31,0.70,\mathrm{and}1.40$ for $h>0$. The inset shows a closer view of the region near the kink in $F\left(h\right)$.

Figure 9

(a) Free-energy landscape in terms of $\theta$ for $\epsilon$ $=$ 0.10 (solid line), 0.31 (dotted line), and 0.62 (dashed line). (b) Heights of energy barriers for all $\epsilon$ in $\theta$ space versus the barrier height in $h$ space, calculated as described in the text. Error bars indicate the uncertainty in local quadratic fits of $n\left(0\right)$ and $n_0$. The dotted line is $F_B\left(\theta \right)=F_B\left(h\right)$.

Figure 10

First derivative of $F\left(\theta \right)$ for $\epsilon$ $=$ 0.10 (solid line), 0.31 (dotted line), and 0.62 (dashed line). All first derivatives exhibit a subtle kink at $\theta =\pm \pi /3$ indicated by the arrow. The horizontal dash-dotted line is at $dF/d\theta$ $=$ 0 and helps illustrate the gradual movement of the minimum to smaller values of $\theta$ for increasing $\epsilon$.

Figure 11

The radius of the solid outer circle is $R_C=R+r$ and is the physical boundary, but the centers of the confined disks can exist only a distance less than or equal to $R$ from the center, indicated by the dashed inner circle.

Figure 12

Case B.

Figure 13

Case C.

Figure 14

Case D.

Figure 15

Configurations where $\theta \approx 0$ (top and middle) and $\theta \approx \pi$ (bottom). Arc lengths are proportional to the number of states $n\left(\delta \theta \right)$ and $n(\pi -\delta \theta )$.