Relation Between Grain Shape and Fractal Properties in Random Apollonian Packing with Grain Rotation

We extend the common theme of random Apollonian packing of circles to consider orientable grains with a noncircular shape. Systems of up to ${10}^6$ grains are examined for a range of polygonal and elliptical shapes using both the random Apollonian packing model and the new rotational random Apollonian packing model which takes into account the extra rotational degree of freedom of noncircular grains. We identify the constraining length $D_c$ that limits growth of the grain during the packing process and find that a universal relation exists between grain shape and the scaling properties of the system.

Figure 1

RAP packings of 500 circles (top left) and triangles (top right). RRAP packing of 500 ellipses with aspect ratio $\lambda =0.7$ (bottom left) and RRAP packing of triangles (bottom right). Grains pack more densely in the RRAP model as a result of being able to rotate during the packing process.

Figure 2

The decay of the pore space volume for triangles and pentagons for $n={10}^6$ in both the RAP (filled symbols) and RRAP (open symbols) models. Inset shows inverse cumulate distribution $N(r)$ of the grain sizes. The clear power-law regions verify the validity of Eqs. (1, 2) in both RAP and RRAP models.

Figure 3

Variation of the packing efficiency exponents $\beta$ (filled symbols) and ${\beta }^{\prime }$ (open symbols) as the number of edges of the packed grains increases.

Figure 4

Variation of packing efficiency exponents $\beta$ (filled symbols) and ${\beta }^{\prime }$ (open symbols) for RAP and RRAP packings of ellipses with ellipticities from $\lambda =1$ (circle) to $\lambda =0.1$ (highly elliptical).

Figure 5

Relative sizes of a triangle, square, hexagon, and circle when each grain’s minimum distance $D_{\min }$ (left) and maximum distance $D_{\max }$ (right) from its center to its edge is the same.

Figure 6

Packing efficiency exponents $\beta$ (filled symbols) and ${\beta }^{\prime }$ (open symbols) versus the area of the grain for the RAP model with $D_{\max }=1$ and for the RRAP model with $D_{\min }=1$. Inset shows data rescaled for the ellipses in the RAP model, using $D_{\mathrm{av}}=1$, following the same universal trend. Dashed line is a best fit to the data (slope $m=0.05$, $y$ intercept $c=0.12$).