# Is there an "accepted" jamming limit for hard spheres placed in the unit cube by random sequential adsorption?

I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" $\theta_{spheres}$ is achieved s.t. no further spheres can be placed inside of the cube. Is there an accepted range of values for $\theta_{spheres}$ in the literature?

For circular discs in the unit square, we can find a value of $\theta_{circles} \approx 0.547 +/- 0.003$ in the literature$^{1,2}$. See this ( Packing density of randomly deposited circles on a plane ) MathOverflow question and:

1. Hinrichsen, E.L., Feder, J., Jøssang, T. Geometry of random sequential adsorption. Journal of Statistical Physics 44(5-6), pp. 793-827 (1986).

2. Cadilhe, A., Araujo, N.A.M., Privman, V. Random sequential adsorption: from continuum to lattice and pre-patterned substrates. J. Phys. Cond. Mat. 19, 065124 (2007).

However, I'm having trouble finding such a range for the 3D case with spheres? Also, is there an accepted value for packing spheres of radius $r$ into a larger unit sphere?

In terms of running simulations, I seem to be hitting a "soft" wall around a value of $\theta_{spheres} \approx 0.335$ for packing $r = 0.02$ radius spheres in a unit cube. I say "soft" because further spheres can still be added with (what seems like) an exponentially growing number of attempts at placement.

The process you describe is usually called Random Sequential Addition (RSA). In this paper, Torquato, Uche, and Stillinger compute the saturation density up to $d=6$ (see Table I).
For $d=3$ they have $\phi_s\approx 0.38278$ as the saturation density.