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.


1 Answer 1


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.

  • $\begingroup$ Also, I don't think anybody would have computed a value for the expected RSA saturation density for any specific size finite box. $\endgroup$ May 19, 2013 at 17:24
  • $\begingroup$ @Yoav Kallus Right, I just mentioned the unit cube for the sake of defining a geometry, and without defining the radius of the spheres being packed therein. $\endgroup$
    – BlueLight
    May 19, 2013 at 18:08
  • 1
    $\begingroup$ The shape of the container should not make a difference in the limit of a very large container. $\endgroup$ May 19, 2013 at 18:45
  • $\begingroup$ @Yoav Kallus Actually that's a good point. $\endgroup$
    – BlueLight
    May 19, 2013 at 19:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.