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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.

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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.

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  • $\begingroup$ Also, I don't think anybody would have computed a value for the expected RSA saturation density for any specific size finite box. $\endgroup$
    – Yoav Kallus
    Commented 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
    Commented May 19, 2013 at 18:08
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    $\begingroup$ The shape of the container should not make a difference in the limit of a very large container. $\endgroup$
    – Yoav Kallus
    Commented May 19, 2013 at 18:45
  • $\begingroup$ @Yoav Kallus Actually that's a good point. $\endgroup$
    – BlueLight
    Commented May 19, 2013 at 19:28

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