It has been recently pointed out in Refs. 1 and 2 that our theoretical result for the jamming density of a binary mixture of hard spheres3 “apparently” violates an upper bound that is obtained by considering the limit where the diameter ratio r=DA/DB. We believe that this apparent contradiction is the consequence of a misunderstanding, which we try to clarify here.

We will follow the notations of our paper3 and denote by η the volume fraction of small particles. Then, a very simple argument1,2 leads to the following result in the limit r:

φj,=min{φj,11η,φj,1φj,1+η(1φj,1)},
(1)

where φj,...

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