  Due to their unique structural and mechanical properties,
randomly-crosslinked polymer networks play an important role in many different
fields, ranging from cellular biology to industrial processes. In order to
elucidate how these properties are controlled by the physical details of the
network (\textit{e.g.} chain-length and end-to-end distributions), we generate
disordered phantom networks with different crosslinker concentrations $C$ and
initial density $\rho_{\rm init}$ and evaluate their elastic properties. We
find that the shear modulus computed at the same strand concentration for
networks with the same $C$, which determines the number of chains and the
chain-length distribution, depends strongly on the preparation protocol of the
network, here controlled by $\rho_{\rm init}$. We rationalise this dependence
by employing a generic stress-strain relation for polymer networks that does
not rely on the specific form of the polymer end-to-end distance distribution.
We find that the shear modulus of the networks is a non-monotonic function of
the density of elastically-active strands, and that this behaviour has a purely
entropic origin. Our results show that if short chains are abundant, as it is
always the case for randomly-crosslinked polymer networks, the knowledge of the
exact chain conformation distribution is essential for predicting correctly the
elastic properties. Finally, we apply our theoretical approach to published
experimental data, qualitatively confirming our interpretations.
