  The partial spatial separation of cold dark matter (DM) and gas is a
ubiquitous feature in the formation of cosmic large-scale structure. This
separation, termed dissociation, is prominent in galaxy clusters that formed
through collisions of massive progenitors, such as the famous `Bullet' cluster.
A direct comparison of the incidence of such dissociated structures with
theoretical predictions is challenged by the rarity of strongly dissociated
systems and the difficulty to quantify dissociation. This paper introduces a
well-defined dimensionless dissociation index $S\in[-1,1]$ that encodes the
quadrupole difference between DM and gas in a custom region. Using a simulation
of cosmic large-scale structure with cold DM and ideal non-radiating gas, in
$\Lambda$CDM cosmology, we find that 90 per cent of the haloes are positively
dissociated ($S>0$), meaning their DM is more elongated than their gas. The
spatial density of highly dissociated massive structures appears consistent
with observations. Through idealised $N$-body+SPH simulations of colliding
gaseous DM haloes, we further explore the details of how ram-pressure causes
dissociation in binary collisions. A suite of 300 such simulations reveals a
scale-free relation between the orbital parameters of binary collisions and the
resulting dissociation. Building on this relation, we conclude that the
frequency of dissociated structures in non-radiative cosmological simulations
is nearly fully accounted for by the major (mass ratio $>1:10$) binary
collisions predicted by such simulations. In principle, our results allow us to
constrain the orbital parameters that produced specific observed dissociated
clusters.
