  The ground state of the uniform antiferromagnetic spin-1/2 Heisenberg chain
can be viewed as a strongly fluctuating liquid of valence bonds, while in
disordered chains these bonds lock into random singlet states on long length
scales. We show that this phenomenon can be studied numerically, even in the
case of weak disorder, by calculating the mean value of the number of valence
bonds leaving a block of $L$ contiguous spins (the valence-bond entanglement
entropy) as well as the fluctuations in this number. These fluctuations show a
clear crossover from a small $L$ regime, in which they behave similar to those
of the uniform model, to a large $L$ regime in which they saturate in a way
consistent with the formation of a random singlet state on long length scales.
A scaling analysis of these fluctuations is used to study the dependence on
disorder strength of the length scale characterizing the crossover between
these two regimes. Results are obtained for a class of models which include, in
addition to the spin-1/2 Heisenberg chain, the uniform and disordered critical
1D transverse-field Ising model and chains of interacting non-Abelian anyons.
