The evolution from a linear temperature gradient to a detonation is investigated for
combustible materials whose chemistry is governed by chain-branching kinetics, using
a combination of high-activation-temperature asymptotics and numerical simulations.
A two-step chemical model is used, which captures the main properties of detonations
in chain-branching fuels. The first step is a thermally neutral induction time, representing
chain initiation and branching, which has a temperature-sensitive Arrhenius
form of the reaction rate. At the end of the induction time is a transition point where
the fuel is instantaneously converted into chain-radicals. The second step is the main
exothermic reaction, representing chain termination, assumed to be temperature insensitive.
Emphasis is on comparing and contrasting the results with previous studies
that used simple one-step kinetics. It is shown that the largest temperature gradient
for which a detonation can be successfully ignited depends on the heat release rate
of the main reaction. The slower the heat release compared to the initial induction
time, the shallower the gradient has to be for successful ignition. For example, when
the rate of heat release is moderate or slow on the initial induction time scale, it was
found that the path of the transition point marking the end of the induction stage
should move supersonically, in which case its speed is determined only by the initial
temperature gradient. For steeper gradients such that the transition point propagates
subsonically from the outset, the rate of heat release must be very high for a detonation
to be ignited. Detonation ignition for the two-step case apparently does not
involve the formation of secondary shocks, unlike some cases when one-step kinetics is used.