We elucidate the effect of the secondary baroclinic vorticity (SBV) on the Richtmyer–Meshkov instability (RMI) accelerated by a weak incident shock and develop a vortex-based model for spike and bubble growth rates. Two major mechanisms of the single-mode RMI, the primary baroclinic vorticity (PBV) and the pressure perturbation, are distinguished by simplified models with the vortex-surface field. We find that the effect of the pressure perturbation can be neglected in the present RMI, and the growth of the interface or vortex surface is first driven by the PBV. Subsequently, the SBV, generated by the misalignment between the density gradient across the interface and the pressure gradient produced by the PBV-induced velocity, leads to the nonlinear growth of the interface with the generation of spikes and bubbles. Inspired by this mechanism, we develop a predictive model of spike and bubble growth rates using the motion of viscous vortex rings. The circulation of the vortex ring is modelled with the SBV effect. This model is validated by five data sets of direct numerical simulations and experiments of the single-mode RMI with various initial conditions.