  In the present work we include the isovector-scalar $\delta$-meson in the
quark-meson coupling model (QMC) and study the properties of asymmetric nuclear
within QMC without and with the $\delta$-meson. Recent constraints set by
isospin diffusion on the slope parameter of the nuclear symmetry energy at
saturation density are used to adjust the model parameters. The thermodynamical
spinodal surfaces are obtained and the instability region at subsaturation
densities within QMC and QMC$\delta$ models are compared with mean-field
relativistic models. The distillation effect in the QMC model is discussed.
