Nucleation barriers, spinodals, and the Ginzburg criterion

Abstract

Nucleation theory is considered in $d$-dimensional systems which undergo a nearly mean-field-like transition, such as Ising magnets or mixtures with a large but finite range $r$ of the interaction, or polymer mixtures with chain lengths ${N}_{A}={N}_{B}=N\ensuremath{\gg}1$. Near two-phase coexistence the nucleation free-energy barriers are $\ensuremath{\Delta}{F}^{*} \ensuremath{\propto}{r}^{d}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}\frac{d}{2}}{(\frac{\ensuremath{\delta}\ensuremath{\psi}}{{\ensuremath{\psi}}_{\mathrm{coex}}})}^{\ensuremath{-}(d\ensuremath{-}1)}$ and $\ensuremath{\Delta}{F}^{*} \ensuremath{\propto}{N}^{(\frac{d}{2}\ensuremath{-}1)}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}\frac{d}{2}}{(\frac{\ensuremath{\delta}\ensuremath{\psi}}{{\ensuremath{\psi}}_{\mathrm{coex}}})}^{\ensuremath{-}(d\ensuremath{-}1)}$, respectively, where ${\ensuremath{\psi}}_{\mathrm{coex}}$ is the order parameter, $\ensuremath{\delta}\ensuremath{\psi}$ the deviation of the order parameter in the metastable state from that at coexistence, and ${T}_{c}$ the critical temperatures where the transition is second order. The crossover to nucleation near ${T}_{c}$ where $\frac{\ensuremath{\Delta}{F}^{*}}{{k}_{B}{T}_{c}}=c{(\frac{\ensuremath{\delta}\ensuremath{\psi}}{{\ensuremath{\psi}}_{\mathrm{coex}}})}^{\ensuremath{-}(d\ensuremath{-}1)}$, $c$ for $d<4$ being a universal constant, is controlled by the same Ginzburg criterion as for the static critical properties, i.e., crossover occurs at ${r}^{d}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}\frac{d}{2}}\ensuremath{\approx}1$, ${N}^{(\frac{d}{2}\ensuremath{-}1)}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}d}\ensuremath{\approx}1$. In polymer mixtures, mean-field-like behavior occurs only for $d>2$. In the mean-field region, metastable states are well defined up to a narrow region near the spinodal curve ${\ensuremath{\psi}}_{\mathrm{sp}} $; the width of this region is given by ${[\frac{(\ensuremath{\psi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{sp}})}{{\ensuremath{\psi}}_{\mathrm{coex}}}]}^{(3\ensuremath{-}\frac{d}{2})}\ensuremath{\approx}{[{r}^{d}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}\frac{d}{2}}]}^{\ensuremath{-}1}$ or ${[{N}^{(\frac{d}{2}\ensuremath{-}1)}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{2\ensuremath{-}\frac{d}{2}}]}^{\ensuremath{-}1}$, respectively. This rounding of the spinodal curve can again be understood by the Ginzburg criterion for the metastable state at $\ensuremath{\psi}$ near ${\ensuremath{\psi}}_{\mathrm{sp}}$. At the unstable side of the mean-field region, the linear theory of spinodal decomposition holds outside of correspondingly narrow regions close to the spinodal curve, too.