Pattern Formation and Glassy Phase in the ${\varphi }^4$ Theory with a Screened Electrostatic Repulsion

We study analytically the structural properties of a system with a short-range attraction and a competing long-range screened repulsion. This model contains the essential features of the effective interaction potential among charged colloids in polymeric solutions and provides novel insights on the equilibrium phase diagram of these systems. Within the self-consistent Hartree approximation and by using a replica approach, we show that varying the parameters of the repulsive potential and the temperature yields a phase coexistence, a lamellar, and a glassy phase. Our results strongly suggest that the cluster phase observed in charged colloids might be the signature of an underlying equilibrium lamellar phase, hidden on experimental time scales.

Figure 1

Temperature ($T$)-repulsion ($4\pi W$) phase diagram of the model for $\lambda =2$ (and $r_0=-1$), showing the relative positions of the paramagnetic (P), ferromagnetic (F), lamellar (L), and glassy (G) phases. The value of $g$ is such that $T_c(W=0)=1$. The continuous (black) and the dashed (blue) curves, found within the Hartree approximation, corresponds, respectively, to the second-order phase transition from P to F, $T_c(W,\lambda )$, and to the first-order transition from P to L, $T_L(W,\lambda )$. The dotted (green) and the dashed-dotted (red) curves, found within the SCSA, identify the dynamical, $T_d(W,\lambda )$, and the ideal, $T_K(W,\lambda )$, transition temperatures to G. The thin dotted (black) line ($\xi =l_m$) marks a crossover temperature below which the system establishes modulated structures, schematically sketched in the figure.

Figure 2

Main frame: Momentum dependence of the correlation function, $G(\mathbf{k})$, for $4\pi W=0.2$ and $\lambda =2$ at $T=T_d$, showing that it is peaked around the typical modulation wave vector $k_m$ with broadening ${\xi }^{-1}$, given by the inverse of the correlation length. Inset: Momentum dependence of the nonergodicity parameter $f_{\mathbf{k}}$ for the same values of $W$, $\lambda$, and $T$.