Charge symmetry broken complex coacervation

Liquid-liquid phase separation has emerged as one of the important paradigms in the chemical physics as well as biophysics of charged macromolecular systems. We elucidate an equilibrium phase separation mechanism based on charge regulation, i.e., protonation-deprotonation equilibria controlled by pH, in an idealized macroion system which can serve as a proxy for simple coacervation. First, a low-density density functional calculation reveals the dominance of two-particle configurations coupled by ion adsorption on neighboring macroions. Then a binary cell model, solved on the Debye-Hückel as well as the full nonlinear Poisson-Boltzmann level, unveils the charge symmetry breaking as inducing the phase separation between low- and high-density phases as a function of pH. These results can be identified as a charge symmetry broken complex coacervation between chemically identical macroions.

Figure 1

Macromolecular solution (a) and the magnified view (b) of a small portion of it. Within a binary cell model, each macroion (indicated by red circles) of radius $R_0$ is surrounded by a cell of radius $R$ and the interaction between a pair such cell-surrounded spheres is considered. The electric field $E_R$ at the cell boundary is assumed to be uniform and as it is the case for interacting planar surfaces, the value of $E_R$ depends on the charge states of the neighboring macroions forming the pair. The two cells of the binary cell model allow for asymmetric charge configurations ($E_R\ne 0$), which are excluded from the standard symmetric charge (single-cell) cell mode ($E_R=0$). In the latter case the binary cell model then reduces to the standard cell model.

Figure 2

Distribution of surface charges ${\sigma }^{*}$ and degrees of protonation $\eta$ as well as the associated bulk packing fraction profile ${\mathrm{\Phi }}_{\text{b}}\left(\eta \right)$ [as defined in Eq. (6)] in a suspension with packing fraction ${\overline{\mathrm{\Phi }}}_{\text{b}}=0.1$ of colloidal macroions with $K=20$ adsorption sites per macroion. For small values of $\chi >0$ a unimodal distribution is observed, whereas for sufficiently large values of $\chi$ the surface-charge distribution becomes bimodal with increasingly large magnitudes of the peak surface charges.

Figure 3

Binodals of the charge-regulation-induced phase separation of colloidal macroions with $K=40$ (blue, light shade), 45 (green, intermediate shade), and 50 (purple, dark shade) adsorption sites per macroion. The charge-regulation parameter $\chi =20$ is chosen arbitrarily and the pH-sensitive parameter $\alpha \in [-13,-7]$ is tuned around the value $-\chi /2=-10$, where oppositely charged colloidal macroions are expected to occur. The interior of the loops corresponds to the two-phase regions, where phase separation into a low- and a high-density phase occurs at the given value of $\alpha$. The two-phase region widens upon increasing the number $K$ of adsorption sites per macroion as a result of an increasing magnitude of the electrostatic interaction.

Figure 4

Structure factors $S_{\text{NN}}\left(q\right)$ (a) and $S_{ZZ}\left(q\right)$ (b) of suspensions with packing fractions ${\overline{\mathrm{\Phi }}}_{\text{b}}=0.1$ (blue, light shade), 0.2 (green, intermediate shade), and 0.3 (purple, dark shade) of charge-regulated colloidal macroions of radius $R_0$ with $K=20$ adsorption sites per macroion and charge-regulation parameters $\alpha =-5, \chi =10$. As defined in Eqs. (7) and (8), the structure factor $S_{\text{NN}}\left(q\right)$ in (a) describes the relative distribution of colloidal macroions irrespective of their charge, whereas $S_{ZZ}\left(q\right)$ in (b) describes the relative distribution of charge within the fluid (see Table 1 for details). The location of the main peak of $S_{ZZ}\left(q\right)$ relative to that of $S_{\text{NN}}\left(q\right)$ indicates an alternating charge structure.

Figure 5

Top panel shows the variation of the grand potential $\beta \mathrm{\Omega }\left({\sigma }_1^{*},{\sigma }_2^{*}\right)/K$ given by Eq. (14) as functions of the surface-charge density variables ${\sigma }_1^{*}$ and ${\sigma }_2^{*}$ for different values of the parameters $\alpha$ and ${\mathrm{\Phi }}_{\text{v}}$. Bottom panel shows cuts through the potential surface at ${\sigma }_1^{*}={\sigma }_2^{*}$ (in yellow, light shade) and ${\sigma }_1^{*}=-{\sigma }_2^{*}$ (in blue, dark shade). (a) Shows the results for $\alpha =-3, \chi =6$, and ${\mathrm{\Phi }}_{\text{v}}=0.6$ which fall inside the phase separation region. Consequently, the asymmetric configuration is the minimum-energy configuration. (b) At the same value of ${\mathrm{\Phi }}_{\text{v}}$, but with $\alpha$ changing from $-3$ to $-2.8$. As it is evident from the curves, the stability has changed toward a configuration with uncoupled cells, thus leaving the phase coexistence region. (c) Again, at $\alpha =-3$ and $\chi =6$ but going down in the particle-cell volume ratio to ${\mathrm{\Phi }}_{\text{v}}=0.05$. As one can see, the grand potential of the symmetric configuration approaches that of the asymmetric configuration. For all the plots, $R_0=10\mathrm{nm}, \kappa =0.1{\mathrm{nm}}^{-1}, {\ell }_B=0.7\mathrm{nm}$, and $K=20$ are used.

Figure 6

Map showing the charge states of the two macroions as functions of the interaction parameter $\alpha$ and the particle-cell volume fraction ${\mathrm{\Phi }}_{\text{v}}$ for $\chi =30$ and two different $K$ values. The solid line corresponds to $K=1257$ whereas the dashed line refers to $K=50$. In each case, one observes a conical-shaped region centered around $\alpha =-\chi /2=-15$ with stretched opening at the top. Inside this region and on the boundary, the two macroion surfaces are oppositely charged ($\left|{\eta }_1-{\eta }_2\right|=1$ or, equivalently, ${\sigma }_1^{*}=-{\sigma }_2^{*}$) whereas outside this region they are identically charged (${\eta }_1={\eta }_2$ or, equivalently, ${\sigma }_1={\sigma }_2$). With increasing $K$ value, this region featuring charge asymmetry widens.

Figure 7

Two-phase surface-charge density coexistence region around the symmetry axis with $\alpha =-3$, at $\chi =6$ for $K=20$.