Charge Regulation Effects in Nanoparticle Self-Assembly

Nanoparticles in solution acquire charge through the dissociation or association of surface groups. Thus, a proper description of their electrostatic interactions requires the use of charge-regulating boundary conditions rather than the commonly employed constant-charge approximation. We implement a hybrid Monte Carlo/molecular dynamics scheme that dynamically adjusts the charges of individual surface groups of objects while evolving their trajectories. Charge regulation effects are shown to qualitatively change self-assembled structures due to global charge redistribution, stabilizing asymmetric constructs. We delineate under which conditions the conventional constant-charge approximation may be employed and clarify the interplay between charge regulation and dielectric polarization.

Figure 1

Effect of charge regulation on pairwise interactions. Top: potential of mean force $V$ between a large particle with radius $R$ and a small particle (red sphere) with constant charge $q_{\mathrm{s}}$ and radius $r$, normalized by the coupling strength $\lambda$ ($\tilde{V}=V/\lambda$). Charge regulation is realized through dissociable sites that can be either neutral or charged (white and blue, respectively, in the inset) and results in a significantly enhanced attraction (blue line) compared to a constant charge $q_{\mathrm{L}}=-q_{\mathrm{s}}$ on the large particle (black line). Bottom: corresponding (normalized) total charge on the larger particle, ${\tilde{q}}_{\mathrm{L}}=q_{\mathrm{L}}/q_{\mathrm{s}}$ (dashed blue line). The parameters employed here correspond to a 6-nm silica particle in deionized water at $\mathrm{pH}=7$ [14]: $n_{\mathrm{ss}}=792$ (${\sigma }_{\mathrm{ss}}\approx 8{\mathrm{nm}}^{-2}$), $\mathrm{\Delta }\mathrm{pK}=-0.5$, $q_{\mathrm{s}}=12q_0$, $R=4l_{\mathrm{B}}$, $2r=l_{\mathrm{B}}$, $\mathrm{pI}=6.7$, $\lambda =32k_{\mathrm{B}}T$.

Figure 2

Self-assembly of binary aggregates. Small particles (red) carry a constant charge $q_{\mathrm{s}}=16q_0$, while large particles either have a constant charge $q_{\mathrm{L}}=-q_{\mathrm{s}}$ [CC, blue spheres in panels (a), (c), and (e)] or are charge regulating [CR, spheres with neutral (white) or charged (blue) surface sites in panels (b), (d), and (f)]. In the CR case, $\mathrm{\Delta }\mathrm{pK}=-1.5$, which results in neutral structures, $⟨q_{\mathrm{L}}⟩\approx -q_{\mathrm{s}}$. Whereas CC conditions give rise to compact structures, CR leads to anisotropic and open assemblies. This observation persists with increasing particle number: $N_{\mathrm{p}}=3$, 4, and 100 large and small particles in panels (a) and (b), (c) and (d), and (e) and (f), respectively, at ion concentration $\mathrm{pI}=5.7$. The CR images are instantaneous realizations; the charge distribution on the large particles is continuously fluctuating.

Figure 3

Dielectric effects on the (normalized) potential of mean force $\tilde{V}$ for the particle pair described in Fig. 1, with the additional condition that the large particle is polarizable. Under constant-charge (CC) conditions, a particle with low dielectric constant ($\widetilde{ϵ}=0.01$) has a diminished attraction (open circles) and a high-dielectric particle ($\widetilde{ϵ}=100$) has an enhanced attraction (asterisks). Both cases are superseded by the attraction strength under charge-regulating (CR) conditions. Here, the high-permittivity case still is slightly stronger, but the effect is barely visible on the scale of the graph. The inset illustrates the CR-BEM model. The BEM layer of polarizable surface patches ($\widetilde{ϵ}=100$) is placed at a distance $R/8$ below the CR layer, with positive (red) or negative (blue) induced dielectric charges. The CR layer is shown as small white (neutral) or charged (blue) spheres.

Figure 4

Schematic diagram showing the applicability of the constant-charge (CC) and constant-potential (CP) approximations of CR. For sufficiently strong charges and intermediate values of the surface capacitance, the use of a full CR solver—as proposed in the main text—is required. In addition, in the two regions denoted by “dielectric,” dielectric polarization effects can be important.