Like-Charge Attraction between Metal Nanoparticles in a $1:1$ Electrolyte Solution

We calculate the force between two spherical metal nanoparticles of charge $Q_1$ and $Q_2$ in a dilute $1:1$ electrolyte solution. Numerically solving the nonlinear Poisson-Boltzmann equation, we find that metal nanoparticles with the same sign of charge can attract one another. This is fundamentally different from what is found for like-charged, nonpolarizable, colloidal particles, the two-body interaction potential for which is always repulsive inside a dilute $1:1$ electrolyte. Furthermore, the existence of like-charge attraction between spherical metal nanoparticles is even more surprising in view of the result that such attraction is impossible between parallel metal slabs, showing the fundamental importance of curvature. To overcome a slow convergence of the numerical solution of the full nonlinear Poisson-Boltzmann equation, we developed a modified Derjaguin approximation which allows us to accurately and rapidly calculate the interaction potential between two metal nanoparticles or between a metal nanoparticle and a phospholipid membrane.

Figure 1

(a) Two infinite metal slabs of width $d$ and total charge density $2{\sigma }_1$ and $2{\sigma }_2$, respectively. ${\sigma }_{a,b,c,d}$ represent the surface charge densities on the faces of the two slabs. The values of ${\sigma }_{a,b,c,d}$ change depending on the separation between the slabs, while the total charge density on each slab remains fixed. (b) Two like-charged spherical metal nanoparticles of charge $Q_1$ and $Q_2$ and radius $a$, separated by a surface-to-surface distance $L$, in an electrolyte solution of concentration ${\rho }_S$.

Figure 2

The pressure between different like-charged metal slabs of width $d=10\AA$ and the charge indicated in the figure, separated by a distance $L$. The pressure is always repulsive, independent of the charge on each slab. The salt concentration is 100 mM.

Figure 3

(a) Electrostatic-entropic force between two like-charged metal nanoparticles of radius $a=50\AA$ and charge $Q_1=-67q$ and $Q_2=-67q$, $-31q$, and $-6q$—orange, violet, and green curves, respectively—in the electrolyte at 40 mM. Positive force is repulsive, and negative is attractive. The squares are forces calculated numerically using the PB equation in cylindrical coordinates and Eq. (4), and the lines are calculated using the modified Derjaguin approximation, Eq. (6). (b) Electrostatic-entropic force—calculated using the modified Derjaguin approximation, Eq. (6)—between two like-charged metallic nanoparticles of radius $a=200\AA$ and charges $Q_1=-4775q$ and various values of $Q_2$. Salt concentrations are as indicated in the figure.

Figure 4

(a) A metal nanoparticle of charge $Q$ and radius $a$, at surface-to-surface distance $L$ from a charged planar membrane, inside an electrolyte solution. (b) Representation of the modified Derjaguin approximation for this system.

Figure 5

Interaction potentials between a spherical metal nanoparticle of radius $a=200\AA$ and charge $Q$, indicated in the figure, and a membrane of charge density $\sigma =-0.26\mathrm{C}/{\mathrm{m}}^2$. The electrolyte concentration is 2 mM. Solid curves are the electrostatic-entropic potentials for different nanoparticle charges, while the dashed curves are the total interaction potentials, which also include the van der Waals dispersion interaction. The membrane was modeled as a decane, with Hamaker constant $A\approx 8.9k_{B}T$.