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Register a new userSingular electrostatic energy of nanoparticle clusters
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The binding of clusters of metal nanoparticles is partly electrostatic. We address difficulties in calculating the electrostatic energy when high charging energies limit the total charge to a single quantum, entailing unequal potentials on the particles. We show that the energy at small separation $h$ has a singular logarithmic dependence on $h$. We derive a general form for this energy in terms of the singular capacitance of two spheres in near contact $c(h)$, together with nonsingular geometric features of the cluster. Using this form, we determine the energies of various clusters, finding that more compact clusters are more stable. These energies are proposed to be significant for metal-semiconductor binary nanoparticle lattices found experimentally. We sketch how these effects should dictate the relative abundances of metal nanoparticle clusters in nonpolar solvents.
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The binding of clusters of metal nanoparticles is partly electrostatic. We address difficulties in calculating the electrostatic energy when high charging energies limit the total charge to a single quantum, entailing unequal potentials on the particles. We show that the energy at small separation $h$ has a strong logarithmic dependence on $h$. We give a general law for the strength of this logarithmic correction in terms of a) the energy at contact ignoring the charge quantization effects and b) an adjacency matrix specifying which spheres of the cluster are in contact and which is charged. We verify the theory by comparing the predicted energies for a tetrahedral cluster with an explicit numerical calculation.
We report on the electrostatic complexation between polyelectrolyte-neutral copolymers and oppositely charged 6 nm-crystalline nanoparticles. For two different dispersions of oxide nanoparticles, the electrostatic complexation gives rise to the formation of stable nanoparticle clusters in the range 20 - 100 nm. It is found that inside the clusters, the particles are pasted together by the polyelectrolyte blocks adsorbed on their surface. Cryo-transmission electronic microscopy allows to visualize the clusters and to determine the probability distributions functions in size and in aggregation number. The comparison between light scattering and cryo-microscopy results suggests the existence of a polymer brush around the clusters.
Aggregation of nanoparticles of given size $R$ induced by addition of a polymer strongly depends on its degree of rigidity. This is shown here on a large variety of silica nanoparticle self-assemblies obtained by electrostatic complexation with carefully selected oppositely charged bio-polyelectrolytes of different rigidity. The effective rigidity is quantified by the total persistence length $L_T$ representing the sum of the intrinsic ($L_p$) and electrostatic ($L_e$) polyelectrolyte persistence length, which depends on the screening, i.e., on ionic strength due to counter-ions and external salt concentrations. We experimentally show for the first time that the ratio L T /R is the main tuning parameter that controls the fractal dimension D f of the nanoparticles self-assemblies, which is determined using small-angle neutron scattering: (i) For $L_T /R<0.3$ (obtained with flexible poly-L-lysine in the presence of an excess of salt), chain flexibility promotes easy wrapping around nanoparticles in excess hence ramified structures with $D_f sim 2$. (ii) For $0.3<L_T /Rle1$ (semiflexible chitosan or hyaluronan complexes), chain stiffness promotes the formation of one-dimensional nanorods (in excess of nanoparticles), in good agreement with computer simulations. (iii) For $L_T /R>1$, $L_e$ is strongly increased due to the absence of salt and repulsions between nanoparticles cannot be compensated by the polyelectrolyte wrapping, which allow a spacing between nanoparticles and the formation of one dimensional pearl necklace complexes. (iv) Finally, electrostatic 2 screening, i.e. ionic strength, turned out to be a reliable way of controlling $D_f$ and the phase diagram behavior. It finely tunes the short-range interparticle potential, resulting in larger fractal dimensions at higher ionic strength.
Electrostatic reaction inhibition in heterogeneous catalysis emerges if charged reactants and products are adsorbed on the catalyst and thus repel the approaching reactants. In this work, we study the effects of electrostatic inhibition on the reaction rate of unimolecular reactions catalyzed on the surface of a spherical model nanoparticle by using particle-based reaction-diffusion simulations. Moreover, we derive closed rate equations based on approximate Debye-Smoluchowski rate theory, valid for diffusion-controlled reactions, and a modified Langmuir adsorption isotherm, relevant for reaction-controlled reactions, to account for electrostatic inhibition in the Debye-Huckel limit. We study the kinetics of reactions ranging from low to high adsorptions on the nanoparticle surface and from the surface- to diffusion-controlled limits for charge valencies 1 and 2. In the diffusion-controlled limit, electrostatic inhibition drastically slows down the reactions for strong adsorption and low ionic concentration, which is well described by our theory. In particular, the rate decreases with adsorption affinity, because in this case the inhibiting products are generated at high rate. In the (slow) reaction-controlled limit, the effect of electrostatic inhibition is much weaker, as semi-quantitatively reproduced by our electrostatic-modified Langmuir theory. We finally propose and verify a simple interpolation formula that describes electrostatic inhibition for all reaction speeds (`diffusion-influenced reactions) in general.
The induced surface charges appear to diverge when dielectric particles form close contacts. Resolving this singularity numerically is prohibitively expensive because high spatial resolution is needed. We show that the strength of this singularity is logarithmic in both inter-particle separation and dielectric permittivity. A regularization scheme is proposed to isolate this singularity, and to calculate the exact cohesive energy for clusters of contacting dielectric particles. The results indicate that polarization energy stabilizes clusters of open configurations when permittivity is high, in agreement with the behavior of conducting particles, but stabilizes the compact configurations when permittivity is low.
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