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New Method Predicts Forces Between Peculiarly Shaped Nanoparticles

At Duke University, materials scientists have developed a simple technique for computing the attractive forces that allow nanoparticles to assemble on their own into bigger structures.

Overview of the approach used for deriving analytical expressions for the interparticle van der Waals interaction potential for faceted nanoparticles. The model goes through a series of simplifications. One block is normalized in a standard position. The other block is then assumed to be a grouping of rods. Any rods outside the boundaries of the first block are assumed to be negligible. The first block is shifted to be centered on each rod of the second block while its forces are being calculated and summed. Image Credit: Duke University.

This latest model, equipped with a graphical user interface that displays its power, will allow scientists to make predictions about how different shapes of nanoparticles will interact with each other—a feat that was not possible before.

The novel technique provides new possibilities for rationally developing these particles for a variety of applications, from driving catalytic reactions to harnessing solar energy. The study results appeared online in the Nanoscale Horizons journal on November 12th, 2020.

Faceted nanoparticles can lead to novel assembly behaviors, which haven’t been explored in the past. Cubes, prisms, rods and so on all exhibit distinct distance- and orientation-dependent interparticle interactions that can be utilized to create unique particle assemblies that one cannot obtain through self-assembly of spherical particles.

Brian Hyun-jong Lee, Study First Author and Mechanical Engineering and Materials Science Graduate Student, Duke University

Gaurav Arya, an associate professor of mechanical engineering and materials science at Duke University, added, “Every time I go through the latest set of published papers in nanotechnology, I see some new application of these types of nanoparticles. But accurately calculating the forces that pull these particles together at very close range is extremely computationally expensive.

Arya continued, “We have now demonstrated an approach that speeds those calculations up by millions of times while only losing a small amount of accuracy.”

The working forces between the nanoparticles are referred to as van der Waals forces. Such forces are caused by slight, fleeting shifts in the density of electrons revolving atoms in accordance with the intricate laws of quantum physics.

Although such forces are weaker compared to other intermolecular interactions, like hydrogen bonds and coulombic forces, they are pervasive and act between all the atoms, usually governing the net interaction that occurs between particles.

To accurately account for these van der Waals forces between particles, it is important to compute these forces that are exerted by all atoms in the particle on all the atoms in a neighboring particle. Even if both the target particles were tiny cubes of sizes smaller than 10 nm, the number of computations that sums up these interatomic interactions would be in the range of tens of millions.

It is easy to see why attempting to do this repeatedly for scores of particles in varied orientations and situated at varied positions in a multi-particle simulation quickly becomes unfeasible.

Lots of work has been done to formulate a summation that gets close to an analytical solution. Some approaches treat particles as made up of infinitesimally small cubes stuck together. Others try to fill space with infinitesimally thin circular rings.

Gaurav Arya, Associate Professor of Mechanical Engineering and Materials Science, Duke University

Arya continued, “While these volume-discretization strategies have allowed researchers to obtain analytical solutions for interactions between simple particle geometries like parallel flat surfaces or spherical particles, such strategies cannot be used to simplify the interactions between faceted particles due to their more complex geometries.”

To overcome this problem, both Lee and Arya adopted a different method by making a number of simplifications. In the initial step, the team demonstrated that instead of cubic elements, the particle is composed of different lengths of rod-shaped elements arranged together.

The model subsequently assumes that rods, the projections of which fall beyond the estimated margin of the other particle, contribute insignificantly to the overall interaction energy. The remaining rods contribute energies that are additionally believed to balance the energies of rods that have even lengths. These rods are situated at the same regular distance as the real rods, but with zero lateral offset.

The ultimate trick is to approximate the distance-reliance of the energy between the rod and the particle by using power-law functions. These functions have closed-form solutions, especially when distances differ linearly with the lateral location of the real rods, as is the case with the flat interacting surfaces of faceted particles.

Once all these simplifications are made, analytical solutions intended for the interparticle energies can be achieved, enabling a computer to run through them. While it may sound like these solutions would introduce a considerable amount of error, the team noted that the outcomes were just 8% off on average from the actual solution for all configurations of particles, and just 25% different at their worst.

While the investigators mainly used cubes for their study, they noted that the method also works with square pyramids, square rods, and triangular prisms.

Based on the material and shape of the nanoparticles, the new modeling method may affect a variety of fields. For instance, gold or silver nanocubes with boundaries proximal to each other can harness and direct the light into minute “hotspots.” This may provide an opportunity for catalyzing chemical reactions or for developing more improved sensors.

This is the first time that anyone has proposed an analytical model for van der Waals interactions between faceted particles. Even though we are yet to apply it for computing interparticle forces or energies within molecular dynamics or Monte Carlo simulations of particle assembly, we expect the model to speed up such simulations by as much as ten orders of magnitude.


Gaurav Arya, Associate Professor of Mechanical Engineering and Materials Science, Duke University

The study was funded by the National Science Foundation (CMMI award 1636356, ACI-1053575).

Journal Reference:

Lee, B. H.-J., (2020) Analytical van der Waals Interaction Potential for Faceted Nanoparticles. Nanoscale Horizons.

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