*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.Courtesy: Gaurav Arya, Duke University.*

Materials
scientists at Duke University have devised a simplified method for calculating
the attractive forces that cause nanoparticles to self-assemble into larger
structures.

With this
new model, accompanied by a graphical user interface that demonstrates its
power, researchers will be able to make previously impossible predictions about
how nanoparticles with a wide variety of shapes will interact with one another.
The new method offers opportunities for rationally designing such particles for
a wide range of applications from harnessing solar energy to driving catalytic
reactions.

The
results appear online on November 12 in the journal Nanoscale Horizons.

"Faceted
nanoparticles can lead to novel assembly behaviors, which haven't been explored
in the past," said Brian Hyun-jong Lee, a mechanical engineering and
materials science graduate student at Duke and first author of the paper.
"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."

"Every
time I go through the latest set of published papers in nanotechnology, I see
some new application of these types of nanoparticles," added Gaurav Arya,
associate professor of mechanical engineering and materials science at Duke.
"But accurately calculating the forces that pull these particles together
at very close range is extremely computationally expensive. We have now
demonstrated an approach that speeds those calculations up by millions of times
while only losing a small amount of accuracy."

The forces
at work between nanoparticles are called van der Waals forces. These forces
arise because of small, temporary shifts in the density of electrons orbiting
atoms according to the complex laws of quantum physics. While these forces are
weaker than other intermolecular interactions such as coulombic forces and
hydrogen bonds, they are ubiquitous and act between each and every atom, often
dominating the net interaction between particles.

To
properly account for such forces between particles, one must calculate the van
der Waals force that every atom in the particle exerts on every atom in a
nearby particle. Even if both of the particles in question were miniscule cubes
of sizes smaller than 10 nanometers , the number of calculations summing all
such interatomic interactions would be in the tens of millions.

It's easy
to see why trying to do this over and over for thousands of particles located
at different positions and in different orientations in a multiparticle
simulation quickly becomes impossible.

"Lots
of work has been done to formulate a summation that gets close to an analytical
solution," said Arya. "Some approaches treat particles as made up of
infinitesimally small cubes stuck together. Others try to fill space with
infinitesimally thin circular rings. 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 skirt
around this issue, Lee and Arya took a different approach by making several
simplifications. The first step involves representing the particle as being
made up not of cubic elements, but of rod-shaped elements of various lengths
stacked together. The model then assumes that rods whose projections fall
outside the projected boundary of the other particle contribute negligibly to
the overall interaction energy.

The
energies contributed by the remaining rods are further assumed to equal the
energies of rods of uniform lengths located the same normal distance as the
actual rods, but with zero lateral offset. The final trick is to approximate
the distance-dependence of the rod-particle energy using power-law functions
that have closed-form solutions when distances vary linearly with the lateral
position of the actual rods, as is case with the flat interacting surfaces of
faceted particles.

After all
of these simplifications are made, analytical solutions for the interparticle
energies can be obtained, allowing a computer to breeze through them. And while
it may sound like they would introduce a large amount of error, the researchers
found that the results were only 8% off on average from the actual answer for
all particle configurations, and only 25% different at their worst.

While the
researchers primarily worked with cubes, they also showed that the approach
works with triangular prisms, square rods and square pyramids. Depending on the
shape and material of the nanoparticles, the modeling approach could impact a
wide range of fields. For example, silver or gold nanocubes with edges close to
one another can harness and focus light into tiny "hotspots,"
creating an opportunity for better sensors or catalyzing chemical reactions.

"This
is the first time that anyone has proposed an analytical model for van der Waals
interactions between faceted particles," said Arya. "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."