Self-Assembly of Nanocrystal Superlattices
We've all played with slap bracelets before. The accessory that, at first, just looks like a curved ruler but with a little bit of force, turns into a bracelet that can fit any size hand. I always had fun with these because it was as if the bracelet was making itself. You didn't have to bend it into place; the laws of physics did it for you.
Sometimes I imagine how cool it would be if we could create a pile of them in such an arrangement, that when you apply force to one, the whole pile self-assembles into a bike, or a car, or even a house. Sadly this will likely stay as just a part of my imagination.
Based on my fascination with slap bracelets, it shouldn't be surprising that I would gain interest in the self-assembly of nanocrystals superlattices.
Now you’re probably thinking, “Woah that was a big leap. What the heck is a self-assembled nanocrystal superlattice?”
Well, first we should define what self-assembly really means. Self-assembly is the tendency for atoms and molecules to form into more complex structures, all through internal and external forces such as Van der Waals forces, electrostatic repulsion forces, entropic forces, etc. (Don't worry if you don't know what these are, we will go over them more in-depth)
A Nanocrystal is a fragment from a metal, superconductor, or insulating crystal. A superlattice is a crystalline structure in which a one, two, or three-dimensional pattern is repeated over and over again.
In this article, I will be summarizing a paper that goes in-depth into the process of making NC superlattices, the forces that determine their structure, and the unique properties that arise from their components and formation.
To check out the full paper, click here “Self-Assembly of Colloidal Nanocrystals: From Intricate Structures to Functional Materials.”
Without further ado, let's talk about nanocrystal superlattices!
How do we Initiate Superlattice Formation?
Nanocrystals consist of a hard inorganic core, and a soft layer of ligands surrounding it. Ligands are chains of molecules that attach to the surface of the NC core.
In order to make and collect these nanocrystals, two things are usually done. First, a solution consisting of the NC core (ex. Gold), the ligands (ex. Hydrocarbon Chains), and a solvent (ex. Water) is mixed together. During this process, the ligands will find their way over to the NC cores and atomically bond to the surface. This process is known as repassivation.
Often times a surfactant (a substance that decreases surface tension) is added to make the repassivation process more efficient. By decreasing the strength at which the solvent molecules attract to each other, the surfactant makes it easier for the ligands to travel through the solvent and onto the NC cores.
The NCs at this stage are of different sizes and shapes, so we need a way to organize them into piles of uniform NCs. There are many ways to do this such as precipitation, which is essentially putting the NCs through nanoscale grates so only particles of a particular size can get through.
Once you got a solution with uniform NCs, you can start the superlattice formation process.
Evaporation
Evaporation is the process of letting the solvent evaporate into the air leaving the nanocrystals behind.
Evaporation is limited in the sense that it can only make thin 2D films of NC superlattices. When the solution is introduced to air, a liquid-air interface forms between them. As the solvent particles rise to the interface to evaporate they pull NCs up with them through intermolecular forces. These NCs then start to cluster at the interface and eventually form a thin, 2D superlattice.
Oftentimes these thin layers can be transferred to a solid surface in which more evaporation can take place resulting in a more structured superlattice.
Despite being limited to 2D structures, this superlattice can be as long as a couple hundred micrometers which is pretty large considering its constituent building blocks are over 1000 times smaller in length. To put that to scale, imagine trying to make an object as big as an entire city using objects as small as a standard guitar.
Gravitational Sedimentation
Gravitational Sedimentation is the process of using gravity’s influence on NCs against its thermal energy. Every NC has the force of gravity pulling down on it with the strength being proportional to the mass of the NC. The thermal energy of each NC causes them to move in random directions.
So if the thermal energy is high enough, that it can cancel out the force of gravity and move the NC a distance greater than its’ diameter, then no sedimentation will happen. However, if the thermal energy of the NC cant move it a distance greater than its’ diameter, then over time, the force of gravity will pull all the NCs to the bottom of the container causing sedimentation which leads to superlattice formation.
While this process allows for 3D nanocrystal superlattices, it is limited in the fact that the influence of gravity becomes negligible as you start to have NCs with lengths below 100nm.
Destabilization
Destabilization occurs when the attractive forces between two NCs are stronger than the attractive forces between NCs and the solvent. In this case, it is more favorable for the NCs to move towards other NCs and form clusters rather than dispersing throughout the solution.
A common way to achieve this is to slowly add a nonsolvent (a substance that cant dissolves another substance) into the solution. This makes the solution increasingly more polar. It is often the case that NCs are capped with hydrocarbon ligands, which are non-polar.
Following the rule that “like dissolves like,” a non-polar solute in a polar substance will cause the solute to form clusters. Through this process, the NCs can finally get close enough to form a superlattice.
Another destabilization technique involves taking a surfactant (specifically dodecyltrimethylammonium bromide) and mixing it in the solution. The surfactant attaches to organic ligands which makes them hydrophilic (attractive to water). This causes a bilayer of water to form around NCs capped with these ligands. These layers attract other layers causing clustering of the water capped NCs.
You can then mix this solution with ethylene glycol which, when heated to the right temperature, causes the bilayer to dissolve allowing the NCs to form into a superlattice.
What Forces Influence Superlattice Formation?
The fascinating part about NC superlattice formation is that these ordered, crystalline structures are all the result of the interparticle forces between NC cores and ligands.
A Balancing Act
The structure and interparticle spacing of a superlattice is determined by the balancing act between attractive and repulsive forces. The forces between two NCs are trying to minimize the amount of free energy (the measure of energy that is available to do work) the system has resulting in stabilization of the superlattice.
First, for a superlattice to form there needs to be steric stabilization (stabilization based on the proximity of the particles).
When the interparticle spacing distance (d) between the ligands of neighboring NCs is less than 2 times the width of the capping layer (layer of ligands surrounding NC) (2L), the attractive forces of the ligands pull on each other, causing exclusion of solvent particles. Once the attractive forces pull the ligands within a distance less than the width of the capping layer (L), the repulsive force (caused by the elastic potential energy from the near touching ligands) causes the ligands to separate. This shows that steric stabilization between NCs is usually at distance d when L<d<2L.
The tendency for the attractive forces of ligands to push out external solvent requires energy. If the solvent is good at dissolving solutes, then it will take more energy to keep the solvent out which is why NCs disperse in good solvents. If the solvent is poor, however, it takes more energy for the solvent to keep the NCs apart. This is why the introduction of unsolvent into a solution is often a reliable way to initiate superlattice formation.
The repulsive forces from the ligands also help to explain why certain shapes can be packed closer than others. As the surface curvature of the NC increases, the repulsive forces between ligands of different NCs decrease. This is due to the fact that a higher curvature means more room in between NCs for the ligands to move around.
Another driving force in superlattice formation comes from electrostatics. If you mix charged ions into a polar NC solution, then the ions will attach to the NCs making them polar. This increases dispersion in the solution due to the “like dissolves like” principle.
This also makes it harder for the NCs to bond together because it will require more energy to keep the polar solvent molecules out than to just let them interfere.
This is why mixing substances into the solution that make the solvent an opposite polarity to the NCs is an effective way to initiate superlattice formation.
What Determines the Structure of a Superlattice?
All NCs have a hard (core) and soft (ligands) component. Examining what structure certain shaped NCs will form in ideal situations allows us to make predictions on what structure these NCs will form in an experiment. In order to do this, we first must look at hard and soft particle models separately.
Density & Entropy Maximizing Formations (Hard Particle Model)
The Hard Particle Model only looks at the NC cores and treats them like solid 3D polyhedral. When hard particles are put into a mixture in which they can aggregate, they will naturally form the most densely packed formations in order to increase overall entropy (a measurement of disorder in a system) in the solution.
This sounds counterproductive at first because putting randomly dispersed particles into an ordered superlattice seems like a way to decrease disorder. The problem with this view is that it doesn't account for the whole system.
While the system loses configurational entropy (disorder from the different ways to arrange particles), the whole system gains rotational (disorder from the different orientation of particles) and translational (disorder from the different positions of particles) entropy due to the increase in volume for the solvent particles to move around.
In other words, by decreasing the space that the NCs take up in the solution, more space is afforded towards the disorder of the solvent.
With this in mind, the next question becomes what formations create the most densely packed lattices for certain shapes?
Densely-Packed Formations
This question is essentially what, German mathematician, David Hilbert tried to answer back in the 18th century. A century after he asked this, it had only been proven for one shape, the sphere.
According to the Kepler−Hales theorem, the most optimal (most dense) formations of spheres are the face-centered cubic (fcc) and hexagonal close-packed (hcp) formations with fcc being slightly more optimal. With these formations, the spheres take up roughly 74.04% of the total volume.
With recent advancements in computing power, we have been able to find the optimal formations for other shapes such as cubes, diamonds, and other polyhedra.
Dense-Packing of Binary Mixtures
Superlattices don't have to just be made of one type of NC. Binary mixtures have been proven to have optimal packing formations that fill more space than those of unitary packings. This is due to the fact that the smaller of the two NCs can fill in the voids in a way that just can't happen with one NC formation. 15 binary sphere packings have been shown to be more optimal than the fcc formation of uniform sphere sizes.
The optimal formation is dependent on the ratio between the two radii of the different spheres which can be seen as Ra/Rb = γ. When γ is over 0.66, the formation of two separate lattices for each sphere forms while for values of γ under 0.2, the attractive forces of the larger spheres tend to push the smaller spheres out of the lattice.
This means a radii ratio between 0.2 and 0.66 is ideal for creating a binary superlattice of spheres.
There are also superlattices that form between NCs with different geometries. As we get into less uniform building blocks, the hard-particle model starts to break down.
For example, in a rod and sphere-shaped binary mixture, it has been shown that certain formations can maximize entropy without maximizing density. More research needs to be done to determine the underlying forces contributing to the formation of multi-shaped superlattices.
Internal Surface Area Minimization (Soft Particle Model)
Just like the hard particle model, the soft particle model strives to maximize entropy. The soft component of an NC is its ligands. They are soft in the sense that they aren't rigid and can change shape, as long as the volume stays the same.
Because of this, the formations soft particles form to maximize entropy is different than those of hard particles. Soft particle formation tries to conserve configurational entropy.
When a ligand is free of external forces, the configurational entropy is very high because there is a large number of configurations the ligands can be in. As you start to add external forces on the ligands, such as those from the ligands of other NCs, the configurational entropy decreases. The amount of possible configurations decreases because the ligand must now conform to an external force.
Because of this, the optimal formations for soft particles are those that reduce the amount of contact ligands have with other ligands.
The Voronoi cell
If you took a point and assigned it to the particle that it was closest to, and then did this for every point in which that same particle is the one closest to it, the summation of these points would be the Voronoi cell of that particle. In other words, the Voronoi cell is the volume that encompasses all the points in which all the points share the same closest particle.
Soft particles form in a way that maximizes the average isoperimetric quotient for its Voronoi cells. The isoperimetric quotient is essentially the ratio between the area of the Voronoi cell and the area of a circle with the same perimeter. A higher isoperimetric quotient means that the Voronoi cell more closely resembles a circle which means the amount of space an NC has is more evenly spread out among its ligands. This helps minimize the overall elastic interaction between ligands on neighboring NCs.
So what formations create the most optimized Voronoi cells?
Optimal Soft Particle Formations
When the problem was first proposed back in the 1800s, it was found that the body-centered cubic (bcc) was the most optimal when it came to packing spheres. This held for 100 years until a more optimal formation was found called the A15 phase.
While the A15 phase is very efficient at minimizing the internal surface area, it does a poor job at densely packing spheres, with only 52% of the available volume being filled up.
Combining the Two Models to Make Predictions
The models we have been looking at are idealized versions of structure formation, which act as if the opposite type of particle doesn't have any influence. In experiments though, it is important to take both models into account to make more reliable predictions.
We can express the ‘softness’ of a crystal by dividing the length of the ligands (L) by the radius of the core (R) or L/R. This ratio helps determine whether the soft or hard particle model will have a larger influence on the formation of the superlattice.
The higher this number is, the more likely the NCs are going to follow a soft or surface area-minimizing formation.
Different Shaped NCs
Not only can NCs be symmetrical shapes, but some research has been done on anisotropic (doesn't look the same from every orientation) NCs such as rods and platelets and the type of superlattices they form.
Anisotropic NCs are more prone to forming defects in their superlattices. This is due to their higher need to be in the right orientation to form a crystal superlattice. While spheres can be in different rotations and not affect the structure drastically, rods being in different orientations would cause noncrystalline structures.
This is why anisotropic NCs often form liquid crystals that have a mix of crystal and liquid properties.
These can either be nematic, in which the orientations are uniform but the positions are random, or smectic, in which the orientations are uniform and the positions are fixed to a certain plane. The diagram below represents the different formations platelet and rod-shaped crystals can form.
Rods
Rod superlattices form tip to tip rather than side to side. This is due to the fact that the tips are more curved and have fewer ligands than the sides resulting in lower repulsive forces.
Due to the anisotropic shape of the rods, the use of surfactants is often necessary to have a crystal superlattice form. The emphasis on uniform orientation between NCs means that more time is required during formation to prevent defects, and a surfactant allows for this by reducing the pressure that the solvent has on the NC clusters.
Platelets
An important detail about platelet formation is that they will form structures in which the plates are face to face. This helps maximize entropy by making the densest formations and makes sense as the attractive forces are strongest between the largest planes of the shape.
When the sizes of plates differ, only 1-dimensional columns will form, while uniform platelet size allows for 2 and 3-dimensional structures to form.
Polyhedral
Polyhedral are any 3D solid shape. Small polyhedral shapes such as tetrahedra and octahedra have been shown to form superlattices that go against the hard packing models by forming tip-tip rather than face to face. This results in voids that could have been avoidable by using face-to-contact.
This comes from the aforementioned ratio between ligand length and core radius. As the core of these NCs get smaller the ligands have more influence causing the surface area-minimizing model to outweigh the dense packing model.
In order to further help with this, Polyhedral shapes often have slightly curved sides to allow more room for the ligands to move around.
Branched Shapes
Branched shapes have been shown to form highly disordered superlattices due to the high degree of uniform orientations they require. The formation process must be very slow or else high-density jamming of the NCs will occur.
Because of their variety of orientations, many formations are possible and are determined by how many branches of the NC make contact with the substrate.
Types of Defects that Form in Superlattices
Just like everything else in life, superlattices are almost never perfect. Usually, some sort of defect gets through.
If the defect is not too large, the superlattice can still be stable despite the increase in free energy. Defects cause configurational entropy to increase which, when not overbearing, can counteract the increase in free energy.
0-Dimensional Defects
These are the simplest defects. It involves a problem with a single NC which can either be a vacancy (missing NC) or an interstitial defect (NC in the wrong spot).
1-Dimensional Defects
These defects occur from an issue with a whole line of NCs
- Edge Dislocation — When an extra line of NCs has been added to the lattice
- Screw Dislocation — When part of the lattice gets more strongly attached to the substrate than the other causing a fault between the two sides.
- Disclination —When the orientation of a line of NCs is different from the other lines
- Vortex Defect — When a line of anisotropic NCs are orientated wrong, causing a runoff effect from the surrounding NCs
2-Dimensional Defects
These types of defects can form for a number of reasons. Two superlattices can form separately and then combine in the wrong way, or an abrupt force during superlattice formation can cause a shift in layers.
- Stacking Fault — When a layered pattern of ABAB or ABCABC gets violated by the addition or deletion of a layer (ie. if an extra C layer was added to an ABC formation then the pattern would be ABCCABC)
- Twin Boundary — A specific stacking fault where the two sides of the defect mirror each other
- Antiphase Boundary — Occurs in binary NC superlattices, where the order of NC layers gets messed up (ie you have an alternating pattern of Cd and Se layers and then one layer of Cd doubles)
- Tilt Boundary — When two superlattices of the same NCs attach but not at parallel causing the apparent tilting of one side of the combined superlattice
- Twist Boundary — Similar to the tilt boundary but, one of the superlattices is more malleable and able to bend around the other
- Phase Boundary — When two superlattices of different NCs attach
3-Dimensional Defects
These defects have the most broken bonds and they go in all directions which makes them the hardest for superlattices to sustain, energy-wise.
- Void — When a vapor bubble gets encased in a superlattice, leaving a big gap with no NCs in its place
- Crack — When the lattice is compressed or stretched too hard causing the readjustment of NCs down an arbitrarily shaped fracture
- Precipitate — When an aggregation of some other substance gets encased in the superlattice, similar to a crumb of food getting stuck in pudding
Chemically Controlling the Structure
One of the exciting things about NC superlattices is that we can have a high degree of control over their structure using the chemical properties of different ligands.
Experiments testing different ligands and their corresponding structures have helped in the characterization of ligands' effects on superlattice structure.
Going back to the ratio L/R, we can determine how much NCs will be influenced by the soft-particle model. With that in mind, we could control the softness of the NCs by giving them longer ligands which would result in more control over the structures that form (fcc, bcc, hcp, A15).
It has also been demonstrated that you can use multicomponent ligands such as block copolymers to have some control over the structure.
For example, when applied to Au cores, block copolymers of hydrophilic and hydrophobic monomers have been shown to form superlattices that, when exposed to water, form vesicles (a substance protected by a surrounding layer of hydrophobic and hydrophilic components). This occurs when the water pulls the hydrophilic part of the NC up, causing it to curve until a complete sphere is formed.
On top of this, we have the ability to open and close the vesicles as we please which can show exciting applications down the line such as guided drug delivery.
Techniques such as tethering molecules onto the ligands, or using multiple different ligands have also been shown to have differing effects on the structure of superlattices although, more research is needed to find the correlations between different ligands and structures.
DNA-Capped NCs
The use of DNA as ligands for nanocrystals has become an increasingly popular idea as it would allow for more direct control over the formation of superlattices.
DNA is the reason the molecules in our body are able to self-assemble into the complex biomolecules and systems that keep us alive. This is due to the instructions for formation being “coded” directly into their molecular structure.
With this in mind, using DNA as ligands would give us the ability to “code” our own instructions into the NCs for superlattice formation.
To do this, a DNA linker strand is mixed in with the NCs which attaches to the surface at one end and leaves the other end open so we can then attach a DNA molecule.
We can also send down DNA that is “coded” to compete with the current ligands on the NCs to remove and replace them.
Once DNA ligands are attached, it has been proven that properties such as reentrant melting (the decreasing of a substance’s melting point beyond a certain pressure), wide gas-solid coexistence (existence of very stable solids and very unstable gases simultaneously), and reversible crystal-crystal transformation (transitions that would be irreversible without added energy) would be possible.
Its also been proven that you can change the formation (fcc, bcc, hcp, etc.) of the structure by changing the DNA molecules you chose for the ligands.
Overall, DNA ligands show the promise of allowing us to create stable superlattice structures that go against equilibrium and free energy reducing principals.
Environmentally Controlling the Structure
We've been talking a lot about how the internal forces from the NCs and superlattice affect the structure but the environment in which the superlattice forms also have a large influence.
Substrate Control
One way we can control superlattice formation is through templated substrates. By putting holes, grooves, or patterns into the substrate we can guide NCs into these spots and then use them to guide the next layer, allowing for templated superlattices.
Another way we can control the structure is to use curved surfaces such as water droplets. By putting droplets of water on the substrate, we can create a superlattice that forms a frame surrounding the drop. In a sense, the droplets are being used as cookie cutters to form hollow circles in the superlattice.
You can also use droplets to form 3D structures in which the NCs surround the droplet. This results in hedgehog-like superlattices.
External Fields
The use of electric, magnetic, and electromagnetic fields have been shown to have an effect on the structure of superlattices, specifically when it comes to the orientation of NCs.
If you attach a charged ion or ligand to the NCs of a lattice, you can give them a uniform orientation by using an external electric field. The field will attract the charged components towards it, causing a uniform alignment of their domains and rotation.
Electromagnetic fields have been shown to have a similar effect. Instead of having photoresponsive NCs, you put them in a photoresponsive medium. The EM field then guides the medium towards it which indirectly guides and aligns the NCs.
Temperature Control
The influence of temperature on superlattice formation does not have as much research as the other techniques but regardless, correlations have been found making it a viable way to control structure.
For example, spherical NCs have been shown to form structures of higher density at higher temperatures which shows a relationship between temperature and density of the structure.
This is due to the increased amount of entropy that can occur at higher temperatures, which encourages more densely packed formations.
Unique Properties that Arise from NC Superlattices
So why is so much research being done on the self-assembly of these superlattices?
Unique electrical, magnetic, luminescent, and plasmonic properties can occur from novel structures of NC superlattices. These properties can be controlled and used by humans for certain tasks if we are able to characterize which structures lead to which properties and why.
Electrical
NCs are often good conductors of electricity as they come from fragments of metal and superconductors. Because common ligands, such as hydrocarbons, are neutral the electrical properties of each NC get localized. This also occurs because of the space ligands force between NCs, preventing the charge to flow freely between neighbors.
One way to fix this is to change the ligands so that they are either conductive or thinner allowing quantum tunneling (the tendency for electrons to “teleport” through very small barriers) to occur. This would allow for current to travel through the whole lattice giving rise to novel properties.
From this, we could theoretically control the flow of electricity through a superlattice by controlling the structure and the types of NCs and ligands used.
Trying to create stabilized superlattices with charged and smaller ligands has been proven difficult but new techniques are being tried.
One recent technique came from the epitaxial layering of quantum dots, by removing the top layer of ligands, and then atomically bonding (or neck bonding) the next layer of NCs on top. It has been theorized, with this method, we could make delocalized superlattices of up to 30nm.
Magnetic
Binary superlattices have novel magnetic properties that form from the combination of each NCs’ individual magnetic properties.
This allows for much more control over the strength of the magnetic field of a superlattice as we can adjust it based on the type of NCs we use and the ratio in which we use them
This level of control shows promise for the use of self-assembled superlattices to store magnetic data more densely.
Luminescent
Through the long-range control of the NCs orientation via Electromagnetic fields, superlattices are a promising way to increase the efficiency of Liquid Crystal Displays (LCDs).
Today's techniques use unpolarized (dispersed, scattered) light which causes the loss of half the photons after going through a polarizer (only lets certain light waves through).
With superlattices, we could absorb the unpolarized light and then polarize the light from multiple angles allowing more photons to reach the viewer.
Plasmonic Resonance
At a high level, plasmonics is the tendency for waves of light, to interact with metal nanoparticles causing oscillations in the electric fields of each particle.
By controlling the oscillations, we can use them to indirectly measure the geometry of different molecules and nanosize particles.
Superlattices can allow us to control the oscillations by adding or removing certain ligands and using different structures.
For example, experiments with Au nanocubes have been used to measure the chirality (asymmetry in a way that an object's mirror image can not be overlapped on it perfectly) of DNA molecules.
Other experiments have shown ways to use NC superlattice to detect prions in blood and serum.
There are many other novel properties that can be discovered through the creation of superlattices and used for real-world applications. This acts as one of the driving forces for continual research into the self-assembly of NC superlattices.
Outlook on Self-Assembled Nanocrystal Superlattices
While self-assembly of nanocrystal superlattices aren't exactly the same as a pile of slap bracelets self-assembling into a house, they do show exciting promise in regards to understanding the world at a nanoscale, while also giving engineers and scientists more control over nanoscale objects and interactions.
I hope this article functioned as a good way for you to understand some of the basic processes and functions of building and using nanocrystal superlattices.
The research paper that this article summarizes goes a lot more in-depth on specific examples and gives a lot more diagrams and photos from specific experiments. If you found this interesting, I definitely suggest you go check it out here.
There's still much more research to be done, but it is clear that self-assembly, even of inorganic nanocrystals, is a very valuable field of research in understanding the processes that makeup us and the things around us.
