Fast Flow in Tiny Tubes

Original Paper: Massive radius-dependent flow slippage in carbon nanotubes


Water is made up of many many molecules of $latex H_2O$. But when you drink it from a glass or take a shower, this doesn’t matter. A typical fluid is so much bigger than an individual molecule that you can just treat it as a continuum: if you shoot water out of a hose with some initial velocity, you can use physics to figure out where the water will land without having the consider the motion of all the molecules. Considering each molecule would get the same result, but would be a vastly more difficult calculation. However, when the fluid is very small, not much larger than the size of individual molecules, then the molecular nature of water starts to matter. The “continuum breakdown” is an intriguing aspect of fluid mechanics and physics in general, but is typically very hard to study experimentally. Recently, a group of researchers based in France overcame these difficulties and managed to study water flowing through carbon nanotubes (Figure 1).

Experiment schematic
Figure 1: Left: an electron microscope image of a carbon nanotube inside a glass microcapillary. Right: a cartoon of water flowing out of the tube. Adapted from Figure 1.

When does it matter that the fluid is made of molecules? One of the implications of the molecular nature of a liquid has to do with what happens when it flows along a wall (e.g. inside a pipe). There is an assumption that the friction between the wall and the fluid will halt the flow right next to the walls, and the fluid will increase in speed farther from the walls. This is called the “no-slip” boundary condition and is a pretty central concept in fluid mechanics that makes the relevant equations much simpler (Figure 2). It is known that this condition is not perfectly true at the molecular scale. There may still be some net motion of fluid right next to the walls leading to slightly faster flow than expected from no-slip conditions because the lack of zero-velocity fluid would mean the average flow rate is higher. The exact nature of the boundary condition depends on the interaction between the molecules of the fluid and the molecules of the wall, and this doesn’t matter when the fluid is so much bigger than the region near the interface.

no slip cow
Figure 2: The slower wind speeds near the surface of the cow are an example of no-slip boundaries in action, where friction between the surface and the air causes the air to slow down. Yes, this was the best demonstration I could find of this.

The ideal way to investigate how molecular interactions affect flow near an interface would be to send fluid through something that is both very narrow and also very long compared to its width (narrow so that a large fraction of the molecules are near the wall, long so that the molecules spend more time in the tube and can be studied more easily). A water molecule is about 0.1 nanometers wide, so ideally the potential conduit wouldn’t be too much wider (for scale, a human hair is about 50,000 nanometers thick). Carbon nanotubes, which are one atom thick like graphene but wrapped into a cylinder instead of a sheet, are pretty close to the ideal: the skinniest is as small as a few nanometers in diameter and up to thousands of times as long. Of course, “just flow water through a carbon nanotube” is easier said than done.

The way to do this is to insert it into a slightly-bigger-but-still-small tube, which in this experiment was done with a glass capillary, which, being from the Latin word capillus meaning hair, is a tube so narrow it looks like a hair. A sharp tip was used to pull a single carbon nanotube out of a big tangled mess of carbon nanotubes called a “forest” (this is how they are arranged when you buy them). Then, the researchers carefully inserted the nanotube into the narrow end of the glass capillary, and the gap between the nanotube and the capillary was sealed. All of this was monitored in real time using a microscope (Figure 3). I recommend reading the Section 1 of the Supplemental Methods, it’s quite fascinating and not too technical. There is also a video of it here.

I met one of the authors of the paper, Derek Stein of Brown University when I visited his lab in 2015. He showed me a prototype of this experiment, and ribaldly described the process of inserting the carbon nanotube into the glass capillary as “nano-sex.” My father is a urologist and might describe this as the world’s smallest catheterization.  

SWNT insertion
Figure 3: A carbon nanotube was pulled from a “forest” and then inserted into a glass capillary tube. Adapted from Supplementary Figures 1 and 3.

Once the tiny tube (the carbon nanotube) was hooked up to the less-tiny tube (the glass capillary), it was simply a matter of connecting two fluid reservoirs with the composite tube, applying pressure to one side, and measuring the rate at which water flows into the other side. To do this, they put fluorescent beads in the water and observed their motion near the exit of the tube. From the speed of the beads near the tube exit, they were able to figure out how fast the fluid must be flowing (Figure 4).

SWNT flow
Figure 4: Top: A microscope image of the nozzle and some beads (I have enhanced the contrast to make the beads more visible. The big black thing at the top right is probably gunk on the microscope or camera). Bottom: The speeds of the beads (no rhyme intended) at different points around the nozzle, as determined from their motion over time. Adapted from Supplementary Figure 9.

Then, by examining the relationship between applied pressure, the measured flow velocity, and the geometry of the nanotubes, the researchers were able to measure something called the fluid permeability of the system. This quantifies how well fluids can flow through the system, analogous to the electrical conductivity of a metal. Since it is known how a fluid behaves in a tube of a given radius with perfect no-slip conditions, the team compared their measurements to those expectations. What they found was that for larger nanotubes, the results were fairly consistent with no-slip, but as the tubes got smaller and a higher proportion of the water molecules come into contact with the interface, the fluid flowed a lot faster than expected (Figure 5). In the smallest tubes, it flowed 25 times faster than expected. The velocity at the walls was not actually zero, and the flow rate was consistent with a tube with twenty times larger in diameter than the one that was actually used — a big enough result to title the paper: Massive radius-dependent flow slippage in carbon nanotubes.

no slip vs. slip
Figure 5: We treat fluids as if they flow like on the left, but in carbon nanotubes, it was more like on the right.

Why does the interaction between carbon and water lead to such massive slipping? This isn’t actually known, but at the atomic scale, friction is due to electrical interactions between the atoms that make up the nanotubes and the water. Carbon nanotubes are fairly conductive, meaning the electrons aren’t that strongly bound to atomic nuclei. The authors hypothesized that an insulating tube with the same chemical structure as a carbon nanotube would have different flow properties. Fortunately, such a thing does exist: boron nitride nanotubes. They did the same type of experiment with the insulating tubes and found that the water flowed much slower than the version with the conducting carbon nanotubes. This actually surprised them— they expected a difference, but not such a big one and they had no explanation for it:

“That these nearly identical channels exhibit very different surface flow dynamics is unexpected… simulations predict that the friction of water on carbon surfaces is lower than on boron nitride surfaces, but even these predictions strongly underestimate the difference observed here.”

Traditional solid-state physics, which deals with the electronic and magnetic properties of crystalline materials, doesn’t usually intersect with soft condensed matter physics, which deals with flowy squishier things. This experiment, showing that the way a fluid flows through a tube depends on the electrical properties of the tube, is taking a step towards bringing them together, even though its results aren’t yet fully explained.

Microcannons firing Nanobullets

Original Paper: Acoustic Microcannons: Toward Advanced Microballistics


Sometimes I read papers that enhance my understanding of how the universe works, and sometimes I read papers about fundamental research leading to promising new technologies. Occasionally though, I read a paper that is just inherently cool. The paper by Fernando Soto, Aida Martin, and friends in ACS Nano, titled “Acoustic Microcannons: Toward Advanced Microballistics” is such a paper.

The grand scheme of this research is developing a tool that can selectively shoot drugs into cells at a microscopic level. This is hard because everything happens really slowly at the microscopic scale in a liquid, in ways that meter-sized beings who live in air would not necessarily expect. For example, it is impossible for small organisms to move through a fluid using a repetitive motion that looks the same in reverse. The way we move our feet back and forth to walk would not work for a tiny aquatic human, because the forward motion in the first phase of movement would be nullified by backwards motion in the second phase. This is why bacteria use things like rotating flagella to move*.

Digressions aside, if you tried to shoot a tiny bullet through a cell wall, it would quickly halt and diffuse away before even hitting the cell wall. Soto, Martin, and collaborators wanted to beat this. Perhaps inspired by the likely unrelated Rodrigo Ruiz Soto, a Costa Rican competitive pistol shooter in the 1968 Olympics, Soto sought to develop a cannon that would change the game in the microscopic world in the same way that gunpowder technology changed things in  the macroscopic world.

The researchers developed a “microcannon,” starting with a thin membrane of polycarbonate plastic studded with small pores, which is a thing you can buy and don’t have to make. The pores would eventually serve as the molds for the barrels of the cannons. They deposited graphene oxide onto the inside of the pores using electrochemistry, and then sputtered gold onto the inside of that graphene layer.  While they were still in the plastic membrane, the cannon pores were filled with a gel (literally gelatin from the supermarket) loaded with micron-sized plastic beads to act as bullets, and the “gunpowder,” which I’ll describe after the next image. The polycarbonate is then washed away with acid, leaving free-floating carbon and gold cannon barrels a few microns in size.

cannon
Figure 1: The microcannons, loaded with nanobullets before and after firing. Adapted from Soto and collaborators

While it is generally difficult to make small things move quickly in a fluid, bubbles are somewhat of an exception to this rule. Their collapse can lead to rapid motion on tiny scales. Taking advantage of this, the authors used perfluorocarbon (molecules with the same structure as hydrocarbons but with fluorine connected to carbon atoms instead of hydrogen) droplets as a propellant, which they turned into bubbles with an ultrasound-induced phase transition (essentially blasting them with soundwaves until they vaporized). When they initiated the collapse of the bubbles, they emitted a pressure wave which drove the nanobullets out of the barrel towards their target**.

cannon2
Figure 2: Composition and operation of the microcannons.

The authors performed two tests to characterize how powerful these things were. First, they embedded the cannons in an agar gel (an algae-based substance that Japanese desserts are made of) and loaded them with fluorescent beads. They looked at where the beads were before firing the ultrasound trigger at the cannon, and after. They observed that the beads had penetrated an average of 17 microns through the gel. However, this is about the thickness of a human cell layer, so this could be used, for example, to shoot a small amount of medication through the layer of cells on the wall of a blood vessel. In some more direct studies of the damage caused by collapsing bubbles (which is a common mechanism of damage to ship propellers), the jets that formed when bubbles collapse were shown with high-speed photography to penetrate about a millimeter into a gel. However, these bubbles were 1000 times bigger than those formed in the microcannons, and it’s not out of the question to assume that the penetration depth scales with bubble size.

jjrqrsy
Figure 3: High-speed photography of a millimeter-sized bubble collapsing near a gel wall and shooting a jet into the gel. The mechanism of nanobullet-firing and penetration is a smaller version of this. From Brujan, Emil-Alexandru, et al. “Dynamics of laser-induced cavitation bubbles near an elastic boundary.” Journal of Fluid Mechanics 433 (2001): 251-281.

The bullets were too fast to record with a microscope camera, so their second test involved recording the motion of the cannon after it fired the bullets. Naively, one would expect to be able to calculate the bullet speed with conservation of momentum from knowing the cannon’s speed, but momentum isn’t conserved in a noisy viscous environment (which brings us back to why it’s so hard for microorganisms to move around). They modeled the fluid dynamical forces acting on the system, measured that the terminal speed of the cannon was about 2 meters per second, and concluded that the initial speed of the bullets is 42 meters per second or 150 kilometers per hour (see appendix). Pretty fast, especially for something so small in a draggy environment.

After finding this paper I emailed the first author, Fernando Soto, saying that I enjoyed his paper, and he responded by saying that he was glad that other people liked his “very sci-fi nanodream.” I don’t know if this technology will succeed in the authors’ goal of localized drug delivery to cells, but I think it’s awesome that they made a functioning microscale cannon.

cannon3
Oh the humanity.

*I recommend reading Life at Low Reynolds Number if this interests you.

**Or just in whatever direction it was pointing, I guess.


Appendix: Velocity calculation

The researchers wanted to figure out how fast the bullets were moving based on their measurement of how fast the cannons were moving. Normally you could just use conservation of momentum, but because of the surrounding fluid, momentum is not necessarily conserved (unless you know the momentum of the fluid as well).

However, we understand how velocity decreases in a fluid based on drag: if the velocity is low, the drag force arises from separating the water molecules from each other, and the force is linear with velocity. If the velocity is high, the force arises mainly from accelerating the water to the speed of object, and the force is quadratic with velocity. To figure out which rule applies you can calculate what’s called the Reynold’s number, Re, which is the ratio of inertial to viscous forces in a fluid. If Re is in the thousands or higher,the flow is turbulent. f it’s below 100, the flow is smooth, or laminar. Specifically, the Reynold’s number is calculated as:

$latex Re=\frac{\rho L v}{\mu}$

where $latex \rho$ is the density of the fluid, L is the length of the object in the flow, v is its velocity, and $latex \mu$ is the viscosity. The microcannon was seen moving at about a micron per second, it was about 15 microns long, and the high speed photograph was done in water (density of 1 kg/L, viscosity of about 0.001 pascal seconds). This means the Reynold’s number was about 13, in the laminar regime, and that drag is due to viscosity and linear.

The equation of motion for a slowing object undergoing viscous drag with an initial velocity is

$latex v(t)=v(0)e^{-kt/m}$

where m is the mass of the cannon (known from stoichiometry) and k is the drag coefficient which depends on the viscosity as well as the geometry of the object experiencing drag. Because they know v(t) (as determined from high speed videography), t (the time since detonation), k, and m, they can find v(0).

Then it is assumed that momentum is conserved during the detonation, so the nanobullets with known mass can have their velocity calculated from

$latex v_{c}m{c}=v_{b}m_{b}$

where the indices c and b refer to cannon and bullet. The velocity was calculated to be 42 m/s. Pretty fast.