Author: Adam Fortais

Adam Fortais is a PhD candidate studying the physics of soft matter at McMaster University in Hamilton, Canada. When he is not squishing things in the lab he likes to write, make art, and sing karaoke.
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Researchers play with elastic bands to understand DNA and protein structures.

Topology, Geometry, and Mechanics of Strongly Stretched and Twisted Filaments: Solenoids, Plectonemes, and Artificial Muscle Fibers

Much of how DNA and proteins function depends on their conformations. Diseases like Alzheimers’ and Parkinsons’ have been linked to misfolding of proteins, and unwinding DNA’s double-helix structure is crucial to the DNA self-copying process. Yet, it’s difficult to study an individual molecule’s mechanical properties. Manipulating objects at such a small scale requires tools like optical and magnetic tweezers that produce forces and torques on the order of pico-Newtons, which are hard to measure accurately. One way around these difficulties is by modeling a complicated molecule as an elastic fiber that deforms in predictable ways due to extension and rotation. However, there are still many things we don’t know about how even a simple elastic fiber behaves when it is stretched and twisted at the same time. Recently, Nicholas Charles and researchers from Harvard published a study that used simulations of elastic fibers to probe their response to stretching and rotation applied simultaneously. The results shed light on how DNA, proteins, and other fibrous materials respond to forces and get their intricate shapes.

Before continuing, I would recommend finding a rubber band. A deep understanding of this work can be gained by playing along with this article.

Long and thin elastic materials, (like DNA, protein, and rubber bands), are a lot like springs. You can stretch or compress them, storing energy in the material proportional to how much you change its length. However, compressing them too much may make the material bend sideways, or “buckle”. It might be more natural to think of this process with a stiff beam like in Figure 1, where a large compressive load can be applied before the beam buckles. But since your rubber band is soft and slender, it buckles almost immediately.

A stick is compressed and at a certain pressure, buckles.
Figure 1. A straight, untwisted stick is compressed and buckles. It’s stiffer and thicker than your rubber band, so it sustains a higher load before buckling. (https://enterfea.com/what-is-buckling-analysis/)

Likewise, twisting your rubber band in either direction will store energy in the band proportional to how much it’s twisted. And, like compression, twisting can also cause it to deform suddenly. Instead of buckling, the result is a double-helix-like braid that grows perpendicular to the fiber’s length, as shown in Figure 2. An important caveat is that the ends of the rubber band are allowed to come together. But what happens when the ends of the band are fixed?

An elastic fiber is twisted into a plectoneme. It looks like a double-helix.
Figure 2. An elastic fiber is held with little to no tension and twisted. A double-helix, braid-like structure is formed. (Mattia Gazzola, https://www.youtube.com/watch?v=5WRkBWXUCNs)

Fixing the ends of a rubber band forces it to stretch as it twists. When this happens, a different kind of deformation can occur that combines extending, twisting, and bending the fiber. By stretching and bending simultaneously, the band forms a solenoid that is oriented along the long-axis of the band, reminiscent of the coil of a spring. An example of the solenoid shape appears in Figure 3.

An elastic fiber is held under tension and twisted. A solenoid structure is produced.
Figure 3. An elastic fiber is held at high tension and twisted. A solenoid structure is formed. (Mattia Gazzola, https://www.youtube.com/watch?v=0LoIwE37aNo)

All of the phenomena described above can be seen by playing with rubber bands, yet a quantitative understanding of how these states form and how to transition between them has remained elusive. To tackle this problem, Charles and coworkers used a computer simulation to calculate the energy stored at each point along an elastic fiber when it is stretched and twisted. The simulated fiber was allowed to deform and search for its lowest energy configuration, a process critical to navigating the system’s instabilities and finding the state you would expect to find in nature.

Figure 4 summarizes some of the different conformations attained by a fiber that is first stretched, then twisted to different degrees. We can see how a fiber with the same tension and different degrees of twist can lead to any one of a wide range of conformations. For instance, a fiber remains straight (yellow dots) when it’s stretched to a length $latex L$ that is 10% longer than its original length $latex L_{0}$ $latex (L/L_{0} = 1.1)$ until it is twisted by $latex \Phi a \approx 1$, where $latex \Phi a$ is the degree of twist multiplied by the fiber’s width divided by its length. Above this value of $latex \Phi a$, the simulated fiber twists into the braided helix structure seen in Figure 2 (blue dots). Likewise, when $latex L/L_{0} = 1.2$, the fiber remains straight until it has a much higher twist, $latex \Phi a \approx 1.5$, where it forms a solenoid (red dots).

Phase diagram of fiber conformations as a function of twist and stretch.
Figure 4. Conformation of a simulated fiber under constant extension $latex L/L_{0}$, twisted by $latex \Phi$ normalized by the fiber dimensions $latex a$. Orange dots are straight, blue dots are double-helix braids, red dots are solenoids, and green dots are mixed states. Black and grey symbols are experimental results from a previous study.

Considering the vast understanding of the universe that physics has given us, it may be surprising that there is so much left to learn from the lowly rubber band. While it’s fun to play with, understanding the way fibers deform could help researchers understand all sorts of biological mysteries. For instance, your DNA is a unique code that contains all of the information needed to create any type of cell you have, but depending on where the cell is in your body, that same DNA only makes some specific cell types. The cell can do this by selectively replicating sections of its DNA while ignoring others. One way it does this is by hiding away certain regions of DNA through folding. Exploring the way simple elastic fibers deform could help explain the way DNA knows how to make the right cells, in the right places.

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Mechanism of Contact between a Droplet and an Atomically Smooth Substrate

Floating droplets shed new light on the flow of fluid at interfaces

Original article: Mechanism of Contact between a Droplet and an Atomically Smooth Substrate

When an experiment doesn’t behave the way we expect, either our understanding of the relevant physics is flawed, or the phenomenon is more complicated than it appears. When a theoretical prediction is off by two orders of magnitude – like what was observed in this recent paper by Hua Yung Lo, Yuan Liu, and Lei Xu of the Chinese University of Hong Kong – something is seriously wrong.

If Lo and colleagues drop a liquid droplet onto a smooth, flat surface, it will take on an equilibrium shape which depends on the properties of the liquid and solid materials at the interface (eg. water on Teflon will form a nearly perfect spherical drop while water on stainless steel will spread out, forming a spherical cap). For low viscosity fluids, the equilibration process happens almost instantly… unless the surface is very flat and very smooth.

If the surface below a droplet is atomically smooth (not a single atom is out of place to roughen the surface), a thin layer of air will form between the droplet and the surface, keeping the droplet from making contact with the surface. Eventually the trapped air will escape, draining out like how a liquid would, allowing the droplet to collapse onto the surface. Traditional fluid dynamics simulations predict that the collapse would take between 10 – 100 seconds. In experiments, however, contact generally happens in less than one second. Lo and coworkers set about investigating this seeming contradiction by observing the flow that happens within the air and liquid at the boundary between a droplet and a smooth surface.

To study this problem, the researchers dropped small spherical oil droplets (1.7 mm diameter) onto a glass surface with a very thin coating of oil which could be tilted. They observed that droplets would compress and bounce as they floated on a pocket of air, before collapsing onto the surface. The contact area was imaged from the bottom and side simultaneously using two high-speed cameras. Side-on sequences are shown in Figure 1 with a slightly tilted surface (a) and a perfectly leveled surface (b). While both droplets collapsed onto the surface far quicker than predicted by simulations, the droplet on the leveled surface was observed to float just above the surface approximately 10 times longer than on the tilted surface before collapsing.

Figure 1. A time sequence of an oil droplet being dropped on an atomically smooth, oil coated, glass surface which is a) slightly tilted (0.3°) and b) leveled (0°). c). Schematic of the droplet and surface. A video of the process can be found here (Figure adapted from the original paper.)

The effect the tilted surface has on this phenomenon became more apparent when viewed from below. On the tilted surface, the droplet would “skid”, observed as a sliding of the droplet’s center from the red point to the blue point in the direction of the green arrow as shown in Figure 2 a) while the size and shape of the air pocket was measured using two-wavelength interferometry [1]. Tilting the surface caused an asymmetric air pocket to develop, with a thinner gap at the front of the droplet and a thicker gap at the back. When a droplet did not skid, it formed a symmetric air pocket like in Figure 2 b). A thinner gap (with difference of just half micrometer) lets the air drain out (and allows contact to be made) much faster than it would for a symmetric air pocket on a flat surface. However, even a flat surface drained 10-times faster than expected.

Figure 2. Images of the bottom of an oil droplet coming in contact with an oil-coated glass slide that is a) slightly tilted, showing a droplet skidding until it reaches full contact with the surface at 109 ms, and b) perfectly leveled, where the droplet still has not contacted the surface at 392 ms. Light and dark bands correspond to the change in thickness of the air pocket. A video of the process in a) and b) can be found here and here.

To understand the flow of air from under the droplet, the researchers modeled it as a low-viscosity fluid. When a low-viscosity fluid flows past a wall (like water through a tube), the friction at the walls may reduce the flow near the walls to something-close-to-zero. This is called a “no-slip boundary condition”. On the other hand, a “plug flow boundary condition” means there is significant slip and therefore flow along the walls. Each of these boundary conditions lead to characteristic velocity profiles like those presented in Figure 3 a). Typically, one would assume that air flowing through the air pocket near the oil interface would have a no-slip boundary condition while something like a sludge or gel would demonstrate plug flow. Yet, it is this assumption that ends up being incorrect.

The researchers measured the velocity of oil within the oil droplet and the surface coating using particle image velocimetry, a technique where small light-reflecting particles are mixed into a material and tracked down as they move along with the surrounding fluid. An image of the oil droplet seeded with the tracer particles is shown in Figure 3. In this way, the researchers were able to directly visualize flow of oil at the air-oil boundaries, finding a sort of “slip layer” along the walls corresponding to the layer of oil being dragged along by the air. This lets larger volumes of air drain from under the droplet, explaining the surprisingly short time it takes for droplets to collapse onto the surface.

Figure 3. a) The velocity profile of air under the droplet (Vr) is a combination of a no-slip (Vp), and slip (Vc) boundary conditions. b) Side-view image of an oil droplet. White dots are reflective particles with velocity shown as yellow arrows. c) Bottom-view of the same oil droplet where the colored streaks (red to purple) trace the flow of the oil on the surface. (Figure adapted from the original paper.)

Despite its apparent simplicity, Lo et al. revealed a fundamental misunderstanding in the way scientists thought about how fluids flow near an interface. Accounting for the effect of slip, the researchers unified both theory and observation and explain why liquid droplets will make contact with a perfectly smooth surface so much faster than originally expected.

[1] a technique that uses light interference to quantify changes in thickness as light and dark bands; narrow bands correspond to rapidly changing thicknesses, much like the lines on a topographic map show changes in elevation. ^

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Elastogranularity and how soil may shape the roots of plants

Original paper: Elastogranular Mechanics: Buckling, Jamming, and Structure Formation

How an elastic beam deforms under load has been a question for as long as there have been engineers to ask it. In some cases, the force on a beam is approximated as a single point. For example, if a diving board is large enough, a diver at the end can be treated as a point mass on the beam. Another common approximation is to consider the force to be a continuous pressure along its length. Treating wind that bends a tree branch as a continuous pressure along the branch’s length is much simpler than adding up the force from every molecule of air on the branch. However, consider the case of a root growing into a granular material like soil. As the root burrows through the soil it will bend due to varying point-like forces along its length. The result is a branching and twisting root system that tries to grow along the path of least resistance. An example of the diversity in plant root morphologies is shown in Figure 1 and gives a sense of how complicated and interesting the physics behind this growth can be.  

Figure 1. An example of the complex and diverse morphology of several types of plants which all live in the same ecosystem [1]

With this in mind, David J. Schunter Jr. et al. from the Holmes group at Boston University have developed a beautiful experiment to study what they call “elastogranular” phenomena. In their experiment, an elastic beam is inserted into a box which is filled at a particular density with uniform beads. An image of a typical experiment is shown in Figure 2. Once the beam reaches the end of the box it will not be able to penetrate any further, and trying to push more of the beam into the box will cause a compression along the beam’s length. This resembles a classic experiment where a beam is compressed in the same manner, but in the absence of beads. Compressing the beam against the end of the box becomes increasingly difficult until eventually the beam “pops” into one large buckle. In this simpler case, the beam was free to buckle with no restrictions. Introducing beads to the system reinforces the beam non-uniformly and constrains the shapes it can take on when it buckles. This complicates the buckling event and leads to interesting new behaviors.

Screenshot 2018-05-11 at 6.13.31 PM.png
Figure 2. An elastic beam is inserted into a box of length $latex L_{0}$ and width $latex 2W_{0}$ filled with beads at an initial packing fraction of $latex \phi_{0} = 0.89$. After a length of beam is inserted equal to $latex L_{0}$, inserting additional length $latex \Delta$ results in buckling. In this experiment the beam takes on two buckles with wavelength $latex \lambda$, and amplitudes of $latex A_{0}$ and $latex A_{1}$ respectively.

In the experiment performed by the Holmes group, a beam is compressed against the end of the box until the beam buckles. If the packing fraction $latex \phi_{0}$ (the fraction of space within the box that is covered in beads) is low, it buckles much like one would expect in the absence of beads—one large buckle, as in Figure 3i. If $latex \phi_{0}$ is higher at the beginning of the experiment, like in Figures 3ii and 3iii, the buckling behavior becomes more complicated. The beam will form one large buckle as before, and as the buckle grows it will take up more area on one side of the box. This forces the beads to reorganize themselves, and the beads on the compressed side of the box become very tightly packed. At this point, they are in a hexagonal arrangement, and they are said to have crystallized [2]. As the beads in one side crystalize, that side becomes stiffer and suppresses further growth of the amplitude of the first buckle, $latex A_{0}$. In order to accommodate the extra length being inserted, $latex \Delta$, an additional buckle forms. The difference in buckling behavior for three values of $latex \phi_{0}$ are shown in Figure 3. [3]

Figure 3. Shape profiles of the inserted rod for various packing fractions ($latex \phi_{0}$) for the same normalized inserted length $latex \Delta/L_{0}$. i) At low $latex \phi_{0}$ the beam forms one large buckle. ii) As $latex \phi_{0}$ increases, the amplitude of the single buckle is suppressed, leading to a second buckle forming on the other side. iii) At even higher $latex \phi_{0}$, the buckles rotate and grow toward each other.

Figures 3 shows that not only are the number and amplitude of buckles significantly affected by the beads that surround the beam, but the orientations of the buckles are changed as well. When the experiment begins at a high $latex \phi_{0}$, the beam finds both sides of the box to be stiff and hard to penetrate. The initial buckle does not grow very much before the second buckle forms, and at high enough $latex \phi_{0}$ both buckles occur nearly simultaneously, forming a twin buckle. Figure 4 shows two systems with different $latex \phi_{0}$ forming twin buckles. In the top sequence where $latex \phi_{0}$ is lower, the buckles maintain a constant distance between each other as they grow since there are plenty of uncrystallized areas (light blue circles) ahead of the buckles into which they can grow. The lower sequence shows that, at higher $latex \phi_{0}$, the majority of uncrystallized beads are found in the wake of a buckle so the buckles instead grow into these regions, as demonstrated by the red lines.

Figure 5. Twin buckles increase in amplitude as $latex \Delta$ increases (left to right) growing into less-dense, uncrystallized regions (light blue circles). For lower $latex \phi_{0}$ (top sequence), uncrystallized beads in front of the buckles can be pushed aside, crystallizing beads away from the buckles (dark blue and yellow circles). At higher $latex \phi_{0}$ (lower sequence), the uncrystallized sections occur behind the buckles, causing the two buckles to grow closer together which is shown by the red lines.

This system bears a striking resemblance to that of plant roots growing into the soil and could be useful in understanding how environmental pressures cause plant root systems to evolve. For example, cacti need to absorb as much water as they can from their environment. One way of accomplishing this is to increase the surface area of the root system by growing wide and close to the surface, rather than deep, in order to collect water from a larger area. By developing thin roots that buckle before they can deeply penetrate the soil, many cacti are able to produce the shallow, wide-reaching roots system they need to find water.

David J. Schunter Jr. and coworkers have shown that combining two well-understood problems—buckling of a beam and reorganization of beads—can lead to unique and interesting bending dynamics. By confining a beam to a box of beads, the buckling of the beam becomes strongly influenced by the packing fraction and reorientation of the beads. This particular system shows a strong resemblance to plant root growth, but also be informative for synthetic applications involving the insertion of flexible filaments into deformable materials.

[1] Michigan Natural Shore Partnership, http://www.mishorelinepartnership.org/plants-for-inland-lakes.html

[2] When spheres crystallize in two dimensions, the hexagonal lattice is the closest possible packing with an area fraction of $latex \phi = 0.9069$. Interestingly, Figure 3iii shows a packing fraction of 0.91, which is higher than this maximum value. This is because the beads are able to pop out of the plane at very high compressions, which can lead to a calculated packing fraction larger than that of the hexagonal lattice. For more information on hexagonal packing, see Wikipedia.

[3] For more information about the specifics of how the deformation of the beam is quantified, a summary of the analysis is available here.