buckling – Softbites

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.

Original paper: Mechanical stability of particle-stabilized droplets under micropipette aspiration

Most of us have had the childhood experience of blowing bubbles. But have you ever wondered how bubbles form and what keeps them stable? The key to making bubbles is surface tension, the tension on the surface of a liquid that comes from the attractive forces between the liquid molecules. Water has a very high surface tension (that’s why bugs can walk on water) making it difficult to stretch to form a thin water layer that we see when bubbles form. By adding soap to water, we can lower the surface tension of the water, allowing us to stretch this water-air interface to form a thin water sheet. As you blow more and more air into a bubble, the bubble will grow larger and larger as the thin layer stretches. Eventually, you’ll reach the limit of the added stretchiness, and the bubble will burst, engraving in your memory its fragile nature.

Fig1-1 — A typical air bubble made out of a water-soap mixture (Figure courtesy of Gilad).

In soft matter, sometimes scientists utilize materials such as solid macroscopic particles instead of soap molecules to reduce the surface tension of an interface. Using particles to stabilize an interface allows them to tailor the mechanical and chemical properties of the interfaces to fabricate capsules. For instance, if a capsule needs to travel in blood-stream for therapeutic purposes, it must be tough enough to withstand blood pressure without rupturing. But if we make such a capsule how can we measure its mechanical response?

In this post, we’ll look into the work by Niveditha Samudrala and her colleagues on measuring the mechanical properties of a particle-stabilized interface. They utilize a direct approach of applying force on such a stabilized interface to study its mechanical response that has eluded earlier techniques. Knowing the stiffness of these particle-coated interfaces, say in the form of capsules, would enable us to use them for different controlled-release applications such as treating a narrowing artery [1] as well as tune them to have different flow properties.

The authors use tiny (smaller than a micrometer!) dumbbell-shaped particles with different surface properties to stabilize an oil-in-water emulsion (see note [2]). Here instead of a thin layer of water sandwiched by the soap molecules, the water-oil interface has been stabilized with micron-sized particles. This stabilization technique will render higher mechanical properties to the interface. Droplets stabilized in this way, known as colloidosomes, have been shown to be capable of encapsulating a wide variety of molecules.

The researchers characterized the particle-stabilized droplets using the micropipette aspiration technique. To understand this technique, imagine picking an air bubble with a straw. What you need to do is to approach the air bubble and then apply a gentle suction (or aspiration) pressure. When the suction pressure becomes larger than the pressure outside of the droplet, then the droplet gets aspirated into the straw forming the aspiration tongue (Figure 1A). Similarly, in the micropipette aspiration technique, a glass pipette (the straw) with an inner diameter of $latex R_p$ is usually used to aspirate squishy stuff, such as cells, vesicles, and here droplets.

To obtain the tension response, therefore the toughness of an aspirated interface, we need to consider the pressures applied to the interface. Let’s consider an aspirated droplet as shown in Fig 1A at mechanical equilibrium (which means the sum of all the forces is zero). We know that each interface has a surface tension acting on it (See Fig 2a). In our bubble example, I mentioned that the soap molecules tend to gather at such interface to decrease the tension (See Fig 2b). But when there are other forces acting on the interface in addition to the presence of the molecules, such as the suction pressure in our case, the tension of the interface now comes from both the surface tension and the suction force. We call this total force the interfacial tension (See Fig 2c). The Young-Laplace equation can be used to relate this interfacial tension to the pressure applied to the interface (Fig 1-B3).

Fig1. Schematic representation of the aspiration technique (A) and the Young-Laplace equations obtained at both interfaces of the outer edge of droplet and tongue inside the pipette (B). $latex P_{atm}$ is the atmospheric pressure set to zero, $latex P_{droplet}$ is the pressure inside the droplet. $latex P_{pip}$ is the suction pressure. $latex R_{v}$ is the radius of droplet outside the pipette and the $latex R_{p}$ is the pipette radius.

When the molecules, or particles in our case, are forced to pack tightly together they oppose the compression force. This opposition is felt at the interface by a pressure called surface pressure (see Fig 2c). Under the interface tension and the surface pressure, the new net interfacial tension is defined as:

$latex \tau=\gamma_{0} – \Pi$.

where $latex \Pi$ is the surface pressure, $latex \gamma_{0}$ is the interface tension which is constant for a given interface.

In this study, Samudrala and her colleagues show that there are two critical pressures after which instabilities form at the interface resulting in droplet dripping into the pipette and buckling respectively (Fig 2d). They conclude that the dripping happens due to the transition of the interface from a particle-stabilized interface to a bare oil-water interface resulting in a sudden suction of tiny oil droplets (basically the droplet drips at this point, see Fig 3B, blue and 3C).

The second instability is the buckling which the researchers propose happens when $latex \tau$ tends to zero. Now let’s see how buckling happens.

The dripping at the first critical pressure decreases the volume of the particle-coated droplet, but note that the surface area is constant because neither particles leave the surface nor the free ones join the droplet (the latter argument is assumed). The continuation of the increase in suction pressure plus the volume lost in the dripping step results in the buckling of the interface (Fig 3b red and 3E, also see note [3]). When the authors aspirate the bare oil droplets as well as droplets stabilized by small molecules, they only see the sudden droplet disappearance with no shape abnormalities due to the fluid nature of the interface rather than solid-like nature for the particle-stabilized case. But why does the buckling happen?

Screen Shot 2017-11-06 at 17.55.45 — Fig 3. Evolution of instabilities of a particle-coated droplet under tension. (a) shows the schematics of the particle-coated droplet being aspirated. (b) Change in aspiration length as a function of suction pressure. Blue line remarks the capillary instabilities. Red line shows the elastic failure of buckling process. (c & d) are the images of capillary and buckling instabilities respectively. (e) shows the case when the suction pressure is above buckling pressure at which the particle coat fails (the figure is adapted with no further change from the original paper).

Recall how we defined the net interfacial tension above; $latex \tau=\gamma – \Pi$. The authors hypothesize that upon suction of a particle-stabilized droplet, particles jam at the interface of the droplet outside of the pipette, creating a high surface pressure. When this surface pressure approaches $latex \gamma$, the net tension becomes zero ($latex \tau=0$, see fig 2d and note how the interface tension is opposed by the surface pressure due to repulsion between particles). When an interface possesses no tension, it means that the interface can no longer bear any loads. Considering any sort of defects or irregularities due to nonuniform particle packing, for such interfaces deformations such as buckling will form. Now, let’s see how the authors test their hypothesis.

The authors observed that at the tip of the tongue, there is a very dilute packing of particles in such a way that the interface to a good approximation resembles the Fig 2a, a bare water-oil interface. With this observation, one can safely assume that the interfacial tension, the $latex \tau$ is equal to the oil/water interface tension, the $latex \gamma$ and write the Young-Laplace equation across the tip of the tongue (see Fig 1B-(1)):

$latex P_{droplet} – P_{pip} = \frac{2\gamma_{0}}{R_{p}}$

where $latex R_{p}$ is the radius of the pipette and is fixed. The authors experimentally show that for a range of droplet size ($latex 10\ \mu m < R_{droplet} < 100 \ \mu m$), the droplet pressure right before buckling varies very close to zero (in above equation all parameters are known except the $latex P_{droplet}$, which is calculated when we put $latex P_{pip} = P_{buckling}$). Therefore, considering the equation (2) in Fig 1B, the net tension would be zero (see note [3]) and with this, the authors correlate that the reason for the formation of buckling is the net-zero tension of the interface.

Taking it all together, we saw that for a droplet with solid-like thin shell, the mechanical response is completely different from the bare or the molecule-stabilized interface. A fairly rigid interface undergoes buckling due to its net tension tending to zero and knowing the threshold of buckling will enable us to tune the mechanical properties of such droplets for different applications from load-caused cargo release (see note [1]) or emulsions with varied flow properties. Imagine if we encapsulate a fragrance in our air bubble, which upon rupturing will release the scent. Now, wouldn’t it be nice if we could control the toughness of this bubble or similar architecture to rupture under a specific condition that we desire (see note [1])?

[1] In a disease called atherosclerosis, the arteries narrow down due to plaque buildup. In this narrow region, the blood pressure is higher than the normal region of the artery. So one can use this pressure difference to crack release the relevant drug from the capsule only in the narrow regions of the artery to dissolve the plaques away. Neat!

[2] If we apply a shear force on a mixture of two or more immiscible liquids in the presence of a stabilizing agent, we produce an emulsion and the stabilizing agent is called an emulsifier. The particles show a significantly higher tendency to gather at an interface in comparison to amphiphilic molecules. Thus, particles are strong emulsifiers. If we mix lemon juice and oil, soon after stopping the mixing, the two solutions will separate. Now, if you add eggs, you stabilize this mixture (egg works as an emulsifier) and you get Mayonnaise!!

[3] The authors report that for particle-stabilized droplets they observed different deformation morphologies such as wrinkles, dimples, folds and in some case complete droplet failure. They attribute this diversity to the non-uniformity of particle packing at the interface. But what is interesting to me is when they decrease the suction pressure, the droplets go back to their original spherical shape and then upon the second aspiration, the deformations happen at the same exact location as were for the first aspiration. This means that during the suction, there is limited particle rearrangement (Watch here).

[4] We can easily set the atmosphere pressure to zero before aspirating the droplets, thus here the $latex P_{atm} = 0$.

Tag: buckling

Elastogranularity and how soil may shape the roots of plants

Dripping, Buckling and Collapsing of a Droplet