Colm P Kelleher – Softbites

2018-07-192018-07-19

Seeing Inside Sand: Visualizing Force Chains with Photoelastic Disks

Original Article: Contact force measurements and stress-induced anisotropy in granular materials

As their name suggests, so-called “granular materials” are made up of “grains” — small (but macroscopic) pieces of sand, glass beads, coffee grounds, or almost any other solid you can think of. Granular materials can flow like a liquid (like sand in an hourglass), resist deformation like a solid (like the sand under your feet at the beach), or quickly transition between these states (like pebbles in a rockslide).

Granular materials have properties that have no equivalent in regular materials like wood, metal, or rubber. In solids like these — the kind we learn about in materials science class — a force applied to the surface propagates through the material smoothly and predictably. If a uniform force is applied to the surface of a material, every equally sized cross-section of that material bears the same amount of load. In granular materials, however, the situation is very different: in a sand pile under stress (that is, when a force is applied to its surface), the force is distributed unevenly — some individual sand grains bear far more load than others. Surprisingly, this remains true even when the sand grains themselves are identical. What’s more, the load-bearing grains connect to one another to make a fractal, lightning-like pattern inside the material, like that shown in Figure 1. These string-like arrangements of load-bearing grains are called force chains.

Screen Shot 2018-07-19 at 12.08.53 PM — **Figure 1 – Force chains in a computer simulation of a sand pile. The thickness of a black line indicates the magnitude of the force at that point inside the sand pile. (From** **Nadukuru & Michalowski (2012)**.)

As Figure 1 shows, force chains are easy to identify in a computer simulation. But can you “see” forces inside a real material? Today’s paper — which is from 2005 but has already proven to be a classic in the field — shows us how it can be done. The secret lies in a clever choice of “grain”: in their experiments, Majmudar and Behringer use about 2,500 transparent plastic disks, each about a centimeter in diameter and half a centimeter tall. These disks are placed in a thin container that confines them to a single plane — this experiment is similar to the board game Connect Four, but without the vertical rails.

Crucially, these plastic disks have a property called photoelasticity: when they are stretched or squeezed, they deform, and when they deform they alter the polarization state of light passing through them. For instance, linearly polarized light might be converted into circularly polarized light, or light that’s still linearly polarized, but along a different axis than before. Thus, placed between crossed (perpendicularly oriented) polarizers, an unstressed disk will appear dark, but a squeezed or stretched disk will appear bright, since any alteration of the polarization state of the incoming light will allow some of it to pass through the second polarizer. What’s more, the pattern of light — like that shown in Figure 2 — can be used to infer the normal and tangential forces acting on each disk.

Screen Shot 2018-07-19 at 12.14.35 PM — Figure 2 – Two plastic disks placed between crossed polarizers [1]. For an unstressed disk (top), the polarization state of the light is unaltered, and no light gets through the second polarizer. For a disk under load (bottom), the polarization state of transmitted light is altered — in this case, the polarization axis is rotated — allowing some light to pass through the second polarizer. The forces on the disk, indicated by thick black arrows, can be inferred from images such as the one on the bottom right

By imaging lots of disks at the same time, photoelasticity can be used to infer the overall stress pattern inside a granular material. Majmudar and Behringer are especially interested in two particularly simple situations: isotropic compression and shear. Under isotropic compression, the collection of disks is squeezed equally from all sides, while under shear, the collection of disks is squeezed on top and bottom, but allowed to expand by an exactly equal amount at the sides.

Interestingly, the system responds very differently to these two types of load: for isotropic compression, the force pattern, shown in the left panel of Figure 3, resembles a random network — short chains of highly stressed disks connect over distances of a few diameters. For shear (Figure 3, right panel), the situation is very different: long force chains, tens of disk diameters in length, extend in the direction along which the system is being squeezed. This phenomenon, where an applied stress causes the material itself to change in a direction-dependent manner, is called stress-induced anisotropy; it is not captured by the linear elasticity theory that students typically learn, even in advanced material science classes.

In the decades since this paper was published, the techniques pioneered by Majmudar and Behringer have allowed scientists to better understand properties of granular materials: under what circumstances force chains form, how they depend on properties of the disk such as shape and friction coefficient, and how they influence behaviors such as jamming – the rapid transition from a flowing state to a rigid one.

Screen Shot 2018-07-19 at 12.18.06 PM — **Figure 3 – A granular material under isotropic compression (left), and shear (right). In the sheared system, long, oriented force chains are clearly visible.**

Postscript: On the day of publication, we learned of the recent death of the PI of this paper, Bob Behringer, at the age of 69. This post highlights just one of the many contributions of this widely respected scientist to the field of soft matter physics. For a more detailed overview of Behringer’s life and work, see here.

Notes:

[1] The experiment described in the paper used crossed (oppositely oriented) circular polarizers rather than the linear ones shown here, but the principle is the same.

2018-01-012018-04-14

Kepler’s New Year’s Gift — On the Six-Cornered Snowflake

Original Booklet: On the Six-Cornered Snowflake: A New Year’s Gift by Johannes Kepler

Some things never change. In winter 1610, Johannes Kepler was stressing out about holiday gifts — in particular, one for his friend and benefactor, the rather grandly-named Johannes Matthaeus Wacker von Wackenfels. Kepler, at the time employed as Imperial Mathematician at the court of Holy Roman Emperor Rudolph II, records his musings on the problem in the opening pages of his now-famous discourse, The Six-Cornered Snowflake.

Kepler sets a high standard for himself. His gift should be of the intellectual variety: an amusing idea or a clever argument. After all, that’s why Wacker keeps him around. Kepler considers several potential topics, dismissing each in turn as being either too serious or too light. Kepler’s intellectual respect for Wacker — who was an accomplished scholar and amateur philosopher in his own right — renders other topics off limits. (An example: given the impressive size of his patron’s zoological library, Kepler jokingly complains that writing a treatise about animals would be “like bringing owls to Athens.”) Wandering around town, feeling guilty about his procrastination, Kepler notices some snowflakes landing on his coat, “all with six corners and feathered radii” [1]. Kepler immediately identifies the perfect topic for his essay:

“‘Pon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year’s gift… the very thing for a mathematician to give.”

The pattern Kepler alludes to here is the six-fold shape of the snowflake [2]. Considering this shape, Kepler alights on what will become the central puzzle of the piece,

“Our question is, why snowflakes in their first falling, before they are entangled in larger plumes, always fall with six corners and with six rods, tufted like feathers.”

In other words, why should snowflakes be six-sided, rather than five-sided, seven-sided, or anything-else-sided? Kepler’s attempts to answer this question are a treasure trove of condensed matter physics: they include the first observation in print that regular shapes can arise from close-packing of identical objects [3] and the famous conjecture that hexagonal packing is the densest way to fill space with spheres [4]. But perhaps the most interesting thing about The Six-Cornered Snowflake is how Kepler sees nature and how he wants the reader to see it.

Untitled.001.jpeg — **Left:** ***Kepler’s sketches of sphere packings.*** **Middle:** ***Thomas Bentley’s classic snowflake photographs, taken around 1902.*** **Right:** ***Pyramidal packing of colloidal spheres gives the beetle*** ***P. argus its characteristic iridescence. (Scale bar is 1 micron.)***

Intellectually and personally, Kepler straddled the gradual transition away from the medieval era of alchemy and astrology [5], and towards the modern age of empirical observation, mathematical models, and testing ideas by experiment. Despite some asides that strike the modern reader as somewhat mystical in character — for instance his numerological obsession with the properties of the natural numbers — we can immediately recognize The Six-Cornered Snowflake as a scientific work. While the whole chain of reasoning is somewhat convoluted [6], The Snowflake includes the following:

1. Kepler imagines matter as being made of tiny, discrete “pieces” that are all identical to one another;

2. He considers (and draws!) how the arrangement of those pieces might influence the material properties of an object, in particular, its shape. In modern parlance, Kepler defines a crystal: a macroscopic object made out of identical pieces arranged into a regular structure called a lattice [7];

3. He tries to articulate a physical principle — such as close-packing — that might explain why the small pieces arrange themselves in such a way as to produce a crystal lattice.

By Kepler’s own admission, none of his arguments adequately explain the shape of snowflakes. (For one thing, he can’t figure out how a three-dimensional process could possibly create two-dimensional crystals.) But, despite this failure, Kepler still manages to suggest a productive direction for future research. Noting that different substances crystallize into different 3D shapes [8], Kepler finishes his essay by kicking the problem over to another branch of the natural sciences: “I have knocked at the door of chemistry and see how much remains to be said before we can get hold of our cause.” In other words, Kepler correctly intuits that understanding the form of crystals necessitates understanding their chemistry.

Today, we know that macroscopic objects are indeed made of tiny identical pieces — atoms and molecules — and that those pieces often arrange themselves in structures that are highly reminiscent of Kepler’s sphere packings. We have learned how to accurately describe the forces that bind atoms together or push them apart. In addition to the shape of crystals, we know that many important material properties — most strikingly rigidity, the ability of a solid to resist deformation — arise because of the regular arrangement of atoms on the micro-scale. We understand (some) statistical physics, which explains how, at high enough temperature, thermal motion overcomes the forces holding the atoms in place, destroying the lattice and melting the solid. Our knowledge of the physics and chemistry of solids has allowed us to engineer with precision the technologies — in particular silicon-based semiconductors — that underpin the modern world.

Although Kepler couldn’t have begun to imagine all this, scientifically speaking, the world of The Snowflake is very modern: a world of material cause and material effect, of microscopic bodies in motion and in contact, a world of forces that are invisible yet comprehensible, and where the properties of the whole can be understood by considering its parts. As physics students, we learn that all the fundamental physical laws can be written on the back of a napkin. And yet, the materials in the world around us exhibit an amazing variety of properties: solid and liquid, conductive and insulating, magnetic and not. How can such a zoo of behaviors and properties arise from physical laws that are fundamentally simple? Kepler’s essay gives us a framework to understand the apparent contradiction. Kepler says: look inside. Look at the pieces. Look at the structure and the symmetry.

A pretty good New Year’s gift for a soft matter enthusiast, even in 2018.

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[1] In today’s world of snowflake wrapping paper, snowflake ornaments, and snowflake emoji, it’s hard to imagine that there was a time when people didn’t know what snowflakes actually look like, but Kepler was apparently the first European to write about the hexagonal symmetry of snow crystals. (The observation is, however, recorded in much older Chinese documents, from the 2nd century BC.)

[2] In fact it isn’t: there are triangular snowflakes too, and how they form is still an active area of research.

[3] Prior to the publication of The Snowflake, English scientist Thomas Harriot privately communicated his ideas on the efficient stacking of cannon balls to Kepler.

[4] This conjecture was only formally proved in 2015, by a group of mathematicians led by Thomas Hales.

[5] Kepler, who was at times employed as an astrologer, thought that astrology, as practiced in 17th century Germany, was mostly nonsense. However, he himself cast horoscopes that he claimed were correct and argued with scholars who wanted to dismiss the discipline entirely.

[6] Despite Kepler’s assurances in the introduction that his piece is “next to nothing,” The Snowflake is 21 pages long — evidence, I think, of the modern tyranny of page limits and copy editors.

[7] Interestingly, this argument correctly predicts the form of so-called “complex materials” like opal (colloidal silica), where the pieces really are (relatively) uniform hard spheres. The water molecules in an ice lattice have much more complicated, directional interactions arising from the hydrogen bonds between them, and so their crystal structure is harder to understand or predict. In fact, it seems that Kepler generally has something like colloidal particles in mind throughout The Snowflake, rather than modern atoms or molecules.

[8] “But the formative faculty of the earth does not take to her heart only one shape; she knows and is practiced in the whole of geometry. I have seen… a panel inlaid with silver ore; from it, a dodecahedron, like a small hazelnut in size, projected to half its depth, as if in flower.”

2017-11-222018-04-14

Termite Climate Control

Original Article: Termite mounds harness diurnal temperature oscillations for ventilation (Non-paywall version here.)

Disclosure: The first author of this paper, Hunter King, is a friend of the present writer (CPK).

Termites are among nature’s most spectacular builders, constructing mounds that can reach heights of several meters. Relative to the size of their bodies, these structures are considerably larger than the tallest skyscrapers constructed by humans [1]. Surprisingly, in many termite species, individual termites don’t spend much time in these mounds. Instead, they live in an underground network of tunnels and chambers that can be home to millions of individual insects. But, if not to live in them, why do termites build such intricate and gigantic above-ground structures [2]?

Scientists have suggested several possibilities: mounds might provide protection from predators, or guard against rain or dramatic changes in temperature. Recent research, however, has focused on the idea that a mound’s main purpose could be to provide ventilation. The problem of ventilation is particularly important for species such as Odontotermes obesus, native to the Indian subcontinent, that “farm” a species of fungus [3]. As human cultivators will no doubt be aware, indoor farming requires careful control of atmospheric conditions. According to this picture, the mound functions like a giant lung, enabling the colony to expel carbon dioxide and exchange it for atmospheric oxygen. But how exactly might this lung work?

Human lungs use a muscle, the diaphragm, to mechanically push out old (carbon-dioxide-rich) air, and suck in fresh (oxygen-rich) air. Obviously, termite mounds don’t have moving parts that would allow them to do this. So what is the physical mechanism that drives gas to flow around the ventilation shafts inside the mound? Over the years, researchers have proposed several ideas, including driving by thermal buoyancy (the tendency of hot air to rise upwards) or external wind. The details of these models are controversial: for instance, thermal-buoyancy-driven flows require temperature differences between different parts of the mound. Are these temperature gradients caused by external heating (that is, from the sun), or by heat generated by the bodies of the termites themselves [4]?

In today’s paper, Hunter King, Samuel Ocko and Lakshminarayanan Mahadevan describe a series of experiments that might help to answer some of these questions. To test the “mound-as-lung” model described above, King and co-workers designed and built directional airflow sensors tailored to the cramped environment and low airspeeds found in the ventilation shafts of mounds built by O. obesus. The mounds, shown in Fig 1A, look a bit like a half-folded umbrella, with ripple-like “flutes” decorating a roughly cone-shaped structure.

King and co-workers measure, as a function of time of day, the air flow velocity in the ventilation conduits near the base of the flutes. These measurements, as the authors put it, are “difficult for several reasons,” in particular the “hostile and dynamic” environment inside the mound — the tendency of termites to aggressively attack anything placed inside their nest, and cover it with “sticky construction material.” As well as measuring the air velocity, King and co-workers use temperature sensors to measure the temperature profile of the surface of the mound, and the carbon dioxide concentration in the nest, underneath the mound, and at the “chimney,” near the top of it. To test the role of heat generated by the bodies of the termites, the researchers also study a “dead” — that is, abandoned — mound.

Figure 2: The top two panels show the air velocity and temperature differential for living mounds (top panel) and one dead mound (middle panel). The bottom panel shows the carbon dioxide concentration in the underground nest, and in the chimney, near the top of the mound. Carbon dioxide in the nest builds up when the temperature differential is small and the air flows slowly. It starts to decrease with increasing temperature differential and increasing flow speed (i.e. more negative flow velocity).

The results of some of these experiments are shown above. In particular, King and co-workers observe similar flow and temperature patterns in the “living” and “dead” mounds and conclude that metabolic heating is not the central mechanism driving ventilation. Noting that the direction of the flow reverses during the night, King concludes that “diurnally driven temperature gradients” — that is, temperature differences caused by the day/night cycle — ventilate the nest. This process is facilitated by the most distinctive architectural feature of the mound, the flutes.

Like fins on a radiator, the flutes efficiently exchange heat with their environment. In the heat of the sun, the flutes heat up faster than the interior, as shown in the IR camera image above. This causes the air in the flutes to rise, thus creating circulation inside the mound. The resulting flow carries oxygen-rich air from the chimney down to the nest. During the night, the flutes cool down faster than the interior, causing the flow pattern to reverse. According to the model that King and his colleagues propose, the termite mound performs the unusual feat of extracting useful work from oscillations in an intensive (in the sense of thermodynamics) environmental parameter.

King and his co-workers speculate that this energy-efficient ventilation strategy, which has evolved over millions of years, might provide inspiration for human designers of environmentally friendly architecture.

Note: After this post was written (but before it was published), the same team published a second paper where they try to find out if the same model applies to mound built by another species of termite that lives on a different continent (spoiler: it does, but some of the details differ).

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Notes:

[1]

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[2] http://www.bbc.com/earth/story/20151210-why-termites-build-such-enormous-skyscrapers

[3] The termites bring partially digested wood back to their nest, where the fungus extracts nutrients and energy from it. In return, the fungus produces fruiting bodies that the termites can eat. The relationship between termite and fungus can be referred to as “farming” or “symbiosis,” depending on your point of view.

[4] The latter mechanism is how honey-bees maintain a constant hive temperature. This ability to preserve “hive homeostasis” is one of the reasons that honeybees can survive in wildly varying climates.

2017-10-272018-04-14

Fluids That Flow Themselves

Original paper: Transition from turbulent to coherent flows in confined three-dimensional active fluids (Non-paywall version here.)

Disclosure: The first author of the paper discussed in this post, Kun-Ta Wu, did his Ph.D. at New York University, in the same research group as the present writer (CPK). At NYU, both Wu and CPK worked on topics unrelated to the research discussed here.

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When we think about fluid flow, we generally think of motion in response to some external force: rivers run downhill because of gravity, while soda moves through a straw because of the pressure difference created by sucking on one end. Recently, however, scientists have become interested in a class of fluids that have the capacity to move all by themselves — the so-called “active fluids.” Active materials — of which active fluids are a subset — are distinct from regular materials because energy is injected into the system at the level of individual molecules. In today’s paper, Kun-Ta Wu and his co-workers explore how such a material can turn its stored chemical energy into useful work: cargo transport.

Why are active materials so interesting? For one thing, many biological systems are active — for example the actin filaments that drive muscle contraction or bacterial swarms. Although active systems are both common and important in our everyday lives, the physical laws that govern their behavior are not well understood [1]. Studying artificial active systems, which are much simpler than living ones, might give us insight into this difficult problem.

As well as helping us to understand basic physics and biology, Wu and his co-workers hope that their research will move us closer to producing artificial materials that transport cargo without adding energy from an external source — a self–powered fluidic conveyer belt [2]. Such a material would be totally different from those that we currently use, and would greatly expand the possibilities available to engineers in fields such as microfluidics and soft robotics.

Wu’s research focuses on a system made up of protein molecules that assemble into cylindrical rods called microtubules. While microtubules are very important in biology [3], Wu uses these tiny rods, suspended in water, to make an artificial active fluid. As well as microtubules, Wu adds two other critical ingredients: kinesin molecular motors, and ATP (adenosine triphosphate), a chemical that many biological systems use as an energy source [4].

fig1 — A sliding force is generated between microtubules by the action of molecular motors. (Adapted from Figure 1 of the original paper.)

A single kinesin molecule attaches to two parallel microtubules and creates a lateral force that slides or “walks” them along each other. A single “step” of this walk involves a chemical reaction that converts one ATP molecule into ADP (adenosine diphosphate), a lower-energy state, thereby converting chemical potential energy into motion. A collection of millions or billions of microtubules (and a similar number of kinesin and ATP molecules) forms a material that writhes and squirms without any forces acting upon it. In the following video, Wu records the motion of both the microtubules themselves (they’re tagged with a fluorescent red dye), and micrometer-sized green particles, which he uses to trace the flow.

Video 1 Using fluorescence microscopy, Wu and colleagues can observe the motion of microtubules (red), as well as test cargo — colloidal particles (green) that are carried along in the flow generated by the motion of microtubules. (Movie 1 of the original paper.)

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But converting energy into useful work doesn’t just require motion; it requires motion that is controlled, directed, and uniform over time — coherent motion. This brings us to the main finding of Wu and coworkers: in the microtubules-motors-ATP system, coherent motion can be produced by controlling the shape of the container. Placed in a large rectangular box, the flow in the middle of the box (“in the bulk”) is turbulent but directionless (see panel A of the below figure). However, when placed in a ring with appropriate dimensions, the flow spontaneously organizes into large-scale circular patterns that are capable of transporting cargo — like fluorescent colloidal particles — over lengths of centimeters or even longer (panel B below).

fig2 — Panel A shows the pattern of flow of a bulk sample of active fluid. The arrows represent the velocity field, and colors represent the normalized *vorticity* of the flow: the extent to which it is rotating clockwise or anticlockwise in a local frame of reference. The left half of the panel shows a snapshot of the flow at a single instant in time, while the right half shows the time average. (This convention is also used in the other flow visualizations shown in this post.) In the time-averaged plot, both velocity and vorticity are almost zero: the flow is turbulent but directionless. Panel B-i shows the ring geometry of one of the sample chambers Wu uses to create coherent flow, and B-ii shows the flow pattern in that chamber. Unlike in the bulk sample, a long-lived circular pattern is generated that pushes the cargo around the ring. (Adapted from Figure 1 of the original paper.)

Interestingly, whether or not this happens is controlled only by the aspect ratio of the container: the channel width divided by its height [5]. Coherent flow is observed when the aspect ratio is between ? and 3; in other words, it disappears if the ring is too flat or too tall. Additionally, Wu shows that the direction of the flow– whether it goes clockwise or counterclockwise — can be controlled by decorating the outside of the container with appropriately shaped notches, which Wu calls ratchets.

Finally, the researchers show that the appearance of directed flow coincides with the onset of nematic order: in circulating samples, the rod-like microtubules tend to align with their neighbors, while in the turbulent samples, they are oriented randomly. According to Wu, this alignment allows the fluid to collectively push itself off the walls of the container, thus generating global circulation.

fig3 — Wu and co-workers use ratchets — small asymmetrical notches on the outside of the ring — to control whether the flow is clockwise (CW) or counterclockwise (CCW). The scale bar shows that flow is coherent over lengths of centimeters. (Adapted from Figure 3 of the original paper.)

Of course, this paper only scratches the surface of the technological potential of active materials. Research on this, and similar ideas, continues both at Brandeis University, where this research was done, and in Worcester Polytechnical Institute, where Wu has recently been appointed professor. Here, according to his website, Wu aims to “advance our understanding of self-organization of active matter as well as to create unprecedented bio-inspired materials.”

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[1] Physical systems at thermodynamic equilibrium obey the Boltzmann distribution — a formula that (in principle) allows us to calculate macroscopic properties of many-body systems, if we know the interactions between the constituent particles. We don’t know of a similar theory that describes the behavior of out-of-equilibrium systems, and active systems are by definition out of equilibrium.

[2] Of course, the energy ultimately has to come from somewhere. In the case of the material studied by Wu et al, the conveyer belt would have to be “charged” with fresh ATP before use.

[3] In particular, microtubules are the most important structural component of the mitotic spindle – the sub-cellular structure that pulls chromosomes copies apart during cell division.

[4] Wu also adds a chemical known as a depletant, which makes the microtubules bundle together, allowing the kinesin to slide them along each other.

[5] Wu also studies cylinders and shows that a similar geometrical parameter controls the appearance of coherent flow.