Metallic Glasses

Glasses have received a lot of attention because of their interesting structure and dynamics (indeed, Nobel Laureate Phil Anderson wrote that “The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition.”) Unlike crystals, they do not possess long-range order — only short or medium-range order, like liquids. Unlike liquids, however, glasses have mechanical properties akin to those of solids. A number of different approaches have been explored to study the physics of glasses, including harnessing the technology of colloidal physics as many in our and other research groups do. Metallic glasses are also model glassy systems, formed when a molten liquid precursor is supercooled at a rate fast enough that glass formation wins over crystal formation.

Here’s a quick description of two recent papers that look at two different aspects of metallic glasses…

1. How easy is it to form a glass? (Li et al., Science 2008)
Intuitively, one might expect that if the liquid precursor to a metallic glass has higher packing density, the atomic subunits making up the liquid have less “free” volume to explore and hence have a lower probability of forming an ordered cluster to nucleate crystal formation: the glass is “easier” to form. (Note that because the density at the glass transition is continuous, unlike the transition between a liquid and crystal, the density of the liquid precursor and the density of the newly-formed glass are the same thing).

Surprisingly, it seems there is very little clear experimental demonstration of this correlation between the density of a metallic glass and the ease with which it is formed. In this paper, Li et al. show a very nice route towards this. They use microfabrication to produce an array of silicon nitride cantilevers, sputter-coated with differing compositions of the binary alloy Cu_xZr_1-x, a popular system for studying metallic glasses. They measure the density difference between the as-deposited glass and the crystal that results from thermal annealing their samples by measuring the deflection of the cantilevers before and after annealing (the density of the resulting crystalline state can be estimated using equilibrium thermodynamics). On the other hand, they measure the ease of glass formation for glasses of differing composition by casting them in wedge-shaped molds; a cross-section of the resulting solid shows a clear boundary between the glassy and the crystalline state, and the thickness of the lower (denser) glassy state is a standard metric for how easily it formed. The beauty of these experiments is that they are quite straightforward, and look at this particular system over a range of relative compositions.

Sure enough, Li et al. see very nice correlation between the glass/crystal density difference and the ease of glass formation over the range of compositions they study. Interestingly, three particular compositions seem to form the glassy state very easily — and surprisingly, only one can be predicted using existing models!

2. What is the medium-range structure of a glass? (Ma et al., Nature Materials 2008)
A good deal of work has focused on understanding the microscopic nature of short-range – that is, on the lengthscale of just a few atoms – order (SRO) in metallic glasses (e.g. Miracle, 2004). A relatively recent model, which is accumulating more and more experimental support, is that alloyed metallic glass are composed of small clusters of majority atoms surrounding a minority atom “seed”. If one is willing to believe this model, the next question is: how do we use this understanding to better understand the nature of medium-range order (MRO) in metallic glasses? It has been suggested that these clusters may closely pack to form the metallic glass. In this paper, Ma et al. suggest another idea: these clusters form a fractal network of dimension 2.31.

The evidence Ma et al. compile to support this notion is compelling. For starters, fractal networks are ubiquitous in materials of interesting microstructure (e.g. see the references in Ma et al.’s paper). One close example is quasicrystals, which also lack translational symmetry, and have been shown to also be described as fractal networks. Second, Raman and neutron-scattering experiments performed in the 1990s suggested the existence of frequency-dependent vibrational excitations in metallic glasses, with a crossover between the phonons that are characteristic of ordered crystals and “fractons”, vibrational excitations of a fractal network. (This is the first time I come across the idea of a “fracton”, and I will have to spend some time rigorizing how I think about them.) In this paper, Ma et al. present their own and others’ neutron and X-ray diffraction data of a number of metallic glasses of different compositions (including the Cu_xZr_1-x mentioned in the previous paper).

In crystalline materials, the momentum-space position of the first Bragg peak in a powder diffraction pattern () is inversely proportional to the largest distance between two atomic planes of the sample — small ’s probe large lengthscales. Representing the atoms making up the crystal as hard spheres of volume , this distance scales as (that is, constant). The key idea in Ma et al.’s paper is that while metallic glasses do not have well-defined Bragg peaks because of their disordered structure, the medium-range order does give rise to a few diffuse scattering “haloes”. Thinking about the atoms of the metallic glass as hard spheres as well, on expects that constant, where mass density/(avogadro’s number * molecular weight) and is the fractal dimension of the network making up the metallic glass. Strikingly, they see this kind of scaling behavior, with . Further analysis of the atomic pair distribution function of their samples (essentially, a measure of how correlated atoms at different distances from each other are) supports this notion of a fractal network over medium-range length scales. It’ll be interesting to see how future work builds on this idea. I’m a bit confused as to what the “atomic volume” as calculated in Ma et al.’s paper physically represents in these alloyed metallic glasses, something the authors don’t go into too much detail on. Naively I would guess this is somehow related to the size of the clusters making up the fractal network — perhaps it would be interesting to use this kind of data to pull out this information and see if it agrees with other work on the structure of these SRO clusters.

Particles in Fluids

I’m back – posts will be much shorter and more paper-centered from now on, as classes and research continue to consume my life.

Three really cool papers recently, all dealing with particles in some kind of flow:

1. Effects of particles in chaotic flow (Ouellette et al., PRL 2008)
Small tracer particles are often used to ‘visualize’ fluid flows, by seeding them into the fluid. If the particles are small enough and have low enough density to match the fluid, they can be considered as infinitesimal fluid elements to a good approximation. This breaks down if the particles are (i) too large, or (ii) too dense. While the effect of having small but dense particles is pretty well studied (since the particles can be taken to be pointlike), the case of large particles is more complicated – one has to solve the relevant Navier-Stokes equation over the surface of each particle. How does a large tracer particle perturb fluid flow?

By imaging the motion of tracer particles of different sizes in a chaotic fluid flow, Ouellette et al. study the flow field around a large tracer particle as well as its own motion. (The smallest particles act as the ‘ideal’ infinitesimal fluid elements that follow the flow well.) The effect of tracer particles being too large or too dense is often thought to be captured by the Stokes number where and are the particle and fluid densities, and are the particle radius and characteristic flow length scale, and is the fluid Reynolds number. It is surprising, then, that the data in these experiments does not seem to solely depend on the Stokes or Reynolds numbers – these dimensionless parameters don’t appear to capture all of the physics associated with inertial effects. Weird.

2. Effects of particles in turbulent flow (Tanaka and Eaton, PRL 2008)
Ok, so the previous paper dealt with non-turbulent flow. This one deals with the case of how particles in a turbulent flow affect the turbulence. Do they make it more turbulent, make it less turbulent, or (unlikely) don’t affect the flow? Can these effects be captured by the Stokes or Reynolds number, unlike the previous case?

Tanaka and Eaton looked at data from many different experiments on this subject, finding (as in Ouellette et al.’s experiments) no systematic dependence on the Stokes or Reynolds numbers. Hm. Instead, they use some very beautiful dimensional analysis to come up with a new dimensionless parameter, what they call the particle momentum number Pa, which seems to capture more of the physics here – for very large and very small values of Pa, the particles augment turbulence, while for an intermediate range of Pa turbulence is attenuated. (Instead of attempting to write the two forms of Pa out, I’m just going to refer the reader to equations 14 and 15 in the paper). This is cool – finally a parameter that yields information about the physics of the situation!

Physically, is there a simple way of seeing what Pa actually means, versus just being a combination of Re, St, and various relevant variables? (The Reynolds number, for example, can be understood as telling one about the relative importance of inertial forces versus viscous forces on a tracer particle; the Stokes number on the other hand tells one about how ‘impactable’ a tracer particle is – it describes how independently the particle can move from the carrier flow.) I wasn’t fully able to decipher this.

Secondly, and I’m not sure if this even makes sense or not, but could this have any relevance to Ouellette et al.’s experiments, in which St or Re on their own were not enough to account for the effects of perturbations due to tracer particles? I did some mindless playing around with Ouellette et al.’s data from figure 4c-d, plotting it as a function of two such possible parameters. The first, ‘Pa1′, is inspired by Tanaka and Eaton’s particle momentum number, and is defined as ; the second, ‘Pa2′, is Tanaka and Eaton’s equation 15: . This is what I’m showing here:

Perhaps unsurprisingly, the two curves (for two tracer particle sizes) still don’t fall on a single curve. Oh well. Again, I’m not entirely sure if it makes sense to ask this question, but is there some combination of St and Re (similar to Pa) that is a more relevant dimensionless parameter for Ouellette et al.’s experiments?

3. Phonons in a 1D microfluidic ‘crystal’ (Beatus et al., Nature Physics 2008)
This is a cute paper that touches on many, many interesting ideas. The basic idea is straightforward: Beatus et al. produced a continuously-flowing array of uniformly-spaced oil drops in a microfluidic channel, surrounded by a continuous oil phase. The drops are disc-like in shape (they are confined in the z-direction), unconstrained in the x-direction (the direction of flow), and the constraint in y (i.e. the width of the channel) is varied, thus varying the friction on the drops.

The cool thing is that these researchers see interesting longitudinal and transverse fluctuations (it’s worth looking at the supplementary movies), and by fourier-transforming their data, they pull out dispersion relations that surprisingly show acoustic phonon propagation. The phonon propagation speed is much smaller than the speed of sound in the surrounding fluid, which leads them to hypothesize that these collective modes arise from dipole-like hydrodynamic interactions between droplets. Very pretty stuff.

My Interest in Biological Physics

Along with moving to a new institution for my Ph.D., I have decided to switch fields, moving from hard condensed matter/nanoscience to soft condensed matter/biological physics. This decision was totally unplanned – even when applying to and visiting graduate schools, I thought I wanted to do some variant of what I did as an undergrad. The thing is, as I visited more and more schools and learned about different people’s work, I found that what really captured my interest and got me excited was the soft matter side of things. Particularly in biological physics, things tend to be messier and less well understood – but this just means that there’s more to learn and quantify. I think physicists are well-positioned to bring something new and useful to the table.

(And recently, I’ve been shifting my focus more and more towards colloids and emulsions – they’re wonderful systems to work with, and allow for what I consider to be some very cool experimental statistical mechanics. My new interest in colloidal physics will have to wait for another post in the future, though.)

While in one sense my interest in soft condensed matter and biological physics has been steadily increasing over the past few years due to some really great classes I took as an undergrad, my interest in the biological side of things was really catalyzed by three key experiments/ideas:

1. Tensegrity and the structure of biological systems
Tensegrity is a mechanical design principle pioneered by Kenneth Snelson and Buckminster Fuller in the 1960’s, in which structures are designed such that the competition between forces – tension versus compression – throughout has a self-stabilizing effect. (A well-known example of this is the geodesic dome.) Among others, Don Ingber has spent a lot of time exploring the application of this idea to the structure of cells. Basically, the idea is that the cytoskeleton of the cell is composed of a network of interconnected units – the microfilaments, microtubules, and intermediate filaments – under tension and compression; that is, it is structured according to the principle of tensegrity. Many groups have explored this idea since it was first proposed, and other theories exist for understanding cellular structure; indeed, many groups, including the one I’m in, currently spend a lot of time trying to better understand the structure and physical properties of cells. (You can read more about this here.) I just thought the idea was so darn cool when I first came across it in this very nice Scientific American article written over a decade ago (look at the cell on page 54!). What’s more, this idea could be used to understand the structure of other assemblies at the micro- or nano-scale, such as buckyballs or nanotubes (e.g. see the chapter by Yakobson on “Carbon Nanotubes: Supramolecular Machines” in the Dekker Encyclopedia of Nanoscience and Nanotechnology), actuated nanocolumns, and even…

2. Viruses from a materials perspective
Yep – reading about tensegrity led to me to Caspar and Klug’s classical work in the 1960’s, in which they attempted to understand the structure of ’spherical’ viral capsids within a tensegrity-inspired framework. Since then, a number of physicists and engineers have spent a good deal of time trying to understand the structure of viral capsids. One framework in particular, developed by David Nelson and co-workers, really appeals to me: I think it’s an elegant combination of ideas from crystallography and continuum mechanics (what they call “spherical crystallography”). Basically, the idea is that if you try to pack a number of particles – be they beads, or the protein subunits of a viral capsid – on the surface of a sphere, the resulting assembly necessarily possesses crystallographic defects resulting from geometrical frustration. I wrote a small review of viral structure and mechanics focused on this work for a nanomechanics class not too long ago, which you can read here, if you want to explore this further. And this is just the tip of the iceberg – people are doing all sorts of crazy things with viruses: playing tug of war with them, watching them spit out their DNA, poking on them, shocking them, and filling them with various cargoes, among other things. Pretty cool stuff.

3. Hitting worms with laser pulses
Again, this is a very broad field in which a lot of great work has been (and continues to be) done. I don’t know enough about it. What first got me excited about biological neural networks and c. elegans was learning about this experiment by Mehmet Fatih Yanik. Basically, Yanik et al. used femtosecond laser pulses to cut single axons in c. elegans worms, observed the resulting phenotypic effects, and watched them grow back within 24 hours. This is pretty neat – after all, being able to perturb these affords researchers quite a lot of control, and could be used to study nerve regeneration processes one axon at a time, among other things. c. elegans is quite the model system, and I’m sure there’s a lot of other cool work going on trying to understand various processes and mechanisms in these worms. For example, in addition to Yanik’s work, my very little reading of research in this field has exposed me to some very interesting papers from Richard Morimoto’s, Ikue Mori’s, Aravi Samuel’s and William Bialek/William Ryu’s groups, to name a few off the top of my head. I still need to learn more about this field, particularly of what the biologists are doing – but again, this femtosecond laser stuff really got my attention when I first came across it.

Back (no, really)

It’s been a while since I last posted, and I’ve been up to quite a bit since then. For starters, I’m officially done with undergrad – apparently now I’m a bachelor of arts and a master of science.

In other news, I finally (!) decided on a grad school – this fall, I will be moving on from Penn and starting my Ph.D. in physics at Harvard. I actually packed up and moved to Cambridge a week ago, and have been getting settled and starting work in my new lab – I’m doing biophysics work in Dave Weitz’s group, which is really exciting. (More details later…)

Physics bloggers have actually been shuffling around quite a bit: for example, fliptomato has come back to the U.S. from the U.K., while in a happy coincidence Mark Trodden will be leaving Syracuse for… Penn!

A nice little sidenote is that one of my main papers stemming from some of the work I did while at Penn was accepted not too long ago – watch out for it in Nano Letters sometime soon…

And lastly, moving has given me a chance to organize some of the clutter in my life. In that spirit, I’ve decided to give my new personal webpage and this blog a quick makeover.

Meme

So, it looks like I’m part of a blog-meme. I normally wouldn’t do this, but I have some time to kill what with all this flying I’m doing (visiting grad schools). So, here goes, slightly modified…

1. Link to the person who tagged you. Done.

2. List 7 a few random things currently on your mind.

• Where should I go for grad school? I have it narrowed down to a few institutions, each of which has its own particular strengths and weaknesses – in terms of the projects I’m very excited about, equipment/facilities, funding, location, the size of the groups, the intellectual culture, etc. The spreadsheet has them dead even to within uncertainty, which means that I’ll have to keep taking data…

• Some of the research groups I’m interested in joining are quite large. This is often cited as a disadvantage, since it could translate into less “face-time” with the advisor, although to be fair – isn’t the more relevant parameter the (postdoc + senior grad student)/new student ratio rather than the faculty/new student ratio?

• Research – just thinking through the details of a number of experiments and simulations that I’m working on. The annoying thing with all this traveling is that it really punches a hole in my productivity (as well as means I’ll be missing the first half of the APS March Meeting!); but then again, talking to all these fascinating people and finding out about all the cool work going on at these different places is invaluable.

• Another thing that traveling makes difficult is staying on top of the literature. I have several tens of papers waiting to be read, and while long flights are great for plowing through them, the rate at which the to-read list grows is impressive.

• I came across this interesting NYT book review on a recent book, Intern by Sandeep Jauhar. The gist is Scrubs-ian in nature – it’s the story of a medical intern trying to deal with the imperfections of day-to-day hospital culture, the meaning of life, etc. – but what really got me was his physics background (he has a Ph.D. from Berkeley) and the analogies he makes: “Life on the wards was like the plasmons I had studied in condensed matter physics… where individual electrons, moving randomly, coalesced into something greater than the sum of their parts. There was a sort of synchronized buzz. … In the midst of this collective excitation, I kept thinking, Why am I so lonely?” Alright, so it’s kind of a stretch, but still – it’s physics.

3. Tag more people at the end of your blog and link to theirs. I’ll suggest Rod, Sam and Travis.

4. Let the tagged people know by leaving a note on their site. Done.