Post mortem: How does the structure of porous media determine the permeability

In my previous blog post, I explain why I would find it useful if researches would shed some light on the history and the circumstances behind their publications. So, this article is about my paper

Christian Scholz, Frank Wirner, Jan Götz, Ulrich Rüde, Gerd E. Schröder-Turk, Klaus Mecke, and Clemens Bechinger
Permeability of porous materials determined from the Euler characteristic, Phys. Rev. Lett. 109, 264504 (2012)

and it’s the closest thing I’ve ever experienced to what enlightenment, if such a thing exists, must feel like.

In life we are always surrounded by porous media, so that it is usually not very difficult to convince anyone of the importance of knowing how such materials behave on a physical level. The properties of ground soil, a classic porous medium, are obviously important to anyone who has ever performed some form of gardening or construction. When you want to brew a coffee it can be quite a challenge to get the grain size, compactification, filter size and flow rate right so that you get the best out of whatever kind of coffee beans you prefer (or you are like me, don’t care too much about the taste as long as it has enough caffeine to keep you going a bit longer).

Now you might ask, what did physicists figure out about porous media that helps us to deal with these questions on an engineering level? One of the many answered to that is a property of fluid flow through porous materials. Quite some time ago a physicist called Henry Darcy realized that when you put twice the pressure on a core of porous material you also double the flow rate, so these two aspects are related linearly. However he also found that the absolute value of the flow rate at a given pressure was quite different, depending on the type of porous material. Today this might seem obvious to us. Getting water through a clogged pipe is obviously more difficult than when it is fresh and clean. Why is that so? Well, again obviously there is an obstacle in the way, so the water needs to flow through the empty spaces within or around it. Clearly this is somehow a property of the empty spaces in a porous medium. But what exactly is it, that determines the ability to let water flow through? Is it the empty volume? Or the surface area? Or could there be something different that we do not intuitively think of?

The first aspect that you need to think about is, what is the minimum requirement for a porous core to let fluid flow through it? Clearly that is, that there must be a path for the fluid to flow through the core. Imagine a piece of polystyrene foam. Polystyrene is a very light-weight porous medium with more than 95% air content. But you will have a hard time pushing water through it in many cases. The reason is, that, despite the high porosity (the relative volume of air) there is typically no nice path for the fluid to flow in such a material. If a material has a well-connected path, which connects two sides, it is called percolating (derived from the coffee percolator). Physicists and mathematicians in the 20th century realized that for porous media that span a certain range of porosities there is typically a threshold that needs to be crossed such that it becomes permeable. Also, and this is the most important aspect, they found that if you know this threshold and also the actual porosity of a material, you can calculate the permeability from a simple equation that has only one free parameter and one parameter that is based entirely on mathematical symmetries, therefore also called universal (because it is the same for all imaginable objects with such symmetries). This is a great result, but there is a crux with the matter. Where exactly is the threshold for a certain porous medium. As it turns out it is not universal at all. The mentioned example, polystyrene foam does is not permeable despite being more than 95% air, but a bucket full of soil with a porosity in the range of 30% to 40% will let water seep away quite easily.

So, close only counts in horseshoes you might think. But in this publication we showed one way on how to get around this by using topological and geometrical properties of an individual porous sample to calculate the permeability. The trick is to find a quantity that behaves similar in dependence on porosity as the permeability, but that is an entirely static topological property of the structure: the Euler characteristic.

The Euler characteristic is a property of geometrical structures that is related to their topology. Imagine a porous structure made of rubber, so that you can easily bend, compress, stretch and twist it. Is there a property of this material that remains the same no matter which of those things you do to the structure? – Yes. It’s the amount of holes and bridges in between hollow spaces inside the material. They might deform, but their number stays the same. In other words, let’s assume you are a tiny molecule in one of these hollow spaces and you can explore all of the volume of this space. No matter how someone might bend or twist the material, you will never be cut of from any part of this pore space (here we assume that you cannot deform the structure so hard that you effectively join or ‘glue’ pore-walls together).

In two dimensions, the Euler characteristic corresponds to the number of hollow parts and solid parts in the material. So when you look at your structure, all we have to count is the number of connected solid and connected hollow (fluid) parts and calculate the difference. How can this be useful to determine the permeability?

Imagine a very hollow porous medium. Just an empty space with a few obstacles obstructing the flow. In such a case, the Euler characteristic is almost equivalent to the number of individual obstacles, for which we assume they are all similar grains of some solid material. When we now increase the number of these obstacles, we know, at some point some of these grains have to join together so we can fit more of them into the pore space. In this case, the number of connected obstacles will become less than the grains that we have put into the material. It is save to assume that at some point all grains will be connected together and we only find very few connected solid parts and hopefully still one connected hollow space. So it is natural to assume that for structures with many densely packed obstacles the difference between connected solid and hollow spaces will not be so different, so the Euler characteristic will be small. For structures with few obstacles the difference between individual obstacles and hollow pore space will be large and the Euler characteristic will be similar to the number of individual grains. Remember the percolation threshold? This was the porosity where we just barely have a connected pore space in the structure. And now we know that close to this threshold the Euler characteristic must be small, while far away it must be large! If we do not know the percolation threshold, maybe we can just use the value of the Euler characteristic instead?

It turns out that for the permeability and certain types of structures this works indeed. Basically the permeability is then proportional to the critical pore cross-section (this gives us the correct physical dimensions) and the Euler characteristic (number of individual obstacles) divided by the number of individual grains.

While in hindsight this seems quite obvious, I remember it was a big a-ha moment for me. I had measurements and simulation results for the permeability of various porous model structures and collected all the geometrical and topological quantities of these structures to see if I could find any meaningful relation. I very vividly remember putting together these numbers, but for most of the relations that I came up with I had some outliers that did not fit my proposed formula. I know today, there were two things that I had to correct for in order to find a relation that the structures were actually obeying. First of all, for the Euler characteristic I had to calculate it only for the accessible or ‘open’ part of the pore space. In particular for structures close to the percolation threshold, many tiny pore spaces that were not accessible from the outside affected the Euler characteristic, but had absolutely no influence on the fluid flow. This ‘repaired’ a lot of nonphysical results that I got initially. Furthermore, the dimension of the problem is given by the cross-section of the ‘critical pore’, which is the smallest pore in the structure that still allows a fully connected pore space. Imagine if you would close up all pores in the structure starting from the smallest pore in the system, at a certain pore size you would eventually fully clog the pore-space so that now fluid can flow through. This is the critical pore space and it’s area defines the dimension of the permeability. When I took these quantities into account, suddenly all my data-points fitted the prediction reasonably. It was a great moment when I made the final plot. I already saw from the values that I calculated that this might work and seeing the final dataset nicely following an equation that I came up with, was very satisfying.

It was a very satisfying moment, not only because of the beauty of the result, but also since the researcher conducting the experiments, i.e. me, had a pretty difficult job. The microfluidic structures used for my experiments had to be manufactured very careful so that they are close enough to the nominal values for which the simulation was conducted (I did not have fancy lithography equipment, everything was optimized ‘by hand’ as good as possible). In particular the critical pore diameter for structures close to the percolation threshold is very sensitive to fabrication uncertainties. Also my samples were very prone to collapse from the capillary forces that arise when injecting fluid into the structure for the first time, due to the small height. It was a bunch of very tedious experiments, but in the end I had a set of 10 samples that properly reflected our numerical models.

The observation of this study came with a pinch of salt though. It should be clear that the formula that I proposed cannot work for arbitrary porous structures. For any kind of model that does not take into account the full microstructure of the porous medium, it is typically easy to find some counter-example structures that do not obey the relation. Typically you can add things like microscopic obstacles and connections somewhere, that clearly do affect the topology, but have almost no influence on the flow. On which length scale do you define pores and open spaces anyway? Do you need to take into account microscopic pores that connect individual pore spaces? Maybe even the individual atoms or the space in between that make up the material? Of course it largely depends on the specific problem. Is it a gas or a fluid that permeates your structure? Is the porous medium saturated or is there gas trapped inside? All this makes a real life porous structure significantly more complicated than the simple models that I studied. However, I still hope that our findings can be incorporated into the multitude of models applied to real porous structures. But even if not, I would say, in this case the famous quote by Richard Feynman applies: ‘Physics is like sex: sure, it may give some practical results, but that’s not why we do it.’

Post Mortem: Measureing Permeability of Microfluidic Porous Media

When you read a scientific publication, in particular papers in natural science, if you are not familiar with the circumstances of scientific reality (I mean what typically happens vs what idealy happens), you might come to the conclusion that the authors had a very specific idea of what they wanted to investigate, before performing the research. However, my experience is, that this is often not true and the outcome of the research, for whatever reason is was performed, largely determines the scientific question that a paper tries to answer.

By this I absolutely don’t want to suggest that the majority of publications do not significantly contribute to our scientific understanding of the world. In fact, I would argue that it is quite the contrary. I would argue that most oft the time we might be unable to ask the right question, if we do not have, at least, some idea of the answer. But in science we’re always trying to search and explain the unknown, so is it even possible to ask a good and non-trivial question before performing the research? There is a famous quote by Johann Wolfgang von Goethe “You only see what you know.”, which points to this problem and reminds us of the danger of sticking to closely to what we already know. Of course, there is always a trade-off. We need to have an idea of what is already known and what is not known (or let’s say, how ‘well’ we know something) in order to understand the relevance of a new observation. Most publications make it quite clear, as a motivation, why a specific question is worth to ask or why a specific system is interesting to investigate. However, the history of how events unfolded and how the authors eventually arrived at the paper that they published, is typically lost.

I believe that for some of my publications this history is actually quite interesting and worth telling. First, because it can sometimes be quite entertaining. Second, because others might learn something useful from it. And third, because there is always a chance that after a long time, some aspects of your research turn out to be really really relevant for mankind in some way or the other in this unlikely case, it would be nice to have some honest documentation about how you ended up doing what you actually did.

So therefore, I am writing this post mortem, about some of my papers. I call it post mortem in analogy to the practice in some industries, where a post mortem refers to a retrospective project wrap-up, typically written for other people familiar with the art.

The very first entry in this series will be about my publication:

Christian Scholz, Frank Wirner, Yujie Li, and Clemens Bechinger
Measurement of flow properties in microfluidic porous media with finite-sized colloidal tracers, Experiments in Fluids 53, 1327 – 1333 (2012) 

This is the very first publication of my academic career. At this moment in time I had already spend about 1.5 years of work on the subject and my impression was that I was far away from discovering anything relevant in the field (this turned out to be not true at all). It was also quite a challenge to manufacture microfluidic samples robustly in our lab. In the beginning, I had a success rate of about 10% and it was not uncommon to hear people shouting in the laboratories behind closed doors, when a sample turned out to be unusable. We regularly suffered from leaking microcluidic cells, collapsing structures, unstable sticking colloidal suspensions as well as contamination (I think this was the first time in my life, where I actually saw some kind of microorganism moving in real time). While this improved in the later stage of my PhD, thanks to the experience that builds up after crafting hundreds of samples, at this point, I was actually thinking: Well, at least we should put together an article about the technical challenges that we had overcome so far.

As the title says the article deals with the measurement of flow in microfluidic porous media using colloidal tracer particles. Typically flow properties such as the permeability in macroscopic porous media are measured by the the external factors around the porous core, such as flow rate and the applied pressure gradient. This is only possible when the flow rates are large enough so that they can be reliably measured.

Permeability is a quantification of the ability of a porous medium to conduct flow of a liquid or gas. The concept is equivalent to the electric conductivity (inverse of the resistance) of a material, which most people are more familiar with.

In this project, however, we wanted to investigate the permeability of microfluidic porous media models. The flow rates in such samples were so small that direct measurement of it seemed far out of scope with conventional laboratory equipment. One solution to, at least, visualize flow in such structures is via the injection of colloidal tracer particles. These particle follow the streamlines of the fluid and can therefore be used to trace the fluid flow. However, even though colloidal particles are relatively small (1 – 5 µm) they were still comparable to the size of the microfluidic structures that we wanted to investigate. In such a situation the speed of the particles does not trace the speed of the fluid at a single infinitesimal point. Instead, particles disturb the flow and also start to rotate. Additionally under a conventional microscope only a 2D projection of the particle motion can be seen, therefore information about the z-Position of each particle is not directly available. Due to the finite size, particles can also not come arbitrarily close to walls, therefore the fluid boundary layer near the wall cannot be accurately resolved and some of the fluid space is ‘shadowed’ of from the particle flow.

Because of this, we had to find a different method to determine permeability in our microfluidic models. We identified two options. First, a constant head method, were the pressure applied to the sample is kept constant as good as possible and the flow rate is measured relative to a reference channel, where we know estimate the flow accuratly from first principles. Second, a falling head method, where the hydrostatic pressure applied to the sample is allowed to slowly relax due to the flow. Since the flow rates are very small, this takes several hours, even for reservoirs with a tiny cross-section.

The first method appeared to work relatively robust, as long as there is no leakage or structure collapse in the samples. However, we wanted to apply a second method as well, in order to have a reference measurement by which we can compare the two different methods, such that we can be confident that we do not have missed systematic effects. For the falling head method, however, it turned out that we had to spend an enormous amount of work to get a working sample. Most samples would suffer from tiny leakage or abundance of tracer particles after some time of measurement. After several months my colleague Frank Wirner finally succeeded in manufacturing one golden sample that was stable enough for an entire measurement. Unfortunately, even for this ideal case, it turned out that the uncertainty of the fit to the measured data did result in a rather large uncertainty of the corrected permeability of the structure. So, while technically we achieved our goal of demonstrating two methods of measuring the permeability in microfluidic samples with extremely small flow rates, we decided that the constant head method was far more trustworthy and robust.

The usefulness of this technique is, in my opinion, surprisingly underestimated. We even had trouble publishing the manuscript in its initial form, since a referee was not convinced that such methods are required at all. I believe, it is easy to underestimated the difficulties of an accurate permeability measurement in thin microfluidic porous channels with ultra low flow rates. Determining the pressure applied to the sample with a differential sensor, while applying a pre-defined constant flow rate might be an alternative, but this will require very expensive ultra accurate syringe pumps and even there parasitic forces might build up in the flexible hoses and joints, such that the flow-rate needs a surprisingly long time to relax (be prepared to wait for an hour or two in between each datapoint). Or solution, I believe, is much more robust and definitely more cost-efficient.

These results are of course also the technical basis of all our later publications in Physical Review and EPL. My feeling is, that this paper shows some very interesting aspects of measuring flow in microfluidic structures, were tracers are really comparable to the typical structure size and recommend it to anyone who would like to attempt doing similar experiments.