Jeroen Rodenburg

Have you ever taken the time to watch a swarm of flying starlings? A swarm, which can consist of as many as 100,000 birds, sometimes forms patterns that are truly mesmerizing to watch. If you have, then perhaps you have also wondered why these patterns form. The behavior is not coordinated by one, or a few, leader bird(s). Instead, the patterns form spontaneously, and result, somehow, from the way that every individual bird flies, and interacts with its neighbours. But how exactly the patterns follow from that individual behavior is not a simple question. Phrased more specifically, the question is: Given that we know exactly how each individual starling moves and how it interacts with other starlings, can we predict the behavior of a large group of starlings?

The behavior of a group of birds falls into a class of systems nowadays known as active matter. Other examples of active matter include schools of fish and crowds of people, but also swimming bacteria and swimming microparticles created in the lab. Active matter is called active because all objects consume energy and usually convert it into motion. Why it is called matter may not be clear at first, but it suggests that we can describe such a system like we describe 'ordinary' matter, as if it were a gas or a liquid. Ordinary matter consists of a large number of units: molecules. The difference is that molecules (usually) do not consume energy and are thus not active but passive.

Passive matter in equilibrium is described by statistical physics. This theory does exactly for passive matter what we hope to achieve for active matter: given how individual molecules interact with each other, statistical physics predicts how the whole of all molecules behaves together. We then describe the collective of molecules simply as one substance, which we then characterize by, for example, pressure and temperature. Pressure and temperature are examples of thermodynamic variables. Statistical physics can also predict at what values of the thermodynamic variables a substance is in a particular phase: when does a substance form a gas, when a liquid, and when a solid? For example, we can calculate at what temperature water boils at the top of Mt. Everest, but ultimately this theory also underlies the engines that power our cars and airplanes.

To what extent can active matter also be described by thermodynamic variables? This question is central to this thesis. We focus on the thermodynamic variables pressure, chemical potential, and surface tension. We have not investigated their use for active matter in general, but for the simple model of active Brownian particles (ABPs). Active Brownian particles are, in short, equal to passive particles, but with the extra ingredient of activity: each particle feels a propulsion force in a direction that changes randomly over time. This propulsion force makes the particles active rather than passive.

The motivation for this research comes largely from the phenomenon known as motility-induced phase separation (MIPS). To explain what this entails, first a short explanation of normal phase separation follows. Passive molecules, such as water molecules, can separate into two phases under the right conditions. These phases then exist simultaneously next to each other. For example, a gas can exist next to a liquid. For such gas-liquid separation to occur, it is essential that the particles attract each other (at a certain distance) - this attractive force ensures that the particles do not just form one gas phase, but also a phase with higher density: the liquid. Particles that only repel each other, on the other hand, form only one gas phase. What turns out now: particles that only repel each other, but are sufficiently active, do exhibit such phase separation. The phase separation is now not caused by an attractive force, but by the fact that the particles are active - in other words, because they have high motility. Therefore, this phenomenon is called motility-induced phase separation (MIPS). Chapter 2 describes MIPS in detail.

MIPS has led to two major questions that this research attempts to answer. To introduce the first question, it is important to know how thermodynamic variables are useful in the description of a 'normal' gas-liquid phase separation. A gas and a liquid that coexist always have, besides the same temperature, also 1) the same pressure and 2) the same chemical potential. These two equalities (at fixed temperature) allow us to predict the densities of the gas and the liquid. The first question is then: can we also define a pressure and chemical potential for active particles, and can we use them to predict the densities of the two phases that coexist in MIPS?

The definition of pressure is discussed in chapters 3, 4, and 5. Each chapter deals with different aspects. What happens if the particles are not spherical (chapter 3)? What is the influence of the solvent in which active particles often swim (chapter 3)? What happens if the propulsion force is not uniform everywhere (chapter 4)? And how do interactions between the particles manifest in the pressure (chapter 5)? The situation is simplest for particles that are spherical and experience a uniform propulsion force everywhere. For these particles, the activity simply ensures that the pressure gets an extra contribution known as the swim pressure.

And for these particles, chapter 5 investigates the definition of the chemical potential. This chapter also examines whether the chemical potential together with the pressure can be used to predict the densities of active phase separations. This works well for phase separations of attractive particles at low activity, but not for the MIPS formed by repulsive particles at high activity.

The second question concerns the interface that separates the two phases in MIPS. Earlier research [26] found that the surface tension of this interface is negative. This raises questions. For example: what exactly does a negative surface tension mean? And: for passive particles, the - always positive - surface tension ensures the stability of the interface, so how can the interface tension of MIPS be negative while the interface is stable?

Chapter 4 addresses these questions. The chapter does not look directly at MIPS, but at a simpler system that resembles it: the interface formed by active particles without interactions between two regions with different propulsion forces. The chapter shows that the stability of this interface is not determined by whether the surface tension is positive or negative. Instead, stability is guaranteed by the Marangoni effect. Upon a perturbation of the interface, this effect leads to a particle flow along the interface in such a way that the interface is restored to its original state.

Chapter 7 presents a concluding discussion. How useful can we say thermodynamic variables are for active Brownian particles? A certain benefit is that they provide extra intuition for the behavior of active matter. They are also quantitatively useful, primarily at low activity. For example, we have seen that pressure and chemical potential can be used to predict the densities of lightly active phase separations.

However, there are still major challenges in the description of active matter. For example, to date, it has not been possible to find thermodynamic variables (with a microscopic expression) that predict the densities of MIPS. Another example is the behavior of active particles without interactions - a so-called active ideal gas - in an external field. While the density profile of a passive ideal gas in a stable state follows directly from the local value of the external potential, chapter 6 shows that this density profile for an active ideal gas depends on the values of the external potential at arbitrarily large distances.

Thus, although thermodynamic variables certainly provide extra insight into active systems, they do not (yet) form a theoretical framework comparable to the statistical physics of passive systems in equilibrium. In the author's modest opinion, this primarily emphasizes how powerful the latter theory is.