Thermodynamic Variables for Active Brownian Particles: Pressure, Surface Tension, and Chemical Potential

In this thesis we studied the thermodynamic variables pressure, chemical potential and surface tension for active Brownian particles (ABPs). The motivation comes largely from the motility-induced phase separation (MIPS) that purely repulsive particles at high enough density can undergo when they are made sufficiently active. MIPS was introduced in chapter 2. We presented phase diagrams, and explained its onset from a stability analysis of the homogeneous isotropic phase. The phenomenon of MIPS inspired two research questions that are central to this thesis. The first question is: can the coexisting densities be found by equating the pressure and chemical potentials of the two phases? Since ABPs are out of equilibrium, a pressure and chemical potential first need to be defined. Chapters 3, 4 and 5 studied the definition of pressure. While complications arise for particles that are anisotropic (chapter 3), or whose propulsion speed is spatially dependent (chapter 4), the situation is relatively straightforward for isotropic particles with homogeneous propulsion speed: the total pressure of the system is the sum of the ‘bare’ pressure, which has the same functional form as the equilibrium pressure, and the swim pressure, which is induced by the activity. If one takes into account the fact that the ABPs swim in a solvent (chapter 3), then the bare pressure is associated with the colloids, while the swim pressure is identified as the pressure of the solvent. Their sum is then indeed the total pressure of the suspension. For these isotropic particles with homogeneous propulsion speed, chapter 5 studied the definition of a chemical potential, and tested whether, together with the pressure, it could be used to predict the coexisting densities. While the densities could be accurately predicted for a coexistence of weakly active Lennard-Jones particles, this was not the case for the highly active MIPS. The discrepancy is a consequence of the fact that the chemical potential-like quantity is not a state function, and that its bulk value depends on the profiles in the interface. The second question concerns the interface of MIPS. In Ref. [26] it was found that its interfacial tension is negative. What does this precisely mean, and why is the interface nonetheless stable? These questions were studied in chapter 4. This chapter did not directly study the MIPS interface, but a simpler system that we showed to share important qualitative features: the interface formed by an active ideal gas in between two bulks with different propulsion speeds. We proposed two possible definitions of the interfacial tension, and investigated the stability of the interface. Remarkably, just like for MIPS, the normal force on a piece of perturbed interface acts in the same direction as the perturbation, and thus seems to have a destabilizing effect. Nonetheless, we found the interface to be stable. The reason is that the interfacial tension depends on the lateral position in the interface. This leads, by a Marangoni-like effect, to tangential currents that restore the interface to its original state.