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We review recent advances in the design, synthesis, and modeling of active fluids. Active fluids have been at the center of many technological innovations and theoretical advances over the past two decades. Research on this new class of fluids has been inspired by the fascinating and remarkably efficient strategies that biological systems employ, leading to the development of biomimetic nano- and micro-machines and -swimmers. The review encompasses active fluids on both the nano- and micro-scale. We start with examples of biological active systems before we discuss how experimentalists leverage novel propulsion mechanisms to power nano- and micro-machines. We then examine how the study of these far-from-equilibrium systems has prompted the development of new simulation methods and theoretical models in nonquilibrium physics to account for their mechanical, thermodynamic and emergent properties. Recent advances in the field have paved the way for the design, synthesis, and modeling of autonomous systems at the nano- and micro-scale and open the door to the development of soft matter robotics.

The entropy change that occurs upon mixing two fluids has remained an intriguing topic since the dawn of statistical mechanics. In this work, we generalize the grand-isobaric ensemble to mixtures, and develop a Monte Carlo algorithm for the rapid determination of entropy in these systems. A key advantage of adiabatic ensembles is the direct connection they provide with entropy. Here, we show how the entropy of a binary mixture A-B can be readily obtained in the adiabatic grand-isobaric $(\mu_{\text{A}}$, $\mu_{\text{B}}, P, R)$ ensemble, in which $\mu_{\text{A}}$ and $\mu_{\text{B}}$ denote the chemical potential of components A and B, respectively, $P$ is the pressure, and $R$ is the heat (Ray) function, that corresponds to the total energy of the system. This, in turn, allows for the evaluation of the entropy of mixing, as well as of the Gibbs free energy of mixing. We also demonstrate that our approach performs very well both on systems modeled with simple potentials and with complex many-body force fields. Finally, this approach provides a direct route to the determination of the thermodynamic properties of mixing, and allows for the efficient detection of departures from ideal behavior in mixtures.

We extend Expanded Wang-Landau (EWL) simulations beyond classical systems and develop the EWL method for systems modeled with a tight-binding Hamiltonian. We then apply the method to determine the partition function and thus all thermodynamic properties, including the Gibbs free energy and entropy, of the fluid phases of Si. We compare the results from quantum many-body (QMB) tight binding models, which explicitly calculate the overlap between the atomic orbitals of neighboring atoms, to those obtained with classical many-body force fields (CMB), which allow to recover the tetrahedral organization in condensed phases of Si through e.g. a repulsive 3-body term that favors the ideal tetrahedral angle. Along the vapor-liquid coexistence, between 3000K and 6000K, the densities for the two coexisting phases are found to vary significantly (by $5$ orders of magnitude for the vapor and by up to 25% for the liquid) and to provide a stringent test of the models. Transitions from vapor to liquid are predicted to occur for chemical potentials that are $10-15$% higher for CMB models than for QMB models, and a ranking of the force fields is provided by comparing the predictions for the vapor pressure to the experimental data. QMB models also reveal the formation of a gap in the electronic density of states of the coexisting liquid at high temperatures. Subjecting Si to a nanoscopic confinement has a dramatic effect on the phase diagram, with e.g. at 6000K a decrease in liquid densities by about 50% for both CMB and QMB models and an increase in vapor densities between 90% (CMB) and 170% (QMB). The results presented here provide a full picture of the impact of the strategy (CMB or QMB) chosen to model many-body effects on the thermodynamic properties of the fluid phases of Si.

The determination of the critical parameters of metals has remained particularly challenging both experimentally, because of the very large temperatures involved, and theoretically, because of the many-body interactions that take place in metals. Moreover, experiments have shown that these systems exhibit an unusually strong asymmetry of their binodal. Recent theoretical work has led to new similarity laws, based on the calculation of the Zeno line and of the underlying Boyle parameters, which provided results for the critical properties of atomic and molecular systems in excellent agreement with experiments. Using the recently developed Expanded Wang-Landau (EWL) simulation method, we evaluate the grand-canonical partition function, over a wide range of conditions, for $11$ $FCC$ and $HCP$ metals ($Ag$, $Al$, $Au$, $Be$, $Cu$, $Ir$, $Ni$, $Pb$, $Pd$, $Pt$ and $Rh$), modeled with a many-body interaction potential. This allows us to calculate the binodal, Zeno line, Boyle parameters and, in turn, obtain the critical properties for these systems. We also propose two scaling laws for the enthalpy and entropy of vaporization, and identify critical exponents of $0.4$ and $1.22$ for these two laws, respectively.

Using molecular simulation, we analyze the capillary condensation and evaporation processes for Argon confined in a cylindrical nanopore. For this purpose, we define the entropy of the adsorbed fluid as a reaction coordinate and determine the free energy associated with both processes along entropic pathways. For capillary condensation, we identify a complex free energy profile resulting from the multi-stage nature of this phenomenon. We find capillary condensation to proceed through the nucleation of a liquid bridge across the nanopore, followed by its expansion throughout the pore to give rise to the stable phase of high density. In the case of capillary evaporation, the free energy profile along the entropy pathway also exhibits different regimes, corresponding to the initial destabilization of the layered structure of the fluid followed by the formation, and subsequent expansion, of a bubble across the nanopore.

Using molecular simulation, we study the nucleation of liquid droplets from binary mixtures and determine the free energy of nucleation along entropic pathways. To this aim, we develop the $\mu_1 \mu_2 VT-S$ method, based on the grand-canonical ensemble modeling the binary mixture, and use the entropy of the system $S$ as the reaction coordinate to drive the formation of the liquid droplet. This approach builds on the advantages of the grand-canonical ensemble, which allows for the direct calculation of the entropy of the system and lets the composition of the system free to vary throughout the nucleation process. Starting from a metastable supersaturated vapor, we are able to form a liquid droplet by gradually decreasing the value of $S$, through a series of umbrella sampling simulations, until a liquid droplet of a critical size has formed. The $\mu_1 \mu_2 VT-S$ method also allows us to calculate the free energy barrier associated with the nucleation process, to shed light on the relation between supersaturation and free energy of nucleation, and to analyze the interplay between the size of the droplet and its composition during the nucleation process.

Using the entropy $S$ as a reaction coordinate, we determine the free energy barrier associated with the formation of a liquid droplet from a supersaturated vapor for atomic and molecular fluids. For this purpose, we develop the $\mu VT-S$ simulation method that combines the advantages of the grand-canonical ensemble, that allows for a direct evaluation of the entropy, and of the umbrella sampling method, that is well suited to the study of an activated process like nucleation. Applying this approach to an atomic system such as $Ar$ allows us to test the method. The results show that the $\mu VT-S$ method gives the correct dependence on supersaturation of the height of the free energy barrier and of the size of the critical droplet, when compared to predictions from classical nucleation theory and to previous simulation results. In addition, it provides insight into the relation between entropy and droplet formation throughout this process. An additional advantage of the $\mu VT-S$ approach is its direct transferability to molecular systems, since it uses the entropy of the system as the reaction coordinate. Applications of the $\mu VT-S$ simulation method to $N_2$ and $CO_2$ are presented and discussed in this work, showing the versatility of the $\mu VT-S$ approach.

Using molecular simulations, we determine the locus of ideal compressibility, or Zeno line, for a series of ionic compounds. We find that the shape of this thermodynamic contour follows a linear law, leading to the determination of the Boyle parameters. We also show that a similarity law, based on the Boyle parameters, yields accurate critical data when compared to the experiment. Furthermore, we show that the Boyle density scales linearly with the size-asymmetry, providing a direct route to establish a correspondence between the thermodynamic properties of different ionic compounds.

Using molecular simulation, we study the nucleation process from supercooled liquid alloys of Cu and Pd. The simulations reveal a complex interplay between the size of the crystal nucleus and its composition that greatly impacts the crystallization process on the nanoscale. In particular, we find that the free energy of nucleation strongly depends on the composition of the alloy, with a free energy barrier for the equimolar alloy that is more than two times larger than for the pure metals. We attribute this increase in free energy to the variations in composition occurring both at the surface and in the core of the nucleus. The local changes in composition are then analyzed by comparing the surface energies of the two metals and by taking into account the competition between crystallization and demixing that takes place at the interface between the nucleus and the surrounding liquid.

Using molecular simulation, we determine Ginzburg-Landau free energy functions for molecular fluids. To this aim, we extend the Expanded Wang-Landau method to calculate the partition functions, number distributions and Landau free energies for $Ar$, $CO_2$ and $H_2O$. We then parametrize a coarse-grained free energy function of the density order parameter and assess the performance of this free energy function on its ability to model the onset of criticality in these systems. The resulting parameters can be readily used in hybrid atomistic/continuum simulations that connect the microscopic and mesoscopic length scales.

We use molecular simulation to compute the thermodynamic properties of 7 rigid models for water (SPC/E, TIP3P, TIP4P, TIP4P/2005, TIP4P/Ew, TIP5P, OPC) over a wide range of temperature and pressure. Carrying out Expanded Wang-Landau simulations, we obtain a high accuracy estimate for the grand-canonical partition function which, in turn, provides access to all properties, including the free energy and entropy, both at the vapor-liquid coexistence and for compressed water. The results at coexistence highlight the close connection between the behavior of the statistical functions and the thermodynamic properties. They show that the subgroup (SPC/E,TIP4P/2005,TIP4P/Ew) provides the most accurate account of the vapor-liquid equilibrium properties. For compressed water, the comparison of the simulation results to the experimental data establishes that the TIP4P/Ew model performs best among the 7 models considered here, and captures the experimental trends for the dependence of entropy and molar Gibbs free energy on pressure.

Using molecular simulation, we assess the impact of an electric field on the properties of water, modeled with the SPC/E potential, over a wide range of states and conditions. Electric fields of the order of $0.1V/$\AA and beyond are found to have a significant impact on the grand-canonical partition function of water, resulting in shifts in the chemical potential at the vapor-liquid coexistence of up to $20$%. This, in turn, leads to increases in the critical temperatures by close to $7$% for a field of $0.2V/$\AA, to lower vapor pressures, and to much larger entropies of vaporization (by up to $35$%). We interpret these results in terms of the greater density change at the transition and of the increased structural order resulting from the applied field. The thermodynamics of compressed liquids and of supercritical water are also analyzed over a wide range of pressures, leading to the determination of the Zeno line and of the curve of ideal enthalpy that span the supercritical region of the phase diagram. Rescaling the phase diagrams obtained for the different field strength by their respective critical properties allows us to draw a correspondence between these systems for fields of up to $0.2V/$\AA.

We determine the impact of the Friedel oscillations on the phase behavior, critical properties and thermodynamic contours in films ($2D$) and bulk phases ($3D$). Using Expanded Wang-Landau simulations, we calculate the grand-canonical partition function and, in turn, the thermodynamic properties of systems modeled with a linear combination of the Lennard-Jones and Dzugutov potentials, weighted by a parameter $X$ ($0

Crystallization often proceeds through successive stages that lead to a gradual increase in organization. Using molecular simulation, we determine the nucleation pathway for solid solutions of copper and gold. We identify a new nucleation mechanism (liquid$\to$$L1_2$~precursor$\to$solid solution), involving a chemically ordered intermediate that is more organized than the end product. This nucleation pathway arises from the low formation energy of $L1_2$ clusters which, in turn, promote crystal nucleation. We also show that this mechanism is composition-dependent since the high formation energy of other ordered phases precludes them from acting as precursors.

Using molecular simulations, we shed light on the mechanism underlying crystal nucleation in metal alloys and unravel the interplay between crystal nucleation and glass transition, as the conditions of crystallization lie close to this transition. While decreasing the temperature of crystallization usually results in a lower free energy barrier, we find an unexpected reversal of behavior for glass-forming alloys as the temperature of crystallization approaches the glass transition. For this purpose, we simulate the crystallization process in two glass-forming Copper alloys, $Ag_6Cu_4$, which has a positive heat of mixing, and in $CuZr$, characterized by a large negative heat of mixing. Our results allow us to identify that this unusual behavior is directly correlated with a non-monotonic temperature dependence for the formation energy of connected icosahedral structures, which are incompatible with crystalline order and impede the development of the crystal nucleus, leading to an unexpectedly larger free energy barrier at low temperature. This, in turn, promotes the formation of a predominantly closed-packed critical nucleus, with fewer defects, thereby suggesting a new way to control the structure of the crystal nucleus, which is of key importance in catalysis.

Using nonequilibrium molecular dynamics (NEMD) simulations, we study the properties of supercooled liquids of Si under shear at T=1060K over a range of densities encompassing the low-density liquid (LDL) and high-density liquid (HDL) forms. This enables us to generate nonequilibrium steady-states of the LDL and HDL polymorphs, that remain stabilized in their liquid forms for as long as the shear is applied. This is unlike the LDL and HDL forms at rest, which are metastable under those conditions and, when at rest, rapidly undergo a transition towards the crystal, i.e. the thermodynamically stable equilibrium phase. In particular, through a detailed analysis of the structural and energetic features of the liquids under shear, we identify the range of densities, as well as the range of shear rates, that give rise to the two forms. We also show how the competition between shear and tetrahedral order impacts the two-body entropy in steady-states of Si under shear. These results open the door to new ways of utilizing shear to stabilize forms that are metastable at rest and can exhibit unique properties, since, for instance, experiments on Si have shown that HDL is metallic, with no band gap, while LDL is semimetallic, with a pseudogap.

Entropy has become increasingly central to characterize, understand and even guide assembly, self-organization and phase transition processes. In this work, we build on the analogous role of partition functions (or free energies) in isothermal ensembles and that of entropy in adiabatic ensembles. In particular, we show that the grand-isobaric adiabatic $(\mu,P,R)$ ensemble, or Ray ensemble, provides a direct route to determine the entropy. This allows us to follow the variations of entropy with the thermodynamic conditions and thus to explore phase transitions. We test this approach by carrying out Monte Carlo simulations on Argon and Copper in bulk phases and at phase boundaries and assess the reliability and accuracy of the method through comparisons with the results from flat-histogram simulations in isothermal ensembles and with the experimental data. Advantages of the approach are multifold and include the direct determination of the $\mu-P$ relation, without any evaluation of pressure via the virial expression, the precise control of the system size and of the number of atoms via the input value of $R$, and the straightforward computation of enthalpy differences for isentropic processes, which are key quantities to determine the efficiency of thermodynamic cycles. A new insight brought by these simulations is the highly symmetric pattern exhibited by both systems along the transition, as shown by scaled temperature-entropy and pressure-entropy plots.

It has been recently shown that nanobubbles exhibit a remarkable and unexpected stability. The lifetime of nanobubbles, formed either within liquids or on hydrophobic surfaces, can exceed by more than 10 orders of magnitude the theoretical expectation, which predicts an almost immediate dissolution due to the very high Laplace internal pressure in such small bubbles. This unexpected property of nanobubbles has made them leading candidates for energy applications, e.g. as high-pressure nanoreactors in fuel cells, and for biological systems, as transport systems for gas delivery to membranes and cells. Here we use molecular simulation to shed light on the molecular mechanisms accounting for the formation and stabilization of nanobubbles under an hydrophobic nanoconfinement. Using an entropic reaction coordinate, we elucidate the nucleation pathway and determine the formation free energy of nanobubbles in water confined in carbon nanotubes. We identify a critical volume for which the existence of nanobubbles is thermodynamically favored, as the free energy profile flattens around this critical volume, and mechanically favored, since the nanoconfined fluid pressure, along the nanotube axis, is positive at this juncture. We also show that the stabilization process is assisted by the hydrophobic nature of the carbon nanotube and by the formation of strong hydrogen bonds at the interface. Caroline Desgranges and Jerome Delhommelle

We combine machine learning (ML) with Monte Carlo (MC) simulations to study the crystal nucleation process. Using ML, we evaluate the canonical partition function of the system over the range of densities and temperatures spanned during crystallization. We achieve this on the example of the Lennard-Jones system by training an artificial neural network using, as a reference dataset, equations of state for the Helmholtz free energy for the liquid and solid phases. The accuracy of the ML predictions is tested over a wide range of thermodynamic conditions, and results are shown to provide an accurate estimate for the canonical partition function, when compared to the results from flat-histogram simulations. Then, the ML predictions are used to calculate the entropy of the system during MC simulations in the isothermal-isobaric ensemble. This approach is shown to yield results in very good agreement with the experimental data for both the liquid and solid phases of Argon. Finally, taking entropy as a reaction coordinate and using the umbrella sampling technique, we are able to determine the Gibbs free energy profile for the crystal nucleation process. In particular, we obtain a free energy barrier in very good agreement with the results from previous simulation studies. The approach developed here can be readily extended to molecular systems and complex fluids, and is especially promising for the study of entropy-driven processes.