Nonequilibrium inertial dynamics of colloidal systems

First-principles nonequilibrium deterministic equation of motion of a Brownian particle and microscopic viscous drag

We present a first-principles thermodynamic approach to provide an alternative to the Langevin equation by identifying the deterministic (no stochastic component) microforce F_{k,BP} acting on a nonequilibrium Brownian particle (BP) in its kth microstate m_{k}. (The prefix "micro" refers to microstate quantities and carry a suffix k.) The deterministic new equation is easier to solve using basic calculus. Being oblivious to the second law, F_{k,BP} does not always oppose motion but viscous dissipation emerges upon ensemble averaging. The equipartition theorem is always satisfied. We reproduce well-known results of the BP in equilibrium. We explain how the microforce is obtained directly from the mutual potential energy of interaction beween the BP and the medium after we average it over the medium so we only have to consider the particles in the BP. Our approach goes beyond the phenomenological and equilibrium approach of Langevin and unifies nonequilibrium viscous dissipation from mesoscopic to macroscopic scales and provides new insight into Brownian motion beyond Langevin's and Einstein's formulation.

Towards a statistical mechanical theory of active fluids

We present a stochastic description of a model of N mutually interacting active particles in the presence of external fields and characterize its steady state behavior in the absence of currents. To reproduce the effects of the experimentally observed persistence of the trajectories of the active particles we consider a Gaussian force having a non-vanishing correlation time τ, whose finiteness is a measure of the activity of the system. With these ingredients we show that it is possible to develop a statistical mechanical approach similar to the one employed in the study of equilibrium liquids and to obtain the explicit form of the many-particle distribution function by means of the multidimensional unified colored noise approximation. Such a distribution plays a role analogous to the Gibbs distribution in equilibrium statistical mechanics and provides complete information about the microscopic state of the system. From here we develop a method to determine the one- and two-particle distribution functions in the spirit of the Born-Green-Yvon (BGY) equations of equilibrium statistical mechanics. The resulting equations which contain extra-correlations induced by the activity allow us to determine the stationary density profiles in the presence of external fields, the pair correlations and the pressure of active fluids. In the low density regime we obtained the effective pair potential ϕ(r) acting between two isolated particles separated by a distance, r, showing the existence of an effective attraction between them induced by activity. Based on these results, in the second half of the paper we propose a mean field theory as an approach simpler than the BGY hierarchy and use it to derive a van der Waals expression of the equation of state.

Force-Field Parameters from the SAFT-γ Equation of State for Use in Coarse-Grained Molecular Simulations

A description of fluid systems with molecular-based algebraic equations of state (EoSs) and by direct molecular simulation is common practice in chemical engineering and the physical sciences, but the two approaches are rarely closely coupled. The key for an integrated representation is through a well-defined force field and Hamiltonian at the molecular level. In developing coarse-grained intermolecular potential functions for the fluid state, one typically starts with a detailed, bottom-up quantum-mechanical or atomic-level description and then integrates out the unwanted degrees of freedom using a variety of techniques; an iterative heuristic simulation procedure is then used to refine the parameters of the model. By contrast, with a top-down technique, one can use an accurate EoS to link the macroscopic properties of the fluid and the force-field parameters. We discuss the latest developments in a top-down representation of fluids, with a particular focus on a group-contribution formulation of the statistical associating fluid theory (SAFT-γ). The accurate SAFT-γ EoS is used to estimate the parameters of the Mie force field, which can then be used with confidence in direct molecular simulations to obtain thermodynamic, structural, interfacial, and dynamical properties that are otherwise inaccessible from the EoS. This is exemplified for several prototypical fluids and mixtures, including carbon dioxide, hydrocarbons, perfluorohydrocarbons, and aqueous surfactants.

Phase-field-crystal modeling of glass-forming liquids: spanning time scales during vitrification, aging, and deformation

Two essential elements required to generate a glass transition within phase-field-crystal (PFC) models are outlined based on observed freezing behaviors in various models of this class. The central dynamic features of glass formation in simple binary liquids are qualitatively reproduced across 12 orders of magnitude in time by applying a physically motivated time scaling to previous PFC simulation results. New aspects of the equilibrium phase behavior of the same binary model system are also outlined, aging behavior is explored in the moderate and deeply supercooled regimes, and aging exponents are extracted. General features of the elastic and plastic responses of amorphous and crystalline PFC solids under deformation are also compared and contrasted.

Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments

Starting from the Kramers equation for the phase-space dynamics of the N-body probability distribution, we derive a dynamical density functional theory (DDFT) for colloidal fluids including the effects of inertia and hydrodynamic interactions (HI). We compare the resulting theory to extensive Langevin dynamics simulations for both hard rod systems and three-dimensional hard sphere systems with radially symmetric external potentials. As well as demonstrating the accuracy of the new DDFT, by comparing with previous DDFTs which neglect inertia, HI, or both, we also scrutinize the significance of including these effects. Close to local equilibrium we derive a continuum equation from the microscopic dynamics which is a generalized Navier-Stokes-like equation with additional non-local terms governing the effects of HI. For the overdamped limit we recover analogues of existing configuration-space DDFTs but with a novel diffusion tensor.

Modeling multiple time scales during glass formation with phase-field crystals

The dynamics of glass formation in monatomic and binary liquids are studied numerically using a microscopic field theory for the evolution of the time-averaged atomic number density. A stochastic framework combining phase-field crystal free energies and dynamic density functional theory is shown to successfully describe several aspects of glass formation over multiple time scales. Agreement with mode coupling theory is demonstrated for underdamped liquids at moderate supercoolings, and a rapidly growing dynamic correlation length is found to be associated with fragile behavior.

Dynamic density functional theory versus kinetic theory of simple fluids

By combining methods of kinetic and density functional theory, we present a description of molecular fluids which accounts for their microscopic structure and thermodynamic properties as well as their hydrodynamic behavior. We focus on the evolution of the one-particle phase space distribution, rather than on the evolution of the average particle density which features in dynamic density functional theory. The resulting equation can be studied in two different physical limits: diffusive dynamics, typical of colloidal fluids without hydrodynamic interaction where particles are subject to overdamped motion resulting from coupling with a solvent at rest, and inertial dynamics, typical of molecular fluids. Finally, we propose an algorithm to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the lattice Boltzmann method.

Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods

Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method.

Beyond the dynamic density functional theory for steady currents: application to driven colloidal particles in a channel

Motivated by recent studies on the dynamics of colloidal solutions in narrow channels, we consider the steady state properties of an assembly of noninteracting particles subject to the action of a traveling potential moving at a constant speed, while the solvent is modeled by a heat bath at rest in the laboratory frame. Here, since the description we propose takes into account the inertia of the colloidal particles, it is necessary to consider the evolution of both positions and momenta and study the governing equation for the one-particle phase-space distribution. First, we derive the asymptotic form of its solutions as an expansion in Hermite polynomials and their generic properties, such as the force and energy balance, and then we particularize our study to the case of an inverted parabolic potential barrier. We numerically obtain the steady state density and temperature profile and show that the expansion is rapidly convergent for large values of the friction constant and small drifting velocities. On the one hand, the present results confirm the previous studies based on the dynamic density functional theory (DDFT): On the other hand, when the friction constant is large, it display effects such as the presence of a wake behind the barrier and a strong inhomogeneity in the temperature field which are beyond the DDFT description.

Dynamical density functional theory for anisotropic colloidal particles

We generalize the formalism of dynamical density functional theory for translational Brownian dynamics toward that of anisotropic colloidal particles which perform both translational and rotational Brownian motion. Using a mean-field approximation for the density functional and a Gaussian-segment model for the rod interaction, the dynamical density functional theory is then applied to a concentrated rod suspension in a confined slab geometry made by two parallel soft walls. The walls are either expanded or compressed and the relaxation behavior is investigated for an equilibrated starting configuration. We find distinctly different orientational ordering during expansion and compression. During expansion we observe preferential parallel ordering of the rods relative to the wall while during compression there is homeotropic ordering perpendicular to the wall. We find a nonexponential relaxation behavior in time. Furthermore, an external field which aligns the rods perpendicular to the walls is turned on or switched off and similar differences in the relaxational dynamics are found. Comparing the theoretical predictions to Brownian dynamics computer simulation data, we find good agreement.

Phase-space approach to dynamical density functional theory

The authors consider a system of interacting particles subjected to Langevin inertial dynamics and derive the governing time-dependent equation for the one-body density. They show that, after suitable truncations of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and a multiple time scale analysis, they obtain a self-consistent equation involving only the one-body density. This study extends to arbitrary dimensions previous work on a one-dimensional fluid and highlights the subtleties of kinetic theory in the derivation of dynamical density functional theory.

Theory of thermostatted inhomogeneous granular fluids: A self-consistent density functional description

The authors present a study of the nonequilibrium statistical properties of a one dimensional hard-rod fluid dissipating energy via inelastic collisions and subject to the action of a Gaussian heat bath, simulating an external driving mechanism. They show that the description of the fluid based on the one-particle phase-space reduced distribution function, in principle necessary because of the presence of velocity dependent collisional dissipation, can be contracted to a simpler description in configurational space. Indeed, by means of a multiple-time-scale method the authors derive a self-consistent governing equation for the particle density distribution function. This equation is similar to the dynamic density functional equation employed in the study of colloids, but contains additional terms taking into account the inelastic nature of the fluid. Such terms cannot be derived from a Liapunov generating functional and contribute not only to the relaxational properties, but also to the nonequilibrium steady state properties. A validation of the theory against molecular dynamics simulations is presented in a series of cases, and good agreement is found.

Multiple time-scale approach for a system of Brownian particles in a nonuniform temperature field

The Smoluchowski equation for a system of Brownian particles in a temperature gradient is derived from the Kramers equation by means of a multiple time-scale method. The interparticle interactions are assumed to be represented by a mean-field description. We present numerical results that compare well with the theoretical prediction together with an extensive discussion on the prescription of the Langevin equation in overdamped systems.

Transport of a heated granular gas in a washboard potential

We study numerically the motion of a one dimensional array of Brownian particles in a washboard potential, driven by an external stochastic force and interacting via short range repulsive forces. In particular, we investigate the role of instantaneous elastic and inelastic collisions on the system dynamics and transport. The system displays a locked regime, where particles may move only via activated processes and a running regime where particles drift along the direction of the applied field. By tuning the value of the friction parameter controlling the Brownian motion we explore both the overdamped dynamics and the underdamped dynamics. In the two regimes we considered the mobility and the diffusivity of the system as functions of the tilt and other relevant control parameters such as coefficient of restitution, particle size, and total number of particles. We find that while in the overdamped regime the results for the interacting systems present similarities with the known noninteracting case, in the underdamped regime the inelastic collisions determine a rich variety of behaviors among which is an unexpected enhancement of the inelastic diffusion.