Active colloidal particles that are propelled by a self-diffusiophoretic mechanism are often described by Langevin equations that are either postulated on physical grounds or derived using the methods of fluctuating hydrodynamics. While these descriptions are appropriate for colloids of micrometric and larger size, they will break down for very small active particles. A fully microscopic derivation of Langevin equations for self-diffusiophoretic particles powered by chemical reactions catalyzed asymmetrically by the colloid is given in this paper. The derivation provides microscopic expressions for the translational and rotational friction tensors, as well as reaction rate coefficients appearing in the Langevin equations. The diffusiophoretic force and torque are expressed in terms of nonequilibrium averages of fluid fields that satisfy generalized transport equations. The results provide a description of active motion on small scales where descriptions in terms of coarse grained continuum fluid equations combined with boundary conditions that account for the presence of the colloid may not be appropriate.
The propulsion of active particles by self-diffusiophoresis is driven by asymmetric catalytic reactions on the particle surface that generate a mechanochemical coupling between the fluid velocity and the concentration fields of fuel and product in the surrounding solution. Because of thermal and molecular fluctuations in the solution, the motion of micrometric or submicrometric active particles is stochastic. Coupled Langevin equations describing the translation, rotation, and reaction of such active particles are deduced from fluctuating chemohydrodynamics and fluctuating boundary conditions …
We investigate the motion of a colloidal particle driven out of equilibrium by an external torque. We use the molecular dynamics simulation that is alternative to the numerical integration approach based on the Langevin equation and is expected to mimic an experiment more realistically. We choose a heat bath composed of thousands of particles interacting to each other through the Lennard-Jones potential and impose the Langevin thermostat to maintain it in equilibrium. We prepare a single colloidal particle to interact with the particles of the heat bath also by the Lennard-Jones potential whil…
Most available theoretical predictions for the self-diffusiophoretic motion of colloidal particles are based on the hydrodynamic thin boundary layer approximation in combination with a solvent body force due to a self-generated local solute gradient. This gradient is enforced through specifying boundary conditions, typically without accounting for the thermodynamic cost to maintain the gradient. Here we present an alternative thermodynamic approach that exploits a direct link between dynamics and entropy production: the local detailed balance condition. We study two cases: First, we revisit se…
In this paper, we explore experimentally the phase behavior of a dense active suspension of self- propelled colloids. In addition to a solid-like and a gas-like phase observed for high and low densities, a novel cluster phase is reported at intermediate densities. This takes the form of a stationary assembly of dense aggregates, with an average size which grows with activity as a linear function of the self-propelling velocity. While different possible scenarii can be considered to account for these observations - such as a generic velocity weakening instability recently put forward -, we show…
We describe a tracer in a bath of soft Brownian colloids by a particle coupled to the density field of the other bath particles. From the Dean equation, we derive an exact equation for the evolution of the whole system, and show that the density field evolution can be linearized in the limit of a dense bath. This linearized Dean equation with a tracer taken apart is validated by the reproduction of previous results on the mean-field liquid structure and transport properties. Then, the tracer is submitted to an external force and we compute the density profile around it, its mobility and its di…
Recent experiments on Brownian colloidal particles have been studied theoretically in terms of overdamped Langevin equations with multiplicative white noise using an unconventional stochastic interpretation. Complementary numerical simulations of the same system are well described using the conventional Stratonovich interpretation. Here we address this dichotomy from a more generic starting point: the underdamped Langevin equation and its corresponding Fokker--Planck equation.
The individual motion of a colloidal particle is described by an overdamped Langevin equation. When rotational degrees of freedom are relevant, these are described by a corresponding Langevin process. Our purpose is to show that the microscopic local density of colloids, in terms of a space and rotation state, also evolves according to a Langevin equation. The latter can then be used as the starting point of a variety of approaches, ranging from dynamical density functional theory to mode-coupling approximations.
The collectivity in the simultaneous diffusion of many particles, i.e. the interdependence of stochastic forces affecting different particles in the same solution, is a largely overlooked phenomenon with no well-established theory. Recently, we have proposed a novel type of thermodynamically consistent Langevin dynamics driven by the Spatially Correlated Noise (SCN) that can contribute to the understanding of this problem. This model draws a link between the theory of effective interactions in binary colloidal mixtures and the properties of SCN. In the current article we review this model from…
We generalise the Langevin equation with Gaussian white noise by replacing the velocity term by a local fractional derivative. The solution of this equation is a Levy process. We further consider the Brownian motion of a fractal particle, for example, a colloidal aggregate or a biological molecule and argue that it leads to a Levy flight. This effect can also be described using the local fractional Langevin equation. The implications of this development to other complex data series are discussed.
Particles that are immersed in a fluid exchange momentum via the fluid, hence their Brownian motion is correlated. By means of multiparticle-collision dynamics simulations we study the interactions between two colloidal beads in a sheared fluid suspension. Recently, this topic has been addressed in experiments on colloidal particles trapped by optical tweezers in a microfluidic device [PRL, 103, 230602 (2009)] and theoretically by means of a Langevin model [Eur. Phys. J E, 33, 313 (2010)]. Although we neglect the rotational degrees of freedom of the colloids, and employ a very simple coupling …
We propose a method to analyze the dynamics of systems exhibiting slow relaxation which is based on mesoscopic non-equilibrium thermodynamics. The method allows us to obtain kinetic equations of the Fokker-Planck type for the probability functional and their corresponding Langevin equations. Our results are compared with the ones obtained by other authors.
Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamic clustering [I. Theurkauff \emph{et al.}, Phys.\ Rev.\ Lett.\ \textbf{108}, 268303 (2012)]. We model this situation by taking into account the translational and rotational diffusiophoretic motion, which the active particles perform in their self-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clustering only when these two phoretic contributions give rise to competing attractive and repulsive interactions, respectively. We identify two dynamic clustering states and…
Hydrodynamic memory in overdamped colloidal dynamics
Princeton Center for Theoretical Science, Princeton University（普林斯顿大学理论物理中心）；Princeton Materials Institute, Princeton University（普林斯顿大学材料研究所）；Department of Mechanical and Aerospace Engineering, Princeton University（普林斯顿大学机械与航空航天工程系）
This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various $n$-time joint distribution functions. Here we present a novel approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Consideri…
We analyze theoretically the dynamics of a single colloidal particle in an externally applied electric field. The thermal motions of microions lead to an anisotropic, nonequilibrium source of noise, pro- portional to the field, in the effective Langevin equation for the colloid. The fluctuation-dissipation ratio depends strongly on frequency, and the colloid if displaced from its steady-state position relaxes with a velocity not proportional to the gradient of the logarithm of the steady-state probability.
We present a new hybrid lattice-Boltzmann and Langevin molecular dynamics scheme for simulating the dynamics of suspensions of spherical colloidal particles. The solvent is modeled on the level of the lattice-Boltzmann method while the molecular dynamics is done for the solute. The coupling between the two is implemented through a frictional force acting both on the solvent and on the solute, which depends on the relative velocity. A spherical colloidal particle is represented by interaction sites at its surface. We demonstrate that this scheme quantitatively reproduces the translational and r…
In this letter, we develop a mode-coupling theory for a class of nonlinear Langevin equations with multiplicative noise using a field theoretic formalism. These equations are simplified models of realistic colloidal suspensions. We prove that the derived equations are consistent with the fluctuation-dissipation theorem. We also discuss the generalization of the result given here to real fluids, and the possible description of supercooled fluids in the aging regime. We demonstrate that the standard idealized mode-coupling theory is not consistent with the FDT in a strict field theoretic sense.
When the motion of a probe strongly disturbs the thermal equilibrium of the solvent or bath, the nonlinear response of the latter must enter the probe's effective evolution equation. We derive that induced stochastic dynamics using second order response around the bath thermal equilibrium. We discuss the nature of the new term in the evolution equation which is not longer purely dissipative, and the appearance of a novel time-scale for the probe related to changes in the dynamical activity of the bath. A major application for the obtained nonlinear generalized Langevin equation is in the study…
A nonequilibrium fluctuation theorem is established for a colloidal particle driven by an external force within the hydrodynamic theory of Brownian motion, describing hydrodynamic memory effects such as the t^(-3/2) power-law decay of the velocity autocorrelation function. The generalized Langevin equation is obtained for the general case of slip boundary conditions between the particle and the fluid. The Gaussian probability distributions for the particle to evolve in position-velocity space are deduced. It is proved that the joint probability distributions of forward and time-reversed paths …
The effective dynamics of a colloidal particle immersed in a complex medium is often described in terms of an overdamped linear Langevin equation for its velocity with a memory kernel which determines the effective (time-dependent) friction and the correlations of fluctuations. Recently, it has been shown in experiments and numerical simulations that this memory may depend on the possible optical confinement the particle is subject to, suggesting that this description does not capture faithfully the actual dynamics of the colloid, even at equilibrium. Here, we propose a different approach in w…
We present the theoretical analysis of the steady state currents and density distributions of particles moving with Langevin dynamics, under the effects of an external potential displaced at constant rate. The Dynamic Density Functional (DDF) formalism is used to introduce the effects of the molecular interactions, from the equilibrium Helmholtz free energy density functional. We analyzed the generic form of the DDF for one-dimensional external potentials and the limits of strong and weak potential barriers. The ideal gas case is solved in a closed form for generic potentials and compared with…
Critical Casimir forces can play an important role for applications in nano-science and nanotechnology, owing to their piconewton strength, nanometric action range, fine tunability as a function of temperature, and exquisite dependence on the surface properties of the involved objects. Here, we investigate the effects of critical Casimir forces on the free dynamics of a pair of colloidal particles dispersed in the bulk of a near-critical binary liquid solvent, using blinking optical tweezers. In particular we measure the time evolution of the distance between the two colloids to determine thei…
The blossoming of interest in colloids and nano-particles has given renewed impulse to the study of hard-body systems. In particular, hard spheres have become a real test system for theories and experiments. It is therefore necessary to study the complex dynamics of such systems in presence of a solvent; disregarding hydrodynamic interactions, the simplest model is the Langevin equation. Unfortunately, standard algorithms for the numerical integration of the Langevin equation require that interactions are slowly varying during an integration timestep. This in not the case for hard-body systems…
Spherical Janus particles are one of the most prominent examples for active Brownian objects. Here, we study the diffusiophoretic motion of such microswimmers in experiment and in theory. Three stages are found: simple Brownian motion at short times, super-diffusion at intermediate times, and finally diffusive behavior again at long times. These three regimes observed in the experiments are compared with a theoretical model for the Langevin dynamics of self-propelled particles with coupled translational and rotational motion. Besides the mean square displacement also higher displacement moment…
We derive generalized Fluctuation-Dissipation Relations (FDR) holding for a general stochastic dynamics that includes as subcases both equilibrium models for passive colloids and non-equilibrium models used to describe active particles. The relations reported here differ from previous formulations of the FDR because of their simplicity: they require only the microscopic knowledge of the dynamics instead of the whole expression of the steady-state probability distribution function that, except for linear interactions, is unknown for systems displaying non-vanishing currents. From the response f…
We derive a closed equation for the empirical concentration of colloidal particles in the presence of both hydrodynamic and direct interactions. The ensemble average of our functional Langevin equation reproduces known deterministic Dynamic Density Functional Theory (DDFT) [Rex and Löwen, Phys. Rev. Lett., 101(14):148302, 2008], and, at the same time, it also describes the microscopic fluctuations around the mean behavior. We suggest separating the ideal (non-interacting) contribution from additional corrections due to pairwise interactions. We find that, for an incompressible fluid and in the…
Analytical models describing the motion of colloidal particles in given velocity fields are presented. In addition to local approaches, leading to well known master equations such as the Langevin and the Fokker-Planck equations, a global description based on path integration is reviewed. This shows that under very broad conditions, during its evolution a dissipative system tends to minimize its energy dissipation in such a way to keep constant the Hamiltonian time rate, equal to the difference between the flux-based and the force-based Rayleigh dissipation functions. At steady state, the Hamil…
Langevin equations for the self-thermophoretic dynamics of Janus motors partially coated with an absorbing layer that is heated by a radiation field are presented. The derivation of these equations is based on fluctuating hydrodynamics and radiative heat transfer theory involving stochastic equations for bulk phases and surface processes that are consistent with microscopic reversibility. Expressions for the self-thermophoretic force and torque for arbitrary slip boundary conditions are obtained. The overdamped Langevin equations for the colloid displacement and radiative heat transfer provide…
Quantitatively understanding of the dynamics of an active Brownian particle (ABP) interacting with a viscoelastic polymer environment is a scientific challenge. It is intimately related to several interdisciplinary topics such as the microrheology of active colloids in a polymer matrix and the athermal dynamics of the in vivo chromosome or cytoskeletal networks. Based on Langevin dynamics simulation and analytic theory, here we explore such a viscoelastic active system in depth using a star polymer of functionality $f$ with the center cross-linker particle being ABP. We observe that the ABP cr…
For stochastic non-equilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both $Δs\t$is shown to obey a fluctuation theorem $<\exp[-Δs\t]> =1$ for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.
We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail wi…
We present results from Langevin dynamics simulations of a glassy active-passive mixture of soft-repulsive binary colloidal disks. Activity on the smaller particles is applied according to the quorum sensing scheme, in which a smaller particle will be active for a persistence time if its local nearest neighbors are equal to or greater than a certain threshold value. We start with a passive glassy state of the system and apply activity to the smaller particles, which shows a nonmonotonous glassy character of the active particles with the persistence time of the active force, from its passive li…
Based on the Fokker-Planck equation we investigate the transport of an overdamped colloidal particle in a static, asymmetric periodic potential supplemented by a time-dependent, delayed feedback force, $F_{\mathrm{fc}}$. For a given time $t$, $F_\mathrm{fc}$ depends on the status of the system at a previous time $t-τ_\mathrm{D}$, with $τ_\mathrm{D}$ being a delay time, specifically on the delayed mean particle displacement (relative to some "switching position"). For non-zero delay times $F_{\mathrm{fc}}(t)$ develops nearly regular oscillations generating a net current in the system. Depending…
By now active Brownian motion is a well-established model to describe the motion of mesoscopic self-propelled particles in a Newtonian fluid. On the basis of the generalized Langevin equation, we present an analytic framework for active Brownian motion with memory delay assuming time-dependent friction kernels for both translational and orientational degrees of freedom to account for the time-delayed response of a viscoelastic medium. Analytical results are obtained for the orientational correlation function, mean displacement and mean-square displacement which we evaluate in particular for a …
