Entropy production of active particles formulated for underdamped dynamics.

The present work investigates the effect of inertia on the entropy production rate Π for all canonical models of active particles for different dimensions and the type of confinement. To calculate Π, the link between the entropy production and dissipation of heat rate is explored, resulting in a simple and intuitive expression. By analyzing the Kramers equation, alternative formulations of Π are obtained and the virial theorem for active particles is derived. Exact results are obtained for particles in an unconfined environment and in a harmonic trap. In both cases, Π is independent of temperature. For the case of a harmonic trap, Π attains a maximal value for τ=ω^{-1}, where τ is the persistence time and ω is the natural frequency of an oscillator. For active particles in one-dimensional box, or other nonharmonic potentials, thermal fluctuations are found to reduce Π.

Theory of Charge Regulation of Colloidal Particles in Electrolyte Solutions.

We present a theory that enables us to (i) calculate the effective surface charge of colloidal particles and (ii) efficiently obtain titration curves for different salt concentrations. The theory accounts for the shift of pH of solution due to the presence of 1:1 electrolyte. It also accounts self-consistently for the electrostatic potential produced by the deprotonated surface groups. To examine the accuracy of the theory, we have performed extensive reactive Monte Carlo simulations, which show excellent agreement between theory and simulations without any adjustable parameters.

Positing the problem of stationary distributions of active particles as third-order differential equation.

In this work, we obtain a third-order linear differential equation for stationary distributions of run-and-tumble particles in two dimensions in a harmonic trap. The equation represents the condition j=0, where j is a flux. Since an analogous equation for passive Brownian particles is first-order, a second- and third-order term are features of active motion. In all cases, the solution has a form of a convolution of two distributions: the Gaussian distribution representing the Boltzmann distribution of passive particles, and the beta distribution representing active motion at zero temperature.

Intuitive view of entropy production of ideal run-and-tumble particles.

This work investigates the entropy production rate, Π, of the run-and-tumble model with a focus on scaling of Π as a function of the persistence time τ. It is determined that (i) Π vanishes in the limit τ→∞, marking it as an equilibrium. Stationary distributions in this limit are represented by a superposition of Boltzmann functions in analogy to a system with quenched disorder. (ii) Optimal Π is attained in the limit τ→0, marking it as a system maximally removed from equilibrium. Paradoxically, the stationary distributions in this limit have the Boltzmann form. The value of Π in this limit is that of an unconfined run-and-tumble particle and is related to the dissipation energy of a sedimenting particle. In addition to these general conclusions, this work derives an exact expression of Π for the run-and-tumble particles in a harmonic trap.

Reactive Monte Carlo simulations for charge regulation of colloidal particles.

We use a reactive Monte Carlo simulation method and the primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different numbers of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium chloride, the titration curves are compared with the experimental measurements obtained using potentiometric titration. A good agreement is found between simulations and experiments. In the case of lithium chloride, the specific ionic adsorption is taken into account through the partial dehydration of lithium ion.

Kuramoto model with run-and-tumble dynamics.

This work considers an extension of the Kuramoto model with run-and-tumble dynamics-a type of self-propelled motion. The difference between the extended and the original model is that in the extended version angular velocity of individual particles is no longer fixed but can change sporadically with a new velocity drawn from a distribution g(ω). Because the Kuramoto model undergoes phase transition, it offers a simple case study for investigating phase transition for a system with self-propelled particles.

Stationary distributions of propelled particles as a system with quenched disorder.

This article is the exploration of the viewpoint within which propelled particles in a steady state are regarded as a system with quenched disorder. The analogy is exact when the rate of the drift orientation vanishes and the linear potential, representing the drift, becomes part of an external potential, resulting in the effective potential u_{eff}. The stationary distribution is then calculated as a disorder-averaged quantity by considering all contributing drift orientations. To extend this viewpoint to the case when a drift orientation evolves in time, we reformulate the relevant Fokker-Planck equation as a self-consistent relation. One interesting aspect of this formulation is that it is represented in terms of the Boltzmann factor e^{-ßu_{eff}}. In the case of a run-and-tumble model, the formulation reveals an effective interaction between particles.

Lattice-gas model of a charge regulated planar surface.

In this work, we consider a lattice-gas model of charge regulation with electrostatic interactions within the Debye-Hückel level of approximation. In addition to long-range electrostatic interactions, the model incorporates the nearest-neighbor interactions for representing non-electrostatic forces between adsorbed ions. The Frumkin-Fowler-Guggenheim isotherm obtained from the mean-field analysis accurately reproduces the simulation data points.

Charge regulation of colloidal particles in aqueous solutions.

We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

Thermodynamic collapse in a lattice-gas model for a two-component system of penetrable particles.

We study a lattice-gas model of penetrable particles on a square-lattice substrate with same-site and nearest-neighbor interactions. Penetrability implies that the number of particles occupying a single lattice site is unlimited and the model itself is intended as a simple representation of penetrable particles encountered in realistic soft-matter systems. Our specific focus is on a binary mixture, where particles of the same species repel and those of the opposite species attract each other. As a consequence of penetrability and the unlimited occupation of each site, the system exhibits thermodynamic collapse, which in simulations is manifested by an emergence of extremely dense clusters scattered throughout the system with energy of a cluster E∝-n^{2}, where n is the number of particles in a cluster. After transforming a particle system into a spin system, in the large density limit the Hamiltonian recovers a simple harmonic form, resulting in the discrete Gaussian model used in the past to model the roughening transition of interfaces. For finite densities, due to the presence of a nonharmonic term, the system is approximated using a variational Gaussian model.

Charge Regulation of Colloidal Particles: Theory and Simulations.

To explore charge regulation (CR) in physicochemical and biophysical systems, we present a model of colloidal particles with sticky adsorption sites which account for the formation of covalent bonds between the hydronium ions and the surface functional groups. Using this model and Monte Carlo simulations, we find that the standard Ninham and Parsegian (NP) theory of CR leads to results which deviate significantly from computer simulations. The problem with the NP approach is traced back to the use of a bulk equilibrium constant to account for surface chemical reactions. To resolve this difficulty we present a new theory of CR. The fundamental ingredient of the new approach is the sticky length, which is nontrivially related to the bulk equilibrium constant. The theory is found to be in excellent agreement with computer simulations, without any adjustable parameters. As an application of the theory we calculate the effective charge of colloidal particles containing carboxyl groups, as a function of pH and salt concentration.

One-dimensional Coulomb system in a sticky wall confinement: Exact results.

This work investigates a one-component one-dimensional Coulomb system in sticky wall confinement. Sticky wall is introduced as an alternative and intuitive depiction of charge regulation, the notion that a surface charge is not a fixed but a fluctuating quantity in dynamic equilibrium with its immediate environment. Emphasis is placed on intuitive derivation and expressions are obtained by observing that the partition function of a charge regulated system can be decomposed into a collection of independent equilibriums with different fixed surface charges. Adsorbed particles behave as ideal-gas particles in a one-dimensional box whose length corresponds to the parameter of stickiness. Among various scenarios considered are a single- and two-wall confinement as well as the case of sticky counterions capable of associating into pairs. Exact solutions provide a view of the role and behavior of surface charge fluctuations, which is an important step in the "beyond-mean-field" analysis. Consequently, the model serves as a simple paradigm of the mechanism that gives rise to the Kirkwood-Shumaker interactions detected in real systems.

General theory of charge regulation within the Poisson-Boltzmann framework: Study of a sticky-charged wall model.

This work introduces a sticky-charge wall model as a simple and intuitive representation of charge regulation. Implemented within the mean-field level of description, the model modifies the boundary conditions without affecting the underlying Poisson-Boltzmann (PB) equation of an electrolyte. Employing various modified PB equations, we are able to assess how various structural details of an electrolyte influence charge regulation.

Mean-field theory of active electrolytes: Dynamic adsorption and overscreening.

We investigate active electrolytes within the mean-field level of description. The focus is on how the double-layer structure of passive, thermalized charges is affected by active dynamics of constituting ions. One feature of active dynamics is that particles adhere to hard surfaces, regardless of chemical properties of a surface and specifically in complete absence of any chemisorption or physisorption. To carry out the mean-field analysis of the system that is out of equilibrium, we develop the "mean-field simulation" technique, where the simulated system consists of charged parallel sheets moving on a line and obeying active dynamics, with the interaction strength rescaled by the number of sheets. The mean-field limit becomes exact in the limit of an infinite number of movable sheets.

Two-component Gaussian core model: Strong-coupling limit, Bjerrum pairs, and gas-liquid phase transition.

In the present work, we investigate a gas-liquid transition in a two-component Gaussian core model, where particles of the same species repel and those of different species attract. Unlike a similar transition in a one-component system with particles having attractive interactions at long separations and repulsive interactions at short separations, a transition in the two-component system is not driven solely by interactions but by a specific feature of the interactions, the correlations. This leads to extremely low critical temperature, as correlations are dominant in the strong-coupling limit. By carrying out various approximations based on standard liquid-state methods, we show that a gas-liquid transition of the two-component system poses a challenging theoretical problem.

Inhomogeneous fluid of penetrable-spheres: Application of the random phase approximation.

The focus of the present work is the application of the random phase approximation (RPA), derived for inhomogeneous fluids [Frydel and Ma, Phys. Rev. E 93, 062112 (2016)], to penetrable-spheres. As penetrable-spheres transform into hard-spheres with increasing interactions, they provide an interesting case for exploring the RPA, its shortcomings, and limitations, the weak- versus the strong-coupling limit. Two scenarios taken up by the present study are a one-component and a two-component fluid with symmetric interactions. In the latter case, the mean-field contributions cancel out and any contributions from particle interactions are accounted for by correlations. The accuracy of the RPA for this case is the result of a somewhat lucky cancellation of errors.

The double-layer structure of overscreened surfaces by smeared-out ions.

The present work focuses on the structure of a double-layer of overscreened charged surfaces by smeared-out charges and probes the link between the structure of a double-layer and the bulk properties of an electrolyte with special view to the role of the Kirkwood crossover. Just as the Kirkwood line divides a bulk solution into a fluid with monotonic and oscillatory decaying correlations, it similarly separates charge inversion into two broad domains, with and without oscillating charge density profile. As initially oscillations may appear like a far-field occurrence, eventually they develop into a full fledged layering of a charge density.

Density functional formulation of the random-phase approximation for inhomogeneous fluids: Application to the Gaussian core and Coulomb particles.

Using the adiabatic connection, we formulate the free energy in terms of the correlation function of a fictitious system, h_{λ}(r,r^{'}), in which interactions λu(r,r^{'}) are gradually switched on as λ changes from 0 to 1. The function h_{λ}(r,r^{'}) is then obtained from the inhomogeneous Ornstein-Zernike equation and the two equations constitute a general liquid-state framework for treating inhomogeneous fluids. The two equations do not yet constitute a closed set. In the present work we use the closure c_{λ}(r,r^{'})≈-λßu(r,r^{'}), known as the random-phase approximation (RPA). We demonstrate that the RPA is identical with the variational Gaussian approximation derived within the field-theoretical framework, originally derived and used for charged particles. We apply our generalized RPA approximation to the Gaussian core model and Coulomb charges.

The double-layer of penetrable ions: an alternative route to charge reversal.

We investigate a double-layer of penetrable ions near a charged wall. We find a new mechanism for charge reversal that occurs in the weak-coupling regime and, accordingly, the system is suitable for the mean-field analysis. The penetrability is achieved by smearing-out the ionic charge inside a sphere, so there is no need to introduce non-electrostatic forces and the system in the low coupling limit can be described by a modified version of the Poisson-Boltzmann equation. The predictions of the theory are compared with the Monte Carlo simulations.

A close look into the excluded volume effects within a double layer.

We explore the effect of steric interaction on the ionic density distribution near a charged hard wall. For weakly charged walls, small particles, and monovalent ions, the mean-field Poisson-Boltzmann equation provides an excellent description of the density profiles. For large ions and large surface charges, however, deviations appear. To explore these, we use the density functional theory. We find that local density functionals are not able to account for steric interactions near a wall. Based on the weighted density approximation, we derive a simple analytical expression for the contact electrostatic potential, which allows us to analytically calculate the differential capacitance of the double layer.