Adsorption on a Spherical Colloidal Particle from a Mixture of Nanoparticles with Competing Interactions.

Adsorption of nanoparticles on a spherical colloidal particle is studied by molecular dynamics simulations. We consider a generic model for a mixture of nanoparticles with energetically favored self-assembly into alternating layers of the two components. When both components are attracted to the colloidal particle, the adsorbed nanoparticles self-assemble either into alternating parallel tori and clusters at the two poles of the colloidal particle, or into alternating spirals wrapped around the spherical surface. The long-lived metastable states obtained in simulations follow from the spherical shape of the adsorbing surface and the requirement that the neighboring chains of the nanoparticles are composed of different components. A geometrical construction leading to all such patterns is presented. When the second component particles are repelled from the colloidal particle and the attraction of the first component is strong, the attracted particles form a monolayer at the surface of the colloidal particle that screens the repulsion of the second component. The subsequent adsorbed alternating spherical layers of the two components form together a thick shell. This structure leads to the adsorption that is larger than in the case of the same attraction of the two components to the colloidal particle.

Pattern Formation in Two-Component Monolayers of Particles with Competing Interactions.

Competing interactions between charged inclusions in membranes of living organisms or charged nanoparticles in near-critical mixtures can lead to self-assembly into various patterns. Motivated by these systems, we developed a simple triangular lattice model for binary mixtures of oppositely charged particles with additional short-range attraction or repulsion between like or different particles, respectively. We determined the ground state for the system in contact with a reservoir of the particles for the whole chemical potentials plane, and the structure of self-assembled conglomerates for fixed numbers of particles. Stability of the low-temperature ordered patterns was verified by Monte Carlo simulations. In addition, we performed molecular dynamics simulations for a continuous model with interactions having similar features, but a larger range and lower strength than in the lattice model. Interactions with and without symmetry between different components were assumed. We investigated both the conglomerate formed in the center of a thin slit with repulsive walls, and the structure of a monolayer adsorbed at an attractive substrate. Both models give the same patterns for large chemical potentials or densities. For low densities, more patterns occur in the lattice model. Different phases coexist with dilute gas on the lattice and in the continuum, leading to different patterns in self-assembled conglomerates ('rafts').

Continuous nonequilibrium transition driven by heat flow.

We discovered an out-of-equilibrium transition in the ideal gas between two walls, divided by an inner, adiabatic, movable wall. The system is driven out-of-equilibrium by supplying energy directly into the volume of the gas. At critical heat flux we have found a continuous transition to the state with a low-density, hot gas on one side of the movable wall and a dense, cold gas on the other side. Molecular dynamic simulations of the soft-sphere fluid confirm the existence of the transition in the interacting system. We introduce a stationary state Helmholtz-like function whose minimum determines the stable positions of the internal wall. This transition can be used as a paradigm of transitions in stationary states and the Helmholtz-like function as a paradigm of the thermodynamic description of these states.

Adsorption in Mixtures with Competing Interactions.

A binary mixture of oppositely charged particles with additional short-range attraction between like particles and short-range repulsion between different ones in the neighborhood of a substrate preferentially adsorbing the first component is studied by molecular dynamics simulations. The studied thermodynamic states correspond to an approach to the gas-crystal coexistence. Dependence of the near-surface structure, adsorption and selective adsorption on the strength of the wall-particle interactions and the gas density is determined. We find that alternating layers or bilayers of particles of the two components are formed, but the number of the adsorbed layers, their orientation and the ordered patterns formed inside these layers could be quite different for different substrates and gas density. Different structures are associated with different numbers of adsorbed layers, and for strong attraction the thickness of the adsorbed film can be as large as seven particle diameters. In all cases, similar amount of particles of the two components is adsorbed, because of the long-range attraction between different particles.

Self-assembly in mixtures with competing interactions.

A binary mixture of particles interacting with spherically-symmetrical potentials leading to microsegregation is studied by theory and molecular dynamics (MD) simulations. We consider spherical particles with equal diameters and volume fractions. Motivated by the mixture of oppositely charged particles with different adsorption preferences immersed in a near-critical binary solvent, we assume short-range attraction long-range repulsion for the interaction between like particles, and short-range repulsion long-range attraction for the interaction between different ones. In order to predict structural and thermodynamic properties of such complex mixtures, we develop a theory combining the density functional and field-theoretical methods. We show that concentration fluctuations in mesoscopic regions lead to a qualitative change of the phase diagram compared to mean-field predictions. Both theory and MD simulations show coexistence of a low-density disordered phase with a high-density phase with alternating layers rich in the first and second components. In these layers, crystalline structure is present in the solid, and absent in the liquid crystals. The density and the degree of order of the ordered phase decrease with increasing temperature, up to a temperature where the theory predicts a narrow two-phase region with increasing density of both phases for increasing temperature. MD simulations show that monocrystals of the solid and liquid crystals have a prolate shape with the axis parallel to the direction of concentration oscillations, and the deviation from the spherical shape increases with increasing periodic order.

Flux and storage of energy in nonequilibrium stationary states.

Systems kept out of equilibrium in stationary states by an external source of energy store an energy ΔU=U-U_{0}. U_{0} is the internal energy at equilibrium state, obtained after the shutdown of energy input. We determine ΔU for two model systems: ideal gas and a Lennard-Jones fluid. ΔU depends not only on the total energy flux, J_{U}, but also on the mode of energy transfer into the system. We use three different modes of energy transfer where the energy flux per unit volume is (i) constant, (ii) proportional to the local temperature, and (iii) proportional to the local density. We show that ΔU/J_{U}=τ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. τ is the characteristic timescale of energy outflow from the system immediately after the shutdown of energy flux. We prove that τ is minimized in stable states of the Rayleigh-Benard cell.

Evaporation of liquid droplets of nano- and micro-meter size as a function of molecular mass and intermolecular interactions: experiments and molecular dynamics simulations.

Transport of heat to the surface of a liquid is a limiting step in the evaporation of liquids into an inert gas. Molecular dynamics (MD) simulations of a two component Lennard-Jones (LJ) fluid revealed two modes of energy transport from a vapour to an interface of an evaporating droplet of liquid. Heat is transported according to the equation of temperature diffusion, far from the droplet of radius R. The heat flux, in this region, is proportional to temperature gradient and heat conductivity in the vapour. However at some distance from the interface, Aλ, (where λ is the mean free path in the gas), the temperature has a discontinuity and heat is transported ballistically i.e. by direct individual collisions of gas molecules with the interface. This ballistic transport reduces the heat flux (and consequently the mass flux) by the factor R/(R + Aλ) in comparison to the flux obtained from temperature diffusion. Thus it slows down the evaporation of droplets of sizes R ∼ Aλ and smaller (practically for sizes from 103 nm down to 1 nm). We analyzed parameter A as a function of interactions between molecules and their masses. The rescaled parameter, A(kBTb/ε11)1/2, is a linear function of the ratio of the molecular mass of the liquid molecules to the molecular mass of the gas molecules, m1/m2 (for a series of chemically similar compounds). Here ε11 is the interaction parameter between molecules in the liquid (proportional to the enthalpy of evaporation) and Tb is the temperature of the gas in the bulk. We tested the predictions of MD simulations in experiments performed on droplets of ethylene glycol, diethylene glycol, triethylene glycol and tetraethylene glycol. They were suspended in an electrodynamic trap and evaporated into dry nitrogen gas. A changes from ∼1 (for ethylene glycol) to approximately 10 (for tetraethylene glycol) and has the same dependence on molecular parameters as obtained for the LJ fluid in MD simulations. The value of x = A(kBTb/ε11)1/2 is of the order of 1 (for water x = 1.8, glycerol x = 1, ethylene glycol x = 0.4, tetraethylene glycol x = 2.1 evaporating into dry nitrogen at room temperature and for Lennard-Jones fluids x = 2 for m1/m2 = 1 and low temperature).

A molecular dynamics test of the Hertz-Knudsen equation for evaporating liquids.

The precise determination of evaporation flux from liquid surfaces gives control over evaporation-driven self-assembly in soft matter systems. The Hertz-Knudsen (HK) equation is commonly used to predict evaporation flux. This equation states that the flux is proportional to the difference between the pressure in the system and the equilibrium pressure for liquid/vapor coexistence. We applied molecular dynamics (MD) simulations of one component Lennard-Jones (LJ) fluid to test the HK equation for a wide range of thermodynamic parameters covering more than one order of magnitude in the values of flux. The flux determined in the simulations was 3.6 times larger than that computed from the HK equation. However, the flux was constant over time while the pressures in the HK equation exhibited strong fluctuations during simulations. This observation suggests that the HK equation may not appropriately grasp the physical mechanism of evaporation. We discuss this issue in the context of momentum flux during evaporation and mechanical equilibrium in this process. Most probably the process of evaporation is driven by a tiny difference between the liquid pressure and the gas pressure. This difference is equal to the momentum flux i.e. momentum carried by the molecules leaving the surface of the liquid during evaporation. The average velocity in the evaporation flux is very small (two to three orders of magnitude smaller than the typical velocity of LJ atoms). Therefore the distribution of velocities of LJ atoms does not deviate from the Maxwell-Boltzmann distribution, even in the interfacial region.

Communication: relative diffusion in two dimensions: breakdown of the standard diffusive model for simple liquids.

Using molecular dynamics simulations for a liquid of identical soft spheres we analyze the relative diffusion constant DΣn(r) and the self diffusion constant Dn where r is the interparticle distance and n = 2, 3 denotes the dimensionality. We demonstrate that for the periodic boundary conditions, Dn is a function of the system size and the relation: DΣn(r = L/2) ≅ 2Dn(L), where L is the length of the cubic box edge, holds both for n = 2 and 3. For n = 2 both DΣ2(r) and D2(L) increase logarithmically with its argument. However, it was found that the diffusive process for large two dimensional systems is very sensitive to perturbations. The sensitivity increases with L and even a very low perturbation limits the increase of D2(L → ∞). Nevertheless, due to the functional form of DΣ2(r) the standard assumption for the Smoluchowski type models of reaction kinetics at three dimensions:DΣn(r) ≈ 2Dn leads to giant errors if applied for n = 2.

Collapse of a nanoscopic void triggered by a spherically symmetric traveling sound wave.

Molecular-dynamics simulations of the Lennard-Jones fluid (up to 10(7) atoms) are used to analyze the collapse of a nanoscopic bubble. The collapse is triggered by a traveling sound wave that forms a shock wave at the interface. The peak temperature T(max) in the focal point of the collapse is approximately ΣR(0)(a), where Σ is the surface density of energy injected at the boundary of the container of radius R(0) and α ≈ 0.4-0.45. For Σ = 1.6 J/m(2) and R(0) = 51 nm, the shock wave velocity, which is proportional to √Σ, reaches 3400 m/s (4 times the speed of sound in the liquid); the pressure at the interface, which is proportional to Σ, reaches 10 GPa; and T(max) reaches 40,000 K. The Rayleigh-Plesset equation together with the time of the collapse can be used to estimate the pressure at the front of the shock wave.

Computer investigations on the asymptotic behavior of the rate coefficient for the annihilation reaction A + A → product and the trapping reaction in three dimensions.

We have performed intensive computer simulations of the irreversible annihilation reaction: A + A → C + C and of the trapping reaction: A + B → C + B for a variety of three-dimensional fluids composed of identical spherical particles. We have found a significant difference in the asymptotic behavior of the rate coefficients for these reactions. Both the rate coefficients converge to the same value with time t going to infinity but the convergence rate is different: the O(t(-1/2)) term for the annihilation reaction is higher than the corresponding term for the trapping reaction. The simulation results suggest that ratio of the terms is a universal quantity with the value equal to 2 or slightly above. A model for the annihilation reaction based on the superposition approximation predicts the difference in the O(t(-1/2)) terms, but overestimates the value for the annihilation reaction by about 30%. We have also performed simulations for the dimerization process: A + A → E, where E stands for a dimer. The dimerization decreases the reaction rate due to the decrease in the diffusion constant for A. The effect is successfully predicted by a simple model.

Large-scale molecular dynamics verification of the Rayleigh-Plesset approximation for collapse of nanobubbles.

We report large-scale (10(7) atoms in an 85-nm-wide container) molecular dynamics simulations of collapse of nanoscopic (5-12 nm in diameter) voids in liquid argon. During the collapse the pressure on the liquid side decreases, and this decrease propagates into liquid at the speed of sound. Despite the nonuniform profile of pressure in the liquid the solutions of the Rayleigh-Plesset equation compares well to the measured evolution of the radius of the void and the velocity of the interface. Evaporation of liquid into the void does not affect the dynamics appreciably.

Evaporation into vacuum: Mass flux from momentum flux and the Hertz-Knudsen relation revisited.

We performed molecular dynamics simulations of liquid film evaporation into vacuum for two cases: free evaporation without external supply of energy and evaporation at constant average liquid temperature. In both cases we found that the pressure inside a liquid film was constant, while temperature decreased and density increased as a function of distance from the middle of the film. The momentum flux in the vapor far from the liquid was equal to the liquid pressure in the evaporating film. Moreover the pseudopressure (stagnation pressure) was found to be constant in the evaporating vapor and equal to the liquid pressure. The momentum flux and its relation to the pressure determined the number of evaporating molecules per unit time and as a consequence the mass evaporation flux. We found a simple formula for the evaporation flux, which much better describes simulation results than the commonly used Hertz-Knudsen relation.

Molecular dynamics study on the influence of quencher concentration on the reaction rate for ionic systems.

The influence of concentrations of reagents on the rate of reaction: A+B-->C+B for low density equimolar mixtures of spherically symmetric ions immersed in the Brownian medium has been investigated by performing large scale molecular dynamics simulations. The Coulomb potential of ion-ion interactions is truncated at the cutoff distance large enough to make the kinetics of the reaction independent of its value. The simulations have been performed at conditions close to that for quenching reactions for fluophores. One of the simulation results is that the excess in the rate coefficient Delta k is always positive and converges to a constant value which is two to three orders in magnitude higher than that for the soft spheres immersed in the Brownian medium [Litniewski, J. Chem. Phys. 124, 114502 (2006)]. Delta k is approximately proportional to c however, if the concentration is high, positive deviations [O(c(2))] are noticeable. The simulation results are compared with simple model that bases on the superposition approximation. The model predicts most of the properties of Delta k. The predicted values are about 30%-40% lower than that from the simulations.

Heat transfer at the nanoscale: evaporation of nanodroplets.

We demonstrate using molecular dynamics simulations of the Lennard-Jones fluid that the evaporation process of nanodroplets at the nanoscale is limited by the heat transfer. The temperature is continuous at the liquid-vapor interface if the liquid/vapor density ratio is small (of the order of 10) and discontinuous otherwise. The temperature in the vapor has a scaling form T(r,t)=T[r/R(t)], where R(t) is the radius of an evaporating droplet at time t and r is the distance from its center. Mechanical equilibrium establishes very quickly, and the pressure difference obeys the Laplace law during evaporation.

The influence of interactions between reagents on the excess in the rate of quenching reaction: molecular dynamics study.

The influence of the interactions between reagents on the excess in the rate coefficient, Deltak, for the instantaneous reaction A+B-->C+B have been investigated by performing large scale molecular dynamics simulations for simple soft spheres. The simulation method has enabled us to determine the contributions to Deltak coming from A-B as well as B-B interactions. The simulations have shown that positive values of Deltak that appear both for the liquid and for the Brownian system [M. Litniewski, J. Chem. Phys. 123, 124506 (2005); 124, 114501 (2006)] result from B-B interactions. If B-B interactions were absent, Deltak was always negative. The influence of B-B interactions was about three times higher for the Brownian system than for the liquid. A qualitative explanation for the effect has been proposed basing on a simple model and analyzing the influence of B-B interactions on fluctuations in concentrations of reagents. The influence of A-B interactions was completely negligible except for the liquid at short times, for which the cancellation of A-B interaction noticeably decreased Deltak.

Computer investigations on the mechanism of the influence of quencher concentration on the rate of simple bimolecular reaction.

Molecular dynamics investigations on the influence of the concentration of B (quencher) on the rate coefficient, k(t), for the reaction A+B-->C+B are continued [M. Litniewski, J. Chem. Phys. 123, 124506 (2005); 124, 114501 (2006)]. The problem is investigated by analyzing the excess in the two-particle probability density function and in its radial moments. The simulations have been performed for the deterministic systems as gas and liquid as well as for the Brownian system. The influence of moderate changes of the reaction radius resulting in changes of the activation energy has been also considered. The most important result is that the excess in k(t) may be not only a direct consequence of fluctuations in concentrations. For the gas, the excess in the mean radial velocity of A towards B dominated over the excess in the value of the probability density function. As a result, the excess in k(t) was negative in spite of the excess in the relative spatial correlations between A and B was positive. The excess in the mean radial velocity was completely unimportant for dense liquids and the Brownian system.

The influence of the quencher concentration on the rate of simple bimolecular reaction: molecular dynamics study. II.

In this paper new results of the simulations [M. Litniewski, J. Chem. Phys. 123, 124506 (2005)] on the influence of the quencher concentration on the reaction A+B-->C+B for the identical soft sphere system are presented. The problem is generalized by considering also the case when the spheres are immersed in the Brownian medium. A significant difference between simple deterministic systems and the Brownian ones is found: the excess in the rate coefficient for the Brownian system is constant and positive, except for very short times. The reaction has been simulated for a very long time, but any tendency to decrease the excess has not been noted. It is also shown that the relative excess in the surviving probability is a universal quadratic function of the quencher concentration for the range of time much longer than the result from the previous simulations. A very strong correlation between the excess in the relative value of spatial correlations between the reagents and the excess in the rate coefficient is shown. It is also shown that the A-A and A-C interactions have some influence on the excess values. A simple model for this effect is presented.

Kinetics of fluorescence quenching for electron transfer and for energy transfer: molecular dynamics tests for spherical molecules.

The Smoluchowski approach to description of fluorescence quenching is tested by comparing the theory with computer simulations for the case of spherical molecules. The distance dependent sink terms describing the electron transfer mechanism and the Forster model for the energy transfer are considered. It is shown that the agreement between the rate coefficient from the model and from simulations depends on the strength of the solute-solvent interactions as well as on the speed of reaction itself. Comparing results of simulations for different quencher concentrations we estimate the strength of quencher concentration dependence effect and the range of times the effect may be significant. In the long time limit the increase in quencher concentration decreased the rate coefficient.

The influence of the quencher concentration on the rate of simple bimolecular reaction: molecular dynamics study.

The paper presents the results of large-scale molecular dynamics simulations of the irreversible bimolecular reaction A+B --> C+B for the simple liquid composed of mechanically identical soft spheres. The systems with the total number of molecules corresponding to 10(7)-10(9) are considered. The influence of the concentration of a quencher (B) on the surviving probability of A and the reaction rate is analyzed for a wide range of the concentrations and for two significantly different reduced densities. It is shown that the quencher concentration dependence effect (QCDE) is, in fact, a composition of two QCDE effects: the short-time QCDE that increases the reaction rate and the long-time QCDE that decreases it. The paper also analyzes the influence of the concentration on the steady-state rate constant, k(ss), obtained by integrating the surviving probability. The excess in k(ss) due to finite quencher concentration changes the sign from negative to positive while going from low to high concentrations. Generally, the excess is extremely weak. It attains a 1% level only if the concentration is very high.