Fluctuation theorem with two independent field parameters: The one-dimensional compressible Ising model.

In this work we consider the nonequilibrium mechanical and magnetic work performed on a one-dimensional compressible Ising model. In the harmonic approximation we easily integrate the mechanical degrees of freedom of the model, and the resulting effective Hamiltonian depends on two external parameters, the magnetic field and the force applied along the chain. As the model is exactly soluble in one dimension we can determine the free energy difference between two arbitrary thermodynamic states of the system. We show the validity of the Jarzynski equality, which relates the free energy difference between two thermodynamic states of the system and the average work performed by external agents in a finite time, through nonequilibrium paths between the same thermodynamic states. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field and the mechanical force applied to the system.

Testing ground for fluctuation theorems: The one-dimensional Ising model.

In this paper we determine the nonequilibrium magnetic work performed on a Ising model and relate it to the fluctuation theorem derived some years ago by Jarzynski. The basic idea behind this theorem is the relationship connecting the free energy difference between two thermodynamic states of a system and the average work performed by an external agent, in a finite time, through nonequilibrium paths between the same thermodynamic states. We test the validity of this theorem by considering the one-dimensional Ising model where the free energy is exactly determined as a function of temperature and magnetic field. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field applied to the system. We have also determined the probability distribution function for the work performed on the system for the forward and reverse processes and verified that predictions based on the Crooks relation are equally correct. We also propose a method to calculate the lag between the current state of the system and that of the equilibrium based on macroscopic variables. We have shown that the lag increases with the sweeping rate of the field at its final value for the reverse process, while it decreases in the case of the forward process. The lag increases linearly with the size of the chain and with a slope decreasing with the inverse of the rate of variation of the field.

Phase diagram and criticality of the two-dimensional prisoner's dilemma model.

The stationary states of the prisoner's dilemma model are studied on a square lattice taking into account the role of a noise parameter in the decision-making process. Only first neighboring players-defectors and cooperators-are considered in each step of the game. Through Monte Carlo simulations we determined the phase diagrams of the model in the plane noise versus the temptation to defect for a large range of values of the noise parameter. We observed three phases: cooperators and defectors absorbing phases, and a coexistence phase between them. The phase transitions as well as the critical exponents associated with them were determined using both static and dynamical scaling laws.

Confinement effects on micellar systems with a hydrogen-bonding solvent.

Space confinement greatly influences the properties of liquids, such as their viscosity and capillary critical point. For aqueous solutions of amphiphiles, this effect is extended to the mobility and micellization properties of these molecules, changing important characteristics of micellar solutions such as the critical micelle concentration (CMC). In the present work, we use a lattice Monte Carlo model, which allows for orientational freedom and hydrogen-bond formation for the water molecules, to investigate confinement effects on a solution of surfactants limited by two parallel walls perpendicular to one of the Cartesian axes. This configuration aims to reproduce a small pore, and walls with a hydrophilic or hydrophobic character are studied. We find that, for hydrophilic walls, there is an increase in the value of the CMC for small pores, caused by space confinement effects and also by the interactions of the amphiphile polar heads with the walls. Micelles are able to adhere to the walls as a whole, and their shape shows little change compared to micelles in the bulk solution. Hydrophobic walls show a more dramatic effect on the properties of the solution, arising mainly from the strong adsorption of the amphiphile tails on the walls, driven by the hydrophobic effect. The process of adsorption of amphiphiles with increasing concentration shows a behavior very similar to the one observed in experiments and simulations of such systems. Micelles adsorbed to the hydrophobic walls also show significant changes in their moments of inertia compared to the bulk ones, which is attributed to the formation of half-micelles that have their tails attached to the walls and the polar heads facing the solution. We extend our analysis to the change in the hydrogen-bonding properties of the solvent caused by the confinement, and how that is directly related to the number of free amphiphiles in our system for different pore sizes. Finally, we test different surfactant sizes and how they affect the micellar shape for different concentrations.

Dynamics of micelle formation from temperature-jump Monte Carlo simulations.

In the present work we perform temperature jumps in a surfactant solution by means of Monte Carlo simulations, investigating the dynamics of micelle formation. We use a lattice model that allows orientational freedom and hydrogen bonding for solvent molecules, which can make a connection between the different time scales of hydrogen bond formation and amphiphilic aggregation. When we perform a large jump between a high-temperature nonmicellized state and a micellized state, there is strong hysteresis between the heating and cooling processes, the latter showing the formation of premicelles that act as nucleation centers for the assembly of larger aggregates and the former is a drive for dissociation of the existing aggregates. Hysteresis is not seen when we perform a small jump between two states that can be both micellized or nonmicellized. Looking for a more detailed analysis of the hydrophobic effect that drives aggregation, we compare the time evolution of the solvent hydrogen bonds in our system close and far from micelles and how that is affected by the formation of large clusters at low temperatures. We find a strong connection between them, with the total number of hydrogen bonds in the system always increasing when micelles are formed. To gain insights into the mechanism of premicellar formation and growth, we measure the lifetime of micellized amphiphiles as a function of the aggregate size and the stage of the aggregation process. Our results indicate that the premicelles are always unstable, quickly exchanging amphiphiles with the solution due to their low probabilty in equilibrium. Furthermore, we find that the stability of individual surfactants in micelles increases with the aggregate size, with the lifetime of amphiphiles in large micelles being as much as 35 times longer than in the case of the unstable premicellar region.

Mean-field theory for the long-range contact process with diffusion.

The effect of diffusion in the one-dimensional long-range contact process is investigated by mean-field calculations. Recent works have shown that diffusion decreases the effectiveness of long-range interactions, affecting the character of the phase transition: for higher values of the diffusion coefficient, stronger long-range interactions are required to enable phase coexistence and first-order behavior. Here we apply a generalized mean-field approximation for the master equation of the model that considers states of an aggregate of L lattice sites. The phase diagram of the model for values of L up to 10 is obtained, and for some values of the diffusion rate extrapolations to infinite-sized systems are given. For low-diffusive systems, approximations with L≥3 are able to reveal the suppression of the phase coexistence induced by diffusion, however, in the high-diffusion regime, larger values of L are necessary to correctly account for the higher range of correlations. We present a very efficient method to study the mean-field equations and determine the nature of the phase transitions that may be of general utility.

Critical properties of the Ziff, Gulari, and Barshad (ZGB) model with inert sites.

We study the model proposed by Ziff, Gulari, and Barshad to mimic the oxidation of carbon monoxide (CO) in the presence of fixed impurities distributed over the catalytic surface. Our focus is on the continuous phase transition between the active phase, where occurs the production of carbon dioxide (CO2), and the inactive phase, where all the non inert sites become filled with oxygen molecules. We employ Monte Carlo simulations to calculate the different ratios between moments of the order parameter at the critical point, as well as, we determine the critical exponents ß and ν⊥ as a function of the concentration of impurities. We show that the presence of impurities over the catalytic surface changes the critical behavior of the system. The critical exponents depend on the concentration of impurities and the model does not belong to the directed percolation universality class.

Effective energy barrier distributions for random and aligned magnetic nanoparticles.

Isothermal magnetic relaxation measurements are widely used to probe energy barriers in systems of magnetic nanoparticles. Here we show that the result of such an experiment can differ greatly for aligned and randomly oriented nanoparticles. For randomly oriented cobalt-doped magnetite nanoparticles we observe a prominent low-energy tail in the energy barrier distribution that is greatly attenuated when the particles are magnetically aligned. Monte Carlo simulations show that this behaviour arises for nanoparticles with both cubic and uniaxial magnetic anisotropy energy terms even though for cubic or uniaxial anisotropy alone the energy barrier distribution is independent of nanoparticle orientation.

Formation of large micellar aggregates before equilibrium in diluted solutions.

We study the formation of premicelles for different values of the concentration of amphiphile molecules in water. Our model consists of a square lattice with water molecules occupying one cell of the lattice while the amphiphilic molecules, represented by chains of five interconnected sites, occupy five cells of the lattice. We perform Monte Carlo simulations in the NVT ensemble, for a fixed temperature and different concentration of amphiphiles, ranging from below to above the critical micelle concentration. We start our simulations from a monomeric state and follow in time all the aggregates sizes until the equilibrium state is reached. We pay particular attention to two aggregate sizes, one related to the minimum and the other to the maximum of the aggregate-size distribution curve obtained at equilibrium. We show that these aggregates evolve in time exhibiting a maximum concentration well before the equilibrium state, revealing the formation of premicelles. The times to reach these maximum concentrations decrease exponentially with the total concentration of the system.

Absorbing states in a catalysis model with anti-Arrhenius behavior.

We study a model of heterogeneous catalysis with competitive reactions between two monomers A and B. We assume that reactions are dependent on temperature and follow an anti-Arrhenius mechanism. In this model, a monomer A can react with a nearest neighbor monomer A or B, however, reactions between monomers of type B are not allowed. We assume attractive interactions between nearest neighbor monomers as well as between monomers and the catalyst. Through mean-field calculations, at the level of site and pair approximations, and extensive Monte Carlo simulations, we determine the phase diagram of the model in the plane y(A) versus temperature, where y(A) is the probability that a monomer A reaches the catalyst. The model exhibits absorbing and active phases separated by lines of continuous phase transitions. We calculate the static, dynamic, and spreading exponents of the model, and despite the absorbing state be represented by many different microscopic configurations, the model belongs to the directed percolation universality class in two dimensions. Both reaction mechanisms, Arrhenius and anti-Arrhenius, give the same set of critical exponents and do not change the nature of the universality class of the catalytic models.

Moment ratios and dynamic critical behavior of a reactive system with several absorbing configurations.

We determine the critical behavior of a reactive model with many absorbing configurations. Monomers A and B land on the sites of a linear lattice and can react depending on the state of their nearest-neighbor sites. The probability of a reaction depends on temperature of the catalyst as well as on the energy coupling between pairs of nearest-neighbor monomers. We employ Monte Carlo simulations to calculate the moments of the order parameter of the model as a function of temperature. Some ratios between pairs of moments are independent of temperature and are in the same universality class of the contact process. We also find the dynamical critical exponents of the model and we show that they are in the directed percolation universality class whatever the values of temperature.

Orientational dynamics for an amphiphilic-solvent solution.

In this work, we performed Monte Carlo simulations on a lattice model for spontaneous amphiphilic aggregation, in order to study the orientational and hydrogen-bonding dynamics of water on different regions inside the micellar solution. We employed an associating lattice gas model that mimics the aqueous solvent, which presents a rich phase diagram with first- and second-order transition lines. Even though this is a simplified model, it makes possible to investigate the orientational dynamics of water in an equilibrium solution of amphiphiles, as well as the influence of the different phases of the solvent in the interfacial and bulk water dynamics. By means of extensive simulations, we showed that, at high temperatures, the behavior of the orientational relaxation and hydrogen bonding of water molecules in the bulk, first, and second hydration shells are considerable different. We observe the appearance of a very slow component for water molecules in the first hydration shell of micelles when the system reaches a high-density phase, consistent with previous theoretical and experimental studies concerning biological water. Also, at high temperatures, we find that water molecules in the second hydration shell of micelles have an orientational decay similar to that of bulk water, but with a generally slower dynamics. Otherwise, at low temperatures, we have two components for the orientational relaxation of bulk water in the low density liquid phase, and only a single component in the high density liquid (HDL) phase, which reflect the symmetry properties of the different phases of the solvent model. In the very dense region of water molecules in the first hydration shell of micelles at low temperatures, we find two components for the orientational relaxation on both liquid phases, one of them much slower than that in the single component of bulk water in the HDL phase. This happens even though our model does not present any hindrance to the water rotational freedom caused by the presence of the amphiphiles.

Thin-film growth by random deposition of linear polymers on a square lattice.

We present some results of Monte Carlo simulations for the deposition of particles of different sizes on a two-dimensional substrate. The particles are linear, height one, and can be deposited randomly only in the two x and y directions of the substrate and occupy an integer number of cells of the lattice. We show there are three different regimes for the temporal evolution of the interface width. At the initial times we observe an uncorrelated growth, with an exponent ß1 characteristic of the random deposition model. At intermediate times, the interface width presents an unusual behavior, described by a growing exponent ß2, which depends on the size of the particles added to the substrate. If the linear size of the particle is two we have ß2 < ß1, otherwise we have ß2 > ß1, for all other particle sizes. After the growth reaches the saturation regime where the interface width becomes constant and is described by the roughness exponent α, which is nearly independent of the size of the particle. Similar results are found in the surface growth due to the electrophoretic deposition of polymer chains. Contrary to one-dimensional results the growth exponents are nonuniversal.

Competing reaction model with many absorbing configurations.

We study a competitive reaction model between two monomers A and B on a linear lattice. We assume that monomer A can react with a nearest-neighbor monomer A or B , but reactions between monomers of type B are prohibited. We include in our model lateral interactions between monomers as well as the effects of temperature of the catalyst. The model is considered in the adsorption controlled limit, where the reaction rate is infinitely larger than the adsorption rate of the monomers. We employ site and pair mean-field approximations as well as static Monte Carlo simulations. We determine the phase diagram of the model in the plane y_{A} versus temperature, where y_{A} is the probability that a monomer of the type A arrives at the surface. This phase diagram shows regions of active and absorbing states separated by a line of continuous phase transitions. Despite the absorbing state of the model to be strongly dependent on temperature, we show that the static critical exponents of the model belong to the same universality class of the directed percolation.

Interplay between micelle formation and waterlike phase transitions.

A lattice model for amphiphilic aggregation in the presence of a structured waterlike solvent is studied through Monte Carlo simulations. We investigate the interplay between the micelle formation and the solvent phase transition in two different regions of temperature-density phase diagram of pure water. A second order phase transition between the gaseous (G) and high density liquid (HDL) phases that occurs at very high temperatures, and a first order phase transition between the low density liquid (LDL) and (HDL) phases that takes place at lower temperatures. In both cases, we find the aggregate size distribution curve and the critical micellar concentration as a function of the solvent density across the transitions. We show that micelle formation drives the LDL-HDL first order phase transition to lower solvent densities, while the transition G-HDL is driven to higher densities, which can be explained by the markedly different degrees of micellization in both cases. The diffusion coefficient of surfactants was also calculated in the LDL and HDL phases, changing abruptly its behavior due to the restructuring of waterlike solvent when we cross the first order LDL-HDL phase transition. To understand such behavior, we calculate the solvent density and the number of hydrogen bonds per water molecule close to micelles. The curves of the interfacial solvent density and the number of hydrogen bonds per water molecule in the first hydration signal a local phase change of the interfacial water, clarifying the diffusion mechanism of free surfactants in the solvent.

Monte Carlo simulations for amphiphilic aggregation near a water phase transition.

In this study we analyze the equilibrium and dynamical properties of a lattice model for amphiphilic aggregation in a waterlike associating solvent. The amphiphiles are described as flexible chains of interconnected sites in a body-centered cubic lattice, with hydrophilic and hydrophobic portions. The solvent molecules occupy a single site and resemble the water tetrahedral molecular structure, with the possibility of hydrogen-bond formation and different densities. Following the phase diagram of the solvent model, we are able to study the effects of a phase transition of the solvent in the micellar dynamics. By carrying out Monte Carlo simulations, we analyze the micelle aggregate size distribution curve, the critical micelle concentration, the surfactant diffusion coefficient, the residence time, and the exit/entering rates of the amphiphiles from/to aggregates of different sizes. We also investigate the dipolar reorientational time correlation function for interfacial water and water molecules in the solvent bulk, as well as the number of hydrogen bonds per molecule in both cases.

Random deposition of particles of different sizes.

We study the surface growth generated by the random deposition of particles of different sizes. A model is proposed where the particles are aggregated on an initially flat surface, giving rise to a rough interface and a porous bulk. By using Monte Carlo simulations, a surface has grown by adding particles of different sizes, as well as identical particles on the substrate in (1+1) dimensions. In the case of deposition of particles of different sizes, they are selected from a Poisson distribution, where the particle sizes may vary by 1 order of magnitude. For the deposition of identical particles, only particles which are larger than one lattice parameter of the substrate are considered. We calculate the usual scaling exponents: the roughness, growth, and dynamic exponents alpha, beta, and z, respectively, as well as, the porosity in the bulk, determining the porosity as a function of the particle size. The results of our simulations show that the roughness evolves in time following three different behaviors. The roughness in the initial times behaves as in the random deposition model. At intermediate times, the surface roughness grows slowly and finally, at long times, it enters into the saturation regime. The bulk formed by depositing large particles reveals a porosity that increases very fast at the initial times and also reaches a saturation value. Excepting the case where particles have the size of one lattice spacing, we always find that the surface roughness and porosity reach limiting values at long times. Surprisingly, we find that the scaling exponents are the same as those predicted by the Villain-Lai-Das Sarma equation.

Three-dimensional square water in the presence of an external electric field.

In this work we study a tridimensional statistical model for the hydrogen-bond (HB) network formed in liquid water in the presence of an external electric field. This model is analogous to the so-called square water, whose ground state gives a good estimate for the residual entropy of the ice. In our case, each water molecule occupies one site of a cubic lattice, and no hole is allowed. The hydrogen atoms of water molecules are disposed at the lines connecting nearest-neighbor sites, in a way that each water can be found in 15 different states. We say that there is a hydrogen bond between two neighboring molecules when only one hydrogen is in the line connecting both molecules. Through Monte Carlo simulations with Metropolis and entropic sampling algorithms, and by exact calculations for small lattices, we determined the dependence of the number of molecules aligned to the field and the number of hydrogen bonds per molecule as a function of temperature and the intensity of the external field. The results for both approaches showed that, different of the two-dimensional case, there is no maximum in the number of HBs as a function of the electric field. However, we observed nonmonotonic behaviors as a function of the temperature of the quantities of interest. We also found the dependence of the entropy on the external electric field at very low temperatures. In this case, the entropy vanishes for the value of the external field for which the contributions to the total energy coming from the HBs and the field become the same.

Short-time dynamics for the spin-3/2 Blume-Capel model.

We employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.

A trimer model for water.

A statistical model for water is studied, where the molecules are represented by trimers in a triangular lattice. Each atom of a water molecule occupies a single site on the lattice, and the HOH bond angle is assumed to be 120 degrees. The molecules can interact via three different potentials: the excluded volume interaction, which prevents two molecules from occupying the same atom site, an attractive potential between any two nearest-neighbor atoms belonging to different molecules (the van der Waals interaction), and the hydrogen bond interaction, which occurs only for a particular orientation and displacement of a pair of molecules. The model is investigated by means of Monte Carlo simulations in the canonical and grand canonical ensembles. The Metropolis and the entropic sampling algorithms are used to obtain the thermodynamics of the system. We find that the entropic sampling prescription is the most efficient algorithm of them, providing information about the entropy and free energy of the system in a straightforward way. The curves for the polarization, number of hydrogen bonds, specific heat, and cumulant of energy were obtained as a function of the temperature and total concentration. In addition, the entropy of the noninteracting version of the model is compared to that of the angular trimers in a square lattice and triangles in a triangular lattice.