Heterogeneous nucleation of a droplet pinned at a chemically inhomogeneous substrate: A simulation study of the two-dimensional Ising case.

Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field Hb favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity are analyzed. Three regimes occur: for small fields, HbHb*), the droplets nucleated at the chemical inhomogeneity grow to the full system size. While the relaxation time for the growth scales as τG∝Hb-1, the nucleation time τN scales as lnτN∝Hb-1. However, the prefactor in the latter relation, as evaluated for our simulations results, is not in accord with an extension of the Volmer-Turnbull theory to two-dimensions, when the theoretical contact angle θc is used.

Numerical evidence of hyperscaling violation in wetting transitions of the random-bond Ising model in d=2 dimensions.

We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.036101] establishes that the average magnetization of the sample, with critical exponent ß, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by γ+2ß=ν_{∥}+ν_{⊥} requires ß=1/2 (γ=4, ν_{∥}=3, and ν_{⊥}=2), the thermodynamic scaling establishes that Δ_{s}=γ+ß, which in contrast requires ß=0 (Δ_{s}=4), where γ, ν_{∥}, ν_{⊥}, and Δ_{s} are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., ß=0) and a direct measurement of the susceptibility critical exponent γ/ν_{⊥}=2.0±0.2, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.

Pinning-depinning transition in a stochastic growth model for the evolution of cell colony fronts in a disordered medium.

We study a stochastic lattice model for cell colony growth, which takes into account proliferation, diffusion, and rotation of cells, in a culture medium with quenched disorder. The medium is composed of sites that inhibit any possible change in the internal state of the cells, representing the disorder, as well as by active medium sites that do not interfere with the cell dynamics. By means of Monte Carlo simulations we find that the velocity of the growing interface, which is taken as the order parameter of the model, strongly depends on the density of active medium sites (ρ_{A}). In fact, the model presents a (continuous) second-order pinning-depinning transition at a certain critical value of ρ_{A}^{crit}, such as, for ρ_{A}>ρ_{A}^{crit}, the interface moves freely across the disordered medium, but for ρ_{A}<ρ_{A}^{crit} the interface becomes irreversible pinned by the disorder. By determining the relevant critical exponents, our study reveals that within the depinned phase the interface can be rationalized in terms of the Kardar-Parisi-Zhang universality class, but when approaching the critical threshold, the nonlinear term of the Kardar-Parisi-Zhang equation tends to vanish and then the pinned interface belongs to the quenched Edwards-Wilkinson universality class.

Droplets pinned at chemically inhomogenous substrates: A simulation study of the two-dimensional Ising case.

As a simplified model of a liquid nanostripe adsorbed on a chemically structured substrate surface, a two-dimensional Ising system with two boundaries at which surface fields act is studied. At the upper boundary, the surface field is uniformly negative, while at the lower boundary (a distance L apart), the surface field is negative only outside a range of extension b, where a positive surface stabilizes a droplet of the phase with positive magnetization for temperatures T exceeding the critical temperature T_{w} of the wetting transition of this model. We investigate the local order parameter profiles across the droplet, both in the directions parallel and perpendicular to the substrate, varying both b and T. Also, precursor effects to droplet formation as T approaches T_{w} from below are studied. In accord with theoretical predictions, for T>T_{w} the droplet is found to have the shape of a semiellipse, where the width (distance of the interface from the substrate) scale is proportional to b (b^{1/2}). So, the area of the droplet is proportional to b^{3/2}, and the temperature dependence of the corresponding prefactor, which also involves the interfacial stiffness, is studied.

Tricritical wetting in the two-dimensional Ising magnet due to the presence of localized non-magnetic impurities.

Fixed vacancies (non-magnetic impurities) are placed along the centre of Ising strips in order to study the wetting behaviour in this confined system, by means of numerical simulations analysed with the aid of finite size scaling and thermodynamic integration methods. By considering strips of size L × M (L << M) where short-range competitive surface fields (H(s)) act along the M-direction, we observe localization-delocalization transitions of the interface between magnetic domains of different orientation (driven by the corresponding surface fields), which are the precursors of the wetting transitions that occur in the thermodynamic limit. By placing vacancies or equivalently non-magnetic impurities along the centre of the sample, we found that for low vacancy densities the wetting transitions are of second order, while by increasing the concentration of vacancies the transitions become of first order. Second- and first-order lines meet in tricritical wetting points (H(tric)(SW), T(tric)(W)), where H(tric)(SW) and T(Tric)(W) are the magnitude of the surface field and the temperature, respectively. In the phase diagram, tricritical points shift from the high temperature and weak surface field regime at large vacancy densities to the T --> 0, H(tric)(SW) --> 1 limit for low vacancy densities. By comparing the locations of the tricritical points with those corresponding to the case of mobile impurities, we conclude that in order to observe similar effects, in the latter the required density of impurities is much smaller (e.g. by a factor 3-5). Furthermore, a proper density of non magnetic impurities placed along the centre of a strip can effectively pin rather flat magnetic interfaces for suitable values of the competing surface fields and temperature.

Far-from-equilibrium growth of magnetic thin films with Blume-Capel impurities.

We investigate the irreversible growth of (2+1)-dimensional magnetic thin films. The spin variable can adopt three states (s(I)=±1,0), and the system is in contact with a thermal bath of temperature T. The deposition process depends on the change of the configuration energy, which, by analogy to the Blume-Capel Hamiltonian in equilibrium systems, depends on Ising-like couplings between neighboring spins (J) and has a crystal field (D) term that controls the density of nonmagnetic impurities (s(I)=0). Once deposited, particles are not allowed to flip, diffuse, or detach. By means of extensive Monte Carlo simulations, we obtain the phase diagram in the crystal field vs temperature parameter space. We show clear evidence of the existence of a tricritical point located at D(t)/J=1.145(10) and k(B)T(t)/J=0.425(10), which separates a first-order transition curve at lower temperatures from a critical second-order transition curve at higher temperatures, in analogy with the previously studied equilibrium Blume-Capel model. Furthermore, we show that, along the second-order transition curve, the critical behavior of the irreversible growth model can be described by means of the critical exponents of the two-dimensional Ising model under equilibrium conditions. Therefore, our findings provide a link between well-known theoretical equilibrium models and nonequilibrium growth processes that are of great interest for many experimental applications, as well as a paradigmatic topic of study in current statistical physics.

First-order and tricritical wetting transitions in the two-dimensional Ising model caused by interfacial pinning at a defect line.

We present a study of the critical behavior of the Blume-Capel model with three spin states (S=±1,0) confined between parallel walls separated by a distance L where competitive surface magnetic fields act. By properly choosing the crystal field (D), which regulates the density of nonmagnetic species (S=0), such that those impurities are excluded from the bulk (where D=-∞) except in the middle of the sample [where D(M)(L/2)≠-∞], we are able to control the presence of a defect line in the middle of the sample and study its influence on the interface between domains of different spin orientations. So essentially we study an Ising model with a defect line but, unlike previous work where defect lines in Ising models were defined via weakened bonds, in the present case the defect line is due to mobile vacancies and hence involves additional entropy. In this way, by drawing phase diagrams, i.e., plots of the wetting critical temperature (T(w)) versus the magnitude of the crystal field at the middle of the sample (D(M)), we observe curves of (first-) second-order wetting transitions for (small) high values of D(M). Theses lines meet in tricritical wetting points, i.e., (T(w)(tc),D(M)(tc)), which also depend on the magnitude of the surface magnetic fields. It is found that second-order wetting transitions satisfy the scaling theory for short-range interactions, while first-order ones do not exhibit hysteresis, provided that small samples are used, since fluctuations wash out hysteretic effects. Since hysteresis is observed in large samples, we performed extensive thermodynamic integrations in order to accurately locate the first-order transition points, and a rather good agreement is found by comparing such results with those obtained just by observing the jump of the order parameter in small samples.

Evidence of a robust universality class in the critical behavior of self-propelled agents: metric versus topological interactions.

The nature of the interactions among self-propelled agents (SPA), i.e., topological versus metric or a combination of both types, is a relevant open question in the field of self-organization phenomena. We studied the critical behavior of a Vicsek-like system of SPA given by a group of agents moving at constant speed and interacting among themselves under the action of a topological rule: each agent aligns itself with the average direction of its seven nearest neighbors, independent of the distance, under the influence of some noise. Based on both stationary and dynamic measurements, we provide strong evidence that both types of interactions are manifestations of the same phenomenon, which defines a robust universality class. Also, the cluster size distribution evaluated at the critical point shows a power-law behavior, and the exponent corresponding to the topological model is in excellent agreement with that of the metric one, further reinforcing our claim. Furthermore, we found that with topological interactions the average distance of influence between agents undergoes large fluctuations that diverge at the critical noise, thus providing clues about a mechanism that could be implemented by the agents to change their moving strategy.

Influence of nonuniform surface magnetic fields in wetting transitions in a confined two-dimensional Ising ferromagnet.

Wetting transitions are studied in the two-dimensional Ising ferromagnet confined between walls where competitive surface fields act. In our finite samples of size L×M, the walls are separated by a distance L, M being the length of the sample. The surface fields are taken to be short-range and nonuniform, i.e., of the form H(1),δH(1),H(1),δH(1),..., where the parameter -1≤δ≤1 allows us to control the nonuniformity of the fields. By performing Monte Carlo simulations we found that those competitive surface fields lead to the occurrence of an interface between magnetic domains of different orientation that runs parallel to the walls. In finite samples, such an interface undergoes a localization-delocalization transition, which is the precursor of a true wetting transition that takes place in the thermodynamic limit. By exactly working out the ground state (T=0), we found that besides the standard nonwet and wet phases, a surface antiferromagnetic-like state emerges for δ<-1/3 and large fields (H(1)>3), H(1)(tr)/J=3, δ(tr)=-1/3,T=0, being a triple point where three phases coexist. By means of Monte Carlo simulations it is shown that these features of the phase diagram remain at higher temperatures; e.g., we examined in detail the case T=0.7×T(cb). Furthermore, we also recorded phase diagrams for fixed values of δ, i.e., plots of the critical field at the wetting transition (H(1w)) versus T showing, on the one hand, that the exact results of Abraham [Abraham, Phys. Rev. Lett. 44, 1165 (1980)] for δ=1 are recovered, and on the other hand, that extrapolations to T→0 are consistent with our exact results. Based on our numerical results we conjectured that the exact result for the phase diagram worked out by Abraham can be extended for the case of nonuniform fields. In fact, by considering a nonuniform surface field of some period λ, with λ<0], one can obtain the effective field H(eff) at a λ coarse-grained level given by H(eff)=1/λ∑(x=1)(λ)H(1)(x,λ). Then we conjectured that the exact solution for the phase diagram is now given by H(eff)/J=F(T), where F(T) is a function of the temperature T that straightforwardly follows from Abraham's solution. The conjecture was exhaustively tested by means of computer simulations. Furthermore, it is found that for δ≠1 the nonwet phase becomes enlarged, at the expense of the wet one, i.e., a phenomenon that we call "surface nonuniformity-induced nonwetting," similar to the already known case of "roughness-induced nonwetting."

Critical behaviour of the Ising ferromagnet confined in quasi-cylindrical pores: a Monte Carlo study.

The critical behaviour of the Ising ferromagnet confined in pores of radius R and length L is studied by means of Monte Carlo computer simulations. Quasi-cylindrical pores are obtained by replicating n-times a triangular lattice disc of radius R, where L = na and a is the spacing between consecutive replications. So, spins placed at the surface of the pores have less nearest-neighbours (NN) as compared to 8 NN for spins in the bulk. These "missing neighbour" effects undergone by surface spins cause a strong suppression of surface ordering, leading to an ordinary surface transition. Also, the effect propagates into the bulk for small tubes (R ≤ 12) and the effective critical temperature of the pores is shifted towards lower values than in the bulk case. By applying the standard finite-size scaling theory, subsequently supported by numerical data, we concluded that data collapse of relevant observables, e.g., magnetization (m), susceptibility, specific heat, etc., can only be observed by comparing simulation results obtained by keeping the aspect ratio C ≡ R∕L constant. Also, by extrapolating "effective" R-dependent critical temperatures to the thermodynamic limit (R → ∞, C fixed), we obtained T(C)(∞) = 6.208(4). As suggested by finite-size scaling arguments, the magnetization is measured at the critical point scales according to [|m|]Tc R(ß/ν) is proportional to [R/L](1/2), where ß and ν are the standard exponents for the order parameter and the correlation length, respectively. Furthermore, it is shown that close to criticality the axial correlation length decreases exponentially with the distance. That result is the signature of the formation of (randomly distributed) alternating domains of different magnetization, which can be directly observed by means of snapshot configurations, whose typical length (ξ) is given by the characteristic length of the exponential decay of correlations. Moreover, we show that at criticality ξ = 0.43(2)R.

Wetting transition in the two-dimensional Blume-Capel model: a Monte Carlo study.

The wetting transition of the Blume-Capel model is studied by a finite-size scaling analysis of L×M lattices where competing boundary fields ±H_{1} act on the first or last row of the L rows in the strip, respectively. We show that using the appropriate anisotropic version of finite-size scaling, critical wetting in d=2 is equivalent to a "bulk" critical phenomenon with exponents α=-1, ß=0, and γ=3. These concepts are also verified for the Ising model. For the Blume-Capel model, it is found that the field strength H_{1c}(T) where critical wetting occurs goes to zero when the bulk second-order transition is approached, while H_{1c}(T) stays nonzero in the region where in the bulk a first-order transition from the ordered phase, with nonzero spontaneous magnetization, to the disordered phase occurs. Interfaces between coexisting phases then show interfacial enrichment of a layer of the disordered phase which exhibits in the second-order case a finite thickness only. A tentative discussion of the scaling behavior of the wetting phase diagram near the tricritical point is also given.

Finite-size scaling approach for critical wetting: rationalization in terms of a bulk transition with an order parameter exponent equal to zero.

Clarification of critical wetting with short-range forces by simulations has been hampered by the lack of accurate methods to locate where the transition occurs. We solve this problem by developing an anisotropic finite-size scaling approach and show that then the wetting transition is a "bulk" critical phenomenon with order parameter exponent equal to zero. For the Ising model in two dimensions, known exact results are straightforwardly reproduced. In three dimensions, it is shown that previous estimates for the location of the transition need revision, but the conclusions about a slow crossover away from mean-field behavior remain unaltered.

Criticality and the onset of ordering in the standard Vicsek model.

Experimental observations of animal collective behaviour have shown stunning evidence for the emergence of large-scale cooperative phenomena resembling phase transitions in physical systems. Indeed, quantitative studies have found scale-free correlations and critical behaviour consistent with the occurrence of continuous, second-order phase transitions. The standard Vicsek model (SVM), a minimal model of self-propelled particles in which their tendency to align with each other competes with perturbations controlled by a noise term, appears to capture the essential ingredients of critical flocking phenomena. In this paper, we review recent finite-size scaling and dynamical studies of the SVM, which present a full characterization of the continuous phase transition through dynamical and critical exponents. We also present a complex network analysis of SVM flocks and discuss the onset of ordering in connection with XY-like spin models.

Far-from-equilibrium growth of thin films in a temperature gradient.

The irreversible growth of thin films under far-from-equilibrium conditions is studied in (2+1)-dimensional strip geometries. Across one of the transverse directions, a temperature gradient is applied by thermal baths at fixed temperatures between T(1) and T(2), where T(1)

Study and characterization of interfaces in a two-dimensional generalized voter model.

We propose and study, by means of numerical simulations, the time evolution of interfaces in a generalized voter model in d=2 dimensions. In this model, a randomly selected voter can change his or her opinion (state) with a certain probability that is an algebraic function of the average opinion of his or her nearest neighbors. By starting with well-defined (sharp) interfaces between two different states of opinion, we measure the time dependence of the interface width (w), which behaves as a power law, i.e., w α t(δ). In this way we characterized three different types of interfaces: (i) between an ordered phase (consensus) and a disordered one (δ=1/2); (ii) between ordered phases having different states of opinion (δ=1/2), which corresponds to interface coarsening without surface tension; and (iii) as in (ii) but considering surface tension. Here, we observe a finite-size induced crossover with exponents δ=1/4 and δ=1/2 for early and longer times, respectively. So, our study allows for the characterization of interfaces of quite different nature in a unified fashion, providing insight into the understanding of interface coarsening with and without surface tension.

Phase diagram and critical behavior of a forest-fire model in a gradient of immunity.

The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl [Physica A 199, 183 (1993)], in which each site of a lattice can be in three possible states: occupied by a tree, empty, or occupied by a burning tree (fire). The trees grow at empty sites with probability p, healthy trees catch fire from adjacent burning trees with probability (1-g), where g is the immunity, and a burning tree becomes an empty site spontaneously. In this paper we study the FFMIT by means of the recently proposed gradient method (GM), considering the immunity as a uniform gradient along the horizontal axis of the lattice. The GM allows the simultaneous treatment of both the active and the inactive phases of the model in the same simulation. In this way, the study of a single-valued interface gives the critical point of the active-absorbing transition, whereas the study of a multivalued interface brings the percolation threshold into the active phase. Therefore we present a complete phase diagram for the FFMIT, for all range of p, where, besides the usual active-absorbing transition of the model, we locate a transition between the active percolating and the active nonpercolating phases. The average location and the width of both interfaces, as well as the absorbing and percolating cluster densities, obey a scaling behavior that is governed by the exponent α=1/(1+ν), where ν is the suitable correlation length exponent (ν(⊥) for the directed percolation transition and ν for the standard percolation transition). We also show that the GM allows us to calculate the critical exponents associated with both the order parameter of the absorbing transition and the number of particles in the multivalued interface. Besides, we show that by using the gradient method, the collapse in a single curve of cluster densities obtained for samples of different side is a very sensitive method in order to obtain the critical points and the percolation thresholds.

Short-time critical dynamics of damage spreading in the two-dimensional Ising model.

The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at T=∞ and magnetization M=0 , an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization M0 in one of the configurations upon quenching the system at T C, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent θ D=1.915(3) , which is much larger than the exponent θ=0.197 characteristic of the initial increase of the magnetization M(t). Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic (R2(t)) grows with an exponent z∗ ≈ η ≈ 1.9, which is the same, within error bars, as the exponent θ D. However, the survival probability of the epidemics reaches a plateau so that δ=0. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at T D ≃ 0.51TC, where all the measured observables exhibit power laws with exponents θ D=1.026(3), δ=0.133(1), and z∗=1.74(3).

Continuous and discrete modeling of the decay of two-dimensional nanostructures.

In this work we review some recent research on the surface diffusion-mediated decay of two-dimensional nanostructures. These results include both a continuous, vectorial model and a discrete kinetic Monte Carlo approach. Predictions from the standard linear continuous theory of surface-diffusion-driven interface decay are contrasted with simulational results both from kinetic and morphological points of view. In particular, we focused our attention on high-aspect-ratio nanostructures, where strong deviations from linear theory take place, including nonexponential amplitude decay and the emergence of several interesting nanostructures such as overhangs developing, nanoislands and nanovoids formation, loss of convexity, nanostructures-pinch off and nanostructures-break off, etc.

Nature of the order-disorder transition in the Vicsek model for the collective motion of self-propelled particles.

One of the most popular approaches to the study of the collective behavior of self-driven individuals is the well-known Vicsek model (VM) [T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995)]. In the VM one has that each individual tends to adopt the direction of motion of its neighbors with the perturbation of some noise. For low enough noise the individuals move in an ordered fashion with net transport of mass; however, when the noise is increased, one observes disordered motion in a gaslike scenario. The nature of the order-disorder transition, i.e., first-versus second-order, has originated an ongoing controversy. Here, we analyze the most used variants of the VM unambiguously establishing those that lead either to first- or second-order behavior. By requesting the invariance of the order of the transition upon rotation of the observational frame, we easily identify artifacts due to the interplay between finite-size and boundary conditions, which had erroneously led some authors to observe first-order transitionlike behavior.

Proposal and applications of a method for the study of irreversible phase transitions.

The gradient method for the study of irreversible phase transitions in far-from-equilibrium lattice systems is proposed and successfully applied to both the archetypical case of the Ziff-Gulari-Barshad model [R. M. Ziff, Phys. Rev. Lett. 56, 2553 (1986)] and a forest-fire cellular automaton. By setting a gradient of the control parameter along one axis of the lattice, one can simultaneously treat both the active and the inactive phases of the system. In this way different interfaces are defined whose study allows us to find the active-inactive phase transition (both of first and second order), as well as the description of the active phase as composed of two further phases: the percolating and the nonpercolating ones. The average location and the width of the interfaces obey standard scaling behavior that is essentially governed by the roughness exponent alpha=1/(1+nu) , where nu is the suitable correlation length exponent.