The interplay of diffusion and heterogeneity in nucleation of the networked Ising model

Nucleation underlies the dynamics of most first-order phase transitions in natural and man-made systems. However, most of the systems of interest are out of equilibrium. Little is known on the effect of nonequilibrium factors on the dynamics of nucleation. Here, we use the forward flux sampling method to investigate the effect of nonequilibrium diffusion on nucleation in small-world Ising networks, wherein spins can be exchanged between nearest-neighboring nodes. We introduce a parameter α to quantify the difference of nucleation rate with and without diffusion. We find that α shows a nonmonotonic dependence on the rewiring probability p of small-world networks. In particular, for different diffusion probability D, a crossover happens at p ≃ 0.17, below which the nucleation rate decreases as D increases, suggesting that the diffusion is against nucleation; while above which the nucleation rate increases with D, indicating that the diffusion is in favor of nucleation. By identifying the distinct features of nucleating clusters along the pathways for different randomness of networks, we reveal the underlying mechanism of such a nontrivial dependence.     

The thermodynamics of cluster formation in nucleation theory

We have derived a precise thermodynamic definition of the standard free energy to form a cluster which is used in nucleation theory. The results [Eq. (9)] have a form differing slightly from the form usually used in nucleation theory and show that the Lothe-Pound correction factor is based on a misconception concerning the standard states involved.     

The quartic oscillator in an external field and the statistical physics of highly anisotropic solids

The statistical mechanics of one- and two-dimensional Ginzburg–Landau systems is evaluated analytically, via the transfer matrix method, using an expression of the ground state energy of the quartic anharmonic oscillator in an external field. In the two-dimensional case, the critical temperature of the order–disorder phase transition is expressed as a Lambert function of the inverse inter-chain coupling constant.     

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring. II. The Charged System and the Two-Component Burgers Equations

We further study the stochastic model discussed in ref. 2 in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These solutions are used to describe the properties of the stationary states of the stochastic model. The values of the currents and of various two-point correlation functions obtained from Monte-Carlo simulations are compared with the mean-field results. Like in the case of equal densities, one finds a pure phase, a mixed phase and a disordered phase.     

Surface pressure in clusters and small particles—is it real?

Using the methods of statistical mechanics and applying the conditions of thermal equilibrium for an ensemble of interacting molecules, it is proved that there is no excess pressure, nor internal and surface stresses, in clusters and small particles that are not subjected to the action of external forces. With this premise taken into account the thermodynamics of spherical and faceted small particles is developed. In a particle-vapour system surrounded by a rigid impervious shell, Kelvin's and Thomson's formulae and Wulff's rule are derived. For a particle-melt system at a constant external pressure it is expected that the melting points of a particle and a bulk solid should be equal. It is noted that, if a particle is not subjected to the action of external fields or bodies, its molecules occupy their natural equilibrium positions, and it is from them that deformations and internal stresses should be counted off. These equilibrium positions differ from those occupied by the same molecules in a bulk solid, which is what usually gives rise to the illusions of the stressed state of a small particle and its deformation caused by the “uncompensated” surface forces.     

The accuracy of the approximations in classical nucleation theory

The mathematical approximations involved in the theory of homogeneous nucleation are shown to produce negligible error in the predicted rates of nucleation.     

Prediction of compositional ordering and separation in alloy nanoclusters

The statistical-mechanical free-energy concentration expansion method (FCEM) in conjunction with semi-empirical coordination-dependent energetic parameters was used for atomistic modeling of Ni-Cu-Al, Ni-Cu-Rh and the corresponding binary Ni-Cu, Ni-Al and Rh-Cu nanocluster systems, containing from 13 to 923 atoms with icosahedral and cuboctahedral shapes. The high efficiency of FCEM enables computations of such relatively large binary or ternary alloy clusters. Remarkable differences, governed by the opposite Ni-Cu and Ni-Al heteroatomic interactions, were noted in the surface segregated “magic-number” ordered compositional patterns of the two ternary clusters. Due to the strong Ni-Al interactions, the compositional ordering extends into the cluster core, and at elevated temperatures a sharp order-disorder transition occurs, depending on the cluster size, shape and composition. The computed site-specific atomic concentrations form the basis for evaluating pertinent thermodynamic functions. For all the alloy clusters a Schottky-type heat capacity anomaly is predicted and attributed to gradual desegregation excitation processes. In addition, inter-cluster compositional separation is computed for Rh-Cu clusters, and transition temperatures estimated from the disappearance of convexity in the free-energy vs. composition curves.     

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics—a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.     

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Topology invariance in percolation thresholds

An universal invariant for site and bond percolation thresholds ( and respectively) is proposed. The invariant writes where and are positive constants, and d the space dimension. It is independent of the coordination number, thus exhibiting a topology invariance at any d. The formula is checked against a large class of percolation problems, including percolation in non-Bravais lattices and in aperiodic lattices as well as rigid percolation. The invariant is satisfied within a relative error of for all the twenty lattices of our sample at d=2, d=3, plus all hypercubes up to d=6. Received: 7 July 1997 / Accepted: 5 November 1997     

On the nucleation of hadronic domains in the quark-hadron transition

We present numerical results on bubble profiles, nucleation rates and time evolution for a weakly first-order quark-hadron phase transition in different expansion scenarios. We confirm the standard picture of a cosmological first-order phase transition, in which the phase transition is entirely dominated by nucleation. We also show that, even for expansion rates much lower than those expected in heavy-ion collisions nucleation is very unlikely, indicating that the main phase conversion mechanism is spinodal decomposition.     

Bulk and surface properties of liquid X-Zr (X=Ag, Cu) compound forming alloys

The phase diagrams of the Ag-Zr and Cu-Zr systems exhibit the existence of different intermetallic compounds in the solid state, and since the structure of a liquid alloy is in some respects similar to that of a crystal, the compound formation phenomenon in these liquid alloy systems has been analysed through the study of surface properties (surface tension and surface composition), dynamic properties (chemical diffusion and viscosity) and microscopic functions (concentration fluctuations in the long-wavelength limit and chemical short-range order parameter) in the frame of the compound formation model (CFM). Moreover, the associative tendency between unlike constituent elements qualitatively expressed by the microscopic functions indicates the glass-forming ability of both systems at higher Zr-concentrations. These results are in agreement with reported experimental data and confirm the applicability of a statistical mechanical theory in conjunction with the CFM to describe the mixing behaviour of compound forming alloys.     

Statistical thermodynamics of soft surfaces

We review the continuum, statistical thermodynamics of surfaces and interfaces in soft matter where both the energy and entropy of the surface are comparable. These systems include complex fluids that are dominated by either surface tension or the interfacial curvature, such as: fluid and solid interfaces, colloidal dispersions, macromolecular solutions, membranes, and other self-assembling aggregates such as micelles, vesicles, and microemulsions. The primary focus is on the theoretical concepts, their universality, and the role of fluctuations and inhomogeneities with connections to relevant experimental systems.     

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring. I. The Neutral System

We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP-invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise, and oppositely charged adjacent particles may swap positions. The model depends on two parameters. Analytic calculations using quadratic algebras, inhomogeneous solutions of the mean-field equations, and Monte Carlo simulations suggest that the model has three phases: (1) a pure phase in which one has three pinned blocks of only positive or negative particles and vacancies and in which translational invariance is broken; (2) a mixed phase in which the current has a linear dependence on one parameter, but is independent of the other one and of the density of the charged particles; in this phase one has a bump and a fluid, the bump (condensate) containing positive and negative particles only, the fluid containing charged particles and vacancies uniformly distributed; and (3) the mixed phase is separated from the disordered phase by a second-order phase transition which has many properties of the Bose–Einstein phase transition observed in equilibrium. Various critical exponents are found.     

The effect of the interaction between the minority phase droplets on the nucleation behavior during the liquid-liquid phase transformation

The microstructure evolution during the liquid-liquid phase transformation of Al-Pb alloy was calculated. The numerical results indicate that the interaction between the minority phase droplets has effect on the nucleation process of the droplets, and the effect increases with the cooling rate and the content of Pb. Supported by the National Natural Science Foundation of China (Grant Nos 50395104, 50671111 and 50620130095)     

Treatment of droplike clusters in nucleation theory using Reiss's method

The expression of the correction factor in nucleation theory is derived by extending the method Reiss used recently. is the factor appearing in the number of critical nuclei (formed as a vapor condenses into liquid drops) as a correction to the conventional theory. It is shown that=p _l / _g /kT, wherep _l is the pressure of the liquid phase inside the drop, _g , is the volume per molecule in the vapor phase,k is the Boltzmann constant, andT is the absolute temperature. The difference between this and Reiss's expression isp _l , which replaces hisp _g (the vapor pressure in equilibrium with the drop). The derived in this paper is compatible with the expression= _g / _l ( _l is the molecular volume in the liquid phase) previously proposed by the present author.     

The effect of spatial correlations on steady-state nucleation kinetics in dense fluid systems

A widely used expression for the steady-state nucleation rate is determined, in part, by the concentration of critical nuclei in a constrained equilibrium state of the system under consideration. We show that when a dense solvent is present, the values of the constrained equilibrium concentrations reflect the spatial correlations that arise from reactant-solvent molecule collisions. We evaluate the effect of such correlations for a simple, model fluid in terms of measurable reaction rate constants; our analysis shows that correlations influence each step in the multistep process of cluster formation and that the overall impact on the nucleation rate is cumulative. We argue that the very low, homogeneous nucleation rates observed in certain miscibility gap experiments are easily understood in the context of our analysis.Supported in part by a grant from the Research Corporation of the Cottrell Foundation.John Simon Guggenheim Memorial Fellow.     

The instability of vicinal electrode surfaces against step bunching II: Theory

It is shown that vicinal surfaces with a regular step array in contact with an electrolyte are intrinsically unstable against a phase separation into areas of flat terraces and areas of step bunches. The effect is caused by the step dipole moment which lowers the potential of zero charge (pzc) of stepped surfaces. Specific calculations performed for vicinal surfaces of silver are qualitatively in keeping with experimental observations. Step bunching is likewise expected for vicinal surfaces of gold and platinum.     

The instability of vicinal electrode surfaces against step bunching I: Experiment

Using electrochemical STM we have studied the stability of arrays of parallel, single atom height steps on vicinal Ag(1 1 1) electrodes in electrolyte. We find that the steps for Ag(1 1 1) electrodes are unstable and form first double-steps and later multiple steps, separated by wide, flat terraces. In this paper denoted as “I: Experiment” we deal with the experimental aspects whereas theoretical aspects are discussed in the following paper “II: Theory”.     

Treatment of droplike clusters by means of the classical phase integral in nucleation theory

The translation inconsistency in the theory of nucleation is discussed in historical perspective. A theory is then developed, beginning with the classical phase integral, which not only allows all approximations to be well defined, but also leads to the most natural droplike model for the cluster. The theory makes it possible to apply, in a consistent manner, the thermodynamics of curved surfaces or, alternatively, moleculardynamic numerical computation schemes to the evaluation of the partition function of the cluster. If the cluster is treated as a macroscopic drop (having the free energy of a macroscopic drop), the result for the distribution of clusters differs in only a minor way from that prescribed by the conventional theory of nucleation. It is concluded that for liquid nuclei the conventional theory is consistent, but that a replacement factor may be necessary for solid nuclei. In general, however, the major problems confronting the theory involve the precise evaluation of the work of cluster formation.