Introduction
A key ingredient in the study of these processes is the mechanism by which two particles exchange positions. From the viewpoint of modelling and simulation purposes, direct particle-particle exchange obviously lead to the simplest codes and coarse-grained equations. Moreover, invoking universality, it is expected that the details of the microscopic mechanisms do not affect large-wavelength, long-time properties, and universal scaling functions. In this spirit, Kawasaki dynamics and the Cahn-Hilliard equation have been extensively used to describe phase ordering in binary alloys. However, in most real solids, microscopic atom-atom exchange can be mediated by a variety of processes [2], direct exchange plays a rather small role since steric hindrance tends to create large energy barrier. The most common mechanisms involve exchange of defects, such as vacancies or interstitial sites. For these reasons, alloys have often been modelled by three-state models [3] whose dynamics is controlled by atom-vacancy exchange. Direct exchange is completely forbidden. Two problems in particular have attracted considerable attention: first, the effect of vacancies on phase separation and domain growth[4,5] , and second, their role in atomic interdiffusion[6,7] . A number of studies has also addressed vacancy-mediated ordering in a variety of antiferromagnetic alloys[8] , as well as surface modes of unstable droplets in a stable vapor phase[9].
In this paper, we will describe a third aspect of defect-mediated dynamics, namely, the inverse of the phase ordering problem. Instead of studying the growth of order in response to a sudden temperature decrease (quenching), we focus on the disordering of a finite system, following a rapid increase in temperature. Starting at zero-temperature ferromagnetic configuration, i.e., a perfectly phase-segregated system with sharp interfaces, we monitor how the interfaces ``roughen'' and how particles of one species are transported into regions dominated by the other species. Clearly, if the final temperature is sufficiently high, the interfaces will eventually disappear, resulting in a homogeneous final state. Several questions emerge quite naturally: Are there characteristic time scales on which the disordering takes place, and how do they depend on the system size, temperature, and other controlling parameters? How do local density profiles and correlation functions evolve with time? Are there any scaling regimes, and what are the appropriate scaling variables? How do these features respond to changes in the relative concentrations of vacancies and alloy components?
In this paper, we address some of these questions with a simple model for defect-mediated interface destruction and bulk disordering. We consider a symmetric (Ising-like) binary alloy of A and B atoms which is diluted by a very small number of vacancies 10-5, found in most real systems). Following an up-quench from zero to a finite temperature, T, the vacancies act as a ``catalyst'' for the disordering process, exchanging with neighboring particles according to the usual energetics of the Ising model. The particles themselves form a passive background whose dynamics are slaved to the defect motion. Thus, this system corresponds to a real material in which the characteristic time scale for vacancy diffusion is much faster than the ordinary bulk diffusion time. While vacancies are typically distributed uniformly in the bulk, certain defects may prefer to accumulate at the interfaces. Thus, the number of defects is not necessarily extensive in system size.
While we allow for some variation in the vacancy number, we consider equal concentrations of A and B atoms. Thus, our work forms a natural complement to the only other study[10] of vacancy-mediated disordering in the literature. There, the alloy composition is chosen highly asymmetric: 95% of A atoms versus only 5% for the B species, with a single vacancy. Thus, the A atoms form a matrix for a B-precipitate. The alloy is first equilibrated at a very low temperature, so that small clusters of B atoms are present. It is then rapidly heated to a higher temperature, and the number and size of B clusters are monitored. Three different scenarios are observed, depending on whether the final temperature is below the miscibility gap, above the miscibility gap but below Tc, or above Tc. In the first case, the precipitates remain compact. They dissolve partially at first, but then equilibrate again by coarsening. In the second case, the precipitates also remain compact but eventually dissolve completely, mostly through ``evaporation'' from their surfaces. In the third case, the clusters decompose rapidly (``exploded'') into a large number of small fragments which then disappear diffusively.
The complete or partial mixing of two materials at an interface plays a key role in many physical processes, such as corrosion or erosion phenomena[11]. We mention just two applications with huge technological potential for device fabrication. The first concerns nanowire etching by electron beam lithography[12] : If a thin film of platinum is deposited on a silicon wafer, interdiffusion of Pt and Si produces a mixing layer. If this layer is heated locally by, e.g., exposure to a conventional electron beam, silicides, such as Pt2Si and PtSi, form. The unexposed platinum can subsequently be etched away, leaving conducting nanoscale structures behind. The second example concerns mesoscopic superlattice structures, consisting of alternating magnetic and nonmagnetic metallic layers. If adjacent magnetic layers couple antiferromagnetically, the application of a large uniform magnetic field to these layered structures results in giant magnetoresistance[13]. However, the performance of these devices requires precisely engineered layer thicknesses and interfaces, and can be significantly affected by disorder[14], including interdiffusion or interfacial fluctuations.
Motivated by these applications, our study can only form a baseline here, for further work on more realistic models. However, it also has some rather fundamental implications. First, it serves as a testing ground for a basic problem in statistical physics, namely, how a system approaches its final steady state, starting from an initial non-stationary configuration. A second view of our study addresses the effect of a random walker on its background medium. Each move of a vacancy slightly rearranges the background atoms, leaving a trail behind like a child running across a sandy beach. In the simplest case, the walker is purely Brownian. In our language, this corresponds to an up-quench to infinite temperature, T= K, where energy barriers are irrelevant. In this case, there is no feedback from the background to the local motion of the vacancy. Nevertheless, each displaced atom displays its own intriguing dynamics (especially in d=2)[15,16]. To study the collective behavior of the atoms, [17] explored a lattice filled with just two species of (indistinguishable) particles. Beyond this simple case is a system heated to finite temperatures. Since the background affects the vacancy through the local energetics associated with the next move, no exact solutions are known. Instead, progress relies mainly on simulations.
The key result of our study is the observation of three distinct temporal regimes, separated by two crossover times, provided the final temperature is not too close to Tc. The intermediate andlate stages of the disordering process exhibit dynamic scaling, with characteristic exponents and scaling functions that are computed analytically. For the system size considered here, a clear breakdown of these scaling forms is observed for temperatures within about 10% of Tc . We argue that this occurs when the correlation length becomes comparable to the system size. In contrast, the early stage of the disordering process consists of interfacial destruction in a highly anisotropic manner[16].
This article is organized as follows. We first introduce our model and define an appropriate disorder parameter. We then give the two-dimensiononal system results starting from the the purely diffusive case, corresponding to T= K and less diffusive at finite tempearture Tc < T < K where particle-particle interactions come into play in Section III. We summarize and conclude with some comments on the effect up-quenches to temperatures, T < Tc, and details concerning interfacial destruction in the Section IV.
