Phase Field Method

The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous smooth fields occupying the entire simulation volume, and dislocations are identified with locations where the field values change rapidly. As we will see later, as an approach to dislocation dynamics simulations PFM holds several advantages. First, it is easier to implement into a computer code than a line DD model. In particular, the complex procedures for making topological changes (Section 10.4) are no longer necessary. Second, the implementation of PFM can take advantage of well-developed and efficient numerical methods for solving partial differential equations (PDEs). Another important merit of PFM is its applicability in a wide range of seemingly different situations. For example, it is possible to simulate the interaction and co-evolution of several types of material microstructures, such as dislocations and alloying impurities, within a unified model. PFM has become popular among physicists and materials scientists over the last 20 years, but as a numerical method it is not new. After all, it is all about solving PDEs on a grid. Numerical integration of PDEs is a vast and mature area of computational mathematics. A number of efficient methods have already been developed, such as the finite difference method [121], the finite element method [122], and spectral methods [123], all of which have been used in PFM simulations. The relatively new aspects of PFM are associated with the method’s formulation and applications, which are partly driven by the growing interest in understanding material microstructures. In Section 11.1, we begin with the general aspects of PFM demonstrated by two simple applications of the method not related to dislocations. Section 11.2 describes the elements required to adapt PFM to dislocation simulations. There we will briefly venture into the field of micromechanics and consider the concept of eigenstrain. The elastic energy of an arbitrary eigenstrain field is derived in Section 11.3. Section 11.4 discusses an example in which the PFM equations for dislocations are solved using the fast Fourier transform method.

Kinetic Monte Carlo Method

The PN model discussed in the preceding chapter is a continuum approach that requires some atomistic input to account for non-linear interactions in the dislocation core. In this chapter, we introduce yet another continuum model that uses atomistic input for a different purpose. The kinetic Monte Carlo (kMC) model does not consider any details of the core structure but instead focuses on dislocation motion on length and time scales far greater than those of the atomistic simulations. The model is especially effective for diamond-cubic semiconductors and other materials in which dislocation motion is too slow to be observed on the time scale of molecular dynamics simulations. The key idea of the kMC approach is to treat dislocation motion as a stochastic sequence of discrete rare events whose mechanisms and rates are computed within the framework of the transition state theory. Built around its unit mechanisms, the kMC model simulates dislocation motion and predicts dislocation velocity as a function of stress and temperature. This data then can be used to construct accurate mobility functions for dislocation dynamics simulations on still larger scales (Chapter 10). In this sense, kMC serves as a link between atomistic models and coarse-grained continuum models of dislocations. The kMC approach is most useful in situations where the system evolves through a stochastic sequence of events with only a few possible event types. The method has been used in a wide variety of applications other than dislocations. For example, the growth of solid thin films from vapor or in solution is known to proceed through attachment and diffusion of adatoms deposited on the surface. Based on a finite set of unit mechanisms of the motion of adatoms, kMC models accurately describe the kinetics of growth and the resulting morphology evolution of the epitaxial films [95, 96, 97]. Similar kMC models have been applied to dislocation motion in crystals with high lattice resistance, such as silicon. In these materials, dislocations consist of long straight segments interspersed with atomic-sized kinks, depicted schematically in Fig. 9.1(a) as short vertical segments. As was explained in Section 1.3, dislocation motion proceeds through nucleation and migration of kink pairs and can be described well by a kMC model.

Free-Energy Calculations

Free energy is of central importance for understanding the properties of physical systems at finite temperatures. While in the zero temperature limit the system should evolve to a state of minimum energy (Section 2.3), this is not necessarily the case at a finite temperature. When an open system exchanges energy with the outside world (a thermostat) and maintains a constant temperature, its evolution proceeds towards minimizing its free energy. For example, a crystal turns into a liquid when the temperature exceeds its melting temperature precisely because the free energy of the liquid state becomes lower than that of the crystalline state. In the context of dislocation simulations, free energy is all important when one has to decide which of the possible core configurations the dislocation is likely to adopt at a given temperature.

Peierls–Nabarro Model Of Dislocations

Chapter 1 introduced dislocations as dual objects permitting both atomistic and continuum descriptions. The subsequent Chapters 2 through 7 discussed various aspects of atomistic simulations and their application to dislocation modeling. In the rest of the book, from Chapter 8 to Chapter 11, we will be treating dislocations as continuum objects. This is a huge simplification that makes it possible to consider dislocation behavior on length and time scales well beyond reach of the atomistic simulations. The following chapters are organized in the order of increasing length and time scales. This particular chapter deals with the famous Peierls–Nabarro continuum model that is most closely related to the atomistic models discussed earlier. Fundamentally, dislocations are line defects producing distortions in an otherwise perfect crystal lattice. While this point of view is entirely correct, the atomistic models of dislocations can deal with only relatively small material volumes where every atom is individually resolved. Furthermore, having to keep track of all these atoms all the time limits the time horizon of atomistic simulations. On the other hand, when the host crystal is viewed as an elastic continuum, the linear elasticity theory of dislocations offers a variety of useful analytical and numerical solutions that are no longer subject to such constraints. Although quite accurate far away from the dislocation center, where the lattice distortions remain small, continuum theory breaks down near the dislocation center, where lattice discreteness and non-linearity of interatomic interactions become important. To obtain a more efficient description of crystal dislocations, some sort of bridging between the atomistic and continuum models is necessary. For example, it would be very useful to have a hybrid continuum–atomistic approach such that it retains the analytic nature of the continuum theory for the long-range elastic fields but also captures the essential non-linear effects in the atomic core. Bearing the names of Rudolf Peierls [86] and Frank Nabarro [87], the celebrated Peierls–Nabarro (PN) model is one such approach. Possibly the most attractive feature of the PN model is its simplicity.

Case Study of Dynamic Simulation

The preceding chapter focused on the dislocation core structure at zero temperature obtained by energy minimization. In this chapter we will discuss a case study of dislocation motion at finite temperature by molecular dynamics (MD) simulations. MD simulations offer unique insights into the mechanistic and quantitative aspects of dislocation mobility because accurate measurements of dislocation velocity are generally difficult, and direct observations of dislocation motion in full atomistic detail are still impossible. The discussion of this case study is complete in terms of relevant details, including boundary and initial conditions, temperature and stress control, and, finally, visualization and data analysis. In Section 3.1 we discussed a method for introducing a dislocation into a simulation cell. It relies on the linear elasticity solutions for dislocation displacement fields. To expand our repertoire, let us try another method here. The idea is to create a planar misfit interface between two crystals, such that subsequent energy minimization would automatically lead to dislocation formation.

Fundamentals of Atomistic Simulations

Fundamentally, materials derive their properties from the interaction between their constituent atoms. These basic interactions make the atoms assemble in a particular crystalline structure. The same interactions also define how the atoms prefer to arrange themselves in the dislocation core. Therefore, to understand the behavior of dislocations, it is necessary and sufficient to study the collective behavior of atoms in crystals populated by dislocations. This chapter introduces the basic methodology of atomistic simulations that will be applied to the studies of dislocations in the following chapters. Section 1 discusses the nature of interatomic interactions and introduces empirical models that describe these interactions with various degrees of accuracy. Section 2 introduces the significance of the Boltzmann distribution that describes statistical properties of a collection of interacting atoms in thermal equilibrium. This section sets the stage for a subsequent discussion of basic computational methods to be used throughout this book. Section 3 covers the methods for energy minimization. Sections 4 and 5 give a concise introduction to Monte Carlo and molecular dynamics methods. When put close together, atoms interact by exerting forces on each other. Depending on the atomic species, some interatomic interactions are relatively easy to describe, while others can be very complicated. This variability stems from the quantum mechanical motion and interaction of electrons [15, 16]. Henceforth, rigorous treatment of interatomic interactions should be based on a solution of Schrödinger’s equation for interacting electrons, which is usually referred to as the first principles or ab initio theory. Numerical calculations based on first principles are computationally very expensive and can only deal with a relatively small number of atoms. In the context of dislocation modelling, relevant behaviors often involve many thousands of atoms and can only be approached using much less sophisticated but more computationally efficient models. Even though we do not use it in this book, it is useful to bear in mind that the first principles theory provides a useful starting point for constructing approximate but efficient models that are needed to study large-scale problems involving many atoms.

Case Study Of Static Simulation

Having discussed the basic concepts of atomistic simulations, we now turn to a case study that demonstrates the use of the static simulation techniques and, along the way, reveals some of the realistic aspects of the dislocation core structure and highlights the coupling between continuum and atomistic descriptions of dislocations. Section 3.1 explains how to use simple solutions of the continuum elasticity theory for setting up initial positions of atoms. The important issue of boundary conditions is then discussed in Section 3.2. Section 3.3 presents several practical methods for visualization of dislocations and other crystal defects in an atomistic configuration. This first case study sets the stage for a subsequent exploration of more complex aspects of dislocation behavior, which demands more advanced methods of atomistic simulations to be discussed in Chapters 4 through 7. In Section 1.2 we already considered the atomistic structure of dislocations in simple cubic crystals. In other crystals, the atomistic structure of dislocations is considerably more complicated but can be revealed through an atomistic simulation. This is the topic of this chapter. An atomistic structure is specified by the positions xi of all atoms. In a perfect crystal, xi ´s are completely determined by the crystal’s Bravais lattice, its atomic basis and its lattice constant (Section 1.1). Now assume a dislocation or some other defect is introduced, distorting the crystal structure and moving atoms to new positions x´i . A good way to describe the new structure is by specifying the displacement vector ui ≡x'i −xi for every atom i. The relationship between ui and xi can be obtained analytically if we approximate the crystal as a continuum linear elastic solid. This is certainly an approximation, but it works well as long as crystal distortions remain small. Let us define x as the position of a material point in the continuum before the dislocation is introduced and x´ as the position of the same material point after the dislocation is introduced. Here, x is a continuous variable and the displacement vector u(x)≡x´−x expressed as a function of x is the displacement field.

Line Dislocation Dynamics

In the preceding chapters we have discussed several computational approaches focused on the structure and motion of single dislocations. Here we turn our attention to collective motion of many dislocations, which is what the method of dislocation dynamics (DD) was designed for. Typical length and time scales of DD simulations are on the order of microns and seconds, similar to in situ transmission electron microscopy (TEM) experiments where dislocations are observed to move in real time. In a way, DD simulations can be regarded as a computational counterpart of in situ TEM experiments. One very valuable aspect of such a “computational experiment” is that one has full control of the simulation conditions and access to the positions of all dislocation lines at any instant of time. Provided the dislocation model is realistic, DD simulations can offer important insights that help answer the fundamental questions in crystal plasticity, such as the origin of the complex dislocation patterns that emerge during plastic deformation and the relationship between microstructure, loading conditions and the mechanical strength of the crystal. So far, two approaches to dislocation dynamics simulations have emerged. In the line DD method to be discussed in this chapter, dislocations are represented as mathematical lines in an otherwise featureless host medium. An alternative approach is to rely on a continuous field of eigenstrains, in which regions of high strain gradients reveal the locations of the dislocation lines. This representation leads to the phase field DD approach, which will be discussed in Chapter 11. Line DD has certain similarities with the models discussed in the previous chapters, but, at the same time, is rather different from all of them. For example, the representation of dislocations by line segments in line DD method is similar to the kinetic Monte Carlo (kMC) model of Chapter 9. However, having to deal with multiple dislocations on large length and times scales necessitates a more economical treatment of dislocations in the line DD method. Thus, line DD usually relies on less detailed discretization of dislocation lines and treats dislocation motion as deterministic.

Finding Transition Pathways

As was discussed in Chapter 2, stable and accurate numerical integration of the MD equations of motion demands a small time step. In MD simulations of solids, the integration step is usually of the order of one femtosecond (10−15 s). For this reason, the time horizon ofMDsimulations of solids rarely exceeds one nanosecond (10−9 s). On the other hand, dislocation behaviors of interest typically occur on time scales of milliseconds (10−3 s) or longer. Such behaviors remain out of reach for direct MD simulations. Time-scale limits of a similar nature also exist in MC simulations. For instance, the magnitude of the atomic displacements in the Metropolis algorithm has to be sufficiently small to ensure a reasonable acceptance ratio, which results in a slow exploration of the configurational space. This disparity of time scales can be traced to certain topographical features of the potential-energy function of the many-body system, typically consisting of deep energy basins separated by high energy barriers. The system spends most of its time wandering around within the energy basins (metastable states) only rarely interrupted by transitions from one basin to another. Whereas the long-term evolution of a solid results from transitions between the metastable states, direct MDand MC simulations spend most of the time faithfully tracing the unimportant fluctuations within the energy basins. In this sense, most of the computing cycles are wasted, leading to very low simulation efficiency. Because the transition rates decrease exponentially with the increasing barrier heights and decreasing temperature, this problem of time-scale disparity can be severe.

Introduction To Crystal Dislocations

Dislocations first appeared as an abstract mathematical concept. In the late 19th century, Italian mathematician Vito Volterra examined mathematical properties of singularities produced by cutting and shifting matter in a continuous solid body [1]. As happened to some other mathematical concepts, dislocations could have remained a curious product of mathematical imagination known only to a handful of devoted mathematicians. In 1934, however, three scientists, Taylor, Polanyi and Orowan, independently proposed that dislocations may be responsible for a crystal’s ability to deform plastically [2, 3, 4]. While successfully explaining most of the puzzling phenomenology of crystal plasticity, crystal dislocations still remained mostly a beautiful hypothesis until the late 1950s when first sightings of them were reported in transmission electron microscopy (TEM) experiments [5]. Since then, the ubiquity and importance of dislocations for crystal plasticity and numerous other aspects of material behavior have been regarded as firmly established as, say, the role of DNA in promulgating life. Dislocations define a great many properties of crystalline materials. In addition to a crystal’s ability to yield and flow under stress, dislocations also control other mechanical behaviors such as creep and fatigue, ductility and brittleness, indentation hardness and friction. Furthermore, dislocations affect how a crystal grows from solution, how a nuclear reactor wall material is damaged by radiation, and whether or not a semiconductor chip in an electronic device will function properly. It can take an entire book just to describe the various roles dislocations play in materials behavior. However, the focus of this book is on the various computational models that have been developed to study dislocations. This chapter is an introduction to the basics of dislocations, setting the stage for subsequent discussions of computational models and associated numerical issues. Like any other crystal defect, dislocations are best defined with respect to the host crystal structure. We begin our discussion by presenting in Section 1.1 the basic elements and common terminology used to describe perfect crystal structures. Section 1.2 introduces the dislocation as a defect in the crystal lattice and discusses some of its essential properties. Section 1.3 discusses forces on dislocations and atomistic mechanisms for dislocation motion.