5R2. Crystals, Defects and Microstructures: Modeling Across Scales. - R Phillips (Brown Univ, Providence RI). Cambridge UP, Cambridge, UK. 2001. 780 pp. Softcover. ISBN 0-521-79357-2. $47.95.
Reviewed by MS Kuczma (Inst of Civil and Env Eng, Univ of Zielona Gora, ul Podgorna 50, Zielona Gora, 65-246, Poland).
This is an invaluable book devoted to the modeling of crystalline materials at different scales. The ultimate Phillips’s aim has been to show the origins of approximate, effective theories of material behavior and how to build such theories that are capable of capturing complex problems involving either multiple length or time scale simultaneously. The book has mixed character, alternating between text and monograph mode, with the same idea presented from a number of different perspectives. The author’s purpose is to present ideas rather than to give an exhaustive description of so many alternative concepts and approaches discussed in the book. Thus, as a prerequisite on the part of the reader, some working knowledge in the fields of physics of solids, continuum mechanics, and differential calculus will be helpful in order that she or he could better focus on the principle thrust of discussions and derivations.
The text is divided into 13 chapters that are grouped in four basic parts. Part I, which is entitled Thinking About the Material World, consists of three chapters and provides an overview of the fundamental ideas that are useful in describing material response and in revealing the link between structure and properties. In particular, emphasis is placed on the notion of material parameter, the significance of phase diagrams to materials science and the role of lattice defects. Next, basic concepts of continuum mechanics are introduced and used in continuum descriptions of deformation and failure. The principle of minimum potential energy is formulated for linear elasticity, and the finite element approach to the corresponding boundary value problem is sketched. Part I ends with a revision of quantum and statistical mechanics. Here, solution of the Schro¨dinger equation is illustrated for a number of cases, including that of the hydrogen atom and the so-called electron gas model, with the aim of demonstrating the analogy between the problem of coupled oscillators and that of bonding in molecules. Finally, statistical mechanics of the Ising model and that of electrons is briefly described.
In Part II, Energetic Description of Cohesion in Solids, the total energy of the system is the starting point for analysis of material behavior. The energy is obtained on the basis of the kinematic measures that can be used to characterize the system’s geometry. On the grounds of microscopic theories, the total energy is a function of the atomic coordinates, whereas in a continuum description the energy is postulated to be a function of the relevant strain measures. The author gives examples of broad classes of total energy functions evaluated from the microscopic perspective, both with a direct incorporation of the electronic degrees of freedom and also their effective representation through the electron density. Starting from the Hamiltonian for a system of interest, which embodies the motions and interactions between all the nuclei and electrons in the system, Phillips explains the logic behind systematic degree of freedom elimination. He exploits the so-called Born-Oppenheimer approximation and constructs some simplified description in terms of effective pair potentials. He also discusses some difficulties with the pair potential description and remedies to cure some of them in the form of potentials with environmental and angular dependence. Further, the tight-binding method is described and its particular realization in the context of periodic solids as so-called k-space methods is illustrated, invoking Bloch’s theorem. Also, the density functional theory and the corresponding Kohn-Sham equations are sketched.
Chapter 5 is concerned with the energetics of both thermal excitations and elastic deformations of crystals. The author shows how the energy methods in conjunction with statistical mechanics can be used in predicting the material properties including the specific heat, thermal expansion and elastic moduli. An interesting analysis of normal modes and phonon dispersion relations is carried out, including the passage to the vibrational density of states. Further, the microscopic derivation of the elastic moduli concept is demonstrated primarily for a generic linear case and with some extension to nonlinear elasticity (the Cauchy-Born rule), linking atomistic models to continuum fields. Chapter 6, which closes Part II, discusses realization of distinct competing crystal structures in the context of elemental and alloy phase diagrams. The issue of structural stability is illustrated by predictions of a number of free energy functions constructed on various levels of sophistication invoked, including an Einstein model for structural change, cluster expansions and the cluster variation method.
Part III, Geometric Structures in Solids: Defects and Microstructures, deals with the various types of defects that populate materials, ranging from point defects (Ch 7) and line defects (Ch 8) to interfacial defects (Ch 9). An analysis of point defects is carried out from both a microscopic and continuum perspective with emphasis being placed on their origins and motion, and also consequences they have on the ultimate macroscopic properties of materials. The process of diffusion is examined, and some effective theories of diffusion are advanced.
Chapter 8 considers dislocations from the perspective of their role as the primary agents of plastic deformation. Both discrete and continuous descriptions are developed. The equilibrium fields associated with typical dislocations are studied within the linear theory of elasticity, which forms the foundation for analysis of interaction energies and configuration forces on dislocations (Peach-Koehler’s formula) and of the Peierls-Nabarro model. Finally, three-dimensional dislocation configurations are discussed, including the cross slip, kinks, and dislocation junctions. Chapter 9 focuses on two-dimensional wall defects (surfaces, stacking faults, grain boundaries) and serves as the basis for the consideration of microstructures within materials. The author highlights their role in producing confinement in the context of elastic waves, mass transport, and plastic action. The notion of an interfacial energy and that of a surface reconstruction are defined and exemplified in a number of representative cases. Chapter 10 analyzes some types of microstructures that exist in materials. First, the author discusses a considerable diversity of microstructures and a possibility of their production by appropriate processing strategies, and what the consequences of such microstructures are to the underlying physical properties of a material. The solution the Eshelby inclusion problem is derived and used in investigating the equilibrium morphologies of second phase particles. Also, the issue of temporal evolution of two-phase microstructures has been considered. Special attention is given to the realization of the compatibility condition in the context of martensitic microstructures. In the consideration of grain growth in polycrystals, the author has made use of the Potts model, phase field models, and sharp interface models.
Part IV, Facing the Multiscale Challenge of Real Material Behavior, constitutes the culmination of preceded developments discussed in the book with the aim to build effective theories based on a reduced set of degrees of freedom. Chapter 11 concerns a mutual interplay between the different types of defects defined earlier. Three groups of problems are considered: diffusion, mass transport assisted deformation, and dis-location-grain boundary interaction. In particular, these include exchange mechanisms for surface diffusion, Nabarro-Herring and Coble mechanisms for creep and the Rice model of dislocation nucleation at acrack tip, and the phenomenon of hardening approached from various perspectives.
Chapter 12 is a crowning part of the book, being explicitly devoted to the problem of bridging scales and effective theory construction. Here, Phillips revisits his previous considerations and generalizes the philosophy behind multi-scale modeling. Indicating to a tradition of multiscale approach, the author records some historic examples of multiscale modeling. He discusses cohesive surface models and mixed approaches in which constitutive models of different sophistication (hierarchical structure) are used, eg, a conventional continuum model in a region far from defects (cracks, dislocations), whereas in the immediate vicinity of defects a more refined analysis pertinent to subscales. In the format of gradient flow dynamics, variational derivations of the Allen-Cahn equation which describes the spatio-temporal evolution of a non-conserved order parameter field, and the Cahn-Hillard equation for such evolution of conserved fields are obtained. He shows how to create a family of effective Hamiltonians by the renormalization procedure and indicates the Monte Carlo method and hyperdynamics methods as a scheme for the hierarchical treatment of diffusive processes, raising the question on how to connect the notion of temperature and time. Phase field models are invoked in the context of both solidification and phase separation, and the finite element interpolation and its consequences in elimination of atomic degrees of freedom. Finally, in Chapter 13, the author has collected his reflections on the universality and specificity in materials and posed a series of intriguing open questions.
The presentation of this difficult and complex subject matter is organized in an accessible and attractive fashion, being pedagogically well balanced as concerns both the level of detail discussed and the notations used. It contains a lot of enlightening descriptions of the current theoretical concepts and many case studies. The included graphical illustrations of numerical results and of experimental findings substantiate the theoretical constructs and, in addition to that, will be provocative to the perspicacious reader. Each chapter closes with a useful section, Further Reading, and a set of problems to solve, which are related to the topics discussed in that chapter. There is a single list of references and a subject index at the end of the book.
A careful reading of the entire book shows that the text is well edited; this reviewer spotted only a few misprints. The various chapters can be read independently, but the logic of the book is intertwined, so in order to better appreciate the message conveyed the book should be treated as a single entity. Although the book is written in the narrative style that may be found by some readers a bit lengthy at times, this reviewer appreciates much the author’s extensive vocabulary.
Phillips has undertaken a very difficult and venturesome enterprise to write a book that encompasses such a wide spectrum of different concepts and issues under one roof. One of the key features this reviewer enjoyed in reading the Phillips’ book is his willingness to explain how macroscopic behavior is built up from microscopic motions, and that there are problems where a proliferation of scales both in space or time (or both) should be accounted for simultaneously. This reviewer believes that the value of this book is in that, on the one hand, it substantiates the need for a multiscale approach and, on the other hand, presents a framework for a mechanism of the information passage from the microscopic scales to those associated with macroscopic observables. It should be noted that the author reveals not only the strengths of various theoretical models, but also their limitations, and shows that the important nonconcepts like nonlinearity, nonlocality, and nonconvexity single out intriguing features of material behavior. For example, nonconvexity implies the existence of multiple wells, whereby admitting of the emergence of complex microstructures. The discussions in this book support this reviewer’s conjecture that the hierarchical modeling via mixed atomistic/continuum models will become the subject of intensive research in the years to come. This kind of approach is in accord with the adage that one should think globally and act locally. One of the critical issues that emerges here is how to effectively control the modeling error in quantities of interest, so that we could determine the proper interfaces and their location which will separate regions where models of different sophistication could be employed within a preset tolerance.
Phillips’ Crystal, Defects and Microstructures makes for most instructive reading. This reviewer very strongly recommends it for graduate students and researchers in science and engineering who wish to gain understanding of the theoretical constructs in the study of materials.
