Computational Materials Science: Surfaces, Interfaces, Crystallization

By A.M. Ovrutsky, A. S Prokhoda and M.S. Rasshchupkyna

Computational Materials Science provides the theoretical basis necessary for understanding atomic surface phenomena and processes of phase transitions, especially crystallization, is given. The most important information concerning computer simulation by different methods and simulation techniques for modeling of physical systems is also presented. A number of results are discussed regarding modern studies of surface processes during crystallization. There is sufficiently full information on experiments, theory, and simulations concerning the surface roughening transition, kinetic roughening, nucleation kinetics, stability of crystal shapes, thin film formation, imperfect structure of small crystals, size dependent growth velocity, distribution coefficient at growth from alloy melts, superstructure ordering in the intermetallic compound.

Computational experiments described in the last chapter allow visualization of the course of many processes and better understanding of many key problems in Materials Science. There is a set of practical steps concerning computational procedures presented. Open access to executable files in the book make it possible for everyone to understand better phenomena and processes described in the book.

• Valuable reference book, but also helpful as a supplement to courses
• Computer programs available to supplement examples
• Presents several new methods of computational materials science and clearly summarizes previous methods and results

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 19, 2013
• ISBN: 9780124202078

Book preview

advices.

Preface

Simulation is one of the main means for development of our ideas of outward things and theoretical description of various phenomena and processes. History of knowledge clearly shows that new, more complicated models come to replace the old, simple ones to provide a better description of the real processes. Simple models such as ideal gas model are easily analyzable. Complication of models leads to the increase of difficulties in their analysis and expects application of advanced mathematical methods.

Mathematical physics and computational mathematics have evolved due to the need for development of analysis and computer techniques. The latter was translated into the language suitable for computers and became a useful instrument for the scientists in different fields of knowledge.

Analysis of the sufficiently realistic models is an extremely hard task, and it is not always possible to reduce results to a form suitable for application of the computational mathematics technique. For example, analytical solutions of the boundary problems of heat and mass transfer could be derived only for bodies of a very simple shape under some certain simplified boundary conditions. At the same time, numerical solution of the initial equations by the finite-difference method (one of the simulation techniques) allows to obtain a full picture of changes in temperature and concentration fields, to take into account movement of the phase boundaries and changes in their shapes. At the same time, simulation program is an analogue of both an analytical solution and its finite expressions. Using calculations provided on computer, it is enough to change the input parameters of the system under consideration in order to obtain corresponding results with complete visualization of the ongoing processes.

An algorithm and a program provided that they are correct and that results of their application are proved at least for simplified models are none the worse for analytical solutions and could be much simpler for usage in practice. For example, now nobody tries to obtain an analytical solution to the many-body problem of celestial bodies, instead appropriate programs for calculations are used.

Hence it is clear why the simulation methods find their place in curricula of famous universities. A good many books are dedicated to the simulation methods at a different level of complexity. Those written by mathematicians are mostly focused on the methods themselves. In textbooks written by theoretical physicists, most attention is given to the phenomenological problems. But those who want to apply simulation methods should bear in mind that in order to be able to do it they need to master the subject itself and to understand the relevant phenomena at the level of latest advances in science and technology. Therefore, it is better not to separate courses in simulation from the main course.

In this book, we yield to the theoretical basis necessary for understanding atomic surface phenomena and processes of phase transitions, especially crystallization. Theoretical basis for computer simulation by different methods and simulation techniques for modeling of physical systems are also presented, as well as additional information concerning their accuracy. A number of results are discussed concerning modern studies of crystallization: processes of thin film formation, kinetics of crystal growth, stability of crystal shapes including crystallization front, and nanocrystal formation during solidification from the supercooled melts.

In the last chapter of this book, several computer experiments from the list proposed to the students of the Dnipropetrovs’k National University are described. Explicit instructions to contents of these works and detailed explanations of the main procedures of programs (Delphi, C++, Visual C# environments, and the Pascal codes of several programs are also included) should help everyone understand the essence of simulations. Open access to executable files (the website of Elsevier http://booksite.elsevier.com/9780124201439/) makes it possible for everyone to achieve a better understanding of the main phenomena described in this book. A description of programs is sufficient for their reconstruction in any programming environments.

Owing to the specific structure of the book, lists of references to its first chapters are considerably reduced. Some educational stuff is given without source references if it was previously presented in some textbooks and it is hard to figure out where it was published for the first time. The following sources were the most often used for the preparation of the book:

D.W. Heermann, Computer Simulation Methods in Theoretical Physics, second ed., Springer, 1990.

Experiment on a Display, Moscow, Science, 1989, 99 p. (in Russian).

M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1989, 385 p.

D. Frenkel, B. Smit, Understanding Molecular Simulation. From Algorithms to Applications, Academic Press, New York, London, Tokyo, 2002, 628 p.

D.K. Belashchenko, Computer Simulation of Liquid and Amorphous Matters MISSIS, Moscow, 2005, 407 p. (in Russian).

H. Gould, J. Tobochnik, An Introduction to Computer Simulation Methods: Applications to Physical Systems Parts 1 and 2, Addison-Wesley, Reading, MA, 1988.

H. Gould, J. Tobochnik, W. Cristian, An Introduction to Computer Simulation Methods: Applications to Physical Systems, third ed., Addison-Wesley, Reading, MA, 2007, 813 p.

V.I. Rashchikov, A.S. Roshal, Numerical Methods in Solution of Physical Problems, Lan’, St. Petersburg, 2005 (in Russian).

Other editions wherein mathematical fundamentals of simulation methods are described in step-by-step fashion.

The book Physics of Surface by A. Zangwill (Cambridge University Press, 1988) remains the most consistent on the subject of surface physics; some materials from this book were used in Chapter 4. More recent researches of the surface structure are represented in the book Introduction to the Physics of Surface by K. Our, V.G. Livshitz, A.A. Saranin, A.V. Zotov, G. Katayama (in Russian, Nauka, Moscow, 2006, 490 p.).

Our book does not cover all aspects of simulations in Materials Science. Simulations of mass crystallization that give information on microstructure formation in materials during crystallization, especially in the high and very high supercooling ranges, are not presented here. Another large direction in modeling, which is of a special importance for production and exploitation of engineering materials, is application of computational methods in continuum mechanics. There are some very useful books dealing with the questions of continuum mechanics. Continuum-based simulation approaches in the continuum scale and atomic scale are described in the book by Dierk Raabe (Computational Materials Science. The Simulation of Materials Microstructures and Properties, Wiley-VCH, Weinheim, New York, Toronto, 1998, 326 p.) and the book edited by Dierk Raabe, Franz Roters, Frederic Barlat, Long-Qing Chen (Continuum Scale Simulation of Engineering Materials: Fundamentals—Microstructures—Process Applications, Wiley-VCH Verlag GmbH & Co. KGaA, 2004, 845 p.). The book of S. Schmauder and L. Mishnaevsky Jr. (Micromechanics and Nanosimulation of Metals and Composites, Springer-Verlag, Berlin Heidelberg, 2009, 421 p.) contains descriptions of different experimental and computational analysis methods of micromechanics of damage and strength of materials.

This book will be useful for everyone who has interest in applying modern simulation techniques for development and analysis of more realistic models of physical processes in Materials Science.

1

Computer Modeling of Physical Phenomena and Processes

This chapter contains the most important information on the essence of computer modeling, in particular, the matter of computational experiments. There is a brief discussion about simulation methods and more detailed introduction to the Monte Carlo (MC) method in statistical physics. The ferromagnetic Ising model is considered as an example. A description of the program, which realized the MC method (within the Metropolis algorithm) for this model, is given in Section 9.6. Execution of computational experiments with this program helps to understand the essence of computer modeling and obtain evidence of action of statistical laws. The main algorithms of integration of equations of movement using the method of molecular dynamics are considered. And the problems of potentials for evaluations of interatomic interactions and boundary conditions, which are important for both methods, are also considered. For these methods, Section 9.13 is also important.

Keywords

Simulations; computational experiments; Monte Carlo method; Ising model; classical interaction potentials; Metropolis algorithm; molecular dynamics; Verlet algorithm; boundary conditions

1.1 Application of Computers in Physics

1.1.1 Role of Models in Theoretical Study

Models of phenomena or processes underlie any physical theory. Such models are simple enough as a rule; their complication hampers the theory by elaborating. If results obtained in a simple model framework are in satisfactory agreement with experimental data, there is no need to complicate it. However, if the essential disagreement in results takes place, it is necessary to choose another model, which would correspond better to a nature of phenomena under consideration.

The ideal gas model is the simplest. Gas is considered as a set of noninteracting mass points, which can move in any direction. It is sufficient to use the ideal gas model in order to find the relationship between the gas pressure on the vessel walls and such characteristic of the molecules movement as their mean-square velocity. We will note that determination of relationships between parameters of the system state and characteristics of the molecule movement is the main subject of the kinetic molecular theory.

In order to determine the pressure of an ideal gas, it is supposed that some velocity distribution of gas molecules exists such that mean-square speed of molecules for the given conditions is a constant value. Hence, the question puzzles, if molecules move freely and do not collide (a mass point has no sizes), how could any certain velocity distribution of molecules be set? Consequently, a better-adjusted model of gas should consider the size of molecules. One of the widely used models of gas considers molecules as solid spheres. This model is used for the description of transport phenomena in gases, such as diffusion, thermal conductivity, and interior friction.

If concentration of gas molecules is high, interaction of molecules mostly defines physical properties. Real gas models consider attraction of molecules. If distance between molecules is small, repulsive forces also should be taken under consideration.

Balance between attractive and repulsive forces determines the average distance between atoms in liquid or in solid body. Resultant forces appear when molecules shift from equilibrium positions; they are in direct proportion with deviation distances. Therefore, the simplest and the most widely used model of a solid body (crystal) is the crystal lattice with atoms disposed in its knots and interacting with each other by elastic forces.

Simple models allow one to perform analytical study easily. Analysis of more realistic models of matters is carried out with application of the special mathematical methods developed by physicists-theorists. As a rule, it is necessary to evaluate complicated integrals to find solutions of algebraic or transcendental equations, their systems, etc., and finally, to compute the matter properties. For example, statistical theories of system ordering (the system of magnetic moments or electric dipole moments, or atoms of different types) are based on searching analytical expressions for the free energy of systems in the framework of the considered model. Calculating the statistical sum is necessary for the Gibbs free-energy determination; the Helmholtz free energy is determined through an internal energy (U) and entropy (S), F=U−TS. The free energy minimum corresponds to the equilibrium state of systems. Minimization of main parameters in the analytical expression for the free energy (order parameter, probability of certain configurations of atoms or dipoles) results in the transcendental equations. Their solutions are usually performed using standard computing procedures. In this case, the computer is still used as a powerful calculator.

1.1.2 Methods of Computer Modeling of Physical Processes

Mathematical models play a great role in the scientific study. With their help, a physical phenomenon is transformed by the means of equations into a discrete algebraic form, which can be used for a numerical analysis. Discrete algebraic equations describe a calculated model. Translation of the latter into machine codes is a computer program. The computer and the program allow exploring evolution of a modeled physical system in computing experiments [1].

Mathematical modeling is a kind of theoretical problem on the numerical solution of the Cauchy boundary value problem. At the instant t=0, the initial state of a system is set in some bounded spatial area (simulated volume) on whose surface some given boundary conditions are retained. Modeling consists of observing evolution of the system state. The basic part of evaluation is a cycle with a certain timestep (Δt), during which the state of physical system progresses over this time. Even the simplest modeling calculation generates a huge amount of data and demands an experimental approach for obtaining desired outcomes (from which the name computing experiment originates). However, even if the amount of information which can be treated by computers is large, their capability is not limitless.

Three methods giving the best performance for modeling physical processes have received the widest application. These methods are: the method of nets for solution of the transport equation (i.e., partial differential equations), the Monte Carlo (MC) method (including its modifications for kinetic modeling), and the method of molecular dynamics (MD method) for modeling of classical statistical and quantum statistical systems. In all cases, it is a question of approximating a continuous environment by a discrete model with local interaction. The choice of method, the search for a model of the substance structure which is adequate to reality, working out algorithms and programs for model performance, carrying out numerical experiments, and analyzing their outcomes comprise the essence of simulation of physical phenomena.

When modeling a large system, the model is loaded into the computer’s memory in a convenient way for calculations, and a parallel execution of noninteracting spatial domains (or interacting is neglected during the timestep) is provided. These evaluations are essentially simple but being distributed over a large volume, they demand many resources. In order to accelerate the execution, the simultaneous work of several computers (a cluster of computers) is organized. The main computer called the host machine rules the cluster.

Modeling by the MC method does not require complicated mathematics because it comes almost from the first principles—probabilities of states or transitions of particles from one state into another are defined by the Boltzmann factor of energies (taken with the negative sign) in units of kT. Modeling by the MC method supposes consideration of the substance models, which are more complicated, than models that are analyzable in the framework of the modern theoretical physics. Statistical modeling by the MC method allows studying equilibrium states of systems. Kinetic modeling by the MC method allows analysis of the course of physical processes.

The MD method develops most intensively now. It is already applied to systems consisting of many thousands of atoms (systems of many millions of atoms are already executed in some research). The method consists of numerical solution of Newton’s equations for all atoms with a timestep smaller than 10−14 s. For this timestep, increments of coordinate values and velocities of all particles are calculated, taking into account their values on the previous timestep. Though the level of adequacy of calculated outcomes to the real physical picture of a yielded process or phenomenon cannot be guaranteed, because dependences of interaction energy of atoms on distance between them are not defined with adequate accuracy; the method is extremely valuable and perspective, due to the exclusively first principles used in it.

1.1.3 Influence of Computers on Methods of Physical Researches

Purposes and means of science were changed due to the computer facilities development. Long-time theoretical physics aspired to analytical solutions of the problems. It seemed to be the single possible method of full description of phenomena. Unfortunately, the most important and actual problems cannot be solved analytically. Computer modeling has proved to be very effective in the case of such problems; its development is connected with efficiency. This progress has now come so far that analytical solutions are not required in many cases. The problem of three bodies—movement of three bodies in the total gravitational field—is not solved analytically yet. However, it does not prevent astronomers from calculating trajectories not only for three but also for any number of bodies by the means of computer modeling. Essentially, an algorithm allowing any accuracy to calculate trajectories using computers is no worse than explicit analytical solution. Numerical solutions allow answering any questions, to which it would possibly answer by means of formulas, when they will be obtained.

Elaboration at the end of twentieth century of software packages that could execute algebraic transformations was unexpected and shocking for some physicists-theorists. It meant that intellectual operations became accessible to computers. Reports became known, in which from the beginning to the end, all the formulas and theorems were obtained (were deduced and proved) by machine. The essence of the conflict was that those machines had inevitably invaded the field of science, which was considered traditionally as belonging to the most qualified scientists, namely, to theorists.

It is also possible to give examples of the lowering of the status of experts owning perfectly some theoretical methods. The qualified experts in the field of the heat and mass transport, which did not master in time new numerical methods, have discovered with surprise that their huge wealth of theoretical knowledge is substantially depreciated. And still it does not allow solving transport equations for complex boundary conditions varying with time; and young researchers, not theorists at all, can do it by means of rather simple programs. The qualified physicists-theorists in the field of statistical physics have discovered that their young colleagues applying the MC method do not simply check its reliability, but obtain already much more powerful outcomes, better mapping structure of substances, and different processes (first of all, the phase transitions), which occur with them. On the other hand, this is a normal phenomenon, that when new people come, they are able to work in a new way. However, they must still learn the process. In addition, the main point is that it is necessary to be a good expert in a certain field of knowledge. The computer can facilitate analyzing of processes or phenomena; it allows one to work with the models as an experimentalist, obtaining outcomes for different initial and boundary conditions (unlike real experiments with the material system, these conditions are known precisely in the case of computer experiment). However, the principle of theoretical work does not vary: it is model development and execution, in this case, with application of the computer.

In cases of application of direct methods, such as the MC or MD methods, based on the most common principles, the core basis of modeling becomes the competent elaboration of algorithms and validation of solutions. For the planning of computing (machine) experiments and fulfilling analysis of outcomes, the core is the knowledge of theoretical researches in this branch of science and of outcomes of the newest experimental research.

The important direction in physics is the modeling of large systems or any system in extreme situations. These are situations when the system differs qualitatively from the total of independent small subsystems, that is, the cases when the radius of correlations is large in it. These are also critical phenomena of different kinds, such as turbulence and wave function collapse. Methods developed for the analysis of such systems find interesting applied applications, sometimes in unexpected branches of knowledge.

Use of modeling in scientific knowledge is caused, as is known, by that circumstance that the immediate object of research is either difficult to access or generally inaccessible for direct research on any physical properties [2]. There is a difference between physical and mathematical modeling. Physical model operation is based on study of the phenomena on models of one physical nature with the original. For example, wind tunnels are used to test small models of airplanes in the air flow. Crystallization of the transparent organic matter (salol) in small vessel works as a model of crystallization of steel ingot. The mathematical model is more generalized than the physical model: it is not required more of physical similarity between the original and the model is not required any more, as the parameters of mathematical model, which have the mathematical description, are only studied, and they are connected by the mathematical relations concerning both the model and the original.

Mathematical models, as they come from the mathematical similarity of the original to the model and are used for studying of the quantitative characteristics and the quantitative correlation of different parameters, may be considered as mathematical computers. On the contrary, the computer (after corresponding programming) is the generalized model of those processes, equations of which can be solved by this machine. The up-to-date computers are used as simulators of objects and processes of the diversified character. Computers with large number of elements (10¹²–10¹⁴), according to Neumann’s theorem, are the universal automatic machines, capable of performing the operation of any automatic machine [1].

It is important to see the COMPUTER as a source of creative pleasure, which can be ensured by an increase of the intellectual (game) part in scientific work. It became possible as a result of the radical increase of labor productivity at so-called routine stages of the working process. Losing time connected with routine operations strongly narrows down creative possibilities of researchers. Besides, many operations inherently are inaccessible to people because of the huge volume of necessary transformations (or logic steps). It is impossible to fulfill them without the participation of machines. For such tasks, the COMPUTER should possess artificial intellect. Its creation includes the software engineering, allowing solving the tasks of intellectual nature by means of computers, for example, proving theorems with application of operations of the formal logic, pattern recognition, and use of natural language for tutoring of robots.

1.1.4 The Basic Aspects of Computer Application in Physics

Generally speaking about the use of computers in physics, it is necessary to discuss four aspects:

1. Numerical analysis (computational mathematics)

2. Symbolic transformations

3. Mathematical modeling

4. Controlling the physical equipment in real time.

In the numerical analysis, evaluations are preceded by the simplifying physical reasons. Solution of many physical problems can be reduced to the solution of a system of linear equations. The analytical solution can be fulfilled for sets of two, three, and four equations. If the number of variables becomes very large, it is necessary to apply numerical methods and computers. In this case, the computer serves as the numerical analysis tool. It is often necessary to calculate a many-dimensional integral to perform operations with big matrixes or to solve a complex differential equation. This stipulates wide application of computers in physics.

This increasing importance gains computer application in theoretical physics for analytical (symbolical) transformations. Analytical transformations are already included in many up-to-date mathematical packages, e.g., Mathcad, Matlab, Maple, and Mathematics. For example, let us suppose that we want to find out the solution of the quadratic equation ax²+bx+c=0. The program of analytical transformations can produce the solution in the formula form x1,2=(−b±(b²−4ac)¹/²)/2a, or in the usual numerical form for definite values a, b, and c. Thus the computer can already deduce equations. It is especially important when equations contain many terms or when their deduction needs many operations. A person most likely will make a mistake but the computer will yield the right answer.

By means of typical programs for analytical transformations, it is possible to fulfill such mathematical operations as differentiation, integration, solution of equations, and series expansion.

Mathematical modeling is characterized by the feature that only the most common physical laws (principles) with minimum analysis are included in algorithms. As an example, let us determine energy distribution in the system with a great number of particles. To answer the question, what is the probability that the value of energy of the particle is in the range from E to E+ΔE? One of the ways to find the answer to such a question is in carrying out the experiment, for example, by definition of velocities of gas molecules. Such experiments were carried out. But they are not easy, and they answer the question of distribution of gas molecules on energies only. The problem can be solved precisely analytically. And it is solved by statistical physics. It is the problem about energy distribution of particles in systems making up the microcanonical ensemble of systems (consisting of a constant number of particles with the constant total energy). However, problems like this cannot always be solved analytically. It is much easier to act differently: to introduce the game rules into the computer program, to simulate a large number of energy exchanges between particles, and to calculate probabilities for distribution determination.

Computers can also be used to answer a what if question. For example, how the distribution of particles on energies would be modified, if the maximum possible value of the exchange energy were varied?; what would be, if exchange of energy occurred in discrete portions? The specified type of modeling finds application for ordering problems in many-particle systems, consisting of dipole or magnetic moments (dielectrics and magnets).

In all varieties of the use of computers in physics, the main purpose is usually understanding not the numbers. Computers have very much influenced physical researchers and the choice of physical systems for study especially. The numerical analysis and modeling are connected with some simplifying approximations, that is, with the choice of the model that allows solving the problem numerically. Thus, a creative work of the researcher is in the foreground.

Computers are also the important tool in experimental physics—controlling the physical equipment. Often they are linked to all phases of laboratory experiment: instrumentation design, controlling of this instrumentation during experiment, obtaining and evaluating data sets. The use of computer facilities not only has allowed experimentalists to sleep better at night but also has made possible experiments which otherwise would be impossible. Some of problems mentioned above, for example, instrumentation design or the data evaluation, are close to problems that scientists come across in theoretical work. However, the tasks connected with controlling and interactive analysis of data differ qualitatively; they demand programming in real time and joining of different types of devices to computing equipment.

1.1.5 Computational Experiments and Their Role in Modern Physics

Why are simulations important for physics? This question was considered in Refs. [1,3,4]. One of the reasons is that the majority of analytical tools, such as differential calculus, suit mostly examination of the linear problems. For example, it is easy to analyze oscillations of one particle solving the equation of its movement (Newton’s second law) in the supposition of the linear restoring force. However, the majority of natural processes are nonlinear, so small changes in one variable can lead to large changes in value of other variables. Nonlinear problems can be solved analytically only in special cases, and the computer gives a possibility of examination of the nonlinear phenomena. Another direction of numerical modeling is the analysis of behavior of systems with many degrees of freedom (consisting of large number of particles or many variables).

Development of computer techniques results in the new sight on physical systems. The statement of the question: How to formulate the task for the computer? has led to the modification of the formulation of some physical laws. Therefore, it is quite practical and it is natural to express laws in the form of rules for the computer, instead of language of the differential equations [1]. Now this new vision of the physical processes leads some physicists to review the computer as a certain physical system and to elaborate the novel architecture of computers, which can simulate natural physical systems more efficiently. Often numerical modeling is termed computational experiment as it has a lot of common with laboratory experiments. Some analogies are shown in Table 1.1, taken from Refs. [1,3].

Table 1.1

Analogies Between Computational and Laboratory Experiments

The advantage of the computational experiment is that the conditions, at which certain process runs, are set precisely in it. As a rule, it is very difficult to define them in real experiments.

The basic point of numerical modeling is creating the model of the idealized physical system. Then it is necessary to elaborate algorithms and procedures for the model realization on the computer. The computer program simulates a physical system and features the computing experiment. Such computational experiment is a bridge between laboratory experiments and theoretical calculations. For example, we can obtain in essence exact outcomes for the idealized model, which does not have a laboratory analog. Comparison of results of modeling with corresponding theoretical calculations stimulates development of computing methods. On the other hand, it is possible to check and improve the model using realistic parameters for more direct comparison of simulation results with the results of the laboratory experiments.

The method of finite differences, the Monte Carlo method or method of molecular dynamics for systems with the great number of particles depends not only on the wish of the researcher. At the choice of the modeling method, the main conditions are: the size (the number of particles in the system) and the environment of the physical system, accuracy of calculation and possibility of interpreting the results obtained with certain reliability; and duration of the computing experiment. For every single case, it is necessary to introduce certain corrective deductions, to choose some correlative coefficients, to consider the nature of particle interaction, to estimate errors. In any case, it is necessary to make many tests of the computing experiments with previously known outcomes. Only under the condition of getting positive outcomes, can you assert that the mathematical model created by you and realized in the computer program is effective and possible for use and prediction of physical properties of such