In-vitro Materials Design
Description
The authors present the physical and mathematical background in sufficient detail for this highly current and important topic, but without unnecessary complications. They focus on approaches with industrial relevance, covering real-life applications taken from concrete projects that range from tribology modeling to performance optimization of integrated circuits.
Following an introduction to the fundamentals, the book describes the most relevant approaches, covering such classical simulation methods as simple and reactive force field methods, as well as highly accurate quantum-mechanical methods ranging from density-functional theory to Hartree-Fock and beyond. A review of the increasingly important multiscale approaches rounds off this section. The last section demonstrates and illustrates the capabilities of the methods previously described using recent real-life examples of industrial applications. As a result, readers gain a heightened user awareness, since the authors clearly state the conditions of applicability for the respective modeling methods so as to avoid fatal mistakes.
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Persons
Philipp Plänitz is CEO of AQcomputare, a company focusing on the calculation of materials properties with ab-initio methods as a service for industrial companies. He received the Diploma and PhD degrees in physics from the Chemnitz University of Technology in 2004 and 2009, respectively. In 2009 he founded AQcomputare, a GWT-TUD spin-off company. His research interests include industrial applications of atomic scale methods for calculating a wide range of material properties.
Michael Schreiber is Full Professor of Physics at Chemnitz University of Technology since 1993. After his PhD in physics, obtained from the Technical University of Dortmund, he moved to Tokyo University for two years. He obtained his first professorship in theoretical chemistry from the University of Mainz in 1990 and was Dean of the Faculty of Science from 1998 to 2001. Michael Schreiber has authored or co-authored more than 330 refereed scientific publications, edited 15 books and contributed to more than 100 books and proceedings.
Content
Newtonian Mechanics and Molecular Dynamics
Operators and Fourier Transformations
Bravais Lattice and k-Space
Introduction to Quantum Mechanical Concepts
II. COMPUTATIONAL METHODS
Classical Simulation Methods:
Simple Force Field Methods
Reactive Force Field Methods
Quantum Mechanical Simulation Methods:
Hartree-Fock (HF) Method
Density Functional Theory (DFT)
Concepts beyond HF and DFT
Multiscale Approaches
III. INDUSTRIAL APPLICATIONS
Tribological Properties
Intercalation of Ions in Electrode Materials
Nano-Structured Materials
Chemical Reactions in Porous Materials
Leakage Current in Integrated Circuits
Chapter 2
Newtonian Mechanics and Thermodynamics
Classical or Newtonian mechanics describes the motion of objects, from small particles to astronomical objects. Newtonian mechanics provides extremely accurate results as long as the domain of study is restricted to macroscopic objects and velocities far below the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to include quantum mechanical effects (see Chapter 4). In the case of velocities close to the speed of light, classical mechanics has to be extended by special or general relativity.
The following section introduces the basic concepts of classical Newtonian mechanics and its application to atomistic objects. At the end of this section, a critical discussion about the restrictions of this approach is given.
2.1 Equation of Motion
Quite often, objects are treated as point particles, that is, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, its mass, and its momentum.
Note:
In reality, all objects have a nonzero size. However, often, they can be treated as point particles, because effects related to the finite size are either not of interest or have to be described by more sophisticated theories such as quantum mechanics.
The position of a point particle can be defined with respect to an arbitrary fixed reference point in space.1 In general, the point particle does not need not be stationary relative to , so is a function of the time t
2.1Without loss of generality, the reference point can always be assumed to be at the origin of the used coordinate system, that is,
2.2Note:
The position of the point particle and all similar quantities are three-dimensional vectors. They must be dealt with using vector analysis. They will be denoted by
where x, y, and z are the Cartesian coordinates of the point particle.
The velocity , or the rate of change of position with time, is defined as the derivative of the position with respect to the time
2.3The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time)
2.4The acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both.
Note:
If only the magnitude of the velocity decreases, this is sometimes referred to as deceleration, but generally, any change in the velocity with time, including deceleration, is simply referred to as acceleration.
As we all know from our everyday life, an acceleration of an object requires the action of a force on it. Sir Isaac Newton was the first who mathematically described this relationship, which is known today as Newton's second law2
2.5The quantity introduced in this equation is called (canonical) momentum. The force acting on a particle is thus equal to the rate of change of the momentum of the particle with time.
As long as the forces acting on a particle are known, Newton's second law is sufficient to completely describe the motion of the particle. Hence, written in a slightly different form, it is also called equation of motion
2.6where the sum of all forces acting on the particle yields the total net force .3
If at a time , the position and the velocity of a point particle are known and all forces acting on that particle are given, then the motion of the particle can be determined for its whole future and past by solving the equation of motion yielding the particle trajectory (see Figure 2.1). This illustrates the deterministic character of Newtonian mechanics.
Figure 2.1 Trajectory of a point particle.
Example: Free particle:
In the case of a free particle, no forces are acting on it. Hence, the equation of motion becomes quite simple
2.7Using Eq. (2.4) and carrying out two integrations over the time t, the trajectory of the particle becomes
2.8with the integration constants (initial velocity) and (initial position). This is the textbook formula well known from basic physics courses.
2.2 Energy Conservation
Imagine a constant force is applied to a point particle and causes a finite displacement . The work done by the force is defined as the scalar product of the force and the displacement vector
2.9In a more general case, the force may vary as a function of position as the particle moves from to along a path C. The work done on the particle is then given by the path integral
2.10In the special case that the work done in moving the particle from to is the same no matter which path is taken, the force is said to be conservative. For example, gravity is a conservative force, as well as the force of an idealized spring (Hooke's law). On the other hand, the force due to friction is nonconservative. All conservative forces can be expressed as the gradient of a scalar function
2.11Except for an arbitrary constant shift c, this function is equal to the potential energy
2.12of the point particle.
Example: Potential energy landscape:
In Figure 2.2, a potential energy landscape is illustrated. The thin solid lines correspond to lines along which the value of the scalar function is constant-the so-called equipotential lines. The force acting on a particle is equal to the gradient of (Eq. (2.11)). The denser the equipotential lines are, the larger the force acting on the particle is.
Two different paths connecting point and point are illustrated.
- The first one runs through a valley, an area with small changes in . Hence, only small forces are acting on a particle along this path.
- The second path crosses a mountain, an area with strong changes in . Hence, large forces are acting on the particle. However, when the particle first climbs up the mountain, but then moves down again, the forces are directed in opposite directions.
Altogether, the work done by moving a particle from point to point is the same for both paths.
Figure 2.2 Potential energy landscape of a conservative force field with two different paths from point to point .
The kinetic4 energy of a point particle5 of mass m and speed v (i.e., the magnitude of the velocity) is given by
2.13The work-energy theorem states that for a point particle of constant mass m, the total work W done on the particle is equal to the change in kinetic energy of the point particle:
2.14If all the forces acting on a particle are conservative and is the total potential energy, the following equalities are satisfied
2.15This result is known as the conservation of energy and states that the total energy
2.16is constant in time. This result is a general (maybe the most general) concept in physics. It holds not only in conservative systems, but also in all physical systems; only the types of energy to be considered must be adapted. In nonconservative open systems, besides the kinetic and potential energy, also the energy exchange with the environment, the change of the internal energy (see Section 2.4), the friction energy, and other energy types have to be taken into account.
2.3 Many Body Systems
Up to now, we have considered only one single point particle and external forces acting on it. In the current subsection we will expand the discussion to a system of N point particles, which may interact with each other. Hereby, interacting particles are those particles that induce forces acting on other particles. As these forces have their source within the considered system of N particles, they are called internal forces (in contrast to external forces that may be applied to the system from outside). The most prominent examples of internal forces are electrostatic forces acting between charged particles or the gravitational force acting between massive particles. However, other types of forces such as van-der-Waals forces or bending and torsion forces also belong to this category.
Many forces are acting pairwise between two different particles i and j. For this type of forces, Newton's third law () holds:
2.17It means that the force induced by particle i on particle j has the same magnitude as force induced by particle j on particle i but acts in opposite direction. Hence, in the case of pairwise acting forces, the sum over all internal forces must vanish:
2.18The total force acting on particle i induced by the remaining particles
2.19is obviously an internal force. With Eq....
