In vitro materials design

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Roman Leitsmann, Philipp Plänitz, and
Michael Schreiber
In-vitro Materials Design


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Roman Leitsmann, Philipp Plänitz, and Michael Schreiber

In-vitro Materials Design
Modern Atomistic Simulation Methods for Engineers


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Authors
Dr. Roman Leitsmann

AQcomputare GmbH

Annaberger Straße 240
09125 Chemnitz
Germany
Dr. Philipp Plänitz

AQcomputare GmbH
Annaberger Straße 240
09125 Chemnitz
Germany
Michael Schreiber

Technische Universität Chemnitz
Institute of Physics
Reichenhainer Str. 70
09126 Chemnitz
Germany
Cover picture courtesy of

Sang-Woo Kim, Ph.D., Professor
School of Advanced Materials Science &
Engineering
SKKU Advanced Institute of Nanotechnology (SAINT)
Sungkyunkwan University (SKKU)
Cheoncheon 300
Suwon 440-746
South Korea

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V

Contents
Preface

IX

Part I

Basic Physical and Mathematical Principles

1

Introduction

1

3

2

Newtonian Mechanics and Thermodynamics 5

2.1
2.2
2.3
2.4

Equation of Motion 5
Energy Conservation 7
Many Body Systems 10
Thermodynamics 11

3

Operators and Fourier Transformations

3.1
3.2
3.3

Complex Numbers 17
Operators 18

Fourier Transformation 20

4

4.1
4.2
4.3

Quantum Mechanical Concepts 25
Heuristic Derivation 25
Stationary Schrödinger Equation 27
Expectation Value and Uncertainty Principle

5

Chemical Properties and Quantum Theory

5.1
5.2

Atomic Model 33
Molecular Orbital Theory 39

6

Crystal Symmetry and Bravais Lattice

47

6.1

6.2
6.3
6.4

Symmetry in Nature 47
Symmetry in Molecules 47
Symmetry in Crystals 49
Bloch Theorem and Band Structure

53

17

33

28


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VI

Contents

57

Part II

Computational Methods

7

Introduction 59

8

Classical Simulation Methods

8.1
8.2
8.3

65
Molecular Mechanics 65
Simple Force-Field Approach 68
Reactive Force-Field Approach 71

9.5.1
9.5.2
9.5.3
9.6
9.6.1
9.6.2
9.6.3
9.6.4
9.7
9.7.1
9.7.2
9.7.3
9.7.4

9.7.5
9.7.5.1
9.7.5.2
9.7.5.3

77
Born–Oppenheimer Approximation and Pseudopotentials 77
Hartree–Fock Method 80
Density Functional Theory 83
Meaning of the Single-Electron Energies within DFT and HF 85
Approximations for the Exchange–Correlation Functional
EXC 88
Local Density Approximation 88
Generalized Gradient Approximation 89
Hybrid Functionals 90
Wave Function Representations 91
Real-Space Representation 91
Plane Wave Representation 92
Local Basis Sets 93
Combined Basis Sets 95
Concepts Beyond HF and DFT 96
Quasiparticle Shift and the GW Approximation 97
Scissors Shift 99
Excitonic Effects 100
TDDFT 100
Post-Hartree–Fock Methods 101
Configuration Interaction (CI) 102
Coupled Cluster (CC) 102
Møller–Plesset Perturbation Theory (MPn) 103

10

Multiscale Approaches

9

9.1
9.2
9.3
9.4
9.5

Quantum Mechanical Simulation Methods

10.1
10.2

105
Coarse-Grained Approaches 105
QM/MM Approaches 108

11

Chemical Reactions

11.1
11.2

111
Transition State Theory 111

Nudged Elastic Band Method 114


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Contents

Part III

Industrial Applications 117

12

Introduction

13

Microelectronic CMOS Technology

13.1
13.2
13.2.1
13.2.2
13.2.3
13.2.4
13.2.5
13.2.6
13.3
13.3.1
13.3.2

13.3.3
13.3.4
13.4
13.4.1
13.4.2
13.4.3
13.4.4
14

14.1
14.2
14.2.1
14.2.2
14.2.3
14.2.4
14.2.5
14.2.6
14.3
14.3.1
14.3.2
14.3.3
14.3.4
14.3.5

119

121
Introduction 121
Work Function Tunability in High-k Gate Stacks 127
Concrete Problem and Goal 127

Simulation Approach 129
Modeling of the Bulk Materials 129
Construction of the HKMG Stack Model 132
Calculation of the Band Alignment 136
Simulation Results and Practical Impact 138
Influence of Defect States in High-k Gate Stacks 141
Concrete Problem and Goal 141
Simulation Approach and Model System 144
Calculation of the Charge Transition Level 145
Simulation Results and Practical Impact 146
Ultra-Low-k Materials in the Back-End-of-Line 149
Concrete Problem and Goal 149
Simulation Approach 151
The Silylation Process: Preliminary Considerations 153
Simulation Results and Practical Impact 155

159
Introduction 159
GaN Crystal Growth 163
Concrete Problem and Goal 163
Simulation Approach 165
ReaxFF Parameter Training Scheme 166
Set of Training Structures: ab initio Modeling 168
Model System for the Growth Simulations 170
Results and Practical Impact 172
Intercalation of Ions into Cathode Materials 174
Concrete Problem and Goal 174
Simulation Approach 176
Calculation of the Cell Voltage 178
Obtained Structural Properties of Lix V2 O5 178

Results for the Cell Voltage 181
Modeling of Chemical Processes

15

Properties of Nanostructured Materials

15.1
15.2

Introduction 183
Embedded PbTe Quantum Dots

187

183

VII


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VIII

Contents

15.2.1
15.2.2
15.2.3
15.2.4

15.2.5
15.2.6
15.3
15.3.1
15.3.2
15.3.3
15.3.4
15.3.5
15.3.6

Concrete Problem and Goal 187
Simulation Approach 188
Equilibrium Crystal Shape and Wulff Construction 190
Modeling of the Embedded PbTe Quantum Dots 191
Obtained Structural Properties 194
Internal Electric Fields and the Quantum Confined Stark Effect 195
Nanomagnetism 199
Concrete Problem and Goal 199
Construction of the Silicon Quantum Dots 200
Ab initio Simulation Approach 203
Calculation of the Formation Energy 204
Resulting Stability Properties 205
Obtained Magnetic Properties 206
References
Index 221

211


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IX

Preface
In many academic and industrial R&D projects, physicists, chemists and engineers
are working together. In particular, the development of advanced functionalized
materials requires an interdisciplinary approach. In the last decades, the size of
common devices and used material structures has become smaller and smaller.
This has led to the emergence of the so-called nanotechnology, that is, a technology that uses material systems with an extent of less than several hundred
nanometers. The enormous technical advances in this field are subject to two
mutually amplifying effects. On the one hand, modern experimental techniques
have been developed that allow the observation, manipulation, and manufacturing of materials at an atomic length scale with an industrially relevant production
rate. On the other hand, the enhancements in the computer technology have led
to a tremendous growth of the scientific field of computational material sciences.
Nowadays, modern simulation methods are indispensable for the design of new
and functionalized nanomaterials. They are essential to understand the chemical
and physical processes beyond many macroscopic effects.
However, the basic concepts of modern atomistic simulation methods are not
very well established in common engineering courses. Furthermore, the existing
literature either deals with very specific problems or is at a very deep physical
or mathematical level of theory. Therefore, the intention of this book is to give a
comprehensive introduction to atomic scale simulation methods at a basic level of
theory and to present some recent examples of applications of these methods in
industrial R&D projects. Thereby, the reader will be provided with many practical
advices for the execution of proper simulation runs and the correct interpretations
of the obtained results.
For those readers who are not familiar with basic modern mathematical
and physical concepts, Part I will give a rough introduction to Newtonian
and quantum mechanics, thermodynamics, and symmetry-related properties.
Furthermore, necessary mathematical concepts will be introduced and the reader

will be provided with the denotation and terminology that will be used later on.
Readers with a fundamental physical and mathematical knowledge may skip this
part and look up certain aspects later, if it is necessary.
Part II gives a brief introduction to important aspects of state-of-the-art atomic
scale simulation techniques. In particular, the basics of classical and reactive


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X

Preface

force field methods, the density functional and Hartree–Fock theory, as well
as multiscale approaches will be discussed. Possible fields of application will be
depicted, and limitations of the methods are illustrated. Furthermore, several
more advanced methods, which are able to overcome some of these limitations,
will be shortly mentioned. The intention of this part is to enable the reader to
decide which simulation method (with which limitations) would be optimal to
investigate a certain problem of interest.
The last part illustrates possible application scenarios of atomic scale simulation techniques for industrially relevant problems. It is divided into three
chapters that consider three different industrial fields: microelectronics, chemical
processes, and nanotechnology. Real industrial problems and the corresponding
contributions of atomic scale simulations will be presented to the reader. Thereby,
the set up, the execution, and the analysis of the results will be discussed in
detail, and many practical hints for potential users of atomic scale simulations
are provided.
Chemnitz
April 2015

Roman Leitsmann


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1

Part I
Basic Physical and Mathematical Principles

In-vitro Materials Design: Modern Atomistic Simulation Methods for Engineers, First Edition.
Roman Leitsmann, Philipp Plänitz, and Michael Schreiber.
© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.


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3

1
Introduction
The scope of this part is to provide the reader with basic physical and mathematical principles that are necessary to understand the discussions in the following
chapters. Furthermore, a notation is introduced, which will be utilized throughout
the remaining book. No special previous knowledge is required from the readership. Nevertheless, a basic scientific knowledge is advantageous. Part I makes no
claim to provide a complete overview. Many things can be discussed only very
briefly. For a more detailed description of special topics and background information, the readers are provided with suitable references.
Those readers who are already familiar with the physical and mathematical concepts can skip this part and look up certain points later if necessary.

In-vitro Materials Design: Modern Atomistic Simulation Methods for Engineers, First Edition.
Roman Leitsmann, Philipp Plänitz, and Michael Schreiber.
© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.


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5

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 𝐫 can be defined with respect to an arbitrary
fixed reference point 𝐑0 in space.1) In general, the point particle does not need not
be stationary relative to 𝐑0 , so 𝐫 is a function of the time t
𝐫 = 𝐫(t).

(2.1)

1) Classical mechanics usually assumes an Euclidean geometry [1] accompanied by a certain threedimensional coordinate system. For simplicity, we use in this book a simple Cartesian coordinate
system.
In-vitro Materials Design: Modern Atomistic Simulation Methods for Engineers, First Edition.
Roman Leitsmann, Philipp Plänitz, and Michael Schreiber.
© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.


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6

2 Newtonian Mechanics and Thermodynamics

Without loss of generality, the reference point can always be assumed to be at the
origin of the used coordinate system, that is,
𝐑0 = (0, 0, 0).

(2.2)

Note: The position of the point particle and all similar quantities are threedimensional vectors. They must be dealt with using vector analysis. They will
be denoted by
𝐫(t) = (x(t), y(t), z(t)),
where x, y, and z are the Cartesian coordinates of the point particle.
The velocity 𝐯, or the rate of change of position with time, is defined as the
derivative of the position with respect to the time
d𝐫
̇
≡ 𝐫.
(2.3)
dt
The 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)
𝐯=

d𝐯
≡ 𝐯̇ = 𝐫̈ .
(2.4)
dt
The acceleration can arise from a change with time of the magnitude of the velocity
or of the direction of the velocity or both.
𝐚=

Note: If only the magnitude v = |𝐯| 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)
d𝐩 d(m𝐯)

=
= m𝐚.
(2.5)
dt
dt
The quantity 𝐩 = m𝐯 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
∑
𝐅i ,
(2.6)
𝐩̇ = m𝐚 =
𝐅=

i

2) The last identity is only true in cases where the mass m of the particle is constant.


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2.2

Energy Conservation

v(t3)
Particle trajectory

v(t1)

v(t2)

r(t1)

r(t2)
r(t3)

R0
Figure 2.1 Trajectory of a point particle.

where the sum of all forces 𝐅i acting on the particle yields the total net force 𝐅.3)
If at a time t0 , the position 𝐫(t0 ) = 𝐫0 and the velocity 𝐯(t0 ) = 𝐯0 of a point particle
are known and all forces 𝐅i 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.
Example: Free particle: In the case of a free particle, no forces are acting on
it. Hence, the equation of motion becomes quite simple
𝐩̇ = m𝐚 = 0.

(2.7)

Using Eq. (2.4) and carrying out two integrations over the time t, the trajectory
of the particle becomes
𝐫(t) = 𝐯0 t + 𝐫0 ,

(2.8)

with the integration constants 𝐯0 (initial velocity) and 𝐫0 (initial position). This

is the textbook formula well known from basic physics courses.

2.2
Energy Conservation

Imagine a constant force 𝐅 is applied to a point particle and causes a finite displacement 𝛿𝐫. The work done by the force is defined as the scalar product of the
force and the displacement vector
W = 𝐅 ⋅ 𝛿𝐫.
3) This is the result of the so-called superposition principle.

(2.9)

7


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8

2 Newtonian Mechanics and Thermodynamics

In a more general case, the force may vary as a function of position as the particle
moves from 𝐫1 to 𝐫2 along a path C. The work done on the particle is then given
by the path integral
W=

∫C

𝐅(𝐫) ⋅ d𝐫.

(2.10)

In the special case that the work done in moving the particle from 𝐫1 to 𝐫2
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 V (𝐫)
𝐅 = − 𝛁 V (𝐫).

(2.11)

Except for an arbitrary constant shift c, this function is equal to the potential
energy
(2.12)

Ep = V (𝐫) + c
of 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 V (𝐫) is constant—the so—called equipotential

High altitude
r2

r1
Low altitude

Figure 2.2 Potential energy landscape of a conservative force field with two different paths
from point 𝐫1 to point 𝐫2 .

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2.2

Energy Conservation

lines. The force acting on a particle is equal to the gradient of V (𝐫) (Eq. (2.11)).
The denser the equipotential lines are, the larger the force acting on the
particle is.4)
Two different paths connecting point 𝐫1 and point 𝐫2 are illustrated.

• The first one runs through a valley, an area with small changes in V (𝐫).
Hence, only small forces are acting on a particle along this path.

• The second path crosses a mountain, an area with strong changes in V (𝐫).
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 𝐫1 to point 𝐫2 is
the same for both paths.
The kinetic energy Ek of a point particle5) of mass m and speed v (i.e., the magnitude of the velocity) is given by
Ek =

1 2
mv .
2

(2.13)

The 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 Ek of
the point particle:
W = 𝛿Ek .

(2.14)

If all the forces acting on a particle are conservative and Ep is the total potential
energy, the following equalities are satisfied
𝐅 ⋅ d𝐫 = − 𝛁 V (𝐫) ⋅ d𝐫 = −𝛿Ep
∫C
∫C
⇒ 𝛿(Ek + Ep ) = 0.

𝛿Ek = W =

(2.15)

This result is known as the conservation of energy and states that the total energy
E = Ek + Ep = const.

(2.16)

is 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.
4) This principle is used in our everyday life in all maps. The denser the contour lines in the map are,

the higher the mountain is and the harder it is to climb it up.
5) For extended objects composed of many particles, the kinetic energy of the composite body is the
sum of the kinetic energies of all particles.

9


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10

2 Newtonian Mechanics and Thermodynamics

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 (𝑎𝑐𝑡𝑖𝑜 = 𝑟𝑒𝑎𝑐𝑡𝑖𝑜) holds:
𝐅𝑖𝑗 = −𝐅𝑗𝑖 .

(2.17)

It means that the force 𝐅𝑖𝑗 induced by particle i on particle j has the same magnitude as force 𝐅𝑗𝑖 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:
N
N
∑
∑

𝐅𝑖𝑗 = 0.

(2.18)

i=1 j=1, j≠i

The total force acting on particle i induced by the remaining N − 1 particles
𝐅i =

N
∑

𝐅𝑖𝑗

(2.19)

j=1, j≠i

is obviously an internal force. With Eq. (2.18), this adds up to
N
∑

𝐅i = 0.

(2.20)

i=1

The last equation holds not only for forces acting pairwise, but also more generally
for all kinds of internal forces. If the forces 𝐅i are moreover conservative forces,
that is, they can be expressed by a scalar potential function V as a generalization
of Eq. (2.11)
𝐅i = −∇i V (𝐫1 , … , 𝐫i , … , 𝐫N ),

(2.21)

the total energy of the N-particle system is conserved. A typical example for such
a system is an infinite one-dimensional chain of particles coupled by ideal springs.
According to Eq. (2.6), a system in which only internal forces are acting between
the particles and no external forces are applied can be described by the following
set of equations of motion:
𝐩̇ i = 𝐅i with i = 1, … , N.

(2.22)


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2.4 Thermodynamics

For real systems, this set of differential equations may get quite complicated and
its solution can be obtained only approximately in most cases. Nevertheless, the

deterministic character of the theory remains. If one knows at a certain time t0
the positions and the velocities of all particles, then the motion of the system is
completely determined. That means, the momentum and position of any particle
in the system can be predicted for any time in the future or past.
With Eq. (2.20), it follows that the total momentum
𝐏=

N
∑

𝐩i

(2.23)

i=1

of such a system is conserved. This law is as important in physics as the conservation of energy, Eq. (2.16). The latter can be written as the sum of the kinetic energy
of the N particles and the potential energy, which can be set equal to the scalar
potential function V from Eq. (2.21), also see Eq. (2.12)
E(𝐯i , 𝐫i ) = Ek + Ep =

N
∑
1
i=1

E(𝐩i , 𝐫i ) = Ek + Ep =

2

mi v2i + V (𝐫1 , … , 𝐫N ),

N
∑
p2i
i=1

2mi

+ V (𝐫1 , … , 𝐫N ).

(2.24)

(2.25)

As can be seen in the second equation, the total energy can be interpreted as a
function of the particle momenta and coordinates. One speaks in this context
often from the so-called Hamilton function (after William Rowan Hamilton) and
uses a “H” as symbol:
E(𝐩i , 𝐫i ) ≡ H(𝐩i , 𝐫i ).

(2.26)

In the general case of nonconservative systems, the Hamilton function can additionally depend implicitly on the time t
H = H(𝐩i , 𝐫i , t).

(2.27)

More details about the Hamiltonian approach to classical mechanics can be found,
for example, in Ref. [2].

Note: As H(𝐩i , 𝐫i ) is a conserved quantity, it plays a central role in the formulation of the quantum mechanical theory and many related simulation methods.

2.4
Thermodynamics

In principle, the motion of N particles can be described by the equations of motion
(2.22) independent of the actual value of N. Irrespective of the practical problems
of solving a large system of differential equations, often the exact motion of each

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12

2 Newtonian Mechanics and Thermodynamics

individual particle is not of interest. Only the behavior of the complete system
of particles matters. A typical example are the macroscopic properties of gases or
liquids. They can be thought of as the characteristics of a system with a huge number (typically more than millions6) ) of particles. Common properties of interest are
the temperature T, the pressure p, the internal energy U, and the entropy S. These
quantities are described by the laws of thermodynamics. A relation between the
microscopic properties of the individual particles and the macroscopic properties
of the whole system can be derived from statistical mechanics.
pressure p: average force exerted to a surface by the particles of the
medium
temperature T: a measure of the averaged kinetic energy of the particles of
the medium
internal energy U: the average of the total energy of the particles of the medium

entropy S: a measure of the disorder of the particles of the medium
The laws of thermodynamics consist essentially of two principles7) :

• First law of thermodynamics: The increase in internal energy U of a closed
system8) is equal to the difference of the heat Q supplied to the system and the
work W done by it: ΔU = Q − W — or in other words, the internal energy of a
closed system is constant (see Figure 2.3(a)).
• Second law of thermodynamics: Heat cannot spontaneously flow from a
colder location to a hotter location — or in other words, the entropy of a closed
system cannot be reduced: ΔS ≥ 0 (see Figure 2.3(b)).
If one takes into account that the work done by the system is equal to the volume
change times the pressure9) W = pΔV and that a heat reversibly supplied to the
system is equal to the temperature times the entropy change caused by that heat
transfer Q = TΔS, one obtains Gibbs’ fundamental equation
ΔU = TΔS − pΔV .

(2.28)

Note: In general, the change of the entropy can be decomposed into the
change due to internal processes ΔSi and the change due to the reversible
exchange of heat with an external system ΔSe = Q∕T. In combination with
6) One mole of a gas or a liquid contains NA = 6.022 ⋅ 1023 particles.
7) In the literature, a zeroth and third law of thermodynamics are also discussed. However, the third
law can be derived from quantum statistics and the zeroth law is a quite general principle valid not
only in thermodynamics.
8) A closed system can exchange energy (heat or work) but not matter with its surrounding environment.
9) This assumption holds only in idealized systems. In real systems, friction energy has also to be taken
into account.

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2.4 Thermodynamics

T1 < T2
Work done
by the
system
W=pΔV

T1

T

p,T

ΔU

T2

ΔQ
irreversible
heat
transfer
until T1 = T2

ΔQ
Reversible heat
exchange with
a reservoir

(a)

(b)

Figure 2.3 Illustration of the first (a) and second (b) thermodynamic law

the second thermodynamic law, one finds
ΔS = ΔSi + ΔSe = ΔSi + Q∕T,
ΔS ≥ Q∕T.

(2.29)

That means, strictly speaking, Gibbs’ fundamental equation in the aforementioned form (2.28) holds only in the case of quasi static, that means, reversible
processes [3].
In the case of an additional change of the number of particles in the system,
Eq. (2.28) can be extended to
∑
ΔU = TΔS − pΔV +
𝜇i ΔNi ,
(2.30)
i

where Ni is the number of particles of the kind i and 𝜇i its chemical potential.
The latter characterizes the stability of the current phase of a particle species. The
lower the chemical potential, the more stable the phase is.
From the form of Eq. (2.30), one can see that the internal energy U is a function
of the variables S, V , and N. One speaks in this context often about a thermodynamic potential
U = U(S, V , N).

(2.31)

Besides the internal energy, one can define seven other thermodynamic potentials,
which can be transformed into each other by a Legendre transformation [3]. The
most important thermodynamic potentials are the enthalpy (not to be confused

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