Molecular dynamics simulation of nanocomposites using BIOVIA Materials Studio, Lammps and Gromacs [1 ed.] 0128169540, 9780128169544

Table of contents :
Contents
Introduction to Molecular Dynamics 1
Overview of BIOVIA Materials Studio LAMMPS and GROMACS 39
Molecular Dynamics Simulation of Metal Matrix Composites Using BIOVIA Materials Studio LAMMPS and GROMACS 101
Molecular Dynamics Simulation of PolymerMatrix Composites Using BIOVIA Materials Studio LAMMPS and GROMACS 141
Molecular Dynamics Simulation of Ceramic Matrix Composites Using BIOVIA Materials Studio LAMMPS and GROMACS 227

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Molecular Dynamics Simulation of Nanocomposites Using BIOVIA Materials Studio, Lammps and Gromacs

Molecular Dynamics Simulation of Nanocomposites Using BIOVIA Materials Studio, Lammps and Gromacs Edited by

Sumit Sharma Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India

Dedicated to My Beloved Parents

Mr. Ashok Sharma and Mrs. Nirmal Sharma and My Guru

Dr. Rakesh Chandra

Contributors Amit Bansal Department of Mechanical Engineering, I.K. Gujral Punjab Technical University, Kapurthala, India Rakesh Chandra Department of Mechanical Engineering, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, India Raj Chawla Department of Mechanical Engineering, Lovely Professional University, Phagwara, India Manish Dhawan Materials Engineering, Department of Mechanical Engineering, Lovely Professional University, Phagwara, Punjab, India Raja Sekhar Dondapati School of Mechanical Engineering, Lovely Professional University, Phagwara, India Pramod Kumar Department of Mechanical Engineering, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, India Apurba Mandal Department of Mechanical Engineering, NIT Uttarakhand, Srinagar, India Pramod Rakt Patel Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India Prince Setia Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur, India Sumit Sharma Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India Gaurav Sharma School of Mechanical Engineering, Lovely Professional University, Phagwara, India S.P. Singh Department of Mechanical Engineering, IIT Delhi, New Delhi, India

xiii

Preface This book is the fruit of the guidance of my supervisors, Dr. Rakesh Chandra and Dr. Pramod Kumar at Dr. B. R. Ambedkar National Institute of Technology, Jalandhar. The editor is highly grateful to Dr. Navin Kumar (IIT Ropar) and Dr. S.P. Singh (IIT Delhi) for their invaluable suggestions. Without their esteemed guidance, support, and motivation, this book might not have materialized. I am very grateful to my parents who have always motivated me during my life. There was an urgent need for such a book because there is currently no book available that caters to the need of students and researchers working in the field of molecular dynamics (MD) simulation. This book will provide the readers with an overview of the three most commonly used tools for MD simulation. Though there are some books on MD such as “Introduction to Practice of Molecular Simulation” by Akira Satoh and “Molecular Modelling for Beginners” by Alan Hinchliffe, none of these books specify the steps used in MD simulation tools such as BIOVIA Materials Studio, Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), and GROningen MAchine for Chemical Simulation (GROMACS). The editor has been working in the field of MD simulation of composites since the last 9 years and, before that, in the field of “finite element modeling” of composites using NISA and MATLAB. BIOVIA Materials Studio, formerly known as Accelrys BIOVIA Materials Studio, is a versatile tool for atomistic modeling. It is widely used in the industry for simulation of proteins, DNA, and various other biological substances and for prediction of the properties of nanomaterials such as carbon nanotubes, polymers, metals, ceramics, and various types of nanocomposites. In this book, the readers will find all the basic steps necessary for simulating any material on BIOVIA Materials Studio. After reading this book, the readers will be able to model their own problems on the above tool for predicting the optical, chemical, mechanical, and electronic properties of any material. The acronym LAMMPS stands for Large-scale Atomic/Molecular Massively Parallel Simulator. It is a classical molecular dynamics simulation package especially designed to run efficiently in parallel mode with only a few particles up to millions or billions. The package is distributed as open source under the term of GNU public license (GPL) by Sandia National Laboratories, which is a lab specializing in defense systems and developed as a Department of Energy (DOE) facility. Sorts of problems that can be addressed using xv

Preface atomistic systems in LAMMPS include water interaction with self-assembled monolayers, ionomer morphologies, nanoparticle coating structures, self-assembly of lipid surfaces, soft material rheology, wetting and surface properties of complex fluids, and nanoindentation, to name a few. GROMACS is a molecular dynamics simulator, which performs calculations for solving molecular behavior under external force fields. GROMACS has emerged to be an extremely useful computational code for computationally replicating the system of molecular cluster and aiding in fetching technical data, which is beneficial for gaining deeper understanding at the atomic scale. Working at the molecular scale aids in optimizing of devices, for creating state-of-the-art equipment. From the modeling of medical devices and implants to simulation of nanoscale flows, GROMACS is in continuous process, toward our understanding at small scale, and the verification of physical principles, which are visually prominent at macroscale. In this book, a list on the applications of GROMACS is presented; however, the readers are advised not to be confined with the discussed utility of the software. A link to various open-source codes on GROMACS is also provided in this book so that the readers find it usable. An attempt has been made here to cover thoroughly all the above three tools so that the users working in the field of MD can chose any tool, out of the abovementioned three, as per the requirement. The editor is highly thankful to all the contributors of this book for their excellent work and to the potential readers for sending their valuable suggestions, if any, so that this book can be improved further. Sumit Sharma Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India February 2019

xvi

CHAPTER 1

Introduction to Molecular Dynamics Sumit Sharma, Pramod Kumar, Rakesh Chandra Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India In the modern nanotechnology age, microscopic analysis methods are necessary to generate new functional materials and investigate physical phenomena on a molecular level. In these methods the constituent species of a system, such as molecules and fine particles, are considered. Macroscopic and microscopic quantities of interest are derived from analyzing the behavior of these species. These approaches, called “molecular simulation methods,” are represented by the Monte Carlo (MC) and molecular dynamics (MD) methods. MC methods exhibit a powerful ability to analyze thermodynamic equilibrium but are unsuitable for investigating dynamic phenomena. MD methods are useful for thermodynamic equilibrium but are more advantageous for investigating the dynamic properties of a system in a nonequilibrium situation. Besides the earlier stated methods, there exist other methods also that can be used to investigate the physical phenomenon on a molecular level. Some of these are as follows: (i) BD methods, which can simulate the Brownian motion of dispersed particles (ii) DPD and lattice Boltzmann methods in which a liquid system is regarded as composed of virtual fluid particles Simulation methods using the concept of virtual fluid particles are generally used for pure liquid systems but are useful for simulating particle dispersions.

1.1 Molecular Dynamics The concept of the MD method is straightforward and logical. The motion of molecules is generally governed by Newton’s equations of motion in classical theory. In MD simulations, particle motion is simulated on a computer according to the equations of motion. If one molecule moves solely on a classical mechanics level, a computer is unnecessary because mathematical calculation with pencil and paper is sufficient to solve the motion of the molecule. However, since molecules in a real system are numerous and interact with each other, such mathematical analysis is impracticable. In this situation therefore computer simulations become a powerful tool for a microscopic analysis. Molecular Dynamics Simulation of Nanocomposites using BIOVIA Materials Studio, Lammps and Gromacs https://doi.org/10.1016/B978-0-12-816954-4.00001-2 # 2019 Elsevier Inc. All rights reserved.

1

2

Chapter 1

In MD simulation, there are different ways through which the position of a particle at a subsequent time can be evaluated. These are listed in the succeeding text: (i) Verlet method: If the velocity terms are not required for determining the molecular position at the next time step, then this scheme is called the “Verlet method.” (ii) Velocity Verlet method: In some cases a scheme using the positions and velocities simultaneously may be more desirable to keep the system temperature constant. If both the current position and velocity of the particle are required to evaluate the position of the particle at the next time step, then this scheme is known as velocity Verlet method. (iii) Leapfrog method: This name arises from the evaluation of the positions and forces and then the velocities, by using time steps in a leapfrog manner. This method is also a significantly superior scheme in regard to stability and accuracy, comparable with the velocity Verlet method. The MD method is applicable to both equilibrium and nonequilibrium physical phenomena, which makes it a powerful computational tool that can be used to simulate many physical phenomena. The main procedure for conducting the MD simulation using the velocity Verlet method is shown in the following steps: (i) (ii) (iii) (iv) (v)

Specify the initial position and velocity of all molecules. Calculate the forces acting on molecules. Evaluate the positions of all molecules at the next time step. Evaluate the velocities of all molecules at the next time step. Repeat the procedures from Step 2.

In the earlier procedure the positions and velocities will be evaluated at every time interval “h” in the MD simulation.

1.2 Monte Carlo Simulation In the MD method the motion of molecules is simulated according to the equations of motion, and therefore it is applicable to both thermodynamic equilibrium and nonequilibrium phenomena. In contrast the MC method generates a series of microscopic states under a certain stochastic law, irrespective of the equations of motion of particles. Since the MC method does not use the equations of motion, it cannot include the concept of explicit time and thus is only a simulation technique for phenomena in thermodynamic equilibrium. Hence it is unsuitable for the MC method to deal with the dynamic properties of a system, which are dependent on time. In this method a series of microscopic states is generated using a probability density function (ρ). The main procedure for the MC simulation of a nonspherical particle system is as follows: (i) Specify the initial position and direction of all particles. (ii) Regard this state as microscopic state i, and calculate the interaction energy Ui. (iii) Choose an arbitrary particle in order or randomly and call this particle “particle α.”

Introduction to Molecular Dynamics 3 (iv) Make particle α move translationally using random numbers and calculate the interaction energy Uj for this new configuration. (v) Adopt this new microscopic state for the case of Uj
Ui and go to Step 7. (vi) Calculate ρj/ρi for the case of Uj > Ui and take a random number R1 from a uniform random number sequence distributed from zero to unity. If R1
ρj/ρi, adopt this microscopic state j and go to Step 7. If R1 > ρj/ρi, reject this microscopic state, regard previous state i as new microscopic state j, and go to Step 7. (vii) Change the direction of particle α using random numbers and calculate the interaction energy Uk for this new state. (viii) If Uk
Uj, adopt this new microscopic state and repeat from Step 2. (ix) If Uk > Uj, calculate ρk/ρj and take a random number R2 from the uniform random number sequence. If R2
ρk/ρj, adopt this new microscopic state k and repeat from Step 2. If R2 > ρk/ρj, reject this new state, regard previous state j as new microscopic state k, and repeat from Step 2.

1.3 Brownian Dynamics A dispersion or suspension composed of fine particles dispersed in a base liquid is a difficult case to be treated by simulations in terms of the MD method, because the characteristic time of the motion of the solvent molecules is considerably different from that of the dispersed particles. Simply speaking, if we observe such dispersion based on the characteristic time of the solvent molecules, we can see only the active motion of solvent molecules around the quiescent dispersed particles. Clearly the MD method is quite unrealistic as a simulation technique for particle dispersions. One approach to overcome this difficulty is to not focus on the motion of each solvent molecule, but regard the solvent molecules as a continuum medium and consider the motion of dispersed particles in such a medium. In this approach the influence of the solvent molecules is included into the equations of motion of the particles as random forces. The BD method simulates the random motion of dispersed particles that is induced by the solvent molecules; thus such particles are called “Brownian particles.” The main procedure for conducting the BD simulation is shown as follows: (i) (ii) (iii) (iv) (v)

Specify the initial position of all particles. Calculate the forces acting on each particle. Generate the random displacements using uniform random numbers. Calculate all the particle positions at the next time step. Return to Step 2 and repeat.

1.4 Dissipative Particle Dynamics It is not realistic to use the MD method to simulate the motion of solvent molecules and dispersed particles simultaneously, since the characteristic time of solvent molecules is much shorter than that of dispersed particles. Hence in the BD method the motion of solvent

4

Chapter 1

molecules is not treated, but a fluid is regarded as a continuum medium. The influence of the molecular motion is combined into the equations of motion of dispersed particles as stochastic random forces. Are there any simulation methods to simulate the motion of both the solvent molecules and the dispersed particles? As far as we treat the motion of real solvent molecules, the development of such simulation methods may be impractical. However, if groups or clusters of solvent molecules are regarded as virtual fluid particles, such that the characteristic time of the motion of such fluid particles is not so different from that of dispersed particles, then it is possible to simulate the motion of the dispersed and the fluid particles simultaneously. These virtual fluid particles are expected to exchange their momentum, exhibit a random motion similar to Brownian particles, and interact with each other by particle-particle potentials. We call these virtual fluid particles “dissipative particles,” and the simulation technique of treating the motion of dissipative particles instead of the solvent molecules is called the “dissipative particle dynamics (DPD) method.” The main procedure for conducting the DPD simulation is quite similar to BD simulations.

1.5 Lattice Boltzmann Method In the lattice Boltzmann method a fluid is assumed to be composed of virtual fluid particles, and such fluid particles move and collide with other fluid particles in a simulation region. A simulation area is regarded as a lattice system, and fluid particles move from site to site; that is, they do not move freely in a region. The most significant difference of this method in relation to the MD method is that the lattice Boltzmann method treats the particle distribution function of velocities rather than the positions and the velocities of the fluid particles.

1.6 Basic Concepts Following are some of the common terms used in MD simulation.

1.6.1 Force Field In the context of molecular modeling, force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles (typically molecules and atoms). Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical calculations. “All-atom” force fields provide parameters for every type of atom in a system, including hydrogen, while “united-atom” force fields treat the hydrogen and carbon atoms in methyl and methylene groups as a single interaction center. “Coarse-grained” force fields, which are frequently used in longtime simulations of proteins, provide even more crude representations for increased computational efficiency. The usage of the term “force field” in chemistry and computational biology differs

Introduction to Molecular Dynamics 5 from the standard usage in physics. In chemistry, it is a system of potential energy functions rather than the gradient of a scalar potential, as defined in physics. The basic functional form of a force field encapsulates both bonded terms relating to atoms that are linked by covalent bonds and nonbonded (also called “noncovalent”) terms describing the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy in an additive force field can be written as (1.1) Etotal ¼ Ebonded + Enonbonded where the components of the covalent and noncovalent contributions are given by the following summations: (1.2) Ebonded ¼ Ebond + Eangle + Edihedral Enonbonded ¼ Eelectrostatic + EvanderWaals

(1.3)

The bond and angle terms are usually modeled as harmonic oscillators in force fields that do not allow bond breaking. A more realistic description of a covalent bond at higher stretching is provided by the more expensive Morse potential. The functional form for the rest of the bonded terms is highly variable. Proper dihedral potentials are usually included. Additionally, “improper torsional” terms may be added to enforce the planarity of aromatic rings and other conjugated systems and “cross terms” that describe coupling of different internal variables, such as angles and bond lengths. Some force fields also include explicit terms for hydrogen bonds. The nonbonded terms are most computationally intensive because they include many more interactions per atom. A popular choice is to limit interactions to pairwise energies. The van der Waals term is usually computed with a Lennard-Jones potential and the electrostatic term with Coulomb’s law, although both can be buffered or scaled by a constant factor to account for electronic polarizability and produce better agreement with experimental observations. In addition to the functional form of the potentials, a force field defines a set of parameters for each type of atom. For example, a force field would include distinct parameters for an oxygen atom in a carbonyl functional group and in a hydroxyl group. The typical parameter set includes values for atomic mass, van der Waals radius, and partial charge for individual atoms; equilibrium values of bond lengths, bond angles, and dihedral angles for pairs, triplets, and quadruplets of bonded atoms; and values corresponding to the effective spring constant for each potential. Parameter sets and functional forms are defined by force field developers to be self-consistent. Because the functional forms of the potential terms vary extensively between even closely related force fields (or successive versions of the same force field), the parameters from one force field should never be used in conjunction with the potential from another. Some of the popular force fields have been discussed in the succeeding text: (i) Assisted Model Building with Energy Refinement (AMBER): AMBER is a family of force fields for MD simulation of biomolecules originally developed by the late Peter Kollman’s [1]

6

Chapter 1 group at the University of California, San Francisco. AMBER is also the name for the MD software package that simulates these force fields. It is maintained by an active collaboration between David Case at Rutgers University, Tom Cheatham at the University of Utah, Tom Darden at NIEHS, Ken Merz at Florida, Carlos Simmerling at Stony Brook University, Ray Luo at UC Irvine, and Junmei Wang at Encysive Pharmaceuticals.

The term “AMBER force field” generally refers to the functional form used by the family of AMBER force fields. This form includes a number of parameters; each member of the family of AMBER force fields provides values for these parameters and has its own name. The functional form of the AMBER force field is shown as follows: X 1
 X 1 kb ðl  l0 Þ2 + ka ðθ  θ0 Þ2 V rN ¼ 2 2 bonds angles +

+

X 1 Vn ð1 + cos ðnω  γ ÞÞ 2 torsions N XN1 X j¼1

i¼j + 1

"  !  6 # r0ij 12 r0ij qi qj + εi, j 2 rij rij 4πε0 rij

(1.4)

where ka and kb are the force constants determined experimentally, l is the bond length after stretching, l0 is the equilibrium bond length, θ is the bond angle after bending, θ0 is the equilibrium bond angle, Vn is the energy term, n is the periodicity parameter, ω is the dihedral bond torsion angle, γ is the phase parameter, εi,j is the well depth, r0ij is the distance at which the interaction energy between the two atoms is zero, rij is the separation between the molecules, qi, qj is the atomic charges on the atoms/molecules, and ε0 is the permittivity of free space. Note that despite the term force field, this equation defines the potential energy of the system; the force is the derivative of this potential with respect to position. The meanings of right-hand side terms are as follows: (a) First term (summing over bonds): represents the energy between covalently bonded atoms. This harmonic (ideal spring) force is a good approximation near the equilibrium bond length but becomes increasingly poor as atoms separate. (b) Second term (summing over angles): represents the energy due to the geometry of electron orbitals involved in covalent bonding. (c) Third term (summing over torsions): represents the energy for twisting a bond due to bond order (e.g., double bonds) and neighboring bonds or lone pairs of electrons. Note that a single bond may have more than one of these terms, such that the total torsional energy is expressed as a Fourier series.

Introduction to Molecular Dynamics 7 (d) Fourth term (double summation over i and j) represents the nonbonded energy between all atom pairs, which can be decomposed into van der Waals (first term of summation) and electrostatic (second term of summation) energies. The form of the van der Waals energy is calculated using the equilibrium distance (r0ij) and well depth (ε). The factor of 2 ensures that the equilibrium distance is r0ij. The energy is sometimes reformulated in terms of σ, where r0ij ¼ 21/6(σ), as used, for example, in the implementation of the soft core potentials. The form of the electrostatic energy used here assumes that the charges due to the protons and electrons in an atom can be represented by a single point charge (or in the case of parameter sets that employ lone pairs a small number of point charges.) To use the AMBER force field, it is necessary to have values for the parameters of the force field (e.g., force constants, equilibrium bond lengths and angles, and charges). A fairly large number of these parameter sets exist and are described in detail in the AMBER software user manual. Each parameter set has a name and provides parameters for certain types of molecules. (a) Peptide, protein, and nucleic acid parameters are provided by parameter sets with names beginning with “ff” and containing a two-digit year number, for instance, “ff99.” (b) Generalized AMBER force field (GAFF) provides parameters for small organic molecules to facilitate simulations of drugs and small-molecule ligands in conjunction with biomolecules. (c) The GLYCAM force fields have been developed by Rob Woods for simulating carbohydrates. (ii) Condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS): COMPASS [2] is a member of the consistent family of force fields (consistent force field (CFF)91, polymer CFF (PCFF), CFF, and COMPASS), which are closely related second-generation force fields. They were parameterized against a wide range of experimental observables for organic compounds containing H, C, N, O, S, P, halogen atoms and ions, alkali metal cations, and several biochemically important divalent metal cations. PCFF is based on CFF91, extended so as to have a broad coverage of organic polymers and (inorganic) metals. COMPASS is the first force field that has been parameterized and validated using condensed-phase properties in addition to various and empirical data for molecules in isolation. Consequently, this force field enables accurate and simultaneous prediction of structural, conformational, vibrational, and thermophysical properties for a broad range of molecules in isolation and in condensed phases.

8

Chapter 1

The COMPASS force field consists of terms for bonds (b), angles (θ), dihedrals (φ), and out-ofplane angles (χ) ) and cross terms and two nonbonded functions, a coulomb function for electrostatic interactions and a 9-6 Lennard-Jones potential for van der Waals interactions: Etotal ¼ Eb + Eθ + Eφ + Eχ + Eb, b0 + Eb, θ + Eb, φ + Eθ, φ + Eθ, θ0 + Eθ, θ0 , φ + Eq + EvdW where Eb ¼

Xh

k2 ðb  b0 Þ2 + k3 ðb  b0 Þ3 + k4 ðb  b0 Þ4

b

Eθ ¼ E∅ ¼

X

Xh

k2 ðθ  θ0 Þ2 + k3 ðθ  θ0 Þ3 + k4 ðθ  θ0 Þ4

(1.5)

i (1.6) i (1.7)

θ

½ k1 ð1  cos ∅Þ + k2 ð1  cos2∅Þ + k3 ð1  cos 3∅Þ

∅

Eχ ¼

X

k2 χ 2

(1.9)

χ

X
 kðb  b0 Þ b0  b00 X Eb, θ ¼ kðb  b0 Þðθ  θ0 Þ b, θ

Eb, b0 ¼

Eb, ∅ ¼

X

(1.8)

(1.10) (1.11)

ðb  b0 Þ½ k1 cos ∅ + k2 cos 2∅ + k3 cos 3∅

(1.12)

ðθ  θ0 Þ½ k1 cos ∅ + k2 cos 2∅ + k3 cos 3∅

(1.13)

b, ∅

Eθ , ∅ ¼

X θ, ∅

Eθ , θ 0 ¼ Eθ , θ 0 , φ ¼

X θ, θ0

X

θ , θ0 , φ


 kðθ  θ0 Þ θ0  θ00


 kðθ  θ0 Þ θ0  θ00 cosφ

Eq ¼

X qi qj ij

EvdW ¼

X ij

2 Eij 42

rij0 rij

3

(1.15)

(1.16)

rij !9

(1.14)

rij0 rij

!6 3 5

(1.17)

where k, k1, k2, k3, and k4 are the force constants determined experimentally; b and θ are the bond length and bond angle after stretching and bending, respectively; b0 and θ0 are the equilibrium bond length and equilibrium bond angle, respectively; φ is the bond torsion angle; χ is the out-of-plane inversion angle; Eb,b0 , Eθ,θ0 , Eb,θ, Eb,φ, Eθ,φ, and Eθ,θ0 ,φ are the cross terms

Introduction to Molecular Dynamics 9 representing the energy due to interaction between bond stretch-bond stretch, bond bend-bond bend, bond stretch-bond bend, bond stretch-bond torsion, bond bend-bond torsion, and bond bend-bond bend-bond torsion, respectively; εi,j is the well depth; r0ij is the distance at which the interaction energy between the two atoms is zero; rij is the separation between the atoms/ molecules; qi, qj are the atomic charges on the atoms/molecules; and ε0 is the permittivity of free space. (iii) CHARMm force field: CHARMm, which derives from CHemistry at HARvard Macromolecular Mechanics, is a highly flexible molecular mechanics and dynamics program originally developed in the laboratory of Dr. Martin Karplus at Harvard University [3]. It was parameterized on the basis of ab initio energies and geometries of small organic models. The CHARMm force fields include the CHARMm (Momany & Rone) force field developed at Accelrys and the academic force fields, primarily developed by Prof. Alex MacKerell and coworkers (charmm19, charmm22, and charmm27). All versions of the CHARMm force fields are available from Accelrys. CHARMm uses a flexible and comprehensive empirical energy function that is a summation of many individual energy terms. The energy function is based on separable internal coordinate terms and pairwise nonbond interaction terms. The total energy is expressed by Eq. (1.18). The electrostatic term can be scaled to mimic solvent effects. The van der Waals combination rules and functional form are derived from rare-gas potentials: X X X kb ðr  r0 Þ2 + kθ ðθ  θ0 Þ2 + Epot ¼ j k∅ j X X qi qj X Aij  2 k∅ cos ðn∅Þ + kχ ðχ  χ 0 Þ + + 4πε0 rij rij 12  
2  Bij 2 2  + Econstraint + Euser sw r , r , r (1.18) ij on off rij 6 where kb, kθ, kφ, and kχ are the force constants for bond stretching, bond bending, bond torsion, and out-of-plane inversion, respectively; r, θ, and χ are the bond length, bond angle, and inversion angle respectively; r0, θ0, and χ 0 are the equilibrium bond length, equilibrium bond angle, and equilibrium inversion angle, respectively; rij are the separation between the atoms/ molecules; n is the periodicity parameter; qi, qj are the atomic charges on the atoms/molecules; ε0 is the permittivity of free space; Aij and Bij are the distances over which the interatomic interaction is zero; and Econstraint and Euser are the energies arising due to constraint (if applied) and certain user-defined terms (to customize the force field), respectively. The r12 term is the repulsive term, describing Pauli repulsion at short ranges due to overlapping electron orbitals, and the r6 term is the attractive long-range term. Hydrogen bond energy is not included as a default energy term. The current CHARMm parameter set has been derived in such a way that hydrogen bond effects are described by the combination of electrostatic and van der Waals forces.

10

Chapter 1

(iv) Consistent valence force field (CVFF): The CVFF, the original force field provided with the Discover program, is a generalized valence force field. Parameters are provided for amino acids, water, and a variety of other functional groups. CVFF [4] also has the ability to use automatic parameters (automatic assignment of values for missing parameters) when no explicit parameters are present. These are noted in the output file from the calculation. CVFF was fit to small organic (amides, carboxylic acids, etc.) crystals and gas-phase structures. It handles peptides, proteins, and a wide range of organic systems. As the default force field in Discover, it has been used extensively for many years. It is primarily intended for studies of structures and binding energies, although it predicts vibrational frequencies and conformational energies reasonably well. The out-of-plane energy for the CVFF force field is calculated as an improper torsion. An improper torsion views three connected atoms and a central atom as if it were torsion. There are three possible improper torsions that can be generated for a particular out of plane, based on permutations of the connected atoms. For CVFF, only one of these improper torsions is used. The analytic form of the energy expression used in CVFF is shown in Eq. (1.19). Terms 1–4 are commonly referred to as the diagonal terms of the valence force field and represent the energy of deformation of bond lengths, bond angles, torsion angles, and out-ofplane interactions, respectively. A Morse potential (Term 1) is used for the bond-stretching term. Discover program also supports a simple harmonic potential for this term:  X X  X Db 1  eαðbb0 + Hθ ð θ  θ 0 Þ 2 + H∅ ð1 + s cos ðn∅ÞÞ Epot ¼ θ

∅

X XX
 XX
 + Hχ χ 2 + Fbb0 ðb  b0 Þ b0  b00 + Fθθ0 ðθ  θ0 Þ θ0  θ00 +

XX b

θ

b

b0

Fbθ ðb  b0 Þðθ  θ0 Þ +

X

θ

θ0


 F∅θθ0 cos ∅ðθ  θ0 Þ θ0  θ00

∅

"  6 # X XX X r∗ 12 qi qj r∗ 0 + + Fχχ 0 χχ + ε 2 r r εrij χ χ0

(1.19)

where Db is the well depth or bond dissociation energy; α is the parameter that controls the width of the potential well (the smaller the “α,” the larger the well); b and b0 are the separation between atoms and equilibrium bond distance, respectively; θ and θ0 are the bond angle and equilibrium bond angle, respectively; χ is the inversion angle; n is the periodicity parameter; qi, qj are the atomic charges on the atoms/molecules; ε is the well depth (in Term 10) and permittivity of free space (in Term 11); Hθ, Hφ, and Hχ are the constants associated with bond bending, torsion, and out-of-plane inversion, respectively; Fb,b0 , Fθ,θ0 , Fb,θ, Fφ,θ,θ0 , and Fχ,χ 0 are the cross terms representing the energy due to interaction between bond stretch-bond stretch, bond bend-bond

Introduction to Molecular Dynamics 11 bend, bond stretch-bond bend, bond torsion-bond bend-bond bend, out of plane/out-of-plane inversion, respectively; r* is the separation at which interatomic potential is zero; and r is the separation between the atoms. The Morse form is computationally more expensive than the harmonic form. Since the number of bond interactions is usually negligible relative to the number of nonbond interactions, the additional cost of using the more accurate Morse potential is insignificant, so this is the default option. When the model being simulated is high in energy (caused, e.g., by overlapping atoms or a high target temperature), a Morse-style function might allow bonded atoms to drift unrealistically far apart. This is not desirable unless we are intending to study bond breakage. Terms 5–9 are off-diagonal (or cross) terms and represent couplings between deformations of internal coordinates. For example, Term 5 describes the coupling between stretching of adjacent bonds. These terms are required to accurately reproduce experimental vibrational frequencies and, therefore, the dynamic properties of molecules. In some cases research has also shown them to be important in accounting for structural deformations. However, cross terms can become unstable when the structure is far from a minimum. Terms 10–11 describe the nonbond interactions. Term 10 represents the van der Waals interactions with a Lennard-Jones function. Term 11 is the coulomb representation of electrostatic interactions. The dielectric constant “ε” can be made distance-dependent (i.e., a function of rij). In the CVFF force field, hydrogen bonds are a natural consequence of the standard van der Waals and electrostatic parameters, and special hydrogen bond functions do not improve the fit of CVFF to experimental data.

1.6.2 Potentials The reliability of atomistic MC and MD simulation techniques that have impacted on areas ranging from drug design to crystal growth depends on the use of appropriate interatomic energies and forces. These interactions are generally described using either analytic potential energy expressions or semiempirical electronic structure methods or obtained from a firstprinciple total-energy calculation. While the latter approach is not subject to errors that can arise from the assumed functional forms and parameter fitting usually required in the first two methods, there remain clear advantages to classical potentials for large systems and long simulation times. In the following sections, we will describe the bond-order potentials, namely, Tersoff model [5] and Brenner model [6], which is also known as reactive empirical bond-order potential (REBO potential). These are not based on a traditional many-body expansion of potential energy in bond lengths and angles; instead, they use a parameterized bond-order function to introduce many-body effects and chemical bonding into a pair potential. The Tersoff and Brenner empirical interatomic potentials (EIPs) are convenient, short-range, bondorder, empirical potentials that are often used in MD simulations and other calculations to model different properties of carbon-based materials. The convenience of these potentials comes from their rather simple, analytic forms and the short-range of atomic interactions. For carbon-based systems the Tersoff model has nine adjustable parameters that are listed in

12

Chapter 1

Table 1.1 that were originally fit to cohesive energies of various carbon systems, the lattice constant of diamond, and the bulk modulus of diamond. The Brenner EIP is based directly on the Tersoff EIP but has additional terms and parameters that allow it to better describe various chemical reactions in hydrocarbons and include nonlocal effects. These parameters are listed in Table 1.2. (a) Tersoff model: The analytic form for the pair potential, Vij, of the Tersoff model [5] is given by the following functions with corresponding parameters listed in Table 1.1.   (1.20) Vij ¼ fijC aij fijR  bij fijA fijR ¼ Aeλ1 rij

(1.21)

fijA ¼ Beλ2 rij

(1.22)

where rij is the distance between atoms i and j; fijA and fijR is the competing attractive and repulsive pairwise terms; fijC is the cutoff term that ensures only nearest-neighbor interactions; and aij is the a range-limiting term on the repulsive potential that is typically set equal to one. The bond angle term, bij, depends on the local coordination of atoms around atom i and the angle between atoms i, j, and k:  1 (1.23) bij ¼ 1 + βn ξnij 2n Table 1.1 Original parameters for Tersoff EIP [5] for carbon-based systems A ¼ 1393.6 eV ˚ 1 λ1 ¼ 3.4879 A ˚ 1 λ3 ¼ 0.0000 A c ¼ 38049.0 d ¼ 4.3484 ˚ R ¼ 1.95 A

B ¼ 346.74 eV ˚ 1 λ2 ¼ 2.2119 A n ¼ 0.72751 β ¼ 1.5724  107 h ¼ 0.57058 ˚ D ¼ 0.15 A

Table 1.2 Original parameters for Brenner EIP [6] for solid-state carbon structures A ¼ 10953.544162170 eV B2 ¼ 17.5674064509 eV ˚ 1 α ¼ 4.746539060 A ˚ 1 λ2 ¼ 1.4332132499 A ˚ Q ¼ 0.3134602960833 A ˚ D ¼ 1.7 A β0 ¼ 0.7073 β2 ¼ 24.0970 β4 ¼ 71.8829

B1 ¼ 12388.79197798 eV B3 ¼ 30.71493208065 eV ˚ 1 λ1 ¼ 4.7204523127 A ˚ 1 λ3 ¼ 1.3826912506 A ˚ R¼2A T0 ¼ 0.00809675 β1 ¼ 5.6774 β3 ¼ 57.5918 β5 ¼ 36.2789

Introduction to Molecular Dynamics 13 ξij ¼

X k#i, j

fikC gijk eλ3 ðrij rik Þ 3

3

c2 c2 
  d2 d 2 + h  cos θijk 2

gijk ¼ 1 +

(1.24)

(1.25)

where θijk is the angle between atoms i, j, and k. This bond angle term allows the Tersoff model to describe the strong covalent bonding that occurs in carbon systems, which cannot be represented by purely central potentials. This angle-dependent term also allows for description of carbon systems that bond in different geometries, such as tetrahedrally bonded diamond and 120-degree tribonded graphene. (b) Brenner model: The Brenner potential [6] for solid-state carbon structures is given by the following functions with corresponding parameters listed in Table 1.2.   (1.26) Vij ¼ fijC fijR  bij fijA fijR

  Q Aeαrij ¼ 1+ rij

fijA ¼

3 X

Bn eλn rij

(1.27)

(1.28)

n¼1

where many of the terms are similar to the Tersoff model described earlier and the bond angle term, bij , is given by  1 σπ DH + ΠRC bij ¼ bσπ + b (1.29) ji ij + bij 2 ij !1=2 X bσπ ¼ 1+ fikC gijk (1.30) ij k#i, j

gijk ¼

5 X

 βi cos i θijk

(1.31)

i¼0

Here, bσπ depends on the local coordination of atoms around atom i and the angle between ij atoms i, j, and k, θijk. The coefficients, βi, in the bond-bending spline function, gijk, fit to experimental data for graphite and diamond and are also listed in Table 1.2. The term, ΠRC ij , accounts for various radical energetics, such as vacancies, which are not considered here; thus this term is taken to be zero. The term, bDH ij , is a dihedral bending function that depends on the local conjugation and is zero for diamond but important for describing graphene and single-

14

Chapter 1

walled carbon nanotubes (SWCNTs). This dihedral function involves third-nearest-neighbor atoms and is given by   T0 X C C
(1.32) ¼ fik fjl 1  cos 2 Θijkl bDH ij 2 k, l#i, j where T0 is a parameter; fijC is the cutoff function; and Θijkl is the dihedral angle of four atoms identified by the indexes, i, j, k, and l and is given by  ! ! (1.33) cos Θijkl ¼ η jik  η ijl ! η jik

!

!

!

r ji  r ik ¼

!

!



r ji

r ik sin θijk

(1.34)

!

where η jik and η ijl are unit vectors normal to the triangles formed by the atoms given by ! the subscripts; r ij is the vector from atom i to atom j; and θijk is the angle between atoms i, j, and k. In flat graphene the dihedral angle, Θijkl, is either 0 or π, and the dihedral term is subsequently zero. Bending of the graphene layer leads to a contribution from this term. Some of the main differences when compared with the Tersoff EIP are as follows: The Brenner EIP includes two additional exponential terms with corresponding adjustable parameters in the attractive pairwise term, it includes a screened Coulomb term in the repulsive pairwise term, it uses a fifth-order polynomial spline between bond orders for diamond and graphite, and it includes a dihedral bending term for bond energies that plays a role in SWCNTs and graphene. (c) Morse potential: The Morse potential [7], named after physicist Philip M. Morse, is a convenient model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Fig. 1.1 shows a comparison between a simple harmonic potential and more advanced Morse potential. Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ℏ ω (ℏ ¼ h/2π), where “h” is the Planck constant, the Morse potential level spacing decreases as the energy approaches the dissociation energy. The dissociation energy De is larger than the true energy required for dissociation D0 due to the zero-point energy of the lowest (v ¼ 0) vibrational level. The Morse potential energy function is of the form  2 (1.35) V ðr Þ ¼ De 1  eaðrre Þ

Introduction to Molecular Dynamics 15

Fig. 1.1 Comparison of harmonic potential and Morse potential.

Here, r is the distance between the atoms, re is the equilibrium bond distance, De is the well depth (defined relative to the dissociated atoms), and a controls the “width” of the potential (the smaller a, the larger the well). The dissociation energy of the bond can be calculated by subtracting the zero-point energy E (0) from the depth of the well. The force constant of the bond can be found by Taylor expansion of V(r) around r ¼ re to the second derivative of the potential energy function from which it can be shown that the parameter, a, is given by Eq. (1.36): a ¼ ðke =2De Þ1=2

(1.36)

where ke is the force constant at the minimum of the well. The zero of potential energy is arbitrary, and the equation for the Morse potential can be rewritten in any number of ways by adding or subtracting a constant value. (d) Lennard-Jones potential: An important concept in simulation methodology is that a significant limitation on the computation of interaction energies or forces between particles leads to an extraordinary reduction of the simulation time. To understand this concept, we consider the interaction

16

Chapter 1 4 3

ULJ / e

2 1 0 –1 –2

0

1 e

2s

2

3

4

r/s

Fig. 1.2 Lennard-Jones potential.

energies between particles or potential curves. The Lennard-Jones potential [8] ULJ is expressed as h i ULJ ¼ 4ε ðσ=rÞ12  ðσ=r Þ6 (1.37) This potential is usually used as a model potential for rare gases such as Ar molecule. Here, ε is the well depth, σ is the quantity corresponding to the particle diameter, and r is the separation between particles/molecules. Fig. 1.2 shows the curve of the Lennard-Jones. potential in which ULJ and r are nondimensionalized by ε and σ. It shows a steep potential barrier in the range of r
σ, which induces such a significant repulsive interaction that particles are prevented from significantly overlapping, and an attractive interaction in the range of r σ, which rapidly decreases to zero. These characteristics of the potential curve indicate that the interaction energy after a distance of approximately r ¼ 3σ can be assumed to be negligible. Hence particle interaction energies or forces do not need to be calculated in the range of r > 3σ in actual simulations. The distance for cutting off the calculation of energies or forces is known as the cutoff distance or cutoff radius, denoted by rcoff.

1.6.3 Ensemble Suppose that we have a macroscopic pure liquid sample, which might consist of 1023 particles, and we want to try and model some simple thermodynamic properties like the pressure p, the internal energy U, or the Gibbs energy G. At room temperature the individual particles making up the sample will be in motion, so at first sight, we ought to try and solve the equations of motion for these particles. In view of the large number of particles present, such an approach would be folly. Just to try and specify the initial positions and momentum of so many particles

Introduction to Molecular Dynamics 17

N, V, T

Fig. 1.3 A box of particles.

is not possible, and in any case such a calculation will give us too much information. For the sake of argument, suppose that the container is a cube. Fig. 1.3 shows a two-dimensional slice through the cube where the size of the particles has been exaggerated by a factor of approximately 1010. The pressure exerted by a gas on a container wall depends on the rate at which the particles collide with the wall. It is not necessary to know which particle underwent a particular collision. What we need to know about is the root-mean-square speed of the particles, their standard deviation about the mean, the temperature, and so on. In statistical thermodynamics, we do not inquire about the behavior of the individual particles that make up a macroscopic sample; we just inquire about their average properties. Classical thermodynamics is essentially particle-free. All that really matters to such a thermodynamicist is bulk properties such as the number of particles N, the temperature T, and the volume of the container V. This information is shown in the right-hand box in Fig. 1.3. Rather than worry about the time development of the particles in the left-hand box in Fig. 1.3, we can make a very large number of copies of the system on the right-hand side. We then calculate average values over this large number of replications, and according to the ergodic theorem the average value we calculate is exactly the same as the time average we would calculate by studying the time evolution of the original system. The two are the same. All the cells in the ensemble are not exact replicas at the molecular level. We just ensure that each cell has a certain number of thermodynamic properties that are the same. There is no mention of molecular properties at this stage. Fig. 1.4 shows an ensemble of cells all with the same values of N, V, and T. This array of cells is said to form a canonical ensemble. There are three other important ensembles in the theory of statistical thermodynamics, and they are classified according to what is kept constant in each cell. Apart from the canonical ensemble, where N, V, and T are kept constant, we have the following ensembles: (i) The microcanonical ensemble where N, the total energy E, and V are kept constant in each cell. In fact, this is a very simple ensemble because energy cannot flow from one cell to another. (ii) In an isothermal-isobaric ensemble, N, T, and the pressure P are kept constant.

18

Chapter 1

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

N, V, T

Fig. 1.4 A canonical ensemble.

(iii) Finally, we have the grand canonical ensemble, where V, T, and the chemical potential are kept constant. The grand canonical ensemble is a fascinating one because the number of particles is allowed to fluctuate.

1.6.4 Thermostat An equilibration procedure may be necessary to obtain the desired system temperature T. In the example of a liquid the temperature Tcal, which is calculated from averaging the assigned velocities of particles, may differ significantly from the desired system temperature T. This may be due to the energy exchange between the kinetic and the potential energies. Hence an equilibration procedure is frequently necessary before starting the main loop in a simulation program. (a) Andersen’s method According to Andersen [9] the temperatures calculated from the translational and angular (r) velocities of particles are denoted by T(t) cal and Tcal, respectively, and written as ðtÞ

Tcal ¼

N N 1 X mv2i ðrÞ 1 X Iω2i ,Tcal ¼ 3N i¼1 k 3N i¼1 k

(1.38)

(r) where N is the total number of particles, assumed to be N≫1. T(t) cal and Tcal calculated from vi and ωi (i ¼ 1, 2,…, N) are generally not equal to the desired temperature T. This equilibration procedure adjusts temperatures calculated from the translational and angular velocities of

Introduction to Molecular Dynamics 19 particles to T during the simulation by using the method of scaling the translational and angular (r) and T(r)ave denote the averaged values of T(t) velocities of each particle. If T(t)ave cal cal cal and Tcal taken, (r) for example, over 50 time steps, then the scaling factors c(t) 0 and c0 are determined as sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi T T ðtÞ ðrÞ c0 ¼ , c0 ¼ (1.39) ðtÞave ðrÞave Tcal Tcal With the scaling factors determined the translational and angular velocities of all particles in a system are scaled as ðtÞ

ðrÞ

v0i ¼ c0 vi , ω0i ¼ c0 ωi ði ¼ 1, 2, …, N Þ

(1.40)

This treatment yields the desired system temperature T. In this example the scaling procedure would be conducted at every 50 time steps, but in practice an appropriate time interval must be adopted for each simulation case. The abovementioned equilibration procedure is repeated to give rise to the desired system temperature with sufficient accuracy. (b) Berendsen thermostat The Berendsen thermostat [10] is an algorithm to rescale the velocities of particles in MD simulations to control the simulation temperature. In this scheme the system is weakly coupled to a heat bath with some temperature. The thermostat suppresses fluctuations of the kinetic energy of the system and therefore cannot produce trajectories consistent with the canonical ensemble. The temperature of the system is corrected such that the deviation exponentially decays with some time constant τ. To maintain the temperature the system is coupled to an external heat bath with fixed temperature T0. The velocities are scaled at each step, such that the rate of change of temperature is proportional to the difference in temperature: dT ðtÞ 1 ¼ ðT0  T ðtÞÞ dt τ

(1.41)

where τ is the coupling parameter that determines how tightly the bath and the system are coupled together. This method gives an exponential decay of the system toward the desired temperature. The change in temperature between successive time steps is 4T ¼

δt ðT0  T ðtÞÞ τ

Thus the scaling factor for the velocities is 0 λ2 ¼ 1 +

(1.42) 1

C δt B B  T 0   1C @ A δt τ T t 2

(1.43)

20

Chapter 1


 The term T t  δt2 is due to the fact that the leapfrog algorithm is used for the time integration. In practice, τ is used as an empirical parameter to adjust the strength of the coupling. Its value has to be chosen with care. In the limit τ ! ∞ the Berendsen thermostat is inactive, and the run is sampling a microcanonical ensemble. The temperature fluctuations will grow until they reach the appropriate value of a microcanonical ensemble. However, they will never reach the appropriate value for a canonical ensemble. On the other hand, too small values of τ will cause unrealistically low-temperature fluctuations. If τ is chosen the same as the time step δt, the Berendsen thermostat is nothing else than the simple velocity scaling. Values of τ  0.1ps are typically used in MD simulations of condensed-phase systems. The ensemble generated when using the Berendsen thermostat is not a canonical ensemble. Though the thermostat does not generate a correct canonical ensemble (especially for small systems), for large systems of the order of hundreds or thousands of atoms/molecules, the approximation yields roughly correct results for most calculated properties. The scheme is widely used due to the efficiency with which it relaxes a system to some target (bath) temperature. In many instances systems are initially equilibrated using the Berendsen scheme, while properties are calculated using the widely known Nos
e-Hoover thermostat, which correctly generates trajectories consistent with a canonical ensemble. (c) Nos
e-Hoover thermostat The Berendsen thermostat is extremely efficient for relaxing a system to the target temperature, but once our system has reached equilibrium, it might be more important to probe a correct canonical ensemble. The extended system method was originally introduced by Nos
e [11] and subsequently developed by Hoover [12]. The idea of the method proposed by Nos
e was to reduce the effect of an external system, acting as heat reservoir, to an additional degree of freedom. This heat reservoir controls the temperature of the given system, that is, the temperature fluctuates around target value. The idea is to consider the heat bath as an integral part of the system by addition of an artificial variable, s , associated with a “mass” Q > 0 and a velocity s_ . The magnitude of Q determines the coupling between the reservoir and the real system and so influences the temperature fluctuations. The artificial variable s plays the role of a timescaling parameter. More precisely the timescale in the extended system is stretched by the factor, s :



d t ¼s dt

(1.44)

The atomic coordinates are identical in both systems. This leads to 1 1 _ s ¼ s, and s_ ¼ s s_ r ¼ r, r_ ¼ s r,



(1.45)

Introduction to Molecular Dynamics 21 The Lagrangian for the extended system is chosen to be L¼

X mi _ 2
 1 s 2 r i  U r + Q s_ 2  gkb T0 ln s 2 2 i

(1.46)

The first two terms of the Lagrangian represent the kinetic energy minus the potential energy of the real system. The additional terms are the kinetic energy of s and the potential, which is chosen to ensure that the algorithm produces a canonical ensemble where g ¼ Ndf in real-time sampling (Nos
e-Hoover formalism) and g ¼ Ndf + 1 for virtual-time sampling (Nos
e formalism). This leads to the Nos
e equations of motion: r€i

Fi 2 s_ r_ i ¼  mi s 2 s

1 X 2 _ 2 mi s r i  gkb T0 s€ ¼ Qs i

(1.47) ! (1.48)



These equations sample a microcanonical ensemble in the extended system (r , p , and t ). However, the energy of the real system is not constant. Accompanying the fluctuations of s , heat transfers occur between the system and a heat bath, which regulate the system temperature. It can be shown that the equations of motion sample a canonical ensemble in the real system. The Nos
e equations of motion are smooth, deterministic, and time reversible. However, because the time evolution of the variable s is described by a second-order equation, heat may flow in and out of the system in an oscillatory fashion, leading to nearly periodic temperature fluctuations. The stretched timescale of the Nos
e equations is not very intuitive, and the sampling of a trajectory at uneven time intervals is rather impractical for the investigation of dynamic properties of a system. However, as shown by Nos
e and Hoover, the Nos
e equations of motion can be reformulated in terms of real system variables. The transformation is achieved through s ¼s , s_ ¼s s_ , s€¼ s 2 s€ + s s_ 2 , r ¼r , r_ ¼s r_ , r€¼ s 2 r€ + s r_ 2

(1.49)

and with substituting γ¼

s_ s

(1.50)

the Lagrangian equations of motion can be written as Fi Fi  γri r€i ¼  γri mi mi   kB Ndf g T0 T ðtÞ 1 γ_ ¼ Q Ndf T ðtÞ r€i ¼

(1.51) (1.52)

22

Chapter 1

In both algorithms, some care must be taken in the choice of the fictitious mass Q and extended system energy Ee. On one hand, too large values of Q (loose coupling) may cause a poor temperature control (Nos
e-Hoover thermostat with Q ! ∞ is MD that generates a microcanonical ensemble). Although any finite (positive) mass is sufficient to guarantee in principle the generation of a canonical ensemble, if Q is too large, the canonical distribution will only be obtained after very long simulation times. On the other hand, too small values (tight coupling) may cause high-frequency temperature oscillations. The variable s may oscillate at a very high frequency. It will tend to be off-resonance with the characteristic frequencies of the real system and effectively decouple from the physical degrees of freedom (slow exchange of kinetic energy). As a more intuitive choice for the coupling strength the Nos
e equations of motion can be expressed as   1 g T0 1 (1.53) γ_ ¼ τNH Ndf T ðtÞ with the effective relaxation time τ2NH ¼

Q Ndf kB T0

(1.54)

The relaxation time can be estimated when calculating the frequency of the oscillations for small deviations δs from the average s .

1.6.5 Boundary Conditions A system of 1-mol-order size, being composed of about 6 1023 particles, never needs to be directly treated in molecular simulations for thermodynamic equilibrium. The use of the periodic boundary condition, explained in the succeeding text, enables us to treat only a relatively small system of about 100–10,000 particles to obtain such reasonable results as to explain the corresponding experimental data accurately. (a) Periodic boundary condition Fig. 1.5 schematically illustrates the concept of the periodic boundary condition for a twodimensional system composed of spherocylinder particles. The central square is a simulation region, and the surrounding squares are virtual simulation boxes, which are made by replicating the main simulation box. As Fig. 1.5 shows, the origin of the xy-coordinate system is taken at the center of the simulation region, and the dimensions of the simulation region in the x- and y-directions are denoted by Lx and Ly. The two specific procedures are necessary in treating the periodic boundary condition. First is the treatment of outgoing particles crossing the boundary surfaces of the simulation region, and second is the calculation of interaction energies or forces with virtual particles being

Introduction to Molecular Dynamics 23

Ly

Lx

Fig. 1.5 Periodic boundary conditions.

in the replicated simulation boxes. As shown in Fig. 1.5 a particle crossing and exiting the left boundary surface has to enter from the right virtual box. When the interaction energy or force of particle i with other particles, for example, particle j, has to be calculated, an appropriate particle j has to be chosen as an object from real and virtual particles j. This may be done in such a way that the distance between particle i and particle j is minimal. (b) Lees-Edwards boundary condition The periodic boundary condition is quite useful for molecular simulations of a system in thermodynamic equilibrium, but this boundary condition is unsuitable for nonequilibrium situations. In treating the dynamic properties of a system in nonequilibrium the most basic and important flow is a simple shear flow, as shown in Fig. 1.6. The velocity profile, linearly varying from U at the lower surface to U at the upper one, can be generated by sliding the lower and upper walls in the left and right directions with the velocity U, respectively. This flow field is called the “simple shear flow.” In generating such a simple shear flow in actual molecular simulations the upper and lower replicated simulation boxes, shown in Fig. 1.5, are made to slide in different directions with a certain constant speed. This sliding periodic boundary condition is called the “Lees-Edwards boundary condition” [13]. Fig. 1.7 schematically depicts the concept of this boundary condition. The replicated boxes in the upper and lower layers slide in each direction by the distance ΔX. If particles move out of the simulation box by crossing the boundary surface normal to the x-axis, as shown in Fig. 1.7, they

24

Chapter 1 U

U

Fig. 1.6 Simple shear flow.

DX

Lx Fig. 1.7 Lees-Edwards boundary condition.

are made to come into the simulation box through the opposite boundary surface, which is exactly the same procedure as the periodic boundary condition. The important treatment in the Lees-Edwards boundary condition concerns the particles crossing the boundary surfaces normal to the y-axis. The same treatment of the periodic boundary condition is applied to the y-coordinate of such particles, but the x-coordinate should be shifted from x to (x ΔX) in the case of Fig. 1.7. In addition, the x-component of velocity vx of these particles needs to be modified to (vx U), but the y-component vy can be used without modification. For the case of evaluating interaction energies or forces, similar procedures have

Introduction to Molecular Dynamics 25 to be conducted for the particles interacting with virtual particles that are in the replicated simulation boxes in the upper or lower layers.

1.7 Molecular Dynamics Methodology If the mass of molecule i is denoted by mi and the force acting on molecule i by the ambient molecules and an external field denoted by fi, then the motion of a particle is described by Newton’s equation of motion: mi

d 2 ri ¼ fi dt2

(1.55)

If a system is composed of N molecules, there are N sets of similar equations, and the motion of N molecules interacts through forces acting among the molecules. Differential equations such as Eq. (1.55) are unsuitable for solving the set of N equations of motion on a computer. Computers readily solve simple equations, such as algebraic ones, but are quite poor at intuitive solving procedures such as a trial-and-error approach to find solutions. Hence Eq. (1.55) will be transformed into an algebraic equation. To do so the second-order differential term in Eq. (1.55) must be expressed as an algebraic expression, using the following Taylor series expansion: xðt + hÞ ¼ xðtÞ + h

dxðtÞ 1 2 d2 xðtÞ 1 3 d 3 xðtÞ + h +… + h dt 2! dt2 3! dt3

(1.56)

Eq. (1.56) implies that x at time (t + h) can be expressed as the sum of x itself, the first-order differential, the second-order differential, and so on multiplied by a constant for each term. If x does not significantly change with time, the higher-order differential terms can be neglected for a sufficiently small value of the time interval h. To approximate the second-order differential term in Eq. (1.55) as an algebraic expression, another form of the Taylor series expansion is necessary: xðt  hÞ ¼ xðtÞ  h

dxðtÞ 1 2 d 2 xðtÞ 1 3 d 3 xðtÞ  h +… + h dt 2! dt2 3! dt3

(1.57)

If the first-order differential term is eliminated from Eqs. (1.56), (1.57), the second-order differential term can be solved as
 d 2 xðtÞ xðt + hÞ  2xðtÞ + xðt  hÞ ¼ + O h2 2 2 dt h

(1.58)

The last term on the right-hand side of this equation implies the accuracy of the approximation, and in this case, terms higher than h2 are neglected, if the second-order differential is approximated as d2 xðtÞ xðt + hÞ  2xðtÞ + xðt  hÞ ¼ dt2 h2

(1.59)

26

Chapter 1

This expression is called the “central difference approximation.” With this approximation and the notation ri ¼ (xi, yi, and zi) for the molecular position and fi ¼ (fxi, fyi, and fzi) for the force acting on particle i the equation of the x-component of Newton’s equation of motion can be written as xi ðt + hÞ ¼ 2xi ðtÞ  xi ðt  hÞ +

h2 fxi ðtÞ mi

(1.60)

Similar equations are satisfied for the other components. Since Eq. (1.60) is a simple algebraic equation, the molecular position at the next time step can be evaluated using the present and previous positions and the present force [14]. If a system is composed of N molecules, there are 3N algebraic equations for specifying the motion of molecules; these numerous equations are solved on a computer, where the motion of the molecules in a system can be pursued with the time variable. Eq. (1.60) does not require the velocity terms for determining the molecular position at the next time step. This scheme is called the “Verlet method.” The velocity, if required, can be evaluated from the central difference approximation as vi ðtÞ ¼

ri ðt + hÞ  ri ðt  hÞ 2h

(1.61)

This approximation can be derived by eliminating the second-order differential terms in Eqs. (1.56), (1.57). It has already been noted that the velocities are unnecessary for evaluating the position at the next time step; however, a scheme using the positions and velocities simultaneously may be more desirable to keep the system temperature constant. If we take into account that the first- and second-order differentials of the position are equal to the velocity and acceleration, respectively, the neglect of differential terms equal to or higher than third-order differential in Eq. (1.56) leads to the following equation: ri ðt + hÞ ¼ ri ðtÞ + hvi ðtÞ +

h2 f ðtÞ 2mi i

(1.62)

This equation determines the position of the molecules, but the velocity term arises on the righthand side, so that another equation is necessary for specifying the velocity. The first-order differential of the velocity is equal to the acceleration: h (1.63) vi ðt + hÞ ¼ vi ðtÞ + f i ðtÞ mi To improve accuracy the force term in Eq. (1.63) is slightly modified and the following equation is obtained: h ðf ðtÞ + f i ðt + hÞÞ (1.64) vi ðt + hÞ ¼ vi ðtÞ + 2mi i The scheme of using Eqs. (1.62), (1.64) for determining the motion of molecules is called the “velocity Verlet method.” It is well known that the velocity Verlet method is significantly

Introduction to Molecular Dynamics 27 superior in regard to the stability and accuracy of a simulation. Consider another representative scheme. Noting that the first-order differential of the position is the velocity and that of the velocity is the acceleration, the application of the central difference approximation to these first-order differentials leads to the following equations: ri ðt + hÞ ¼ ri ðtÞ + hvi ðt + h=2Þ vi ðt + h=2Þ ¼ vi ðt  h=2Þ +

h f ðtÞ mi i

(1.65) (1.66)

The scheme of pursuing the positions and velocities of the molecules with Eqs. (1.65), (1.66) is called the “leapfrog method.” This name arises from the evaluation of the positions and forces and then the velocities, by using time steps in a leapfrog manner. This method is also a significantly superior scheme in regard to stability and accuracy, comparable with the velocity Verlet method. The MD method is applicable to both equilibrium and nonequilibrium physical phenomena, which makes it a powerful computational tool that can be used to simulate many physical phenomena if computing power is sufficient. The main procedure for conducting the MD simulation using the velocity Verlet method has the following steps: (i) (ii) (iii) (iv) (v)

Specify the initial position and velocity of all molecules. Calculate the forces acting on molecules. Evaluate the positions of all molecules at the next time step from Eq. (1.62). Evaluate the velocities of all molecules at the next time step from Eq. (1.64). Repeat the procedures from Step 2.

In this procedure, positions and velocities will be evaluated at every time interval h in the MD simulation. The method of specifying the initial positions and velocities is as follows.

1.7.1 Initial Positions To develop a simulation program, it is necessary to have an overview of the general methodology, which should include the assignment of the initial configuration and velocities, the treatment of boundary conditions, and techniques for reducing computation time. An appropriate initial configuration has to be set with careful consideration given to the physical property of interest, so that the essential phenomena can be grasped. For example, if nonspherical molecules or particles are known to incline in a preferred direction, there may be some advantages to using a parallelepiped rectangular simulation region rather than a cubic one. The periodic boundary condition is a representative model to manage the boundary of a simulation region. It is almost always used for systems in thermodynamic equilibrium. On the other hand, for investigating the dynamic properties of a system, the simple shear flow is frequently treated, and in this case the Lees-Edwards boundary condition is available. Techniques for reducing computation time become very important in large-scale threedimensional simulations, and methods of tracking particle neighbors, such as the cell index method, are indispensable.

28

Chapter 1

(a) Spherical systems Setting an initial configuration of particles is an indispensable procedure for both MD and MC methods. Although it is possible to assign randomly the initial position of particles in a simulation region, a regular configuration, such as a simple cubic lattice or a face-centered cubic lattice, is handled in a more straightforward manner. The random allocation suffers from the problem of the undesirable overlap of particles and from possible difficulties in achieving high packing fractions. Lattice assignments are almost free from the overlap problem and can achieve high packing fractions. However, the lattice packing may be too perfect for some simulations, requiring the adjustment of a small random perturbation. In the following paragraphs the method of setting the initial configuration in a regular lattice formation for a two-dimensional configuration has been discussed, which will be followed by a threedimensional configuration. Fig. 1.8 shows several lattice systems that may be used to assign an initial configuration for a two-dimensional system. A basic lattice form is expanded to fill the whole simulation region, and particles are then located at each lattice point. y

y

a

a x

x a

a y

2a a x 3a

Fig. 1.8 Initial conditions for a two-dimensional system.

Introduction to Molecular Dynamics 29 Fig. 1.8A, the simplest lattice model, may be suitable for a gaseous system. However, even if the particle-particle distance “a” is equal to the particle diameter, a high packing fraction cannot be obtained by using this simple lattice model. Hence it is inappropriate for the simulations of a liquid or solid system. Since there is only one particle in the unit cell shown in Fig. 1.8A, a system with total particle number N (¼ Q2) can be generated by replicating the unit cell (Q1) times in each direction to make a square simulation region of side length L ¼ Qa. So for the use of this lattice system as the initial configuration the particle number N has to be taken from N ¼ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. The number density of particles n is given by n ¼ N/L2, and the area fraction is given by Φs ¼ π(d/2)2 N/L2, where d is the particle diameter. In practice the number of particles N and the area fraction Φs are first chosen; then the values of Q and L are evaluated from which the value of “a” can be determined. With these values the initial configuration of particles can be assigned according to the simple lattice system shown in Fig. 1.8A. The lattice system shown in Fig. 1.8B can yield a higher packing fraction and therefore may be applicable for an initial configuration of a gaseous or liquid state, but it has limited application to a solid state. Since there are two particles in the unit cell of this lattice, a system with total particle number N ¼ 2Q2 can be generated by replicating the unit cell (Q1) times in each direction. In this case the simulation region is also a square of side length L ¼ Qa, and the possible value of N is taken from 2, 8, 18, 32, 50, 72, 98, 128, 162, 200, and so on. The number density of particles n is given by n ¼ N/L2, and the area fraction Φs is given by Φs ¼ π(d/2)2 N/L2. Fig. 1.8C shows the most compact lattice for a two-dimensional system. This lattice model may also be applicable to a solid system. If the dark particles are assumed to constitute the unit lattice, it follows that there are four particles in this unit lattice. Hence by replicating the unit lattice (Q1) times in each direction the simulation region becomes a rectangle of side lengths Lx ¼ 31/2aQ and Ly ¼ 2aQ, with a total number of particles N ¼ 4Q2, where the possible value of N is taken from 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, and so on. The particle number density n is given by n ¼ N/LxLy, and the area fraction Φs is given by Φs ¼ π(d/2)2 N/LxLy. The actual assignment of the abovementioned quantities for simulations is similar to that for Fig. 1.8A. Fig. 1.9 shows several lattice models for a three-dimensional system. Fig. 1.9A is the simple cubic lattice model, which is suitable as an initial configuration mainly for a gaseous or liquid system. Since there is only one particle in the unit cell, the number of particles in a system is given by N ¼ Q3 by replicating the unit cell (Q1) times in each direction. In this case the possible value of N is taken from N ¼ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, and so on. The simulation region is a cube of side length L ¼ Qa. The number density n and the volumetric fraction ΦV are given by n ¼ N/L3 and ΦV ¼ 4π(d/2)3 N/3L3, respectively. The facecentered cubic lattice model shown in Fig. 1.9B is one of the close-packed lattices and therefore may be applicable as an initial configuration of a solid state. Since there are four particles in the unit cell, the total number of particles in the simulation region is given by N ¼ 4Q3 by replicating the unit cell (Q1) times in each direction. In this case the total number of particles

30

Chapter 1

Fig. 1.9 Initial positions for a three-dimensional system.

is taken from N ¼ 4, 32, 108, 256, 500, 864, 1372, and so on. The number density and the volumetric fraction are given by n ¼ N/L3 and ΦV ¼ 4π (d/2)3 N/3L3, respectively. As in a twodimensional system, for the actual assignment of the abovementioned quantities, the particle number N and the volumetric fraction ΦV are first chosen, then Q and L are evaluated, and finally the lattice distance “a” is determined. For a gaseous or liquid system the simple lattice models shown in Figs. 1.8A and 1.9A are applicable in a straightforward manner for developing a simulation program. In contrast, for the case of a solid system, the choice of an appropriate lattice used for the initial configuration of particles is usually determined by the known physical properties of the solid. (b) Nonspherical systems For a nonspherical particle system the orientation of the particles must be assigned in addition to their position, so that the technique for setting the initial configuration is a little more difficult than that for a spherical particle system. For this purpose a versatile technique whereby a wide range of initial configurations can be assigned is desirable. If particle-particle interactions are

Introduction to Molecular Dynamics 31

z

y

y

x z

x

Fig. 1.10 Initial conditions for spherocylinder particles.

large enough to induce the cluster formation of particles in a preferred direction, then an appropriately large initial configuration has to be adopted for the simulation to capture such characteristic aggregate structures. We consider the example of a system composed of spherocylinder particles, as shown in Fig. 1.10, with a magnetic moment at the particle center normal to the long particle axis. The spherocylinder is a cylinder with hemisphere caps at both ends. An ensemble of these particles can be expected to aggregate to form raft-like clusters with the magnetic moments inclining in the applied magnetic field direction. Hence a simulation region with sufficient length in the direction of the cluster formation has to be taken for the simulation particles to aggregate in a reasonable manner. The spherocylinder can be characterized by the ratio of the particle length l to the diameter d of the cylindrical part, known as the aspect ratio rp ¼ l/d. For Fig. 1.10 where rp ¼ 3 the particles are placed in contact with three and nine rows in the x- and y-directions, respectively, leading to a configuration of 27 particles in a square region in the xy-plane. Extending this configuration to 18 layers in the z-direction yields an initial configuration of spherocylinder particles with a simulation region (Lx, Ly, and Lz) ¼ (3rpd, 3rpd, and 6rpd) with total number of particles N ¼ 486; if four rows are arranged in the x-direction, then a simulation region larger than the present case can be adopted with a simulation region (Lx, Ly, and Lz) ¼ (4rpd, 4rpd, and 8rpd). If the particle-particle distances are expanded equally in each direction to yield a desired volumetric fraction of particles ΦV, then this expanded system may be used as an initial configuration for simulations. Such an expansion with a factor α of particle-particle distances

32

Chapter 1

Fig. 1.11 Aggregation for spherocylinder particles in (A) short axis direction and (B) long axis direction.

gives rise to the system volume V ¼ 54r3pd3α3. The volumetric fraction ΦV is related to the system volume as ΦV ¼ NVp/V in which Vp is the volume of a spherocylinder particle, expressed as Vp ¼ πd3(3rp 1)/12. From these expressions the expansion ratio α can be obtained as  1 
1 3π 3rp  1 3 α¼ rp 4ΦV

(1.67)

This initial configuration is applicable for a system in which particles are expected to aggregate in the direction of the particle short axis, as shown in Fig. 1.11A. If particles are expected to aggregate in the direction of the particle long axis, as shown in Fig. 1.11B, it is straightforward to follow a similar procedure with the spherocylinder particles aligned in the z-direction in Fig. 1.10. The main procedure for setting the initial configuration is summarized as follows: (i) Consider an appropriate initial configuration, with sufficient consideration given to the physical phenomenon of interest. (ii) Set a nearly close-packed situation as an initial configuration. (iii) Calculate the total number of particles N. (iv) Evaluate the expansion ratio α from Eq. (1.67) to give rise to the desired volumetric fraction ΦV. (v) Expand particle-particle distances equally by the factor α. (vi) Perturb the particle positions by small distances using random numbers to destroy the regularity of the initial configuration; otherwise, all particle-particle interactions may be zero, and therefore the particles may not move with time.

Introduction to Molecular Dynamics 33

1.7.2 Initial Velocities There is a difference in the method of assigning the initial velocities for spherical and nonspherical systems. For nonspherical systems, we have to assign angular velocities in addition to translational velocities. Methods for assigning the initial velocities for these systems have been discussed in the succeeding text. (a) Spherical systems In the MD method the motion of particles is described by pursuing their position and velocity over time, so these factors have to be specified as an initial condition. If the system of interest is in thermodynamic equilibrium with temperature T, the particle velocities are described by the following Maxwellian distribution [15]: n m  o  m 3=2 (1.68) exp  v2ix + v2iy + v2iz f ðvi Þ ¼ 2πkT 2kT in which k is Boltzmann’s constant, m is the mass of particles, and vi ¼ (vix, viy, viz) is the velocity vector of particle i. Since the Maxwellian distribution f is the probability density distribution function, the probability of particle i being found in the infinitesimal velocity range between vi and (vi + dvi) becomes f(vi)dvi. Characteristics of this function can be understood more straightforwardly by treating the distribution function fx as the x-velocity component. The probability density distribution function χ(vi) for the speed vi ¼ (v2ix + v2iy + v2iz) of particle i can be derived from Eq. (1.68) as n m o  m 3=2 (1.69) v2i exp  v2 χ ðvi Þ ¼ 4π 2πkT 2kT i For a given system temperature T the initial velocities of particles for simulations can be assigned according to the probability density function in Eq. (1.68) or (1.69). To set the initial velocities of particles in MD simulations or to generate random displacements in BD and DPD simulations, it is necessary to generate random numbers according to a particular probability distribution. The probability distributions of interest here are the Gaussian distribution (also known as the normal distribution) and the Maxwell-Boltzmann distribution (or Maxwellian distribution). Since the velocity of particles theoretically has the Maxwellian velocity distribution for thermodynamic equilibrium, the initial velocity of particles in simulations must have such a velocity distribution. The method of setting the initial velocity of particles according to the Maxwellian distribution has been discussed in the following paragraph. It has been assumed that the stochastic variable x, such as the particle velocity or a random displacement, obeys the following normal distribution ρ(x): ( ) 1 ðx  xÞ2 (1.70) exp  ρðxÞ ¼ 2σ 2 ð2π Þ1=2 σ

34

Chapter 1

in which σ 2 is the variance and x is the average of the stochastic variable x. To generate the stochastic variable x according to this normal distribution, the following equations are used together with a uniform random number sequence ranging from zero to unity:
1=2
1=2 cos ð2πR2 Þ or x ¼ x + 2σ 2 ln R1 sin ð2πR2 Þ x ¼ x + 2σ 2 ln R1

(1.71)

According to either equation of Eq. (1.71), the required number of values of the stochastic variable are generated using a series of random numbers, such as R1 and R2, taken from a uniform random number sequence. In this way the initial velocities of particles and random displacements can be assigned. The technique in Eq. (1.71) is called the Box-Muller method [16]. For generating a uniform random number sequence, there is an arithmetic method and a machine-generated method. The arithmetic method is reproducible, and the same random number sequence can be obtained at any time in the simulations. In contrast the machinegenerated method is generally not a reproducible sequence, and a different sequence of random numbers is generated each time a simulation is run. For the case of the Maxwellian velocity distribution, the velocity components of particle i can be assigned using the random numbers R1, R2, …, R6 taken from a uniform random number sequence as 

1=2   kT vix ¼ 2 cos ð2πR2 Þ lnR1 m    1=2 kT cos ð2πR4 Þ lnR3 viy ¼ 2 m

(1.72)

   1=2 kT cos ð2πR6 Þ viz ¼ 2 lnR5 m In this way, all the initial velocity components can be assigned using random numbers. Each particle requires a new, that is, a different, set of random numbers. The temperature that is evaluated from the initial particle velocities assigned by the abovementioned method is approximately equal to the desired system temperature, but may not necessarily be satisfactory. Hence an equilibration procedure is usually necessary before starting the main loop in an actual simulation program. This will be explained in the section that follows. (b) Nonspherical systems For a nonspherical particle system the initial angular velocities need to be assigned in addition to the translational velocities. Similar to the translational velocity v ¼ (vx, vy, and vz) discussed

Introduction to Molecular Dynamics 35 earlier the angular velocity ω ¼ (ωx, ωy, and ωz) is also governed by the Maxwellian distribution fω (ω). The expression for fω (ω) is   3=2  I I  2 2 2 (1.73) exp  f ω ðωÞ ¼ ω + ωy + ωz 2πkT 2kT x where I is the inertia moment of a particle. The characteristics of the exponential function in Eq. (1.68) or (1.73) demonstrate that the probability of particles appearing with larger translational and angular velocities increases with the system temperature. The method for setting the initial translational velocities using uniform random numbers, explained in the previous subsection, is applicable to the present angular velocity case. Here, m and (vix, viy, and viz) in Eq. (1.72) are replaced by I and (ωix, ωiy, and ωiz). New uniform random numbers need to be used for each particle.

1.8 Molecular Potential Energy Surface The complete mathematical description of a molecule, including both quantum mechanical and relativistic effects, is a formidable problem, due to the small scales and large velocities. However, for this discussion, these intricacies are ignored, and the focus is on general concepts, because molecular mechanics and dynamics are based on empirical data that implicitly incorporate all the relativistic and quantum effects. Since no complete relativistic quantum mechanical theory is suitable for the description of molecules, we start with the nonrelativistic, time-independent form of the Schr€ odinger [17] description: H ψ ðR, rÞ ¼ E ψ ðR, r Þ

(1.74)

where H is the Hamiltonian for the system, ψ is the wave function, and E is the energy. In general, ψ is a function of the coordinates of the nuclei (R) and of the electrons (r). Although the Eq. (1.74) is quite general, it is too complex for any practical use, so approximations are made. Noting that the electrons are several thousands of times lighter than the nuclei and therefore move much faster, Born and Oppenheimer [18] proposed what is known as the Born-Oppenheimer approximation: the motion of the electrons can be decoupled from that of the nuclei, giving two separate equations. The first equation describes the electronic motion or the potential energy surface: H ψ ðr; RÞ ¼ E ψ ðr; RÞ

(1.75)

This depends only parametrically on the positions of the nuclei. Note that this equation defines energy E(R), which is a function of only the coordinates of the nuclei. This energy is usually called the potential energy surface. The second equation then describes the motion of the nuclei on this potential energy surface E(R): H ϕ ðRÞ ¼ E ϕ ðRÞ

(1.76)

36

Chapter 1

Solving Eq. (1.76) is important if we are interested in the structure or time evolution of a model. Eq. (1.76) is the Schr€ odinger equation for the motion of the nuclei on the potential energy surface. In principle, Eq. (1.75) can be solved for the potential energy E, and then, Eq. (1.76) could be solved. However, the effort required to solve Eq. (1.75) is extremely large, so usually an empirical fit to the potential energy surface, commonly called a force field, is used. Since the nuclei are relatively heavy objects, quantum mechanical effects are often insignificant in which case Eq. (1.76) can be replaced by Newton’s equation of motion: 

dV d2 R ¼m 2 dR dt

(1.77)

The solution of Eq. (1.77) using an empirical fit to the potential energy surface E(R) is called MD. Molecular mechanics ignores the time evolution of the system and instead focuses on finding particular geometries and their associated energies or other static properties. This includes finding equilibrium structures, transition states, relative energies, and harmonic vibrational frequencies. Materials’ properties result from quantum mechanical interactions described by solutions of a Schr€ odinger equation. Solving the full many-body problem was first addressed by Hohenberg, Kohn, and Sham (HKS) [19,20] who formulated a theory known as “density functional theory” (DFT), where electrons are replaced by effective electrons with the same total density moving in the potential generated by the other electrons and ion cores. DFT is a quantum mechanical modeling method used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory the properties of a many-electron system can be determined by using functionals, that is, functions of another function, which in this case is the spatially dependent electron density. Hence the name DFT comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensedmatter physics, computational physics, and computational chemistry. DFT was put on a firm theoretical footing by the two Hohenberg-Kohn (H-K) theorems. The original H-K theorems held only for nondegenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these. The first H-K theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to three spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the time-dependent domain to develop time-dependent DFT (TDDFT), which can be used to describe excited states. The second H-K theorem defines energy functional for the system and proves that the correct ground-state electron density minimizes this energy functional.

Introduction to Molecular Dynamics 37 Within the framework of Kohn-Sham DFT (KS DFT) the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons. Modeling the interactions becomes the difficulty within KS DFT. The simplest approximation is the local density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas-Fermi model, and from fits to the correlation energy for a uniform electron gas. Noninteracting systems are relatively easy to solve as the wave function can be represented as a Slater determinant of orbitals. In a system of N electrons, the total density is expressed as a sum of the orbitals ψ i as ρðr Þ ¼

N X

jψ i ðrÞj2

(1.78)

i¼1

where ψ i are solutions of the Kohn-Sham equations   ħ2 2  r + Veff ψ i ðrÞ ¼ εi ψ i ðrÞ 2m

(1.79)

where ℏ is the reduced Planck constant; Veff is the effective potential due to Coulomb and exchange-correlation contributions; and εi is the orbital energy of the corresponding KohnSham orbital, ψ i. However, the HKS theory usually called DFT does not provide an algorithm of determining this functional. The solution came from the LDA that assumes that the functional depends only on the local electron density. LDA is an accurate approximation for systems with slowly varying charge densities, like many metals; however, it has some limitations in systems far from equilibrium and for nonuniform charge densities. Development of efficient algorithms has made DFT the primary method for calculating properties such as band structures, cohesive energies, and activation barriers. However, DFT is inefficient if the atoms are allowed to move, for example, during the geometric optimization of a molecule.

References [1] W.D. Cornell, P. Cieplak, C.I. Bayly, et al., A second generation force field for the simulation of proteins, nucleic acids, and organic molecules, J. Am. Chem. Soc. 117 (1995) 5179–5197. [2] H. Sun, COMPASS: an ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds, J. Phys. Chem. 5647 (98) (1998) 7338–7364. [3] B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, M. Karplus, CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem. 4 (2) (1983) 187–217. [4] P. Dauber-Osguthorpe, V.A. Roberts, D.J. Osguthorpe, J. Wolff, M. Genest, A.T. Hagler, Structure and energetics of ligand binding to proteins: Escherichia coli dihydrofolate reductase-trimethoprim, a drugreceptor system, Proteins Struct. Funct. Genet. 4 (1988) 31–47.

38

Chapter 1

[5] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B 37 (1988) 6991. [6] D.W. Brenner, Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films, Phys. Rev. B 42 (15) (1990) 9458. [7] P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels, Phys. Rev. 34 (1929) 57–64. [8] J.E. Lennard-Jones, On the determination of molecular fields, Proc. R. Soc. Lond. A 106 (738) (1924) 463–477. [9] H.C. Andersen, Molecular dynamics simulations at constant pressure and/or temperature, J. Chem. Phys. 72 (4) (1980) 2384. [10] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, J.R. Haak, Molecular-dynamics with coupling to an external bath, J. Chem. Phys. 81 (8) (1984) 3684–3690. [11] S. Nos
e, A unified formulation of the constant temperature molecular-dynamics methods, J. Chem. Phys. 81 (1) (1984) 511–519. [12] W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A 31 (3) (1985) 1695–1697. [13] A.W. Lees, S.F. Edwards, The computer study of transport processes under extreme conditions, J. Phys. C Solid State Phys. 5 (1972) 1921. [14] A. Satoh, Introduction to Practice of Molecular Simulation, first ed., Elsevier, London, 2011 (Chapter 2). [15] J.C. Maxwell, Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres, The London, Edinburgh, and Dublin, Philos. Mag. J. Sci. 19 (4) (1860) 19–32. [16] G.E.P. Box, M.E. Muller, A note on the generation of random normal deviates, Ann. Math. Stat. 29 (2) (1958) 610–611. [17] E. Schr€odinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev. 28 (6) (1926) 1049–1070. [18] M. Born, J.R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 389 (20) (1927) 457–484. [19] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (3B) (1964) B864–B871. [20] W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (4A) (1965) A1133–A1138.

CHAPTER 2

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS This chapter has been divided into three parts as follows: Chapter 2.1: Overview of BIOVIA Materials Studio Chapter 2.2: Overview of LAMMPS Chapter 2.3: Overview of GROMACS

Chapter 2.1

Overview of BIOVIA Materials Studio Sumit Sharma, Pramod Kumar, Rakesh Chandra Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India

BIOVIA Materials Studio [1] is software for simulating and modeling materials. It is developed and distributed by BIOVIA (formerly Accelrys), a firm specializing in research software for computational chemistry, bioinformatics, cheminformatics, molecular dynamics simulation, and quantum mechanics. This software is used in advanced research of various materials, such as polymers, carbon nanotubes, catalysts, metals, and ceramics, by top universities, research centers, and high-tech companies. BIOVIA Materials Studio is a client-server model software package with Microsoft Windows-based PC clients and Windows- and Linux-based servers running on PCs, Linux IA-64 workstations, and HP XC clusters. BIOVIA Materials Studio [1] offers a rich set of features for material modeling at the atomic level. There are various modules available in BIOVIA Materials Studio for modeling, visualization, and analysis of material systems. Its graphic user interface is very clear and intuitive, which offers a high-quality, Windows-standard environment into which we can plug any BIOVIA Materials Studio product. In the following sections, various modules of BIOVIA Materials Studio have been discussed along with suitable examples wherever required. Molecular Dynamics Simulation of Nanocomposites using BIOVIA Materials Studio, Lammps and Gromacs https://doi.org/10.1016/B978-0-12-816954-4.00002-4 # 2019 Elsevier Inc. All rights reserved.

39

40

Chapter 2

2.1.1 Modules The various modules of BIOVIA Materials Studio have been discussed here. The readers can easily follow these steps for modeling in BIOVIA Materials Studio: (a) Materials Visualizer Materials Visualizer provides fast, interactive tools that enable you to construct graphic models of molecules, crystalline materials, surfaces, interfaces, layers, and polymers. You can manipulate, view, and analyze these models. Materials Visualizer also handles graph, tabular, and textual data and provides the software infrastructure and analysis tools to support the full range of BIOVIA Materials Studio products. Materials Visualizer can be run as a standalone tool for building, visualizing, and editing structures. Interaction with Windows productivity tools allows easy sharing and reporting of results and data. (b) Adsorption Locator Adsorption Locator enables us to simulate a substrate loaded with an adsorbate or an adsorbate mixture of a fixed composition. Adsorption Locator is designed for the study of individual systems, allowing you to find low-energy adsorption sites on both periodic and nonperiodic substrates or to investigate the preferential adsorption of mixtures of adsorbate components, for example. Adsorbates are typically molecular gases or liquids, and substrates are usually porous crystals or surfaces, such as zeolites or CNTs, or amorphous structures, such as silica gel or activated carbon. Adsorption Locator identifies possible adsorption configurations by carrying out Monte Carlo searches of the configurational space of the substrate-adsorbate system as the temperature is slowly decreased. (c) Amorphous Cell Amorphous Cell is a suite of computational tools that allow us to construct representative models of complex amorphous systems and to predict key properties. Among the properties that we can predict and investigate are cohesive energy density, equation-of-state behavior, chain packing, and localized chain motions. The methodology of Amorphous Cell construction is based on an extension of well-established methods for generating bulk-disordered systems containing chain molecules in realistic equilibrium conformations. Other features include provision for construction of arbitrary mixture systems containing any combination of small molecules and polymers, in addition to special capabilities for producing ordered nematic mesophases and slabs of amorphous material suitable for use in creating models of interphases, as would be required to study adhesion or lubrication. (d) COMPASS COMPASS is a powerful force field supporting atomistic simulations of condensed-phase materials. COMPASS stands for condensed-phase optimized molecular potentials for atomistic

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 41 simulation studies. COMPASS is the first ab initio force field that has been parameterized and validated using condensed-phase properties, in addition to various ab initio and empirical data for molecules in isolation. Consequently, this force field enables accurate and simultaneous prediction of structural, conformational, vibrational, and thermophysical properties for a broad range of molecules in isolation and in condensed phases and under a wide range of conditions of temperature and pressure. The COMPASS force field is under continuous review and is updated frequently. The most recent version is denoted as COMPASS on the user interface dialogs. (e) COMPASS II COMPASS II is a significant extension to the COMPASS force field. COMPASS II extends the existing coverage of COMPASS to include a significantly larger number of drugs and compounds of interest to researchers investigating ionic liquids. COMPASS II was developed in collaboration with Prof. Huai Sun of Jiao Tong University, Shanghai. Using tools developed by Prof. Sun the existing COMPASS force field was combined with existing quantum mechanical calculations to obtain new parameters and to resolve a small number of inconsistencies in the existing COMPASS force field. New valance parameters were derived by comparison of equilibrium structures and vibrational frequencies. The training set used to determine these parameters was extended as follows: (i) A polymer database consisting of 430 homo- and copolymers was analyzed using COMPASS, and 454 missing terms were found, represented by 105 fragments. These species were added to the training set. (ii) The NIST ionic liquid molecule database was analyzed in a similar way, and 40 species were added to the training set. (iii) Maybridge database consisting of 59,465 molecules was scanned using COMPASS, and 1257 fragments were added to the training set. (f) Forcite Forcite is a molecular mechanics module for potential energy and geometry optimization calculations of arbitrary molecular and periodic systems using classical mechanics. Forcite offers support for the COMPASS, UFF, and DREIDING force fields. With this wide range of force fields, Forcite can handle essentially any material. The geometry optimization algorithm offers steepest descent, conjugate gradient, and quasi-Newton methods, in addition to the Smart algorithm, which uses these methods sequentially. This allows very accurate energy minimizations to be performed. The Forcite module allows us to perform a wide range of molecular mechanic calculations on both molecular and periodic systems using classical force field-based simulation techniques. Forcite can perform many different tasks such as singlepoint energy calculation, geometry optimization, molecular dynamics, quench dynamics, anneal dynamics, shear, confined shear, cohesive energy density calculation, mechanical property calculation, and solvation free energy calculation.

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The Forcite energy task allows us to calculate the total energy of the specified system and report contributions to the energy from different force field terms, such as bond and van der Waals terms. The energy task is useful for assessing whether a force field is applicable to a given system. It can also indicate if the structure has a reasonable geometry and, hence, if a geometry optimization should be performed. The Forcite geometry optimization task allows us to refine the geometry of a structure until it satisfies certain specified criteria. This is done using an iterative process in which the atomic coordinates, and possibly the cell parameters, are adjusted until the total energy of the structure is minimized. In general, therefore, the optimized structure corresponds to a minimum in the potential energy surface. Forcite geometry optimization is based on reducing the magnitude of calculated forces and (where appropriate) stresses until they become smaller than defined convergence tolerances. It is also possible to specify an external model to represent the behavior of the system under tension or compression. The forces on an atom are calculated from the potential energy expression and will, therefore, depend on the force field that is selected. For crystal structures, Forcite geometry optimization honors any symmetry elements defined for the system. The optimization therefore amounts to a constrained optimization with a reduced number of degrees of freedom. The Forcite dynamics task allows you to simulate how the atoms in a structure will move under the influence of computed forces. Before performing a Forcite dynamics calculation a thermodynamic ensemble should be selected, and the associated parameters such as the simulation time step and the simulation temperature should be chosen. It is often difficult to perform experiments for fixed NVE conditions, so making comparisons between experiment and simulation is difficult. Application of classical Legendre transforms allows alternative ensembles to be derived, an example being the canonical, or NVT, ensemble, where a system can exchange heat with the environment and so remain at constant temperature (T). Such an ensemble can be simulated by altering Newton’s equations via the application of a thermostat. Forcite offers a choice of five thermostats: velocity scale, Nos
e, Nos
e-Hoover-Langevin, Andersen, and Berendsen. The time step is an important parameter in an integration algorithm. To make the best use of computation time, a large time step should be used. However, if the time step is too large, it may lead to instability and inaccuracy in the integration process. Typically, this is manifested as a systematic drift in the constant of motion, but it can also lead to the job failing unexpectedly due to a large energy deviation between steps. The Forcite mechanical property task allows us to calculate mechanical properties for a single structure or a trajectory of structures. A Forcite mechanical property calculation may be performed on either a single structure or a series of structures generated, for example, by a dynamics run and stored in a trajectory file (.arc, .his, .trj, and .xtd). The mechanical properties are then calculated in the succeeding text, averaged over all valid configurations, and reported in the output text document.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 43

2.1.2 Simulation Strategy Here, we have explained the general simulation strategy for a CNT-reinforced polymer composite using BIOVIA Materials Studio. The same strategy can be extended for modeling other types of fiber-reinforced composites. The general simulation strategy used in BIOVIA Materials Studio is highlighted in the succeeding text: 1. The CNT is built by using the “Build” nanostructure tool in MS 7.0. Here, we can specify the configuration of the CNT, that is, whether it is armchair (n,n) or zigzag (n,0) or chiral (n,m). 2. Then using “Build symmetry” option under “Build” tool, we increase the length of the CNT to the desired value by entering a numerical value in the options provided. For example, when we click the build symmetry option, we will be asked to enter three values of a, b, and c. Enter the value in “c,” and press enter. 3. A CNT will appear on the screen. 4. To make a polymer, we first model a monomer on the software. This can be done as shown in the succeeding text. 5. Creating and using repeat units. The repeat unit represents the structure of the monomer after it is chemically bonded into a polymer. In BIOVIA Materials Studio the head of the repeat unit is highlighted with a cyan wireframe, and the tail is highlighted with a magenta wireframe. When the polymer is constructed, terminal dangling bonds are replaced with the original atoms (usually hydrogens) or terminal repeat units. 6. Fragments and repeat units. A repeat unit can be created from any fragment that is either sketched or imported into BIOVIA Materials Studio. A repeat unit is different from a fragment in that it has a defined head, tail, and backbone. Initiators and terminators are special types of repeat units. They have a head atom and a backbone defined but no designated tail. The initiators and terminators will bond to the existing polymer chain at the endpoints, removing their head atoms in the process. To create a repeat unit from a fragment: (a) Sketch a fragment or open a 3-D atomistic document containing one. (b) Choose Build j Build Polymers j Repeat Unit from the menu bar to open the Repeat Unit dialog. (c) Select the desired head atom and click the Head Atom button. A cyan wireframe is displayed around the atom to indicate that it is the head atom. (d) Select a different atom and click the Tail Atom button. A magenta wireframe is displayed around the atom to indicate that it is the tail atom. A set of atoms are now selected; this is the automatically defined backbone. Whenever you define a head and a tail, the backbone is automatically defined as the shortest connecting path between the two atoms. To modify

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the backbone path, select a set of atoms, and click the Set Backbone Atoms button. The Clear Backbone Atoms button removes all of the selected atoms from the backbone definition. (e) Select a pseudo-chiral backbone atom, with different nonbackbone substituents, and click the Chiral Center button to assign the atom as a chiral center. Chiral inversion and tacticity pertain to repeat units that contain pseudo-chiral centers. 7. Existing and sketched repeat units. Simple polymer chains can be constructed from repeat units stored in repeat unit libraries or from sketched repeat units. To use an existing repeat unit to build a polymer chain: (a) Choose Build j Build Polymers j Homopolymer from the menu bar to open the Homopolymer dialog. (b) Select a repeat unit library from the Library dropdown list. (c) Select a repeat unit from the Repeat unit dropdown list. (d) Set other construction parameters on the Polymerize and Advanced tabs. (e) Click the Build button to create the polymer chain. The new chain will be displayed in a new 3D Atomistic Document. To use a user-defined repeat unit to build a polymer chain: (a) With the desired repeat unit displayed in the 3D Viewer, choose Build j Build Polymers j Homopolymer from the menu bar to open the Homopolymer dialog. (b) Select the Current project entry in the Library dropdown list. This indicates that you want to use a repeat unit that is in your project, rather than one of the repeat units in the set of libraries that is provided. (c) Select the name of the document that contains your repeat unit in the Repeat unit dropdown list. If the document you have selected does not contain a valid repeat unit, you will be notified. (d) Set any construction parameters as earlier. (e) Click the Build button to construct the polymer chain. The constructed chain will be displayed in a new 3D Atomistic Document. 8. After modeling the polymer and CNT, the next step is to pack the polymer around the CNT. This is done using the “Amorphous Cell” module. Under “Amorphous Cell,” click on the “packing” option, keeping the “quality” as “fine.” Specify the required density. Specify the “force field” under “energy” option. Specify the file name of the polymer that is to be packed outside the CNT by selecting from the pull-down menu under “Option” in “Amorphous Cell.” Finally, click “Run.” At the end of this step, we will get a periodic cell in which polymer will be packed around the CNT.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 45 9. Next step is performing the geometry optimization using the “Forcite module.” There is “More” option in the Forcite module. On clicking this option, we can specify the type of algorithm that is to be used for optimization, and we also specify the number of iterations for which the simulation will run. Parameters for optimization of the geometry have been listed in every chapter. Run the optimization till the energy is minimized. 10. After geometry optimization, next step is “Dynamics” run. Again go to the “Forcite” module and select “dynamics” run instead of geometry optimization. Clicking the “More” option will open a new window in which we specify the ensemble, velocities, temperature, time step, and total number of steps. Select the quality as “fine” and click “run.” After the completion of this step, we can view the energy plot and the temperature plot to see whether the system has been stabilized or not. 11. Next step is calculating the mechanical properties. Under “Forcite” module, select “mechanical property” task. Specify the number of steps and maximum strain amplitude. Here, we can again specify whether we want to optimize our structure or not. Click “Run” to begin the task of mechanical property calculation. At the end of this step, we will get a stiffness matrix and the values of Young’s moduli. The whole procedure can be represented as a flow diagram shown in Fig. 2.1.1. In molecular dynamics, mutual atomic interactions are described by force potentials associated with bonding and nonbonding phenomena. The interatomic potential energy is the sum of bonding energy and nonbonding energy: U ¼ Ubonding + Unonbonding

(2.1.1)

For CNTs, the nonbonding term is mainly the energy of van der Waals force, which normally has a weak influence on the overall mechanical behavior among the atomic interactions of the carbon microstructure. The dominant part of the total potential energy, the bonding energy, is a

Fig. 2.1.1 Flowchart of simulation strategy.

46

Chapter 2

sum of three different interactions among atoms: bond stretching, bond bending, and bond torsion: Ubonded ¼ Ubondstretch + Uanglebend + Utorsion

(2.1.2)

Here, Ubondstretch ¼

Xh

k2 ðb  b0 Þ2 + k3 ðb  b0 Þ3 + k4 ðb  b0 Þ4

i (2.1.3)

b

Uanglebend ¼

Xh

k2 ðθ  θ0 Þ2 + k3 ðθ  θ0 Þ3 + k4 ðθ  θ0 Þ4

i (2.1.4)

θ

Utorsion ¼

Xh

i k1 ð1  cos∅Þ + k2 ð1  cos 2∅Þ + k3 ð1  cos 3∅Þ

(2.1.5)

∅

where k1, k2, k3, and k4 are the force constants determined experimentally; b and θ are the bond length and bond angle after stretching and bending, respectively; b0 and θ0 are the equilibrium bond length and equilibrium bond angle, respectively; and φ is the bond torsion angle. The elastic moduli are calculated by directly computing the average mechanical forces developed between carbon atoms in the nanotube. The effective elastic moduli determination using the force approach can be calculated directly from the virial theorem given by Swenson [2] in which the expression of the stress tensor in a macroscopic system is given as the function of atom coordinates and interatomic forces. The method provides a continuum measure of the internal mechanical interactions between atoms. In an atomistic calculation, the internal stress tensor can be obtained using the so-called virial expression given by Swenson [2], which is as shown in the succeeding text: 1 σ¼ V0

"

n X

!# ! X  + mi vi vTi rij fijT 

(2.1.6)

i 0, by > 0, cz > 0

(2.3.4)

  1 1 1 jbx j  ax , jcx j  ax , cy   by 2 2 2

(2.3.5)

2.3.4.2 The Group Concept The GROMACS MD uses user-defined groups of atoms for simulation purposes. The maximum of 256 groups are allowed; however, each atom can be subjected to six different groups: i. Temperature-coupling group The temperature coupling parameters, such as reference temperature, time constant, and number of degrees of freedom, are considered [44]. This parameter allows to define a state of temperature, where there could exist a differential temperature range. This group is essential in implementing state of the systems, where the constituent structures have different temperature magnitude. For example, temperature coupling group is beneficial in modeling systems, where the surface has to be kept at a lower temperature, than the adsorbing molecule, as shown in Fig. 2.3.4. ii. Freeze group The atoms belonging to freeze group are kept stationary during the simulation process. This methodology is particularly useful to model equilibration [45]. Moreover, to avoid

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Fig. 2.3.4 Illustration of heat surface and adsorbing molecule in thermal contact.

unreasonable movements of molecules during MD simulations, this methodology is useful. Also, an atom has six degrees of freedom, and its motion can be constrained such that the atom has the provision to move either in a particular plane or line. In other words, user-defined directional constraint can be imposed to the collections of atoms, to restrict its motion. Hence, a fully frozen atom cannot be moved by any constraint. iii. Accelerate group When external force is to be imposed to the system, “accelerate group” is useful to achieve the required acceleration of the atoms. This methodology is equivalent to the system subjected to external force condition. Each atom in an “accelerate group” is imposed with an acceleration ag condition. Hence, it allows to develop nonequilibrium simulation [46]. Moreover, this methodology is useful in obtaining transport properties, like thermal conductivity. iv. Energy-monitor group Mutual interactions among all the energy groups are compiled during the simulation process. All the nonbonded interactions between the pairs of energy-monitor group can be excluded. v. Center-of-mass group This classification of group is useful when low-viscous medium is to be modeled; an example would be a gas system. Low viscosity would refer to a state when the intermolecular distance is large, such that the collision between the atoms doesn’t occur, which could have given rise to frictional forces. Owing to this frictional forces, the center of mass motion is prevented [47]. vi. Compressed position output group To reduce the size of the trajectory file (.xtc or .tng), a subset of all particles can also be stored.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 79

2.3.5 Molecular Dynamics A global flow scheme of MD is shown in Fig. 2.3.5. Each simulation requires a set of initial coordinates. Additionally, for transient simulations, initial velocities of all the particles are also required as input parameter.

2.3.5.1 Initial Conditions i. Topology and force field

1. Input initial conditions Potential interaction V of atoms Position r of atoms Velocities n of atoms

2. Compute forces The force on atom is defined as, Fi =–

JV Jri

3. Update configuration The movement of atoms is simulated by numerically solving Newton’s equations of motion F d2ri = i mi dt2

4. Output step (if required) Write positions, velocities, energies, temperature, pressure, etc.

Fig. 2.3.5 Global molecular dynamics (MD) algorithm.

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The system topology and the description of the force field are required for simulation intent. According to the desired condition, the molecules constituting in a cell might be subjected to external force. Under such circumstances, this methodology is useful. ii. Coordinates and velocities Before the start of the simulation, the size of the cell, coordinates, and the velocities of all the particles are needed to be assigned. The box size is determined by three vectors b1, b2, and b3. If the velocities remain unknown, the program uses the initial atomic velocities vi, where i ¼ 1,…,3 N, at a given absolute temperature T, defined as rffiffiffiffiffiffiffiffiffiffiffi   mi mi v2i exp  (2.3.6) pðvi Þ ¼ 2πkT 2kT where k is Boltzmann’s constant. To retrieve the velocity, normally random distributed numbers as shown in Fig. 2.3.6 are generated by adding 12 random numbers Rk, defined as 0  Rk < 1

(2.3.7)

iii. Center-of-mass motion The center of mass is usually kept stationary at each step to carry out simulation under no external forces, that is, the velocity of the center of mass should remain constant. However, in 3

×10–6

Probability distribution (-)

2.5

2

1.5

1

0.5

0 0

1

2

3

4 5 6 Velocity (m/s)

7

8

9

10

Fig. 2.3.6 Maxwell-Boltzmann velocity distribution curve generated from random numbers.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 81 practice the algorithm introduces a small slow change in the center of mass; as a result the total kinetic energy of the system remains unfixed. This could lead to large deviation in the required simulation results in the long run, especially when the temperature parameter is coupled to the cell.

2.3.5.2 Neighbour Searching Internal forces are either generated from fixed (static) lists or from dynamic lists. The forces from dynamic lists contain nonbonded interactions between any pair of particles. When nonbonded forces are considered, it is convenient to have all particles in a rectangular box. A triclinic box can be transformed into a rectangular box. It may be necessary because output coordinates are always in a rectangular box, even when a dodecahedron or triclinic box is used for the simulation purposes.

2.3.5.3 Pair Lists Generation The nonbonded pair forces are needed to be calculated for those pairs i and j for which the distance rij is less than the given cutoff radius Rc. When the interaction between the particle pairs is fully accounted by bonded interactions, then the particles are excluded. A pair list is employed by GROMACS for which the nonbonded interactions are calculated. The pair list would consist of particles i, a displacement vector for particle i, including the particles j, that are within “rlist” of this particular image of particle i. The list is updated in every “nstlist” steps, where “nstlist” is typically 10. A provision is there to calculate the total nonbonded force on each particle, as all the particles are in a shell around the list cutoff, that is, at a distance between “rlist” and “rlistlong.” In order a make a neighbor list, all particles that are close, that is, within the neighbor list cutoff, to a given particle must be found. This methodology of searching is generally referred as neighbor search (NS) or pair search, which involves period boundary condition and determination of image. The search algorithm is O (N), along with a simpler O (N2) that is also available under some conditions.

2.3.5.4 Cut-Off Schemes: Group Versus Verlet GROMACS supports two different cutoff scheme setups, from its version 4.6 release. The original one is based on particle group and the other on Verlet buffer. Fundamental differences are present between the two schemes, which affect the results, performance, and feature support. The group scheme can almost work like the Verlet scheme; however, a decrease in performance is expected. When the simulation deals with molecules, which are commonly abundant in many simulations, such as water molecules, the group scheme is especially fast. A neighbor list is generated in the group scheme, which consists of pair of groups at one particle. Charge groups were originally present but with proper treatment of long-range order of electrostatics. These simulations were performed in unbuffered status, which was the only

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advantage. The center of geometry is within the cutoff distance when groups are put into the neighbor list. MD steps are performed until the interactions between all particles are calculated, until the neighbor list is updated. This setup is efficient, as the neighbor search only check distance between charge-group pair and not particle pairs. This setup is efficient as the neighbor search only checks the distance between the charge-pair group and not particle pairs. Without explicit buffering, this setup leads to the phenomenon of energy drift, as some particle pairs that are within the cutoff do not react and some outside the cutoff do react. Such situations occurred when (i) particles move across the cutoff neighbor search steps and (ii) when more than one particle or particle pairs move in/out of the cutoff, with their geometric center distance outside/inside of the cutoff. The intervention of user in adding a buffer to the neighbor list removes such problems; however, high computational cost is involved. The severity of such problems depends on the system, the properties in which we are interested, and the cutoff setup. A buffered pair list is present in Verlet cutoff scheme by default. However, the cluster of particles in this scheme is not static, as in the case of group scheme. To ensure that the particles move between pair search steps, forces between almost all the particles within the cutoff distance are calculated. However, the size of the applied buffer depends upon various issues, such as the implementation of temperature parameter.

2.3.5.5 Energy Drift and Pair-List Buffering When dealing with canonical (NVT) ensemble, the atomic displacement and the shape of the potential at the cutoff can be used to determine the average error caused by the finite Verlet buffer size. The displacement distribution obtained along one dimension for a particle moving freely, with mass m, in a time period t at temperature T is Gaussian with zero mean and variance σ 2 ¼ tκBT/m. The variance changes to σ 2 ¼ σ 212 ¼ tκ BT(1/m1 + 1/m2), for the distance between two particles. However, in actual practice, particles do interact with other particle over the specified time period, and hence the real displacement distribution is much narrower. When a nonbonded interaction cutoff distance of rc and a pair-list cutoff rl ¼ rc + rb are provided, then the average error after time t for pair interactions between one particle of type 1 surrounded by particles of type 2 with number density ρ2, when the particle distance changes from r0 to rt, can be expressed as ð ðrc ∞ 4πr02 ρ2 V ðrt ÞG

hΔV i ¼ 0 rl

r  r t 0 dr 0 dr t σ

(2.3.8)

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 83 ð ðrc ∞ 4πr02 ρ2

 ∞ rl

ðrc ∞ ð

2

 4π ðrl + σ Þ ρ2 ∞ rl

 r  r 1 t 0 2 0 00 V ðrc Þðrt  rc Þ + V ðrc Þ ðrt  rc Þ G dr 0 dr t σ 2

(2.3.9)

  1 1 rt  r0 dr 0 dr t V 0 ðrc Þðrt  rc Þ + V 00 ðrc Þ ðrt  rc Þ2 + V 000 ðrc Þ ðrt  rc Þ3 G σ 2 6

8 9 h r   rb i 1 0 b 2 2 > > > > E ð r Þ r σG + σ  r V c b b > > > > σ σ 2 > > > > > > > > < = h   i     1 r r 2 b b 00 2 2 2 2 ¼ 4π ðrl + σ Þ ρ2 + V ðrc Þ σ rb + 2σ G  rb rb + 3σ E > > σ σ 6 > > > > > > > > > > h   i > >     1 r r > : + V 000 ðrc Þ rb σ r2 + 5σ 2 G b  r 4 + 6r 2 σ 2 + 3σ 4 E b > ; b b b σ σ 24 (2.3.10) where G is a Gaussian distribution, along with zero mean and unit variance. It is always desired to obtain least energy error, so σ will be small compared with rc and rl. The energy error would be averaged over all particles and weighted with particle counts. However, when dealing with condensed matter phase, the estimated error will be much greater than in actual practice, as the displacement will be much smaller than for freely moving particles. When constraints are present, such as bonded molecules, some particles will have fewer degree of freedom, which will tend to reduce the energy error. Furthermore, the displacement in an arbitrary direction of a particle with two degrees of freedom is not Gaussian, rather it follows the complimentary error function:   pffiffiffi π jr j pffiffiffi erfc pffiffiffi (2.3.11) 2σ 2 2σ where σ 2 is κBT/m. However, this distribution cannot be integrated analytically to obtain the energy error. One important implementation is present that reduces the energy error caused by the finite Verlet buffer list size. The earlier derivation assumes a particle pair list. However, for more efficiency, GROMACS uses a cluster pair list. The list consists of clusters of four particles (in most cases), called a 4  4 list. Slightly beyond the pair-list cutoff, there will still be a large fraction of particle pairs present. This unknown fraction can be determined in a simulation and accurately predicted under reasonable assumptions.

2.3.5.6 Cut-Off Artifacts and Switched Interactions Using the Verlet scheme, the pair potentials are shifted to be zero at the cutoff, making the potential the integral of the force. Such situation is possible in the group scheme, if the shape of the potential corresponds to zero at the cutoff distance. Moreover, energy drift is possible when

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the forces are nonzero at the cutoff. Such effects are negligibly small and may be even skipped, under other dominant factors, like integration errors from constants. To completely avoid such cutoff artifacts, the nonbonded forces can be completely switched to zero at distances smaller than the neighbor list cutoff.

2.3.5.7 Grid Search All particles are put on a grid, with the smallest spacing Rc/2 in each of the directions. Along the direction of each box vector, a particle i has three images. Each direction of the image is associated with 1, 0, and 1, associated to a transition of each box vector. The searching method involves the construction of images first and then searching the neighbors corresponding to the image of i. On such occasion, some grid cells may be searched more than once. The grid search methodology is equally fast for rectangular and triclinic boxes.

2.3.5.8 Charge Groups For the reduction of cutoff artifacts of coulomb interactions, charge groups were introduced. When dealing with plain cutoff, significant jumps in the potential and forces arise when atoms with (partial) charges translates in and out of cutoff radius. These jumps can be reduced by moving groups of atoms with net charge zero, called charge group, in and out of the neighbor list. Such condition reduces the cutoff effects from charge-charge level to the dipoledipole level. However, under this circumstance, a slight negative result may be produced, depending on the neighbor list generated and the process of calculating interactions. Some important reasons are present for using the “charge group.” In molecular topology the neighbor searching is carried out on the basis of charge groups. If the image distance between the geometric centers of the atoms is less than the cutoff radius, all atom pairs are included in the pair list. Furthermore, the charge groups are ignored, when the Verlet cutoff scheme is implemented.

2.3.6 Compute Forces In this section, the methodology for calculation of the forces is discussed, along with the various mathematical models involved in this scheme.

2.3.6.1 Potential Energy When the forces are computed, the potential energy resulting from each interaction is computed as well. The summation of all the potential energy arises from various contributors, such as Lennard-Jones, coulomb, and bonded terms.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 85

2.3.6.2 Kinetic Energy and Temperature The temperature is calculated using the total kinetic energy of the N particle: Ekin ¼

N 1X mi v2i 2 i¼1

(2.3.12)

The absolute temperature can be computed using 1 Ndf kT ¼ Ekin 2

(2.3.13)

where k is Boltzmann’s constant and Ndf is the number of degrees of freedom, which is given by Ndf ¼ 3N  Nc  Ncom

(2.3.14)

Here the number of constraints on the system is given by Nc. When more than one temperature coupling group is used the number of degrees of freedom for group i is given as   3N  Nc  Ncom i ¼ 3N i  Nci Ndf 3N  Nc

(2.3.15)

When pressure calculation is required, the kinetic energy can also be written in the form of tensor, defined as Ekin ¼

N 1X mi vi vi 2 i

(2.3.16)

2.3.6.3 Pressure and Virial The difference between the kinetic energy Ekin and the virial gives the pressure tensor P as 2 P ¼ ðEkin  ΞÞ V

(2.3.17)

where V represents volume of the computational box. The scalar pressure P, which can be utilized for pressure coupling for isotropic systems, is given by P ¼ traceðPÞ=3

(2.3.18)

The virial Ξ is defined as Ξ¼

1X rij Fij 2 i > > > =7 6 nTC Δt < T 0 6  1 7 (2.3.32) λ ¼ 41 + 5 > 1 τT > > > :T t  Δt ; 2 τ ¼ 2CV τT =Ndf k

(2.3.33)

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 89

2.3.10.2 Velocity-Rescaling Temperature Coupling A velocity-rescaling thermostat is essentially a Berendsen thermostat, with the add-on stochastic term, which helps in predicting the correct canonical ensemble, given by sffiffiffiffiffiffiffiffiffi dt KK 0 dW (2.3.34) dK ¼ ðK0  K Þ + 2 pffiffiffiffiffi Nf τ T τT Moreover, this thermostat possesses the advantage of Berendsen thermostat, which is firstorder decay, with no oscillations. Furthermore, when the NVT assemble is employed, the energy quantity that is conserved is written to the energy and log file.

2.3.10.3 Andersen Thermostat Various methods are available that can be utilized for maintaining a constant temperature. One scheme is to take an NVE integrator and then reselecting the particles from a MaxwellBoltzmann distribution periodically. This is achieved by randomizing all the velocity simultaneously, at every τ/Δt steps, or by initializing every particle with small probability at every time step of Δt/τ, where Δt represents the time step. Due to the predefined condition on the way constraints operate, simultaneous randomization should be done of all the particles. However, due to parallelization issues, the Andersen version is not currently utilized. However, the Andersen-massive can be implemented despite of the involved constraint. Furthermore, this thermostat can be utilized only with velocity Verlet algorithms, as at each time step, it operates directly on velocities. Additionally, this algorithm avoids some ergodicity issues of other thermosetting algorithms. It is a consequence of the fact that between energetically decoupled components of a system, energy cannot oscillate back and forth, as in velocity scaling motion. Moreover, the kinetics of system slows down by randomizing correlated motions of the system.

2.3.10.4 Nose-Hoover Temperature Coupling For maintaining a targeted temperature, Berendsen weak coupling is exceptionally efficient for the relaxation of a system. However, once the equilibrium is reached, it is also important to explore correct canonical ensemble. To enable canonical ensemble at such equilibrium condition, GROMACS further supports the extended-ensemble approach, proposed by Nos
e [48], which was modified later by Hoover [49]. The Hamiltonian system is continued by the introduction of thermal reservoir and a friction term, in the equations of motion.

90

Chapter 2

The friction factor is calculated as the product friction parameter and velocity of each particle. The friction parameter, or “heat bath” variable, is defined as a fully dynamic quantity, along with its momentum and equation of motion. The difference between the current kinetic energy and reference temperature yields the time derivative. In this context, the equations of motion of particle are replaced by d 2 ri Fi pξ dr i ¼  dt2 mi Q dt

(2.3.35)

where the equation of motion of friction factor parameter is given by dpξ ¼ ðT  T0 Þ dt

(2.3.36)

where T0 denotes the reference temperature and T is the current instantaneous temperature of the system. The constant Q is usually termed as “mass parameter of the reservoir.” The conserved quantity for the Nose-Hoover equations of motion is defined as H¼

N X p2ξ pi + U ðri , r2 , …, rN Þ + + Nf kTξ 2mi 2Q i¼1

(2.3.37)

where Nf is the total number of degrees of freedom. However, due to the dependence of mass parameter on reference temperature, it is some undesirable for describing the coupling strength. For maintaining the coupling strength, Q must be changed with reference temperature. Hence, it is preferable to work with the period τT of the oscillations of kinetic energy between the system and the reservoir. Its mathematical formulation is given by Eq. (2.3.38): Q¼

τ2T T0 4π 2

(2.3.38)

A strongly damped exponential relaxation can be achieved using weak coupling scheme, whereas an oscillatory relaxation is achieved using Nose-Hoover approach. The realistic time in the relaxation using Nose-Hoover coupling is several times larger than the period of the oscillations selected. Such oscillations also indicate that the time constant should be 4–5 times larger compared with relaxation time with weak coupling. Nose-Hoover dynamics consists of a system with collections of harmonic oscillators, where only a subsection of phase space is ever sampled, even if infinite long simulation process exists. Under such circumstances, the Nose-Hoover chain approach was developed, with each having its own Nose-Hoover thermostat. In the present scenario, the default number of chains is 10,

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 91 which can be modified by user. To include a chain of thermosetting particles, the existing equations can be modified as follows: d 2 ri Fi pξ1 dr i ¼  dt2 mi Q1 dt dpξ1 pξ ¼ ðT  T0 Þ  pξ1 2 dt Q2 ! p2ξi1 dpξi¼2…N pξ  kT  pξi i + 1 ¼ dt Qi1 Qi + 1 ! p2ξN1 dpξN  kT ¼ dt QN1

(2.3.39) (2.3.40)

(2.3.41)

(2.3.42)

The quantity that is conserved for Nose-Hoover chains is H¼

N M p2 M X X X pi ξk + U ðr1 , r2 , …, rN Þ + + N kTξ + kT ξk f 1 2mi 2Q0k i¼1 k¼1 k¼2

(2.3.43)

In the output the velocity data of Nose-Hoover thermostat are not included, due to the considerable consumption of space. In velocity Verlet and leapfrog dynamics, calculations for matching the temperature with the reference temperature are different. Examining the Trotter decomposition for better understanding the difference between constanttemperature integrators, consider the case of Nose-Hoover dynamics, where the splitting of Liouville operator as iL ¼ iL1 + iL2 + iLNHC

(2.3.44)

where iL1 ¼

N  X pi i¼1

iL2 ¼

N X i¼1

iLNHC ¼

mi Fi :



∂ ∂ri

∂ ∂pi

N X pξ pξ ∂ ∂  vi rvi + + ðT  T0 Þ Q Q ∂ξ ∂p ξ i¼1

(2.3.45)

(2.3.46)

(2.3.47)

When using standard velocity Verlet with Nose-Hoover temperature control the expression becomes   exp ðiLΔtÞ ¼ exp ðiLNHC Þ exp ðiL2 Δt=2Þ exp ðiL1 ΔtÞexp ðiLNHC Δt=2Þ + O Δt3 (2.3.48)

92

Chapter 2

With half-step-averaged temperature control using md-vv-avek, this decomposition will not work. This is attributed to the fact that until the second velocity step, we do not have the fullstep temperature. An alternate decomposition can be constructed, which will be reversible, by changing the place of N-heterocyclic carbenes (NHC) and velocity portions of the decomposition:   exp ðiLΔtÞ ¼ exp ðiL2 Δt=2Þ exp ðiLNHC Δt=2Þ exp ðiL1 ΔtÞexp ðiLNHC Δt=2Þ exp ðiL2 Δt=2Þ + O Δt3 (2.3.49) This formulation eases to visualize the difference between the difference genres of velocity Verlet integrator.

2.3.10.5 Group Temperature Coupling Using GROMACS, a temperature coupling can be achieved on groups of atoms, which are typically a protein or solvent. Due to imperfect energy exchange at molecular level due to effects including cutoffs, such algorithms were introduced. In this scheme the whole system is coupled to one heat bath. An additional feature of this methodology is the assignment of temperature partially to the system. In other words a part of the simulation cell can be temperature coupled while other parts not. It is achieved by specifying 1 for the time constant τT for the group, which is not to be maintained to a reference temperature. When a part of the system is assigned to a temperature, the system will still be converging to an NVT system. To minimize the errors in the temperature, arose due to discretized time step, only those molecules should be thermostatic and not the remaining molecules.

2.3.11 Pressure Coupling As the simulation cell can be coupled to a “temperature bath,” similarly, a “pressure bath” can also be assigned to it. GROMACS supports both the extended-ensemble Parrinello-Rahman approach [50,51] and the Berendsen algorithm [52] and, for velocity Verlet, the MartynaTuckerman-Tobias-Klein (MTTK) [53]. Moreover, any coupling method can be utilized for combining Parrinello-Rahman and Berendsen. However, MTTK can only be applied with Nose-Hoover temperature control.

2.3.11.1 Berendsen Pressure Coupling The box vectors and the coordinates are rescaled by the Berendsen algorithm at every step or nPC steps, with a matrix μ, which consists of the influence of first-order kinetic relaxation of the pressure toward a given pressure P0, according to

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 93 dP P0  P ¼ dt τp

(2.3.50)

nPC Δt βij P0ij  Pij ðtÞ 3τp

(2.3.51)

The scaling matrix μ is defined as μij ¼ δij 

where β is the isothermal compressibility of the system. In most likely cases, it would correspond to be a diagonal matrix. When the system is completely anisotropic, it has to be rotated. Such rotation, in the first order in the scaling, is approximated, which is typically less than 10-4. However, the actual scaling matrix μ’ is given by 0 1 μxx μxy + μyx μxz + μzx (2.3.52) μyy μyz + μzy A μ0 ¼ @ 0 0 0 μzz Here the velocities are neither rotated nor scaled. The Berendsen scaling can be implemented isotropically, which implies that a diagonal matrix with elements of size trace (P)/3 is used. Semiisotropic scaling can be used for systems with interfaces. In this particular case, isotropically scaling is achieved for x/y-directions and independent scaling for z-direction. Furthermore, compressibility in the x/y- or z-directions can be initialized to zero, for scaling in the remaining direction(s). However, if full anisotropic deformations are allowed, along with the usage of constraints, then either scaling must be allowed slowly or decrease the time step for avoiding errors from constraint algorithms. Moreover the Berendsen pressure control algorithm does not yield the exact NPT ensemble, even though the algorithm produces correct average pressure.

2.3.11.2 Parrinello-Rahman Pressure Coupling When the cases deal with fluctuation in pressure or volume, for example, calculating the thermodynamic properties preferably for small systems, it poses a problem that the exact ensemble may not be properly defined, and the true NPT ensemble is not simulated. In this context a constant-pressure simulation is also supported by GROMACS, using the ParrinelloRahman approach, which has similarity with Nose-Hoover coupling and theoretically provides true NPT ensemble. Implementing the Parrinello-Rahman barostat the box vectors represented by matrix b obey the matrix equation of motion: db2 1 ¼ VW 1 b0 ðP  Pref Þ 2 dt

(2.3.53)

94

Chapter 2

where V denotes the volume of the box and W represents the matrix parameter, which determines the strength of the coupling. The matrices P and Pref are the current and reference pressures, respectively. The governing equations of motion are also changed. The Parrinello-Rahman modification is d 2 r Fi dr i ¼ M 2 mi dt dt

(2.3.54)

 db0 db 0 0 1 + b b M¼b b dt dt

(2.3.55)

1

The inverse parameter matrix W1 specifies the strength of the coupling and the nature of box deformation. When the elements of W1 are zero, the box restriction will be automatically fulfilled. The coupling strength can be automatically calculated in GROMACS, and the approximate isothermal compressibility β and the pressure time constant τp must be provided in the input file:  1  4π 2 βij W ij ¼ 2 3τp L

(2.3.56)

If the pressure is very far from equilibrium, very large box oscillations may be resulted from Parrinello-Rahman coupling, which could even crash the run. In such situations, either the time constant have to be increased or the weak coupling scheme may be utilized to reach the target pressure, and then the Parrinello-Rahman coupling can be implemented, once the system is in equilibrium. In addition, using the leapfrog algorithm, the pressure at time t is not available, until the time step has completed. Hence the pressure from the previous step must be utilized, which makes the algorithm not directly reversible. Such situations may be inappropriate for high precision thermodynamic calculations.

2.3.11.3 Surface-Tension Coupling Considering a system that consists of more than one distinguishable phase, which are separated by surfaces, parallel to the x-y plane, then the surface tension and the z-component of the pressure cab are coupled to a pressure bath. Till the present version of GROMACS, this scheme only works with Berendsen pressure coupling. The average surface tension γ(t) can be calculated from the difference between the normal and lateral pressure: 1 γ ðtÞ ¼ n

L ðz 

 Pxx ðz, tÞ + Pyy ðz, tÞ Pzz ðz, tÞ  dz 2

0

(2.3.57)

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 95   Pxx ðtÞ + Pyy ðtÞ L ¼ Pzz ðtÞ  2 n

(2.3.58)

where Lz represents the height of the box and n is the number of surfaces. Moreover, by scaling the height of the box with μzz, the pressure in the z-direction can be corrected, given by ΔPzz ¼

Δt fP0zz  Pzz ðtÞg τp

μzz ¼ 1 + βzz ΔPzz

(2.3.59) (2.3.60)

This formulation is similar to the normal pressure coupling, expect the omission of 1/3 factor. The pressure correction in the z-direction is then utilized to get the correct convergence for the surface tension value γ 0. The correction factor for the box length, in the x/y-direction, is given by    Pxx ðtÞ + Pyy ðtÞ Δt nγ 0 (2.3.61) β  Pzz ðtÞ + ΔPzz  μx=y ¼ 1 + 2 2τp x=y μzz Lz In most cases, the τp values will scale, under incorrect compressibility. The convergence of the surface tension is affected by compressibility, when surface tension coupling is applied. Moreover, under constant box length (βzz ¼ 0), the ΔPzz is also set to zero, which is necessary for obtaining correct surface tension.

2.3.12 The Complete Update Algorithm For velocities and coordinate update, the complete algorithm is given using leapfrog. GROMACS provides the provision of preventing the motion of selected particles, which are defined as a “freeze group.” Under such situations, a freeze factor fg is a vector quantity. Hence, this freeze factor and the acceleration ah must be taken into account for updating the algorithm of the velocities, which becomes

     Δt Δt FðtÞ (2.3.62) ¼ fg λ v t + Δt + ah Δt v t+ 2 2 m

2.3.13 Output Step The most important step of the MD run is the creation of trajectory file, which contains the coordinates of the particles and (optionally) the velocities at fixed intervals. The trajectory file contains the following information: positions, velocities, forces, dimensions of the simulation cell, integration step, integration time, etc. When velocity Verlet integrators are used, velocity

96

Chapter 2

labeled at time t is for that time, whereas for other integrators (e.g., stochastic dynamics), velocities at time t are for time t  1/2 Δt. Moreover, the trajectory files are lengthy in nature; therefore it is preferable to omit steps. To retain information, it is adequate to write a frame every 15 steps, since for the highest frequency in the system, 30 steps are made per period.

2.3.14 Advantage and Functional Characteristics The following advantages are offered by the GROMACS software: i. GROMACS provides high performance for computational purpose. ii. GROMACS is user-friendly, along with the clear text formatting of topologies and parameter files. A number of consistency checking are involved, and unambiguous error messages are issued, when something is wrong. iii. With no scripting language involved, all programs use an interface, along with command line options for input and output files. iv. Elapsed simulation time will be continuously available in the interface, along with the expected time for the final completion and logging of data. v. GROMACS is a free software, available under Lesser General Public License (LGPL), version 2.1. The GROMACS software consists of a preprocessor, a parallel MD, and an energy minimization program, which can utilize an arbitrary number of processors, an optional monitor, and several analysis tools. The programming for the language is written in ANSI C. The following are the functional characteristics of the software: i. Nature of physical problem Analysis and prediction of dynamic behavior of macromolecules can be made under external driving forces. ii. Method of solution GROMACS uses classical Newtonian equations of motion and force fields, based on variable or fixed bonded or nonbonded molecular interaction. The developed system of molecules is coupled to external bath of constant temperature and pressure. Periodic conditions (rectangular) can be applied to the simulation, which would provide an ease to establishing repetitive behavior of molecular structure. iii. Restriction on the complexity of the problem The computational file size, limited by the memory and the number of processors, depends on the complexities of the molecular interactions.

Overview of BIOVIA Materials Studio, LAMMPS, and GROMACS 97 iv. Running time A typical small biomolecule (a peptide of 20 residues) in water (800 atoms) runs in 100-time steps (0.2 ps) in 1 min on a 32-i860 processor system. In the past years, GROMACS has evolved and responded to rapid growth, to be a large project, with almost two million lines of code. From the first release of GROMACS the most significant change in the software is that it has moved to C++. While many parts of the code are still coded using C algorithm, it will take a couple of years to have a complete transition to C++. Incorporating the transition to a more versatile language has led to the involvement of modularity in the code. Partitioning of functionality and consequently in the memory has led to better handling of memory and errors. Within the simulation, parallel computer functionality can be enabled, which would lead to the splitting of work, into independent entity of modules, for faster transaction toward less computational time lapse.

2.3.15 Application of GROMACS GROMACS has emerged to be an extremely useful computational code for computationally replicating the system of molecular cluster and aiding in fetching technical data, which is beneficial for gaining deeper understanding at the atomic scale. Working at the molecular scale aids in optimizing of devices, for creating state-of-the-art equipment. From the modeling of medical devices and implants to simulation of nanoscale flows, GROMACS is in continuous process, toward our understanding at small scale, and the verification of physical principles, which are visually prominent at macroscale. In this section, a list on the applications of GROMACS is presented; however, this is a small note on the applications, and the readers are suggested not to be confined with the discussed utility of the software.

2.3.15.1 Biochip Devices GROMACS provides molecular dynamics simulations for the construction and optimization of biochip devices, with miniature passages as shown in Fig. 2.3.7. With the molecular dynamics approach, the flow problem can be accurately measured and resolved. The thermal flow problems is inherent to the gradients of both velocity and thermal profile; however, interlink between both the gradient can be understood on the basis of molecular approach. Physical conditions of frostbite or burns suffered by the patients can also be studied by the simulation approach. The simulations mainly focus on the effect of effective microchannel width and the thermal boundary conditions at the solid-fluid interface on the velocity profile. GROMACS provides useful insights on the influence of global and local effects on bioflow models, important for configuration and the operation of biochip devices.

98

Chapter 2 Temperature boundary (T) Wall atom

Direction of flow

Effective channel width (D) Molecule of fluid

y

x

Fig. 2.3.7 Schematic diagram of a simulation geometry.

2.3.15.2 Molecular Modeling of Biomolecules Calculating the molecular dynamics data and solving the energy equation are an essential feature of GROMACS. Modeling of complex biomolecule like protein, which is the most important nutrient present in the food apart from carbohydrates and fats, can be simulated with molecular dynamics (MD). The functional properties exhibited by the molecules vary at different processing and storage stages. During processing of various molecules, under various external thermal fields, like boiling and roasting. or nonthermal fields, like pulsed electric fields and pressure processing, external stresses are applied, which would affect the shelf life of the product. Therefore efforts are made by the researchers in understanding the complex dynamics of processing protein structures in recent years. Usage of magnetic resonance imaging (MRI) and X-ray diffraction has been for studying the dynamics of molecular structure; however, due to various limitations depending upon the technique adopted and their intrinsic expensive nature, MD simulation techniques are emerging as a viable alternative to overcome the standard issues existing due to the prevailing techniques.

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CHAPTER 3

Molecular Dynamics Simulation of Metal Matrix Composites Using BIOVIA Materials Studio, LAMMPS, and GROMACS This chapter has been divided into three parts as follows: Chapter 3.1: Prediction of Mechanical Properties of Graphene/Silicon Carbide-Reinforced Aluminum Composites Using BIOVIA Materials Studio Chapter 3.2: Prediction of Mechanical Properties of Graphene/Copper Nanolayered Composites Using LAMMPS Chapter 3.3: Molecular Dynamics Simulation of Lithium Metal/Polymer Electrolyte Interfacial Properties Using GROMACS

Chapter 3.1

Prediction of Mechanical Properties of Graphene/ Silicon Carbide-Reinforced Aluminum Composites Using BIOVIA Materials Studio Sumit Sharma*, Pramod Kumar*, Rakesh Chandra*, Gaurav Sharma† *

Department of Mechanical Engineering, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar, India, †School of Mechanical Engineering, Lovely Professional University, Phagwara, India

Metal matrix composites (MMCs) are nowadays widely used in various areas such as automotive, aerospace, and electronics. Moreover the particle-reinforced MMCs are of particular interest because of their excellent mechanical and physical properties [1]. Recently, particles of SiC were Molecular Dynamics Simulation of Nanocomposites using BIOVIA Materials Studio, Lammps and Gromacs https://doi.org/10.1016/B978-0-12-816954-4.00003-6 # 2019 Elsevier Inc. All rights reserved.

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102 Chapter 3 used as reinforcement for aluminum matrix [2,3] that helped in increasing the elastic modulus and tensile strength of the composites. A number of studies have been conducted on SiC particlereinforced aluminum composites (SiC-Al) in the last few decades [4,5]. All of these studies suggested that the deformation behavior of MMCs is dominated by the reinforcing particles. Experimental studies showed that the mechanical properties of these composites depend on the volume fraction (Vf), size, and distribution of the SiC-reinforced particles [6–8]. The particle-matrix interface has also been found to play an important role in the mechanical properties of SiC-Al composites. Though a number of experimental studies have been conducted on SiC-Al composites, still the microstructural changes in these composites have not been understood fully because of the difficulties with in situ detection using the laboratory apparatus. A number of simulation methods have been developed for analyzing the mechanical behavior of SiC-Al composites. Finite element analysis (FEA) has been performed to study the effect of particle shape on the deformation behavior of SiC-Al composites [9]. The effect of particle shape on thermal residual stress and strain distributions in composites was studied by Qin et al. [9]. For this purpose, five two-dimensional SiCp/6061Al model composites reinforced with spherical, hexagonal, square, triangular, and shuttle-shaped particles, respectively, were analyzed by finite element method (FEM). The results showed that the particle shape has a great effect on thermal residual stress and strain fields in composites. There were a residual plastic strain concentration in the matrix around a pointed particle corner and a serious residual stress concentration in the pointed particle corner. The two concentrations increased with decreasing pointed corner degree. When an external stress was applied, the plastic strain concentration in the matrix around the pointed particle corner and the serious stress concentration in the pointed particle corner that would fracture on a relative low level of applied stress could decrease the ductility of the composite. Two 6061 Al alloy matrix composites reinforced with general SiC particles and blunted SiC particles were studied on the basis of the FEM analyses. It was found that most very pointed particle corners were eliminated after SiC particles were blunted. Replacing general SiC particles with blunted ones reduced the residual plastic strain concentrations in the matrix and serious residual stress concentration in the particle. Therefore blunted SiC particle-reinforced composite showed a higher ductility than the general one. The effect of particle cracking and interface failure has also been studied [10,11]. In contrast, molecular dynamics (MD) simulations have been rarely used for studying the effect of particle reinforcement on Al. Dandekar and Shin [12] studied the interface failure in SiC-Al composites and developed the traction-separation law using MD for high-strain-rate loading. This law was then used in FEA to get the stress-strain curve of SiC-Al composites. An embedded atom model (EAM) and a Tersoff potential were used to simulate aluminum and SiC, respectively, while a Morse potential was successfully parameterized from ab initio data to represent the Al-SiC interface. The subsequent traction-separation relationships for Mode I and Mode II failure at high temperatures have been developed through MD simulations. The parameterized traction-separation law based on the MD results was found to be consistent with the existing continuum-based cohesive zone models. The maximum stress in shear was approximately 27% smaller than the maximum stress in tensile mode for all simulated temperatures. A hierarchical multiscale model was proposed to simulate

Molecular Dynamics Simulation of Metal Matrix Composites 103 high-strain-rate tensile behavior of the composite. Simulation results showed good agreement in reproducing the experimental stress-strain curve of a split-Hopkinson pressure bar test, further validating the empirical traction-separation laws for the cohesive zone. Song et al. [13] predicted the combined effects of particle size and distribution on the mechanical properties of the SiC particle-reinforced Al-Cu alloy composites. It was shown that small ratio between matrix/ reinforcement particle sizes resulted in more uniform distribution of the SiC particles in the matrix. The SiC particles distributed more uniformly in the matrix with increasing in mixing time. It was also shown that homogenous distribution of the SiC particles resulted in higher yield strength, ultimate tensile strength, and elongation. Yield strength and ultimate tensile strength of the composite reinforced by 4.7-μm-sized SiC particles were higher than those of composite reinforced by 77-μm-sized SiC particles, while the elongation showed opposite trend with yield strength and ultimate tensile strength. Fracture surface observations showed that the dominant fracture mechanism of the composites with small SiC particle size (4.7 μm) was ductile fracture of the matrix, accompanied by the “pull out” of the particles from the matrix, while the dominant fracture mechanism of the composites with large SiC particle size (77 μm) was ductile fracture of the matrix, accompanied by the SiC particle fracture. Boostani et al. [14] dispersed SiC nanoparticles in Al matrix making use of encapsulation capacity of graphene sheets, semisolid stirring of the Al melt, ultrasonic treatment, and pressure application during solidification. A new solidification model taking into account the alteration of the solidification mechanism from particle pushing to particle engulfment, making use of at least 40% enhancement in higher thermal conductivity and diminished repelling forces of SiC nanoparticles tuned by encapsulating graphene sheets, was suggested. This nanostructure manipulation can make about 350% and 258% enhancement in yield strength and tensile ductility, respectively, compared with that of unreinforced Al alloy. The results achieved based on the devised analytic model have shown the significant effect of thermal activated dislocation in strengthening due to considerable mismatch between thermal expansion coefficient of graphene sheets and Al matrix. Fractographic observations disclosed a dimple fracture surface for the semisolid-processed Al-matrix composite reinforced by the nanoparticles that were encapsulated by graphene sheets using ball milling process compared with the cleavage fracture surface of those fortified without the application of graphene. The mechanical properties of the SiC/graphene composites under tensile were studied via MD methods by Yao et al. [15]. The SiC/graphene composites with monolayer graphene and multilayer graphene were built and compared. As a result the composites with different interface structures presented different tensile behaviors. The single graphene sheet in contact with Siterminated SiC (1 1 1) surface showed the higher failure strain and strength. It was also found that the elastic modulus of the composites increased with the increasing graphene Vf. As for the composites fabricated with the multilayer graphene, the failure strain of the whole system decreased with increasing graphene layers in the case that graphene served as a single layer. In contrast, when the graphene sheets served as continuous layers, the failure strain did not change significantly, and the strength was increased monotonically with the increasing number of graphene layers. Chattopadhyay et al. [16] reported the manufacturing of a multilayer

104 Chapter 3 graphene-embedded composite of Al alloys by direct exfoliation of graphite into graphene with the help of friction stir alloying (FSA). The formation of this nanocomposite and optimization of the process parameters led to an approximately twofold increase in the strength, without loss in ductility, due to the dispersion of the graphene in Al. The manufacturing process was scalable and cost-effective as it used the graphite powder and Al sheets as the raw materials. The presence of graphene layers in the metal matrix was confirmed using Raman spectroscopy and transmission electron microscopy (TEM). The graphene sheet thickness was measured using AFM after extracting it from the composite. MD simulation results revealed the evolution of newer structures and defects that have resulted in the enhanced properties of the nanocomposite. Boostani et al. [17] dispersed SiC nanoparticles in Al matrix making use of encapsulation capacity of graphene sheets. The analytic model devised by the researchers demonstrated the significant role of shear lag and thermally activated dislocation mechanisms in strengthening Al MMCs due to the exceptional negative thermal expansion coefficient of graphene sheets. This, in turn, triggers the pinning capacity of nanosized rodlike Al carbide, prompting strong interface bonding for SiC nanoparticles with the matrix, thereby enhancing tensile elongation. From the abovementioned studies, it could be inferred that the effect of particle size and Vf on the mechanical properties of Al matrix has not been studied at the atomic level, when the size of the particle is in nanometers. Thus it becomes necessary to perform the MD simulations for predicting the effect of particle reinforcement on the mechanical behavior of Al-matrix composites when subjected to a given loading. Recently a new Al-based composite reinforced with SiC nanoparticles and encapsulated by graphene was synthesized using the powder metallurgy technique [14]. The researchers reported an increase in ductility and yield strength. This could be attributed to the weakening of particle agglomeration, improved interface strength, and the effect of other strengthening mechanisms [17]. For further development of these composites, the effect of graphene and SiC on the mechanical properties of Al-matrix composites needs to be studied using MD, which could serve as a powerful technique for studying the interactions at an atomic level. In this study the effect of SiC nanoparticle has been studied at the atomic level by using MD approach. For this purpose the BIOVIA Materials Studio [18] software has been used for ˚. modeling and analysis of the composites. The diameter of the SiC nanoparticle was taken as 10 A Firstly the pure Al matrix was modeled for predicting its properties. This was followed by reinforcing the Al matrix with varying Vf of SiC nanoparticle. The Vf was varied from 0% to 20% for studying the variation in the properties of the SiC-Al composite. The effect of graphene was also studied by reinforcing SiC-Al with graphene. The properties of this hybrid composite were then predicted using the Forcite module of the BIOVIA Materials Studio [18] software.

3.1.1 MD Methodology All the MD simulations have been performed using BIOVIA Materials Studio [18]. Fig. 3.1.1 ˚ 3 containing 4631 atoms. For shows the MD model of pure Al of size 40.5
40.5
40.5 A studying the effect of SiC and graphene on the mechanical properties of Al a number of MD

Molecular Dynamics Simulation of Metal Matrix Composites 105 models of SiC-reinforced, graphene-reinforced, and SiC-graphene-reinforced Al composites were modeled using BIOVIA Materials Studio [18]. Fig. 3.1.2 shows the size of a SiC nanoparticle and a number of SiC-reinforced Al composites with different Vf of SiC nanoparticles. Similarly, for studying the effect of graphene reinforcement on the properties of Al a number of MD models with different Vf of graphene were simulated. Fig. 3.1.3 shows the different MD models of graphene-reinforced Al composites. Finally a hybrid composite containing SiC-graphene-Al was modeled as shown in Fig. 3.1.4. A number of such models were simulated for predicting the effect of varying Vf on the properties of Al-based composites.

Fig. 3.1.1 MD model of pure Al containing 4631 atoms.

Fig. 3.1.2 See figure legend on next page.

Fig. 3.1.2—cont’d ˚ , (B) SiC-reinforced Al composite with SiC Vf of 1%, MD model of (A) SiC nanoparticle of diameter 10 A (C) SiC-reinforced Al composite with SiC Vf of 3%, and (D) SiC-reinforced Al composite with SiC Vf of 5%.

Fig. 3.1.3 MD model of (A) pure graphene and graphene-reinforced Al composites with different Vf of graphene as (B) 13% and (C) 30%.

108 Chapter 3

Fig. 3.1.4 MD model of SiC-graphene-reinforced Al composite with Vf of graphene as 13% and that of SiC nanoparticle as 2%.

The SiC nanoparticles and graphene were randomly placed in Al matrix. Before applying any strain on these models, they were subjected to thermal stabilization at 300 K and zero pressure using the constant number of atoms (N), constant pressure (P), and constant temperature ensemble (T), that is, the NPT ensemble. The periodic boundary conditions (PBCs) were used in all the simulations. All the samples were then subjected to a strain rate of 1
109 s1 along the y-axis. The virial stresses in y-direction were calculated at each strain level. The engineering strain was calculated as the change in length of the volume element along the y-direction after loading, divided by the original length of the element. For Al matrix the embedded atom method (EAM)-based potential [19] has been used for describing the interatomic interactions. The Tersoff potential [20] has been used for describing the atomic interactions within SiC and between SiC and graphene. The Morse potential [12] has been used for the interactions between the SiC nanoparticles and the Al matrix.

3.1.2 Results and Discussion (a) Effect of SiC volume fraction The Vf of SiC nanoparticles in Al matrix was varied from 0% to 20%. Fig. 3.1.5 shows the stress-strain curves of SiC nanoparticle-reinforced Al composites during tensile loading. It could be seen from Fig. 3.1.5 that there was a significant effect of Vf of SiC nanoparticles on the mechanical properties of SiC-Al composites. The peak stress was found to increase with increase of Vf. The ductility was reduced considerably when Vf of SiC nanoparticles was

Molecular Dynamics Simulation of Metal Matrix Composites 109 2.50 Pure A1 2.00

Stress (GPa)

5 vol.% SiC 1.50 10 vol.% SiC 1.00 15 vol.% SiC 0.50 20 vol.% SiC 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Strain

Fig. 3.1.5 Stress-strain variation of pure Al and SiC-reinforced Al composites with different Vf of SiC nanoparticles. Table 3.1.1 Mechanical properties of SiC-Al composites SiC Nanoparticle Volume Fraction (Vf) % 0 5 10 15 20

Elastic Modulus (GPa)

Tensile Strength (GPa)

30.00 33.24 35.67 37.45 40.02

1.07 1.20 1.32 1.42 1.50

increased from 15% to 20%. For the models with high Vf, the stress was reduced suddenly at a strain of 0.20, indicating the failure through tensile mode, whereas for the models having low values of Vf of SiC nanoparticles, the stress was found to remain steady and the strain at failure was higher than 0.40. A similar trend was observed in another study [21] for MMCs. The increase in tensile strength and elastic moduli could be attributed to the presence of SiC nanoparticles in the Al matrix. Table 3.1.1 shows the variation of elastic modulus and tensile strength with SiC nanoparticle Vf. Both the elastic moduli and tensile strength were found to increase with an increase in Vf of SiC. With an increase in Vf of SiC in Al matrix, the elastic modulus increases because of the increased load-bearing capability of SiC nanoparticles. The decrease in ductility with increase in Vf could be explained more clearly with a discussion on fracture mechanisms of particle-reinforced MMCs. Generally, there are three types of fracture

110 Chapter 3 mechanisms for such composites. First is the formation, growth, and coalescence of voids in the metal matrix, which leads to the failure of the composite. Second mechanism consists of brittle failure of the nanoparticles. The third mechanism is of interfacial debonding between the metal matrix and the nanoparticles because of the formation of voids at the interface. Out of the abovementioned three mechanisms, the most dominant has been found to be the formation, growth, and coalescence of voids. This mechanism has been found to be dominant when the size of the particle reinforcement is less than 5 μm [22]. The breakage of nanoparticles has not been considered in this study because of the particle size in nanometers/angstrom. (b) Effect of particle size For studying the effect of SiC particle size on the mechanical properties of Al-matrix ˚ were used. Fig. 3.1.6 shows composites, two nanoparticles of SiC having diameters 10 and 20 A the stress-strain variation of Al composites reinforced with two different SiC nanoparticles, one ˚ and the other having the diameter of 20 A ˚ . The elastic moduli and having the diameter of 10 A tensile strength of these composites for the same Vf (Vf ¼ 20%) have been shown in Table 3.1.2. It could be inferred from Fig. 3.1.6 and Table 3.1.2 that SiC-Al composites reinforced with 10˚ -diameter SiC nanoparticles show higher tensile strength and elastic modulus in comparison A ˚ -diameter SiC nanoparticles. Similar results have with the composites reinforced with 20-A been reported by other researchers [23,24]. This could be attributed to the increased contact area between the particles and the metal matrix in the case of small size SiC nanoparticle 2.50

SiC nanoparticle diameter = 10 angstrom

Stress (GPa)

2.00

1.50

1.00 SiC nanoparticle diameter = 20 angstrom

0.50

0.00 0.00

0.10

0.20 Strain

0.30

0.40

Fig. 3.1.6 Stress-strain variation of SiC-reinforced Al composites with different size of SiC nanoparticle reinforcement.

Molecular Dynamics Simulation of Metal Matrix Composites 111 reinforcement. This large contact area helps in promoting the load transfer easily from the matrix to the SiC particles. Thus the small size nanoparticles bear higher stress at a given value of strain, resulting in a higher value of peak stress. On the contrary the tensile strength of the composites reinforced with large size SiC particles was found to be reduced. Thus it could be said that the composites with small size particle reinforcement exhibit higher strength because of a larger number of particle-matrix interfaces, whereas for composites with large size particles, the ductility was found to be improved because of larger interparticle spacing. (c) Effect of graphene reinforcement The effect of graphene reinforcement in Al matrix has also been predicted using MD simulation. For this purpose, two Vf of graphene were chosen, 13% and 30%. Fig. 3.1.7 shows the stress-strain variation of graphene-reinforced Al composites for the two Vf. From Fig. 3.1.7 and Table 3.1.3, it could be seen that the graphene reinforcement significantly enhances the elastic moduli and tensile strength of SiC-Al composites. This was found to be in agreement with a previous study [17]. Table 3.1.2 Effect of particle size on the mechanical properties of SiC-Al composites SiC Nanoparticle Volume Fraction (Vf) % 20.0

SiC Nanoparticle ˚) Diameter (A

Elastic Modulus (GPa)

Tensile Strength (GPa)

10.0 20.0

40.02 44.45

1.50 1.20

3.00

2.50

13% Graphene87% A1

Stress (GPa)

2.00

1.50

30% Graphene60% A1

1.00

0.50

13% Graphene2% SiC-85% A1

0.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Strain

Fig. 3.1.7 Effect of graphene reinforcement on the stress-strain behavior of SiC-reinforced Al composites.

112 Chapter 3 Table 3.1.3 Mechanical properties of graphene-Al composites Graphene Volume Fraction (Vf) % 0 13 30

Elastic Modulus (GPa)

Tensile Strength (GPa)

30.00 43.46 50.88

1.07 1.42 1.70

The increase in tensile properties of graphene-reinforced Al composites could be attributed to better bonding in the interfacial region between graphene and Al matrix in comparison with the bonding between SiC and Al matrix. The effect of combined SiC-graphene reinforcement on the stress-strain behavior of Al-matrix composites has also been shown in Fig. 3.1.7. It could be seen that the elastic modulus and tensile strength of SiC-graphene-reinforced Al hybrid composite were significantly higher in comparison with the SiC-Al composite. This was because of better elastic performance of SiC-graphene reinforcement in comparison with the SiC reinforcement alone. It has been found in another study [25] that the energy around the interface of graphene-Al is much lower in comparison with that of SiC-Al interface, thus showing that the Al-graphene-SiC composite has a better bonding performance, which leads to reduced energy dissipation and better load transmission between the matrix and the reinforcements. It has also been shown that the local strain in graphene-SiC is much lower than that in SiC particles at a given global strain, since the graphene-SiC has the higher elastic modulus. Lower local strain in particles induces higher local strain in the metal matrix that accelerates the nucleation and propagation of shear bands and finally leads to the fracture of the metal matrix. It is not expected but reasonable that graphene has negative effect on the ductility of composites. Thus it can be summarized that the enhancement of the Al-graphene-SiC composites mainly owes to the improved interface bonding performance and improvement for the elastic modulus of particles, which leads to the higher strength along with the weaker elongation capacity.

3.1.3 Conclusion The present study investigates the mechanical properties of SiC nanoparticle-reinforced Almatrix composites using MD simulations and can provide useful information for the designing of these composites. The effect of Vf and size of nanoparticles on the properties of SiC-Al composites has been predicted. The effect of SiC-graphene as reinforcement on the mechanical properties of Al-based composites has also been modeled. The main findings of the study can be highlighted as given in the following: (i) Increase in the Vf of SiC nanoparticles leads to an increase in both the yield strength and the elastic modulus of the Al-matrix composites. (ii) For a given Vf the composites with small-sized SiC nanoparticles show higher strength because of more number of interfaces. The composites with large-sized particles have higher ductility owing to the larger interparticle spacing.

Molecular Dynamics Simulation of Metal Matrix Composites 113 (iii) The addition of graphene improves the bonding between the SiC nanoparticle and Al matrix and also leads to an enhancement in elastic modulus and tensile strength but leads to a reduction in ductility.

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Chapter 3.2

Prediction of Mechanical Properties of Graphene/ Copper Nanolayered Composites Using LAMMPS Amit Bansal*, Prince Setia†, Raj Chawla‡ *

Department of Mechanical Engineering, I.K. Gujral Punjab Technical University, Kapurthala, India, Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur, India, ‡ Department of Mechanical Engineering, Lovely Professional University, Phagwara, India †

Among the various nanoreinforcements available today, graphene is the strongest one [1], which is more than 200 times stronger than steel. Hence it has been used recently [2–4] for strengthening of different classes of materials such as glass, ceramic, polymers, and metals. Metal-graphene layered composites (MGLCs) consist of alternate layers of metal and graphene sheets. They exhibit superior mechanical properties [5–8]. The development of MGLC was first reported by Kim et al. [6] by using monolayer graphene sheets. The nanopillar compression test was conducted for testing the properties of the synthesized MGLC. A significant increment in the strengths of Ni/Cu-graphene composites was observed because of the constraint on the propagation of dislocations across the metal-graphene interface. Gao et al. [3] provided a method for uniform dispersion of graphene in Al-matrix composites, while protecting the structure of graphene (Gr). A modified Hummers’ method was used for preparing graphene oxide (GO) sheets with a negative charge. Al powders were coated with hexadecyl trimethyl ammonium bromide (CTAB) to obtain the surface positive charge. Powder metallurgy route was adopted for fabrication of the samples. The effect of graphene content on the tensile properties and fracture mechanisms of Gr/Al composite was also investigated. The fracture mode of the Gr/Al composites changes from ductile fracture to brittle fracture with increasing the graphene contents. Kim et al. [6] demonstrated a new material design in the form of a nanolayered composite consisting of alternating layers of metal (copper or nickel) and

Molecular Dynamics Simulation of Metal Matrix Composites 115 monolayer graphene that has ultrahigh strengths of 1.5 and 4.0 GPa for copper-graphene with 70-nm repeat layer spacing and nickel-graphene with 100-nm repeat layer spacing, respectively. The ultrahigh strengths of these metal-graphene nanolayered structures indicated the effectiveness of graphene in blocking dislocation propagation across the metal-graphene interface. Ex situ and in situ TEM compression tests and MD simulations confirm a buildup of dislocations at the graphene interface. Li et al. [8] used a bioinspired nanolaminated microstructure for fabricating a bulk graphenereinforced Al-matrix composite through a composite powder assembly method. The development of bioinspired reduced GO (RGO)-Al nanolaminated composites that utilized the key advantages of graphene, including its high strength, modulus, and two-dimensional geometry, was reported. Tensile test revealed that graphene in the nanolaminated composites has remarkably higher strengthening and stiffening efficiencies than those of other reinforcements, and the composites maintained a similar or even slightly higher total elongation than the unreinforced Al matrix. The deformation behavior of the composites was characterized by a significant strain hardening at low strain values, followed by a steady-state flow. These findings were rationalized in terms of a balance between dislocation accumulation and dynamic recovery at the graphene/Al interfaces. Zu et al. [9] investigated the shock response of Cu/graphene nanolayered composite using MD simulation. The spall damage and deformation of Cu, delamination of the nanolaminates, and wrinkling and fracture of graphene were also studied for normal and parallel shock loading. It was found that the Cu (111)/graphene interface was the source of dislocations in Cu and it was also acting as a barrier to their propagation. The nucleation sites were found to follow the Moire pattern. When subjected to shock loading, graphene was found to form wrinkles and fracture when released and stretched under parallel shocks. Song and Zha [10] investigated the mechanical behavior of Ni-coated single-walled carbon nanotubes (SWCNTs) and Al/SWCNT composites using MD simulation. Pullout behavior of Ni-coated SWCNT/Al composite was studied. The Ni-coated SWCNT/Al composite was found to have higher Young’s modulus in comparison with the uncoated SWCNT/Al composite. The Ni coating was found to be improving the interfacial bonding between the SWCNT and the matrix. Lee et al. [11] investigated the tensile behavior of CNT-Al composites using MD. The study showed that compared with pure Al the Young’s moduli of CNT-Al composites increased by 30%–40% and the toughness was found to improve by 37%–100%. Component analysis revealed that even a small amount of SWCNT could play an important role in improving the mechanical properties. SWCNTs were found to increase the fracture strain of the composite by almost two times. A detailed fracture mechanism of the CNT-Al composites was obtained using MD. The lattice structure change, stacking faults, and microvoid nucleation were observed during the MD simulations. Silvestre et al. [12] performed compression studies of CNT/Al composites using MD simulation for analyzing the mechanical behavior of these composites. The variation of

116 Chapter 3 energies of Al, CNT, and interface with the imposed displacement was studied. The modes of failure of CNT-Al composites were also predicted. Two cases were discussed and analyzed: case A with free CNT boundaries and case B with fixed CNT boundaries. Young’s modulus was found to increase by 50% in case A and 100% in case B. It was also seen that there was no improvement in both the yield stress and yield strain of the CNT-Al composite. Premature failure of CNT-Al composite was attributed to the buckling of CNT. Liu et al. [13] used MD simulations to study the strengthening effect of graphene-metal nanolayered composites under shock loading. It was inferred that the graphene interface plays two roles to strengthen the composites under shock loading. The graphene interfaces could be approximately considered as free boundary due to its relatively small bending stiffness, which led to interlayer reflections and weakening the shock wave. The strong in-plane strength of graphene impeded the dislocation propagation, prevented melting, and healed the metal layer. The abovementioned studies suggest that there are very few studies related to the nanolayered composites. The deformation mechanism of few-atom-thick metal layers has not been explored yet. The present study investigates the deformation mechanism and mechanical behavior of MGLCs using MD simulations. The elastic properties could then be predicted using the elastic region of stress-strain diagrams. The effect of thickness of metal layers on the properties of the composites was also investigated using various samples containing alternate layers of graphene and Cu metal and subjected to tensile loading.

3.2.1 MD Simulation (a) Inter-atomic potential The potential used in this study for describing the interaction of Cu atoms was the EAM potential [14]. The covalent bonding between the C atoms of each graphene layer was represented using the Tersoff potential [15]. The 12-6 Lennard-Jones (LJ) potential was used for defining the interactions between the metal and graphene layers. The LJ potential has been used widely by researchers [6,13] for describing the weak van der Waals interaction between the reinforcement (CNTs/graphene) and the matrix in MD simulations. A detailed description of the LJ potential and its parameters could be found in another work [2]. An expression for the total potential energy of the Cu-graphene composite could be obtained by adding various functions as given by Eq. (3.2.1): CCu ETotal ¼ EEAM + ETersoff + ECC LJ + ELJ

(3.2.1)

Here EEAM is the EAM potential of copper matrix; ETersoff is the Tersoff potential of graphene is the van der Waals interactions of carbon atoms between metal and reinforcement; ECC LJ is the van der Waals interactions of carbon and copper atoms between graphene; and ECCu LJ metal and graphene.

Molecular Dynamics Simulation of Metal Matrix Composites 117 (b) MD simulation The mechanical properties of MGLCs were compared with the pure metal matrix. For this, several models of pure and layered composites of different thicknesses were constructed. The pure metal matrix was taken as a face-centered cubic (fcc) crystalline-structured Cu. Fig. 3.2.1 shows a pure Cu matrix. A number of similar pure Cu structures were modeled using open visualization tool (OVITO) package [16]. The layered composites were composed of alternate layers of Cu and graphene. The top and bottom layers were of Cu, and in between these layers of Cu, the graphene layers were embedded in alternate positions as shown in Fig. 3.2.2. The ˚ in the c-direction. In thickness of the layered composite samples was varied from 100 to 25 A ˚. Fig. 3.2.2D the thickness of the three-layered graphene-reinforced Cu composite was 100 A ˚ in Fig. 3.2.2C, 50 A ˚ in Fig. 3.2.2B, and Similarly the thickness of the same composite was 75 A ˚ 25 A in Fig. 3.2.2A. These composite models were named as Composite1, Composite2, Composite3, and Composite4, respectively. Similarly the pure Cu metal models were denoted as Pure Cu1, Pure Cu2, Pure Cu3, and Pure Cu4, respectively. PBCs were assumed in x- and y-directions for both pure Cu and layered composites. In z-direction the models were free for both the cases. The LAMMPS was used for performing all the simulations. An isothermal-isobaric ensemble (NPT) was firstly used for equilibration of all the models. This was followed by loading the models along the x-axis under a canonical ensemble (NVT). The velocity Verlet algorithm was used for integrating the equations of motion. The models were loaded with a strain rate of 0.0001 (1/ps) at a fixed temperature of 300 K. This strain rate was finalized keeping in mind the size of the models and the available computational capacity. Various strain rates of 0.00005, 0.0005, and 0.001 were also applied to all the models, and the results were compared with the main strain rate of 0.0001.

Fig. 3.2.1 Atomic model of pure Cu matrix.

118 Chapter 3

Fig. 3.2.2 Atomic model of Cu-graphene layered composites with thickness (A) 25, (B) 50, ˚. (C) 75, and (D) 100 A

3.2.2 Results and Discussion Table 3.2.1 shows the predicted values of Young’s modulus (Ex) and Poisson’s ratio (νxy) of Composite1 at different strain rates. It could be seen that the mechanical properties were almost independent of the strain rate, within the elastic region. All the pure and layered composite models were subjected to an NPT ensemble for the duration of 60 ps so that the structured models get equilibrated. These were then stretched in the x-direction by applying a uniaxial tensile load. In this section the mechanical properties of both pure Cu and Cu-graphene layered composites have been predicted. The stress-strain curves obtained from the MD simulations have been used for the analysis of the obtained results. Finally the deformation mechanisms of all the samples at different strains have also been discussed. (a) Stress-strain plots Fig. 3.2.3 highlights the stress-strain curves for pure Cu models of different thickness denoted as Pure Cu1, Pure Cu2, Pure Cu3, and Pure Cu4. It could be inferred from Fig. 3.2.3 that thickness has insignificant effect on the properties. The models were stretched elastically under the applied load. This was followed by yielding of the pure Cu models. The models then Table 3.2.1 Mechanical properties of composite1 at different strain rate Strain Rate Ex (GPa) νxy

0.00005 208 0.27

0.0001 208 0.27

0.0002 208 0.27

0.0005 209 0.28

0.001 209 0.28

Molecular Dynamics Simulation of Metal Matrix Composites 119

Fig. 3.2.3 Stress-strain diagrams of pure Cu models.

reached a maximum stress value that was followed by plastic deformation. Fig. 3.2.4 depicts the stress-strain curves for all MGLCs, namely, Composite1, Composite2, Composite3, and Composite4. The models were found to deform elastically under the applied load. When the load was increased, yielding was observed (at the first drop on the stress-strain diagram) at point A (shown in Figs. 3.2.5–3.2.8). This was found to occur at a strain less than that of the pure Cu model. This was attributed to the effect of C-Cu interactions at the metal-graphene interfaces. 110 100

Composite1

90 80 sx (GPa)

70

Composite2

60 50 Composite3

40 30 20

Composite4

10 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ex

Fig. 3.2.4 Stress-strain diagrams of Cu-graphene layered composites.

120 Chapter 3 20 C

B

Composite1

σx (GPa)

A

D

10

Pure Cu1

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ex

Fig. 3.2.5 Effect of graphene layers on mechanical properties of Composite1 in comparison with its Pure Cu1 counterpart. 30 C B Composite2 sx (GPa)

20 A

D

10 Pure Cu2

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ex

Fig. 3.2.6 Effect of graphene layers on mechanical properties of Composite2 in comparison with its Pure Cu2 counterpart.

These interactions resulted in load transformation through the interfaces. Point B in Figs. 3.2.5– 3.2.8 highlights the fact that the MGLC models could bear the applied load successfully up to this point, when one of the graphene layers cracked. This was followed by the failure of the other two graphene sheets, denoted by points C and D in Figs. 3.2.5–3.2.8. Finally the composites failed because of the applied tensile load.

Molecular Dynamics Simulation of Metal Matrix Composites 121 40 B Composite3

30 sx (GPa)

C

D

20 A

Pure Cu3

10

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ex

Fig. 3.2.7 Effect of graphene layers on mechanical properties of Composite3 in comparison with its Pure Cu3 counterpart.

100 C Composite4

80

B

sx (GPa)

D 60

40 Pure Cu4

20

A

0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ex

Fig. 3.2.8 Effect of graphene layers on mechanical properties of Composite4 in comparison with its Pure Cu4 counterpart.

It could be seen that the graphene reinforcement significantly improved the strength and the failure strain of MGLCs. Increasing the thickness of the Cu layers from Fig. 3.2.2A–D, the strengthening effect of graphene reduced considerably. Or it can be stated that increasing the graphene percentage (Fig. 3.2.2A–D) from Composite1 to Composite4, the mechanical

122 Chapter 3 Table 3.2.2 Mechanical properties of pure Cu and MGLCs Ex (GPa) νxy Tensile strength (GPa)

Pure Cu1

Pure Cu2

Pure Cu3

Pure Cu4

Composite1

Composite2

Composite3

Composite4

114 0.5 9.7

112 0.49 9.5

120 0.52 9.9

128 0.51 10.8

208 0.27 18.0

240 0.30 24.5

330 0.17 32.4

596 0.08 92.8

properties were found to improve. Further, to compare the mechanical properties of MGLCs with graphene, a single graphene sheet was modeled and was subjected to the same boundary conditions and similar loading. The elastic modulus of pure graphene was found as 1.07 TPa, whereas its tensile strength was found to be 190 GPa. The ultimate strain was found to be 34%. It could be observed from Fig. 3.2.4 and Table 3.2.2 that the tensile strength of the MGLCs was comparatively lower in comparison with the single graphene, specifically for those layered models that had higher percentage of Cu. But the graphene reinforcement was found to be significantly increasing the mechanical properties of MGLCs. This could be inferred from Table 3.2.2 and Figs. 3.2.5–3.2.8. Thus both Cu and graphene layers were found to be playing an important role in load transfer through them. Also the increase in tensile strength of MGLCs with the addition of graphene nanolayers was found to be in good agreement with an experimental work by Kim et al. [6]. Figs. 3.2.5–3.2.8 show the stress-strain curves for each layered composite in comparison with the pure Cu counterpart. The stiffness and strength of the MGLCs were found to be improved significantly with the addition of graphene reinforcement. Figs. 3.2.5–3.2.8 clearly showed the excellent effect of graphene nanolayers on both elastic and plastic zones of the layered composites. The yielding of pure Cu was indicated by the first big fall on the stress-strain diagrams of pure Cu models. This was followed by plastic deformation under the applied tensile load. Similarly the elastic deformation of MGLCs was denoted by point A, where the first small fall of stress occurred. After this point the MGLCs continued to resist the applied load because of the strong graphene layers until the next large drop of stress occurred at point B. In the region between points A and B, the Cu metal layers deformed plastically, while the graphene layers undergo elastic deformation. One of the layers of graphene was found to fracture at point B, between the strain values of 20% and 25%. After this the stress remained almost constant until point C, where the second graphene layer also failed. Upon increasing the load further, the third graphene layer also fractured at point D, thus showing that the composite as a whole had failed. The toughness of the material improved considerably with the addition of graphene layers in Cu. The ultimate strain of MGLCs was higher in comparison with that of pure graphene. The trend of the stress-strain diagrams was found to be similar to the experimental work by Kim et al. [6]. (b) Elastic modulus and Poisson’s ratio Young’s modulus Ex and Poisson’s ratio νxy of the modeled pure Cu and MGLCs were determined using the elastic zone of the stress-strain curves from the origin to point A in

Molecular Dynamics Simulation of Metal Matrix Composites 123 Figs. 3.2.5–3.2.8. For obtaining a single value the values were obtained for each time step of MD simulation, and then an average value was obtained on the linear limit of the stress-strain diagrams. From Table 3.2.2, it could be observed that the graphene reinforcement considerably increased the elastic modulus (Ex) of MGLCs, namely, the value of Ex for Composite1 increased by 82% in comparison with its pure Cu counterpart. Similarly for Composite2, Composite3, and Composite4, the percentage increase in Ex were found to be 114%, 175%, and 365%, respectively, with respect to their pure Cu counterparts. The percentage decrease in νxy for Composite1, Composite2, Composite3, and Composite4 with respect to their pure Cu counterparts was found to be 46%, 39%, 67%, and 84%, respectively. The tensile strength of MGLCs was found to increase by 86%, 158%, 227%, and 759% for Composite1, Composite2, Composite3, and Composite4, respectively. (c) Mechanism of deformation The models of pure Cu, namely, Pure Cu1, Pure Cu2, Pure Cu3, and Pure Cu4, were found to deform plastically under the applied strains of 0.1, 0.2, and 0.4. During the plastic deformation, dislocations were formed that propagated within the lattice along the planes of maximum shear stress. These dislocations nucleated because of stress concentration at the yield point (highest point in Fig. 3.2.3). The plastic deformation continued with the help of slip plane movement, which corresponds to the fall in stress beyond the yield point in Fig. 3.2.3. In the case of MGLCs, the graphene layers could block the movement of dislocations and thus could prevent the failure of these composites. Because of the restricted movement of dislocations across the composite, strain hardening occurred, which resulted in high strengths for MGLCs as could be inferred from Fig. 3.2.4. The yielding of Composite1 occurred at the yield point (point A in Fig. 3.2.5). In the region between points A and B, plastic deformation continued with the buildup of dislocations at various locations near the graphene-Cu interface. This continued until one of the graphene layers fractured at point somewhere between points B and C in Fig. 3.2.5–3.2.8. Finally the fracture of the MGLCs occurred beyond point D when all the graphene layers had fractured. Similar deformation mechanism was applicable for all the MGLCs, namely, Composite2, Composite3, and Composite4. The restricted movement of the dislocations could be due to the strong covalent network of graphene. Kim et al. [6] observed the same effect during the compression tests on Cu and Ni/graphene layered composites. The graphene layers caused accumulation of the dislocations at the Cu-graphene interface resulting in enhancement of strength and modulus of MGLCs.

3.2.3 Conclusion The mechanical behavior of pure Cu and MGLCs was analyzed using uniaxial tensile load by making use of MD simulations. The deformation mechanism was also studied. The graphene layers were found to significantly enhance the strength, stiffness, and failure strain of MGLCs.

124 Chapter 3 The value of Ex for Composite1 increased by 82% in comparison with its pure Cu counterpart. Similarly for Composite2, Composite3, and Composite4, the percentage increase in Ex was found to be 114%, 175%, and 365%, respectively, with respect to their pure Cu counterparts. Reducing the thickness of the pure Cu layers, the mechanical properties of MGLCs were found to improve. Because of the strong in-plane strength of graphene layers, the deformation pattern of Cu layers was strongly influenced. The graphene layers were found to be providing efficient barrier to the propagating dislocations and slip planes inside the MGLCs. This study has resulted in the development of novel MGLCs that could be used in the design of advanced structural materials having high tensile strength and being lightweight at the same time.

References [1] I.A. Ovid’ko, Mechanical properties of graphene, Rev. Adv. Mater. Sci. 34 (2013) 1–11. [2] R. Rezaei, M. Shariati, H. Tavakoli-Anbaran, C. Deng, Mechanical characteristics of CNT-reinforced metallic glass nanocomposites by molecular dynamics simulations, Comput. Mater. Sci. 119 (2016) 19–26. [3] X. Gao, H. Yue, E. Guo, et al., Preparation and tensile properties of homogeneously dispersed graphene reinforced aluminum matrix composites, Mater. Des. 94 (2016) 54–60. [4] S. Sharma, R. Chandra, P. Kumar, Mechanical and thermal properties of graphene-carbon nanotube-reinforced metal matrix composites: a molecular dynamics study, J. Compos. Mater. 51 (23) (2017) 3299–3313. [5] R. Rezaei, C. Deng, H. Tavakoli-Anbaran, M. Shariati, Deformation twinning mediated pseudoelasticity in metal–graphene nanolayered membrane, Philos. Mag. Lett. 96 (2016) 322–329. [6] Y. Kim, J. Lee, M.S. Yeom, et al., Strengthening effect of single-atomic-layer graphene in metal-graphene nanolayered composites, Nat. Commun. 4 (2013) 2114. [7] H.G.P. Kumar, M.A. Xavior, Graphene reinforced metal matrix composite (GRMMC): a review, Procedia Eng. 97 (2014) 1033–1040. [8] Z. Li, Q. Guo, Z. Li, et al., Enhanced mechanical properties of graphene (reduced graphene oxide)/aluminum composites with a bioinspired nanolaminated structure, Nano Lett. 15 (2015) 8077–8083. [9] J. Zhu, S.N. Luo, J.Y. Huang, L. Wang, B. Li, X.J. Long, Shock response of Cu/graphene nanolayered composites, Carbon 103 (2016) 457–463. [10] H.Y. Song, X.W. Zha, Influence of nickel coating on the interfacial bonding characteristics of carbon nanotube-aluminum composites, Comput. Mater. Sci. 49 (2010) 899–903. [11] S. Lee, B.K. Choi, G.H. Yoon, Molecular dynamics studies of CNT-reinforced aluminum composites under uniaxial tensile loading, Compos. Part B 91 (2016) 119–125. [12] N. Silvestre, B. Faria, J.N. Canongia Lopes, Compressive behavior of CNT-reinforced aluminum composites using molecular dynamics, Compos. Sci. Technol. 90 (2014) 16–24. [13] X.Y. Liu, F.C. Wang, H.A. Wu, W.Q. Wang, Strengthening metal nanolaminates under shock compression through dual effect of strong and weak graphene interface, Appl. Phys. Lett. 104 (2014) 23190 (1–4). [14] H.W. Sheng, M.J. Kramer, A. Cadien, T. Fujita, M.W. Chen, Highly optimized embedded-atom-method potentials for fourteen fcc metals, Phys. Rev. B 83 (2011) 134118. [15] J. Tersoff, Modeling solid-state chemistry: interatomic potentials for multicomponent systems, Phys. Rev. B 39 (1989) 5566. [16] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO—the open visualization tool modeling, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012.

Further Reading [17] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19.

Molecular Dynamics Simulation of Metal Matrix Composites 125 Chapter 3.3

Molecular Dynamics Simulation of Lithium Metal/Polymer Electrolyte Interfacial Properties Using GROMACS Amit Bansal*, Prince Setia†, Raj Chawla‡ *

Department of Mechanical Engineering, I.K. Gujral Punjab Technical University, Kapurthala, India, Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur, India, ‡ Department of Mechanical Engineering, Lovely Professional University, Phagwara, India †

Li metal has been widely used as an anode material for high-energy density electrochemical storage devices. The commercialization of Li metal as an anode material started in 1970s [1]. Initially, it had to face a lot of difficulties because of several factors such as the growth of dendrites, safety risks, and low efficiencies. In the early 1990s Li-ion batteries replaced the Li metal ones in most of the products. These Li-ion batteries were based on intercalation compounds such as graphite. However, the specific capacity of Li metal is approximately 10 times of that of graphite. Also, graphite-based batteries suffer from aging difficulty [2]. The next generation of Li batteries such as Li-air or Li-S batteries uses Li metal as the anode because of the increased safety due to reduction in the formation of dendrites. It has been reported [3] that to limit the growth of dendrites, the shear modulus of the electrolyte should be nearly twice of that of the Li metal. If the dendrites grow rapidly, the Li-metal batteries may become unstable and thus cause safety concerns. The key to stabilize the Li-metal anode is the electrolyte [1]. The commonly used electrolyte solvents for Li-ion batteries such as the liquid organic carbonates are highly reactive on Li-metal surface. These solvents can form an electrically insulating and ionically conductive layer on the anode surface during the first charge/discharge cycles, known as the solid electrolyte interphase (SEI). This SEI layer suppresses the growth of dendrites in Li-ion batteries, while in the case of Li-metal batteries, this SEI layer is unstable leading to a rapid growth of dendrites. In comparison with the SEI, the solid polymer electrolyte (SPE) has a higher mechanical strength that can result in the suppression of the dendritic growth. The use of SPEs as electrolyte was initiated by Wright et al. [4]. It was shown by [4] that the ether-based polymers have high conductivity in the salt solution of PEO. This area was further explored significantly by Armand et al. [5,6] showing the potential use of PEO in Li batteries. In using this liquid-free polymer electrolyte in a working battery cell, it must have high ionic conductivity at ambient temperature, high value of lithiumion transfer number, and sufficient dissolution and dissociation properties of the lithium salt [7].

126 Chapter 3 Most of the solid-state polymer batteries are unable to achieve these properties, thus limiting their widespread usage. The polymer electrolytes can be distinguished as true SPEs [6], SPEs with small amount of low-molar-mass polar compounds known as plasticized systems [8], gel polymer electrolytes (GPEs) [9], rubbery systems or polymers in salt [10,11], and composite electrolytes [8,12,13]. The commonly used SPEs are obtained by the dissolution of lithium salt in a polymer matrix. There are various factors that govern the lithium-ion conductivity of these electrolytes such as the molecular weight of polymer and the nature of the lithium salt anion [9,14]. There are certain limitations of both high- and low-molecular-weight polymer electrolytes. The highmolecular-weight polymer electrolytes generally have poor ionic conductivity [15], while the low-molecular-weight polymer electrolytes are generally unable to reduce the growth of dendrites on the Li-metal anode, although the systems with high concentrations have shown the property of reducing the dendritic growth [16,17]. MD simulations have been widely used as a computational technique for studying SPE systems. This is because the dynamic properties such as ionic motion can be fully explained with respect to the molecular structure of the systems. The results can also be compared with the experimental studies. Most of the MD studies have been conducted on high-molecular-weight PEO with a variety of Li salts [18]. All these MD simulations have shown that the bonding between Li cations in PEO electrolyte is very strong with an average coordination of 5–6 ether oxygen atoms [19]. This has a profound influence on the mechanism of Li+ conduction. A number of force fields have been used for conducting MD studies of SPEs [20–23] including the polarizable force fields used for modeling of branched and linear PEO of varying molecular weight at different temperatures, with different Li salts and concentrations. MD simulations have been performed on crystalline PEO-based electrolytes [24], polyelectrolytes with tethered anions [25], and SPEs containing ceramic nanoparticles [26]. Recently, also alternative polymer hosts to PEO have been explored in MD studies [27]. Most of these studies have been conducted using bulk polymer as electrolytes in SPE materials. There exist some studies on SPE/electrode interfaces in Li-ion batteries. For example, PEO/V2O5 interface has been studied by MD simulations [28]. The structural and dynamic properties of the interfacial regions have been compared with the bulk-like parts of the electrolyte. Borodin et al. [29] have investigated the interface between PEO and TiO2 using a quantum chemistry-based force field. It was observed that the TiO2 surfaces have significant effects on the PEO density. The polymer located in the interface regions displayed reduced conformational and structural relaxations as compared with the bulk-like regions. From the literature review, it can be said that there are very few studies on the PEO electrolyte/Li-metal interfaces using MD simulations. In this work, MD simulations on LiTFSI-doped PEO polymer at the surface of a lithium metal anode have been performed using GROMACS 5.1.1 [30]. The effects of the Li-metal surface on the structural properties and

Molecular Dynamics Simulation of Metal Matrix Composites 127 on the ionic diffusion of the SPE have been studied, and the results have been compared with those of the bulk SPE. An atomistic picture on how the structure-dynamic properties of this material changes close to the electrode surface, which will be decisive for the operation of Li-metal SPE battery cells, has been presented.

3.3.1 MD Simulation In this study, all the MD simulations have been performed using GROMACS 5.1.1 [30]. The effect of a Li surface on the structural and dynamic properties of the electrolyte has been studied using two different models. First model was that of an analytic wall potential model in which the Li atoms were (implicitly described) on both sides of the polymer electrolyte. This model has been shown in Fig. 3.3.1A. The second model was an explicit Li atom surface model, which forms an interface with the electrolyte. This has been shown in Fig. 3.3.1B. To compare the results a bulk polymer electrolyte box has also been simulated. The simulations have been performed using a nonpolarized Li-metal model and without any applied external field, which corresponds to a situation when the battery is not in operation. The bulk electrolyte was composed of PEO and LiTFSI salt. The polymer electrolyte box was constructed from 20 PEO chains, each containing 30 ethylene oxide (EO) units. Steepest-descent algorithm was used for

Fig. 3.3.1 Equilibrated MD models of (A) Li walls and (B) Li slab.

128 Chapter 3 minimizing the energy. MD simulations were performed for 250 ns for all the systems with a time step of 0.001 ps using velocity Verlet algorithm for pure polymer electrolyte and a leapfrog integrator for Li-slab and Li-wall models. These were decided keeping in mind the stability and computational time. The MD simulations were performed using a constant number of atoms (N), pressure (P), and temperature (T) ensemble, that is, NPT ensemble. The thermostat used for pure polymer electrolyte was Berendsen [31], whereas for Li walls the Parrinello-Rahman [32] thermostat was used. Temperature in both the thermostats was taken as 400 K. The final configuration, obtained after equilibration of the pure polymer electrolyte, was then used for making the simulation models of the MD boxes such as the Li-wall model or the Li-slab model. PBCs have been used in all the directions for Li slab and bulk polymer electrolyte. For Li-wall model the PBCs were applied in x- and y-directions only. A 12-6 LJ potential was used for Li walls, having the force constants similar as that of a Li-slab model. Lorentz-Berthelot combining rules have been used for describing the interactions with all other particles. A constant number of atoms (N), volume (V), and temperature (T) ensemble, that is, NVT ensemble at 400 K, was applied to the Li-slab model. This was done to prevent it from changing to an fcc structure during the calculations. To study the effects of the Li surface on SPE and not the other way round, a LJ model was used for Li atoms. The elevated temperature was chosen because of the poor conductivity of Li+ [33] and also because the same temperature was used in the previous MD simulation of PEO/TFSI [34]. All the molecular systems were prepared using the Packmol package. This package helps in keeping the particles in the box with the closest distance greater than a chosen tolerance level between the molecules. Except Li+ particles, all the particle interactions were modeled using the LJ potential from optimized potentials for liquid simulations-all atom (OPLS-AA) [35] force field. This potential has been shown in Eq. (3.3.1): 

r 6  ro 12 o (3.3.1) 2 E ¼ εo r r εo is the equilibrium nonbonded energy and ro is the equilibrium nonbonded distance between two similar atoms. For the pair interactions of Li ion, the LJ force field parameters were based on the generalized Amber force field (GAFF) [36]. The procedure of obtaining the LJ parameters for bodycentered cubic (bcc) Li metal was the same as that for fcc metals in a previous MD simulation study [37]. In this method the target was to achieve the experimental density and surface tension of Li metal at 298 K. For this purpose a 5
5 bcc Li slab was modeled from the experimental ˚ 3. This has data of [38] containing 10 layers and having dimensions of 17.5
17.5
17.5 A been shown in Fig. 3.3.2A. The deviation of density of Li during the simulations was limited to 0.05%. To achieve this deviation the simulations were performed using NPT ensemble at 298 K. The surface tension was investigated using two Li (100) slabs that were modeled by cleaving the Li bulk cell. These have been shown in Fig. 3.3.2B and C.

Molecular Dynamics Simulation of Metal Matrix Composites 129 The Li (100) slab has been found to be the most stable surface orientation [39]. The surface energies were calculated by using Eq. (3.3.2) using the NVT MD simulations under normal conditions: Esurf ¼

Es  Eu 2A

(3.3.2)

Es is the total energy of the separated slabs; Eu is the total energy of the unified slabs; and A is the surface area of the slabs. The final surface energy obtained using MD simulations was found to be in agreement with the experimental results of [40], with a deviation of only 0.6%. The values of εo and ro of the LJ parameters were found to be 10 kJ/mol and 0.29 nm for bcc Li, respectively. The final MD model was generated by using 9
9 Li (100) slab with 10 layers along with the equilibrated polymer electrolyte. To avoid the periodic image interactions, vacuum region has been employed in the super cell.

3.3.2 Results and Discussion (a) Structural properties To study the structural properties of SPE interacting with the Li wall or slab, the radial distribution functions (RDFs) have been used as shown in Fig. 3.3.3. The RDF plots in

Fig. 3.3.2 See figure legend on next page.

Fig. 3.3.2—cont’d MD model of (A) bcc Li super cell, (B) unified surface (C) separated surfaces obtained by cleaving in (100) direction.

Molecular Dynamics Simulation of Metal Matrix Composites 131

Li+ -O(TFSI) 35

Radial distribution function, g(r)

30

Li slab + PE

25 20 Li wall + PE 15 10 Bulk PE

5 0 0

(A)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r (nm)

1

Li+ -O (PEO) 50

Radial distribution function, g(r)

45 Li slab + PE

40 35 30 25

Li wall + PE

20 15 10

Bulk PE

5 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r (nm)

1

(B) Fig. 3.3.3 RDFs of (A) Li+-OTFSI and (B) Li+-OPEO for the systems under consideration.

132 Chapter 3 Fig. 3.3.3 show that the highest peak occurs at 0.202 nm for both Li+-OPEO and Li+-OTFSI pairs. The first coordination shell displays sharp Li+-O peaks in the Li-slab model but slightly smoother for the Li walls and the bulk SPE systems. This trend indicates a more well-defined structure in the electrolyte by introducing the Li slab. The second and third coordination shells, which occur at Li+-OTFSI distances of 0.44 and 0.6 nm, were found to be more distinct in the Li-slab model and suggest an intermediate range order (IRO). For longer distances the peaks vanish in the bulk SPE, indicating only very local ordering in this system. The coordination numbers (CNs) were calculated from integrations over the RDFs. These have been shown in Fig. 3.3.4. Fig. 3.3.4A shows the CN for Li+-OTFSI interactions, while Fig. 3.3.4B shows the CN for Li+-OPEO interactions. Generally, all CN functions were found to be similar in all investigated systems, but still, some notable differences could be observed. The average CN for ether oxygen in the first shell around the Li+ ion was about 5 and for oxygen atoms of TFSI around 0.9, indicating an average CN of 6 of which one was an oxygen from the anion. Such a high degree of ion pairing was not unexpected considering the rather high salt concentration. The first Li+-O peak in Fig. 3.3.3 was in good agreement with neutron diffraction isotopic substitution (NDIS) experiments on PEO/LiTFSI at 296 K [41] and also displayed similarities with MD studies on PEO/LiTFSI at 393 K by Borodin and Smith [19], although the CN obtained here was somewhat higher. Raman spectroscopy studies [42] and MD simulations [19] also showed that ion aggregation of PEO/LiTFSI did not change remarkably with temperature, hence validating this comparison. The CN of OTFSI for Li+ was found to be higher in the case of Li slab, followed by Li-wall and Li bulk SPE. This shows a highly ordered structure of anions in the case of electrolyte, when a surface model of Li slab was introduced. An exactly opposite trend was seen for OPEO around the cation, thereby maintaining the total CN for three systems. This could be attributed to the attraction of charged species by the surfaces followed by an interaction with each other resulting in more ordered structures. To understand the effect of a surface on the SPE, the structural properties of the electrolyte closer to the surface of Li metal have been investigated and have also been compared with the regions that were further away in a region called “bulk-like” region. For this purpose, two ˚ from the Li slab in the so-called different distances in the z-direction, first within 10 A ˚ from the Li slab in the bulk region, were interfacial region and second in the range of 10–25 A selected. Fig. 3.3.5 shows the obtained results. It can be observed that both Li+-OPEO and Li+OTFSI RDFs show sharper peaks in the interfacial region (Fig. 3.3.5B) than in the bulk (Fig. 3.3.5B), thus confirming that the introduction of an interface imposes local structure on the ions in this region, while more disordered regions seem to appear further from the surface. This picture was confirmed by the CN functions (Fig. 3.3.6). There were a smaller number of TFSI anions around the Li+ in the bulk area of the electrolyte, indicating a depletion of ions in this region.

Molecular Dynamics Simulation of Metal Matrix Composites 133

Li+ -O (TFSI) 1.50 Li slab + PE

Coordination number, n(r)

1.25 1.00

Li wall + PE

0.75 0.50

Bulk PE

0.25

(A)

0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 r (nm)

Li+ -O (PEO) 7

Coordination number, n(r)

6 5 4

Li wall + PE 3 2 1

(B)

Bulk PE

Li slab + PE

0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 r (nm)

Fig. 3.3.4 CNs of (A) Li+-OTFSI and (B) Li+-OPEO for the systems under consideration.

134 Chapter 3

Bulk region

Radial distribution function, g(r)

200

(Li+) O(PEO)

150

100

(Li+) - O(TFSI)

50

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r (nm)

1

(A)

Interfacial region

Radial distribution function, g(r)

400 350 300

(Li+) -O(PEO)

250 200 150 100

(Li+) -O(TFSI)

50 0

(B)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r (nm)

Fig. 3.3.5 RDFs of Li+-OTFSI and Li+-OPEO for (A) the bulk region and (B) the interfacial region.

Molecular Dynamics Simulation of Metal Matrix Composites 135

Bulk region

Coordination number, n(r)

6.00 5.00

(Li+) O(PEO)

4.00 3.00 2.00

(Li+) O(TFSI)

1.00

0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 r (nm) (A)

Interfacial region

Coordination number, n(r)

6

(B)

5 4

(Li+) O(PEO)

3 2 1

(Li+) O(TFSI)

0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 r (nm)

Fig. 3.3.6 CNs of Li+-OTFSI and Li+-OPEO in (A) the bulk region and (B) the interfacial region.

136 Chapter 3 It could be seen from Fig. 3.3.6 that the densities of Li+ and TFSI ions were significantly higher closer to the interfaces in both the wall and slab models. Moving from the interfaces to the center of the box led to a clear reduction in the probability of finding either species, but primarily the ions and not least the Li cations, which experienced the most dramatic redistribution when surfaces were introduced. This result indicated the formation of a double layer close to the electrode surface, which introduced a resistance in the ionic transport over this boundary layer. This observation was in agreement with the previous experimental works [43,44]. Moreover, in both systems, PEO and anion particles were closer to the surface of Li metal than the Li ions, thus indicating a preference for direct interactions with the electron-rich compounds. To monitor the changes in the PEO chain due to the redistribution of salt in SPE systems and also due to possible interactions with the Li surfaces, the mean-squared radius of gyration hhs2ii and the mean-squared end-to-end distance hhr2ii were calculated in the three systems as given by Eqs. (3.3.3), (3.3.4). The results obtained have been tabulated in Table 3.3.1: ** ++  2  1X (3.3.3) s ¼ kri  rcm k2 N i EE  2  DD r (3.3.4) ¼ krn  ro k2 rcm is the coordinate of the center of mass of a PEO chain; N is the number of atoms in a chain; ro is the coordinates of the first atom of the chain; and rn is the coordinates of the last atom of the chain. The results in Table 3.3.1 show that the PEO structure was slightly more extended in the system with Li surfaces than in the bulk SPE, where the PEO chain had the highest compactness. This was likely due to that there was less ion pairing in the bulk SPE system, as seen in Fig. 3.3.4. The more intense coordination between PEO and Li caused physical cross-links in the system and consequently decreased the radius of gyration. The mean-squared radius of gyration hhs2ii and the mean-squared end-to-end distance hhr2ii were also calculated for PEO in the interface and bulk-like regions for Li-slab model at 400 K. These have been shown in Fig. 3.3.2. It could be observed that all values decrease as compared with the analysis of the larger box. This was an artifact of the method applied for the estimation of radius of gyration, where the smaller volume Table 3.3.1 Average values of radius of gyration hhs2ii1/2 and end-to-end distance hhr2ii1/2 in bulk PE, Li walls, and Li slab at 400 K (units in nm) Bulk SPE Li walls Li slab

hhs2ii1/2

hhr2ii1/2

0.74 0.80 0.78

2.01 2.16 2.10

Molecular Dynamics Simulation of Metal Matrix Composites 137 of the investigated box only allows detection of more limited polymeric sequences. Second the values show the reversed order in comparison with the values of Table 3.3.1. The value of hhs2ii1/2 was now larger in the bulk region in comparison with the interfacial region. This could be because PEO backbone was more contracted close to the Li surface, while the depletion of ions in the bulk region allowed larger polymer flexibility and thus a larger radius of gyration. This was in accordance with the higher density of the Li ions in the interfacial regions and consequently a stronger coordination of the PEO chains to the ionic species. Another effect could be the surface interactions between the Li slab and the PEO chain itself. This corresponded well with the results obtained by Borodin et al. [29], which saw a flattening of PEO close to the interface in MD simulations of SPE/TiO2 systems (Table 3.3.2). (b) Dynamics of diffusion To study the dynamics of bulk PE and the Li-wall and Li-slab models, the self-diffusion coefficients (Di) of the particles have been calculated using the Einstein equation from the diffusive regions using the mean-square displacements (MSD) given by Eq. (3.3.5). The brackets signify the ensemble average at 400 K. ½Ri ðtÞ  Ri ð0Þ2 t!∞ 6t

Di ¼ lim

(3.3.5)

Ri(t) is the vector position of species i at time t. The results obtained using Eq. (3.3.5) have been given in Table 3.3.3. The diffusion constants for the particles in the bulk clearly show a correlation between the Li-ion and PEO dynamics. However, the dynamics of TFSI ion was not connected to them and generally displayed a higher diffusion constant, indicating a high negative transport number (t-). In the Li-wall model the three particles do not exhibit significant differences in their dynamics and possess similar Table 3.3.2 Average values of radius of gyration hhs2ii1/2 and end-to-end distance hhr2ii1/2 in interface and bulk-like regions for Li-slab model at 400 K (units in nm) ˚ from Interfacial region (