Introduction to classical molecular dynamics (MD) simulation

Chapter 3 Methodology

3.3 Introduction to classical molecular dynamics (MD) simulation

MD is based on laws of thermal dynamics, statistics and Newtonian mechanics to predict various properties of N-body systems by simulating the motion of atoms, which is broadly applied in materials science and chemical physics nowadays. Since the classical MD calculation was successfully used to simulate the phase transition process for a hard sphere system in 1957,³²⁹ it is acknowledged as one of the most powerful methods to study the structures of materials at the atomic level.

Figure 3.2 shows the schematics of a classical MD simulation, and there are basically four steps: (1) Build an initial configuration and set preliminary conditions, e.g., atomic positions, atomic velocities, interactions between atoms, boundary conditions, and set temperature and pressure conditions;

(2) Optimize the initial configuration to avoid possibly unreasonable structures, in general, a relative stable configuration with a low total energy is the basis for executing a MD simulation;

(3) Perform the time integration of MD calculations under the pre-defined constraints, including calculating the forces exerted on each atom by differentiating the potential energy surface, and then solving the motion equations of the system to obtain the atomic positions and velocities at the next moment;

(4) Analyze and extract relevant physical quantities of the system.

59 Figure 3.2 The schematic of the classical MD simulations.

3.3.1 Interactions between atoms

During MD simulations, choosing an appropriate potential to describe the interactions between atoms is critical for the accuracy of the simulated results. In fact, it is impossible to determine a potential function that universally describes different systems because the interactions between different atoms are quite different, and therefore, various empirical or semi-empirical potentials have been proposed.

Initially, pair potentials that treat the atomic interactions as only a function of the relative position between two atoms were popular, however, in the actual multi-atom system, the existence of one atom must affect the interactions of other atoms. Therefore, potential energy surfaces between atoms have been gradually developed from the pair ones to many-body ones, and currently various many-body potentials have been proposed to describe the interactions between atoms for various system. In this thesis, MD simulation was performed in studying the formation mechanisms of the Cu(111) foil and embedded atom model (EAM) potential was adopted to describe Cu-Cu interactions.³³⁰

The EAM potential is widely used in metal systems, the potential energy of the ith atom is:

𝐸_𝑖 = 𝐹_𝛼(∑ 𝜌_𝛽(𝑟_𝑖𝑗)

𝑗≠𝑖

) +1

2∑ 𝜙_𝛼𝛽(𝑟_𝑖𝑗)

𝑗≠𝑖

(3.27)

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where 𝑟_𝑖𝑗 is the distance between the ith and jth atoms, 𝜌_𝛽 represents the contribution to the electron charge density from the jth atom of type 𝛽 at the location of the ith atom, the embedding function 𝐹𝛼

represents the energy required to place the ith atom of type 𝛼 into the electron cloud, and 𝜙𝛼𝛽 is a pair potential function. Clearly, EAM potential is a many-body potential because the electron charge density is a summation over many atoms. A cutoff radius is needed in this potential, beyond which the interactions between atoms can be ignored.

3.3.2 Boundary conditions

Since the macroscopic properties of materials are determined by a huge number of particles, to accurately simulate the behaviors of a real material, a very large system is required. However, due to the limitations of computer memory and computing power, MD simulations can only deal with a limited number of atoms, and hence an appropriate boundary condition is necessary to mimic the bulk materials.

At present, there are mainly two boundary conditions, which are periodic boundary conditions and non- periodic boundary conditions.

Periodic boundary condition is very useful for simulating bulk materials via a system with a limited number of atoms in MD simulations. A simulation box containing dozens to tens of thousands of atoms is built as a simulation unit cell, the periodic boundary condition then creates an infinite number of identical boxes around this unit cell, and in this case, once there are atoms moving out of the simulation box from one side, there must be a corresponding number of atoms entering the box from the opposite side. Periodic boundary condition is broadly used to simulate the macroscopic properties of real materials by using a limited number of atoms, greatly saving the calculation time and improving the calculation efficiency. However, the periodic boundary condition is not always applicable to every direction of the system, especially the low-dimensional materials. For instance, nonperiodic boundary condition is usually required in investigating zero-dimensional nanoparticles, one-dimensional nanowires or 2D materials. In fact, the condition of combining periodic and nonperiodic boundaries is commonly adopted in the simulated system depending on the specific requirements of the studied materials.

3.3.3 Ensemble

Ensemble is a statistical concept, which is a collection of systems with the same state. Ensemble controls thermodynamic quantities of the studied system, such as atomic number (N), pressure (P), volume (V), temperature (T), energy (E) and enthalpy (H), and it can be classified into microcanonical ensemble (NVE), canonical ensemble (NVT), grand canonical ensemble (μVT), Isothermal–isobaric ensemble (NPT) and Isoenthalpic–isobaric ensemble (NPH), etc.

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NVE ensemble is the one with constant N, V and E, which corresponds to an adiabatic process and has no heat and mass exchange with the outside environment. In NVT ensemble, N, V and T of the system are conserved, which can be regarded as an isolated system placed in a thermostat and is generally suitable for the MD simulation at a constant temperature. To ensure a constant temperature, thermal exchange with the environment is allowed and a variety of thermostat algorithms have been developed, such as Nosé-Hoover thermostat, Berendsen thermostat, etc. For μVT ensemble, μ, V and T of the system are kept constant, and is usually used for the system that can exchange both energy and mass with the environment. NPT ensemble keeps N, P and T of the system unchanged, and therefore, besides a thermostat, a barostat is also needed to keep a constant pressure. NPH ensemble requires N, P and H of the systems to be constant, which is difficult to realize and therefore is not common in practical MD simulations.

3.3.4 Time integration algorithm

MD simulations require time integration of the motion equation. During simulation, the motion equation is discretized into finite difference equations by the finite difference algorithm, and common finite difference algorithms include Verlet and Gear predictor-corrector methods. The Verlet integration is the mostly frequently adopted one in MD simulations, which is based on computing the Taylor expansions to the third order for the atomic position at different time directions:³³¹

𝑟⃑(𝑡 − Δ𝑡) = 𝑟⃑(𝑡) − 𝑣⃑(𝑡)Δ𝑡 +1

2𝑎⃑(𝑡)Δ𝑡²−1

6𝑏⃑⃑(𝑡)Δ𝑡³+ 𝒪(Δ𝑡⁴) (3.28)

𝑟⃑(𝑡 + Δ𝑡) = 𝑟⃑(𝑡) + 𝑣⃑(𝑡)Δ𝑡 +1

2𝑎⃑(𝑡)Δ𝑡²+1

6𝑏⃑⃑(𝑡)Δ𝑡³+ 𝒪(Δ𝑡⁴) (3.29)

where 𝑣⃑(𝑡) = 𝑟⃑^′(𝑡) is the velocity, 𝑎⃑(𝑡) = 𝑟⃑^′′(𝑡) is the acceleration, and 𝑏⃑⃑(𝑡) = 𝑟⃑^′′′(𝑡) is the third derivative of the position regarding to the time.

Adding the two expansions, we get:

𝑟⃑(𝑡 + Δ𝑡) = 2𝑟⃑(𝑡) − 𝑟⃑(𝑡 − Δ𝑡) + 𝑎⃑(𝑡)Δ𝑡²+ 𝒪(Δ𝑡⁴) (3.30) It can be seen that the Verlet integration only gives the atomic positions but not the velocities, and so some improved algorithms were proposed, among them, the Velocity-Verlet algorithm is more commonly used and the expression of position and velocity are as follows:³³²

𝑟⃑(𝑡 + Δ𝑡) = 𝑟⃑(𝑡) + 𝑣⃑(𝑡)Δ𝑡 +1

2𝑎⃑(𝑡)Δ𝑡² (3.31)

𝑣⃑(𝑡 + 𝛥𝑡) = 𝑣⃑(𝑡) +1

2(𝑎⃑(𝑡) + 𝑎⃑(𝑡 + 𝛥𝑡))𝛥𝑡 (3.32)

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It should be noted that the choice of time step is very important for MD simulations, which significantly affects the efficiency and accuracy of the calculations. A smaller step can reduce the calculation error and obtain more accurate results, but it also increases the simulation time and reduces the calculation efficiency. A safe time step is usually set to be about 1/10 of the vibration period of the atoms in the simulated system.

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Chapter 4 A general theory of 2D materials alignment on