In this chapter, the literatures available on the nanoindentation using MD and FEM simulations are broadly discussed. From the discussion, it is clear that the substrates of the indentation, indenter shape, indenter size, indentation depth, indentation speed, etc., are the key parameters of the nanoindentation simulations and their variation can significantly affect the results. The dislocation activities are mainly dependent on the materials and their activated slip systems during the indentation process. The MD simulations of nanoindentation can provide a better insight for the materials deformation at the nanoscale; however, implementing the FEM simulations of nanoindentation can be useful as it is computationally more feasible and provides much analogous results with MD which can be further verified using the experiment.
Chapter 2. Literature Review 18
Table2.1:LiteraturereviewonFEMstudyofnanoindentation. AuthorModelingMaterialandSub- stratesizeIndentersize& IndentationdepthResults ShanandSitara- man(2003)3DFEMandEx- perimentalAlalloy,Ti, 600nmBerkovich,50nmMaterialselasticmodulusdoesnotchangeat thenanoscalebutthehardeningexponentsand yieldstressesarehigheratthenanoscalecom- paretothebulkscale. Bressanetal. (2005)2Daxi-symmetric modelCu,Ti,Fe,18µm height,18µmra- diuscylinder
Conical,300nmThenanoindentationresultsdependhighlyon meshsize,indentertipradiusandimposedhard- eninglaw. Khanetal.(2010)2Daxi-symmetric modelAl−2024alloy, 2mmby2mmConical,2000- 4000nmTheexperimentalresultsshowgoodagreement withtheFEMmodel. Gˆırleanuetal. (2011)2Daxi-symmetric modelAlN,TiN,1200 nmand1800nmConical,540nm, and1200nmLessplasticdeformationofthesubstrateinAlN andAlrichfilmsthanforTiN. Kotetal.(2013)2Daxi-symmetric modelTiNcoatingon Steel,200µmSpherical,20µm, and2.5µmFirstcrackinacoatingappearsontheouter surfaceofthecoating-substrateinterface. Pandureetal. (2014)2Dand3Daxi- symmetricmodelHighspeed steel,12 mm×4mm×2mm
Spherical,200 µm,and150nmFEMmodelingcanproduceloading-unloading curvestopredicttheplasticdeformationduring indentation. Akatsuetal. (2016)2Daxi-symmetric modelBrass,Duralumin 50µmConical,2µmPlasticpropertiesofthematerialsdependonthe elasticmodulus,residualdepthtoindentation depthratio,andindenterangle. LofajandN´emeth (2017)2Daxi-symmetric modelW−Ccoatingon Steel,1µmcoat- ingon250µm substrate
Conical,1500nmSubstratewithdifferentcoatingthicknessesre- vealsthestronginfluenceofthetipradiusand coatingthicknessontheshapeofhardnessand indentationmodulusdepthprofiles. Zhangetal. (2017b)2Daxi-symmetric model(CA4XR element)
Fusedquartz, 500nm×500nmSpherical,100nm, and25nmIntroducedcontactdepthcorrectionforim- provedresultsinthesub-micron(≤20nm) depthindentation. Chenetal.(2017)IFEM,2Dax- isymmetricmodelAluminumalloy, 2000µmdiameter cylinder,1000µm inheight Conical,4µmYieldstressishighlydependentontheindenta- tiondepth,youngmodulusandpoisson’sratio andisnotswayedbythelengthscale.
Chapter 3
Computational Methodology
In this chapter, the computational methodology for the nanoindentation problem is pre- sented with the necessary theoretical frameworks. For the present problem, molecular dynamics and Finite Element simulations have been adopted. Therefore, the fundamen- tal backgrounds of these computational approaches are provided and how the nanoin- dentation problem are modeled using these methodology are presented. The parameters of interest for present computations are also discussed. Finally the validations of these methodologies are performed to ensure the accuracy of the presented results.
3.1 Molecular Dynamics Method
3.1.1 Basic concepts
The Molecular Dynamics (MD) method is a well developed technique to study the properties of material comprised of few to billions of atoms with empirical inter-atomic potentials. This is a classical method where the position of atoms and their change with time is recorded. The electronic contributions of the atoms are mostly neglected and it is assumed that the electrons are glued to the nuclei. The interactions between the two atoms are calculated using a potential function that solely depends on the atomic position and the local environment. This potential function is proposed based on ex- perimental measurement of specific properties or from electronic interaction from the atoms. Since the calculation of atomic interactions in electronic level is time consuming
19
Chapter 3. Computational Methodology 20 and costly, these empirical potentials are calculated from a very small sample and then tested if these are capable of predicting the specific properties within a good agree- ment of the experiment. Using this empirical force potential, Newtonian mechanics is applied where the individual atoms are considered as separate particle and their dy- namic evolutions are determined form the numerical integration of equation of motions.
By performing this integration, the atom position and velocity is obtained in the finite ensemble of the simulation cell. From this process, it is possible to extract all of the information about thermodynamic properties of interest. The success of a MD simula- tion depend on the following factors: proper computational implementation of the MD method, a well suited inter-atomic potential, post-processing of data obtained from the simulations. The dynamic evolution of the system is governed by classical Newtonian mechanics. For each atom i, the equation of motion can be expressed in the following form:
Mid2Ri
dt2 =Fi=−∇RiΦ (3.1)
which is derived from the classical Hamiltonian of the system:
Hˆ =XMiVi2
2 + Φ (3.2)
An atom of massMi moves as a rigid particle at the velocityVi in the effective potential of other particles, φ(Ri). The atomic force Fi is obtained as the negative gradient of the effective potential, Fi=−∇RiΦ. Solving these second order ordinary differential equations, all the atoms in a simulation cell may appear simplistic.Those atoms can be regarded as particles obeying Newton’s 2nd law as following equation:
F(x) =md2x
dt2 (3.3)
Now, the trajectory, x(t) of that atom can be found by integrating the above equation.
If at any time t, the atom is at position x1, with velocity v1 = dxdt1 ; and acceleration a1= ddt2x2 =m−1F(x), afterδttime the position of the atom and the velocity of the atom can be expressed by
x2=x1+v1δt (3.4)
Chapter 3. Computational Methodology 21
v2 =v1+a1δt=v1+m−1F(x)δt=v1−m−1 dV
dx
x1
δt (3.5)
In general, if the initial positionx1, the velocityv1of the atom, and the potential energy V(x) are given to compute positionsx1(t1), x2(t2), x3(t3),... by the following generalized equations:
xi=xi−1+vi−1δt;i= 1,2,3... (3.6)
vi=vi−1+ai−1δt=vi−1+m−1F(xi−1)δt=vi−1−m−1 dV
dx
xi−1
δt (3.7)
Using this equation of velocity, its trajectory x(t) can be computed for a particle by integrating the equations of motion . Molecular Statics is a modified version of the molecular dynamics where the atoms are considered at zero temperature. This is widely used method to find the minimum energy configuration by moving atoms to its minimum potential energy state. This idea is similar to quenching of the system quickly to reach the state of 0K temperature so that the system will reach a local energy minimization configuration. The algorithm of the quenching is:
Ifv.f >0; let v=v Ifv.f <0; let v= 0
Wherev and f represents the velocity and forces of an individual atom. The algorithm of molecular dynamics simulation is shown in Fig. 3.1.
3.1.2 Integration algorithm
Integration of the equation of motion is one of the major challenges during the imple- mentation of the Molecular dynamics approach. During the integration of equations, the process must follow the conservation of energy principle and must be reversible at any stage of the computation. The algorithm used for the computation should be com- putationally efficient and should be able to implement a large integration time step to reduce the computation time. Only one force is evaluated per time step for integra- tion. There are different integrators available in MD. Commonly used integrators are
Chapter 3. Computational Methodology 22
Figure 3.1: Flow chart of the basic MD algorithm.
Verlet, Velocity Verlet, Predictor-Corrector, Gear predictor-Corrector, etc. For present simulations Velocity-Verlet algorithm are considered which has improved accuracy over the Verlet algorithm . Velocity-Verlet algorithm starts with the position and velocity expansion as follows
r(t+δt) =r(t) +v(t)δt+1
2a(t)δt2+... (3.8)
v(t+δt) =v(t) + 1
2δt[a(t) +a(t+δt) +... (3.9)
Chapter 3. Computational Methodology 23 afterwards, each integration cycle calculates the velocities at the mid-step using:
v(t+δt
2) =v(t) +1
2a(t)δt (3.10)
the positions are calculated at the next step using the following equation:
r(t+ δt
2) =r(t) +v(t+δt
2)δt (3.11)
The accelerations of the atoms are calculated at next step from the potential. Finally, it updates the atom velocities as
v(t+δt) =v(t+δt 2) +1
2a(t+δt)δt (3.12)
The timestep chosen for the integration is very important as choosing a small time step will cover the trajectory for a limited part of the simulation and too large time step will result in numerical instability.
3.1.3 Temperature and pressure control
In MD simulations, temperature of a system is controlled using the thermostats. Veloc- ity scaling, Langevin dynamics, Nose-Hoover, Berendsen, Andersen are some common thermostat to control the temperature in MD. Nose-Hoover is a widely used algorithm which includes a heat bath explicitly as an additional degree of freedom. It introduces an artificial variable “s” that basically serves as a time scaling parameter. The Hamiltonian form of the Nose-Hoover thermostat is expressed by the following equations
HˆN ose−Hoover =
N
X
i=1
−→Pi 2
2mis2 +U(−→r N) +Ps2
2Q + (f+ 1)kbTlns (3.13)
Here, the magnitude of Q controls the coupling strength. If the value of Q is too large, it will have loose coupling and show poor temperature control. On the other hand, if the value of Q is small, it will have a tight coupling and show high frequency temperature. The pressure of a system is controlled by using the barostats. Basically, a macroscopic system maintains it pressure by changing its volume. This is important
Chapter 3. Computational Methodology 24 because most experiments are performed at constant pressure instead of at constant volume. For constant box shape, Berendsen. Andersen barostat are applicable whereas when the volume of simulation box is changed, Parrinello-Rahman barostat are used. For solids where the shape of materials is changed after the deformation, Parrinello-Rahman barostats should be used for better accuracy.
3.1.4 Boundary conditions
Boundary conditions play a significant role for implementation of a physical system through the molecular dynamics approach. Fixed simulation cell boundary and the periodic boundary conditions are most common type of boundary conditions. For fixed wall, the boundary can be repulsive, atomistic rigid walls or semi rigid walls. Periodic boundary conditions are particularly useful as this enables the features to calculate the macroscopic properties from only a fewer particles. Therefore, it is possible to model a bulk material using this kind of boundary condition. In this boundary condition, when an atom is moving out from one boundary of the simulation cell, it appears using the opposite boundary of the same simulation cell. Therefore, the primary cell is replicated in all simulated direction as image cells. The primary and image cells have the same number, position, momentum, size and shapes. An illustration of the periodic boundary conditions are shown in Fig. 3.2.
The forces between a primary and image cell can be represented by the following equa- tions. If iand j are in the primary cell,
−→Fi =−∂U(−r→ij)
∂−→ri
(3.14)
Ifj is an image cell,
−→Fi =−X
−→α
∂U(−r→ij − −→α L)
∂−→ri (3.15)
3.1.5 Inter-atomic potential
MD simulations use the Newtonian mechanics and solve the equations of motion to describe the position of an atom. If the mass and force acting on any pair of atoms are
Chapter 3. Computational Methodology 25
(a) (b)
Figure 3.2: Periodic boundary condition of the MD simulation.
identified their position and velocity with time. This interaction force is basically the spatial derivative of the potential energy. This potential energy can be of two forms:
Inter-atomic and Intra-molecular. Based on this, the interaction models are developed.
The chart shown in Fig. 3.3presents the classification of potential energy.
Some common inter-atomic potentials are: Lennard-Jones(LJ), Tersoff, AIREBO, EAM, REAXFF etc. The metals are well defined using the Embedded Atom Method (EAM) inter-atomic potential. In metals, there are ionized atom core with a sea of delocalized valence electrons which is defined well by EAM potential. The EAM has the following formulation:
Etotal=X
i
F(ρi) +1 2
X
i,j(i6=j)
φ(rij) (3.16)
ρi =X
j
f(rij) (3.17)
Where, ρi is the electron density at atom i, F(ρi) is the embedding function, φ(rij)is the pair potential between atoms i and j, f(rij) is the electron density function at atom idue to atom j. In LAMMPS software, eam/fs pair potential computes pairwise interactions for metals and metal alloys using a generalized form of EAM potentials with some modification incorporated by Finnis and Sinclair. This potential can accurately
Chapter 3. Computational Methodology 26
Figure 3.3: Classification of inter-atomic potential used for the MD simulations.
predict the lattice constant, bulk modulus and surface energy as band character of metallic cohesion is considered. The total energyEi of an atomiis given by
Ei =Fα
X
j6=i
ρβ(rij)
+1 2
X
j6=i
φαβ(rij) (3.18)
The above equation has the same form as the EAM formula above, except thatρindicates a functional specific to the atomic types of both atoms iand j. In this way, different elements can contribute in different ways to the total electron density at an atomic site depending on the identity of the element at that atomic site.
3.1.6 Advantages and limitations of Molecular Dynamics (MD)
Alder and Wainwright first introduced MD in the late 1950’s to study the interactions of hard spheres. With the advent of time and development of computational tool such as modern computer and software, this method has proved its immense potential. Nowa- days, it is possible to perform calculations for a very complex shape of material and sophisticated physical phenomena. The advantage of MD simulations are:
• Physically meaningful as the potential energy is used.
• No approximation is considered in treating the N-body problem.
• Thermodynamic, structural, mechanical, dynamic and transport properties of a system can be modeled in a unified way.
Chapter 3. Computational Methodology 27
• Millions of atoms can be structured for representation of a bulk system.
• Complete control over input, initial and boundary conditions of the physical sys- tem.
• Atomic trajectories can be visualized for further processing of the data.
The current limitations of the MD are the lack of inter-atomic potential for all the materials and time-scale of the simulation is too high. With the development of MD, it is now possible to model any physical phenomena by multiscale approach, where it can be easily overcome these shortcomings of the MD and still get useful results both at nanoscale and bulk scale of the material.