Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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968 Part D Materials Performance Testing

i 3 4 5 6 7

5 50 95 5 50 95 5 50 95 5 50 95 5 50 95

1 1.7 20.6 63.2 1.3 15.9 52.7 1 12.9 45.1 0.8 10.9 39.3 0.7 9.4 34.8

2 13.5 50 86.5 9.8 38.6 75.1 7.6 31.4 65.7 6.3 26.4 58.2 5.3 22.8 52.1

3 36.8 79.4 98.3 24.9 61.4 90.2 18.9 50 81.1 15.3 42.1 72.9 12.9 36.4 65.9

4 47.3 84.1 98.7 34.3 68.6 92.4 27.1 57.9 84.7 22.5 50 77.5

5 54.9 87.1 99 41.8 73.6 93.7 34.1 63.6 87.1

6 60.7 89.1 99.2 47.9 77.2 94.7

7 65.2 90.6 99.3

i 8 9 10 11 12

5 50 95 5 50 95 5 50 95 5 50 95 5 50 95

1 0.64 8.3 31.2 0.6 7.4 28.3 0.5 6.7 25.9 0.5 6.1 23.8 0.4 5.6 22.1

2 4.6 20.1 47.1 4.1 18 42.9 3.7 16.2 39.4 3.3 14.8 36.4 3 13.6 33.9

3 11.1 32.1 59.9 9.8 28.6 55 8.7 25.9 50.7 7.9 23.6 47 7.2 21.7 43.8

4 19.3 44 71.1 16.9 39.3 65.5 15 35.5 60.7 13.5 32.4 56.4 12.3 29.8 52.7

5 28.9 55.9 80.7 25.1 50 74.9 22.2 45.2 69.6 20 41.2 65 18.1 37.9 60.9

6 40 67.9 88.9 34.5 60.7 83.1 30.4 54.8 77.8 27.1 50 72.9 24.5 46 68.5

7 52.9 79.8 95.4 45 71.4 90.2 39.3 64.5 85 35 58.8 80 31.5 54 75.5

8 68.8 91.7 99.4 57.1 82 95.9 49.3 74.1 91.3 43.6 67.6 86.5 39.1 62.1 81.9

9 71.7 92.6 99.4 60.6 83.8 96.3 53 76.4 92.1 47.3 70.2 87.7

10 74.1 93.3 99.5 63.6 85.2 96.7 56.2 78.3 92.8

11 76.2 93.9 99.5 66.1 86.4 97

12 77.9 94.4 99.6

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Part D 16

973

Modeling

Part E

Part E Modeling and Simulation Methods

17 Molecular Dynamics

Masato Shimono, Tsukuba, Japan

18 Continuum Constitutive Modeling

Shoji Imatani, Kyoto, Japan

19 Finite Element and Finite Difference Methods

Akira Tezuka, Tsukuba, Japan

20 The CALPHAD Method

Hiroshi Ohtani, Kitakyushu, Japan

21 Phase Field Approach

Toshiyuki Koyama, Nagoya, Japan

22 Monte Carlo Simulation

Xiao Hu, Tsukuba, Japan

Yoshihiko Nonomura, Ibaraki, Japan

Masanori Kohno, Tsukuba, Japan

975

Molecular Dyn

17. Molecular Dynamics

This chapter gives an overview of the molecu-

lar dynamics (MD) simulation method. In the

ﬁrst section, the basics techniques of MD will

be introduced so that the reader is able to

start actual calculations. In the following three

sections, several applications of MD simula-

tion, some adapted for actual materials and

others concerned with idealized model sys-

tems, will be presented. In the second section,

a simulation study of diffusionless transforma-

tions such as the martensitic transformation and

solid-state amorphization is presented. Anal-

ysis of amorphous and crystalline structures

is described in Chap. 5 in this handbook. MD

calculations of the processes involved in the

preparation of amorphous alloys by rapid so-

lidiﬁcation as well as annealing processes in

these alloys are discussed in the third sec-

tion. In the last section, the investigation of

atomic diffusion processes in liquid and solid

(crystalline and amorphous) phases by MD sim-

ulation is explored. Readers may also refer to

Chap. 7 concerning atomic diffusion processes in

materials.

17.1 Basic Idea of Molecular Dynamics........... 975

17.1.1 Time Evolution of the Equations

of Motion .................................... 975

17.1.2 Constraints

on the Simulation Systems ............ 978

17.1.3 Control of Temperature

and Pressure ............................... 980

17.1.4 Interaction Potentials ................... 985

17.1.5 Physical Observables..................... 987

17.2 Diffusionless Transformation ................. 988

17.2.1 Martensitic Transformation............ 988

17.2.2 Transformations in Nanoclusters .... 991

17.2.3 Solid-State Amorphization ............ 993

17.3 Rapid Solidiﬁcation .............................. 995

17.3.1 Glass-Formation

by Liquid Quenching..................... 995

17.3.2 Annealing of Amorphous Alloys...... 999

17.3.3 Glass-Forming Ability

of Alloy Systems ...........................1004

17.4 Diffusion .............................................1006

17.4.1 Diffusion in Crystalline Phases .......1006

17.4.2 Diffusion in Liquid

and Glassy Phases ........................1008

17.5 Summary .............................................1010

References ..................................................1010

17.1 Basic Idea of Molecular Dynamics

In this section, the basic formalism of MD simulation

will be introduced, and some useful methods invented

for MD calculations will then be presented.

17.1.1 Time Evolution of the Equations

of Motion

The essential task in MD simulation is to solve the

Newtonian equation of motion numerically. Hence,

techniques for numerical solution of differential equa-

tions will be explained in this section. The main topics

are numerical recipes used for this purpose such as the

Verlet and the Gear methods.

The basic idea of molecular dynamics is simple: to

follow the motion of all atoms in the system by solving

the Newtonian equations of motion

(t)/dt

= F

(t)/m

, i = 1, 2, 3,...,N ,

(17.1)

iα

(t)/dt

=−

∂φ/∂r

iα

(t) ,α=1, 2, 3

(17.2)

if the Lagrangian of the system has the following form



−φ(r

, r

,...) . (17.3)

In the numerical integration of the differential equa-

tion (17.1), we should calculate the coordinates of

Part E 17

976 Part E Modeling and Simulation Methods

v (t + 2Δt)

v (t + Δt)

v (t)

r (t)

r (t + 2Δt)

r (t + Δt)

Fig. 17.1 An illustrative view of the integration steps of

a particle trajectory in MD simulation

all particles in the system step by step in time, as

depicted in Fig. 17.1. For simplicity, let us use a one-

dimensional expression x(t) for the particle coordinates.

The basic equation for the purpose of predicting the co-

ordinates x(t +Δt) at a future time t +Δt is the Taylor

expansion

x(t +Δt) = x(t)+x



(t)Δt +x



(t)(Δt)



(t)(Δt)

/3!+···, (17.4)

where the prime denotes the functional derivative. For

a practical calculation, we should truncate the Taylor

expansion at a ﬁnite order of Δt, and also include the

information of the Newtonian equation of motion. To

achieve the latter, we use the following relation, which

holds for the second time derivative of the particle co-

ordinates at any time



(t) = F(t)/m . (17.5)

Based on this idea, a large number of algorithms

have been suggested. We shall explore some of the

popular algorithms together with their merits and

drawbacks.

Verlet Method

The ﬁrst example of the popular algorithms is the Ver-

let method [17.1], which can be derived as follows. Let

us write the Taylor series of a forward time step and

a backward time step up to third order

x(t +Δt) = x(t)+x



(t)Δt +x



(t)(Δt)



(t)(Δt)

/3!+O



(Δt)



, (17.6)

x(t −Δt) = x(t)−x



(t)Δt +x



(t)(Δt)

−x



(t)(Δt)

/3!+O



(Δt)



. (17.7)

By adding the above expansions, the third-order terms

cancel and, using the equation of motion (17.5), we

get

x(t +Δt) =2x(t) −x(t −Δt) + F(t)(Δt)

/m ,

(17.8)

which holds up to third order in the time step Δt. This

relation is called the Verlet leap-frog algorithm.

An advantage of this method is its simple form,

which allows us to code a program very easily. For

example, the coding processes can be

Step 1 Specify the initial position x(t

) and the position

at the ﬁrst step x(t

+Δt)

Step 2 Calculate the force F(t

+Δt) at the ﬁrst step

Step 3 Compute x(t

+2Δt) according to (17.8)

Step 4 Repeat steps 2 and 3 with t

→t

+Δt.

Throughout the whole calculation processes, we

only need to record the two latest positions of the

particles in the Verlet method, which saves computer

memory space.

The form of (17.8) includes only particle coordi-

nates and no velocity terms. The velocity v(t) can be

evaluated by rewriting (17.8) as follows

x(t +Δt) = x(t) +v(t)Δt +F(t)(Δt)

/2m , (17.9)

v(t) =[x(t +Δt)]−x[(t −Δt)]/2Δt . (17.10)

Although the Verlet method has merit due to its

simple form, we encounter a difﬁculty when the force

F(t) depends on the velocity v(t), because the vel-

ocity (17.10) can be computed only after calculating

the forward-step coordinate x(t +Δt), which is itself

calculated using the velocity v(t). This is something of

a vicious circle. Since the force term rarely includes the

velocity for usual atomic interactions, this deﬁciency is

not always a limitation. However, as we shall discuss

later, when we try to control the temperature or the pres-

sure of the simulation system by introducing a ﬁctitious

variable, we inevitably encounter this problem caused

by a velocity-dependent force term.

To avoid this situation, a slightly more complicated

prescription, called the predictor–corrector scheme is

commonly used, where the coordinates

x(t +Δt)at

a forward step is temporarily calculated, and then sub-

sequently corrected using the force term F(t +Δt)

calculated at the next time step.

Gear Method

Among the methods using the predictor–corrector

scheme, the most popular method is one called the Gear

predictor–corrector method [17.2]. The calculation pro-

cedure consists of two parts. The ﬁrst step is called

Part E 17.1

Molecular Dynamics 17.1 Basic Idea of Molecular Dynamics 977

the predictor step, which is just a Taylor expansion ex-

pressed in vector representation for a set of coordinates

x(t) and their time derivatives x



(t), x



(t), x



(t),...

q(t +Δt) =Gq(t) , where

⎛

⎜

⎝

1111···

0123···

0013···

0001···

⎞

⎟

⎠

, q(t)=

⎛

⎜

⎝

x(t)



(t)Δt



(t)(Δt)

/2!



(t)(Δt)

/3!

⎞

⎟

⎠

(17.11)

One advantage of the Gear method is that we can choose

the order of time step Δt in the Taylor expansion to

take into account. If we choose to calculate up to or-

der (Δt)

p−1

, or equivalently we choose the coordinate

vector q(t)asap-component vector, then we call it the

p-th order Gear method.

Using the predicted coordinates

q(t +Δt), we then

calculate the force F(t +Δt) in the second, corrector

step as follows

q(t +Δt) =

q(t +Δt)+A(Δt)



F(t +Δt) −



(t +Δt)



2 ,

(17.12)

where A is a constant vector called the correction vector.

The forms of the matrix G and vector A depend

on the order of the method. Whereas G is uniquely de-

termined by the order p, there is no unique choice for

the correction vector A that make the third component

unity and thereby satisfy the equation of motion at

t +Δt. Gear determined [17.2] the vector A so that the

numerical errors do not accumulate monotonically. For

the fourth-, ﬁfth- and sixth-order algorithms, the pre-

scribed values are listed below

G =

⎛

⎜

⎝

1111

0123

0013

0001

⎞

⎟

⎠

, A =

⎛

⎜

⎝

1/6

5/6

1/3

⎞

⎟

⎠

for p =4 , (17.13)

G =

⎛

⎜

⎝

11111

01234

00136

00014

00001

⎞

⎟

⎠

, A =

⎛

⎜

⎝

19/120

3/4

1/2

1/12

⎞

⎟

⎠

for p =5 ,

(17.14)

G =

⎛

⎜

⎝

11111 1

01234 5

0013610

0001410

00001 5

00000 1

⎞

⎟

⎠

, A =

⎛

⎜

⎝

3/20

251/360

11/18

1/6

1/60

⎞

⎟

⎠

for p =6 . (17.15)

Consequently, the coding process is as follows

Step 1 For the p-th order method, specify the initial po-

sition x(t

) and its derivatives x



), x



),...,

and x

(p−1)

Step 2 Calculate the predicted coordinates

q(t

+Δt)at

the ﬁrst step according to (17.11).

Step 3 Calculate the force F(t

+Δt) at the ﬁrst step

using

q(t

+Δt).

Step 4 Compute the corrected q(t

+Δt) according to

(17.12).

Step 5 Repeat steps 2 to 4 with t

→t

+Δt.

In the course of numerical integration, it is in-

evitable that numerical errors will accumulate. The size

of these errors depends on the numerical integration

method used, as well as the time step Δt. To estimate the

order of the numerical errors, we calculate the trajectory

x(t) of a simple harmonic oscillator of unit mass



(t) =−x(t) , (17.16)

and evaluate the deviation from the analytical solu-

tion for different integration methods and different time

steps.

Figure 17.2 shows the results for the deviation of

the numerically integrated trajectory from the exact so-

lution after 100 oscillation periods (200π). As one can

see from the slope of a linear approximation to the plots

depicted in the ﬁgure, the numerical error of the p-th

Gear method scales as (Δt)

. This behavior is just what

one would expect from (17.11), but it is not straight-

forward, because the number of calculation steps grows

as (Δt)

−p

for a constant interval T, although the er-

ror for each step is assured to be as low as (Δt)

the p-th Gear method. In this context, it is interesting

that the numerical error of the Verlet method behaves as

(Δt)

, even though the error at each step is kept as low

as (Δt)

. In this sense, the Verlet method is considered

to be a second-order formalism.

The tendency in Fig. 17.2 tells us that we can

keep the numerical errors low if we use the Gear

method with a sufﬁciently small time step compared

to a typical time scale for characteristic motion of

Part E 17.1