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

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998 Part E Modeling and Simulation Methods

–4.4

–4.5

–4.6

–4.7

–4.8

–4.9

Energy (eV)

0 500 1000 1500 2000

Temperature (K)

Fig. 17.25 The temperature dependence of the internal en-

ergy of the Ti-25 at. % Al system with 4096 atoms in

a heating process from an amorphous phase. The heating

rate is 1.0K/ps

0.7

0.6

0.5

0.4

1000

800

600

400

200

Concentration (at.% Al)

0 20 40 60 80 100

0 20406080100

(K)

Fig. 17.26 The composition dependence of the glass-

transition temperature (upper column) and the crystalliza-

tion temperature (lower column) observed in the Ti-Al

system. The glass-transition temperature is rescaled by

the melting point at each composition. The crystalliza-

tion temperatures taken from the heating procedure with

the rate of 1.0K/ps are denoted by the closed trian-

gles, those from a long time annealing are denoted

by the open triangles, and those from the experimen-

tal observation (after [17.36]) are denoted by the closed

circles

1000

100

0.1

0.01

0 20406080100

Cooling rate (K/ps)

Concentation (at.%Al)

Crystal

Amorphous

Fig. 17.27 The solidiﬁed phases after the cooling process

mapped onto the cooling rate versus composition plane.

The squares and the circles correspond to the crystalline

and amorphous phases, respectively

a split-shaped second peak in the amorphous phase and

a shell structure in the crystalline phase.

In glass transitions it is a general feature [17.37]that

a lower cooling rate gives a lower T

. This tendency is

also observed in the simulation. By comparing the re-

sults between the case with the cooling rate 10 K/ps and

that with the rate 100 K/ps in Fig. 17.24,weﬁndthat

the lower cooling rate gives the lower energy amorph-

ous state and, consequently, gives the lower T

. The

dependence of T

on the cooling rate is approximately

estimated as that cooling rate that is 10 times higher

gives a 50 K increase in T

Figure 17.25 shows an example of the time evo-

lution of the internal energy in a heating procedure

from an amorphous phase for the Ti-25 at. % Al case.

The drop at around 800 K corresponds to amorphous-to-

crystal transition, while the subsequent jump at 1900 K

corresponds to melting. Thus we can deﬁne the crystal-

lization temperature T

and the melting point T

of this

composition.

By performing this procedure for different com-

positions, we can obtain [17.38] the composition

dependence of the glass-transition temperature T

and

the crystallization temperature T

in the Ti-Al system.

The results are shown in Fig. 17.26.SinceT

as well as

depends on the cooling/heating rate, the plotted data

are rescaled values corresponding to a constant rate of

1.0K/ps. In the upper graph, the compositional depen-

dence of the glass-transition temperature T

are plotted,

where the value of T

is normalized by the observed

Part E 17.3

Molecular Dynamics 17.3 Rapid Solidiﬁcation 999

melting point T

at each composition. In the lower

graph, the composition dependence of the crystalliza-

tion temperature T

taken from the heating procedure

with constant rate are denoted by the closed triangle,

while those from long-time annealing are denoted by

the open triangle, and those from experimental obser-

vations [17.36] are denoted by the closed circles. We

ﬁnd an agreement between the simulation results and

the experimental observations.

We can also map the obtained phase after quenching

onto the cooling rate–composition plane in Fig. 17.27,

where the open triangles denote crystalline phases,

while the closed circles denote amorphous phases. By

using these plots, we can estimate the amorphous-

forming composition range at a given cooling rate.

The results are shown in Fig. 17.28, where the

upper two results are from the sputtering deposit ex-

periments [17.31], the middle two are from the MD

simulations with the EAM potentials, and the lower

two are from the MD simulations with the LJ po-

tentials. Considering that the effective cooling rate in

the sputtering deposition experiments is the order of

–10

K/s, the MD simulations with EAM poten-

tials show better agreement with the experiments than

those with the LJ potentials.

As mentioned before, there is a risk that the peri-

odic boundary condition might harm the dynamics of

the simulation system. Hence we investigate the effect

of the system size on the critical cooling rate by vary-

ing N from 218 to 8000 atoms for the pure Al system.

In Fig. 17.29, we have mapped the ﬁnal phases on the

cooling rate–system size plane using the same symbols

as used in Fig. 17.27. This shows that the effect is sig-

niﬁcant in small systems with less than 1000 atoms.

The similar effect of size on the thermodynamical ob-

servables such as the melting point, the crystallization

temperature, and the glass-transition temperature, are

also calculated and plotted in Fig. 17.30, where the ef-

fect appears smaller than on the critical cooling rate,

but with a similar nature. So, we conclude that we need

at least a few thousand atoms in simulations using pe-

riodic boundary conditions to reduce the effect of the

boundary conditions.

17.3.2 Annealing of Amorphous Alloys

The annealing procedure of the Ti-Al amorphous alloys

is simulated in this subsection. The atomistic mechan-

ism of the structural relaxation in amorphous alloys as

well as that of crystallization is investigated by MD

simulation with special attention to the local atomic

0 20 40 60 80 100

Al concentration (at.%)

Sputtering

experiments

simulations

(EAM)

simulations

(LJ)

α(hcp) fcc

(Thickness: 2–3 μm)

α(hcp) Amorphous

fcc

(Thickness: 0.05–0.2 μm)

Crystal Amorphous

(Cooling rate: 10

K/s)

(Cooling rate: 10

K/s)

Crystal

Crystal (fcc + hcp)

(Cooling rate: 10

K/s)

Crystal Amorphous

(Cooling rate: 10

K/s)

Crystal

Fig. 17.28 Glass-forming ranges of the Ti-Al system given by ex-

periments and simulations. The upper two are given from the

sputtering deposition experiments (after [17.31]), the middle two

are given from the MD simulations with the EAM potentials, and

the lower two are given from MD simulations with the LJ poten-

tials, respectively. The crystal structure found in the simulations is

fcc, hcp, or their mixture in any case

1000

100

100 1000 10 000

Cooling rate (K/ps)

Crystal

Amorphous

Fig. 17.29 The system-size dependence of the critical cool-

ing rate when varying the total number N of atoms in

the simulation for the pure Al system. The quenched

phases are denoted by the same symbols as used

in Fig. 17.27

Part E 17.3

1000 Part E Modeling and Simulation Methods

1000

800

600

400

200

100 1000 10000

, T

(K)

Fig. 17.30 The system-size dependence of the melting

point T

, the crystallization temperature T

, and the glass-

transition temperature T

when varying the total number N

of atoms in the simulation for the pure Al system

structure of the amorphous phases and the vibrational

modes characteristic of the amorphous states.

In amorphous alloys, neither the local atomistic

structure nor the mechanism of the structural relaxation

is clearly understood. To clarify these problems, the

structural change at the atomistic level will be inves-

tigated in the relaxation and crystallization process of

amorphous phases in the Ti-Al binary system by using

the MD simulation discussed in the previous subsection.

To analyze the local structure in the amorphous phases

obtained in the simulations, we shall introduce some

local order parameters, which will shed light on the ba-

sic feature of the atomistic structure of the amorphous

phases from different angles.

Icosahedral Symmetry

From experimental observations [17.39]andMD simu-

lations [17.19], it has been established that icosahedral

Fig. 17.31 Icosahedral clusters found in Ti-Al amorphous

alloys obtained in the simulations. The brown spheres and

the white spheres denote the Ti atoms and the Al atoms,

respectively

1 nm

Fig. 17.32 The medium-range structure in a Ti-50 at. %Al

amorphous alloy with 8000 atoms after the classiﬁcation

of the atoms according to the symmetry of the surrounding

atoms. The dark gray, light gray,andwhite spheres de-

note the atoms with icosahedral symmetry, the atoms with

crystalline symmetry, and the other atoms with somewhat

distorted surroundings, respectively

structures are the basic building blocks of the amorph-

ous state in monatomic systems. Since the atomic size

difference between Ti and Al is small (a few %),

Ti-Al amorphous alloys might include such icosahe-

dral structures. Indeed, we can ﬁnd [17.34] many types

of icosahedral clusters in the Ti-Al amorphous phases

obtained by the simulations, as depicted in Fig. 17.31.

Therefore, we shall investigate the local atomic struc-

ture while paying particular attention to this icosahedral

symmetry to analyze the microstructure of the phases

emerging in the heating and annealing processes.

For this purpose, the Voronoi tessellation tech-

nique [17.19] is useful to analyze the local symmetry

of atomic conﬁguration around each atom. Dividing the

whole conﬁguration space by the bisecting planes for

all atomic pairs, we can assign the Voronoi polyhedron

or the Wigner–Seitz cell to each atom. The shape of the

Voronoi polyhedron reﬂects the local structure around

the corresponding atom. Thus we can assign [17.19]

the environmental character to the atoms from the dis-

tribution of the faces of the corresponding Voronoi

polyhedron.

Part E 17.3

Molecular Dynamics 17.3 Rapid Solidiﬁcation 1001

By evaluating the shape of the Voronoi polyhedra,

we can classify the atoms into three groups accord-

ing to their surrounding structure [17.38]: atoms with

an environment of icosahedral symmetry, atoms with

an environment of crystalline symmetry, and other

atoms with somewhat distorted surroundings. Fig-

ure 17.32 shows the above classiﬁcation applied to a Ti-

50 at. % Al amorphous alloy with 8000 atoms, where

the icosahedral-like atoms, the crystal-like atoms, and

the other atoms are depicted by the dark gray, light gray,

and white spheres, respectively. The remarkable feature

is that there is a region with crystalline symmetry even

in the amorphous phase and it forms a medium-range

ordered structure on the nanometer scale together with

a region with icosahedral symmetry. The geometrical

origin of the medium scale is understood as follows.

The essence of glass-formation is the competition

between the locally stable state (icosahedral cluster)

and the globally stable state (crystalline ordering) and

the domination of the former. However, the icosahedral

clusters have ﬁvefold symmetry so they cannot ﬁll up all

space by themselves and there is some limit of order at

the nanometer scale for such icosahedral packing. That

is why the inhomogeneity on a medium scale exists in

the amorphous states.

Thus we can take the fraction X

ico

of the atoms with

icosahedral symmetry and the fraction X

cry

of the atoms

with crystalline symmetry as order parameters which

reﬂect the local symmetry.

In a similar sense, we use the fraction X

penta

of the

pentagons in the total Voronoi faces as an order par-

ameter, since all the faces of the Voronoi polyhedron

with icosahedral packing consist of pentagons, while

those with bcc, fcc,orhcp crystal consist of squares and

hexagons.

DRP Structure

Historically, the ﬁrst model of atomistic structure for

amorphous alloys is the dense random packing (DRP)

model [17.40]. In the DRP model, the basic structure

is the closest packing of hard spheres and the basic

unit is the tetrahedron made of four mutually contacting

spheres. Since such closest packing is almost ideally re-

alized in the icosahedral clusters, the abundance of the

icosahedral clusters in the amorphous phase reﬂects the

existence of the DRP structure and hence the tetrahedral

packing in its basis. Therefore, by counting the number

of tetrahedra made of four mutually neighboring atoms

in the amorphous alloys, we can use the number N

tetra

of tetrahedral clusters per atom as an order parameter to

characterize the order of the DRP structure.

Free Volume

Next we introduce the notion of free volume fol-

lowing the idea of Cohen and Grest [17.41]. The

atoms surrounded by enough free volume can move

by the length of the order of atomic distance, while

those having little space around them can only make

oscillatory moves around their equilibrium positions.

Those atoms having enough free volume are called

liquid-like atoms. In the free volume theory, the glass-

to-liquid transition can be understood as a percolation

of free volume or the liquid-like atoms. In this con-

text, we take a simple deﬁnition for the free volume as

follows.

Firstly we deﬁne the nearest neighbors of an atom

by the atoms within a distance of 1.4 times of its atomic

size, which corresponds to the ﬁrst minimum in the

radial distribution. Then we deﬁne an atom as having

enough free volume if it has fewer than 12 neighbors.

Under this deﬁnition, atoms surrounded by a crystal

packing such as fcc, hcp,orbcc are not atoms with free

volume, and neither do atoms surrounded by the icosa-

hedral structure. On the other hand, even in a crystal, the

atoms neighboring a defect structure such as a vacancy

are atoms with free volume. Thus we have another par-

ameter X

free

which is the fraction of atoms with enough

free volume.

Figure 17.33 shows the time evolution of the above

order parameters in the annealing process of amorphous

alloys. The left column corresponds to a Ti-25 at. %Al

amorphous alloy annealed at 520 K and the right col-

umn corresponds to a Ti-50 at. % Al amorphous alloy

annealed at 810 K. The common feature is the de-

crease in the atomic volume and the energy, as well

as that in the free volume. On the other hand, the

striking difference is in the evolution of X

penta

and

cry

, that is, an increase in X

penta

and X

ico

together

with an decrease in X

cry

are observed in the right

column, while a decrease in X

penta

and X

ico

together

with an increase in X

cry

are observed in the left

column. This indicates that a more stable amorph-

ous phase with less free volume would form in the

annealing procedure for the system with the compos-

ition 50 at. % Al, while embryos of crystalline phases

would form in the annealing period for the system

with the composition 25 at. % Al. Consequently, in the

relaxation period, the microscopic process proceeds

differently depending on the Al concentration of the

system. At the midway composition, where the glass-

forming ability is high, less free volume and hence

more stable amorphous phase forms, while the early

stages of crystallization take place at low Ti or low Al

Part E 17.3

1002 Part E Modeling and Simulation Methods

–3.8

–3.805

–3.81

–3.815

Annealing time (ns)

0246810

–3.625

–3.63

–3.635

Annealing time (ns)

0246810

0.27

0.22

Annealing time (ns)

0246810

0.3

0.26

Annealing time (ns)

0246810

2.6

2.55

Annealing time (ns)

0246810

2.9

2.7

Annealing time (ns)

0246810

0.54

0.5

0.46

Annealing time (ns)

0246810

0.62

0.61

0.6

Annealing time (ns)

0246810

0.46

0.55

0.45

Annealing time (ns)

0246810

0.3

0.25

Annealing time (ns)

0246810

E (eV) E (eV)

free

tetra

penta

cry

Fig. 17.33 The time evolution of the internal energy E and the struc-

ture parameters, X

free

, N

tetra

, X

penta

,andX

cry

. The left column

corresponds to the Ti-25 at. % Al system with 8000 atoms annealed

at 520 K, while the right column corresponds to the Ti-50 at. %Al

system of the same size annealed at 810 K

concentration where the glass-forming ability is rather

low.

Next we investigate how the above parameters that

show local structures change in crystallization pro-

cesses. Figure 17.34 shows the time evolution of the

internal energy E, the fraction X

penta

of the pen-

tagons, the fraction X

cry

of atoms with crystalline

symmetry, the fraction X

free

of free volume atoms,

and the number N

tetra

of tetragonal clusters in the

annealing process at 580 K of a Ti-25 at. % Al amorph-

ous alloy with 8000 atoms. The drastic change in

the parameters at t = 11 ns apparently corresponds to

the amorphous-to-crystal transition. This can also be

seen from the snapshots of atomic conﬁguration of

a) the as-quenched state (annealing time t = 0ns),

b) a relaxed amorphous state (t = 10.3ns), c) a par-

tially crystallized state (t = 11.4 ns), and d) a fully

crystallized state (t = 15 ns), which are depicted in

the right of Fig. 17.34. By using this type of analy-

sis, we can depict a time–temperature-transformation

(TTT) curve of the simulation system, as shown in

Fig. 17.35, where the tentative TTT-curve of the Ti-

25 at. % Al amorphous alloy is shown by the bold

line.

Finally we proceed to investigate the vibration

modes in the amorphous state. The vibrational state of

thermal atomic motion is closely related to the local

atomic structure. In this sense, an excess of vibra-

tion states experimentally observed [17.42]inmany

types of amorphous phases is thought to play an

important role. This is called the boson peak and

seems to be a universal feature of amorphous states.

However, its structural origin is not clearly under-

stood yet. Indeed, some pioneering MD studies have

shown that these modes are closely related to the

local atomistic structure [17.43] and a cooperative

motion on the namometer scale [17.44]. Hence we

here try to uncover the microscopic origin of the bo-

son peak by investigating the atomic vibrational state

in the Ti-Al amorphous phases during the annealing

procedure. The investigation of the microscopic ori-

gin of the boson peak helps us to understand the

physics of the glass transition and the amorphous state

itself.

As mentioned in Sect. 17.1.5, we can calculate the

power spectrum f (ω) of the atomic vibration energy

from the velocity autocorrelation function of the atoms

f (ω) =

∞



dt cos ωt



v(0)v(t)





, (17.72)

where the brackets means the average over all atoms.

In Fig. 17.36, the calculated vibration energy spec-

trum of three different phases of Ti-25 at. %Al al-

loys in the annealing process at 580 K shown in

Fig. 17.34, are depicted, where the full line, dashed

line, and dotted line correspond to the as-quenched

amorphous state (annealing time t = 0 ns), a relaxed

amorphous state (t = 10.3 ns), and a crystallized state

(t = 15 ns), respectively. We can easily see the ex-

cess of states in both the low- and high-frequency

Part E 17.3

Molecular Dynamics 17.3 Rapid Solidiﬁcation 1003

Annealing time (ns)

0 5 10 15

–3.81

–3.84

–3.87

Annealing time (ns)

0 5 10 15

–3.87

0.8

0.7

0.6

0.5

0.4

Annealing time (ns)

0 5 10 15

0.2

0.1

Annealing time (ns)

0 5 10 15

2.6

2.4

2.2

1.8

a) b)

t = 0 t = 10.3

c) d)

t = 11.4 t = 15

a) b) c) d)

E (eV)

penta

, X

cry

penta

cry

free

tetra

Fig. 17.34a–d The time evolution of the internal energy E, the fraction X

penta

of the pentagons, the fraction X

cry

atoms with crystalline symmetry, the fraction X

free

of free volume atoms, and the number N

tetra

of tetragonal clusters in

the annealing process at 580 K of the Ti-25 at. % Al amorphous alloy with 8000 atoms. The right graphs are snapshots of

atomic conﬁguration at the annealing times (a) t = 0ns,(b) t = 10.3ns,(c) t =11.4ns,and(d) t = 15 ns

tail in the amorphous states and that the amount

of the excess decreases with time of annealing. Es-

pecially, the excess in the low-frequency tail that

1200

1000

800

600

400

200

0.1 1 10

Annealing time (ns)

Temperature (K)

corresponds to the boson peak is shown in the inset of

Fig. 17.36.

To clarify the microscopic origin of the boson peak,

we next investigate the differences between the con-

tributions made to the excess of vibration modes by

different local structures. For this purpose, the classi-

ﬁcation into three regions based on the atomic local

symmetry illustrated in Fig. 17.32 is used, and the con-

tribution to the vibrational spectrum from each region

is calculated. The results calculated at 10 K for a Ti-

50 at. % Al amorphous alloy with 8000 atoms are shown

in Fig. 17.37, where the solid line, dashed line, and

dotted line correspond to the icosahedral-like atoms,

Fig. 17.35 The calculated TTT curve of an annealed Ti-

25 at. %Al amorphous alloy (bold line). The dotted lines

denote the histories of the heat treatments performed in

the simulations and the triangles denote the points where

crystallization is observed



Part E 17.3

1004 Part E Modeling and Simulation Methods

1.5

0.5

f(φ)

2πφ (THz)

0 20406080100

As quenched

Annealed (10.3)

Crystallized

Fig. 17.36 The time evolution of the vibration spectrum

in the annealing process at 580 K of the Ti-25 at. %Al

amorphous alloy with 8000 atoms. The solid line,the

dashed line,andthedotted line correspond to the as-

quenched amorphous phase, the relaxed amorphous phase

after 10 ns annealing, and the crystallized phase after 15 ns

annealing, respectively. The inset denotes the low fre-

quency tails

the crystalline-like atoms, and the other atoms, respec-

tively, and the inset shows the lower frequency tails.

This means that the contribution to the boson peak from

atoms with more free volume and fewer neighbors is the

largest, while that from the icosahedral atoms is next,

and that from crystalline atoms is the smallest. This is

consistent with the fact that the decrease in the boson

peak during the annealing process is accompanied by

a decrease in free volume. But the quantitative differ-

ence is so small that we could only discern it in the inset

of Fig. 17.37.

17.3.3 Glass-Forming Ability

of Alloy Systems

In order to acquire the high glass-forming ability

of alloy systems, there are well-known empirical

rules [17.28]: large atomic size difference and nega-

tive heat of mixing between the alloying elements. In

this section, the microscopic origin of these rules is

explored by using MD calculations for a model sys-

tem in which the atomic size ratio and the heat of

mixing between the constitute elements can be varied

independently.

Here we again choose a binary system where

the interaction is described by the 8–4 LJ poten-

1.5

0.5

f(φ)

2πφ (THz)

0 20406080

100

Icosahedral

Disordered

Crystalline

Fig. 17.37 The contribution to the vibration spectrum from

the atoms with different local structures calculated at

10 K for a Ti-50 at. % Al amorphous alloy with 8000

atoms. The solid line,thedashed line,andthedot-

ted line correspond to atoms with icosahedral symmetry,

atoms with crystalline symmetry, and other atoms with

distorted surrounding, respectively. The inset denotes the

low-frequency tails

tial as a model for binary alloys, which is discussed

in Sect. 17.2.3. There are two reasons for using this

model. The ﬁrst is that we can independently vary the

atomic size ratio and the heat of mixing by changing

the parameter r

and e

, respectively. The second

reason can be found in Fig. 17.28, which shows the

calculated amorphous-forming range in the Ti-Al sys-

tem both for the EAM potential and the LJ potential.

Quantitatively, the results from the LJ potential are

poorer than those from the EAM potential compared

to the experimental observations, but qualitatively, the

predicted composition range that has maximum glass-

forming ability is almost the same for both potentials.

In addition, the predicted critical cooling rate for the

LJ potential is a thousand times higher than that

for the EAM potential, which means the necessary

calculation time is much lower for the LJ poten-

tial. In this sense, the MD simulation using the LJ

potential is a counterpart of the acceleration experi-

ment, if we are only interested in the glass-forming

range.

The model system used in the simulation is almost

the same as that used in Sect. 17.2.3,exceptthatwe

assume that e

=1 and that the heat of mixing

is controlled by varying the interaction parameter e

That is, e

< 1 means a positive heat of mixing, while

Part E 17.3

Molecular Dynamics 17.3 Rapid Solidiﬁcation 1005

> 1 means a negative heat of mixing. All physi-

cal quantities are expressed in the above units in this

section. Note that the unit of cooling rate is around

5×10

K/s for typical alloy systems.

To simulate rapid solidiﬁcation processes, we ﬁrst

prepare a liquid phase in the same manner as mentioned

in Sect. 17.3.1, and then rapidly quench the system to

solidify. Depending on the cooling rate as well as the

atomic size ratio and the heat of mixing of the system,

we have a crystalline phase or an amorphous phase.

By varying the cooling rate, we can estimate the crit-

ical cooling rate needed for amorphization by melt

quenching. Alternatively, by varying the atomic size

ratio r

, the chemical interaction e

, and the concen-

tration x

of the solute element, we can calculate the

glass-forming range by melt quenching under a constant

cooling rate.

First we show the pure geometrical effect on the

glass-forming ability. In Fig. 17.38a, the shaded regions

denote the glass-forming ranges in the rapid solidiﬁca-

tion on the



, r



plane at ﬁxed chemical interaction

of e

=1. The darker shading corresponds to the lower

cooling rate. For the midway compositions, the glass-

forming ability is extremely high if the atomic size ratio

is less than 0.9.

We note that the glass-forming ranges are not

symmetric in composition and are slightly shifted to

higher solute concentration. This suggests that the

glass-forming ability should be higher when the adding

element has a larger atomic size than the main con-

stituent than when the adding element has a smaller

size, assuming that the solute element is added in the

same amount.

To investigate the chemical effect on the glass-

forming ability for melt quenching, we next ﬁx the

atomic size ratio at r

=1 and vary the chemical

bond strengths between 0.5 < e

< 1.5. In this case,

however, we cannot observe any obvious change in

the critical cooling rate compared to the system with

=1.

Then we turn to the system with a ﬁxed atomic size

ratio of 0.95 and vary e

. Figure 17.38bshowsthe

glass-forming range in this case, where the heat of mix-

ing ΔH is deﬁned as ΔH =





/2−e

. The

larger negative ΔH gives a greater glass-forming abil-

ity, although no amorphization is observed in the cases

at low cooling rates less than 1×10

−4

The above results show the following two features:

1) The effect of the chemical factor on the glass-forming

ability is rather smaller than that of the geometrical fac-

1.00

0.95

0.90

0.85

0.80

0.75

0.2

0.1

0.0

–0.1

–0.2

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

Cooling

rate:

Heat of mixing

1×10

–3

5×10

–4

1×10

–4

2×10

–5

Atomic size ratio

Concentration Concentration

Fig. 17.38a,b Composition ranges in which amorphous phases

are formed by melt quenching in the model binary sys-

tems. (a) Glass-forming ranges under the variation of r

with

=1; (b) glass-forming ranges under the variation of e

with

= 0.95. The darker shading corresponds to the lower cooling

rate

Fig. 17.39 An amorphous phase (left) and a nano-crys-

talline phase (right) prepared by rapid solidiﬁcation

tor and it becomes signiﬁcant if it is combined with

the geometrical effect. 2) Both the geometrical and the

chemical factor act asymmetrically, in other words, the

glass-forming ability is different whether we add the

larger atomic size element to the main constituent el-

ement or add the smaller one.

Finally, we show an interesting possibility for con-

trol of the nanostructure of rapidly solidiﬁed phases.

Figure 17.39 shows snapshots of an amorphous phase

(left) and a nanocrystalline phase (right) obtained

by rapid solidiﬁcation processes with different condi-

tions. That is, they are obtained in different binary

systems with different atomic size ratios under dif-

ferent cooling rates. These snapshots indicates that,

to choose the parameters such as the atomic size

ratio properly, we could control the grain size of

nanocrystalline phases by rapid solidiﬁcation, as well

as we could control the formation of the glassy

phases.

Part E 17.3

1006 Part E Modeling and Simulation Methods

17.4 Diffusion

In this section, we shall focus on atomic diffusion. The

atomistic mechanism for diffusion is investigated by the

MD simulation in a number of cases: the diffusion via

vacancies or interstitials in crystalline phases including

ordered phases, and the diffusion in the liquid and the

glassy phases.

17.4.1 Diffusion in Crystalline Phases

In this section, the atomic diffusion process in crys-

talline phases is investigated by MD simulation. The

atomic-level processes are clariﬁed in the cases of diffu-

sion via vacancies and that via interstitials. The differ-

ence between the diffusion properties in pure metallic

systems and those in ordered alloy systems is discussed.

To investigate diffusion properties in crystalline

phases, here we choose the alloy model discussed in

Sect. 17.2.1 where the atomic interactions are described

by the 8–4 LJ potential. Therefore, all units used in this

section are the same as those found in Sect. 17.2.1. The

advantage of using this model is that we can ﬁnd three

ordered phases in this system, that is, the L1



and the



(fcc-based) phases at low temperatures, and the B2

(bcc-based) phase at high temperatures. So we can com-

pare the diffusion properties of the ordered alloys with

those of pure metals.

The atomic diffusion in crystalline phases is mainly

due to vacancies and interstitials. In MD simulations,

the atomic diffusivity can easily be calculated from

the atomic mean square displacement as in (17.58),

while diffusion experiments are always accompanied

with many difﬁculties. We introduce a single vacancy

0.06

0.04

0.02

0.00

0 10000 20000 30000 40000 50000

Mean square displacement

Time steps

Fig. 17.40 Time evolution of average mean square dis-

placement of atoms in an fcc phase introduced one vacancy

into N =4000 atoms at T =0.6

into an fcc crystalline phase consisting of N = 4000

atoms of element 1 at T = 0.60. Figure 17.40 shows

a time evolution of the atomic mean square displace-

ment in the fcc phase introduced a single vacancy. The

slope of the time dependence denoted by the dotted line

in Fig. 17.40 corresponds to the atomic diffusivity D,

while the intersection point of the dotted line with the

vertical axis corresponds to the average value of thermal

vibration. Note that the temperature T =0.6 amounts to

90% of the melting temperature of the fcc structure of

elements 1, and the drawback of the MD calculation of

the diffusivity is that we can barely estimate the diffu-

sivity at low temperatures far away from the melting

point within the available calculation time.

According to the prescription discussed above, we

can estimate the diffusivity via vacancies in the fcc

phase of element 1 as well as the alloy phases of the

element 1 and the element 2, that is, the B2 phase and

the L1



phase by introducing a pair of vacancies of the

element 1 and 2 into a perfect crystalline phase. The

results are shown in Fig. 17.41. The temperature de-

pendence of the atomic diffusivity in each phase are

denoted by closed circles (fcc), open triangles (B2), and

closed triangles (L1



). In the calculation, a pair of va-

cancies is introduced into an N = 4000 system for the

fcc phase, and into an N = 3456 system for the or-

dered phases. The values T

−1

on the horizontal axis are

normalized by the melting temperature of each phase,

–04

–05

–06

–07

1 1.2 1.4 1.6

Diffusivity

FCC (intst)

FCC (vac)

L10''

Fig. 17.41 The temperature dependencies of the diffusiv-

ity in a B2 phase (open triangles), an L1



phase (closed

triangles), and an fcc phase (closed circles) with the va-

cancy concentration ≈0.0005 and that of diffusivity in an

fcc phase with the same interstitial concentration (open

squares)

Part E 17.4

Molecular Dynamics 17.4 Diffusion 1007

Fig. 17.42 Trajectories of largely moved atoms in an fcc

phase with two vacancies at T =0.30. The balls denote the

ﬁnal positions of atoms and the sticks connect them to the

initial positions

except the L1



phase, which is normalized by the melt-

ing temperature of the B2 phase.

Although the calculated values include a consid-

erable amount of statistical errors, especially at low

temperatures, we can recognize the following proper-

ties. Comparing the diffusivity in the B2 phase with

that in the L1



phase, the atomic diffusion is sup-

pressed in the latter phase, mainly because the L1



phase has a structure based on a close-packed struc-

ture. Comparing the diffusivity in the fcc phase with

that in the L1



phase, both of which are based on

close-packed structure, the atomic diffusion is also sup-

pressed in the latter phase, which suggests that the

atomic diffusion would be suppressed in ordered alloys.

We have also calculated the diffusivity in the fcc phase

by introducing a pair of interstitials into an N = 4000

system; the results are plotted in Fig. 17.41. The re-

sults indicate that diffusion via interstitials has a much

higher rate and lower activation energy than that via

vacancies.

In the MD simulations, we can easily pick up the

atoms that contribute the diffusion. In Fig. 17.42,we

have depicted the trajectories of atoms which have

moved a distance larger than half of the nearest neigh-

bor distance within 10

time steps in an fcc phase

of element 1 having introduced a pair of vacancies at

T =0.30, where the balls denote the ﬁnal positions of

Fig. 17.43 Trajectories of largely moved atoms in an fcc

phase with two interstitials at T = 0.25. The balls denote

the ﬁnal positions of atoms and the sticks connect them to

the initial positions

such atoms and the sticks connect them to their initial

positions. Two strings of atom trajectories can be iden-

tiﬁed, one of which looks broken into three parts due

to the periodic boundary conditions. These two strings

correspond to two vacancies introduced into a perfect

fcc crystal.

In Fig. 17.43, similar trajectories of atoms that have

moved signiﬁcant distances after 10

time steps are

depicted for an fcc phase, having introduced a pair of in-

terstitials at T =0.25. The two ﬁgures look similar, but

there are two differences between them. One is the his-

tory of the string of the sequence of the moving atoms

has been made. Each string looks like a row of tadpoles

and has two ends: one end is the head of the leading tad-

pole, and the other end is the tail of the tadpole at the

opposite end of row. So we call the former the head of

the string and the latter the tail of the string. For the case

of diffusion via vacancies, the atomic jump starts from

the head of the string and ends with the tail of the string,

while the direction of the time sequence is opposite for

the case of diffusion via interstitials. The second differ-

ence is in the distance of each atomic jump. For the case

of diffusion via vacancies, the distance of each atomic

jump is nearly always the nearest-neighbor distance,

while that for the case of the diffusion via interstitials is

less than the nearest-neighbor distance. These charac-

teristics of atomic diffusion will be revisited in the next

Part E 17.4