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162 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

remains open, these preliminary results give hope that continued atomistic

simulations of this type could soon provide a satisfactory answer.

It is both possible and desirable to apply the same simulation methods

to other intermetallic compounds that can reveal different diffusion mech-

anisms. The fact that NiAl supports NNN vacancy jumps is due to its rel-

atively open (bcc-based) structure and a large atomic size difference, with

Ni atoms being significantly smaller than Al. The same factors are likely

to be responsible for the collective nature of many diffusive jumps in this

compound. Preliminary MD simulations for L1

-TiAl (close-packed

structure with a small atomic size difference) found no NNN vacancy

jumps or collective transitions and presented a relatively simple picture of

first-neighbor vacancy jumps in a manner reasonably consistent with cur-

rent KMC models.

3.8.4 Accelerated Molecular Dynamics

Many diffusion applications of MD are hampered by the time-scale

limitation inherent in this method. Indeed, atomic vibration frequencies in

solids are typically on the order of 10 THz. To resolve such vibrations

properly, the integration time step of MD cannot be larger than a few fem-

toseconds. Therefore, even after a million MD steps, the accumulated

time is only about a nanosecond. For relatively small systems (a few hun-

dred atoms), the time can be pushed as far as to the microsecond range. In

many systems, however, the number of diffusion jumps that can happen

over this time is not enough for a reliable evaluation of the diffusion coef-

ficient, even at high temperatures. It could be argued that this time scale

can still be suitable for the exploration of unknown diffusion mechanisms.

This is true, but again, only at relatively high temperatures. We cannot

exclude, however, the possibility that the dominant diffusion mechanisms

at low temperatures can be different from those at high temperatures. High

temperatures favor mechanisms with relatively high barriers and large

attempt frequencies, while low temperatures are typically dominated by

low barrier mechanisms with small attempt frequencies. Moreover, many

structures only exist at low temperatures, due to structural transforma-

tions, atomic disorder, melting, or any other factors that may prevent the

structure of interest from being studied at high temperatures. In such sit-

uations, even the identification of diffusion mechanisms by MD becomes

a difficult computational task.

In recent years, a number of new methods have been developed that

expand the time scale accessible by MD.

[89]

Such methods, referred to

collectively as accelerated dynamics methods, include parallel replica

dynamics,

[165]

hyperdynamics,

[166]

and temperature-accelerated dynamics.

[167]

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ATOMISTIC COMPUTER SIMULATION OF DIFFUSION, MISHIN 163

The common idea of these methods is to accelerate the rate at which the

system escapes from the energy basins corresponding to its equilibrium

states, but to do so in a way that preserves the relative probabilities of

different escape paths.

In parallel replica dynamics,

[165]

the system is replicated on M parallel

computer processors used in the simulation. The states of the replicas are

randomized, and independent MD runs are implemented on the processors

in a parallel mode. The Mindependent dynamic trajectories simulated this

way explore the phase space a factor of M faster than a single trajectory

would, which is the key to the accelerated escape rates. Whenever a

processor detects a transition, simulations on all other processors are

stopped, and the clock is advanced by the time summed over all trajecto-

ries. The trajectory that experienced a transition continues to run for some

time to allow for possible recrossing events, and the clock is advanced by

that time. The final state is then replicated on all M processors, random-

ized, and the simulation continues. This method is very general; in partic-

ular, it does not require that the system follow TST. The only assumption

made is that the transitions follow first-order kinetics with an exponential

decay of the transition time. This assumption guarantees an unbiased

state-to-state evolution of the system. Although the ideal acceleration fac-

tor M is never achieved, the boost can be very significant.

[89]

The effi-

ciency of the method is reduced by the randomization time as well as the

time required to include possible dynamic correlations after each transi-

tion. There is also an overhead associated with the automated detection of

transitions. The latter is implemented by relaxing the system at regular

time intervals and comparing the relaxed state with the initial one.

The hyperdynamics method

[166]

takes a different approach. Namely,

the potential energy surface within each energy basin is modified by

adding a specially designed boost potential. The latter should be positive

near the equilibrium state and go to zero near the dividing surfaces. By

raising the energy around the equilibrium, the boost potential reduces all

transition barriers and accelerates the system’s evolution from state to

state. The boost potential distorts the atomic vibrations in each equilib-

rium state and makes them totally unphysical. However, it can be con-

structed in such a way that the probabilities of different escape paths

from each state, determined within TST, are not affected. As a result, the

system will follow the same state-to-state evolution as it would without

the boost potential, but will do so much faster. In contrast to parallel

replica dynamics, the hyperdynamics method relies on TST. The method

has been used in a number of simulation studies relating to metal surfaces,

and impressively high boost factors (~10

) have been demonstrated.

[89, 166]

An attractive idea is to combine hyperdynamics with parallel replica

dynamics, which could lead to even higher boost factors.

[89]

The critical

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164 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

step and the major challenge of the method is the construction of the boost

potential. Ideally, the latter should not rely on any prior information

regarding possible transitions, so that the method could be applied to an

exploration of unknown structures and diffusion mechanisms. An opti-

mum should also be found between the acceleration of transition rates and

the computational overhead associated with the construction of the boost

potential. This is a nontrivial problem, even in relatively simple sys-

tems.

[89, 166]

In complex situations, the efficiency of the method can be

impeded by difficulties in constructing a boost potential for a wrinkled

energy surface with multiple shallow minima.

The temperature-accelerated dynamics (TAD) method

[167, 168]

relies on

more approximations than the two previous methods, but the boost factors

can be much higher. The idea is that an MD simulation is run at a high

temperature, T

, but the system is advanced from state to state with tran-

sition probabilities corresponding to a lower temperature, T

. At each step

of the procedure, an MD simulation starts from an equilibrium state and

is run at temperature T

until a transition is detected. As mentioned previ-

ously, the algorithm checks for possible transitions by periodically relax-

ing the system and comparing the relaxed structure with the initial one.

Once a transition is detected, its energy barrier e

is evaluated by the NEB

method. Assuming that the harmonic TST is valid at both temperatures (T

and T

), the transition rate should follow the Arrhenius temperature

dependence, Eq. (42). Then the escape time t

detected at temperature T

can be extrapolated to the escape time t

at temperature T

 t

exp







. (51)

Notice that only the transition barrier, not the attempt frequency, is needed

for this extrapolation. The MD trajectory is then reflected back to the

original basin and the simulation continues. This procedure produces a set

of transitions that can happen from the initial state at temperature T

together with the extrapolated escape times t

at temperature T

. The

transition corresponding to the shortest value of t

is considered to be the

one that would happen if MD simulations were run at temperature T

Assuming also that there is a lower bound on possible attempt frequen-

cies, the time can be estimated at which the high-temperature simulation

can be safely stopped. At that point, the transition with the shortest t

implemented, the clock is advanced by t

, and the simulation continues

from the new state. As in the previous methods, no prior information on

possible diffusion mechanisms is used in TAD, which makes it suitable

for the exploration of new structures and unknown diffusion mechanisms.



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ATOMISTIC COMPUTER SIMULATION OF DIFFUSION, MISHIN 165

The procedure carries a significant computational overhead, since a cer-

tain number of possible transitions from each state needs to be detected

before one of them is accepted. This overhead limits the number of tran-

sitions that are actually implemented in comparison with running regular

MD at temperature T

. However, if the difference between the tempera-

tures T

and T

is large enough, the extrapolated MD time at T

can reach

spectacularly large values (for example, up to several months

[169]

) and lead

to boost factors as high as 10

[169, 170]

To alleviate the possible trapping

effect arising from low transition barriers, the repeated transitions can be

treated in the KMC mode.

[167, 168]

Overall, the accelerated dynamics methods have a potential to make a

significant difference to modern atomistic studies of diffusion in solids.

So far, they have mainly been applied to surface diffusion and surface

growth of metals. However, the coming years may see other interesting

applications, including diffusion in grain boundaries, dislocation cores,

ordered intermetallic compounds, and other systems.

3.9 Conclusions

Atomistic computer simulations present a powerful approach to gain-

ing new knowledge on diffusion processes in materials. The capabilities

of such simulations have drastically improved in recent years due to the

development of new simulation methods reinforced by increased com-

puter power. More reliable models of atomic interaction in materials have

been developed, allowing a description of point defect properties on a

quantitative level. The impact of first-principles calculations on diffusion

science is already quite significant and will increase in the future. At the

same time, the use of semi-empirical interatomic potentials will continue

to be the mainstream of diffusion simulations. Since such simulations

require statistical averaging and involve rare events, fast methods that

provide access to increasingly larger systems and longer simulation times

will always be in demand. The traditional MD and KMC methods are and

will remain the main tools of diffusion simulations. At the same time, such

simulations may greatly benefit from adopting the new approaches devel-

oped in recent years, such as the accelerated dynamics methods and the

emerging on-the-fly Monte Carlo schemes.

We believe that two trends are playing a fundamental role in shaping

modern diffusion science. First, diffusion theory is turning more towards

specific materials and more complex systems. Although the traditional

analysis of vacancy diffusion in an abstract alloy A-B is still an important and

respectable subject that well deserves further efforts, the new challenge is to

understand, describe, and calculate diffusion coefficients in a particular

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166 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

metal, alloy, or compound. In some (but not all) of the rare cases when this

was done, the agreement with experiment was encouraging. There is notable

progress in the modeling of surface and GB diffusion, but diffusion along

interphase boundaries and dislocations remains an almost unexplored area.

The second trend is the growing interest in understanding diffusion

mechanisms on the atomic level. While most of the traditional diffusion the-

ory conveniently assumes the simple vacancy mechanism, the recent stud-

ies of surface and GB diffusion reveal a rich variety of other mechanisms,

which can be mediated either by vacancies or interstitials and can involve

collective atomic displacements. Many of the new mechanisms originate

from previously unknown structural forms of point defects, such as delo-

calized vacancies and interstitials, and unstable vacancies. In fact, unstable

vacancies and collective jumps have now been found, by atomistic simula-

tion, even in bulk alloys and compounds. Why should we be so interested

in diffusion mechanisms? Clearly, gaining knowledge of the relevant atomic

mechanisms is the first step in any predictive calculation of diffusion coef-

ficients in a particular material. Besides this practical aspect, however, there

is a basic question lying in the very foundation of diffusion science: how

exactly do atoms move in solids? It is the inexhaustible desire to find a sat-

isfactory answer to this question that motivates the pursuit of diffusion

mechanisms, a pursuit that will never be over.

Acknowledgment

Work on this chapter was supported by the U.S. Air Force Office

of Scientific Research (Metallic Materials Program) under Grant

No. F49620-01-0025.

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