Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications

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Titanium alloys: modelling of microstructure288

processes of decohesion (along

(1210)

plane in our case). Here, the trace of

a macrocrack plane coincides with the

[1321]

line direction. In a Ti

single crystal with operative basal slip, cracks propagate along macroscopic

habit planes parallel to

{1012}, {21 13} and {4311}

. Taking into account

only these types of planes, it is possible to demonstrate that the

[1321]

line

direction coincides with the

(2113)

plane projection into the surface foil

(1011)

. Thus, the accumulation of shear microcracks in (0001) slip bands,

being the initiating mechanism of macrocleavage, ultimately determines the

low plasticity of a single crystal Ti

Al with operative basal slip.

It is noted that stresses generated by superdislocations in a slip band

might promote both microcrack nucleation and transversal slip of screw

superdislocations in the prism plane. Formation of plastic zones in the prism

plane leads to microcrack blunting and coming to a stop. These processes are

observed experimentally (Fig. 11.6). Therefore, in the slip band within several

parallel basal planes, a microcrack system forms. A main crack forms as a

result of the coalescence of these microcracks.

The type of the forming macrocleavage surface is determined by the

values of applied stresses and decohesion energy for the crack plane. The

experimental data prove that decohesion occurs in pyramidal planes. Normals

to the experimentally observed pyramidal planes lie on the stereographical

projection near the arc of the big circle which depicts the trace of the

(1011)

plane with the

[1012]

normal. When a single crystal is compressed along

the

[1012]

direction, the maximum normal tensile stress appears in the

planes whose normals lie in the

(1011)

plane, i.e. they belong to the above

arc of the big circle. Thus, a cleavage in the pyramidal planes with maximum

normal stress is observed experimentally. Although external normal stress is

high, we did not observe microcracks in prism planes of the

{1210}

type,

for instance, suggesting that they are caused by the difference in cohesion

characteristic of these planes (cleavage stress, surface energy, etc.). This

matter requires additional investigation.

There is one more significant element which facilitates microcrack

nucleation. In this section, we have described the model of shear microcrack

nucleation in the basal plane. According to this model, a microcrack nucleates

as a result of coalescence of several a-superdislocations of screw orientation,

i.e. the top of a formed microcrack is parallel to the Burgers vector

1120〈〉

of superdislocation. This direction <

1120

> belongs to cleavage microfacets

on the (0001),

{1012}

and

{1011}

planes of the zigzag cracks, observed

experimentally (Fig. 11.4 and Table 11.2). The

{1 123}

plane is often the

main plane of a crack. One can suppose that in the top of the crack in the

planes of

{1 123}

type, a plastic zone does not form by the dislocations of

prism slip systems. This prevents the blunting of the crack and promotes its

spreading through the bulk of the crystal.

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Crystallographic and fracture behaviour of titanium aluminide 289

11.6 Summary

Transmission electron microscopy shows that the (0001) basal slip is

accompanied by short bands of

{1100}

prism slip formation. The

macrocleavage surface is a step-like one, consisting of microcracks of shear

type along the (0001) basal planes and decohesion in the

{1210}

prism

planes. In a Ti

Al single crystal orientated for basal slip, the planes of micro-

and macrocracks cleavage are the basal and pyramidal planes of

{1011}

{1012}

and

{1 123}

types. The shear microcracks’ coalescence in the basal

plane results in the formation of the step-like fracture surface, consisting of

(0001),

{1011}

and

{1012}

planes. The most probable plane of the main

crack is the

{1 123}

plane. The microcrack stopping and blunting in the

(0001) plane is due to plastic zone formation in the prism slip systems.

A model of shear microcracks nucleation in basal planes, with screw a-

superdislocations coalescence, has demonstrated that long-range stress of

dislocations in a slip band plays an important part in the process of microcrack

nucleation. Both theoretically and experimentally, brittle fracture is initiated

by nucleation of microcracks of shear type along basal slip planes.

11.7 References

Kar’kina L E, Yakovenkova L I and Rabovskaya M Y (2002), ‘Dissociation of 1/3

<1120> superdislocations upon basal slip in Ti

Al’, Fizika Metallov i Metallovedenie,

93 (1), 32–41.

Yakovenkova L I, Kar’kina L E, Kirsanov V V, Balashov A N and Rabovskaya M Y

(2000), ‘N-particle potentials of interatomic interaction in Ti

Al and simulation of

planar defects in (0001), {1100}, {2021}, and {1121} planes’, Fizika Metallov i

Metallovedenie, 89 (3), 237–45.

Yakovenkova L I, Karkina L E and Rabovskaya M Y (2003a), ‘The atomic structure of

the 1/3

[21 10] superdislocation core and prismatic slip in Ti

Al’, Tech Phys, 48 (1),

56–62.

Yakovenkova L I, Karkina L E and Rabovskaya M Y (2003b), ‘Superdislocation core

structure in pyramidal slip planes and temperature anomalies of Ti

Al intermetallic

deformation’, Tech Phys, 48 (10), 1289–95.

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290

Atomistic simulations of interfaces and

dislocations relevant to TiAl

Abstract: TiAl alloys for high temperature applications usually have the

form of lamellar structures composed of γ-TiAl and α

-Ti

Al phases.

Refinement of the lamellar size is a promising way of generally improving

the yield strength of such alloys, but this does not work when the lamellae

are too thin. This chapter takes the first steps towards an understanding of

the variation of yield strength with lamellar size based on atomistic

modelling. Topics discussed include bond-order potentials for the Ti–Al

system, the structures and energies of γ/α

interfaces, simulations with and

without misfit dislocations, and with stoichiometry variations at the

interface, and the structures of dislocation cores.

Key words: interatomic potential, dislocations, dislocation pile-ups,

molecular statics, lamellae.

12.1 Introduction

Regarding the mechanical properties of lamellar TiAl alloys, both a Hall–

Petch relation between yield stress σ

and lamellar thickness λ, and saturation

of yield stress with decreasing lamellae width have been observed

experimentally (Maruyama et al., 2001, 2002). The Hall–Petch relation and

the upper limit to the hardening are predicted on the basis of a pile-up model

of dislocations. The multiple pile-up model considers that, during the early

stage of deformation, a dislocation pile-up is formed in a lamella due to the

blockage by an interlamellar interface. To eventually defeat the interface, the

applied shear stress τ is raised, enabling more dislocations to enter the pile-

up. Yielding occurs when the leading dislocation in a pile-up can overcome

the barrier stress at an interface.

However, the σ

– λ curve of the lamellar TiAl alloy exhibits complex

behaviour in the transition region from the Hall–Petch relation to the upper

limit, which cannot be explained by the simple pile-up model. Maruyama

et al. (2004) considered the distribution of lamellar thickness based on

measurement of a real lamellar microstructure. However, the distribution of

lamellar thickness by itself cannot explain the complex transition from the

Hall–Petch relation to the saturation.

To understand the complex σ

– λ relation, we have to take account of the

change in the structure of a γ/α

boundary in the transition region. In a thick

lamellar structure, misfit dislocations are introduced into γ/α

lamellar

boundaries to relieve the lattice misfit between γ and α

and are found on the

lamellar boundaries (Maruyama et al., 2004). We refer to these boundaries

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Atomistic simulations of interfaces and dislocations 291

as semi-coherent. On the other hand, in an alloy with a thin lamellar structure,

the γ/α

boundaries are found to be perfectly coherent (Maruyama et al.,

2004).

The entire curve of yield stress versus lamellar thickness can be explained

by taking account of the difference in lamellar boundary structure between

a dislocated boundary and a coherent one (Maruyama et al., 2004). The

difference in lamellar boundary structures, specifically the absence or presence

of misfit dislocations, may be associated with a difference in the barrier

stress at an interface. The σ

– λ curve was explained assuming the boundary

resistance τ* to be 440 MPa for γ lamellae thicker than 50 nm, and 260 MPa

for those thinner than 50 nm (Maruyama et al., 2004). It was conjectured that

dislocation sources in lamellar boundaries interacted with the misfit

dislocations, and the misfit dislocations acted as obstacles to the operation of

the dislocation sources. The dislocated boundaries are more resistant to

dislocation motion across the boundaries.

In its present form, the Hall–Petch model is far from providing a complete

theory of yield in a multilayer, since it pays no attention to the operation of

dislocation sources, and it uses estimates, mostly based on elasticity, for the

barrier stress. It is important to understand the mechanisms of deformation

and transition of dislocation across the γ/α

interface at the atomic scale. In

particular, in order to improve the model, a key requirement is a more accurate

calculation of barrier strengths in a multilayer.

Modelling of the dislocation–γ/α

interface interactions in lamellar TiAl

has been generally carried out using the continuum approach in which no

account is taken of the material’s microstructure. Recent advances in

computational materials science have enabled atomistic modelling of the

dislocation–interface interactions to become a respectable alternative for

elucidation of the role of atomic-scale effects on the transmission of plastic

deformation across the interface. Based on such calculations, one hopes that

ultimately it will be possible to design new, lighter alloys with both room-

temperature ductility, and strength that increases with temperature as close

to the melting point as possible.

12.2 Tasks

Many simulations can be carried out with the Oxford order–N (OXON)

program package using the bond order potential (BOP), discussed in more

detail below.

The following research tasks are of increasing complexity and will require

an increasing number of atoms in the simulation, with a corresponding increase

in the cost of computation, visualization and analysis:

(i) possible γ/α

boundary structures (coherent and partially coherent) by

molecular statics, including the role of local stoichiometry variations

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Titanium alloys: modelling of microstructure292

(ii) mobile dislocation core structures in γ and α

crystals by molecular

statics

(iii) application of shear stress to the dislocation model to study the friction

barrier

(iv) molecular statics and dynamics study of the blocking strength of a

coherent γ/α

interface using BOP

(v) molecular statics simulations of the interaction of misfit dislocations

with mobile dislocations and evaluation of the influence of the misfit

dislocations on the blocking strength of the γ/α

interface

(vi) introduction of a queue of mobile dislocations in γ or α

layers, which

will be pushed against the γ/α

interface in order to study the mechanisms

of pile-up formation and eventually the penetration of the interface.

Detailed methodologies are discussed in the next two sections.

12.3 Computational procedure

The calculations can be carried out using a molecular statics method, whereby

the internal energy of the system studied is minimised by moving the particles

according to the conjugate gradients algorithm so as to attain zero forces on

each atom. An initial phase of molecular dynamics simulation and quenching

may be carried out in order to avoid trapping in local minima.

The microstructure of TiAl lamellar alloys is formed by a succession of γ

(L1

structure) and α

(DO

structure) parallel lamellae related by

(0001) //(111) ,

αγ

and

〈〉〈〉2//110.

110

αγ

The γ lamellae have three

orientation groups with respect to the interface plane (two orientation variants

in each group) corresponding to the three possible positions of the 〈110〉

direction in the habit plane. We refer to them as Groups I, II and III.

A first set of simulations should be carried out to explore the structures

and energies of these interfaces. They can be constructed initially to be

perfectly coherent, and then relaxed energies and structures are obtained.

With much larger supercells, the corresponding semi-coherent interfaces can

be studied by introducing misfit dislocations, the nature of which is described

below. The boundary structures obtained will be used in simulations of the

dislocation–boundary interactions.

Since several types of dislocation can be activated in the α

and

γ phases,

numerous crystallographic situations can be encountered. They differ in the

nature of the interface, and in the incident system, which depends on the

orientations of γ lamellae with respect to the interface plane. The events

occurring at an interface can also be different for a given crystallographic

situation as a function of the orientation of the applied stress. Atomistic

simulations of the structures of the main dislocations observed in α

and γ

can be used to investigate their interaction and subsequently to study the

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Atomistic simulations of interfaces and dislocations 293

transfer of deformation across the interface between α

and each of the three

γ lamellae orientation groups.

The simulations should be designed in the light of experimental work. In

-Ti

Al, deformation occurs by glide of dislocations with

b = 1120

〈〉

{1010}

, and less favourably on (0001), and with

b = 1126

〈〉

on either

{1121}

{2021}

. The

110 {111}〈〉

ordinary slip system is dominant in

the γ phase.

The transfer mechanisms of the deformation at the γ/α

interface have

been studied by in situ straining experiments. Different types of behaviour

have been found, depending on the nature of the interface, on the involved

deformation systems and/or on the lamella orientation. The results of these

studies may be summarised as follows:

(i) Transmission of shears through γ/α

interfaces at the intersection of

these interfaces and certain cross-lamellar twins is effected by slip

of dislocations with

b = 1120

〈〉

in the α

phase, which may glide on

either

{1010}

{2021}

(ii) Such shear transmission events that result in dislocations with

b = 1126

〈〉

appear to be very difficult.

(iii) Dislocations with

b = 1120

〈〉

and

b = 1126

〈〉

can be generated by

stress-induced activation of sources in γ/α

interfaces. The latter

dislocations glide on either

{1121}

{2021}

, although mostly slip

occurs on

{1121}

(iv) In γ orientation variants of Group I, the Burgers vector

b = 110

〈〉

the ordinary dislocation is parallel to the Burgers vector

b = 2110

〈〉

in the α

phase. The movement of dislocations from the α

lamella to

the next γ (I) lamella and further to the other α

lamella occurs by a

cross slip process. The γ/α

boundaries between an α

lamella and the

γ variant of Group II and III behave in a similar way to usual grain

boundaries because 〈110〉 is not parallel with

〈〉2110

. One of the

following two slip systems may operate in the four γ variants of Groups

II and III – the activation of a

10〈〉1

dislocation whose glide plane is

continuous with the primary slip plane of α

lamella or the activation

of a

10〈〉1

dislocation and its glide on a {111} plane not continuous

with the primary slip plane. This slip in γ may activate a

120 {1010}〈〉1

slip system in the next α

lamella.

Based on these results, the focus of attention is on the movement of

b = 1120

〈〉

dislocation in the α

phase and the ordinary dislocation

b = 110

〈〉

in the orientation variants of the γ

phase.

Besides the coherent and incoherent interfaces, other issues include the

core structures, and possibly also the friction stresses, of the mobile dislocations.

A simulation cell with at least one lattice repeat distance along the dislocation

line is necessary. The dislocations are introduced into the centre of the

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Titanium alloys: modelling of microstructure294

simulation cell by displacing all the atoms from their perfect lattice positions

according to their anisotropic elastic displacement field. All atoms in an

outer shell with radius R

out

in width (greater than 2R

cut

, where R

cut

is the cut-

off range of the BOP) should be fixed at their original elasticity positions

during the relaxation. Atoms within an inner radius are then allowed to relax,

using the BOP interactions employing periodic boundary conditions along

the dislocation line direction. The relaxation calculations are performed by

minimising the energy of the atomistic model using a molecular statics method.

To study the motion of dislocations, we apply homogenous shear stresses

to our atomistic model. The stresses are actually imposed via homogeneous

shear strains by displacement boundary conditions and are increased in a

stepwise manner. After every step, the entire system is relaxed and examined

for changes in the core structure using differential displacement plots. The

minimum stress at which the dislocation moved by a translational unit is

defined to be the friction stress of the dislocation. Increments corresponding

to about a tenth of the estimated strain at the friction stress are applied until

the dislocation core moves by at least one lattice repeat distance normal to

the dislocation line in the glide plane.

A rectangular parallelepiped consisting of a two-part computational cell

is used in all simulations of interfaces and dislocation–interface interactions.

Before introducing an incident dislocation in the γ or α

crystals,

the interface

simulation cells are relaxed using periodic boundary conditions along the

γ/α

interface directions. In the direction perpendicular to the interface, the

boundary conditions are expected to simulate the bulk media on either side

of the interface. Along the interface normal, atoms within two slabs of width

cut

, one at the upper surface and the other at the lower surface, are fixed

at their perfect lattice positions. The atoms between these slabs are allowed

to relax.

After this initial relaxation of the γ/α

interface, the corresponding

dislocations are introduced into the simulation cell by displacing all atoms

according to the anisotropic elastic displacement field. As before, atoms

within an inner radius are relaxed using the BOP interactions with the outer

shell fixed. The relaxed structures of the dislocation cores are examined

using differential displacement plots. To determine the blocking strength of

the interface to dislocations, a pure shear strain is applied on the previously

relaxed cell. The sense of the shear strain is such that the force on the

dislocation pushes it towards the interface from the corresponding γ or α

layer. Increasing amounts of shear strain are applied to the initially relaxed

configurations. When slip dislocations collide with the interface, there are

three fundamental reactions that may take place – the dislocation may be

absorbed, transmitted, or rejected. In order to understand the nature of the

nucleation of the dislocations and their interaction with the lamellar interfaces,

we need to answer the following questions: (i) will the dislocation be completely

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Atomistic simulations of interfaces and dislocations 295

transmitted through the interface into the adjacent lamella or order domain;

(ii) will it retain its integrity in the interface but undergo core reconstruction;

(iii) will the dislocation dissociate into discrete interfacial defects, which

may subsequently disperse along the interface due to their mutual repulsion.

The stress at which the dislocations overcome the interface obstacle (the

dislocation moves freely through the bilayer simulation cell) is the barrier

strength of the interface.

To investigate the blocking strength of the γ/α

interface against a dislocation

pile-up, we introduce a queue of dislocations in our atomistic model. After

the initial relaxation in the absence of external applied stress, the pile-up is

pushed against the interface by a shear stress, which is increased in a stepwise

manner. The incremental shear stress enables more dislocations to enter the

pile-up, until the total stress at the head of the pile-up is so concentrated that

slip occurs in the adjacent lamella and one or more dislocations are driven

away from the head of the existing pile-up. Stresses required for the above

process are evaluated.

The blocking strength of the γ/α

interfaces depends on the presence of

interfacial dislocations. Crystal lattice images of a γ/α

lamellar boundary

observed in a coarse lamellar structure show that the interface contains several

ledges of two atomic plane heights. Analyses of their Burgers vector indicate

that the ledges correspond to a

110 112〈〉〈〉

()

1010 21 1〈〉〈〉

()

dislocation. All the ledges have the same sign on a lamellar boundary and are

distributed regularly with similar spacing, indicating that they are misfit

dislocations introduced to accommodate the lattice misfit.

Guided by the experimental observations, we introduce

10 112〈〉〈〉

()

misfit dislocation into our atomistic model and simulate the interaction of

the misfit dislocation in γ/α

boundaries with mobile dislocations. The

dislocation is created by removal or addition of two layers of atoms. In order

to analyse the role of the misfit dislocation, increasing amounts of shear

strain are applied to the initially relaxed configurations both in the presence

and in the absence of mobile dislocations in γ or α

crystals. This will show

how the misfit dislocation affects the resistance of the γ/α

interface. The

purposes of such simulations are: (i) to determine the critical load required

to induce or transfer plastic deformation at γ/α

interfaces; and (ii) to elucidate

the mechanisms of such deformations. Questions to be answered include

which particular lamellar boundary and orientation of the applied stress

leads to the emission or transfer of dislocations at the boundary, and a migration

of the boundary itself due to the stresses.

12.4 Choice of interatomic potential

Since titanium is a transition element with a partially filled d-band, it can be

expected that bonding in Ti–Al alloys will possess a significant covalent

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Titanium alloys: modelling of microstructure296

component. Atomistic calculations of γ/α

interfaces were carried out using

the Finnis–Sinclair type many-body potentials. Such central force schemes

are not predictive as far as the energies of the planar faults are concerned and

underestimate both the antiphase boundary (APB) energy on {100} planes

and the energy of superlattice intrinsic stacking faults (SISF). While this

could be corrected empirically by introducing longer ranged (third and further

neighbours) functions, the physical reason for the problem is the neglect of

covalent contributions. A higher value is achieved if the non-central character

of atomic bonding is considered, which is the case in BOPs. Embedded-atom

method (EAM) interatomic potentials were constructed for the Ti–Al system

and employed in studies of dislocations, interfaces and point defects. These

calculations describe a number of general features of lattice defects but

neglect the strong covalent and directional bonding that is characteristic for

the Ti–Al alloys. Another telling feature of TiAl is the negative value of

Cauchy pressures, which are the combinations of elastic constants C

– C

and C

– C

. With potentials of the EAM or Finnis–Sinclair type, these

pressures are always positive, and even the earlier BOP formulations could

not correct this problem, which was solved only within the new BOP scheme

in which the environmental dependence of the repulsive energy is taken into

account.

A general description of BOPs and the tight-binding model on which they

are based is given by Finnis (2003). BOPs were first formulated by Pettifor

and coworkers and represent a numerically efficient scheme that works within

the orthogonal tight-binding approximation. A further advantage of BOPs is

that they allow the evaluation of the energy in a time that scales linearly with

the number of atoms, so that it is feasible to simulate the tens of thousands

of atoms needed to study the interaction between dislocations and γ/α

interfaces. Variants of the BOP method were implemented in the OXON.

A remarkable property of the BOP for L1

TiAl developed by Znam et al.

(2003) is that it can be used not only for 1:1 stoichiometry of TiAl but shows

excellent transferability to the 3:1 stoichiometry of Ti

Al which is vital for

the investigation of γ/α

interface. All the parameters of this model were

fitted only to TiAl, which makes its transferability a genuine test that it is

physically meaningful. The most important test of the potential was the

calculation of the stacking-fault-like defects APB, complex stacking fault

and SISF on {111} planes in L1

TiAl. The calculated energies of these

faults are in good agreement with those evaluated by ab initio methods.

The characteristics of the BOP constructed by Znam et al. (2003) mentioned

above show that it is the most suitable atomistic model for atomistic study of

dislocations not only in TiAl but also in Ti

Al, as well as the interactions

between them and the interfaces between these phases.

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Atomistic simulations of interfaces and dislocations 297

12.5 References

Finnis M (2003), Interatomic Forces in Condensed Matter, Oxford: Oxford University

Press.

Maruyama K, Yamada N and Sato H (2001), ‘Effects of lamellar spacing on mechanical

properties of fully lamellar Ti–39.4mol%Al alloy’, Mater Sci Eng A, 319–321, 360–

63.

Maruyama K, Suzuki G, Kim H Y, Suzuki M and Sato H (2002), ‘Saturation of yield

stress and embrittlement in fine lamellar TiAl alloy’, Mater Sci Eng A, 329–331, 190–

95.

Maruyama K, Yamaguchi M, Suzuki G, Zhu H, Kim H Y and Yoo M H (2004), ‘Effects

of lamellar boundary structural change on lamellar size hardening in TiAl alloy’, Acta

Mater, 52 (17), 5185–94.

Znam S, Nguyen-Manh D, Pettifor D G and Vitek V (2003), ‘Atomistic modelling of TiAl

– I. Bond-order potentials with environmental dependence’, Phil Mag, 83 (4), 415–

38.

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