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Dislocation glide in model Ni„Al…solid solutions by molecular dynamics
E. Rodary,1D. Rodney,2L. Proville,1Y. Bréchet,3and G. Martin1
1Service de Recherches de Métallurgie Physique, CEA-Saclay, 91191 Gif sur Yvette, France
2GPM2/ENSPG, Domaine Universitaire BP 46, 38402 St Martin d’Hères Cedex, France
3LTPCM/ENSEEG, Domaine Universitaire BP 75, 38402 St Martin d’Hères Cedex, France
(Received 23 March 2004; published 31 August 2004)
The glide of an edge dislocation, in a random solid solution, Ni (1to8at%Al), is simulated by molecular
dynamics (MD). An embedded atom method potential has been optimized to reproduce the relevant properties
of the face centered cubic solid solution and of the L12Ni3Al phase. Glide is studied at fixed temperature and
applied stress. Three parameters are found to be necessary to describe the rate of shear as a function of applied
shear stress:
sis the static threshold stress, below which the glide distance of the dislocation is not sufficient
to insure sustained shearing;
dis the dynamical threshold stress, which reflects the friction of the pinning
potential on the moving dislocation; Bis the friction coefficient, which relates the effective stress 共
−
d兲to
the glide velocity. We also find that the obstacles are made of specific configurations of the Al atoms, which are
brought in positions of strong mutual repulsion in course of the glide process. The solute-solute short range
repulsion, rather than the usually assumed dislocation-solute interaction, is thus argued to be the main mecha-
nism responsible for chemical hardening in the present concentrated random solid solution. The use of the
above results in the frame of multi-scale modeling is exemplified.
DOI: 10.1103/PhysRevB.70.054111 PACS number(s): 62.20.Fe
I. INTRODUCTION
While the elastic theory is well suited for most problems
related to dislocations, atomic scale modeling is unique for
describing the core of dislocations [e.g., Ref. 1], the effect of
electronic structure on the latter (materials chemistry),2and
ultimately the selection of the glide systems.3Also unex-
pected features are revealed by atomic scale modeling, when
studying the interaction of dislocations with defect clusters
(e.g., self-interstitials,4–6 stacking fault tetrahedra,7cavities
and precipitates8), or dislocation nucleation at locations of
strong stress concentration, e.g., in nano-indentation,9,10 at
crack tips11–13 or in small confined devices.14
Here we address the question of dislocation glide in con-
centrated solid solutions, with Ni共Al兲as a prototype. The
classical scheme for describing hardening in random solid
solutions, is inherited from that of precipitate- and of dilute
solution- hardening:15–17 The glide of the dislocation line
proceeds under the combined effect of the external stress, the
line tension and the dislocation -precipitate or -solute atom
interaction. In the case of a concentrated solid solution, the
potential energy surface, along which the dislocation moves,
results from the superposition of two contributions: A far
field one, with undulations of weak amplitude, resulting from
the long range strain field of the solute atom population, and
a short range contribution due to the solute atoms close to the
dislocation. The equilibrium shape of a dislocation in such a
strain field results from the balance between the line tension
of the dislocation (in a medium with elastic constants which
depend on the solute content), and the interaction energy of
the dislocation with the strain field of the solute population.
The glide of the dislocation follows a complex path, made of
obstacle unpinning events, which trigger cascades of cross-
ings of lower potential energy barriers. Adetailed simulation
of such a process at zero and finite temperatures can be
found in Refs. 18–21 with application to internal friction in
solid solutions.22 Note that, in such models, when calculating
the potential energy surface along which the dislocation
movement proceeds, the position of the solute atoms is taken
as fixed (frozen), unaffected by the glide process.
Here we want to explore whether this classical scheme,
which is quite efficient for dilute random solid solutions or
for precipitates, is realistic in the case of more concentrated
solid solutions, i.e., whether this scheme is appropriate for
what is known as chemical hardening. We consider here ran-
dom solid solutions; the effect of short range order is beyond
the scope of the present paper. For this purpose, we simulate
the glide process of a dislocation at the atomic scale and
study the glide distance as a function of time under a broad
range of applied stresses, and for a range of compositions.
In the following, we first develop an energy model for
Ni共Al兲solid solutions and give some details on the molecu-
lar dynamics technique we use to simulate the glide of an
edge dislocation (Sec. II). We then describe the shear rate
observed in our simulations, as a function of the applied
stress, for solute contents ranging from 1 to 8 at %, at 300 K
(Sec. III). Several “in situ” observations show that disloca-
tion pinning results from specific configurations of the Al
atoms which are brought in positions of strong mutual repul-
sion in course of the glide process. This is the main finding
of this study. In the discussion section, we outline the pos-
sible strategies to incorporate the present findings into a mi-
cromechanical model.
II. ENERGY MODEL AND METHODOLOGY
Dislocation glide is simulated by molecular dynamics
(MD)at constant temperature and applied stress in a way
similar to that described in Refs. 4 and 5. Following the
pioneering work by Daw and Baskes,11,23 we employ an em-
PHYSICAL REVIEW B 70, 054111 (2004)
1098-0121/2004/70(5)/054111(11)/$22.50 ©2004 The American Physical Society70 054111-1
bedded atom method (EAM)interatomic potential which
provides a good compromise between the realism of disloca-
tion simulation and the computational load necessary to
handle a system large enough to keep at least part of the
complexity of the real alloy and to minimize the influence of
the boundary conditions. In order to simulate dislocation
glide in a Ni-rich Ni-Al solid solution, the energy model had
to be based on a Ni potential adapted to such simulations.
The Ni-Al potentials published in the literature (e.g., Refs.
24 and 25)use Ni potentials with stacking fault energies too
low to simulate realistic dislocation dissociations. We thus
had to optimize a new one. The requirement for this potential
is to provide a reasonable description of the elastic properties
and stacking fault energies and of their dependence with
composition.
A. Energy model for the Ni„Al…solid solution
In the framework of the EAM, the total energy of the
alloy is additive.24,26 The energy of each atom is the sum of
two terms: A repulsive part, written as a sum of pairwise
interactions ⌽
␣
(with
␣
,

=Ni,Al), and an embedding en-
ergy, F
␣
关
共r兲兴, which depends only on the nature,
␣
,ofthe
embedded atom and on the electronic density,
共r兲,atthe
position, r, of that atom. The electronic density is the sum of
single atom functions:
共r兲=兺


共r兲. As a consequence, the
EAM model for an alloy implies 7 functions, 6 of which
describe the pure elements 共⌽
␣␣
,F
␣
,
␣
,
␣
=Ni,Al兲, and only
one, ⌽
␣
,
␣
⫽

, is specific to the alloy. For the sake of
simplicity, we started from homo-atomic functions taken
from the literature and constructed only ⌽NiAl.
For pure nickel we chose the potential developed by An-
gelo et al. which has proved to be well adapted to the atomic
scale simulation of dislocations.5,26 In particular, the disso-
ciation width of the edge dislocation is quantitatively well
reproduced by this potential. For pure aluminum we chose
the potential proposed by Voter and Chen,24 which has the
drawback of yielding a low stacking fault energy (compared,
e.g., to the potential of Ercolessi et al.27), but has the advan-
tage of being written in the very same functional form as that
for Ni, which makes the optimization procedure, to be de-
scribed below, more tractable.
As for ⌽NiAl, we keep the form used for pure metals (first
term in the equation below, using Angelo’s notation26)aug-
mented by a short distance repulsive term (last term):
⌽NiAl共r兲=c1关exp共−2c2共r−c3兲兲 −c2共r−c3兲兴
⫻exp共1/共r−rcut
NiAl兲兲 +c4共rinf −r兲3H共rinf −r兲.
共1兲
In Eq. (1),H共x兲=0, or 1, respectively, for x⬍0or艌0.
The last term in the right-hand side (RHS)of Eq. 1, of the
same form as proposed by Ludwig and Gumbsch25 was
found useful in view of the large size effect of Al in Ni and
of the existence of very large strains in the dislocation core
region. The range, rinf, was not optimized but fixed to
0.15 nm.
In order to give more flexibility to the optimization of the
potential, we introduced three more parameters in the homo-
atomic functions, in a way not to alter the quality of the fit to
the properties of the pure metals (see Ref. 24 for more de-
tails). Keeping Voter’s notation, these are: sAl, which scales
the electronic density function of Al, gAl and gNi, which shift
the repulsive and embedding energy contributions of pure Al
and Ni while, keeping the balance constant.
These eight parameters were optimized by an iterative
procedure: First, a simulated annealing technique is used to
adjust the parameters with a merit function written as the
sum of the squares of the relative deviations of material
properties, calculated with the potential, from their values
known experimentally. These properties are evaluated from a
rigid lattice model and are relative to the L12Ni3Al com-
pound (lattice parameter, cohesive energy, C11,C
12,C
44, and
C44
⬘elastic constants)and to the solid solution (the lattice
parameter with 7.3 and 10.5 at % Al, for which precise mea-
surements are available28); when modeling the solid solution,
local order, as measured in Ref. 28 is taken into account.
Second, the potential so obtained is used to simulate the fully
relaxed solid solution at finite temperature, using a Monte
Carlo (MC)technique. The volume, atomic positions, and
local chemical order are explored using Metropolis algo-
rithm, varying the volume, displacing atoms and permuting
the latter, with respective probabilities of 1/8, 3/4, and 1/8.
The parameters are accordingly adapted, in order to improve
the realism of the simulation. The values of the parameters
obtained by this procedure are given in Table I.
B. Assessment of the energy model
The model for pure nickel yields the correct value of the
lattice parameter, a reasonable shear modulus (within 1% of
the experimental data)and acceptable stacking fault energy
(too small by 28%, when compared to the mean value of the
scattered existing experimental data).When1to8at%Al
are added in a fully random manner and the solid solution is
simulated by the MC technique, we find, in agreement with
existing experimental data, a roughly linear increase of the
lattice parameter and a decrease of the shear modulus and of
the stacking fault energy. The quantitative assessment is
shown in Table II.
The melting temperature is of the right order of magni-
tude. On the other hand, the predicted solubility limit is too
small by one order of magnitude. In the following, disloca-
tion glide is studied by MD at temperatures and time scales
TABLE I. Parameter values chosen for the EAM potential (see text).
c1共eV兲2.7277 c4共keV nm−3兲87.5715 sAL 3.8804
c2共nm−1兲16.765 rcut
NiAl共nm兲0.46168 gAL共meV nm3兲−9.5130
c3共nm兲0.19915 rinf共nm兲0.15 gNi共meV nm3兲6.0178
RODARY et al. PHYSICAL REVIEW B 70, 054111 (2004)
054111-2
where thermally activated atomic diffusion cannot proceed.
As a consequence, highly supersaturated solutions can be
handled without problem. However, the underestimate of the
solubility limit by our potential is a consequence of too large
ordering energies, which may result in an overestimate of the
short range repulsion among Al atoms. This, we expect,
might reinforce the chemical hardening we are studying.
C. Technical details
The computation cell we use is depicted in Fig. 1. The X,
Y,Zdirections are parallel, respectively, to 关110兴,关1
¯
12兴,
关11
¯
1兴. The 共11
¯
1兲glide plane is perpendicular to Z. Periodic
boundary conditions prevail in the Xand Ydirections, while
the movements of the atoms in the upper and lower free
surfaces parallel to the glide plane are confined in the planes
±Zmax. The effect of imposing periodic boundary conditions
in the Xdirection has been discussed in Ref. 5.
A typical simulation proceeds as follows. Solute and sol-
vent atoms are first located at random on the lattice, with the
appropriate concentration. The precise dimensions of the box
are adjusted in order to match the temperature and composi-
tion dependence of the lattice parameter of the random solid
solution. The positions of the atoms are then set according to
the elastic theory of dislocations, the edge dislocation being
placed in the center of the box. The configuration so obtained
is then relaxed, using a conjugate gradient routine, which
permits the dissociation of the dislocation. Initial velocities
are taken from a Maxwell distribution and the MD simula-
tion is started. Time integration is performed using Verlet’s
algorithm with a time step of 2.10−15 s. Temperature is main-
tained at a constant value by rescaling the atomic velocities
every 102time steps. After thermalization, the external stress
is applied, i.e., a constant force, parallel to the glide direc-
tion, is added to each atom in the outer Zsurfaces parallel to
the glide plane. The intensity of the force is proportional to
the inverse of the number of atoms in the plane, in order to
make sure the force per unit area is the same (with opposite
sign)in the upper and lower surfaces.
A standard method to identify atoms in dislocation cores
in pure metals is to select the ones with highest energies.12,13
In the present simulations, since the energy per atom is much
contrasted in the solid solution, the energy scale is of no
help. We rather identify the atoms in the dislocation cores as
the ones having an environment, which exhibits neither a
face centered cubic-, nor a hexagonal closed packed
structure.5
The optimum size of the computational cell is a compro-
mise between the duration of the computations, and the re-
alism of the simulations. We mainly worked with a box con-
taining 232 320 atoms. Some studies implied up to 867 600
atoms. The exact values of the box dimensions are adjusted
in order to account for the dependence of the lattice param-
eter on temperature and composition. Several box widths Ly,
in the Ydirection (see Fig. 1)along the dislocation line,
ranging from Ly=8.6 to 43.12 nm have been probed. Most
simulations were performed with a box width of 17.2 and
21.56 nm. More details on the dependence of dislocation be-
havior on the value of Lyare given in Sec. III C. Several box
lengths in the glide direction (direction Xin Fig. 1)have
been probed, ranging from 10 to 30.1 nm. For 15 nm and
more, the dissociation width of the dislocation is almost in-
sensitive to the length of the box. The interaction of the
dislocation with its periodic images is screened by the inter-
action with the solute atoms. As for the thickness in the Z
direction, we found 10 nm to be sufficient to screen the ef-
fect of the two-dimensional dynamics imposed to the atoms
at the free surfaces.
TABLE II. Properties of (a)the Ni共Al兲solid solution and (b)the
Ni3Al
␥
⬘phase, as given by the EAM potential compared to experi-
mental values. In (a),c,a,
,Esf,Tm,cmax stand, respectively, for
the Al concentration, the lattice constant, the shear modulus, the
stacking fault energy, the melting temperature and the solubility
limit at 1500 K. In (b),E0, stands for the cohesive energy and the
Cij’s for the elastic constants with the classical notation.
Ni共Al兲Solid SolutionaThis Model Experiment
a/a
c共Al at%兲−1 ⬃0.01 ⬃0.0068a
/
c共Al at%兲−1 ⬃−0.02 ⬃−0.01b
Esf/Esf
c共Al at%兲−1 ⬃−0.125 ⬃−0.017c
Tm共K兲1750±200 1725→1700d
cmax共1500 K兲共Al at%兲⬃110
d
Ni3Al
␥
⬘phasebThis Model Experiment
a共nm兲0.3571 0.3567e
E0共eV兲−4.52 −4.57f
C11共GPa兲231 223e
C44
⬘共GPa兲70 67e
C12共GPa兲131 148e
C44共GPa兲111 125e
aReference 28.
bReference 29.
cReference 30.
dReference 31.
eReference 32.
fReference 33.
FIG. 1. Computational cell: The dislocation has decomposed
into two Shockley partials in its glide plane (dashed line); atoms in
the partial dislocation cores are shown, as well as the Al atoms of
the solid solution Lx=30 nm, Ly=43.12 nm, Lz= 7.32 nm. This cell
comprises 867 000 atoms.
DISLOCATION GLIDE IN MODEL Ni共Al兲SOLID…PHYSICAL REVIEW B 70, 054111 (2004)
054111-3
Figure 2 gives a typical image of the dissociated disloca-
tion, together with the Al atoms, at successive steps of the
glide process in the 共11
¯
1兲plane. In what follows, the posi-
tion of the dislocation as a function of time, X共t兲, will be
extensively used. The latter is defined as the coordinate,
along the glide direction, of the median of the staking fault
ribbon; it is monitored every 2 ps (see Fig. 2).
III. DISLOCATION GLIDE VELOCITY AND SHEAR RATE
The quantity physically relevant to be extracted from the
simulations is the dislocation velocity. The quantity of inter-
est for macroscopic plasticity is the shear rate. It is related to
the dislocation velocity via Orowan’s equation: the shear
rate,
␥
˙is proportional to the dislocation- velocity, V, and
density,
d:
␥
˙=
dbV, with b, the Burgers vector. In the simu-
lations, the dislocation density is 1/hL, with hand L, respec-
tively, for the thickness (Zdirection)and the length (Xdirec-
tion)of the computational cell. Since hand Lare small,
dis
large (of the order of 5 1015 m−2), and Orowan’s equation
would give very large shear rates although the velocity of
individual dislocation segments is reasonable. In addition,
for macroscopic shear to be observed, the dislocation seg-
FIG. 2. Edge dislocation glid-
ing in a 3 at % Al random solid
solution, under 70 MPa at increas-
ing times. The computational cell
is the same as in Fig. 1, as seen
along the 关11
¯
1兴direction. The ar-
row defines the position of the
dislocation.
RODARY et al. PHYSICAL REVIEW B 70, 054111 (2004)
054111-4
ment must glide on a distance large enough to promote the
critical bending for unpinning from the forest dislocations.
This latter process might be sensitive to the value of Ly(Fig.
1)which imposes the periodicity of the atomic configuration
along the dislocation line. This point is discussed at the end
of the present section. Before, we first describe our results on
the dislocation glide velocity and the associated shear rate.
A. Dislocation glide velocity
All results given in the present section were obtained in
random solid solutions, with Ly=17.2 nm. The dependence
of the results on this parameter is discussed in Sec. III C.
Under a given applied stress, the dislocation glides in a more
or less smooth manner. Figure 3 illustrates typical behaviors
as observed in a solution with 3 at % Al, at 300 K, for four
distinct values of the applied stress.
As can be seen, for an applied stress of 100 MPa, the
dislocation glides for 300 nm within about 375 ps. The mean
glide velocity 共⬃0.8 nm.ps−1兲is thus well below the sound
velocity 共Cs⬃2nm.ps
−1兲. Notice that the glide distance of
300 nm is 20 times the length of the cell: The upper half of
the cell has thus been shifted by 20 Burgers vectors with
respect to the lower half. Each shift by one Burgers vector
creates a new configuration of the Al atoms across the glide
plane. To a first approximation, since we deal here with ran-
dom solid solutions, we may consider that the sequence of
configurations so produced is representative of the configu-
rations one single dislocation would explore on gliding a
distance of 300 nm.
For an applied stress of 70 MPa, the dislocation still
glides for 300 nm, but with a lower mean velocity
共⬃0.1 Cs兲, and in a much less smooth manner. For 65 MPa,
the glide stops, for at least 50 ps, after about 5 sweeps of the
cell. The delay of 50 ps, we use, is long enough for the sound
to travel more than 5 times the length of the dislocation in
the cell.
For an applied stress of 60 MPa, the glide is stopped in
the course of the fourth sweep. At still lower an applied
stress 共⬍45 MPa兲, no glide is observed at all (at least within
50 ps). Some glide would probably be observed if very long
simulations were performed. In a preliminary study, we
found a large scatter of the time to wait to observe some
glide, which prevented us from doing an extensive study of
this lower stress threshold.
The latter threshold of 45 MPa, which depends on the
duration of observation, corresponds to the onset stress for
micro-creep, not to be confused with the yield stress. Indeed,
for macroscopic shear to proceed, the dislocation segment
must glide on a distance large enough to promote the critical
bending for unpinning from the forest dislocations. In order
to stay on the conservative side, we chose a minimum glide
distance of 300 nm (corresponding to a dislocation density
⬃2.8 1012 m−2). With such a criterion, the shear rate is zero
up to a minimum applied stress, which we name hereafter the
static threshold stress,
s. From Fig. 3, e.g., in Ni 3 at % Al
at 300 K, we find 65 MPa⬍
s⬍70 MPa.
B. Shear rate
Because of the boundary conditions imposed in the Zdi-
rection, the thickness hof the box is a constant and the shear
rate of the computational cell scales with the dislocation ve-
locity with a constant, b/hL. For that reason, in the follow-
ing, we skip this transparent b/hL factor and define the ef-
fective dislocation velocity as the velocity the dislocation
should have in order to account for the shear rate. Above the
static threshold stress, the effective dislocation velocity is
identical to the velocity measured in the simulation. Below
the static threshold stress, although at the scale of the simu-
lation, the dislocation can travel some distance, this distance
is too small to allow for unpinning from the forest disloca-
tions and to contribute significantly to the macroscopic strain
(it would rather give rise to recoverable microplasticity):In
this situation the effective dislocation velocity is zero.
Figure 4 shows the (effective)dislocation velocity, as a
function of the applied stress, as observed in Ni共Al兲random
solid solutions with increasing Al contents (3, 5, 8 at%); pure
Ni is shown for the sake of comparison. For very high values
of the applied stress, the saturation of the dislocation velocity
shows up. At intermediate stress values, a viscous regime is
observed: The velocity is a linear function of the applied
stress, with a friction coefficient (per unit length)B. Below
the static threshold stress
s, the effective velocity drops to
zero. The linear portion of the velocity versus stress curve
extrapolates to zero velocity at the dynamical threshold
stress
d. In the stress range of practical interest (i.e., before
saturation), the effective dislocation velocity is thus defined
by three parameters:
V=0if
⬍
s;V=共
−
d兲b/Bif
s⬍
,
as long as VⰆCs.共2兲
In Eq. (2),
s,
d, and Bare composition and temperature
dependent.
The three parameters are found to increase linearly with
the composition (Fig. 5): Starting from pure Ni where
s
=
d=5 MPa and B=11 10−6 Pa.s, the static threshold stress
increases by ⬃25 MPa/at% Al, the dynamical one by
⬃15 MPa/at% and the friction by ⬃210
−6 Pa.s/at%, at
FIG. 3. Mean position of the dislocation, as a function of time,
in a random 3% Al solid solution at 300 K with four distinct values
of the applied stress: 60, 65, 70, and 100 MPa.
DISLOCATION GLIDE IN MODEL Ni共Al兲SOLID…PHYSICAL REVIEW B 70, 054111 (2004)
054111-5
300 K. The magnitude of the increase of
swith Al content
compares well with reported values for the increase of the
0.2% elastic limit (10 to 20 MPa/at % Al), according to Ref.
34.The temperature effect has not been studied in details.
Preliminary results show that increasing the temperature
from 300 to 500 K, in a random solid solution with 5 at %
Al, results in a decrease of the static threshold from
⬃130 MPa to ⬃120 MPa; the dynamical threshold does not
vary significantly, and the friction coefficient increases from
⬃20 to ⬃30 10−6 Pa.s.
C. Effect of the periodic boundary condition along the
dislocation line
We found that close to the static threshold, the glide dis-
tance of the dislocation depends on the box size LYin the Y
direction. Figure 6(a)shows the position of the dislocation as
a function of time in a random solid solution with 3 at % Al
under a shear stress of 65 MPa, for two distinct values of LY.
As can be seen, for the smallest value of LY共17.3 nm兲,
65 MPa are not enough to produce sustained glide: The dis-
location glides a fraction of the box length and stops (for at
least 100 ps). With LY=21.56 nm and above, the glide dis-
tance is larger than four box lengths.
A second effect of the value of LYis as follows. All the
above simulations were performed with fully random solid
solutions. Despite this randomness, it is found that, close to
the static threshold stress, the glide distance of the disloca-
tion depends on the seed of the random number generator
used to construct the initial distribution of solute atoms. Fig-
ure 6(b)gives an example thereof; the effect is larger for
smaller values of LY(compare the plain lines in Figs. 6(a)
and 6(b). Notice that the very same simulations performed at
70 MPa with various LYvalues, exhibit extensive dislocation
glide, while no glide at all ever occurs at 50 MPa. The value
of the static threshold is thus defined within ±10 MPa.
Although a more detailed analysis is necessary to under-
stand the effect of the box width, two possible origins may
be proposed. Firstly, if the pinning centers are rare (less than
one on a distance of Ly), the periodic boundary condition in
FIG. 4. (a)Effective glide velocity as a function of applied
stress; (b)definition of the three parameters
s,
d, and B(see text).
FIG. 5. Concentration dependence at 300 K of (a)the static
(triangles)and dynamic (squares)threshold stresses, and (b)the
friction coefficient per unit length.
RODARY et al. PHYSICAL REVIEW B 70, 054111 (2004)
054111-6
the Ydirection imposes Lyas the distance between pinning
centers: the larger Ly, i.e., the distance between pinning cen-
ters, and the lower the critical stress to escape. In addition,
increasing Lywidens the range of available deformation
modes for the dislocation line, which amounts to increase its
flexibility, hence to decrease the effective line tension. Since,
as will be shown below, the pinning centers are well sepa-
rated and short ranged, an increased flexibility makes the
depinning event easier, and the threshold stress lower. Also,
as noticed, close to the static threshold stress, the smaller Ly,
the more sensitive the glide distance to the initial configura-
tion of the random solution: this suggests that the pinning
centers only sample a small fraction of the Al atoms con-
tained in the simulation cell.
IV. DISCUSSION
As seen on Fig. 3, the position of the dislocation segment
as a function of time reveals a jerky glide motion. This sug-
gests a “stop and go” type of analysis. Let us stress that we
describe, here, the motion of the dislocation segment as a
whole, as defined in Sec. IIC, not the change of shape of the
dislocation and of the staking fault ribbon. Consider the po-
sition of the dislocation as a function of time, e.g., in a 3 at
% random solid solution, at 300 K, under 70 MPa (Fig. 3).
The time required to glide a distance of 300 nm is approxi-
mately 700 ps. The same curve can be analyzed at a finer
scale: we may ask what is the time ⌬trequired to glide a
distance ⌬Xof 50, 20, or 10 nm. The time intervals ⌬tex-
hibit a complicated distribution, an example of which is
given in Fig. 7. It contains information on the distribution of
pinning strengths, which may be analyzed using a statistical
approach. This work will be discussed in a future work.
We now summarize our main findings, and outline a way
of incorporating these results, obtained at the atomistic level,
in a micromechanical model to describe the macroscopic
plastic behavior.
The dislocation glide proceeds as follows:
— under large shear stresses, the dislocation glides
smoothly at a velocity which saturates close to the shear
wave speed;
— under lower stresses, the mean glide velocity is a lin-
ear function of the stress and extrapolates to zero at the dy-
namical threshold stress,
d;
— in the latter regime, a static threshold stress,
s, exists
below which the dislocation stops for a “long” period of time
after gliding “some” distance. Sustained macroscopic shear
only occurs above
s.
An intuitive interpretation of the static and dynamic
thresholds is as follows: The strong obstacles control the
static threshold below which no long range dislocation mo-
tion is possible. Once these strong obstacles are overcome,
the moving dislocation still feels from the weaker obstacles a
pinning force, which slows down its motion. This is the
physical origin of the dynamic threshold.
There is some arbitrariness in the definition of
s, since it
depends on the value we choose for the observation time and
for the minimum glide distance. We chose 300 nm for the
minimum glide distance, since it guarantees that the disloca-
FIG. 6. Glide distance of the dislocation as a function of time in
an ideal solid solution with 3 at % Al under 70 MPa; (a)for Ly
=17.3 nm (solid line)and Ly=21.56 nm (dashed line)and the same
seed of the random number generator for the initial configuration of
the ideal solid; (b)same as (a)with another seed for the random
number generator.
FIG. 7. Distribution of the time required to glide a distance
⌬X共10,20,50 nm兲, in a random solid solution with 3 at % Al,
under an applied stress of 70 MPa, at 300 K.
DISLOCATION GLIDE IN MODEL Ni共Al兲SOLID…PHYSICAL REVIEW B 70, 054111 (2004)
054111-7
tion might propagate through a low forest density. We chose
50 ps as the observation time (i.e., maximum allowed wait-
ing time)for computational reasons. We found, however, that
within such a crude definition, the static threshold value is
defined within ±10 MPa. Indeed a detailed study of the
stress, temperature and composition dependence of the dis-
tribution of the waiting time would be of great interest21 but
the very large scatter of the waiting times at lower stresses,
and their sensitivity to initial conditions, prevented us from
making such a study.
Careful examination of the glide process at the atomic
scale, revealed the following dislocation pinning mechanism:
When a partial dislocation glides one step forward, it shifts
the atoms above the glide plane with respect to those below
the glide plane, by a vector 1/6具112典. In the process of this
displacement, some Al atoms (large compared to Ni atoms)
are brought in close contact, a strongly repulsive configura-
tion (this repulsion accounts for part of the stability of the
L12structure of Ni3Al). On the contrary, if no such nearest
neighbor Al-Al pair can ever form, the dislocation moves
more easily. As an example, to test this idea, we generated a
“constrained random” 8 at % Al solid solution, forbidding
the shear induced formation of nearest neighbor Al-Al pairs
around the glide plane: the way to achieve this, is to impose
zero Al content in one of the two 共11
¯
1兲planes across to the
glide plane. Under 150 MPa, the glide velocity, in the con-
strained solution, is larger than in a fully random solution, as
shown in Fig. 8(a). Similarly, glide is observed at 100 MPa,
Fig. 8(b), much below the static threshold stress in the fully
random solution 共180 MPa兲.
We also studied the effect of the external stress on the
glide velocity of the dislocation in a 3% solid solution con-
structed with the same constraint as above (Fig. 9), and com-
pared with a fully, unconstrained, random solid solution. The
Al-Al pairs across the glide plane are found to play an im-
portant role in the glide process. First they are, indeed, the
obstacles, which determine the value of static threshold. Fig-
ure 9 clearly shows that in the absence of shear induced
Al-Al pairs, the static threshold is much lower, close to its
value in pure Ni. Second, when the external stress is much
larger than the static threshold of the solid solution, the glide
velocity is larger in the constrained solution (i.e., in the ab-
sence of shear induced Al-Al pairs)as compared to the un-
constrained random solution: The shear induced Al-Al pairs
form obstacles which slow down the dislocation glide and
are responsible for the dynamic threshold stress.
This proves indeed that the origin of the chemical hard-
ening process is the repulsive Al-Al interaction, and not the
interaction between the Al atoms and the gliding disloca-
tions. The waiting time on one of these obstacles would be
controlled, we suggest, by the collective vibration of the few
atoms the obstacle is made of.
The dependence with concentration of the static and the
dynamic thresholds has been shown to be linear, in the range
of concentrations investigated. This is indeed consistent with
the fact that, if the dominant obstacles are solute pairs, their
density scales as the square of the solute concentration and
the elastic limit increases, according to Friedel statistics,35,40
as the square root of the density of obstacles, i.e., as the
solute concentration itself.41
It is worth noticing that, in one of the very few systematic
experimental studies of yield stress as function of solute con-
tent in a large range of composition, Wille et al.,36 in Cu共Mn兲
solid solutions, reach the following conclusion: Using the
standard theory of solute dislocation interaction to account
for the observed hardening, the density of pinning centers is
lower and their strength larger than expected for single solute
pinning. They argue that MnMn doublets are responsible for
the hardening.
The relevant quantity provided by our approach is the
dislocation velocity for a given applied stress. It should be
stressed that the dislocation velocities, which we generated,
overlap the range accessible by experiments for similar ap-
plied stresses, despite the fact that the shear rates are several
order of magnitude larger than experimental ones. The rea-
son for this is the small size of the computational cell, which
makes the dislocation density artificially large. In the spirit of
multi-scale modeling, the quantity to be transferred from this
study into larger scale models is the law for the glide veloc-
ity of a dislocation segment, i.e., Eq. (2), coupled with a
FIG. 8. (a)Glide distance of the dislocation as a function of
time, under 150 MPa, ina8at%Alfully random solution, i.e.,
close to the static threshold stress (solid line)and for the same
solution, constrained by removing Al atoms from one of the two
共11
¯
1兲planes contiguous to the glide plane (dashed line).(b)Same
as (a)for 100 MPa, i.e., much below the static threshold stress.
RODARY et al. PHYSICAL REVIEW B 70, 054111 (2004)
054111-8
realistic description of the dislocation density.
In the following derivation, we have chosen the simplest
classical evolution equation for the dislocation density to il-
lustrate the feasibility of this approach. We do not claim that
the macroscopic behavior is accurately described at this
stage, since the present study deals only with the solute ef-
fects on dislocation glide. For instance, the present paper
does not address the question of the solute effects on dy-
namic recovery.
Following Rodney et al.,37 we take advantage of the shear
rate as obtained from atomic scale simulations [Eq. (2)],to
compute stress-strain curves as predicted by a simple micro-
mechanical model. The rate of plastic shear,
␥
˙pis given by
Orowan’s law:
␥
˙p=
dbV,共3a兲
and the dislocation density
dchanges in time because of the
competition between multiplication and annihilation:38
˙d=共A冑
d−D
d兲
␥
˙p.共3b兲
The first term in this equation describes the storage of dislo-
cation on the so-called “forest dislocations,” and the second
terms accounts for the annihilation distance between disloca-
tions of opposite sign. The parameter Ais related to stage II
hardening, and the parameter Dis a measure of the efficiency
of dynamic recovery. It is usually observed that Ais inde-
pendent of solute content, whereas D, which results from
extra degrees of freedom such as cross slip or climb, depends
on solute elements.39 A quantitative description of the influ-
ence of solute elements on the stress strain curve would re-
quire a study of the effect of solutes on cross slip and climb,
paralleling the one we have performed here for glide.
The dislocation velocity in Eq. (3a)is given by Eq. (2),
except for the flow stress, which must now include the resis-
tance opposed by the forest dislocations, on top of solute
hardening depicted by
s. The spacing of obstacles respon-
sible for solute hardening being much smaller that the spac-
ing between forest dislocations, a linear superposition can be
safely assumed:40
y=
s+
␣
b冑
d.共3c兲
The
␣
parameter represents the pinning force of the forest
dislocations.
The set of Eqs. (3a)–(3c)and (2)can be integrated with a
given loading condition, namely a constant total strain rate
test. Figure 10 shows typical stress-plastic strain curves com-
puted with the values of the parameters as given by the
present atomic scale simulation. The phenomenological pa-
rameters
␣
,A,Dintroduced by the micromechanical model
are given classical values.37 The initial dislocation density is
1014 m−2, and the imposed strain rate is 10−3 s−1. As can be
seen, the model qualitatively reproduces the increase of the
flow stress with the solute content, but fails to reproduce the
solute effect on the hardening which is observed, e.g., in
Cu共Al兲solid solutions. This is not a surprise, since the sen-
sitivity of the parameter Dto the solute content has been
ignored.
V. CONCLUSION
Based on an EAM potential, which we have adapted, we
have simulated the glide of an edge dislocation in random
Ni共Al兲solid solutions (1to8at%Al)at 300 K. Within
reasonable computer time, we could study the glide mecha-
nism in the range of dislocation velocities of metallurgical
interest.
Three parameters,
s,
d, and B, are found necessary to
describe the rate of shear:
sis the static threshold stress,
FIG. 9. Glide velocity as a function of the external stress, for a
3 at % solid solution with a fully random distribution of Al (solid
line)and with a solute distribution, where one of the two 共11
¯
1兲
planes contiguous to the glide plane, is constrained to be free of Al
atoms (dashed line). The error bars are estimated by performing a
sampling with respect to the seed of the random number generator
used to construct the initial configuration of the solid solution.
FIG. 10. Stress versus plastic strain as computed from the mi-
cromechanical model for 4 distinct Al contents: 0, 1, 5, 10 at %.
The imposed strain rate is 10−3 s−1; A=4.108m−1, D=6.5,
␣
=0.3;
the initial dislocation density is 1014 m−2. The values of all the other
parameters are taken from the present atomic scale simulation.
DISLOCATION GLIDE IN MODEL Ni共Al兲SOLID…PHYSICAL REVIEW B 70, 054111 (2004)
054111-9
below which the glide distance of the dislocation is not suf-
ficient to insure sustained mesoscopic shearing;
d,isthe
dynamical threshold stress, which reflects the friction of the
pinning potential on the moving dislocation; B, is the friction
coefficient which relates the effective stress 共
−
d兲to the
glide velocity. Both
s,
d, and Bincrease linearly with the
solute content. Preliminary results suggest that
sdecreases,
and Bincreases with temperature, while
dremains unaf-
fected.
The main conclusion of this study is that chemical hard-
ening is not simply the result of a direct dislocation solute
共Al兲interaction, but the result of close solute-solute (Al-Al)
repulsion, when the passage of the dislocation forces them
one against the other, which opposes the glide of one half
crystal, with respect to the other.
The qualitative analysis of the geometry of glide, as well
as the preliminary results on the effect of temperature, sug-
gest the need to revisit the concept of thermal activation of
dislocation glide in solid solutions. The modification of the
atomic position due to the moving dislocation, and its depen-
dence with temperature have to be taken into account to de-
velop a sound description of the thermally activated motion.
The classical picture of the thermally activated motion of a
dislocation in a potential computed with frozen atomic posi-
tions is likely to be of limited applicability when the solute
concentration is such that the above described mechanism is
mainly responsible for the hardening effect. In this spirit, the
effect of short range order on dislocation motion deserves to
be revisited.
Finally, we have outlined a method for integrating the
present findings from the atomistic level into a microme-
chanical approach in order to predict macroscopic stress
strain curves. While the increase of the yield stress with the
composition is correctly described, an atomic scale study of
cross slip and climb is needed to account for the influence of
solute on dynamic recovery which controls further work
hardening.
ACKNOWLEDGMENTS
Useful discussions and technical advice by Dr. E. Adam,
Dr. J.L. Bocquet, Dr. D. Caillard, Dr. N.V. Doan, and Dr. M.
Fivel are gratefully acknowledged, as well as enlightening
comments by Professor. J. Friedel and Professor F.R. Na-
barro. G.M. thanks Northwestern University (Department of
Materials Science and Engineering)for hosting him as a visi-
tor; part of this paper was written during this visit.
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