Strongly Non-Arrhenius Self-Interstitial Diffusion in Vanadium

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by  Luis A. Zepeda-Ruiz

Strongly Non-Arrhenius Self-Interstitial Diffusion in Vanadium
LuisA.Zepeda-Ruiz,1,2,∗ J¨orgRottler,1 SeungwuHan†,1 GraemeJ.Ackland,3 RobertoCar,1 andDavidJ.Srolovitz1
1Princeton Institute for the Science and Technology of Materials (PRISM), Princeton University, Princeton, NJ 08544
2Chemistry and Materials Science Directorate, Lawrence Livermore
National Laboratory, P.O. Box 808, L-371, Livermore, CA 94550
4 3School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
0 (Dated: February 6, 2008)
0
We study diffusion of self-interstitial atoms (SIAs) in vanadium via molecular dynamics simu-
2
lations. The h111i-split interstitials are observed to diffuse one-dimensionally at low temperature,
n but rotate into other h111i directions as the temperature is increased. The SIA diffusion is highly
a
non-Arrhenius. At T < 600 K, this behavior arises from temperature-dependent correlations. At
J
T > 600 K, the Arrhenius expression for thermally activated diffusion breaks down when the mi-
6 gration barriers become small compared to the thermal energy. This leads to Arrhenius diffusion
2
kinetics at low T and diffusivity proportional to temperature at high T.

]
PACSnumbers: 66.30.Fq,61.72.Bb,61.82.Bg
i
c
s
-
l The creation and migration of self-interstitial atoms ulation of SIA migration in V using an improved inter-
r
t (SIAs) arecritical for microstructuralevolutionof mate- atomicpotentialforV[13](refittoexperimentalandfirst
m
rialsina varietyofsituations, suchas inthe highenergy principles data [11] to reproduce the stable interstitial
. radiation environment of nuclear reactors [1] and in ion configuration)to address this discrepancy. Inparticular,
t
a implantation [2]. Although SIA formation energies are we examine SIA diffusion as a function of temperature
m
much larger than typical thermal energies, they form in to determine the SIA migration mechanisms. We find
- abundance during collision cascades induced by imping- thatwhileSIAmigrationinVissimilartothatinbccFe
d
ing energetic particles. SIAs in metals are typically very in many respects, its temperature dependence is highly
n
o mobile (i.e.,their migrationbarriersarerelativelysmall) unusual,exhibitingstronglynon-Arrheniusbehaviorand
c andhence playanimportantroleincontrollingthe rates correlationeffects. Itisthisnon-Arrheniusbehaviorthat
[
of many microstructural processes in such applications, is the focus of the present Letter.

1 in particular the phenomenon of void swelling. The SIA is introduced in the form of a stable 111 -
v h i
SinceSIApropertiesandmobilitiesareverydifficultto split interstitial and equilibrated for 10 ps at fixed tem-
5
2 determine experimentally, one often employs computer perature using a Langevin thermostat. The simulation
5 simulations [3, 4, 5, 6]. For example, simulations of was then switched to a microcanonical (NVE) ensemble
1 body centered cubic (bcc) iron (and several other bcc in order to study SIA migration dynamics. Simulations
0 metals), have shown that SIAs preferentially lie along were run at temperatures between 100 and 2000 K in a
4
0 110 orientationsbutrotateinto 111 -directions,where cubic simulation cell of edge length 10a0, where a0 is
/ htheyican migrate easily using thhe criowdion configura- the temperature dependent-lattice parameter (the linear
at tion as transition state. Other simulations have sug- thermal expansion coefficient was 8.4 10−6 K−1). The
×
m gested that SIA migration in vanadium is very similar diffusivity was measured by averaging over several 1 ns
- to that in Fe [7, 8, 9, 10]. However, these empirical in- simulations. Theinterstitialpositionwasidentifiedbydi-
d teratomic potential-based simulations are not consistent viding the space into Wigner-Seitz (W-S) cells centered
n
with recentfirstprinciples calculationsthat clearlyshow around each perfect crystal lattice site. Interstitials are
o
c that the lowestenergy SIAconfigurationin V is a 111 - located in W-S cells containing more than one atom.
h i
v: splitinterstitial,ratherthantheh110i-splitconfiguration Representative trajectories of the h111i-split intersti-
i found in Fe [11]. Interestingly, the first principles calcu- tial center of mass, collected over the whole 1 ns simula-
X
lations also revealed that the 111 -oriented SIA migra- tions, are shown in Fig. 1. For each temperature, more
h i
r tion energy is extraordinarily small ( 0.01eV), which than 1000 jumps were observed, where an SIA jump is
a ≤
explains the experimental observation of diffusion down the exchange of anatom between neighboringW-S cells.
to 4 K [12]. SIA migration in Fe and V must therefore AsseeninFig.1,the interstitialmigrationmechanismis
differ in their microscopic mechanisms. stronglytemperaturedependent. Forlowandintermedi-
ate temperatures (100-600 K) the 111 -split interstitial
We perform a series of moleculardynamics (MD) sim- h i
executesafullyone-dimensional(1D)randomwalkalong
a 111 -directionduring the 1 ns simulation,as shownin
h i
Fig. 1(a). At T 700 K, the 111 -split interstitial be-
∼ h i
†current address: Department of Physics, Ewha Womans Univer- gins to make infrequent rotations from one 111 - to an-
h i
sity,Seoul 120-750,Korea. other 111 -direction. Thisresultsinathree-dimensional
h i
2
T @KD
2000 1200 800 600
1012
D 1011
1
-
s
@
Ωr
1010
6 8 10 12 14 16 18 20
1(cid:144)k T @eV-1D
B
FIG. 2: Frequencyof rotations as a function of temperature.
FIG. 1: Typical trajectories of migrating SIAs for tempera- ThesolidlineisanArrheniusfittothedatawithslope∆Er =
tures of (a) 300 K,(b) 700 K, (c) 900 K and (d) 1400 K. 0.44 eV, preexponentialfactor ν0 =1.3×1013s−1.
(3D)trajectorythatconsistsoflong1Drandomwalkseg- using standard averaging procedures [17]. Note that
ments with abrupt reorientations, as seen in Fig. 1(b). although the rotations at higher T change the dimen-
As the temperature increases, the frequency of the rota- sionality of the diffusion path, they do not contribute
tioneventsincreasesandthelengthsofthe1Dtrajectory to transport and hence the diffusion mechanism is al-
segments decrease. At high temperatures, the rotation ways one-dimensional. If the diffusivity were Arrhenius
events become very frequent, leading to nearly isotropic (D/a2 = ν exp[ ∆E /k T]), the data in Fig. 3 would
0 0 − d B
diffusion (Fig. 1(d)). liealongastraightline. Thisisclearlynotthecase;Fig.3
Althoughthesetrajectoriesappeartobequalitatitively showspronouncedcurvature- especiallyat hightemper-
similar to those reported for other bcc metals (i.e. Fe ature. Although Arrhenius behavior is widely expected
and Mo) [4, 14, 15, 16] (cf. Fig. 1(d) and Fig. 5 in [15]), for diffusion in the solid state, it is clearly inapplicable
they differ in the elementary migration mechanism. The here.
stable interstitial is the 111 -split configuration in V, Non-Arrheniusbehavior canhaveseveraldifferentori-
h i
but the 110 -split configuration in Fe and Mo. In the gins. The energy barrier could simply change with tem-
h i
Fe and Mo cases, the split interstitial sits in the 110 - perature as a result of thermal expansion, as has been
h i
orientation until it is thermally activated into one of the argued for the self-diffusion in bcc metals via a vacancy
111 -directions where it can migrate easily before re- mechanism [18, 19]. The magnitude of the observed de-
h i
turning to a 110 -orientation [15]. There are no relax- viationsfromArrheniusbehavioristoolargetoattribute
h i
ation events of this type in interstitial migration in V. to such a mechanism. The existence of multiple reaction
Here, the stable 111 -splitinterstitialmigrates long dis- pathways with different energy barriers can also lead to
h i
tancesandonlyrequiressignificantthermalactivationto curvatureinFig.3. However,detailedexaminationofthe
reorient or rotate. atomicconfigurationduring diffusionshowsthatthere is
The temperature dependence of the rate of rotation no change in diffusion mechanism over the entire tem-
of the split interstitials from one 111 direction to an- perature range. Although rotations are first observed at
h i
other, ω , in V is shown in Fig. 2. The data is well 700 K within the 1 ns duration of the simulations,
r ∼
described by a conventional Arrhenius fit of the form strong deviations from Arrhenius behavior are observed
ω = ν exp[ ∆E /k T], suggesting that rotation is a alreadyatlowertemperatures. Athirdpossibilityisthat
r 0 r B
−
simple thermally activated process. The activation en- the degree of correlations in the diffusion process (i.e.,
ergy, ∆E = 0.44 eV, is consistent with the energy dif- particle jumps retain memory and the random walk is
r
ference between the 111 and 110 configurations com- non-Markovian) is temperature dependent. Indeed, ex-
h i h i
puted in first-principles(0.35 eV) andstatic calculations amination of the SIA trajectories show that the 111 -
h i
using the new interatomic potential (0.4 eV) [13]. interstitial has a higher probability of jumping back in
The diffusivity D (solid symbols) of the 111 -split the direction from whence it came, rather than forward
interstitial is shown in Fig. 3 for a temperathureirange along the same direction. At high temperatures, by con-
between 100 K and 2000 K. D was determined from trast, this effect appears to be reversed.
D = R2(t) /2tfor1Ddiffusion,wherethemeansquared We quantified this observation by measuring a corre-
h i
displacement R2(t) was calculated from the trajectory lation factor for split interstitial diffusion f, defined as
h i
3
T@KD 50 101
2000 500 200 100
100
50 1.5 40 10-1
(cid:144)Ds20 f 1.0 2(cid:144)Dms30 1100--32
2m10 0.5 c 0 1 2 3 4 5 6
c -4 Ε(cid:144)k T
-4 5 0 50 100 1020 B
@10 1(cid:144)kBT@eV-1D @D
f 2 10
(cid:144)
D
D, 1
0 2 4 6 8 10
k T(cid:144)DE
0 20 40 60 80 100 120 B d
1(cid:144)k T@eV-1D
B FIG. 4: The diffusivity D ((cid:4)) from Fig. 3 as a linear func-
tionofT,normalizedby∆Ed. Thethicksolidlinerepresents
FIG. 3: A plot of thediffusivity of the h111i-split interstitial the diffusivity of a particle in a sinusoidal potential at finite
in the form suggested by the Arrhenius relation. The filled temperatureasdescribedbyEq.(1)foravalueofγ =0.1τ−1
symbols ((cid:4)) correspond to the measured diffusivity D and whereτ =pmσ2/ǫ. Thedashedlineisthefreeparticlelimit
open symbols (⋄) to D normalized by the correlation factor
of the same model, D = kBT/ǫmγ. The inset shows an Ar-
f (see text). The straight line is a low temperature fit to
rheniusplot of thediffusivityoftheparticle in thesinusoidal
tehxepoDn/enftidaaltfaa,ctcoorrroefsνp0on=d1in.5g×t1o0∆12Es−d1=in0t.h0e18AerrVheannidusafoprrme-. pdaosthenedtialilnaeshaafsunslcotpioenoonfeǫ./kBT asasolidlineandthestraight
Statistical errors are of order the symbol size. The inset to
thefigure shows the variation of f as a function of 1/kBT.
temperature regime (where the behavior is Arrhenius)is
smaller than the thermal energy. This is the source of
f = 2D/D , where D is the “bare” diffusion constant the multiple interstitial hops at high T. Conventional
b b
defined as D = l2n. Here, n is the mean number of derivations of activated escape over barriers [20] usually
b 0
jumps/secondandl0 =√3a0/2isthejumplength(near- assume ∆Ed kBT .
≫
est neighbor distance in the bcc lattice). If the intersti- Rather thantrying to apply the Arrhenius description
tial trajectory is described by a sequence of jump vec- tointerstitialself-diffusioninthissystem,amoregeneral
tors~l , the correlation factor f is alternatively given by modelisthemotionofaparticleinaperiodicpotentialat
i
f =1+2Pin−1h~li·~l0i/l02, i.e. f =1 for an uncorrelated all temperatures (from kBT ≪ ∆ED to kBT ≫ ∆ED).
random walk. The inset to Fig. 3 shows the variation In the limit that the barrierheight is completely negligi-
of f with 1/k T. At low T (T 600 K), SIA motion blerelativetotheenergyoftheheatbath(i.e.,afreepar-
B
≤
is anticorrelated (f < 1). The effect of the correlations ticle), standard arguments predict D = kBT/mγ, where
on the diffusivity can be isolated by plotting D/f rather misthemassoftheparticleandγarelaxationtimescale
than D in Fig. 3 (open symbols). The low temperature associatedwithavelocity-dependentfrictionforce. Since
D/f data (T 500K) lie along a nearly straight line this free particle diffusion coefficient is expected to be a
≤
with slope ∆E =0.018 eV. Hence, the temperature de- linear function of T, we replot the data from Fig. 3 on a
d
pendent correlation factor explains the relatively weak linear temperature scale (Fig. 4). In this representation,
deviations from Arrhenius behavior at low T. The un- the data is nearly linear, albeit with weak curvature at
usually smallmigrationenergy is consistentwith experi- low temperature. This suggests that SIA diffusion in V
mental observations [12] and the first principle estimate is free particle like at high temperature (D T), but
∼
[11]. One possible origin of the anticorrelations is that follows the normal Arrhenius, hopping dynamics at low
thefiniterelaxationtimeofthelocalenvironmentaround T (D e−∆ED/kBT). The deviation from linearity at
∼
the SIAbecomeslessimportantasthe thermalenergyof low temperature and the deviation from Arrhenius be-
the system increases. havior at high temperature suggests that a cross-over is
occuring between the free and hopping particle limits.
As the temperature is increased beyond 300 K, the
Inordertobetterunderstandthistransition,weexplic-
correlation factor rises quickly to a value greater than
itlyconsiderasimpleone-dimensionalparticleofmassm
unity. f > 1 is very unusual and may be thought of as
diffusing in a sinusoidal potential by numerically solving
correlatedinterstitialhopping overseveralbarrierswith-
the Langevin equation
outcompletelythermalizinginbetween. Thisinterpreta-
tion is consistent with the fact that the correlation cor-
mx¨ γx˙ =ǫ/2cos[x/σ]+η, (1)
rected diffusivity, D/f, only yields a straight line at low −
T but not at high T. We note that where D/f is ris- where η is a Brownian white noise. Inserting values of
ing quickly, the energy barrier ∆E obtained in the low m and σ for V and ǫ = ∆E , this model yields nearly
d d
4
linear diffusivity for 1 < k T/ǫ < 10 for reasonable val- of T rather than Arrhenius, as usually assumed, could
B
ues of γ, followed by a crossover into Arrhenius behav- have important implications for predicting the lifetimes
ior (see inset of Fig. 4) for k T/ǫ . 1. Changing the ofreactorcomponentsinvanadiumandotherbccmetals.
B
value of γ shifts the temperature at which the transi- We thank A. F. Voter, G. H. Gilmer, and J. A. Caro
tion from the particle hopping to free particle behavior for useful discussions. This work was performed under
is observed. This excellent agreement between the MD the auspices of the U. S. Department of Energy, Of-
results and model predictions demonstrates that the ob- ficeofFusionEnerySciences(DE-FG02-01ER54628)and
served strongly non-Arrhenius interstitial diffusion in V Lawrence Livermore National Laboratory under Con-
is a direct consequence of the relative magnitudes of the tract No. W-7405-Eng-48.
activation energy and the thermal energy.
Itisinterestingtocomparethesituationdescribedhere
for V to that for bcc Fe. The crowdion mechanism en-
abling the easy interstitial migration along 111 direc-
h i ∗ corresponding author: [email protected]
tions in V is also available in Fe, and estimates for the
[1] F. W. Young, J. R. Cost, A. Nowick, and J. O. Stiegler,
migrationbarrier∆E =0.04eV[4]aresimilarto the V
d Mater. Sci. Eng. 35, 91 (1978).
case. However,measurements of the apparent activation
[2] K.Nordlund,J.Keinonen,M.Ghaly,andR.S.Averback,
energyfordiffusivity,analogoustotheonepresentedhere Nature 398, 49 (1999).
(albeit over a smaller temperature range between 700 K [3] T.D´ıazdelaRubiaandM.W.Guinan,Phys.Rev.Lett.
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d
eV [16] or ∆E = 0.17 eV [21]. This larger effective [4] B. D. Wirth, G. R. Odette, D. Maroudas, and G. E.
d Lucas, J. Nucl. Mater. 244, 185 (1997).
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h i [5] N. Soneda and T. D´ıaz de la Rubia, Phil. Mag. A 78,
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h i 995 (1998).
easy-diffusionconfigurationis not populated at all times
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h i
splitinterstitialisavailablefortransportisgivenbyF = [7] A.M.MinashinandV.A.Ryabov,J.Nucl.Mater.233-
P /(P +P ),whereP exp[ ∆E /k T] 237, 996 (1996).
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and P exp[ ∆E /k T] are p∼robabi−lities for the [8] K. Morishita, N. Sekimura, and T. D´ıaz de la Rubia, J.
h110i ∼ − b B Nucl. Mater. 248, 400 (1997).
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h i h i [9] K. Morishita, T. D´ıaz de la Rubia, E. Alonso,
state and back, respectively. Therefore, the interstitial
N. Sekimura, and N. Yoshida, J. Nucl. Mater. 283, 753
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(2000).
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h i
entation (like in V) and F. For ∆Ef ∆Eb, the effec- Instr. Meth. Phys.Res. B 180, 66 (2001).
≫
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(∆E +∆E ), but for ∆E ∆E , it is ∆E . D. J. Srolovitz, Phys.Rev. B 66, 220101 (2002).
D f b f D
∼ ≫ ∼ [12] R. R. Coltman Jr., C. E. Klabunde, J. K. Redman, and
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f ≫ b J. M. Williams, Radiat. Eff. 24, 69 (1975).
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h i
dimensionallyalong 111 directionsduringthe1nssim- [15] R. C. Pasianot, A. M. Monti, G. Simonelli, and E. J.
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described by Arrhenius kinetics with activation energy B 15, 699 (1977).
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d
[20] P. Haenggi, P. Talkner, and M. Borkovec, Rev. Mod.
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331 (2001).
rhenius to free particle type diffusion with increasing T.
The fact that this type of diffusion is a linear function