Point defect absorption by grain boundaries in α -iron by atomic density function modeling
O. Kapikranian, H. Zapolsky, R. Patte, C. Pareige, B. Radiguet, P. Pareige
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Point defect absorption by grain boundaries in α -iron by atomic density function modeling O. Kapikranian, ∗ H. Zapolsky, R. Patte, C. Pareige, B. Radiguet, and P. Pareige
Groupe de Physique des Mat´eriaux, Universit´e et INSA de Rouen, UMR CNRS 6634,Av. de l’universit´e, BP 12, 76801 Saint Etienne du Rouvray, France (Dated: August 10, 2018)Using the atomic density function theory (ADFT), we examine the point defect absorption at [110] symmet-rical tilt grain boundaries in body-centered cubic iron. It is found that the sink strength strongly depends onmisorientation angle. We also show that the ADFT is able to reproduce reasonably well the elastic propertiesand the point defect formation volume in α -iron. PACS numbers: 61.50.Ah, 61.72–y, 61.82.Bg, 62.20.D–
I. INTRODUCTION
Point defect sinks, such as individual dislocations or grainboundaries, play a crucial role in embrittlement, swelling,or non-equilibrium solute segregation driven by the pointdefect fluxes . These phenomena are especially importantin irradiated materials. Fine grain polycrystalline materialscan exhibit enhanced resistance to irradiation as they possesshigh concentration of point defect sinks in the form of graininterfaces . A few experimental studies have been reportedabout irradiation effects on nanocrystalline materials, whichconfirm good resistance against irradiation .Usually, theoretical determination of GB sink strength isbased on the dislocation representation of grain boundariesand, consequently, on the consideration of elastic interactionbetween the latter and point self defects . This approachis hardly applicable to an arbitrary GB geometry that canbe found in real materials. Molecular statics results for thevacancy and self-interstitial atom formation energies at GBshave appeared in literature recently . Atomistic dynamicssimulations remain numerically too costly, and thus limited toindividual nanograins , since one is dealing with a diffusionalphenomena. The atomic density function theory (ADFT),which is not atomistic in the strict meaning of the term, butkeeps track of the atomic microstructure, seems to be a goodcandidate for this task as it operates on diffusional time scale.It was shown in Ref. [12] that the ADFT gives correct atomicconfigurations for various GB geometries. Basic features ofgrain boundaries as well as dislocation emission from grainboundaries have also been examined in literature using thephase-field-crystal model, which is very close in spirit to theADFT. There is, however, a question how to incorporate selfpoint defects in this type of model.In this paper, a way of describing self point defects in theearlier introduced atomic density function model is pro-posed. Using this new development of the ADFT, both va-cancy and self-interstitial absorption by grain boundaries ismodeled. It is shown that simulation results give access to the ∗ Now at ENSICAEN, 6 Bd Mar´echal Juin, 14050 Caen, France; Alsoat Institute for condensed matter physics of the National Academyof Sciences of Ukraine, 79011 Lviv, Ukraine; Electronic address:[email protected] sink strengths of GBs of different misorientations. Thus, theapproach proposed in the present paper opens a possibility ofa unified modeling of polycrystalline materials sink strength.
II. EQUILIBRIUM STATE AND ELASTIC PROPERTIES
To model the point defect absorption at grain boundaries theADFT has been used. A theoretical foundation of the ADFTis based on the nonequilibrium Helmholtz free energy of asystem that is a functional F [ ρ ] of an atomic density function, ρ ( r ) . This function is an occupation probability to find anatom at the site r of the underlying Ising lattice. FollowingRefs. [13] and [12], the free energy functional can be writtenas: F [ ρ ] = 12 Z d rV Z d r ′ V W ( | r − r ′ | ) ρ ( r ) ρ ( r ′ ) (1) + k B T Z d rV [ ρ ( r ) ln ρ ( r ) + (1 − ρ ( r )) ln(1 − ρ ( r ))] , where W ( | r − r ′ | ) is the atomic interaction potential and V the system volume. The form of the Fourier transform V ( k ) of W ( | r − r ′ | ) was chosen to reproduce the form of the firstpeak of the structural factor of iron, calculated by moleculardynamics at the melting point : V ( k ) = V (cid:0) − k / (cid:0) ( k − k ) + k (cid:1)(cid:1) (2)with k ≃ . k , and k ≃ . k , where k is the mini-mum position of the potential V ( k ) , and V defines the energyscale.To model the evolution of the atomic density function ρ ( r ) , the kinetic microscopic diffusion equation proposed inRef. [13] has been used. The general form of such equation,assuming that the relaxation rate is linearly proportional to thetransformation driving force, is: ∂ρ ( r ) /∂t = − L ∇ δF/δρ, (3)where L is a constant. Equation (3) is conservative with re-spect to the mean atomic density ρ = R d rρ ( r ) /V . Non-conservative kinetics is obtained if the Laplace operator isdropped in Eq. (3).To model the kinetics of defect absorption at grain bound-aries, Eq. (3) in a reduced form was numerically solved us-ing the semi-implicit Fourier scheme . In our simulationsthe next set of reduced variables was used: reduced distance x ∗ = x/a ( a is the second-nearest-neighbor distance in abcc lattice), reduced time t ∗ = tLV /a , reduced energy F ∗ = F/V , and temperature T ∗ = k B T /V .To validate our potential, the elastic constants of the bcciron have been evaluated. For this purpose, the three char-acteristic deformations were used: (a) uniform compres-sion/expansion, x, y, z → (1 − ξ ) x, (1 − ξ ) y, (1 − ξ ) z , (b)equal contraction/expansion along two cube edges, x, y, z → (1+ ξ ) x, (1 − ξ ) y, z , and (c) pure shear, x, y, z → x + ξy, y, z .The elastic constants C , C ′ = ( C − C ) / , and the bulkmodulus B = ( C + 2 C ) / have been obtained numeri-cally from the second derivative of the free energy, Eq. (1),with respect to ξ .The temperature dependence of the experimental elasticconstants of iron is sometimes fitted using the semi-empiricalVarshni expression . The evolution of the elastic constantsof our model with the reduced temperature T ∗ is presented inFig. 1 along with the Varshni fit from Ref. [16]. The compar-ison between the experimental and the ADFT data (presentedin reduced units) in Fig. 1 is done by matching two differenttemperature points in each scale. First, the reduced temper-ature of solid phase instability in ADFT model is associatedwith the iron melting temperature (right side of the abscissaaxes). Second, we matched the lowest reduced temperatureexplored in the present study (0.015) and the ambient temper-ature (300K). This choice, together with the ordinate scaling,has been done to have the best simultaneous fit of the bulkmodulus, C and Zener anisotropy parameter.Our results follow the general trend of the experimentaldata, but since our model does not take into account the mag-netic ordering, the Zener anisotropy parameter A = C /C ′ remains nearly constant through the entire temperature range,contrary to the experiment . It is argued, for example inRef. [17], based on DFT calculations, that the temperature de-pendence of the elastic anisotropy parameter of iron is of mag-netic origin. The observed nonlinearity of our results (whichleaves the anisotropy parameter A unchanged, though) is dueto the vicinity of the stability limit of the solid phases and notdue to the magnetic effects.The Zener anisotropy parameter that we have obtained inour simulations is A ≃ . , which is very close to the ex-perimental value of . for the bcc iron at room tempera-ture . It is important to mention that while an arbitrary GPascale of the elastic constants can be chosen by assigning aphysical value to V in the potential Eq. (2), their ratios, and,notably, the anisotropy parameter A are independent from V and uniquely determined by the form of the potential derivedfrom the structure factor. III. POINT DEFECTS IN THE ADFT
To understand how point defects can be introduced in theADFT, first, the influence of the mean atomic density ρ on V un i t s G P a reduced temperature T * KelvinBulk mod. (exp.)C (exp.)C’ (exp.)Bulk mod.C C’ FIG. 1. Bcc iron elastic constants in the ADFT as functions of re-duced temperature (points). For comparison, the Varshni fits of ex-perimental data from Ref. [16] are plotted with lines with respect tothe upper and right axes. the form of atomic peaks should be considered. Due to theform of the local free energy in Eq. (1) the atomic densityfunction is strictly confined in the interval [0 , . In fact, thelocal free energy term impedes the growth of inhomogeneitiesmore and more as one approaches 0 or 1, so those values arenever actually reached. During a BCC crystal growth the am-plitudes of the peaks related to the atomic positions reach theirequilibrium value (close to 1), whereas in between the peaks,the atomic density is approaching zero. This is the reason whywith varying the temperature or the density, only the width ofthe atomic density peaks varies significantly, which is differ-ent from the regular phase field crystal (PFC) model. In thePFC description the local term of the free energy is approxi-mated by a Landau polynomial. In this case the amplitude ofthe atomic peaks is not confined between 0 and 1 but growswith the varying parameters and reaches negative values at itsminimum. This difference, as we will show below, is crucialfor the elastic properties of the model and vacancy diffusion.Let us choose as an approximation a Gaussian form ofatomic peaks of a fixed height: ρ ( r ) = X i e − ( r − r i ) /σ , (4)with the parameter σ controlling the “width” of the atom; thesummation in Eq. (4) is carried out on the sites of a bcc lat-tice of spacing a . It should be noted that the integral of asingle Gaussian peak in its Wigner-Seitz primitive cell, nor-malized to the volume of the cell, gives the mean density ρ . If the atomic density profile vanishes at the borders ofthe Wigner-Seitz cell, the integration interval can be extendedto infinity and Gaussian integration can be applied, leading to σ = a (cid:0) ρ / (2 π / ) (cid:1) / .The Fourier transform of Eq. (4) is ρ k = ρ δ k , + X q δ k , q e − σ k / , (5)where the sum is over the first Brillouin zone of the reciprocallattice face-centered cubic lattice. Rewriting Eq. (1) in theFourier variables and using Eq. (5), the non-local part, whichrepresents the internal energy F int , can be written as F int = V ρ + X q V ( q ) e − σ q / . (6)We will call the n -mode the contribution of all the terms inEq. (6) with | q | = q n , and q < q < q < . . . . The firstthree modes correspond to q = 2 √ π/a , q = 4 π/a , and q = 4 √ π/a . When the density ρ ≪ , which is the casein our simulation, the exponential prefactor in (6) is nonva-nishing for q - s corresponding to higher modes. So far, wehave made no assumption about the lattice parameter a , let uschoose a = 2 / π/k [where k is the minimum positionof the potential V ( k ) ] as the reference. The function V ( k ) ispresented in Fig. 2 (a) where the wave vectors of the first threemodes are indicated with arrows (dashed for a = a and solidfor a = 1 . a ). Note that, since σ ∼ a and q ∼ a − , theexponential prefactor in Eq. (6) does not depend on a . Theaforementioned figure gives an idea about the energy gain as-sociated with the first mode and its loss associated with highermodes.The energy given by Eq. (6) was calculated in one-, two-and three-mode approximations and is plotted in Fig. 2(b) as afunction of the ratio a/a . It can be seen that the minimum ofthe free energy for two and three modes approximation corre-sponds to a min > a . Since σ ∼ ρ / , the exponential prefac-tor decreases more slowly with the mode number for lower ρ ,and, consequently, a min increases with decreasing ρ . Thenthe equilibrium lattice parameter in the ADFT depends on themean value of the atomic density ρ and, consequently, is notuniquely determined by the position k of the minimum of theinteraction potential V ( k ) . From now on, we will be referringto a min as just a .The higher modes contribution is determinant for the elas-tic properties of the ADFT. If the atomic density profile werenot confined between 0 and 1, the first mode would pre-vail over the following ones and the one mode approxima-tion could have been used. This is equivalent to consid-ering only the lowest wave vectors in Eq. (6) and putting σ = 0 (the first mode amplitude being normalized to 1).This would lead to C = 16 α , C = C = 8 α , with α = V (cid:2) ( k /k ) + ( k /k ) (cid:3) , and anisotropy parameter A = 2 , which differ significantly from the iron elastic con-stants relations . The fact that the ADFT reproduces signif-icantly better the elastic constants of bcc iron, is due to theparticular form of the local free energy term in Eq. (1) thatmakes the higher modes contribution important.In Fig. 3, the free energy as a function of ρ is presented fortwo different temperatures. The free energy of the liquid stateis given by F liq . = V ρ / k B T ( ρ ln ρ + (1 − ρ ) ln(1 − ρ )) , whereas that of the solid phase is computed numerically.The fit of the structure factor used in this paper correspondsto the temperature T ∗ ≃ . [Fig. 3(a)]. The equilibriumvalues of the atomic density in solid and liquid phases werefound using a common tangent construction. Assuming thatat this temperature the atomic volume of liquid at coexistence -0.26-0.24-0.22-0.2-0.18-0.16-0.14-0.12-0.1-0.08 0.8 1 1.2 1.4 1.6 1.8 2 2.2 V ( q ) / V qa /(2 π )a=a a>a -0.67-0.66-0.65-0.64-0.63-0.62-0.61-0.6-0.59-0.58 0.9 0.95 1 1.05 1.1 F i n t ( V = ) a/a one mode2 modes3 modes (a) (b)FIG. 2. (a) The Fourier transform V ( k ) of the interatomic interactionenergy as a function of the distance. (b) Illustration to the equilib-rium lattice parameter a following from Eq.(6) for ρ ≃ . .The first three modes positions corresponding to a bcc structure arehighlighted with arrows in (a). -16-14-12-10-8-6 0.04 0.06 0.08 0.1 0.12 F * ρ ρ ρ BCCLIQUID -16-14-12-10-8-6 0.04 0.06 0.08 0.1 0.12 ρ BCCLIQUID (a) (b)FIG. 3. The solid and liquid state free energies for reduced tempera-tures T ∗ = 0 . (a) and . (b). with solid is the same as the atomic volume of the metastablesolid at the same ρ , one can estimate the melting volumechange.Thus, instead of the lattice expansion ∆ a/a = ( a ( ρ ) − a ( ρ min0 )) /a ( ρ min0 ) , we will rather speak of the relative vol-ume change ∆ V /V = 3∆ a/a . We have found numericallythe dependence of ∆ V /V on ρ , by relaxing the lattice con-stant a for each value of the average atomic density (see Fig.4). As one would expect, due to thermal expansion, a ( ρ ) increases when going from T ∗ = 0 . to 0.025. At the sametime, ρ min0 slightly decreases (see the legend of Fig. 4). Unfor-tunately, this trend does not persist for the whole temperaturerange, so we cannot make a real comparison to the experi-mental data on thermal expansion of iron (for example, seeRef. [18]). From the data corresponding to T ∗ = 0 . in Fig.4, and the extreme points of the liquid-solid coexistence re-gion ρ and ρ in Fig. 3, one gets a relative volume changeon melting, (∆ V /V ) melt . = 3( a ( ρ ) − a ( ρ )) /a ( ρ ) [where a ( ρ ) corresponds to a metastable solid at ρ ] ofaround . It is consistent with the general result fromRef. [19] for bcc lattices, following from random packingsand the Goldschmidt premise.Using these results at lower temperatures, when the solidstate becomes absolutely stable, the case when ρ < ρ can be considered as a crystal with some (non-equilibrium)concentration of vacancies. In the case when ρ > ρ the crystal will be said to contain some (non-equilibrium) -0.1-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 r e l a t i v e v o l u m e c hange ∆ V / V ( ρ - ρ )/ ρ T*=0.025T*=0.02
FIG. 4. The relative volume change as a function of the averageatomic density profile ρ . The values ρ corresponding to thesolid free energy minimum position in Figs. 3(a) and 3(b) are 0.0837and 0.0834, respectively. concentration of self-interstitial atoms (SIA). For example, ifone atomic peak removal is associated with a single vacancy, | ρ − ρ | /ρ gives the vacancy concentration c v . Therelative volume change due to the vacancies is given by theformula ∆ V /V = c v V F / Ω , where V F is the vacancy for-mation volume and Ω the atomic volume. The linear fit forsmall | ρ − ρ | /ρ (approaching 0 from below) in Fig.4, which corresponds to small c v , leads to V F ≃ . at T ∗ = 0 . . The latter corresponds to a relaxation volume V rel = V F − Ω ≃ − . . The data available for com-parison are all low or zero temperature results. The most re-liable, to our knowledge, experimental relaxation volume forvacancies in iron is of –0.05 at 6 K . First-principle calcula-tions at 0 K gave a value of –0.45 in Ref. [21] that the authorsthemselves assumed being overestimated due to the absenceof local relaxation in their calculations. However, there ex-ist experimental data suggesting a thermal expansion of pointdefects that is up to 15 times that of the matrix . Apply-ing a factor of 15 would put the experimental 6K data in goodagreement with our estimation. Our result can thus be consid-ered as physically reasonable, but in any case the point defectformation volume will be rather considered as a phenomeno-logical parameter in our model. IV. POINT DEFECT ABSORPTION BY GRAINBOUNDARIES
According to the previous considerations the introductionof point defects in a perfect crystal with a fixed size of thesimulation box will simply increase the energy of the system.However, when point defect sinks are present, such as dislo-cations or grain boundaries, the system will tend to decreaseits energy by pushing the point defects to the sinks.To model this phenomena, we have performed simulationswith [110] low- and high-angle symmetric tilt grain bound-aries by decreasing or increasing the initial equilibrium aver-age density profile value ρ . The crystal with grain boundarieswas constructed using the procedure described in Ref. [12].The point defect absorption, manifest in atoms disappearingor appearing at the GB, can be seen directly from Fig. 5, where darker colors correspond to higher values of the atomicdensity function. The relative volume changes used were of ∼ − for vacancies and of ∼ for SIA. These values cor-respond to unrealistically high concentrations of point defects,and compensate for the fact that the grain size is rather smallin our simulations ( ∼ Vacancies SIA(a) (b)FIG. 5. Point defect absorption kinetics at a [110] edge dislocation(indicated with white dashed lines) in a system with (a) vacancy and(b) self-interstitial supersaturation. The dislocation is part of a GBwhich plane is highlighted with green color. The [110] cross-sectionsof the atomic density function profile are given in logarithmic grayscale in order to make visible the variations of the ADF in betweenthe atomic peaks. The time axis is pointing downwards.
To characterize the volume of an atom, the integration ofthe ADF in the Voronoi cell associated with this atom has beendone. The Voronoi cell is defined as the part of the space situ- (a) (b)FIG. 6. The intrinsic strain [dilatation (a) and compression (b)] dueto the edge dislocation at the 4.24 ◦ [110] GB, as given by the localatomic density normalized to its bulk value. (a) -0.5-0.48-0.46-0.44-0.42-0.4-0.38 160 180 200 220 240 l o c a l a t o m . den s . pe r un i t g r a i n i n t e r f a c e simulation time t * ρ =0.060.06250.0650.0675 0.070.07250.075 (b) m odu l u s o f ab s o r p t i on r a t e ( t * − a − ) relative volume change ∆ V/V
FIG. 7. The absorption kinetics at the 4.24 ◦ [110] GB (a), as given bythe total decrease of the normalized local atomic density per a graininterface. The absolute value of the slope of the curves presented in(a) is plotted in (b) as a function of the relative volume change. ated closer to a given atom than to any other one. This quantitywill be referred to as the local atomic density. Then, the rel-ative deviation of the local atomic density from its bulk valuewill reflect the local strain (compression-expansion) field. Thelatter is represented in Fig. 6 for a low-angle GB fragmentfrom Fig. 5. For better perception of the local atomic den-sity variation, in Fig. 6, colors on equal-sized spheres are usedto visualize the deviation of the local atomic density from itsbulk value (red color associated to this value is used as theupper rendering threshold for compression and the lower ren-dering threshold for expansion). The total relative decrease(per unit interface) of the local atomic density in time is plot-ted in Fig. 7(a) for different tilt angle GBs. The curves arelinear, so their slope, plotted in Fig. 7(b) versus the relativevolume change, can be taken as the measure of the vacancyabsorption rate ∂c v /∂t . The volume change is itself propor-tional to the vacancy concentration c v : ∆ V /V = c v V F / Ω .The fact that the dependence in Fig. 7(b) is nearly linear isconsistent with the linear rate equations commonly used todescribe point defect absorption by extended defects .There is no vacancy production during our simulation,so the rate equation for the vacancy absorption reads as: ∂c v /∂t = − k D v c v . Since the bulk vacancy diffusion coef-ficient D v does not depend on the GB geometry, the vacancyabsorption rates of different [110] symmetric tilt grain bound- m odu l u s o f v a c an cy ab s o r p t i on r a t e ( t * − a − ) tilt angle (°) Σ Σ FIG. 8. The modulus of the vacancy absorption rate per unit graininterface ( a ) as a function of the GB tilt angle. aries plotted in Fig. 8 reveal their relative sink strengths k .The sink strength of the low-angle GBs increases with the tiltangle due to the increasing of the dislocation density like itwas found in Ref. [8]. In Ref. [9], however, it was obtainedthat the sink strength remains almost constant above ◦ tilt,due to the mutually annihilating elastic fields of neighboringdislocations. It was assumed therein that the long-range dif-fusion of point defects toward the GB is the rate-limiting stepfor the sink strength determination. This assumption does nothold in our model and it is rather the interaction of point de-fects with the dislocation cores and not with the far-reachingelastic field that is determinant for the absorption rates in oursimulations.It is notable that a low sink strength is obtained for the Σ9 and Σ3 grain boundaries. This result is in agreement with theexperimental study presented in Ref. [25]. The molecular stat-ics modeling done in Ref. [10] is also in favor of our resultsas it shows a general trend of a mean vacancy formation en-ergy decreasing with Σ (that is, the low- Σ GBs being the lessenergetically favorable for absorbing vacancies). Finally, it iscoherent with the other properties of special GBs approachingthose of the bulk material (energy, excess volume) due to theirbulk-like atomic arrangements.
V. CONCLUSIONS
It has been previously shown that the atomic density func-tion model reproduces well the atomic structures of [100] and[110] symmetric tilt GBs .In the present paper, we have demonstrated that point de-fects can be also adequately described by this model as well asthe point defect absorption by GBs. It comes possible from theadequate description of the elastic properties of given materi-als. We show that the form of the local term of the free-energyfunctional plays a crucial role in the description of these prop-erties.This new development of the ADFT gives a new insight intothe GB sink strength for vacancy annihilation. Accurate fit ofthe interaction potential to a structural factor allows us to givea quantitative description of vacancy migration to GBs. Thisgives access to sink strengths of GBs of complex geometriesor of polycrystalline materials.As a perspective, first, Frenkel pairs annihilation in irradi-ated materials can be modeled with two coupled atomic den-sity functions, one for the vacancies and another one for theself-interstitials. Second, introducing another atomic species,with elastic properties different from the α -iron matrix, willallow us to describe non-equilibrium solute segregation at theGBs driven by the fluxes of points defects. This work is in progress. VI. ACKNOWLEDGMENTS
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