Relationship between grain boundary segregation and grain boundary diffusion in Cu-Ag alloys
RRelationship between grain boundary segregation and grainboundary diffusion in Cu-Ag alloys
R. K. Koju and Y. Mishin Department of Physics and Astronomy, MSN 3F3,George Mason University, Fairfax, Virginia 22030, USA (Dated: July 8, 2020)While it is known that alloy components can segregate to grain boundaries (GBs),and that the atomic mobility in GBs greatly exceeds the atomic mobility in the lat-tice, little is known about the effect of GB segregation on GB diffusion. Atomisticcomputer simulations offer a means of gaining insights into the segregation-diffusionrelationship by computing the GB diffusion coefficients of the alloy components as afunction of their segregated amounts. In such simulations, thermodynamically equi-librium GB segregation is prepared by a semi-grand canonical Monte Carlo method,followed by calculation of the diffusion coefficients of all alloy components by molec-ular dynamics. As a demonstration, the proposed methodology is applied to a GBis the Cu-Ag system. The GB diffusivities obtained exhibit non-trivial compositiondependencies that can be explained by site blocking, site competition, and the onsetof GB disordering due to the premelting effect.
Keywords: Atomistic modeling; alloys; grain boundary segregation; grain boundary diffusion. a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l I. INTRODUCTION
Solute segregation to grain boundaries (GBs) can affect many mechanical, thermody-namic and kinetic properties of materials [1]. Once at GBs, the solute atoms can diffusethrough the material much faster than they would by regular, lattice diffusion mecha-nisms [2]. The accelerated atomic transport along GBs, often referred to as “short-circuit”diffusion, can control the kinetics of processes such as creep deformation [3–5], phase pre-cipitation [6, 7], as well as complex kinetic phenomena such as dynamic strain aging [8, 9].It is well-established that the self-diffusion and solute diffusion coefficients in GBs canexceed the lattice diffusion coefficients by many orders of magnitude, especially at lowhomologous temperatures [2]. What remains poorly understood is how the amount of GBsegregation can affect the rate of GB diffusion. For example, in a binary alloy A-B one mustconsider GB diffusion coefficients of both the solute component B ( D B ) as well as the hostcomponent A ( D A ). Several questions arise. For example, as the amount of GB segregationof B increases, do the diffusion coefficients D A and D B both increase, both decrease, orcan show opposing trends? Which physical factors control the effect of segregation on GBdiffusion? Can the segregation-diffusion relation change with temperature and/or alloycomposition? How does the frequently occurring disordering of the GB structure at hightemperatures can affect the segregation-diffusion relation?To our knowledge, these questions remain largely open. Answering them by experimentis not impossible in principle but is hampered by technical obstacles. One of them beingthat, to keep track of both diffusion coefficients ( D A and D B ) in the same GB, a co-diffusionexperiment is required with concurrent monitoring of the segregated amounts. The experi-ment would then have to be repeated for a set of alloy compositions and/or temperatures.Such experiments are technically challenging and, to our knowledge, have not been per-formed so far. Another challenge is related to the fact that GB diffusion experiments arepredominantly carried out at relatively high temperatures at which a significant fraction ofthe atoms diffusing along the GB leaks into the surrounding lattice regions [2]. Under suchconditions, called the type-B kinetic regime, one can only extract from the experiment thetriple product sD A,B δ , s being the segregation factor and δ the GB width. Separate deter-mination of the GB diffusion coefficients D A,B requires specially designed low-temperatureexperiments conducted in the so-called type-C regime. C-regime measurements are muchmore difficult and have only been performed for a small number of systems [2, 10–15]. Suchsystems do not include alloys with a varied chemical composition. Furthermore, only thesolute diffusivity D B has been measured in the C-regime.Given the experimental challenges mentioned above, a meaningful alternative approachis offered by atomistic computer simulations. It has recently been demonstrated that GBdiffusion coefficients can be reliably computed in pure metals as well as dilute binary alloys(in the latter case, for solute diffusion only) [16–19]. This methodology can serve as astarting point from which to launch a systematic study of the effect of GB segregation onGB diffusion of both chemical components in binary, and in the future multicomponent,alloy systems.The goal of this paper is to initiate work in the outlined direction by performing a seriesof simulations of GB segregation and GB diffusion in Cu-rich Cu-Ag solid solutions chosenhere as a model system. The Cu-Ag system has the advantage of exhibiting a limited solidsolubility of the two elements and a strong GB segregation trend. Its choice also puts uson a familiar ground since much information has already been obtained for this systemin previous work [20–23]. In particular, a reliable interatomic potential is available [20],and the phase diagram predicted by this potential has been accurately computed [20, 23].GBs in Cu have been studied extensively [17–19, 21, 24–37]. One typical GB was chosenhere as an example, with the intent of extending this work to a larger set of boundariesin the future. We perform a detailed study of Ag GB segregation in a wide temperature-composition domain of the Cu-Ag system, followed by a similarly detailed study of GBdiffusion of both Ag and Cu and its correlation with the segregation behavior. II. METHODOLOGY
Atomic interactions in the Cu-Ag system were modeled using an embedded atom poten-tial [20] that accurately reproduces a large number of physical properties of both Cu andAg. The potential was fitted to first-principles energies of Cu-Ag compounds and predictsthe Cu-Ag phase diagram in reasonable agreement with experiment. Molecular dynamics(MD) simulations were performed using the Large-scale Atomic/Molecular Massively Par-allel Simulator (LAMMPS) [38]. The Monte-Carlo (MC) simulations utilized the parallelMC code ParaGrandMC developed by V. Yamakov at NASA [39–41].As a representative high-angle GB, we chose the symmetrical tilt Σ17(530)[001] GBwith the misorientation angle of 61 . ◦ . Here, Σ is the reciprocal density of coincidentsites, [001] is the tilt axis, and (530) is the GB plane. The boundary was created in arectangular periodic simulation block whose edges were, respectively, parallel to the tilt axis( x -direction), normal to the tilt axis ( y -direction), and normal to the GB plane ( z -direction).The block had the approximate dimensions of 10 . × . × .
13 nm and contained1 . × atoms. The ground-state structure of the GB in pure Cu was obtained by the γ -surface method [24, 25, 42]. The structure consists of identical kite-shaped structuralunits arranged in a zig-zag array as shown in Fig. 1. The rows of structural units runningparallel to the tilt axis can be interpreted as closely spaced edge dislocations forming theGB core. The same structure of this GB was previously obtained in Cu [17, 28] and Ni [43].The GB energy was found to be 856 mJ/m in agreement with previous reports [17, 28].A prescribed amount of Ag was introduced into Cu by semi-grand canonical MC simu-lations implemented at a chosen temperature T and a fixed value of the chemical potentialdifference between Ag and Cu. The trial moves of the MC process included random dis-placements of randomly selected atoms with a random re-assignment of their chemicalspecies to either Ag or Cu. The trial move additionally included random changes in thedimensions of the simulation block with rescaling of the atomic coordinates to achieve zeropressure conditions in all three directions. The trial move was accepted or rejected by theMetropolis algorithm. The simulation produced a thermodynamically equilibrium distribu-tion of Ag atoms in the GB region and inside the grains for the targeted alloy composition.The simulations covered the temperature range between 600 K and 1100 K, with the alloycompositions varying from pure Cu to the solidus line.The amount of Ag segregation was quantified by the excess number of Ag atoms perunit GB area at a fixed total number of atoms:[ N Ag ] = N Ag − N N (cid:48) Ag N (cid:48) , (1)where N Ag and N (cid:48) Ag are the numbers of Ag atoms per unit area in two regions with andwithout the GB, respectively, and N and N (cid:48) are the respective total numbers of Cu andAg atoms. Both regions were large enough to include both the GB and the interiors of thegrains.The degree of structural disorder in the GB was measured by the layer-averaged structurefactor S ( k ). The simulation block was divided into 0.1 nm thin layers parallel to the GBplane and numbered by index i . The structure factor corresponding to layer i is defined by S i ( k ) = 1 N i (cid:118)(cid:117)(cid:117)(cid:116) N i (cid:88) j =1 cos ( k · r j ) + N i (cid:88) j =1 sin ( k · r j ) , (2)where k = 2 π [2 /a, ,
0] is the chosen reciprocal lattice vector, r j is the position of atom j within the layer i , a is the cubic lattice parameter, and N i is the total number of atoms inthe layer. The structure factor so defined equals one in the perfect lattice at 0 K, has avalue S ∞ ( k ) < ϕ ( z i ) = S i ( k ) − S ∞ ( k ), is definedas the order parameter at position z i = λi in the GB region ( λ being the layer thickness).Furthermore, the width w of the order parameter minimum can be taken as the structuralwidth of the GB. Specifically, w was defined as twice the standard deviation of the Gaussianfitted to the order parameter profile ϕ ( z i ) across the GB. Knowing the GB width, the Agconcentration in the GB can be found by averaging the atomic fraction of Ag over the layerof width w centered at the Gaussian peak. This concentration provides a complementarymeasure of the GB segregation in addition to [ N Ag ].GB diffusion coefficients were computed from MD simulations performed on GBs pre-equilibrated by MC simulations. First, the potential energy peak across the current GBposition was constructed by averaging the potential energy over thin layers parallel to theboundary plane. The peak width was typically around 1 nm or larger. Mean-square atomicdisplacements, (cid:104) x (cid:105) and (cid:104) y (cid:105) , parallel to the GB plane were computed as functions of time forboth Ag and Cu atoms. The calculations only included atoms within a 1 nm thick windowcentered at the boundary position. The mean-square displacements were monitored overa period of time ∆ t ranging from 24 ns to 60 ns, depending on the alloy composition andtemperature. The GB diffusion coefficients of Ag and Cu in both directions were obtainedfrom the Einstein relations D x = (cid:104) x (cid:105) / t and D y = (cid:104) y (cid:105) / t , respectively. Due to thestructural anisotropy of the GB, the diffusion coefficients parallel ( D x ) and normal ( D y ) tothe tilt axis are generally different. To account for slight variations in the GB position withtime due to thermal fluctuations, the 1 nm layer in which the mean-square displacementswere calculated was periodically re-centered to the current GB position identified with thepotential energy peak. III. RESULTSA. Grain boundary segregation
Fig. 2 illustrates typical equilibrium segregation profiles in the Cu-2 at.% alloy at varioustemperatures. The profiles were obtained by averaging the atomic fraction of Ag over thinlayers parallel to the GB and then averaging over multiple snapshots saved during the MCsimulations. Note that the segregation peak grows higher with decreasing temperature andbroadens with increasing composition. As will be discussed below, the width of the segre-gation zone drastically increases near the solidus line as the GB undergoes the premeltingtransformation.Representative order parameter profiles ϕ ( z ) are shown in Fig. 3. At a fixed temper-ature (1100 K in this case), the minimum becomes deeper as Ag concentration increases,indicating the accumulation of structural disorder in the GB core. As the alloy compositionapproaches the solidus line, the order parameter in the GB tends to zero ( ϕ (0) → w rapidly increases and eventually spreads across the entire simulation block(Fig. 4). This behavior is a clear manifestation of GB melting and a sign that the alloycomposition has reached the solidus line at the given temperature.Isotherms of GB segregation are plotted in Fig. 5 using two measures of segregation: thetotal segregated amount [ N Ag ] (number of excess Ag atoms per unit area) and the chemicalcomposition (at.%Ag) within the GB core. Both segregation parameters increase, in a non-linear manner, with increase in the alloy concentration and decrease in temperature. Larger[ N Ag ] values result from both the increase in the GB concentration and the GB broadeningeffect (Fig. 5a). By contrast, the isotherms shown in Fig. 5b capture the behavior of the GBcomposition alone. Note that, at temperatures above the eutectic temperature predictedby the interatomic potential ( T E = 935 K [20]), the GB composition reaches the liquiduscomposition on the computed phase diagram [20]. Thus, at temperatures above T E , theGB transforms into a liquid layer of the liquidus composition when the grain compositionapproaches the solidus line. GB melting behavior in the Cu-Ag system was also noted inprevious simulation studies [23, 44].Distribution of the segregated Ag atoms inside the GB was examined in detail usingthe OVITO visualization software [45]. In dilute compositions, the GB remained highlyordered and the segregated Ag atoms substituted for the host Cu atoms at particularpositions within the GB structural units (Fig. 6a). As the alloy concentration increased,the GB structure grew increasingly disordered (Fig. 6b) until the structural units could nolonger be distinguished (Fig. 6c). We emphasize that this disordering effect was entirelycaused by the Ag segregation. In pure Cu, the GB structure remained well-ordered untilhigh temperatures approaching the Cu melting point (1326 K [46]). B. Grain boundary diffusion
The GB diffusion coefficients were computed at temperatures and alloy compositionslying within the Cu-based solid solution domain on the Cu-Ag phase diagram. For thechosen GB, the diffusion coefficients parallel ( D x ) and normal ( D y ) to the tilt axis werefound to be nearly equal. Thus, only the average values D = ( D x + D y ) / D versus 1 /T , shown Fig. 7a (Cu diffusion) and Fig. 7b (Ag diffusion). The alloy compositionsare limited to 2 at.%Ag to avoid close proximity of the solidus line. While diffusion in highlypremelted GBs representing liquid layers could also be measured, the results would not berelevant to the segregation-diffusion relationship pursued in this work.The diffusion coefficients in Fig. 7 reasonably follow the Arrhenius relation D = D exp (cid:18) − EkT (cid:19) (3)at all temperatures. The plots demonstrate that the diffusion coefficients of both com-ponents depend on the alloy composition. To display the composition dependence moreclearly, we plot the diffusion coefficients as a function of at.%Ag in Fig. 8a. Two trends areobvious: • Ag atoms diffuse in the GB slower than the host Cu atoms at low concentrations butfaster at higher concentrations. The crossover occurs at about 1 at.%Ag. • While the Ag diffusion coefficients increase with Ag concentration monotonically, theCu diffusion coefficients display a non-monotonic composition dependence, with alocal minimum occurring at about 1 at.%Ag.Note that the diffusion coefficients are shown in Fig. 8a on the logarithmic scale, meaningthat the trends described are quite significant. The following explanation of these trendscan be proposed. At low temperatures, the Ag atoms tend to segregate to particular GBsites offering the largest segregation energy. Due to this energetic preference, the Ag atomsspend most of the time occupying such favorable sites. They are reluctant to jump toalternate sites (i.e., against the driving force) to participate in the diffusion process, whichresults in slower diffusion rates. As additional Ag atoms segregate to the GB, they areforced to occupy less favorable (higher energy) sites and are more likely to contribute tothe diffusion flux. In other words, the trapping effect weakens and Ag diffusion acceleratesas the alloy concentration increases. At the same time, the Cu atoms diffuse slower withthe addition of Ag due to the site blocking effect: the less mobile Ag atoms disrupt thefast diffusion pathways for Cu diffusion within the GB structure. As a result, the Ag andCu diffusivities display opposite trends, converging toward each other as clearly observedin Fig. 8a.This explanation only applies as long as the GB maintains an ordered structure withwell-defined structural units offering distinct types of segregation site. This is certainly truefor dilute alloy compositions as illustrated in Fig. 6a. At higher Ag concentrations whenthe GB develops a significant disorder (Fig. 6b) and eventually transforms into a liquid-likestate (Fig. 6c), the situation changes. Diffusion in disordered GBs is governed by differentatomic mechanisms from those in ordered structures [17, 47], hence a change in the diffusiontrend with composition can be expected. This change can explain the crossover of the Agand Cu diffusivities and the existence of a local minimum of the Cu GB diffusivity at about1 at.%Ag. This is the approximate composition at which the GB disordering commencesat the temperatures studied here (Fig. 6b).The crossover effect also manifests itself in the composition dependence of the activationenergy E of GB diffusion appearing in Eq.(3). While Ag GB diffusion is characterized by ahigher activation energy in comparison with Cu below about 1 at.%Ag, the two activationenergies converge to each other in more concentrated alloys in which the GB loses theordered structure (Fig. 8b).For validation of our methodology, we can compare the activation energies computed inthis work with experimental data available in the literature (Table I). For GB self-diffusionin Cu, only data for polycrystals is available [48]. The reported activation energy variesbetween E = 0.751 eV and 0.878 eV, depending on the chemical purity of the material [48].Our calculations predict E = 0.828 eV, which we consider a good agreement given that thepolycrystalline value of E represents an average over many GB types. For Ag GB diffusion,the experiments give E = 1.126 eV (in pure Cu [14]) and 1.128 eV (in Cu-0.2 at.%Ag [15]),in both cases for polycrystalline samples. The closest chemical compositions studied in thiswork are Cu-0.12 at.%Ag and Cu-0.25 at.%Ag. The respective activation energies, 0.918 eVand 0.967 eV, compare well with the experiment considering that they were obtained forone particular GB. Another piece of experimental information comes from a recent study ofAg diffusion in a Cu bicrystal with the Σ5(310)[001] GB [15]. Even though this boundary isdifferent from ours and is considered special, the experimental activation energy (0.983 eV or1.067 eV, depending on the diffusion direction) is close to our results for the Σ17(530)[001]boundary in the dilute limit. Thus, the comparison with experiment is very encouragingand lends confidence to the simulation results reported in this paper. IV. CONCLUSIONS
The goal of this work was to demonstrate that it is now possible to probe the effect of GBsegregation on GB diffusion of both the solute and solvent components in alloys by meansof atomistic computer simulations. The methodology proposed combines MC simulationsto create an equilibration GB segregation with MD simulations to extract the GB diffusioncoefficients. A reliable interatomic potential is required, and the relevant part of the phasediagram must be known or computed.As an example, we have studied diffusion in a representative GB in the Cu-Ag system inthe temperature-composition domain of Cu-based solid solutions. Our results indicate thatthe GB diffusivities of the solute (Ag) and solvent (Cu) atoms can exhibit quite different andnon-trivial composition/temperature dependencies. They can correlate with each other,anti-correlate, cross, or have local minima. These behaviors reflect intricate interplaysbetween different diffusion mechanisms and physical effects, such as site blocking and sitecompetition. One factor that is more crucial in alloys than it is in elemental solids is thedisordering of the GB structure. When the alloy composition and/or temperature approachthe solidus line on the phase diagram, GBs can become atomically disordered at relativelylow temperatures, eventually transforming to a liquid film [17, 23, 44, 49]. This disorderingis fueled by GB segregation and can drastically alter the GB diffusion mechanisms and thusthe segregation-diffusion relationship in comparison with ordered GB structures prevailingin elemental solids and/or solid dilute solutions.This work was performed on one particular GB in one binary system. Future studies inthe proposed direction may include larger GB sets, multicomponent systems, and a moredetailed analysis of the underlying diffusion mechanisms.
Acknowledgement:
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Distance normal to the grain boundary ( ) A g ( a t . % )
900 K950 K1000 K1050 K1100 K
Figure 2: Ag GB segregation profiles in the Cu-2 at.%Ag alloy at various temperatures. −40 −20 0 20 40 Distance normal to the grain boundary (Å) O r d e r p a r a m e t e r Figure 3: Representative profiles of the order parameter ϕ ( z ) across the GB for three alloy com-positions at the temperature of 1100 K. The curves represent Gaussian fits of the local minimumoccurring at the GB position. (a) G B w i d t h ( Å ) Alloy composition (at.%Ag) (b) − G B o r de r pa r a m e t e r Alloy composition (at.%Ag)
Figure 4: (a) GB width w and (b) GB order parameter ϕ (0) as functions of alloy composition atthe temperature of 1100 K. Note that w diverges to infinity while ϕ (0) tends to zero at the soliduscomposition of about 4 at.%Ag. (a) G B S eg r ega t i on [ N A g ] ( n m − ) Alloy composition (at.%Ag)
600 K700 K800 K900 K 950 K1000 K1050 K1100 K (b)
950 K1000 K1050 K1100 K G B c on c en t r a t i on ( a t. % A g ) Alloy composition (at.%Ag)
600 K700 K800 K900 K 950 K1000 K1050 K1100 K
Figure 5: (a) Amount of Ag GB segregation [ N Ag ] and (b) GB composition (atomic percentageof Ag atoms) as functions of alloy composition at different temperatures. Each curve ends at thesolidus line on the phase diagram. In (b), the dashed lines represent the liquidus compositionsobtained from the phase diagram at temperatures ≥
950 K. (a)(b)(c)Figure 6: Distribution of Ag atoms in the GB at the alloy compositions of (1) Cu-0.12 at.%, (b)Cu-1 at.%Ag, and (c) Cu-2 at.%Ag at the temperature of 900 K. The Ag and Cu atoms are shownin blue and pink, respectively. Note the accumulation of GB disorder with increase in the GBsegregation. (a) − − − D ( m / s ) − K) T (K)Pure Cu0.12% Ag0.25% Ag0.50% Ag1.00% Ag1.50% Ag1.75% Ag2.00% Ag (b) − − − D ( m / s ) − K) T (K)0.12% Ag0.75% Ag1.25% Ag1.50% Ag1.75% Ag2.00% Ag
Figure 7: Arrhenius diagrams of GB diffusion coefficients of (a) Cu and (b) Ag in Cu-Ag alloyswith different chemical compositions. (a) − − − D ( m / s ) Alloy composition (at.%Ag)
Cu 1100 KCu 1050 KCu 1000 K Cu 950 KCu 900 KAg 1100 K Ag 1050 KAg 1000 KAg 950 K Ag 900 K (b) A c t i v a t i on ene r g y ( e v ) Alloy composition (at.%Ag)