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Because of this difficulty, we have inal grain boundaries can be thought of as high symmetry taken a step back. We seek to systematically explore the grain boundaries with added flaws, the energy cusps at cohesive laws for mode I fracture for all possible grain special grain boundaries have the same θ log θ form. A boundary geometries in 2D that can be simulated in peri- similar argument will apply to the peak fracture stress. odic boundary conditions for a particular size and strain. Just as adding dislocations to the perfect crystal adds We will initially be focusing on symmetric grain bound- a nucleation point for fracture, and therefore a discon- aries and then expanding the picture to look at asym- tinuity (jump down) in fracture strength, adding a flaw metric 2D grain boundaries. One use of such data would to a high symmetry, high angle grain boundary abruptly be for finite element simulations of polycrystals6, as de- changes the local fracture strength by adding a potential scribed above. For this purpose, it would be nice to find nucleation site for fracture. a functional form which describes the fracture strength of the grain boundary as a function of geometry. We find, however, that the fracture strength as a func- II. PROCEDURE FOR CALCULATING GRAIN tion of tilt angles is discontinuous everywhere, with par- BOUNDARY ENERGY AND COHESIVE STRENGTH ticularly large jumps at special, high symmetry grain boundaries composed of a simple arrangement of struc- tural units with a low repeat distance. These special We calculate energies and cohesive strengths for grain 21 boundaries are also associated with cusps in the grain boundaries using the DigitalMaterial package to per- boundary energy. We will describe dependence of the en- form atomistic simulations. The potential we are using ergy and the fracture strength near these special bound- for 2D simulations is the Lennard Jones potential with a aries by treating them as perfect, albeit complex, crystals smooth, fourth order cutoff between 2.41 and 2.7 A.˚ The with added dislocations to break the symmetry. ground state is a triangular lattice with a lattice constant ˚ Most studies use the coincidence site model12 to of 1.11 A. construct and classify special grain boundaries5,7,8,9,10 while a few mention the significance of structural units7,9,13,14,15. The only 2D study we are aware of, the “bubble-raft” model, observes the structural units of sev- eral of the special, high angle grain boundaries for the tri- angular lattice and how patterns of structural units com- bine to create vicinal geometries14. Other 3D studies dis- cuss how grain boundaries with geometries close to those of special grain boundaries can be decomposed into the structural unit of the special grain boundary with added flaws13,15. We develop a systematic way of finding high symmetry geometries and show that the combinations of patterns of structural units at vicinal grain boundaries are key to understanding the dependence of energy and fracture strength on geometry. Sansoz and Molinari find the grain boundary energy by allowing the grains to relax together from an initial sepa- ration of a few angstroms9 while others perform a conju- gate gradients search7. We use a systematic method for explicitly imposing a relative shift between the grain and using atomistic relaxation for finding the global energy FIG. 2: An Example of a Grain Boundary. The tilt 8,15 minimum similar to that used in . It is well estab- angles are given by θ1 and θ2 and the repeat distances are lished that there exist cusps in the grain boundary energy D1 = |(a1,b1)| and D2 = |(a2,b2)|. We have assumed the ◦ for special high angle grain boundaries7,9,10,15,16,17,18,19. convention where 0 indicates an orientation with Miller in- Recent studies of grain boundary constitutive properties dices (2, 1) for the vertical face of the grain boundary. focus on the response to shear5,9 or compare shear and tension8,11. Warner et al. claim that the tensile response We set up the grain boundary by initializing two rect- does not depend on the geometry of the grain boundary5. angular grains with the given rotations that define the Others have seen jumps in the tensile fracture strength grain boundary geometry we wish to measure. An exam- at high angle grain boundaries geometries7,10. ple of the configuration of atoms that make up a grain We will explain the cusps in energy and jumps in frac- boundary simulation is shown in figure 2. We use peri- ture strength by drawing an analogy between perfect odic boundary conditions in the y-direction in order to crystals and high symmetry, high angle grain boundaries. avoid edge effects which could potentially cause cracks to The dislocation model of low angle grain boundaries gives nucleate at the intersection with the surface27. This has a θ log θ form for the grain boundary energy. Because vic- the disadvantage of only allowing geometries that have fi- 3 nite repeat distances. A method for finding these geome- tries is discussed in detail in section IIIB. If necessary, both grains are strained slightly by equal and opposite amounts in order to have both grains fit into a periodic box. We use a constrained layer of atoms to impose fixed boundary conditions in the x-direction. These are repre- sented by the darker atoms in figure 2. The constrained layer of atoms has a width equal to twice the cutoff dis- FIG. 3: The Necessary Search Range for y- tance of the potential to eliminate surface effects from displacements. The minimum range we must search over the free atoms. to find the most natural configuration is equal to the minimum Besides the tilt angles, there are other factors to con- shift that gives an equivalent configuration of atoms along the sider in constructing the grain boundary geometries. For surface. Assuming that the edges of the repeat cells line up at commensurate grain boundaries, there will be an ideal the bottom, the minimum shift to produce an equivalent con- relative displacement in the direction parallel to the grain figuration is done by lining up the top edge of the first repeat boundary (the y-direction in figure 2). This ideal dis- cell in the grain with a larger repeat (the grain on the right placement will correspond to the lowest energy and thus in the diagram) with the nearest edge of a repeat cell in the is the most natural configuration for a given pair of tilt grain with a smaller repeat distance (the grain on the left). If D is equal to an integer number of repeats of D , the mini- angles.28 2 1 mum search range is equal to D1. ⌊D2/D1⌋ gives the number We find the displacement along the boundary which of repeats of the left grain that fit within a single repeat of gives the lowest energy by initializing the two grains the right grain and ⌈D2/D1⌉ gives the number of repeats of with a small displacement in the x-direction and vary- the grain on the left that contain one repeat of the grain on ing displacements in the y-direction, relaxing the atoms the right. If D2/D1 is not an integer, the minimum search (with the boundary layers contrained to be rigid and non- range is then min (|D2 − ⌊D2/D1⌋D1| , |D2 − ⌈D2/D1⌉D1|) rotating), and measuring the grain boundary energy per length. The range of displacements in the y-direction that we must search over is given by we intend to measure. We wish to measure the fracture toughness of all possible pairs of tilt angles, even if those D1 D2 mod D1 =0 pairs of tilt angles happen to be unstable. In a few cases, ∆= min ( D2 D2/D1 D1 , the minimization procedure described above produced a | −⌊ ⌋ | D2 D2/D1 D1 ) D2 mod D1 =0 curved grain boundary. In these cases, we constrained | −⌈ ⌉ | 6 (1) the displacements in the y-direction such that a consis- where we assume D1
15
10
5 Final Displacement (Å)
0 0 5 10 15 Initial Displacement (Å)
1.6
1.4
1.2
1
Grain Boundary Energy (Lennard-Jones Units) 0.8
0 5 10 15 Initial Displacement (Å) FIG. 5: Finding the Minimum Necessary Range of Angles. For each point in the shaded region, there is an equivalent grain boundary in each of the other regions. FIG. 4: Finding the Energy Minimizing Grain Bound- ary Configuration. Each plateau in the top plot and each flat region in the bottom plot correspond to the basin of at- Since the basis vectors are at a 60◦ angle to one another, traction of a local minimum. The most natural grain bound- the repeat distance of a particular orientation of a 2D ary configuration corresponds to the global energy minimum. triangular lattice is given by
1 √3 izontal directions, resulting in the same grain bound- D = (a + b)2 + ( b)2 = a2 + b2 + ab. (3) ary. Reflecting through the (30, 30) point takes (θ1,θ2) s 2 2 to (60 θ1, 60 θ2), reversing the sense of rotation of p each grain.− As− shown in figure 5, this is equivalent to The tilt angle is then given by flipping the grain boundary in the vertical direction. Re- −1 a + b/2 flecting through the θ2 = 60 θ1 line takes (θ1,θ2) to θ = sin . (4) − √a2 + b2 + ab (60 θ2, 60 θ1), both switching the grains and reversing the− senses of− rotation. This is equivalent to flipping the In order to expand the number of possible geometries, grain boundary in the horizontal direction, also shown we have also considered geometries for which applying in figure 5. Thus, we only need to consider the pairs of a small strain to each grain allows us to fit both grains tilt angles in the triangle enclosed by the θ = 0 line, the 2 inside the same periodic box. For any pair of orientations θ = θ line, and the θ = 60 θ line – the shaded region 2 1 2 1 that have repeat distances that are not commensurate, in figure 5. − we can find a continued fraction approximation to the ratios of their lengths, p/q D /D . We can strain each ≈ 1 2 grain equally into a box of size L = (1/2)(pD2 + qD1), B. Finding All Possible Geometries for Periodic where the strain required is L pD2 /L. For a strain of Boundary Conditions 0.05%, the grain boundary would| − have| to be 1,000 A˚ long before the strain would alter the structure of the grain In order to better simulate the grain boundary in the boundary. We find all pairs of tilt angles that correspond bulk as described in section II, we would like to use peri- to commensurate or near commensurate grain boundaries odic boundary conditions along the direction of the grain by looping over pairs of surface vectors and comparing boundary (y-direction). In order to simulate 2D grain the repeat distances. boundaries in periodic boundary conditions, we need not Figure 6 shows all possible geometries that can be sim- only for both grain orientations to have finite repeat dis- ulated with a periodic length of 70 lattice constants or tances, but also for the repeat distances to be commen- less, and a strain of 0.05% or less. The θ1 = θ2 line cor- surate with one another. Let (a,b) be the lattice vector responds to perfect crystals, while the θ1 = 60 θ2 line that is parallel to the edge of a 2D triangular lattice. corresponds to symmetric grain boundaries. There− is a 5
gle flaw added. Because of this hierarchical procedure for constructing lower-symmetry boundaries, the grain boundary energy and fracture strength as a function of tilt angle will have a self similar nature.
TABLE I: High Symmetry Grain Boundary Geome- tries θ Miller Repeat Distance Σ Structure Indices (lattice constants)
49.10 (1,4) 2.64 7 FIG. 6: All Possible Geometries. The set of points above represent all 2D grain boundary geometries that can be simulated in a periodic box of 70 lattice constants or less, 43.89 (2,5) 3.61 13 with a strain of 0.05% or less. There are gaps near per- fect crystals, symmetric grain boundaries, and high symmetry grain boundaries (discussed in section III C) because creating 53.41 (1,7) 4.35 19 a new, nearby geometry requires adding flaws at large dis- tances. Lines radiating from high symmetry geometries rep- resent flaws added at closer and closer distances as you move 40.89 (1,2) 4.58 21 away from the point representing the original geometry. Lines of slope ±1 represent a constant misorientation between the two grains. 38.94 (4,7) 5.56 31 gap near each of these lines because creating very small 51.78 (2,11) 7.0 49 angle grain boundaries, or a geometry very close to sym- metric grain boundaries, requires adding flaws that are separated by large distances. We see gaps near other The coincidence site lattice model (CSL) describes high symmetry grain boundaries for the same reason. grain boundaries in terms of Σ, the inverse density of High symmetry grain boundaries are discussed in sec- lattice sites that are shared by the two grain orienta- tion III C. The lines radiating out from high symmetry tions when rotated about a common lattice point12. In grain boundaries represent adding single flaws to those 2D, CSL grain boundaries are necessarily commensurate high symmetry grain boundaries at larger and larger dis- and vice versa. For the triangular lattice, each common tances, as you approach the high symmetry boundary. lattice point for a given pair of grain orientations has a Each line represents a different type of flaw. The sym- repeat cell that is an equilateral triangle with one edge metry about the θ1 = 30 line is due to the fact that defined by the surface vector with length D. Σ is equal grains with tilt angles θ and 60 θ have the same re- − to the number of lattice points inside this cell and is peat distance. Therefore, the grain boundaries given by given by the area of the cell divided by the area of one the tilt angles (θ ,θ ) and (60 θ ,θ ) have the same 2 1 2 − 1 2 lattice triangle, Σ = D . While the CSL formulation overall repeat distance, though these are different grain gives a simple method for finding commensurate grain boundaries. boundaries, it is misleading to suggest that the actual coincidence of sites plays a physical role. In fact, shift- ing the grains in the y-direction so as to minimize the C. High Symmetry Grain Boundaries grain boundary energy (described in section II) generally causes the atoms to no longer coincide. Without this Certain grain boundary geometries have particularly shift, the two free surfaces will have atomic planes that low repeat distances. These grain boundaries have spe- meet at the same point. The elastic energy is lowered by cial properties. They mark the center of cusps in the staggering the dislocations20. Wolf has pointed out that grain boundary energy and discontinuous increases in the for 3D, the CSL formulation of grain boundaries involves fracture strength as a function of tilt angle. Table I show a redundant number of parameters18. examples of high symmetry grain boundaries and figure 7 Bishop and Chalmers introduced the concept of “struc- shows examples of geometries in between, which consti- tural units” - polygonal structures of atoms along high tute adding a single flaw per supercell repeat distance to angle grain boundaries13. We find that the patterns of the high symmetry grain boundary. Note how the added “structural units” are more relevant to the physical prop- flaws constitute a compromise between the two high sym- erties of the grain boundaries. When relaxed, the struc- metry geometries. Furthermore, a less high symmetry tural units of grain boundaries in 2D, triangular lattices grain boundary can be repeated, and again have a sin- are 5 atoms forming a regular pentagon or a pentagon 6
have found the surface lattice vectors of the two sides of the grain boundary to be a useful description. We sug- gest that the structure of high angle grain boundaries can best be described in terms of a dislocation model. The “extra flaws” added to create vicinal grain boundaries can be described as partial dislocations. Figure 7 shows how to find the Burger’s vectors for the added flaws by examining the flaw in the pattern of structural units.29 We will show in section V B that this model gives excel- lent agreement in the stress field due to the added flaws, outside the background of the original flaw structure. We will show in sections IV and V that the dislocation model provides the most powerful framework for understand- ing the geometry dependence of the properties of grain boundaries.
IV. GRAIN BOUNDARY ENERGY FIG. 7: Finding the Burger’s Vector for Flaws Along Vicinal Grain Boundaries. Grain boundaries with tilt an- gles in the vicinity of high symmetry grain boundaries are We have measured the grain boundary energy and peak comprised of the pattern of structural units of the high sym- stress for all symmetric grain boundaries with repeat dis- metry grain boundary with an element from the neighboring tances under 20 lattice constants and asymmetric grain high symmetry geometry. When defining the Burger’s vector boundaries with repeat distances under 30 lattice con- of a flaw added to a high symmetry grain boundary we are no stants. In order to explore the regions close to high sym- longer comparing the flaw to the perfect lattice but to the re- metry grain boundaries, we have added a few geometries peating pattern of structural units. Therefore, we must cross with longer repeat distances, close to the high symmetry the grain boundary at equivalent portions of the structural grain boundaries. unit in our reference loop and in our loop around the flaw. The energy30 associated with the series of flaws that The loop on the bottom is the reference loop around the re- gion of the grain boundary without the added flaw. The loop make up the grain boundary is defined in equation 2. on the top surrounds the added flaw, which we model as a As stated earlier, and found by several earlier stud- 7,9,16,17,18,19,22 partial grain boundary dislocation. The arrows indicate the ies , cusps appear at high symmetry bound- segments needed to make the top loop match the bottom loop; aries. For small angle grain boundaries, the grain bound- for example, the arrow on the upper right indicates that the ary energy has the form path around the defect is one atomic length shorter on that leg than the corresponding leg of the reference loop. The ar- µb eα rows sum to the Burger’s vector of the added grain boundary EGB = θ log θ log θ . (5) ◦ 4π(1 ν) | | 2π θ ∝ | | | | dislocation. This grain boundary has a tilt angle of 44.3 − | | and the added partial dislocation has a Burger’s vector with where µ is the shear modulus, b is the Burger’s vector, ν a length of 0.495 lattice constants and a direction along the is the Poisson ratio, and α is a factor that includes the negative x-axis. core energy20. One can now imagine the same scenario applied to high symmetry boundaries. We can take a high symmetry grain boundary and add or subtract flaws that is slightly stretched. Each high symmetry grain a distance d apart as shown in figure 8. By the same boundary is comprised of a simple pattern of pentago- reasoning as used for the low angle grain boundaries, the nal structural units (table I). When a high symmetry energy near the high symmetry grain boundary will have grain boundary is perturbed, an element of the pattern the form of structural units from the neighboring high symmetry grain boundary is introduced (figure 7). As the tilt an- µb eα′ EGB = E0 + θ θ0 log (6) gles move closer to the neighboring high symmetry grain 4π(1 ν) | − | 2π θ θ0 boundary, the flaws become closer together until they − | − | outnumber the original structure. The roles of flaw and where E0 is the energy of the high symmetry grain ′ original structure are then reversed. In this manner, com- boundary which occurs at the angle θ0 and α incorpo- binations of patterns of structural units can be used to rates the core energy of the flaw within the pattern of build up any commensurate, high angle grain boundary. flaws. We find the structural units a useful way of concep- The grain boundary energies for all of the symmetric tualizing the different grain boundary geometries. We grain boundaries that we have measured are shown in have not found the coincidence site lattice model useful in figure 9. Note that cusps occur at the angles listed in our investigations. In systematically constructing grain table I. We are able to fit the data for symmetric grain boundaries with small repeat distances (section III B), we boundaries to a function of the form 7
1 cos6θi log was chosen as a fitting function | | cos 6θ−cos 6θi| because it asymptotically gives a θ log(1/θ) shaped cusp in the near vicinity of θi and because it has the correct symmetry: even mirror symmetry at 0◦ and 30◦ and an overall period of 60◦. We use one term that is antisym- FIG. 8: Adding a Flaw to a High Symmetry Grain metric about θi and one term that is symmetric about θi Boundary. The lighter dislocations represent flaws added a so that we can fit the shape on either side of the cusp in- distance d apart, to an existing pattern of dislocations, shown dependently. We do not expect the slope of the curve on in black, with a short repeat distance. The added flaws can also move, screen, or cancel the flaws that make up the high either side of the cusp to be the same since the Burger’s symmetry boundary. vectors of the additional flaws for the geometries on either side of the high symmetry geometry may differ (figure 7).
The θi can be any angles that have the shortest re- peat distances, such as those given in table I. The curve in figure 9 is the result of fitting equation 7 to the data shown in the same figure. We have used ◦ ◦ ◦ ◦ θi = (49.10 , 43.89 , 40.89 , 38.94 ) since these angles have particularly prominent cusps and short Miller in- dices (given in table I). Three smooth sinusoidal terms were used in the final sum over cos 6jθ. The result is analogous to a devil’s staircase, with a cusp singularity at each angle that corresponds to a spe- cial rational number. In principle, the energy has a loga- rithmic cusp at a dense set of points but (as is typical for devil’s staircases) the high-order cusps rapidly diminish in size.
FIG. 9: Grain Boundary Energies for Symmetric Coefficients for Fitting Grain Boundary Geometries. Cusps appear at high symmetry grain bound- TABLE II: Energy to Eq. 7. aries (listed in table I) and have the same θ log θ shape as the (s) (a) (s) (a) energy of low angle grain boundaries. The line is the fit given i ai ai bi bi − by equation 7. Notice that hints of smaller cusps are visible 1 0.76 7.4 × 10 4 0.196 13.6 in the data. − 2 0.68 -0.023 0.292 9.19 × 10 7 3 0.22 −8.9 × 10−3 0.104 4.76 4 0.44 0.039 0.154 3.93 × 10−3
b0 a0 a30 b0 b30 EGB(θ) = a sin3θ log + 0| | sin3θ 1.83 1.65 1.72 4.14 | | b c1 c2 c3 d + a sin 3(θ 30) log 30 30| − | sin 3(θ 30) -0.14 1.71 -0.070 0.094 n | − | (s) + a cos6θ cos6θi i | − | i=0 X The results for the energy and peak stress of all of b(s) log i the grain boundaries we have simulated are shown in fig- cos6θ cos6θi | − | ure 10. As with the symmetric grain boundaries, we see (a) energy cusps for grain boundaries with particularly low + a (cos6θ cos6θi) i − (a) repeat distances. For example, one of the cusps in fig- bi ure 10 is centered around the grain boundary with tilt log ◦ ◦ cos6θ cos6θi angles 24.79 and 3.00 which has a repeat distance of | − |! m 9.53 lattice constants. Figure 11 shows geometries in the neighborhood of this high symmetry boundary with a + cj cos(6jθ)+ d j=0 constant misorientation. We see a similar, θ log θ shape X to the cusp centered around the high symmetry geom- The first two terms fit the cusps at 0◦ and 30◦, where etry, suggesting that the same concept of adding flaws the cusps are symmetrical about their respective center to the short-repeat distance grain boundary applies to points. The next set of terms in the sum fit the cusps asymmetric 2D grain boundaries along constant misori- at high symmetry tilt angles. The function cos6θ entation lines. | − 8
location per repeat distance after it is relaxed, we must choose surface vectors that have a difference equal to a basis vector and have the same repeat distance. One pair of surface vectors is (2n +1, n) and (2n +1, n 1). This gives a symmetric grain boundary− with a single− − dis- location with tilt angles close to 0◦, a Burger’s vector equal to (0, 1) and a repeat distance of √3n2 +3n + 1. Our simulations show that such grain boundaries fail via intragranular fracture rather than intergranular fracture as shown in figure 12.
FIG. 10: Grain Boundary Energies for Asymmet- ric Geometries. The surface and contour represents a FIG. 12: Intragranular Fracture for Low Angle Grain smooth interpolation between the particular geometries that Boundaries. The figure on the left shows a grain boundary have been simulated, represented by the black dots in the ◦ with symmetric tilt angle 0.81 at 0 strain. The same grain θ θ plane. Perfect crystals have zero grain boundary energy, 1 2 boundary is shown on the right with a strain of 3.125%. Sym- shown by the groove along the θ = θ line. ◦ 1 2 metric low angle grain boundaries centered around the 0 ori- entation (Miller indices (0, 1)) fail via intragranular fracture rather than intergranular fracture.
◦ The apex of the triangle θ1 = θ2 = 30 in figure 6 is a 1.15 perfect crystal oriented at 30◦, which is the end point of a series of symmetric tilt boundaries. At exactly θ1 = θ2 = 30◦, there is an abrupt jump up in fracture strength since 1.1 the perfect crystal has no “extra dislocations” at which to nucleate fracture. Our simulations find a peak stress of 4.31 (Lennard-Jones Units) for the perfect crystal. ◦ 1.05 The low angle grain boundaries near the 30 lattice orientation have surface vectors (1,n) and (n, 1), repeat distances √1+ n + n2, and total Burger’s vector ( 1, 1), − Grain Boundary Energy (Lennard-Jones Units) which splits into two flaws with Burger’s vectors (0, 1) 1 22 23 24 25 26 27 28 and ( 1, 0) shown in figure 13. For wide enough sim- θ 1 ulations,− these dislocations glide in opposite directions until they are restricted by the constrained zones on ei- FIG. 11: Asymmetric Grain Boundary Energies in ther side. For narrower simulations, the dislocations do Neighborhood of High Symmetry Geometry. The grain not glide but form nucleation points for grain boundary boundary geometries considered in the above plot each have ◦ a misorientation of 21.79 and are in the vicinity of the high fracture causing an abrupt jump down in the peak stress symmetry grain boundary with tilt angles 24.79 and 3.00. The compared to the peak stress of the perfect crystal. lines are fit to θ log θ for the points on either side of the cusp. For narrow simulations of low angle grain boundaries in this region, we find that the peak stress has a parabolic dependence on angle, shown in figure 14. We can explain this parabolic dependence as due to a partial screening of V. FRACTURE STRENGTH the external stress σext by the neighboring dislocations on the grain boundary. Assume that the dislocation has a critical stress for nucleating fracture equal to σc. The A. Low Angle Symmetric Grain Boundaries dislocation feels a stress due to its neighboring disloca- tions, each a distance d apart, in addition to the external, The Frank conditions state that the total Burger’s vec- applied stress. The total stress felt by each dislocation tor for the dislocations making up a low angle grain can be written boundary is equal to the difference of the surface vec- ∞ a tors that define the orientation of each grain. In order to σ = σ + n (7) ext dn guarantee that the grain boundary will have only one dis- neigh. disloc. n=0 X X 9
µb x(x2 y2) σ (x, y) = − (10) xy 2π(1 ν) (x2 + y2)2 − where the x-direction is the direction of the Burger’s vec- tor. Since the Volterra solution is odd, the n = 1 term of the stress at each dislocation vanishes as we sum over the neighboring dislocations on either side. The first nonva- nishing term in equation 7 is the n = 2 term which has three contributions. (1) The n > 1 terms are the mul- tiple expansions of the stress field23 as well as nonlinear terms. (2) The nonlinear term in strain field has the 2 FIG. 13: Intergranular Fracture for Low Angle Grain form du/dx du/dx, giving a power law of 1/r . (3) ∗ Boundaries. The first figure shows a grain boundary with Geometrical restrictions cause some grain boundaries to symmetric tilt angle of 30.96 at 0 strain. The second figure have flaws unequally spaced in the y-direction (though shows the same grain boundary at 3.625% strain. In this for the results given in figure 14 we have only explored wider simulation, the dislocations glide apart (thin diagonal geometries with equally spaced flaws). The grain bound- white stripes). The third and fourth figures shows a narrower aries geometries used in figure 14 do have flaws that are simulation of the same tilt angle where the intergranular frac- not aligned perfectly in the x-direction. In each of these ture nucleates at each dislocation. cases, shifting the dislocation constitutes adding a dislo- cation dipole (adding one positive and one negative, can- celing a dislocation and adding a new one), and therefore is another contribution to the 1/r2 term. The external
2.3 stress needed to produce a stress equal to σc at each flaw is then
2.2 a2 2 σpeak = σc = σc A(θ θ ) (11) − d2 − − 0 2.1 where a2 combines the three contributions described above. This parabolic dependence on θ 30◦ is depicted 2 on the left-hand side of figure 15. −
1.9
Peak Stress (Lennard-Jones Units) B. High Angle Grain Boundaries 1.8 30 30.5 31 31.5 Symmetric Tilt Angle Figure 15 shows the results of the peak stress mea- surements for high angle symmetric grain boundaries. At the same points for which we had cusps in energy, FIG. 14: Peak Stress vs. Tilt Angle For Low An- gle Grain Boundaries. The peak stress has a large jump we have discontinuous increases in fracture strength. By ◦ downward as soon as a grain boundary deviates from θ = 30 drawing the same analogy between adding dislocations (perfect crystal). Here the peak stress for the perfect crystal, to perfect crystals and adding flaws to high symmetry ◦ at the 30 orientation is 4.31 (Lennard-Jones units), which boundaries as described in figure 8 we can understand would require a vertical scale 11 times as big. The disloca- the discontinuities in the fracture strength at high sym- tions forming the boundary act as nucleation sites for frac- metry grain boundaries and the angular dependence of ture no matter how far apart they are. After this jump the fracture strength near the high symmetry geometries. peak stress has a parabolic dependence on angle for low angle For high angle grain boundaries, the added flaw is no grain boundaries that are constrained by width to fracture in longer the sole nucleation site for fracture and fracture qualitatively similar ways. Fracture nucleates exactly at the does not necessarily nucleate in the core of the added dislocation and the first non-vanishing term in the stress at flaw. The added dislocation creates a stress field given this point due to neighboring dislocations goes as 1/d2, where d is the distance between dislocations. roughly by the Volterra solution (equation 10) with a pos- itive stress on one side, negative stress on the other, and a singularity at the center shown in figure 16. The stress field differs slightly from the Volterra solution because where σext is the external stress. The n = 1 term is the Volterra solution given by the elastic constants of the material at the grain bound- ary vary from those of the perfect crystal. The fracture nucleates along the boundary in the region where the µb y(3x2 + y2) σxx(x, y) = (8) stress due to the added flaw is positive. Because frac- −2π(1 ν) (x2 + y2)2 − ture does not nucleate at the center of the added flaw, µb y(x2 y2) the Volterra solution as summed over the neighboring, σ (x, y) = − (9) yy 2π(1 ν) (x2 + y2)2 added flaws does not cancel at the nucleation site. This − 10
2.1
2.05 2.5
2
1.95
2 1.9
1.85 Peak Stress (Lennard-Jones Units) Peak Stress (Lennard-Jones Units)
1.5 1.8 30 35 40 45 50 55 60 47 48 49 50 51 52 53 Symmetric Tilt Angle Symmetric Tilt Angle
FIG. 15: Peak Stress vs. Tilt Angle For High Angle FIG. 17: Peak Stress vs. Tilt Angle Near High Sym- Grain Boundaries. The peak stress as a function of tilt metry Grain Boundaries. There is a discontinuity in peak angle is discontinuous everywhere, with higher values at spe- stress at tilt angles close to high symmetry grain boundaries. cial tilt angles representing high symmetry grain boundary The plot above shows the peak stress for the grain bound- geometries. The dependence of peak stress on angle near the ary with tilt angle 49.1 (described in table I) and the nearby ◦ high symmetry grain boundaries (parabolic near 30 , linear geometries. near 49.11◦) depends on the structure of the additional flaws that make up the nearby geometries.
µ(3 2√3y)(θ0 θ) 3 − − + O((θ0 θ)) (13) ≈ 2πy(1 ν)(y √3) − leads to a linear law for fracture strength as a function − − of tilt angle, for grain boundaries near high symmetry where θ0 is the tilt angle of the high symmetry grain geometries. boundary. We need to look at simulations in the fixed displacement (narrow width) regime in order to observe where fracture nucleates.31 We find that for geometries with this pattern of flaws, fracture nucleates at the same distance above the added flaw. The external stress needed to nucleate fracture at a distance y from the added flaw, along the grain boundary is then approxi- mately
µ(3 2√3y)(θ0 θ) σpeak = σc − − (14) − 2πy(1 ν)(y √3) − − FIG. 16: Stress Fields Due to Dislocations. The figure which explains the linear dependence on angle shown in on the left shows the stress fields surrounding two dislocations figures 16 and 17. according to formula 10. The figure on the right shows the The results for the peak stress of all grain bound- stress fields surrounding the added dislocations as calculated aries with repeat lengths less than 30 lattice constants according to the virial definition of atomic stress24. are shown in figure 18. We also note spikes in peak stress for grain boundaries with particularly low repeat Consider the grain boundaries with tilt angles ranging distances. Consider again the grain boundary with tilt from 49.39◦ to 53.41◦, which are close to the high symme- angles 24.79◦ and 3.00◦ which has a repeat distance of try grain boundary at 49.11◦. The additional flaws that 9.53 lattice constants. Figure 19 shows the peak stress characterize these grain boundaries have Burger’s vectors of this grain boundary and grain boundaries in its neigh- equal to ( √3 sin θ, sin θ) and ( √3 sin θ, sin θ), where borhood, along a line of constant misorientation. θ is the symmetric− tilt− angle of the− lattice. The norms are 2 sin θ and the angles are 210 and 150◦. For these geometries,| | the xx component of the stress field (due to VI. CONCLUSION two dislocations a distance D b/2(θ θ)) along the ≈ 0 − y-axis is We have systematically explored the space of 2D grain boundaries, the patterns of structural units that make µ(2√3y(θ0 θ) 3 sin(θ0 θ)) sin(θ0 θ) up short repeat distance grain boundaries, and the flaws σxx(y) = − − − −(12) 2π(1 ν)(y(θ θ) √3 sin(θ θ)) in the patterns of structural units that make up vicinal − 0 − − 0 − 11
tilt angle dependence of peak stress near special grain boundaries. In principle, with enough computer time it would be possible to conduct a similar study for the 5D space of idealized 3D grain boundaries. We suspect that the per- fect crystal analogy could also explain the cusps in energy and discontinuous spikes in peak stress for special 3D grain boundaries. The geometry dependence of energy and peak stress surrounding these special grain bound- aries will be more complicated since the grain boundaries can have a pure tilt, pure twist, or a mixed type of geome- try. In real polycrystals, the fracture strength and energy are further complicated by impuritites at the interface, emission of dislocations during fracture, and more com- plex geometries such as triple junctions of grains, which are often the nucleation site for fracture. Systematic studies of such systems are infeasible and better suited FIG. 18: Grain Boundary Peak Stresses for Asym- by on the fly simulations of local regions of interest. metric Geometries As with figure 10, the surface and con- tours represent a smooth interpolation between particular ge- ometries. Due to the interpolation, each high stress peak is Acknowledgments rendered as a finite-width cone. The peak stress of perfect crystals is much higher than that of grain boundaries. In or- der to use a smaller vertical scale, we have not included these This work was supported by NSF Grants No. data points which would otherwise show a ridge along the ITR/ASP ACI0085969 and No. DMR-0218475. We diagonal. also wish to thank Nicholas Bailey, Drew Dolgert, Gerd Heber, Anthony Ingraffea, Surachute Limkumnerd, Chris Myers, and Paul Wawrzynek. 2
1.8
1.6
1.4
1.2 Peak Stress (Lennard-Jones Units)
1 22 23 24 25 26 27 28 θ 1
FIG. 19: Asymmectric Grain Boundary Peak Stresses In Neighborhood of High Symmetry Geometry. The grain boundary geometries plotted above are the same as those in figure 11, along a line of constant misorientation in the vicinity of a high symmetry geometry. The lines show a possible linear fit to the points on either side of the low repeat distance grain boundary.
grain boundaries. We have shown that the patterns of structural units are key to understanding the singularities in energy and peak stress at special grain boundaries by drawing an analogy to perfect crystals. We have used this insight to find a functional form for the energies of 2D symmetric grain boundaries and to understand the 12
1 A. Needleman, Journal of the Mechanics and Physics of level structure and properties (Chapman & Hall, 1992), Solids 38, 289 (1990). chap. Special Properties of Σ grain boundaries., pp. 190– 2 M. Falk, A. Needleman, and J. Rice, in 207. Pro-ceedings of the 5th European Mechanics 23 N. Bailey, Ph.D. thesis, Cornell (2002). of Materials Conference (Delft, 2001), URL 24 M. P. Allen and D. J. Tildesley, Computer Simulation of http://citeseer.ist.psu.edu/490781.html. Liquids (Oxford Science Publications, 1987). 3 E. Iesulauro, A. R. Ingraffea, S. Arwade, and P. A. 25 Y. I. Frenkel and K. T., Zhurnal tekhnicheskoi fiziki 8, Wawrzynek, in Fatigue and Fracture Mechanics, edited by 1340 (1938). W. G. Reuter and R. S. Piascik (American Society for Test- 26 For cracks propagating through a grain boundary, the di- ing and Materials, West Conshohocken, PA, 2002), vol. 33. rection of the crack front in relation to the structure of the 4 E. Iesulauro, A. R. Ingraffea, G. Heber, and P. A. grain boundary may affect the fracture toughness, but this Wawrzynek, in 44th AIAA/ASME/ASCE/AHS Struc- dependence is not included in cohesive laws. tures, Structural Dynamics, and Materials Conference 27 Another alternative is to use a layer of constrained atoms. (AIAA, Norfolk, VA, 2003). Typically, in a finite element simulation of a subsection of 5 D. H. Warner, F. Sansoz, and J. F. Molinari, International a material, the nodes on the surfaces of the model are con- Journal of Plasticity 22, 754 (2006). strained to not move in a direction perpendicular to the 6 V. R. Coffman, J. P. Sethna, G. Heber, A. Liu, A. Ingraf- surface. This is known as “rollered” boundary conditions. fea, and E. I. Barker (2007), in preparation. We can imitate these boundary conditions in an atomistic 7 S. P. Chen, D. J. Srolovitz, and A. F. Voter, Journal of simulation by constraining a layer of atoms at the surface Materials Research 4, 62 (1989). or edge of the simulation to not move perpendicular to that 8 F. Sansoz and J. F. Molinari, Scripta Materialia 50, 1283 surface or edge. This will partially ameiliorate the edge ef- (2004). fects, suppress the Poisson effect, and has the advantage 9 F. Sansoz and J. F. Molinari, Acta Materialia 53, 1931 of allowing us to simulate grain boundaries of any geome- (2005). try6. We have compared simulations of different sizes with 10 O. A. Shenderova, D. W. Brenner, A. Omeltchenko, X. Su, rollered boundary conditions in the y-direction to a simu- and L. H. Yang, Phys. Rev. B 61, 3877 (2000). lation of the same geometry, but with periodic boundary 11 D. E. Spearot, K. I. Jacob, and D. L. McDowell, Mechanics conditions in the y-direction. We see that if we use rollered of Materials 36, 825 (2004). boundary conditions, we need a simulation that is 48 times 12 W. Bollmann, Crystal Defects and Crystalline Interfaces longer in the y-direction in order to converge to within 2% (Springer-Verlag, New York, 1970). of the results found by a simulation with periodic boundary 13 G. H. Bishop and B. Chalmers, Scripta Metallurgica 2, 133 conditions. Hence, for computational efficiency, we choose (1968). to use periodic boundary conditions in the y-direction. 14 Y. Ishida, Grain Boundary Structure and Properties (Aca- 28 The Frenkel-Kontorova model25 describes a one dimen- demic Press Inc., London, 1976), chap. The Bubble Raft sional chain of atoms, connected by springs, subject to an as a Model for Grain Boundary Structure, pp. 93–106. external, sinusoidal potential. If the relaxed length of the 15 D. Wolf and K. L. Merkle, Materials Interfaces: Atomic- spring and the width of the potential wells are commen- level structure and properties (Chapman & Hall, 1992), surate a minimum energy state exists. For the incommen- chap. Correlation between the structure and energy of surate case, there will be a pinned phase with many local grain boundaries in metals, pp. 88–150. minima and an unpinned phase, depending on the depth 16 R. J. Harrison, G. A. Bruggeman, and G. H. Bishop, Grain of the wells in relation to the stiffness of the springs. Boundary Structure and Properties (Academic Press Inc., 29 Since this Burger’s vector may be a sum of lattice vectors London, 1976), chap. Computer Simulation Methods ap- from either side of the grain boundary, the total may not plied to Grain Boundaries, pp. 45–91. be a full translation vector of the triangular lattice. 17 P. H. Pumphrey, Grain Boundary Structure and Properties 30 We use ’energy’ for ’energy per unit length’ throughout. (Academic Press Inc., London, 1976), chap. Special High 31 For simulations of a few angstroms wide, the crack opens Angle Grain Boundaries, pp. 139–200. up along the grain boundary with a slow unzipping mech- 18 D. Wolf, Materials Interfaces: Atomic-level structure and anism, starting at particular flaws. For wider simulations, properties (Chapman & Hall, 1992), chap. Atomic-level ge- the crack snaps open, soon after a peak stress is reached. ometry of crystalline interfaces, pp. 1–52. The slow unzipping of the narrow simulations is charac- 19 D. Wolf and J. A. Jaszczak, Materials Interfaces: Atomic- teristic of fixed displacement boundary conditions - since level structure and properties (Chapman & Hall, 1992), the displacements of a region close to the interface is fixed, chap. Role of interface dislocations and surface steps in the opening of the crack is controlled, allowing us to see the work of adhesion, pp. 662–690. the details of how the crack opens. The snapping open, 20 J. P. Hirth and J. Lothe, Theory of Dislocations (John seen with a wider simulation, is characteristic of fixed force Wiley & Sons, New York, 1982). boundary conditions. In the wider simulations, the mate- 21 N. Bailey, T. Cretegny, J. P. Sethna, V. R. Coffman, A. J. rial on either side of the grain boundary acts as a spring, Dolgert, C. R. Myers, J. Schiotz, and J. J. Mortensen, Dig- which effectively softens the boundary conditions, approx- ital material: a flexible atomistic simulation code (2006), imating fixed-force boundaries for long lengths. When the URL http://arxiv.org/abs/cond-mat/0601236. constrained region is far away, the interface snaps open 22 G. Palumbo and K. T. Aust, Materials Interfaces: Atomic- once it reaches the maximum stress that it can sustain.