Dislocation-Controlled Formation and Kinetics of Grain Boundary Loops In

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Roel and , 1073/pnas.1804352115/-/DCSupplemental oioe sn pia ie irsoy(3 ( (33) microscopy free is video loop then GB optical is the using grain of is vor- monitored inner evolution misorientation the subsequent the The and as desired forces. off external long the from turned as When is rotates vortex S1). the grain (Movie reached, inner applied The is S1). tex Fig. shown and as 2 a and tweezing of 1 optical portion Fig. holographic circular in by opti- an a generated with vortex rotating counterclockwise cal grain) by (inner created crystal colloidal are 2D loops GB The Loops GB of Creation lattices. hexagonal visualize 2D of directly in structure loops dislocation to GB and us kinetics allow shrinkage time time-resolved (21–29) and cova- the crystals length strong colloidal accessible the of conveniently to scales due The stable (20). are bonding The which to lent size. graphene, contrast in in and loops spontaneously misorientation shrink GB loops loops well-defined GB GB colloidal a create obtained with to vortices demand, optical on using them manipulate and in challenging remain GB misorientation of the structure systems. of these dislocation function sys- and a 19), kinetics as (2, loops the graphene of electron in by studies achieved driven tematic recently shrinkage been their has and loops irradiation GB of creation the ntesrnaekntc oprdwt Eq. with compared kinetics anomalies shrinkage with together the simulations, in in found been have 16) (15, aiu ioinaini eaoa rsa,adteinitial the and crystal, hexagonal radius, a in misorientation maximum misorientation, initial hsatcecnan uprigifrainoln at online information supporting contains article This 1 the under Published Submission. Direct PNAS a is article This interest. of conflict no declare authors The wrote R.P.A.D. and F.A.L. and protocols; paper. results; and the the stage interpreted R.P.A.D. performed tweezing and optical A.C. D.G.A.L.A., the A.C., and F.A.L., developed F.A.L. A.C. research; data; analyzed designed F.A.L. R.P.A.D. research; and F.A.L. contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To op n eaei otecmlxdsoainsrcueand structure experiments. our dislocation in complex visualize directly the we to which reactions, it boundary relate grain and of other- kinetics this loops shrinkage while measure the We form, from restored. value to elastically patch the universal loop is on the boundary deformation only of the grain depends wise a size that for the value type universal the and lattice a rotation that exceed of find to of angle needs we monolayer the tweezers, crystalline of optical a product using deforming By particles defects of. the colloidal and made of boundaries are properties topological grain they the loop-shaped to due resulting is the always This not bicrystal. does a crystal to large lead a inside patch circular a Twisting Significance ee eepotteihrn oteso olia crystals colloidal of softness inherent the exploit we Here, R 0 A–C iswti h range the within lies , 3–2 frdtis see details, (for (30–32) NSlicense. 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APPLIED PHYSICAL SCIENCES A BCHolographic tweezing Hologram

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Fig. 1. GB loop formation and elastic-to-plastic deformation crossover. (A) Creation of a GB loop in a colloidal crystal by rotation of a circular grain counterclockwise (arrow), using an optical vortex. (Scale bar, 25 µm. This scale bar also applies to D and E.) (B) Schematic of the sample cell containing the 2D colloidal crystal, mounted on a holographic optical tweezing and video-microscopy setup. (C) Schematic of the hologram, phase front, intensity proﬁle, c and direction of propagation of the optical vortex used to create the GB loop. (D) Shrinkage of a GB loop of initial misorientation θ0 > θ , after the vortex c is turned off. The local crystal orientation is indicated by the color bar below E. (E) Same as in D in the case θ0 < θ , where the deformation is restored by grain rotation clockwise (arrow). (F) Representative particle trajectories (labeled 1, 2, and 3) color coded according to the time elapsed during the exposure to the vortex (period ON to OFF on the color bar) and the subsequent evolution of the grain (period OFF to END on the color bar) in the case of plastic c c deformation (θ0 > θ ). (G) Same as in F but now in the case of elastic deformation (θ0 < θ ).

spacing. Crucially, the use of optical vortices enables us to cre- the initial misorientation, θc , which marks a crossover between c ate relatively large GB loops in comparison with a previously plastic deformation followed by shrinkage for θ0 > θ and elastic c used approach based on single-particle trapping (32). Moreover, deformation followed by rotation at constant radius for θ0 < θ . a recently developed method based on “optical blasting” (34) enables the creation of arbitrarily shaped GB loops but without Kinetics of Shrinkage control of the misorientation, which is a key parameter in the We now characterize the shrinkage kinetics of the GB loops cre- c structural stability and kinetics of GB loops. ated by plastic deformation (θ0 > θ ). To this end, we monitor Fig. 1D shows the crystal orientation maps (for details see SI the time evolution of the area A = πR2 of GB loops (for details Appendix, section 3 and Fig. S2) corresponding to the evolution see SI Appendix, section 3 and Fig. S2) with R0 = 10.8 ± 0.5σ for ◦ ◦ ◦ of a GB loop with θ0 = 16.4 and R0 = 10.3σ after the vortex is initial misorientations θ0 in the range 9 −28 (Fig. 2A), well c ◦ turned off. For this relatively large initial misorientation, a GB above the critical value θ ' 5 for this R0. In all cases, A(t) loop is obtained, which then spontaneously shrinks until a sin- decreases monotonically and, importantly, it is clearly observed gle crystal is left (Movie S1). The trajectories of the particles that the higher the initial misorientation is, the longer the shrink- confirm that after complete shrinkage, they are located at dif- age takes. To test whether the GB loop shrinks at constant rate, ferent lattice sites than before the deformation (Fig. 1F). This we integrate Eq. 1 as A/A0 = 1 − t/tf , where the total shrinkage ∗ irreversibility indicates that the deformation induced by the vor- time, tf = A0/(2πM ), is directly measured from the data in Fig. tex is plastic. Surprisingly, when we repeat the experiment with a 2A. Plotting A/A0 vs. t/tf shows a good collapse of the data (Fig. ◦ misorientation of θ0 = 4.5 , the grain merely rotates back clock- 2A, Inset), which indicates that the shrinkage occurs at a constant wise with a fixed radius, until merging with the outer crystal (Fig. rate and is thus effectively characterized by a single kinetic coef- 1E). The trajectories of the particles clearly confirm that they ficient, the reduced mobility M ∗. Its variation as a function of the return to their original crystal sites (Fig. 1G). This reversibility initial misorientation θ0 is shown in Fig. 2B, where it can be seen shows that the deformation is elastic in this case. These obser- that M ∗ decreases upon increasing initial misorientation, consis- vations thus show that for a given R0, there is a critical value of tent with Fig. 2A. This trend is particularly notable as it is the

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APPLIED PHYSICAL SCIENCES AB

Fig. 3. Dislocation origin of elastic-to-plastic deformation crossover. (A, Left) Schematic view of an isolated dislocation induced by the insertion of two semiinfinite crystal lines (gray) with its Burgers vector ~b expressed in crystal coordinates (blue and red arrows). (A, Center) Schematic of the simplest straight GB configuration, which contains identical dislocations with their Burgers vectors along the normal to the GB (~n) and spaced by σ/θ.(A, Right) Schematic of the simplest GB loop configuration, which consists of six equally spaced dislocations with their Burgers vectors along the local normal (~n). (B) Examples of GB loops obtained in the experiments, with Rθ near (Left) and well above (Right) [Rθ]c.(C, Left) Local view of a portion of GB loop (dashed line) during c rotation of the grain by the optical vortex (white arrow). The dislocation structure is shown in the elastic regime (R0θ0 < [Rθ] ; C, Left), at the crossover c c (R0θ0 = [Rθ] ; C, Center), and in the plastic regime (R0θ0 > [Rθ] ; C, Right).

[Rθ]c , so that the GB loop is composed of many closely spaced Dislocation Reactions dislocations. Finally, we show that grain shrinkage is achieved by complex dis- We now elucidate how the elastic-to-plastic crossover origi- location reactions (13, 32) that we directly visualize for a GB nates from the dislocation structure of GB loops. In the case loop with R0θ0 = 1.7σ in Fig. 4 and Movie S3. Dislocation reac- of elastic deformation, the initial misorientation of a GB loop P tions must preserve the total Burgers vector ~bi while lowering with a given initial size is so small that Eq. 3 is violated; i.e., i the elastic strain energy (7, 13, 32). We first observe that the R0θ0 < 3σ/π. Therefore, the dislocation spacing σ/θ is larger number of dislocations along the GB loop can be reduced by a than πR/3 and as a result, no GB loop can be formed as the dis- so-called dislocation recombination (Fig. 4A and Movie S4). For locations cannot be fitted on it. The elastic deformation is then ~ ~ ~ ~ marked by the dissociation of pairs of dislocations with opposite two neighboring dislocations with |b1| = |b2| = σ and |b1 + b2| > Burgers vectors parallel to the GB loop (Fig. 3C) and not per- σ, as is the case for the flower-defect–like configuration in Fig. pendicular as required for a stable GB. Hence, once the vortex is 4A, recombinations do not occur due to the large increase in the turned off, the inner grain rotates back until the original crystal is elastic strain energy. Instead, we observe a second process con- recovered. This is simply achieved via the glide [motion parallel sisting of a dislocation dissociation (Fig. 4B and Movie S5) and to the Burgers vector (7)] of dislocations with opposite Burgers two subsequent recombinations between the dislocations result- vector toward each other. In the case of plastic deformation, the ing from the dissociation and their neighboring dislocations (Fig. misorientation of a GB loop with a fixed size is large enough so 4C). Note that glide allows the dislocations to meet their neigh- that Eq. 3 is satisfied, i.e., R0θ0 ≥ 3σ/π. In this case, dislocations bors. Similar dislocation reactions occur until a defect structure are created such that the sum of their Burgers vectors is locally corresponding to a pair of dislocations with opposite Burgers perpendicular to the GB loop (Fig. 3C), and hence a stable GB vectors, analogous to the Stone–Wales defect in graphene (20, loop is formed. This corresponds to a plastic deformation as the 37, 38), is reached (Fig. 4D). The perfect crystal is recovered original crystal can be recovered only via grain shrinkage. by a third elementary process, reminiscent of the Stone–Wales It is important to stress that [Rθ]c = 3σ/π is a general lower transformation (20, 39), namely a simple edge rotation of 90◦ in bound that solely stems from the hexagonal lattice structure and the Voronoi diagram (Fig. 4D), topologically equivalent to the the loop shape of the GB. It means that no GB loop with Rθ bond rotation in graphene (20, 37–39). While in graphene this is below 3σ/π can be created by any continuous local deforma- energetically very costly (20), in our colloidal crystal it simply cor- tion of any 2D hexagonal crystal. In colloidal crystals, however, responds to the two fivefold coordinated (sevenfold coordinated) we have shown that this lower bound has a clear dynamic signa- particles moving toward (away from) each other, as illustrated ture in that it manifests itself as a diverging reduced mobility of in Fig. 4D. This shows how very small particle displacements shrinking GB loops. in colloidal crystals lead to topological changes in the lattice to

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Fig. aegee al. et Lavergne .vndrMe ,e l 21)Hgl oprtv tesrlxto ntwo-dimensional in relaxation stress cooperative Highly two-dimensional (2014) al. other et and B, Meer graphene der van Polycrystalline 9. (2014) YP Chen OV, Yazyev 8. (2002) DR Nelson (2001) 7. DJ shape, Grain Bacon (2004) D, DJ Srolovitz Hull M, Haataja 6. MI, Mendelev A, Karma boundaries. AE, grain Lobkovsky idealized 5. of motion Two-dimensional (1956) WW Mullins 4. temperature (1997) high situ JM In Howe (2016) JH 3. Warner graphene. AW, in Robertson Q, loops Chen K, boundary He Grain C, Gong (2011) 2. al. et E, Cockayne 1. B A otclodlcrystals. colloidal soft materials. UK). Cambridge, Press, Univ Oxford). relation. Herring the and mobility boundary grain Phys Interfaces Solid-Solid and Solid-Liquid Solid-Vapor, of graphene. Kinetics in loops boundary grain closed large 10:9165–9173. of studies level atomic 195425. systems. ∗ t=177s t=407s oeydpnson depends solely 27:900–904. w ilcto eobntoswt h orsodn aac fBresvcosfraGB a for vectors Burgers of balance corresponding the with recombinations dislocation Two (A) loops. GB of shrinkage during reactions Dislocation R hsRvB Rev Phys 0 a Nanotechnol Nat recombinations θ 0 Dislocation = 1.7σ nefcsi aeil:Aoi tutr,Temdnmc and Thermodynamics Structure, Atomic Materials: in Interfaces eet n emtyi odne atrPhysics Matter Condensed in Geometry and Defects 26:6239–6253. rcNt cdSiUSA Sci Acad Natl Proc (0,-1) ( ilcto iscaineetfloe ytegieo h rdcsto products the of glide the by followed event dissociation dislocation A ( B) shrinkage. of start the from counted is Time S3). Movie Dislocation dissociation (-1,0) nrdcint Dislocations to Introduction 9:755–767. R ilcto arta niiae olwn rcs analogous process a following annihilates that pair dislocation A (D) shrinkage. further enable to recombinations Subsequent ) 0 θ 0 hsdmntaigta h kinet- the that demonstrating thus , + (-1,1) (1,-1) (1,-1) t=412s + 111:15356–15361. + (1,0) ◦ caMater Acta (-1,0) dertto nteVrnidarm hc sahee ysalatprle atcedisplacements. particle antiparallel small by achieved is which diagram, Voronoi the in rotation edge (0,-1) (Butterworth-Heinemann, Wly e York). New (Wiley, (0,1) 52:285–292. hsRvB Rev Phys t=247s (Cambridge C Nano ACS Appl J neighbours 83: Glide to 0 iL ec ,RbrsnJ(05 eeteege fgaht:Density-functional graphite: of energies Defect (2005) J Robertson in S, migration and Reich boundary motion L, grain of Li boundary observation 20. grain Atom-by-atom (2012) on al. geometry et S, of Kurasch influence 19. sphere-forming The two-dimensional (2008) in N mechanisms Bernstein Ordering 18. (2005) al. et DA, grain Simultaneous Vega (2006) W 17. Carter J, polycrystalline Warren for A, Lobkovsky framework D, theoretical Srolovitz M, Unified Upmanyu (2013) 16. A Karma Y, grain Xu nanocrystalline of A, simulations Adland crystal 15. field Phase by (2012) studied PW rotation Voorhees grain KA, and Wu migration con- 14. boundary Grain thermal (2012) the Y on Mishin loop ZT, Trautt boundary grain 13. a of Effects solid (2013) electron al. two-dimensional et the N, in Khosravian Defects (1979) 12. R Morf BI, Halperin DS, Fisher 11. upr ECSatn rn 279541-IMCOLMAT). financial Grant for Starting acknowledged (ERC is support (ERC) Council these Research European engineer- of The cussions. properties in the ACKNOWLEDGMENTS. help thereby may and the content loops controlling loop GB materials. addition, mathemati- GB of In the with kinetics ing 40–42). materials and (8, 2D formation structures other colloidal We dual to hexagonal formation misorientation. in cally loop relevant structure GB and are dislocation of size their crystals topology fully to the initial relation on are their in results loops quantified our of GB that as product believe structure, of dislocation the kinetics initial by shrinking their the by determined that show also C D calculations. graphene. rotation. copolymers. block rotation. grain and migration boundary evolution. pattern dimensions. two in growth dynamics. molecular study. dynamics molecular A 135. graphene: of ductivity melting. for implications and t=421s (-1,0) caMater Acta aoLett Nano (0,0) hsRvB Rev Phys (1,0) hsRvE Rev Phys hsRvLett Rev Phys caMater Acta 56:1106–1113. 12:3168–3173. t=452s t=448s (-1,0) 72:184109. etakPu hii n ui s o sfldis- useful for Isa Lucio and Chaikin Paul thank We caMater Acta 71:061803. hsRvB Rev Phys + (-1,0) 110:265504. 60:2407–2424. (1,-1) 60:407–419. + caMater Acta 20:4692–4712. (1,-1) (0,-1) 90° NSLts Articles Latest PNAS 54:1707–1719. optMtrSci Mater Comput (0,-1) t=432s | 79:132– f6 of 5

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