Dynamical processes of interstitial diffusion in a two-dimensional colloidal crystal

Sung-Cheol Kima,1 , Lichao Yua,2, Alexandros Pertsinidisa,3, and Xinsheng Sean Linga,4

aDepartment of Physics, Brown University, Providence, RI 02912

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 20, 2020 (received for review October 16, 2019)

In two-dimensional (2D) solids, point defects, i.e., vacancies and heated crystals (15). Previous studies of point defects in atomic interstitials, are bound states of topological defects of edge dislo- solids are mostly based on indirect techniques (16). As a result, cations and disclinations. They are expected to play an important the details of the dynamical processes of this type of point defects role in the thermodynamics of the system. Yet very little is remain elusive. known about the detailed dynamical processes of these defects. Here we report a direct video imaging study of the local ﬂuc- Two-dimensional colloidal crystals of submicrometer microspheres tuations in interstitial defects during diffusion in a 2D colloidal provide a convenient model solid system in which the microscopic crystal. The microscopic origins that determine how fast an inter- dynamics of these defects can be studied in real time using video stitial defect diffuses in the lattice are determined. One can microscopy. Here we report a study of the dynamical processes directly visualize the equilibrium dynamics of a point defect and of interstitials in a 2D colloidal crystal. The diffusion constants see how nonequilibrium behavior can emerge from defect–lattice of both mono- and diinterstitials are measured and found to interactions. be signiﬁcantly larger than those of vacancies. Diinterstitials are clearly slower than monointerstitials. We found that, by plotting Results the accumulative positions of ﬁve- and sevenfold disclinations Our experiment was performed in 2D colloidal crystals made of relative to the center-of-mass position of the defect, a sixfold sym- a ∼1% aqueous suspension of 0.36 µm diameter polystyrene- metric pattern emerges for monointerstitials. This is indicative of sulfate microspheres (Duke Scientiﬁc No. 5036). The aque- an equilibrium behavior that satisﬁes local detailed balance that ous solution was thoroughly deionized to achieve an estimated the lattice remains elastic and can be thermally excited between Debye screening length κ−1 ≈ 390 nm, at room temperature lattice conﬁgurations reversibly. However, for diinterstitials the (22 ◦C). The colloidal spheres, being negatively charged, form sixfold symmetry is not observed in the same time window, and a crystal due to the repulsive screened Coulomb potential. Two the local lattice distortions are too severe to recover quickly. This fused silica substrates (a thin coverslip and a thick disk) sepa- observation suggests a possible route to creating local melting of rated by ∼2 µm conﬁned the suspension to form a single-layer a lattice (similarly one can create local melting by creating diva- colloidal crystal with lattice constant a ≈ 1.1 µm. The details of cancies). This work opens up an avenue for microscopic studies of the experimental setup can be found in ref. 17. the dynamics of melting in colloidal model systems.

2D colloidal crystal | interstitial defects | diffusion | detailed balance Signiﬁcance

rystallization is a spontaneous symmetry-breaking process Defects in crystalline materials are of broad and fundamental Cduring which the many-body system acquires the emer- interest. In condensed-matter physics, defect dynamics con- gent properties of shear rigidity and long-range order (1). In tain essential information about the microscopic processes of two dimensions at ﬁnite temperatures, true long-range order crystal formation and melting. Two-dimensional melting (2D) is absent due to the accumulative effects of long-wavelength is widely accepted to be mediated by the proliferation of phonons (2). Nevertheless, the orientational (or topological) edge dislocations. In 2D crystals, vacancies and interstitials order survives. The melting of a two-dimensional (2D) crys- are, in fact, bound pairs of edge dislocations and disclina- tal can be either ﬁrst order or continuous via the well- tions. They are expected to play critical roles in 2D melting. We known mechanism of defect unbinding proposed by Kosterlitz demonstrated signiﬁcant progress in quantifying the dynam- and Thouless (3), Halperin and Nelson (4), and Young ical processes of interstitials. We also propose a simple yet (5). In well-controlled experiments, it is apparent that the powerful method in visualizing the time-averaged conﬁgura- Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) mecha- tions of the defects, providing a direct tool to assess whether nism of melting has been conﬁrmed (6). However, in real mate- the detailed balance is obeyed or violated in the ﬂuctuating rials, defects are always present, and their roles in the melting processes in a lattice. transition are less understood. There are growing interests in recent years in the dynamics of various kinds of defects in 2D Author contributions: A.P. and X.S.L. designed research; S.-C.K., A.P., and X.S.L. per- colloidal crystals (7, 8) and how they may change the physics of formed research; S.-C.K., L.Y., and X.S.L. analyzed data; and S.-C.K. and X.S.L. wrote the 2D melting. paper. y In two dimensions, point defects, vacancies, and interstitials The authors declare no competing interest.y are nontrivial as they are composites of topological defects, edge This article is a PNAS Direct Submission.y dislocations, and disclinations (9–11). Thus it is expected that Published under the PNAS license.y they will play important roles in the melting processes. Near 1IBM T. J. Watson Research Center, Yorktown Heights, NY 10598.y melting, vacancies will be essentially indistinguishable from ther- 2 Google LLC, Cambridge, MA 02142.y mally excited dislocation pairs (7). The physics of interstitials are 3Structural Biology Program, Memorial Sloan Kettering Cancer Center, New York, NY less clear. In fact, interstitials have long been of importance to 10065.y the study of solids, e.g., in the understanding of acoustic absorp- 4 To whom correspondence may be addressed. Email: Xinsheng [email protected] tion in metals (12, 13) and for understanding the properties of This article contains supporting information online at https://www.pnas.org/lookup/suppl/ crystalline solids near melting (14). There have been suggestions doi:10.1073/pnas.1918097117/-/DCSupplemental.y that interstitials may play a critical role in the melting of super- First published May 28, 2020.

13220–13226 | PNAS | June 16, 2020 | vol. 117 | no. 24 www.pnas.org/cgi/doi/10.1073/pnas.1918097117 Downloaded by guest on October 3, 2021 Downloaded by guest on October 3, 2021 nesiil omn tteohredo h o.Ms fthe of Most row. al. et the Kim in of resulting end displace, other to the particles axes, several at crystalline of forming three row the interstitials its a of off cause that one can particle found along we we “good” particle Furthermore, the a (17). dragging vacancy drag by a can create pro- to we the site In that lattice force). discovered Waals we der particles van cess, two (by mostly irreversibly particles; together impurity removing stuck for tool a as ers (NI-1409) grabber (18). frame algorithm particle-tracking a a by using captured recorder. analyzed and were SVO-9500MD images Sony CCD a (CCD) The and frames device 30 SSC-M370) charge-coupled of (Sony monochrome rate camera a a at using videos second as per recorded were images real-time il n oei costeﬁl fve (∼50 par- a view capture of can ﬁeld the we across that it so move beam and the ticle of position the control to a with 135) Neoﬂuar; Axiovert (Zeiss scope 90; INNOVA and imn a 0prilswe tcnan h oeo inesiil (C diinterstitial. a of core the contains it when particles 10 has diamond 6 5 The disclinations. 1. Fig. nor2 oli xeiet ntal eue pia tweez- optical used we initially experiment, colloid 2D our In notcltezr ytmwt an with system tweezers optical An D, h ugr etr r enda h ipaeetbtendslntos rmlwt ihcoordinations. high to low from disclinations, between displacement the as deﬁned are vectors Burgers The Insets) (A and N.A CD AB inesiili Dclodlcytl e n ledt niae5 n 7-coordinate and 5- indicate dots blue and Red crystal. colloidal 2D a in diinterstitial (B) and mono- (A) isolated an of images Video B) λ 1 = 514 = × imn eincnan ieprilsi efc atc u 0prilswe tecoe h oeo oonesiil h 4 The monointerstitial. a of core the encloses it when particles 10 but lattice perfect a in particles nine contains region diamond 5 i meso) eue oaal mirrors rotatable used We immersion). oil .3, m a osrce na netdmicro- inverted an on constructed was nm) 100× Ar betv ZisPlan (Zeiss objective + µm ae (Coherent laser × ) The 50µm). 1 μ m and eanytinuaindarm fcrepnigiae in images corresponding of diagrams triangulation Delaunay D) ~ d pt eoi ahcngrto (7). conﬁguration each in vectors zero Burgers to up The add 7 vector. 5, Burgers point of net n a pairs no bound have neutral, several should topologically defect of Being combination conﬁgurations disclinations. a Interstitial coordinated as particles. viewed indi- coordinate dots be more blue can and or particles 7 coordinate Red fewer cate experiment. or real inter- 5 a indicate single in dots a observed directly see diinterstitials can line 1A shows one dashed Fig. counting, the in inside simple stitial particles by of Clearly, number identi- loops. the are conﬁgurations counting their by and ﬁed Delaunay defects corresponding point represen- their The two and diagrams. shows images 1 video Fig. captured dynamics. tative micro- defect the of into processes insights scopic unprecedented provides diffusion stitial of subject main the diinterstitial, two a study. ﬁnd region, this also that we frequency in less particles with extra monointerstitial; a is it time, ij ietvdoiaigo h eltm eairo h inter- the of behavior real-time the of imaging video Direct pitn rmrddtt ledti i.1, Fig. in dot blue to dot red from (pointing hsexample This 1B. Fig. in interstitials two and PNAS | ue1,2020 16, June | o.117 vol. 1 μ m | indeed Insets) o 24 no. A and | 13221 (C B. ×

APPLIED PHYSICAL SCIENCES A B C

SI I2 I2d

DFE

I2a I3 I4

G H I

SDI DI2d,a DI2d,b

J K L

DI2a DI3d DI4d

Fig. 2. Delaunay triangulation diagrams of mono- and diinterstitial conﬁgurations. (A–F) Monointerstitials (A), a split interstitial (SI); (B), a twofold symmet- ric interstitial (I2); (C), a disjoint twofold symmetric interstitial (I2d); (D), a twofold symmetric interstitial with distorted lattice (I2a); (E), a threefold symmetric interstitial (I3); (F), and a fourfold symmetric interstitial (I4); (G–L) diinterstitials; (G), a split diinterstitial (SDI); (H and I), disjoint twofold symmetric diintersti- tial (DI2d,a, DI2d,b); (J), a twofold symmetric diinterstitial with distorted lattice (DI2a); and (K and L), disjoint threefold and fourfold symmetric diinterstitials (DI3d, DI4d).

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Mean square displacement (µm²) (18), method triangulation Delaunay standard the Following 10 15 20 25 30 0 5 63 04 446 44 42 40 38 36 0.0 52 25 20 15 di-interstitial mono-interstitial 0.5 1.0 Time (sec) 1.5 2.0 mono-vacancy h oiino on eeti enda h center-of-mass the as deﬁned 20. is and defect 19 point refs. a from of adopted position is The notation the diinterstitials; for 2 Fig. in observed di-vacancy 2.5 rmteﬁl fve.Bu icei trigpito rc n black and trace away of interstitial. point of diffused starting centers is interstitial are circle dots the Blue view. on after of ﬁeld superimposed seconds the from diagram, few triangulation a trajectories a the show lines yellow The for trajectory terstitial 3. Fig. Monoin- (A) defects. diinterstitial and mono- of Trajectories A–F PNAS inesiil oeoto h Dsml region sample 2D the of quickly. out move diinterstitials iey n t r for are ﬁts respec- diinterstitials, and and tively, mono- for ﬁts linear are of function from mean- culated a The as displacement diinterstitial. square and mono- placement: 4. Fig. o oonesiil n nFg 2 Fig. in and monointerstitials for | inesiiltaetre for trajectories Diinterstitial (B) s. ∼30 ue1,2020 16, June iedpnec fma-qaedis- mean-square of dependence Time - rjcois e n lelines blue and Red trajectories. ∼6-s | o.117 vol. δ t auso 1 of values | o 24 no. ∼ δ t .Most s. 3 a cal- was | 0s. ∼10 13223 G–L

APPLIED PHYSICAL SCIENCES position from all 5, 7 coordinated disclinations within a green aged them for each interstitial type. We found that these loop (Fig. 2). mean-square displacements show linear dependence on time To study the diffusion dynamics of the defects, we constructed lag for both mono- and diinterstitials in Fig. 4, conﬁrm- the time-elapsed trajectories of the interstitials as shown in Fig. 3, ing their behavior as random walkers. The diffusion con- superimposed on the underlying colloidal lattice. The colloidal stants, which are obtained from the linear ﬁtting, are Dmono = 2 2 lattice is obtained after the interstitial moved away from the ﬁeld 10.95 ± 0.04 µm /s and Ddi = 7.53 ± 0.05 µm /s, respectively. of view. The ﬁrst observation is that mono- and diinterstitials in The minor discrepancy between linear line and the mean- 2D colloidal crystals behave differently in diffusion. The proce- square displacement of the diinterstitial can be explained by dures by which the defect movements are tracked are described lack of enough sampling because one of disclinations in the in SI Appendix, Fig. S1. Fig. 3A shows that a monointerstitial diinterstitial quickly disappears out of the microscopic view moves randomly in a 2D triangular lattice, while the diinterstitial window. in Fig. 3B diffuses along a certain lattice axis and seldom changes As predicted in a numerical calculation (20), we found that its direction. the diffusion constant of interstitials is indeed larger than that From the trajectories of multiple interstitials, we calculated of mono- and divacancies (21). Surprisingly, the diffusion con- the square displacements with different time lags and aver- stant of monointerstitials is signiﬁcantly larger than that of

AB

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R

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Fig. 5. Dynamics of interstitial diffusion. (A) Delaunay triangulation diagram of a monointerstitial with the center marked as an “x” and the 5/7 disclina- tions. The red lines indicate edge dislocations. (B) The lattice conﬁguration from a sequential video frame following that in A with the center marked as an “o.” (Inset) The displacement vector ~R of the interstitial is deﬁned from x to o. (C and D) ~R vs. time for a monointerstitial and a diinterstitial, respectively, over 8 s.

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y (µm) −3 −2 −1

−3 −2 −1 y (µm) 0 1 2 3 0 1 2 3 −3 aepo fpril oiin f(C of positions particle of plot Same D) −3 (A and mono-i. di-i. -iciainpstosoe ,clrdb ahcngrto n etrdaottemnitrtta eetcore. defect monointerstitial the about centered and conﬁguration each by colored s, 8 over positions 7-disclination B) ( and 5- (A) of Plot B) −2 −2 5-fold disclinations −1 −1 x ( x ( 0 0 µ µ m) m) -oriaedslntosoelpe ntpo h etrddfc oeo h diinterstitial the of core defect centered the of top on overlapped disclinations 7-coordinate (D) and 5- ) 1 0ltieconstants lattice ∼10 1 2 2 DI4d DI3d DI2d DI2 I4 I2d I3 I2 3 3 vrg oiino l vfl n eefl iciain.The disclinations. sevenfold and ﬁvefold the as all interstitial of the and position of mono- center average both hop- the for deﬁne time the we of Here, function of diinterstitials. a as ﬂuctuations shown The are distance diffusion. ping interstitial of dynamics 24.) ref. in described residence is approximately the dure be estimated to and We interstitials monointerstitials the (23). of barrier time activation and attempt regarding rate information the contains conﬁguration ticular D B i.5poie diinlisgtit h hral activated thermally the into insight additional provides 5 Fig. par- a in time residence the processes, activated thermally For y (µm) −3 −2 −1 −3 −2 −1 y (µm) 3 0 1 2 0 1 2 3 −3 −3 mono-i. di-i. −2 −2 PNAS 7-fold disclinations o oedtis see details, more for S2; Fig. Appendix, SI τ di | 0 = −1 −1 ue1,2020 16, June .02 x ( o inesiil.(h proce- (The diinterstitials. for s x ( 0 µ 0 µ m) m) | o.117 vol. 1 1 τ mono | o 24 no. 0 = 2 2 .04 DI4d DI3d DI2d DI2 I4 I2d I3 I2 | for s 13225 3 3

APPLIED PHYSICAL SCIENCES displacement vector R~ is deﬁned as the displacement between shown in Fig. 6 C and D, it appears that a diinterstitial seems to the center positions from two sequential frames, as shown in live in a twofold symmetric conﬁguration for a long period. In Fig. 5 A and B. The monointerstitials tend to have longer our present experiment, we cannot conclude whether the same residence time and smaller hopping distance. With this observa- sixfold symmetry patterns in Fig. 6 A and B can be recovered if tion, if the processes are purely stochastic, one would expect the we were able to trap the diinterstitial and observe it for a long diinterstitials to diffuse faster than monointerstitials, contrary to period. the result in Fig. 4. A hint of the solution to this mystery can be found in Fig. 3. Discussion For the monointerstitial the directions of the hopping process We report a detailed study of interstitial diffusion dynamics in are diverse. However, the diinterstitial has only one or two direc- a 2D colloidal crystal. We determined the diffusion coefﬁcients tions as if it has a memory of a previous direction. A detailed for both mono- and diinterstitials and discovered that the diffu- analysis (SI Appendix, Fig. S3) of the ﬂuctuations between conﬁg- sion coefﬁcient of the monointerstitial is signiﬁcantly larger. By urations leads to the conclusion that what we have observed is a examining the microscopic transitions between conﬁgurations, strong memory effect for diinterstitials. The angular distribution we found that the diinterstitials have a strong memory effect. of mono- and diinterstitial core displacements in SI Appendix, Namely, monointerstitials were found to diffuse as 2D random Fig. S3 conﬁrms this behavior. As a result, diinterstitials diffuse walkers, while diinterstitial diffusion is quasi–one-dimensional. as a quasi–one-dimensional random walker. We developed a simple method to ascertain the equilibrium The above analyses are based on the standard procedures of dynamics by looking at the conﬁgurational ﬂuctuations rela- studying individual defect diffusion (21). However, these anal- tive to the center-of-mass position of the defect. Thus, we can yses may have missed a deeper physical origin. In Fig. 6, we directly visualize the equilibrium and nonequilibrium dynamics plot the ﬁve- and sevenfold disclinations positions relative to of mono- and diinterstitials, respectively. The apparent break- the defect core over a period of 8 s (before it diffuses out of down of local detailed balance in the ﬂuctuating conﬁgurations the ﬁeld of view). The locations of ﬁve- and sevenfold coordi- of diinterstitial diffusion is particularly striking. It deserves more nated disclinations of interstitials are shifted and plotted on top future studies as it may serve as a model system for studying of the defect core grouped by each conﬁguration in Fig. 2. We nonequilibrium dynamics in nondriven systems (26). Given the ﬁnd that there is a striking sixfold symmetry in the distribution signiﬁcant local distortions created by diinterstitials that seem to of the disclination locations in the four different conﬁgurations persist in time, we suggest that it may be a feasible way to create I2, I3, I4, and I2d of monointerstitials in Fig. 6A. In Fig. 6B, local melting using such defects. It is interesting to note that the the distributions of sevenfold disclinations exhibit a similar disordered local structure surrounding a diinterstitial is rem- behavior. iniscent of the local melting phenomenon when a large par- The simple interpretation is that the disclinations associated ticle is dragged through a 2D colloidal crystal (27, 28). The with an interstitial locally destroy the orientational order at a results presented here should be of interest for many areas of given instant in time. Thermal ﬂuctuations, which cause transi- condensed-matter physics (29). tions between different conﬁgurations, restore this order over time. Namely, thermal ﬂuctuations restore the symmetry that Data Availability. The corresponding video ﬁles in MPEG for- was spontaneously broken by the topological defects (disclina- mat for Fig. 3 are provided in Movies S1 and S2. The original tions). This is possible only when ﬂuctuations locally preserve video data, recorded on VHS tapes using a Sony SVO-9500MD detailed balance (25). However, this equilibrium behavior is not recorder, are available upon request. preserved for diinterstitials even though they are in the same lat- ACKNOWLEDGMENTS. This work was supported by the National Science tice, under the same conditions (temperature, ionic strength). As Foundation Division of Materials Research (DMR) Grant 1005705.

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