Dynamical processes of interstitial diffusion in a two-dimensional colloidal

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

aDepartment of , 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) , point defects, i.e., vacancies and heated (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 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 . 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 of 0.36 µm diameter - 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 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 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- 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 (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.

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SI I2 I2d


I2a I3 I4


SDI DI2d,a DI2d,b


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




CD mono-interstitial di-interstitial 3 3 m) 2 m) µ 2 µ ( ( R R

1 1

0 0

0246802468 Time (sec) Time (sec)

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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