Ordering of Binary Colloidal Crystals by Random Potentials
Andre S. Nunes, Sabareesh K. P. Velu, Iryna Kasianiuk, Denys Kasyanyuk, Agnese Callegari, Giorgio Volpe, Margarida M. Telo da Gama, Giovanni Volpe, Nuno A.M. Araújo
OOrdering of Binary Colloidal Crystals by Random Potentials
Andr´e S. Nunes, ∗ Sabareesh K. P. Velu, ∗ Iryna Kasianiuk, Denys Kasyanyuk, Agnese Callegari, Giorgio Volpe, Margarida M. Telo da Gama, Giovanni Volpe,
2, 5 and Nuno A. M. Ara´ujo † Centro de F´ısica Te´orica e Computacional and Departamento de F´ısica,Faculdade de Ciˆencias, Universidade de Lisboa, P-1749-016 Lisboa, Portugal, EU Department of Physics, Bilkent University, Cankaya, 06800 Ankara, Turkey Department of Physics, Bilkent University and UNAM, Cankaya, 06800 Ankara, Turkey Department of Chemistry, University College London,20 Gordon Street, London WC1H 0AJ, United Kingdom, EU Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden, EU
Structural defects are ubiquitous in condensed matter, and not always a nuisance. For example,they underlie phenomena such as Anderson localization and hyperuniformity, and they are nowbeing exploited to engineer novel materials. Here, we show experimentally that the density ofstructural defects in a 2D binary colloidal crystal can be engineered with a random potential. Wegenerate the random potential using an optical speckle pattern, whose induced forces act stronglyon one species of particles (strong particles) and weakly on the other (weak particles). Thus, thestrong particles are more attracted to the randomly distributed local minima of the optical potential,leaving a trail of defects in the crystalline structure of the colloidal crystal. While, as expected,the crystalline ordering initially decreases with increasing fraction of strong particles, the crystallineorder is surprisingly recovered for sufficiently large fractions. We confirm our experimental resultswith particle-based simulations, which permit us to elucidate how this non-monotonic behaviorresults from the competition between the particle-potential and particle-particle interactions.
Perfect crystalline structures are not commonly foundin Nature, because, even in the absence of impurities,structural defects occur spontaneously and disrupt theperiodicity of the crystalline lattice [1]. For example,when a melt is cooled down, multiple crystallites growwith degenerate orientations [2]. Since the coarseningtime of these crystallites diverges with size, structuraldefects appear and prevent the emergence of global or-der [3, 4]. While the existence of these defects is a chal-lenge when growing single crystals, it can also be an op-portunity when engineering the properties of materials;
FIG. 1.
Colloidal crystals with tunable degree of dis-order.
Final configurations obtained in (a-c) experimentsand (d-f) simulations, for different molar fractions χ of strongparticles. The weak (silica) particles are light gray, and thestrong (polystyrene) particles are dark gray. indeed, control over defects enables the development ofsolid-state devices with fine-tuned mechanical resilience,optical properties, and heat and electrical conductivity[5–9]. In atomic crystals, engineering structural defectsis an experimental challenge for two reasons [10]: first,current visualization techniques at the atomic scale donot provide a high spatial or time resolution [11, 12];second, no current technique can control the density ofdefects in a systematic manner [13]. The first challengecan be overcome studying colloidal crystals as modelsfor atomic systems [14, 15], where colloidal particles canbe individually tracked using standard digital video mi-croscopy techniques [16–18]. Here, we demonstrate thatthe second challenge can be solved combining a binarycolloidal mixture and an optical random potential gener-ated by a speckle light pattern. This permits us to controlthe density of structural defects in the resulting 2D col-loidal crystal and to explore a surprising non-monotonicbehavior of their ordering and stability.We use a binary colloidal suspension of equally-sizedpolystyrene (refractive index n ps ≈ .
59) and silica( n si ≈ .
42) spherical particles with diameters d PS =4 . ± . µ m and d SiO = 3 . ± . µ m, respectively.To characterize the composition of the mixture, we usethe molar fraction of polystyrene particles defined as χ = N ps /N t where N ps is the number of polystyreneparticles and N t is the total number of particles. Welet these particles sediment at the bottom surface of ahomemade sample chamber so that they are effectivelyconfined in a quasi-2D space (see Supplemental Mate-rial for details [19]). We illuminate from above with aspeckle pattern, which we generate by mode-mixing acoherent laser beam in a multimode optical fibre (see a r X i v : . [ c ond - m a t . s o f t ] M a r supplementary Fig. S1 and Supplemental Material fordetails [19]) [20–22]. Speckle patterns form rough, dis-ordered optical potentials characterized by wells whosedepths are exponentially distributed and whose averagewidth is given by diffraction (here, average grain size σ = 3 . ± . µ m). Furthermore, the fibre imposes aGaussian envelope (beam waist σ G = 72 . ± . µ m)to the speckle pattern, which attracts the particles to-wards the center of the speckle pattern effectively confin-ing them in space. Since the optical forces acting on theparticles increase for larger mismatches between their re-fractive index and that of the surrounding medium (herewater, n w ≈ .
33) [23], the optical forces acting on thepolystyrene (strong) particles are about 2 × higher thanthose exerted on silica (weak) particles. Importantly, theoptical forces at the deepest local minima of the specklepotential are strong enough to trap the strong particles,but not the weak ones.We start with a low concentration of particles (1 . · mL − ) and switch on the optical potential. The par-ticles are attracted towards its center by the Gaussianenvelope. When only weak particles are present ( χ = 0),they eventually form an (almost) perfect hexagonal col-loidal crystal, as shown in Fig. 1a. When we introducestrong particles as χ increases, these get trapped in thelocal minima of the disordered potential and introducedefects that reduce the hexagonal order. Already withonly 20% of strong particles ( χ = 0 . χ = 0 .
5) are strongly interacting with the potential(Fig. 1c). Thus, strong particles act as defects in the crys-talline structure of the weak ones, compromising globalorder. These results are confirmed by particle-based sim-ulations, as shown in Figs. 1d-f (see supplemental infor-mation [19]). As we will see in more detail below, we cancontrol the density of defects by adjusting χ as well asthe intensity and grain size of the pattern.To quantify the order of the crystalline structure, wemeasured the six-fold bond-order parameter, (cid:104) φ (cid:105) , de-fined as [24] (cid:104) φ (cid:105) = 16 N c N c (cid:88) l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N b (cid:88) j e i θ lj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1)where the outer sum is over the N c particles within 7.5particle diameters from the center of the potential (whichis the area where the aggregate is formed), the inner sumis over the N b neighbors of a particle in the Voronoi tes-sellation, and θ lj is the angle between the x -axis and theline connecting the centers of particles j and l . (cid:104) φ (cid:105) = 1for perfect hexagonal crystals (in practice, it is neverexactly one, because of thermal fluctuations and othertransient perturbations to the periodic order) and it de-creases with the number of structural defects. Figure 2shows (cid:104) φ (cid:105) obtained experimentally and numerically as FIG. 2.
Crystalline order for different molar fractionsof strong particles.
Six-fold bond order parameter (cid:104) φ (cid:105) asa function of the molar fraction χ obtained experimentally(triangles) and numerically (squares; the blue line connectsthe symbols for visual guidance). The numerical results areaverages over 100 samples. The top snapshots show the finalconfigurations in the experiments (first row), the Voronoi tes-selation (second row), and the spatial Fourier transform (thirdrow) for χ = 0, 0 .
22, 0 .
76, and 1. The filled (empty) circlesat the center of the Voronoi cells indicate strong (weak) par-ticles. The cells are colored by the number of nearest neigh-bors, namely, equal (green), lower (red), greater (blue) thansix. See also supplementary video 1. a function of the molar fraction χ . For χ = 0, (cid:104) φ (cid:105) ≈ χ increases, the value of (cid:104) φ (cid:105) decreases due to the formation of structural defects. Thesnapshots in the top rows of Fig. 2 show the final con-figurations (first row), the corresponding Voronoi tessel-lations (second row), and the spatial Fourier transform(third row), for different values of χ .Surprisingly, the data reported in Fig. 2 show that (cid:104) φ (cid:105) reaches its minimum value for χ min ≈ .
6, and then the
FIG. 3.
Local dynamics of the interaction between particles and minima in the random potential. (a) Examples oftrajectories of weak (light gray) and strong (dark gray) particles in the presence of a speckle obtained numerically for differentvalues of the molar fraction χ . The particle density is 10 × lower than that of maximal packing and the Gaussian envelopeis absent. The four simulations were preformed under exactly the same conditions, including the same sequence of randomnumbers for the thermostat (see Supplemental Material [19]). The black circles on the top left corner indicate the particle size.The random potential intensities are in units of k B T and σ is one particle diameter. (b) When a weak particle (light gray)is located at a potential minimum and a strong particle (dark gray) is in its vicinity, it is energetically favorable to exchangethe two, but the opposite process (c) is not. (d) The free energy may be significantly reduced when two particles of the samespecies share the same potential minimum. See also supplementary video 2. global order increases for χ > χ min . In particular, for χ = 1, the strong particles self-assemble into an hexag-onal crystal, despite the presence of the underlying ran-dom potential. This result is corroborated by the Voronoitessellation of the final configurations and by the respec-tive spatial Fourier transforms. From this analysis, wecan see that the number of Voronoi cells with a numberof neighbors different from six becomes higher near theminimum of (cid:104) φ (cid:105) , and that the Fourier transforms dis-play dimmer intensity peaks near the same value. Thisobservation suggests a change in the effective interactionbetween the strong particles and the underlying poten-tial: from one that favors disorder at a low χ to onefavoring order at larger χ .In order to elucidate the microscopic mechanisms un-derlying this behavior, we employ trajectories obtainedby particle-based simulations to study the interactionsbetween the two particle’s species and the local minimain the potential. Figure 3(a) shows some trajectories ofweak (light gray) and strong (dark gray) particles at var-ious χ . We performed these simulations using a randompotential without the Gaussian envelope to highlight thedynamics of the interaction between the particle and thelocal minima. In all cases, the weak particles can hop between minima, while the strong particles get readilytrapped in them; in fact, the effective diffusion coefficientof the strong particles is significantly lower than that ofthe weak particles (see supplementary Fig. S2 [19]). Atlow χ , the strong particles quickly populate the minimathat are sufficiently deep to prevent their escape and re-main there for the entire simulation time, because thisconfiguration is energetically favorable (Figs. 3b and 3c);therefore, the number of spatial defects increases mono-tonically with the number of the trapped strong parti-cles, leading to a decrease of (cid:104) φ (cid:105) with increasing χ . Atlarge χ , the number of strong particles is greater thanthe potential minima and thus it becomes energeticallyfavorable to have more than one strong particle in oneminimum (Figs. 3d). This allows the spatial rearrange-ment of the particles since the energy of the interactionwith the speckle is no longer strong enough to localize theparticles, a large scale crystalline structure is favorable,consistent with the increase in (cid:104) φ (cid:105) observed in Fig. 2.When χ = 1, all particles are strong and thus the hexag-onal crystalline structure is recovered.In order to explore how robust this effect is, we stud-ied numerically how it depends on the properties of theunderlying speckle pattern. The speckle is characterized FIG. 4.
Dependence of the order parameter on the speckle properties.
Six-fold bond order parameter as a functionof the molar fraction ( χ ) obtained numerically, for different values of the speckle (a) strength and (b) spatial correlation σ .Results in (a) were obtained for σ = 1 and in (b) for V = 15 .
1, and are averages over 100 samples. by a strength V corresponding to the average potentialdepth (in units of k B T , where k B is the Boltzmann con-stant and T is the absolute temperature of the sample)and by a spatial correlation σ (in units of the particlediameter), which corresponds to the average grain size.Figure 4(a) shows (cid:104) φ (cid:105) for different V . Although thecurves in the range 1 . < V (cid:54) . V : when V increases, the number of strong particles that can betrapped increases monotonically and, consequently, χ min shifts to the right and the minimum becomes deeper. For V > .
8, the behaviour seems to become independentof the molar fraction (and always disordered), becausethe weak particles are also strongly trapped. Figure 4(b)shows (cid:104) φ (cid:105) for different values of σ . A pronounced mini-mum is only observed for intermediate values of σ , closeto unity (particle diameter). If σ (cid:29) σ (cid:28)
1, the op-tical forces are negligible for different reasons: for σ (cid:29) σ (cid:28)
1, the optical poten-tial varies on a length scale smaller than the particle sizeand thus its gradient averages to zero over the particlecross-section (see supplementary Fig. S3 [19]). In the lat-ter case, the optical force on a particle is the sum of thecontributions over the particle’s cross-section, which canbe described by an effective random potential that differsfrom the one originally applied (Supplemental Materialand supplementary Fig. S4 and S5 [19]).In conclusion, we have shown that the order in a two-dimensional binary colloidal crystal can be controlledby an underlying random optical potential, when eachspecies experiences distinct optical forces. Since the in-tensity of the optical forces depends on the mismatch ofthe indices of refraction of the particles and the surround- ing medium, the particles with the larger index mismatchare more responsive (strong particles) than those withthe lower mismatch (weak particles). For the parame-ters of the optical potential that were considered, onlythe strong particles respond significantly to the poten-tial. Thus, strong particles tend to occupy the minimaof the potential and nucleate structural defects in the,otherwise, periodic hexagonal structure of the weak par-ticles. The density of defects is controlled by the frac-tion of strong particles and the statistical properties ofthe underlying potential. When the number of strongparticles increases beyond the number of local minimathat can trap them, the trapping mechanism becomesless effective and the hexagonal order is recovered as thefraction of strong particles increases. Here, we have con-sidered a random optical potential with Gaussian spatialcorrelations and with a characteristic length given by thestandard deviation σ . However, it is technically possibleto generate other optical potentials, e.g. periodic [23] orwith different spatial correlations [25, 26]. Thus, one cancontrol not only the density of defects, but also their spa-tial distribution. Time-varying optical potentials couldalso be employed to change the position of strong par-ticles and defects in time, affecting the overall dynamics[18, 21, 27]. Understanding how the spatial distributionof defects influences the physical properties of materialsis a question of both scientific curiosity and technologicalinterest that can now be addressed in a systematic way.Andr´e S. Nunes, Margarida M. Telo daGama and Nuno A. M. Ara´ujo acknowledge fi-nancial support from the Portuguese Founda-tion for Science and Technology (FCT) un-der Contracts nos. EXCL/FIS-NAN/0083/2012,UID/FIS/00618/2019, SFRH/BD/119240/2016 andPTDC/FIS-MAC/28146/2017 (LISBOA-01-0145-FEDER-028146). Margarida Telo da Gama and NunoArajo would like to thank the Isaac Newton Institutefor Mathematical Sciences for support and hospitalityduring the program ”The mathematical design of newmaterials” where the final version of this manuscriptwas completed. This program was supported by EP-SRC Grant Number: EP/R014604/1. Giorgio Volpeacknowledges support from the Royal Society undergrant RG150514. We also acknowledge Parviz Elahi forhis help with the experimental setup. ∗ contributed equally. † [email protected][1] W. Bollmann, Crystal defects and crystalline interfaces (Springer Science & Business Media, 2012).[2] K. Pyka, J. Keller, H. L. Partner, R. Nigmatullin,T. Burgermeister, D. M. Meier, K. Kuhlmann, A. Ret-zker, M. B. Plenio, W. H. Zurek, A. del Campo, andT. E. Mehlst¨aubler, “Topological defect formation andspontaneous symmetry breaking in ion coulomb crys-tals,” Nat. Commun. , 2291 (2013).[3] W. H. Zurek, “Causality in condensates: Gray solitons asrelics of BEC formation,” Phys. 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