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9-17-2007

Bubble Raft Model for a Paraboloidal Crystal

Mark Bowick Department of , Syracuse University, Syracuse, NY

Luca Giomi Syracuse University

Homin Shin Syracuse University

Creighton K. Thomas Syracuse University

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Recommended Citation Bowick, Mark; Giomi, Luca; Shin, Homin; and Thomas, Creighton K., "Bubble Raft Model for a Paraboloidal Crystal" (2007). Physics. 144. https://surface.syr.edu/phy/144

This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. 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−1 FIG. 1: (color online) Lateral and top view of a computer reconstruction of two paraboloidal rafts with κ1 ≈ 0.15 cm (left) −1 and κ2 ≈ 0.32 cm (right). The number of bubbles is N1 = 3813 and N2 = 3299 respectively. The color scheme highlights the 5−fold (red) and 7−fold (blue) disclinations over 6−fold coordinated bubbles (yellow). we first make the bubbles and only later spin the ves- cave meniscus due to the surface tension combined with sel to make the paraboloid (cf. Bragg and Nye [1], who the native curvature of the paraboloid was observed to spun their system in order to generate smaller bubbles produce a stacking of bubbles in a narrow double layer but stopped the spinning to look at the bubbles on a flat surrounding the perimeter of the vessel. surface). To image the bubbles, we mount a CCD digital To characterize the order of the crystalline raft, we camera on the top of the vessel, with lighting from a ring measure the translational and orientational correlation around the (clear) vessel to eliminate glare. The camera functions g(r) and g6(r). The former gives the prob- rotates along with the whole system so that the shutter ability of finding a particle at distance r from a sec- speed is unimportant in imaging the bubbles. We use a ond particle located at the origin. The function is nor- second camera to find the aspect ratio of the paraboloid. malized with the density of an equivalent homogeneous We equilibrate the system and eliminate stacking of bub- system in order to ensure g(r) = 1 for a system with bles by imposing small perturbations of the angular fre- no structure. Interactions between particles build up quency to mimic the role of thermal noise. The vessel has correlations in their position and g(r) exhibits decay- radius R = 5 cm; the height of paraboloids varies from ing oscillations, asymptotically approaching one. For h = 0–4 cm. The bubble diameter, extracted from the a two-dimensional with a triangular struc- Delaunay triangulation of our images, is a =0.84(1) mm ture the radial correlation function is expected to ex- with monodispersity ∆a/a 0.003. The normal curva- hibit sharp peaks in correspondence with the sequence ≈ ture κ of the paraboloid at the origin varies from 0–0.32 r/a = √n2 + nm + m2 = 1, √3, 2, 2√3 . . . while the −1 cm . In addition to the flat disk, we observe two differ- amplitude of the peaks decays algebraically as r−η with −1 ent curvature regimes: small curvature κ1 0.15 cm η = 1/3 [16] (dashed line in Fig. 2). Within the preci- −1 ≈ and high curvature κ2 0.32 cm . sion of our data, the positional order of the paraboloidal ≈ Figure 1 shows a computer reconstruction of two bub- crystals assembled with the bubble raft model reflects ble rafts with these κ = κ1, κ2. We extract two di- this behavior, although more accurate measurements are mensional coordinates from the images with a brightness required in order to clarify a possible dependence of the based particle location algorithm [15]. Data sets are then exponent η on the curvature. processed to correct possible imprecisions and finally De- The orientational correlation function g6(r) is calcu- launay triangulated. We choose to exclude from the tri- lated as the average of the product ψ(0)ψ∗(r) of the angulation the first 3–4 bubble rings formed along the hexatic order parameter over the wholeh sample.i For boundary of the cylindrical vessel, where the sharp con- each bubble (labeled j) that has two or more neighbors, 3

form of a consisting of one-dimensional arrays of tightly bound (5, 7) fold disclinations pairs. In a planar confined system, grain− boundaries typically span the entire length of the crystal, but if a non-zero Gaus- sian curvature is added to the medium, they can appear in the form of scars carrying a net +1 topological charge and terminating in the bulk of the crystal. Prominent examples of grain boundaries are visible in the two lattices shown in Fig. 1. For a gently curved paraboloid (with κ 0.15 cm−1), grain boundaries form long (possibly branched)≈ chains running from one side to the other and passing through the center. As the cur- vature of the paraboloid is increased, however, this long grain boundary is observed to terminate in the center (see Fig. 3). For R = 5 cm, the elastic theory of de- −1 fects predicts a structural transition at κc = 0.27 cm FIG. 2: (color online) Translational and orientational corre- in the limit of large core [11]. In this limit the lation functions (g and g6, respectively) for rafts with (a,b) creation of defects is strongly penalized and the lattice − κ ≈ 0.32 cm 1, a = (0.8410 ± 0.0025) mm, and (c,d) κ ≈ 0.15 has the minimum number of disclinations required by the −1 cm , a = (0.9071 ± 0.0037) mm. All the curves are plotted topology of the embedding surface. In a low curvature as functions of r/a, where r is the planar distance from the paraboloid (κ<κc) these disclinations are preferentially center and a is the bubble radius. The envelope for the crys- located along the boundary to reduce the elastic talline solid decays algebraically (dashed line), while the ori- entational correlation function approaches the constant value of the system. When the aspect ratio of the paraboloid 0.8. exceeds a critical value κc(R), however, the curvature at the origin is enough to support the existence of a 5 fold disclination and the system undergoes a struc- Zj − ψj (r) = (1/Zj ) Pk=1 exp(6iθjk), where Zj is the num- tural transition. In the limit of large core energies, when ber neighbors of i and θjk is the angle between the j k only six disclinations are present, such a transition im- − bonds and a reference axis. One expects g6(r) to decay plies a change from the C6v to the C5v rotational symme- exponentially in a disordered , algebraically in a try group. Together with our experimental observations, hexatic phase and to approach a non zero value in the these considerations point to the following mechanism for case of a crystalline solid. In the systems studied we find scar nucleation in a paraboloidal crystal. In the regime that g6(r) to approaches value 0.8 in the distance of 5–6 in which the creation of defects is energetically inexpen- lattice spacings. sive, geometrical frustration due to the confinement of Of particular interest is the structure of the grain the lattice in a simply connected region is responsible for boundaries appearing in the paraboloidal lattice for dif- the formation of a long side-to-side grain boundary. But ferent values of the curvature parameter κ. Grain bound- aries form in the bubble array during the growing process as a consequence of geometrical frustration. As noted, any triangular lattice confined in a simply connected re- gion with the topology of the disk is required to have a net disclination charge Q = 6. Each disclination has an energy associated with long-range elastic distortion of the lattice and a short range core-energy. While the former is responsible for the emergent of a geomet- rically frustrated crystal, the latter plays the role of the energy-penalty required for the creation of a single discli- nation defect. Although dependent on the interparticle interactions, and so different from system to system, the disclination core-energy is generally much smaller than the elastic energy, especially in the case of a bubble array where the particle-particle interaction is dominated by hard-core repulsion. Defects are then favored to prolifer- FIG. 3: (color online) Example of terminating grain boundary ate throughout the crystal. In the flat plane, however, the scar for a system with large Gaussian curvature. The scar elastic due to an isolated disclination is extremely starts from the circular perimeter of the vessel and terminates high and defects are energetically favored to cluster in the roughly in the center carrying a net +1 topological charge. 4

theories and provide system sizes that are prohibitive for numerical simulations. Future experiments might explore varying the shape of the vessel to investigate the role of the boundary on the bulk order. This setup is also suitable for studying dynamical phenomena such as the glide of dislocations in the process as well as the formation of vacan- cies and interstitials (e.g., following Ritacco et al. [17], looking at a cascade of bursting bubbles on a paraboloid). We acknowledge David Nelson for stimulating dis- FIG. 4: (color online) Delaunay triangulation of a portion of cussions and Philip Arnold of the Syracuse University ≈ −1 the paraboloidal lattice with κ 0.32 cm near the center Physics Department machine shop for assisting in the (left) and the boundary (right). Red and blue lines repre- sent the geodesics directed toward first and second neighbors, construction of the experimental apparatus. The work respectively. of MJB, LG and HS was supported by the NSF through grants DMR-0219292 and DMR-0305407, and through funds provided by Syracuse University. The work of CKT when the curvature of the paraboloid exceeds a critical was supported in part by NSF grant DMR 0606424. value (dependent on the radius of the circular bound- ary), the existence of a +1 disclination near the center is energetically favored. Such a disclination serves as a nucleation site for 5 7 dislocations and the side-to-side [1] W. L. Bragg and J. F. Nye, Proc. Roy. Soc. A 190, 474 grain boundary is replaced− by a terminating center-to- (1947). side scar. For intermediate curvature the theory also [2] C. Kittel, Introduction to solid state physics, (Wiley, New predicts a regime of coexistence of isolated disclinations York, 1966). and scars due to the variable Gaussian curvature. For [3] R. P. Feynman, R. B. Leighton and M. , Feyn- dense systems (i.e. number of subunits larger than a few man lectures on physics Vol. 2, (Addison-Wesley, Read- hundred for our geometry), the coexistence is suppressed ing, Ma., 1963) Chapter 30. 237 because the embedding surface will appear nearly flat at [4] A. W. Simpson and P. H. Hodkinson, Nature , 320 the length scale of a lattice spacing. The bulk of the (1972). [5] A. Gouldstone, K. J. Van Vliet and S. Suresh, Nature system is thus populated uniquely by scars. Our data 411, 656 (2001). confirm this prediction. [6] Y. Wang, K. Krishan and M. Dennin, Phys. Rev. E 73, Away from the center of the paraboloid, we have com- 031401 (2006). pared the crystalline directions with the geodesics start- [7] M. J. Bowick, D. R. Nelson and A. Travesset, Phys. Rev. ing from a given reference point (see Fig. 4). Near B 62, 8738 (2000); M. Bowick, A. Cacciuto, D. R. Nelson, 89 the boundary, the directions of both first and second and A. Travesset, Phys. Rev. Lett. , 185502 (2002); A. Perez-Garrido, M. J. W. Dodgson, and M. A. Moore, neighbors (in red and blue respectively), are reason- Phys. Rev. B 56, 3640 (1997). ably aligned with the geodesics. The alignment becomes [8] M. Bowick, D. R. Nelson, and A. Travesset, Phys. Rev. decorrelated after roughly five lattice spacings with the E 69, 041102 (2004). decorrelation more pronounced in the radial direction [9] V. Vitelli and D. R. Nelson, Phys. Rev. E 70, 051105 (maximal principal curvature) where the normal curva- (2004). ture is largest. As one gets closer to the center, the [10] V. Vitelli, J. B. Lucks, and D. R. Nelson, Proc. Natl. Acad. Sci. USA 103, 12323 (2006). geodesic correlation becomes weaker and almost com- 76 pletely vanishes along the radial direction. Along the [11] L. Giomi and M. J. Bowick, Phys. Rev. B , 054106 (2007). angular direction (minimal principal curvature), on the [12] A. A. Koulakov and B. I. Shklovskii, Phys. Rev. B 57, other hand, the crystalline axes appear aligned with the 2352 (1998); M. Kong, B. Partoens, A. Matulis and F. geodesic directions. M. Peeters, Phys. Rev E 69, 036412 (2004); A. Mughal In this paper we have demonstrated a visualizable ex- and M. A. Moore, Phys. Rev E 76, 011606 (2007). ample of a non-Euclidean, non-spherical crystal. Despite [13] A. R. Bausch et al., Science 299, 1716 (2003). 21 the simplicity of our technique we found good agreement [14] T. Einert et al., Langmuir , 12076 (2005). [15] J. C. Crocker and D. G. Grier, J. Interface Sci. with the elastic theory of defects in curved space. Bub- 179, 298 (1996). ble rafts are shown to be effective models for the study of [16] P. M. Chaikin and T. C. Lubensky, Principles of Con- non-Euclidean crystallography. Bubble assemblages pro- densed Physics, (Cambridge University Press, 3 vide a large number of particles (order 10 ) with very Cambridge, 2000). simple and inexpensive equipment. They give access to [17] H. Ritacco, F. Kiefer and D Langevin, Phys. Rev. Lett. details that are necessarily unavailable to continuous field 98, 244501 (2007).