Entropic Crystal-Crystal Transitions of Brownian Squares K. Zhao, R. Bruinsma, and T.G. Mason This supplementary material contains the following sections: image processing methods, measurements of Brownian dynamics, Fourier transform analysis, lattice angle determination, calculation of order parameters and selected correlation functions, and microscopic observations of systems of larger square particles. Video Particle Tracking: Positions and Orientations of Squares Figure S1 is an example showing centers and orientations of squares found by our modified Interactive Data Language (IDL) routine based on the particle tracking programs of Crocker and Grier (1). The uncertainty in the orientation of a square found by this method is about ±7 for a typical image in which about 400 squares are observed. This angular uncertainty can be improved by using higher magnification, yet this also reduces the number of squares in the field of view. A conservative estimate of the uncertainty in the center position of a square is ± 0.1 µm (i.e. in either coordinate). Measurements of Brownian Dynamics By performing positional and rotational tracking of a sequence of images in movies taken at various φ A, we have analyzed the dynamics of individual squares in several different phases using an IDL particle tracking program. The 2D mean square displacement (MSD), <Δr 2 (t)>, corresponding to the translational motion of the center of a square, is shown in Fig. S2(a); the 1D mean square angular displacement, <Δθ 2 (t)>, page 1 of 14
corresponding to rotational motion of a square, is shown in Fig. S2(b). Here, we use angle brackets to denote both time and ensemble averages over all particles in the movies. At lower area fractions, corresponding to the isotropic phase, a square diffuses by Brownian motion both translationally and rotationally; collisions of a square with neighboring squares are relatively infrequent. However, at higher area fractions, both translational diffusion and rotational diffusion of a square become more confined, since a given square collides more frequently with neighboring squares around it. At such higher φ A, local collective motion of squares plays a more important role in the system's dynamics. In the RX phase, squares are translationally confined to their lattice sites but still can rotate freely to explore all angles, so the rotational MSD for the RX phase still increases with time, indicating that the rotational motion remains ergodic. However, at even higher φ A in the RB phase, not only do the squares exhibit highly restricted translational Brownian fluctuations about lattice sites due to crowding, but they also exhibit highly restricted rotational fluctuations, too. In the RB phase, tip-tip passage events of neighboring squares are rarely observed in a given field of view over a long time scale (i.e. many hours), and such rare events are usually associated with very infrequent local collective translational fluctuations of at least several squares within the lattice. This indicates that the rotational motion of the squares at high φ A well inside the RB phase is effectively non-ergodic, and the rotational MSD curve consequently exhibits a long-time plateau. Fourier Transforms of Microscope Images For each image, using standard procedures, we crop a 512 pixel x 512 pixel page 2 of 14
portion of the image from the center of the raw 550 pixel x 550 pixel image, convert to grayscale, and use the standard FFT command in IDL to perform the Fast Fourier Transform. This command returns a 512 pixel x 512 pixel grayscale FFT intensity having 8-bit dynamic range. Since the primary features of interest in the FFT are within a 100 pixel x 100 pixel region at the center, we crop the FFT and only show this smaller region. We block the center of the FFT using a small black square (7 pixel x 7 pixel). We use a color look-up table to color-code the intensity values over the range from zero to 255. Lattice Angle Determination In the RX and the RB phases, the lattice angle α is directly obtained from the peaks of the FFT patterns using ImageJ. In the CE region, since both hexagonal crystallites and rhombic crystallites appear in the same image, the FFT pattern is a mixture reflecting both structures and is therefore difficult to use. Instead, to obtain the average lattice angle for each type of crystallite, first we identify rhombic crystallites and hexagonal crystallites using the following criteria: rhombic crystallites: ϕ 4 > 0.65 hexagonal crystallites: ϕ 4 0.65 and ϕ 6 > 0.5, where ϕ 4 and ϕ 6 are the local 4-fold and 6-fold bond-orientational order parameters, respectively (see section below for definitions). Subsequently, for each crystallite, we directly estimate the lattice angle from the center positions of particles in the crystallite. Order Parameters and Correlation Functions The order parameters and correlation functions that we calculate are based on page 3 of 14
standard definitions (2-5). For each particle, identified by j, we define a complex number, the local m-fold bond-orientational order parameter ϕ m :. (S1) Here, N j is the number of nearest neighbors of particle j, and θ jk is the angle between an arbitrary fixed axis and the line connecting the centers of particles j and k. There are several possible ways of calculating N j. For 6-fold bond-orientational order, N j is obtained from video images by Voronoi construction of the positions of particles; whereas, for 4- fold bond-orientational order, we calculate ϕ 4 using the first 4 nearest neighbors (i.e. N j = 4). The m-fold positional order parameter ζ m for each particle j is defined as:, (S2) where is the reciprocal lattice vector of an appropriate lattice. For example, to calculate the hexagonal positional order parameter, the reciprocal hexagonal lattice is used, and similarly to calculate the square positional order, the reciprocal square lattice vector is used. For the isotropic phase (I), there is no periodic lattice structure, so we choose the reciprocal lattice vector of a corresponding hexagonal lattice structure at the same area fraction. After calculating ϕ m and ζ m for all N particles at a fixed particle area fraction, we determine the global m-fold bond-orientational order parameter ψ m :, (S3) where ω represents a global phase. Similarly, the global positional order parameter S m is: page 4 of 14
(S4) In order to calculate S m, the reciprocal lattice vector must be first obtained. This can be determined from the spacing and orientation of the lattice. The lattice spacing can be inferred from the density, and the lattice orientation is the phase ω from the calculation of global bond orientation order parameter. In practice, we permit the lattice orientation to vary in the range between 0 and 2π, calculate a set of possible S m, and then choose the maximum of this set as the order parameter s actual value. Without correction, this numerical approach can lead to non-zero values of S m even in the isotropic phase. Since both positional and orientation order parameters approach zero in the dilute isotropic phase, we ensure that the order parameters go to zero by appropriately correcting for a constant offset introduced by this numerical approach, by subtracting using an average of first three points determined at low φ A. Since a regular square is anisometric, it is necessary to calculate the global m-fold molecular orientational order parameter Φ m :. (S5) Here, γ j is the angle of orientation of particle j. In order to remove the degeneracy of molecular orientation due to the 4-fold symmetry of squares, we use. For m = 2, Φ 2 is the nematic order parameter; for m = 4, Φ 4 is the tetratic order parameter; and, for m = 6, Φ 6 is the hexatic order parameter. In addition to order parameters, correlation functions also reveal important details about the spatial extent of order in 2-d thermal systems of platelets. Using standard page 5 of 14
conventions, the bond-orientational correlation function is defined as:, (S6) the spatial correlation function relating to positional order is defined to be:, (S7) and the molecular orientational correlation function is defined as:. (S8) Here, Re represents an operator returning the real part of the value, and < > means taking the ensemble average. Changes in the order parameters are consistent with the assignment of the phases and area fractions associated with the phase transitions. As the system is compressed from I to RX phases, both S 6 and ψ 6 increase rapidly and are highly correlated while S 4 and ψ 4 remain small [Fig. S3(a), (b)]. S 6 reaches a maximum within RX. The maximum value of S 6 is significantly below unity due to defects in the lattice structure and longerrange decorrelations in position that give rise to the smearing in Fig. 2 of the main text. In the phase coexistence region (CE), as crystallites of rhombic lattice grow, S 4 and ψ 4 increase while S 6 drops very quickly. As φ A further increases, the squares are in the pure RB phase, where S 4 and ψ 4 become bigger and are also highly correlated. Since a single lattice cannot be precisely defined in CE, we refrain from plotting the calculated 6-fold order parameters above the pure RX phase. Although the 4-fold order parameters are not well defined below the pure RB phase, either, we show the results of our calculations there to display the growth in 4-fold order through and above CE as φ A is increased, recognizing that the values of these order parameters are strictly meaningful in the pure page 6 of 14
RB phase. We also show a rise in Φ 4 (φ A ) [Fig. S3(c)] that is highly correlated with the rises in ψ 4 (φ A ) and S 4 (φ A ), indicating that the average orientations of individual squares have a strong degree of correlation with the lattice directions and the development of 4- fold bond-orientational order that arises in the RB lattice. This reveals that the overall lattice structure formed and shapes of the constituent particles are typically highly interdependent at high φ A. From our observations, we have also determined the 4-fold and 6-fold correlation functions, in addition to the simple g(r/d) shown in the main text in Fig. 2. For each of the images in Fig. 1 of the main text, we have calculated 6-fold and 4-fold bond orientational correlation functions g 6 (r/d) and g 4 (r/d), and hexagonal translational order and square translational order correlation functions g s 6 (r/d) and g s 4 (r/d). In addition, we have calculated the correlation function for square molecular orientation g mo 4 (r/d). Due to the limited dynamic range in r/d, limited by our field of view, some of the order parameters have limited utility, whereas others provide additional insight into the spatial range of order. As an example, in Fig. S4(a) and S4(b), we show g 6 and g 4. In the I phase ( ), both correlation functions decay exponentially. In the pure RX phase ( ), g 6 has an algebraic decay rather than exponential decay, and clearly g 6 has a decay exponent that is closer to zero than -1/4, which is the slope of dashed line in Fig. S4(a). By contrast, in the pure RX phase, g 4 (r) decays exponentially. At higher φ A, in the pure RB phase ( ), g 4 basically shows no decay, indicating nearly perfect crystalline order over our limited viewing area. We can estimate correlation lengths ξ by fitting the correlation functions using simple exponential decay form exp(-r/ξ). Although not all decays are exponential, we use page 7 of 14
this simplistic approach to determine some bounds on ξ. We find that, for the I phase, all correlation functions decay to a background level within one diameter of a square, so all correlation lengths are no more than D. For the RX phase, the correlation length for g 6 is ξ 6 > 200D, and the correlation length for g s 6 is ξ s 6 ~ 30D. In the pure RX phase, all other correlation lengths are ~ D, so there are no significant 4-fold correlations. In the pure RB phase, we find that ξ 4 > 200D, ξ s 4 ~ 180D, and ξ mo 4 > 200D. Larger Square Particles: Smaller Relative Corner-Rounding Similar 2-d osmotic compression experiments using larger square particles (outer edge length L = 4.5 µm; corner-rounding-to-length ratio ζ = 0.06) further confirm the high-density phases observed for smaller square particles. Since the radius of curvature associated with the corner-rounding is set by the lithographic fidelity of the stepper system, independent of the mask design, larger particles have proportionately less cornerrounding than smaller particles. They also enable exploration of slightly larger particle densities, although reaching equilibrium becomes more difficult. Examples of images for the larger square particles in rhombic structure are shown in Fig. S5. We find that the rhombic structure can persist up to quite large φ A even in the limit where the corner rounding parameter is small: ζ << 1. The positions of the squares self-adjust into a rhombic lattice to enable a larger range of rotational angles (i.e. more rotational entropy), and this contributes to a strong degree of spatial correlation in RB while reducing the effects of possible sliding degeneracy that could be present in the SQ phase. page 8 of 14
References: 1. Crocker JC, Grier DG (1996) Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci. 179: 298-310. 2. Murray CA, Winkle DHV (1987) Experimental observation of two-stage melting in a classical two-dimensional screened coulomb system. Phys. Rev. Lett. 58: 1200-1203. 3. Strandburg KJ (1988) Two-dimensional melting. Rev. Mod. Phys. 60: 161-207. 4. Nelson DR (2002) Defects and geometry in condensed matter (Cambridge Univesity Press, Cambridge). 5. Wojciechowski KW, Frenkel D (2004) Tetratic phase in the planar hard square system. Comput. Meth. Sci. Technol. 10: 235-255. page 9 of 14
Figures: Figure S1. Example analysis results for a system of squares (L = 2.4 µm) in a rhombic lattice using IDL image processing. The area fraction shown is φ A = 0.74. The center (+) and one vertex (*) of each square are determined, yielding its position and orientation. page 10 of 14
Figure S2. Time- and ensemble-averaged mean square displacements (MSDs) of square particles as a function of the time interval, t, at particle area fractions φ A : 0.57 (circles, I phase), 0.63 (squares, RX phase), and 0.77 (triangles, RB phase) for (a) translational motion and (b) rotational motion. Here, D is the distance between the centers of a pair of squares when their vertices touch: D = L, where L is the length of a square's edge. page 11 of 14
Figure S3. Measures of spatial and orientational order as a function of square particle area fraction φ A : (a) 6-fold translational order S 6 (circles) and 4-fold translational order S 4 (squares); (b) 6-fold bond-orientational order ψ 6 (circles) and 4-fold bond-orientational order ψ 4 (squares); (c) 4-fold molecular-orientational order Φ 4 (squares). page 12 of 14
Figure S4. Calculated correlation functions at area fractions φ A : 0.52 (, I), 0.62 (, RX), 0.65 (,CE) and 0.74 (, RB). (a) 6-fold bond-orientational correlation function g 6 (r/d). For reference, the dashed line is g 6 (r/d) ~ (r/d) -1/4. (b) 4-fold bond-orientational correlation function g 4 (r/d). Dashed line: g 4 (r/d) ~ (r/d) -1/4. page 13 of 14
Figure S5. Optical microscope images of larger square particles, which have reduced corner-rounding (square's outer edge length: L = 4.5 µm, thickness: h = 1 µm), in the rhombic phase after 2-d osmotic compression to high area fractions φ A. The internal square hole does not affect the phase behavior and is ignored when determining φ A. (a) φ A 0.70, rhombic angle α 79, (b) φ A 0.85, α 88. page 14 of 14