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Specific Heat in Two-Dimensional Melting week ending PRL 113, 127801 (2014) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2014 Specific Heat in Two-Dimensional Melting Sven Deutschländer,1 Antonio M. Puertas,2 Georg Maret,1 and Peter Keim1 1Physics Department, University of Konstanz, 78464 Konstanz, Germany 2Department of Applied Physics, University of Almeria, 04120 Almeria, Spain (Received 19 December 2013; published 18 September 2014) We report the specific heat cN around the melting transition(s) of micrometer-sized superparamagnetic particles confined in two dimensions, calculated from fluctuations of positions and internal energy, and corresponding Monte Carlo simulations. Since colloidal systems provide single particle resolution, they offer the unique possibility to compare the experimental temperatures of the peak position of cNðTÞ and symmetry breaking, respectively. While order parameter correlation functions confirm the Kosterlitz- Thouless-Halperin-Nelson-Young melting scenario where translational and orientational order symmetries are broken at different temperatures with an intermediate so called hexatic phase, we observe a single peak of the specific heat within the hexatic phase, with excellent agreement between experiment and simulation. Thus, the peak is not associated with broken symmetries but can be explained with the total defect density, which correlates with the maximum increase of isolated dislocations. The absence of a latent heat strongly supports the continuous character of both transitions. DOI: 10.1103/PhysRevLett.113.127801 PACS numbers: 61.20.Ja, 64.70.D−, 64.70.pv, 65.40.Ba KTHNY theory, a microscopic melting scenario for two- suggested that the nature and number of transitions in two dimensional solids developed by Kosterlitz, Thouless, dimensions either depends on the dislocation core energy Halperin, Nelson, and Young [1–3], motivated extended [6,34] (which might implicitly depend on the particle pair analytic theories [4–8], numerous experimental studies interaction being short or long range) or the angular [9–24], and simulations [25–44] to clarify the detailed stiffness of the crystal being lower than a critical value [8]. melting mechanism and the order of phase transitions While first-order phase transitions are known to show a in two dimensions. The KTHNY melting is mediated by discontinuity in the free energy and a δ-like divergence in the dissociation of two kinds of topological defects, the specific heat at the transition temperature, the defect dislocations and disclinations. This scenario predicts two free energy and specific heat of the two-dimensional continuous phase transitions, where translational and ori- Coulomb gas have only discontinuities and no divergence entational order is broken at different temperatures by for both transitions [2,46,47]. Thus, the behavior of the the unbinding of pairs of dislocations and disclinations. specific heat can be used to identify the order of the In a triangular lattice, a disclination is a five- or sevenfold transition. On the experimental side, there have been only oriented site and a dislocation consists of oppositely calorimetric measurements so far, e.g., on atomic mono- charged disclinations, namely, a pair of bound five- and layers on graphite which show different results concerning sevenfold sites. Both types of topological defects can be the number of peaks in the specific heat, their position treated as a Coulomb gas obeying a logarithmic interaction and magnitude [48–52]. These experiments lack a precise potential in two dimensions [45]. For the dislocation determination of symmetry switching points, leaving the unbinding, a vector charge description becomes necessary correlation to occurring phase transitions still elusive. due to the directional character of the defects. The self- Simulations of interacting dislocations show that for small screening of the vector gas is taken into account by a dislocation core energies, the specific heat has a large renormalization group analysis. A scalar gas of charged discontinuity consistent with a first-order transition while defects describes the disclination unbinding in a second for large core energies, a single moderate peak was step. The intermediate thermodynamic phase is named observed, pointing to a continuous character [34]. hexatic. Several experiments [12–15,18–22] and simula- Laplacian roughening models [26] (which are dual to tions [25–33] clearly show the existence of the hexatic 2D melting) and Lennard-Jones systems [33] display one phase, but some studies additionally report first-order nondivergent peak along the two-step KTHNY scenario characteristics [16,35–40] versus continuous order whereas a nonideal Yukawa system shows two singularities [18,19,22]. Continuous solid-hexatic and first-order hex- associated with two transitions [53]. However, in contrast atic-isotropic characteristics have been observed for hard to the atomic or molecular systems mentioned above, discs [41]. Alternatively the two-step melting can be microscopy of individual particles in colloidal systems preempted by a single first-order transition when the pair allows a direct comparison between specific heat and potential contains two length scales [42,43]. It has been symmetry switching points. 0031-9007=14=113(12)=127801(6) 127801-1 © 2014 American Physical Society week ending PRL 113, 127801 (2014) PHYSICAL REVIEW LETTERS 19 SEPTEMBER 2014 In this work, we present a melting study of super- temperature (NVT ensemble), with N ¼ 2500 particles paramagnetic colloidal spheres confined in two dimensions interacting with a dipolar potential: βVðrÞ¼Γ=r3, with and corresponding Monte Carlo simulations. The precise distances measured in units of ðπnÞ−1=2. The interactions ¼ 9 knowledge of the particle pair potential together with high are cut off at Rcut a0, which is large enough to avoid precision single particle resolution and long term stability effects from the truncation [29]. The systempffiffiffi is simulated in of the sample allows us to measure an anomaly in the a rectangular box with a size ratio 2∶ 3 and a (hard disc) specific heat and compare it with simulations. Using order 2D area fraction ϕ ¼ 0.07 to mimic the experimental condi- parameter correlation functions, we confirm in the experi- tions. Cycles of increasing Γ from the fluid to the crystal, ments and simulations the two-step KTHNY melting and subsequent decrease to the fluid again were used to scenario from a solid phase through a hexatic fluid to an confirm that there is no hysteresis within our Γ resolution. isotropic fluid, yet we find a single peak in the specific To determine the respective symmetry breaking tem- heat. Remarkably, this peak does not coincide with either peratures, we analyze the spatial correlation function ð Þ¼hψ Ãð Þψ ð0Þi transition temperature but lies within the hexatic phase. g6 r P 6 r 6 of the orientational order parameter 1 We show that it is connected to a sharp increase of the ψ 6 ¼ exp ði6θ Þ, where the sum runs over all n nj k jk j number of topological defects associated with a progressive nearest neighbors of particle j, and θ is the angle of the unbinding of dislocation pairs on heating above the solid- jk kth bond with respect to a certain reference axis. According hexatic transition temperature. Further, we do not find an to the KTHNY theory, g6ðrÞ approaches a constant value additional peak correlated to the disclination unbinding in the solid phase (long-range order), decays algebraically which might not be resolvable due to the very small ∼r−η6 in the hexatic fluid (quasi-long-range), and expo- concentration of single disclinations < 5‰ in the back- ∼ −r=ξ6 ground of a large overall defect density at the hexatic- nentially e in the isotropic fluid (short-range), with ξ isotropic transition. an orientational correlation length 6. The orientational η The experimental system consists of an ensemble of exponent 6 is inversely proportional to the orientational spherical superparamagnetic polystyrene beads, with diam- stiffness of the system: infinite in the solid, zero in the eter d ¼ 4.5 μm and mass density 1.7 kg=dm3, dissolved isotropic fluid, and finite in the hexatic phase, approaching η ∼ 1 4 and sterically stabilized with sodium dodecyl sulfate in a value of 6 = at the hexatic-isotropic transition [2,18]. ð Þ water. The colloidal suspension is sealed in a millimeter This behavior of g6 r has successfully been probed – sized glass cell where sedimentation leads to the formation and verified in experiments [13,18 21] and simulations 5 – of a monolayer (> 10 particles) on the bottom glass plate. [27,29 32]. The results for our system are shown in Fig. 1: The whole sample is under steady control and stable for for both experiment and simulation, we clearly observe the more than 20 months which allows sufficient equilibration characteristic behavior of g6ðrÞ for the different phases, times and provides ideal sample conditions, e.g., vanishing verifying the stability of the orientational quasi-long-range density gradients or drifts. The ensemble is kept at room ordered hexatic fluid. For the solid-hexatic transition, exp temperature and a highly homogeneous, finely tunable we find the (inverse) transition temperatures Γm ≈ 70.3 sim external magnetic field H perpendicular to the colloidal and Γm ¼ 69.25 and for the hexatic-isotropic transition Γexp ≈ 67 3 Γsim ¼ 68 25 layer induces a repulsive dipole-dipole interaction between i . and i . [54]. These values are the particles. This is quantified by the inverse system extracted from Fig. 1. Since the solid-hexatic transition temperature that is defined as the ratio of the mean is more difficult to locate by means of g6ðrÞ, we present magnetic energy between two neighboring particles Emag a finite-size analysis of the translational order parameter and the thermal energy, and the 2D Lindemann parameter in the Supplemental Material confirming these values [55]. It is well known that 3=2 2 Emag μ0ðπnÞ ðχHÞ the width of the hexatic phase is affected by the system size Γ ¼ ¼ ; ð1Þ kBT 4πkBT which might explain the small differences in the melting temperatures in experiment and simulation.
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