Specific Heat in Two-Dimensional Melting

We report the specific heat $c_N$ 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 $c_N(T)$ 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.

Figure 1

Spatial orientational correlation $g_6(r)=⟨{\psi }_6^{*}(r){\psi }_6(0)⟩$ at different system temperatures $\mathrm{\Gamma }$ for experiment (a) and simulation (b). The data are plotted on a log-log scale in reduced coordinates, where $a_0=(n{)}^{-1/2}$ is the mean particle distance in the respective system. The decay behavior of $g_6(r)$ has distinct characteristics in the solid (constant), the hexatic fluid (algebraic decay), and in the isotropic fluid (exponential decay). An algebraic exponent of $-1/4$ marks the hexatic-isotropic transition [2].

Figure 2

Filled symbols correspond to experiment, open symbols to simulation. (a) Specific heat $c_N/k_B$ as a function of $\mathrm{\Gamma }$ calculated via the derivative approach. The inset shows the energy per particle. (b) $c_N/k_B$ for experiment (right side of the right scale) and simulation (left side of the right scale) from energy fluctuations and the total defect number density (left scale), counting all defects ($\rho$) and isolated disclinations (${\rho }_{\text{disc}}$). (c) and (d) The behavior of isolated dislocations and disclinations analyzed with tanh fit to show the steepest increase. The background color of (b),(c),(d) is organized as follows: the solid and isotropic fluid found by the experiment are colored blue and red (dark gray), respectively. The hexatic phase is colored green (light gray). The color gradient shows the estimated error in determining the transition temperatures.