In a recent commentary, J. M. Kosterlitz described how D. Thouless and he got motivated to investigate melting and suprafluidity in two dimensions [Kosterlitz JM (2016) J Phys Condens Matter 28:481001]. It was due to the lack of broken translational symmetry in two dimensions-doubting the existence of 2D crystalsand the first computer simulations foretelling 2D crystals (at least in tiny systems). The lack of broken symmetries proposed by D. Mermin and H. Wagner is caused by long wavelength density fluctuations. Those fluctuations do not only have structural impact, but additionally a dynamical one: They cause the Lindemann criterion to fail in 2D in the sense that the mean squared displacement of atoms is not limited. Comparing experimental data from 3D and 2D amorphous solids with 2D crystals, we disentangle Mermin-Wagner fluctuations from glassy structural relaxations. Furthermore, we demonstrate with computer simulations the logarithmic increase of displacements with system size: Periodicity is not a requirement for Mermin-Wagner fluctuations, which conserve the homogeneity of space on long scales.Mermin-Wagner fluctuations | 2D ensembles | glass transition | phase transition | confined geometry F or structural phase transitions, it is well known that the microscopic mechanisms breaking symmetry are not the same in two and in three dimensions. Whereas 3D systems typically show first-order transitions with phase equilibrium and latent heat, 2D crystals melt via two steps with an intermediate hexatic phase. Unlike in 3D, translational and orientational symmetry are not broken at the same temperature in 2D. The scenario is described within the Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) theory (1-5), which was confirmed (e.g., in colloidal monolayers) (6, 7). However, for the glass transition, it is usually assumed that dimensionality does not play a role for the characteristics of the transition, and 2D and 3D systems are frequently used synonymously (8-12), whereas differences between the 2D and 3D glass transition are reported in ref. 13.In the present work, we compare data from colloidal crystals and glasses and show that Mermin-Wagner fluctuations, well known from 2D crystals, are also present in amorphous solids (14, 15). Mermin-Wagner fluctuations are usually discussed in the framework of long-range order (magnetic or structural). However, in the context of 2D crystals, they have also had an impact on dynamic quantities like mean squared displacements (MSDs). Long before 2D melting scenarios were discussed, there was an intense debate as to whether crystals and perfect longrange order (including magnetic order) can exist in 1D or 2D at all (16)(17)(18)(19). A beautiful heuristic argument was given by Peierls (17): Consider a 1D chain of particles with nearest neighbor interaction. The relative distance fluctuation between particle n and particle n + 1 at finite temperature may be ξ. Similar is the fluctuation between particle n + 1 and n + 2. The relative fluctuation between second nearest neig...
Using positional data from videomicroscopy and applying the equipartition theorem for harmonic Hamiltonians, we determine the wave-vector-dependent normal mode spring constants of a two-dimensional colloidal model crystal and compare the measured band structure to predictions of the harmonic lattice theory. We find good agreement for both the transversal and the longitudinal modes. For q-->0, the measured spring constants are consistent with the elastic moduli of the crystal.
Using videomicroscopy data of a two-dimensional colloidal system the bond-order correlation function G{6} is calculated and used to determine both the orientational correlation length xi{6} in the liquid phase and the modulus of orientational stiffness, Frank's constant F{A}, in the hexatic phase. The latter is an anisotropic fluid phase between the crystalline and the isotropic liquid phase. F{A} is found to be finite within the hexatic phase, takes the value 72/pi at the hexatic<-->isotropic liquid phase transition, and diverges at the hexatic<-->crystal transition as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. This is a quantitative test of the mechanism of breaking the orientational symmetry by disclination unbinding.
While the melting of crystals is in general not understood in detail on a microscopic scale, there is a microscopic theory for a class of two-dimensional crystals, which is based on the formation and unbinding of topological defects. Herein, we review experimental work on a colloidal two-dimensional model system with tunable interactions that has given the first conclusive evidence for the validity of this theory on a microscopic level. Furthermore, we show how the mechanism of melting depends on the particle interaction and that a strong anisotropy of the interaction leads to a changed melting scenario.
Using positional data from video-microscopy we determine the elastic moduli of two-dimensional colloidal crystals as a function of temperature. The moduli are extracted from the wave-vectordependent normal mode spring constants in the limit q → 0 and are compared to the renormalized Young's modulus of the KTHNY theory. An essential element of this theory is the universal prediction that Young's modulus must approach 16π at the melting temperature. This is indeed observed in our experiment.PACS numbers: 64.70. Dv,61.72.Lk,82.70.Dd In the early 70th Kosterlitz and Thouless [1] developed a theory of melting for two-dimensional systems. In their model the phase transition from a system with quasi-long-range order [2] is mediated by the unbinding of topological defects like vortices or dislocation pairs in the case of 2D crystals. They showed that the phase with higher symmetry has short-range translational order. Halperin and Nelson [3,4] pointed out that this phase still exhibits quasi-long-range orientational order and proposed a second phase transition now mediated by the unbinding of disclinations to an isotropic liquid. The intermediate phase is called hexatic phase. This theory, being based also on the work of Young [5], is known as KTHNY theory (Kosterlitz, Thouless, Halperin, Nelson and Young); it describes the temperature-dependent behavior of the elastic constants, the correlation lengths, the specific heat and the structure factor (for a review see [6,7]). Experiments with electrons on helium [8,9] and with 2D interfacial colloidal systems [10,11,12,13,14] as well as computer simulations [15,16,17] have been performed to test the essential elements of this theory. But research has mainly focused on the behavior of the correlation functions (an illustrative example is the work of Murray and van Winkle [11]). Only a few works can be found that deal with the elastic constants, especially the shear modulus, [9,18,19,20] even though the Lamé coefficients and their renormalization near melting take a central place in the KTHNY theory.A very strong prediction of the KTHNY theory has never been verified experimentally. It states that the renormalized Young's Modulus K R (T ), being related just to the renormalized Lamé coefficients µ R and λ R , must approach the value 16π at the melting temperature [4],which is obviously an universal property of 2D systems at the melting transition. This Letter presents experimental data for elastic moduli of a two-dimensional colloidal model system, ranging from deep in the crystalline phase via the hexatic to the fluid phase. These data, indeed, confirm the theoretical prediction expressed by eq. (1).The experimental setup is the same as already described in [23]. The system is known to be an almost perfect 2D system; it has been successfully tested, and explored in great detail, in a number of studies [14,21,22,23]. Therefore we only briefly summarize the essentials here: Spherical colloids (diameter d = 4.5 µm) are confined by gravity to a water/air interface formed by a ...
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