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Register a new userLindemann melting criterion in two dimensions
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Sergey Khrapak
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It is demonstrated that the Lindemanns criterion of melting can be formulated for two-dimensional classical solids using statistical mechanics arguments. With this formulation the expressions for the melting temperature are equivalent in three and two dimensions. Moreover, in two dimensions the Lindemanns melting criterion essentially coincides with the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young melting condition of dislocation unbinding.
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The force-level Elastically Collective Nonlinear Langevin Equation theory of activated relaxation in glass-forming free-standing thin films is re-visited to improve its treatment of collective elasticity effects. The naive cut off of the isotropic bulk displacement field approximation is improved to explicitly include spatial anisotropy with a modified boundary condition consistent with a step function liquid-vapor interface. The consequences of this improvement on dynamical predictions are quantitative but of significant magnitude and in the direction of further speeding up dynamics and further suppressing Tg. The theory is applied to thin films and also thick films to address new questions for three different polymers of different dynamic fragility. Variation of the vitrification time scale criterion over many orders of magnitude is found to have a minor effect on changes of the film-averaged Tg relative to its bulk value. The mobile layer length scale grows strongly with cooling and correlates in a nearly linear manner with the effective barrier deduced from the corresponding bulk isotropic liquid alpha relaxation time. The theory predicts a new type of spatially inhomogeneous dynamic decoupling corresponding to an effective factorization of the total barrier into its bulk temperature-dependent value multiplied by a function that only depends on location in the film. The effective decoupling exponent grows as the vapor surface is approached. Larger reductions of the absolute value of Tg shifts in thin polymer films are predicted for longer time vitrification criteria and more fragile polymers. Quantitative no-fit-parameter comparisons with experiment and simulation for film-thickness-dependent Tg shifts of PS and PC are in reasonable accord with the theory, including a nearly 100 K suppression of Tg in 4 nm PC films. Predictions are made for polyisobutylene thin films.
Magneto-rheological elastomers (MREs) are functional materials that can be actuated by applying an external magnetic field. MREs comprise a composite of hard magnetic particles dispersed into a nonmagnetic elastomeric matrix. By applying a strong magnetic field, one can magnetize the structure to program its deformation under the subsequent application of an external field. Hard MREs, whose coercivities are large, have been receiving particular attention because the programmed magnetization remains unchanged upon actuation. Hence, once a structure made of a hard MRE is magnetized, it can be regarded as magnetized permanently. Motivated by a new realm of applications, there have been significant theoretical developments in the continuum description of hard MREs. By reducing the 3D description into 1D or 2D via dimensional reduction, several theories of hard magnetic slender structures such as linear beams, elastica, and shells have been recently proposed. In this paper, we derive an effective theory for MRE rods under geometrically nonlinear 3D deformation. Our theory is based on reducing the 3D magneto-elastic energy functional for the hard MREs into a 1D Kirchhoff-like description. Restricting the theory to 2D, we reproduce previous works on planar deformations. For further validation in the general case of 3D deformation, we perform precision experiments with both naturally straight and curved rods under either constant or constant-gradient magnetic fields. Our theoretical predictions are in excellent agreement with both discrete simulations and precision-model experiments. Finally, we discuss some limitations of our framework, as highlighted by the experiments, where long-range dipole interactions, which are neglected in the theory, can play a role.
We present a detailed numerical simulation study of a two dimensional system of particles interacting via the Weeks-Chandler-Anderson potential, the repulsive part of the Lennard-Jones potential. With reduction of density, the system shows a two-step melting: a continuous melting from solid to hexatic phase, followed by a a first order melting of hexatic to liquid. The solid-hexatic melting is consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario and shows dislocation unbinding. The first order melting of hexatic to fluid phase, on the other hand, is dominated by formation of string of defects at the hexatic-fluid interfaces.
It is well-known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function $S_2(mathbf{r})$ is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function $S_2(r)$. In this work, we consider three different classes of Debye random media. First, we generate the most probable class using the Yeong-Torquato construction algorithm. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.
The hexatic fluid refers to a phase in between a solid and a liquid which has short range positional order but quasi-long range orientational order. In the celebrated theory of Berezinskii, Kosterlitz and Thouless and subsequently refined by Halperin, Nelson and Young, it was predicted that a 2-dimensional hexagonal solid can melt in two steps: first, through a transformation from a solid to a hexatic fluid which retains quasi long range orientational order and then from a hexatic fluid to an isotropic liquid. In this paper, using a combination of real space imaging and transport measurements we show that the 2-dimensional vortex lattice in a-MoGe thin film follows this sequence of melting as the magnetic field is increased. Identifying the signatures of various transitions on the bulk transport properties of the superconductor, we construct a vortex phase diagram for a two dimensional superconductor.
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