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Register a new userDestruction of ultra-slow diffusion in a three dimensional cylindrical comb structure
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We present a rigorous result on ultra-slow diffusion by solving a Fokker-Planck equation, which describes anomalous transport in a three dimensional (3D) comb. This 3D cylindrical comb consists of a cylinder of discs threaten on a backbone. It is shown that the ultra-slow contaminant spreading along the backbone is described by the mean squared displacement (MSD) of the order of $ln (t)$. This phenomenon takes place only for normal two dimensional diffusion inside the infinite secondary branches (discs). When the secondary branches have finite boundaries, the ultra-slow motion is a transient process and the asymptotic behavior is normal diffusion. In another example, when anomalous diffusion takes place in the secondary branches, a destruction of ultra-slow (logarithmic) diffusion takes place as well. As the result, one observes enhanced subdiffusion with the MSD $sim t^{1-alpha}ln t$, where $0<alpha<1$.
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Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of $e^t$. The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.
We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.
We present numerical simulations of a model of cellulose consisting of long stiff rods, representing cellulose microfibrils, connected by stretchable crosslinks, representing xyloglucan molecules, hydrogen bonded to the microfibrils. Within a broad range of temperature the competing interactions in the resulting network give rise to a slow glassy dynamics. In particular, the structural relaxation described by orientational correlation functions shows a logarithmic time dependence. The glassy dynamics is found to be due to the frustration introduced by the network of xyloglucan molecules. Weakening of interactions between rod and xyloglucan molecules results in a more marked reorientation of cellulose microfibrils, suggesting a possible mechanism to modify the dynamics of the plant cell wall.
Amorphous solids such as glass are ubiquitous in our daily life and have found broad applications ranging from window glass and solar cells to telecommunications and transformer cores. However, due to the lack of long-range order, the three-dimensional (3D) atomic structure of amorphous solids have thus far defied any direct experimental determination without model fitting. Here, using a multi-component metallic glass as a proof-of-principle, we advance atomic electron tomography to determine the 3D atomic positions in an amorphous solid for the first time. We quantitatively characterize the short-range order (SRO) and medium-range order (MRO) of the 3D atomic arrangement. We find that although the 3D atomic packing of the SRO is geometrically disordered, some SRO connect with each other to form crystal-like networks and give rise to MRO. We identify four crystal-like MRO networks - face-centred cubic, hexagonal close-packed, body-centered cubic and simple cubic - coexisting in the sample, which show translational but no orientational order. These observations confirm that the 3D atomic structure in some parts of the sample is consistent with the efficient cluster packing model. Looking forward, we anticipate this experiment will open the door to determining the 3D atomic coordinates of various amorphous solids, whose impact on non-crystalline solids may be comparable to the first 3D crystal structure solved by x-ray crystallography over a century ago.
We study the phase ordering dynamics of a two dimensional model colloidal solid using molecular dynamics simulations. The colloid particles interact with each other with a Hamaker potential modified by the presence of equatorial patches of attractive and negative regions. The total interaction potential between two such colloids is, therefore, strongly directional and has three-fold symmetry. Working in the canonical ensemble, we determine the tentative phase diagram in the density-temperature plane which features three distinct crystalline ground states viz, a low density honeycomb solid followed by a rectangular solid at higher density, which eventually transforms to a close packed triangular structure as the density is increased further. We show that when cooled rapidly from the liquid phase along isochores, the system undergoes a transition to a strong glass while slow cooling gives rise to crystalline phases. We claim that geometrical frustration arising from the presence of many crystalline ground states causes glassy ordering and dynamics in this solid. Our results may be easily confirmed by suitable experiments on patchy colloids.
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