Hierarchical thermoelectrics: crystal grain boundaries as scalable phonon scatterers

Thermoelectric materials are strategically valuable for sustainable development, as they allow for the generation of electrical energy from wasted heat. In recent years several strategies have demonstrated some efficiency in improving thermoelectric properties. Dopants affect carrier concentration, while thermal conductivity can be influenced by alloying and nanostructuring. Features at the nanoscale positively contribute to scattering phonons, however those with long mean free paths remain difficult to alter. Here we use the concept of hierarchical nano-grains to demonstrate thermal conductivity reduction in rocksalt lead chalcogenides. We demonstrate that grains can be obtained by taking advantage of the reconstructions along the phase transition path that connects the rocksalt structure to its high-pressure form. Since grain features naturally change as a function of size, they impact thermal conductivity over different length scales. To understand this effect we use a combination of advanced molecular dynamics techniques to engineer grains and to evaluate thermal conductivity in PbSe. By affecting grain morphologies only, i.e. at constant chemistry, two distinct effects emerge: the lattice thermal conductivity is significantly lowered with respect to the perfect crystal, and its temperature dependence is markedly suppressed. This is due to an increased scattering of low-frequency phonons by grain boundaries over different size scales. Along this line we propose a viable process to produce hierarchical thermoelectric materials by applying pressure via a mechanical load or a shockwave as a novel paradigm for material design

A lower bound for nodal count on discrete and metric graphs

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference

On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure

Dynamics of nodal points and the nodal count on a family of quantum graphs

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr\"odinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure

Overview of Stabilizing Ligands for Biocompatible Quantum Dot Nanocrystals

Luminescent colloidal quantum dots (QDs) possess numerous advantages as fluorophores in biological applications. However, a principal challenge is how to retain the desirable optical properties of quantum dots in aqueous media while maintaining biocompatibility. Because QD photophysical properties are directly related to surface states, it is critical to control the surface chemistry that renders QDs biocompatible while maintaining electronic passivation. For more than a decade, investigators have used diverse strategies for altering the QD surface. This review summarizes the most successful approaches for preparing biocompatible QDs using various chemical ligands

Динамика насаждений сосны крымской (Pinus pallasiana L. ) в горном Крыму

За период 1938 – 2000 гг. произошло "смещение" мест произрастания сосновых древостоев в более богатые и влажные условия. Увеличилась площадь насаждений сосны крымской. Площадь сосновых культур больше чем в 3 раза превысила площадь естественных лесов. Средний запас сосновых лесов составляет 136 м³/га. Древостои ІІ и высших классов бонитета занимают лишь 12 % сосновых лесов.За період 1938 – 2000 рр. відбувся "зсув" місць виростання соснових деревостанів у багатші й вологіші умови. Збільшилася площа насаджень сосни кримської. Площа соснових культур у понад 3 рази перевершила площу природних лісів. Середній запас соснових лісів становить 136 м³/га. Деревостани ІІ і вищих класів бонітету займають лише 12 % соснових лісів.For 1938 – 2000 "displacement" of pine stands to more rich and moist sites has occurred. Area of P. pallasiana has increased. Pine plantation area has exceeded area of natural pine forests more than 3 times. Mean stock of pine forests is 136 m³/ha. Stands of the ІІ and higher growth classes take only 12 % of pine forests