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Related Concept Videos

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Recrystallization is a purification technique used to separate impurities from solid compounds. In this technique, no chemical reactions occur. Instead, it exploits physical properties only, specifically, the solubility differences between the desired compound and impurities, either at a single temperature or at different temperatures, and under other selected conditions. The solid-solution equilibrium (solubility equilibrium) of each component in the solution represents a binary phase...
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Crystallization is a phase transformation process in which crystals are precipitated from a supersaturated solution or formed from other sources. During crystallization, atoms or molecules arrange themselves into a well-defined, rigid crystal lattice to minimize energy.
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Unlike ionic or small covalent molecules, polymers do not form crystalline solids due to the diffusion limitations of their long-chain structures. However, polymers contain microscopic crystalline domains separated by amorphous domains.
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The mechanism for anionic chain-growth polymerization involves initiation, propagation, and termination steps. In the initiation step, a nucleophilic anion, such as butyl lithium, initiates the polymerization process by attacking the π bond of the vinylic monomer. As a result, a carbanion, stabilized by the electron‐withdrawing group, is generated. The resulting carbanion acts as a Michael donor in the propagation step and attacks the second vinylic monomer, which acts as a Michael...
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Metallic Solids

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Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
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The radical chain-growth polymerization mechanism consists of three steps: initiation, propagation, and termination of polymerization. The polymerization initiates when a free radical generated from the radical initiator adds to the unsaturated bond in the monomer. The unpaired electron of the free radical and one π electron in the unsaturated bond creates a σ bond between the free radical and the monomer. As a result, the other π electron in the unsaturated bond converts this species into...
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Growing Protein Crystals with Distinct Dimensions Using Automated Crystallization Coupled with In Situ Dynamic Light Scattering
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Note on Crystallization for Alternating Particle Chains.

Laurent Bétermin1, Hans Knüpfer2, Florian Nolte2

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Statistical Physics
|October 22, 2020
PubMed
Summary
This summary is machine-generated.

We prove that particles in one-dimensional chains form equidistant configurations, known as crystallization, for various interactions. This finding applies to neutral and non-neutral systems, including those with Coulomb potentials.

Keywords:
ConvexityCrystallizationEnergy minimizationIonic crystals

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Investigating particle arrangements in periodic chains is crucial for understanding material properties.
  • Crystallization, or the formation of equidistant configurations, is a fundamental phenomenon in many-body systems.

Purpose of the Study:

  • To demonstrate the optimality of equidistant configurations (crystallization) in one-dimensional periodic chains with alternating particle types.
  • To analyze systems with mirror-symmetric potentials, including inverse power laws and Coulomb interactions.

Main Methods:

  • Analytical proofs for crystallization at any scale.
  • Analysis of high-density systems with specific potential forms (inverse power laws with origin repulsion).
  • Derivation of necessary conditions for crystallization using Fourier transforms.

Main Results:

  • Proved crystallization at any scale for neutral and non-neutral systems with inverse power law interactions, including the 3D Coulomb potential.
  • Showed the minimality of equidistant configurations at high density for systems with inverse power laws and repulsion at the origin.
  • Derived a necessary condition for high-density crystallization based on the Fourier transform of potential sums.

Conclusions:

  • The equidistant configuration is optimal for one-dimensional periodic chains under various conditions.
  • The study provides a mathematical framework for understanding crystallization in diverse physical systems.
  • Positivity of the Fourier transform of interaction potentials is a key indicator for high-density crystallization.