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Phase Transitions: Vaporization and Condensation02:39

Phase Transitions: Vaporization and Condensation

The physical form of a substance changes on changing its temperature. For example, raising the temperature of a liquid causes the liquid to vaporize (convert into vapor). The process is called vaporization—a surface phenomenon. Vaporization occurs when the thermal motion of the molecules overcome the intermolecular forces, and the molecules (at the surface) escape into the gaseous state. When a liquid vaporizes in a closed container, gas molecules cannot escape. As these gas phase molecules...
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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to occupy...
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A phase transition is the process in which a substance changes from one state of matter to another, like from a solid to a liquid, liquid to gas, or vice versa, at a specific temperature and under given pressure conditions. This change is spontaneous and is affected by alterations in temperature and pressure. These parameters impact the strength of the forces between molecules (intermolecular forces) in the substance.During a phase transition, both the initial and final phases of the substance...
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Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...
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The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).

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Research and Development of High-performance Explosives
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Phase transitions in supercritical explosive percolation.

Wei Chen1, Jan Nagler, Xueqi Cheng

  • 1School of Mathematical Sciences, Peking University, Beijing, China. chenwei2012@ict.ac.cn

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 18, 2013
PubMed
Summary
This summary is machine-generated.

In percolation theory, the Bohman-Frieze-Wormald model shows a second giant component (C(2)) can emerge after the first (C(1)) stops growing. A critical threshold, α(c), determines the delay and size of these components.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Graph Theory

Background:

  • Percolation theory studies the formation of large-scale connectivity in networks as connections are added.
  • The Bohman-Frieze-Wormald (BFW) model investigates discontinuous phase transitions in percolation.
  • Previous research established the emergence of a single giant component (C(1)) in the BFW model for α values between 0.6 and 0.95.

Purpose of the Study:

  • To investigate the emergence of a second giant component (C(2)) in the BFW percolation model.
  • To analyze the dependence of C(1) and C(2) emergence and growth on the parameter α.
  • To identify a critical threshold (α(c)) and characterize the continuous percolation of C(2).

Main Methods:

  • Analysis of the Bohman-Frieze-Wormald (BFW) percolation model within the α range of [0.6, 0.95].
  • Focus on the transition from a single giant component (C(1)) to the emergence of a second giant component (C(2)).
  • Mathematical techniques to determine the bifurcation point α(c) and analyze scaling exponents for C(2).

Main Results:

  • A second giant component (C(2)) emerges in the supercritical regime, following the initial giant component (C(1)).
  • A bifurcation point α(c) = 0.763 ± 0.002 was identified, influencing the delay and growth dynamics of C(1) and C(2).
  • For α < α(c), C(1) stops growing immediately, and the delay to C(2) decreases with increasing α. For α > α(c), C(1) continues growing, and the delay increases with α.

Conclusions:

  • The BFW model exhibits a continuous percolation transition for C(2) beyond a critical α value.
  • α(c) represents the minimum delay between C(1) and C(2) emergence, signifying the threshold for C(2) existence.
  • Detailed characterization of C(2)'s continuous percolation, including scaling exponents and relations, was established.