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Related Concept Videos

Phase Changes01:19

Phase Changes

Phase transitions play an important theoretical and practical role in the study of heat flow. In melting or fusion, a solid turns into a liquid; the opposite process is freezing. In evaporation, a liquid turns into a gas; the opposite process is condensation.
A substance melts or freezes at a temperature called its melting point and boils or condenses at its boiling point. These temperatures depend on pressure. High pressure favors the denser form of the substance, so typically, high pressure...
Time and frequency -Domain Interpretation of Phase-lead Control01:24

Time and frequency -Domain Interpretation of Phase-lead Control

Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
The design of phase-lead control involves the strategic placement of poles and zeros to balance steady-state error and system...
Time and frequency -Domain Interpretation of Phase-lag Control01:21

Time and frequency -Domain Interpretation of Phase-lag Control

Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
Phase-lag controllers do not place a pole at zero, but instead influence the steady-state error by amplifying any finite,...
Pole and System Stability01:24

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.
Phase Transitions02:31

Phase Transitions

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...
Phase Transitions01:21

Phase Transitions

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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Related Experiment Video

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Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
08:39

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Stabilizing the blue phases.

G P Alexander1, J M Yeomans

  • 1Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2007
PubMed
Summary

This study explores cholesteric liquid crystal phase diagrams using Landau-de Gennes theory. Elastic constants significantly influence blue phase stability, especially with helix inversion in the cholesteric phase.

Area of Science:

  • Materials Science
  • Condensed Matter Physics
  • Physical Chemistry

Background:

  • Cholesteric liquid crystals exhibit complex phase behavior.
  • Landau-de Gennes theory provides a framework for understanding liquid crystal phases.
  • The influence of elastic constants on phase stability requires detailed investigation.

Purpose of the Study:

  • To investigate the phase diagram of cholesteric liquid crystals using Landau-de Gennes theory.
  • To explore the impact of Frank elastic constants on the stability of cubic blue phases.
  • To analyze the effect of temperature-dependent pitch and helix inversion on the phase diagram.

Main Methods:

  • Utilizing Landau-de Gennes theory to model the free energy of cholesteric liquid crystals.
  • Incorporating all three Frank elastic constants (splay, twist, bend) into the free energy functional.

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  • Allowing for a temperature-dependent pitch in the cholesteric phase.

    • Main Results:

    • The stability region of cubic blue phases is highly sensitive to the values of elastic constants.
    • Increased bend elastic constant relative to splay reduces blue phase stability.
    • Reduced twist elastic constant relative to other constants also diminishes blue phase stability.
    • Helix inversion in the cholesteric phase dramatically increases the stability of blue phase I and alters the overall phase diagram.

    Conclusions:

    • Elastic constants play a critical role in determining the phase behavior of cholesteric liquid crystals.
    • The Landau-de Gennes framework, with modified free energy, accurately captures these dependencies.
    • Systems exhibiting helix inversion present unique phase diagrams with enhanced blue phase I stability.