Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Parameter, order

Several examples of order parameters are shown in Fig. 2.5. Some of them, e.g., the torsional angles a, are dynamical variables, which means that they can be fully [Pg.50]

If in addition to a thermodynamic driving force, a system has kinetic mechanisms available to produce a phase transformation (e.g., diffusion or atomic structural relaxation), the rate and characteristics of phase transformations can be modeled through combinations of their cause (thermodynamic driving forces) and their kinetic mechanisms. Analysis begins with identification of parameters (i.e., order parameters) that characterize the internal variations in state that accompany the transformation. For example, site fraction and magnetization can serve as order parameters for a ferromagnetic crystalline phase. [Pg.420]

Model energy functionals will be obtained through consideration of the energetic contribution of order parameter fields, and this is preceded by a survey of order parameters. [Pg.420]

The exponent j for the temperature dependence is characteristic of mean field behaviour. [Pg.17]

The second-order nature of the transition is confirmed since the entropy change at the transition is zero. This can be shown by calculating the entropy density (at constant volume), s  [Pg.17]

Above the phase transition this takes the value [Pg.17]

This shows that the entropy density decreases continuously to zero as the phase transition is approached, i.e. there is no discontinuity in entropy. [Pg.17]

A similar analysis can be made for the Landau free energy of a weakly first-order phase transition, for example the nematic-isotropic transition exhibited by some liquid crystals (Section 5.7.1). The free energy (Eq. 1.13), is supplemented by an additional cubic term C T)xlr if the transition is first order. The first-order nature of the transition can be confirmed by calculating the entropy density change at the transition, which turns out to be [Pg.17]


At r = 0, all the spins are either aligned up or down. The magnetization per site is an order parameter whieh vanishes at the eritieal point. Along the eoexistenee eiirve at zero field... [Pg.521]

Assume that the free energy can be expanded in powers of the magnetization m which is the order parameter. At zero field, only even powers of m appear in the expansion, due to the up-down symmetry of the system, and... [Pg.536]

Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal. Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal.
Figure A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ... Figure A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ...
In general the width of the coexistence line (Ap, Ax, or AM) is proportional to an order parameter s, and its absolute value may be written as... [Pg.639]

For the kind of transition above which the order parameter is zero and below which other values are stable, the coefficient 2 iiiust change sign at the transition point and must remain positive. As we have seen, the dependence of s on temperature is detemiined by requiring the free energy to be a miniimuii (i.e. by setting its derivative with respect to s equal to zero). Thus... [Pg.643]

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. [Pg.649]

Another problem can be the choice of an order parameter for the detemiination of p and of the departure from... [Pg.651]

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... [Pg.656]

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches. Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.
In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... [Pg.732]

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. [Pg.732]

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... [Pg.732]

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. [Pg.735]

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence... Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...
Figure A3.3.6 Free energy as a function of the order parameter cji for the homogeneous single phase (a) and for the two-phase regions (b), 0. Figure A3.3.6 Free energy as a function of the order parameter cji for the homogeneous single phase (a) and for the two-phase regions (b), 0.
For critical quench experiments there is a synnnetry < )q = 0 and from equation (A3,3,50) S( ) = ( ), leading to a syimnetric local free energy ( figure A3,3,6) and a scaled order parameter whose average is zero, 8v / = i /. For off-critical quenches this synnnetry is lost. One has 8( ) = ( ) -t ( ) which scales to 5 j/ = with... [Pg.739]

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... [Pg.739]

For a conserved order parameter, the interface dynamics and late-stage domain growth involve the evapomtion-diffusion-condensation mechanism whereby large droplets (small curvature) grow at tlie expense of small droplets (large curvature). This is also the basis for the Lifshitz-Slyozov analysis which is discussed in section A3.3.4. [Pg.745]

Again consider a single spherical droplet of minority phase ( [/ = -1) of radius R innnersed m a sea of majority phase. But now let the majority phase have an order parameter at infinity that is (slightly) smaller than +1, i.e. [i( ) = < 1. The majority phase is now supersaturated with the dissolved minority species,... [Pg.749]

Global and local correlation times, generalized order parameter, S... [Pg.1505]


See other pages where Parameter, order is mentioned: [Pg.87]    [Pg.628]    [Pg.633]    [Pg.643]    [Pg.651]    [Pg.651]    [Pg.652]    [Pg.657]    [Pg.732]    [Pg.733]    [Pg.733]    [Pg.735]    [Pg.735]    [Pg.736]    [Pg.737]    [Pg.737]    [Pg.737]    [Pg.738]    [Pg.739]    [Pg.739]    [Pg.741]    [Pg.742]    [Pg.743]    [Pg.745]    [Pg.745]    [Pg.746]    [Pg.750]    [Pg.753]    [Pg.1505]   
See also in sourсe #XX -- [ Pg.84 , Pg.115 , Pg.116 , Pg.119 , Pg.393 , Pg.394 , Pg.510 , Pg.513 , Pg.638 , Pg.639 , Pg.666 ]

See also in sourсe #XX -- [ Pg.420 ]

See also in sourсe #XX -- [ Pg.3 , Pg.122 , Pg.125 , Pg.126 , Pg.127 , Pg.148 , Pg.150 , Pg.155 , Pg.174 , Pg.184 , Pg.264 ]

See also in sourсe #XX -- [ Pg.64 , Pg.65 , Pg.200 , Pg.311 ]

See also in sourсe #XX -- [ Pg.233 ]

See also in sourсe #XX -- [ Pg.182 , Pg.194 ]

See also in sourсe #XX -- [ Pg.41 ]

See also in sourсe #XX -- [ Pg.71 ]

See also in sourсe #XX -- [ Pg.203 , Pg.214 ]

See also in sourсe #XX -- [ Pg.12 , Pg.17 , Pg.243 , Pg.246 , Pg.247 , Pg.248 , Pg.249 , Pg.250 , Pg.251 , Pg.252 ]

See also in sourсe #XX -- [ Pg.41 ]

See also in sourсe #XX -- [ Pg.17 ]

See also in sourсe #XX -- [ Pg.6 , Pg.10 , Pg.19 , Pg.59 , Pg.60 , Pg.62 , Pg.63 , Pg.68 , Pg.72 , Pg.86 , Pg.87 , Pg.89 , Pg.105 , Pg.107 , Pg.181 , Pg.183 , Pg.194 , Pg.272 , Pg.275 ]

See also in sourсe #XX -- [ Pg.290 , Pg.291 , Pg.313 , Pg.316 ]

See also in sourсe #XX -- [ Pg.12 ]




SEARCH



Absorbance, Order Parameter, and Dichroic Ratio Measurement

Acentric order parameter

Alloy order parameter

Anisotropy decays order parameters

Apparent Order Parameters for Flexible Chains

Aqueous solutions order parameter

Bilayer internal dynamics order parameters, membrane thickness, sterols

Bond orientational order paramete

Bond orientational order parameter

Chain order parameter

Chiral nematics order parameters

Chiralic molecules, order parameter

Chirality order parameter

Concentration-Related Parameters Order of Reaction

Conformational order parameter

Connected order parameter

Conserved order parameter

Continuity equation order parameter

Critical parameters in order

Crystal phase order parameter

D-wave order parameter

Definition of an Order Parameter

Degree of Freedom Selection State Variables, Order Parameters and Configurational Coordinates

Density anomalies order parameter

Direct evaluation of the order parameter fluctuations

Distributions and Order Parameters

Double first-order model parameters

Dynamics of Order Parameter

Edwards order parameter

Elastic constants order parameter dependence

Equilibrium Formulation and Order Parameter

Estimating order parameters by MQNMR

Evaluation of the order parameter

Evolution Equations for Order Parameters

Field Dependence of Order Parameter Hysteresis Loops

First-order absorption models model parameter estimation

Fluctuations of the Order Parameter

Fluctuations of the order parameter in chemical reactions

Generalized order parameter

Ginzburg Landau order parameter

Glass order parameter

Global orientational order parameter

Hard order parameter

Hard order parameter transitions

Identification second order parameters

Impact of the Order Parameter

In-plane order parameter

Induced polar alignment order parameter

Integral order desorption with coverage-independent parameters

Kinematics order parameters

Kinetics order parameter models

Landau order parameter

Landau theory order parameter

Leucine residues, order parameters

Liquid crystal media order parameters

Local bond order parameter

Local order parameter

Long-distance order parameter

Long-range order parameter

Macroscopic Description of Order Parameters

Macroscopic order parameter

Mean field model order parameter, temperature dependence

Membranes order parameters

Microscopic order-macroscopic disorder parameters

Molecular disorder structural order parameter

Molecular order parameter

Multi-component order parameter

Multiple order parameter model

NMR Parameters Defined as Second-Order Energy Perturbations

Nematic liquid crystals order parameter

Nematic order parameters

Non-conserved order parameter

Nuclear magnetic resonance order parameters

One-component order parameter

Operational test parameters order

Optical order parameter

Order Parameter Fluctuations in the Nematic Phase

Order Parameter and Dichroic Ratio of Dyes

Order Parameter, Phase Transition, and Free Energies

Order Parameters Based on DCs

Order Parameters, Reaction Coordinates, and Extended Ensembles

Order electricity parameter

Order parameter Bessel function

Order parameter adsorption

Order parameter aggregation

Order parameter angular overlap

Order parameter at the surface

Order parameter biaxiality

Order parameter columnar phase)

Order parameter complex

Order parameter component

Order parameter consistency relation

Order parameter curves

Order parameter definition, 38-41 experimental

Order parameter density

Order parameter dependence

Order parameter determination

Order parameter dimensionality

Order parameter director field

Order parameter discontinuity

Order parameter distribution

Order parameter domain, temperature

Order parameter ferroelectricity

Order parameter field

Order parameter field theory

Order parameter function

Order parameter icosahedral

Order parameter mean field functional

Order parameter microscopic

Order parameter miscibility

Order parameter models

Order parameter numerical values

Order parameter of aggregation and fluctuations

Order parameter of nematics

Order parameter overall

Order parameter partial

Order parameter phase shift

Order parameter polymer orientation

Order parameter profile

Order parameter radial

Order parameter reaction coordinates

Order parameter reduction

Order parameter relaxation theory

Order parameter response time

Order parameter similarity

Order parameter space

Order parameter surfaces

Order parameter temperature dependence

Order parameter tensor

Order parameter theory

Order parameter wave (smectic

Order parameter, definition

Order parameter, equilibrium phase diagrams

Order parameter, susceptibilities

Order parameters against observed temperature

Order parameters biaxial

Order parameters defined

Order parameters directors

Order parameters fourth rank

Order parameters measurement

Order parameters simulation

Order parameters transitional

Order parameters, aliphatic

Order parameters, aliphatic hydrocarbons

Order parameters, convergence

Order parameters, convergence characteristics

Order parameters, mesoscopic polymer

Order parameters, thermotropic liquid crystals

Order-parameter critical exponent

Order-parameter fluctuations

Order-parameter fluctuations decay rate

Order-parameter matrix

Order-parameter scaling Monte Carlo

Ordered alloys order parameter

Ordering models interaction parameters

Ordering parameter

Ordering parameter, molecular glasses

Ordering parameters calculation

Orientational order paramete

Orientational order paramete from birefringence

Orientational order parameter from birefringence

Orientational order parameters

Parameters long-range order parameter

Parameters order parameter

Phase separating/ordering systems conserved order parameter

Pikin Indenbom order parameter

Polar alignment order parameter

Polarization order parameter

Poly , order paramete

Poly order parameter

Principal Orientational Order Parameter (Microscopic Approach)

Proper order parameter

Reaction order, rate equation and Arrhenius parameters

Relationship Between Microscopic and Macroscopic Order Parameters

Relaxation of the order parameter

Relaxation order parameter model

Residual Couplings and Dynamic Order Parameters

Saupe Theory Order Parameter Near Tc

Saupe order parameters

Scalar Orientational Order Parameter

Scalar and Tensor Order Parameters

Scalar order parameter

Second order parameter correlation

Second-order crystal-field parameters

Secondary order parameter

Sequence order parameter

Short-order parameter

Short-range order parameter

Similarity measure and order parameter

Smectic order parameters

Soft order parameters

Soft order parameters transitions

Spatial correlations and the order parameter

Structural order parameters

Structural order parameters bond-orientational

Structural order parameters crystal-independent measures

Structural order parameters ordering phase diagram

Structural order parameters specific bond-orientational

Structural order parameters terminology

Subject order parameter

Superconducting order parameters

Superconducting order parameters temperature dependence

Superconductors order parameter

Surface excess order parameter

Surface layer order parameter

Surface ordering parameter

Surface-Induced Changes in the Orientational Order Parameter

Symmetry and the Order Parameter

Symmetry of the superconducting order paramete

Symmetry, Structure and Order Parameters

Temperature Dependence of the Nematic Order Parameter

Tensorial order parameter

The Pikin-Indenbom Order Parameter

The Smectic C Order Parameter

The intermediate order parameter

The order parameter

Theory perturbation, ordering parameter

Translational order parameter

Two-order parameter model of liquid

Viscosity coefficients order parameter dependence

Warren and Cowleys order parameter

Warren-Cowley order parameter

What is the order parameter

Zero-order absorption models model parameter estimation

Zero-order electro-optical parameters

© 2024 chempedia.info