Statistical Mechanics of Phase Coarsening

1983 ◽  
Vol 21 ◽  
Author(s):  
M.E. Glicksman ◽  
P.W. Voorhees
Keyword(s):  
Statistical Mechanics ◽  
Particle Distribution ◽  
Ostwald Ripening ◽  
Volume Fraction ◽  
High Volume ◽  
Distribution Functions ◽  
Average Particle ◽  
Two Phase ◽  
Phase Coarsening ◽  
Lsw Theory

ABSTRACTPhase coarsening, also termed Ostwald ripening, is generally thought to be a slow, diffusion-controlled process which occurs subsequent to phase separation under extremely small under- or over-saturation levels. The theory due to Lifshitz, Slyzov, and Wagner (LSW), which predicts the coarsening kinetics and the particle distribution function is applicable to dilute systems only, in which particle-particle interactions are unimportant. Most practical systems, however, have large enough volume fractions of the dispersed phase to violate the essential assumptions of LSW theory. Recent progress will be described on simulating Ostwald ripening in randomly dispersed, high volume fraction systems. A fast algorithm for solving the multiparticle diffusion problem (MDP) will be described, permitting simulation of coarsening dynamics by cyclic time-stepping and updating the diffusion solution for large random particle arrays. The rate constants, controlling the growth of the average particle, and the particle distribution functions were obtained by numerical simulations up to a volume fraction of 0.55. A new statistical mechanics theory has now been developed which reproduces the MDP simulation data accurately, and finally makes clear how the linear mean-field approximations employed by LSW theory must be modified to describe real systems. The new theory provides a comprehensive approach to understanding microstructural coarsening in two-phase systems.

Phase Coarsening as a Relaxation Phenomenon

1999 ◽  
Vol 580 ◽  
Author(s):  
S. P. Marsh
Keyword(s):  
Ostwald Ripening ◽  
Accurate Determination ◽  
Average Particle Size ◽  
Particle Morphology ◽  
Relaxation Phenomenon ◽  
Average Particle ◽  
Two Phase ◽  
Phase Coarsening ◽  
Growth Phenomenon

AbstractPhase coarsening, also known as Ostwald ripening, is generally treated as a growth phenomenon in which the average particle size increases during isothermal aging. However, coarsening is a relaxation process driven by a reduction in the excess interfacial energy of a two-phase structure. This viewpoint introduces a single temporal offset parameter that relates the experimental clock to the asymptotic time scale. Analysis of ripening as a relaxation phenomenon permits accurate determination of the kinetics that is less sensitive to particle morphology. Implications of this approach on measurement and interpretation of ripening kinetics will be discussed.

scholarly journals Quantitative Phase-Field Simulation of Composition Partition and Separation Kinetics of Nanoscale Phase in Fe-Cr-Al Alloy

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Shi Chen ◽  
Yongsheng Li ◽  
Shujing Shi ◽  
Shengshun Jin
Keyword(s):  
Phase Separation ◽  
Phase Field ◽  
Ostwald Ripening ◽  
Volume Fraction ◽  
Phase Interface ◽  
Al Alloy ◽  
Average Particle ◽  
Phase Field Simulation ◽  
Field Simulation ◽  
Separation Kinetics

Phase separation of the Cr-enriched nanoscale α′ phase in the Fe-38 at.% Cr-10 at.% Al alloy is studied by utilizing phase-field simulation. The partition of elements in the α and α′ phases is clarified with the composition evolution through the α/α′ phase interface, and the separation kinetics is quantitatively investigated by the temporal evolution of the size and volume fraction of the α′ phase. Aluminum partitions into the Fe-enriched α phase and depletes in the α′ phase, and the partition coefficient decreases as the temperature changes from 720 K to 760 K for the steady-state coarsening stage. As the temperature increases, the initial change rate of the volume fraction of the α′ phase is faster, indicating an accelerated phase separation. At the coarsening stage, the average particle distance and coarsening rate constant of the α′ phase increase with increased temperature, and the ratio of the Ostwald ripening is dominating compared with coalescence coarsening. The element partition and kinetics evolution of the α′ phase with temperature are helpful for the morphology and property predication of nanoscale precipitates.

Simulating Phase Coarsening of Ultra-High Volume Fractions

2010 ◽  
Vol 638-642 ◽  
pp. 3925-3930 ◽  
Author(s):  
K.G. Wang ◽  
X. Ding
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Volume Fraction ◽  
High Volume ◽  
Triple Junctions ◽  
Ginzburg Landau ◽  
Volume Fractions ◽  
Phase Coarsening ◽  
Kinetics Of

The dynamics of phase coarsening at ultra-high volume fractions is studied based on two-dimensional phase-field simulations by numerically solving the time-dependent Ginzburg-Landau and Cahn-Hilliard equations. The kinetics of phase coarsening at ultra-high volume fractions is discovered. The microstructural evolutions for different ultra-high volume fractions are shown. The scaled particle size distribution as functions of the dispersoid volume fraction is presented. The particle size distribution derived from our simulation at ultra-high volume fractions is close to Wagner's particle size distribution due to interface-controlled ripening rather than Hillert's grain size distribution in grain growth. The changes of shapes of particles are carefully studied with increase of volume fraction. It is found that more liquid-filled triple junctions are formed as a result of particle shape accommodation with increase of volume fraction at the regime of ultra-high volume fraction.

EQUILIBRIUM VELOCITY DISTRIBUTION FUNCTIONS FOR A KINETIC MODEL OF TWO-PHASE FLOWS

1995 ◽  
Vol 05 (02) ◽  
pp. 191-211 ◽  
Author(s):  
LIONEL SAINSAULIEU
Keyword(s):  
Velocity Distribution ◽  
Gas Flow ◽  
Stokes Equations ◽  
Volume Fraction ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Solid Particles ◽  
Navier Stokes ◽  
Two Phase ◽  
Entropy Balance

We consider a cloud of solid particles in a gas flow. The cloud is described by a probability density function which satisfies a kinetic equation. The gas flow is modeled by Navier-Stokes equations. The two phases exchange momentum and energy. We obtain the entropy balance of the gas flow and deduce some bounds for the volume fraction of the gas phase. Writing the entropy balance for the dispersed phase enables one to determine the particles equilibrium velocity distribution function when the gas flow is known.

Numerical simulation of in-pipe particle transportation in CO2 foam

2017 ◽  
Vol 5 (2) ◽  
pp. SF243-SF249
Author(s):  
Cheng Huang ◽  
Xiao Sun ◽  
Jinqiao Wu
Keyword(s):  
Numerical Simulation ◽  
Critical Point ◽  
Cross Section ◽  
Particle Distribution ◽  
Settling Velocity ◽  
Volume Fraction ◽  
Two Phase ◽  
Multiple Phase ◽  
Full Development ◽  
Fracturing Fluids

Proppant-carrying foam fracturing fluids have complex rheological and transportation properties. Current studies on these fluids often focus on experimental phenomena. However, due to the limitation of experimental research, only macroscopic properties, such as the critical settling velocity, can be obtained. Transportation mechanisms and volume fraction distributions are poorly understood as well. In our study, the liquid-solid drag coefficient is corrected, and the mathematical physical model of non-Newtonian fluids of the particle-foam multiple phase is established by using a two-phase model. Proppant settling and transport properties in foam fracturing fluids are numerically studied, particle distribution on pipe cross section is obtained at various conditions, and a criterion for full development of fluid flow in pipe is set. We also find that when the Reynolds number (Re) is less than 190, the critical point of full development of flow increases with Re, whereas when Re is greater than 190, the critical point of full development decreases exponentially with the increasing of Re before stabilizing at approximately 45.

Phase Transformation Textures in Ti-6Al-4V Alloy

2005 ◽  
Vol 495-497 ◽  
pp. 681-686 ◽  
Author(s):  
Sven C. Vogel ◽  
D. Bhattacharyya ◽  
G.B. Viswanathan ◽  
D.J. Williams ◽  
H.L. Fraser
Keyword(s):  
Orientation Relationship ◽  
Room Temperature ◽  
Volume Fraction ◽  
Turbine Blades ◽  
Large Body ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Transformation Mechanism ◽  
Two Phase ◽  
Crystal Direction

Titanium alloys are widely used in various industrial, domestic, and medical applications such as turbine blades, bicycle frames, knee implants, etc. The two-phase titanium alloy Ti-6Al-4V (wt. percent) is considered to be a workhorse alloy for many applications in these diverse fields. Despite the large body of work on this alloy, the question of the transformation mechanism from the hcp a to the bcc b phase, occurring on heating to temperatures above the a/b transus at ~980°C, is still unresolved. Due to experimental difficulties, it has not yet been clearly determined whether the increase in b volume fraction occurs by fresh nucleation of b crystals within a phase grains or the growth of preexisting b grains. Since the Burgers orientation relationship holds only if the b grains are nucleated within the a grains, the outcome of this question greatly affects texture-modeling efforts for this system. The Burgers orientation relationship predicts that the {0001} crystal direction in a grain of the a phase becomes a {110} crystal direction in a grain of the b phase after the transformation. In this work we present experimental results from in-situ texture measurements performed on the HIPPO neutron diffractometer at LANSCE. Using the combination of time-offlight neutrons and full-pattern Rietveld analysis allowed us to determine the orientation distribution functions of both phases at room temperature, 800°C, 1020°C and again at room temperature. We found strong indications that the b phase indeed grows from grains preexisting at room temperature. Upon re-transformation from b to a we found that the Burgers relationship is followed.

Phase-field simulation of 2-D Ostwald ripening in the high volume fraction regime

2002 ◽  
Vol 50 (8) ◽  
pp. 1895-1907 ◽  
Author(s):  
Danan Fan ◽  
S.P. Chen ◽  
Long-Qing Chen ◽  
P.W. Voorhees
Keyword(s):  
Phase Field ◽  
Ostwald Ripening ◽  
Volume Fraction ◽  
High Volume ◽  
High Volume Fraction ◽  
Phase Field Simulation ◽  
Field Simulation

Small-Angle X-ray Scattering Size Parameters and Higher Moments of the Particle-Size Distribution Function in the Asymptotic Stage of Ostwald Ripening

1995 ◽  
Vol 28 (5) ◽  
pp. 553-560 ◽  
Author(s):  
J. Möller ◽  
R. Kranold ◽  
J. Schmelzer ◽  
U. Lembke
Keyword(s):  
Particle Size ◽  
Size Distribution ◽  
Ostwald Ripening ◽  
Distribution Functions ◽  
Higher Moments ◽  
X Ray ◽  
Diffusion Limited ◽  
X Ray Scattering ◽  
Lsw Theory ◽  
Kinetics Of

Small-angle X-ray scattering (SAXS) is a powerful tool to study the kinetics of phase separation in materials. A simple procedure is presented that allows one to prove if the particle-size distribution established in a system in the late stages of phase separation corresponds to the predictions of the classical Lifshitz–Slyozov–Wagner (LSW) theory for the asymptotic stage of Ostwald ripening. The method is based on the correlations between certain SAXS size parameters and the higher moments of the LSW size distribution functions for diffusion-limited or reaction-limited ripening. It is suggested that the use of these size parameters, which can be obtained with high accuracy from the scattering curve, is frequently more advantageous than a direct comparison of the experimentally obtained size distributions with the asymptotic size-distribution functions predicted by the LSW theory. The method is applicable if the suppositions made in the LSW theory that the precipitated particles should be homogeneous spheres with volume fraction tending to zero are fulfilled. The method is applied to a photochromic glass; although the silver-halide precipitates contained in the glass develop according to the power law of diffusion-limited Ostwald ripening, their size distribution is shown not to correspond to the features of the LSW size distribution. Consequently, in this case the LSW theory cannot describe quantitatively the kinetics of ripening.

Degree of Crystallinity and Strain in B4C and SiC Thin Films as a Function of Processing Conditions

1998 ◽  
Vol 524 ◽  
Author(s):  
J. Hershberger ◽  
Z. U. Rek ◽  
F. Kustas ◽  
S. M. Yalisove ◽  
J. C. Bilello
Keyword(s):  
Thin Films ◽  
Particle Size ◽  
Nearest Neighbor ◽  
Volume Fraction ◽  
Average Particle Size ◽  
Grazing Incidence ◽  
Quantitative Information ◽  
Distribution Functions ◽  
Degree Of Crystallinity ◽  
Average Particle

ABSTRACTAmorphous and crystalline content in sputtered B4C and SiC thin films has been analyzed by synchrotron grazing incidence x-ray scattering (GIXS). GIXS provided quantitative information on the average structure while TEM was used to find inhomogeneities such as small volume fraction phases. GIXS results were compared to simulations to determine average particle size or bond length for crystalline or amorphous phases respectively. In this work, we compared results from films deposited with, and without, an RF bias applied to the substrate during deposition. Results indicated that SiC can be described as strained polycrystalline material with particle size of approximately 13 Å for biased samples and 9Å for unbiased samples. Boron carbide deposited without bias was completely crystalline with a particle size of approximately 30 Å, while the data suggested that B4C deposited with bias is amorphous. The scattering from the biased materials was Fourier transformed to yield radial distribution functions (RDF). This provided nearest neighbor distances, and it was demonstrated that the technique can be used to determine full three-dimensional strain tensors in amorphous thin films.

Composition Fluctuations in the Ostwald Ripening

2008 ◽  
Vol 277 ◽  
pp. 187-192
Author(s):  
G.V. Lutsenko ◽  
Andriy Gusak
Keyword(s):  
Computer Simulation ◽  
Particle Size ◽  
Large Volume ◽  
Ostwald Ripening ◽  
Volume Fraction ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Two Phase ◽  
Large Volume Fraction ◽  
Composition Fluctuations

The Ostwald ripening of a two-phase binary alloy has been considered for case of “large” volume fraction of precipitating phase. The approach is proposed in which the composition fluctuations into the vicinity of particles are considered. In this approach the evolution of particle size distributions is analyzed using the computer simulation.

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