An Extended Cahn-Hilliard Model for Interfaces with Cubic Anisotropy

2001 ◽  
Vol 701 ◽  
Author(s):  
T.A. Abinandanan ◽  
F. Haider
Keyword(s):  
Interfacial Energy ◽  
Scaling Function ◽  
Two Dimensions ◽  
One Dimensional ◽  
Universal Scaling ◽  
Main Effect ◽  
The One ◽  
Universal Scaling Function ◽  
Particle Shapes ◽  
Cubic Anisotropy

ABSTRACTFor studying systems with a cubic anisotropy in interfacial energy σ, we extend the Cahn-Hilliard model by including in it a fourth rank term, which leads to an additional linear term in the evolution equation for the compositioneld. It also leads to an orientation-dependent effective fourth rank coeffcient γ(hkl) in the governing equation for the one-dimensional composition prole across a planar interface. The main effect of a non-negative γ(hkl) is to increase both σ and interfacial width w, each of which, upon suitable scaling, is related to γ(hkl) through a universal scaling function. The anisotropy in the interfacial energy can be large enough to give rise to corners in the Wul. shapes in two dimensions. In particles of finite sizes, the corners get rounded, and their shapes tend towards the Wul. shape with increasing particle size. In the study of unmixing of concentrated alloys, the anisotropy not only leads to non-spherical particle shapes, but also to strongly elongated morphologies.

An extended Cahn-Hilliard model for interfaces with cubic anisotropy

2001 ◽  
Vol 696 ◽  
Author(s):  
T.A. Abinandanan ◽  
F. Haider
Keyword(s):  
Interfacial Energy ◽  
Scaling Function ◽  
Two Dimensions ◽  
Wulff Shape ◽  
One Dimensional ◽  
Main Effect ◽  
The One ◽  
Universal Scaling Function ◽  
Particle Shapes ◽  
Cubic Anisotropy

AbstractFor studying systems with a cubic anisotropy in interfacial energy σ, we extend the CahnHilliard model by including in it a fourth rank term, which leads to an additional linear term in the evolution equation for the composition field. It also leads to an orientation-dependent effective fourth rank coeficient γ(hkl) in the governing equation for the one-dimensional composition profile across a planar interface. The main effect of a non-negative γ(hkl) is to increase both σ and interfacial width w, each of which, upon suitable scaling, is related to γ(hkl) through a universal scaling function. The anisotropy in the interfacial energy can be large enough to give rise to corners in the Wulff shapes in two dimensions. In particles of finite sizes, the corners get rounded, and their shapes tend towards the Wulff shape with increasing particle size. In the study of unmixing of concentrated alloys, the anisotropy nt only leads to non-spherical particle shapes, but also to strongly elongated morphologies.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

SOLVING THE TWO-DIMENSIONAL INVERSE FRACTAL PROBLEM WITH THE WAVELET TRANSFORM

Fractals ◽  
1996 ◽  
Vol 04 (04) ◽  
pp. 469-475 ◽  
Author(s):  
ZBIGNIEW R. STRUZIK
Keyword(s):  
Wavelet Transform ◽  
Scale Invariance ◽  
Wavelet Decomposition ◽  
Two Dimensions ◽  
The Self ◽  
Continuous Wavelet ◽  
Two Dimensional ◽  
One Dimensional ◽  
Affine Functions ◽  
The One

The methodology of the solution to the inverse fractal problem with the wavelet transform1,2 is extended to two-dimensional self-affine functions. Similar to the one-dimensional case, the two-dimensional wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition. It possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. As many fractals are naturally defined on two-dimensions, this extension constitutes an important step towards solving the related inverse fractal problem for a variety of fractal types.

The True Proportions of Gay's Acis and Galatea

PMLA ◽  
1965 ◽  
Vol 80 (4-Part1) ◽  
pp. 325-331
Author(s):  
Bertrand H. Bronson
Keyword(s):  
Eighteenth Century ◽  
Two Dimensions ◽  
One Dimensional ◽  
John Gay ◽  
The Arts ◽  
The One ◽  
The Mind ◽  
Theatrical Criticism

The Masque, or serenata, or pastoral opera, Acis and Galatea—in eighteenth-century printings it was indifferently categorized—has been not so much neglected as quite ignored by the biographers and critics of John Gay. In its entirety, words and music, it is a masterpiece, and the reasons for its lying unregarded, except by historians of music, deserve to be scrutinized because they signalize a recurrent failing on the part of those who write on the arts, when a work exists simultaneously in more than one medium. Only lately, in truth, has criticism begun to cope with Shakespeare himself as drama existent in and for living embodiment on a physical stage and nowhere else, not even in the mind of Coleridge. (Theatrical criticism is by habit only piecemeal commentary on separate productions.) Similarly, to set small matters beside great ones, only of late has the ballad of tradition begun to be considered as song and not as a literary or pseudo-literary genre, sufficient and self-sustaining in its text alone. And the bardic tradition of the Ugo-Slavs is teaching us much about the Homeric epics of which former generations were unaware. Signs, in fact, are here and there beginning to appear of an unwillingness to rest content with the one-dimensional conception of arts which are only half-fulfilled until they are realized in two dimensions or more. A drawing of a sculpture is not enough; a sculpture of an action is not enough; pantomime does not suffice the spoken scene; the verbal text of a musical scena will not satisfy. Nor can any of these be adequately criticized on a basis of missing dimensions.

Universal scaling function for domain growth in the Glauber-Ising chain

1992 ◽  
Vol 45 (14) ◽  
pp. 8131-8133 ◽  
Author(s):  
Zhifang Lin ◽  
Xiaomin Wang ◽  
Ruibao Tao
Keyword(s):  
Scaling Function ◽  
Domain Growth ◽  
Ising Chain ◽  
Universal Scaling ◽  
Universal Scaling Function

A Comparison of Diffusion Flame Stability in One and Two Spatial Dimensions Near Cold, Inert Surfaces

2001 ◽  
Author(s):  
Robert Vance ◽  
Indrek S. Wichman
Keyword(s):  
Two Dimensions ◽  
Low Flow ◽  
Dimensional System ◽  
Flame Stability ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Cellular Flames ◽  
The One ◽  
Inert Surfaces

Abstract A linear stability analysis is performed on two simplified models representing a one-dimensional flame between oxidizer and fuel reservoirs and a two-dimensional “edge-flame” between the same reservoirs but above a cold, inert wall. Comparison of the eigenvalue spectra for both models is performed to discern the validity of extending the results from the one-dimensional problem to the two-dimensional problem. Of primary interest is the influence on flame stability of thermal-diffusive imbalances, i.e. non-unity Lewis numbers. Flame oscillations are observed when Le > 1, and cellular flames are witnessed when Le < 1. It is found that when Le > 1 the characteristics of flame behavior are consistent between the two models. Furthermore, when Le < 1, the models are found to be in good agreement with respect to the magnitude of the critical wave numbers. Results from the coarse mesh analysis of the two-dimensional system are presented and compared to the one-dimensional eigenvalue spectra. Additionally, an examination of low reactant convection is undertaken. It is concluded that for low flow rates the behavior in one and two dimensions are similar qualitatively and quantitatively.

Comparison of two numerical methods for the integration of the Takagi–Taupin equations

1981 ◽  
Vol 14 (6) ◽  
pp. 432-436 ◽  
Author(s):  
C. Nourtier ◽  
D. Taupin
Keyword(s):  
Numerical Calculation ◽  
Numerical Methods ◽  
Two Dimensions ◽  
Integration Step ◽  
Order Method ◽  
Third Order ◽  
Acta Cryst ◽  
One Dimensional ◽  
Numerical Resolution ◽  
The One

Two methods for the numerical resolution of the Takagi-Taupin equations are compared. It is shown that for a small integration step Taupin's [Acta Cryst. (1967), 23, 25–35] extension to two dimensions of the one-dimensional Runge–Kutta third-order method is more accurate than the algorithm of Authier, Malgrange & Tournarie [Acta Cryst. (1968), A24, 126–136] but, for a given precision, Authier, Malgrange & Tournarie's method is faster than Taupin's so the former will usually be preferred for numerical calculation.

SOME EXACT CALCULATIONS OF AN ALTERNATING TRANSVERSE ISING MODEL

2002 ◽  
Vol 16 (26) ◽  
pp. 3871-3881 ◽  
Author(s):  
HIDENORI SUZUKI ◽  
MASUO SUZUKI
Keyword(s):  
Ising Model ◽  
Transverse Field ◽  
Two Dimensions ◽  
Honeycomb Lattice ◽  
Temperature Phase ◽  
One Dimensional ◽  
Transverse Ising Model ◽  
Temperature Phase Transition ◽  
Exact Calculations ◽  
The One

The alternating transverse Ising model with A and B sublattices is solved exactly in one and two dimensions, when a transverse field applied only to the A sublattice. The critical point of the honeycomb lattice is given as a function of the alternating transverse field. Moreover, the zero-temperature phase transition in the one-dimensional model with another alternating transverse field is discussed rigorously.

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