Anisotropic functions: a genericity result with crystallographic implications

Victor J. Mizel; Alexander J. Zaslavski

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 624-633
  • ISSN: 1292-8119

Abstract

top
In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments that the stored energy density (surface tension) along a step edge was a smooth symmetric function β of the azimuthal angle θ to the step, and that the positive function β attains its minimum value at θ = π / 2 and its maximum value at θ = 0 . The function β provided the crucial thermodynamic parameters needed for the engineering of these materials. Moreover the minimal energy configuration of the step is determined by the values of the stiffness function β ' ' + β which ultimately leads to the magnitude and direction of surface mass flow for these materials. In the 1990's there was a dramatic improvement in electron microscopy which permitted real time observation of the meanderings of a step edge under Brownian heat oscillations. These observations provided much more rapid determination of the relevant thermodynamic parameters for the step edge, even for crystals at temperatures below their roughening temperature. Use of these tools led J. Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive manner as their temperature is varied. The present article is devoted to a model described by a class of variational problems. The main result of the paper describes the solutions of the corresponding problem for a generic integrand.

How to cite

top

Mizel, Victor J., and Zaslavski, Alexander J.. "Anisotropic functions: a genericity result with crystallographic implications." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 624-633. <http://eudml.org/doc/90747>.

@article{Mizel2010,
abstract = { In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments that the stored energy density (surface tension) along a step edge was a smooth symmetric function β of the azimuthal angle θ to the step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$. The function β provided the crucial thermodynamic parameters needed for the engineering of these materials. Moreover the minimal energy configuration of the step is determined by the values of the stiffness function$\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for these materials. In the 1990's there was a dramatic improvement in electron microscopy which permitted real time observation of the meanderings of a step edge under Brownian heat oscillations. These observations provided much more rapid determination of the relevant thermodynamic parameters for the step edge, even for crystals at temperatures below their roughening temperature. Use of these tools led J. Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive manner as their temperature is varied. The present article is devoted to a model described by a class of variational problems. The main result of the paper describes the solutions of the corresponding problem for a generic integrand. },
author = {Mizel, Victor J., Zaslavski, Alexander J.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Complete metric space; generic property; variational problem.; complete metric space; variational problem},
language = {eng},
month = {3},
number = {4},
pages = {624-633},
publisher = {EDP Sciences},
title = {Anisotropic functions: a genericity result with crystallographic implications},
url = {http://eudml.org/doc/90747},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Mizel, Victor J.
AU - Zaslavski, Alexander J.
TI - Anisotropic functions: a genericity result with crystallographic implications
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 624
EP - 633
AB - In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments that the stored energy density (surface tension) along a step edge was a smooth symmetric function β of the azimuthal angle θ to the step, and that the positive function β attains its minimum value at $\theta = \pi/2$ and its maximum value at $\theta = 0$. The function β provided the crucial thermodynamic parameters needed for the engineering of these materials. Moreover the minimal energy configuration of the step is determined by the values of the stiffness function$\beta'' + \beta$ which ultimately leads to the magnitude and direction of surface mass flow for these materials. In the 1990's there was a dramatic improvement in electron microscopy which permitted real time observation of the meanderings of a step edge under Brownian heat oscillations. These observations provided much more rapid determination of the relevant thermodynamic parameters for the step edge, even for crystals at temperatures below their roughening temperature. Use of these tools led J. Hannon and his coexperimenters to discover that some crystals behave in a highly anti-intuitive manner as their temperature is varied. The present article is devoted to a model described by a class of variational problems. The main result of the paper describes the solutions of the corresponding problem for a generic integrand.
LA - eng
KW - Complete metric space; generic property; variational problem.; complete metric space; variational problem
UR - http://eudml.org/doc/90747
ER -

References

top
  1. B. Dacorogna and C.E. Pfister, Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl.71 (1992) 97-118.  Zbl0676.46031
  2. I. Fonseca, The Wulff theorem revisited. Proc. R. Soc. Lond. A432 (1991) 125-145.  Zbl0725.49017
  3. J. Hannonet al., Step faceting at the (001) surface of boron doped silicon. Phys. Rev. Lett.79 (1997) 4226-4229.  
  4. J. Hannon, M. Marcus and V.J. Mizel, A variational problem modelling behavior of unorthodox silicon crystals. ESAIM: COCV9 (2003) 145-149.  Zbl1066.49011
  5. H.C. Jeng and E.D. Williams, Steps on surfaces: experiment and theory. Surface Science Reports34 (1999) 175-294.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.