iMechanica - coarsening
https://imechanica.org/taxonomy/term/232
enGoing beyond 2D Neumann-Mullins (or, what is popularly known as, solving the beer froth structure)
https://imechanica.org/node/1302
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/232">coarsening</a></div><div class="field-item odd"><a href="/taxonomy/term/899">grain growth</a></div><div class="field-item even"><a href="/taxonomy/term/900">microstrctural evolution</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Introduction</span></p>
<p> The <a href="http://dsc.discovery.com/news/2007/04/25/beerfoam_hum.html?category=human&guid=20070425153000" title="Discovery -- beer foam structure solved">blogosphere is abuzz</a> with the latest report of the generalisation of the von Neumann-Mullins grain growth relation to 3 (and N) dimensions by <a href="http://www.math.ias.edu/%7Ephares/macpherson/RDM.html" title="MacPherson homepage">MacPherson</a> and <a href="http://prism.princeton.edu/bios/SrolovitzBio.htm" title="Srolovita homepage">Srolovitz</a> (As an interesting aside, almost all the reports say mathematical structure of beer foam structure resolved, or words to that effect --hence, I also decided to join the bandwagon on that one). I heard Prof. Srolovitz describe the work in a seminar nearly six months ago. Based on my notes of the talk, I would like the explain their work in this post. <a href="http://en.wikipedia.org/wiki/Curvature" title="Curvature wiki">Curvature</a> in the following refers to mean curvature (and not Gaussian).</p>
<p> <span>2D von Neumann-Mullins grain growth relation</span></p>
<p> Consider a cell structure in 2 dimensions.</p>
<ol><li>Using <a href="http://en.wikipedia.org/wiki/Euler_characteristic" title="Euler's polyhedron formula">Euler's polyhedron formula</a>, which states that for a convex polyhedron, V-E+F = 2, where V, E, and F are the vertices, edges and faces of the polyhedron, </li>
<li>assuming isotropic energy (and equal mobility) for all the cell walls, and </li>
<li>assuming that triple junctions are points at which the cell walls make 120 degrees with each other (in other words, achieve equilibrium),</li>
</ol><p> <a href="http://en.wikipedia.org/wiki/John_von_Neumann" title="Neumann wiki">von Neumann</a> showed that six sided cells are stable; the grains with sides greater than six sides grow while those with less than six sides shrink. Neumann made the assumption that the curvature of each of the walls is a constant. <a href="http://www.materials.cmu.edu/rohrer/mullins/wwmullins.html" title="Mullins homepage">Mullins</a> relaxed the curvature condition, and showed that the result holds even if the curvature is not a constant--mean curvature is what matters. Thus, Neumann's results are valid for soap froths, while that of Mullins is valid even for grain boundaries--albeit in two dimensions. This is a purely topological result and can also be derived using <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss-Bonnet theorem">Gauss-Bonnet theorem</a> -- see the derivation given by <a href="http://www.imm.rwth-aachen.de/hp/person/gg/gg_en.htm" title="Gottstein webpage">Guenter Gottstein</a> and Lasar S Svindlerman in <a href="http://www.amazon.com/Grain-Boundary-Migration-Metals-Thermodynamics/dp/084938222X" title="Grain boundary migration book">Grain boundary migration in metals: Thermodynamics, Kinetics</a>, pp. 309-310, CRC Press, New York 1999, for example.</p>
<p> <span>3D (and N) generalization to von Neumann-Mullins relation<br /></span><br /> Apparently, there had been several attempts to generalize the Neumann-Mullins to 3D in vain, so far. MacPherson and Srolovitz manage to do just that--in the process, they also obtain an N-dimensional<span> </span>generalization. The idea behind the derivation of such a generalized result is as follows:</p>
<ol><li>Generalize Neumann-Mullins to multiply connected domains in 2D; and,</li>
<li>Volume integrate the result.</li>
</ol><p> Obviously, the two step process is nothing but considering all possible 2D sections of the given 3D structure, doing a 2D Neumann-Mullins analysis on each of them, and putting the results of all these analyses together. And, this is also the point where things become a bit (too) mathematical. </p>
<p> In any case, I understand that the net result of the two step process described above is the introduction of a natural measure of length--called mean width--and this measure is a <a href="http://en.wikipedia.org/wiki/Hadwiger%27s_theorem" title="Hadwiger measure">Hadwiger measure</a>. And, the Neumann-Mullins result can be stated in terms of the Hadwiger measures in N dimensions, of which, the 2D and 3D results become a special case. And, the result also shows that in 3- and higher dimensions, the result is not purely topological.</p>
<p> <span>What next?</span></p>
<p> In real systems, say, a grain boundary, for example, the boundary energies are anisotropic; the mobilities are not constant; the triple junctions induce drag on the boundary motion -- Or, in other words, each of the assumptions made by Neumann, Mullins, MacPherson, and Srolovitz are to be relaxed. Thus, this is but a first step in the search for an understanding of grain growth and coarsening studies.</p>
<p> <span>Relevant links</span></p>
<p> <a href="http://mogadalai.wordpress.com/2006/10/30/grain-growth-beyond-von-neumann-mullins/" title="Beyond Neuymann-Mullins blogpost">Most of this post is based on this blog post of mine</a>. I have also <a href="http://http//mogadalai.wordpress.com/2007/04/26/grain-growth-beyond-von-neumann-mullins-edition-2/" title="Beyond Neumann-Mullins --2nd edition">collected the links</a> to the <a href="http://dx.doi.org/10.1038/nature05745" title="Paper of MacPherson and Srolovitz">paper of MacPherson and Srolovitz</a>, the <a href="http://www.nature.com/nature/journal/v446/n7139/suppinfo/nature05745.html" title="Supplementary information to MacPherson and Srolovitz paper">supplementary information to the paper</a>, the <a href="http://dx.doi.org/10.1038/446995a" title="News and Views piece on the Nature paper">News and Views piece on the work by David Kinderlehrer</a>, and the <a href="http://www.sciam.com/article.cfm?chanID=sa003&articleID=29A52001-E7F2-99DF-34AF7AB8412D860A&ref=rss" title="SciAm on the beer froth paper">Scientific American news report</a>.</p>
<p> Have fun!</p>
</div></div></div>Fri, 27 Apr 2007 08:12:20 +0000Mogadalai Gururajan1302 at https://imechanica.orghttps://imechanica.org/node/1302#commentshttps://imechanica.org/crss/node/1302Dynamics of wrinkle growth and coarsening in stressed thin films
https://imechanica.org/node/258
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/17">thin film</a></div><div class="field-item odd"><a href="/taxonomy/term/218">buckling</a></div><div class="field-item even"><a href="/taxonomy/term/219">wrinkling</a></div><div class="field-item odd"><a href="/taxonomy/term/220">R. Huang Group Research</a></div><div class="field-item even"><a href="/taxonomy/term/231">dynamics</a></div><div class="field-item odd"><a href="/taxonomy/term/232">coarsening</a></div><div class="field-item even"><a href="/taxonomy/term/233">power law</a></div><div class="field-item odd"><a href="/taxonomy/term/234">patterns</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><a href="http://link.aps.org/abstract/PRE/v74/e026214" target="_blank" title="wrinkle dynamics">Rui Huang and Se Hyuk Im, Physical Review E 74, 026214 (2006).</a></p>
<p>A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkles can be analyzed using an energy approach. In between, the wrinkle pattern undergoes a coarsening process with a peculiar dynamics. By using a proper scaling and two-dimensional numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening till equilibrium. It is found that, during the initial growth, a stress-dependent wavelength is selected and the wrinkle amplitude grows exponentially over time. During coarsening, both the wrinkle wavelength and amplitude increases, following a simple scaling law under uniaxial compression. Slightly different dynamics is observed under equi-biaxial stresses, which starts with a faster coarsening rate before asymptotically approaching the same scaling under uniaxial stresses. At equilibrium, a parallel stripe pattern is obtained under uniaxial stresses and a labyrinth pattern under equi-biaxial stresses. Both have the same wavelength, independent of the initial stress. On the other hand, the wrinkle amplitude depends on the initial stress state, which is higher under an equi-biaxial stress than that under a uniaxial stress of the same magnitude.</p>
</div></div></div>Tue, 26 Sep 2006 16:05:41 +0000Sehyuk Im258 at https://imechanica.orghttps://imechanica.org/node/258#commentshttps://imechanica.org/crss/node/258