A rigorous derivation of mean-field models describing 2D micro phase separation

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<jats:title>Abstract</jats:title><jats:p>We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction<jats:inline-formula><jats:alternatives><jats:tex-math>$$\rho $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>such that the micro phase separation results in an ensemble of small disks of one component. We consider the two dimensional case in this paper, whereas the three dimensional case was already considered in Niethammer and Oshita (Calc Var PDE 39:273–305, 2010). Starting from the free boundary problem restricted to disks we rigorously derive the heterogeneous mean-field equations on a time scale of the order of<jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {R}}^{3}\ln (1/\rho )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>ln</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, where<jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {R}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math></jats:alternatives></jats:inline-formula>is the mean radius of disks. On this time scale, the evolution is dominated by coarsening and stabilization of the radii of the disks, whereas migration of disks becomes only relevant on a larger time scale.</jats:p>

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