Results 1  10
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19
STRIPE PATTERNS IN A MODEL For Block Copolymers
, 2010
"... We consider a patternforming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1gvalued functions; the values 0 and 1 correspond to the A and B phases. The paramete ..."
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Cited by 12 (5 self)
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We consider a patternforming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1gvalued functions; the values 0 and 1 correspond to the A and B phases. The parameter " is the ratio between the intrinsic, material lengthscale and the scale of the domain. We show that in the limit " ! 0 any sequence u " of patterns with uniformly bounded energy G "ðu"Þ becomes stripelike: the pattern becomes locally onedimensional and resembles a periodic stripe pattern of periodicity Oð"Þ. In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional G " Gammaconverges to a limit functional G0. This limit functional is defined on fields of rankone projections, which represent the local direction of the stripe pattern. The functional G0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2norm of the divergence of the projection field, or equivalently the L2norm of the curvature of the field. At the level of patterns the converging objects are the jump measures jru "j combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and R€oger, Arch. Rational Mech. Anal. 193 (2009) 475 537, provides the initial estimate and leads to weak measurefunction pair convergence. We obtain strong convergence by exploiting the nonintersection property of the jump set.
Approximation of the Helfrich’s functional via diffuse interfaces
 SIAM J. Math. Anal
, 2010
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Copolymerhomopolymer blends: global energy minimisation and global energy bounds
 Calc. Var. Partial Differential Equations
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Stability of monolayers and bilayers in a copolymerhomopolymer blend model
, 2009
"... We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface OhtaKawasaki energy for straight mono and bilay ..."
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Cited by 7 (5 self)
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We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharpinterface OhtaKawasaki energy for straight mono and bilayers and determining the latter’s sign. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Both for the monolayer and for the bilayer there exist parameter regions where these structures are unstable. For strong repulsive interaction between the monomer types in the diblock copolymer the bilayer is always stable with respect to interface perturbations, irrespective of the domain size. The monolayer is only stable for small domain size. In the course of our computations we also give a Green’s function for the Laplacian on a twodimensional periodic strip.
ΓCONVERGENCE OF GRAPH GINZBURG–LANDAU FUNCTIONALS
"... Abstract. We study Γconvergence of graphbased Ginzburg–Landau functionals, both the limit for zero diffusive interface parameter ε → 0 and the limit for infinite nodes in the graph m → ∞. For general graphs we prove that in the limit ε → 0 the graph cut objective function is recovered. We show tha ..."
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Cited by 6 (2 self)
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Abstract. We study Γconvergence of graphbased Ginzburg–Landau functionals, both the limit for zero diffusive interface parameter ε → 0 and the limit for infinite nodes in the graph m → ∞. For general graphs we prove that in the limit ε → 0 the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg–Landau functional. For both functionals we also study the simultaneous limit ε → 0 and m → ∞, by expressing ε as a power of m and taking m → ∞. Finally we investigate the continuum limit for a nonlocal meanstype functional on a completely connected graph. 1.
Stripe patterns and a projectionvalued formulation of the eikonal equation
 Phil. Trans. R. Soc. A
, 2012
"... We describe recent work on striped patterns in a system of block copolymers. A byproduct of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation the unknown is a field of projection matrices of the form P = e ⊗ e, where e is a unit vector field. We ..."
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Cited by 1 (0 self)
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We describe recent work on striped patterns in a system of block copolymers. A byproduct of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation the unknown is a field of projection matrices of the form P = e ⊗ e, where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.
Copolymerhomopolymer blends: global energy minimisation and global energy bounds
, 2007
"... We study a variational model for a diblockcopolymer/homopolymer blend. The energy functional is a sharpinterface limit of a generalisation of the OhtaKawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in comple ..."
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We study a variational model for a diblockcopolymer/homopolymer blend. The energy functional is a sharpinterface limit of a generalisation of the OhtaKawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be nonunique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.
unknown title
, 2013
"... A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limit ..."
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A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limit