This paper is concerned with the diffuse interface Ohta-Kawasaki energy in three space dimensions, in a periodic setting, in the parameter regime corresponding to the onset of non-trivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to non-trivial minimizers takes place as the small parameter characterizing the width of the interfaces between the two phases goes to zero, while the volume fraction of the minority phases vanishes at an appropriate rate. The value of the threshold is shown to be related to the optimal binding energy solution of Gamow's liquid drop model of the atomic nucleus. Beyond the threshold the average volume fraction of the minority phase is demonstrated to grow linearly with the distance to the threshold. In addition to these results, we establish a number of properties of the minimizers of the sharp interface screened Ohta-Kawasaki energy in the considered parameter regime. We also establish rather tight upper and lower bounds on the value of the transition threshold.
We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problem…
In this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem, arising in the modeling of diblock copolymers. In the main result, by means of a purely variational procedure, we show that it is possible to construct (locally minimizing) periodic critical points whose shape resemble that of any given strictly stable constant mean curvature (periodic) hypersurface. Along the way, we establish several auxiliary results of independent interest.
We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers. We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial sy…
We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, Rätz, Röger and the second author. The model is an extended Cahn-Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinit…
We consider the problem of rigorously computing periodic minimizers to the Ohta-Kawasaki energy. We develop a method to prove existence of solutions and determine rigorous bounds on the distance between our numerical approximations and the true infinite dimensional solution and also on the energy. We use a method with prescribed symmetries to explore the phase space, computing candidate minimizers both with and without experimentally observed symmetries. We find qualitative differences between the phase diagram of the Ohta-Kawasaki energy and self consistent field theory when well away form th…
We study a 3D ternary system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers, which combines an interface energy with a long range interaction term. Although both the binary case in 2D and 3D, and the ternary case in 2D, are quite well studied, very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by some quantity depending…
We study a non-local Cahn-Hilliard energy arising in the study of di-block copolymer melts, often referred to as the Ohta-Kawasaki energy in that context. In this model, two phases appear, which interact via a Coulombic energy. As in our previous work, we focus on the regime where one of the phases has a very small volume fraction, thus creating "droplets" of the minority phase in a "sea" of the majority phase. In this paper, we address the asymptotic behavior of non-minimizing stationary points in dimensions $n \geq 2$ left open by the study of the $Γ$-convergence of the energy established in…
In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions $N\leq 3$. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are $(i)$ well-preparedness of the initial data and $(ii)$ smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the …
This is the second in a series of papers in which we derive a $Γ$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the $Γ$-limit of the leading…
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
Institute of Computer Science and Computational Mathematics, Jagiellonian University（雅盖隆大学计算机科学与计算数学研究所）；Institute of Applied Mathematics and Mechanics, University of Warsaw（华沙大学应用数学与力学研究所）；Department of Mathematics, Rutgers, The State University of New Jersey（新泽西州立大学罗格斯大学数学系）；Department of Mathematical Sciences, George Mason University（乔治梅森大学数学科学系）
We study a two-dimensional ternary inhibitory system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which two species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geom…
We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. In the course of our computations we also give the Green's function for the Laplacian on a periodic strip and explain the heuristic method by which we found it.
The formation and dynamics of free-surface structures, such as steps or terraces and their interplay with the phase separation in the bulk are key features of diblock copolymer films. We present a phase-field model with an obstacle potential which follows naturally from derivations of the Ohta-Kawasaki energy functional via self-consistent field theory. The free surface of the film is incorporated into the phase-field model by including a third phase for the void. The resulting model and its sharp interface limit are shown to capture the energetics of films with steps in two dimensions. For th…
We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation t…
This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minim…
Using area-preserving curve shortening flow, and a new inequality relating the potential generated by a set to its curvature, we study a non-local isoperimetric problem which arises in the study of di-block copolymer melts, also referred to as the Ohta-Kawasaki energy. We are able to show that the only connected critical point is the ball under mild assumptions on the boundary, in the small energy/mass regime. In particular this class includes all rectifiable, connected 1-manifolds in $\mathbb{R}^2$. We also classify the simply connected critical points on the torus in this regime, showing the…
