Related papers: Lamellar phase solutions for diblock copolymers wi…
We study functionals \begin{equation*} F_\varepsilon (u,\rho) := \frac{1}{\varepsilon} \int_\Omega W(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_\Omega \int_\Omega \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx +…
We consider the N-body problem in a layered geometry containing cold polar molecules with dipole moments that are polarized perpendicular to the layers. A harmonic approximation is used to simplify the hamiltonian and bound state properties…
We consider a one-phase free boundary problem of the minimizer of the energy \[ J_{\gamma}(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^{\gamma}dx}, \] with constants $0<s,\gamma<1$. It is…
We present a generalized theory of microphase separation for charged-neutral diblock copolymer melt. Stability limit of the disordered phase for salt-free melt has been calculated using Random Phase Approximation (RPA) and self-consistent…
We present numerical calculations of lamellar phases of block copolymers at patterned surfaces. We model symmetric di-block copolymer films forming lamellar phases and the effect of geometrical and chemical surface patterning on the…
In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with…
We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $\sigma\in(0,1)$. We obtain regularity results for…
This paper investigates the homogenization, dimension reduction, and linearization of a composite plate subjected to external loading within the framework of non-linear elasticity problem. The total elastic energy of the problem is of order…
We study energy relaxation in a phenomenological model for polymer built from rheological considerations: a one dimensional nonlinear lattice with dissipative couplings. These couplings are well known in polymer's community to be possibly…
We present several ordering mechanisms in diblock copolymers. For temperatures above the order-disorder temperature and in the weak segregation regime, a linear response theory is presented which gives the polymer density in the vicinity of…
In this paper we consider a two-dimensional model of a copolymer consisting of a random concatenation of hydrophilic and hydrophobic monomers, immersed in a micro-emulsion of random droplets of oil and water. The copolymer interacts with…
For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):=…
The morphology and dynamics of polymerization-induced phase separation in the initially homogeneous solution of a non-reactive component in reactive monomers are investigated by incorporating the reaction kinetics into the time-dependent…
The original Thomson problem of "spherical crystallography" seeks the ground state of electron shells interacting via the Coulomb potential; however one can also study crystalline ground states of particles interacting with other…
A soft particle model for diblock (AB) copolymer melts is proposed. Each molecule is mapped onto two soft spheres built by Gaussian A- and B-monomer distributions. An approximate analytical expression for the joint distribution function for…
In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The…
An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions.…
We study the regularity of minimizers of the functional $\mathcal E(u):= [u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$. More precisely,…
We show that topological phases with fractional excitations can occur in two-dimensional ultracold dipolar gases on a particular class of optical lattices. Due to the dipolar interaction and lattice confinement, a quantum dimer model…
We investigate the effect of chemically patterned surfaces on the morphology of diblock copolymers below the order-disorder transition. Profiles for lamellar phases in contact with one surface, or confined between two surfaces are obtained…