No abstract

We propose efficient pseudospectral numerical schemes for solving the self-consistent, mean-field equations for inhomogeneous polymers. In particular, we introduce a robust class of semi-implicit methods that employ asymptotic small scale information about the nonlocal density operators. The relaxation schemes are further embedded in a multilevel strategy resulting in a method that can cut down the computational cost by an order of magnitude. Three illustrative problems are used to test the numerical methods: (i) the problem of finding the mean chemical potential field for a prescribed inhomogeneous density of homopolymers; (ii) an incompressible melt blend of two chemically distinct homopolymers; and (iii) an incompressible melt of AB diblock copolymers. Introduction.Field-theoretic models and approaches have proven to be very useful in the study of inhomogeneous polymer and complex fluid phases. The application of such methods to dense phases, such as melts and concentrated solutions of homo-, block-, and graft copolymers, has been particularly fruitful in unraveling the complexities of equilibrium self-assembly in such systems [15,18,16,21,9,19]. Normally the statistical field theory models are solved in the mean-field (saddle point) approximation, although recent developments enable direct numerical simulation of field theories without any simplifying approximations [13,11,7]. Here we shall be concerned with the development of efficient numerical algorithms for solving equilibrium field theories in the mean-field approximation, the so-called self-consistent field theory (SCFT), that is well known in the fields of polymer and colloid science.The SCFT equations are a highly nonlinear set of equations in one or more chemical potential fields. The equations have also a strong nonlocality that emerges from the solution of a modified diffusion equation and reflects the connected nature of a polymer over distances of order, its radius-of-gyration. Solutions for the potential fields, in turn, uniquely specify the equilibrium monomer densities of the different species and other thermodynamic and structural quantities of interest. Two strategies for solving the SCFT equations have emerged: a spectral approach heavily exploited by Matsen and Schick [20,21] and a real-space approach followed by Scheutjens and Fleer [25], Fraaije [8], Fraaije et al. [9], Shi, Noolandi, and Desai [26], Whitmore and Vavasour [29], and Drolet and Fredrickson [5,6], among others. A disadvantage of the fully spectral approach is that the computational effort scales poorly [as O(N 3 x )] with the number of spectral elements N x . It is therefore not well suited to highresolution simulations where there is no advanced knowledge of the symmetries of *

We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic PDEs for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications. The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can follow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

Solvent evaporation has proven to be a remarkably successful tool for directing self-assembly in block copolymers, yet the microscopic mechanisms, processing history dependence and macroscopic control parameters influencing pattern selection remain poorly understood. Here, we leverage dynamical field theory simulations to clarify how copolymer self-assembly proceeds during evaporation. We find that cylinders in the vertical orientation tend to form under modest evaporation rates and relatively weak segregation strengths, and link this behavior to nontrivial, morphology-dependent density correlations present at the ordering front.

The coalescence of two equal-sized deformable drops in an axisymmetric flow is studied, using a boundary-integral method. An adaptive mesh refinement method is used to resolve the local small-scale dynamics in the gap and to retain a reasonable speed of computation. The thin film dynamics is successfully simulated, with sufficient stability and accuracy, up to a film thickness of times the undeformed drop radius, for a range of capillary numbers, Ca, from and viscosity ratios from 4

Recently there has been a strong interest in the area of defect formation in ordered structures on curved surfaces. Here we explore the closely related topic of self-assembly in thin block copolymer melt films confined to the surface of a sphere. Our study is based on a self-consistent field theory (SCFT) model of block copolymers that is numerically simulated by spectral collocation with a spherical harmonic basis and an extension of the Rasmussen-Kalosakas operator splitting algorithm [J. Polym. Sci. Part B: Polym. Phys. 40, 1777 (2002)]. In this model, we assume that the composition of the thin block copolymer film varies only in longitude and colatitude and is constant in the radial direction. Using this approach we are able to study the formation of defects in the lamellar and cylindrical phases, and their dependence on sphere radius. Specifically, we compute ground-state (i.e., lowest-energy) configurations on the sphere for both the cylindrical and lamellar phases. Grain boundary scars are also observed in our simulations of the cylindrical phase when the sphere radius surpasses a threshold value R_{c} approximately 5d , where d is the natural lattice spacing of the cylindrical phase, which is consistent with theoretical predictions [Bowick, Phys. Rev. B 62, 8738 (2000); Bausch, Science 299, 1716 (2003)]. A strong segregation limit approximate free energy is also presented, along with simple microdomain packing arguments, to shed light on the observed SCFT simulation results.

We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle ͑cusp͒ of the zero-viscosity-ratio solution.

No abstract