There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The flow of homogeneous fluids through porous media: analogies with other physical problems

A mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented. Our calculation, which traverses the weak- to strong-segregation regimes, is free of traditional approximations. Regions of stability are determined for disordered (DIS) melts and for ordered structures including lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (QIm3m), close-packed spheres (CPS), and the bicontinuous cubic network with Ia3d symmetry (QIa3d). The CPS phase exists in narrow regions along the order−disorder transition for χN ≥ 17.67. Results suggest that the QIa3d phase is not stable above χN ∼ 60. Along the L/QIa3d phase boundaries, a hexagonally perforated lamellar (HPL) phase is found to be nearly stable. Our results for the bicontinuous Pn3m cubic (QPn3m) phase, known as the OBDD, indicate that it is an unstable structure in diblock melts. Earlier approximation schemes used to examine mean-field behavior are reviewed, and compa...

Unifying Weak- and Strong-Segregation Block Copolymer Theories

____ : ..... '"i-t.-

Droplet based microfluidics is a rapidly growing interdisciplinary field of research combining soft matter physics, biochemistry and microsystems engineering. Its applications range from fast analytical systems or the synthesis of advanced materials to protein crystallization and biological assays for living cells. Precise control of droplet volumes and reliable manipulation of individual droplets such as coalescence, mixing of their contents, and sorting in combination with fast analysis tools allow us to perform chemical reactions inside the droplets under defined conditions. In this paper, we will review available drop generation and manipulation techniques. The main focus of this review is not to be comprehensive and explain all techniques in great detail but to identify and shed light on similarities and underlying physical principles. Since geometry and wetting properties of the microfluidic channels are crucial factors for droplet generation, we also briefly describe typical device fabrication methods in droplet based microfluidics. Examples of applications and reaction schemes which rely on the discussed manipulation techniques are also presented, such as the fabrication of special materials and biophysical experiments.

https://iopscience.iop.org/article/10.1088/0034-4885/75/1/016601/pdf

Droplet based microfluidics

https://web.math.ucsb.edu/~hdc/public/JCP2003BCB.pdf

Computation of multiphase systems with phase field models

Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.

Computation of Multiphase Systems with Phase Field Models

An efficient dynamically adaptive mesh for potentially singular solutions

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.

Numerical Solution of Polymer Self-Consistent Field Theory

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 O(10−4) times the undeformed drop radius, for a range of capillary numbers, Ca, from O(10−4–10−1) and viscosity ratios from O(0.1–10). The results are compared with experimental results from our earlier studies as well as the simple scaling theory for film drainage. The collisions for time-independent flow simulating head-on collisions in the experimental studies show two distinctively different regimes. At lower capillary numbers, the interfaces of the thin film between the colliding drops remain almost spherical up to the point of film rupture, and the dimensionless drainage time scales as tdG∼Ca. At higher capillary numbers, t...

Hector D. Ceniceros

Papers

Computation of multiphase systems with phase field models

Computation of Multiphase Systems with Phase Field Models

An efficient dynamically adaptive mesh for potentially singular solutions

Numerical Solution of Polymer Self-Consistent Field Theory

Coalescence of two equal-sized deformable drops in an axisymmetric flow