ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

March 2006 , Volume 6 , Issue 2

Special Issue On

Advances in Materials Modeling: Analysis and Simulations

Guest Editor: Carlos J. Garcia-Cervera

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2006, 6(2): i-ii
doi: 10.3934/dcdsb.2006.6.2i

*+*[Abstract](518)*+*[PDF](26.9KB)**Abstract:**

The field of Materials Science is probably the most multidisciplinary among the scientific and engineering fields. It relates to the study of matter, and the interactions between the atomic, nano-, micro-, and meso- scales. The focus of this special issue of DCDS-B is the modeling, analysis, and simulations of complex systems in Materials Science. In this special issue we present a collection of articles covering a wide range of topics, including ferromagnetism, liquid crystals, electro-kinetic hydrodynamics, and mass transport, among others. The articles have undergone a rigorous peer review process.

Understanding processes such as crack propagation and fracture, or solid state phase transformations is fundamental in the manufacture of new metallic alloys, designed for specific applications. In many cases, this involves relating the desired properties of a material, to the structure of the atoms and phases in that material. Advances in this area are presented in the papers of Alikakos, Bates, Cahn, Fife, Fusco, and Tanoglu, and in the paper of Yu and Du.

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2006, 6(2): 237-255
doi: 10.3934/dcdsb.2006.6.237

*+*[Abstract](586)*+*[PDF](337.5KB)**Abstract:**

We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.

2006, 6(2): 257-272
doi: 10.3934/dcdsb.2006.6.257

*+*[Abstract](576)*+*[PDF](368.1KB)**Abstract:**

Varistor ceramics are very heterogeneous nonlinear conductors, used in devices to protect electrical equipment against voltage surges in power lines. The fine structure in the material induces highly oscillating coefficients in the elliptic electrostatic equation as well as in the Maxwell equations. We suggest how the properties of ceramic varistors can be simulated by solving the homogenized problems, i.e. the corresponding homogenized elliptic problem and the homogenized Maxwell equations. The fine scales in the model yield local equations coupled with the global homogenized equations. Lower and upper bounds are also given for the overall electric conductivity of varistor ceramics. These two bounds are associated with two types of failures in varistor ceramics. The upper bound corresponds to thermal heating and the puncture failure due to localization of strong currents. The lower bound corresponds to fracturing of the varistor, due to charge build up at the grain boundaries resulting in stress caused by the piezoelectric property of the varistor.

2006, 6(2): 273-290
doi: 10.3934/dcdsb.2006.6.273

*+*[Abstract](579)*+*[PDF](269.3KB)**Abstract:**

The goal of this article is to analyze the time asymptotic stability of one dimensional Bloch walls in ferromagnetic materials. The equation involved in modelling such materials is the Landau-Lifchitz system which is non-linear and parabolic. We demonstrate that the equilibrium states called Bloch walls are asymptotically stable modulo a rotation and a translation transverse to the wall. The linear part of the perturbed equation admits zero as an eigenvalue forbidding a direct proof.

2006, 6(2): 291-310
doi: 10.3934/dcdsb.2006.6.291

*+*[Abstract](457)*+*[PDF](526.6KB)**Abstract:**

We revisit the permeation flow issue in weakly sheared cholesteric liquid crystal polymers in plane Couette and Poiseuille flow geometries using a mesoscopic theory obtained from the kinetic theory for flows of cholesteric liquid crystal polymers [2]. We first present two classes of equilibrium solutions due to the order parameter variation and the director variation, respectively; then, study the permeation mode in weakly sheared flows of cholesteric liquid crystal polymers employing a coarse-grain approximation. We show that in order to solve the permeation flow problem correctly using the coarse-grain approximation, secondary flows must be considered, resolving a long standing inconsistency in the study of cholesteric liquid crystal flows [7]. Asymptotic solutions are sought in Deborah number expansions. The primary and secondary flow as well as the director dynamics are shown to dominate at leading order while the local nematic order fluctuations are higher order effects. The leading order solutions are obtained explicitly and analyzed with respect to the cholesteric pitch and other material parameters. The role of the anisotropic elasticity in equilibrium phase transition and permeation flows is investigated as well.

2006, 6(2): 311-338
doi: 10.3934/dcdsb.2006.6.311

*+*[Abstract](767)*+*[PDF](361.9KB)**Abstract:**

Weak topology implicit schemes based on Monge-Kantorovich or Wasserstein metrics have become prominent for their ability to solve a variety of diffusion and diffusion-like equations. They are very flexible, encompassing a wide range of nonlinear effects. They have interesting interpretations as descent algorithms in an infinite dimensional manifold setting or as dissipation principles for motion in a highly viscous environment. Transport plays a fundamental role in these schemes, as noted by Brenier and Benamou and reviewed below. The reverse implication is less explored and, at least at the outset, less obvious. Here we discuss the simplest situations in the context of systems of transport equations. We show how arbitrary Fokker-Planck Equations in one dimension conform to the mass transport paradigm. Finally, we provide some additional examples, including a simple existence result for velocity-jump processes.

2006, 6(2): 339-356
doi: 10.3934/dcdsb.2006.6.339

*+*[Abstract](702)*+*[PDF](285.4KB)**Abstract:**

Monolayer films of liquid crystalline polymers (LCPs) are modelled with a mesoscopic two-dimensional (2D) analogue of the Doi-Hess (1981, 1976) rigid rod model. One aim is to establish a more complete solution to the classical problem of how orientational degeneracy of quiescent nematic equilibria breaks in weak shear. We exploit the simplicity of 2D liquids to extend results of Kuzuu and Doi (1983,1984), Marrucci and Maffetone (1989-1991), See, Doi and Larson (1990), Forest

*et al.*(2003-2004). We recall the distinction between two versus three dimensional quiescent phase diagrams and the isotropic-nematic phase transition, then analyze the deformation of these respective bifurcation diagrams in shear flow. We give a simple proof that limit cycles, known as tumbling orbits, must arise beyond the parameter boundary for the steady-unsteady transition. Finally, we show the shear-perturbed 2D phase diagram is significantly more robust to closure approximations than the 3D system.

2006, 6(2): 357-371
doi: 10.3934/dcdsb.2006.6.357

*+*[Abstract](652)*+*[PDF](269.7KB)**Abstract:**

Electro-kinetic fluids can be modeled by hydrodynamic systems describing the coupling between fluids and electric charges. The system consists of a momentum equation together with transport equations of charges. In the dynamics, the special coupling between the Lorentz force in the velocity equation and the material transport in the charge equation gives an energy dissipation law. In stationary situations, the system reduces to a Poisson-Boltzmann type of equation. In particular, under the no flux boundary conditions, the conservation of the total charge densities gives nonlocal integral terms in the equation. In this paper, we analyze the qualitative properties of solutions to such an equation, especially when the Debye constant $\epsilon$ approaches zero. Explicit properties can be derived for the one dimensional case while some may be generalized to higher dimensions. We also present some numerical simulation results of the system.

2006, 6(2): 373-389
doi: 10.3934/dcdsb.2006.6.373

*+*[Abstract](694)*+*[PDF](3954.6KB)**Abstract:**

Using the Gauss-Seidel projection method developed in [4] and [17], we simulate the three dimensional domain wall structures for thin films at various thickness. We observe transition from Néel wall to cross-tie wall and to Bloch wall as the thickness is increased. Periodic structures for cross-tie wall are also studied. The results are in good agreement with the experimental observations. Hysteresis loops are calculated for samples of various sizes. In particular, we study the effect of cross-tie wall in the switching process. These simulations have demonstrated high efficiency of the Gauss-Seidel projection method.

2006, 6(2): 391-406
doi: 10.3934/dcdsb.2006.6.391

*+*[Abstract](609)*+*[PDF](818.9KB)**Abstract:**

In the phase-field modeling of the mezoscopic morphology and microstructure evolution in many material processes, an anisotropic mobility is often needed that depends on the interfacial normal direction. It is a challenge to define the anisotropic mobility function on the whole simulation domain while the interfacial normal can only be meaningfully determined on the interface. We propose a variational approach for the construction of a smoothened mobility function that mimics the prescribed anisotropic mobility on the interface and extends smoothly to the whole simulation domain. Some theoretical analysis of the proposed method are made to ensure its validity and to provide hints on the effects and the choices of various parameters. An iterative scheme for the numerical solution of the variational problem is also described. Several numerical tests are presented to illustrate the effect of a smoother anisotropic mobility on the interfacial dynamics, and the advantage over using a cutoff mobility.

2006, 6(2): 407-425
doi: 10.3934/dcdsb.2006.6.407

*+*[Abstract](656)*+*[PDF](546.0KB)**Abstract:**

The aim of this work is to model and simulate processing-induced heterogeneity in rigid, rod-like nematic polymers in viscous solvents. We employ a mesoscopic orientation tensor model due to Doi, Marrucci and Greco which extends the small molecule, liquid crystal theory of Leslie-Ericksen-Frank to nematic polymers. The morphology has various physical realizations, all coupled through the model equations: the orientational distribution of the ensemble of rods, anisotropic viscoelastic stresses, and flow feedback. Previous studies in plane Couette & Poiseuille flow (with the exception of [7]) have focused on the coupling between hydrodynamics and the orientational distribution of rigid rod polymers with identical anchoring conditions at solid boundaries; without flow, the equilibrium structure is homogeneous. Here we explore steady structures that emerge with mismatch anchoring conditions at the walls, which couple an equilibrium elastic distortion across the channel, short and long range elasticity potentials, and hydrodynamics. In plane Couette (where moving plates drive the flow) and Poiseuille flow (where a pressure gradient drives the flow), in the regime of weak flow and strong distortional elasticity, asymptotic analysis yields closed-form steady solutions and scaling laws with identical wall conditions. We focus simulations in this regime to expose the effects due to wall anchoring conflicts, and illustrate the induced morphology of the orientational distribution, stored viscoelastic stresses, and non-Newtonian flow. A remarkably simple diagnostic emerges in this physical parameter regime, in which salient morphology features are controlled by the amplitude and sign of the difference in plate anchoring angles of the director field at the two plates.

2017 Impact Factor: 0.972

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