Semilinear parabolic problems in thin domains with a highly oscillatory boundary

In this paper, we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment (thin domain) which has an oscillating boundary. We combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations. (C) 2011 Elsevier Ltd. All rights reserved.

Semilinear parabolic problems in thin domains with a highly oscillatory boundary

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Error estimates for a neumann problem in highly oscillating thin domains

In this work we analyze the convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain with highly oscillatory behavior. We consider the case where the height of the domain, amplitude and period of the oscillations are all of the same order, and given by a small parameter e > 0. Using an appropriate corrector approach, we show strong convergence and give error estimates when we replace the original solutions by the first-order expansion through the Multiple-Scale Method.

Homogenization of elliptic problems in thin domains with oscillatory boundaries

Los dominios finos, es decir, dominios sustancialmente más pequeños en alguna o varias de sus direcciones que en el resto, aparecen en muchos campos de la ciencia. Por ejemplo, dinámica de fluídos (lubricación, conducción de fluídos en tubos delgados, dinámica de oceanos...), mecánica de sólidos (barras delgadas, placas o cáscaras) o incluso en fisiología (circulación de la sangre). Así, el amplio número de posibles aplicaciones a situaciones reales ha hecho que la investigación de modelos de ecuaciones en derivadas parciales en dominios finos se convierta en un tema muy estudiado en los últimos años. Desde un punto de vista matemático, el estudio de las soluciones de una EDP en un dominio fino es un caso particular de la cuestión general relativa a cómo la variación de los dominios afecta al comportamiento de las soluciones de la EDP. En este marco, obtener la ecuación límite del modelo considerado, comparar la solución de la ecuación límite y las soluciones del problema en el dominio fino, analizar los coeficientes de la ecuación límite y comprender cómo la geometría del dominio afecta a la ecuación límite son algunos de los objetivos que deberían ser alcanzados. De hecho, es importante señalar que este tipo de cuestiones no sólo proporcionan importantes resultados teóricos sino que son muy relevantes desde el punto de vista de las aplicaciones. Por ejemplo, ser capaz de reducir el problema original a un problema mucho más sencillo, problema límite, que refleje las principales características del problema de partida puede ser muy útil para ingenieros y físicos...

Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary

We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media.

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, AN Carvalho. G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz. Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are Upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in L(p) and H(1) norms. (C) 2008 Elsevier Inc. All rights reserved.

A NOTE ON RESOLVENT CONVERGENCE ON A THIN DOMAIN

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bundles of polytwins as meta-elastic domains in the thin polycrystalline simple multi-ferroic system PZT.

We used enhanced piezo-response force microscopy (E-PFM) to investigate both ferroelastic and ferroelectric nanodomains in thin films of the simple multi-ferroic system PbZr(0.3)Ti(0.7)O(3) (PZT). We show how the grains are organized into a new type of elastic domain bundles of the well-known periodic elastic twins. Here we present these bundle domains and discuss their stability and origin. Moreover, we show that they can arrange in such a way as to release strain in a more effective way than simple twinning. Finally, we show that these bundle domains can arrange to form the macroscopic ferroelectric domains that constitute the basis of ferroelectric-based memory devices.

Nucleation, growth, and control of ferroelectric-ferroelastic domains in thin polycrystalline films

The unique response of ferroic materials to external excitations facilitates them for diverse technologies, such as nonvolatile memory devices. The primary driving force behind this response is encoded in domain switching. In bulk ferroics, domains switch in a two-step process: nucleation and growth. For ferroelectrics, this can be explained by the Kolmogorov-Avrami-Ishibashi (KAI) model. Nevertheless, it is unclear whether domains remain correlated in finite geometries, as required by the KAI model. Moreover, although ferroelastic domains exist in many ferroelectrics, experimental limitations have hindered the study of their switching mechanisms. This uncertainty limits our understanding of domain switching and controllability, preventing thin-film and polycrystalline ferroelectrics from reaching their full technological potential. Here we used piezoresponse force microscopy to study the switching mechanisms of ferroelectric-ferroelastic domains in thin polycrystalline Pb 0.7Zr0.3TiO3 films at the nanometer scale. We have found that switched biferroic domains can nucleate at multiple sites with a coherence length that may span several grains, and that nucleators merge to form mesoscale domains, in a manner consistent with that expected from the KAI model. © 2012 American Physical Society.

Formation of mesophase mono-domains and micro-structures in thin films of a series of copolyethers containing both odd-numbered methylene units

The structural and morphological evolution of mono-domains in thin films has been investigated for a series of liquid crystalline (LC) copolyethers. The copolyethers studied were synthesized by the reaction of 1-(4-hydroxy-4 ' -biphenylyl)-2-(4-hydroxyl-phenyl)propane (TPP) with 1,7-dibromoheptane and 1,11-undecane at different compositions (coTPPs-7/11). In contrast to the solution-cast thin films without annealing, which exhibit the isotropic homogeneous molecular orientation, mono-domains with a homeotropic alignment were found in coTPP-7/11(5/5) after the thin films were annealed in the high-temperature columnar phase (Phi '). Similar to the nucleation process in polymer crystallization, transmission electron microscopic observations show that small mono-domains appear in the initial stage of annealing, where molecules form a uniaxial in-plane chain orientation. With increasing annealing time, the molecular orientation gradually became tilted with respect to the substrate surface, and finally, a uniaxial homeotropic molecular orientation was achieved after a prolonged annealing time. The lateral size of mono-domains was found to increase continuously with annealing time and grew into a circular shape, indicating an isotropic lateral growth scheme which implies a hexagonal molecular packing proved by the electron diffraction experiments.

High mobility diketopyrrolopyrrole (DPP)-based organic semiconductor materials for organic thin film transistors and photovoltaics

In recent years, the electron-accepting diketopyrrolopyrrole (DPP) moiety has been receiving considerable attention for constructing donor-acceptor (D-A) type organic semiconductors for a variety of applications, particularly for organic thin film transistors (OTFTs) and organic photovoltaics (OPVs). Through association of the DPP unit with appropriate electron donating building blocks, the resulting D-A molecules interact strongly in the solid state through intermolecular D-A and π-π interactions, leading to highly ordered structures at the molecular and microscopic levels. The closely packed molecules and crystalline domains are beneficial for intermolecular and interdomain (or intergranular) charge transport. Furthermore, the energy levels can be readily adjusted, affording p-type, n-type, or ambipolar organic semiconductors with highly efficient charge transport properties in OTFTs. In the past few years, a number of DPP-based small molecular and polymeric semiconductors have been reported to show mobility close to or greater than 1 cm2 V -1 s-1. DPP-based polymer semiconductors have achieved record high mobility values for p-type (hole mobility: 10.5 cm2 V-1 s-1), n-type (electron mobility: 3 cm2 V-1 s-1), and ambipolar (hole/electron mobilities: 1.18/1.86 cm2 V-1 s-1) OTFTs among the known polymer semiconductors. Many DPP-based organic semiconductors have favourable energy levels and band gaps along with high hole mobility, which enable them as promising donor materials for OPVs. Power conversion efficiencies (PCE) of up to 6.05% were achieved for OPVs using DPP-based polymers, demonstrating their potential usefulness for the organic solar cell technology. This article provides an overview of the recent exciting progress made in DPP-containing polymers and small molecules that have shown high charge carrier mobility, around 0.1 cm2 V-1 s-1 or greater. It focuses on the structural design, optoelectronic properties, molecular organization, morphology, as well as performances in OTFTs and OPVs of these high mobility DPP-based materials.

3,6-Di(furan-2-yl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione and bithiophene copolymer with rather disordered chain orientation showing high mobility in organic thin film transistors

Pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione or diketopyrrolopyrrole (DPP) is a useful electron-withdrawing fused aromatic moiety for the preparation of donor-acceptor polymers as active semiconductors for organic electronics. This study uses a DPP-furan-containing building block, 3,6-di(furan-2-yl)pyrrolo[3,4- c]pyrrole-1,4(2H,5H)-dione (DBF), to couple with a 2,2′-bithiophene unit, forming a new donor-acceptor copolymer, PDBFBT. Compared to its structural analogue, 3,6-di(thiophen-2-yl)pyrrolo[3,4-c]pyrrole-1,4(2H,5H)-dione (DBT), DBF is found to cause blue shifts of the absorption spectra both in solution and in thin films and a slight reduction of the highest occupied molecular orbital (HOMO) energy level of the resulting PDBFBT. Despite the fact that its thin films are less crystalline and have a rather disordered chain orientation in the crystalline domains, PDBFBT shows very high hole mobility up to 1.54 cm 2 V-1 s-1 in bottom-gate, top-contact organic thin film transistors.

Sensing Shape Recovery Using Conductivity Noise in Thin Films of NiTi Shape Memory Alloys

Low frequency fluctuations in the electrical resistivity, or noise, have been used as a sensitive tool to probe into the temperature driven martensite transition in dc magnetron sputtered thin films of nickel titanium shape-memory alloys. Even in the equilibrium or static case, the noise magnitude was more than nine orders of magnitude larger than conventional metallic thin films and had a characteristic dependence on temperature. We observe that the noise while the temperature is being ramped is far larger as compared to the equilibrium noise indicating the sensitivity of electrical resistivity to the nucleation and propagation of domains during the shape recovery. Further, the higher order statistics suggests the existence of long range correlations during the transition. This new characterization is based on the kinetics of disorder in the system and separate from existing techniques and can be integrated to many device applications of shape memory alloys for in-situ shape recovery sensing.

High energy oxygen ion induced modifications in lead based perovskite thin films

The lead based ferroelectric PbZr0.53Ti0.47O3 (PZT), (Pb0.90La0.10)TiO3 (PLT10) and (Pb0.80La0.20)TiO3 (PLT20) thin films, prepared by pulsed laser ablation technique, were studied for their response to the 70 MeV oxygen ion irradiation. The dielectric analysis, capacitance-voltage (C- V) and DC leakage current measurements were performed before and after the irradiation to high-energy oxygen ions. The irradiation produced considerable changes in the dielectric, C-V, leakage characteristics and induced some amount of amorphization. The PZT films showed partial recrystallization after a thermal annealing at 400 degrees C for 10 min. The phase transition temperature [T-c] of PLT20 increased from 115 degrees C to 120 degrees C. The DC conductivity measurements showed a shift in the onset of non-linear conduction region. The current density decreased by two orders of magnitude after irradiation. After annealing the irradiated films at a temperature of 400 degrees C for 10 min, the films partially regained the dielectric and electrical properties. The results are discussed in terms of the irradiation-induced amorphization, the pinning of the ferroelectric domains by trapped charges and the thermal annealing of the defects generated during the irradiation. (C) 2007 Elsevier B.V. All rights reserved.