On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

A rheological and SAXS study of the lamellar order in a side-on liquid crystalline block copolymer

We study the structure and shear flow behavior of a side-on liquid crystalline triblock copolymer, named PBA-b-PA444-b-PBA (PBA is poly(butyl acrylate) and PA444 is a poly(acrylate) with a nematic liquid crystal side-on mesogen), in the self-assembled lamellar phase and in the disordered phase. Simultaneous oscillatory shear and small-angle X-ray scattering experiments show that shearing PBA-b-PA444-b-PBA at high frequency and strain amplitudes leads to the alignment of the lamellae with normals perpendicular to the shear direction and to the velocity gradient direction, i.e., in the perpendicular orientation. The order-to-disorder transition temperature (T-ODT) is independent of the applied strain, in contrast to results reported in the literature for coil-coil diblock copolymers, which show an increase in T-ODT with shear rate. It is possible that in our system, T-ODT does not depend on the applied strain because the fluctuations are weaker than those present in coil-coil diblock copolymer systems.

Reversible sphere-to-lamellar wetting transition at the interface of a diblock copolymer system

We use ellipsometry to investigate a transition in the morphology of a sphere-forming diblock copolymer thin-film system. At an interface the diblock morphology may differ from the bulk when the interfacial tension favours wetting of the minority domain, thereby inducing a sphere-to-lamella transition. In a small, favourable window in energetics, one may observe this transition simply by adjusting the temperature. Ellipsometry is ideally suited to the study of the transition because the additional interface created by the wetting layer affects the polarisation of light reflected from the sample. Here we study thin films of poly(butadiene-ethylene oxide) (PB-PEO), which order to form PEO minority spheres in a PB matrix. As temperature is varied, the reversible transition from a partially wetting layer of PEO spheres to a full wetting layer at the substrate is investigated.

Kinetics of layer hopping in a diblock copolymer lamellar phase

In the ordered state, symmetric diblock copolymers self-assemble into an anisotropic lamellar morphology. The equilibrium thickness of the lamellae is the result of a delicate balance between enthalpic and entropic energies, which can be tuned by controlling the temperature. Here we devise a simple yet powerful method of detecting tiny changes in the lamellar thickness using optical microscopy. From such measurements we characterize the enthalpic interaction as well as the kinetics of molecules as they hop from one layer to the next in order to adjust the lamellar thickness in response to a temperature jump. The resolution of the measurements facilitate a direct comparison to predictions from self-consistent field theory.

Local gratings due to angular hole burning in a photorefractive polymer

We investigate the processes involved in writing real-time holographic gratings in a photorefractive polymer (PRP) that incorporates an azo-dye. In such systems there may be gratings due to mechanisms associated with trans–cis isomerization (angular hole burning (AHB) and/or angular redistribution), which appear in addition to those arising from the photorefractive (PR) effect. The work presented here helps to understand the interactions which may occur between these different gratings. The formation of local gratings due to mechanisms associated with photoisomerization is studied, in a new PRP based on the photoconductor poly(N-vinylcarbazole):2, 4, 7-trinitro-9-fluorenone, plasticized with N-ethylcarbazole. The polymer includes the azo-dye 4-nitro-4'-pentyloxy-azobenzene and we observe both PR and photoisomerization gratings. The gratings are shown to be both polarization-sensitive and reversible. The presence of the photoisomerization gratings (which diffract almost as strongly as the PR gratings) significantly affects the field-dependent diffractive behaviour of the composite. A measurement of the lifetime of the cis state is made (τcis = 38 s) using photoinduced dichroism. This is close to the decay time constant of the local gratings (τdecay = 42 s), and it is suggested that the local grating mechanism is AHB of the azo-dye. This is the first time (to the knowledge of the authors) that a local grating due to AHB has been demonstrated in a PRP.

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

Step edges in thin films of lamellar-forming diblock copolymer

Self-consistent field theory (SCFT) is used to study the step edges that occur in thin films of lamellar-forming diblock copolymer, when the surfaces each have an affinity for one of the polymer components. We examine film morphologies consisting of a stack of ν continuous monolayers and one semi-infinite bilayer, the edge of which creates the step. The line tension of each step morphology is evaluated and phase diagrams are constructed showing the conditions under which the various morphologies are stable. The predicted behavior is then compared to experiment. Interestingly, our atomic force microscopy (AFM) images of terraced films reveal a distinct change in the character of the steps with increasing ν, which is qualitatively consistent with our SCFT phase diagrams. Direct quantitative comparisons are not possible because the SCFT is not yet able to probe the large polymer/air surface tensions characteristic of experiment.

Melting Regime of the Anionic Phospholipid DMPG: New Lamellar Phase and Porous Bilayer Model

Aqueous dispersions of the anionic phospholipid dimyristoyl phosphatidylglycerol (DMPG) at pH above the apparent pK of DMPG and concentrations in the interval 70-300 mM have been investigated by small (SAXS) and wide-angle X-ray scattering, differential scanning calorimetry, and polarized optical microscopy. The order. disorder transition of the hydrocarbon chains occurs along an interval of about 10 degrees C (between T(m)(on) similar to 20 degrees C and T(m)(off) similar to 30 degrees C). Such melting regime was previously characterized at lower concentrations, up to 70 mM DMPG, when sample transparency was correlated with the presence of pores across the bilayer. At higher concentrations considered here, the melting regime persists but is not transparent. Defined SAXS peaks appear and a new lamellar phase L(p) with pores is proposed to exist above 70 mM DMPG, starting at similar to 23 degrees C (similar to 3 degrees C above T(m)(on)) and losing correlation after T(m)(off). A new model for describing the X-ray scattering of bilayers with pores, presented here, is able to explain the broad band attributed to in-plane correlation between pores. The majority of cell membranes have a net negative charge, and the opening of pores across the membrane tuned by ionic strength, temperature, and lipid composition is likely to have biological relevance.

Laser microstructuring of azopolymers via surface relief gratings: controlling hydrophobicity

In this paper the large-scale mass transport mechanism is used to microstructure azopolymeric films, aiming at controllable hydrophobic surfaces. Using an Ar(+) laser with intensity of 70 mW/cm(2), we produced egg-crate-like surfaces with periods from 1.0 to 3.5 mu m that present distinct wetting properties. The static contact angle of water was measured on the microstructured surfaces, and the results revealed an increase of approximately 9 degrees for a surface pattern period of 2 mu m. Our results indicate the use of the microstructuring method described here for the fabrication of devices with controllable hydrophobicity.

NMR study of slow motions of HDA hydrocarbons chains inside lamellar structures

Measurements of H-1 and C-13 Nuclear Magnetic Resonance (NMR) for the nano-composite materials formed by the intercalation of hexadecylamine (HDA) in metal oxides (TiO2, V2O5 and MoO3), are reported. The H-1 NMR spin-lattice relaxation in the rotating frame was described by using the spectral density due to Davidson and Cole, which incorporates a distribution of correlation times characterized by a width parameter epsilon. The fitting of the data was obtained for epsilon = 0.74, indicating that the correlation times are distributed over a narrow range in this system. High-resolution C-13 NMR techniques were used to resolve the NMR lines of middle-chain methylene groups in the spectra and variable contact time cross-polarization {H-1-}C-13 experiments were employed to analyze the reorientation dynamics of the CH3 and CH2 groups in the HDA chains.

Morphological abnormalities and apoptosis in lamellar tissue of equines after intestinal obstruction and treatment with hydrocortisone

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

Repair of lamellar scleral lesions in dogs with preserved equine renal capsule: short report

OBJETIVO: Avaliar o uso da cápsula renal de eqüino preservada em glicerina 98% no reparo de lesões lamelares esclerais em cães. MÉTODOS: Foram utilizados 12 cães, machos e fêmeas, com peso médio de 12kg. Foram realizadas avaliações clínica e morfológica aos 1, 3, 7, 15, 30 e 60 dias de pós-operatório. Após anestesia geral e procedimentos padrões de preparo do campo operatório, foi realizada cantotomia temporal, seguida de incisão conjutival e escleral com área de 0,5x0,5 cm na posição de 1hora, próxima ao limbo. em seguida, um fragmento de mesma dimensão de cápsula renal de eqüino preservada em glicerina, previamente hidratado em solução salina, foi aplicado ao defeito escleral criado sendo fixado com pontos simples isolados com vicryl 7-0®. RESULTADOS: A avaliação clínica revelou blefaroespasmo/fotofobia até o sétimo dia de pós-operatório. Foi observado edema conjuntival até o quinto dia, acompanhado de secreção ocular mucóide, que persistiu até o décimo dia de pós-operatório. Não foram observados sinais clínicos de rejeição do enxerto em todos os animais, em todos os períodos avaliados. Os segmentos anterior e posterior do bulbo ocular não apresentaram sinais de inflamação. A análise morfológica revelou exsudação inflamatória aguda nos períodos precoces e intermediários da avaliação e inflamação crônica nos períodos tardios da observação. Houve incorporação do enxerto ao leito receptor. CONCLUSÃO: Os resultados sugerem que a cápsula renal de eqüino preservada pode ser mais uma alternativa de membrana biológica para o reparo de lesões esclerais lamelares em cães e no homem.