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We discuss the study of the sphericity of the real shape of round thin-film membranes when it changes during the bulge testing. Membrane structures: SiN x /SiO 2 /SiN x /SiO 2 , pSi* /SiN x /SiO 2 , Al, etc. We described the technique for determining the areas of deviation of the membrane surface shape from a spherical one, estimating the magnitude and peculiarities of the distribution of the radius of curvature along the membrane diameter. It is shown that the shape of the membranes differs from spherical closer to the edge (perimeter), and in many cases also to the area of the top (center) of the membrane. A trend was found: an increase in the radius of curvature as it approaches the center of the membrane.

The closed-form solution of the problem regarding elastic contact between a transversely, uniformly loaded circular membrane and a spring-reset rigid flat circular plate has potential application value in sensor developments or bending-free shell designs, but it still needs to be further improved. In this paper, on the basis of existing studies, the plate/membrane elastic contact problem is reformulated by improving the system of differential equations governing the elastic behavior of a large deflection of a circular membrane. Specifically, the radial geometric equation used in the existing studies is improved by giving up the assumption of a small rotation angle for the membrane, and an improved closed-form solution to the plate/membrane elastic contact problem is presented. The convergence and validity of the improved closed-form solution are analyzed, and the difference between the closed-form solutions before and after improvement is graphically shown. In addition, the effect of changing some important geometric and physical parameters on the improved closed-form solution is investigated.

The features of thin-film membranes, which are formed above round holes in silicon substrates using the Bosch-process are considered. The membrane has a complex shape due to the presence of the stress state of the initial films. The analysis of the dependence of the membrane deflection w on the supplied overpressure P is used to calculate the mechanical characteristics of the membranes. In this case, it is necessary to determine directly on the membrane its diameter, the thickness of the constituent layers, the change in the topography of the membrane surface over its entire area as the overpressure increases. Determination of the membrane diameter and the thicknesses of the constituent layers is shown by the example of p -Si*/SiN x /SiO 2 and SiN x /SiO 2 /SiN x /SiO 2 membranes. We used spectral ellipsometry, energy-dispersive X-ray spectroscopy, optical profilometry, optical microscopy. The influence of the peculiarities of the fixing conditions on the stress-strain state of membranes is shown, and the assessment is carried out by means of numerical modeling. A technique has been developed for measuring and calculating the mechanical characteristics of membranes that have an initial deflection. The calculation result is shown on the example of a membrane with an initial deflection of 2 µm—SiN x /SiO 2 /SiN x /SiO 2 and a membrane with an initial deflection of 30 µm—Al/SiO 2 /Al.

We discuss a technique for investigating changes in complex topography and shape of structures using geomorphometric methods to study surfaces of wafers and membranes formed by the Bosch process. The wafers were analyzed before and after the deposition of the SiO 2 layer. The membranes were analyzed during the bulge testing. The study was carried out using maps of the catchment area and principal curvatures taking into account artifacts of the approximation of experimental data. We found a correspondence between the distribution of lines connecting the highest surface areas before and after the deposition of the SiO 2 layer on the wafers. For membranes with structure: Al(0 . 8 μm)/SiO 2 (0 . 6 μm)/Al(1 . 1 μm), pSi*(0 . 8 μm)/SiN x (0.13 μm)/SiO 2 , Al(0 . 6 μm) we also found that features of membrane boundaries are mainly caused by their initial shape rather than change under the action of an applied pressure. The advantages of geomorphometric methods for studying changes in the shape of wafers and thin-film membranes in technological processes for the manufacturing of microelectronic devices are shown in comparison with traditional methods for analyzing surface topography maps.

The anticipated use of elastic membranes for deflection-based rain gauges has provided an impetus for this paper to revisit the large deflection problem of a peripherally fixed circular membrane subjected to liquid weight loading, a statics problem when the fluid–structure interaction of membrane and liquid reaches static equilibrium. The closed-form solution of this statics problem of fluid–structure interaction is necessary for the design of such membrane deflection-based rain gauges, while the existing closed-form solution, due to the use of the small rotation angle assumption of the membrane, cannot meet the design requirements for computational accuracy. In this paper, the problem under consideration is reformulated by giving up the small rotation angle assumption, which gives rise to a new and somewhat intractable nonlinear integro-differential equation of the governing out-of-plane equilibrium. The power series method has played an irreplaceable role in analytically solving membrane equations involving both integral and differential operations, and a new and more refined closed-form solution without the small rotation angle assumption is finally presented. Numerical examples conducted show that the new and more refined closed-form solution presented has satisfactory convergence, and the effect of giving up the small rotation angle assumption is also investigated numerically. The application of the closed-form solution presented in designing such membrane deflection-based rain gauges is illustrated, and the reliability of the new and more refined closed-form solution presented was confirmed by conducting a confirmatory experiment.

In this study, the problem of axisymmetric deformation of peripherally fixed and uniformly laterally loaded circular membranes with arbitrary initial stress is solved analytically. This problem could be called the generalized Föppl–Hencky membrane problem as the case where the initial stress in the membrane is equal to zero is the well-known Föppl–Hencky membrane problem. The problem can be mathematically modeled only in terms of radial coordinate owing to its axial symmetry, and in the present work, it is reformulated by considering an arbitrary initial stress (tensile, compressive, or zero) and by simultaneously improving the out-of-plane equilibrium equation and geometric equation, while the formulation was previously considered to fail to improve the geometric equation. The power-series method is used to solve the reformulated boundary value problem, and a new and more refined analytic solution of the problem is presented. This solution is actually observed to be able to regress into the well-known Hencky solution of zero initial stress, allowing the considered initial stress to be zero. Moreover, the numerical example conducted shows that the obtained power-series solutions for stress and deflection converge very well, and have higher computational accuracy in comparison with the existing solutions.

Adhesion between coatings and substrates is an important parameter determining the integrity and reliability of film/substrate systems. In this paper, a new and more refined theory for characterizing adhesion between elastic coatings and rigid substrates is developed based on a previously proposed pressurized blister method. A compressed air driven by liquid potential energy is applied to the suspended circular coating film through a circular hole in the substrate, forcing the suspended film to bulge, and then to debond slowly from the edge of the hole as the air pressure intensifies, and finally to form a blister with a certain circular delamination area. The problem from the initially flat coating to the stable blistering film under a prescribed pressure is simplified as a problem of axisymmetric deformation of peripherally fixed and transversely uniformly loaded circular membranes. The adhesion strength depends on the delamination area and is quantified in terms of the energy released on per unit delamination area, the so-called energy release rate. In the present work, the problem of axisymmetric deformation is reformulated with out-of-plane and in-plane equilibrium equations and geometric equations, simultaneously improved, and a new closed-form solution is presented, resulting in the new and more refined adhesion characterization theory.

The resonant frequency shifts of a circular membrane caused by an adsorbate are the sensing mechanism for a drum resonator. The adsorbate mass and position are the two major (unknown) parameters determining the resonant frequency shifts. There are infinite combinations of mass and position which can cause the same shift of one resonant frequency. Finding the mass and position of an adsorbate from the experimentally measured resonant frequencies forms an inverse problem. This study presents a straightforward method to determine the adsorbate mass and position by using the changes of two resonant frequencies. Because detecting the position of an adsorbate can be extremely difficult, especially when the adsorbate is as small as an atom or a molecule, this new inverse problem-solving method should be of some help to the mass resonator sensor application of detecting a single adsorbate. How to apply this method to the case of multiple adsorbates is also discussed.

Textile wastewater, which often contains dye contamination and other pollutants, can harm and detrimental effects on the environment. In this research, the characteristics of circular disc-shaped ceramic membranes were investigated by fabricating them and studying the impact of sintering temperature. The sintering process of membranes was conducted at four different temperatures ranging 600–900 °C. The membranes were characterized using Fourier transform infrared spectroscopy, thermogravimetric analysis, field emission scanning electron microscopy, zeta potential, and X-ray diffraction. The constructed membranes displayed remarkable chemical stability in both acidic and basic solutions. They exhibited porosity ranging from 15.44 to 28.22% and an average pore size ranging from 0.416 to 4.32 µm. The membranes were sintered at 700 °C showed highest permeability, with a range of 0.04412 to 0.13628 L/h m 2 for transmembrane pressures of 0.5 to 4 bar, respectively. These membranes demonstrated potential for effective separation of dye methylene blue. The surface of membrane exhibited a negative charge between pH 3 and pH 11, indicating that adsorption was primary mechanism for removal of cationic dye. At a pressure of 0.5 bar, the efficiency of dye removal decreased from 99.61% at a feed concentration of 10 mg/L to 99.37% at 100 mg/L. The experiment design and analysis were examined and optimized transmembrane pressure, feed concentration, flux, and rejection using response surface methodology via central composite design. This study highlights the excellent potential of these membranes for textile dye treatment and various other membrane-based applications. Graphical abstract

The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular membrane before deflection. However, there is still a lack of an effective closed-form solution for the large deflection problem with initial uniform in-plane stress. In this study, the problem is formulated and is solved analytically. The initial uniform in-plane stress is first modelled by stretching or compressing an initially flat, stress-free circular membrane radially in the plane in which the initially flat circular membrane is located, and based on this, the boundary conditions, under which the large deflection problem of an initially flat circular membrane under in-plane radial stretching or compressing and out-of-plane gas pressure loading can be solved, are determined. Therefore, the closed-form solution presented in this paper can be applied to the case where the initially flat circular membrane may, or may not, have a uniform in-plane stress before deflection, and the in-plane stress can be either tensile or compressive. The numerical example conducted shows that the closed-form solution presented has satisfactory convergence.