FEM exploration of the potential of silica diatom frustules for vibrational MEMS applications

https://doi.org/10.1016/j.sna.2020.112270Get rights and content

Highlights

  • Numerical analysis of silica diatom frustule vibration was carried out by FEA.

  • Correlations derived for natural frequencies depending on morphology and mechanical properties.

  • Comparative modal analysis performed for centric (Coscinodiscus acus) and pennate (Synedra acus) diatoms.

Abstract

Numerical simulations were carried out using the Finite Element Method (FEM) to determine the frequency characteristics of mechanical vibration of diatom silica frustules under the conditions and at frequencies that are not readily accessible to experimental measurement. The results revealed the influence of the frustule morphology on the natural frequency spectra. The effect of frustule density, stiffness, dimensions, pore size, and wall thickness on the eigenfrequencies and the corresponding modal shapes were studied in detail. Diatom frustules have natural frequencies in the range between several MHz and tens of MHz that make them a promising candidate for future MEMS applications. Eigenfrequencies depend linearly on the speed of sound in the frustule wall and decrease parabolically with the diameter and the pore size to diameter ratio. Dimensional analysis allowed obtaining functional correlations that encapsulate the various dependencies in compact analytical form. The satisfactory nature of our calculations and correlations derived from them is confirmed through an agreement with the analytical solutions from the literature.

Introduction

The biological class of diatomic algae is an example of nature’s fascinating collection of intricate hierarchical architectures of exoskeletons (known as frustules) that are optimized for functional and structural performance at multiple length scales, and produced in a cheap and environmentally friendly way in the volume that vastly exceeds that of artificially manufactured nanostructured materials [1,2]. Diatoms, a class of single-cell algae found in aquatic environments have millions of unique species with distinct morphologies of frustules made from amorphous hydrated silica [3]. Depending on symmetry diatoms are usually classified into two groups: centric (often circular) and pennate (elongated) (Fig. 1). Extraordinary mechanical properties, such as high deformability and high specific strength [4], are combined with the ability of biological reproduction even in Antarctic temperatures (below freezing) [5], making diatoms a unique instance of natural nano-fabrication at the global scale. The manipulation and genetic control of diatom frustule nano-structuring could revolutionize the device fabrication routes for energy storage, optoelectronics, solar cells [6], and batteries [7]. Diatoms have intricate intrinsic features: biocompatibility, unique nano-pore arrangement, specific surface area and mechanical strength. In recent decades they have been proposed for use in drug delivery [[6], [7], [8]], microfluids [9,10], batteries [11,12], templates [[13], [14], [15]], biosensors [16,17], for energy conversion and storage [18]. Diatoms used as a platform for nanotechnology hold a crucial advantage over photolithography because they multiply in geometric progression, i.e. at a much faster rate than is possible for any technology of fabrication for MEMS [18] and in which is faster than the manufacturing speed of MEMS. This bioinspired silica can be used directly or in modified form in design for specific properties. Diatoms have rigid surfaces in relative motion, and their tribology has a high potential regarding 3D MEMS [19]. Some studies of the mechanical properties of diatom frustules have been reported in the literature. The mechanical vibration behaviour of diatom frustules is important because their small size, light weight and high stiffness lead to the expectation of them having high natural vibration frequencies with potential applications in MEMS, especially considering the diversity of sizes and shapes that diatoms offer. The Finite Element Method offers a suitable analysis technique for the study of mechanical properties of small biomaterials. The mechanical properties of diatoms are widely studied in the literature [4]. In particular, computer modelling has become an increasingly popular approach to the study of biomaterials with complex structure at the nano- and micro- scales. Hamm et al. performed Finite Element Analysis of Fragilariopsis kerguelensis frustules and showed that diaphragmatic grooves have high rigidity of about 9.8 GPa [4]. The authors considered that diatoms obtained their unique architecture as a result of a long evolutionary process under the influence of environmental factors, in order to optimize such properties as protection from predators. Several studies attempted to analyze diatom frustule structere in the context of hierarchical design by Nature. Moreno et al. studied the relationship between porosity and mechanical properties [21]. Similarly, Gutiérrez et al. investigated the effect of morphological features (such as the diameter of diatoms, pore size and thickness of individual walls) on the deformation response of centric diatoms [22]. It remains extremely challenging, if not impossible to simulate the real structure of diatoms. Several studies have shown the most considerable importance of pore size for the properties of diatom frustules. Yuan Xing proposed the use of Focused Ion Beam - Scanning Electron Microscopy (FIB-SEM) as a powerful tool for the systematic 3D study of the morphological features of diatoms on the basis of a series of high-resolution images [23]. Lu el. reported high precision simulation of Coscinodiscus sp. valve, starting with a unit cell that included three layers of the valve (hole, cribrum and cribellum) and even taking into account the curvature of the pore walls of the hole (areola chambers) [24]. Meza et al. made a significant advance in additive manufacturing complex microarchitectural materials down to the nanometer precision. in situ nanomechanical testing showed high strength and stiffness, and near complete recovery (up to 98 %) after large compression up to ≥50 % [25]. More recently, E. Topal et al. used the combination of X-ray computed tomography (XCT), FEM analysis and micro-scale testing to study the mechanical response to Didymosphenia geminata. They were able to obtain accurate 3D morphological data for model input. The limited XCT resolution of 130 nm meant that SEM was used to define nanopore size and shape. Effective Young’s Modulus of the frustule was determined to be 31.8 GPa, significantly lower than bio-silica (70 GPa) due to porosity, as is the case for many biological materials with hierarchical structure.

In the present study, we focus our attention on the numerical study of the effect of morphology and material properties of diatom frustules on the mechanical vibration behaviour. The type of diatoms was selected based on the ordered arrangement of pores and symmetry. Symmetry alleviates simulation analysis and visualization. Coscinodiscus sp. (centric) and Synedra acus (pennate) diatom species satisfy the criteria above. Their slim nanostructures can be integrated into nanodevices, which is the strategic focus of our study. The main objective of our research is to explore diatom’s potential for MEMS, namely, oscillators or vibrational sensors able to resonate at eigenfrequencies detecting specific external vibration. Our scientific group recently managed to make other steps towards this goal, namely, to achieve guided colonization of Si wafers with diatoms [21] and to reduce natural diatom opal to obtain nanostructured Si objects while retaining neat nanostructure [22]. These steps are being theoretically and conceptually supported by the modelling presented in this manuscript.

Coscinodiscus sp. and Synedra acus diatoms represent the extremes in terms of taxonomy and, as one can see, topology as well. Centric diatoms can be considered as quasi-spherical thin shells (or domes) fundamentally and thoroughly studied in continuous mechanics. The issues of porosity and height-to-diameter aspect become crucial for eigenfrequencies. Pennate diatoms depending on the width-to-length aspect may be treated as close to centric or, like for the considered Synedra, as slender rod or beam, or even as elastic continuous string. Theoretical models for long rods or elastic strings are also well documented in scientific literature and simplest formulae can be applied for fast estimations. Porosity can be accounted for as reduction factors for density and stiffness. Diatom frustules consist of 10–70% amorphous silica with the density not exceeding 2600 kg/m3, and the remaining organic components such as proteins with the density of 1300 kg/m3, and polysaccharides of 1070 kg/m3. Therefore, frustule densities can range from 1400 kg/m3 to 2200 kg/m3, according to literature [24]. The frustules of diatom algae consist of two halves, known as valves. In the case of Coscinodiscus sp. with centric symmetry (Fig. 1a) the shape of each valve is remindful of a Petri dish with the diameter ranging from 50 to 200 μm, and the dome height of 12 μm. A system of holes or pores with the diameters of 0.4 μm–2.6 μm penetrate the frustule from the concave to the convex surface. The valves that are joined by the surrounding circular girdle band of about 3 μm width. The thickness of the silica frustule walls varies from 0.2 to 2 μm [20]. In previous studies, nanoindentation of Coscinodiscus sp. revealed the material stiffness of 22.4 GPa [4], which is comparable to that of the cortical bone (20 GPa) [25], but is significantly lower than that of bulk silica glass (73 GPa) [26] perhaps due to the presence of both organic binder and nanometer size porosity. Synedra acus with the structure illustrated in Fig. 1b is a class Fragilariophyceae that is abundant in ground water. It was chosen as representative of the pennate group of diatoms that display bi-fold symmetry. Pennate diatoms typically have the form of elongated ellipsoids consisting of the upper valve (epitheca) and lower valve (hypotheca) joined by the girdle band. Each half shell consists of ribs (costae) with orifices classified according to their location and size into raphe (central longitudinal ridge), striae and areolae. The cell culture was isolated from the natural population endemic in Lake Baikal and was allowed to grow naturally in a plastic contained placed on a laboratory windowsill.

Section snippets

3D CAD and FEA modelling of hierarchically structured frustules

The numerical simulations carried out in the current project studied the dependence of vibration properties on a range of parameters, namely, the overall frustule geometry, the dimension of characteristic features, and material properties. Additionally, the mesh size effect was considered to ensure the reliability of results. At the start of the simulation, a three-dimensional CAD model was created using SolidWorks software version SP5.0. The model was constructed based on the actual morphology

Analysis of the results

For the purposes of evaluating the suitability of diatom frustules for MEMS applications as vibration elements, the key requirement concerns the ability to predict the eigenfrequency values based on the frustule size, shape, and mechanical characteristics.

Modal shapes

The shapes of first six vibration modes are illustrated in Fig. 3. For Coscinodiscus sp. the displacements mapped correspond to the direction along the symmetry axis of the valve. For Synedra acus the displacements shown are normal to the

Comparison between numerical simulations and analytical solution

To confirm the reliability of our results, we used the analytical results, particularly those of Shang [31,[32], [33], [34], [35], [36], [37], [38], [39], [40]]. Using the Naghdi-Reissner shell theory and Legendre functions, an analytical solution was obtained for vibration frequencies of spherical-cylindrical shells that was expressed in the form:

f=12πRλE/ρ(1-v2), where λ is a dimensionless scaling parameter, that can be re-written in our notation as ω1DEρλt,d and compared with our result: ω

Conclusions

The natural vibration frequencies of two different diatom frustule shapes were analysed, centric Coscinodiscus sp. and pennate Synedra acus. For the former centric shape in particular it was revealed how the natural frequencies depend on the frustule morphological features, such as material stiffness and density, valve diameter, wall thickness, and pore diameter. Diatom frustules are expected to have vibration frequencies in the MHz range. Bio-silica low density and high Young’s Modulus make

Author statement

Bakhodur Abdusatorov – simulation, interpretation. PhD student, Skoltech CEST

Alexey I. Salimon – conceptualization, supervision. PhD, Senior Research Engineer, Skoltech CEST

Yekaterina D. Bedoshvili – conceptualization, supervision. PhD, specialization in limno-biology

Yelena V. Likhoshway – conceptualization, supervision. D. Sci., Professor, specialization in limno-biology.

Alexander M. Korsunsky – conceptualization, supervision, intellectual leadership. Professor, specialization in materials

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

AMK wishes to acknowledge funding support from the Royal Society (UK) under project IEC/R2/170223.

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