#### Abstract

Tribological performances of the diatom frustule are investigated with the liquid-solid interaction (FSI) method. Take, for example, the representative *Coscinodiscus* sp. shell; the diatom frustule with the porous structure is achieved by the scanning electron microscope (SEM). Based on the frustule, the representative diatom frustule is modeled. Further, tribological performances of the diatom at its different geometry sizes and velocities are solved with FSI method and compared with corresponding values for the nonporous structure. The numerical result shows that the existence of the porous structure of the diatom helps to reduce friction between it and ambient water and to increase its load-carrying capacity.

#### 1. Introduction

A diatom is a single-celled organism, which is widely distributed in the sea water or freshwater. So far, over hundreds of thousands of species of diatoms have been found in the world [1]. The most striking feature of the diatom is its elaborate frustule made up of grotesque amorphous silicon. Considerable researches available show that the main cause that the diatom has not been eliminated in the past tens of millions of years [2–6] is due to superior mechanical properties of its frustule, which is attracting the intense attention of researchers. Kröger [7] observed through SEM that the different diatom frustules exhibit diversity in their shapes and sizes. Michels et al. [8] found through the experiment that the elaborately porous structure of the diatom can greatly reduce its damage probability in the sea water. Meanwhile, fruitful porous structures of the* Coscinodiscus *sp. and* Thalassiosira eccentrica* are found by Losic et al. through SEM [9]. Besides, an experiment result about* F. kerguelensis* from Hamm et al. [10] shows that this kind of diatom frustules have the elastic modulus as high as 22.4 GPa, with strong ability to restore the shell deformation. Almqvist et al. [11] pointed out that the high hardness and elastic modulus of the* Navicula pelliculosa* depend on its species. By imitating the porous structure of the diatom frustule, Jang et al. [12] created a three-dimensional hollow nanometer ceramic structure, whose tensile strength is as high as 1.75 GPa. Based on the molecular dynamics theory, Sen and Buehler [13] found that the toughness for every layer of the porous structure of the diatom ranges from to J/m³. By comparison of the bending strengths for the specified four different diatoms, Young et al. [14] revealed that the bending strength varies due to the discrepancy in their morphologies. Another comparative study about the ductility, elasticity modulus, and toughness of the diatom silica shell was conducted by Garcia et al. [15]. They found that the above three values are up to 120%, 36 GPa, and 1.29 GJ/m³, respectively.

However, the above researches are mainly centralized on biological and mechanical properties of the diatom. So far, a very few researches of tribological performances for the diatom have been conducted. Gebeshuber et al. [16, 17] found that the self-lubrication of the diatom can significantly reduce its autologous friction and wear. Toyoda et al. [18, 19] pointed out that different tribological behaviors of the diatom are due to the diversity of its morphology structures. However, these tribological properties have been not well understood to date, especially the mechanism of lubrication and friction for the diatom frustule [20]. Therefore, it is necessary to explore this tribological mechanism. In the present work, the morphology structure of the representative* Coscinodiscus* sp. is observed by SEM and its 3D model is established with Pro/E software based on the observed sample. Then tribological performances for the porous structure of the diatom frustule are simulated with FSI method through the CFX module of the commercial software ANSYS 14.5, in which the two-way coupling between the liquid and solid is considered simultaneously in solving the associated fluid or constitutive equations. Finally, the associated conclusions are drawn.

#### 2. Model

##### 2.1. Image Collection

To clearly observe the pores on the surface of the* Coscinodiscus* sp. by means of SEM, it is necessary to make the* Coscinodiscus *sp. sample have good electrical conductivity. For this, the authors make an ultrasonic treatment of the sample with TOAB (tetraoctyl ammonium bromide) and ethanol, as surface active agents, and then make a jetting gold on the sample. After this, the image collection is realized by means of ZEISS AURIGA dual beam system with the resolution of 1 nm, where the diatom images collected under condition of normal temperature are shown in Figure 1. The image of the whole* Coscinodiscus* sp. at 1720 magnification is presented in Figure 1(a), while the porous frustule of the* Coscinodiscus *sp., obtained at magnification of 1.119 × 10^{4} times, is shown in Figure 1(b). From Figure 1(b), one can observe that elaborate porous structures are distributed on the* Coscinodiscus *sp. frustule, in which case its tribological mechanism will be revealed in what follows.

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##### 2.2. Simulation Model

In the simulation, representative* Coscinodiscus* sp. frustule is used. The model of an elementary cell taken from this diatom frustule is established, as shown in Figure 2(a). In the model, the cell thickness along the direction is set to be 900 nm. Around the geometry center of each unit, 19 pores are assumed to exist distributed in the form of hexagon with side length of 1.2 × 10^{3} nm. The symbols and represent the diameter and depth of the pore, respectively, and is the interval between two closely adjacent pores. In the solution, some boundary conditions need to be determined for the simulation, exhibited in Figure 2(b) where the displacement for the bottom surface of the diatom is constrained along the - and -axes. Meanwhile, arrow I indicates the water moving with the velocity along the positive direction, while arrow II stands for the* Coscinodiscus *sp. frustule moving with the velocity along the direction. Besides these, the elastic modulus, Poisson ratio, and density of the frustule are 73 GPa, 0.17, and 2500 Kg/m^{3}, respectively. And the viscosity, specific heat, and density of the water are set to be 0.00105 Pa·s, 3890 J/Kg/K, and 1025 Kg/m^{3}, respectively.

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#### 3. Governing Equations

During the FSI analysis, the inflow and outflow of the water through the* Coscinodiscus *sp. frustule are identical in the mass, and thus the continuous equation describing the variation of the water can be used and is given below:
where denotes the water density and are the velocity vectors in the , , and directions, among which the velocity in direction is stated previously.

Based on an assumption that lubricant is isothermal, laminar, and continuous, the Navier-Stokes equation used for describing the hydrodynamic action of the water is expressed as where , , and denote the pressure, viscosity, and velocity of the water, respectively. is the transposition symbol.

For the* Coscinodiscus *sp. frustule, its force equilibrium equation is expressed as
where , , and are the stress, body force, and acceleration vectors of the* Coscinodiscus *sp. frustule in the , , and directions, respectively. is the diatom frustule density.

At the interface between the* Coscinodiscus *sp. frustule and water, the stress equilibrium and displacement compatibility conditions should be satisfied simultaneously; that is,
where and separately denote the stresses of the water and* Coscinodiscus *sp. frustule at the interface between them, and their corresponding unit normal vectors are denoted by and , respectively. and denote displacements of the water and* Coscinodiscus *sp. frustule at the interface, respectively.

It should be pointed out that (1)–(3) need to be discretized into algebraic equations and then are solved with FSI method at a grid system. For (1) and (2), they are solved for the water velocity and it pressure and so forth by simple algorithm.

According to the obtained pressure, (3) and (4) are computed for getting the stress and displacement of the frustule. For this, the finite element method is usually employed due to the complexity in the porous frustule of the diatom, although there are other efficient methods for computing solid performances such as FFT one [21–23]. When the displacement is obtained, it, as boundary input, is transmitted to (1) and (2) and then repeat the above solution process until the given convergent standards are met. This process is accomplished by the CFX, FEA, and ANSYS-Multifield solvers in the commercial software ANSYS14.5 version.

Once the water pressure is achieved according to (1)–(4), the load-carrying capacity (i.e., the ability to resist water pressure) for the* Coscinodiscus* sp. frustule can be computed by integrating the pressure over the interface; that is,
where is the interface area.

The frictional force can be calculated by the following integral expression:
where denotes the relative velocity between the water and* Coscinodiscus* sp. frustule in the direction.

With the calculated and , the friction coefficient can be obtained by

In what follows, the tribological performances, mainly the load-carrying capacity, frictional force, and friction coefficient, for the* Coscinodiscus* sp. frustule will be analyzed.

#### 4. Results and Discussions

##### 4.1. Check of Mesh

In the simulation, numerical results may be affected by the used mesh density in discretizing the imitated* Coscinodiscus *sp. frustule. Therefore, it is necessary to conduct an analysis of the mesh sensitivity. This analysis is made under the condition of the pore diameter of 195 nm, pore depth of 200 nm, and the 375 nm interval between closely adjacent pores. Meanwhile, the water velocity of 2 m/s and the* Coscinodiscus* sp. frustule velocity of 1 m/s are employed. Based on the above parameters, the maximum water pressures at different mesh sizes are calculated. The calculation shows that when the mesh size is 50 nm, the relative percentage difference (*Dif*.%) of is 1.08%. In this case, if the mesh density continues to be added, the difference continues to become smaller. Therefore, the critical mesh size of 50 nm is chosen for the following simulation about tribological performances for the porous structure of the* Coscinodiscus *sp. frustule.

##### 4.2. Geometry Sizes and Velocity Effects

Figure 3 gives the load-carrying capacity () as a function of pore diameter () of the* Coscinodiscus* sp. frustule with varying pore depth () and interval between the pores. In each subfigure, the dotted line represents the value for the nonporous frustule. As shown in these subfigures,* Coscinodiscus* sp. frustule has the value larger than that for the nonporous frustule in the same . Moreover, increases with increasing or . This increment can be explained by the water film pressures applied on the frustule, as illustrated in Figures 4(a) and 4(b). Figure 4(a) exhibits the pressure without the pore consideration, while that for the* Coscinodiscus* sp. frustule is presented in Figure 4(b). Compared with the relative smooth pressure distribution with the smaller amplitude in Figure 4(a), there exist many pressure peaks for the* Coscinodiscus* sp. structure shown in Figure 4(b), which is due mainly to the phenomenon that every pore acts as an oil reservoir [24]. Thus, it is understandable that the load-carrying capacity of the latter is larger than that for the former according to (5). The above discrepancy can be observed by the comparison between the pressures marked off with point in Figure 4(a) and point in Figure 4(b).

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Friction is an important care in tribology researches. Figures 5 and 6 separately exhibit the discrepancy in the frictional force and friction coefficient between the* Coscinodiscus* sp. frustule and its corresponding nonporous structure. In the two figures, the dotted lines represent and for the nonporous frustule, respectively. Opposite to the variation trend of in Figure 3, both and further for the porous structure decrease compared with values of the nonporous structure, if or is fixed, which is consistent with the experimental results from Gebeshuber and Crawford [16]. This can be explained with the velocity gradient. Figure 7(a) exhibits the velocity gradient for the nonporous structure, while it fluctuates significantly near the pores for the* Coscinodiscus *sp. frustule (e.g., marked off with ) in Figure 7(b). Meanwhile, it can be seen from Figures 7(a) and 7(b) that the existence of the porous structure results in the decrement in the amplitude for the velocity gradient. In this case, the frictional force for the porous frustule decreases according to (6) and further its friction coefficient decreases because of its enhanced load-carrying capacity, which can be observed from (7).

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The influences of the interval on the load-carrying capacity, frictional force, and friction coefficient of the* Coscinodiscus* sp. frustule with varying pore diameter and pore depth are presented in Figures 8, 9, and 10, respectively. As shown in these figures, the existence of the pores results in the larger load-carrying capacity, smaller frictional force, and further smaller friction coefficient, if and are fixed simultaneously. Meanwhile, in the cases of geometry sizes given, has little effect on the load-carrying capacity and friction coefficient.

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The above results were obtained at the* Coscinodiscus* sp. frustule velocity = 1 m/s. To explore effects on tribological performances, keep the water velocity equal to 2 m/s and interval at 375 nm. Figure 11 gives the friction coefficients as a function of with varying pore depth . As shown in Figure 11, the friction coefficients of the* Coscinodiscus* sp. frustules are less than those of the nonporous structure frustules, which is consistent with the conclusion of Figures 6 and 10. It should be pointed out that the bigger friction coefficients exhibited in the present study are due mainly to the comparatively small load-carrying capacity of the water because of its parallel flow through the* Coscinodiscus* sp. frustule.

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In this case, the frictional force and load-carrying capacity separately exhibit a relative reduction and increment if the frustule pores are considered, omitted here.

#### 5. Summary

Take, for example, the* Coscinodiscus* sp. moving in the water; tribological performances of the elaborate pores on the diatom frustule were investigated with FSI method. The simulation shows that these pores can enhance the load-carrying capacity for the diatom to resist the water pressure. And they also contribute to the reduction in the frictional force between the diatom frustule and water and further friction coefficient. Within the range of geometry sizes used, the larger pore diameter or depth will help to improve the load-carrying capacity but reduce the friction coefficient. This porous structure for the diatom frustule is promising in engineering applications.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to express their gratitude for the support from the Natural Science Foundation of China (Contract no. 51375509), the Fundamental Research Funds for the Central Universities of China (Contract no. CDJZR12248801), and Program for New Century Excellent Talents of Ministry of Education of China (Project no. NCET-10-0881).