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Micro-cantilever sensors are widely used to detect biomolecules, chemical gases, and ionic species. However, the theoretical descriptions and predictive modeling of these devices are not well developed, and lag behind advances in fabrication and applications. In this paper, we present a novel multiscale simulation framework for nanomechanical sensors. This framework, combining density functional theory (DFT) calculations and finite element method (FEM) analysis, is capable of analyzing molecular adsorption-induced deformation and stress fields in the sensors from the molecular scale to the device scale. Adsorption of alkanethiolate self-assembled monolayer (SAM) on the Au(111) surface of the micro-cantilever sensor is studied in detail to demonstrate the applicability of this framework. DFT calculations are employed to investigate the molecular adsorption-induced surface stress upon the gold surface. The 3D shell elements with initial stresses obtained from the DFT calculations serve as SAM domains in the adsorption layer, while FEM is employed to analyze the deformation and stress of the sensor devices. We find that the micro-cantilever tip deflection has a linear relationship with the coverage of the SAM domains. With full coverage, the tip deflection decreases as the molecular chain length increases. The multiscale simulation framework provides a quantitative analysis of the displacement and stress fields, and can be used to predict the response of nanomechanical sensors subjected to complex molecular adsorption.

Nanomechanical sensors have attracted considerable interest, as they are a promising tool for real-time and label-free detection of chemical gases and biomolecules [

The sensitivity of the induced surface stress dominates the performance of cantilever-shaped nanomechanical sensors. Understanding the physical mechanisms of adsorption-induced surface stress and their influences on the overall displacement and stress fields are the key to designing next-generation nanomechanical sensors. Surface stresses due to molecular adsorption often arise from two main sources: weak inter-adsorbate interactions and strong adsorbate–substrate interactions [

Quantitative analysis of the displacement and stress fields of a nanomechanical sensor due to the adsorption-induced surface stress remains a theoretical challenge [

In this study, we propose a multiscale simulation framework for nanomechanical cantilever sensors based on DFT calculations and finite element method (FEM) analysis. DFT calculations are used to compute the induced surface stress of molecular adsorption on the molecular recognition layer. The calculated surface stresses are then used in the FEM analysis to resolve the deformation and stress fields of the nanomechanical sensors. A gold-coated cantilever sensor exposed to alkanethiolate self-assembled monolayers (SAM) is used to demonstrate the applicability of the proposed multiscale framework.

_{11} and g_{22}, as shown in

Firstly, we describe the theoretical background of this framework. The surface stress tensor is defined as:
_{αβ}_{s}_{b}

The surface stress tensor for the clean gold surface can be obtained from

According to several experimental observations, alkanethiolate molecules self-assemble into well-ordered, poly-crystalline monolayers on the Au(111) surface. For FEM analysis at the device level, we assume that each film element represents an alkanethiolate SAM domain on a gold surface, whose initial stress is equal to the induced surface stress of the representative volume obtained from DFT calculations. The nodal loads of each film element are the work balance of the body forces {_{0}}, and the initial stress{_{αβ}

In general, the SAM orientations on the gold surface are complex and induced surface stresses from the SAM adsorption are anisotropic [_{11} and _{22}) into normal stress components (_{xx} and _{yy}_{xy}

DFT calculations have been widely used to predict and estimate a great variety of material and molecular properties [

To obtain the induced surface stresses, we applied DFT calculations in two unit cells: An alkanethiolate-coated Au(111) surface and a clean Au(111) surface. For the clean gold surface, the unit cell consists of a

The projector augmented wave [

The unit cells, whose calculated equilibrium lattice constant is 4.14 Å, were kept fixed during atomic relaxation, and the optimized structures were obtained until the force on each atom was less than 0.01 eV/Å. After obtaining the optimized structures, we calculated the stress tensor of the slabs using the Hellmann–Feynman theorem [

FEM analysis was employed in the device level simulation. A commercial finite element software, ABAQUS, with a thin shell/solid modeling technique was used to capture the deformation and stress fields of the sensors [

We note that scanning tunneling microscope (STM) images suggest the adsorption layer of alkanethiol SAMs typically characterized by SAM domains with areas of close-packed molecules separated by domain boundaries [^{2} SAM domain. The number of shell elements is well beyond the needs of solution convergence for a micro-cantilever. For completeness, a convergence study was carried.

The surface stresses obtained from the DFT calculations were applied as initial stresses in the shell elements. A surface-to-surface tie constraint was used to attach the shell section to the top surface of the solid section, so that two sections could be built separately but simulated together. Another advantage of this modeling technique is that it is not necessary to match the number of elements of the solid part with that of the shell section. This not only reduces computing time considerably, but also allows us to study the effects of SAM coverage and distribution on the deformation and stress distribution of the sensors. The support block's bottom and top faces along the surface normal were kept fixed, except for the face connected to the beam. The nonlinear geometry procedure in ABAQUS was applied in the static stress/strain analysis. Applying nonlinear geometry procedure in ABAQUS was meant for completeness but not mandatory since the typical deflections of a micro-cantilever are very small and the geometric nonlinearity effects can be neglected.

Molecular configurations of alkanethiolate adsorbed on Au(111) were optimized in our DFT calculations. The optimized structural parameters have a good agreement with those in the literature [

The surface stresses from DFT calculations are listed in

In contrast, anisotropic surface stresses are found on the alkanethiolate-covered gold surface. For example, the induced surface stresses of hexanethiolate adsorbed on Au(111) are a compressive stress of −1.54 N/m along the molecular chain direction and a tensile stress of 0.28 N/m along a direction perpendicular to the molecular chain. These anisotropic surface stresses can be understood by the following behaviors.

First, the sulfur atom of alkanethiolate attracts electrons from the surface Au atoms to form covalent-like Au–S bonds. Therefore, this charge removal in the gold surface yields a compressive surface stress. The surface stresses of the alkanethiolate with shorter chains are much smaller than those of the clean Au surface, indicating a larger relief of tensile stress or a compressive stress during molecular adsorption. In addition, the formation of two Au–S bonds on the three-hollow site breaks the symmetric structure on the gold surface and yields a huge anisotropic compressive stress.

Second, molecular chains generate attractive forces in this molecular configuration on Au(111). Due to the orientation of the long molecular chain, the attractive force along the direction of the chain is smaller than that in the perpendicular direction. Therefore, larger attractive forces in a direction perpendicular to the molecular chain could yield greater tensile stress and compensate for the compressive stress produced by the charge redistribution. For example, the induced surface stress of −1.54 N/m for hexanethiolate on Au(111) along the direction perpendicular to the molecular chain is much more compressive than the 0.28 N/m stress in the direction parallel to the molecular chain. The average surface stress of hexanethiolate adsorption on the Au(111) surface is a compressive stress of −0.63 N/m and has a very good agreement with experimental data within ±0.05 N/m [

The principal surface stresses from DFT calculations were uniformly transformed into normal and shear stresses, and then randomly applied to the shell elements in our FEM models as shown in _{11} and Δ_{22}) divided by the film height, which is the distance between the center of the gold slab and the last carbon in the chain of the optimized alkanethiolate in the DFT unit cell, in a direction normal to the surface. The stress/strain analysis was carried out in ABAQUS.

The longitudinal stress distribution in the silicon nitride layer of the micro-cantilever subjected to a full coverage of hexanethiolate SAMs on Au(111) is shown in _{11} stress occurs on the upper (bottom) surface of the silicon nitride layer due to the overall compressive surface stress in the shell section. The S_{11} stresses concentrate near the corners of the cantilever, close to the supporting block, as shown in the inset of

Based on a multiscale simulation framework, we can further study the effects of the alkanethiolate SAM coverage or the chain length at the molecular level on the deformation of the micro-cantilever sensor at the device level. For different coverages of SAM domains, an FEM analysis of ten samples of randomly distributed SAM domains was conducted and the average of the tip deflections of these ten samples was used for further analysis. The result is shown in

A multiscale simulation framework for micro-cantilever sensors has been developed. It successfully connected density functional theory (DFT) calculations and finite element method (FEM) analysis to predict the device-level deformation and stress fields. DFT calculations were used to resolve the induced surface stresses of molecular adsorption upon the recognition layer while FEM analysis was conducted to analyze the device response of the nanomechanical sensors due to molecular adsorption. Alkanethiolate self-assembled monolayer (SAM) adsorption upon the Au(111) surface of a micro-cantilever sensor was used as an example to demonstrate the applicability of this multiscale framework. The tip deflection of the micro-cantilever sensors increase linearly as the coverage of SAM domains increases. Under full coverage, the tip deflection decreases as the molecular chain length increases.

The present multiscale simulation framework allows us to predict the overall displacement and stress fields of the device. One immediate benefit by using the present scheme is that molecular calculations of surface stresses such as those from DFT now become relevant for analyzing deformation and stress fields of a microcantilever device. The multiscale framework provides a quantitative analysis of the displacement and stress fields of micro-cantilever sensors, and can be used to predict the responses of nanomechanical sensors subjected to complex molecular adsorption.

The authors gratefully acknowledge the computational support from the National Center for High-Performance Computing in Taiwan. CSC would like to acknowledge the funding support from the National Science Council of Taiwan (NSC 100-2628-E-002-004, NSC 100-2627-E-002-001) and National Taiwan University (10R80920-05).

The authors declare no conflict of interest.

(_{11} and g_{22}) are substituted as initial stresses in a section of film depicted as a 2D circle. The thickness _{s}

An illustration of the supercell model for a clean gold surface.

(

(

(

(_{11} (MPa) stress contour of the shell surface subjected to a 30% coverage of hexanethiolate SAMs on Au(111). The white color indicates no initial stress; (

Top view of the S_{11} stress distribution of the silicon nitride layer in the micro-cantilever subjected to a full coverage of hexanethiolate SAMs on Au(111).

(

Material properties and thicknesses of each layer of the micro-cantilever in μMKS units.

Surface Film Layer | 60,000 | 0.4 | 0.00066–0.00127 |

Gold | 60,000 | 0.4 | 0.02 |

Silicon Nitride | 280,000 | 0.2 | 0.5 |

Silicon | 160,000 | 0.3 | 0.5 |

Induced surface stresses of the clean and alkanethiolate-covered Au surfaces. C1, C2, C3, C4, C5, and C6 denote methanethiolate, ethanethiolate, propanethiolate, butanethiolate, pentanethiolate, and hexanethiolate, respectively.

_{11}(N/m) |
_{22}(N/m) |
_{11}(N/m) |
_{22}(N/m) | |
---|---|---|---|---|

clean | 3.02 | 3.02 | ||

C1-covered | 1.72 | 0.59 | −1.30 | −2.43 |

C2-covered | 2.01 | 0.85 | −1.01 | −2.17 |

C3-covered | 2.28 | 0.86 | −0.74 | −2.16 |

C4-covered | 2.61 | 1.23 | −0.41 | −1.79 |

C5-covered | 2.92 | 1.27 | −0.10 | −1.75 |

C6-covered | 3.30 | 1.48 | 0.28 | −1.54 |