The Rayleigh-Ritz Solution for DNA-Microcantilevers

LIU Cheng; HE Xiaobing; SHEN Xudong

doi:10.21656/1000-0887.460067

Volume 47 Issue 5

May 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(5): 577-588

LIU Cheng, HE Xiaobing, SHEN Xudong. The Rayleigh-Ritz Solution for DNA-Microcantilevers[J]. Applied Mathematics and Mechanics, 2026, 47(5): 577-588. doi: 10.21656/1000-0887.460067

Citation:

LIU Cheng, HE Xiaobing, SHEN Xudong. The Rayleigh-Ritz Solution for DNA-Microcantilevers[J]. Applied Mathematics and Mechanics, 2026, 47(5): 577-588. doi: 10.21656/1000-0887.460067

Citation:

LIU Cheng, HE Xiaobing, SHEN Xudong. The Rayleigh-Ritz Solution for DNA-Microcantilevers[J]. Applied Mathematics and Mechanics, 2026, 47(5): 577-588. doi: 10.21656/1000-0887.460067

PDF( 3474 KB)

The Rayleigh-Ritz Solution for DNA-Microcantilevers

doi: 10.21656/1000-0887.460067

1. School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R. China;
2. Construction Engineering School, Zhejiang College of Construction, Hangzhou 311231, P.R. China;
3. Center for Soft Machines and Smart Devices, Huanjiang Laboratory,Zhuji， Zhejiang 311816, P.R. China

Received Date: 2025-04-09
Rev Recd Date: 2025-06-09

Available Online: 2026-06-04

Publish Date: 2026-05-01

Abstract

Abstract

DNA-microcantilevers hold significant potential in biosensing due to their nanomechanical properties, yet the assumption regarding uniform curvature and axial strain of the neutral axis remains unverified. To address this, a functional potential energy model coupling the free energy of liquid crystalline DNA molecular layers with the composite beam deformation, was established. An innovative triangular series modal superposition approach was used to characterize displacement fields, and enable the variational solution via the Rayleigh-Ritz method, and a segmented integration strategy was employed to realize the full-domain free energy computation. Then the nonlinear equations governing Ritz coefficients were solved through the fixed-point iteration to yield microcantilever deformations and DNA strand spacing distributions. The results indicate that, the deformed curvature exhibits pronounced uniformity, providing theoretical support for the uniformity assumption. Convergence analysis demonstrates that, the triangular series expansion order ≥10 and the number of beam segments divided ≥2 500 ensure solution accuracy for deflection and stress, effectively validating the correctness of the uniformity assumption within the coupled model.
- DNA-microcantilever,
- nano-mechanics,
- free energy,
- Rayleigh-Ritz

FullText(HTML)

References(30)

References

FRITZ J, BALLER M K, LANG H P, et al. Translating biomolecular recognition into nanomechanics[J].Science,2000,288(5464): 316-318.

[2]MUNTEANU S, GAM-DEROUICH S, FLAMMIER C, et al. Scanning electrochemical microscopy monitoring in microcantilever platforms[J].Analytical Chemistry,2012,84(17): 7449-7455.

[3]YUE M, LIN H, DEDRICK D E, et al. A 2-D microcantilever array for multiplexed biomolecular analysis[J].Journal of Microelectromechanical Systems,2004,13(2): 290-299.

[4]LVAREZ M, CARRASCOSA L G, MORENO M, et al. Nanomechanics of the formation of DNA self-assembled monolayers and hybridization on microcantilevers[J].Langmuir,2004,20(22): 9663-9668.

[5]ILIC B, YANG Y, CRAIGHEAD H G. Virus detection using nanoelectromechanical devices[J].Applied Physics Letters,2004,85(13): 2604-2606.

[6]TAN Z Q, ZHANG N H, MENG W L, et al. Mechanism for invalid detection of microcantilever-DNA biosensors due to environmental changes[J].Journal of Physics D:Applied Physics,2016,49(22): 225402.

[7]MCKENDRY R, ZHANG J, ARNTZ Y, et al. Multiple label-free biodetection and quantitative DNA-binding assays on a nanomechanical cantilever array[J].Proceedings of the National Academy of Sciences of the United States of America,2002,99(15): 9783-9788.

[8]FRITZ J, BALLER M K, LANG H P, et al. Stress at the solid-liquid interface of self-assembled monolayers on gold investigated with a nanomechanical sensor[J].Langmuir,2000,16(25): 9694-9696.

[9]STACHOWIAK J C, YUE M, CASTELINO K, et al. Chemomechanics of surface stresses induced by DNA hybridization[J].Langmuir,2006,22(1): 263-268.

[10]WATARI M, GALBRAITH J, LANG H P, et al. Investigating the molecular mechanisms of in-plane mechanochemistry on cantilever arrays[J].Journal of the American Chemical Society,2007,129(3): 601-609.

[11]GODIN M, TABARD-COSSA V, MIYAHARA Y, et al. Cantilever-based sensing: the origin of surface stress and optimization strategies[J].Nanotechnology,2010,21(7): 075501.

[12]MERTENS J, CALLEJA M, RAMOS D, et al. Role of the gold film nanostructure on the nanomechanical response of microcantilever sensors[J].Journal of Applied Physics,2007,101(3): 034904.

[13]TABARD-COSSA V, GODIN M, BURGESS I J, et al. Microcantilever-based sensors: effect of morphology, adhesion, and cleanliness of the sensing surface on surface stress[J].Analytical Chemistry,2007,79(21): 8136-8143.

[14]ARROYO-HERNNDEZ M, TAMAYO J, COSTA-KRMER J L. Stress and DNA assembly differences on cantilevers gold coated by resistive and E-beam evaporation techniques[J].Langmuir,2009,25(18): 10633-10638.

[15]DOMNGUEZ C M, KOSAKA P M, MOKRY G, et al. Hydration induced stress on DNA monolayers grafted on microcantilevers[J].Langmuir,2014,30(36): 10962-10969.

[16]STREY H H, PARSEGIAN V A, PODGORNIK R. Equation of state for DNA liquid crystals: fluctuation enhanced electrostatic double layer repulsion[J].Physical Review Letters,1997,78(5): 895-898.

[17]STREY H H, PARSEGIAN V A, PODGORNIK R. Equation of state for polymer liquid crystals: theory and experiment[J].Physical Review E,1999,59(1): 999-1008.

[18]ZHANG N H, SHAN J Y. An energy model for nanomechanical deflection of cantilever-DNA chip[J].Journal of the Mechanics and Physics of Solids,2008,56(6): 2328-2337.

[19]张能辉, 单金英. 基因芯片纳米力学行为的能量模型[J]. 力学季刊, 2007,28(1): 54-57.(ZHANG Nenghui, SHAN Jinying. An energy model for nanomechanical behavior of gene chip[J].Chinese Quarterly of Mechanics,2007,28(1): 54-57. (in Chinese))

[20]李晶晶, 谭邹卿, 张能辉. 基因芯片纳米挠度和表面应力的解析预测[J]. 力学季刊, 2010,31(1): 113-117.(LI Jingjing, TAN Zouqing, ZHANG Nenghui. An analytical prediction for nanomechanical deflection and surface stress of gene chip[J].Chinese Quarterly of Mechanics,2010,31(1): 113-117. (in Chinese))

[21]王德鹏, 王记增. 可识别单一核苷酸错配的微悬臂基因检测技术力学模型[J]. 兰州大学学报(自然科学版), 2017,53(4): 558-568.(WANG Depeng, WANG Jizeng. A mechanic model for the micro-cantilever-based gene detection of single nucleotide mismatches[J].Journal of Lanzhou University (Natural Sciences), 2017,53(4): 558-568. (in Chinese))

[22]TAN Z Q, CHEN Y C, ZHANG N H. Theoretical analysis for bending of single-stranded DNA adsorption on microcantilever sensors[J].Sensors,2018,18(9): 2812.

[23]YANG Y, ZHANG N, LIU H, et al. Piezoelectric and flexoelectric effects of DNA adsorbed films on microcantilevers[J].Applied Mathematics and Mechanics (English Edition), 2023,44(9): 1547-1562.

[24]WU J, ZHANG Y, ZHANG N. Anomalous elastic properties of attraction-dominated DNA self-assembled 2D films and the resultant dynamic biodetection signals of microbeam sensors[J].Nanomaterials,2019,9(4): 543.

[25]WU J, ZHANG N. Clamped-end effect on static detection signals of DNA-microcantilever[J].Applied Mathematics and Mechanics (English Edition), 2021,42(10): 1423-1438.

[26]TAN Z, FENG Y, SHI X, et al. A mechano-electro-chemical coupling model for bending analysis of single-stranded DNA microbeam biosensors due to flexoelectricity[J].Journal of Applied Mechanics,2024,91(4): 041003.

[27]TAN Z, FENG Y, SHI X, et al. A unified model for mechanical deformations of single stranded DNA-microbeam biosensors considering surface charge effects[J].Applied Mathematical Modelling,2024,135: 559-577.

[28]MIYATANI T, FUJIHIRA M. Calibration of surface stress measurements with atomic force microscopy[J].Journal of Applied Physics,1997,81(11): 7099-7115.

[29]WU G, JI H, HANSEN K, et al. Origin of nanomechanical cantilever motion generated from biomolecular interactions[J].Proceedings of the National Academy of Sciences of the United States of America,2001,98(4): 1560-1564.

[30]PETERLINZ K A, GEORGIADIS R M, HERNE T M, et al. Observation of hybridization and dehybridization of thiol-tethered DNA using two-color surface plasmon resonance spectroscopy[J].Journal of the American Chemical Society,1997,119(14): 3401-3402.

Relative Articles

Supplements(0)

Cited By

Proportional views