Computational methods for nano-mechanical sensors S. Adhikari 1 1 Chair of Aerospace Engineering, College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP , UK 3rd International Conference on Innovations in Automation and Mechatronics Engineering - ICIAME2016, Gujarat, India Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 1
Swansea University Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 2
Swansea University Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 3
My research interests Development of fundamental computational methods for structural dynamics and uncertainty quantification A. Dynamics of complex systems B. Inverse problems for linear and non-linear dynamics C. Uncertainty quantification in computational mechanics Applications of computational mechanics to emerging multidisciplinary research areas D. Vibration energy harvesting / dynamics of wind turbines E. Computational nanomechanics Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 4
Outline Introduction 1 One-dimensional sensors - classical approach 2 Static deformation approximation Dynamic mode approximation Overview of nonlocal continuum mechanics 3 One-dimensional sensors - nonlocal approach 4 Attached biomolecules as point mass Attached biomolecules as distributed mass Two-dimensional sensors - classical approach 5 Two-dimensional sensors - nonlocal approach 6 Conclusions 7 Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 5
Introduction Nanoscale systems Nanoscale systems have length-scale in the order of O ( 10 − 9 ) m. Nanoscale systems, such as those fabricated from simple and complex nanorods, nanobeams and nanoplates have attracted keen interest among scientists and engineers. Examples of one-dimensional nanoscale objects include (nanorod and nanobeam) carbon nanotubes (Ijima, 1993), zinc oxide (ZnO) nanowires and boron nitride (BN) nanotubes, while two-dimensional nanoscale objects include graphene sheets and BN nanosheets. These nanostructures are found to have exciting mechanical, chemical, electrical, optical and electronic properties. Nanostructures are being used in the field of nanoelectronics, nanodevices, nanosensors, nano-oscillators, nano-actuators, nanobearings, and micromechanical resonators, transporter of drugs, hydrogen storage, electrical batteries, solar cells, nanocomposites and nanooptomechanical systems (NOMS). Understanding the dynamics of nanostructures is crucial for the development of future generation applications in these areas. Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 6
Introduction Nanoscale systems (b) Zinc Oxide (� ZnO�)�nanowire� (d) Protein� (a) DNA� (� c�) Boron Nitride� nanotube� (�BNNT� )� Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 7
Introduction General approaches for studying nanostructures Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 8
Introduction Nano mechanical sensors Progress in nanotechnologies has brought about a number of highly sensitive label-free biosensors. These include electronic biosensors based on nanowires and nanotubes, optical biosensors based on nanoparticles and mechanical biosensors based on resonant micro- and nanomechanical suspended structures. In these devices, molecular receptors such as antibodies or short DNA molecules are immobilized on the surface of the micro-nanostructures. The operation principle is that molecular recognition between the targeted molecules present in a sample solution and the sensor-anchored receptors gives rise to a change of the optical, electrical or mechanical properties depending on the class of sensor used. These sensors can be arranged in dense arrays by using established micro- and nanofabrication tools. Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 9
Introduction Cantilever nano-sensor Array of cantilever nano sensors (from http://www.bio-nano-consulting.com) Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 10
Introduction Cantilever nano-sensor Carbon nanotube with attached molecules Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 11
Introduction The mechanics behind nano-sensors Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 12
Introduction Mass sensing - an inverse problem This talk will focus on the detection of mass based on shift in frequency. Mass sensing is an inverse problem. The “answer” in general in non-unique. An added mass at a certain point on the sensor will produce an unique frequency shift. However, for a given frequency shift, there can be many possible combinations of mass values and locations. Therefore, predicting the frequency shift - the so called “forward problem” is not enough for sensor development. Advanced modelling and computation methods are available for the forward problem. However, they may not be always readily suitable for the inverse problem if the formulation is “complex” to start with. Often, a carefully formulated simplified computational approach could be more suitable for the inverse problem and consequently for reliable sensing. Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 13
Introduction The need for “instant” calculation Sensing calculations must be performed very quickly - almost in real time with very little computational power (fast and cheap devices). Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 14
One-dimensional sensors - classical approach Static deformation approximation Single-walled carbon nanotube based sensors Cantilevered nanotube resonator with an attached mass at the tip of nanotube length: (a) Original configuration; (b) Mathematical idealization. Unit deflection under the mass is considered for the calculation of kinetic energy of the nanotube. Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 15
One-dimensional sensors - classical approach Static deformation approximation Single-walled carbon nanotube based sensors - bridged case Bridged nanotube resonator with an attached mass at the center of nanotube length: (a) Original configuration; (b) Mathematical idealization. Unit Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 16
One-dimensional sensors - classical approach Static deformation approximation Resonant frequencies of SWCNT with attached mass In order to obtain simple analytical expressions of the mass of attached biochemical entities, we model a single walled CNT using a uniform beam based on classical Euler-Bernoulli beam theory: EI ∂ 4 y ( x , t ) + ρ A ∂ 2 y ( x , t ) = 0 (1) ∂ x 4 ∂ t 2 where E the Youngs modulus, I the second moment of the cross-sectional area A , and ρ is the density of the material. Suppose the length of the SWCNT is L . Depending on the boundary condition of the SWCNT and the location of the attached mass, the resonant frequency of the combined system can be derived. We only consider the fundamental resonant frequency, which can be expresses as � f n = 1 k eq (2) 2 π m eq Here k eq and m eq are respectively equivalent stiffness and mass of SWCNT with attached mass in the first mode of vibration. Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 17
One-dimensional sensors - classical approach Static deformation approximation Cantilevered SWCNT with mass at the tip Suppose the value of the added mass is M . We give a virtual force at the location of the mass so that the deflection under the mass becomes unity. For this case F eq = 3 EI / L 3 so that k eq = 3 EI (3) L 3 The deflection shape along the length of the SWCNT for this case can be obtained as Y ( x ) = x 2 ( 3 L − x ) (4) 2 L 3 Assuming harmonic motion, i.e., y ( x , t ) = Y ( x ) exp ( i ω t ) , where ω is the frequency, the kinetic energy of the SWCNT can be obtained as � L T = ω 2 ρ AY 2 ( x ) dx + ω 2 2 MY 2 ( L ) 2 0 (5) � 33 � L � = ρ A ω 2 Y 2 ( x ) dx + ω 2 2 M 1 2 = ω 2 140 ρ AL + M 2 2 0 Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 18
One-dimensional sensors - classical approach Static deformation approximation Cantilevered SWCNT with mass at the tip Therefore m eq = 33 (6) 140 ρ AL + M The resonant frequency can be obtained using equation (54) as � � 3 EI / L 3 f n = 1 k eq = 1 33 2 π m eq 2 π 140 ρ AL + M (7) � � � α 2 β = 1 140 EI 1 = 1 √ 2 π 11 ρ AL 4 M 140 2 π 1 + 1 + ∆ M ρ AL 33 where � 140 α 2 = or α = 1 . 888 (8) 11 � EI β = (9) ρ AL 4 M µ = 140 and ∆ M = ρ AL µ, (10) 33 Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 19
One-dimensional sensors - classical approach Static deformation approximation Cantilevered SWCNT with mass at the tip Clearly the resonant frequency for a cantilevered SWCNT with no added tip mass is obtained by substituting ∆ M = 0 in equation (7) as f 0 n = 1 2 π α 2 β (11) Combining equations (7) and (11) one obtains the relationship between the resonant frequencies as f 0 n f n = √ (12) 1 + ∆ M Adhikari (Swansea) Computational methods for nano sensors February 5, 2016 20
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