Modeling and simulation of a biosensor
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III. Mathematical modeling
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The mathematical model describes a composite structure consisting of three coupled layers: two solid layers with different elastic and electric properties and a liquid layer treated as a compressible viscous fluid. The full coupling between deformations and electric fields is assumed.
In Figure, a schematic picture of the biosensor is presented. A thin film (isotropic guiding layer) is deposited on a substrate made of a quartz crystal. The input and output interdigital transducers (IDTs) are located between the substrate and film. To obtain purely shear polarized modes, the direction of the wave propagation is chosen to be orthogonal to the crystalline X-axis. The choice of the film and substrate materials must provide that the wave velocity in the film is less than the one in the substrate so that the waves will be transferred into the film.
The model should meet the following requirements: 1) high sensitivity with respect to very small mass loadings, 2) careful modeling of actors and sensors taking into account their massiveness and electro-conductivity, 3) adequate modeling of the liquid layer and the fluid-structure interaction. Also the influence of acoustic absorbers, which reduce the reflection of waves at the ends of the device, have to be considered.
The model is realized on the basis of finite element method and uses automatic derivation of the model equations for various crystal cuts. We use a reduction of 3D-model either to a front-face model where the averaging in x2-direction is done or to a top-face model obtained by the averaging in x3-direction under assumptions about the exponential attenuation of wave amplitudes with the depth.
Such simplified models are used for the computation of the sensitivity and optimal operating frequency of the biosensor. The results obtained (see section "Simulation") are consistent with physical experiments. However, these models are still time- and resource-consuming so that numerous series of computer experiments where geometrical and material parameters are being varied can hardly be performed.
A way to speed up numerical simulations consists in the usage of dispersion relations which express the dependence of the surface shear wave velocity on the operating frequency. Many characteristics of the biosensor can be computed on the base of dispersion relations.
We proposed a semi-analytic method.
The one for the computation of dispersion relations for multi-layered
structures similar to that in Figure. Thereby, each of the solid layers can
be anisotropic and the number of solid layers is not restricted. A fluid
and a bristle layer (aptamer) on the top of the upper solid layer are accounted.
The method is based on the consideration of a plane wave propagating in x1-direction. The wave satisfies elasticity equations in every solid layer and linearized Navier-Stokes equations in the fluid. The dispersion relations are computed from the continuity of displacements and normal pressures on the interfaces between the solid layers, continuity of velocities and pressures on the solid-liquid interface, and the exponential attenuation of the wave amplitude in the substrate. The aptamer layer is accounted by the calculation of the averaged horizontal force produced by the liquid, which results in some modification of the normal pressure continuity condition on the solid-liquid interface. Numerically, the problem is reduced to the statement and solution of a system of transcendental equations whose roots represent propagating velocities for different wave types. A computer program that implements the computation of the wave propagation velocity depending on the operating frequency allows us to compute very interesting dependencies (see section Simulation).
