only in the dimension that governs boundary layer development simultaneously maximizes both surface ux of analyte and the eective sensing area of the device.
This work was supported by funding from the Jacobs Institute for Molecular Engineering for Medicine at the California Institute of Technology.
boundary conditions commonly imposed when solving dierential equations analytically, while those in the interior experience a continuity boundary condition" that applies to the dependent variables as well as their derivatives.
Figure 5.7 shows the ow cell used for the present work, comprising a rectangular ow cell with a symmetry plane that bisects both the cell as well as the sensor. This half cell is 2.5 mm in the ow direction, and 1.5 mm x 1.5 mm in cross section (3 mm wide by 1.5 mm tall for the full cell). These dimensions were determined to be suciently large that increasing them led to less than 1% change in integrated analyte adsorption rates to the sensor for all ow rates considered here.
The sensor surface features a no-slip" boundary condition typical of solid walls and the analyte is assumed to adsorb instantaneously there. Figure 4.7 shows additional boundary conditions. We assume a at prole for the inlet ow velocity, Uinlet, and an inlet analyte concentration of Cinlet = 1 fM. The top, bottom, and side boundaries are given no-slip" and no ux" conditions, while the ow outlet is set to atmospheric pressure p0 = 101,325 Pa. The diusion coecient of Interleukin-2 (IL2, Mw 15.5 kDa) was approximated to beD≈10−10m2/s[104].
The array of elements used in an FE solver is called the mesh, and its design can signicantly impact the accuracy of the mathematical solution. A balance must be struck between making the mesh elements small enough to capture local changes in dependent variables and making them prohibitively small that the computer has insucient resources to nd a solution. We evaluate the dependence of our results on surface mesh element size, lmesh, by studying ow around a spherical sensor of radius Rsphere = 42.5µm andUinlet= 0.01m/s. We setlmesh to be a maximum ofRsphere/N and calculate the steady-state binding rate to the entire sensor surface. These data are reported in Fig. 4.8 as a relative binding rate with respect to that calculated for the caseN = 500where
ΦN=500= BindingrateforN
BindingrateforN = 500. (4.4)
This limiting condition (N = 500) represents the mesh size beyond which solutions take prohibitively long
to converge. The value ofΦN=500is greater than 0.98 for N > 80, indicating an acceptable 2% relative error.
Therefore, we set the surface mesh size in all models for the current work as 1/80 of the critical dimension (Rsphere for a sphere andrminorfor a toroid).
While analyte is assumed to adsorb to the entire surface of the resonator, only a fraction of that surface is eective for sensing. COMSOL solutions for the mode in a spherical and toroidal WGM resonator, calculated using methods described by Oxborrow [103], were used to identify the functional region of the sensor dened by where the mode intensity at the resonator surface is >10% of the surface maximum. These results are shown in Fig. 4.9. The eective mode height for the toroid modeled here (hmode= 2.84µm) is less than half that for a sphere of identical outer radius (hmode = 6.00µm) due to mode compression by the minor radius of the toroid.
Single-Molecule Binding Data
We compare the predictions of our model with single-molecule binding data observed experimentally by Armani and colleagues [1]. Those data are transient measurements monitoring the resonant wavelength,λR , of the mode as interleukin-2 binds to the resonator, which has been functionalized with anti-IL2. The sensor exhibits a step-wise response inλRfor a range of IL2 concentrations regardless of whether buer or bovine blood serum was used as the solvent. The experimental data included in Fig. 4.5 represents the average time between binding events detected during the rst 60 seconds of the experiment. This time window was chosen in accordance with the assumptions made in the computational model. A step was dened as any time |∆λR|>2σ, whereσ is the noise of the signal inλR within a single step and averaged over the rst ve steps in an experiment. Typical values ofσwere between 0.15 fm (0.15× 10-15 M) and 0.25 fm.
Figure 4.6: Flow cell geometry used in COMSOL Multiphysics simulation of ow around a WGM sensor.
The near plane is a symmetry plane that bisects the cell and the resonator. The sensors are cut out of the cell, and their surfaces feature no-slip ow and Ci = 0 (i.e., instantaneous surface reaction) boundary conditions.
Figure 4.7: Flow cell boundary conditions used in COMSOL Multiphysics simulations of ow around WGM sensors. (a) Symmetry plane. (b) No-slip" and no ux" conditions. (c) Uniform ow inlet velocity U =Uinletand inlet concentrationCinlet= 1fM. (d) Flow outlet at pressurep0= 101,325 Pa.
Figure 4.8: The relative surface binding rate to a sphere withRsphere= 42.5µmandUinlet= 0.01m/s as a function of the surface mesh element size (lmesh=Rsphere/N), calculated with respect to the caseN = 500 (lmesh = 85nm). This quantity converges with increasing N and achieves a relative error of less than 2%
for N > 80 (see inset for detail).
Figure 4.9: Solutions for the normalized mode intensity (NMI) of a (a) toroidal and (c) spherical optical whispering gallery mode resonator. The eective sensing area is determined by where the NMI is greater than 10% of the surface maximum as indicated by dotted lines for (b) a toroid and (d) a sphere.
Chapter 5
The Physics of Extreme Sensitivity in WGM Optical Resonator Biosensors
5.1 Abstract
Whispering gallery mode (WGM) optical biosensors are capable of extraordinarily sensitive specic and non- specic detection of species suspended in a gas or uid. Recent experimental results suggest that these devices may attain single-molecule sensitivity to protein solutions in the form of stepwise shifts in their resonance wavelength, λR, but present sensor models predict much smaller steps than were reported. This study examines the physical interaction between a WGM sensor and a molecule adsorbed to its surface, exploring assumptions made in previous eorts to model WGM sensor behavior, and describing computational schemes that model the experiments for which single protein sensitivity was reported. The resulting model is used to simulate sensor performance, within constraints imposed by the limited material property data. On this basis, we conclude that nonlinear optical eects would be needed to attain the reported sensitivity, and that, in the experiments for which extreme sensitivity was reported, a bound protein experiences optical energy uxes too high for such eects to be ignored.