Computational nanoplasmonics in the quasistatic limit for biosensing applications

The phenomenon of localized surface plasmon resonance (LSPR) provides high sensitivity in detecting biomolecules through shifts in resonance frequency when a target is present. Computational studies in this field have used the full Maxwell equations with simplified models of a sensor-analyte system, or they neglected the analyte altogether. In the long-wavelength limit, one can simplify the theory via an electrostatics approximation while adding geometrical detail in the sensor and analytes (at moderate computational cost). This work uses the latter approach, expanding the open-source PyGBe code to compute the extinction cross section of metallic nanoparticles in the presence of any target for sensing. The target molecule is represented by a surface mesh, based on its crystal structure. PyGBe is research software for continuum electrostatics, written in python with computationally expensive parts accelerated on GPU hardware, via PyCUDA. It is also accelerated algorithmically via a treecode that offers $O(NlogN)$ computational complexity. These features allow PyGBe to handle problems with half a million boundary elements or more. In this work, we demonstrate the suitability of PyGBe, extended to compute LSPR response in the electrostatic limit, for biosensing applications. Using a model problem consisting of an isolated silver nanosphere in an electric field, our results show grid convergence as $1/N$, and accurate computation of the extinction cross section as a function of wavelength (compared with an analytical solution). For a model of a sensor-analyte system, consisting of a spherical silver nanoparticle and a set of bovine serum albumin (BSA) proteins, our results again obtain grid convergence as $1/N$ (with respect to the Richardson extrapolated value). Computing the LSPR response as a function of wavelength in the presence of BSA proteins captures a redshift of 0.5 nm in the resonance frequency due to the presence of the analytes at 1-nm distance. The final result is a sensitivity study of the biosensor model, obtaining the shift in resonance frequency for various distances between the proteins and the nanoparticle. All results in this paper are fully reproducible, and we have deposited in archival data repositories all the materials needed to run the computations again and recreate the figures. PyGBe is open source under a permissive license and openly developed. Documentation is available at http://pygbe.github.io/pygbe/docs/.

Figure 1

Illustration of the localized surface plasmon resonance (LSPR) effect of a metallic nanoparticle under an electromagnetic field.

Figure 2

Nanoparticle interacting with an electromagnetic wave.

Figure 3

Analyte-sensor system under electric field.

Figure 4

Spherical nanoparticle in a constant electric field.

Figure 5

Grid-convergence study for the extinction cross section of a spherical silver nanoparticle, computed with PyGBe. The figure, plotting script, and auxiliary files are available under cc-by [41].

Figure 6

Extinction cross section as a function of wavelength for an 8-nm silver sphere immersed in water. The peak in the values of extinction cross section corresponds to the plasmon resonance of the metallic nanoparticle under the incoming electric field. The figure, plotting script, and auxiliary files are available under cc-by [42].

Figure 7

Grid-convergence study of extinction cross section of a spherical silver nanoparticle with a BSA dimer at $d=1\mathrm{nm}$. The figure, plotting script, and auxiliary files are available under cc-by [41].

Figure 8

Sensor protein display: BSA dimers located at $\pm 1\mathrm{nm}$ of the nanoparticle in the $z$ direction. The figure, plotting script, and auxiliary files are available under cc-by [44].

Figure 9

Extinction cross section as a function of wavelength for an $8\mathrm{nm}$ silver sphere immersed in water with two BSA dimers placed $\pm 1\mathrm{nm}$ away from the surface in the $z$ direction, and at infinity (no protein).

Figure 10

Extinction cross section as a function of wavelength for an 8-nm silver sphere immersed in water with two BSA dimers placed at $\pm 1\mathrm{nm}$ away from the surface in the $x$-direction (a) and $y$-direction (b), and at infinity (no protein).

Figure 11

Extinction cross section as a function of wavelength for an 8-nm silver sphere immersed in water with two BSA dimers placed at 2, 1, and $0.5\mathrm{nm}$ away from the surface in the $z$ direction, and at infinity (no protein). The test case with $d=0.5\mathrm{nm}$ is close to the limit where quantum tunneling might happen. Such effects are not captured by our classical model.

Figure 12

Sensor protein display: BSA dimers located at $\pm 1\mathrm{nm}$ of the nanoparticle in the $x$ direction (a) and $y$ direction (b). The figure, plotting script, and auxiliary files are available under cc-by [44].