Numerical tool for estimating the dielectric constant, the thickness, and the coverage of immobilized inhomogeneous protein films on gold in aqueous solution

Tiago A. T. Sousa; Leiva C. Oliveira; Franz H. Neff; Hervé M. Laborde; Antonio M. N. Lima

doi:10.1364/AO.57.006866

Applied Optics
Vol. 57,
Issue 24,
pp. 6866-6875
(2018)
•https://doi.org/10.1364/AO.57.006866

Numerical tool for estimating the dielectric constant, the thickness, and the coverage of immobilized inhomogeneous protein films on gold in aqueous solution

Tiago A. T. Sousa, Leiva C. Oliveira, Franz H. Neff, Hervé M. Laborde, and Antonio M. N. Lima

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Tiago A. T. Sousa, Leiva C. Oliveira, Franz H. Neff, Hervé M. Laborde, and Antonio M. N. Lima, "Numerical tool for estimating the dielectric constant, the thickness, and the coverage of immobilized inhomogeneous protein films on gold in aqueous solution," Appl. Opt. 57, 6866-6875 (2018)

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- Instrumentation and Measurements
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About this Article
History
- Original Manuscript: April 18, 2018
- Revised Manuscript: June 15, 2018
- Manuscript Accepted: July 9, 2018
- Published: August 13, 2018

Abstract

A numerical simulation tool is reported for nanometer thin and inhomogeneous immobilized protein films on gold in aqueous solution. It allows for the first time, to the best of our knowledge, the simultaneous assessment of refractive index, film thickness, and surface coverage. The model relies on and combines the convective diffusion equation, the Langmuir adsorption isotherm, and the Helmholtz equation, with appropriate boundary conditions. These three differential equations were jointly solved using a multiphysics software. The physical film parameters were extracted employing an optimization procedure for immobilized bovine serum albumin, hemoglobin, and neutravidin films. The relatively good agreement between the extracted values for the refractive index, film thickness, and surface coverage and the corresponding values reported in the open literature show the correctness of the proposed methodology.

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Parameter	Symbol	Value
Protein		neutravidin
Shape		spherical
Radius	$r$	2.9 nm
Diffusion coefficient	$D$	$6 \times 10^{- 7} {cm}^{2} / s$
Flow velocity	$u$	8.3 mm/s
Adsorption constant	$k^{on}$	$9.4 m^{3} / s / mol$
Desorption constant	$k^{off}$	0.0078/s
Light wavelength	$λ$	670 nm
Light incident angle range	$[θ_{I}, θ_{F}]$	[65°,72°]
Dielectric constant
Optical substrate	$ϵ_{1}$	2.3104
Gold film	$ϵ_{2}$	$- 14.379 + 1.0084 j$
Flow cell solution	$ϵ_{4}$	1.7876
Thickness
Optical substrate	$d_{1}$	10,000 nm
Gold film	$d_{2}$	50 nm
Flow cell solution	$d_{4}$	10,000 nm

	Reference Values from Literature		Estimated (Laborde et al. [16])	Estimated Proposed Method
	$n_{3}$	$d_{3}$ (nm)	$Θ$	$n_{3}$	$d_{3}$ (nm)	$Θ$
Neutravidin	1.55 [25]	$5.8 \pm 1.8$ [26]	0.39	1.58	4.4	0.46
BSA	1.57 [27,28]	$4 \pm 2$ [18]	0.32	1.53	4.9	0.29
Hemoglobin	1.485 [28]	4 [28]	0.41	1.60	5	0.39

Parameter	Symbol	Value
Protein		neutravidin
Shape		spherical
Radius	$r$	2.9 nm
Diffusion coefficient	$D$	$6 \times 10^{- 7} {cm}^{2} / s$
Flow velocity	$u$	8.3 mm/s
Adsorption constant	$k^{on}$	$9.4 m^{3} / s / mol$
Desorption constant	$k^{off}$	0.0078/s
Light wavelength	$λ$	670 nm
Light incident angle range	$[θ_{I}, θ_{F}]$	[65°,72°]
Dielectric constant
Optical substrate	$ϵ_{1}$	2.3104
Gold film	$ϵ_{2}$	$- 14.379 + 1.0084 j$
Flow cell solution	$ϵ_{4}$	1.7876
Thickness
Optical substrate	$d_{1}$	10,000 nm
Gold film	$d_{2}$	50 nm
Flow cell solution	$d_{4}$	10,000 nm

	Reference Values from Literature		Estimated (Laborde et al. [16])	Estimated Proposed Method
	$n_{3}$	$d_{3}$ (nm)	$Θ$	$n_{3}$	$d_{3}$ (nm)	$Θ$
Neutravidin	1.55 [25]	$5.8 \pm 1.8$ [26]	0.39	1.58	4.4	0.46
BSA	1.57 [27,28]	$4 \pm 2$ [18]	0.32	1.53	4.9	0.29
Hemoglobin	1.485 [28]	4 [28]	0.41	1.60	5	0.39

Tiago A. T. Sousa	https://orcid.org/0000-0002-7903-5315
Leiva C. Oliveira	https://orcid.org/0000-0001-9147-4604
Hervé M. Laborde	https://orcid.org/0000-0002-7024-3038
Antonio M. N. Lima	https://orcid.org/0000-0002-4568-9126

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Applied Optics