Description of the slightly modified extended Mie theory For: ‘Quantitative Interpretation of Gold Nanoparticle-based Bioassays Designed for Detection of Immunocomplex formation’ By: Ye Zhou, Hongxing Xu, Andreas Dahlin, Jacob Vallkil, Carl AK Borrebaeck, Christer Wingren, Bo Liedberg and Fredrik Höök We applied the extended Mie theory with fundamental assumptions of spherical Au NPs and homogeneous dielectric layers characterized by unique thicknesses, di, and effective refractive indexes, ni, where the subscript, i, refers to the different layers. To theoretically mimic the experimental situation, the homogenously coated spheres are placed on a substrate with refractive index ns. For loosely packed Au NPs, the couplings among the particles and between the substrate that influence the total light scattering are dominated by the dipole couplings. The large inter-particle distances of ~125 nm and the particle size of ~40 nm used in this study justify this approach. Hence, the extinction cross section is given by: C ext 2 '' ( 3 Re( a b ) ( 2 n 1 ) Re( a n bn )) 1 k2 n2 [1] where a n and bn are the Mie coefficients of a coated sphere, which can be calculated as follows 1: an U nI bn VnI 1 where U nI and VnI can be recursively deduced from the following equations: U ni i n ( xi ) n' ( yi ) U ni 1 n' ( yi ) i n' ( xi ) n ( yi ) U ni 1 n ( yi ) i n' ( xi ) n ( yi ) U ni 1 n ( yi ) i n ( xi ) n' ( yi ) U ni 1 n' ( yi ) [2] Vni i n ( xi ) n' ( yi ) Vni 1 n' ( yi ) i n' ( xi ) n ( yi ) Vni 1 n ( yi ) i n' ( xi ) n ( yi ) Vni 1 n ( yi ) i n ( xi ) n' ( yi ) Vni 1 n' ( yi ) [3] with the initial conditions: U ni 0 0 , Vni 0 0 , and n ( z ) zjn ( z ) , n ( z ) zhn(1) ( z ) , xi ki 1ri , yi ki ri , i ki 1 / ki , i i 1 / i , where jn (z ) and hn(1) ( z ) are the spherical Bessel function and the spherical Hankel function of the first kind, respectively, ki is the absolute value of the wave vector, i i is the magnetic permeability and ri r0 dj is the radius of the outer surface of the ith layer. For j 1 bare spheres, i = 0 is used in the above sub-/superscriptions. Furthermore, a '' in eq. 2 is the modified dipolar term due to the couplings among NPs and between the substrate. In order to calculate a '' , the inter-particle couplings are first considered using an approach similar to that described in the work by Xu et. al2. In brief, the effective dipolar coefficient due to the inter-particle couplings a1' a1 / 1 j ALj , where 3ia1 /( 2k 3 ) is the dipole polarizability, and ALj is the retarded dipole- coupling element of Lth sphere to other spheres (see work performed by Lazarides, et al.3 for details). For simplicity, a symmetric triangular particle arrangement is assumed, and j ALj is assumed to be identical for each particle. The dipole coupling between a particle coated with dielectric shells and a flat substrate can then be approximately treated as the coupling between a dipole and a mirror image dipole on the flat surface. Hence, the total dipolar term becomes: 2 2 ns nm 2 3 a a / 1 2 / 4 n ( 2 r ) m 2 ns nm '' 1 ' 1 [4] where n m is the refractive index of the surrounding medium (nm=1.333 for water and nm=1.36 for ethanol) and r is the radius of the coated sphere. 2 A curve fitting using the above formulas to a typical experimental spectrum of immobilized gold particles is shown in Figure 1b. The radius distribution of nanospheres obtained from the fitting was a Gaussian distribution of the radius r0 = 18 nm with a standard deviation σ= 5.5 nm (corresponding to a size distribution of 36±5.5 nm, similar to the value obtained using AFM), and an inter-particle centre to centre distance of 125 nm. Interestingly, the density of the nanospheres required to get an extinction value comparable to that experimentally observed is 61/μm2, which is close to that obtained using AFM (Fig. 1a). Although the distribution of the inter-particle distance can influence the obtained results, the calculated vary only slightly (0.07nm for the first layer in this case) when the density of nanoparticles was doubled, i.e. an inter-particle separation of ~90nm. A variation of 0.07nm is insignificant also with respect to the observed peak-position change of >2.5nm for the first layer (Table 1 in main text). Hence the distribution of the inter-particle separation does not influence the main conclusions of this study. 3 REFERENCES 1. Xu, H. X., Multilayered metal core-shell nanostructures for inducing a large and tunable local optical field. Phys Rev B 2005, 72, 073405. 2. Xu, H. X.; Käll, M., Modeling the optical response of nanoparticle-based surface plasmon resonance sensors. Sens. Actuator B-Chem. 2002, 87, 244-249. 3. Lazarides, A. A.; Schatz, G. C., DNA-linked metal nanosphere materials: Fourier-transform solutions for the optical responase. J Chem Phys 2000, 112, (6), 2987-2993. 4