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
n2
[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.
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