Electronic Supporting Information
Self-assembly of like-charged nanoparticles into
Voronoi diagram
Dániel Zámbóa, Kohta Suzunob, Szilárd Pothorszkya, Dóra Bárdfalvyc, Gábor Hollóc,
Hideyuki Nakanishid, Dawei Wange, Daishin Ueyamab, András Deáka, István Lagzic*
aCentre for Energy Research, Institute of Technical Physics and Materials Science, Budapest,
Hungary
bMeiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1
Nakano, Nakano-ku, Tokyo, Japan
cDepartment of Physics, Budapest University of Technology and Economics, 1111, Budafoki
út 8, Budapest, Hungary
dDepartment of Macromolecular Science and Engineering, Graduate School of Science and
Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
eLehn Institute of Functional Materials, School of Chemistry and Chemical Engineering, Sun
Yat-Sen University, Guangzhou 510275, China
Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics.This journal is © the Owner Societies 2016
Figure S1. Comparison of a chemically generated Voronoi diagram based on nanoparticles
aggregation and a geometrically generated Voronoi diagram. Black crosses and black lines cor-
respond to the seeds/generators of the Voronoi diagram and the boundaries of the
geometrically generated Voronoi cells, respectively.
Figure S2. Determination of the distance between two pieces of crystals/seeds (L, blue lines)
and the edges of the aggregation-free gaps (marked by red crosses).
Figure S3. Grayscale analysis of the pixels along a line between two seeds (upper graph). The
edges of the aggregation-free gaps were determined where the spatial gradients of the smoothed
grayscale curve had extremes (below graph).
Figure S4 Evolution of a Voronoi diagram formed in an experiment (a) and in a numerical
simulation (b) using bigger (~ 4 mm) sodium chloride crystals placed randomly on top of the
solid hydrogel. Black regions in (b) correspond to the positions and shape of the crystals.
Figure S5 Evolution of a Voronoi diagram formed in an experiment (a) and in a numerical
simulation (b) using smaller (~ 0.2 mm) sodium chloride crystals placed randomly on top of the
solid hydrogel. Black regions in (b) correspond to the positions of the crystals.
Figure S6 Calculated Voronoi patterns with different threshold concentrations for aggregation
of nanoparticles (a) a* = 0.03 M and (b) a* = 0.15 M. The relative average error between two
patterns is less than 2%. In the numerical simulations all model parameters were identical ex-
pect the threshold concentration.
Calculation of van der Waals and electrostatic interaction potentials
The vdW interaction of MUA coated AuNPs can be calculated using the following equation
( )( ) ( ) ( )
2 2 2
2 2
41ln 1 ,
3 4 22 2
vdW a a a
B a a a
U d r r rA
k T d r d r d r d
= − + + −
+ + +
(1)
where A is the Hamaker constant for AuMUA system (4.52×10–19J),1 ra is the radius of the
particle and d is the separation distance between the particles surfaces. The radius ra of the
nanoparticle was calculated as a sum of the gold core’s radius (3.25 nm) and the thickness of
the MUA layer (1.60 nm).
The electric double layer repulsion was estimated as2
( ) 2
,2
EDL da
B a
U d rZe
k T r
κ− =
(2)
where κ is the reciprocal Debye length and Z is the interaction constant. The interaction constant
can be calculated as follows
2
2 0
064 ,
4
B
B
zek TZ tanh
e k T
ψπε ε
=
(3)
where ε0 is the vacuum permittivity (8.854×10–12 Fm–1), ε is the relative permittivity for water
(80.1 at 20 °C), z is the valence of the ions and ψ0 is the surface potential of nanoparticle, and
it is calculated as
( )
1
0 1
20
2,
8
Bk T
sinhze
RT c
σψ
ε ε
−
∞
=
(4)
where c∞ is the bulk concentration of the electrolyte. The surface potential can be calculated
from the surface charge density (σ), which is a function of ζ-potential
0 42
2 4
B
B a B
k T ze zesinh tanh .
ze k T r k T
εε ζ ζσ κ
κ
= +
(5)
The zeta potential of the particles (ζ = – 45 mV) was obtained from electrokinetic measurements
using a Malvern Zetasizer NanoZS. The total nanoparticle-nanoparticle interaction energy was
obtained from Equations (1) and (2)
( ) ( ) ( ).
total EDL vdW
B B B
U d U d U d
k T k T k T= +
(6)
References
1 T. Laaksonen, P. Ahonen, C. Johans and K. Kontturi, ChemPhysChem, 2006, 7, 2143–
2149.
2 K. J. M. Bishop, C. E. Wilmer, S. Soh and B. A. Grzybowski, Small, 2009, 5, 1600–
1630.
