experimentally using spherical dendrimers to produce core–shell tecto(dendrimers)
and are described later (see Sects. 6.3.3 and 6.4.3).
Mathematically [167], these core–shell relationships have been analyzed as a
function of the ratio of the core spheroid (r 1 ) and shell spheroid (r 2 ) radii [167],
wherein the core spheroid size is systematic increased relative to the shell spheroid.
Quite remarkably, this treatment produces many important symmetries and geometries that appear to mimic those observed for atoms at the picoscale level in the
context of the VSEPR theory. For example, at an r 1 /r 2 value of 0.155, a valence of
3 shell spheroids and a trigonal geometry (D 3h ) is observed. At values for
r 1 /r 2 ¼ 0.255–0.414, one observes a valency of 4 with tetrahedral (T h ) symmetry,
and at r 1 /r 2 ¼ 0.255 one observes a valency of 8 with octahedral (O h ) symmetry
(see also Sect. 6.4.3). In essence, these valencies and geometries represent spacesaturated values around core atoms or core spheroids, respectively. These space
saturation values around a core may be engineered by simply tuning the relative
core and shell radii. This provides a powerful and useful strategy for defining
valency for all surface-reactive spheroidal nano-objects. It can be seen that when
the core reagent is small and the shell reagent is large, only a very limited number of
shell-type reagents can be attached to saturate the space surrounding the core
(i.e., r 1 /r 2 ¼ 0.155–1.20). Quite remarkably, when r 1 /r 2 ¼ 1, as would be the
case for metal nanoclusters, a valency of 12 and an icosahedral (I h ) symmetry is
observed (see Fig. 21 and Sect. 6.4.3). This is consistent for core–shell-type metal
nanoclusters (i.e., gold nanoclusters), as reported by Schmidt et al. [152, 153]
(Fig. 19). However, when r 1 /r 2 ! 1.20 more space surrounding the core allows
the attachment of more spheroidal shell reagents p to discrete saturation values
(N max ). This saturation value (N max ) is discrete and can be determined from the
general expression described by the Mansfield–Tomalia–Rakesh equation [167]
(described later in Sect. 6.4.3).
6.4 Combining Soft and Hard Nano-element Categories
to Create Combinatorial Libraries of Nanocompounds
and Nano-assemblies
6.4.1 Recent Literature Examples Fulfilling and Verifying Atom
Mimicry and Superatom Behavior by Forming 3D Nanoscale
Lattices, Nanocompounds, and Nano-assemblies Reminiscent
of Atomic Elements
Very recently, important examples describing the chemical combination and
assembly of these proposed hard and soft nano-element categories (i.e., superatoms)
as described in Fig. 24 have now appeared in the literature and are referred to as
“nanoscale atom mimicry” at the nanoscale. In each case, our early concept has
been fulfilled and validated by these authors, who have referred to these nanoscale
Twenty-First Century Polymer Science After Staudinger: The Emergence of. . .
363
and are described later (see Sects. 6.3.3 and 6.4.3).
Mathematically [167], these core–shell relationships have been analyzed as a
function of the ratio of the core spheroid (r 1 ) and shell spheroid (r 2 ) radii [167],
wherein the core spheroid size is systematic increased relative to the shell spheroid.
Quite remarkably, this treatment produces many important symmetries and geometries that appear to mimic those observed for atoms at the picoscale level in the
context of the VSEPR theory. For example, at an r 1 /r 2 value of 0.155, a valence of
3 shell spheroids and a trigonal geometry (D 3h ) is observed. At values for
r 1 /r 2 ¼ 0.255–0.414, one observes a valency of 4 with tetrahedral (T h ) symmetry,
and at r 1 /r 2 ¼ 0.255 one observes a valency of 8 with octahedral (O h ) symmetry
(see also Sect. 6.4.3). In essence, these valencies and geometries represent spacesaturated values around core atoms or core spheroids, respectively. These space
saturation values around a core may be engineered by simply tuning the relative
core and shell radii. This provides a powerful and useful strategy for defining
valency for all surface-reactive spheroidal nano-objects. It can be seen that when
the core reagent is small and the shell reagent is large, only a very limited number of
shell-type reagents can be attached to saturate the space surrounding the core
(i.e., r 1 /r 2 ¼ 0.155–1.20). Quite remarkably, when r 1 /r 2 ¼ 1, as would be the
case for metal nanoclusters, a valency of 12 and an icosahedral (I h ) symmetry is
observed (see Fig. 21 and Sect. 6.4.3). This is consistent for core–shell-type metal
nanoclusters (i.e., gold nanoclusters), as reported by Schmidt et al. [152, 153]
(Fig. 19). However, when r 1 /r 2 ! 1.20 more space surrounding the core allows
the attachment of more spheroidal shell reagents p to discrete saturation values
(N max ). This saturation value (N max ) is discrete and can be determined from the
general expression described by the Mansfield–Tomalia–Rakesh equation [167]
(described later in Sect. 6.4.3).
6.4 Combining Soft and Hard Nano-element Categories
to Create Combinatorial Libraries of Nanocompounds
and Nano-assemblies
6.4.1 Recent Literature Examples Fulfilling and Verifying Atom
Mimicry and Superatom Behavior by Forming 3D Nanoscale
Lattices, Nanocompounds, and Nano-assemblies Reminiscent
of Atomic Elements
Very recently, important examples describing the chemical combination and
assembly of these proposed hard and soft nano-element categories (i.e., superatoms)
as described in Fig. 24 have now appeared in the literature and are referred to as
“nanoscale atom mimicry” at the nanoscale. In each case, our early concept has
been fulfilled and validated by these authors, who have referred to these nanoscale
Twenty-First Century Polymer Science After Staudinger: The Emergence of. . .
363