which flexible polymers are grafted are interesting building blocks for various
nanostructured materials [114, 115]. Since it is possible to vary parameters such
as the radius (denoted as R c ) of the solid nanoparticle that forms the core of the
spherical polymer brush, the grafting density of the macromolecules, their chemical
nature, their degree of polymerization, and the chemical nature of the matrix
(solvent or polymer melt) in which the spherical polymer brushes are embedded,
a rich behavior can be expected [116, 117].
In the theoretical modeling [117–120] to be briefly described here, only limited
aspects of this broad field were addressed, namely the case where the density of the
nanoparticles in the system is sufficiently dilute such that only the structure of
isolated spherical brushes or their pairwise interactions are of interest; dense
aggregates formed from more than two spherical brushes were not considered.
Two complementary techniques were used, molecular dynamics computer simulation [121] and analytical calculations based on density functional methods [122]
and the self-consistent field theory of polymers [123]. We also restrict attention to
the case where the grafting density is high enough such that under good solvent
conditions the close approach of two nanoparticles (to a distance of a few diameters
of the effective monomers) is prohibited by the large entropic penalties; hence, the
direct van der Waals attraction of the nanoparticles can always be neglected. Note
that for the standard bead-spring model of flexible polymers that is used here [121],
every effective monomer represents 3–5 chemical monomers, so one can associate
the effective monomer diameter (which is taken as length unit here a ¼ 1) to a
physical size of 1 nm, while the radius R c of the nanoparticle typically was about 8a.
Choosing a number N of effective monomeric subunits of the grafted chains from
N ¼ 20 to N ¼ 80, the thickness of the brush coating on the nanoparticles also is a
few nanometers, comparable to R c . Thus, our studies are neither in the limit where
the spherical polymer brush can be considered as a many-arm star polymer, nor in
the limit R c ! 1 where the brush coating can be described in terms of a planar
polymer brush.
Figure 35 [118] shows typical configurations of two nanoparticles (with a total
number of 92 grafted chains containing N ¼ 40 effective monomers per chain)
under good solvent conditions, for distances between them of r ¼ 55 and r ¼ 20.
In this case, the potential of mean force W(r) between the two particles is uniformly
repulsive: although it is practically zero in the first case, i.e., the particles are
essentially noninteracting, it is 200 k B T for r ¼ 20. Such a close approach of the
brushes, where the two polymeric shells strongly interpenetrate, therefore cannot
occur simply by thermal fluctuations.
However, the situation is different when the solvent quality varies (Fig. 36).
Note that we have treated only implicit solvent in the simulations, whereas in the
analytical calculations spherical brushes embedded in concentrated polymer
solutions could also be treated, to check the extent to which this case resembles a
dilute solution under Theta solvent conditions [120]. Figure 36 shows that in poor
solvents conditions, for moderate grafting density, the nanoparticle is no longer
coated by a uniform polymer layer but the system prefers to form a dumbbellshaped object. At both the “north pole” and the “south pole” of the nanoparticle
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
153
nanostructured materials [114, 115]. Since it is possible to vary parameters such
as the radius (denoted as R c ) of the solid nanoparticle that forms the core of the
spherical polymer brush, the grafting density of the macromolecules, their chemical
nature, their degree of polymerization, and the chemical nature of the matrix
(solvent or polymer melt) in which the spherical polymer brushes are embedded,
a rich behavior can be expected [116, 117].
In the theoretical modeling [117–120] to be briefly described here, only limited
aspects of this broad field were addressed, namely the case where the density of the
nanoparticles in the system is sufficiently dilute such that only the structure of
isolated spherical brushes or their pairwise interactions are of interest; dense
aggregates formed from more than two spherical brushes were not considered.
Two complementary techniques were used, molecular dynamics computer simulation [121] and analytical calculations based on density functional methods [122]
and the self-consistent field theory of polymers [123]. We also restrict attention to
the case where the grafting density is high enough such that under good solvent
conditions the close approach of two nanoparticles (to a distance of a few diameters
of the effective monomers) is prohibited by the large entropic penalties; hence, the
direct van der Waals attraction of the nanoparticles can always be neglected. Note
that for the standard bead-spring model of flexible polymers that is used here [121],
every effective monomer represents 3–5 chemical monomers, so one can associate
the effective monomer diameter (which is taken as length unit here a ¼ 1) to a
physical size of 1 nm, while the radius R c of the nanoparticle typically was about 8a.
Choosing a number N of effective monomeric subunits of the grafted chains from
N ¼ 20 to N ¼ 80, the thickness of the brush coating on the nanoparticles also is a
few nanometers, comparable to R c . Thus, our studies are neither in the limit where
the spherical polymer brush can be considered as a many-arm star polymer, nor in
the limit R c ! 1 where the brush coating can be described in terms of a planar
polymer brush.
Figure 35 [118] shows typical configurations of two nanoparticles (with a total
number of 92 grafted chains containing N ¼ 40 effective monomers per chain)
under good solvent conditions, for distances between them of r ¼ 55 and r ¼ 20.
In this case, the potential of mean force W(r) between the two particles is uniformly
repulsive: although it is practically zero in the first case, i.e., the particles are
essentially noninteracting, it is 200 k B T for r ¼ 20. Such a close approach of the
brushes, where the two polymeric shells strongly interpenetrate, therefore cannot
occur simply by thermal fluctuations.
However, the situation is different when the solvent quality varies (Fig. 36).
Note that we have treated only implicit solvent in the simulations, whereas in the
analytical calculations spherical brushes embedded in concentrated polymer
solutions could also be treated, to check the extent to which this case resembles a
dilute solution under Theta solvent conditions [120]. Figure 36 shows that in poor
solvents conditions, for moderate grafting density, the nanoparticle is no longer
coated by a uniform polymer layer but the system prefers to form a dumbbellshaped object. At both the “north pole” and the “south pole” of the nanoparticle
Structure Formation of Polymeric Building Blocks: Complex Polymer Architectures
153