Macromol. Theory. Simul. 13
, 487 (2004)
In this study we extend our previous work concerning the Rouse dynamics of linear
alternating copolymers (Macromolecules 36
, 486 (2003)), to
tree-like structures and focus on copolymeric dendrimers built from monomers
of two kinds
; as before, we let the monomers differ in their interaction
with the solvent. In the framework of generalized Gaussian structures (GGS) we
consider alternating arrangements of monomers over the dendritic structures.
We develop a semi-analytical method to determine for such structures
(of arbitrary functionality,
, and number of generations,
the eigenfrequencies (relaxation times).
A dendrimer is characterized by its
and by the number
of generations. The number
of beads of the dendrimer is then:
We should note that
is the lowest
value which leads to non-trivial dendrimers; in general we take
. Furthermore, here all beads of even
generations are assumed to have
as friction constant, whereas all beads of odd
generations have friction constant
. Figure 1 shows a particular example for an alternating copolymeric dendrimer, for which
Figure 1: Dendrimer with alternating generations
The normal modes of the dendrimer are fall into two groups: (i) modes in which
the core is mobile and (ii) modes in which the core is immobile.
The advantage here is that our algorithm
bypasses the direct diagonalization of the corresponding (asymmetric)
connectivity matrix. The method is especially suitable for very large
, where, because of the exponential increase of
, limits on computer time and, especially, on
computer memory render soon a brute-force diagonalisation (note that we
the eigenvalues) unfeasible. Even for moderately large
our method has advantages, because it can be readily implemented on
small PCs using existing programs, such as MATHEMATICA or MAPLE.
The storage and the loss moduli, show here again a multitude of features, which mainly depend on
the difference in the mobilities, or, equivalently, in the friction coefficients
and of the - and -beads. These features range from
the presence of large plateau-type regions in
to the appearance
of double-peaks in
In contrast to linear alternating copolymers, the behavior of the
dynamic moduli of copolymeric systems with dendritic topology can shed
light into their composition, i.e. into the relative numbers
of - and -beads.
Figure 2: The reduced storage modulus
and the reduced loss modulus
plotted in double logarithmic scales versus the reduced frequency
. Shown are results for alternating copolymeric dendrimers. The parameters are and , while ranges from to .
Dynamics of end-linked star polymers
Authors: Cristian Satmarel, Christian von Ferber, and
J. Chem. Phys. 123
, 034907 (2005)
In this work we focus on the dynamics of macromolecular networks formed
end-linking identical polymer stars. The resulting macromolecular
network can then be
viewed as consisting of spacers which connect branching
points (the cores of the stars).
Figure 1: Network consisting of end-linked polymer stars, of which one is highlighted by a circle. Here, the number of stars is , the functionality of each star is, and the length of each arm is ; the total number of beads in the system is
. Note that the structure is very flexible so that in many realizations it will appear considerably less ordered than here.
We succeed in analyzing exactly, in
the framework of the generalized Gaussian model, the eigenvalue
spectrum of such networks. As applications we focus on several
topologies, such as regular networks and dendrimers. The central bead may differ (chemically and functionally)
from the other beads, a fact which we take into account by assigning it
a different friction coefficient. In Fig. 1 we show such a network
formed from end-linked stars. Corresponding to the coloring used in
Fig. 1, we call in the following the core bead "black" and the chain
procedure which we use involves an exact real-space renormalization,
which allows to relate the star-network to a (much simpler) network, in
which each star is reduced to its core. It turns out that the
eigenvalue spectrum of the star-polymer structure consists of two
parts: One follows in terms of polynomial equations from the relaxation
spectrum of the corresponding renormalized structure, while the second
part involves the motion of the spacer chains themselves. We find a set
of eigenmodes, belonging to different classes, that fully determine the
eigenvalue spectrum of the system. In the first class, the coordinates
of some of
the black beads are non-vanishing and the black beads do not move in
parallel. The second
class is that in which all the black beads move in
parallel. The next two classes, are such
that all the black beads are at rest.
This two-step approach of first finding the eigenvalue spectrum
of the reduced black system, and only then calculating the eigenvalue
spectrum of the full system,
considerably reduces the efforts of solving the problem.
As a specific example of branched polymers, we treat here star-burst dendrimers, which are built from
stars, linked in such a way as to describe a regular Cayley tree, see
Figure 2. One may remark that our renormalization procedure reduces the
dendrimer with spacers and dangling bonds of Fig. 2 to a simple Cayley
A second generation dendrimer with spacers. Here, and each spacer chain contains two beads (). The dash-dotted line indicates one of the stars from which the structure is built.
The effects of varying the parameters
which determine the length of the spacers and of the dangling chains,
and the ratio of the friction coefficients, respectively are presented
in Figs. 3 and 4. The number of white beads (which are more mobile
) is larger than the number of black beads, and their relaxation dominates the spectrum. For
the situation is reversed: The black beads are more mobile and the
minor peak corresponding to their relaxation is shifted to higher
frequencies, while the major peak induced by the white beads keeps its
position. Corresponding effects are seen also in the behavior of the
, which for
display a plateau resulting from a gap in the relaxation. The plateau
is formed between the domains of high and of low frequency relaxation.
the white beads dominate the high frequency relaxation, leading to a lower level of the plateau than for
, where the high frequency relaxation of the black beads is only weakly evident.
Figure 3. The storage
and the loss moduli for a dendrimer of generation g=6, functionality f=3, with spacer chains of length s=4 between the branching points and dangling bonds of length k=2. Here, varies from to .
The influence in and of increasing the length of the spacers
and of the dangling chains, can be observed in Fig. 4. The classical dendrimer () shows a typical, non-scaling behavior of the relaxation part of the spectrum, both for
and for . Increasing the relaxation moduli display a mixture of dendrimer and chain-like behavior. For short chains we observe in
a second minor peak at lower frequencies, which, for longer chain
lengths becomes less and less pronounced, eventually remaining visible only as a shoulder. For long chains both
are dominated by the chain like relaxation behavior of the white beads.
Figure 4: The storage and the loss moduli for a dendrimer of generation , when the number of spacers between the branching points grows from to . In all the situations .