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Analysis of the Spatial Dimensions of Fully Aromatic Dendrimers.

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Dense-Shell Dendrimers
Analysis of the Spatial Dimensions of Fully
Aromatic Dendrimers
Sabine Rosenfeldt,* Nico Dingenouts,
Dominic Ptschke, Matthias Ballauff,
Alexander J. Berresheim, Klaus M"llen, and
Peter Lindner
Dendrimers present a fascinating class of molecules because
of their regularly branched structure and their unusual
properties.[1–4] Most of the dendrimers described so far are
composed of flexible repeating units. Such a molecule can
assume a vast number of conformations that are generated by
rotation about the bonds between these units (Figure 1 b). In
particular, the terminal groups can fold back into the interior
of the molecule. A totally different situation arises when
considering the stiff, fully aromatic dendrimer G4-M (Figure 1 a): In this case the benzene rings are either linear or
form an angle of 608 or 1208 to each other. Therefore, rotation
about the bonds between the rings cannot change the
structure of the molecule profoundly. Most importantly, no
backfolding of the terminal groups is possible in such a
The structure shown in Figure 1 b suggests a “dense-shell”
structure of flexible dendrimers, which consists of a hollow
core closed by a densely packed shell of end groups. Such a
dendritic box could be used, for example, as a molecular
container. Hence, many ideas for possible applications of
dendrimers originated from this picture.[1–3] These ideas were
further encouraged by an early publication by de Gennes and
Hervet in which the first theoretical treatment of such denseshell structures was presented.[5] Subsequent theoretical work,
however, has demonstrated convincingly that flexible dendrimers exhibit their maximum density in the core of the
molecule (“dense-core model”) and that there is a finite
probability for the end groups to be located at any position
within the molecule.[6–9] Experimental studies by small-angle
neutron scattering (SANS[10]) have fully corroborated this
result (see the review of this work in reference [11]). In
[*] Dr. S. Rosenfeldt,+ Dr. N. Dingenouts, D. Ptschke,
Prof. Dr. M. Ballauff +
Universit%t Karlsruhe
Kaiserstrasse 12, 76128 Karlsruhe (Germany)
Fax: (+ 49) 721-608-3153
Dr. A. J. Berresheim, Prof. Dr. K. M<llen
Max-Planck-Institut f<r Polymerforschung
Postfach 3148, 55021 Mainz (Germany)
Dr. P. Lindner
Institut Laue-Langevin
B.P. 156X, 38042 Grenoble Cedex (France)
[+] Present Address:
Physikalische Chemie I
Universit%t Bayreuth
95440 Bayreuth (Germany)
Angew. Chem. Int. Ed. 2004, 43, 109 –112
particular, the distribution of end groups in a flexible fourthgeneration dendrimer was determined experimentally for the
first time by use of SANS.[12]
Thus a change of paradigm has occurred, as the previous
assumption of the dense-shell model has been replaced by a
model based on well-established results of polymer physics.
Herein we elucidate this point further by analyzing the spatial
structure of the stiff, fully aromatic dendrimer G4-M.[12–17] It
has been demonstrated by SANS[11, 18–22] that such a stiff
dendrimer has a dense-shell structure as expected. In
accordance with results obtained by NMR spectroscopy[17] it
is shown that a high segment density must result in the
periphery of the molecule if the number of generations is high
The analysis employed herein consists of measuring the
SANS intensities I0(q) at different contrast. This method,
termed contrast variation, has been applied frequently to
colloidal systems.[10, 11] Hence, it suffices to delineate the main
points of this analysis:[11, 24] The intensity I0(q) of an isolated
molecule dissolved in a suitable solvent is governed by the
contrast 1̄1m, where 1̄ is the average scattering-length
density of the molecule and 1m is the scattering-length density
of the solvent. The scattering length defines the strength of
the scattered radiation emanating from a given atom or
segment. Since the scattering intensity is always related to the
corresponding scattering volume, the scattering-length densities are the decisive quantities. The quantity q is the
magnitude of the scattering vector and can be calculated
from the scattering angle q and the wavelength l of the
radiation employed (q = (4p/l)sin(q/2)).[10] The contrast in
the SANS experiments can be changed by using mixtures of
deuterated and nondeuterated solvents, because the scattering-length density of the solvent depends strongly on the
degree of deuteration.[10]
The average scattering-length density 1̄ can be obtained
directly from experimental data because I0(q) can be approximated by the Guinier law for small scattering angles, that is,
small q values [Eq. (1)]. In this equation, N/V is the number of
I0 ðqÞ ffi
N 2
V pð
11m Þ2 exp q2
dissolved molecules, Vp their volume, and Rg the radius of
gyration.[10, 11] The volume Vp can be used to calculate the
molecular weight M according to M = NL Vp v̄, where v̄ is the
partial specific volume determined experimentally.[18–20] First,
however, the experimental scattering intensities must be
extrapolated carefully to vanishing concentration, and the
incoherent contribution always present in neutron scattering
must be subtracted before Equation (1) is applied.[22]
Figure 2 shows the Guinier plot of the SANS data for the
dissolved dendrimer G4-M according to Equation (1). The
curves differ with respect to the contrast 1̄1m obtained
through mixtures of deuterated and nondeuterated toluene. It
is evident that the extrapolation to q = 0 provides no
difficulty, so that Vp (= 30.3 nm3) and 1̄ (= 2.42 = 1010 cm2)
can be obtained. From these data and the measured specific
volume the molecular weight M was determined to be
22 250 g mol1, which is in agreement, within experimental
DOI: 10.1002/anie.200351810
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 1. a) The stiff dendrimer G4-M.[12, 13] b) A flexible dendrimer that has been studied by small-angle neutron scattering.[8, 18, 21–23]
dendrimer G4-M is displayed in Figure 3. A weak side
maximum is visible at about 1.7 nm1, which already points
to a well-defined structure of G4-M.
Figure 2. Guinier plot[10] [see Eq. (1)] of the scattering intensities
obtained at different contrast. The contrast between the dissolved dendrimer G4-M and the solvent was adjusted through the use of mixtures
of deuterated and nondeuterated toluene. The ratio of the deuterated
to nondeuterated toluene is given. All data have been extrapolated to
vanishing concentration, and the incoherent background has been
error ( 600 g mol1), with the calculated value
(22 927 g mol1) and the value determined by mass spectrometry. Moreover, the radius of gyration Rg, which also depends
on contrast, can be determined from this plot and extrapolated to infinite contrast by an appropriate procedure.[11] For
the dendrimer G4-M Rg turns out to be 2.6 nm.
If 1̄ is known, the measured scattering intensity I0(q) can
be expanded in powers of the contrast 1̄1m
[Eq. (2)],[11, 18, 20, 21] which provides the partial intensities
I0 ðqÞ ¼ ½
11m 2 IS ðqÞ þ 2 ½
11m ISI ðqÞ þ II ðqÞ
IS(q), ISI(q), and II(q). The first term IS(q) contains information on the spatial structure of the dissolved molecule. In
contrast, the second and third terms are related to the
variation of the scattering-length density within the molecule[25] and therefore cannot be used for analyzing the shape
of the molecule. The resulting partial intensity IS(q) of the stiff
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 3. Modeling of the partial scattering intensity IS(q) (triangles)
obtained from the experimental SANS data by contrast variation. Solid
lines: fit obtained by the model described in Figure 4. Inset: number
of spheres N(r) located within the distance r from the center of the
molecule. The distribution of the end groups is shown by a dashed
The transformation of IS(q) into real space can not be
done by the method employed for flexible dendrimers as
shown in Figure 1 b. Simulations demonstrated that the direct
Fourier transformation of IS(q) only leads to correct results
for fully flexible dendrimers, that is, if the structure is
averaged over a sufficiently high number of conformers.[8, 9]
The dendrimer G4-M under consideration in this case,
however, hardly has any flexibility as a result of its rigid
structure. Moreover, it has no radial symmetry.[26]
Therefore, IS(q) was analyzed by the model shown in
Figure 4. In general, small-angle scattering does not have
atomic resolution and parts of the molecule such as the
benzene rings can be rendered in a suitably coarse-grained
model when evaluating IS(q). Thus the central biphenyl unit is
represented by two spheres, and each of the four dendrons
emanating from this central unit is modeled by eight spheres
Angew. Chem. Int. Ed. 2004, 43, 109 –112
(Figure 4). Hence, each sphere represents five benzene rings.
The distance between the centers of these spheres must
approximately be the dimensions of two benzene rings, and
Figure 4. Model used to determine the partial scattering intensity IS(q)
[Eq. (2)]. The molecular structure of the dendrimer is represented by
spheres connected by freely rotating bonds. The central biphenyl unit
is modeled by two small spheres (diameter: 0.4 nm). Each dendron
emanating from this center is modeled by eight spheres whose
distance (0.87 nm) and radius (0.8 nm) are treated as fit parameters in
the simulation.
this parameter therefore determines the overall dimensions of
the molecule. Since each sphere must provide the scattering
power of five benzene rings, the diameter was increased
appropriately. Hence, neighboring spheres overlap. For the
sake of clarity, however, this overlap is not shown in Figure 4.
Finally, the bonds between all spheres can be rotated freely.
The scattering intensity can now be calculated as follows:
Different conformers are generated by a set of torsional
angles chosen at random. If a given torsional angle leads to an
overlap between two spheres that do not belong to the same
dendron, this angle is dismissed and a new one is generated.
To obtain sufficient accuracy, about 500 conformers were
generated in this way. The scattering intensity IS(q)conformer of
each conformer results from the scattering contribution of all
the spheres and their positions in the conformer according to
Equation (3),[10] where dij denotes the distance between the
IS ðqÞconformer ¼
Ai ðqÞ Aj ðqÞ
sin q dij
q dij
spheres i and j with radius R in the corresponding conformer.
The scattering amplitudes Aj(q) of the spheres are given by
Equation (4).[10] The average over all conformations results
Ai ðqÞ ¼
4p 3 sin q Rq R cos q R
from the arithmetic mean of all IS(q)conformer values calculated
in this way. The only fit parameters are the distance of the
spheres and their radius R [see Eq. (4)].
From this modeling of IS(q) (optimal fit, solid line in
Figure 3) the distance between the spheres d turns out to be
0.87 nm and the radius R is 0.8 nm. These values lead to full
agreement between theory and experiment within the limits
of error. The distance between the spheres corresponds
approximately to the length of a biphenyl unit. The diameter
Angew. Chem. Int. Ed. 2004, 43, 109 –112
of the spheres (1.6 nm) fits well to the average dimensions of
the subunits, which consist of five benzene rings each.
The inset in Figure 3 shows the number N(r) of spheres
located within distance r from the center of the molecule.
While the first generation is still strictly localized, higher
generations are spread out over larger distances due to the
averaging of different conformers. Therefore, SANS shows
that the dendrimer G4-M has a stiff structure in which the
segments have a well-defined distance from the center.
The above conclusions derived from SANS measurements
were corroborated by an investigation by Saalw?chter and coworkers[17] by using solid-state NMR spectroscopy. In this case
the marked slowing of the terminal phenyl groups could be
proved directly and traced back to their mutual steric
hindrance. Moreover, this study demonstrated that the
dendrons can not reorient even at high temperatures. This
again points to a strong mutual steric hindrance between the
The conclusions drawn from the present work are related
to our basic understanding of dendrimers and their molecular
shape: The dendrimer G4-M exhibits a molecular structure
that can be described by the dense-shell model hitherto
applied to all dendrimers. Such a stiff shape, however, in
which all end groups are located in the periphery of the
molecule (see Figure 3, inset) can only be realized by a
concomitantly stiff chemical structure. Flexible dendrimers,
on the other hand, are characterized by a segmental
distribution that exhibits its maximum in the center of the
molecule.[11, 18–23] in such a structure the end groups can fold
back easily.[18] Consequently, flexible dendrimers exhibit the
dense-core structure discussed above that was predicted by
theory quite some time ago. On the other hand, dense-shell
dendrimers can only be formed from stiff units, as shown for
the dendrimer G4-M.
Experimental Section
The dendrimer was synthesized as previously described.[12, 13] Toluene
(Merck, p.a.) and deuterated toluene (C7H8 ; Euriso-top, degree of
deuteration: 99.96 %) were used as received. The specific volume
((0.82 0.04) cm3 g1) of G4-M was determined by using a DMA-60
densitometer (Paar, Graz, Austria). All SANS data were obtained by
using the beamline D11 of the Institut Laue-Langevin in Grenoble.
The data were corrected to obtain the absolute intensities by use of
the software directly available at the beamline. The extrapolation of
the data to vanishing concentration and the subtraction of the
incoherent background were carried out as described in reference [22].
Received: May 6, 2003 [Z51810]
Keywords: dendrimers · small-angle neutron scattering ·
structure elucidation
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2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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[25] Both terms need to be subtracted carefully before evaluating the
scattering intensity to deduce the spatial structure. This problem
is of central importance when analyzing dendrimers, which are
rather small objects. The analysis of I0(q) without this correction
may lead to incorrect conclusions.[11, 24]
[26] Central to this analysis is the description of flexible dendrimers
by an average segment density termed T(r), which is of central
symmetry and leads to IS(q) upon Fourier transformation.[8] For
stiff systems, however, the deviations of an averaged intensity
would be too high, so that no meaningful description of IS(q)
would be possible. This can be seen from the fact that the
description through an averaged T(r) would lead to pronounced
minima (zeros) of I0(q). The experimental scattering intensity,
however, exhibits only a weak side maximum (see Figure 3).
Moreover, this demonstrates that G4-M does not have radial
symmetry, and models that assume distributions with central
symmetry can not be applied.
2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Angew. Chem. Int. Ed. 2004, 43, 109 –112
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