6. DENDRIMERS:
6.2. X- RAY DIFFRACTION OF DENDRIMER SOLUTIONS
Molecular sizes, shapes, and intermolecular interferences contribute to scattering curves obtained from a polymer solution. According to equation (2-7), the experimentally recorded scattering intensity I(q) consists of contributions of the form factor of the scattering entities, F(q), and the structure factor, S(q). The latter accounts for inter-particle interactions. For sufficiently diluted solutions, inter-molecular correlations can effectively be ignored and S(q) approaches unity. Thus, it becomes possible to obtain the scattering intensity at scattering vectors q, whose coherent part is directly proportional to the form factor of the observed macromolecules,
) ( )
(
~
0
q NF q
I
c→
. (6-3)
For semi-dilute solutions however, S(q) ≈ 1 is no longer true for very low q values leading to a slight depression of I(q). The advantage of experiments in the semi-dilute regime over a more concentrated one is that inter-particle interactions, which affect changes in the scattering function, are unlikely to exist. Therefore, the form factor F(q) is almost unchanged but the quality of the scattering data and is simultaneously
6. Dendrimers
increased due to the higher concentration. Solutions with a polymer mass fraction of 1% can be considered to be in the dilute regime, while mass fractions of 5% exhibit a small downturn at low-q with no significant effect for q > 0.5nm-1.159
The overall particle size is typically expressed by the radius of gyration Rg. Rg gives measure of the mean square distance of scattering centers from the molecule’s center of mass,
In the above equation, ri, i = 1…N, denote the positions of the N monomers comprising the polymer. r is the position of the center of mass. The brackets denote averaging over all measurable microscopic states, each weighted with the corresponding Boltzmann factor for the energy of the instantaneous configuration. For spherical particles, the radius of gyration is connected to the radius of the spheres R according to the following equation:167
Rg
R 3
= 5 (6-5)
The most common methods for determining the radius of gyration are graphical methods. There are primarily two graphical methods known as Guinier and Zimm plot, respectively.168 Zimm fits become accurate for systems that can be described as long Gaussian chains. Guinier fits represent a method suited for objects, which can be described as spheres. Based on the dendrimer molecular architecture, it is reasonable to assume that dendrimers are rather spherical objects. Therefore, their particle scattering factor in the low q limit (qRg << 1) can be approximated by the Guinier approximation for the scattering intensity69, 159, 167 with the functional form:
(
/3)
exp )
(q a1 a2 q2Rg2
I = + − . (6-6)
Performing fits to scattering profiles in the low q limit (qRg << 1) with above equation yields the radius of gyration Rg. a1 and a2 are used to account for the absolute scale and the baseline of experimental data.
However, this way of determining Rg is limited when experimental data for sufficient low q values are not available. In this study, the smallest accessible region of the particle scattering term varies from qminRg ≈ 0.22 for PPI dendrimers generation 3 to qminRg ≈ 0.9 for PAMAM dendrimers generation 8. Therefore, estimating Rg by the Guinier approximation, which is based on expansions around q = 0, is assumed to be increasingly inappropriate with increasing dendrimer size. Consequently, the Guinier
6. Dendrimers
approximation is only used for the smaller dendrimers PPI generation 3, 4 and PAMAM generation 3. For dendrimers of higher generations, a different approach has to be used, which is described below.
Owing to their precise architecture, dendrimers are monodispersely synthesizable. The degree of dendrimer polydispersity has been measured by atomic force microscopy to be less than 1.08 for generations 5-10.169 Accordingly, no polydispersity of the system has to be considered. Different dendrimeric entities are contributing to the dendrimer scattering spectra at q ranges corresponding to their length scales. The overall dendrimer shape on larger length scales determines the scattering intensity in the low q region. Beside this, the loose, polymeric character of dendrimers gives rise to internal density variations with a correlation length ξ. The so-called “blob scattering”
originating from these contributes significantly to the overall scattering intensity at larger scattering vectors. Accordingly, the dendrimer form factor F(q) can be described by the sum of scattering arising form the overall shape, Fshape(q), and the blob scattering contribution, Fblob(q):170
( )
q F( )
q a F( )
qF = shape + rel blob . (6-7)
arel gives the relative weight of the two terms.
F(q) is the Fourier transform of the density correlation function γ(r). Taking the sum of both terms in equation (6-7) instead of their convolution is strictly valid only if the q ranges of both scattering contributions are clearly separated, allowing to neglect interference terms. According to the model introduced by Beaucage,171 this can be ascertained by replacing q by q in Fblob(q):
( )
(
erf qRg / 6)
3q = q . (6-8)
This results in a cut-off to low q values, where shape contributions become dominant.
A so-called ‘blob’ is a spherical volume with a radius ξ. Within a blob, sections of the dendrons have to be described as self-avoiding (sub) walks and excluded volume interaction has to be taken into account. Accordingly, scattering from length scale r ≤ ξ is therefore analogous to that of a dilute polymer solution. Other than in semi-dilute solutions, where the blob size depends only on the polymer concentration, in the Daoud–Cotton model,172 which was originally developed for star polymers, it is postulated that ξ is expected to increase with radial distance from the center. However in the q range of interest, contributions from the largest, outermost blobs should significantly dominate the scattering signal. Therefore, the r dependence of ξ can be neglected.160
6. Dendrimers
Figure 6-3: Schematic representation of the two contributions Fshape and Fblob
stemming from the overall dendrimer shape and from the loose polymeric character of dendrimers, respectively.
Fblob(q) describes density variations on length scales smaller than the correlation length ξ of internal density variations. On length scales larger than ξ, γ(r) is equal to zero.
Determining Fblob(q), the integration in equation (6-9) is therefore limited to r ≤ ξ,
( )
= π∫
ξ γThe density correlation function can be described by a power-law dependence, whose exponent can be derived from the Flory-Huggins parameter ν = 3/5 for good solvent conditions, which are fulfilled for the DNA-water system:173
1
In order to obtain an analytic expression for Fblob(q), the factor exp(-r/ξ) is introduced into the integral in equation (6-9) as cut-off and the integration is extended to infinity.
Taking into account equations (6-8), (6-9), and (6-10), the following expression can be derived for the scattering contribution of internal density variations:160
( ) ( ( ) )
According to the discussion of dendrimer properties in chapter 6.1, the dendrimer density profile is modeled by a convolution of a homogeneous sphere with radius R and a Gaussian distribution with standard deviation σ (Figure 6-4). Using the fact that the Fourier transformation splits a convolution into the product of the Fourier transform of the multipliers, the scattering amplitude Ashape(q) can be expressed as follows:174
6. Dendrimers
Figure 6-4: Modeled dendrimer density profile.
( ) ( ) ( )
ρ is the dendrimer segment density profile. The form factor can be calculated from the scattering amplitude according to the relation
( ) ( ) ( ) [ ( ) ( ) ]
⎟⎟In summery, the model for fitting the scattering intensity of dendrimer solutions described above includes four adjustable parameters: the sphere radius R, the width of the smeared surface region determined by the standard deviation σ of the assumed Gaussian distribution, the correlation length ξ of density variations, and the relative weighting factor arel. Analog models accounting for the contribution of the loose, polymeric character as well as the overall compact shape have been already successfully applied to dense polymer systems such as star polymers,170, 175, 176 diblock-copolymer micelles,177 and dendrimers.160, 174, 178
6. Dendrimers