6. DENDRIMERS:
6.3. G ENERATION DEPENDENCE OF THE DENDRIMER CONFORMATION
Small angle X-ray scattering data of all dendrimers are obtained at low polymer concentrations (≈ 2wt%) at pH = 8.5. Since further dilution had no effect on the scattering intensity profile, the shown scattering data represent dendrimer form factors F(q). Any influence of inter-particle interaction can be considered to be negligible at those concentrations.
In Figure 6-5a, scattering intensities are plotted versus the dimensionless, generalized variable qRg. This allows for direct comparison of generation dependent changes in the general shape, independent of particle dimensions. The commonly used logarithmic representation ensures a very sensitive visualization over a broad qRg range. Successive data sets are separated by a multiplicative factor of 10 for the sake of clarity. Recorded scattering intensities are fitted by the fuzzy surface model developed above. Results are listed in Table 6.2.
The form factor F(q) shows the development of oscillations for high q values with increasing generation number. For PAMAM dendrimers generation 6 and 8 secondary and ternary (only G = 8) maxima are clearly visible. As an example, the two different contributions Fblob and Fshape to the overall scattering are separately plotted for PAMAM generation 8 dendrimers in Figure 6-5b. The shape contribution Fshape(q) obtained from the density profile of a fuzzy sphere is dominating the scattering at low q values. Furthermore, Fshape(q) is reflecting the typical behavior expected for spherical objects with the occurrence of higher order maxima in the high q region.
Leaving all four fit parameters adjustable, it is important to emphasize that the fuzzy sphere model presented above describes experimental scattering profiles in a very satisfactory way. Initially, it is important to briefly discuss mutual effects of the fit parameters used. The correlation length ξ has been introduced as a fit parameter of the blob scattering term Fblob(q) in above model, whereas the parameters R and σ are included in the contribution resulting from the overall shape Fshape(q). The effect of ξ on R and σ is negligible, since Fblob(q) and Fshape(q) are dominant on different length scales and therefore in different q regions. However, ξ and arel are mainly affecting one another. The model derived above is most sensitive to distinguish between effects arising from the blob scattering and the detailed character of the surface, respectively, for data obtained from dendrimers of highest generations (PAMAM 6 and 8).
Concerning in particular scattering profiles obtained from PAMAM dendrimers
6. Dendrimers
Figure 6-5: (a) Double-logarithmic plot of scattering intensities obtained from aqueous dendrimer solutions (pH = 8.5) versus the generalized variable qRg. Data are separated by a constant factor of 10 for better visibility.
Solid lines represent fits to the data using the fuzzy surface model.
Positions of minimum intensity are annotated by dashed lines. (b) Comparison of modeling and experimental data for PAMAM generation 8 in double-logarithmic representation. The two contributions Fblob and Fshape are shown.
generation 6 and 8, the sensitivity of the fit routine can be even more increased when considering only q ranges including the intermediate drop off and the second local maximum (q = 0.6-1.5nm-1). Separate analysis of data collected from PAMAM 6 and 8 at different pH conditions are initially analyzed in this q region, leaving all four parameters adjustable. The obtained results show that the correlation length ξ is not changing significantly or systematically around an average value of ξ = (1.54 ± 0.09)nm. In what follows, the correlation length is kept fixed to 1.54nm, thereby reducing the amount of fit parameters to three (R, σ, arel). This finding is supported by data on PAMAM dendrimers of high generations presented in literature, where the correlation length has been determined to be ξ ≈ (1.56 ± 0.1)nm, however without considering pH.160, 178
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Table 6-2: Summary of the results obtained from fitting the SAXS data (pH = 8.5).
The minima in intensity at qminRg, as shown in Figure 6-5, are most sensitive to the size of the particles. For spheres, values of
(
+)
⋅ = ∈ℵ= n n
R
qmin g 3/5 2 1 π/2 3.65,6.08,8.52,...; (6-14) are expected (Mie scattering).160 Measured values of qminRg = 3.62, 6.09, 8.52 are in very good agreement with the expected values. The fact that the minima are more pronounced with increasing generation number indicates a tendency of the molecules to be become more compact and spherical having sharper boundaries.159, 160, 179 For smaller dendrimers, the radii of gyration Rg are additionally obtained by Guinier fits (equation (6-6)) of the low q region (Table 6.2). Interestingly, the ratio of the radius of gyration and the particle radius, Rg/R, varies between 0.772 (PAMAM 3) and 0.81 (PPI 4). This is close to the theoretical value of 5/3 ≈0.775 expected for an ideal, homogenous sphere.
According to equation (6-2), the molecular weight MW of a dendrimer is proportional to 3
2G+3 − for PAMAM dendrimers and 2G+2 −3 for PPI dendrimers. Assuming a constant density in the core region r ≤ R, R is expected to obey to the following relation since the volume is proportional to the molecular weight:
( )
Fitting experimental data, R obtained for both types of dendrimers follows above dependence very well (Figure 6-6a), independent of the chemical character of dendrimers. The relation R~MW1/3 is indicative of a compact (space-filling) dendrimer structure with a fractal dimensionality of approximately 3. This scaling exponent is in excellent agreement with experimental160 and theoretical results.43, 156
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Figure 6-6: (a) Dependence of the dendrimer radius R on the generation number G.
Solid lines represent fits to the data according to equation (6-15). (b) σ/R obtained from modeling the dendrimer surface fuzziness in dependence of the generation number G.
In Figure 6-6b, the ratio σ /R of the width of the surface region relative to the radius of the corresponding solid sphere is plotted in dependence of the generation number. σ/R decreases significantly with increasing G. Consistently, this again reflects the tendency of the molecules to extend their homogenous interior with increasing generation number.
Figure 6-5b shows that at high scattering vectors, the dendrimer form factor is significantly determined by the superposed contribution Fblob(q) from density inhomogeneities on length scales smaller than the correlation length ξ. Accordingly, Fblob(q) leads to a smearing out of higher order maxima stemming from Fshape(q).160 Fblob(q) is expected to result in a typical power-law dependence q-5/3. The exponent can be derived from the Flory–Huggins parameter ν analogous to that obtained from a semi-dilute polymer solution under good solvent conditions.
In Figure 6-7, scattering intensities of different dendrimers are shown for high scattering vectors in a double-logarithmic representation. The limiting power-law behavior of dendrimers is still a matter of discussion in literature. From star polymers a q-5/3 power-law behavior is known,180 while particles with sharp interfaces between particle and solvent exhibit q-4 power-law behavior.181
6. Dendrimers
Figure 6-7: Double-logarithmic representation of the power-law dependence in the high q region. Black lines represent linear fits to the experimental data.
q-5/3 and q-4 power-law decays are shown for comparison as gray lines.
A transition from a q-4 to q-5/3 power-law decay with decreasing dendrimer generation for the high q region, which is accompanied by a disappearance of higher order maxima, is reported from SAXS analysis of PPI and PAMAM dendrimers.159, 182 Accordingly, a transition from a rather compact, spherical shape of high-generation dendrimers to a much looser, star-like shape of low-generation dendrimers is assumed.159, 182 Contrary to these findings, Rathgeber et al. observe a q-5/3 power-law decay for PAMAM dendrimers independent of generation number.160, 174
Experimental data shown in Figure 6-7 exhibit a slight progression in power-law behavior with generation number. This finding is valid for PPI as well as PAMAM dendrimers. Exponent values range from -1.7 for dendrimers of generation 3 to -2.1 for dendrimers of generation 8. Observed changes in limiting power-law exponent with generation number are small compared to SAXS results presented in reference 159 and 182. Supported by the appearance of oscillation in the high q region with increasing generation number, which are typical for spherical objects, these experimental findings could be interpreted as resulting from a conformational transition towards a more compact shape as this has been done in references 159 and 182. However, such a conclusion is not mandatory, as is demonstrated in what follows.
In Figure 6-8, values obtained for the relative weighing factor of the two scattering contributions arel are plotted in dependence of generation number in semi-logarithmic representation. arel can be consider in analogy to considerations of Richter et al. for star polymers183 as the ratio of intensities arel = I(1/ξ)/I(0) of the two scattering contributions. I(1/ξ) is the scattering intensity on length scale q ≈ 1/ξ of the blob size.
6. Dendrimers
Figure 6-8: Relative weighing factor arel of the two scattering contributions as a function of the generation number, G. Solid lines represent fits to the data according to equation (6-16).
At these high q values, an incoherent superposition of the coherent scattering from within each blob occurs. Accordingly, the scattering intensity can be expressed as I(1/ξ) ~ NblobNseg2. Nblob is the number of blobs and Nseg is the number of segments per blob. In the Guinier regime at q = 0, scattering contributions of all segments add up coherently and the scattering intensity of the whole molecule becomes proportional to I(0) ~ (NblobNseg)2= Ntot2 with Ntot being the total number of segments in the whole dendrimer. arel is therefore given by
i.e. arel is the ratio of the number of scatterers in one blob to the total number of scatterers Ntot given by equation (6-2). Since the blob size does not depend on the generation of dendrimers, it is reasonable to assume that the number of segments within a blob Nseg is independent of generation number, too. Fits to the experimental data according to above equations show good agreement within the experimental accuracy (Figure 6-8). The number of segments within a blob is determined to be NsegPAMAM ≈27 and NsegPPI ≈32 for PAMAM and PPI dendrimers, respectively. This is in good
6. Dendrimers
agreement with small angle scattering results reported in literature for PAMAM dendrimers (Nseg ≈ 22).160
It is important to notice that arel increases significantly with decreasing generation number for both types of dendrimers. This indicates that the limiting power-law behavior is increasingly dominated by scattering from the internal, loose polymeric structure expressed by Fblob leading to a q-5/3 power-law behavior. Therefore, vanishing of higher-order maxima in the sphere form factor is not necessarily a consequence of changes in the overall shape of dendrimers and therefore it is not a sensitive measure for potential structural changes.