Selected SAXS data from PAs 5–8 at 0.25% in 4 mM.
Size distribution is an important structural aspect in order to rationalize relationship between structure and property of materials utilizing polydisperse nanoparticles. One may come to mind the use of dynamic light scattering (DLS) for the characterization of the size distribution of particles. However, only solution samples can be analyzed and even for those, the solution should be transparent or translucent because of using visible light. It is needless to say that solid samples are out of range. Furthermore, the size distribution only in the range of several tens of nanometers can be characterized, so DLS is useless for particles in the range of several nanometers. Therefore, the small-angle X-ray scattering (SAXS) technique is much superior when considering the determination of the size distribution in several nanometers length scale for opaque solutions and for solid specimens. Furthermore, the SAXS technique is applicable not only for the spherical particle but also for platelet (lamellar) and rod-like (cylindrical) particles. In this chapter, we focus on the form factor of a variety of nanostructures (spheres, prolates, core-shell spheres, core-shell cylinders and lamellae). Also getting started with a monodisperse distribution of the size of the nanostructure, to unimodal distribution with a narrow standard deviation or wide-spreading distribution and finally to the discrete distribution can be evaluated by the computational parameter fitting to the experimentally obtained SAXS profile. In particular, for systems forming complicated aggregations, this methodology is useful. Not only the size distribution of ‘a bunch of grapes’ but also the size distribution of all ‘grains of grapes in the bunch’ can be evaluated according to this methodology. This is very much contrasted to the case of the DLS technique by which only ‘a bunch of grapes’ is analyzed but ‘grains of grapes in the bunch’ cannot be. It is because the DLS technique in principle evaluates diffusion constants of particles and all of the grains in the same bunch of grapes diffuse as a whole. Thus, the methodology is important to highlight versatility and diversity in real materials, especially in soft matter, both in the liquid and in the solid states.
- form factor
- core-shell sphere
- core-shell cylinder
- discrete distribution function
In recent years, controlling of nanostructures has been more significantly considered in the field of materials science, especially relating to the soft matter . Versatile properties or functions can be obtained through designing nanostructures in solid-state materials, as well as nanomaterials dispersed in liquid-state substance. Even for contradictory properties such as hard and soft, they may be coexistent in one material when fabricating so-called inclined nanostructures (for instance, nanoparticle size is gradually changing as a function of the position in material). This in turn indicates that size distribution of the nanostructures should be rigorously evaluated for better understanding effects of nanostructure on properties and functions. For biological systems or supramolecular organizations, situation is very much contrast to the other ubiquitous materials as described above because they form spontaneously a regular aggregation. Therefore, the size distribution is narrow and follows a simple mathematical function with a comparatively small standard deviation. By contrast, discrete distribution of the size is required to determine for the ubiquitous materials. However, even for regular nanostructures, the determination of the discrete distribution of the nanostructure size is needed to reveal a transient state upon transition from the state 1 to the state 2, being triggered by sudden change in temperature, pH, or other external parameters.
It is well known that the size distribution of particles can be evaluated by the use of dynamic light scattering (DLS). However, only solution samples can be analyzed and even for those, the solution should be transparent or translucent because of using visible light. It is needless to say that solid samples are out of range. Furthermore, the size distribution only in the range of several tens of nanometers can be characterized, so DLS is useless for particles in the range of several nanometers. Therefore, the small-angle X-ray scattering (SAXS) technique is much superior when considering the determination of the size distribution in several nanometers length scale for opaque solutions and for solid specimens . Furthermore, the SAXS technique is applicable, not only for the spherical particle but also for platelet (lamellar) and rod-like (cylindrical) particles and it enables us to determine the thickness distribution of lamellae or the cross-sectional radius distribution of cylinders. Namely, the SAXS technique does not matter types of particle shape even for hallow cylinders or hollow spheres .
The principle is simple. Scattering comprises not only contribution from regularity of space-filling ordering (the lattice factor) of particles but also from a single particle (the form factor). The particle scattering can be mathematically formulated depending on the type of particle shape (lamella, cylinder or sphere). In the block copolymer microdomain systems, the Gauss distribution of the particle size has been assumed. Only recently, direct determination of the discrete size distribution has been available by conducting fitting theoretical scattering function to the experimentally obtained SAXS profile (the plot of the scattering intensity as a function of the magnitude of the scattering vector,
2. Nanoparticles with a narrow size distribution
First of all, some typical examples of the experimentally observed form factor are demonstrated. The samples are self-assembly of proteins, block copolymer microdomains and peptide amphiphiles. Apoferritin is a protein having ability to store iron atoms and it is referred to as ferritin when iron atoms are bound. Apoferritin forms a spherical shell as a self-assembled nanostructure with a very uniform size. As indicated in Figure 1 (pH-dependence of SAXS profiles), its SAXS profiles (apoferritin, 24-mer) exhibit characteristic features with many peaks due to its uniform shape for pH ≥3.40 . Dramatic change in the SAXS profile is detected between pH = 1.90 and 3.40. This means that apoferritin is disassembled for acidic condition. Time-resolved SAXS measurements have been utilized to study disassembling and reassembling process upon the change in pH [4, 5]. In Figure 1, the curve shows the result of the SAXS modeling by the scattering program GNOM . Since protein molecules produce the typical form factor, it is frequently used to obtain commissioning data for newly launched SAXS beamline or apparatus [7–9].
It is known that block copolymer spontaneously forms a regular nanostructure with a narrow size distribution. Figure 2 shows examples of the SAXS profiles for sphere-forming block copolymer (SEBS; polystyrene-block-poly(ethylene-co-butylene)-block-polystyrene triblock copolymer) having Mn = 6.7 × 104, Mw/Mn = 1.04, PS volume fraction = 0.084) , where Mn and Mw denote number-average and weight-average molecular weights, respectively. In Figure 2, the solid curve is the results of the model calculation for the spherical particle, but not only the form factor, but also the lattice factor of BCC (body-centered cubic) is taken into account. The full equation is as follows [11–14]:
where <x> is the average of the quantity of x.
for the bcc lattice and
for the fcc lattice. In Eqs. (3) and (4),
As clearly observed in Figure 2, the broad peak around
Very recently, it has been found that PS spherical microdomains were deformed upon the uniaxial stretching of the SEBS-8 film specimens . Since SEBS triblock copolymer with the glassy PS spherical microdomains can be used as a thermoplastic elastomer (TPE), the film specimen can be stretched. In Figure 4, 2D-SAXS patterns are displayed to recognize the deformation of the round shape form factor upon the uniaxial stretching. Figure 4a shows the 2D-SAXS pattern for the SEBS-8 film specimen. Here, it is clearly observed that the round shape form factor appears at
The results for the 1D-SAXS profiles in
if the homogeneous densities in the core and in the shell can be assumed with
As for core-shell cylinders, the form factor is:
Matson et al.  have reported the SAXS modeling of the form factor of the core-shell cylinder for self-assembling peptide amphiphiles (PAs) as shown in Figure 8A and B. The molecules self-assembled into the core-shell cylinder are illustrated in Figure 8C. Such cylinders can be detected with cryogenic TEM as shown in Figure 9. The SAXS profile is shown in Figure 10 with the model curve, where the size distribution in the core radius is modeled using a log-normal distribution with the polydispersity being around 27–30% (see Table 1 for the structural parameter determined by the SAXS modeling), while the radial shell thickness is assumed to be monodisperse. Although the modeling results explain very well the experimentally obtained SAXS profiles, the fact that the radial shell thickness is assumed to be monodisperse means it is difficult to determine individually two distributions in inner and outer radius. For more detailed structure analyses, more experimental variations are required to gather information from different kinds of aspects, like the example shown in Figure 6a and b (parallel and perpendicular to SD).
|Mean core radius (A)||12||12||12||12|
|Radial shell thickness (A)||29||27||25||25|
|Total diameter (nm)||8.2||7.8||7.4||7.4|
|Radial polydispersity (σ)||0.3||2.7||0.3||0.28|
3. Concept of evaluation of discrete distribution by SAXS
In this section, the concept of evaluation of the discrete distribution of the size of the nanostructure is explained. As an example, the lamellar model calculations are displayed in Figure 11, for which the mathematical equation is :
Figure 12c shows one of the typical results of the SAXS profiles for poly(oxyethylene) (PEG), which forms lamellar crystallites. The exact sample used for the result of Figure 12c was a polymer blend of PEG with poly (D, L-lactide) (PDLLA), which is a racemic copolymer and therefore amorphous. The compositions of PEG/PDLLA were 80/20 (DL20) by weight. Figure 12c shows the result of the SAXS measurement at 64.0°C in the heating process . At the temperature of 64.0°C slightly below the melting temperature of PEG (64.5°C), the typical form factor of lamellar particle was observed first time for the crystalline polymer. It was expected that the thinner lamellar which has a lower melting temperature melted away in the heating process. The thickest lamellae can only survive at the highest temperature and therefore, the thickness distribution became sharp. This may be the reason of the observation of the typical form factor of lamellar particle. As a matter of fact, a very sharp distribution was evaluated as shown in Figure 12d by the method described below.
Hereafter, the data analysis method for the direct determination of the thickness distribution of lamellar particle is described. The model particle scattering intensity,
Thus, the thickness distribution as shown in Figure 12d was also evaluated. Although such a sharp distribution around
4. Widely spread discrete distribution evaluated by SAXS
4.1. Lamellar case
Tien et al. have reported results of comprehensive studies of the higher-order crystalline structure of PEG in blends with PDLLA [22, 24, 25]. For several blend compositions, they have discussed the effects of blending PDLLA on the structural formation of PEG. It is remarkable that they found more regular higher-order structure for PEG 20 wt % composition (DL20) as compared to the PEG 100% sample in the as-cast blend sample (cast from a dichloromethane solution). More interestingly, they reported that the 1D-SAXS profile markedly changed from lattice peak dominant type to particle scattering dominant type when heating the as-cast sample, as shown in Figure 13. The compositions of PEG/PDLLA were 100/0, 95/5 (DL5), 90/10 (DL10) and 80/20 (DL20) by weight. Figure 13 shows the results of the SAXS measurements in the heating process. Based on the results, we have conducted the evaluation of the lamellar thickness distribution in the heating process from the as-cast state up to 64°C and succeeded in showing that the distribution became sharper with the average thickness becoming larger, as shown in Figure 14. That study is the first showing quantitative evidence of the well-known concept of ‘lamellar thickening’ when a crystalline polymer is thermally annealed just below its melting temperature. Tien et al. have also conducted the same evaluation under higher pressure (5 and 50 MPa) .
Jia et al.  have recently evaluated thickness distribution of the hard segment domains for supramolecular elastomers (starblocks of soft polyisobutylene and hard oligo(
4.2. Spherical case
In this subsection, the size distribution of nanoparticles is described. Fujii et al. [28, 29] have synthesized novel for sterically stabilized polypyrrole-palladium (PPy-Pd) nanocomposite particles. Such a characteristic particle containing heavy element has recently been attracting intensively general interests of researchers in many fields under the name of element-blocks . Figure 17 shows a TEM image of these particles with a schematic of the structure. The ordinary 1D-SAXS profiles of 1, 2 and 3% aqueous dispersions of the nanocomposite particles are shown together in Figure 18a as a plot of log[
The form factor,
In Eq. (24),
Bimodal size distribution was clearly obtained with two peaks at approximately
Kuroiwa et al. [31, 32] have synthesized novel amphiphilic
5. Concluding remarks
In this chapter, we focused on the form factor of a variety of nanostructures (spheres, prolates, core-shell spheres, core-shell cylinders and lamellae). Also getting started with a mono-disperse distribution of the size of the nanostructure, to unimodal distribution with a narrow standard deviation or wide-spreading distribution and finally to the discrete distribution can be evaluated by the computational parameter fitting to the experimentally obtained SAXS profile. In particular, for systems forming complicated aggregations, this methodology is useful. Not only the size distribution of ‘a bunch of grapes’ but also the size distribution of all ‘grains of grapes in the bunch’ can be evaluated according to this methodology. This is very much contrasted to the case of the DLS technique by which only ‘a bunch of grapes’ is analyzed but ‘grains of grapes in the bunch’ cannot be. It is because the DLS technique in principle evaluates diffusion constants of particles and all of the grains in the same bunch of grapes diffuse as a whole. Thus, the methodology is important to highlight versatility and diversity in real materials, especially in soft matter, both in the liquid and in the solid states. At present, however, the shape of the nanostructure is limited in spherical or lamellar. On extending the methodology to the complicated structures such as cylinder, prolate, oblate, or core-shell type, there are tremendous difficulties. For cylinder, prolate, or oblate, difference in the degree of orientation of such particles spoils the methodology such that the size distributions for two principal directions (height and radius for the cylinder case/long axis radius and short axis radius for the prolate and oblate cases) cannot be uniquely evaluated. As for core-shell type particles, the inner and outer radii couple to alter its form factor, so that the size distributions for them cannot be uniquely evaluated either. As a matter of fact, the size distribution is introduced with keeping constant of the ratio of the inner and outer radii for the core-shell spheres . Similarly, for the core-shell cylinders , the size distribution in the core radius is incorporated, while the radial shell thickness is assumed to be monodisperse. For more detailed structure analyses, more experimental variations are required to gather information from different kinds of aspects, like the example shown in Figure 6a and b (parallel and perpendicular to SD). These difficulties should be overcome.
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