Simplified Polydispersion Analysis of Small-Angle Scattering Data
<p>SDFs used to simulate the SCSs of polydisperse inhomogeneities; <math display="inline"><semantics> <msub> <mi>N</mi> <mi>o</mi> </msub> </semantics></math> are normalized to ensure a volume fraction of 1%; <math display="inline"><semantics> <msub> <mi>R</mi> <mi>o</mi> </msub> </semantics></math> = 20 Å; four values of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> are considered (0.1, 0.2, 0.3, and 0.4).</p> "> Figure 2
<p>Simulated SCSs of a lognormal SDF of spherical inhomogeneities; <math display="inline"><semantics> <msub> <mi>N</mi> <mi>o</mi> </msub> </semantics></math> are normalized to ensure a volume fraction of 1%; <math display="inline"><semantics> <msub> <mi>R</mi> <mi>o</mi> </msub> </semantics></math> = 20 Å; four values of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> are considered (0.1, 0.2, 0.3, and 0.4). The Guinier and the Porod approximations are present in the investigated Q range.</p> "> Figure 3
<p>Guinier approximation of the simulated SCS by using the SDF with <math display="inline"><semantics> <mi>σ</mi> </semantics></math> = 0.2. The points used for the Guinier approximation are shown with black symbols, while the others are red. The fit is shown with a red line.</p> "> Figure 4
<p>Porod approximation of the simulated SCS by using the SDF with <math display="inline"><semantics> <mi>σ</mi> </semantics></math> = 0.2. The points used for the fit are shown with black symbols, while the others are red. The fit is shown with a red line.</p> "> Figure 5
<p>SDFs reconstructed with the parameters (<math display="inline"><semantics> <msub> <mi>N</mi> <mi>o</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>R</mi> <mi>o</mi> </msub> </semantics></math> and <math display="inline"><semantics> <mi>σ</mi> </semantics></math>) calculated with the SPA for the four cases (<math display="inline"><semantics> <mi>σ</mi> </semantics></math> = 0.1, 0.2, 0.3 and 0.4). The original SDFs are shown in <a href="#applsci-12-10677-f001" class="html-fig">Figure 1</a>.</p> "> Figure 6
<p>Guinier approximation of the simulated SCS by using the SDF with <math display="inline"><semantics> <mi>σ</mi> </semantics></math> = 0.4 and an incoherent background of 1.0 × 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> cm<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. The points used for the Guinier approximation are shown with black symbols, while the others are red. The fit is shown with a red line.</p> "> Figure 7
<p>Porod approximation of the simulated SCS by using the SDF with <math display="inline"><semantics> <mi>σ</mi> </semantics></math> = 0.4 and an incoherent background of 1.0 × 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> cm<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>. The points used are shown with black symbols, while the others are red. The fit is shown with a red line.</p> "> Figure 8
<p>SDFs reconstructed with the parameters (<math display="inline"><semantics> <msub> <mi>N</mi> <mi>o</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>R</mi> <mi>o</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <mi>σ</mi> </semantics></math>) calculated with the SPA for the case <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math> without (red line) and with additional backgrounds of 1.0 × 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math> (B1—blue line), 1.0 × 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math> (B2—green line), and 1.0 × 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math> cm<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> (B3—yellow line). The original SDF is also shown for comparison (black line).</p> "> Figure 9
<p>Analysis of experimental SCS of Ludox silica particles HS30 with a 0.3% volume fraction: (<b>a</b>) The SPA with the fits of the Guinier (red line), as well as of the Porod (blue line) approximations along with the values of the optimized parameters and their uncertainties. The points used for both fits are shown with black symbols, while the others are red. (<b>b</b>) A fitting procedure with a Weibull SDF, where corrections for the wavelength spread (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mo>/</mo> <mi>λ</mi> </mrow> </semantics></math> = 18%) and for the multiple scattering were applied [<a href="#B29-applsci-12-10677" class="html-bibr">29</a>].</p> "> Figure 10
<p>Ludox silica particles HS30 with a 0.3% volume fraction: the Weibull SDF (blue line) obtained with the fitting procedure, with corrections for the wavelength spread (<math display="inline"><semantics> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mo>/</mo> <mi>λ</mi> </mrow> </semantics></math> = 18%) and for the multiple scattering, and the lognormal SDF (red line) calculated with the SPA.</p> ">
Abstract
:1. Introduction
2. SAS Theoretical Background
- •
- The inhomogeneities, as well as the sample matrix, or solvent, are homogeneous: the so-called two-phase system.
- •
- The scattering is isotropic, and the SCS depends on the modulus of the scattering vector Q.
- •
- The system is diluted, i.e., the concentration of the inhomogeneities is so low that the coherence between neutrons scattered by different inhomogeneities is negligible
2.1. Monodisperse Inhomogeneities
2.2. Polydisperse Inhomogeneities
2.3. Spheres
3. Lognormal Distribution Function
4. Simplified Polydispersion Analysis
5. Simulations
Background
6. SANS Experiment
7. Conclusions
- •
- Two-phase system;
- •
- Isotropic scattering;
- •
- Sharp interfaces of the inhomogeneities;
- •
- The Q-range includes both the Guinier and Porod approximations;
- •
- Presence of one family of inhomogeneities, described by a lognormal SDF;
- •
- The presence formalism has been developed for spherical inhomogeneities; however, it can be extended to other shapes.
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Abbreviations
ED | electron density |
RID | refraction index density |
SAS | small-angle scattering |
SCS | scattering cross-section |
SDF | size distribution function |
SLD | scattering length density |
SPA | simplified polydispersion analysis |
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Carsughi, F. Simplified Polydispersion Analysis of Small-Angle Scattering Data. Appl. Sci. 2022, 12, 10677. https://doi.org/10.3390/app122010677
Carsughi F. Simplified Polydispersion Analysis of Small-Angle Scattering Data. Applied Sciences. 2022; 12(20):10677. https://doi.org/10.3390/app122010677
Chicago/Turabian StyleCarsughi, Flavio. 2022. "Simplified Polydispersion Analysis of Small-Angle Scattering Data" Applied Sciences 12, no. 20: 10677. https://doi.org/10.3390/app122010677