Choice of Starting Values of Particle-Size Distribution Parameters for Their Calculation from Small-Angle X-ray Scattering Data

Abstract

Simple methods are proposed for determining the starting values of the parameters of particle-size distribution models (mean radius and its standard deviation), calculated from small-angle X-ray scattering curves. Estimates of these parameters from above based on the obtained analytical expression for the Guinier region of the scattering curve from a polydisperse system obeying the Schultz distribution are proposed for systems with narrow distributions. It is proposed to estimate the parameters and range of sizes from below based on the obtained expression of the Porod asymptotics for a polydisperse system. A method for calculating the generalized Guinier–Porod approximation in Kratky coordinates, from which independent estimates of the average size and variance can also be obtained, is proposed. The efficiency of the developed approach is demonstrated by an example of analyzing the scattering intensity from aqueous solutions of silicasol nanoparticles.

Appendices

APPENDIX 1

Average Volume and Radius for Particles Obeying the Schulz Distribution

Using formula (3) for the Schulz distribution, we obtain the average volume of a single particle in the form

$$\begin{gathered} \left\langle {{{V}_{1}}} \right\rangle = \int\limits_0^\infty {V(R){{f}_{{{\text{Sch}}}}}(z,{{R}_{0}},R)dR = \frac{{4\pi }}{3}} \frac{{z_{1}^{{{{z}_{1}}}}}}{{{{R}_{0}}\Gamma ({{z}_{1}})}} \\ \times \;\int\limits_0^\infty {{{{{R}^{3}}(R} \mathord{\left/ {\vphantom {{{{R}^{3}}(R} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}}{{)}^{z}}} \exp ( - {{z}_{1}}{R \mathord{\left/ {\vphantom {R {{{R}_{0}})}}} \right. \kern-0em} {{{R}_{0}})}}dR, \\ \end{gathered} $$

$$\left\langle {{{V}_{1}}} \right\rangle = \frac{{4\pi }}{3}\frac{{({{{{{{{z}_{1}}} \mathord{\left/ {\vphantom {{{{z}_{1}}} {{{R}_{0}})}}} \right. \kern-0em} {{{R}_{0}})}}}}^{{z{{{\kern 1pt} }_{1}}}}}}}{{\Gamma ({{z}_{1}})}}\int\limits_0^\infty {{{R}^{{z + 3}}}} \exp ( - {{z}_{1}}{R \mathord{\left/ {\vphantom {R {{{R}_{0}})}}} \right. \kern-0em} {{{R}_{0}})}}dr,$$

with introduced designation $\lambda \equiv {{z{{{\kern 1pt} }_{1}}} \mathord{\left/ {\vphantom {{z{{{\kern 1pt} }_{1}}} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}}$; $z + 3 = {{z}_{{{\kern 1pt} 1}}} + 3 - 1$,

$$\left\langle {{{V}_{1}}} \right\rangle = \frac{{4\pi }}{3}\frac{{{{\lambda }^{{{{z}_{{{\kern 1pt} 1}}}}}}}}{{\Gamma ({{z}_{{{\kern 1pt} 1}}})}}\int\limits_0^\infty {\exp ( - \lambda R){{R}^{{{{\beta }_{V}} - 1}}}dR} ,$$

where $\beta {{{\kern 1pt} }_{V}}\; = {{z}_{1}} + 3.$

Using the tabular integral

$$J(\lambda ,\beta ) = \int\limits_0^\infty {\exp ( - \lambda x){{x}^{{\beta - 1}}}dx = {{\Gamma (\beta )} \mathord{\left/ {\vphantom {{\Gamma (\beta )} {{{\lambda }^{\beta }}}}} \right. \kern-0em} {{{\lambda }^{\beta }}}}} ,$$

(A.1.1)

we obtain

$$\begin{gathered} \left\langle {{{V}_{1}}} \right\rangle = \frac{{4\pi }}{3}\frac{{{{\lambda }^{{{{z}_{{{\kern 1pt} 1}}}}}}}}{{\Gamma ({{z}_{{{\kern 1pt} 1}}})}}\frac{{\Gamma ({{\beta }_{V}})}}{{{{\lambda }^{{{{\beta }_{V}}}}}}} = \frac{{4\pi }}{3}\frac{{{{\lambda }^{{{{z}_{{{\kern 1pt} 1}}}}}}}}{{\Gamma ({{z}_{1}})}}\frac{{\Gamma ({{z}_{{{\kern 1pt} 1}}} + 3)}}{{{{\lambda }^{{{{z}_{{{\kern 1pt} 1}}} + 3}}}}} \\ = \frac{{4\pi }}{3}\frac{{\Gamma ({{z}_{1}} + 3)}}{{\Gamma ({{z}_{1}})}}\frac{1}{{{{\lambda }^{3}}}}, \\ \end{gathered} $$

using the property

$$\begin{gathered} \Gamma (p + n) \\ = (p + n - 1)(p + n - 2) \ldots (p + 1)p\Gamma (p), \\ \end{gathered} $$

(A.1.2)

we arrive at

$$\left\langle {{{V}_{1}}} \right\rangle = {{(4\pi } \mathord{\left/ {\vphantom {{(4\pi } {3)}}} \right. \kern-0em} {3)}}{{({{z}_{1}} + 2)({{z}_{1}} + 1){{z}_{{{\kern 1pt} 1}}}} \mathord{\left/ {\vphantom {{({{z}_{1}} + 2)({{z}_{1}} + 1){{z}_{{{\kern 1pt} 1}}}} {{{\lambda }^{3}}}}} \right. \kern-0em} {{{\lambda }^{3}}}}.$$

Thus, the average volumes of the particles whose radii can be described by the Schulz distribution can be calculated from the formula

$$\left\langle {{{V}_{1}}} \right\rangle = \frac{{4\pi }}{3}\frac{{({{z}_{1}} + 2)({{z}_{1}} + 1)}}{{z_{1}^{2}}}R_{0}^{3},$$

(A.1.3)

where ${{R}_{0}}$ is a parameter in the Schulz distribution. Let us show that R0 coincides with the mean value for the Schulz distribution:

$$\begin{gathered} \left\langle R \right\rangle = \int\limits_0^\infty {Rf(z,{{R}_{0}},r)dR} , \\ \left\langle R \right\rangle = \frac{{{{z}_{{{\kern 1pt} 1}}}^{{{{z}_{1}}}}}}{{{{R}_{0}}\Gamma ({{z}_{1}})}}\int\limits_0^\infty {{{R(R} \mathord{\left/ {\vphantom {{R(R} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}}{{)}^{z}}} \exp ( - {{z}_{1}}{R \mathord{\left/ {\vphantom {R {{{R}_{0}})}}} \right. \kern-0em} {{{R}_{0}})}}dR. \\ \end{gathered} $$

(A.1.4)

To reduce the integral to the standard form, we introduce the variable ${{\beta }_{0}} = {{z}_{{{\kern 1pt} 1}}} + 1{\kern 1pt}$; then, using $\Gamma ({{z}_{{{\kern 1pt} 1}}} + 1) = {{z}_{{{\kern 1pt} 1}}}\Gamma ({{z}_{{{\kern 1pt} 1}}})$, we arrive at

$$\begin{gathered} \left\langle R \right\rangle = {{{{\lambda }^{{{{z}_{1}}}}}} \mathord{\left/ {\vphantom {{{{\lambda }^{{{{z}_{1}}}}}} {\Gamma ({{z}_{1}})}}} \right. \kern-0em} {\Gamma ({{z}_{1}})}}\int\limits_0^\infty {\exp ( - \lambda r){{r}^{{{{\beta }_{0}} - 1}}}dr = \left[ {{{{{\lambda }^{{{{z}_{1}}}}}} \mathord{\left/ {\vphantom {{{{\lambda }^{{{{z}_{1}}}}}} {\Gamma ({{z}_{1}})}}} \right. \kern-0em} {\Gamma ({{z}_{1}})}}} \right]} \\ \times \;\left[ {{{\Gamma ({{z}_{1}} + 1)} \mathord{\left/ {\vphantom {{\Gamma ({{z}_{1}} + 1)} {{{\lambda }^{{{{z}_{1}} + 1}}}}}} \right. \kern-0em} {{{\lambda }^{{{{z}_{1}} + 1}}}}}} \right] = {{{{z}_{1}}} \mathord{\left/ {\vphantom {{{{z}_{1}}} {\lambda = {{R}_{0}}}}} \right. \kern-0em} {\lambda = {{R}_{0}}}}. \\ \end{gathered} $$

Thus, the average radius for a system of particles is exactly equal to one of the parameters in the Schulz distribution:

$$\left\langle R \right\rangle = \int\limits_0^\infty {rf(z,{{R}_{0}},r)dr = {{R}_{0}}} .$$

(A.1.5)

The standard deviation σ0 is calculated similarly. To this end, we use the formula for the variance

$$D(r) = \left\langle {{{R}^{2}}} \right\rangle - {{\left( {\left\langle R \right\rangle } \right)}^{2}},$$

where

$$\left\langle {{{R}^{n}}} \right\rangle = \int\limits_0^\infty {{{r}^{n}}f\left( {z,{{R}_{0}},R} \right)dR} .$$

Calculation for the case with $n = 1$ was already performed ((A1.4) and (A1.5)): $\left\langle R \right\rangle = R_{0}^{{}}$. Let us calculate $\left\langle {{{R}^{2}}} \right\rangle$:

$$\left\langle {{{R}^{2}}} \right\rangle = \frac{{{{z}_{1}}^{{{{z}_{1}}}}}}{{{{R}_{0}}\Gamma ({{z}_{1}})}}\int\limits_0^\infty {{{{{R}^{2}}(R} \mathord{\left/ {\vphantom {{{{R}^{2}}(R} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}}{{)}^{z}}} \exp ( - {{z}_{1}}R{\text{/}}{{R}_{0}})dR{\kern 1pt} .$$

(A.1.6)

Making the same substitution $\lambda = {{{{z}_{1}}} \mathord{\left/ {\vphantom {{{{z}_{1}}} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}},$ as when calculating the particle volume and assuming that β = z₁ + 2, we obtain expression (A.1.6) in the form of a tabular integral:

$$\left\langle {{{R}^{2}}} \right\rangle = {{{{\lambda }^{{{{z}_{1}}}}}} \mathord{\left/ {\vphantom {{{{\lambda }^{{{{z}_{1}}}}}} {\Gamma ({{z}_{1}})}}} \right. \kern-0em} {\Gamma ({{z}_{1}})}}\int\limits_0^\infty {\exp ( - \lambda r){{r}^{{\beta - 1}}}dr} .$$

Using (A.1.1), we obtain

$$\begin{gathered} \left\langle {{{R}^{2}}} \right\rangle = \left[ {{{{{\lambda }^{{{{z}_{1}}}}}} \mathord{\left/ {\vphantom {{{{\lambda }^{{{{z}_{1}}}}}} {\Gamma ({{z}_{1}})}}} \right. \kern-0em} {\Gamma ({{z}_{1}})}}} \right]\left[ {{{\Gamma ({{z}_{1}} + 2)} \mathord{\left/ {\vphantom {{\Gamma ({{z}_{1}} + 2)} {{{\lambda }^{{{{z}_{1}} + 2}}}}}} \right. \kern-0em} {{{\lambda }^{{{{z}_{1}} + 2}}}}}} \right] \\ = {{({{z}_{1}} + 1){{z}_{1}}} \mathord{\left/ {\vphantom {{({{z}_{1}} + 1){{z}_{1}}} {{{\lambda }^{2}}}}} \right. \kern-0em} {{{\lambda }^{2}}}}. \\ \end{gathered} $$

(A.1.7)

Returning to the initial parameters $\lambda = {{z{{{\kern 1pt} }_{1}}} \mathord{\left/ {\vphantom {{z{{{\kern 1pt} }_{1}}} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}},$ we can write $\left\langle {{{R}^{2}}} \right\rangle = R_{0}^{2}({{z}_{1}} + 1){\text{/}}{{z}_{1}}$. Then the variance is $D(r) = \left[ {({{\mu }_{{{\kern 1pt} 1}}} + 1){\text{/}}{{\mu }_{1}} - 1} \right]R_{0}^{2} = {{R_{0}^{2}} \mathord{\left/ {\vphantom {{R_{0}^{2}} {{{z}_{1}}}}} \right. \kern-0em} {{{z}_{1}}}},$ and, hence, the standard deviation σ₀ is

$${{\sigma }_{0}} = {{{{R}_{0}}} \mathord{\left/ {\vphantom {{{{R}_{0}}} {\sqrt {{{z}_{{{\kern 1pt} 1}}}} }}} \right. \kern-0em} {\sqrt {{{z}_{{{\kern 1pt} 1}}}} }}.$$

(A.1.8)

APPENDIX 2

2.1. Guinier Approximation for the Scattering Intensity from a Polydisperse System of Spheres Whose Radii Obey the Schulz Distribution

The scattering intensity of a polydisperse diluted system with the particle–particle interference disregarded can be written in the form

$$\begin{gathered} {{I}_{{i\;{\text{Poly}}}}} = \left\langle N \right\rangle \left\langle {{{V}_{i}}} \right\rangle {{\left\langle {\Phi _{{{\text{Poly}}}}^{2}} \right\rangle }_{\Omega }} \\ = \left\langle {{{N}_{i}}} \right\rangle \int\limits_0^\infty {V(R){{f}_{{{\text{Sch}}}}}(z,{{R}_{{0i}}},R){\kern 1pt} {{{\left\langle {\Phi _{{{\text{mono}}}}^{2}(sR)} \right\rangle }}_{\Omega }}dR} , \\ \end{gathered} $$

(A.2.1)

where ${{\left\langle {\Phi _{{{\text{poly}}}}^{2}} \right\rangle }_{\Omega }}$ and ${{\left\langle {\Phi _{{{\text{mono}}}}^{2}} \right\rangle }_{\Omega }}$ are the scattering form factors of the poly- and monodisperse systems of particles, respectively; ${{\left\langle {} \right\rangle }_{\Omega }}$ is the sign of spatial averaging over a solid angle Ω for the sphere form factor (is omitted below in view of the spherical symmetry); $\left\langle {{{N}_{i}}} \right\rangle$ is the mean number of particles in the irradiated volume of the sample for the ith fraction; and $\left\langle {{{V}_{i}}} \right\rangle$ is the mean particle volume, which is determined in Appendix 1 (A.1.3). Using Guinier approximation $\Phi _{{{\text{mono}}}}^{2} = 1 - {{(sR_{g}^{*})}^{2}}{\text{/}}3$ ($R_{g}^{*}$ is the particle radius of gyration; in particular, for a monodisperse system of spherical particles, $R_{g}^{*} = R_{0}^{*}\sqrt {({3 \mathord{\left/ {\vphantom {3 {5)}}} \right. \kern-0em} {5)}}} {\kern 1pt}$), we arrive at the well-known expression

$$\Phi _{{{\text{mono}}}}^{2} = 1 - {{(sR_{g}^{*})}^{2}}{\text{/}}5,$$

(A.2.2)

where $R_{0}^{*}$ is the sphere radius for a monodisperse system. We assume the particle radius to be a variable value ${{R}_{0}} = r$, obeying the Schulz distribution. Then we rewrite (A.2.2) in the form $\Phi _{{{\text{mono}}}}^{2}(sR) = 1 - {{(sR)}^{2}}{\text{/}}5$, as a result of which expression (A.2.1) takes the form

$$\begin{gathered} {{I}_{{{\text{Poly}}}}} = \left\langle N \right\rangle \left\langle {{{V}_{1}}} \right\rangle \Phi _{{{\text{Poly}}}}^{2}(z,{{R}_{0}},s) \\ = \left\langle N \right\rangle \int\limits_0^\infty {V(R){{f}_{{{\text{Sch}}}}}(z,{{R}_{0}},R)\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)dR} , \\ \end{gathered} $$

(A.2.3a)

where

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2}(z,{{R}_{{0{\kern 1pt} }}},s) \\ = \frac{1}{{\left\langle {{{V}_{1}}} \right\rangle }}\int\limits_0^\infty {V(R){{f}_{{{\text{Sch}}}}}\left( {z,{{R}_{0}},R} \right)\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)dR} , \\ \end{gathered} $$

(A.2.3b)

is the form factor normalized to unity; at $s = 0$, $\Phi _{{{\text{Poly}}}}^{2}(z,{{R}_{0}},0) = 1$ for a polydisperse system. If we substitute the mean particle volume from (A.1.3) and Schulz distribution from (3) into (A.2.3b), using the designation ${{z}_{1}} = z + 1$, (A.2.3b) can be written in the expanded form:

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{{3z_{{1'}}^{2}}}{{4\pi ({{z}_{1}} + 2)({{z}_{1}} + 1)R_{0}^{3}}}\frac{{z_{1}^{{{{z}_{1}}}}}}{{\Gamma ({{z}_{1}}){{R}_{0}}}} \\ \times \;\int\limits_0^\infty {{{(4\pi } \mathord{\left/ {\vphantom {{(4\pi } {3){{R}^{3}}}}} \right. \kern-0em} {3){{R}^{3}}}}({R \mathord{\left/ {\vphantom {R {{{R}_{0}}{{)}^{z}}{\kern 1pt} \exp ( - {{z}_{1}}{R \mathord{\left/ {\vphantom {R {{{R}_{0}})\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)}}} \right. \kern-0em} {{{R}_{0}})\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)}}}}} \right. \kern-0em} {{{R}_{0}}{{)}^{z}}{\kern 1pt} \exp ( - {{z}_{1}}{R \mathord{\left/ {\vphantom {R {{{R}_{0}})\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)}}} \right. \kern-0em} {{{R}_{0}})\left( {1 - {{{(sR)}}^{2}}{\text{/}}5} \right)}}}}} {\kern 1pt} {\kern 1pt} dR, \\ \end{gathered} $$

making the substitution $sR = x,$ $sR_{0}^{{}} = {{x}_{0}},$ $R = {x \mathord{\left/ {\vphantom {x s}} \right. \kern-0em} s},$ $dR = {{dx} \mathord{\left/ {\vphantom {{dx} {s,}}} \right. \kern-0em} {s,}}$ ${R \mathord{\left/ {\vphantom {R {R_{0}^{{}} = {x \mathord{\left/ {\vphantom {x {{{x}_{0}}}}} \right. \kern-0em} {{{x}_{0}}}}}}} \right. \kern-0em} {R_{0}^{{}} = {x \mathord{\left/ {\vphantom {x {{{x}_{0}}}}} \right. \kern-0em} {{{x}_{0}}}}}}$, we arrive at

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{{z_{{1'}}^{2}}}{{({{z}_{1}} + 2)({{z}_{1}} + 1)}}\frac{{z_{1}^{{{{z}_{1}}}}}}{{x_{0}^{4}\Gamma ({{z}_{1}})}} \\ \times \;\int\limits_0^\infty {{{x}^{3}}({x \mathord{\left/ {\vphantom {x {{{x}_{0}}{{)}^{z}}\exp ( - {{z}_{1}}{x \mathord{\left/ {\vphantom {x {{{x}_{0}})}}} \right. \kern-0em} {{{x}_{0}})}}}}} \right. \kern-0em} {{{x}_{0}}{{)}^{z}}\exp ( - {{z}_{1}}{x \mathord{\left/ {\vphantom {x {{{x}_{0}})}}} \right. \kern-0em} {{{x}_{0}})}}}}} \left( {1 - {{x}^{2}}{\text{/}}5} \right)dx,\; \\ \end{gathered} $$

$$\begin{gathered} \Rightarrow \;\;\Phi _{{{\text{Ploy}}}}^{2} = \frac{{z_{{1'}}^{2}}}{{({{z}_{1}} + 2)({{z}_{1}} + 1)}}\frac{1}{{x_{0}^{3}\Gamma ({{z}_{1}})}}{{({{z}_{1}}{\text{/}}{{x}_{0}})}^{{{{z}_{1}}}}} \\ \times \;\int\limits_0^\infty {{{x}^{{z + 3}}}\exp ( - {{z}_{1}}x{\text{/}}{{x}_{0}}{{x}_{0}})\left( {1 - {{x}^{2}}{\text{/}}5} \right)} dx, \\ \end{gathered} $$

denoting ${{\lambda }_{{{\kern 1pt} s}}} \equiv {{{{z}_{1}}} \mathord{\left/ {\vphantom {{{{z}_{1}}} {{{x}_{0}}}}} \right. \kern-0em} {{{x}_{0}}}}$, we have

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{{z_{{1'}}^{2}}}{{({{z}_{1}} + 2)({{z}_{1}} + 1)}}\frac{{\lambda _{s}^{{{{z}_{1}}}}}}{{x_{0}^{3}\Gamma ({{z}_{1}})}} \\ \times \;\int\limits_0^\infty {{{x}^{{z + 3}}}\exp ( - } {{\lambda }_{s}}x)\left( {1 - {{x}^{2}}{\text{/}}5} \right)dx,\;\; \\ \end{gathered} $$

$$\begin{gathered} \Rightarrow \;\;\Phi _{{{\text{Poly}}}}^{2} = \frac{1}{{({{z}_{1}} + 2)({{z}_{1}} + 1){{z}_{1}}}}\frac{{\lambda _{s}^{{{{z}_{1}} + 3}}}}{{\Gamma ({{z}_{1}})}} \\ \times \;\int\limits_0^\infty {{{x}^{{z + 3}}}\exp ( - } {{\lambda }_{s}}x)\left( {1 - {{x}^{2}}{\text{/}}5} \right)dx, \\ \end{gathered} $$

denoting ${{\beta }_{{{\kern 1pt} 1}}} = {{z}_{1}} + 3;\;{{\beta }_{2}} = {{z}_{1}} + 5,$ we have

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{1}{{({{z}_{1}} + 2)({{z}_{{{\kern 1pt} 1}}} + 1){\kern 1pt} {{z}_{1}}}}\frac{{\lambda _{s}^{{{{\mu }_{1}} + 3}}}}{{\Gamma ({{z}_{1}})}} \\ \times \;\int\limits_0^\infty {\exp ( - } {{\lambda }_{s}}x)\left( {{{x}^{{{{\beta }_{1}} - 1}}} - {{{{x}^{{{{\beta }_{2}} - 1}}}} \mathord{\left/ {\vphantom {{{{x}^{{{{\beta }_{2}} - 1}}}} 5}} \right. \kern-0em} 5}} \right)dx. \\ \end{gathered} $$

(A.2.4)

Then, using the tabular integral (A.1.1), we can write the last expression in the form

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{1}{{({{z}_{1}} + 2)({{z}_{{{\kern 1pt} 1}}} + 1){\kern 1pt} {{z}_{1}}}}\frac{{\lambda _{s}^{{{{z}_{1}} + 3}}}}{{\Gamma ({{z}_{1}})}} \\ \times \;[J({{\lambda }_{s}},{{\beta }_{1}}) - {{J({{\lambda }_{{{\kern 1pt} s}}},{{\beta }_{2}})} \mathord{\left/ {\vphantom {{J({{\lambda }_{{{\kern 1pt} s}}},{{\beta }_{2}})} 5}} \right. \kern-0em} 5}], \\ \end{gathered} $$

factorizing the first integral and taking into account that, according to the property of the gamma-function (A.1.2), $({{z}_{{{\kern 1pt} 1}}} + 2)({{z}_{{{\kern 1pt} 1}}} + 1){\kern 1pt} {\kern 1pt} {{z}_{{{\kern 1pt} 1}}}\Gamma ({{z}_{{{\kern 1pt} 1}}}) = \Gamma ({{z}_{{{\kern 1pt} 1}}} + 3)$, we rewrite (A.2.4) in the form

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2} = \frac{{\lambda _{s}^{{{{z}_{1}} + 3}}}}{{\Gamma ({{z}_{1}} + 3)}}J({{\lambda }_{s}},{{\beta }_{1}}) \\ \times \;[1 - {{({1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0em} 5})J({{\lambda }_{s}},{{\beta }_{2}})} \mathord{\left/ {\vphantom {{({1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0em} 5})J({{\lambda }_{s}},{{\beta }_{2}})} {J({{\lambda }_{s}},{{\beta }_{1}})}}} \right. \kern-0em} {J({{\lambda }_{s}},{{\beta }_{1}})}}], \\ \end{gathered} $$

(A.2.5)

where ${{\lambda }_{{{\kern 1pt} s}}} \equiv {{{{z}_{1}}} \mathord{\left/ {\vphantom {{{{z}_{1}}} {{{x}_{0}}}}} \right. \kern-0em} {{{x}_{0}}}}$ and ${{\beta }_{{{\kern 1pt} 1}}} = {{z}_{1}} + 3;$ ${{\beta }_{2}} = {{z}_{1}} + 5.$

With allowance for the fact that $J({{\lambda }_{{{\kern 1pt} s}}},{{\beta }_{1}}) = {{\Gamma ({{\beta }_{1}})} \mathord{\left/ {\vphantom {{\Gamma ({{\beta }_{1}})} {\lambda _{s}^{{{{\beta }_{1}}}}}}} \right. \kern-0em} {\lambda _{s}^{{{{\beta }_{1}}}}}} = {{\Gamma ({{z}_{1}} + 3)} \mathord{\left/ {\vphantom {{\Gamma ({{z}_{1}} + 3)} {\lambda _{s}^{{{{z}_{1}} + 3}}}}} \right. \kern-0em} {\lambda _{s}^{{{{z}_{1}} + 3}}}}$ (A.1.1), expression (A.2.5) takes the form

$$\Phi _{{{\text{Poly}}}}^{2} = 1 - {{({1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0em} 5})J({{\lambda }_{s}},{{\beta }_{2}})} \mathord{\left/ {\vphantom {{({1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0em} 5})J({{\lambda }_{s}},{{\beta }_{2}})} {J({{\lambda }_{s}},{{\beta }_{1}})}}} \right. \kern-0em} {J({{\lambda }_{s}},{{\beta }_{1}})}}.$$

(A.2.6)

Now we only have to calculate (using the same tabular integral (A.1.1)) the ratio of integrals in (A.2.6):

$$\begin{gathered} \frac{{J({{\lambda }_{{{\kern 1pt} s}}},{{\beta }_{2}})}}{{J({{\lambda }_{{{\kern 1pt} s}}},{{\beta }_{1}})}} = \frac{{\Gamma ({{z}_{1}} + 5)}}{{\Gamma ({{z}_{1}} + 3)}}\frac{{\lambda _{s}^{{{{z}_{1}} + 3}}}}{{\lambda _{s}^{{{{z}_{1}} + 5}}}} \\ = \frac{{({{z}_{1}} + 4)({{z}_{1}} + 3)}}{{\lambda _{s}^{2}}} = \frac{{({{z}_{1}} + 4)({{z}_{1}} + 3)}}{{z_{1}^{2}}}x_{0}^{2}. \\ \end{gathered} $$

Recalling that ${{x}_{0}} = sR_{0}^{{}}$, we finally obtain the Guinier approximation for the particles whose radii obey the Schulz distribution:

$$\Phi _{{{\text{Poly}}}}^{2} = 1 - \frac{1}{5}\frac{{({{z}_{1}} + 4)({{z}_{1}} + 3)}}{{z_{1}^{2}}}{{(s{{R}_{0}})}^{2}},$$

(A.2.7)

where, as was shown in (A.1.5), the parameter ${{R}_{0}}$ in the Schulz distribution $f(z,\,{{R}_{0}},\,r)$ determines the mean particle radius in this distribution. Having comparing the obtained Guinier approximation with the similar approximation for a monodisperse system (A.2.2), one can conclude that the presence of polydispersity underestimates the particle radius by a factor of $\sqrt \alpha$, where

$$\alpha = {{(R_{0}^{*}{\text{/}}{{R}_{0}})}^{2}} = {{({{z}_{1}} + 4)({{z}_{1}} + 3)} \mathord{\left/ {\vphantom {{({{z}_{1}} + 4)({{z}_{1}} + 3)} {z_{1}^{2}}}} \right. \kern-0em} {z_{1}^{2}}}.$$

(A.2.8)

In the limit for “strongly polydisperse” systems, when ${{z}_{{{\kern 1pt} 1}}} = 1 \Rightarrow \alpha = 20$, $\sqrt \alpha = \sqrt {20} \approx 4.47$; for “narrow polydisperse” systems, having calculated the limit ${{z}_{{{\kern 1pt} 1}}} \to \infty$, we arrive at $\alpha = 1$.

The dependence of the parameter z₁ on α is derived from (A.2.8) by solving the square equation $(\alpha - 1)\,z_{1}^{2} - 7{{z}_{1}} - 12 = 0$, which has a single positive root at $\alpha \geqslant 1$:

$${{z}_{1}} = \frac{{7 + \sqrt {1 + 48{\kern 1pt} \alpha } }}{{2(\alpha - 1)}}.$$

(A.2.9)

According to (A.1.8), the standard deviation ${{\sigma }_{0}}$ will tend to zero (${{\sigma }_{0}} \to 0$) at ${{z}_{1}} \to \infty \,,\;$ i.e., when the parameter $\alpha \to 1$. In this case (A.2.8) the average particle radius in the polydisperse system will tend to the particle radius in the monodisperse system: $R_{0}^{*} = {{R}_{0}}$.

2.2. Calculation of the Intensity and Porod Asymptotics for Homogeneous Spherical Particles Whose Radii Obey the Schulz Distribution

Using (A.2.1), we can write

$$\begin{gathered} \Phi _{{{\text{Poly}}}}^{2}(z,{{R}_{0}},s) \\ = \frac{1}{{\left\langle {{{V}_{1}}} \right\rangle }}\int\limits_0^\infty {V(r)f(z,{{R}_{0}},r)\Phi _{{{\text{mono}}}}^{2}(sr)dr} , \\ \end{gathered} $$

(A.2.10)

where $\Phi _{{{\text{mono}}}}^{2}$ is the form factor of a homogeneous sphere [12, 13]:

$$\Phi _{{{\text{mono}}}}^{2}(sr) = 9{{\left( {\frac{{\sin (sr) - sr\cos (sr)}}{{{{{(sr)}}^{3}}}}} \right)}^{2}},$$

(A.2.11)

and the mean (in the Schulz distribution) particle volume is determined by formula (A.1.3). With allowance for this fact, expression (A.2.10) can be rewritten in the form

$$\Phi _{{{\text{Poly}}}}^{2}(z,{{R}_{0}},s) = \frac{{z_{1}^{2}}}{{({{z}_{1}} + 2)({{z}_{1}} + 1){\kern 1pt} R_{0}^{3}}}\frac{{z_{1}^{{{{z}_{1}}}}}}{{{{R}_{0}}\Gamma ({{z}_{1}})}}$$

$$ \times \,\int\limits_0^\infty {{{(r} \mathord{\left/ {\vphantom {{(r} {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}}{{)}^{z}}\exp ( - {{z}_{1}}{r \mathord{\left/ {\vphantom {r {{{R}_{0}})}}} \right. \kern-0em} {{{R}_{0}})}}{{r}^{3}}9} $$

(A.2.12)

$$ \times \;{{\left( {{{\sin (sr) - sr\cos (sr)} \mathord{\left/ {\vphantom {{\sin (sr) - sr\cos (sr)} {{{{(sr)}}^{3}}}}} \right. \kern-0em} {{{{(sr)}}^{3}}}}} \right)}^{2}}dr,$$

where ${{z}_{1}} = z + 1$. Introducing the variables $t = {r \mathord{\left/ {\vphantom {r {{{R}_{0}}}}} \right. \kern-0em} {{{R}_{0}}}},$ $dr = {{R}_{0}}dt,$ $s{{R}_{0}} = x$, we arrive at

$$\begin{gathered} \Phi _{{{\text{poly}}}}^{2}({{z}_{1}},x) = C({{z}_{1}})\int\limits_0^\infty {{{t}^{{{{z}_{1}} + 2}}}\exp ( - {{z}_{1}}t} ) \\ \times \;{{\left[ {{{\sin (xt) - xt\cos (xt)} \mathord{\left/ {\vphantom {{\sin (xt) - xt\cos (xt)} {{{{(xt)}}^{3}}}}} \right. \kern-0em} {{{{(xt)}}^{3}}}}} \right]}^{2}}dt, \\ \end{gathered} $$

(A.2.13)

where $C({{z}_{1}}) = \left[ {{{9z_{1}^{{{{z}_{1}} + 3}}} \mathord{\left/ {\vphantom {{9z_{1}^{{{{z}_{1}} + 3}}} {\Gamma ({{z}_{1}} + 3)}}} \right. \kern-0em} {\Gamma ({{z}_{1}} + 3)}}} \right]$. Expanding the squared difference and integrating each term, we finally obtain (provided that z₁ > 1)

$$\begin{gathered} \Phi _{{{\text{poly}}}}^{2}({{z}_{1}},s{{R}_{0}}) = \frac{{144}}{{\Gamma ({{z}_{1}} + 3)}}\frac{1}{{{{\xi }^{4}}}} \\ \times \;\left[ {A({{z}_{1}},\xi ) - B({{z}_{1}},\xi ) + C({{z}_{1}},\xi )} \right], \\ \end{gathered} $$

(A.2.14a)

where $\xi = {{2s{{R}_{0}}} \mathord{\left/ {\vphantom {{2s{{R}_{0}}} {{{z}_{1}}}}} \right. \kern-0em} {{{z}_{1}}}}$, and the terms A, B, and C, entering (A.2.14а), are defined as

$$A({{z}_{1}},\xi ) = \frac{2}{{{{\xi }^{2}}}}\Gamma ({{z}_{1}} - 3)\left[ {1 - \frac{{\cos [({{z}_{1}} - 3)\arctan (\xi )]}}{{{{{(1 + {{\xi }^{2}})}}^{{{{({{z}_{1}} - 3)} \mathord{\left/ {\vphantom {{({{z}_{1}} - 3)} 2}} \right. \kern-0em} 2}}}}}}} \right],$$

$$B({{z}_{1}},\xi ) = \frac{2}{\xi }\Gamma ({{z}_{1}} - 2)\frac{{\sin [({{z}_{1}} - 2)\arctan (\xi )]}}{{{{{(1 + {{\xi }^{2}})}}^{{{{({{z}_{1}} - 2)} \mathord{\left/ {\vphantom {{({{z}_{1}} - 2)} 2}} \right. \kern-0em} 2}}}}}},$$

(A.2.14b)

$$C({{z}_{1}},\xi ) = \frac{1}{2}\Gamma ({{z}_{1}} - 1)\left[ {1 + \frac{{\cos [({{z}_{1}} - 1)\arctan (\xi )]}}{{{{{(1 + {{\xi }^{2}})}}^{{{{({{z}_{1}} - 1)} \mathord{\left/ {\vphantom {{({{z}_{1}} - 1)} 2}} \right. \kern-0em} 2}}}}}}} \right].$$

At $\xi = {{2s{{R}_{0}}} \mathord{\left/ {\vphantom {{2s{{R}_{0}}} {{{z}_{1}}}}} \right. \kern-0em} {{{z}_{1}}}} \gg 1$ and z₁ > 1 we obtain the Porod asymptotics:

$$\mathop {\Phi _{{{\text{poly}}}}^{2}(\xi )}\limits_{\xi \gg 1} \approx \frac{{{{C}_{0}}}}{{{{\xi }^{4}}}}\left( {1 + \frac{1}{{{{{\left( {1 + {{\xi }^{2}}} \right)}}^{{{z \mathord{\left/ {\vphantom {z 2}} \right. \kern-0em} 2}}}}}}} \right),$$

(A.2.15)

where С₀ is a constant.

APPENDIX 3

The Padé approximation is a rational function in the form

$$\left[ {{L \mathord{\left/ {\vphantom {L M}} \right. \kern-0em} M}} \right] = \frac{{{{a}_{0}} + {{a}_{1}}x + {{a}_{2}}{{x}^{2}} + \ldots + {{a}_{L}}{{x}^{L}}}}{{{{b}_{0}} + {{b}_{1}}x + {{b}_{2}}{{x}^{2}} + \ldots + {{b}_{M}}{{x}^{M}}}}.$$

(A.3.1)

To obtain the generalized Guinier–Porod approximation, it is sufficient to expand the scattering intensity for a monodisperse sphere ${{\Phi }^{2}}(s{{R}_{0}})$ in the Padé series P[L/M] in powers of the denominator at L = 0, i.e., to obtain a fractional rational expression of the form s²I_mod = P(Ф²(sR₀)) [0/M]:

$${{s}^{2}}I = {{{{s}^{2}}{{I}_{0}}} \mathord{\left/ {\vphantom {{{{s}^{2}}{{I}_{0}}} {\sum\limits_{k = 0}^M {{{b}_{k}}{{z}^{k}}} ,}}} \right. \kern-0em} {\sum\limits_{k = 0}^M {{{b}_{k}}{{z}^{k}}} ,}}$$

(A.3.2)

where $z = s{{R}_{0}}$. In particular, we obtain

$${{s}^{2}}{{I}_{{\bmod }}} = \frac{{{{I}_{0}}{{s}^{2}}}}{{1 + \left( {s{{R}_{0}}} \right){{^{2}} \mathord{\left/ {\vphantom {{^{2}} {5 + {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} \mathord{\left/ {\vphantom {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} {175}}} \right. \kern-0em} {175}}}}} \right. \kern-0em} {5 + {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} \mathord{\left/ {\vphantom {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} {175}}} \right. \kern-0em} {175}}}}}},$$

(A.3.2a)

$${{s}^{2}}{{I}_{{\bmod }}} = \frac{{{{I}_{0}}{{s}^{2}}}}{{1 + {{{{{\left( {s{{R}_{0}}} \right)}}^{2}}} \mathord{\left/ {\vphantom {{{{{\left( {s{{R}_{0}}} \right)}}^{2}}} {5 + {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} \mathord{\left/ {\vphantom {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} {175 + {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} \mathord{\left/ {\vphantom {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} {23\,625}}} \right. \kern-0em} {23\,625}}}}} \right. \kern-0em} {175 + {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} \mathord{\left/ {\vphantom {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} {23\,625}}} \right. \kern-0em} {23\,625}}}}}}} \right. \kern-0em} {5 + {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} \mathord{\left/ {\vphantom {{4{{{\left( {s{{R}_{0}}} \right)}}^{4}}} {175 + {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} \mathord{\left/ {\vphantom {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} {23\,625}}} \right. \kern-0em} {23\,625}}}}} \right. \kern-0em} {175 + {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} \mathord{\left/ {\vphantom {{47{{{\left( {s{{R}_{0}}} \right)}}^{6}}} {23\,625}}} \right. \kern-0em} {23\,625}}}}}}}}.$$

(A.3.2b)

The result of calculating the generalized Guinier–Porod approximation from the scattering intensity approximation for a homogeneous sphere is presented in Fig. 7. It can be seen in Fig. 7a that the maximum in exponential approximation (7) is located between [0/4] and [0/6] of the Padé approximation. It can be seen in Fig. 7b that the position of the maximum shifts with a rise in the power of denominator M; correspondingly, the ratio $\alpha = {{({{R}_{0}}{\text{/}}R_{0}^{*})}^{2}}$ tends to unity. Since s_max = 0.029 nm^–1 for the experimental curve, it follows from s_max R₀ = $\sqrt 5$ that R₀ = 7.71 nm. In this case, for M = 4, according to Table 5, ${{s}_{{\max }}}R_{0}^{*} = 2.572$; hence, $R_{0}^{*} = 10.7$ nm, and for M = 6 ${{s}_{{\max }}}R_{0}^{*} = 2.206$ and $R_{0}^{*} = 7.91$ nm. Correspondingly, for $M = 4$ and 6, we have $\alpha = 1.92$ and 1.05, respectively. In practice the $R_{0}^{*}$ values are found as fitting parameters according to formulas (A.3.2a) or (A.3.2b) for $M = 4$ or 6, respectively. The result of fitting for these М values is presented in Fig. 7b. Obviously, now there is no need to increase the polynomial power in the denominator, because even at $M = 6$ we have $\alpha = 1.05\,,$ hence, ${{z}_{1}} = 141.7$. This yields a fairly narrow starting Schulz distribution with a standard deviation $\sigma = {{{{R}_{0}}} \mathord{\left/ {\vphantom {{{{R}_{0}}} {\sqrt {{{z}_{1}}} }}} \right. \kern-0em} {\sqrt {{{z}_{1}}} }} = 0.65\,$ for R₀ = 7.71 nm. Thus, with an increase in the approximation order M the position of the maximum in the model curve shifts towards the experimental one, and the fitting quality increases. Note that s_maxR₀ = $\sqrt 5 = 2.236$ for the exponential approximation; therefore, as was noted above, the exponential approximation is located between [0/4] and [0/6] of the Padé approximation for the generalized Guinier–Porod approximation, the numerical values of the positions of maxima for which are listed in Table 5.

Table 5.   Dependence of the position of maxima in the model curves in dependence of the power in the Padé approximation for the scattering intensity from a homogeneous sphere s²Ф²(sR₀) in dimensionless Kratky coordinates