Combining Small-Angle X-ray Scattering and X-ray Powder Diffraction to Investigate Size, Shape and Crystallinity of Silver, Gold and Alloyed Silver-Gold Nanoparticles

Abstract

The combination of simultaneous measurements of small and wide angle X-ray scattering (SAXS/WAXS–SWAXS) to investigate the overall size, shape, and crystallinity of silver nanoparticles and alloyed silver-gold nanoparticles in the size range of 8 to 80 nm is shown. The obtained results for overall size are in agreement with the particle size obtained by differential centrifugal sedimentation (DCS) and transmission electron microscopy (TEM). In addition to the overall size, SWAXS provided precise information about the crystallographic internal structure of the particles, providing a powerful multi-scale tool for structural characterization of the studied systems.

Appendix

In the following the theoretical models used to fit the experimental SAXS data are described. Detailed information about these models can be found in the book of Lindner (1991) [20]. The non-linear least squares method was used to determine the optimal values of the parameters. The uncertainties of these parameters were determined from the correlation matrix.

1.1 Long Cylinders

For a system composed of long cylinders with polydispersity in radius as used, the scattering intensity is given by:

$$ {I}_{rod}(q)= Sc.{P}_{MOD}(q){P}_{ROD}(q)+ Back $$

(1)

where S_C is an overall constant factor, Back a constant background, P_ROD(q) is the form factor for an infinitely thin rod with length L [21] and e P_MOD(q) is the form factor of a polydisperse circular cross section given by:

$$ {P}_{MOD}(q)=\frac{\int_0^{\infty }D(r)m{(r)}^2{F}_S\left(q,r\right) dr}{\int_0^{\infty }D(r)m{(r)}^2 dr} $$

(2)

m(r) is the area of the circular cross section (π r²) and F_S(q,R) is the intensity form factor for a circular cross section of radius R given by,

$$ {F}_S\left(q,R\right)={\left(\frac{J_1(qR)}{qR}\right)}^2 $$

(3)

The polydispersity in radius D(R,R₀,σ) is described as a Schulz-Zimm function [20]:

$$ D\left(R,{R}_0,\sigma \right)={\left(\frac{z+1}{R_0}\right)}^{z+1}\frac{R^z}{\varGamma \left(z+1\right)}\exp \left(-\left(z+1\right)\frac{R}{R_0}\right) $$

(4)

where R₀ is the central value of the distribution, Γ is the gamma function and z = 1/(σ/ R₀)²–1 (σ is the polidispersity). This distribution is very useful because it allows asymmetric distributions.

1.2 Cylinder with elliptical cross section

For Ag Plates a model composed of cylinders with elliptical cross section was used. The scattering intensity is given by:

$$ {I}_{ell}(q)= Sc.P(q)+ Back $$

(5)

where S_C is an overall scale factor, Back a constant background and P(q) is the form factor of cylinders with elliptical cross section given by:

$$ P(q)=\frac{2}{\pi}\underset{0}{\overset{\pi /2}{\int }}\underset{0}{\overset{\pi /2}{\int }}\left[\frac{{2\mathrm{B}}_1\left( qr\left(A,B,\varphi, \alpha \right)\right)}{qr\left(A,B,\varphi, \alpha \right)}\frac{\sin \left(\left( qL\cos \alpha \right)/2\right)}{\left( qL\cos \alpha \right)/2}\right]\mathrm{d}\varphi\ \sin \alpha\ d\alpha $$

(6)

with $r\left(A,B,\varphi, \alpha \right)=0.5\left(\sqrt{A^2{\sin}^2\varphi +{B}^2{\cos}^2\varphi}\right)\sin \alpha$. In this equation, L is the length of the cylinders, B₁ is the Bessel function of first order and A and B are the major and minor axis of the elliptical cross section respectively.

1.3 Polydisperse system of spheres with aggregates

In this case, the scattering intensity is given by:

$$ I(q)= Sc\left(\underset{0}{\overset{\infty }{\int }}{V}^2(R)D\left(R,\sigma \right){F}_{sphere}{\left(q,R\right)}^2 dR\right){S}_{PY}\left(q,{R}_{HS}\right)\cdot {S}_G\left(q,R{G}_1\right)+ Back $$

(7)

where Sc is the global scale factor, R is the radius of the sphere, V(R) is sphere volume, σ is the size polydispersity, RG is the average radius of gyration of the aggregates, S_G is the Guinier structure factor, R_HS is the hard sphere interaction radius and Back is a constant background.

The polydispersity in radius D(R,σ) is described as a Schulz-Zimm function shown previously in Eq. 4.

In order to take into account the interaction among the spherical particles (repulsion), the hard-sphere PercusYevick structure factor S_PY(q,R_HS) was used [18]:

$$ {S}_{PY}\left(q,{R}_{HS}\right)=\frac{1}{1+24\eta G\left(2{R}_{HS}q\right)/\left(2{R}_{HS}q\right)} $$

(8)

Where R_HS is the hard sphere radius, η is volume fraction of hard spheres and G is given by:

$$ {\displaystyle \begin{array}{l}G(A)=\frac{a\left(\sin A-A\cos A\right)}{A^2}+\frac{\beta \left(2A\sin A+\left(2-{A}^2\right)\cos A-2\right)}{A^3}+\\ {}\kern4em \frac{\gamma \left[-{A}^4 cpsA+4\left\{\left(3{A}^2-6\right)\cos A+\left({A}^3-6A\right)\sin A+6\right\}\right]}{A^5}\end{array}} $$

(9)

with,

$$ \alpha =\frac{{\left(1+2\eta \right)}^2}{{\left(1-\eta \right)}^4},\beta =\frac{-6\eta {\left(1+\eta /2\right)}^2}{{\left(1-\eta \right)}^4},\gamma =\frac{\eta \alpha}{2} $$

(10)

The contribution of aggregates and/or large particles is described in terms of their average gyration radius RG and the scale factors Sc^G [10]. The expression for this structure factor SG is given by:

$$ {S}_G\left(q, RG\right)=1+S{c}^G{e}^{\frac{-{q}^2R{G}^2}{3}} $$

(11)

where RG_sph = R_sph(3/5)^1/2.