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Wikiadmin moved page Problem: Polydisperse spheres to Problem:Polydisperse spheres

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ucph>Tommy: Created page with "We imagine a solution of polydisperse spheres with volume fraction $\phi$. When we can assume $S(q)=1$ (typically for low concentrations, $\phi \ll 1$), the total SANS cro..."

2019-07-14T21:29:56Z

Created page with "We imagine a solution of polydisperse spheres with volume fraction $\phi$. When we can assume $S(q)=1$ (typically for low concentrations, $\phi \ll 1$), the total SANS cro..."

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We imagine a solution of polydisperse spheres with volume fraction $\phi$. When we can assume $S(q)=1$ (typically for low concentrations, $\phi \ll 1$), the total SANS cross section per sample volume is given by

:$ I(q)= \phi V (\Delta \rho)^2 P(q), \,$

where $V$ is the volume of the particle.

=====Question 1=====
Assume that the volume fraction is $\phi= 0.01$ and that the excess scattering length density is $\Delta\rho=3\cdot 10^{10} \rm{cm} / \rm{c}m^3$. Using the form factors found in the problem [[Exercises in Small angle neutron scattering#Problem: Scattering form factor for spheres|Scattering form factor for spheres]], calculate and plot the volume specific cross section from spherical particles of radii $R=20$ Å, $R=40$ Å, $R=60$ Å and $R=80$ Å respectively if they are considered monodisperse and dilute.

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
<figure id="fig:XS_monodisp_hard_spheres_phi0p01_deltarho3e10"> [[File:XS monodisp hard spheres phi0p01 deltarho3e10.png| thumb | 200px | <caption> The total SANS scattering cross section per sample volume for dilute monodisperse hard spheres.</caption>]]</figure>

<figtable id="tab:dilute_monodisperse_spheres">[[File:Dilute monodisperse spheres.png|frame|<caption> The volume of selected spherical particles and the total SANS scattering cross section per sample volume for diluted particles of volumefraction $\phi$=0.01 in the limit of $q\to 0$.</caption>]]</figtable>

The total SANS scattering cross section per sample volume for dilute monodisperse hard spheres of selected sizes at fixed volumefraction is shown in <xr id="fig:XS_monodisp_hard_spheres_phi0p01_deltarho3e10">Figure %i</xr>.
It is seen that it is similar to the form factors shown in Question 3 of [[Exercises in Small angle neutron scattering#Problem: Scattering form factor for spheres|Scattering form factor for spheres]], the only difference is the scaling due to the prefactors of the formfactor.

The interpolated values for $I(q=0)$ with a fixed volumefraction of $\phi=0.01$ are shown in <xr id="tab:dilute_monodisperse_spheres"> Table %i </xr>.

{{hidden end}}

=====Question 2=====
Now, assume instead that your sample of spheres is weakly polydisperse, and that it basically follows a Gaussian distribution, with $R_{\rm av}=R$ and with a relative standard deviation of 20 % giving the volume specific scattering cross-section

:$ I(q) = \phi V (\Delta\rho)^2 \displaystyle\int {\rm d}R \ D(R)\cdot P(q,R) ,$

where $V=4\pi R_{av}^3 /3$ is the average volume of the particles with average radius $R_{av}$ but distributed by the size distribution $D(R)$. The formfactor amplitude is denoted $P(q,R)$. An appropriate integration range might be $R\in[R_{av}-3\sigma;R_{av}+3\sigma]$.

Plot the volume specific scattering cross-section in the polydisperse case for (at least) the $R=60$ Å sample.

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
<figure id="fig:dilute_polydisperse_spheres"> [[File:XS polydisp hard spheres phi0p01 deltarho3e10.png| thumb | 200px | <caption> The total SANS scattering cross section per sample volume for dilute polydisperse hard spheres.</caption>]] </figure>

We consider now a polydispersity of the spheres of 20% in a Gaussian distribution, i.e.

:$ D(R)=\dfrac{1}{\sqrt{2\pi}\sigma} e^{-\tfrac{(R-R_{av})}{2\sigma^2}} $

with $\sigma=0.2R$.

The cross-section per sample volume for dilute polydisperse hard spheres is shown in <xr id="fig:dilute_polydisperse_spheres"> Figure %i </xr>. The 20% smearing has now completely removed the ripples on the curve in the high-$q$ tail and replaced it by a power law. The integration over sizes has been performed numerically in the range $[R_{av}-3\sigma;R_{av}+3\sigma]$.

{{hidden end}}

=====Question 3=====
For higher particle concentrations, the particle-particle interactions generally have to be taken into account in the modeling of the scattering intensity via the structure factor, $S(q)=|S(\mathbf{q})|^2$. The particles interact with a hard-sphere volume fraction $\phi$.

This structure factor may be combined with the form factor to yield the total volume specific scattering cross section.

Use the locally monodisperse approximation ([[Small angle neutron scattering, SANS#label-eq:local_mono|this equation]]). An appropriate integration range for the Gaussian distribution might be $R\in[R_{av}-3\sigma;R_{av}+3\sigma]$.

The expression for $S(q)$ has been calculated (using the Percus-Yevick approximation for the closure relation<ref name="pedersen">J.S. Pedersen, ''Neutrons, X-rays, and Light'' (Elsevier, 2002)</ref>)

:$ S(q,R) = \dfrac{1}{1+\dfrac{24\phi}{2Rq}\cdot G(2Rq) } ,$

where in this equation

:$ \begin{align*}
G(A)
&=\dfrac{\alpha(\sin{A}-A\cos{A})}{A^2}+\dfrac{\beta(2A\sin{A}+(2-A^2)\cos{A}-2)}{A^3}\\
&\quad +\dfrac{\gamma(-A^4\cos{A}+4((3A^2-6)\cos{A}+(A^3-6A)\sin{A}+6))}{A^5}
\end{align*} $

and

:$ \alpha=\dfrac{(1+2\phi)^2}{(1-\phi)^4}, \quad \beta=\dfrac{-6\phi(1+\phi/2)^2}{(1-\phi)^4}, \quad \gamma=\dfrac{\phi \alpha}{2}. $

Calculate and plot the scattering cross-section per sample volume for the $R=60$ Å polydisperse spheres for volume fractions of 1 %, 5 %, 10 %, 25 % and 50 % using the expression above for the structure factor.

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
<figure id="fig:interacting_polydisperse_spheres"> [[File:XS interacting hard spheres R60AA deltarho3e10.png| thumb | 200px | <caption> The total SANS scattering cross section per sample volume for interacting polydisperse hard spheres with average radius $R=60$ Å. Also shown in one case is interacting monodisperse spheres.</caption>]] </figure>
The cross-section per sample volume for interacting hard spheres is shown in <xr id="fig:interacting_polydisperse_spheres">Figure %i </xr> . From the figure it is seen that increasing volumefraction of the spheres, and thereby increasing interaction between the spheres, gives an increasingly well-defined peak at $q\approx \pi / R=0.05$ Å$^{-1}$. Notice also how the low-$q$ intensity decreases with increasing volumefraction (interaction via the structurefactor).

{{hidden end}}

<references/>

Problem:Polydisperse spheres - Revision history

Wikiadmin: Wikiadmin moved page Problem: Polydisperse spheres to Problem:Polydisperse spheres

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ucph>Tommy: Created page with "We imagine a solution of polydisperse spheres with volume fraction \(\phi\). When we can assume $S(q)=1$ (typically for low concentrations, \(\phi \ll 1\)), the total SANS cro..."