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Polydispersity Distributions

With some models in sasmodels we can calculate the average form factor for a population of particles that exhibit size and/or orientational polydispersity. The resultant form factor is normalized by the average particle volume such that

P(q) = scaleF*F⟩ ⁄ V + background

where $F$ is the scattering amplitude and $langlecdotrangle$ denotes an average over the size distribution.

Each distribution is characterized by its center $bar x$, its width $sigma$, the number of sigmas $N_sigma$ to include from the tails, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. Volume parameters have polydispersity PD (not to be confused with a molecular weight distributions in polymer science) leading to a size distribution of width $text{PD} = sigma / bar x$, but orientation parameters use an angular distributions of width $sigma$. $N_sigma$ determines how far into the tails to evaluate the distribution, with larger values of $N_sigma$ required for heavier tailed distributions. The scattering in general falls rapidly with $qr$ so the usual assumption that $G(r - 3sigma_r)$ is tiny and therefore $f(r - 3sigma_r)G(r - 3sigma_r)$ will not contribute much to the average may not hold when particles are large. This, too, will require increasing $N_sigma$.

Users should note that the averaging computation is very intensive. Applying polydispersion to multiple parameters at the same time or increasing the number of points in the distribution will require patience! However, the calculations are generally more robust with more data points or more angles.

The following five distribution functions are provided:

  • Rectangular Distribution
  • Gaussian Distribution
  • Lognormal Distribution
  • Schulz Distribution
  • Array Distribution

These are all implemented as number-average distributions.

Rectangular Distribution

The Rectangular Distribution is defined as

f(x) = (1)/( Norm) 1   for |x − x| ≤ w       0   for |x − x| > w

where $bar x$ is the mean of the distribution, $w$ is the half-width, and Norm is a normalization factor which is determined during the numerical calculation.

Note that the standard deviation and the half width $w$ are different!

The standard deviation is

σ = w ⁄ (3)

whilst the polydispersity is

PD = σ ⁄ x
pd_rectangular.jpg

Rectangular distribution.

Gaussian Distribution

The Gaussian Distribution is defined as

f(x) = (1)/( Norm)exp − ((x − x)2)/(2σ2)

where $bar x$ is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.

The polydispersity is

PD = σ ⁄ x
pd_gaussian.jpg

Normal distribution.

Lognormal Distribution

The Lognormal Distribution is defined as

f(x) = (1)/( Norm)(1)/(xp)exp − ((ln(x) − μ)2)/(2p2)

where $mu=ln(x_text{med})$ when $x_text{med}$ is the median value of the distribution, and Norm is a normalization factor which will be determined during the numerical calculation.

The median value for the distribution will be the value given for the respective size parameter, for example, radius=60.

The polydispersity is given by $sigma$

PD = p

For the angular distribution

p = σ ⁄ xmed

The mean value is given by $bar x = exp(mu+ p^2/2)$. The peak value is given by $max x = exp(mu - p^2)$.

pd_lognormal.jpg

Lognormal distribution.

This distribution function spreads more, and the peak shifts to the left, as $p$ increases, so it requires higher values of $N_sigma$ and more points in the distribution.

Schulz Distribution

The Schulz distribution is defined as

f(x) = (1)/( Norm)(z + 1)z + 1(x ⁄ x)z(exp[ − (z + 1)x ⁄ x])/(xΓ(z + 1))

where $bar x$ is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation, and $z$ is a measure of the width of the distribution such that

z = (1 − p2) ⁄ p2

The polydispersity is

p = σ ⁄ x

Note that larger values of PD might need larger number of points and $N_sigma$. For example, at PD=0.7 and radius=60 |Ang|, Npts>=160 and Nsigmas>=15 at least.

pd_schulz.jpg

Schulz distribution.

For further information on the Schulz distribution see: M Kotlarchyk & S-H Chen, J Chem Phys, (1983), 79, 2461.

Array Distribution

This user-definable distribution should be given as as a simple ASCII text file where the array is defined by two columns of numbers: $x$ and $f(x)$. The $f(x)$ will be normalized to 1 during the computation.

Example of what an array distribution file should look like:

30 0.1
32 0.3
35 0.4
36 0.5
37 0.6
39 0.7
41 0.9

Only these array values are used computation, therefore the parameter value given for the model will have no affect, and will be ignored when computing the average. This means that any parameter with an array distribution will not be fittable.

Note about DLS polydispersity

Many commercial Dynamic Light Scattering (DLS) instruments produce a size polydispersity parameter, sometimes even given the symbol $p$! This parameter is defined as the relative standard deviation coefficient of variation of the size distribution and is NOT the same as the polydispersity parameters in the Lognormal and Schulz distributions above (though they all related) except when the DLS polydispersity parameter is <0.13.

For more information see: S King, C Washington & R Heenan, Phys Chem Chem Phys, (2005), 7, 143

Document History

2015-05-01 Steve King
2017-05-08 Paul Kienzle

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