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Polydispersity Distributions

With some models in sasmodels we can calculate the average intensity for a population of particles that exhibit size and/or orientational polydispersity. The resultant intensity 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 a center value $bar x$ or $x_text{med}$, a width parameter $sigma$ (note this is not necessarily the standard deviation, so read the description carefully), the number of sigmas $N_sigma$ to include from the tails of the distribution, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. The meaning of a polydispersity parameter PD (not to be confused with a molecular weight distributions in polymer science) in a model depends on the type of parameter it is being applied too.

The distribution width applied to volume (ie, shape-describing) parameters is relative to the center value such that $sigma = mathrm{PD} cdot bar x$. However, the distribution width applied to orientation (ie, angle-describing) parameters is just $sigma = mathrm{PD}$.

$N_sigma$ 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 distribution functions are provided:

  • Uniform Distribution
  • Rectangular Distribution
  • Gaussian Distribution
  • Boltzmann Distribution
  • Lognormal Distribution
  • Schulz Distribution
  • Array Distribution
  • User-defined Distributions

These are all implemented as number-average distributions.

Suggested Applications

If applying polydispersion to parameters describing particle sizes, use the Lognormal or Schulz distributions.

If applying polydispersion to parameters describing interfacial thicknesses or angular orientations, use the Gaussian or Boltzmann distributions.

If applying polydispersion to parameters describing angles, use the Uniform distribution. Beware of using distributions that are always positive (eg, the Lognormal) because angles can be negative!

The array distribution provides a very simple means of implementing a user- defined distribution, but without any fittable parameters. Greater flexibility is conferred by the user-defined distribution.

Uniform Distribution

The Uniform Distribution is defined as

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

where $bar x$ ($x_text{mean}$ in the figure) is the mean of the distribution, $sigma$ is the half-width, and Norm is a normalization factor which is determined during the numerical calculation.

The polydispersity in sasmodels is given by

PD = σ ⁄ x
pd_uniform.jpg

Uniform distribution.

The value $N_sigma$ is ignored for this distribution.

Rectangular Distribution

The Rectangular Distribution is defined as

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

where $bar x$ ($x_text{mean}$ in the figure) 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 in sasmodels is given by

PD = σ ⁄ x
pd_rectangular.jpg

Rectangular distribution.

Note

The Rectangular Distribution is deprecated in favour of the Uniform Distribution above and is described here for backwards compatibility with earlier versions of SasView only.

Gaussian Distribution

The Gaussian Distribution is defined as

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

where $bar x$ ($x_text{mean}$ in the figure) is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.

The polydispersity in sasmodels is given by

PD = σ ⁄ x
pd_gaussian.jpg

Normal distribution.

Boltzmann Distribution

The Boltzmann Distribution is defined as

f(x) = (1)/( Norm)exp − (|x − x|)/(σ)

where $bar x$ ($x_text{mean}$ in the figure) is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.

The width is defined as

σ = (kT)/(E)

which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a characteristic energy per particle.

pd_boltzmann.jpg

Boltzmann distribution.

Lognormal Distribution

The Lognormal Distribution describes a function of $x$ where $ln (x)$ has a normal distribution. The result is a distribution that is skewed towards larger values of $x$.

The Lognormal Distribution is defined as

f(x) = (1)/( Norm)(1)/(xσ)exp − (1)/(2)((ln(x) − μ)/(σ))2

where Norm is a normalization factor which will be determined during the numerical calculation, $mu=ln(x_text{med})$ and $x_text{med}$ is the median value of the lognormal distribution, but $sigma$ is a parameter describing the width of the underlying normal distribution.

$x_text{med}$ will be the value given for the respective size parameter in sasmodels, for example, radius=60.

The polydispersity in sasmodels is given by

PD = σ = p ⁄ xmed

The mean value of the distribution is given by $bar x = exp(mu+ sigma^2/2)$ and the peak value by $max x = exp(mu - sigma^2)$.

The variance (the square of the standard deviation) of the lognormal distribution is given by

ν = [exp(σ2) − 1]exp(2μ + σ2)

Note that larger values of PD might need a larger number of points and $N_sigma$.

pd_lognormal.jpg

Lognormal distribution for PD=0.1.

For further information on the Lognormal distribution see: http://en.wikipedia.org/wiki/Log-normal_distribution and http://mathworld.wolfram.com/LogNormalDistribution.html

Schulz Distribution

The Schulz (sometimes written Schultz) distribution is similar to the Lognormal distribution, in that it is also skewed towards larger values of $x$, but which has computational advantages over the Lognormal 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 $bar x$ ($x_text{mean}$ in the figure) is the mean of the distribution, 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

where $p$ is the polydispersity in sasmodels given by

PD = p = σ ⁄ x

and $sigma$ is the RMS deviation from $bar x$.

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

pd_schulz.jpg

Schulz distribution.

For further information on the Schulz distribution see: M Kotlarchyk & S-H Chen, J Chem Phys, (1983), 79, 2461 and M Kotlarchyk, RB Stephens, and JS Huang, J Phys Chem, (1988), 92, 1533

Array Distribution

This user-definable distribution should be given 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 fitable.

User-defined Distributions

You can also define your own distribution by creating a python file defining a Distribution object with a _weights method. The _weights method takes center, sigma, lb and ub as arguments, and can access self.npts and self.nsigmas from the distribution. They are interpreted as follows:

  • center the value of the shape parameter (for size dispersity) or zero if it is an angular dispersity. This parameter may be fitted.
  • sigma the width of the distribution, which is the polydispersity parameter times the center for size dispersity, or the polydispersity parameter alone for angular dispersity. This parameter may be fitted.
  • lb, ub are the parameter limits (lower & upper bounds) given in the model definition file. For example, a radius parameter has lb equal to zero. A volume fraction parameter would have lb equal to zero and ub equal to one.
  • self.nsigmas the distance to go into the tails when evaluating the distribution. For a two parameter distribution, this value could be co-opted to use for the second parameter, though it will not be available for fitting.
  • self.npts the number of points to use when evaluating the distribution. The user will adjust this to trade calculation time for accuracy, but the distribution code is free to return more or fewer, or use it for the third parameter in a three parameter distribution.

As an example, the code following wraps the Laplace distribution from scipy stats:

import numpy as np
from scipy.stats import laplace
from sasmodels import weights
class Dispersion(weights.Dispersion):
    r"""
    Laplace distribution
    .. math::
        w(x) = e^{-\sigma |x - \mu|}
    """
    type = "laplace"
    default = dict(npts=35, width=0, nsigmas=3)  # default values
    def _weights(self, center, sigma, lb, ub):
        x = self._linspace(center, sigma, lb, ub)
        wx = laplace.pdf(x, center, sigma)
        return x, wx

You can plot the weights for a given value and width using the following:

from numpy import inf
from matplotlib import pyplot as plt
from sasmodels import weights
# reload the user-defined weights
weights.load_weights()
x, wx = weights.get_weights('laplace', n=35, width=0.1, nsigmas=3, value=50,
                            limits=[0, inf], relative=True)
# plot the weights
plt.interactive(True)
plt.plot(x, wx, 'x')

The self.nsigmas and self.npts parameters are normally used to control the accuracy of the distribution integral. The self._linspace function uses them to define the x values (along with the center, sigma, lb, and ub which are passed as parameters). If you repurpose npts or nsigmas you will need to generate your own x. Be sure to honour the limits lb and ub, for example to disallow a negative radius or constrain the volume fraction to lie between zero and one.

To activate a user-defined distribution, set the following environment variable:

SASMODELS_WEIGHTS=path/to/folder/name_of_distribution.py

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.

pDLS = (()ν ⁄ x2)

where $nu$ is the variance of the distribution and $bar x$ is the mean value of $x$.

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
2018-03-20 Steve King
2018-04-04 Steve King

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