Changeset 97d172c in sasmodels


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Timestamp:
Aug 9, 2018 10:17:24 AM (2 months ago)
Author:
smk78
Branches:
master, F1F2models_grethe, beta_approx, cuda-test, py3, ticket-1074-gammainc, ticket-1084, ticket-1102-pinhole, ticket-1142-plugin-reload, ticket-1155BE_PolyElectrolyte, ticket-1157, ticket-608-user-defined-weights, ticket_1156
Children:
86bb5df
Parents:
e9b17b18
Message:

Overhaul of polydispersity.rst Intro and DLS sections

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1 edited

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  • doc/guide/pd/polydispersity.rst

    rd712a0f r97d172c  
    88.. _polydispersityhelp: 
    99 
    10 Polydispersity Distributions 
    11 ---------------------------- 
    12  
    13 With some models in sasmodels we can calculate the average intensity for a 
    14 population of particles that exhibit size and/or orientational 
    15 polydispersity. The resultant intensity is normalized by the average 
    16 particle volume such that 
     10Polydispersity & Orientational Distributions 
     11-------------------------------------------- 
     12 
     13For some models we can calculate the average intensity for a population of  
     14particles that possess size and/or orientational (ie, angular) distributions.  
     15In SasView we call the former *polydispersity* but use the parameter *PD* to  
     16parameterise both. In other words, the meaning of *PD* in a model depends on  
     17the actual parameter it is being applied too. 
     18 
     19The resultant intensity is then normalized by the average particle volume such  
     20that 
    1721 
    1822.. math:: 
     
    2125 
    2226where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an  
    23 average over the size distribution $f(x; \bar x, \sigma)$, giving 
     27average over the distribution $f(x; \bar x, \sigma)$, giving 
    2428 
    2529.. math:: 
     
    3034Each distribution is characterized by a center value $\bar x$ or 
    3135$x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 
    32 the standard deviation, so read the description carefully), the number of 
    33 sigmas $N_\sigma$ to include from the tails of the distribution, and the 
    34 number of points used to compute the average. The center of the distribution 
    35 is set by the value of the model parameter. The meaning of a polydispersity  
    36 parameter *PD* (not to be confused with a molecular weight distributions  
    37 in polymer science) in a model depends on the type of parameter it is being  
    38 applied too. 
     36the standard deviation, so read the description of the distribution carefully),  
     37the number of sigmas $N_\sigma$ to include from the tails of the distribution,  
     38and the number of points used to compute the average. The center of the  
     39distribution is set by the value of the model parameter. 
    3940 
    4041The distribution width applied to *volume* (ie, shape-describing) parameters  
    4142is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$.  
    42 However, the distribution width applied to *orientation* (ie, angle-describing)  
    43 parameters is just $\sigma = \mathrm{PD}$. 
     43However, the distribution width applied to *orientation* parameters is just  
     44$\sigma = \mathrm{PD}$. 
    4445 
    4546$N_\sigma$ determines how far into the tails to evaluate the distribution, 
     
    5152 
    5253Users should note that the averaging computation is very intensive. Applying 
    53 polydispersion to multiple parameters at the same time or increasing the 
    54 number of points in the distribution will require patience! However, the 
    55 calculations are generally more robust with more data points or more angles. 
     54polydispersion and/or orientational distributions to multiple parameters at  
     55the same time, or increasing the number of points in the distribution, will  
     56require patience! However, the calculations are generally more robust with  
     57more data points or more angles. 
    5658 
    5759The following distribution functions are provided: 
     
    7274           the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2),  
    7375           351-353 <http://media.iupac.org/publications/pac/2009/pdf/8102x0351.pdf>`_  
    74            in order to make the terminology describing distributions of properties  
    75            unambiguous. Throughout the SasView documentation we continue to use the  
    76            term polydispersity because one of the consequences of the IUPAC change is  
    77            that orientational polydispersity would not meet their new criteria (which  
    78            requires dispersity to be dimensionless). 
     76           in order to make the terminology describing distributions of chemical  
     77           properties unambiguous. However, these terms are unrelated to the  
     78           proportional size distributions and orientational distributions used in  
     79           SasView models. 
    7980 
    8081Suggested Applications 
    8182^^^^^^^^^^^^^^^^^^^^^^ 
    8283 
    83 If applying polydispersion to parameters describing particle sizes, use 
     84If applying polydispersion to parameters describing particle sizes, consider using 
    8485the Lognormal or Schulz distributions. 
    8586 
    8687If applying polydispersion to parameters describing interfacial thicknesses 
    87 or angular orientations, use the Gaussian or Boltzmann distributions. 
     88or angular orientations, consider using the Gaussian or Boltzmann distributions. 
    8889 
    8990If applying polydispersion to parameters describing angles, use the Uniform  
     
    332333^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
    333334 
    334 Many commercial Dynamic Light Scattering (DLS) instruments produce a size 
    335 polydispersity parameter, sometimes even given the symbol $p$\ ! This 
    336 parameter is defined as the relative standard deviation coefficient of 
    337 variation of the size distribution and is NOT the same as the polydispersity 
    338 parameters in the Lognormal and Schulz distributions above (though they all 
    339 related) except when the DLS polydispersity parameter is <0.13. 
    340  
    341 .. math:: 
    342  
    343     p_{DLS} = \sqrt(\nu / \bar x^2) 
    344  
    345 where $\nu$ is the variance of the distribution and $\bar x$ is the mean 
    346 value of $x$. 
     335Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and  
     336it should not be assumed that any of the following can be simply equated with  
     337the polydispersity *PD* parameter used in SasView. 
     338 
     339The dimensionless *Polydispersity Index (PI)* is a measure of the width of the  
     340distribution of autocorrelation function decay rates (*not* the distribution of  
     341particle sizes itself, though the two are inversely related) and is defined by  
     342ISO 22412:2017 as 
     343 
     344.. math:: 
     345 
     346    PI = \mu_{2} / \bar \Gamma^2 
     347 
     348where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the  
     349intensity-weighted average value, of the distribution of decay rates. 
     350 
     351*If the distribution of decay rates is Gaussian* then 
     352 
     353.. math:: 
     354 
     355    PI = \sigma^2 / 2\bar \Gamma^2 
     356 
     357where $\sigma$ is the standard deviation, allowing a *Relative Polydispersity (RP)*  
     358to be defined as 
     359 
     360.. math:: 
     361 
     362    RP = \sigma / \bar \Gamma = \sqrt{2.PI} 
     363 
     364PI values smaller than 0.05 indicate a highly monodisperse system. Values  
     365greater than 0.7 indicate significant polydispersity. 
     366 
     367The *size polydispersity P-parameter* is defined as the relative standard  
     368deviation coefficient of variation   
     369 
     370.. math:: 
     371 
     372    P = \sqrt\nu / \bar R 
     373 
     374where $\nu$ is the variance of the distribution and $\bar R$ is the mean 
     375value of $R$. Here, the product $P \bar R$ is *equal* to the standard  
     376deviation of the Lognormal distribution. 
     377 
     378P values smaller than 0.13 indicate a monodisperse system. 
    347379 
    348380For more information see: 
    349 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 
     381`ISO 22412:2017, International Standards Organisation (2017) <https://www.iso.org/standard/65410.html>`_.  
     382`Polydispersity: What does it mean for DLS and Chromatography <http://www.materials-talks.com/blog/2014/10/23/polydispersity-what-does-it-mean-for-dls-and-chromatography/>`_.  
     383`Dynamic Light Scattering: Common Terms Defined, Whitepaper WP111214. Malvern Instruments (2011) <http://www.biophysics.bioc.cam.ac.uk/wp-content/uploads/2011/02/DLS_Terms_defined_Malvern.pdf>`_.  
     384S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143.  
     385T Allen, in *Particle Size Measurement*, 4th Edition, Chapman & Hall, London (1990). 
    350386 
    351387.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    357393| 2018-03-20 Steve King 
    358394| 2018-04-04 Steve King 
     395| 2018-08-09 Steve King 
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