Changeset 01dba26 in sasmodels for doc/guide/pd/polydispersity.rst
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- Sep 12, 2018 5:17:58 PM (6 years ago)
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doc/guide/pd/polydispersity.rst (modified) (8 diffs)
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doc/guide/pd/polydispersity.rst
rf41027b r01dba26 8 8 .. _polydispersityhelp: 9 9 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 10 Polydispersity & Orientational Distributions 11 -------------------------------------------- 12 13 For some models we can calculate the average intensity for a population of 14 particles that possess size and/or orientational (ie, angular) distributions. 15 In SasView we call the former *polydispersity* but use the parameter *PD* to 16 parameterise both. In other words, the meaning of *PD* in a model depends on 17 the actual parameter it is being applied too. 18 19 The resultant intensity is then normalized by the average particle volume such 20 that 17 21 18 22 .. math:: … … 21 25 22 26 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an 23 average over the sizedistribution $f(x; \bar x, \sigma)$, giving27 average over the distribution $f(x; \bar x, \sigma)$, giving 24 28 25 29 .. math:: … … 30 34 Each distribution is characterized by a center value $\bar x$ or 31 35 $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 36 <<<<<<< HEAD 32 37 the standard deviation, so read the description carefully), the number of 33 38 sigmas $N_\sigma$ to include from the tails of the distribution, and the … … 42 47 However, the distribution width applied to *orientation* (ie, angle-describing) 43 48 parameters is just $\sigma = \mathrm{PD}$. 49 ======= 50 the standard deviation, so read the description of the distribution carefully), 51 the number of sigmas $N_\sigma$ to include from the tails of the distribution, 52 and the number of points used to compute the average. The center of the 53 distribution is set by the value of the model parameter. 54 55 The distribution width applied to *volume* (ie, shape-describing) parameters 56 is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$. 57 However, the distribution width applied to *orientation* parameters is just 58 $\sigma = \mathrm{PD}$. 59 >>>>>>> master 44 60 45 61 $N_\sigma$ determines how far into the tails to evaluate the distribution, … … 51 67 52 68 Users 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. 69 polydispersion and/or orientational distributions to multiple parameters at 70 the same time, or increasing the number of points in the distribution, will 71 require patience! However, the calculations are generally more robust with 72 more data points or more angles. 56 73 57 74 The following distribution functions are provided: … … 69 86 70 87 88 **Beware: when the Polydispersity & Orientational Distribution panel in SasView is** 89 **first opened, the default distribution for all parameters is the Gaussian Distribution.** 90 **This may not be suitable. See Suggested Applications below.** 91 71 92 .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace 72 93 the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2), 73 94 351-353 <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). 95 in order to make the terminology describing distributions of chemical 96 properties unambiguous. However, these terms are unrelated to the 97 proportional size distributions and orientational distributions used in 98 SasView models. 79 99 80 100 Suggested Applications 81 101 ^^^^^^^^^^^^^^^^^^^^^^ 82 102 83 If applying polydispersion to parameters describing particle sizes, use103 If applying polydispersion to parameters describing particle sizes, consider using 84 104 the Lognormal or Schulz distributions. 85 105 86 106 If applying polydispersion to parameters describing interfacial thicknesses 87 or angular orientations, usethe Gaussian or Boltzmann distributions.107 or angular orientations, consider using the Gaussian or Boltzmann distributions. 88 108 89 109 If applying polydispersion to parameters describing angles, use the Uniform … … 422 442 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 423 443 424 Many commercial Dynamic Light Scattering (DLS) instruments produce a size 425 polydispersity parameter, sometimes even given the symbol $p$\ ! This 426 parameter is defined as the relative standard deviation coefficient of 427 variation of the size distribution and is NOT the same as the polydispersity 428 parameters in the Lognormal and Schulz distributions above (though they all 429 related) except when the DLS polydispersity parameter is <0.13. 430 431 .. math:: 432 433 p_{DLS} = \sqrt(\nu / \bar x^2) 434 435 where $\nu$ is the variance of the distribution and $\bar x$ is the mean 436 value of $x$. 444 Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and 445 it should not be assumed that any of the following can be simply equated with 446 the polydispersity *PD* parameter used in SasView. 447 448 The dimensionless **Polydispersity Index (PI)** is a measure of the width of the 449 distribution of autocorrelation function decay rates (*not* the distribution of 450 particle sizes itself, though the two are inversely related) and is defined by 451 ISO 22412:2017 as 452 453 .. math:: 454 455 PI = \mu_{2} / \bar \Gamma^2 456 457 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the 458 intensity-weighted average value, of the distribution of decay rates. 459 460 *If the distribution of decay rates is Gaussian* then 461 462 .. math:: 463 464 PI = \sigma^2 / 2\bar \Gamma^2 465 466 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** 467 to be defined as 468 469 .. math:: 470 471 RP = \sigma / \bar \Gamma = \sqrt{2 \cdot PI} 472 473 PI values smaller than 0.05 indicate a highly monodisperse system. Values 474 greater than 0.7 indicate significant polydispersity. 475 476 The **size polydispersity P-parameter** is defined as the relative standard 477 deviation coefficient of variation 478 479 .. math:: 480 481 P = \sqrt\nu / \bar R 482 483 where $\nu$ is the variance of the distribution and $\bar R$ is the mean 484 value of $R$. Here, the product $P \bar R$ is *equal* to the standard 485 deviation of the Lognormal distribution. 486 487 P values smaller than 0.13 indicate a monodisperse system. 437 488 438 489 For more information see: 439 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 490 491 `ISO 22412:2017, International Standards Organisation (2017) <https://www.iso.org/standard/65410.html>`_. 492 493 `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/>`_. 494 495 `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>`_. 496 497 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143. 498 499 T Allen, in *Particle Size Measurement*, 4th Edition, Chapman & Hall, London (1990). 440 500 441 501 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 447 507 | 2018-03-20 Steve King 448 508 | 2018-04-04 Steve King 509 | 2018-08-09 Steve King
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