Difference between revisions of "Circular orientation distribution function"
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| − | In assessing the orientation of aligned materials, one can use the [[orientation order parameter]] to quantify order. Another possibility is to fit [[scattering]] data using an equation that has 'circular wrapping' (i.e. periodic along <math>\scriptstyle 2 \pi</math>). Ruland ''et al.'' present such an equation: | + | In assessing the orientation of aligned materials, one can use the [[orientation order parameter]] to quantify order. Another possibility is to fit [[scattering]] data using an equation that has 'circular wrapping' (i.e. periodic along <math>\scriptstyle 2 \pi</math>). |
| + | |||
| + | =<math>\eta </math> function= | ||
| + | Ruland ''et al.'' present such an equation: | ||
:<math> | :<math> | ||
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Where <math>\scriptstyle \chi </math> is the angle along the arc of the [[scattering features|scattering ring/feature]]. The single fit parameter (<math>\scriptstyle \eta </math>) is convenient in that it behaves in a similar way to an order parameter: a value close to 1.0 indicates strong alignment, while progressively smaller values indicate lesser alignment. For a random sample, the scattering is isotropic and <math>\scriptstyle \eta = 0</math>. | Where <math>\scriptstyle \chi </math> is the angle along the arc of the [[scattering features|scattering ring/feature]]. The single fit parameter (<math>\scriptstyle \eta </math>) is convenient in that it behaves in a similar way to an order parameter: a value close to 1.0 indicates strong alignment, while progressively smaller values indicate lesser alignment. For a random sample, the scattering is isotropic and <math>\scriptstyle \eta = 0</math>. | ||
| + | [[Image:Eta func-I chi.png|500px]] | ||
| + | |||
| + | ==Normalized== | ||
| + | The function normalized so that the maximum is always at 1 would be: | ||
| + | :<math> | ||
| + | I_{\mathrm{norm}} (\chi) = \frac{(1+\eta)^2 - 4 \eta}{(1+\eta)^2 - 4 \eta \cos^2 \chi} | ||
| + | </math> | ||
| + | |||
| + | ==References== | ||
| + | * Ruland, W.; Tompa, H., The Effect of Preferred Orientation on the Intensity Distribution of (Hk) Interferences. Acta Crystallographica Section A 1968, 24, 93-99. [http://dx.doi.org/10.1107/S0567739468000112 10.1107/S0567739468000112] | ||
| + | * Ruland, W.; Smarsly, B., Saxs of Self-Assembled Oriented Lamellar Nanocomposite Films: An Advanced Method of Evaluation. J. Appl. Crystallogr. 2004, 37, 575-584. [http://dx.doi.org/10.1107/S0021889804011288 doi: 10.1107/S0021889804011288] | ||
| + | * [[Kevin G. Yager]], Christopher Forrey, Gurpreet Singh, Sushil K. Satija, Kirt A. Page, Derek L. Patton, Jack F. Douglas, Ronald L. Jones and Alamgir Karim [http://pubs.rsc.org/en/Content/ArticleLanding/2015/SM/C5SM00896D#!divAbstract Thermally-induced transition of lamellae orientation in block-copolymer films on ‘neutral’ nanoparticle-coated substrates] ''Soft Matter'' '''2015''' [http://dx.doi.org/10.1039/C5SM00896D doi: 10.1039/C5SM00896D] | ||
| + | * De France, K.J.; Yager, K.G.; Hoare, T.; Cranston, E.D. "Cooperative Ordering and Kinetics of Cellulose Nanocrystal Alignment in a Magnetic Field" Langmuir 2016, 32, 7564–7571. [http://dx.doi.org/10.1021/acs.langmuir.6b01827 doi: 10.1021/acs.langmuir.6b01827] | ||
| + | |||
| + | =Maier-Saupe distribution parameter= | ||
| + | |||
| + | :<math> | ||
| + | I(\chi) = \frac{1}{c} \exp \left [ m \cos ^2 \chi \right ] | ||
| + | </math> | ||
| + | |||
| + | Where <math>\scriptstyle m</math> is a parameter that can be related to the [[orientation order parameter|order parameter]] <math>\scriptstyle S</math>; specifically <math>\scriptstyle m =0 </math> is for an isotropic distribution (<math>\scriptstyle S = 0</math>), while <math>\scriptstyle m \to \infty</math> is for a well-aligned system (<math>\scriptstyle S \to 1</math>). | ||
| + | |||
| + | [[Image:MaierSaupe-ODF-01.png|500px]] | ||
| + | [[Image:MaierSaupe-ODF-021.png|500px]] | ||
| + | |||
| + | The parameter ''c'' can be used to normalize: | ||
| + | [[Image:MaierSaupe-ODF-01b.png|500px]] | ||
| + | [[Image:MaierSaupe-ODF-02b.png|500px]] | ||
| − | == | + | ==References== |
| − | * | + | * B. J. Lemaire, P. Panine, J. C. P. Gabriel and P. Davidson [http://iopscience.iop.org/article/10.1209/epl/i2002-00159-8/meta The measurement by SAXS of the nematic order parameter of laponite gels] ''Europhysics Letters'' '''2002''', 59 (1), 55-61. |
| − | * | + | * Maier W. and Saupe A., Z. Naturforsch. A, 13 ([http://www.znaturforsch.com/aa/v13a/c13a.htm 1958]) [http://zfn.mpdl.mpg.de/data/Reihe_A/13/ZNA-1958-13a-0564_n.pdf 564]; 14 ([http://www.znaturforsch.com/aa/v14a/c14a.htm 1959]) [http://zfn.mpdl.mpg.de/data/Reihe_A/14/ZNA-1959-14a-0882.pdf 882]; 15 ([http://www.znaturforsch.com/aa/v15a/c15a.htm 1960]) [http://zfn.mpdl.mpg.de/data/Reihe_A/15/ZNA-1960-15a-0287.pdf 287]. |
Latest revision as of 14:22, 15 April 2022
In assessing the orientation of aligned materials, one can use the orientation order parameter to quantify order. Another possibility is to fit scattering data using an equation that has 'circular wrapping' (i.e. periodic along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 2 \pi} ).
Contents
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta } function
Ruland et al. present such an equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I (\chi) = \frac{1 - \eta^2}{(1+\eta)^2 - 4 \eta \cos^2 \chi} }
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi } is the angle along the arc of the scattering ring/feature. The single fit parameter (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \eta } ) is convenient in that it behaves in a similar way to an order parameter: a value close to 1.0 indicates strong alignment, while progressively smaller values indicate lesser alignment. For a random sample, the scattering is isotropic and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \eta = 0} .
Normalized
The function normalized so that the maximum is always at 1 would be:
References
- Ruland, W.; Tompa, H., The Effect of Preferred Orientation on the Intensity Distribution of (Hk) Interferences. Acta Crystallographica Section A 1968, 24, 93-99. 10.1107/S0567739468000112
- Ruland, W.; Smarsly, B., Saxs of Self-Assembled Oriented Lamellar Nanocomposite Films: An Advanced Method of Evaluation. J. Appl. Crystallogr. 2004, 37, 575-584. doi: 10.1107/S0021889804011288
- Kevin G. Yager, Christopher Forrey, Gurpreet Singh, Sushil K. Satija, Kirt A. Page, Derek L. Patton, Jack F. Douglas, Ronald L. Jones and Alamgir Karim Thermally-induced transition of lamellae orientation in block-copolymer films on ‘neutral’ nanoparticle-coated substrates Soft Matter 2015 doi: 10.1039/C5SM00896D
- De France, K.J.; Yager, K.G.; Hoare, T.; Cranston, E.D. "Cooperative Ordering and Kinetics of Cellulose Nanocrystal Alignment in a Magnetic Field" Langmuir 2016, 32, 7564–7571. doi: 10.1021/acs.langmuir.6b01827
Maier-Saupe distribution parameter
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\chi) = \frac{1}{c} \exp \left [ m \cos ^2 \chi \right ] }
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle m} is a parameter that can be related to the order parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle S} ; specifically Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle m =0 } is for an isotropic distribution (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle S = 0} ), while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle m \to \infty} is for a well-aligned system (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle S \to 1} ).
The parameter c can be used to normalize:
