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A '''biaxial nematic''' is a spatially homogeneous [[liquid crystal]] with three distinct optical axes. This is to be contrasted to a simple [[nematic]], which has a single preferred axis, around which the system is rotationally symmetric. The [[symmetry group]] of a biaxial nematic is <math>D_{2h}</math> i.e. that of a rectangular right parallelepiped, having 3 orthogonal <math>C_2</math> axes and three orthogonal mirror planes. In a frame co-aligned with optical axes the second rank [[order parameter]] [[tensor]] of a biaxial nematic has the form
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
:<math>
Q=
\begin{pmatrix}
-\frac{1}{2}S+T & 0 &0 \\
0 &-\frac{1}{2}S-T & 0 \\
0 & 0& S\\
\end{pmatrix}
</math>


where
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


<math>S</math> is the standard nematic scalar order parameter
Registered users will be able to choose between the following three rendering modes:


<math>T</math> a measure of the biaxiality.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The first report of a biaxial nematic appeared in 2004<ref>
<!--'''PNG''' (currently default in production)
{{cite journal
:<math forcemathmode="png">E=mc^2</math>
|last1=Madsen |first1=L. A.
|last2=Dingemans |first2=T. J.
|last3=Nakata |first3=M.
|last4=Samulski |first4=E. T.
|year=2004
|title=Thermotropic Biaxial Nematic Liquid Crystals
|journal=[[Physical Review Letters]]
|volume=92 |pages=145505
|doi=10.1103/PhysRevLett.92.145505 |pmid=15089552 |bibcode=2004PhRvL..92n5505M
|issue=14
}}</ref><ref>
{{cite journal
|last1=Prasad |first1=V.
|last2=Kang |first2=S.-Woong
|last3=Suresh |first3=K. A.
|last4=Joshi |first4=Leela
|last5=Wang |first5=Qingbing
|last6=Kumar |first6=Satyendra
|year=2005
  |title=Thermotropic Uniaxial and Biaxial Nematic and Smectic Phases in Bent-Core Mesogens
|journal=[[Journal of the American Chemical Society]]
|volume=127 |pages=17224
|doi=10.1021/ja052769n
|issue=49
}}</ref> based on a [[boomerang]] shaped [[oxadiazole]] '''bent-core mesogen'''. The biaxial nematic phase for this particular compound only occurs at temperatures around 200 °C and is preceded by as yet unidentified [[smectic]] phases.


[[Image:Biaxialnematic.gif|center|Biaxial nematic boomerang liquid crystal]]
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


It is also found that this material can segregate into [[Chirality (chemistry)|chiral]] domains of opposite handedness.<ref>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
{{cite journal
|last1=Görtz |first1=V.
|last2=Goodby |first2=J. W.
|year=2005
|title=Enantioselective segregation in achiral nematic liquid crystals
|journal=[[Chemical Communications]]
|pages=3262
|doi=10.1039/B503846D
|issue=26
}}</ref> For this to happen the boomerang shaped molecules adopt a helical superstructure.


In one azo bent-core mesogen a thermal transition is found from a uniaxial N<sub>u</sub> to a  biaxial nematic N<sub>b</sub> mesophase,<ref>
==Demos==
{{cite journal
|last1=Prasad |first1=V.
|last2=Kang |first2=S.-W.
|last3=Suresh |first3=K. A.
|last4=Joshi |first4=L.
|last5=Wang |first5=Q.
|last6=Kumar |first6=S.
|year=2005
|title=Thermotropic Uniaxial and Biaxial Nematic and Smectic Phases in Bent-Core Mesogens
|journal=[[Journal of the American Chemical Society]]
|volume=127 |pages=17224
|doi=10.1021/ja052769n
|issue=49
}}</ref> as predicted by theory and simulation.<ref>
{{cite journal
|last1=Bates |first1=M.
|last2=Luckhurst |first2=G.
|year=2005
|title=Biaxial nematic phases and V-shaped molecules: A Monte Carlo simulation study
|journal=[[Physical Review E]]
|volume=72 |pages=051702
|doi=10.1103/PhysRevE.72.051702
|bibcode = 2005PhRvE..72e1702B
|issue=5 }}</ref> This transition is observed on heating from the N<sub>u</sub> phase with [[Polarizing optical microscopy]] as a change in [[Schlieren texture]] and increased light transmittance and from [[x-ray diffraction]] as the splitting of the nematic reflection. The transition is a [[Phase transition|second order transition]] with low energy content and therefore not observed in [[differential scanning calorimetry]]. The positional order parameter for the uniaxial nematic phase is 0.75 to 1.5 times the mesogen length and for the biaxial nematic phase 2 to 3.3 times the mesogen length.


[[Image:Biaxialnematic2005.png|center|600px|Azo bent-core mesogen thermal transitions in °C: K 82.8 Sy 93.4 Sx 104.3 Sc 118.5 Nb 149 Nu 176.5 I]]
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Another strategy towards biaxial nematics is the use of mixtures of classical rodlike mesogens and disklike [[discotic]] mesogens.  The biaxial nematic phase is expected to be located below the minimum in the rod-disk phase diagram. In one study<ref>
{{cite journal
|last1=Apreutesei |first1=D.
|last2=Mehl |first2=G. H.
|year=2006
|title=Completely miscible disc and rod shaped molecules in the nematic phase
|journal=[[Chemical Communications]]
|pages=609
|doi=10.1039/b512120e
|issue=6
}}</ref> a miscible system of rods and disks is actually found although the biaxial nematic phase remains elusive.


==See also==
* accessibility:
* [[Chromonic]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
* [[Liquid crystal]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
* [[Liquid crystal display]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
* [[Liquid crystal polymer]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
* [[Lyotropic liquid crystal]]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
* [[Plastic crystallinity]]
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
* [[Smart glass]]
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
* [[Thermochromics]]


== References ==
==Test pages ==
{{reflist}}


[[Category:Phases of matter]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
[[Category:Crystallography]]
*[[Displaystyle]]
[[Category:Liquid crystals]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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