Editing Liquid crystal (section)

==Theoretical treatment of liquid crystals==
Microscopic theoretical treatment of fluid phases can become quite complicated, owing to the high material density, meaning that strong interactions, hard-core repulsions, and many-body correlations cannot be ignored. In the case of liquid crystals, anisotropy in all of these interactions further complicates analysis. There are a number of fairly simple theories, however, that can at least predict the general behavior of the phase transitions in liquid crystal systems.

===Director===

As we already saw above, the nematic liquid crystals are composed of rod-like molecules with the long axes of neighboring molecules aligned approximately to one another. To describe this anisotropic structure, a dimensionless unit vector '''''n''''' called the ''director'', is introduced to represent the direction of preferred orientation of molecules in the neighborhood of any point. Because there is no physical polarity along the director axis, '''''n''''' and '''''-n''''' are fully equivalent.<ref name=b1/>

===Order parameter===

[[File:Nematic-Director.png|thumb|The ''local nematic director'', which is also the ''local optical axis'', is given by the spatial and temporal average of the long molecular axes.]]

The description of liquid crystals involves an analysis of order. A second rank symmetric traceless tensor order parameter, the [[Q tensor]] is used to describe the orientational order of the most general [[biaxial nematic]] liquid crystal. However, to describe the more common case of uniaxial nematic liquid crystals, a scalar order parameter is sufficient.<ref>{{cite book |last1=Chaikin |first1=P. M. |last2=Lubensky |first2=T. C. |title=Principles of condensed matter physics |date=1995 |location=Cambridge |isbn=9780521794503 |page=168 |publisher= Cambridge University Press }}</ref> To make this quantitative, an orientational order parameter is usually defined based on the average of the second [[Legendre polynomial]]:

:<math>S = \langle P_2(\cos \theta) \rangle = \left \langle \frac{3 \cos^2(\theta) - 1}{2} \right \rangle</math>

where <math>\theta</math> is the angle between the liquid-crystal molecular axis and the ''local director'' (which is the 'preferred direction' in a volume element of a liquid crystal sample, also representing its ''[[optical axis|local optical axis]]''). The brackets denote both a temporal and spatial average. This definition is convenient, since for a completely random and isotropic sample, ''S''&nbsp;=&nbsp;0, whereas for a perfectly aligned sample S=1. For a typical liquid crystal sample, ''S'' is on the order of 0.3 to 0.8, and generally decreases as the temperature is raised. In particular, a sharp drop of the order parameter to 0 is observed when the system undergoes a phase transition from an LC phase into the isotropic phase.<ref>{{cite journal|title = A model for the orientational order in liquid crystals| doi =10.1007/BF02453342|journal=Il Nuovo Cimento D|volume = 4| issue =3|date =1984| page = 229|bibcode = 1984NCimD...4..229G| vauthors = Ghosh SK | s2cid =121078315}}</ref> The order parameter can be measured experimentally in a number of ways; for instance, [[diamagnetism]], [[birefringence]], [[Raman scattering]], [[Nuclear magnetic resonance|NMR]] and [[Electron Paramagnetic Resonance|EPR]] can be used to determine S.<ref name=b5/>

The order of a liquid crystal could also be characterized by using other even Legendre polynomials (all the odd polynomials average to zero since the director can point in either of two antiparallel directions). These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.<ref name=b2/>

A positional order parameter is also used to describe the ordering of a liquid crystal. It is characterized by the variation of the density of the center of mass of the liquid crystal molecules along a given vector. In the case of positional variation along the ''z''-axis the density <math>\rho (z)</math> is often given by:

:<math>\rho (\mathbf{r}) = \rho (z) = \rho_0 + \rho_1\cos(q_sz - \varphi) + \cdots \, </math>

The complex positional order parameter is defined as <math>\psi (\mathbf{r}) = \rho_1 (\mathbf{r})e^{i\varphi(\mathbf{r})}</math> and <math>\rho_0</math> the average density. Typically only the first two terms are kept and higher order terms are ignored since most phases can be described adequately using sinusoidal functions. For a perfect nematic <math>\psi = 0</math> and for a smectic phase <math>\psi</math> will take on complex values. The complex nature of this order parameter allows for many parallels between nematic to smectic phase transitions and conductor to superconductor transitions.<ref name=b1/>

===Onsager hard-rod model===
{{unsolved|physics|Can the nematic to smectic (A) phase transition in liquid crystal states be characterized as a [[background independence|universal]] phase transition?}}
A simple model which predicts lyotropic phase transitions is the hard-rod model proposed by [[Lars Onsager]]. This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder which the approaching cylinder's center-of-mass cannot enter (due to the hard-rod repulsion between the two idealized objects). Thus, this angular arrangement sees a ''decrease'' in the net positional [[entropy]] of the approaching cylinder (there are fewer states available to it).<ref>{{cite journal| journal=Annals of the New York Academy of Sciences|volume = 51| issue =4|date = 1949| page = 627| doi =10.1111/j.1749-6632.1949.tb27296.x|title=The effects of shape on the interaction of colloidal particles|bibcode = 1949NYASA..51..627O| last1=Onsager| first1=Lars |s2cid = 84562683| name-list-style = vanc }}</ref><ref name=vroege>{{cite journal| title = Phase transitions in lyotropic colloidal and polymer liquid crystals| doi= 10.1088/0034-4885/55/8/003|journal=Rep. Prog. Phys.|volume = 55| issue =8| date = 1992| page = 1241|bibcode = 1992RPPh...55.1241V| vauthors = Vroege GJ, Lekkerkerker HN | url = https://dspace.library.uu.nl/bitstream/1874/22348/1/lekkerkerker_92_phase_transition_lyotropic_colloidal.pdf| hdl= 1874/22348| s2cid= 250865818}}</ref>

The fundamental insight here is that, whilst parallel arrangements of anisotropic objects lead to a decrease in orientational entropy, there is an increase in positional entropy. Thus in some case greater positional order will be entropically favorable. This theory thus predicts that a solution of rod-shaped objects will undergo a phase transition, at sufficient concentration, into a nematic phase. Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.<ref name=vroege/> An extension of Onsager Theory was proposed by Flory to account for non entropic effects.

===Maier–Saupe mean field theory===
This statistical theory, proposed by [[Alfred Saupe]] and Wilhelm Maier, includes contributions from an attractive intermolecular potential from an induced dipole moment between adjacent rod-like liquid crystal molecules. The anisotropic attraction stabilizes parallel alignment of neighboring molecules, and the theory then considers a [[mean-field theory|mean-field]] average of the interaction. Solved self-consistently, this theory predicts thermotropic nematic-isotropic phase transitions, consistent with experiment.<ref>{{cite journal| vauthors = Maier W, Saupe A | s2cid=93402217| journal=Z. Naturforsch. A|volume = 13| issue=7|language=de|title=Eine einfache molekulare theorie des nematischen kristallinflussigen zustandes|page = 564|date =1958|bibcode = 1958ZNatA..13..564M|doi = 10.1515/zna-1958-0716 |doi-access =free}}</ref><ref>{{cite journal| vauthors = Maier W, Saupe A |s2cid=201840526|title=Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase .1|language=de|journal=Z. Naturforsch. A|volume = 14|issue=10| page = 882|date =1959|bibcode = 1959ZNatA..14..882M |doi = 10.1515/zna-1959-1005 |doi-access =free}}</ref><ref>{{cite journal| vauthors = Maier W, Saupe A | s2cid=97407506| journal=Z. Naturforsch. A|volume = 15| issue=4|language=de|title=Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase .2|page = 287|date =1960|bibcode = 1960ZNatA..15..287M|doi = 10.1515/zna-1960-0401 |doi-access =free}}</ref> Maier-Saupe mean field theory is extended to high molecular weight liquid crystals by incorporating the [[bending stiffness]] of the molecules and using the method of [[path integrals in polymer science]].<ref>{{cite book |last1=Ciferri |first1=Alberto | name-list-style = vanc |title=Liquid crystallinity in polymers : principles and fundamental properties |date=1991 |publisher=VCH Publishers |location=Weinheim |isbn=3-527-27922-9}}</ref>

===McMillan's model===
McMillan's model, proposed by William McMillan,<ref>{{cite journal|title= Simple Molecular Model for the Smectic A Phase of Liquid Crystals|journal=Phys. Rev. A|volume= 4 |date=1971|issue=3|page=1238|doi=10.1103/PhysRevA.4.1238|bibcode = 1971PhRvA...4.1238M| vauthors = McMillan W }}</ref> is an extension of the Maier–Saupe mean field theory used to describe the phase transition of a liquid crystal from a nematic to a smectic A phase. It predicts that the phase transition can be either continuous or discontinuous depending on the strength of the short-range interaction between the molecules. As a result, it allows for a triple critical point where the nematic, isotropic, and smectic A phase meet. Although it predicts the existence of a triple critical point, it does not successfully predict its value. The model utilizes two order parameters that describe the orientational and positional order of the liquid crystal. The first is simply the average of the second [[Legendre polynomials|Legendre polynomial]] and the second order parameter is given by:

: <math>\sigma = \left\langle\cos\left(\frac{2\pi z_i}{d}\right)\left(\frac{3}{2}\cos^2\left(\theta_i\right) - \frac{1}{2}\right)\right\rangle</math>

The values ''z<sub>i</sub>, θ<sub>i</sub>'', and ''d'' are the position of the molecule, the angle between the molecular axis and director, and the layer spacing. The postulated potential energy of a single molecule is given by:

:<math>U_i(\theta_i, z_i) = -U_0\left(S + \alpha\sigma\cos\left(\frac{2\pi z_i}{d}\right)\right)\left(\frac{3}{2}\cos^2\left(\theta_i\right) - \frac{1}{2}\right)</math>

Here constant α quantifies the strength of the interaction between adjacent molecules. The potential is then used to derive the thermodynamic properties of the system assuming thermal equilibrium. It results in two self-consistency equations that must be solved numerically, the solutions of which are the three stable phases of the liquid crystal.<ref name=b5/>

===Elastic continuum theory===
{{See also|Distortion free energy density}}
In this formalism, a liquid crystal material is treated as a continuum; molecular details are entirely ignored. Rather, this theory considers perturbations to a presumed oriented sample. The distortions of the liquid crystal are commonly described by the [[Frank free energy density]]. One can identify three types of distortions that could occur in an oriented sample: (1) twists of the material, where neighboring molecules are forced to be angled with respect to one another, rather than aligned; (2) splay of the material, where bending occurs perpendicular to the director; and (3) bend of the material, where the distortion is parallel to the director and molecular axis. All three of these types of distortions incur an energy penalty. They are distortions that are induced by the boundary conditions at domain walls or the enclosing container. The response of the material can then be decomposed into terms based on the elastic constants corresponding to the three types of distortions. Elastic continuum theory is an effective tool for modeling liquid crystal devices and lipid bilayers.<ref>{{cite journal|title = Continuum theory for nematic liquid crystals| journal=Continuum Mechanics and Thermodynamics|volume = 4|date= 1992| issue =3| page = 167| doi =10.1007/BF01130288|bibcode = 1992CMT.....4..167L| vauthors = Leslie FM | s2cid=120908851}}</ref><ref>{{cite journal | vauthors = Watson MC, Brandt EG, Welch PM, Brown FL | title = Determining biomembrane bending rigidities from simulations of modest size | journal = Physical Review Letters | volume = 109 | issue = 2 | pages = 028102 | date = July 2012 | pmid = 23030207 | doi = 10.1103/PhysRevLett.109.028102 | bibcode = 2012PhRvL.109b8102W | doi-access = free }}</ref>