Editing Liquid crystal (section)

===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.