Fredericks Transition

The Fredericks transition , or the Fredericks effect , is a transition from a configuration with a homogeneous director (a unit vector that defines the orientation of the optical axis of the liquid crystal) to a configuration with a deformed director when a sufficiently strong magnetic or electric field is applied. This transition is not a phase transition , since at any point in the liquid crystal the degree of ordering of the molecules relative to each other remains unchanged. Below a certain threshold value of the field, the director remains undeformed. When the field value gradually increases from the threshold value, the director begins to spin around the direction of the field until it is aligned with it in the same direction. Thus, the Fredericks transition can occur in three different configurations known as torsion geometry, longitudinal bend geometry, and lateral bend geometry. The first to observe this transition was V.K. Fredericks and Repyeva in 1927 ^[1] . The name was proposed by the Nobel laureate in physics, Pierre-Gilles de Gennes .

Content

Usage

An example of the use of the Fredericks effect in liquid crystal displays. LC - lcd molecule, I - image, G - glass plate; E - electrode, P - polarizer

The Fredericks Transition is widely used in LCD displays of portable battery-powered devices such as calculators and wristwatches. Each pixel of such a display contains a cell with a liquid crystal oriented in a certain way due to surface forces (left picture). Applying voltage to such a cell changes the orientation of the molecules in the gap between the surfaces (right figure). As a result, the optical activity of the cell changes, and, therefore, its ability to transmit polarized light, creating the ability to display the desired information.

Deriving Relationships

Torsion geometry

If a nematic liquid crystal bounded by two parallel plates that orient the director parallel to the plates is placed in a sufficiently strong constant electric field, then the director will be distorted. If at zero field the director is directed along the x axis, then when an electric field is applied along the y axis, it will be described by the formulas:

\mathbf {\hat {n}} =n_{x}\mathbf {\hat {x}} +n_{y}\mathbf {\hat {y}}

{\ displaystyle \ mathbf {\ hat {n}} = n_ {x} \ mathbf {\ hat {x}} + n_ {y} \ mathbf {\ hat {y}}}

n_{x}=\cos {\theta (z)}

{\ displaystyle n_ {x} = \ cos {\ theta (z)}}

n_{y}=\sin {\theta (z)}

{\ displaystyle n_ {y} = \ sin {\ theta (z)}}

.

Under these conditions, the free energy density of Frank is written in the form:

{\mathcal {F}}_{d}={\frac {1}{2}}K_{2}\left({\frac {d\theta }{dz}}\right)^{2}

{\ displaystyle {\ mathcal {F}} _ {d} = {\ frac {1} {2}} K_ {2} \ left ({\ frac {d \ theta} {dz}} \ right) ^ {2 }}

Total energy of distortion and electric field per unit volume:

U={\frac {1}{2}}K_{2}\left({\frac {d\theta }{dz}}\right)^{2}-{\frac {1}{2}}\epsilon _{0}\Delta \chi _{e}E^{2}\sin ^{2}{\theta }

{\ displaystyle U = {\ frac {1} {2}} K_ {2} \ left ({\ frac {d \ theta} {dz}} \ right) ^ {2} - {\ frac {1} {2 }} \ epsilon _ {0} \ Delta \ chi _ {e} E ^ {2} \ sin ^ {2} {\ theta}}

Then free energy per unit area:

F_{A}=\int _{0}^{d}{\frac {1}{2}}K_{2}\left({\frac {d\theta }{dz}}\right)^{2}-{\frac {1}{2}}\epsilon _{0}\Delta \chi _{e}E^{2}\sin ^{2}{\theta }\,dz

{\ displaystyle F_ {A} = \ int _ {0} ^ {d} {\ frac {1} {2}} K_ {2} \ left ({\ frac {d \ theta} {dz}} \ right) ^ {2} - {\ frac {1} {2}} \ epsilon _ {0} \ Delta \ chi _ {e} E ^ {2} \ sin ^ {2} {\ theta} \, dz}

Minimizing it, we get:

\left({\frac {\partial U}{\partial \theta }}\right)-{\frac {d}{dz}}\left({\frac {\partial U}{\partial \left({\frac {d\theta }{dz}}\right)}}\right)=0

{\ displaystyle \ left ({\ frac {\ partial U} {\ partial \ theta}} \ right) - {\ frac {d} {dz}} \ left ({\ frac {\ partial U} {\ partial \ left ({\ frac {d \ theta} {dz}} \ right)}} \ right) = 0}

K_{2}\left({\frac {d^{2}\theta }{dz^{2}}}\right)+\epsilon _{0}\Delta \chi _{e}E^{2}\sin {\theta }\cos {\theta }=0

{\ displaystyle K_ {2} \ left ({\ frac {d ^ {2} \ theta} {dz ^ {2}}} \ right) + \ epsilon _ {0} \ Delta \ chi _ {e} E ^ {2} \ sin {\ theta} \ cos {\ theta} = 0}

Rewriting through $\zeta ={\frac {z}{d}}$ ${\ displaystyle \ zeta = {\ frac {z} {d}}}$ and $\xi _{d}=d^{-1}{\sqrt {\frac {K_{2}}{\epsilon _{0}\Delta \chi _{e}E^{2}}}}$ ${\ displaystyle \ xi _ {d} = d ^ {- 1} {\ sqrt {\ frac {K_ {2}} {\ epsilon _ {0} \ Delta \ chi _ {e} E ^ {2}}} }}$ Where $d$ ${\ displaystyle d}$ the distance between the two plates, we get:

\xi _{d}^{2}\left({\frac {d^{2}\theta }{d\zeta ^{2}}}\right)+\sin {\theta }\cos {\theta }=0

{\ displaystyle \ xi _ {d} ^ {2} \ left ({\ frac {d ^ {2} \ theta} {d \ zeta ^ {2}}} \ right) + \ sin {\ theta} \ cos {\ theta} = 0}

Multiplying both sides of the differential equation by ${\frac {d\theta }{d\zeta }}$ ${\ displaystyle {\ frac {d \ theta} {d \ zeta}}}$ simplify this equation:

{\frac {d\theta }{d\zeta }}\xi _{d}^{2}\left({\frac {d^{2}\theta }{d\zeta ^{2}}}\right)+{\frac {d\theta }{d\zeta }}\sin {\theta }\cos {\theta }={\frac {1}{2}}\xi _{d}^{2}{\frac {d}{d\zeta }}\left(\left({\frac {d\theta }{d\zeta }}\right)^{2}\right)+{\frac {1}{2}}{\frac {d}{d\zeta }}\left(\sin ^{2}{\theta }\right)=0

{\ displaystyle {\ frac {d \ theta} {d \ zeta}} \ xi _ {d} ^ {2} \ left ({\ frac {d ^ {2} \ theta} {d \ zeta ^ {2} }} \ right) + {\ frac {d \ theta} {d \ zeta}} \ sin {\ theta} \ cos {\ theta} = {\ frac {1} {2}} \ xi _ {d} ^ {2} {\ frac {d} {d \ zeta}} \ left (\ left ({\ frac {d \ theta} {d \ zeta}} \ right) ^ {2} \ right) + {\ frac { 1} {2}} {\ frac {d} {d \ zeta}} \ left (\ sin ^ {2} {\ theta} \ right) = 0}

\int {\frac {1}{2}}\xi _{d}^{2}{\frac {d}{d\zeta }}\left(\left({\frac {d\theta }{d\zeta }}\right)^{2}\right)+{\frac {1}{2}}{\frac {d}{d\zeta }}\left(\sin ^{2}{\theta }\right)\,d\zeta \,=0

{\ displaystyle \ int {\ frac {1} {2}} \ xi _ {d} ^ {2} {\ frac {d} {d \ zeta}} \ left (\ left ({\ frac {d \ theta } {d \ zeta}} \ right) ^ {2} \ right) + {\ frac {1} {2}} {\ frac {d} {d \ zeta}} \ left (\ sin ^ {2} { \ theta} \ right) \, d \ zeta \, = 0}

{\frac {d\theta }{d\zeta }}={\frac {1}{\xi _{d}}}{\sqrt {\sin ^{2}{\theta _{m}}-\sin ^{2}{\theta }}}

{\ displaystyle {\ frac {d \ theta} {d \ zeta}} = {\ frac {1} {\ xi _ {d}}} {\ sqrt {\ sin ^ {2} {\ theta _ {m} } - \ sin ^ {2} {\ theta}}}}

Value $\theta _{m}$ ${\ displaystyle \ theta _ {m}}$ - value $\theta$ ${\ displaystyle \ theta}$ at $\zeta =1/2$ ${\ displaystyle \ zeta = 1/2}$ . We introduce $k=\sin {\theta _{m}}$ ${\ displaystyle k = \ sin {\ theta _ {m}}}$ and $t={\frac {\sin {\theta }}{\sin {\theta _{m}}}}$ ${\ displaystyle t = {\ frac {\ sin {\ theta}} {\ sin {\ theta _ {m}}}}}$ and integrate over $t$ ${\ displaystyle t}$ from 0 to 1:

\int _{0}^{1}{\frac {1}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}\,dt\,\equiv K(k)={\frac {1}{2\xi _{d}}}

{\ displaystyle \ int _ {0} ^ {1} {\ frac {1} {\ sqrt {(1-t ^ {2}) (1-k ^ {2} t ^ {2})}}} \ , dt \, \ equiv K (k) = {\ frac {1} {2 \ xi _ {d}}}}

The quantity K (k) is a complete elliptic integral of the first kind. Given that $K(0)={\frac {\pi }{2}}$ ${\ displaystyle K (0) = {\ frac {\ pi} {2}}}$ we get the threshold value of the field $E_{t}$ ${\ displaystyle E_ {t}}$ .

E_{t}={\frac {\pi }{d}}{\sqrt {\frac {K_{2}}{\epsilon _{0}\Delta \chi _{e}}}}

{\ displaystyle E_ {t} = {\ frac {\ pi} {d}} {\ sqrt {\ frac {K_ {2}} {\ epsilon _ {0} \ Delta \ chi _ {e}}}}}

Notes

↑ Fréedericksz, Repiewa, 1927 .

Literature

Pikin S.A., Blinov L.M. Fredericks effect // Liquid crystals / Ed. L. G. Aslamazova. - M .: Nauka , 1982. - S. 51-84. - 208 p. - ( The Quantum Library . Issue 20). - 150,000 copies.
Collings, Peter J., Hird, Michael. Introduction to Liquid Crystals: Chemistry and Physics. - Taylor & Francis Ltd., 1997. - ISBN 0-7484-0643-3 .
de Gennes, Pierre-Gilles, Prost, J. The Physics of Liquid Crystals. - 2nd. - Oxford University Press. - ISBN 0-19-851785-8 .
Fréedericksz, V., Repiewa, A. Theoretisches und Experimentelles zur Frage nach der Natur der anisotropen Flüssigkeiten (Eng.) // Zeitschrift für Physik Society. - 1927. - Vol. 42 , no. 7 . - P. 532-546 . - DOI : 10.1007 / BF01397711 .
Fréedericksz, V., Zolina, V. Forces causing the orientation of an anisotropic liquid (English) // Trans. Faraday Soc .. - 1933. - Vol. 29 . - P. 919–930 . - DOI : 10.1039 / TF9332900919 .
Priestley, EB, Wojtowicz, Peter J., Sheng, Ping. Introduction to Liquid Crystals. - Plenum Press, 1975 .-- ISBN 0-306-30858-4 .
Zöcher, H. The effect of a magnetic field on the nematic state ( Transactions ) // Transactions of the Faraday Society. - 1933. - Vol. 29 . - P. 945–957 . - DOI : 10.1039 / TF9332900945 .

[_d691fab9b9cdedc1-1] Fréedericksz, Repiewa, 1927 .