Fréedericksz transition
Encyclopedia
The Fréedericksz transition is a phase transition
in liquid crystals produced when a sufficiently strong electric
or magnetic field
is applied to a liquid crystal in an undistorted state. Below a certain field threshold the director remains undistorted. As the field value is gradually increased from this threshold, the director begins to twist until its aligned with the field. In this fashion the Fréedericksz transition can occur in three different configurations known as the twist, bend, and splay geometries. The phase transition
was first observed by Fréedericksz and Repiewa in 1927. In this first experiment of theirs, one of the walls of the cell was concave so as to produce a variation in thickness along the cell. The phase transition is named in honor of the Russian physicist Vsevolod Frederiks
.
Under this arrangement the distortion free energy density
becomes:
The total energy per unit volume stored in the distortion and the electric field is given by:
The free energy per unit area is then:
Minimizing this using calculus of variations
gives:
Rewriting this in terms of
and 
where 
is the separation distance between the two plates results in the equation simplifying to:
By multiplying both sides of the differential equation by
this equation can be simplified further as follows:
The value
is the value of 
when 
. Substituting 
and 
into the equation above and integrating with respect to 
from 0 to 1 gives:
The value K(k) is the complete elliptic integral of the first kind. By noting that
one finally obtains the threshold electric field 
.
As a result, by measuring the threshold electric field one can effectively measure the twist Frank constant so long as the anisotropy in the electric susceptibility and plate separation is known.
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
in liquid crystals produced when a sufficiently strong electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
or magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
is applied to a liquid crystal in an undistorted state. Below a certain field threshold the director remains undistorted. As the field value is gradually increased from this threshold, the director begins to twist until its aligned with the field. In this fashion the Fréedericksz transition can occur in three different configurations known as the twist, bend, and splay geometries. The phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
was first observed by Fréedericksz and Repiewa in 1927. In this first experiment of theirs, one of the walls of the cell was concave so as to produce a variation in thickness along the cell. The phase transition is named in honor of the Russian physicist Vsevolod Frederiks
Vsevolod Frederiks
Vsevolod Konstantinovich Frederiks was a Russian/Soviet physicist. His primary contribution to physics was in the field of liquid crystals...
.
Twist Geometry
If a nematic liquid crystal that is confined between two parallel plates that induce a planar anchoring is placed in a sufficiently high constant electric field then the director will be distorted. If under zero field the director aligns along the x-axis then upon application of an electric field along the y-axis the director will be given by:Under this arrangement the distortion free energy density
Distortion free energy density
The Distortion free energy density is a quantity that describes the distortion of a liquid crystal from its preferred state in which all of the liquid crystal molecules are aligned parallel to one common axis...
becomes:
The total energy per unit volume stored in the distortion and the electric field is given by:
The free energy per unit area is then:
Minimizing this using calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
gives:
Rewriting this in terms of
and where is the separation distance between the two plates results in the equation simplifying to:By multiplying both sides of the differential equation by
this equation can be simplified further as follows:The value
is the value of when . Substituting and into the equation above and integrating with respect to from 0 to 1 gives:The value K(k) is the complete elliptic integral of the first kind. By noting that
one finally obtains the threshold electric field .As a result, by measuring the threshold electric field one can effectively measure the twist Frank constant so long as the anisotropy in the electric susceptibility and plate separation is known.