Liquid Crystals

In this example, I apply Tensorial and TContinuumMechanics, to the case of distortions in a nematic single liquid crystal.

I first recall that in a nematic single crystal, the molecules are aligned on average along one common vector denoted t in the following. But the molecules are not oriented, so the direction ±t are equivalent.

The order parameter is a tensor, called Q which must be invariant by the change t⇔- t.

A nematic can be deformed, and these elastic deformations modify the free energy of the system. Three main deformations can be considered :

- the splay (where div t ≠ 0)

- the bend (where t . curl t = 0)

- the twist (where t // curl t)

This leads to introduce three elastic constants (, , ).

A systematic way to found explicit relations between Q and t, curl t and div t, is described here.

Initialization

Conventionally, the "black" flavor is used for the canonical orthonormal basis :

Nematics : Microscopic approach

Simple rods

The rods have in average the direction t.

A single rod is charaterized by a unit vector J in spherical coordinates. We will use the average direction t= of the rods, as Z-axis:

In spherical coordinates, the unit vector J is given by,

Distribution function of the state of alignment of the rods:

For nematics, f is (1) independent of φ, and (2) m and -m are equivalent so that f(θ)==f(π-θ)

so that,

The first multipole which gives a non zero average is the quadrupole,

Order parameter Q: Three dimensional situation

The system is located in an orthonormal (black) basis e, and we consider two molecules or fiber elements with principal axis and , close from one another. The principal axis may correspond to a magnetic moment, or to identify a steric constrain...

The hamiltonian of interaction is,

The uniaxial nematic tensor order parameter is (we can verify that its trace is zero) :

It is convenient to introduce the usual parameter q by,

The red basis will be here a monoclinic basis :

Rotation of the basis:

We recall first the expression of a rotation matrix using Euler angles (α, β, γ)

R is the product of the three rotations,

We call Β the basis change :

and check that,

For a general rotation, we would have,

t in the x-y plane (=0):

t makes an angle φ with the first axis :

{td[1]→Abs[t] Cos[φ],td[2]→Abs[t] Sin[φ]}

In a basis {X, Y, Z} :

Free energy in terms of q and φ (one-dimensional)

"Tensorization" of the parameters : Tensor Notations

It is convenient in the following to use scalar tensors in place of the usual parameters. We will denote with an UnderBar the tensor associated with each parameter.

A. Forms , , ... and {}, without derivatives

B. Forms with first order covariant derivatives

I. Forms

Warning : EinsteinSum[] does not works :

1) on contravariant derivatives (we apply StandardDownIndices[g]).

2) on expression in an hold form (we apply ReleaseHoldD).

The coefficients of the partial derivatives and is given by,

and we can check that they are equal to,

Introducing the projection of ∇φ on the t-plane we find the expression for :

II. Forms

with a similar approach, we find

III. Form

IV. Form

V. Form

VI. Form

VII. Form

VIII. Form

IX. Form

X. Form

XI. Combinations and

Simplifications

Some of the above expression may be simplified. This leads to a simplification rule (Simplifrule) :

C. RESUME

We summarize below all the results :

Another question is : what are the expressions involving ∇q and ∇φ as functions of the and its covariant derivatives.

To do this we have to invert the above expressions.

Inversion

We first introduce shortened notations for the various terms, and find how some of them are related :

we check the validity of these relations :

... and obtain the expected inverse relation :

D. RESUME (inversion)

B3 q-B4 q==0 |

-4 B10+2 B6+B7+2 B8-2 B9==0 |

All the results are summarized here :

A more general approach

The Landau free energy of a nematic-isotropic system may be written up to fourth order in Q :

these rotational invariants only depend on q :

and the corresponding free energy is,

fw will be chosen in the form: fq = . This fixes A,B, and C (rulABC):

Now the free energy has the general expression :

List of invariants built with t, div[t], curl[t], grad[], grad[t], Δ[t]

We can now generate the invariants built with t.

We use the operators defined in TContinuumMechanics (TDiv,TGrad,TCurl,TLaplacian).

List of terms of order one and three in t, with zero, and first order derivatives,

List of terms of order two in t, with zero, first and second order derivatives,

List of terms of order four in t, with zero and first order derivatives,

In listQb, the factors ≡ are already in listQa, so we have cancelled the corresponding terms

In addition, the following terms are duplicated in the list listQa :

So in the general case, there exists seven invariants of this type :

Notations for the invariants : Sq for order two, Qu for order four.

Correspondence between the list of invariant and the Divergence, Curl, Laplacian... forms,

while the inverse relation is,

Sq[3]→grad[t].grad[t] |

Sq[4]→tΔ[t] |

Od[1]→div[t] |

There exists the following three identities :

What is the meaning of the term "grad[t].grad[t]" ?

Now we showx that gradtgradt is the sum of a divergence div[...] plus curl[t\!\(\(]\^2\)\)+div[t\!\(\(]\^2\)\):

We start from the two last identities (valid for arbitrary t) :

and eliminate tΔ[t] :

Now the expression grad[t].grad[t] can be rewritten, using,

Explicit form of K in the case arbitrary)

Created by Mathematica (November 27, 2007) |