A brief introduction into the Q-tensor theory of nematic ...users.uoa.gr › ~nalikako ›...

A brief introductioninto the Q-tensor theory of nematic 1 liquid crystals

Arghir Dani Zarnescu

Basque Center for Applied Mathematics , Spainand

“Simion Stoilow” Institute of Mathematicsof the Romanian Academy, Romania

January 12, 2017

1νηµαArghir Dani Zarnescu Q-tensor theory January 12, 2017 1 / 38

The plan:

An introduction to the Q-tensor theory of nematic liquid crystals

Defects: 2d and 3d, stability

Dynamics without flow: cubic instability and statistically self-similar dynamics

Arghir Dani Zarnescu Q-tensor theory January 12, 2017 2 / 38

An introduction

to the Q-tensor theory

of nematic liquid crystals

2

2Simulation by C. Zannoni groupArghir Dani Zarnescu Q-tensor theory January 12, 2017 3 / 38

Liquid crystals: physics

A measure µ such that 0 ≤ µ(A) ≤ 1 ∀A ⊂ S2

The probability that the molecules are pointing in a direction contained in thesurface A ⊂ S2 is µ(A)

Physical requirement µ(A) = µ(−A) ∀A ⊂ S2


Landau-de Gennes Q-tensor reduction and the physicalQ-tensors

Q =

∫S2

p ⊗ p dµ(p) −13

Id

Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor

If ei , i = 1, 2, 3 are eigenvectors of Q , with corresponding eigenvaluesλi = 1, 2, 3, we have

−13≤ λi =

∫S2

(p · ei)2dµ(p) dp −

13≤

23

for i = 1, 2, 3, since∫S2 dµ(p) = 1.


Landau-de Gennes Q-tensor reduction and earliertheories

The Q-tensor has 5 degrees of freedom and is :I isotropic is Q = 0I uniaxial if it has two equal eigenvaluesI biaxial otherwise

Ericksen’s theory (1991) for uniaxial Q-tensors which can be written as

Q(x) = s(x)

(n(x) ⊗ n(x) −

13

Id), s ∈ R, n ∈ S2

hence 3 degrees of freedom

Oseen-Frank theory (1958) take s in the uniaxial representation to be afixed constant s+ thus obtaining an object with 2 degrees of freedom


The heart of the matter:defects and their cores


Q-tensors: beyond liquid crystals

Carbon nanotubes:

LC states of DNA:

Active LC: cytoskeletal filaments and motor proteins


So what are the equations? The big picture

Full system-strongly coupled equations fortemperature, Q-tensors and fluid: weak solutions

Coupled Navier-Stokes and Q-tensor system(weaksolutions in 2D and 3D, with regularity andphysicality in 2D)

Dynamics for the Q-tensor system only-statisticaldynamics

Stationary elliptic systemI singular perturbation problem and qualitative

description of solutionsI existence and energetic stability in 2D for index

k2 -defects

I existence and energetic stability for the “meltinghedgehog” solution

The constrained Q tensor theory: topological issues


The constrained Q-tensor theory


The constrained Q-tensor theory

Use functions Q : Ω→ s+

(n ⊗ n − 1

3 Id) with s+ , 0 and n ∈ S2.

Defects are defined as discontinuities of such functions Q .


Line fields versus vector fields and defects I


Line fields versus vector fields and defects II

Bad choice: we have generated vector field defects that did not exist in the line field


Line fields versus vector fields and defects III


Defects of line fields: regularity


Defects of line fields: topology


A complex topology

Theorem (JM Ball, AZ) Let G be a domain with holes in the plane. A line field inW1,p for p ≥ 2 is orientable if and only if its restriction to the boundary is orientable.


Towards obtaining equations

The simplest way to obtain physically relevant configurations is by minimizing anenergy functional:

F [Q ,D] =

∫Ωψ(Q(x),D(x)) dx

where (Qij(x))i,j=1,...,d is a Q-tensor, i.e. symmetric and traceless d × d matrix(d = 2, 3) and ‘D ∼ ∇Q‘, is a third order tensor. Here Q(x) takes values into theconstrained space s+

(n ⊗ n − 1

3 Id) with s+ , 0 and n ∈ S2.

Simplest example: ∫Ω

3∑i,j,k=1

∣∣∣∣∣∣∂Qij

∂xk

∂Qij

∂xk

∣∣∣∣∣∣ dx =

∫Ω|∇Q |2


A domain with holes and partial boundary conditions

E(Q) =

∫Ω

3∑i,j=1

2∑k=1

∂Qij

∂xkdx =

∫Ω|∇Q |2 dx


Comparison between line fields and vector fields globalenergy minimizers

E(Q) =

∫Ω

3∑i,j=1

2∑k=1

∂Qij

∂xkdx =

∫Ω|∇Q |2 dx


An analytic results about orientable vs non-orientableminimisers


Beyond constrained Q-tensors

Energy functionals in the three theories:

Landau-de Gennes:

FLG[Q] =

∫Ω

L2

Qij,k (x)Qij,k (x) + fB(Q(x)) dx

fB(Q) =α(T − T ∗)

2tr

(Q2

)−

b3

tr(Q3

)+

c4

(trQ2

)2

with Q(x) : Ω→ M ∈ R3,M = Mt , trM = 0 a Q-tensor

Ericksen’s theory:

FE [s, n] =

∫Ω

s(x)2|∇n(x)|2 + k |∇s(x)|2 + W0(s(x)) dx

with (s, n) ∈ R × S2

Oseen-Frank:

FOF [n] =

∫Ω

ni,k (x)ni,k (x) dx, n ∈ S2


More general energy functionals

The general form of an energy functional:

F [Q ,D] =

∫Ωψ(Q(x),D(x)) dx

where (Qij(x))i,j=1,...,d is a Q-tensor, i.e. symmetric and traceless d × d matrix(d = 2, 3) and ‘D ∼ ∇Q‘, is a third order tensor.

Physical invariances require that:

ψ(Q ,D) = ψ(Q∗,D∗)

where Q∗ = RQR t and D∗ijk = RilRjmRknDlmn for any R ∈ O(3).


The Q-tensor energy functionals II

We can decompose:

ψ(Q ,D) = ψ(Q , 0) + ψ(Q ,D) − ψ(Q , 0) = ψB(Q)︸︷︷︸bulk

+ψE(Q ,D)︸︷︷︸elastic

Then ψB(Q) = ψB(RQR t ) for R ∈ O(3) implies that there exists ψB so that

ψB(Q) = ψB(tr(Q2), tr(Q3)).

Example of elastic terms that respect the physical invariances:

I1 = Qij,k Qij,k , I2 = Qij,jQik ,k ,

I3 = Qij,k Qik ,j , I4 = Qij,lQij,k Qkl

Note that I2 − I3 = (QijQik ,k ),j − (QijQik ,j),k .


The Oseen-Frank energy

In the case of the Oseen-Frank theory similar considerations provide the energy functional:

G[n,∇n] =

∫Ω

K1(divn)2 + K2(n · curl n)2 + K3(n × curln)2 dx

+

∫Ω

(K2 + K4)(tr(∇n)2 − (divn)2)] dx


Q-tensors versus directors energies

TakeQ(x) = s(n(x) ⊗ n(x) −

13

Id)

with s fixed and n(x) ∈ S2. Let

K1 := 2L1s2 + L2s2 + L3s2 −23

L4s3, K2 := 2L1s2 −23

L4s3

K3 = 2L1s2 + L2s2 + L3s2 +43

L4s3,K4 = L3s2

Then F [Q ,D] = G[n,∇n] with

G[n,∇n] =

∫Ω

K1(divn)2 + K2(n · curl n)2 + K3(n × curln)2 dx

+

∫Ω

(K2 + K4)(tr(∇n)2 − (divn)2)] dx

the Oseen-Frank energy functional.


The cubic instability

Take an elastic energy:

ψE(Q ,D) =4∑

j=1

Lj Ij

where Lj , j = 1, . . . , 4 are elastic constants.

If L4 , 0 (corresponding to the cubic term) thenF [Q ,D] =

∫ΩψB(Q(x)) + ψE(Q(x),D(x)) dx is seen to be unbounded from

below by taking (J.M. Ball): Q = s(x)(

x|x | ⊗

x|x | −

13 Id

)for suitable s(x) (then

I4 = 49 s(s′2 − 3

r2 s2)).

If L4 = 0 then for ψB ≡ 0 we have:

F [Q] ≥ µ‖∇Q‖2L2

for some µ > 0 if and only if (Longa, Monselesan, and Trebin, 1987):

L3 > 0, −L3 < L2 < 2L3, −35

L3 −110

L2 < L1


A way of “solving” the cubic instability I: the singularpotential

Define a bulk potential that enforces that the eigenvalues of Q are between − 13 and 1 − 1

3 .

fsing(Q) =

infρ∈AQ

∫S2 ρ(p) log(ρ(p)) dp if λi [Q] ∈ (−1/3, 2/3), i = 1, 2, 3,

+∞ otherwise,

where

AQ =ρ : S2 → [0,∞)

∣∣∣∣ ρ ∈ L1(S2),

∫S2ρ(p) dp = 1;

Q =

∫S2

(p ⊗ p −

13

Id3

)ρ(p) dp

.

We will refer to it as the singular potential.

Approach considered by different authors:

I J. Katriel, G.F. Kventsel, G.R. Luckhurst and T.J. Sluckin, Liquid Crystals 1,337-355 (1986)

I I. Fatkullin, V. Slastikov, Phys. D, 237 (2008), no. 20, 2577-2586I J.M. Ball and A. Majumdar, Mol. Cryst. Liq. Cryst. 525 (2010) 1-11


A way of “solving” the cubic instability II: the singularpotential

This time the energy functional with singular potential as before:

Fsing[Q] :=3∑

i,jk=1

∫Ω

L1Qij,k Qij,k + L2Qij,jQik ,k + L3Qij,k Qik ,j

+ L4QklQij,lQij,k + fsing(Q) dx

is bounded from below provided that the ellipticity constant associated to the firstthree terms is large enough to absorb the bounded part of the cubic term.


The strict physicality issue

Note that Fsing[Q] < ∞ implies also that −13 < λi(Q(x)) < 2

3 for almost allx ∈ Ω.

Major open problem: Is it true that for a global minimiser of Fsign (undersuitable boundary constraints) there is an ε > 0 such that

−13

+ ε < λi(Q(x)) <23− ε (1)

for almost any x ∈ Ω?

Note that a Q that satisfies (1) is referred to as being strictly physical.

Some partial results:I For one elastic constant L2 = L3 = 0 (Ball and Majumdar)I in 2D and with L3 = 0-Baumann and PhillipsI Partial regularity in 3D -L.C. Evans and H. Tran


THANK YOU!


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