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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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
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The heart of the matter:defects and their cores
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Q-tensors: beyond liquid crystals
Carbon nanotubes:
LC states of DNA:
Active LC: cytoskeletal filaments and motor proteins
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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
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The constrained Q-tensor theory
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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 .
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Line fields versus vector fields and defects I
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Line fields versus vector fields and defects II
Bad choice: we have generated vector field defects that did not exist in the line field
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Line fields versus vector fields and defects II
Bad choice: we have generated vector field defects that did not exist in the line field
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Line fields versus vector fields and defects III
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Defects of line fields: regularity
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Defects of line fields: topology
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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.
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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
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A domain with holes and partial boundary conditions
E(Q) =
∫Ω
3∑i,j=1
2∑k=1
∂Qij
∂xkdx =
∫Ω|∇Q |2 dx
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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
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An analytic results about orientable vs non-orientableminimisers
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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
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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
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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
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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).
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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).
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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 .
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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 .
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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
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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.
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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
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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
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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
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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.
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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
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THANK YOU!
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