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The Landau-de Gennes theory for nematic liquid crystals.
APALA MAJUMDAR Department of Mathematical Sciences
University of Bath, UK
Summer School on “Frontiers of Applied and Computational Mathematics”
9th-21st July 2018
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Nematic Liquid Crystals
The nematic phase is the simplest liquid crystalline phase of matter with properties that are intermediate between a solid and a liquid.
The constituent molecules have
• no positional order (flow about freely) but
• tend to align along certain locally preferred directions i.e. exhibit long-range orientational ordering.
Nematic – Greek word for `thread’.
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Important mathematical parameters -
Nematic liquid crystals are anisotropic liquids with preferred directions of
molecular alignment. The preferred alignment directions constitute the first set
of important parameters.
n (r)
n (r) : preferred direction
of orientation of the long
molecular axes.
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Scalar order parameter “S”:
measure of the degree of alignment about preferred directions.
n
n n
S = 0; no alignment S = 1; perfect
alignment
S 0.5; typical liquid crystal.
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The Oseen-Frank Theory for Nematic Liquid Crystals
• assume constant value of scalar order parameter
• describe preferred direction by a unit-vector field n(r)
• just two degrees of freedom to describe a three-dimensional unit-vector field
Limitations: • Cannot describe secondary direction of orientational ordering/preferred
alignment • Cannot describe higher-dimensional defects such as line and surface defects • In fact, powerful theorem by Hardt, Kinderlehrer and Lin states that Hausdorff
dimension of singular set is less than one even with elastic anisotropy
dVnnWnE ,][
22
42
2
3
2
2
2
1,
nntrKKnnK
nnKnKnnW
421423212;||;0,, KKKKKKKK
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The Landau-de Gennes Theory
The Nobel Prize in Physics in 1991 was awarded to Pierre-Gilles de Gennes for "for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers“.
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The Landau-De Gennes Theory
• General continuum theory that can account for all nematic phases and physically observable singularities.
•Define macroscopic order parameter that distinguishes nematic liquid crystals from conventional liquids, in terms of anisotropic macroscopic quantities such as the magnetic susceptibility and dielectric anisotropy.
• The Q – tensor order parameter is a symmetric, traceless 3×3 matrix.
22112313
232212
131211
QQQQ
QQQ
QQQ
Q
Five degrees of freedom.
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Eigenvalues of the Q-tensor and LC Phases
0λ
ppλmmλnnλQ
3
1i
i
321
• isotropic – triad of zero eigenvalues
• uniaxial – a pair of equal non-zero eigenvalues; OF theory is a special uniaxial case with constant eigenvalues
• biaxial – three distinct eigenvalues and two locally preferred directions of molecular alignment.
0Q0λλλ321
I
3
1nn3λQ2λλλ;λλ
132
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The Landau-de Gennes energy functional The physically observable configurations are modelled by minimizers of the Landau-de Gennes free energy functional subject to the imposed boundary conditions.
In the absence of any external fields and surface effects, the simplest form of the Landau-de Gennes energy is given by
The thermotropic potential : -
• non-convex , non-negative potential with multiple critical points
• dictates preferred phase of liquid crystal – isotropic/ uniaxial/ biaxial?
dVQ,QwQfQIB
0T c,,b,a
,,tr4
ctr
3
btr
2
aQf
**
2232
B
TT
cbaCQQQ
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The Thermotropic Potential
0T c,,b,a
,,tr4
ctr
3
btr
2
aQf
**
2232
B
TT
cbaCQQQ
• Compute critical points of this quartic polynomial as a function of the temperature
ji
cbaC
i
i
i
i
i
i
i
i
jii
i
321B
3
1
23
1
2
3
1
3
3
1
2
321B
0,,f
0
),,(4
c
3
b
2
a,,f
All critical points are either uniaxial or isotropic for all temperatures.
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The Thermotropic Potential
All critical points are either uniaxial or isotropic for all temperatures • Compute minimizers of bulk potential by looking at the critical points of the
thermotropic potential restricted to uniaxial tensors.
2
3Sn
InnSQ
cbaCSSS ,,9
c
27
2b
3
aSf
432
B
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cbaCSSS ,,9
c
27
2b
3
aSf
432
B
Introduction to Q-tensor theory 2014
Nigel J. Mottram, Christopher J.P. Newton
Also see Professor Sir John Ball 2015 Lyon Notes. A.Majumdar 2010 In : European Journal of Applied Mathematics. 21, 2, p. 181-203
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The Elastic Energy Density
• Make correspondence with Oseen-Frank energy density by using the substitution
3exp
InnSQ
OF
dVnntrkk
nnknnknknE
22
2422
2
33
2
22
2
11][
Introduction to Q-tensor theory 2014 Nigel J. Mottram, Christopher J.P. Newton
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John Ball 2015 Lyon Lecture Notes. L.Longa et.al Liquid Crystals Volume 2, 1987 - Issue 6.
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John Ball 2015 Lyon Lecture Notes. Davis & Gartland 1998 FINITE ELEMENT ANALYSIS OF THE LANDAU-DE GENNES MINIMIZATION PROBLEM FOR LIQUID CRYSTALS L.Longa et.al Liquid Crystals Volume 2, 1987 - Issue 6.
Key Steps in Proof • Coercivity
• Lower semicontinuity
• Non-empty admissible space
• Existence of minimizer from direct methods in the calculus of variations
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J. Ball & A. Majumdar, 2010 Nematic liquid crystals: from Maier-Saupe to a continuum theory. Molecular Crystals and Liquid Crystals, 525, 1 - 11.
• Proof by explicit construction based on localized perturbation inside a unit ball
likkijlkjikkijkikjijkijkij
QQQLQQLQQLQQLQQw,,4,,3,,2,,1
,
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likkijlkjikkijkikjijkijkij
QQQLQQLQQLQQLQQw,,4,,3,,2,,1
,
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Boundary Conditions • Strong anchoring
• Weak anchoring
• recover strong anchoring as the anchoring coefficient “W” becomes very large
• Planar degenerate anchoring (where the eigenvectors are preferentially anchored on the surfaces)
xxQxQS
2
2SS
QQtrW
QI
Nigel J. Mottram, Christopher J.P. Newton, 2014 Introduction to Q-tensor theory
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Some problems with the Landau-de Gennes Theory: • Lack of physically relevant bounds for the order parameter; interpret Q as the second
moment of the probability distribution function for the molecular orientations
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A. Majumdar, 2010 Equilibrium order parameters of liquid crystals in the Landau-de Gennes theory. European Journal of Applied Mathematics, 21, 181 - 203.
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Some problems with the Landau-de Gennes Theory: • When do the Landau-de Gennes minimizers respect the second moment bounds?
• Need to consider minimizers of the thermotropic potential for low temperatures
3
),,(min
InncbasQ
0T c,,b,a
,,tr4
ctr
3
btr
2
aQf
**
2232
B
TT
cbaCQQQ
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Some problems with the Landau-de Gennes Theory: • Lack of coercivity with 4 elastic constants, perhaps rectified with higher-order terms,
where do we stop?
• Need 4 elastic constants to match with Oseen-Frank theory.
• Validity for small length scales?
• Physically relevant values of the order parameter?
• Quantitative estimates for validity of Landau-de Gennes theory in terms of temperature, geometrical and material properties?
Schlieren texture with defects in a nematic sample.
(www.lci.kent.edu/defect.html )
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The Isotropic-Nematic Transition Temperature
0
432
2232
**
B
c, T,b,αTTαa
a,b,cCQtrc
Qtrb
Qtra
Qf
c
ba
27
2
0;;33
2
min
QSn
Inn
C
BQ
• The special temperature
• The bulk energy minimizers – the isotropic state and a continuum of uniaxial states, both of which have equal energies
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The Gradient – Flow Dynamic Model
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Front propagation on a nematic droplet…
• Radial uniaxial Dirichlet condition
• Initial conditions with an interface
structure
• Look for solutions of the form
3ˆˆ
3
Irr
C
BQ
b
3ˆˆ,
IrrtrhQ
C
Bhthth
3,10,0
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Front propagation by mean curvature as L 0
3ˆˆ,
IrrtrhQ
C
Bhthth
3,10,0
• Prove front propagation by mean curvature in an asymptotic limit
2
dt
d
Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.
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Comments on proof
• Proof only works for initial conditions with an interface structure with suitably bounded weighted energy
)(1)(),(, RttRhRw
Key references (including concept of weighted energy): • Lia Bronsard and Robert V Kohn. Motion by mean curvature as the singular limit of Ginzburg-Landau
dynamics. Journal of differential equations, 90, 1991. • Lia Bronsard and Barbara Stoth. The singular limit of a vector-valued reaction-diffusion process. Transactions of the American Mathematical Society, 350, 1998. Technical differences due to additional term in evolution equation and the director field is a solution of the harmonic map equations and not the Laplace equation.
3ˆˆ,
IrrtrhQ
Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.
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Numerical Examples
• Uniaxial initial conditions with and without an interface structure
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Numerical Examples
• Biaxial initial conditions
• Symmetry-breaking initial conditions that depend on angular variable
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Front propagation on a disc with 5D Q-tensors…
• Uniaxial Dirichlet conditions with a two-dimensional radial director field
• Can describe long-time behaviour in terms of two unit-vector fields
3311
Inn
C
BQ
b
• The corresponding Q-tensors are
Planar initial conditions; isotropic core at centre for all time
Non- planar initial conditions; perfectly ordered uniaxial state
22112313
232212
131211
QQQQ
QQQ
QQQ
Q
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Numerical Examples on a Disc – Solving LdG Gradient Flow System
G Di Fratta et. al Half-integer point defects in the q-tensor theory of nematic liquid crystals. Journal of Nonlinear Science , 26, 2016.
1,0,00,cos,sin;sin,cos pmn
• Planar biaxial initial conditions
rQtrQt 1
,
Majumdar, A., Milewski, P. A. & Spicer, A. 2016 , SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.
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Numerical Examples on a Disc
• Planar symmetry-breaking biaxial initial conditions
rQtrQt 1
,
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Numerical Examples on a Disc
• Non-Planar initial conditions
rQtrQt 2
,
Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.
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Non-Planar Initial Conditions continued
rQtrQt 2
,
Eigenvalue evolution at origin
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Loss of Interface Structure
rQtrQt 2
,
10
*logt
Majumdar, A., Milewski, P. A. & Spicer, A. 19 Jul 2016 In : SIAM Journal on Applied Mathematics. 76, 4, p. 1296-1320 25 p.
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Gradient-Flow Dynamics with Forcing
Dirr, N. P. and Yip, N. K. 2006. Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces and Free Boundaries 8(1), pp. 79-109.
Joint work with Amy
Spicer, Nicolas Dirr and
Patrick Dondl.
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Gradient-Flow Dynamics with Forcing
Joint work with Amy Spicer and Nicolas Dirr
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Gradient-Flow Dynamics with Forcing
Joint work with Amy Spicer and Nicolas Dirr
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This research is supported by
• EPSRC Career Acceleration Fellowship EP/J001686/1 and
EP/J001686/2.
• Royal Society Newton Advanced Fellowship
• Shanghai Jiao Tong University
• Oxford Centre for Nonlinear PDEs
• OCIAM Visiting Fellowship
Thank you for your attention!