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Controling Motile Disclinations with an Electric Field
Funding:
DST-INSPIRE program
December 17, 2017
Department of Physics,
(a) Asutosh College (C.U.),
(b) IISc Bangalore.
Ψb
Ψt
ε = ? =
Ψt
Ψb
ε
CompFlu 17
Amit Kumar Bhattacharjee

Point defects in nematics
n
Amit Bhattacharjee 1
Prologue GLdG model Electrokinetics UN Electrokinetics BN Conclusion
 Mesophases consist of anisotropic molecules (e.g. rods, discs,
V-shape) with long range orientational order without translational
order. uniaxial (UN), biaxial (BN) phase rotational symmetry
about direction of order described by headless vector n (director)
and m (secondary director).
 Orientational order
 In a continuous symmetry breaking phase transition by (a) rapid quench,
rigidity and topological defects are inevitable during solidification, (b)
shallow quench, anisotropic droplets nucleation kinetics[1]
.
[1] Bhattacharjee, Scientific Reports (2017).
ℚ=
3
2
S(n⊗n−
1
3
δ)+B2(l⊗l−m⊗m).
CompFlu 17

 Mesophases consist of anisotropic molecules (e.g. rods, discs,
V-shape) with long range orientational order without translational
order. uniaxial (UN), biaxial (BN) phase rotational symmetry
about direction of order described by headless vector n (director)
and m (secondary director).
 Orientational order
 In a continuous symmetry breaking phase transition by (a) rapid quench,
rigidity and topological defects are inevitable during solidification, (b)
shallow quench, anisotropic droplets nucleation kinetics[1]
.
 Under crossed polarizers, charged point defects in
UN & BN display 2 or 4 brushes texture with core
structure of enhanced/reduced degree of ordering.
[1] Bhattacharjee, Scientific Reports (2017).
sin2
[2θ]
spatial
extent
Sandn
±1,±
1
2
C y
Cz
ℚ=
3
2
S(n⊗n−
1
3
δ)+B2(l⊗l−m⊗m).
CompFlu 17
1
0
n

 Topological classification[1]
: (a) UN phase (b) BN phase From
Homotopy theory, UN phase have one class and BN phase have five class of defects.
 However, energetics and direct numerical simulation[2]
showed (a) dissociation of integer defect
into a pair of defects, (b) existence of 2 class of defects in BN.
Energy of single point defect :
 Identify and classify defects using Burger’s circuit[2]
:
(a) in UN, traversing contour rotates by
(b) in BN defect class rotates by no rotation.
rotates by no rotation.
both rotates by
sin2
[2θ]
spatial
extent
Sandn
C y
Cz
π1(P2)=Z2 , π1(P3)=Q8 .
±
1
2
±
1
2
±
1
2
ℱ discl=π K k2
ln(
R
rc
)+ℱ discl
c
.
γ, n ±π.
Cx , n ±π, l
Cy , l ±π, n
Cz ,{n ,l } ±π.
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1
0
[1] Mermin, Rev.Mod.Phys (1979), Kleman-Lavrentovich, Soft Mat (2002).
[2] Kobdaj et al, Nucl.Phys.B (1994), Zapotoky et al, PRE (1996).

Line disclinations in nematics
±
1
2
 Points in 2D correspond to strings in 3D.
 Tensor field visualization: (i) evaluate singularity using Burger’s circuit in
the vector field, (ii) glyph based techniques like Muller and Westin matrices[1]
,
(iii) hyperstreamline seeding method[2]
, (iv) streamtubes formed in the
singularity of easy identification and classification scheme[3]
.
CompFlu 17
S ,B2
[1] McLoughlin et al, Comput.Graph.Forum, (2010).
[2] Abukhdeir et al, IEEE Visual.Comp.Graph, (2015).
[3] Bhattacharjee, Ph.D. Thesis, HBNI (2010).
ℚ=
3
2
S(n⊗n−
1
3
δ).

Line disclinations in nematics
S
±
1
2
 Points in 2D correspond to strings in 3D.
 Kinetics : (i) formation of reduced nematic domains,
(ii) domain coarsening generating to disclinations,
(iii) line extinction kinetics through intercommutation and ring formation.
 Three subsequent stage in kinetics:
Initial diffusive regime : for 5CB, T*
= 34.20o
C
Porod’s law scaling regime : Tc
= 34.44o
C
String diffusion regime : [Reference T = 33.65o
C] B2
C y
Cz
L(t)t
0.5
,
L(t)t
0.4
,
L(t)t
0.5
.
UN
±
1
2
BN
CompFlu 17
C y
Cz

Importance in Science and Technology
Cosmic strings,
Turok et al, Science, '91
Abrikosov Lines in
Superconductor,
Smørgrav et al, PRL, '05
Vortex Lines in
BoseEinstein Condensate,
Henn et al, PRL, '09
Flux line Vortex in He3
,
Grzybowsk et al,
PNAS, '02
String networks in
Ecology,
Avelino et al, PLA, '14
Trefoil knot particle,
Martinez et al,
Nat. Mat. '14
Cholesteric colloidal Knot,
Musevic et al, Science '11
Self-assembly Amphiphiles
Wang et al, Nat. Mat. '15 Nematic Caps, Uchida et al,
Soft Matter '15
CompFlu 17

Central question to ponder
 How to control disclinations using external forces such as thermal fluctuations,
electro-magnetic field, shear forces and so on …
 Electric field driven alignment of the director : Fréedericksz transition.
CompFlu 17
[1] Yeomans et al, PRE (2001), PRL (2002), Vella et al, PRE (2005),
Tanaka et al, PRL (2006), Avelino SoftMat. (2011).
[2] Nikkhou et al, Nat.Phy. (2015).

 Disclinations under intense electric field[1,2]
: key-components are (i) backflow, (ii) uniform
electric field, (iii) elastic anisotropy and (iv) nematic tensor. Both for point and line defects,
it is shown that defect merging speed are different.
CompFlu 17
±
1
2
[1] Yeomans et al, PRE (2001), PRL (2002), Vella et al, PRE (2005),
Tanaka et al, PRL (2006), Avelino SoftMat. (2011).

 Disclinations under intense electric field : key-components are (i) backflow, (ii) uniform
 Disclinations under low to moderate electric field ? In this limit, backflow probably neglected.
Important questions are: (i) role of thermal fluctuations, (ii) nonuniformity in the electric field,
(iii) elastic anisotropy and (iii) nematic tensor.
CompFlu 17
±
1
2

 Disclinations under intense electric field : key-components are (i) backflow, (ii) uniform
 Disclinations under low to moderate electric field ? In this limit, backflow probably neglected.
Important questions are: (i) role of thermal fluctuations, (ii) nonuniformity in the electric field,
(iii) elastic anisotropy and (iii) nematic tensor.
 Progress were limited[1,2]
, “as it is not easy to solve the Poisson equation with an inhomogeneous
dielectric constant to calculate the local electric field”. Central claims were nematic caps form
on colloidal surface due to nonuniform electric field intensity distribution.
CompFlu 17
±
1
2
[1] Onuki et al, PRE (2006), EPJE (2009), Soft Matter (2015).
[2] Cummings et al, PRE (2014).

 In previous example, is the nonuniformity is
manifest due to the colloidal inclusion or
inherently the field structure is nonuniform in
pure sample?
 Whether long-lived loops are due to nonuniform
electric field or symmetry-breaking boundaries?
CompFlu 17
±
1
2

Amit Bhattacharjee
Outline
 Ginzburg-Landau-de Gennes theory and “Fluctuating Electronematics” method.
 Effect of thermal fluctuations on disclination kinetics.
 Effect of nonuniform electric field on disclination kinetics of uniaxial nematics.
 Effect of nonuniform electric field on disclination kinetics of biaxial nematics.
 Conclusions.
Controling Motile Disclinations with an Electric Field
CompFlu 17

ℱ bulk=[1
2
ATrℚ2
+
1
3
BTr ℚ3
+
1
4
C Tr(ℚ2
)2
+E ' Tr(ℚ3
)2
],
Amit Bhattacharjee
ℚ .
ℱ elastic= L1[1
2
(∂ ℚ)2
+
1
2
κ(∂⋅ℚ)2
+
1
2
Θℚ⋅(∂ℚ)2
],
 Ground state free energy[1]
as a polynomial expansion of
at 25o
C[2]
.
Landau – de Gennes formalism
ℱ total=∫d
3
x( ℱ bulk+ ℱ elastic +ℱ dielec). A=A0 (1−
T
T
*
) ,
B= size disparity .
uniaxial biaxial
{κ=L2/L1, Θ=L3/ L1}.
ℱ dielec=−
ϵ0
8π
D⋅∂ Ψ , where D=−ϵ⋅∂ Ψ=−(ϵs δ+ϵa ℚ)⋅∂ Ψ .
CompFlu 17 8
κ=Θ=1 for MBBA, κ=40,Θ=1for 5CB
[1] Gramsbergen etal, Phys.Rep. (1986).
[2] Blinov & Chigrinov, Electrooptic displays (1994).
[3] Bhattacharjee, arXiv:1707.09703 (2017).

ℱ bulk=[1
2
ATrℚ2
+
1
3
BTr ℚ3
+
1
4
C Tr(ℚ2
)2
+E ' Tr(ℚ3
)2
],
Amit Bhattacharjee
ℚ .
ℱ dielec=−
ϵ0
8π
D⋅∂ Ψ , where D=−ϵ⋅∂ Ψ=−(ϵs δ+ϵa ℚ)⋅∂ Ψ .
 Ground state free energy as a polynomial expansion of
 Fréedericksz threshold
Landau – de Gennes formalism
EF = π
Lx
[9S2
L1 {1+2(L2+L3)/3 L1 }
2ϵ0 ϵa
]
1/2
; ΨF =Lx EF .
ℱ total=∫d
3
x( ℱ bulk+ ℱ elastic +ℱ dielec).
uniaxial biaxial
ℱ elastic= L1[1
2
(∂ ℚ)
2
+
1
2
κ(∂⋅ℚ)
2
+
1
2
Θℚ⋅(∂ℚ)
2
],
CompFlu 17
{κ=L2/L1, Θ=L3/ L1}.
8

Fluctuating Electronematics
∂t ℚαβ(x ,t)=−Γ
[δαμ δβν+δαν δβμ−
2
3
δαβ δμ ν]δ ℱ total
δℚμ ν
+ ζαβ(x ,t).
⟨ζα β( x ,t)⟩ = 0 ,
⟨ζαβ( x ,t)ζμ ν(x ' ,t ')⟩ = 2kB T Γ[δαμ δβν +δαν δβμ−
2
3
δαβδμ ν]δ(x−x')δ(t−t').
[1] Stratonovich, Zh.Eksp.Teor.Fiz (1976).
[2] Bhattacharjee et al, J. Chem. Phys. (2010).
[3] Vella et al, PRE (2005).
 Thermal kinetics[1,2]
as for moderate to low electric field,
 For intense electric field,
Note that hydrodynamic flow must be accounted for in this limit, as electric drag can be
balanced by the backflow[3]
.
 Thermal force sets the temperature scale via FDT
 In dry limit, Stokes-Einstein equation dictates the viscosity.
∂t Ψ(x ,t)=∂⋅D ,
O(Ψ/ΨF )~O(ℚ).
O(Ψ/ΨF )≫O(ℚ).
2
3
δℚμ ν
+ ζαβ(x ,t).
kB T
K η
= constant , K=Frank constant .
=ϵ0(ϵs ∂2
Ψ+ϵa ∂α ℚαβ ∂β Ψ)
=...− ϵ0 ϵa ∂α Ψ∂β Ψ/8 π
CompFlu 17

Fluctuating Electronematics
2
3
δℚμ ν
+ ζαβ(x ,t).
⟨ζα β( x ,t)⟩ = 0 ,
⟨ζαβ( x ,t)ζμ ν(x ' ,t ')⟩ = 2kB T Γ[δαμ δβν +δαν δβμ−
2
3
δαβδμ ν]δ(x−x')δ(t−t').
 Thermal kinetics[1,2]
as for moderate to low electric field,
 For intense electric field,
Note that hydrodynamic flow must be accounted for in this limit, as electric drag can be
balanced by the backflow[3]
.
 Thermal force sets the temperature scale via FDT
 Stochastic MOL approach : change of basis.
∂t Ψ(x ,t)=∂⋅D ,
O(Ψ/ΨF )~O(ℚ).
O(Ψ/ΨF )≫O(ℚ).
∂t ℚαβ(x ,t)=−Γ[δαμ δβν+δαν δβμ−
2
3
δℚμ ν
+ ζαβ(x ,t).
=ϵ0(ϵs ∂2
Ψ+ϵa ∂α ℚαβ ∂β Ψ)
=...− ϵ0 ϵa ∂α Ψ∂β Ψ/8 π
CompFlu 17
[1] Stratonovich, Zh.Eksp.Teor.Fiz (1976).
[2] Bhattacharjee et al, J. Chem. Phys. (2010).
[3] Vella et al, PRE (2005).

 Droplet morphology & nucleation kinetics in 5CB material.
Experimental reproducibility of SMOL
Experiment Theory
CompFlu 17

[1] Chen et al, Langmuir (2007),
Bhattacharjee, Sci.Rep. (2017).
[2] Turok etal, PRL(1991), Science(1994).
 Droplet morphology & nucleation kinetics[1]
in 5CB material.
 Intercommutation events and,
Kinetics of disclination surface density[2]
 Both thermal fluctuation and elastic anisotropy tend to
increase string density without affecting the physical laws.
ρ = t−1∓0.001
.
Experimental reproducibility of SMOL
Experiment Theory
CompFlu 17

Amit Bhattacharjee CompFlu 17 11
 Electric field is switched on at and switched off at
 Disclinations are long lived[1]
in the presence of electric field.
Electrokinetics in uniaxial nematic
ton toff .
ton toff
disclinationdensity(inμm−2
)
t(in ms)
10
0
10
1
10−2
10−1
10−2
EF
10−1
EF
100
EF
no field
10−2
EF
10−1
EF
100
EF
ϵa>0ϵa<0

 Electric field is switched on at and switched off at
 Disclinations are long lived[1]
in the presence of electric field.
Energy of a planar disclination:
elastic energy / unit area = drag force / unit area
equilibrium kinetics
electric field effect

= π K k2
ln(
ξ
ζ
)−
π ϵ0 ϵa
4k
E2
(ξ2
−ζ2
)+ ℱ discl
c
.
ℱ discl = ∬d2
x[K (∂ f )2
−ϵ0 ϵa E2
sin2
f ]/2 ,
ton toff .
ton toff
(with f =k ϕ+c ,0≤ϕ≤2 π , k=±
1
2
).
)
t(in ms)
10
0
10
1
10−2
10−1
10−2
EF
10−1
EF
100
EF
no field
10−2
EF
10−1
EF
100
EF
ϵa>0ϵa<0
ℱ discl
ξ
−η∂t ξ
ξ = t−1/ 2
.
ξ = e
νt
, ν=
π ϵ0 ϵa E2
8k η
.
Thus ,
ϵa
k
>0 holds for both sign of k and ϵa ,leading to a slowed down kinetics .

 Electric flux lines are inherently nonuniform for
 String cores remain with sufficiently-reduced nematic
order, such that they aren’t influenced much by
the electric forces, even if the director is aligned.
 The medium exhibits memory in between an
elastic response to the field.
[1] Vella et al, Phys.Rev.E (2005).
E≤EF for both ϵa<0and ϵa>0.
10
−1
EF10
−2
EF 10
0
EF
100μ m
S&∂Ψ(inVμm−1
)
 We qualitatively find BL lines[1]
connecting strings for that we
don’t quantify in devoid of backflow.
E≥2 EF

Comparison between uniform/nonuniform scenario with/without one elastic approximation.
100μ m
S&∂Ψ(inVμm−1
)

Application: electrokinetics in biaxial nematic
CompFlu 17
t=0.96ms t=1.5 ms t=4.5 ms t=7.5 ms
 In Biaxial nematic media also the flux
lines are highly nonuniform and electric
field dilates the disclination kinetics.
 Depending on the sign of , electric
field filters out strings of different
topology by inducing a kinetic
asymmetry.
ϵa>0ϵa<0
ϵa
Cz
C y
toff =∞
ton=1ms

CompFlu 17
asymmetry.
 Recall that for a planar string,

ϵa>0ϵa<0
ϵa
Cz
C y
ξ = eνt
, ν=
π ϵ0 ϵa E
2
8k η
.
Thus ,
ϵa
k
yields ± sign for different topological string, leading to a faster decay of either class .
)
t(in ms)

CompFlu 17
asymmetry.
 Recall that for a planar string,

 This is attributed to an increase (decrease) of total free energy for positive (negative)
ϵa
ξ = eνt
, ν=
π ϵ0 ϵa E
2
8k η
.
Thus ,
ϵa
k
yields ± sign for different topological string, leading to a faster decay of either class .
ϵa .
no field
ϵa>0
ϵa<0
−1.86
−1.76
−1.66
0.05
0.10
0.15
−1.90
−1.89
−1.88
energydensity(inJcm
−3
)
bulkisotropictotal
t(in ms)

Conclusions
CompFlu 17
Unlike droplet nucleation kinetics, thermal fluctuations do not play a dramatic role other than
slowing down the athermal kinetics and increasing the disclination density.
Electric field is inherently nonuniform in the pure sample [while ] and it’s not
the colloidal impurity that brings nonuniformity in the electric field structure.
Uniformity in the electric field is obtained as the Fréedericksz limit is reached, thus a term
proportional to “EQE” in the GLdG free energy is valid only in the intense electric field limit
where backflow plays a crucial role.
Combination of thermal fluctuation and nonuniform electric field results into a time dilated
disclination kinetics.
Electric field cannot influence the disclinations to orient along/perpendicular to the field
direction, even when the director is oriented.
The nonuniform electric field induces a memory to the system within the elastic response and
induces a kinetic asymmetry between the different class of disclinations in biaxial nematics.
O( Ψ
ΨF
)~ O(ℚ)
Results

Conclusions
Amit Bhattacharjee
CompFlu 17
Without using traditional methods, we directly identify defect location from the structure of
Q tensor & classify them by computing a Burger’s circuit integral.
Fluctuating Electronematics is an efficient, 2nd
order accurate, numerical scheme that
brings control over various key-components of this complex problem.
Without calculating the traditional correlators, a dynamic length scale is extracted from the
geometry of the disclinations using surface triangulation method.
Methods
Conclusions
C. Dasgupta (IISc),
S. Ramaswamy (IISc),
P.B.S. Kumar (IITPkd),
N.V. Madhusudana (RRI).
Thanks
Session open to questions
arXiv:1707.09703

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