Nucleating Nematic Droplets

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Distinct kinetic pathways of nucleated microdroplets in a thermal freely suspended nematogenic film Amit Bhattacharjee Funding: DST-INSPIRE program December 13, 2016 Department of Physics IISc Bangalore CompFlu 2016

Transcript of Nucleating Nematic Droplets

Page 1: Nucleating Nematic Droplets

Distinct kinetic pathways of nucleated microdroplets in a thermal freely suspended nematogenic film

Amit Bhattacharjee

Funding: DST-INSPIRE program

December 13, 2016

Department of PhysicsIISc Bangalore

CompFlu 2016

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“mesophases” - the 4th state of matter

Consist of anisotropic molecules (e.g. rods, discs, V-shape) with long range

orientational order, devoid of translational order.

Uniaxial/biaxial phase rotational symmetry about direction of order described

by one/two headless vector n (director) and m (secondary director).

Liquid - nematic transition is weakly first order.

n

Amit Bhattacharjee IISc Bangalore 1

Distinct kinetic pathways of nucleated microdroplets in a thermal freely suspended nematogenic film

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Technological and scientific necessity

Threaded colloids, Orlandini et al, PRL '15

PDLC droplet, Raina et al, Opt.Mat. '04

Inkjet printing, Macdonald et al, Lab Chip, '15

2,4-brush defectsChandrasekhar et al

Curr.Sci. '98

Cosmic strings, Turok et al, Science, '91

MD nanodroplets,Zannoni et al, Soft Matter, '12

MC nanodroplets,Dijkstra et al, PRL '07

Amit Bhattacharjee IISc Bangalore 2

Emulsions, Poulin et al, Science '97

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Central question to ponder

Fast cooling / heating (quench) brings rigidity & topological defect in the process of solidification / forming vapour. A deterministic dynamics can capture such zero-temperature kinetics, in liquids, in magnetic systems, in liquid crystals & so on ...

What is the kinetic pathway for shallow quench (nucleation)?

Amit Bhattacharjee IISc Bangalore 3

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Central question to ponder

Fast cooling / heating (quench) brings rigidity & topological defect in the process of solidification / forming vapour. A deterministic dynamics can capture such zero-temperature kinetics, in liquids, in magnetic systems, in liquid crystals & so on ...

What is the kinetic pathway for shallow quench (nucleation)? Complexity in experiments (i) probing narrow temperature

window (for 5CB, T* = 34.2oC, Tc = 34.44oC , T**= 34.47oC.), (ii) avoiding heterogeneous nucleation.

Amit Bhattacharjee IISc Bangalore 3[1] A.Q. Shen et al, Langmuir (2008).

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Central question to ponder

Fast cooling / heating (quench) brings rigidity & topological defect in the process of solidification / forming vapour. A deterministic dynamics can capture such zero-temperature kinetics, in liquids, in magnetic systems, in liquid crystals & so on...

What is the kinetic pathway for shallow quench (nucleation)? Complexity in experiments (i) probing narrow temperature

window (for 5CB, T* = 34.2o C, Tc = 34.44o C , T**= 34.47o C.), (ii) avoiding heterogeneous nucleation.

Complexity in simulations (i) sampling algorithms for rare events, (ii) pseudo-dynamics (MC), (iii) probed length and time scales are in nm and ns only.

Amit Bhattacharjee IISc Bangalore 3

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Central question to ponder

Fast cooling / heating (quench) brings rigidity & topological defect in the process of solidification / forming vapour. A deterministic dynamics can capture such zero-temperature kinetics, in liquids, in magnetic systems, in liquid crystals & so on...

What is the kinetic pathway for shallow quench (nucleation)? Complexity in experiments (i) probing narrow temperature

window (for 5CB, T* = 34.2o C, Tc = 34.44o C , T**= 34.47o C.), (ii) avoiding heterogeneous nucleation.

Complexity in simulations (i) sampling algorithms for rare events, (ii) pseudo-dynamics (MC), (iii) probed length and time scales are in nm and ns only.

Need a top-down approach from “stochastic mesoscopic theory”.

Amit Bhattacharjee IISc Bangalore 3

Prologue Introduction Nematic drop Isotropic drop Conclusion

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Mesoscopic theory Ground state free energy as polynomial

expansion of

F bulk=[ 12 ATrℚ2+13BTrℚ3+

14CTr (ℚ2)2] .

Athermal phase diagramAthermal energy diagram

L. Landau(Laureate '62)

P.G. de Gennes

(Laureate '91)

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B =disparity

A=A0 (1−T

T *) .

ℚαβ(x , t)=[3 S (nα nβ−δαβ) + T (lα l β−mαmβ)] /2 .

F (S )=0,F ' (S )=0,F ' ' (S )=0,

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Mesoscopic theory Elastic distortions - splay ,

bend , twist .

EL theory

Full Q-tensor theory

How GLdG theory can be useful to experimental mesures ?

F elastic=[ 12 K1(∂⋅n)2+12K 2(n⋅∂×n)2+

12K 3(n×∂×n)2] .

F elastic=[ 12 L1(∂ℚ)2+12L2(∂⋅ℚ)

2+12L3ℚ⋅(∂ℚ)

2] .

K 1=9S 2 L14

(2+κ−ΘS ) , K 2=9S 2 L14

(2−Θ S ) , K 3=9S 2 L14

(2+κ+2ΘS ) ;

(K 1,K 2, K 3)=(6.4,3,10)×10−7 dyn ,(B ,C )=(7.2,8 .8) Jcm−3 .

(K 1,K 2, K 3)=(6,4,7.5)×10−7dyn ,(B ,C )=(2.66,2 .76) Jcm−3 .

κ=L2/ L1 ,

L1=0.6, κ=40.7 .L1=8.6,κ=1.2 .

Θ=L3/L1 .

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(∂⋅n≠0)(∂×n⊥n) (∂×n∥n)

splay twist bend

5CB:

MBBA:

At

25o C

[1] Schiele Trimper, Phys.Stat.Solidi (1983).

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Mesoscopic stochastic theory

∂tℚαβ=−Γ [(A+CTrℚ2)ℚαβ+Bℚαβ2 −L1∂

2ℚαβ−L2∂α∂γℚβγ ] + ζαβ .

∂tℚαβ(x , t )=−Γ[δαμ δβν+δαν δβμ− 23 δαβδμ ν]δ Fδℚμ ν

+ ζαβ(x , t ) .

GldG dynamics (model A) for non-conserved order[1]

Property of noise:

Equation of motion

Route to equilibrium: Spinodal kinetics below T* & above T**.

Nucleation kinetics in [T*,Tc] & [Tc,T**].

⟨ ζαβ( x , t)⟩ = 0

⟨ ζαβ( x , t) ζμ ν(x ' , t ' )⟩ = 2 k BT Γ[δαμδβν+δανδβμ− 23 δαβδμ ν]δ(x−x ' )δ(t−t ' )

[1] Stratonovich, Zh.Eksp.Teor.Fiz (1976).

R. Stratonovich

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Nematic droplets in isotropic background

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34.2o C < 34.27o C < 34.44o C

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Nucleation of droplets in metastable medium

Free energy of a circular droplet[1]

Maximize F critical radius barrier heightnucleation rate

Energy of nematic droplet:

Morphology of droplet is unlikely circular – inapplicability of Becker-Doering theory (CNT) if director structure isn't uniform.

No Rapini-Papoular term[2,3], resulting to exact estimate of GLdG energy.

F (R)=−43π R3ρN Δμ + 4π R2σ .

Rc=2σ

ρN|Δμ|, F c=

16πσ3

3ρN2 (Δμ)2

,

NI

F (R)=−∫Vd 3 x [ρN Δμ(ℚ)] + ∫∂ S

d 2 x [σ (∂ℚ)] .

τ−1=A e−F c /k BT .

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[1] Rayleigh, Scientific Papers (1899), [2] [3] Meissa et al, Phys.Rev. E. (1995), Bhattacharjee et al, Phys.Rev.E (2008).

F=∫∂ S

d 2 x [1+W (q⋅n)2 ] .

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Nucleation of droplets in metastable mediumn nn

(κ<0) (κ=0) (κ>0)

Athermal case [1]: ellipsoidal droplet shape with uniform director orientation.

Thermal case: (i) in weak anchoringlimit, results agree with athermal scenario.

F elastic=[ 12 L1(∂ℚ)2+12L2(∂⋅ℚ)

2] , ∂tℚ∼L1∂2ℚ+L2∂(∂⋅ℚ) , κ=L2 /L1 .

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[1] Bhattacharjee et al, Phys.Rev.E (2008).

(K 2=2K 1=2K 3) (K 2=K1=K 3) (K 2=23K1=

23K 3)

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Surface interfacial anchoring Does de Gennes ansatz hold for curved interface ?

For planar interface[1], & So planar or homeotropic anchoring is favoured for or For curved interface in quasi-2D , when no fluctuations

Doesn't hold when fluctuations are taken into account,

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F elasticθ=π /2=

18κ(∂z S+∂zT )

2 F elasticθ=0 =

12κ(∂ z S )

2 .

κ>0 κ<0 .

F elasticθ=π /2=κ

2 [(∂ x S )2+14sin 2(2ϕ)(∂ yT )

2+{−∂ y S+(−12+cosϕ)∂ yT }

2] .F elasticθ=0 =κ

2 [{∂ yS

2+12sin (2ϕ)∂xT +(

12−cosϕ)∂ yT }

2+14{∂ xS+cos(2ϕ)∂xT +sin(2ϕ)∂ yT }

2] .

(∂ z S ,∂ zT=0)

(∂ θ ,∂ ϕ=0) ,

Hom

eo

Pl a

nar

trop

i c

(∂ θ ,∂ ϕ≠0) .

[1] P.G. de Gennes, Mol.Cryst.Liq.Cryst (1973).

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Nucleation of droplets in metastable mediumn nn

(κ<0) (κ=0) (κ>0)

(κ≫0)

Athermal case: ellipsoidal droplet shape with uniform director orientation.

Thermal case: (i) in weak anchoring limit, results agree with athermal scenario. de Gennes ansatz does not hold at curved interface. (ii) in strong anchoring limit, ellipsoidal droplet morphology with hyperbolic hedgehog defect, biaxial ring at outer core is seen.

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[Movie](K 2=K1 /10)

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Droplet growth process

Growth process for small surface energy Nucleation of several droplets coalescence defect annealing kinetics.

Growth process for large surface energyNucleation of few droplets coalsecence no defect kinetics.

(L1>0):

(L1≫0):

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F elastic=[ 12 L1(∂ℚ)2+12L2(∂⋅ℚ)

2] , ∂tℚ∼L1∂2ℚ+L2∂(∂⋅ℚ) , κ=L2 /L1 .

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Droplet growth process

JMAK equation: orx (t)=1−e(t /T )m

, ln [−ln {1−x(t )}]=mln(t )−mlnT .

<S

><

T>

m > 2

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ln[−ln{1−

x(t)} ]

=

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Droplet growth process

JMAK equation: orIntial growth law for tagged cluster & average cluster

x (t)=1−e(t /T )m

, ln [−ln {1−x (t)}]=mln(t )−mlnT .

<S

><

T>

Thermal limitedregime vol

drivenregime

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L(t) ∼ √(at 2+bt+c).

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Droplet growth process

Initial growth process dictated by curvature elasticity and capillary forces (higher Laplace pressure). Volume driving force or difference in free energy drives late stage growth, leading to ballistic motion[2].

ηv = −σR

+σ ' ' (θ)

R+ Δ F .

Herring's eqn[1]:

viscosityx growth velocity

Laplace pressure

anisotropiccapillary pressure

volume drivingforce

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[1] A.D. Rey et al, Continuum Mech.Therm. (2007).[2] Huisman Fasolino, Phys.Rev.E. (2007).

JMAK equation: orIntial growth law for tagged cluster & average cluster

x (t)=1−e(t /T )m

, ln [−ln {1−x (t)}]=mln(t )−mlnT .

L(t) ∼ √(at 2+bt+c).

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Transient free energy during growth

JMAK equation: orIntial growth law for tagged cluster & average cluster

Monotonic decrease in energy with overshoot in elastic energy.

x(t)=1−e(t /T )m

, ln [−ln {1−x (t)}]=mln(t )−mlnT .

<S

><

T>

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L(t) ∼ √(at 2+bt+c).

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Applicability of classical theory

x(t)=1−e(t /T )m

, ln [−ln {1−x(t)}]=mln(t )−mlnT .

L(t) ∼ √(at 2+bt+c).

JMAK equation: orIntial growth law for tagged cluster & average cluster

Monotonic decrease in energy with overshoot in elastic energy. Breakdown of CNT & JMAK equation.

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Applicability of classical theory

x(t)=1−e(t /T )m

, ln [−ln {1−x(t)}]=mln(t )−mlnT .

L(t) ∼ √(at 2+bt+c).

JMAK equation: orIntial growth law for tagged cluster & average cluster

Monotonic decrease in energy with overshoot in elastic energy. Breakdown of CNT & JMAK equation. Consecutive events spatially proximate, temporally independent

due to absence of long range ordering in isotropic media.

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Summary: nematic droplets in isotropic medium[1]

Droplets are circular in one-elastic approximation , in weak anchoring limit noncircular shape observed with uniform director field, while hyperbolic hedgehog defects with biaxial ring is seen in strong anchoring limit.

(κ=0)

Prologue Introduction Nematic drop Isotropic drop Conclusion

[1] Bhattacharjee, Sci. Rep. (2016).

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Experiment Theory

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Summary: nematic droplets in isotropic medium[1]

Droplets are circular in one-elastic approximation , in weak anchoring limit noncircular shape observed with uniform director field, while hyperbolic hedgehog defects with biaxial ring is seen in strong anchoring limit.

de Gennes ansatz doesn't hold for curved interface. Nucleation followed by growth fits to the JMAK model, but not

exponent. Unusual growth law at initial stage, crossing over to ballistic

motion at late stage, signifying latent heat is expelled at much faster rate from interface.

Energy monotonically decreases, with an overshoot in elastic energy.

Becker-Doering theory does not hold, displaying a violation of CNT.

Spatially proximate temporal events are independent of each other.

(κ=0)

Prologue Introduction Nematic drop Isotropic drop Conclusion

[1] Bhattacharjee, Sci. Rep. (2016).

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Thanks:

R. Adhikari (IMSc),G. Menon (IMSc), C. Dasgupta (IISc),S. Dhara (UoH), V.A. Raghunathan (RRI),D. Frenkel (U.Cambridge),A. Laskar (U. Pittsburgh).

Session open to questions ...