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
“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
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
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
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
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
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
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)
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 4
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,
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 .
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 5
(∂⋅n≠0)(∂×n⊥n) (∂×n∥n)
splay twist bend
5CB:
MBBA:
At
25o C
[1] Schiele Trimper, Phys.Stat.Solidi (1983).
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
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 6
Nematic droplets in isotropic background
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore
34.2o C < 34.27o C < 34.44o C
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 .
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 7
[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 ] .
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 .
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 8
[1] Bhattacharjee et al, Phys.Rev.E (2008).
(K 2=2K 1=2K 3) (K 2=K1=K 3) (K 2=23K1=
23K 3)
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,
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 9
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).
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.
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 10
[Movie](K 2=K1 /10)
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):
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 11
F elastic=[ 12 L1(∂ℚ)2+12L2(∂⋅ℚ)
2] , ∂tℚ∼L1∂2ℚ+L2∂(∂⋅ℚ) , κ=L2 /L1 .
Droplet growth process
JMAK equation: orx (t)=1−e(t /T )m
, ln [−ln {1−x(t )}]=mln(t )−mlnT .
<S
><
T>
m > 2
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 12
ln[−ln{1−
x(t)} ]
=
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
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 13
L(t) ∼ √(at 2+bt+c).
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
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 14
[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).
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>
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 15
L(t) ∼ √(at 2+bt+c).
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.
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 16
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.
Prologue Introduction Nematic drop Isotropic drop Conclusion
Amit Bhattacharjee IISc Bangalore 16
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).
Amit Bhattacharjee IISc Bangalore 18
Experiment Theory
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).
Amit Bhattacharjee IISc Bangalore 18
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 ...
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