Particle and field based methods for complex fluids and soft materials

Structural properties of a binary colloidal
mixture under shear reversal
Amit e
Workshop Bartholomäberg
Particle and field based methods for
complex fluids and soft materials
Amit Kumar Bhattacharjee
Courant Institute of Mathematical Sciences
New York University, New York
IISERM SeminarApril 13, 2015

Amit Bhattacharjee
Collaborators and advisors
Aleksandar Donev (New York)
Andy Nonaka (Berkeley)
Alejandro Garcia (San Jose)
John B. Bell (Berkeley)
Juergen Horbach (Duesseldorf)
Matthias Fuchs (Konstanz)
Thomas Voigtmann (Koeln)
Gautam I. Menon (Chennai)
Ronojoy Adhikari (Chennai)
Fluctuating hydrodynamics of
multi-component non-ideal liquids
and chemically reactive fluids.
“Bauschinger effect” in dense
supercooled colloidal melt under
instantaneous shear reversal.
Inhomogeneous phenomena in
nematic liquid crystals.
USA
Germany
USA
India
1/38Amit Bhattacharjee Courant Institute (NYU)

Amit Bhattacharjee
Prologue
 Solid, liquid, gas, plasma.
 F = E – TS; Hard matter (crystals) = E dominated phases (minimize E);
Soft matter (liquids) = S dominated phases (maximize S).
 Changes of phase – order of transition (e.g. liquid to solid, paramagnet to
ferromagnet).
 Soft to touch, easily malleable, can't withhold shear.
 Examples: milk, paint (colloid), rubber, tissues (polymer), toothpaste (gels),
LCD devices (liquid crystal) ….
States of matter
Complex fluids
2/38
Introduction Dense colloids Multispecies mixtures Liquid Crystals Conclusion
Amit Bhattacharjee Courant Institute (NYU)

Amit Bhattacharjee
Prologue
 Multistage transition process in fluids composed with
anisotropic particles: mesophases (Nematic, Discotic, Cholesteric,
Smectic A – C, Columnar liquid crystals).
 Glass transition – a non thermodynamic transition :
a) no consumption/expulsion of latent heat,
b) no changes in structural properties,
c) (almost) no change in thermodynamic properties,
d) drastic change in transport properties (viscosity, diffusion-constant etc).
 Complex physico-chemical processes in multicomponent gases and liquids
leading to macroscale structure.
Complexity in complex fluids
3/38
n

Prologue
Necessity for studying
Images © [1] Vinayak Industries, Mumbai,
[2] Schott AG, Mainz, [3] Wagner (Delaware)
 Technological applications:
Thermometers, laptop & mobile screens,
Casting, cooling and solidification,1,2
Body armour (STF enabled Kevlar).3
 Medical examples:
Sub cellular structures, blood flow,
joint lubricants, pharmaceuticals.
4/38

Prologue
Theoretical methods
 Atomistic description:
a) Ignore electronic d.o.f. classical N-particle Newton's equation.
b) approximation: 2-body interations in central forcefield (e.g. LJ,
Yukawa, WCA etc).
 Mesoscopic description:
a) Identify order parameter, broken symmetry, conservation laws,
type of transition of the phase.
b) Construct a free energy functional and spatial coarse-graining.
c) Temporal coarse graining.
 Measurement of the equilibrium and non-equilibrium properties.
5/38

Prologue
LLNS
TDGL
KMC
DPD
SRD
LBM
LD
BD
DFT
MD Meso-scale
Micro-scales
Length
Time
Computational methods
Macro-scales
6/38

Outline
Bauschinger effect in vitrifying colloidal melt
 Simulation methods.
 Rheological properties in forward shear.
 Response to instantaneous shear reversal.
 Structural properties and interconnection with stresses.
 Nonequilibrium thermodynamics of diffusion.
 Low Mach number equations.
 Numerical methods and benchmarks.
 Applications: Giant nonEQ concentration fluctuations.
 Compressible hydrodynamics of reactive gas.
 Comparison of particle/field based methods for homogeneous systems.
Fluctuating hydrodynamics of multispecies mixtures

Bauschinger effect in binary supercooled colloidal
glass-forming melt
=
8/38
Amit Bhattacharjee
+ = ?
=
Courant Institute (NYU)
Zausch et al,
J. Phys. Cond. Matt. (2008).
Brader et al,
Phys. Rev. E (2010).

Simulation method
 WCA pair potential1
(soft, purely repulsive)
 Thermostat : dissipative particle dynamics[2,3]
(DPD) local
conservation of momentum.
 Solve N-particle Newton's equation with Lees Edwards BC.
mi
˙⃗ri = ⃗pi ; ˙⃗pi=−∑i≠ j
⃗∇ U ij (⃗r )−∑i≠ j
ζω
2
( ⃗rij )( ̂rij⋅⃗vij ) ̂rij+√2kB T ζω( ⃗rij)N ij ̂rij .
conservative dissipative stochastic
N=2NA=2NB=1300, σ AA=1.0, σBB=5/6, σAB=(σAA+σBB)/2, ϵ=1, L=10.
[1] Chandler et al, J. Chem. Phys. (1971).
[2] Espanol et al, Euro. Phys. Lett. (1996).
[3] Peters, Euro. Phys. Lett. (2004).
UWCA (r)=
{4 ϵ[( σ
r
)
12
−(σ
r
)
6
]+ϵ, r<2
1/6
σ
0, r≥21/6
σ
〈N ij (t)〉 = 0,
〈 N ij(t)N kl (,t ')〉 = (δik δjl +δil δjk )δ(t−t ')δ(r−r ')
9/38

S (q)=
1
N
〈ρ(q)ρ(−q)〉
g(r)=
V
N
2
〈∑i
N
∑j≠i
N
δ(ri−r j−r)〉
Fs
α
(q ,t)=
1
N α
∑i
N α
〈ρi (q ,t)ρi(q ,0)〉
Δrα
2
(t)=〈∣rα(t)−rα (0)∣2
〉
t2
t
caging
caging
~
~
Equilibrium: structure and dynamics
 Pair correlation .
 Structure function .
 Density autocorrelator (SISF) .
 Mean squared displacement (MSD) .
10/38

Out-of-equilibrium scenario
 Shear applied through Lees-Edwards BC.
 Planar Couette flow is established within a few
NEMD steps (no shear banding).
 Shear rate perturbs the interplay between
intrinsic single particle time & structural
relaxation time shear thinning:
linear response breaks down.
˙
˙
x
y
z
gradient
vorticity
Newtonian
T=0.4
0

0
˙−1
11/38
T=0.4, ˙γ=0.005
τ0=0.48, τα=2.5 x103
Pe0=2.4 x10−3
, Peα=12.5.

Properties in forward shear: dynamics
 Stress tensor1
:
 Visco-elastic response.
 Local stress:
 Jump in local stress variance at 10% strain amplitude.
 Overshoot in stress[2,3]
: shear induced local melting
of glass (breaking of cage structure): superdiffusive
intermediate motion2
.
elastic
plastic
T=0.4
EQ
〈 r
2
〉~t

 ˙tw
0

xy=〈 xy 〉=−1/V 〈∑i=1
N
[mi vi , x vi , y∑j≠i
rij , x Fij , y ]〉.
kinetic virial
xy=−1/V ∑j≠i
rij , x Fij , y .
12/38
[1] Kirkwood, J. Chem Phys. (1946).
[2] Horbach et al, J. Phys. Cond. Mat. (2008).
[3] Bhattacharjee, Soft Matter (accepted).

Properties in forward shear: structure
 Pair correlation shows no signature of shear.
 Projection onto spherical harmonics:
real and imaginary component of
is sensitive to shear.
 Interconnection between stress and structure[1,3]
 Maximum extension-compression exhibited
near overshoot2
seen in .
 No shear banding3
found (planar Couette flow
is established for all steps).
g22
αβ
(r)
σxy=K cα
2
∫
0
∞
drr3 ∂V
αβ
∂r
ℑ(g22
αβ
(r))
g(r)=∑l=0
∞
∑m=−l
l
glm (∣r∣)Y
lm
(θ,ϕ).
N 1=K cα
2
∫
0
∞
dr r3 ∂V
αβ
∂r
ℜ(g22
αβ
(r))
σ=
ρ2
2
∫
0
∞
d r∑α ,β
cα cβ
rr
r
∂V αβ
∂r
g
αβ
(r)
g(r ,θ)
γ=0.025 γ=0.25γ=0.1
[1] Kirkwood, J Chem Phys. (1946).
[2] Hess et al, Phys. Rev. A, (1987).
[3] Bhattacharjee, Soft Matter (accepted)
13/38
extension
compression

Instantaneous shear reversal: dynamics
 Strong history dependence: preparation state-dependent response.
 Bauschinger effect[1,2]
: less yield strength when reversed from plastic deformed state.
 No signature of strong resistance to the back flow, shear banding2
, STZs or
channelized stress relaxation.
 No overshoot in stresses[2,3]
and absence of superdiffusive motion.
−γw
el
−γw
max −γw
s
[1] Karmakar et al, Phys. Rev. E (2010).
[2] Bhattacharjee et al, J. Chem. Phys. (2013).
[3] Bhattacharjee, Soft Matter (accepted).
14/38

Properties after shear reversal
γ=75−0.025 γ=75−0.25
 Absence of superdiffusive motion due to cage-removal.
 Osmotic pressure and local stress variance stays
unchanged.
 Isotropic evolution of structure in reversal with
attainment of Couette flow in few MD steps.
Amit Bhattacharjee 15/38Courant Institute (NYU)

Summary: rheology of dense colloidal melt
 Response to forward shear: Shear and normal stress overshoot with step jump in
osmotic pressure and local stress variance at 10% strain amplitude with
super-diffusive particle motion.
 Response to shear reversal : history (strain) dependent flow effect, lesser yield
strength and elastic constants, absence of overshoot and super-diffusive motion.
 Local structure (projected onto spherical harmonics) is sensitive to flow, without
any shape distortion at equal stress at late times. No cluttering in structure found
while reversing the flow direction.
 Findings in par with experiments1
and the MCT-ITT theoretical framework2
.
16/38Amit Bhattacharjee
[1] Egelhaaf lab, Univ. Düsseldorf.
[2] Fuchs group, Univ. Konstanz.

Fluctuating hydrodynamics of non-ideal multispecies mixtures
Aim: To formulate theory and accurate computation for n-component
miscible liquid at finite temperature in flow.
[1] Vailati et al, Nature Comm. (2011).
17/38
Amit Bhattacharjee
 Soret effect induced giant non-equilibrium concentration fluctuations in
microgravity1
.
5mm side
1mmthick

18/38
Amit Bhattacharjee
Theoretical prescription
 Assumption of local equilibrium, balance equations
(mass, momentum, energy & entropy).
 Constitutive relations, OCRR, odd and even order processes.
 Relation between diffusive flux and concentration gradient in 1-component
gas/liquid: Fick's law.
 In binary system, chemical potential gradient drives diffusion process
(Einstein-Teorell approach, Chapman-Enskog approach). Also, diffusion
can be induced by temperature gradient (Soret effect), heat exchange
(Dufour effect), barodiffusion and external forces.
 Ideal and non-ideal systems of gas and liquid.
 Straightforward generalization in multicomponent diffusion:
Maxwell-Stefan and Fickian description.
 Thermal fluctuation can be added to deterministic flux (LLNS) satisfying
discrete-FDT.

Low Mach number hydrodynamics
 Sound waves are faster than momentum diffusion in liquids (Ma=0 limit).
 EOS constraint .
 Low Mach number equations
 ensures continuity equation . EOS constraint leads to (1).
 Constitutive flux-force relation obtained from non-equilibrium TD of
diffusion for nonideal liquids comparing diffusion driving force to frictional
force
 Non-ideality parameter .
 is SPD, zero row and column sums, so as .
 where .
∂t ρi =−∇⋅(ρi v)−∇⋅Fi , (i=1,2,..., N )
∂t (ρ v)+∇ π =−∇⋅(ρv v
T
)+∇⋅(η(∇ +∇
T
)v+Σ)+ρ g ,
∇⋅v =−∇⋅(Σi=1
N
Fi / ̄ρi) Σ = √ηkB T (W +WT
)
〈Wij (r ,t)Wkl (r ' ,t ')〉 = δik δjl δ(t−t ')δ(r−r ')
Σi=1
N
ρi/ ̄ρi=1
Σi=1
N
Fi=0 ∂t ρ=−∇⋅(ρv)
…. (1)
F= ̄F+̃F (determinstic + stochastic)
̄F=−L(
∇T μ
T
+ξ
∇ T
T
2
)=−ρW χ[Γ ∇ x+(ϕ−w)
∇ P
nkB T
+ζ
∇ T
T
]
L,χ ξ
19/38
ϕi=ρi/ ̄ρi
wi=ρi/ρ
Γ=Ι+(X−xx
T
)Η
xi=
wi /mi
Σi=1
N
wi/miχ=(Λ+TrΛ w w
T
)
−1
−(Tr Λ)
−1
11
T
Λij=−xi xj/Dij

Low Mach number fluctuating hydrodynamics
 Comparing MS and Onsager expression gives the stochastic contribution.
 Complete equation for mass fraction:
 Numerical scheme: staggered grid, finite-volume method implemented on
BoxLib: scalars live on centres, vectors live on faces and edges ensuring
Einstein's discrete FDT.
 Benchmarks: static and dynamic correlators: &
20/38
∂t (ρ w)+∇⋅(ρw v) = ∇⋅
{ρW
[χ
(Γ ∇ x+(ϕ−w)
∇ P
nkBT
+ζ
∇ T
T )]+√2kB L1
2
Ζ
}
̃F=√2kB L1
2
Ζ
Sw
(i , j)
(k) Sρ(k)
〈Ζi (r ,t)Ζj (r ' ,t ')〉=δij δ(t−t ' )δ(r−r ' )
Sρ(kx ,ky)
m1=1,m2=2,m3=3,
ρ1=0.6,ρ2=1.05,ρ3=1.35,
̄ρ1=2, ̄ρ2=3, ̄ρ3=3.857,
Lx=Ly=32,Δx=Δ y=1,
Sρ
eq
=0.3.

Non-equilibrium fluctuations
21/38
 In presence of weak concentration gradient,
correlations in non-eq fluctuations occur by
coupling to velocity fluctuations : power law
spectrum[1,2]
~ .
 For theoretical calculations, we create diffusion
barrier for first species and deal with ideal( ), isothermal( ),
incompressible( ) mixture with stochastic mass flux .
 Barodiffusion gives ordinary equilibrium
fluctuations while thermo-diffusion (Soret
effect) gives correct enhanced spectrum as
usually done in experiments.
k−4
̃F=0
Amit Bhattacharjee
[1] Bhattacharjee et al, Phys. of Fluids (2015). [2] Donev et al CAMCOS, 9-1, 47 (2014).
H=0 ∇ T =0
̄ρ1,2,3=1
Lx=128,Ly=64
Δx=Δy=1

Compressible hydrodynamics of multispecies reactive mixture
22/38
 Elementary reaction , mass conservation
 Compressible FNS with law of mass action(LMA): chemical hydrodynamics
 Stochastic momentum
flux
 Number density evolution for homogeneous well-mixed system,
Log mean equation (LME):
Chemical Langevin equation (CLE):
 LMA:
 For ideal gas,
Σs=1
Ns
νsr Μs ⇔Σs=1
Ns
νrs Μs
∂t ρs =−∇⋅(ρs v)−∇⋅Fs +ms Ωs , (s=1,2,..., Ns)
∂t (ρ v) =−∇⋅Π−∇⋅(ρv v
T
+pΙ)+ρ g ,
∂t (ρ E) =−∇⋅[(ρ E+p)v]−∇⋅[ϑ+Π⋅v]+ρ v⋅g
̃Π = √ηkB T (W +WT
)+(
√kB κT
6
−
√kB ηT
3
Tr(W +WT
))
〈Wij (r ,t)Wkl (r ' ,t ')〉 = δik δjl δ(t−t ')δ(r−r ')
Π=−η(∇ +∇
T
)v−(κ−
2
3
η)I (∇⋅v)+̃Π
Amit Bhattacharjee
∑s
Fs=0, ∑s
ms Ωs=0.
Ω=̄Ω+̂Ω.
dns/dt= ̄Ωs+∑r
νsr √2 Dr
LM
/dV oWr (t)
dns/dt= ̄Ωs+∑r
νsr √2 Dr
CL
/dV W r (t)
+ -
Σs(νsr−νrs)mr=0.Nr
+ -
̄Ωs=Σr νsr
p
τr kB T
[exp(Σs νsr ms μs /kB T)−exp(Σs νsr ms μs/kB T)],
̄Ωs=Σr νsr (kf Πs' ns'
νs' r
−kr Πs' ns'
νs' r
),
- Dr
LM
=logmean[k f Πs ns
νsr
, kr Πs ns
νsr
],
Dr
CL
=arthmean[k f Πs ns
νsr
,kr Πs ns
νsr
].
+ -
+

Homogeneous dimerization reaction
23/38
 For ,
 Production rate factors
 At equilibrium , mass fraction .
 Comparison with particle based methods (SSA) at EQ: LME is closest truth
to CME, while CLE has it's usual shortcoming (unphysical negative values).
 At out-of-EQ states: noise covariance of CLE agrees more to SSA/CME, but
while distribution is not Gaussian, CLE is no better than LME.
Amit Bhattacharjee
[1] Bhattacharjee et al, arXiv:1503.07478 (2015).
̄Ω1=−2(kf n1
2
−kr n2), ̄Ω2=− ̄Ω1/2, m2=2m1, n1+2n2=n02A⇔ A2
kf /kr=1/n0 Y1=n1/n0=0.5
DLM
=
kf n1
2
−kr n2
ln(kf n1
2
)−ln(kr n2)
, DCL
=
1
2
(kf n1
2
+kr n2)
100 monomer
+ 4 dimer
〈N1〉≈54
kf =2.78x10
−4
kr=0.3
Δt=0.005
〈N1〉≈16
kf =0.00625
kr=0.2
16 monomer
+ 8 dimer
kf
kr

Non-equilibrium fluctuations in flow
24/38
 Effect of chemical reaction ,
is penetration depth that controls switch to spectrum
for small wave numbers ( ), long wave numbers still exhibit
spectrum[1,2]
. At very small wave numbers saturates.
 Linear concentration profile is only established at no-reaction limit.
 Validity to couple SSA & hydrodynamics: work in progress.
k−2
S(k)=
kB T (∇ Y1)2
ηχk
4 (1+(dk)−2
)−1
k
−4
Amit Bhattacharjee
[1] Bedeaux et al, J.Chem Phys (2008).
[2] Bhattacharjee et al, arXiv:1503.07478 (2015)
d=√χ/3 kr
k≪
1
d
S(k)
RD RD + Hydro

Summary: Multispecies diffusive liquids and reactive gas
 We formulated a complete theory amenable to computers for studying n-component
ideal/non-ideal liquid at finite temperature from first principles of NEQ-TD in
conjunction to low-Mach (quasi-incompressible) formulation. This is first direct solution
of the full LLNS equations, maintaining 2nd
order accuracy.
 We find non-equilibrium power law spectra in the presence of concentration gradient
that is put either by hand or derived via temperature gradient (Soret effect) as
incorporated in experiments.
 Chemical reactions affect the spectra by truncating the low Fourier modes, giving clear
distinction between diffusion and reaction dominated regime.
 Different formalism for chemical reaction hints that SSA gives correct distribution of
CME (poisson process) while SODE's (diffusion process) are not quantitatively accurate.
LME is better than CLE for close to equilibrium while in out-of-EQ, both are worse.
Hint for improving LME/CLE: Poisson noise (Tau-leaping).
25/38

Amit Bhattacharjee
Inhomogeneous phenomena in nematic liquid crystals

Nematic “mesophase”
 Consist of anisotropic molecules (e.g. rods and discs) 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 l (co-director).
 Liquid-nematic solid transition is weakly first order.
Motivating examples
Topological defect entanglement
in NLC film of width 790mm after
temperature quench, showing
monopoles, boojums and various
integer/half-integer defects [Turok
et al, Science, '91]
Schlieren textures with two
and four brushes exhibited
by a uniaxial NLC film at 118
deg celsius [Chandrasekhar,
et al, Current Science, '98]
Nucleation of ellipsoidal NLC
droplet with aspect ratio 1.7
and homogeneous director
field in MC simulation. [Cuetos
et al, Phys.Rev.Lett, '07]
27/38

 Quantified through a symmetric traceless tensor field with
five degrees of freedom.
Definition of order
Qαβ
molecular frame:
principal frame:
Qαβ( x ,t)=∫d u f ( x ,u ,t)uu
Qαβ(x ,t)=
3
2
S (nα nβ−
1
3
δαβ)+
T
2
{lα lβ−(n×l)α(n×l)β }
 Principal values represent strength of uniaxial and biaxial order (S,T)
 Principal axes denote director, codirector and joint normal ( ).
correspond to isotropic liquid phase.
correspond to uniaxial nematic phase.
correspond to biaxial nematic phase.
 Statics:
n ,l , n×l
S=T =0
S=
2
3,
T =0
T ≠0
FGLdG=∫d3
x[
1
2
ATr Q2
+
1
3
BTr Q3
+
1
4
C (TrQ2
)2
+E'
(TrQ3
)2
+
1
2
L1(∂α Qβ γ)(∂α Qβγ)+
1
2
L2(∂α Qαβ)(∂γ Qβγ)]
28/38

Statics (contd.) and kinetics
∂t Qαβ(x ,t)=−Γαβμ ν
δ FGLdG
δQμ ν
+ζαβ(x ,t)
 Landau-Ginzburg model-A kinetics for non-conserved order
 is a stochastic thermal force satisfying the structure of .Qαβ
ζαβ
Free energy
diagram
Phase diagram
29/38

Numerical recipe and benchmarks
 Projection to orthonormal basis .
 Numerical integration of a equations and back transformation to
principal frame extraction of eigenvalues/eigenvectors to get back .
 Integrator benchmarks: OU process, static and dynamic correlator in
isotropic phase, angle-angle correlator in uniaxial nematic phase.
Qαβ=∑
i=1
5
ai ( x ,t)T αβ
i
,ζαβ=∑
i=1
5
ai (x ,t)ζαβ
i
Qαβ
Determinstic problems:
 Method of Lines (MOL).
 Spectral collocation method (SCM).
 High performance computing (HPC).
 Stochastic Method of Lines (SMOL).
Stochastic problems:
Applications
 Structure of isotropic-nematic
interface.
 Spinodal coarsening kinetics.
 Nucleation kinetics.
30/38

Application-I
Local/nonlocal properties
of isotropic-nematic interface
31/38

Application-I : Local/nonlocal properties of I-N interface
Isotropic IsotropicNematic
z
x
ya
Lx
Free energy per unit length of having a strip:
 Nature of I-N interface? de Gennes ansatz1
no
anisotropic elasticity reducing to scalar equation
in S. Later works tackled planar anchoring
problem with three variables. No results known
for oblique anchoring.
 Finding the ansatz to be valid at limit.
F=−a Lx(FN −F I )+Lx σ
L2=0
1) P.G. de Gennes, Mol.Cryst.Liq.Cryst. (1971).
L2=0
32/38
Fdistortion=∫d3
x {
1
2
L1(∂α Qβ γ)(∂α Qβ γ)+
1
2
L2(∂α Qα β)(∂γQβ γ)}

L2=18L1
L2
 Local biaxiality of uniaxial interface
with planar anchoring (using SCM).
 In oblique anchoring, director alignment
favours sign of .
L2=−L1
L2=36L1
33/38

ζαβ=0Fluctuating interface ζαβ≠0
34/38

Application-II
Phase ordering spinodal kinetics
35/38

Application-II : Phase ordering spinodal kinetics [2D (d=2; s=3)]
 Topological classification and visualization of point defects1
–
(Uniaxial) and (Biaxial).
 Defects visualized via scalar and vector order that shows all class of
defect classes partially absent in schlieren texture measured in
experiments ( ).
π1(S
2
/ℤ2)=Z2 π1(S
3
/ D2)=ℚ8
intensity ∝sin2
[2θ]
S (x ,t)
sin
2
(2θ)[ x ,t]
T (x ,t)
Uniaxial defect
Biaxial
defect
1) Mermin, Rev. Mod. Phys. (1979).
36/38

Application-II : Intercommutation of line defects
 Line defects annihilate by intercommuting (exchanging segements)
and forming loops1
.
 Competion between energetics and Topology no topological rigidity
found in Biaxial nematics.
Uniaxial defect
Biaxial defect
[3D (d=3; s=3)]
1) Turok et al, Science (1991)
37/38

 Formulation of fluctuating kinetics of uniaxial/biaxial nematic in GLdG
framework, leading to novel visualization techniques and HPC ( in 2D,
lattice in 3D).
 Validated de Gennes ansatz at limit of the problem.
 Isotropic-uniaxial nematic interface obtains biaxiality for .
 Oblique director anchoring: scalars remain local to the interface and
anchoring follows a linear profile: planar for and homeotropic for
 Classification and visualization of all defect classes.
 Controversy in growth exponent been settled with clear time-scale
separation at different stages of phase ordering.
 Minimal GLdG framework incapable of topological rigidity in BN.
Methods
Summary: Inhomogeneous phenomena in nematics
1024
2
256
3
I-N Interface
L2=0
L2≠0
L2>0 L2<0.
Coarsening kinetics

 Thermal fluctuation inducded nucleation of NLC phase in 3D.
 Uniaxial and Biaxial NLC under electric field in 3D.
 Coupling incompressible flow[1,2]
to Q equations and
fluctuations 3D line defects in flow.
 Rheology in nematics.
 Multicomponent phase flow with fluctuations2
.
Field based methods
Research proposal(s)
2D uniaxial defects in
electric field [Oliveira
et al, Phys.Rev.E, '10]
2D slice of ellipsoidal
nematic phase in isotropic
phase [Bhattacharjee,
PhD Thesis, '10]
Velocity field of
defects pair in LB
simulations [Yeomans
et al, Phys.Rev.Lett., '02]
±
1
2
[1] Berris & Edwards, Thermodynamics of
flowing system, Oxford (1994).
[2] Donev et al, Phys. Rev. E (2014).
Nucleation and
growth of nematic
phase of 5CB at
cooling rate 0.001
degC/min
[Sun et al,
Phys.Rev.E, '09]
Amit Bhattacharjee Courant Institute (NYU) 1/2
Proposal

 Dense colloidal rheology: role of size disparity3
, flow
geometry (planar Couette[4,5]
, uniaxial extension, mixed),
flow history (shear cessation6
, non-instantaneous flow
reversal and LAOS7
), coarse graining.
 Microrheology of colloid-nematic mixture8
.
 Glassy nematic rheology8
.
Particle based methods
Research proposal(s)
 Chemical reaction-diffusion systems: GENERIC / CLE / CME coupled
to compressible NS equations: Schlögl model1
, Dimerization reaction2
.
[1] Lubensky et al, Phys. Rev. E. (2012).
[2] Bedeaux et al J. Chem. Phys (2011).
[3] Voigtmann and Horbach, Phys. Rev. Lett. (2009).
[4] Bhattacharjee et al, J. Chem. Phys. (2013).
[5] Bhattacharjee, arXiv 1410.8115 (2014).
[6] Zausch Horbach, Euro. Phys. Lett. (2010).
[7] Brader et al, Phys. Rev. E., (2010).
[8] Onuki et al, Phys. Rev. E., (2014).
Elastic map with small and
large elastic constant at
strain=0.1 [Bhattacharjee,
unpublished]

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