MS_Thesis_Presentation

Phase Field Modelling of droplet coalescence coupled
with chemical reaction
Prashant Kumar
Ecole Polytechnique, France
30/08/2013
Kumar (Ecole Polytechnique, France) Phase Field Modelling of droplet coalescence coupled with chemical reaction30/08/2013 1 / 30

Aim of the project
The aim of the project was to study the effect of fluid flow on the mixing
of the components and the rate of total product formation during the
coalescence of the droplets.

Experimental Setup
(a) Experimental Setup (b) Processes in steps
Pancake shape

Phenomena

Literature for the droplet coalescence
For droplets with equal radii, initially, when the rneck → 0, the neck
grows as t and this regimes is known as the viscous-dominated
regime. 1,2
As the rneck increases, the Reynold number also increases and the
neck grows as t0.5 and this regime is known as the inertia-dominated
regime. 1,2

Literature: Coalescence regimes

Contour of the neck during the coalescence

Initiation of the coalescence process

Phase Field approach

Symbols
Symbol Physical quantity Unit
R0, R1, R2 Droplet radius m
µ Viscosity Pa s
γ Surface Tension N m−1
ρ Density Kg m−3
Ux ,Uy x and y direction velocity m s−1
CA, CB, CC Componenet concentration M
kAB rate constant of the chemical reactiom M−1 s−1
D Component diﬀusivity m2 s−1
w Interface width m
τ Phase ﬁeld relaxation time s

Phase Field approach: Double well potential
We assume a double well potential φ2(1 − φ)2 which gives φ a
tangent hyperbolic shape: φ = 0.5(1 − tanh(r−Ro√
2w
))

Governing Equations: Phase Field evolution
equation
Temporal evolution equation for the order parameter
τ(
∂φ
∂t
+ U · φ) = w2 2
φ − φ(1 − φ)(2 − φ) − w2
k(φ)| φ| (1)
In the polar coordinates, w2 2φ reduces to w2 1
r
∂
∂r (r ∂φ
∂r ) which drives
the motion by curvature. In order to counter this unphysical process,
an artiﬁcial term is added to the above equation.
w2 2φ − w2k(φ)| φ| is a modiﬁed Laplace operator.
φ(1 − φ)(2 − φ) is the negative of the double well potential
U · φ advects the order parameter

Governing Equations: Fluid flow
Navier Stokes is used to solve for the fluid flow.
ρ(
∂U
∂t
+ U. U) = − P + · (µ U) + γk(φ) φ (2)
Vorticity - Streamfunction formulation is used to solve the Navier
Stokes equation
ρ(
∂ω
∂t
+ U. ω) = µ( 2
ω) + γ × (k φ) (3)
Streamfunction is solved from the vorticity using the Poisson equation
2
ψ = −ω (4)
The velocity field is obtained from the streamfunction using the
relation
Ux = −
∂ψ
∂y
, Uy =
∂ψ
∂x
(5)

Governing Equations: Reaction diffusion system
We study a simple irreversible chemical reaction: A + B → C
∂Ci
∂t
+ U · Ci = · (Di Ci ) ± Kr CACB (6)
We cut off the fluxes at the boundary of the droplet by multiplying φ
with the fluxes.
∂Ci
∂t
+ · (φUCi ) = · (φDi Ci ) ± Kr CACB (7)

System size and the simulation domain
Diﬀusive time scale is given by: τD =
R2
0
D
Viscous time scale: τviscous = µR0
γ
Inertial time scale:τinertia = ρR3
γ
Taking the following values for the parameters (ρ = 1000
Kgm−3,µ = 0.005 Pa s, γ = 0.02 Nm−1, R0 = 100 µm), we get
τD = 0.1s
τviscous = 2.5 ∗ 10−5s, and
τinertia = 2.236 ∗ 10−4s
Keeping in mind the diﬀerence in the time scales and the minimum
grid spacing h = 0.025 µm, we ran simulations on systems with
R0 = 1 µm.

Droplets with equal radii - Coalescence
R1(µm) R2(µm) µ1(Pa s) µ2(Pa s) µ3(Pa s) γ(Nm−1)
1 1 0.01 0.01 0.01 0.05

Rate of growth of the common neck with time in
droplets with equal radii

Reactant evolution- droplets with equal radii

Product evolution - droplets with equal radii

Droplets with unequal radii - Coalescence

Reactant evolution- Less viscous outer ﬂuid
1 1 0.01 0.01 0.01 0.05

Product evolution - Less viscous outer ﬂuid

Reactant evolution - Highly viscous outer ﬂuid
1 1 0.01 0.01 0.01 0.05

Product evolution - Highly viscous outer ﬂuid

Rate of total product formation with time: Droplets
with equal radii

Rate of total product formation with time: Droplets
with unequal radii
-22
-20
-18
-16
-14
-12
-10
-8
-6
-12 -11 -10 -9 -8 -7 -6
log10(Product)
log10(time (sec))
a
b
c
d
e
a=2.7 +/- 0.1
b=2.0 +/- 0.1
c=2.5 +/- 0.1
d=2.0 +/- 0.1
e=1.5 +/- 0.1
Slope
µr=1
µr=0.001
µr=100

Morphology Diagram
µr1 = µ2
µ1
and µr2 = µ3
µ1

References
JENNS EGGERS, JOHN R. LISTER & HOWARD A. STONE, 1999
Coalescence of liquid drops, J. Fluid. Mech. 401, 293-310.
DIRK G.A.L. AARTS, HENK N.W. LEKKERKERKER, HUA GUO,
GERARD H. WEGDAM & DANIEL BONN, 2005 Hydrodynamics of
Droplet Coalescence, Phy. Rev. L 95, 164503.
MARTIN Z. BAZANT & H.A. STONE, 2000 Asymptotics of
reaction-diffusion fronts with one static and one diffusing reactant,
Physica D 147, 95-121.
KNUT ERIK TEIGEN, XIANGRONG LI, JOHN LOWENGRUB, FAN
WANG & AXEL VOIGT, 2009 A diffuse-interface approach for
modelling transport, diffusion and adsorption/ desorption of material
quantities on a deformable interface, Commun. Math Sci. 4(7),
1009-1037.
X. LI, J. LOWENGRUB, A. RATZ & A. VOIGT, 2009 Solving PDEs
in complex geometries: a diffuse domain approach, Commun. Math
Sci. 7(1), 81-107.

References
SEBASTIEN NGUYEN, R. FOLCH, VIJAY K. VERMA, HERVE
HENRY & MATHIS PLAPP, 2009 Phase-field simulations of viscous
fingering in shear-thinning fluids, Physics of Fluids 22, 103102.
Y. SUN & C. BECKERMANN, 2004 Diffuse inteface modeling of
two-phase flows based on averaging: mass and momentum equations,
Physica D 198, 281-308.
SAMUEL M. ALLEN & JOHN W. CAHN, 1978 A microscopic theory
for antiphase boundary motion and its application to antiphase
domain coarsening, Acta Metallurgica 27, 1085-1095.
JOSEPH D. PAULSEN, JUSTIN C. BURTON, SIDNEY R. NAGEL,
SANTOSH APPATHURAI, MICHAEL T. HARRIS & OSMAN A.
BASARAN, 2012 The inexorable resistance of inertia determines the
initial regime of drop coalescence, PNAS 109(18):6857-61 22511714
T. BIBEN, C. MISBAH, A. LEYRAT & C. VERDIER, 2003 An
advected field approach to the dynamics of fluid interfaces, Euro.
Phys. Lett. 63, 623.

References
E. FRADET, C. McDOUGALL, P. ABBYAD, R. DANGLA, D.
McGLOIN & C.N. BAROUD, 2011 Combining rails and anchors with
laser forcing for selective manipulation within 2D droplet arrays Lab
on a Chip , 11, 4228-4234.
A. HUEBNER, C. ABELL, W. HUCK, C.N. BAROUD & F.
HOLLFELDER, 2011 Monitoring a Reaction at Submillisecond
Resolution in Picoliter Volumes, Anal. Chem., 83, 1462-1468.
DAVID M. YOUNG, 1950 Iterative methods for solving partial
diﬀerential equations of elliptical type, PhD thesis, Harvard University
ETIENNE FRADET, 2013 PhD thesis, Ecole Polytechnique, France

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