Spontaneous motion of droplets during the demixing transition
in binary fluids
V. Kumaran
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
The convective interaction between a pair of droplets coarsening during the demixing transition of
a binary fluid is examined. The starting point is the model H equation for binary fluids, and the
droplet sizes are considered to be large enough that thermal fluctuations are neglected. Droplet
motion is induced by the convective coupling in the concentration equation, where there is a flux of
concentration due to the fluid velocity, and a reciprocal effect in the momentum equation. The effect
of the convective force density is separated into two parts—one due to the sharp concentration
gradients at the droplet interface, and the other due to the variation in the matrix. It is shown that
the dominant contribution to the fluid velocity field is due to the sharp concentration variation at the
interface, and this is proportional to the square of the droplet flux at the surface. The surface flux is
determined by solving the diffusion equation in the matrix between the droplets, and matching the
solution to that in the interfacial region. The analysis indicates that there is an attractive interaction
if the two droplets have radii larger or smaller than the critical radius, while the interaction is
repulsive if the radius of one droplet is larger and the other smaller than the critical radius. The
magnitude of the induced droplet velocity is estimated.
I. INTRODUCTION
There are two different types of morphologies that could
develop during the late stages of the demixing transition in
binary fluids. In a nearly symmetric quench, demixing proceeds due to the formation and coarsening of random interfaces which separate the two phases. In an asymmetric
quench, droplets of the minority phase nucleate in the majority phase, and then grow due to diffusive transport or Brownian motion. The evolution of the morphology is very different in the two cases; the characteristic length scale of a
random interface increases proportional to t ⫺1 , while the
characteristic droplet radius during droplet growth increases
proportional to t ⫺1/3 in the case of diffusive growth as well
as Brownian motion, where t is the time after the quench.
There have been many theoretical studies1–7 and computer
simulations8,9 which have shown that there is a qualitative
difference in the droplet growth process between metal alloys and fluid mixtures. This difference is because droplet
coarsening could be enhanced by convective transport of material, in addition to the usual diffusion process that is responsible for the transport of material in solid lattices.10,11
Experimental studies12–15 have also shown that the growth
process is very different from the diffusion controlled growth
processes in metal alloys and magnetic systems.
In addition to diffusive transport and Brownian motion,
another mechanism for the coalescence of droplets has been
identified in recent experiments.12,13 In these experiments, it
was observed that even when the droplets are sufficiently
large that Brownian motion is negligible, coalescence takes
place due to the spontaneous motion of droplets towards
each other. It was suggested that this could be due to the
force exerted by the convective term in the momentum con-
servation equation, which is the reciprocal of the convective
transport term in the mass conservation equation in the
model H16 equations for a binary fluid. Further, it was also
suggested that this force is caused by the sharp gradient in
the concentration at the droplet interface. An analysis of the
motion of a droplet in a steady concentration gradient was
carried out by Tanaka17 to illustrate the effect of concentration gradient on droplet motion. Simulations in a two dimensional system18 have also demonstrated the attractive interaction between growing droplets in a supersaturated matrix.
Though experimental systems involve a large number of
droplets simultaneously interacting with each other, the basic
physics of the interaction is better illustrated by first considering a pair of droplets growing in a supersaturated matrix.
Moreover, the multipole expansion method used here for calculating the interactions can be easily extended to a system
with many droplets. The interaction between a pair of droplets of equal radius was analyzed earlier,5 and it was found
that there is an attractive interaction between the droplets due
to convective effects. In the present case, the analysis is extended to droplets of different sizes, and the droplet deformation is also examined. The calculation of the flow induced
due to concentration variations is carried out in two steps. In
the first step, it is shown that the sharp concentration gradients at the interface provide the dominant contribution to be
convective force density and the induced fluid velocity, and
the fluid velocity field is related to the flux at the surface. In
the second step, the diffusion equation for the concentration
field around a pair of droplets is solved to obtain the surface
flux, and the motion of the droplets. The principal result of
this analysis is that there is spontaneous droplet motion due
to a contribution to the surface force density when the surface flux is nonzero, and this contribution is large compared
to surface tension effects under nonequilibrium conditions.
II. ANALYSIS
The system consists of two droplets A and B with radii
R A and R B separated by a vector distance L growing due to
diffusion in a matrix with a concentration c ⬁ which is higher
than the saturation concentration. The free energy of the system is assumed to be of the usual Cahn–Hilliard square gradient form with an additional contribution due to the kinetic
energy of the fluid
F关c兴⫽
冕 冋
dx f 关 c 兴 ⫹
册
␳
K
共 ⳵ c 兲 2 ⫹ v 2i ,
2 i
2
␦F
⫺ v i ⳵ i c⫹ ␰
␦c
冋
册
共2兲
␦F
⫹␨i ,
␦c
共3兲
⫽⌳ ⳵ 2i ⫺K ⳵ 2j c⫹
␳⳵ t v i ⫽ ␮ ⳵ 2j v i ⫹ ⳵ i c
␦ f 关c兴
⫺ v i ⳵ i c⫹ ␰ ,
␦c
where ␮ is the fluid viscosity, and ␰ and ␨ i are the appropriate noise sources required to satisfy the fluctuation dissipation theorem. The convective force density, which is the second term on the right side of 共3兲, is reciprocal to the
convective term in the mass conservation Eq. 共2兲, and these
two terms are required to ensure that the Poisson bracket
relations are satisfied. The Langevin noise sources are neglected in the late stages of spinodal decomposition when the
Brownian motion is negligible. In the concentration equation, the convective transport is considered to be small compared to the diffusion of solute, and the conditions for this
assumption to be valid are examined a little later. With this
simplification, the concentration equation is
⳵ t c⫽⌳ ⳵ 2i
␦F
.
␦c
共4兲
The momentum equation is further simplified by neglecting
fluid inertia, which is small in realistic systems5 to give
冋 册
␮ ⳵ 2j v i ⫹ ⳵ i c
␦F
⫽0,
␦c ⬜
共5兲
where 关 ¯ 兴⬜ represents the transverse component. The above
equation can be easily solved to give
v i 共 x兲 ⫽
冕
冋
dx⬘ J i j 共 x⫺x⬘ 兲 ⳵ j c 共 x⬘ 兲
where the Oseen tensor J i j is
␦F
␦ c 共 x⬘ 兲
冋
册
⬜
,
共6兲
册
1 ␦ i j x ix j
⫹
.
8 ␲ ␮ 兩 x兩 兩 x兩 3
共7兲
A further simplification is possible in the expression for the
velocity 共5兲 by realizing that the convective force density due
to the contribution proportional to f 关 c 兴 in the expression of
the free energy is longitudinal
⳵ ic
␦ f 关c兴
⫽⳵i f 关c兴
␦c
共8兲
and therefore does not contribute to the transverse part of the
velocity field. The equation for the velocity in the fluid is
then given by
共1兲
where indicial notation has been used for the vectors, ⳵ i
⬅( ⳵ / ⳵ x i ), ␳ is the fluid density and v i is the velocity. It is
not necessary to specify f 关 c 兴 because, as discussed later, the
motion of the droplets depends only on the square gradient
term in the free energy. The conventional model H
equations16 are used for the concentration and velocity field
in the fluid
⳵ t c⫽⌳ ⳵ 2i
J i j 共 x兲 ⫽
v i 共 x兲 ⫽⫺K
冕
dx⬘ J i j 共 x⫺x⬘ 兲 ⳵ j c 共 x⬘ 兲 ⳵ 2k c 共 x⬘ 兲 .
共9兲
There are two length scales of interest in the problem, the
magnitude of the droplet radius R and the interfacial thickness h. The fluid velocity field can be separated into two
contributions—one due to the sharp gradients in the concentration field at the droplet surface and the second due to the
variation in the concentration in the matrix between the two
droplets. It can be shown as follows that the velocity due to
the nonequilibrium correction to the concentration field at
the interface is large compared to that due to the variation in
the matrix. For this purpose, it is useful to consider a specific
model for the free energy f 关 c 兴 ,
f 关c兴⫽
␹
共 c⫺c d 兲 2 共 c⫺c m 兲 2 ,
2
共10兲
where ␹ is a constant, and c d and c m are the equilibrium
droplet and matrix concentrations, respectively. The flux of
material from the matrix to the droplet causes interface motion, and the concentration in the reference frame fixed on
the interface is determined from the concentration equation
⫺u 共 xs 兲 ⳵ z c⫽⌳ ⳵ z2
冋 册
␦F
␦c
⫽⌳ ⳵ z2 关 ⫺K ⳵ z2 c⫹2 ␹ 共 c⫺c d 兲
⫻ 共 c⫺c m 兲共 2c⫺c d ⫺c m 兲兴 ,
共11兲
where z is the coordinate normal to the interface, and u(xs )
is the normal velocity of the interface due to the flux of
solute. The equilibrium concentration profile is
c e⫽
冉冊
z
c d ⫹c m c d ⫺c m
⫺
tanh
,
2
2
h
共12兲
where h⫽ 关 2K/ ␹ (c d ⫺c m ) 2 兴 1/2 is the interfacial thickness. In
the above expression, curvature affects have been neglected
since the interface thickness is small compared to the radius
of the droplet. It is useful to express the concentration profile
as c⫽(c d ⫹c m )/2⫹ ␺ (c d ⫺c m )/2. The equilibrium interface
concentration is then
␺ e ⫽⫺tanh
冉冊
z
.
h
共13兲
the length scale for the variation of c * is R. With this simplification, the concentration equation reduces to
⳵ t c * ⫽D ⳵ 2j c * ,
共18兲
where the diffusion coefficient D⫽⌳ ␹ (c d ⫺c m ) . If the motion of the interface is due to diffusion, the left-hand side of
the above equation can be estimated as (uc * /R), where u,
the normal velocity of the interface, is O(Dc * /R(c d ⫺c m )).
It is easily seen that in the late stage when the supersaturation is small and c * Ⰶ(c d ⫺c m ), the left-hand side of 共18兲 is
small compared to the right-hand side. However, there is
droplet motion due to convective effects as well, and it is
shown a little later that the velocity field in this case scales as
(K(c ⬁ ⫺c m ) 2 R/L 2 ␮ ), where ␮ is the fluid viscosity and L is
the distance between the interacting droplets. It can easily be
verified from 共1兲 that the surface tension ␴ ⬃K(c d ⫺c m ) 2 /h,
and therefore the fluid velocity due to convective effects
scales as ( ␴ (c ⬁ ⫺c m ) 2 Rh/L 2 (c d ⫺c m ) 2 ␩ ). This velocity
could be significant even though the ratio (c ⬁ ⫺c m )/(c d
⫺c m ) and 共h/L兲 are small, because the ratio 共␴/␩兲 could be
quite large. For example, the surface tension and viscosity
for water are O(10⫺2 kg/s2 ) and 10⫺3 kg/m/s, respectively,
and so the ratio 共␴/␩兲 is O(10 m/s). Therefore, even if the
ratio (Rh/L 2 )(c ⬁ ⫺c m ) 2 /(c d ⫺c m ) 2 is O(10⫺4 ), the convective velocity is still of O(1 mm/s). The ratio of the convective and diffusive terms in the conservation equation is
O( ␴ hR 2 /L 2 ␩ D)(c ⬁ ⫺c m ) 2 /(c d ⫺c m ) 2 . In the present analysis, this parameter is considered to be small, and diffusion is
considered to be dominant in the leading approximation. In
the absence of convection, the concentration equation reduces to
2
FIG. 1. Schematic of the variation of the concentration field on the scale of
the droplet radius 共a兲 and on the scale of the interfacial thickness 共b兲.
The nonequilibrium correction to the interfacial concentration profile is determined by linearizing the concentration
equation in the departure from equilibrium ␾ ⫽ ␺ ⫺ ␺ e ,
⫺
冋
冉
冊册
3 ␺ 2e ⫺1
D 2
1 2
u 共 xs 兲
⳵ z*␺ e⫽ 2 ⳵ z* ⫺ ⳵ z*␾ ⫹
␾ , 共14兲
h
h
2
2
where the dimensionless coordinate z * ⫽(z/h), and D
⫽⌳ ␹ (c d ⫺c m ) 2 is the bulk diffusion coefficient. The above
fourth order equation can be integrated twice to obtain
冋
冉
冊册
3 ␺ 2e ⫺1
1 2
⫺ ⳵ z*␾ ⫹
␾
2
2
⫽
uh
D
冋冕
z*
0
册
dy 共 ⫺ ␺ e 共 y 兲兲 ⫹z * ⫺log共 2 兲 ,
共15兲
where the constants of integration are determined from the
requirement that the ␾⫽0 and d z * ␾ ⫽0 in the droplet phase
z * →⫺⬁, where the concentration is equal to the equilibrium concentration. It is not necessary to obtain the numerical solution to Eq. 共14兲 for ␾, but is sufficient to recognize
the following limiting behavior of the function ␾ (z * ),
␾ ⬀exp共 2z * 兲
2u 共 xs 兲 h
z*
␾⫽
D
for z * →⫺⬁,
共16兲
for z * →⬁.
The above limiting behavior is accurate only for z⬃h at the
interface, and this is not applicable for z⬃R because there is
a correction due to curvature effects. A schematic of the
variation in the concentration field on the scale of the droplet
radius is shown in Fig. 1共a兲, and on the scale of the interface
thickness h is shown in Fig. 1共b兲.
The concentration field in the matrix is determined by
solving the diffusion equation 共4兲. It can be shown, as follows, that this reduces to the Laplace equation for the concentration field in the matrix. The concentration field in the
matrix is expressed as c⫽c ⬁ ⫹c * , where c * is the difference
between the local concentration and the background concentration c ⬁ at a large distance from the droplet. The concentration equation 共4兲 is expressed in terms of c * , and linearized in the limit (c ⬁ ⫺c m )Ⰶ(c d ⫺c m ) appropriate for late
stage growth, to obtain
⳵ t c * ⫽⌳ ⳵ 2i 关 ⫺K ⳵ 2j c * ⫹ ␹ 共 c d ⫺c m 兲 2 c * 兴 .
共17兲
In the above equation, it can easily be verified that the first
term on the right is O(h 2 /R 2 ) smaller than the second, since
⳵ 2j c * ⫽0.
共19兲
It can also be inferred that the convective force density in the
matrix, which is proportional to ⳵ 2j c, is also zero in the leading approximation. Consequently, it is sufficient to consider
the fluid flow due to the convective force density at the interface alone.
For sharp interfaces, the integral in 共9兲 can be separated
into the product of integrals over the coordinate normal to
the interface z and the area of the droplet A:
v i 共 x兲 ⫽⫺K
冕
dA J i j 共 x⫺xs 兲
冕
dz ⳵ i c 共 x⬘ 兲 ⳵ 2j c 共 x⬘ 兲 ,
共20兲
where xs is the position along the surface of the droplet. In
deriving the above equation, the Oseen tensor J i j (x⫺x⬘ )
⫽J i j (x⫺xs ⫺zn) has been approximated by J i j (x⫺xs ). The
error due to this approximation is O(h/R), since the concentration gradients are significant only over the interfacial region of thickness h. At equilibrium, it can be shown that the
term ⫺K 兰 dz ⳵ i c(x⬘ ) ⳵ 2j c(x⬘ ) gives the normal force exerted
by the interface, which is the product of the surface tension,
mean curvature and the unit normal. The dominant contribution to the term ⳵ i c at equilibrium is n i ⳵ z c, since there are no
concentration variations in the tangential direction. The term
⳵ 2j c can be written as
⳵ 2j c⫽ ⳵ z2 c⫹
冉
冊
冉 冊
1
h2
1
⫹
⳵ z c⫹O 2 ,
R1 R2
R
共21兲
where R 1 and R 2 are the principal radii of curvature at a
point on the interface, and the second term on the right-hand
side of 共21兲 is because the coordinate z is measured from a
curved surface. Using the above relations, the term
⫺K
冕
dz ⳵ i z ⳵ 2j z
⫽⫺Kn i
冕 冋⳵
dz
2
z c ⳵ z c⫹
冋
⬁
⫽⫺Kn i 共 ⳵ z c 兲 2 兩 ⫺⬁
⫹
冉
冉
冊 册
1
1
⫹
共 ⳵ zc 兲2
R1 R2
1
1
⫹
R1 R2
冊冕
册
dz 共 ⳵ z c 兲 2 .
共22兲
At equilibrium, it is easily verified that the first term on the
right-hand side of 共22兲 is identically zero, because ⳵ z c,
which is the flux of solute at the interface, is zero. The second term on the right-hand side of 共22兲 is the product of the
unit normal (n i ), the curvature 关 (1/R 1 )⫹(1/R 2 ) 兴 , and the
surface tension K 兰 dz( ⳵ z c) 2 ; and the negative sign in 共22兲
indicates that the force due to surface tension is directed
opposite to the outward unit normal n i . In a nonequilibrium
system, the flux at the interface is nonzero, and consequently
the first term on the right-hand side of 共22兲 is nonzero. Moreover, this term is O(R/h) larger than the second term, and
consequently provides the dominant contribution to the convective force density at the interface, and the second term on
the right-hand side can be neglected in comparison to the
first. With this approximation, the velocity field is
v i 共 x兲 ⫽⫺K
⫻
冕
⫽⫺K
冕
dA J i j 共 x⫺xs 兲 n j 共 xs 兲
dz ⳵ z c 共 xs ⫹nz 兲 ⳵ z2 c 共 xs ⫹nz 兲
冕
dA J i j 共 x⫺xs 兲 n j 共 xs 兲
冋
共 ⳵ z c 共 xs ⫹nz 兲兲 2
2
册
⬁
.
c A ⫽C e ⫹
⫺⬁
共23兲
Since the droplet concentration has attained its equilibrium
value, ⳵ z c⫽0 in the limit z→⫺⬁. The gradient of the interfacial concentration profile in the region z→⬁ is determined
by matching the interfacial solution valid for z⬃h and the
outer solution in the matrix valid for z⬃R in the intermediate region zⰇh and zⰆR. It is shown a little later that this
matching condition reduces to the requirement that the concentration at the interface obtained from the outer solution
should be equal to the equilibrium matrix concentration, and
the concentration gradient ⳵ z c⫽( j n /D), where j n is the normal flux at the interface obtained from the outer solution.
With these simplifications, the expression for the fluid velocity reduces to
v i 共 x兲 ⫽⫺
K
2D 2
冕
dA J i j 共 x⫺xs 兲关 j n 共 xs 兲兴 2 .
matching the inner solution in the interfacial region 共16兲 with
the outer solution obtained by solving 共19兲 in the intermediate region hⰆzⰆR. In this region, the inner solution for the
concentration field scales as c m ⫹O(uz(c d ⫺c m )/D) from
共16兲, which is c m ⫹O(O(z(c ⬁ ⫺c m ) /R)) since the velocity
of the interface u is O((D/R)(c ⬁ ⫺c m )/(c d ⫺c m )). Thus, the
deviation of the concentration field from its equilibrium
value c m in the matrix is O(z(c ⬁ ⫺c m )/R), which is small
compared to the magnitude of the concentration variation in
the outer region (c ⬁ ⫺c m ) in the limit hⰆzⰆR. Consequently, it is appropriate to set the concentration at the interface equal to the equilibrium concentration c m while obtaining the outer solution.
The radius of the droplet in the absence of Brownian
motion or convection is determined by the Lifshitz–Slyozov
theory,11 where the droplet concentration is assumed to have
reached its equilibrium value, but the matrix is supersaturated. The chemical potential for the droplet phase contains
an additional contribution due to the Laplace pressure arising
from the surface tension between the droplet and matrix,
while the chemical potential of the matrix is higher than the
equilibrium chemical potential. When the droplet size is
smaller than a critical value, the chemical potential of the
droplet 共which contains a contribution proportional to the
inverse of the radius due to the Laplace pressure兲 is higher
than that of the matrix, and the droplet shrinks. When the
droplet size is larger than the critical value, the chemical
potential of the droplet phase is lower than that of the supersaturated matrix and the droplet grows. The critical radius
共for a given supersaturation of the matrix兲 is the radius at
which the droplet neither shrinks nor grows. The equilibrium
concentration of the matrix at the droplet surface depends on
the radius of the droplet, and the equilibrium concentrations
for two interacting droplets with radius R A and R B are11
共24兲
It can be seen from 共24兲 that the fluid velocity field depends on the flux of solute at the interface of the droplet. The
concentration field and the interfacial flux due to the interaction between a pair of droplets is examined next. The boundary conditions at the droplet surfaces are determined by
C⬘
C⬘
, c B ⫽C e ⫹
,
RA
RB
共25兲
where C e is the equilibrium concentration above a plane surface, and (C ⬘ /R A ) and (C ⬘ /R B ) are the corrections to the
equilibrium concentration due to the curvature of the droplet.
It is now necessary to solve the Laplace equation for the
concentration field 共19兲 with the boundary conditions c *
⫽⫺c A* ⫽c ⬁ ⫺c A at the surface of droplet A and c * ⫽⫺c B*
⫽c ⬁ ⫺c B at the surface of droplet B. The critical radius11 is
given by R c ⫽C ⬘ /(c ⬁ ⫺C e ), and c * ⫽0 at the interface for
droplets with the critical radius. In the theory for late stage
droplet growth, droplets with radius larger than the critical
radius grow due to diffusion of solute into the droplet, while
droplets with radius smaller than the critical radius shrink
due to diffusion from the droplet to the matrix.
The solution is obtained using multipole expansions,
where the concentration field is determined as an expansion
in spherical harmonics in two coordinate systems with origins at the centers of the two droplets. The solution for the
concentration field is obtained as an asymptotic expansion in
the parameters (R A /L) and (R B /L), where L is the distance
between the two droplets. Two spherical coordinate systems
(r A ,␪A , ␾ A ) and (r B ,␪B , ␾ B ) with origins at the centers of the
two droplets are chosen as shown in Fig. 2. The concentra-
A 0m ⫽⫺c A* ␦ 0m ,
A nm ⫽⫺
冉 冊 冋兺
n⫺m m⫺n⫺1
RA
RB
B q 共 m⫺n⫺q⫺1 兲
q⫽0
冉 冊册
n⫹q
q
for n⬎0,
共29兲
B 0m ⫽⫺c B* ␦ 0m ,
B nm ⫽⫺
FIG. 2. Coordinate systems used for calculating the concentration field
around a pair of droplets.
tion field is axisymmetric about the line joining the centers
of the two droplets, and the solution for the concentration
field is given by
冉
⬁
c *⫽
兺
n⫽0
An
n⫹1 P n 共 cos共 ␪ A 兲兲 ⫹
rA
Bn
冊
n⫹1 P n 共 cos共 ␪ B 兲兲 ,
rB
共26兲
where P n (x) are Legendre polynomials. The coefficients A n
and B n are expressed as an asymptotic expansion in the parameters (R A /L) and (R B /L),
⬁
A n⫽
兺
m⫽0
A nm
⬁
B n⫽
兺
m⫽0
B nm
冉 冊
RA
L
冉 冊
RB
L
1
rA
n⫹1
P n 共 cos共 ␪ A 兲兲
冉冊
1
⫽
L
冉 冊
1
rB
n⫹1 ⬁
兺
q⫽0
j A ⫽⫺D ⳵ rA c * 兩 r A ⫽R A
⬁
⫽⫺D
⫻
冉
⬁
兺 兺
n⫽0 m⫽0
冋冉
R Bm⫹q⫹1 R An
L n⫹m⫹q⫹1
冊冉 冊册
⫽
⬁
⫽⫺D
⬁
兺 兺
n⫽0 m⫽0
兺
q⫽0
冉 冊冉 冊
n⫹q
q
rB
L
P q 共 cos共 ␪ B 兲兲 ,
共28兲
1
L
兺
q⫽0
冋冉
⫺
m⫺n⫺1
m
⫹
P n 共 cos共 ␪ A 兲兲 ,
共 n⫹1 兲 B nm
RB
冉 冊冉
nA qm
RB
冊冉 冊
RA
L
R Am⫹q⫹1 R Bn
nB qm
RA
m
n⫹q
q
L n⫹m⫹q⫹1
冉 冊
共30兲
冊冉 冊
冊冉 冊册
RB
L
兺
q⫽0
P n 共 cos共 ␪ B 兲兲 .
The fluid velocity field can now be calculated using Eq.
共24兲, and the velocity of the surface of the droplet thereby
evaluated. It is convenient to determine the radial velocity of
the droplet surface using an expansion in Legendre polynomials
⬁
v Ar 共 ␪ I 兲 ⫽
兺 V 共An 兲 P n共 cos共 ␪ A 兲兲 .
共32兲
i⫽n
The radial velocity does not contain a component proportional to P 0 (cos( ␪ A )) due to the incompressibility require(n)
are determined using Eq. 共24兲
ment. The coefficients V Ar
and the orthogonality condition for Legendre polynomials
冕
K
2D 2
兺
J⫽A,B
冕
⬘ n j 共 xJs
⬘ 兲关 j 共 xJs
⬘ 兲兴 2
dxJs
⬘兲
dxAs n i 共 xAs 兲 J i j 共 xAs ⫺xJs
共 2n⫹1 兲
4 ␲ R A2
P n 共 cos共 ␪ A 兲兲 .
共33兲
冉 冊冉 冊
n⫹q
q
n⫹q
q
共31兲
V 共An 兲 ⫽⫺
q
P n 共 cos共 ␪ B 兲兲
冉冊
n⫹q
n
j B ⫽⫺D ⳵ rB c * 兩 r B ⫽R B
n⫹1
n⫹1 ⬁
共 n⫹1 兲 A nm
⫺
RA
共27兲
m
冉 冊册
The normal flux at the surfaces of droplet A and B are then
obtained from the above solution
⫹
,
q⫽0
A q 共 m⫺n⫺q⫺1 兲
for n⬎0.
m⫺n⫺1
m
and the coefficients A nm and B nm are chosen to satisfy the
boundary conditions 共25兲 at the surfaces of the two droplets.
For this purpose, it is necessary to express the spherical harmonics in the coordinate system centered at droplet A in
terms of those centered at droplet B and vice versa. These
relations are provided by Hobson19
冉 冊
冉 冊 冋兺
n⫺m m⫺n⫺1
RB
RA
rA
L
(1)
V Ar
q
P q 共 cos共 ␪ A 兲兲
for r A ⬍L and r B ⬍L. Using the above relations, the coefficients A nm and B nm are
provides the translational velocity of the
The coefficient
droplet, while the higher order terms give the rate of deformation. The details of the calculation of the integrals in 共33兲
are provided in the Appendix, and the results for V A(n) for 0
⭐n⭐5 correct to O(1/L 5 ) are also listed in the Appendix.
Due to symmetry considerations, the droplets are constrained to move along the line joining their centers, and the
velocity perpendicular to the line joining their centers is
zero. The translational velocity of the droplet along the line
joining the centers is determined as the average of the velocity over along the line joining the centers over the surface of
the droplet:
V A⫽
1
S
冕
dS cos共 ␪ A 兲v Ar ,
共34兲
where S is the surface of the droplet, and the factor cos( ␪ A )
provides the component of the radial along the line joining
the centers. It can be easily verified that when the velocity is
expanded in a Legendre polynomial series 共32兲, the integral
共34兲 provides
V A ⫽V 共A1 兲 .
共35兲
It is useful to examine the motion and deformation of the
droplets when the distance of separation is large compared to
the droplet radius. The leading order contribution to the
mean velocity of the droplet, V A(1) , proportional to (1/L 2 ), is
V 共A1 兲 ⫽
4Kc A* c B* R B
5L 2
⫹O 共 1/L 3 兲 .
共36兲
In the limit of large L, this velocity is positive if the product
(c A* c B* ) is positive, and the droplets approach each other
along their line of centers. Thus, two droplets with radii
greater than or less than the critical radius approach each
other, while a pair of droplets with one larger and the other
smaller than the critical radius move away. This physical
reason for this behavior is shown in Fig. 3. When the two
droplets are either larger or smaller than the critical radius,
the flux on the part of the droplet surface facing the other
droplet is smaller than that on the part not facing the other
droplet. Since the velocity at a point on the surface depends
on the square of the normal flux at that point from 共24兲, this
variation of the flux results in the two droplets coming towards each other. When one of the droplets is larger and the
other smaller than the critical radius, the flux in between the
two droplets is larger than that on the parts of the surface not
facing the other droplet. This results in the droplets moving
away from each other.
The leading order contribution to the deformation rate
V A(2) , proportional to (1/L 3 ), is
V 共A2 兲 ⫽
4Kc A* c B* R A R B
L3
⫹O 共 1/L 4 兲 .
共37兲
This coefficient is also positive for c A* c B* ⬎0, indicating that
droplets which approach each other deform into prolate
spheroids which are elongated along the line joining the centers, while the coefficient is negative for c A* c B* ⬍0 indicating
that droplets which move apart deform into oblate spheroids
which are compressed along the line joining their centers.
III. CONCLUSIONS
The interaction between a pair of droplets in the demixing transition in a binary fluid has been examined in the late
stages where the Brownian diffusion is not important. The
model H concentration and momentum equations were used
as the starting point for the analysis, and inertial effects and
FIG. 3. Variation in the surface flux due to the interaction between two
droplets 共a兲 both of which have radii larger than the critical radius; 共b兲 both
of which have radii smaller than the critical radius; 共c兲 one has radius larger
and the other smaller than the critical radius.
Brownian motion were neglected in this analysis. The convective interaction arises due to the presence of the convective terms in the concentration equation, and the reciprocal
term in the momentum equation required for satisfying the
fluctuation dissipation theorem. The effect of the convective
term in the momentum equation was separated into two parts
—one due to the sharp concentration gradients at the interface and the other due to the variations in the matrix between
two droplets. It was shown that the dominant effect is due to
the sharp concentration variations at the interface under nonequilibrium conditions, and this effect was proportional to
the square of the solute flux at the interface. The flux at the
interface was determined by solving the diffusion equation in
the matrix between the two droplets, and matching the solution with that in the interfacial region.
The results of the analysis indicate that convective effects do result in spontaneous motion of the droplets. If both
droplets have radii that are larger or smaller than the critical
radius, then the droplets approach each other, while if one
has a larger radius and the other has a smaller radius, then
there is a repulsive interaction. The physical reason for this is
shown in Fig. 3. In addition, the deformation of the droplets
could also be determined in the present analysis, and it was
found that droplets attracted to each other deform into prolate spheroids along the line of centers, while droplets that
are repelled deform into oblate spheroids.
It is useful to estimate the magnitude of the droplet velocities. It can easily be verified from 共1兲 that the surface
tension ␴ ⬃K(c d ⫺c m ) 2 /h, where c d is the droplet concentration and c m is the concentration in the matrix which is in
equilibrium with the droplet phase. Therefore the fluid velocity due to convective effects scales as ( ␴ (c ⬁
⫺c m ) 2 Rh/L 2 (c d ⫺c m ) 2 ␩ ). The ratio (R/L)⬃1, and (h/L)
⬃10⫺3 for typical values of R⬃1 ␮ m and h⬃1 nm. The
surface tension and viscosity for water are O(10⫺2 kg/s2 )
and 10⫺3 kg/m/s, respectively, and so the ratio ( ␴ / ␩ ) is
O(10 m/s). Therefore, even if the ratio (c ⬁ ⫺c m ) 2 /(c d
⫺c m ) 2 is in the range 10⫺2 – 10⫺4 , the velocity is
1 – 100 ␮ m/s, which could be significant.
cos共 ␪ A 兲 ⫽cos共 ␪ * 兲 cos共 ␪ A⬘ 兲 ⫺sin共 ␪ * 兲 sin共 ␪ A⬘ 兲 cos共 ␾ * 兲 .
共A2兲
⬘ )n j (xAs
⬘ ) can be expressed
The function n i (xAs )J i j (xAs ⫺xAs
in terms of the angles ␪ A⬘ , ␪ * , and ␾*:
⬘ 兲 n j 共 xAs
⬘ 兲
n 共 xAs 兲 J i j 共 xAs ⫺xAs
⫽
3 cos共 ␪ * 兲 ⫺1
8 ␲ ␮ R A 共 2 冑2 兲关 1⫺cos共 ␪ * 兲兴 1/2
(n)
Using the above expressions, the component V AA
is
APPENDIX
The calculation of the integrals in 共33兲 is discussed in
this section. The expression for V A(n) can be separated into
(n)
(n)
and V AB
, which represent the Legendre
two parts, V AA
modes of the velocity induced at the surface of droplet A due
to the flux at the surface of droplet A and B, respectively,
n兲
⫽⫺
V 共AA
⫻
n兲
⫽⫺
V 共AB
K
2D 2
冕
冕
⬘ n j 共 xAs
⬘ 兲关 j 共 xAs
⬘ 兲兴
dxAs
⬘ 兲
dxAs n i 共 xAs 兲 J i j 共 xAs ⫺xAs
K
2D 2
冕
⬘ ⫺L兲
⫻J i j 共 xAs ⫺xAs
共 2n⫹1 兲
4 ␲ R A2
n兲
⫽
V 共AA
h n⫽
4 ␲ R A2
2
冕
P n 共 cos共 ␪ A 兲兲 ,
共A1兲
dxAs n i 共 xAs 兲
P n 共 cos共 ␪ A 兲兲 .
⫺2R B L cos共 ␪ B⬘ 兲兲 1/2
2n⫹1
P n 共 cos共 ␪ A⬘ 兲兲 2
冕
␲
0
共A5兲
Using 共A3兲, the coefficients h n for 0⭐n⭐5 are h 0 ⫽0, h 1
⫽(8/5), h 2 ⫽(8/7), h 3 ⫽(16/15), h 4 ⫽(80/7), and h 5
⫽(40/39), respectively.
(n)
The second component V AB
is determined using an
asymptotic expansion in the parameters (R A /L) and (R B /L)
1
8␲␮
冕
⬘ n j 共 xBs
⬘ 兲
dxBs
⬘ 兲兴 2
⫻关共 xBs
兺n 兺m
g mn
P 共 cos共 ␪ B⬘ 兲兲 ,
Lm n
共A8兲
⬘
is the distance between the position xAs on droplet A and xBs
on droplet B, and
cos共 ␪ * 兲 ⫽⫺cos共 ␪ A 兲 cos共 ␪ B⬘ 兲 ⫺sin共 ␪ A 兲 sin共 ␪ B⬘ 兲 cos共 ␾ A 兲
共A9兲
共A6兲
where the coefficients g mn determined as follows. The coordinate system shown in Fig. 2 is used for the analysis, and
the product
冊
cos共 ␪ * 兲 ⫺2R A L cos共 ␪ A 兲
共A4兲
d ␪ A n i 共 xAs 兲
cos共 ␪ * 兲 共 R A ⫺R B cos共 ␪ * 兲 ⫺L cos共 ␪ A 兲兲共 R A cos共 ␪ * 兲 ⫺R B ⫹L cos共 ␪ B⬘ 兲兲
,
⫹
r
r3
⬘ in a spheriwhere ␪ B⬘ is the azimuthal angle of the vector xBs
cal coordinate system centered at droplet B, and ␪ A and ␾ A
are the azimuthal and meridional angles of the vector xAs in
a coordinate system centered at droplet A,
r⫽ 共 R A2 ⫹R B2 ⫹L 2 ⫺2R A R B
* 兲兴 2 h n P n 共 cos共 ␪ A⬘ 兲兲 ,
⬘ 关 j 共 xAs
dxAs
1
n兲
⫽
V 共AB
The first component
is determined as follows. The
coordinate system shown in Fig. 2 is employed, where the
⬘ and the line joining the centers
azimuthal angle between xAs
of the two droplets is ␪ A⬘ , and the azimuthal and meridional
⬘ are ( ␪ * , ␾ * ), respectively. The
angles between xAs and xAs
azimuthal angle between the vector xAs and the line joining
the centers of the droplets is then given by
冉
冕
⬘ 兲 n j 共 xAs 兲 P n 共 cos共 ␪ A 兲兲 .
⫻J i j 共 xAs ⫺xAs
(n)
V AA
⬘ 兲 n j 共 xBs
⬘ 兲⫽
n i 共 xAs 兲 J i j 共 xAs ⫺xBs
1
8␲␮
where
2
共 2n⫹1 兲
⬘ n j 共 xBs
⬘ 兲关 j 共 xBs
⬘ 兲兴
dxBs
共A3兲
.
共A7兲
⬘ .
is the cosine of the angle between the vectors xAs and xBs
The expression 共A7兲 is expanded in a Taylor series in the
parameter 共1/L兲
⬘ 兲 n j 共 xBs
⬘ 兲⫽
n i 共 xAs 兲 J i j 共 xAs ⫺xBs
l m 共 ␪ A , ␾ A ,␪B 兲
Lm
共A10兲
and the coefficients g mn are evaluated from the coefficients
lm
g mn ⫽
共 2n⫹1 兲
4 ␲ R A2
冕
dxAs l m 共 ␪ A , ␾ A ,␪B 兲 P n 共 cos共 ␪ A 兲兲 .
共A11兲
The following coefficients g mn for (0⭐m⭐5) and (0⭐n
⭐5) are not zero:
g 11⫽⫺2 cos共 ␪ B⬘ 兲 ,
2兲
V 共AA
⫽
g 21⫽R B ⫺3 cos共 ␪ B⬘ 兲 2 R B ,
g 22⫽⫺2 cos共 ␪ B⬘ 兲 R A ,
⫹
g 31⫽ 共 2 cos共 ␪ B⬘ 兲 R A2 兲 /5⫹4 cos共 ␪ B⬘ 兲 R B2 ⫺6 cos共 ␪ B⬘ 兲 3 R B2 ,
g 32⫽2R A R B ⫺6 cos共 ␪ B⬘ 兲 2 R A R B ,
g 33⫽ 共 ⫺12 cos共 ␪ B⬘ 兲 R A2 兲 /5,
g 41⫽ 共 ⫺3R A2 R B 兲 /5⫹ 共 9 cos共 ␪ B⬘ 兲 2 R A2 R B 兲 /5⫺ 共 3R B3 兲 /2
⫹12 cos共 ␪ B⬘ 兲
g 42⫽ 共 6
2
R B3 ⫺ 共 25
cos共 ␪ B⬘ 兲 R A3 兲 /7⫹12
⫺18 cos共 ␪ B⬘ 兲
3
cos共 ␪ B⬘ 兲
4
共A12兲
K 40c A c B R A3 R B
,
␮
77L 5
1兲
V 共AB
⫽
⫹ 共 65 cos共 ␪ B⬘ 兲 3 R B4 兲 /2⫺ 共 105 cos共 ␪ B⬘ 兲 5 R B4 兲 /4,
g 52⫽ 共 ⫺12R A3 R B 兲 /7⫹ 共 36 cos共 ␪ B⬘ 兲 2 R A3 R B 兲 /7⫺6R A R B3
⫹48 cos共 ␪ B⬘ 兲 2 R A R B3 ⫺50 cos共 ␪ B⬘ 兲 4 R A R B3 ,
g 53⫽ 共 4 cos共 ␪ B⬘ 兲 R A4 兲 /3⫹ 共 138 cos共 ␪ B⬘ 兲 R A2 R B2 兲 /5
R A2 R B2 ,
g 54⫽ 共 40R A3 R B 兲 /7⫺ 共 120 cos共 ␪ B⬘ 兲 2 R A3 R B 兲 /7,
g 55⫽ 共 ⫺10 cos共 ␪ B⬘ 兲 R A4 兲 /3.
(n)
(n)
and V AB
for
Using the above relations, the results for V AA
5
0⭐n⭐5, correct to O(1/L ), are
冋
K 4c A* c B* R B
4
⫺ 3 共 c A* 2 R A R B ⫹c B* 2 R B2 兲
2
␮
5L
5L
⫹
⫺
12c A* c B* R B2 R A
5L 4
册
8
„共 c A* 2 ⫹c B* 2 兲 R B2 R A 共 R A ⫹R B 兲 … ,
5L 5
共A13兲
冋
c A* c B* R A R B R A R B 共 c A* 2 R A ⫹c B* 2 R B 兲
K
⫺
⫹
␮
L3
L4
⫺
⫺ 共 33 cos共 ␪ B⬘ 兲 R B4 兲 /4,
1兲
V 共AA
⫽
册
4兲
V 共AA
⫽
0兲
⫽0,
V 共AB
cos共 ␪ B⬘ 兲 R A R B2
g 51⫽ 共 ⫺18 cos共 ␪ B⬘ 兲 R A2 R B2 兲 /5⫹6 cos共 ␪ B⬘ 兲 3 R A2 R B2
0兲
⫽0,
V 共AA
,
K 8c A* c B* R A2 R B 8R A2 R B
⫺
共 7c A* 2 R A ⫹16c B* 2 R B 兲 ,
␮
15L 4
105L 5
g 44⫽ 共 ⫺20 cos共 ␪ B⬘ 兲 R A3 兲 /7,
⫺42 cos共 ␪ B⬘ 兲
冋
5L 5
5兲
⫽0,
V 共AA
R B3 兲 /2,
R A R B2 ,
12c A* c B* R A2 R B2
3兲
⫽
V 共AA
g 43⫽ 共 18R A2 R B 兲 /5⫺ 共 54 cos共 ␪ B⬘ 兲 2 R A2 R B 兲 /5,
3
冋
K 4c A* c B* R A R B 4R A R B
⫺
共 5c A* 2 R A ⫹8c B* 2 R B 兲
␮
L3
35L 4
c A* c B* R A R B 共 20R A R B ⫺R A2 R B2 兲
冋
冋
5L 5
册
,
册
2兲
V 共AB
⫽
c A* c B* R A2 R B R A2
K
⫺
⫹ 5 共 c A* 2 R A R B ⫺c B* 2 R B2 兲 ,
␮
L4
L
3兲
⫽
V 共AB
6c A* c B* R A3 R B
K
⫺
,
␮
5L 5
册
4兲
⫽0,
V 共AB
5兲
V 共AB
⫽0.
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2