PRINCIPLES OF EMULSION FORMATION

Chemical Engimavfng
Science.
Printed in Great Britain.

Vol.

48. No.

2, pp. 333-349.

PRINCIPLES

OCW-2507/93
Is.OO+O.KI
Pergamon Press Ltd

1993.

OF EMULSION

Pieter

FORMATION

Walstra

Department
of Food Science, Wageningen
Agricultural
University,
Wageningen,

the Netherlands

ABSTRACT
The phenomena
occurring
during emulsion
formation
are briefly
reviewed.
Droplet break-up
in laminar and in turbulent
flow is
discussed
and quantitative
relations
are given.
The roles of
the surfactant
are considered,
i.e. lowering
the interfacial
facilitating
break-up)
and
preventing
tension
(and
thereby
recoalescence
(via the Gibbs-Marangoni

effect),
in relation
to
the time scales of the various processes
occurring.
INTRODUCTION
This article concerns
the formation
of classical
emulsions,
so
not micro-emulsions,
multiple
emulsions
or high-internal
phase
emulsions
(HIPEs).
This subject
was reviewed
earlier
in some
detail by the author
(Walstra,
1983), from which we will take
information,
literature
listed
most
without

referring
to
there.
Since this review was written
- in 1978 - new results
and some of these
and considerations
have become
available,
will be given
here, in addition
to reviewing
briefly
the most
salient points.
To make an emulsion,
we need oil and water (or more general
an
oily
and
an aqueous
phase),
a surfactant
and
energy.
The
essential
characteristics
of the resulting
emulsion

are:
- The emulsion
type:
oil-in-water
or water-in-oil.
This
is
primarily
determined
by the type of surfactant
(see further
on).
- The droplet
size distribution,
since smaller
droplets
are
nearly
always
more
stable
against
creaming,
coalescence
and
often also flocculation.
It is easy to make droplets
(gentle
shaking
suffices),

but
it
may
be
difficult
to
make
the
droplets
small enough.
This means that the essential
process
is not droplet
formation
but droplet break-up.
Moreover,
newly
formed droplets
may coalesce
again during emulsification,
and
this should be avoided as much as possible.
ENERGY

RELATIONS

is energy needed?
first be deformed
and

Why

In order to break up a droplet
it must
this is opposed by the Laplace pressure,

333


PIETER WA~STRA

334

which is the difference
in pressure
between the convex
concave side of a curved interface
and is given by

and

the

PL = y (l/R, + l/R,)
where
y is the
interfacial
tension
and R, and R, are the
principal

radii of curvature.
For a spherical
drop of radius r
we thus have pL = 2 y / r and taking,
for example
r = 0.2 urn
(which
is often
desired)
end
y = 0.01
N m-l (which
is a
reasonable
value),
we have
a Laplace
pressure
of 10' Pa (1
bar).
In order
to deform
the drop, a larger
external
stress
has
to
be
applied;
this

pressure
implies
a
very
large
gradient,
since
the stress
difference
has to occur
over
a
distance
of the
order
of r. The
stress
can
be due
to a
velocity
gradient
and then is a shear stress, or it can be due
to
a
pressure
difference
arising
from
inertial

effects
(chaotic motion of the liquid).
To achieve
the very high shearing
stress or the very intense
velocity
fluctuations
small
needed
to deform
and
break
up
droplets,
very much energy has to be dissipated
in the liquid.
with a radius
of 1 urn have to be
Assume
that oil droplets
formed
in water
and that the volume
fraction
of oil 4 = 0.1
and y = 0.01 N m-l, we obtain a specific
surface area A of 3 x
lo5 m-' and the net surface
free energy
needed

to create
that
surface
Ay = 3 kJ rnm3. In practice,
we need about 3 MJ ma3 to
make the emulsion,
which means that by far the greatest
amount
of the energy supplied
is dissipated
into heat.
It is seen from eq. (1) that the stress - and consequently
the
amount
of energy
- needed
to deform and thereby
break up the
droplets
is less if the interfacial
tension
is lower,
which
can be achieved
by adding
a sufficient
amount
of a suitable
surfactant.
This is one role of the surfactant,

but not the
immediate
most
essential
one,
which
is
to
prevent
the
formed
This
will
be
recoalescence
of
the
newly
drops.
discussed
further
on. We will consider
first the break-up
of
This
can be achieved
in laminar
flow
due to shear
drops.

inertial
effects
stresses,
or
in
turbulent
flow,
where
(pressure
fluctuations)
are
predominant,
although
shear
For inertial
stresses
may be of importance
in some cases.
for instance
caused
by ultrasonic
effects
due to cavitation,
refer
earlier
review;
since,
some
new
waves,

we
to
our
literature
has appeared
(e.g. Li and Fogler,
1978; Reddy and
Fogler, 1980).
DROPLET

BREAK-UP

IN LAMINAR

FLOW

flow field equals
The stress
exerted
on a drop in a laminar
gradient
and qC the viscosity
of
slcG, where G is the velocity
This
is counteracted
by
the
the
continuous

phase.
stress
Laplace pressure
and the ratio is called the Weber number:
We

= q, G r

/

Y


Principles of emulsion formation

335

We,,

Fig.

number
for
disruption
of
critical
1 The
Weber
droplets
in simple shear flow (curve, results

by
average
resulting
for
the
Grace,
1982)
and
size
in a colloid
mill
(hatched
area,
droplet
1990) as a function
of the
results
by Ambruster,
viscosity
ratio disperse to continuous
phase.

If We exceeds
a critical
value We_ (of the order of one), the
drop bursts. We_ depends on the type of flow and on the ratio
of the drop viscosity
to that of the continuous
phase r),/l),.
Break-up

of single
droplets
in simple
shear
flow
(velocity
gradient
in the direction
normal to that of the flow and thus
equal
has
been
well
studied
and ,some
to the
shear
rate)
results
are shown as the curve in Fig. 1. These results
agree
well with theory.
Others
have often found somewhat
different
results,
WeGr showing
the same trend but being at a slightly
probably
is that

lower or at a higher
level; the explanation
break-up
of the drop also depends
on the rate at which G is
attained
and on the time during which G lasts (Torza et al.,
2972).
(2) shows
that
for a low viscosity
of the continuous
Eq.
phase,
deformation
of
requires
small
drops
extremely
high
if y = 0.005 N m-l and ?jC =
velocity
gradients.
For example,
s-1 to obtain
low3 Pa s (water),
it would
need
G = 25*106

gradients
can
droplets
of r = 0.2 urn (We_ = 1). Such velocity
usually
not be produced
except over very small distances.
It
is also
seen
that
no break-up
occurs
for qo/qC > 4. The
explanation
is roughly
that the drop cannot deform as fast as
the simple shear flow induces deformation.
Deformation
time of
stress
a drop
is proportional
to its
visaosity
over
the
applied,
time according
that is %/rlcG, whereas the deformation

to the flow would
be l/G. So if rlo/qC >> 1, the drop does
but the deformed
drop starts
rotating
deform
to some extent,
at a rate of G/2. (For a low viscosity
drop, the liquid in it
rotates
while the drop keeps its orientation
with respect
to
the
direction
of
flow:
it can
deform
to a
consequently,
greater
extent.)
The viscosity
ratio above which no break-up
occurs how ever large We, turns out to be 4, both from theory
and experiment.


PIETER WALSTRA

336

In elongational
flow
shear,
velocity
gradient
in
the
(no
direction
of the flow) no rotation
is induced,
and even very
viscous
drops can be deformed
and broken up, if the velocity
gradient
lasts long enough;
the latter may be a problem
since
in
most
situations
elongational
flow
is
a
transient

phenomenon.
For the range of viscosity
ratios given in Fig. 1,
and for plane hyperbolic
flow, We,, is almost constant
at about
0.3. Thus, elongational
flow is more efficient
in breaking
up
drops, especially
at a high viscosity
ratio.
Up till fairly recently,
the theory
for droplet
disruption
in
simple
shear
had only
been
tested
for the deformation
and
burst
of single
drops.
In practice,
however,

drops
will be
disrupted
many times until they have reached
a critical
size
and conditions,
i.e. We_, will vary within
the apparatus
and
during
the process.
a spread
in droplet
size will
Anyway,
result. The theory has now been tested in a colloid mill, made
in such a way as to cause true
simple shear: conditions
as to
composition
of
both
phases
and
and
concentration
of
type
surfactant

were varied
widely
(Ambruster,
1990; Schubert
and
Ambruster,
1989). The average droplet size rX2 (being the third
over the second moment of the frequency
distribution
of r) was
determined
and used for calculating
We_. Results
are indicated
in Fig.
1 and it is seen that a reasonable
agreement
with
prediction
was
obtained.
Break-up
was
observed
at somewhat
higher values of the viscosity
ratio than 4, up to qD/qc = 10.
This may have been due to some elongational
component
in the

flow
in the apparatus,
but another
explanation
may be the
presence
of a surfactant.
This allows
the development
of an
interfacial
tension gradient
on the surface of a sheared drop.
The latter
causes
the tangential
stress
to be not any more
continuous
across
the
boundary
(which
is
a
droplet
prerequisite
in the theories
applied
to drop

disruption
in
laminar
flow) and this hinders
the development
of flow of the
liquid
in the drop. This may, in turn, make deformation
and
thereby
break-up
easier.
It would
be useful
to study
this
aspect in more detail.
Up till now, we have tacitly
assumed
that both
liquids
are
Newtonian.
If they are visco-elastic,
the situation
becomes
much more complicated.
If the disperse
phase is visco-elastic,
droplet

break-up
is in general
more difficult,
especially
if
the relaxation
time is considerable.
If the continuous
phase
IS
visco-elastic,
it
realize
high
becomes
difficult
to
elongational
velocity
have
been
gradients.
These
aspects
studied in some detail fairly recently
(Han and Funatso,
1978;
Chin and Han, 1979, 1980).
DROPLET

BREAK-UP

IN TURBULENT

FLOW

It will be clear from the previous
section
that laminar
flow
is mostly
not suitable
for breaking
up drops
suspended
in
water or another
low viscosity
liquid. Flow conditions
have to
be (intensely)
turbulent.
In turbulent
flow (see e.g. Davies,
1972), the local flow velocity
u varies
in a chaotic
way and

Principles of emulsion formation

the fluctuations
often are characterized
by u', i.e. the rootmean-square
average
difference
between
u and the overall
flow
If the turbulence
is isotropic
(which
is more
or
velocity.
less the case if the Reynolds
number
is high and the length
the flow can be characterized
in a
scale considered
small),
There
is a
according
to the Kolmogorov
theory.
simple
way,

they are the higher
spectrum
of eddy sizes, and the smaller
so high that
their velocity
gradient
(u'/x), until it becomes
the eddies dissipate
their kinetic energy into heat; the size
scale and
of the smallest
eddies x0 is called the Kolmogorov
droplets
smaller
than this are usually
not greatly
deformed.
are called
energy-bearing
eddies
and
Somewhat
larger
eddies
For these
they are mainly
responsible
for droplet
break-up.
eddies we have

u’(x)

=

c

El/3

x1/3

p-1/3

where
x is eddy
size
(or distance
over
which
u' is considered),
p is mass density
and C a constant
of the order of
unity. The power density
E (often called the energy density),
i.e. the average amount of energy dissipated
per unit time and
unit
main
characterizing
the

volume,
is
the
parameter
turbulence.
The
eddies
cause
pressure
fluctuations
of the
order
of p{u'(x)j2
and if these are larger
than the Laplace
4y/x of a neighbouring
droplet
of diameter
x, the
pressure
droplet
may be broken
up_ This results
in a largest
diameter
of droplets
that can remain in the turbulent
field of
d PaX = xmax = C ~-2/sy3/5

p-1/5

The turbulent
field is mostly
not quite homogeneous
and the
resulting
droplets
will thus show a spread
in size. Since in
many
oases
the
resulting
droplet
size
distribution
has
a
fairly
constant
shape
for variable
E, eq.
holds
(4) mostly
also
for an average
droplet
size

(e.g. d,,), albeit
with
a
different
constant.
It is a very useful equation
that has been
shown to hold remarkably
well for a wide range of conditions,
provided
that recoalescence
of droplets
is limited;
see the
earlier
review,
where
also
some
additional
conditions
are
discussed.
Some results are given in Fig. 2. It is seen that the stirrer
is much
less
effective
than
the high
pressure

homogenizer
(although
the stirrer would have produced
smaller droplets
for
the
same
energy
consumption
in a flow-through
arrangement,
presumably
by a factor
of about
5). This
is because
the
homogenizer
dissipates
the energy in a much shorter time, thus
causing
E to be higher.
The stirrer dissipates
much energy at
a level where it cannot break-up
small droplets.
Note that the
stirrer
and the ultrasonic
transducer

(which
also
produces
pressure
fluctuations)
show
the
expected
-0.4,
slope
of
predicted
by eq. (4): here power density
is proportional
to
consumption.
In
homogenizer
the
the
net
energy
energy
consumption
is given by the homogenizing
pressure
p, but here
the
power
density

is proportional
to p312, since
the
time
during
which
the
energy
is
dissipated
is
inversely

331


338

PlETER

WALSTRA

proportional
to the liquid velocity
through
the homogenizing
valve, which is, in turn, proportional
to pl". Consequently,
the slops of log droplet size versus log energy input is -0.6,
an additional

reason why high-pressure
homogenizers
are very
effective
in producing
very small droplets.
1
20

lJltm Tut-t-ax
(batch. 2 min

10

)

5

2

1

0.5

-

Fig.

lqP(MJ

t+

:

2 Average
droplet diameter
xgg as a function
of net
energy input P (varied by varying
intensity,
not
duration
of treatment)
for dilute
paraffin
oilin-water
emulsions
produced
In various
machines.
From Walstra,
1983.

(4) predicts
that changing
qc does not result In a change
Eq.
in droplet
size. This is indeed
often roughly

the case, but
there
are some exceptions,
in that
a slight
dependency
is
observed,
average droplet size mostly somewhat decreasing
with
increasing
viscosity.
Presumably,
pressure
fluctuations
may
not in all cases be the only cause for droplet break-up.
On a
droplet
caught
between
eddies
with a size
S> droplet
size,
shear stresses will act and these may be sufficient
for breakpresumably,
the
flow
is

similar
to
up:
type
of
plane
hyperbolic
flow.
If this
is the
mechanism,
resulting
the
relation is
(5)
If the situation
is in between true inertial
and true viscous
forces,
viscosity
thus
have
some
effect.
Increasing
may
viscosity
also means
decreasing
Reynolds

number,
hence
less
intense turbulence,
hence on average larger eddies, hence more


Principles of emulsion

formation

shear.
This would
also imply that the spread
in conditions,
for
spread
in
droplet
size,
becomes
larger
hence
the
this
has
indeed
been
observed
increasing

viscosity,
and
the effect of viscosity
of the
(Walstra,
1974). Nevertheless,
continuous
phase on the resulting
droplet size distribution
is
mostly slight.
this
If a soluble
polymer
is added to the continuous
phase,
of
causes
some increase
in qC, but it also has the effect
smaller
eddies
are
turbulence
depression:
especially
the
This results
in a larger
average

removed
from the spectrum.
droplet
size (up to a factor 2) and a narrower
droplet
size
distribution,
If a liquid contains
many particles,
they also
depress turbulence,
but the effect on emulsion
formation
has to the author's
knowledge
- not been studied,
It may well be,
however,
that turbulence
depression
is one of the reasons why
a high internal
phase volume
fraction
causes
larger
droplets
to be formed:
nevertheless,
other

factors
are probably
more
important
(see the next section).

Fig.

phase
(q:, _in
3 Effect
of viscosity
of disperse
mPa s) on average
droplet
size (d,,, in pm) for
various
machines
(turbomfxer,
circles;
ultrasonic
generator,
crosses;
homogenizers,
other symbols).
From Walstra,
1974.

(4) also
predicts

no effect
of the viscosity
of the
Eq.
disperse
phase
on the resulting
droplet
size,
and this
is
clearly
not in agreement
with experiments.
Fig. 3 shows some
results
and
it is seen
that
for constant
E, log
average
droplet
size versus log I), gave straight
lines with a slope of
0.35
to 0.39.
The viscosity
effect
has been

discussed
by
Davies
(1985). He added in the derivation
of eq. (4) a viscous
stress
term = q&'/d
to the Laplace
pressure
4y/d, where d =
droplet diameter.
This leads to

339


mETEIt

340

d max

=

c

E-2/5

(y

WhLSTRh

lj,u'/4)3'5P-1'5

+

This equation
does not agree with the constant
and virtually
parallel
Slope8 in Fig. 3. D&vies assumed the flow velocity
in
the droplet to be equal to the external
u' and this may not be
true anymore
for q. >> q, and it is certainly
not the case in
the
presence
of
surfactant,
which
makes
that
the
droplet
surface
can
withstand
a certain

shearing
In other
stress.
words, the viscous
stress term to be added should not contain
U'
but
the internal
velocity
Us,. Because
u,,/d equals
the
external
stress
over
qn and because
this
stress
is at the
prevailing
conditions
of the order
of the Laplace
pressure,
additional
the
term
is of
the
same

form
as
Laplace
the
pressure
and the result is merely a different
constant
in eq.
(4). independent
of Q~.
In this connection,
it is useful
to consider
the time needed
for deformation
of the droplet
rdcf, which may be defined
as q,,
over the stress.
The latter equals the external
stress minus
the Laplace pressure.
We thus obtain
=dmi

J

q,

/

(C E2'3 d"' p"' - 4 y/d)

(6)

The constant
C is unknown,
but in order to obtain
reasonable
values
for the resulting
droplet
size, we have taken it to be
are depicted
in Fig. 4. It should be noticed
5; some results
that eq. (6) is different
from the one given
before:
rdeI =
the latter
relation
is based
on the
qDd/y
(Walstra,
1983);
spontaneous
relaxation
of the droplet

shape after it has been
deformed,
but this
is not realistic.
The dependence
of the
deformation
time on d is according
to eq. (6) even reversed:
now
the
smaller
the
droplet
becomes,
the
longer
the
deformation
time.
This
is
because
according
to
the
Kolmogorov
theory
- the size of a droplet
disrupting

eddy is
roughly
the
same
as
that
of
the
drop,
and
the
pressure
difference
caused
by an eddy increases
with size. All these
relations
only hold within
certain
bounds
and are not quite
exact, but they serve to illustrate
trends.
The life
yields
t

time
=

eddy

x

of
/

u

an
’ ( x

eddy
)

r

x2/3

can
fe3

be

derived

p1’3

from

eq.

(3),

which
(7)

several
Results
are
shown
in
Fig.
4,
which
gives
emulsification
process
characteristic
times
throughout
the
the finally
resulting
drop
(ever
decreasing
droplet
size);
It is seen

size according
to eq. (4) would be about 0.3 pm.
time is mostly
that for a relatively
low v&, the deformation
shorter
than the life time of the eddies of the size of the
which would imply that the pressure
fluctuations
last
droplet,
long enough
for the droplets
to be disrupted
by these eddies.
this is not any more the case.
For higher droplet viscosities,
This implies
that larger
eddies
are responsible
for droplet
break-up.
Hence,
for a larger droplet viscosity
the resulting
of the greater
drops
are on average
larger

and - because


Principles of emulsion formation

341

spread
in flow conditions
for larger
eddies
- show more
spread
This
has
indeed
been
observed
(Walstra,
1974).
It
in size.
quantitative
theory
along
would
be useful
to develop
a more
these

lines
and, of course,
test it.
droplet diameter (pm1
time

specific
Fig.

4

THE

ROLE

surface area (m-‘I

Calculated
characteristic
times
for
deformation
duration
of
surfactant,
adsorption
of
droplets,
during
of eddies

and rate
of droplet
encounters,
emulsification
process
(decreasing
droplet
the
from
turbulence
size).
Calculated
isotropic
theory
for E = 10" W m-', qc = 1 mPa s, qn = 0.1 or
F =
10 Pa s, cp = 0.2, m, = the cmc (corresponding
2 mol m-j
2.5 umol m-', and y 5 mN m-l) or initially
(broken
lines);
y without
surfactant
= 35 mN m-l.
Surfactant
is sodium
dodecyl
sulphate.

OF

THE

SURFACTANT

During
emulsification
three
main processes
occur:
1.
Droplets
are deformed
and possibly
broken
up.
2
and
adsorbed
onto
the
Surfactant
is
transported
to
deformed
and the newly
formed
droplets.
3.

Droplets
encounter
each other
and possibly
coalesce.
It
these
processes
occur
should
be
realized
that
simultaneously
and
also
that
they
each
occur
numerous
times
during
emulsion
formation,
which
implies
that
a steady
state

is not necessarily
reached.
In other
words,
if emulsification
would
be continued,
smaller
droplets
may
possibly
result.
Of
fairly
course,
conditions
change
during
emulsification
and a


PETER WALSTRA

342

obvious
change
is that
area - the concentration

- due to the increasing
droplet
surface
of surfactant
in solution
decreases.

The
surfactant
has
two
main
roles
to
play:
it
lowers
interfacial
tension,
thereby
facilitating
droplet
break-up:
and it prevents
(to a varying
degree)
recoalescence.
Moreover,
if

surfactant
concentration
is
high
and
the
resulting
interfacial
tension
very
low,
it may
under
some
conditions
cause
nspontaneous
emulsification"
due
to
strong
the
interfacial
tension
gradients
induced.
Such
a droplet
break-up
without

putting
in
much
mechanical
only
of
energy
is
importance
in the earlier
stages
of emulsion
formation
and has
little
to
do
with
the
final
droplet
size
obtained,
unless
emulsification
is achieved
by simple
shaking
or when
we come

into
the
realm
of microemulsions.
These
aspects
will
not
be
considered
here.
Different
surfactants
lower
y to a different
degree
and
this
should
affect
the
final
droplet
size
according
to eq.
(2) or
For a surfactant
giving
a lower

y less
energy
is thus
eq.(4).
needed
to obtain
a certain
droplet
size.
As seen
in Fig.
5,
obtained
in
the
predicted
relations
are
indeed
roughly
turbulent
flow,
provided
that
there
is
of
surfactant.
Similar
relations

have
been
foundanin
ezczloid
mill
(Ambruster,
1990).
But
there
are
some
exceptions
(e.g.
Ambruster,
1990)
and the course
of the curves
in Fig.
5 is not
readily
explained.
Naturally,
for
a lower
total
surfactant
concentration,
speaking
the
the

surface
excess
l? (loosely
concentration
of surfactant
in the interface)
during
break-up
will
be lower
and correspondingly
the effective
y higher,
but
that does not explain
the different
shapes
of the curves.

droplet

surface area/pm-’
“non-ionic.”

IS-

-G
Ucrm

-6

‘Ol.?Ira~P

l.O-

-8
- IO

o.s-

-15
-30

0

0

I

,

5

10

I

[surfactantl/kg.mb3

Fig.

5 Effect
of
total
surfactant
concentration
on
the
resulting
average
droplet
size,
other
conditions
Interfacial
tension
at high
being
equal;
4 = 0.2.
concentration
for
the
non-ionic
2, caseinate
10
and
the
PVA's
20

mN.m",
approximately.
Approximate
results
from various
sources.


Principles of emulsion formation

Fig.

6 Diagram
of the Gibbs-Marangoni
effect
acting
on
two approaching
droplets
during
emulsification.
Surfactant
molecules
indicated by Y. See text.

There
must
therefore
be
differences

in
the
degree
of
recoalescence.
.That recoalescence
can occur, also in the case
of polymer
surfactants,
has been shown
in experiments
where
after emulsification
the surfactant
concentration
is lowered
emulsion
is
then
again
subjected
to
the
same
and
the
emulsifying
treatment:
the
average

droplet
size
is indeed
observed
to
increase.
Prevention
of
coalescence
of
newly
formed drops is presumably
due to the Gibbs-Marangoni
effect.
This is illustrated
in Fig. 6. If two drops move towards each
other
(which happens
very frequently)
and if they still are
they will acquire
more
insufficiently
covered
by surfactant,
surfactant
at their
surface
during
their

approach,
but the
amount of surfactant
available
for adsorption
will be lowest
where the film between the droplets
is thinnest.
This leads to
an interfacial
tension
gradient,
r being
highest
where
the
film is thinnest.
The gradient
causes the surface
to move in
the direction
of the highest y or, in other words, surfactant
moves
in the interface
towards
the site of lowest
surface
excess. This gradient
causes streaming
of the liquid along the

surface
(the Marangoni
effect),
thus
will
drive
which
the
droplets
away
from
each
other.
stabilizing
Hence,
a self
mechanism.
Note
that
the
mechanism
only
works
if
the
surfactant
is in the
continuous
This
phase.

must
be
the
explanation
of Bancroft's
rule:
when
making
an emulsion
of
oil, water and surfactant,
the phase in which the surfactant
is (best) soluble
becomes
the continuous
one; and, in turn,
the fact that Bancroft's
rule is never violated
(unless 4 is
extreme)
is
a
indication
strong
that
the
Gibbs-Marangoni

343

DETER WALSTRA

344

effect is responsible
for preventing
recoalescence.
Note also
that the mechanism
only works in a non-equilibrium
situation:
after
the
available
surfactant
molecules
are
evenly
distributed
over
the
droplet
surface
(as in a "finished"
emulsion)
it does
not act. Then,
the colloidal
interaction

forces primarily
determine
the coalescence
stability.
Whether the Gibbs-Marangoni
effect is strong enough depends on
the Gibbs
elasticity
E of the film between
the approaching
droplets.
E is defined as twice the surface dilational
modulus
(E=2dy/dlnA,
where A is surface area) and is given by
(Lucassen,
1981)
E=

-2dy/dlnr
1+
(h/2)dm,/dr

(8)

where
h is film thickness.
If E is high,
the stabilizing
mechanism

works
- because
now a strong
interfacial
tension
gradient
can
develop
and
if
it
is
low,
it
may
be
insufficient.
A sample calculation
for E as a function
of the
molar
surfactant
concentration
in the continuous
phase m, is
given in Fig. 7. It follows that E Is higher if h is smaller
and, for most situations,
if m, is higher. For most polymers,
E
is much

lower
than for small molecule
surfactants.
This
is
because
for
the
same
mass
concentration
the
molar
concentration
of a polymer
is much lower, which also implies
that IY (expressed
in moles per unit surface
area) is mostly
low during emulsification,
which, in turn, causes -dy/d lnr to
be almost zero. In fact, the values of -dy/d 1r-C obtained
from
experimental
results will be too high, because these have been
obtained at equilibrium
conditions.
f (mN.m-‘I

Fig.

7 Effect
of
sodium
dodecyl
concentration
of
sulphate
in the liquid on the Gibbs elasticity
of
a film of 1 urn thick (after Lucassen,
1981). The
broken line roughly
indicates
the relation
for a
mixture of surfactants.
From Walstra,
1989.


Principles

According
to
state
for
approximation
Yo -

of emulsion

formation

and
Benjamins
(1982)
in
an
interface

de Feijter
surfactants
given
by

345

the
is

equation
of
in
first

(9)

Y =I-RT/(l--8)2

meaning
and 8 is the
where
yO is y for l? = 0, RT has its usual
surface
fraction
covered
by
surfactant,
e-ggiven
by
nnr',
where
n is the number
of molecules
per unit
surface
area
and r
a
polymer
molecule,
their
radius.
If
say
a
protein,
is
adsorbed

it
changes
conformation,
thereby
commonly
its
increasing
the effective
r, thus
increasing
8, thus
decreasing
effect
may
be considerable
(de Feijter
and
Benjamins,
Y: the
the
magnitude
of
the
Gibbs
elasticity,
1982).
Consequently,
and
thereby
the

extent
to which
recoalescence
is prevented,
on
the
time
scale
of
the
rearrangement
of
the
will
depend
molecule
in
interface.
This
dependence
Will
polymer
the
polymers,
undoubtedly
but
at
present
no
vary

among
experimental
results
seem
to
be
available.
In
the
author's
this
effect
and
the molecular
weight
of the
polymer
opinion,
may
be
the
main
variables
causing
differences
among
polymer
in
the
resulting

average
droplet
size,
if
the
surfactants
polymer
is present
in a relatively
low concentration.
If its
"equilibrium"
concentration
is high,
value
of y will
be
the
determinant,
and
differences
polymers
in
resulting
among
droplet
size are indeed
far smaller;
see Fig. 5.
It

is
consider
needed
useful
to
the
time
for
the
also
surfactant
to
reach
the
droplet
surface,
This
not
=&I%*
at the prevailing
conditions
of very
determined
by diffusion;
high
velocity
gradients
or very
intense
turbulence,

transport
droplet
is
almost
towards
the
entirely
determined
by
convection
(Levich,
1962).
This
implies,
in
author's
the
that
experiments
in which
the
decrease
in
surface
opinion,
tension
at
a macroscopic
surface
above

a solution
of
the
surfactant
is measured
as a function
of time,
are
irrelevant
emulsification.
to
That
correlations
are
sometimes
found
between
the
rate
of
lowering
y and
the
effectivity
of
the
surfactant
in producing
small
droplets,

is presumably
due
to
the
fact
that
high
molecular
weight
surfactants
diffuse
more
slowly
to the
interface
in the
tests
performed,
while
they
also
give
rise
to a relatively
low Gibbs
elasticity
under
the
conditions
of emulsification.

The actual
situation
may be more
complicated
because
of the association
of many
surfactants
in
solution,
especially
the formation
of micelles.
It may be that
in some
cases
the rate of dissociation
of individual
molecules
from
a micelle
is
rate
determinant;
on
the
other
hand,
it
cannot

be
ruled
out
that
micelles
as such
collide
with
the
droplet,
considering
the
strong
inertial
forces
in turbulent
flow.
We
will
here
only
consider
free
molecules
being
transported
by convection
and we then
obtain
approximately

in
laminar
flow
=ad,

and

in

~20

turbulent

r/dm,
flow

G

(loa)


346

PIETER WALSTRA

T rd.

-

10 I' II,"~ / d m, @I2

(lob)

Some results are given in Fig. 4. It is seen that in turbulent
flow, 'cadsis definitely
longer
than fdci, the more so as the
droplets
become
smaller
and
for
surractant
concentration.
Similar
relations
hold
f%
l~~~~ification
in
laminar
flow. Consequently,
during droplet
break-up
r will be
lower
and
r higher
than
their

"equilibrium"
values.
This
points to the Gibbs-Marangoni
effect being essential
and also
to a situation
in which break-up
and coalescence
go along for
some time until
the smallest
droplet
size is attained:
even
then,
break-up
and coalescence
presumably
go on,
balancing
each other.
Fig. 4 also gives some results
for the average
time elapsing
before
a droplet
encounters
another
one, assuming

them to be
randomly
oriented
throughout
the
available
volume.
The
relation
is in isotropic
turbulence
‘cnc tl d=
l

p1'3 / 15 4 E1'3

(11)

It is seen that
for high 4 and small droplets,
T,,, becomes
(much) shorter
than 'G,~..This certainly
points
to coalescence
then becoming
important.
Some authors hold that break-up
would
be a first order

rate process
and coalescence
second
order,
and that the change in the number of globules
N would be given
by dN/dt = K,N - K,N', but such a relation
is usually
not in
agreement
with experimental
results.
In the authors
opinion,
most of the recoalescence
occurs with newly formed drops, that
probably
originate
from one parent drop, and are thus close to
each other anyway.
In other words, for these drops r_
is even
shorter than shown in Fig. 4.
log (d,/pm)
0

-0.2

-0.4

1.4
loq(pHIPa)
8 Effect
of
homogenizing
pressure
the
(P)
on
resulting
average
droplet
size (a,,) when varying
fraction
of oil
(indicated
on
the
the
volume
continuous
phase
(a
curves)
and
leaving
the
protein solution)
the same. From Walstra,
1988.

0.6

Fig.

1


Principles of emulsion formation

Nevertheless,
at higher
volume
fractions,
recoalescence
is
probably
more important.
Fig. 8 gives
some results
from the
author's
laboratory
(Walstra,
1988). Qualitatively,
the higher
average
droplet
size at a higher 4, and the relatively
larger

effect of 4 for a higher power density,
are probably
due to
- the smaller
amount of surfactant
available
per unit surface
this causes
a higher
effective
y and a lower
area
created:
Gibbs elasticity
(more coalescence);
- the higher encounter
frequency
of droplets,
particularly
of
droplets
onto which
yet little
surfactant
has adsorbed:
this
causes more frequent
coalescence;
- turbulence
depression,

causing
droplet
break-up
to be less
efficient.
CONCLUSION
The
break-up
of
drops
in
laminar
and
turbulent
flow
is
reasonably
well understood,
although
quantitative
explanation
of the effect
of the viscosity
of the disperse
phase
in the
case of turbulence
would need further
study.
It appears

as if
droplet
disruption
due
to
cavitation
(in
an
ultrasonic
generator)
is much like that in turbulent
flow. To achieve
a
high
efficiency
it is necessary
to dissipate
the available
mechanical
energy in the shortest
time possible.
The role of the surfactant
is qualitatively
understood,
but
quantitative
relations
are hardly
available,
especially

for
polymer
surfactants,
like
proteins.
The
development
of
simulation
models
for
adsorption
of surfactant
and
for the
phenomena
occurring
during a close approach
of droplets,
would
be very useful.
The role of the Gibbs elasticity
needs further
study,
especially
its dependence
on time scale
(of the order
of microseconds)
for different

surfactants.
LIST
:

E
G
m,
P
r

I

ie
X

Y
r
E
rlc
tlD
P
t
cp

OF FREQUENTLY

USED

SYMBOLS

constant
(-)
droplet diameter
(m)
Gibbs elasticity
of film (N m-l)
velocity
gradient
(s-l)
concentration
of surfactant
(mol mm3 or kg m-')
pressure
(Pa)
radius (m)
average velocity
in turbulent
eddy (m s-l)
Weber number
(Pa Pa-')
distance,
such as eddy size (m)
interfacial
tension
(N m-l)
surface excess
(mol rnm2)
power density
(W rnm3)
viscosity

of continuous
phase (Pa s)
viscosity
of disperse
phase (Pa s)
mass density
(kg rnm3)
characteristic
time (5)
volume fraction
of disperse
phase (m3 m-')

347


PIETERWALSTRA

348

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und

der Strbmungsverhaltnisse
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J.A. and J. Benjamins
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Principles of emulsion formation

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349