Thin Liquid Films (Part II) - NPTELThin Liquid Films (Part II) Dr. Pallab Ghosh ... Let us consider a thin liquid film formed between two emulsion drops as shown in the following...

NPTEL Chemical Engineering Interfacial Engineering Module 5: Lecture 4

Joint Initiative of IITs and IISc Funded by MHRD 1/22

Thin Liquid Films (Part II)

Dr. Pallab Ghosh

Associate Professor

Department of Chemical Engineering

IIT Guwahati, Guwahati–781039

India



Table of Contents

Section/Subsection Page No. 5.4.1 Hydrostatics of thin liquid films 3

5.4.2 Drainage of liquid films 420

5.4.2.1 Lubrication model 7

5.4.2.2 Film drainage time 11

5.4.2.3 Stability of thin liquid films 15

5.4.2.4 Black films 18

Exercise 21

Suggested reading 22



5.4.1 Hydrostatics of thin liquid films

Let us consider a thin liquid film formed between two emulsion drops as shown

in the following figure. Half of the film is shown in Fig. 5.4.1.

Fig. 5.4.1 Disjoining pressure model of thin liquid film.

The film is assumed to be plane parallel. The liquid in the film exists in contact

with the bulk liquid in the Plateau border. The film liquid would exist in

hydrostatic equilibrium with the surrounding liquid if an external force directed

perpendicular to the surfaces balances the internal force. The internal force in the

film per unit area is known as disjoining pressure .

The origin of disjoining pressure is van der Waals, electrostatic double layer and

steric forces (see Lectures 1–5 of module 3) acting between the surfaces of the

thin film.

The drop phase pressure is dp , bulk liquid phase pressure is lp , interfacial

tension is and the radius of the drop is dR . In the meniscus region outside the

film we have,

2d l

dp p

R

(5.4.1)

Since dp is larger than lp , the equilibrium at the film surface is ensured by the

action of the disjoining pressure as,

d lp p (5.4.2)

The following relationship also holds at equilibrium.

f dp p (5.4.3)



The disjoining pressure is a function of the film thickness. The disjoining

pressure manifests as a normal surface excess force, sF , as shown in Fig. 5.4.1. It

is taken to be positive when it acts to disjoin (i.e., separate) the film surfaces. sF

can be calculated by using the DLVO theory (see Lecture 5 of Module 3). The

hydrostatic equilibrium model of thin liquid film is known as the disjoining

pressure model.

5.4.2 Drainage of liquid films

Thinning of soap films by the drainage of liquid has been extensively studied

since the time of Newton. The qualitative observations made by Newton and

Gibbs revealed that the walls of soap bubbles grow thinner in time and pass

through thicknesses of the order of the wavelength of the visible light. Colors of

soap films is illustrated in Fig. 5.4.2.

(a) (b) (c) Fig. 5.4.2 Color of soap films: (a) colorful soap bubbles, (b) color in circular soap film, and (c) color bands in soap film: the top left portion is the thinnest

and appears black.

If a bubble is blown from an aqueous surfactant solution, it is a common

observation that after some time, it appears tinged with a great variety of colors.

Isaac Newton (1704) has described the change in color of the bubble as follows:

“to defend these Bubbles from being agitated by the external Air …, as soon as I

had blown any of them, I cover’d it with a clear Glass, and by that means its

Colours emerged in a very regular order, like so many concentrated Rings

encompassing the top of the Bubble. And as the Bubble grew thinner by the

continual subsiding of the Water, these Rings dilated slowly and overspread the

whole Bubble, descending in order to the bottom of it, where they vanish’d



successively. In the mean while, after all the Colours were emerged at the top,

there grew in the center of the Rings a small round Black Spot … Besides the

aforesaid colour’d Rings, there would often appear small Spots of Colours,

ascending and descending up and down the sides of the Bubble, by reason of

some Inequalities in the subsiding Water. And sometimes small Black Spots

generated at the sides would ascend up to the larger Black Spot at the top of the

Bubble and unite with it”.

Different colors appear due to the interference of light reflected from the two

surfaces of the film. At the advanced stages of the thinning process, thin black

spots are formed, which are sometimes very unstable. The interest in the thinning

process of films has grown considerably due to their importance in the

understanding of coalescence of drops and bubbles, and the stability of emulsions

and foams.

Mysels and co-workers (circa 1959) were the first to report in detail the different

types of drainage process, concentrating on vertical films formed by withdrawal

of glass frames from pools of surfactant solutions. They observed mobile films

which drained in minutes and showed turbulent motions along the edges, and

rigid films which took hours to drain and showed little or no motion. They

proposed that the rapid drainage and turbulence observed for the mobile films

were the result of ‘marginal regeneration’. In this phenomenon, a thick film

flowed into the Plateau borders near the legs of the frame at some elevations

owing to the greater suction force exerted on it by the low pressure in the borders.

Simultaneously, a thin film was pulled out of the borders at other elevations to

maintain constant surface area for the overall film.

Curvature of the interface plays an important role in the drainage process. For

example, the film formed between two droplets in an emulsion is actually not flat.

The thickness of the film and curvature of the interfaces change with time. When

two emulsion droplets approach each other, a dimple forms as shown in Fig.

5.4.3.

Due to the trapped liquid in the central region of the film, the film is thicker there

as compared to the peripheral region. The interfaces deform during the drainage



of the film and a narrow region is developed near the rim of the film, which is

known as the barrier ring. It is named so because it causes constriction to the

outward flow of the film liquid, which slows down the drainage process.

The shape of the film changes with time as the film thins. The presence of

surfactants is believed to influence the film drainage process.

The shape of the film can be studied by interferometry and video microscopy.

The various stages of thinning of a film formed between a drop and an initially

flat waterorganic interface are shown in Fig. 5.4.3.

Fig. 5.4.3 Schematic of a dimpled thin film formed between two drops,

symmetric drainage of the film with time for a 3 mm diameter anisole drop at wateranisole interface in presence of 3 106 kg/m3 SDS and 10 mol/m3 KCl,

and asymmetric drainage of the film for a 3 mm diameter anisole drop at wateranisole interface in presence of 2 105 kg/m3 SDS and 10 mol/m3 KCl

(Hodgson and Woods, 1969) (adapted by permission from Elsevier Ltd., 1969).

The drainage can be symmetric as well as asymmetric, as shown in these figures.

The dimple sometimes flattens with time and finally a near-flat thin film can be

obtained. The theoretical analyses of the thinning process often make simplifying

assumptions such as, axisymmetric interfaces (Slattery, 1990) and flat film

(Edwards et al., 1991).

Asymmetric drainage in foam films has been analyzed by Joye et al. (1994).



5.4.2.1 Lubrication model

Let us consider the approach of two drops in a liquid under Stokes flow

conditions as shown in Fig. 5.4.4.

(a) (b) (c)

Fig. 5.4.4 Schematic diagram of the drainage of the liquid film trapped between two drops: (a) drops approach each other, (b) exaggregated view of the near-

contact region, and (c) sketches of the streamlines.

As the drops move towards each other along the line of their centers, the

hydrodynamic force opposing their motion increases tremendously as the distance

between the drops decreases. The exaggerated view of the contact region is

shown in the Fig. 5.4.4 (b). A similar phenomenon occurs when a drop moves

towards a flat interface.

As the two drops approach, the resulting radially outward flow in the narrow gap

separating the drops exerts a tangential shear stress on the drop surfaces. This

causes a tangential motion of the drop surfaces and drives a flow inside the drops.

A sketch of the streamline patterns is shown in Fig. 5.4.4 (c). This flow has a

significant effect on the force that resists the approach of the drops. The flow in

the narrow gap and the flow within the drops are discussed in this section.



Let the radii of the drops be 1a and 2a . The viscosity of the continuous phase is

and that of the drop liquid is . The relative velocity is 1 2U v v . The

closest separation between the drop surfaces is 0h .

Since we have assumed axisymmetric profile, 0h is the located at the center of

the film. The force that resists the relative motion of the drops and slows down

the film thinning process is dominated by a small region located near the axis of

symmetry. This region is known as the lubrication region. Our objective is to

derive the velocity profile in this region.

The radial velocity profile, ,u r z , can be decomposed into two parts as,

, ,t pu r z u r u r z (5.4.4)

where tu r is the tangential velocity, and ,pu r z is the velocity for the

parabolic portion of the flow driven by the local pressure gradient, which is zero

at the drop surfaces (i.e., at 1z and 2z ).

The velocity, pu , is given by,

1 21

2pp

u z z z zr

(5.4.5)

The tangential stress exerted by the fluid in the gap on the drop surfaces is given

by,

1 22t

z z z z

u u h pf

z z r

(5.4.6)

A mass balance on the fluid flowing out of the film gives the following equation.

2

1

2 2 , dz

z

r U r u r z z (5.4.7)

Substituting ,u r z from Eq. (5.4.4) into Eq. (5.4.7), carrying out the integration,

and substituting 2 1h z z , we get,

22 2

6t

th f

r U r hu

(5.4.8)



The magnitude of the hydrodynamic force that resists the relative motion of the

drops is given by,

0

2 dlF p r r r

(5.4.9)

Since 0h is small compared to the radius of the drops, 0 1h a , and the

pressure is large compared to the z-component of the shear stress on the drop

surfaces in the close contact region.

The film thickness, h , is a function of the radial position, r . Therefore, the radial

length-scale in the close contact region is 1 20ah , where a is the reduced radius

[ 1 2 1 2 a a a a ], because this is the radial distance for which h r changes by

0O h .

The pressure reduces significantly at radial positions far away from the

lubrication region, i.e., 0p for 1 20r ah . Therefore, the upper limit of

integration in Eq. (5.4.9) can be taken as .

The spherical surfaces of the drops may be approximated as paraboloids near the

axis of symmetry. Therefore, assuming the drops to be undeformed, the variation

of film thickness, h , with r can be expressed as,

2

0 2

rh r h

a (5.4.10)

Equation (5.4.10) gives one relationship between the tangential velocity, tu , and

tangential shear stress, tf . At the drop surfaces, tu and tf are significant only

within an 1 20O ah radius of the axis of symmetry. Since this region is rather

small compared with the radii of the drops because 0 h a , the interfaces may

be treated as flat.

Davis et al. (1989) have used the boundary integral form of the Stokes flow

equations to calculate the tangential velocity as,



0

1,t tu r r r f r dr

(5.4.11)

where ,r r is an elliptic-type Green’s function kernel. It has an integrable

(logarithmic) singularity at r r .

1 2 1 22 2 0

2 2

cos d,

2 cos2 1

rr r

rrr rr r

(5.4.12)

Equation (5.4.11) gives the second relationship between tu and tf . Equations

(5.4.8) and (5.4.11) are solved simultaneously to obtain the tangential velocity

distribution, tu r , and tangential stress distribution, tf r .

Then Eqs. (5.4.6) and (5.4.9) are used to find the dynamic pressure distribution,

p r , and the hydrodynamic lubrication force, lF .

When the viscosity of the drop liquid is much higher than the viscosity of the

continuous phase 0 a h , the drops behave like rigid spheres. If the

viscosity of the drop liquid is comparable to the viscosity of the continuous phase

or smaller 0 a h then the drops offer little resistance to the radial flow in

the gap.

When is of the order of 0a h , the drops offer significant resistance to the

radial flow in the gap, but does not exhibit the rigid sphere behavior. In this case,

the velocity scales for both the uniform and parabolic portion of the radial flow in

the gap are 0O U a h . The drop flow and gap flow are fully coupled. Davis et

al. (1989) have presented the solutions in these cases.

The mobility considered here is purely a hydrodynamic effect involving the

viscosities of the liquids constituting the drop and the dispersion medium. The

effects of surfactants are not considered in this model.



5.4.2.2 Film drainage time

The time required for thinning of the film is important in coalescence of drops

and bubbles, and for the stability of emulsions and foams. When the thickness of

the film reduces to the critical film thickness, it ruptures leading to coalescence.

In a separation system such as the gravity settler, the smaller droplets grow bigger

by dropdrop coalescence. Ultimately, the bigger drops coalesce with their parent

phase by coalescence at the flat bulk interface.

The thinning of a uniform liquid film of thickness h and viscosity with time t

under the action of a constant force, F , is given by the Reynolds equation,

2 2

2 2

3 1 1

16f

i

n at

F h h

(5.4.13)

where fa is the area of the film of initial thickness ih , and n is the number of

immobile surfaces.

When 2n , the velocity at both the surfaces is zero. On the other hand, when

1n , the velocity at one of the surfaces is zero and the velocity gradient at the

other surface is zero. Under these conditions, the drainage is very slow. Other

values of n are also possible, which correspond to different surface velocities and

velocity gradients.

The time required to reach the critical film thickness can be calculated from Eq.

(5.4.13) by putting ch h . If c ih h , Eq. (5.4.13) may be simplified (putting

2n ) to,

2

2

3

4

f

c

at

Fh

(5.4.14)

Example 5.4.1: Calculate the time required for thinning of a flat film from 1 m to 100

nm if a 50 N force is applied on it. The area of the film is 63 10 m2 and the viscosity

of the film liquid is 0.8 mPa s. Take 2n .

Solution: The time required for thinning of the film is given by,



2

2 2

3 1 1

4f

i

at

F h h

Given: 30.8 10 Pa s, 61 10ih m, 71 10fh m and 63 10fa m2

Therefore,

23 63

6 2 27 6

3 0.8 10 3 10 1 13.4 10

4 50 10 1 10 1 10t

s

Let us consider the drainage of an axisymmetric film of uniform thickness

trapped between a drop and an initially flat bulk liquidliquid interface as shown

in Fig. 5.4.5.

Fig. 5.4.5 Schematic of a drop approaching a bulk liquidliquid interface.

The viscosity of the drop-liquid is d and the viscosity of the continuous phase is

. The applied constant force is the gravitational force, gF . The radial pressure

gradient during the drainage of the film is given by,

2

2p u

r z

(5.4.15)

The volumetric rate of outflow of liquid at any radial position, r , must be equal

to the rate of decrease in the volume of the film due to the approach of the two

interfaces with the relative velocity, U dh dt . Therefore,



2

02

hr U r udz (5.4.16)

The integral of the pressure acting on the surface of the film of radius, fR , is

equal to the applied constant force gF .

0

2fR

gF p r rdr (5.4.17)

Let us assume that the interfaces are mobile such that 0u u at 0z and hu u

at z h , and the circulation lengths in the two phases are nearly equal and

similar in magnitude to the radius of the drop. The time required for the draining

of the film is given by (Jeelani and Hartland, 1994),

2 4

2 2

3 1 1

16f

g f i

n Rt

F h h

(5.4.18)

where 2n is given by,

2 43

1 d

d i

nR

h

(5.4.19)

The above equation was derived for ‘clean interfaces’, i.e., when there is no

surfactant adsorbed at the interfaces. If surfactant is present in the system,

interfacial tension gradient can be present. The interfacial tension gradient

decreases the surface shear stress and the surface velocity, which retard the

draining of the film (Slattery, 1990; Cristini et al., 1998).

In presence of an interfacial tension gradient, d dr , the drainage time is still

given by Eq. (5.4.18). However, 2n is given by (Jeelani and Hartland, 1994),

23

fi

4

31 1

2fd

d i g i

nRR

h F h r

(5.4.20)



where fir is the initial value of the interfacial tension gradient at the

periphery of the film. It is difficult to predict the value of this parameter.

However, it can be calculated from the experimental data on film drainage.

Jeelani and Hartland (1994) calculated the value of fir from the time

required by a film to drain from 1 m thickness to its critical thickness ch by

using the coalescence time of drops. It was assumed that when the thickness of

the film reached the critical thickness ch , the effects of van der Waals force

caused rupture of the film. However, the values of fir obtained by this

method vary by several orders of magnitude.

These film drainage equations were derived ignoring the electrostatic double

layer and other non-DLVO repulsive forces, which can be important when

surfactant molecules are adsorbed at the interfaces.

Example 5.4.2: Consider the drainage of a toluene film trapped between a water drop

and toluenewater interface. The radius of the drop is 1.5 mm. Calculate the time

required for the film to drain from an initial thickness of 1 m to 50 nm. Given:

water 1000 kg/m3, toluene 870 kg/m3, 0.6 mPa s, 1d mPa s and

36 mN/m.

Solution: From Eq. (5.4.20), 2n is given by,

2 33 3

3 6

4 41.481 10

3 3 0.6 10 1.5 101 11 10 1 10

d

d i

nR

h

33 3 54 41.5 10 130 9.8 1.801 10

3 3g dF R g N

The radius of the film is calculated from the following equation.

1 2 1 222 3 4

3130 9.8

2 2 1.5 10 4.89 103 3 36 10

f dg

R R

m



2 4

2 2

43 3 4

5 2 29 6

3 1 1

16

3 0.6 10 1.481 10 4.89 10 1 1

16 1.801 10 50 10 1 10

0.664 s

f

g f i

n Rt

F h h

The film drainage time given by Eq. (5.4.18) assumes uniform thickness of the

film. However, this is an approximation because in many cases, the thickness of

the film varies with the radial position.

Slattery (1990) has considered the variation of film thickness with r . The

drainage can be non-axisymmetric. Non-axisymmetric film drainage has been

modeled by Joye et al. (1994).

Many film drainage models avoid consideration of interfacial rheological

properties by invoking assumption that the film surfaces are either completely

mobile or immobile. However, the drainage times differ considerably in these two

limiting cases. Malhotra and Wasan (1987) have studied the effects of interfacial

rheological properties on the drainage of foam and emulsion films.

5.4.2.3 Stability of thin liquid films

The stability analysis of thin liquid films involves the solution of equation of

continuity, Maxwell and NavierStokes equations. The intermolecular forces in

the film are assigned to a body-force acting within the film and incorporated in

the NavierStokes equation of motion.

This approach may be viewed as an alternative to the disjoining pressure model

described in Section 5.4.1. The body-force model can be applied easily to

nonsymmetric and nonplanar films.

The force is derived from a potential which describes the energy of

interaction of molecules due to van der Waals force in an infinitesimal volume

with respect to the entire ensemble of molecules in the system. This potential can



be calculated by a microscopic Hamaker method integrating the intermolecular

potential over the system volume. The body-forces are absent within the adjacent

bulk phases A and B shown in Fig. 5.4.6.

Fig. 5.4.6 Schematic diagram of pressure and body-force interaction potential distributions in a planar film.

The tangential components of the body-force are significant near the Plateau

borders. In equilibrium, the body forces are counterbalanced by pressure

gradients and the total equilibrium potentials p in the film and bulk phases

are constant. Therefore, the disjoining pressure can be expressed as,

at :z h A Af f b bp p (5.4.21)

and

at :z h B Bf f b bp p (5.4.22)

From the condition of continuity of pressure across the planar surface in

equilibrium we have,

at :z h Af bp p (5.4.23)

and

at :z h Bf bp p (5.4.24)

Therefore, we have,

at :z h Af b (5.4.25)

and



at :z h Bf b (5.4.26)

The body-force is defined as, d dF n z , where n is a unit vector tangent

to the coordinate z and normal to the surfaces of the film, as shown in Fig. 5.4.6.

Therefore, the magnitude of the surface contribution of the body-force acting at

the film surfaces is d dh . The disjoining pressure, , can be expressed as the

sum of disjoining pressures due to the van der Waals, electrostatic double layer

and short range repulsive forces.

The two modes in which instability occurs in thin liquid films formed between

two liquid phases are ‘squeezing’ and ‘stretching’ modes as shown in Fig. 5.4.7

(when the film is supported on a solid surface, only one surface is corrugated).

Fig. 5.4.7 Modes of instability of a thin liquid film: squeezing mode and

stretching mode.

In the squeezing mode, the corrugations at one surface of the film are completely

out of phase with the corrugations at the other surface. In the stretching mode,

this phase difference is zero. The symmetric mode of vibration leads to the

thinning of the film causing it to rupture. The anti-symmetric mode of instability

usually occurs for the low-tension film surfaces.

The small dynamic fluctuations in the equilibrium thin film engender velocity and

pressure fluctuations in the film. The growth coefficient of these fluctuations

determines the stability. If the growth coefficient is positive, the disturbances

grow and the system becomes unstable. If the growth coefficient is zero, the

stability is ‘marginal’. The disturbances are damped if the growth coefficient is

negative.

The stability of a thin film depends upon the wavelength of the disturbance.

Marginal stability occurs at a ‘critical’ wavelength. The time required for a small



disturbance to destabilize the thin film is of the order of the inverse of the largest

growth coefficient.

5.4.2.4 Black films

Black films are important because their properties are governed mainly by van

der Waals and electrostatic double layer forces between the surfactant layers. The

black films are divided into two categories: common black films (CB-films) and

Newton black films (NB-films). These films are also known as first and second

black films, respectively.

The thickness of the CB-films lies between 6 nm and 100 nm depending on the

electrolyte concentration of the soap solution from which the film is drawn. The

van der Waals attraction and electrostatic double layer repulsion balance the

hydrostatic force in the film at equilibrium.

Their equilibrium thickness decreases with increasing electrolyte concentration

because the action range of the double layer repulsion decreases with increasing

electrolyte concentration. The NB-films are ~5 nm thick and the thickness of the

film is almost independent of the electrolyte concentration. These films are

formed when the electrolyte concentration exceeds some critical value (de Feijter

and Vrij, 1978).

The continuous thinning of a soap lamella leads to the common black film. For

the formation of the Newton black film, it is necessary that a spot of such film

nucleates and grows at the expense of thicker film.

The nucleation of Newton black film can occur during the thinning of the lamella

as well as after the formation of the common black film. The transition from CB-

film to the NB-film is a sort of phase transition in the film surface. According to

Ibbotson and Jones (1969), the transition from the CB-film to NB-film is similar

to the flocculation of a hydrophilic colloid.

The Newton black films are quite ordered. The ‘sandwich structure’ of the NB-

film proposed by de Feijter and Vrij (1979) is shown in Fig. 5.4.8.



Fig. 5.4.8 Proposed structure of Newton black soap film made from an aqueous solution of sodium dodecyl sulfate and NaCl (de Feijter and Vrij, 1979). = 0.48

nm (adapted by permission from Elsevier Ltd., 1979).

In this structure, the NB-film consists of an aqueous core covered with two

monolayers of dodecyl sulfate ions. Assuming that the three layers are

homogeneous, the equivalent thickness of the film, eql , is given by (Frankel and

Mysels, 1966),

21

eq 2 1 22

12

1

nl l l

n

(5.4.27)

where 1l is the thickness of the hydrocarbon layer of refractive index 1n , and 2l

is the thickness of the aqueous core of refractive index 2n .

1l can be calculated from the surface excess concentration of the surfactant in the

NB-film by the equation,

1M

l

(5.4.28)

where M is the molecular weight of the hydrocarbon tail of the surfactant and

is the density of the hydrocarbon layer.

Kashchiev and Exerowa (1980) have proposed that the rupture of these films

occurs by the nucleation-of-hole mechanism. The film was described by



regarding its two monolayers as mutually adsorbed on each other and the

molecules within each of the monolayers were packed in a two-dimensional

lattice. They have derived expressions for the nucleation work, steady-state

nucleation rate and a formula for the mean time for the rupture of the film as a

function of concentration of surfactant in the bulk solution.



Exercise

Exercise 5.4.1: Two small glycerol drops are approaching each other along their center

at a relative velocity of 1 mm/s in carbon tetrachloride medium. The radii of the drops

are 10 m and 15 m. Calculate the lubrication force when the minimum separation

between the drop surfaces is 100 nm. Given: viscosity of carbon tetrachloride = 0.83 mPa

s.

Exercise 5.4.2: For a Newton black film made from the aqueous solution of sodium

dodecyl sulfate and 0.5 mol/dm3 NaCl, one dodecyl sulfate ion occupies 2036 10 m2 at

298 K. Determine the thickness of the aqueous core if the equivalent thickness of the film

is 4.4 nm.

Exercise 5.4.3: A flat aqueous film of 65 10 m2 area is thinning from an initial

thickness of 1 m at 298 K. Calculate the time that it will take to thin down to 50 nm by

using Reynolds equation. The magnitude of the applied force is 60 N. Assume that the

velocity at both the film surface is zero.

Exercise 5.4.4: Answer the following questions clearly.

1. What is lubrication force? How does it originate?

2. Explain the significance of Reynolds equation.

3. Explain the body-force model.

4. What do you understand by squeezing mode and stretching mode of instability?

5. Explain how different color-bands are developed in a draining soap film.

6. What is a black film? Why is it called so?

7. What is the difference between a common black film and a Newton black film?

Explain how electrolytes affect the thickness of these two types of black film.

8. Explain how a Newton black film ruptures.



Suggested reading

Textbooks

P. Ghosh, Colloid and Interface Science, PHI Learning, New Delhi, 2009,

Chapter 8.

Reference books

D. A. Edwards, H. Brenner, and D. T. Wasan, Interfacial Transport Processes

and Rheology, Butterworth-Heinemann, Boston, 1991, Chapters 11 & 12.

J. C. Slattery, Interfacial Transport Phenomena, Springer-Verlag, New York,

1990, Chapter 3.

Journal articles

A. K. Malhotra and D. T. Wasan, Chem. Eng. Commun., 55, 95 (1987).

D. Kashchiev and D. Exerowa, J. Colloid Interface Sci., 77, 501 (1980).

G. Ibbotson and M. N. Jones, Trans. Faraday Soc., 65, 1146 (1969).

J. A. de Feijter and A. Vrij, J. Colloid Interface Sci., 64, 269 (1978).

J. -L. Joye, G. J. Hirasaki, and C. A. Miller, Langmuir, 10, 3174 (1994).

R. H. Davis, J. A. Schonberg, and J. M. Rallison, Phys. Fluids A, 1, 77 (1989).

S. A. K. Jeelani and S. Hartland, J. Colloid Interface Sci., 164, 296 (1994).

S. P. Frankel and K. J. Mysels, J. Appl. Phys., 37, 3725 (1966).

T. D. Hodgson and D. R. Woods, J. Colloid Interface Sci., 30, 429 (1969).

V. C. Cristini, J. Blawzdziewicz, and M. Loewenberg, J. Fluid Mech., 366, 259

(1998).

Thin Liquid Films (Part II) - NPTELThin Liquid Films (Part II) Dr. Pallab Ghosh ... Let us consider a thin liquid film formed between two emulsion drops as shown in the following...

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