coalescence of bubbles and stability of foams in aqueous solutions of tween surfactants

chemical engineering research and design 8 9 ( 2 0 1 1 ) 2344–2355

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Chemical Engineering Research and Design

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Coalescence of bubbles and stability of foams in aqueoussolutions of Tween surfactants

Sayantan Samanta, Pallab Ghosh ∗

Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India

a b s t r a c t

Coalescence of air bubbles and stability of foams in aqueous solutions of Tween 20, 40, 60 and 80 are reported in

this work. Adsorption of the surfactants at air–water interface was studied by measuring the surface tension of the

surfactant solutions. The surface tension profiles were fitted using a surface equation of state derived from the Gibbs

and Langmuir adsorption equations. The critical micelle concentration and surface tension at this concentration

were determined. The effect of surfactant concentration on coalescence of air bubbles at flat air–water interface

was studied. The role of steric force on coalescence time was investigated. The coalescence time distributions were

fitted by the stochastic model. The mean values of the distributions were compared with the predictions of seven

film-drainage models. Stability of foams was analyzed by the Ross–Miles test. The initial and residual foam heights

were measured at different surfactant concentrations. The stability of foams was compared with the coalescence

time of the bubbles.

© 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Bubble; Coalescence; Film-drainage theory; Foam; Ross–Miles test; Stochastic theory; Tween surfactant

1. Introduction

Coalescence of bubbles is a very important factor in the sta-bility of gas–liquid dispersions such as foams. Foams haveapplications in many industrial, environmental and biologi-cal processes. Stable gas bubbles are sometimes useful (e.g.,heavy metal ions can be efficiently removed from wastewa-ter stream using gas bubbles, and oil can be removed fromindustrial wastewater by microbubbles). However, at timesthe presence of bubbles may be undesirable (e.g., the pres-ence of gas bubbles affects the quality of finished products inphotographic, paper, rubber and glass industries). The under-standing of bubble growth, stability and coalescence is crucialin many of these processes. Thus, a sound understanding ofcoalescence of bubbles in aqueous solutions is essential fromfundamental as well as economical point of view.

Coalescence of air bubbles is greatly influenced by the type
and concentration of surfactants present in the solution. Inpure water, air bubbles coalesce almost instantly (Lessard and
Abbreviations: CMC, critical micelle concentration (mol/m3);HLB, hydrophilic–lipophilic balance.

∗ Corresponding author. Tel.: +91 361 2582253;fax: +91 361 2690762.

E-mail address: [email protected] (P. Ghosh).Received 12 January 2011; Received in revised form 16 March 2011; Ac

0263-8762/$ – see front matter © 2011 The Institution of Chemical Engidoi:10.1016/j.cherd.2011.04.006

Zieminski, 1971). However, coalescence is not instantaneousin aqueous solutions of surfactants. The main factors thatdetermine the coalescence time in these solutions are the sur-face excess concentration of the surfactant and the repulsivesurface forces such as electrostatic double layer, hydrationand steric forces. The role of these forces in the stability ofthin liquid films has been discussed in the literature (Ghosh,2009a). Some authors have suggested that the interfacial ten-sion gradient and surface viscosities play important roles incoalescence (Edwards et al., 1991). Ionic surfactants stabilizethe thin liquid films by electrostatic double layer and hydrationforces whereas, the non-ionic surfactants stabilize the films bysteric force.

The Tween-series1 of non-ionic surfactants have diverseapplications in bioscience, foods, pharmaceuticals and cos-metics. These surfactants are polyoxyethylene sorbitan estersof aliphatic fatty acids. They are rather non-toxic and possess

cepted 11 April 2011

an extremely compatible set of physical properties that allowfor widespread use along with other surfactants. Apart from

1 ‘Tween’ is a registered trademark of Uniqema, a unit of ICIAmericas Inc.

neers. Published by Elsevier B.V. All rights reserved.

http://www.sciencedirect.com/science/journal/02638762

www.elsevier.com/locate/cherd

mailto:[email protected]

dx.doi.org/10.1016/j.cherd.2011.04.006

chemical engineering research and design 8 9 ( 2 0 1 1 ) 2344–2355 2345

Nomenclature

a radius of bubble (m)af area of the film (m2)A surface area (m2)AH Hamaker constant (J)Am minimum surface area occupied by a surfactant

molecule (m2)B modified Hamaker constant (J m)c concentration of surfactant in the solution

(mol/m3)c* surfactant concentration at which the film elas-

ticity reaches its maximum (mol/m3)Ci (i = 1, 2) constants in Eq. (9)D� surface diffusivity of the surfactant molecules

(m2/s)Ef elasticity of foam film (N/m)EM Marangoni elasticity (N/m)fr repulsive force generated by one mole of sur-

factant at the barrier ring (N/mol)F (�) cumulative probability distribution of coales-

cence timeg acceleration due to gravity (m/s2)h film thickness (m)hc critical film thickness (m)H initial foam height (m)H̃ steady state foam height (m)k Boltzmann’s constant (J/K)KL parameter of Langmuir adsorption equation

(m3/mol)L thickness of polymer brush layer (m)NA Avogadro’s number (mol−1)P� dimensionless coalescence thresholdr̄ average radius of the bubbles in foam (m)R gas constant (J mol–1 K–1)Rb radius of barrier ring (m)s mean distance between the attachment points

(m)S� normalized standard deviationt time (s)t̄ characteristic diffusion time (s)tc,i coalescence time predicted by model i (s)T temperature (K)wb width of the barrier ring (m)x number of data points

Greek letters˛ fraction of the total amount of surfactant at

the air–water interface that remains at the bar-rier ring after the displacement of surfactantmolecules to the barrier ring

� surface tension of surfactant solution (N/m)�̃ dynamic surface tension of surfactant solution

(N/m)�0 surface tension of pure water (N/m)�CMC surface tension at the CMC (N/m)�exp experimental value of surface tension (N/m)�model value of surface tension predicted by the EOS

(N/m)� surface excess concentration of the surfactant

(mol/m2)� ∞ adsorption capacity of the surfactant (mol/m2)

�̄ mean value of the distribution of surfaceexcess, � (mol/m2)

� m minimum value of the surfactant concentra-tion at the barrier ring required to prevent thebubble from coalescence (mol/m2)

ı separation between two surfaces (m)� mean square average deviation (N/m)�p excess pressure in the film (Pa)�� difference in density between the two phases

(kg/m3)�i roots of the Bessel function of first kind and

order one� viscosity of the aqueous phase (Pa s)˘ repulsive disjoining pressure due to steric force

(Pa)� standard deviation in the distribution of �

(mol/m2)� dimensionless coalescence time

extensive use as a detergent and an emulsifier in a numberof domestic, scientific and industrial applications, the Tweensurfactants have found use in cell lysis, nuclei isolation andcell fractionation. The properties of Tween surfactants used inthe present study are presented in Table 1.

Several workers have reported the surface properties ofmixed systems of proteins and Tween surfactants (Krägelet al., 1995), and studies on the stability of foams using Tweensurfactants (Bezelgues et al., 2008; Cox et al., 2009; Maldonado-Valderrama and Langevin, 2008; Ruíz-Henestrosa et al., 2008).However, very little comparative study has been reported inthe literature on the foaming characteristics of the Tweensurfactants. Also, there is hardly any work reported in the liter-ature on coalescence of air bubbles in aqueous solutions of theTween surfactants. With this background, the main objectiveof the present work was to investigate the coalescence of airbubbles and stability of foams in aqueous solutions of Tween20, 40, 60 and 80, and correlate these results, because thebubble coalescence time is often considered as an indicationof the stability of foam. The compositions of the surfactantsolutions were selected from the detailed surface tensionprofiles of their aqueous solutions. These profiles were fit-ted using a surface equation of state based on Gibbs andLangmuir adsorption equations. The coalescence time distri-butions were fitted using the stochastic model (Ghosh andJuvekar, 2002) and the parameters of the model were analyzedwith the physical properties of the systems. The mean val-ues of the coalescence time distributions were compared withthe predictions of seven film-drainage models (Slattery, 1990).The foam stability was determined by the Ross–Miles test. Theinitial and residual foam heights were analyzed.

2. Theory

2.1. Adsorption of non-ionic surfactant at air–waterinterface

For a non-ionic surfactant, the variation of surface tension ofits solution with its concentration in the solution is given bythe Szyskowski equation (Ghosh, 2009b).

� = �0 − RT�∞ ln (1 + KLc) (1)

2346 chemical engineering research and design 8 9 ( 2 0 1 1 ) 2344–2355

Table 1 – Properties of the Tween surfactants.

Name ofsurfactant

Chemical name HLBa CMCb

(mol/m3)CMC reported in theliterature (mol/m3)c

�CMC (mN/m)

Tween 20 Polyoxyethylene (20) sorbitan monolaurate 16.7 0.060 0.059 38.5Tween 40 Polyoxyethylene (20) sorbitan monopalmitate 15.6 0.030 0.027 43.0Tween 60 Polyoxyethylene (20) sorbitan monostearate 14.9 0.022 0.025 43.8Tween 80 Polyoxyethylene (20) sorbitan monooleate 15.0 0.012 0.012 46.2

a HLB: hydrophilic–lipophilic balance [values reported in the literature (Hait and Moulik, 2001)].b CMC: critical micelle concentration.c Values reported in the literature (Schwarz et al., 2007).

Eq. (1) can be derived from Gibbs and Langmuir adsorptionequations assuming that the surface excess concentration ofthe surfactant is equal to its surface concentration. This sur-face EOS has two unknown parameters, viz. � ∞ and KL, whichcan be obtained by fitting the experimental surface tensionversus concentration profile. From the value of � ∞, the min-imum surface area per adsorbed molecule can be obtainedas,

Am = 1�∞NA

(2)

The value of Am depends on the adsorption characteristics ofthe surfactant at the interface. Since the value of � ∞ is asymp-totic because � ∞ is obtained by fitting Eq. (1) to the � versus cprofiles in the region where c is less than the critical micelleconcentration (CMC), the value of Am is approximate. However,these values of Am are not far from the true minimum areabecause the variation of � with c becomes small well beforethe surfactant concentration approaches the CMC.

2.2. Film-drainage theories of coalescence

The film-drainage models of coalescence time of bubbles insurfactant solutions given by Slattery (1990) are presentedin Table 2. The equations presented in Table 2 were devel-oped for the buoyancy-driven coalescence of a bubble at a flatgas–liquid interface. A liquid film forms between the bubbleand the flat interface when the bubble approaches the latter.This film thins with time by the drainage of liquid by creepingflow. When the thickness of the thin film reaches its criticalvalue, the van der Waals forces cause it to rupture.

Several assumptions were made while developing the mod-
els presented in Table 2. The two interfaces bounding thedraining liquid film were assumed to be axisymmetric. The
Table 2 – Film-drainage models of coalescence time.

Equation for coalescence timea Based on the model of

tc,1 = 1.07 �a3.4(��g)0.6

�1.2B0.4 Chen et al. (1984)

tc,2 = 0.705 �a3.4(��g)0.6

�1.2B0.4

tc,3 = 1.046 �a4.5��g

�1.5B0.5 Mackay and Mason (1963)

tc,4 = 0.37 �a4.5��g

�1.5B0.5

tc,5 = 5.202 �a1.75

�0.75B0.25 Hodgson and Woods (1969)

tc,6 = 0.79 �a4.06(��g)0.84

�1.38B0.46 Slattery (1990)

tc,7 = 0.44 �a4.06(��g)0.84

�1.38B0.46

a B = 1 × 10−28 J m.

deformation of the flat interface was assumed to be small.The bubble was assumed to be small so that the Bond num-ber (≡ ��ga2/�) was much smaller than unity. Thus, thedeformation of the bubble was negligible and the bubbleremained spherical. The Reynolds lubrication approximationwas applied. It was assumed that the surfactant moleculesadsorbed at the interfaces such that the resulting interfacialtension gradients were sufficiently large and the tangentialcomponents of velocity were zero. The effects of mass transferwere neglected. The film liquid was assumed to be Newtonian.All inertial effects were neglected. The disjoining pressuresdue to electrostatic double layer and steric repulsive forceswere neglected, but the attractive force due to van der Waalsinteraction between the surfaces was taken into account. Theequations for coalescence time given in Table 2 were obtainedby a stability analysis of film-drainage models, which weredeveloped on the basis of different shapes of the interfaces(Chen et al., 1984; Hodgson and Woods, 1969; Mackay andMason, 1963).

Some works on film-drainage theory suggest that the sur-face viscosities and surface tension gradient play an importantrole in the coalescence process (Edwards et al., 1991; Tambeand Sharma, 1991). However, these models have not givenexplicit equations for coalescence time that can be used tocompare with the experimental data. A film-drainage modelincorporating the effect of interfacial tension gradient hasbeen developed (Jeelani and Hartland, 1994). However, thereis hardly any method by which the value of the interfacialtension gradient can be predicted. Moreover, the values ofinterfacial tension gradient calculated from the coalescencetime data indicate that the gradients can vary over a widerange. Therefore, this model cannot be used to predict coa-lescence time.

It has been suggested in the literature (Nikolov and Wasan,1995) that the thermodynamic stability of the thin liquidfilm is the main factor that decides the coalescence timein surfactant-stabilized systems. Thin liquid films rupturedue to the growth of instabilities, which are caused by thethermal fluctuations. A film can rupture after its thicknessreduces to the critical value, hc. The fluctuations can corru-gate a deformable interface, and in certain cases the van derWaals force can be strong enough to cause thermodynamicinstability in the film (Vrij and Overbeek, 1968). The criticalfilm thickness may be calculated from the following equation.

hc = 0.267

(af A2

H

6��p

)1/7

(3)

A stochastic distribution of hc is always observed (Gurkovet al., 2010; Scheludko and Manev, 1968). In a recent work
(Vakarelski et al., 2010), it has been suggested that the thermalfluctuations in film thickness have no effect on coalescence.


2

Itestitefctvtsstt(ptcdJ

F

wt

t

Tttisoca

t

R

i

.3. Stochastic theory of coalescence

t has been experimentally observed by several workers thathe coalescence time of bubbles does not have a single valueven under identical experimental conditions (i.e., when theize of the bubbles, temperature of the system and composi-ion of the solution remain constant), but a wide distributions omnipresent (Kumar and Ghosh, 2006). In order to explainhis, and to explain some other observations that cannot bexplained by the film-drainage theory (e.g., the effects of sur-actant and salt on coalescence time), the stochastic model ofoalescence was developed (Ghosh and Juvekar, 2002). In thisheory, the variation in coalescence time was attributed to theariation in the surface excess concentration of the surfac-ant molecules at the air–water interface. The distribution ofurface excess was assumed to be Gaussian. The variation inurface excess is mainly caused by the hydrodynamic fluc-uations in the region of contact where the bubble strikeshe flat interface with a high velocity. Ghosh and Juvekar2002) have provided several experimental evidences to sup-ort the hypothesis that the fluctuation in the coalescenceime is caused by a random variation in the surface excessoncentration (� ). The following equation for the cumulativeistribution of coalescence time was developed by Ghosh and

uvekar (2002).

(�) = 12

⎡⎢⎢⎢⎢⎣erf

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1

S�

√2

⎛⎜⎜⎜⎜⎝

P�(1 +

∞∑i=1

e−�2

i�

) − 1

⎞⎟⎟⎟⎟⎠

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+erf

(1

S�

√2

)⎤⎥⎥⎥⎥⎦ (4)

In Eq. (4), � represents the dimensionless coalescence time,hich is defined as, � = t/t̄, where t̄ is the characteristic diffusion

ime, which is given by,

¯ = Rb2

D�(5)

he characteristic diffusion time is usually much longer thanhe coalescence time because coalescence occurs much beforehe entire build-up of surfactant molecules at the barrier rings depleted to the initial uniform concentration. The value ofurface diffusivity (D� ) of many surfactants at various statesf the monolayer (e.g., gaseous, liquid-expanded and liquid-ondensed states) has been reported in the literature (Agrawalnd Neuman, 1988).

The radius of the barrier ring, Rb, may be estimated fromhe equation (Princen, 1963),

b = 2a2(

��g

3�

)1/2(6)

This equation was derived by considering the buoyancy-nduced deformation of a bubble on a flat deformable

air−water interface. P� is the dimensionless coalescencethreshold, which is given by,

P� = �m

˛�̄= a

(wbfr˛) �̄

(��g�

3

)1/2(7)

The normalized standard deviation in surface excess, S� ,is defined as,

S� = �

�̄(8)

It depicts the broadness of the coalescence time distribu-tion. Therefore, this model has two unknown parameters,viz. P� and S� , which may be obtained by fitting Eq. (4) tothe experimental coalescence time distribution. The varia-tion of P� with the physical properties of the system can beexplained from Eq. (7). The fit of the stochastic model to theexperimental distributions of coalescence time of bubbles hasbeen good. The variation of the dimensionless coalescencethreshold with bubble size, surface tension, surface excessconcentration and surface forces has been explained in theseworks semi-quantitatively.

2.4. Correlations for the stability of foams

Several attempts have been reported in the literature to cor-relate the initial foam height (H) with the physicochemicalproperties of the foaming systems (e.g., surface tension, CMCof the surfactant, and density and viscosity of the liquid). Alinear relationship between the initial foam height and 1/�

has been reported from static foam test (Rosen and Solash,1969). Similar attempts have been made for dynamic foam-ing systems as well. A correlation between the steady-statefoam height

(H̃)

, average radius of the bubbles in the foam(r̄) and surface tension has been developed (Pilon et al., 2001),which predicts that H̃ ∝ �/r̄2.6. The retention time (defined asthe slope of the linear part of the plot of volumetric gas flowrate versus the gas volume contained in the system) varieslinearly with the Marangoni elasticity (which is defined as,EM = d�/dln A, where dln A denotes the relative change in sur-face area and d� denotes the corresponding change in surfacetension) (Malysa et al., 1981). Some authors have suggestedthat the interfacial tension gradient plays an important rolein the formation and stability of foams (Rosen, 2004; Tamuraet al., 1995). However, no quantitative relationship has beenproposed with either initial or residual foam heights withthese parameters.

3. Experimental

3.1. Materials

Tween 20, 40 and 60 were purchased from Sigma−Aldrich(India). Tween 80 was purchased from Merck (India). All thesesurfactants had 99% purity. They were used as received fromtheir manufacturers. The water used in this study was purifiedfrom a Millipore® water purification system. Its conductiv-ity was 1 × 10−5 S/m and the surface tension was 72.5 mN/m(298 K).

3.2. Measurement of surface tension

Surface tension was measured using a computer controlledtensiometer [manufacturer: GBX (France), model: ILMS-3, pre-


Concentration (mol/m3)

3.50.0250.020.0150.010.0050

70

80

2.560

70

(mol

/m2 )

mN

/m)

Experimental data

Fit of surface EOS

Surface excess profile

1.5

40

50

Γx

106

γ(m

0.5300.080.060.040.020

Concentration (mol/m3)Concentration (mol/m )

Fig. 1 – Variation of surface tension of aqueous solution ofTween 20 with the concentration of surfactant. The fit of thesurface EOS [Eq. (1)] and the variation of � with surfactantconcentration are shown in the figure.

cision: 0.01 mN/m]. The Wilhelmy plate method was used tomeasure the surface tension. The sample vessels and the platewere methodically cleaned before each measurement. Thecleaning was done by following the procedure described in theASTM Standard D1331-89 (2001). The platinum–iridium platewas burned to red hot conditions in the blue flame of a Bunsenburner. The sample vessel was moved with a very slow speed(∼200 �m/s) during the measurements.

The entire range of surfactant concentration under studywas divided into several intervals. This enabled us to detectthe subtle changes in surface tension accurately. The sur-factant solutions were prepared by dissolving the surfactantin water and subsequently diluting the stock solutions. Theaqueous solution of the surfactant was kept at rest for 1 h afterpreparation, and then the surface tension was measured. Thevalues of surface tension measured by the procedure men-tioned above were highly reproducible. Average values of threereadings are reported.

3.3. Measurement of surface shear viscosity

An interfacial rheometer (Physica MCR 301, Anton–Paar,Germany) equipped with a bi-conical bob was used to measurethe surface shear viscosity. The temperature was maintainedwith the help of a Peltier element. A procedure similar to thatemployed for cleaning the vessels during the measurement ofsurface tension was used for the measurement of interfacialshear viscosity. The viscosity was measured in the rotationmode employing a shear rate of 0.1 s−1. The average valuesobtained by three measurements are reported.

3.4. Study of coalescence of bubbles at air–waterinterface

Coalescence time of bubbles was studied by following theprocedure reported in the literature (Fu and Slattery, 2007;Giribabu et al., 2008). The bubbles were formed in a speciallydesigned coalescence cell made of glass (manufacturer: SchottDuran, Germany). The diameter of the cell was 10 cm. Thevessel had a hole on its wall near the bottom where a Teflon-coated rubber septum was fixed. Air bubbles were formed bya syringe inserted through the septum. As the surface tensionvaried with the change in surfactant concentration, the size ofthe bubbles also varied. The variation in the size of the bubbleswas minimized by using appropriate needles.

The bubbles were released a few centimeters away fromthe flat air−water interface. After the bubble struck the flatair−water interface, an up-and-down motion similar to thatreported by Ghosh and Juvekar (2002) was observed. The timeduring which a bubble rested on the flat air−water interface(i.e., the coalescence time) was measured by a digital videocamera [manufacturer: Sony (Japan), model: DCR-HC32E, opti-cal zoom: 20×]. It had a timer having a resolution of 0.1 sfitted with it. The time count began as soon as the bubblestruck the flat air−water interface. In some cases, especiallyat the low surfactant concentrations, some bubbles coalescedas soon as they struck the interface, which we call instanta-neous coalescence. After a bubble coalesced, the next bubblewas released after the visible disturbances at the flat interfacehad subsided. Coalescence times of one hundred bubbles werestudied in each experiment. A cumulative distribution of coa-lescence time was prepared from these measurements. The
distribution was highly reproducible from one experiment toanother. The distance between the point of release of the bub-
bles and the flat air–water interface did not have any effecton the coalescence time distributions. The size of the bub-ble was determined by image analysis using ImageJ software(Abramoff et al., 2004).

3.5. Static test of foam stability

The Ross–Miles foam test was performed by following the pro-cedure described in the ASTM Standard D1173–07 (2007). Thefoam was generated by filling a pipette with 200 cm3 of the sur-factant solution. The solution was allowed to fall a specifieddistance (90 cm) into 50 cm3 of the same solution that was con-tained in a receiver. The height of the foam that was producedimmediately upon draining of the pipette was measured byusing the digital video camera. The decay in foam height withtime was also measured. The average values of foam heightsobtained in three experiments are reported.

3.6. Experimental conditions

All experiments on measurement of surface tension, interfa-cial shear viscosity, coalescence time and foam stability werecarried out in an air-conditioned room where the tempera-ture was maintained at 298 K. The variation of temperature inthe room was within 0.5 K. This small fluctuation in tempera-ture did not have any effect on the physical properties of thesystems.

4. Results and discussion

4.1. Adsorption of surfactants at air–water interface

The adsorption of Tween surfactants at air–water interfacewas studied by measuring the variation of surface tensionwith surfactant concentration. The surface tension profiles forTween 20, 40, 60 and 80 are shown in Figs. 1–4. The surface ten-sion of the surfactant solutions decayed gradually after theinitial rapid decrease. The concentration at which the surfacetension ceased to decay with surfactant concentration wastaken as the CMC. The CMCs of Tween 20, 40, 60 and 80 are 0.06,0.03, 0.022 and 0.012 mol/m3, respectively, which are indicated
in Figs. 1–4 by vertical lines. These values agree well with theCMCs reported in the literature (Schwarz et al., 2007).



30 0.005 0.01 0.015

70

80

Experimental data

2

2.5

60

6(m

ol/m

2 )

mN

/m)

Fit of surface EOS


1.540

50

Γx

106

γ(m

1300.040.030.020.010



0.004 0.006


30 0.002

70

80

)

Experimental data

260

06(m

ol/m

2 )

mN

/m)

Fit of surface EOS


1

40

50

Γx

10γ(

0300.030.020.010



0 0.001 0.002 0.003 0.004


3

70

80

)Experimental data

260

06(m

ol/m

2 )

mN

/m)

Experimental data

Fit of surface EOS


1

40

50

Γx

10γ(m

0300.0150.010.0050



The free energy of formation of micelles depends on thehydrophobic and hydrophilic parts of the surfactants. Thehydrophilic parts of the four surfactants are same. The onlydifference in their structure lies in the aliphatic hydrocarbonchain. In the same homologous series, increase in the length ofthe hydrocarbon chain usually leads to a reduction in the CMC,because formation of micelle becomes easier with increase inhydrophobicity (Rosen, 2004). Therefore, the sequence men-tioned above for Tween 20, 40 and 60 is expected. The lowerCMC of Tween 80 may be due to the unsaturation present in thealiphatic chain which restricts the conformation of the chain.The surface tension of Tween 80 solution ceases to decreasewith surfactant concentration at a lower value of the latterdue to the saturation of the air–water interface, and hence theCMC is reached early. It has been reported in the literature(Hait and Moulik, 2001) that the CMC of the Tween surfactantscan be correlated with the HLB by the equation,

ln (CMC) = C1 + C2 (HLB) (9)

where C1 and C2 are constants. However, the HLB valuesof Tween 60 and Tween 80 are very similar (see Table 1).Therefore, the significantly lower CMC of Tween 80 cannot beexplained by Eq. (9). The different adsorption of Tween 80 atthe air–water interface, as discussed above, is the likely reasonfor the difference in CMC.

The surface EOS fitted the surface tension profiles well.The part of the profile where the variation of surface ten-sion with concentration was largest was used for fitting theEOS because the air–water interface was unsaturated and thesteric repulsion between the adsorbed surfactant moleculeswas rather small in this concentration range. The two parame-ters of the surface EOS, viz. � ∞ and KL, are presented in Table 3.These parameters were obtained by minimization of the meansquare average deviation (�) in surface tension.

� =[∑

(�exp − �model)2

x

]0.5

(10)

The values of � are reported in Table 3. The surface area occu-pied by a surfactant molecule at the air–water interface isgiven in the last column of Table 3.

The surface tension attained at CMC, �CMC, is reported inTable 1. The values of �CMC for the surfactants follow thesequence: Tween 20 < Tween 40 < Tween 60 < Tween 80. Thevalues of � ∞ follow the reverse sequence. It is evident fromthese values that there is difference in the composition andstructure of the surface phase among these surfactants. Thesurfactant molecules orient in the monolayer at the air–waterinterface depending on their structure. The length of thehydrocarbon chain follows the sequence: Tween 20 < Tween40 < Tween 60 ≈ Tween 80. The hydrophilic parts of these sur-factants are same. Surface tension of aqueous surfactantsolution is mainly determined by the groups and moleculespresent in the outermost layer of the surface. The contributionof water to the surface energy is much higher than the methy-lene or methyl groups (Huang et al., 1999). The water-content issmaller (and hydrocarbon-content is higher) in the monolayerof Tween 20 than the other surfactants. It has been reportedin the literature that the Tween surfactant molecules stand atangles less than /2 rad on hydrophobic surfaces immersed in
water (Graca et al., 2007). It is likely that the hydrocarbon tailsof Tween 40 and Tween 60 stand at angles well below /2 rad


Table 3 – Parameters of surface EOS.

Surfactant KL (m3/mol) � ∞ (mol/m2) × 106 � (mN/m) Am (m2) × 1020

Tween 20 3353 3.05 0.7 54.4Tween 40 3968 3.00 0.8 55.3Tween 60 7019 2.96 0.8 56.1

1.1 59.5

1

0.8

1

0.6

F

00073l/

0.2

0.40.007

0.02

0.05

Fit of the stochastic model

3mol/m3mol/m3mol/m

0806040200

Coalescence Time (s)

Fig. 5 – Variation of coalescence time with the

The approximate calculation presented below illustratesthe effect of steric repulsion. Suppose that the surface is mod-

Dimpled central part

of the thin liquid film

Water

Air

Adsorbed polymeric

surfactant

Air bubble

Tween 80 7658 2.79

on the air–water interface, and the Tween 20 molecules standstraighter than these molecules. The lowest � ∞ and highest�CMC of Tween 80 may be attributed to the double bond presentin its aliphatic chain, as discussed before. The surface areaoccupied by non-ionic surfactants is generally large becauseof the hydration of their hydrophilic part (Myers, 2006). Thevalues of Am (reported in Table 3) indicate that a Tween 20molecule occupies less area at the air–water interface (at sat-uration) than the other surfactants, which is expected as perthe discussion presented before.

The variation of surface excess concentration, � , with sur-factant concentration is also shown in Figs. 1–4. It can beobserved from these figures that the most rapid increase in �

with surfactant concentration occurs at very dilute surfactantconcentrations, and the value of � approaches � ∞ graduallyas the surfactant concentration approaches the CMC. Simi-lar observations can be made from the results reported in theliterature (Tajima et al., 1970). The equilibrium constants (KL)presented in Table 3 represent the relative rates of adsorp-tion and desorption of the surfactant molecules. A high valueof KL indicates either a fast rate of adsorption of the surfac-tant molecules on the air–water interface and/or slow rate ofdesorption of the molecules from the interface at dynamicequilibrium. As the value of KL becomes larger, smaller bulkconcentrations are needed to achieve the same degree of sur-face coverage (subject to the value of � ∞).

4.2. Coalescence of air bubbles at flat air–waterinterface

Coalescence of air bubbles was studied at three concentra-tions for each surfactant. These concentrations were selectedbased on the coalescence times of the bubbles. The compo-sitions of the coalescence systems are presented in Table 4.The characteristic diffusion time, t̄, of the stochastic modelwas calculated from Eq. (5) and the radius of the barrier ring,Rb, was calculated from Eq. (6). The calculation of t̄ requiresthe value of the surface diffusivity (D� ) of the surfactantmolecules. The value of surface diffusivity, however, is approx-imate. In this study, a value of D� = 1 × 10−10 m2/s was used forall surfactants. Ghosh and Juvekar (2002) have pointed out thatthe stochastic model parameters, t̄ and P� , are autocorrelated.Therefore, a different choice of surface diffusivity would leadto a different value of t̄, and hence P� . However, in a given set ofexperiments where the surface diffusivity is constant, the fitof the model and the trend of P� are insensitive to the assumedvalue of D� . This implies that, although we cannot accuratelypredict the value of P� by this analysis, we can correctly predictits trend.

The bubble coalescence time distributions for the Tween 20system are shown in Fig. 5. In dilute surfactant solution (viz.0.007 mol/m3), all bubbles coalesced within 27 s. However, asthe surfactant concentration was increased, the coalescencetime increased. The stability of the bubbles is attributed to the
steric force imparted by the hydrated polyoxyethylene groupsof the surfactant molecules. The polymeric adsorbed layer
concentration of Tween 20.

encounters a reduction in entropy when confined in a verysmall space as the bubble approaches the flat air–water inter-face. This is illustrated in Fig. 6. Since the reduction in entropyis thermodynamically unfavorable, their approach is inhib-ited. For the surfactants used in the present study, the stericrepulsion originated from the overlap of the adsorbed layersmay be expressed by the de Gennes equation (deGennes, 1987;Israelachvili and Wennerström, 1992). The repulsive disjoiningpressure (˘) between the two surfaces is given by,

˘ ≈ kT

s3

[(2L

ı

)9/4

−(

ı

2L

)3/4]

, ı < 2L (11)

where s is related to the surface excess concentration, � , bythe equation,

s =(

1�NA

)1/2(12)

Eq. (11) assumes that L > s. The first term in Eq. (11) arisesfrom the osmotic repulsion between the coils and the secondterm comes from the elastic energy of the chains.

Fig. 6 – Steric repulsion between the bubble and the flatair–water interface.


Table 4 – Predictions from the film-drainage models.

Surfactant c (mol/m3) � (mN/m) tc,1 (s) tc,2 (s) tc,3(s) tc,4 (s) tc,5 (s) tc,6 (s) tc,7 (s) tExptc (s)

Tween 20 0.007 48.1 246.6 162.5 9971.7 3527.3 4.5 1699.2 946.4 13.20.020 41.3 259.2 170.7 10506.5 3716.4 4.7 1788.5 996.1 31.40.050 39.1 276.4 182.2 11388.5 4028.4 4.9 1926.2 1072.8 41.8

Tween 40 0.008 47.1 287.6 189.5 12199.9 4315.4 4.8 2039.5 1135.9 2.60.010 45.7 262.3 172.8 10767.8 3808.8 4.6 1823.6 1015.7 8.40.015 44.9 267.2 176.1 11020.2 3898.2 4.7 1862.9 1037.5 23.5

Tween 60 0.007 46.2 293.8 193.6 12529.9 4432.2 4.9 2090.2 1164.1 5.20.015 44.8 268.3 176.8 11079.3 3919.1 4.7 1872.1 1042.7 15.50.022 43.8 275.8 181.7 11468.5 4056.7 4.8 1932.5 1076.4 28.9

Tween 80 0.005 49.0 310.2 204.3 13528.6 4785.5 5.1 2236.5 1245.6 1.80.010 46.8 289.3 190.6 12289.9 4347.3 4.9 2053.3 1143.6 2.30.012 46.2 294.0 193.7 12538.1 4435.1 4.9 2091.5 1164.9 2.9

egsbd�

r3nibbogfflhtai

cpswci

interesting aspect about the Tween 80 system was that thecoalescence times were very small even at surfactant concen-

1

0.8

0.4

0.6

F

0.0083mol/m

0.2

.

0.01

0.015


3mol/m3mol/m

050403020100

0.023 46.0 295.7 194.8

rately covered such that � = 1 × 10−6 mol/m2, then Eq. (11)ives s = 1.3 nm. For ı/2L = 0.9, from Eq. (11), we obtain repul-ive pressure, ˘ = 6.5 × 105 N/m2. This pressure acts along thearrier ring, which has a small width, say, 50 nm. A 2.7 mmiameter air bubble has a barrier ring of ∼1 mm radius for= 45 mN/m. For such a barrier ring, the area on the barrier

ing that provides the major part of the repulsion is, 2Rbwb =.1 × 10−10 m2. The magnitude of the repulsive force origi-ated from the overlap of the adsorbed surfactant monolayers

s, 2Rbwb˘ i.e., 2 × 10−4 N. This repulsive force is sufficient toalance the buoyancy acting on a 2.7 mm diameter air bub-le (viz. 1 × 10−4 N). Eqs. (11) and (12) were developed basedn the assumption that each of the polymer molecules israfted at one end to the surface, which may be quite differentrom the adsorption of the polymeric surfactant molecules atuid–fluid interfaces. Nonetheless, the calculations presentedere demonstrate that the steric repulsive force exerted byhe surfactant molecules is sufficient to balance the buoy-ncy force by which the bubble is pressed to the flat air–waternterface.

The fit of the stochastic model [i.e., Eq. (4)] to the coales-ence time distributions is shown by the lines in Fig. 5. Thearameters of the stochastic model (i.e., t̄, P� and S� ) are pre-ented in Table 5. It is observed from these values that thereas a reduction in the value of the dimensionless coales-

ence threshold, P� , when the surfactant concentration was
ncreased from 0.007 to 0.05 mol/m3. From Eq. (7), P� ∝ √
�/�̄ ,

Table 5 – Parameters of the stochastic model.

Surfactant Bubblediameter(mm)

c(

mol/m3)

t̄ (s) P� S�

Tween 20 2.6 0.007 7737.7 8.6 0.202.5 0.020 7703.8 5.6 0.072.5 0.050 8129.2 5.1 0.07

Tween 40 2.7 0.008 9191.7 18.5 0.182.6 0.010 8144.3 10.2 0.152.6 0.015 8271.1 6.9 0.12

Tween 60 2.7 0.007 9356.7 12.6 0.152.6 0.015 8300.5 7.3 0.112.6 0.022 8493.9 6.1 0.09

Tween 80 2.8 0.005 10212.3 20.5 0.112.7 0.010 9236.8 18.9 0.102.7 0.012 9360.8 17.3 0.102.7 0.023 9405.5 13.3 0.08

12628.1 4466.9 4.9 2105.3 1172.5 5.2

because the other parameters in this equation do not varymuch with surfactant concentration. The reduction in thevalue of P� may be attributed to the reduction in surface ten-sion (�) and increase in surface excess concentration

(�̄ ≡ �

)with increase in surfactant concentration.

A similar increase in coalescence time with increase insurfactant concentration was observed in the Tween 40 and60 systems, which are depicted in Figs. 7 and 8, respec-tively. In these systems, however, the value of P� decreasedsignificantly with increase in surfactant concentration. The


Fig. 7 – Variation of coalescence time with theconcentration of Tween 40.

1

0.8

0.4

0.6

F

0 0073mol/m

0.2

0.007

0.015

0.022


mol/m3mol/m3mol/m

06040200


Fig. 8 – Variation of coalescence time with theconcentration of Tween 60.


1

0.8

0.4

0.6

F

0.005 mol/m3

0.2

0.01 mol/m

0.012 mol/m

0.023 mol/m


3

3

3

01086420


Fig. 9 – Variation of coalescence time with the

48

)

0.05 mol/m

0.02 mol/m

3

3

36

eigh

t (m

m) 0.007 mol/m3

24

Foa

m H

e

122000150010005000

(s)Time

Fig. 10 – Variation of foam height with time in Tween 20solutions.

360.015 mol/m

0.01 mol/m

3

3

20

28 0.008 mol/m3

12Foa

m H

eigh

t (m

m)

42000150010005000

Time (s)

Fig. 11 – Variation of foam height with time in Tween 40

increased with increase in surfactant concentration. The ini-tial foam height followed the sequence: Tween 20 > Tween

20

24

0.022 mol/m

0.015 mol/m

3

3

16

200.007 mol/m3

8

12

Foa

m H

eigh

t (m

m)

42000150010005000

Time (s)

concentration of Tween 80.

trations close to the CMC (see Fig. 9). To check the coalescencebehavior at high-surfactant concentrations, experiment wasperformed at 0.023 mol/m3 surfactant concentration, which iswell above the CMC. The coalescence times slightly increaseddue to the steric force imparted by the micelles. However,the mean value of the coalescence time distribution was 5.2 sonly. The surface excess concentration of Tween 80 (i.e., �̄ )is smaller than the other surfactants. The less concentrationof the surfactant molecules at the interface leads to fastercoalescence of the bubbles.

The values of the normalized standard deviation, S� , weresmall for all the surfactant systems. Its value decreased withincrease in surfactant concentration. A small value of S�

reflects that the ratio of the standard deviation (� ) to themean of the Gaussian distribution of surface excess concen-tration

(�̄)

was small. In other words, the fluctuations in� from one bubble to another were small. Therefore, withincrease in surfactant concentration, the monolayer at theinterface was more stable and showed less fluctuation.

The coalescence times predicted by the seven film-drainage models are presented in Table 4. The mean valuesof the coalescence time distributions are given in the last col-umn of this table. According to these models, coalescence timeis correlated to the surface tension as, tc ∝ �−p, where p variesbetween 0.75 and 1.5. The values predicted by all the mod-els are quite different from the experimental values. Only thevalues predicted by Model 5 (for which the coalescence time isrepresented as tc,5) are relatively close to the experimental val-ues for those coalescence systems in which the coalescencetimes were small. None of these film-drainage models takeinto account the disjoining pressure due to the steric force.The seven film-drainage models widely differ in depicting thedependence of coalescence time on the size of bubble, densitydifference between the phases, surface tension of the solu-tion and van der Waals attractive force. This is the reason whythese models predict different coalescence times for the samesurfactant system.

According to the hydrophobic-force theory (Wang and Yoon,2006), the air bubble is most hydrophobic in the absence ofsurfactant, and its hydrophobicity decreases with increase insurfactant concentration. The decrease in hydrophobic forceis related to the adsorption of surfactant at the air–water inter-face. The hydrophobic force weakens with increase in theconcentration of surfactant at the air–water interface. Coales-
cence of air bubbles is instantaneous in pure water, i.e., whenthe air–water interface is most hydrophobic. From the results
solutions.

obtained in this study, it is apparent that the coalescence timedata for a given surfactant qualitatively obey this theory.

4.3. Stability of foams in aqueous solutions of Tween20, 40, 60 and 80

The variation of foam height with time at different surfactantconcentrations is depicted in Figs. 10–13 for the four surfac-tant systems. The initial foam height is a parameter of foamformation because the foam generated by the Ross–Miles tech-nique is a dynamic phenomenon involving rapid entrainmentof air. The other parameter studied in the Ross–Miles test isthe residual foam height.

It is observed from Figs. 10–13 that the initial foam height



18

240.023 mol/m

0.012 mol/m

0 01 mol/m3

3

3

12

0.01 mol/m

0.005 mol/m3

6Foa

m H

eigh

t (m

m)

02000150010005000

Time (s)


4fntfdfitwiatttfrsFsteTelbicsiii

dsfwaTocsccsaha

40

50

(μN

s/m

)

20

30

ear

Vis

cosi

ty

10

20

nter

faci

al S

he

Tween 20Tween 40Tween 60Tween 80

00.080.060.040.020

In

Surfactant Concentration (mol/m3)

Fig. 14 – Variation of surface shear viscosity with the

0 > Tween 60 > Tween 80. However, the difference in the initialoam height between the Tween 60 and Tween 80 systems wasot large. The foam height decreased rapidly with time duringhe initial period (i.e., first hundred seconds), and the residualoam height changed slowly with time thereafter. The rapidrainage can be attributed to the dynamic adsorption of sur-actant molecules at air–water interface. It has been reportedn the literature (Tamura et al., 1995) that the rate of adsorp-ion of a non-ionic surfactant is related to its CMC: a surfactantith higher CMC generally undergoes faster adsorption at the

nterface. Therefore, Tween 20, which has the highest CMCmong the four surfactants studied, produces the highest ini-ial foam height. The initial and residual foam heights forhe surfactants increased with increase in their concentra-ion. This clearly indicates that the formation and stability ofoam increased with increase in surfactant concentration. Theesidual foam heights in all these surfactant systems followimilar trends as the coalescence times of bubbles shown inigs. 5 and 7–9. However, the bubble coalescence time was con-iderably lower in the Tween 80 system than the same in otherhree surfactant systems, which was not entirely reflected inither initial or residual foam height of this surfactant system.he reason behind this apparent anomaly could be the differ-nce in the mechanism of formation and rupture of the thiniquid film and the surfactant distribution in it. For the bub-le coalescence experiments, the bubble strikes the air–waternterface with some impact (even if it is released from a pointlose to the flat interface), which can displace some of theurfactant molecules from the interface. On the other hand,n the Ross–Miles foam test, the foam drains without any suchmpact. It appears that the surfactant monolayer for Tween 80s more sensitive to such impact than the other surfactants.

As per the film-drainage theory, the stability of foamepends on the Marangoni effect, surface shear viscosity andurface elasticity. The relation between foam stability and sur-ace viscosity is not clear because there are stable foams inhich the surface viscosity is not particularly high and there

re viscous monolayers which do not produce stable foams.he foam is unstable if the surface viscosity is either too highr too low (Rosen, 2004). The variation of surface shear vis-osity with surfactant concentration is shown in Fig. 14. Theurface viscosity increases with increase in surfactant con-entration. It decreases with the decrease in surface excessoncentration of the surfactant. The main contribution to theurface viscosity comes from the hydrated hydrophilic groupst the air–water interface. With the increase in length of the
ydrocarbon chain, the area occupied by a surfactant moleculet the air–water interface increases (see Table 3) and its sur-
concentration of the surfactants.

face excess concentration decreases. The variation in surfaceshear viscosity shown in Fig. 14 agrees with the trend observedin the stability of foams.

Surface elasticity is an important parameter in the for-mation and stability of foams (Georgieva et al., 2009). Theformation of foam is favored when the surface elasticity ishigh. Therefore, the initial foam height would be high if thesurface elasticity is high. The Gibbs elasticity of foam films,with the assumption that the surface concentration of surfac-tant is equal to its surface excess concentration, is given by(Rosen, 2004),

Ef = 4RT� 2

hc + 2� (1 − �/�∞)(13)

We can use the Langmuir adsorption equation to substitutefor � in Eq. (13).

� = �∞KLc

1 + KLc(14)

Therefore, Eq. (13) becomes,

Ef = 4RT� 2∞K2Lc

h(1 + KLc)2 + 2�∞KL

(15)

The unknown quantity in Eq. (15) is the foam film thickness,h. The thickness of the film may depend on the structure ofthe surfactant and its concentration (Clark et al., 1991; Gracaet al., 2007). Eq. (15) predicts that the film elasticity increaseswith increase in surfactant concentration, reaches maximum,and then decreases. The surfactant concentration at which theelasticity reaches its maximum is given by,

c∗ = (1 + 2�∞KL/h)0.5

KL(16)

The maximum shifts to a lower concentration with increasein the film thickness. The results shown in Figs. 10–13 qualita-tively agree with the predictions of Eq. (15) that the elasticity,and hence the initial foam height, increases with increase insurfactant concentration for a given surfactant.

The Marangoni effect is significant when the maximumrate of decrease in surface tension, (d�̃/dt)max (where �̃ isthe dynamic surface tension) is large. The value of (d�̃/dt)maxdepends on the structure of surfactant and its concentration.
For a constant hydrophilic part, it decreases with increase inthe number of carbon atoms present in the aliphatic chain


(Tamura et al., 1998). Therefore, the values of (d�̃/dt)max areexpected to follow the sequence: Tween 20 > Tween 40 > Tween60 ≈ Tween 80. The magnitude of (d�̃/dt)max increases withincrease in surfactant concentration as well.

To summarize, the film-drainage theory predicts that thestability of foams should follow the order: Tween 20 > Tween40 > Tween 60 > Tween 80. It also predicts that the foam stabil-ity would increase with increase in surfactant concentrationfor a given surfactant. These predictions qualitatively agreewith the results shown in Figs. 10–13.

5. Conclusions

On the basis of the experimental results and the analysis of thetheoretical models, we can draw the following conclusions.

(1) The CMCs of the Tween surfactants studied in thiswork follow the sequence: Tween 20 > Tween 40 > Tween60 > Tween 80. The values of � ∞ follow the same sequence.The surface tension attained at the CMC follows thereverse sequence. These trends are due to different struc-tures of the hydrophobic parts of these surfactants whichcause difference in their adsorption at the air–water inter-face.

(2) The surface tension profiles are fitted well by the surfaceEOS developed from the Gibbs and Langmuir adsorptionequations. The parameters of the surface EOS are consis-tent with the adsorption patterns of the surfactants. Thearea occupied by a surfactant molecule at air–water inter-face, calculated from the EOS parameter, � ∞, follows thesequence: Tween 20 < Tween 40 < Tween 60 < Tween 80.

(3) The air bubbles are stable to coalescence in the aqueoussolutions of Tween 20, 40 and 60, however, they coales-cence quickly in Tween 80 solutions. Tween 80 is not ableto impart stability to the bubbles even at very high con-centrations.

(4) The coalescence time of air bubbles increases withincrease in surfactant concentration. The high stabilityof the air bubbles is attributed to the repulsive disjoiningpressure due to the steric and hydration forces.

(5) Stochastic distributions of coalescence time are observedin all surfactant systems. The stochastic model of coales-cence fits these distributions well and the parameters ofthe model are consistent with the properties of the system.The predictions of film-drainage models do not comparewell with the mean values of the coalescence time distri-butions.

(6) The initial and residual foam heights obtained from theRoss–Miles static foam test agree well with the trendsobserved in the coalescence of bubbles. The predictionsfrom the film-drainage theories qualitatively agree withthe stability of foams.

Acknowledgment

The authors thank the Department of Science and Technology(Government of India) for partial financial support of the workreported in this article.

References

Abramoff, M.D., Magelhaes, P.J., Ram, S.J., 2004. Image processingwith ImageJ. Biophotonics Int. 11, 36–42.

Agrawal, M.L., Neuman, R.D., 1988. Surface diffusion inmonomolecular films. II. Experiment and theory. J. ColloidInterface Sci. 121, 366–380.

Bezelgues, J.-B., Serieye, S., Crosset-Perrotin, L., Leser, M.E., 2008.Interfacial and foaming properties of some food grade lowmolecular weight surfactants. Colloids Surf. (A) 331, 56–62.

Chen, J.-D., Hahn, P.S., Slattery, J.C., 1984. Coalescence time for asmall drop or bubble at a fluid–fluid interface. AIChE J. 30,622–630.

Clark, D.C., Wilde, P.J., Wilson, D.R., 1991. Destabilization of�-lactalbumin foams by competitive adsorption of thesurfactant Tween 20. Colloids Surf. 59, 209–223.

Cox, A.R., Aldred, D.L., Russell, A.B., 2009. Exceptional stability offood foams using class II hydrophobin HFBII. FoodHydrocolloids 23, 366–376.

deGennes, P.G., 1987. Polymers at an interface; a simplified view.Adv. Colloid Interface Sci. 27, 189–209.

Edwards, D.A., Brenner, H., Wasan, D.T., 1991. InterfacialTransport Processes and Rheology. Butterworth-Heinemann,Boston.

Fu, K.-B., Slattery, J.C., 2007. Coalescence of a bubble at afluid–fluid interface: comparison of theory and experiment. J.Colloid Interface Sci. 315, 569–579.

Georgieva, D., Cagna, A., Langevin, D., 2009. Link between surfaceelasticity and foam stability. Soft Matter 5, 2063–2071.

Ghosh, P., 2009a. Coalescence of bubbles in liquid. Bubble Sci.Eng. Technol. 1, 75–87.

Ghosh, P., 2009b. Colloid and Interface Science. PHI Learning, NewDelhi.

Ghosh, P., Juvekar, V.A., 2002. Analysis of the drop restphenomenon. Trans. IChemE (A) 80, 715–728.

Giribabu, K., Reddy, M.L.N., Ghosh, P., 2008. Coalescence of airbubbles in surfactant solutions: role of salts containingmono-, di-, and trivalent ions. Chem. Eng. Commun. 195,336–351.

Graca, M., Bongaerts, J.H.H., Stokes, J.R., Granick, S., 2007. Frictionand adsorption of aqueous poloxyethylene (Tween)surfactants at hydrophobic surfaces. J. Colloid and InterfaceSci. 315, 662–670.

Gurkov, T.D., Angarska, J.K., Tachev, K.D., Gaschler, W., 2010.Statistics of rupture in relation to the stability of thin liquidfilms with different size. Colloids Surf. (A),doi:10.1016/j.colsurfa.2010.12.010.

Hait, S.K., Moulik, S.P., 2001. Determination of critical micelleconcentration (CMC) of nonionic surfactants bydonor–acceptor interaction with iodine and correlation ofCMC with hydrophile–lipophile balance and other parametersof the surfactants. J. Surfactants Detergents 4, 303–309.

Hodgson, T.D., Woods, D.R., 1969. The effect of surfactants on thecoalescence of a drop at an interface. J. Colloid Interface Sci.30, 429–446.

Huang, J.-B., Mao, M., Zhu, B.-Y., 1999. The surfacephysico-chemical properties of surfactants in ethanol–watermixtures. Colloids Surf. (A) 155, 339–348.

Israelachvili, J.N., Wennerström, H., 1992. Entropic forces betweenamphiphilic surfaces in liquids. J. Phys. Chem. 96, 520–531.

Jeelani, S.A.K., Hartland, S., 1994. Effect of interfacial mobility onthin film drainage. J. Colloid Interface Sci. 164, 296–308.

Krägel, J., Wüstneck, R., Clark, D., Wilde, P., Miller, R., 1995.Dynamic surface tension and surface shear rheology studiesof mixed �-lactoglobulin/Tween 20 systems. Colloids Surf. (A)98, 127–135.

Kumar, M.K., Ghosh, P., 2006. Coalescence of air bubbles inaqueous solutions of ionic surfactants in presence ofinorganic salt. Chem. Eng. Res. Des. 84, 703–710.

Lessard, R.R., Zieminski, S.A., 1971. Bubble coalescence and gastransfer in aqueous electrolyte solutions. Ind. Eng. Chem.Fundam. 10, 260–269.

Mackay, G.D.M., Mason, S.G., 1963. The gravity approach andcoalescence of fluid drops at liquid interfaces. Can. J. Chem.Eng. 41, 203–212.

Maldonado-Valderrama, J., Langevin, D., 2008. On the differencebetween foams stabilized by surfactants and whole casein or

http://dx.doi.org/10.1016/j.colsurfa.2010.12.010


M

M

N

P

P

R

R

R

S

S

ˇ-Casein. Comparison of foams, foam films, and liquidsurfaces studies. J. Phys. Chem. (B) 112, 3989–3996.

alysa, K., Lunkenheimer, K., Miller, R., Hartenstein, C., 1981.Surface elasticity and frothability of n-octanol and n-octanoicacid solutions. Colloids Surf. 3, 329–338.

yers, D., 2006. Surfactant Science and Technology. John Wileyand Sons, Hoboken, New Jersey.

ikolov, A.D., Wasan, D.T., 1995. Effects of surfactant on multiplestepwise coalescence of single drops at liquid–liquidinterfaces. Ind. Eng. Chem. Res. 34, 3653–3661.

ilon, L., Fedorov, A.G., Viskanta, R., 2001. Steady-state thicknessof liquid–gas foams. J. Colloid Interface Sci. 242, 425–436.

rincen, H.M., 1963. Shape of a fluid drop at a liquid–liquidinterface. J. Colloid Sci. 18, 178–195.

osen, M.J., 2004. Surfactants and Interfacial Phenomena. JohnWiley and Sons, Hoboken, New Jersey.

osen, M.J., Solash, J., 1969. Factors affecting initial foam heightin the Ross–Miles foam test. J. Am. Oil Chem Soc. 46, 399–402.

uíz-Henestrosa, V.P., Sánchez, C.C., Patino, J.M.R., 2008.Adsorption and foaming characteristics of soy globulins andTween 20 mixed systems. Ind. Eng. Chem. Res. 47, 2876–2885.

cheludko, A., Manev, E., 1968. Critical thickness of rupture ofchlorobenzene and aniline films. Trans. Faraday Soc. 64,1123–1134.

chwarz, D., Junge, F., Durst, F., Frölich, N., Schneider, B., Reckel,
S., Sobhanifar, S., Dötsch, V., Bernhard, F., 2007. Preparativescale expression of membrane proteins in escherichia
coli-based continuous exchange cell-free systems. Nat.Protocols 2, 2945–2957.

Slattery, J.C., 1990. Interfacial Transport Phenomena.Springer-Verlag, New York.

Tajima, K., Muramatsu, M., Sasaki, T., 1970. Radiotracer studieson adsorption of surface active substance at aqueous surface.I. Accurate measurement of adsorption of tritiated sodiumdodecylsulfate. Bull. Chem. Soc. Japan 43, 1991–1998.

Tambe, D.E., Sharma, M.M., 1991. Hydrodynamics of thin liquidfilms bounded by viscoelastic interfaces. J. Colloid InterfaceSci. 147, 137–151.

Tamura, T., Kaneko, Y., Ohyama, M., 1995. Dynamic surfacetension and foaming properties of aqueous polyoxyethylenen-dodecyl ether solutions. J. Colloid Interface Sci. 173, 493–499.

Tamura, T., Takeuchi, Y., Kaneko, Y., 1998. Influence of surfactantstructure on the drainage of nonionic surfactant foam films. J.Colloid Interface Sci. 206, 112–121.

Vakarelski, I.U., Manica, R., Tang, X., O’Shea, S.J., Stevens, G.W.,Grieser, F., Dagastine, R.R., Chan, D.Y.C., 2010. Dynamicinteractions between microbubbles in water. Proc. Natl. Acad.Sci. 107, 11177–11182.

Vrij, A., Overbeek, J.T.G., 1968. Rupture of thin liquid films due tospontaneous fluctuations in thickness. J. Am. Chem Soc. 90,3074–3078.

Wang, L., Yoon, R.-H., 2006. Role of hydrophobic force in the
thinning of foam films containing a nonionic surfactant.Colloids Surf. (A) 282–283, 84–91.

coalescence of bubbles and stability of foams in aqueous solutions of tween surfactants

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