Colloids and Surfaces

A: Physicochemical and Engineering Aspects 170 (2000) 45–50

Interfacial tension measurement by the rotating meniscus

Pablo Contreras a,* , Mihaela Olteanu b

a PDVSA-INTEVEP, Apartado 76343-Caracas 1070A, Venezuelab Department of Physical Chemistry, Uni6ersity of Bucharest, Bucharest, Romania

Received 16 July 1999; accepted 8 November 1999

Abstract

The interfacial tension of some pure liquid–liquid systems was calculated from the shape of a rotating meniscus.This technique is presented as an alternative method to determine the interfacial tension when the liquid–liquid ratioduring the test is expected to affect the relative rates of desorption of the surfactant from the interface. The profileof each meniscus has been studied by analysis of each image with the use of a video camera and a computer. Thepressure difference across the interface, combined with Laplace’s equation was used to obtain the general differentialequation. The approach developed by Bashforth and Adams and extended by Sugden was used to obtain, for the firsttime, a real value of the ratio of curvature at the apex. This parameter was necessary to obtain an approximation ofthe solution and thus the interfacial tension for each system. A computer program and a mathematical convergencemethod were used to reach the solution. The values obtained were comparable with the calculated data according tothe theoretical equation of Fowkes and the values reached by other experimental techniques. Ultra low interfacialtensions have been obtained from crude oil/alkaline solutions systems too. The possibility of studying the interfacialrheological behavior using this technique has been analyzed © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Interfacial tension; Ultra low tensions; Interfacial rheology

www.elsevier.nl/locate/colsurfa

1. Introduction

Most of the problems of surface chemistry arerelated more or less with the study of surfacetension (ST) or interfacial tension (IT), in static ordynamic conditions. Thus, an important numberof techniques have been developed in order toobtain this thermodynamic property. Practicallyall existing methods rely on a detailed analysis ofinterfacial shape [1,2]. A recent review made by

Franses et al. presents a description of the dy-namic techniques involving drops, bubbles, liquidjets and menisci [3]. Rusanov and Prokhorov havepresented an extended study about methods formeasuring ST and IT [1]. These authors classifiedthe techniques based on: profiles of menisci withfixed shape, measuring extreme values of parame-ter of menisci, methods using menisci at stability,dynamic methods and methods using rotatingfield. This last group includes the spinning droptechnique, which is one of the most common andis recommended for systems with ultralow interfa-cial tensions particularly (10−6 mN m−1). Limita-

* Corresponding author.E-mail address: contreraspa@pdvsa.com (P. Contreras)

0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.

PII: S0927 -7757 (99 )00499 -9

P. Contreras, M. Olteanu / Colloids and Surfaces A: Physicochem. Eng. Aspects 170 (2000) 45–5046

tions of this method have been discussed by Tay-lor [4].

In this work the rotating meniscus (RM) ispresented as an option to obtain IT values instatic conditions without the limitation of theratio between the phases, one of the main prob-lems of the methods mentioned above. It is basedon a detailed analysis of interface shape in equi-librium condition. The curvature radius at themeniscus bottom has been calculated, as an inde-pendent value for each system, without the con-sideration of any simplification. The IT isobtained directly from a differential equation. Theability to calculate IT values and to obtain theinterfacial area as a function of time with the useof a high precision video camera constitutes analternative method to study the interfacialrheology.

Theoretical aspects of RM technique are dis-cussed as follows. If a cylindrical tube of radius R(Fig. 1), containing two liquids of densities r1 andr2, is rotated with an angular velocity v, about itsvertical axis, the meniscus between the fluidsreaches an elevation z, as a consequence of itsdeformation. After an equilibrium time, the elon-gation ceases when the IT g, balances the elonga-

tion force due to the centrifugal field. Theequation describing the meniscus shape has beenobtained from the pressure drop in the axis ofrotation. Here, P1 is the pressure at any point inphase 1, and P2 at any point in phase 2,

P1−P2= (r2−r1)�

gz−v2x2

2n

+2g

b(1)

where x is the distance from the centre on thetube toward the wall, b is the radius of curvatureon the bottom of the meniscus and g is the gravityconstant [5,6].

This last expression is equivalent with the ex-pression formulated by Bird et al. to describe theshape of the surface in a liquid which rotated in atube, but without the influence of the IT (the lastterm, 2g/b). Bird uses the equations of variationfor isothermal systems, applying the expressionsof movement in cylindrical coordinates [7].

For equilibrium of a non-spherical meniscus(Fig. 1), the analysis must take into account thedeviation from the sphericity, in order to obtainthe exact solution of Eq. (1), at zero contactangle,

z¦(1+z %2)3/2+

z %x(1+z %2)1/2

=(r2−r1)

g

�gz−

v2x2

2n

+2b

(2)

where the two left terms are the inverse of R1 andR2, the two radii of curvature.

Unfortunately, it has not been possible to ob-tain an explicit solution to Eq. (2) in terms of z,the usual experimental parameter. Only approxi-mate solutions of entirely adequate accuracy havebeen performed.

The approach developed by Bashforth andAdams, to study the capillary rise problem, wasemployed to obtain a real curvature ratio value atthe apex b. These authors simultaneously withSugden said that in the case of a figure of revolu-tion, the two radii of curvature, R1 and R2, mustbe equal at the apex and equal to b. Taking intoaccount this idea, it is possible to estimate b forany system [2,8].

The function f(x)=z is obtained by video im-ages of the interface in equilibrium condition,which can be processed directly in a computer(Fig. 2).

Fig. 1. Equilibrium of a non-spherical meniscus of height z,formatted between two liquids of densities r1 and r2 whichrotate with an angular velocity v, in a tube of radius R. R1

and R2 are the two radii of curvature.

P. Contreras, M. Olteanu / Colloids and Surfaces A: Physicochem. Eng. Aspects 170 (2000) 45–50 47

Fig. 2. Rotating meniscus technique: (A) glass tubing, (B)meniscus, (C) motor, (D) tachometer, (E) video camera and(F) computer.

cr=(r2−r1)v2

4g, (5)

cg=(r2−r1)g

g(6)

where cr is the rotating constant and cg the gravi-tational content.

2. Experimental

The rotating cell used is displayed in Fig. 2. Acapillary tube of 0.4 cm internal radius and ap-proximately 25 cm length, previously washed anddried in a furnace, was arranged in a verticalmotor, of 30–2500 RPM with 0.1 KW. The veloc-ity of rotation v was fixed and measured with adigital tachometer.

Four model systems were studied: n-hexane/wa-ter, glycerin/n-hexane, water/decane and water/toluene. Distilled water was used, the rest of thefluids were of analytical grade of Merck. Thecapillaries were filled with 4 ml of each phase andleft at rest vertically for 24 h at 24°C.

To measure systems of low or ultra low interfa-cial tensions, two systems were selected. A cleansystem, acid oil (heptane-oleic acid) in contactwith alkaline solutions and Urdaneta crude oil ofVenezuela (19 °API, 3.5 of total acid number), incontact with alkaline solution. In this last case, anonionic tensoactive, Plantaren 1200, an alkylpolyglycoside of Henkel, was added at two of thealkaline solutions. A reference tube with crudeoil/distilled water was analyzed too. All theseexperiments were run at 20°C. Tubes were left inrepose for 30 days.

The procedure of IT determination can be re-sumed briefly. Each tube was inserted and fixed inthe motor, rotation velocity was adjusted andfinally verified with the tachometer. After theequilibrium, 20 min for the model systems, and 90min for the crude oil, meniscus images were takento be analyzed directly in a computer. A videocamera, a capture and video editor programs wereused. Three of these images for systems of rela-tively high, medium and low interfacial tensionsare displayed in Fig. 3. The IT was calculatedaccording to theoretical procedure.

Fig. 3. Pictures of the meniscus scanned of glycerin/n-hexanesystem at 24°C, Urdaneta crude oil/water and Urdaneta crudeoil/alkaline solution system at 20°C and 1500 RPM.

A mathematical adjustment of each interfacialcurve was made and the coefficients a, b, c,…g off(x) were estimated. The form of the meniscus ispresented as the function f(x)=z, therewith apolynomial equation of sixth degree could bewritten. With the b value and the rest of the Eq.(2) parameters, the IT was calculated. A methodof convergence was used to reach a solution ap-proximation. The dimensionless parameters a andb (Eqs. (3)–(6)), defined by Princen in order tocharacterize the shape of the meniscus were calcu-lated too.

a=2crb3, (3)

b=cgb2 (4)

P. Contreras, M. Olteanu / Colloids and Surfaces A: Physicochem. Eng. Aspects 170 (2000) 45–5048

Table 1Results for the models systems

System Dr (gr cm−3) RPM z (cm) b (cm) a (10−2) b (10−3) g (mN m−1) Experimental/calculated(RM)

1200n-Hexane/water 0.70.34 0.26 2.3 4.4 51.1* @20°C/48.5@28°C

0.60 1550 3.6 0.22 6.3Glycerin/n-hex- 8.4 33.8* @24°C/34.9@24°C

aneDecane/water 0.27 1535 1.3 0.21 1.6 2.3 51.2 [9] @20°C/45.2@24°C

1550 2.6 0.22Toluene/water 1.30.13 1.7 35.7 [11] @20°C/31.4@24°C

* This work with Wilhelmy plate.

3. Results and discussion

A summary of results for the model systems isshown in Table 1. First of all, the radius ofcurvature at the apex, b, is in accordance withthat observed in the images between 0.21 and 0.26cm. Likewise, the theoretical approximation valid-ity is confirmed. The values of the dimensionlessparameters a and b allows the relation betweenthe parameters involved to be checked. In generala and b tend to be high if the inertial effect of Dr

is bigger. The maximum value observed is forglycerin/n-hexane and the minimum for thetoluene/water. At the same time, the results cor-roborate that z depends basically on the Dr val-ues, for systems with similar IT and at the samerotating velocity.

As we can see in Table 1, the IT obtained bythe RM is comparable with the values of theWilhelmy plate. For glycerin/n-hexane, the differ-ence is around 1.1 mN m−1 and for the otherliquids the difference is a direct consequence oftemperature variance.

Additionally the IT values presented in Table 2,calculated with RM, are similar to the theoreticalvalues obtained with the equation of Fowkes[10,11]. It should be pointed out that the ITcalculated with the RM is close to the values ofFowkes, except for decane/water, maybe due tothe experimental errors.

The results for the clean system, acidic oil(heptane-oleic acid) are in Fig. 4. The main aspecthere is the fact that is possible to obtain IT valuesof until 10−4 mN m−1 as with other knowntechniques. Additionally, these results put in evi-dence the effect of the liquid–liquid ratio, when it

effects the relative rate of desorption of the sur-factant from the interface. The IT (Fig. 4) remainsas an ultra low IT after 0.1% of NaOH, againstthe observation of other authors [12,13]. Theyhave found that the IT for high alkalinity (\0.1%) grows to reach 1 mN m−1 using a spinningdrop tensiometer.

Table 2Experimental results obtained by RM and theoretical valuescalculated by the Fowkes equation

System g (mN m−1) Fowkesg (mN m−1)equationCalculated RM

n-Hexane/water 48.5 @28°C 50.5 @28°C

34.4 @24°CGlycerin/n-hex- 34.9 @24°C

ane45.2 @24°C 50.8 @24°CDecane/water

Fig. 4. Interfacial tension with RM for the system: heptane-oleic acid (13 mol m−3), in contact with alkaline solution at20°C. The ionic strength was kept constant, 171 mol m−3 withthe addition of NaCl.

P. Contreras, M. Olteanu / Colloids and Surfaces A: Physicochem. Eng. Aspects 170 (2000) 45–50 49

Table 3Experimental results obtained by RM for Urdaneta crude oil/water, alkaline solutions and alkaline solutions+nonionic surfactant(Plantaren 1200), 1:1, at 1500 RPM and 20°C, after 30 days of rest

System b (cm)Z (cm) g (mN m−1)

Oil phase Aqueous phase

1.8Urdaneta crude oil 0.21Water 26.13.3 2.71×10−2Alkaly, 0.04% NaOH 3.4×10−3Urdaneta crude oil

Alkaly, 0.07% NaOHUrdaneta crude oil 3.9 1.41×10−2 1.9×10−3

4.0 7.45×10−2Urdaneta crude oil 3.4×10−3Alkaly, 0.04% NaOH +plantaren 1200a

5.0 6.82×10−2Alkaly, 0.07% NaOH +plantaren 1200a 6.6×10−4Urdaneta crude oil

a Alkyl polyglycoside, 3.25×10−4 M.

This behavior has been the subject of numerousdiscussions due to the importance in enhanced oilrecovery. Wade et al. suggests that the minimumis a consequence of a middle phase formation,which is too small in extent to be detectable [14].The use of a spinning drop tensiometer with aphase volume ratio of 1:200 (oil:water), producesquestionable results. In our proposition, the RMtechnique is an alternative to this kind of study.Only at high alkalinity, (more than 1% of NaOH),the phase volume ratio does not change apprecia-bly [15].

The simultaneous adsorption of ionized andunionized acid upon the interface and its syn-ergetic effect is another of the explanations of thisbehavior [13]. After the minimum, the IT in-creases because the unionized acid becomes ion-ized, but again in this case, an inappropriatetechnique to measure the IT produces this ques-tionable conclusion.

The results for crude oil systems are in Table 3.The radius of curvature was calculated in eachcase. As observed, b changes considerably, beingsmaller when the interfacial tension decreases.Thus, the crude oil/water system has a b of 0.21cm and IT of 26.1 mN m−1, in the case of crudeoil/alkaline solutions, b is of the order of 10−2 cmdue to the fact that IT is smaller. The effect of thenonionic tensoactive was evaluated as well. Valuesof b of 10−2 cm and IT of 10−3–10−4 mN m−1

were obtained for systems with a meniscus of 4and 5 cm in height.

It is remarkable that the IT reported in all thiswork reflects exactly the values of the system

studied due to the fact that we do not take only asmall portion of the oil, like practically all theknown techniques.

One parallel aspect derived from this work isthe possibility to estimate IT in dynamic condi-tions and some rheological properties of one in-terface. The Marangoni viscosity zM, of oil/watersoluble films, could be estimated from the varia-tion of the interfacial area S, with the time (S/(t.This property expressed in cm s−1, analogous tothe viscosity of three-dimensional systems, thatdepends simultaneously upon the properties ofthis phase and upon the rate of attainment ofequilibrium between the substrate and the ad-sorbed film, could be obtained from the relationof the surface tension gradient Dg [16].

In fact, this viscosity is the parameter that weare using to study the presence of mixed film insome particular cases of interest. The results willbe published later.

4. Conclusion

We suggest that the rotating meniscus might beused to measure the IT, keeping in mind thepossibility of volume changes of phases. The re-sults are comparable with theoretical and experi-mental values published. Ultra low IT can bemeasured too. The option of determining IT andthe area generated as a function of time opens thepossibility to employ the technique to study somerheological properties.

P. Contreras, M. Olteanu / Colloids and Surfaces A: Physicochem. Eng. Aspects 170 (2000) 45–5050

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