A CRITICAL REVIEW: SURFACE AND INTERFACIAL TENSION MEASUREMENT BY THE DROP WEIGHT METHOD

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A CRITICAL REVIEW: SURFACE AND INTERFACIALTENSION MEASUREMENT BY THE DROP WEIGHTMETHODBoon-Beng Lee a; Pogaku Ravindra a; Eng-Seng Chan aa Centre of Materials and Minerals, School of Engineering and InformationTechnology, Universiti Malaysia Sabah, Kota Kinabalu, Sabah, Malaysia

Online Publication Date: 01 August 2008To cite this Article: Lee, Boon-Beng, Ravindra, Pogaku and Chan, Eng-Seng (2008)'A CRITICAL REVIEW: SURFACE AND INTERFACIAL TENSION MEASUREMENTBY THE DROP WEIGHT METHOD', Chemical Engineering Communications, 195:8,

889 - 924To link to this article: DOI: 10.1080/00986440801905056URL: http://dx.doi.org/10.1080/00986440801905056

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A Critical Review: Surface and Interfacial TensionMeasurement by the Drop Weight Method

BOON-BENG LEE, POGAKU RAVINDRA,AND ENG-SENG CHAN

Centre of Materials and Minerals, School of Engineering and InformationTechnology, Universiti Malaysia Sabah, Kota Kinabalu, Sabah, Malaysia

The drop weight method has been used as a standard method for surface andinterfacial tension measurement. However, lack of appropriate guidelines in usingthis method has resulted in errors. The specific objective of this critical review isto present the experimental setup, the limitations on the correction factors, andthe principle of the drop weight method. Mathematical models of correction factorswere evaluated by using a proposed error analysis. The use of the proposedLee-Chan-Pogaku model and HG-Equation 2 for correction factor determinationis suggested. However, further investigations would be required to justify the validityof the correction factors at low r=V1=3 range and their use for viscous fluids. Thephysics of drop detachment is complicated; more investigations would be requiredto form a rigid theory of this method.

Keywords Drop weight correction factors; Drop weight method; Error analysis;Interfacial tension; Mathematical model; Surface tension

Introduction

The knowledge of surface and interfacial tension is of great utility for fundamentalresearch and industrial engineering applications. Surface and interfacial tensiondetermines the quality of many industrial products such as coatings, paints, inkjet printing, detergents, cosmetics, pharmaceuticals, lubricants, pesticides, foodproducts and agrochemicals (Ali et al., 2006; Queimada et al., 2004). Furthermore,interfacial tension has an effect on some important steps in production processessuch as catalysis, adsorption, distillation, and extraction phenomena (Queimadaet al., 2004). The effects of surface tension and interfacial tension have been widelystudied (Ayirala et al., 2006). In many advanced countries, surface tension measure-ment is used as a standard for environmental pollution control to monitor thequality of wastewater (Gunde et al., 1992) and the atmosphere (Capel et al., 1990).

The drop weight method (also known as drop volume method) is one of thepreferred methods for surface and interfacial tension measurement due to its sim-plicity of handling, easy temperature control, small sample size, good reproducibilityand its applicability for either liquid-air or liquid-liquid systems. A precision of�0.01 mN=m can be achieved under favorable circumstances (Earnshaw et al.,

Address correspondence to Eng-Seng Chan, Centre of Materials and Minerals, School ofEngineering and Information Technology, Universiti Malaysia Sabah, Locked bag No. 2073,88999 Kota Kinabalu, Sabah, Malaysia. E-mail: engseng.chan@gmail.com

Chem. Eng. Comm., 195:889–924, 2008Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print/1563-5201 onlineDOI: 10.1080/00986440801905056

1996). Presently, advanced automated drop volume apparatus is commercially avail-able in the market from the Sinterface, Kruss, and Lauda companies. However, it isprohibitively expensive, and it may not be economically feasible for infrequent usersto own automated drop volume apparatus.

Without clear guidelines, serious errors (as high as 10%) could be encounteredduring measurement (Mori, 1991). Moreover, there is a lack of intensive errorevaluation on various drop weight correction factor models in the users’ selectionguides. Hence, a guideline on the use of the drop weight method to measure surfaceor interfacial tension is useful for investigators. Accordingly, the principle of thedrop weight method, the experimental setup, the operating parameters of the dropweight apparatus, and error analysis on various mathematical models are discussedin the present review.

Drop Weight or Drop Volume Apparatus

A basic requirement for drop weight apparatus is a dripping tip and a weighingbalance for surface or interfacial tension measurement. However, an additionalinstrument such as temperature and drop formation controller is used to giveimproved accuracy and reproducibility.

With the development and availability of micrometer syringes and hypodermicneedles, drop weight apparatus advanced in precision over the previous design.(Parreira, 1965; Wilkinson, 1972; Doyle and Carroll, 1989). Drop weight apparatushas been further improved in the recent past (Gunde et al., 1992; Miller et al., 1994;Matsuki and Kaneshina, 1994; Fainerman and Miller, 1995; Li et al., 1996): a high-precision syringe pump is used for better drop formation rate control, photoelectricsensors or laser beams are used for drop number count, a microprocessor device isused for drop volume measurement in small time intervals, a computer is used formeasurement control and analysis of data, and an ultrahigh speed camera is usedfor drop formation observation at small time intervals. Nowadays, the drop volumeapparatus can be operated in automated on-line mode. It is very effective forinterfacial tension measurement for chemical production processes (Alexander andMatteson, 1987; Doyle and Carroll, 1989).

The use of drop volume apparatus has been extended to measure interfacialtension of non-transparent fluids. For instance, interfacial tension measurementbetween asphalted materials (e.g., bitumen) and alkaline solutions by the dropweight method was not possible in the early 1930 s due to the difficulty in dropcounting (Traxler and Pittman, 1932). However, it is now made possible with theuse of a photoelectric sensor (Xu, 1995).

Drop Weight or Drop Volume Determination Parameters

Operation Mode

Generally, the drop weight method can be operated in three modes, static, dynamic,or quasi-static. The static mode is the evacuated way of drop formation at infinitelyslow rate. The weight or volume of the detached drop is measured after a certain per-iod of drop formation time to determine the static surface tension. Surface tension ofpure liquids is often measured by the static mode.

890 B.-B. Lee et al.

On the other hand, the dynamic mode involves drop formation under constantflow rate and measures the volume of each detached drop, instead of drop weight, atevery definite time interval. The dynamic mode is usually used for dilute solutionscontaining surface-active agents (e.g., surfactant). The reason is that surfactant solu-tions take a certain length of time (from a few milliseconds to few seconds) to reachtheir equilibrium or static surface tension values due to surface adsorption kinetics(Pierson and Whitaker, 1976a, b; Tornberg, 1977; Jho and Burke, 1983; Van Hunselet al., 1986; MacLeod and Radke, 1993; Fainerman and Miller, 1995; Fainermanet al., 1997).

Likewise, the quasi-static mode is also used for dynamic surface tensionmeasurement. It consists of growing a drop of a defined volume at a dripping tipin the shortest possible formation time. Therefore, the quasi-static mode providesdynamic surface tension data at a smaller time interval than the dynamic mode.The quasi-static mode is usually used for slow adsorption kinetic solutions (froma few minutes to a few hours) such as protein solution and polymer solution(Tornberg, 1977; Tornberg and Lundh, 1981; Miller and Schano, 1986; Van Hunseland Joos, 1989; Razafindralambo et al., 1995). To date, drop volume apparatus isable to measure dynamic surface or interfacial tension in the time range of about0.2 to 1000 s.

The choice of the operating mode is dependent on the properties of the fluid andthe sensitivity of the drop weight or drop volume apparatus.

Drop Number

Typically, in static mode, drop weight measurement involves weighing accumulatedliquid or solution from a large number of drops to determine the average weight perdrop (Thiessen and Man, 2000). The procedure of drop weight measurement alwaysinvolves discarding the first few drops (Thiessen and Man, 2000). A drop number inthe range of 10 to 30 has been proposed by several investigators to obtain consistentdrop weight (Morgan and Bole, 1913; Harkins and Brown, 1919; Harkins andZollman, 1926; Boucher et al., 1967; Wilkinson, 1972; Jomsurang and Sakamon,2005; Hoorfar et al., 2006). Statistically, 30 drops of a sample is adequate to giveaccurate and reproducible measurement.

Because advanced instrumentations are usually used for the dynamic mode, thevolume of each drop can be calculated from the flow rate and drop frequency(drop number per time) of the tested sample (Alexander and Matteson, 1987; Hooland Schuchardt, 1992; Mollet et al., 1996). Therefore, on-line determination ofsurface and interfacial tension can be conducted (Miller et al., 1994; Matsuki andKaneshina, 1994; Fainerman and Miller, 1995; Li et al., 1996).

Sample Delivery

Liquid sample is generally delivered in three different options. First, the air in thedrop collection chamber is evacuated to reduce the pressure so that the liquid orsolution drips from the dripping tip. Second, the pressure in the liquid or solutionis increased with a pump. Third, one end of the dripping tip is left open and theliquid or solution is allowed to drip by its own weight due to gravitational force(Rusanov and Prokhorov, 1996).

Surface Tension Measurement by Drop Weight Method 891

By using the first two delivery options, a change in the uniformity of the flow ofsample can be minimized. However, with the use of the third delivery option (opendripping tip), nonuniform flow of sample is commonly encountered. The problemcan be avoided with the aid of a joining tube (often a ball in the middle) placedbetween a sample reservoir and the dripping tip (Rusanov and Prokhorov, 1996).Furthermore, the problem can also be eliminated with a large volume sample reser-voir of a larger diameter than the diameter of the dripping tip (Jho and Burke, 1983).

Evaporation Effect

Although the accuracy gained by slow drop formation is important, possible errorsdue to evaporation could be significant, especially for volatile liquids and aqueoussolutions. Therefore, procedures to determine drop weight with slow drop formationtime and also eliminate evaporation effect are suggested by Morgan and Bole (1913).Evaporation effect is usually minimized by using fast drop formation procedures: thedrops are drawn to nearly full size (about 95%), and only the last fraction of eachdrop formed is turned off slowly (Harkins and Brown, 1919; Parreira, 1965) or aftera certain waiting time (about one minute) (Wilkinson, 1972; Gunde et al., 1992).

Hydrodynamic Effects

It was found that the drop weight of a sample is different with the variations of dropformation times from a dripping tip (Harkins and Brown, 1919; Edward, 1929;Ferguson, 1929; Campbell, 1970; Hoorfar et al., 2006). This indicates that the hydro-dynamic effects have to be considered in drop weight measurement. Even a pureliquid exhibits hydrodynamic effects in a drop formation time interval of 1 to 30 s(Miller et al., 1998). Therefore, surface and interfacial tension measurementsare to be performed for drop times of above 30 s to avoid hydrodynamic effects(Miller et al., 1998).

It has been observed that the hydrodynamic effects are influenced by the nozzletip size, the surface tension, and the density difference of the sample (Kloubek, 1975;Jho and Burke, 1983; Jho and Carreras, 1984; Van Hunsel and Joos, 1989; Milleret al., 1994). Jho and Burke (1983) proposed an empirical correction equation tooffset the hydrodynamic effects, which are a linear relationship between McGeeslope and equilibrium drop mass. The validity of Jho and Burke’s equation has beenreported for interfacial systems (Van Hansel and Joos, 1989; Deshiikan et al., 1998).

The hydrodynamic effect due to the influence of surface rheological behavior ofa fluid has been reported (Whitaker, 1976; Jho and Burke, 1983; Henderson andMicale, 1993; Miller et al., 1998). The influence of rheological behavior of a fluidcan be identified by surface tension number, Nc (Whitaker, 1976). Surface tensionnumber, Nc, must remain much larger than unity in order to eliminate the influenceof rheological behavior of a fluid (Whitaker, 1976). The surface tension number isdefined as:

Nc ¼rcg2

Conversely, the hydrodynamic effects can be nullified for liquids with Nc greaterthan 100 or viscosity up to 15 cP by applying Jho and Burke’s (1983) equation(Jho and Carreras, 1984).

Miller et al. (1998) introduced two empirical correction equations to offset thehydrodynamic effects by taking into consideration the viscosity of liquids.

For low viscous liquids (about 1 cP):

c ¼ mg

2prwðr=V1=3Þ

� �1� 0:008þ 0:0041r

� �ð2Þ

For high viscous liquids (>1 cP):

c ¼ mg

2prwðr=V 1=3Þ

� �1� ½ð0:028rþ 0:0145Þ lnðnÞ þ ð0:0134rþ 0:0096Þ�

� �ð3Þ

However, the validity of Equations (2) and (3) beyond the studied conditions(kinematics viscosity, n>30 mm2=s) requires further investigation.

Dripping Tip

The dripping tip is the most critical part of a drop weight or drop volume apparatus.It will directly influence the accuracy of surface and interfacial tension measurementsof liquids and solutions.

Dripping Tip PositionIt is advisable to place the dripping tip perpendicular to the datum line. It was foundthat the inclination of the dripping tip could cause deviation in drop weight determi-nation (Gans and Harkins, 1930).

For liquid-liquid systems, contrasting results have been reported on the depen-dency of whether the drop is hanging ‘‘pendant up,’’ with the heavier fluid inside thelighter or vice versa (Andreas et al., 1938; Campanelli and Wang, 1997). Hence,further investigations are required to justify the dripping tip position.

Dripping Tip TypeDripping tips made from different materials such as glass, brass, stainless steel,Teflon, and tungsten carbide have been used for drop weight determination; nosignificant difference is observed (Harkins and Brown, 1919; Henderson and Micale,1993; Xu, 1995; Pu and Chen, 2001a). Furthermore, it is a good practice to treat theexternal surface of dripping tips with octadecyltricholorosilane to prevent wetting ofthe sample at the outer surface of tips, especially for interfacial systems (Hendersonand Micale, 1993). As an alternative, a drop of the first liquid should be forcedthrough the nozzle to cover the tip to its circumference before immersion into thesecond liquid (Kaufman, 1976).

Dripping Tip SizeIt was suggested that the tip size should be selected in such a way that the r=V1=3

value lies between 0.65 and 0.95, a region of minimum correction factor boundaryðWðr=V1=3Þ � 0:60Þ for an air-liquid system (Harkins and Brown, 1919; Wilkinsonand Kidwell, 1971). However, for a liquid-liquid system, the dripping tip shouldbe chosen so that r=V1=3 lies between 0.50 and 0.65 (Harkins and Brown, 1919).In addition, guidelines on the selection of dripping tips have been suggested to obtaincorrection factors at the above suggested r=V1=3 range based on the density and

surface tension of liquid (Wilkinson and Kidwell, 1971; Kaufman, 1976). Therefore,the radius of 0.30 cm should be selected, without exceeding the recommended rangefor liquid or aqueous solution with water-like properties (q � 1 g=mL andc � 72 mN=m).

Furthermore, by using a micrometer syringe, the error in surface tensionmeasurement is decreased with increment of the dripping tip size (Parreira, 1965;Zholob et al., 1997). This is due to the formation of large drop size from a largedripping tip, as large drop size is desired for a less sensitive measuring device. How-ever, it was observed that as the dripping tip size increased, the detached drop weightbecame less dependent on the dripping tip size. This is because the gravitational forceis more predominant in the drop detachment process than the surface tension force(Boucher and Evans, 1975). It is also difficult to completely wet the plane surface oflarge tip (Edgerton et al., 1937). Consequently, a correlation is proposed to estimatethe maximum dripping tip radius that can be used for a particular fluid system as afunction of density and surface tension (Halligan and Agrawal, 1971).

Dripping Tip ConditionsThe tips should be ground to ensure that the ends are perfectly flat and perpendicularto the bore with no chipping (Harkins and Brown, 1919; Wilkinson, 1972; Alexanderand Matteson, 1987; Pu and Chen, 2001a). The plane surface of the tip is roughened toensure complete wetting by the dripping fluid (Harkins and Brown, 1919; Eversole andDedrick, 1933; Pu and Chen, 2001a).

Dripping Tip GeometryThe geometry of a dripping tip can be basically classified into two types: capillarytips and sharp-edge tips, as shown in Figure 1. The main concern in the use ofdifferent geometry dripping tips is the deviation in the contact angle of the drippingfluid with the tip’s plane surface (Harkins and Brown, 1919; Bartell et al., 1933;

Figure 1. (Above) Dripping tip geometry: (A) thick-wall capillary tip, (B) thin-wall capillarytip, (C) sharp-edge tip. (Below) Drop detachment according to the dripping tip geometry.

Edgerton et al., 1937; Vacek and Nekovar, 1973; Hozawa et al., 1981; Pu and Chan,2001a). Therefore, it is important to note the wetting degree of the dripping fluid onthe surface of the dripping tip; the drops are formed either at the inner diameter or atthe outer diameter of the tip (Bartell et al., 1933; Skelland and Slaymaker, 1990;Miller et al., 1994; Garandet et al., 1994). If the drops are formed at the innerdiameter, then the dimension of the inner diameter should be used for surface orinterfacial tension calculation and vice versa if the drops are formed at the outerdiameter. The effects of different dripping tip geometries on the determination ofdrop weight correction factors are discussed in later sections.

Dripping Tip Length-to-Diameter RatioA suitable selection of the ratio of the dripping tip length to its diameter (L=D) isrecommended. L=D ratio is an important criterion to ensure a fully developedvelocity profile at the tip exit and minimum flow disturbances in the tip (Scheeleand Meister, 1968; Skelland and Slaymaker, 1990). Therefore, an L=D more than0.035 Re (Bird et al., 2001) should be considered for measuring interfacial tensionas well as for drop volume determination from volumetric flow rate.

The Principle of the Drop Weight Method

Tate’s Law

The basis of the drop weight method can be traced back to Tate, who postulated thatthe weight of a detached drop was proportional to the diameter of the dripping tipand to the weight of the liquid that would rise up into a tube of the same diameter bycapillary action (Yildirim et al., 2005). Consequently, Tate’s conclusions haveresulted in the following equation, commonly known as Tate’s law:

mg ¼ 2prc ð4Þ

In other words, Tate’s law is based on ad hoc force balance at the dripping tip. Ifa liquid is allowed to flow out from the bottom of a tip to form a drop, the drop fallsfrom the tip when it reaches a critical dimension or when the gravitational force(determined by the mass of the drop) is no longer balanced by the surface tension,as illustrated in Figure 2. The weight of the falling drop is measured, and then thesurface tension of a liquid can be determined from the following equation:

c ¼ mg

2prð5Þ

Figure 2. Force balance at a dripping tip based on Tate’s law.

However, Tate’s law is only a poor approximation. This is because only a fractionof the drop is actually detached from the dripping tip, and liquid residue is left at thedripping tip after drop detachment. Therefore, Tate’s law normally underestimates thevalue of surface tension (Morgan and Stevenson, 1908; Garandet et al., 1994).

Lohnstein’s Theory

A set of correction factors f (r=a) in terms of r=a value is introduced to correct themeasured quantities (weight or volume) of a detached drop in order to obtain theactual surface tension value (Harkins and Humphrey, 1916; Harkins and Brown,1919; Hauser et al., 1936; Hartland and Srinivasan, 1974):

2prf ðr=aÞ ¼Vqg

2prf ðr=aÞ ð6Þ

The correction factors are calculated based on the assumption that the drop residuehas the same contact angle with the tip as that of the drop of maximum volume (idealdrop) that could be supported by the tip.

Lohnstein proposed (Figure 3) a correlation that deviates more than 4% for r=agreater than 0.4 (Harkins and Brown, 1919; Wilkinson, 1972). Accordingly,Lohnstein’s experiment has been repeated in a more systematic and precise mannerwith liquid-air and liquid-liquid systems (Harkins and Humphrey, 1915, 1916; Vacekand Nekov�aar, 1979). However, the determination of correction factors fromLohnstein’s theory is tedious because it involves surface tension value in both sidesof Equation (6). Therefore, a series of approximations is required to determine thesurface or interfacial tension value.

Harkins and Brown’s Drop Weight Correction Factors

In order to simplify the calculation of surface tension, Harkins and Brown (1919)suggested another set of correction factors based on the function wðr=V 1=3Þ, where

Figure 3. Drop weight correction factors, f(r=a). Note: the data of Lohnstein (1906) werecollected from Harkins and Humphrey (1916).

the determination of the surface tension involves the knowledge only of tip radiusand drop weight:

c ¼ mg

2prwðr=V1=3Þ ¼VDqg

2prwðr=V 1=3Þ ð7Þ

Equation (7) shows that the drop weight does not depend only on the dripping tipradius and surface tension of the dripping liquid but also on the shape of the drop(as indicated by the cubic root of drop volume). Moreover, for the ease of usingthe two sets of correction factors, correlations have been introduced to convertthe r=a value to r=V 1=3 and vice versa (Lunn, 1919; Wilkinson, 1972).

In reference to Figure 4, Harkins and Brown’s (1919) work included a r=V 1=3

range from 0.31 to 1.225 and wðr=V 1=3Þ from 0.535 to 0.721. The range was laterexpanded by other investigators, for r=V1=3 from 0.0638 to 4.45 and wðr=V 1=3Þcorrection factors from 0.19 to 0.94 (Dunken, 1942 (From Vacek and Nekovar,1973); Brown and McCormick, 1948 (From Campbell, 1970); Campbell, 1970; Wilk-inson, 1972). In addition, Harkins and Brown (1919) considered only the densityrange of 0:87 � q � 2:18 g=mL, surface tension range of 26:6 � c � 71:96 mN=m,and viscosity range of g � 1:5cP. Therefore, further investigations have been con-ducted with liquid metals that have density up to 20 g=ml and surface and interfacialtension up to 2500 mN=m (Bartell et al., 1933; Dunken, 1942 (From Campbell 1970);Brown and McCormick, 1948 (From Vacek and Nekovar, 1973); Campbell, 1970;Wilkinson and Aronson, 1973; Vinet et al., 1993; Garandet et al., 1994).

Figure 4. Drop weight correction factors, Wðr=V1=3Þ. Note: the data of Lohnstein (1906) werecollected from Vacek and Nekovar (1973). The data of Rayleigh (1896) were collectedfrom Harkins and Brown (1919). The data of Dunken (1942) were collected from Vacekand Nekovar (1973) and Wilkinson and Aronson (1973).

Furthermore, the viscosity range has been expanded to 60 000cp and above (Pu andChen, 2001b). Details of the above-mentioned works are presented in Table I.

Mathematical Models of Drop Weight Correction Factors

In the past, there has been interest in developing mathematical models for thecorrection factors. A review of the models is summarized in Table II. Generally, cor-rection factor mathematical models are developed based on Harkins and Brown’s(1919) and=or Wilkinson’s (1972) experimental data at a certain range of ðr=V1=3Þ.At an earlier stage, the experimental data were best fitted into a polynomial equation(Lando and Oakley, 1967; Strenge, 1969; Wilkinson and Kidwell, 1971; Clift et al., 1978(From Skelland and Slaymaker, 1990); Gunde and Hartland, 1984 (From Pu and Chen,2001a), Deshiikan et al., 1998). Figure 5 shows that most studies focused on the r=V1=3

range between 0.0 and 1.2 or the ‘‘concave’’ part of Harkins and Brown-Wilkinson’scurve. This is due to their wide applications and high accuracy. On the other hand,a simpler model is proposed by Zhang and Mori (1993), which covers a r=V 1=3 rangefrom 0 to 0.95 or nearly the ‘‘linear’’ part of Harkins and Brown-Wilkinson’s curve.

Earnshaw et al. (1996) felt that the use of a polynomial equation in the correc-tion factor determination might lead to deviation, which will be either increased ordecreased in the final calculated surface or interfacial tension. To eliminate that, acubic spline approximation is used to fit Harkins and Brown’s (1919) as well asWilkinson’s (1972) experimental data, and the value of the correction factors is givenin a table form. The table gives correction factors with a wide range of r=V1=3, from 0to 1.598, at an interval of 0.002 units.

However, among the proposed mathematical models, not all models have uni-versal applications. Figure 6 shows that some of the mathematical models work onlywithin a specific range. A detailed discussion on their limitations is given in Table II.Moreover, some correction models in terms of linear equations have been developedbecause of the specific use of the correction factors within a very small range ofr=V 1=3 (Suggitt et al., 1949; Vinet et al., 1993). In addition, some models thatdescribe the correlation of r=a and f(r=a) and the correlation of wðr=V1=3Þ andrðwðr=V 1=3Þ=VÞ are given in Table III. These models are seldom used because theyinvolve complicated calculations for surface or interfacial tension determination.

Selection of Mathematical Model of Drop Weight Correction Factors

Over the past 40 years, many mathematical models of drop weight correction factorshave been proposed, and some have been widely applied. Due to a lack of propererror analysis of the models, a large deviation in the final calculation of surface orinterfacial tension could be encountered. Therefore, the error analysis would bebenefit users in selecting appropriate mathematical models.

In the past, different researchers have applied different approaches to conducterror analysis on the correction factors models, such as mean error (ME) analysis,coefficient of variation (CV) analysis, and average percentage error (APE) analysis(Strenge, 1969; Wilkinson and Kidwell, 1971; Babu, 1987). As a result, variousstatistical analyses may confuse users when making decisions on the model selection.

In this review, average absolute deviation (AAD) and maximum absolute devi-ation (MAD) are proposed as a more appropriate approach to evaluate the accuracyof the existing correction factor models. AAD analysis indicates the degree of best fit

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of the experimental data with the models as a whole, while MAD analysis indicatesthe consistency of the best fit of the experimental data with the models. As acomparison, ME analysis reflects just the best fit of the experimental data withthe model as a whole but does not show the consistency of individual experimental

Figure 5. Mathematical model of drop weight correction factors.

Figure 6. Limitation ranges of several correction factor models.

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data. CV analysis is usually used to provide normalized measure of the data distri-bution, if the correction factors are not distributed normally.

With a total of 131 experimental data from Harkins and Brown’s (1919) andWilkinson’s (1972) work, the error analysis was conducted selectively based onthe range of r=V 1=3 for specific models. The summary of the analysis is shown inTable IV. Based on the AAD and MAD analyses, the choice of the best model ishighlighted in Table IV.

It is always preferable and convenient to use a mathematical equation instead ofcorrection factors in tabulated form or equations with a certain limited r=V 1=3 range.With reference to Tables II and IV, an accurate mathematical equation for a widerr=V 1=3 range is still lacking. As a result, a new drop weight correction factors modelwas developed by using MATLAB (Version 6.5, release 13, MathWorks). The r=V 1=3

range included was between 0.0 and 1.2. The seventh order of polynomial fit wasfound to be the best fit out of 131 experimental data from Harkins and Brown(1919) and Wilkinson (1972). Furthermore, the error analysis was conducted usingthe new model suggested. It was found that the model gave 0.47% of average devi-ation and about 1.95% of maximum deviation from the experimental data. The newcorrection factors model, known as the Lee-Chan-Pogaku drop weight correctionfactors model, is proposed:

wðr=V1=3Þ ¼ 1:000� 0:9121ðr=V 1=3Þ � 2:109ðr=V1=3Þ2 þ 13:38ðr=V1=3Þ3

� 27:29ðr=V1=3Þ4 þ 27:53ðr=V1=3Þ5 � 13:58ðr=V1=3Þ6

þ 2:593ðr=V1=3Þ7 ð8Þ

In addition, it is suggested that the Lee-Chan-Pogaku model should be usedin the range of 0:00 � r=V1=3 � 1:20 and HG Equation 2 in the range of1:20 � r=V 1=3 � 1:60 in future investigations for surface and interfacial tensionmeasurements.

In-Depth Investigations on the Drop Weight Correction Factors Curve

Pattern of the Curve

An extensive investigation was conducted by Edgerton et al. (1937) to understandthe interesting pattern of Harkins and Brown’s drop weight correction curve. Thedripping tips and experimental conditions were simulated as reported by Harkinsand Brown (1919). Drop formation at different sizes of dripping tip is observed byusing a high-speed camera. Figure 7 shows that the curve can be divided into threebroad regions:

. Region I: A large portion of the drop is actually detached from the tip. Within thisregion, a ‘‘bulging’’ drop can be defined as the drop with a maximum diameterlarger than the diameter of the tip. It is formed at the tip when r=V1=3 is less than0.5977 (Boucher and Evans, 1975). As r=V1=3 increases, the correction factordecreases to a minimum value. This is due to the increment in diameter of the dropbeing relatively less than the increment of dripping tip diameter (Edgerton et al.,1937).

. Region II: The increment of correction factors between the minimum and themaximum in this region is speculated to be influenced by the increase in thickness

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffi

Pn i¼1

tedÞ2=ðn�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffi

Pn i¼1

tedÞ2=n�ðk�

��

)¼Pn i¼

ent�

tedÞ=

of the stem length and the portion of the pendant drop detached. The increment ofthe correction factors is continued until it reached a maximum value (Edgertonet al., 1937).

. Region III: At the maximum value, the maximum size of pendant drop is attained.The curve falls off rapidly as the tip diameter and stem length are increased. More-over, the increase in the size of secondary drops is small compared to the increasein the circumference of the tip (Edgerton et al., 1937).

Applicability of the Curve for Various Types of Fluids

According to Lunn (1919) and Freud and Harkins (1929), the correction factorscould not be universally valid for all liquids due to the differences in density andviscosity. Consequently, the question of the applicability of the curve for aqueoussolutions, surfactant solutions, and viscous liquids or solutions was raised.

The applicability of the correction factor curve for aqueous and protein solu-tions has been validated. Comparison between the surface tension value determinedby using the drop weight method and the values determined using other methods(e.g., maximum bubble pressure method and Wilhelmy plate method) has beenreported (Suggitt et al., 1949; Boucher et al., 1967; Tornberg and Lundh, 1981).

In addition, the applicability of the curve for surfactant solutions can beadopted. Evidence, including experimental results, theoretical analysis, and imagesof drop detachment of surfactant solution, has been reported (Pierson and Whitaker,1976a, b; Whitaker, 1976). Furthermore, investigations on the applicability of thedrop weight method to determine the dynamic surface tension of surface-active agentsolutions have been conducted by Miller, Fainerman, and coworkers (1980s to date).As mentioned earlier, the dynamic mode is usually used to measure the surfacetension of surfactant solution. There is a problem of slowness in attaining theequilibrium and determining the surface age compared to the static mode (Millerand Schano, 1986; Rusanov and Prokhorov, 1996). Detailed discussions on the

Figure 7. Drop formation at different dripping tip sizes.

interpretation of the measured results in comparison with the data obtained usingother methods (the maximum bubble pressure method and the ring method) aregiven by Miller et al. (1997).

It was reported that the curve can be used for surface tension measurement ofviscous liquid (silicone oil) with viscosity up to 60,000 cP and above (Pu and Chen,2001b), although the validation of the results is incomplete without comparisonwith other methods. Furthermore, it was found that the drop detachment processof viscous liquid is different from that of non-viscous liquid (Hauser et al., 1936;Edgerton et al., 1937; Shi et al., 1994). The drop detachment of shear-thinningliquid has also been numerically and experimentally investigated (Davidson andCooper-White, 2003, 2006). It was reported that the drop detachment is influencedby the rheological behavior of the fluid (Davidson and Cooper-White, 2003, 2006).Therefore, the volume of detached drops and residual drop of viscous liquid is defi-nitely different from those liquids studied by Harkins and Brown (1919). As a result,further investigation on the influence of rheological properties of liquids or solutionson the drop weight correction factors is required.

Furthermore, Figure 8 shows that the published correction factors for liquidmetals at r=V 1=3 less than 0.35 exhibit great discrepancy with Wilkinson’s (1972)data. The discrepancy is due to the high density value of liquid metals and the differ-ences in the wetting behavior of liquids at the plane surface of the dripping tip duringdrop detachment (Wilkinson and Aronson, 1973). On the other hand, Figure 8shows that the correction factors of liquid metals from Vinet et al. (1993) andGarandet et al. (1994) are in good agreement with Lohnstein’s data. Therefore, a

Figure 8. Drop weight correction factors at r=V1=3 less than 0.40. Note: the data of Lohnstein(1906) were collected from Vacek and Nekovar (1973). The data of Rayleigh (1896) werecollected from Harkins and Brown (1919). The data of Dunken (1942) were collected fromVacek and Nekovar (1973) and Wilkinson and Aronson (1973).

systematic study is required to further justify the applicability of the current dropweight correction factors at lower r=V 1=3 values.

Effect of Dripping Tip Geometry on Drop Weight Correction Factors

In earlier sections of this review, the effect of dripping tip geometry on the dropweight determination was discussed. It was shown that thick-wall dripping tipsprovide better results than thin-wall dripping tips (Harkins and Brown, 1919). Withreference to Figure 4, more scattered data are shown by Rayleigh (1896; as refer-enced by Harkins and Brown, 1919) for a thin-wall dripping tip. This discussion isvalid for large dripping tips. Gunde et al. (2001) proved that no apparent differenceis exhibited with the use of thin-wall small-size dripping tips on the determination ofHarkins and Brown’s correction factors.

In addition, drop detachment from sharp edge tips and capillary tips is found tobe different, as illustrated in Figure 1 (Hozawa et al., 1981). It was found that theimportant factor governing the maximum stable pendant drop volume for sharpedge tip is Bond number (density and surface tension of the fluid) and for capillarytip it is the advancing contact angle (Hozawa et al., 1981). On the contrary, it wasfound that the residual drop volume is not dependent on the dripping tip geometry(Vacek and Nekovar, 1973). In other words, the geometry of dripping tip plays animportant role in the formation of the maximum pendant drop, but the surfacetension force is responsible for the formation of the residual drop. Therefore, theinfluence of dripping tip geometry on Harkins and Brown’s drop weight correctionfactors requires further investigations.

On the other hand, studies have been conducted to design a small-size sharp-edge tip that allows interfacial tension measurement without any correction factors:

c ¼ VDqg

pD¼ mg

pDð9Þ

Under precisely controlled conditions at low flow rates, using small dripping tipswith 0.025 cm diameter and wall thickness of 0.0005 to 0.002 cm, the drop neckformation problem that is encountered with conventional tips (the thinnest wallthickness is about 0.01 cm) can be eliminated (Hool and Schuchardt, 1992; Xu,1995; Campanelli and Wang, 1997; Terzieva et al., 1999). To date, this approachis applied only to liquid-oil or solution-oil system for interfacial measurement, andit usually involves the introduction of lighter phase into heavy phase (Campanelliand Wang, 1997). It was found that the viscosity of heavy phase has a significanteffect on the detached drop volume (Izard, 1972). Therefore, the use of such a systemfor interfacial tension measurement of viscous liquids requires extra attention.

The Concept of ‘‘Ideal Drop’’

Actually, Harkins and Brown’s drop weight correction factor equation also repre-sents the ratio of the volume of falling drop to the volume of an ‘‘ideal drop.’’ Thus,Equation (7) can be rewritten as

wðr=V 1=3Þ ¼ V

Videalð10Þ

In other words, the ‘‘ideal drop’’ that obeyed Tate’s law would be true only if theouter boundary of the drop is vertical, the liquid joins the tip, compressive forcesdo not exist across the plane, and the entire pendant drop is detached (Edgertonet al., 1937). These ideal conditions are never achieved in actual practice in manysystems (Boucher and Evans, 1975; Boucher and Kent, 1978). It has been reportedthat even when the pendant drop is entirely detached, the weight of the ‘‘ideal drop’’still cannot be attained (Freud and Harkins, 1929; Boucher and Evans, 1975; Gundeet al., 2001; Pu and Chen, 2001b). The reason is that the stability of any system isbroken earlier than its ideal or theoretical case due to fluctuations (Rusanov andProkhorov, 1996). Therefore, Equation (10) is merely empirical and an exact theoryto describe the drop weight method does not exist yet.

However, the ‘‘ideal drop’’ is successfully determined only in two systems withthe use of a small-size sharp-edge tip. They are the formation of a lighter phase drop-let in a heavy phase system (oil-liquid system) and the formation of a gas bubble in aliquid system. The occurrence of ‘‘ideal drop’’ in such systems is because the buoy-ancy force has more pronounced impact than the interfacial tension during the dropbreakup. Hence, no visible neck is formed or residual drop left at the dripping tipafter detachment.

Other Alternative Correlations

Over the years, several other correlations have been proposed as an alternative toHarkins and Brown’s drop weight correction factors to determine surface or interfa-cial tension by using the drop weight method.

Wilkinson (1972) recommended the relationship of r=a and r=V 1=3 be used forsurface tension calculation. Figure 9 shows that these two dimensionless termsexhibit an exclusive feature, that is, for a given liquid there is only one unique valueof r=a for each value of r=V 1=3. Therefore, the surface tension of a liquid can becalculated from Equation (11) after the r=a value was determined from Figure 9.

Figure 9. The relationship between r=a and r=V1=3. The curve was plotted from the data ofHarkins and Brown (1919) and Wilkinson (1972).

c ¼ r2qg

ðr=aÞ2ð11Þ

However, in the use of Equation (11) for surface tension calculation, (r=aÞ2 isinvolved. Therefore, magnification of error could be encountered due to the smalldeviation in the determination of the value of r=a.

Furthermore, in another dimensionless number correlation model based ondimensionless maximum drop volume, Vd , the value of r=V 1=3 is suggested (Babu,1987). With reference to Figure 10, the Vd value can be determined from r=V 1=3

values. Then, the surface tension can be calculated from the following equation:

c ¼ DqgVd

� �2=3

ð12Þ

However, the validity of Equation (12) was limited to r=V1=3 of less than 1.20. Thecorrelation became independent of r=V1=3 when r=V 1=3 was more than 1.20.

Recently, a dimensionless correlation with more theoretical basis analysis isproposed by Yildirim et al. (2005). Based on dimensionless number and computationanalysis by using numerical solution of one-dimensional slender jet equations, asimple correlation is proposed:

BO ¼ 3:60ðr=V1=3Þ2:81 ð13Þ

Figure 11 shows that the experimental data of Harkins and Brown (1919) andWilkinson (1972) are well fitted in Equation (13). Equation (13) is claimed to haveless than 1% error provided that the effect of hydrodynamic is not significant(we �10�7 and Oh ¼ 10) (Yildirim et al., 2005). Due to the limitation of one-dimensional slender jet equations, this correlation was simulated only with liquidthat has a viscosity up to 50 cP. Although the data of liquid with a viscosity of1,500 cP from Wilkinson (1972) were well fitted in the correlation, it is importantto extend the analysis for higher viscosity fluids, as this simple correlation couldbe useful for surface or interfacial tension measurement by the drop weight method.

Figure 10. The relationship between Vd and r=V1=3. The curve was plotted from the data ofHarkins and Brown (1919) and Wilkinson (1972).

Drop Detachment Studies

The drop weight method involves the collection of the combined weight of primaryand satellite drops that detach from a dripping tip for surface or interfacial tensionmeasurement. Figure 12 illustrates the shape evolution of a drop; it changes from asingle mass of fluid into two or more drops during drop detachment. In the past,there have been many studies on drop detachment, but few studies have been directlycarried out toward understanding the theoretical basis of the drop weight method(Freud and Harkins, 1929; Hauser et al., 1936; Edgerton et al., 1937; Paddy and Pitt,1973; Boucher and Evans, 1975; Whitaker, 1976; Tornberg and Lundh, 1981;Rusanov and Prokhorov, 1996; Yildirim et al., 2005).

Since there are speculations on the effect of viscous force on drop detachment,an investigation on drop detachment of viscous liquid at slow flow rate was conduc-ted to confirm applicability of the method (Wilson, 1988). Tate’s law is re-modifiedto take into account viscous forces in the force balance by incorporating one-dimensional theory (Wilson, 1988). Wilson’s model shows that a viscous drop forms

Figure 11. The correlation between Bond number, BO, and r=V1=3. The curve was plottedfrom the data of Harkins and Brown (1919) and Wilkinson (1972).

Figure 12. Drop detachment at low velocity: drop growth (Stage 1), necking (Stage 2), anddetachment (Stage 3).

and grows slowly into a short thread or column; this thread then stretches under itsown weight, and finally it ruptures halfway to form a drop. Although it was foundthat the trend of the theoretical drop size is similar to that of experimental drop size,the theoretical drop size has been underestimated (Weis et al., 1992). However, theresults show that viscous force is involved in drop detachment.

Furthermore, dimensionless numerical analysis is favorable in several theoreticalstudies of the drop weight method (Schulkes, 1994; Yildirim et al., 2005). Based onHarkins and Brown’s (1919) experimental data; it was found that the satellite dropvolume is only a small fraction of the volume of the primary drop for r=V 1=3 lessthan 0.80 (Schulkes, 1994). The satellite drop is less than 2% of the volume of fluidthat breaks away. Conversely, more than 10% of the fluid that breaks away from thenozzle is taken up by the secondary drop for r=V 1=3 greater than 1.20. This explainsthe claim made by Harkins and Brown (1919) that their results are less accurate forr=V 1=3 greater than 1.20.

Generally, understanding of pendant drop formation (Stage 1 in Figure 12) hasbeen well studied and discussed in many reported investigations (Paddy and Pitt,1973; Hartland and Srinivasan, 1974; Boucher and Evans, 1975; Rusanov andProkhorov, 1996). To date, studies on understanding drop necking phenomena(Stage 2 in Figure 12) are still in progress. More investigations are required to deter-mine the principle of the drop weight method.

Conclusions

In general, there is no restriction rule on the type of sample fluid that can be testedby using the drop weight method. As a rule of thumb, proper design of the dropweight apparatus will enable the users to determine accurate and reproducible data.The key factor for a good design is the physical properties of the sample fluid. Exten-sive discussions on the experimental setup of the drop weight method, whichincluded the operation mode, drop number, sample delivery, and dripping tips, werepresented in this review. To further improve the measurements, elimination of theevaporation and hydrodynamic effects is important. Various mathematical modelsof correction factors were proposed by several investigators. Based on the resultof the proposed error analysis, the Lee-Chan-Pogaku model was proposed for theuse of 0:0 < r=V1=3 < 1:20 and HG equation 2 for 1:20 < r=V 1=3 < 1:60. The advan-tage of using these models is to eliminate the difficulty of making comparisons ofdata among different investigations. Although the use of the correction factors formany fluids has been reported, precautions should be taken when using the correctionfactors for viscous fluids. The validation of the correction factors at low r=V 1=3 rangewould require further investigations. More investigations are needed to form a rigidtheory of the drop weight method despite the fact that it has been used as a standardprocedure to measure surface and interfacial tension for more than a century.

Acknowledgments

The authors would like to thank the Ministry of Higher Education, Malaysia(Fundamental Research Grant Scheme: FRG 0075-TK-1=2006), for providing thefinancial support for this study. The authors also wish to express their gratitude tothe colleagues and collaborators who have contributed to the development of this study.

Nomenclature

a capillary constant, a ¼ ð2c=DqgÞ1=2

BO Bond number, BO ¼ r2qg=cD diameter of dripping tip, cmg gravitational force, 981 cm=s2

L length of dripping tip, cmm mass of falling drop, gOh Ohnesorge number, Oh ¼ g=ðqrcÞ1=2

r radius of dripping tip, cmf(r=a) drop weight correction factor in function of (r=a)wðr=V=1=3Þ drop weight correction factor in function of ðr=V=1=3ÞRe Reynolds number, Re ¼ quD=gt drop formation time, su velocity, cm=sV volume of detached drop, cm3

Vd dimensionless maximum volume, Vd ¼ vðDqg=cÞ2=3

Videal volume of detached drop that obeys Tate’s law (ideal drop volume)We Weber number, We ¼ qQ2=p2r3c

Greek letters

c surface=interfacial tension, mN=mg viscosity, cPn kinematic viscosity, mm2=sDq density difference between two immiscible phasesq density, g=mL

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A CRITICAL REVIEW: SURFACE AND INTERFACIAL TENSION MEASUREMENT BY THE DROP WEIGHT METHOD

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