Measurement 45 (2012) 317–324

Contents lists available at SciVerse ScienceDirect

Measurement

journal homepage: www.elsevier .com/ locate/measurement

An easy way to measure surface free energy by drop shape analysis

Luca Mazzola ⇑, Edoardo Bemporad, Fabio CarassitiMechanical and Industrial Engineering Department, University ‘‘Roma Tre’’, Via della Vasca Navale 79, 00146 Rome, Italy

a r t i c l e i n f o

Article history:Received 14 July 2011Received in revised form 4 November 2011Accepted 22 November 2011Available online 1 December 2011

Keywords:Surface free energyWettabilityTadmor equationAdhesion phenomenon

0263-2241/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.measurement.2011.11.016

⇑ Corresponding author. Tel.: +39 0657333496; faE-mail address: luca.mazzola@stm.uniroma3.it (

a b s t r a c t

The drop shape analysis is a conventional method to evaluate wettability and surface freeenergy. The analysis of drop profile is commonly used for the contact angle measurementbetween the liquid drop and sample surface.

In many cases such as confined surfaces, it is difficult to carry out this measure, thereforeit is necessary to evaluate the shape of the drop from top view point.

The aim of this work was to develop a procedure in order to measure wettability and sur-face free energy values by means of the analysis of the drop shape from top view point.Starting from Tadmor equation, a polynomial function was obtained to correlate the dropdiameter to its contact angle on a flat horizontal sample surface.

In addition, analyses were carried out in order to study the shape of liquid drop varyingthe tilt of sample surface. In this case was shown a correlation between the tilt grade andthe maximum characteristic lengths (parallel and orthogonal to pitch) of a drop shape on atilt surface sample.

Using the maximum characteristic lengths and a polynomial function previouslyobtained, it is possible to establish values of pitch, surface free energy and wettability ofsample surface.

In order to confirm the developed procedure, tests on three different materials have beenperformed obtaining these three performance indexes.

All results obtained by means of this alternative analysis have been compared and cor-roborated by the classical method using the contact angles of the drop analyzed from a lat-eral view.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. State of the art

The recent development of self-cleaning surfaces,anti-bacterial surfaces, biocompatible or decorative coat-ings and engineering of desired nano- and micro-patternsin thin films increases the importance of studying theirchemical and physical properties such as wettability andsurface free energy [1–6].

Various methodologies are available to measure wetta-bility and surface free energy as outlined in Refs. [7,8].

. All rights reserved.

x: +39 0677333256.L. Mazzola).

These methods are considered to be most powerful, be-cause of their accuracy, easy to use, and versatility [9,10].

Contact angle of liquid drop applied on sample surface,represents an important parameter in adhesion science. Itprovides consistent valuable information about surface en-ergy, hydrophobicity, roughness and chemical heterogene-ity [11].

However, in such cases where a surface is rough and/orheterogeneous or where the contact angle to be measuredis low, most techniques that use a meridian drop profileface have some difficulties [12,13]. In such cases, knowingthe volume, the diameter of sessile drop and its surfacetension [14–16], the contact angles measurements ob-tained by the analyses of the drop from top view pointcan be used with great advantage.

Fig. 2. Tadmor trend taking into account the liquid drop of 6 ll.

318 L. Mazzola et al. / Measurement 45 (2012) 317–324

Bikerman founded, discovered more than 60 years ago[17], that for water droplets below a certain size, the con-tact angle could be found by measuring simply the dropletdiameter, if the drop volume was known [18].

Later Tadmor [19] obtained a more accurate relations(Eq. (1) and (2) between the contact angle of the dropand its diameter, knowing a volume on a flat horizontalsurface sample, by means of geometrical consideration.Obviously this simplified approach, similarly to Bikerman,would only be applicable to small drops and/or very largeliquid surface tension because, only under these circum-stances, the effects of gravity might be neglected [20].

In case of contact angle smaller or greater than 90�, twodifferent relations were obtained:

For h < 90�

d ¼ 2ffiffiffiffiV3p

sin hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip3 ½2� 3 cosðhÞ þ ðcosððhÞÞ3�3

q ð1Þ

While for h > 90�

d ¼ 2ffiffiffiffiV3p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip3 ð2þ 3 cosð180� hÞ � ðcosð180� hÞÞ3Þ3

q ð2Þ

It is highlighted that for angles h < 90� the measureddiameter is the real one; this diameter is named ‘‘basediameter’’. On the contrary in the case of h > 90�, the mea-sured diameter from top view point represents the maxi-mum diameter of the drop, named ‘‘apparent diameter’’,which has not been measured on the contact between li-quid and solid. The ‘‘apparent diameter’’ is larger thanthe ‘‘base diameter’’ as shown in Fig. 1.

Plotting the Tadmor’s equations in the same graph, it ispossible to see that the relation between drop diameterand contact angle has a hyperbolic trend (Fig. 2).

The main limits of these equations are that they cannotbe inverted using maths, therefore it is not possible to ob-tain the contact angle values from measured diameters.

In addition, it is known that the technique of drop analy-sis from top view point, using Tadmor equations, is expectedto give an accurate result only when axisymmetric approx-imation for the drop shape can be used. Therefore a horizon-tal and flat surface must be taken into account, in fact, this

Fig. 1. Possible configuration of the liquid drop on a surface (a). Exp

method cannot be used in the case of a not completely hor-izontal surface [19].

However, on such surfaces the drops are not axisym-metric anymore and the contact angle varies along thecontact line [21].

The purpose of this work is to identify the equation thathas a trend similar to that of Tadmor equations, where,knowing the diameter, it is possible to obtain the contactangle. Furthermore, using a measured diameter value, aprocedure has been developed in order to make usablethe revised Tadmor equation in the case of tilted surface.

The new algorithm has been tested on different materi-als; the outcomes of the surface free energy and wettabilityobtained by these experiments have been compared withthe values obtained using a classical method of contact an-gle measurement.

2. Materials and methods

2.1. Materials

In this work, in order to show the high efficiency of thenew procedure to measure surface free energy, fourdifferent materials were used. In order to confirm a largefield of application, we chose to use three types of polymers:a commercial plastic such as Polycarbonate (PC) Lexan, apure poly(tetrafluoroethylene) (PTFE) and poly(methyl-methacrylate) (PMMA) supplied by GoodFellow and onemetalloid material, Silicon (Si).

lanation of the apparent diameter and the base diameter (b).

L. Mazzola et al. / Measurement 45 (2012) 317–324 319

All samples were cleaned in ultrasonic bath (ethanolsolution) for 5 min.

2.2. Wettability and surface free energy calculation

Wettability and surface free energy of the materialshave been investigated using a contact angle meter devel-oped in laboratories of Roma Tre University. This instru-ment was developed according to UNI EN 828, UNI 9752,ASTMD-5725-99.

The liquids used to measure surface free energy in thiswork were methylene iodide, water and formamide. Tendrops (volume of 6 ll) of each liquid were deposited onthe samples surface. These measurements should be donerepeatedly in order to determine a mean arithmetic value.The contact angle measurements were performed in thestandard conditions of room temperature and humidity(22 �C and 30% respectively).

Images of the drop were elaborated with an Analysissoftware (Soft Imaging System). All images acquired wereoptimized and binarized before the contact angle anddiameter were measured in order to improve the accuracyand the reliability of the measures.

The surface free energy was calculated with the Girifal-co model [22] and with the more complex model ofOwens–Wendt [23–25].

The first model provides the value of surface free energyusing only one test liquid (in this case iodide methylene),according to:

csl ¼ cs þ cl � 2 �ffiffiffiffiffiffiffiffiffiffiffiffics � cl

pð3Þ

where csl is the interfacial energy (known), cs the surfaceenergy of solid (unknown), and cl is the surface energy ofliquid (known).

Considering Young’s equation:

csl ¼ cs � cl � cos h ð4Þ

where h represents the measure of contact angle.Combining the Eqs. (3) and (4), the surface free energy

value is obtained from:

cs ¼cl

4

� �� ð1þ 2 � cos hÞ2 ð5Þ

Note that this method can be used only to evaluate thesurface free energy of materials purely dispersive in nat-ure, because a dispersion liquid (iodide methylene) is used.However it is known that the total surface free energy ofmaterials is due to the dispersive component of surface en-ergy for the most part, therefore this method can be con-sidered suitable to know the order of magnitude of totalsurface free energy.

To have the exact value of surface free energy (sum ofdispersion and polar components), the Owens–Wendtmethod was used. The Owens–Wendt approach is one ofthe most commonly method used for calculating the sur-face free energy of the materials [23,26]. The principalassumption of the OW method is that the surface free en-ergy is a sum of the two components: dispersion and polarcomponents [27]. The liquid used for this model were twopolar liquids (water and formamide) and one non polar(methylene iodide).

The Owens–Wendt model is represented by the geo-metric mean relationship:

12ð1þ cos hÞcL ¼ ðcD

s � cDl Þ

12 þ ðcP

S � cpl Þ

12 ð6Þ

where h is the contact angle between the liquid droplet andsurface, cL the liquid total surface tension, cD

l the dispersioncomponent of liquid surface tension, and cP

l is the polarcomponent of liquid surface tension.

The unknown terms in this equation are:

cDS = dispersion component of the solid surface free

energy.cP

S = polar component of the solid surface free energy.

The value of total surface free energy of the solid is ob-tained using the following equation:

cS ¼ cDS þ cP

S ð7Þ

To solve Eq. (6), dividing by ðcpl Þ

12, it can be rewritten as

[28]:

12 ð1þ cos hÞcL

ðcPl Þ

12

¼ cDl

cPl

� �12

� ðcDS Þ

12 þ ðcP

S Þ12 ð8Þ

The left-hand side of Eq. (8) contains quantities whichare measured experimentally (h) or which are availablein the literature ðcP

l ; clÞ, so that a plot of the left-hand side

of Eq. (8) versus quantity ðcPL

clÞ

12 gives a straight line with

slope ðcDS Þ

12 and intercept ðcP

S Þ12.

For the calculations, the surface tension and its compo-nents of the liquids are taken from literature [29–31].

3. Results

At the beginning, in order to assess if the Tadmor equa-tion fits with the real trend of the experimental diameter–angle correlation, a several measures of both parameterswere done; Fig. 3 shows a good overlap, therefore since itis necessary to know the contact angle value from mea-sured diameter and considering that the Tadmor equationsare not inverted, a new function was realized.

The new function is a polynomial curve obtained bymeans of ‘‘Table Curve 2D’’ software. It fits accurately(r2 = 0.99997) both the Tadmor equations (Fig. 4).

The polynomial function has the subsequent expression:

Y ¼ ðaþ cx0;5 þ exþ gx1;5Þð1þ bx0;5 þ dxþ fx1;5 þ hx2Þ

ð9Þ

where y represent a diameter value whereas x the contactangle; the coefficients of polynomial are: a = 76.043548,b = 4.2148674, c = �0.10433018, d = �0.58916401, e = �1.7477936, f = 0.0050410895, g = 0.11302929, h =0.0015744692

Using the same software, this polynomial equation canbe inverted in order to obtain the contact angle from mea-sured diameters. The same function was obtained but withdifferent coefficients:

Fig. 3. Fitting between experimental data (drop diameter–contact angle) and Tadmor curve.

Fig. 4. Fitting between polynomial function and Tadmor equation.

320 L. Mazzola et al. / Measurement 45 (2012) 317–324

X ¼ ðaþ cy0;5 þ eyþ gy1;5Þð1þ by0;5 þ dyþ fy1;5 þ hy2Þ

ð10Þ

a = 8.4141334, b = �2.4575302, c = �14.330117, d =2.2830703, e = 7.4069892, f = 0.95486294, g = � 1.061563,h = 0.15250538

In order to estimate the reliability of this equation,several tests were performed; in particular using this in-verted function, it was possible to obtain the contact anglefrom measured diameter. These results were comparedwith a classical method of contact angle measurementsby means of the analysis of the drop profile. The results

are summarized in the Table 1 where it is highlighted thatthe difference percentage is under 5.5%, therefore this newfunction, ‘‘derived to Tadmor’’, can be applied to obtain areliable measure of contact angle.

A further combined analysis, using two methods, wasdone to evaluate wettability and surface free energy onthree different materials: polycarbonate, poly(methyl-methacrylate) and silicon.

As shown in Table 2, a good correspondence of wettabil-ity and surface free energy (by using the two methods) wasobtained, in addition the data are in agreement with liter-ature [23,24].

Table 1Comparison between water contact angle measured by the classicalmethod (by analyzing the drop profile) and the new method using theequation derived to Tadmor. In order to have a wide range of contact angle,it was changed the roughness of the four materials using sandpapers.

Contact angle(lateral view) (�)

Equation derived to Tadmor(from on high view) (�)

Differencepercentage (%)

26.56 26.25 1.1727.61 26.68 3.3342.04 42.69 1.5564.13 64.11 0.1575.98 78.06 2.6186.41 83.61 3.24105.38 99.78 5.31108.91 114.34 4.99133.43 131.88 1.16140.67 144.74 2.89

Table 2Surface free energy (measured using iodide methylene) and wettability(measured using water) obtained by both, classical and new method.

Materials Properties Lateralview

From on highview ‘‘derivedto Tadmor’’

Percentagedifference (%)

PC Wettability (�) 72.45 71.32 1.55SFE (mJ/m2) –Girifalco

43.65 44.86 2.68

SFE (mJ/m2) – OW 46.31 44.95 2.93PMMA Wettability (�) 58.67 59.43 1.29

SFE (mJ/m2) –Girifalco

44.46 44.02 0.98

SFE (mJ/m2) – OW 48.08 46.86 2.3Si Wettability (�) 64.29 65.28 1.55

SFE (mJ/m2) –Girifalco

40.46 41.02 1.38

SFE (mJ/m2) – OW 42.08 42.86 1.85

Fig. 5. Instrument configuration to tilt the angle (a) of surface sample.

L. Mazzola et al. / Measurement 45 (2012) 317–324 321

From Table 2 it is evident that the difference betweenthe wettability calculated by the top view point and thosemeasured by lateral view for the three materials is under1.6%. On the contrary the difference percentage of surfacefree energy according to the two model (Girifalco andOwens–Wendt) is under 3%. Therefore, it was demon-strated the reliability and strength of the new methodol-ogy of analysis from the top view point of the drop.

Therefore, this new equation (derived from Tadmorequations) represents a safe and new method to obtainthe contact angle measurements from the analysis of liquiddrop from on top view point.

However this method shows a few sensitiveness forhydrophobic or super-hydrophobic surfaces, in fact inthese cases for small variation of the diameter, very differ-ent values of contact angles are obtained (as shown inFig. 4).

A further limitation of this measurement method, is dueto the perfect horizontality of the sample surface. Indeed, itcould be difficult to take horizontally the sample surface,therefore it is necessary to identify a procedure which per-mits to overcome this limitation.

In order to simulate different slope of surfaces, the flatsample surface was applied on the sample holder and thewhole device was tilted; the tilt grade was measured by

digital bevel (Fig. 5). Captures of drops are done both usinga vertical camera, in order to see the shape of drops fromtop view point, and lateral camera, in order to see theadvancing and receding angles (drop profile).

Analyses performed on tilted surface are addressed toidentify a procedure to evaluate both the tilt of surfaceand its surface free energy, since the shape of the drop isshown on top view point.

Tests were carried out using a PTFE sample and initiallythe liquid used was a iodide methylene. The images werecaptured both laterally and from above, in order to mea-sure the advancing and receding contact angle. It is possi-ble to see from Fig. 6 and from Table 3 that increasing tiltgrade, the lateral view of drop shape becomes asymmetric.The tilting device method is used to determine the advanc-ing and the receding contact angle; it is clear that, increas-ing the tilt grade, the hysteresis increase, however themean value of contact angle is the same.

From Fig. 6 it is possible to see that increasing tilt grade,the drop become stretched along the descent plane and itshrinks orthogonally on the same plane. In addition itwas proven that there are two characteristic lengths ofthe drop shape that change with tilt grade; these lengthsare represented as the maximum lengths of the drop alongtwo orthogonal directions (Fig. 7).

The shape factor can be identified as:

M ¼ ba

ð11Þ

The Eq. (11) represents the aspect ratio of the drop tilt-ing the surface; the value of the shape factor M is equal orminor to one unit (for M = 1 the drop is on a horizontal flatsurface, therefore it is spherical and symmetrical). Experi-mental data (Fig. 8) show that there is a linear correlationbetween the shape factor M and the tilt grade. From theseresults, it is possible to carry out a linear regression thatfits the experimental data with the r2 = 0.98.

The equation which describes this regression is:

Y ¼ �0:0142 � xþ 1:0257 ð12Þ

where y represents the M factor and x represents a tiltgrade; therefore knowing the shape factor M, it is possible

Fig. 6. Different shapes of the liquid drops obtained varying the tilt angle from lateral view and from top point of view.

Table 3Data of advancing and receding angle obtained varying the tilt grade.

Tiltangle(�)

Advancingcontactangle (�)

Recedingcontactangle (�)

Hysteresis(�)

Mean valuebetweenadvancing andrecedingcontact angles(�)

0 64.63 ± 2.03 64.63 ± 1.75 0 64.63 ± 1.755 70.78 ± 3.53 53.34 ± 3.09 17.44 62.06 ± 4.61

10 71.31 ± 1.89 51.78 ± 3.42 19.53 61.55 ± 1.5713 76.75 ± 2.62 46.11 ± 1.97 30.64 61.43 ± 2.8815 77.94 ± 3.28 44.09 ± 1.35 33.85 61.01 ± 1.8518 79.48 ± 2.78 42.41 ± 1.60 37.08 60.94 ± 3.6320 79.73 ± 3.65 37.89 ± 0.98 41.84 58.81 ± 2.0222 83.03 ± 2.32 35.07 ± 4.23 47.96 59.05 ± 2.3925 87.95 ± 2.85 35.18 ± 1.98 52.76 61.56 ± 1.31

Fig. 7. The maximum lengths of drop along two orthogonal directions.The factor M is obtained as the ratio b/a.

Fig. 8. Correlation between shape factor and tilt angle by using themethylene iodide.

322 L. Mazzola et al. / Measurement 45 (2012) 317–324

to obtain the tilt grade (degree) of surface by the followingequation:

a ¼ 72:23� 70:42 �M ð13Þ

To check the effectiveness of this equation a correlationbetween factor M and tilt grade was calculated on two dif-ferent sample surfaces. The results show a good agreementof the data, so the equation can be used to identify if thesample surface is horizontal or has a tilt grade.

However it is highlighted that this equation it is reliableonly for one liquid (the iodide methylene) on all types ofmaterials. Therefore each liquid has the specific equationto determine the tilt grade of the surface.

Since the surface free energy was measured both withGirifalco method and with Owens Wendt method, othertwo liquids (water and formamide) are used, besidesiodide methylene, in order to determine the equation tocalculate the tilt grade. In this way it is possible to deter-mine the tilt grade (degree) of any materials by usingone of these three liquids. In the following system, are re-ported the equations of each liquid.

a ¼ 72;23� 70;42 �M ðmethylene iodideÞa ¼ 169:49� 141:44 �M ðwaterÞa ¼ 140:85� 141:48 �M ðformamideÞ

8><>: ð14Þ

It is evident that the two equations regarding water andformamide are similar, in particular the slope of thestraight line is the same and the intercept is quite different.The same behavior of the two liquid trends probably is dueto the fact that they are polar, in fact the first equation con-cerning iodide methylene (which is a dispersive liquid) iscompletely different.

The next step was to calculate the surface free energy,knowing the shape of the drop; from the two maximumlengths of the drop (Fig. 7) it is possible to consider anequivalent diameter (De) as if the drop is axisymmetric,therefore it can be identified by means of only one diame-ter. The relation is:

Fig. 9. Evaluation of corrective factor. The red point represent a diameterof horizontal surface sample.

Fig. 10. The flow diagram for the new algorithm of the non-axisymmetricdrop shape analysis.

L. Mazzola et al. / Measurement 45 (2012) 317–324 323

De ¼ ðaþ bÞ2

ð15Þ

Nevertheless tests (Fig. 9) showed that there is an offsetbetween the real diameter of a horizontal surface and anequivalent diameters obtained analyzing sample with dif-ferent tilt grade. This discrepancy is likely due to simplegeometry.

Therefore, considering the difference between the meanvalue of equivalent diameters and the mean value of thereal diameters, it is possible to obtain a corrective factorL = 0.4263; consequently the equivalent diameter cor-rected is:

Dec ¼ De � L ð16Þ

The value of corrected equivalent diameter can be re-placed in the Eq. (10) in order to calculate the surface freeenergy.

Note that the corrective factor is the same changing thematerials, the liquid and the volume of the drop (in partic-ular volumes from 2 to 6 ll were investigated).

Subsequently, experimental tests were performed in or-der to compare results obtained on both horizontal samplesurface and tilted, using the procedure of non-axisymmet-ric shape analysis, developed in this work. These results arecompared with those obtained by means of classical meth-ods (analysis of drop profile) measuring contact angle onhorizontal surface. Results (reported in Table 4) are thesame for the two techniques and they allow us to affirmthat the new procedure is correct, reliable and sensitive.

Table 4Comparison of surface free energy results, measured using both newalgorithm and classical method.

Material Tiltangle(�)

SFE (lateralview) (caseof tiltedsurface) (mJ/m2)

SFE (from onhigh view)(case of tiltedsurface) (mJ/m2)

SFE (lateralview) (case ofhorizontalsurface) (mJ/m2)

PC 5 43.485 44.679 44.52510 43.711 45.339

Si 5 42.842 38.345 41.85310 41.339 40.804

PTFE 5 26.839 24.316 25.93210 26.555 25.326

4. Conclusions

The classical method to measure wettability and surfacefree energy by contact angle measurement involves somelimitations, in fact in the case of a large or confined surface,it is impossible to measure the contact angle. In additionsample surfaces are not always horizontal, therefore it isnecessary to identify a simple and quick algorithm, to mea-sure these two performance indexes (wettability and sur-face free energy).

A way back, different studies were carried out in orderto solve these problems. In particular Tadmor identifiedtwo equations in order to correlate the contact angle witha drop diameter.

This work confirms experimentally the effectiveness ofTadmor equation, however this equation does not permitto obtain the contact angle of the drop out of its diameterbecause of the difficulties to invert this equation mathe-matically. However, a polynomial, which has the sametrend of Tadmor curves, was developed. Since this equationcan be inverted to obtain the contact angle, we demon-strated that it is possible to measure wettability and sur-face free energy analyzing the drop shape from on topview point.

Further result obtained in this work consists in theapplication of this equation in the case of tilted surface.For this purpose several tests with different tilt grade werecarried out. A shape factor, defined as a ratio between thetwo maximum dimensions of the shape drop, was identi-fied. A linear correlation between the shape factor and tiltgrade was achieved using a iodide methylene, water andformamide.

The shape factor is shown to be an useful measure in or-der to analyze the drop applied on surface sample that isnot totally horizontal.

Using a polynomial function and a shape factor, a stan-dard procedure was developed.

A new developed algorithm (Fig. 10), based on analysis ofthe shape of non-axisymmetric drop is shown to be very use-ful not only to evaluate the tilt grade of the surface but also todetermine the wettability and surface free energy of sessile

324 L. Mazzola et al. / Measurement 45 (2012) 317–324

drop. The success of the new algorithm is confirmed by therepeatability and reliability of the experimental results.

References

[1] H. Gau, S. Herminghaus, P. Lenz, R. Lipowsky, Liquid morphologieson structured surfaces: from microchannels to microchips, Science283 (1999) 46–49.

[2] D.E. Kataoka, S.M. Troyan, Patterning liquid flow on the microscopicscale, Nature 402 (1999) 794–797.

[3] C.S. Chen, M. Mrksich, S. Huang, G.M. Whitesides, D.E. Ingber,Geometric control of cell life and death, Science 276 (1997) 1425.

[4] E. Delamarche, A. Bernard, H. Schmidt, B. Michel, H.A. Biebuyck,Patterned delivery of immunoglobulinus to surfaces usingmicrofluidic networks, Science 276 (1997) 779.

[5] M.A. Rodriguez-Valverde, M.A. Cabrerizo-Vilchez, P. Rosales-Lopez,A. Paez-Duenas, R. Hidalgo-Alvarez, Contact angle measurements ontwo (wood and stone) non-ideal surfaces, Colloids and Surf. A:Physicochem. Eng. Aspects 206 (2002) 485–495.

[6] M. Brugnara, C. Della Volpe, S. Siboni, D. Zeni, Contact angle analysison polymethylmethacrylate and commercial wax by using anenvironmental scanning electron microscope, Scanning 28 (2006)267–273.

[7] J.K. Spelt, E.I. Vargha-Butler, Contact angle and liquid surface tensionmeasurements, in: A.W. Neumann, J.K. Spelt (Eds.), Applied SurfaceThermodynamics, Marcel Dekker, New York, 1996, p. 379.

[8] A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, sixth ed.,John Wiley & Sons, New York, 1997.

[9] O.I. del Rio, A.W. Neumann, Axisymmetric drop shape analysis:computational methods for the measurement of interfacialproperties from the shape and dimensions of pendant and sessiledrops, J. Colloid Interface Sci. 196 (1997) 136–147.

[10] D. Li, P. Cheng, A.W. Neumann, Contact Angle Measurement byAxisymmetric Drop Shape Analysis (ADSA)’’, Adv. Colloid InterfaceSci. 39 (1992) 347–382.

[11] H. Tavana, C.N.C. Lam, K. Grundke, P. Friedel, D.Y. Kwok, D.Y. Hair,A.W. Neumann, Contact angle measurements with liquids consistingof bulky molecules, J. Colloid Interface Sci. 279 (2004) 493–502.

[12] F.K. Skinner, Y. Rotenberg, A.W. Neumann, Contact anglemeasurements from the contact diameter of sessile drops bymeans of a modified axisymmetric drop shape analysis, J. ColloidInterface Sci. 130 (1989) 25–35.

[13] E. Moy, P. Cheng, Z. Policova, S. Treppo, D.Y. Kwok, D.R. Mack, P.M.Sherman, A.W. Neumann, Measurement of contact angles from themaximum diameter of non-wetting drops by means of a modifiedaxisymmetric drop shape analysis, Colloids Surf. 58 (1991) 215–227.

[14] S. Lahooti, O.I. del Rio, A.W. Neumann, P. Cheng, Axisymmetric dropshape analysis (ADSA), in: A.W. Neumann, J.K. Spelt (Eds.), AppliedSurface Thermodynamics, Marcel Dekker, New York, 1996. pp. 379.

[15] J. Graham-Eagle, S. Pennell, Contact angle calculations from thecontact/maximum diameter of sessile drops, Int. J. Numer. Method.Fluids 32 (2000) 851–861.

[16] A.S. Dimitrov, P.A. Kralchevsky, A.D. Nikolov, H. Noshi, M.Matsumoto, Contact angle measurements with sessile drops andbubbles, J. Colloid Interface Sci. 145 (1991) 279–282.

[17] J.J. Bikerman, Electrokinetic equations and surface conductance: asurvey of the diffusive double layer theory of colloidal solutions,Trans. Faraday Soc. 36 (1940) 154.

[18] J.J. Bikerman, A method of measuring contact angles, Ind. Eng. Chem.13 (1941) 443–444.

[19] R. Tadmor, Line energy and the relation between advancing,receding and Young contact angles, Langmuir 20 (2004) 7659–7664.

[20] E. Moy, A.W. Neumann, Prediction of particle behaviour atsolidification fronts by Lifshitz Theory: deficiencies of the surfacetension components approach for estimating interfacial tensions,Colloids Surf. 43 (1990) 151–167.

[21] S. Iliev, N. Pesheva, Nonaxisymmetric drop shape analysis and itsapplication for determination of the local contact angles, J. ColloidInterface Sci. 301 (2006) 677–684.

[22] M. Zenkiewicz, Methods for the calculation of surface free energy ofsolids, J. Achiev. Mater. Manuf. Eng. 24 (2007) 137–145.

[23] D.K. Owens, R.C. Wendt, J. Appl. Polym. Sci. 13 (1969) 1741.[24] M. Zenkiewicz, Polymer Test. 26 (2007) 14–19.[25] L. Cerne, B. Simoncic, Textile Res. J. 75 (2004) 426–432.[26] A. Rudawska, E. Jacniacka, Inter. J. Adhes. Adhes. 29 (2009) 451–457.[27] L. Ponsonnet, K. Reybier, N. Jeffrzic, V. Comte, C. Lagneau, M. Lissac,

C. Martelet, Mater. Sci. Eng. C23 (2003) 551–560.[28] L. Boulangè-Pettermann, J. Joud, B. Baradoux, Contact Angle, in: K.L.

Mittal (Ed.), Wettability and Adhesion, vol. 5, VSP/Brill, Leiden, 2008,pp. 139–151.

[29] L. Holysz, E. Chibowski, K. Terpilowsi, Contact Angle, in: K.L. Mittal(Ed.), Wettability and Adhesion, vol. 5, VSP/Brill, Leiden, 2008, pp.95–111.

[30] C. Della Volpe, S. Siboni, J. Colloid Interface Sci. 195 (1997) 121–136.[31] E. McCafferty, J. Adhesion Sci. Technol. 16 (2002) 239–255.