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Comparative study on the surface free energy of a solid calculated by different methods
ARTICLE IN PRESS
POLYMER TESTING Polymer Testing 26 (2007) 14–19 www.elsevier.com/locate/polytest
Test Method
Comparative study on the surface free energy of a solid calculated by different methods Marian Z˙ enkiewicz Kazimierz Wielki University, ul. Chodkiewicza 30, Bydgoszcz 85-064, Poland Received 15 June 2006; accepted 4 August 2006
Abstract This paper presents analysis of differences in the surface free energy (SFE) values of a solid, calculated using the methods of Owens–Wendt (OW), van Oss–Chaudhury–Good (vOCG), and Neumann with three measuring liquids: water, formamide, and diiodomethane. The concept of the analysis has been based on the differences in SFE, which occur objectively and regardless of both the precision and the performing conditions of the contact angle (CA) measurements. These differences result from utilizing different mathematical relations between CA and SFE for each of the methods. The results obtained with these three methods are compared with one another over the SFE range common for polymers (20–50 mJ/m2). It is concluded that the relative difference in SFE between the results from the Neumann and vOCG (or OW) methods can reach 21%, while that between the results from the vOCG and OW methods is considerably lower and does not exceed 3%. r 2006 Elsevier Ltd. All rights reserved. Keywords: Polymeric materials; Surface free energy; Contact angles; Methods of calculation
1. Introduction Adhesion is one of the basic physical phenomena appearing in various fields of technology and is significant for many industrial processes. It plays an important role in gluing, printing and coating of polymeric materials. In order to obtain high strength of adhesive joints, the substrate surface has to be fully wetted by the glue, ink, or coating. Different techniques for surface layer (SL) modification of polymeric materials are commonly applied to improve the wettability and, thus, adhesion [1–3]. The physical values such as a contact angle (CA) and surface free energy (SFE) are used to charTel.: +48 52 3419 208; fax: +48 56 3414 773.
E-mail address: [email protected]. 0142-9418/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2006.08.005
acterize and predict adhesive properties of the polymeric materials. Therefore measurements of CA of sessile drops deposited on surfaces of various materials, and different methods of calculation of SFE are powerful analysis tools widely used in many industrial sectors and for many practical applications, especially for coating, gluing and printing. It is evident that if the SFE of polymeric material is raised by any kind of surface treatment to a higher level and wettability is improved, then adhesion properties can be improved also. For these reasons, the methods used for the SFE determination are intensively investigated [4–9]. The Young and Laplace equations, well known for 200 years, constitute principles for the determination of SFE of solids. However, in the second half of the 20th century, rapid progress in the interface
ARTICLE IN PRESS M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
science and wettability processes occurred. As a result, new calculation methods for the SFE determination were elaborated, which are very important for polymer surface investigations. These methods are formulated on the basis of a previously measured contact angle or of the drop-shape analysis for various liquids [2,4,5]. While measuring CA for the SFE determination, many technical restrictions and serious scientific dilemmas occur. Some of these concern all the methods for the SFE determination and include such problems as thermodynamic equilibrium, metastable state of the shape and CA of the examined drop, physical interpretation of advancing and receding contact angles, and origin of differences in the SFE values for various liquids applied and different methods used. These problems point out directions of further interface research of polymeric materials [6–9]. The main methods for the determination of SFE are those of Zisman [10], Owens–Wendt (OW, known also as the Kaelble method) [11], and a relatively new method of van Oss–Chaudhury– Good (vOCG) [12,13]. The last one is a subject of interest for many researchers and heated scientific discussions. The main issue of these discussions concerns determination of acid and base components of SFE of measuring liquids, defining acid and base levels of SL of various polymers or polymeric materials, and rules for appropriate choice of measuring liquids. A method based on the so-called state equation which was modified to the Neumann equation [14,15] is a subject of a detailed analysis in the literature. One of the controversies dealing with this equation is due to the nature of the b constant (see Section 2.3) which is, for some authors, a universal material constant, whereas for others, it is only an equation parameter [8]. There is also a new, very interesting proposition for the SFE determination which is based on measuring the advancing and receding contact angles with use of one liquid only [16]. The most often applied method for polymeric materials so far is the OW method, in which water and diiodomethane are used. A growing interest in the vOCG method, based on the results of CA measurements performed with use of three appropriately selected liquids, is an inspiration for new investigations in this field. These investigations include analysis of parameters of measuring liquids used to measure CA and construction of a set of
15
mathematical equations utilized to determine SFE of investigated materials [17–19]. A similar study has also been performed while using the Neumann method [8]. This study is a continuation of previous works of the author [20,21]. Its aim was to analyze differences in the values of SFE of a solid, calculated by means of the OW, vOCG, and Neumann methods with use of commonly applied measuring liquids such as water, formamide, and diiodomethane. The analysis was made in a new way, consisting of separation and determination of some differences between the SFE values calculated with these three methods. These differences occur objectively and regardless of the precision and performing conditions of the CA measurements and result from different mathematical formulae valid for each method. 2. Characteristics of contact-angle measuring methods 2.1. Method of OW The principal assumption of the OW method is that SFE is a sum of two components: dispersion (SFED) and polar (SFEP) [11]. SFED reflects the dispersion interaction occurring on an interface and SFEP is a sum of polar, hydrogen, inductive and acid–base interactions. The analysis of this method was carried out using the following notation:
W (water, a polar liquid), F (formamide, a polar liquid), and D (diiodomethane, a dispersion liquid) for measuring liquids used in the CA measurements; gS (SFE), gSd (SFED), and gSp (SFEP) for SFE and its components of a solid, respectively; gL (SFE), gLd (SFED), and gLp (SFEP) for SFE and its components of a measuring liquid, respectively (L refers to W, F or D); YL for a contact angle (L means the same as above).
Then, the set of equations in the OW method, with water and diiodomethane used as measuring liquids, is of the form ðgSd gWd Þ0:5 þ ðgSp gWp Þ0:5 ¼ 0:5gW ð1 þ cos YW Þ,
(1)
ðgSd gDd Þ0:5 þ ðgSp gDp Þ0:5 ¼ 0:5gD ð1 þ cos YD Þ,
(2)
gS ¼ gSd gSp .
(3)
ARTICLE IN PRESS 16
M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
The gS value of an investigated material is calculated as follows. First, gW, gWd, gWp, gD, gDd, and gDp are to be determined. These values can be measured by one of the commonly used methods [2,4] or just be accepted from the literature. The next step is to measure CA, using a set of measuring liquids. These measurements should be done repeatedly (more than 7 times) in order to determine a mean arithmetic value and its standard deviation. The values of gSd and gSp are calculated from Eqs. (1) and (2) and gS, from Eq. (3). To measure CA, a set of two measuring liquids (dispersion and polar) has to be applied. Thus, two combinations of three measuring liquids are possible: (WD) or (FD). The (WF) set cannot be used because both water and formamide are polar. In the SFE measurements, sets of other liquids can be applied, providing that there is a polar liquid and dispersion one. A requirement that none of the liquids should react with the surface layer of an investigated material is another important limitation in using measuring liquids. This means that penetration of a liquid into or solving of the material cannot occur as well.
To calculate gS, it is necessary to know all the SFE components for particular measuring liquids. Therefore, one has to (i) measure CA, using three different liquids, (ii) solve the set of Eqs. (4)–(6), and (iii) put the obtained results into Eq. (7) and, then, into Eq. (8). The set of the measuring liquids has to include two bipolar liquids (here, W and F) and one apolar (here, D). Other restrictions in using the measuring liquids are similar as those in the OW method.
2.2. Method of vOCG
cos YL ¼ 2ðgS =gL Þ0:5 exp½bðgL gS Þ2 1
As in the OW method, SFE of a solid and liquids in the vOCG method is composed of two parts: gSLW, which is called the Lifshitz–van der Waals (LW) component and gSAB, called the acid–base (AB) component [12,13]. The latter is equal to a doubled – mean geometrical value of acid (g+ S ) and base (gS) interactions. Generally, the SFE value of a solid and liquids is composed of three elements. Thus, three measuring liquids are required to measure CA. Using the same notation as in the OW method and the same measuring liquids, the set of equations is as follows:
2.3. Method of Neumann The Neumann method is based on the assumption that there is a relation between SFE of a solid (gS), SFE of a liquid wetting the solid surface (gL), and SFE of the solid–liquid interface (gSL). The relation can be expressed in a general form by the following state equation: F ðgS ; gL ; gSL Þ ¼ 0.
(9)
Using Eq. (9), followed by appropriate substitutions and equation transformations, a formula for the Neumann method is obtained in the following form [15]: (10)
with b ¼ 0.0001247. In spite of the controversy over Eq. (10), this formula is still of interest to researchers as a very convenient tool for the determination of SFE. The main advantage of the method applying this equation is using only one measuring liquid in the CA measurements. It makes the measurements easier to perform and limits the number of errors. The determination of gS from Eq. (10), when gL and YL are known, requires numerical computations, being easy as well.
0:5 þ 0:5 ðgSLW gWLW Þ0:5 þ ðgþ þ ðg S gW Þ S gW Þ
¼ 0:5gW ð1 þ cos YW Þ,
ð4Þ
0:5 þ 0:5 ðgSLW gFLW Þ0:5 þ ðgþ þ ðg S gF Þ S gF Þ
¼ 0:5gF ð1 þ cos YF Þ,
ð5Þ
0:5 þ 0:5 ðgSLW gDLW Þ0:5 þ ðgþ þ ðg S gD Þ S gD Þ ¼ 0:5gD ð1 þ cos YD Þ,
ð6Þ
0:5 gSAB ¼ 2ðgþ S gS Þ ,
(7)
gS ¼ gSLW þ gSAB .
(8)
3. Assumptions for and run of the analysis Because the SFE values obtained with the three methods are different, the reasons causing the SFE differences should be discussed. Generally, these differences may be due to the reasons classed into the following four groups: (a) errors of the CA measurements (surface roughness and heterogeneity, CA hysteresis), (b) wrong measuring liquids selected (ill-conditioning of equation set, swelling effects),
ARTICLE IN PRESS M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
The reasons included in groups a–c are humanrelated and also depend on the measuring apparatus, measurement conditions, and measurement precision. As to group d, performing the CA measurements does not influence the differences in the SFE values. Since the latter depend on the mathematical relation used, detailed consideration of these objectively occurring differences is needed. To discuss the differences in the SFE values obtained with the OW, vOCG, and Neumann methods, a new approach to the analysis of the methods was adopted. In this approach, three commonly used measuring liquids (W, F, and D) were considered and two basic assumptions were made: (1) The analysis should enable determination of objectively occurring differences in the SFE values calculated with use of the three abovementioned methods, regardless of the kind of materials studied. The considered SFE values of these materials are in the range of 20–50 mJ/m2, being typical of the large majority of polymeric materials with modified or non-modified surface layer. (2) The results obtained with the Neumann method and with water as a measuring liquid are considered as reference values, which is based on the fact that this method requires only one measuring liquid for the CA measurements. This has an essential meaning for the determination of CA with other measuring liquids while using the OW and vOCG methods. The comparative analysis has been based on the results obtained with an appropriate computer program and can be divided into the following steps: (1) Determination of characteristic curves from Eq. (10) being transformed numerically gS ¼ f ðYL Þ
(11)
with L ¼ W, F, D. (2) Determination of gS from Eq. (11) for YW varying from 1051 to 551 with the decrement
of 51. Thus, the obtained gS values vary in the range of 20.0–50.8 mJ/m2. (3) From Eq. (10), calculation of YF and YD relating to each gS value determined in the previous step. (4) By means of OW (Eqs. (1)–(3)) and vOCG (Eqs. (4)–(8)) methods, calculation of SFE using appropriate sets of CA accepted in step 2 and computed in step 3. (5) Comparison of the results obtained with various methods. Intervals between neighboring measurement points (51), accepted in step 2, seem to be sufficient in view of both the aim of the calculations and the course of each characteristic curve (Eq. (11)). 4. Results and discussion The characteristic curves for each measuring liquid (W, F, and D) are presented in Fig. 1. As seen, the plots are similar in shape to one another and the gS values decrease monotonically with increasing CA. Moreover, the values of gS increase with gL of each liquid over the entire range of YL considered. In this range, the characteristics shown in Fig. 1 can be approximated with a high accuracy, using the following polynomial: gS ¼ aY3L þ bY2L þ cYL þ d.
(12) 2
The determination coefficient, R , for each polynomial function approximated in that way, exceeds 0.999. Values of the polynomial coefficients for the measuring liquids are listed in Table 1. The SFE values calculated for varying YW and the corresponding values of YF and YD are listed in Table 2, with the accuracy of 0.1 mJ/m2 for SFE and
SFE [mJ/m2]
(c) wrong values of SFE and its components for measuring liquids, accepted from the literature, and (d) different mathematical relations between CA and SFE, used in each method applied.
17
80 70 60 50 40 30 20 10 0
W F D
0
20
40
60 80 ΘL [deg]
100
120
140
Fig. 1. Surface free energy (SFE) of a solid versus the contact angle (YL) of water (W), formamide (F), and diiodomethane (D), calculated with the Neumann method.
ARTICLE IN PRESS Table 1 Polynomial coefficients used for approximation of characteristic curves (cf. Fig. 1) Measuring liquid
Water Formamide Diiodomethane
Polynomial coefficient (Eq. (12)) a
b
c
d
0.5 10–3 0.5 10–3 0.5 10–3
6.5 10–3 6.5 10–3 6.5 10–3
0.1328 0.0790 0.0637
73.41 58.29 51.00
Table 2 Surface free energies (SFE) of a solid, calculated with the Neumann method while using contact angles of water (YW), formamide (YF), and diiodomethane (YD) SFE (mJ/m2)
20.0 23.0 26.1 29.2 32.4 35.5 38.6 41.7 44.8 47.8 50.8
SFE [mJ/m2]
M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
18
60.0 55.0 50.0 45.0 40.0 N 35.0 vOCG 30.0 OWw OWf 25.0 20.0 15.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 SFE (N) [mJ/m2]
Fig. 2. Comparison of surface free energy (SFE) values of a solid, calculated with the Neumann (N), van Oss–Chaudhury-Good (vOCG), and Owens–Wendt (OWw or OWf) methods (w and f stand for water and formamide, used as polar measuring liquids).
Contact angle (deg) YW
YF
YD
105 100 95 90 85 80 75 70 65 60 55
91 85 79 74 68 62 56 50 44 38 31
83 77 71 64 58 51 45 37 30 20 1
11 for YL. The values in Table 2 are the starting data for calculations conducted with use of the OW and vOCG methods and for comparing the obtained results. The obtained SFE values are compared in Fig. 2. The data calculated with the Neumann method are based on the CA measurements carried out with use of W (parameters for chemically pure water were applied), those calculated with the vOCG method, on the CA measurements with use of W, F, and D, and those calculated with the OW method, on the CA measurements with use of W and D or F and D. A reference scale of the horizontal axis refers to the SFE values computed with the Neumann method; therefore, it is denoted as SFE(N). The plot corresponding to this method is a straight line at a slope of 451 to the horizontal axis. The smallest differences in the results obtained with use of the three methods are in the middle of the SFE range analyzed, i.e. between 32 and 42 mJ/m2. Differences between the SFE values calculated with the methods of vOCG and OW and those from the Neumann method are presented in Table 3. As
Table 3 Differences between the surface free energy (SFE) values calculated with the Neumann method and those obtained using the methods of van Oss–Chaudhury–Good (vOCG) and Owens–Wendt with water (OWw) and formamide (OWf) as polar measuring liquids SFE calculated with the Neumann method (mJ/m2)
20.0 23.0 26.1 29.2 32.4 35.5 38.6 41.7 44.8 47.8 50.8
Differences in SFE (%) vOCG
OWw
OWf
17.8 13.8 10.0 6.4 3.1 0.1 2.4 4.5 6.1 7.2 7.8
19.9 15.9 12.0 8.2 4.6 1.2 1.9 4.6 7.0 9.0 10.6
20.9 17.2 13.7 10.3 7.3 4.5 2.2 0.2 1.4 2.5 3.2
seen, the largest differences (ca. 6–21%) appear over the SFE range of 20–30 mJ/m2. They are relatively large (6–11%) also for gSX43 mJ/m2. The relative differences between the results obtained with the vOCG method (W, F, and D applied) and OW method (W and D) are considerably smaller than those mentioned above. They do not exceed 3% over the SFE range of 20–50 mJ/m2. However, the relative differences between the results obtained with the OW method for two sets of measuring liquids (W and D or F and D) are slightly larger, reaching 7% in the same range of the SFE values. It is interesting to note that the higher the SFE values of the material, the higher the differences discussed above.
ARTICLE IN PRESS M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
5. Conclusions
The SFE values calculated with use of various methods differ not only due to errors and different performing conditions of the CA measurements but also due to objectively existing differences in the mathematical formulae used. In the SFE range of 20–50 mJ/m2, being characteristic of most polymeric materials, the differences between the SFE values calculated with the Neumann and vOCG (or OW) methods reach 21%. On the other hand, the relative differences between the results obtained with the vOCG and OW methods are considerably smaller, not exceeding 3% over the entire range of SFE. The differences in the SFE values, calculated with various methods, depend also on the kind of measuring liquids used in the CA measurements.
References [1] R.D. Adams (Ed.), Adhesive Bonding. Science, Technology and Applications, first ed, Woodhead Publishing Limited, 2005. [2] M. Z˙ enkiewicz, Adhesion and Modification of Surface Layer of Macromolecular Materials, first ed, Scientific-Technical Publishers, 2000. [3] A.V. Pocius, Adhesion and Adhesives Technology. An Introduction, first ed, Hanser Publishers, 1997. [4] F. Garbassi, M. Morra, E. Occhiello, Polymer Surfaces. From Physics to Technology, first ed, Wiley, New York, 1998. [5] K.L. Mittal (Ed.), Contact Angle, Wettability and Adhesion, VSP, 2002. [6] R.J. Good, Contact angle, wetting, and adhesion: a critical review, J. Adhes. Sci. Technol. 6 (12) (1992) 1269. [7] L.H. Lee, Surface hydrogen-bond components and linear solvation energy relationship parameters, J. Adhes. 63 (1997) 187.
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[8] C. Della Volpe, D. Maniglio, M. Brugnara, S. Siboni, M. Morra, The solid surface free energy calculation I. In defense of multicomponent approach, J. Colloid Interface Sci. 271 (2004) 434. [9] S. Siboni, C. Della Volpe, D. Maniglio, M. Brugnara, The solid surface free energy calculation II. The limits of the Zisman and of the ‘‘equation-of-state’’ approaches, J. Colloid Interface Sci. 271 (2004) 454. [10] W.A. Zisman, Relation of equilibrium contact angle to liquid and solid constitution, ACS Adv. Chem. Ser. 43 (1961) 1. [11] D.K. Owens, R.C. Wendt, Estimation of the surface free energy of polymers, J. Appl. Polym. Sci. 13 (1969) 1741. [12] C.J. van Oss, R.J. Good, M.K. Chaudhury, The role of van der Waals forces and hydrogen bonds in ‘‘hydrophobic interactions’’ between biopolymers and low energy surfaces, J. Colloid Interface Sci. 111 (2) (1986) 378. [13] C.J. van Oss, M.K. Chaudhury, R.J. Good, Interfacial Lifschitz–van der Waals and polar interactions in macroscopic systems, Chem. Rev. 88 (1988) 927. [14] A.W. Neumann, R.J. Good, C.J. Hope, M. Sejpal, An equation-of-state approach to determine surface tensions of low-energy solids from contact angles, J. Colloid Interface Sci. 49 (2) (1974) 291. [15] D. Li, A.W. Neumann, Equation of state for interfacial tensions of solid–liquid systems, Adv. Colloid Interface Sci. 39 (1992) 299. [16] E. Chibowski, Surface free energy of a solid form contact angle hysteresis, Adv. Colloid Interface Sci. 103 (2003) 149. [17] P. Dalet, E. Papon, J.-J. Villenave, Surface free energy of polymeric materials: relevancy of conventional contact angle data analysis, J. Adhes. Sci. Technol. 13 (8) (1999) 857. [18] C. Della Volpe, D. Maniglio, S. Siboni, M. Morra, Recent theoretical and experimental advances in the application of van Oss–Chaudhury–Good acid–base theory to analysis of polymer surfaces I. General aspects, J. Adhes. Sci. Technol. 17 (11) (2003) 1477. [19] E. McCafferty, Acid–base effects in polymer adhesion at metal surfaces, J. Adhes. Sci. Technol. 16 (3) (2002) 239. ˙ enkiewicz, Effects of electron-beam irradiation on [20] M. Z wettability and surface free energy of polypropylene film, Int. J. Adhes. Adhes. 25 (2005) 61. [21] M. Z˙ enkiewicz, The analysis of principal conditions of van Oss–Chaudhury–Good method in investigations of surface layers of polymeric materials, Polimery 51 (3) (2006) 169.
POLYMER TESTING Polymer Testing 26 (2007) 14–19 www.elsevier.com/locate/polytest
Test Method
Comparative study on the surface free energy of a solid calculated by different methods Marian Z˙ enkiewicz Kazimierz Wielki University, ul. Chodkiewicza 30, Bydgoszcz 85-064, Poland Received 15 June 2006; accepted 4 August 2006
Abstract This paper presents analysis of differences in the surface free energy (SFE) values of a solid, calculated using the methods of Owens–Wendt (OW), van Oss–Chaudhury–Good (vOCG), and Neumann with three measuring liquids: water, formamide, and diiodomethane. The concept of the analysis has been based on the differences in SFE, which occur objectively and regardless of both the precision and the performing conditions of the contact angle (CA) measurements. These differences result from utilizing different mathematical relations between CA and SFE for each of the methods. The results obtained with these three methods are compared with one another over the SFE range common for polymers (20–50 mJ/m2). It is concluded that the relative difference in SFE between the results from the Neumann and vOCG (or OW) methods can reach 21%, while that between the results from the vOCG and OW methods is considerably lower and does not exceed 3%. r 2006 Elsevier Ltd. All rights reserved. Keywords: Polymeric materials; Surface free energy; Contact angles; Methods of calculation
1. Introduction Adhesion is one of the basic physical phenomena appearing in various fields of technology and is significant for many industrial processes. It plays an important role in gluing, printing and coating of polymeric materials. In order to obtain high strength of adhesive joints, the substrate surface has to be fully wetted by the glue, ink, or coating. Different techniques for surface layer (SL) modification of polymeric materials are commonly applied to improve the wettability and, thus, adhesion [1–3]. The physical values such as a contact angle (CA) and surface free energy (SFE) are used to charTel.: +48 52 3419 208; fax: +48 56 3414 773.
E-mail address: [email protected]. 0142-9418/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2006.08.005
acterize and predict adhesive properties of the polymeric materials. Therefore measurements of CA of sessile drops deposited on surfaces of various materials, and different methods of calculation of SFE are powerful analysis tools widely used in many industrial sectors and for many practical applications, especially for coating, gluing and printing. It is evident that if the SFE of polymeric material is raised by any kind of surface treatment to a higher level and wettability is improved, then adhesion properties can be improved also. For these reasons, the methods used for the SFE determination are intensively investigated [4–9]. The Young and Laplace equations, well known for 200 years, constitute principles for the determination of SFE of solids. However, in the second half of the 20th century, rapid progress in the interface
ARTICLE IN PRESS M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
science and wettability processes occurred. As a result, new calculation methods for the SFE determination were elaborated, which are very important for polymer surface investigations. These methods are formulated on the basis of a previously measured contact angle or of the drop-shape analysis for various liquids [2,4,5]. While measuring CA for the SFE determination, many technical restrictions and serious scientific dilemmas occur. Some of these concern all the methods for the SFE determination and include such problems as thermodynamic equilibrium, metastable state of the shape and CA of the examined drop, physical interpretation of advancing and receding contact angles, and origin of differences in the SFE values for various liquids applied and different methods used. These problems point out directions of further interface research of polymeric materials [6–9]. The main methods for the determination of SFE are those of Zisman [10], Owens–Wendt (OW, known also as the Kaelble method) [11], and a relatively new method of van Oss–Chaudhury– Good (vOCG) [12,13]. The last one is a subject of interest for many researchers and heated scientific discussions. The main issue of these discussions concerns determination of acid and base components of SFE of measuring liquids, defining acid and base levels of SL of various polymers or polymeric materials, and rules for appropriate choice of measuring liquids. A method based on the so-called state equation which was modified to the Neumann equation [14,15] is a subject of a detailed analysis in the literature. One of the controversies dealing with this equation is due to the nature of the b constant (see Section 2.3) which is, for some authors, a universal material constant, whereas for others, it is only an equation parameter [8]. There is also a new, very interesting proposition for the SFE determination which is based on measuring the advancing and receding contact angles with use of one liquid only [16]. The most often applied method for polymeric materials so far is the OW method, in which water and diiodomethane are used. A growing interest in the vOCG method, based on the results of CA measurements performed with use of three appropriately selected liquids, is an inspiration for new investigations in this field. These investigations include analysis of parameters of measuring liquids used to measure CA and construction of a set of
15
mathematical equations utilized to determine SFE of investigated materials [17–19]. A similar study has also been performed while using the Neumann method [8]. This study is a continuation of previous works of the author [20,21]. Its aim was to analyze differences in the values of SFE of a solid, calculated by means of the OW, vOCG, and Neumann methods with use of commonly applied measuring liquids such as water, formamide, and diiodomethane. The analysis was made in a new way, consisting of separation and determination of some differences between the SFE values calculated with these three methods. These differences occur objectively and regardless of the precision and performing conditions of the CA measurements and result from different mathematical formulae valid for each method. 2. Characteristics of contact-angle measuring methods 2.1. Method of OW The principal assumption of the OW method is that SFE is a sum of two components: dispersion (SFED) and polar (SFEP) [11]. SFED reflects the dispersion interaction occurring on an interface and SFEP is a sum of polar, hydrogen, inductive and acid–base interactions. The analysis of this method was carried out using the following notation:
W (water, a polar liquid), F (formamide, a polar liquid), and D (diiodomethane, a dispersion liquid) for measuring liquids used in the CA measurements; gS (SFE), gSd (SFED), and gSp (SFEP) for SFE and its components of a solid, respectively; gL (SFE), gLd (SFED), and gLp (SFEP) for SFE and its components of a measuring liquid, respectively (L refers to W, F or D); YL for a contact angle (L means the same as above).
Then, the set of equations in the OW method, with water and diiodomethane used as measuring liquids, is of the form ðgSd gWd Þ0:5 þ ðgSp gWp Þ0:5 ¼ 0:5gW ð1 þ cos YW Þ,
(1)
ðgSd gDd Þ0:5 þ ðgSp gDp Þ0:5 ¼ 0:5gD ð1 þ cos YD Þ,
(2)
gS ¼ gSd gSp .
(3)
ARTICLE IN PRESS 16
M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
The gS value of an investigated material is calculated as follows. First, gW, gWd, gWp, gD, gDd, and gDp are to be determined. These values can be measured by one of the commonly used methods [2,4] or just be accepted from the literature. The next step is to measure CA, using a set of measuring liquids. These measurements should be done repeatedly (more than 7 times) in order to determine a mean arithmetic value and its standard deviation. The values of gSd and gSp are calculated from Eqs. (1) and (2) and gS, from Eq. (3). To measure CA, a set of two measuring liquids (dispersion and polar) has to be applied. Thus, two combinations of three measuring liquids are possible: (WD) or (FD). The (WF) set cannot be used because both water and formamide are polar. In the SFE measurements, sets of other liquids can be applied, providing that there is a polar liquid and dispersion one. A requirement that none of the liquids should react with the surface layer of an investigated material is another important limitation in using measuring liquids. This means that penetration of a liquid into or solving of the material cannot occur as well.
To calculate gS, it is necessary to know all the SFE components for particular measuring liquids. Therefore, one has to (i) measure CA, using three different liquids, (ii) solve the set of Eqs. (4)–(6), and (iii) put the obtained results into Eq. (7) and, then, into Eq. (8). The set of the measuring liquids has to include two bipolar liquids (here, W and F) and one apolar (here, D). Other restrictions in using the measuring liquids are similar as those in the OW method.
2.2. Method of vOCG
cos YL ¼ 2ðgS =gL Þ0:5 exp½bðgL gS Þ2 1
As in the OW method, SFE of a solid and liquids in the vOCG method is composed of two parts: gSLW, which is called the Lifshitz–van der Waals (LW) component and gSAB, called the acid–base (AB) component [12,13]. The latter is equal to a doubled – mean geometrical value of acid (g+ S ) and base (gS) interactions. Generally, the SFE value of a solid and liquids is composed of three elements. Thus, three measuring liquids are required to measure CA. Using the same notation as in the OW method and the same measuring liquids, the set of equations is as follows:
2.3. Method of Neumann The Neumann method is based on the assumption that there is a relation between SFE of a solid (gS), SFE of a liquid wetting the solid surface (gL), and SFE of the solid–liquid interface (gSL). The relation can be expressed in a general form by the following state equation: F ðgS ; gL ; gSL Þ ¼ 0.
(9)
Using Eq. (9), followed by appropriate substitutions and equation transformations, a formula for the Neumann method is obtained in the following form [15]: (10)
with b ¼ 0.0001247. In spite of the controversy over Eq. (10), this formula is still of interest to researchers as a very convenient tool for the determination of SFE. The main advantage of the method applying this equation is using only one measuring liquid in the CA measurements. It makes the measurements easier to perform and limits the number of errors. The determination of gS from Eq. (10), when gL and YL are known, requires numerical computations, being easy as well.
0:5 þ 0:5 ðgSLW gWLW Þ0:5 þ ðgþ þ ðg S gW Þ S gW Þ
¼ 0:5gW ð1 þ cos YW Þ,
ð4Þ
0:5 þ 0:5 ðgSLW gFLW Þ0:5 þ ðgþ þ ðg S gF Þ S gF Þ
¼ 0:5gF ð1 þ cos YF Þ,
ð5Þ
0:5 þ 0:5 ðgSLW gDLW Þ0:5 þ ðgþ þ ðg S gD Þ S gD Þ ¼ 0:5gD ð1 þ cos YD Þ,
ð6Þ
0:5 gSAB ¼ 2ðgþ S gS Þ ,
(7)
gS ¼ gSLW þ gSAB .
(8)
3. Assumptions for and run of the analysis Because the SFE values obtained with the three methods are different, the reasons causing the SFE differences should be discussed. Generally, these differences may be due to the reasons classed into the following four groups: (a) errors of the CA measurements (surface roughness and heterogeneity, CA hysteresis), (b) wrong measuring liquids selected (ill-conditioning of equation set, swelling effects),
ARTICLE IN PRESS M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
The reasons included in groups a–c are humanrelated and also depend on the measuring apparatus, measurement conditions, and measurement precision. As to group d, performing the CA measurements does not influence the differences in the SFE values. Since the latter depend on the mathematical relation used, detailed consideration of these objectively occurring differences is needed. To discuss the differences in the SFE values obtained with the OW, vOCG, and Neumann methods, a new approach to the analysis of the methods was adopted. In this approach, three commonly used measuring liquids (W, F, and D) were considered and two basic assumptions were made: (1) The analysis should enable determination of objectively occurring differences in the SFE values calculated with use of the three abovementioned methods, regardless of the kind of materials studied. The considered SFE values of these materials are in the range of 20–50 mJ/m2, being typical of the large majority of polymeric materials with modified or non-modified surface layer. (2) The results obtained with the Neumann method and with water as a measuring liquid are considered as reference values, which is based on the fact that this method requires only one measuring liquid for the CA measurements. This has an essential meaning for the determination of CA with other measuring liquids while using the OW and vOCG methods. The comparative analysis has been based on the results obtained with an appropriate computer program and can be divided into the following steps: (1) Determination of characteristic curves from Eq. (10) being transformed numerically gS ¼ f ðYL Þ
(11)
with L ¼ W, F, D. (2) Determination of gS from Eq. (11) for YW varying from 1051 to 551 with the decrement
of 51. Thus, the obtained gS values vary in the range of 20.0–50.8 mJ/m2. (3) From Eq. (10), calculation of YF and YD relating to each gS value determined in the previous step. (4) By means of OW (Eqs. (1)–(3)) and vOCG (Eqs. (4)–(8)) methods, calculation of SFE using appropriate sets of CA accepted in step 2 and computed in step 3. (5) Comparison of the results obtained with various methods. Intervals between neighboring measurement points (51), accepted in step 2, seem to be sufficient in view of both the aim of the calculations and the course of each characteristic curve (Eq. (11)). 4. Results and discussion The characteristic curves for each measuring liquid (W, F, and D) are presented in Fig. 1. As seen, the plots are similar in shape to one another and the gS values decrease monotonically with increasing CA. Moreover, the values of gS increase with gL of each liquid over the entire range of YL considered. In this range, the characteristics shown in Fig. 1 can be approximated with a high accuracy, using the following polynomial: gS ¼ aY3L þ bY2L þ cYL þ d.
(12) 2
The determination coefficient, R , for each polynomial function approximated in that way, exceeds 0.999. Values of the polynomial coefficients for the measuring liquids are listed in Table 1. The SFE values calculated for varying YW and the corresponding values of YF and YD are listed in Table 2, with the accuracy of 0.1 mJ/m2 for SFE and
SFE [mJ/m2]
(c) wrong values of SFE and its components for measuring liquids, accepted from the literature, and (d) different mathematical relations between CA and SFE, used in each method applied.
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80 70 60 50 40 30 20 10 0
W F D
0
20
40
60 80 ΘL [deg]
100
120
140
Fig. 1. Surface free energy (SFE) of a solid versus the contact angle (YL) of water (W), formamide (F), and diiodomethane (D), calculated with the Neumann method.
ARTICLE IN PRESS Table 1 Polynomial coefficients used for approximation of characteristic curves (cf. Fig. 1) Measuring liquid
Water Formamide Diiodomethane
Polynomial coefficient (Eq. (12)) a
b
c
d
0.5 10–3 0.5 10–3 0.5 10–3
6.5 10–3 6.5 10–3 6.5 10–3
0.1328 0.0790 0.0637
73.41 58.29 51.00
Table 2 Surface free energies (SFE) of a solid, calculated with the Neumann method while using contact angles of water (YW), formamide (YF), and diiodomethane (YD) SFE (mJ/m2)
20.0 23.0 26.1 29.2 32.4 35.5 38.6 41.7 44.8 47.8 50.8
SFE [mJ/m2]
M. Z˙enkiewicz / Polymer Testing 26 (2007) 14–19
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60.0 55.0 50.0 45.0 40.0 N 35.0 vOCG 30.0 OWw OWf 25.0 20.0 15.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 SFE (N) [mJ/m2]
Fig. 2. Comparison of surface free energy (SFE) values of a solid, calculated with the Neumann (N), van Oss–Chaudhury-Good (vOCG), and Owens–Wendt (OWw or OWf) methods (w and f stand for water and formamide, used as polar measuring liquids).
Contact angle (deg) YW
YF
YD
105 100 95 90 85 80 75 70 65 60 55
91 85 79 74 68 62 56 50 44 38 31
83 77 71 64 58 51 45 37 30 20 1
11 for YL. The values in Table 2 are the starting data for calculations conducted with use of the OW and vOCG methods and for comparing the obtained results. The obtained SFE values are compared in Fig. 2. The data calculated with the Neumann method are based on the CA measurements carried out with use of W (parameters for chemically pure water were applied), those calculated with the vOCG method, on the CA measurements with use of W, F, and D, and those calculated with the OW method, on the CA measurements with use of W and D or F and D. A reference scale of the horizontal axis refers to the SFE values computed with the Neumann method; therefore, it is denoted as SFE(N). The plot corresponding to this method is a straight line at a slope of 451 to the horizontal axis. The smallest differences in the results obtained with use of the three methods are in the middle of the SFE range analyzed, i.e. between 32 and 42 mJ/m2. Differences between the SFE values calculated with the methods of vOCG and OW and those from the Neumann method are presented in Table 3. As
Table 3 Differences between the surface free energy (SFE) values calculated with the Neumann method and those obtained using the methods of van Oss–Chaudhury–Good (vOCG) and Owens–Wendt with water (OWw) and formamide (OWf) as polar measuring liquids SFE calculated with the Neumann method (mJ/m2)
20.0 23.0 26.1 29.2 32.4 35.5 38.6 41.7 44.8 47.8 50.8
Differences in SFE (%) vOCG
OWw
OWf
17.8 13.8 10.0 6.4 3.1 0.1 2.4 4.5 6.1 7.2 7.8
19.9 15.9 12.0 8.2 4.6 1.2 1.9 4.6 7.0 9.0 10.6
20.9 17.2 13.7 10.3 7.3 4.5 2.2 0.2 1.4 2.5 3.2
seen, the largest differences (ca. 6–21%) appear over the SFE range of 20–30 mJ/m2. They are relatively large (6–11%) also for gSX43 mJ/m2. The relative differences between the results obtained with the vOCG method (W, F, and D applied) and OW method (W and D) are considerably smaller than those mentioned above. They do not exceed 3% over the SFE range of 20–50 mJ/m2. However, the relative differences between the results obtained with the OW method for two sets of measuring liquids (W and D or F and D) are slightly larger, reaching 7% in the same range of the SFE values. It is interesting to note that the higher the SFE values of the material, the higher the differences discussed above.
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5. Conclusions
The SFE values calculated with use of various methods differ not only due to errors and different performing conditions of the CA measurements but also due to objectively existing differences in the mathematical formulae used. In the SFE range of 20–50 mJ/m2, being characteristic of most polymeric materials, the differences between the SFE values calculated with the Neumann and vOCG (or OW) methods reach 21%. On the other hand, the relative differences between the results obtained with the vOCG and OW methods are considerably smaller, not exceeding 3% over the entire range of SFE. The differences in the SFE values, calculated with various methods, depend also on the kind of measuring liquids used in the CA measurements.
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