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Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good method
Author’s Accepted Manuscript Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good method Anna Rudawska, Elżbieta Jacniacka www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(18)30008-3 https://doi.org/10.1016/j.ijadhadh.2018.01.006 JAAD2112
To appear in: International Journal of Adhesion and Adhesives Accepted date: 4 January 2018 Cite this article as: Anna Rudawska and Elżbieta Jacniacka, Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good m e t h o d , International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2018.01.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anna RUDAWSKA1, Elżbieta JACNIACKA2 1
Lublin University of Technology, Faculty of Mechanical Engineering, Nadbystrzycka 36 Str, 20-618 Lublin, Poland, e-mail: [email protected], +48 81 5384232 2 Lublin University of Technology, Faculty of Mechanical Engineering, Nadbystrzycka 36 Str, 20-618 Lublin, Poland, e-mail: [email protected], +48 81 5384228 Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good method Abstract. Among the great number of methods for measuring the surface free energy of solids, all originate in contact angle measurements, which are carried out using various probe liquids, and employ certain relationships and assumptions. Contact angle measurement can be distorted by a number of factors that occur during the measurement and distort obtained values, and therefore must be accounted for. The present work describes the application of the van Oss-Chaudhury-Good approach in determining the surface free energy of aluminium sheets with the use of three probe liquids: water, diiodomethane and glycerol, with assumed values of the surface free energy components. The study employed direct measurement of the contact angle with the goniometer. Subsequently, measurement uncertainty with the van Oss-Chaudhury-Good method was evaluated, which revealed that the major source of uncertainty is the discrepancy between constant values describing properties of probe liquids. Keywords: surface free energy, van Oss-Chaudhury-Good method, contact angle, uncertainty
Introduction Adhesion is a surface phenomenon, which is harnessed in a wide range of technical spheres, such as, inter alia, adhesive bonding, painting, varnishing, printing, coating with various layers and production technology [1-4]. To learn the adhesive properties of materials is crucial to assessing their suitability for a given application, reaction under given conditions or in the processes highlighted above. The surface characteristics of a substrate can influence the level of adhesion between the substrate and an adhesive and thus the strength of a bonded joint[4]. The adhesion properties of materials can be determined using the surface free energy (SFE). The surface free energy (SFE) is a characteristic parameter of a material strongly associated to the state of equilibrium of the atoms in the surface whose value depends on the nature and the aggregate state of material [5]. Determination of the SFE of solids requires employing indirect methods, since direct methods are exclusively applicable in liquids. A number of indirect methods are applied in the SFE determination, e.g. the approaches of Fowkes, Owens-Wendt, van Oss-Chaudhury-Good, Zisman, Wu or Newman [6-11]. Since these approaches represent indirect methods, the measured value of the SFE may be distorted by a number of external factors, concurrent in the process, as well as the assumptions made in calculations [5, 6, 12-15]. The contact angle measurement of probe liquids on a solid surface lies at the heart of a number of methods for calculating the SFE from different assumptions [16-19]. The major limitation of these methods is the existence of a number of factors that may distort the calculated value of the SFE [20, 21], among which we ought to mention the probe liquid drop volume, drop shape [22,23], drop spreading time on the analysed surface (thermodynamical equilibrium in reference to metastability of the drop and of the contact angle, and physical interpretation of advancing and receding angle, and also contact angle hysteresis) [24, 25, 26], the type of probe liquid [27], surface roughness and topography [1,28], physical and chemical uniformity of the analysed surface or the temperature of the test [14, 29-33] and also methods to measure contact angle [34,35]. Moreover the contact angle is 1
sensitive to actual physico-chemical conditions of the solid-liquid interface [36]. Consequently, these factors should be considered during experiments. Furthermore, certain simplifications and assumptions are introduced in the SFE measurement, which indicates the need for evaluation of measurement uncertainty. In [37] the authors show that the type of probe liquid applied in tests affects the obtained values of the SFE. At the same time, assuming different relationships for water in determining the three components of the SFE, which is the foundation of the van Oss-Chaudhury-Good approach, may affect the SFE values obtained in tests [10]. Extrand [38] presented the uncertainty in contact angle measurements for the tangent method. In [39] the authors presented general information regarding measurement error in the SFE determination with the van Oss-Chaudhury-Good method, and juxtaposed the calculated error with the error produced with the Owens-Wendt approach. Nevertheless, one issue is that the error was calculated from the scatter of contact angles obtained for different probe liquids. Our technique has involved evaluation of uncertainty of assumed constant values for probe liquids and uncertainty of the measuring chain. Our study employs a technique to assess the type of material used as substrate in adhesive joints in further sections of this paper, and obtained test results. We have also exploited a different methodology for measuring the uncertainty in a contact-angle-based technique for SFE determination. Uncertainty of surface free energy of the solid depends not only on uncertainty of the contact angle of the model liquid on a given solid surface but also on that of both surface tension and contact angle measurements on the reference solids which were used for determination of components and parameters of the model liquid surface free energies. It should be remembered that van Oss et al. assumed that the LW component of water is equal to 21.8 mN/m. For the first time this was determined by Fowkes as the dispersion component from the interfacial tension of water-n-alkane. Next for the other liquids, van Oss et al. determined the LW component from the contact angle on an apolar solid whose surface free energy results were obtained only from LW intermolecular interactions. Knowing LW values determined in such a way and assuming that acid and base parameters of water have the same values as well as the contact angle of water, diiodomethane and for example glycerol on a monopolar solid (for example PMMA), van Oss et al. determined relative values of acid and base parameters of acid-base components of the polar liquids surface tension. The following paper employs the van Oss-Chaudhury-Good approach to determine the SFE, the results of which were analysed with respect to the measurement uncertainty of contact angles of selected probe liquids. The van Oss-Chaudhury-Good method (vOCG) The Lifshitz-van de Waals/acid-base approach proposed by van Oss, Chaudhury and Good (the van Oss-Chaudhury-Good method) is one of the indirect methods for solid surface free energy determination [1, 19, 40]. The method draws on the assumption proposed by Fowkes [32], which states that the surface free energy of any liquid and solid may be expressed as the sum of its surface free energy components. The components are distinguished based on different interactions occurring at the solid-liquid interface: dispersive, dipole (dipole-dipole, dipole-induced dipole), electrostatic, hydrogen bonds, -type bonds or electron donor-acceptor bonds [10]. The fundamental equation which relates the interfacial tension between solid and liquid (SL) to solid surface tension (S), liquid surface tension (L), and the contact angle () between a drop of liquid and a chemically homogeneous, non-adsorbing, smooth and horizontal solid surface, is given by the well-known Young’s equation (1) [12]: SL = S - Lcos (assumption: > 0)
(1)
According to the approach of van Oss et al. [6, 12, 41], the surface free energy () of a liquid or solid can be expressed by two components (2):
2
γ γ LW γ AB
(2)
where is the polar component, which results from Lifshitz-van der Waals (London dispersion, Debye induction and Keesom orientation forces), whereas AB is the polar component, which results from complementary acid–base interactions (electron donor, -, and electron acceptor, +). LW
Due to their mutual compensation, dipole-dipole interactions are weak, therefore their involvement in the surface free energy is relatively insignificant (about 2%), hence may be omitted. Accordingly the component LW results largely from dispersive interactions of London [9]. AB interactions most frequently occur as a result of hydrogen bonding interactions and can be expressed by the following relationship (3) [9, 42]:
γ AB 2(γ i γ i )0,5
(3)
where: i+ – denotes the parameter of the surface free energy of a Lewis acid, i- – is the parameter representing the surface free energy of a Lewis base, indexed i denotes the subsequent solids or probe liquids. According to the vOCG method, the measurement of the SFE of analysed materials consists in measuring the contact angle of three probe liquids on their surface and calculating the S for the set of three equations (4) [9]: ) 0,5 (γ γ ) 0,5 γ (1 cosΘ )/2 (γ LW γ LW ) 0,5 (γ γ Li S S Li i S Li S
(4)
where: i=1,2,3. The contact angle measurement requires the application of two polar and one apolar test liquids, and in order to solve equation (4), knowledge of particular values of applied test liquids is required. Frequently used polar liquids include water, glycerol, formamide or ethylene glycol, whereas apolar liquids (which do not show characteristics of either Lewis acids or bases) are predominantly represented by diiodomethane or α-bromonaphthalene. The method used in this study is detailed in [2, 7, 8, 19]. Using the van Oss-Chaudhury-Good method is rather uncomplicated, but results must be carefully considered on account of the following problems [6, 8, 19, 37, 40, 42]: base components of analysed surfaces are systematically greater than the acidic components, and are therefore generally apt to be much more basic than acidic, the results strongly depend on the choice of the three test liquids, and what is more, an inappropriate liquid may interfere with the structure of the test material, since interaction with the surface of a test material (forming chemical bonds) may lead to its dissolving or swelling, the probe liquid may penetrate the material and therefore the contact angle value of the test liquid may be distorted as a result of the drop departing from the circular, the unknowns of calculations are, in practice, the roots of surface tension components, which in certain cases assume negative values, test errors resulting from the contact angle measurement technique employed, which may be connected with: incorrectly determined tangent to drop at the three-phase contact point [10,15], time difference between depositing the drop on the surface and the measurement of the contact angle [24], disregarding the deformation of the drop from the circular, as well as certain inconsistencies regarding the volume of the test liquid drop discrepancies;
3
the topic of selecting an incorrect reference scale for the determination of the acidic and base components of the surface free energy is currently under investigation and discussion [6]. The values of the coefficient used for acid-base properties of test liquids are arbitrary. For water, the acid-base properties ratio at 20C may be set to 1 (+/- = 1) where each parameter contributes 25.5 mJ/m2 [6, 29, 42, 44]. However this does not mean that water is as equally strong as a Lewis acid and as a Lewis base. The acid and base parameters of the same solvent cannot be compared, however, the acid (or base) parameters of different solvents can be . The methodology used in the study established an arbitrary reference scale, denoting relationships between the parameters of the surface free energy (acidic and base) of particular probe liquids, therefore also of solids analysed in the study with the use of the liquids in question. The employed reference scale, however, may be replaced by a different scale. The form of the equation, +/- = 1, is unrestricted (mathematically), and therefore any reference scale for water may be applied, such as +/- = k, in which case the following assumption for water may be made: + = 25.5 (k)0.5 mJ/m2 and - = 25.5 (k)0.5 mJ/m2. The aforementioned means that with an increase in k, the value of + increases while - decreases; at the same time, the values of SAB, SLW and S remain constant. For instance, Lee [30, 43] and Żenkiewicz [42] assume the following relationship for water: +/= 1.8; this value was based on the results of analysis of solvatochromic parameters of different liquids. The values of + and - calculated from this relationship were equal to 34.2 mJ/m2 and 19.0 mJ/m2 respectively. Although this relationship shows domination of the basic character in the surface layer of polymer material, which confirms test results, this method is rarely included in the SFE measurement with the vOCG method. Della Volpe and Siboni [6], however, apply a different, equally seldom employed, scale +/- = 4.35, where the values of + and - for water amounted to 48.5 mJ/m2 and 11.16 mJ/m2 respectively. The implementation of Volpe and Siboni’s scale results in the change of the SFE of other test liquids, and more precisely the values of the acidic and base parameters (+ and - ) , and to a lesser extent the component of the Lifshitz–van der Waals (LAB). Methodology and subject of the study The study was conducted on 15 specimens made of 100x25 mm 7075PLT0 aluminium alloy sheet, specimen thickness was equal to g = 0.1 mm. Prior to contact angle measurement, the surface of specimens was degreased with acetone, which consisted in wiping the specimen three times with an acetone-soaked cloth. Upon third application, acetone was left to dry. Degreasing was carried out at a temperature of 21±1oC and air humidity of 30±1%. The wetting contact angle measurement was determined 7 minutes after the surface was pre-treated. The van Oss-Chaudhury-Good method for calculation of solid surface free energy was applied, and the measurements were taken using three set test liquids: polar (distilled water and glycerol) and apolar (diiodomethane), with known surface free energies and components. Values of the surface free energy S and its components and parameters as determined with the above measuring liquids are given in Table 1.
4
Table 1. Values of the surface free energy L and its components determined with test liquids [7, 10, 30, 42]. Wetting liquids
Surface free energy and its components and parameters
Water
L [mJ/m2] 72.8
LLW [mJ/m2] 21.8
LAB [mJ/m2] 51.0
L+ [mJ/m2] 25.5
L[mJ/m2] 25.5
Glycerol
64.0
34.0
30.0
3.92
57.4
Diiodomethane
50.8
50.8
0
0
0
The contact angle measurements were made with wetting liquids, taking an average of 8-12 drops with each type of liquid. In total, 50 measurements were executed on 5 specimens of aluminium alloy sheet for each liquid. Approximately 2 l of distilled water and glycerol and 1 l of diiodomethane was the volume of the drop used for each measurement taken at a temperature of 21±1oC and air humidity of 30±1%. The contact angle reading was taken within 5 s of drop formation. Final contact angle measurements were recorded (usually within 10-15 s) once their values remained constant upon drop deposition. Repeated measurements of a given contact angle were all within 2. The tests were conducted using a PGX goniometer, manufactured by the Swedish company Fibro System, and computer image analysis software PG v.3.4.0.1 [45]. Some of the contact angle measurement conditions are derived from the ASTM D5946 standard [46]. The surface free energy of aluminium alloy sheet was calculated only apparently because the contact angle values used for calculating the SFE did not fulfil the Young equation. To calculate the surface free energy of aluminium alloy sheet, in terms of dispersive and polar components, equation 3 was employed. Test results Surface free energy Table 2 and Table 3 show the results of the contact angle measurements for different probe liquids on the surface of aluminium alloy specimen, their surface free energy and its components and parameters. Table 2. Values of the contact angle Contact angle
Θw [°] 78.03
Θg [°] 59.65
Θd [°] 38.66
Standard deviation of contact angle
=3,26
=4.37
=8.78
Table 3. Values of the surface free energy S and its components Surface free energy and its components and parameters
S [mJ/m2] 47.9
SLW [mJ/m2] 40.3
SAB [mJ/m2] 7.7
S+ [mJ/m2] 1.3
S[mJ/m2] 11.2
Each contact angle measurement was repeated 50 times, for each of the applied probe liquids. Presented below are the histograms of obtained values.
5
Observation number
Water
Contact angle, ° Fig. 1. Contact angle of water – histogram
Observation number
Diidomethane
Contact angle, ° Fig. 2. Contact angle of diiodomethane – histogram
6
Observation number
Glycerol
Contact angle, ° Fig. 3. Contact angle of glycerol - histogram Comparative analysis of scatter of the contact angle measurement results for different test liquids indicates that the values obtained for diiodomethane show the biggest difference from the normal distribution, consequently the dispersion of results is the highest, as is the standard deviation. A possible cause of this observation is that the apolar liquid displays the most irregular drop shape on the analysed surfaces, hence resulting in differences in contact angle values.
Surface free energy measurement uncertainty The van Oss-Chaudhury-Good approach is an indirect method for calculation of solid surface free energy, where the estimate of measured values (output) is obtained from the equation for measurement function, by replacing the measured values with estimates of input values [47]. To obtain the SFE measurement equation, we must solve the system of equations (4) with respect to contact angles w, g and d. Using the equations from [39] we obtain the following:
LW 0,5 s
0,5 s
0, 5 s
where:
A1 1 cos w B1 1 cos g C1 1 cos d
2D
A2 1 cos w B2 1 cos g C2 1 cosd
2D A3 1 cos w B3 1 cos g C3 1 cos d
2D
B C A B C A
A1 w d g 1
1
2
2
2
3
0, 5
g
w
0, 5 d
d
g
0, 5 w
w
d
LW 0, 5 g
w
LW 0, 5 d
d
g
LW 0, 5 w
w
d
LW 0, 5 g
g
g d
(5a) (5b) (5c)
0, 5
(6a)
d
0, 5 w
(6b)
w
0, 5 g
(6c)
g
LW 0, 5 d
(6d)
d
LW 0, 5 w
(6e)
w
LW 0, 5 g
(6f)
g
LW 0, 5 d
(6g) 7
C D B3 g w dLW 3
d
0, 5
LW 0 , 5 w
w
LW 0 , 5 g
LW 0, 5 w
g
LW 0 , 5 w
0,5 g
d
d
g
0,5 d
LW 0 , 5 g
w
0,5 d
d
0,5 w
LW 0 , 5 d
g
0,5 w
w
0,5 g
(6h) (6i) (6j)
where the indexed signs denote: S- the SFE and its components for a given material, w - for water, g for glycerol, d- for diiodomethane. Given the relationships (2), (3), (5) and the fact that for apolar liquids (diiodomethane) equation takes the form: C1 1 cos d 2D
d d 0 the
2
S
A2 1 cos w B2 1 cos g C 2 1 cos d D A3 1 cos w B3 1 cos g C3 1 cos d 2D
(7)
Estimated input values were derived from arithmetic means of the wetting angle measurement results (Table 3). Type A standard evaluation of measurement uncertainty was employed. This approach was chosen due to its suitability for studies where a large number of repeated measurements are performed. The empirical model of the mean value is an estimator of the standard deviation of the distribution [47, 48, 49] and is taken as standard uncertainty uA.
w
nw
u Aw ( w )
i 1
2
iw
nw 1 nw
n 1 n nw
u Ag ( g )
2
ig
i 1
g
g
i 1
0.62 0
g
nd
u Ad ( d )
0.460
d
2
id
nd 1 nd
1.24 0
(8)
Apart from uncertainty regarding the scatter of the contact angle measurement results uA, another uncertainty analysed in the study is the uncertainty of the measuring chain instrumentation. A study of measuring chain [50] shows the maximum uncertainty of calibration in measurement of angles max 10 0,017 rad , while its major component is systematic uncertainty. In such cases the authors of [47] suggest that type B evaluation of uncertainty (a uniform or rectangular distribution of possible values) should be applied. uB t
where:
u Bt
max 0.580 3
(9)
- standard uncertainty of measuring chain.
When various uncertainties occur in one study, combined standard uncertainty uc, which uses the law of propagation of uncertainty [47, 48] is calculated, we have that:
8
2 w uBt2 0.740 0.013rad uw w u Aw 2 g uBt2 0.850 0.019rad ug g u Ag 2 d uBt2 1.370 0.024rad ud d u Ad
(10)
For uncorrelated input quantities, combined standard uncertainty of the estimate of input takes the following form [47, 48]: N u 2 y ci2u 2 xi i 1 (11) u(xi)- combined standard uncertainty of the estimate xi ci
y xi
(12)
ci - sensitivity coefficient of the estimate of input x.i The contact angle measurements for different test liquids constitute separate experiments, therefore the danger of correlation is non-existent. Employing (7) and (11) produces the equation of combined standard uncertainty uc(L) of the surface free energy measurement with the van Oss-Chaudhury-Good method: 2
u c ( S ) S u w w S u g g S u d d w g d 2
2
(13)
Standard uncertainty of the contact angle measurement is not an only component of the surface free energy measurement uncertainty. The functions of relationship measurement (7) and equations of coefficients (6a-6j) make use of experimentally determined constants of the surface free energy and its components for the employed test liquids. A key problem with the available literature in the field [2, 6, 7, 9, 10, 27, 29, 42, 51] is that it does not undertake the problem of measuring uncertainty regarding determination of these constants. What can be observed, however, are the discrepancies in the values of the acid-base parameters of the surface free energy, + and -, of test liquids given the constant component LW and total surface free energy. Calculated from the equations (5a-5c), (6a-6j) and (7), the surface free energy and its acid-base component assumed different values, depending on the accepted values of constants; for the following values of parameters of the surface free energy w w 25.5mJ / mm 2 , g 3.92mJ / mm 2 , g 57.4mJ / mm 2 the discrepancies amounted to: S 2.69 mJ / m 2 SLW 0 mJ / m 2 sAB max 2.69 mJ / m 2
For the component of the SF resulting from the long-distance interactions sLW the difference is equal to 0 because in all referenced works the values of this component are identical. In accordance with instructions in [47], in the case when only two quantities occur, the antimodal distribution model U should be employed, and the standard uncertainty ought to be calculated from the relationship: u s s
u s SAB
2
S
5 3
2.07mJ / m 2
AB 2 s
5 3
2.07mJ / m 2
(14)
9
Considering standard uncertainty of variance of constants, we may determine combined uncertainty for the total free energy: (15) u ( ) u u u u 2
2
S
c
w
S
2
S
w
w
g
S
g
g
d
d
d
2 s
s
Evaluation of measurement accuracy required employing expanded uncertainty, calculated from:
U ( S ) k uc S
(16)
where: k- coverage factor. The value of coverage factor responds to the distribution of measurement quantities representing random variables [47, 48, 49, 52]. The hypothesis of normality of distribution of w,g and d was based on the Epps-Pulley test [53], which is employed where the number of values is higher than 8 and lower than 200 (8n200). The distribution of values was identified. The software application used to analyse the data was qs-STAT 3.2. The confidence level was 1-=0.95. Table 4 shows results of statistical analysis. Table 4. Epps-Pulley test results and identified type of distribution Variable
Standard value
Critical free energy
Type of Regression distribution coefficient
w
0.240
0.376
normal
0.990
g
0.215
0.376
normal
0.996
d
1.654
0.376
mixed
0.999
For diiodomethane the standard value of the contact angle is higher than critical (Table 4). Therefore, the null hypothesis for the population of normal distribution at the confidence level 0.95 was rejected. For distilled water and glycol, the standard value is lower than the critical value of the SFE, and therefore it may be assumed that the measurement results are derived from the population of normal distribution. Table 5 shows the uncertainty budget. Table 5. Standard uncertainty budget for the surface free energy Symbol of quantity w
[rad] g
[rad] d
[rad] γS [mJ/m2] γS [mJ/m2]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
6.87
0.09
1.04
0.019
normal
-27.11
- 0.40
0.67
0.024
mixed
42.84
1.02
2.69
2.070
U-type
1.00
2.07
47.99
-
-
-
2.35
Using the algorithm for the surface free energy measurement we can calculate the budget for LW AB components S i S . Sensitivity coefficients for long distance interactions were established from 10
the relationship (5a), whereas for the acid-base interaction from (5b) and (5c). The equation (17) LW AB shows the relationship for component S , and (18) for S . 2
LW LW LW u c ( SLW ) S u w w S u g g S u d d w g d
2
2
2
(17)
AB AB AB uc ( LAB ) S uw w S u g g S ud d us2 SAB w g d 2
2
(18)
Results of calculations are shown in Tables 6 and 7. Table 6. Budget for combined standard uncertainty Symbol of quantity
w [rad]
g [rad]
d [rad]
SLW [mJ/m2]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
0.00
0.00
1.04
0.019
normal
0.00
0.00
0.67
0.024
mixed
28.26
0.68
40.28
-
-
-
0.68
Table 7. Budget for combined standard uncertainty Symbol of quantity
w
SLW
SAB
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
6.85
0.09
1.04
0.019
normal
-27.11
- 0.40
0.67
0.024
mixed
14.58
0.35
2.69
2.070
antimodal U-type
1.00
2.07
7.71
-
-
-
2.15
[rad]
g [rad]
d [rad] SAB 2
[mJ/m ] SAB
[mJ/m2]
The analysis of standard uncertainty budgets, shown in Tables 5-7, indicates that for the SFE measured with the van Oss-Chaudhury-Good method the sensitivity coefficient for the contact angle measurement of diiodomethane is the highest, as is standard uncertainty of this measurement. This
11
result indicates that the share of the diiodomethane contact angle measurement uncertainty in the combined measurement uncertainty of the SFE is considerably higher than of the other test liquids. Therefore, in order to reduce uncertainty in measurement of the surface free energy and its components we need to reduce the standard uncertainty of the diiodomethane contact angle measurement by, e.g. increasing the number of repetitions of this measurement compared to the other LW test liquids. In measurements of the S component of the SFE (long distance interactions – Table 6) standard uncertainty results solely from uncertainty of the contact angle measurement of an apolar test liquid, i.e. diiodomethane, which appears unsurprising, having analysed the equations (5a), (6a), (6b) and (7). For apolar liquids, coefficients A1 and B1 are equal to 0, whereas in the case of the acid-base AB component of the surface free energy, S , the sensitivity coefficients acquire the highest values in the measurement of the contact angle of glycol (Table 7). Nevertheless, what constitutes the highest share in the uncertainty of the total surface free energy and its acid-base component is the uncertainty resulting from the discrepancy in values of constants + and - which describes the properties of probe liquids. These constant quantities are obtained experimentally. The parameter that allows quantification of measurement reliability is expanded uncertainty. Authors of [49] show different methods of approximation to estimate expanded uncertainty. However, works [47] and [48] suggest that it is the method described with the relationship (15), and for standard accuracy measurement k = 2, which should be employed. Table 8 shows the results of evaluating expanded uncertainty of measurement of the SFE and its components.
Table 8. Expanded measurement uncertainty of the SFE and its components Symbol of quantity γS [mJ/m2]
γ [mJ/m2] 47.99
u [mJ/m2] 2.35
k 2
U [mJ/m2] 4.70
SLW
40.28
0.68
2
1.36
7.71
2.15
2
4.30
[mJ/m2] SAB
[mJ/m2]
The results in Table 8 show clearly that, although its value is significantly lower than that of the component LW, it is the correct determination of the acid-base component of the SFE AB that is critical for accuracy of measurement of the total free energy. For comparison, the results of the contact angles of distilled water and diiodomethane were used in the determination of the SFE with the Owens-Wendt method. Similarly, as in the previous analyses, the uncertainty budget was established according to the same procedure, which we have described in detail in our previous work [50]. The Owens-Wendt method assumes that the surface free energy (S) is a sum of two components: polar (Sp) and dispersive (Sd), and that they all are characterised by the following relationship (19) [2, 19, 40, 44]: S = Sd + Sp
(19)
The polar component contains the dipole-induced dipole (Debye-van der Waals interactions) and dipole-dipole (Keesom-van der Waals) interactions as well as the hydrogen bond and pi-electron interactions [2, 13, 15, 41]. Dispersive forces are the second component of the surface free energy and the London forces describe dispersive intermolecular interactions (included dipole-induced dipole interactions). 12
To determine polar and dispersive components of the surface free energy it is necessary to measure the contact angle [14, 31, 45] of surfaces of tested materials with two test liquids. Probe liquids used to this aim are liquids whose surface free energy and its polar and dispersive components are known. The components Sd and Sp of the tested materials can be determined using relevant formulas given in the studies [2, 3, 10, 51, 54]. Table 9 shows the uncertainty budget for the measurement of the surface free energy with the Owens-Wend
method
for
given
constant
p d [29]: w 72.8; w 51.0; w 21.8 ,
quantities
d 50.8; dp 0; dd 50.8 .
Table 9. Budget for combined standard uncertainty S for the O-W method Symbol of quantity
w
[rad] d
[rad] S [mJ/m2] 2 S [mJ/m ]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
- 25.78
- 0.33
0.67
0.024
mixed
40.13
0.96
3.06
2.360
antimodal u-type
1.00
44.29
-
-
-
2.36 2.54
The uncertainty budgets for the SFE determined with the Owens-Wendt and the van OssChaudhury-Good methods bear certain resemblances. In both cases the highest proportion of uncertainty was represented by different values of constants characterising the test liquids. In [19] the authors give different values of the dispersive and the polar component at constant surface free energy. Uncertainty calculated from the formula (15), where the coverage factor k = 2 takes the following form: U SOW 5.08mJ / m 2 Authors of [18] evidenced that different methods for measuring the surface free energy produce different results, which is confirmed by the fact that the results in Table 5 and 9 similarly show certain differences. To determine whether these are statistically significant, the Student’s t-test was conducted at the level of significance = 0.05, and the degree of freedom equal to f =(n2+n1-2)=98. The calculations based on the following relationship: t , f
SOG SOW u 2 SOG u 2 SOW
(20)
where: SOG - the SFE measured with the van Oss-Chaudhury-Good method u 2 SOG - standard uncertainty of the SFE measured with the van Oss-Chaudhury-Good SOW - the SFE measured with the Owens-Wendt method u 2 SOW - standard uncertainty of the SFE measured with the Owens-Wendt method Relationship (20) describes large samples and approximated values of standard uncertainty, and both these conditions are in this case met. The test statistics t,f = 1.068 is lower than the critical 13
value, t,fkryt = 1.984, therefore it may be concluded that the SFE calculated with the van OssChaudhury-Good and the Owens-Wendt methods are equivalent. Summary Measurement uncertainty with the van Oss-Chaudhury-Good method revealed that the major source of uncertainty is the discrepancy between constant values describing properties of probe liquids. Expanded uncertainty of the SFE measured with the van Oss-Chaudhury-Good method is equal to 4.30 mJ/m2, which amounts to 9% of the measured quantity. The uncertainty of the measurement may be optimised by introducing additional series of measurement repetitions. However, in the presented case the expected effect will not be reached as the results in Tables 8, 9, 10 and 11 indicate that it is the uncertainty of constants characterising the test liquids that represents the major proportion of total uncertainty, both for the van Oss-Chaudhury-Good and the Owens-Wendt methods. As long as no standards specifying these constants are introduced, the results of measurements will not be equivalent or credible, and the SFE measurement may be included merely for comparison; an issue which has been raised by researchers in the field. The test results obtained from this study lend support to previous findings in the literature and highlight the demand for further investigations in the field. Our study evidences that it is critical for research papers to include the assumptions that were made in measuring the surface free energy and its components, and particularly to apply the same assumptions to different methods for the surface free energy measurement. References [1] Baldan A. Adhesion phenomena in bonded joints. Int J Adhes Adhes 2012; 38: 95–116. [2] Żenkiewicz M. Adhesion and modification of the high-molecular materials surface layer. Warsaw: WNT, Poland; 2000. [3] Kinloch AJ. Adhesion and Adhesives. New York: Chapman and Hall; 1987. [4] Adams RD. Adhesive boding. Science, technology and applications. UK: Woodhead Publishing Limited; 2010. [5] Schuster JM, Schvezov CE, Rosenberger MR. Influence of experimental variables on the measure of contact angle in metals using the sessile drop method. Proc Mat Sci 2015; 8: 742–751. [6] Della Volpe C, Siboni S. Some reflections on acid-base solid surface free energy theories. J Coll Interface Sci 1997; 195: 121–136. [7] Michalski M-C, Hardy J, Saramago BJ. On the surface free energy of PVC/EVA polymer blends: comparison of different calculation methods. J Coll Interface Sci 1998; 208: 319–328. [8] Greiveldedinger M, Shananhan MER. A critique of the mathematical coherence of acid/base interfacial free energy theory. J Coll Interface Sci 1999; 215: 170–178. [9] Hołysz L. Surface free energy components of silica gel determined by the thin layer wicking method for different layer thicknesses of gel. J Mat Sci 1998; 30: 445–452. [10] Żenkiewicz M. The analysis of principal conditions of van Oss-Chaunhury-Good’s method in investigations of surface layer of polymer materials. Polimery 2006; 51: 169–176. [11] Kaelble DH. Dispersion-polar surface tension properties of organic solids. J Adhes 1970; 2: 66–81. [12] Faibish RS, Yoshida W, Cohen Y. Contact angle study on polymer-grafted silicon wafers. J Coll Interface Sci 2002; 256: 341–350. [13] Fernández V, Khayet M. Evaluation of the surface free energy of plant surfaces: toward standardizing the procedure. Front Plant Sci 2015; 6: 510 (doi:10.3389/fpls.2015.00510). [14] Barberis F, Capurro M. Wetting in the nanoscale: A continuum mechanics approach. J Coll Interface Sci 2008; 326: 201–210. [15] Żenkiewicz M. New method of analysis of the surface free energy of polymeric materials calculated with Owens-Wendt and Neumann methods. Polimery 2006; 51: 584–587.
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[16] Yuk SH, Jhon MS. Contact angles on deformable solids. J Coll Interface Sci 1986; 110: 252– 257. [17] Vafaei S, Podowski MZ. Analysis of the relationship between liquid droplet size and contact angle. Adv Coll Interface Sci 2005; 113: 133–146. [18] Park S-J, Cho M-S, Lee J-R. Studies on the surface free energy of carbon-carbon composites: effect of filler addition on the ILSS of composites. J Coll Interface Sci 2000; 226: 60-64. [19] Kwok DY, Neumann AW. Contact angle measurement and contact angle interpretation. Adv Coll Interface Sci 1999; 81: 167–249. [20] Sommers AD, Jacobi AM. Wetting phenomena on mirco-grooved aluminum surfaces and modeling of the critical droplet size. J Coll Interface Sci 2008; 328: 402–411. [21] Godd RJ. Contact angle, wetting, and adhesion: a critical review. J Adhes Sci Technol 1992; 6:1269-1302. [22] Schmitt M, Heib F. High-precision drop shape analysis on inclining flat surfaces: Introduction and comparison of this special method with commercial contact angle analysis. J Chem Phys 2013; 139: 134–201. [23] Drelich J, The significance and magnitude of the line tension in three-phase (solid-liquidfluid) system. Coll Surface A 1996; 116: 43–54. [24] Diaz ME, Fuentes J, Cerro RL, Savage MD. Hysteresis during contact angle measurement. J Coll Interface Sci 2010; 343: 574–583. [25] Schmitt M, Hempelmann R, Ingebrandt S, Munief W, Durneata D, Groß F, Heib F. Statistical approach for contact angle determination on inkling surfaces: “slow-moving: analyses of nonaxisymmetric drops on a flat silanozed silicon wafer. Int J Adhes Adhes 2014; 55: 123–131. [26] Vendantam S, Panchagnula MV. Constitutive modeling of contact angle hysteresis. J Coll Interface Sci 2008; 321: 393–400. [27] Extrand CW. A thermodynamic Model for Contact Angle Hysteresis. J Coll Interface Sci 1998;207:11–19. [28] Packham DE. Surface energy, surface topography and adhesion. Int J Adhes Adhes 2003; 23: 437–448. [29] Kwok DY. The usefulness of the Lifshitz–van der Waals:acid–base approach for surface tension components and interfacial tensions. Coll Surf A 1999; 156: 191–200. [30] Lee L-H. Roles of molecular interactions in adhesion, adsorption, contac angle and wettability. J Adhes Sci Technol 1993; 7: 583–634. [31] Van Oss CJ, Ju L, Chaudhyry MK, Good RJ. Estimation of polar parameters of the surface tension of liquids by contact angle measurements on gels. J Coll Interface Sci 1989; 128: 313–319. [32] Chinnam J, Das D, Vajjha R, Satti J. Measurements of the contact angle of nanofluids and development of a new correlation. I Com Heat Mass Trans 2015; 62: 1–12. [33] Ajaev V, Gambaryan-Roisman T, Stephan P. Static and dynamic contact angles of evaporating liquids on heated surfaces, J Coll Interface Sci 2010; 342: 550–558. [34] Shang J, Flury M, Harsh JB, Zollars RL. Comparison of different methods to measure contact angle of soil colloids. J Coll Interface Sci 2008; 328: 229–307. [35] Packham DE. Work of adhesion: contact angle and contact mechanics. Int J Adhes Adhes 1996;16: 121–128. [36] Yamamoto K, Ogata S.: 3-D thermodynamic analysis of superhydrophobic surfaces. J Coll Interface Sci 2008; 326: 471-477. [37] Della Volpe C, Maniglio D, Brugnara M, Siboni S, Morra M. The solid surface free energy calculation. I. In defense of the multicomponent approach. J Mat Sci 2004; 271: 434–453. [38] Extrand CW. Uncertainty in contact angle measurements from the tangent metod. J Adhes Sci Technol 2016; 30: 1597–1601. [39] Shael-Levanon S, Marmur A. Validity and accuracy in evaluating surface tension of solid by additive approaches. J Coll Interface Sci 2003; 262: 489–499. [40] Żenkiewicz M, Gołębiewski J, Lutomirski S. Experimental verification of certain van OssGood method elements. Polimery 1999; 44: 212–217. 15
[41] Jańczuk B, Białopiotrowicz T, Zdziennicka A. Some remarks on the components of the liquid surface free energy. J Coll Interface Sci 1999; 211: 96–103. [42] Żenkiewicz M. Experimental verification of certain van Oss-Good methods elements. Polimery 1999; 44: 212–217. [43] Lee L-H. Correlation between Lewis acid-base surface interaction components and linear salvation energy relationship solvatochromic and parameters. Langmuir 1996; 12: 1681–1687. [44] Hołysz L. Investigation of the effect of substrata on the surface free energy components of silica gel determined by thin layer wicking method. J Coll Interface Sci 2000; 35: 6081–6091. [45] Rudawska A, Danczak I, Müller M, Valasek P. The effect of sandblasting on surface properties for adhesion. Int J Adhes Adhes 2016, 70, 176–190. [46] ASTM D 5946-04. Standard Test Method for Corona-Treated Polymer Films Using Water Contact Angle Measurements. [47] Expression of uncertainty in calibration. Document EA-4/02 GUM Warsaw, Poland 1999 (available on: http://bip.gum.gov.pl/pl/bip/px_ea_4_02.pdf, (access on: December 2016) [48] http://physics.nist.gov (access on: December 2016) [49] Kubisa S., Moskowicz S. Uncertainty of measurement. Attempt to systematize concepts and methods of calculation. Measurement, Automation, Control 1 (2004) 32–36 (in Polish). [50] Rudawska A, Jacniacka E. Analysis of determining surface free energy uncertainty with the Owens-Wendt method. Int J Adhes Adhes 2009; 29: 45–457. [51] Schuster JM, Schvezov CE, Rosenberger MR. Analysis of the results of surface free energy measurement ofTi6Al4V by different methods. Proc Mat Sci 2015; 8: 732–741. [52] Otomański P. Utilization of the approximate methods to determine the uncertainty of measurement results in indirect measurements. Measurement Automation Monitoring 2005; 9: 5–8 (in Polish). [53] Dietrich E, Schulze A. Methods of staying in the field of processing, processing and processing of products. Notika System. Poland, Warsaw 2000 (in Polish). [54] Rudawska A. Surface free energy and geometric structure of some epoxy composites. Polimery 2008; 53: 452–456.
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PII: DOI: Reference:
S0143-7496(18)30008-3 https://doi.org/10.1016/j.ijadhadh.2018.01.006 JAAD2112
To appear in: International Journal of Adhesion and Adhesives Accepted date: 4 January 2018 Cite this article as: Anna Rudawska and Elżbieta Jacniacka, Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good m e t h o d , International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2018.01.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Anna RUDAWSKA1, Elżbieta JACNIACKA2 1
Lublin University of Technology, Faculty of Mechanical Engineering, Nadbystrzycka 36 Str, 20-618 Lublin, Poland, e-mail: [email protected], +48 81 5384232 2 Lublin University of Technology, Faculty of Mechanical Engineering, Nadbystrzycka 36 Str, 20-618 Lublin, Poland, e-mail: [email protected], +48 81 5384228 Evaluating uncertainty of surface free energy measurement by the van Oss-Chaudhury-Good method Abstract. Among the great number of methods for measuring the surface free energy of solids, all originate in contact angle measurements, which are carried out using various probe liquids, and employ certain relationships and assumptions. Contact angle measurement can be distorted by a number of factors that occur during the measurement and distort obtained values, and therefore must be accounted for. The present work describes the application of the van Oss-Chaudhury-Good approach in determining the surface free energy of aluminium sheets with the use of three probe liquids: water, diiodomethane and glycerol, with assumed values of the surface free energy components. The study employed direct measurement of the contact angle with the goniometer. Subsequently, measurement uncertainty with the van Oss-Chaudhury-Good method was evaluated, which revealed that the major source of uncertainty is the discrepancy between constant values describing properties of probe liquids. Keywords: surface free energy, van Oss-Chaudhury-Good method, contact angle, uncertainty
Introduction Adhesion is a surface phenomenon, which is harnessed in a wide range of technical spheres, such as, inter alia, adhesive bonding, painting, varnishing, printing, coating with various layers and production technology [1-4]. To learn the adhesive properties of materials is crucial to assessing their suitability for a given application, reaction under given conditions or in the processes highlighted above. The surface characteristics of a substrate can influence the level of adhesion between the substrate and an adhesive and thus the strength of a bonded joint[4]. The adhesion properties of materials can be determined using the surface free energy (SFE). The surface free energy (SFE) is a characteristic parameter of a material strongly associated to the state of equilibrium of the atoms in the surface whose value depends on the nature and the aggregate state of material [5]. Determination of the SFE of solids requires employing indirect methods, since direct methods are exclusively applicable in liquids. A number of indirect methods are applied in the SFE determination, e.g. the approaches of Fowkes, Owens-Wendt, van Oss-Chaudhury-Good, Zisman, Wu or Newman [6-11]. Since these approaches represent indirect methods, the measured value of the SFE may be distorted by a number of external factors, concurrent in the process, as well as the assumptions made in calculations [5, 6, 12-15]. The contact angle measurement of probe liquids on a solid surface lies at the heart of a number of methods for calculating the SFE from different assumptions [16-19]. The major limitation of these methods is the existence of a number of factors that may distort the calculated value of the SFE [20, 21], among which we ought to mention the probe liquid drop volume, drop shape [22,23], drop spreading time on the analysed surface (thermodynamical equilibrium in reference to metastability of the drop and of the contact angle, and physical interpretation of advancing and receding angle, and also contact angle hysteresis) [24, 25, 26], the type of probe liquid [27], surface roughness and topography [1,28], physical and chemical uniformity of the analysed surface or the temperature of the test [14, 29-33] and also methods to measure contact angle [34,35]. Moreover the contact angle is 1
sensitive to actual physico-chemical conditions of the solid-liquid interface [36]. Consequently, these factors should be considered during experiments. Furthermore, certain simplifications and assumptions are introduced in the SFE measurement, which indicates the need for evaluation of measurement uncertainty. In [37] the authors show that the type of probe liquid applied in tests affects the obtained values of the SFE. At the same time, assuming different relationships for water in determining the three components of the SFE, which is the foundation of the van Oss-Chaudhury-Good approach, may affect the SFE values obtained in tests [10]. Extrand [38] presented the uncertainty in contact angle measurements for the tangent method. In [39] the authors presented general information regarding measurement error in the SFE determination with the van Oss-Chaudhury-Good method, and juxtaposed the calculated error with the error produced with the Owens-Wendt approach. Nevertheless, one issue is that the error was calculated from the scatter of contact angles obtained for different probe liquids. Our technique has involved evaluation of uncertainty of assumed constant values for probe liquids and uncertainty of the measuring chain. Our study employs a technique to assess the type of material used as substrate in adhesive joints in further sections of this paper, and obtained test results. We have also exploited a different methodology for measuring the uncertainty in a contact-angle-based technique for SFE determination. Uncertainty of surface free energy of the solid depends not only on uncertainty of the contact angle of the model liquid on a given solid surface but also on that of both surface tension and contact angle measurements on the reference solids which were used for determination of components and parameters of the model liquid surface free energies. It should be remembered that van Oss et al. assumed that the LW component of water is equal to 21.8 mN/m. For the first time this was determined by Fowkes as the dispersion component from the interfacial tension of water-n-alkane. Next for the other liquids, van Oss et al. determined the LW component from the contact angle on an apolar solid whose surface free energy results were obtained only from LW intermolecular interactions. Knowing LW values determined in such a way and assuming that acid and base parameters of water have the same values as well as the contact angle of water, diiodomethane and for example glycerol on a monopolar solid (for example PMMA), van Oss et al. determined relative values of acid and base parameters of acid-base components of the polar liquids surface tension. The following paper employs the van Oss-Chaudhury-Good approach to determine the SFE, the results of which were analysed with respect to the measurement uncertainty of contact angles of selected probe liquids. The van Oss-Chaudhury-Good method (vOCG) The Lifshitz-van de Waals/acid-base approach proposed by van Oss, Chaudhury and Good (the van Oss-Chaudhury-Good method) is one of the indirect methods for solid surface free energy determination [1, 19, 40]. The method draws on the assumption proposed by Fowkes [32], which states that the surface free energy of any liquid and solid may be expressed as the sum of its surface free energy components. The components are distinguished based on different interactions occurring at the solid-liquid interface: dispersive, dipole (dipole-dipole, dipole-induced dipole), electrostatic, hydrogen bonds, -type bonds or electron donor-acceptor bonds [10]. The fundamental equation which relates the interfacial tension between solid and liquid (SL) to solid surface tension (S), liquid surface tension (L), and the contact angle () between a drop of liquid and a chemically homogeneous, non-adsorbing, smooth and horizontal solid surface, is given by the well-known Young’s equation (1) [12]: SL = S - Lcos (assumption: > 0)
(1)
According to the approach of van Oss et al. [6, 12, 41], the surface free energy () of a liquid or solid can be expressed by two components (2):
2
γ γ LW γ AB
(2)
where is the polar component, which results from Lifshitz-van der Waals (London dispersion, Debye induction and Keesom orientation forces), whereas AB is the polar component, which results from complementary acid–base interactions (electron donor, -, and electron acceptor, +). LW
Due to their mutual compensation, dipole-dipole interactions are weak, therefore their involvement in the surface free energy is relatively insignificant (about 2%), hence may be omitted. Accordingly the component LW results largely from dispersive interactions of London [9]. AB interactions most frequently occur as a result of hydrogen bonding interactions and can be expressed by the following relationship (3) [9, 42]:
γ AB 2(γ i γ i )0,5
(3)
where: i+ – denotes the parameter of the surface free energy of a Lewis acid, i- – is the parameter representing the surface free energy of a Lewis base, indexed i denotes the subsequent solids or probe liquids. According to the vOCG method, the measurement of the SFE of analysed materials consists in measuring the contact angle of three probe liquids on their surface and calculating the S for the set of three equations (4) [9]: ) 0,5 (γ γ ) 0,5 γ (1 cosΘ )/2 (γ LW γ LW ) 0,5 (γ γ Li S S Li i S Li S
(4)
where: i=1,2,3. The contact angle measurement requires the application of two polar and one apolar test liquids, and in order to solve equation (4), knowledge of particular values of applied test liquids is required. Frequently used polar liquids include water, glycerol, formamide or ethylene glycol, whereas apolar liquids (which do not show characteristics of either Lewis acids or bases) are predominantly represented by diiodomethane or α-bromonaphthalene. The method used in this study is detailed in [2, 7, 8, 19]. Using the van Oss-Chaudhury-Good method is rather uncomplicated, but results must be carefully considered on account of the following problems [6, 8, 19, 37, 40, 42]: base components of analysed surfaces are systematically greater than the acidic components, and are therefore generally apt to be much more basic than acidic, the results strongly depend on the choice of the three test liquids, and what is more, an inappropriate liquid may interfere with the structure of the test material, since interaction with the surface of a test material (forming chemical bonds) may lead to its dissolving or swelling, the probe liquid may penetrate the material and therefore the contact angle value of the test liquid may be distorted as a result of the drop departing from the circular, the unknowns of calculations are, in practice, the roots of surface tension components, which in certain cases assume negative values, test errors resulting from the contact angle measurement technique employed, which may be connected with: incorrectly determined tangent to drop at the three-phase contact point [10,15], time difference between depositing the drop on the surface and the measurement of the contact angle [24], disregarding the deformation of the drop from the circular, as well as certain inconsistencies regarding the volume of the test liquid drop discrepancies;
3
the topic of selecting an incorrect reference scale for the determination of the acidic and base components of the surface free energy is currently under investigation and discussion [6]. The values of the coefficient used for acid-base properties of test liquids are arbitrary. For water, the acid-base properties ratio at 20C may be set to 1 (+/- = 1) where each parameter contributes 25.5 mJ/m2 [6, 29, 42, 44]. However this does not mean that water is as equally strong as a Lewis acid and as a Lewis base. The acid and base parameters of the same solvent cannot be compared, however, the acid (or base) parameters of different solvents can be . The methodology used in the study established an arbitrary reference scale, denoting relationships between the parameters of the surface free energy (acidic and base) of particular probe liquids, therefore also of solids analysed in the study with the use of the liquids in question. The employed reference scale, however, may be replaced by a different scale. The form of the equation, +/- = 1, is unrestricted (mathematically), and therefore any reference scale for water may be applied, such as +/- = k, in which case the following assumption for water may be made: + = 25.5 (k)0.5 mJ/m2 and - = 25.5 (k)0.5 mJ/m2. The aforementioned means that with an increase in k, the value of + increases while - decreases; at the same time, the values of SAB, SLW and S remain constant. For instance, Lee [30, 43] and Żenkiewicz [42] assume the following relationship for water: +/= 1.8; this value was based on the results of analysis of solvatochromic parameters of different liquids. The values of + and - calculated from this relationship were equal to 34.2 mJ/m2 and 19.0 mJ/m2 respectively. Although this relationship shows domination of the basic character in the surface layer of polymer material, which confirms test results, this method is rarely included in the SFE measurement with the vOCG method. Della Volpe and Siboni [6], however, apply a different, equally seldom employed, scale +/- = 4.35, where the values of + and - for water amounted to 48.5 mJ/m2 and 11.16 mJ/m2 respectively. The implementation of Volpe and Siboni’s scale results in the change of the SFE of other test liquids, and more precisely the values of the acidic and base parameters (+ and - ) , and to a lesser extent the component of the Lifshitz–van der Waals (LAB). Methodology and subject of the study The study was conducted on 15 specimens made of 100x25 mm 7075PLT0 aluminium alloy sheet, specimen thickness was equal to g = 0.1 mm. Prior to contact angle measurement, the surface of specimens was degreased with acetone, which consisted in wiping the specimen three times with an acetone-soaked cloth. Upon third application, acetone was left to dry. Degreasing was carried out at a temperature of 21±1oC and air humidity of 30±1%. The wetting contact angle measurement was determined 7 minutes after the surface was pre-treated. The van Oss-Chaudhury-Good method for calculation of solid surface free energy was applied, and the measurements were taken using three set test liquids: polar (distilled water and glycerol) and apolar (diiodomethane), with known surface free energies and components. Values of the surface free energy S and its components and parameters as determined with the above measuring liquids are given in Table 1.
4
Table 1. Values of the surface free energy L and its components determined with test liquids [7, 10, 30, 42]. Wetting liquids
Surface free energy and its components and parameters
Water
L [mJ/m2] 72.8
LLW [mJ/m2] 21.8
LAB [mJ/m2] 51.0
L+ [mJ/m2] 25.5
L[mJ/m2] 25.5
Glycerol
64.0
34.0
30.0
3.92
57.4
Diiodomethane
50.8
50.8
0
0
0
The contact angle measurements were made with wetting liquids, taking an average of 8-12 drops with each type of liquid. In total, 50 measurements were executed on 5 specimens of aluminium alloy sheet for each liquid. Approximately 2 l of distilled water and glycerol and 1 l of diiodomethane was the volume of the drop used for each measurement taken at a temperature of 21±1oC and air humidity of 30±1%. The contact angle reading was taken within 5 s of drop formation. Final contact angle measurements were recorded (usually within 10-15 s) once their values remained constant upon drop deposition. Repeated measurements of a given contact angle were all within 2. The tests were conducted using a PGX goniometer, manufactured by the Swedish company Fibro System, and computer image analysis software PG v.3.4.0.1 [45]. Some of the contact angle measurement conditions are derived from the ASTM D5946 standard [46]. The surface free energy of aluminium alloy sheet was calculated only apparently because the contact angle values used for calculating the SFE did not fulfil the Young equation. To calculate the surface free energy of aluminium alloy sheet, in terms of dispersive and polar components, equation 3 was employed. Test results Surface free energy Table 2 and Table 3 show the results of the contact angle measurements for different probe liquids on the surface of aluminium alloy specimen, their surface free energy and its components and parameters. Table 2. Values of the contact angle Contact angle
Θw [°] 78.03
Θg [°] 59.65
Θd [°] 38.66
Standard deviation of contact angle
=3,26
=4.37
=8.78
Table 3. Values of the surface free energy S and its components Surface free energy and its components and parameters
S [mJ/m2] 47.9
SLW [mJ/m2] 40.3
SAB [mJ/m2] 7.7
S+ [mJ/m2] 1.3
S[mJ/m2] 11.2
Each contact angle measurement was repeated 50 times, for each of the applied probe liquids. Presented below are the histograms of obtained values.
5
Observation number
Water
Contact angle, ° Fig. 1. Contact angle of water – histogram
Observation number
Diidomethane
Contact angle, ° Fig. 2. Contact angle of diiodomethane – histogram
6
Observation number
Glycerol
Contact angle, ° Fig. 3. Contact angle of glycerol - histogram Comparative analysis of scatter of the contact angle measurement results for different test liquids indicates that the values obtained for diiodomethane show the biggest difference from the normal distribution, consequently the dispersion of results is the highest, as is the standard deviation. A possible cause of this observation is that the apolar liquid displays the most irregular drop shape on the analysed surfaces, hence resulting in differences in contact angle values.
Surface free energy measurement uncertainty The van Oss-Chaudhury-Good approach is an indirect method for calculation of solid surface free energy, where the estimate of measured values (output) is obtained from the equation for measurement function, by replacing the measured values with estimates of input values [47]. To obtain the SFE measurement equation, we must solve the system of equations (4) with respect to contact angles w, g and d. Using the equations from [39] we obtain the following:
LW 0,5 s
0,5 s
0, 5 s
where:
A1 1 cos w B1 1 cos g C1 1 cos d
2D
A2 1 cos w B2 1 cos g C2 1 cosd
2D A3 1 cos w B3 1 cos g C3 1 cos d
2D
B C A B C A
A1 w d g 1
1
2
2
2
3
0, 5
g
w
0, 5 d
d
g
0, 5 w
w
d
LW 0, 5 g
w
LW 0, 5 d
d
g
LW 0, 5 w
w
d
LW 0, 5 g
g
g d
(5a) (5b) (5c)
0, 5
(6a)
d
0, 5 w
(6b)
w
0, 5 g
(6c)
g
LW 0, 5 d
(6d)
d
LW 0, 5 w
(6e)
w
LW 0, 5 g
(6f)
g
LW 0, 5 d
(6g) 7
C D B3 g w dLW 3
d
0, 5
LW 0 , 5 w
w
LW 0 , 5 g
LW 0, 5 w
g
LW 0 , 5 w
0,5 g
d
d
g
0,5 d
LW 0 , 5 g
w
0,5 d
d
0,5 w
LW 0 , 5 d
g
0,5 w
w
0,5 g
(6h) (6i) (6j)
where the indexed signs denote: S- the SFE and its components for a given material, w - for water, g for glycerol, d- for diiodomethane. Given the relationships (2), (3), (5) and the fact that for apolar liquids (diiodomethane) equation takes the form: C1 1 cos d 2D
d d 0 the
2
S
A2 1 cos w B2 1 cos g C 2 1 cos d D A3 1 cos w B3 1 cos g C3 1 cos d 2D
(7)
Estimated input values were derived from arithmetic means of the wetting angle measurement results (Table 3). Type A standard evaluation of measurement uncertainty was employed. This approach was chosen due to its suitability for studies where a large number of repeated measurements are performed. The empirical model of the mean value is an estimator of the standard deviation of the distribution [47, 48, 49] and is taken as standard uncertainty uA.
w
nw
u Aw ( w )
i 1
2
iw
nw 1 nw
n 1 n nw
u Ag ( g )
2
ig
i 1
g
g
i 1
0.62 0
g
nd
u Ad ( d )
0.460
d
2
id
nd 1 nd
1.24 0
(8)
Apart from uncertainty regarding the scatter of the contact angle measurement results uA, another uncertainty analysed in the study is the uncertainty of the measuring chain instrumentation. A study of measuring chain [50] shows the maximum uncertainty of calibration in measurement of angles max 10 0,017 rad , while its major component is systematic uncertainty. In such cases the authors of [47] suggest that type B evaluation of uncertainty (a uniform or rectangular distribution of possible values) should be applied. uB t
where:
u Bt
max 0.580 3
(9)
- standard uncertainty of measuring chain.
When various uncertainties occur in one study, combined standard uncertainty uc, which uses the law of propagation of uncertainty [47, 48] is calculated, we have that:
8
2 w uBt2 0.740 0.013rad uw w u Aw 2 g uBt2 0.850 0.019rad ug g u Ag 2 d uBt2 1.370 0.024rad ud d u Ad
(10)
For uncorrelated input quantities, combined standard uncertainty of the estimate of input takes the following form [47, 48]: N u 2 y ci2u 2 xi i 1 (11) u(xi)- combined standard uncertainty of the estimate xi ci
y xi
(12)
ci - sensitivity coefficient of the estimate of input x.i The contact angle measurements for different test liquids constitute separate experiments, therefore the danger of correlation is non-existent. Employing (7) and (11) produces the equation of combined standard uncertainty uc(L) of the surface free energy measurement with the van Oss-Chaudhury-Good method: 2
u c ( S ) S u w w S u g g S u d d w g d 2
2
(13)
Standard uncertainty of the contact angle measurement is not an only component of the surface free energy measurement uncertainty. The functions of relationship measurement (7) and equations of coefficients (6a-6j) make use of experimentally determined constants of the surface free energy and its components for the employed test liquids. A key problem with the available literature in the field [2, 6, 7, 9, 10, 27, 29, 42, 51] is that it does not undertake the problem of measuring uncertainty regarding determination of these constants. What can be observed, however, are the discrepancies in the values of the acid-base parameters of the surface free energy, + and -, of test liquids given the constant component LW and total surface free energy. Calculated from the equations (5a-5c), (6a-6j) and (7), the surface free energy and its acid-base component assumed different values, depending on the accepted values of constants; for the following values of parameters of the surface free energy w w 25.5mJ / mm 2 , g 3.92mJ / mm 2 , g 57.4mJ / mm 2 the discrepancies amounted to: S 2.69 mJ / m 2 SLW 0 mJ / m 2 sAB max 2.69 mJ / m 2
For the component of the SF resulting from the long-distance interactions sLW the difference is equal to 0 because in all referenced works the values of this component are identical. In accordance with instructions in [47], in the case when only two quantities occur, the antimodal distribution model U should be employed, and the standard uncertainty ought to be calculated from the relationship: u s s
u s SAB
2
S
5 3
2.07mJ / m 2
AB 2 s
5 3
2.07mJ / m 2
(14)
9
Considering standard uncertainty of variance of constants, we may determine combined uncertainty for the total free energy: (15) u ( ) u u u u 2
2
S
c
w
S
2
S
w
w
g
S
g
g
d
d
d
2 s
s
Evaluation of measurement accuracy required employing expanded uncertainty, calculated from:
U ( S ) k uc S
(16)
where: k- coverage factor. The value of coverage factor responds to the distribution of measurement quantities representing random variables [47, 48, 49, 52]. The hypothesis of normality of distribution of w,g and d was based on the Epps-Pulley test [53], which is employed where the number of values is higher than 8 and lower than 200 (8n200). The distribution of values was identified. The software application used to analyse the data was qs-STAT 3.2. The confidence level was 1-=0.95. Table 4 shows results of statistical analysis. Table 4. Epps-Pulley test results and identified type of distribution Variable
Standard value
Critical free energy
Type of Regression distribution coefficient
w
0.240
0.376
normal
0.990
g
0.215
0.376
normal
0.996
d
1.654
0.376
mixed
0.999
For diiodomethane the standard value of the contact angle is higher than critical (Table 4). Therefore, the null hypothesis for the population of normal distribution at the confidence level 0.95 was rejected. For distilled water and glycol, the standard value is lower than the critical value of the SFE, and therefore it may be assumed that the measurement results are derived from the population of normal distribution. Table 5 shows the uncertainty budget. Table 5. Standard uncertainty budget for the surface free energy Symbol of quantity w
[rad] g
[rad] d
[rad] γS [mJ/m2] γS [mJ/m2]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
6.87
0.09
1.04
0.019
normal
-27.11
- 0.40
0.67
0.024
mixed
42.84
1.02
2.69
2.070
U-type
1.00
2.07
47.99
-
-
-
2.35
Using the algorithm for the surface free energy measurement we can calculate the budget for LW AB components S i S . Sensitivity coefficients for long distance interactions were established from 10
the relationship (5a), whereas for the acid-base interaction from (5b) and (5c). The equation (17) LW AB shows the relationship for component S , and (18) for S . 2
LW LW LW u c ( SLW ) S u w w S u g g S u d d w g d
2
2
2
(17)
AB AB AB uc ( LAB ) S uw w S u g g S ud d us2 SAB w g d 2
2
(18)
Results of calculations are shown in Tables 6 and 7. Table 6. Budget for combined standard uncertainty Symbol of quantity
w [rad]
g [rad]
d [rad]
SLW [mJ/m2]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
0.00
0.00
1.04
0.019
normal
0.00
0.00
0.67
0.024
mixed
28.26
0.68
40.28
-
-
-
0.68
Table 7. Budget for combined standard uncertainty Symbol of quantity
w
SLW
SAB
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
6.85
0.09
1.04
0.019
normal
-27.11
- 0.40
0.67
0.024
mixed
14.58
0.35
2.69
2.070
antimodal U-type
1.00
2.07
7.71
-
-
-
2.15
[rad]
g [rad]
d [rad] SAB 2
[mJ/m ] SAB
[mJ/m2]
The analysis of standard uncertainty budgets, shown in Tables 5-7, indicates that for the SFE measured with the van Oss-Chaudhury-Good method the sensitivity coefficient for the contact angle measurement of diiodomethane is the highest, as is standard uncertainty of this measurement. This
11
result indicates that the share of the diiodomethane contact angle measurement uncertainty in the combined measurement uncertainty of the SFE is considerably higher than of the other test liquids. Therefore, in order to reduce uncertainty in measurement of the surface free energy and its components we need to reduce the standard uncertainty of the diiodomethane contact angle measurement by, e.g. increasing the number of repetitions of this measurement compared to the other LW test liquids. In measurements of the S component of the SFE (long distance interactions – Table 6) standard uncertainty results solely from uncertainty of the contact angle measurement of an apolar test liquid, i.e. diiodomethane, which appears unsurprising, having analysed the equations (5a), (6a), (6b) and (7). For apolar liquids, coefficients A1 and B1 are equal to 0, whereas in the case of the acid-base AB component of the surface free energy, S , the sensitivity coefficients acquire the highest values in the measurement of the contact angle of glycol (Table 7). Nevertheless, what constitutes the highest share in the uncertainty of the total surface free energy and its acid-base component is the uncertainty resulting from the discrepancy in values of constants + and - which describes the properties of probe liquids. These constant quantities are obtained experimentally. The parameter that allows quantification of measurement reliability is expanded uncertainty. Authors of [49] show different methods of approximation to estimate expanded uncertainty. However, works [47] and [48] suggest that it is the method described with the relationship (15), and for standard accuracy measurement k = 2, which should be employed. Table 8 shows the results of evaluating expanded uncertainty of measurement of the SFE and its components.
Table 8. Expanded measurement uncertainty of the SFE and its components Symbol of quantity γS [mJ/m2]
γ [mJ/m2] 47.99
u [mJ/m2] 2.35
k 2
U [mJ/m2] 4.70
SLW
40.28
0.68
2
1.36
7.71
2.15
2
4.30
[mJ/m2] SAB
[mJ/m2]
The results in Table 8 show clearly that, although its value is significantly lower than that of the component LW, it is the correct determination of the acid-base component of the SFE AB that is critical for accuracy of measurement of the total free energy. For comparison, the results of the contact angles of distilled water and diiodomethane were used in the determination of the SFE with the Owens-Wendt method. Similarly, as in the previous analyses, the uncertainty budget was established according to the same procedure, which we have described in detail in our previous work [50]. The Owens-Wendt method assumes that the surface free energy (S) is a sum of two components: polar (Sp) and dispersive (Sd), and that they all are characterised by the following relationship (19) [2, 19, 40, 44]: S = Sd + Sp
(19)
The polar component contains the dipole-induced dipole (Debye-van der Waals interactions) and dipole-dipole (Keesom-van der Waals) interactions as well as the hydrogen bond and pi-electron interactions [2, 13, 15, 41]. Dispersive forces are the second component of the surface free energy and the London forces describe dispersive intermolecular interactions (included dipole-induced dipole interactions). 12
To determine polar and dispersive components of the surface free energy it is necessary to measure the contact angle [14, 31, 45] of surfaces of tested materials with two test liquids. Probe liquids used to this aim are liquids whose surface free energy and its polar and dispersive components are known. The components Sd and Sp of the tested materials can be determined using relevant formulas given in the studies [2, 3, 10, 51, 54]. Table 9 shows the uncertainty budget for the measurement of the surface free energy with the Owens-Wend
method
for
given
constant
p d [29]: w 72.8; w 51.0; w 21.8 ,
quantities
d 50.8; dp 0; dd 50.8 .
Table 9. Budget for combined standard uncertainty S for the O-W method Symbol of quantity
w
[rad] d
[rad] S [mJ/m2] 2 S [mJ/m ]
Quantity estimate
Standard uncertainty
Type of distribution
Sensitivity coefficient
Share in combined uncertainty
1.36
0.013
normal
- 25.78
- 0.33
0.67
0.024
mixed
40.13
0.96
3.06
2.360
antimodal u-type
1.00
44.29
-
-
-
2.36 2.54
The uncertainty budgets for the SFE determined with the Owens-Wendt and the van OssChaudhury-Good methods bear certain resemblances. In both cases the highest proportion of uncertainty was represented by different values of constants characterising the test liquids. In [19] the authors give different values of the dispersive and the polar component at constant surface free energy. Uncertainty calculated from the formula (15), where the coverage factor k = 2 takes the following form: U SOW 5.08mJ / m 2 Authors of [18] evidenced that different methods for measuring the surface free energy produce different results, which is confirmed by the fact that the results in Table 5 and 9 similarly show certain differences. To determine whether these are statistically significant, the Student’s t-test was conducted at the level of significance = 0.05, and the degree of freedom equal to f =(n2+n1-2)=98. The calculations based on the following relationship: t , f
SOG SOW u 2 SOG u 2 SOW
(20)
where: SOG - the SFE measured with the van Oss-Chaudhury-Good method u 2 SOG - standard uncertainty of the SFE measured with the van Oss-Chaudhury-Good SOW - the SFE measured with the Owens-Wendt method u 2 SOW - standard uncertainty of the SFE measured with the Owens-Wendt method Relationship (20) describes large samples and approximated values of standard uncertainty, and both these conditions are in this case met. The test statistics t,f = 1.068 is lower than the critical 13
value, t,fkryt = 1.984, therefore it may be concluded that the SFE calculated with the van OssChaudhury-Good and the Owens-Wendt methods are equivalent. Summary Measurement uncertainty with the van Oss-Chaudhury-Good method revealed that the major source of uncertainty is the discrepancy between constant values describing properties of probe liquids. Expanded uncertainty of the SFE measured with the van Oss-Chaudhury-Good method is equal to 4.30 mJ/m2, which amounts to 9% of the measured quantity. The uncertainty of the measurement may be optimised by introducing additional series of measurement repetitions. However, in the presented case the expected effect will not be reached as the results in Tables 8, 9, 10 and 11 indicate that it is the uncertainty of constants characterising the test liquids that represents the major proportion of total uncertainty, both for the van Oss-Chaudhury-Good and the Owens-Wendt methods. As long as no standards specifying these constants are introduced, the results of measurements will not be equivalent or credible, and the SFE measurement may be included merely for comparison; an issue which has been raised by researchers in the field. The test results obtained from this study lend support to previous findings in the literature and highlight the demand for further investigations in the field. Our study evidences that it is critical for research papers to include the assumptions that were made in measuring the surface free energy and its components, and particularly to apply the same assumptions to different methods for the surface free energy measurement. References [1] Baldan A. Adhesion phenomena in bonded joints. Int J Adhes Adhes 2012; 38: 95–116. [2] Żenkiewicz M. Adhesion and modification of the high-molecular materials surface layer. Warsaw: WNT, Poland; 2000. [3] Kinloch AJ. Adhesion and Adhesives. New York: Chapman and Hall; 1987. [4] Adams RD. Adhesive boding. Science, technology and applications. UK: Woodhead Publishing Limited; 2010. [5] Schuster JM, Schvezov CE, Rosenberger MR. Influence of experimental variables on the measure of contact angle in metals using the sessile drop method. Proc Mat Sci 2015; 8: 742–751. [6] Della Volpe C, Siboni S. Some reflections on acid-base solid surface free energy theories. J Coll Interface Sci 1997; 195: 121–136. [7] Michalski M-C, Hardy J, Saramago BJ. On the surface free energy of PVC/EVA polymer blends: comparison of different calculation methods. J Coll Interface Sci 1998; 208: 319–328. [8] Greiveldedinger M, Shananhan MER. A critique of the mathematical coherence of acid/base interfacial free energy theory. J Coll Interface Sci 1999; 215: 170–178. [9] Hołysz L. Surface free energy components of silica gel determined by the thin layer wicking method for different layer thicknesses of gel. J Mat Sci 1998; 30: 445–452. [10] Żenkiewicz M. The analysis of principal conditions of van Oss-Chaunhury-Good’s method in investigations of surface layer of polymer materials. Polimery 2006; 51: 169–176. [11] Kaelble DH. Dispersion-polar surface tension properties of organic solids. J Adhes 1970; 2: 66–81. [12] Faibish RS, Yoshida W, Cohen Y. Contact angle study on polymer-grafted silicon wafers. J Coll Interface Sci 2002; 256: 341–350. [13] Fernández V, Khayet M. Evaluation of the surface free energy of plant surfaces: toward standardizing the procedure. Front Plant Sci 2015; 6: 510 (doi:10.3389/fpls.2015.00510). [14] Barberis F, Capurro M. Wetting in the nanoscale: A continuum mechanics approach. J Coll Interface Sci 2008; 326: 201–210. [15] Żenkiewicz M. New method of analysis of the surface free energy of polymeric materials calculated with Owens-Wendt and Neumann methods. Polimery 2006; 51: 584–587.
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