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Abstract

During the processes occurring at a solid surface, the changes in the surface free energy take place. The knowledge about surface free energy is very helpful for understanding the processes taking place on the surface. However, experimental determination of solid surface free energy is still not a fully solved problem. In this paper, some problems dealing with calculation of solid surface free energy from contact angle are discussed based on literature values of advancing and receding contact angles measured on four different fluoropolymers surface. The four approaches most often used for the calculation are described and especial focus on the approach in which both the advancing and receding contact angles is paid. It is concluded that using probing liquids the absolute value of solid surface free energy cannot be determined. However, the determined apparent values of the energy are very helpful to understand the conditions necessary for a given process to occur.

Keywords

Fluoropolymers contact angles surface free energy

Introduction

The processes occurring on a solid surface are accompanying with the surface free energy changes. Some important surface processes are wetting, cleaning, washing, anticorrosion coverages, painting, chemical plant protection, mineral flotation, adsorption, etc. The knowledge about surface free energy is very helpful for understanding the processes taking place on the surface. However, while there are several good methods for determination of surface free energy of a liquid (surface tension of liquid), so experimental determination of surface free energy of a solid is still not a fully solved problem (Bormashenko, 2013; Chibowski, 2007; Chibowski and Terpiłowski, 2009; Della Volpe and Siboni, 2008; Etzler, 2003; Marmur, 2006, 2009; Marmur et al., 2017, and the references therein).

The most often used method relies on measurement of wetting contact angles and then application of the existing theoretical approaches to calculate the energy, although inverse gas chromatography technique is sometimes applied too (Mohammadi-Jam and Waters, 2014; Yao et al., 2015). If behind the liquid droplet the solid surface is bare, the contact angle formed by the droplet is termed as the advancing contact angle θ_a. Then, when the three-phase line has retreated, e.g. by sucking a volume of the liquid drop into the syringe, the contact angle of this “new equilibrium” is smaller, and it is called receding contact angle θ_r. The difference between the advancing and receding contact angles is defined as contact angle hysteresis (CAH), H (Chibowski and Jurak, 2013; Kuchin and Starov, 2015a, 2015b). Actually, there are also other definitions of appearing contact angles (Marmur, 2006, 2009; Marmur et al., 2017). Most often used probe liquids for contact angle measurement are polar water and formamide and apolar diiodomethane. The theoretical models of solid surface free energy are in fact based on Young’s equation (Young, 1805)

γ SV = γ 1 v \cos θ + γ s 1

(1) where γ_sv is the solid surface free energy in equilibrium with the liquid vapor, γ_lv is the liquid surface free energy (surface tension), γ_sl is the solid/liquid interfacial free energy, and θ is the contact angle at three-phase solid/liquid/gas contact line.

There are different theoretical approaches and the most often applied for determination of a solid surface free energy are the following:

1. Lifshitz–van der Waals/acid base which was proposed by Van Oss et al. (1986, 1988). In this approach, the total surface free energy consists of the apolar Lifshitz–van der Waals component $γ_{s}^{LW}$ and the polar Lewis acid–base $γ_{s}^{AB}$ component which is geometric mean of the electron donor $γ_{s}^{-}$ and electron acceptor $γ_{s}^{+}$ parameters

γ s = γ_{s}^{LW} + γ_{s}^{AB} = γ_{s}^{LW} + 2 (γ_{s}^{+} γ_{s}^{-}) 1 / 2

(2)

And the interfacial free energy is expressed

γ sl = {(γ_{s}^{LW}) 1 / 2 - (γ_{l}^{LW}) 1 / 2} 2 + 2 {(γ_{s}^{+}) 1 / 2 - (γ_{l}^{+}) 1 / 2} \cdot {(γ_{s}^{-}) 1 / 2 - (γ_{l}^{-}) 1 / 2}

(3)

The apolar $γ_{s}^{LW}$ component besides main London dispersion interaction includes Keesom permanent dipole force and Debye dipole induction force. However, according to Van Oss et al. (1986, 1988), the contribution of the Keesom and Debye forces amount only ca. 5% to the total Lifshitz–van der Waals interactions operating at a surface. The Lewis acid–base interaction, $γ_{s}^{AB}$ , in most systems results from the hydrogen bonding, i.e. proton donor (electron acceptor) $γ_{s}^{+}$ and proton acceptor (electron donor) $γ_{s}^{-}$ . Moreover, most of the substances characterize small value of $γ_{s}^{+}$ in comparison to $γ_{s}^{-}$ one, because they are basic (Van Oss et al., 1997). In this approach, the work of adhesion W_A is expressed as follows

W A = γ l (1 + \cos θ) = 2 (γ_{s}^{LW} γ_{l}^{LW}) 1 / 2 + 2 (γ_{s}^{+} γ_{l}^{-}) 1 / 2 + 2 (γ_{s}^{-} γ_{l}^{+}) 1 / 2

(4)

To determine the solid surface free energy components, one has to solve simultaneously three equations of type equation (4) using the advancing contact angle of three different probe liquids whose surface tension components are known and at least two of them are polar. From the energy components total surface free energy can be calculated.

2. Owens–Wendt (O–W) (Owens and Wendt, 1969) based on Fowkes’ (1963) pioneering ideas assumed that the surface free energy can be expressed as a sum of two components: London dispersion interactions $γ_{s}^{d}$ and polar $γ_{s}^{p}$ component

γ l (1 + cos θ) = 2 (γ_{s}^{d} γ_{l}^{d}) 1 / 2 + 2 (γ_{s}^{p} γ_{l}^{p}) 1 / 2

(5) where

γ_{1}^{d}

and

γ_{1}^{p}

are mean London dispersion component (superscript d) and polar component (p) of liquid (l), respectively. The equation allows determination of a solid surface free energy from advancing contact angle, for example, of polar water and apolar diiodomethane. The sum of

γ_{s}^{d}

and

γ_{s}^{p}

consists of the total solid surface free energy γ_s.

3. Neumann’s equation of state (Li and Neumann, 1990, 1992; Neumann et al., 1974; Tavana and Neumann, 2007) allows calculation of a solid total surface free energy using measured advancing contact angle and numerical solution of equation (6) in which the coefficient β (0.0001247 m²/mJ) was determined experimentally

\cos θ = - 1 + 2 \sqrt{\frac{γ SV}{γ LV}} e - β (γ LV - γ SV) 2

(6)

4. Measuring the advancing $θ a$ and receding $θ r$ contact angles of a probe liquid and applying the CAH approach proposed by Chibowski (2003, 2007) and Chibowski and Perea-Carpio (2002), the total surface free energy γ_s can be calculated by equation (7)

γ s = \frac{γ L (1 + \cos θ a) 2}{2 + \cos θ r + \cos θ a}

(7)

This is the only approach in which both the advancing and receding contact angles are applied in calculation of the free energy and the privilege is that only one liquid is needed to evaluate the energy. It should be stressed that determination of surface free energy of solids is not a fully solved problem. The values determined by different approaches, even using the same probing liquids, may differ more or less depending on the kind of solid surface. Therefore, application of different approaches allows verification of the apparent surface free energy of investigated surface (Chibowski and Jurak, 2013; Chibowski and Terpiłowski, 2009). It should be stressed that all the above-mentioned methods are debatable and were criticized in the numerous published papers which would be difficult to cite here. Most of the problems are discussed in the above cited papers and in the references therein quoted. However, because there is not any other method for solid surface free energy determination, the existing ones are useful for better understanding of the processes occurring at interfaces. Because the purpose of this paper was not a general critical review of the existing problems, therefore it is mainly focused on application of CAH for solid surface free energy determination and to point out importance of orientation of the probe liquid molecules for the calculated values of solid surface free energy. For this purpose using the literature data of advancing and receding contact angles of homological series of n-alkanes and other liquids, the surface free energy of four polymeric solids is discussed.

Surface free energy of some fluoropolymers

Tavana et al. (2007) published a very precisely measured advancing and receding contact angles of above-mentioned high purity liquids measured on four different fluoropolymer films deposited on silicon surface by dip coating whose repeat units are shown in Figure 1. As is seen the polymers differ in their structure and number of flour atoms present in the units. This reflects in the advancing and receding contact angles which are presented in Table 1 for n-alkanes and in Table 2 for other liquids which were adapted from Tavana et al.’s (2007) paper. In the tables, there are given density of the liquids which will be used later.

Figure 1.

The polymer units: (a) poly[4,5-difluoro-2,2bis (trifluoromethyl)-1,3-dioxole-co-tetrafluoroethyene], 65 mol% dioxole (Teflon AF 1600); (b) poly(2,2,3,3,4,4,4-heptafluoro-butyl methacrylate) (EGC-1700); (c) poly(ethene-alt-N-(4(perfluoroheptyl carbonyl)amino-butyl) maleimide) (ETMF); and (d) poly(octadecene-alt-N-(4-(perfluoro heptylcarbonyl) aminobutyl)maleimide) (ODMF) (from Tavana et al. (2007)).

Table 1.

Advancing and receding (extrapolated) contact angles (degrees) of n-alkanes on the polymers in degrees and the liquids surface tension (mN/m) (adapted from Tavana et al., 2007), and their density (g/cm³).

n-Alkane	Solid
	n-Alkane surface tension (mN/m)	n-Alkane density (g/cm³)	Teflon AF 1600	EGC-1700	ETMF	ODMF
	n-Alkane surface tension (mN/m)	n-Alkane density (g/cm³)	θ_A/θ_R	θ_A/θ_R	θ_A/θ_R	θ_A/θ_R
Heptane	20.03	0.684	47.2/39.5	48.7/35.1	60.3/58.2	–
Octane	21.53	0.703	52.5/45.8	53.4/46.2	64.0/61.7	–
Nonane	22.64	0.718	56.2/49.9	57.0/50.0	66.6/64.4	–
Decane	23.54	0.730	59.3/53.6	59.8/52.6	68.8/66.7	67.1/63.1
Undecane	24.47	0.740	61.6/56.0	62.0/55.1	70.7/68.3	68.7/64.2
Dodecane	25.49	0.749	63.8/58.4	64.0/58.0	72.2/69.8	70.0/65.2
Tridecane	26.04	0.756	65.7/59.7	65.6/59.1	73.4/71.5	71.1/66.9
Tetradecane	26.58	0.763	67.2/61.6	67.0/60.8	74.4/72.2	72.7/68.3
Pentadecane	27.07	0.768	68.1/62.5	68.3/62.2	75.7/73.7	74.3/70.0
Hexadecane	27.30	0.773	69.5/64.2	69.3/63.5	76.9/73.7	74.7/70.0

EGC-1700: poly(2,2,3,3,4,4,4-heptafluoro-butyl methacrylate); ETMF: poly(ethene-alt-N-(4(perfluoroheptyl carbonyl)amino-butyl) maleimide); ODMF: poly(octadecene-alt-N-(4-(perfluoro heptylcarbonyl) aminobutyl)maleimide).

Table 2.

Advancing and receding (extrapolated) contact angles (degrees) of probing liquids on the polymers in degrees and the liquids surface tension (mN/m) (adapted from Tavana et al. (2007)), and their density (g/cm³).

Liquid	Solid
	Liquid surface tension (mN/m)	Liquid density (g/cm³)	Teflon AF 1600	EGC-1700	ETMF	ODMF
	Liquid surface tension (mN/m)	Liquid density (g/cm³)	θ_A/θ_R	θ_A/θ_R	θ_A/θ_R	θ_A/θ_R
OMCTS	18.2	0.956	43.7/38.4	42.2/36.8	57.7/54.7	53.6/50.2
DMCPS	18.77	0.958	45.7/39.5	44.5/37.5	58.2/54.7	55.2/51.3
p-Xylene	27.9	0.861	68.7/63.3	64.8/–	75.6/71.5	–/–
o-Xylene	29.3	0.880	71.2/64.7	66.4/–	77.5/74.0	–/–
cis-Decalin	32.2	0.896	75.5/67.9	72.8/63.5	80.9/77.2	–/–
CDDT	33.9	0.89	78.3/70.8	73.7/65.0	82.7/78.5	–/–
Met. silicate	38.7	1.174	83.7/76.5	–/–	86.4/76.8	–/–
Lepidyne	43.2	1.099	89.5/83.1	–/–	91.4/81.9	–/–
1-Bromo-naphtalene	43.7	1.481	84.0/68.1	84.0/68.1	92.1/83.0	–/–

CDDT: trans,trans,cis-1,5,9-cyclododecatriene; DMCPS: decamethylcyclopentasiloxane; EGC-1700: poly(2,2,3,3,4,4,4-heptafluoro-butyl methacrylate); ETMF: poly(ethene-alt-N-(4(perfluoroheptyl carbonyl)amino-butyl) maleimide); ODMF: poly(octadecene-alt-N-(4-(perfluoro heptylcarbonyl) aminobutyl)maleimide); OMCTS: octamethylcyclotetrasiloxane.

The contact angles were measured by sessile drop method. The advancing angles were determined at a very slowly advancing three-phase contact line (0.5 mm/min) and using drop shape analysis profile. Although the advancing contact angles were constant, the receding ones, measured in a similar manner but during the line receding, appeared to decrease on timescale. The decrease depended on the kind of polymer and liquid. Therefore, the authors extrapolated the receding contact angles values to zero time when the three-phase line started to recede. The detailed description of the method and discussion of the reasons of the observed behavior can be found in Tavana et al. (2007). Because purpose of this paper is to analyze the polymer surface free energy, therefore the contact angles values are used as published in the paper but rounded to decimal of degree. First from the contact angles of the liquids the surface free energy values of the polymers were calculated by CAH approach (equation (7)) and they are plotted in Figure 2 versus the surface tension of the probing liquids.

Figure 2.

Surface free energy of fluoropolymers (see Figure 1) calculated from contact angle hysteresis of the liquids whose advancing and receding contact angles were measured on the polymers versus the liquid surface tension.

Analyzing the results in Figure 2 first of all it is seen that the calculated values of the energy increase with increasing surface tension of the probing liquid even up to 6–7 mmJ/m². The surface free energy of the polymers depends also on the number of flour atoms and their arrangements in the unit and therefore Teflon AF 1600 and EGC-1700 possess bigger free energy than ETMF and ODMF polymers, which is completely understandable (see also discussion in Tavana et al. (2007)). However, an interesting question is why the values of energy calculated by CAH model increase with increasing surface tension of the probing liquids. Moreover, even for n-alkanes series this is also the case. As can be seen in Figure 2 the energy increases by ca. 2 mJ/m² if calculated from n-heptane and n-hexadecane contact angles. To easily find an explanation, only the results obtained for Teflon AF 1600 will be analyzed. Obviously the same reasoning would be applied to the rest of the investigated polymers. In Figure 3, the energy of Teflon calculated from CAH and Fowkes’ model (equation (5), note the polar term amounts zero in these systems) is plotted. For comparison the energy was also calculated from the receding contact angles. While the values calculated from CAH increase from 16.3 to 18.0, if calculated from n-heptane to n-hexadecane, so those calculated from advancing contact angle decrease from 14.1 to 12.5 mJ/m². The values calculated from receding contact angles θ_R run parallel to those calculated from θ_A but they are larger by 1.6 mJ/m² because θ_R is smaller (Figure 3).

Figure 3.

Surface free energy of Teflon AF 1600 calculated from CAH approach and from advancing and receding contact angles of n-alkanes. CAH: contact angle hysteresis.

Although the fluoropolymers have low surface free energy indeed, the values determined from θ_A seem to be evidently too small. More appropriate are the values calculated from CAH (Figure 3). The literature values for Teflon PTFE (–CF₂–CF₂–)_n range from 18.0 to 25.8 mJ/m² (Surface Energy Data for PTFE, http://www.accudynetest.com/polymer_surface_data/ptfe.pdf). The question is still open why the values increase or decrease monotonically with increasing the chain length of n-alkane. A light can be shed if one takes into account the anisotropic ( $γ_{h}^{a}$ ) and isotropic ( $γ_{h}^{d}$ ) London dispersion terms of n-alkane chain, which was observed starting from n-heptane and discussed by Fowkes (1980). The anisotropic term results from anisotropy of polarizability of the adjacent chains oriented parallel, which causes an increase in the cohesion energy. This was deduced from the light scattering studies and heats of mixing (Fowkes, 1980). Moreover, these anisotropic forces can interact only with anisotropic polarizable molecules, which is not the case for water, whose molecules are anisotropic at the interface and do not sense the contribution of correlated molecular orientation (CMO). In consequence, the work of water adhesion is weaker than one expect if calculated from the total surface tension of n-alkane.

Figure 4 presents original and CMO surface tension of n-alkanes versus their chain length (Fowkes, 1980). Note that the results for n-tridecane and n-pentadecane are extrapolated because of lack of data. It is seen the surface “activity” of the alkanes significantly decreases with the increasing chain length. Based on these results the correlation factor has been calculated and the values are plotted in Figure 5.

Figure 4.

Original and corrected for correlated molecular orientation surface tension of n-alkanes.

Figure 5.

Correlating factor for n-alkanes calculated on the basis of the results in Figure 4.

In the case of solid hydrocarbons their surface free energy is lower if the chains are oriented normal than parallel to the surface. This weaker interaction is due to the lower surface concentration of interacting groups than in case of parallel orientation. The interacting potential of CH₃ groups is 82% greater than for CH₂ and the ratio of the work of adhesion of the two orientation amounts 3.23 (Fowkes, 1969).

Taking into account the above data it was possible to recalculate the surface free energy of Teflon AF 1600 using only the “active” part of the n-alkanes surface tension. The correlating factor values from Figure 5 and the surface tension of n-alkanes from Tavana et al.’s (2007) paper were applied. The results are shown in Figure 6 both calculated from CAH approach and from the advancing contact angles. For comparison also the values calculated with total surface tension are plotted. The results obtained with the correlated surface tension values are amazing. The values are practically constant in 0.5 mJ/m² range and are not dependent on the n-alkane chain length. Moreover, the values calculated with CMO corrected surface tension and with the advancing contact angles do not decrease with the chain length as they do if calculated with total surface tension (Figure 6). However, the results obtained by CAH model look more consistent and the values are more pleasurable than those, rather too low, calculated from the advancing contact angles.

Figure 6.

Surface free energy of Teflon AF 1600 calculated with total surface tension and CMO surface tension of n-alkanes versus their total surface tension. CMO: correlated molecular orientation.

To better depict the surface free energy of Teflon AF 1600 in Figure 7 are plotted the values, which were calculated by different approaches and with or without the correlation of n-alkanes surface tension, versus the surface tension of n-alkanes used as the probing liquids. As it is seen the most changing values are those calculated by equation of state (equation (6)) which varies from ca. 19 to 27 mJ/m². In this figure there are also plotted the energy values calculated by CAH (equation (7)) but using the n-alkanes surface tension corrected with their density, i.e. the surface tension of given alkane was divided by its density. Thus, calculated values of the energy are only slightly dependent on the n-alkane chain length and the values change only within 1 mJ/m² between 24 and 23 mJ/m² decreasing with increasing n-alkane chain length. However, the values are the biggest among those calculated by other approaches except for those calculated by equation of state, whose mean value agrees with those calculated with corrected surface tension for n-alkane density. The reasoning to use the corrected surface tension of n-alkanes with their density originates from the well-known dependency of these two parameters

γ = c (ρ c - ρ p) 4

(8) where c is the constant, ρ_c is the liquid density, and ρ_p is its vapor density.

Figure 7.

Surface free energy of Teflon AF 1600 calculated using total and CMO surface tension of n-alkanes by different approaches versus their total surface tension. CMO: correlated molecular orientation.

Because for rest probing liquids no correlation of molecular orientation data could be found, contrary to their densities, therefore to find whether calculated values of the Teflon surface free energy could be more coherent, the surface tension values of the liquids from Table 2 were corrected with their density, i.e. divided by density of respective n-alkane. Then they were used for calculation of surface free energy of the Teflon AF 1600. The results are plotted in Figure 8. Unfortunately, thus “corrected” values of the apparent surface free energy of Teflon are still scattered in the range 14–24 mJ/m², which is even a bit larger than in the case of “uncorrected “surface tensions plotted in Figure 1. To mention, the attempt to correct with molecular volume also did not improve coherency of the calculated values.

Figure 8.

Surface free energy of Teflon AF 1600 calculated by CAH approach using “corrected” with density surface tension of the probing liquids γ_l/d. CAH: contact angle hysteresis.

Summary and conclusion

In the light of the above results, it can be concluded that the experimentally determined surface free energy of solids depend to some extent on the probing liquid used for the contact angle measurement and the theoretical approach applied for calculation. It means that thus determined values are not absolute but apparent ones.

The total surface tension of a probing liquid is not appropriate one for determination of real value of a solid surface free energy. This is also true even in the case of such relatively simple systems like nonpolar polymer/apolar n-alkane where only London dispersion forces interact. This is because of molecular orientation of longer n-alkanes causing an increase in the cohesion forces between parallel oriented chains of the molecules. This effect is eliminated if the correlated surface tensions of the alkanes are used for calculations.

One may conclude that no absolute value of solid surface free energy can be determined because its value depends on the kind and strength of forces acting on the solid surface from the probing liquid phase, and the distance of interacting moieties (groups or atoms), and/or the molecule orientation. Nevertheless, these apparent values of the energy are useful for better understanding of processes occurring at the surface (interface).

CAH method gives comparable results of the surface free energy to those obtained by other approaches and possesses convenience that the energy can be calculated from CAH and surface tension of only one liquid.

Footnotes

Acknowledgements

This work has been partially carried out in the frame of Marie Curie Initial Training Network “Complex Wetting Phenomena” (Project number 607861).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Some problems of characterization of a solid surface via the surface free energy changes

Abstract

Keywords

Introduction

Surface free energy of some fluoropolymers

Summary and conclusion

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

References