Article pubs.acs.org/Langmuir
Further Reflections on the Geometric Mean Combining Rule for Interfacial Tension Sylvain Chevalier and Manoj K. Chaudhury* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, United States ABSTRACT: Wettability is a widely used method to estimate the surface (free) energies of solids. The measured contact angles are usually processed within the framework of Fowkes and Good that uses a geometric mean combining rule of interfacial interactions. Recently, this method of calculating the interfacial tension has been questioned as it appears to yield somewhat unphysical results of interfacial energetics in certain situations. We would like to demonstrate that these unphysical results are consequences of the neglect of the preferential enrichment or depletion of the most surface-active functionalities of a molecule composed of various chemical groups at the liquid−air, liquid−liquid, and liquid−solid interfaces that the quintessential Fowkes−Good analysis does not account for. When the base state of the surface energy is estimated using Lifshitz theory and the preferential segregation of the functional groups at the interface is taken into account, the difficulty associated with the Fowkes−Good approach seems to disappear. This, however, raises new challenges and opportunities related to the estimation of surface energetics based on wettability.
1. INTRODUCTION Zisman et al.1−3 pioneered the method of wettability to estimate the surface free energies of low-energy solids. In that method, the cosines of the contact angles (θ) of various liquids measured on a solid surface are plotted against their surface tensions. When the resultant plot is extrapolated to the cos θ = 1 axis, the corresponding surface tension of the putative liquid that would just spread on the surface as a thin continuous film is called the critical surface tension (γc) of wetting. While γc is a measure of the surface free energy (γs) of the solid, one cannot necessarily equate it with γs as there is no guarantee that this would lead to a zero interfacial tension at the solid−liquid interface, a requirement for γc to be equal to γs in Young’s equation. The next major impetus in this field of research was provided by Fowkes4,5 as well as by Good and Girifalco.6 In their treatments, the equations of Young and Dupre7 are combined with the assumption that the work of adhesion of two different materials interacting via dispersion forces can be expressed as a geometric mean of the work of cohesion of the two pure phases. This treatment thus allows an estimation of the interfacial tension of two nonpolar condensed phases as γ12 = γ1 + γ2 − 2 γ1γ2
mean combining rule, bromonaphthalene is soluble in alkane. Furthermore, the interfacial tensions at the interfaces of various hydrogen-bonding liquids and alkanes as predicted by the geometric mean combining rule of dispersion interaction deviate somewhat from their experimental values. The subject of this paper is an attempt to resolve some of the issues raised by Kwok, Lee, and Neumann8 by further processing the geometric combining rule of interfacial interaction within the framework of another important principle of interfacial interaction, namely, the principle of independent surface action, which was conceived long ago by Hardy,9 Langmuir,10,11 and Harkins.12,13 We begin our discussion by pointing out that several methods have evolved in the last few decades to estimate the surface as well as interfacial tensions in condensed phases, especially liquids. One such method is derived from equilibrium statistical mechanics14 that relates the surface or the interfacial tension to the integral of excess transverse stress at an interface; the integral is to be performed along a direction normal to the interface. Another method15−17 relates interfacial tension to the interfacial density profile within the square gradient approximation. A third method18,19 uses Dupre’s relation between surface tension and the work of cohesion, in which it is now common to estimate the latter using Lifshitz’s theory of electrodynamic fluctuations. The popularity of this method stems from the fact that the limitation of the pairwise additivity approximation of intermolecular interactions in a condensed phase is avoided as the continuum level dielectric susceptibilities of the materials are used to estimate interfacial interaction.19−22 Lifshitz theory, however,
(1)
Recently, Kwok, Lee, and Neumann8 published a very interesting and useful paper in which certain flaws inherent in the above method for estimating interfacial tension have been pointed out. Among a list of anomalies pointed out in that paper, one is that the interfacial tension at a bromonaphthalene−alkane interface using eq 1 is found to be positive. Since an interface with a positive excess of surface free energy should be stable, it was pointed out that contrary to the prediction of the geometric © XXXX American Chemical Society
Received: August 15, 2015 Revised: September 17, 2015
A
DOI: 10.1021/acs.langmuir.5b03054 Langmuir XXXX, XXX, XXX−XXX
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disordered region. This experiment, along with that reported in ref 25, thus provide strong evidence that the intrinsic surface energy of a surface comprising methylene groups is higher than that of the methyl groups. The role of the functional groups in wetting and adhesion has also become eminently clear in recent experiments27,28 that combined the methods of contact mechanics and contact angles. The former method studies the area of contact of a deformable hemisphere of an elastomer with a smooth, flat surface of the same material. Johnson−Kendall−Roberts29 (JKR) realized that such a deformation in solid is caused by the same wetting forces as those leading to the spreading of one liquid over a solid. When the JKR method of contact mechanics is used, it is possible to estimate the adhesion energy W of two similar surfaces that, according to Dupre’s equation, should be twice its surface free energy γs. Since the same surface energy can also be deduced from the contact angle of a nonwetting liquid in conjunction with a specific model of interfacial adhesion,7 a comparative study of contact mechanics and wettability is quite ideal in systematic studies of interfacial interactions, not only in air but also in a liquid that allows a measurement of how a solid−liquid interfacial tension30 is modified when a surfactant is injected into a liquid. The directly measured surface free energies23,24 (Appendix A2) of various surfaces in air as obtained from contact mechanics correlate rather well with those obtained from contact angles. Certain interesting anomalies have, however, been observed27,28 in that the estimated value of γs depends, to some degree, on the choice of the test liquid. These studies, nevertheless, clearly illustrated that γs depends on the outermost chemical constitution of a surface and demonstrated that the geometric mean combining rule of Fowkes4 rests upon a rather solid foundation, as far as a first-order estimation of surface energy is concerned. A surface comprising multiple functionalities can, however, display more complex behaviors, for which it is not uncommon to observe the contact angle of a lower-surfacetension liquid (e.g., hexadecane) being larger than that of a higher-surface-tension liquid (e.g., water).31 1.1. Roles of End Groups in the Surface Tension of End-Functional Alkanes. In this paper, we do not intend to treat the subject of the solid−liquid interface; we rather focus our attention on pure liquids and their pairs for which all of the surface and interfacial tensions can be directly measured. However, before we develop this story any further, we elaborate upon the dependency of the surface tension of a liquid on its end groups in the same vein as was done by Owen.32 The data summarized in Figure 1 show how the surface tensions of various alkyl functional liquids depend on their chain length as well as upon the end group functionalities. For example, the surface tension of an alkane decreases with its molecular length when its terminal group is hydroxyl,33 whereas it increases when it is methyl.33−35 The base state of a liquid depends upon its compositional densities that remain uniform all the way from the surface to the bulk. In order to minimize the excess free energy, the surface tension of a liquid should be somewhat lower than this base state. Statistically, there should be a slight enrichment of one of the functionalities over the other. From the viewpoint of ISA, the results in Figure 1 would be easy to understand if the intrinsic surface energy of the terminal hydroxyl group is greater than that of the methylene group of the main chain, and in juxtaposition, the surface energy of the methyl group is lower than that of the CH2 group. The increase or decrease in the surface tension of these alkanes with chain length then is a consequence of the dilution of the end
applies for interactions of two materials at long separation distances in which their molecular details are lost. A major pitfall of this method is that the interaction energy diverges at zero separation distance, which can be bounded only by using an empirical cutoff length19−22 for interfacial distance. Fortunately, this cutoff length is fairly constant for a wide range of materials that allows appropriate conversion between the coefficient of interaction (i.e., the calculated Hamaker constant) and the measured surface tension. Schwinger et al.23 calculated the same interaction using the celebrated source theory in which this cutoff appears in the form of the transverse momentum of the photon contributing to the surface tension. When a homologous series of liquids is considered, the cutoff length19−22 is found to depend somewhat on its molecular weight. Thus, within the above caveat, while the Lifshitz theory provides almost a quantitative estimate of the surface tension of a nonpolar liquid or a solid in its base state, it lacks some of the insightful details of the fine structure of a molecule that make surface chemistry even richer and, perhaps, more interesting. The role of this fine structure is inherent in yet another (semiempirical) method of estimating surface tension that is known as the principle of independent surface action (ISA). Various authors9−12,24 recognized very early in the development of surface science that the intrinsic surface energies of the constituent functional groups of a molecule are not always the same. For alkanes, if the intrinsic surface energy of a methyl group is taken to be lower than that of the methylene groups, then many mysterious behaviors of alkanes can readily be resolved. Owing to its lower intrinsic surface energy, ISA suggests that the CH3 groups of an alkane would be preferentially oriented toward the air−liquid interface, with a concomitant depletion of the CH2 groups. In this configuration, the surface tension of a shorterchain alkane should be closer to that (20 mJ/m2) of the methyl group for a low-molecular-weight alkane, but it should increase and reach the limiting value (∼35 mJ/m2) of the methylene groups as its chain length increases. This situation should, however, change when an alkane is in contact with a highersurface-tension liquid. In that case, more of the CH2 groups should face the liquid displacing the methyl groups from it, thus increasing the energy of adhesion between the two condensed phases. A continuum-mechanics-based theory of interfacial interaction that does not consider this type of surface enrichment of one functionality over the other thus remains incomplete in accurately predicting the interfacial tension of two condensed phases. This idea that the surface or the adhesion energy of a condensed phase is determined by the outermost functional group of a surface is already quite prevalent in the surface chemistry literature.25−28 An influential paper on this subject was published by Bain and Whitesides.25 A related study was carried out later by Lestelius et al.,26 who prepared a surface with chemiadsorbed alkylthiol monolayers. A shorter-chain alkanethiol was made to diffuse from one side of a surface, whereas a longer-chain alkanethiol was made to diffuse from its other side in such a way that a silicon wafer coated with a thin film of gold became imagewise modulated by reacting with these thiols when placed parallel to their diffusing fronts. With this method, very dense monolayers were formed on the two opposite sides of the wafer that projected mainly the methyl groups outward, whereas a disordered phase was created toward the center of the wafer, exposing a number of methylene groups (Appendix A1). A probe liquid such as hexadecane or water exhibits higher contact angles over the ordered regions of the surface than that on the B
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two phases (1 and 2) is calculated using the following equation of Dupre:7 W12 = γ1 + γ2 − γ12
(2)
Here we select the pairs of liquids in such a way that they interact with each other only via dispersion forces so that the measured values of W12 can be correlated to what is obtained from the Lifshitz theory19−22 W12 =
A12 12πl0 2
(3)
Here, l0 is a cutoff length that prevents the divergence of the energy of interaction when the phases are in intimate contact and A12 is the Hamaker constant19−22 of interaction between two phases in air:
Figure 1. End groups dictate how the surface tension of a functional alkane3,32−35 varies with the chain length, which is represented here by the number (n) of carbon atoms in the alkane. The solid lines are guides to the eye.
A12 =
functionality as the chain length increases. Within this scenario, the near independence of the surface tension of bromoalkanes33 with respesct to the chain length of the alkanes is a consequence of the intrinsic surface tension of CH2Br being not all that different from that of the CH2CH2 groups. Polydimethylsiloxanes, where the methyl groups are distributed all throughout the molecule, also exhibit a weak dependence of its surface tension3,32 on its chain length. One theoretical method in which the orientation of functional groups near a surface is accounted for is based on the self-consistent lattice field calculation such as those pioneered by Scheutjens and Fleer,36 Sanchez and Lacombe,37 Theodorou,38 Schlangen et al.,39 and Jalbert et al.40 Here the main chain is considered to be composed of segments linked to each other that can freely orient themselves on a defined lattice, leading to a given conformation. Schlangen and co-workers39 completed this statistical description to describe the evolution of surface tension with the length of the alkane backbone. Koberstein41 nicely summarized these principles in the context of the end-group effects on the surface segregation and surface reorganization of polymers and proposed a simple and elegant idea of how to account for the surface effects of specific end groups, which we will elaborate below. All of these treatments, as we see them, are exemplifications of the principle of independent surface action as proposed by Hardy,9 Langmuir,10,11 and Harkin,12,13 and later by Adam.24 Adam,24 in fact, went one step further by stating that the shielding of the highly polarizable carbon atom by the hydrogen atoms in such a group as CH3 is an important factor underlying its greater surface activity in contact with air in comparison to a CH2 group, where the carbon atom is less shielded. Against the backdrop of the above developments, we combine here Lifshitz’s method of calculating the surface tension of a liquid in its base state, i.e., when there is no preferred orientation of any functional group on the surface, and then take into account the specific surface activity of the terminal group in light of the principle of independent surface action.
Δi =
∞ ∞ ⎡ (Δ Δ )s ⎤ 3 kBT ∑ ′ ∑ ⎢ 1 3 2 ⎥ ⎣ s ⎦ 2 n=0 s=1
(4)
εi(iωn) − 1 εi(iωn) + 1
(5)
In eq 5, ε(iωn) is the dielectric susceptibility that is to be estimated along the complex frequency axis iωn. For dielectric liquids,19,42 this function is ε∞ − ε0
ε(iωn) = 1 + 1+
(1 − β)
( ) ωn ωMW
ε0 − n0 2
+
1+
2
( ) ωn ω IR
+
n0 2 − 1 1+
2
( ) ωn ω UV
(6)
where ε∞ is the static dielectric constant, ε0 is the dielectric constant at the onset of infrared relaxation, n0 is the refractive index in the visible range, and β is the Cole−Cole parameter that is equal to 0.3 for glycerol and 0 for the other H-bonding liquids used to compute the Hamaker constants. ωMW, ωIR, and ωUV are the characteristic absorption frequencies19,20,43,44 in the microwave, infrared, and ultraviolet regions of the dielectric spectra, respectively. For a metallic liquid, mercury,45,46 the dielectric permeability results from a low-frequency plasma oscillation and from the core electron excitation (J = 3/2 and 5/2 states of Hg(II) in the 5d96s2 configuration). ε(iωn) is thus expressed as47 ε(iωn) = 1 + +
η 38.95 + ωn(ωnτ + 1) π
18.38 π
∞
∫10
∞
dω ∫7.94 ω(ωω2−+1.46 ω 2) n
ω − 3.16 dω ω(ω2 + ωn 2)
(7)
with η = 4πσ0 where σ0 represents the dc conductivity and τ is the relaxation time of plasma oscillation. It should be pointed out that the prime in eq 4 indicates that the zero-frequency term of the interaction has to be multiplied by 1/2 and the interaction is sampled at the thermal photon frequencies: ωn = n2πkBT/ℏ with n varying from zero to infinity. In our calculation, the maximum value of n is 2000 when all of the dielectric permeabilities approach unity. Using the dielectric and the optical data available in the literature, we followed the well-established procedures19,42 to calculate the Hamaker constants (Table 1) of different liquids using eq 4. Figure 2 shows that the thermodynamic work of adhesion W12 is directly proportional to A12 (presented in Table 2) over a range of liquid pairs. From the slope of this plot and eq 3 we estimate that l0 = 0.159 (±0.001) nm. Such a universal
2. RESULTS Before embarking upon the role of surface segregation in the surface tension of pure alkanes in contact with air, we examine the relationship between the work of adhesion of these alkanes against various liquids, for which the surface and interfacial tensions have already been measured and reported in the literature including the recent valuable results of Kwok et al.8 The thermodynamic work of adhesion between C
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Table 1. Hamaker Constants of Various Liquids Calculated Using Lifshitz Theory Based on Their Spectroscopic Data and the Surface Tension of Each Liquida characteristic frequencies (rad/s) ε∞
material
n02
pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane hexadecane water glycerol formamide mercury
1.819 1.864 1.899 1.925 1.947 1.965 1.979 1.991 2.002 2.011 2.026 1.755 2.14 2.1
ε0
ωIR (1014)
ωUV (1016)
γ (mJ/m2)
(1020 J)
1060 8 276
5.540 5.540 5.540 5.540 5.540 5.540 5.540 5.540 5.540 5.540 5.540 5.66 3.28 5.50
1.877 1.873 1.870 1.863 1.864 1.873 1.853 1.877 1.852 1.846 1.848 1.793 1.895 1.500
16.05 18.40 20.14 21.62 22.85 23.83 24.66 25.35 25.99 26.56 27.45 72.80 64.00 58.80 484
3.75 4.07 4.32 4.50 4.66 4.82 4.88 5.03 5.05 5.10 5.23 4.20 6.60 6.00 33.00
1.844 1.890 1.948 1.972 1.991 2.005 2.014
80.1 42.5 110
5.2 4.1 5.3
Hamaker constant
ωMW (108)
a The spectroscopic data were obtained from refs 19, 20, and 42−47. The sources for the surface tension values are cited in the text. Here, the Hamaker constants are computed using Lifshitz theory, whether the tabulated surface tensions of the corresponding liquids, the source of which are cited in text, are experimentally measured values.
correlating it with the surface tension of the pure components for reasons that will be obvious below. We also noted that the difference between the Hamaker constant computed for two phases using eq 4 and that deduced from a geometric mean rule is less than 5%, all the way from alkane−water to the alkane−mercury interfaces. These results, coupled with the observation that W12 is directly proportional to A12 over a range of liquid pairs, lead us to conclude that W12 = (W11W22)1/2 or 2(γ1γ2)1/2, which is the classical geometric mean equation of Fowkes.4 The surface tension of n-alkanes in its base state could thus be calculated from γ = A/(24πl02) with the value of l0 = 0.159 nm. It is interesting that the surface tensions of the alkanes calculated using this approach (Figure 3) deviate from their experimental values for smaller alkanes, but the disagreement virtually disappears for higher alkanes. Here, we examine if the above discrepancy can be eliminated by taking into account the effect of the preferential surface segregation of the end groups at the interface. In order to carry out such an analysis, we follow the lead of Koberstein,41 who proposed a unique modification of the
Figure 2. Correlation between the work of adhesion (W12) of various liquids against alkanes as obtained from Dupre’s equation (eq 2) with the Hamaker constant (A12) calculated from Lifshitz theory (eq 4). The fitted line is forced to go through the origin.
behavior of the cutoff length had already been observed in the past19−21 from the ratio of the surface tension to the Hamaker constant. Here, we correlate the Hamaker constant with the work of adhesion between two different materials instead of
Table 2. Work of Adhesion and the Hamaker Constant for Various Pairs of Liquidsa A12 (10−20 J)
γ12 (mJ/m2)
a
material
water
glycerol
formamide
pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane hexadecane water
50.90 51.40 51.90 52.50 52.40 53.20 53.10 53.70 54.00 54.50 55.20
26.50 29.93 29.91 29.42 30.22 28.03 31.12 24.74 26.40 31.40 31.90
25.97 27.50 26.98 25.41 27.22 27.73 26.92 26.10 24.82 29.14 27.25
mercury 378.00 375.00 372.00
water
glycerol
formamide
3.97 4.13 4.26 4.35 4.42 4.50 4.53 4.60 4.61 4.63 4.69
4.97 5.18 5.34 5.45 5.55 5.64 5.67 5.76 5.77 5.80 5.88
4.74 4.94 5.09 5.19 5.29 5.38 5.41 5.49 5.50 5.53 5.60
426.00
W12 (mJ/m2) mercury 11.59 12.24 12.40
water
glycerol
formamide
37.95 39.80 41.04 41.92 43.25 43.43 44.36 44.45 44.79 44.86 45.07
53.55 52.47 54.23 56.20 56.63 59.80 57.54 64.61 63.59 59.16 59.57
48.88 49.70 51.96 55.01 54.43 54.90 56.54 58.05 59.97 56.22 59.02
11.67
mercury 124.40 130.62 134.85
130.80
The interfacial tensions were obtained from refs 4, 8, 32−34, 47, and 48. The surface tensions of the liquids in air are given in ref 1. D
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the terms in eq 10 are divided by the Hamaker constant and re-expressed as follows
RTC b γ = C1 − λ A A
(11)
where C1 = 1/24l02. Using the calculated Hamaker constants and the experimental values of the surface tensions of n-alkanes, we expect that a plot of γ/A versus RTCb/A should be a straight line, providing the values of two distance parameters: λ and l0 from the slopes and the intercepts of the ensuing plot. Figure 4 demonstrates that such a plot is indeed linear, thereby Figure 3. Surface tension of n-alkanes as obtained from the Lifshitz theory (red circles) as a function of the total carbon atoms in the molecule. Blue squares are the experimental values.29 The solid lines are guides to the eye.
Gibbs−Duhem equation of adsorption of a complex molecule in accounting for the end group effect by assigning disparate chemical potentials to the end functionality and the backbone of a long-chain molecule. A similar idea was also used previously in order to elucidate the thermodynamics of the micellization49,50 of surfactant molecules in water. It should be pointed out that neither the chemical potentials nor the number densities of the end and midportion of the chains are independent of each other. Hence, there is no extra degree of freedom beyond what is obtained from the standard Gibbs phase rule. This treatment stipulates only that the standard chemical potential of the end group (and thus the main chain) depends on its distance from the free surface so that μ0(x) + RT ln(C(x)) = μb0 + RT ln(Cb) (where index b stands for bulk); thus, the concentration of the end group follows a profile given by the Boltzmann equation. On the basis of the previous studies, we surmise that the surface excess (or depletion) of the end groups leads to a change in the surface tension of the liquid following the Gibbs adsorption equation:
dγ = −Γ dμ
Figure 4. Plot of γ/A versus RTCb/A for n-alkanes, the slope and the intercept of which yield the values of the Gibbs length λ as well as a molecular cutoff length l0.
yielding values of λ and l0 as 0.118(±0.008) nm and 0.154 (±0.001) nm, respectively. According to this analysis, the surface tension should be separated into two parts: the first one is the base value as related to the Hamaker constant, and the second one arises directly from the enrichment of the CH3 groups on the surface. Previous studies39 based on lattice field theory led to an equation for the surface tension of n-alkanes as a function of its molecular weight (Mw) that conforms to an earlier equation proposed by LeGrand and Gaines:51
(8)
where μ is the chemical potential of the end functionality and Γ is the relative surface excess of the end group that is related to the bulk concentration (Cb = 2ρ/Mw) of the end group as Γ=
∫ (C(x) − C b) dx = λC b
γ = γ∞ −
(12)
Equation 12 uses the base value of γ∞ as 38 mJ/m to calculate the molecular-weight-dependent surface tension of a specific alkane using a value of the exponent α as 2/3. Equation 11, on the other hand, uses a specific base state for a specific alkane. Surface tensions calculated using eqs 11 and 12, however, overlap each other (Figure 5). 2.1. Effect of Temperature. An important test regarding the validity of eq 11 would be to examine how it is able to account for the effect of temperature on the surface tension of a liquid. In order to examine the temperature dependence of surface tension,52 we use eq 11 in the following form: 2
(9)
where ρ and Mw are the density and the molecular weight of an alkane. As C(x) = cb exp(−ϕ(x)/kBT), the value of λ, the Gibbsian distance parameter, should be in the range of the fraction of the diameter of a methyl group, i.e., in the range of 0.1 to 0.2 nm. Using eqs 8 and 9, we write Δγ = γ − γ ′ = −λRTC b
k Mwα
(10)
Here γ′ is the surface tension of an alkane in its base state (i.e., in the absence of the end group effect). Equation 10 may also be viewed as a manifestation of the balance of the osmotic [RT(Cb − C(x))] and van der Waals [π(x)] stresses near the interfacial region in that the excess free energy that reduces the surface tension is given by ∫ ∞ o π(x) dx = −λRTCb. Our premise is that this γ′ can be estimated using the Lifshitz theory. On the other hand, γ is the reduced value of the surface tension due to the preferential adsorption of the methyl groups at the air−alkane interface. Now summarizing the above points,
RTC b(T ) γ (T ) = C1 − λ(T ) A (T ) A (T )
(13)
Two issues, however, arise when studying eq 13. First, we are not able to predict a priori how exactly λ depends on temperature. But the a posteriori analysis of the data presented below suggests that it is weakly temperature-dependent. Second, the Hamaker constants of n-alkanes implicitly depend on temperature through the function ε(iωn) by virtue of its dependence on E
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made available. These data along with the well-known literature values of the interfacial tensions of various pairs of immiscible liquids allowed us to re-examine the relationship between the thermodynamic works of adhesion of two condensed phases (interacting only via dispersion forces) and their Hamaker constants. The nice correlation between W12 and A12 as reported here further substantiates the already existing philosophy underlying the direct relationship between Dupre’s energy of adhesion and surface tension. This new calculation clarifies certain important points, which are highlighted below. For example, the standard geometric mean combining rule predicts the work of adhesion between glycerol and pentane8 to be 46 mJ/m2. This leads to the interfacial tension of this pair to be about 34 mJ/m2, which is higher than the experimental value of 26.5 mJ/m2 (Kwok et al.8). The correlation between W12 and A12 and the application of eq 2, on the other hand, predict this value to be 28.5 mJ/m2. Thus, for the geometric mean combining rule to be valid, the effective surface tension (20.6 mJ/m2) of pentane has to be larger than 16 mJ/m2 (as measured in air), when it is in contact with glycerol. This is consistent with the discussion associated with Figure 3 in that the surface tension of pentane in contact with air is lowered due to the enrichment of the surface by the methyl groups below its base value estimated using Lifshitz theory. When the same liquid is in contact with glycerol, more of the CH2 groups populate the interface in order to maximize the work of adhesion; the interface is thus depleted of the CH3 groups (Figure 7).
Figure 5. Comparison of the relationship between surface tension and the molecular weight of n-alkanes as obtained from eqs 11 and 12, respectively.
the polarization density and thus the temperature. One of the simplest ways to estimate the temperature dependency of the Hamaker constant is to follow the lead of Castellanos et al.,52 who rewrote the Hamaker constant by considering mainly the contributions arising from the static and the ultraviolet components of the interactions: 2 3k bT ⎛ ε0 − 1 ⎞ 3ℏωUV (n2 − 1)2 A= ⎜ ⎟ + 4 ⎝ ε0 + 1 ⎠ 32π 2 (n2 + 1)3/2
(14)
We will not elaborate on this approach as all the details can be found in ref 52. Here we use the temperature-dependent experimental values33 of the surface tension, the Hamaker constants, and the bulk concentrations52,53 Cb to construct a plot of γ(T)/A(T) versus RTCb(T)/A(T). The data assimilated in this manner for different alkanes cluster nicely around a line at each of the three temperatures (20, 60, and 90 °C) as shown in Figure 6.
Figure 7. Schematic of a plausible reorientation of the alkane molecules as the contacting phase of the alkane is changed from air to a liquid, especially with a strong dispersion interaction.
It is quite plausible that all of the alkanes used to construct Figure 2 maximize their interactions when in contact with another liquid in the above manner, which would explain the nice correlation between W12 and A12 observed with various alkanes. In this context, let us also consider the case of bromonaphthalene and pentane. The geometric mean combining rule predicts that the interfacial tension8 of this interface, based on the surface tensions of bromonaphthalene (44 mJ/m2) and pentane (16 mJ/m2), is 7.0 mJ/m2. Kwok, Lee, and Neumann8 pointed out that such a positive interfacial tension goes against the observation that these two liquids are miscible with each other. Within the scenario described as above, if we take the surface tension of pentane to be close to 20.6 mJ/m2 when it is in contact with bromonapthalene, then the interfacial tension calculated using Fowkes’ equation turns out to be −0.08 mJ/m2. The interface of these two liquids, therefore, must disintegrate via dissipative transport, which is consistent with experimental observation. Thus, it appears that if the surface tension of an alkane in contact with air is γ, then its value would be higher than that when it comes in contact with another liquid by virtue of the depletion of the surface-active functionality (eq 10). The interfacial tension between two dispersive liquids may thus be calculated using the following modification of Fowkes’ equation:
Figure 6. Plot similar to that shown in Figure 4. However, the calculations here were performed at three different temperatures.
From the intercepts of the three curves, the value of l0 is estimated to be 0.154 (±0.001) nm at 20 °C, 0.153 (±0.001) nm at 60 °C, and 0.151 (±0.002) nm at 90 °C and appears to be fairly a temperature-independent parameter. The slopes inform that the distance λ increases with temperature: at 20 °C, λ = 0.118 (±0.009) nm; at 60 °C, λ = 0.163 (±0.019) nm; and finally at 90 °C, λ = 0.213 (±0.024) nm. It is gratifying to note that the value of l0 (0.154 nm) obtained from this temperature-dependent study of the surface tension is fairly close to that obtained from Figure 2.
γ12 ≈ γ1 + γ2 − 2[(γ1 + λ1RTC b1)(γ2 + λ 2RTC b2)]0.5
3. DISCUSSION Thanks to Neumann’s group,8 the precise interfacial tension data of various H-bonding liquids in contact with n-alkanes was
(15)
In the above, allowance is given for γ1 to be modified as well. It is highly noteworthy that this equation allows for a negative F
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Langmuir
Another important point that needs future consideration is that we used here a simple expression for the chemical potential by taking the activity coefficient to be unity. In a general way,58 however, chemical potential depends more explicitly on activity than concentration, which would have further bearing on the estimation of surface and interfacial tensions. In summary, what we presented here should not be considered to be more than a minimal but useful model for understanding the interfacial tension of dispersive liquids on the basis of the celebrated Lifshitz theory and the principle of independent surface action. We exercise caution in overstating our conclusions regarding interfacial tension in view of some observations made by Dwight et al.,59 who proposed that a transition region exists at a solid−liquid or a liquid−liquid interface that is not explicitly taken into account in our current analysis. Well-established phase field-15−17 and statistical mechanical14-based models, in fact, estimate surface tensions using density and the transverse pressure profiles at the interphase of two liquids. An estimation of surface tension using Lifshitz theory, on the other hand, assumes a geometrically sharp interface, although recently some effort has been made to relax this assumption.60 In this context, it is worth pointing out that Schwinger et al.23 developed a method to calculate surface tension (also using a geometrically sharp boundary) that is consistent with its conventional definition in that the surface tension is the variation of the free energy of the system with respect to its area. These authors used the celebrated “source theory” to obtain an expression for the surface tension of a condensed phase, such as liquid helium in terms of its dielectric function as follows:
interfacial tension at an interface of two liquids as a result of the surface reorganization upon contact. At this juncture, we point out that a new method to estimate the interfacial tension has been developed by Marmur and Valal54 using a Gibbsian approach of defining an interphase in which the interfacial tension appears as an excess of the internal energies of the bulk phases and an interfacial-positiondependent internal energy. It seems possible that the treatment used here can be merged with that of Marmur and Valal54 that, so far, does not consider the end group effects, although it has other interesting features. Steinberg55 envisaged 50 years ago the possibility that the work of adhesion (W12) of two different phases can be larger than half of their cohesions (W11 + W22)/2. He successfully used this idea to develop a phase map of the miscibility or immiscibility of biological cells that is central to understanding morphogenesis, movement, and sorting phenomena. That W12 can be larger than (W11 + W22)/2 leads immediately to the possibility of a negative interfacial tension (Dupre’s equation) that amounts to a sufficient condition for the solubility of one liquid in another. In a legendary paper9 published about 100 years ago, Hardy first planted the seed of what is known today as the principle of independent surface action with this statement: “If the stray field of a molecule, that is of a complex of these atomic systems, be unsymmetrical, the surface layer of fluids and solids, which are close packed states of matter, must differ from the interior mass in the orientation of the axes of the fields...”. In this remarkable paper, he also envisaged the possibility that the interfacial tension can be negative as he wrote, “fluids mix when TAB (interfacial tension) is negative.” This important insight has been overlooked in surface chemistry for too long. Hardy even elevated the status9 of a negative interfacial tension to a “purely chemical theory of the miscibility of fluids”. For phases undergoing donor−acceptor20 interactions, it is relatively easy to figure out how such an interface could satisfy Hardy’s conjecture. Our analysis, as presented here, illustrates how a negative interfacial tension is plausible even in systems undergoing dispersion interactions. The simple semiempirical phenomenology presented here seems to capture how surface reorganization can influence the surface tensions of n-alkanes not only at room temperature but also at other temperatures (albeit semiquantitatively) as well. The requirement for a positive Gibbs parameter (λ = ∫ ∞ l exp(ϕ(x) − 1)dx, with l being a cutoff length) for the methyl group is that its self-energy56 ϕ(x) is negative ϕ(x) = −
kT 2x 3
∞
∑′ n=0
* (iωn) ⎡ 1 − ε(iω ) ⎤ αCH 3 n ⎢ ⎥ ε(iωn) ⎣ 1 + ε(iωn) ⎦
γ=
1 16π 2
∫0
∞
dζ[ε(iζ ) − 1]2
∫k
∞ c
⎡ ζ2 ζ6 ⎤ d k 2 ⎢2 − 2 − 6 ⎥ 2k ⎣ 2k ⎦ (17)
Here, k is the transverse momentum of a photon, the minimum value of which is kc = ℏωc/c, where ℏ is the Planck’s constant, c is the speed of light, and ωc is a frequency corresponding to atomic separation, which should be related to l0 (eq 3) as ωc ≈ c/l0 ≈ 1018 s−1. This was indeed the type of value used by Schwinger et al.23 to make estimates of the surface tension as well as the latent heat of vaporization of liquid helium that approximately coincided with the experimental measurements. However, by noting the empirical nature of ωc, the authors stated, “We are evidently lacking a quantitative theory, which would require a detailed microscopic consideration.” We hope that the overall philosophy presented here will motivate research in developing more comprehensive analyses of interfacial tension in condensed phases, perhaps within the scopes of the source theory, the phase field, and/or statistical mechanical models. The problems that require amicable resolutions are (1) the well-known asymmetry of the fluorocarbon− hydrocarbon interaction at the interfaces, (2) the quantification of interactions of phases comprising the donor−acceptor groups, and (3) understanding the reorganization of surfaces comprising multiple functionalities.
(16)
where α*CH3 is the excess polarizability of the methyl group, which is roughly the difference between the polarizability of a methyl group and that of the bulk consisting of the methylene groups having equivalent volumes of a methyl group. The static excess polarizability α*CH3 estimated using the approach of bond polarizability57 is in the range of −7 Å3, which suggests via eq 16 that the methyl groups should be positively adsorbed at the air−alkane interface. The exact calculation of λ(T), however, requires αCH * 3to be expressed as a function of frequency (iωn), as is the case with the dielectric permeability (ε(iωn)) of the bulk phase, which is beyond the scope of the current paper.
4. CONCLUSIONS The main conclusion of this paper is that the estimation of the surface tension of liquids requires a consideration of the surface activity of the constitutive molecules at air and liquid−liquid interfaces. The geometric mean combining rule of interfacial tension, as is currently used, requires a correction by taking into account the roles of such surface reorganizations. G
DOI: 10.1021/acs.langmuir.5b03054 Langmuir XXXX, XXX, XXX−XXX
Langmuir
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APPENDIX
Article
AUTHOR INFORMATION
Corresponding Author
A1. Role of End Groups in the Wettability of Solids
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. (S.C.) Research Intern from ESPCI Paris Tech.
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REFERENCES
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Figure A1. Schematic of the chemical modification of a solid surface by the counter diffusion of alkyl thiols of dissimilar chain lengths on gold. On the two opposite ends of the surface, the molecules selfassemble closely, thus exposing mainly the methyl groups outward. In between, where the diffusive fronts meet, the surface exhibits disorder, thus exposing some of the methylene groups. The contact angle of hexadecane probes this disorder. On the ordered parts of the surface composed of mainly the methyl groups, the contact angle is high. The angle, however, decreases as the degree of disorder increases. Here, C10, C12, and so forth indicate the number of carbon atoms along the length of the chain. This figure is plotted on the basis of the data published in ref 26.
A2. Correlation between the Directly Measured Surface Energies Using Contact Mechanics and Those Estimated Indirectly Using Contact Angles
Figure A2. Comparison of the surface energy of a surface-modified PDMS elastomer obtained from the contact angle and from contact mechanics and the contact angle.27,28 H
DOI: 10.1021/acs.langmuir.5b03054 Langmuir XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.langmuir.5b03054 Langmuir XXXX, XXX, XXX−XXX
