The surface energy of montmorillonite

Journal of Colloid and Interface Science 268 (2003) 263–265 www.elsevier.com/locate/jcis

Note

The surface energy of montmorillonite A.K. Helmy,a,∗ E.A. Ferreiro,a,b and S.G. de Bussetti a a Universidad Nacional del Sur, B8000 Bahía Blanca, Argentina b Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina

Received 31 January 2003; accepted 29 July 2003

Abstract By combining the relation that describes pair interaction in binary mixtures with the Young equation, a formula is obtained for calculating the surface energy of montmorillonite as a function of the surface pressure, the surface tension of water, and the liquid/solid contact angle. The formula is an equation of an inverted parabola, which could be represented by a polynomial function. Roots of the polynomial gave one real value of 205.066 ± 2.764 mJ m−2 for the surface energy of montmorillonite. The value obtained is of the expected magnitude and probably is better than those obtained by previous approaches.  2003 Elsevier Inc. All rights reserved. Keywords: Surface energy; Surface tension; Montmorillonite

1. Introduction Adsorption, wetting, swelling, and other phenomena are examples of interactions at interfaces. A property of solids involved in these interactions is the surface free energy. No direct method is available for its measurement as is the case for liquids. Therefore this property of solid surfaces is evaluated from knowledge about solid/liquid interfacial interactions and measurements of contact angles and surface pressures. So far the surface energy of montmorillonite has been determined from contact angles using two liquids. Thus a second liquid phase is present on the solid when the sessile drop of the first liquid is made to contact the solid surface [1–6]. As is shown below, the surface free energy of a solid (γS ), as well as other related properties, can also be obtained from measurements using only one liquid, provided that values of the surface tension of the liquid (γL ), the solid/liquid contact angle (α), and the surface pressure (Π) can be determined. Other approaches that require data on one liquid for the determination of γS suffer from some inaccuracies. In the equation of state approach for low-surface-energy solids [7] an adjustable solid/vapor tension (γSV ) is adopted. Similarly, in the acid–base theory [8] for surface energies of polar sur* Corresponding author.

E-mail address: [email protected] (A.K. Helmy). 0021-9797/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.07.007

faces, where only one liquid is used, γSV is replaced by γS in the Young equation, thus neglecting the surface pressure, which is generally not valid [9]. All methods used for the determination of the surface energy of solids and are based on liquid/solid interactions, require the use of an extra relation in addition to the Young equation. Several approximate relations are in use (see review [6]). We will develop below an alternative approach using a general relation that describes pairs interaction in binary mixtures.

2. Background The system under study is a solid surface in contact with an immiscible liquid and its vapor. We attempt to describe γS in terms of the surface pressure, the surface tension of a liquid (γL ), and the solid/liquid contact angle (α). In binary mixtures where interaction of pairs takes place, the expression B11 + B22 − 2B12 = B

(1)

is obeyed with B a positive quantity [10,11]. Usually the value of B12 is intermediate between those of B11 and B22 [10]. The B’s may represent one of various properties such as virial coefficients [10], Hamaker constants [11], or surface

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A.K. Helmy et al. / Journal of Colloid and Interface Science 268 (2003) 263–265

tension [12,13]. The subscripts refer to interactions between similar or nonsimilar partners. Specializing for the system under consideration we write Eq. (1) as

The values set for γS in Eq. (5) were chosen to fall around the value given by the equation

γS + γL − 2γSL = γ ,

A factor greater than 2 was set in Eq. (7) because a factor of 2 is insufficient to produce nullity as Eq. (2) would indicate. For the experimental values of Π (60 mJ m−2 ), γL (72.8 mJ m−2 ), and α(0), γS = 235.2 mJ m−2 and hence the range was between 150 and 230 mJ m−2 . Also taken into account is the range of published values of the same and similar minerals.

(2)

where γSL is the solid/liquid interfacial tension and γ is a positive quantity. We may, however, dispense with γ in Eq. (2) and write γS + γL − kγSL = 0,

(3)

where k is an interaction parameter. Proceeding, we write the Young equation, γSL = γS − γL cos α − Π.

(4)

(5)

γS + γL . γS − γL cos α − Π

(6)

Equation (5) contains two unknowns, γS and k. Hence one extra relation is necessary. Since Eq. (5) is an inverted parabola it can be represented by a rational integral function (a polynomial). This function was obtained by the following procedure. Values are set for γS in Eq. (5) and the equation is solved for k values, which are then used in the calculation of the constants in the polynomial k = a0 + a1 γS + a2 γS2 + a3 γS3 + a4 γS4 .

For systems with a similar type of interaction force the most frequently used method for the calculation of γS is based on the Fowkes relation, γSL = γS + γL − 2(γS γL )1/2.

and also k=

(8)

3. The Fowkes–Good approach

From (3) and (4), γS (k − 1) = k(γL cos α + Π) + γL

γS + γL − 3γSL = 0.

(7)

Eliminating k between (6) and (7), we obtain an equation of the fifth order in γS , which is then solved for γS . Fig. 1 shows a plot of k as parameter of the values set to γS in Eq. (5).

(9)

Good [14,15] proposed a formula with an interaction parameter φ, γSL = γS + γL − 2φ(γS γL )1/2 ,

(10)

and defined φ by φ = φV φA ,

(11)

where φV is a molecular volume correction which was later withdrawn, and φA = A12 /(A11 A22 )1/2 ,

(12)

where the A’s are Hamaker constants. Combining Eq. (10) with the Young equation (4) gives
 1/2 1/2 φγS = Π + γL (1 + cos α) 2γL (13) and also
2  φ 2 = Π + γL (1 + cos α) 4γS γL .

(14)

For our system Eq. (12) gives φ = 1 since the Hamaker constants of water and of the montmorillonite are 4.4 and 5.28 × 10−13 erg, respectively [16], as well as A12 = (A11 A22 ), as shown by Gregory [17]. Hence the φ parameter remains underterminable in this approach. It is worth mentioning that all data for clays were obtained by the two-liquid method; the parameter φ is not considered, i.e., taken equal to unity.

4. Results and discussion

Fig. 1. Variation of k as a function of the surface energy of montmorillonite (γS ) according to Eq. (15). The error bars are the 95% confidence interval from averaging, and the line is the corresponding fits.

For the calculation of the value of γS we used literature data for Π , γL , and α. According to Brooks [18], α is equal zero, γL = 72.8 mJ m−2 [7]. From data given for montmorillonite Π = 62.8 mJ m−2 [19] at monolayer coverage with

A.K. Helmy et al. / Journal of Colloid and Interface Science 268 (2003) 263–265

water while Chassin et al. [4] obtained a value of 55 mJ m−2 at p/p0 = 0.2 and calculated 67 mJ m−2 from heat of immersion data of Green–Kelley [20] (H = 140 mJ m−2 ≈ γL cos α + Π). In the present calculation we used a value of 60 mJ m−2 for Π . Using the above-mentioned values and the procedure mentioned before, we obtained the polynomial for k as k = 1455.1 − 28.577γS + 0.2117γS2 − 0.0007γS3 + 8.725 × 10−7 γS4 ,

(15)

with r 2 = 0.999, which in combination with Eq. (6) gives

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5. Conclusions 1. Values of surface pressure, contact angle, and surface tension of one liquid are sufficient for the calculation of the surface energy of montmorillonite, using a relation obtained by combining the Young equation with a relation that describes pair interaction in binary systems. 2. The surface energy of montmorillonite is found equal to 205.066 ± 2.764 mJ m−2 . 3. A novel procedure (a polynomial expansion) was adopted to replace the lack of an independent expression for the interaction tension parameter k. 4. The surface energy is not obtained as summed components.

0 = −193310.08 + 5249.1256γS − 56.69076γS2 + 0.30466γS3 − 8.15868 × 10−4 γS4 + 8.725 × 10−7 γS5 .

Acknowledgments (16)

Roots of the γS obtained using the software Matlab V 5.3 gave a real value of γS = 205.066 and 149 + 10.64i, 149 − 10.64i, 216.01 + 41.92i, and 216.92 − 41.92i. Setting this value for γS in Eq. (15) gave k = 3.845 ± 0.096. With the value of γS = 205.066 ± 2.764 mJ m−2 we find from Eq. (14) that φ 2 = 0.708 and hence φ in Eq. (13) equals 0.841. It is worth mentioning that all data obtained by the two liquid method regard φ as equal to unity. A few authors determined γS of montmorillonite using the two-liquid method, for example, a value of 186 mJ m−2 in [4], while in [19] the values are 178.9 and 212.0 mJ m−2 , depending on the value of the contact angle of water under the organic liquid. Equation (14) with φ = 1 gives a value for γS = 145.16 mJ m−2 . The values obtained by the above-mentioned authors are thus similar to the value found in the present work. This value, in contrast to the others, was obtained without any assumption regarding the value of a mixed tension in a binary system. Finally, γSL can be calculated from Eq. (4) since γS is now known. It is equal to 72.3 mJ m−2 , a value almost equal to γL . This result reflects the fact that in adsorption systems such as clay in the presence of water molecules, the clay surface is converted progressively to a water surface and γ SL then has the value of γL . This result is in agreement with Eqs. (1) and (2) since for the interaction between similar pairs γLL = γL , i.e., mixing water with water, the tension does not vanish. This results also follows from the heat of immersion of water in water which is equal to zero and is given by H = γL − γLL = 0 [21]. It is therefore a conceptual error that γSL could ever have a value of zero [6,7]. It is obvious therefore that all formulas such as Eq. (9) that gives γSL = 0 for partners with equal tensions are in trouble.

The rational integral function was suggested by Dr. Amina Helmi, Faculty of Physics and Astronomy, Utrecht University, the Netherlands. The present work was financed by the Secretaría General de Ciencia y Técnica of the Universidad Nacional del Sur (Research Group Project, Expte. No. 1876/00, Res. CSU 190/00).

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