LETTER pubs.acs.org/Langmuir
Relationship between Surface Tension and Surface Coverage Fredric M. Menger* and Syed A. A. Rizvi† Department of Chemistry, Emory University, 1515 Dickey Drive, Atlanta, Georgia 30322, United States ABSTRACT: Surfactant action is caused in part by a dramatic reduction in surface tension. Using surface excess measurements from a radioactive surfactant, it was possible to show that (a) the surface tension declines only slightly when the occupancy of the air/water interface increases from 0 to 60% of the maximum and (b) the steep drop in surface tension in region B (Figure 1), frequently observed to be linear, begins at about 80% occupancy. Surfactant continues to enter the interface cooperatively up to and past the critical micelle concentration. Linearity in region B is not indicative of surface saturation despite a seemingly constant surface excess throughout the region. The disparity between interfacial areas determined by surface tension and by other methods is discussed in terms of these results.
A
plot of surface tension versus ln [surfactant], shown in Figure 1, can be divided into three regions: (a) a rather level section at low concentrations (region A), (2) a steep, often linear decline (region B), and (3) a leveling off again when the critical micelle concentration (cmc) is reached (region C). Despite the prevalence of surface tension versus ln [surfactant] plots throughout the colloid and surface literature,1 the composition of the air/ water interface within critical region B is actually not known. Thus, we are largely ignorant of the relationship between surface tension and the percent surfactant coverage of the interface, a fundamental gap in our understanding that we hope to alleviate partially in this letter. The Gibbs equation is relevant to the situation as seen in eq 1, where Γ is the surface excess (equivalent to the interfacial concentration in units of mol/cm2), c is the bulk surfactant concentration, γ is the surface tension, and dγ/d(ln c) is the slope of the surface tension plot. Γ¼
dγ=dðln cÞ nRT
ð1Þ
Now there are two possibilities when the slope dγ/d(ln c) is seemingly constant, as is often the case for region B: (a) Because the slopes at all points are identical, the apparent surface excess Γ must also be constant in the region. Such a condition would pertain if the interface were saturated with surfactant throughout region B. When the area/molecule at saturation (given by 1016/ NΓ where N is Avogadro’s number) is computed from the slope of region B, as is common practice,2 saturation within region B is tacitly implied. Otherwise, a single area would not be obtainable from the entire B region. (b) The second (somewhat vague) possibility is that the linearity found so frequently in region B is fortuitous or an artifact and thus irrelevant to saturation. However, if the interface were not saturated and if the surfactant occupancy were in fact continually increasing as region B declined, r 2011 American Chemical Society
then the calculation of unique minimum interfacial areas from region B, or a portion of region B, would be questionable. Even restricting the Gibbs analysis to points immediately below the cmc could give unreliable data if the interface had not yet become saturated. No law of nature states that the interface must be fully saturated prior to micelle formation. Of course, analyses using points near the cmc are complicated by premicellar aggregates3 and uncertainties in the activity. Nonlinear curve fitting to linear region B adds little to the information. The main point here is that for a long time there has been a pressing need to deal with these issues using hard data that relate surface tension to surface occupancy. Rosen’s book Surfactants and Interfacial Phenomena4 includes a scholarly list (16 pages long!) of Gibbs-determined minimum areas per molecule. The areas are surprisingly insensitive to rather sizable structural changes. For example, n-C10H21SO4Na+, (n-C7H15)2CHSO4Na+, and n-C12H25CH25CH(COO)N+(CH3)3 have areas of 57, 51, and 54 Å2/molecule, respectively. One would intuitively have expected a much greater response of the packing area to the surfactant bulk and charge type. The Gibbs-based method is by no means disqualified by the data, but a concern about its reliability is aroused. We have already argued on the basis of four experimental approaches (surface tension, monolayer studies, conductivity, and NMR-based diffusivity) that the interface is, in fact, not saturated in region B.57 Doubts about this conclusion8 were readily refuted,9 yet a need did remain for an experimentally based, quantitative understanding of exactly how surface tension depends on the percent of interfacial coverage. This understanding was best derived from surface excess measurements independent Received: August 2, 2011 Revised: October 24, 2011 Published: October 26, 2011 13975
dx.doi.org/10.1021/la203009m | Langmuir 2011, 27, 13975–13977
Langmuir
Figure 1. Three regions of a typical of surface tension vs ln [surfactant] plot. Region B has in the past reflected an unknown coverage of the air/ water interface. Region C reflects the fact that the surfactants reach a critical micelle concentration (cmc) where added surfactant molecules preferentially enter the micelles relative to the interface. cmc's commonly vary from submillimolar to 10 mM.
LETTER
Figure 3. Plot of the percent surface coverage vs ln [SDS] for pre-cmc points. The line is drawn for visual purposes only. The plot shows clearly how the occupancy of the interface increases smoothly up to the cmc in the absence of pre-cmc saturation.
As will be shown presently, the interface is not entirely filled with surfactant even at its cmc of 8 mM.12 The percent surface coverage can be obtained from the Γ values in Figure 2, assuming that saturation occurs at 4.8 mol/cm2. A plot of the percent surface coverage versus ln [SDS] is linear as seen in Figure 3. It remains only to relate the surface tension to the percent surface coverage, and this was accomplished with eq 2 published 14 years ago by Lin et al. 13 In this equation, γ and γo are the surface tensions of the surfactant solution and pure water, respectively, Γ∞ is the maximum surface concentration, and x = Γ/Γ∞ (i.e., the fraction of surface coverage). γ γo ¼ Γ∞ RT½lnð1 xÞ
Figure 2. Plot of surface excess Γ in mol/cm2 vs the pre-cmc concentration of SDS surfactant in solution. The plot is based on the data reported by Nilsson.11 The surface excess Γ (equivalent to the interfacial concentration) was determined directly via the use of tritiated SDS whose interfacial adsorption was monitored by a flow-proportional counter.
of surface tension and its attendant difficulties. A method that serves that purpose is now described. The key to our approach is a brilliant but largely forgotten paper10 by Nilsson published from Stockholm over 50 years ago.11 He prepared tritiated SDS and determined its adsorption isotherm at the air/water interface with the aid of a specially constructed windowless flow-proportional counter. Adsorbed molecules were so close to the surface that β-particle absorption by the solution could be neglected. The use of a weak β emitter (1/10 the energy of C14) favored the accuracy and range of the experiments. Nilsson’s work was selected, among the meager number of similar studies, for the special care taken in acquiring the data. Even the relative humidity above the interface was controlled. The adsorption isotherm of tritiated SDS in water at 25 °C is given in Figure 2. It is seen that the surface excess Γ increases smoothly as a function of the bulk SDS concentration.
kx2 2
ð2Þ
Parameter k is a cooperativity term, where k < 0 if cohesive intermolecular forces increase the surface coverage and k > 0 if adsorption becomes more difficult as the surface becomes crowded. Initially, we assumed that k = 4.0 (the same value used for n-octanol by Lin et al.)13 and evaluated the Γ∞RT constant (30.5) from the known γ of SDS at its cmc (40 mN/m)14 and from x = 0.94 as taken from the Nilsson data. A plot of γ versus ln [SDS] is given in Figure 4, where cooperativity constant k in eq 2 was again assumed to equal 4.0. When the cooperativity was removed by setting k = 0, much the same shaped curve was obtained except that the slope of region B is not as steep. It is, of course, reasonable that SDS selfassembles cooperatively at the interface, similar to cooperative micelle formation at the cmc, because both processes are driven by hydrophobic interactions among a collection of chains. Above each point in Figure 4 is placed the percent surface coverage at that particular concentration. It is those key numbers, the crux of this letter, that reveal valuable insights into the packing status of region B, as described in the next paragraph. It is seen that the surface tension is only slightly affected by low surface coverages (80%). Although by no means reflecting saturation, region B is nonetheless linear. Thus, areas from a Gibbs-based analysis, while erroneous, depart from reality by a factor of only 2 or less as compared with direct area measurements via radio isotope methods.17 A comparison of Gibbsbased data with those obtained from monolayers spread on a Langmuir trough shows a similar disparity.18 Accurate neutron reflection-derived areas have also shown that indirect Gibbs-based
Department of Pharmaceutical Sciences, College of Pharmacy, Nova Southeastern University, 3200 South University Drive, Ft. Lauderdale, Florida 33328, United States.
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