J. Phys. Chem. B 2000, 104, 10339-10347
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Critical Coagulation of Langmuir Monolayers: 2D Schulze-Hardy Rule Masahito Sano,* Ayumi Kamino, and Seiji Shinkai Chemotransfiguration Project-JST, 2432 Aikawa, Kurume, Fukuoka 839-0861, Japan ReceiVed: July 3, 2000; In Final Form: August 31, 2000
The Schulze-Hardy rule, a well-known relation on colloidal dispersions between the critical coagulation concentration (ccc) and ionic valency of added electrolytes, is shown to hold in monolayer systems. A carboxyazobenzene derivative with a relatively short alkyl chain forms only a kinetically stable monolayer at the air-water interface without developing measurable surface pressure. It features the novel band structure on mica as Langmuir-Blodgett films, 1.5 nm thick, 60 nm wide, and can be longer than 100 µm. Atomic force microscopy reveals that the bands coagulate critically when the ion concentration in the water phase reaches a critical value. For monovalent ions, Na+ and Li+ exhibit critical coagulation, while K+ and Cs+ develop destabilizing hydration forces on mica before critical points are reached. Both La3+ and Ce3+ cause critical coagulation at much lower concentrations. For divalent ions, ccc’s are in the order Cd2+ , Mg2+ ∼ Ba2+ < Ca2+ < Sr2+. By comparing the experimental adsorption isotherms with a Poisson-Boltzmann theory that assumes charge regulation by metal-carboxylate complex formation, Ba2+ is identified to interact anomalously among alkaline earth metals. For hard Lewis acids, it is empirically found that ccc is related to the ionic radius by a power law relationship, which is used to take into account charge regulation on the film. Then, ccc is shown to be inversely proportional to the ninth power of the ratio of the ionic radius to valency.
Introduction It has been known for a long time that a trace amount of multivalent ions added to water has significant effects on Langmuir (L) monolayers of carboxylic acids at the air-water interface.1-3 Because divalent ions induce condensation and crystallization of monolayers leading to better stabilization of Langmuir-Blodgett (LB) films, they have been the subjects of intense studies.4-6 Previous studies, mostly based on surface pressure-area isotherm and X-ray diffraction, establish that condensations occur by changes in molecular packing and in electrostatic interactions between the polar heads as well as the heads and ions. There are also numerous studies of the condensed morphology by atomic force microscopy (AFM), fluorescent microscopy, and Brewster angle microscopy, revealing morphological changes as the ions are added to the water subphase. In solution, electrolyte ions are known to coagulate certain colloidal sols. These sols are not thermodynamically stable, although it may takes a very long time to coagulate them. Nearly 100 years ago, Schulze7 and Hardy8 reported that the electrolyte counterions with the larger valency coagulate kinetically stable colloidal dispersions more effectively. The minimum concentration of ions necessary to cause rapid coagulation of colloids is known as the critical coagulation concentration (ccc). The Schulze-Hardy rule states that
ccc ∼ (1/z)n
(1)
where z is the valency of the electrolyte counterions. Typically, n is 6 or 2 in solution.9 The first quantitative explanation of this rule by Derjaguin-Landau-Verwey-Overbeek (DLVO) 50 years later granted DLVO theory a status of prominent importance in colloid science.10,11 * Corresponding author. E-mail:
[email protected]. Fax: +81-94239-9012.
Although the monolayer condensation and the sol coagulation are well-known, there has been no report to analyze the condensation behaviors of monolayers through a perspective of colloidal coagulation. In this paper, we show that the Schulze-Hardy rule holds in two dimensions (2D) by a Langmuir monolayer that is only kinetically stable. An “ordinary” monolayer develops sufficient surface pressure to stabilize the film thermodynamically. Recently, a Langmuir monolayer of a weakly amphiphilic compound was found to exhibit the properties of neither ordinary monolayers nor 2D crystals.12 The alkyl chain length of 4-heptoxy-4′-carboxyazobenzene is too short to form a stable monolayer but too long to condense into 3D aggregates at the air-water interface. The film can exist as a monolayer without developing measurable surface pressure. The LB film on mica features the mesoscopic band structure 1.5 nm thick, 60 nm wide, and possibly over 100 µm long. The bands are formed spontaneously upon spreading and are not produced by the LB processes, such as compression and lifting. The mesoscopic bands are formed as long as the molecular area at spreading is greater than 8 nm2/molecule at 20 °C. AFM demonstrates that the band is an association of monolayer clusters in a few tens of nanometers. Moreover, the bands are not thermodynamically stable. Elevating temperatures or aging on the water surface cause slow reorientation of molecules to result in disintegration of the bands. These properties suggest that the mesoscopic bands represent a kinetically stable 2D colloidal system. The present experiment is motivated by a possibility that the coagulation analogous to 3D systems may take place in Langmuir monolayers. Effects of mono-, di-, and trivalent ions on the band morphology are followed directly by AFM, UVvis reflection spectroscopy, and X-ray photoelectron spectroscopy (XPS). In this series of experiments, we have found that some monovalent ions interact with the mica substrate in a specific manner and hinder the film property. Thus, a detailed
10.1021/jp002387y CCC: $19.00 © 2000 American Chemical Society Published on Web 10/19/2000
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Figure 1. AFM image of the mesoscopic bands on mica transferred from (A) 10-7 M, (B) 7.0 × 10-6 M, (C) 10-5 M, and (D) 10-4 M Ca2+ at pH 7.0. The bands in (A) are 1-1.5 nm high, 60 nm wide, and can be longer than 100 µm. In (D), multilayers are formed.
account of monovalent ions is given in a separate paper,13 and only a brief discussion is presented in this paper. For divalent ions, the adsorption isotherms were measured using XPS. The results were analyzed by a Poisson-Boltzmann theory to show that the charge regulation by the metal-carboxylate complex formation is effective. By phenomenologically introducing the ionic radius to represent the effect of charge regulation, a relationship between ccc and z is obtained. Experimental Section Reagent-grade salts MClz (M ) Na, Li, K, Cs for z ) 1; Mg, Ca, Sr, Ba, Cd for z ) 2; La, Ce for z ) 3) were used as purchased. The reported concentrations are the added amounts that are calculated but not determined specifically. The azobenzene compounds was synthesized as described previously.14 The subphase water was purified by Nanopure II and Fi-streem system (Barnstead), and its pH was adjusted to 7.0 by NaHCO3 (about 0.06 mM for alkaline earth metals and 0.006 mM for Cd). Thus, the “salt-free condition” contains 0.06 mM Na+. All LB processes were carried out in air for monovalent and alkaline earth metals and in N2 atmosphere for trivalent ions and Cd2+. The compound dissolved in chloroform was spread on an electrolyte solution at 20 °C so that the molecular area became 2.5 nm2/molecule, above the critical aggregation concentration.
After being aged for 15 min, the film was compressed to 0.5 nm2/molecule. UV-vis reflection spectra (MCPD-110, Otsuka Electronics) were obtained during compression. A single upstroke transfer was made on muscovite mica without moving the barrier. The LB film morphology was examined by AFM (TopoMetrix) operating in a noncontact mode in ambient conditions. For divalent ions, the adsorption isotherms were measured by XPS (Perkin-Elmer ESCA 5300 System). The fraction of ions per molecule in the LB film was obtained by measuring the areas under the peaks of Mg2p, Ca2p1/2 and Ca2p3/2, Sr3d3/2 and Sr3d5/2, Ba3d5/2, and Cd3d5/2. To minimize an effect due to the vacuum instability of the film,15,16 the samples were cooled by liquid N2 and the data collected for the first 1 h was used for all analyses. The atomic ratios of C1s or N1s to one of the mica elements (K2p3/2, Al2p, or Si2p) were monitored to make sure that each sample has the same film coverage. The atomic concentration ratios of a divalent ion to C or N are calibrated by measuring the EDTA-metal complexes of known compositions (purchased from Doujindo Laboratory) for each ion. Results Morphological Characteristics. Figure 1 is a series of AFM images observed at various Ca2+ concentrations and illustrates
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Figure 2. AFM images of the bands transferred from (A) 10-6 M and (B) 10-3 M Mg2+, (C) 3 × 10-5 M and (D) 10-3 M Sr2+, (E) 10-6 M and (F) 10-3 M Ba2+, (G) 10-6 M, and (H) 10-5 M Cd2+. The heights of multilayers vary with ions and are not drawn with the same color scale.
10342 J. Phys. Chem. B, Vol. 104, No. 44, 2000 a typical change of morphology upon additions of salts. At low concentrations, the bands are 1-1.5 nm thick and straight, have the constant width of 60 nm, and are separated from each other (Figure 1A). When the ion concentration is increased, the bands show no morphological changes until the ion concentration reaches a threshold value. At this concentration, most of the bands suddenly coagulate along the long axis (Figure 1B). A slight increase in the ion concentration induces coalescence of the coagulated bands (Figure 1C). No change in the band thickness indicates that coagulation so far is strictly within the monolayer plane and does not lead to 3D aggregation. In the case of Ca2+, further increases cause the multilayer formation (Figure 1D). The multilayer islands consist of long strips formed by random associations of many short and narrow plates. Each strip lacks any geometrically defined regularity. The step heights of multilayers are not the regular increments of a monolayer or a bilayer, suggesting complex molecular packing in the multilayer states. This multilayer formation by ions is easily differentiated from the collapsed region caused by overcompression, which gives very local 3D aggregation without definite shapes. For monovalent ions, (Li+, Na+, K+, and Cs+), Li+ and Na+ exhibit the same coagulation behavior as that of Ca2+ with an exception of multilayer formation.13 The concentrations over 10-3 M of these ions do not produce multilayers. The coagulated morphologies are also similar to that of Ca2+. For K+ and Cs+, even the lowest concentrations necessary as a buffer cause the bands to shape irregularly. The examples of morphology near the threshold and higher concentrations by other divalent ions are given in Figure 2. Near the threshold, most ions exhibit local coagulation randomly over the entire length of the bands. Within the variation of observed images, we could not detect any significant differences in the coagulated morphology of Mg2+, Ca2+, Ba2+, and Cd2+ near the threshold. Only the coagulated morphology of the Sr2+ film has a unique feature of segregation of the straight bands and the large round plates. Mg2+ differs from other divalent ions by the fact that the multilayer is not formed at the concentrations as high as 10-3 M. For Sr2+, the segregated morphology seen in the monolayer state continues to the multilayer state; the uncoagulated bands and the multilayers coexist on the same surface. For Ba2+, multilayers are dominated by the wide plates rather than strips. Only Cd2+ gives the multilayer strips that have geometrically defined edges and consist of long, regular rectangles. Furthermore, although there is no regularity in the step heights of multilayers for most divalent ions, Cd2+ produces a regular step height of 3 nm corresponding to a bilayer thickness. The trivalent ions, La3+ and Ce3+, also induce rapid coagulation at the threshold concentrations. A characteristic feature of coagulation by the trivalent ions is that the multilayers are formed as soon as they coagulate (Figure 3). To see the effect of ions on the band morphology more quantitatively, the fractions of areas occupied by the uncoagulated segments of bands over the total film area (including defects) are measured directly from the AFM images. A large number of AFM images were collected from several independently prepared samples for each concentration. We arbitrary define the uncoagulated segment to be the part of a band that has a constant 60 nm width over about 1 µm. For the trivalent ions, the fraction was calculated with respect to the total scan area from a large number of images. As shown below, the chosen value of 1 µm and the exact relation between the band fraction and the area fraction are not relevant for the present analysis because a change from all uncoagulated to all coagu-
Sano et al.
Figure 3. AFM images of the bands transferred from 3 × 10-7 M La3+. In this particular image, the upper section is mainly occupied by the unaffected bands while the lower section contains coagulated, multilayered bands.
Figure 4. Normalized fractions of the uncoagulated band area over the whole film area plotted against the bulk ion concentrations for the monovalent ions.
Figure 5. Normalized fractions of the uncoagulated band area over the whole film area plotted against the bulk ion concentrations for the divalent ions. The smooth curves are used to connect each data point for clearer presentation.
lated bands takes place within a very small region of concentrations. Figures 4-6 show the uncoagulated fraction as a function of the bulk ion concentration for mono-, di-, and trivalent ions, respectively. For easier comparison, the fractions are normalized
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Figure 6. Normalized fractions of the uncoagulated band area over the whole scan area plotted against the bulk ion concentrations for the trivalent ions.
TABLE 1: Critical Coagulation Concentrations (ccc’s)
Figure 7. UV-vis reflection spectra of the monolayers at the airelectrolyte interface containing ions as indicated. The “salt-free” condition contains 0.06 mM Na+. The collapsed film was made by overcompression to 0.15 nm2/molecule. Vertical scales are not the same for each ion, and each curve is shifted vertically for clear presentation.
ion Li+
Na+
Mg2+
Ca2+
ccc 1600 3000 (10-6 M)
0.60
4.0
Sr2+ Ba2+ Cd2+ La3+ Ce3+ 20
0.70 0.03 0.35 0.26
by the largest value (over 0.9) for each ion. The large standard deviation around the transition region is mainly due to statistically insufficient sampling of bands in an image with small scanning ranges (typically 5-20 µm, 400 × 400 pixels to maintain molecular resolutions) of AFM compared with the length of bands (over 100 µm). Despite the errors, coagulation is seen to take place abruptly for all ions except K+ and Cs+. It has been shown that these monovalent ions induce strong hydration forces on the mica surface at very low concentrations.13,17 This results in destabilization of the LB film well before the coagulation by ion-amphiphile interactions takes place. Thus, the thresholds cannot be determined for these ions. Cd2+ had an additional technical difficulty in performing LB in N2 atmosphere. We think that the true coagulation is much sharper than the plots indicate. The concentrations at which the fraction becomes 0.5 are summarized in Table 1. Considering the errors discussed above, the coagulation curves of all ions are similar, and the larger valency ions tend to have small threshold values. Within the divalent ions, the order is Cd2+ , Mg2+ ∼ Ba2+ < Ca2+ < Sr2+. Spectral Characteristics. Changes in molecular packing are followed by the long-wavelength π-π* transition of azobenzene chromophore. The selected UV-vis reflection spectra from the electrolyte surface are displayed in Figure 7. The spectral shape was totally independent of the area per molecular for all ions. A salt-free film produces the spectra with two peaks at 250 and 305 nm. The peak at 305 nm corresponds to H-like aggregates of azobenzene chromophore, indicating a high orientational order of the dipole moments.18,19 Although we only show the spectra obtained at 10-6 M Ba2+ (near the coagulation point), all other ions give the same spectra as that of the salt-free film when the films are in the monolayer states. Thus, coagulation has no effect on the spectral shape as long as the film stays as a monolayer. The films in multilayer states, however, produce different spectra depending on the ions. Sr2+ (not shown) gave the identical spectra as Ca2+. For alkaline earth metals, the peak at 250 nm is shifted to 265 nm. The high Ba2+ concentrations produce an apparent peak at 295 nm. This is actually the 305 nm peak, appeared to be shifted owing to a long shoulder of the 265 nm peak. The intensity at 265 nm is definitely larger than the one at 305 nm, despite the relatively large experimental
Figure 8. Fraction of Ca2+ per molecule as measured by XPS (filled and empty circles) and calculated by the PBS model (solid curve) as a function of the bulk Ca2+ concentration. The coagulation curve of Figure 5 is replotted as the dotted line.
error in the short-wavelength region. The high Ca2+ (Sr2+) concentration has also an intense 265 nm peak. Additionally, Ca2+ (Sr2+) gives a broad band from 300 to 350 nm. Most interestingly, Cd2+ produces no change even after the multilayer formation and yields the spectra identical to the monolayer state. In all cases, these spectral changes are not due to collapsing of the film by overcompression, as the overcompressed film develops a characteristic peak around 350 nm as shown in the top spectrum.20 Ion-Binding Characteristics. Figures 8-11 display the fraction of divalent ions per molecule bound in the LB film as a function of the bulk ion concentration. Each point is the average of six to eight independently prepared samples. As for Mg2+, the fraction appears to increase rapidly at 10-6 M of the bulk Mg2+ concentration (not shown). However, the low atomic sensitivity for photoelectrons and the uneven baseline of Mg2p made the standard deviation of each point larger than the change due to the ion concentration. Thus, the Mg2+ films were not analyzed further. For other ions, the filled circle is the value referenced to C1s and the empty circle denotes that referenced to N1s (with an exception of Cd2+ that has a peak too close to N1s). There was no significant difference in the atomic concentration ratios of ion regardless of using C1s and N1s as a reference. This indicates that all films have nearly the same quality, as the films with variable surface coverage and many defects tend to give different values. For Ca2+, Ba2+, and Cd2+, the ion binding curves are increasing functions of the bulk ion
10344 J. Phys. Chem. B, Vol. 104, No. 44, 2000
Sano et al. noticeable change of the binding fraction when the coagulation occurs. An extremely small amount of Cd2+ is required for the bands to coagulate. For other alkaline earth metals Ca2+, Sr2+, and Ba2+, the coagulation takes place when approximately one ion binds for every four amphiphilic molecules, although this condition has been already satisfied at much lower bulk concentrations of Sr2+. Discussion
Figure 9. Fraction of Sr2+ per molecule as measured by XPS (filled and empty circles) and calculated by the PBS model (solid curve) as a function of the bulk Sr2+ concentration. The coagulation curve of Figure 5 is replotted as the dotted line.
Figure 10. Fraction of Ba2+ per molecule as measured by XPS (filled and empty circles) and calculated by the PBS model (solid and dashed curves) as a function of the bulk Ba2+ concentration. The solid curve is obtained using the original stability constant (pKBa ) 0.341), while the dashed curve is the best fit by varying the stability constant (pKBa ) 1.3). The coagulation curve of Figure 5 is replotted as the dotted line.
Figure 11. Fraction of Cd2+ per molecule as measured by XPS (filled circles) and calculated by the PBS model (solid curve) as a function of the bulk Cd2+ concentration. The coagulation curve of Figure 5 is replotted as the dotted line.
concentrations. Only Sr2+ shows flattening of the adsorption isotherm above 10-6 M. Also plotted as the dotted lines are the coagulation curves from AFM data. For all ions, there is no
Multilayer Formation. The di- and trivalent ions induce multilayers at high concentrations. It is not known if the multilayer is formed spontaneously during solvent evaporation when monolayer is spread or produced as a result of instabilities of coagulated monolayers that easily become multilayers by external disturbances during LB processes. Only Cd2+ gives the multilayers that have the regular step height of a bilayer and are in a H-aggregate state. This is consistent with the common observation that Cd2+ forms a stable 1:2 complex with the carboxylates in the LB films. Two layers are joined by Cd2+ without altering molecular packing and stacked simply to multilayers. Other ions cause the morphology, the layer thickness, and the UV-vis spectra to vary from the monolayer states, indicating that the bands are significantly disturbed in the multilayer states. On the basis of many previous studies on LB films, we speculate that stoichiometry, lattice matching, and hydration are important to make these differences. Further discussion in the multilayer states is not a scope of this paper, and we limit our analysis on the monolayer states in the following. Critical Coagulation. The results clearly show the existence of ion-specific threshold concentrations that cause abrupt coagulation of the bands. Since the film hardly develops measurable surface pressure, the molecules tend to stay within the bands and do not spread out over the free water area. Then, each band can be treated as a self-contained, distinct object floating on the electrolyte surface. The UV-vis peak at 305 nm is sensitive to the orientation of azobenzene chromophores. The shifts to 350 or 405 nm were observed for all homologues of the amphiphile when the films collapsed.20 A lack of these peaks indicates that the coagulation is not collapsing of the film. Since there is no change in the spectral shape for all ions as long as the film remains as a monolayer, the molecular structure of the bands remains the same during coagulation. Thus, the coagulation is between the bands or the constituent clusters and not at the molecular level. This indicates that the role of ions is not bridging of the polar heads, but shielding of the electric potentials between the bands. It is consistent with the observation that the monovalent ions also produce the same coagulation morphology as the multivalent ions. Since a change in pH produces a different type of morphology (wavy bands),21 a possibility that the coagulation by the monovalent ions is due to shifting of pKa is not likely. We know that, at pH 7.0, nearly a half of the carboxylic groups in the bands are ionized.21 Under salt-free conditions, the bands do not stick together even at very small molecular area (the excess parts simply collapse to 3D aggregates usually at the band ends, like a rolled carpet). As we show in the following section, an ordinary PoissonBoltzmann-Stern model assuming an infinite, uniform monolayer describes the interaction between the polar heads and the counterions in the direction perpendicular to the monolayer surface averaged over a large area. The average electric field in the perpendicular direction falls off very sharply. The observed coagulation is caused by shielding of the parallel component of electrostatic repulsion between the band edges
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by electrolyte ions, since the edges locally break the continuity in the film plane. Furthermore, we show here that the SchulzeHardy rule is obeyed between the threshold concentrations and the ionic valency. Therefore, the observed coagulation is analogous to 3D sols, and the threshold values are the critical coagulation concentrations. Charge Regulation. If the electrostatic interaction between colloidal particles is described by valency only, then the ccc should be nearly constant among the ions of the same valency. The result as summarized in Table 1 clearly indicates that ccc varies within the same valency. Among the divalent ions, Cd2+ has a much lower ccc than other divalent ions. This is in good agreement with the binding properties of divalent ions on ordinary fatty acid monolayers. Cd2+ can form a stable complex with carboxylates by covalent-like bonding, whereas other ions bind by weaker electrostatic interactions. Within alkaline earth metals, however, the order of critical concentrations is Mg2+ ∼ Ba2+ < Ca2+ < Sr2+, which does not follow the periodic table. Unlike monovalent ions where the ion-dependent exchange property of mica strongly affects the band stability, the divalent ions cause no such complication even at the highest concentration examined.22 Thus, this order has no relation to the mica substrate. In 3D systems, it has been known that the effective charge depends on the chemical equilibrium between carboxylic groups and specific metal ion.23 It is possible that this charge regulation is effective for the present carboxylic amphiphile as well. From the availability of existing data and experimental feasibility, we have chosen the divalent ions for the detailed examination of ion binding. The adsorption curves are analyzed by a PoissonBoltzmann-Stern (PBS) model developed by Bloch and Yun.24 Their model has been applied successfully to Na+, Ca2+, Ba2+, and Cd2+ binding of fatty acid monolayers in L-films24 as well as in some LB films.25 Briefly, their PBS model consists of an infinite monolayer with a uniform Stern layer whose charge density is completely determined by the chemical equilibrium between carboxylates and ions
R-COO- + i z h R-COOi z-1 (i ) H, Na, M, with z ) 1, 1, 2)
(2)
with the equilibrium (stability) constants
Ki )
[R-COOi z-1]S [R-COO-]S[i z]S
(3)
where M is a divalent ion and [θ]S is the surface concentration of θ. After integration and some algebra, the present problem is reduced to solving the Grahame equation for P0
[(P0 - 1)/P0]{UHXH + UNaXNa + UMXM[(P0 + 2)/P0]}1/2 ) (1 - XM)/(1 + XH + XNa + XM) (4) where P0 relates the surface and bulk concentrations
[i z]S ) P0z[i z]
(5)
Here, Ui ) A2/(BKi), Xi ) Ki[i]S, B ) 4.37 × 10-7 cm is a constant, and A is the area occupied by a molecule. Thus, for the given constants Ki, the surface concentrations of all relevant species are obtained as a function of the bulk ion concentrations by numerically solving eq 4. It has been demonstrated24 that the final result is not sensitive to the exact value of A, and here A is fixed to 25 Å2, a value suggested by the X-ray analysis of the present amphiphile.14
TABLE 2: Stability Constants Used To Calculate Binding Fractions ion pKi (L/mol)
H+
Na+
Ca2+
Sr2+
Ba2+
Cd2+
4.870a
-0.771a
0.505a
0.43b
0.341a
1.913a
1.3c
a Block and Yun,24 originally from refs 26 and 27. b Extrapolated from the ionic radius. c The best fit with the XPS data.
Following their method, Ki is assumed to be intrinsic to each metal ion and is taken from the value obtained in 3D solution studies.26,27 The calculated ion fractions are shown as the solid curves in Figures 8-11. The values of Ki and their sources are listed in Table 2. Considering no adjustable parameters and rather crude estimates of Ki, the theory reproduces the experimental data of Ca2+ very well. The systematic deviations of Sr2+, however, cannot be accounted for by the uncertainties in the experimental data or the estimation of KSr. The curve connecting the experimental points shows a monotonic increase until 10-6 M, flattening at about 0.25 for higher concentrations. Such a leveling-off can only be possible at 0.5 with the PBS model using unrealistic values of KSr. The PBS model also deviates considerably for Ba2+, although the curve shape is correctly reflected. The calculated values of Cd2+ fraction are significantly higher than the experimental data at low concentrations but show a good agreement for the concentrations higher than 10-6 M. At higher concentrations, the PBS model describes the Ca2+ and Cd2+ binding behaviors very well. In solution, over the wide ranges of pH and ionic strength, Ca2+ shows stronger binding with the carboxylate groups among alkaline earth metals. Also, Cd2+ binds stronger than alkaline earth metals. This result indicates that the coagulated film with strongly binding ions can be represented by a continuous charge layer. On the assumption that the PBS model describes Ca2+ correctly for the entire concentration range, the surface concentrations of various species were calculated. There is, however, no discontinuous or abrupt change in the concentrations of any species at the ccc. For other cases, the original PBS model fails to describe the ion-binding behavior. The stability constants used for H+ and Na+ are 4.87 and -0.771 L/mol, respectively. Since the concentration of Na+ is so low, it is not relevant for the present case. Even if we use another value of pKH ) 5.5, reported as the intrinsic pKa of stearic acid monolayer,28 the curve is only shifted slightly to the right, and the mismatch becomes even worse. In the model, the chemical equilibrium is considered only up to the 1:1 complex. Bloch and Yun claim only a small contribution from the 1:2 complex and list a number of reasons.24 The 1:1 stoichiometry is reasonable for the present single layer on mica and is consistent with our spectroscopic data. Additionally, in the case that 1:2 is relevant, its effect should be more pronounced with strongly binding Ca2+ and Cd2+. There is interesting consistency that Sr2+ giving different coagulation morphology from others deviates from the model most. Although the present values of the stability constants are obtained from averaging the values listed in the tables found in solution studies, the experimentally obtained value varies somewhat with pH, ionic strength, temperature, and packing structures of carboxylate compounds. In the case of Ba2+, the similar functional dependence of the experimental points and the PBS model suggests that we may treat KBa as an adjustable parameter to fit the curve. A dashed curve in Figure 10
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Figure 12. Double-logarithmic plot of the critical coagulation concentration against the ionic radius. Note that the alkaline earth metals are on a straight line with a slope of 8.0 with the exception of Ba2+, which exhibits anomalous binding.
represents the result of best fitting, which gives pKBa) 1.3. In electrolyte solution, the value of 1.3 is well over the range of ordinary variation from the average at 0.341. This stability constant is not typical of alkaline earth metals and is closer to that of Cd2+. Here, we see consistency that Ba2+ having the unusually large stability constant for the adsorption isotherm possesses the small ccc. This strongly suggests that Ba2+ behaves anomalously in the ordering sequence of the critical concentrations Mg2+ ∼ Ba2+ < Ca2+ < Sr2+. The anomalous behavior of Ba2+ has been noted by a previous infrared study of stearic acid monolayers29 as well as a crystalline structure study of Ba-stearate.30,31 Schulze-Hardy Rule. While the model does not reproduce the adsorption isotherms completely, it shows that the stability constants reflect charge regulation partially. The stability constants, however, vary with the experimental conditions, and more invariant quantity is desirable for a description of charge regulation. It is well-known that, for the hard Lewis acids such as alkaline earth metals forming complexes with the hard Lewis bases such as carboxylates,32 the stability constants are inversely related to the ionic radius. Empirically, we find that a doublelogarithmic plot of ccc against the ionic radius ri of the divalent ions forms two straight lines, (Mg2+, Ca2+, Sr2+) and (Ba2+, Cd2+), with a slope of 8.0 (Figure 12). This suggests that a power law relation ccc ∼ rim may be introduced to take into account the specific interaction between each ion and the carboxylate group, provided that the ions belong to the same Lewis acids group. All ccc data are summarized as a function of the ratio of ionic radius ri to the valency z in Figure 13. Those ions that show specific adsorptions to the substrate (K+ and Cs+) and the soft Lewis acids (Ba2+ and Cd2+) are excluded. Assuming a functional form ccc ∼ rim/zn, the best fit that gives the least residue (R ) 0.91) yields m ) n ) 9.0. The Schulze-Hardy rule is, then, given by
ccc ∼ (ri/z)9
(6)
in 2D. The valency exponent of 9 is larger than 6 or 2 commonly found in 3D systems. Presently, we are not aware of any theory that explains this result. The largest uncertainty in this rule is the functional form that the ionic radius enters. The radius dependence was obtained empirically only on the divalent ions. The experimental condi-
Figure 13. Double-logarithmic plot of the critical coagulation concentration against the ratio of ionic radius to valency. The slope changes slightly depending on the source of ionic radii. The least residue is obtained with Shannon s value33 giving the slope of 9.0.
tions limited the number of other valency ions available for testing. It is probably necessary to find other amphiphiles to confirm the relation. With the present power law relation, the agreement between the absolute exponents on radius and valency seems to be accidental. If the best-fit exponent of the divalent ions, m ) 8.0, is preassumed, the final result (R ) 0.81) ccc ∼ ri8.0/z9.1 is obtained, indicating that the exponent of z is hardly affected. The excellent linear fit over 5 decades of ccc ensures that eq 6, although it may not be exact, is good to the first order. Conclusions Using the highly anisotropic shape of monolayers of the shortchain carboxyazobenzene compound, critical coagulation caused by the electrolyte ions is shown to follow the Schulze-Hardy rule with the exponent of 9. This study also demonstrates that the mesoscopic bands are 2D colloidal objects. Presently, it is not known how the small clusters of a few tens of nanometers self-organize into a structure extending more than 100 000 nm long but only 60 nm wide. Development of 2D colloidal theory to explain the 2D Schulze-Hardy rule may provide a clue to the mesoscopic self-organization mechanism of monolayer clusters. References and Notes (1) Langmuir, I.; Schaefer, V. J. J. Am. Chem. Soc. 1937, 59, 2400. (2) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Wiley: New York, 1966. (3) Yamauchi, A.; Matsubara, A.; Kimizuka, H.; Abood, L. G. Biochim. Biophys. Acta 1968, 150, 181. (4) Ulman, A. An Introduction to Ultrahin Organic Films: From Langmuir-Blodgett to Self-Assembly; Academic: New York, 1991. (5) Binks, B. P. AdV. Colloid Interface Sci. 1991, 34, 343. (6) Yazdanian, M.; Yu, H.; Zografi Langmuir 1990, 6, 1093. (7) Schulze, H. J. Prakt. Chem. 1882, 25, 431. (8) Hardy, W. B. Proc. R. Soc. London 1900, 66, 110. (9) Overbeek, J. Th. G. Pure Appl. Chem. 1980, 52, 1151. (10) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (11) Israelachvili, J. N. Intermolceular and Surface Forces; Academic: London, 1992. (12) Sano, M.; Kamino, A.; Shinkai, S. Langmuir 1999, 15, 13. (13) Sano, M.; Kamino, A.; Shinkai, S. J. Colloid Interface Sci. 1999, 220, 24. (14) Sano, M.; Sasaki, D. Y.; Isayama, M.; Kunitake, T. Langmuir 1992, 8, 1893. (15) Kobayashi, K.; Takaoka, K.; Ochiai, S. Thin Solid Films 1988, 159, 267. (16) Sastry, M.; Pal, S.; Paranjape, D. V.; Rajagopal, A.; Adhi, S.; Kulkarni, S. K. J. Chem. Phys. 1993, 99, 4799.
2D Schulze-Hardy Rule (17) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (18) Hochstrasser, R. M.; Kasha, M. Photochem. Photobiol. 1964, 3, 317. (19) Kawai, T.; Umemura, J.; Takenaka, T. Langmuir 1989, 5, 1378. (20) Sano, M.; Kunitake, T. Paper presented at the 44th meeting of Colloid and Surface Chemistry; The Chemical Society of Japan: Saitama, 1991; p 56. (21) Sano, M.; Kamino, A.; Shinkai, S. Chem. Lett. 1999, 1091. (22) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 97, 446. (23) Metcalfe, I. M.; Healy, T. W. Faraday Discuss. Chem. Soc. 1990, 90, 335. (24) Bloch, J. M.; Yun, W. Phys. ReV. A 1990, 41, 844.
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