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Breakdown of the Poisson-Boltzmann description for electrical double layers involving large multivalent ions
Thin Solid Films 327–329 (1998) 19–23
Breakdown of the Poisson-Boltzmann description for electrical double layers involving large multivalent ions N. Cuvillier*, F. Rondelez Laboratoire PCC, Institut Curie, 11 Rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
Abstract Langmuir films of amphiphilic molecules with ionizable head groups are very convenient model systems for basic electrostatic studies. Indeed their surface charge density can be continuously adjusted by a simple mechanical compression of the monolayer. Being planar, they also lend themselves to the direct observation of the ion distribution in the aqueous subphase by Brewster angle optical microscopy and by scattering techniques. Using X-ray reflectivity, we have measured the electrical double-layer profile for eicosylamine Langmuir monolayers in contact with a subphase containing large trivalent phosphotungstate ions. The profile observed with a vertical resolution of a few angstroms is markedly different from the diffuse exponential layer described by the Poisson–Boltzmann equation and valid for small monovalent ions. The trivalent anions appear to be condensed in a narrow layer of monomolecular width and of high density. The Brewster images are compatible with this finding and show in addition that their in-plane distribution is homogeneous. Their volume fraction is locally so large, typically 50–70%, that the total number of counter-ions in this layer exceeds what is required by electroneutrality. The interface seen by the X-ray technique appears thus as globally negative. We postulate that the electric compensation is provided by an accumulation of positive hydronium ions near the interface. 1998 Elsevier Science S.A. All rights reserved Keywords: Langmuir films; X-ray reflectivity; Electrostatic
1. Introduction Electrostatic interactions often play a key role in many colloidal systems when the particles are electrical charged (e.g. in biological membranes or colloidal dispersions). The main reason is that these interactions are long range and therefore overcome the Van der Waals forces, except near contact. Their spatial range is controlled by the distribution profile of the counter-ions in the fluid phase surrounding the charged object. For monovalent ions and small surface charge densities, the linearized Poisson–Boltzmann equation predicts a diffuse double layer, with an exponential decay length l which scales as the square root of the solution ionic strength. Detailed experimental tests of the GouyChapman-Stern model by two different methods have been recently published and the results are in excellent agreement with the theory. The first is based on measurements of the forces exerted between two mica plates bearing known sur* Corresponding author. Service de Physique de l’E´tat Condense CEA Saclay, 91191 Gif Sur Yvette Cedex. Fax: +33 1 69088786; e-mail: [email protected]
0040-6090/98/$ - see front matter PII S0040-6090 (98 )0 0579-3
face charges and maintained at controlled distances [1]. The second consists of detecting optically the equilibrium distances between ferrofluid droplets in solutions: in this case, the repulsive electrostatic forces are exactly balanced by the attractive magnetic dipole forces [2]. For large multivalent ions it has been realized for a long time that the ionic profile should markedly deviate from the classical picture. However there is not yet clear-cut experimental data nor widely approved theoretical models in the literature which describe this situation [3]. In this paper, we present X-ray reflectivity measurements on monolayers of charged amphiphiles which allow a direct determination of the ionic profile with atomic resolution. As a model system, we have selected a monolayer of eicosyl amine spread over an aqueous subphase of hydrochloric acid or phosphotungstic acid. The amine head groups are positively charged at all pHs below 6 whereas the chlorine and phosphotungstate anions provide archetypes of monovalent and trivalent counter-ions respectively. The X-ray data have been completed by surface pressure isotherm measurements and also by optical observations using Brewster angle microscopy.
1998 Elsevier Science S.A. All rights reserved
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N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
electron densities due to the water molecules and to the counter-ions:
2. Experimental details 2.1. Monolayer preparation
el el rel ci = nPta cPta + nw
The eicosylamine (C20H43-NH2) monolayers were spread from chloroform solutions on dilute hydrochloric acid (HCl) or phosphotungstic acid (H3PW12O40) subphases in a laboratory-made rectangular trough (40 × 10 × 1 cm3). The pH of the bulk subphase was fixed at 2 in all experiments by adjusting the acid content of the electrolyte. At this pH, the amine group of the amphiphilic molecule is always fully ionized [5] and the phosphotungstate anions (PW12 O340− ) are known to be chemically stable. Their size is ˚ 3) [4]. The surface pressure approximately (10 × 10 × 10 A was measured by a force transducer connected to a paper blade (10 × 4 × 0.2 mm3). The monolayer surface, and therefore the monolayer surface charge density, was varied by using a computer-controlled compression barrier made of Teflon. The range of mean molecular areas per amine, A, ˚ 2. This gives approxiwas typically between 150 and 20 A mately a 10-fold of variation in the surface charge density j, from 0.1 to 1 C/m2. 2.2. X-ray reflectivity The X-ray reflectivity experiments have been performed on a home-built reflectometer, equipped with a 1.5-kW Cu tube, a graphite monochromator and two NaI detectors to monitor simultaneously the incident and reflected beam intensities. The electron density profile r(z) perpendicular to the air-water interface is related to the reflectivity R(Qz) measured at each scattering wave vector Qz [6] by: 1 ∂r(z) 2 dzj expiQz z (1) R(Qz ) = R f (Qz )j rs ∂z
1 − cPta nPta nw
(3)
Phosphotungstate anions (Pta) and water (w) molecules el have nel Pta and nw electrons, respectively, whereas their volumes are vPta and vw, and cPta is the local volume concentration in phosphotungstate anions. For the counter-ion layer, we will see in the following that a box of constant electronic density gives an excellent fit to the data. Finally, ˚ at each interface we have imposed a fixed roughness of 3 A for a more realistic description. This particular value corresponds to the amplitude of the thermally induced fluctuations.
3. Results and discussion 3.1. Pressure-area isotherms Fig. 1 shows the pressure-area isotherms for a monolayer of eicosylamine spread on the two acidic subphases, both at pH 2. The dotted curve corresponds to the hydrochloric acid solution at a concentration of 10−2 M. It is characterized by a wide plateau region of approximately 1 mN/m all the way ˚ 2/ from the highest dilutions of the monolayer down to 30 A molecule, and by a steep increase in pressure, up to more than 60 mN/m, at the lowest areas per molecule. This behavior is typical of a monolayer which undergoes a first-order gas-liquid condensed (G/LC) transition upon compression. What is remarkable is the area at which the monolayer reaches the low compressibility liquid-condensed state. The limiting area (extrapolated at zero pressure) is found ˚ 2/molecule, which is significantly larger than the to be 30 A
Here R f is the Fresnel reflectivity of the bare interface (i.e. without the monolayer) while rs is the electronic density of the semi-infinite aqueous subphase. Since it is difficult to inverse this equation to obtain r(z), the common practice is to postulate a profile and to perform a best fit using a minimum number of adjustable parameters [7,8]. The vertical structure of the air-water interface can thus be ˚ resolution [9]. For our system investigated with a 2 A we have chosen a two-slabs model, one for the eicosylamine monolayer and one for the counter-ion layer. The electron density in the first slab can be taken as uniform since it is mainly due to the long alkyl chains and the head group can be neglected in the first approximation. It is given by: rel a =
nel a Ata
(2)
where nel a is the number of electrons in a single amine molecule, A is the average area per amine, and ta is the monolayer thickness. The electron density in the second slab is given by the weight concentration average of the
Fig. 1. Pressure–area isotherms for a monolayer of eicosylamine C20H43NH+3 spread onto a solution of hydrochloric acid (HCl 10−2 M) (dotted curve) and of phosphotungstic acid (H3PW12O40 3 × 10−3 M) (solid line). ˚ 2/mol per min), the subphase pH is In both cases the compression rate is 2 A 2, and T = 20°C.
N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
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3.2. Brewster angle microscopy
˚ 2/amine), the tion. At the end of the coexistence region (30 A field of view becomes uniform and one observes a homogeneous bright phase. ˚ 2/ Fig. 3 shows the corresponding image observed at 132 A molecule in the case of the phosphotungstic acid subphase. Quite surprisingly, the whole surface of the trough appears uniformly bright. One possible explanation is that no domains are formed, even though the mean area per molecule falls in the plateau region of the phase diagram. Were this be true however, the surface density of the amphiphiles then becomes identical to the mean surface density. This value is too low to create the large refractive index necessary to justify the high brightness of the recorded pictures. A more plausible interpretation is that the signal is due to the PW12O340− counter-ions and that they form a dense and homogeneous layer below the monolayer of charged amines. Phosphotungstate anions are indeed very electron dense and of large enough size to provide a detectable optical index change if adsorbed at the air-water interface. We have checked that the field of view remains uniformly bright as the monolayer is expanded across the coexistence region ˚ 2/amine, there is simply a reducof the isotherm. Until 150 A tion in the general intensity level as the surface density of the amphiphile is lowered. On the other hand, the interface remains dark if there is no amine monolayer. This eliminates the possibility that the optical brightness could be due to the spontaneous adsorption of the phosphotungstate anions at the bare air-water interface. Similarly no optical contrast has been detected if the phosphotungstate anions are put in contact with a monolayer of amine which is uncharged or with a monolayer of long chain alcohol of the same alkyl length. This suggests that non-electrostatic interactions such as attractive Van der Waals forces between the counter-ions and the hydrocarbon chains of the amphiphiles are not important in our experiments. One last argu-
Brewster angle microscopy (BAM) is a convenient technique to investigate the coexistence regions of the phase diagram for Langmuir monolayers. The local changes in density are coupled to variations in the refractive index, which then modulate the intensity of the light reflected by the monolayer [11]. Fig. 2 shows the ellipsometric image ˚ 2, and in the observed for a mean area per molecule of 100 A case of the hydrochloric acid subphase. For this area value, the monolayer is well into the plateau region of the isotherm and bright domains dispersed in a dark background are observed. Their typical size is 1500 mm in length by 50 mm in width (the full screen is 3 × 3 mm2). They correspond to high density, liquid condensed phase, islands in coexistence with the low density gas phase. Their elongated shape results from the competition between the LC/G line tension, which tends to reduce the boundary length, and the repulsive dipolar interaction between the charged head groups, which favors the elongation of the domains [12]. The proportion of LC domains increases when the layer is compressed, as expected from the lever rule for a first-order phase transi-
Fig. 2. Brewster angle microscopy image of an eicosylamine monolayer ˚ 2/molecule. The monospread onto a hydrochloric acid subphase at 100 A layer is in its liquid condensed-gas (LC/G) coexistence region. The liquid condensed domains appear as bright filaments dispersed in the dark gaseous phase. The field of view is 3 × 3 mm.
hard core area of the aliphatic chain, which is of the order of ˚ 2. This long range repulsion is due to the strong elec20 A trostatic interactions between the charged amine head groups. The surface pK value for long chain amines at the air-water interface has been measured to be 10.1 (compared with 10.6 in bulk aqueous solutions) [10] and it is therefore reasonable to assume that the head groups are fully ionized at pH 2. The solid curve corresponds to the surface pressure isotherm for the same eicosylamine monolayer but spread over an 3 × 10−3 M aqueous solution of phosphotungstic acid. In that case, the plateau region does not extend as low as before and the increase in pressure as the monolayer is compressed further is also not as steep. This behavior is the signature of a gas to liquid expanded first-order transition. The liquid expanded phase is observed between 60 and ˚ 2/molecule, which is typical for amphiphiles with a 40 A single alkyl chain. Its compressibility is roughly constant over that range of areas and is of the order of 2 mN/(A2/ molecule). When the pressure reaches 44 mN/m, a sudden drop is observed in the measured pressure value which then ˚ 2/molecule, relaxes down to 40 mN/m. Between 40 and 20 A the exact shape of the isotherm depends on the compression speed, and the monolayer is clearly not in thermal equilibrium state. This metastability is however not due to a dissolution of the amphiphilic molecules in the subphase. At 20 ˚ 2/molecule, the pressure re-increases again as the aliphatic A chains experience hard core steric repulsion. The fact that the two isotherms are markedly different reflects the influence of the nature (size and valence) of the counter-ions present in the subphase. The interactions between the charged amine head groups ‘dressed’ by the counter-ions are obviously different depending if the ions are small and monovalent or large and trivalent.
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Fig. 3. Brewster angle microscopy image as in Fig. 2, but for a phospho˚ 2. Surpristungstic acid subphase. The mean area per molecule is 132 A ingly enough, no liquid expanded domains are detected in the image even though the above conditions correspond to the liquid expanded-gas (LE/G) two-phase region of the monolayer. The uniform brightness is due to a dense layer of phosphotungstate counter-ions present at the air–water interface. This layer is spatially homogeneous and of sufficiently high optical index to be detected by imaging ellipsometry.
ment which has to be considered is related to the low lateral resolution of our BAM set-up. Since it is not better than 10 mm, domains of that size or below will remain undetected if they are randomly distributed in the plane of the interface. Such a possibility is not easy to disprove using optical ellipsometry and should be checked in future work, by using near-field, high spatial resolution, probes such as atomic force microscopy.
˚, in the inset. The amine layer has a thickness ta = 9.8 A which is consistent with the thickness t calculated for tilted ˚ [13] and of alkyl chains of extended length tmax = 24 A ˚ 2 (one has t = tmax). geometric cross section A0 = 20 A From the fit, the electron density of the amine layer is ˚ −3. This last value compare well with the found to be 0.3 A ˚ −3, taking the known surface dencalculated value of 0.35 A sity and a total number of electrons of 299 for the long chain amines. As the monolayer surface density increases from ˚ 2/amine), we have found that ta increases from 5140 to 40 A ˚ . The chains reorient themselves towards the ver6 to 20 A tical axis when they are forced to pack more closely. The counter-ions are all contained in a narrow region of thick˚ and of volume fraction 58 ± 3%. They are ness of 10 ± 2 A therefore located in a single layer of monomolecular width and in immediate contact with the charged amine head groups. The electrostatic interaction thus appear to be very large and the electrical double layer for these trivalent anions is certainly not as diffuse as in the classical Gouy– Chapman picture. The ions stay concentrated in a Stern layer of almost close-packed density. Actually, we have even observed volume fractions as high as 70% for amine ˚ 2. This X-ray result is in full agreesurface densities of 40 A ment with the BAM optical observation of a dense uniform layer of counter-ions. In Fig. 5, the ratio R between the number of counter-ions detected in this layer and the number of amine head groups at the air-water interface has been plotted as a function of the mean area per eicosylamine, A. We observe that this ratio does not vary as would be expected from straightforward electroneutrality. Since the amine head groups are
3.3. X-ray reflectivity The electron density profile perpendicular to the interface has been measured by surface X-ray reflectometry in the case of the phosphotungstic acid subphase at pH 2. All ˚ 2/amine have simireflectivity curves between 140 and 40 A lar shapes and Fig. 4 shows a typical reflectivity curve cor˚ 2/amine, and normalized to the Fresnel responding to 75 A reflectivity of a bare air-water interface. The data points go ˚ −1, and through a marked primary maximum near 0.15 A there is a second, although weaker, maximum around 0.38 ˚ −1 which appears as a shoulder on the main broad peak. A The peak values of the reflected intensity are both well above unity, eight and three times, respectively. This overall shape indicates that the interface is sharp and that a layer of much higher electron density than water is present below the amine monolayer. The solid line is the result of a least square fit using the two-slabs model described previously. The parameters are the thicknesses of the amine layer and of the counter-ions layer, and their respective electron densities. The agreement is truly excellent and confirms our representation of the interface by two independent layers, each with constant, and spatially homogeneous, electron ˚ 2/amine and a bulk densities. The detailed profile for 75 A phosphotungstic acid concentration of 3 × 10−3 M, is shown
Fig. 4. Normalized X-ray reflectivity for a monolayer of eicosylamine spread onto a subphase of phosphotungstic acid H3PW12O40. The mean ˚ 2. The solid line is the best fit to the experimental area per molecule is 75 A data using a two-slabs model to describe the eicosylamine and the counterion layers respectively. In addition we have introduced a fixed roughness ˚ at each interface. The inset shows the actual electron density profile of 3 A derived from the fit, and the smearing effect due to the roughness (dotted line).
N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
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˚ 2/amine (after the pressure drop The region below 40 A observed in Fig. 1) corresponds to the formation of multilayers of adsorbed counter-ions. (as proven as by additional X-ray reflectivity measurements which will be published elsewhere).
4. Conclusion
Fig. 5. Ratio of the number of trivalent counter-ions detected by X-ray reflectivity to the number of amine head groups for different areas per eicosylamine molecule. The data points have been obtained over many independent experiments. The dotted line indicates that the variation is approximately linear in the investigated range. The measured R values are characteristically higher than the 0.3 value predicted by a simple electroneutrality criterion.
monovalent and the phosphotungstate anions trivalent, it would be normal to find a fixed value R = 0.3, independent of A. Rather, R increases monotonously with A and changes ˚ 2/amphiphile by as much as 0.3–1.3 between 40 and 140 A 2 ˚ , there is thus an excess of molecule. Except for A = 40 A negative counter-ions in the vicinity of the charged monolayer. If there was a deficit of negative charges, we could have thought that some of the counter-ions have not been measured in the X-ray experiments. Indeed the reflected intensity depends on the square of the local concentration gradients and therefore smooth variations are not easily detected. However this possibility is clearly ruled out here since the effect is in the opposite direction. We can similarly eliminate the possibility that the counter-ions bear a smaller charge than expected or that they are chemically degraded. Phosphotungstic acid is a strong acid which is completely dissociated and also known to be chemically stable against hydrolysis at pH 2. In addition, products of hydrolytic degradation are all more highly charged [14]. If we accept the fact that the positive surface charges of the monolayer are overcompensated by the phosphotungstate anions, the only way to satisfy global electroneutrality is to admit that there is an accumulation of extra cations near the interface which bring the required positive charges. The only other positively charged species present in solution are the hydronium H3O+ cations. They are present in large quantities in the aqueous subphase and they will be attracted by the net charge of the eicosylamine-phosphotungstate counter-ions system. Since their electron density is equal to that of water, there is no contrast for the X-rays and they remain undetected [15]).
We have measured the electrical double layer profile of large multivalent counter-ions in the vicinity of a positively charged Langmuir monolayer. Even at low surface charge ˚ 2/charge), the counter-ions are density (and down to 40 A located in the immediate vicinity of the monolayer and forms a dense layer of molecular thickness. They actually overcompensate the positive surface charge of the Langmuir monolayer. The surface seen by the X-rays seems to bear a net negative charge. We surmise that the global electroneutrality of the system is achieved by the hydronium cations present in the aqueous subphase and which are attracted to the interface.
Acknowledgements We wish to thank P. Ganguly for introducing us to the heteropolyanions and D. Andelman and R. Orland for helpful theoretical discussions. We gratefully acknowledge the financial support of IFCPAR-CEFIPRA, the Indo-French Centre for the promotion of Advanced Research, under contract No. 907-2-94-2036.
References [1] J.N. Israelachvili, Accounts Chem. Res. 20 (1987) 415. [2] O. Mondain-Monval, F. Leal-Calderon, J. Bibette, J. Phys. II (France) 6 (1996) 1313. [3] I. Borukhov, D. Andelman, H. Orland, Phys. Rev. Lett. 79 (3) (1997) 435. [4] M.R. Spirlet, W.R. Busing, Acta Cryst. 154 (1987) 137. [5] X. Zhao, S. Ong, H. Wang, K.B. Eisenthal, Chem. Phys. Lett. 214 (1993) 203. [6] A. Braslau, P.S. Pershan, G. Swislow, B.M. Ocko, J. Als-Nielsen, Phys. Rev. A 45 (12) (1992) 8894. [7] L.G. Parrat, Phys. Rev. Lett. 95 (1954) 359. [8] D. Bouzida, S. Kuma, R.H. Swendsen, Phys. Rev., A 45 (12) (1988) 8894. [9] L. Bosio, J.J. Benattar, F. Rieutord, Rev. Phys. Appl. 22 (1987) 775. [10] J.J. Betts, B.A. Pethica, Trans. Faraday Soc. 52 (1956) 1581. [11] S. Hennon, J. Meunier, Rev. Sci. Instrum. 62 (1991) 936. [12] K.J. Stine, D. Statmann, Langmuir 8 (10) (1992) 2510. [13] M. Deutsch, Europhys. Lett. 30 (1995) 283. [14] M.T. Pope, Heteropoly and Isopoly Oxometalates, Springer, Berlin, 1983. [15] N. Cuvillier, M. Bonnier, F. Rondelez, D. Paranjape, M. Sastry, P. Ganguly, Prog. Colloid. Polym. Sci. 105 (1997) 118.
Breakdown of the Poisson-Boltzmann description for electrical double layers involving large multivalent ions N. Cuvillier*, F. Rondelez Laboratoire PCC, Institut Curie, 11 Rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
Abstract Langmuir films of amphiphilic molecules with ionizable head groups are very convenient model systems for basic electrostatic studies. Indeed their surface charge density can be continuously adjusted by a simple mechanical compression of the monolayer. Being planar, they also lend themselves to the direct observation of the ion distribution in the aqueous subphase by Brewster angle optical microscopy and by scattering techniques. Using X-ray reflectivity, we have measured the electrical double-layer profile for eicosylamine Langmuir monolayers in contact with a subphase containing large trivalent phosphotungstate ions. The profile observed with a vertical resolution of a few angstroms is markedly different from the diffuse exponential layer described by the Poisson–Boltzmann equation and valid for small monovalent ions. The trivalent anions appear to be condensed in a narrow layer of monomolecular width and of high density. The Brewster images are compatible with this finding and show in addition that their in-plane distribution is homogeneous. Their volume fraction is locally so large, typically 50–70%, that the total number of counter-ions in this layer exceeds what is required by electroneutrality. The interface seen by the X-ray technique appears thus as globally negative. We postulate that the electric compensation is provided by an accumulation of positive hydronium ions near the interface. 1998 Elsevier Science S.A. All rights reserved Keywords: Langmuir films; X-ray reflectivity; Electrostatic
1. Introduction Electrostatic interactions often play a key role in many colloidal systems when the particles are electrical charged (e.g. in biological membranes or colloidal dispersions). The main reason is that these interactions are long range and therefore overcome the Van der Waals forces, except near contact. Their spatial range is controlled by the distribution profile of the counter-ions in the fluid phase surrounding the charged object. For monovalent ions and small surface charge densities, the linearized Poisson–Boltzmann equation predicts a diffuse double layer, with an exponential decay length l which scales as the square root of the solution ionic strength. Detailed experimental tests of the GouyChapman-Stern model by two different methods have been recently published and the results are in excellent agreement with the theory. The first is based on measurements of the forces exerted between two mica plates bearing known sur* Corresponding author. Service de Physique de l’E´tat Condense CEA Saclay, 91191 Gif Sur Yvette Cedex. Fax: +33 1 69088786; e-mail: [email protected]
0040-6090/98/$ - see front matter PII S0040-6090 (98 )0 0579-3
face charges and maintained at controlled distances [1]. The second consists of detecting optically the equilibrium distances between ferrofluid droplets in solutions: in this case, the repulsive electrostatic forces are exactly balanced by the attractive magnetic dipole forces [2]. For large multivalent ions it has been realized for a long time that the ionic profile should markedly deviate from the classical picture. However there is not yet clear-cut experimental data nor widely approved theoretical models in the literature which describe this situation [3]. In this paper, we present X-ray reflectivity measurements on monolayers of charged amphiphiles which allow a direct determination of the ionic profile with atomic resolution. As a model system, we have selected a monolayer of eicosyl amine spread over an aqueous subphase of hydrochloric acid or phosphotungstic acid. The amine head groups are positively charged at all pHs below 6 whereas the chlorine and phosphotungstate anions provide archetypes of monovalent and trivalent counter-ions respectively. The X-ray data have been completed by surface pressure isotherm measurements and also by optical observations using Brewster angle microscopy.
1998 Elsevier Science S.A. All rights reserved
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N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
electron densities due to the water molecules and to the counter-ions:
2. Experimental details 2.1. Monolayer preparation
el el rel ci = nPta cPta + nw
The eicosylamine (C20H43-NH2) monolayers were spread from chloroform solutions on dilute hydrochloric acid (HCl) or phosphotungstic acid (H3PW12O40) subphases in a laboratory-made rectangular trough (40 × 10 × 1 cm3). The pH of the bulk subphase was fixed at 2 in all experiments by adjusting the acid content of the electrolyte. At this pH, the amine group of the amphiphilic molecule is always fully ionized [5] and the phosphotungstate anions (PW12 O340− ) are known to be chemically stable. Their size is ˚ 3) [4]. The surface pressure approximately (10 × 10 × 10 A was measured by a force transducer connected to a paper blade (10 × 4 × 0.2 mm3). The monolayer surface, and therefore the monolayer surface charge density, was varied by using a computer-controlled compression barrier made of Teflon. The range of mean molecular areas per amine, A, ˚ 2. This gives approxiwas typically between 150 and 20 A mately a 10-fold of variation in the surface charge density j, from 0.1 to 1 C/m2. 2.2. X-ray reflectivity The X-ray reflectivity experiments have been performed on a home-built reflectometer, equipped with a 1.5-kW Cu tube, a graphite monochromator and two NaI detectors to monitor simultaneously the incident and reflected beam intensities. The electron density profile r(z) perpendicular to the air-water interface is related to the reflectivity R(Qz) measured at each scattering wave vector Qz [6] by: 1 ∂r(z) 2 dzj expiQz z (1) R(Qz ) = R f (Qz )j rs ∂z
1 − cPta nPta nw
(3)
Phosphotungstate anions (Pta) and water (w) molecules el have nel Pta and nw electrons, respectively, whereas their volumes are vPta and vw, and cPta is the local volume concentration in phosphotungstate anions. For the counter-ion layer, we will see in the following that a box of constant electronic density gives an excellent fit to the data. Finally, ˚ at each interface we have imposed a fixed roughness of 3 A for a more realistic description. This particular value corresponds to the amplitude of the thermally induced fluctuations.
3. Results and discussion 3.1. Pressure-area isotherms Fig. 1 shows the pressure-area isotherms for a monolayer of eicosylamine spread on the two acidic subphases, both at pH 2. The dotted curve corresponds to the hydrochloric acid solution at a concentration of 10−2 M. It is characterized by a wide plateau region of approximately 1 mN/m all the way ˚ 2/ from the highest dilutions of the monolayer down to 30 A molecule, and by a steep increase in pressure, up to more than 60 mN/m, at the lowest areas per molecule. This behavior is typical of a monolayer which undergoes a first-order gas-liquid condensed (G/LC) transition upon compression. What is remarkable is the area at which the monolayer reaches the low compressibility liquid-condensed state. The limiting area (extrapolated at zero pressure) is found ˚ 2/molecule, which is significantly larger than the to be 30 A
Here R f is the Fresnel reflectivity of the bare interface (i.e. without the monolayer) while rs is the electronic density of the semi-infinite aqueous subphase. Since it is difficult to inverse this equation to obtain r(z), the common practice is to postulate a profile and to perform a best fit using a minimum number of adjustable parameters [7,8]. The vertical structure of the air-water interface can thus be ˚ resolution [9]. For our system investigated with a 2 A we have chosen a two-slabs model, one for the eicosylamine monolayer and one for the counter-ion layer. The electron density in the first slab can be taken as uniform since it is mainly due to the long alkyl chains and the head group can be neglected in the first approximation. It is given by: rel a =
nel a Ata
(2)
where nel a is the number of electrons in a single amine molecule, A is the average area per amine, and ta is the monolayer thickness. The electron density in the second slab is given by the weight concentration average of the
Fig. 1. Pressure–area isotherms for a monolayer of eicosylamine C20H43NH+3 spread onto a solution of hydrochloric acid (HCl 10−2 M) (dotted curve) and of phosphotungstic acid (H3PW12O40 3 × 10−3 M) (solid line). ˚ 2/mol per min), the subphase pH is In both cases the compression rate is 2 A 2, and T = 20°C.
N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
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3.2. Brewster angle microscopy
˚ 2/amine), the tion. At the end of the coexistence region (30 A field of view becomes uniform and one observes a homogeneous bright phase. ˚ 2/ Fig. 3 shows the corresponding image observed at 132 A molecule in the case of the phosphotungstic acid subphase. Quite surprisingly, the whole surface of the trough appears uniformly bright. One possible explanation is that no domains are formed, even though the mean area per molecule falls in the plateau region of the phase diagram. Were this be true however, the surface density of the amphiphiles then becomes identical to the mean surface density. This value is too low to create the large refractive index necessary to justify the high brightness of the recorded pictures. A more plausible interpretation is that the signal is due to the PW12O340− counter-ions and that they form a dense and homogeneous layer below the monolayer of charged amines. Phosphotungstate anions are indeed very electron dense and of large enough size to provide a detectable optical index change if adsorbed at the air-water interface. We have checked that the field of view remains uniformly bright as the monolayer is expanded across the coexistence region ˚ 2/amine, there is simply a reducof the isotherm. Until 150 A tion in the general intensity level as the surface density of the amphiphile is lowered. On the other hand, the interface remains dark if there is no amine monolayer. This eliminates the possibility that the optical brightness could be due to the spontaneous adsorption of the phosphotungstate anions at the bare air-water interface. Similarly no optical contrast has been detected if the phosphotungstate anions are put in contact with a monolayer of amine which is uncharged or with a monolayer of long chain alcohol of the same alkyl length. This suggests that non-electrostatic interactions such as attractive Van der Waals forces between the counter-ions and the hydrocarbon chains of the amphiphiles are not important in our experiments. One last argu-
Brewster angle microscopy (BAM) is a convenient technique to investigate the coexistence regions of the phase diagram for Langmuir monolayers. The local changes in density are coupled to variations in the refractive index, which then modulate the intensity of the light reflected by the monolayer [11]. Fig. 2 shows the ellipsometric image ˚ 2, and in the observed for a mean area per molecule of 100 A case of the hydrochloric acid subphase. For this area value, the monolayer is well into the plateau region of the isotherm and bright domains dispersed in a dark background are observed. Their typical size is 1500 mm in length by 50 mm in width (the full screen is 3 × 3 mm2). They correspond to high density, liquid condensed phase, islands in coexistence with the low density gas phase. Their elongated shape results from the competition between the LC/G line tension, which tends to reduce the boundary length, and the repulsive dipolar interaction between the charged head groups, which favors the elongation of the domains [12]. The proportion of LC domains increases when the layer is compressed, as expected from the lever rule for a first-order phase transi-
Fig. 2. Brewster angle microscopy image of an eicosylamine monolayer ˚ 2/molecule. The monospread onto a hydrochloric acid subphase at 100 A layer is in its liquid condensed-gas (LC/G) coexistence region. The liquid condensed domains appear as bright filaments dispersed in the dark gaseous phase. The field of view is 3 × 3 mm.
hard core area of the aliphatic chain, which is of the order of ˚ 2. This long range repulsion is due to the strong elec20 A trostatic interactions between the charged amine head groups. The surface pK value for long chain amines at the air-water interface has been measured to be 10.1 (compared with 10.6 in bulk aqueous solutions) [10] and it is therefore reasonable to assume that the head groups are fully ionized at pH 2. The solid curve corresponds to the surface pressure isotherm for the same eicosylamine monolayer but spread over an 3 × 10−3 M aqueous solution of phosphotungstic acid. In that case, the plateau region does not extend as low as before and the increase in pressure as the monolayer is compressed further is also not as steep. This behavior is the signature of a gas to liquid expanded first-order transition. The liquid expanded phase is observed between 60 and ˚ 2/molecule, which is typical for amphiphiles with a 40 A single alkyl chain. Its compressibility is roughly constant over that range of areas and is of the order of 2 mN/(A2/ molecule). When the pressure reaches 44 mN/m, a sudden drop is observed in the measured pressure value which then ˚ 2/molecule, relaxes down to 40 mN/m. Between 40 and 20 A the exact shape of the isotherm depends on the compression speed, and the monolayer is clearly not in thermal equilibrium state. This metastability is however not due to a dissolution of the amphiphilic molecules in the subphase. At 20 ˚ 2/molecule, the pressure re-increases again as the aliphatic A chains experience hard core steric repulsion. The fact that the two isotherms are markedly different reflects the influence of the nature (size and valence) of the counter-ions present in the subphase. The interactions between the charged amine head groups ‘dressed’ by the counter-ions are obviously different depending if the ions are small and monovalent or large and trivalent.
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Fig. 3. Brewster angle microscopy image as in Fig. 2, but for a phospho˚ 2. Surpristungstic acid subphase. The mean area per molecule is 132 A ingly enough, no liquid expanded domains are detected in the image even though the above conditions correspond to the liquid expanded-gas (LE/G) two-phase region of the monolayer. The uniform brightness is due to a dense layer of phosphotungstate counter-ions present at the air–water interface. This layer is spatially homogeneous and of sufficiently high optical index to be detected by imaging ellipsometry.
ment which has to be considered is related to the low lateral resolution of our BAM set-up. Since it is not better than 10 mm, domains of that size or below will remain undetected if they are randomly distributed in the plane of the interface. Such a possibility is not easy to disprove using optical ellipsometry and should be checked in future work, by using near-field, high spatial resolution, probes such as atomic force microscopy.
˚, in the inset. The amine layer has a thickness ta = 9.8 A which is consistent with the thickness t calculated for tilted ˚ [13] and of alkyl chains of extended length tmax = 24 A ˚ 2 (one has t = tmax). geometric cross section A0 = 20 A From the fit, the electron density of the amine layer is ˚ −3. This last value compare well with the found to be 0.3 A ˚ −3, taking the known surface dencalculated value of 0.35 A sity and a total number of electrons of 299 for the long chain amines. As the monolayer surface density increases from ˚ 2/amine), we have found that ta increases from 5140 to 40 A ˚ . The chains reorient themselves towards the ver6 to 20 A tical axis when they are forced to pack more closely. The counter-ions are all contained in a narrow region of thick˚ and of volume fraction 58 ± 3%. They are ness of 10 ± 2 A therefore located in a single layer of monomolecular width and in immediate contact with the charged amine head groups. The electrostatic interaction thus appear to be very large and the electrical double layer for these trivalent anions is certainly not as diffuse as in the classical Gouy– Chapman picture. The ions stay concentrated in a Stern layer of almost close-packed density. Actually, we have even observed volume fractions as high as 70% for amine ˚ 2. This X-ray result is in full agreesurface densities of 40 A ment with the BAM optical observation of a dense uniform layer of counter-ions. In Fig. 5, the ratio R between the number of counter-ions detected in this layer and the number of amine head groups at the air-water interface has been plotted as a function of the mean area per eicosylamine, A. We observe that this ratio does not vary as would be expected from straightforward electroneutrality. Since the amine head groups are
3.3. X-ray reflectivity The electron density profile perpendicular to the interface has been measured by surface X-ray reflectometry in the case of the phosphotungstic acid subphase at pH 2. All ˚ 2/amine have simireflectivity curves between 140 and 40 A lar shapes and Fig. 4 shows a typical reflectivity curve cor˚ 2/amine, and normalized to the Fresnel responding to 75 A reflectivity of a bare air-water interface. The data points go ˚ −1, and through a marked primary maximum near 0.15 A there is a second, although weaker, maximum around 0.38 ˚ −1 which appears as a shoulder on the main broad peak. A The peak values of the reflected intensity are both well above unity, eight and three times, respectively. This overall shape indicates that the interface is sharp and that a layer of much higher electron density than water is present below the amine monolayer. The solid line is the result of a least square fit using the two-slabs model described previously. The parameters are the thicknesses of the amine layer and of the counter-ions layer, and their respective electron densities. The agreement is truly excellent and confirms our representation of the interface by two independent layers, each with constant, and spatially homogeneous, electron ˚ 2/amine and a bulk densities. The detailed profile for 75 A phosphotungstic acid concentration of 3 × 10−3 M, is shown
Fig. 4. Normalized X-ray reflectivity for a monolayer of eicosylamine spread onto a subphase of phosphotungstic acid H3PW12O40. The mean ˚ 2. The solid line is the best fit to the experimental area per molecule is 75 A data using a two-slabs model to describe the eicosylamine and the counterion layers respectively. In addition we have introduced a fixed roughness ˚ at each interface. The inset shows the actual electron density profile of 3 A derived from the fit, and the smearing effect due to the roughness (dotted line).
N. Cuvillier, F. Rondelez / Thin Solid Films 327–329 (1998) 19–23
23
˚ 2/amine (after the pressure drop The region below 40 A observed in Fig. 1) corresponds to the formation of multilayers of adsorbed counter-ions. (as proven as by additional X-ray reflectivity measurements which will be published elsewhere).
4. Conclusion
Fig. 5. Ratio of the number of trivalent counter-ions detected by X-ray reflectivity to the number of amine head groups for different areas per eicosylamine molecule. The data points have been obtained over many independent experiments. The dotted line indicates that the variation is approximately linear in the investigated range. The measured R values are characteristically higher than the 0.3 value predicted by a simple electroneutrality criterion.
monovalent and the phosphotungstate anions trivalent, it would be normal to find a fixed value R = 0.3, independent of A. Rather, R increases monotonously with A and changes ˚ 2/amphiphile by as much as 0.3–1.3 between 40 and 140 A 2 ˚ , there is thus an excess of molecule. Except for A = 40 A negative counter-ions in the vicinity of the charged monolayer. If there was a deficit of negative charges, we could have thought that some of the counter-ions have not been measured in the X-ray experiments. Indeed the reflected intensity depends on the square of the local concentration gradients and therefore smooth variations are not easily detected. However this possibility is clearly ruled out here since the effect is in the opposite direction. We can similarly eliminate the possibility that the counter-ions bear a smaller charge than expected or that they are chemically degraded. Phosphotungstic acid is a strong acid which is completely dissociated and also known to be chemically stable against hydrolysis at pH 2. In addition, products of hydrolytic degradation are all more highly charged [14]. If we accept the fact that the positive surface charges of the monolayer are overcompensated by the phosphotungstate anions, the only way to satisfy global electroneutrality is to admit that there is an accumulation of extra cations near the interface which bring the required positive charges. The only other positively charged species present in solution are the hydronium H3O+ cations. They are present in large quantities in the aqueous subphase and they will be attracted by the net charge of the eicosylamine-phosphotungstate counter-ions system. Since their electron density is equal to that of water, there is no contrast for the X-rays and they remain undetected [15]).
We have measured the electrical double layer profile of large multivalent counter-ions in the vicinity of a positively charged Langmuir monolayer. Even at low surface charge ˚ 2/charge), the counter-ions are density (and down to 40 A located in the immediate vicinity of the monolayer and forms a dense layer of molecular thickness. They actually overcompensate the positive surface charge of the Langmuir monolayer. The surface seen by the X-rays seems to bear a net negative charge. We surmise that the global electroneutrality of the system is achieved by the hydronium cations present in the aqueous subphase and which are attracted to the interface.
Acknowledgements We wish to thank P. Ganguly for introducing us to the heteropolyanions and D. Andelman and R. Orland for helpful theoretical discussions. We gratefully acknowledge the financial support of IFCPAR-CEFIPRA, the Indo-French Centre for the promotion of Advanced Research, under contract No. 907-2-94-2036.
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