Formation of Stable Singularities in Mixed Monolayers of Porphyrins

Langmuir 2000, 16, 1299-1305

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Formation of Stable Singularities in Mixed Monolayers of Porphyrins and Tetracosanoic Acid upon SFM Tapping Christian Messerschmidt,† Andrea Schulz,† Ju¨rgen P. Rabe,‡ Arnold Simon,§ Othmar Marti,§ and Ju¨rgen-Hinrich Fuhrhop*,† Institut fu¨ r Organische Chemie der Freien Universita¨ t Berlin, Takustrasse 3, D-14195 Berlin, Germany, Institut fu¨ r Physik der Humboldt-Universita¨ t zu Berlin, Invalidenstrasse 110, D-10115 Berlin, Germany, and Fakulta¨ t fu¨ r Naturwissenschaften der Universita¨ t Ulm, Abteilung Experimentelle Physik, Albert-Einstein-Allee 11, D-89069 Ulm, Germany Received June 21, 1999. In Final Form: October 6, 1999 Langmuir films of tetracosanoic acid have been transferred at 10 mN/m from water to mica surfaces and were characterized by scanning force microscopy (SFM) in the tapping mode at first as a smooth monolayer. Then, upon repeated tapping cycles, many singularities appeared in form of 2.5 nm high pairs of peaks in a plane, which stretched only 1.8 nm above the mica subphase. These peaks are attributed to islands of upright-standing molecules in a layer of molecules tilted at an angle of 35°. Transferring the films thus results first in a nonequilibrated film on mica, which undergoes relaxation upon tapping to a nonhomogeneous equilibrium phase. The same phenomenon was observed in films made of rigid porphyrin and tetracosanoic acid domains at 20 mN/m. The formation of fluid fatty acid structures at pressures where strong ordering prevails in pure fatty acid films was related to a reorientation in the rigid porphyrin domains after the transfer to mica. SFM phase shift images were applied to different hard and soft parts of the mixed monolayer, and scanning near-field optical microscopy was used to confirm the assignment of the porphyrin domains on the basis of their fluorescence.

1. Introduction Stiff porphyrin chromophores dissolve well in fluid phospholipid monolayers and bilayers. The solubility of magnesium octaethylporphyrin in fluid phospholipids is, for example, 10 mol % and drops practically to zero after transition from the gas to the liquid phase of these layers.1 Fluorescence microscopy on the water surface reveals large fluorescent spots at low surface pressures, which disappear upon increasing the surface pressure.1-3 In cases where the dye and the lipid phase of the host monolayer mix, one observes smaller areas per molecule4 and the shape of the isotherms changes at different mixing ratios. We report here on combined scanning force microscopy (SFM) and scanning near-field optical microscopy (SNOM) studies of mixed porphyrin 1/tetracosanoic acid monolayers, where 1 forms rigid domains even at very low surface pressures.5 The hard porphyrin and soft fatty acid monolayers should differ strongly in their Young modulus and represent a good system for the interpretation of SFM phase shift images at different forces. Recently a model to interpret phase shifts on the basis of stiffness and contact radii has been proposed.6 SNOM studies of the mixed monolayers were carried out in order to assign the domains. They were possible because porphyrin 1 is known to form fluorescent monolayers without self-quenching.5 * To whom correspondence should be addressed. † Freien Universita ¨ t Berlin. ‡ Humboldt-Universita ¨ t zu Berlin. § Universita ¨ t Ulm. (1) Mo¨hwald, H.; Miller, A.; Stich, W.; Knoll, W.; Ruaudel-Teixier, A.; Lehmann, T.; Fuhrhop, J.-H. Thin Solid Films 1986, 141, 261. (2) Fischer, A.; Heithier, H.; Knoll, W.; Mo¨hwald, H. Ber. BunsenGes. Phys. Chem. 1981, 85, 195. (3) Heithier, H.; Mo¨hwald, H. Z. Naturforsch. 1983, 38C, 1003. (4) Mo¨bius, D. Z. Phys. Chem., Neue Folge 1987, 154, 121. (5) Endisch, C.; Bo¨ttcher, C.; Fuhrhop, J.-H. J. Am. Chem. Soc. 1995, 117, 8273. (6) Magonov, S. N.; Elings, V.; Whangbo, M.-H. Surf. Sci. 1997, 375, 385.

A new equilibrium state showing stable vortices was formed and detected by tapping SFM. 2. Experimental Section The synthesis of β-tetraethyl-β-tetrapyridinylporphyrin has been described earlier.7 Tetracosanoic acid was purchased from Aldrich and used as received. The mixtures were cosolubilized in chloroform and prepared as monolayers on the air-water interface on a LAUDA Langmuir film balance FW2 (LaudaKo¨nigshofen, Germany). The subphase water was purified by a Millipore system, adjusted in the case of pH ) 2.5 with hydrochloric acid. The LB films were transferred onto freshly cleaved mica (Plano, Wetzlar/Germany) at a speed of 1 mm/min. Scanning force microscopy (SFM) measurements were performed using a Digital Instruments Nanoscope IIIa (Santa Barbara, CA) in tapping mode. Height and phase images were recorded simultaneously. The tapping mode phase offset was always zeroed before scanning. Silicon cantilevers (Digital Instruments) with a spring constant of 45-60 N/m and a resonance frequency in the range of 270-350 kHz were used. The scanning rate was usually 1.5 Hz. Scanning near-field optical microscopy (SNOM) was performed using a home-built microscope. The topographical images were obtained by using a piezoelectric shear force distance control.8,9 For the optical image, the fluorescence was excited with the 488 nm argon ion laser line and an output power of the tip around 1 nW. The fluorescence is detected in transmission (Nikon Plan Fluor 60 × 0.85, 160 mm) by means of an avalanche photodiode (EG&G single photon counting module SPCM-200 PQ, dark counts 2/s), protected by an edgefilter (Schott OG 590 nm).

3. Results and Discussion The porphyrin bolaamphiphile 1 has hydrophilic north and south edges (four pyridine rings) and hydrophobic (7) Endisch, C.; Fuhrhop, J.-H.; Buschmann, J.; Luger, P.; Siggel, U. J. Am. Chem. Soc. 1996, 118, 6671. (8) Hsu, J. W. P.; Lee, M.; Deaver, B. S. Rev. Sci. Instrum. 1995, 66, 3177. (9) Brunner, R.; Bietsch, A.; Hollricher, O.; Marti, O. Rev. Sci. Instrum. 1997, 68, 1769.

10.1021/la990792e CCC: $19.00 © 2000 American Chemical Society Published on Web 12/01/1999

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Langmuir, Vol. 16, No. 3, 2000

east and west edges. The four β-ethyl groups also prevent precipitation of stacks. This porphyrin is known to form stable monolayers on bulk water.5 LB films were either formed from subphases at pH ) 7.0 or at pH ) 2.5. The area/molecule at pH ) 7.0 was 0.87 nm2, which is about half of the area occupied by one porphyrin lying flat on the water surface (∼1.70 nm2). Since β-ethylporphyrins are known to form dimers in solution,7 we propose a bilayer of porphyrins. In the SFM image of a film produced at pH 7.0 and a pressure of 20 mN/m, a very regular thin film was found (not shown). At pH ) 2.5 porphyrin 1 is known to produce fluorescent monolayers of upright-standing porphyrins.5,10 All attempts to achieve a film in uprightstanding porphyrins on mica failed, however. Neither allowing several hours for two-dimensional crystallization on the water surface nor slow transfer speed was successful. At pH ) 2.5 we observed upright-standing stripes (1.8 nm height) with a width of ca. 50 nm, which were surrounded by flat-lying porphyrins dimers (0.8 nm height; Figure 1). The stripes are arranged in a parallel way with uniform distances of about 100 nm. The area/molecule at pH ) 2.5 is only slightly smaller than the one at pH ) 7.0 and amounts to 0.83 nm2. In both cases leaving the film in the compressed state for 2 h diminished the molecular area to 0.76 nm2. Similar slow ordering in rigid films is a common phenomenon.11-13 Since the area per molecule is not very different at both pH 7.0 and 2.5, we propose that the stripes are formed when the film is transferred from water to the mica surface. The rearrangement may be caused by the interaction of the two protonated pyridine rings with the negatively charged mica surface. This assumption is supported by the observation that no such stripes occurred on a gold surface. Two positively charged pyridine rings on the mica surface and two neutral pyridine units pointing to the gas phase would lead to a strong dipole moment. In growing aggregates the unfavorable lateral alignment of these dipoles competes with stabilization associated with decreasing line tension between the domains.14,15 The interplay between these two driving forces determines the size of the aggregate. Aggregation leads to an addition of the single dipole moments to a net dipole moment. The net dipole moments of neighboring aggregates repel each other and account for the distance between the stripes. The high ratio of length/width for these fibers indicates a preferential direction for aggregation, which was already found in the aqueous aggregates of this compound.5,10 The stacking process between the porphyrins occurs faster than lateral aggregation in the aggregates. Some holes were also observed in the porphyrin film next to the upright standing porphyrins. They should relate to the loss of covered surface area upon rearrangement of the porphyrins on the solid mica subphase. If the changes in the porphyrin domain took place on the water surface, these holes would not appear. Tetracosanoic acid (C23H47COOH) forms stable Langmuir monolayers on an aqueous subphase of pH 2.5. Figure 2a displays pressure-area isotherms at T ≈ 20.5 °C, T ≈ 17.5 °C, and T ≈ 15.5 °C, recorded with a barrier moving at constant speed (compression from about 0.6 to 0.2 nm2/ molecule within 5 min). Langmuir-Blodgett (LB) layers (10) Donner, D.; Bo¨ttcher, C.; Messerschmidt, C.; Siggel, U.; Fuhrhop, J.-H. Langmuir 1999, 15, 5029. (11) Gaines, G. L. Insoluble Monolayers at a Liquid Gas Interface; Wiley: New York, 1966. (12) Rabinovitch, W.; Robertson, R. F.; Mason, S. G. Can. J. Chem. 1960, 38, 1881. (13) Pallas, N. R.; Pethica, B. A. Langmuir 1985, 1, 509. (14) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. (15) Mo¨hwald, H. Rep. Prog. Phys. 1993, 56, 653.

Messerschmidt et al.

Figure 1. Langmuir films of porphyrin 1 at pH ) 2.5: (a) SFM height image after transfer to mica at T ) 20.5 °C and 20 N/m. Scan size ) 10µm. Characteristic stripes of upright-standing porphyrins can be seen. (b) Height profile of a smaller scan of the same film (scan size: 900 nm) showing two prominent heights: 0.8 nm for the flat-lying dimers and 1.8 nm for the upright-standing porphyrins. (c) Isotherm of flat-lying dimers of 1 on an aqueous subphase (pH 2.5) (see text).

have been transferred to mica at 20, 15 and 10 mN/m. Tapping-mode SFM images reveal quite smooth layers with characteristic holes (Figure 2c, d). Repeated imaging does not change the appearance of the layer transferred at 20 mN/m (Figure 2c), while layers transferred at 10 mN/m exhibit an increasing number of islands, which are on the order of 50 nm in diameter and whose twodimensional density saturates after a few scans (Figure 2e). The layer transferred at 15 mN/m exhibits two kinds of domains, A and B, which do not differ significantly in height but whose contrast in phase images differs markedly (Figure 2f,g). Isotherms of tetracosanoic acid have been interpreted in terms of five regions I-V (Table 1).16 The transfers at 20, 15, and 10 mN/m were from regions V, IV and between I and II, respectively. The layers transferred from region (16) Schwartz, D. K.; Schlossman, M. L.; Pershan, P. S. J. Chem. Phys. 1992, 96, 2356.

Mixed Porphyrin/Tetracosanoic Acid Monolayers

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Figure 2. Langmuir and Langmuir-Blodgett films of tetracosanoic acid at pH 2.5: (a) Isotherms at different temperatures. (b) Enlargement of a part of the isotherms at T ) 20.5 °C and T ) 15.5 °C. The isotherms are subdivided into different phases according to Schwartz et al.16 Arrows pointing downward correspond to the three transferred films in Figure 2c-g, implying that a transfer to a solid surface corresponds to a decrease in motional freedom and thus to a decrease in temperature as compared to the water surface. (c) SFM height image of a film transferred at 20 mN/m and T ) 20.5 °C. (d) SFM height image of a film transferred at 10 mN/m and T ) 20.5 °C, first scan of a size of 10 µm. (e) SFM height image of the same film after 5 scans at 10 µm at a scan size of 13 µm; singularities have developed in the scanned area. (f) SFM height image of a film transferred at 15 mN/m showing no homogeneous film. (g) SFM phase image recorded simultaneously as in (f). The darker area corresponds to a region with lower phase shift (weaker) and the brighter area to a region with a higher phase shift (harder area).

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Figure 3. Schematic illustration of the proposed mechanism of the development of the vortices. Pairs of vortices are energetically favorable17 because the overall tilt direction does not change. Table 1. Structural Assignment to Regions in Isotherms of Tetracosanoic Acid (According to Schwartz et al.16)

region

homogeneous (hom.) or inhomogeneous (inhom.)

I

inhom.

II III

hom. inhom.

IV

inhom.

V

hom.

ordered tilted (35°) chains (phase I) + disordered phase ordered tilted (35°...27°) chains ordered upright chains (phase U) + disordered phase of lower density ordered tilted (27°) chains (phase I) + disordered phase of higher density ordered upright chains (phase U)

V exhibit very few holes and do not alter upon tapping. We therefore assign also the transferred layer to ordered upright chains (phase U). Layers transferred from the border between regions I and II should be also homogeneous, and indeed the layer imaged “as prepared” appears homogeneous. However, upon repeated imaging islands appear, which are attributed to a second phase. Their two-dimensional density saturates upon prolonged scanning, indicating the establishment of an equilibrium. This implies that the homogeneous film after transfer is in a metastable state, which relaxes upon tapping into a coexisting two-phase region. If one assumes that the substrate effectively lowers the temperature of the layer, the state of the transferred layer corresponds to region III or IV. Layers transferred from region IV exhibit two distinct layer phases with indistinguishable height but different contrast in phase imaging. This indicates an inhomogeneous layer as expected for region IV. The formation of the islands can be interpreted in terms of the theory of Kosterlitz and Thouless17 for phase transitions in two-dimensional systems via singularities.


The singularities are typically formed at high temperatures, where the entropy term dominates over the internal energy term. In our case the singularities are induced by the tapping tip, which obviously provides the required activation energy for the relaxation process either by a local rise of the temperature or by the force exerted by the tip. During light tapping the tip may effectively pull the fatty acid molecules into an upright position, which finally leads to stable vortices. The surrounding molecules have to change their tilt, and on opposite sides of this vortex the molecules will be tilted in opposite directions, which means that the vortices can only appear as pairs (Figure 3). Heating to 45 °C did not produce the new phase, presumably because the necessary activation energy was not provided. We refrained from a quantitative determination of the activation energy provided during the SFM tapping experiment, since the energy dissipation in the Langmuir-Blodgett layer is difficult to quantify. Interestingly, the vortices are not formed close to the edges, indicating a range across which the singularities interact. A 1:1 mixture of porphyrin 1 and the C24-acid in chloroform was then spread onto a water subphase of pH 2.5. The experimental molecular area of 1.06 nm2 corresponds to the sum of both components, indicating formation of independent domains18,19 (Figure 4a). For domain formation a plot of the mole percent of one component in different mixing ratios vs the area/molecule based on average molecular weight gives a straight line, as the molecules assume their usual area/molecule values, whereas for miscible components no linearity can be found.20 Figure 4b shows a straight line for our case. The SFM images identify different domains (Figure 5a), which can be distinguished by their appearance and height. The porphyrin domains show the characteristic stripes, and the height values found correspond well to the monolayer of pure porphyrin (Figure 1). The height of the fatty acid domains is (2.4 ( 0.3) nm. Force modulation measurements support our interpretation of the different domains. Indenting the sample with a constant force leads to a higher deflection of the cantilever for stiffer surfaces. In the Nanoscope software, however, lower deflection (weaker surfaces) gives bright areas. The striped porphyrin domains were found to be clearly harder than the bright fatty acid domains (Figure 5b). Figure 5d shows a mixed film transferred from a subphase at pH ) 7 (pure water). In contrast to the film at pH ) 2.5 (Figure 5c), no upright-standing porphyrins (1.6 nm) can be observed. At moderate tapping conditions (with a ratio of

Figure 4. Mixed monolayers of porphyrin 1 and tetracosanoic acid (molar ratio 50:50) at pH ) 2.5, T ) 20.5 °C and 20 mN/m: (a) Pressure/area isotherm. The extrapolated area/molecule is 1.06 nm2. (b) Calculated (- -) and experimental (s) plot of additive molecular area vs mole percent of flat-lying porphyrin 1 dimers and tetracosanoic acid at a tilt angle of 35°.


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Figure 5. (a) Typical SFM height image of the mixed film of tetracosanoic acid and porphyrin 1 (molar ratio 50:50). (b) SFM force modulation image of the mixed film. The bright areas consist of soft tetracosanoic acid and the dark areas and stripes of porphyrin 1. (c) Right: height profile along the line marked in the SFM height image on the left. The heights can be assigned as follows: The gray stripes are upright-standing porphyrins 1 (1.6 nm) in a plane of flat-lying porphyrin 1 dimers (0.7 nm). The white islands consist of tetracosanoic acid (2.4 nm). (d) height profile of a mixed monolayer transferred at pH ) 7 (T ) 20.5 °C, 20 mN/m). Only porphyrin dimers (0.8 nm) and tetracosanoic acid (2.2 nm) occur, but no upright-standing porphyrins (1.6 nm).

set-point amplitude to free amplitude rSp ) ASp/A0 ) 0.96), higher phase shifts were found for the porphyrin domains (Figure 6a). Recently, a model to interpret phase shifts has been proposed:6 Under conditions where the forces between tip and sample are repulsive but still moderate, phase shifts are proportional to the Young modulus and thus the stiffness of the sample. In accordance with this prediction, we found higher phase shifts for the stiffer porphyrin domain than for the softer fatty acid domain. At higher tapping forces (rSp ) 0.77), the phase image is inverted (Figure 6d). In this region only the contact area is decisive for the phase shift.6 The contact area should (17) Kosterlitz, J. M.; Thouless, D. J. J. Phys. C 1973, 6, 1181. (18) Bu¨cher, H.; v. Elsner, O.; Mo¨bius, D.; Tillmann, P.; Wiegand, J. Z. Phys. Chem., Neue Folge 1969, 65, 152. (19) Mo¨bius, D. Ber. Bunsen-Ges. Phys. Chem. 1978, 82, 848. (20) Cheung, J. H.; Rubner, M. F. Thin Solid Films 1994, 244, 990.

be higher in the fatty acid domains as the tip indents further into the sample. Applying a moderate force to the mixed monolayers at pH ) 2.5 induces the formation of the same kind of islands in the tetracosanoic acid domains that we observed in the pure tetracosanoic acid films. The development of these vortices can clearly be seen in Figure 6a, b. Figure 6b was taken five scans after Figure 6a. In Figure 6a smooth tetracosanoic acid domains appear, whereas in Figure 6b a large number of islands resembling those of the pure tetracosanoic acid film have been induced. Figure 6c shows an enlargement of the tetracosanoic acid domain after the second scan, illustrating that the vortices are formed in pairs according to the mechanism shown in Figure 3: For every vortex a partner at a distance of approximately 100 nm can be found. In Figure 6b there are no islands at the edge of a tetracosanoic acid domain. The first vortices

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Figure 6. 2 µm scan of a mixed film of porphyrin 1 and tetracosanoic acid (molar ratio 50:50): (a) Left: SFM phase image of the first scan at low tapping force. The soft fatty acid domain appears darker than the hard porphyrin domain. (b) SFM phase image five scans later. The scan conditions remained unchanged. (c) SFM height image of the second scan. Only a few vortices have developed. The 100 nm long dashes connect pairs of neighboring vortices. (d) SFM images after increasing the tapping force. Left: height images showing no more vortices, right: inverted phase image. (e) SFM images 10 scans later without changing the scan conditions. Part of the fatty acid domain has been removed, while the porphyrin domain remained unchanged. (f) 5 µm scan at lower tapping force. The first vortices appear again at the bottom of the image.

appear 100 nm inside the domain, corresponding to the distance between the pairs. The islands were produced in a film, which was transferred at a pressure of 20 mN/m.


Figure 7. Scanning near-field optical microscopy of the mixed film (molar ratio 50:50): (a) Fluorescence image. The white arrow indicates a nonfluorescent domain, and the four black arrows point to fluorescent stripes corresponding to the uprightstanding porphyrins. (b) height image of the same area. The white areas are assigned to tetracosanoic acid. Inset: height and fluorescence intensity profile along the white lines in (a) and (b). The gray arrow points to a tetracosanoic acid domain, and the four porphyrin stripes appear in the fluorescence intensity profile.

A pure film of tetracosanoic acid at this transfer pressure showed no island formation (Figure 2c). This can only be caused by special interactions with the neighboring porphyrin domains. As outlined above, the orientation of the porphyrin molecules within the domains changes upon transfer from water to mica. Upright-standing stripes needing less space than the flat-lying dimers are formed. The fatty acid domains therefore get a larger space


Langmuir, Vol. 16, No. 3, 2000 1305

corresponding to a lower net pressure for the transfer of the tetracosanoic acid domain. Therefore, one has to compare the fatty acid domains in the mixed film with the pure domains at a lower pressure, which were shown to produce islands. Scanning at higher force (where the phase image is inverted), however, leads to the formation of holes in the fatty acid domain (Figure 6d, e). Ultimately the whole fatty acid domain can be removed by such a scraping procedure while the porphyrin domains stay untouched. Figure 6f finally was taken at low force again (rSp ) 0.96), showing the phase shifts for moderate tapping and induction of microdomains again at the bottom of the image. These images prove furthermore that there are no multilayers involved in the studied system as the height of the domains can easily be measured at the sites of the induced holes. The SFM measurements detected two domains containing porphyrins in different orientations with respect to the subphase, but did not characterize them. Scanning near-field optical microscopy (SNOM),21 however, showed that both the upright-standing and the flat-lying porphyrin domains fluoresced strongly. The intensity ratio was found to be 2.3, which roughly corresponds to the density ratio. Within the limits of error of the estimates of the porphyrin density, the fluorescence is thus the same for both domains. The expected self-quenching effects in the leaflets of upright-standing porphyrins22 could not be detected. SNOM measurements with a resolution of about 100 nm in the stripe region (see inset in Figure 7b) are, however, useful for the detection of quenching effects of additives10 or for the detection of differences between the domains in electrochemical reactions with metallic subphases.

equilibrium phases is known from the work of Schwartz.16 We also observed the phenomenon first in the mixed porphyrin-tetracosanoic acid system. Here, the shrinking of the rigid porphyrin domains upon transfer from water to mica produces empty space at the edges of the fatty acid domains. The emerging gap cannot be closed by movement of the domain platelets because the gaps appear all around the domains. Both domains therefore develop new equilibrium states. The flat-lying porphyrin dimers equilibrate with stripes of upright-standing porphyrins, and the whole domain remains rigid. The C24-chains of the fatty acid lose their contact with the porphyrin domains and fluidize at the edges. Singularities develop in pairs with an average distance of 100 nm. The same distance of the islands of upright-standing molecules is also kept to the fluid edges. This observation is, to the best of our knowledge, the first example of the change of the equilibrium state of a fluid membrane by neighboring rigid domains. In contrast, one knows many examples of effects of membrane viscosity on rigid solutes, in particular of membrane lipid-protein interactions. Helices may convert to sheets, and membrane-crossing helices may be forced on the membrane forces by a change from a fluid to a liquid crystalline phase.23 The surface-bound monolayer is, of course, not a good model for fluid cell membranes. Membranes connected with an inner cytoskeleton, on the other hand, could very well react in a similar manner. SFM on solid surfaces may then be a useful tool in addition to fluorescence measurements in solution.

4. Conclusion

Acknowledgment. We thank Prof. W. Helfrich for helpful discussions about the development of the singularities, Dr. W. Stocker for discussions about SFM-related issues, and O. Hollricher for help concerning the SNOM experiments. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 448 “Mesoscopically structured systems” and SFB 312 “Vectorial Membrane Processes”).

The discovery of singularities in fatty acid monolayers after SFM tapping came as a surprise because this technique has been applied so often to similar preparations. At room temperature and on a mica surface, the phenomenon may be unique to C24-systems, which have not been studied as often, although their richness in (21) Betzig, J.; Trautman, J. K.; Harris, T. D.; Weiner, J. S.; Kostelak, R. L. Science 1991, 251, 1468. (22) Choudhury, B.; Weedon, A. C.; Bolton, J. R. Langmuir 1998, 14, 6192.

LA990792E (23) Parola, A. H. Biomembranes; VCH: Weinheim, 1993.

Formation of Stable Singularities in Mixed Monolayers of Porphyrins

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