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Formation of two-dimensional herringbone aggregates in Langmuir films of amphiphilic merocyanine dye derivatives
Colloids and Surfaces A: Physicochem. Eng. Aspects 284–285 (2006) 112–118
Formation of two-dimensional herringbone aggregates in Langmuir films of amphiphilic merocyanine dye derivatives Keiichi Ikegami ∗ Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Received 13 July 2005; received in revised form 10 October 2005; accepted 28 October 2005 Available online 6 December 2005
Abstract Doublet absorption bands are seen in visible spectra of Langmuir films of amphiphilic merocyanine dyes, DS, DO and DSe, prepared on pure water. In the DS and DO cases, the both maxima of the doublet bands exhibit red shifts upon the compression of the films. These phenomena cannot be explained by assuming that isolated monomers and/or dimers are the majority in the films. Contrastively, they can be well explained by assuming that large herringbone aggregates are the majority. In the latter model, the surface pressure-induced spectral change is interpreted as a structural phase transition. A numerical simulation for herringbone aggregates of DS reproduces the observed excitation wavenumbers. Infrared absorption spectra have been observed for Langmuir–Blodgett films of the dyes transferred from pure-water surfaces. They indicate the formation of mutual hydrogen bonds between the carboxylic groups in adjacent dye molecules and then suggest that the interaction between those groups is a part of the driving force of the aggregation. © 2005 Elsevier B.V. All rights reserved. Keywords: Dye aggregate; J-aggregate; Langmuir film; Langmuir–Blodgett film; Merocyanine dye; Phase transition; Spectroscopy
1. Introduction Dye aggregates are self-assembled nanostructures whose functions are useful for optical sensitizing, chemical sensing and information storage. According to Kuhn and Kuhn [1], “studying the interplay between dye molecules in different geometries is of interest in approaching the goal of constructing supramolecular machines.” In the light of this general question, J-aggregates formed in Langmuir (L) and Langmuir–Blodgett (LB) films of amphiphilic merocyanine dyes have been studied extensively [2–19]. For example, the importance of structure–function correlation has been recognized [18]. That is, it has been indicated that the alkyl chain and carboxylic group, both of which were introduced to make the dyes amphiphilic, and the shape of the chromophore itself play important roles in the J-aggregation. However, J-aggregates are not always formed in L-films of merocyanine dyes [6–10]. In fact, when 3-carboxymethyl-5-[2(3-octadecyl-2(3H)-benzothiazolylidene)ethylidene]-2-thioxo-
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4-thiazolidinone (DS in Fig. 1) and its derivatives DSe and DO (Fig. 1) are spread on pure water as pure L-films, J-aggregates are hardly formed. Instead, doublet absorption bands appear immediately after spreading the materials (Fig. 1). The doublet band of DS was already reported for L-films upon pure water [10] and cast films on SnO2 electrodes [20], but has not carefully been examined because it was assigned to a sum of monomeric and dimeric absorption bands [20] and then has been considered not very interesting. However, apart from cast films, it is quite questionable whether isolated monomers or dimers can be the major composition of pure L-films. (Since the absorption due to a gas phase is negligible, the doublet bands are assignable to a condensed phase.) Whether the monomeric band of a dye is observable in its L and LB films is a crucial problem in analyzing the optical properties of its J-aggregate formed in the films because, to evaluate the band shift due to aggregation, it is desirable to compare the absorption bands observed for J-aggregated and monomeric dyes in similar environments. Furthermore, if the doublet bands originate from some aggregates, the interplay between the dye molecules in them is directly related with the Kuhn and Kuhn’s general question. Another interesting point is the large total
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Fig. 1. Chemical structures of the merocyanine dyes used in this study (upper pane) and the visible absorption spectra of their L-films on pure water observed immediately after spreading the materials (lower pane). The spectra are normalized by their largest maxima, respectively.
widths of the doublet bands. When dyes are used as sensitizers for imaging, narrow absorption bands are advantageous for enhancing the color purity. On the contrary, broad absorption bands are advantageous for solar power conversion. Elucidation of the mechanism leading to the doublet band in L-films of those merocyanine dyes may provide a guideline for broadening the total bandwidth of sensitizers. In this context, L-films of DS, DO and DSe have been prepared upon pure water and their optical properties have been characterized. In addition to the appearance of the doublet bands, their surface pressure-induced changes have been detected. Infrared (IR) spectroscopic study of transferred films has been performed. These experimental results will be described in Section 3. In Section 4, the optical properties of DS films and their changes upon compression will be semi-quantitatively interpreted based on a model structure, i.e. a herringbone aggregate.
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Co. Ltd. attached to the trough. The optical configuration was the same as that used in the previous studies [16,18]. Baseline tilts of the spectra were corrected so that the absorbance data at 25,000 and 15,390 cm−1 are null. A rather high compression speed (−5.6 × 10−3 nm2 /s) was adopted to suppress baseline drift and film collapse. Absorption spectra of DS films prepared upon Mg2+ - and Ca2+ -containing subphases are also recorded for comparison. In these cases, aqueous solutions of MgCl2 and CaCl2 (4 × 10−4 mol/l) were used after their pH values being adjusted to 6–7 by adding a small amount of KHCO3 . In order to perform IR measurements, L-films on pure water under 25 mN/m as well as those on the Mg2+ - and Ca2+ containing subphases were transferred with the conventional vertical immersing-withdrawing method [21] onto plates of CaF2 precoated by three monolayers of deuteride cadmium arachidate. Visible and IR absorption spectra of the LB films were recorded by Perkin-Elmer Lambda-900 and System-2000 spectrometers, respectively. A precoated CaF2 plate was used as a reference. Absorption spectra of chloroform and methanol solutions of the dyes were also recorded by those spectrometers, though the concentrations of dilute solutions (<10−5 mol/l) were not precisely controlled. 3. Results Pure L-films of DS on a pure-water subphase show a doublet absorption band immediately after the spreading (Fig. 1). The lineshape of this doublet band is unchanged until the surface pressure becomes detectable (Fig. 2) and quite different from those observed for L-films prepared on a Mg2+ -containing
2. Experimental Pure water with a resistivity greater than 1.8 × 107 cm was prepared by using a Millipore Milli-Q system. The merocyanine dyes, i.e. DS, DO and DSe, were purchased from the Japanese Research Institute for Photosensitizing Dyes, Co. and used without further purification. Solutions of the dyes with the concentration of 5 × 10−4 mol/l were made using spectroscopic grade chloroform as the solvent. These solutions were spread on pure water contained in a Lauda Filmwaage trough under air at 20 ◦ C to form L-films. Since the initial areas of the L-films were large enough (ca. 1.2 nm2 per molecule), compression of the films was started 10 s after the end of the spreading; and it was stopped when the surface pressure reached to 25 or 40 mN/m. On compression of L-films, their visible absorption spectra were recorded by a WRM-10TP polychrometer of Jasco
Fig. 2. Visible absorption spectra of a dilute chloroform solution (ca. 4 × 10−6 mol/l) of DS and those of L-films of DS prepared on pure-water and Mg2+ -containing subphases observed when the molecular areas are ca. 0.8 nm2 . The surface pressure–molecular area isotherms observed for these DS films are indicated in Inset (the isotherm measurements are stopped at 40 or 25 mN/m so as to keep these surface pressures).
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subphase (Fig. 2). This doublet band has one maximum at each of the high and low energy sides of the monomeric band of DS observed in a dilute chloroform solution (Fig. 2, maximum at νchl = 18940 cm−1 ) and in a dilute methanol solution (νMeOH = 19120 cm−1 ). The absorption intensities of the two maxima are nearly identical with each other. Doublet bands are also seen in DO and DSe films upon pure water. The absorption spectrum observed for the DO films immediately after the spreading (Fig. 1) is quite similar to that observed in the DS case. The only difference is the positions of the absorption bands and it well corresponds to the difference between the monomeric excitation energies observed for chloroform solutions of the dyes, 1220 cm−1 . As for the DSe films immediately after the spreading, the doublet band is located in a region of smaller wavenumbers compared with the DS case (Fig. 1). The difference in the absorption energy between the DS and DSe solutions is small and cannot cause that between the DS and DSe films. The latter may be caused by different dimeric or aggregate structures. The ordinate of Fig. 2 is molar absorption coefficient, ε, estimated from the obtained transmittance data assuming the three-dimensional random orientation of the transition dipole moments in each samples. With this ordinate, scaling of the L-films’ spectra by a factor of 2/3 makes their areas ( εdν) comparable with that of the solution spectrum. This fact implies that the transition moments in L-films are nearly parallel to the water surface at least under null surface pressure, regardless whether the subphase is pure water or aqueous solution of Mg2+ . DS molecules spread upon a Mg2+ -containing subphase show a limited area of 0.64 nm2 and form a stable L-film (Fig. 2 inset). The absorption spectrum of the L-film is not affected by the surface pressure up to 25 mN/m. DS molecules spread upon a pure-water subphase show completely different behavior. Their limited area is 0.71 nm2 , larger than that in the Mg2+ -containing case, and the formed L-film shows anomalously large compressibility (Fig. 2 inset). Moreover, compression of the L-film is associated with the change in the absorption spectrum (Fig. 3). Both of the minima of the doublet band in the second derivative (d2 ε/dν2 ) spectrum discontinuously shift to the low energy side as the surface pressure is increased. Namely, the excitation wavenumbers in the low-pressure phase, νLPx = 19610 cm−1 and νLPy = 17990 cm−1 , respectively, shift to those in the highpressure phase, νHPx = 19050 cm−1 and νHPy = 17670 cm−1 . The apparent continuous shift of the ε spectrum reflects a continuous increase in the portion of the high-pressure phase that may partly cause the anomalous compressibility. The intensities of the two maxima in ε are nearly identical with each other through out the surface pressure-induced spectral change. In order to know whether the observed surface pressureinduced spectral change is reversible, a compression-expansion experiment has been performed (Fig. 4). Although the area of the L-film observed during the expansion process is considerably smaller than that observed during the compression process, the absorption spectrum observed for the expanded film (c) well corresponds to that observed at the initial stage (a), except for the absolute ε value. This result suggests that the surface pressureinduced change is intrinsically reversible. However, at the same
Fig. 3. Variation of the absorption spectrum of the DS film on pure water (the same film as was measured for Fig. 2) due to compression (upper pane), together with the second derivative spectra (lower pane).
time, it suggests that L-films are irreversibly changed during the compression: some DS molecules may be squeezed out from the films (collapse). This irreversible change may partly cause the anomalous compressibility. The absolute ε values of the spectra b and c are, accordingly, underestimated.
Fig. 4. Surface pressure–molecular area relationship observed during a compression–expansion cycle of a DS film on pure water (upper pane) and the absorption spectra measured for the film at the initial stage, after compression and after expansion, respectively (lower pane).
K. Ikegami / Colloids and Surfaces A: Physicochem. Eng. Aspects 284–285 (2006) 112–118
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hydrogen bonds between the carboxylic groups in adjacent dye molecules. 4. Discussion One of the simplest models for an L-film of a dye with two absorption maxima is a mixture of isolated monomers and (parallel) dimers [20]. However, application of this model to the present cases encounters difficulties: (i) How can the monomers and dimers be isolated in a pure and condensed L-film? (ii) How can the surface pressure induce a shift of the monomeric band? The other simplest model is isolated dimers in which the dye chromophores are not parallel to each other. As is well known [1], the excitation energy of the dyes in this case splits as: λ± = E ± J1,2
Fig. 5. IR absorption spectra of LB films of DO, DS and DSe prepared from pure-water surfaces. Their visible spectra are indicated in inset.
Surface pressure-induced spectral changes are detected for the DO and DSe films, too. In the DO case, it takes place under low surface pressure (<5 mN/m). Both of the maxima of the double band shift to the low energy side and the maximum at the high energy side becomes dominant (data not shown). In the DSe case, it takes place under higher surface pressure (>15 mN/m). The intensity of the maximum at the low energy side is decreased to a certain extent, but no shifts of the maxima are detectable (data not shown). LB films of the dyes have been prepared by transferring their L-films on pure water under 25 mN/m. The doublet lineshapes of the visible absorption bands observed for the LB films (Fig. 5 inset) imply the similarity between the molecular arrangements in L and LB films of the dyes. Namely, the visible absorption spectrum of DS LB films is compatible with that observed for DS L-films under 25 mN/m. The spectra of DO and DSe LB films exhibit intermediate lineshapes between their counterparts observed for L-films under 0 and 25 mN/m, respectively. Hence, the IR absorption spectra of the LB films are useful in revealing the nature of the visible doublet band. The IR spectra of the DS, DO and DSe films have common features (Fig. 5). Firstly, bands due to hydrogen-bonded carboxylic groups are observed around 1730 cm−1 . Secondly, bands due to free keto groups are observed around 1670 cm−1 . Thirdly, bands due to stretching modes of the central conjugated system of the chromophore (observed around 1520, 1540 and 1510 cm−1 for DS, DO and DSe, respectively) are split or broadened in comparison with those in the solution spectra. All of these features are contrastive to the characteristics of the IR absorption spectra observed for J-aggregates of the dyes [15,18,19] and therefore they can be considered characteristic for the phases exhibiting the doublet visible bands. Particularly, the spectral features observed for the C O stretching bands imply the formation of mutual
(1)
where E is the monomeric excitation energy and J1,2 is the interaction between transition dipole moments in the dimer. One of the problems of this model is the same as (i) in the above case and the other is that (ii) the average of the observable excitation energies, λ± , of a dimer is always E and hardly modified by surface pressure. On the contrary, a large herringbone aggregate, which is the next simplest model, creates no qualitative difficulties. In fact, the excitation energy of the dyes in this case split as: λ± = E + J±
(2)
where J± are determined by the arrangement of transition dipole moments and J+ = −J− (see Appendix A). Consequently, an increase in the surface pressure can induce shifts of λ± toward the same direction through a structural deformation. Moreover, this model is consistent with those observed for L-films of 6MeDS, which is an analogue of DS, prepared on a pure-water subphase [18]. In those films, decay of a J-band and growth of a doublet band that is similar to those dealt with in this study proceed simultaneously and an isosbestic point was clearly observed, showing that the both maxima of the doublet band originates from the same phase. It is difficult to eliminate other complicated models rigorously, but the following numerical simulations will support the herringbone aggregate model. Furthermore, it will provide a sketch of the aggregate structure. In this semi-quantitative discussion, DS is dealt with because the doublet band in its L-films on pure water shows surface pressure-induced shift and J-bands are observed for its L-films on aqueous solutions of divalent metals. To simulate the excitation wavenumbers numerically, the structure of the herringbone aggregate should be expressed by a few parameters. Since the elemental unit of the herringbone aggregate contains two molecules that are not parallel to each other, two adjustable parameters, β and lc , are introduced to describe such a “dimeric” unit shown in Fig. 6(a), where β is a half of the angle between the transition dipole moments and lc is the distance between the apex and the center of the transition dipole moments. (This “dimeric” unit is not necessarily a true dimer.) As for the arrangement of the dimeric units, a simple
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Fig. 6. Structures of the dimeric unit (a) and its arrangement (b) assumed for the major composition of the L-films of DS formed on a pure-water subphase, and that of the J-aggregates assumed for DS films on Mg2+ - and Ca2+ -containing subphases (c). The closed circles in (a) stand for the point charges introduced by the extended dipole approximation. The dotted lines in (b) represent planes of symmetry in a large aggregate.
and symmetric one is assumed (Fig. 6(b)) without introducing any additional adjustable parameters. The thickness of the chromophore (= side by side distance between chromophores) is estimated at t = 0.35 nm from a density-functional calculation [17]. The E/hc value is set to vchl = 18940 cm−1 because the solvent effect for the monomeric bands causing the difference between vMeOH and vchl is small. (It seems too small to cause the difference between vLPy and vchl or that between vLPx and vchl . This is another difficulty in attributing vLPy or vLPx to the isolated monomer.) The interaction between the nth and mth transition dipole moments in the aggregate, Jn,m , is estimated based on the extended dipole approximation [1] because its rigorous evaluation is very difficult. That is, Jn,m is approximated at Coulomb interaction between two pairs of positive and negative point charges that are fixed at the molecules n and m, respectively. The vector connecting the charges ±qtr at the molecule n, denoted by ltrn , is parallel to µtrn and its length ltrn corresponds to the effective length of the chromophore (Fig. 6(a)). The strength of the charge qtr satisfies the relation of qtr ltrn = µtrn . This approximation leads to better results, especially when the dye molecules are closely packed, than the point dipole approximation, in which Jn,m is approximated at the Coulomb interaction between two point dipole moments. This comes from the fact that the former takes into account tr the charge distribution to a l is estimated at 0.52 nm [18]. certain extent. In the DS case, n tr µ is treated as an adjustable parameter, while the dielectric n constant is fixed at 2.3 [22]. J-aggregates of DS in L-films prepared upon Mg2+ containing and Ca2+ -containing subphases, respectively, show their absorption maxima at vMg = 16260 and vCa = 16640 cm−1 . These observations are also incorporated into the discussion. (The L-films of DS prepared on a Cd2+ -containing subphase also show a red-shifted absorption band, but that band is
much broader than those observed in the L-films on Mg2+ and Ca2+ -containing subphases and the Cd2+ -containing “Jaggregate” of DS may not be appropriate to incorporate into the present discussion.) As for the molecular arrangements of these J-aggregates, the brickstone model [6–9,11] illustrated in Fig. 6(c) is adopted and two sets of the structural parameters are suggested from geometrical considerations based on the densityfunctional optimization of the molecular structure [17], like in the case of the previous work [18]: Type (I) L = 1.62 nm and s = 0.81 nm, and Type (II) L = 1.56 nm and s = 0.55 nm. Through numerical simulations, it was found that β = 41.92◦ ◦ and and lc = 0.855 nm give rise to νLPx and νLPy and β = 39.15 tr lc = 0.898 nm give rise to νHPx and νHPy , when µn is set to 10.6 D. This µtrn value leads to 16,600 and 16,300 cm−1 for the Type (I) and (II) brickstone arrangements, respectively, and these values are comparable to vCa and vMg , respectively. The estimated β values are not far from 45◦ and consistent with the fact that the two maxima in the doublet band have nearly the same intensity in both the low and high-pressure cases (see Appendix A). The lc values slightly larger than L/2 and the negative correlation between this and the β values are compatible with the geometry of the dimeric unit. In addition, the estimated µtrn value is comparable to that evaluated from the solution spectrum (ca. 9.6 D). Thus, the herringbone-type molecular arrangement illustrated in Fig. 6(b) semi-quantitatively explains the visible absorption of DS films prepared on pure water under both low and high surface pressures. It should be emphasized that changes in the structural parameters of the aggregate, β and lc , can cause simultaneous decreases in the both excitation energies, λ± . Therefore, the red shifts of the both maxima of the double band observed upon compressing the L-films can be interpreted as a surface pressure-induced structural phase transition. The IR measurements of the dye’s LB films prepared from pure-water surfaces have implied the formation of mutual hydrogen bonds between the carboxylic groups of adjacent dye molecules. Although the environments around the carboxylic groups in the LB and L-films are largely different, the basic structures of the herringbone aggregates seem to be kept in LB films. Therefore, it is plausible to assign the interaction between the carboxylic groups to a part of the driving force of the aggregation. In conclusion, the doublet absorption bands observed for the L-films of DS, DO and DSe prepared upon a pure-water subphase suggest that herringbone aggregates of the dyes are formed immediately after the spreading. The observed surface pressure-induced spectral change in the DS cases, i.e. the red shifts of the both maxima, is well explained by deformation of the aggregates. This finding implies that when the J-aggregates of the dyes are discussed, the J-shifts should not be measured from one of the maxima in such doublet bands. Rather, they should be measured from the maxima in the solution spectrum or, if possible, both of the doublet bands and J-bands should be explained systematically in the same framework. The obtained IR spectroscopic data imply that the carboxylic group connected to the dye chromophores plays an important role in the aggregation. Since the carboxylic group may not directly modify the chromophores’ optical
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functions, it may be useful as a “connector” in the bottom-up nanotechnology. Appendix A The theory of the optical properties of dye aggregates has been well established, but the general formula is very complicated. Therefore, a brief summary specialized to the case of two-dimensional herringbone aggregates is given in order to clarify the basis of the present discussion, especially that of Eq. (2), which is not a simple extension of Eq. (1). The Hamiltonian of the singly excited states of an aggregate containing N → ∞ molecules can be represented with the basis set of wavefunctions |ψn > = |. . . gn−1 en gn+1 . . . >as: ⎞ ⎛ .. . ⎟ ⎜ ⎟ ⎜ E Jn−1,n Jn−1,n+1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Jn,n−1 E Jn,n+1 (A.1) H =⎜ ⎟ ⎟ ⎜ Jn+1,n−1 Jn+1,n E ⎟ ⎜ ⎠ ⎝ .. . where |en > and |gn > are the nth dye’s ground and excited state wavefunctions, respectively, the diagonal term E is the excitation energy of an isolated monomeric dye, and the offdiagonals Jn,m = <ψn |V(n,m)|ψm > are caused by the Coulomb interaction between the nth and mth molecules, V(n,m). With this Hamiltonian, wavefunctions |±> defined as: |± = N −1/2 (±1)n |ψn , (A.2) n
can be used as trial eigenfunctions and H|±> are calculated as: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ H |± = N −1/2 (±1)n ⎣E + (±1)m Jn,m ⎦ |ψn . m=n
n
Hence, if J± (n) =
(A.3) (±1)m Jn,m are independent of n, |±> are
m=n
really eigenfunctions of H. Two-dimensional herringbone arrangements like illustrated in Fig. 6(b) and the absence of edge effects due to the large N introduce planes of symmetry in the aggregate structure (dotted lines in Fig. 6(b)). Here, the two-dimensionality confines the transition dipole moment of each dye in the aggregate plane. The plane of symmetry of each dimeric unit (zx-plane in Fig. 6(a)) coincides with one of those planes of symmetry of the aggregate. When dyes n and n + 1 belong to the same dimer, therefore, Jn,m always has its counterpart of Jn+1,m = Jn,m . In addition, all the dimers in the aggregate are equivalent. Consequently, J± (n) is independent of n, and then: λ± = E + J± (n) E + J±
(A.4)
are the eigenvalues of H, i.e. the excitation energies corresponding to the excited states of |±> and it should be emphasized that J+ = − J− . This equation is the same as Eq. (2) in the text.
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The transition dipole moments µtr± that correspond to the excitations of the aggregate from the ground state |G> to |±>, respectively, can be calculated by putting the electrons’ position operators rn,k = (xn,k , yn,k ) between these wavefunctions, where k indexes the electrons in the nth molecule. Setting the phases of µtrn gn | rn,k |en so that the x components of all µtrn are real and positive, the x and y components of µtr± of large herringbone aggregates are readily obtained as: 2 2 tr 2 µ = e2 G| xn,k |± = N −1 (±1)n µtrnx ±x n k n 2 N µtrn cos2 β for |+ (A.5) = |− 0 and
2 2 tr 2 2 −1 n tr yn,k |± = N (±1) µny µ±y = e G| n k n 0 for |+ = . (A.6) 2 |− N µtr sin2 β n
2 2 2 The sum of µtr+ and µtr− coincides with N µtrn (note that tr 2 µ is independent of n), showing that singly excited states n of the aggregate other than |±> have null transition dipole moments. More importantly, the intensity ratio between the two transition modes is tan2 β, being exactly the same as in the dimer case. At last, the special case of β = 0◦ should be mentioned. In this case, the condition that J± (n) is independent of n is fulfilled by the brickstone arrangement. Hence, (A.4)–(A.6) are applicable to two-dimensional large J-aggregates, too, and in fact they coincide with the discussion in Ref. [1]. (Note that Jn,m in this paper corresponds to 2Jij in Ref. [1]. This difference comes from the different definitions of the transition dipole moment. Namely, it is defined based on one-electron wavefunctions in Ref. [1], while it is defined based on molecular wavefunctions in this paper. The latter is more directly connected to the areas of experimentally observable absorption bands.) References [1] H. Kuhn, C. Kuhn, in: T. Kobayashi (Ed.), J-Aggregates, Word Scientific, Singapore, 1996, Chapter 1. [2] M. Sugi, S. Iizima, Thin Solid Films 68 (1980) 199. [3] K. Iriyama, M. Yoshiura, F. Mizutani, Thin Solid Films 68 (1980) 47. [4] S. Kuroda, M. Sugi, S. Iizima, Thin solid Films 99 (1983) 21. [5] S. Kuroda, K. Ikegami, Y. Tabe, K. Saito, M. Saito, M. Sugi, Phys. Rev. B 43 (1991) 2531. [6] H. Nakahara, D. M¨obius, J. Colloid Surf. Sci. 114 (1986) 363. [7] H. Nakahara, K. Fukuda, D. M¨obius, H. Kuhn, J. Phys. Chem. 90 (1986) 614. [8] T. Kawaguchi, K. Iwata, Thin Solid Films 165 (1988) 323. [9] T. Kawaguchi, K. Iwata, Thin Solid Films 191 (1990) 173. [10] M. Yoneyama, T. Nagao, T. Murayama, Chem. Lett. (1989) 397. [11] L. Wolthaus, A. Schaper, D. M¨obius, Chem. Phys. Lett. 225 (1994) 322. [12] N. Kato, K. Saito, T. Serata, H. Aida, Y. Uesu, J. Chem. Phys. 115 (2001) 1473.
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[13] N. Kato, M. Yamamoto, K. Itoh, Y. Uesu, J. Phys. Chem. B 107 (2003) 11917. [14] Y. Hamanaka, H. Kurosawa, A. Nakamura, Y. Uchiyama, K. Marumoto, S. Kuroda, Chem. Phys. Lett. 363 (2002) 233. [15] K. Ikegami, C. Mingotaud, M. Lan, J. Phys. Chem. B 103 (1999) 11261. [16] K. Ikegami, Jpn. J. Appl. Phys. 41 (2002) 5444. [17] K. Ikegami, S. Kuroda, Chem. Phys. 295 (2003) 205.
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Formation of two-dimensional herringbone aggregates in Langmuir films of amphiphilic merocyanine dye derivatives Keiichi Ikegami ∗ Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Received 13 July 2005; received in revised form 10 October 2005; accepted 28 October 2005 Available online 6 December 2005
Abstract Doublet absorption bands are seen in visible spectra of Langmuir films of amphiphilic merocyanine dyes, DS, DO and DSe, prepared on pure water. In the DS and DO cases, the both maxima of the doublet bands exhibit red shifts upon the compression of the films. These phenomena cannot be explained by assuming that isolated monomers and/or dimers are the majority in the films. Contrastively, they can be well explained by assuming that large herringbone aggregates are the majority. In the latter model, the surface pressure-induced spectral change is interpreted as a structural phase transition. A numerical simulation for herringbone aggregates of DS reproduces the observed excitation wavenumbers. Infrared absorption spectra have been observed for Langmuir–Blodgett films of the dyes transferred from pure-water surfaces. They indicate the formation of mutual hydrogen bonds between the carboxylic groups in adjacent dye molecules and then suggest that the interaction between those groups is a part of the driving force of the aggregation. © 2005 Elsevier B.V. All rights reserved. Keywords: Dye aggregate; J-aggregate; Langmuir film; Langmuir–Blodgett film; Merocyanine dye; Phase transition; Spectroscopy
1. Introduction Dye aggregates are self-assembled nanostructures whose functions are useful for optical sensitizing, chemical sensing and information storage. According to Kuhn and Kuhn [1], “studying the interplay between dye molecules in different geometries is of interest in approaching the goal of constructing supramolecular machines.” In the light of this general question, J-aggregates formed in Langmuir (L) and Langmuir–Blodgett (LB) films of amphiphilic merocyanine dyes have been studied extensively [2–19]. For example, the importance of structure–function correlation has been recognized [18]. That is, it has been indicated that the alkyl chain and carboxylic group, both of which were introduced to make the dyes amphiphilic, and the shape of the chromophore itself play important roles in the J-aggregation. However, J-aggregates are not always formed in L-films of merocyanine dyes [6–10]. In fact, when 3-carboxymethyl-5-[2(3-octadecyl-2(3H)-benzothiazolylidene)ethylidene]-2-thioxo-
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4-thiazolidinone (DS in Fig. 1) and its derivatives DSe and DO (Fig. 1) are spread on pure water as pure L-films, J-aggregates are hardly formed. Instead, doublet absorption bands appear immediately after spreading the materials (Fig. 1). The doublet band of DS was already reported for L-films upon pure water [10] and cast films on SnO2 electrodes [20], but has not carefully been examined because it was assigned to a sum of monomeric and dimeric absorption bands [20] and then has been considered not very interesting. However, apart from cast films, it is quite questionable whether isolated monomers or dimers can be the major composition of pure L-films. (Since the absorption due to a gas phase is negligible, the doublet bands are assignable to a condensed phase.) Whether the monomeric band of a dye is observable in its L and LB films is a crucial problem in analyzing the optical properties of its J-aggregate formed in the films because, to evaluate the band shift due to aggregation, it is desirable to compare the absorption bands observed for J-aggregated and monomeric dyes in similar environments. Furthermore, if the doublet bands originate from some aggregates, the interplay between the dye molecules in them is directly related with the Kuhn and Kuhn’s general question. Another interesting point is the large total
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Fig. 1. Chemical structures of the merocyanine dyes used in this study (upper pane) and the visible absorption spectra of their L-films on pure water observed immediately after spreading the materials (lower pane). The spectra are normalized by their largest maxima, respectively.
widths of the doublet bands. When dyes are used as sensitizers for imaging, narrow absorption bands are advantageous for enhancing the color purity. On the contrary, broad absorption bands are advantageous for solar power conversion. Elucidation of the mechanism leading to the doublet band in L-films of those merocyanine dyes may provide a guideline for broadening the total bandwidth of sensitizers. In this context, L-films of DS, DO and DSe have been prepared upon pure water and their optical properties have been characterized. In addition to the appearance of the doublet bands, their surface pressure-induced changes have been detected. Infrared (IR) spectroscopic study of transferred films has been performed. These experimental results will be described in Section 3. In Section 4, the optical properties of DS films and their changes upon compression will be semi-quantitatively interpreted based on a model structure, i.e. a herringbone aggregate.
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Co. Ltd. attached to the trough. The optical configuration was the same as that used in the previous studies [16,18]. Baseline tilts of the spectra were corrected so that the absorbance data at 25,000 and 15,390 cm−1 are null. A rather high compression speed (−5.6 × 10−3 nm2 /s) was adopted to suppress baseline drift and film collapse. Absorption spectra of DS films prepared upon Mg2+ - and Ca2+ -containing subphases are also recorded for comparison. In these cases, aqueous solutions of MgCl2 and CaCl2 (4 × 10−4 mol/l) were used after their pH values being adjusted to 6–7 by adding a small amount of KHCO3 . In order to perform IR measurements, L-films on pure water under 25 mN/m as well as those on the Mg2+ - and Ca2+ containing subphases were transferred with the conventional vertical immersing-withdrawing method [21] onto plates of CaF2 precoated by three monolayers of deuteride cadmium arachidate. Visible and IR absorption spectra of the LB films were recorded by Perkin-Elmer Lambda-900 and System-2000 spectrometers, respectively. A precoated CaF2 plate was used as a reference. Absorption spectra of chloroform and methanol solutions of the dyes were also recorded by those spectrometers, though the concentrations of dilute solutions (<10−5 mol/l) were not precisely controlled. 3. Results Pure L-films of DS on a pure-water subphase show a doublet absorption band immediately after the spreading (Fig. 1). The lineshape of this doublet band is unchanged until the surface pressure becomes detectable (Fig. 2) and quite different from those observed for L-films prepared on a Mg2+ -containing
2. Experimental Pure water with a resistivity greater than 1.8 × 107 cm was prepared by using a Millipore Milli-Q system. The merocyanine dyes, i.e. DS, DO and DSe, were purchased from the Japanese Research Institute for Photosensitizing Dyes, Co. and used without further purification. Solutions of the dyes with the concentration of 5 × 10−4 mol/l were made using spectroscopic grade chloroform as the solvent. These solutions were spread on pure water contained in a Lauda Filmwaage trough under air at 20 ◦ C to form L-films. Since the initial areas of the L-films were large enough (ca. 1.2 nm2 per molecule), compression of the films was started 10 s after the end of the spreading; and it was stopped when the surface pressure reached to 25 or 40 mN/m. On compression of L-films, their visible absorption spectra were recorded by a WRM-10TP polychrometer of Jasco
Fig. 2. Visible absorption spectra of a dilute chloroform solution (ca. 4 × 10−6 mol/l) of DS and those of L-films of DS prepared on pure-water and Mg2+ -containing subphases observed when the molecular areas are ca. 0.8 nm2 . The surface pressure–molecular area isotherms observed for these DS films are indicated in Inset (the isotherm measurements are stopped at 40 or 25 mN/m so as to keep these surface pressures).
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subphase (Fig. 2). This doublet band has one maximum at each of the high and low energy sides of the monomeric band of DS observed in a dilute chloroform solution (Fig. 2, maximum at νchl = 18940 cm−1 ) and in a dilute methanol solution (νMeOH = 19120 cm−1 ). The absorption intensities of the two maxima are nearly identical with each other. Doublet bands are also seen in DO and DSe films upon pure water. The absorption spectrum observed for the DO films immediately after the spreading (Fig. 1) is quite similar to that observed in the DS case. The only difference is the positions of the absorption bands and it well corresponds to the difference between the monomeric excitation energies observed for chloroform solutions of the dyes, 1220 cm−1 . As for the DSe films immediately after the spreading, the doublet band is located in a region of smaller wavenumbers compared with the DS case (Fig. 1). The difference in the absorption energy between the DS and DSe solutions is small and cannot cause that between the DS and DSe films. The latter may be caused by different dimeric or aggregate structures. The ordinate of Fig. 2 is molar absorption coefficient, ε, estimated from the obtained transmittance data assuming the three-dimensional random orientation of the transition dipole moments in each samples. With this ordinate, scaling of the L-films’ spectra by a factor of 2/3 makes their areas ( εdν) comparable with that of the solution spectrum. This fact implies that the transition moments in L-films are nearly parallel to the water surface at least under null surface pressure, regardless whether the subphase is pure water or aqueous solution of Mg2+ . DS molecules spread upon a Mg2+ -containing subphase show a limited area of 0.64 nm2 and form a stable L-film (Fig. 2 inset). The absorption spectrum of the L-film is not affected by the surface pressure up to 25 mN/m. DS molecules spread upon a pure-water subphase show completely different behavior. Their limited area is 0.71 nm2 , larger than that in the Mg2+ -containing case, and the formed L-film shows anomalously large compressibility (Fig. 2 inset). Moreover, compression of the L-film is associated with the change in the absorption spectrum (Fig. 3). Both of the minima of the doublet band in the second derivative (d2 ε/dν2 ) spectrum discontinuously shift to the low energy side as the surface pressure is increased. Namely, the excitation wavenumbers in the low-pressure phase, νLPx = 19610 cm−1 and νLPy = 17990 cm−1 , respectively, shift to those in the highpressure phase, νHPx = 19050 cm−1 and νHPy = 17670 cm−1 . The apparent continuous shift of the ε spectrum reflects a continuous increase in the portion of the high-pressure phase that may partly cause the anomalous compressibility. The intensities of the two maxima in ε are nearly identical with each other through out the surface pressure-induced spectral change. In order to know whether the observed surface pressureinduced spectral change is reversible, a compression-expansion experiment has been performed (Fig. 4). Although the area of the L-film observed during the expansion process is considerably smaller than that observed during the compression process, the absorption spectrum observed for the expanded film (c) well corresponds to that observed at the initial stage (a), except for the absolute ε value. This result suggests that the surface pressureinduced change is intrinsically reversible. However, at the same
Fig. 3. Variation of the absorption spectrum of the DS film on pure water (the same film as was measured for Fig. 2) due to compression (upper pane), together with the second derivative spectra (lower pane).
time, it suggests that L-films are irreversibly changed during the compression: some DS molecules may be squeezed out from the films (collapse). This irreversible change may partly cause the anomalous compressibility. The absolute ε values of the spectra b and c are, accordingly, underestimated.
Fig. 4. Surface pressure–molecular area relationship observed during a compression–expansion cycle of a DS film on pure water (upper pane) and the absorption spectra measured for the film at the initial stage, after compression and after expansion, respectively (lower pane).
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hydrogen bonds between the carboxylic groups in adjacent dye molecules. 4. Discussion One of the simplest models for an L-film of a dye with two absorption maxima is a mixture of isolated monomers and (parallel) dimers [20]. However, application of this model to the present cases encounters difficulties: (i) How can the monomers and dimers be isolated in a pure and condensed L-film? (ii) How can the surface pressure induce a shift of the monomeric band? The other simplest model is isolated dimers in which the dye chromophores are not parallel to each other. As is well known [1], the excitation energy of the dyes in this case splits as: λ± = E ± J1,2
Fig. 5. IR absorption spectra of LB films of DO, DS and DSe prepared from pure-water surfaces. Their visible spectra are indicated in inset.
Surface pressure-induced spectral changes are detected for the DO and DSe films, too. In the DO case, it takes place under low surface pressure (<5 mN/m). Both of the maxima of the double band shift to the low energy side and the maximum at the high energy side becomes dominant (data not shown). In the DSe case, it takes place under higher surface pressure (>15 mN/m). The intensity of the maximum at the low energy side is decreased to a certain extent, but no shifts of the maxima are detectable (data not shown). LB films of the dyes have been prepared by transferring their L-films on pure water under 25 mN/m. The doublet lineshapes of the visible absorption bands observed for the LB films (Fig. 5 inset) imply the similarity between the molecular arrangements in L and LB films of the dyes. Namely, the visible absorption spectrum of DS LB films is compatible with that observed for DS L-films under 25 mN/m. The spectra of DO and DSe LB films exhibit intermediate lineshapes between their counterparts observed for L-films under 0 and 25 mN/m, respectively. Hence, the IR absorption spectra of the LB films are useful in revealing the nature of the visible doublet band. The IR spectra of the DS, DO and DSe films have common features (Fig. 5). Firstly, bands due to hydrogen-bonded carboxylic groups are observed around 1730 cm−1 . Secondly, bands due to free keto groups are observed around 1670 cm−1 . Thirdly, bands due to stretching modes of the central conjugated system of the chromophore (observed around 1520, 1540 and 1510 cm−1 for DS, DO and DSe, respectively) are split or broadened in comparison with those in the solution spectra. All of these features are contrastive to the characteristics of the IR absorption spectra observed for J-aggregates of the dyes [15,18,19] and therefore they can be considered characteristic for the phases exhibiting the doublet visible bands. Particularly, the spectral features observed for the C O stretching bands imply the formation of mutual
(1)
where E is the monomeric excitation energy and J1,2 is the interaction between transition dipole moments in the dimer. One of the problems of this model is the same as (i) in the above case and the other is that (ii) the average of the observable excitation energies, λ± , of a dimer is always E and hardly modified by surface pressure. On the contrary, a large herringbone aggregate, which is the next simplest model, creates no qualitative difficulties. In fact, the excitation energy of the dyes in this case split as: λ± = E + J±
(2)
where J± are determined by the arrangement of transition dipole moments and J+ = −J− (see Appendix A). Consequently, an increase in the surface pressure can induce shifts of λ± toward the same direction through a structural deformation. Moreover, this model is consistent with those observed for L-films of 6MeDS, which is an analogue of DS, prepared on a pure-water subphase [18]. In those films, decay of a J-band and growth of a doublet band that is similar to those dealt with in this study proceed simultaneously and an isosbestic point was clearly observed, showing that the both maxima of the doublet band originates from the same phase. It is difficult to eliminate other complicated models rigorously, but the following numerical simulations will support the herringbone aggregate model. Furthermore, it will provide a sketch of the aggregate structure. In this semi-quantitative discussion, DS is dealt with because the doublet band in its L-films on pure water shows surface pressure-induced shift and J-bands are observed for its L-films on aqueous solutions of divalent metals. To simulate the excitation wavenumbers numerically, the structure of the herringbone aggregate should be expressed by a few parameters. Since the elemental unit of the herringbone aggregate contains two molecules that are not parallel to each other, two adjustable parameters, β and lc , are introduced to describe such a “dimeric” unit shown in Fig. 6(a), where β is a half of the angle between the transition dipole moments and lc is the distance between the apex and the center of the transition dipole moments. (This “dimeric” unit is not necessarily a true dimer.) As for the arrangement of the dimeric units, a simple
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Fig. 6. Structures of the dimeric unit (a) and its arrangement (b) assumed for the major composition of the L-films of DS formed on a pure-water subphase, and that of the J-aggregates assumed for DS films on Mg2+ - and Ca2+ -containing subphases (c). The closed circles in (a) stand for the point charges introduced by the extended dipole approximation. The dotted lines in (b) represent planes of symmetry in a large aggregate.
and symmetric one is assumed (Fig. 6(b)) without introducing any additional adjustable parameters. The thickness of the chromophore (= side by side distance between chromophores) is estimated at t = 0.35 nm from a density-functional calculation [17]. The E/hc value is set to vchl = 18940 cm−1 because the solvent effect for the monomeric bands causing the difference between vMeOH and vchl is small. (It seems too small to cause the difference between vLPy and vchl or that between vLPx and vchl . This is another difficulty in attributing vLPy or vLPx to the isolated monomer.) The interaction between the nth and mth transition dipole moments in the aggregate, Jn,m , is estimated based on the extended dipole approximation [1] because its rigorous evaluation is very difficult. That is, Jn,m is approximated at Coulomb interaction between two pairs of positive and negative point charges that are fixed at the molecules n and m, respectively. The vector connecting the charges ±qtr at the molecule n, denoted by ltrn , is parallel to µtrn and its length ltrn corresponds to the effective length of the chromophore (Fig. 6(a)). The strength of the charge qtr satisfies the relation of qtr ltrn = µtrn . This approximation leads to better results, especially when the dye molecules are closely packed, than the point dipole approximation, in which Jn,m is approximated at the Coulomb interaction between two point dipole moments. This comes from the fact that the former takes into account tr the charge distribution to a l is estimated at 0.52 nm [18]. certain extent. In the DS case, n tr µ is treated as an adjustable parameter, while the dielectric n constant is fixed at 2.3 [22]. J-aggregates of DS in L-films prepared upon Mg2+ containing and Ca2+ -containing subphases, respectively, show their absorption maxima at vMg = 16260 and vCa = 16640 cm−1 . These observations are also incorporated into the discussion. (The L-films of DS prepared on a Cd2+ -containing subphase also show a red-shifted absorption band, but that band is
much broader than those observed in the L-films on Mg2+ and Ca2+ -containing subphases and the Cd2+ -containing “Jaggregate” of DS may not be appropriate to incorporate into the present discussion.) As for the molecular arrangements of these J-aggregates, the brickstone model [6–9,11] illustrated in Fig. 6(c) is adopted and two sets of the structural parameters are suggested from geometrical considerations based on the densityfunctional optimization of the molecular structure [17], like in the case of the previous work [18]: Type (I) L = 1.62 nm and s = 0.81 nm, and Type (II) L = 1.56 nm and s = 0.55 nm. Through numerical simulations, it was found that β = 41.92◦ ◦ and and lc = 0.855 nm give rise to νLPx and νLPy and β = 39.15 tr lc = 0.898 nm give rise to νHPx and νHPy , when µn is set to 10.6 D. This µtrn value leads to 16,600 and 16,300 cm−1 for the Type (I) and (II) brickstone arrangements, respectively, and these values are comparable to vCa and vMg , respectively. The estimated β values are not far from 45◦ and consistent with the fact that the two maxima in the doublet band have nearly the same intensity in both the low and high-pressure cases (see Appendix A). The lc values slightly larger than L/2 and the negative correlation between this and the β values are compatible with the geometry of the dimeric unit. In addition, the estimated µtrn value is comparable to that evaluated from the solution spectrum (ca. 9.6 D). Thus, the herringbone-type molecular arrangement illustrated in Fig. 6(b) semi-quantitatively explains the visible absorption of DS films prepared on pure water under both low and high surface pressures. It should be emphasized that changes in the structural parameters of the aggregate, β and lc , can cause simultaneous decreases in the both excitation energies, λ± . Therefore, the red shifts of the both maxima of the double band observed upon compressing the L-films can be interpreted as a surface pressure-induced structural phase transition. The IR measurements of the dye’s LB films prepared from pure-water surfaces have implied the formation of mutual hydrogen bonds between the carboxylic groups of adjacent dye molecules. Although the environments around the carboxylic groups in the LB and L-films are largely different, the basic structures of the herringbone aggregates seem to be kept in LB films. Therefore, it is plausible to assign the interaction between the carboxylic groups to a part of the driving force of the aggregation. In conclusion, the doublet absorption bands observed for the L-films of DS, DO and DSe prepared upon a pure-water subphase suggest that herringbone aggregates of the dyes are formed immediately after the spreading. The observed surface pressure-induced spectral change in the DS cases, i.e. the red shifts of the both maxima, is well explained by deformation of the aggregates. This finding implies that when the J-aggregates of the dyes are discussed, the J-shifts should not be measured from one of the maxima in such doublet bands. Rather, they should be measured from the maxima in the solution spectrum or, if possible, both of the doublet bands and J-bands should be explained systematically in the same framework. The obtained IR spectroscopic data imply that the carboxylic group connected to the dye chromophores plays an important role in the aggregation. Since the carboxylic group may not directly modify the chromophores’ optical
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functions, it may be useful as a “connector” in the bottom-up nanotechnology. Appendix A The theory of the optical properties of dye aggregates has been well established, but the general formula is very complicated. Therefore, a brief summary specialized to the case of two-dimensional herringbone aggregates is given in order to clarify the basis of the present discussion, especially that of Eq. (2), which is not a simple extension of Eq. (1). The Hamiltonian of the singly excited states of an aggregate containing N → ∞ molecules can be represented with the basis set of wavefunctions |ψn > = |. . . gn−1 en gn+1 . . . >as: ⎞ ⎛ .. . ⎟ ⎜ ⎟ ⎜ E Jn−1,n Jn−1,n+1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Jn,n−1 E Jn,n+1 (A.1) H =⎜ ⎟ ⎟ ⎜ Jn+1,n−1 Jn+1,n E ⎟ ⎜ ⎠ ⎝ .. . where |en > and |gn > are the nth dye’s ground and excited state wavefunctions, respectively, the diagonal term E is the excitation energy of an isolated monomeric dye, and the offdiagonals Jn,m = <ψn |V(n,m)|ψm > are caused by the Coulomb interaction between the nth and mth molecules, V(n,m). With this Hamiltonian, wavefunctions |±> defined as: |± = N −1/2 (±1)n |ψn , (A.2) n
can be used as trial eigenfunctions and H|±> are calculated as: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ H |± = N −1/2 (±1)n ⎣E + (±1)m Jn,m ⎦ |ψn . m=n
n
Hence, if J± (n) =
(A.3) (±1)m Jn,m are independent of n, |±> are
m=n
really eigenfunctions of H. Two-dimensional herringbone arrangements like illustrated in Fig. 6(b) and the absence of edge effects due to the large N introduce planes of symmetry in the aggregate structure (dotted lines in Fig. 6(b)). Here, the two-dimensionality confines the transition dipole moment of each dye in the aggregate plane. The plane of symmetry of each dimeric unit (zx-plane in Fig. 6(a)) coincides with one of those planes of symmetry of the aggregate. When dyes n and n + 1 belong to the same dimer, therefore, Jn,m always has its counterpart of Jn+1,m = Jn,m . In addition, all the dimers in the aggregate are equivalent. Consequently, J± (n) is independent of n, and then: λ± = E + J± (n) E + J±
(A.4)
are the eigenvalues of H, i.e. the excitation energies corresponding to the excited states of |±> and it should be emphasized that J+ = − J− . This equation is the same as Eq. (2) in the text.
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The transition dipole moments µtr± that correspond to the excitations of the aggregate from the ground state |G> to |±>, respectively, can be calculated by putting the electrons’ position operators rn,k = (xn,k , yn,k ) between these wavefunctions, where k indexes the electrons in the nth molecule. Setting the phases of µtrn gn | rn,k |en so that the x components of all µtrn are real and positive, the x and y components of µtr± of large herringbone aggregates are readily obtained as: 2 2 tr 2 µ = e2 G| xn,k |± = N −1 (±1)n µtrnx ±x n k n 2 N µtrn cos2 β for |+ (A.5) = |− 0 and
2 2 tr 2 2 −1 n tr yn,k |± = N (±1) µny µ±y = e G| n k n 0 for |+ = . (A.6) 2 |− N µtr sin2 β n
2 2 2 The sum of µtr+ and µtr− coincides with N µtrn (note that tr 2 µ is independent of n), showing that singly excited states n of the aggregate other than |±> have null transition dipole moments. More importantly, the intensity ratio between the two transition modes is tan2 β, being exactly the same as in the dimer case. At last, the special case of β = 0◦ should be mentioned. In this case, the condition that J± (n) is independent of n is fulfilled by the brickstone arrangement. Hence, (A.4)–(A.6) are applicable to two-dimensional large J-aggregates, too, and in fact they coincide with the discussion in Ref. [1]. (Note that Jn,m in this paper corresponds to 2Jij in Ref. [1]. This difference comes from the different definitions of the transition dipole moment. Namely, it is defined based on one-electron wavefunctions in Ref. [1], while it is defined based on molecular wavefunctions in this paper. The latter is more directly connected to the areas of experimentally observable absorption bands.) References [1] H. Kuhn, C. Kuhn, in: T. Kobayashi (Ed.), J-Aggregates, Word Scientific, Singapore, 1996, Chapter 1. [2] M. Sugi, S. Iizima, Thin Solid Films 68 (1980) 199. [3] K. Iriyama, M. Yoshiura, F. Mizutani, Thin Solid Films 68 (1980) 47. [4] S. Kuroda, M. Sugi, S. Iizima, Thin solid Films 99 (1983) 21. [5] S. Kuroda, K. Ikegami, Y. Tabe, K. Saito, M. Saito, M. Sugi, Phys. Rev. B 43 (1991) 2531. [6] H. Nakahara, D. M¨obius, J. Colloid Surf. Sci. 114 (1986) 363. [7] H. Nakahara, K. Fukuda, D. M¨obius, H. Kuhn, J. Phys. Chem. 90 (1986) 614. [8] T. Kawaguchi, K. Iwata, Thin Solid Films 165 (1988) 323. [9] T. Kawaguchi, K. Iwata, Thin Solid Films 191 (1990) 173. [10] M. Yoneyama, T. Nagao, T. Murayama, Chem. Lett. (1989) 397. [11] L. Wolthaus, A. Schaper, D. M¨obius, Chem. Phys. Lett. 225 (1994) 322. [12] N. Kato, K. Saito, T. Serata, H. Aida, Y. Uesu, J. Chem. Phys. 115 (2001) 1473.
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