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Molecular Arrangement of Alkylated Fullerenes in the Liquid Crystalline Phase Studied with X-ray Diffraction Paulo A. L. Fernandes,*,† Shiki Yagai,‡,§ Helmuth M€ohwald,† and Takashi Nakanishi*,†,§,
Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germany, ‡Chiba University, 1-33 Yayoi-cho, Inageku, Chiba 263-8522, Japan, §PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama, Japan, and National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba 305-0047, Japan )
†
Received September 11, 2009. Revised Manuscript Received October 9, 2009 Fullerenes (C60) with attached aliphatic chains are shown to form fluid bilayer structures that are distinguished by very characteristic and well-resolved X-ray diffraction patterns. Since, in addition, we vary systematically the number and length of the chains, detailed understanding of the structures can be achieved. To make the analysis transparent, simple boxlike electronic density profile models are proposed to explain the relative intensity of the several Bragg peaks present in the X-ray patterns. The models allow detailed characterization of the molecular organization. The molecules arrange themselves in bilayers with their long axis on average perpendicular to the plane of the layers. Considering the bilayers composed of three sections with different electronic density, the C60 heads occupy a fixed length of approximately 17 A˚, corresponding to almost no interdigitation, the connector section around 4 A˚, and the carbon chains’ perpendicular length depends on the number and length of the chains. The analysis reveals that reducing the number of carbons per chain (from 20 to 16) results in a shorter unit cell, while reducing the number of chains (from 3 to 2) results in a shorter but also slightly thinner unit cell, in agreement with known molecular packing volumes.
1. Introduction Facile and economical production of highly ordered structures with superior photoconductivity, of great interest in optoelectronic applications, remains a challenge for materials science.1-3 Organic materials due to their relatively low-cost, easy processing, and tailoring are emerging as an alternative to the inorganic semiconductor-based photovoltaics.4-6 Fullerene (C60), exhibiting relatively high carrier mobility,7,8 is a very promising organic n-type semiconductor with applications in solar cells.9,10 However, imperfect molecular arrangement and defects acting as electron-hole recombination sites greatly diminish the photovoltaic efficiency of the system. The self-assembling nature of liquid crystals (LCs)11-13 can provide an elegant solution to induce a higher degree of ordering in the molecular arrangement, *Corresponding authors. E-mail:
[email protected] (P.A.L.F.);
[email protected] (T.N.). (1) Yamamoto, Y.; Fukushima, T.; Suna, Y.; Ishii, N.; Saeki, A.; Seki, S.; Tagawa, S.; Taniguchi, M.; Kawai, T.; Aida, T. Science 2006, 314, 1761–1764. (2) Feng, X. L.; Marcon, V.; Pisula, W.; Hansen, M. R.; Kirkpatrick, J.; Grozema, F.; Andrienko, D.; Kremer, K.; M€ullen, K. Nat. Mater. 2009, 8, 421– 426. (3) Funahashi, M.; Hanna, J. I. Adv. Mater. 2005, 17, 594–598. (4) Tang, C. W. Appl. Phys. Lett. 1986, 48, 183–185. (5) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J. Science 1995, 270, 1789–1791. (6) Bredas, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. Rev. 2004, 104, 4971–5003. (7) Frankevich, E.; Maruyama, Y.; Ogata, H. Chem. Phys. Lett. 1993, 214, 39– 44. (8) Anthopoulos, T. D.; Singh, B.; Marjanovic, N.; Sariciftci, N. S.; Ramil, A. M.; Sitter, H.; Colle, M.; de Leeuw, D. M. Appl. Phys. Lett. 2006, 89, 213504–3. (9) Hoppe, H.; Sariciftci, N. S. J. Mater. Chem. 2006, 16, 45–61. (10) Kim, J. Y.; Lee, K.; Coates, N. E.; Moses, D.; Nguyen, T. Q.; Dante, M.; Heeger, A. J. Science 2007, 317, 222–225. (11) Kato, T.; Mizoshita, N.; Kishimoto, K. Angew. Chem., Int. Ed. 2006, 45, 38–68. (12) Goodby, J. W.; Saez, I. M.; Cowling, S. J.; Gortz, V.; Draper, M.; Hall, A. W.; Sia, S.; Cosquer, G.; Lee, S. E.; Raynes, E. P. Angew. Chem., Int. Ed. 2008, 47, 2754–2787. (13) Tschierske, C. Chem. Soc. Rev. 2007, 36, 1930–1970.
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enabling the wet processing of effective carrier transport systems of π-conjugated cores.14-17 Fullerene-based LCs have been developed, and their optoelectronic properties have been measured in the columnar18,19 and smectic18,20,21 phases. Detailed knowledge of the molecular arrangement, position, and orientation of the π-conjugated cores in the liquid crystalline phases can provide fundamental feedback to improve optoelectronic performance.22-25 In our previous work,26 long-range smectic ordering of high C60 content fullerene derivatives bearing long alkyl chains was evidenced, together with a relatively high electron mobility determined by the time-of-flight technique. In this Article, we discuss in detail the molecular arrangement of three samples composed of a C60 head and hydrocarbon tail with different (14) Adam, D.; Schuhmacher, P.; Simmerer, J.; Haussling, L.; Siemensmeyer, K.; Etzbach, K. H.; Ringsdorf, H.; Haarer, D. Nature 1994, 371, 141–143. (15) Percec, V.; Glodde, M.; Bera, T. K.; Miura, Y.; Shiyanovskaya, I.; Singer, K. D.; Balagurusamy, V. S. K.; Heiney, P. A.; Schnell, I.; Rapp, A.; Spiess, H. W.; Hudson, S. D.; Duan, H. Nature 2002, 419, 384–387. (16) Sergeyev, S.; Pisula, W.; Geerts, Y. H. Chem. Soc. Rev. 2007, 36, 1902–1929. (17) Shimizu, Y.; Oikawa, K.; Nakayama, K. I.; Guillon, D. J. Mater. Chem. 2007, 17, 4223–4229. (18) Lenoble, J.; Campidelli, S.; Maringa, N.; Donnio, B.; Guillon, D.; Yevlampieva, N.; Deschenaux, R. J. Am. Chem. Soc. 2007, 129, 9941–9952. (19) Sawamura, M.; Kawai, K.; Matsuo, Y.; Kanie, K.; Kato, T.; Nakamura, E. Nature 2002, 419, 702–705. (20) Felder-Flesch, D.; Rupnicki, L.; Bourgogne, C.; Donnio, B.; Guillon, D. J. Mater. Chem. 2006, 16, 304–309. (21) Li, W. S.; Yamamoto, Y.; Fukushima, T.; Saeki, A.; Seki, S.; Tagawa, S.; Masunaga, H.; Sasaki, S.; Takata, M.; Aida, T. J. Am. Chem. Soc. 2008, 130, 8886– 8887. (22) Deschenaux, R.; Donnio, B.; Guillon, D. New J. Chem. 2007, 31, 1064– 1073. (23) Peroukidis, S. D.; Vanakaras, A. G.; Photinos, D. J. J. Phys. Chem. B 2008, 112, 12761–12767. (24) Sazonovas, A.; Orlandi, S.; Ricci, M.; Zannoni, C.; Gorecka, E. Chem. Phys. Lett. 2006, 430, 297–302. (25) Kouwer, P. H. J.; Mehl, G. H. J. Mater. Chem. 2009, 19, 1564–1575. (26) Nakanishi, T.; Shen, Y.; Wang, J.; Yagai, S.; Funahashi, M.; Kato, T.; Fernandes, P.; M€ohwald, H.; Kurth, D. G. J. Am. Chem. Soc. 2008, 130, 9236– 9237.
Published on Web 10/26/2009
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Figure 2. Schematic drawing of X-ray specular reflectivity from a thin ordered film. between two glass slides and heating at around 200 °C while rubbing gently to obtain a smooth homogeneous surface and then removing the top glass slide and cooling down. After placing the samples in the diffraction setup and heating to the mesophase temperature range, they behaved similar to very viscous fluids. The reflectivity and diffraction experiments were carried out with a Rigaku Rint-2200 X-ray diffractometer with monochromated Cu KR, 8.1 keV, radiation and a temperature controlled heating stage. The intensity of the incident beam was around 106 counts per second (c/s), the acquisition time was 1 s, the resolution of the X-ray diffraction setup measured as the full width at half-maximum of the direct beam was approximately 10-2 A˚-1, and the background radiation was around 10-5 c/s. The sample was roughly estimated to be several tens to several hundred micrometers thick, well beyond the resolution imposed limit of ca. 600 A˚ to allow detection of Kiessig fringes. The width of the observed Bragg peaks was limited by the experimental resolution. Figure 1. Molecular structures and polarized optical microscopy images of three alkylated C60 derivatives with different carbonchain number or length: (3,4,5)C20C60 (a, b at 185 °C), (3,4,5)C16C60 (c, d at 202 °C), and (3,4)C20C60 (e, f at 200 °C).
number or length of the carbon chains26 in the smectic phase (see Figure 1). The combination of C60 electronic properties, high C60 content, smectic self-organization, and very rich X-ray diffraction spectra presenting several Bragg peaks (up to 14th order and systematic absences) indicative of a highly ordered phase and also capable of acting as a fingerprint of the detailed molecular arrangement inspires the present study. The main motivation is to understand the interplay of tail moieties and C60 packing on the self-organized, liquid crystalline, molecular structure. This analysis could stimulate studies on novel molecular designs, providing feedback for the synthesis of advanced semiconducting liquid crystalline systems.
2. Materials and Experimental Details The synthesis of the C60 derivatives, fulleropyrrolidines functionalized with a multi(alkyloxy)phenyl group ((3,4,5)C20C60,27 (3,4,5)C16C60,28 and (3,4)C20C6029), has been previously described and the products unambiguously characterized. Polarized optical microscopy observation was carried out using an Olympus BX51 optical microscopy system with a Linkam temperature-controlled heating stage. Sample preparation for X-ray diffraction was carried out by first sandwiching a small amount of material (27) Nakanishi, T.; Miyashita, N.; Michinobu, T.; Wakayama, Y.; Tsuruoka, T.; Ariga, K.; Kurth, D. G. J. Am. Chem. Soc. 2006, 128, 6328–6329. (28) Nakanishi, T.; Schmitt, W.; Michinobu, T.; Kurth, D. G.; Ariga, K. Chem. Commun. 2005, 5982–5984. (29) Nakanishi, T.; Takahashi, H.; Michinobu, T.; Takeuchi, M.; Teranishi, T.; Ariga, K. Colloids Surf., A 2008, 321, 99–105.
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3. Theoretical Description of X-ray Analysis X-rays are electromagnetic radiation with a wavelength range corresponding to typical molecular and atomic distances in materials. X-rays interact mostly with the electron clouds surrounding the atoms and can be used to probe the molecular and atomic structure of materials. Indeed, by scanning the incident angle (θi) of an X-ray beam projected onto a thin sample while measuring the intensity reflected at the same angle (θr), in the socalled X-ray reflectivity geometry (Figure 2), information-rich spectra are obtained that depend on the properties of the film, for example, roughness, electronic density, and film (dfilm) and layer (dlayer) thickness with atomic resolution. As with all electromagnetic radiation, the incident beam’s interaction with matter is governed by the index of refraction (n) which, for X-rays, is n ¼ 1 - δ - iβabs
ð1Þ
where δ = Fereλ2/2π, βabs = μλ/4π, Fe is the electronic density of the sample, re is the classical electron radius (2.81 10-5 A˚), λ is the radiation wavelength in vacuum (1.54 A˚ in our experiments), and μ is the absorption coefficient of the sample.30 Because most of the scattering is elastic, the incident and reflected wave vectors are related by |KBi| = |KBd| = 2π/λ. The wave vector transfer, q, is then simply related to the incident angle by |qB| = |KBd - KBi| = (4π/λ)sin θi. It is important to notice that, in this geometry, q is always perpendicular to the plane of the film, meaning that, by scanning the incident angle, the properties of the film are probed along the z direction and averaged over the beam imprint along (30) de Jeu, W. H.; Ostrovskii, B. I.; Shalaginov, A. N. Rev. Mod. Phys. 2003, 75, 181–235.
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the other two directions (x and y). In our work, this implies that n n(z). In X-ray diffraction presentations, it is usual to present the Bragg expression 2d sin θi = mλ or equivalently q = (2π/d)m, which gives the angular (or q) position of intensity maxima, socalled Bragg peaks, resulting from the constructive interference when the path difference traveled by the waves between scattering centers is equal to an integer number (m) of wavelengths. Although intuitively helpful, Bragg’s law gives only the approximate position of the peaks, making no prediction about their relative intensities. To make a deeper analysis of the results, a more complex model is needed. Such a model was developed by Parratt,31 consisting of a recursive numerical algorithm that divides the sample in a defined number (N þ 1) of slabs of constant index of refraction (or electronic density) and, starting from the bottom, iteratively calculates the reflectivity from each slab (eq 2) based on the Fresnel reflectivity contribution (eq 3)32 of the interface between each pair of adjacent slabs j, j þ 1. The total reflectivity of the film corresponds to the reflectivity of the top slab, R = |R1|2.33 Rj ¼
rj , jþ1 þ RJþ1 e - iqjþ1 djþ1 1 þ rj, jþ1 RJþ1 e - iqjþ1 djþ1 rj , jþ1 ¼
qj - qjþ1 qj þ qjþ1
ð2Þ
ð3Þ
A reflectivity spectrum is very rich in information and is sensitive to several sample parameters such as homogeneity, ordering, thickness, roughness, and so on. Because the interest of this work is to characterize the molecular structure or arrangement of the samples, we have focused our analysis on the Bragg peaks, trying to explain semiquantitatively their relative intensities. Therefore and to simplify the models, we have in some cases34-37 assumed simple or easy to handle values when they are not expected to have a major influence in the Bragg peaks’ relative intensities. The Parratt -algorithm-based spectra presented in this work were performed with a homemade software and checked against a standard “Parratt” software.31 All the theoretical tools necessary to determine, with the help of a computer, electronic density models that explain experimental reflectivity data have been introduced above. However, to understand why or which box models should be tested, it is important to gain a deeper intuition of the reflection mechanism causing the (31) Parratt, L. G. Phys. Rev. 1954, 95, 359–369. (32) The Fresnel reflectivity represented by eq 3 is strictly speaking only valid for the polarization component perpendicular to the plane of incidence, the plane that contains both incident and reflected waves; however, for typical X-ray index of refraction values and at the low incidence angles of the experiments, both components are similar and eq 3 constitutes therefore a good approximation. (33) The X-ray beam finite size is taken into account by convoluting the theoretical reflectivity with a Gaussian function with full width at half-maximum corresponding to the resolution of the beam. Background radiation is considered in the form of a constant added to the reflectivity profile. (34) Typically, in nonconducting materials, βabs is orders of magnitude lower than δ, meaning that it can be ignored in a first approximation. Therefore, the absorption of the films is taken to be zero, or equivalently the index of refraction is always treated as a real number in the models. (35) Because the glass slide substrate mainly influences the reflectivity part of the spectra, it is neglected in the models. (36) Because the number of reflecting layers mainly influences the broadness of the Bragg peaks, which are actually resolution limited, an arbitrary number of 90 reflecting layers is used in the models for all samples. (37) It is possible to model the roughness or fluctuations of the film by introducing a decay factor in eq 3. However, because this is not the main objective of this work and for simplification, no film roughness or fluctuations are considered in the box models proposed. The presence of several peaks in the experimental X-ray diffraction spectra suggests indeed that the roughness should be small and small enough fluctuations should not change the main results.
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Figure 3. Schematic drawings of first and second order Bragg reflections when the electronic density profile presents two opposite-sign (a) or same-sign (b) slopes at half the layer thickness, d.
so-called diffraction or Bragg peaks. As explained above, these peaks correspond to the constructive interference of the waves from different reflecting planes when the path difference equals an integer number of wavelengths. From this definition, one would expect to observe a fundamental peak corresponding to one wavelength path difference followed by the higher harmonics. This is however not always the case as, depending on the symmetry of the electronic density profile, some peaks might be absent, socalled forbidden reflections. As an example, imagine the electronic density profile (Fe) of one layer composed of one box with width equal to exactly half the layer thickness as depicted in Figure 3a, left. In this case, when the sample is irradiated at an incidence angle corresponding to the first order Bragg peak, the reflection from the middle of the layer (d/2, with only half a wavelength relative path difference) suffers a 180° phase change, because of the negative slope of the electronic density profile at this point, in accordance with eq 3. It then interferes constructively with the waves reflected from the plane z = 0 to form an observable first order Bragg peak, as expected (Figure 3a). Conversely, when the sample is irradiated at an incidence angle corresponding to the second order Bragg peak, the reflection from the middle of the layer continues to be phase shifted by 180° but because the relative path difference is one wavelength it will interfere destructively with the wave reflected from the plane z = 0, leading to the absence of the second order Bragg peak (as depicted in Figure 3a, right). It is easy to see that all the even order Bragg peaks will be absent in this case. If the electronic density slopes have the same sign (Figure 3b), it is the odd order Bragg peaks that will be absent. This can be easily understood as, in this simple example, it is equivalent to a sample with half the layer thickness corresponding to double, or even order, peak positions. An example of such a case is the well-known forbidden reflections in the antiferroelectric phase.38,39 It is fairly straightforward to extend the reasoning used in the example of Figure 3 to the case of electronic density profiles presenting two slopes with relative distance different from half the layer thickness. If we define the spacing between slopes, s, then no peaks should be observed at angular positions m = q/q0 given by d opposite-sign slopes: m ¼ ð1 þ iÞ s d 1 same sign slopes: m ¼ þi s 2
ð4aÞ ð4bÞ
(38) Fernandes, P.; Barois, P.; Grelet, E.; Nallet, F.; Goodby, J. W.; Hird, M.; Micha, J.-S. Eur. Phys. J. E 2006, 20, 81–87. (39) Fernandes, P.; Barois, P.; Wang, S. T.; Liu, Z. Q.; McCoy, B. K.; Huang, C. C.; Pindak, R.; Caliebe, W.; Nguyen, H. T. Phys. Rev. Lett. 2007, 99, 227801-4.
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Figure 4. Schematic diagram of a layer with thickness dlayer. dcc is the carbon-chain length, dcon is the connector length, and d2C60 is the perpendicular length occupied by the two C60 molecules (inner, darker circles). The lighter, outer circles represent the exclusion volume of each C60 molecule.
where i is any integer.40 The value of m in eq 44 can be a noninteger indicating an attenuated reflectivity zone. The degree of destructive interference depends on the distance, height, and relative sign between electronic density slopes. There will only be complete destructive interference leading to a peak absence if the distance between electronic density slopes is exactly an integer fraction of the layer thickness and if the reflected waves have the same intensity. In general, electronic density slopes of real samples will have different heights and noninteger fraction distances, and therefore, there will be no absent or forbidden peaks but rather attenuated peaks. In addition, the electronic density of real samples will often be better described by several boxes, instead of the simple case presented above, becoming too complex to calculate the contributions from all the slopes. The reasoning presented can, however, be applied to the more prominent electronic density box slopes which have a greater influence on the reflectivity. This reflectivity attenuation mechanism can therefore provide a general guide to find electronic density profiles that match the reflectivity or diffraction spectra of arbitrary samples. Finally, we note that in reflectivity experiments only the intensity of the reflected waves is measured and phase information is lost. This can lead to ambiguity in the interpretation of the results; that is, several electronic density profiles can explain the same data. It is therefore fundamental to use our knowledge about the sample to trial through the possible models and obtain a credible electronic density that explains the experimental data.
4. Results and Discussion The observation of multiple Bragg peaks in the diffraction patterns of the fluid samples presented below is indicative of an unusually high-order condensed matter phase where the molecules arrange themselves in layers parallel to the substrate. The Bragg peak positions correspond to layer thicknesses that are larger than one molecular length. This suggests that the samples form bilayer structures, as reported before,26 with the C60 moieties in a head to head configuration. The Bragg peak positions of samples (3,4,5)C20C60, (3,4,5)C16C60, and (3,4)C20C60 correspond, respectively, to layer thicknesses of d(3,4,5)C20C60 ≈ 53.0 A˚, d(3,4,5)C16C60 ≈ 46.4 A˚, and d(3,4)C20C60 ≈ 45.4 A˚ in this revised detailed analysis.41 The molecules are assumed to arrange their longer axis on average perpendicular to the plane of the layers, in accordance with the absence of plane birefringence and the textures observed with polarized optical microscopy (see (40) In the case of Figure 3, where the prefactor of eq 44 is equal to 2, it is easy to see that absent peaks are expected at even and odd order Bragg peaks for oppositeand same-sign slopes, respectively. (41) In the previous study,26 the d-spacings were calculated only from the (001) peak positions and found to be 55.9 A˚ for (3,4,5)C20C60, 45.4 A˚ for (3,4,5)C16C60, and 46.4 A˚ for (3,4)C20C60.
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Figure 5. Electronic density profiles (a,c) and comparison (b,d) between corresponding model (red line) and experimental (blue line) X-ray diffraction pattern of sample (3,4,5)C20C60 in the mesophase. Molecular arrangement cartoon drawn to help visualization. (a) One-box model representing complete C60 interdigitation. (c) Two-box model representing no C60 interdigitation. The red crosses in the XRD (b,d) highlight the experimental peaks whose intensity is not satisfactorily reproduced by the model.
Figure 1).26 The presence of a broad halo in all the spectra26 (subtracted from the data presented here) at q ≈ 1.34 A˚-1 corresponding to a distance of approximately 4.7 A˚ indicates that the carbon chains are in the liquid state. A schematic of the molecular arrangement in the layer can be seen in Figure 4, where the thickness, dlayer, is considered as the sum of three terms: dlayer ¼ d2C60 þ 2dcc þ 2dcon
ð5Þ
dcc is the length of the carbon chains, d2C60 is the length of the two C60 heads, and dcon is the length of the connector between them. In the following, the length of these sections will be estimated. The strategy followed consists of first checking several electronic density models against the data from sample (3,4,5)C20C60 and subsequently comparing the successful model against the diffraction data of samples (3,4,5)C16C60 and (3,4)C20C60, after changing the carbon chains’ section length (dcc) accordingly, as explained below. a. Electronic Density Profile. The perpendicular interdigitation of the C60 heads is critical to determine the electronic density profile. Two extreme scenarios are considered: in the case of complete C60 interdigitation, the electronic density profile of one layer can be represented in a first approximation by a 6.85 A˚ wide, higher density single box corresponding to one C60 molecule42 in a flat, lower density surrounding, corresponding to the carbon chains (Figure 5a). The X-ray diffraction spectrum of such an electronic density profile is compared with the experimental data in Figure 5b, and the red crosses serve to highlight the experimental peaks whose intensity is not satisfactorily reproduced by the proposed density profile. It is easy to realize that completely interdigitated C60 heads cannot explain the experimental data. In the theoretical section, it is explained how the slope distance in the electronic density profile can have an effect on the intensity (42) SES Research, http://sesres.com/PhysicalProperties.asp (accessed 07/07/ 2009).
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Figure 6. Three-box electronic density profile (a) and comparison (b) between corresponding model (red line) and experimental (blue line) X-ray diffraction pattern for sample (3,4,5)C20C60 in the mesophase. Molecular arrangement cartoon drawn to help visualization.
of the diffraction peaks. Examining the experimental peak intensities of sample (3,4,5)C20C60 (Figure 5b and d, blue line) reveals that the third order peak is weakened and that the ninth order peak is almost nonobservable when compared to surrounding peaks. According to eq 4b, this suggests that the electronic density profile of the sample should present same-sign slopes separated by approximately one-sixth of the layer thickness. A profile with two 6.85 A˚ wide boxes corresponding to the C60 molecules separated by ∼2 A˚, compatible with this requirement, is presented in Figure 5c. Such a profile represents the case of noninterdigitated C60 moieties. This model does not reproduce all the peaks’ intensities, as can be seen in Figure 5d. Small deviations in the parameters describing the two previous electronic density models also cannot explain the experimental data. This is a fundamental result indicating that more complex models must be considered. Taking into account a lower density connector section, the electronic density profile presented in Figure 6a, characterized by d2C60 = 17.0 A˚, dcc = 14.0 A˚, and dcon = 4.0 A˚, is found to explain the diffraction data, following the peaks’ intensities reasonably well (Figure 6b). We note that the length of the box corresponding to the two C60 moieties, d2C60, separating the two dominating slopes, is very close to one-third of the layer thickness (53.0 A˚), as required by eq 4a to produce a weaker third order peak intensity. Several similar models with different box heights and widths were tested and do not provide better compatibility with the experimental data. b. Planar Molecular Area. Although a reflectivity experiment mainly probes the perpendicular electronic density profile, it is also sensitive to the average planar electronic density over the beam imprint which will depend on the planar molecular packing. The planar arrangement can also influence C60 perpendicular interdigitation; a more loosely packed planar arrangement will in general result in higher interdigitation (lower d2C60). It is therefore possible, in principle, to estimate the molecular area based on Xray reflectivity data. However, the intensity of the Bragg peaks is mainly independent of the absolute value of the electronic density, which essentially determines the reflectivity from the first and last interface, but is sensitive to electronic density differences. This means that similar electronic density profiles obtained by adding an arbitrary, small constant can equally reproduce the intensity of the Bragg peaks. Therefore, it is important to have a reference value to determine the physically realistic electronic density profile. In our case, this reference was taken to be the electronic density of carbon chains which, taking into account the sample temperature of 185 °C and the typical thermal expansion coefficient of carbon chains, is expected to be approximately 0.26 e-/A˚3. Because the electronic density profile of Figure 6a must be compatible with the number of electrons (720 from the Langmuir 2010, 26(6), 4339–4345
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two C60, 186 from the connectors, and 966 from the carbon chains for sample (3,4,5)C20C60) and volume occupied by one unit cell, consisting of two molecules, it is possible to estimate the area occupied by one unit cell, ∼114.5 A˚2, corresponding to a square with edge ∼10.7 A˚. It is known that C60 forms fcc lattice crystals with unit spacing of 14.17 A˚, corresponding to a minimum C60 distance of approximately 10.0 A˚.42 Because the samples studied in this work are higher temperature fluids, it is reasonable to assume that the inter-C60 distance should therefore be larger than 10 A˚, in agreement with the value proposed in our model. The unit cell volume of sample (3,4,5)C20C60 is then ∼6068 A˚3, and the average mass and electronic density are ∼0.95 g/cm3 and ∼0.31 e-/A˚3, respectively. For comparison, it is known that at room temperature similar carbon chains can have 0.7 g/cm3 and C60 fcc crystals have a mass density of ∼1.68 g/cm3 and electronic density of ∼0.51 e-/A˚3.42 Because the C60 moieties only occupy ca. 30% of the molecular volume and the sample is in a fluid phase, the lower mass density seems reasonable. A very small interdigitation (Zint ≈ 0.6 A˚) of the C60 heads must be taken into account to explain the 17.0 A˚ of the respective section. It is also instructive to compare the number of electrons in each section of the sample, calculated as the product of volume and electronic density, with the number of electrons in the C60 heads, connector, or carbon chains. From the density profile of Figure 6a, the carbon-chain section has ∼833 electrons and the C60 and connector sections have ∼1039 electrons as compared to 966 and 906 electrons, respectively. This reveals that the carbon chain-section must have fewer electrons than expected, suggesting that part of the carbon chains occupy the other sections, in accordance with the concept that they are in the liquid state and attempt to occupy all the volume available to them. c. Effect of Carbon-Chain Length and Number. The molecules of the three samples studied in this work differ only in the number and length of the carbon chains (Figure 1). It is therefore reasonable to propose that the electronic density profile of the samples should be similar except for the carbon-chain section length (dcc). Indeed, keeping the C60 and connector length constant and changing the carbon-chain length (dcc) to 10.7 and 10.2 A˚ in order to accommodate for the different layer thickness of samples (3,4,5)C16C60 and (3,4)C20C60, respectively, this model is also able to explain quite well their experimental diffraction patterns, as can be seen in Figure 7. According to eq 4a, this good agreement can be roughly understood because the ratio between the reduced layer length and the constant C60 section length is in this case slightly lower than 3, causing a weaker third and eighth order peak. Although the background signal of sample (3,4)C20C60 is higher than that of the other two samples, probably due to a spurious reflection, we remark that the peak intensities still follow reasonably close the experimental data. Table 1 displays the electronic density and length of each section of the density profiles shown in Figures 6 and 7. The main parameters resulting from the analysis based on the X-ray diffraction data of the three samples are resumed in Table 2. It is remarkable that the model proposed, correcting only the carbon-chain section length, is capable of reproducing so well the diffraction data of the three samples. Because the change in the number of carbons in each sample is known, it is possible to determine the expected change in the length of the carbon-chain section. Extrapolating linearly from the thickness dcc = 14.0 A˚ measured for sample (3,4,5)C20C60 with 3 20 = 60 carbonchain segments to one with ncc carbon-chain segments, one expects dcc = 14.0(ncc/60) A˚. Substituting the number of chains and carbons per chain of samples (3,4,5)C16C60 and (3,4)C20C60 gives ∼11.2 and ∼9.3 A˚, respectively, for the length DOI: 10.1021/la903429j
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Fernandes et al. Table 2. Main Sample Parameters Determined from the Analysis of the X-ray Diffraction Data for Samples (3,4,5)C20C60, (3,4,5)C16C60, and (3,4)C20C60
dlayer unit cell area unit cell volume unit cell electrons electronic density, Fe
Figure 7. Three-box electronic density profiles (a,c) and comparison (b,d) between corresponding model (red line) and experimental (blue line) X-ray diffraction pattern for sample (3,4,5)C16C60 (a,b) and (3,4)C20C60 (c,d) in the mesophase. Molecular arrangement cartoon drawn to help visualization. The density profiles (a,c) differ only in the length of the carbon-chain section (dcc) which is 10.7 and 10.2 A˚, respectively, for samples (3,4,5)C16C60 and (3,4)C20C60. Table 1. Parameters Describing Each Section of the Electronic Density Profiles That Explain the Experimental X-ray Diffraction Data of the Three Samples connector section
two C60 section -
0.43 e /A˚ electronic density, Fe perpendicular length, d 17.0 A˚
3
-
carbon-chain section -
0.26 e /A˚
4.0 A˚
14.0 A˚, (3,4,5)C20C60 10.7 A˚, (3,4,5)C16C60 10.2 A˚, (3,4)C20C60
3
of the carbon-chain section, deviating slightly from the values obtained for the models that explain the diffraction patterns of the two samples, 10.7 and 10.2 A˚, respectively. We have seen above that out of 60 carbon-chain segments, roughly 8 (corresponding to 66.5 electrons, considering each chain segment has 8 electrons, 6 from the carbon and 2 from the hydrogen atoms) are in the C60 and connector sections and therefore should not contribute to the ratio. The corrected formula is then dcc ¼ 14:0
ncc - 8 Å 60 - 8
ð6Þ
Using the previous equation we get dcc_(3,4,5)C16C60 = 10.8 A˚, very close to the 10.7 A˚ obtained from the diffraction data. This good agreement suggests that the unit cell planar area is the same for samples (3,4,5)C20C60 and (3,4,5)C16C60, as expected because the samples differ only slightly in the length of the carbon chains. As required by charge conservation, the 96 electron difference (corresponding to four chain segments per chain) between the two samples is compatible with the number of electrons obtained by the product of the molecular area, carbon-chains density, and half the layer length difference (3.3 A˚). Calculating eq 6 for sample (3,4)C20C60 gives an even smaller value dcc_(3,4)C20C60 = 8.6 A˚, in disagreement with 4344 DOI: 10.1021/la903429j
(3,4,5)C16C60
(3,4)C20C60
53.0 A˚ 114.5 A˚2 6068.5 A˚3 1872 0.309 e-/A˚3
46.4 A˚ 114.5 A˚2 5312.8 A˚3 1680 0.316 e-/A˚3
45.4 A˚ 107.8 A˚2 4898.7 A˚3 1550 0.316 e-/A˚3
the longer length, ∼10.2 A˚, obtained from the diffraction data. Assuming the density of the carbon chains remains constant, this relative elongation must be compensated by an area decrease. In fact, eq 6 should only be valid if the molecular area does not change, which is not the case when the samples have one chain less and the C60 attractive interaction is able to pull the molecules closer together, winning against the reduced steric repulsion from the fewer carbon chains. Based on the density profile and the number of electrons, the molecular area of sample (3,4)C20C60 is ∼107.8 A˚2, 6% smaller than that for the other two samples and corresponding to a square with an ∼10.4 A˚ edge. We note further that eq 6 is expected to be a reasonable approximation for the carbonchain length of similar samples varying only slightly in the number of carbon-chain segments. The number of electrons obtained by calculating the product of the electronic density of each section by its length and area for samples (3,4,5)C20C60 (1872 e-), (3,4,5)C16C60 (1676 e-), and (3,4)C20C60 (1550 e-) is very close to the number obtained by counting the electrons in two molecules, constituting a unit cell, see Table 2, as required.
5. Conclusions
0.22 e /A˚
3
(3,4,5)C20C60
Analysis and modelization of highly resolved X-ray diffraction patterns of alkylated C60 liquid crystalline samples differing in number and length of the alkyl tails enabled the detailed characterization of their molecular organization. The molecules arrange themselves in bilayers with a fluidlike homeotropic tail alignment relative to the plane of the layers. A simple, three-box model is proposed for the perpendicular electronic density profile that, inspired by the molecular structure, includes the division of the layer profile in three sections (Figure 4), with the C60 heads occupying a length of ca. 17.0 A˚, the connector section occupying 4.0 A˚, and the carbonchains section occupying a length that depends on the number and length of the carbon chains in a liquid state. The corresponding X-ray diffraction spectra, determined with the Parratt algorithm, are able to reproduce reasonably well the major features, namely, the relative peak intensities of the very rich experimental X-ray diffraction data. Density profile analysis also indicates that the liquid carbon chains partially occupy the other sections of the profile, as expected. Detailed analysis indicates that reducing the length of the chains from 20 ((3,4,5)C20C60) to 16 ((3,4,5)C16C60) results in shorter unit cells with the same planar area. Reducing the number of chains from 3 ((3,4,5)C20C60) to 2 ((3,4)C20C60) results in a shorter but also slightly thinner unit cell. This suggests that the planar area of the unit cell results from the balance between the entropic, repulsive carbon-chain interactions and the attractive C60 interactions and that it can be controlled by adjusting the number and/or length of the carbon chains. This is an important point, as more densely packed C60 moieties are expected to result in higher electron carrier mobility, of fundamental importance in applications. Further experimental Langmuir 2010, 26(6), 4339–4345
Fernandes et al.
work should be performed to study in more detail the planar C60 arrangement and its dependence on the number and length of the carbon chains. More importantly, the analytical approach followed can be also applicable to other systems to obtain detailed characterization of the molecular arrangement.
Langmuir 2010, 26(6), 4339–4345
Article
Acknowledgment. This work was supported, in part, by a Grant-in-Aid from the Ministry of Education, Sciences, Sports, and Culture, Japan, and PRESTO, JST, Japan (T.N.). We thank Dr. Y. Shen (MPI), Prof. D. G. Kurth (W€urzburg Univ.), and Dr. M. Takeuchi (NIMS) for helpful discussions.
DOI: 10.1021/la903429j
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