Analysis of the Size of Two-Component C60-C70 Fullerene Whiskers

Trans. Mat. Res. Soc. Japan 43[4] 229-232 (2018)

Toshio Konno1, Takatsugu Wakahara2, Chika Hirata2, Kun’ichi Miyazawa3 and Kazuhiro Marumoto1,4* 1 Division of Materials Science, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan 2 National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan 3 Imaging Section (IMG), Okinawa Institute of Science and Technology Graduate University (OIST), 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan. 4 Tsukuba Research Center for Energy Materials Science (TREMS), University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan *Corresponding author. Fax: 81-29-853-4490. E-mail address: [email protected]

C60-C70 two-component fullerene whiskers were synthesized by a liquid-liquid interfacial precipitation (LLIP) method using various ratios of C70 to C60, and were analyzed for their dimensions using scanning electron microscopy (SEM). The diameters and the lengths of the C60- C70 fullerene whiskers show ever-increasing W-type graphs which have two local minimal values, respectively. The shape of the graphs were resulted from changing of crystal growth rate with varying of fullerene compositions of mother solution. The carcinogenicity of C60-C70 fullerene whiskers can be evaluated using the C60-C70 fullerene whiskers whose dimensions and physical factors are specified, and they will be applied to various products such as the active layer materials of solar cells and the anchor materials. Key words: C60-C70 two-component fullerene whisker, Solid solutions, Liquid-liquid interfacial precipitation method, Toxicity, Size analysis

1. INTRODUCTION or hearts [911]. It was reported that the toxicity of multi- Fullerene needle-like crystals with the aspect ratios of 3 walled carbon nanotubes is changed by some factors or greater and the diameters less than 1000 nm are called including diameter, length, rigidity, and surface fullerene nanowhiskers (FNWs) [1,2]. The FNWs are modification [12]. Furthermore, carcinogenicity of composed of fullerene molecules such as C60, C70, fullerene nanofibers are considered to depend on their dimensions derivatives, and endohedral metallofullerenes [14]. and biodegradability according to the theory by Stanton and FNWs can be applied to bulk-hetero-junction type Pott [13,14]. Hence, it is important to reveal dimensions and electrode materials and quantum-dot type electrode biodegradability of FNWs for evaluation of their toxicity. materials as thin, long and efficient n-type semiconductors With regard to biodegradability, although untreated FNWs or light and flexible superconducting wire rods by alkali- composed of C60 (C60 NWs) were reported to show high metal doping [5,6]. Fullerene whiskers composed of C60 and biodegradability and weak cytotoxic effects [15,16], the C70 (C60-C70 fullerene whiskers) are characterized by the long C60 NWs hardened by a heat treatment under vacuum structure analysis and composition analysis using a focused conditions were reported to exhibit lower biodegradability ion beam processing apparatus (FIB-SEM), scanning than that of the untreated C60 NWs and abilities to induce electron microscopy (SEM), X-ray diffraction (XRD), the Nod-like receptor pyrin domain containing 3 (Nlrp3)- high-performance liquid chromatography (HPLC), Raman mediated IL-1b secretion which leads to inflammation spectroscopy and ultraviolet–visible (UV–vis) [17,18]. With regard to dimensions, we have investigated spectroscopy, and found to be useful semiconductor the diameters and lengths of C60 NWs by varying the materials with the composite effects of high Young’s solution volume, the ratio between good and poor solvents, moduli, large specific strength, small amount of and the water amount [19,20]. rhombohedral phase, the solid solubility limit of 14 mass% It is necessary to control diameters and lengths of C60- C70 or 27 mass% C60, and optical properties of both C60 and C70 fullerene whiskers for evaluation of their C70 [7,8]. They can be applied to various products such as carcinogenicity by living-body test using C60-C70 fullerene the active layer materials of solar cells featuring the optical whiskers whose dimensions and physical factors are properties of both C60 and C70 and the anchor materials revealed. In present paper, we investigated the variation of featuring the large specific strength and the high Young’s the dimension of the C60-C70 fullerene whiskers as a moduli. function of the fullerene compositions in the mother On the other hand, it is necessary to evaluate the toxicity solutions in order to consolidate science of size control of of FNWs for their application. The toxicity of carbon the whiskers. They were demonstrated to exhibit the materials has been evaluated. In human bodies, there are a diameters more than 250 nm and the lengths more than 2 lot of black spots observed in both lungs due to the µm and can be used as standard samples for evaluation of inhalation of carbon particles from the air. Several studies their carcinogenicity by living-body test and could be suggest that accumulated carbon particles in lungs are applied to various products such as the active layer proportionally associated with mortality by cancers of lungs materials of solar cells and the anchor materials.

229 230 Analysis of the Size of Two-Component C60-C70 Fullerene Whiskers

2. EXPERIMENTAL The C60-C70 fullerene whiskers were synthesized by a modified liquid-liquid interfacial precipitation (LLIP) method [21]. Fullerene powders (MTR Ltd. C60 99.5%, MTR Ltd. C70 98+%) with various ratios between C60 and C70 were dissolved in toluene (WAKO JIS special grade) by ultrasonic agitation (Iuchi VS-150). Total fullerene concentration was 2.5 mg to 1 ml of toluene when the fullerene compositions in the mother solutions ranged from 0 to 50 wt% C70. On the other hand, total fullerene concentration was 1.4 mg ml−1 when the fullerene compositions in the mother solutions ranged from 70 to 100 wt% C70. The solutions were filtered using syringe filters with 450 nm pores (MITSUBA HIGH GRADE SYRINGE, Whatman 25 mm GD/X) to generate the C60-C70 toluene solutions. 4 ml of each mother solution was stored in glass bottles (volume: 10 ml, inner diameter: 18 mm) at 15 °C using a water bath (AS ONE UCT-1000). An equal volume of 2- propanol (WAKO JIS special grade), whose temperature was set to be 15 °C (SANYO MIR-153) in advance, was slowly layered along the inside wall of the bottle onto a mother solution to form a liquid-liquid interface. After forming the interface, the bottle was manually mixed 30 times to achieve homogeneous precipitation, then kept at 15 °C for 2 days to make C60-C70 fullerene whiskers grown. Then, the supernatant solution was removed, and 2- propanol was poured into the glass bottle to stabilize the precipitates. After vacuum filtration (KIRIYAMA 5B-21, ULVAC DTC-21), the precipitates were vacuum-dried (AS ONE VO-300) at 100 °C. The measurements of the size of specimens were performed using a field-emission scanning electron microscope (FE-SEM; Hitachi NB5000). Fullerene composition in the mother solution is determined using high-performance liquid chromatography (HPLC; Figure 1. The SEM images of the C60-C70 fullerene JASCOUV-2070, PU2089, CO-2065, LC-NetII/ADC). whiskers. The compositions of mother solutions were (a) C60-0 wt% C70, (b) C60-10 wt% C70, (c) C60-20 wt% C70, 3. RESULTS AND DISCUSSION (d) C60-30 wt% C70, (e) C60-40 wt% C70, (f) C60-50 wt% C70, (g) C60-70 wt% C70, (h) C60-80 wt% C70, (i) C60-90 Figure 1 shows the SEM images of the synthesized C60- wt% C70 and (j) 100 wt% C70, respectively. C70 fullerene whiskers with various morphologies. The images show the formation of fine needle-like crystals (nanowhiskers) in (a)(d), and much larger needle-like FNWs and those of C70 FNWs when the fullerene crystals in (e)-(j), respectively. composition in the mother solutions is 535 wt% C70, and Figures 2 and 3 show the diameters and the lengths of the the measured diameters of the C60-C70 fullerene whiskers are more than both the critical diameters of C60 FNWs and C60-C70 fullerene whiskers as a function of the fullerene those of C70 FNWs when the fullerene composition in the composition in the mother solutions (wt% C70), respectively.   The C60-C70 fullerene whiskers exhibited diameters of 250 mother solutions is 45 50 wt% C70 or 80 100 wt% C70. nm to 7 µm and lengths of 2 µm to 50 µm. They can be used These results from Figure 4 approximately correspond to to living-body test following the theory by Stanton and Pott the measured values of lengths of the C60-C70 fullerene for evaluating their carcinogenicity. whiskers in Figure 3. Thus, the lengths of the C60-C70 Here, there are the critical diameters of fullerene fullerene whiskers were affected by the diameters of the whiskers, which are the threshold values and determine the C60-C70 fullerene whiskers. lengths of fullerene whiskers [20]. It was reported that the Figures 2 and 3 show ever-increasing W-type behaviors lengths of fullerene whiskers are a few µm when the which have two local minimal values and one local diameters of fullerene whiskers are below the critical maximal value, respectively. diameter, and are unlimited and very large when the Degree of supersaturation (S) is defined as bellows. diameters of fullerene whiskers exceed the critical diameter. Figure 4 shows the relationship among the measured values S = , (1) 𝐶𝐶 − 𝐶𝐶eq of diameters of the C60-C70 fullerene whiskers, theoretical 𝐶𝐶eq critical diameters of C60 FNWs, and those of C70 FNWs. In where C means concentration of fullerenes in the mother Figure 4, the measured diameters of the C60-C70 fullerene solutions and Ceq means concentration of fullerenes in the whiskers are less than both the critical diameters of C60 saturated solutions. In this paper, as S = 1425 using C of

Toshio Konno et al. Trans. Mat. Res. Soc. Japan 43[4] 229-232 (2018) 231

2. EXPERIMENTAL The C60-C70 fullerene whiskers were synthesized by a modified liquid-liquid interfacial precipitation (LLIP) method [21]. Fullerene powders (MTR Ltd. C60 99.5%, MTR Ltd. C70 98+%) with various ratios between C60 and C70 were dissolved in toluene (WAKO JIS special grade) by ultrasonic agitation (Iuchi VS-150). Total fullerene concentration was 2.5 mg to 1 ml of toluene when the fullerene compositions in the mother solutions ranged from 0 to 50 wt% C70. On the other hand, total fullerene concentration was 1.4 mg ml−1 when the fullerene compositions in the mother solutions ranged from 70 to 100 wt% C70. The solutions were filtered using syringe filters with 450 nm pores (MITSUBA HIGH GRADE SYRINGE, Whatman 25 mm GD/X) to generate the C60-C70 toluene solutions. 4 ml of each mother solution was stored in glass bottles Figure 2. The diameters of the C60-C70 fullerene Figure 5. The number density of the C60-C70 fullerene (volume: 10 ml, inner diameter: 18 mm) at 15 °C using a whiskers as a function of the fullerene composition in whiskers in the solution as a function of fullerene water bath (AS ONE UCT-1000). An equal volume of 2- the mother solutions (wt% C70). composition in the mother solutions (wt% C70). propanol (WAKO JIS special grade), whose temperature was set to be 15 °C (SANYO MIR-153) in advance, was slowly layered along the inside wall of the bottle onto a mother solution to form a liquid-liquid interface. After forming the interface, the bottle was manually mixed 30 times to achieve homogeneous precipitation, then kept at 15 °C for 2 days to make C60-C70 fullerene whiskers grown. Then, the supernatant solution was removed, and 2- propanol was poured into the glass bottle to stabilize the precipitates. After vacuum filtration (KIRIYAMA 5B-21, ULVAC DTC-21), the precipitates were vacuum-dried (AS ONE VO-300) at 100 °C. The measurements of the size of specimens were performed using a field-emission scanning electron microscope (FE-SEM; Hitachi NB5000). Fullerene composition in the mother solution is determined using high-performance liquid chromatography (HPLC; Figure 1. The SEM images of the C60-C70 fullerene JASCOUV-2070, PU2089, CO-2065, LC-NetII/ADC). whiskers. The compositions of mother solutions were (a) C60-0 wt% C70, (b) C60-10 wt% C70, (c) C60-20 wt% C70, Figure 3. The lengths of the C60-C70 fullerene whiskers Figure 6. Theoretical value of crystal growth rate of C60 3. RESULTS AND DISCUSSION (d) C60-30 wt% C70, (e) C60-40 wt% C70, (f) C60-50 wt% as a function of the fullerene composition in the mother and C70 as a function of the fullerene composition in the C70, (g) C60-70 wt% C70, (h) C60-80 wt% C70, (i) C60-90 70 Figure 1 shows the SEM images of the synthesized C60- solutions (wt% C70). mother solutions (wt% C ). wt% C70 and (j) 100 wt% C70, respectively. C70 fullerene whiskers with various morphologies. The images show the formation of fine needle-like crystals (nanowhiskers) in (a)(d), and much larger needle-like FNWs and those of C70 FNWs when the fullerene crystals in (e)-(j), respectively. composition in the mother solutions is 535 wt% C70, and Figures 2 and 3 show the diameters and the lengths of the the measured diameters of the C60-C70 fullerene whiskers are more than both the critical diameters of C60 FNWs and C60-C70 fullerene whiskers as a function of the fullerene those of C70 FNWs when the fullerene composition in the composition in the mother solutions (wt% C70), respectively.   The C60-C70 fullerene whiskers exhibited diameters of 250 mother solutions is 45 50 wt% C70 or 80 100 wt% C70. nm to 7 µm and lengths of 2 µm to 50 µm. They can be used These results from Figure 4 approximately correspond to to living-body test following the theory by Stanton and Pott the measured values of lengths of the C60-C70 fullerene for evaluating their carcinogenicity. whiskers in Figure 3. Thus, the lengths of the C60-C70 Here, there are the critical diameters of fullerene fullerene whiskers were affected by the diameters of the whiskers, which are the threshold values and determine the C60-C70 fullerene whiskers. lengths of fullerene whiskers [20]. It was reported that the Figures 2 and 3 show ever-increasing W-type behaviors lengths of fullerene whiskers are a few µm when the which have two local minimal values and one local diameters of fullerene whiskers are below the critical maximal value, respectively. diameter, and are unlimited and very large when the Degree of supersaturation (S) is defined as bellows. diameters of fullerene whiskers exceed the critical diameter. Figure 4. The diameters of the C60-C70 fullerene Figure 7. Theoretical value of crystal growth rate of the Figure 4 shows the relationship among the measured values S = , (1) whiskers and critical diameters of both C60 FNWs and C60-C70 fullerene whiskers as a function of the fullerene 𝐶𝐶 − 𝐶𝐶eq of diameters of the C60-C70 fullerene whiskers, theoretical C70 FNWs as a function of the fullerene composition in composition in the mother solutions (wt% C70). 𝐶𝐶eq critical diameters of C60 FNWs, and those of C70 FNWs. In where C means concentration of fullerenes in the mother the mother solutions (wt% C70). FNWs means fullerene nanowhiskers. Figure 4, the measured diameters of the C60-C70 fullerene solutions and Ceq means concentration of fullerenes in the whiskers are less than both the critical diameters of C60 saturated solutions. In this paper, as S = 1425 using C of

2 3

232 Analysis of the Size of Two-Component C60-C70 Fullerene Whiskers

-1 -1 1.42.5 g l and Ceq of 0.1 g l , crystal nucleus were stable References after nucleation [22]. Hence, crystal size, especially [1] K. Miyazawa (Ed.), Fullerene Nanowhiskers, Pan Extended Configurational Polyhedra Based on Graph Representation for Crystalline Solids diameters of C60-C70 fullerene whiskers was dependent of Stanford Publishing Pte. Ltd., Singapore, 2011. number density of C60-C70 fullerene whiskers [23]. The [2] K. Miyazawa, J. Nanosci. Nanotechnol., 9, 41-50 Koretaka Yuge1 number density of the C60-C70 fullerene whiskers in the (2009). 1 Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan solution is shown in Figure 5. Figure 5 exhibit ever- [3] K. Miyazawa, Y. Kuwasaki, A. Obayashi, and M. decreasing M-type behaviors which have two local Kuwabara, J. Mater. Res., 17, 83-88 (2002). We propose theoretical approach based on combination of graph theory and generalized Ising model (GIM), maximal values and one local minimal value, respectively. [4] T. Wakahara, Y. Nemoto, M. Xu, K. Miyazawa, and which enables systematic determination of extremal structures for crystalline solids without any information The number density of C60-C70 fullerene whiskers was D. Fujita, Carbon, 48, 3359-3363 (2010) about interactions or constituent elements. The conventional approach to find such set of structure typically considered to depend on crystal growth rate (R) at the early [5] H. Takeya, K. Miyazawa, R. Kato, T. Wakahara , T. employs configurational polyhedra (CP) on configuration space based on GIM description, whose vertices can time of the precipitation, which is defined as followings. always be candidates to exhibit maximum or minimum physical quantities. We demonstrate that the present [24] Ozaki, H. Okazaki, T. Yamaguchi, and Y. Takano, Molecules, 17, 4851-4859 (2012). approach can construct extended CP whose vertices not only include those found in conventional CP, but also [6] H. Takeya, R. Kato, T. Wakahara , K. Miyazawa, T. include other topologically and/or configurationally characteristic structures on the same dimensional configu- R = a Vs v (C - Ceq) exp( ), (2) ration space with the same set of figures composed of underlying lattice points, which therefore has significant − 𝐸𝐸des Yamaguchi, T. Ozaki, H. Okazaki, and Y. Takano, B advantage over the conventional approach. where a means height of fullerene𝑘𝑘 𝑇𝑇 molecules, Vs means Mater. Res. Bull., 48, 343-345 (2013). volume of surfaces of crystals available for precipitation, v [7] D. Matsuura, T. Konno, T. Wakahara, K. Miyazawa, and T. Kizuka “Young’s Modulus of C60/C70 Alloy means thermal vibration frequency of fullerene molecules, I. INTRODUCTION formation about vertices that are originally found at low- Edes means desolvation energy of fullerene molecules, kB Nanowhiskers”, Abstracts of 2014 Tsukuba dimensional space based on low-dimensional figures. In order means Boltzmann constant, and T means temperature. In Nanotechnology Symposium (TNS’14), July 25-26, to overcome this problem in GIM-based conventional CP, the this paper, a is approximately 1 nm, v is about 109 s-1, (C – 2014, University of Tsukuba, Tsukuba, Japan. In classical systems where physical quantity (especially, 24 -3 present study proposes construction of CP based on extended Ceq) is 13 × 10 m at the early time of the precipitation, [8] T. Konno, T. Wakahara, and K. Miyazawa, J. Cryst. dynamical variables) can be a linear map for structures, a set 20 -1 graph representation unified with GIM-based description for Edes per one molecule is 1.2× 10 J, kB is 1.38 J K , T is Growth, 416, 41-46 (2014). of special structure that has maximum or minimum physical crystalline solids, whose spectrum not only contains informa- 288 K, and Vs at the early time of the precipitation greatly [9] C. A. Pope, R. T. Burnett, M. J. Thun, E. E. Calle, D. quantity should always be restricted by spatial constraint on -15 -11 3 tion about GIM pair correlations, but also includes higher- changed from 10 to 10 m with varying of fullerene Krewski, K. Ito, G. D. Thurston, , JAMA, 287, 1132– constituents of given system. This can be quantitatively deter- compositions of mother solution. Using above values, R of dimensional figures (or links) consisting of the corresponding 1141 (2002). mined by the landscape of so-called configurational polyhe- C60 and C70 are calculated to be shown in Figure 6. Vs of pairs. Such information of higher-dimensional (higher-order) [10] D. Wang, Y. Minami, Y. Shu, S. Konno, T. Iijima, Y. dra (CP),1 which determines maximum and minimum value C70 increased at 13.7 wt% C70 which is the solid solubility Morishita, M. Noguchi, , Cancer Sci., 94, 707–711 links can be considered of great significance to further ad- limit of C70 in the matrix phase of C60 because C70 was able that basis functions for describing structures (typically, gen- (2003). 2 dress macroscopic properties that do not simply corresponds to precipitate at surfaces of most crystals. By the same eralized Ising model (GIM) description is employed) can [11] M. C. Turner, D. Krewski, C. A. Pope, Y. Chen, S. M. to linear map with respect to microscopic states on configura- token, Vs of C60 increased at 73.4 wt% C70 (26.6 wt% C60) take: Such special structures always locate on vertices of the Gapstur, M. J. Thun, Am. J. Respir. Crit. Care Med., tion space, including diffusion and deformation behavior for which is the solid solubility limit of C60 in the matrix phase CP. Since the landscape of the CP can be determined without 184, 1374–1381 (2011). crystalline solids.6 We will demonstrate that the proposed CP of C70 [8,24]. Thus, R of C70 have the maximal value at 13.7 any information about energy or constituents, we can a priori [12] S. Toyokuni, Adv. Drug. Deliv. Rev., 65, 2098–2110 retains characteristic vertices found for conventional CP, and wt% C70 and R of C60 exhibit the maximal value at 73.4 know a set of candidate structure to exhibit extremal phys- (2013). also can figure out other special structures in terms of graph, wt% C70. As theoretical value of R of the C60-C70 fullerene ical quantities when condition of spatial constraint on con- [13] M. F. Stanton, M. Layard, A. Tegeris, E. Miller, M. on the same low-dimensional space considered. whiskers is determined by sum of R of C60 and R of C70 to stituents of the system is given. Using this characteristics, ef- be shown in Figure 7, R of the C60-C70 fullerene whiskers May, E. Morgan, and A. Smith, J. Natl. Cancer Inst., ficient prediction of alloy ground-state structures have amply exhibits local maximum at around 13.7 wt% C70 and 73.4 67, 965-975 (1981). wt% C70. We consider that the behavior of theoretical [14] F. Pott, U. Ziem, F. J. Reiffer, F. Huth, H. Ernst, and been investigated so far, and the concept of the CP has re- 3 II. METHODOLOGY values of R corresponds to the behavior of the number U. Mohr, Exp. Toxcol. Pathol., 32, 129-152 (1987). cently been extended to finite-temperature properties, where density of C60-C70 fullerene whiskers. Therefore, we configurational density of states (CDOS) along CP-based spe- [15] S. Nudejima and K. Miyazawa, J. Okuda, A. Very recently, we successfully construct unified representa- consider that the diameters of the C60-C70 fullerene cial coordination well characterize temperature dependence of Taniguchi, Adv. Biomed. Res., 89-94 (2010). tion of microscopic structure on periodic lattice (i.e., atomic whiskers in Figure 2 depended on R of the C60-C70 fullerene internal energy near order-disorder phase transition. Very re- [16] J. Okuda-Shimazaki, S. Nudejima, S. Takaku, K. configuration) in terms of both GIM and graph description. whiskers, and the lengths of the C60-C70 fullerene whiskers Kanehira, S. Sonezaki, and A. Taniguchi, Health, 2, cently, importance of the role of spatial constraint is also con- were dependent of R of the C60-C70 fullerene whiskers and Explicit relationship between GIM and graph representation 1456-1459 (2010). firmed for finite-temperature properties by the authors, where 7 the diameters of the C60-C70 fullerene whiskers. for given figure R on given lattice is expressed by [17] K. Miyazawa, T. Mashino, T. Suga, J. Mater. Res., 18, dynamical variables in thermodynamically equilibrium state

2730-2735 (2003). of multicomponent alloys can be well-characterized by a few N R [18] H. Cui, W. Wu, K. Okuhira, K. Miyazawa, T. Hattori, specially-selected microscopic states established from the in- 1 t 4. CONCLUSION ψ (⃗σ) = ρ− Cm Tr Bl (⃗σ) Bl (⃗σ) , (1) K. Sai, M. Naito, K. Suzuki, T. Nishimura, Y. formation of spatial constraint.4,5 ⟨ ⟩R R ∑ ·  ∑ ±  We have synthesized the C60-C70 fullerene whiskers, which m (l R ) Sakamoto, A. Ogata, T. Maeno, A. Inomata, D. Nakae, ∈ were investigated by the SEM method to conduct the size The practical problem for using CP is that number of A. Hirose, and T. Nishimaki-Mogami, Biochem.   analysis. They exhibited diameters of 250 nm to 7 µm and its vertices exponentially increases at high-dimensional con- where left-hand side corresponds to multisite correlation for lengths of 2 µm to 50 µm and can be used as standard Biophys. Res. Commun., 452, 593–599 (2014). figure R in GIM description. Matrix B denotes upper triangu- [19] K. Miyazawa, C. Hirata, and T. Wakahara, J. Cryst. figuration space considered. It is thus fundamentally im- samples to living-body test for evaluating their lar part of extended graph Laplacian, defined by Growth, 405, 68-72 (2014). portant to find out as many extremal structures as possi- carcinogenicity or will be applied as the active layer ble at low-dimensional space based on geometrically low- materials of solar cells and the anchor materials. The [20] K. Miyazawa and K. Hotta, J. Cryst. Growth, 312, dimensional figures, such as symmetry-nonequivalent pairs (α) ϕp (σi)ϕq (σ j)(p,q α,i, j R,i < j), diameters and the lengths of the C60-C70 fullerene whiskers 2764-2770 (2010). B (i, j)= ∈ ∈ (2) on lattice. However, it has been shown1 that a set of struc- R ( ) as a function of the fullerene composition in the mother [21] K. Miyazawa, A. Obayashi, and M. Kuwabara, J. Am. {0√ otherwise solutions showed the ever-increasing W-type behavior, Ceram. Soc., 84, 3037-3039 (2001). ture at vertices of CP is invariant with linear transformation of which were ascribed to be caused by the nucleation with the [22] N. Ueda, J of JSCM, 43, 612-621 (1970) (in Japanese). the coordination within given subspace, which directly means where p and q denotes conventional GIM basis function in- crystal growth rate which exhibits local maximum at around [23] P. P. Weimarn, Chem. Rev., 2, 217-242 (1925). that when we would like to further figure out extremal struc- dex on each lattice point i and j, and σk is the pseudo-spin t 13.7 wt% C70 and 73.4 wt% C70. [24] Y. Saito, “Kessyou seityou”, Syoukabou, Tokyo, tures based on the conventional CP approach, we should ex- variable to specify atomic occupation. A = B + B corre- (2002). plicitly include information about longer-range and/or higher- sponds to adjacency matrix. NR is the dimension of figure,

dimensional figures on lattice, but such CP projected onto Cm is integer depending on the type of figure, and ρR denotes (Received March 2, 2018; Accepted June 7, 2018; desired low-dimensional space typically lose significant in- number of the possible path to construct considered figure R. 4 Published Online August 1, 2018)