2 FIG. 1: : n FIG. 2: : n (Ch h ) N T B Ch h n(z) = (sin ϵ cos ω(z), sin ϵ sin ω(z), cos ϵ), (1) 1968 Meyer [5] 50 N T B Ch h [4] N T B 10 nm Ch h 1 µ

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1 : (Dated: February 5, 2016), (Ch), (Oblique Helicoidal) (Ch H ), Twist-bend (N T B ) I. (chiral: ) (achiral) (n) (Ch) (N ) 1996 [1] [2] 2013 (N T B ) [3] 2014 [4] (oblique helicoid) Electronic address: URL:

2 2 FIG. 1: : n FIG. 2: : n (Ch h ) N T B Ch h n(z) = (sin ϵ cos ω(z), sin ϵ sin ω(z), cos ϵ), (1) 1968 Meyer [5] 50 N T B Ch h [4] N T B 10 nm Ch h 1 µm

3 3 FIG. 3: N T B Ch h [4] II. F = F ani + F co. (2) 1 2 F ani = k B T f L (n(r 1 ) Ω 1 ) ln 4πf L (n(r 1 ) Ω 1 )dr 1 dω f L (r 1, Ω 1 )f L (r 2, Ω 2 )U LL (r 1, Ω 1 ; r 2, Ω 2 )dr, (3) 2 r Ω f L (cos θ) U LL [6, 7] U LL (r 1, Ω 1 ; r 2, Ω 2 )/k B T = c L (Ω 1 Ω 2 ˆr 12 )P 1 (Ω 1 Ω 2 ) +ν L P 2 (Ω 1 Ω 2 ), (4) P 1 (x) = x, P 2 (x) = (3/2)(x 2 1/3), r 12 r 1 r 2, ˆr 12 r 12 / r c L 2 c L

4 4 (Ω 1 Ω 2 ˆr 12 ) c L (4) 1 Twist c L (4) 1 Twist N T B Ch h ( (2)) 2 (n) p = (0, 0, 1) : (n p) 2 F co = γ p α Q αβ (r)p β dr. (5) Q αβ p α p (α(= x, y, z)) γ [J/m 3 ] γ (E) E 1[V/µm] Twisting Power, S L y = sin 2 ϵ, (6) (I) S L = 0, (N) S L > 0, y = 0, (N ) S L > 0, y = 1, N T B S L > 0, 0 < y < 1, III. 4 (γ L a 3 γ/k B T ) (T ) 2 1 (CP) (TCP) pn(i) N T B N [3] N T B N N

5 5 FIG. 4: (γ L a 3 γ/k B T ) (T ) N T B p K 33 ±3.6d 0 S L a 2 γ, (7) d 0 K 33 a N T B p = 10 nm K 33 /a 2 γ 100 (a 2 γ) 1/100 N T B Ch h Q 0 Q 0 c L /ν L = 2πd 0 /p 0 p 0 N p 0 = 10 nm, d 0 = 0.3 nm Q 0 = 0.1 Q 0 6 N T B Ch h 5 n H: ( ) 2 n p N T B Ch h Q 0 N T B Q 0 > 0.1 Ch h Q 0 < 0.01

6 6 FIG. 5: IV. N T B N T B K 22 K 33 K 22 /K 33 > 0 N T B [8] Ch h Meyer [5] Dozov N T B K 33 < 0 [9] N T B [1] T. Niori, T. Sekine, J. Watanabe, T. Furukawa, and H. Takezoe, J. Mater. Chem. 6, 1231 (1996). [2] J. Thisayukta, H. Niwano, H. Takezoe, and J. Watanabe, J. Am. Chem. Soc. 124, 3354 (2002). [3] V. Borshch, Y. -K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, Nat. Commun. 4, 2635 (2013). [4] J. Xiang, S. V. Shiyanovskii, C. Imrie, and O. D. Lavrentovich, Phys. Rev. Lett. 112, (2014). [5] R. B. Meyer, Appl. Phys. Lett. 12, 281 (1968). [6] Y. R. Lin-Liu, Y. M. Shih, C. W. Woo, and H. T. Tan, Phys. Rev. A 14, 445 (1976). [7] A. Matsuyama, J. Chem. Phys. 139, (2013); accepted in Liq. Cryst. (2016). [8] G. Barbero, L. R. Evangelista, M. P. Rosseto, R. S. Zola, and I. Lelidis, Phys. Rev. E, 92, (R) (2015). [9] I. Dozov, Europhys. Lett. 56, 247 (2001).

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