Superhelix Structure in Helical Conjugated Polymers Synthesized in

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Superhelix Structure in Helical Conjugated Polymers Synthesized in an Asymmetric Reaction Field Taizo Mori and Kazuo Akagi* Department of Polymer Chemistry, Kyoto University, Kyoto 615-8510, Japan ABSTRACT: The superhelix structure of helical polyacetylene (H-PA) synthesized in a chiral nematic liquid crystal (N*LC) used as an asymmetric reaction field was mathematically specified and presented as a chemically graspable feature. Fibril and spiral morphologies of H-PA were clarified as forming cylindrical and conical helices, respectively. The torsion of the H-PA main chain as the primary structure induced the formation of higher-order helical structures, such as fibril and spiral morphologies. The formation mechanism of H-PA in N*-LC was investigated in detail from the relationship between the hierarchical helical structure of H-PA and N*-LC. The superhelix structure was also used to rationalize the different formation mechanisms for helical aromatic conjugated polymers, such as helical poly(m-phenylene) (PMP). It was found that the curvature and torsion are essential for distinguishing the helical structures of H-PA and PMP. The present approach, based on conventional mathematical formulas, is useful for describing the helical structures of several types of conjugated polymers.

1. INTRODUCTION The left and right hands are mirror images to each other, but they are not superimposable. Objects of opposite chirality are called enantiomers. Numerous objects that range from elementary particles to the cosmos have been confirmed to be chiral.1 The physical processes involved in natural phenomena usually result in bilateral symmetry. The mirror image of a reaction appears to occur at the same rate as the original reaction, but only L-amino acids are found in proteins.2 In particle physics, charge conjugation and parity (CP) violations violate the postulated CP symmetry observed during the beta decay of cobalt-60 nuclei.3 The origin of chirality is an interesting topic in science. A chiral curve is called a spiral or helix. Specifically, a spiral is a curve that emanates from a central point.4 Hurricanes and typhoons show spiral patterns, and their rotation directions are mirror images in the Northern and Southern Hemispheres, respectively. A helix is a line that curves and rises around a central line,5 such as in the nucleic acid double helix.6 An almost univalve shell has a right-handed shell, but this mechanism has not been explained yet.7 Recently, the handedness of the univalve shell has been clarified to be controlled through the addition of mechanical stimulation to the embryo.8 Although a spiral and a helix are distinct technical terms, they are commonly confused with each other. In this study, the spiral is defined as a twodimensional curve and the helix is defined as a three-dimensional curve. Archimedean and logarithmic spirals are representative examples.9 The Archimedean spiral, such as the groove of gramophone records, has a constant separation distance. The logarithmic spiral is a special type of spiral curve that often appears in nature. The shell of a nautilus and the alignment of sunflower seeds show logarithmic spirals. Clothoids are also © 2013 American Chemical Society

widely used as transition curves in railroad engineering or highway engineering for connecting and transiting the geometry between tangent and circular curves.10 In the nucleic acid double helix, alpha helices in the secondary structure of proteins and coil springs are called cylindrical/circular helices. A conical helix is made by a univalve shell. It is possible to provide mathematical descriptions of spirals and helices. The DNA double helix is known to form a superhelix structure in which a helix is itself coiled into a helix.6 The superhelix structure of DNA can be evaluated from the twist rate of a nucleic acid. The hierarchic helical structure of a macromolecule might be mathematically specified; however, few reports have appeared in the literature of macromolecules having higher-order helical structures. We have synthesized helical polyacetylene (H-PA) in a chiral nematic liquid crystal (N*-LC) as an asymmetric reaction field.11,12 The main chains, fibrils, and fibril bundles of H-PA are twisted in a one-handed direction. The spiral morphology of HPA is the higher-order structure. Hierarchical structures are rarely observed in synthetic polymers. In this work, we devise a mathematical description of the hierarchical structure of H-PA and consider in more detail the formation mechanism of the HPA in a N*-LC. We expect that this study may be useful in elucidating the formation mechanism of not only helical conjugated polymers but also other types of helical polymers. Received: May 25, 2013 Revised: July 31, 2013 Published: August 22, 2013 6699

dx.doi.org/10.1021/ma401092v | Macromolecules 2013, 46, 6699−6711

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2. GENERAL 2.1. Spiral. A spiral is defined as a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. A spiral can be easily described in polar coordinates. Some well-known spirals include the Archimedean spiral and the logarithmic spiral. The Archimedean spiral is described by eq 1-1. It is the locus of points that correspond to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. The spiral has a constant separation distance equal to 2πa. A concentric circle is not included in the Archimedean spiral. A logarithmic spiral can be described by eq 1-2 in a polar coordinate system. The distance between the turnings of a logarithmic spiral increases in geometric progression. The logarithmic spirals are self-similar in that they are self-congruent under all similarity transformations. When b = 0 in eq 1-2, the spiral becomes a circle of radius a. r = aθ

(1-1)

r = abθ

(1-2)

x = a cos t , y = a sin t , z = bt

(1-3)

A cylindrical helix is formed when a wire is wound around a cylinder. The wire is bent tangent to the cylinder. The crosssectional surface of the wire is twisted around the tangent lines as the axis of rotation. The curvature (κ) describes the degree to which something is curved. The torsion (τ) represents the degree of the twisted angle per unit length of a helix. A helix can be described by its curvature and torsion. The curvature and torsion are expressed with the Frenet−Serret formulas, shown in eq 1-4: ... det(r ̇, r ̈, r ) |r ̇ × r |̈ κ= , τ = (1-4) |r |̇ 3 |r ̇ × r |̈ 2 Figure 2 illustrates the curvature and torsion using rectangular blocks. Each block is considered to be a unit length of the helix.

The rotation direction of a spiral spreading out from a center is opposite to that of a spiral approaching the center. The handedness of the spiral is determined by the rotation direction of the spiral. If the rotation direction of the spiral is unknown or if the spiral remains stationary, the handedness of the spiral cannot be defined. In this study, the handedness of a static spiral is defined by viewing the spiral that spreads out from the center. In a two-dimensional plane, a spiral is enantiomorphic. In three-dimensional space, one spiral exhibits a left-handed direction when viewed from the top but exhibits a right-handed direction when viewed from the opposite side because the spiral can be turned over, as shown in Figure 1a. Consequently, spirals do not exhibit chirality in three-dimensional space.

Figure 2. (a−d) Schematic illustrations of curvature and torsion and of (e) a cylindrical helix.

The curvature and torsion are positive for a right-handed helix and are negative for a left-handed one. Figure 2a shows the arranged 10 blocks whose curvature and torsion are both 0° per step. Figure 2b shows a block twisted in a positive or righthanded direction by 20° from one end to the other; its curvature κ is 0 and its torsion is τ = 2° per step over 10 steps. When these blocks are extended to 360°, a right-handed ribbon is built. Figure 2c shows a block curved 20° to the left; its curvature is −2° and its torsion is 0 per step over 10 steps. When the blocks are copied by curvature κ = −2° and torsion τ = 0° per step over 360°, the block returns to an initial position with left-handed rotation, and the locus draws a cycle. Figure 2d shows a block that is both twisted and curved, i.e., curvature κ = −2° and torsion τ = 2° per step over 10 steps. Figure 2d shows the combination of Figures 2b,c. Figures 2a,c, which show a torsion τ = 0, show the same plane and are two-dimensional curves. However, Figures 2b,d, which show cases where the torsions are not equal to 0°, are three-dimensional curves. Figure 2d is bent over out-of-plane by several degrees. When the block of Figure 2d is copied with a curvature κ = −2° and torsion τ = 2° per step, a right-handed cylindrical helix is formed, as shown in Figure 2e. The curvature and torsion of the cylindrical helix defined by eq 1-3 are derived in eq 1-5 from the previously mentioned Frenet−Serret formulas (eq 1-4).

Figure 1. Schematic views of (a) spiral and (b) left-handed helices from both sides.

2.1. Helix. A helix is defined as a three-dimensional curve that turns around an axis at a varying distance while moving parallel to the axis. The screw direction of a helix is resolved by the righthand rule. Notably, the handedness of a helix has a different definition in certain areas, such as botany, optics, and electrical engineering. A right-handed helix cannot be turned or flipped to look like a left-handed one unless it is viewed in a mirror. One of most general helices is the cylindrical helix, which intersects the elements of a cylinder at a constant angle. The cylindrical helix can be described by eq 1-3. In nature, DNA and the α helix form cylindrical helices. Spiral steps and a helical coil spring are examples of cylindrical helices found in artifacts.

κ=

a , a + b2 2

τ=

b a + b2 2

(1-5)

When the radius r and gradient angle α of a cylindrical helix are defined by eq 1-3, the correlation between the parameters a, b, r, 6700


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and α is described by eq 1-6. As a result, the curvature and torsion of the cylindrical helix are rewritten as shown in eq 1-7: r = a , tan α = b/a κ=

(cos α)2 sin α cos α , τ= r r

and torsion of the conical helix 1-8 are given by eq 1-9. The screwed direction of the conical helix torsion depends on the parameter b. A right-handed conical helix has a positive torsion and a positive parameter b, which means that the handedness of the conical helix is the same direction as the torsion of the helix, as already shown for a cylindrical helix. As shown in Figure 3b, a right-handed conical helix, when viewed from the top site, shows a right-handed Archimedean spiral. However, this helix, when viewed from the bottom side, shows a left-handed Archimedean spiral. Thus, the handedness of the conical helix is the same as the spiral when viewed from the top site and exhibits the opposite handedness when viewed from the bottom side. In addition, the rotation direction of a cylindrical helix depends on the point of view. For instance, two opposite historical conventions are used in circular polarization. As shown in Figure 1d, the left-handed cylindrical helix, if regarded as circular polarization, is observed to rotate in the lefthand direction from the point of view of the source, but from the right-hand direction when viewed from the standpoint of the detector. Notably, the handedness of polarization is defined from the point of view of the source in electrical engineering, whereas it is viewed from the point of view of the receiver in optics. In this study, the handedness of a helix is defined according to the righthand rule used in mathematics and physics.

(1-6)

(1-7)

The gradient angle α is described in Figure 3a. When 0 < α < π/2 and π/2 < α < π, the handedness of the cylindrical helix is right-

Figure 3. (a) A cylindrical helix and (b) a conical helix.

and left-handed, respectively. The handedness of the cylindrical helix only depends on the torsion because the radius r is positive. The right-handed cylindrical helix has positive torsion, and the left-handed helix has negative torsion, which means that the handedness of the cylindrical helix is decided from the screw direction, i.e., the torsion. A conical helix is formed when a wire is wound around a circular cone. A spiral can be observed from the top of a conical helix. From the top view, univalve shells show the logarithmic spiral, and blades of drills show the Archimedean spiral. The conical helix is described by eq 1-8, which shows the Archimedean spiral from the top of the helix. The curvature

x = at cos t , y = at sin t , z = bt

(1-8)

⎡ a 2(2 − t 2)2 + b2(4 + t 2) ⎤1/2 κ = a⎢ ⎥ ⎣ (a 2(4 + t 2) + b2)3 ⎦ ⎡ a 2(1 + t 2) ⎤ τ = b⎢ 2 ⎥ ⎣ a (4 + t 2) + b2 ⎦

(1-9)

Figure 4. (a) SEM image of H-PA fibrils; (b) a cylindrical helix; (c) a schematic illustration of the formation mechanism of H-PA in N*-LC; (d) a model structure of the half-pitch in the N*-LC. 6701


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Figure 5. (a) SEM image of H-PA and (b) a model structure of H-PA.

3. RESULTS AND DISCUSSION 3.1. Correlation between the Helical Pitch of N*-LC and H-PA. As shown in Figure 4a, the fibril of H-PA with a springlike screw pattern has a cylindrical helix. The radius r, the gradient angle α, and the height ph of the cylindrical helix are shown in Figure 4b. The value of ph is equal to the helical pitch of the helix. The correlations between the parameters r, α, and ph are described in eq 2-1: p tan α = h (2-1) 2πr

The helical pitch ph of H-PA can be calculated using only the diameter (d) and the distance (L) of the fibril bundles of H-PA. Experimental and calculated values for the helical pitches of HPA (systems from 1 to 7) are summarized in Table 1. Figure 6 Table 1. Diameter (d) and Distance (L) of the Fibril Bundles and the Experimental and Calculated Helical Pitches (ph) of H-PA exptl value

We have previously discussed the formation mechanism of the H-PA in a N*-LC.12 The PA fibrils grow along the LC molecules, which direct the growth. The PA chain grows along the twisted LC molecules, particularly around the LC molecules arranged horizontal to the substrate, to give a helical chain. The angle of the N*-LC director, which is perpendicular to the substrate, is set to 0. In Figure 4c, the angle of the LC director in the center of the cylindrical helix is rotated by π/2 degrees. In Figure 4d, the rotation angle θ of the LC with radius r is equal to the gradient angle α. The correlation between the helical pitch p of the N*-LC and the radius r and gradient angle α is shown in eq 2-3, which is derived from eq 2-2, because the helical pitch is defined as the distance along the helical axis that results in one full turn (2π) of the helix:

α:

⎛p ⎞ p π = ⎜ − r ⎟: ⎝ ⎠ 4 2 4

α=

(2-3)

The helical pitch p of the N*-LC is corrected by the height ph of the cylindrical helix that corresponds to the helical pitch of HPA, as shown in eq 2-4: ⎛π 2π ⎞ 2πr ph = 2πr tan⎜ − ⎟= 2π r p⎠ ⎝2 tan p

( )

(2-4)

( d2πL )

L (μm)

ph (μm)

ph (μm)

0.68 ± 0.11 0.35 ± 0.02 0.53 ± 0.07 0.69 ± 0.04 0.45 ± 0.04 0.76 ± 0.07 0.34 ± 0.04

1.90 ± 0.13 1.08 ± 0.08 1.23 ± 0.06 1.44 ± 0.15 1.16 ± 0.13 1.53 ± 0.05 1.30 ± 0.13

3.51 ± 0.23 2.13 ± 0.02 2.01 ± 0.09 2.60 ± 0.27 2.15 ± 0.12 2.52 ± 0.27 2.26 ± 0.15

3.40 1.96 2.07 2.31 2.02 2.41 2.45

= [1.4772 + 1.3272 − 2 × 1.477 × 1.327 × cos(124.2°)]1/2 (2-6)

= 2.479

1 unit = 1.329 + 1.480 × sin(126.8° − 90°) = 2.216 (2-7)

The curvature and the torsion of H-PA were calculated on the basis of system 3. The radius and helical pitch of the fibril bundle in system 3 were 265 and 2000 nm, respectively. The contour length N* of the helix per one pitch was calculated to be 2600 nm

dπ tan

d (μm)

1 2 3 4 5 6 7

1 unit

The diameter d of the H-PA fibril bundles and the distance L between the fibril bundles are defined in Figure 5. As shown in Figure 5b, the half-helical pitch p/2 of N*-LC is approximately equal to the distance between the fibril bundles of H-PA; i.e., p/2 is equal to L. As a result, the helical pitch of H-PA ph is described as eq 2-5: ph =

system

shows scanning electron microscope (SEM) images of the HPAs of systems 1 to 7. The H-PA of systems 3 to 6 and that of 7 were synthesized using gravity flow and magnetic field methods, respectively. The calculated H-PA helical pitches coincide well with the experimental values. The fibril bundle of H-PA is confirmed to form a cylindrical helix. 3.2. Curvature and Torsion of H-PA. The curvature and the torsion of PA per unit cell can be calculated from the pitch and the radius of the fibril bundles of the H-PA fibrils. As shown in Figure 7, the C−C and CC bond distances are 1.477 and 1.327 Å, respectively, and the C−CC bond angle is 124.2° in the trans-transoid PA. The C−C and CC bond distances are 1.480 and 1.329 Å, and the C−CC bond angle is 126.8° in the cis-transoid PA. The unit cell length of the trans-transoid PA is 2.479 Å, as determined from eq 2-6, and that of the cis-transoid PA is 2.216 Å, as determined from eq 2-7. The unit-cell length of PA is assumed to be 2.3 Å (0.23 nm) because the cis-transoid content of the PA sample synthesized at room temperature is approximately 50%.

(2-2)

π 2πr − 2 p

calcd value

(2-5) 6702


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Figure 6. SEM images of H-PAs (systems 1 to 7) and molecular structures of nematic liquid crystals and chiral dopants.

Figure 7. (a) A cylindrical helix, (b) a developed figure of a cylindrical helix, and (c) the vector sum of curvature and torsion. Chemical structures of (d) trans-transoid and (e) cis-transoid PA.

using eq 2-8. Thus, system 3 has 1.1 × 104 (2600 nm/0.23 nm) units per one pitch. The curvature κ = 0.020° and torsion τ = 0.025° per unit cell were evaluated using eqs 1-7 and 2-9. The HPA curvatures and torsions in systems 1−7 are summarized in Table 2. These results indicate that the helicity of H-PA is quite

small; hence, H-PA can be regarded as having a very slightly twisted structure in its main chain.

Table 2. Radius (r) and Helical Pitch (ph), Gradient Angle (α) of the Fibril Bundle, Contour Length (N*), Units per One Pitch, Curvature (κ), and Torsion (τ) per Unit Cell system 1 2 3 4 5 6 7

r (μm) 0.34 0.18 0.27 0.35 0.23 0.38 0.17

ph (μm) 3.51 2.13 2.01 2.60 2.15 2.52 2.26

α (deg) 58.6 62.9 50.5 50.4 56.5 46.4 64.8

N* (μm)

unit

4.11 2.39 2.60 3.38 2.57 3.47 2.50

1.79 × 10 1.04 × 104 1.13 × 104 1.47 × 104 1.12 × 104 1.51 × 104 1.09 × 104 4

κ (deg)

τ (deg)

0.010 0.016 0.020 0.016 0.018 0.016 0.014

0.017 0.031 0.025 0.019 0.027 0.017 0.030

N* =

(2πr )2 + ph 2

2π = N*

k2 + τ 2

(2-8)

(2-9)

The radius (r) and helical pitch (ph) of the fibril bundle can be expressed with the curvature (κ) and torsion (τ) using eqs 1-6, 17, and 2-1 and Figure 7, as follows: r=

k τ , ph = 2π 2 2 k +τ k + τ2 2

(2-10)

Three-dimensional plots of the radius and helical pitch of the fibril bundle in terms of the curvature and torsion are shown in Figure 8. Note that curvature and torsion in Table 2 are expressed in degrees per unit cell, while those in eq 2-10 are in radians per unit 6703


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Figure 8. Three-dimensional plots of (a) radius, r, and (b) helical pitch, ph, as functions of curvature, κ and torsion, τ, by using eq 2-10.

Figure 9. Plots of radius r as functions of curvature, κ and torsion, τ, in the ranges of (a) [0 < κ, τ < 0.04] and (b) [0.01 < κ, τ < 0.035], by using eq 2-11. The gradient angles of 45° and 65°, evaluated from the equation of tan α = τ/ κ, are indicated in yellow lines. The radius of the fibril bundle vary in the range of tan α < 0.64, as shown in blue region of (a).

Table 2 shows that the value of the gradient angle (α) is in the range of 45°−65°. It is clear from eq 1-5, eq 1-6, and Figure 8 that the gradient angle can be expressed as a function of the curvature and torsion, tan α = τ/κ. The values corresponding to α = 45° (τ/ κ = 1) and 65° (τ/κ = 2.1) are drawn in yellow lines in Figures 10a and 11b. The relation 0 < p/2r < π holds true for H-PA, so the gradient (α) of H-PA is evaluated to be 32.7° < α < 90° using eq 2-3. This finding means that the radius and helical pitch of the HPA fibril bundle vary in the range of tan α (= τ/κ) < 0.64, which is in the blue regions in Figures 9a and 10a. In the region of 0.010 < κ < 0.020 and 0.017 < τ < 0.031, the radius of the fibril bundle decreases as the torsion increases, and the helical pitch decreases as the curvature increases, as shown in Figures 9b and 10b, respectively. In other words, when the fibril bundle is strongly twisted (the torsion is large), the fibril bundle becomes fine.

length. It is therefore required to transform eq 2-10 into eqs 2-11 and 2-12 when the values in Table 2 are used. r=

k 180 0.23 k 2 + τ 2 π 103

ph = 2π

τ 180 0.23 k 2 + τ 2 π 103

(2-11)

(2-12)

Plots of the radius and helical pitch of the fibril bundle, derived from eqs 2-11 and 2-12, are shown in Figures 9 and 10, respectively. The curvature and torsion in Figures 9a and 10a are in the range of 0 < κ, τ < 0.04, and those in Figures 9b and 10b are 0.01 < κ, τ < 0.035. The radius and helical pitch of the fibril bundles in systems 1−7 are indicated by green circles in the κ−τ planes of Figures 9 and 10. 6704


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Figure 10. Plots of helical pitch, ph, as functions of curvature, κ and torsion, τ, in the ranges of (a) [0 < κ, τ < 0.04] and (b) [0.01 < κ, τ < 0.035] (b), by using eq 2-12. The gradient angles of 45° and 65°, evaluated from the equation of tan α = τ/ κ, are indicated in yellow lines. The helical pitch of the fibril bundle vary in the range of tan α < 0.64, as shown in blue region of (a).

Figure 11. (a) POM image and (b) a model structure of N*-LC, (c) an SEM image, and (d) a schematic illustration of H-PA.

When the fibril bundle is fine (the curvature is large), the helical pitch of the fibril bundle becomes short. 3.3. Conical Helix of H-PA. Figure 11a shows POM images of the spiral morphology of the N*-LC that was prepared through the addition of (S)-(PCH506)2-Binol as a chiral dopant to an equal-molar mixture of the N-LCs, PCH302 and PCH304. This N*-LC is abbreviated as (S)-N*-LC. The (S)-N*-LC showed a right-handed morphology. Figure 11c shows an SEM image of the spiral morphology of H-PA synthesized in the (S)N*-LC, abbreviated as (S)-H-PA. The (S)-N*-LC showed a right-handed spiral morphology. The spiral direction of the (S)H-PA is the same as that of the (S)-N*-LC. The morphology of

H-PA appears to replicate that of the N*-LC during interfacial acetylene polymerization. The spiral morphology of N*-LC appears to form an Archimedean spiral because the spatial period of the N*-LC is equal to one-half the pitch. The spiral morphology of the N*-LC shows a double-spiral, as described by eq 3-1. The constant separation distance of the spiral is equal to πa. The helical pitch of the N*-LC is defined as p*, so that a = p*/2π. x = at cos|t |, y = at sin|t | 6705

(3-1)


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Figure 12. (a, d) AFM images of the spiral morphology of H-PA; (b, d) schematic illustrations of the spiral morphology of H-PA; and (c) a POM image of the spiral structures of N*-LC.

handed, as are the fibrils. The screw direction of the fibril (as torsion) and the handedness of the spiral are the same direction. The torsion of the conical helix derived from eq 1-9 coincides with the handedness of the helix. The (R)-N*-LC has a lefthanded double-spiral structure at the free surface. The (R)-N*LC forms a left-handed conical helix, but the screw direction of (R)-N*-LC is right-handed. The screw direction of (R)-N*-LC is opposite the handedness of the conical helix because the helical axis of H-PA is perpendicular to that of the N*-LC. As evident in Figure 11, the helical axis of H-PA is parallel to the PA chain, and the helical axis of N*-LC is perpendicular to the striae of the N*LC. As shown in Figure 13, the ribbon-like twisted structures are arranged perpendicular to the helical axes of the N*-LC. The helical pitch of the N*-LC and the width of the ribbon are defined as p and r, respectively. The stacked ribbons form a cylindrical helix. When p/2r < π, the cylindrical helix becomes right-handed (Figure 13a). However, when p/2r > π, the cylindrical helix is left-handed (Figure 13b). The N*-LC has a periodic structure with half-pitch (p/2). Thus, the p/2r is constantly smaller than π (p/2r < π) in the N*-LC. In this case, the handedness of the cylindrical helix that is formed by the N*LC is opposite the screw direction of the N*-LC. If the handedness of the cylindrical helix (p/2r < π) is assumed to be due to the torsion of the N*-LC spiral, the torsion is coincident with the handedness of the N*-LC conical helix. The screw directions of N*-LC and H-PA are summarized in Table 3. 3.4. Primary and Higher-Ordered Structures of H-PA. Figure 14 shows superhierarchical helical structures in H-PA. The H-PA film exhibits a Cotton effect in the region from 450 to 800 nm, which corresponds to a π → π* transition of the conjugated chain. The H-PA chain has been indicated to be twisted with a one-handed direction. A screw-like structure and a bundle of fibrils with the same screw direction as that of the fibril also exist. If the H-PA chain is assumed to be the primary

A two-dimensional curve such as an Archimedean spiral has no enantiomer in three-dimensional space. If the spiral of the N*LC consisted of two-dimensional curves, the N*-LC would show both right- and left-handed morphologies in the POM images. However, the (S)-N*-LC showed almost right-handed spirals. These results therefore suggest that the spiral morphologies of both the N*-LC and the H-PA are three-dimensional curves. Figure 12 shows AFM images of H-PA synthesized in the N*LC. The center of the H-PA spiral morphology is higher than the outer area. The difference in height between the center and outer part of the spiral morphology is approximately 2.5 μm, which suggests that the spiral morphology of H-PA has a conical helix that is convex upward at the surface. The H-PA fibrils are twisted around the conical helices, as shown in Figure 12b. These results suggest that the spiral of N*-LC also forms a conical helix because the H-PA replicates the morphology of the N*-LC. Pieranski et al. reported that cholesteric (Ch)-LC forms a focal conic structure with a conical helix at the free surface.13 They synthesized the siloxane oligomer, which induced the formation of side-chain-type Ch LC. The siloxane oligomer showed a Ch phase that transitioned to a glass-phase state when it was rapidly cooled. As a result, the spiral of the Ch-LC formed a conical helix that was convex upward at the surface. The N*-LC with a shorter helical pitch shows a focal conic structure similar to that of a smectic (Sm)-LC phase, which is characterized by fan-shaped and polygonal structures in POM images. The focal conic structure consists of a concentric conical formed with the layer structure of a Sm-LC. The half-helical pitch of the N*-LC has a pseudolayer structure similar to the layer structure of Sm-LCs. The focal conic structure of the N*-LC has a conical helix, and the layer of half-helical pitch of the helix shows a spiral morphology. The H-PA forms the conical helix, as in the case of N*-LC, because the H-PA is synthesized in the N*-LC. The spiral direction of the H-PA synthesized in the (R)-N*-LC is left6706


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Figure 13. Schematic illustrations of the cylindrical helix formed in N*-LC.

handedness of the H-PA chain correspond to the curvature and the torsion, respectively. The neighboring helical chains spontaneously form a helical fibril through van der Waals interactions. In addition, the neighboring helical fibrils are gathered through van der Waals integration to produce a bundle of fibrils with a helically twisted structure. Because the horizontally aligned LC molecules exhibit a spiral structure, the bundle of fibrils also forms a spiral morphology with domains of various size. The H-PA spiral with the conical helix corresponds to a quaternary structure. The various sizes of domains correspond to quinary structures. The higher-order structures, such as the secondary to quinary structures, give the characteristic spiral morphology, which is rarely observed in synthetic polymers. Such a hierarchical helical structure is attributed to the

Table 3. Screw Directions of Helices of N*-LC and H-PA N*-LC

chiral dopant (R)(PCH506)2Binol (S)(PCH506)2Binol

H-PA

twisted direction

screwed direction of cylindrical helix

screwed direction of conical helix

screwed direction of fibrils

screwed direction of conical helix

right

left

left

left

left

left

right

right

right

right

structure, the fibril and fibril bundle of the H-PA are regarded as secondary and tertiary structures, respectively. The bentness and 6707


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Figure 15. Model structures of helical poly(p-phenylene)s [PPPs].

as film, must be considered. As shown in Figure 16, when the right-handed PPPs are helically π-stacked in a right-handed manner, the phenylene rings are slid in a manner similar to the opening of a folded fan. However, when the right-handed PPPs are helically π-stacked in a left-handed manner, the phenylene rings are arranged parallel to each other. The letter arrangement is much more than stereospecifically favorable because “face-toface” interchain π−π stacking integrations between the phenylene rings effectively stabilize the self-assembled structures. The handedness of the primary structures is not the same as that of the higher-order structures in aromatic conjugated polymers. PMP has two types of structures: zigzag planar and helical structures. The PMP-bearing chiral side chains have been shown to form a cylindrical helix.15 As shown in Figure 17, the phenylene rings are twisted in a right-handed manner in the right-handed cylindrical helix of PMP. The torsion of PMP is the same as the handedness of PMP. When a single helical aromatic conjugated polymer forms a higher-order structure, such as a cylindrical helix, the handedness of the primary structure is the same as that of the higher-order structure. This relationship between torsion and handedness of a helical aromatic conjugated polymer such as PMP is coincident with that of H-PA. Clarification of how the torsion that determines the twisting direction of a primary structure affects the higher-order structure to result in a screwed or spiral morphology is important. The curvature and torsion of a PMP15b sample having a radius of 4.2 Å and a helical pitch of 3.7 Å were evaluated. Because PMP has an intrachain 61 helical structure where 6 phenylenes form one helix, the curvature (κ) of 59.4° and the torsion (τ) of 8.3° per repeat unit of 4.4 Å in length are calculated using eqs 1-7, 2-8, and 2-9. The schematic structure of PMP is shown in Figure 18. It is interesting to note that the large curvature and small torsion of PMP imply that it is a helicene-type helix. In fact, PMP forms an intrachain helically π-stacked structure where the torsion of the phenylene unit is small enough to allow effective π-stacking between the phenylene units. The results of the curvature and torsion calculations for PMP are consistent with the experimentally confirmed polymer structure. Assuming that the length (4.4 Å) of a repeat unit for PMP does not change, the radius and helical pitch of PMP can be expressed by eqs 4-1 and 4-2, respectively. Plots of the radius and helical pitch in terms of the κ−τ plane are shown in Figure 19.

Figure 14. Primary and higher-ordered structures of H-PA.

asymmetrical polymerization that results from the use of the N*LC as a reaction field. 3.5. Curvature and Torsion of Aromatic Conjugated Polymers. We next discuss helical aromatic conjugated polymers, such as poly(p-phenylene) [PPP]14 and poly(mphenylene) [PMP],15 focusing on the curvature and torsion. The handedness of PPP depends on the dihedral angle between the two phenylene rings. In this case, the left- and right-handed PPPs are defined as M(−)- and P(+)-helicities, respectively. Figure 15 shows ribbon-like twisted PPPs with dihedral angles of 30° per phenylene unit. The examination of a single chain of PPP in a dispersed state, such as dilute solution, may be sufficient for its characterization. However, not only a single chain of PPP but also a self-assembled stacked structure in a condensed state, such

r= 6708

k 180 × 4.4 k2 + τ 2 π

(4-1)


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Figure 16. Model structures of helically π-stacked right-handed PPPs.

Figure 17. Model structures of helically π-stacked poly(m-phenylene)s [PMPs].

When PMP has no torsion (τ = 0), the curvature is calculated to be 60° from eq 4-1 and the radius is 4.2 Å from eq 4-2. Therefore, PMP with τ > 0 should have a curvature and radius less than 60° and 4.2 Å, respectively. Figure 19 shows how the radius and helical pitch of PMP change in the region of κ < 60° and r < 4.2 Å. The blue area of the figure highlights these regions. When the torsion is large, the radius of the helix is small, although the change itself is relatively small, as shown in Figure 19a. Meanwhile, when the torsion is large, the helical pitch is also large, indicating that the twisting of the phenylene ring increases as the helical pitch becomes greater. This result suggests that as the helical pitch increases, the conjugation of the polymer is weakened, making it difficult to form the intrachain helically πstacked structure. Thus, when the torsion of PMP is large, the radius of the helix is small and the helical pitch is large. Similarly, when the torsion of H-PA is large, the radius of the fibril bundle is small, and when the curvature is large, the helical pitch is small. It is common for H-PA and PMP with large torsions to have large helical pitches. However, there is a difference in the dependence of the helical pitches between H-PA and PMP. This is attributed to the difference between the H-PA and PMP mechanisms of helical structure formation. It can be assumed from Figures 9 and 10 that H-PA has a gradient angle (α) in the range of 32.7° < α < 90° and that the radius and helical pitch of the fibril bundle vary in this range of α. On the other

Figure 18. Schematic illustration of poly(m-phenylene) [PMP] with 61 helix. The curvature (κ) of 59.4° and torsion (τ) of 8.3° are evaluated from the radius of 4.2 Å and helical pitch of 3.7 Å.

ph = 2π

180 τ × 4.4 2 k +τ π 2

(4-2)

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Figure 19. Plots of (a) radius, r and (b) helical pitch, ph, as functions of curvature, κ and torsion, τ, in the ranges of [50 < κ < 62, 0 < τ < 20], by using eqs 4-1 and 4-2.

hand, the helical structure of PMP is formed by intrachain πstacking between phenylene moieties that are substituted by chiral groups. Therefore, the unit length (4.4 Å) and the number of units per helical pitch (6) can be fixed, and the radius and helical pitch vary in the region of κ < 60° and r < 4.2 Å (see Figure 19). Because the regions for the curvature and torsion are different between H-PA and PMP, those for the radius and helical pitch are also different. In other words, while the gradient angle is fixed in H-PA, the unit length and the number of units per helical pitch are fixed in PMP. Such a difference gives rise to a differential dependence of the helical pitch on the curvature and torsion between H-PA and PMP. Here, it is worthy to emphasize that the curvature (κ ≈ 0.02°) and torsion (τ ≈ 0.03°) per repeat unit in H-PA are much smaller than the corresponding values (κ ≈ 59.4° and τ ≈ 8.3°) for PMP. This result can be rationalized by the difference in the helical pitch, i.e., the degree of helical twisting. Because H-PA is synthesized in N*-LC bearing a helical pitch on the micrometer order, the helical pitch of H-PA is similar in magnitude to that of N*-LC. As a result, the helical pitch of H-PA has approximately 10 000 repeat units (see Table 2). Meanwhile, the meta-linkage of the phenylene moieties and the chiral side chains in PMP allow for substantial bending and twisting of the phenylene moieties, respectively, and also stabilize the intrachain helical π-stacking between the phenylene moieties. Thus, PMP has a helical pitch on the order of angstroms, and the helical pitch of PMP has 6 repeat units (61 helix). This explains why the curvature and torsion per repeat unit in PMP are much larger than those in HPA. It can be argued that the values of curvature and torsion are useful for characterizing the shapes of helical structures of various types of conjugated polymers.

The superhelix structure of H-PA was used to rationalize the formation mechanism of helical aromatic conjugated polymers, such as helical PMP. The difference between H-PA and PMP formation mechanisms of helical structures is elucidated. The curvature and torsion were essential for distinguishing the helical structure of H-PA from that of PMP. The present approach based on conventional mathematical formulas is useful for describing the helical structures of various types of conjugated polymers and might be feasible for other types of helical polymers.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (K.A.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Dr. Mutsumasa Kyotani, Tsukuba Research Center for Interdisciplinary Materials Science, University of Tsukuba, for measurements of SEM. This work was supported by a Grants-in-Aid for Science Research (S) (No. 20225007) and (A) (No. 25246002) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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REFERENCES

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4. SUMMARY The superhelix structures of conjugated polymers were investigated through SEM observations of H-PAs. We found that the fibril bundles of H-PAs form a cylindrical helix. The cylindrical helix of H-PA coincides with the formation mechanism of H-PA in the N*-LC used as an asymmetric reaction field. We clarified that the spiral morphology of H-PA forms a conical helix, similar to the case of N*-LC. The H-PA has hierarchical helical structures, including primary, secondary, tertiary, and even higher-order structures. The torsion of the HPA chain that occurs in the primary structure induces higherorder helical structures, such as fibrils, fibril bundles and spiral morphologies. Such helical structures result from the use of the N*-LC as a reaction field in the asymmetrical polymerization. 6710


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