Various Dodecagon Assembly Manners as the Basis of Dodecagonal Quasicrystalline Tiling from Pentablock Quarterpolymers of the <i>AB</i>₁<i>CB</i>₂<i>D</i> Type

Various assembly manners of dodecagons composed of triangles and squares have been observed in the transmission electron microscope images of microphase-separated ternary blends of pentablock quarterpolymers of the <i>AB</i>₁<i>CB</i>₂<i>D</i> type in the bulk. <i>N</i>₃/<i>N</i>₄ ratios, where <i>N</i>₃ and <i>N</i>₄ are the numbers of triangles and squares, respectively, in the observed area, are all confirmed to be close enough to the theoretical value of 4/<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:math> ≑ 2.31 for the dodecagonal quasicrystalline state irrespective of the dodecagon assembly fashion and also of the number density of dodecagons. The appearance of the hierarchically scaled-up tiles is recognized for several types of assembly, that is, the 2 + <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:math> ≑ 3.7₂ times and the (2 + <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:math>)² ≑ 11.3₇ times magnified tiles in their edge lengths are found, where the scale factor of 2 + <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:math> is associated with the inflation operation. However, the areas in which the self-similarity holds are fairly narrow and limited, whereas their phason strains are reasonably small. Thus, the dodecagon assembly manners of the present sample have turned out to be far from the theoretically ideal tiling, while they rather have the feature of random tilings.