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Quasicrystals (materials with long range order but without the usual spatialperiodicity of crystals) were discovered in several soft matter systems in thelast twenty years. The stability of quasicrystals has been attributed to thepresence of two prominent length scales in a specific ratio, which is 1.93 forthe twelve-fold quasicrystals most commonly found in soft matter. We proposedesign criteria for block copolymers such that quasicrystal-friendly lengthscales emerge at the point of phase separation from a melt, basing ourcalculations on the Random Phase Approximation. We consider two block copolymerfamilies: linear chains containing two different monomer types in blocks ofdifferent lengths, and ABC star terpolymers. In all examples, we are able toidentify parameter windows with the two length scales having a ratio of 1.93.The models that we consider that are simplest for polymer synthesis are, first,a monodisperse A_L B A_S B melt and, second, a model based on random reactionsfrom a mixture of A_L, A_S and B chains: both feature the length scale ratio of1.93 and should be relatively easy to synthesise.

Design of linear block copolymers and ABC star terpolymers that produce two length scales at phase separation