Scaling Law for Sequence-Induced Demixing of Compositionally Identical Copolymers

The critical incompatibility of polymers with different compositions scales inversely with their length. For instance, a mixture of A and B homopolymers of length <i>N</i> segregates at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:math>. But what if the difference between the blend components is subtler? We demonstrate that a mixture of AB copolymers with identical composition ─ equal amounts of A and B monomers ─ but different primary sequences can still phase separate. Incompatibility arises from distinct positional correlations between monomers of different chains. Calculating the Gaussian fluctuation correction to the free energy reveals that critical incompatibility from sequence differences follows a distinct yet universal scaling with chain length, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msqrt><mml:mi>N</mml:mi></mml:msqrt></mml:math>. This power law holds for both regular-sequence and statistical copolymers. A closed-form expression is derived for blends of block-alternating chains. The new theoretical scaling is confirmed by coarse-grained simulations, offering important insights into multiphase coexistence in biomolecular condensates.