Investigating the kinetics of single-chain expansion upon release in theta conditions

The free expansion of a confined chain in theta solvents following a sudden removal of the confining constraint is investigated using Langevin dynamics simulations in both two- and three-dimensional spaces. The average evolution of the chain size exhibits a sigmoidal transition between the confined and the free states on a logarithmic timescale, indicating a two-stage expansion, each characterized by its own timescale. A kinetic theory is developed by applying Onsager's variational principle, which balances the change in free energy with energy dissipation. Through scaling analysis, the characteristic time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi> <mml:mn>1</mml:mn></mml:msub> </mml:math> for the first expansion stage is shown to scale as the cube of the initial chain size, while the chain size increases according to a power law with an exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>α</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> , independent of the spatial dimension. In the second stage, the timescale <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:math> is found to be proportional to the square of the chain length, and the evolution of the chain size follows an exponential recovery function powered by an exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>α</mml:mi> <mml:mn>2</mml:mn></mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn></mml:mrow> </mml:math> . These results are further validated by a direct analysis of the kinetic equations via simulations. Moreover, the general forms of the free energy for the two expansion stages are established through the integration of the kinetic equations. Finally, physical interpretations are proposed, employing a radial expansion model and a diffusive mechanism to explain the observed scaling behaviors. This work explores a model system under the specific solvent condition, providing foundational theory and enhancing our understanding of the expansion-upon-release phenomenon.