Studying the Expansion Kinetics of a Single Bead-Spring Chain Under Theta Conditions in 3D and 2D Spaces

Expansion of a polymer chain released from a confined state under theta conditions is investigated by means of Langevin dynamics simulations in both three- and two-dimensional geometries. Using a bead-spring chain representation, the θ-temperature is first identified through a scaling analysis of the chain size that exhibits the expected power-law behavior. Extensive simulations then carried out at this temperature reveal that the time evolution of the average chain size during expansion displays sigmoidal curves on log-log plots, indicating a two-stage process: an initial rapid power-law expansion from the confining size, followed by a slower exponential relaxation toward the final equilibrium size. By examining the regularity of these curves, we determine the scaling laws that govern the characteristic times and expansion exponents in each stage. The 2D expansion is found to proceed with slower dynamics than the 3D case, owing to the restriction of the available chain motion on a plane. Plotting the expansion rate against chain size provides a direct test of the kinetic equations, producing master evolutionary trajectories for both stages. Integrating these kinetic equations yields the scaling forms of the free energy for each stage, from which several relationships between the scaling exponents are obtained. This study demonstrates that incorporating local steric constraints and prohibiting segment crossing is essential for accurately describing chain expansion kinetics at the theta point, thereby exposing the limitations of idealized freely jointed chain models.