Compressing Θ-Chain in Slit Geometry

When compressed in a slit of width D, a Θ-chain that displays the scaling of size R₀ (diameter) with respect to the number of monomers N, R₀ ∼ aN^1/2, expands in the lateral direction as R_|| ∼ aN^ν(a/D)^2ν-1. Provided that the Θ condition is strictly maintained throughout the compression, the well-known scaling exponent of Θ-chain in two dimensions, ν = 4/7, is anticipated in a perfect confinement. However, numerics shows that upon increasing compression from R₀/D < 1 to R₀/D ≫ 1, ν gradually deviates from ν = 1/2 and plateaus at ν = 3/4, the exponent associated with the self-avoiding walk in two dimensions. Using both theoretical considerations and numerics, we argue that it is highly nontrivial to maintain the Θ condition under confinement because of two major effects. First, as the dimension is reduced from three to two dimensions, the contributions of higher order virial terms, which can be ignored in three dimensions at large N, become significant, making the perturbative expansion used in Flory-type approach inherently problematic. Second and more importantly, the geometrical confinement, which is regarded as an applied external field, alters the second virial coefficient (B₂) changes from B₂ = 0 (Θ condition) in free space to B₂ > 0 (good-solvent condition) in confinement. Our study provides practical insight into how confinement affects the conformation of a single polymer chain.