Polymer chains in confined geometries: Massive field theory approach

The massive field theory approach in fixed space dimensions $d<4$ is applied to investigate a dilute solution of long-flexible polymer chains in a good solvent between two parallel repulsive walls, two inert walls, and for the mixed case of one inert and one repulsive wall. The well-known correspondence between the field theoretical ${\varphi }^4$ $O\left(n\right)$-vector model in the limit $n\to 0$ and the behavior of long-flexible polymer chains in a good solvent is used to calculate the depletion interaction potential and the depletion force up to one-loop order. In order to make the theory UV finite in renormalization-group sense in $3\le d<4$ dimensions we performed the standard mass renormalization and additional surface-enhancement constants renormalization. Besides, our investigations include modification of renormalization scheme for the case of two inert walls. The obtained results confirm that the depletion interaction potential and the resulting depletion force between two repulsive walls are weaker for chains with excluded volume interaction (EVI) than for ideal chains because the EVI effectively reduces the depletion effect near the walls. Our results are in qualitative agreement with previous theoretical investigations, experimental results, and with the results of Monte Carlo simulations.

Figure 1

The functions $\Theta \left(y\right)$ and $\Gamma \left(y\right)$ for two repulsive walls.

Figure 2

The functions $\Theta \left(y\right)$ and $\Gamma \left(y\right)$ for one repulsive and one inert wall with and without excluded volume interactions (EVI).

Figure 3

The functions $\Theta \left(y\right)$ and $\Gamma \left(y\right)$ for two inert walls with and without EVI. Here we introduced notations: $\delta c_{semi}=\delta c_i$ and $\delta c_i^{S-S}=\delta c_{slit}$, with $i=1,2$.

Figure 4

The functions $\Theta \left(y\right)$ and $\Gamma \left(y\right)$ for two repulsive walls in comparison to [14].

Figure 5

Comparison of our theoretical results with Monte Carlo simulations for a trapped chain between two repulsive walls. The plots Ideal chain (exact) and EVI (wide slit) represent the results of our calculations. RW (MC) and the first SAW (MC) are due to the estimated asymptotic behavior in the narrow slit limit by [20] for random walks and self avoiding walks. The second SAW (MC) are the results obtained by [19].

Figure 6

Comparison of approximated theoretical results with experimental observation due to [5].