1. Introduction
Statistical physics properties of single polymer chains can be significantly influenced or even determined by geometrical confinements and applied external forces [
1,
2,
3,
4]. A detailed understanding of the behaviors of polymers under such circumstances is still considered to be an unsolved problem in polymer physics after more than half a century. However, even so, advances in the study of geometrically and potentially constrained polymers are playing a role in the development of many existing nanotechnologies of genomics and materials science, etc. [
5,
6,
7].
For polymers under confinement, the effects of constraints have usually been classified into three regimes (weak, moderate, and strong confinements), which are distinguished in terms of the comparison between the polymer’s unconfined radius of gyration and the Kuhn length of the typical confinement length scale. In the regime of weak confinement, Casassa [
8] has discussed the free energy of ideal chains trapped in pores with different shapes based on the theory of diffusion. Then, de Gennes and his coworkers [
9,
10] developed the so-called blob model to describe the confinement of a long, flexible polymer. According to this model, the free energy of confinement is given by
Here,
H is the length scale of confinement,
is the Boltzmann constant, and
T is the absolute temperature.
is the radius of gyration of the unconfined polymer,
N is the number of monomers,
is of the order of the monomer-monomer separation, and
v equals 1/2 for ideal polymers and approximately 3/5 for polymers with excluded volume. A widely used model of a polymer with bending energy is the wormlike chain (WLC) model, characterized by the contour length
L and persistence length
Lp, which was first proposed by Kratky and Porod in 1949 [
11]. In the strong confinement regime, Odijk [
12,
13] has argued that polymer behavior can be interpreted in terms of
statistically independent segments with deflection length
. Based on this picture, he [
12,
13,
14] obtained expressions of the confinement free energy,
F, and the average extension of the chain,
, in terms of
as
where
represent the free energy and extension-associated deflection lengths, respectively:
[
12] has been suggested for the confinement of a cylindrical tube with diameter
D. For the confinement of a rectangular tube with height and width
Hh and
Hw, the deflection length associated with the free energy calculation has been suggested as [
15,
16]
In contrast, this deflection length associated with the average extension is given as
We note that Equations (4) and (5) have different expressions and prefactors. The prefactors in Equations (4) and (5) were determined by using various numerical techniques and theoretical derivations, such as the Monte Carlo simulations [
17,
18] and eigenvalue technique associated with the Fokker–Planck equations [
15,
19]. Examples of the determined prefactors are illustrated in
Table 1. We can see that the prefactors
and
, respectively determined from free energy and extension, have an almost 10 times difference in quantity.
In addition, a slit of separation
H can be regarded as a rectangular tube with height
Hh =
H and infinite width
. Statistical properties of polymer chains confined in the slit have been extensively studied [
20,
21,
22,
23,
24,
25] based on Monte Carlo simulations and eigenvalue analysis. The deflection length in a strong confinement regime has been confirmed to follow the Odijk scaling law,
Although the deflection length in Equation (5) can be viewed as the combination of that for two slits with heights
Hh and
Hw, respectively [
15], it can be observed from Equations (4)–(6) that the deflection length in Equation (5) is not consistent with Equation (6) as
.
Beyond the Odijk regime, Chen [
24] numerically calculated the confinement free energy by treating the problem of a confined polymer as an eigenvalue problem. He also suggested an interpolating formula that can have very good agreement with that of the numerical calculations for polymers under both strong and weak confinements. In addition, an extended de Gennes regime [
20,
26,
27] has also been identified based on the Monte Carlo simulations.
Interestingly, external forces can pose similar effects to the statistical behaviors of single polymer chains as geometrical confinements. For a polymer chain to be stretched by a sufficiently large force,
, a deflection length also exists and can be expressed as [
3,
28]
, where
, so that the force-extension relation can be expressed as
Polymers in real microenvironments are usually subjected to both geometrical constraints and external forces. Wang and Gao [
29] have revealed that the average extension of a strongly tube-confined and force-stretched polymer chain can be equivalent to that of an unconfined chain subjected to an effective stretching force. Li and Wang [
30] later confirmed that this equivalence property is still valid for the tube-confined polymers in a much more extended Odijk regime. Therefore, for a semiflexible polymer chain in the deflection confinement regime, one can generally have
where one can set
or
as the normalized effective force due to the existence of strong confinement: For the confinements of rectangular tubes,
,
are given by Equations (4) and (5).
However, as shown above, Odijk deflection lengths based on free energy and extension are different. Then a critical question arises. Which deflection length should be used if the polymer chain is under both geometrical confinement and force stretching? Therefore, there are still open questions on how the Odijk length can be uniquely and precisely defined for polymer chains confined in rectangular tubes.
In this study, for semiflexible polymer chains confined in rectangular tubes and slits, we derived a modified deflection length, which was expected to be valid for a more extended range than the classical Odijk deflection length. This extended deflection length was directly used to quantitatively formulate both energy and extension. We will present numerical calculations based on the eigenvalue technique developed by Chen and coworkers [
19,
31,
32] and Brownian dynamics simulations in terms of the generalized bead-rod (GBR) model [
30,
33,
34] to justify our theoretical predictions.