and where \( \lambda \) = 0.69 is a critical exponent.1 This expres-sion can easily be modified to account for the case of a channel in a wall of finite thickness.45 To account for the effect of confinement of the monomers inside the cavity, we consider the case where this part of the polymer is in the semi-dilute regime. Here, the confined portion of the polymer can be viewed as a collection \( n _ { \mathrm { t } } \) blobs, each of size \( \xi \sim \sigma \phi ^ { \nu / ( 1 - 3 \nu ) } \), where \( \nu \) is the Flory exponent. It is easily shown that the confinement free energy scales as23 \( \Delta F _ { \mathrm { c } } / k T = n _ { \mathrm { b } } \approx ( V / \sigma ^ { 3 } ) ^ { - 1 / ( 3 \nu - 1 ) } m ^ { 3 \nu / ( 3 \nu - 1 ) } \), where m is the number of monomers in the cavity. For a finite length nanochannel that is spanned by an average of \( n _ { \mathrm { p } } \) bonds, we note that \( m = s - n _ { \mathrm { p } } / 2 \). For the pore length \( L \) = 1.3 used here, we estimate \( n _ { \mathrm { p } } \, = \, 2 \) when the pore is filled. Thus, the total free energy, F, is expected to satisfy

\[ ( F - F _ { 0 } ) / k T \approx ( V / \sigma ^ { 3 } ) ^ { - 1 . 2 5 } ( s - n _ { \mathrm { p } } / 2 ) ^ { 2 . 2 5 } , \qquad ( 8 ) \]

where we have used \( \nu \) =35. This approximation is valid when \( n _ { \mathrm { p } } / 2 \, < \, s \, < \, N \, - \, 1 \, - \, n _ { \mathrm { p } } / 2 \). Over most of the range of s, \( n _ { \mathrm { p } } \) is constant, the logarithmic term \( F _ { 0 } \) is negligible. Only near the upper and lower bounds of s is the variation of \( F _ { 0 } \) appreciable. This feature, as well as a partial emptying of the pore, accounts for the dips in F near \( s = N - \) 1. Otherwise F is dominated by the confinement free energy of the cavity. In the case of fixed V , F is predicted to increase with s independent of the polymer length N. In addition, the variation of F with s has positive curvature, and F increases with decreasing V . These predictions are qualitatively consistent with the data.

The use of the semi-dilute approximation to estimate free energy of confinement in the cavity is expected to be valid only over a restricted range of densities. In Ref. 48 it was shown that the predictions are valid only up to packing fractions of \( \phi \, \approx \) 0.15, where \( \phi ~ \equiv ~ \pi N \sigma ^ { 3 } / 6 V \). Beyond this value the number of monomers per blob is unacceptably low. A lower limit on \( \phi \) is imposed by the condition that the number of blobs, \( n _ { \mathrm { b } } = N \dot { \phi } ^ { 1 / ( 3 \nu - \ddot { 1 } ) } \), be sufficiently large. For the polymer lengths considered in this work, it is difficult to find a range of s that satisfies both conditions simultaneously. To analyze the data, we use a more relaxed condition for low density and consider the case where \( n _ { \mathrm { b } } \ge 3 \).

Figure 3 shows the results of fits using Eq. (8) for \( N \)= 121 and various cavity volumes. The free energy functions have been shifted by \( F _ { 0 } ( s ) \), which is the free energy function for a flat wall that was calculated ex-plicitly by simulation. We use a fitting function of the form \( F - F _ { 0 } = c _ { 0 } + c _ { 1 } ( s - 1 ) ^ { - \alpha } \). The lower limit of the range of the fit is indicated in the plot, while the fitting function is deliberately extrapolated beyond the upper limit of the fitting range to illustrate the divergence of the prediction from the simulation results at high den-sity. The upper limit itself is explicitly labeled for two of the functions. Note that scaling predictions underes-timate the free energy in the region where \( \phi \geq \) 0.15. This is consistent with the observation noted in Ref. 48 that the confinement free energy crosses over into a concen-trated regime where the excluded volume interactions are screened, which leads to a higher value of \( \alpha \). From the fit to the data in the valid range, it was found that \( ^ { \prime \prime } \)=1.9, 2.0 and 1.9 for V =150, 250 and 500, respectively. This is somewhat below the predicted value of \( ^ { o } \) = 2.25. Not-ing the expected dependence of \( c _ { 1 } \propto V ^ { - \beta } \) where \( \beta \)=1.25, the ratio of \( c _ { 1 } \) measured for V =500 and \( V \)=250 yields \( \beta \)= 0.9, while the ratio for V =500 and V =150 yields \( \beta \)=1.1. Thus, the measured values of the exponents \( \alpha \) and \( \beta \) both deviate from the predicted values. Undoubt-edly, this is arises from failing to properly satisfy the condition that \( n _ { \mathrm { b } } \gg 1 \). We speculate that better sat-isfying this condition would yield improved agreement. However, this would necessitate using polymer chains at least an order of magnitude larger than those considered here, which is not feasible for us at present.

FIG. 3. Free energy functions for polymers of length \( N \) = 121 and a spherical cavity of various volumes. The dashed black lines show fits to the data in the region for which \( n _ { \mathrm { b } } \ge 3 \) and \( \phi < \) 0.15. The lower limit of this range is evident from the minimum s for the fitting curves, and the upper end of this range is explicitly labeled for two of the curves. For V =500, the upper limit extends beyond the range of the data.

Next we consider the effects of the anisometry of the confining cavity. Figure 4 shows free energy functions for chains of length \( N \)= 101 in a cavity of volume \( V \)= 500. Figure 4(a) shows results for prolate ellipsoidal cavities (\( r > \) 1) and Fig. 4(b) shows results for oblate cavities (\( r < \) 1). In each case, the result for spherical cavities is also shown, for comparison. The most notable feature here is the fact that deviations from spherical symmetry in either direction lead to an increase in the free energy. Note, however, that the curves for oblate cavities follow the opposite trend at low s (see the inset of Fig. 4(b)), in contrast to the case for prolate cavities. The explana-tion for this difference is straightforward. As the first few monomers enter the cavity, the effects of confinement are felt mainly by the curvature of the cavity wall near the