“window potential” of a chosen functional form. The form of this potential is given by:

\[ W _ { i } ( s ) = \left\{ \begin{array} { l l } { \infty , \qquad s < s _ { i } ^ { \mathrm { m i n } } } & { } \\ { 0 , \qquad s _ { i } ^ { \mathrm { m i n } } < s < s _ { i } ^ { \mathrm { m a x } } } & { } \\ { \infty , \qquad s > s _ { i } ^ { \mathrm { m a x } } } & { } \end{array} \right. \qquad ( 7 ) \]

where \( s _ { i } ^ { \mathrm { m i n } } \) and \( s _ { i } ^ { \mathrm { m a x } } \) are the limits that define the range of s for the i-th window. Within each “window” of s, a probability distribution \( p _ { i } ( s ) \) is calculated in the simu-lation. The window potential width, \( \Delta s \equiv s _ { i } ^ { \mathrm { m a x } } - s _ { i } ^ { \mathrm { m i n } } \), is chosen to be sufficiently small that the variation in F does not exceed a few \( k _ { \mathrm { B } } T \). Adjacent windows overlap, and the SCH algorithm uses the \( p _ { i } ( s ) \) histograms to re-construct the unbiased distribution, \( \mathcal { P } ( s ) \). The details of the histogram reconstruction algorithm are given in Ref. 46. A description for an application to a physical system comparable to that studied here is presented in Ref. 32.

Polymer configurations were generated carrying out single-monomer moves using a combination of trans-lational displacements and crankshaft rotations. The trial moves were accepted with a probability \( \begin{array} { r l } { p _ { \mathrm { a c c } } } & { { } = } \end{array} \)\( \operatorname* { m i n } ( 1 , e ^ { - \Delta E / k _ { \mathrm { B } } T } ) \), where \( \Delta F \) is the energy difference be-tween the trial and current states. Initial polymer con-figurations were generated such that s was within the allowed range for a given window potential. Prior to data sampling, the system was equilibrated. As an il-lustration, for a \( N \) = 121 polymer chain, the system was equilibrated for typically \( \sim \mathrm { 1 0 ^ { 7 } } \) MC cycles, following which a production run of \( \sim 1 0 ^ { 8 } \) MC cycles was carried out. During each MC cycle a move for each monomer is attempted once, on average.

The windows are chosen to overlap with half of the adjacent window, such that \( s _ { i } ^ { \mathrm { m a x } } = s _ { i + 2 } ^ { \mathrm { m i n } } \). The window width was typically \( \Delta s \, = \, \sigma \). Thus, a calculation for \( N \)= 121, where the translocation coordinate spans a range of \( s \, \in \, [ 0 , 1 2 0 ] \), required separate simulations for 239 different window potentials. For each simulation, individual probability histograms were constructed using the binning technique with 10 bins per histogram.

The free energy functions obtained from the MC sim-ulations were used with the procedure summarized in Sec. III to study the translocation dynamics under the as-sumption that conformational quasi-equilibrium is main-tained during the process. The translocation probability \( \mathcal { W } ( s , t ; s _ { 0 } , 0 ) \) was determined by solving Eq. (1) for a cho-sen value of \( s _ { 0 } \). Typically, we used \( s _ { 0 } = L / ( 2 \sigma ) \), at which point the first monomer isj ust on the verge of exiting the pore into the cavity. The equation was solved using standard numerical methods with a “spatial” grid size of \( \Delta s \) = 0.01 and a time increment of \( \Delta t = 0 . 0 0 2 \mathcal { D } ^ { - 1 } \). The distribution \( P ( \tau ) \) was calculated using Eq. (3), where a standard five-point method was used to evaluate the derivatives in Eq. (2). The translocation probability, \( p _ { b } \), and mean first passage time, \( \langle \tau \rangle \), were calculated by nu-merical integration of Eqs. (4) and (6), respectively, using Simpson’s rule.

In the results presented below, quantities of length are measured in units of \( \sigma \), energy in units of \( k _ { \mathrm { B } } T \), force in units of \( k _ { \mathrm { B } } T / \sigma \) and time in units of \( \mathcal { D } ^ { - 1 } \).

# V. RESULTS

We consider first the case of spherical cavities in the absence of a driving force or adsorption potential, i.e. \( r \) = 1, \( f _ { \mathrm { d } } \, = \, 0 \) and \( \epsilon \) = 0. Figure 2 shows free energy functions for cavity volumes of V =150, 250 and 500, each for polymer lengths ranging from N=31 to N=141. As expected, the free energy cost of confining the polymer in the cavity increases as the confinement volume decreases. In addition, the curves all have positive curvature every-where except at near the upper and lower bounds. This indicates that free energy cost of inserting each monomer into the cavity increases as the number (and hence den-sity) of monomers inside increases. This feature is con-sistent with results from previous MC studies27,32,33 and theoretical studies16,26 that explicitly account for repul-sion between monomers, in contrast to the case for ideal polymers.15 Another noteworthy feature is the strong de-gree of overlap between the curves for different N and the same V . Thus, the free energy cost of transferring one monomer from the outside to the inside of the cavity de-pends approximately only on the density of monomers in the cavity. Deviations from this trend are evident where F dips abruptly near \( s = N - \) 1.

FIG. 2. Free energy functions for several different polymer lengths for a cavity aspect ratio of \( r \) = 1. Results for three different cavity volumes are shown.

To provide a quantitative analysis of the free en-ergy functions, we employ a standard scaling theory approach.23 We first recall that the standard form of the entropic free energy barrier for translocation of a polymer through a narrow pore in a flat wall of negligible thick-ness is \( F _ { 0 } / k T = ( 1 - \lambda ) \ln [ m ( N - m ) ] \), where m segments lie on one side of the pore, and \( N - m \) lie on the other

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