b Instrucciones de Diseño para la Ordenanza 1.

While the excitation-driven movement of the solute within the pore can be quanti- fied in several ways, a particularly instructive way is to examine the time-dependent displacement of the solute relative to the pore interface, as implied by the re- sults in Chapter 4. In each of four thousand 200-ps NEMD simulations, the minimum distance to the pore is determined at each time. To address period- icity in z when determining the nearest pore atom, replicas in the −z and +z direction were constructed. To save calculation time, the periodic images were only considered when the solute z position in the primary image was z < −7 or z > +7 Å, respectively. Accordingly, for each configuration, the distances, d = p∆x2+ ∆y2+ ∆z2, between each site of the Stockmayer solute and each pore atom within (∆x, ∆y, ∆z) = (15.0, 15.0, 8.0) Å were calculated (keeping in mind periodicity in z), and the minimum distance was determined. In addition to determining the minimum solute-silica distance at each time, the displace- ment relative to the pore interface was calcuated. That is, for each simulation, ∆dmin, rel(t) = dmin(t) − dmin(0) was calculated. The average over the 4000 NEMD

trajectories is presented in the top panel of Figure 5.2. The result is a small, -0.5 Å net displacement over the course of the trajectory. Notably, the net displacement is just outside of statistical uncertainty. Because ∆dmin, rel(t) = dmin(t) − dmin(0),

negative displacements indicate that dmin(t) < dmin(0), and on average and at long

times, the solute has moved closer to the pore wall.

-2 -1 0 1 2 ∆ d min,rel (t) (Å) 0 50 100 150 200 Time (ps) -2 -1 0 1 2 ∆ d min,rel (t) (Å)

Figure 5.2: The Stockmayer solute in a hydrophilic pore shows a small (0.5 Å) net displacement toward the silica interface by 200 ps after excitation, as shown in the top panel. This can be decomposed (bottom panel) into displacements toward (black) or away from (red) the silica interface. Neither direction shows, on average, displacements much more than 1.5 Å.

d_{min, rel}(t) curves by the direction of their overall displacement. For example, trajec- tories showing net solute displacements over 200 ps that are negative (i.e., toward the pore wall) are averaged. Similarly, trajectories in which the net displacement of the solute is away from the wall are considered together. The result of such a calculation is shown in the bottom panel of Figure 5.2. The black curve represents the average of all trajectories with solutes that move toward the wall. If the solute moves toward the pore wall, on average it moves approximately 1.5 Å. Similarly, if the solute moves away from the wall, on average it moves approximately 1 Å. Thus net effect (shown in the top panel), can be thought of as the weighted sum of molecules moving to and away from the interface. The result is the previously discussed weak, but statistically significant, average displacement of approximately 0.5 Å toward the pore interface.

Because the equilibrium MD simulations discussed in Chapter 4 indicated that spectra are a function of solute position within the pore, and because of the weak (albeit statistically significant) post-excitation displacement observed here, it is reasonable to speculate that displacement is also a function of position at the time of excitation (initial position). To this end, the displacement relative to the pore interface as a function of time has been parsed by the minimum distance to the pore interface at t = 0 by 1 Å increments. Additionally, the minimum distance to the pore interface averaged over the initial 10% of the trajectory (20 ps) has also been used to parse the relative displacements. The results of the former method are presented in Figure 5.3. The latter was motivated by noting that the solute displacements (relative to the minimum distance to the pore wall) at early times are

0 50 100 150 200 Time (ps) -4 -3 -2 -1 0 1 2 ∆ d min,rel (t) (Å)

Figure 5.3: The solute displacement as a function of time is shown for initial minimum distances to the pore of 5 (black), 7 (red), 9 (blue), and 11 (purple) Å. Solutes near the interface travel slightly away from it, whereas solutes in the pore interior are strongly drawn toward the pore interface.

often unpredictable and sometimes change sign. It was suspected that averaging the position over the initial 10% and using that value for decomposition would result in “proper binning,” with the result that the curves would become smoother. This, however, was not the case. In fact, this allows curves with different t = 0 positions to be averaged together, which results in erratic displacement profiles as a function of time.

In Figure 5.3, the minimum distance to the pore interface at t = 0 distinguishes the curves. The black, red, blue, and purple curves represent dmin(0) =5, 7, 9,

and 11 Å from the wall, respectively. A schematic interpretation of these results is provided in Figure 5.4. Because the solute molecule is large, dmin(0) = 5

Å is already near the pore interface. The results of Chapter 4 suggest that the energy to displace solvent near the pore interface is large. Additionally, increased direct contact with the interface is eventually disfavored by strong Lennard-Jones repulsions. It is speculated that for these reasons, the black curve shows a net displacement over 200 ps that is slightly positive or away from the pore interface. For initial solute positions further from the interface, the net displacement is larger and toward the wall. Notably, this is at odds with a simple physical model based strictly on the Coulombic potential, in which the larger charges associated with the excited state solute result in stronger attraction to the pore interface, where movement would be most dramatic. A simple explanation (implied previously) may be that solutes already at the interface at time t = 0 have only one direction to move—away from the interface. Currently, the role of the solvent in displacement of the solute after excitation is unclear. The results presented here, however,

Figure 5.4: The curves of Figure 5.3 are schematically interpreted. For solute t0

minimum distances to the pore interface (dots) of 5 (black), 7 (red), 9 (blue), and 11 (purple) Å, displacements relative to the pore interface (arrows) over 200 ps are shown.

indicate that for the Stockmayer solute in the hydrophilic pore, a change in charge distribution can lead to a net displacement–in this case toward the pore interface. Thus, the first requirement in the solute diffusion hypothesis is satisfied in these NEMD simulations.