^{1}and Richard M. Stratt

^{1,a)}

### Abstract

It is not obvious that many-body phenomena as collective as solute energy relaxation in liquid solution should ever have identifiable molecular mechanisms, at least not in the sense of the well-defined sequence of molecular events one often attributes to chemical reactions. What can define such mechanisms, though, are the most efficient relaxation paths that solutions take through their potential energy landscapes. When liquid dynamics is dominated by slow diffusive processes, there are mathematically precise and computationally accessible routes to searching for such paths. We apply this observation to the dynamics of preferential solvation, the relaxation around a newly excited solute by a solvent composed of different components with different solvating abilities. The slow solvation seen experimentally in these mixtures stems from the dual needs to compress the solvent and to do solvent-solvent exchanges near the solute. By studying the geodesic (most efficient) paths for this combined process in a simple atomic liquid mixture, we show that the mechanism for preferential solvation features a reasonably sharp onset for slow diffusion, and that this diffusion involves a sequential, rather than concerted, series of solvent exchanges.

We thank Dr. Chengju Wang for a number of helpful suggestions regarding the implementation of this landscape analysis. This work was supported by NSF Grant No. CHE-0809385.

I. INTRODUCTION

II. OUR MODEL AND ITS BASIC DYNAMICS

A. Model and simulation protocols

B. Molecular dynamics results

III. DETERMINING THE MOST EFFICIENT SOLVATION PATHWAYS

A. Geodesic pathways in the potential energy landscape ensemble

B. Computational approaches

IV. DETERMINING MECHANISTIC INFORMATION

A. Lengths of geodesic pathways

B. The number of solvent atoms participating in the geodesic pathways

V. CONCLUDING REMARKS

### Key Topics

- Solvents
- 108.0
- Diffusion
- 21.0
- Solution processes
- 18.0
- Excited states
- 16.0
- Molecular dynamics
- 14.0

## Figures

The dependence of solvation dynamics on the solvent composition for the model used in this paper. The nonequilibrium solvation response function S(t) is shown for solutions consisting of 0%, 10%, 50%, 80%, and 100% S solvents. The larger scale figure depicts the longer-time-scale diffusive relaxation; the insert shows the initial (subpicosecond) inertial relaxation for the same solutions.

The dependence of solvation dynamics on the solvent composition for the model used in this paper. The nonequilibrium solvation response function S(t) is shown for solutions consisting of 0%, 10%, 50%, 80%, and 100% S solvents. The larger scale figure depicts the longer-time-scale diffusive relaxation; the insert shows the initial (subpicosecond) inertial relaxation for the same solutions.

The time evolution of solvent structure around the excited-state solute for the 10% S case. What we plot here are both the solute/strong solvent radial distribution function and the solute/weak solvent radial distribution function following a excitation of the solute from the ground to the excited-state. The insert in the lower panel highlights the nonmonotonic weak solvent behavior seen in the first shell peak: although the peak height eventually ends up being smaller than its initial value, during the initial (solvent compression) phase, this height actually grows. It is not until after the first 1–2 ps that the peak height begins to shrink.

The time evolution of solvent structure around the excited-state solute for the 10% S case. What we plot here are both the solute/strong solvent radial distribution function and the solute/weak solvent radial distribution function following a excitation of the solute from the ground to the excited-state. The insert in the lower panel highlights the nonmonotonic weak solvent behavior seen in the first shell peak: although the peak height eventually ends up being smaller than its initial value, during the initial (solvent compression) phase, this height actually grows. It is not until after the first 1–2 ps that the peak height begins to shrink.

The time evolution of the first solvation shell population for the 10% S case. The curves drawn here are derived by integrating the radial distributions functions shown in Fig. 2 to the first minimum.

The time evolution of the first solvation shell population for the 10% S case. The curves drawn here are derived by integrating the radial distributions functions shown in Fig. 2 to the first minimum.

Relationship between the average net configuration space (3N-dimensional) distance traveled and the average extent of solvation accomplished for the 10% S case. Both the average net distance (plotted in units of ) and the average solvation progress measure depend parametrically on the elapsed time t; the graph shown combines the two averages to determine their time-independent relation.

Relationship between the average net configuration space (3N-dimensional) distance traveled and the average extent of solvation accomplished for the 10% S case. Both the average net distance (plotted in units of ) and the average solvation progress measure depend parametrically on the elapsed time t; the graph shown combines the two averages to determine their time-independent relation.

The growth of the most efficient (geodesic) path length during preferential solvation. Unlike the earlier figures, which reported molecular dynamics, this figure looks solely at the geometry of the solution potential energy landscape. A set of geodesic paths is found, and the average conductivity ratios (the square of the direct-to-geodesic-path-lengths quotient, ) are shown as a function of average solvation progress, s, for 10%, 50%, 80%, and 100% S solvents. (In addition, for the 10% S case, we plot the results for both atom and atom solutions, demonstrating that finite size effects are negligible.) The color-coded arrows drawn above the figure indicate the locations we identify (from left to right) as the onsets of diffusive pathways for the 10%, 50%, and 80% S solvents, respectively.

The growth of the most efficient (geodesic) path length during preferential solvation. Unlike the earlier figures, which reported molecular dynamics, this figure looks solely at the geometry of the solution potential energy landscape. A set of geodesic paths is found, and the average conductivity ratios (the square of the direct-to-geodesic-path-lengths quotient, ) are shown as a function of average solvation progress, s, for 10%, 50%, 80%, and 100% S solvents. (In addition, for the 10% S case, we plot the results for both atom and atom solutions, demonstrating that finite size effects are negligible.) The color-coded arrows drawn above the figure indicate the locations we identify (from left to right) as the onsets of diffusive pathways for the 10%, 50%, and 80% S solvents, respectively.

The evolution of the first-solvation-shell populations plotted as a function of the solvation progress variable S(t) for the three solvents shown. The figure, which simply replots the kinds of molecular dynamics results portrayed in Figs. 1 and 3 on a single graph, emphasizes that the first shell numbers of strong solvent (S), weak solvent (W), and total solvent begin to exhibit noticeable changes at different points in the solvation process. Vertical lines, drawn at the same locations as the arrows in Fig. 5, indicate the points at which the geodesic analysis predicts that the dynamics is becoming strongly diffusive.

The evolution of the first-solvation-shell populations plotted as a function of the solvation progress variable S(t) for the three solvents shown. The figure, which simply replots the kinds of molecular dynamics results portrayed in Figs. 1 and 3 on a single graph, emphasizes that the first shell numbers of strong solvent (S), weak solvent (W), and total solvent begin to exhibit noticeable changes at different points in the solvation process. Vertical lines, drawn at the same locations as the arrows in Fig. 5, indicate the points at which the geodesic analysis predicts that the dynamics is becoming strongly diffusive.

Extracting mechanistic information from the number of degrees of freedom waiting to participate. Whether we would say that the hypothetical chemical reaction illustrated here proceeds in a single concerted step, or as a sequence of two steps, depends on whether the shortening of the BC distance r(BC) takes place at the same time as (upper panel), or after (lower panel), the lengthening of the AB distance r(AB).

Extracting mechanistic information from the number of degrees of freedom waiting to participate. Whether we would say that the hypothetical chemical reaction illustrated here proceeds in a single concerted step, or as a sequence of two steps, depends on whether the shortening of the BC distance r(BC) takes place at the same time as (upper panel), or after (lower panel), the lengthening of the AB distance r(AB).

The number of solvent atoms waiting to move, , in preferential and ordinary solvation, as determined from the participation numbers associated with the landscape geodesics. The three panels present the waiting numbers of weak, strong, and total solvent atoms for the 10%, 50%, and 80% S solutions, each as a function of the solvation progress variable s. For comparison, each panel also shows (as dots practically indistinguishable from the horizontal axes) the results from the pure solvent (100% S) case. As in Fig. 6, we draw vertical lines where the growing lengths of the geodesic paths indicate that the dynamics is starting to become strongly diffusive.

The number of solvent atoms waiting to move, , in preferential and ordinary solvation, as determined from the participation numbers associated with the landscape geodesics. The three panels present the waiting numbers of weak, strong, and total solvent atoms for the 10%, 50%, and 80% S solutions, each as a function of the solvation progress variable s. For comparison, each panel also shows (as dots practically indistinguishable from the horizontal axes) the results from the pure solvent (100% S) case. As in Fig. 6, we draw vertical lines where the growing lengths of the geodesic paths indicate that the dynamics is starting to become strongly diffusive.

## Tables

Potential energy parameters (Lennard-Jones well depths between species and ).

Potential energy parameters (Lennard-Jones well depths between species and ).

Thermodynamics properties of solutions with an excited solute. [For each of the excited-state-solute/solvent mixtures studied in this paper, we give the temperature T, landscape energy , and average solute energy gap , all reported in units of the reference well depth . The mixtures with a ground-state solute all have , , and (independently of the solvent composition). To help calibrate the differences in the solvation thermodynamics, we also report the equilibrium Stokes shift for each mixture.]

Thermodynamics properties of solutions with an excited solute. [For each of the excited-state-solute/solvent mixtures studied in this paper, we give the temperature T, landscape energy , and average solute energy gap , all reported in units of the reference well depth . The mixtures with a ground-state solute all have , , and (independently of the solvent composition). To help calibrate the differences in the solvation thermodynamics, we also report the equilibrium Stokes shift for each mixture.]

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