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Ion Dissociation Dynamics in an Aqueous Premelting Layer Samuel P. Niblett* and David T. Limmer*

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ABSTRACT: Using molecular dynamics simulations and methods of importance sampling, we study the thermodynamics and dynamics of sodium chloride in the aqueous premelting layer formed spontaneously at the interface between ice and its vapor. We uncover a hierarchy of time scales that characterize the relaxation dynamics of this system, spanning the picoseconds of ionic motion to the tens or hundreds of nanoseconds associated with fluctuations of the liquid−crystal interface in their presence. We find that distort both local interfaces, incurring restoring forces that result in the ions preferentially residing in the middle of the layer. While pair dissociation is thermodynamically favorable, these structural and dynamic effects cause its rate to vary by over an order of magnitude through the layer, with a maximum rate significantly depressed from the corresponding bulk value. The environment of ions in the premelting layer is distinct from that in a bulk liquid, being dominated by slow reorganization of water molecules and a water structure intermediate between ice and its melt.

■ INTRODUCTION challenging due to the complex structure of these interfaces. The mechanisms and rates of reactions at extended interfaces At typical polar temperatures, the surface of an ice crystal is ff covered by a thin film of disordered water molecules called the can be dramatically di erent from those of the homogeneous 14,15 materials that make them up.1,2 This is especially true of premelting layer or quasi-liquid layer. This layer forms to reactions in water, as their mechanisms are often sensitive to the mitigate the large surface tension of an exposed ice crystal since hydrogen-bonding network, which is heavily disrupted by an the liquid-like surface structure has fewer dangling hydrogen bonds and a smaller surface dipole.16 There are several interface. Considerable advances have been made in under- 2−5 outstanding questions surrounding premelting layers, partic- standing reactions at water−air interfaces, but ice−air ularly regarding their lateral extent and dynamics of surfaces remain poorly studied despite their importance in the 16−19 6 formation. Here, we consider how this complex, fluctuating of the polar atmosphere. Here, we use molecular environment affects reactivity. dynamics (MD) simulations to study the paradigmatic model Specifically, we probe the dynamics of a prototypical reaction, system of ion pair dissociation within the premelting layer of ice. the dissociation of sodium chloride, in the aqueous premelting We find that emergent structural and dynamic properties of the layer using classical simulation and theory. Separation of singly interface depress the overall relative to a bulk Downloaded via UNIV OF CALIFORNIA BERKELEY on February 22, 2021 at 17:49:30 (UTC).

See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. charged atomic ions has long been a preferred case study to liquid, with a strong dependence on where in the layer the understand -phase reaction dynamics computation- dissociation event occurs. ally,4,20,21 partly because the forces between the ions and Reactions at ice interfaces contribute significantly to the solvent molecules are simple and easy to model and partly chemical composition of the atmosphere.7 This influence arises because the dynamics of the reaction are intrinsically connected from various settings and chemical cycles. Some reactions to those of the solvent. It is now well established that the release atmospherically active species, such as the accelerated dissociation reaction mechanism has a significant contribution conversion of HOBr and HOCl into reactive Br /Cl at the 2 2 from solvent degrees of freedom,21 although how best to surface of ice particles in stratospheric clouds8,9 and on sea ice.10 describe this contribution in a simple reaction coordinate These halogen molecules subsequently photolyze into halogen − remains a subject of debate.22 26 NaCl dissociation is a radicals that contribute to ozone depletion. Other surface reactions remove trace gases, for example, the oxidation of SO2 fi 11,12 Received: December 18, 2020 to H2SO4 that contributes to ice and snow acidi cation. Finally, ice particles in urban settings catalyze photolysis of Revised: February 7, 2021 organic pollutant molecules, often resulting in more toxic byproducts.13 Understanding how ice−air interfaces affect chemical reactivity is therefore an important task, which remains

© XXXX American Chemical Society https://dx.doi.org/10.1021/acs.jpcb.0c11286 A J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B pubs.acs.org/JPCB Article

Figure 1. Typical configurations of a contact ion pair (CIP) from molecular simulations. The sodium ion is shown in yellow, and the chloride is shown in green. Dark and light blue surfaces represent the liquid−crystal and liquid−vapor interfaces, respectively. From left to right, the three panels show the ionic centers of mass at z̅=0,S/2, and S. particularly suitable test case for our purposes because the short-range interactions truncated at 8.5 Å and electrostatic optimal reaction coordinate depends on water molecules up to interactions computed by the particle mesh Ewald method.33 8 Å away from the ions themselves.22 This wide interaction To obtain a nanometer-thick premelting layer, we performed sphere will be frustrated by the confines of the premelting layer our calculations at 2 K below the ambient pressure melting to create a distinct dynamic environment, as observed in earlier temperature of TIP4P-2005. This implies a simulation temper- simulations.27 ature of 250 K.28 Prior to inserting the ion pair, the surface was The paper is organized as follows. We first introduce the equilibrated using a 50 ns MD simulation, where the model and simulation details used, giving particular attention to temperature increased from 200 to 250 K at a rate of 1 K/ns, methods for identifying the boundaries of the premelting layer and a further 100 ns simulation with a constant temperature of and discussion of the time scales exhibited by the system. We 250 K. The outermost ice unit cell on each surface became then discuss the equilibrium and energetic properties of the disordered, producing a premelting layer approximately 7 Å ion−interface system, focusing on the response of the interface thick. The remaining block of ice was 6 unit cells thick on to the solutes and the free-energy landscape experienced by the average. Except where otherwise noted, a Langevin thermostat ions. Finally, we evaluate the rate of ion dissociation, revealing was used to control the temperature, with a damping time of how the rate behavior depends on the local solvation 10 ps where we wished to study dynamics or 1 ps if only static environment, and interpret these results using reaction rate properties were being considered. Simulations were performed theory. using the LAMMPS package.34 To describe the premelting layer, we require a means of ■ METHODS: SIMULATING ICE−VAPOR INTERFACES identifying the interface between the liquid and crystal (lc) and that between the liquid and vapor (lv). On mesoscopic length We studied ion pair dissociation in the premelting layer using scales, phases are defined through structural order parameters molecular dynamics simulations. Water was represented by the 28 that manifest the globally broken symmetry associated with the TIP4P-2005 model, which accurately reproduces various phase transition. On molecular scales, an interface between two properties of liquid water, particularly the liquid−vapor surface 29 phases can be associated with the rapid spatial variation of a tension. Sodium and chloride ions were modeled as charged similar suitable local order parameter. Using the method of Lennard-Jones atoms with interaction parameters taken from Willard and Chandler, we define interfaces using isosurfaces of a the all-atom optimized potentials for liquid simulations (OPLS- 35,36 continuous coarse-grained order parameter field. AA) force field.30 The OPLS-AA force field predicts an ion The coarse-grained fieldisobtainedbydefining an and hydration free energy similar to other appropriate local order parameter ϕα(r)atafield point r to fixed-charge models,31,32 and so we expect that our results are distinguish the α = {lv, lc} interface. Due to the discreteness of qualitatively insensitive to the specific model. While we neglect atoms and molecules, any function defined directly from atomic the polarizability of the ions, its impact on the thermodynamics positions would be rapidly varying. Therefore, we convolute of the relatively nonpolarizable, surface-inactive NaCl species is ϕα(r) with a Gaussian smoothing function and arrive at a expected to be minimal. Further, the large separation of time smoothly varying order parameter field, scales between the polarization fluctuations and ion pair nuclear motion suggests a minor role for polarizability in the dynamics of −−′()/2rr22ξ ion pair dissociation. e ϕξαα̅(;rrr )=′′∫ d ϕ ( ) Most of our calculations were performed using an Ih ice (2πξ )3/2 (1) crystal containing 8x6x8 primitive unit cells with the crystallo- graphic c axis parallel to the z direction. An additional vacuum whose variation can encode the location of long-lived interfaces. region was added in contact with the basal (0001) plane of the The smoothing function is parameterized by a coarse-graining crystal, increasing the length of the z dimension to 170 Å. This length scale ξ, chosen to be comparable to a molecular diameter. slab geometry contains two ice−vapor interfaces on the high- The construction of our model system ensures that both and low-z faces of the crystal. A single NaCl pair was equilibrated interfaces are roughly planar. Crystalline periodicity means that at the large-z interface. This geometry is illustrated in Figure 1. most properties of the system should be translationally invariant Periodic boundary conditions were applied in all directions, with in the plane parallel to the interfaces, so for convenience, we will

B https://dx.doi.org/10.1021/acs.jpcb.0c11286 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B pubs.acs.org/JPCB Article write simulation coordinates as r ={x, z}. The interface α is ■ RESULTS AND DISCUSSION fi x de ned locally at as Relaxation within the Premelting Layer. The typical c dynamics of ions within the premelting layer are marked by a htα(,xx )=[− maxδϕαα ϕ̅ (, z ; ξ )] z (2) hierarchy of distinct relaxation time scales. Figure 2 summarizes several measures of local relaxation within the premelting layer. ϕc ⟨ϕ̅⟩ ⟨ϕ̅⟩ ⟨ ⟩ where α =( α i + α j)/2, ... i denotes an average over Each panel shows time correlation functions for the deviation of phase i, and δ is Dirac’s delta function. The maximization over z a local quantity from its mean. The local height autocorrelation acts to select the upper interface, where the ions are located. In function practice, we evaluate the order parameter field on a regular Cartesian grid and locate the isosurface by linear interpolation Cth()=⟨δδ hαα (,0)xx h (, t )⟩ (5) between grid points. Further coarse-graining can provide information on longer decays to 0.1 of its initial value within 0.1 ns for the lv interface length scales. We define a semilocal interface height and within 0.5 ns for the lc interface. The faster decorrelation of the lv interface is consistent with expectations from the capillary 1 wave theory in that the relaxation time scale is inversely Hαα(,xxxxxtht )= ∫ d(′Θ Δ−| ′− | )(,) ′ 39 πΔ2 (3) proportional to the surface tension. Figure 2 also shows the autocorrelation function of the local where Θ is the Heaviside step function. This height, Hα(x, t), is mean height the instantaneous mean height of the α interface in a circle of radius Δ centered at x. Both hα(x, t) and Hα(x, t) fluctuate in time as molecules change their configurations. For the liquid−vapor interface, the appropriate order ϕ ∑Nw δ − parameter, lv(r)= i=1 (r ri), is just the local number density computable as a sum over all of the water molecules. ’ Here, ri is the position vector of the water molecule i s center of ξ ϕc −3 mass. We adopted the parameters = 2.4 Å and lv = 0.016 Å , which were previously shown to give an accurate and convenient definition of the lv interface.35 The liquid−crystal interface requires a more complex order parameter. The 6th-degree Steinhardt−Nelson−Ronchetti order parameter is convenient − for this use.36 38 This function is defined as ϕ ∑Nw δ − lc(r)= i=1 q(ri) (r ri), where q(ri) is a projection of the local density onto the 6th-degree spherical harmonic.

2 1/2 6

qY()ri = ∑∑∑6m (ϕθjk ,jk ) mjikj=−6 ∈nn( ) ∈ nn( ) (4) ji zy ϕj θ z where Ylm(j ij, ij) is the l-m spherical harmonicz function of the j − z angular coordinatesj for vector (ri rj), and nn(zi) indicates the four nearestk neighbors of atom i. A typical water{ molecule in the premelted layer has q = 0.22, very close to the equilibrium value in pure water.36 Previous work has shown36 that appropriate ξ ϕc −3 parameters for the lc interface are = 2.5 Å and lc = 0.0167 Å . Figure 1 depicts typical configurations of an ion pair in the premelting layer. The two interfaces separate the system into three clear phases as expected. Both interfaces are perpendicular to z on average but undergo substantial fluctuations out of that ⟨δ 2 ⟩ ≈ 2 plane with mean-squared deviations hlc 2Å and ⟨δ 2⟩ ≈ 2 δ − ⟨ ⟩ hl 1Å, where h = h h . The premelting layer is defined as the volume enclosed by the lc and lv interfaces on an exposed surface of the ice block. The layer thickness is given ⟨ − ⟩ by S = hlv(x; t) hlc(x; t) . In the following, we choose the ⟨ ⟩ origin z = 0 such that hlc = 0 at 250 K in the absence of ions. ⟨ ⟩ Figure 2. Relaxation time scales for the premelting layer. The top panel The z coordinates will be expressed as fractions of S = hlv under the same conditions. shows the autocorrelation function for the height of an individual site fi ≈ on the interface, hα(x, t). The middle panel shows the autocorrelation of We nd that S(250 K) 7 Å and this value does not change Δ fi fl a spatially averaged interface height Hα(x, t), with = 1 nm. In both signi cantly in the presence of the ion pair, re ecting the panels, the shaded regions indicate the statistical error of the correlation negligible freezing point depression for the concentration function. The dashed line in the middle panel represents a simulation considered. Our observed value of S agrees with a wide range subject to a harmonic restraint on the ice molecules, as explained in the of other water models and experimental data when expressed as text. The last panel shows the autocorrelation function for the z − 36 ff a function of (T Tm)/Tm, as shown in the previous work. component velocity of di erent ionic species.

C https://dx.doi.org/10.1021/acs.jpcb.0c11286 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B pubs.acs.org/JPCB Article | | 36,45,46 CtH()=⟨δδ Hαα (,0)xx H (, t )⟩ (6) function on the magnitude x = x . Capillary wave theory predicts long-ranged spatial fluctuations in the position of an Δ for = 1 nm, representing a large enough radius to encompass interface, with no characteristic length scale. To check the the three solvation shells of the NaCl ion pair, which are believed impact of finite size effects from these fluctuations, we to influence dissociation dynamics.22 The lv-interface autocor- performed these calculations on a system size twice as large in relation decays within approximately 0.1 ns, comparable to the fi time required for a single site to decorrelate. The lc-interface each of the x directions as de ned in the Methods section. ≈ S autocorrelation, on the other hand, decorrelates much more When z̅ /2 and the ions lie near the center of the slowly, at least 10 times slower than the corresponding single- premelting layer, the average deviation of the interface from its site function. The average interface height retains some memory equilibrium position is negligible. Figure 3 shows how each of its position for tens to hundreds of nanoseconds. This slow interface responds when the ions approach. In all panels, decay is due to rare, discontinuous jumps in the mean interface Gα(x) → 0 at large x, showing that the influence of the ions has a height, where a layer of water molecules melts or freezes across finite range. The correlation lengths associated with the height large regions of the ice interface.40,41 The long waiting time deviations are up to 1 nm for the lv interface and up to 2 nm for between jumps makes averaging over these fluctuations the lc interface, equivalent to several molecular diameters. challenging, resulting in the large error estimate for CH(t). As the ions approach z̅= S, the lc interface retains its To mitigate the difficulty of studying the structure and equilibrium statistics, while the lv interface locally rises by up to ff dynamics of ions in the presence of slow di usion of the lc 2 Å. This increase results from water molecules being drawn interface, after equilibration we restrained the water molecules upwards to cover the ions and screen their charge from the − S located below z = 0.4 , attaching them to their equilibrium vacuum. Its magnitude generally increases with increasing z.A̅ positions using a harmonic potential with spring constant − notable exception occurs at z̅= 0.9S, where the ion pair actually k = 5 kcal Å 2. This spring constant permits vibrations of magnitude approximately 0.5 Å but no diffusion of the ice depresses hlv relative to its instantaneous position. At this molecules, so that melting is suppressed. The dashed line in the particular z,̅ the ions are frequently situated parallel to the mean middle panel of Figure 2 demonstrates that height deviations of lv interface and are left uncovered by the water. The ions the lc interface decay quickly in the absence of the melting event, themselves do not contribute to the interface calculation, so the − recovering the behavior of Ch(t). The z = 0.4S limit ensures uncovered ions lead to a depression in the interface. that the top bilayer of ice molecules is not affected by the restraint, so they can still freeze and melt locally over the course of thermal fluctuations without nucleating the global diffusion of the interface. Finally, in Figure 2, we consider the vertical motion of the ions. The z-directional velocity autocorrelation function Ct()=⟨ v (0)() vt⟩ vzzz (7) is largely decayed within 10 ps for each ion individually and for the center of mass of a bound ion pair. We computed the z- ff − directional di usivities of the ions from Cvz(t) using the Green 42,43 × −7 2 −1 Kubo formula, obtaining values of Dz = 2.6 10 cm s for a free cation, 2.7 × 10−7 cm2 s−1 for a free anion, and 8.9 × 10−7 cm2 s−1 for the ion pair. These diffusion constants are extremely slow, approximately an order of magnitude slower than bulk ionic diffusion at 250 K and 2 orders of magnitude smaller than bulk water diffusion at ambient temperature.44 These values indicate that the ionic species require approx- imately 10 ns to diffuse over a distance equal to their molecular diameter. Such slow diffusivities imply a very sluggish dynamic environment and illustrate the challenges of studying rare events in the premelting layer. Similar slow ion dynamics were previously observed by Hudait et al. in a different model of ice premelting.27 Ion−Interface Spatial Correlations. While the dynamics of the ions and interfaces are not strongly correlated, we expect that the ions and interfaces will influence each other statically. To quantify these spatial correlations, we examined the local shape of each interface near an ion. We calculated the mean deviation of the interface height from its average interface height, Figure 3. Average distortion of the interfaces, Gα(x), when one or two ions are nearby. Each panel shows results for a single sodium ion, single Gx()=⟨ hx () ⟩ −⟨ hx ()⟩ ααri α (8) chloride ion, and a sodium chloride pair constrained to their equilibrium separation. Different panels show different values of the ’ conditioned on a solute present at ri = {0, z},̅ where z̅is the ion s ionic center-of-mass z̅coordinate. Alongside each panel is the center of mass. We consider the dependence of this correlation corresponding typical configuration of the ions and nearest interface.

D https://dx.doi.org/10.1021/acs.jpcb.0c11286 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B pubs.acs.org/JPCB Article

Figure 4. Free energy as a function of ionic separation R and center-of-mass z coordinate, z.̅ Three characteristic z̅values are indicated with dashed lines. The dash-dotted line approximately indicates the dissociation energy barrier, R‡(z).̅

Conversely, when the ions approach z̅= 0, the lv interface is toward each interface. This increase represents an interplay unaffected, while the lc interface is locally depressed. This between reduced solvation, entropic penalties from pinning the depression indicates local disordering of the ice structure leading interfaces, and the energy cost of disrupting the crystal structure. to improved ion solvation. The magnitude of the deviation varies Approaching the lv interface, the average coordination numbers with the number of ions and z̅but usually falls between 1 and for both bound and separated ion pairs decrease4 and the 4 Å. This magnitude is larger than the equivalent deviations in average electrostatic potential energy on the ions from the water hlv, indicating that the lc interface is softer and easier to deform increases in a manner anticipated from Figure 3a,b. Upon consistently with its lower surface tension. approaching the lc interface, ion solvating water molecules An ion pair or free sodium ion at z̅≈−0.1S creates a significant reorganize, disrupting the ice structure and melting the crystal, depression in the lc interface, but a free chloride ion does not. as anticipated from Figure 3c,d. Instead, the anion incorporates into the ice surface structure, For all z̅values, the free energy displays two clear basins of replacing one of the water molecules. Disruption of the periodic attraction, one for the contact ion pair (CIP) state in the range of crystal structure is therefore minimized, and the height of the 2.5 < R < 3.4 Å and another for the solvent-separated pair (SSP) interface hardly changes. The cation is unable to incorporate in at around 4.5 < R < 5.5 Å. These states are separated by a barrier this way. Indeed, in general, the influence of the cation on both that is approximately 6.5kBT at z̅= 0.5S and varies by up to 3kBT interfaces is greater than that of the anion due to its greater over the premelting layer. The free energy also exhibits a broad, charge density, which provides a strong bias to reorganize the shallow minimum for R > 6.5 Å, corresponding to completely surrounding water molecules. With the exception of z̅= 0.9S, the dissociated ions. The energy barrier separating this minimum effect of the ion pair is intermediate between the two free ions. from the SSP state is very small. The free-energy change to None of the distortions observed in these calculations are dissociate the ions in the premelting layer implies a 6-fold significantly larger than the instantaneous capillary wave decrease in the dissociation constant relative to bulk water fluctuations for this system. The influence of ions appears to around the same temperature and pressure, based on an consist of pinning the equilibrium capillary waves in place. This independent bulk free-energy calculation. pinning is associated with an entropic cost of moving the ions Dynamics of Ion Pair Dissociation. We have shown that near to the interfaces, which influences the ion thermodynamics both static and dynamic fluctuations vary strongly through the considerably47 as detailed below. premelting layer. Analogously, the rate of ion pair dissociation Thermodynamics of Ion Pair Dissociation. To under- need not be constant and could similarly depend on where in the stand the thermodynamics of ion pairing in the premelting layer, layer the rare event occurs.49 Since the vertical motion of the we computed the free energy as a function of the ionic separation ions is so slow, we will assume that dissociation events occur at R and center-of-mass vertical coordinate z.̅ We employed fixed ionic z coordinates. We have computed transition rates umbrella sampling using a series of simulations from the CIP to SSP as a continuous function of z,̅ using the with an additional harmonic potential, Bennett−Chandler approach.50 − * 2 − * 2 fi Vb(R, z)=̅ kr/2(R R ) + kz/2(z̅ z ) . These simulations We have de ned the CIP reactant state and SSP product state were combined using the weighted histogram analysis method48 using a dividing surface along the ionic separation distance, − ‡ fi ‡ to compute an unbiased free energy, F(R, z)=̅ kBT lnP(R, z),̅ denoted R (z).̅ For each xed value of z,̅R (z)̅ is chosen as the where P(R, z)̅ is the probability of observing the ions at location of the saddle point along R, which is close to R = 3.7 Å ’ − separation R and height z,̅ and kBT is Boltzmann s constant times for all z.̅ The Bennett Chandler rate is the product of a temperature. We used 478 distinct simulations, R* varying from transition-state theory estimate along this dividing surface, ffi κ 2.0 to 6.0 Å and z* varying from approximately −0.2l to 1.0S. kTST(z),̅ and the transmission coe cient, (z).̅ The transition- −1 2 Half of the simulations used kr = kz = 1 kcal mol Å , and the state theory estimate is the product of an attempt frequency and remainder employed stronger springs (up to 80 kcal mol−1 Å2) the probability of reaching the dividing surface from the reactant to ensure adequate sampling. Simulation length was varied for state, the same reason, with most data sets representing 5 ns of ‡ simulation time. kT e−βFR(,) z̅ kz()= B Figure 4 shows the resulting free-energy landscape alongside a TST ̅ ‡ 2μ R −βFRz(,) ∫ deR ̅ typical snapshot to illustrate its connection to the system −∞ (9) geometry. For all R values, the free energy is lowest near the middle of the premelting layer and increases monotonically where μ is the reduced mass of the ion pair.

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The transmission coefficient is defined as the long time limit We expect R to be a poor descriptor of reaction dynamics in a of the correlation function premelting layer. Even in bulk water, previous investigations 21,23−25 ̇ ‡ have found that solvent degrees of freedom must be ⟨RRtRz(0)θ[− ( ) (̅ )]⟩‡ κ()z = lim included in the reaction coordinate to well describe the ̅ ̇ fi t→∞ ⟨|R(0)|⟩‡ /2 (10) transition-state ensemble. Nevertheless, the rate pro le is largely determined by the z̅dependence of the free energy to move fl which is the ux through the dividing surface conditioned on along R, as encoded in the transition-state theory estimate. This ⟨···⟩ ending in the SSP state. The notation ‡ indicates an average suggests that the recrossing dynamics do not greatly affect the over an equilibrium ensemble of initial conditions for which competition between different environments for dissociation, R = R‡(z),̅ obtained from constrained simulations. In practice, ̇ despite depressing the rate by up to a factor of 100. this quantity is averaged independently for R(0) > 0 and fi ̇ To understand the overall scale of the rate pro le, we have R(0) < 0. For each initial condition, an ensemble of initial compared k(z)̅ to that estimated from Kramers theory in the − velocities is generated from the Maxwell Boltzmann distribu- spatial diffusion limit.51,52 In bulk water at ambient temperature, tion and the corresponding trajectories are propagated for 10 ps ion pair dissociation is known to have inertial components to its ffi with microcanonical trajectories, which is su cient for the dynamics.23 However, at low temperatures within the system to commit to either the reactant or product basin. The premelting layer, small diffusivity of the ions in Figure 2 ensemble averages in eq 10 converge slowly, in our case suggests that momentum correlations are damped out on the requiring approximately 5000 trajectories. The transmission time scales of dissociation. Assuming that the dynamics are ffi − coe cients are very small, in the range of 0.01 0.05 for the overdamped, an approximate transmission coefficient ff di erent z̅values. can be evaluated in the harmonic barrier limit, The dissociation rate, k(z)=̅ k (z)̅κ(z),̅ as a function of z̅is κ ≈ μω ‡ ω TST K(z)̅ bD(R , z)/̅kBT, where b is the imaginary frequency shown in Figure 5. The rate is slow near the lc interface where associated with the free-energy barrier, and D(R‡, z)̅ is the diffusion constant along R evaluated at the top of the barrier at fixed z.̅ We estimate D(R‡, z)̅ following a procedure due to Hummer, valid for simulations where the reaction coordinate R is restrained to a particular window with a harmonic bias potential.53 Specifically, an autocorrelation function of R(t)is computed under a harmonic bias to remain on top of the barrier. Its value at t = 0 and decay time can be used to compute D(R‡, z).̅ Both values are obtained from 10 ns constrained simulations using the biasing potentials employed in the umbrella sampling calculations. Consistent with our assumption that dissociation occurs at a fixed z̅value, we found that ‡ ≫ D(R , z)̅ Dz. Therefore, reactive events proceed much faster than vertical ionic motion, and the dissociation processes at values of z̅can be considered independent.54 κ The Kramers rate, kK(z)=̅ kTST(z)̅K(z),̅ accounts for Figure 5. Variation of the ion dissociation rate with the position in the recrossing due to friction along the reaction coordinate. In premelting layer. Rate constants computed according to the TST − Figure 5, we see that this estimate agrees fairly well with k(z),̅ theory, Kramers theory, and Bennett Chandler theory are all shown. particularly near the two interfaces. This observation suggests Lines shown are guides for the eye. that most of the observed recrossing is due to slow dynamics along the reaction coordinate, caused by the sluggish environ- ment of the premelting layer. The biggest gap between the − the ions are essentially frozen in place. The rate increases toward Kramers and Bennett Chandler rate estimates comes in the the layer center, where improved solvation stabilizes the range of 0.1 < z̅< 0.5, perhaps indicating a more complex transition-state configurations and then drops off toward the lv reaction mechanism associated with ions attaching to the ice interface. The total variation over the premelting layer is over an crystal surface. order of magnitude. At its peak in the middle of the layer, the dissociation rate is an order of magnitude smaller than it would ■ CONCLUSIONS be in a bulk liquid at these same conditions, kbulk. This The picture of the premelting layer that emerges from our depression of the rate reflects how different the solvation simulations is that of a confined, sluggish environment, where environment is from a bulk solution, even in the middle of the water molecules adopt liquid-like configurations but with liquid-like layer. significant residual ice structure. Both interfaces affect the At z̅= 0 and below, the ions must either melt large regions of dynamics of nearby reactions through energetic and dynamic ice or incorporate into the crystal structure. Both processes are factors, and in the case of ion dissociation, both factors act to associated with slow and position-dependent dynamics. More- slow the reaction. The ice interface appears to be much more over, describing ionic behavior in the crystal itself would significant for determining reaction dynamics, in both probably require relaxing our harmonic restraint on the ice magnitude and range of interaction. molecules and hence averaging our calculations over rare layer- We found that the ions significantly alter the properties of the melting events. These considerations meant that we were unable nearby interfaces. The dynamics of both the lc and lv interfaces to obtain converged rate calculations for z̅≤ 0. However, we are dominated by capillary wave fluctuations decaying in note that the free-energy cost of moving an ion pair this deep approximately 1 ns. When an ion pair resides near the interface, into the crystal is large. these waves are pinned in place, creating a long-lived distortion

F https://dx.doi.org/10.1021/acs.jpcb.0c11286 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B pubs.acs.org/JPCB Article of the interface shape. In the range of ion positions that are (7) Domine, F.; Shepson, P. B. Air-snow interactions and atmospheric energetically feasible, the distortions do not exceed the chemistry. Science 2002, 297, 1506−1510. equilibrium fluctuation size, but the entropic cost of pinning (8) Abbatt, J. P. Heterogeneous reaction of HOBr with HBr and HCl − the wave contributes to the energy barrier, which keeps the ions on ice surfaces at 228K. Geophys. Res. Lett. 1994, 21, 665 668. near the layer center. (9) Bianco, R.; Hynes, J. T. Heterogeneous reactions important in The dissociation dynamics of water molecules in the atmospheric ozone depletion: A theoretical perspective. Acc. Chem. Res. ff fi 2006, 39, 159−165. premelting layer di er signi cantly from those in bulk solution, (10) Domine, F. Can we monitor snow properties on sea ice to even at the layer center where the water structure appears mostly fi investigate its role in tropospheric ozone depletion? J. Geophys. Res.: liquid-like. Partially dissociated con gurations are harder to Atmos. 2017, 122, 11,107−11,111. solvate in the premelting layer than contact ion pairs, leading to a (11) Abbatt, J. P. Interactions of atmospheric trace gases with ice significantly increased dissociation energy barrier. Kinetics also surfaces: Adsorption and reaction. Chem. Rev. 2003, 103, 4783−4800. play a more significant role in the premelting layer, mostly due to (12) Conklin, M. H.; Sommerfeld, R. A.; Laird, S. K.; Villinski, J. E. high friction from the slow water dynamics and residual ice-like Sulfur dioxide reactions on ice surfaces: Implications for dry deposition structure. The implications of this slow, fluctuating environment to snow. Atmos. 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Email: [email protected] Sci. U.S.A. 2016, 113, 12347−12349. David T. Limmer − (18) Qiu, Y.; Molinero, V. Why is it so difficult to identify the onset of Chemistry Department, University of − California, Berkeley, California 94720, United States; ice premelting? J. Phys. Chem. Lett. 2018, 9, 5179 5182. Chemical Science Division, Lawrence Berkeley Laboratory, (19) Llombart, P.; Noya, E. G.; MacDowell, L. G. Surface phase transitions and crystal habits of ice in the atmosphere. Sci. Adv. 2020, 6, Berkeley, California 94720, United States; Kavli Energy No. eaay9322. Nanosciences Institute, University of California, Berkeley, (20) Jorgensen, W. L.; Buckner, J. K.; Huston, S. E.; Rossky, P. J. California 94720, United States; orcid.org/0000-0002- Hydration and energetics for tert-butyl chloride ion pairs in aqueous 2766-0688; Email: [email protected] solution. J. Am. Chem. Soc. 1987, 109, 1891−1899. (21) Geissler, P. L.; Dellago, C.; Chandler, D. Kinetic pathways of ion Complete contact information is available at: − https://pubs.acs.org/10.1021/acs.jpcb.0c11286 pair dissociation in water. J. Phys. Chem. B 1999, 103, 3706 3710. (22) Ballard, A. J.; Dellago, C. Toward the mechanism of ionic dissociation in water. J. Phys. Chem. B 2012, 116, 13490−13497. Notes fi (23) Mullen, R. G.; Shea, J. E.; Peters, B. Transmission coefficients, The authors declare no competing nancial interest. committors, and solvent coordinates in ion-pair dissociation. J. Chem. Theory Comput. 2014, 10, 659−667. ■ ACKNOWLEDGMENTS (24) Yonetani, Y. Solvent-coordinate free-energy landscape view of This material is based on work supported by the U.S. water-mediated ion-pair dissociation. Mol. Phys. 2017, 8976, 2987− Department of Energy, Office of Science, Office of Advanced 2998. 2+ − Scientific Computing Research, Scientific Discovery through (25) Salanne, M.; Tazi, S.; Vuilleumier, R.; Rotenberg, B. 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