Ion-Specific Solvation Water Dynamics: Single Water versus Collective

Article pubs.acs.org/JPCA

Ion-Specific Solvation Water Dynamics: Single Water versus Collective Water Effects Klaus F. Rinne,† Stephan Gekle,‡ and Roland R. Netz*,† †

Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

‡

S Supporting Information *

ABSTRACT: Recent femtosecond-resolved spectroscopy experiments demonstrate the single-water orientational dynamics in the first solvation shell around monatomic ions to be slowed down. In contrast, dielectric spectroscopy experiments exhibit a blue shift of the water dielectric relaxation time with rising salt concentration, indicative of faster water dynamics. Using molecular dynamics simulations employing nonpolarizable and thermodynamically optimized ion force fields, we reproduce both experimental trends and resolve these conflicting experimental findings by the simultaneous analysis of singlewater and collective-water dynamics in the ion solvation shells. While the single-molecule reorientational dynamics of first solvation shell water around ions indeed slows down, the collective dynamics, which furnishes the dominant contribution to the dielectric response, accelerates. This collective acceleration is rationalized by a dramatically decreasing water cooperativity around ions when compared to bulk water, quantified by the Kirkwood dielectric enhancement factor. The static dielectric decrement of salt solutions is thus reinterpreted as a dielectric structure breaking rather than a water alignment effect. Both the dielectric blue shift and the dielectric decrement become stronger with increasing anion size, meaning larger halide ions such as iodide are more efficient dielectric structure breakers than small halide ions such as fluoride.

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INTRODUCTION The dynamic and structural properties of water in the immediate vicinity of ions and charged surface groups of proteins and other biomolecules are important for a number of technological and biological processes1,2 such as protein folding, coagulation and flocculation, and nanofiltering. In addition to nuclear magnetic resonance (NMR) studies,3,4 over the past several years, novel spectroscopic methods such as two-dimensional (2D) IR vibrational echo spectroscopy,5,6 femtosecond polarization-resolved IR pump−probe experiments,7,8 and polarization-resolved terahertz time-domain Raman spectroscopy9 have greatly enhanced the understanding of the hydrogen bond dynamics in the water solvation shell around charged and uncharged solutes. These experimental advances were paralleled by molecular simulations that reproduced and explained many of the spectroscopic features.10−14 Solvation water in alkali halide salt solutions constitutes without doubt the simplest system for the investigation of the water dynamics in the presence of charged groups, and therefore many exemplary studies exist for such model systems.15−17 Yet a number of open and puzzling questions were raised even in these seemingly simple systems, which motivate the current study. Many spectroscopic studies demonstrated that solvation water next to hydrophobic solutes and ions at elevated concentrations is generally slowed down in its orientational motion and clear ion-specific trends were observed. The slowing down of the water self-relaxation has been understood in terms of the extended jump model.11,12,14 © 2014 American Chemical Society

For dilute solutions of some large ions the effect on the water dynamics is small or even reverse, showing an acceleration of the reorientation dynamics3,4,14 (which is not the topic of this paper). In contrast to these methods, spectroscopic experiments in the gigahertz18 and the terahertz19 regimes probe the collective water motion. By a careful analysis of the concentration dependent spectroscopic shifts, the measured spectra can be decomposed into bulk water and solvation water contributions. Interestingly, in dielectric spectroscopy studies20−22 the dielectric relaxation times of the salt solutions NaF, NaCl, NaBr, and NaI were found to decrease with increasing salt concentration and with increasing anion size. The speedup of the dielectric relaxation with increasing NaCl concentration was also seen in simulations.23−25 This finding is at odds with the above-mentioned ultrafast spectroscopic studies where the same ions induce a slowing down of the solvation water dynamics. In this paper we resolve this apparent contradiction by molecular dynamics simulations of NaF, NaCl, NaBr, and NaI salt solutions, for which we simultaneously determine the dielectric spectra as well as the single-water orientation dynamics. The dielectric spectra show a significant blue shift and a simultaneous decrease of the static dielectric constant with increasing salt concentration and increasing anion size, in Received: July 4, 2014 Revised: November 19, 2014 Published: December 4, 2014 11667

dx.doi.org/10.1021/jp5066874 | J. Phys. Chem. A 2014, 118, 11667−11677

The Journal of Physical Chemistry A

Article

almost quantitative agreement with experimental data. This serves as a validation of the force fields employed and of the technique we use in order to extract the spectral information from the simulation trajectories. By a further analysis of the simulation trajectories, we show that the individual water orientation dynamics indeed slows down in the vicinity of all halide ions, so that the blue shift observed in the dielectric spectra is solely due to collective dynamic effects, i.e., the coupling of the dynamics of one water molecule to its neighbors. By a separate analysis of the water dynamics in consecutive solvation shells around ions, we furthermore demonstrate that the dynamic effects we are discussing primarily stem from the first solvation shell. The acceleration of the collective water dynamics in the first solvation shell is rationalized by a dramatic decrease of the water cooperativity. This effect is in line with the Madden−Kivelson theory that relates the dielectric constant and relaxation time of homogeneous liquids and was derived on the basis of linear response theory. The decrease of the water cooperativity in the first solvation shell both accelerates the water motion and at the same time reduces the dielectric constant, which can be quantified in terms of a dramatically decreased Kirkwood factor in the first solvation shell. This shows that the dielectric decrement and the blue shift in the dielectric spectra of salt solutions are in fact connected and the consequence of a dielectric structure breaking influence of ions on the first solvation shell of water, in line with the Madden−Kivelson theory. We conclude that whenever collective polarization contributions are picked up by an experimental technique, the connection to the underlying molecular dynamics is indirect and can only be elucidated by a careful decomposition into collective and single-molecular dynamic effects, for which molecular simulations are often suitable.

σ (f ) =

∫0

∞

→̇ ⎯ ⃗ · P (t )⟩ d t e−2π ift ⟨P (0)

∞

→̇ ⎯ e−2π ift ⟨JI⃗ (0) ·P (t )⟩ dt

(2)

ϕ W (t ) =

⃗ (0) ·PW ⃗ (t )⟩ ⟨PW 3VkBT ϵ0

(3)

ϕIW (t ) =

⃗ (0) ·J ⃗ (t ) − J ⃗ (0) ·PW ⃗ (t )⟩ 1 ⟨PW I I 2 3VkBT ϵ0

(4)

ϕI(t ) =

⟨JI⃗ (0) ·JI⃗ (t )⟩ 3VkBT ϵ0

(5)

we can express the ion conductivity as the sum of a purely ionic term and an ion−water cross-correlation term, σ( f) = −2πift −2πift ∫∞ ϕI(t) dt − 2πif∫ ∞ ϕIW(t) dt, and it transpires 0 e 0 e ∞ that σ(f=0) = ∫ 0 ϕI(t) dt. We decompose the regularized susceptibility Δχ(f) into three separate terms according to28 Δχ( f) = χW( f) + χIW( f) + ΔχI(f):

∫0

χW (f ) = ϕW (0) − 2π if χIW (f ) = −2 ΔχI (f ) = −

∫0

i 2πf

∞

∞

e−2π ift ϕW (t ) dt

e−2π ift ϕIW (t ) dt

∫0

∞

(e−2π ift − 1)ϕI(t ) dt

(6) (7)

(8)

The full derivation is given in the Supporting Information. Spectral Decomposition. We split the water polarization correlation functions into different solvation shell contributions23,29−32 distinguished by the index k according to

■

1 3VkBT ϵ0

∫0

Defining the autocorrelation and cross-correlation functions of water polarization and ion current as

ϕWk (t ) =

METHODS Dielectric Response Functions. The complex frequencydependent electric susceptibility χ( f) = χ′( f) − iχ″(f) connects the total system polarization P⃗( f) to the electric field E⃗ ( f) via the linear-response relation P⃗( f) = χ(f)ϵ0E⃗ ( f), where ϵ0 is the vacuum permittivity. According to the fluctuation dissipation theorem,26,27 χ( f) follows from equilibrium polarization correlations via χ (f ) = −

1 3VkBT ϵ0

1 3VkBT ϵ0

NW

∑ ∑ ⟨pi ⃗ (0)·pj⃗ (t )⟩ i ∈ {Nk} j = 1

(9)

where p⃗i is the polarization of water molecule i and {Nk} is the set of water molecules in the solvation shell indexed by k; NW is the total number of water molecules. Note that this decomposition is not unique, but it allows discussion of spectral contributions in an insightful way. Following refs 33−35, we further split the correlation functions into selfcorrelations ϕkW,self and collective correlations ϕkW,coll according to

(1)

k ϕW,self (t ) =

→̇ ⎯ where P (t ) denotes the time derivative of the time-dependent total polarization P⃗ (t), V is the system volume, and kBT is the thermal energy. For a salt solution P⃗ consists of the water polarization P⃗ W and the ion polarization P⃗ I according to P⃗ = P⃗ W + P⃗I; in a periodic system it is convenient to express P⃗I via the →̇ ⎯ ion current JI⃗ according to JI⃗ (t) = PI(t ). χ( f) has a lowfrequency diverging imaginary part due to the ion dc conductivity, so what is experimentally typically reported is the regularized expression Δχ( f) = χ( f) + iσ0/(2πf), where σ0 = σ(f=0) is static conductivity of the ions. The frequency-dependent ion conductivity σ(f) = σ′( f) − iσ″(f) relates the ion current to the applied electric field according to JI⃗ ( f) = σ( f)ϵ0E⃗ ( f) and follows from the polarization−ion current cross-correlations via

k ϕW,coll (t ) =

1 3VkBT ϵ0

1 3VkBT ϵ0

∑

⟨pi ⃗ (0) ·pi ⃗ (t )⟩

i ∈ {Nk}

(10)

NW

∑ ∑ ⟨pi ⃗ (0)·pj⃗ (t )⟩ i ∈ {Nk} j ≠ i

(11)

By construction, the sum over the solvation shell index k of the self and collective terms returns the total water polarization autocorrelation function: ϕ W (t ) =

k k (t ) + ϕW,coll (t )) ∑ (ϕW,self k

(12)

The decomposed spectral contributions follow as χWk (f ) = ϕWk (0) − 2π if 11668

∫0

∞

e−2π ift ϕWk (t ) dt

(13)



Article

Figure 1. (A) The radial distribution functions gNa,O(r) between sodium and water oxygen are very similar for all studied 1 M salt solutions; the curves almost perfectly superimpose. The black dashed lines at the minima separate first, second, and third solvation shell water. (B) The first maximum of the radial distribution function g−,O(r) between anions and water oxygen moves to smaller separation as the anion size decreases. The orange (black) dashed lines separate the solvation shells for F (Cl); the dividing separations for Br and I are similar to those for Cl and not shown. (C) Water number fractions in the different solvation shells nkW for all studied salt solutions; the first solvation shell is further divided into cationic and anionic solvation water W1+ and W1−, respectively. (D) Static susceptibility contributions χ′kW( f=0) of the different water solvation shells. (E, F) Decomposition of the imaginary and real parts of the dielectric spectrum Δχ(f) for 2 M NaCl into the water contributions W1+, W1−, W2, and W3, the ion−water cross-contribution IW, and the ion contribution I (which is multiplied by a factor of 5 for better visibility). The dashed lines denote single Debye fits, and the solid lines are Cole−Cole fits to the simulation data. k k χW,self (f ) = ϕW,self (0) − 2π if

∫0

∞

simulated densities of different salt solutions NaF, NaCl, NaBr, and NaI at 1 M as well as NaCl at 0.5 and 2 M compare well with experimental data, which serves as an additional validation of our ionic force fields (see the Supporting Information). Because of memory limitations, all trajectories are split in segments of 20 ns length for data analysis. The polarization components of each segment are Fourier transformed via fast Fourier transformation (FFT), and correlations are calculated by multiplication in Fourier space. After back transformation into the time domain, correlation functions are averaged over all segments. A time cutoff is used during calculation of the spectra via Fourier transformation of the correlation functions as discussed in the Supporting Information. For the calculation of total polarization correlation functions, we update the decomposition of water into different solvation shells every 10 fs according to the current configuration; consequently, the water partitioning changes in time. For the calculation of the self-polarization correlations, the trajectories are split in segments of 100 ps length. The dipolar autocorrelation function of each water molecule is calculated and then clustered into the different solvation shells depending on the water configuration at the beginning of each trajectory segment. A time of 100 ps is sufficiently long to obtain correlations in the relevant time range; on the other hand, it is short enough that switching of the solvation shell is infrequent and thus negligible. The collective correlations are obtained by subtracting the selfcorrelation from the total correlation.

k e−2π ift ϕW,self (t ) d t

(14) k k χW,coll (f ) = ϕW,coll (0) − 2π if

∫0

∞

k e−2π ift ϕW,coll (t ) d t

(15)

Therefore, the water dielectric contribution χW(f) can thus be split into a maximal number of eight different terms: χW( f) = ∑k (χkW,self( f) + χkW,coll(f)), where the index k further separates the third, second, and first cationic and first anionic solvation shell water. Simulation Methods. We use GROMACS 4.5.436 to simulate three separate trajectories with a duration of 100 ns each. For pure water only one trajectory and for NaF five trajectories are simulated. The simulation box contains about 7000 water molecules in the NPT ensemble at 300 K with a pressure coupling time of 1 ps. We use the Nose−Hoover thermostat with a coupling time of 1 ps and a 2 fs integration time step. The choice of the simulation ensemble (NPT versus NVT) and the barostat and thermostat coupling times influence the results only negligibly, as shown in the Supporting Information. The neighbor list is updated every 20 fs, and trajectories are saved every 10 fs. Electrostatics are computed by particle mesh Ewald methods, and the Lennard-Jones interactions are subject to a switch cutoff between 1.1 and 1.2 nm. We use the SPC/E water model37 and our previously optimized ion force fields with standard mixing rules for chloride and bromide and nonstandard mixing rules for fluoride and iodide; for iodide we choose the I(4) force field.38 The 11669



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Figure 2. (A, B) Real and imaginary parts of the dielectric susceptibility Δχ(f) of NaCl solutions at different concentrations. Dots are simulation data; solid lines denote Cole−Cole fits. The inset in (B) shows the Cole−Cole exponent α for different NaCl concentrations from simulations (circles) and experiments20−22 (stars). (C) Static dielectric constant ϵCC obtained from Cole−Cole fits to the dielectric spectra from simulations (circles) and experiments20−22 (stars). Besides the general offset of about 10 between simulations and experiments, the simulations reproduce the experimental trends. (D, E) Real and imaginary parts of the dielectric susceptibility Δχ( f) of different 1 M sodium halide solutions. Dots are simulation data; solid lines denote Cole−Cole fits. The inset in (E) compares the corresponding Cole−Cole exponent α from simulations (circles) and experiments20−22 (stars). (F) Cole−Cole relaxation time τ from simulations (circles) and experiments20−22 (stars). Besides the overall shift by about 2 ps, the experimental ion specificity and concentration dependence are reproduced.

■

ion volume and nonideal mixing effects, the sum ∑k nkW does not equal unity, so when dividing the spectral water contributions by nkW we eliminate trivial water dilution effects and can therefore extract the spectral contribution per water molecule in a meaningful way. Figure 1C shows that for all 1 M salt solutions studied by us about 20% of the water is first solvation shell water (W1), with a rather even distribution among cationic and anionic solvation water, and the remaining water is roughly equally partitioned between the second and third (and beyond) solvation shells. For NaCl we see that increasing salt concentration moves third solvation shell water, W3, to first solvation shell water, W1, andto a lesser degreealso to second solvation shell water, W2. Comparison with Experimental Dielectric Spectra. In Figure 2A,B dots show simulation data for the real and imaginary dielectric functions for NaCl solutions at three different concentrations; for comparison we also include the pure water spectra. The solid lines are fits according to the Cole−Cole function:

RESULTS AND DISCUSSION Water Hydration Shells. We first discuss structural properties of our simulations of alkali halide solutions. Note that we simulated NaF, NaCl, NaBr, and NaI solutions at the concentration of 1 M, while for NaCl we additionally performed simulations at 0.5 and 2 M. The radial distribution functions gNa,O(r) between sodium and water oxygen in Figure 1A and g−,O(r) between anions and water oxygen in Figure 1B exhibit well-defined maxima and minima. In agreement with previous simulations,39−41 the positions of the local minima in the radial distributions, indicated by orange vertical dashed lines when F is the anion and by black dashed lines when Cl is the anion (Br and I give positions very similar to Cl and are therefore not shown), allow distinguishing the first solvation shell water, W1, second solvation shell water, W2, and third solvation shell water (including all remaining water molecules), W3, as indicated in the figures. Note that the state k of a water molecule is defined by the distance to the closest ion. The first solvation shell we further decompose into cationic and anionic solvation water, W1+ and W1−, depending on whether water is closer to sodium or an anion. The water number fraction in the different solvation layers, denoted by nkW, where the index k distinguishes different solvation shells, is normalized with respect to the equivalent number of water molecules in bulk water, ∑k nkW = NW/Nbulk, where NW is the number of water molecules in the simulation at a given salt concentration and Nbulk = Vρbulk is the corresponding number of water molecules in the absence of salt, where V is the volume of the simulation box and ρbulk = 33.4 nm−3 is the pure water number density. Due to the finite

Δχ (f ) + 1 =

ϵCC − ϵ∞ 1 + (2π ifτ )1 − α

+ ϵ∞

(16)

with the dielectric constant ϵCC, the dielectric relaxation time τ, and the exponent α as free fitting parameters. We fix ϵ∞ = 1 since our force fields do not include high-frequency atomic polarization effects. Real and imaginary parts are fitted simultaneously with logarithmically distributed data points in the frequency range between 0.1 and 100 GHz, and the imaginary part is doubly weighted because of its lower absolute value (the maximum of which is smaller than the maximum of 11670



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Figure 3. Normalized polarization correlation functions of the different water solvation shells for 1 M NaCl for (A) the self-term ϕkW,self(t)/ϕkW,self(0), (B) the collective term ϕkW,coll(t)/ϕkW,coll(0), and (C) the sum of both terms ϕkW(t)/ϕkW(0). The corresponding results for pure bulk water are indicated by red lines. Simulation results are shown by solid lines; dashed lines are double Debye fits in the range [0, 25 ps]. (D) Self-part of the spectral absorption per water molecule of different solvation shells χ″kW,self( f)/nkW. Simulation results are shown by solid lines; the dashed lines are Cole−Cole fits to the spectra. (E) Collective part of the spectral absorption per water molecule of different solvation shells χ″kW,coll( f)/nkW. (F) Contribution of the different solvation shells to the total spectral absorption per water molecule χ″kW( f)/nkW. The insets in (D)−(F) show the static dielectric contribution per water molecule in the different water shells. The color labeling is identical in all subfigures.

simulated and experimental static dielectric constant ϵCC in Figure 2C and the relaxation time in Figure 2F is primarily due to SPC/E water itself. The changes inflicted by different salt types and different concentrations are reproduced very nicely in simulations when compared to experimental data. In the Supporting Information we plot experimental and simulated spectra on top of each other: because of the shifts of relaxation times and dielectric constant the comparison is not straightforward. In addition, in the insets of Figure 2B,E we compare the exponent α obtained from Cole−Cole fits of the simulation and experimental data. This exponent quantifies the spectral deviations from a single Debye form that is recovered for α = 0; the agreement for NaCl and different concentrations in Figure 2B and for different salt solutions at 1 M in Figure 2E is remarkable. We conclude that the simulation model describes the experimental ion-specific dielectric spectra on a satisfactory level, which validates both the simulation methods and the nonpolarizable optimized ion force fields employed in the present study. In the following we use spectral decomposition into different water solvation shells (which is experimentally difficult but straightforward in simulations) and into singlewater and collective water contributions (which reflects different experimental techniques used to study solvation water dynamics) in order to gain insight into the mechanism behind the dielectric changes we see in Figure 2 as a function of both salt concentration and ion type. Spectral Decomposition into Water Solvation Shells. In Figure 1E,F we show a decomposition of the real and imaginary dielectric spectra for 2 M NaCl into the ion−ion ΔχI( f), the ion−water χIW( f), and the four different water solvation shell contributions: χW1+( f), χW1−( f), χW2(f), and

the real part roughly by a factor of 2). One clearly sees that the dielectric response of the NaCl solutions decreases with increasing salt concentration, a well-known effect that is known as dielectric saturation.42−45 Ion-specific dielectric effects are addressed in Figure 2D,E, where we compare simulated spectra for NaF, NaCl, NaBr, and NaI solutions at the fixed concentration of 1 M. Again, simulation data are shown by dots and corresponding Cole−Cole fits are shown by solid lines. Here we see that with increasing anion size the dielectric response diminishes: the NaF solution (orange line) has the strongest response while NaI (pink line) has the weakest response. In Figure 2C we compare the dielectric constant ϵCC from the Cole−Cole fits of our simulation data (circles) with corresponding experimental results (stars).20−22 Besides a shift by about 10−15 we also observe for pure water, for different salt types, as well as for NaCl at different concentrations, the experimental trends are very well captured. This in particular holds for the decrease of the dielectric constant as the NaCl concentration increases and as the anion size increases. In Figure 2F we compare the relaxation time τ obtained from Cole−Cole fits of our simulation results with corresponding experimental data.20−22 Also here clear trends are discerned: In simulations as well as in experiments, 1 M NaF shows the longest correlation time of all 1 M salt solutions, while 1 M NaI has the shortest correlation time. The ordering of the three anions Cl, Br, and I at 1 M is somewhat peculiar, as Br has the largest relaxation time of these three ions, an anomaly that is nicely brought out by the simulations. We see that all simulation relaxation times are shifted up by about 2 ps when compared with experimental data; this in particular holds for pure water. We conclude that the deviations between the 11671



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Figure 4. Parameters of the Cole−Cole fits to the simulated spectral water contributions for all salt types and salt concentrations, sorted with respect to the different solvation shells. (A) Relaxation times of the self-correlation part τW,self, (B) relaxation times of the collective correlation part τW,coll, and (C) relaxation times of the sum of self part and collective part τW. (D−F) Cole−Cole exponent α of the self-correlation part, the collective correlation part, and the sum of both. The results for pure water are denoted by broken horizontal lines. The dashed black lines are guides to the eye to separate results for different solution shells (in the middle) from the cationic/anionic first solvation shell data (left) and the total water results (right). The color labeling is identical in all subfigures.

χW3( f). Solid lines present Cole−Cole fits, while broken lines are single Debye fits to the simulation data. This comparison is interesting, since experimental results are often decomposed into a sum of single Debye processes. Note that the sums of all six contributions equal the total spectra shown as green lines in Figure 2A,B. The ion−ion contribution ΔχI( f), which is multiplied by a factor of 5 for better visibility, is significantly blue shifted as compared to the other contributions. It is the weakest contribution and only has a negligible influence on the total spectrum. The ion−water cross-contribution χIW(f) is in fact negative and thus suppresses the dielectric response, a wellknown effect predicted by the Hubbard−Onsager theory and due to a kinetic ion−water coupling.46 The second solvation shell water contribution χW2( f) (blue line) by far constitutes the dominating contribution for 2 M NaCl, while χW1+( f), χW1−(f), and χW3( f) give smaller and comparable contributions. In Figure 1D we present the dielectric constants for the different solvation water contributions. The data roughly follow the water number fractions nkW presented in Figure 1C with the exception of first solvation shell water, which has a smaller dielectric constant than expected based on the number fraction, as will be discussed in more detail further below. As the water contribution to the dielectric spectrum is the dominant one, we will in the following concentrate on analyzing the water part in more detail and in particular investigate the question of what causes the change of the water dielectric response in the ion solvation layers (when compared with bulk water) and what we can learn about the solvation water dynamics from that. Self versus Collective Dielectric Contributions. In Figure 3A we compare the time evolution of the normalized polarization self-correlation function ϕkW,self(t) of different water

solvation shells defined in eq 10 for 1 M NaCl; solid lines are simulation data and broken lines are double-exponential fits in the time range [0, 25 ps]. The self-correlation function of the entire water ensemble ϕW,self(t) is denoted by a black line. The cationic and anionic first solvation shell correlations ϕW1+,self(t) and ϕW1−,self(t) show the slowest decay, consistent with a recent simulation study,14 while the third solvation shell contribution ϕW3,self(t) is the fastest self-correlation function but still slower than the corresponding self-correlation function for pure water (red line). A simple-minded explanation is that the ion electric field orients the water dipole and thereby slows down reorientational motion; a more refined theory takes collective water effects into account and can in particular explain ionconcentration effects on the water dynamics.11,12,14 The slowing down of water is stronger for inner solvation shells but still sizable for third solvation shell water. The general trends are in line with 2D IR vibrational echo spectroscopy experiments where single water dynamic retardation in NaBr solutions was observed.15 Similar retardation was measured experimentally in trimethylamino-N-oxide (TMAO) solutions.6 How can the retardation of the single-water polarization correlation functions in salt solution in Figure 3A be brought in harmony with the pronounced acceleration of the dielectric relaxation shown in Figure 2F? In Figure 3B we show the normalized collective polarization correlation functions ϕkW,coll(t) for the different water solvation shells defined in eq 11 in 1 M NaCl, again compared with pure water (red line). We see exactly the opposite trend as for the self-term: The correlations in the inner solvation shells decay fastest, while the third solvation shell correlations W3 decay similarly to pure water. In fact, water in the vicinity of cations shows a quite abrupt drop of collective correlations after about 20 ps. The 11672



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Figure 5. (A) Plot of the static dielectric contribution per water molecule χ′kW(f=0)/nkW versus the Cole−Cole relaxation time τkW for the different water solvation shells of 1 M NaCl. The result for pure water is indicated by a red circle. The broken lines are straight lines fitted separately to pure water, to the second (W2) and third solvation shell (W3) results, and to the first solvation shell results. (B) Plot of the Kirkwood factor gK = (2ϵ + 1)χ′kW(f=0)/(3ϵχ′kW,self( f=0) ) (black circles), the ratio of the dielectric water relaxation time and the single water relaxation time τkW/τkW,self (blue triangles), the rescaled dielectric relaxation time τkW/τW (green squares), and the rescaled single-water relaxation time τkW,self/τW,self (red stars) for different water solvation shells in 1 M NaCl. τkW is normalized with respect to the dielectric relaxation time of the entire water ensemble τW, and τkW,self is normalized with respect to the single-water relaxation time of the entire water ensemble τW,self. (C) Plot of the ratio χ′kW( f=0)τkW,self/ (χ′kW,self( f=0)τkW) for the different water solvation shells in 1 M NaCl, which constitutes a test of the simplified Madden−Kivelson equation, eq 17. The horizontal dashed line denotes the corresponding result for pure bulk water.

total polarization correlation function ϕkW(t) for the different solvation shells k is, according to eq 12, the sum of the self part ϕkW,self(t) and the collective part ϕkW,coll(t). Our results in Figure 3C show that ϕkW(t) is dominated by the collective part and shows similar qualitative features. In conclusion, although ions slow down the individual ion reorientation dynamics in the solvation shells, the collective dynamics is accelerated so that the total polarization correlations (which are dominated by collective or cross-correlation contributions between different water molecules) decay faster in the inner solvation shells compared to the outer shells. Before we discuss the mechanism behind this counterintuitive collective acceleration of the solvation-water dynamics, we display the corresponding behavior in the spectral representation. In Figure 3D−F we show the dielectric absorption (i.e., the imaginary part of the dielectric function) per water molecule due to self-correlations, collective correlations, and the sum of both correlations, respectively, all for 1 M NaCl. Solid lines denote simulation data, dashed lines are Cole−Cole fits for logarithmically distributed data points in a frequency range between 0.1 and 100 GHz, and the insets show the static susceptibilities per molecule. The self spectra for the different water solvation shells in Figure 3D are quite similar to each other, which is remarkable in light of the pronounced differences seen in the corresponding correlation functions as a function of time in Figure 3A. This just stresses the often subtle connection between real-time data and its spectral representation. The static self contribution in the inset in Figure 3D is independent of the solvation shell, reflecting the trivial fact that the polarizability of a single water molecule is constant and proportional to the dipole moment squared. The spectra are weakly red-shifted for the inner solvation shells, which is more easily discerned in Figure 4A where we plot the relaxation times for all salt solutions and all water solvation shells as extracted from Cole−Cole fits to the spectral contributions. In Figure 4A it is seen that the spectral selfrelaxation time increases also with increasing salt concentration and with decreasing anion size and is strictly larger than the pure water relaxation time (denoted by a horizontal red dashed line).

The collective spectra for the different water solvation shells in Figure 3E differ dramatically from each other. The absorption per water molecule decreases substantially for the inner solvation shells and is weakest for the first solvation shell around Na (shown as a purple line). The static contribution per water molecule shown in the inset is next to sodium, W1+, reduced by about 80% compared to the pure water value indicated by a red dashed line. Next to chloride W1− the reduction is still about 50%. The plot of the collective relaxation times in Figure 4B demonstrates a pronounced blue shift of the collective spectra as one goes from the outer solvation shells to the inner shells; this blue shift is most dramatic for the cationic first solvation shell water W1+. The reason for the pronounced symmetry breaking between the cationic and anionic first solvation shells is twofold: first Na is smaller than all anions and second the partial charge on the water oxygen is more localized than on the water hydrogens, leading to a stronger orientation effect of water around Na. The properties of the total dielectric spectra in Figure 3F, which are the sum of the self and collective spectra in Figure 3D,E and which are measured experimentally, are dominated by the collective spectral contributions; i.e., they are attenuated (see inset in Figure 3F) and blue shifted (see Figure 4C) for the inner solvation shells. We finally want to address the question as to what causes the rather nonintuitive blue shift of the spectral contributions from the inner solvation shells demonstrated in Figure 4C, which is responsible for the blue shift of the total dielectric spectra in Figure 2F in the presence of salt when compared to pure water. Since the spectral contributions are dominated by the collective parts, as clearly seen by a comparison of the amplitudes of the spectra in Figure 3D,E, this question is relegated to the nature of the collective polarization dynamics. The Madden−Kivelson theory for one-component liquids in its simplified version predicts a linear relation between the static dielectric constant χ′(f=0) and the dielectric relaxation time τ:47 gK

χ ′(f = 0) 3ϵ τ ≡ = 2ϵ + 1 χ ′self (f = 0) τself

(17)

where the proportionality factor is the inverse single-molecule (or self) polarization relaxation time τself and ϵ is the static 11673



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show marked deviations from the pure water behavior. Supposedly here the orientational electric field of the ions constitutes an additional mechanism that influences both the water dielectric constant and the relaxation time, and which is not captured by the Madden−Kivelson theory. The second and third solvation results denoted by W2 and W3 in Figure 5C are rather close to the pure water results. Since the deviations of the different solvation shells from the horizontal red dashed line almost cancel, the average results for the 1 M NaCl solution denoted by W almost perfectly agree with the pure water results. As a consequence, this means that the effects we are analyzing here are easily missed if only the entire water ensemble is considered. Finally, in Figure 4D−F we display the Cole−Cole exponent α of the various water solvation shell contributions for the different salts, separately for the self contribution in Figure 4D and the collective contribution in Figure 4E, and the sum is shown in Figure 4F. This parameter quantifies the spectral deviations from a simple single Debye-like process. We see that the self-relaxation process in Figure 4D is characterized by a large positive exponent for all salt types and for all solvation shells, while the collective contribution in Figure 4E shows (except for the first solvation shell around Na denoted by W1+) a slightly negative exponent. Although the collective contribution dominates the sum, the resulting exponent α in Figure 4F turns out to be positive for all water solvation layers, with the cationic first solvation shell showing the largest exponent. This means that a decomposition of the total dielectric signal into contributions coming from different solvation shells does not make the separate processes more Debye-like; conversely, the proliferating non-Debye-like character of the dielectric spectra of, e.g., NaCl solutions with increasing concentration demonstrated in Figure 2B seems not to be indicative of two distinct ionic and water-like contributions at different time scales (an interpretation that is often used in the discussion of experimental spectra19,20). Rather, the individual dielectric relaxation processes in a salt solution are intrinsically non-Debye-like. Indeed, our decomposition into self and collective contributions in Figure 4D−F shows that typically massive cancellation between different contributions takes place: Even for pure bulk water, indicated by horizontal broken lines, the separate self and collective contributions are characterized by relatively large exponents of opposite signs. The sum of the two spectra in Figure 4F happens to be almost perfectly described by a pure Debye spectrum. For the inner solvation shells this cancellation is less perfect, so that the effective exponents of the sums of self and collective contributions show larger deviations from Debye behavior. As a consequence, water in the first cationic solvation shell is shown to be predominantly responsible for the nonDebye-like character of concentrated ionic solutions, as observed in our simulations and in agreement with dielectric resonance spectroscopy experiments.20−22 Results for the Second Legendre Polynomials. Polarization-selective pump−probe techniques measure the orientational relaxation of the OD bond pseudovector of single D2O or HOD molecules in solution. By careful analysis the signal of solvation water can be separated from the bulk water contribution.15,17,52,53 What experimentally is measured corresponds to the decay of the orientational correlation function P2(⟨u⃗OH(0)·u⃗OH(t)⟩), where P2(x) = (3x2 − 1)/2 is the second Legendre polynomial and u⃗OH(t) is the normalized OH-bond vector. The dipolar correlation function considered so far and

dielectric constant of the medium. Here we introduced the Kirkwood factor gK which can be viewed as the dielectric enhancement factor which quantifies by how much correlation effects enhance (or decrease) the dielectric constant of a liquid over the value of an isolated molecule.48 Note that χ′self(f=0) is a constant and only depends on the molecular dipole moment. For water, gK ≃ 2.3 in previous simulations49 and gK ≃ 2.7 in experiments,50 which is rationalized by the presence of temporary water clusters with parallel aligned dipole moments, which gives rise to an effectively larger dipole moment and thus an enhanced polarizability. Note that for some liquids, where antiparallel dipolar alignment is preferred, the Kirkwood factor can also be smaller than unity.51 It is intuitively clear that clusters of molecules with aligned dipoles will relax slower compared to uncorrelated single molecules, which on a simplified level captures the essence of eq 17. We will now apply the Madden−Kivelson theory, originally derived for homogeneous single-component liquids, to the present scenario of water that is partitioned into different solvation shells. In Figure 5A we test eq 17 by plotting the contribution of the different water solvation shells to the static dielectric constant per water molecule χ′kW(f=0)/nkW of the different water solvation shells for a 1 M NaCl solution versus the respective Cole−Cole relaxation time τkW. We see that pure water, third solvation shell water W3, and second solvation shell water W2 lie nicely on a straight line. The cationic and anionic first solvation shell data also lie on a straight line albeit with a different prefactor, which according to eq 17 would suggest that water in the first solvation shell around ions is characterized by a larger self-relaxation time than second or third solvation shell water. Indeed, this is in line with the trends seen for the selfrelaxation times for different solvation shells in Figure 4A. The Madden−Kivelson-type scaling exhibited by the data in Figure 5A suggests that the dielectric spectra of inner solvation shell water are blue shifted not because individual water molecules themselves move faster (which we know they do not, as demonstrated in Figure 3A), but rather because the solvation structure breaks and decreases the cooperative nature of the dipolar ordering in the first solvation shell. To corroborate this reasoning, in Figure 5B we plot the various quantities that appear in eq 17 for the different water solvation shells in a 1 M NaCl solution. The black circles denote the Kirkwood factor gK, which in the first solvation is considerably smaller than in the second or third solvation shell. This shows that the presence of an ion almost completely destroys the cooperative water structure that leads to the large value of gK ≃ 2.5 in pure bulk water, which we determined in separate simulations; this is why ions can be called very efficient dielectric structure breakers. Note that our value of gK ≃ 2.5 differs from the previous result gK ≃ 2.349 and is in fact closer to the accepted experimental value of gK ≃ 2.7.50 The ratio of dielectric relaxation and selfrelaxation times τkW/τkW,self (denoted by blue triangles in Figure 5B) qualitatively follows the Kirkwood factor gK, but shows less drastic variations. Finally, the self-relaxation time τkW,self (red stars) indeed is slightly larger in the first solvation shell, reflecting that the water reorientation is hindered, in line with our above reasoning, while the dielectric relaxation time τkW (green squares) is slightly smaller in the first solvation shell compared with the outer shells. We finally test the Madden− Kivelson equation in Figure 5C by plotting the ratio χ′kW(f=0)τkW,self/(χ′kW,self(f=0)τkW); deviations from unity indicate deviations from eq 17. The horizontal broken line denotes the ratio for pure water. We see that the first solvation shell results 11674



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Figure 6. (A) Results for the second Legendre polynomial of the water OH-bond autocorrelation function for different solvation shells in 1 M NaCl. Solid lines are simulation data and broken lines are double Debye fits in the range [0.2, 25 ps]. Results for pure bulk water are denoted by red lines. Black circles denote denote experimental results for 1 M NaCl,17 and red diamonds denote experimental results for pure water.15 The short-time behavior is shown in (B). (C) Slower relaxation time τslow of the double Debye fits in (A) for all different salt solutions and concentrations, sorted according to the different solvation shells. All relaxation times are slower than the pure water value (denoted by a horizontal red dashed line).

earlier simulations for fluoride,13,54 chloride,11,14,54 bromide,14,54−56 and iodide.14,54 Our results are not able to resolve the experimental trend of increasing relaxation times of P2 with decreasing anion size from iodide to chloride.57 Recent experiments demonstrate slowing down of the relaxation time of P2 with increasing concentration of NaI17 consistent with our simulations, but in contrast to an earlier simulation study for NaI using polarizable force fields.13 The second Legendre polynomials of the dipole vector and the hydrogen−hydrogen vector of single water molecules are shown in the Supporting Information and decay similarly to the OH-bond vector in Figure 6. Summarizing this section, our results obtained in the main part of this paper for the self and collective decays of polarization or dipolar correlations also pertain to orientational correlations, but the ion-specific trends and the effects seen in the inner solvation shell are somewhat weaker.

presented in Figure 3A (corresponding to the first Legendre polynomial) and the second Legendre polynomial show the same relaxation only for single Debye processes. Since our process is more complex, it seems warranted to check whether P2(⟨u⃗OH(0)·u⃗OH(t)⟩) shows behavior similar to that of the dipole correlation. In Figure 6A we present P2(⟨uO⃗ H(0)·u⃗OH(t)⟩) for all different solvation water components for a 1 M NaCl solution and also for pure water. Simulation data are shown by solid lines, while the broken lines denote double Debye fits in the range [0.2, 25 ps]. We observe a fast decay within the first 200 fs which is independent of the presence and type of ions; this is more clearly demonstrated in Figure 6B, where pure water and the data for all water solvation shells in 1 M NaCl are shown to exhibit identical decays at short times (all solid lines superimpose perfectly). At longer times the different solvation shells in Figure 6A show distinct decay times: the first solvation shells W1+ and W1− exhibit a slower decay than the second solvation shell W2, which itself decays more slowly than the third solvation shell W3. In agreement with our simulations, in experiments P2 of the OH vector decreases faster for pure water15 (red diamonds) than for 1 M NaCl denoted by black circles.17 Note, however, that the experimental data for water and 1 M NaCl in fact decay considerably slower than our simulation results. In fact, this discrepancy occurs in the opposite direction than for the dielectric relaxation time presented in Figure 2F, where experimental relaxation is seen to be faster than simulations. Weighting of simulation results with the vibration excitation probability and rescaling of the experimental data has been shown to improve the consistency between experimental and simulation results of concentrated NaCl aqueous solutions,11 but this was not done in the present work. To display ion-specific effects, we present the slow relaxation time of the double Debye fit of P2(⟨u⃗OH(0)·u⃗OH(t)⟩) in Figure 6C for all different salt types and concentrations sorted with respect to the different solvation shells. Similar to our results for the single-dipole relaxation times shown in Figure 4A, relaxation times slightly increase as one moves from the outer solvation shell W3 to the first solvation shell W1− and W1+, but the magnitude of these changes are somewhat smaller for P2 when compared to the dipolar correlations discussed before. Water in all salt solutions exhibits slower relaxation than in pure bulk water. Here we refer to the results summed over all solvation shells denoted by W in Figure 6C, in agreement with

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CONCLUSIONS Dielectric spectroscopy experiments on concentrated sodium halide solutions demonstrate a characteristic blue shift of the main dielectric relaxation when compared with the pure water spectrum. This blue shift increases with increasing salt concentration and also with increasing anion size (except Br, which shows a weak anomaly). On the other hand, ultrafast spectroscopic methods that directly obtain the single-water molecule reorientation dynamics17 demonstrate that solvation water is slower than bulk water for elevated salt concentrations. We resolve this puzzle by a decomposition of the dielectric response of solvation water into the self and the collective parts: the self part indeed is slower, while the collective part, which constitutes the dominating contribution to the total dielectric spectrum, is faster compared with bulk water. The acceleration of the collective dynamics of solvation water is rationalized by a decrease of water cooperativity in the first solvation shell around ions, which at the same time explains the drastically reduced static dielectric constant of salt solutions in terms of the dielectric structure breaking influence of salt ions. A similar connection between a blue shift in the absorption spectrum and a corresponding decrease of the static dielectric constant has recently been observed in simulations for thin water slabs.58 The self (local) dynamics of solvation water and the collective water dynamics as studied in dielectric spectroscopy are two extreme views of water motion. Experimental 11675



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measurements of femtosecond-resolved fluorescence shifts of excited probe molecules allow determination of the spectral response of solvation water averaged over an intermediate spatial scale;59 such experiments thus serve to bridge self and collective water dynamics. A corresponding comparison with the results reported in this paper would be worthwhile. In our work we use our recently optimized nonpolarizable ionic force fields in conjunction with the SPC/E water model. These force fields are the result of thermodynamic optimization with respect to experimental single-ion hydration free energies and with respect to activity coefficients of ion pairs.38 Our results show good to perfect agreement with experimental dielectric spectra and reproduce the experimentally observed ion-specific trends effortlessly. In many recent simulation studies polarizable force fields are used.14 We are not disclaiming the importance of polarizability effects, in particular when solvation effects around ions are involved. Our results indicate that such polarizability effects to a certain degree are accounted for in properly optimized force fields, in particular with force fields that are benchmarked with respect to singleion solvation free energies, since polarization effects are an important part of the solvation free energies. In the future, it would be certainly useful to redo the current simulation study with polarizable force fields that have been optimized with respect to thermodynamic observables relevant for solution properties. The additional advantage and challenge would be that a polarizable force field is determined by three parameters and thus in principle allows the simultaneous optimization with respect to three different experimental observables. Likewise, redoing similar simulations with a water model that more accurately reproduces the dielectric spectrum of pure water would be desirable. Clearly, for this the ionic force fields would have to be optimized from scratch.

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ASSOCIATED CONTENT

S Supporting Information *

Derivation of dielectric response functions and a test of the Madden−Kivelson equation for NaF, NaBr, and NaI. Furthermore, the second Legendre polynomial for the dipole vector and the hydrogen−hydrogen vector correlations is shown for 1 M NaCl. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank R. Buchner for discussions and for supplying additional unpublished data. We gratefully acknowledge financial support from the DFG via SFB 1078 and computing time on the HPC cluster at ZEDAT, Freie Universität Berlin. S.G. thanks the Volkswagen Foundation for financial support.

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REFERENCES

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Ion-Specific Solvation Water Dynamics: Single Water versus Collective

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