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Extrapolating Single Organic Ion Solvation Thermochemistry from Simulated Water Nanodroplets Jonathan Coles, Céline Houriez, Michael Mautner, and Michel Masella J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b02486 • Publication Date (Web): 15 Jul 2016 Downloaded from http://pubs.acs.org on July 19, 2016

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The Journal of Physical Chemistry

Extrapolating Single Organic Ion Solvation Thermochemistry from Simulated Water Nanodroplets Jonathan P. Coles,∗,†,‡ Céline Houriez,¶ Michael Meot-Ner (Mautner),§ and Michel Masellak Exascale Computing Research Lab, Campus Teratec, 2 rue de la Piquetterie, 91680 Bruyères-le-Chˆ atel, France, Université de Versailles Saint-Quentin-en-Yvelines, 55 avenue de Paris, 78000 Versailles, France, MINES ParisTech, PSL Research University, CTP Centre Thermodynamique des Procédés, 35 rue Saint-Honoré, 77300 Fontainebleau, France, Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284-2006, United States, and Department of Chemistry, University of Canterbury, Christchurch, New Zealand 8001, and Laboratoire de Biologie Structurale et Radiobiologie, Service de Bioénergétique, Biologie Structurale et Mécanismes, Institut de Biologie et de Technologies de Saclay, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France E-mail: [email protected]

∗

To whom correspondence should be addressed Exascale Computing Research Lab, Campus Teratec, 2 rue de la Piquetterie, 91680 Bruyères-le-Châtel, France ‡ Université de Versailles Saint-Quentin-en-Yvelines, 55 avenue de Paris, 78000 Versailles, France ¶ MINES ParisTech, PSL Research University, CTP - Centre Thermodynamique des Procédés, 35 rue Saint-Honoré, 77300 Fontainebleau, France § Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284-2006, United States, and Department of Chemistry, University of Canterbury, Christchurch, New Zealand 8001 k Laboratoire de Biologie Structurale et Radiobiologie, Service de Bioénergétique, Biologie Structurale et Mécanismes, Institut de Biologie et de Technologies de Saclay, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France †

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Abstract We compute the ion/water interaction energies of methylated ammonium cations and alkylated carboxylate anions solvated in large nanodroplets of 10,000 water molecules using 10 ns molecular dynamics simulations and an all-atom polarizable force-field approach. Together with our earlier results concerning the solvation of these organic ions in nanodroplets whose molecular sizes range from 50 to 1,000, these new data allow us to discuss the reliability of extrapolating absolute single-ion bulk solvation energies from small ion/water droplets using common power-law functions of the cluster size. We show that reliable estimates of these energies can be extrapolated from a small data set comprising the results of three droplets whose sizes are between 100 and 1,000 using a basic power-law function of the droplet size. This agrees with an earlier conclusion drawn from a model built within the mean-spherical framework and paves the road towards a theoretical protocol to systematically compute the solvation energies of complex organic ions.

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1

Introduction

Bulk hydration properties of chemical species may be estimated by extrapolating data derived from the solvation process of water nanodroplets of varying size. This method has, for instance, been used experimentally to investigate ion and single-electron hydration properties. 1–6 One may also approach the problem computationally by means of molecular simulations. The main difficulty there is to prevent evaporation phenomena along the trajectories, but this can readily be achieved by embedding the droplets within a sufficiently large cavity. That protocol was used, for instance, by Caleman et al 7 to simulate mono-atomic ions as well as in our own studies focusing on methylated ammonium cations 8 and alkylated carboxylate anions 9 solvated individually in water nanodroplets of between 50 to 1,000 water molecules. In particular, from our droplet simulations, we extrapolated the absolute ion solvation enthalpy of the latter organic ion allowing us to compute a theoretical estimate of the absolute proton solvation enthalpy ∆Hsolv (H+ ) using standard thermodynamical cycles. Extrapolation by means of a power-law function must be done with care, however. Clauset et al 10 discussed in detail, from a numerical point of view, the main parameters affecting the quality of power-law extrapolation. These authors highlighted the difficulty of identifying the range of data over which the power-law approximation holds. To illustrate the problem, consider a physical quantity A, such as the ion solvation enthalpy in an aqueous environment. The most common power-law function used to extrapolate bulk properties from droplet data is F = A∞ + B/R, where A∞ is the bulk converged value of A and R ∼ n1/3 is the droplet radius. The function F corresponds to a finite-size formulation of the ion solvation electrostatic energy in the Born continuum dielectric model 11–13 (note that other F -like functions corresponding to a larger expansion in R−1 have also been proposed, in particular by Coe et al 6 ). According to the discussions of Rips and Jortner, 11 the function F represents well the size dependence of the ion solvation energy in droplets with at least n > 125, 11 regardless of the location of the solute within them. However, experimentally deriving a data set covering a large range of droplet sizes is 3



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far from being always possible. For instance, based on an F -like function, the (H2 O)− n cluster photoelectron lineshape was extrapolated to bulk from n ≤ 60 data 5,6 and the proton solvation enthalpy ∆Hsolv (H+ ) was extrapolated from experimental Eu3+ /(H2 O)n data where 55 ≤ n ≤ 140. 1 Through computational simulations, it is possible to study a large range of droplet systems. For instance, in our previous studies focusing on the solvation of organic ions, 8,9 we simulated five different droplets whose sizes span a larger range of 50 ≤ n ≤ 1000. Although the data corresponding to these droplet sizes are well suited to perform F -based extrapolation according to the discussions of Rips and Jortner, 11 these authors showed also that a true F behavior of the ion solvation energy is expected only for n ≥ 1000. Hence, one may argue that our data set was still too small to ensure a highly reliable extrapolation in our former studies. 8,9 To assess the reliability of extrapolating bulk ion hydration energy values from a set of droplets whose size n ≤ 1000, we performed 11 new simulations with methylated ammonium cations and linear chain carboxylate anions, each embedded alone in water droplets with n = 10, 000. These simulations were performed at the 10 ns scale by considering sophisticated polarizable force fields to model both ion/water and water/water interactions. They provide new data that are considered together with our earlier ones to extrapolate new sets of bulk values, in particular by considering different power-law functions equal to or converging to the power-law function F as n → ∞. This allows us to assess the effects of the droplet data set size on the extrapolated values as well as the relevance of considering functions of greater or lesser sophistication to perform the extrapolation. This leads us to define the basic requirements of a transferable and reliable simulation procedure based on droplet simulations (in terms of fitting process, fitting function and data set size) that can be used to estimate the single-ion solvation energy of any kind of monoatomic and small organic ion.

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2

Theoretical details

In this section we discuss the physical model used to describe ion/water droplet systems, the simulations we performed, and details of the power-law functions used to extrapolate to bulk properties. For simplicity, we denote 1000 as 1k and 10,000 as 10k.

2.1

The model

The ion/water droplet systems are described using the same total potential energy U as in our former studies 8,9 that deal with methylated ammonium cations and linear alkylated carboxylates. For water/water interactions, we use the polarizable rigid water model TCPE/2013 that was shown to accurately model water in liquid and gas phase, as well as the water interactions in ionic first hydration shells. 9,14 To handle the ion/water interactions, we consider the following energy term ′

U iw = U rep + U qq + U pol + U disp + U shb

(1)

′

where U rep is a short range repulsive term and U qq is the standard Coulombic interaction term with the electrostatic charges qi located on the atomic centers. U disp is a standard dispersion term ∗

U

disp

=−

N N X X σij i=1 j>i

rij6

.

(2)

where rij is the distance between two atoms i and j and the superscript

∗

indicates that the

sum includes only pairs of atoms separated by more than two chemical bonds. The ion/water polarization term corresponds to the one from the water TCPE/2013 model where N

U

pol

N

N

N∗

µ µ µ µ X 1X p2i X 1X q pi Tij pj . = − pi · Ei − 2 i=1 αi i=1 2 i=1 j=1

5


(3)


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Only non-hydrogen atoms are considered as polarizable centers (i.e., Nµ < N ); their polarizabilities αi are taken as isotropic and their induced dipole moments pi obey

pi = αi ·



Eq i

N∗

+

µ X

j=1



Tij · pj 

(4)

where Tij is the dipolar interaction tensor and Eqi is the electric field generated on the polarizable center i by the surrounding charges qj . Both Tij and Eqi include a short-range damping component. Ion intramolecular degrees of freedom are handled by the term U rel that includes standard stretching, bending and torsional terms, as well as an improper torsional term for carboxylates. For ammonium ion/water interactions, we showed that the binding energies of small ion/water clusters computed from high level quantum computations are accurately reproduced by using only the above energy terms. 8 For carboxylate/water interactions, we need the additional strong hydrogen bond (SHB) term U shb to meet such a condition for small carboxylate/water clusters 9 U shb =

X

De (ρoh )f (rshb )g(ψ) .

(5)

The sum runs over all of the carboxylate/water strong hydrogen bonds. The Gaussian functions f and g are centered at the equilibrium values of the SHB length rshb and O − H · · · Ocarboxylate angle ψ, respectively. Environmental effects on the strength of a local SHB are accounted for by considering De as a linear function of the local density ρoh of water O-H bonds in the anion vicinity. More details about the above energy components and their parameters may be found in our former studies. 8,9 Importantly, we showed that the above ion/water energy term U iw is able to accurately reproduce the potential energy surface of small ion/water clusters in gas phase, as well as the water structural properties in the ion vicinity in bulk water. 6


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2.2

Simulation details

+ + + + In this study we considered the ions NH+ 4 , CH3 NH3 , (CH3 )2 NH2 and (CH3 )3 NH ); K ;

and the six linear alkylated carboxylate anions (from methanoate to hexanoate). We placed each ion alone in a 10k water molecule droplet and ran a single MD simulation of each ion/droplet system for 10 ns in the NVT ensemble using the same protocol as in our former studies (see also Supporting Information). The ions were initially located at the droplet centers but were then free to move under the action of the microscopic system forces. To speed up the simulations, we computed all the electrostatic and polarization interactions within the droplets using the Fast Multiple Method (FMM) that we recently proposed 15 for polarizable force-fields based on the induced dipole moment approach and as implemented in our parallel simulation code POLARIS(MD). The precision of our FMM approach is controlled by a parameter θ that determines when a multipole expansion approximation of two spatial domains is accurate enough to be used instead of the standard O(N 2 ) pairwise algorithm. Here, we set θ = 0.5 to ensure that the electrostatic and polarization FMM forces to differ on average by 0.1% compared to those computed from the O(N 2 ) scheme. To prevent water evaporation phenomena along the trajectories, the water molecules feel a smooth repulsive potential which is zeroed a few angstroms before the spherical domain where the atomic density is the highest (thus, the repulsive term only marginally affects the dynamical properties of the droplet systems). The cavity radius Rcavity is defined to be 12 ˚ A greater than the largest distance between a droplet atom and the droplet center of mass (COM) in the initial conditions. The smooth repulsive potential is zeroed at Rcavity − 4 ˚ A using a B-spline function. This is in contrast to our previous studies, where the water molecules underwent an elastic collision when they reached the boundary of the spherical cavity. This approach avoids having to remove the kinetic COM and global rotational energy after an elastic collision event.

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2.3

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Extrapolation to bulk ion/water interaction energies

iw We extrapolate the mean bulk ion/water interaction energy U¯∞ from a given droplet data

set (denoted DS) by considering four different power-law functions of the form iw Fa,ǫdisp (n) = U¯∞ +

ǫ0 ǫdisp . + 1/3 + a) (n + a)3

(n1/3

(6)

The functions Fa,ǫdisp (n) measure the ion/water interaction energy U¯ iw (n) in a droplet of size n. The simplest function F0,0 is the most commonly used to account for the ion/water electrostatic interactions in the Liquid Drop Model. 16 The parameter a was introduced to account for the ion size when estimating the droplet volume. 13 This parameter is expressed in ˚ A in the following discussion by scaling its value by the water molecular radius rs = (3/4πρs )1/3 ≈ 1.93 once the fitting process completes (ρs is the bulk liquid water density). In our former study dealing with the carboxylate hydration, 9 we considered a third term scaling as n−2/3 to account for the organic ion charge distribution effects on ion/water interactions. Such a term was also previously proposed by Coe et al 6 to extrapolate water cluster adsorption data, mainly for numerical reasons (i.e., to better account for the data curvatures corresponding to small clusters). Here, we rewrite that third term to account for ion/water long tail dispersion effect, which represents a non-negligible part of the organic ion/water interaction. The new term scales as n−1 (see Supporting Information), and in line with the arguments of Coe et al, 6 it is expected to better model the data corresponding to small droplets. Note that ǫdisp can be roughly estimated from the dispersion parameters σij corresponding to ion/water interactions in our microscopic models according to

ǫdisp ∼

X 16 2 2 π ρs σij . 9 i∈ion,j∈water

(7)

From our force-field dispersion parameter set, ǫdisp is expected to range from +30 to +100 kcal mol−1 . The parameters of Fa,ǫdisp are computed based on a standard mean least square fit by

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numerically minimizing the function

G(a, ǫdisp ) =

X

n∈DS

wn |U¯ iw (n) − Fa,ǫdisp (n)|2 ,

(8)

where wn is a positive weight. In the particular case of the function F0,0 , and as the exponent of n is set to a constant value, it is worth noting that same values of the F0,0 parameters are derived either by numerically minimizing the function G(0, 0) with all the weights wn set to unity or by using the linear regression formula where F0,0 is considered as a linear function of n−1/3 . The “ionic radius” a is expected to be a few ˚ A. Its effect on the droplet ion/water interaction energy Fa,ǫdisp (n) vanishes slowly as the droplet size n increases. For instance, the ratio a/n1/3 is still larger than 0.1 for n = 1000 and a = 4 ˚ A. Nevertheless, in an attempt to account for its expected stronger effect on small droplets, we minimize the functions Ga6=0,ǫdisp by setting the weight wn to a/n1/3 , with a = 4 ˚ A. However this leads to no significant difference in the best parameter set compared with all the weights wn set to unity. In the following discussion, we take the best parameters for Fa6=0,ǫdisp when using wn = a/n1/3 . Using the functions Fa,ǫdisp to measure the ion/water interaction energies in droplets implicitly assumes that the droplet is spherical. In our previous studies, we showed that the shape of all the droplets for n between 50 and 1k is quasi-spherical on average. 8,9 As discussed below, this is also the case for 10k droplets. Note that a small error was made when computing carboxylate/water interaction energies in droplets in our former carboxylate study, 9 which leads to small differences of at most ±2 kcal mol−1 compared with the corrected ion/water solvation energies discussed below. However, this represents only at most 1.5% of the extrapolation bulk carboxylate/water interaction energies and that has no incidence on all our former conclusions concerning carboxylates.

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3

Results and discussion

In this section we discuss the results of our ion/water simulations. The distance r is the separation between the nitrogen or the COO− carbon and the droplet COM. The full set of simulation data FS corresponds to droplets whose size is between 50 and 10k, while the reduced set RS corresponds only to sizes n ≤ 1k.

3.1

Structural properties of 10k droplet systems

From the droplet water density profiles extracted along the 10k trajectories (see Supporting Information on the K+ system), the 10k droplets are quasi-spherical on average. Their mean radius is about 41 ˚ A and the air/water interface thickness is about 4 ˚ A, regardless of the ion. The water density within the 10k droplets is slightly larger than in bulk liquid at ambient conditions (0.0331 molecules per ˚ A3 ) by at most a few percent. This is in line with our former results regarding smaller droplets. 8 We computed the distance r during the trajectories for each ion. After 1 ns of simulation, and with the exception of ethanoate and pentanoate, the temporal evolution of r for most of the ions was usually within the corresponding most probable distance r domain as computed from umbrella sampling simulations at the air/liquid water interface 8,9 (see Supporting Information, the ion location probability densities P∞ (r) at the air/liquid water interface are also plotted there). This explains the overall good agreement between the ion location densities P10k for eight organic ions computed from the last 9 ns segment of the 10k trajectories and P∞ , scaled by a factor r2 to account for the quasi-spherical shape of the 10k droplets (see Supporting Information). We may thus argue that these eight ion/10k droplet systems reached their equilibrium regime after the first 1 ns simulation segment. For ethanoate and pentanoate, we note a strong discrepancy between the latter 9 nsbased P10k and P∞ probability densities. We suspect that the chaotic effects inherent to MD simulations are responsible for the delay in reaching the equilibrium regime for these two

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ions as compared to the other ones. This is supported by the densities P10k computed from the last 1 ns trajectory segment, which are in a much better agreement with the densities P∞ for both these anions (cf. Supporting Information). Hence, with the exception of the above two carboxylates for which no conclusion can still be drawn, the present 10k results are in line with our previous ones concerning the ion affinity − for the air/water interface in large droplets: small ions, like NH+ 4 or HCOO , have a weaker

propensity for the droplet boundary than larger ions, namely (CH3 )2 NH+ 2 /propanaote and above.

3.2

Accuracy of the mean ion/water interaction energies computed from a single 10k droplet simulation

iw We computed the mean ion/water interaction energies U¯10k within a 10k droplet according

to iw = U¯10k

Z

iw (r) × P10k (r)dr , U¯10k

(9)

iw where U¯10k (r) and P10k (r) are the mean ion/water interaction energy and the ion location

probability density at a distance r, respectively. While we performed in our former studies an exhaustive sampling of the ion/water potential energy surface of small droplets (n ≤ 1k) by means of 40 umbrella sampling simulations iw of 10 ns, 8,9 here we estimate the values U¯10k only from the data extracted along a single

10 ns simulation for each 10k system, without applying any geometrical constraint. From the above discussion, a reliable density P10k might not be computed from a single 10 ns scale iw simulation. However, that has a weak effect on the accuracy of the mean energies U¯10k , as iw (r) profiles are relatively flat for all the 10k systems, even if they slightly increase the U¯10k

from the droplet core to the air/water interface, as for smaller droplets. 8,9 For instance, as iw (r) varies from −141 to −136 kcal mol−1 from near seen in Figure 1, the pentanoate U¯10k

the droplet core to the droplet boundary (here we have ignored the data corresponding to

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r ≤ 15 ˚ A, as the ion presence in that region is very rare). This agrees with the U¯niw (r) profiles that we reported for smaller droplet systems 8,9 as well as with earlier simulation results of Hagberg et al 17 about the interactions of halide anions with small water droplets iw (studied using a polarizable approach). That remarkable feature of a relatively flat U¯10k (r) iw leads to U¯10k values that depend only marginally on the probability densities P10k . For iniw stance, by scaling the pentanoate U10k (r) by two different P10k densities corresponding to

a weak or a strong ion affinity for the droplet boundary (see Supporting Information), we iw find pentanoate U¯10k values that differ by at most 0.8 kcal mol−1 . Moreover, the standard iw deviations σU of the values U10k (r) are about 6 kcal mol−1 for all the ions. As each set of iw U10k (r) values comprises about n = 150 different points and assuming the values of each set

to be temporally uncorrelated, the statistical uncertainty affecting the average for each set √ iw iw is σU / n ≈ 0.5 kcal mol−1 . Hence, the total uncertainty δ U¯10k affecting the U¯10k estimate is about 1.3 kcal mol−1 , regardless of the ion. We estimated the uncertainty δ U¯niw for the smaller droplet systems (n ≤ 1k) to be of the same order of magnitude, about 0.9 kcal mol−1 . For the sake of simplicity, we will consider the same uncertainty value δ U¯ iw = 1 kcal mol−1 for all the droplet systems, regardless of their size. iw iw It is worth noting that the U¯10k values are about 7 kcal mol−1 more negative than the U¯1k iw ones, regardless of the ion. That still represents about half of the difference between U¯100 iw and U¯1k values, 8,9 regardless of the ion (see also the U¯ iw values summarized in Supporting

Information). This shows the slow convergence of the droplet ion/water interaction energy to the bulk limit.

3.3

Extrapolation to absolute single ion and proton bulk solvation energies

To discuss the effect of the droplet data set and of the choice of the fitting function on the iw extrapolated values U¯∞ , we will consider below the corresponding proton solvation energies, iw ∆Hsolv (H+ ), which linearly depend on the U¯∞ values. According to a standard thermody-

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namical cycle, the absolute proton solvation energy can be computed from the absolute ion solvation energy and experimental thermochemical data as follows 0 ∆Hsolv (H+ ) = ∆Hg→aq (PH) − ∆Hg→aq (P) + ∆Hgprot (P) + ∆Haq,dissociation (PH) ,

(10)

where PH and P are the protonated and unprotonated form of a particular chemical species. 0 ∆Hgprot (P) measures the gas phase proton affinity of P and ∆Haq,dissociation (PH) measures the

energy in aqueous phase of the acidic dissociation reaction PH → P + H+ . We take their values from Ref. 18. As no experimental data are available for hexanoate, we take in this case 0 the same experimental data for ∆Hgprot (P) and ∆Haq,dissociation (PH) as for pentanoate. The

absolute ion solvation energies ∆Hg→aq (I) (I = PH for methylated ammonium ions and I = P iw for carboxylates) are computed from the extrapolated bulk ion/water interaction energy U¯∞

according to iw ww ∆Hg→aq (I) = U¯∞ + ∆U¯bulk + ∆U¯ rel − kB T ,

(11)

ww where ∆U¯bulk and ∆U¯ rel are the mean water destabilization energies induced by the ion

in liquid water and the mean ion intramolecular relaxation energy from gas phase to the ww aqueous phase, respectively. We consider the ∆U¯bulk and ∆U¯ rel values reported in our ww former studies. 8,9 The components ∆U¯bulk were shown to be very close for each kind of

ion and the ∆U¯ rel ones to be small (at most 1.2 kcal mol−1 for hexanoate). Assuming the energy data extracted from our 10 ns simulations are temporally uncorrelated, the statistical uncertainties affecting the latter two values are small, about 0.1 kcal mol−1 . The results of the linear regression when considering F0,0 as a linear function of n−1/3 and from the full droplet data set FS are shown in Figure 2. The linear regression coefficient R is greater than 0.996 regardless of the ion, demonstrating the reliability of the power-law function F0,0 to model ion/water interaction energy in droplets, regardless of their size. This iw is also supported by the uncertainty affecting the values U¯∞ linearly extrapolated for the

data set FS, which is small, about 0.4 kcal mol−1 (cf. Supporting Information for details).

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These R values also suggest that droplet data sets smaller than FS may be considered to iw compute accurate enough estimates of U¯∞ . In Figure 3, we plot the ∆Hsolv (H+ ) values iw computed from U¯∞ , linearly extrapolated from data sets comprising results from two to

six droplet. With the exception of the ∆Hsolv (H+ ) estimates computed from the smallest data set DS2 (comprising only the data of the droplets n = 50 and n = 100), all these new ∆Hsolv (H+ ) values are close to those computed from the full data set FS within less than 1 kcal mol−1 on average, regardless of the ion. Moreover, the values ∆Hsolv (H+ ) averaged on methylammonium data and on carboxylate ones, respectively, are also very close regardless of the droplet data set, about 272±2 kcal mol−1 (cations) and 268±3 kcal mol−1 (carboxylates). The “ionic radii” a computed by fitting the droplet data set FS to the function Fa6=0,ǫdisp range from −2.3 to +2.7 ˚ A and there is no relation between the magnitude of a and the ion size. The optimized values of the parameter ǫdisp (accounting for ion/water dispersion in functions Fa,ǫdisp 6=0 ) vary from −319 to +596 kcal mol−1 . Although these parameters are expected to be positive according to the relation (7), their orders of magnitude (in absolute value) are in good overall agreement with the ideal ones computed from that relation. For both of these parameters a and ǫdisp , we may argue that the overall small size of the full droplet data set FS and the error δ U¯ iw affecting the ion/water interaction energies (about 1 kcal mol−1 ) prevents us from accurately estimating their values. Nevertheless, the values a are small, as expected, and the order of magnitude of ǫdisp shows that the ion/water dispersion energy converges by n =1k (within less than 0.6 kcal mol−1 ), while the total ion/water interaction energy is still overestimated in 1k droplets by about 10% (i.e., by more iw than about 10 kcal mol−1 ) compared to the bulk value U¯∞ . iw We plot the ∆Hsolv (H+ ) estimates computed from the extrapolated U¯∞ values derived

from the four different functions Fa,ǫdisp and from the full data set FS in Figure 4. For methylated ammonium ions, the four functions Fa,ǫdisp lead to very close ∆Hsolv (H+ ) estimates: they are within 1 kcal mol−1 . For carboxylates, we note an overall similar agreement among the ∆Hsolv (H+ ) estimates, with the exception of pentanoate, whose ∆Hsolv (H+ ) es-

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timate varies from −270 kcal mol−1 (F0,0 ) to −266 kcal mol−1 (Fa6=0,ǫdisp 6=0 ). Nevertheless, the ∆Hsolv (H+ ) values averaged on cation and anion data are unaffected by the choice of the function Fa,ǫdisp : about 272.5 ± 1.5 for cations and 268.0 ± 2.5 kcal mol−1 for carboxylates. The uncertainty is the root mean square deviation of the ∆Hsolv (H+ ) values for each kind of ion. For Fa,ǫdisp other than F0,0 , the ∆Hsolv (H+ ) values computed from the reduced data set RS can differ up to 2 kcal mol−1 for each cation and up to 7 kcal mol−1 for each carboxylate compared to FS-based data. This exhibits the necessity of large enough data sets when using power-law functions more complex than F0,0 . This also explains the large dispersion of our previously published ∆Hsolv (H+ ) values computed from the data of the different carboxylates solvated in droplets smaller than 10k. In particular, we note here that we get the same disagreement for the carboxylate ∆Hsolv (H+ ) values when extrapolating the data set RS and by using our former function F whose third component scales as n−2/3 . 9 The above results show that we may consider the most basic function F0,0 to extrapolate reliable bulk ion/water interaction energies, even by considering smaller droplet data sets than both the RS and FS ones. As simulating small ion/water droplets is not an intensive computational task, we can apply our computational protocol to estimate reliable absolute single-ion solvation energies for any kind of small ion at the day scale using reasonable computational resources (once the model parameters have been assigned). In Table 1, we summarize our results of the cations and carboxylates. We present there the absolute single-ion solvation energies ∆Hg→aq (I) computed according to Equation (11) iw values and the full droplet data set FS, together with from the different extrapolated U¯∞

experiment-based values. 18,19 As expected from the above discussion regarding ∆Hsolv (H+ ), the methylated ammonium ∆Hg→aq (I) values are in very good agreement with experiment (within less than 2 kcal mol−1 ), while the carboxylate ones are systematically overestimated by about 5 kcal mol−1 on average compared to experiment. As discussed in our former study, 9 there is still some room to improve the model for carboxylates. In particular, it is not obvious

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how to estimate the ratio between the additive dispersion and the anti-cooperative strong hydrogen bond energy term by considering only the data of small carboxylate/water clusters. This may lead to a few kcal mol−1 of error when computing bulk carboxylate solvation energies and thus explain the above disagreement for the carboxylate-based ∆Hsolv (H+ ) estimates compared to experiment.

4

Conclusions

We have computed the ion/water interaction energies of methylated ammonium cations, K+ and alkylated carboxylate anions solvated in large water nanodrops of about n = 10k water molecules, in order to discuss the quality of our previously extrapolated estimates of these energies in bulk water from 50 ≤ n ≤ 1k droplet data. In particular, we extrapolated different bulk ion/water energy values by considering all the possible data sets made from the full data set corresponding to 50 ≤ n ≤ 10k and by means of four different power-law functions of the droplet size n. Our extrapolated bulk ion/water energies are slightly affected by the droplet data set when using the most basic power-law function based on a simple linear function of n−1/3 . For instance, if we consider all the data sets larger than the smallest one corresponding to n = (50, 100), the differences in the extrapolated ion/water interaction energies are at most 1 kcal mol−1 on average among all the possible data sets and for the two kinds of ions. This agrees with the earlier conclusion of Rips and Jortner 11 drawn within the theoretical framework of the mean-spherical approximation (MSA) that accurate bulk properties can be extrapolated from droplet data with sizes n ≥ 125. However, when considering more sophisticated power-law functions of n−1/3 , which account for the ion effect on the droplet size and/or explicitly for ion/water dispersion, the extrapolated bulk ion/water energy values can vary largely based on the data set considered for the extrapolation. Although these bulk energies are slightly affected by the function

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choice when considering droplet data up to 10k droplets (within less than 2 kcal mol−1 on average), much larger differences (up to 7 kcal mol−1 ) are observed when considering only the results of droplets whose sizes are n ≤ 1k. This may arise from the overall small size of our data sets (at most 5 droplet systems per ion) that prevents capturing the details of the data curvature in the domain 100 ≤ n ≤ 1k (predicted to be a pivotal transition domain between the cluster and the continuum regime in Rips and Jortner’s MSA framework 11 ). Hence, we recommend the use of the most basic power-law function, which is robust enough to consider reduced droplet data sets to extrapolate reliable bulk ion/water energies. From our extrapolated bulk ion/water interaction energies, we have computed the corresponding absolute proton solvation energies. The mean value for methylated ammonium ions reported here agrees with our earlier one 9 of 272 ± 2 kcal mol−1 and matches the experimental estimate (ranging from 271 to 275 kcal mol−1 18,20,21 ), regardless of the extrapolation protocol used (fitting function and droplet data set). For carboxylates, their corresponding mean proton solvation energy of about −268 ± 3 kcal mol−1 is slightly underestimated compared to ammonium cations and experiment even when considering the largest droplet data set (including 10k results). It is important to stress that our purely theoretical estimates of ∆Hsolv (H+ ) are truly single-ion results, while experimental cluster-based methods use cycles that involve ion pair solvation energies. 22,23 As our extrapolation to bulk appears to be numerically reliable, the proton solvation energy overestimation for carboxylates must arise mainly from drawbacks in the ion/water model. As already discussed, 9 this model considers two short-range energy terms to model intermolecular ion/water interactions, namely the additive dispersion and the anti-cooperative strong hydrogen bond terms. It is not obvious how to estimate the optimal ratio of these terms by considering only small carboxylate/water cluster quantum results. Nevertheless, as we have demonstrated that accurate enough bulk ion/water energies can be extrapolated using the basic F0,0 function and from a small droplet data set (comprising just the data of water droplets of size 100, 300 and 600), the effect of this ratio on the

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carboxylate-based estimate of the absolute proton solvation energy may thus be further and systematically investigated using a reasonable amount of computational resources. The model for methylated ammonium ions appears to be more robust than for the carboxylate ions, as it considers in particular less energy terms. Hence we recommend to only consider our mean ammonium-based value as a new theoretical proton solvation energy ∆Hsolv (H+ ), about 272 ± 2 kcal mol−1 , which is in particularly good agreement with experimental estimates ranging from 271 to 275 kcal mol−1 .

Supporting Information Available Raw U¯niw data, simulation details, water density profile for the K+ /10k droplet system, ion/droplet center of mass distance temporal evolutions, ion location probability densities, and a video showing the trajectory of ethanoate along the 10 ns simulation of the ethanoate/10k water droplet are provided in Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgments We thank Othman Bouizi and Emmanuel Oseret (Exascale Computing Research Laboratory, a joint INTEL/CEA/UVSQ/GENCI laboratory) for their help in optimizing the code POLARIS(MD). This work was granted access to the HPC resources of [CCRT/CINES/IDRIS] under the allocation 2015-[6100] by GENCI (Grand Equipement National de Calcul Intensif).

References (1) Donald, W. A.; Leib, R. D.; Demireva, M.; O’Brien, J. T.; Prell, J. S.; Williams, E. R. Directly Relating Reduction Energies of Gaseous Eu(H2 O)3+ n , n = 55-140, to Aqueous

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Solution: The Absolute SHE Potential and Real Proton Solvation Energy. J. Am. Chem. Soc. 2009, 131, 13328–13337. (2) Donald, W. A.; Leib, R. D.; Demireva, M.; Williams, E. R. Ions in Size-Selected Aqueous Nanodrops: Sequential Water Molecule Binding Energies and Effects of Water on Ion Fluorescence. J. Am. Chem. Soc. 2011, 133, 18940–18949. (3) O’Brien, J. T.; Williams, E. R. Effects of Ions on Hydrogen-Bonding Water Networks in Large Aqueous Nanodrops. J. Am. Chem. Soc. 2012, 134, 10228–10236. (4) Bragg, A. E.; Verlet, J. R. R.; Kammrath, A.; Cheshnovsky, O.; Neumark, D. M. Hydrated Electron Dynamics: From Clusters to Bulk. Science 2004, 306, 669–675. (5) Coe, J. V. Connecting Cluster Anion Properties to Bulk: Ion Solvation Free Energy Trends with Cluster Size and the Surface vs Internal Nature of Iodide in Water Clusters. J. Phys. Chem. A 1997, 101, 2055–2063. (6) Coe, J. V.; Williams, S. M.; Bowen, K. H. Photoelectron Spectra of Hydrated Electron Clusters vs. Cluster Size: Connecting to Bulk. Int. Rev. Phys. Chem. 2008, 27, 27–51. (7) Caleman, C.; Hub, J.; van Maaren, P.; van der Spoel, D. Atomistic Simulation of Ion Solvation in Water Explains Surface Preference of Halides. Proc. Natl. Acad. Sci. USA. 2011, 108, 6838–6842. (8) Houriez, C.; Meot-Ner (Mautner), M.; Masella, M. Simulated Solvation of Organic Ions: Protonated Methylamines in Water Nanodroplets. Convergence toward Bulk Properties and the Absolute Proton Solvation Enthalpy. J. Phys. Chem. B 2014, 118, 6222–6233. (9) Houriez, C.; Meot-Ner (Mautner), M.; Masella, M. Simulated Solvation of Organic Ions II: Study of Linear Alkylated Carboxylate Ions in Water Nanodrops and in Liquid Water. Propensity for Air/Water Interface and Convergence to Bulk Solvation Properties. J. Phys. Chem. B 2015, 119, 12094–12107. 19



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(10) Clauset, A.; Shalizi, C. R.; Newman, M. E. J. Power-Law Distributions in Empirical Data. SIAM Review 2009, 51, 661–703. (11) Rips, I.; Jortner, J. Ion Solvation in Clusters. J. Chem. Phys. 1992, 97, 536–546. (12) Makov, G.; Nitzan, A. Solvation and Ionization near a Dielectric Surface. J. Phys. Chem. 1994, 98, 3459–3466. (13) Peslherbe, G. H.; Ladanyi, B. M.; Hynes, J. T. Cluster Ion Thermodynamic Properties: The Liquid Drop Model Revisited. J. Phys. Chem. A 1999, 103, 2561–2571. (14) Réal, F.; Vallet, V.; Flament, J.-P.; Masella, M. Revisiting a Many-Body Model for Water Based on a Single Polarizable Site. From Gas Phase Clusters to Liquid and Air/Liquid Water Systems. J. Chem. Phys. 2013, 139, 114502. (15) Coles, J. P.; Masella, M. The Fast Multipole Method and Point Dipole Moment Polarizable Force Fields. J. Chem. Phys. 2015, 142, 024109. (16) Lee, N.; Keesee, R.; Castleman, A. On the Correlation of Total and Partial Enthalpies of Ion Solvation and the Relationship to the Energy Barrier to Nucleation. J. Colloid Interface Sci. 1980, 75, 555 – 565. (17) Hagberg, D.; Brdarski, S.; Gunnar Karlström, G. On the Solvation of Ions in Small Water Droplets. J. Phys. Chem. B 2005, 109, 4111–4117. (18) Meot-Ner (Mautner), M. The Ionic Hydrogen Bond. Chem. Rev. 2005, 105, 213–284. (19) Wilson, B.; Georgiadis, R.; Bartmess, J. E. Enthalpies of Solvation of Ions. Aliphatic Carboxylic Acids: Steric Hindrance to Solvation. J. Am. Chem. Soc. 1991, 113, 1762– 1766. (20) Meot-Ner, M. Heats of Hydration of Organic Ions: Predictive Relations and Analysis of Solvation Factors Based on Ion Clustering. J. Phys. Chem. 1987, 91, 417–426. 20


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(21) Tuttle, T. R.; Malaxos, S.; Coe, J. V. A New Cluster Pair Method of Determining Absolute Single Ion Solvation Energies Demonstrated in Water and Applied to Ammonia. J. Phys. Chem. A 2002, 106, 925–932. (22) Klots, C. E. Solubility of Protons in Water. J. Phys. Chem. 1981, 85, 3585–3588. (23) Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.; Earhart, A. D.; ; Coe, J. V.; Tuttle, T. R. T. J. The Proton’s Absolute Aqueous Enthalpy and Gibbs Free Energy of Solvation from Cluster-Ion Solvation Data. J. Phys. Chem. A 1998, 102, 7787–7794.

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Figure 1: Mean pentanoate/water interaction energy U¯ iw (r) as a function of the pentanoate/droplet center of mass distance r. (Dashed line) Raw averages, (solid black line) U¯ iw (r) interpolation using an 8th order polynomial function. (Grey bar, right axis) The number of snapshots where the ion was at location r along a 9 ns trajectory segment of the pentanoate/10k water droplet system. For a given r, U¯ iw (r) is the mean of m values, with m the grey bar height.

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Figure 2: Extrapolation of bulk ion/water interaction energies from the full droplet data set FS by means of the power-law function F0,0 (Equation 6).

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Figure 3: Absolute proton solvation energy ∆Hsolv (H+ ) computed from bulk ion/water interaction energy linearly extrapolated from different droplet data sets and using the function F0,0 . (Green) Results from the full droplet data set FS. (Red circles) Results from the smallest data set DS2 comprising only the droplet n = (50, 100) data. (Blue) Results from the data set DS3 comprising the droplet n = (50, 100, 300) data. (Black circles) Results from all the other data subsets that can be drawn from FS. The two horizontal dashed lines correspond to the two most common experimental estimates of ∆Hsolv (H+ ). The error bars are omitted with the exception of the full droplet data set FS results.

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Figure 4: Absolute proton solvation energy ∆Hsolv (H+ ) computed from bulk ion/water interaction energy extrapolated from the full data set FS and the four different functions Fa,ǫdisp . The two horizontal dashed lines correspond to the two most common experimental estimates of ∆Hsolv (H+ ). The error bars are omitted with the exception of the full droplet data set FS results.

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Table 1: Absolute single-ion solvation energies −∆Hg→aq (I) from bulk ion/water interaction energies extrapolated using the four functions Fa,ǫdisp and by using the full droplet data set FS, in kcal mol−1 . In parentheses, the optimized values of the parameters a and ǫdisp in ˚ A and kcal mol−1 , respectively. Cation experimental −∆Hg→aq (I) values from Ref. 18. Carboxylates experimental −∆Hg→aq (I) values from Ref. 19.

−

HCOO CH3 COO− C2 H5 COO− C3 H7 COO− C4 H9 COO− C5 H11 COO− NH+ 4 CH3 NH+ 3 (CH3 )2 NH+ 2 (CH3 )3 NH+ K+

(0,0) 90.7 97.2 96.6 95.3 96.2 93.7 87.1 79.1 72.7 68.1 89.7

(a,0) 90.1 (-0.51) 95.4 (-1.15) 97.0 (0.25) 95.1 (-0.22) 99.9 (2.27) 95.1 (0.84) 88.0 (0.43) 81.4 (-0.03) 72.7 (0.68) 69.3 (0.43) 91.1 (0.66)

(0,ǫdisp ) 90.2 (77) 95.7 (215) 97.0 (-58) 95.3 (17) 98.6 (-319) 94.7 (-127) 87.8 (-116) 80.6 (15) 72.7 (-118) 69.0 (-86) 90.6 (-119)

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(a,ǫdisp ) 90.4 (1.48,441) 95.6 (0.15,254) 95.8 (-2.11,-158) 93.9 (-2.31,-125) 100.0 (2.70,151) 95.1 (0.95,24) 87.6 (-0.25,-111) 79.6 (0.44,-62) 73.0 (0.93,439) 68.5 (-0.55,-166) 91.4 (1.77,596)


Exp. 85.9 89.7 90.5 92.9 95.2 88.6 81.0 74.6 66.8 88.0

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