Modeling Tetraalkylammonium Halide Salts in Water: How

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J. Phys. Chem. B 2009, 113, 4360–4371

Modeling Tetraalkylammonium Halide Salts in Water: How Hydrophobic and Electrostatic Interactions Shape the Thermodynamic Properties Hartmut Krienke,*,† Vojko Vlachy,‡ Gudrun Ahn-Ercan,† and Imre Bako´§ Institute of Physical and Theoretical Chemistry, UniVersity of Regensburg, D-93040 Regensburg, Germany, Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, AsˇkercˇeVa 5, SI-1000 Ljubljana, SloVenija, and Institute of Structural Chemistry, Chemical Research Center of the Hungarian Academy of Sciences, Pusztaszeri u. 59-67, H-1025 Budapest, Hungary ReceiVed: September 8, 2008; ReVised Manuscript ReceiVed: January 15, 2009

The explicit water molecular dynamics simulation was used to study tetramethylammonium and tetraethylammonium chloride and bromide solutions in water at 298 K. The outcome of the simulations in the form of various distribution functions was used to construct the solvent-averaged potentials between interacting molecules. In the next step, which involved the Ornstein-Zernike integral equation theory in the hypernetted chain approximation, these potentials were used to calculate the osmotic coefficients. We showed that this approach is able to explain the experimental results for the osmotic pressure of these salts. 1. Introduction In the last 50 years aqueous tetraalkylammonium halide solutions have received much theoretical and experimental attention. One reason probably lies in the fact that these solutions are connected with the development of an important concept, the so-called “iceberg formation”, proposed by Frank and Evans.1 Earlier studies of these solutions are summarized in an excellent review paper of Wen.2 There had been substantial criticism to this model, and more advanced models of ion hydration had been put forward, including those by Frank and Wen, Samoilov, and others. The authors are aware of this criticism, see for example, ref 18. In addition to the thermodynamic and transport properties of these solutions studied over several decades, there is a growing number of investigations where other techniques were applied.3-17 Among these the small angle neutron scattering data 6-10 in particular should provide guidance for theoretical modeling. Understanding the hydration of molecular ions containing both hydrophobic and ionic groups is of major importance for the biological sciences. In the last 12 years, explicit solvent simulations of tetraalkylammonium halide solutions19-21,23-26 became available. Among earlier papers on this subject we found the work of Slusher and Cummings21 very informative. The authors provide in the introductory part of their paper a mini-review of theoretical (and in part also experimental) work published prior to 1997. The conclusion they arrived at was that experimental and theoretical studies suggested the notion of an enhanced water-water interaction, which some supported and others opposed. Of more recent date is the simulation study of Shinto and co-workers,26 who reexamined the calculations of the potential of mean force (pmf) for the ion pairs in tetramethylammonium chloride water solution. These authors provided new data for the hydration of the tetramethylammonium ion and for the potential of mean force between two such ions in the presence of chloride as the * To whom correspondence should be addressed. E-mail: hartmut.krienke@ chemie.uni-regensburg.de. † University of Regensburg. ‡ University of Ljubljana. § Chemical Research Center of the Hungarian Academy of Sciences.

counterion. Their results apply to the infinite dilution limit and were critically compared with other molecular dynamics (MD) simulation results published in the literature. The tetramethylammonium group is also contained in some ionic surfactants,13 polyelectrolytes (see for example ref 27) and some biologically important molecules. It is known to possess a capacity to destabilize proteins. The theoretical studies listed above provide valuable information about the solvation of these molecules. But at this stage an examination of their thermodynamic properties does not seem to be feasible and it is important to develop different, more approximate approaches. The work presented in this contribution is motivated by the experimental observations of Lindenbaum and Boyd.28,29 These authors measured osmotic coefficients and heats of dilution of various alkylammonium halides in water at 298 K. To complement these experimental studies Wen and co-workers30 measured the mean activity and osmotic coefficients of tetraalkylammonium fluorides. They found these two thermodynamic coefficients to be much higher in the case of fluorides than in the case of the other halide ions. In the present work we shall focus on the osmotic coefficient results of tetramethylammonium and tetraethylammonium bromides and chlorides,28,29 as shown in Figure 1. The concentration dependence of these electrolytes is as expected: the osmotic coefficient Φ decreases from unity at low concentration and then increases at higher concentration. The latter effect is ascribed to strong hydration in the higher concentration range but also to the excluded volume effect of the solute species. As we see, the osmotic coefficient of tetramethylammonium chloride at concentrations above 0.5 mol dm-3 is substantially higher than that of tetramethylammonium bromide. An explanation one could invoke in such a situation is the following: the chloride ion is more strongly solvated than the bromide ion, and this fact is reflected in the higher osmotic coefficient of the former salt. The same experimental behavior holds true also for the corresponding tetraethylammonium salts. An interesting feature of the measurements presented in Figure 1 is the following. As we see, the osmotic coefficient curve for tetraethylammonium chloride lies above the corresponding curve for tetramethylammonium chloride. One is tempted to explain this result as a consequence of the greater

10.1021/jp8079588 CCC: $40.75  2009 American Chemical Society Published on Web 03/02/2009

Modeling Tetraalkylammonium Halide Salts in Water

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4361 TABLE 1: Parameters of the H2O Model for T ) 298.15Ka

a

Figure 1. Osmotic coefficient of tetramethylammonium (Me4N+) and tetraethylammonium (Et4N+) chlorides and bromides as a function of the molar concentration. Original experimental data in ref 28 and 29 aretransformedfromtheexperimentalP,TconditionstotheMcMillan-Mayer system of variables as suggested in ref 45. The molality of the original data has been converted to the molarity.

size of the tetraethylammonium ion. Accepting for a moment this suggestion, we proceed to the bromide salts of the same cations. Here, however, the situation is reversed; tetraethylammonium bromide has a somewhat lower osmotic coefficient than tetramethylammonium bromide. Expressed in other words; the addition of four extra CH2 groups, which change the tetramethylammonium into the more hydrophobic tetraethylammonium ion, makes the latter respond differently to the exchange of the chloride ion with the bromide ion. Experimental results for other solutions indicate that the osmotic coefficients of tetraalkylammonium fluorides30 and chlorides increase with increasing number of CH2 groups, while the corresponding bromides and iodides show a decrease of the osmotic coefficient with increasing size of the cation. This interesting effect has been noticed before.2,31 In particular, Desnoyers and co-workers31 (see their Figure 2) analyzed deviations of the activity coefficients from the Debye-Hu¨ckel theory. They showed that in the case of the I- anion these deviations decrease in the order from the lithium ion to the tetrapropylammonium ion. On the contrary, if F- is chosen as an anion, the order is exactly reversed. In our view, the above examples illustrate the complexity of those systems where we have present simultaneously hydrophobic groups as well as charged groups in aqueous solution. Such or even more complicated situations occur in chemical and biological systems. We have to mention here the work of Rozycka-Roszak and co-workers32-34 concerning the influence of the counterions on the hydration of alkylammonium salt micelles. An explanation of such and similar effects on the atomic level is clearly of substantial importance for the chemical and biological sciences. The study presented in this paper can be divided into two stages. In the first we present new results for the molecular dynamics investigations of the tetramethylammonium and tetraethylammonium cations in combination with chloride and bromide ions in water, following important preceding studies.19-21,23-26 A detailed description of the molecules based on the atomic structure is used to obtain these results. This is called the Born-Oppenheimer (BO) approach. In the all-atom simulations the outcome critically depends on the values of the input parameters, that is, on the modeling of the interparticle interactions. For this reason we compared our resulting pair distribution functions (pdf) with previous calculations (obtained for different models of ions and/or water) wherever it was possible. The results guided us in constructing the solventaveraged potentials between the various ionic species needed

parameter

name

value

distance (Å) angle (deg) charge (e0) charge (e0) L-J distance (Å) L-J distance (Å) L-J energy (J/mol) L-J energy (J/mol)

r(O-H) a(H-O-H) qH qO σHH σOO εHH εOO

0.999 107.2 0.40 -0.80 0.40 3.166 732.5 648.9

Geometry, charges, and interaction parameters from ref 35.

for the second step of the study. At this stage we utilized the Ornstein-Zernike (OZ) integral equation in the hypernetted chain (HNC) approximation to calculate the osmotic coefficients of the tetramethylammonium and tetraethylammonium chlorides and bromides based on the solvent-averaged McMillan-Mayer (MM) level of description. In this part of the calculation water is no longer considered explicitly but rather treated as a dielectric continuum. The numerical results for the osmotic coefficients obtained in this way were finally compared with experimental data available in literature. 2. Explicit Solvent Simulations 2.1. Molecular Models. The potential energy U(1, ..., N) of a N particle system can be decomposed into the sum of intermolecular pair interaction potentials, URβ(12) ) U(R, β), between molecule R and molecule β.

U(1, ..., N) )

∑ ∑ U(R, β) R

(1)

β>R

The system contains Nk molecules of species k where the total number of particles is

N)

∑ Nk

(2)

k

The intermolecular pair interaction potentials are given by the superposition of all i-j site-site interaction potentials U(rRi, βj) U(R, β) )

∑ ∑ U(r

Ri,βj)

i

)

j

∑ ∑ [U

(rRi,βj) + UC(rRi,βj)]

LJ

i

j

(3) with the Lennard-Jones (LJ) part

ULJ(rRi,βj) ) 4εRi,βj

[( ) ( ) ] σRi,βj rRi,βj

12

-

σRi,βj rRi,βj

6

(4)

and the Coulomb part

UC(rRi,βj) )

zRizβje2 4πε0rRi,βj

(5)

where rRi, βj is the distance between site i of molecule R and site j of molecule β. The intermolecular potentials used in this work were published previously. First in Table 1 we present pair potential

4362 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Krienke et al.

TABLE 2: Lennard-Jones Parameters of Halide Ions37

TABLE 4: Parameters of the Et4N+ Ion (from ref 38)

ion

F-

Cl-

Br-

I-

Parameter

Et4N+

Value

σii (Å) εii/kB (K)

4.00 5.9

4.86 20.2

5.04 32.5

5.40 49.1

distance (Å) distance (Å) distance (Å) angle (deg) charge (e0) charge (e0) charge (e0) L-J distance (Å) L-J distance (Å) L-J distance (Å) L-J energy (J/mol) L-J energy (J/mol) L-J energy (J/mol)

r(Men-N) r(Men-Me) r(Me-N) R(Men-N-Men) qN qMen qMe σNN σMeMe σMenMen εNN εMeMe εMenMen

1.53 1.53 2.60 109.4 -0.028 0.177 0.08 3.25 3.96 3.80 818 698 494

parameters for water as derived by Krienke and Schmeer.35 This water model based on the SCF-SSOZ calculations was shown to yield good results for the structural, thermodynamic, and dielectric properties and is of comparable accuracy to the SPC/E model of water. The Lennard-Jones parameters for halide ions given in Table 2, introduced by Pa´linka´s et al.,36 were used recently.37 The model of the tetramethylammonium ion used in our calculations is a rigid tetrahedral one with a united atom description of the CH3 group denoted Me (see Figure 2). The parameters for this ion are taken from the works of Jorgensen19,20 and are given in Table 3. The same potential parameters were used before by several other researchers.21,25,26 The parameters for tetraethylammonium ions were more difficult to find in the literature. We decided to use a rigid 9 site interaction site model where the CH2 and CH3 groups are each taken as one site and denoted Men and Me, respectively. An additional complication lies in the fact that this ion can exist in different conformations. Following the recent study of Luzhkov et al.,38 we chose the S4 conformation with geometry parameters and partial charges defined from the RESP HF/6-31G(d) calculations in that work. OPLS parameters were taken for the Lennard-Jones interactions of the sites.19,20 The parameters are summarized in Table 4. The dissimilar sites of the Lennard-Jones parameters were obtained using the combining rules in the form of σij ) 1/2 (σii + σjj) and εij ) (εiiεjj)1/2. 2.2. Molecular Dynamics Simulation. All classical MD simulations were performed with DLPOLY software.39,40 The investigated mixtures contained 1024 water molecules; 12 cations and 12 anions in a cubic box at a temperature T ) 298.15 K, with periodic boundaries and the minimum-image convention. The box lengths were about 31.3 Å. Energetically optimized initial configurations of the different mixtures were generated from Monte Carlo (MC) simulations with MCFLUID software41 developed in our laboratory. For the MC simulations the standard isothermal-isochoric (NVT) method with Me-

tropolis sampling was used. Intermolecular interactions were spherically truncated at nearly half the box length and the longrange electrostatic interactions were treated using the Ewald summation method. After this pretreatment, the MD simulations were carried out in the NVT ensemble using the Nose´-Hoover thermostat with a relaxation time of 1.0 ps. To calculate electrostatic interactions the Ewald technique with automatic parameter optimization and a real space cutoff of nearly half of the box length was employed. The bonds of the molecules in the mixtures were maintained by the SHAKE algorithm. Total simulation time was 900 ps with time steps of 1 fs. From the trajectory files of the MD runs, the site-site pair distribution functions gi-j(r) were obtained. These functions are defined through the molecular pair distribution functions gRβ(12) as

gi-j(r) )

1 (8π2)2

∫ ∫ gRβ(12)δ[rij(12) - r] d(1) d(2) (6)

while the definition of the molecular pair distribution functions is

g(12) )

V2Ω2 QN

[

∫ · · · ∫ exp - U(1,kB...,T N)

]

d(3)...d(N) (7)

with U(1, ..., N) given by eq 1. kB is the Boltzmann constant and T is the temperature. Ω ) 4π for linear molecules, Ω ) 8π2 for nonlinear molecules, and QN is the configuration integral of the molecular system.

QN ) Figure 2. The united atom representation of the tetramethylammonium ion and the tetraethylammonium ion used in the simulations: CH3-site (Me; red), CH2-site (Men; gray), N-site (blue).

TABLE 3: Parameters of the Me4N+ Ion (from refs 19, 20, 25) parameter

Me4N+

value

distance (Å) angle (deg) charge (e0) charge (e0) L-J distance (Å) L-J distance (Å) L-J energy (J/mol) L-J energy (J/mol)

r(Me-N) R(Me-N-Me) qN qMe σNN σMeMe εNN εMeMe

1.51 109.4 0.00 0.25 3.25 3.96 818 698

[

∫ · · · ∫ exp -

]

UN(1, ..., N) d(1)...d(N) kBT

(8)

2.3. Solvation of Ions. We begin the discussion with the ion-water distribution functions shown in Figures 3 and 4. First in Figure 3 we present the results for the site-site distribution functions between the water molecule and the nitrogen of the tetramethylammonium ion (Me4NCl solution) and the tetraethylammonium ion (Et4NCl solution). In panel a we show the site-site distribution function between the oxygen atom of water and the nitrogen, gO-N(r), and in panel b the corresponding distribution function for the hydrogen atom of water and nitrogen, gH-N(r). A difference between the O-N site-site distribution functions of the Me4NCl and Et4NCl solutions is clearly visible in panel a. In the case of the Me4N+ ion the oxygens can approach closer to the nitrogen and the interaction


J. Phys. Chem. B, Vol. 113, No. 13, 2009 4363

Figure 3. Molecular dynamics (MD) explicit water simulations for Me4NCl and Et4NCl in water (cf. eq 6 and Tables 1-4): (a) water(oxygen)-cation center(nitrogen) correlations; (b) water(hydrogen)cation center(nitrogen) correlations; both at c ) 0.65 mol dm-3.

Figure 4. Molecular dynamics (MD) explicit water simulations for Me4NBr and Et4NBr in water (cf. eq 6 and Tables 1-4): (a) water(oxygen)-cation center(nitrogen) correlations; (b) water(hydrogen)cation center(nitrogen) correlations; both at c ) 0.65 mol dm-3.

is stronger. In contrast to this, the peak describing the Et4N+-O correlation is smaller and considerably broader, reflecting the larger size of the ion and the different distribution of charges associated with the Et4N+ model. Similar information can be extracted from panel b of Figure 3, where the water(H)-N correlation is displayed. The equivalent results for the Me4NBr and Et4NBr solutions are shown in Figure 4, and the conclusions arrived at are similar to the above ones since the two figures are very close to each other. Some information about the water-cation correlation can already be extracted from these figures. The oxygens are located slightly closer to the N atoms of the tetraalkylammonium groups than the hydrogens of water molecules. However, the difference is very small and together with the nearly equal heights of the first peak of gO-N(r) and gH-N(r) in each system, this supports the idea of an (almost) tangential arrangement of water molecules around the cation. Solvation of tetramethylammonium cations in water has been studied theoretically and experimentally in several papers and here we wish to compare our results with previous works. The cation-water(oxygen) distribution function for tetramethylammonium chloride solutions in water has been presented by Slusher and Cummings.21 The result for this quantity, presented in Figure 4 of ref 21, agrees quite well with ours as far as the position and the height of the cation-oxygen(water) peak is concerned (cf. our Figure 3a). Note that different models of water were used in the two cases. The cation-water(center of mass) distribution function for tetramethylammonium chloride solutions has also been obtained experimentally9 using neutron diffraction scattering. The molecular dynamics results21 are in semiquantitative agreement with the experimental results of Turner.9 In another relevant paper23 a solution of tetramethy-

lammonium chloride was studied at a concentration of 0.555 mol dm-3. The authors used a different water model as well as different parameters to model tetramethylammonium and chloride ions (for details see Table 1 of ref 23). The result for the tetralkylammonium ion-water(O) distribution function obtained by their model has a peak at a similar distance (∼4.4 Å) as the curve shown in Figure 3a, but it is substantially higher. For the purpose of this study it is also important to analyze the Cl--water and the Br--water distribution functions, which are shown in Figure 5. In panel a we display the distribution functions for the water(O)-Cl- and the water(H)-Cl- pairs in an aqueous solution of tetramethylammonium ions. The corresponding information about water(O)-Br- and water(H)-Brinteraction is shown in panel b of this figure. As we can see from Figure 5, the difference between the oxygen(water)-anion and hydrogen(water)-anion curves is considerably larger, in contrast to the water-cation curves. The hydrogen of water is located closer to the anion than the corresponding oxygen. This fact reflects strong interaction of water molecules with the relatively small anions. The corresponding anion-water distribution functions for the tetraethylammonium solutions are very similar to the ones shown in Figure 5 and therefore not shown here. 2.4. Ion-Ion Correlations. The water-water pair distribution functions do not show big differences in going from one solution to another. This is because water is added in great excess and the structure should be similar to that of pure water in all cases. In general effects on the water structure caused by the different solutes are small and within the uncertainties of simulations. The same conclusion has been arrived at by other researchers.21


Figure 5. MD explicit water simulations for Me4NCl and Me4NBr in water: (a) water(oxygen)-Cl- anion correlations and water(hydrogen)Cl- anion correlations; (b) water(oxygen)-Br- anion correlations and water(hydrogen)-Br- anion correlations; both at c ) 0.65 mol dm-3.

But the water-water distributions of solvent around different solutes are different in terms of three-body functions, even at infinite dilution, as shown in a recent study by integral equation theory for inhomogeneous molecular liquids.22 These effects can have substantial effect on solvation in biosystems, and this aspect deserves further studies. For the study of thermodynamic parameters, which we are aiming at in the next sections, information about the ion-ion distribution functions is the most important. These functions, especially for like-charged ions, are difficult to obtain and rather long simulations are needed for good results. Before presenting our simulations we wish to discuss the rather scarce data available in the literature. Among previous studies Slusher and Cummings21 calculated the cation-anion pair distribution functions for some tetraalkylammonium ions (among others also for the tetramethylammonium ion) with Cl- or Br- ions as counterions at finite concentration. The center of mass radial potential between cation and anion in Me4NCl solution has a relatively high peak for the distance r ≈ 4.6 Å (cf. Figure 3 of ref 21). The recent contribution of Shinto and co-workers26 is concerned with the potential of mean force between the various ions at infinite dilution. Results were presented for the potential of mean force between ions in aqueous Me4NCl solution (cf. Table 1 of their paper). The simulations were performed for the same Me4N+ model19 as we used but for a different representation of Cl-. The SPC/E model of water was used in their simulations. The resulting potential of mean force between tetramethylammonium ions is repulsive with a small ”dip” for a distance around 9.0 Å. The result was accordingly interpreted as representing a solvent separated pair. In addition, the chloride-

Krienke et al. chloride correlation was studied and compared with previous data, which were obtained either with different potentials or other different simulation protocols. The conclusion was that the Cl--Cl- potential of mean force has two minima corresponding to the different solvent separated solute pairs. The cation-anion potential of mean force was also calculated. This function showed that the Me4N+-Cl- potential of mean force has two broad minima corresponding to the contact solute pair and the solvent separated solute pair. The authors summarized their findings in Table 1 of ref 26. Our results for the corresponding ion-ion distribution functions, are shown in Figures 6 and 7. In Figure 6 we present in the left panels the curve of the Me4N+-Cl- (panel c1), the Cl--Cl- (panel b1) and the Me4N+-Me4N+ (panel a1) pair distribution. Separately in the right panels we provide the equivalent information for Me4NBr solution. The results shown in Figure 6 indicate strong interaction between the cation and anion. In the left panel c1, the height of the peak of the Me4N+-Cl- pair distribution function is about 2.5 and it is positioned at r ≈ 4.9 Å. From the right panel (c2), where the result for the Me4N+-Brinteraction is shown, we see that the peak value is about 3.4 while its position is slightly less than 5.0 Å. This indicates that Me4N+-Br- attraction is much stronger than the cation-anion interaction in the case of the Me4NCl solution. Unfortunately, we could not find in the literature any previous results for Me4NBr solutions of finite concentrations for comparison with our simulations. The results for the Me4N+-Me4N+ correlation as obtained in ref 26 belong to infinite dilution and are therefore independent of the nature of the counterion. Our calculation applies to c ) 0.65 mol dm-3, and as demonstrated in Figure 6a1,a2, it slightly depends on the nature of the partner anion in solution. The differences, however, are small. The positions of the first maximum in our cation-cation pair distribution functions are around 8 Å, and the minimum around 11 Å. These numbers agree reasonably well with the data given in Table 1 of ref 26, where the values reported are 9.0 and 11.2 Å, respectively. Both calculations suggest a weak attraction between the Me4N+ ions; this finding is important for the continuation of this analysis. Next in Figure 7 we present the distribution functions for Et4NCl solution in the left panels and for Et4NBr solution in the right ones. To the best of our knowledge there has been no previous simulation of these solutions so we do not have any data to compare with our calculations. In the left panels (c1, b1, and a1) we show the Et4N+-Cl-, Cl--Cl-, and Et4N+-Et4N+ pair distribution functions. The results for Et4NBr solution are displayed in an analogous way. We compare first the Et4N+-Cland Et4N+-Br- results. In this case the two pair distribution functions are quite close to each other; the peak for the Et4N+-Br- pair distribution function is only slightly higher. The other two pair distribution functions also exhibit very small differences. The anion-anion pair distributions shown in the panels b1 and b2 are very similar to each other, but the Br- ions seem to be more strongly correlated. There seems to be a slightly greater difference in the shape of the two Et4N+-Et4N+ distribution functions; the pair distribution function becomes less structured in the presence of Br- ions. The differences in the behavior of these four solutions, as evidenced on the molecular level in Figures 6 and 7, should be responsible for their different macroscopic behavior. We can get additional findings of the structure of water and of the ion distribution around the TAA ions with the help of a three dimensional probability analysis of our MD simulations.



Figure 6. Ion-ion center-center distribution functions of Me4NCl (left panels) and Me4NBr (right panels) in water as obtained by explicit water solvent MD simulations; c ) 0.65 mol dm-3. (a1,a2) Cation-cation center-center distribution functions; (b1,b2): anion-anion center-center distribution functions; (c1,c2) cation-anion center-center distribution functions.

Additionally to the site-site correlation the probability of finding another water molecule or ion in a selected distance interval from the central particle is shown depending also on the relative spatial orientation of the particles. The results of such an analysis are shown in the Figures 8-12. The observation of a first water shell around the TAA ions was restricted to r(N–O) ≈ 5 Å, a second shell was selected with r(N–O) between ∼5 Å and 6.5 Å. In Figure 8 the first (gray) and second (gold) water shells around the Me4N+ ion are shown for the 0.65 molar Me4NCl model solution. The hydrophobic methyl groups are avoided by the water in the first shell, the water prefers the space between these methyl groups of the TAA ions. The spatial analysis depicted in Figure 9 shows, that the anions also like to stay in the space between the methyl groups (their probability distribution is shown with blue color). The anions prefer the same orientational distributions around the cation as the water molecules, but they do not approach to the center of the cation. One finds them between the first and the second water shell which is shown for comparison (gold). From Figures 10 and 11 it can be seen, that the same findings are valid for the distribution of water and anions around the Et4N+ ion. Figure 10a shows the probability of finding a first shell of water molecules (gray) around this cation in a 0.65 molar Et4NCl solution. The geometry of the Et4N+ ion is fixed as the S4 conformation discussed in section 2.1. Figure 10b shows the orientational distribution of water molecules in the first (gray) and in the second shell (gold) around the cation in a 0.65 molar

Et4NBr solution. In connection with the cation-water site-site distributions (Figures 3 and 4) one finds that the regions to find the first water molecules around the Et4N+ ion are at greater distances from the center than in the case of the Me4N+ ion, and the distributions are broader. The distribution of the Cl- ions around the cation in the Et4NCl solution are shown in Figure 11 (gold) together with the distribution of the water molecules of the first shell (gray). As in the case of the Me4N+ ion the chloride ions appear at the same orientational positions as the water molecules but at larger distances. This corresponds to the site-site correlations shown in Figures 4 and in Figures 6 and 7. For the cation-cation interaction there is a different picture, as can be seen from Figure 12. For the spatial arrangement of two Me4N+ ions in a Me4NBr solution the most probable correlations are seen between the corners of the tetrahedrons circumscribing the Me4N+ ions (cf., the red colored probability regions in that figure). Similar conclusions can be drawn for the Et4N+-Et4N+ interactions. 3. The Solvent-Averaged Approach 3.1. McMillan-Mayer Theorem. In this section we use the information obtained from the detailed atomic simulations to construct simplified solvent-averaged ionic interaction potentials,42-45 which should yield correct thermodynamic results for the systems under study. Our goal is to understand the


Krienke et al.

Figure 7. Ion-ion center-center distribution functions of Et4NCl (left panels) and Et4NBr (right panels) in water as obtained by explicit water solvent MD simulations; c ) 0.65 mol dm-3. (a1,a2) Cation-cation center-center distribution functions; (b1,b2): anion-anion center-center distribution functions; (c1,c2) cation-anion center-center distribution functions.

Figure 8. The first water shell (gray) and the second water shell (gold) around the Me4N+ ion in a Me4NCl solution.

underlying physics, which yields the results displayed in our Figure 1 and those shown in Figure 2 of Desnoyers and coworkers.31

Figure 9. The distribution of the first water shell (gray), of the chloride anions (blue) and of the second water shell (gold) around the Me4N+ ion in a Me4NCl solution.

The theoretical basis of the interionic theory is given by the famous McMillan-Mayer theorem.46 The theory is based on



Figure 11. The distribution of the first water shell (gray) and of the chloride anion (gold) around the Et4N+ ion at 0.65 molar aqueous solution of Et4NCl.

implementation it is usually assumed that the N-particle potential W∞N can be decomposed into a sum of pair interactions W∞ij ,

W N∞(r1, r2, ..., rN) )

∑ W ij∞(ri, rj)

(10)

i,j

This potential of mean force between two ions W∞ij (ri, rj) is connected with the infinite dilution limit of the corresponding ion-ion pair distribution function at the Born-Oppenheimer level

W ij∞(ri, rj) ) -kBT ln(lim gij(ri, rj)) ) -kBT ln(gij∞(ri, rj)) Fionf0

(11) An example of the application is the molar osmotic coefficient of an electrolyte solution, which is given by the relationship Figure 10. (a) The distribution of the first water shell (gray) around the Et4N+ ion at 0.65 molar aqueous solution of Et4NCl. (b) The distribution of the first water shell (gray) and of the second water shell (gold) around the Et4N+ ion at 0.65 molar aqueous solution of Et4NBr.

ΦMM )

2πFion posm )1+ FkBT 3

∑ x˜ x˜ ∫ i j

i,j

o

∞

∂[-βW ∞ij (r)] 3 r gij(r) dr ∂r

(12) the grand partition function, Ξ(UN′) ) exp(βpV) of a multicomponent N′ particle system made up from (N′-N) solvent molecules and N solute particles. As usual β ) 1/kBT, pressure is denoted by p and volume by V. The partition function Ξ(UN′) can be rigorously transformed into an effective grand partition function Ξ(WN) involving only the N solute particles, related to the osmotic pressure posm

¯ (W N∞) ) exp(βposmV) Ξ

(9)

Notice that the solvent-averaged potential W∞N applies to the infinite dilution limit of the solute particles. For practical

where x˜i is the ion fraction, Fi is the number density of species i and Fion is the total number density of the ions

x˜i )

Fi ; Fion

Fion )

∑ Fj

(13)

j

gij(r) ) 1 + hij(r) is the center-center pair distribution function between the ionic species i and j at finite ion concentration. ΦMM differs from the experimental (P,T) osmotic coefficient, ΦLR, due to the different equilibrium pressures, that is, ambient atmospheric pressure for ΦLR and osmotic pressure for ΦMM.


ΦMM ) ΦLR(1 + mM)

c)

ds d

Krienke et al.

(14)

md exp(γposm) (1 + mM)

(15)

posm ) νcRTΦLR

(16)

In these equations c is the molarity and m is the molality of the solution, ds and d are the analytical densities of solvent and solution, respectively, M is the molecular weight of the electrolyte under study, γ is the isothermal compressibility, and ν is the number of ions of a single solute. 3.2. Connection between the Explicit Solvent and McMillan-Mayer Descriptions. The connection between the explicit solvent approach as described in section 2 (also called the Born-Oppenheimer level, BO) and the McMillan-Mayer level of description was first presented by McMillan and Mayer.46 A derivation of the osmotic pressure formula in terms of the N-particle potentials of mean forces of the solute particles is to be found in the book by Hill.47 In terms of the two-particle potentials of mean forces and of the two-particle correlation function hij(r) between ionic species i and j, the connection between the two levels was first given by Adelman48 and subsequently used in the work of Levesque et al.49 and Friedman.50 According to the studies mentioned above, the system of Ornstein-Zernike equations for solvent and solute particles R, β, and γ at BO level reads

hRβ(1, 2) ) cRβ(1, 2) +

∑ Fγ ∫ cRγ(1, 3) hγβ(3, 2) d(3)

Figure 12. Additionally to the probability distributions of Figure 9 the distribution of other Me4N+ ions around a central Me4N+ ion is shown (red) in a Me4NBr solution.

A closure relation corresponding to equation 18, given in terms of the ion-ion correlations only, can be derived from subsequent cluster analysis48,51

ln[1 + hij(r)] ) -βW ij∞(r) + hij(r) - cijeff(r)

(21)

γ

(17) This system of equations can be used for calculation of hRβ(1, 2) together with suitable closure relations between hRβ(1, 2) and cRβ(1, 2) of the form

-βW Rγ(1, 2) ) ln(1 + hRγ(1, 2)) ) -βURγ(1, 2) + hRγ(1, 2) - cRγ(1, 2) + BRγ(1, 2) (18) BRγ(1, 2) are the so-called bridge functions, which are often neglected (hypernetted chain approximation) in these integral equation theories. The system of eq 17 can be reformulated by introduction of the “effective” direct correlation functions cijeff(1, 2) between ionic species i and j48

hij(1, 2) ) cijeff(1, 2) +

∑ Fk ∫ cikeff(1, 3) hkj(3, 2) d(3)

Equations 19 and 21 comprise the integral equation theory for the ion-ion correlation functions at finite ion concentrations Fion at MM level in the hypernetted chain approximation. This set of coupled integral equations has to be solved numerically. 3.3. Solvent-Averaged Potential between Ions. Calculations of excess functions at the McMillan-Mayer level of description start with the potential of mean force of two ions W∞ij (1, 2) ≡ W∞ij (r) in pure water. This solvent-averaged interaction potential is approximated by the superposition of a Coulomb and a yet unspecified short-range term WSR ij (r).

W ij∞(r) ) W ijC(r) + W ijSR(r)

Solvent averaging leads to the occurrence of the dielectric constant ε of the pure solvent in the Coulomb term.

k

(19) ceff ij (1,

An explicit expression for 2) in terms of solvent-solvent (css) and ion-solvent (cis) direct correlations is given as

cijeff(1, 2) ) cij(1, 2) + Fs2

∫ ∫ cis(1, 3)

×

[1 - css(3, 4)]-1 csj(3, 2) d(3) d(4) (20) where Fs is the solvent number density. Notice that all the solvent effects are now subsumed in ceff ij .

(22)

W ijC(r) )

zizje2 1 4πε0ε r

(23)

The MD simulations with the water model defined in Table 1 give a static dielectric constant of ε ) 76 at T ) 298 .15 K. Therefore we used the experimental dielectric constant of water in eq 23 for the Coulomb part of W∞ij (r). ∞ The short-range contribution WSR ij (r) to Wij (r) contains the direct Lennard-Jones interactions between the ionic sites in an averaged form, as well as contributions caused by the penetration and altering of the ionic solvation shells.



TABLE 5: Parameters of the Model Potential (Charged Hard Spheres + Square Well) for the HNC Calculations on the McMillan-Mayer Level (Contact Distances, Rij; Width of Solvation Shell, Sij; Square Well Depth in Units of kBT, dij) ions i-j

Ri

Sij

dij

me-me et-et cl-cl br-br me-cl me-br et-cl et-br

7.0 7.5 5.0 5.0 4.5 4.5 4.5 4.5

4.0 4.0 1.0 1.0 2.5 2.5 3.0 3.0

-0.10 -0.20 -0.1 -0.2 -0.52 -0.65 -0.3 -0.53

Simple representations of the short-range potential, WSR ij (r), can be provided with the help of the step potential. In this paper the ions are modeled as charged hard spheres with a nonadditive hardsphere interaction in a continuous solvent represented by its dielectric constant ε. The contribution of the short-range potential to W∞ij (r) is modeled by a square-well potential added to the Coulomb interaction in the continuous medium. The potential of mean force at the infinite dilution limit is therefore given by

W ij∞(r) ) ∞; W ij∞(r) )

r e Rij

zizje2 1 + dij ; 4πε0ε r

W ij∞(r) )

zizje2 1 ; 4πε0ε r

(24)

Rij < r e R′ij r > R′ij

where Rij is the contact distance, Sij is the width of the solvation shell with R′ij ) Rij + Sij and dij is the depth of the square-well reflecting the short-range interaction. The parameters for such potential models were in earlier work43obtained from fitting the transport and thermodynamic excess functions, such as for example the osmotic coefficient. An important novelty of the present study is that we use information from the explicit solvent MD calculation to estimate the model parameters defining eq 24. In constructing the potential of mean force at the infinite dilution limit, W∞ij (r), we were guided by information extracted from our explicit water MD simulations (sections 2.3 and 2.4): (i) the strong increase of the pair distribution function among ions of opposite sign was used to estimate the size parameter Rij; (ii) the width of the first peak of the pair distribution function provides an estimate for Sij; and (iii) the height of the first peak of the pair distribution function was used as a measure for the depth dij of the potential well. The values of the parameters Rij, Sij, and dij applied in this paper are collected in Table 5. Numerical results for gij(r), needed to calculate ΦMM, can be obtained from W∞ij (r) by iteratively solving the Ornstein-Zernike integral equation in the hypernetted chain approximation, that is combination of eqs 19 and 21. We allowed optimization of parameters Rij, Sij, and dij with regard to the best agreement (judged by the eye) of the calculated osmotic coefficients (eq 11) with the experimental values in the concentration range from zero to 0.9 mol dm-3. Notice that in order to apply the McMillan-Mayer theory consistently (see, for example, ref 42) we need to require for equal ions (i ) j) that the interaction parameters be the same (regardless of the ion partner) in all solutions. As an example, we present in Figure 13 the model pair distribution functions among the ions in Me4NBr solution of the same molarity as used in the MD calculation. The solvent-

Figure 13. Me4NBr in water. Ion-ion correlations at concentration c ) 0.65 mol dm-3; HNC calculation with charged square-well potentials.

averaged potentials, W∞ij (r), used in the integral equation calculations are shown in Figure 14. Finally, in Figure 15 we present the comparison between the experimental data for the osmotic coefficient and the quantity as obtained from the (McMillan-Mayer) hypernetted chain calculations. Note that the experimentally obtained values of the osmotic coefficient were transformed from the conditions of measurement (constant P and T) to the McMillan-Mayer conditions42-45 as first proposed by Friedman.45 3.4. Thermodynamic Results: Theory versus Experiment. Let us first discuss the values of the model parameters leading to the relatively good agreement between theory and experiment indicated in Figure 15. One interesting feature emerging from the MD simulations is the fact that the Rij values for tetramethylammonium and tetraethylammonium cation interactions with Br- and Cl- ions are very close to each other. This looks surprising at first sight; because of the extra CH2 groups the tetraethylammonium ion is expected to be larger than tetramethylammonium. Notice, however, that we are talking about the effective size of the molecular ions in aqueous solution reflecting hydration effects. This result is consistent with previous simulations; see for example, results shown in Figure 3 of ref 21. In Table 5 these values are for simplicity taken to be exactly the same, that is 4.5 Å. Let us focus now on the Me4NCl-Et4NCl comparison. The value of parameter dij of the potential function prescribed by eq 24 and leading to the curves shown in Figure 15 is more negative for Me4N+-Cl- than for the Et4N+-Cl- interaction pair. This suggests that an additional short-range attraction between the two species, caused by restructuring of the water


Figure 14. Model potentials of mean force at infinite dilution: (a) Et4N+-Et4N+ and Me4N+-Me4N+ cation-cation interactions; (b) Cl--Cl- and Br--Br- anion-anion interactions; (c) cation-anion interactions.

molecules between the Me4N+ and Cl- ion, takes place. The combined effect of all interactions (note that d+- has the strongest impact on thermodynamics) yields lower values of osmotic coefficient of the Me4NCl solutions (in comparison with Et4NCl) for all the concentrations examined in this study. Next we look more closely at the Me4NBr and Et4NBr osmotic coefficients. Here the situation seems to be different: the potential well (d+-) for the Me4N+-Br- interaction is only slightly deeper than the one in the case of the Et4N+-Br- interaction. In such a situation the like-ion contributions (++ and - -) have to be considered in more detail. Both respective parameters, that is, dij for the Br--Br- and Et4 N+-Et4N+ pairs, are more negative than the corresponding parameters for the Cl--Cl- and Me4+-Me4+ interaction, indicating some association of the equally charged species. All these effects roughly compensate in our McMillanMayer model calculation, while in the experimental values the likeion association seems to prevail, leading to the slightly lower osmotic coefficient of Et4NBr solutions in comparison with the Me4NBr case. The picture outlined here is consistent with the results presented in Figures 6 and 7. The results presented in Figure 15 can be summarized as follows: the osmotic coefficients depend strongly on the nature of the anions. For chlorides the electrostatic part of the effective interaction is dominant in the concentration dependence of the excess functions; for bromides we already seem to have some hydrophobic effect. Different cations seem to amplify this behavior, which should be even more visible in the case of the tetrapropylammonium cations. A small hydrophobic attraction is noticed in the case of the Me4N+-Me4N+ interaction and it is enlarged for the Et4N+-Et4N+ pair.

Krienke et al.

Figure 15. Osmotic coefficients of (a) of tetramethylammonium chlorides and bromides and (b) tetraethylammonium chlorides and bromides. Experimental data (symbols) and HNC calculations (lines). Original experimental data in refs 28, 29 are transformed from the experimental P,T conditions to the McMillan-Mayer system of variables as suggested in ref 45. The molality of the original data has been converted to the molarity.

4. Conclusions This paper represents an attempt to understand the thermodynamic behavior of some alkylammonium salts. More precisely four different solutions, tetramethylammonium and tetraethylammonium chlorides and bromides were studied using the explicit water molecular dynamics approach. This is the first more systematic comparison of these solutions. The study was motivated by the intriguing behavior of the osmotic coefficient (see Figure 1) and other thermodynamic properties of these solutions. It is clear from the experimental data that the presence of hydrophobic groups strongly influences the concentration dependence of the osmotic coefficient and other thermodynamic parameters. In the first part of the study we presented the various site-site distribution functions, all obtained for an electrolyte concentration equal to 0.65 mol dm-3. The results for water-ion and ion-ion pair distribution functions were calculated and compared for these four different systems. The authors are aware that different molecular descriptions (force fields) may yield different results. It is therefore worth mentioning that the results of this work are in good agreement with the previous all atom simulation results for the methylammonium and ethylammonium salt solutions available in the literature. In the second part of the study we used the structural information gathered to construct the short-range solvent-averaged potentials with the capacity to fit the osmotic coefficient measurements. For this purpose the Ornstein-Zernike integral equation in the hypernetted chain approximation was utilized. A reasonably good agreement between the measurements and the McMillan-Mayer model was obtained for all four solutions.

Modeling Tetraalkylammonium Halide Salts in Water The approach presented here is similar to that adopted by other researchers, see for example refs 52-54 to mention only the most recent studies. It represents an attempt to bridge the different length scales, one, exploring the neighborhood of the charged group or ion, that extends a few Å into solution and the other, the macroscopic scale (hundreds of Å), that defines the thermodynamic properties. Our approach is slightly different than that used in the papers mentioned above, but it has similar weaknesses. We may say that the idea of using the all atom simulations to construct the potential of mean force between the ions and calculate the thermodynamic properties of the solution is only partially successful. In our case the structural differences between the solutions containing chloride and bromide ions are relatively small and almost within the precision of the calculations. We believe that calculation involving the fluoride and/or iodide ions would be more revealing since the differences in thermodynamic properties are much larger for these ions. Such a calculation is currently underway. A similar situation applies to experimental data of alkylammonium salt solutions. On one hand we have the measurements probing the vicinity of the charged groups. Such approaches are, for example, offered by dielectric relaxation measurements and small angle neutron scattering. It seems that these methods reveal relatively small differences between different electrolyte solutions; the magnitudes of the ion-specific effects could be within the experimental errors. On the other hand we have thermodynamic information in the form of the osmotic and activity coefficients revealing drastic differences for these same solutions; ion specific effects are noticeable. In other words, the relatively small differences caused by the watermediated interaction between charged groups seem to cause significant differences in thermodynamic properties. Theoretical attempts to connect the two length scales, such as the one presented in this study, seem to be only partially successful so far. Acknowledgment. This work was done while Vojko Vlachy was Visiting Professor at the Institute of Physical and Theoretical Chemistry at University of Regensburg, Germany; generous support of the German Research Foundation through the Mercator Programme is gratefully acknowledged. V.V. also acknowledges partial support from the Slovenian Research Agency through grant P1-0103-0201. Imre Bako´ acknowledges support through the grant 436UNG113/184/0-1 from the DFG. References and Notes (1) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507–532. (2) Wen, W. Y. In Water and Aqueous Solutions; Horne, R. A., Ed.; Wiley-Interscience: New York, 1972; pp 613-661. (3) Green, J. L.; Sceats, M. G.; Lacey, A. R. J. Chem. Phys. 1987, 87, 3603–3610. (4) Eriksson, P.-O.; Lindblom, G.; Burnell, E. E.; Tiddy, G. J. T. J. Chem. Soc., Faraday Trans. 1 1988, 84, 3129–3139. (5) Keçki, Z.; Dryjański, P. J. Mol. Struct. 1992, 275, 135–143. (6) Turner, J.; Soper, A. K.; Finney, J. L. Mol. Phys. 1992, 77, 411– 429. (7) Turner, J.; Soper, A. K.; Finney, J. L. Mol. Phys. 1992, 77, 431– 437. (8) Calmettes, P.; Kunz, W.; Turq, P. Phys. B 1992, 180-181, 868– 870. (9) Turner, J. Z.; Soper, A. K.; Finney, J. L. J. Chem. Phys. 1995, 102, 5438–5443. (10) Polydorou, N. G.; Wicks, J. D.; Turner, J. Z. J. Chem. Phys. 1997, 107, 197–204.

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4371 (11) Liegl, B.; Bradl, S.; Scha¨tz, T.; Lang, E. W. J. Phys. Chem. 1996, 100, 897–904. (12) Baar, C.; Buchner, R.; Kunz, W. J. Phys. Chem. B 2001, 105, 2906– 2913. (13) Buchner, R.; Ho¨lzl, C.; Stauber, J.; Barthel, J. Phys. Chem. Chem. Phys. 2002, 4, 2169–2179. (14) Saielli, G.; Scorrano, G.; Bagno, A.; Wakisaka, A. ChemPhysChem 2005, 6, 1307–1315. (15) Takekiyo, T.; Yoshimura, Y. J. Phys. Chem. A 2006, 110, 10829– 10833. (16) Takekiyo, T.; Yoshimura, Y. Chem. Phys. Lett. 2006, 420, 8–11. (17) Chong, S. H.; Hirata, F. J. Phys. Chem. B 1997, 101, 3209–3220. (18) Dill, K. A.; Truskett, T. M.; Vlachy, V.; Hribar-Lee, B. Annu. ReV. Biophys. Biomol. Struct. 2005, 34, 173–199. (19) Jorgensen, W. L.; Gao, J. J. Phys. Chem. 1986, 90, 2174–2182. (20) Carlson, H. A.; Nguyen, T. B.; Orozco, M.; Jorgensen, W. L. J. Comput. Chem. 1993, 14, 1240–1249. (21) Slusher, J. T.; Cummings, P. T. J. Phys. Chem. B 1997, 101, 3818– 3826. (22) Ishizuka, R.; Chong, S.-H.; Hirata, F. J. Chem. Phys. 2008, 128, 034504-1–034504-10. (23) Hawlicka, E.; Dlugoborski, T. Chem. Phys. Lett. 1997, 268, 325– 330. (24) Garcıá-Tarre´s, L.; Guardia, E. J. Phys. Chem. B 1998, 102, 7448– 7454. (25) Garde, S.; Hummer, G.; Paulaitis, M. E. J. Chem. Phys. 1998, 108, 1552–1561. (26) Shinto, H.; Morisada, S.; Higashitani, K. J. Chem. Eng. Jpn. 2004, 37, 1345–1356. (27) Arh, K.; Pohar, C.; Vlachy, V. J. Phys. Chem. B 2002, 106, 9967– 9973. (28) Lindenbaum, C. E.; Boyd, E. J. Phys. Chem. 1964, 68, 9ll–917. (29) Lindenbaum, C. E. J. Phys. Chem. 1966, 70, 814–820. (30) Wen, W. Y.; Saito, S.; Lee, C. M. J. Phys. Chem. 1966, 70, 1244– 1248. (31) Desnoyers, J. E.; Arel, M.; Perron, G.; Jolicoer, C. J. Phys. Chem. 1969, 73, 3346–3351. (32) Rozycka-Roszak, B.; Zylka, R.; Sarapuk, J. Z. Naturforsch. 2000, 55c, 413–417. (33) Rozycka-Roszak, B.; Zylka, R.; Sarapuk, J. Z. Naturforsch. 2001, 56c, 154–157. (34) Rozycka-Roszak, B.; Zylka, R.; Kral, T.; Przyczyna, A. Z. Naturforsch. 2001, 56c, 407–412. (35) Krienke, H.; Schmeer, G. Z. Phys. Chem. 2004, 218, 749–764. (36) Pa´linka´s, G.; Riede, O.; Heinzinger, K. Z. Naturforsch. 1977, 32a, 1197. (37) Ahn-Ercan, G.; Krienke, H.; Kunz, W. Curr. Opin. Colloid Interface Sci. 2004, 9, 92–96. (38) Luzhkov, V.; Oesterberg, F.; Acharya, P.; Chattopadhyaya, J.; Aqvist, J. Phys. Chem. Chem. Phys. 2002, 4, 4640–4647. (39) Smith, W.; Leslie, M.; Forester, T. R. The DL_POLY_2.14 User Manual; CCLRC, Daresbury Laboratory: Daresbury, Warrington WA 4 4AD, England, 2003. (40) Krienke, H.; Opalka, D. J. Phys. Chem. C 2007, 111, 15935–15941. (41) Krienke, H.; Fischer, R.; Barthel, J. J. Mol. Liq. 2002, 98-99, 331– 356. (42) Ramanathan, P. S.; Krishnan, C. V.; Friedman, H. L. J. Sol. Chem. 1972, 1, 237–262. (43) Ebeling, W.; Krienke, H. Stud. Phys. Theor. Chem., Part C 1988, 38, 113–160. (44) Faigl, P. Ein Beitrag zur Statistischen Thermodynamic verduennter Waessriger Elektrolytloesungen. Thesis. University of Rostock, Germany, 1978. (45) Friedman, H. L. J. Sol. Chem. 1972, 1, 387–412. (46) McMillan, W. G.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276–305. (47) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956. (48) Adelman, S. A. J. Chem. Phys. 1976, 64, 724–731. (49) Levesque, D.; Weis, J. J.; Patey, G. N. J. Chem. Phys. 1980, 72, 1887–1899. (50) Friedman, H. L. J. Chem. Phys. 1982, 76, 1092–1105. (51) Barthel, J.; Krienke, H.; Kunz, W. Physical Chemistry of Electrolyte Solutions-Modern Aspects; Steinkopff, Darmstadt and Springer: New York, 1998. (52) Gavryushov, S.; Linse, P. J. Phys. Chem. B 2006, 110, 10878– 10887. (53) Gavryushov, S. J. Phys. Chem. B 2006, 110, 10888–10895. (54) Hess, B.; Holm, C.; van der Vegt, N. J. Chem. Phys. 2006, 124, 164509–164516.

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Modeling Tetraalkylammonium Halide Salts in Water: How

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