Systematic Evaluation of Ion Diffusion and Water Exchange

Abstract

As a fundamental property of all fluids, diffusion plays myriad roles in both science and our daily lives. Diffusive properties of many liquids including water have been extensively studied both experimentally and theoretically, while for transition metal ions, there exists significant experimental data that has not been extensively studied theoretically. Hence, high confidence predictions for challenging systems like radioactive ions that are biohazardous can’t be reliably generated. In this work, a workflow named ISAIAH (Ion Simulation using AMBER for dIffusion Action when Hydrated) was designed to accurately simulate the diffusion coefficients of 15 monoatomic ions with charges varying from −1 to +3 in four water models. As the results indicate, good agreement with experimental values was achieved leading us to select ²³⁹Pu⁴⁺ (for which no experimental data is available) as a candidate ion to make a theoretical prediction of its diffusion coefficient in water. Among all the force field parameter sets, the ones parametrized using an augmented 12-6-4 Lennard-Jones (LJ) potential showed lower Average Unsigned Errors (AUE) for ions of various radii and electron configurations relative to some 12-6 LJ parameters. This observation agrees well with the fact that diffusion is affected by both the hydration free energy (HFE) and the ion-oxygen distance (IOD) between solute and solvent molecules both of which the 12-6-4 model handles well.

Graphical Abstract

Introduction

Diffusion has been both theoretically and experimentally studied for more than a century, from first being described by Fick’s Laws in 1855,¹ to a steady-state model in 1935,² and then to the more advanced frame-of-reference model in 1960³. In 1969, the first ion diffusion coefficient was experimentally measured for Th⁺ and Th³⁺ using isotopic exchange radiochemical techniques.⁴ Later in 1989, microfluidic devices and direct compositional analysis were employed to obtain higher accuracy for both water and ion diffusion coefficient measurements under different solute concentrations.^5-6 NMR is also a powerful tool to experimentally measure diffusion coefficients of open shell systems, including molecules containing isotopes with non-zero spin quantum numbers.⁷ However, in contrast to the continuous development of experimental methods for measuring diffusion coefficients, computational evaluation of diffusion coefficients are challenging especially for proteins and other macromolecules which require long simulation timescales largely due to sampling issues and the lack of well-defined algorithms.⁸ Dufrêche et al. developed a self-consistent microscopic theory to compute self-diffusion coefficient of LiCl, NaCl and KCl as a continuous function of concentration up to 1M.⁹ Wang et al. designed a molecular dynamics workflow for simulating and calculating diffusion coefficients of many solvents using Green’s functions.¹⁰ Ohba et al. used the QM/MM method to simulate Li⁺ diffusion in graphite,¹¹ and a similar quantum-classical hybrid model was later used by Tomanek et al. for simulating water desalination by an all-carbon membrane.¹² However, a pure molecular mechanics model that represents all the physics of ions in solution (e.g., polarization, charge transfer, ion-induced dipole interactions) has not been fully developed for simulating single ions, especially divalent and trivalent ion diffusion in bulk water systems, due to the nature of these highly charged ions.

The strong polarizability of highly charged ions will generate charge-induced dipoles on interacting ligand molecules. Many molecular mechanical models have been developed to take polarization and related effects into account.¹³ For example, Åqvist and Warshel¹⁴ and Peng¹⁵ derived a cation dummy-atom model to better represent the coordination sphere of an ion. Li et al. developed a 12-6-4 Lennard-Jones model to incorporate ion-induced dipole interactions with ligand molecules.¹⁶ Considering the simplicity, accuracy and accessibility of the latter model for many ions of interest, we have explored in detail the ability of the 12-6-4 LJ model (relative to the 12-6 model) to predict the diffusion coefficients of 15 ions with charges varying from −1 to +3.

Mathematically, the 12-6-4 LJ model can be described by Eq. (1)

$$U_{ij}\left(r_{ij}\right)=\frac{e^2{Q}_i{Q}_j}{r_{ij}}+\frac{C_{12}^{ij}}{r_{ij}^{12}}-\frac{C_6^{ij}}{r_{ij}^6}-\frac{C_4^{ij}}{r_{ij}^4}=\frac{e^2{Q}_i{Q}_j}{r_{ij}}+{\epsilon }_{ij}\left[{\left(\frac{R_{min,ij}}{r_{ij}}\right)}^{12}-2{\left(\frac{R_{min,ij}}{r_{ij}}\right)}^6\right]-\frac{C_4^{ij}}{r_4^{ij}}$$ (1)

Compared to the 12-6 LJ potential form, an augmented $\frac{{\mathrm{C}}_4^{ij}}{r_4^{ij}}$ term was added to describe the interactions between point charges and induced dipoles of polarizable ligands. Overall, the 12-6-4 LJ model has two advantages. First, it is physically meaningful and can be mathematically derived.¹⁷ Second, by using the noble gas curve (NGC) ¹⁸ to couple R_min,ij and ε_ij, only R_min,ij and $C_4^{ij}$ are the only variables involved in the parametrization process.¹⁹

In previous work,^19-21 we successfully parametrized the R_min,ij and $C_4^{ij}$ terms for various ions with charges ranging from −1 to +4 in four new water models: OPC3,²² OPC,²³ TIP3P-FB (TIP3P-FB) and TIP4P-FB (TIP4P-FB).^24-25 These new water models were developed for better bulk properties (density, viscosity, etc.) under various temperature, pressure and solute conditions, which makes them ideal candidates for predicting diffusion coefficients of ions in aqueous environments. Moreover, because diffusion is an effect involving both geometric changes in the ion coordination sphere and the concomitant energy changes, our reported parameter sets are useful starting points to estimate diffusion coefficients using ISAIAH (Ion Simulation using AMBER for dIffusion Action when Hydrated). Details of our approach are provided below.

Theoretical Background and Methods

Obtaining diffusion coefficients under specific concentrations

The ISAIAH workflow functionality takes the ion mass, ion charge, water model, parameter set (i.e., HFE, IOD or 12-6-4) as inputs, and outputs their diffusion coefficients at infinite dilution. Each parameter set is named after the targeted physical property. For example, the HFE set is aimed to reproduce the experimental HFE value. The method used by the ISAIAH workflow was adapted from Panteva et al.,²⁶ modified as suggested by Bullerjahn et al.²⁷ to reduce the standard deviation by changing the sampling step from 1 fs to 0.2 ps. In the AMBER package,²⁸ the diffusion coefficient was calculated using Eq. (2) and (3) in the CPPTRAJ program:²⁹

$$MSD=\frac{{\sum }_{i=0}^N\mid x_{i+1}-x_i{\mid }^2}{(N-1)}$$ (2)

$$D=\underset{t\to \infty }{\text{lim}}\frac{MSD}{6t}$$ (3)

Where MSD stands for the mean square displacement of a single particle. In Eq. (2), N is the total number of frames taken from the production phases of the simulations, which is equal to the total simulation time divided by the sampling window. x_i is the coordinates of the particle at the i-th step and ∣x_i+1 − x_i∣² is the squared distance that the particle has traveled between step i and i+1. Eq. (3) was then used to calculate the final diffusion coefficient D using the MSD value.

Because the diffusion coefficient is a concentration-dependent property, the diffusion coefficient of an ion in an infinitely dilute solution was calculated using the workflow in ISAIAH and then compared to experimental values.³⁰ First, four identical ions were solvated individually in four boxes with box lengths of 40, 50, 58, and 62 Å, which refers to concentrations of 26.0mM, 13.2mM, 8.5mM and 7.0mM respectively. For each system, 20 independent simulations were performed as described below: (1) 5000 steps of minimization using the steepest descent algorithm followed by 5000 steps of minimization using the conjugate gradient algorithm; (2) 360 ps simulation in the NVT ensemble to gradually heat the system from 0 K to 298.15 K; (3) 2 ns of equilibration in the NPT ensemble at 298.15 K and 1 atm; (4) 1 ns NVT simulation at 298.15 K to further equilibrate the system; (5) 2 ns of simulation was performed in 80 successive cycles in order to guarantee MSD vs. time linearity. Each cycle consists of 5 ps of NPT equilibration followed by 20 ps of NVE production whose snapshots were saved every 0.2 ps. Over the 80 cycles, the water box sizes change slightly during the NPT step but by less than ±1%, so we used the box size of the last NPT cycle for linear extrapolation. For all these minimizations and MD simulations, periodic boundary conditions (PBCs) were used. The particle mesh Ewald (PME) method was applied to handle the long-range electrostatic interactions.³¹ The nonbonded cutoff was set to 10 Å in these simulations. The “three-point” SHAKE algorithm was used to constrain the geometries of the water molecules.³² The Langevin thermostat with a collision frequency of 2.0 ps-1 was used to control the temperature in all MD simulations that required temperature control.³³ The Berendsen barostat with a relaxation time of 1.0 ps was used to control the pressure in the NPT ensembles.³⁴ To validate the workflow further, a benchmark on the number of NPT-NVE cycles was conducted on Al³⁺ with 12-6-4 parameter set in OPC water and the result is shown in Figure S1A. The MSD vs. time plot for one of the 8000-water simulation is also presented in Figure S1B, which demonstrates that 20ps is long enough for the system to converge.

Calculating diffusion coefficients of ions in infinitely diluted conditions

Using Eq. (2) and Eq. (3), the intrinsic diffusion coefficients D_I of the ions for each of the 20 ps NVE production runs was calculated. Then the D_I values of the 80 NVE production runs were averaged to obtain the real diffusion coefficient D_R of the ion for the individual run over the 1.6 ns of production simulation following Eq. (4).

$$D_R=\frac{{\sum }_{i=1}^{80}{D}_I^i}{80}$$ (4)

Afterwards, D_R values of all 20 independent runs were averaged to get the concentration-dependent diffusion coefficient D_C for each water box respectively using Eq. (5). The standard deviations s.d._C of these 20 independent runs were calculated in this step as well using Eq. (6).

$$D_C=\frac{{\sum }_{j=1}^{20}{D}_R^j}{20}$$ (5)

$$s.d{.}_C=\sqrt{\frac{{\sum }_{j=1}^{20}{\left(D_R^j-D_C\right)}^2}{19}}$$ (6)

Finally, to achieve a more reasonable comparison with the experimental data, each of the D_C values, as well as s.d._C values, were then scaled by a factor of (2.3×10⁻⁵ cm²/s) /D_W to get D_L and s.d._L, using Eq. (7) and (8),

$$D_L=D_C\times \frac{2.3\times {10}^{5c{m}^2}∕s}{D_W}$$ (7)

$$s.d{.}_L=s.d{.}_C\times \frac{2.3\times {10}^{5c{m}^2}∕s}{D_W}$$ (8)

where 2.3×10⁻⁵ cm²/s is the experimentally determined diffusion constant for water molecules, and D_W is the averaged diffusion coefficient for water molecules inside that same water box. Note here that the Ls are the lengths of simulation boxes (40, 50, 58, and 62 Å), so D_L and s.d._L are both functions of the box sizes. Lastly, the ion diffusion coefficient at infinite dilution $D_0^{sim}$ was obtained using Eq. (9) through extrapolation based on the four scaled size-dependent diffusion coefficients D_L, while the standard deviation $s.d{.}_0^{sim}$ were obtained by averaging the error ratio and multiplying that by the final $D_0^{sim}$value, as elaborated in Eq. (10).

$$D_0^{sim}=(D_L{)}_{L\to +\infty }={\left(\widehat{a}\times \frac{1}{L}+\widehat{b}\right)}_{L\to \infty }$$ (9)

$$s.d{.}_0^{sim}=\frac{{\sum }_{k=1}^4\frac{s.d{.}_L^k}{s.d{.}_L^k}}{4}\times D_0^{sim}$$ (10)

Herein $\widehat{a}$ and $\widehat{b}$ values are the slope and intercept from the linear regression. $\widehat{a}$ is related to viscosity and $\widehat{b}$ is related to the water molecule size, while L is length of the side of the water box. The whole process has been automated in the ISAIAH workflow. The open-source code is available at https://github.com/lizhen62017/ISAIAH along with a tutorial on how to run this workflow.

Analysis of average values for each ion

Differences between water models and parameter sets will result in variations in the computed diffusion constants so we computed the average diffusion coefficient and average standard deviation over all water models and parameter sets and compared them with experimental values. This was done to get a sense of trends within a water model to make ion diffusion predictions. The relative differences and their standard deviations are given by Eq. (11) and Eq. (12).

$$P.E.\%=\frac{\langle D_0^{sim}\rangle }{D_0^{exp}}\times 100\%$$ (11)

$$S.D.\%=\langle s.d{.}_0^{sim}\rangle \times \frac{\langle D_0^{sim}\rangle }{D_0^{exp}}\times 100\%$$ (12)

Where $\langle D_0^{sim}\rangle$ and $\langle s.d{.}_0^{sim}\rangle$ are the expected diffusion coefficient and standard deviation of that infinitely diluted ion, averaged over all water models and parameter sets. $D_0^{exp}$ is the experimental diffusion coefficient of that ion under infinite dilution conditions as well. These quantities are discussed below.

Calculation and ligand exchange rate for five representative ions

To further investigate the molecular-level factors affecting diffusion, the ligand-exchange rate was calculated for five representative ions, Ca²⁺, Li⁺, Na⁺, K⁺ and Mg²⁺ in 40 Å OPC water boxes using the 12-6-4 LJ parameter sets. The reasons for selecting these five ions is because they are ubiquitous in materials and biological sciences, while having a wide range of exchange rates. The method is derived from Grotz et al.³⁵ To start, the 80 simulations previously described were concatenated to form a 1.6 ns (80*20ps) trajectory containing 8000 frames. Then, using the “hbond” function in CPPTRAJ,²⁹ a screening was conducted over all water oxygens to only select ions that approached the metal ion below a defined cutoff distance, where this cutoff was defined as the average value between the first and second peak of the RDF (see Figure 1), i.e. the boundary of the first solvation shell. Next, the “distance” function in CPPTRAJ was used to build the distance vs. frame relationship for all the selected water oxygens. Lastly, a python script was used to count the total number of water insertion/deletions (denoted as N). An insertion was defined as a water molecule entering from beyond the cutoff distance (first local minimum in the RDF) to less than 90% of it and then staying within that threshold for at least 4ps (20 frames), while deletion was defined as a water molecule leaving from below the cutoff distance to more than 110% of it and staying beyond that threshold for at least 4ps (20frames).

Figure 1.

Definition of the cutoff distance for the water exchange rate calculation on Ca²⁺, Li⁺, Na⁺, K⁺ and Mg²⁺. 90% of r_cutoff is the threshold for defining insertion, while 110% of r_cutoff is the threshold for defining deletion.

After obtaining the total ligand insertion/deletion count N, this number was then inserted into Eq. (13) to calculate the ligand exchange rate in the unit of M/s.

$$rate=\frac{CN\times \left[M(H_{2}O{{)}_{CN}}^{n+}\right]\times N}{2\times t_B\times N_{H_{2}O}}$$ (13)

Where N_H2O is the total number of water (herein it is 1477, 1501, 1549, 1607, 1466 respectively for Ca²⁺, Li⁺, Na⁺, K⁺ and Mg²⁺), while t_B is described by Eq. (14)

$$t_B=\frac{CN\times N_M\times t_{sim}}{N_{H_{2}O}-CN}$$ (14)

In Eq. (14), t_sim is the time of simulation, which is 1.6ns. N_M is the number of metal ions, i.e., one in all our simulations and CN is the coordination number of the metal ion. By inserting our computed values into these two equations, the ligand exchange rate can be determined and compared with experimental values.

Experimental values

The experimental diffusion coefficients of all 15 ions are given in Table 1,³⁰ together with their coordination numbers (CNs),¹³ which has been hypothesized to be a key factor affecting the diffusion rate.

Table 1.

Experimental diffusion coefficients and CNs for all 15 ions.

Ion	Electronic structure	Diffusion coefficient (10−5 cm2/s)30	CN13
F−	[Ne]	1.48	4.1–6.8
Cl−	[Ar]	2.03	6–8.5
Br−	[Kr]	2.08	6
Li+	[He]	1.029	4–6
Na+	[Ne]	1.334	4–8
K+	[Ar]	1.957	6–8
Ag+	[Kr] 4d10	1.648	2–4
Be2+	[He]	.599	4
Mg2+	[Ne]	.706	6
Ca2+	[Ar]	.792	8
Cu2+	[Ar]3d9	.714	6
Zn2+	[Ar]3d10	.703	6
Al3+	[Ne]	.541	6
Cr3+	[Ar]3d3	.595	6
Fe3+	[Ar]3d5	.604	6

From Table 1, we can confirm the previous theory that diffusion coefficients strongly correlate with the charge and size of specific ions.³⁶ Many computational works on halides and monovalent cations also validate this theory.^37-39 Similarly, the ISAIAH workflow also calculates the diffusion coefficients as outputs controlled by variables like Q_i, Q_j and $C_4^{ij}$ (charge related), as well as R_min,ij and ε_ij (radius-related) allowing us to thoroughly consider as many factors as possible.

Linear extrapolation and final diffusion coefficients

To illustrate the details of the diffusion coefficient calculation, the original linear extrapolation plots for Mg²⁺ and Ca²⁺ in all four water models are shown in Figures 1A and 1B. Similar linear extrapolation plots for the other 13 ions can be found in Figures S2~S14.

Herein the hydration free energy (HFE) set means the parameters R_min,ij and ε_ij can successfully reproduce the HFE of that ion, but not the ion-oxygen distance (IOD), the IOD set means R_min,ij and ε_ij can successfully reproduce the IOD but not the HFE. CM means the compromise set, where R_min,ij and ε_ij underestimate the IOD and overestimate HFE, but within an acceptable error range, only divalent ions had their CM sets determined in our previous work.²⁰ The12-6-4 set means using the augmented LJ model, where R_min,ij, ε_ij and $C_4^{ij}$ can successfully reproduce both the HFE and IOD at the same time.

Overall, the linearity of each extrapolation is acceptable. Similar extrapolation plots are available in the Supplementary information, Figures S2-S14. The overall results for the diffusion coefficients, as well as their standard deviations, are listed in Table 2.

Table 2.

Simulated diffusion coefficients and standard deviations as well as percent error relative to experiment of 15 ions in four water models. The numbers below each element name are their experimental diffusion coefficients. All units are in 10⁻⁵cm²/s.

Table 2 also displays the percentage differences between simulations and experiment as a heat plot, where red indicates an overestimation and blue indicates an underestimation, respectively. A detailed color scale is provided below the table. The comparison results showed a very systematic trend of relative differences.

Analysis of differences between water models and parameter sets

An AUE (average of unsigned error) analysis was conducted over all ions for each combination of water model and parameter set. The result is presented in Table 3.

Table 3.

AUE of each combination of water model and parameter set. Value in each cell is calculated by averaging differences between 100% and the calculated percentage (from Table 2) of all 15 (5 if the parameter set is CM) ions.

Water Models	HFE	IOD	CM (divalent only)	12-6-4
OPC3	16.85%	18.69%	15.41%	17.46%
OPC	16.24%	23.08%	13.16%	21.08%
TIP3P-FB	13.36%	17.50%	11.00%	14.58%
TIP4P-FB	17.92%	21.94%	9.35%	17.98%

According to Table 2 and 3, between different parameter sets, HFE sets are generally robust, but deviate from the average values significantly for Cu²⁺ and Zn²⁺, while the IOD sets are usually outliers for the other 13 ions. In comparison, CM (if applicable) and 12-6-4 sets are closer to the average value. As discussed above, the better behavior of the CM and 12-6-4 parameter sets can still be explained by the interaction energy. In previous work,²⁰ interaction energies played a significant role between different water models and parameter sets in conjunction with the same ion. Taking Mg²⁺ in the OPC water model as the example again, the interaction energy for HFE, IOD, CM and 12-6-4 sets are −102.92, −80.43, −86.58 and −84.50 kcal/mol respectively, showing a trend in the diffusion coefficients of IOD > 1264 > CM > HFE, which matches what is shown in Table 2. In contrast, the variations between different water models are not that significant when compared to those between parameter sets for the same ion, except that most four-point water models (OPC and TIP4P-FB) generally have higher diffusion coefficients than three-point water models (OPC3 and TIP3P-FB). This cannot be explained solely by the ion-water interaction energy, since the interaction energy for four-point water models are generally more negative than three-point water models.²² A possible explanation lies in the water-water interaction,⁴⁰ which was given as an explanation for the overestimation of Be²⁺ and Al³⁺. In general, three-point water models have a larger volume per molecule when compared to four-point water models.⁴¹ This makes the water exchange favor a S_N1-like reaction to undergo a dissociative exchange mechanism, where the water being replaced will leave first, then the new water will fill in the cavity in the first hydration shell. In comparison, four-point water models tend to undergo S_N2-like, or associative mechanisms while exchanging water molecules.⁴² This is demonstrated by the fact that three-point water models usually have lower coordination numbers compared to four-point water models, indicating the transition state for three-point water exchange is under-coordinated, while over-coordinated for four-point water exchange transitions.

Analysis of overall diffusion coefficients for each ion

According to Table 2, the overall simulated diffusion coefficients are highly correlated with the charges and radii of each ion tested. This can be analyzed individually using two factors: energetic and geometric. For example, the diffusion coefficient is highly dependent on the interaction energy according to previous research.^19-20 Using the comparison between Mg²⁺ and Ca²⁺ as an example, the interaction between Mg²⁺ and water has been shown to be stronger than the interaction between Ca²⁺ and water by multiple methods.^43-44 Therefore, the ion-water interaction will be stronger for Mg²⁺ relative to Ca²⁺. As expected, the results of both simulation and experiment showed a lower diffusion coefficient for Mg²⁺ compared to Ca²⁺. The only exception, according to Table 2, is the OPC3 water model in conjunction with HFE parameter set which gives a higher diffusion coefficient for Mg²⁺ than Ca²⁺, this may be due to the observed standard deviation of the infinitely diluted diffusion coefficient in this instance. From the method section, standard deviations of each infinitely diluted diffusion coefficient are highly dependent on each water box’s standard deviation before the extrapolation, meanwhile each water box’s standard deviation is dependent on the uniformity between 20 individual runs. The first factor that may affect the simulation outcome is intrinsic, where the ion box size does not have an exact linear relationship because the experimental water diffusion coefficient will not always be 2.3×10⁻⁵ cm²/s. Most of the simulations yield water diffusion coefficients slightly lower than what has been reported by previous research listed in Table 4, likely due to the presence of the ion.

Table 4.

Water model diffusion coefficient averaged over all ions and box sizes (10⁻⁵ cm²/s); all values are collected at 298.15K.

Water Models	Average diffusion coefficients in this research	Diffusion coefficients without ions
OPC3	2.28	2.3045
OPC	2.34	2.3523
TIP3P-FB	2.27	2.2824
TIP4P-FB	2.19	2.2124

Other previous research also suggests that the existence of ions will further decrease the water diffusion coefficient when the ion concentration increases.⁴⁶ The second factor affecting the standard deviation is variations in the water-water interaction in either the 12-6 model or the augmented 12-6-4 model. This uncertainty may affect the diffusion coefficient, especially when the water box sizes are changing during the linear extrapolation.⁴⁷ For example, in a previous study,^48-49 it was shown for a long water wire, increasing the wire length will significantly enlarge the standard deviation and percent error of H⁺ and OH⁻ diffusion coefficients. A similar situation might be occurring here, where the ions are in 3D water boxes of various sizes.

Nonetheless, energetic factors alone cannot explain a significant error in the simulation result, where Be²⁺ diffuses faster than both Ca²⁺ and Mg²⁺. In fact, the simulation of Be²⁺ also gives the highest over-estimation percentage of the overall diffusion coefficient compared to experiment. This can likely be explained by the second factor, i.e., the geometric influences on diffusion cannot be correctly simulated due to the fact that for some ions, their parameters do not reproduce the CN value well.²⁰ Specifically, the geometric factor caused by different CN values on diffusion can be further dissected into two parts: The diffusion pattern and the ligand exchange pattern.²⁷

Different diffusion patterns may lead to different errors between simulation and experiment. It has been proposed by previous research that there are three types of diffusion patterns:^50-51 independent diffusion with a very short residence time scale for the first solvation shell, co-diffusion with a clear separation between the solvation shell and bulk water (illustrated by Figure 3A), and intermediate diffusion, with an unstable first shell of solvation that keeps exchanging waters with the solvent (illustrated by Figure 3B). For the intermediate diffusion case, two exchange mechanisms have been described^{42, 52}, where the associative mechanism resembles a S_N2-like exchange between water molecules, while the dissociative case resembles a S_N1-like exchange. In the present research, these three diffusion patterns and two exchange mechanisms were also observed. Be²⁺ undergoes co-diffusion with one stable hydration shell layer, but the second hydration layer undergoes dissociative exchange between water molecules. This however, does not reflect the real situation, as Be²⁺ and water may form a stable two-layer [Be(OH₂)₆₊₁₂]²⁺ cluster complex to slow down the diffusion speed even more.⁵³ Hence, we can hypothesize that Be²⁺ is diffusing too fast since it is not pulling along two solvation shells in our case. A similar explanation is also applicable to Al³⁺, where the diffusion coefficient is overestimated by about 20% compared to experiment, since Al³⁺ is known for forming large ion-water clusters or even a cross-linking gel structure.⁵⁴ The simulation indicates a stable first solvation shell for Al³⁺, but a second solvation shell following an associative exchange mechanism. This may not reflect the real situation, where both the first and second solvation shells of Al³⁺ are stable.⁵⁵ To successfully simulate this effect, we hypothesize that the forcefield may need to include a more realistic description of water-water interaction.

Figure 3.

Illustration of diffusion models: the co-diffusing model (A) and exchange model (B). Green arrows and circles depict the first solvation shell diffusion, while blue arrows and circles depict the water exchange of first solvation shell. The process indicated by the blue arrow can be further classified as an associative or dissociative mechanism.

Systematically, various diffusion patterns can be the result of different levels of the relationship between ion-water interaction energies and diffusion coefficients. Based on Table 2, it is reasonable to propose that as the charge of an ion increases, the interaction energy between ion and water is getting stronger, which makes the ion more likely to undergo a co-diffusion pattern or intermediate diffusion with an associative mechanism, i.e., surrounding water molecules are strongly attracted to the ion during the entire simulation. This was observed for all the trivalent ions during our simulations, as well as most of the divalent ions. However, for all the halides and some monovalent ions, they are more favored to undergo an independent diffusion pattern or intermediate diffusion with a dissociative mechanism, with the first solvation shell water weakly attracted by the ion during the simulation. Hence, we observe that if an ion undergoes co-diffusion or intermediate diffusion with an associative mechanism, the simulation usually overestimates the diffusion coefficients, while if an ion undergoes independent diffusion or intermediate diffusion with a dissociative pattern, the simulation usually underestimates the diffusion coefficients. For the overestimation of the divalent/trivalent ion diffusion coefficients (except Ca²⁺), a possible molecular level explanation is the overestimated ligand-exchange rate we observed in our simulations,⁵⁶ because the exchange of water is believed to increase the diffusion speed (see Table 5, experimental values of Mg²⁺ and Ca²⁺). However, the underestimation of the halide/monovalent ion diffusion coefficients, except Li⁺, is likely a complex outcome of ligand exchange and steric hindrance. Since the ligand exchange rate itself has little effect on the diffusion rate if water molecules are weakly attracted by ions (see Table 5, experimental values of Li⁺ and Na⁺). Under this circumstance, more exchange might block the diffusion transit pathway of the ion, rather than increase the diffusion rate.

Table 5.

Ligand exchange rate results for Ca²⁺, Li⁺, Na⁺, K⁺ and Mg²⁺, in a 40 Å OPC water boxes in conjunction with 12-6-4 LJ parameter sets.

Ion Name	Total insertion/ deletion in 1.6ns (N) (1 μs for Mg2+)	Exchange frequency (k) (1/ns) (Experiment)56 (1/μs for Mg2+)	Exchange rate k × CN × [M(H2O)CNn+] (M/ns) (Experiment)56 (M/μs for Mg2+)	Charge over radius ratio (e*Å) (Calculated from Couture and Laidler)57	Diffusion Coefficients 10−5cm2/s (Experimental)
Ca2+	15	0.58 (0.32)	0.12 (0.07)	1.89	0.77 (0.79)
Li+	100	6.22 (1.25)	0.81 (0.16)	1.28	1.15 (1.03)
Na+	114	5.91 (1.25)	0.92 (0.19)	1.02	1.09 (1.33)
K+	199	8.84 (1.99)	1.61 (0.36)	0.75	1.68 (1.96)
Mg2+	22	1.82 (0.51)	0.28 (0.08)	2.56	0.76 (0.71)

Calculation and discussion on ligand exchange rate for five representative ions.

In order to further explore the theory proposed in Figure 3, Eqs. (13) and (14) given in the method section were applied to calculate the ligand exchange rate for Ca²⁺, Li⁺, Na⁺, K⁺ and Mg²⁺ in 40 Å OPC water boxes using the 12-6-4 LJ parameter sets. The results are organized in Table 5. Note here due to the extreme low ligand exchange rate of Mg²⁺, the simulation duration was adjusted to 1μs following methods mentioned in the Supporting Information.

The results in Table 5 shows that Ca²⁺ and Mg²⁺ undergo a co-diffusion pattern while their first solvation shell seldom exchanges with other water molecules. The relative differences between five representative ions also agree with the theory described by Figure 3. Also, from the last column of Table 5, it can be concluded that larger charge over radius ratio will lead to less rapid water exchange in the first solvation shell, and finally lower diffusion coefficients. However, when comparing with the results from Marcus,⁵⁶ it is clear that all five k values are overestimated especially for the monovalent ions.

The overestimation could be due to the chosen distance cutoff. In the work of Grotz et al., two cutoffs were selected according to the energy profile, and was shown to have little effect while varying both cutoffs slightly for one ion (Figure S1 of ref. 33). In light of this, we chose to use the RDF to avoid doing PMF profiles for multiple ions with the expectation it would only play a minor role as seen in ref. 33.

Debate exists in whether 90%~110% of the average between the first and second peak of the RDF is too arbitrary when defining “insertion” and “deletion”, as shown in Figure 1. In previous research on the convergence behaviour of solvation shells in condensed phases, the cutoffs were also selected arbitrarily and scaled up as the solvent molecule radii increases.⁵⁸ Indeed, there is sensitivity to the choice of insertion and deletion, but we feel our choice is reasonable and reproducible given that RDFs can be easily generated. Overall, this strategy of defining water insertion/deletion cutoffs has been shown control the error in a reasonable range for multiple ions.

Other than the kinetic analysis of the ligand exchange rate, the exchange effect can also be analyzed in a thermodynamic way, with the free energy ΔG_ex being estimated by Eq. (15).⁵⁹

$$\frac{dln{\mu }_{ion}}{dT}+\frac{1}{T}-\frac{dln{D}_{wat}}{dT}=\frac{\frac{\Delta G_{ex}}{R{T}^2}}{1+0.0655e^{\frac{\Delta G_{ex}}{RT}}}$$ (15)

$$D_{ion}={\mu }_{ion}\times k_{B}T$$ (16)

Where μ_ion is the ion mobility that can be converted to a diffusion coefficient using Eq. (16), i.e. the Einstein relation.⁶⁰ ΔG_ex will be the free energy of ligand exchange (in this case, water exchange). By analyzing Eq. (15) qualitatively, it is reasonable to derive that if the diffusion of an ion is less temperature sensitive, the ion will tend to have a higher exchange free energy, meaning the exchange ratio will be slower. According to Wang et al., the AMBER forcefield tends to underestimate the temperature sensitivity of the diffusion coefficient, for both solvents and small solutes,¹⁰ which explains why Ca²⁺, Li⁺, Na⁺ and K⁺ mostly underestimate diffusion coefficients compared to experimental values according to Table 2.

Prediction of the highly charged, radioactive ion ²³⁹Pu⁴⁺

With our understanding that we can predict diffusion coefficients with an overall relative difference no more than ±30%, the prediction of the diffusion coefficient for ²³⁹Pu⁴⁺ was undertaken. ²³⁹Pu⁴⁺ is highly radioactive and is difficult to extract using PUREX,⁶¹ which has diffusion-related steps in its process. Due to the lack of reliable experimental data for ²³⁹Pu⁴⁺ diffusion in pure water, we decided to computationally estimate this value. In this computational experiment only OPC water was used. Simulations conducted in the OPC water model are believed to have a strong transferability between bulky water environments and water-gel mixture conditions making it a good choice.²²

According to Cusnir et al., the diffusion coefficient of ²³⁹Pu⁴⁺ in 10mM MOPS buffer inside a polyacrylamide (PAM) gel (pH=6.50) at room temperature is 0.229±0.015×10⁻⁵ cm²/s.⁶² According to a previous study it was shown that the PAM gel can slow down the diffusion of a large ion (like SeCN⁻) by a factor of about two.⁶³ Hence, the final simulation result of 0.5579±0.04×10⁻⁵ cm²/s using the 12-6-4 parameter set in OPC water is a reasonable prediction for the experimental diffusion rate of ²³⁹Pu⁴⁺ in a pure aqueous solution. This claim can be further validated by a strong agreement with metadynamic simulations.⁶⁴

Conclusion and Future Perspective

In this project, the ISAIAH workflow was created to automate the workflow for the determination of diffusion coefficients, and it was then applied to the study of the diffusion coefficient of 15 ions in conjunction with a variety of water box and parameter set conditions. The overall results suggest that the maximum deviation between theory and experiment was <30%, but with most ions/water model combinations having lower uncertainties. A prediction for the diffusion coefficient of a highly charged, radioactive ²³⁹Pu⁴⁺ was also conducted, and the results provide qualitative guidance on experiments regarding ²³⁹Pu⁴⁺ diffusion in water or even in PUREX extractions.

In the future we will optimization the ISAIAH workflow to further speed the workflow, increase its accuracy and storage costs. The determination of ion diffusion constants provides an interesting probe to understand the effect of parameter choices on molecular hydration. Hence, future work will also focus on exploring the molecular-level details of why some ions/water model combinations have large errors and how these can be reduced in future model development. For example, the water-water interaction is something to look at further to improve computed diffusion constants. Finally, developing a better understanding of the ion diffusion mechanism and its variation among ions may better help understand the functional role of biologically relevant ions. ³⁵

Figure 2.

Linear extrapolation of Mg²⁺ (1A) and Ca²⁺ (1B) diffusion in different sizes of water boxes, with different water models and different parameter sets. Vertical bars indicate standard deviations among 20 replicate simulations for each ion-water combination. Final intercepts of each extrapolation are the final diffusion coefficients (10⁻⁵ cm²/s). Experimental values are 0.706*10⁻⁵cm²/s for Mg²⁺ and 0.792*10⁻⁵cm²/s for Ca²⁺.

Figure 4.

Prediction of the ²³⁹Pu⁴⁺ diffusion coefficient. Only the OPC water model was used and parameter sets are from previous work.¹⁹