Molecular Dynamic Simulations of Strontium Chloride Nanoparticle

Dec 3, 2008 - The nucleation kinetics for strontium chloride nanoparticles in supercritical water is studied by molecular dynamic simulations...
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J. Phys. Chem. C 2008, 112, 20181–20189

20181

Molecular Dynamic Simulations of Strontium Chloride Nanoparticle Nucleation in Supercritical Water Igor M. Svishchev,* Alexander Y. Zasetsky, and Istok G. Nahtigal Department of Chemistry, Trent UniVersity, Peterborough, Ontario, Canada K9J 7B8 ReceiVed: April 28, 2008; ReVised Manuscript ReceiVed: September 22, 2008

The nucleation kinetics for strontium chloride nanoparticles in supercritical water is studied by molecular dynamic simulations. The thermodynamic states probed in this work span the temperature range of 673 to 1073 K and the density range of 0.17 to 0.34 g/cm3. The cluster size distributions and size of critical nuclei are reported. The size distribution of strontium chloride clusters shows a very strong dependence on the system density, with larger clusters forming faster at lower densities. The clusters composed of 15-30 ions appear critical over the temperature and density ranges examined. The nucleation rate values obtained in this work are on the order of 3 × 1028 to 9 × 1028 cm-3 s-1. Water is found in a substantial amount on the surface of larger SrCl2 clusters, with some water molecules being trapped for nanoseconds inside cages formed by surrounding ions in the interior of clusters. 1. Introduction In recent years, the properties of aqueous systems under supercritical conditions have been receiving increasing interest because of their importance in natural geothermal processes,1 prospective in the management of hazardous waste,2 and potential in the design of new materials.3 The uniqueness of supercritical water (SCW) as a solvent is that the solubility of a material, to a large extent, can be controlled (small changes in pressure result in substantial changes in the density and solvent power) and thus can be used in materials processing. From this viewpoint, of particular interest is the production of crystalline or amorphous powders with controllable size distributions of nanoparticles. The solvation structures and dynamics of aqueous electrolyte solutions under supercritical conditions have been studied both experimentally and computationally. The extended X-ray absorption fine-structure technique4,5 has revealed many important properties of the solutes in SCW, with main focus on ion coordination numbers. From a computational standpoint, the alkali and alkaline earth metal chloride solutions have been particularly well-studied;5-11 the properties examined range from the hydrogen bond statistics and coordination numbers to the residence time of water molecules near the ions. For many of these systems, especially sodium chloride solutions, extensive computational studies on the ion association (pairing) under supercritical conditions have been carried out (see, for example, ref 10 and 11 and references therein). Furthermore, the nucleation rates for NaCl nanoparticles in SCW have been recently probed by the molecular dynamic (MD) simulation method.12,13 It turns out, however, that the data on particle nucleation kinetics in supercritical water for materials other than NaCl are still lacking, despite their importance for applied and fundamental research (nucleation theory). For example, strontium is known to be a major uranium fission product, resulting in the accumulation and precipitation of radioactive strontium salts in the liquid nuclear waste, as well as, to a lesser extent, in the * Corresponding author. Phone: (705) 748-1011ext. 7063. Fax: (705) 748-1625. E-mail: [email protected].

water coolant systems of nuclear power plants.14,15 In the present work, we aim to append the data and report the properties and nucleation kinetics for strontium chloride clusters. The present results are obtained by large-scale MD simulations. In order to study the effect of temperature and density on the formation of SrCl2 clusters, we have submitted a series of MD simulations over a range of the thermodynamic states relevant to the conditions found in hydrothermal applications using supercritical water (such as materials synthesis and power cycles). The cluster size statistics as well as their formation and induction (lag) times are directly obtained from molecular trajectories. The estimates for macroscopic properties such as the nucleation rate are also computed and reported in the following. 2. Nucleation in Multicomponent Systems Within the framework of the classical nucleation theory (CNT), the total (Gibbs) free energy of the formation of a condensed (new phase) nucleus is a combination of the energy “gain” (the new phase is energetically beneficial) for transforming unit volume of the system to the new phase and the energy “loss” due to the interface formation.16-18 For a multicomponent system, with a number of simplifications imposed, the expression for the formation free energy of nuclei reads n

∆G )

∑ ni(µil - µiv) + Aσ

(1)

i)1

where σ is the surface tension of the vapor- liquid (old phasenew phase) interface, µli is the chemical potential of the component i in the condense (new) phase, µvi is the chemical potential in vapor (old phase), ni is the number of molecules of the component i, and A is the surface area of the particle (which is equal to πR2 in the case of spherical particles). The set of corresponding Gibbs-Thomson equations, that also define the size of critical nuclei in multicomponent systems, can be written as19

(µil - µiv) +

2συi )0 R

(2)

where υi is the partial molecular volume of component i, and R

10.1021/jp803705z CCC: $40.75  2008 American Chemical Society Published on Web 12/03/2008

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TABLE 1: Simulation Details and Properties of SrCl2 in SCW T (K)

density (g/cm3)

H2O

Sr2+

Cl-

simulation time (ns)

largest cluster (atoms)

coord. no., Cl-

coord. no., Sr2+

673 873 1073 673 873 1073 673 873 1073

0.17 0.17 0.17 0.25 0.25 0.25 0.36 0.36 0.36

4000 4000 4000 4000 4000 4000 4000 4000 4000

46 46 46 46 46 46 46 46 46

92 92 92 92 92 92 92 92 92

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

99 138 138 126 138 138 135 138 138

4.2 3.0 2.8 6.1 4.5 3.5 6.5 5.5 4.5

2.0 1.3 1.2 2.5 1.8 1.3 2.6 2.2 1.7

is the particle radius (nuclei are assumed to have spherical shape). In theory, once the size of critical nucleus (radius R*) is determined, one can compare the experimental value for nucleation rate with that obtained from the following equation

J ) Kexp(∆G*/kbT)

(3)

where the value ∆G*(R*) corresponds to the free energy of formation of the critical nucleus R*, kb is the Boltzmann constant, T is temperature, and K is the pre-exponential factor, which depends on specific characteristics of molecular systems under consideration. The difficulties in applying the CNT approach to a real multicomponent system can be brought best by listing the simplifications that are used in deriving eqs 1-3: (a) two subsystems (i.e., vapor (old) and condense (new) phase) are uniform; in other words, they are a uniform mixture of the components in both the old and new phase; (b) this same uniformity assumption applies to the interface (c) that is considered to be infinitely sharp; (d) condense particles have spherical shape; and (e) σ is taken to be the surface tension value for planar interfaces. It is therefore not surprising that the estimates obtained by CNT can differ by many orders of magnitude from those obtained experimentally. The deviations may very likely arise from the inhomogeneity of forming condensed (new) phases. It is believed, however, that the differences result primarily from an inadequate account of the interface effect. Real interfaces may have a complex density profile, markedly nonuniform structure, composition, and finite thickness. All these restrictions make it extremely difficult to quantitatively describe the nucleation process on a CNT-based approach alone. The main difficulty in the present case of H2O/ Sr2+/Cl- system is to obtain the excess surface free energy for amorphous SrCl2 clusters. To the best of our knowledge, the surface tension is not known for amorphous SrCl2 in the first place and the clusters themselves (as will be shown in the following) are covered by a layer of liquid water, making the CNT estimates even more problematic. Analytical approximations for nucleation based on the kinetics rather than the Gibbsian thermodynamics have also been developed and a considerable progress has been made in this direction.20,21 Even so, the derivation of an accurate analytical model for the nucleation rate in multicomponent systems based on kinetic approaches is far from being completed. For example, after rather complicated algebra, the authors of the work21 have derived an analytical expression for the steady-state rate of binary nucleation. The quantitative estimates of nucleation parameters, however, require extensive calculations using density functional theory (DFT), necessary to obtain “real density profiles” for components at the interface. Alternatively, the molecular dynamic and Monte-Carlo computer simulation methods can be used to study nucleation in multicomponent systems. These techniques are free of many

of the above-mentioned shortages, not free of their own though. Based on model potentials that effectively, in comparison to an “exact” description of quantum mechanics, and yet efficiently approximate intermolecular interactions, the methods enable an accurate description (prediction) of macroscopic properties for many body systems. In particular, by using the MD simulation method, we can look at the process of nucleus formation and growth in complex molecular systems in “real” time as well as examine its size, shape, and composition. The main limitation of molecular dynamic simulations is relatively short length and time scales. The system size and time length typically accessible by the MD method at present are several thousand molecules over the time period of 1-100 ns. This is, nevertheless, sufficient for the present study, where we aim at probing the process of SrCl2 cluster formation at higher temperatures and pressures, and where the value of nucleation rate is significantly higher and thus requires reasonable computer power/time. 3. Simulation Details We have employed the following potential models: water molecules are considered rigid and interact through the simple point charge extended (SPC/E) pair potential by Berendsen et al.22 and the interaction functions for Sr2+-Sr2+, Sr2+-Cl-, Cl--Cl-, which are consistent with the SPC/E water model, are taken from the Smith and Dang work.23 The Lorentz-Berthelot combination rule for the Lennard-Jones part of the potential is applied in the case of H2O-Cl- and H2O-Sr2+ interactions. These models are capable of a sufficiently accurate description of the properties under study. In the present simulation work, where the central property of interest is the particle nucleation rate, the use of bond flexibility and polarizability appears expensive computationally. Note also that the simulation study by Sangster and Dixon24 has shown that the addition of polarizability does not significantly affect the spatial distribution of ions and their dynamical properties. The equations of motion are integrated using the Verlet algorithm with the SHAKE constraints technique25 and a time step of 2 fs. All the simulations are carried out in a cubic simulation cell replicated in 3D by the periodic boundary conditions. The NVT ensemble (isochoric-isothermal conditions) with temperature controlled by the Nose´-Hoover thermostat26 with the relaxation time of 20 fs was used. The reaction field method with conducting boundary (the infinitely large value of dielectric loss, which holds for electrolytes) conditions was used to treat long-range electrostatic interactions. The parallel molecular dynamics code for arbitrary molecular mixtures (M.DynaMix) by Lyubartsev and Laaksonen27 was used. Simulations have been carried out on a Linux cluster that enables one to perform calculations on the length/time scale required for studying cluster formation in SCW. The system size, trajectory lengths, temperatures, and densities are specified in Table 1. In each simulation 4000 water molecules were

MD Simulation of SrCl2 Nanoparticle Nucleation in SCW

Figure 1. Sr2+--Cl- radial distribution function for 673, 873, and 1073 K.

initially equilibrated at the temperature of interest. Next, 138 ions were randomly distributed over the simulation cell in such a way to prevent the formation of any contact ion pairs at the beginning and overlap with water molecules. The solution concentration was 9.19 wt % SrCl2. The density and temperature ranges of 0.17-0.36 g/cm3 and 673-1073 K were examined. Note that the critical temperature of the SPC/E water was estimated to be 641 K by Hayward and Svishchev.28 We have sampled the trajectories of atoms, following ion insertion, without any equilibration periods, because (evidently) nonequilibrium processes were being examined. The geometric criterion (similar to that proposed in the Stillinger’s work29) for the analysis of clustering was used, specifically, any two neighboring atoms belong to the same cluster if they are separated by a distance less than 0.37 nm (which corresponds to the average value between the minima in the Sr2+-Cl- radial distribution functions (shown in Figure 1) at different temperatures and there is a continuous path connecting them. The cluster formation and evolution have been sampled at each hundredth step during a simulation run with the clustering analysis being performed on-the-fly. 4. Results and Discussion 4.1. Cluster Size Distribution. The statistics of cluster sizes have been obtained directly by tracing the number of clusters and their sizes during the course of simulations. The evolution of the system during 3 ns and the formation of clusters are illustrated in Figure 2, in which the system undergoes the transformation from virtually uniformly scattered ions and ionpairs at 20 ps (left upper panel) to a few larger clusters consisting of several tens of ions after 3 ns of simulation time (bottom right panel). In the plot, water molecules (4000) are shown by red/white rods, and strontium and chloride ions are shown by the blue and green spheres, respectively. During the first stage, the clusters grow primarily by the flux of monomers (single atoms), thus forming a number of small ion associations/clusters that can be seen as continuous regions in the smaller particle region in the size distribution diagrams in Figure 3. Further growth progresses through the fusion of these intermediate size clusters, which is illustrated via the gaps in the larger particle region in the size distributions shown in Figure 3. From an empirical viewpoint, the clusters whose sizes lie near the threshold between these two very different in nature processes, the cluster formation by the monomer flux and the growth by fusion may be considered as critical. In our case, the critical

J. Phys. Chem. C, Vol. 112, No. 51, 2008 20183 sizes are in the range of 15-30 atoms. Note that the size of these critical clusters depends strongly on the system density. The density effect is illustrated in Figure 3, which presents the size distributions obtained for three different densities at the temperature of 1073 K. Because the size distributions are obtained as a sum over the entire length of simulated trajectories, they characterize the frequency of occurrence of different size clusters. Similar results were obtained at other temperatures studied. In particular, we have observed that at 873 K and at the lowest density of 0.17 g/cm3, the system forms the largest particles of 138 atoms without any transitional long-living smaller clusters. For the density of 0.25 g/cm3, two clusters with sizes of 64 and 74 atoms form first, and then fuse, forming the particle of 138 atoms. The process is more complex for the highest density of 0.36 g/cm3. In this case, the formation of clusters composed of 10, 12, 24, and 36 ions occurs at the first stage. Some of these particles join together to form the particle with intermediate size of 100 ions, and then of 129 ions, before reaching the largest possible size (for the system size used in this work). Coagulation of two clusters with 64 and 74 atoms is shown in Figure 4, where the stages of the fusion process are shown by three snapshots at 1.04, 2.3, and 2.6 ns. As can be seen, it requires about 1 ns for these two nanoparticles to find each other, join together, and to form a compact nearspherical droplet. 4.2. Radius of Gyration. A convenient measure of particle size is the gyration radius,30 Rg, which is calculated in our work as follows

R2g )

1 2N2

∑ Rij2

(4)

i,j

where Rij is the root-mean-square distance between ions i and j, and N is the number of ions comprising the cluster. It should be noted that the gyration radius can, in principle, be determined experimentally in light scattering experiments as well as with the small-angle neutron diffraction technique, allowing an experimental validation of computer simulation results. With regard to the present work, the gyration radii for several state points are shown in Figure 5. This Figure presents the radius of gyration as a function of the number of ions, up to the maximum possible cluster size. We note that at lower densities there appears to be a wider distribution of radii, whereas at 0.36 g/cm3, there appears to be a convergence trend in Rg, to the value of 1.2 nm. Large fluctuations in the radius of gyration at lower densities can be attributed to the fact that these clusters are loosely bound and chainlike (which can be confirmed by visual inspection of cluster configurations). The computed gyration radii for the SrCl2 clusters tend to be larger than those obtained for NaCl clusters made up of the same number of ions,13 presumably because of the larger size of the cation and higher proportion of chlorine species in the cluster. 4.3. Shape of Clusters. Useful characteristics on the cluster morphology (shape) and its dynamics during fusion events can be obtained by computing the inertia tensor. By definition, the components of inertia tensor are computed as M

Iij )

∑ mk(r2kδij - rkirkj), i, j ) 1...3

(5)

k)1

where mk is the mass of ion k, rk is the distance between the cluster center-of-mass and ion k, δij is the Kronecker delta. Computing the eigenvalues of the matrix I, we can obtain the principal moments (i ) j) of the inertia tensor. The simplest way to show the time evolution of a cluster shape is to represent

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Figure 2. Nucleation of Sr2+/Cl- clusters at the temperature of 673 K and density of 0.17 g/cm3. Snapshots of the system at 0.02, 0.14, 0.26, 0.5, 1.0, and 3.0 ns are shown (from left to right and top to bottom). Water molecules are shown by red/white rods, and Sr2+ and Cl- ions are shown by the blue and green spheres, respectively.

it as a graph of the temporal evolution of the eigenvalues of its inertia tensor. The three principal moments for the largest cluster are shown in Figure 6 as a function of time. It is seen that the shape of the cluster is close to a symmetrical top, as two principal inertia moments are about the same and the third is somewhat lower. This graph also captures a fusion event (the snapshots of which are shown in Figure 4) when two particles of 64 and 74 atoms collide and form the particle of 138 atoms. At the moment of collision the Ixx and Iyy increase sharply, because the formed particle has a marked needlelike shape at the beginning. Then it becomes more and more compact (or spheroidlike) and the principal moments of inertia converge to the same value. This technique can provide the estimates for the “shape relaxation”

time; specifically, the time required for the formation of a compact near-spheroid-shaped nanoparticle after a collision of two or more smaller clusters. Roughly, our estimate of shape relaxation time for the fusion event shown in Figure 4 is 0.5 ns. Two sequential cluster fusion events at the temperature of 1073 K and density of 0.17 g/cm3 are shown in Figure 7. It is seen that the shape relaxation time after the fusion events at higher temperature of 1073 K is somewhat shorter than it is at 873 K (see Figure 6). The values of principal moment of inertia also provide the information on the “equilibrium” shape of clusters. Figures 6 and 7 show that two moments are about the same and the third has a somewhat lower value. This implies that the particles are not exactly spherical but rather elongated spheroids, and as such

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Figure 3. Size distribution (formation frequency) of SrCl clusters. Histograms on the left are obtained at the temperature of 673 K, diagrams on the right correspond to the temperature of 1073 K. (a, d) Density of 0.17 g/cm3; (b, e) density of 0.25 g/cm3; (c, f) density of 0.36 g/cm3.

they could be roughly approximated by a prolate spheroid. The components of inertia tensor for a spheroid with z-axis along the axis of symmetry are given by the following relations: Ixx ) Iyy ) 1/5M(a2+c2) and Izz ) 2/5Ma2, where two distinct axis lengths are denoted as a and c, and M is the mass. A specific cluster can then be visualized by finding the values a and c and drawing corresponding spheroid. For instance, the spheroid axis lengths a and c for the cluster shown in the bottom plot of Figure 4 are equal to 1.75 and 1.55 nm, respectively. It should be noted that the above relations are valid for the uniformly solid (“ideal”) spheroids, whereas in real clusters there is a substantial amount of void space because of the finite size of ions. The effect of which is significant enough that it cannot be ignored, prompting that the inertia constants (1/5 and 2/5 in the above relations) be different than that for a uniform solid. For example, in order to match our calculated gyration radii with axis length estimates, the inertia constants for our ionic cluster case would have to be increased by a factor of 3, from 1/5 and 2/5 to 3/5 and 6/5, respectively. 4.4. Nucleation Rate. To characterize the nucleation rate J quantitatively, we look at the growth-decay dynamics of the strontium chloride clusters. Specifically, the nucleation rate values have been obtained using the kinetic approach introduced by Yasuoka and Matsumoto,31 where the temporal development of the formation of nuclei larger than some threshold size is used. In other words, the nucleation rate J is being defined as the number of nuclei larger than the critical size generated per unit volume per unit time. The nucleation rate value is calculated from the slope of the kinetic curve (see Figure 8), after appropriate normalization to the volume of simulation cell.12

The size of the critical nucleus is, in turn, estimated from the above analysis by making a series of observations for different threshold sizes. With increasing threshold size, the inclination of the growth curve decreases, at the critical size the inclination stops decreasing (and remains constant for larger threshold sizes). From a kinetic perspective, this is where growth and decay are balanced, yielding the size of the critical nucleus. Note that there is also an induction (lag) time before the system reaches a metastable equilibrium32 and no nuclei larger than the critical size are observed. The nucleation time lag is measured from the kinetic curve of a critical nucleus, and is defined as the time required to form this nucleus. The critical clusters size, lag times and the nucleation rates for each of the temperatures and densities studied are given in Table 2. The critical sizes range from 15 (at 0.36 g/cm3) to 30 (at 0.17 g/cm3). The cluster nucleation rates for SrCl2 appear to be on the order of 3 × 1028 to 9 × 1028 cm-3 s-1, which are very similar (within 1 order of magnitude) to the nucleation rates obtained for NaCl clusters in supercritical water.12,13 At this point we would like to recall Figure 3, which shows the frequency of occurrence of the clusters with different sizes formed on the duration of simulation. This Figure can be very helpful in identification of the size of critical nucleus. In particular, voids in the distributions shown in Figure 3 are indicative of the sizes where the critical size cluster can be found. As the critical sized cluster is unstable, a transient species which either grows or decays upon its formation, it has a shortlived existence (on the order of the Sr-Cl bond vibration time), so that it is seldom caught within our sampling time. Therefore these critical clusters result in voids or valleys in size distribu-

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Figure 4. Final stage of formation (fusion) of the largest Sr2+/Clcluster from two clusters of 64 and 74 atoms at the temperature of 873 K and density of 0.25 g/cm3 (snapshots at 1.04, 2.3, and 2.6 ns are shown from top to bottom). Water molecules are shown by red/white rods, and Sr2+ and Cl- ions are shown by the blue and green spheres, respectively.

tions such as in Figure 3. In our work, we first inspect these size distributions to identify possible critical clusters, and then analyze select growth-decay curves (see Figure 8) to obtain more accurate estimates for the critical nucleus size and to calculate the associated nucleation rate. To ensure the reproducibility of the results, we have carried out four additional simulation runs, starting from different initial configurations, for the system with the density of 0.17 g cm-3 at 873 K and obtained very similar values for the nucleation rate, namely, 3.8, 3.9, 4.0, and 4.0 × 1028 cm-3 s-1. Also, another simulation has been performed at these conditions for the system half as big (2000 waters, 46 and 23 ions). The value of 3.8 × 1028 cm-3 s-1 was obtained for the nucleation rate, illustrating the (seeming) absence of the system size effect. 4.5. Ion Hydration in SCW. We performed the standard analysis of the ion hydration based on the radial distribution functions (RDF). Note that the RDFs used in this analysis are

Svishchev et al. computed during final half of the MD trajectories in order to exclude contributions from strongly nonequilibrium processes that dominate at the beginning. The Sr2+-O and Cl--O radial distribution functions with their corresponding integrals obtained from our simulations are plotted in Figure 9, and the characteristic values (first shell coordination numbers) are summarized in Table 1. The position of the first peak on the Sr2+-O radial distribution function at 873 K and pressure of about 250 bar is found to be at 2.6 Å. This is in a good agreement with the X-ray experimental data for supercritical water at 260-340 bar.4 Interestingly, the pair Sr2+-O correlations for nearest neighbors in normal liquid water (2.62 Å) and supercritical water (2.59 Å) are very close. This indicates that enhanced thermal motion near the critical point has a weak effect on interatomic distances in the first hydration shell of strontium ions.12 The value of the water coordination number near the Sr2+ ion is in the range from 1.2 molecules at 1073 K and 0.17 g/cm3 to 2.6 molecules at 673 K and 0.36 g/cm3 (see Table 1). The coordination number for the Cl- ion is in the range of 2.9 (at 673 K and 0.17 g/cm3) to 6.9 (at 673 K and 0.36 g/cm3). This is in agreement with the MD simulation data for the Cl- ion, where the value of coordination number reported is 7.0 at the temperature of 683.15 K and the density of 0.36 g/cm3.11 Ab initio molecular dynamics studies of the Sr2+ hydration shell properties33 at lower temperatures have shown that the hydration number value decreases with increasing temperature (7.5 at 25 °C and 6.7 at 350 °C). The authors33 then predict the 6-fold coordination of Sr2+ ion at 600 K for a dilute solution. Our simulations give the coordination number of 1.2 to 3, depending on the density. We may note that in the case of present simulations, ions form aggregates in a metastable solution. As a result, most of them reside in the cluster interior and do not have a direct contact with water molecules. Although all the ions in supercritical water salt out to form the largest possible (in our model system) ionic cluster during about one nanosecond, water molecules at the same time form a substantial layer on the surface. This is illustrated in Figure 10, where the cluster composed of Sr2+ and Cl- ions is surrounded by water molecules, some of which being trapped inside the cluster. It is clear that water plays an important role in the process of nucleation and growth of SrCl2 particles, the mechanism is not as clear though. The density of SCW is relatively high (it is not a gas phase) and the ions are hydrated prior to nucleation. The separation between water and ion occurs during the cluster formation. Hydrating water molecules form a layer on the surface of the clusters, whereas the ions reside in the interior. Therefore, the nucleation rate should be affected as a consequence of the changes in the surface free energy of clusters. The growth kinetics should, in turn, be affected via the changes in the ion accommodation coefficient. Note also that the occasional trapping of water molecules inside clusters is a rare event. Such trapped molecules found in the interior of large clusters only, 100 ions and larger, whereas water molecules form a layer on the surface of all the ionic clusters. Interestingly, trapped water molecules can reside in the interior of clusters for a substantial time period. In other words, water molecules and ions can form relatively stable associations in SCW on the time scale of nanoseconds. The residence time for water molecules that reside inside clusters significantly exceeds their residence time in the hydration shell of single ions.11

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Figure 5. Gyration radius as a function of the number of ions. Plots are shown for the following conditions: (a) 673 K, 0.17 g/cm3; (b) 673 K, 0.25 g/cm3; (c) 673 K, 0.36 g/cm3; (d) 873 K, 0.36 g/cm3; (e) 1073 K, 0.36 g/cm3.

Figure 6. Time evolution of principle components (eigenvalues) of moments of inertia tensor for the largest cluster at the temperature of 873 K and density of 0.25 g/cm3.

5. Concluding Remarks In closing our discussion, we would like to point out that by using molecular simulations, specifically the MD method, the

Figure 7. Time evolution of principle components (eigenvalues) of moments of inertia tensor for the largest cluster at the temperature of 1073 K and density of 0.17 g/cm3.

entire pathway of cluster formation can be traced starting from the initially uniformly distributed atoms (species) to formed cluster populations in equilibrium with supercritical water. The

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Figure 9. Sr2+-O and Cl--O radial distribution functions (solid lines) and their integrals (dashed lines) computed for the system with the density of 0.25 g/cm3 at 873 K.

Figure 10. “Hydrated” Sr2+/Cl- cluster formed in supercritical water at the density of 0.25 g/cm3 and temperature of 1073 K. Figure 8. Time evolution (growth-decay dynamics) of clusters equal to or larger than the critical size at specific state points: (a) 1073 K, 0.17 g/cm3, NC ) 14; (b) 1073 K, 0.25 g/cm3, NC ) 15; (c) 1073 K, 0.36 g/cm3, NC ) 15. Dashed lines indicate the slopes used in estimation of the nucleation rates.

underlying processes responsible for the formation of the new phase nuclei can then be specified. Supplementary and worthy information on the cluster shape, relaxation times, ion hydration, etc., is also readily obtainable from the trajectories of molecules and atoms generated in MD simulations. The size distribution of emerging SrCl2 clusters in supercritical water shows a very strong dependence on the system’s density, with larger clusters forming faster at lower solution densities. The clusters consisting of 15-30 ions appear critical in the nucleation dynamics over the temperature and density range examined. We have calculated the SrCl2 nucleation rates, which are on the order of 3 × 1028 to 9 × 1028 cm-3 s-1; these kinetics are very similar (within 1 order of magnitude) to those obtained for the NaCl in supercritical water. Water clearly plays

an important role in the nucleation of ionic clusters. We have observed that water molecules may reside both on the surface and in the interior of SrCl2 clusters, with some water molecules being trapped for nanoseconds inside the cages formed by neighboring ions. TABLE 2: Nucleation Rates and Critical Nucleus Size

T (K) 673 873 1073

density F (g/cm3)

critical nucleus N (atoms)

induction period τc (ps)

nucleation rate J (× 1028 cm-3 s-1)

0.17 0.25 0.36 0.17 0.25 0.36 0.17 0.25 0.36

31 21 14 30 21 15 30 24 15

181 121 71 151 91 51 121 71 51

3.0 3.9 7.4 3.8 4.4 8.3 3.8 4.6 8.4

MD Simulation of SrCl2 Nanoparticle Nucleation in SCW Acknowledgment. We express our gratitude to the Natural Sciences and Engineering Research Council of Canada and SHARCNET for their financial support. SHARCNET (www.sharcnet.ca) is a consortium of universities and colleges operating a network of high-performance computer clusters across south western, central, and northern Ontario. References and Notes (1) Roedder E., Fluid Inclusions As Samples of Ore Fluids. In Geochemistry of Hydrothermal Ore Deposits; Barnes, H. L., Ed.; Holt, Rinehart,& Winston: New York, 1967. (2) Shaw, R. W.; Brill, T. B.; Clifford, A. A.; Eckert, C. A.; Franck, E. U. Chem. Eng. News. 1991, 69, 26–39. (3) Reverchon, E.; Adami, R. J. Supercrit. Fluids 2006, 37, 1–22. (4) Pfund, D. M.; Darab, J. G.; Fulton, J. L.; Ma, Y. J. Phys. Chem. 1994, 98, 13102–13107. (5) Palmer, B. J.; Pfund, D. M.; Fulton, J. L. J. Phys. Chem. 1996, 100, 13393–13398. (6) Balbuena, P. B.; Johnston, K. P.; Rossky, P. J. J. Am. Chem. Soc. 1994, 116, 2689–2690. (7) Balbuena, P. B.; Johnston, K. P.; Rossky, P. J. J. Phys. Chem. 1996, 100 (7), 2706–2715. (8) Zahn, D. Phys. ReV. Lett. 2004, 92, 040801. (9) Yang, Y.; Meng, S. J. Chem. Phys. 2007, 126, 044708. (10) Sherman, D. M.; Collings, M. D. Geochem.Trans. 2002, 3, 102. (11) Zhu, Y.; Lu, X.; Ding, H.; Wang, Y. Mol. Simul. 2003, 29 (12), 767–772. (12) Lummen, N.; Kvamme, B. Phys. Chem. Chem. Phys. 2007, 9 (25), 3251–3260.

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