Aggregation of Heteropolyanions in Aqueous ... - ACS Publications

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Aggregation of Heteropolyanions in Aqueous Solutions Exhibiting Short-Range Attractions and Long-Range Repulsions Mrinal K. Bera,*,†,○ Baofu Qiao,†,○ Soenke Seifert,‡ Benjamin P. Burton-Pye,§,∥ Monica Olvera de la Cruz,⊥,#,∇ and Mark R. Antonio*,† †

Chemical Sciences and Engineering Division and ‡APS X-ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, United States § Department of Chemistry, Lehman College of the City University of New York, 250 Bedford Park Boulevard West, Bronx, New York 10468, United States ∥ Ph.D. Program in Chemistry, The Graduate Center of the City University of New York, New York, New York 10016, United States ⊥ Department of Chemistry, #Department of Physics and Astronomy, and ∇Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *

ABSTRACT: Charged colloids and proteins in aqueous solutions interact via short-range attractions and long-range repulsions (SALR) and exhibit complex structural phases. These include homogeneously dispersed monomers, percolated monomers, clusters, and percolated clusters. We report the structural architectures of simple charged systems in the form of spherical, Keggin-type heteropolyanions (HPAs) by small-angle X-ray scattering (SAXS) and molecular dynamics (MD) simulations. Structure factors obtained from the SAXS measurements show that the HPAs interact via SALR. Concentration and temperature dependences of the structure factors for HPAs with −3e (e is the charge of an electron) charge are consistent with a mixture of nonassociated monomers and associated randomly percolated monomers, whereas those for HPAs with −4e and −5e charges exhibit only nonassociated monomers in aqueous solutions. Our experiments show that the increase in magnitude of the charge of the HPAs increases their repulsive interactions and inhibits their aggregation in aqueous solutions. MD simulations were done to reveal the atomistic scale origins of SALR between HPAs. The short-range attractions result from water or proton-mediated hydrogen bonds between neighboring HPAs, whereas the long-range repulsions are due to the distributions of ions surrounding the HPAs.

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INTRODUCTION Short-range attractions and long-range repulsions (SALR) between charged particles play critical roles in processes of biological and industrial interest and are crucial to understand the structure of soft materials containing proteins, DNA, colloids, and polymers.1 The delicate balance between shortrange attractions and long-range repulsions governs the stability of charged particles in solutions: the former favors aggregation of the particles to form either well-defined clusters or percolated structures, and the later prevents the particles from aggregation. Aggregation occurs when the magnitude of the attractive interaction energy exceeds both the thermal energy (kBT, where kB is the Boltzmann constant, and T denotes the temperature) and the repulsive potential energy barrier. Small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) experiments 2−8 along with simulations9−13 on particles with SALR have exhibited two correlation peaks at different reciprocal lattice vectors, Q, as shown in Figure 1a. The so-called “high-Q peak” or “aggregation peak” is generally assumed to be due to shortrange correlations between associated monomeric particles © XXXX American Chemical Society

within well-defined clusters or percolated structures, as shown in Figure 1c−e. In contrast, as summarized below, the “low-Q peak” has been interpreted in different ways by different groups of experimentalists and theorists. Through SAXS, SANS, and molecular dynamics (MD) simulations on lysozyme proteins and colloids with SALR, Stradner et al. argued2,4,14 that the proteins and colloids underwent aggregation to form stable clusters with a concentration-dependent cluster size. Their experiments showed two correlation peaks, similar to the peaks in Figure 1a, corresponding to ordering in the proteins and colloids at two different length scales. They argued that the lysozyme molecules with SALR underwent aggregation to form stable well-defined clusters; the position of the high-Q peak, which was found to be independent of concentration, corresponds to associated molecules within the clusters as shown in Figure 1d. The position of the low-Q peak was also found to be Received: October 29, 2015 Revised: December 14, 2015

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DOI: 10.1021/acs.jpcc.5b10609 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

particles in solutions instead of stable clusters as claimed by Stradner et al.2 Subsequent studies15 revealed that the dynamic properties of clusters in systems exhibiting SALR interactions are time-scale dependent. Hence, the observation of peaks in the structure factor response is not sufficient on its own to determine the nature of the aggregation of particles with SALR. Through a rigorous combination of simulations and theoretical calculations, Godfrin et al.12 have recently developed a helpful “rule-of-thumb” to identify the correlations in an aggregation process. According to the rule, if the normalized magnitude of the low-Q structure factor peak is larger than 3, then the aggregates must consist of either clusters (Figure 1d) or percolated clusters (Figure 1e). At this point, it is important to note that all the experimental studies of SALR discussed so far have involved either solutions of polydisperse (approximately 5%) micrometer-sized charged colloids or complex protein molecules. Both are known to possess heterogeneous charge distributions and/or patchy hydrophobic/hydrophilic regions that elevate the system complexities16 and add complications to the interpretation of experimental data. In order to arrive at a better understanding of the aggregation of charged systems with SALR in solutions, we have identified monodisperse molecular systems of heteropolyacids in aqueous solutions that do not have the complexities mentioned above for colloids and proteins. Heteropolyacids are well-known17−19 for their high solubility in water and their Brønsted acidity. Upon dissolution, heteropolyacids dissociate into negatively charged heteropolyanions (HPAs) and protons. HPAs have garnered a great deal of interest due to their applications in the fields of acid catalysis, biochemistry, sensor applications, and new energy materials.20,21 Because of their well-defined structures, sizes, and pHindependent anionic charges over a wide range of solution conditions, HPAs are also widely used as model systems for fundamental research.17 HPAs, especially those constructed of 5d-transition metal oxide frameworks, provide excellent electron density contrast and particle scattering;22 both of which are desirable for obtaining high-quality SAXS data. Moreover, the aggregation behaviors of HPAs have recently attracted considerable attention due to the observation23,24 of the formation of spherical shell-like aggregates (dubbed “blackberries”) in solutions. Although the reasons for such aggregation processes are still unclear,25 recent experiments26,27 and MD simulations28,29 have shown evidence of enhanced associations of cations on the surfaces of HPAs. These interactions are suggested to lead to the formation of blackberries. In order to understand structural phases in solutions, we have explored the aggregation behaviors of Keggin-type HPAs with different charges: α-[PW12O40]3− (abbreviated P-HPA hereafter), α-[SiW12O40]4− (abbreviated Si-HPA), and α-[AlW12O40]5−(abbreviated Al-HPA). Each of these Keggin anions has a sphere-like structure as shown in Figure 2a. The interatomic distances between opposite terminal oxygen atoms (10.4 Å) and between opposite corner-shared oxygen atoms (7.4 Å) provide a means to measure the average molecular diameter, which is approximately 8.8 Å and, most important, is independent of charge. As such, Keggin-HPAs are one of the most studied and well-known polyoxometalate systems, particularly in terms of electrochemistry and energyrelated applications.19 Furthermore, aspects of the aggregation behaviors of these HPAs have been studied through MD simulations28 that suggest an enhancement of aggregation with

Figure 1. (a) A typical SAXS or SANS structure factor, S(Q), showing two distinct peaks (a low-Q peak designated P1 and a high-Q aggregation peak designated P2) corresponding to different correlations in the solution of particles interacting via short-range attractions and long-range repulsions (SALR). S(Q) approaches 1 at large Q as shown by the solid horizontal line. Recent simulations and experimental studies have illustrated that these two peaks arise from correlations in the structures formed by (b) dissociated monomers, (c) randomly percolated monomers, (d) clusters of contacting monomers, and (e) randomly percolated clusters of contacting monomers.

independent of concentration at high values; Stradner et al. envisioned it to be due to “long-range cluster−cluster correlations” as shown in Figure 1d. In contrast, Shukla et al.,5 by repeating the experiments of Stradner et al.,2 observed a concentration-dependence of the low-Q peak, which confounded the picture of the formation of stable clusters by proteins in solutions and, hence, contradicts the assignment that the low-Q peak is due to long-range cluster−cluster correlations. Recently, Godfrin et al.9,12 and Liu et al.3,6,10 have revisited the issue of SALR through experimentation and coarse-grained simulations. Both groups found evidence for the existence of two peaks in the scattering patterns (similar to Stradner et al.2 and Shukla et al.5). But the low-Q peak was not interpreted as sole evidence for the presence of long-range correlations between stable clusters. Rather, it was attributed to the formation of structures ranging from nonassociated monomers (Figure 1b), randomly percolated monomers (Figure 1c), clusters (Figure 1d), to randomly percolated clusters (Figure 1e), depending upon solution conditions. Furthermore, through neutron spin echo experiments, Porcar et al.7 showed the formation of dynamic clusters of charged B


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Figure 2. (a) Crystal structure of Keggin-type heteropolyanions (HPAs) with different ionic charges (−3e, −4e, −5e) corresponding to different heteroatoms at the center (P, Si, and Al, respectively). (b) SAXS patterns (colored solid lines) collected from different concentrations of phosphotungstic acid (P-HPA), as noted in the inset, in 100 mM HCl solutions. The SAXS pattern collected from the neat 100 mM HCl solution is shown as the solid black line. The dashed black lines are the incoherent X-ray scattering backgrounds that have been calculated using atomic form factors and the SAXS pattern of the neat 100 mM HCl solution according to eq 1. (c) The incoherent background-subtracted and concentrationnormalized SAXS patterns that have been extracted from the primary data shown in panel b. The calculated form factor response is shown as the black dashed line. The experimental responses for the different P-HPA concentrations (shown as solid colored lines) are the products of the P-HPA form factor, F(Q), and the effective structure factors, Seff(Q), in the solutions. (d) The Seff(Q) for all of the P-HPA concentrations obtained by normalization of the corresponding data of panel c with F(Q).

ID-C beamline of the Advanced Photon Source at Argonne National Laboratory, U.S.A. Figure 2b shows a set of azimuthally averaged concentration-dependent SAXS data collected from aqueous solutions of phosphotungstic acid in 100 mM HCl. A gradual increase in the intensity (I) near Q = 0 is observed with the increase in the P-HPA concentration. In addition, the growth of a broad peak is observed as a function of increasing P-HPA solution concentration indicating that the correlations between P-HPAs develop as their concentrations increase. Background Correction and Data Reduction. As background correction plays a crucial role in extracting quantitative metrical information from SAXS data, we have taken a comprehensive approach to estimate the incoherent Xray scattering background of the solutions employed for our experiments. In general, most SAXS experiments are performed on dilute solutions of solutes. In these, the direct subtraction of the solvent scattering is sufficient to provide suitably background-corrected data. In our case, however, we are not only measuring SAXS from concentrated solutions but, also, using strong scatterers in the form of HPAs. As such, elevated concentrations of HPAs not only increase the scattering intensity at Q = 0 but, also, increase the overall incoherent background, which is separate from the solvent scattering. In all the SAXS patterns of Figure 2b, a sharp dip is observed at approximately 0.9 Å−1. This feature corresponds to the minima

the increase in magnitude of the charge on the HPA ions. Because there is no experimental verification of those results, we performed experiments with Keggin-type anions not only for comparison with the simulation results but also to develop a better understanding about the aggregation behaviors of charged particles in general. SAXS data were acquired for aqueous solutions of P-, Si-, and Al-HPAs with −3e, −4e, and −5e charges, respectively, as a function of HPA concentrations and solution temperatures. In combination with our atomistic MD simulations, this work has led to a new understanding of the origins of SALR in the aggregation of HPAs.

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EXPERIMENTS AND SIMULATION METHODS SAXS Measurements. Aqueous solutions of phosphotungstic acid (α-H 3 PW 12 O 40 ), silicotungstic acid (αH4SiW12O40), and aluminotungstic acid (α-H5AlW12O40) were prepared in 100 mM hydrochloric acid (pH = 1). This low pH initial condition was necessary to maintain the integrity of the P-HPA framework, which is not stable at pH values higher than approximately 1.5.20 We made the solutions of Si-HPA and AlHPA with 100 mM HCl to facilitate direct comparison with PHPA. Concentration-dependent SAXS measurements were performed with a pendant drop system, whereas temperaturedependent SAXS studies were done with capillary tubes of 1.5 mm diameter (Figure S1 in the Supporting Information). All SAXS data were acquired with 28 keV incident X-rays at the 12C


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The Journal of Physical Chemistry C of the HPA form factors, F(Q). A gradual increase in the intensity of the dip region with the increase in the HPA concentration is diagnostic of increasing incoherent scattering. To address this issue, we have developed a means to estimate the background by using the solvent (and the capillary tube) scattering and calculating the incoherent scattering contributions from the HPAs. The solvent (and the capillary tube) scattering were measured by collecting the SAXS patterns from neat 100 mM HCl solutions with the pendant drop (and the capillary tube) configurations. For HPA-containing solutions, the atomic form factors of all the atoms in the HPAs contribute to the incoherent scattering background in addition to the solvent (and the capillary tube) scattering. Hence we calculated the total background intensity using the following equation Ibac(Q ) = nsolIsol(Q ) + nHPA FHPA(Q )

Figure 3. Fully normalized structure factors obtained from SAXS measurements performed on the acidic (100 mM HCl) solutions of PHPA, Si-HPA, and Al-HPA with concentrations of 380, 330, and 380 mM, respectively.

(1)

Here, nsol and nHPA are the molar fractions of solvent and HPAs in the solutions, respectively. Isol(Q) is the SAXS pattern obtained from the solvent (i.e., 100 mM HCl solution). FHPA(Q) is the stoichiometery-summed over form factors of all the atoms of the HPAs. For instance, FHPA(Q) for P-HPA (PW12O403−) would be FP(Q) + 12FW(Q) + 40FO(Q), where FP(Q), FW(Q), and FO(Q) are the atomic form-factors of phosphorus (P), tungsten (W), and oxygen (O), respectively, computed using the International Tables of Crystallography.30 The backgrounds estimated for different concentrations of HPAs, using the above formalism, are shown in Figure 2b as the black dashed lines. Figure 2c shows the corresponding SAXS patterns after the background intensity subtraction followed by normalization with the HPA concentrations. The collapse of all the processed SAXS patterns in the high-Q region indicates that we have successfully subtracted the background contributions. It also indicates that the contributions from HPA-solvent correlations are negligible in the experimental Q-range probed in our measurements. Thus, the concentration-normalized SAXS patterns are equivalent to the products of the form factors, F(Q), and the effective structure factors, Seff(Q), due to the distributions of HPAs in the solutions. As the HPAs are monodisperse by their very nature, we have further normalized the background-subtracted and concentration-normalized SAXS data with the HPA form factors to obtain Seff(Q), as shown in Figure 2d. The HPA form factors were calculated from single crystal structures31 using the SolX32 software. Despite the concentration normalization, the data shown in Figure 2d exhibit an overall reduction of Seff(Q) as a function of concentration. A similar trend was also observed for Seff(Q) obtained from concentrated lysozyme solutions by Stradner et al.2 We attribute this reduction in overall intensity to absorption effects, which increase as a function of increasing concentration due to the presence of the heavy element W (Z = 74) in the molecular HPA anions. In order to compensate for absorption effects (which have no effective influence on peak positions and relative peak intensities), we have performed one last normalization of the Seff(Q) data with constant factors to obtain S(Q), as shown in Figures 3 and 4a. This normalization also serves to make the experimental structure factors comparable with the theoretically defined structure factors (i.e., S(Q) ≈ 1 for large-Q values) and most important, facilitates direct comparisons among all the structure factors collected for different HPA concentrations. The SAXS data discussed in all the subsequent sections are in the form of fully, unity-normalized structure factors, S(Q), computed by the above-mentioned method.

Simulations. As the main aim of this work is to understand the aggregation behaviors of Keggin heteropolyanions and the origin of SALR, we performed classical all-atom MD simulaitons instead of fitting the SAXS data with theoretical models.10 In contrast to proteins and colloids, which are nanometers to micrometers in sizes, HPAs are much smaller (approximately 8.8 Å in diameter, see Figure 2a). Such a small particle size makes it feasible to investigate the long-range correlations by means of simulations at the all-atom resolution. The simulations were performed for two different [PW12O40]3− concentrations, 60 and 200 mM, with 100 mM HCl. The proton ions were modeled using H3O+ exclusively. Table S1 in the Supporting Information provides the full components of the two systems. The force-field parameters of P-HPA have been reported by Lopez et al.33 under the framework of the AMBER force field. This is the only available force field for atomistic simulations on HPA systems. In this force field, atoms interact with each other in a pairwise additive fashion. As the two solutions involve charged species, we distributed the charge of each species among all the atoms of that species by use of partial charges. For instance, we distributed the −3e charge of the P-HPA anions over all 53 atoms, that is, 1-P, 12-W, and 40O. The pairwise Coulomb interactions were calculated for each atom up to a given cutoff distance (i.e., 1.2 nm herein) with the smooth Particle Mesh Ewald algorithm employed for calculating contributions from Coulomb interactions beyond 1.2 nm. The pairwise van der Waals interactions were calculated up to 1.2 nm employing the long-range correction on the dispersion interactions. The initial structures were obtained by randomly mixing all the molecules in a simulation box with edge lengths of 8 nm. After the energy minimization, an annealing simulation was performed to speed up the aggregation of the HPA molecular anions. Once the aggregation of the P-HPA ions converged, the production simulation was performed under the isobaric−isothermal ensemble (NTP) for the duration of 150 ns for data collection and analysis. See the Supporting Information (section S4) for more details regarding the force field parameters and the simulation methodology.

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RESULTS AND DISCUSSION Charge Dependence of HPA Aggregation. Figure 3 shows the experimental structure factors, S(Q), obtained from aqueous solutions of HPAs with different charges and with essentially identical concentrations. P-HPAs with a −3e charge D


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Figure 4. (a) The fully normalized structure factors, S(Q), of aqueous P-HPA solutions (with 100 mM HCl) with different concentrations (in mM) obtained from the corresponding data in Figure 2d. The peaks are marked in two categories, P1 and P2, corresponding to different correlations depicted in (c) and (d). Ranges of the peak positions are shown as blue vertical slabs. (b) The inter-HPA separations determined from peak P1 of S(Q) are plotted (red circles) as a function of P-HPA concentration. The error-bars in the inter-HPA separations estimated from the structure factors are smaller than the symbols. The average inter-HPA separations estimated by assuming their random distribution in solution (eq 2) are shown as blue stars. (c) A schematic illustration showing a mixture of percolated associated (contacting) monomers and nonassociated (noncontacting) monomers in the concentrated P-HPA solutions, leading to the observation of two SAXS peaks, P1 and P2. (d) A schematic illustration showing correlations between nonassociated monomers observed in dilute solutions of HPAs. Their sole presence results in a single SAXS correlation peak, P1, such as observed for Si-HPA and Al-HPA.

exhibit two correlation peaks, P1 and P2, whereas both Si-HPAs and Al-HPAs with −4e and −5e charges, respectively, show only the P1 correlation peak. This difference directly signifies that there are contrasts between the interactions exhibited by PHPAs and those by Si-HPAs and Al-HPAs. The two peaks in the structure factor for P-HPA correspond to two different length scales. This system response with two structure factor peaks resembles that for particles with SALR as shown in Figure 1a. The high-Q peak, P2, at 0.73 Å−1 corresponds to 8.6 Å, which is the center-of-mass to center-of-mass (P−P) distance between individual P-HPAs obtained from 2π/QP2, where QP2 is peak position of P2. The direct correspondence between the length scale obtained from the high-Q peak (8.6 Å) for P-HPA and its average diameter (8.8 Å, see Figure 2a) provides evidence of aggregation by direct contact. In contrast, the absence of P2 for Si-HPA and Al-HPA indicates the absence of contact associations in aggregated entities in solution. The exclusive presence of the low-Q peak, P1, at 0.43 Å−1 for SiHPA and Al-HPA corresponds to 14.6 Å. This distance, which is approximately 6 Å longer than the average cluster diameter, suggests noncontact associations between Si-HPAs and AlHPAs. Although the concentration of Si-HPA (330 mM) was slightly lower than that (380 mM) of Al-HPA, both solutions provide peaks, P1, at the same Q-value (see Figure 3). Upon the basis of concentration and charge considerations, the peak for

the Si-HPA (−4e anion) may be predicted to occur at a lowerQ value compared to that for the Al-HPA (−5e anion). Their equivalent positions indicate that in the absence of associated HPAs, as observed for the P-HPA (−3e anion), the Si-HPA and Al-HPA systems have charge-independent noncontact longrange associations in solutions. It is noteworthy that by means of atomistic MD simulations using the hydronium ion (H3O+) as the only countercation and in the absence of an acidic electrolyte (i.e., a near-neutral pH solution) Chaumont and Wipff28 reported contact aggregation behaviors that decreased in the following order: Al-HPA > P-HPA > Si-HPA. The inclusion of pH (as in our experiments with 100 mM HCl electrolytes) and other types of protons (e.g., Zundel, H5O2+, and Eigen, H9O4+, ions) can certainly impact the outcome of the simulations. In this regard, the leading position of Al-HPA as the strongest contact-aggregating system in the simulation trend stands in contrast with our experimental findings. Furthermore, one control simulation by us (Table S1 in the Supporting Information) has indicated that the absence of an acidic pH (i.e., no extra HCl) leads to different aggregation behaviors of P-HPAs (Figure S13 in the Supporting Information). In addition to the high-Q peak P2 for P-HPA, the low-Q peak P1 is observed at 0.31 Å−1 (see Figure 3). This Q-value corresponds to 20.2 Å, signifying long-range interactions typical of SALR that can lead to the different types of correlated E


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Figure 5. Temperature dependence of the fully normalized structure factors, S(Q), for (a) P-HPA with −3e charge, (b) Si-HPA with −4e charge, and (c) Al-HPA with −5e charge in acidic (100 mM HCl) aqueous solutions of concentrations 380, 330, and 380 mM, respectively. The data obtained at different temperatures are shifted vertically for clarity, and increasing temperatures are shown by vertical arrows in each of the slabs. (d) The average inter-HPA separations obtained from peak P1 are shown as colored circular symbols. The colors of the circles in (d) correspond to the colors used in the individual panels of (a−c). The error bars in (d) are smaller than the symbols.

mers, clusters, and randomly percolated clusters, as shown in Figure 1. From our experiments, the intensities of the low-Q peaks obtained for all P-HPA concentrations are less than 2 and, hence, using the rule of thumb of Godfrin et al.12 we can directly eliminate the possibility of cluster formation in our aqueous solutions. In the absence of clustered states, the possible correlations that can give rise to the low-Q peaks are due to nonassociated monomers (Figure 1b) and randomly percolated monomers (Figure 1c). Now, the question arises: Which state, either the randomized nonassociated monomers or the monomer-percolated structures, is responsible for the low-Q peak in the S(Q) obtained from the aqueous solutions of P-HPA? This can be answered by following the high-Q correlation peak, P2, whose presence for the concentrations higher than 47.7 mM indicates the presence of contactassociated P-HPAs. This scenario is only possible for a system of percolated monomers as shown in Figure 1c. In contrast, when the concentration is lower than 47.7 mM, the high-Q peak is not observed and the P-HPAs remain only as nonassociated monomers, as shown in Figure 4d, interacting via the long-range repulsive part of SALR and, hence, only the low-Q peak prevails. Regarding the presence of both peaks for P-HPAs at concentrations higher than 47.7 mM, it is not possible to conclude from concentration-dependent studies alone that the solutions consist solely of percolated monomers or, alternatively, as mixtures of nonassociated monomers and randomly percolated monomers as shown in Figure 4c. Considering the assignment of peak P1 as arising from homogeneous distributions and random dispersions of nonassociating monomers, we further explored its dependence with P-HPA concentrations by computing the average center-ofmass to center-of-mass (i.e., P−P) separations using 2π/QP1, where QP1 is the peak position of P1. The results are shown in

structures discussed in the Introduction and illustrated in Figure 1b−d. In order to understand the aggregation behavior of P-HPA and to pin-down the particular correlations responsible for the low-Q peak, P1, we have explored the solution concentration and temperature dependencies as discussed below. Concentration Dependence of HPA Aggregation. In line with the preceding discussion, the structure factors, S(Q), of P-HPA solutions with different concentrations obtained from SAXS reveal two correlation peaks, P1 and P2, as shown in Figure 4a. The position of the high-Q peaks, P2, at 0.73 Å−1 is independent of the P-HPA concentrations. In contrast, the position of the low-Q peaks, P1, shifts to lower Q values with decreasing concentrations. Furthermore, a correlation between the P1 and P2 peak intensities is observed. As the intensity of P2 decreases monotonically with decreasing P-HPA solution concentration (from 381.7 to 0.7 mM), the behavior of P1 exhibits a divided response that pivots around the concentration of 95.4 mM for which there is a maximum in the P1 intensity. From the maximum concentration of 381.7−95.4 mM, the P1 peak intensities increase with decreasing P-HPA concentrations, whereas for decreasing concentrations below 95.4 mM (down to the minimum value of 0.7 mM), the intensities decrease. Another way to describe the trend in the low-Q peak intensities is with regard to the high-Q peak intensities. When the intensity of peak P2 flattens out to 1 at the 47.7 mM concentration and for all subsequent dilutions, the intensity of peak P1 decreases monotonically with decreasing concentrations. The initial synchronized intensity decrease of the high-Q peak and the increase of the low-Q peak with the decrease in P-HPA solution concentrations is a signature of particle correlations via SALR.3,6,9,10,12 According to Godfrin et al.,12 the low-Q peak can arise due to correlations between monomers, randomly percolated monoF


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Figure 6. (a) Snapshot of the atomistic simulation of the system with the P-HPA concentration of 200 mM with 100 mM HCl. Solvent (water) molecules are omitted from the display. (b) SAXS structure factors for the two P-HPA solution concentrations (60 and 200 mM) calculated from the atomistic simulations. (c) Radial distribution functions of the P-HPAs.

understand the interaction landscape of HPAs, we have further performed temperature-dependent SAXS studies. Temperature Dependence of HPA Aggregation. The structure factors, S(Q), obtained for P-HPA, Si-HPA, and AlHPA solutions with concentrations of 380, 330, 380 mM, respectively, as a function of temperature, from 10 to 70 °C, are shown in Figure 5a−c. As evident from Figure 5a, peak P2 for P-HPA stays fixed at Q = 0.73 Å−1, whereas peak P1 shifts to higher-Q values with increasing solution temperatures. In addition, slight increases in the widths of peaks P1 and P2 are observed. These two temperature dependencies for P-HPA can be explained by the presence of a mixture of nonassociated monomers and randomly percolated monomers, as shown in Figure 4c. With the increase in temperature, randomly percolated P-HPA monomers break up into nonassociated monomers thereby increasing their concentrations and, hence, decreasing the average separation between them. This explains the movement of peak P1 to higher-Q values as well. The breaking up of randomly percolated monomers is supported by the slight reduction in the intensity of peak P2 as shown in Figure S3 (Section S2 in the Supporting Information). Also accompanying the increase in temperature, disorder increases in both the distributions of randomly percolated monomers and nonassociated monomers, leading to slight increases in the widths of both peaks (0.284 and 0.286 Å−1 for P1 and 0.394 and 0.443 Å−1 for P2 with temperatures of 10 and 70 °C, respectively). Hence, through temperature-dependent measurements, we can conclusively say that as a result of SALR, PHPAs form mixtures of nonassociated monomers and randomly percolated monomers. In such mixtures, there is a dynamicequilibrium between the associated percolated structures and the nonassociated monomers, a scenario that is confirmed in our MD simulations discussed immediately below. In contrast to the P-HPA system, the single peak (P1) observed for the SiHPA and Al-HPA solutions (Figure 5b,c, respectively) exhibits no temperature dependence, suggesting the existence of only noncontact-associated monomers. This confirms our experimental finding that increasing the magnitude of the negative charge on HPAs (in the form of Si-HPA, −4e, and Al-HPA, −5e) induces electrostatic repulsions that inhibit the random percolation process and that are stronger (i.e., more repulsive) than those for P-HPA (−3e).

Figure 4b. If we consider that the HPAs interact repulsively (or not at all), an assumption that may or may not be valid under our solution conditions and given the influence of cations, vide infra, then they should distribute themselves uniformly in the entire volume of the solution. In this case, the average separation between P-HPAs can be calculated34 as ⟨a⟩ = 1.104 × [NAC ]−1/3

(2)

Here, C is the P-HPA concentration and NA is Avogadro’s number. The calculated average separations between P-HPAs as a function of their concentrations are also shown in Figure 4b. It is obvious that the average experimental separation between the P-HPA monomers increases with the decrease in their concentration but the increase is not as large as calculated from eq 2. The average experimental separation converges to 42 Å for very dilute systems (50 mM), the observed separation is slightly higher than the calculated value; this is attributed to the presence of randomly percolated monomers that reduce the effective number of noncontact-associated P-HPA monomers available in the solution. Because of the absence of aggregation in the concentrated solutions of Si-HPA and Al-HPA, as shown in Figure 3, the concentration dependences of Si-HPA and AlHPA aggregation were not studied further. In order to better G


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The Journal of Physical Chemistry C Atomistic MD Simulations. A snapshot of the simulation box for the P-HPA concentration of 200 mM is shown in Figure 6a. A corresponding simulation snapshot for the 60 mM concentration is provided in Figure S6a (Supporting Information). The coexistence of randomly percolated PHPAs and nonassociated P-HPA monomers is observed in the simulation boxes for both concentrations (see Figure S6 in the Supporting Information for the structures of some of the percolated P-HPA monomers). The structure factors, S(Q), were computed using the coordinates of the phosphorus atoms, which denote the center-of-mass of the P-HPA molecule, and are presented in Figure 6b. For both P-HPA concentrations, two correlation peaks were observed. The high-Q peak is located at 0.73 Å−1, which is independent of the P-HPA concentration and is in close agreement with the experimental peak P2, as shown in Figure 4a. The low-Q peak is slightly shifted from 0.21 Å−1 (60 mM) to 0.23 Å−1 (200 mM). This shift is consistent with the experimental observations of peak P1 at 0.18 Å−1 (47.7 mM) and 0.25 Å−1 (190.8 mM), respectively. Most importantly, the relative reduction of the P2 peak intensity with the decrease in the HPA concentration is also observed in the simulations. Despite the use of limited simulation box sizes (approximately 8 nm in each dimension), which led to large fluctuations in the low-Q region in the structure factor calculations, the qualitative agreement of the simulation results with the experiments is noteworthy. For quantitative agreements, larger simulation boxes are required. Another agreement between the experiments and the simulations is the constancy of the P2 peak widths, which are of the order of 0.3 Å−1. This corresponds to a correlation length of 21 Å, comparable to two HPA diameters, and indicates that the aggregated structures of P-HPAs responsible for peak P2 are not ordered beyond the first nearest-neighbor distances, as observed in Figure S7 in the Supporting Information. Aggregation of P-HPAs. To understand the aggregation behaviors of the P-HPA system, the radial distribution functions of phosphorus atoms, g(r), were obtained for both solution concentrations. As shown in Figure 6c, the g(r) plots show two sharp peaks at 9.6 and 10.3 Å. These indicate the formation of randomly percolated monomers (see Figure S10 in the Supporting Information for some snapshots of contact HPA−HPA interactions with different distances). These percolating P-HPA entities contribute to the calculated structure factor peak P2 at Q = 0.73 Å−1 as shown in Figure 6b and is consistent with the experimental peaks P2, as shown in Figure 4a. It is to be kept in mind that the SAXS experiments access only average information about the separation (8.6 Å) between the P-HPAs in the percolated monomer structures. The simulations provide accurate spatial distributions in the form of g(r) as long as the calculated S(Q) matches the experimental response (as it does here). In this regard, it is gratifying to note that the separation values obtained from simulations for the percolating-monomer structures are within the range of distances expected, 7.4−10.4 Å, based solely upon the diameter of the molecular anions as shown in Figure 2a. We further computed the probability of aggregates containing different numbers of P-HPAs for both the solution concentrations of 60 and 200 mM. The results are provided in Table 1. The probabilities of the nonassociated P-HPA monomers are quite high (69.8% for 60 mM, and 25.1% for 200 mM) compared to randomly percolated monomers (2-, 3-, 4-mers, etc.) that arise from contact associations. Furthermore, the probabilities of the percolated monomers decrease with

Table 1. Probabilities (%) of Aggregates of P-HPAs with Different Numbers of Monomers at Two Different P-HPA Concentrations (mM) Obtained from MD Simulationsa [PHPA]

1-mer

2-mer

3-mer

4mer

5mer

6mer

7mer

8mer

...b

60 200

69.8 25.1

22.6 26.0

6.6 14.0

1.0 9.5

5.2

4.5

3.7

2.5

...

a

The P-HPAs that are closer than the cutoff distance (rcut) are defined as aggregated, where rcut = 11.6 Å denotes the first coordination shell (see g(r) in Figure 6c). bHigher level aggregates are not listed; their probabilities decrease monotonically.

their increasing sizes, ruling out the formation of monodisperse aggregates of a particular size. Such observations collectively exclude the possibility of the formation of uniformly distributed clusters (Figure 1d) and percolated-clusters (Figure 1e). Instead, the calculations of the distributions of aggregates of different sizes, in addition to the simulation snapshot in Figure 6a support the presence of a mixture of nonassociated monomers and randomly percolated associated-monomers, consistent with our experimental observations. Moreover, for the high (200 mM) P-HPA concentration, the probabilities of the percolated-monomers of all possible sizes increase (Table 1). This results in the relative enhancement of the P2 peak intensity (corresponding to contacting monomers) for the 200 mM concentration (Figure 6b). Also, the picture of randomly percolated associated-monomers is supported by the fact that the ordering of the HPAs responsible for peak P2 only extends up to the first nearest neighbor and the peak intensity is less than 3 (in accord with the rule-of-thumb by Godfrin et al.12). As shown in Movies S3 and S4 in the Supporting Information, the randomly percolated monomers are in dynamical equilibrium with the nonassociated monomers. We calculated the stability of randomly percolated monomers (Figure S8 in the Supporting Information) and found that their lifetimes are on the picoseconds time scale. The high P-HPA solution concentration results in the formation of less stable randomly percolated monomeric structures than in the 60 mM solution. Our results agree well with the concept of dynamic aggregation processes observed in lysozyme protein solutions proposed by Porcar et al., 7 who found lifetimes of approximately tens of nanoseconds for aggregated structures. The larger lifetimes for proteins are attributed to stronger short-range attraction potentials (approximately −4 kBT)7 compared to those (approximately −2 kBT) for HPAs in our work (Figure 7a). Short-Range Attractions and Long-Range Repulsions. On the basis of the calculated g(r), the potential of mean force (PMF) between P-HPA particles can be computed by PMF(r) = −kBT ln[g(r)].36,37 Figure 7a shows the computed PMF: Two regions are observed. (1) PMF(r) < 0 with P-HPA separations of 0 describes the long-range repulsions between P-HPAs. These persist up to approximately 30 Å of the inter-HPA separations. This distance agrees with the low-Q correlation peak P1 of the calculated SAXS (Figure 6b). Even though such long-range repulsion is expected to originate from the screened Coulomb interaction40 predicted by Debye−Huckle theory,35 it has not been investigated to the best of our knowledge at the atomistic resolution. To understand the origin of the long-range repulsions in the second region where PMF(r) > 0, the distribution of all the ions (i.e., H3O+, Cl−, and the constituent P-HPA anions) and water molecules were calculated (Figure S12 in the Supporting Information). The total amount of charge from the ions, and ions and water, surrounding the reference P-HPA anion are plotted in Figure 7c. Because each P-HPA carries a −3e charge, the total amount of charge should converge to +3e for a neutral system. As observed in Figure 7c for both P-HPA concentrations, the total ion charge converges to +3e at the distance of approximately 30 Å, which agrees with the distance where the long-range repulsion disappears (Figure 7a), and with the position of the low-Q peak in the SAXS (Figure 6b). In I


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mean force; SALR, short-range attractions and long-range repulsions; SAXS, small-angle X-ray scattering; Si-HPA, α[SiW12O40]4−

formation of small dynamic randomly percolated monomers. The energetic landscape here is a far weaker one than that leading to gelation in colloidal systems interacting via shortrange attractions.41,42 The long-range repulsions between PHPAs result from electrostatic interactions that are due to the distribution of ions surrounding them. The results presented here are crucial in improving our understanding of the aggregation of charged systems, the sizes of which range from inorganic ions of a few angstroms, to complex protein molecules of several nanometers, up to colloidal particles of hundreds of nanometers.

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(1) Kovalchuk, N.; Starov, V.; Langston, P.; Hilal, N. Formation of Stable Clusters in Colloidal Suspensions. Adv. Colloid Interface Sci. 2009, 147−48, 144−154. (2) Stradner, A.; Sedgwick, H.; Cardinaux, F.; Poon, W. C. K.; Egelhaaf, S. U.; Schurtenberger, P. Equilibrium Cluster Formation in Concentrated Protein Solutions and Colloids. Nature 2004, 432, 492− 495. (3) Liu, Y.; Porcar, L.; Chen, J. H.; Chen, W. R.; Falus, P.; Faraone, A.; Fratini, E.; Hong, K. L.; Baglioni, P. Lysozyme Protein Solution with an Intermediate Range Order Structure. J. Phys. Chem. B 2011, 115, 7238−7247. (4) Stradner, A.; Cardinaux, F.; Schurtenberger, P. A Small-Angle Scattering Study on Equilibrium Clusters in Lysozyme Solutions. J. Phys. Chem. B 2006, 110, 21222−21231. (5) Shukla, A.; Mylonas, E.; Di Cola, E.; Finet, S.; Timmins, P.; Narayanan, T.; Svergun, D. I. Absence of Equilibrium Cluster Phase in Concentrated Lysozyme Solutions. Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 5075−5080. (6) Liu, Y.; Fratini, E.; Baglioni, P.; Chen, W. R.; Chen, S. H. Effective Long-Range Attraction between Protein Molecules in Solutions Studied by Small Angle Neutron Scattering. Phys. Rev. Lett. 2005, 95, 118102. (7) Porcar, L.; Falus, P.; Chen, W. R.; Faraone, A.; Fratini, E.; Hong, K. L.; Baglioni, P.; Liu, Y. Formation of the Dynamic Clusters in Concentrated Lysozyme Protein Solutions. J. Phys. Chem. Lett. 2010, 1, 126−129. (8) Min, Y. J.; Akbulut, M.; Kristiansen, K.; Golan, Y.; Israelachvili, J. The Role of Interparticle and External Forces in Nanoparticle Assembly. Nat. Mater. 2008, 7, 527−538. (9) Godfrin, P. D.; Valadez-Perez, N. E.; Castaneda-Priego, R.; Wagner, N. J.; Liu, Y. Generalized Phase Behavior of Cluster Formation in Colloidal Dispersions with Competing Interactions. Soft Matter 2014, 10, 5061−5071. (10) Liu, Y.; Chen, W. R.; Chen, S. H. Cluster Formation in TwoYukawa Fluids. J. Chem. Phys. 2005, 122, 044507. (11) Sciortino, F.; Mossa, S.; Zaccarelli, E.; Tartaglia, P. Equilibrium Cluster Phases and Low-Density Arrested Disordered States: The Role of Short-Range Attraction and Long-Range Repulsion. Phys. Rev. Lett. 2004, 93, 055701. (12) Godfrin, P. D.; Castaneda-Priego, R.; Liu, Y.; Wagner, N. J. Intermediate Range Order and Structure in Colloidal Dispersions with Competing Interactions. J. Chem. Phys. 2013, 139, 154904. (13) Valadez-Perez, N. E.; Castaneda-Priego, R.; Liu, Y. Percolation in Colloidal Systems with Competing Interactions: The Role of LongRange Repulsion. RSC Adv. 2013, 3, 25110−25119. (14) Cardinaux, F.; Stradner, A.; Schurtenberger, P.; Sciortino, F.; Zaccarelli, E. Modeling Equilibrium Clusters in Lysozyme Solutions. Europhys. Lett. 2007, 77, 48004. (15) Riest, J.; Eckert, T.; Richtering, W.; Nagele, G. Dynamics of Suspensions of Hydrodynamically Structured Particles: Analytic Theory and Applications to Experiments. Soft Matter 2015, 11, 2821−2843. (16) Derot, C.; Porcar, L.; Lee, Y. J.; Pincus, P. A.; Jho, Y. S.; In, M. Electrostatic Interaction between Nonuniformly Charged Colloids: Experimental and Numerical Study. Langmuir 2015, 31, 1649−1659. (17) Kozhevnikov, I. V. Catalysis by Heteropoly Acids and Multicomponent Polyoxometalates in Liquid-Phase Reactions. Chem. Rev. 1998, 98, 171−198. (18) Jeannin, Y. P. The Nomenclature of Polyoxometalates: How to Connect a Name and a Structure. Chem. Rev. 1998, 98, 51−76. (19) Kourasi, M.; Wills, R. G. A.; Shah, A. A.; Walsh, F. C. Heteropolyacids for Fuel Cell Applications. Electrochim. Acta 2014, 127, 454−466.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b10609. Details about the experimental SAXS data acquisition and MD simulations along with supporting figures. (PDF) Supporting movies. Movie S1: Movie of the simulation trajectory of the system with [PW12O40]3− = 60 mM with 100 mM HCl. Water molecules are omitted from the display. Movie S2: Rotation movie showing the 3D structure of one P-HPA oligomer (a 15-mer) in the system with [PW12O40]3− = 200 mM with 100 mM HCl. Movie S3: Movie of the simulation trajectory of the system with [PW12O40]3− = 60 mM with 100 mM HCl showing the dynamically-stable P-HPA aggregates (green solid lines). Water molecules are omitted from the display. Movie S4: Movie of the simulation trajectory of the system with [PW12O40]3− = 200 mM with 100 mM HCl showing the dynamically-stable PHPA aggregates (green solid lines). Water molecules are omitted from the display. (ZIP)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] or [email protected]. *E-mail: [email protected]. Author Contributions ○

M.K.B. and B.Q. contributed equally to this work and should be considered cofirst authors.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work and the use of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility at Argonne National Laboratory, is based upon work supported by the U.S. DOE, Office of Science, Office of Basic Energy Science, Division of Chemical Sciences, Biosciences and Geosciences, under contract No DE-AC02-06CH11357. B.Q. gratefully acknowledges the computing resources provided on Blues, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. M.O.d.l.C. acknowledges the support from U.S. Department of Energy Award DE-SC0000989.

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ABBREVIATIONS Al-HPA, α-[AlW12O40]5−; HPA, heteropolyanion; MD, molecular dynamics; P-HPA, α-[PW12O40]3−; PMF, potential of J


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The Journal of Physical Chemistry C (20) Ammam, M. Polyoxometalates: Formation, Structures, Principal Properties, Main Deposition Methods and Application in Sensing. J. Mater. Chem. A 2013, 1, 6291−6312. (21) Long, D. L.; Tsunashima, R.; Cronin, L. Polyoxometalates: Building Blocks for Functional Nanoscale Systems. Angew. Chem., Int. Ed. 2010, 49, 1736−1758. (22) Bera, M. K.; Ellis, R. J.; Burton-Pye, B. P.; Antonio, M. R. Structural Aspects of Heteropolyacid Microemulsions. Phys. Chem. Chem. Phys. 2014, 16, 22566−22574. (23) Liu, T. B.; Diemann, E.; Li, H. L.; Dress, A. W. M.; Muller, A. Self-Assembly in Aqueous Solution of Wheel-Shaped Mo154 Oxide Clusters into Vesicles. Nature 2003, 426, 59−62. (24) Yin, P. C.; Li, D.; Liu, T. B. Solution Behaviors and SelfAssembly of Polyoxometalates as Models of Macroions and Amphiphilic Polyoxometalate-Organic Hybrids as Novel Surfactants. Chem. Soc. Rev. 2012, 41, 7368−7383. (25) Verhoeff, A. A.; Kistler, M. L.; Bhatt, A.; Pigga, J.; Groenewold, J.; Klokkenburg, M.; Veen, S.; Roy, S.; Liu, T. B.; Kegel, W. K. Charge Regulation as a Stabilization Mechanism for Shell-Like Assemblies of Polyoxometalates. Phys. Rev. Lett. 2007, 99, 066104. (26) Antonio, M. R.; Nyman, M.; Anderson, T. M. Direct Observation of Contact Ion-Pair Formation in Aqueous Solution. Angew. Chem., Int. Ed. 2009, 48, 6136−6140. (27) Pigga, J. M.; Kistler, M. L.; Shew, C. Y.; Antonio, M. R.; Liu, T. B. Counterion Distribution around Hydrophilic Molecular Macroanions: The Source of the Attractive Force in Self-Assembly. Angew. Chem., Int. Ed. 2009, 48, 6538−6542. (28) Chaumont, A.; Wipff, G. Interactions between Keggin Anions in Water: The Higher Their Charge, the Higher Their Condensation? A Simulation Study. Eur. J. Inorg. Chem. 2013, 2013, 1835−1853. (29) Leroy, F.; Miro, P.; Poblet, J. M.; Bo, C.; Bonet Avalos, J. Keggin Polyoxoanions in Aqueous Solution: Ion Pairing and Its Effect on Dynamic Properties by Molecular Dynamics Simulations. J. Phys. Chem. B 2008, 112, 8591−8599. (30) Brown, P. J.; Fox, A. G.; Maslen, E. N.; O’Keefe, M. A.; Willis, B. T. M. Intensity of Diffracted Intensities. In International Tables of Crystallography; Prince, E., Ed.; International Union of Crystallography: 2006; Vol. C, pp 554−595.10.1107/97809553602060000600 (31) Brown, G. M.; Noe-Spirlet, M. R.; Busing, W. R.; Levy, H. A. Dodecatungstophosphoric Acid Hexahydrate, (H5O2+)3(PW12O403‑)True Structure of Keggin’s ’Pentahydrate’ from Single-Crystal X-Ray and Neutron-Diffraction Data. Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 1977, 33, 1038−1046. (32) Zuo, X. B.; Cui, G. L.; Merz, K. M.; Zhang, L. G.; Lewis, F. D.; Tiede, D. M. X-Ray Diffraction ″Fingerprinting″ of DNA Structure in Solution for Quantitative Evaluation of Molecular Dynamics Simulation. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 3534−3539. (33) Lopez, X.; Nieto-Draghi, C.; Bo, C.; Avalos, J. B.; Poblet, J. M. Polyoxometalates in Solution: Molecular Dynamics Simulations on the α-PW12O403‑ Keggin Anion in Aqueous Media. J. Phys. Chem. A 2005, 109, 1216−1222. (34) Lavrik, N. L.; Voloshin, V. P. Calculation of Mean Distances between the Randomly Distributed Particles in the Model of Points and Hard Spheres (the Method of Voronoi Polyhedra). J. Chem. Phys. 2001, 114, 9489−9491. (35) Brydges, D. C.; Martin, P. A. Coulomb Systems at Low Density: A Review. J. Stat. Phys. 1999, 96, 1163−1330. (36) Leung, C. Y.; et al. Molecular Crystallization Controlled by pH Regulates Mesoscopic Membrane Morphology. ACS Nano 2012, 6, 10901−10909. (37) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (38) Qiao, B.; Demars, T.; Olivera de la Cruz, M.; Ellis, R. J. How Hydrogen Bonds Affect the Growth of Reverse Micelles around Coordinating Metal Ions. J. Phys. Chem. Lett. 2014, 5, 1440−1444. (39) Chaumont, A.; Wipff, G. Ion Aggregation in Concentrated Aqueous and Methanol Solutions of Polyoxometallates Keggin Anions: The Effect of Counterions Investigated by Molecular Dynamics Simulations. Phys. Chem. Chem. Phys. 2008, 10, 6940−6953.

(40) Leunissen, M. E.; Christova, C. G.; Hynninen, A. P.; Royall, C. P.; Campbell, A. I.; Imhof, A.; Dijkstra, M.; van Roij, R.; van Blaaderen, A. Ionic Colloidal Crystals of Oppositely Charged Particles. Nature 2005, 437, 235−240. (41) Lu, P. J.; Zaccarelli, E.; Ciulla, F.; Schofield, A. B.; Sciortino, F.; Weitz, D. A. Gelation of Particles with Short-Range Attraction. Nature 2008, 453, 499−503. (42) Luo, J. H.; Yuan, G. C.; Zhao, C. Z.; Han, C. C.; Chen, J.; Liu, Y. Gelation of Large Hard Particles with Short-Range Attraction Induced by Bridging of Small Soft Microgels. Soft Matter 2015, 11, 2494−2503.

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