English

• 0.   Introduction

  Understanding the chemistry and physicochemical properties of aqueous solutions of metal ions, which play significant roles in the fields of basic and applied chemistry, is critical. Metal ion hydrolysis^[1] in aqueous solutions has received extensive attention, as it is ubiquitous. Numerous cations generally aggregate after hydrolysis and form polymeric aggregates that are linked by hydroxyl or oxygen bridges^[2]. Large polymeric networks are formed, which may reach colloidal sizes, and several remain suspended in the aqueous medium as sols or precipitate in gel form^[1]. The colloidal polymer significantly affects the chemistry of processes that involve heavy tetravalent cations (Th(Ⅳ), Pu(Ⅳ), and Zr(Ⅳ))^[2-4]. The polymerization reactions that occur during the formation of colloidal polymers in solutions of numerous heavy tetravalent cations are affected by numerous factors, such as pH, reaction time, temperature, and concentration, and thus, these reactions are experimentally challenging. Zr(Ⅳ), as a typical heavy tetravalent cation that may form colloidal polymers in solution, is commonly the subject of numerous studies^[5]. In an aqueous solution, Zr(Ⅳ) may adopt a wide variety of structures, due to its significant charge, large size, and ready ligand acceptance^[6]. Zr may more often adopt a polymeric rather than monomeric form in solutions^[7-8], and Zr(Ⅳ) hydrolyzes in acidic solutions to first form the tetrameric unit^[9] [Zr₄(OH)₈(H₂O)₁₆]⁸⁺ before more extensive aggregates of the tetrameric unit^[7]. The hydrolysis of Zr in aqueous solutions has been previously investigated using various techniques, including Raman spectroscopy^[10], wide‐ and small‐angle X‐ray scattering (SAXS)^{[2, 9, 11-12]}, ¹⁷O nuclear magnetic resonance spectroscopy^[13], extended X‐ray adsorption fine structure spectroscopy^[14], and ultracentrifugation^[15]. The presence of [Zr₄(OH)₈(H₂O)₁₆]⁸⁺ in highly acidic aqueous solutions has been confirmed by hydrolysis studies^{[2, 9]} and comparison with the crystal structure of ZrOCl₂·8H₂O (ZOC)^[15-16], which contains discrete tetrameric units. In addition, an equilibrium between the tetra‑ and octameric forms of [Zr₈(OH)₁₉]¹³⁺, which affects the pH values of chloride solutions, has been suggested^[17-22].

  As a low‐toxicity, readily available, low‐cost Zr(Ⅳ) salt^[23], ZOC is a widely used feedstock in preparing Zr‐based catalysts. Generally, the preparation processes of catalysts and catalytic reaction processes are separate. The efficient catalysis of the transfer hydrogenation of biomass‐derived molecules by the feedstock (i.e., ZOC) used in catalyst preparation should be time‐ and energy‐saving. Additionally, ZOC serves as an efficient catalyst in numerous crucial organic reactions, including the Bachmann condensation reaction, Michael additions of amines and indoles, syntheses of 1, 8‐dioxo‐octahydroxanthene derivatives, and biomass valorization, due to its Brønsted (due to the in situ‐formed HCl) and Lewis acidities (due to the in situ‐generated ZrO(OH)₂·xH₂O) in organic media^[24-27]. ZOC serves as a critical raw material in producing various Zr compounds, such as stabilized ZrO₂ and mixed rare‐earth oxides, as it exhibits resistance to environmental factors and various chemicals and confers superior stability^[28-33]. However, clarifying the dominant species of Zr(Ⅳ) that are formed under different conditions is challenging. Therefore, this study investigated ZOC and examines Zr(Ⅳ) solutions in detail using SAXS to determine the dominant species in Zr(Ⅳ) solutions under different experimental conditions. Meanwhile, the corresponding structures of the dominant species were calculated and designed using the simple program CYLview^[34].

  1.   Experimental

  The aqueous Zr(Ⅳ) stock solutions were prepared by dissolving ZOC crystalline material (supplied by Jiangxi Jing′an High Tech, Nanchang, China) in deionized water or different concentrations of HCl to explore the effects of four factors on the dominant species within the solutions, i.e. ρ_ZOC: 10, 50, or 300 g·L^-1; Heating treatment: 298 or 353 K; Standing time: 1, 30, or 60 d; c_HCl: 0.5, 1, 2, 4, or 5 mol·L^-1.

  These samples were analyzed using the Guinier law of SAXS and the structures of the dominant species within the Zr(Ⅳ) solutions were modeled using the program CYLview. Small‐angle scattering is widely applied in determining particle sizes and, in several cases, their geometries, and thus, SAXS was adopted in this study. Based on the SAXS measurements, determining the electronic radii of gyration (R_g) of particles and the absolute intensity at zero angles, which was not established in previous experimental studies, is possible^[25]. With these two physical quantities, novel structural data regarding the polymeric Zr species in the solutions could be obtained. CYLview enables a rigorous, automatic method of determining the parameters of a cylinder by continuously adjusting them so that the cylinder encloses all atoms, minimizing the size of the cylinder.

  SAXS was conducted at Anton Paar using a Cu Kα radiation source, and scattering patterns were collected using an Eiger R 1M Horizontal detector with a resolution of 1030×1065 pixels. The image exposure time and X‐ray wavelength were 3×300 s and 0.154 2 nm, respectively, and the distance between the detector and sample was 197.034 mm. The sample scattering was corrected for detector sensitivity, dark current, and the background solvent. The background scattering using an HCl solution (or deionized water) was measured in the same cell with the same concentration of HCl as that in the sample. Moreover, the results of SAXS were identical, regardless of whether HCl solution or deionized water was used, and thus, when we analyzed the results, the background scattering was ignored. The Origin program was used to produce intensity contour plots and perform standard analyses (e.g., Guinier plots).

  2.   Simulation theory

  X‐ray scattering depends on the number of particles, the electron contrast between the particle and medium, and the sizes of the particles^[35]:

  $ I(q)=I_{\mathrm{g}}(q) N_{\mathrm{P}}(\Delta \rho V)^2 \exp \left(-q^2 R_{\mathrm{g}}^2 / 3\right) $

  (1)

  where V is the volume of the particle, N_P is the number of particles per unit volume, R_g is the radius of gyration, I_g is the scattering intensity due to a single electron, Δρ is the electron density difference between the particle and medium, and q is the scattered wave vector, q=4πsin θ/λ, where 2θ is the scattering angle and λ is the X‐ray wavelength.

  Based on Eq.1, the plot of ln I vs q² should be linear and the R_g of the particle may be determined using the slope of this line. The Guinier law is at least valid for q≤1/R_g, and it is not a function of the particle shape in this range of q. The Guinier law is shown in Eq.2:

  $ \ln I=I(0) \exp \left(-q^2 R_{\mathrm{g}}^2 / 3\right) $

  (2)

  The intensity at zero angles, I(0), which is obtained by the extrapolation of the Guinier law to q=0, yields data regarding the numbers of particles and electrons in each particle:

  $ I(0)=I_{\mathrm{g}}(q) N_{\mathrm{p}}(\Delta \rho V)^2 $

  (3)

  Eq.1 and 3 are originally developed for monodisperse particles. In polydisperse systems, I(0) and the observed R_g may be expressed as follows^[35]:

  $ R_{\mathrm{g}}^2=\sum\limits_{i=1}^n\left[N_{\mathrm{P}, i}\left(\Delta \rho_i V_i\right)^2 R_{\mathrm{g}, i}^2\right] / \sum\limits_{i=1}^n N_{\mathrm{P}, i}\left(\Delta \rho_i V_i\right)^2 $

  (4)

  $ I(0)=I_{\mathrm{g}}(0) N_{\mathrm{P}, i}\left(\Delta \rho_i V_i\right)^2 $

  (5)

  Where N_{P, i} is the number density of the ith particle, V_i is the volume of the particle, R_{g, i} is the radius of gyration of the ith particle, and Δρ_i is the difference in the electron densities of the particle and medium.

  The Guinier law does not yield data regarding the shape of the particle, and thus, this should be determined via the calculation of a complete scattering curve. There are three potential geometries within the ZOC solution: a sphere, a cylinder, and an ellipsoid of revolution^[2]. Overall, the three geometries exhibit few differences, as the aspect ratio is relatively small, and the cylinder is more closely related to the shape of the particle^[17]. Therefore, the structures were modeled with the assumption that the particle shapes were cylindrical. The R_g values of all common shapes may be calculated using the formulas shown in Table 1 after the parameters are obtained using CYLview.

  Table 1

  

  Table 1.  Formulas used to calculate the R^g values of various common shapes

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  Shape	Formula
  Ball (radius: r)	(3/5)0.5r
  Spherical shell (outer radius: r, inner radius: rin)	(3/5)0.5{[1-(rin/r)5]/[1-(rin/r)3]}0.5r
  Ellipsoid (radius: r, r, rv)	{[2+(rv/r)2]/5}0.5r
  Cylinder (length: 2l, diameter: 2r)	[(r2/2)+(l2/3)]0.5
  Thin disk (radius: r)	20.5r
  Fine fiber (length: 2l)	30.5l
  Parallelepiped (side length: 2a, 2b, 2c)	[(a2+b2+c2)/3]0.5
  Cube (side length: 2a)	a

  3.   Results and discussion

  The Guinier plots of ln I as a function of q² of all samples yielded the R_g values. The Guinier plots of the HCl solutions (or deionized water) were virtually straight lines, which were not included in the analysis. Furthermore, the R_g values of the models were obtained using CYLview.

  3.1   Effect of mass concentration of ZOC

  The typical plots of ZOC solutions with concentrations of 10, 50, and 300 g·L^-1 in deionized water are shown in Fig. 1. Clearly, at ρ_ZOC=10 and 50 g·L^-1, linear plots were obtained (except at extremely low I, where the beam stop interferes with proper intensity determination). The linearity of the plot further indicates the monodispersity of the Zr(Ⅳ) system. The R_g values were not very different, and thus, the dominant species did not change with ρ_ZOC. However, a nonlinear region was observed at 300 g·L^-1 of ρ_ZOC, which may be due to the contribution of interparticle interference. The linear regions exhibited slopes that yield R_g values of 0.550 nm. Despite the wide variance in concentration, the constant R_g indicates that the cause of the nonlinear region does not significantly influence the linear region of the Guinier plot. Therefore, the most direct approach considers the linear region as indicative of the behaviors of the independent particles.

  Figure 1

  

  Figure 1.  Guinier plots for different ZOC concentrations in one day

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  In addition, at very low q² values, the observed small dome shape was due to the mutual repulsion between particles in the solution, which is generally caused by electrostatic interactions. As the Cl^- in ZOC is free, the repulsion effects of the particles at 300 g·L^-1 of ρ_ZOC are clearer, whereas, at the medium and low concentrations (10 and 50 g·L^-1), the interparticle repulsion effects only manifest at low q² values. However, the dominant species are consistent in the aqueous ZOC solutions with different concentrations.

  3.2   Effect of heating treatment

  As shown in Fig. 2, the R_g values and shapes of the Guinier plots differed considerably at ρ_ZOC of 10 and 50 g·L^-1 after heating. The upper dome shape is due to the repulsion between the particles, whereas after heating, concave shapes were observed, and thus, heating may cause particle aggregation. The particles gain additional energy due to the heating, which is partially used to counteract the repulsive forces between the particles, and the remaining energy should also induce particle aggregation. As a result, the quality of the dominant particles in the solution should increase considerably, accelerating particle aggregation in the solution. Due to the distinction between the results obtained at ρ_ZOC=300 and 10 or 50 g·L^-1, the Guinier plot obtained at 300 g·L^-1, which is shown in Fig. 3, is analyzed separately.

  Figure 2

  

  Figure 2.  Guinier plots for heating treatment (Left: 10 g•L^-1; Right: 50 g•L^-1)

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  Figure 3

  

  Figure 3.  Guinier plots for heating treatment at 300 g•L^-1 of ρ_ZOC

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  The Guinier plots at 298 and 353 K displayed virtually no changes at ρ_ZOC=300 g·L^-1 after heating. As shown in Fig. 3, the upper dome shape was still observed, i.e., the repulsion between particles is so high that the extra energy provided to the particles via heating may not overcome the repulsion at the high concentration. Numerous particles occur in a unit volume at ρ_ZOC=300 g·L^-1, and thus, the repulsion between particles is further increased. Compared with those at medium and low concentrations, the types of particles present in the high‐concentration solution are more complex (the nonlinear region in the Guinier plot was larger), and thus, the energy provided via heating is insufficient to overcome the interparticle repulsion and cause particle aggregation. Thus, the shape of the plot at 353 K was the same as that at 298 K.

  3.3   Effect of standing time

  According to the concentration of ZOC, the analysis of standing time should also be divided into two cases (Fig. 4 and 5). As shown in Fig. 4, the R_g values increased slightly with the standing time, which is mainly due to an increase in the free Cl^- in the solution. As the standing time increases, the Cl^- within the solution should be incorporated into the particle structure and the overall charge should decrease, and the charges of several particles should even change from positive to neutral. Neutral particles are also more stable in solution.

  Figure 4

  

  Figure 4.  Guinier plots for standing time at 10 g•L^-1 (left) and 50 g•L^-1 (right) of ρ_ZOC

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  In the Guinier plot at the high concentration, the standing time exhibited virtually no effect on the R_g values, as shown in Fig. 5. The plots at 30 and 60 d appeared to be parallel, and thus, the tolerance of the species toward Cl^- has reached a saturated state at 300 g·L^-1. Generally, the standing time displays little effect on the general structures of the dominant species in ZOC solutions, and the effect is mainly reflected in terms of the free Cl^- at medium and low concentrations.

  Figure 5

  

  Figure 5.  Guinier plots for standing time at ρ_ZOC=300 g•L^-1

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  3.4   Effect of concentration of HCl

  The influences of the HCl concentration were analyzed in three cases: 10, 50, and 300 g·L^-1 of ZOC, and all samples were allowed to stand for 30 d. Notably, when ρ_ZOC=300 g·L^-1 and c_HCl=5 mol·L^-1, crystals were formed, and thus, this case is not analyzed here.

  As shown in Fig. 6, the Guinier plots of all samples were linear, and thus, the dominant species in the solutions were homogeneous at 10 g·L^-1. Remarkably, a horizontal line was observed at c_HCl > 2 mol·L^-1, potentially because Zr⁴⁺ is the dominant species under these conditions. This experimental phenomenon is consistent with the conclusions of Clearfield^[7] and Devia^[8], who reported the existence of the monomeric species Zr⁴⁺ under specific conditions, with a low ρ_ZOC and high c_HCl value. Remarkably, the R_g values in deionized water and c_HCl=0.5 mol·L^-1 were 0.632 and 0.250 nm, respectively, and the former was much larger than the latter. Therefore, Cl^- inhibits the formation of multimers of the dominant species. No Zr—Cl bonds are present in ZOC crystals, and thus, the electrostatic forces generated by Cl^- are larger than the binding forces between the Zr polymers.

  Figure 6

  

  Figure 6.  Guinier plots for cHCl at ρ_ZOC=10 g•L^-1

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  As shown in Fig. 7, R_g decreased regularly with increasing c_HCl at ρ_ZOC=50 g·L^-1. Overall, the degree of polymerization of the dominant species in the solution decreased with increasing c_HCl. The concentration of Zr increases, and thus, even at a high c_HCl, the dominant species is still present in aggregates rather than in the ionic state. Therefore, the Zr salt is essentially a polymer in solution.

  Figure 7

  

  Figure 7.  Guinier plots for cHCl at ρ_ZOC=50 g•L^-1

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  The situation is even more complex at ρ_ZOC=300 g·L^-1, as shown in Fig. 8. As c_HCl was varied from 0 to 4 mol·L^-1, the Guinier plot changed from arched to linear, i.e., the linear region of the Guinier plot increased with increasing c_HCl, which indicates that increasing c_HCl promotes homogenization of the species in the ZOC solution. However, R_g still decreased with increasing c_HCl, which is mainly due to the inhibition of the polymerization of the Zr‐containing species by Cl^-.

  Figure 8

  

  Figure 8.  Guinier plots for cHCl at ρ_ZOC=300 g•L^-1

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  In summary, among the four influencing factors, the ρ_ZOC and c_HCl exhibit the highest influences on the dominant species in the ZOC solution. Furthermore, the R_g values of all experimental samples were obtained using the Guinier plots. However, the structures of the dominant species in the solutions were not analyzed, and thus, they were simulated and calculated.

  3.5   Modeling the structures of the dominant Zr(Ⅳ) species

  The R_g values indicating the dominant Zr(Ⅳ) species under different experimental conditions were obtained via the Guinier analyses of the results of SAXS. Under the conditions of low ρ_ZOC and high c_HCl, the ions are the dominant species in the solution, according to the results of the experimental analyses and the corresponding literature. Under other conditions, the dominant species occur as polymers, but the types of polymers in the solutions are unclear. Therefore, several possible structures, which are based on the crystal structure data of the ZOC single crystal^[7], were constructed, as shown in Fig. 9. In these models, the H atoms may be ignored, as they are always combined with O atoms. The O atoms shown in Fig. 9 represent two formations: OH (double hydroxyl bridging bonds, ) and H₂O. The construction of the octamer was based on the related research of Singhal^[17], but unfortunately, the details of the octamer species were not previously analyzed. Therefore, the number of chlorides in the octamer was calculated based on the octamer data. The octamer is formed by two stacked tetramers, [Zr₄(OH)₈(H₂O)₁₆]⁸⁺, which are connected by —OH—, and Cl^- is pivotal in the formation of the dominant species within the ZOC solutions, particularly at a high c_HCl.

  Figure 9

  

  Figure 9.  Possible chemical structures of the dominant species fo ZOC

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  As a Cl^- occurs around each bond that connects two tetramers due to electrostatic neutralization, the octamers contain up to 20 chloride ions in their structures. The Cl^- close to the H₂O (denoted Cl1) attached to the Zr atom is further analyzed. Compared to the Cl^- close to the double oxygen bridge (denoted Cl2), Cl1 is more likely to be free in the water, i.e., the Cl1 site is likely not part of an octamer or a tetramer in the solution, which is partly due to the high dielectric constant of the solute. Therefore, the number of Cl^- in the octameric form ranges from 0 to 12, and thus, all possible structures are designed.

  3.6   Calculation of the R_g values of the structures and analysis of the dominant species in the solutions

  The raw data were obtained from the coordinates of each atom in the crystal structure, and the Supporting information contains more details regarding the atomic coordinates of each structure. The parameters of the simulated structures were then calculated in CYLview, using the conversion formula of the cylinder shown in Table 1 to yield the R_g of each simulated structure. To match the experimental value to the simulated value more closely, several structures with additional chloride ions were calculated via interpolation based on the original structure. Table 2 shows the calculated R_g of each chemical structure. There are only three boundary conditions, according to a theoretical analysis, in terms of the number of Cl^- in an octamer: 0, 8, and 12. Therefore, based on the boundary conditions, an increase in Cl^- was considered to obtain more accurate structures of the octamer, as shown in Table 2.

  Table 2

  

  Table 2.  Calculated R_g of each chemical structure

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  Code of structure	Chemical structure of species model	Rg / nm
  a	[Zr(H2O)8]4+	0.166
  b	[Zr2(OH)2(H2O)12]6+	0.278
  c	[Zr3(OH)4(H2O)16]8+	0.321
  c	[Zr3(OH)4(H2O)16Cl2]6+	0.367
  d	[Zr4(OH)8(H2O)16]8+	0.371
  e	[Zr4(OH)8(H2O)16Cl4]4+	0.414
  f	Zr4(OH)8(H2O)16Cl8	0.459
  g	[Zr8(OH)20(H2O)24]12+	0.510
  g	[Zr8(OH)20(H2O)24Cl4]8+	0.556
  g	[Zr8(OH)20(H2O)24Cl6]6+	0.609
  h	[Zr8(OH)20(H2O)24Cl8]4+	0.640
  h	Zr8(OH)20(H2O)24Cl12	0.664

  As shown in Table 2, the calculated R_g values were in a range of 0.166-0.664 nm, whereas the measured R_g values were between 0.004 and 1.032 nm. To realize the accuracies of the designed structures, we adopted a novel method: the effective sampling rate. It may be defined as the number of samples within the calculated range as a percentage of the total number of samples measured. However, several samples may be excluded from the total number of samples measured: R_g=0.004, 0.021, 0.933, 1.032 nm. The forms of the ions at a high c_HCl and low ρ_ZOC yield measured R_g values under these conditions of approximately 0 nm, which are consistent with values of 0.004 and 0.021 nm. The effective sampling rate obtained by excluding those samples was 95%. The other measured values were all perfectly consistent with the constructed chemical models within the error range.

  Here, several explanations for the larger values are provided. Firstly, the experimental conditions should be noted: fresh, heating, 10 and 50 g·L^-1. Remarkably, under these conditions without heating, the measured R_g values are 0.556 and 0.547 nm, which are consistent with the structure of the octamer with a certain amount of Cl^-. However, other particles occur in the solution, and the solution of dissolved ZOC crystals in deionized water is acidic. Additionally, at a lower acidity, the dominant form in the ZOC solution is the octamer. Therefore, numerous octamers occur in the solution under these conditions without heating, and heating may cause partial aggregation. Thus, polymerization is very likely to occur when large amounts of the two species occur in a solution.

  A few changes in the R_g values at ρ_ZOC=10 and 50 g·L^-1 are observed after standing for a few days. These changes are mainly because the number of chloride ions in the dominant species also changes slightly with increasing standing time, and the change may be expressed by the following chemical formula (ρ_ZOC=10 or 50 g·L^-1):

  $ \begin{aligned} 30\;\mathrm{~d}:\left[\mathrm{Zr}_8(\mathrm{OH})_{20}\left(\mathrm{H}_2 \mathrm{O}\right)_{24} \mathrm{Cl}_4\right]^{8+} &+4 \mathrm{Cl}^{-} \rightarrow \\ & {\left[\mathrm{Zr}_8(\mathrm{OH})_{20}\left(\mathrm{H}_2 \mathrm{O}\right)_{24} \mathrm{Cl}_8\right]^{4+}} \end{aligned} $

  (6)

  $ \begin{aligned} 60\;\mathrm{~d}:\left[\mathrm{Zr}_8(\mathrm{OH})_{20}\left(\mathrm{H}_2 \mathrm{O}\right)_{24} \mathrm{Cl}_8\right]^{4+} &+4 \mathrm{Cl}^{-} \rightarrow \\ &\mathrm{Zr}_8(\mathrm{OH})_{20}\left(\mathrm{H}_2 \mathrm{O}\right)_{24} \mathrm{Cl}_{12} \end{aligned} $

  (7)

  Additionally, the effect of HCl at different mass concentrations of ZOC may be explained by the chemical equations shown in Fig. 10. As the concentration of HCl increases, at almost all ρ_ZOC values, the degree of polymerization of the dominant species in the solution decreases. Only at very high c_HCl and very low ρ_ZOC should the Zr in solution occur in an ionic state, whereas under other conditions, it should appear as a polymer.

  Figure 10

  

  Figure 10.  Chemical equations representing the effect of HCl

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  4.   Conclusions

  SAXS has been used to determine the sizes and geometries of small polymeric structures in ZrOCl₂·8H₂O (ZOC) solutions under different conditions, which was impossible using other experimental techniques. In this study, Guinier analysis was used to calculate the R_g values under various experimental conditions, and the chemical structures of various species were modeled according to the crystal structure of ZOC. The spatial coordinates of each atom corresponding to the chemical structure were then obtained, and the CYLview program was used to calculate and convert the R_g of each simulated structure. Upon comparing the measured R_g with the simulated R_g to determine the structure of the main species occurring under each condition, the degree of matching was approximately 95%. We also identified two extreme cases via experimental measurements and analyzed the causes of these phenomena: the smaller R_g values are due to the ionic forms in the ZOC solutions, and the larger R_g values are due to the polymerization of tetra‐ and octamers. This study covers the dominant species within ZOC solutions under various conditions, which are of considerable significance in theoretical research in the Zr industry.

  Supporting information is available at http://www.wjhxxb.cn

  
  Acknowledgments: We acknowledge financial support from the cooperative project between universities in Chongqing and institutes affiliated with the Chinese Academy of Sciences (Grant No. HZ2021013), the sub‐project of the National Program on Key Research Project (Grant No.2019YFC1907702), and the Double Thousand Plan of Jiangxi Province (Grant No. jxsq2020105012).
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  Figure 1  Guinier plots for different ZOC concentrations in one day

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  Figure 2  Guinier plots for heating treatment (Left: 10 g•L^-1; Right: 50 g•L^-1)

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  Figure 3  Guinier plots for heating treatment at 300 g•L^-1 of ρ_ZOC

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  Figure 4  Guinier plots for standing time at 10 g•L^-1 (left) and 50 g•L^-1 (right) of ρ_ZOC

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  Figure 5  Guinier plots for standing time at ρ_ZOC=300 g•L^-1

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  Figure 6  Guinier plots for cHCl at ρ_ZOC=10 g•L^-1

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  Figure 7  Guinier plots for cHCl at ρ_ZOC=50 g•L^-1

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  Figure 8  Guinier plots for cHCl at ρ_ZOC=300 g•L^-1

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  Figure 9  Possible chemical structures of the dominant species fo ZOC

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  Figure 10  Chemical equations representing the effect of HCl

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  Table 1.  Formulas used to calculate the R^g values of various common shapes

  Shape	Formula
  Ball (radius: r)	(3/5)0.5r
  Spherical shell (outer radius: r, inner radius: rin)	(3/5)0.5{[1-(rin/r)5]/[1-(rin/r)3]}0.5r
  Ellipsoid (radius: r, r, rv)	{[2+(rv/r)2]/5}0.5r
  Cylinder (length: 2l, diameter: 2r)	[(r2/2)+(l2/3)]0.5
  Thin disk (radius: r)	20.5r
  Fine fiber (length: 2l)	30.5l
  Parallelepiped (side length: 2a, 2b, 2c)	[(a2+b2+c2)/3]0.5
  Cube (side length: 2a)	a

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  Table 2.  Calculated R_g of each chemical structure

  Code of structure	Chemical structure of species model	Rg / nm
  a	[Zr(H2O)8]4+	0.166
  b	[Zr2(OH)2(H2O)12]6+	0.278
  c	[Zr3(OH)4(H2O)16]8+	0.321
  c	[Zr3(OH)4(H2O)16Cl2]6+	0.367
  d	[Zr4(OH)8(H2O)16]8+	0.371
  e	[Zr4(OH)8(H2O)16Cl4]4+	0.414
  f	Zr4(OH)8(H2O)16Cl8	0.459
  g	[Zr8(OH)20(H2O)24]12+	0.510
  g	[Zr8(OH)20(H2O)24Cl4]8+	0.556
  g	[Zr8(OH)20(H2O)24Cl6]6+	0.609
  h	[Zr8(OH)20(H2O)24Cl8]4+	0.640
  h	Zr8(OH)20(H2O)24Cl12	0.664

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