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**Results**

**Syntheses of nanosized aluminosilicates**

A pH and visual study of the synthesis process suggested three distinct growth phases as displayed in Figure 1. NaOH addition during the initial minutes of the experiment (Stage 1) results in a rapid rise in pH. The pH begins to decrease as soon as NaOH addition is finished, which is approximately at the maximum between 0-5 min. During this period a translucent gel-like phase appears in the solution. Stage 2 is marked by a period of rapid pH decrease where the solution, depending on starting conditions, either returns to clear or develops a white cloudy precipitate, which persists for the rest of the synthesis. Stage 3 is the period in which the pH becomes approximately stable. The solution was then placed in anaerobic bottles and heated for 7 days at 95C, after which a cloudy white precipitate would often form. When dried, the solution would produce a fine white powder.

**Phase identification**

The final solids of each synthesis were characterized by HR-TEM imaging and/or pXRD analysis. Three distinct phases were identified. Figure 2 presents pXRD profiles for samples consisting of pure imogolite and proto-imogolite, as well as for an amorphous silica reference. HR-TEM images of imogolite and proto-imogolite are also given in Figure 2, along with an example of amorphous silica observed in one of the synthesis products. All three pXRD profiles show a series of one or more broad, weak peaks consistent with materials lacking long-range periodicity (i.e., short-range ordered). The pXRD patterns of each agree with previously reported spectra (Arancibia-Miranda et al., 2013; Levard et al., 2012; Musić et al., 2011). HR-TEM images of the imogolite and proto-imogolite endmembers generally showed highly aggregated nanoparticles with different morphologies (Figure 2A and B). The imogolite sample exhibited fairly distinct and elongated nanotube-like shapes. The sample that was expected to be allophane based on its pXRD characteristics showed no evidence of well-formed spherical nanoparticles. Thus, we assumed that the morphological characteristics of this sample were consistent with proto-imogolite. Amorphous silica consisted of aggregates of globular with varying particle sizes in the range of 10’s to 100’s of nms (Figure 2C). This was confirmed through EDS analysis which showed the sample contained only silicon (need to add this to methods now but not sure on specs yet). Overall, the sizes and morphologies observed in HR-TEM were generally consistent with what has been reported previously for synthetic imogolite, proto-imogolite, and amorphous silica.

The majority of the other products of the different synthesis experiments did not result in a pure single phase as commonly found in natural samples, (Harsh, 2002), but instead contained a mixture of the three different endmembers. Qualitative evidence for these mixtures was provided by pXRD, which showed combinations of distinct pXRD peaks for imogolite, proto-imogolite and/or amorphous silica all present in a single sample (Table 1).

**Linear combination fitting**

Linear combination fitting of pXRD was used to quantify the abundances of imogolite, proto-imogolite, and amorphous silica in a set of 23 synthesis products. An example of the LCF fit of a pXRD pattern from a synthesized product is shown in figure 2. Table 1 shows the results of LCF analysis and summarizes the starting conditions used for each synthesis. LCF results show that the abundances of proto-imogolite and imogolite varied from approximately 0 up to 100% for the different synthesis conditions. The abundance of amorphous silica, which has been reported previously in studies of imogolite and proto-imogolite synthesis (Wada et al., 1979), varied between approximately 0 up to 31%. The data illustrate how variations in the abundances of proto-imogolite, imogolite, and amorphous silica occur with differences in synthesis conditions.

**Multivariate regression modeling of data set 1**

The data in Table 1 was analyzed using multivariate regression, which produced a model of imogolite proportion that followed the following relationships:

= 009 + 0.35*hydro* −1.82*c* + 0.11*Al* : *Si* [Eq. 1]. The model was used to produce data of predicted imogolite proportion, which was compared to the experimentally determined proportions in Figure 3.

The model demonstrates that imogolite proportion increases with increasing hydrolysis ratio and Al:Si ratio, and that the proportion of imogolite decreases as concentration of starting reagents increases. The model was found to be significant, with a p-value of 1.3×10^{-9}, and an R^{2} of 0.86, which means that the model explains 86% of the variance in the data. The starting concentration and hydrolysis ratio factors were both statistically significant, with >99.99% and >99% confidence respectively. Both the intercept and elemental ratio factor both failed a 0.05 p value significance test.

The RMSE is 0.13, meaning the model can predict the imogolite proportion to within ±13%. The DW statistic for the residuals of this model is 2.26, which with a 99% confidence rate the null hypothesis can be rejected, suggesting there is no autocorrelation in the residuals.

A model for determining the proto-imogolite proportion was also developed, and is described as follows: *PI* =1.01− 0.30*hydro* + 0.93*C* − 0.092*Al* : *Si* [Eq. 2]

The model describes that as concentration of starting regents increases, the proportion of proto -imogolite in final the product also increases. It also suggests that as the hydrolysis ratio and elemental ratio increase the proto-imogolite proportion is decreased, directly opposed to the imogolite relationships.

This model was found to be significant, with a p-value of 1.62×10^{-9}. It has an R^{2} of 0.88; the model does not explain 12% of the variance of the data. The intercept and hydrolysis ratio factor were found to be significant with a >99.99% confidence. The concentration factor was also found to be significant to a p value of 0.05. As with the other model, the elemental ratio was insignificant, with a p-value of 0.22. The RMSE is similar to the imogolite model, but slightly lower at 0.12. The DW statistic was found to be 0.5, which rejects the null hypothesis and suggests positive autocorrelation in the residuals, meaning there is some predictive measure in the data that is not captured by the model.

Amorphous silica was found to have significant variance and limited predictive power, and so a full model is not reported. The model is, based on p-values, 9 orders of magnitude less significant than the proto-imogolite and imogolite models. The overall significance is 99%, but the average confidence of all of the individual factors is 65%. None of the variables were found to be significant. Intuitively however, the Al: Si ratio suggested that as Si increases relative to Al, total amorphous silica proportion increases at 85% confidence. The p-value of the model is 0.01, but the R^{2} value is 0.32, meaning the model explains only 32% of the variance of the amorphous silica proportion. The DW statistic for the residuals is 0.68, indicating positive autocorrelation.

**In situ Dv(R) particle size data**

We performed *in situ* SAXS studies where we repeated the outlined synthesis procedures and measured average and median particle size (Figure 6). We have confirmed that increasing concentrations of starting reagents reduces the average precursor particle size, following equation : *R** _{avg}* = 2.3 − 9.6

*C*− 0.88

*s*+ 0.68

*H*− 0.33

*Al*:

*Si*[Eq. 3]

This model explains average precursor particle size where C is starting concentration of reagents, s is speed of NaOH addition, H is hydrolysis ratio, and Al:Si is elemental ratio. The model explained 91% of the variance in the data, with a confidence >99.99%, and had random residuals as evidenced by a DW statistic of 2.08. Increasing concentration decreases average precursor particle size, and has the largest influence on the final size distribution of the particles.

Tracking Dv(R) results

*in situ*through the first hour of synthesis yielded similar results to the pH study shown in Figure 1, and is shown in Figure 6. Three stages emerged again, where stage 1 shows rapid growth and nucleation of particles, with increasing average particle size. Stage 2 shows minimal growth in particle size but continued nucleation, and by stage 3 the Dv(R) analysis shows no growth over time in size or significant nucleation of particles

**Multivariate regression modeling of data set 2**

A second set of syntheses was performed *in situ*, and using Dv(R) analysis average and most frequent particle size data were added to the system. These experiments aimed to quantify the sizes of the particles that would be heated and aged to obtain the final synthesis products. Once stage 3 was reached, and there was no longer evidence of nucleation or growth, the particle size data was recorded, and added to the set of synthesis conditions (table 2). The data was then analyzed identically to the previous data set to produce proportion abundance models. The imogolite proportion was modeled as follows:

= −0.046 + 0.34*hydro* − 0.40*C* − 0.073*Al* : *Si* + 0.11*R _{avg}* − 0.38

*R*[eq. 4]

_{med}The model demonstrates that as average particle size increases, the imogolite proportion also increases, and that the inverse is true for the median particle size. The hydrolysis ratio constant was calculated to > 99.99% confidence, the elemental ratio constant was calculated to 99% confidence, and the concentration and mean radius size constants were calculated to >98% confidence. The p-value for the entire model is 9×10

^{-14}. The intercept of the model and the median particle size were found to be statistically insignificant with a p-value of 0.15 and 0.07 respectively. The rate of NaOH addition was found to decrease the overall confidence of the model, with a p-value that suggested addition rate did not influence endmember proportion, and so was omitted. The adjusted R

^{2}of the model is 0.98. The RMSE of the model is 0.05, meaning that the model can predict the imogolite proportion to within ±5%.

The three overlapping factors for both equations 1 and 4 (concentration Al:Si ratio, OH:Al ratio) shared identical direction, and the coefficients were all within a standard error of one another. The similarity of the models is described well by the residuals of the data shown in Figure 7. The DW statistic for equation 4 was 1.84, which failed to reject the null hypothesis at 99% confidence. Visually, there does appear to be a linear trend in the residuals at lower proportions of imogolite, with both models.

The proto-imogolite proportion of this second data set was modeled by:

*PI*=1.08 − 0.34

*hydro*+ 0.37

*C*− 0.0007

*Al*:

*Si*− 0.034

*R*

*+ 0.050*

_{avg}*R*

_{med}*[Eq. 5].*

This model shares the same trends as the model developed through the data in Table 1, and the absolute factors are within a standard deviation. The model describes relationships that are opposite those of the imogolite proportion, including the particle size where mean particle size growth leads to less proto-imogolite.

The p-value of the overall model is 3×10

^{-8}, along with an R

^{2}value of 0.91, indicating a high degree of confidence that the model is explaining 91% of the variance in the data. Unlike the imogolite proportion model, the individual variables are less significant than when combined into the general model. The coefficient for the intercept, hydrolysis, and concentration were significant to >95%, but the rest of the factors ranged from p-values of 0.1 to 0.8. The RMSE of this model is also higher than that of the imogolite model, at 0.09; this model can predict the proto-imogolite proportion, on average, to within 9%.

The DW statistic for the residuals of this model is 0.5, which confirms the alternative hypothesis that there is positive autocorrelation in the residuals. Visual interpretation of the residuals shown in figure 8 also shows both the similarity of the models and the linear, predictable trend at high concentrations of proto-imogolite.

**Discussion**

**Growth stages of aluminosilicate nanoparticles**

Results from prior studies conducted at similar condition suggest that Stage 1 involves the rapid nucleation and growth of gibbsite-like Al(OH)3 nanoparticles (Wada et al., 1979; Ohashi et al., 2002) (Figure 1) . Stage 2 is thought to involve silica attachment to the sheets resulting in the formation of the local imogolite structure (Du et al., 2018). This occurs through the oxolation mechanism where some of the –OH groups on the gibbsite-like sheets are substituted by the silica tetrahedra, releasing 3 moles of H3O^{+} into solution per mole of of Si attached. (Arancibia-Miranda et al., 2013). The stabilization in pH and particle nucleation and growth with time suggests that the development of the imogolite structure slows and eventually ceases during Stage 3.

**Influence of synthesis conditions on phase abundance**

The models developed in equations 1-5 describe the relationship between the system of physical and chemical conditions used for synthesis, and the abundances of imogolite, proto-imogolite, and amorphous silica the final synthesized products.

The hydrolysis ratio was determined to be the most important factor controlling morphology. As the hydrolysis ratio increased from 1 to 3, the proportion of imogolite increased, and the proportion of proto-imogolite formed decreased. A consequence of increasing the hydrolysis ratio is that the initial pH of solution also increases. With a hydrolysis ratio of 1, the pH ranged from 3.3-4.2, while when the hydrolysis ratio was set to 3, the pH ranged from 9.2 to 10. Allophanes have been shown to be unstable in alkaline conditions, where defect pores develop, enlarge, and break down the structure (Wang et al., 2018) . A similar mechanism would likely prevent proto-imogolites from forming in alkaline conditions, explaining the decrease of proto-imogolite proportion with increasing hydrolysis ratio.

Previous studies of imogolite synthesis have found that at hydrolysis ratios below 1.5, tubular structures were unable to form, and significant structural defects below 2.0 were observed (Levard et al., 2011b). The imogolite model supports that experimental result; a decreasing hydrolysis ratio resulted in significantly decreased imogolite proportion. Experimentally, the average imogolite portion of samples with a hydrolysis ratio below 1.5 was 9%, which agrees well with the previously reported findings.

The Al:Si ratio was also found to be a significant factor that influences final phase composition. As the relative abundance of Al increases, the imogolite proportion increases. The idealized formula for imogolite is Al2(OH)3SiO3OH, with a 2:1 Al:Si ratio. As the ratio is deviated from, and the Si increases, the amount of amorphous silica increases as well. This amorphous silica likely interferes with the growth kinetics of imogolite. Proto-imogolite particles have been shown to be able to incorporate silica into its structure, polymerizing chains of Si branching off from the tetrahedral sites, creating a Si-rich local structure (Levard et al., 2012). This is reflected in the model equations, where as the relative amount of Si increases, the proto-imogolite proportion also increases.

The models show that increasing concentration of starting reagents leads to a lower proportion of imogolite. Based on reported literature results, producing high concentrations of nucleated particles have been suggested to impede imogolite growth kinetics, especially growing longer tube structures (Maillet et al., 2011).

Based on results from numerical modeling, it has been proposed that there is a size threshold at 4 nm for precursor particles where the energetically favorable morphology for growth transitions from spherical to tubular (Thill et al., 2017). This result agrees with the *in situ* studies that are described be equations 4 and 5. The precursor particles that were >3.6 nm in diameter had, on average, an increase of imogolite final phase composition of 0.92. This confirms the result of the numerical modeling experimentally, with a slight adjustment to the morphological transition to 3.6nm.

**Table Of Contents**

**Abstract**

**General Audience Abstract**

**Acknowledgements**

**List of Figures**

**List of Tables**

**1.Introduction**

**2.Methods**

2.1 Synthesis

2.2 Powder X-Ray Diffraction analysis

2.3 Small Angle X-Ray Scattering

2.4 Transmission Electron Microscopy

2.5 Multivariate Linear Regression Analysis

2.6 Statistical interpretation

**3. Results**

3.1 Syntheses of nanosized aluminosilicate

3.2 Phase identification

3.3 Linear combination fitting

3.4 Multivariate regression modeling of data set 1

3.5 In situ Dv(R) particle size data

3.6 Multivariate regression modeling of data set 2

**4. Discussion**

4.1 Growth stages of aluminosilicate nanoparticles

4.2 Influence of synthesis conditions on phase abundance

4.3 Evidence of allophane versus proto-imogolite production in the literature

4.4 Modeled versus experimental pXRD patterns produced in literature

4.5 The significance of the in situ data

4.6 Multivariate approach to explaining geochemical systems

**5. Conclusions**

Works Cited

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Multivariate Analysis of Factors Regulating the Formation of Synthetic Allophane and Imogolite Nanoparticles