Temperature Behavior of Precursor Clusters at the Pre-Crystallization Phase of K(H₂PO₄) Studied by SAXS

Abstract

Elementary building blocks for the growth of KDP crystals were established. The solution of potassium dihydrogen phosphate (KH2PO4–KDP) has been experimentally studied by the small-angle X-ray scattering (SAXS) method. The analysis of SAXS data in the temperature range of 2.5–90 °C using a set of models of 3D fragments of the crystal structure showed that the saturated solution contains above K⁺, H₂PO₄⁻ and KH₂PO₄ monomers, as well as mainly octamers. The 3D model of the octamer isolated from the crystal structure has dimensions of 17.443 Å along the [001] axis and 5.963 Å along the [100] and [010] axes. As the temperature is decreased, starting from the saturation temperature of the solution, the volume fraction of octamers sharply increases while the volume fraction of monomers decreases. The results indicate that the monomers and octamers represent major components in the solution with the presence of minor populations of other oligomers. The significant dominance of octamers in the supersaturated solution indicates that they are elementary building blocks for the growth of KDP crystals of tetragonal modification.

1. Introduction

Crystals play a key role as basic parts of electronics [1,2] and photonics [3], and so much attention is paid to the mechanisms of crystal growth [4,5,6,7,8]. The greatest success has been achieved at the stage of growing large crystals [9,10,11,12], but there is no generally accepted point of view in the description of the initial state of crystal growth—the nucleation [13].

Here, first of all, two models compete. According to one model, the initial formation of crystals occurs by the sequential connection of elements [14]. The second model assumes that crystallization occurs through the formation of an intermediate phase of clusters of precursors (Figure 1) [15,16].

This model gained particular weight after the suggestion that the above precursor clusters represent 3D fragments of the crystal structure and the experimental confirmation of this hypothesis for lysozyme solutions using small-angle X-ray scattering (SAXS) and neutrons (SANS) [17,18,19,20], as well as the calculation of the stability of protein oligomers by molecular dynamics methods. Sometimes a two-step model of crystallization is distinguished. There is some research related to the realization of a two-step model of crystallization in proteins [21,22] or inorganic substances [23]. According to the review [24], a crystallization process consists of two steps: (1) a disordered precursor forms, and (2) a crystal nucleus forms inside precursor.

Recently, such precursor clusters have also been found in an inorganic (non-protein) solution of potassium dihydrogen phosphate [25]. Unfortunately, in this work the measurements were carried out in an insufficiently strong beam, and the low concentration of KDP made it possible to detect only the very fact of the appearance of oligomers by imposing sharp temperature changes. KDP is known as an excellent electro-optical nonlinear optical crystal with a large electro-optical nonlinear coefficient, high laser damage threshold, and a laser frequency doubling effect, electro-optical effect, piezoelectric effects and other special features. It is widely used in inertial confinement fusion engineering (ICF) and electro-optical switching devices [26]. Therefore, it is necessary to understand the process of the crystallization of KDP.

As a rule, crystallization processes begin to be considered from the stage of nucleation. However, this process can be significantly affected by the nature of the elementary building blocks of the crystals forming the overgrowth. This article is devoted to the experimental determination of such blocks in solutions of KDP.

In this work, systematic temperature SAXS measurements of KDP solutions were carried out, which revealed the regularities in the formation of precursor clusters simultaneously with crystal structure analysis. For the investigated solution, small KDP crystals were grown and changes in solution structure were measured in this solution.

2. Materials and Methods

To prepare the solutions, KDP crystals were used, which were dissolved in Millipore ultrapure water (water resistance 18 MΩ*cm). The sample concentration was 335 mg/mL, which corresponded to the saturation temperature of KDP 40 °C [27]. Solutions with crystals were initially heated in a thermostat up to 90 °C and kept at this temperature until the crystals were completely dissolved. Before measurements, quartz capillaries with an outer diameter of 2 mm were filled with the solutions. Since filling took place at room temperature, the capillaries were returned to the thermostat until visible crystals in the capillary were dissolved. After that, the capillaries were fixed on a sample holder, for which the temperature was controlled, and the temperature was set to 90 °C.

Any impurity, dust, interface, cuvette or surface can serve as a heterogeneous center of the formation of a nucleus [28,29], but it can only participate in the formation of the nucleus. The size of such a nucleus is much larger than KDP octamers, and it cannot affect the formation of precursor clusters. Extra pure K(H₂PO₄) without impurities, ultrapure water with final filtration (the content of particles ˃ 0.22 µm in water is 1 particle/mL), was used in the preparation of samples. At each stage, chemically clean disposable tubes were used, which also does not contain impurities and dust. Samples were also loaded into disposable quartz capillaries.

SAXS data for a KDP solution were collected on the “BioMUR” beamline of the Kurchatov Synchrotron Radiation Source (NRC Kurchatov Institute) [30,31]. The KDP solution (335 mg/mL) and corresponding matched solvent blank were enclosed in quartz capillaries with an external diameter of 2 mm. Monochromatic radiation with a wavelength of 0.1445 nm (photon energy 8.1 keV) was used. The sample–detector distance was 700 mm corresponding to the angular range of the scattering vector s = 0.8 ÷ 6.0 nm^–1, where s = 4πsin(θ)/λ, 2θ is the scattering angle, and λ is the X-ray wavelength.

During the measurements, the temperature of the capillary was changed from 90 to 2.5 °C using a JULABOFP-89HL circulating thermostat in varying steps: from 90 to 10 °C the step was 10 °C, and from 10 to 5 °C step was 5 °C, from 5 to 2.5 °C step was 2.5 °C; at a rate of 0.5 deg/min, the exposure time was 10 min at each point, and the temperature accuracy was 0.01 °C. The signal was recorded using a PILATUS 1M two-dimensional pixel detector (Dectris, Baden, Switzerland). The powder of silver behenate was used for s-axis calibration. The two-dimensional scattering pattern was averaged over the radial direction using the program FIT2D [32]. Subtraction of the buffer scattering contributions was performed by using the program PRIMUS [33].

The radius of gyration R_g of solute was estimated using the Guinier approximation [34]:

$$\mathrm{I}\left(\mathrm{s}\right)\cong \mathrm{I}\left(0\right)\ast \exp \left(-\frac{{\left({\mathrm{s}\ast \mathrm{R}}_{\mathrm{g}}\right)}^2}{3}\right),{\mathrm{s}\ast \mathrm{R}}_{\mathrm{g}}<1.3$$

(1)

The scattering intensity in a multicomponent system (case of diluted solution) can be presented as a linear combination of scattering components in the form:

$$\mathrm{I}\left(\mathrm{s}\right)={\displaystyle \sum }_{\mathrm{k}=1}^{\mathrm{N}}{\mathsf{\nu }}_{\mathrm{k}}{\mathrm{I}}_{\mathrm{k}}\left(\mathrm{s}\right)$$

(2)

where N is the number of components in the system, and ν_k and I_k(s) are the relative volume fraction and the intensity of the kth component, respectively. The program OLIGOMER [33] was used to determine the volume fractions of precursor clusters in the KDP solution using the non-negative linear least-squares algorithm. Crystallographic models of oligomeric KDP clusters were built by the program Diamond 3.0 [35] based on the tetragonal KDP atomic structure determined in [36]. The theoretical scattering curves of oligomeric KDP clusters were calculated using the program CRYSOL [37].

The program MIXTURE [33] that analyses multicomponent polydisperse systems in terms of mixture of simple geometric objects was used to independently verify the results of OLIGOMER modelling. The system was represented by a mixture of three components (spheres, ellipsoids, cylinders) with the sizes corresponding to monomeric, tetrameric and octameric KDP molecules.

The quality of the approximation was estimated by minimizing the discrepancy χ² between the experimental and calculated scattering curves using the formulae:

$${\mathsf{\chi }}^2=\frac{1}{\mathrm{N}-1}{\displaystyle \sum }_{\mathrm{j}}{\left[\frac{\mathrm{I}\left({\mathrm{s}}_{\mathrm{j}}\right)-{\mathrm{cI}}_{\mathrm{calc}}\left({\mathrm{s}}_{\mathrm{j}}\right)}{\mathsf{\sigma }\left({\mathrm{s}}_{\mathrm{j}}\right)}\right]}^2$$

(3)

$$\mathrm{c}=\left[{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\frac{\mathrm{I}\left({\mathrm{s}}_{\mathrm{j}}\right){\mathrm{I}}_{\mathrm{calc}}\left({\mathrm{s}}_{\mathrm{j}}\right)}{\mathsf{\sigma }{\left({\mathrm{s}}_{\mathrm{j}}\right)}^2}\right]{\left[{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\frac{{\mathrm{I}}_{\mathrm{calc}}\left({\mathrm{s}}_{\mathrm{j}}\right)}{\mathsf{\sigma }{\left({\mathrm{s}}_{\mathrm{j}}\right)}^2}\right]}^{-1}$$

(4)

where N is the number of experimental points, I(s_j) and σ(s_j) are the experimental scattering intensities and associated errors, c is the scaling factor and I_calc(s_j) is the calculated intensity from the oligomeric mixture.

3. Results

The KDP structure is described by the tetragonal space group of symmetry I4 ̅2d (No. 122), unit cell parameters a = 7.4606(4) Å, c = 6.9800(4) Å [36]. Phosphorus, oxygen, potassium and hydrogen atoms occupy crystallographic positions 4a, 16e, 4b and 16e, respectively, in the structure. Phosphorus atoms are located in the H₂PO₄ tetrahedron. The six nearest tetrahedrons are at distances of ≈3.5–4.1 Å from the potassium atom, and the distances between the potassium atom and the other four nearest potassium atoms are ≈4.1 Å.

To analyze the SAXS data, we used models of precursor clusters (3D fragments), from which a KDP crystal can be formed in a saturated solution (Figure 2). Models of precursor clusters were determined by the results of combinatorial topological modeling: monomers, dimers, tetramers and octamers [24]. The 2K[H₂PO₄] cluster-dimer is formed from two bound monomers. Bonds exist between the vertex of the [H₂PO₄]⁻ tetrahedron and the K+ ion from the neighboring monomer. The 4K[H₂PO₄] tetramers are formed from two dimers and also have two bonds between the vertices of the [H₂PO₄]⁻ tetramer and K+ ions from neighboring dimers. The 8K[H₂PO₄] octamers are built from two tetramers (Figure 2).

The experimental and theoretical SAXS curves calculated by the program OLIGOMER (using isolated oligomeric clusters) are shown in Figure 3. The results are presented in

Table 1. The volume fractions of monomers, dimers, tetramers and octamers; the average radius of gyration R_g of the solute; and the fit quality values χ² obtained by OLIGOMER for a KDP solution with a concentration of 335 mg/mL.

T, °C	χ2	Fraction of Monomers, %	Fraction of Dimers, %	Fraction of Tetramers, %	Fraction of Octamers, %	Rg, Å
90	1.02	66.1 ± 1.1	0	0	33.9 ± 0.3	3.15 ± 0.05
80	0.90	69.6 ± 0.9	0	0	30.4 ± 0.2	3.13 ± 0.04
70	0.96	63.6 ± 1.0	0	0	36.4 ± 0.3	3.20 ± 0.04
60	1.02	57.4 ± 0.9	0	0	42.6 ± 0.3	3.30 ± 0.05
50	1.5	55.8 ± 1.1	0	0	44.2 ± 0.4	3.51 ± 0.07
40	1.2	50.8 ± 0.8	0	0	49.2 ± 0.3	3.40 ± 0.06
30	1.14	45.8 ± 0.6	0	0	54.2 ± 0.4	3.40 ± 0.05
20	1.5	43.9 ± 0.7	0	0	56.1 ± 0.6	3.70 ± 0.07
10	1.32	34.4 ± 0.4	0	0	65.6 ± 0.6	3.65 ± 0.06
5	1.38	27.5 ± 0.5	0	0	72.5 ± 0.6	3.75 ± 0.07
2.5	1.38	22.0 ± 0.5	0	0	78.0 ± 0.6	3.63 ± 0.06
2.5 *	0.90	30.0 ± 0.5	0	0	70.0 ± 0.6	3.67 ± 0.06

The measurement at T = 2.5 °C * was taken the next day after exposure in the sample holder at 2.5 degrees for about 18 h.

.

The slope of the SAXS curves becomes sharper as the temperature is decreased (Figure 3a). This behavior points to the formation of associates and/or an increase of their volume fractions compared to the initial state of the system. Indeed, for the initial state of the system at 90 °C, the average radius of gyration R_g was equal to (0.31 ± 0.01) nm, whereas it significantly increases to (0.37 ± 0.01) nm as the temperature decreases to 2.5 °C.

The calculated SAXS curves for oligomeric clusters for a supersaturated KDP solution coincide with the experimental data over the entire angular range (Figure 1b). The χ² values do not exceed 1.4, which indicates a good fit quality. It was found that the solution contains only monomers and octamers, and the content of octamers increases with the temperature decrease from 30% (T = 80 °C) to 78% (T = 2.5 °C). It is in agreement with the singular value decomposition analysis of the scattering curves that found only two significant singular values corresponding to two nonrandomly oscillating singular vectors (Figure 4).

According to the obtained data (Table 1), three stages of crystallization can be distinguished:

(1)

A major fraction of monomers and a minor fraction of octamers (90 °C < T< 50 °C).

(2)

A significant increase in the volume fraction of octamers (50 °C < T< 5 °C).

(3)

The appearance of crystals, which is also confirmed by a decrease in the radius of gyration, as well as a decrease in the volume fraction of octamers (at T = 2.5 °C measured the next day).

One can see from Figure 5 that the volume fraction of octamers is observed to increase as the temperature is decreased. It can be distinguished, based on Table 1 and Figure 5, that for the studied KDP solution, the crystallization point was between 50 and 40 °C, which slightly differs from the previously reported data [24] where the crystallization temperature is estimated as 40 °C for a given KDP concentration. On the other hand, looking at the R_g values in Table 1 and taking into account the associated errors, the crystallization temperature can be closer to 40 °C, where a significant increase of average R_g is observed.

To verify the results, we have used an alternative approach where the system was approximated in term of simple geometric bodies (spheres, ellipsoids and cylinders) corresponding to the sizes of monomers, tetramers and octamers of KDP atomic structures, respectively. The volume fractions of the components obtained by the program MIXTURE are shown in

Table 2. The volume fractions of spheres (with radius 2.0 Å), ellipsoids (with semiaxes 2.5 Å, 2.5 Å and 5.0 Å) and cylinders (with radius 2.25 Å and length 17 Å) corresponding to the sizes of monomers, tetramers and octamers, and the fit quality values χ² obtained by the program MIXTURE for a KDP solution with a concentration of 335 mg/mL. A moderate polydispersity on the radius of the components was allowed during the modelling.

T, °C	χ2	Fraction of Spheres, %	Fraction of Ellipsoids, %	Fraction of Octamers, %
90	1.42	55.0 ± 1.3	12.0 ± 0.5	33.0 ± 0.8
80	1.35	54.0 ± 1.4	13.0 ± 0.4	33.0 ± 0.8
70	1.46	54.0 ± 1.3	13.0 ± 0.5	32.0 ± 0.8
60	1.14	59.0 ± 1.5	9.0 ± 0.4	32.0 ± 0.8
50	1.10	53.0 ± 1.3	10.0 ± 0.5	37.0 ± 0.7
40	1.44	50.0 ± 1.2	10.0 ± 0.5	40.0 ± 0.8
30	1.08	49.0 ± 1.2	9.0 ± 0.5	42.0 ± 0.7
20	1.02	47.0 ± 1.1	9.0 ± 0.4	44.0 ± 0.7
10	1.05	43.0 ± 1.0	8.0 ± 0.4	49.0 ± 0.8
5	1.38	44.0 ± 1.0	6.0 ± 0.3	50.0 ± 0.8
2.5	1.07	45.0 ± 1.1	5.0 ± 0.3	50.0 ± 0.8
2.5 *	1.40	46.0 ± 1.0	6.0 ± 0.3	48.0 ± 0.9

The measurement at T = 2.5 °C * was taken the next day after exposure in the sample holder at 2.5 degrees for about 18 h.

. As can be seen, the obtained results are in a qualitive agreement with the OLIGOMER modelling—namely, the main components in the system are represented by spheres (monomers) and cylinders (octamers), whereas the volume fraction of the ellipsoids (tetramers) does not exceed 13% at all temperatures.

After the completion of measurements at 2.5 °C, the capillary with the solution was kept for 18 h in a thermostat at a constant temperature. As a result, crystals grew in the capillary (Figure 6), which reflects a transition to the third stage of crystallization (Figure 1). It is also in agreement with the data from Table 1, where a decrease in the radius of gyration and in the volume fraction of octamers is observed.

4. Discussion

SAXS is a powerful technique that provides structural information in nanoscale range and is actively used for studying time-dependent evolutions of macromolecules and precipitation reactions for inorganic substances [38].

As a result of SAXS data analysis, one can see from Figure 5 that, starting from a temperature of 80 °C (the temperature of formation of a saturated solution), the volume fraction of KDP monomers sharply decreases and the volume fraction of KDP octamers increases as the temperature is decreased. At the same time, other types of oligomers seem to represent minor components in solution. It clearly indicates the leading role of octamers in the formation of crystals. At the same time, one has to note that the octamers represent just one of the possible precursor cluster intermediates of the pre-crystallization phase of KDP solution. One cannot exclude the presence of octamers with different crystal lattice types that may nucleate faster than the tetragonal one.

Since octamers are precursor clusters [24], the results obtained by SAXS allow us to conclude that they are the 3D fragments from which the crystal was grown from a saturated KDP solution (Figure 7).

In general, it can be noted that this work confirms the three-stage growth model (Figure 8): namely, at high temperatures, the monomers represent the major fraction of the unsaturated solution and as the temperature is decreased, the volume fraction of octamers is significantly increased with increasing values of saturation. Finally, the measurement performed the next day at a temperature of 2.5 °C found that the third stage of crystallization began to be observed and is accompanied by the crystal formation (Figure 6).

The present work shows that the monomers and octamers are the main components in pre-crystallization solution with possibly minor populations of other oligomers, and indicates that crystallization occurs on the basis of the formation of the intermediate phase of precursor clusters.

Thus, the present work confirms the concept of the formation of precursor clusters from 3D fragments of the crystal structure in the crystallized supersaturated solution of the intermediate phase. The formation of this phase precedes nucleation [39], which represents a separate process. Consideration of crystallization usually begins with the process of nucleation, which essentially depends on the structure of the ruler-precursor. The process of nucleation may be strongly spaced in time with the formation of clusters, but they are undoubtedly interconnected. Therefore, the procedure for determining precursor clusters described in this paper can significantly facilitate and speed up the search for optimal conditions for crystal growth.