Study of the Formation of Precursor Clusters in an Aqueous Solution of KH₂PO₄ by Small-Angle X-ray Scattering and Molecular Dynamics

Abstract

The structure of an aqueous solution of potassium dihydrogen phosphate (KH₂PO₄, KDP) was studied by small-angle X-ray scattering (SAXS) and molecular dynamics (MD). According to SAXS data, the octameric species (KH₂PO₄)₈ are formed in solution in addition to K⁺, (H₂PO₄)⁻, and KH₂PO₄, while the presence of other types of oligomers is not observed. When the temperature drops below the saturation temperature (~60 °C), the volume fraction of octamers increases sharply, reaching 50% at 4 °C. The results of MD calculations of the temporal stability relationships of dimers (KH₂PO₄)₂, tetramers (KH₂PO₄)₄, and octamers (KH₂PO₄)₈ show that the dimers and tetramers disintegrate rapidly (50–100 ps), while the octamers remain stable. A comparative analysis of the bonds between the octamers and the KDP crystal lattice was carried out when the octamer was inserted during crystal growth in the directions [001] and [100]. The possible relationship of the obtained results with the changes in the anisotropy of growth rates (habitus) of KDP crystals at different degrees of supersaturation is discussed.

1. Introduction

The transition from the liquid state to the solid state is one of the fields of condensed state physics for which, to date, there are no generally recognized transition mechanisms. Of particular difficulty are models of the formation of crystalline embryos, and on a scale of tens of nanometers, they are limited to assumptions.

Recently, three-stage crystallization models [1,2] have gained particular popularity when intermediate precursor clusters are expected to form. In a series of papers [3,4,5], a number of physical and computational experiments were carried out, confirming that such precursor clusters are 3D fragments of the crystal structure, which rely on “building bricks” when crystals originate and grow.

These precursor clusters were experimentally detected for crystallization solutions of a number of proteins by small-angle X-ray scattering (SAXS) and neutron scattering (SANS) [3]. Molecular dynamics methods have confirmed that only precursor clusters are stable, while other possible clusters (3D fragments) disintegrate rapidly [4].

Recently, a study was conducted on the association of the concentration of lysozyme protein precursor clusters with the peculiarities of solid phase formation [5]. The results of this work show that with the small volume fractions of octamers (up to 1%), the crystal growth was not observed, the single crystal formation was observed from 1% to 5%, the aggregation or denaturation was observed from 5% to 7%, and only the denaturation was observed above.

The three-step model described above applies to all compounds, but the main work on the role of 3D fragments as precursor clusters has been carried out on crystallization solutions of proteins. Recently, the existence of precursor clusters for inorganic compounds was confirmed by KH₂PO₄ (KDP) [6], where by the SAXS technique it was found that in an aqueous solution of potassium dihydrogen phosphate with a concentration of 335 mg/mL, in addition to K⁺, (H₂PO₄)⁻, and precursor clusters, octamer molecules were found.

To suggest and develop the above results, we first repeated the temperature dependencies of SAXS curves of crystallization solutions of potassium dihydrogen phosphate (KDP, KH₂PO₄) with other concentrations of 501 mg/mL.

It is much more important that the stability calculations of three types of KDP oligomers (the dimer, tetramer, and octamer) be carried out by the molecular dynamics (MD) method.

The features of the inter-atomic bonds of the octamer when the octamer is inserted into the crystal in the directions [100] and [001] are analyzed, and the influence of these features on the anisotropy of the growth of KDP crystals is discussed.

2. Materials and Methods

2.1. Sample Preparation and SAXS Measurements

The KDP crystals were grown at the Institute of Crystallography. As the solvent for the KDP crystals, Millipore ultrafine water was used (water resistance 18 MΩ × cm). The concentration of KDP in an aqueous solution was 501 mg/mL, which corresponds to the saturation temperature of the KDP solution at 60 °C [7]. The crystals were dissolved in water at 80 °C in a thermostat.

SAXS measurements in a pre-crystallization (supersaturated) solution of KDP using a plane-parallel cell were carried out at the BioMUR station of the Kurchatov synchrotron radiation source (National Research Center “Kurchatov Institute”) [8]. Monochromatic radiation with a wavelength of 0.1445 nm (the radiation energy of 8.1 keV) was used. The sample-to-detector distance was 700 mm, corresponding to the angular range of the scattering vector s = 1.0 ÷ 6.0 nm⁻¹, where s = 4πsin(θ)/λ, 2θ is the scattering angle, and λ is the X-ray wavelength.

During the measurement, the capillary temperature was varied using the circulating thermostat JULABOFP-89HL(JULABO GmbH, Seelbach, Germany) from 90 °C to 2.5 °C for a 501 mg/mL KDP crystallization solution at a rate of 0.5 deg/min, and the exposure time was 10 min at each point. The signal was recorded using a two-dimensional PILATUS3 1M pixel detector (Dectris, Baden-Daetwill Switzerland). The angular axis calibration was made using silver behenate. The two-dimensional scattering pattern was averaged along the radial direction using the FIT2D program version 17.006 [9] while masking the non-active gap regions of the detector. The scattering of the buffer solution was subtracted from the KDP scattering curves using the PRIMUS program (version from ATSAS 3.2.0 release) [10]. The average radius of gyration R_g of the particles was estimated using the Guinier approximation [11]:

$$I\left(s\right)=I\left(0\right)\times \mathit{exp}\left(-\frac{{\left(s*R_g\right)}^2}{3}\right), s\times R_g<1.3,$$

(1)

where $R_g$ is the gyration radius.

The scattering intensity in a multicomponent system of diluted solution can be represented as a linear combination of its components as follows:

$$I\left(s\right)={\sum }_{k=1}^N{\nu }_k{I}_k\left(s\right),$$

(2)

where N is the number of components, ${\nu }_k$ and I_k(s) are the volume fraction and intensity of the k-th component, respectively. In order to quantitatively characterize the composition of precursor clusters in the KDP solution, the OLIGOMER program (version from ATSAS 3.2.0 release) [10] was used, which finds the volume fractions of components by solving a system of linear equations using the theoretical SAXS curves from the atomic structures of KDP monomers, dimers, tetramers, and octamers calculated using the CRYSOL program (version from ATSAS 3.2.0 release) [10].

The fit quality was evaluated by minimizing the discrepancy value χ² between the experimental data and the calculated scattering curves using the following formula:

$$\begin{gathered} {\chi }^2=\frac{1}{N-1}{\sum }_j{\left[\frac{I\left(s_j\right)-c{I}_{calc}\left(s_j\right)}{\sigma \left(s_j\right)}\right]}^2 \\ c=\left[\sum _{i=1}^{N_p}\frac{I\left(s_j\right)I_{calc}\left(s_j\right)}{{\sigma \left(s_j\right)}^2}\right]{\left[\sum _{i=1}^{N_p}\frac{I_{calc}\left(s_j\right)}{{\sigma \left(s_j\right)}^2}\right]}^{-1} \end{gathered}$$

(3)

where N is the number of experimental points, c is the scaling factor, and $I_{calc}\left(s_j\right)$ and $\sigma \left(s_j\right)$ are the calculated intensity and the experimental intensity error estimates, respectively.

In order to analyze the chemical bonds between precursor clusters and the KDP crystal, the model of self-assembly of the KDP structure from the octamer clusters proposed in [12] based on the results of the topological analysis of the KDP structure was used. The bond lengths were analyzed based on the KDP structure defined in [13].

2.2. Molecular Dynamic Simulations

The dimeric and tetrameric structures of KDP were obtained from the cluster-octamer model [14]. To prepare the MD simulation files, the Amber [14,15] MD software package was used. KH₂PO₄ molecule topology files were generated by Antechamer in the General AMBER Force Field [16]. The TIP4P model [17] was used for the water molecule. The KDP oligomers were placed in a cube-shaped simulation cell. The minimum distance from the KDP molecules to the cell faces was 1.5 nm. The cell was filled with water molecules. At the first stage, the energy was minimized for each of the 4 systems, corresponding to the monomer, dimer, tetramer, and octamer. Further, the systems were equilibrated at 303 K and 1 bar pressure by modeling the dynamics in NVT and NPT ensembles with a duration of 50 ps and 100 ps, respectively. The modified Langevin thermostat [18] and Berendsen barostat [19] with time constants ɣLn = 3 ps and τP = 2 ps, respectively, were used for thermostating and barostating the systems. A productive 250 ps MD simulation for each of the systems was carried out in an NPT ensemble with an integration step of 2 ps and the use of a semi-empirical PM6 of the Hamiltonian [20] for KDP molecules. Long-range electrostatic interactions were calculated using the Ewald summation scheme in PME [21,22]. Coulomb and Lennard-Jones potentials were truncated to 1 nm.

3. Results

3.1. Study of Precursor Cluster Formation by SAXS

The SAXS data permitted us to estimate the volume fractions of the pre-crystallization precursor clusters for the 501 mg/mL KDP crystallization solution. For the processing of the experimental SAXS data, the models of monomer, dimer, tetramer, and octamer were taken into account (see Section 2.1 for details). The mixtures of monomers and octamers were found to neatly fit the SAXS data, as can be seen in Figures S1 and S2.

It can be seen from Figure 1 that the volume fraction of octamers began to grow sharply when the temperature dropped below 60 °C in the case of a crystallization solution with a concentration of 501 mg/mL.

Based on the data shown in Figure 1 and

Table 1. The volume fractions of pre-crystallization precursor clusters for the 501 mg/mL KDP crystallization solution as estimated by SAXS data. The volume fractions of dimers and tetramers were found to be equal to zero.

T, °C	χ2	Volume Fractions of Monomers, %	Volume Fractions of Octamers, %
90	1.44	74.5 ± 1.2	25.5 ± 0.3
80	1.23	74.5 ± 1.3	25.5 ± 0.3
70	1.56	73.6 ± 1.2	26.4 ± 0.3
60	1.56	75.0 ± 1.3	25.0 ± 0.3
50	1.68	71.8 ± 1.3	28.2 ± 0.3
40	1.62	64.4 ± 1.2	35.6 ± 0.3
30	1.86	62.5 ± 1.2	37.5 ± 0.4
20	1.92	57.7 ± 1.1	42.3 ± 0.4
10	1.8	49.7 ± 0.9	50.3 ± 0.5
5	2.04	48.8 ± 0.8	51.2 ± 0.5
2.5	2.04	45.9 ± 0.9	54.1 ± 0.6

, the following temperature ranges can be distinguished, which describe the initial stage of KDP crystallization:

(1) The stable behavior of the volume fractions of monomers and octamers (temperature above 60 °C).

(2) The sharp increase in the volume fraction of octamers with an increase in supersaturation and a decrease in the volume fraction of monomers (temperature below 60 °C).

At the supersaturated solution stage, crystal growth begins. With the growth of the volume fraction of octamers, supersaturation increases. As is known, the crystal begins its growth at a temperature lower than the saturation temperature. In turn, this means that the growth of the KDP crystal begins at least with the monomer/octamer ratio of 3:1 and the growth dynamics of the volume fraction of the octamer.

In the next step, crystal growth is observed in the saturated solution.

The data shown in Table 1 and Figure 1 point also to the absence of the formation of dimers and tetramers, which agrees well with the results of the work [6].

3.2. Calculation of the Stability of KDP Oligomers Using the MD Method

In order to estimate the stability of dimers, tetramers, and octamers of KDP in solution, a computational experiment was conducted using the method of molecular dynamics. Figure 2 shows images of dimer, tetramer, and octamer structures at time points equal to 0 ps, 50 ps, 150 ps, and 250 ps.

It can be clearly seen from Figure 2 that the KDP octamer retains its structure throughout the calculation interval, while the tetramer and dimer begin to decay after 150 picoseconds and almost completely decay after 250 picoseconds.

It can be seen from Figure 2 that dimers decompose over time not into monomers but rather into ions of potassium and dihydrogen phosphate residue, which is clearly visible at t = 150 ps and 250 ps. The tetramers, in turn, disintegrate into monomers, which is clearly visible at t = 250 ps. The octamers remained stable as a result of molecular dynamics, which is consistent with the results of the analysis of SAXS data, which point to the octamer as a building block of KDP crystal growth. Molecular dynamics shows that fluctuations in the distance between phosphates for dimers and tetramers are more than five angstroms. The results obtained confirm the results of the study [6].

3.3. Analysis of Chemical Bonds between Precursor Clusters from Which the Structure of KDP Crystals Is Formed and the Crystal Structure

The KDP structure in the tetragonal phase belongs to the spatial symmetry group I4-2d (No. 122), with the parameters of the unit cell a = 7.4606 (4) Å and c = 6.9800 (4) Å [13].

The size of the octamer cluster is ≈ 17.443 Å in the direction of the crystallographic axis c and ≈5.963 Å in the direction of the crystallographic axes a and b [6].

There are three kinds of chemical bonds between adjacent octamer clusters: two types of K1-O1 (≈2.914 Å and ≈2.825 Å) between potassium atoms and oxygen atoms included in the H₂PO₄, tetrahedron, and hydrogen bonds O1-H1…O1 (

Table 2. The chemical bond lengths of octamer clusters in the KDP structure.

№	Bond Type	Bond Length, Å
Direction c
1	K1-O1 × 8	2.914
2	K1-O1 × 4	2.825
3	O1-H1…O1× 6
	O1-H1	0.775
	H1…O1	1.729
	O1-O1	2.502
Direction a
1	K1-O1 × 8	2.825
2	O1-H1…O1 × 4
	O1-H1	0.775
	H1…O1	1.729
	O1-O1	2.502
Direction b
1	K1-O1 × 6	2.825
2	O1-H1…O1 × 4
	O1-H1	0.775
	H1…O1	1.729
	O1-O1	2.502

). The crystallographic position of hydrogen atoms is populated by ½. The number of chemical bonds between octamers in different directions and their length are not the same, which may indicate a different degree of bond strength between octamers in different crystallographic directions.

The formation of chemical bonds between a separate octamer cluster joining from solution to the surface of a KDP crystal formed of octamers is shown below. A situation is shown where the crystal surface does not contain “steps” of embedded octamers, i.e., the octamer is attached to a layer parallel to planes [001] and [100].

Figure 3 and Figure 4 show the chemical bonds of the octamer with the crystal surfaces parallel to the faces [001]—Figure 3 and [100]—Figure 4. Table 2 shows the types of bonds and their lengths.

In the direction of the crystallographic axis c, the octamer has the largest size. In this direction between the attaching octamer and the surface of the crystal consisting of octamers (corresponding to the crystallographic plane [001]), there are eight bonds K1-O1 with a length of ≈2.914 Å, four bonds K1-O1 with a length of ≈2.825 Å, and six hydrogen bonds O1-H1…O1 (taking into account the population of H1 equal to 0.5) with a total length of donor-acceptor ≈ 2.502 Å.

If the octamer is attached to the face [100] of the crystal, as shown in Figure 4, eight bonds are formed between the embedding octamer and the crystal K1-O1 with a length of ≈2.825 Å and four hydrogen bonds O1-H1...O1 ≈ 2.504 Å.

The total number of links K1-O1 is maximized in the directions of the c and a axes, and the average link length is greater in the axis c (≈2.884 Å).

It can be seen from Table 2 and Figure 3 and Figure 4 that in the direction of the crystallographic axis c there are long K1-O1 bonds (2.914 Å), which are less strong than the shorter chemical bonds K1-O1 in the direction of the crystallographic axes a and b (2.825 Å). Thus, it can be assumed that the binding of the octamer to the crystal in the direction of the crystallographic axis c is weaker than in the directions a and b.

It is known that with small supersaturations, the growth of the KDP crystal occurs in the direction [001], while with large supersaturations, the formation of prismatic steps in the directions perpendicular to [001] [23,24] is observed.

A more important circumstance, with normal crystal growth (small supersaturation), is the crystal growth step size factor, which is larger in the direction [001]. The size of the step in this direction corresponds to the size of the octamer ≈17.4 Å, from which it follows that the octamer is a pre-crystallization precursor cluster, which was confirmed both in the framework of SAXS experiments [6] and in the framework of calculations using molecular dynamics in Section 3.2.

Competition between octamers in solution at large supersaturations begins when embedded in a crystal. Since the bond of the octamer to the crystal in the [001] direction is weaker than in the perpendicular directions (Table 2), the growth rate of the KDP crystal in the [100] and [010] directions, where the bonds are stronger, increases [24]. Prismatic steps are formed in these directions.

4. Discussion

The SAXS results show that the precursor clusters for KDP crystals are octamers of size ≈ 17.4 × 5.9 Å, embedded in the crystal in the direction [001] by a step of size ≈ 17.4 Å and in the direction [100] by a step of size ≈ 5.9 Å. In our opinion, this explains the habit of the crystal in low supersaturation (Figure 5) [25].

As it is known [26], with an increase in supersaturation from 0.02 to 0.1, the growth rate along the direction [100] increases by more than an order of magnitude, in contrast to the growth rate in the direction [001], and the crystal habit changes significantly.

Based on the results of this work, this can be explained by the emergence of competition between octamers embedded in the crystal. In this case, not only the configuration of the octamer (its dimensions in different directions) begins to play a role, but also the strength of the connection between the octamer and the crystal, which, according to Section 3.3, is greater when the octamer is embedded on the face [100] rather than on the face [001].

5. Conclusions

The present work confirms that crystallization solution precursor clusters can be determined on small amounts of crystallization solutions by SAXS and computational experiments by molecular dynamics methods, both for high molecular weight and inorganic substances. In particular, the conditions for the formation of a particular crystalline phase can be determined with a great economy of time and resources. For example, for KDP and DKDP, several tetragonal and monoclinic phases are possible. Which phase will be formed can be determined by the precursor cluster. For the tetragonal phase, the formation of octamers will be an indication.