# Millisecond-time-scale controlled freeze-quench for solute-intermediate analysis by solid-state NMR

Ieva Goldberga, Trevys Chanal, Tristan Georges, Guillaume Laurent, Thierry Azaïs

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Ieva Goldberga, Trevys Chanal, Tristan Georges, Guillaume Laurent, Thierry Azaïs. Millisecond-time-scale controlled freeze-quench for solute-intermediate analysis by solid-state NMR. Faraday Discussions, 2025, Advanced imaging techniques in biomineralisation research, 261, pp.99-115. 10.1039/D5FD00032G. hal-05462812

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https://hal.sorbonne-universite.fr/hal-05462812v1

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Milliseconds time-scale controlled freeze-quench for solute intermediates
analysis by solid-state NMR

Ieva Goldberga,¹ Trevys Chanal,¹ Tristan Georges,¹,² Guillaume Laurent¹ and Thierry Azaïs¹*

¹ Sorbonne Université, CNRS, Chimie de la Matière Condensée de Paris, LCMCP, F-75005 Paris, France

² Present address : Laboratory of Magnetic Resonance, Institut des Sciences et Ingénierie Chimiques, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

* To whom correspondence should be addressed: thierry.azais@sorbonne-universite.fr

**Abstract**

Crystallization is a fundamental process in biomineralization, yet its complexity increases significantly in non-classical pathways of nucleation and growth, where numerous intermediates exist between free ions in solution and the final crystalline phase. Characterization of such intermediates can be delicate as some processes can proceed very fast, inducing a short temporal window for their characterization. The problem is particularly true for the so-called prenucleation species, for which this difficulty is made worse due to their solubility, nanometric size and highly dynamic nature. In this communication, we introduce an innovative approach based on the "trapping" of reactive intermediates in a vitreous matrix, inspired by cryo-TEM sample preparation but adapted specifically to solid-state NMR characterization. This approach enables time-resolved analysis since the aging time of the mineralizing reaction is controlled on the milliseconds time-scale using a dedicated stopped-flow device. The cryo-fixation is achieved by spraying the reacting solution in cold liquid isopentane at -145°C, and the NMR rotors are filled with specifically designed packing tools, ensuring the control of the low temperature during the whole process. Finally, the cryo-fixed solution can be studied by low-temperature solid-state NMR by acquiring time-resolved NMR spectra as snapshots of the ongoing reaction. First, we show that phosphate solutions can be efficiently vitrified using this protocol and studied by low-temperature ³¹P solid-state NMR at -120 °C. We demonstrate that the ³¹P chemical shift interaction of cryo-quenched solutions varies with the pH, allowing us to distinguish the different phosphate species coexisting in solution (PO₄³⁻, HPO₄²⁻, H₂PO₄⁻ and H₃PO₄) based on their chemical shift anisotropy patterns, which is characteristic of their protonation degree. The determination of the proportion of each species at varying pH levels enables us to construct the speciation diagram under our experimental conditions. We observe that the "apparent pKa," i.e., the pH values for each chemical equilibrium, are slightly influenced by the low temperature, and possibly by the preparation conditions. Finally, we demonstrate that our method can be applied to study fast calcium phosphate crystallization, revealing the early stages of amorphous calcium phosphate (ACP) nucleation within just 20 ms of reaction time, as shown by ³¹P ssNMR. Importantly, the described methodology is the first step towards studying fast out-of-equilibrium solutions through trapping and then studying by solid-state NMR of transient intermediate species. Indeed, proper freeze quenching prevents the transient species from transforming and preserves their native environment, such as hydration, pH, or ionic strength. We demonstrated applications for biomineralization-relevant reactions, but in principle, any aqueous reaction can be studied.
Introduction

Crystallization is a fundamental process that underlies many natural and industrial processes, from biomineralization and ice formation to the production of pharmaceuticals, manufacturing materials, and food products. Understanding how crystallization occurs and progresses is crucial for advancing research in biomineralization, conservation, environmental and materials sciences. The early stages of crystallization, particularly how atoms coalesce to form a stable nucleus that can grow into a larger crystal, have always been of significant interest. Traditionally, crystallization has been described by the classical nucleation and growth theory (CNT)¹ where first a nucleus of a defined critical size nucleates from solution followed by its growth from monomers addition until the final crystal. However, in the past decade, this theory has been challenged by the emergence of the so-called non-classical crystallization pathway (NCP) due to the observation of stable solute transient species that form in solution before nucleation.² In the context of biomineralization, such PNS have been reported for calcium phosphates,³,⁴ carbonates⁵,⁶ and oxalates.⁷,⁸ The characterization of these PNS is particularly challenging due to their solubility, nanometric size, dynamic nature, and short lifespan. Numerous analytical approaches have been used to characterize these species, as cryo-TEM approaches,³⁻⁵,⁹ SAXS,⁹,¹⁰ monitoring the evolution of calcium potentiometry (for calcium-containing biominerals),⁴,⁶,¹¹⁻¹³ DLS,⁹ ultracentrifugation,⁶ computational studies¹¹,¹⁴,¹⁵ and by electron ionization mass spectrometry.¹⁶ Despite the number of methodologies used to study these species, there is still a lack of information available in terms of composition, structural arrangement, and kinetic evolution, necessitating the development of innovative time-resolved monitoring techniques.

Nuclear Magnetic Resonance (NMR) spectroscopy is a very powerful technique in studying the structure and dynamics at the atomic and molecular level of liquid or solid matter.¹⁷ Thus, NMR spectroscopy can potentially provide precious insights into the various transient species formed during the nucleation and crystallization processes. For example, the CLASSIC NMR approach exploits alternate acquisition of both liquid-state and solid-state NMR (ssNMR) spectra as a function of time to obtain complementary information on the time-evolution of both the solid phase and the solute species during a crystallization process.¹⁸ However, conventional NMR approaches often face limitations due to the poor intrinsic sensitivity of NMR spectroscopy, which implies long acquisition times that prevent the study of fast reaction processes. The use of hyperpolarization methods such as dynamic nuclear polarization (DNP) can be a partial solution to track transient solute species such as PNS. Indeed, dissolution DNP (dDNP) allows the acquisition of NMR spectra on a very short time scale (~0.5 s⁻¹) and has been successfully used to monitor the formation and the evolution of PNS for calcium phosphates¹⁹,²⁰ and carbonates²¹,²² in fast mineralizing solutions. The rates of formation, aggregation and kinetic exchange with free ions involving PNS can be determined using this approach.¹⁹ However, limitations exist as dDNP equipment is rare, and 2D experiments cannot be recorded with dDNP (unless by combining with single-scan methodologies).

Thus, ideally, it would be beneficial to "stop" the reaction and characterize the system at specific time points as long as needed to record 1D and 2D NMR spectra. To tackle this, we introduce a new low-temperature ssNMR analysis approach coupled with an innovative sample preparation consisting of the cryo-fixation of aqueous solutions controlled on milliseconds time scale.

More precisely, a stopped-flow device allows the precise control of mixing and aging of mineralizing aqueous solutions. Then, a freeze-quench accessory allows the cryo-fixation of the mineralizing solutions and the filling of NMR rotors. The transient species and phases, including PNS,
are stable in the vitreous matrix, and ssNMR analysis can provide time-resolved "snapshots" of a mineralizing reaction.

The thermal quenching approach of two mixed solutions was first introduced in the 1960s, where the technique was tested by using a model reaction of the reduction of Fe³⁺ to Fe²⁺ ions of quinol by EPR spectroscopy.²³ For biological and water-containing samples, precise cryo-fixation requires the formation of vitreous ice at freezing rates of ≥10⁵ K/s and/or the use of cryoprotectants (e.g., glycerol) to prevent crystalline ice formation.²⁴ Cryo-fixation has become a routine technique to preserve materials and their structure at a native state and has been widely adopted across various experimental techniques, with significant advances in cryo-electron microscopy (cryo-EM) where sample preparation is generally achieved *via* plunge-freezing using liquid ethane or an ethane/propane mixture as a cryogen.²⁵⁻²⁷

Furthermore, similar approaches have been adapted for ssNMR, where cryo-quenched samples have been prepared by spraying the reaction mixture into a cryogenic medium, either propane (at -188°C)²⁸⁻³⁰ or isopentane (at -140 °C)³¹,³² or onto the surface of rotating copper plate cooled to approximately to -193 °C.³³,³⁴ However, the reported sample preparation and rotor packing vary and can be complicated. For samples fixed in a cold propane,²⁸,²⁹ the following rotor packing was done under inert conditions with a cryogenic solvent removal. Where methodologies reported by Tycko *et al.*,³¹⁻³⁴ use specifically home-built equipment, one of the experimental setups³³ uses a rotating copper plate that is placed in a custom-designed glove box to avoid ice formation on top of the plate. The sprayed mixture creates a thin film that is scraped off the surface of the plate and transferred to the NMR rotor. Another setup used by Tycko lab performed freeze-trapping by spraying the reaction mixture into stirred cold isopentane, which is removed *via* suction for 5-15 s, leaving a dense slurry of frozen solution and some residual isopentane.³² The mixture and holder are placed in liquid nitrogen, then the frozen sample is transferred to a packing funnel with a rotor inside. The temperature is lowered to -160 °C, melting isopentane, and the frozen mixture is broken up and packed using precooled packing tools.

In all previously reported examples, sample preparation requires a very specific experimental setup that can be inaccessible to some laboratories. Therefore, a more user-friendly approach regarding sample preparation and accessibility is required. Here, we show that the use of a commercial stopped-flow device connected to a freeze-quench accessory and dedicated cryogenic sample packing allows low-temperature NMR analysis of reacting solutions, with potential use for DNP magic angle spinning (MAS) approach.

The use of the commercially available system enables precise sample preparation protocols by controlling the volumes and the mixing times and increasing the rate of reproducibility and reliability of results. Similar equipment has already been used to prepare samples for EPR²⁵,³⁶ and solid-state NMR²⁸⁻³⁰ measurements. It has been shown by EPR studies the freezing time of the droplets is between 5-10 ms,³⁶,³⁷ with a freezing rate estimated to be ~10⁵ K/s.²⁹

A similar approach, but without automatization of mixing and aging, has already been used for solid-state NMR combined with DNP. It becomes possible to probe millimolar concentrations in inorganic transient species and progress in describing their structure and composition.³⁸,³⁹ For example, ¹³C MAS DNP was used to probe CaCO₃ dense liquid droplets for which the ionic speciation (i.e., carbonate/bicarbonate ratio) inside the droplets was found independent from the global pH.³⁸ In the case of CaCO₃ prenucleation clusters, 2D ¹H-¹³C DNP NMR shows that the Ca²⁺···HCO₃⁻ ion pair can be stabilized by an aspartic acid molecule.³⁸ Beyond a biomineralization context, ¹³C MAS-DNP was also used to trap and evidence metastable polymorphs, including amorphous precursors in the context
of the crystallization of organic molecules<sup>40-43</sup> such as glycine or aminobenzoic acid.<sup>44</sup> However, the total sample preparation time—including mixing the solutions, aging, rotor packing, and insertion into the NMR probe—is crucial. Minimizing this time is essential to capture transient intermediates before they evolve into more stable states. Manual sample preparation, where the sampling is done by hand, takes at least several minutes before starting the NMR acquisition, which is unsuitable for fast reactions. Therefore, implementing automated systems for sample mixing, aging, and rapid quenching would enable the effective capture of short-lived species and improve the repeatability of experimental studies. Furthermore, optimizing the rotor packing process and its insertion into the magnet is essential to address practical aspects of the workflow.

## NMR sample preparation procedure

The sample preparation setup is displayed in Figure 1. The stopped-flow device (BioLogic SFM-4000, Figure 1a) is equipped with three main components: (1.) four independent syringes, (2.) mixers, and (3.) spray. Up to four solutions can be loaded and then mixed in three different compartments where solutions can be aged from a minimum of 9 ms.<sup>45</sup> After mixing, the resulting solution is transferred through the last mixer and the ejection delay line (EDL) to a nozzle (3.) with an internal diameter of 0.5 mm and sprayed into a 50 mL falcon tube that is filled and stored inside cold isopentane (-145 °C) (Figure 1b). Cold isopentane is prepared by gradually adding liquid nitrogen under stirring in a dewar.

The amount of aged solution sprayed depends on the EDL volume used. Four volumes are available: 19, 41.6, 93.6, 141.8 and 189.5 µL. In the present work, for short aging times (20 and 100 ms), medium-size EDL was used (93.6 µL), delivering a total of 90 µL per spray; for longer aging times, the biggest EDL was used (189.5 µL), collecting 180 µL per spray. In all experiments, the total volume of the reaction mixture collected was 360 µL to ensure enough sample for filling the rotor. The filling volume of a standard 4 mm Zirconia solid-state NMR rotor is ~80 µL. More details of the mixing system are provided in the Supplementary Information Figure SI1.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** (a) Time-controlled freeze-quench device with highlighted components for (1.) syringes, (2.) mixing chambers, and (3.) spray. In (b), the spraying setup is shown with a polystyrene box filled with cold isopentane (not shown here for simplicity) containing an adapted 50 mL Falcon tube for particle</figcaption></figure>
collection. The sprayed particles are collected in the Falcon tube before packing the NMR rotor. (c) General scheme of the solution preparation and cryo-fixation procedure.

The resulting frozen particles in cold isopentane are shown in Figure 2. Once the liquid droplets hit the cold isopentane, they solidify and sink immediately to the bottom of the Falcon tube. Sub-centimeter particles can be observed with the naked eye with some aggregation and deformation (Figure 2a). Most of the particles adopt a disc-like shape. Higher magnification observations were performed using a custom-built setup (Figure S12.) designed for imaging under temperature-controlled conditions (approximately -100 °C). Around 2 mL of cold isopentane containing frozen particles was loaded into the sample holder and placed on a cold stage to keep the mixture chilled during imaging. The chamber was sealed with a cover and filled with cold nitrogen gas, which was used to maintain a sufficiently low temperature, ensuring that no particles melted during imaging. Round-shaped frozen particles are observed with an average diameter of 300-400 µm (Figure 2b-d and Figure S13). Similar morphological observations have been made before for frozen droplets containing lysozymes prepared by similar spraying in cold isopentane.⁴⁶

<figure><img src="image_3.png" /><figcaption>**Figure 2.** (a) Image showing the appearance of the frozen particles in isopentane straight after spraying (the purple color is due to a dye added to the sprayed solution). (b)-(c) Optical microscopy images of frozen particles shown (using a home-built light microscope).⁴⁷ More images are available in the Supplementary Information in Figure S13. (d) Histogram representing the average size of the frozen particles. In total, 120 images were processed, and 544 particles were measured. Note that the microscope setup could only measure particle size of < 1.5 mm.</figcaption></figure>

Maintaining the cold chain up to the NMR analysis is crucial, and thus, the NMR rotor packing is a critical step. A general solid-state NMR rotor packing scheme is shown in Figure 3. In detail, frozen particles are transferred to a homemade copper packing tool containing an NMR rotor (see details in Figure S14 and S15) and kept in a cold isopentane bath at -145 °C. To prevent any increase in sample temperature during packing, it is crucial to keep all packing tools cold, maintaining them below -145 °C. Once the rotor is filled, the packing funnel (holding the rotor) is placed on top of the Teflon holder
containing a needle, allowing the rotor to be pushed up to a level of the top of the filling funnel while still submerged in cold isopentane (Figure SI6). The excess of solid and isopentane is then removed using a plastic pipette, and the level of rotor filling is checked to ensure sufficient space for the cap. The rotor is carefully removed from the funnel using cold tweezers and kept in the cold isopentane (Figure SI6a). A zirconia cap is placed inside the cold copper packing funnel to mount it on a rotor. The rotor is then turned around and reinserted into the packing tool, where the cap is secured by gently pressing the rotor inside the filling funnel using cold tweezers (Figure SI6b). Alternatively, the rotor cap can be mounted using a cold nitrogen gas setup (<-145 °C) (Figure SI7). Once the rotor cap is securely mounted, the rotor is removed from the copper funnel and stored in cold isopentane until further use in ssNMR analysis.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** General scheme of sample packing into solid-state NMR rotor. The rotor filling process (i) is done in a cold isopentane (-145 °C) bath. Cap mounting (ii) can be performed in the same cold isopentane bath or done under a cold nitrogen gas outside. Packed rotor can be stored in cold isopentane (iii). See Supplementary Information Figures SI4-SI7 for more detailed images.</figcaption></figure>

Then, the rotor is transferred directly into the 4mm NMR probe cooled down at -120 °C through vertical insertion from the top of the magnet. Such an approach enables reliable, successful insertions and imminent magic angle spinning of the rotor. We observed that excess isopentane from the outside of the rotor evaporates due to the strong flux of the insertion as well as bearing and drive gas into the coil.

## NMR analysis

In the context of calcium phosphate mineralization, ³¹P NMR is a useful probe for mineral phase analysis,⁴⁸ allowing the understanding of subtle structural modifications induced by ionic substitutions or growth underconfinement, for instance.⁴⁹ But before the study CaP mineralization process using the above described freeze-quench device, the behavior of pure phosphate solutions must be evaluated in these preparation conditions to check the correct cryofixation of the solutions and to get ³¹P NMR reference spectra at -120 °C since the low temperature can impact NMR chemical shifts.

In solution, the ³¹P isotropic chemical shift ($\delta_{iso}$) is highly sensitive to its local environment, particularly the protonation state of phosphate, as shown in Figure 4a. Since the first report by Moon *et al.*,⁵⁰ ³¹P
NMR has been a standard noninvasive method for determining intracellular pH in various biological models. A single sharp resonance is observed at each pH due to fast chemical exchange between phosphate species and rapid molecular tumbling, which averages out anisotropic interactions like dipolar couplings and chemical shift anisotropy (CSA). By plotting chemical shift versus the pH, a sigmoidal curve is typically observed for each protonation equilibrium (Figure 4c), where the inflection points correspond to the pKa values of the phosphate species, i.e., the pH at which two protonation states are present in equal proportions. However, the fast chemical exchange in solution prevents the individual analysis of different phosphate species involved in the exchange. Lowering the temperature slows down the exchange kinetics and would lead, in principle, to separate ³¹P resonances for the various phosphate species. However, in the case of freezing, the reintroduction of anisotropic interactions such as CSA or dipolar couplings requires MAS to access high-resolution spectra.

Thus, 400 mM phosphate-containing solutions at different pH were cryo-fixed using the setup described above and analyzed at low temperature by ³¹P MAS ssNMR at -120°C. The resulting ³¹P isotropic resonances (Figure 4b) differ significantly from solution-state NMR spectra. They are ~100 times broader (average LWHM of 4.5 ppm) and display a Gaussian line-shape. Importantly, no reprecipitation of the initial phosphate salt or crystallization of a phosphate phase is observed. These observations support the efficient cryofixation of the solutions and the dispersion of the phosphates in a vitreous matrix. However, the distinct contribution of individual ³¹P resonance of the different phosphate species is not observed contrary to carbonate solutions where the distinct contributions of HCO₃⁻ and CO₃²⁻ resonances are seen in the low-temperature (100K) ¹³C NMR spectrum of a vitrified matrix.³⁸ Indeed, the contributions from more or less protonated phosphate species (H₃PO₄, H₂PO₄⁻, HPO₄²⁻ and PO₄³⁻) are spread over a smaller range compared to HCO₃⁻ and CO₃²⁻ (5 vs 15 ppm). We also observe an increase in δᵢₘₛ with increasing pH, similar to liquid NMR, with a marked transition around pH 7.4 and 12 corresponding to the expected pKa₂ and pKa₃ of phosphate (Figure 4c). However, contrary to what is anticipated, δᵢₘₛ stays constant around pH 2 (pKa₁).
<figure><img src="image_5.png" /><figcaption>**Figure 4.** (a) $^{31}$P solution state NMR spectra of sodium phosphate dodecahydrate solutions (25 mM) at different pH. (b) $^{31}$P CP MAS ssNMR spectra ($v_{MAS}$ = 5 kHz; $t_{CP}$ = 1.5 ms; T = -120 °C) of cryo-fixed phosphate solutions (400 mM) at different pH. (c) variation of the $^{31}$P chemical shift according to pH values.</figcaption></figure>

At slow MAS spinning ($v_{MAS}$ = 2 kHz), the resulting spectra (Figure 5a) differ significantly from the solution-state NMR spectra, as isotropic resonances (orange dashed line in Figure 5a) are flanked by spinning side bands (SSB). The relative intensities of the SSB manifolds vary according to the pH value. Indeed, for phosphates, the SSB manifold is mainly determined by the CSA, which depends on the local symmetry around the $^{31}$P that is changing with phosphate protonation. Thus, phosphate speciation determines CSA parameters and SSB manifold line shapes.

The $^{31}$P ssNMR MAS spectra fitting allows the determination of the characteristic CSA parameters ($\Delta_{CSA}$ and $\eta_{CSA}$) of each phosphate species. It was found that ssNMR spectra recorded at pH 12, 10, 4, and 0.5 can predominantly be fitted using a single site corresponding to the following phosphate species: $PO_4^{3-}$, $HPO_4^{2-}$, $H_2PO_4^-$ and $H_3PO_4$, respectively (Figure 5b). As a result, $^{31}$P CSA parameters ($\Delta_{CSA}$ and $\eta_{CSA}$) of these individual phosphate species are extracted (Figure 5b and Table SI1 and SI2), and their extracted CSA parameters are in good agreement with their expected symmetries$^{51}$ and with experimental values found in sodium or potassium salts.$^{52}$ In particular, $PO_4^{3-}$ exhibits a very small $\Delta_{CSA}$ magnitude compared to other species, which is in agreement with the high symmetric tetrahedral geometry expected around the phosphorus.$^{51,53}$ We also noticed the small $\eta_{CSA}$ values for $HPO_4^{2-}$ and $H_3PO_4$ (~0.3-0.4), which are in agreement with their expected $C_3$ symmetry.$^{51}$ A characteristic change in sign for $\Delta_{CSA}$ from $H_2PO_4^-$ (negative) to $HPO_4^{2-}$ (positive) was also observed and is in accordance with the literature.$^{51,52,54}$
<figure><img src="image_6.png" /><figcaption>**Figure 5.** (a) $^{31}$P CP MAS ssNMR spectra ($V_{MAS}$ = 2 kHz; $t_{CP}$ = 1.5 ms; T = -120°C) of cryo-fixed phosphate solutions (400 mM) at different pH. The orange dashed line is used as a guide to show the position of the isotropic resonances. (b) Decomposition of the $^{31}$P CP MAS ssNMR spectra with CSA components. Spectra at pH 0.5, 4, 10 and 12 can predominantly be fitted using one single site corresponding to the following phosphate species: $PO_4^{3-}$, $HPO_4^{2-}$, $H_2PO_4^-$ and $H_3PO_4$, respectively. Other spectra were fitted by the combination of two individual components. Extracted CSA parameters are listed in Table SI1 and SI2.</figcaption></figure>

The $^{31}$P ssNMR spectra recorded at other pH values correspond to an equilibrium between two phosphate species. Thus, they were fitted by a combination of two individual CSA components previously determined. The resulting fits can be seen in Figure 5b, and the extracted CSA values in Table S1 and SI2.

Importantly, it has been previously noticed for carbonate solutions that low-temperature CP MAS experiments of vitrified solutions can be considered semi-quantitative since, surprisingly, the relative intensities of $CO_3^{2-}$ and $HCO_3^-$ resonances measured by CP MAS were similar, whatever the contact time, to those measured through direct polarization.38 Thus, $CO_3^{2-}$ and $HCO_3^-$ proportions in the vitrified matrix can be evaluated, and a speciation diagram at low temperatures can be drawn from the CP MAS experiment. We checked that assumption for the present study, and a similar behavior
was observed for phosphate cryo-fixed solutions. At pH 7.4, ³¹P direct polarization and CP MAS spectra exhibit identical signals (position and LW, for contact times ranging from 0.5 to 10 ms) despite the presence of two species having two different protonation states (HPO₄²⁻ and H₂PO₄⁻) (Figure SI8). The relative intensities of HPO₄²⁻ and H₂PO₄⁻ at pH 7.4 are identical, regardless of the acquisition sequence used (CP MAS or direct polarization). Thus, ³¹P CP MAS experiments can be used to quantify the proportions of phosphate species in the cryo-fixed solutions.

Phosphate species proportions were extracted from the decompositions of the ³¹P CP MAS spectra (Table S2) and reported as a function of the pH. Hence, phosphate equilibria (i.e., the apparent pKa) at -120 °C in the vitrified matrix can be determined after simulation of the evolution of the speciation with the pH and compared to theoretical pKa in solution (Figure 6).

The apparent pKa of a solution can depend upon many factors, including temperature, pressure, and ionic strength.⁵⁵ To better understand the temperature effect on phosphate equilibria in the solutions, phosphate concentrations, and pKa values were fitted at 25°C and 0°C to estimate the effect of temperature lowering (Figure 6a). The following modified Van't Hoff equation was employed,⁵⁵ which accounts for the temperature dependence of the equilibrium constant by incorporating enthalpy and heat capacity changes:

$$lnK_2 = \frac{\Delta H_1 - T_1\Delta C_p}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) + \frac{\Delta C_p}{R} \ln\left(\frac{T_2}{T_1}\right) + lnK_1$$

where K₁ and K₂ represent the equilibrium constants at reference and target temperatures (T₁ and T₂), respectively, ΔH₁ is the enthalpy change at the reference temperature (-8.0, 3.6 and 16.0 kJ/mol for pKa₁, pKa₂ and pKa₃, respectively), ΔCₚ denotes the change in heat capacity at constant pressure (-141, -230 and -242 J/(mol·K) for pKa₁, pKa₂ and pKa₃, respectively),⁵⁶ and R is the universal gas constant (8.314 J/(mol·K). The calculations show that lowering the temperature impacts theoretical pKa in solution; at T = 0°C, pKa₂ and pKa₃ values slightly increase while pKa₁ slightly decreases.

Concerning experimental equilibria determined from ³¹P ssNMR, only the first equilibrium (H₃PO₄ ⇌ H₂PO₄⁻; pKa₁) is consistent with this trend as it is lowered to pH ~1.7. In opposition, the second (H₂PO₄⁻ ⇌ HPO₄²⁻; pKa₂) and the third (HPO₄²⁻ ⇌ PO₄³⁻; pKa₃) equilibria are found lower than expected at pH ~6.7 and ~11.4, respectively. However, equilibria determination at -120 °C must be taken with caution, as ³¹P spinning side band manifold decomposition is particularly sensitive and can induce important variations in the determined phosphate proportions (±10%). The use of higher static magnetic fields would help in this determination as the CSA contribution in the NMR spectra increases with the magnetic field. Moreover, we have to keep in mind that in our samples, phosphates are not dispersed in liquid solutions but in vitrified solids and that chemical equilibria may be strongly influenced. In any case, it is certain that the final temperature alone does not control the equilibria in the cryofixed solutions, since at -120°C it would induce the following pKa: pKa₁ = 2.89, pKa₂ = 11.18 and pKa₃ = 18.55. For comparison, CO₃²⁻/HCO₃⁻ equilibria in carbonate vitrified solutions determined by ¹³C DNP MAS at 100K are much more impacted as the apparent pKa₂ (HCO₃⁻ ⇌ CO₃²⁻) is shifted down from 10.3 to ~6.³⁸
<figure><img src="image_7.png" /><figcaption>**Figure 6.** (a) Theoretical phosphate speciation diagram at 25 and 0 °C with their corresponding pKa values. (b) Speciation fractions were obtained from fitted ³¹P ssNMR data at -120°C and fitted using the Henderson-Hasselbach equation.⁵⁷ The resulting apparent pKa values are indicated.</figcaption></figure>

Now that the behavior of phosphate species in cryo-fixed solutions is determined, CaP mineralization can be studied with our sample preparation approach combined with low-temperature ³¹P ssNMR. In particular, we investigated a case of fast CaP formation by mixing calcium and phosphate-containing solutions with equal initial concentrations of 50 mM at pH 7.4. The reaction proceeds rapidly, as precipitation of a white solid is seen immediately with the naked eye upon mixing. To investigate the very first steps of the reaction and their evolution, cryofixation of the mineralizing solution has been prepared after 20, 100 ms, and 1 and 60 seconds and studied by low-temperature ³¹P CP MAS ssNMR (Figure 7).

The free phosphate solution displays a unique broad resonance centered at 2.7 ppm, which is in agreement with the reference Pi solution at the same pH (Figure 7a). By increasing the reaction mixing time from 20 ms to 60 sec, we observe the progressive broadening of the resonance on the right side (~ +2 ppm) associated with a shift of the maximum toward lower frequency (~ -1.8 ppm) (Figure 7b). Such signal evolution is characteristic of the interaction of phosphate with Ca²⁺ as supported by calcium potentiometry and pH monitoring where a dramatic drop in [Ca²⁺]ᵣfree is evidenced (down to ~7.5 mM) associated to a decrease in pH (down to ~6.5) within the first seconds of reaction (Figure SI9). This process leads to the formation of amorphous calcium phosphate (ACP) as evidenced by cryo-TEM
observation showing the formation of solid spheres after a few seconds of reaction (~30-40 s) (Figure S10).

Slow MAS spinning ($v_{MAS}$ = 2 kHz) was also used to record the SSB manifold after 1 minute of reaction (Figure 7c). Fitting the resulting CSA pattern with contributions from only $HPO_4^{2-}$ and $H_2PO_4^-$ (where a ~50/50 proportion is expected at pH 6.6 – see Figure 6b) cannot account for the full SSB signal intensity justifying the presence of an extra contribution in agreement with the presence of ACP. To further analyze these samples, advanced experiments such as two-dimensional (2D) $^1H-^31P$ or $^{31P-}^{31P}$ correlation NMR could provide valuable insights into the assignment of species present in the mixtures.

After 2 hours of reaction (Figure 7a), $^{31}$P solid-state NMR reveals the crystallization of the system as a sharp resonance corresponding to brushite $CaH_2PO_4 \cdot 2H_2O$ is observed. Note that the temperature strongly influences the $^{31}$P chemical shift of brushite as it decreases from 1.6 to 0.8 ppm when the temperature is lowered from ambient to -120°C (Figure S11). Hence, preliminary recording of solid-state NMR spectra of reference crystalline phases at low temperatures is essential to identify the final crystalline phases unambiguously in mineralizing solutions. Brushite $^{31}$P resonance is observed together with a broad resonance centered around 2 ppm (in agreement with the final pH ~5.9), indicating the presence of remaining free phosphate ions in the solution. The crystallization is supported by calcium potential monitoring where a second transition is observed circa 300s, corresponding to the transformation of ACP into brushite, leading to a final $[Ca^{2+}]_{free}$ around 5 mM (Figure S19).

Importantly, the freeze quench device described above allows us to capture the transition from free phosphate to crystalline phase through condensed amorphous intermediates in solution within seconds with solid-state NMR.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** $^{31}$P CP MAS ssNMR spectra of cryo-fixed CaP mineralizing solutions recorded after various reaction times. (a) From bottom to top: initial 25 mM phosphate (Pi) solution (black); 50 mM Pi and 50 mM Ca mixed solutions aged for 20 ms (blue), 100 ms (green), 1 s (red) and 60 s (purple) before cryofixation; reaction mixture after 2 hours of reaction (frozen in liquid nitrogen ($LN_2$)). Spectra were recorded using $v_{MAS}$ = 5 kHz, $t_{CP}$ = 1.5 ms and T = -120°C. (b) Overlay of the spectra highlighted in (a) shows the signal width increase on the right side of the $^{31}$P resonance. (c) $^{31}$P CP MAS ssNMR recorded at 2 kHz MAS for the 1-minute aged CaP reacting solution together with spectral decomposition using CSA parameters from Table S1.</figcaption></figure>
## Conclusion

In this communication, we introduced an innovative method to study out-of-equilibrium aqueous solutions using time-resolved low-temperature solid-state NMR. The method relies on the cryofixation of reacting solutions in order to trap and preserve transient reaction intermediates. The aging time is controlled on a tens of milliseconds time scale by a stopped-flow device connected to a freeze-quench accessory, where the cryofixation is ensured by spraying the solutions in cold liquid isopentane at -145°C. Then, the filling of NMR rotors is achieved with specifically designed packing tools ensuring the control of the low temperature during the whole process.

We demonstrate that phosphate solutions can be efficiently vitrified using this protocol and studied by low-temperature ³¹P solid-state NMR. We show that the different phosphate species existing in aqueous solution (PO₄³⁻, HPO₄²⁻, H₂PO₄⁻ and H₃PO₄) can be distinguished depending on the pH. If the isotropic components of these species severely overlap at B₀ = 7T and T = -120°C, it is possible to differentiate them based on their CSA parameters, which are characteristic of their protonation degree. The ³¹P CP MAS experiments allow the determination of the proportion of each species as a function of pH and, consequently, enable to plot the speciation diagram in our conditions of analysis. We observe that the "apparent pKa," i.e., the pH values for each chemical equilibrium, is slightly influenced by the low temperature and possibly the preparation conditions.

We also demonstrate that our method can be applied to fast precipitating reactions, particularly calcium phosphate crystallizations, where the very first stages of ACP nucleation are revealed by low-temperature ³¹P ssNMR after 20 ms of reaction.

Importantly, the described methodology is a first step toward studying fast out-of-equilibrium solutions using solid-state NMR. This approach can provide access to transient intermediates of a short lifespan. Indeed, proper freeze quenching prevents transient species from transformation and preserves their native environment, such as hydration, pH, or ionic strength. Such a native state is inaccessible with most *ex-situ* or isolation methods that can induce deleterious structural modifications. We demonstrated applications for biomineralization-relevant reactions, but in principle, any aqueous reaction can be studied.

Finally, introducing radicals in the solution should enable us to combine our methodology with DNP without using glycerol, which accounts for 60 vol% of DNP juice and can modify the reaction kinetics and the nature of the reaction intermediates or the reaction itself. Experiments are currently in progress.

## Experimental

*Solutions.* Pure phosphate [Pi] solutions (400 mM) were prepared by dissolving K₂HPO₄ (≥98 %, Sigma-Aldrich) in deionized water, and pH was adjusted using aqueous HCl or NaOH.

For CaP mineralizing solutions, Pi solution (50 mM) was prepared by dissolving K₂HPO₄ (≥98 %, Sigma-Aldrich) in a buffer of 50 mM Tris (≥99 %, Sigma-Aldrich) and 150 mM KCl (≥99 %, Sigma-Aldrich) at pH 7.4. Calcium [Ca] containing solution (50 mM) was prepared analogously using CaCl₂ 2H₂O (≥99 %, Sigma-Aldrich).

*Time-resolved cryofixation.* The fast-freezing samples were prepared using a stopped-flow apparatus (BioLogic SFM-4000) connected with a freeze-quench accessory. Solutions were loaded into separate
syringes (10 mL each) in the unit and kept at room temperature for all experiments. BIO-KINE software (BioLogic) was used to program the mixing, aging, and spraying sequence. For NMR experiments, per each spray, calcium and phosphate-containing solutions were mixed in equal quantities with a total volume of 100 µL and aged for 10 ms, 100 ms, 1s, and 1 min, then directly sprayed in cold isopentane (at -140 °C) with a flow rate of 10 mL/ s. Frozen solid was then packed in a 4 mm zirconia rotor using a home-built packing tools (Figure SI1-SI5), maintaining the temperature < -140 °C. Isopentane was mixed with liquid nitrogen to reach -140 °C in a nitrogen dewar. The solution aged for 2h was frozen by immersion in liquid N₂.

Samples for optical imaging were prepared with deionized water mixed with small quantities of dye (2,6-Dichlorophenolindophenol, 1mg/ 50mL) for better visual contrast and to help frozen droplets manipulation in the colorless cold isopentane bath.

*Solid-state NMR.* Solid-state NMR experiments were recorded on an AVANCE 300 Bruker spectrometer operating at *v*(¹H) = 300.13 MHz and *v*(³¹P) = 121.50 MHz, using a 4 mm double resonance wide variable-temperature probe head. Samples were packed in 4 mm zirconia rotors and sealed with zirconia caps. Magic angle spinning (MAS) frequency was manually adjusted and maintained at 2 or 5 kHz (± 20Hz). Sample spinning and cooling were achieved by passing nitrogen gas through an exchange dewar filled with liquid nitrogen. The ¹H-³¹P Cross-Polarization (CP) MAS NMR experiments were recorded using an excitation pulse t90⁰(¹H) = 3.2 µs followed by a contact time tCP = 1.5 ms for all experiments presented in the main text (except variable tCP experiments). Each spectrum was recorded with 2560 scans and a recycle delay of 6 s. The ³¹P direct polarization experiment was performed using a 30⁰ excitation pulse with a recycle delay of 120 s to ensure full relaxation with a total of 16 or 32 scans acquired. During the acquisition of all NMR experiments, SPINAL-64 decoupling was used (*v*RF(¹H) = 70 kHz).⁵⁸ All spectra were fitted using DMFit software, employing 'CSA mas' fitting model.⁵⁹ The chemical shifts were referenced to commercial hydroxyapatite at room temperature with an δiso centered at 2.7 ppm.

*Optical microscopy.* Optical images of sprayed particles were acquired using a homemade setup composed of a CDD camera (Applied Vision Technologies) interfaced with a long-distance optical system (Optem Fusion). The light source was a VSI 220 from Optem. The images were acquired with Vimba software. The system was cooled down by using a Linkam chamber, as reported in the reference.⁴⁷

Histograms for particle size determination were obtained using Origin(Pro), Version 2018.⁶⁰ Speciation plots to fit theoretical and experimental pKa values were done using MATLAB Version R2022b.⁶¹

## Acknowledgment

TA acknowledges Nadine Nassif for support and fruitful scientific discussions. Marco Faustini is acknowledged for providing access to the home-build optical microscope. The project received financial support from the French National Agency of Research ANR for funding (grant number ANR-21-CE29-0025-01).

## Data availability

Data for this article, including the raw data, are available upon request to the authors.

## Conflicts of interest
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