Iurii Chubak

Leeor Alon

Emilia V Silletta

Guillaume Madelin

Alexej Jerschow
email: alexej.jerschow@nyu.edu

Benjamin Rotenberg
email: benjamin.rotenberg@sorbonne-universite.fr

Quadrupolar 23 Na + NMR Relaxation as a Probe of Ultrafast Collective Dynamics in Aqueous Electrolyte Solutions

Nuclear magnetic resonance relaxometry represents a powerful tool for extracting dynamic information. Yet, obtaining links to molecular motion is challenging for many ions that relax through the quadrupolar mechanism, which is mediated by electric field gradient fluctuations and lacks a detailed microscopic description. For sodium ions in dilute and concentrated aqueous electrolytes, we show that combining ab initio calculations and classical molecular dynamics to account for electron cloud effects and long-time sampling of fluctuations enables reaching a good agreement between the computed and experimental relaxation rates over a broad range of temperatures. While the commonly-assumed rotational models fail to rationalize the measured effects, we find the subpicosecond timescale of electric field gradient relaxation to be comparable to that of solution structural rearrangements. This work thus shows that quadrupolar sodium relaxometry can sense the fast collective dynamics around the solute, thereby elevating its role in exploring electrolyte or bioelectrolytic systems.

INTRODUCTION

The proper characterization and modeling of the solvation structure of alkaline cations (e.g., Li + , Na + , and K + ) in aqueous solution is of considerable interest both in physiological systems [START_REF] Ohtaki | Structure and dynamics of hydrated ions[END_REF][START_REF] Inglese | Brain tissue sodium concentration in multiple sclerosis: A sodium imaging study at 3 Tesla[END_REF][START_REF] Ouwerkerk | Tissue sodium concentration in human brain tumors as measured with 23 Na MR imaging[END_REF][START_REF] Madelin | Sodium MRI: Methods and applications[END_REF][START_REF] Guermazi | Compositional MRI techniques for evaluation of cartilage degeneration in osteoarthritis[END_REF][START_REF] Silletta | Multinuclear absolute magnetic resonance thermometry[END_REF][START_REF] Hu | X-nuclei imaging: Current state, technical challenges, and future directions[END_REF][START_REF] Madelin | X-Nuclei Magnetic Resonance Imaging[END_REF] and electrolytes used for electrochemical devices [START_REF] Slater | Sodiumion batteries[END_REF][START_REF] Yabuuchi | Research development on sodium-ion batteries[END_REF][START_REF] Hwang | Sodium-ion batteries: Present and future[END_REF][START_REF] Chandrashekar | 7 Li MRI of Li batteries reveals location of microstructural lithium[END_REF][START_REF] Pecher | Materials' methods: NMR in battery research[END_REF]. Nuclear magnetic resonance (NMR) spectroscopy provides an excellent source of dynamic and structural information for a number of nuclear species, including 23 Na with a nuclear spin 3/2 and close to 100% natural abundance that produces the second strongest NMR signal after protons in biological tissues [START_REF] Wertz | Nuclear spin resonance of aqueous sodium ion[END_REF]. The NMR sensitivity of sodium is 9.2% of that of proton, while its typical concentration can be three, or more, orders of magnitude lower. Thus, in biological systems the sodium signal-to-noise ratio is 3,000-12,000 times lower than that of 1 H [START_REF] Madelin | Sodium MRI: Methods and applications[END_REF]. Nonetheless, the longitudinal relaxation time T 1 of 23 Na (typically 40 ms and below) is short compared to that of 1 H (on the order of seconds) [START_REF] Madelin | Sodium MRI: Methods and applications[END_REF], allowing for rapid averaging of the signals such that quantitative analysis is made possible within reasonable time scales [START_REF] Headley | Nuclear magnetic resonance relaxation of 23 Na in porous media containing NaCl solution[END_REF].

The shortness of the 23 Na NMR relaxation times is due to a fluctuating quadrupole interaction related to the changes in the solvation shell and the proximity of other ions [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. The relaxation rate is determined from a combination of the strength of the electric field gradient (EFG) at the nucleus quantified by means of the quadrupolar coupling constant (QCC) C Q and the characteristic correlation time τ c with which the memory of fluctuations is lost. While the knowledge of C Q and τ c can potentially provide information about the hydration sphere structure [START_REF] Versmold | Interaction induced magnetic relaxation of quadrupolar ionic nuclei in electrolyte solutions[END_REF][START_REF] Aidas | A quantum mechanics/molecular dynamics study of electric field gradient fluctuations in the liquid phase. the case of Na + in aqueous solution[END_REF] and useful dynamic properties (e.g., diffusion coefficients, viscosity, or conductivity [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF]), respectively, their unambiguous determination from the experimentally-measured rates in solution has remained essentially impossible [START_REF] Woessner | NMR relaxation of spin-3/2 nuclei: Effects of structure, order, and dynamics in aqueous heterogeneous systems[END_REF].

Different models have been suggested to rationalize quadrupolar relaxation using dielectric description [START_REF] Hynes | A continuum theory for quadrupole relaxation of ions in solution[END_REF][START_REF] Perng | A dielectric theory of spin-lattice relaxation for nuclei with electric quadrupole moments[END_REF], mode-coupling analysis [START_REF] Bosse | Mode-coupling theory of field-gradient correlation functions: The quadrupolar relaxation rate in liquids[END_REF], definite molecular processes (e.g., water reorientation [START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part I: Limiting values for infinite dilution[END_REF][START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part II: Relaxation at finite ion concentrations[END_REF][START_REF] Hertz | Nuclear magnetic relaxation of alkali metal ions in aqueous solutions[END_REF] and collective symmetry-breaking fluctuations [START_REF] Carof | Collective water dynamics in the first solvation shell drive the NMR relaxation of aqueous quadrupolar cations[END_REF]), or Brownian rotational diffusion [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF]. Ab initio [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Schmidt | Beyond isotropic tumbling models: Nuclear spin relaxation in liquids from first principles[END_REF][START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF][START_REF] Philips | Quadrupolar 14 N NMR relaxation from force-field and ab initio molecular dynamics in different solvents[END_REF][START_REF] Philips | Quadrupolar NMR relaxation of aqueous 127 I -, 131 Xe + , and 133 Cs + : A firstprinciples approach from dynamics to properties[END_REF] and classical [START_REF] Carof | Collective water dynamics in the first solvation shell drive the NMR relaxation of aqueous quadrupolar cations[END_REF][START_REF] Engström | A molecular approach to quadrupole relaxation. Monte Carlo simulations of dilute Li + , Na + , and Claqueous solutions[END_REF][START_REF] Engström | Molecular dynamic simulation of quadrupole relaxation of atomic ions in aqueous solution[END_REF][START_REF] Linse | in heterogeneous aqueous systems[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF][START_REF] Carof | Accurate quadrupolar NMR relaxation rates of aqueous cations from classical molecular dynamics[END_REF][START_REF] Carof | On the microscopic fluctuations driving the NMR relaxation of quadrupolar ions in water[END_REF][START_REF] Mohammadi | Nuclear magnetic resonance spin-lattice relaxation of lithium ions in aqueous solution by NMR and molecular dynamics[END_REF][START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF] molecular dynamics (MD) simulations have become indispensable tools in assessing predictions of such theories, and invalidated the isotropic monoexponential character of the quadrupolar relaxation that is often assumed under a continuous solvent description. A pronounced role of intermolecular cross-correlations on the relaxation was emphasized [START_REF] Versmold | Interaction induced magnetic relaxation of quadrupolar ionic nuclei in electrolyte solutions[END_REF][START_REF] Carof | Collective water dynamics in the first solvation shell drive the NMR relaxation of aqueous quadrupolar cations[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF]. Contrary to classical MD that relies on approximations [START_REF] Carof | Accurate quadrupolar NMR relaxation rates of aqueous cations from classical molecular dynamics[END_REF][START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF][START_REF] Sternheimer | On nuclear quadrupole moments[END_REF][START_REF] Sternheimer | Shielding and antishielding effects for various ions and atomic systems[END_REF], ab initio methods provide the best accuracy of the computed EFG at the ion position that incorporates the electron cloud contribution [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF][START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Autschbach | Analysis of electric field gradient tensors at quadrupolar nuclei in common structural motifs[END_REF][START_REF] Charpentier | The PAW/GIPAW approach for computing NMR parameters: A new dimension added to NMR study of solids[END_REF]. However, the associated high computational cost often impedes the long-time sampling of EFG fluctuations [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Schmidt | Beyond isotropic tumbling models: Nuclear spin relaxation in liquids from first principles[END_REF] and, hence, the ac-curacy of the correlation time estimates, even for aqueous ions at infinite dilution [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF]. Therefore, the fully first-principles approach may not be appropriate in concentrated electrolytes, for which the quadrupolar NMR relaxation slows down with increasing salt concentration [26-28, 48, 49].

Here we show that C Q parametrized at the ab initio level combined with τ c evaluated classically allows reaching good agreement between the calculated and experimentally-obtained quadrupolar rates of 23 Na + in electrolyte solutions at multiple salt concentrations and temperatures. We find that the main effect of increased relaxivity is due to a lengthening of the correlation times, rather than a change of the average quadrupolar coupling constant. Contrary to the commonly-assumed picture, we conclude that rotational models based on the water dipole reorientation or Stokes-Einstein-Debye relation significantly overestimate the EFG correlation times. Rather, our results indicate that the EFG relaxation occurs over a time scale comparable to that of solution structural rearrangements, suggesting that the shorttime collective dynamics of the liquid primarily drives the quadrupolar relaxation at the sodium ion position.

RESULTS

Electron cloud contribution to electric field gradients

We perform density functional theory (DFT) calculations to determine quantum EFGs at the Na + position in aqueous sodium chloride (NaCl) solutions at varying salt concentrations c = 1-5 molal (denoted with mol•kg -1 or m) at T = 25 °C (see Methods for details). The projectoraugmented wave (PAW) method [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Charpentier | The PAW/GIPAW approach for computing NMR parameters: A new dimension added to NMR study of solids[END_REF][START_REF] Petrilli | Electric-field-gradient calculations using the projector augmented wave method[END_REF] is used to reconstruct the all-electron charge density at the nucleus. A configuration of a NaCl solution at 5 mol•kg -1 with converged charge densities is highlighted in Fig. 1a.

In classical MD, the electron cloud contribution to the EFG can be incorporated by means of the Sternheimer approximation [START_REF] Sternheimer | On nuclear quadrupole moments[END_REF][START_REF] Sternheimer | Shielding and antishielding effects for various ions and atomic systems[END_REF], in which the full EFG at the nucleus V is proportional to the EFG created by the external charge distribution

V ext : V ≃ (1 + γ)V ext .
Here, the electron cloud polarization is included via the simple rescaling factor 1 + γ, with the Sternheimer (anti)shielding factor γ being typically large γ ≫ 1 [START_REF] Sternheimer | Shielding and antishielding effects for various ions and atomic systems[END_REF]. To validate the Sternheimer approximation for present systems, we have compared ab initio, V AI , against classical, V ext , EFGs at the Na + position, as determined on the same set of classically-generated solution configurations (see Methods). Consistently with aqueous ions at infinite dilution [START_REF] Carof | Accurate quadrupolar NMR relaxation rates of aqueous cations from classical molecular dynamics[END_REF][START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF], we find a strong correlation between V AI and V ext for all c = 1-5 mol•kg -1 , as seen in Fig. 1b γ eff feature a small increase with c (less than 5% compared to the infinite dilution value γ eff,0 = 10.54 ± 0.11 [START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF]) associated with the modifications of the ion's solvation sphere (see Supplementary Note II B and II C).

Despite the small changes of γ eff with increasing c, the Sternheimer approximation for the EFG variance, (1+γ eff ) 2 ⟨V 2 ext ⟩, that is necessary for the NMR rate computation (Eq. (1) in Methods) underestimates the ab initio value ⟨V 2 AI ⟩ by more than 20% (highlighted in Fig. 1d using γ eff,0 ). This again underlines the deficiencies of the Sternheimer approximation [START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF] that does not take into account non-electrostatic electron cloud polarization effects, such as short-range repulsion [START_REF] Aidas | A quantum mechanics/molecular dynamics study of electric field gradient fluctuations in the liquid phase. the case of Na + in aqueous solution[END_REF][START_REF] Calandra | Nuclear quadrupole coupling of 17 O and 33 S in ionic solids: Invalidation of the Sternheimer model by short-range corrections[END_REF][START_REF] Gambuzzi | Computational interpretation of 23 Na MQMAS NMR spectra: A comprehensive investigation of the Na environment in silicate glasses[END_REF]. To improve upon the variance predictions, we formally define the Sternheimer factor γ ′ eff as eff,0 = 12.09 ± 0.14 at infinite dilution (Fig. 1c). The EFG variance prediction (1+γ ′ eff,0 ) 2 ⟨V 2 ext ⟩ using γ ′ eff,0 at infinite dilution is within 5% accuracy of ⟨V 2 AI ⟩ within the considered concentration range, a much better estimate in comparison to the simple Sternheimer approximation (Fig. 1d). While not capturing all condensed-phase effects that arise with increasing c, the estimate (1 + γ ′ eff,0 ) 2 ⟨V 2 ext ⟩ provides a fair accuracy, reproduces the trend of ⟨V 2 AI ⟩ to decrease with the salt concentration (see Fig. 1d), and permits to avoid computationally expensive DFT calculations at multiple system state points of interest. As discussed below, in combination with the EFG relaxation dynamics captured at the classical level, this approach provides a good description of the quadrupolar 23 Na + NMR rates in aqueous solutions.

(1 + γ ′ eff ) 2 = ⟨V 2 AI ⟩/⟨V

Relaxation of electric field gradient fluctuations

We perform classical MD simulations employing the Madrid-2019 force field (FF) [START_REF] Zeron | A force field of Li + , Na + , K + , Mg 2+ , Ca 2+ , Cl -, and SO 2-4 in aqueous solution based on the TIP4P/2005 water model and scaled charges for the ions[END_REF][START_REF] Blazquez | The Madrid-2019 force field for electrolytes in water using TIP4P/2005 and scaled charges: Extension to the ions F -, Br -, I -, Rb + , and Cs +[END_REF] to facilitate the long-time sampling of EFG fluctuations and to investigate the mechanisms behind the concentration and temperature behavior of the quadrupolar 23 Na + NMR rate in aqueous sodium chloride, bromide (NaBr), and fluoride (NaF) solutions (see Methods for simulation details). Two facts give confidence in this approach: (i) a very strong correlation between the full and classical (external) EFGs (Fig. 1b), indicating that the dynamics of the former should be largely determined by that of the latter; (ii) while classical MD with rigid water molecules do not quantitatively reproduce the librational or hydrogenbond stretching water dynamics that occur at very short times below ∼50 fs [START_REF] Carlson | Exploring the absorption spectrum of simulated water from MHz to infrared[END_REF], it is expected that these high frequency motions do not significantly affect the dominating long-time (∼1 ps) EFG relaxation mode (e.g., see Ref. [START_REF] Carof | On the microscopic fluctuations driving the NMR relaxation of quadrupolar ions in water[END_REF] and below).

Increasing salt concentration c or decreasing temperature T causes a profound slow-down of the EFG fluctuations at the ion position (Fig. 2). Due to a qualitative similarity of the EFG relaxation in the solutions considered, here we will focus on the case of NaCl; see Supplementary Information (SI) for NaBr and NaF. Fig. 2a shows the ACFs of the classical EFG at the Na + position, C EFG (t) ≡ ⟨V ext (0):V ext (t)⟩, as a function of c at T = 25 °C (see Supplementary Fig. S8 for other T ). Similarly to a single Na + in water [START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF][START_REF] Carof | Accurate quadrupolar NMR relaxation rates of aqueous cations from classical molecular dynamics[END_REF][START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF], C EFG (t) relaxes in two steps: (i) a rapid initial decay happening at t ≲ 0.2 ps that corresponds to ≈70% of the EFG decorrelation. This is in good agreement with the ab initio MD results for Na + at infinite dilution [START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF], highlighting the validity of the classical approach; (ii) a much slower secondary decay occurring in the picosecond regime. As seen in Fig. 2a, the increase in c leaves the initial fast de- cay practically unchanged, while causing a pronounced slow-down of the second decay mode. The latter is highlighted in the inset of Fig. 2a showing the EFG ACFs for t < 1 ps for different c at T = 25 °C (see also Supplementary Fig. S9). A qualitatively similar trend is found with decreasing temperature, as we show in Fig. 2b at c = 4 m and in Supplementary Fig. S8 for other c.

The form of the EFG ACF decay in Fig. 2 suggests a collective pathway behind the relaxation. After the initial fast decay that can be described with an exponential ∼e -t/τ f with τ f ≈ 62 fs, we find a development of a much slower relaxation mode that profoundly depends on c and T . Compared to earlier results [START_REF] Carof | On the microscopic fluctuations driving the NMR relaxation of quadrupolar ions in water[END_REF], our long-time sampling reveals that the slow part of the EFG ACF is not exponential, as clearly seen from the behavior of C EFG (t) on a semi-logarithmic scale in Figs. 2a-b and as we show with explicit fits in Supplementary Note II E. Except at very long times, we find that the slow decay can be modeled either with a two-exponential (Supplementary Fig. S10) or a stretched exponential fit ∼e -(t/τs) β with β = 0.67 ± 0.05 (Supplementary Fig. S11 andS12), which suggests a broad distribution of contributing relaxation modes (Supplementary Fig. S13). Although observed over a limited time range (up to a decade), we find that the long-time tail of the EFG ACFs is consistent with a power law ∼t -5/2 , as shown with C EFG (t) on a log-log scale for c = 4 m in Fig. 2c. Such a hydrodynamic tail was predicted by a mode-coupling theory of Bosse et al. for the EFG ACF in molten salts [START_REF] Bosse | Mode-coupling theory of field-gradient correlation functions: The quadrupolar relaxation rate in liquids[END_REF]. It originates from the coupling between the ion motion and shear excitations in the liquid, a mechanism causing the well-known ∼t -3/2 tail of the velocity ACF [START_REF] Alder | Decay of the velocity autocorrelation function[END_REF]. While sampling of the EFG fluctuations at even longer time scales is necessary to decisively confirm to presence of ∼t -5/2 regime, our results for Na + in Fig. 2c suggest that its relative contribution may be marginal because the apparent onset of the algebraic decay occurs at times when the ACF has decayed considerably.

Quadrupolar relaxation rates

The combination of EFG fluctuations captured at the classical level and a consistent inclusion of the electron cloud contribution to the EFG enables reaching a good quantitative agreement between the calculated and experimentally-measured quadrupolar NMR rate for 23 Na + in aqueous NaCl, as we compare in Fig. 3 with filled and open symbols, respectively. As seen in Eq. [START_REF] Ohtaki | Structure and dynamics of hydrated ions[END_REF] in Methods, the quadrupolar rate is proportional to the product of the effective correlation time of EFG fluctuations,

τ c = C -1 EFG (0) ∞ 0 dt C EFG (t)
, and the EFG variance, which we approximate as

⟨V 2 ⟩ = (1+γ ′ eff,0 ) 2 ⟨V 2 ext ⟩ with γ ′
eff,0 = 12.09 and ⟨V 2 ext ⟩ = C EFG (0). The integration of C EFG (t) over tens of picoseconds is necessary to obtain well-converged correlation times τ c (Supplementary Fig. S14), notably at high salt concentrations and low temperatures (Fig. 2). Finally, based on estimates in Supplementary Note I C, here we assume that the 23 Na + rate 1/T 1 is almost entirely determined by the quadrupolar interaction, whereas possible contributions from the dipole-dipole coupling between the spin of the 23 Na nucleus and that of 1 H, 23 Na, 35 Cl are marginal.

The NMR rate 1/T 1 grows with increasing salt concentration c and with reducing temperature T (Fig. 3), suggesting that the slowing down of EFG fluctuations determines the rate behavior (Fig. 2). Experimentally, 1/T 1 rises by about 50% within the considered range of concentrations c = 0.17 50 °C, in line with the previous results [START_REF] Woessner | NMR relaxation of spin-3/2 nuclei: Effects of structure, order, and dynamics in aqueous heterogeneous systems[END_REF][START_REF] Hertz | Nuclear magnetic relaxation of alkali metal ions in aqueous solutions[END_REF][START_REF] Eisenstadt | Nuclear magnetic relaxation in ionic solution. I. Relaxation of 23 Na in aqueous solutions of NaCl and NaClO4[END_REF]. At T = 30 °C, 1/T 1 increases from around 15.9 s -1 at c = 0.17 m to 25.2 s -1 at c = 5.1 m. With increasing T from 25 to 50 °C, 1/T 1 reduces by more than 25% for considered salt concentrations. In general, our computational results for 1/T 1 of 23 Na + agree well with the experimental data, especially at lower salt concentrations c ≲ 2.5 m, reproducing both the concentration and temperature behavior. For higher salt concentrations, we find that 1/T 1 grows systematically faster with increasing c as compared to the experiments, yet the relative error remains less than 15% over the considered range of conditions. The latter difference is likely caused by the shortcomings of the employed FF in capturing dynamic properties of the solution for c ≳ 2 m [START_REF] Blazquez | The Madrid-2019 force field for electrolytes in water using TIP4P/2005 and scaled charges: Extension to the ions F -, Br -, I -, Rb + , and Cs +[END_REF].

Microscopic parameters of the relaxation

We find that the slowing down of EFG fluctuations at the Na + position primarily causes a marked increase in the quadrupolar NMR relaxation rate with increasing c and decreasing T (Fig. 3). In Fig. 4, we quantify the role of dynamic and static effects that are reflected in the changes of τ c and ⟨V 2 ⟩, respectively, with varying salt concentration and temperature, as obtained in MD simulations of aqueous NaCl (see Supplementary Fig. S18 for other electrolyte solutions). While τ c increases by a factor of ∼1.5-2.5 with increasing c and decreasing T within the considered range of parameters (Fig. 4a), the value of ⟨V 2 ⟩ reduces concurrently by up to 10% (Fig. 4c), indicating that the augmented correlation times are mainly responsible for the rate behavior.

For considered c and T , τ c of Na + is quite short and below 1 ps (Fig. 4a), a feature already pointed out in previous classical [START_REF] Engström | A molecular approach to quadrupole relaxation. Monte Carlo simulations of dilute Li + , Na + , and Claqueous solutions[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF][START_REF] Carof | Accurate quadrupolar NMR relaxation rates of aqueous cations from classical molecular dynamics[END_REF][START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF] and ab initio [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF] MD studies at infinite dilution. At T = 25 °C, we find that τ c increases from 0.41 ps at c ≈ 0.06 m to 0.65 ps at c = 4 m. Despite the rapid decorrelation of EFG ACFs for t ≲ 0.2 ps (Fig. 2), we find that the contribution of the slow relaxation process to τ c yields more than 85% of its overall value and also grows with increasing c and decreasing T (Fig. 4b). The dominance of the slow non-exponential decay of EFG ACFs over the τ c behavior again exemplifies the governing role of collective processes behind the quadrupolar Na + relaxation.

While the EFG variance at the Na + position is largely determined by the first solvation shell contribution (Fig. 4d), a quantitative understanding of the QCC is only achieved if we take into account point charges within a radius of r ≳ 8 Å around the central ion, approximately the length scale of pronounced ion-ion and ion-solvent correlations (Supplementary Figs. S3 andS4). We find that ⟨V 2 ⟩ is reduced in bipyramidal complexes with octahedral symmetry, coordinated by six water molecules, yet only by 10% compared to the ensemble average (Supplementary Note II D). Contrary to the full EFG variance, the contribution of the first solvation shell features an increase with c (Fig. 4d), correlated with the fact that the six-coordinated state becomes less likely with increasing the salt concentration (Supplementary Fig. S7). Our consistently calculated QCC for 23 Na + in aqueous NaCl is in the range between 19•10 6 and 20.6•10 6 rad•s -1 for considered c and T (Supplementary Fig. S15), a value approximately 3-4 times larger than previous estimates based on the assumption that the EFG primarily decorrelates by translational and reorientational water dynamics with τ c ≈ 3-7 ps [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF][START_REF] Woessner | NMR relaxation of spin-3/2 nuclei: Effects of structure, order, and dynamics in aqueous heterogeneous systems[END_REF][START_REF] Civan | NMR of Sodium-23 and Potassium-39 in Biological Systems[END_REF]. We thus conclude that the aforementioned modes of motion provide only a minor contribution to the observed relaxation.

Assessment of the relaxation models

We utilize information available in experiments and molecular simulations in Fig. 5 to shed light on the mechanisms behind the quadrupolar relaxation. First, we focus on the possibility to model the EFG correlation time τ c using the commonly-used Stokes-Einstein-Debye (SED) relation τ SED c = 4πηr 3 0 /3k B T , where η is the dynamic viscosity of the solution, r 0 is the sodium's hydrodynamic (Stokes) radius, and k B is the Boltzmann constant. Within the SED picture, the EFG relaxation at the Na + position is governed by the Brownian rotational diffusion, likely to be related with collective reorientations of ion-water solvation complexes [START_REF] Eisenstadt | Nuclear magnetic relaxation in ionic solution. I. Relaxation of 23 Na in aqueous solutions of NaCl and NaClO4[END_REF]. While the SED model assumptions are not expected to hold down to the molecular scale [START_REF] Laage | A molecular jump mechanism of water reorientation[END_REF][START_REF] Turton | Stokes-Einstein-Debye failure in molecular orientational diffusion: Exception or rule?[END_REF], we systematically explore τ SED c in relation to τ c , as it is often exploited to rationalize quadrupolar relaxation dynamics of 23 Na + [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF].

We use the translational Stokes-Einstein relation D = k B T /6πηr 0 to determine the concentration-and temperature-dependent values of the Stokes radius from the experimental Na + diffusion coefficients (Fig. 5a) and highly accurate NaCl viscosity values provided by Kestin et al. [START_REF] Kestin | Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20-150 °C and the pressure range 0.1-35 MPa[END_REF] (Fig. 5b). D and η calculated in our MD simulations (see Methods) are in good agreement with the experiments, especially for c ≲ 2 m, capturing both the concentration and temperature behavior (compare filled and open symbols in Figs. 5a-b). The viscosity η in MD was obtained via the Green-Kubo formula using the stress ACFs (Fig. 5c), as detailed in Eqs. ( 3) and (4) in Methods. In Fig. 5d, we compare τ SED c calculated from state-dependent Stokes radii r 0 (c, T ) against the effective EFG correlation time τ exp c obtained from the experimental NMR rates and the sodium's QCC from simulations (Supplementary Fig. S15) rather than those from previous estimates [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF][START_REF] Woessner | NMR relaxation of spin-3/2 nuclei: Effects of structure, order, and dynamics in aqueous heterogeneous systems[END_REF][START_REF] Civan | NMR of Sodium-23 and Potassium-39 in Biological Systems[END_REF]. r 0 assumes values between 1.5 and 2.0 Å for considered parameters (Supplementary Fig. S17) While both τ SED c and τ exp c generally lengthen with increasing c and decreasing T , τ SED c exceeds τ exp c by a factor of 8-17 (Fig. 5d). Similar results are obtained in our simulations (Supplementary Fig. S17). At T = 25 °C, τ SED c increases from around 7.5 to 8.2 ps, larger by more than one order of magnitude than τ exp c that grows from 0.44 to 0.66 ps for increasing c from 0.17 to 5.1 m. Thus, we conclude that the EFG correlation times cannot be understood on the basis of the SED relation parameterized using the translational hydrodynamic radius of sodium ions r 0 = k B T /6πηD. [START_REF] Turton | Stokes-Einstein-Debye failure in molecular orientational diffusion: Exception or rule?[END_REF]. We obtain a similar value of r eff 0 ≈ 0.68 Å from our MD simulations (Supplementary Fig. S17). Therefore, the validity of the relation τ exp c ∝ η/k B T explains the correlation between τ exp c and D -1 reported in Refs. [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF], rather than simplified assumptions of the rotational Brownian diffusion that yield much larger estimates of τ exp c (Fig. 5d). We now return to microscopic time scales of molecular motion in relation to that of EFG fluctuations. The average water dipole reorientation time τ dip = ∞ 0 dt ⟨P 1 [u(t)•u(0)]⟩ assumed to drive the quadrupolar relaxation within the Hertz model [START_REF] Versmold | Interaction induced magnetic relaxation of quadrupolar ionic nuclei in electrolyte solutions[END_REF][START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part I: Limiting values for infinite dilution[END_REF][START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part II: Relaxation at finite ion concentrations[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF] is 11-14 times larger compared to τ c , as extracted in our simulations (Fig. 5f and Supplementary Fig. S16). Above, u is a unit vector pointing along the HOH bisector of a water molecule and P 1 (x) = x is the first Legendre polynomial. This indicates that the single molecule reorientation with neglected intermolecule cross-correlations cannot explain the EFG relaxation dynamics. Yet, as seen in Fig. 5f, both τ dip and τ c increase similarly with increasing c and decreasing T , suggesting that the overall deceleration of the electrolyte dynamics, marked by an enhanced viscosity, impacts in a similar way both the motions that drive water reorientation as well as those that cause the EFG relaxation at the ion position.

This is further illustrated in

To illustrate the relationship between these effects, in our MD simulations we extract a typical time scale of solution structural relaxation τ struct using the stress tensor ACFs (Fig. 5c). While the short-time behavior of C stress (t) corresponding to elastic, vibrational contributions features little changes with varying c and T [START_REF] Hansen | Identification of structural relaxation in the dielectric response of water[END_REF], its long-time tail slows down with increasing c and decreasing T , indicating an overall deceleration of the viscous dynamics of the liquid. We find that the long-time tail can be modeled well using a stretched exponential decay, ∼e -(t/τK) β K , with β K ≈ 0.61 ± 0.04 (consistent with earlier simulations of pure water [START_REF] Hansen | Identification of structural relaxation in the dielectric response of water[END_REF] and time-resolved spectroscopy experiments [START_REF] Torre | Structural relaxation in supercooled water by time-resolved spectroscopy[END_REF]). τ K is in the range between 0.13 and 0.70 ps for considered parameters. The mean structural relaxation time τ struct = τ K β -1 K Γ(β -1 K ), defined through the integral of the stretched exponential expression, is strongly correlated and comparable to the subpicosecond EFG correlation time τ c (Fig. 5g). While the stress and EFG tensors are not directly related to each other, both quantities are inherently collective, the relaxation of their fluctuations is mainly driven by manybody correlations and features a similar stretched decay for t ≳ 0.4 ps. All these observations suggest that the fast collective dynamics of the liquid that drive its structural rearrangements are also responsible for the quadrupolar NMR relaxation.

CONCLUSIONS AND PERSPECTIVES

We have shown that the multiscale methodology combining DFT PAW calculations to parameterize the QCC and classical MD simulations to sample long-time EFG fluctuations enables an accurate description of the concentration and temperature behavior of the quadrupolar NMR relaxation rates of 23 Na + in aqueous electrolyte solutions. The resulting NMR rates are in very good agreement with the experimental data, especially at low salt concentrations, as validated in aqueous NaCl at multiple concentrations and temperatures. We find that the growth of the relaxation rate T -1 1 with increasing c and decreasing T is primarily due to the slowing down of the EFG fluctuations reflected in the augmented EFG correlation time τ c , while the concurrent changes in the QCC are rather small. We have demonstrated that the commonly-assumed rotational relaxation models based on either the water dipole reorientation [START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part I: Limiting values for infinite dilution[END_REF][START_REF] Hertz | Magnetic relaxation by quadrupole interaction of ionic nuclei in electrolyte solutions part II: Relaxation at finite ion concentrations[END_REF][START_REF] Hertz | Nuclear magnetic relaxation of alkali metal ions in aqueous solutions[END_REF] or on the Stokes-Einstein-Debye relation [START_REF] Price | Correlation of viscosity and conductance with 23 Na + NMR T1 measurement[END_REF][START_REF] Mitchell | Can sodium NMR provide more than a tracer for brine in petrophysics?[END_REF][START_REF] D'agostino | 23 Na NMR T1 relaxation measurements as a probe for diffusion and dynamics of sodium ions in salt-glycerol mixtures[END_REF] overestimate the consistently-determined τ c by at least an order of magnitude. This disagreement is understandable as these models restrict the relaxation description to one-or twobody correlations, oversimplifying the inherently collective character of the EFG at the ion position [START_REF] Versmold | Interaction induced magnetic relaxation of quadrupolar ionic nuclei in electrolyte solutions[END_REF][START_REF] Carof | Collective water dynamics in the first solvation shell drive the NMR relaxation of aqueous quadrupolar cations[END_REF][START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF]. The quantitative interpretation of the EFG correlation times in terms of such simple isotropic models should therefore be used with caution. Instead, our results indicate that the Na + EFG relaxation occurs over a subpicosecond time scale comparable to that of solution structural rearrangements τ struct . This again invalidates a continuous-solvent hydrodynamic description assuming that τ c ≫ τ struct .

Our results suggest that the quadrupolar NMR relaxometry of 23 Na + may be used as a complementary tool to analyze electrolyte dynamics in the THz domain. As the quadrupolar relaxation is largely determined by the processes in the immediate vicinity of the solute, it can provide supplementary information on the fast, collective, molecular motions in ionic solvation cages. The ability to capture the NMR rates by means of classical MD allows elucidating the quadrupolar phenomena that occur in multicomponent systems, such as concentrated aqueous solutions of multiple salts [START_REF] Dubouis | Chasing aqueous biphasic systems from simple salts by exploring the LiTFSI/LiCl/H2O phase diagram[END_REF], mixtures of salts with glycerol [START_REF] Abbott | Do group 1 metal salts form deep eutectic solvents?[END_REF], or polyelectrolytes [START_REF] Becher | From local to diffusive dynamics in polymer electrolytes: NMR studies on coupling of polymer and ion dynamics across length and time scales[END_REF], where the relaxation dynamics may be influenced by environment heterogeneity, interface formation, microphase separation, or ion binding to polyelectrolyte chains. Future work could also focus on developing mesoscopic approaches that would allow a quantitative description of quadrupolar relaxation in complex biological-type compartments, characterized either by slow-motion conditions with dynamics within intracellular and extracellular spaces in biological tissues that may include structural anisotropy with residual quadrupolar coupling and a distribution of correlation times (as, for example, in articular cartilage where sodium ions are surrounded by a collagen matrix [START_REF] Madelin | Sodium MRI: Methods and applications[END_REF]).

METHODS

Quadrupolar NMR relaxation rates

The quadrupolar mechanism dominates the relaxation of nuclei with spin I > 1/2 and is due to the coupling between their quadrupolar moment eQ with the EFG tensor V at the nucleus position [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. While the NMR relaxation of spin components is generally bi-exponential for 23 Na with I = 3/2 [START_REF] Woessner | NMR relaxation of spin-3/2 nuclei: Effects of structure, order, and dynamics in aqueous heterogeneous systems[END_REF][START_REF] Hubbard | Nonexponential nuclear magnetic relaxation by quadrupole interactions[END_REF], it is possible to define effective longitudinal and transverse quadrupolar relaxation rates, 1/T 1 and 1/T 2 , respectively, provided that the "fast motion" or "extreme narrowing" regime holds [START_REF] Badu | Quadrupolar NMR spin relaxation calculated using ab initio molecular dynamics: Group 1 and group 17 ions in aqueous solution[END_REF][START_REF] Spiess | Rotation of molecules and nuclear spin relaxation[END_REF][START_REF] Cowan | Nuclear Magnetic Resonance and Relaxation[END_REF]. In this case, the characteristic EFG correlation time τ c is much smaller than the Larmor period ω -1 0 of the nucleus, ω 0 τ c ≪ 1. The latter can be shown to be fulfilled for all cases considered in this work as the relevant correlation times for 23 Na in electrolyte solutions are below 100 ps, and the magnetic field used in the experiments is 11.7 T that corresponds to ω -1 0 ≈ 7.6 ns. As we show in more detail in the SI, the two quadrupolar relaxation rates become equal in the fast motion regime and, combined with the rotational invariance of the system, can be expressed as [START_REF] Roberts | Ionic quadrupolar relaxation in aqueous solution: Dynamics of the hydration sphere[END_REF] 1

T 1 = 2I + 3 20I 2 (2I -1) eQ ℏ 2 V 2 τ c ( 1
)
where ℏ is the reduced Planck constant, τ c is an effective correlation time of EFG fluctuations

τ c = V 2 -1 ∞ 0 dt ⟨V(0):V(t)⟩ , (2)
where ⟨V(0):V(t)⟩ = α,β ⟨V αβ (0)V αβ (t)⟩ with α, β = x, y, z and the brackets ⟨. . . ⟩ denoting an ensemble average, and V 2 ≡ ⟨V(0):V(0)⟩ is the EFG variance at the ion position. For a 23 Na nucleus with I = 3/2 and Q = 104•10 -31 m 2 [START_REF] Pyykkö | Year-2017 nuclear quadrupole moments[END_REF], the rate constant 1/T 1 can be recast as 1/T 1 = C 2 Q τ c /10 with the quadrupolar coupling constant (QCC) defined as

C 2 Q = 2 3 eQ ℏ 2 ⟨V 2 ⟩ [16, 30].
Finally, Eq. ( 1), which follows from linear response theory, allows to calculate the quadrupolar spin-lattice relaxation rate 1/T 1 from the EFG fluctuations in equilibrium MD simulations without an imposed magnetic field.

NMR experiments

Solution samples with 9 different NaCl concentrations were prepared by mixing x mg of NaCl in (y-x) mg of deionized water in a beaker, with x = 0.1, 0.2, 0.5, 0.8, 1.1, 1.4, 1.7, 2.0, 2.3 mg and y = 10 mg, to make solutions of concentrations 0.173, 0.349, 0.901, 1.488, 2.115, 2.786, 3.505, 4.278, and 5.111 mol•kg -1 to 5 mm NMR tubes (sample volume = 0.5 mL). All mass measurements were performed on a Mettler Toledo ME204E balance with a resolution of 0.1 mg. The solution at 26% weight corresponds to NaCl saturation in water at 20 °C [START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF].

NMR experiments were performed on an 11.7 T NMR Bruker Avance I spectrometer operating at 132.3 MHz for 23 Na, using a 5 mm double resonance broadband probe. The test tubes with the solutions were placed inside the spectrometer where the sample temperature could be controlled using gas flow and a temperature sensor providing a precise, stable and reliable temperature regulation. After each desired temperature reached steady state, a standard free induction decay was acquired followed by a longitudinal relaxation time T 1 mapping sequence, and a diffusion pulse sequence. At each temperature, the tuning and matching was checked. The duration of the 90°pulse was 9.6 µs, whereas that for the 180°pulse was 19.6 µs. A standard inversion-recovery pulse sequence was used to acquire T 1 with 32 logarithmically spaced steps. The delay was varied from 1 ms to 400 ms for 23 Na. Diffusion coefficients were measured using a Pulsed-Gradient-Spin-Echo in 32 steps with a maximum b-value of 2200 s•mm -2 . The maximum diffusion gradient was 1 T•m -1 and the duration was 4 ms.

Molecular dynamics simulations

Aqueous sodium chloride (NaCl), bromide (NaBr), and fluoride (NaF) solutions were simulated using classical MD employing the Madrid-2019 FF [START_REF] Zeron | A force field of Li + , Na + , K + , Mg 2+ , Ca 2+ , Cl -, and SO 2-4 in aqueous solution based on the TIP4P/2005 water model and scaled charges for the ions[END_REF] that is based on the TIP4P/2005 water model [START_REF] Abascal | A general purpose model for the condensed phases of water: TIP4P/2005[END_REF] and uses scaled charges of +0.85e and -0.85e (e is the fundamental unit of charge) for Na + cations and Cl -, Br -, and F -anions, respectively. The FF parameters are listed in the SI. The scaled ionic charges aim at taking into account the electronic contribution to the dielectric constant at high frequencies in a mean-field fashion [START_REF] Kirby | Charge scaling manifesto: A way of reconciling the inherently macroscopic and microscopic natures of molecular simulations[END_REF]. At a moderate computational cost in comparison to fully polarizable models, the EFG relaxation within the Madrid-2019 FF [START_REF] Zeron | A force field of Li + , Na + , K + , Mg 2+ , Ca 2+ , Cl -, and SO 2-4 in aqueous solution based on the TIP4P/2005 water model and scaled charges for the ions[END_REF] has recently been shown to accurately describe the quadrupolar NMR relaxation rates of alkali metal ions at infinite dilution [START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF], in particular that of Na + . Solutions comprised of N = 1000 water molecules and N p ion pairs were initialized at different salt concentrations c between 0.06 m (N p = 1) and 4 m (N p = 72) in a cubic box at the equilibrium solution density ρ(c, T ) obtained in N P T simulations at P = 1 bar. The densities are in excellent agreement with the experimental ones, as discussed in Supplementary Note I B.

The equilibrated electrolyte systems were then simulated in the N V T ensemble. Both N P T and N V T simulation runs were carried out in the open-source Met-alWalls package on graphics processing units [START_REF] Marin-Laflèche | Metalwalls: A classical molecular dynamics software dedicated to the simulation of electrochemical systems[END_REF] with electrostatic interactions computed with Ewald summation [START_REF] Aguado | Ewald summation of electrostatic multipole interactions up to the quadrupolar level[END_REF] and a short-range cutoff of 1.24 nm. The constant temperature was maintained using the Nose-Hoover chains thermostat with a time constant of 1 ps. System temperatures in range from 10 °C to 50 °C were considered. The equations of motion were integrated using the velocity Verlet algorithm and an integration time step of 1 fs. The effective rigidity of water molecules was imposed with the help of the RATTLE algorithm with a precision of 10 -9 . For each (c, T ) state point, at least five independent runs of length 5 ns were performed to measure the EFG at the ion positions (sampled every 50 fs). Full Ewald summation expressions [START_REF] Aguado | Ewald summation of electrostatic multipole interactions up to the quadrupolar level[END_REF] were used in the computation of the EFGs, as recently implemented in MetalWalls [START_REF] Chubak | NMR relaxation rates of quadrupolar aqueous ions from classical molecular dynamics using force-field specific Sternheimer factors[END_REF]. For the considered system parameters, the relaxation of EFG fluctuations was found not to be affected by the finite box size, as we discuss in Supplementary Note. II A.

Ab initio calculations

To determine EFGs with the electron cloud contribution, smaller systems containing 55 water molecules and N p = 1, 2, 3, 4, and 5 NaCl ions pairs, corresponding to the salt concentrations c = 1, 2, 3, 4, 5 mol•kg -1 , were simulated in the same way as the larger ones using the Madrid-2019 FF. In a single N V T simulation run at T = 25 °C, 2000 configurations were sampled with a period of 10 ps, and were later used in DFT-based EFG calculations with periodic boundary conditions in the Quantum Espresso (QE) package [START_REF] Giannozzi | QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials[END_REF]. No additional geometry optimization of the configurations was performed in the DFT calculations. The pseudopotential-based projectoraugmented wave (PAW) method [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Charpentier | The PAW/GIPAW approach for computing NMR parameters: A new dimension added to NMR study of solids[END_REF][START_REF] Petrilli | Electric-field-gradient calculations using the projector augmented wave method[END_REF] was used to reconstruct the all-electron charge density in the vicinity of the nucleus using the QE-GIPAW package [START_REF] Varini | Enhancement of DFT-calculations at petascale: Nuclear magnetic resonance, hybrid density functional theory and Car-Parrinello calculations[END_REF]. The self-consistent electron densities were calculated using the PBE functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF], a kinetic energy cutoff of 80 Ry, and norm-conserving pseudopotentials of the GI-PAW package [START_REF]GIPAW Norm-Conserving Pseudopotentials[END_REF]. In the case of Na + ions, the EFGs obtained with the PBE functional were shown to be in good agreement [START_REF] Philips | Quadrupolar NMR relaxation from ab initio molecular dynamics: Improved sampling and cluster models versus periodic calculations[END_REF] with those obtained with the hybrid PBE0 functional [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF].

Dynamical properties of electrolyte solutions

The shear viscosity of aqueous electrolyte solutions was obtained using the Green-Kubo relation [START_REF] Chen | Are pressure fluctuationbased equilibrium methods really worse than nonequilibrium methods for calculating viscosities?[END_REF]:

η = V k B T +∞ 0 dt C stress (t), (3)
with V being the system volume and k B standing for the Boltzmann constant. The stress tensor ACF C stress (t) was computed as [START_REF] Chen | Are pressure fluctuationbased equilibrium methods really worse than nonequilibrium methods for calculating viscosities?[END_REF] C stress (t) = 1 10

α,β ⟨P αβ (t)P αβ (0)⟩,

where α, β run over the three Cartesian components and P αβ is the traceless symmetrized part of the stress tensor σ αβ : P αβ = 1 2 (σ αβ + σ βα ) -1 3 δ αβ γ σ γγ . For each salt concentration, the viscosity was measured over more than 5 independent simulation runs of length 5 ns with the stress tensor sampled every integration time step (1 fs).

The Na + diffusion coefficients were extracted from the long time limit of the ion's mean-square displacement:

D = lim t→∞ 1 6N p t Np i=1 [r i (t) -r i (0)] 2 , (5)
where N p is the number of sodium ions in the system, r i (t) is the position of the i-th ion at time t, and the brackets ⟨• • • ⟩ stand for ensemble averaging. The obtained diffusion coefficients were corrected for finite-size effects using the Yeh-Hummer relation [START_REF] Yeh | System-size dependence of diffusion coefficients and viscosities from molecular dynamics simulations with periodic boundary conditions[END_REF]:

D ∞ = D + k B T ξ 6πηL (6)
with the diffusion coefficient D ∞ corresponding to a macroscopic system, D being obtained in a cubic simulation box with side length L, and ξ ≈ 2.837297. The calculated values of viscosity η in Eq. ( 3) were used for evaluating D ∞ in Eq. ( 6). The finite-size correction term corresponded to 17-22 % of the measured value D.

FIG. 1 .

 1 FIG. 1. Electron cloud contribution to the EFG at the Na + position. a Representative system configuration of a NaCl solution at 5 m. The gray opaque regions around water molecules show charge densities obtained with DFT PAW calculations (see Methods). b Component-wise comparison of ab initio EFGs, V AI αβ , against classical EFGs, V ext αβ , at the position of Na + ions on the same set of configurations for different salt concentrations c. The solid lines indicate the fit for an effective Sternheimer factor γ eff : V AI αβ = (1 + γ eff )V ext αβ . c Effective Sternheimer factors for Na + obtained from the linear fit (yellow squares) or from the ratio (1+γ ′ eff ) 2 = ⟨V 2 AI ⟩/⟨V 2 ext ⟩ (blue circles) at different c. d EFG variance at the Na + position for different c as obtained directly with ab initio calculations (green triangles), or using the value of γ eff (yellow squares) and γ ′ eff (blue circles) at infinite dilution. The error bars in (c) and (d) were calculated using bootstrapping.

FIG. 2 .

 2 FIG. 2. Relaxation of EFG fluctuations. a Normalized autocorrelation functions CEFG(t)/CEFG(0) of the EFG at the position of a Na + ion obtained using classical MD simulations for different salt concentrations c at T = 25 °C in aqueous NaCl solutions. Qualitatively similar trends are found for other concentrations and temperatures (see Supplementary Fig. S8). Insets in (a) highlight the short-time behavior of the ACFs for t < 1 ps and a typical system configuration at c = 4 m (Na + and Cl -ions are blue and green, respectively). b Temperature behavior of CEFG(t) at c = 4 m. c Long-time behavior of CEFG(t) plotted on a double logarithmic scale at c = 4 m for different temperatures (the legend shown in (b)). The black solid and dashed lines highlight a powerlaw scaling ∼t α with α = -5/2 and α = -3/2, respectively. Shaded regions in (a), (b), and (c) indicate standard errors from multiple independent simulation runs.

  FIG. 3. Concentration and temperature dependence of the quadrupolar relaxation rate. T -1 1 of 23 Na + in aqueous NaCl as a function of the salt concentration c as obtained in experiments (E, solid lines and open symbols) and simulations (S, filled symbols) for different temperatures. The error bars for simulation results are associated with the approximation for incorporating the electron cloud contribution to the EFG.

FIG. 4 .

 4 FIG. 4. Microscopic parameters of the EFG relaxation. a Effective correlation time τc of the EFG fluctuations at the Na + position as a function of salt concentration c for different temperatures T . b Relative contribution of the slow EFG relaxation mode to τc for different c and T . The contribution was estimated using the stretched exponential fit of the normalized EFG ACFs (Supplementary Note II E). c Variance of the total EFG at the ion position ⟨V 2 ⟩ = (1+γ ′ eff,0 ) 2 ⟨V 2 ext ⟩ as a function of c for different temperatures. The legend in (c) is the same as in (a). d EFG variance, ⟨V 2 ⟩, evaluated from water molecules and ions located within a different number of solvation shells around the central Na + ion as a function of c at T = 25 °C. The error bars from multiple independent simulation runs are either explicitly shown or do not exceed the symbol size.

FIG. 5 .

 5 FIG. 5. Assessing models of the quadrupolar relaxation. a Na + diffusion coefficient as a function of salt concentration c for different temperatures T in experiments (solid lines with open symbols, legend shown in (a)) and simulations (filled symbols, legend shown in (b)). b Dynamic solution viscosity as a function of c for different T in experiments (solid lines with open symbols, legend shown in (a)) and simulations (filled symbols, legend shown in (b)). Experimental viscosities were taken from Ref. [58]. c Stress tensor ACFs Cstress(t) normalized by the system volume V and kBT for increasing C at T = 25 °C. The black dotted line shows the stretched exponential fit of the long-time decay at c = 4 m. Inset: the time scale of solution structural relaxation τstruct, as extracted from the long-time decay of Cstress(t) (see main text), as a function of c at multiple temperatures. T decreases from top to bottom, the legend is shown in (f) and (c). d, Stokes-Einstein-Debye time plotted versus the EFG correlation time τ exp c , as extracted from experimental data for different c and T . e τ exp c as a function of η/kBT for different temperatures. r eff 0 is the effective hydrodynamic radius of a Na + ion extracted under assumption that τ exp c can be modeled by a SED relation (the gray line shows the best fit). d and e share the same legend. f Mean water dipole reorientation time τ dip plotted versus the EFG correlation time τc, as extracted in simulations for different c and T . g τstruct plotted versus τc for different c and T . The gray line indicates the linear dependence, τstruct = τc.

= 4πη r eff 0 3 /

 3 Fig. 5e showing τ exp c plotted against η/k B T for various temperatures. While a Stokes-Einstein-like relation holds for τ exp c , that is a strong correlation τ exp c ∝ η/k B T exists for the considered range of parameters, the effective Stokes radius r eff 0 = 0.69 Å that would correspond to the EFG correlation time τ exp c within the SED model is clearly unphysical and smaller than the ionic radius 1.02 Å. r eff 0 was obtained from the fit τ exp c 3k B T + τ eff 0 with an additional intercept τ eff 0 = 0.11 ps needed for the best data representation

ACKNOWLEDGEMENTS

We would like to thank Dr. Seena Dehkharghani for the thoughtful discussions on the influence of temperature on water dynamics and NMR relaxation parameters and Dr. Antoine Carof for useful discussions on the modeling of quadrupolar NMR relaxation. This work was supported in part by the National Institutes of Health (NIH): grant no. R01EB026456. This project received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program (grant agreement no. 863473). The authors acknowledge access to HPC resources from GENCI (grant no. A0110912966) and AJ wishes to acknowledge the HPC resources of NYU. AJ aknowledges funding from the US National Science Foundation under award no. CHE2108205.


Van Long

Nguyen Huu

Cache management in MASCARA-FPGA: from coalescing heuristic to replacement policy

We presented ModulAr Semantic CAching fRAmework (MASCARA) that deployed Semantic Caching (SC) to perform a fast query processing based on Field Programmable Gate Arrays (FPGAs) accelerators. In addition of the accelerators, cache management plays an important role to address coalescing strategy and replacement policy so as to maximize the performance of FPGA caching. Therefore, in this paper, we present a coalescing heuristic with a new replacement function that leverages advantages of traditional strategies and overcomes their drawbacks. The proposed heuristic reduces response time, improves data availability, and saves cache space with respect to the semantic locality of query workload.

INTRODUCTION

Recently, acceleration for the data management system (e.g., Spark [START_REF] Armbrust | Spark SQL: Relational Data Processing in Spark[END_REF]) with a specific computing hardware (e.g., Field Programmable Gate Array FPGA), has declined by both industry and academia, such as, [START_REF] Salami | AxleDB: A novel programmable query processing platform on FPGA[END_REF], [START_REF] Mueller | Glacier: A Query-to-Hardware Compiler[END_REF], [START_REF] Owaida | Centaur: A Framework for Hybrid CPU-FPGA Databases[END_REF], [START_REF] Woods | Less Watts, More Performance: An Intelligent Storage Engine for Data Appliances[END_REF], [START_REF] Sukhwani | A Hardware/Software Approach for Database Query Acceleration with FPGAs[END_REF], [START_REF] Gupta | Accelerating datacenter workloads[END_REF], and [START_REF] Crockett | The Zynq Book: Embedded Processing with the Arm Cortex-A9 on the Xilinx Zynq-7000 All Programmable Soc[END_REF]. Moreover, using a caching technique, for example, Semantic Caching (SC) [START_REF] Keller | A Predicate-Based Caching Scheme for Client-Server Database Architectures[END_REF], [START_REF] Orazio | Semantic Caching Framework: An FPGA-Based Application for IoT Security Monitoring[END_REF] which could finely exploit data and knowledge in a query or reduce data transfer, should also be considered. Therefore, we have proposed a Modular Semantic Caching framework (MASCARA) [START_REF] Van Long Nguyen Huu | MASCARA (ModulAr Semantic CAching fRAmework) towards FPGA Acceleration for IoT Security Monitoring[END_REF] that relies on the acceleration via FPGA kernels (e.g., Query Trimming) [START_REF] Long | MASCARA-FPGA Cooperation Model: Query Trimming through Accelerators[END_REF].

State of the art. MASCARA-FPGA speeds up the execution of a query or a sequence of queries. To reach this goal, in addition to the appropriate accelerators, dealing with cache management is important to maximize the performance. In particular, cache management in SC consists of two main procedures: the coalescing strategy and the replacement policy. The first one determines how the data regions are formed, merged and partitioned, to ensure the optimal granularity of the cached elements while minimizing the overhead of using the cache space. Meanwhile, the second one determines how semantic information is added to and removed from the cache. The coalescing strategy could be solved by the straightforward approaches, such as: Nerver Coalescing or Always Coalescing ( [START_REF] Dar | Semantic Data Caching and Replacement[END_REF], [START_REF] Björn Þór Jónsson | Performance and Overhead of Semantic Cache Management[END_REF], [START_REF] Godfrey | Answering Queries by Semantic Caches[END_REF], [START_REF] Vancea | A Scalable Cooperative Semantic Caching (CoopSC) Approach to Improve Range Queries[END_REF], or [START_REF] Qun Ren | Semantic caching and query processing[END_REF]). Since the above solutions are limited by the capabilities of CPU, the contributions were not interested in studying the effects of different strategies. In general, both of them have major side effects to FPGA caching (i.e., MASCARA-FPGA), such as, cache space utilization, the granularity problem, or the responses times of the system. Furthermore, they can be exacerbated if we do not have an appropriate replacement policy that should be adapted for query workloads with general semantic locality.

Contribution. Therefore, we propose a heuristic solution that leverages their advantages and minimize their side effects. In particular, it decides whether to coalesce data regions based on the recency of usage (temporal locality) and percentage of response contribution (spatial locality) that are presented through a new replacement function. It is worth to note that our solution is flexible when the semantic locality of query workload changes according to their applications. Experiments with TPC-H benchmark [START_REF]TPC-H Benchmark revision 3.0.1[END_REF] validates that MASCARA-FPGA with the heuristic has: a fast response time, a high cache efficiency, and a low overhead in memory usage. For example, with two kernels, the response time of MASCARA-FPGA is faster up to 9.23 times than the MASCARA-Server (baseline) with 1𝐺𝐵 of dataset.

COALESCING IN MASCARA-FPGA

We presented the SC concept and prototype of MASCARA-FPGA in [START_REF] Van Long Nguyen Huu | MASCARA (ModulAr Semantic CAching fRAmework) towards FPGA Acceleration for IoT Security Monitoring[END_REF][START_REF] Long | MASCARA-FPGA Cooperation Model: Query Trimming through Accelerators[END_REF]. If we have 𝑁 data region 𝐷𝑅 and the answer of an incoming query 𝑄 overlaps them, the result could be the formation of 2𝑁 + 1 new 𝐷𝑅, where 𝑁 + 1 , Vol. of them are the answer to the query. The following question arises: should we combine some, all, or none of the 𝑁 + 1 𝐷𝑅 into one or more larger 𝐷𝑅 [START_REF] Björn Þór Jónsson | Performance and Overhead of Semantic Cache Management[END_REF] (as shown in Figure 1).

Conventional strategies

Always Coalescing will coalesce all the 𝐷𝑅 which contributes to the answer of the most recent 𝑄, so that only one 𝐷𝑅 corresponds to the query. Without generating a large number of 𝑆, it results to a good performance because we can avoid the overhead of Query Trimming accelerators. The worst case occurs when the response of 𝑄 takes a large portion of cache, resulting to a poor space utilization. In contrast, Never Coalescing does not coalesce any data region that contributes to the most recent queries, resulting in a drastically increase of number and complexity of 𝑆 and 𝐷𝑅. Thus, this solution could have a negative impact on responses times even if we have multiple parallel accelerators on FPGA.

Obviously, applying individually the two solutions raises the issues of efficiency, space utilization, or end-to-end response time of cache. Therefore, it seems reasonable to use them alternatively by checking the current situation of data regions and/or their future contributions in query answers. In particular, the segments 𝑆 representing corresponding 𝐷𝑅 should be measured and indexed by their "profit" 𝑆 𝑉 in the cache. The function of calculating 𝑆 𝑉 is based on temporal (e.g., LRU) and spatial locality (e.g., contribution to query answering). In other words, 𝑆 𝑉 is now considered to use in both coalescing and replacement.

Heuristic: from coalescing to replacement

In this section, we present how we manage the cache through a heuristic which considers both coalescing and replacement. Given the recency of usage, we assume that the most recent coalescing/replacement value is 𝑉 𝑚𝑎𝑥 , which is increased by one for each new 𝑄. Meanwhile, the coalescing/replacement value for each data region is 𝑆 𝑉 . The process of finding the profit of a 𝐷𝑅 is divided into two steps: 1) computing the intermediate profit and deciding whether to merge based on its value, and 2) computing future profit of the its remaining part (as shown in Algorithm 1).

Algorithm 1: Cache management with heuristic

Input: Input: cache with list of 𝑆, 𝐷𝑅 and a query 𝑄 Output: Output: cache updated with coalescing heuristic

𝑝𝑑𝑖𝑠 := 𝑆 𝑉 _𝑖𝑛𝑡𝑒𝑟 -𝑆 𝑉 _𝑜𝑟𝑖 24 𝑆 𝑉 := 𝑝𝑑𝑖𝑠 * (1 -𝑝) + 𝑆 𝑉 _𝑜𝑟𝑖 25 end
From row 10 to 21 in Algorithm 1, we measure the percentage of the data region that contributes to the query answer: 𝑝 = 𝑅 𝑄 /𝑅, where 𝑅 𝑄 is the number of records that match the query answer and 𝑅 is the total number of records in the region. Then, the new replacement/coalescing value is temporarily updated as follows:

𝑆 𝑉 = 𝑆 𝑉 + (𝑉 𝑚𝑎𝑥 -𝑆 𝑉 ) * 𝑝.
Note that (𝑉 𝑚𝑎𝑥 -𝑆 𝑉 ) ensures that the new 𝑆 𝑉 does not have a higher value than 𝑉 𝑚𝑎𝑥 . In other words, the new data region with respect to the last query 𝑄 will have the highest value 𝑆 𝑉 . This function is adaptable for all regions in SC, regardless of whether the region contributes to answering the query or not.

Based on the updated 𝑆 𝑉 for all regions contributing to the query answer, we propose a threshold 𝑇 as a part of "coalescing filter" 𝑆 𝑉 < 𝑄 𝑉 * 𝑇 that decides whether to merge all, some or none of them. In general, 𝑇 can be scaled from 0 (Never Coalescing) to 1 (Always Coalescing). Meanwhile, 𝑄 𝑉 = 𝑉 𝑚𝑎𝑥 is the value of the new 𝐷𝑅 with respect to the new query appears. If 𝑆 𝑉 < 𝑄 𝑉 * 𝑇 , the overlapping part between existed 𝐷𝑅 and new 𝐷𝑅 of 𝑄 will be merged (coalesced) into the old one. Thus, the number of generated or cached segments are stored, resulting in an efficient response time of MASCARA-FPGA. Otherwise, if 𝑆 𝑉 >= 𝑄 𝑉 * 𝑇 , the new 𝐷𝑅 of 𝑄 will be cached which has the same value 𝑆 𝑉 as its predecessor 𝐷𝑅. By this way, this decision increases the data granularity of the cache, resulting in higher efficiency. Although we have not yet explored the cost model related to the "coalescing filter" as well as the optimization problem (e.g., Knapsack or Dynamic Programming), we can adjust the threshold 𝑇 in practical to find a "reasonable coalescing filter" based on cache performance, cache efficiency, and cache space usage.

As the second step (from row 22 to 25 in Algorithm 1), after merging some of the 𝐷𝑅, the remaining ones might shrink into the new parts. Therefore, their future profit that could be evaluated for the next queries should be recalculated as follows: 𝑆 𝑉 = 𝑝𝑑𝑖𝑠 * (1 -𝑝) + 𝑆 𝑉 _𝑜𝑟𝑖 where 𝑆 𝑉 _𝑜𝑟𝑖 is the original profit of the 𝐷𝑅 before starting the procedure. 𝑝𝑑𝑖𝑠 = 𝑆 𝑉 _𝑖𝑛𝑡𝑒𝑟 -𝑆 𝑉 _𝑜𝑟𝑖 , consists of the profit distance, is the gap between the intermediate profit 𝑆 𝑉 _𝑖𝑛𝑡𝑒𝑟 and the original profit 𝑆 𝑉 _𝑜𝑟𝑖 .

Example 2.1. As shown in Figure 2, we assume that the contribution of 𝐷𝑅𝑖 and 𝐷𝑅 𝑗 to the query answer is 𝑝 𝑖 = 0.75 and 𝑝 𝑗 = 0.71, respectively. The last value 𝑉 𝑚𝑎𝑥 = 35 for the appearance of query 𝑄. Although 𝑝 𝑖 , 𝑝 𝑗 are nearly equal, their contributions to the answer are different in size (i.e., 𝐷𝑅𝑖 > 𝐷𝑅 𝑗). From (a) to (b), the 𝑆 𝑉 of 𝑆 𝑖 has increased significantly from 10.3 to 18.4. If we choose 𝑇 = 0.5, then 𝑇 * 𝑄 𝑉 = 0.5 * 35 = 17.5 does not pass the condition of "coalescing filter" (17.5 < 18.4). Thus, cache will keep the overlapping part between 𝐷𝑅 𝑖 and 𝐷𝑅 𝑄 as a totally new one. In contrast, with the same procedure, 𝐷𝑅 𝑗 does not pass the "filter". It means that cache will merge the overlapping part of 𝐷𝑅 𝑗 into 𝐷𝑅 of 𝑄. At the end of the procedure, in (c), we reevaluate the future profit of 𝐷𝑅 𝑖 , 𝐷𝑅 𝑗 to prepare for the next 𝑄. For example, using the formula consists of 𝑝𝑑𝑖𝑠, the future profit of the remaining 𝐷𝑅𝑖 is: 𝑆 𝑉 = (18.4 -10.3) * (1 -0.75) + 10.3 = 12.325.

Fig. 2. The coalescing heuristic in cache management

Since 𝑆 𝑉 is used for both coalescing and replacement, our heuristic overcomes the limitation of LRU in the context of SC. Indeed, considering that 𝐷𝑅 can vary in size, removing it from the cache should depend not only on its contribution to the last query response, but also on its actual size. In other words, we calculate the ratio 𝑟 between the actual coalescing/replacement 𝑆 𝑉 and its size 𝑠 in the cache: 𝑟 = 𝑆 𝑉 /𝑠𝑖𝑧𝑒_𝑜 𝑓 _𝑆 𝐷 (from row 3 to 9 in Algorithm 1). Thanks to the real representation of 𝑟 (e.g., 𝑆 𝑉 = 12.325 in the above example) if the cache needs space for a new data region 𝐷𝑅, the selection of a victim would be more accurate than LRU. It should be noted that 𝐷𝑅 which overlaps the query response are excluded from this procedure. In summary, by approximating both temporal and spatial locality, the impact of query workload (i.e., semantic locality) could be alleviated in general applications.

EXPERIMENTS

Configuration. A server is equipped with an Intel® Xeon® Gold 5118 CPU running at 2.30 GHz and 64 GB RAM. The FPGA platform is an Alveo U200 board with three 16GB DDR4.

Query workload. Experiments are performed with only one relation (i.e., lineitem) of TPC-H [START_REF]TPC-H Benchmark revision 3.0.1[END_REF]. Based on the query Q6 of TPC-H, we generate the range query with the select-project format.

Influencing parameters The parameters that could affect the performance of MASCARA-FPGA are presented in [START_REF] Long | MASCARA-FPGA Cooperation Model: Query Trimming through Accelerators[END_REF]. Moreover, we consider also Skew (a fixed fraction of the queries that has semantic centerpoint within a Hot Region).

Evaluated prototypes. We evaluate our contribution by comparing several prototypes: This experiment is performed by measuring the processing time of Query Trimming over 1000 segments 𝑆. Obviously, as shown in Figure 3a, the Query Trimming accelerator has a higher 𝑆𝑃𝑆𝑒𝑐 than similar procedure on Server. When 𝑁𝑏_𝑃𝑟𝑒𝑑_𝑆𝑒𝑔 increases, 𝑆𝑃𝑆𝑒𝑐 decreases significantly. In particular, 𝑆𝑃𝑆𝑒𝑐 of server drops to 102 𝑆𝑃𝑆𝑒𝑐, meanwhile, 𝑆𝑃𝑆𝑒𝑐 of kernel is 661 when 𝑁𝑏_𝑃𝑟𝑒𝑑_𝑆𝑒𝑔 = 4. If we deploy two kernels in parallel, 𝑆𝑃𝑆𝑒𝑐 increases from 661 𝑆𝑃𝑆𝑒𝑐 to 1196 𝑆𝑃𝑆𝑒𝑐. To resume, it can be seen that the Query Trimming on FPGA outperforms the server based on CPU only when the query complexity (e.g., 𝑁𝑏_𝑃𝑟𝑒𝑑_𝑆𝑒𝑔) increases. Moreover, with high 𝑆𝑃𝑆𝑒𝑐 of accelerators, the main issue of Never Coalescing now is alleviated while keeping its benefits in MASCARA-FPGA.

A large number of 𝑁𝑏_𝑆𝑒𝑔 could also cause a bottleneck in Query Trimming [START_REF] Long | MASCARA-FPGA Cooperation Model: Query Trimming through Accelerators[END_REF]. As shown in Figure 3b, when the query size increases, 𝑁𝑏_𝑆𝑒𝑔 grows rapidly, especially with Never Coalescing (e.g., 191 𝑆 for query size = 10%). With 𝑇 = 0.5, 𝑁 𝐵_𝑆𝑒𝑔 of our heuristic is slightly higher than the best one Always Coalescing (e.g., 133 𝑆 for query size = 10%). In summary, we consider that MASCARA-Server could also apply the heuristic but the performance might not be significant as Always Coalescing due to the limited throughput of Query Trimming on server. Therefore, it is more preferable to MASCARA-FPGA in which we have high throughput Query Trimming kernels.

Cache performance

As it can be seen in Figure 4a, the FPGA prototypes (B, C, and D) have lower response time than (A) on server. Obviously, (C) is the fastest because using Always Coalescing reduces the number of generated segments, which leads to fast execution of Query Trimming. Meanwhile, our heuristic (D) finds a good balance between (B) and (C), and shows a remarkable speed-up (i.e., 4.84 with 𝑆𝐹 = 1𝐺𝐵). We also found that the speed up of the FPGA prototypes decrease when 𝑆𝐹 increases. For example, at 𝑆𝐹 = 10𝐺𝐵, (B) is 2.07, (C) is 3.26, and (D) is 2.81 times faster than (A). As shown in [START_REF] Van Long Nguyen Huu | MASCARA (ModulAr Semantic CAching fRAmework) towards FPGA Acceleration for IoT Security Monitoring[END_REF][START_REF] Long | MASCARA-FPGA Cooperation Model: Query Trimming through Accelerators[END_REF], in this case we need to execute the 𝑅𝑄 if they appear over a large and full (i.e., 10𝐺𝐵) dataset on the server. Fortunately, with reasonable hardware consumption (presented in Section 3.5), we can use multiple accelerators (i.e., two kernels) to mitigate this problem. In Figure 4b, we change the semantic locality through Skew, to show that our heuristic is preferable for general applications. Indeed, the speed up of FPGA prototypes increases since most of the (probe) queries are executed on FPGA, which is faster than the server thanks to its accelerator Probe Query Executing. For example, at 𝑆𝐹 = 1𝐺𝐵, we observe that (B), (C), and (D) are faster than (A) 4.78, 7.22 and 5.73 times, respectively.

Cache space utilization

We run the experiment through 50 queries with 𝑁𝑏_𝑃𝑟𝑒𝑑_𝑄𝑢𝑒𝑟𝑦 = 4, which has the same response size (as show in Figure 5a). Meanwhile, the cache should be initialized with almost 1000 segments 𝑆 that have 𝑁𝑏_𝑃𝑟𝑒𝑑_𝑆𝑒𝑔 >= 4 thanks to the warm-up queries. Obviously, with 𝑁 + 1 new 𝐷𝑅 that are not merged, Never Coalescing will store the duplicated key attributes that leads to significant overhead of cache memory. For example, with a workload of 1% queries, we need more space in prototype (B) than usual (23.38%) to store the fragmented 𝐷𝑅. Meanwhile, in (C), the cache space requirement has only 4.83%. (D) is slightly higher than (C) (7.37%) because we merge only the 𝐷𝑅 that satisfies our heuristic condition. Moreover, when the query size increases, the space overheads of all strategies decrease and converge at a certain query size (i.e., query size workload = 10%).

The other influencing factor of our heuristic is the threshold 𝑇 . It is obvious that when 𝑇 increases, for example 𝑇 = 0.7, the possibility of merging in (D) is high, which allows saving more space in the cache memory. In contrast, reducing 𝑇 means that the cache prefers to keep the data region independent to increase the data granularity. Depending on the status of the cache, 𝑇 should be changed dynamically to maintain system performance.

Cache efficiency

We measure the cache efficiency of our heuristic by varying the size on average between 10 and 200 times of queries in workload (as shown in Figure 5b). Our solution allows to have a highest cache efficiency compared to the conventional approaches. For example, when cache size is small (e.g., ten times the query size), prototype (D) with 𝑇 = 0.3 could answer almost 68% of the queries (i.e., data regions of 𝑃𝑄s are 68%). In detail, our heuristic evaluates the contribution-to-size ratio that considers both temporal and spatial locality. However, when cache size increases (i.e., is more than 50 times the query size), the results of the prototypes tend to saturate. For example, when the cache size is 100 times of the query size, the cache efficiencies are 74%, 66%, and 88% with respect to (B), (C), and (D), with 𝑇 = 0.3. Besides cache space utilization, 𝑇 also changes cache efficiency of (D). For example, if 𝑇 changes from 0.3 to 0.5, the cache efficiency decreases from 68% to 53%, where the cache size being times larger than the query size. This efficiency could decrease even further if we increase 𝑇 from 0.5 to 0.7. Obviously, the ability to answer queries does not differ too much between the prototype when cache size is large enough.

Hardware resources

The hardware resource consumption of MASCARA-FPGA is shown in Table 1. Cache Manager: Heuristic has a small footprint in our prototype thanks to the arbitrary precision fixed-point data type of computing 𝑆 𝑉 and 𝑟 . Another positive point is that we only need one Cache Manager: Heuristic in MASCARA-FPGA.

CONCLUSION

In this paper, we propose a coalescing heuristic with new replacement function that takes advantage of the two straightforward approaches: Always and Never Coalescing. In particular, this heuristic takes into account the temporal and spatial locality of the data regions with respect to the query response by computing their profit in the cache. To summarize, we improved the performance of FPGA caching in three ways: end-to-end response time, cache efficiency, and cache space utilization. As future works, we are interested in: i) Investigating a cost model for automatically selecting the optimal coalescing threshold 𝑇 based on the cache state in real-time. ii) Extending MASCARA-FPGA towards join queries with fragmented data regions.

Fig. 1 .

 1 Fig. 1. Data Region in Always and Never Coalescing

  (A) MASCARA-Server with Always Coalescing LRU, (B) MASCARA-FPGA with Never Coalescing LRU, (C) MASCARA-FPGA with Always Coalescing LRU, and (D) MASCARA-FPGA with heuristic.

3. 1

 1 Segments -Query Trimming throughput (a) Throughput of kernel . (b) Generated segment by strategies

.

Fig. 3 .

 3 Fig. 3. Impact of coalescing strategies

9 .Fig. 4 .

 94 Fig. 4. Response Time when changing semantic locality. 50 queries workload. Cache has 1000 𝑆. Hot Region = 20%.

  (a) Cache space utilization . (b) Cache efficiency

.

Fig. 5 .

 5 Fig. 5. Space overhead and data availability of cache. Hot Region = 20% and Skew = 0.8.

1

  Pass Query Trimming, outputs are: 𝑃𝑄, 𝑅𝑄 2 Execute 𝑃𝑄 and 𝑅𝑄 /* Replacement with the minimal ratio */

	20	end
	21 end
	/* Heuristic of coalescing: step 2	*/

3 while 𝑟 ≠ End Of List do 4 𝑟 := 𝑆 𝑉 /𝑠𝑖𝑧𝑒_𝑜 𝑓 _𝑆 𝐷 5 Finding the minimal 𝑟 6 end 7 𝑉 𝑖𝑐𝑡𝑖𝑚 := 𝑆 with 𝐷𝑅 that has minimal 𝑟 8 Remove 𝑉 𝑖𝑐𝑡𝑖𝑚 9 Add 𝑅𝑄 with new 𝐷𝑅 /* Heuristic of coalescing: step 1 */ 10 Choosing threshold 𝑇 11 Assign 𝑄 𝑉 := 𝑉𝑚𝑎𝑥 12 while 𝑆 ≠ End Of List do 13 𝑝 := 𝑅 𝑄 /𝑅 14 𝑆 𝑉 _𝑖𝑛𝑡𝑒𝑟 := 𝑆 𝑉 _𝑜𝑟𝑖 + (𝑉 𝑚𝑎𝑥 -𝑆 𝑉 _𝑜𝑟𝑖 ) * 𝑝 15 if 𝑆 𝑉 < 𝑄 𝑉 * 𝑇 then 16 Coalescing between 𝑆 𝐷 and 𝑄 𝐷 17 end 18 else if 𝑆 𝑉 >= 𝑄 𝑉 * 𝑇 then 19 No-coalescing between 𝑆 𝐷 and 𝑄 𝐷 22 while 𝑆 ≠ End Of List do 23

Table 1 .

 1 Hardware resources of MASCARA-FPGA

	Component	Look-Up Tables (LUTs) Flip Flops (FFs) 36Kb BRAM
	Attribute Matching	4476	2914	11
	Predicate Matching	17668	16621	84
	Semantic Extracting	6838	3682	17
	Probe Query execution	7739	5436	31
	Query Process Controller	5394	4379	10
	Cache Manager: Heuristic	6627	5461	44
	Total resources	48712	38493	197
	Alveo-U200 usage (%)	5.46	2.1	11.15

ACKNOWLEDGMENTS

This work is funded by N°2019/0477 CIFRE scholarship of ANRT within the project of B<>Com and Nokia Bell Labs, France.