The discovery emerged from a high-dimensional, charge-recursive neural network force field, or QRNN, trained exclusively on quantum-chemical gas-phase cluster calculations. Traditional molecular mechanics force fields, the kind that have served protein simulation so well, struggle with the peculiar physics of concentrated polar organic electrolytes, while ab initio molecular dynamics is limited to tiny systems and short timescales. Machine-learning force fields promise the accuracy of quantum chemistry at a cost many orders of magnitude lower, and the team’s refined model delivered a picture that conventional force fields simply cannot see. In simulations of lithium hexafluorophosphate in ethylene carbonate, the QRNN trajectories showed a sharp peak in the lithium-lithium radial distribution function near three angstroms, a signature of cations sitting in close proximity. The widely used OPLS4 force field, run on the same systems, produced no such peak and placed more than ninety-nine percent of lithium ions in single-ion clusters.

The statistics are striking. At 1.2 molar salt concentration, roughly forty percent of lithium ions were found in multi-lithium clusters during the QRNN trajectories, corresponding to an average of about 1.3 lithium ions per complex. At 2 molar, that proportion climbed above sixty-eight percent, while at 0.5 molar it fell to around twenty-three percent, showing that clustering scales strongly with concentration. Some clusters contained two or three lithium ions, and at the highest concentrations a smaller number contained four or even five. Crucially, these aggregates consisted of lithium cations bridged by ethylene carbonate molecules through their carbonyl oxygens, with the hexafluorophosphate anions pushed away. The ethylene carbonate molecules act as molecular glue, their high dielectric constant and polarizability stabilizing the otherwise repulsive approach of multiple positive charges, something that would be impossible in the gas phase or in an ordinary organic solvent.

Because such a radical departure from accepted solvation pictures demands extraordinary evidence, the team subjected their model to a battery of validation tests. Association energies computed with the QRNN force field agreed with density functional theory calculations at the ωB97X-D3BJ/def2-TZVPD level to within chemical accuracy, meaning errors below one kilocalorie per mole, for structures drawn from both QRNN and OPLS4 trajectories. The OPLS4 force field, by contrast, erred by ten to one hundred times more, and performed especially poorly on the multi-lithium clusters. Lithium-lithium distance scans in the presence of solvent molecules revealed a clear energy minimum between 2.4 and 3.0 angstroms, which the QRNN model reproduced faithfully, and an ab initio molecular dynamics simulation of a nearly five-hundred-atom system confirmed that a di-lithium cluster remains stable in that minimum region over the course of the simulation.

The model also passed demanding condensed-phase tests without ever having seen experimental data during training. Predicted densities for seven common electrolyte solvents, including ethylene carbonate, propylene carbonate, vinylene carbonate, fluoroethylene carbonate and three linear carbonates, matched measured values closely, and viscosity predictions for lithium hexafluorophosphate in a three-to-seven ethylene carbonate and dimethyl carbonate mixture were accurate up to the highest practical salt concentrations, a marked improvement over the team’s earlier work. Ionic conductivities from the simulations agreed strongly with experimental values at working temperatures. Because the potential energy surface was parameterized purely against quantum-chemical calculations, these comparisons constitute a fully independent test, and the results make a compelling case that first-principles machine-learning force fields can represent a complex chemical system to atomic resolution.

Experimental corroboration came from solid-state nuclear magnetic resonance spectroscopy performed on flash-frozen electrolyte samples. Freezing the liquid preserves the local solvation structures that fast molecular exchange would otherwise blur at room temperature. Direct-polarization lithium-7 spectra captured the full lithium population, while proton-to-lithium cross-polarization experiments showed attenuated signal consistent with lithium binding to carbonyl oxygens with protons facing away. Most tellingly, fluorine-to-lithium cross-polarization produced almost no signal, even after fivefold amplification. Both contact ion pairs and anion-containing aggregates would place lithium and fluorine atoms close enough to generate strong dipolar couplings and enhance such transfer. The absence of that spectroscopic signature indicates that the frozen samples contain predominantly solvent-separated species, exactly the single- and multi-lithium clusters the simulations predicted.

Simulated carbon-13 spectra built from clusters extracted from the QRNN trajectories matched the experimental line shapes across three salt concentrations, reproducing both the spread in chemical shifts of ethylene carbonate molecules bound in lithium clusters and the gradual growth of a shoulder on the carbonyl peak as concentration increased. When the identical protocol was applied to OPLS4 trajectories, the simulation adequately predicted spectra for dilute samples dominated by single-lithium structures but deviated from experiment once multi-lithium clusters became prevalent, particularly at carbon sites closest to the lithium ions. The agreement between a purely computational prediction and cryogenic solid-state NMR measurements provides some of the strongest evidence yet that these unusual cation aggregates are real features of the working electrolyte, not artifacts of a neural network’s imagination.

Why does this matter for the battery itself? The reduction potentials of electrolyte species determine what happens when the cell is first charged, when the anode draws electrons and the solid electrolyte interphase begins to form. Quantum-chemical calculations of one-electron reduction potentials, referenced to the lithium/lithium-ion couple through the ferrocenium/ferrocene standard, showed that the most common single-lithium complex, LiEC4, reduces at a much lower voltage than the family of di-lithium complexes such as Li2EC7. Experimentally, interphase formation begins at approximately 0.8 volts versus lithium/lithium-ion on both graphite and silicon anodes. The single-lithium clusters would not be spontaneously reduced at that voltage, but the di-lithium species fall within a tenth of a volt of it, and they appear to undergo even more favorable second one-electron reductions afterward. In other words, the multi-lithium clusters may not merely participate in the initial formation of the interphase; they may govern it, with the excess electron localizing on a carbonate moiety of an ethylene carbonate ligand shared between two lithium centers.

The finding finally offers a mechanistic explanation for one of battery chemistry’s oldest empirical choices: why ethylene carbonate has been indispensable in commercial electrolytes since the 1990s. Its unusual combination of high dielectric constant and polarizability in an aprotic solvent appears to be exactly what enables multiple lithium ions to co-complex around bridging carbonyl oxygens, creating the species that reduce at the right voltage to build a protective interphase. Beyond the specific chemistry, the work is a proof of concept that machine-learning force fields trained without any experimental input can uncover surprising phenomena in systems far more complex than liquid water, spanning elements across the periodic table. The team now plans to extend the platform to other solvents, salts and additives, and to add electrode models, reactive chemistry and applied voltage, aiming for an atomic-level picture of interphase formation. If successful, large-scale graphics-processing-unit-based simulation could become a genuine engineering tool, allowing electrolytes to be designed computationally rather than discovered by trial and error, with consequences for battery efficiency, cost and the broader electrification of transport and grid storage.

Subject of Research: Multi-lithium-ion clustering in lithium-ion battery electrolytes revealed by machine-learning molecular simulations and solid-state NMR spectroscopy

Article Title: Evidence for multi-Li+ clustering in lithium-ion battery electrolytes

Article References: Stevenson, J. M., Agarwal, G., Debnath, S., Goldsmith, Z. K., Mondal, M., Svirinovsky-Arbeli, A., Yeu, I. W., Metlay, A. S., Kwon, Y., Bhai, L., Guha, R. D., Keeler, E. G., Weber, J. L., Steingart, D. A., Marbella, L. E., Urban, A., Jacobson, L. D., Halls, M. D., & Friesner, R. A. (2026). Evidence for multi-Li+ clustering in lithium-ion battery electrolytes. Nature, 658(8136), 636-642. https://doi.org/10.1038/s41586-026-11138-7

Image Credits: AI Generated

DOI: 10.1038/s41586-026-11138-7

Keywords: lithium-ion batteries, electrolytes, machine-learning force fields, molecular dynamics, solid-state NMR, solvation structure, solid electrolyte interphase, ethylene carbonate, quantum chemistry, reduction potentials, computational chemistry, energy storage

The study, published in the Journal of the American Chemical Society, was led by Professor Hou Guangjin and Associate Professor Liang Lixin from the Dalian Institute of Chemical Physics of the Chinese Academy of Sciences, in collaboration with Professor Dai Weili of Nankai University. Rather than examining finished crystals and inferring how they formed, the team watched the crystallization as it happened, using an advanced form of solid-state nuclear magnetic resonance spectroscopy that could track the fleeting interactions between inorganic building blocks and the organic molecules that guide their assembly. What they found was a proton-mediated, dual-channel gating mechanism in which two distinct chemical pathways alternately steer the reorganization of precursors and the birth of crystal nuclei.

To appreciate why this matters, it helps to understand the peculiar chemistry of molecular sieve synthesis. Aluminophosphate molecular sieves, designated AlPO4-n, and their silicon-containing cousins, SAPO-n, are typically crystallized under acidic, hydrothermal conditions from a gel containing aluminum, phosphorus, and, in the case of SAPOs, silicon sources. Suspended in this reactive soup are organic structure-directing agents, or OSDAs — carefully chosen organic molecules that act as templates around which the inorganic framework organizes itself. The final framework topology, and with it the catalytic properties that make these materials commercially valuable, depends intimately on how these organic and inorganic partners interact during nucleation and growth. But those interactions are transient, heterogeneous, and notoriously difficult to observe directly.

Conventional analytical methods have long struggled with this problem. X-ray diffraction, the standard tool for determining crystal structures, only becomes informative once crystals are large and ordered enough to diffract, by which point the critical early stages of nucleation are already history. Liquid-phase spectroscopic techniques, meanwhile, lose sensitivity or resolution when applied to the viscous, heterogeneous gels from which molecular sieves grow. The DICP-led team recognized that solid-state nuclear magnetic resonance offered a way through, provided they could push the technique’s sensitivity and speed far beyond what was previously possible. Their solution was a new ultrafast two-dimensional experiment correlating hydrogen-1 and aluminum-27 nuclei.

The technical innovation at the heart of the study is an ultrafast two-dimensional 1H-27Al heteronuclear multiple-quantum coherence, or HMQC, NMR experiment built on an innovative adiabatic SR4 dipolar-recoupling pulse sequence. In plain terms, the experiment works by deliberately reintroducing the weak magnetic interactions between hydrogen nuclei and nearby aluminum nuclei — interactions that are normally averaged away by the rapid spinning of the sample at the so-called magic angle. Recoupling these dipolar interactions allows the spectrometer to detect which protons sit close to which aluminum sites, producing a two-dimensional correlation map that reveals the local chemical neighborhoods within the reacting gel. The catch has always been that such recoupling sequences are inefficient and slow, often requiring hours of signal averaging to produce usable spectra.

The researchers overcame this bottleneck by introducing tanh/tan adiabatic inversion pulses into the recoupling sequence. Adiabatic pulses sweep through their frequency range in a way that makes them remarkably tolerant of imperfections in radiofrequency field strength and of resonance offsets, meaning they invert spins uniformly across a broad bandwidth. The result was a dramatic improvement in recoupling efficiency, enabling high-quality two-dimensional correlation spectra to be acquired in just 30 seconds. That speed is not merely a convenience; it is what makes the experiment truly operando. Crystallization is a dynamic process, and a technique that requires hours per spectrum can only offer static snapshots. With spectra in half a minute, the team could follow the evolving host-guest interactions in near real time as the gel transformed into crystal.

Combined with theoretical simulations, the spectroscopic data revealed a mechanism with an elegant two-channel structure. In the first channel, hydroxyl-driven hydrolysis takes center stage: protons attack and reorganize the inorganic precursor species, breaking and reforming bonds in the aluminophosphate network. In the second channel, the organic structure-directing agents take over, mediating templating interactions that lock the reorganized precursors into the geometry required for the target framework. Crucially, these two channels do not operate independently or simultaneously in a simple additive way. Instead, they alternate, gating each other’s activity in a rhythm that regulates precursor reorganization and nucleation. The protons, by toggling the system between hydrolysis-dominated and templating-dominated regimes, effectively conduct the entire assembly process.

The team went on to quantify how protonation accelerates the condensation reactions that build the framework itself. Their findings showed that protonation lowers the activation energy of P-O-Al condensation — the chemical step in which phosphorus and aluminum species link together to form the crystalline network — while simultaneously enhancing the thermodynamic driving force for framework formation. In other words, acidity does not merely speed the reaction along a pre-existing path; it changes both the kinetic barrier and the energetic payoff, making framework formation both easier to initiate and more favorable to complete. This dual effect provides a concrete, mechanistic explanation for a long-observed but poorly understood phenomenon: the strong dependence of molecular sieve crystallization rates and outcomes on the acidity of the synthesis medium.

Perhaps the most visually striking consequence of the mechanism lies in crystal morphology. The researchers found that the synergistic effects of proton-promoted nucleation and facet-selective stabilization — the preferential protection of certain crystal faces during growth — enabled the formation of nanosheet-like and nanorod-like molecular sieves across different framework topologies. Morphology control is far from a cosmetic concern in this field. The shape and aspect ratio of molecular sieve crystals determine their external surface area, the accessibility of their pore systems, and their diffusion characteristics, all of which feed directly into catalytic performance. A rational handle on morphology, grounded in an understanding of proton-mediated gating, could therefore translate into better-designed catalysts and adsorbents without the trial-and-error empiricism that has traditionally dominated synthesis optimization.

Professor Hou summarized the significance of the work by noting that the study provides new insights into the role of acidity in molecular sieve crystallization and establishes an advanced NMR approach for investigating crystallization mechanisms and guiding the rational synthesis of porous materials. The implications extend well beyond the aluminophosphate family. The operando solid-state MAS NMR methodology, with its ultrafast 2D correlation capability, is a general-purpose probe of host-guest interactions in reacting systems, and the conceptual framework of proton-gated dual-channel assembly may find echoes in other solution-grown functional materials. For a field that has spent half a century perfecting the products of crystallization while remaining largely blind to the process itself, the ability to watch protons orchestrate the birth of a crystal — one 30-second spectrum at a time — marks a genuine step change in how molecular sieves will be understood, and ultimately designed, in the years ahead.

Subject of Research: Proton-mediated crystallization mechanisms of aluminophosphate molecular sieves studied by operando solid-state NMR spectroscopy

Article Title: Scientists reveal how protons control molecular sieve crystallization

Article References: Scientists reveal how protons control molecular sieve crystallization. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: molecular sieves, aluminophosphate, SAPO, crystallization, protons, acidity, solid-state NMR, HMQC, nucleation, structure-directing agents, catalysis, porous materials

The technique in question, magic-angle spinning NMR, spins a solid sample at an angle of 54.74 degrees relative to the magnetic field, the so-called magic angle at which the anisotropic interactions that broaden spectral lines in solids are averaged away. For proteins, this opens a window onto structure and dynamics in states that are difficult to study by other means: microcrystals, membrane proteins embedded in lipid bilayers, and enormous assemblies such as virus capsids. In recent years, researchers have increasingly combined this approach with fluorine labelling, introducing atoms of the isotope fluorine-19 into proteins at chosen positions. Because natural biological molecules contain no fluorine at all, the fluorine signal stands out against a silent background, and the nucleus itself has favourable magnetic properties, including a high sensitivity and a wide chemical shift range that makes individual sites easy to distinguish.

Fluorine labelling also brings spectroscopic bonuses. Pairing fluorine-19 with carbon-13 at the same position in an aromatic ring enables two-dimensional experiments and allows scientists to exploit the TROSY effect, which preserves sharp lines in large molecules. Deuterating the labelled amino acids further reduces unwanted relaxation pathways and the need for proton decoupling. The high chemical shift anisotropy of fluorine and its strong dipolar couplings, once formidable obstacles in solid samples, have been tamed by modern probes capable of spinning samples at more than 55 kilohertz per second, making biomolecular fluorine MAS NMR increasingly practical for measuring long-range distances of up to roughly 20 angstroms and mapping structural features of proteins that would otherwise remain invisible.

Yet the method has a stubborn bottleneck. After each scan, the nuclear spins must relax back to their equilibrium polarisation before the next scan can begin, and for fluorine-19 in proteins this longitudinal relaxation time, T1, is often several seconds long. The optimal recycle delay for maximising the signal-to-noise ratio per unit time is directly proportional to T1, which means that most of the experiment is spent simply waiting for the spins to recover rather than collecting data. Ironically, the very strategy used to sharpen fluorine spectra makes the problem worse: deuterated labelling precursors remove the short-range proton-fluorine dipolar couplings that would otherwise help the spins relax quickly, stretching T1 even further.

The Austrian team, led by Lea Becker and Paul Schanda, attacked this bottleneck with paramagnetic doping, a technique long used to speed up proton-detected solid-state NMR. The idea is to add a chelated paramagnetic ion, typically copper(II) or gadolinium(III) bound to a carrier molecule, to the sample buffer. The unpaired electrons of the metal act as relaxation catalysts: through dipolar interactions with nearby nuclei, they accelerate the return of spin polarisation to equilibrium. Because the effect falls off with the sixth power of the electron-nucleus distance, only nuclei close to the metal are directly relaxed, but in protonated samples the enhancement spreads throughout the protein via proton-proton spin diffusion. The art lies in choosing a concentration that shortens T1 substantially without shortening T2, the transverse relaxation time, which would blur the spectral lines and destroy resolution.

What makes the new study notable is that it extends this approach to fluorine for the first time in a biological solid-state sample. The strength of the paramagnetic relaxation enhancement scales with the square of the gyromagnetic ratio of the observed nucleus, and fluorine-19 has one of the highest gyromagnetic ratios of any stable nucleus. The researchers therefore expected a strong effect at concentrations similar to those used in proton experiments. Their test bed was TET2, a dodecameric aminopeptidase from the hyperthermophilic archaeon Pyrococcus horikoshii, a 468-kilodalton assembly of twelve identical 39-kilodalton subunits that has served as a benchmark system in the Schanda laboratory for years. Each subunit carries four tryptophan residues, which the team labelled with fluorine by feeding the bacteria a specially synthesised deuterated 5-fluoroanthranilic acid precursor that the cells convert into 5-fluorotryptophan.

Before testing the dopants, the researchers first had to know which spectral peak belonged to which tryptophan. The fluorine spectrum of the labelled protein showed four distinct signals, one per tryptophan site, but assigning them required a clever genetic workaround. The team constructed five mutant proteins in which combinations of tryptophans were replaced by phenylalanines, leaving either a single fluorinated tryptophan or three of the four in place. Comparing the spectra of these mutants pinned down every signal. The measured fluorine T1 values without any dopant ranged from about 1.7 seconds to 6 seconds depending on the site, implying that an optimally sensitive experiment would demand recycle delays of up to 7.6 seconds between scans, an enormous expenditure of instrument time for a multi-hour or multi-day measurement.

The team then compared two gadolinium chelates, Gd(DTPA-BMA), the agent known clinically as Omniscan, and Gd(DTPA), at concentrations from 2 to 16 millimolar. Both compounds accelerated fluorine relaxation, but their behaviour differed strikingly. Gd(DTPA) produced a longitudinal enhancement roughly seven times stronger per millimolar concentration than Gd(DTPA-BMA), and a transverse enhancement more than fourteen times stronger, differences the authors attribute to the distinct physicochemical properties of the two chelates, such as their water exchange rates. That raw power came at a cost: at 8 millimolar, Gd(DTPA) broadened the fluorine lines so severely that the individual peaks merged, and it also altered the crystallisation behaviour of the protein, likely by binding to its surface. Gd(DTPA-BMA) proved the better compromise. At 8 millimolar, it cut the average fluorine T1 from about 3.2 seconds to 0.92 seconds, reducing the optimal recycle delay from roughly 4 seconds to 1.2 seconds, a more than threefold acceleration with no significant loss of resolution.

Perhaps the most intriguing findings concern the residue-by-residue pattern of the enhancement. The four tryptophan sites responded differently to the dopants, and the pattern depended on which relaxation parameter was measured, which compound was used, and whether fluorine or carbon was observed. The team tested whether these differences could be explained by simple structural parameters, such as the solvent accessibility of each tryptophan or the density of surrounding protons, but found no direct correlation. One site, W164, showed an unusually strong longitudinal enhancement relative to its transverse enhancement for both compounds, a signature the authors suggest could reflect a locally altered rotational correlation time of the gadolinium complex, perhaps through weak binding to the protein. Because the longitudinal and transverse enhancements sample the spectral density at different frequencies, they respond differently to such changes, offering a sensitive but complicated probe of the molecular environment.

The implications reach well beyond one protein. Faster fluorine MAS NMR means more signal per unit time for the experiments that benefit most from fluorine labelling: distance measurements, studies of conformational dynamics, and structural work on large assemblies where sensitivity is chronically scarce. The authors anticipate that the approach can be transferred to many sample types and experimental designs, while cautioning that the optimal dopant and concentration will depend on the specific system, since the balance between accelerating T1 and preserving T2 is delicate and compound-dependent. For a field that has watched fluorine NMR swing back into fashion on the strength of new labelling chemistry and faster spinning hardware, the message of this study is that one of the oldest sensitivity tricks in the solid-state NMR toolbox now works for fluorine too, and it may make the difference between an experiment that takes days and one that fits into an overnight run.

Subject of Research: Paramagnetic doping to accelerate fluorine-19 magic-angle spinning NMR of proteins

Article Title: Accelerated 19F biomolecular magic-angle spinning NMR with paramagnetic dopants

Article References: Becker, L. M., Toscano, G., Kapitonova, A., Singh, R., Guillerm, U., Lichtenecker, R. J., & Schanda, P. (2026). Accelerated 19 F biomolecular magic-angle spinning NMR with paramagnetic dopants. Magnetic Resonance, 7(1), 29-37. https://doi.org/10.5194/mr-7-29-2026

Image Credits: AI Generated

DOI: 10.5194/mr-7-29-2026

Keywords: fluorine-19 NMR, magic-angle spinning, paramagnetic doping, gadolinium chelates, protein structure, solid-state NMR, relaxation enhancement, TET2 aminopeptidase, fluorotryptophan labelling, T1 relaxation, structural biology, spectroscopy

The central question the authors tackle is deceptively simple: when two perturbative schemes claim to describe the same experiment, are they actually computing the same thing? The answer, worked out in detail in their manuscript posted on 11 September 2026 as a discussion preprint under review for the journal Magnetic Resonance, is a qualified yes. Bock, Ernst and Uhrig identify the Floquet–Magnus expansion as the essential bridge between the two formalisms. By expressing the Floquet solution in the language of the Magnus expansion, the standard tool for time-dependent perturbation theory, they are able to prove the mathematical equivalence of Floquet theory and average Hamiltonian theory term by term. The equivalence, however, is formal rather than practical. Two calculational routes that lead to identical expressions in principle can still differ enormously in how error-prone, transparent and numerically stable they are along the way, and it is precisely here that the new work makes its mark.

Average Hamiltonian theory has been the workhorse of solid-state NMR since the 1960s. Its logic is intuitive: if the Hamiltonian repeats with a period that is short compared with the timescale on which the interesting dynamics unfold, then over many cycles the system behaves, to a good approximation, as if it evolved under a single constant, effective Hamiltonian. The trick is to compute that effective Hamiltonian as a series of corrections, each involving nested commutators of the Fourier components of the periodic Hamiltonian. The method has an Achilles heel, though. At higher orders it demands explicit time integrations over products of oscillating terms, and those integrations are notorious breeding grounds for algebraic mistakes. Anyone who has carried an average Hamiltonian calculation to third or fourth order knows the tangle of trigonometric factors and commutator chains that must be evaluated without a single slip.

Floquet theory approaches the same physics from a different direction, treating the periodic drive not as a perturbation to be averaged away but as a structural feature of the problem. In the Floquet picture, the Hilbert space of the quantum system is enlarged to include sideband indices that count quanta of the driving frequency, a step closely analogous to how quasi-momentum appears for electrons in a crystal lattice. The time-dependent problem then becomes a time-independent one in the extended space, and standard perturbative machinery can be brought to bear. The authors advocate a specific variant, the Floquet–Van Vleck approach, in which the effective Hamiltonian is accompanied by a second object called the kick operator. The kick operator accounts for the micromotion within each driving cycle, the fast wiggles that the effective Hamiltonian deliberately ignores, and keeping it in the formalism turns out to be essential for a clean and consistent description.

One of the paper’s key technical contributions is a calculation scheme that avoids explicit integration altogether. By working with the Fourier components of the periodic Hamiltonian and separating secular contributions, those terms that survive averaging because they commute with the dominant part of the Hamiltonian, from non-secular ones, the authors obtain a hierarchy of corrections that is far less prone to algebraic error. On this foundation they derive the first four orders of both average Hamiltonian theory and the Floquet expansion, providing the community with explicit analytical expressions at orders where previous treatments often stopped or grew unwieldy. The consistent separation of secular and non-secular pieces, they report, is especially advantageous for numerical robustness, because it keeps the calculation stable even when the perturbative parameters are not particularly small, a situation that arises routinely in real spectrometers where spinning frequencies and radiofrequency amplitudes push against the limits of simple approximations.

The payoff of this rigor is quantified in a striking comparison. Despite the formal equivalence of the two frameworks, the accuracy of the Floquet–Van Vleck approach is about three times better than that of average Hamiltonian theory at the orders studied. The reason is subtle but important: the two methods organize the same physics differently, and the Floquet organization, with its clean split between the effective Hamiltonian and the kick operator, distributes truncation errors more favorably. For experimentalists designing pulse sequences or interpreting high-resolution spectra of solids, a threefold improvement in the accuracy of predicted frequencies and evolution is not an academic nicety. It can determine whether a calculated spectrum matches the measured one well enough to extract structural information about a protein, a catalyst or a battery material, or whether discrepancies are blamed on the theory when they are actually artifacts of the calculational route.

To ground the mathematics in real experiments, the authors examine two canonical test cases. The first is the Bloch–Siegert shift, a small displacement of resonance frequencies caused by the counter-rotating component of an oscillating radiofrequency field, a phenomenon known since the early days of magnetic resonance. Their calculation of this shift out to third order, highlighted in the open discussion accompanying the preprint as particularly revealing, shows how higher-order terms become observable when strong decoupling fields are applied. The second test case involves dipolar coupled spin systems under magic-angle spinning, the technique in which a sample is spun at precisely the angle where the anisotropic dipole-dipole interactions average most effectively, forming the backbone of modern solid-state NMR. Here the periodic drive is mechanical, set by the rotor frequency, and the interplay between spinning, chemical shifts and spin-spin couplings is exactly the kind of problem where the choice of theoretical framework matters most.

The discussion thread attached to the preprint illustrates why this work resonates with practitioners. Commenter Tom Barbara, a veteran of micro-coil NMR, noted that groups working with radiofrequency fields producing nutation frequencies on the order of a megahertz had long wondered why such Bloch–Siegert shifts went unnoticed in spectra. Matthias Ernst’s reply explains the puzzle: all lines in a spectrum shift by the same amount, and since spectroscopists typically measure only relative frequencies, a uniform shift is invisible unless an undecoupled reference is recorded. He points out that recent publications have not only observed but exploited the heteronuclear Bloch–Siegert shift, using it to calibrate the radiofrequency field amplitude of insensitive nuclei, and he recalls personally noticing the significant proton shift that appears when decoupling nitrogen-15 during acquisition. The exchange also touches on a terminological subtlety: the frequency shifts used in MRI to map radiofrequency fields share the Bloch–Siegert name but arise from a different mechanism with a different symmetry under rotations, a confusion that has trailed the field since the early papers by Bodenhausen and Emsley.

What elevates this preprint beyond a methodological housekeeping exercise is its timing. Solid-state NMR is pushing into regimes of ever-faster magic-angle spinning, with rotors now exceeding spinning frequencies of 70 kilohertz in some applications, and into decoupling schemes for nuclei that are difficult to observe directly. In these regimes the perturbative parameters are strained, the boundary between secular and non-secular terms shifts, and the silent assumptions baked into older treatments begin to creak. By proving exactly where Floquet theory and average Hamiltonian theory agree, identifying the Floquet–Magnus expansion as the hinge connecting them, and demonstrating that the Floquet–Van Vleck route with its explicit kick operator delivers superior numerical accuracy, Bock, Ernst and Uhrig have given the magnetic resonance community both a conceptual map and a practical recommendation. The authors’ bottom line is unambiguous: for analyzing magic-angle spinning experiments, Floquet theory is the tool of choice, prized for its numerical robustness, efficiency and accuracy. As the preprint moves through peer review, its four-order analytical results are already positioned to become a reference point for anyone who needs to know precisely what a periodically driven spin system is really doing, cycle after cycle, beneath the measured spectrum.

Subject of Research: Mathematical equivalence of Floquet theory and average Hamiltonian theory and their application to solid-state NMR spectroscopy

Article Title: Floquet Theory and Average Hamiltonian Theory Revisited: Equivalence, Convergence and Applications to NMR

Article References: Bock, A. J., Ernst, M., & Uhrig, G. S. (2026). Floquet Theory and Average Hamiltonian Theory Revisited: Equivalence, Convergence and Applications to NMR. https://doi.org/10.5194/mr-2026-10

Image Credits: AI Generated

DOI: 10.5194/mr-2026-10

Keywords: Floquet theory, average Hamiltonian theory, NMR spectroscopy, magic-angle spinning, Bloch-Siegert shift, spin dynamics, perturbation theory, Floquet-Magnus expansion, kick operator, solid-state NMR, decoupling, effective Hamiltonian

The review, led by a team at Guangdong University of Technology with collaborators at Walailak University and Shenzhen University, synthesizes recent advances in multiscale characterization and computational modeling of geopolymer materials. Its central message is that the field is undergoing a structural transition, documented through bibliometric analysis, away from conventional phase and composition identification and toward precise atomic-scale analysis, dynamic in situ monitoring, and data-driven material design. In other words, geopolymer science is maturing from a descriptive craft into a quantitative, predictive discipline, and the tools driving that shift are the same ones that transformed semiconductor physics and structural biology: synchrotron radiation, neutron scattering, advanced electron microscopy, molecular simulation, and machine learning.

What makes geopolymers so difficult to characterize is their peculiar structural personality. Unlike crystalline ceramics, they are ordered over short and medium ranges but completely disordered over long ranges, forming three-dimensional aluminosilicate networks in which alkali cations balance the negative charge of aluminum tetrahedra. This amorphous nature defeats the workhorse technique of traditional materials science, X-ray diffraction, which relies on periodic lattice order to produce interpretable patterns. The situation is compounded by the growing trend of blending multiple industrial solid wastes into a single binder, which introduces substantial compositional complexity, heterogeneous gel formation, and dynamic structural reconstruction as the material cures and ages.

To pierce this amorphous fog, the review highlights the power of advanced spectroscopic techniques. Solid-state nuclear magnetic resonance spectroscopy, in particular, has become indispensable for resolving local atomic configurations, distinguishing silicon coordination environments, quantifying network connectivity, and identifying how heteroatoms such as iron, magnesium, and heavy metals are incorporated into the gel framework. Infrared spectroscopy, long used to track the progressive shift of silicon-oxygen-aluminum stretching bands during geopolymerization, now operates in time-resolved and spatially resolved modes that can watch gel nucleation unfold in real time. For iron-rich precursors such as red mud and volcanic ash, Mössbauer spectroscopy reveals the oxidation state and coordination of iron species, a critical variable because iron can either participate in the binder network or remain as inert inclusions that dictate strength and durability.

Perhaps the most striking technical development the review documents is the rise of pair distribution function analysis, a total-scattering method that extracts structural information from the diffuse scattering that conventional diffraction discards. Applied with synchrotron X-rays and neutrons, pair distribution function analysis has allowed researchers to reconstruct the atomic structure of geopolymer gels directly, resolving debates about the role of charge-balancing extra-framework aluminum and tracking how local structure evolves during the earliest stages of gel formation. In situ neutron pair distribution function experiments have even captured the structural evolution of geopolymer gels as they form, while related work has mapped how local structure changes with temperature, carbonation, sulfate attack, and high-temperature exposure, connecting atomic-scale chemistry to the degradation mechanisms that matter in service.

At larger length scales, advanced microscopy is revealing the hierarchical architecture that atomic spectroscopy cannot see. Scanning and transmission electron microscopy, increasingly combined with focused ion beam sample preparation and cryogenic transfer, expose the nanoscale morphology of gel particles and the interfacial transition zones that govern composite behavior. Cryo-electron microscopy has delivered a genuinely viral result in the cement science community: direct observation of multistep nucleation and growth of aluminosilicate gel, showing that these amorphous binders assemble through intermediate phases rather than simple precipitation. Meanwhile, atomic force microscopy and nanoindentation map mechanical heterogeneity across gel phases and interfaces, and quantitative backscattered electron imaging paired with energy-dispersive spectroscopy now allows full-component characterization of interfacial zones in recycled and fiber-reinforced geopolymer concretes.

Pore structure, the hidden skeleton that controls transport, durability, and freeze-thaw resistance, has likewise come into sharper focus. X-ray and neutron tomography reconstruct three-dimensional pore networks non-destructively, from laboratory micro-computed tomography to hard X-ray nanotomography capable of resolving features tens of nanometers across. Low-field nuclear magnetic resonance relaxometry has emerged as a rapid, non-destructive probe of pore size distribution and water states, tracking setting, reaction kinetics, and even the influence of paramagnetic iron in waste-derived binders. Small-angle neutron scattering resolves bimodal pore evolution during early curing, and neutron radiography and tomography visualize capillary water absorption and moisture transport in real time, exploiting the exceptional sensitivity of neutrons to hydrogen.

The review also emphasizes that simulation and artificial intelligence are no longer peripheral add-ons but central pillars of the field. Molecular dynamics simulations of sodium aluminosilicate hydrate and calcium aluminosilicate hydrate gels now illuminate dissolution mechanisms, adsorption of heavy metals and radionuclides, ion migration in nanopores, and the interfacial bonding between gels and aggregates or fibers, often in direct dialogue with experimental scattering data. On the data side, machine learning models predict compressive strength, freeze-thaw degradation, bond strength, and multi-objective performance trade-offs involving cost and carbon dioxide emissions, with interpretable and transfer-learning frameworks extending predictions to ultra-high-performance geopolymer systems. Emerging machine learning interatomic potentials promise to bring quantum-level accuracy to simulations of cementitious systems at unprecedented scale.

Why does all this matter beyond the laboratory? Geopolymers sit at the intersection of two global imperatives: decarbonizing construction and managing mountains of industrial waste. They have demonstrated potential in marine structures, road construction, nuclear and heavy metal waste immobilization, fire-resistant panels, 3D-printed elements, and even neutron-shielding composites for radiation protection. But every one of those applications depends on reproducible performance, and reproducibility demands structure-property relationships grounded in measurement rather than recipe. The characterization toolkit assembled in this review is precisely what converts a variable waste stream into an engineering material whose atomic network, gel hierarchy, and pore architecture can be tuned deliberately.

The authors’ synthesis points toward a future in which synchrotron beamlines, neutron facilities, automated microscopy, and machine learning pipelines operate as an integrated design engine, closing the loop from composition to structure to property to prediction. Their bibliometric evidence shows the literature itself shifting toward dynamic monitoring and data-driven design, a signal that the community is already building that engine. For a material class once dismissed as too messy for fundamental science, geopolymers are becoming a showcase for how modern multiscale characterization can tame even the most disordered matter, and the payoff may be a generation of sustainable, high-performance binders designed on the computer before they are ever mixed in the lab.

Subject of Research: Multiscale structural characterization and structure-property relationships of geopolymer materials

Article Title: Review: advanced characterization of geopolymer materials-from multiscale structural understanding to structure-property relationships

Article References: He, X., Pantongsuk, T., Chen, W., Deng, R., Yu, T., & Zhang, B. (2026). Review: advanced characterization of geopolymer materials-from multiscale structural understanding to structure-property relationships. Journal of Materials Science. https://doi.org/10.1007/s10853-026-13826-1

Image Credits: AI Generated

DOI: 10.1007/s10853-026-13826-1

Keywords: geopolymers, aluminosilicate binders, low-carbon cement, solid-state NMR, pair distribution function, synchrotron radiation, neutron scattering, electron microscopy, molecular dynamics, machine learning, industrial solid waste, structure-property relationships