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At least 217 records · Page 12

Sunspots and the physics of magnetic flux tubes. VIII - Overstability in a magnetic field in a downdraft

The dynamical properties of convective overstability in a vertical magnetic field with a downdraft are considered. A variety of effects is illustrated. The overstability produces Alfven waves propagating both upward or downward along the magnetic field. The favored direction of emission may be upward or downward depending upon the magnitude of the heat transport coefficient. The largest asymmetry is produced by a difference in reflectivity between the upper and lower boundaries. It is shown that a very modest reflection coefficient of the upper boundary, with no reflection at the lower boundary, causes most of the waves to be emitted downward, and vice versa. Applying these results to the flux tubes extending up through the convective zone of the Sun, it follows that those flux tubes are dynamically active beneath the surface, as suggested earlier by ourselves and others, but there is no reason to expect any significant wave flux to appear in the field above the surface. The waves propagate downward into the Sun and are presumably dispersed there by the nonlinear interaction with the turbulent convection, etc.

Parker, E. N.

Fundamental Studies of the Vibrational, Electronic, and Photophysical Properties of Tetrapyrrolic Architectures

The ability to capture and utilize light in the near-ultraviolet (NUV), visible and near-infrared (NIR-I and NIR-II) spectral regions (i.e., 320–400, 400–700, 700–1000, 1000–1700 nm) is essential for any solar-energy conversion scheme. Nature employs chlorophylls and bacteriochlorophylls in light-harvesting architectures to absorb light in the blue and red/NIR regions. Accessory pigments (carotenoids, bilins) augment absorption of the (bacterio)chlorophylls in the green region. The harvested energy is funneled to a reaction center protein, where charge separation occurs. Subsequent migration of the electron and the hole stabilizes and stores the energy from light via redox chemistry. The long-term objective of the Bocian/Holten&Kirmaier/Lindsey research program under this DOE grant has been to develop tetrapyrrole-based molecular architectures that absorb sunlight, funnel energy and separate charge with high efficiency. Integral to the program has been iterative cycles of design, synthesis and characterization that provided deep insights into the relationships between chemical composition, electronic structure, and key static and dynamic properties (vibrational, redox, photophysical, energy/charge transfer) of tetrapyrrolic systems. Such architectures included monomers, dyads, larger arrays, and complexes with accessory components. The objective was to develop molecular designs and guiding principles to enhance current and future energy-conversion schemes. Molecular arrays targeted to address one or more fundamental questions concerning light harvesting and energy/charge transfer were constructed from analogues of the naturally occurring hemes, chlorophylls and bacteriochlorophylls. Diverse, tunable synthetic building blocks were prepared that spanned the three respective tetrapyrrole families, which are the porphyrins, chlorins and bacteriochlorins. Thus, the research focused on porphyrins as well as synthetic surrogates for chlorophylls (chlorins, 13 1 -oxophorbines and chlorin-imides) and bacteriochlorophylls (bacteriochlorins, bacterio-13 1 -oxophorbines and bacteriochlorin-imides), generically termed hydroporphyrins. Although the three tetrapyrrole classes (porphyrins, chlorins and bacteriochlorins) absorb light strongly in the violet-blue spectral region, the long-wavelength absorption band typically lies in the green-orange, red, and NIR regions, respectively, with increasing intensity. Understanding the spectra, electronic structure, and energy/charge-transfer properties of such tetrapyrrolic macrocycles is of central importance for the rational design of molecular architectures for solar-energy conversion. Our integrated program of molecular design and synthesis coupled with a variety of spectroscopic, electrochemical, and computational studies have probed from first principles how structural and electronic properties of tetrapyrrolic macrocycles dictate spectral properties as well as the rates of ground-state hole/electron transfer and excited-state energy flow in multicomponent architectures. Individual molecules and multicomponent architectures were designed to test ideas of fundamental importance, often requiring the development of new synthetic methodology. The members of the collaborative team had almost daily discussions by phone and/or e-mail concerning design of molecules, flow of compounds between the labs, planning of physical characterization studies, discussing results and analysis and integrating into design of next generation architectures, and the preparation of manuscripts. Furthermore, students and postdocs in the different labs routinely communicated with one another to facilitate the advancement of the research activities. In short, a highly integrated and collaborative research program was well established among the groups. The research effort involved molecular design and synthesis of synthetic molecular architectures by the Lindsey group integrated with physicochemical and photophysical characterization by the Bocian group and the Holten&Kirmaier group (Figure 2). The Bocian group carried out electrochemical, electron paramagnetic resonance (EPR), resonance Raman (RR), and Fourier-transform infrared (FT-IR) studies, as well as density functional theory (DFT) calculations and the time-dependent extension (TDDFT) to gain insight into excited-state properties. The Holten&Kirmaier group carried out static and time-resolved absorption and fluorescence spectroscopy studies and simulated absorption spectra using molecular orbital (MO) energies from DFT as input to the four-orbital model to complement the TDDFT calculations. The combined measurements provided understanding of the vibrational/electronic properties of the individual molecules and the changes that occur upon incorporation into multicomponent architectures. This information underpinned elucidating the mechanisms and timescales of ground-state hole/electron transfer and excited-state energy and charge transfer.

14 SOLAR ENERGY

Alternative descriptions of wave and particle aspects of the harmonic oscillator

The dynamical properties of the wave and particle aspects of the harmonic oscillator can be studied with the help of the time-dependent Schroedinger equation (SE). Especially the time-dependence of maximum and width of Gaussian wave packet solutions allow to show the evolution and connections of those two complementary aspects. The investigation of the relations between the equations describing wave and particle aspects leads to an alternative description of the considered systems. This can be achieved by means of a Newtonian equation for a complex variable in connection with a conservation law for a nonclassical angular momentum-type quantity. With the help of this complex variable, it is also possible to develop a Hamiltonian formalism for the wave aspect contained in the SE, which allows to describe the dynamics of the position and momentum uncertainties. In this case the Hamiltonian function is equivalent to the difference between the mean value of the Hamiltonian operator and the classical Hamiltonian function.

Schuch, Dieter

Mesophyll conductance and cell wall composition: insights from a meta‐analysis across species

Mesophyll conductance (g m ) plays a critical role in plant photosynthesis by regulating the diffusion of CO 2 from substomatal cavities to the site of carbon fixation within the chloroplasts. Despite its importance, our understanding of the factors influencing g m , particularly the role of cell wall properties, remains incomplete. In this issue of New Phytologist , Roig-Oliver et al. ( 2025 ; pp. 2384–2391) address this knowledge gap through a comprehensive meta-analysis across multiple species from the major clades of land plants. By analyzing trends in g m and two other key physiological parameters – the ratio of pectin to cellulose and hemicellulose in the cell wall by weight (P/(C + H)), and cell wall thickness (T cw ) – their findings reveal strong correlations of gm with the combination of P/(C + H) and T cw , where g m is directly proportional to P/(C + H) and inversely proportional to T cw . This work not only underscores the influence of cell wall composition on CO 2 diffusion but also provides a valuable framework for further experimental and modeling studies. By shedding light on the dynamic properties of cell walls, shaped by both composition and environmental regulation, this study opens new frontiers for understanding and improving photosynthesis.

59 BASIC BIOLOGICAL SCIENCES

Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts.

43 PARTICLE ACCELERATORS

Delving into the depths of NGC 3783 with XRISM V. Broad-band modelling of ionized outflows

The Seyfert 1 galaxy NGC 3783 hosts a multi-phase warm absorber (WA) that has been extensively studied in the X-ray band. High-resolution spectra from 2000−2001 revealed a complex outflow with multiple ionization and velocity components. Two decades later, new XMM-Newton and XRISM observations allow us to investigate the long-term evolution of these outflows. We performed joint spectral modelling of the XMM-Newton /RGS and XRISM/Resolve time-averaged spectra using the pion photoionization code within SPEX. We derived the ionization parameter, column density, turbulent velocity, and outflow velocity for each absorption component, and investigated their thermal stability and absorption measure distribution (AMD) to characterize the physical and dynamical properties of the WA in NGC 3783 in 2024. We compared these results with the 2000−2001 epoch to assess long-term variability, stability, and possible changes in the absorber population. We identify eight WA components spanning log ξ = 1.08 − 3.38 and outflow velocities of 480−1230 km s −1 . The ranges of column densities and turbulent velocities remain broadly consistent with the WAs from 2000−2001, but the earlier data contained more low-ionization, high-velocity components. The total column density of all the outflows in 2024 is 1.5 times larger than in 2000−2001, which means that it has been replenished by fresh material. The dominant unresolved transition array (UTA) absorber (component B3) has increased its column density by a factor of three while maintaining a similar ionization parameter. The WAs in NGC 3783 have undergone significant structural and dynamical evolution over the past 24 years.

79 ASTRONOMY AND ASTROPHYSICS

Emergent topological quasiparticle kinetics in constricted nanomagnets

The ubiquitous domain wall kinetics under magnetic field or current application describes the dynamic properties in nanostructured magnets. However, when the geometrical size of a nanomagnetic system is constricted to the limiting domain wall length scale, the competing energetics between anisotropy, exchange, and dipolar interactions can cause emergent kinetics due to quasiparticle relaxation, similar to bulk magnets of atomic origin. This paper presents a joint experimental and theoretical study to support this argument: constricted nanomagnets, made of antiferromagnetic and paramagnetic neodymium thin film with honeycomb motif, reveal fast kinetic events at picosecond timescales due to the relaxation of topological quasiparticles that persist to low temperature in the absence of any external stimuli. This discovery is especially important considering the fact that paramagnets or antiferromagnets have no net magnetization. Yet, the kinetics in neodymium nanostructures is quantitatively similar to that found in ferromagnetic counterparts and only varies with the thickness of the specimen. This suggests that a universal, topological quasiparticle-mediated dynamical behavior can be prevalent in nanoscopic magnets, irrespective of the nature of the underlying magnetic material.

Guo, Jiasen

Experimental characterization of deployable trusses and joints

The structural dynamic properties of trusses are strongly affected by the characteristics of joints connecting the individual beam elements. Joints are particularly significant in that they are often the source of nonlinearities and energy dissipation. While the joints themselves may be physically simple, direct measurement is often necessary to obtain a mathematical description suitable for inclusion in a system model. Force state mapping is a flexible, practical test method for obtaining such a description, particularly when significant nonlinear effects are present. It involves measurement of the relationship, nonlinear or linear, between force transmitted through a joint and the relative displacement and velocity across it. An apparatus and procedure for force state mapping are described. Results are presented from tests of joints used in a lightweight, composite, deployable truss built by the Boeing Aerospace Company. The results from the joint tests are used to develop a model of a full 4-bay truss segment. The truss segment was statically and dynamically tested. The results of the truss tests are presented and compared with the analytical predictions from the model.

Ikegami, R.

Response properties of pigeon otolith afferents to linear acceleration

In the present study, the sensitivity to sinusoidal linear accelerations in the plane of the utricular macula was tested in afferents. The head orientation relative to the translation axis was varied in order to determine the head position that elicited the maximal and minimal responses for each afferent. The response gain and phase values obtained to 0.5-Hz and 2-Hz linear acceleration stimuli were then plotted as a function of head orientation and a modified cosine function was fit to the data. From the best-fit cosine function, the predicted head orientations that would produce the maximal and minimal response gains were estimated. The estimated maximum response gains to linear acceleration in the utricular plane for the afferents varied between 75 and 1420 spikes s-1 g-1. The mean maximal gains for all afferents to 0.5-Hz and 2-Hz sinusoidal linear acceleration stimuli were 282 and 367 spikes s-1 g-1, respectively. The minimal response gains were essentially zero for most units. The response phases always led linear acceleration and remained constant for each afferent, regardless of head orientation. These response characteristics indicate that otolith afferents are cosine tuned and behave as one-dimensional linear accelerometers. The directions of maximal sensitivity to linear acceleration for the afferents varied throughout the plane of the utricle; however, most vectors were directed out of the opposite ear near the interaural axis. The response dynamics of the afferents were tested using stimulus frequencies ranging between 0.25 Hz and 10 Hz (0.1 g peak acceleration). Across stimulus frequencies, most afferents had increasing gains and constant phase values. These dynamic properties for individual afferents were fit with a simple transfer function that included three parameters: a mechanical time constant, a gain constant, and a fractional order distributed adaptation operator.

Non-NASA Center

Unravelling the orbits of cluster galaxy populations according to their dominant gas ionization source

ABSTRACT We investigate the kinematical and dynamical properties of cluster galaxy populations classified according to their dominant source of gas ionization, namely: star-forming (SF) galaxies, optical active galactic nuclei (AGNs), mixed SF plus AGN ionization (transition objects, T), and quiescent (Q) galaxies. We stack 8892 member galaxies from 336 relaxed galaxy clusters to build an ensemble cluster and estimate the observed projected profiles of numerical density and velocity dispersion, $\sigma _P(R)$, of each galaxy population. The MAMPOSSt code and the Jeans equations inversion technique are used to constrain the velocity anisotropy profiles of the galaxy populations in both parametric and non-parametric ways. We find that Q (SF) galaxies display the lowest (highest) typical cluster-centric distances and velocity dispersion values. Transition galaxies are more concentrated and tend to exhibit lower velocity dispersion values than SF galaxies. Galaxies that host an optical AGN are as concentrated as Q galaxies but display velocity dispersion values similar to those of the SF population. MAMPOSSt is able to find equilibrium solutions that successfully recover the observed $\sigma _P(R)$ profile only for the Q, T, and AGN populations. We find that the orbits of all populations are consistent with isotropy in the inner regions, becoming increasingly radial with the distance from the cluster centre. These results suggest that Q galaxies are in equilibrium within their clusters, while SF galaxies have more recently arrived in the cluster environment. Finally, the T and AGN populations appear to be in an intermediate dynamical state between those of the SF and Q populations.

Valk, Greique A. (ORCID:0009000827731299)

Dynamically hot galaxies. I - Structural properties

Results are reported from an analysis of the structural properties of dynamically hot galaxies which combines central velocity dispersion, effective surface brightness, and effective radius into a new 3-space (k), in which the axes are parameters that are physically meaningful. Hot galaxies are found to divide into groups in k-space that closely parallel conventional morphological classifications, namely, luminous ellipticals, compacts, bulges, bright dwarfs, and dwarf spheroidals. A major sequence is defined by luminous ellipticals, bulges, and most compacts, which together constitute a smooth continuum in k-space. Several properties vary smoothly with mass along this continuum, including bulge-to-disk ratio, radio properties, rotation, degree of velocity anisotropy, and 'unrelaxed'. A second major sequence is comprised of dwarf ellipticals and dwarf spheroidals. It is suggested that mass loss is a major factor in hot dwarf galaxies, but the dwarf sequence cannot be simply a mass-loss sequence, as it has the wrong direction in k-space.

Bender, Ralf

Quantum tensor network algorithms for evaluation of spectral functions on quantum computers

We investigate quantum algorithms derived from tensor networks to simulate the static and dynamic properties of quantum many-body systems. Using a sequentially prepared quantum circuit representation of a matrix product state (MPS) that we call a quantum tensor network (QTN), we demonstrate algorithms to prepare ground and excited states on a quantum computer and apply them to molecular nanomagnets (MNMs) as a paradigmatic example. In this setting, we develop two approaches for extracting the spectral correlation functions measured in neutron-scattering experiments: (a) a generalization of the SWAP test for computing wave function overlaps and, (b) a generalization of the notion of matrix product operators to the QTN setting which generates a linear combination of unitaries. The latter method is discussed in detail for translationally invariant spin-half systems, where it is shown to reduce the qubit resource requirements compared with the SWAP method and may be generalized to other systems. We demonstrate the versatility of our approaches by simulating spin-1/2 and spin-3/2 MNMs, with the latter being an experimentally relevant model of a Cr$^{3+}_{8}$ ring. Here, our approach has qubit requirements that are independent of the number of constituents of the many-body system and scale only logarithmically with the bond dimension of the MPS representation, making them appealing for implementation on near-term quantum hardware with mid-circuit measurement and reset.

Neutron scattering

Electronic Excitation‐Driven β‐Ga 2 O 3 Metastability Transformation and Self‐Organization Mechanism: β→κ/γ/δ Phases

Here, irradiation-driven multiphase self-organization presents emergent opportunities for the customization of nanoscale engineering properties, dynamically tuning strain-field distributions and interfacial electronic structures. Responding to intense electronic excitation-induced energy deposition, the dominant phase transformations, with varying Gibbs free energy $Δ,G^o_f$ are confirmed as β → κ → γ → δ that are located in specific microregions for Gallium (III) oxide (Ga 2 O 3 ), as follows: (i) Surface-localized interstitial accumulation under compressive stress triggers β → δ via semi-coherent interface formation. (ii) Tensile stress within latent tracks drives vacancy-mediated oxygen layer truncation (4/12 periodicity along ⟨0001⟩), stabilizing coherent 4H (ABCB) κ and 3C (ABC) β (ABC) interfaces through strain-compensated octahedral distortion. (iii) Screw dislocation-mediated lattice relaxation induces β → γ via cation disordering (Ga 3+ occupancy at β-interstitial sites), forming metastable spinel γ with mixed occupancy across 16d/8a Wyckoff sites. Irradiation-driven β-Ga 2 O 3 →κ/γ/δ transitions, as mechanistically revealed via inelastic thermal spike (i-TS) calculations and molecular dynamics simulations, induce defect-mediated nonlinear photoresponse, critical for optoelectronic engineering.

Han, Xinqing [Shandong Univ., Jinan (China)]

phosaa14SB and phosaa19SB: Updated Amber Force Field Parameters for Phosphorylated Amino Acids

Phosphorylated amino acids are involved in many cell regulatory networks; proteins containing these post-translational modifications are widely studied both experimentally and computationally. Simulations are used to investigate a wide range of structural and dynamic properties of biomolecules, such as ligand binding, enzyme-reaction mechanisms, and protein folding. However, the development of force field parameters for the simulation of proteins containing phosphorylated amino acids using the Amber program has not kept pace with the development of parameters for standard amino acids, and it is challenging to model these modified amino acids with accuracy comparable to proteins containing only standard amino acids. In particular, the popular ff14SB and ff19SB models do not contain parameters for phosphorylated amino acids. Here, the dihedral parameters for the side chains of the most common phosphorylated amino acids are trained against reference data from QM calculations adopting the ff14SB approach, followed by validation against experimental data. Finally, library files and corresponding parameter files are provided, with versions that are compatible with both ff14SB and ff19SB.

77 NANOSCIENCE AND NANOTECHNOLOGY

Accurate Force Field for Carbon Dioxide–Silica Interactions Based on Density Functional Theory

Fluid–silica interfaces are ubiquitous in chemistry, occurring in both natural geochemical environments and practical applications ranging from separations to catalysis. Simulations of these interfaces have been, and continue to be, a significant avenue for understanding their behavior. A constraining factor, however, is the availability of accurate force fields. Most simulations use traditional “mixing rules” to determine nonbonded dispersion interactions, an approach that has not been critically examined. Here, in this study, we present Lennard-Jones parameters for the interaction of carbon dioxide with silica interfaces that are optimized to reproduce density functional theory (DFT)-based binding energies. The modeling is based on the recently developed silica-DDEC force field, whose atomic charges are consistent with DFT calculations. Standard mixing rules are found to predict weaker CO 2 binding to silica than that obtained from DFT, an effect corrected by the optimized parameters given here. This behavior extends to other silica force fields (Clayff and Gulmen-Thompson), and the present Lennard-Jones parameters improve their performance as well. The effects of improved Lennard-Jones parameters on the structural and dynamical properties of condensed CO 2 in silica slit pores are also examined.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Atomistic Insights into Lithium–Glyme Solvate Ionic Liquids: Effects of Chain Length and Anion Coordination

For this study, mixtures of lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) in diglyme (G2), triglyme (G3) and tetraglyme (G4) at solvate ionic liquid (SIL) concentrations were investigated using classical molecular dynamics (cMD) simulations with a physically motivated force-field specifically developed for modeling these systems. The structural and dynamical properties of the mixtures were computed and analyzed. Lithium solvation shells, radial distribution functions, and X-ray structure factors were studied across the different SIL systems. Translational diffusion and rotational relaxation times were also evaluated, exhibiting similar trends with increasing glyme chain length. The results are consistent with experimental data and in good agreement with previous computational studies on G3 and G4. These findings validate the accuracy of the force field in modeling glyme systems and its use for describing the [Li(G2) 4/3 ][TFSI] mixture. Additionally, the thermal and electrochemical stability of these electrolytes were systematically examined. The thermal stability appears to be governed by cooperative interactions among glyme molecules, while the electrochemical stability is primarily influenced by Li + -anion interactions, which vary significantly with glyme chain length. Overall, the study sheds light on the crucial role of the anion in these glyme-based SILs and offers valuable insights into Li + -glyme systems at SIL concentrations, highlighting their promise as potential Li-ion battery electrolytes.

anions

The Polarity of Co-solvents Regulates the Charge Storage Mechanisms in Supercapacitors with Concentrated Electrolytes

Developing better energy storage devices depends on comprehending the underlying mechanisms involved in charge storage. With the continuous conception of new electrolytes, this task becomes progressively more urgent and complex. An example is the utilization of co-solvated concentrated solutions. While these show promising electrochemical responses, their dynamic properties (especially under confinement) and their relationships with performance are not fully understood. Here, we combined modified step potential electrochemical spectroscopy and quasielastic neutron scattering to investigate systems composed of activated mesoporous carbon (AMC) and concentrated solutions of lithium bis(trifluoromethanesulfonyl)imide in acetonitrile co-solvated with either toluene or acetone. Further, we report that acetone does not impair surface-controlled mechanisms, contrary to the case with toluene, which competes with charged species to populate the AMC’s pores without contributing to charge storage. In turn, toluene promotes a greater overall capacitance owing to Faradaic processes, which may be related to changes in the solvation structures under confinement.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Signatures of Thermoreversible Associations in X-ray and Neutron Scattering from Dilute Polyzwitterion Solutions

In aqueous solutions of polyzwitterions (PZs), an interplay between dipole–dipole interactions and hydration of zwitterionic groups can lead to thermoreversible associations, which have been difficult to detect in experiments. Here, in this study, we investigated dilute aqueous solutions of poly(1-(3-sulphonatopropyl)-2-vinylpyridinium) (P2VPPS) using small-angle X-ray and neutron scattering (SAXS and SANS) to probe the structure and neutron spin-echo (NSE) spectroscopy to probe dynamics. The SAXS and SANS data show that the correlation length increases with an increase in the concentration of P2VPPS for three different molecular weights. The addition of 0.1 M NaCl to one of the solutions led to almost no dependence of the correlation length on the concentration. Such a concentration dependence of the correlation length suggests the formation of clusters driven by thermoreversible associations in the solutions. The NSE measurements show that the solutions with a larger correlation length display slower relaxation, reflecting the reduced mobility of larger clusters. These results should be considered as signatures of thermoreversible associations in the solutions of P2VPPS. To establish a quantitative link between local structure (clusters) and dynamics in dilute solutions of P2VPPS, we combined a thermoreversible gelation theory for the structure of PZ solutions (Li, S.-F.; Muthukumar, M. Theory of Thermoreversible Gelation and Anomalous Concentration Fluctuations in Polyzwitterion Solutions. J. Chem. Phys. 2024, 161, 024903) with a model for the dynamics of the clusters (generalized Zimm model), developed in this work. Using such a theoretical framework, we have predicted the distribution of clusters in the solutions probed with SANS and SAXS. With the distributions, the generalized Zimm model has been used to extract the diffusion constant of the clusters and their characteristic size from the NSE data, where the latter agrees with the values estimated from the SAXS/SANS data. These findings confirm the presence of thermoreversible associations and establish a quantitative link between local structure (clusters) and dynamics in dilute solutions of P2VPPS. Furthermore, with growing interest in technological applications, this work can provide useful insights into the structural and dynamical properties of other PZ solutions.

field theory