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DOE OSTI2023

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive nonasymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. Finally, the proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

convergence↗

DOE OSTI2022

Complex pH-Dependent Interactions between Weak Polyelectrolyte Block Copolymer Micelles and Molecular Fluorophores

Amphiphilic block copolymers with weak polyelectrolyte blocks can assemble stimulus-responsive nanostructures and interfaces. Applications of these materials in drug delivery, biomimetics, and sensing largely rely on the well-understood swelling of polyelectrolyte chains upon deprotonation, often induced by changes in pH or ionic strength. This deprotonation can also tune interfacial interactions between the polyelectrolyte blocks and surrounding solution, an effect which is less studied than morphological swelling of polyelectrolytes but can be just as critical for intended function. Here, we investigate whether the pH-driven morphological response of polyelectrolyte-bearing nanostructures also affects the interactions of these nanostructures with molecules in solution, using micelles of a short-chain polybutadiene-block-poly(acrylic acid) (pBd–pAA) as a model system. Here we introduce a Förster resonance energy transfer (FRET) approach to probe interactions between micelles and fluorescent molecular solutes as a function of solution pH. As expected, the pAA corona of these pBd–pAA micelles increases in thickness monotonically as a function of pH. However, FRET efficiency, which provides a metric of the spatial proximity of fluorescently labeled micelles and freely diffusing fluorophores, exhibits complex nonmonotonic behavior as a function of pH, indicating that the average separation of micelles and acceptor fluorophores is not strictly correlated with micelle swelling. Dialysis experiments quantify the affinity of fluorophores for micelles as a function of pH, confirming that changes in FRET are driven almost entirely by the pH-dependent affinity of the pAA block for the investigated molecular fluorophores, not simply by a shape change of the pAA corona. This study provides key insights into the interfacial interactions between weak-polyelectrolyte-bearing nanostructures and molecular solutes, of importance for the development of their stimulus-responsive applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Temperature-Dependent Configurational Entropy Calculations for Refractory High-Entropy Alloys

The cluster expansion formalism for alloys is used to construct surrogate models for three refractory high-entropy alloys (NbTiVZr, HfNbTaTiZr, and AlHfNbTaTiZr). These cluster expansion models are then used along with Monte Carlo methods and thermodynamic integration to calculate the configurational entropy of these refractory high-entropy alloys as a function of temperature. Many solid solution alloy design guidelines are based on the ideal entropy of mixing, which increases monotonically with N , the number of elements in the alloy. However, our results show that at low temperatures, the configurational entropy of these materials is largely independent of N , and the assumption described above only holds in the high-temperature limit. This suggests that alloy design guidelines based on the ideal entropy of mixing require further examination.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Computational thermomechanics for crystalline rock. Part II: Chemo-damage-plasticity and healing in strongly anisotropic polycrystals

We present a thermal–mechanical–chemical-phase field model that captures the multi-physical coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure. This migration has a significant implication on the permeability evolution, creep deformation, and crack healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with computational homogenization. As such, we introduce a thermodynamic framework and a corresponding computational implementation that explicitly captures the brine diffusion along the grain boundary and crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order phase field that represents the regularized crack region in which a newly derived non-monotonic driving force is used to capture the fracture and healing due to the solution–precipitation. Numerical examples are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics material behaviors of rock salt.

42 ENGINEERING↗

DOE OSTI2022

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Abstract Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, H I . Here, we point to the existence of an analytical solution for such a problem for an arbitrary H I beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of H I , the solution also shows a considerable quantum speedup in computations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Characterizing the Impacts of Multi-Scale Heterogeneity on Solute Transport in Fracture Networks

We model flow and transport in fracture networks with varying degrees of fracture-to-fracture aperture heterogeneity and network intensity to show how changes in these properties can cause the emergence of anomalous flow and transport behavior. If fracture-to-fracture aperture heterogeneity is increased in sparse networks, velocity fluctuations inhibit high flow rates and solute transport can be delayed. Surprisingly, transport can be slowed even in cases where hydraulic aperture is monotonically increased. As the intensity of the networks is increased, more connected pathways allow for particles to bypass these effects. There exists transition behavior where with relatively few connected pathways in a network, first arrival times of particles are not heavily affected by fracture-to-fracture aperture heterogeneity, but the scaling behavior of the tails is strongly influenced. These results reinforce the importance of considering multi-scale effects in fractured systems and can inform flow and transport processes in both natural and engineered fractured systems.

58 GEOSCIENCES↗

DOE OSTI2026

Explicit Monotone Stable Super-Time-stepping Methods for Finite Time Singularities

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge–Kutta–Legendre (RKL) and Runge–Kutta–Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multistage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the time-step size for stability scales with stage number 𝑠 as $\mathcal{O}$⁡(𝑠 2 ). Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit time-step methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive time steps. Here, in this study, we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller time steps. Compared to commonly used implicit numerical methods, we achieve a significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev and the orthogonal Runge-Kutta-Chebyshev methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

On the design of stable, consistent, and conservative high-order methods for multi-material hydrodynamics

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Electrochemistry at Back-Gated, Ultrathin ZnO Electrodes: Field-Effect Modulation of Heterogeneous Electron Transfer Rate Constants by 30× with Enhanced Gate Capacitance

We report steady-state voltammetry of outer-sphere redox species at back-gated ultrathin ZnO working electrodes in order to determine electron transfer rate constants k ET as a function of independently-controlled gate bias, V G . We demonstrate that k ET can be modulated as much as 30-fold by application of V G ≤ 8 V. Key to this demonstration was integrating the ultrathin (5 nm) ZnO on a high dielectric constant (k) insulator, HfO 2 (30 nm), which was grown on a Pd metal gate. The high-k HfO 2 dramatically decreased the required V G values and increased the gate-induced charge in ZnO compared to previous studies. Importantly, the enhanced gating power of the Pd/HfO 2 /ZnO stack meant it was possible to observe a non-monotonic dependence of k ET on V G , which reflects the inherent density of redox acceptor states in solution. Furthermore, this work adds to the growing body of literature demonstrating that electrochemical kinetics (i.e., rate constants and overpotentials) at ultrathin working electrodes can be tuned by V G , independent of the conventional electrochemical working electrode potential.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Lyapunov exponent as a signature of dissipative many-body quantum chaos

A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large- q limit of a q -body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of q ≥ 4 based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment. Published by the American Physical Society 2024

García-García, Antonio M.↗

DOE OSTI2025

Concentration-Driven Slowdown of Chain-Exchange Kinetics in Block Copolymer Micelle Solutions

Here, we investigate the dependence of chain-exchange kinetics on polymer concentration for spherical micelles prepared from a polystyrene-b-poly(ethylene-alt-propylene) (SEP) diblock copolymer ( M n = 75 kDa, Ð = 1.05, and f PS = 0.18) in the PEP-selective solvent squalane. Solutions with concentrations of 15 vol% or more adopt a body-centered-cubic (BCC) packing of micelles, or a dense liquid-like packing (LLP), as confirmed by small-angle X-ray scattering. Concurrently, the mean aggregation number of the micelles grows steadily from about 60 in dilute solution to nearly 200 by 50 vol%. Time-resolved small-angle neutron scattering (TR-SANS) analysis reveals a monotonic and substantial (i.e., more than four orders of magnitude) decrease in the rate of chain exchange as the polymer concentration increases from 1 to 50 vol%. The TR-SANS relaxation functions are analyzed, taking into account both changes in contrast (due to chain exchange) and evolution in the structure factor (due to refinement of micelle packing). The activation energy for chain pullout is extracted using an established model and can be expressed as a linear function of polymer concentration, indicating that, in addition to the enthalpic barrier for extracting the core block, the evolution of micelle characteristics with increasing concentration imposes further constraints on chain exchange. Consistent with a theory of Halperin, these results suggest that the dominant factor reflects steric penalties during core block pullout arising from a reduced interfacial area per chain and associated increased corona crowding, due to the increase in aggregation number. The rate of exchange is independent of whether the micelles adopt a BCC or LLP arrangement, confirming that the rate-limiting step is extraction of the core block.

chemical structure↗

DOE OSTI2025

A simple centrifuge cell method for ex situ quantification of electrical conductivity of slurry electrode materials

We present the design, experimental procedure, and experimental evaluation of a system for fast, simple, and ex situ characterization of electrical conductivity of slurry electrode materials. The system uses a custom-designed electrochemical cell compatible with centrifugation in a swing-bucket centrifuge. The cell features cylindrical graphite electrodes that are partially sheathed so as to expose only 2 mm of the electrode surface to the bottom region of the packed particulate pellet. Also presented is a conduction model that provides a shape factor for estimating effective conductivity. We tested aqueous solutions of carbon black (CB), activated carbon (AC), and mixtures thereof. These particles were dispersed in 0.0 and 0.5 M NaCl solutions. Measurements show that the effective conductivity initially increases linearly with pellet mass and then saturates at higher masses. Notably, CB exhibited a fivefold increase in conductivity than AC at equal pellet masses. CB/AC mixtures at a fixed pellet mass were tested with CB mass fractions of 0 to 100%. Interestingly, the mixture conductivity was found to be a non-monotonic function of CB mass fraction, with a maximum conductivity at about 60 % CB mass fraction. At this maximum, the mixture conductivity is approximately 30 % higher than pure CB. NaCl concentration in the slurry solution had no effects on conductivity. These results highlight the interactions between slurry electrode composition and compaction, offering insights for optimizing slurry electrodes. Furthermore, the system and results may also be applicable to evaluation of particulate materials (including slurries) used for Li-ion batteries, capacitive deionization, fuel cells, and flow electrodes.

Capacitive deionization↗

DOE OSTI2023

Dissolution and reprecipitation of amorphous silica in silica Rich shales induces Non-Monotonic evolution of porosity in acidic reactive environments

Advances in sustainable subsurface energy technologies are crucial for meeting our energy and resource needs for a climate-resilient future. Novel strategies to harness subsurface shale reservoirs for recovering valuable metals and for enabling CO 2 storage are influenced by the morphological and mineralogical heterogeneities of these materials. In this context, delineating the interactions of highly acidic solutions such as wet supercritical CO 2 on shales with varying mineralogy is crucial to inform the stability of caprock seal for CO 2 storage and enhancements in permeability for fluid transport, reactivity, and storage. The feedback chemical effects associated with the interactions of acidic solutions on the morphologies and mineralogies of shales have not been extensively investigated. These insights are crucial for assessing temporal changes in the reactivity and the fate of the fluids in subsurface environments. In this study, we investigate the effect of 1M HCl solution on the chemistry and morphology of three different shale samples with varying carbonate, clay and silica contents. An increase in the amorphous content, from 37% to 41.3%, of silica-rich and carbonate/clay lean shale is noted due to reactions with an acidic solution which is attributed to the dissolution of Si-bearing phases such as clays, accompanied by SiO 2 precipitation. In shales bearing high content of clays and carbonates, significant increase in the pore volumes and surface areas are noted. Non-monotonic changes in the micron-scale porosity of silica rich–carbonate/clay lean (e.g., Mowry shale) are noted using in-situ X-ray microtomography experiments. Due to the initial mobilization of silica and dissolution of carbonate/clay phases, the total porosity slightly increases from 6.7% to 10.7% followed by a decrease to similar to 4% caused by SiO 2 reprecipitation. In conclusion, these findings suggest that even though silica is less reactive in acidic environments, the changes in the amorphous and crystalline content due to dissolution and reprecipitation alter the porosity and fluid flow paths.

04 OIL SHALES AND TAR SANDS↗

DOE OSTI2025

Improved Guarantees for Optimal Nash Equilibrium Seeking and Bilevel Variational Inequalities

We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other problem classes, including the optimal solution selection problem and the optimal Nash equilibrium (NE) seeking problem. Our main contribution is threefold. (i) We consider bilevel VIs with monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method, named IR-EG 𝚖,𝚖 . We improve the existing iteration complexity results for addressing both bilevel VI and VI-constrained convex optimization problems. (ii) Under the strong monotonicity of the outer-level mapping, we develop a method named IR-EG 𝚜,𝚖 and derive faster guarantees than those in (i). We also study the iteration complexity of this method under a constant regularization parameter. These results appear to be new for both bilevel VIs and VI-constrained optimization. (iii) To our knowledge, complexity guarantees for computing the optimal NE in nonconvex settings do not exist. Motivated by this lacuna, we consider VI-constrained nonconvex optimization problems and devise an inexactly projected gradient method, named IPR-EG, where the projection onto the unknown set of equilibria is performed using IR-EG 𝚜,𝚖 with a prescribed termination criterion and an adaptive regularization parameter. We obtain new complexity guarantees in terms of a residual map and an infeasibility metric for computing a stationary point. Here, we validate the theoretical findings using preliminary numerical experiments for computing the best and the worst NEs.

bilevel optimization↗

DOE OSTI2022

Understanding effects of chemical complexity on helium bubble formation in Ni-based concentrated solid solution alloys based on elemental segregation measurements

Here, helium bubble formation and swelling were systematically studied in Ni-based concentrated solid solution alloys containing different numbers and types of elements. Our microscopy analysis showed that although increasing the alloy chemical complexity helps suppress bubble formation in general, there is no monotonic relationship between the bubble growth rate and the number of alloying elements. Certain elements (e.g., Fe and Pd) are more effective in suppressing bubble growth than others (e.g., Cr and Mn). Atom probe tomography was applied to accurately measure elemental segregation around bubbles, revealing unique effects of certain alloying elements on vacancy migration towards bubbles. More specifically, the high vacancy mobility via Cr sites leads to a large vacancy flux and an increased bubble size, while the high degree of atomic size mismatch introduced by Pd helps deflect vacancy flow away from bubbles and decrease the amount of swelling. The effects identified in this study provide new strategies to design concentrated solid solutions with superior resistance to swelling.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Superconductivity and phase diagrams of CaK(Fe 1- x Mn x ) 4 As 4 single crystals

In order to study the effects of Mn substitution on the superconducting and magnetic ground state of CaKFe 4 As 4 (T c = 35 K), members of the CaK (Fe 1–x Mn x ) 4 As 4 series have been synthesized by high-temperature solution growth in single-crystalline form and characterized by elemental analysis, thermodynamic, and transport measurements. These measurements show that the superconducting transition temperature decreases monotonically and is finally suppressed below 1.8 K as x is increased from 0 to 0.036. For x values greater than 0.016, signatures of a magnetic transition can be detected in both thermodynamic and transport measurements in which kinklike features allow for the determination of the transition temperature T* that increases as Mn substitution increases. Here, a temperature-composition (T–x) phase diagram is constructed, revealing a half-dome of superconductivity with the magnetic transition temperature T* appearing near 26 K for x ~ 0.017 and rising slowly up to 33 K for x ~ 0.036. In addition to the creation of the T–x phase diagram for CaK(Fe 1–x Mn x ) 4 As 4 , specific-heat data are used to track the jump in specific heat at T c ; the CaK(Fe 1–x Mn x ) 4 As 4 data do not follow the scaling of ΔC p with $T$$^{3}_{c}$ as many of the other Fe-based superconducting systems do. These data suggest that, as magnetic pair breaking is present, the jump in C p for a given T c is reduced. Elastoresistivity coefficients 2m 66 and m 11 – m 12 as a function of temperature are also measured. 2m 66 and m 11 – m 12 are qualitatively similar to CaK(Fe 1–x Ni x ) 4 As 4 . This may indicate that the magnetic order in Mn-substituted system may be still the same as CaK(Fe 1–x Mn x ) 4 As 4 . Superconductivity of CaK(Fe 1–x Mn x ) 4 As 4 is also studied as a function of magnetic field. A clear change in H' c2 (T)/T c , where H' c2 (T) is dH' c2 (T)/dT, at x ~ 0.015 is observed and probably is related to change of the Fermi surface due to magnetic order. Coherence lengths and the London penetration depths are also calculated based on H c1 and H c2 data. Coherence lengths as the function of x also show the changes near x = 0.015, again consistent with Fermi-surface changes associated with the magnetic ordering seen for higher-x values.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO 2 Leakage through a Compromised Well

Carbon capture, utilization, and storage (CCUS) describes a set of technically viable processes to separate carbon dioxide (CO 2 ) from industrial byproduct streams and inject it into deep geologic formations for long-term storage. Legacy wells located within the spatial domain of new injection and production activities represent potential pathways for fluids (i.e., CO 2 and aqueous phase) to leak through compromised components (e.g., through fractures or micro-annulus pathways). The finite element (FE) method is a well-established numerical approach to simulate the coupling between multi-phase fluid flow and solid phase deformation interactions that occur in a compromised well system. We assumed the spatial domain consists of a three-phases system: a solid, liquid, and gas phase. For flow in the two fluids phases, we considered two sets of primary variables: the first considering capillary pressure and gas pressure (PP) scheme, and the second considering liquid pressure and gas saturation (PS) scheme. Fluid phases were coupled with the solid phase using the full coupling (i.e., monolithic coupling) and iterative coupling (i.e., sequential coupling) approaches. The challenge of achieving numerical stability in the coupled formulation in heterogeneous media was addressed using the mass lumping and the upwinding techniques. Numerical results were compared with three benchmark problems to assess the performance of coupled FE solutions: 1D Terzaghi’s consolidation, Liakopoulos experiments, and the Kueper and Frind experiments. We found good agreement between our results and the three benchmark problems. For the Kueper and Frind test, the PP scheme successfully captured the observed experimental response of the non-aqueous phase infiltration, in contrast to the PS scheme. These exercises demonstrate the importance of fluid phase primary variable selection for heterogeneous porous media. We then applied the developed model to the hypothetical case of leakage along a compromised well representing a heterogeneous media. Considering the mass lumping and the upwinding techniques, both the monotonic and the sequential coupling provided identical results, but mass lumping was needed to avoid numerical instabilities in the sequential coupling. Additionally, in the monolithic coupling, the magnitude of primary variables in the coupled solution without mass lumping and the upwinding is higher, which is essential for the risk-based analyses.

deformation flow↗

DOE OSTI2024

Dynamics of inspiraling dark energy

Abstract We investigate the dynamics of a multifield dark energy model, which arises from certain rapid-turning and inspiraling trajectories in field space. We find the speed of sound $$c_s$$ c s of the dark energy perturbations around the background and show that $$c_s$$ c s is monotonically decreasing with time. Furthermore, it has a positive-definite lower bound that implies a certain clustering scale. We also extend the previously known background solution for dark energy to an exact solution that includes matter. This allows us to address the implications of our model for two cosmological tensions. More precisely, we argue that the $$\sigma _8$$ σ 8 tension can be alleviated generically, while reducing the Hubble tension requires certain constraints on the parameter space of the model. Notably, a necessary condition for alleviating the Hubble tension is that the transition from matter domination to the dark energy epoch begins earlier than in $$\Lambda $$ Λ CDM.

Anguelova, Lilia↗