Pairing Force Research Articles

S01 hyperon superfluidity in neutron stars from a separable pairing force of finite range

Starting from a separable pairing force of finite range of nucleons, a new set of the pairing strengths for the $^{1}{S}_{0}$ hyperon-hyperon pairing based on the quark model is proposed and the $^{1}{S}_{0}$ hyperon superfluidity in neutron stars with the new hyperon pairing strengths is investigated. The reliable $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}$ pairing strength is 2/3 times $NN$ pairing strength. The upper limit of $\mathrm{\ensuremath{\Sigma}}\mathrm{\ensuremath{\Sigma}}$ pairing strength is 2/3 times $NN$ pairing strength. The lower limit of the ratio of $\mathrm{\ensuremath{\Xi}}\mathrm{\ensuremath{\Xi}}$ pairing strength to $NN$ pairing strength is 1. With these pairing strengths, the maximum $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}$ pairing gap is several tenths of a MeV which can be compared with the results calculated with the widely used pairing force ESC00; because of a low fraction of ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{-}}$ hyperon, the $\mathrm{\ensuremath{\Sigma}}$ superfluidity is very weak or even absent in neutron stars; the $\mathrm{\ensuremath{\Xi}}$ superfluidity is stronger than the results calculated with ESC08c and the ${\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ pairing may potentially affect the cooling properties of massive neutron stars.

High-spin levels, β− decay and monopole effects in A = 128 hole nuclei near 132Sn

The high-spin levels and β− decay in neutron-rich hole nuclei of A=128 are studied by large-scale shell-model calculations with the extended pairing plus multipole-multipole force model. The excited energy and logft values of β− decay are predicted in the final states of 128Ag, 128Cd, and 128In by the cascaded decays calculated in A=128 hole nuclei. The 16+ level of 128Cd is predicted as a spin-trap isomer feeding the existing high-spin 16+ of 128In by β− decay. The monopole effects are investigated carefully in the β− decay process from the ground state 0+ of 128Cd to the 1+ levels of 128In, while the half-life is affected by monopole corrections acting on the configuration of final states.

Effect of Ti5Si3 on the wear properties of Ti–3Si-1.5Fe–1Mo titanium alloy with ultrahigh hardness

In this work, Ti–3Si-1.5Fe–1Mo titanium alloy reinforced with in-situ synthesized Ti5Si3 was prepared, and ultrahigh hardness (55.6 HRC) was achieved by solution aging treatment. The effects of Ti5Si3 on the friction and wear behavior of the alloy were investigated. Results show that the wear rate of the alloy after hot treatment increases from 3.51 × 10−4 mm3 N−1 m−1 to 3.93 × 10−4 mm3 N−1 m−1, yet the ultrahigh hardness alloy exhibits poor wear resistance. A series of the wear tracks characterizations and thermodynamical calculations were performed to elucidate its wear mechanism: during the reciprocating sliding, the reduction of the Ti5Si3 content leads to the increase of the friction pair force on the matrix, TiO2 and SiO2 wear debris are generated then fall off, both the oxidative wear and adhesive wear aggravated, which seriously impair the wear resistance of the alloy.

Statistical correlations of nuclear quadrupole deformations and charge radii

Shape deformations and charge radii, basic properties of atomic nuclei, are influenced by both the global features of the nuclear force and the nucleonic shell structure. As functions of proton and neutron number, both quantities show regular patterns and, for nuclei away from magic numbers, they change very smoothly from nucleus to nucleus. In this paper, we explain how the local shell effects are impacting the statistical correlations between quadrupole deformations and charge radii in well-deformed even-even Er, Yb, and Hf isotopes. This implies, in turn, that sudden changes in correlations can be useful indicators of underlying shell effects. Our theoretical analysis is performed in the framework of self-consistent mean-field theory using quantified energy density functionals and density-dependent pairing forces. The statistical analysis is carried out by means of the linear least-square regression. The local variations of nuclear quadrupole deformations and charge radii, explained in terms of occupations individual deformed Hartree-Fock orbits, make and imprint on statistical correlations of computed observables. While the calculated deformations or charge radii are, in some cases, correlated with those of their even-even neighbors, the correlations seem to deteriorate rapidly with particle number. The statistical correlations between nuclear deformations and charge radii of different nuclei are affected by the underlying shell structure. Even for well deformed and superfluid nuclei for which these observables change smoothly, the correlation range usually does not exceed $\Delta N=4$ and $\Delta Z=4$, i.e., it is rather short. This result suggests that the frequently made assumption of reduced statistical errors for the differences between smoothly-varying observables cannot be generally justified.

Seniority and configurations in neutron-rich Nickel isotopes * *Supported by the Interdisciplinary Scientific and Educational School of Moscow University "Fundamental and Applied Space Research"

Excited states in low-energy spectra of Ni are considered. Accordingly, pairing forces in the form of surface delta interaction are employed to account for the formation of the ground state multiplet with seniority states. The multiplet splitting is described with mass relationships of masses of neighboring nuclei. Subsequently, the seniority model is adopted to reproduce or predict the states in odd-even isotopes and in even-even isotopes. The correct account of the state should allow for the description of the reversed order of states with and observed in experiments. The results obtained are compared with the structure of similar multiplets in isotones.

Two-Hole Ground State: Dichotomy in Pairing Symmetry

A single-hole ground state Ansatz for the two-dimensional t-J model has been recently studied by the variational Monte Carlo (VMC) method. Such a doped hole behaves like a "twisted" non-Landau quasiparticle characterized by an emergent quantum number in agreement with exact numerics. In this work, we further investigate the ground state of two holes by VMC. It is found that the two holes strongly attract each other to form a pairing state with a new quantum number the same as obtained by the numerical exact diagonalization and density matrix renormalization group (DMRG) calculations. A unique feature of this pairing state is a dichotomy in the pairing symmetry, i.e., a d-wave in terms of the electron c operators and an s-wave in terms of the new quasiparticles, as explicitly illustrated in the ground state wave function. A similar VMC study of a two-hole wave function for the t-J two-leg ladder also yields a good agreement with the DMRG result. We demonstrate that the pairing mechanism responsible for the strong binding here is not due to the long-range antiferromagnetic order nor the resonating-valence-bound pairing in the spin background but is the consequence of the quantum phase-strings created by the hopping of holes. The resulting spin-current pattern mediating the pairing force is explicitly illustrated in the VMC calculation. Physical implications to superconductivity at finite doping are also discussed.

Relativistic Hartree-Fock-Bogoliubov model for axially deformed nuclei

Background: At the forefront of nuclear science, unstable nuclei are of special significance, in which the deformation and pairing correlations play an essential role in determining the nuclear structure. Under the mean field approach, the pairing and deformation effects have been uniformly considered within the deformed nonrelativistic Hartree-Fock-Bogoliubov model and the deformed relativistic Hartree-Bogoliubov (D-RHB) model. However, due to the limitation of the Hartree approach, an important ingredient of the nuclear force---the tensor force that contributes via the Fock diagram---is missing in the D-RHB model. Recently, the axially deformed relativistic Hartree-Fock (D-RHF) model was established for deformed nuclei, in which the $\ensuremath{\pi}$-pseudovector ($\ensuremath{\pi}$-PV) coupling can take into account the tensor force effects naturally. While limited by the BCS pairing, it cannot be safely applied to describe unstable nuclei.Purpose: The aim of this work is to develop the axially deformed relativistic Hartree-Fock-Bogoliubov (D-RHFB) model for the reliable description of a wide range of unstable nuclei, by utilizing the spherical Dirac Woods-Saxon (DWS) base.Method: Staring from the Lagrangian density that is based on the meson-propagated picture of nuclear force, the full Hamiltonian, that contains both mean field and pairing contributions, is derived by quantizing the Dirac spinor field in the Bogoliubov quasiparticle space, and the expectation with respect to the Bogoliubov ground state gives the full energy functional. As an extension of the D-RHF model, the degree of freedom associated with the $\ensuremath{\rho}$-tensor ($\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$) coupling is implemented, and incorporating with the Bogoliubov scheme the finite-range Gogny force D1S is utilized as the pairing force. Moreover, qualitative analyses on the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings are presented to better understand their enhancements on the deformation effects.Results: Space convergence related to the spherical DWS base is confirmed for the D-RHFB model by taking light nucleus $^{24}\mathrm{Mg}$ and mid-heavy one $^{156}\mathrm{Sm}$ as candidates. Compared to light nuclei, extraordinarily more negative energy states are necessitated to keep the expansion completeness on the spherical DWS base for mid-heavy and heavy nuclei, due to the enhanced correlations between the expansion components with large $\ensuremath{\kappa}$ quantity, as indicated by the nature of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings. Furthermore, because of the enhanced deformation effects by the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, the RHF Lagrangian PKA1 presents a deeper bound ground state for $^{24}\mathrm{Mg}$ than the other selected Lagrangians, in addition to predicting a fairly deep bound local minimum with large oblate deformation.Conclusions: The axially deformed relativistic Hartree-Fock-Bogoliubov model has been established with confirmed convergence checks. The effects of the $\ensuremath{\pi}$-PV and $\ensuremath{\rho}\text{\ensuremath{-}}\mathrm{T}$ couplings, coupled with nuclear deformation, are analyzed in both light and mid-heavy nuclear systems, and are expected to be manifested in a wide range of unstable nuclei.

Proton-neutron pairing and binding energies of nuclei close to the N=Z line

We analyze the contribution of isovector and isoscalar proton-neutron pairing to the binding energies of even-even nuclei with $N\ensuremath{-}Z=0, 2, 4$ and atomic mass $20<A<100$. The binding energies are calculated in the mean-field approach by coupling a Skyrme-type functional to an isovector-isoscalar pairing force of zero range. The latter is treated in the framework of quartet condensation model (QCM), which conserves exactly the particle number and the isospin. The interdependence of pairing and deformation is taken into account by performing self-consistent Skyrme-HF $+$ QCM calculations in the intrinsic system. It is shown that the binding energies are not changing much when the isoscalar pairing is switched on. This fact is related to the off-diagonal matrix elements of the pairing force, which are less attractive for the isoscalar force, and to the competition between the isoscalar and isovector pairing channels.

ΛΛ pairing effects in spherical and deformed multi- Λ hyperisotopes

The $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Lambda}}$ pairing effects in spherical and deformed multi-$\mathrm{\ensuremath{\Lambda}}$ hyperisotopes are investigated in the framework of the Skyrme-Hartree-Fock approach employing a $\ensuremath{\delta}$ pairing force with the pairing strength of $\mathrm{\ensuremath{\Lambda}}$ hyperons being $4/9$ of that for nucleons. For spherical hyperisotopes, the occurrences of magic numbers $\ensuremath{-}S=2$, 8, 18, 20, 34, 58, 68, and 70, which are attributed to a Woods-Saxon-like $\mathrm{\ensuremath{\Lambda}}$ hyperon potential, are evinced by the sudden drop of $2\mathrm{\ensuremath{\Lambda}}$ separation energies and the vanishing pairing gaps and pairing energies. The results are compared with equivalent ones in recent Hartree-Fock-Bogoliubov and relativistic-Hartree-Bogoliubov calculations. For the deformed hyperisotopes, more possible $\mathrm{\ensuremath{\Lambda}}$ hyperon magic numbers $\ensuremath{-}S=4$, 6, 10, 14, 26, 30, and 32 are found based on the analysis of the single-particle energy levels, and are all sensitive to the quadrupole deformation ${\ensuremath{\beta}}_{2}$. The steps of the $2\mathrm{\ensuremath{\Lambda}}$ separation energies are accordingly smaller than in spherical hyperisotopes, and the possibilities for pairing are consistently reduced.

The band structure of the whole spectrum of an N-body cold system containing atoms with arbitrary integer spin and dominated by singlet pairing force

The spectra of N-boson systems with arbitrary nonzero spin have been studied. Firstly, only the singlet pairing interaction is considered, a set of exact eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin textures of various states different in N and/or , and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller N. Some features and notable phenomena that might emerge in high- systems, say, the ground band might have extremely high level density, have been discussed.

A modified Jarzynski free-energy estimator to eliminate non-conservative forces and its application in nanoparticle-membrane interactions.

Computational methods to understand interactions in bio-complex systems are however limited to time-scales typically much shorter than in Nature. For example, on the nanoscale level, interactions between nanoparticles (NPs)/molecules/peptides and membranes are central in complex biomolecular processes such as membrane-coated NPs or cellular uptake. This can be remedied by the application of e.g. Jarzynski's equality where thermodynamic properties are extracted from non-equilibrium simulations. Although, the out of equilibrium work leads to non-conservative forces. We here propose a correction Pair Forces method, that removes these forces. Our proposed method is based on the calculation of pulling forces in backward and forward directions for the Jarzynski free-energy estimator using steered molecular dynamics simulation. Our results show that this leads to much improvement for NP-membrane translocation free energies. Although here we have demonstrated the application of the method in molecular dynamics simulation, it could be applied for experimental approaches.

Two-proton emission from excited states of proton-rich nuclei

The effective liquid drop model (ELDM) and the generalized liquid drop model (GLDM) are extended to the case of studying the two-proton (2p) radioactivity from the excited states of proton-rich nuclei. It is shown that the experimental 2p decay half-lives are reproduced well by the ELDM and the GLDM. Then, the 2p decay half-lives of excited states of some nuclei that are not yet available experimentally are predicted by the two models, which are useful for searching for the new 2p decay candidates in future. Meanwhile, the above predicted half-lives are analyzed and compared with those given by the unified fission model (UFM). Next, the influence of the uncertainties of the decay energy and the angular momentum on the half-lives are analyzed in the frame of the two models by taking the 2p radioactivity of the 21<sup>+</sup> isomeric state of <sup>94</sup>Ag for example. It is found that the half-lives go up with the increase of the angular momentum, following the law of the quadratic function. Furthermore, the strong dependence of the half-lives on the decay energy suggests that it is important and necessary to measure accurately the mass value of the parent nucleus and the daughter nucleus and the excitation energy. Finally, it is necessary to point out that the existence of the 2p radioactivity in the 21<sup>+</sup> isomeric state of<sup> 94</sup>Ag remains to be a mystery. Moreover, although the 2p radioactivity is observed from the higher excited states of <sup>17</sup>Ne and <sup>18</sup>Ne, the relevant hypotheses have not yet been further tested experimentally. The construction of a new generation of radioactive ion beam facilities, such as the high intensity heavy-ion accelerator facility (HIAF), is expected to be used to uncover the nature of the 2p radioactivity in the 21<sup>+</sup> isomeric state of <sup>94</sup>Ag and further test the hypotheses of the 2p decay from the higher excited states of <sup>17</sup>Ne and <sup>18</sup>Ne. On the other hand, some microscopic models, such as the shell model, need to be further developed by including some necessary physical factors, such as the tensor force, three-body force and accurate pairing force, to describe the mechanism of the 2p emission of the excited states more reasonably. In summary, more nuclear structure information can be extracted by studying the 2p radioactivity of the excited states. It is worth studying further although it is rather difficult to observe.

Research on dynamic load spectrum of electric vehicle transmission system based on underdamping characteristic

Overall, the electric vehicle transmission system shows an underdamped characteristic. Under changeable road conditions and high-frequency response for the motor, the resulting dynamic load environment may cause multiple failure modes, such as contact fatigue failure and bending fatigue fracture, for the transmission component, which limits the electric vehicle transmission component lifespan and system reliability. To reflect the dynamic load characteristics of the electric vehicle transmission system using a permanent magnet synchronous motor as a power source and accurately calculate the dynamic load of the transmission system, a high-speed helical gear-rotor-bearing coupling mechanical model for the electric vehicle transmission system was built based on simulating actual operation working conditions of the electric vehicle and considering the external load excitation caused by the Electromagnetic torque of the permanent magnet synchronous motor and vehicle driving resistant change as well as internal excitation caused by gear time-varying meshing rigidness and meshing error. Through simulation calculation of the mechanical model, the dynamic meshing force of the gear pair and dynamic contact force of the support bearing was obtained. Based on the Hertz contact theory, the stress-time history was obtained for the key parts, the rain flow counting method was adopted for the statistics collection and analysis of the stress-time history, and the fatigue load spectrum for various key parts of the transmission system was obtained. The result lays a foundation for the fatigue life prediction and reliability analysis for the pure electric vehicle transmission system.

Pairing strength in the relativistic mean-field theory determined from the fission barrier heights of actinide nuclei and verified by pairing rotation and binding energies

Abstract We have studied the strength of the Bardeen–Cooper–Schrieffer (BCS) pairing force, used as a residual interaction to the relativistic mean-field approach, to reproduce the height of the inner fission barriers for actinide nuclei. It was found that increasing the pairing strength by about $13\%$ makes the reproduction of the inner fission barriers better over a wide range of actinide nuclei. This result was verified by using the moment of inertia of the pairing rotational energy, which was introduced to avoid mean-field and odd-mass effects in the pairing interaction, to deduce purely the pairing strength. The pairing interaction thus determined could also improve the description of the binding energy of heavy nuclei. As a result, a consistent picture among inner fission barrier, binding energy, and pairing moment of inertia could be obtained in terms of the relativistic mean-field + BCS theory for a broad region of the actinide nuclei.

Pairing effects in nuclear pasta phase within the relativistic Thomas–Fermi formalism

Pairing effects in non-uniform nuclear matter, surrounded by electrons, are studied in the protoneutron star early stage and in other conditions. The so-called nuclear pasta phases at subsaturation densities are solved in a Wigner–Seitz cell, within the Thomas–Fermi approximation. The solution of this problem is important for the understanding of the physics of a newly born neutron star after a supernova explosion. It is shown that the pasta phase is more stable than uniform nuclear matter on some conditions and the pairing force relevance is studied in the determination of these stable phases.

Finite-temperature linear response theory based on relativistic Hartree Bogoliubov model with point-coupling interaction

The finite-temperature linear response theory based on the finite-temperature relativistic Hartree-Bogoliubov (FT-RHB) model is developed in the charge-exchange channel to study the temperature evolution of spin-isospin excitations. Calculations are performed self-consistently with relativistic point-coupling interactions DD-PC1 and DD-PCX. In the charge-exchange channel, the pairing interaction can be split into isovector ($T = 1$) and isoscalar ($T = 0$) parts. For the isovector component, the same separable form of the Gogny D1S pairing interaction is used both for the ground-state calculation as well as for the residual interaction, while the strength of the isoscalar pairing in the residual interaction is determined by comparison with experimental data on Gamow-Teller resonance (GTR) and Isobaric analog resonance (IAR) centroid energy differences in even-even tin isotopes. The temperature effects are introduced by treating Bogoliubov quasiparticles within a grand-canonical ensemble. Thus, unlike the conventional formulation of the quasiparticle random-phase approximation (QRPA) based on the Bardeen-Cooper-Schrieffer (BCS) basis, our model is formulated within the Hartree-Fock-Bogoliubov (HFB) quasiparticle basis. Implementing a relativistic point-coupling interaction and a separable pairing force allows for the reduction of complicated two-body residual interaction matrix elements, which considerably decreases the dimension of the problem in the coordinate space. The main advantage of this method is to avoid the diagonalization of a large QRPA matrix, especially at finite temperature where the size of configuration space is significantly increased. The implementation of the linear response code is used to study the temperature evolution of IAR, GTR, and spin-dipole resonance (SDR) in even-even tin isotopes in the temperature range $T = 0 - 1.5$ MeV.

Study of Wear and Redistribution Dynamic Forces of Wheel Pairs Restored by a Wear-Resistant Coating 15Cr17Ni12V3F

The relevance of this study lies in the need to research the wear process of the elements of dynamic systems and to establish the dependence of the geometric and structural characteristics of wheels restored by plasma surfacing with a coating of 15Cr17Ni12V3F on dynamic, cyclically varying loads. The study was aimed to establish the dependencies between the deviation of the wear area, the formation of contact and fatigue stresses, and the change in the phase structure of the wheel defect. It is important to justify the permissible limits of wear of the transverse profile in the contact zone of a wheel and ridge. The object of the study was the dynamic interaction of the “wheel–ridge–rail contact surface” system. To achieve the goal, the following methodology was adopted: kinematic analysis, strength calculation, the use of mathematical analysis in dynamic system modelling, virtual modelling in the SOLIDWORKS software environment of the GearTrax application, experiment planning, and model correction through the results of metallographic studies. The results of the study are presented as reasonable prediction criteria that consider contact cycles during the formation of fatigue stresses at the stage of defect origin. The process of the dynamic interaction of the contact worn profile of a wheel with a railway rail is explored. Polynomial equations are proposed to substantiate the optimal design and technological parameters of designing a railway carriage wheel. The permissible limits of wear of the transverse profile in the contact zone of the wheel and the ridge are justified while taking the coefficient of the reduction of contact stresses in the metal into account. The dependences of the change in static load on the utilization factor of the railway carriage load capacity are established. The dependences of changes in fatigue stresses on the design deviation of the contact area of wheel wear are established. It is confirmed that the stress concentration under cyclic loads is formed in the ferritic layers of the material structure before the appearance of wear.

Phonon-based thermal configurational forces: Definitions and applications in rupture of semiconductors

A general framework encompassing spatial and material settings of atomistic crystalline systems is presented. Theoretical and computational analysis of configurational forces in the context of atomistic thermomechanics constitutes the central topic of the presented paper. We distinguish between atomistic configurational forces stemming from the changes in static energy density and the changes in vibrational energy density. To this end, we first propose a novel atomistic phonon-based thermal configurational forces rendering infinitesimal changes in vibrational energy density under configurational variations. Next, we employ pair and triplet configurational forces resulting from the changes in static energy density due to two-body stretch and three-body stretch-bending. Using thermal configurational forces, we study numerically the problem of temperature rise, bond stretching and bending occurring at the crack tip induced by brittle crack propagation in crystalline lattices. Energy release caused by large harmonic vibrations and deformations is estimated by the magnitude of configurational forces at the crack tip. The atomistic temperature is shown to be governed by atom-wise frequency through local entropy. Finally, we demonstrate suitability and versatility of atomistic configurational forces in analyzing fracture of silicon diamond lattice subjected to mode-I loading at different finite temperatures.

High fidelity, discrete element method simulation of magnetorheological fluids using accurate particle size distributions in LIGGGHTS extended with mutual dipole method

We simulate magnetorheological fluids (MRF) using open source LIGGGHTS soft sphere discrete element method code, extended by us to include a mutual dipole magnetic model. Our simulations take advantage of the many pair forces available in the LIGGGHTS framework, including SJKR cohesion, friction, and rolling resistance. In addition, we have included an uncoupled, Couette flow background carrier fluid. The simulated particles in this work are polydisperse, with distributions made to match the distributions used to produce magnetorheological fluids in literature, increasing the fidelity of the simulations. Using the accurate particle size distributions, high heritage contact models, and an uncoupled fluid model, we are able to match experimental MRF yield stress results more closely than with monodisperse simulations.

Molecular Forces in Liquid-Liquid Extraction.

The phase transfer of ions is driven by gradients of chemical potentials rather than concentrations alone (i.e., by both the molecular forces and entropy). Extraction is a combination of high-energy interactions that correspond to short-range forces in the first solvation shell such as ion pairing or complexation forces, with supramolecular and nanoscale organization. While the latter are similar to the long-range solvent-averaged interactions in the colloidal world, in solvent extraction they are associated with lower characteristic lengths of the nanometric domain. Modeling of such complex systems is especially complicated because the two domains are coupled, whereas the resulting free energy of extraction is around kBT to guarantee the reversibility of the practical process. Nevertheless, quantification is possible by considering a partitioning of space among the polar cores, interfacial film, and solvent. The resulting free energy of transfer can be rationalized by utilizing a combination of terms which represent strong complexation energies, counterbalanced by various entropic effects and the confinement of polar solutes in nanodomains dispersed in the diluent, together with interfacial extractant terms. We describe here this ienaics approach in the context of solvent extraction systems; it can also be applied to further complex ionic systems, such as membranes and biological interfaces.