Dusty (Complex) Plasmas

The field of complex (dusty) plasmas-low-temperature plasmas containing charged microparticles-is reviewed: The major types of experimental complex plasmas are briefly discussed. Various elementary processes, including grain charging in different regimes, interaction between charged particles, and momentum exchange between different species are investigated. The major forces on microparticles and features of the particle dynamics in complex plasmas are highlighted. An overview of the wave properties in different phase states, as well as recent results on the phase transitions between different crystalline and liquid states are presented. Fluid behaviour of complex plasmas and the onset of cooperative phenomena are discussed. Properties of the magnetized complex plasmas and plasmas with nonspherical particles are briefly mentioned. In conclusion, possible applications of complex plasmas, interdisciplinary aspects, and perspectives are discussed.

This doctoral thesis reports analytical and numerical results for the electrostatic response of a dusty plasma to a moving test charge. Two important physical aspects of dusty plasmas, namely grain size distribution and grain charging dynamics were taken into account. In the first case, a dusty plasma in thermal equilibrium and with a distribution of grain sizes is considered. A size distribution is assumed which decreases exponentially with the grain mass for large sizes and gives a simple smooth reduction for small sizes. The electrostatic response to a slowly moving test charge, using a second order approximation is found and the effects of collisions are also investigated. It turns out that for this particular size distribution, there is a remarkably simple result that the resulting effective distribution for the electrostatic response is a kappa (generalized Lorentzian) distribution. In the second case, we present an analytical model for the shielding of a slowly moving test charge in a dusty plasma with dynamical grain charging for cases both with and without the collision effects. The response potential is treated as a power series in test charge velocity. Analytical expressions for the response potential are found up to second order in test charge velocity. The first-order dynamical charging term is shown to be the consequence of the delay in the shielding due to the dynamics of the charging process. It is concluded that the dynamical charging of the grains in a dusty plasma enhances the shielding of a test charge. To clarify the physics, a separate study is made where the charging is approximated by using a time delay. The resulting potential shows the delayed shielding effect explicitly. The terms in the potential that depend on the charging dynamics involve a spatial shift given by the test charge velocity and the charging time. The wake potential of a fast moving test charge in the case of grain charging dynamics was also found. It was observed that the grain charging dynamics leads to a spatial damping and a phase shift in the potential response. Finally, combining these two physical aspects, generalized results for the electrostatic potential were found incorporating the terms from both grain size distribution and grain charging dynamics. The generalized results contain the previous work where these two effects were studied separately and which can now be found as special limiting cases. This kind of work has relevance both in space and astrophysical plasmas.

The nonlinear aspects of longitudinal motion of interacting point masses in a lattice are revisited, with emphasis on the paradigm of charged dust grains in a dusty plasma (DP) crystal. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed. Making use of a general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or else) analytic potential form $\phi(r)$ and arbitrarily long site-to-site range of interactions, it is shown that dust-crystals support nonlinear kink-shaped localized excitations propagating at velocities above the characteristic DP lattice sound speed v 0. Both compressive and rarefactive kink-type excitations are predicted, depending on the physical parameter values, which represent pulse- (shock-)like coherent structures for the dust grain relative displacement. Furthermore, the existence of breather-type localized oscillations, envelope-modulated wavepackets and shocks is established. The relation to previous results on atomic chains as well as to experimental results on strongly-coupled dust layers in gas discharge plasmas is discussed.

This review surveys the present state of the art of diagnostics and removal techniques for particles suspended in plasma. Research on "Dusty Plasmas" is necessary for ITER and the fusion reactors of the future. The aim is to transfer the knowledge acquired in complex (dusty) plasma research and in the low temperature plasma community to the broad fusion community, which is being mobilized to understand/quantify this phenomenon and find solutions. The introductory style makes it accessible also to safety personnel and other interested readers with only basic knowledge of plasma. In the transfer from research to technology we have tried to assess the applicability of the methods proposed and we have identified further, much needed, R&D.

The nonlinear aspects of longitudinal motion of interacting point masses in a lattice are revisited, with emphasis on the paradigm of charged dust grains in a dusty plasma (DP) crystal. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed. Making use of a general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or else) analytic potential form φ(r) and arbitrarily long site-to-site range of interactions, it is shown that dust-crystals support nonlinear kink-shaped localized excitations propagating at velocities above the characteristic DP lattice sound speed v0. Both compressive and rarefactive kink-type excitations are predicted, depending on the physical parameter values, which represent pulse-(shock-)like coherent structures for the dust grain relative displacement. Furthermore, the existence of breather-type localized oscillations, envelope-modulated wavepackets and shocks is established. The relation to previous results on atomic chains as well as to experimental results on strongly-coupled dust layers in gas discharge plasmas is discussed.

Theoretical and numerical investigations are carried out for the amplitude modulation of dust-ion acoustic waves (DIAW) propagating in an unmagnetized weakly coupled collisionless fully ionized plasma consisting of isothermal electrons, warm ions and charged dust grains. Modulation oblique (by an angle $\theta$ ) to the carrier wave propagation direction is considered. The stability analysis, based on a nonlinear Schrödinger-type equation (NLSE), exhibits a sensitivity of the instability region to the modulation angle $\theta$ , the dust concentration and the ion temperature. It is found that the ion temperature may strongly modify the wave’s stability profile, in qualitative agreement with previous results, obtained for an electron-ion plasma. The effect of the ion temperature on the formation of DIAW envelope excitations (envelope solitons) is also discussed.

The Galileo spacecraft was the first man-made satellite of Jupiter, orbiting the 8 planet between December 1995 and September 2003. The spacecraft was equipped 9 with a highly sensitive dust detector that monitored the jovian dust environment be-10

Results are presented that have been obtained while operating the graphite hollow cathode duoplasmatron ion source in dual mode under constant discharge current. This dual mode operation enabled us to obtain the mass and emission spectra simultaneously. In mass spectra C3 is the main feature but C4 and C5 are also prominent, whereas in emission spectra C2 is also there and its presence shows that it is in an excited state rather than in an ionic state. These facts provide evidence that C3 is produced due to the regeneration of a soot forming sequence and leave it in ionic state. C3 is a stable molecule and the only dominant species among the carbon clusters that survives in a regenerative sooting environment at high-pressure discharges.

Some properties of nonlinear dust acoustic waves in magnetized dusty plasma with variable charges by reductive perturbation technique have been studied. The effect of adiabatic dust charge variations under the assumption that the ratio of dust charging time to the dust hydrodynamical time is very small, and the nonadiabatic dust charges variations under the assumption that the same ratio is small but finite, are also incorporated. It is seen that the magnetic field and the dust charge variations significantly modify the wave amplitude. It is also seen that in case of adiabatic charge variations, the Korteweg-de Vries (KdV) equation governs the nonlinear dust acoustic wave, whereas in case of nonadiabatic dust charge variations, the wave is governed by the KdV Burger equation. Nonadiabaticity generated anomalous dissipative effect causes generation of the dust acoustic shock wave. Numerical integration of KdV Burger equation shows that the dust acoustic wave admits oscillatory (dispersion dominant) or monotone (dissipation dominant) shock solutions depending on the magnitude of the coefficient of the Burger term. Index Terms-Charge fluctuations, magnetized dusty plasma, nonlinear dust acoustic waves.

Experiments with micropareticles flow were conducted in a DC discharge. A sharp threshold in the neutral gas pressure for the onset of an unstable dust wave mode was observed. Highly space and time resolved measurements of the microparticle flow combined with probe measurements of the plasma parameters have allowed detailed comparison with a theoretical model. The model demonstrates good qualitative and quantitative agreement with the experimental data providing accurate estimates of the particle charge.

Knowledge of the real EEDF is of great importance in understanding the underlying physics of processes occurring at the magnetized plasma, such as the formation of transport barriers, cross-field diffusion coefficients and plasma–substrate interactions. In the present
experiment, the application of Langmuir probe to evaluate EEPF in presence of magnetic fields within the range 594 –32 G is investigated. The data recorded for EEPFs in magnetic fields and
in dust is acquired using current–voltage characteristics measured in low pressure hydrogen plasma. The values of plasma density, electron temperature and EEPF are evaluated with a single cylindrical Langmuir probe at different axial positions (1 cm to 6 cm) from the magnet.
From the recent EEPF observations in presence of magnetic field, it shows a bi-Maxwellian EEPF structure at different magnetic fields. But at different magnetic field, it is observed that the low energy electron population changes whereas the high-energy electron population
remains almost constant. EEPF measurement shows almost identical behaviour with the unmagnetized plasma when the larmour radius of electron is greater than or equal to 10 times of the probe radius.

The Galileo spacecraft was the first man-made satellite of Jupiter, orbiting the 8 planet between December 1995 and September 2003. The spacecraft was equipped 9 with a highly sensitive dust detector that monitored the jovian dust environment be-10

An interesting aspect in the research of complex (dusty) plasmas is the experimental study of the interaction of microparticles with the surrounding plasma for diagnostic purposes. Local electric fields can be determined from the behavior of particles in the plasma, i.e. particles may serve as electrostatic probes. From particle trajectories, also the determination of momentum flux in beams is possible, and the particles serve as force probes. Recently, temperature sensitive features of particular phosphors were utilized for measuring the surface temperature of microparticles confined in the sheath of a rf plasma. The experiments were performed under variation of gas pressure and rf power of the process plasma and offer a promising approach for the improvement of process plasmas and the understanding of plasma-particle interaction.

On the basis of a dynamic analysis the effectiveness of dust suspension for the reduction and enhancement of electron density in the E-layer of the ionosphere has been investigated in this paper. The analysis is based on the modelling of the E-layer as the Chapman α layer (validated earlier); the electron/ion production function, arrived at by Chapman and effective electron temperature-dependent electron-ion recombination coefficients in agreement with observations have been used. The balance of the charge on the particles and the number/ energy balance of the constituents have been taken into account. The following is the physics of the change in electron density in the ionosphere by the suspension of dust. First, the dust provides a source (emission) and sink (accretion) of electrons. Second, the dust emits photoelectrons with energies much higher than those of ambient electrons, which enhances the electron temperature, leading to a reduced electron-ion recombination coefficient, and thus to a higher electron density. An interplay of these processes and the natural processes of electron production/annihilation determines the electron density and temperature in the dust suspension in the ionosphere. The numerical results, corresponding to suspension of dust of silicate (high work function) and Cs coated bronze (low work function) in the E-layer at 105 km are presented and discussed.

Effects of nonadiabaticity of variable dust charge, dust fluid temperature, trapped electrons as well as nonisothermality of ions on the amplitude modulation of dust acoustic waves in an unmagnetized dusty plasma are investigated. A modified nonlinear Schrödinger equation (NLSE) is obtained by the standard reductive perturbation technique and is solved numerically by the split-step Fourier method. The modulational instability and the envelope solitary wave structure are found to be modified somewhat by the effects of nonthermally distributed ions and trapped electrons.

ARTICLE scitation.org/journal/adv Dust ion acoustic multi-shock wave excitations in the weakly relativistic plasmas with nonthermal nonextensive electrons and positrons ABSTRACT This article investigates the dust ion acoustic multi-shock wave excitations in weakly relativistic multi-component plasma by assuming non-thermal, nonextensive electrons and positrons, relativistic ion fluid having kinetic viscosity and immobile dust. Burgers equation is derived to investigate such excitations by applying the reductive perturbation method. The exponential functions are directly implemented to determine the novel multi-shock wave solution of Burgers equation. The dust ion acoustic (DIA) multi-shock wave excitations are investigated systematically to reveal the effects of parameters, namely, viscosity coefficient of ions, positron to electron density ratio, immobile dust to electron density ratio, ion to electron temperature ratio, electron to positron temperature ratio, and relativistic streaming factor of ions in the presence of nonthermal, nonextensive, and concurrently acting nonthermal and nonextensive electrons as well as positrons. It is found that the amplitudes and widths of not only single, but also multi-shock wave compressive and rarefactive electrostatic potential structures are changed with the influence of all plasma parameters. The obtained results may be useful to analyze the nature of DIA multi-shock wave phenomena in various astrophysical as well as space environments (particularly, in pulsar relativistic winds with supernova ejecta) and future studies in plasma laboratory.

The charge of dust particles is determined experimentally in a bulk dc discharge plasma in the pressure range 20 -100 Pa. The charge is obtained by two independent methods: one based on an analysis of the particle motion in a stable particle flow and another on an analysis of the transition of the flow to an unstable regime. Molecular-dynamics simulations of the particle charging for conditions similar to those of the experiment are also performed. The results of both experimental methods and the simulations demonstrate good agreement. The charge obtained is several times smaller than predicted by the collisionless orbital motion theory, and thus the results serve as an experimental indication that ion-neutral collisions significantly affect particle charging.

Thus, generation of the laserinduced dust gas
plasma flows is possible in the case when the spectral
energy ionization threshold for the matrix substance
proves to be below the dustcomponent evaporation
threshold. Such conditions are possible under the
action of shortwave radiation on easily ablating poly
meric media containing fine transparentdielectric
particles. The electron concentration in plasma and
the dustparticle charge achievable in this way enable
us to carry out investigations in a previously unstudied
range of parameters. The comparative analysis of the
known data on the dustparticle charge with those
obtained in this work shows that there exists a depen
dence of the dustparticle charge on the electron con
centration in the plasma carrier, which is close to that
for the power (with the superscript equal to ~0.25).
The effect of photoexcitation and step photoioniza
tion of dust particles can be used for increasing the
absorption coefficient for dust gasplasma flows with
the purpose of increasing the efficiency of their further
heating by the coherent IR radiation, which is of prac
tical interest for solving urgent problems of designing
plasma–photon energy installations of the highden
sity power of various purposes with active plasma–dust
media.

Scattering in the attractive screened Coulomb (Yukawa) potential is investigated. The momentum-transfer cross section is numerically calculated and analytical approximations are presented. The results are applied to estimate the ion-drag force acting on an isolated micron-sized grain in low-pressure bulk plasmas.

In this paper, the characteristics of dust-acoustic solitary waves in dusty plasma are studied. Dust charge and temperature are treated as variables. The authors have used the pseudopotential method to investigate the possibility of compressive as well as rarefactive solitons. An expression for the pseudopotential has been derived. The pseudopotential is a function of the Mach number, the relative temperature of low and high ion components, the relative ion concentration of dust charge and the temperature. Numerical computation shows that for the chosen set of parameters, only compressive solitons exist and their amplitudes increase with increasing Mach number. An increase in the dust temperature results in the disappearance of the compressive soliton. It is the only small parameter regime where compressive as well rarefactive solitons coexist. The effect of the relative ion temperature on solitons is also investigated. In the small amplitude limit, an increase in the dust temperatu...

Some properties of nonlinear dust acoustic waves in magnetized dusty plasma with variable charges by reductive perturbation technique have been studied. The effect of adiabatic dust charge variations under the assumption that the ratio of dust charging time to the dust hydrodynamical time is very small, and the nonadiabatic dust charges variations under the assumption that the same ratio is small but finite, are also incorporated. It is seen that the magnetic field and the dust charge variations significantly modify the wave amplitude. It is also seen that in case of adiabatic charge variations, the Korteweg-de Vries (KdV) equation governs the nonlinear dust acoustic wave, whereas in case of nonadiabatic dust charge variations, the wave is governed by the KdV Burger equation. Nonadiabaticity generated anomalous dissipative effect causes generation of the dust acoustic shock wave. Numerical integration of KdV Burger equation shows that the dust acoustic wave admits oscillatory (dispersion dominant) or monotone (dissipation dominant) shock solutions depending on the magnitude of the coefficient of the Burger term.

The Galileo spacecraft was the first man-made satellite of Jupiter, orbiting the planet between December 1995 and September 2003. The spacecraft was equipped with a highly sensitive dust detector that monitored the jovian dust environment between approximately 2 and 370 R J (jovian radius R J = 71492 km). The Galileo dust detector was a twin of the one flying on board the Ulysses spacecraft. This is the tenth in a series of papers dedicated to presenting Galileo and Ulysses dust data. Here we present data from the Galileo dust instrument for the period January 2000 to September 2003 until Galileo was destroyed in a planned impact with Jupiter. The previous Galileo dust data set contains data of 2883 particles detected during Galileo's interplanetary cruise and 12978 particles detected in the jovian system between 1996 and 1999. In this paper we report on the data of additional 5389 particles measured between 2000 and the end of the mission in 2003. The majority of the 21250 particles for which the full set of measured impact parameters (impact time, impact direction, charge rise times, charge amplitudes, etc.) was transmitted to Earth were tiny grains (about 10 nm in radius), most of them originating from Jupiter's innermost Galilean moon Io. They were detected throughout the jovian system and the impact rates frequently exceeded 10 min −1. Surprisingly large impact rates up to 100 min −1 occurred in August/September 2000 when Galileo was far away (≈ 280 R J) from Jupiter, implying dust ejection rates in excess of 100 kg s −1. This peak in dust emission appears to coincide with strong changes in the release of neutral gas from the Io torus. Strong variability in the Io dust flux was measured on timescales of days to weeks, indicating large variations in the dust release from Io or the Io torus or both on such short timescales. Galileo has detected a large number of bigger micron-sized particles mostly in the region between the Galilean moons. A surprisingly large number of such bigger grains was measured in March 2003 within a 4-day interval when Galileo was outside Jupiter's magnetosphere at approximately 350 R J jovicentric distance. Two passages of Jupiter's gossamer rings in 2002 and 2003 provided the first actual comparison of in-situ dust data from a planetary ring with the results inferred from inverting optical images. Strong electronics degradation of the dust instrument due to the harsh radiation environment of Jupiter led to increased calibration uncertainties of the dust data.

The Galileo spacecraft was the first man-made satellite of Jupiter, orbiting the planet between December 1995 and September 2003. The spacecraft was equipped with a highly sensitive dust detector that monitored the jovian dust environment between approximately 2 and 370 R J (jovian radius R J = 71492 km). The Galileo dust detector was a twin of the one flying on board the Ulysses spacecraft. This is the tenth in a series of papers dedicated to presenting Galileo and Ulysses dust data. Here we present data from the Galileo dust instrument for the period January 2000 to September 2003 until Galileo was destroyed in a planned impact with Jupiter. The previous Galileo dust data set contains data of 2883 particles detected during Galileo's interplanetary cruise and 12978 particles detected in the jovian system between 1996 and 1999. In this paper we report on the data of additional 5389 particles measured between 2000 and the end of the mission in 2003. The majority of the 21250 particles for which the full set of measured impact parameters (impact time, impact direction, charge rise times, charge amplitudes, etc.) was transmitted to Earth were tiny grains (about 10 nm in radius), most of them originating from Jupiter's innermost Galilean moon Io. They were detected throughout the jovian system and the impact rates frequently exceeded 10 min −1. Surprisingly large impact rates up to 100 min −1 occurred in August/September 2000 when Galileo was far away (≈ 280 R J) from Jupiter, implying dust ejection rates in excess of 100 kg s −1. This peak in dust emission appears to coincide with strong changes in the release of neutral gas from the Io torus. Strong variability in the Io dust flux was measured on timescales of days to weeks, indicating large variations in the dust release from Io or the Io torus or both on such short timescales. Galileo has detected a large number of bigger micron-sized particles mostly in the region between the Galilean moons. A surprisingly large number of such bigger grains was measured in March 2003 within a 4-day interval when Galileo was outside Jupiter's magnetosphere at approximately 350 R J jovicentric distance. Two passages of Jupiter's gossamer rings in 2002 and 2003 provided the first actual comparison of in-situ dust data from a planetary ring with the results inferred from inverting optical images. Strong electronics degradation of the dust instrument due to the harsh radiation environment of Jupiter led to increased calibration uncertainties of the dust data.

The existence of Bernstein–Greene–Kruskal (BGK)-like trapped ion modes in dusty plasmas is investigated by means of the pseudo-potential method applied to the Vlasov–Poisson system. The nonlinear dispersion relation, determining the phase velocity, and the pseudo-potential, representing the wave form and hence its spectral decomposition, are derived and analysed with respect to the effect of dust. Dust is found to diminish the region in ω, k-space, where periodic wave solutions of fast and slow mode character exist. Localized wave solutions in the form of solitary ion holes, owing their existence to the presence of dust, coexist as well, but turn into slow ion acoustic double layers in the limit of vanishing dust.

In this review, an overview of structural properties and phase transitions in finite spherical dusty (complex) plasma crystals -so-called Yukawa balls -is given. These novel kinds of Wigner crystals can be directly analyzed experimentally with video cameras. The experiments clearly reveal a shell structure and allow to determine the shell populations, to observe metastable states and transitions between configurations as well as phase transitions. The experimental observations of the static properties are well explained by a rather simple theoretical model which treats the dust particles as being confined by a parabolic potential and interacting via an isotropic Yukawa pair potential. The excitation properties of the Yukawa balls such as normal modes and the dynamic behavior, including the time-dependent formation of the crystal requires, in addition, to include the effect of friction between the dust particles and the neutral gas. Aside from first-principle molecular dynamics and Monte Carlo simulations several analytical approaches are reviewed which include shell models and a continuum theory. A summary of recent results and theory-experiment comparisons is given and questions for future research activities are outlined.

The wave dispersion relation in a two-dimensional strongly coupled plasma crystal is studied by theoretical analysis and molecular dynamics simulation taking into account a constant magnetic field parallel to the crystal normal. The expression for the wave dispersion relation clearly shows that high-frequency and low-frequency branches exist as a result of the coupling of longitudinal and transverse modes due to the Lorenz force acting on the dust particles. The high-frequency and the low-frequency branches are found to belong to right-hand and left-hand polarized waves, respectively.

Two applications of microparticles (micron-size particles) for laboratory plasma diagnosis are discussed. The first application is about injecting hypervelocity microparticles [(HDI) for hypervelocity dust injection] for internal magnetic field measurement in high-temperature plasmas. Since the concept of HDI has already been examined in details in our previous works, the primary focus here is to compare different schemes of microparticle acceleration. A new design of HDI based on plasma-dynamic accelerator is described to inject multiple microparticles to velocities around 10 km/s simultaneously. The other application is about using microparticles to measure plasma flow [(mPTV) for microparticle tracer velocimetry]. Directions of plasma flow at multiple locations can be measured simultaneously using mPTV because ion drag dominates over other forces inside laboratory plasmas of order 10 19 m 3 in density and a few electron volts in temperature. In addition to complex interactions between a microparticle with plasma, the magnitude of plasma flow may not be obtained directly from the microparticle velocity because of the time it takes for each microparticle to relax to local plasma velocity. In summary, microparticles are naturally small objects in all three dimensions and can, therefore, become useful diagnostics for laboratory plasmas with minimal perturbation.

Two applications of microparticles (micron-size particles) for laboratory plasma diagnosis are discussed. The first application is about injecting hypervelocity microparticles [(HDI) for hypervelocity dust injection] for internal magnetic field measurement in high-temperature plasmas. Since the concept of HDI has already been examined in details in our previous works, the primary focus here is to compare different schemes of microparticle acceleration. A new design of HDI based on plasma-dynamic accelerator is described to inject multiple microparticles to velocities around 10 km/s simultaneously. The other application is about using microparticles to measure plasma flow [(mPTV) for microparticle tracer velocimetry]. Directions of plasma flow at multiple locations can be measured simultaneously using mPTV because ion drag dominates over other forces inside laboratory plasmas of order 10 19 m 3 in density and a few electron volts in temperature. In addition to complex interactions between a microparticle with plasma, the magnitude of plasma flow may not be obtained directly from the microparticle velocity because of the time it takes for each microparticle to relax to local plasma velocity. In summary, microparticles are naturally small objects in all three dimensions and can, therefore, become useful diagnostics for laboratory plasmas with minimal perturbation.

In this paper, we consider the nonlinearity in the propagation of electromagnetic (e.m.) waves in a
plasma caused by the electron temperature dependence of the coefficient of recombination of
electrons with ions; specifically, the ionospheric E layer has been investigated. The enhancement in
electron temperature by an intense electromagnetic wave causes reduction of the electron-ion
recombination coefficient and thereby enhancement of electron density, the electron collision frequency
also gets enhanced. The equations for number and energy balance of electrons and the
wave equation have been used to predict the dependence of electron density/collision frequency
and the nonlinear refractive index and absorption coefficient on aE20
(proportional to wave irradiance).
The dependence of the propagation parameters on aE20
has been used to investigate the nonlinear
electromagnetic wave propagation in the ionosphere. The study concludes that the electron
temperature dependence of the recombination coefficient should be considered in all analyses of
nonlinear plasma-e.m. wave interaction.