Orbital-motion-limited theory of dust charging and plasma response

Two years before Irving Langmuir coined the term “plasma” as in plasma physics,¹ he and Mott-Smith laid down a foundational theory on the charging of a spherical and cylindrical probe in a laboratory plasma,² which was necessary for interpreting the measurement of what became later known as Langmuir probes.³ In the ensuing decades, the charging and dynamics of solid particulates immersed in plasmas built the foundation for a new discipline in plasma physics—dusty plasmas,^4–6 in which the collective behavior of a group of dust particles in a plasma environment is studied. The physics of both dust in a plasma and dusty plasmas finds applications in space,^7,8 astrophysics,^9,10 and laboratory.^11–13 The fundamental dust-plasma interaction, which is also essential to understand the collective behavior in a dusty plasma, includes (1) charging of the dust by absorption of plasma particles so it becomes subject to the electric field (⁠$E$⁠) via the electrical force $QdE$ (⁠$Qd$ is the dust charge); (2) heating of the dust via the collection of plasma electron and ion energy fluxes so the dust particulate can melt, evaporate, or simply sublimate; (3) dragging of the dust via a frictional force enhanced by the Coulomb interaction with the flowing background plasma.

To understand the dust dynamics and its change of state, one must resort to a dust charging theory. The most-widely used of such is the Orbital-Motion-Limited (OML) theory, which dates back to the work of Mott-Smith and Langmuir.² In the 1960s, Al'pert et al.¹⁴ and Laframboise¹⁵ completed the current formulation. The OML theory is known to predict accurately the dust potential for a small^16,26 and not so small dust grain,^17–19 despite its simplifying assumption on collisionless ion orbit that misses the absorption radius effect away from the dust surface.^20,21 Surprisingly, the existing formulation cannot be used for predicting the plasma response, namely, solving the plasma potential $ϕ$⁠, as shown in Allen et al.²² As the result, OML theory to date cannot predict the fundamental quantity of dust charge. In practice, one has been using an idealized dust charge-potential relationship due to Whipple,⁷ who computed the dust capacitance using the conventional Debye shielding potential that does not take into account the constraint of angular momentum conservation in setting the plasma density near the dust surface.

The purpose of this paper is to present a revised OML formulation that, for the first time, is able to predict the plasma response and hence the dust charge. The OML predictions will then be contrasted with the Whipple approximation to elucidate the missing physics in dust charging for a spherical dust of comparable size with the Debye length, which is of special importance to dust transport/survivability in tokamaks.^23,24 This new OML formulation is also important for electron-emitting dust particulates as it provides the basis for extending the OML theory to positively charged dust.²⁵ 

As an approximation of the Orbital Motion (OM) theory which follows the collisionless particle orbit via conservation of energy and angular momentum, the OML theory will inevitably introduce discrepancies in the evaluation of dust current/heat collection, and in the plasma ion density evaluation. The former will give rise to a discrepancy in dust potential, while the latter in plasma potential and hence dust charge, in comparison with those predicted by the OM theory. The usefulness of the OML formulation is due to its simplicity, and the relatively high accuracy for a variety of applications where the dust size is not large compared with the plasma Debye length. It must be clarified that the fatal breakdown of the previous OML formulation in the form of an imaginary ion density when applied to calculate the plasma potential and dust potential²² is not due to the OML approximation of ignoring the ion absorption radius effect. The root cause is the OML ion density formula originally given in Ref. 14, which we find will produce an imaginary density as long as Debye shielding of a dust particle is in place. In the Appendix, after we present the corrected OML theory and its application in calculating the plasma potential and dust charge, we give a detailed account of how the previous OML formulation would break even in the regime of small dust size compared to Debye length, a limit commonly accepted for OML approximation to be highly accurate.

The rest of the paper is organized as follows. In Sec. II, we briefly recall the key components of the OML theory and its inability to predict dust charge in its current formulation. A corrected formulation on the OML ion density is shown in Sec. III, which completes the OML formulation for calculating both dust potential and dust change. In Sec. IV, we apply the new OML model to evaluate the plasma response and the dust charge, and contrast the results with the widely used Whipple approximation. The findings are summarized in Sec. V.

There are three essential ideas underlying OML theory. First, the collection of plasma electrons and ions by the dust is governed by their collisionless orbit. These are subject to two conservation laws: energy [$E=mα(vr2+vt2)/2+qαϕ(r)$] conservation and angular momentum (⁠$J=mvtr$⁠) conservation. Here, m (q) and $ϕ$ are the mass (charge) of the particles of species α (⁠$α=e, i$ labels electrons and ions, respectively) and the plasma potential, while r, v_r, and v_t are the radial distance, radial velocity, and tangential velocity in a spherical reference frame centered on the dust grain. The radial motion of the plasma particles is governed by a one degree-of-freedom Hamiltonian with effective potential $Φeff$⁠,

Second, whether the plasma particle reaches the dust is governed by $Φeff$⁠. For a typical negatively charged dust, $ϕ$ is negative and monotonically increasing with r, so that $Φeff$ is positive and monotonically decreasing with r for electrons. As the result, only electrons with $E>Φeff(r=rd)$ from far away can reach the dust of radius $rd$⁠. The final and third idea, which is a simplifying approximation, is what delineates OML from the more complete but rarely used OM theory.^15,26 The OML theory approximates the $Φeff$ for ions also as a monotonic function of r, which results in the simplification that ions with $vr2>−2Φeff(rd)/mi$ from far away can reach and charge the dust. This is a remarkable simplification since one does not need $ϕ(r)$ to evaluate the ion and the electron current collected by the dust, which are

Here, e is the elementary charge, k_B is the Boltzmann constant, $ne0$ (⁠$ni0$⁠) is the electron (ion) density away from the dust, $ne0=Zni0, Te, (i)$ is the electron (ion) temperature, $qe=−e$⁠, q_i = Ze, and $ϕd=ϕ(rd).$ Setting $Ii+Ie=0$⁠, one can solve for the dust potential as a function of $Te/Ti$⁠, ion charge state Z, and $me/mi$⁠. For an electron-proton plasma with $Te=Ti$⁠, OML predicts $ϕd=−2.5(kBTe/e)$⁠, which is in remarkable agreement with particle-in-cell simulations¹⁹ that do not make the OML assumption of ion $Φeff$ being monotonic.

The OML approximation of a monotonic ion $Φeff$ is known to be violated for ions with a certain range of J. This comes about²¹ because the centrifugal potential energy $J2/(2mir2)$ decreases with r at precisely $1/r2$⁠, but the electrical potential energy $Zeϕ(r)$ increases at a faster rate (exponential in r) in the Debye shielding region, before it eventually asymptotes to $1/r2$ for large r. This can produce one or multiple extrema in $Φeff$ away from the dust surface, Fig. 1, which can turn back some ions in a range of J at $rm>rd$⁠. In the literature, this is known as the absorption radius (at r_m > r_d) effect for certain ions. By neglecting this subtlety, OML approximation would over-estimate the ion current to the dust. Interestingly, for dust size small and even comparable to Debye length, this correction appears to be small and OML prediction of dust potential remains reliable.

A far more serious problem was identified by Allen, Annaratone, and de Angelis in 2000 (Ref. 22) that the OML theory cannot predict the plasma response to the presence of a dust particle, which requires the solution of the OML Poisson equation for the plasma potential

where $ε0$ is vacuum permittivity. Allen et al.²² used the well-known OML ion and electron density given by Al'pert, Gurevich, and Pitaevskii in 1965,¹⁴ 

where

The failure of OML theory manifests in an imaginary ion density when $ϕ>ϕd/z2$⁠. For the initially faster than $1/r2$ increase in $ϕ(r)$ due to Debye shielding, which is the cause for ion absorption radius at r_m > r_d as noted previously for a negatively charged dust, Allen et al. concluded that the contradiction between $ϕ(r)$ and an imaginary n_i in Eq. (6) would be an intrinsic defect, which prevents the OML theory from predicting the plasma response. Since the dust charge is related to the normal electric field at the dust surface, which requires the solution of $ϕ(r)$⁠, one reaches the inevitable position from Ref. 22 that OML theory cannot predict the dust charge, despite its success on dust potential as re-affirmed in Refs. 16 and 18.

Intuitively, it is quite puzzling that the OML simplification of ignoring the absorption radius effect, which contributes a small error in the ion charging current, would produce an imaginary ion density, as revealed in the analysis of Allen et al. Physically, a maxima in ion $Φeff(r)$ at r_m > r_d would pose a barrier that turn back ions with a range of J. Ignoring this effect with the OML approximation should manifest in an over-estimation of the ion density for $r<rm$⁠. So why Allen et al. discovered an apparently inherent contradiction in the OML theory between $ϕ(r)$ and $niOML(r)?$

Following the physical picture just given, one is tempted to conclude that the resolution has to come from a revision of the OML expression for $niOML(r)$ as given in Eq. (6). We find that this is indeed the case. The cause is a change in integration bound for the OML ion density when the plasma potential transitions from $ϕ(r)<ϕd/z2$ to $ϕ(r)>ϕd/z2$⁠. To understand this subtlety, which has been evidently elusive for the past five decades, we recall that since OML assumes a monotonically varying $Φeff(r)$⁠, the ions at r have a collisionless orbit that will either intercept the dust particle or be reflected by the effective potential before it can reach the dust surface.

In the canonical case that $limr→∞ϕ(r)=0$⁠, the birth energy of the background plasma ion far away must have

If this ion reaches r, it must have, at r, that

For a negatively charged dust which has $ϕ(r)<0$⁠, a plasma ion of such unbounded orbit (meaning that the ion orbit connects to infinity) must have higher kinetic energy as it approaches the dust,

As illustrated in Fig. 2, this is outside a circle in (v_r, v_t) space, which intercepts the v_r = 0 axis at

Not all of these ions can reach the dust surface (r = r_d) due to angular momentum conservation. In the OML approximation, the effective potential $Φeff(r)$ peaks at $r=rd$⁠,

The ions with $E<Φeff(rd)$ will be reflected by the effective potential before they can reach $rd$⁠. At $r>rd$⁠, these reflected ions satisfy, after explicitly writing out $E<Φeff(rd)$⁠,

For a negatively charged dust with $ϕd−ϕ(r)≤0$⁠, the reflected ions are bounded by a parabola in (v_r, v_t) space,

As illustrated in Fig. 2, it intercepts the v_r = 0 axis at

As we shall see, this should be compared with the intercept of Eq. (13) with v_r = 0 axis, i.e., $vtb$ in Eq. (14).

To evaluate the ion density at $r≥rd$⁠, which is the solution of the Vlasov equation, we integrate the Maxwellian distribution, which is assumed for the background plasma far away, in the velocity space region where they are allowed.¹⁴ For ions with $vr<0$⁠, their population lies in the region of the (v_r, v_t) velocity space given by

For ions with $vr>0$⁠, only the reflected ions are present so they are in a region of the velocity space given by

These give rise to integration bounds for $vr<0$ and $vr>0$ separately. The one for $vr<0$ is straightforward, see Fig. 2, and the previous OML form¹⁴ is correct.

The one for $vr>0$ is complicated by the possibility that $vtr2<vtb2$⁠, which implies

If this is the case, the integration has two zones, separated by a set of critical $vrc$ and $vtc$⁠, which are the intercept of the two constraints, $vr2+vt2=−2Zeϕ(r)/mi$ and $(z2−1)vt2−vr2=2Ze(ϕ−ϕd)/mi$⁠,

As shown in the left-side diagram of Fig. 2, the integration can be carried out in two zones, depending on whether $vr>vrc$ or not. The lower zone (denoted as I) is

The upper zone (denoted as II) is

The ion density is given by

which yields Eq. (6), now valid only for$ϕ(r)<ϕd/z2$⁠.

In the case that $vtr2>vtb2$⁠, i.e.,

the integration is restricted to a single zone given by

which is illustrated in the right-side diagram of Fig. 2. The ion density then takes the form

where $ϕ(z)≥ϕd/z2$⁠, while Eq. (6) should be used for $ϕ(z)<ϕd/z2$⁠. Contrasting Eq. (28) with Eq. (6), one sees that the previously known imaginary OML ion density for $ϕ>ϕd/z2$ is removed in the corrected OML theory for $ni$⁠.

With the corrected OML ion density, the OML Poisson equation (4) can be solved for the plasma response. It is important to note that the radial electric field at the dust surface $Ed=−dφ/dz(z=1)$ is related to the dust charge according to Gauss's law,

Since the OML theory to date cannot predict $Qd$⁠, users of the OML charging theory have been using a simple relation for spherical dust capacitance due to Whipple,⁷ 

with

the electron Debye length. Whipple's idea is to expand the plasma density at $z≫1$⁠, as in the standard Debye shielding calculation. Applying this approach to OML's Poisson equation, one finds

and hence

Introducing the so-called linearized Debye length

and defining the normalized dust radius as

one finds the solution to the Poisson equation,

Hence, the dust charge is

Unlike the original Whipple formula [Eq. (30)], which assumes a uniform ion density, Eq. (38) takes into account the ion density response. It is interesting to note that for a small dust $rd≪λD$⁠, the Debye shielding contribution, which is usually small, can be enhanced substantially if $Z≫1$ and/or $Te≫Ti$⁠. We will call the charge-potential relationship given in Eq. (38) the generalized Whipple approximation, to distinguish it from the well-known Whipple formula in Eq. (30).

The actual OML density for $r−rd<λD$ can deviate significantly from the asymptotic expansion valid when $r−rd≫λD$⁠, see Fig. 3. This is already evident from the analytical form of n_e and n_i in OML theory. Since the dust charge can be alternatively computed by integrating the net charge of the plasma,

due to overall charge conservation, one suspects that the OML prediction of dust charge can be significantly different from that in Eq. (38). From the numerically evaluated n_e and n_i shown in Fig. 3, we find that sharper deviation occurs closer to r_d as the dust size becomes smaller, but there is a greater spatial extent of the deviation for large dust size. Since the dust charge corresponds to the spatial integration of electron and ion density, one finds that such deviation from the simple Debye shielding calculation is proportional to $rd/λD$⁠. In Fig. 4, we plot

as a function of $rd/λD$⁠. The specific case has Z = 1 and $β=Te/Ti=1$⁠, so the generalized Whipple approximation, Eq. (38), is simply

The deviation of the OML prediction of dust charge from Whipple approximation can be significantly greater if the dust size is approaching the Debye shielding length, which is consistent with the results of Ref. 27. This is ultimately due to the effect of angular momentum conservation on the plasma density near the dust particle. Obviously, a significant correction in Q_d implies a substantial change in the electrical force $QdE$⁠, which can impact the dust dynamics, for example, in a tokamak reactor.^23,24

In conclusion, we have resolved a long-standing issue in the OML charging theory, and by doing so, obtained a complete OML theory that predicts both dust potential and dust charge. This is enabled by a revised OML ion density formula for the case of $ϕ>ϕd/z2$⁠, which is given in Eq. (28). We also provide the first calculation of the plasma potential and the dust charge using the OML theory. Our results show that for applications where the dust particulate radius is much smaller than the plasma Debye length, the Whipple approximation is in good agreement with the OML prediction of the dust charge-potential relationship. When the dust size is comparable to or larger than the Debye shielding length, which is a case of importance to magnetic fusion, we discover significant deviation from the Whipple approximation of the dust charge-potential relationship. This is attributed to the fundamental role of angular momentum conservation in setting the plasma electron and ion density near the dust particle.

Since OML is an approximation of the OM theory by ignoring the ion absorption radius effect, one expects that its prediction of dust potential and charge should deviate from that of the OM theory. The usefulness of OML charging theory traditionally lies with its surprisingly good agreement with OM on dust potential.^16–19,26 With the new OML formulation that is able to calculate dust charge and plasma response, it is an important next step to carry out a detailed comparison of the OML dust charge prediction with OM theory for a range of dust sizes.

This research was supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under Contract No. DE-AC52-06NA25396.

As pointed out by Allen et al.,²² the OML ion density can become imaginary if one is to solve the Poisson equation, Eq. (4), for the plasma potential using the previously known OML ion density formula by Al'pert et al.¹⁴ This suggests a fundamental breakdown of the OML theory for plasma potential and hence dust charge calculation. Allen et al. investigated this problem from the angle of the OML approximation, i.e., the neglect of the ion absorption radius effect. In the main text, we give a physics argument that ignoring the ion absorption radius effect should only introduce a discrepancy in ion density compared to OM prediction, but not an unphysical imaginary number. Our corrected calculation of the OML ion density removes the possibility of an imaginary density. Here, we give a more detailed account of how the imaginary ion density comes about in the previous OML formulation. The objective is to further clarify the contrast between (1) a physical approximation (OML) that introduces quantitative discrepancy, which vanishes in its limit of applicability, and (2) an invalid theoretical formulation that produces unphysical results.

The condition for $ni(z)$ to become imaginary (hence unphysical) in Eq. (6) is to have

For a negatively charged dust (⁠$φ<0$ and $φd<0$⁠), this implies a plasma potential $φ(z)$ that rises faster than $φd/z2$ with z – 1 the normalized distance from the dust surface. As long as Debye shielding is in action (i.e., the plasma transport to the dust surface is not intrinsically ambipolar), we know that $φ$ would rise exponentially as a function of z within the Debye shielding sphere. The cross-over point after which Eq. (A1) is satisfied can be illustrated using the Debye shielding potential solution of Eq. (37) by setting $φd/z2φ=1$⁠. The cross-over point z_c is the solution of

Here, $rd̂$ is the dust radius normalized against the Debye length and z – 1 is the radial distance from the dust surface normalized against the dust size. For $rd̂=0.01$⁠, one finds z_c = 648 or 6.48 Debye length away. If $rd̂=0.0001$⁠, we have z_c = 116 672 or 11.7 Debye length away. In other words, the breakdown of the ion density formula, Eq. (6), is the direct result of the Debye shielding physics, which is independent of how small the dust size is compared to the Debye length.

Of course, the Debye shielding potential as given in Eq. (37) is an analytical approximation, so the actual cross-over point z_c would likely differ in its exact location from that predicted by Eq. (A2). When the OML Poisson equation is numerically solved, the failure of the OML formulation using the ion density given in Eq. (6) manifests in a not-a-number floating point violation for all small $rd̂$s we have attempted, as one would expect by taking the squared root of a negative number. This is to be contrasted with the corrected OML formulation, which produces the numerical solutions shown in Sec. IV.