On the charge of nanograins in cold environments and Enceladus dust
OOn the charge of nanograins in cold environmentsand Enceladus dust
N. Meyer-Vernet
LESIA, Observatoire de Paris - CNRS - Universit´e Pierre et Marie Curie - Universit´eDenis Diderot, Meudon, France.
Abstract
In very-low energy plasmas, the size of nanograins is comparable to the dis-tance (the so-called Landau length) at which the interaction energy of twoelectrons equals their thermal energy. In that case, the grain’s polarizationinduced by approaching charged particles increases their fluxes and reducesthe charging time scales. Furthermore, for grains of radius smaller than theLandau length, the electric charge no longer decreases linearly with size, buthas a most probable equilibrium value close to one electron charge. We giveanalytical results that can be used for nanograins in cold dense planetary en-vironments of the outer solar system. Application to the nanodust observedin the plume of Saturn’s moon Enceladus shows that most grains of radiusabout 1 nm should carry one electron, whereas an appreciable fraction ofthem are positively charged by ion impacts. The corresponding electrostaticstresses should destroy smaller grains, which anyway may not exist as crys-tals since their number of molecules is close to the minimum required forcrystallization.
Keywords:
Nanostructures and nanoparticles, Ices, Enceladus, Saturn,Satellites, Interplanetary dust
1. Introduction
Dust particles of nanometric size have been detected in situ in variousparts of the solar system, e.g. near comets (Utterback and Kissel, 1990),
Email address: [email protected] () URL: () Preprint submitted to Icarus October 31, 2018 a r X i v : . [ a s t r o - ph . E P ] J un n the Earth low ionosphere (e.g. Friedrich and Rapp (2009) and referencestherein), in the solar wind near 1 AU (Meyer-Vernet et al., 2009), in streamsejected by Jupiter (Zook et al., 1996) and Saturn (Kempf et al., 2005), inthe atmosphere of Saturn’s moon Titan (Coates et al., 2007), and in theplume ejected by the icy moon Enceladus (Jones et al., 2009). Nanograins,which make the transition between molecules and bulk materials, can be pro-duced by condensation of gases and aggregation of molecules (Kimura, 2012),and/or by fragmentation of larger dust (e.g. (Mann and Czechowski, 2012)).The large surface-to-volume ratio of nanoparticles makes the proportion ofsurface atoms significant, so that their characteristic properties often differfrom those of bulk materials and they are major agents for interactions withparticles and fields.Nanograins play an important role in magnetized environments becausetheir interaction with electromagnetic fields varies in proportion of their elec-tric charge, which varies more slowly with size than do the friction forces(proportional to surface) and the gravitational forces (proportional to vol-ume). Hence nanograins are generally driven by electromagnetic forces asare plasma particles, so that their electric charge governs their dynamics(e.g. Burns et al. (2001); Horanyi (1996); Mann and Czechowski (2012);Mann et al. (2013)). The electric charge can also determine the grain’sminimum size via the electrostatic stresses producing fracture, and it alsoaffects the grains’ growth and coalescence. At larger scales, it determines theLarmor frequency and thus the time scale of grains’ pick-up.At nanometric sizes, several effects make the charging processes differentfrom the classical charging of larger objects. First, it is well known that theparticle sticking coefficients and photoelectric and secondary emission yieldscan change (Watson, 1972; Chow et al., 1993; Weingartner and Draine, 2001;Abbas et al., 2010), essentially because the electron free path in matter is ofthe order of (or larger than) 1 nm below ∼
10 eV (Fitting et al., 2001).Two further effects appear when the grain radius becomes comparable to r L = e / (4 π(cid:15) k B T ) (1)in a plasma of temperature T . Since r L (nm) (cid:39) . /T eV , this concernsnanograins in plasmas of temperature (cid:39) ∼ r L orsmaller induce polarization charges whose Coulomb attraction increases thecollected fluxes, thereby decreasing the charging time scales. Furthermore,since at this scale the charging becomes discretized, the equilibrium chargeon a grain no longer varies in proportion of its size, but becomes comparableto one electron charge in a wide range of sizes, because the probability thatan uncharged grain collects an electron exceeds the probability that a neutralor negatively charged grain collects an ion.The latter phenomena have been studied in the contexts of the Earth’sionosphere (e.g. Jensen and Thomas (1991); Rapp and L¨ubken (2001) andreferences therein) and of the interstellar medium (e.g. Draine and Sutin(1987); Weingartner and Draine (2001)). In this paper, we consider theseeffects for cold dense planetary environments in the outer solar system, whichare subjected to different constraints. We derive analytical results that canbe used in these contexts, and apply them to the nanograins detected inEnceladus plume (Jones et al., 2009; Hill et al., 2012), where the electrons arecold enough (Shafiq et al., 2011) to put the Landau radius in the nano range,the plasma is dense enough (Morooka et al., 2011) for the photoelectronemission to be negligible, and the (larger) dust concentration is high enoughto deplete the electrons by a large amount.These calculations will enable us to estimate the grains’ size limit set byelectrostatic disruption and to compare it with other physical processes.Units are SI, unless otherwise indicated explicitly.
2. Basic impact charging
Before considering nanograins, let us briefly summarize the classical elec-tric charging by collection and emission of particles for a dust grain of radius a (cid:29) r L in a plasma whose electron and ion densities may be different becauseof the possible presence of dust. The charging of a grain changes its electric potential, which changes theparticle fluxes until an equilibrium is reached when the different charge fluxesbalance each other. The electron flux tends to exceed that of ions becauseof the faster electron speeds (except in the case of strong electron depletion3iscussed in Sect. 2.3); hence, when the charging is mainly due to electronand ion impacts, the body charges negatively until it repels sufficiently theelectrons for their flux to balance that of positive ions. For this to be so, theelectron potential energy at the body’s surface e Φ must exceed sufficiently(but not too much) the particle thermal energy ∼ k B T . Thus the equilibriumgrains’ potential with respect to the ambient plasma is Φ (cid:39) − ηk B T /e , with η of order of magnitude unity. For a sphere of radius a much smaller thanboth the Debye length L D and the grains’ separation, the electric charge is Q = 4 π(cid:15) a Φ. Substituting the above value of Φ yields the number of chargeunits
Q/e at equilibrium Z = − ηa/r L (2)with η ∼ a/r L (cid:29) η is easily calculated since in that case the particles aresubjected to the Coulomb potential of the grain without intervening barriersof potential (the so-called orbit-limited condition (Laframboise and Parker,1973; Whipple, 1981)). When the plasma particles are singly charged andhave isotropic Maxwellian velocity distributions, the classical fluxes of eachparticle species can then be expressed straightforwardly as N = N e −| η | = N e −| Z | r L /a repelled particles (3) N = N (1 + | η | ) = N (1 + | Z | r L /a ) attracted particles (4)per unit grain’s surface, where r L is the Landau radius corresponding to thetemperature of the species considered and N is the flux of that species onan uncharged grain N = s n (cid:104) v (cid:105) / s n ( k B T / πm ) / (5)Here s , n , T , m , and (cid:104) v (cid:105) are the sticking probability, number density, tem-perature, mass, and mean speed of the species concerned in the unperturbedplasma. For ions of mass m i and same temperature T as electrons of mass m e , we have N e /N i = µ ( n e /n i ) with µ = ( s e /s i )( m i /m e ) / (6)At equilibrium, the electron and ion fluxes balance, and η is the solution ofthe equation η = ln[ µ ( n e /n i ) / (1 + η )] (7)4his confirms that η is of order of magnitude unity, except if µ ( n e /n i ) (cid:39)
1- a case that we will discuss later. Note that we have not assumed n e = n i ,in order for the results to be applicable in dusty environments. Therefore,although µ (cid:29) µ ( n e /n i (cid:29)
1, but only the weaker inequality µ ( n e /n i ) > s i (cid:39)
1, but the sticking probability of electrons may be smaller because thefree path of electrons in solids (which decreases as energy decreases at energiesexceeding a few 100 eV) reaches a minimum generally smaller than 1 nm inthe vicinity of tens eV, and increases as energy decreases again to valuescomparable to or greater than 1 nm around 1 eV, taking into account elasticand inelastic scattering (Fitting et al., 2001). A conservative assumption is s e (cid:39) . − s e may be much smalleras the number of atoms decreases (Michaud and Sanche, 1987), essentiallybecause the limited number of degrees of freedom precludes the conservationof energy and momentum in the collision.For water-group incident ions µ (cid:39) × s e , so that Eq.(7) yields η (cid:39) s e n e /n i = 0 .
01, 0.1, or 1.
The above estimates assume that photoemission (including photodetach-ment) and secondary emission are negligible. The photoelectron emission onuncharged grains at heliospheric distance r AU (in astronomical units) can beapproximated by (Grard, 1973) N ph (cid:39) . × χ/r χ ∼ . − πa - to facilitate comparison withother fluxes (we have taken into account that the projected sunlit area isone-quarter of the grain’s surface area). The smaller value of χ correspondsfor example to materials such as graphite or ice, the larger to silicates.For nanograins χ may be different for two main reasons which act inopposite senses. First, since the photon attenuation length generally exceedsthe photoelectron escape length by a large amount, a small grain size limitsthe distance from the excitation region to the surface, which tends to increasethe yield compared to that of bulk materials (Watson, 1972; Weingartner and5raine, 2001). Second, the photon absorption cross-section (normalized tothe cross-sectional area) at the relevant wave lengths λ ∼ . µ m variesroughly as 2 πa/λ (cid:39) × − a nm when this size parameter is much smallerthan unity; this is expected to decrease χ significantly. Because of the largeuncertainties in these properties, we will use the conservative assumption χ ≤ . Am p ) collection foruncharged grains at 10 AU heliocentric distance ( (cid:39) Saturn’s orbit), N ph /N i ≤ A / / ( n i (cm ) T i (eV) ) (9)Photoelectron emission is thus expected to be of minor importance forwater-group incident ions of density n i ≥
50 cm − and temperature ∼ ∼ Finally, let us discuss briefly how the plasma electron depletion n e /n i isrelated to the charged dust. Strictly speaking, the above calculations holdwhen the dust grains are “isolated”, thus when the grains’ number density n d is small enough that their separation exceeds twice the Debye length, i.e.2 n / d L D < (3 / π ) / . With the grains’ charge (2) and the Debye length L D = [4 πr L ( n e + n i )] − / (10)the ratio of the charge carried by the grains to that available in the mediumis | Z | n d / ( n e + n i ) (cid:39) ηP with P ≡ πn d aL D (11)6 −2 −1 μ μ =180 η n e /n i −1/P P n e / n i − η = Z r L / a Figure 1: Electron-to-ion number density n e /n i (left axis, solid line) and normalizedgrain’s charge − η = Zr L /a (right axis, dashed) versus the parameter P = 4 πn d (cid:104) a (cid:105) L D ,from (12) with µ = 180 (water-group ions and electron sticking probability s e (cid:39)
1) fordust grains of radius a > r L . The limits n e /n i ∼ /µ and η ∼ /P for P → ∞ are plottedfor comparison (thin lines). Since
P < (2 n / d L D ) because the grains’ separation necessarily exceeds theirdiameter, we have P < L D , so that in thatcase the grains’ charge is not expected to perturb significantly the plasma.In the opposite case P >
1, the grains’ Debye spheres overlap and theelectrons are depleted since many of them rest on the grain’s surface, whichin turn reduces the grain’s charge (Havnes et al., 1984; Whipple et al., 1985).However, if the grain size is much smaller than the grains’ separation, theincrease of the grain-to-plasma capacitance due to the neighbouring grainscan be neglected and Eqs.(3)-(4) still hold (Whipple et al., 1985), with η ≡− e Φ /k B T = − Zr L /a where Φ is the grain-to-plasma potential. Therefore (7)holds in that case and using the quasi-neutrality condition n i − n e = − n d Z and rewriting (10) as n i + n e = n d a/P r L , one deduces straightforwardly n e n i = (1 + η ) e η µ = 1 − P η
P η (12)From (12), one can deduce two of the three parameters n e /n i , η , and P , from7ne of them. Fig. 1 shows n e /n i versus P , which enables one to deduce thegrains’ properties from the electron depletion or vice-versa. For completenesswe have also plotted η versus P , first published by Whipple et al. (1985) andreproduced in several papers. Note that the pioneering results by Havnes etal. (1984, 1990) and references therein use a different definition of P , basedon the plasma properties outside the dusty region - which is relevant forstudying the local properties of a dust cloud; both definitions agree in thelimit of small potentials (when the Boltzmann factors can be linearized).In the limit P → ∞ , (12) yields n e /n i (cid:39) /µ η (cid:39) /P → µn e /n i → n e /n i (cid:39) /µ (Mendis and Rosenberg, 1994).Of course, when the grains have a continuous size distribution dn d /da (for a min < a < a max ), P must be calculated by replacing in (11) a by itsmean value (cid:104) a (cid:105) = (cid:82) a max a min da a ( dn d /da ) /n d (Havnes et al., 1990). In generalthe minimum a min is unknown, but dust analyzers can measure the numberdensity n d ( a ) of grains larger than some radius a > a min . In that case, onecan derive a min from the measured value of n e /n i without having to make ahypothesis on the grains’ potential. For example, with dn d /da ∝ a − p (with p >
2) so that the number density of grains larger than a is n d ( a ) ∝ a − p ,we substitute (cid:104) a (cid:105) for a in (11) and get a min = (cid:18) πL D P p − p − n d ( a ) a p − (cid:19) / ( p − (14)which together with (12) yields a min as a function of n e /n i . This will be usedin Sect. 5.
3. Grains’ polarization
The results of Sect. 2 hold for grains of size much larger than the Landauradius; how much larger will be determined later (Eq.(39) and Fig. 3). Forsmaller grains, the polarization charges induced on a grain by approachingcharged particles produce an electric potential which perturbs significantlytheir trajectories, as first shown in the context of ion capture by aerosols8n the Earth’s ionosphere (Natanson, 1960); hence the particle fluxes aremodified.Consider an ion or an electron (charge ± e ) approaching at distance r = ax from the centre of a spherical grain of radius a and electric charge Ze . Theapproaching particle is subjected to an electrostatic field that can be derivedfrom the potentialΦ( r ) = e π(cid:15) a (cid:20) Zx ∓ x ( x − (cid:21) x = r/a (15)obtained by adding to the Coulomb potential of the grain’s charge that ofthe induced image (Jackson, 1999). Consider first an uncharged grain ( Z = 0). Plasma particles of charge ± e are subjected to the image term of the potential (15), so that they areall attracted and their trajectories are bent towards the grain. Considerparticles arriving isotropically from large distances at speed v . The impactparameter p of the trajectory which barely grazes a sphere of radius r is givenby conservation of energy and momentum (in spherical coordinates) as p /a = x + x v / [2( x − x = r/a (16)where x v = [ e / (4 π(cid:15) mv / /a (17)For r/a → ∞ , (16) yields p → r , as expected since the image potential be-comes negligible at large distances and produces straight lines trajectories; atsmall distances, the trajectories are bent by the image force and for r/a → p/a → ∞ because of the increasing bending of the trajectories as theimage term in (15) increases. Between these two extremes, p has a minimum p obtained from (16) by noting that dp/dx = 0 for ( x − = x v /
2, whence p /a = 1 + √ x v . The particle random flux snv/ p /a , which yields the flux N v = sn (cid:104) v + e ( π(cid:15) ma ) − / (cid:105) / (cid:104) v (cid:105) = (8 k B T /πm ) / yields the flux N = N F for Z = 0 (18) F = 1 + ( πr L / a ) / (19)This exceeds the flux N given by (5) by the factor F , which may be verylarge when a (cid:28) r L . 9 .2. Repelling grains Now, consider particles which are repelled at distances r (cid:29) a (electronsif Z <
Z > r given by2 x − | Z | x ( x − x = r /a (20)For Z = 1 and 2, (20) yields x (cid:39) .
62 and 1.42 respectively, whereas in thelimit | Z | → ∞ we have x (cid:39) . | Z | − / . At the maximum of the barrierof potential, the potential energy of the charge ± e is ± e Φ( r ) = | Z | e π(cid:15) a (cid:20) x − | Z | − x ( x − (cid:21) (21)For Z → + ∞ , the expansion of the bracket in (21) to first order in | Z | − / yields [1 − | Z | − / ], whereas for Z = 1 and 2 the bracket (cid:39) / [1 + | Z | − / ].Hence a reasonable approximation of (21) is ± e Φ( r ) (cid:39) π(cid:15) a | Z | e | Z | − / (22)As soon as the approaching particles come closer than r , they are at-tracted. Hence the effective collection radius is increased by a factor y , with y (cid:39) x for x (cid:29) r . In the general case we have y < x be-cause of the repelling electric force farther than r . Since only particles ofkinetic energy exceeding | e Φ( r ) | can reach this distance, the flux of repelledparticles with a Maxwellian distribution of temperature T is given by N = N y e −| e Φ( r ) | /k B T (cid:39) N F r ( Z ) repelled particles (23) F r ( Z ) (cid:39) (cid:2) | Z | + 4 a/r L ) − / (cid:3) e − (cid:18) | Z | rL/a | Z |− / (cid:19) (24)where we have used an approximation of y derived by Draine and Sutin(1987). Comparing with (3), one sees that the polarization increases signif-icantly the flux of repelled particles, by a factor (cid:39) . × e r L / a for | Z | = 1and a/r L (cid:28)
1, which can be quite large for very small grains.10 .3. Attracting grains
Finally, consider particles which are attracted at distances r (cid:29) a (positiveions if Z <
Z > Ze which would yield the flux (4) inthe absence of polarization and acts far from the grain; second, the imagecontribution which would yield the flux (18) when Z = 0 and acts close tothe grain. By comparing (4) and (18), one sees that the attraction of thegrain’s charge generally dominates, so that the flux is given by (4) with acorrecting factor. Using an approximation for this factor (Draine and Sutin,1987), one obtains for a Maxwellian distribution at temperature TN = N F a ( Z ) attracted particles (25) F a ( Z ) = (1 + | Z | r L /a ) (cid:2) | Z | + a/ r L ) − / (cid:3) (26)We conclude that when the grain’s size does not exceed the Landau radiusby a large amount, the polarization increases the fluxes whatever the grain’scharge, thereby decreasing the charging time scales. Furthermore, becausethe flux of repelled particles is increased by a larger factor than the otherones because of the exponential term, the negative equilibrium charge tendsto increase. However, since in that case | Z | is not large, a statistical treatmentof the grain charge distribution is needed.
4. Charge probability distribution
Let f ( Z ) be the probability that a grain carries the charge Ze . Thepopulation of grains of charge Ze is depleted by collecting electrons andions and replenished when grains of charge ( Z + 1) e collect electrons andwhen grains of charge ( Z − e collect ions (as discussed in Sect. 2, weneglect secondary or photoelectron emission). Under stationary conditions,this yields the simple recurrence relation (Draine and Sutin, 1987) f ( Z ) N i ( Z ) = f ( Z + 1) N e ( Z + 1) (27)equivalent to a more complicated relation used by (Rapp and L¨ubken, 2001).Applying (27) iteratively with N e ( Z ) and N i ( Z ) given by (18) for Z = 0 and11espectively by (23) and (25) for Z <
Z >
0, we obtain f ( Z ) /f (0) = ( µ n e /n i ) | Z | F − (cid:89) Z (cid:48) = Z (cid:20) F r ( Z (cid:48) + 1) F a ( Z (cid:48) ) (cid:21) Z < f ( Z ) /f (0) = ( µ n e /n i ) − Z F Z (cid:89) Z (cid:48) =1 (cid:20) F r ( Z (cid:48) − F a ( Z (cid:48) ) (cid:21) Z > F , F r , F a are defined respectively in (19), (24), (26), and we set F r (0) = 1 in (28). This can be solved by using (cid:80) + ∞−∞ f ( Z ) = 1. Eqs.(28)-(29) yield in particular f ( − f (0) (cid:39) ( µ n e /n i ) 1 + ( πr L / a ) / (1 + r L /a ) [1 + (1 + a/ r L ) − / ] (30) f ( − f (+1) (cid:39) ( µ n e /n i ) (31)Two important consequences emerge. First, since µ (cid:29) a (cid:28) r L , (30) yields f ( − /f (0) (cid:39) ( µ n e /n i ) ( πa/ r L ) / ,an approximation which turns out to be accurate to within 5% in the wholerange a/r L ≤ Z = 0and -1 since the number of negative charges is not only limited by the ex-ponential factor in (23), but also by electron field emission (e.g. (Mendisand Axford, 1974; Draine and Sutin, 1987)). Indeed, for nanograins we have e / π(cid:15) a (cid:39) . | E | > V/m (Gomer, 1961). An ejected electron near the surface of a grain of charge
Ze < | E | (cid:39) ( Z +1) e/ π(cid:15) a . Hencethe condition | E | < V/m for electron field emission not to occur limitsthe grain charge state to
Z > − (cid:0) . a nm (cid:1) Field emission limit (32)12 −2 −1 L = 1a/r L = 0.2 f(+1)/f(−1)f(−1)/ Σ f− < Z > − < Z > n e /n i −2 −1 −4 −3 −2 −1 c h a r g e p r ob a b iliti e s Figure 2: Mean number of grain charge units (cid:104) Z (cid:105) (left axis), proportion of negativelycharged grains f ( − / (cid:80) f (right axis) and ratio of positively to negatively charged grains(dash-dotted, right axis) versus the electron-to-ion number density n e /n i for two valuesof a/r L with µ = 180 (water-group ions with s e (cid:39) We do not consider ion field emission which limits the positive charging, sinceit requires a much higher field.For a ≤ Q/e to Z ≥ −
1, as noted by Hill et al. (2012). Hence, theprobability is concentrated on the states Z = 0 and -1, and the mean grains’charge number at equilibrium is (cid:104) Z (cid:105) (cid:39) − f ( −
1) = − / [1 + f (0) /f ( − (cid:104) Z (cid:105) (cid:39) −
11 + (8 r L /πa ) / / ( µ n e /n i ) (33) (cid:39) − (cid:20) − n i /s e n e ( a nm T eV ) / (cid:21) − (34)Therefore, the mean equilibrium charge of grains of radius a (cid:39) (cid:39) s e n e /n i (cid:29) − ,13nd the ratio of positive to negative grains is according to (30) f (+1) /f ( − (cid:39) (5 × − n i /s e n e ) (35)These values are plotted in Fig. 2. They hold at equilibrium. The charg-ing time scales can be estimated from the electron and ion flux on an un-charged grain, respectively, given from (18)-(19), which yield τ − (cid:39) (cid:34) πa n e s e (cid:18) k B T e πm e (cid:19) / (cid:18) (cid:16) πr L a (cid:17) / (cid:19)(cid:35) − (36) (cid:39) (cid:20) × − s e n e cm − a (cid:18) T / + 1 . a / (cid:19)(cid:21) − s (37) τ +1 (cid:39) ( µn e /n i ) × τ − (38) Using (33) (valid for small grains) and (2) with (7) (valid for large grains),we obtain the general approximation (cid:104) Z (cid:105) (cid:39) − η a/r L −
11 + 0 . × − ( n i /n e )( r L /a ) / (39)for water-group incident ions, with η solution of e η (1 + η ) (cid:39) n e /n i and s e = 1; if s e < n e /n i must be multiplied by s e in these expressions. Asimilar approximation was studied by Draine and Sutin (1987) in the specialcase n e = n i . Equation (39) is plotted in Fig 3 for several values of n e /n i and compared to the classical result valid for a (cid:29) r L and n e (cid:39) n i (dotted).Two further consequences emerge. First, the quasi-neutrality condition n e − n i = n d Z no longer yields (12) for grains of radius a (cid:46) r L since in thatcase (cid:104) Z (cid:105) is no longer given by the first term of (39); one can neverthelessapply (12) in the Enceladus plume for larger grains, since they carry most ofthe dust total charge (Dong and Hill, 2012). Second, the grains’ charge-to-mass ratio, which governs their dynamics, varies faster with mass for smallergrains. Indeed, whereas the charge-to-mass ratio of large grains varies inproportion of Z/a ∝ /a (from the dominant first term in (39)), the charge-to-mass ratio of smaller grains varies faster when the second term in (39) isdominant; if n e /n i (cid:39) − this occurs as soon as a (cid:46) r L , whereas if n e /n i (cid:39)
1, the charge-to-mass ratio goes as 1 /a for a < r L .14 −2 −1 −2 −1 − < Z > a/r L n e /n i = 1n e /n i = 0.1n e /n i = 0.01H O + ions Figure 3: Mean equilibrium number of electrons carried by a grain from (39) versus itsnormalized radius, for different values of the electron-to-ion number density and µ = 180(water-group ions with electron sticking coefficient s e (cid:39) −(cid:104) Z (cid:105) =3 . a/r L is plotted (light dotted) for comparison. These values do not take into accountthe electron field emission limit, which prevents multiple charging for grains of radiussmaller than about 1 nm.
5. Enceladus nanograins
Consider nanograins in the Enceladus plume, almost at rest (Tokar et al.,2009) with respect to Enceladus as well as with respect to the plasma, oftypical parameters: n i ∼ − × cm − , n e /n i ∼ − − − , T ∼ O + (Cravens etal., 2009) and with a neutral gas density n ∼ × cm − (Waite et al.,2006). This yields the Landau radius r L (cid:39) . L D (cid:39) . ↔ .
23 m for n i ∼ ↔ cm − respectively. We deduce from (34) (see also Fig 2) that more than 50% of the grainsof radius a (cid:39) e /n i > − /s e ( s e being the electron sticking probability), which confirmsthe value inferred for these grains (Hill et al., 2012).The corresponding charging time scale is given by (37), which yields τ − (cid:39) ↔ n e s e ∼ ↔ cm − respectively. Note that neglectingthe grains’ polarization would yield charging time scales roughly twice largerthan these values. With a speed (cid:39) . (cid:39) . ↔ R E (Enceladus radius R E (cid:39)
250 km). This confirms the order-of-magnitude estimate by Hill et al.(2012) indicating that these nanograins are charged by the ambient plasma.
Consider now the observed positively charged grains (Jones et al., 2009;Hill et al., 2012), whose origin is under debate. Hill et al. (2012) suggestedthree possible mechanisms: secondary electron emission, impacts of positiveions, and triboelectric charging (the latter first suggested by Jones et al.(2009)). The results of Sect. 4 enable us to estimate the grains’ chargingstates resulting from the flux of ambient electrons and ions (taking into ac-count the grains’ polarization and charge discretization). According to (35),we have f (+1) /f ( − ∼ (5 × − n i /s e n e ) ∼ × − ↔ × − for n e /n i ∼ − ↔ − respectively (for s e ∼ a (cid:39) f (+1) /f ( −
1) yields the ratio of positively to negatively charged grains. How-ever this is an equilibrium value, which holds when the time involved exceedsthe larger charging time scale τ +1 ; otherwise, the ratio f (+1) /f ( −
1) shouldbe smaller since τ +1 > τ − . With n i (cid:39) × cm − , (38) yields τ +1 (cid:39) (cid:39) . (cid:39) R E . This value is of the order of the involved paths, which suggests thata significant proportion of these nanograins have their equilibrium charge.These estimates are a strong indication that the impacts of ambient plasmaparticles can explain both the negatively and positively charged grains, with-out having to rely on other charging processes. The charge of these nanograins cannot compensate for the strong ob-served electron depletion, and larger grains act to achieve plasma quasi-neutrality (Yaroshenko et al., 2009; Farrell et al., 2010). When those “largegrains”represent the major contribution to the total dust charge, Fig. 116Eq.(12)) enables one to deduce directly their properties from the observedelectron depletion or vice-versa, via P . For example, let us assume dn d /da ∝ a − p for a > a min with p = 4 . n d ( a ) (cid:39) . − for grains of radius a > µ m, according to typical measurements (Spahnet al., 2006; Kempf et al., 2008). Eqs.(12) and (14) show that with an ob-served electron depletion n e /n i (cid:39) − , quasi-neutrality can be achieved with a min (cid:39) . µ m. Note that this estimate does not require the grain potentialas an input since it is calculated in parallel, contrary to estimates using themeasured spacecraft potential (e.g. Shafiq et al. (2011)).Finally, according to Eq.(13), the electrons cannot be more depleted than n e /n i (cid:39) /µ (cid:39) × − /s e ≥ × − (since s e ≤ Finally, consider grains of radius smaller than about 1 nm. Accordingto (34), their charge should decrease below one charge unit when a nm < (10 − n i /s e n e ) /T eV , whereas their charging time scale increases. This mightpossibly contribute to the decrease observed in the flux of charged grainsbelow 1 nm (Hill et al., 2012; Jones et al., 2012).Several other effects are expected to act at such small sizes. First, thehigh electric field E = Ze/ π(cid:15) a at the grain’s surface can make it explodeif the electrostatic stress (cid:15) E exceeds the maximum grain’s tensile strengthagainst fracture S (e.g. (Hill and Mendis, 1979; Draine and Salpeter, 1979)).The condition (cid:15) E < S yields the limiting grain radius a nm > . × | Z | / ( S/ Nm − ) − / (40)The maximum tensile strength S of sub-nanometric ice grains is highly un-certain. Since the tensile strength of macroscopic materials is determined ina large part by cracks and dislocations, S is expected to exceed by a largeamount the (highly temperature dependent) value of macroscopic ice S ∼ Nm − (Croft et al., 1979), for sub-nanometric grains having a compact struc-ture.Given these uncertainties and the weak dependence on S of the size limit(40), we make below a tentative order-of-magnitude estimate. With thebonding strength energy (cid:39) . × − eV per hydrogen bond, 4 hydrogenbonds per water molecule, and assuming bulk water ice density i.e. about3 × water molecules/m , we obtain ∼ × Jm − (equivalent to force17er unit area), from which we deduce the tentative strength against fracture S ∼ × Nm − .Substituting this value in (40) with Z = 1 yields the minimum grainradius a min (cid:39) . S as S − / , so that varying S by a factor of 5 would produce a variation in a min of about 50 %. Note also that 0.7 nm is roughly the size of a unit cell of aIh ice crystal and twice the width of an elementary step of ice crystallization(Sazaki et al., 2010).Another size limitation might be produced by the centrifugal stress dueto the grain’s spin induced by impacts of molecules (Spitzer, 1978; Draineand Salpeter, 1979). Contrary to the above electrostatic limit, it also actson uncharged grains. At equilibrium (justified by Eq. (42)), the rms angularspeed ω of a grain due to collisions with neutrals of temperature T satisfies Iω (cid:39) k B T where I (cid:39) (8 / πρa . A spinning grain of mass density ρ will be destroyed if ( π/ ρa ω > S , which yields the survival condition (e.g.(Meyer-Vernet, 1984)) a nm > . × T /
30 eV ( S/ Nm − ) − / (41)Substituting T (cid:39) .
02 eV (Waite et al., 2006) and the order of magnitude S ∼ × Nm − determined above, we obtain a > . T (although this would yielda size limit higher by the factor ( T /T ) / ), because the ion number densityis too small for inducing a significant grain’s spin during the time scalesinvolved. Indeed, for the grains to acquire a spin governed by the thermalenergy of a particle species, they should have been struck by their own massof these particles. Applying (5) to H O molecules of mass m (cid:39) m p yieldsthe time scale τ spin (cid:39) (2 π ) / ρa n ( m k B T ) / ∼ a nm / ( n − T /
20 eV ) s (42)whence τ spin ∼ . × s, which is of the order of magnitude of the timescales involved and smaller by more than two orders of magnitude than thevalue for ion impacts (which confirms that the latter do not affect the grains’spin). 18 . Concluding remarks We have derived analytical expressions for the charge of nanograins incold dense and dusty environments, under conditions relevant in the outersolar system, and applied them to Enceladus nanograins. This analysis showsthat a large proportion of nanograins should be charged with one electron,as assumed in previous studies and argued by Hill et al. (2012), and that theimpacts of ambient ions should explain the observed positively charged grainswithout having to assume other charging processes. Electrostatic stresses areexpected to limit the size of charged grains to a minimum radius of about0.7 nm - a value which should be taken with caution since it assumes acompact structure and varies with the badly known grain tensile strength S in proportion of S − / . This effect might contribute to the strong decreasein the grain number density observed at radii below about 1 nm (Jones etal., 2009; Hill et al., 2012).However, subnanometric ice grains fall into the uncertain transition regionbetween macroscopic and microscopic behavior. In particular, the electronsticking coefficient s e is expected to decrease nearly linearly with radius for a (cid:46) (cid:39) (cid:39) O molecules, more than 50 % of which should lie at the surface. This num-ber is smaller than the recently determined minimum number of 250 − O molecules required for crystallization of a water cluster (Pradzynski,2012), which corresponds to a radius of about 1.3 µ m with the density ofice. This radius of 1.3 µ m (whose dependence on temperature is unknownand which should be taken with caution in the context of Enceladus plume),is close to the observed onset of flux decrease (Jones et al., 2009; Hill et al.,2012). Acknowledgments
I thank Tom Hill and another reviewer for their helpful comments on themanuscript. 19 eferences