Corresponding author: C. M. Hartzell, Department of Aerospace Engineering Sciences, University of Colorado at Boulder, UCB 431, Boulder, CO 80309, USA. (firstname.lastname@example.org)
 The cohesion between small dust particles plays an important role in determining the electrostatic force required to loft charged dust off a surface. On airless, celestial bodies, the cohesive bond between dust particles can be stronger than the gravitational force. Assuming that the charge on dust particles is given by Gauss' law, a theoretical model considering both cohesive and gravitational forces has predicted that intermediate-sized particles require the smallest electric field strength to loft. We experimentally confirm that, for a given electric field, intermediate-sized particles are lofted, while smaller and larger particles do not move.
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 Electrostatically dominated dust motion has been hypothesized to occur since the Lunar Horizon Glow was observed by the Surveyor spacecraft [Rennilson and Criswell, 1974]. It was thought that the Lunar Horizon Glow was caused by light scattered off of 10 μm diameter dust particles above the surface of the Moon at sunset [Rennilson and Criswell, 1974]. One possible mechanism of separating dust particles from the surface is electrostatic lofting, where, due to the interaction of the Moon's surface with the solar wind plasma and solar UV radiation, dust particles on the lunar surface are charged and feel a strong upward force. Since the gravity on asteroids is much less than on the Moon, it was natural to extend the hypothesis of electrostatic lofting to these bodies [Lee, 1996]. The main supporting evidence for electrostatic dust lofting on asteroids comes from the observation of “dust ponds” on the asteroid Eros by the Near Earth Asteroid Rendezvous mission [Robinson et al.2001]. The spokes in Saturn's rings have also been attributed to electrostatic dust motion [Goertz and Morfill, 1983]. However, none of these phenomena have been definitively linked to electrostatic lofting. Electrostatic dust lofting continues to be a controversial topic [Laursen, 2011], due to our lack of conclusive observational data and the implications for our understanding of the evolution of planetary bodies. Additionally, dust motion poses a hazard to future human and robotic missions to explore these bodies.
 Electrostatic dust lofting remains poorly understood by the dusty plasma physics community. Prior evaluations of the feasibility of electrostatic dust lofting have assumed that it will occur if the electrostatic force on the dust is greater than the gravitational force holding the grain on the surface [Lee, 1996]. However, this model neglects the cohesion between dust grains, which is significant for small grains [Hartzell and Scheeres, 2011]. The theory presented in Hartzell and Scheeres  shows that cohesion drives the electric field required to loft small grains. In this paper, we experimentally demonstrate the preferential lofting of intermediate-sized grains, as predicted in the theory.
2 Theory of Cohesion in Electrostatic Lofting
Hartzell and Scheeres  present a theory for the electric field required for a given dust particle to be electrostatically lofted, which will be briefly reviewed here. The cohesive force (in N) between two equally sized spherical dust particles [Perko et al.2001] is the following:
where C = D / 3.3 × 10 − 18 m2, D is the Hamaker constant, S is a nondimensional index of the powder's cleanliness, and d is the diameter of the dust particles. The Hamaker constant is a property of the material ( ∼ 1 × 10 − 21 J for polystyrene [Lin et al.1995] and 4.3 × 10 − 20 J for lunar soil [Perko et al., 2001]). S ranges from zero to one and is a measure of the thickness of the adsorbed molecules between the dust grains [Perko et al.2001]. Scheeres et al.  discuss the influence of cohesion on asteroids.
 The gravitational force acting on a grain is given by the following:
where ρ is the dust particle density and gs is the gravitational acceleration at the surface of the body.
 The electrostatic force acting on a grain is the product of the grain's charge and the local electric field. We assume that the charge on a grain is given by Gauss' law (Q = EAϵ0) [Wang et al.2007; Sheridan and Goree, 1992; Flanagan and Goree, 2006; Hartzell and Scheeres, 2011; Sheridan and Hayes, 2011], where Q is the charge on the grain, E is the local electric field in the plasma, A is the surface area of the grain, and ϵ0 is the permittivity constant. Note that this expression of the charge on a dust grain does not take into account charge discretization and the time-varying nature of grain charging. Essentially, given some electric field, the required charge density is applied to the grain surface area of interest. Given this expression for the grain's charge, the electrostatic force felt by the grain is
Thus, electrostatic lofting will occur if:
Substituting in the expressions for the forces and solving for the electric field required for lofting gives the following:
By examining equation (5), it can be seen that for small particle sizes, the cohesive term ( ∝ 1 / d) dominates, while the gravity term ( ∝ d) dominates for large particle sizes. Thus, there will be an intermediate particle size where the electric field required for lofting is minimized. This work will experimentally demonstrate the validity of this theory.
3 Experimental Setup
 The experimental setup developed by Wang et al.  was used. Piles of uniformly sized polystyrene microspheres were placed on a biased, conducting plate in an argon plasma. A plasma sheath forms above the biased plate. Since the plate is biased negatively, the insulating dust grains have a positive charge at their equilibrium state to repel the ion current. At the edge of the pile, the electric field points both upwards and outwards, away from the center of the pile. When the electrostatic force is large enough to overcome gravity and cohesion, dust grains will be lofted and redeposited away from the center of the pile. Levitating dust was not observed. Wang et al.  has shown that dust spreads through ballistic trajectories with both vertical and horizontal motion (as opposed to rolling) in this experimental setup. As the dust spreads, the electric field at the edge of the central pile weakens. The pile will stop spreading when the electric field at the edge of the pile is too weak to overcome the gravitational and cohesive forces. Particles may be lofted a multiple times, “hopping” away from the original pile. The forces acting on the dust grain at the beginning of each “hop” are the same as those of the initial hop. Grains directly in contact with the plate (a monolayer of grains), where the adhesion force between the grains and the plate would be present, will not be lofted since the positively charged grains would be attracted to the negatively biased plate. Thus, the extent of the pile spreading is an indirect, but valid measure of the conditions required to launch grains.
 In our experiment, the vacuum chamber was pumped down to a pressure of 1 × 10 − 6 torr for approximately 24 h, in order to allow outgassing of the dust samples that increases the strength of the cohesion between grains. The argon pressure in the chamber was 1 × 10 − 3 torr, and the plate was biased to − 60 V. The plasma was created by an emissive filament. The bulk plasma potential was 2.6 V, the total electron density was 6.1 × 1013 m − 3, and the effective electron temperature was 1.50 eV. The resulting Debye length of the plasma was approximately 1 mm. Identical cylindrical piles of polystyrene microspheres were created by filling 4 mm diameter holes drilled into a 1.5 mm thick sheet of metal. Extreme care was taken to ensure that no dust particles were scattered away from the central pile prior to the experiment. The polystyrene microspheres used were very uniform in size (Table 1).
Table 1. Size Distribution of the Polystyrene Microspheres Used. Dust Sizes will be Referred to by Their Integer Values: “15 micron dust pile” as Opposed to “14.9 micron dust pile”. The Density of the Dust is 1.05 g/cm 3
 A CCD camera was used to automatically photograph the dust piles at set intervals. An emissive probe measured the plasma potential above the dust and the plate.
 Three dust piles (with 5, 15, and 25 μm particles, respectively) were placed on the conducting plate in the vacuum chamber. Figures 1 and 2 show the dust piles before and after exposure to the plasma for 1 hr. The 15 μm pile (center in Figures 1 and 2) spreads, while the other two piles (5 and 25 μm) do not appear to spread based on visual inspection. Emissive probe measurements (discussed later) show that the initial electric field at the edge of all three piles was the same. Thus, the electrostatic force required to loft the 15 μm particles was less than the electrostatic force supplied, resulting in spreading of the dust. Since the 5 and 25 μm particles did not spread, we conclude that the electric field required to loft the 15 μm particles is less than that required to loft 5 and 25 μm particles.
 This experiment was repeated three times with 5, 10, 20, and 25 μm grains and nine times with 15 μm grains. In order to quantitatively assess the spreading of the dust piles, a script was written to compare the final image of the piles to the initial image taken immediately after the plate was biased. The brightness values of the image at the start of the experiment were subtracted from those at the end of the experiment. Residual reflectances of less than 1% were set to zero. The resulting “negative” showed the pixels that increased in brightness over the course of the experiment, indicating spreading of the piles. The resulting average extent of radial spreading is given in Figure 3.
 Figure 3 shows that there is a significant difference in the spreading of the 15 μm dust as compared to the other dust sizes tested, indicating that a weaker electric field is required to loft the 15 μm particles. The 10 and 20 μm piles show moderate amounts of spreading.
 Figure 4 shows the spreading of the 15 μm dust as a function of position (left, center, or right) on the biased plate. The slight variation in spreading with pile placement (less than one standard deviation and less than 10% deviation from the average spreading in the central pile location) could be due to a slight tilt of the camera or a nonuniformity in the plasma. The spreading of the 15 μm dust shown in Figure 3 is taken from the full data set, with the pile in all three positions on the biased plate. The 20 and 25 μm dust were always placed in the right position. The 5 and 10 μm dust samples were always placed in the left position, which Figure 4 shows to be the most favorable position for spreading. Despite being placed in the position predisposed to spreading, the 5 and 10 μm dust spread less than the 15 μm dust. If the experiment was repeated with the 20 and 25 μm dust in the left pile position, then they may exhibit more spreading than the results shown in Figure 4. However, they are unlikely to exhibit more spreading than the pile of 15 μm dust, given the small extent of position-dependent bias observed. Even if the 20 and 25 μm dust exhibited more spreading than the 15 μm dust when in the left pile position, our hypothesis of the preferential lofting of intermediate-sized grains would still hold since the 5 and 10 μm grains exhibit very limited spreading in the left pile position. Additionally, the trend of the 25 μm grains spreading less than the 20 μm grains will not be changed by placement in the left position. Extensive spreading of the 20 and 25 μm grains when placed in the left pile position would only serve to increase the size of the particle that is easiest to loft. Thus, we conclude that our results are not significantly influenced by pile position bias.
 There is no evidence that the observed spreading is due to the lofting of clumps of dust grains. Clumping is most often observed in small grains. A clump of 5 μm grains having approximately the same mass as a single 15 μm grain (and thus, the same gravitational force) can be formed. The strength of the cohesive force on the clump would be at least as much as the cohesive force on a single 5 μm grain (Fco ∝ d, from equation (1)). Assuming that the clump has the same charge as the single 15 μm grain, then the clump of 5 μm grains would be expected to spread as far as the 15 μm grains. However, we do not observe any significant spreading of the 5 μm dust. Thus, the observed dust spreading is not due to the motion of clumps of dust.
 Figure 5 shows the electric field required to loft the polystyrene microspheres used in this experiment, as predicted by equation (5). The product of the cleanliness and Hamaker constant of the sample (CS2) determines the particle size that requires the minimum electric field to loft. The cleanliness and Hamaker constant are specific to the material and sample preparation used. From the experimental results, we know that the particle size requiring the weakest electric field to loft must be between 10 and 20 μm because these sizes bound the particle size of maximum spreading. Thus, we can constrain the value of CS2 to be between 5.4 × 10 − 7 and 2.2 × 10 − 6 kg/s 2. Figure 5 shows curves for these values of CS2.
 Emissive probe measurements of the horizontal variation in the plasma potential above the dust piles were taken after the dust spreading stopped (Figure 6). Each of the peaks in Figure 6 corresponds to a pile of dust. The full width at half maximum (FWHM) of the peak over the 15 μm pile is ∼ 1.6 cm, which is twice as large as the FWHMs of the peaks over 5 and 25 μm piles, ∼ 0.7 and ∼ 0.8 cm, respectively. This confirms our visual observation that the pile of 15 μm dust spreads more than either the 5 or 25 μm dust. The potentials measured by the probe are influenced by the two-dimensional nature of the emitting wire (which is approximately the same length as the initial diameter of the dust piles). Thus, the probe measures both the potential in the sheath above the dust pile as well as the sheath above the conducting plate. Since the 15 μm pile has spread, the potential measurement at the center of the pile is less influenced by the more negative potential of the plate sheath than the potential measurement at the center of the other two piles. Thus, the maximum potential of the central peak is larger than that of the other two peaks. This explanation is supported by the observation that the maximum potentials of the other two peaks are approximately equal, as expected since they both have the same diameter.
 At the beginning of the experiment, the electric field at the edges of each of the piles is the same since all the piles have the same form and experience the same plasma environment. Additionally, Figure 6 shows that the potential profiles over the 5 and 25 μm piles (which did not spread) are virtually identical, indicating that pile shape and not dust size dictates the potential profile above the pile. The electric field at the edge of the dust pile weakens as the dust pile spreads, and the spreading stops when the electrostatic force is too weak to overcome gravity and cohesion. Since the pile of 15 μm dust spreads more than the others, we expect the electric field at the edge of the pile of 15 μm dust to be less than that at the edge of the 5 and 25 μm piles. Calculated by numerically differentiating the smoothed potential data in Figure 6, Figure 7 shows the horizontal electric field at a height of 3 mm above the conducting plate. Figure 7 shows that the electric fields at the edges of the pile of 5 μm and pile of 25 μm dust are similar and are greater than the electric field at the edges of the pile of 15 μm dust, as expected.
 Using the emissive probe, we attempted to measure the vertical potential profile above the dust piles in order to obtain the electric field required to loft dust particles. A nonmonotonic sheath in this experimental setup has been previously observed by Wang et al. , resulting in an upward electrostatic force on the positively charged particles. In our experiment, a nonmonotonic sheath was not detected above the dust piles. The potential minimum was likely very close to the dust piles due to the thinner sheath for our plasma conditions as compared to those in [Wang et al., 2009] and thus was not possible to observe.
 Although the horizontal electric field is not directly responsible for electrostatic dust lofting, it is involved in breaking the cohesive bonds between grains. Thus, we estimate that the vertical electric fields are the same order of magnitude as the horizontal electric fields ( ∼ 20 V/cm). This approximation is supported by measurements in Wang et al. . The theoretically predicted electric fields required to loft dust (Figure 5) are about an order of magnitude larger than the experimentally observed electric fields (Figure 7). The time-varying charge of the dust particles could be an order of magnitude larger than the Gauss' law approximation, which predicts an average value (see discussion in Hartzell and Scheeres ; Sheridan and Hayes ). Although the level of charging of dust grains remains uncertain, the preferential lofting of intermediate-sized grains confirms that the grain charging is proportional to d2.
5 Discussion and Conclusions
 We have experimentally demonstrated the validity of a theory predicting the preferential lofting of intermediate-sized grains. The theory has been applied to the gravity environments of the Moon and asteroids Eros and Itokawa in Hartzell and Scheeres . The asphericity of in situ dust grains may cause our model of spherical grains to overestimate or underestimate the strength of the cohesive force, depending on the packing of the regolith. However, given the Hamaker constant for lunar regolith [Perko et al.2001] and cleanliness values of the magnitude observed here (), we predict that cohesion will dictate the electric field required for lofting for particles smaller than 1 mm on Itokawa (100 μm on Eros and 10 μm on the Moon).
 We have presented a theory that shows that strong intergrain cohesion will dictate the electrostatic force required to loft small dust particles [Hartzell and Scheeres, 2011]. Considering cohesion and gravity, we see that there exists a grain size that requires the smallest electric field to be lofted. This theory has been confirmed experimentally by placing piles of different sized grains on a biased plate in a plasma. We have observed the significant spreading of 15 μm particles, moderate spreading of 10 and 20 μm particles and negligible movement of 5 and 25 μm particles. We confirmed that the electric field required to loft 15 μm grains is less than that required by smaller and larger particles by measuring the horizontal electric field at the edge of these piles. Thus, we demonstrated the preferential lofting of intermediate-sized grains in a plasma sheath, as predicted by theory.
 C. M. H. acknowledges support from the NASA Earth and Space Science Fellowship, Grant NNX09AR51H. D. J. S. acknowledges support from the NASA Discovery Data Analysis Program and Planetary Geology and Geophysics program. This work was partially supported by the NASA Lunar Science Institute's Colorado Center for Lunar Dust and Atmospheric Studies (CCLDAS).