1 Introduction: Planetary Formation and the Problem of Accretion

The protoplanetary disk forms from the gravitational collapse and subsequent cooling of the molecular cloud, resulting in the condensation of dust (metals and silicates) and ice (hydrogen bearing molecules) grains. Grains begin to collide and stick to one another temporarily, but most of the gas atoms and molecules remain unbound and unaccreted. Ninety eight percent of the gas, hydrogen and helium, never condenses.

Processes that are involved in planet formation include (a) gravitational collapse of the proto-sun and the protoplanetary nebular disk. (b) Temperature-dependent condensation of gases to form solids; from refractory silicates at higher temperatures nearer the center of the nebula to lightweight volatiles at lower temperatures in the opposite direction. (c) Accretion, the process through which cooled, solidified grains of dust and ice orbiting the protosun in a protoplanetary disk collide, stick together, and gradually grow to meters in size to form planetesimals. Gravitational interaction modifies the orbits of planetesimals above a few 100 km in size. With crossing orbits, planetesimals collide violently, often shattering into smaller pieces. Only the largest of planetesimals survive these high-energy collisions with lower mass planetesimals, and can continue to grow. The process of formation of protoplanets from such planetesimals has been investigated and credibly modeled in recent years by Kokubo and Ida (1995, 1996, 1998, 2000). Bypassing planetesimal formation issues, they describe a two-stage process, starting with the runaway growth of planetesimals already kilometers in size to form protoplanets, followed by oligarchic growth of the larger part of the protoplanet population to form planets.

The net ‘sticking’ of grains to form planetesimals is the most poorly understood process in the protoplanetary nebular accretion model. As grain aggregates grow in size, mean collision velocity also increases, increasing the probability of fragmentation. We attempt to simulate a mechanism for the charging and coalescence of <20 micron sized particles into 1 km planetesimals, using weak electrostatic discharge. The relative presence or lack of such a mechanism may well explain the variations in the abundance and size distribution of planets in solar systems (Youdin and Shu 2002).

2 Context: Accretion Mechanisms

A number of hypotheses have been proposed to explain the enhanced ‘stickiness’ of grains, including (Blum and Wurm 2008) inelastic, adhesive collisions, and magneto-rotational induced turbulence generating sufficiently strong viscosity to push grains along and enhance their potential for collision (Balbus and Hawley 1991; Krauss et al. 2007). Van der Waals attraction would result from surface induced dipoles and electrostatic attraction from a net dipole moment on each grain, due to a non-uniform charge distribution on the grain. Evidence we describe below indicates that the attractive force between pairs of dust grains or between dust grains and surfaces is apparently caused by the dipole–dipole interaction between polarized molecules.

The agglomeration of micron sized grains under simulated early solar nebular conditions has been observed by a number of workers (e.g., Blum et al. 2000; Marshall et al. 2005; Abrahamson and Marshall 2002). Although this phenomenon has been interpreted as the result of Van der Waals attraction resulting from Brownian motion induced collisions for non-magnetic particles (Blum et al. 2000), electrostatic forces have been demonstrated to be, for grains in the microns to tens of microns size range under consideration here, 100–1,000 times more effective than van der Waals forces in the ‘sticking’ of silicate grains (Dominik and Tielens 1997; Marshall and Cuzzi 2001). In addition, Nuth and coworkers (Withey and Nuth 1999; Nuth and Wilkinson 1995; Nuth et al. 1994) reported production of single domain magnetic iron grains during simulations of nebular vapor phase nucleation even in the absence of an ambient magnetic field. They suggested that such grains, acting as stable saturated magnetic dipoles could greatly enhance the coagulation of submicrometer grains in the primitive protoplanetary nebula.

A number of workers have considered the potential role of strong electrostatic discharge in the accretion process (e.g., Tunyi et al. 2004; Guttler et al. 2008; McBreen et al. 2005). A strong discharge would occur as a result of a temporally isolated surge of charged particles, analogous to terrestrial lightning (Desch and Cuzzi 2000). As in terrestrial lightning, an intense current can generate a magnetic field around an instantaneous virtual electric conductor and magnetize ferromagnetic components in the grains, resulting in magnetic attraction between grains and form aggregates and compact clusters with permanent magnetic moments (Blum and Wurm 2008). We are not discussing the more exceptional strong discharges here, but relatively weak discharges induced by solar flares.

When grains are in motion and collisions occur, charges of either sign can be exchanged as a result of differences in size, orientation, or composition, a process called tribocharging (Marshall et al. 2005). The result will be a spotty distribution of charges on grain surfaces, especially those with rough surfaces and high surface area/volume as a result of frequent collisions, resulting in local net negative or positive charging on individual grains and the attraction of oppositely charged grains, to form dipoles (Marshall et al. 2005). Poppe and coworkers (2000) have demonstrated that surface morphology plays a role in increasing sticking probability of irregular-shaped dust particles even at higher velocities.

Over the last decade, Marshall and coworkers have observed rapid aggregation of charged grains into filaments or cohesive masses under simulated protoplanetary nebular conditions in a series of experiments (Marshall 1998a, b; Marshall and Cuzzi 2001; Abrahamson and Marshall 2002; Marshall and Freund 1996; Marshall and Sauke 1999; Marshall et al. 1999; Bagga et al. 2009); thus, they have demonstrated that coulombic forces may play a considerable role in particle aggregation in a variety of dust cloud environments. Regardless of composition, shape, or size, particles formed stable aggregates in the presence of randomly distributed positive and negative charges and weak gravity. The rate of aggregation and the extent to which coulombic forces dominate were demonstrated to be correlated with the density of dust particles. The linear nature of the aggregation implies that charge exchange during contact creates an irregular charge distribution and a dipolar mechanism for particle attraction. Marshall and coworkers have speculated that the finer grain component may transport charge to the larger particles or clumps, causing differential charging and ‘stickiness’ of aggregate surfaces. Coulombic forces apparently dominate over gravitational forces in bringing grains together at these scales, with grain spheres of influence up to ten times their diameters (Marshall and Cuzzi 2001), until clumps are large enough for gravitational attraction to play major role. When only positive or only negative charges were added to the dusty environment, and the environment became distinctively dominated by positive or negative charges, respectively, disintegration of aggregates occurred.

In the work presented here, we begin by investigating the behavior of an already formed grain layer in the presence of excess charge, a weak electron discharge equivalent to a solar flare. We generate sufficient discharge to cause grains to be repelled from the loose aggregate with sufficient velocity to stick on the nearest oppositely charged surface, analogous to powder coating. Our observations strengthen the hypothesis that an environment with weak discharges is critical for grain coalescence, and allow us to explore the range of velocities and binding energies involved in forming and disrupting aggregates of small grains in the micron to tens of micron size range in a simulated proto-planetary nebular environment. Taken together, these experiments provide the potential for an electrostatically based model for planetesimal formation in protoplanetary disks, and potentially for understanding the variations in the numbers and size distributions of planets in the resulting solar systems.

3 Experiment

Our experiments are designed to test the interaction between a weak electron beam and dust particles on a surface in the presence of an electric field in the equivalent of a simulated protoplanetary disk solar nebular environment where dust grains are embedded in a gas that forms the basis of a weak discharge. Thus, we used environmental chamber with an intermediate vacuum (10−4 to 10−5 torr) (Figs. 1, 2). The dust sample used was lunar regolith simulant, JSC-1AVF, a special split of JSC-1AF (Taggart 2002) sieved to <20 microns as confirmed by size particle distribution measurements performed in the lab (Clark et al. 2009). The experimental set up consists of a charged particle (electron) gun operating in the presence of a Penning Discharge initiated by the application of a weak electric field in the electron beam through the use of a small conducting object just above the non-conducting dust-covered surface (Curtis et al. 2006; Clark et al. 2007). The cylindrical containment wall is positively charged. At this stage, repulsion (negatively charged dust on an increasingly negatively charged surface) accelerates dust away from the originally dusty surface to the containment wall. A positively charged coil is then introduced, generating repulsion (relative to positively charged containment wall), removal of the dust from the containment wall and recollection on the original, still negatively charged, surface.

Fig. 1
figure 1

Experimental set up

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Fig. 2
figure 2

Schematic of experiment. From left to right, a initial set up of electron gun with pin probe and negative charging of dust covered surface (stage 1), b disruption and tangential acceleration of dust to positively charged containment cylinder wall (stage 2), and c introduction of positively charged coil causing dust detachment and collection on the original negatively charged surface (stage 3)

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4 Results

We successfully focused the relatively weak electron beam generated using a <1mA current -electron gun set to 1,750 volts at low power (<100 Watts) using a −900 volt VDC grid and a +1,000 volt pin probe 2 mm above the dusty surface (Fig. 3) to control the electrostatic potential of the surface in the presence of a moderate electric field of 500 to 1,000 volts through the onset of a discharge. The introduction of a relatively weak electric field initiated a cascade of electrons in the beam by ionizing surrounding low density gas, converting the relatively high energy, low density electron flux from the gun to a low energy, high density electron flux. The negative charge to mass ratio of the initially neutral dust grains rapidly increased, causing sufficient inter-train electrostatic repulsion to accelerate grains tangentially and rapidly away from the negatively charged surface and to cause grains to stick to surrounding containment cylinder walls charged to +1,000 VDC (Fig. 3). Once sufficient charge build up occurred, in a matter of seconds to tens of seconds, the tangential transfer of all dust to the containment cylinder walls occurred almost instantaneously. Then, the addition of a similarly positively charged coil resulted in dust repulsion and recollection at the foot of the containment wall (Fig. 3).

Fig. 3
figure 3

Demonstration of dust movement in experiment and modeled behavior parameters. Left: dusty surface (stage 1); Dust on containment cylinder wall (left) (stage 2) and dust returned to initial surface (stage 3)

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5 Discussion

To first approximation, the behavior of dust grains in relationship to other grains or surfaces is subject to three combined forces (F TOT) (1). Coulombic force (F C) (2) causes displacement on the non-uniformly charged particle as a dipole in a non-uniform electric field, where q is the charge, E is the electric field, y is the vertical axis, θ is the angle of displacement relative to the vertical axis, and d is the effective length of the dipole. Van der Waals force (F VdW) (3), operating as the inverse square of the distance, causes attraction of the particle to surrounding particles, opposing the coulombic force. A is the Hamaker’s atomic density–dependent coefficient measure of interaction between the particle and particles surrounding it, R is the particle radius, and H is the distance between the particle and its surroundings. Gravitation (F G) (4) is an opposing force, where m P is the particle mass and G the Earth’s gravitational constant.

$$ F_{\text{TOT }} = F_{\text{C}} - F_{\text{VdW}} - F_{\text{G}} $$
(1)
$$ F_{\text{C}} = q\left( {{\text{d}}E} \right)/\left( {{\text{d}}y} \right){\text{d}}({ \cos }\,{{\uptheta}}) $$
(2)
$$ F_{\text{VdW}} = A \, R/\left( { 6 {\text{H}}^{ 2} } \right) $$
(3)
$$ F_{\text{G }} = m_{\text{P}} G $$
(4)

Van der Waals depends on the inverse square of the distance. Gravity is dependent on the mass of a very small particle. Coulombic force depends neither on the mass nor on the physical distance per se but on the large charge differential, causing this force to dominate, and allowing gravity and Van der Waals attractions to the initial surface to be overcome.

Marshall and Cuzzi (2001) modeled the observed interaction of similar grains in a charged environment to account for coulombic forces, determining that minimum binding energies were 100 (for microns) to 1,000 (for hundreds of microns) times greater than those predicted by Dominik and Tielens (1997) based on Van der Waals interactions for grains of comparable size. According to Marshall and Cuzzi (2001), the critical binding energy to dislodge grains in the <20 microns size range should be on the order of 10−4 to 10−5 ergs, or 10−11 to 10−12 joules. Under our experimental conditions (1,000 volts surface potential, 1 mA equivalent to 6.2 × 1015 charges/second), we should be able to generate sufficient charge for disruption to occur in seconds to tens of seconds, as observed. In mixtures, the smaller velocity and critical binding energy of the largest grain apparently applies because smaller grains tend to stick to larger ones (Dominik and Tielens 1997). Disruption of an entire aggregate should occur at ten times the critical binding energy. Once the critical energy is exceeded, removal of grains or disintegration of an aggregate occurred in fractions of a second (Dominik and Tielens 1997).

Disruption of the initial stable dipole-aligned aggregate by net negative charging in the presence of a field resulted in the repulsion, torque, tangential acceleration, and encounter with relatively positively charged containment cylinder. Grains were observed to stick to the containment wall. In order for this to happen, negative charge was redistributed and grains reformed meta-stable dipoles. Smaller charged grains are accelerated and stick to less charged surfaces or grains, forming a coating on surfaces or larger grains (Fig. 4). Charged grains continue to form a thick coating on less charged surfaces or larger grains. Continued collisional compression of grains (Dominik and Tielens 1997) produces a feedstock of smaller grains to be continually charged to coat surfaces or larger grains, initiating a chain reaction.

Fig. 4
figure 4

Model of dust behavior. Smaller charged grains are accelerated and stick to less charged surfaces or grains, forming a coating on surfaces or larger grains. Charged grains continue to form a thick coating or to stick to the larger and larger grains. Continued collisional compression of grains produces a feedstock of smaller grains to be continually charged to coat surfaces or larger grains, initiating a chain reaction

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Successful electrostatic aggregation for repelled grains in this size range implies the achievement of negative discharge induced velocities above 10 cm/s (minimum for sticking) but below 100 cm/s (leading to disruption) (Marshall and Cuzzi 2001; Dominik and Tielens 1997), as well as charge to mass ratios (q E/m) for individual dust particles of between 1016 and 1017 charges (or 1/100 to 1/10 of a coulomb) per cm2. The positive potential of the containment cylinder wall had to be increased to generate repulsion sufficient enough to overcome the critical binding energy to dislodge grains, as indicated above. Free fall in the presence of the gravitational field then resulted in collection of grains at the foot of the containment walls.

6 Conclusions: Implications for Protoplanetary Nebulae

Our work, by demonstrating the ability of differentially charged grains exposed to weak discharge to overcome gravity and other physical ‘sticking’ forces and to be repelled and accelerated sufficiently to form stable coatings on less charged surfaces, supports the contention that coulombic forces play an important role in grain accretion. We demonstrate an electrostatic mechanism for sustainable disruption, acceleration, and growth of coated grains or surfaces. The periodic introduction of net negative charges from solar discharge events (flares) to the net balance of positive charges from solar wind (proton) and negative charges from interplanetary plasma (electrons) would result in acceleration of grains with sufficient velocity to stick and form larger aggregates. Rapid aggregation of surface dust layers would allow growth and formation of bodies that eventually become planetesimals. Too frequent and strong net charge introduction (over-active proto-sun) would result in a solar system with few large planets close to the sun. Conversely, an under-active proto-sun would tend to produce fewer coulombic interactions (relative to random Brownian motion dominated interactions), less energetic collisions, more loosely bound and distributed grains and, in extreme cases, no planets. In order for a solar system like ours to form, proto-sun frequency, timing, and level of output would have to be ‘just right’ for sufficient charge exchange, or tribocharging, to dominate periodically and form a larger number of strongly-bound, more widely distributed stably coated aggregates.