Electrostatic Barrier Against Dust Growth in Protoplanetary Disks. I. Classifying the Evolution of Size Distribution
Abstract
Collisional growth of submicronsized dust grains into macroscopic aggregates is the first step of planet formation in protoplanetary disks. These grains are expected to carry nonzero negative charges in the weakly ionized disks, but its effect on their collisional growth has not been fully understood so far. In this paper, we investigate how the charging affects the evolution of the dust size distribution properly taking into account the charging mechanism in a weakly ionized gas as well as porosity evolution through lowenergy collisions. To clarify the role of the size distribution, we divide our analysis into two steps. First, we analyze the collisional growth of charged aggregates assuming a monodisperse (i.e., narrow) size distribution. We show that the monodisperse growth stalls due to the electrostatic repulsion when a certain condition is met, as is already expected in the previous work. Second, we numerically simulate dust coagulation using Smoluchowski’s method to see how the outcome changes when the size distribution is allowed to freely evolve. We find that, under certain conditions, the dust undergoes bimodal growth where only a limited number of aggregates continue to grow carrying the major part of the dust mass in the system. This occurs because remaining small aggregates efficiently sweep up free electrons to prevent the larger aggregates from being strongly charged. We obtain a set of simple criteria that allows us to predict how the size distribution evolves for a given condition. In Paper II, we apply these criteria to dust growth in protoplanetary disks.
Subject headings:
dust, extinction — planetary systems: formation — planetary systems: protoplanetary disks1. Introduction
The standard coreaccretion scenario for planet formation (Mizuno, 1980; Pollack et al., 1996) is based on the socalled planetesimal hypothesis. This hypothesis assumes that solid bodies of size larger than kilometers (called “planetesimals”) form in a protoplanetary disk prior to planet formation. However, the typical size of solid particles in interstellar space is as small as a micron or even smaller (Mathis et al., 1977). It is still open how the submicronsized grains evolved into kilometersized planetesimals.
The simplest picture for dust evolution towards planetesimals can be summarized into the following steps. (1) Initially, submicronsized particles coagulate into larger but highly porous, fractal aggregates through lowvelocity collisions driven by Brownian motion and differential settling towards the midplane of the disk (Wurm & Blum, 1998; Blum et al., 1998; Kempf et al., 1999). (2) As the aggregates grow to “macroscopic” (mm to cm) sizes, the collisional energy becomes high enough to cause the compaction of the aggregates (Blum, 2004; Suyama et al., 2008; Paszun & Dominik, 2009). (3)The compaction cause the increase in the stopping times of the aggregates, allowing them to concentrate in the midplane of the disk (Safronov, 1969; Goldreich & Ward, 1973), the center of vortices (Barge & Sommeria, 1995), or turbulent eddies (Johansen et al., 2007). (4) Planetesimals may form within such dense regions through gravitational instability (Safronov, 1969; Goldreich & Ward, 1973) or through further collisional growth (Weidenschilling & Cuzzi, 1993; Weidenschilling, 1995).
However, there is great uncertainty on how large dust aggregates can grow through mutual collisions (see, e.g., Blum & Wurm, 2008; Güttler et al., 2010). As the collisional compaction proceeds, the aggregates decouple from the ambient gas and obtain higher and higher relative velocities driven by radial drift (Weidenschilling, 1977) and gas turbulence (Völk et al., 1980). The collision velocity can exceed even without turbulence, but it is uncertain whether such highspeed collisions lead to the sticking or fragmentation of the aggregates (Blum & Wurm, 2008; Wada et al., 2009; Teiser & Wurm, 2009; Güttler et al., 2010). In addition, collisional compaction itself can cause the reduction of sticking efficiency (Blum & Wurm, 2008; Güttler et al., 2010). This may terminates the collisional growth before the fragmentation occurs (Zsom et al., 2010).
By contrast, it is generally believed that dust coagulation proceeds rapidly until the aggregates grow beyond the initial, fractal growth stage since the collision velocity is too low to cause the reduction of sticking efficiency (Dominik & Tielens, 1997; Blum & Wurm, 2008; Güttler et al., 2010). However, one of the authors has recently pointed out that electric charging of aggregates could halt dust growth before the aggregates leave this stage (Okuzumi, 2009, hereafter O09). Protoplanetary disks are expected to be weakly ionized by a various kinds of highenergy sources, such as cosmic rays (Umebayashi & Nakano, 1981) and Xrays from the central star (Glassgold et al., 1997). In such an ionized environment, dust particles charge up by capturing ions and electrons, as is well known in plasma physics (Shukla & Mamun, 2002). In equilibrium, dust particles acquire nonzero negative net charges because electrons have higher thermal velocities than ions. This “asymmetric” charging causes a repulsive force between colliding aggregates, but this effect has been ignored in previous studies on protoplanetary dust growth. O09 has found that the dust charge in a weakly ionized disk can be considerably smaller than in a fully ionized plasma but can nevertheless inhibit dust coagulation in a wide region of the disk. It is also found that the electrostatic barrier becomes significant when the dust grows into fractal aggregates, i.e., much earlier than the growth barriers mentioned above emerge. Thus, the dust charging can greatly modify the current picture of dust evolution towards planetesimals.
The analysis of the electrostatic barrier by O09 is based on the assumption that dust aggregates obey a narrow size distribution. In reality, however, size distribution is determined as a result of the coagulation process, and it has been unclear how the distribution evolves when the dust charging is present. The purpose of this study is to clarify how the size distribution of dust aggregates evolves when the aggregates are charged in a weakly ionized gas.
According to O09, the effect of dust charging can become already significant before the collisional compaction of aggregates becomes effective. In this stage, dust aggregates are expected to have lower and lower internal density (i.e., higher and higher porosity) as they grow, as is suggested by laboratory experiments and –body simulations (Wurm & Blum, 1998; Blum et al., 1998; Kempf et al., 1999). This porosity evolution has been ignored in most theoretical studies on dust coagulation (e.g., Nakagawa et al., 1981; Tanaka et al., 2005; Brauer et al., 2008), in which aggregates are simplified as compact spheres. However, when analyzing the electrostatic barrier, the porosity evolution must be accurately taken into account; in fact, as we will see later, the ignorance of the porosity evolution leads to considerable underestimation of the electrostatic barrier, because compact spheres are generally less coupled to the ambient gas and hence have higher collision energies than porous aggregates. In this study, we use the fractal dust model recently proposed by Okuzumi et al. (2009, hereafter OTS09). Classically, fractal dust growth has been only modeled with either of its two extreme limits, namely, ballistic clustercluster and particlecluster aggregation (BCCA and BPCA; e.g., Ossenkopf, 1993; Dullemond & Dominik, 2005). To fill the gap between the two limits, OTS09 introduced a new aggregation model (called the quasiBCCA model) in which aggregates grow through unequalsized collisions. OTS09 found from –body simulations that the resultant aggregates tend to have a fractal dimension close to even if the size ratio deviates from unity. This explains why fractal aggregates with are universally observed in various lowvelocity coagulation processes (Wurm & Blum, 1998; Blum et al., 1998; Kempf et al., 1999). OTS09 summarized the results of their –body simulations into a simple analytic formula giving the increase in the porosity (volume) for general hitandstick collisions. This formula together with the Smoluchowski equation extended for porous dust coagulation (OTS09) enables us to follow the evolution of size distribution and porosity consistently with dust charging.
As we will see later, our problem involves many model parameters, such as the initial grain size and the gas ionization rate. To fully understand the dependence of the results on these parameters, we do not assume any protoplanetary disk model but seek to find general criteria determining the outcome of dust evolution. This approach allows us to investigate the effect of the electrostatic barrier with any protoplanetary disk models. Application of the growth criteria to particular disk models will be done in Paper II (Okuzumi et al., 2011).
This paper is structured as follows. In Section 2, we describe the dust growth model used in this study. In Section 3, we examine the case of monodisperse growth in which all the aggregates grow into equalsized ones. The monodisperse model allows us to introduce several important quantities governing the outcome of the growth. We analytically derive a criterion in which the “freezeout” of monodisperse growth occurs. In Section 4, we present numerical simulations including the evolution of the size distribution to show how the outcome of the growth differs from the prediction of the monodisperse theory. We discuss the validity of our dust growth model in Section 5. A summary of this paper is presented in Section 6.
2. Dust Growth Model
In this section, we describe the dust growth model considered in this study.
We consider collisional growth of dust starting from an ensemble of equalsized spherical grains (“monomers”). Each aggregate is characterized by its mass, radius, projected area, and charge. For simplicity, we assume “local” growth, i.e., we neglect global transport of dust within a disk.
We focus on the first stage of dust evolution in protoplanetary disks and assume that aggregates grow through “hitandstick” collisions, i.e., collisions with perfect sticking efficiency and no compaction. It is known theoretically (e.g. Kempf et al., 1999) and experimentally (e.g. Wurm & Blum, 1998) that hitandstick collisions lead to highly porous aggregates. To take into account the porosity evolution, we adopt the fractal dust model proposed by OTS09. This model characterizes each aggregate with its mass and “characteristic radius” (see OTS09 and Section 2.3 for the definition of the characteristic radius), and treat the two quantities as independent parameters. Another important parameter is the projected area This determines how the aggregates are frictionally coupled to the gas. In the OTS09 model, is not treated as an independent parameter but is given as a function of and . Note that is not generally equal to a naive “cross section” , especially when the aggregates is highly porous (Figure 1; see also Figure 4 of OTS09). Distinction between and allows us to avoid overestimation of the gas drag force to dust aggregates. In Section 2.3, We will describe the porosity model in more detail.
The collision probability between two aggregates 1 and 2 is proportional to their relative speed times the collisional cross section given by (e.g., Landau & Lifshitz, 1976)
(1) 
where is the kinetic energy associated with the relative motion, is the reduced mass, and is the energy needed for the aggregates to collide with each other. In this paper, is called “the electrostatic energy” for colliding aggregates. Below, we describe how to determine and .
2.1. Charging
We adopt the dust charging model developed by O09. In this model, dust aggregates are surrounded by a weakly ionized gas and charge up by capturing free electrons and ions. These ionized particles are created by the nonthermal ionization of the neutral gas and are removed from the gas phase through the adsorption to the dust as well as the gasphase recombination. The dust charge and the number densities of ions and electrons are thus determined by the balance among the ionization, recombination, and dust charging. In equilibrium, the average charge of aggregates with radius is given by (see Equation (23) of O09)
(2) 
where is the Boltzmann constant, is the gas temperature, is the elementary charge, and is a dimensionless parameter characterizing the charge state of the gasdust mixture. O09 has analytically shown that the equilibrium conditions are reduced to a single equation for . When the adsorption to the dust dominates the removal of the ionized gas, the equation for is written as (see Equation (34) of O09)
(3) 
where is the mass of ions (electrons), is their sticking probability onto a dust monomer, and
(4) 
is a dimensionless quantity depending on
the total projected area and
total radius of aggregates,
and the ionization rate and number density of neutral gas particles.
Equation (3) originates from the quasineutrality condition,
, where and are the
number density of ions and electrons, and
is the total charge
carried by dust in a unit volume.
Physically, is related to the surface potential of aggregates. For an aggregate with charge and radius , the surface potential is given by . Equation (2) implies that , namely, is the surface potential averaged over aggregates of radius and normalized by . Note that is apparently independent of , but is actually not because depends on the size distribution of aggregates through and . It should be also noted that the radius can be interpreted as the electric capacitance (i.e., ). This is the reason why we have denoted the total radius as .
As shown in O09, asymptotically behaves as (see Section 2.3 of O09)
(5) 
where is the solution to
(6) 
Equation (6) is known as the equation for the equilibrium charge of a dust particle embedded in a fully ionized plasma (Spitzer, 1941; Shukla & Mamun, 2002). Equation (5) suggests that the charge state of dust particles in a weakly ionized gas is characterized by two limiting cases. If , the total negative charge carried by dust aggregates is negligibly small compared to , and the quasineutrality condition approximately hold in the gas phase, i.e., . If , by contrast, most of the negative charge in the system is carried by aggregates, and the quasineutrality condition approximately holds between ions and negatively charged dust. For this reason, O09 referred to the former phase as the ionelectron plasma (IEP), and to the latter as the iondust plasma (IDP). Figure 2 schematically shows the difference between the two plasma states.
For given and , Equation (3) determines as a function of . In typical protoplanetary disks, the dominant ion species are molecular ions (e.g., ) or metal ions (e.g., ) depending on the abundance of metal atoms in the gas phase (Sano et al., 2000; Ilgner & Nelson, 2006). Although is likely to be close to unity (Umebayashi & Nakano, 1980; Draine & Sutin, 1987), at low temperatures is poorly understood. Umebayashi (1983) estimated using a semiclassical phonon theory to obtain for . However, the uncertainty in does not strongly affect the evaluation of . For example, assuming (the mass of Mg) and , is 3.78 for , and is even for .
Figure 3 illustrates the dependence of on for fixed and with various (=1.0, 0.3, 0.1, 0.03). We find that can be well approximated by
(7) 
In Figure 3, we compare Equation (7)
with the numerical solutions to the original equation.
The approximate formula recovers all the numerical solutions within an error of .
This means that is well approximated as a function of
for this parameter range.
Up to here, we have considered only the mean value of the charge . In fact, there always exists a finite value of the charge dispersion , and moreover, the mean value is not necessarily larger than (O09). Nevertheless, we will assume below that the dust charge is always equal to . The validity of this assumption will be discussed in Section 5.2.
2.2. Dust Dynamics
As found from Equation (1), the relative velocity between aggregates determines whether they can overcome the electrostatic barrier to collide. In this study, we model the motion of dust aggregates in the following way. We assume that the motion of each aggregate relative to the ambient gas consists of random Brownian motion and systematic drift due to spatially uniform acceleration (e.g., uniform gravity). Then, the probability density function for the relative velocity between two aggregates 1 and 2 is given by
(8) 
where is the difference of the drift velocities between the two aggregates. Here, we have assumed that the systematic motion has no fluctuating component, that is, the velocity dispersion is thermal even when . We will discuss the effect of adopting a different velocity distribution in Section 5.3.
We further assume that aggregates are frictionally coupled to the ambient gas, and give as
(9) 
where is the stopping time of each aggregate and is the uniform acceleration. In this study, we focus on small aggregates and give according to Epstein’s law,
(10) 
where is the gas density and is the mass of the gas particles. Epstein’s law is valid when the size of the aggregate is smaller the mean free path of gas particles.
In a protoplanetary disk, relative motion like Equation (9) is driven by several processes. For example, the gravity of the central star causes acceleration towards the midplane of the disk, where is the Kepler rotational frequency and is the distance from the midplane. Another example is the acceleration driven by gas turbulence in the strong coupling limit. When both of two colliding aggregates are frictionally well coupled to the turbulent eddies of all scales, the relative velocity between the aggregates is approximately given by , where and are the characteristic velocity and turnover time for the smallest eddies, respectively (Weidenschilling, 1984; Ormel & Cuzzi, 2007). This means that turbulence behaves as an effective acceleration field of for strongly coupled aggregates.
As the collisional cross section depends on the stochastic variable , it is useful to treat collision events statistically. To do so, we introduce the collisional rate coefficient
(11) 
With Equations (1) and (8), the integration can be analytically performed. Using , we have (Shull, 1978)
(12)  
where is the error function, and and are defined as
(13) 
with
(14) 
(15) 
Note that and are the relative kinetic energy associated with differential drift and the electrostatic energy normalized by , respectively.
Equation (12) has the following simple asymptotic forms:
(16) 
where is the mean thermal speed between the colliding aggregates. The exponential factor originates from the highenergy tail of the Maxwell distribution. This factor guarantees nonvanishing even for large .
2.3. Porosity Model
As shown by O09, the charging affects dust growth before the collisional compaction becomes effective. In this early stage, aggregates have a highly porous structure (Wurm & Blum, 1998; Kempf et al., 1999). The porosity influences their collisional growth through the collisional and aerodynamical cross sections. It also affects dust charging through the capacity (=radius) and the capture cross section for ions and electrons. Therefore, it is important to adopt a realistic model for the porosity of aggregates.
In this study, we adopt the porosity model developed by OTS09. This model is based on –body simulations of successive collisions between aggregates of various sizes. This model provides a natural extension of the classical hitandstick aggregation models (see OTS09 and references therein). Collisional fragmentation and restructuring is not taken into account, so the porosity increase only depends on the physical sizes of colliding aggregates. This assumption is valid as long as the collisional energy is sufficiently lower than the critical energy for the onset of collisional compaction. The validity of this assumption will be discussed in Section 5.4.
Porosity Increase After Collision
Our porosity model measures the size of a porous aggregate with the characteristic radius , where is the number of constituent monomers within the aggregate, is the coordinate of the th constituent monomer, and is the center of mass. Figure 1 shows the characteristic radius as well as the projected area of a numerically created porous aggregate. In our model, the porosity of each aggregate is characterized by and , while the projected area is assumed to be a function of them. In the following subsections, we summarize how and are calculated in this model.
The porosity evolution of aggregates after a collision is expressed in terms of the increase in the porous volume . For a collision between aggregates with volumes and , the volume of the resulting aggregate, , can be generally written as
(17) 
where is a dimensionless factor depending on and . We refer to as the “void factor” since it identically vanishes for compact aggregation.
It is known that there are two limiting cases for hitandstick collisions (see, e.g., Mukai et al., 1992; Kozasa et al., 1993). One is called the ballistic cluster–cluster aggregation (BCCA) where aggregates grow only through equalsized collisions. On average, the characteristic radius of BCCA clusters is related to the monomer number as
(18) 
where is the radius of monomers and is the fractal dimension of BCCA clusters (e.g., Mukai et al., 1992). The void factor for the BCCA growth can be calculated from Equation (18) as (OTS09). The opposite limit is called the ballistic particle–cluster aggregation (BPCA), in which an aggregate grows by colliding with monomers. On average, the characteristic radius of BPCA clusters is given by , where is the porosity of BPCA clusters (e.g., Kozasa et al., 1993). The void factor is found to be (OTS09). Note that both and are constant.
To obtain for more general cases, OTS09 presented a new aggregation model called the “quasiBCCA” (QBCCA). In the QBCCA, an aggregate grows through unequalsized collisions with a fixed mass ratio , where and are the monomer numbers of the target and projectile, respectively. The projectile is chosen among the outcomes of earlier collisions, so that the resultant aggregate has a selfsimilar structure. OTS09 performed body simulations of aggregate collisions with various size ratios and found that the void factor for QBCCA is approximately given by
(19) 
Note that approaches to in the BCCA limit () as must be by the definition of BCCA.
Unfortunately, Equation (19) does not reproduce the void factor in the BPCA limit (). To bridge the gap between the BCCA and BPCA limit, OTS09 considered a formula
(20) 
It is easy to check that Eqaution (20) approaches to and in the BCCA and BPCA limits, respectively. Equation (20) will be used in the numerical simulations presented in Section 4 to determine the porosity (volume) of aggregates after collisions.
Projected Area
The projected area is another key property of porous aggregates. This does not affect only the charge state of the gasdust mixture (Equation (4)) but also the drift velocity of individual aggregates (Equation (10)).
For BCCA clusters, the projected area averaged for fixed is well approximated by (Minato et al., 2006)
(21) 
For BPCA clusters, the averaged projected area is simply related to the radius as . For more general porous aggregates, including QBCCA clusters, the averaged projected area is well approximated by (Equation (47) of (OTS09))
(22) 
where and are is the characteristic radius and monomer number of the aggregate considered, and and are the characteristic radius and projected area of BCCA clusters with the same monomer number (i.e., Equations (18) and (21)), respectively. Note that the above formula reduces to Equation (21) in the BCCA limit () and to in the BPCA limit (, ).
It should be noted that the above formulae can be only used for the average value of . This does not bother us when we compute the charge state of aggregates, since it only depends on the total projected area . However, we cannot ignore the dispersion of when we calculate the differential drift velocity between aggregates, especially between BCCAlike clusters. For example, let us consider two BCCA clusters with different masses and . As Equation (21) suggests, the mean masstoarea ratio of BCCA clusters approaches to a constant value in the limit of large . Hence, if we ignored the area dispersion, we would have a differential drift velocity vanishing for very large and even if . Clearly, this would lead to underestimation of and overestimation of the electrostatic repulsion.
To avoid this problem, we should replace in with , not with , where the overlines denote the statistical average. In particular, if the standard deviation of scales linearly with its mean, we can write as (see Appendix)
(23) 
where is the ratio of the standard deviation to the mean of . In the Appendix, we evaluate from the numerical data on the projected area of sample BCCA clusters. We find that can be well approximated as for . In the following sections, we will assume for all aggregates, since the area dispersion is only important for collision between BCCAlike clusters.
2.4. Nondimensionalization
As seen above, our dust model is characterized by a number of model parameters. To find a truly independent set of model parameters, we scale all the physical quantities involved into dimensionless ones.
We introduce the dimensionless radius and mean projected area,
(24) 
(25) 
Also, we scale the mass with the the monomer number , where is the mass of monomers. The normalized drift energy and electrostatic energy are already given by Equations (14) and (15), respectively. Using (, , ) instead of (, , ), we have
(26) 
(27) 
where the dimensionless coefficients and are defined as
(28)  
(29) 
with the monomer material density .
We also introduce the normalized distribution function
(30) 
where is the number density of monomers in the initial state. Note that the mass conservation ensures . Using , we rewrite the ionization parameter as
(31) 
where and are the normalized total projected area and capacitance, and is a dimensionless ionization rate defined by
(32)  
The surface potential is determined as a function of by Equation (3), or
(33) 
where we have eliminated using Equation (6).
From the above scaling, we find the collisional growth of charged dust aggregates can be characterized by five dimensionless parameters (, , , , ).
3. Monodisperse Growth Model
Before proceeding to the full simulations, we consider simplified situations where dust grows into monodisperse aggregates, i.e., where all the aggregates have the same monomer number at each moment. This greatly helps us to understand the results of the numerical simulations shown in the following section.
Within the framework of the hitandstick aggregation model, the monodisperse growth is equivalent to the BCCA growth. Thus, the assumption of the monodisperse growth is expressed by the following relations:
(34) 
(35) 
(36) 
where is the fractal dimension of BCCA clusters and is the delta function. Since is close to 2 (see Section 2.3.1), we simply set in the following calculation. Note that the factor appearing in Equation (36) accounts for the mass conservation .
Under the monodisperse approximation, the drift and electrostatic energies ( and ) can be given as a function of . Substituting Equations (34) and (35) into Equation (26), the drift energy can be written as
(37) 
Thus, under the monodisperse approximation, and degenerate into a single parameter . Similarly, the electrostatic energy is written as , where is given by Equation (33) with and . The expression for can be further simplified using the approximate formula for (Equation (7)) to eliminate . The result is
(38) 
Note that this expression no longer involves . From Equations (37) and (38), we find that the outcome of the monodisperse growth is (approximately) determined by three parameters , , and .
For later convenience, we define the “effective kinetic energy” as
(39) 
or equivalently, . The first term in the right hand side of Equation (39) accounts for the contribution of Brownian motion to the collisional energy (). We expect that the monodisperse growth is strongly suppressed when exceeds .
Here, we give some examples to show how and depends on the parameters. Figure 4 shows as a function of for . As found from this figure, the kinetic energy is constant at due to Brownian motion (), and increases with mass at due to the differential drift () . The qualitative behavior is the same for every . The value of only determines the mass at which the differential drift starts to dominate over Brownian motion in the kinetic energy. In figure 4, we also plot for with varying the value of , , ). For all the cases, quickly increases with and finally becomes proportional to . This reflects the transition of the plasma state from the IDP () to the IEP (). In the IEP limit, depends on but is independent of . An important difference among the three examples is the timing of the plasma transition: for smaller , approaches the IEP limit at larger . This difference makes the ratio between and qualitatively different among the three cases. For , exceeds when the relative motion is dominated by Brownian motion. For , by contrast, exceeds when the relative motion is dominated by the differential drift. For , does not exceed for arbitrary . As we see in Section 4, this difference is a key to understand the collisional growth of dust aggregates with size distribution.
To quantify these differences for general cases, we introduce the following quantities:

The drift mass . This is defined as the mass at which the relative motion starts to be dominated by the differential drift.

The plasma transition mass . This is defined as the mass at which the plasma state shifts from the IDP to the IEP.

The maximum energy ratio . This is the maximum value of the ratio in the monodisperse growth. If , the electrostatic energy exceeds the kinetic energy at a certain mass.

The freezeout mass . This is the mass at which starts to exceed . Note that the freezeout mass is only defined when .
In the following subsection, we describe how these quantities are related to the parameters (, , ).
3.1. : the Drift Mass
The first and second terms in the right hand side of Equation (39) represents Brownian motion and the differential drift. Since the second term monotonically increases with , there exists a critical mass at which the dominant relative motion changes from the Brownian motion to the differential drift. We define as the critical mass satisfying . Using Equation (37), the equation for is written as
(40) 
This equation implicitly determines as a function of . For example, when (see Figure 4).
Figure 5 shows the solution to Equation (40). When , is well approximated as
(41) 
where is the masstoarea ratio in the limit of . Using Equation (41), is simply rewritten as
(42) 
which asymptotically behaves as for and for . The asymptotic form of is schematically illustrated in Figure 6(a).
3.2. : the Plasma Transition Mass
3.3. : the Maximum Energy Ratio
The maximum energy ratio determines whether the electrostatic energy exceeds the kinetic energy during the monodisperse growth. Since scales linearly with , the quantity depends only on and .
Figure 7 plots as a function of and . It is seen that the maximum energy ratio behaves differently across the line . This can be easily understood from Figure 8, which schematically illustrates the mass dependence of and (Equations (42) and (45)). If (), the energy ratio reaches the maximum at . Since and , we obtain independently of . If (), by contrast, reaches the maximum at . Using and , we have , which depends on both and .
3.4. : the Freezeout Mass
When , there exists a critical mass at which the electrostatic energy takes over the kinetic energy . As we will see in Section 3.5, the monodisperse growth is strongly suppressed at . For this reason, we refer to as the “freezeout mass.”
The freezeout mass can be calculated from the condition