We utilize the dimensionless discrete-sectional model described by McMurry and Li (2017) to simulate evolution of particle size distribution for a system with a single condensing species. We assume that the condensing species is produced at a constant rate by gas-phase reaction. Our code uses 200 discrete bins and 250 sectional bins, with a geometric volume amplification factor of 1.0718 for neighboring sections.

Physical processes that affect particle growth, including wall deposition, loss to preexisting particles, cluster evaporation and dilution, can be characterized by dimensionless parameters in this model. In the present study, however, not all aforementioned processes are discussed. Our previous work shows that wall losses, scavenging by preexisting particles and dilution have qualitatively similar effects on aerosol dynamics. Therefore, in this work we focus on preexisting aerosols and dilution to illustrate factors that contribute to errors in measured GRs and do not explicitly discuss wall deposition. A single dimensionless parameter, L, is used to indicate the abundance of preexisting particles, with larger L values representing a higher concentration of preexisting particles (or, equivalently, a slower rate at which the nucleating species is produced by chemical reaction). L is calculated with the equation L=148kbTπm11/2AFuchsRβ11fm, where AFuchs is the Fuchs surface area concentration (Fuchs and Sutugin, 1971), kb is the Boltzmann constant, m1 is the mass of the monomer, R is the condensing species production rate and β11fm is the monomer collision frequency function. The loss rate for particles containing k monomers is L/k1/2. This size dependence is included when solving the coupled differential equations for time-dependent cluster concentrations. Similarly, the dimensionless quantity M that characterizes dilution is given by the expression M=Qdil/VRβ11fm, where Qdil is the dilution flow rate and V is the volume of the system. Note the fractional dilution loss is independent of particle size. In addition to loss to preexisting particles and dilution, we consider the effect of cluster evaporation on particle growth with the assumption that evaporation follows the classical liquid droplet model. Two dimensionless parameters, E and Ω, are needed to fully describe the evaporation process. The dimensionless evaporation parameter, E, is proportional to the saturation vapor concentration of the nucleating species, while Ω is the dimensionless surface tension (Rao and McMurry, 1989; McMurry and Li, 2017). The evaporation rate for particles containing k monomers, Ek, is calculated with a discretized equation of the form Ek=Ec1kexp⁡32Ωk23-k-123, where c(i,k) is the dimensionless collision frequency between a monomer and a particle containing k monomers. To simplify our discussion, Ω is fixed to be 16 throughout this work (a representative value for the surface tension of sulfuric acid aqueous solutions), while the value of E is varied.

The solution to the GDE for a constant rate system (R = constant) depends on dimensionless time, cluster size and the dimensionless variables L, M, E, Ω, etc., but is independent of the rate at which condensing vapor is produced by chemical reaction. That rate is required to transform the computed nondimensional solutions to dimensional results using simple multiplicative expressions given by McMurry and Li (2017): Nk=Rβ11fm1/2Ñk;t=1Rβ11fm1/2τ;dp=v11/3d̃p. In the above equations, Ñk is the dimensionless concentration of particle-containing k monomers, τ is the dimensionless time, d̃p is the dimensionless particle size and v1 is the monomer volume. Assuming a monomer volume of 1.62×10-22 cm3 (volume of one sulfuric acid plus one dimethylamine molecule with a density of 1.47 g cm-3), d̃p=30 would be equivalent to a dimensional particle size of 16.4 nm.

At time t1 and t2, if two particle sizes dp,t1 and dp,t2 are used to represent the particle size distribution, the measured GR can be calculated using the following equation as a first-order approximation: GRmdp,t1+dp,t22,t2+t12=dp,t2-dp,t1t2-t1. If dp,ti is available for a time series {ti}i=1,2,…, GR can also be obtained by derivatizing a fitting function dp=dp(t) to obtain GR at any time ta: GRmdp,ta=ddp(t)dtt=ta. To implement Eqs. (7) or (8), it is necessary to choose a particle size that is representative of the particle size distribution at a given time. The choice of this representative size varies among publications and can depend on the types of available data. Based on previous studies (Kulmala et al., 2012; Lehtipalo et al., 2014; Stolzenburg et al., 2005; Yli-Juuti et al., 2011), we have selected four representative sizes for discussion: d̃p,mode, d̃p,sr100, d̃p,sr50 and d̃p,tot50. At a given time τ, d̃p,mode is the particle size at which dÑ(τ)/dlog⁡10d̃p reaches its local maximum. If the shape of the mode is lognormal, d̃p,mode is equal to the geometric mean of the distribution. As suggested by Kulmala et al. (2012), the lognormal distribution method involves calculating GRs from observed time-dependent trends of d̃p,mode. The maximum concentration method is based on the time when particles in a given size bin, d̃p,sr100, pass through their maximum (100 %) concentration (Lehtinen and Kulmala, 2003). The appearance time method is based on the time when particle concentrations in a bin, d̃p,sr50, pass through a specified percentage of its maximum (we have used 50 %). GRs are sometimes based on total concentrations of particles larger than a specified size. We refer to the particle size above, for which the total number concentration of particles reaches 50 % of its maximum value as d̃p,tot50. This approach is especially useful when measurements are carried out with a battery of CPCs with differing cutoff sizes. For simplicity, in this paper we assume that CPC detection efficiencies increase from 0 to 100 % at a given cutoff size. In practice, measured size-dependent detection efficiencies are typically used when analyzing CPC battery data. Figure 1 shows the location of these representative sizes at τ=20, 60 and 100 for two nucleation scenarios in the absence of preexisting particles. d̃p,mode, d̃p,sr100, d̃p,sr50 and d̃p,tot50 are marked as points, with their y coordinates representing particle concentrations at corresponding sizes.

As will be shown later, values of GRm obtained with d̃p,mode, d̃p,sr100, d̃p,sr50 or d̃p,tot50 are not equal. To differentiate these cases, GRm are notated as GRm,mode, GRm,sr100, GRm,sr50 and GRm,tot50 accordingly.

The true GR (GRtrue) defined in this paper follows the Lagrangian approach (Olenius et al., 2014), i.e., tracking the volume change of individual particles, and only including molecular species exchange by condensation and evaporation. It is calculated with the following expression: GRtrue=dd̃pdτ=2πd̃p2dṼdτ=2πd̃p2⋅Ṽ+ci,kÑ1⋅dτ-Ek⋅dτ-Ṽdτ=2ci,kÑ1-Ekπd̃p2, where d̃p is the representative size, Ñ1 is the concentration of monomers and Ek is the particle evaporation rate given by Eq. (5).

If evaporation is negligible (E=0) and Ñ1 is constant, Eq. (9) leads to a higher GR for smaller particles, mainly because of the increased monomer collision frequency relative to particle size (Tröstl et al., 2016). Throughout this work Eq. (9) is used to evaluate true particle GR. Note GRtrue is calculated from dimensionless size and time, and is therefore dimensionless. Since we focus on relative values of true and measured GRs, our conclusions are unaffected by the dimensionality of GR. However, dimensionless GRs can be converted to dimensional values with Eq. (6).

As mode diameter (d̃p,mode) is often employed to derive particle GR, in this section we discuss the error of using GRm,mode as a substitute for GRtrue in the absence of preexisting particles. The effect of preexisting particles is discussed in Sect. 3.3.

Both condensation and coagulation lead to growth of d̃p,mode. To understand their relative importance, we attribute GRm,mode to three processes: monomer condensation minus evaporation (GRtrue), coagulation of the mode with clusters (GRm,cluster) and self-coagulation of the mode (GRm,self). The last two processes are the main causes of the discrepancy between GRm,mode and GRtrue. To evaluate GRm,cluster and GRm,self, the range of “clusters” and “mode” are defined as illustrated in Fig. 1 by the two shaded regions at τ=100: clusters (beige) and nucleation mode (light blue). Clusters and nucleation mode are separated by d̃p,min, where dÑ/dlog⁡10d̃p is at a local minimum. Stolzenburg et al. (2005) assumed the nucleation mode is lognormal and calculated GRtrue and GRm,self with the method of moments. In this work, since the mode for collision-controlled nucleation deviates significantly from lognormal (see Fig. 1a), no assumption regarding the shape of the nucleation mode is made. Instead, GRm,cluster and GRm,self are calculated with the first-order numerical approximation method outlined in Appendix A.

The calculation results are summarized by Fig. 2. We first consider collision-controlled nucleation (E=0). For this nucleation scenario, Fig. 2a shows d̃p,mode on the left y axis and GR values on the right. A third-order polynomial is used for fitting d̃p,mode=d̃p,mode(τ) and is plotted as a solid black line. Differentiating the fitted polynomial with respect to time gives the value of GRm,mode. It is clear that GRtrue only accounts for a small fraction (17–20 %) of GRm and is on par with the contribution of GRm,cluster (15–22 %). Self-coagulation is the major contributor (62–78 %) to GRm. Thus, using GRm,mode as a substitute for GRtrue leads to an overestimation by as much as a factor of about 6. We believe collision-controlled nucleation (E=0) in the absence of other particle loss mechanisms such as wall deposition (W=0) and scavenging by preexisting particles (L=0) provides an upper limit for overestimation of GRtruefor a constant rate system (R = constant). This is because these conditions lead to the maximum number of particles that can be produced by nucleation. High concentrations lead to high coagulation rates, and it is coagulation that is primarily responsible for errors in GRm. Furthermore, as is discussed below, the absence of evaporation and scavenging by nucleated particles keeps monomer concentrations low relative to values achieved when E≠0 (see Fig. 2a). Low monomer concentrations reduce the value of GRtrue, thereby increasing relative errors in GRm.

Distinctive features of particle growth emerge when cluster evaporation is included by setting E=1×10-3. Figure 2b shows results for this nucleation scenario. Most noticeably, particles grow considerably faster at early stages of simulation. This occurs because evaporation depletes clusters and correspondingly increases monomer concentration. In the absence of preexisting particles, monomer concentration accumulates until the supersaturation is high enough for nucleation to take place (see Fig. 2c). The accumulated monomers then rapidly condense on the nucleated particles, leading to the rapid particle growth shown in Fig. 2b. To capture this rapid growth, two third-order polynomials are used to fit d̃p,mode values for τ<40 and τ>35, respectively, with an overlapping region for 35<τ<40. Furthermore, in comparison to collision-controlled nucleation, contribution of GRm,cluster to GRm,mode becomes negligible, due to decreased cluster concentration by evaporation. For τ>30, GRtrue accounts for about 40–55 % of GRm,mode, larger than that of collision-controlled nucleation; for τ<25, GRtrue almost entirely accounts for GRm,mode and even exceeds GRm,mode at the very beginning of the nucleation. GRtrue / GRm,mode > 1 indicates a rapidly forming nucleation mode, in which freshly nucleated particles enter the mode and skew the mode distribution toward smaller sizes, slowing down the shift of the mode peak towards larger values.

Increase of GRtrue / GRm,mode by evaporation is explained by the elevated monomer concentration due to particle volatility and the smaller number of particles formed by nucleation: the former increases GRtrue, and the latter decreases GRm,self and GRm,cluster. Figure 2c plots monomer concentration Ñ1 as a function of time for several values of E. Noticeably, monomer concentration increases with E since higher cluster evaporation rates require higher monomer concentrations (i.e., higher supersaturation) to overcome the energy barrier of nucleation. Once nucleation takes place, high monomer concentration leads to rapid nanoparticle GRs.

Figure 2d shows GRtrue / GRm,mode at τ=30, 50, 100 and 150 for several E values. At a given time, GRtrue / GRm,mode clearly increases with E: when evaporation rates are not negligible (i.e., E≠0), GRm,mode is closer to GRtrue than occurs when E=0. Again, this is because the elevated monomer concentrations increase GRtrue and the lowered concentrations of clusters and nucleated particles decrease GRm,cluster and GRm,self. As E approaches 0, the value of GRtrue / GRm,mode converges to that of the collision-controlled nucleation (∼ 0.2). One data point, corresponding to E=5×10-3 and τ=30, with a value of 1.8, is not shown in Fig. 2d. It has a value significantly greater than unity because of the large quantities of nucleated particles entering the mode, skewing the mode peak toward smaller sizes.

In this section we examine how observed GR depends on the choice of a representative size. The application of GRm,mode to deduce GRtrue, though convenient in practice, depends on the existence of a nucleation mode. However, the nucleation mode is usually not well defined in the early stage of nucleation. In contrast, GRs based on other representative sizes (d̃p,sr50, d̃p,sr100 and d̃p,tot50) are not dependent on mode formation and are available for all particle sizes. In light of this, GRm,sr100, GRm,sr50 and GRm,tot50 have often been employed to describe the GR of small particles (< 5 nm). The effects of preexisting particles are neglected in this section (i.e., L=0) but are discussed in Sect. 3.3.

For collision-controlled nucleation, d̃p,mode, d̃p,sr50, d̃p,sr100 and d̃p,tot50 are plotted as functions of time in Fig. 3a. The magnitude of the representative sizes follow d̃p,mode < d̃p,bin100 < d̃p,tot50 < d̃p,bin50, as was previously illustrated in Fig. 1a. d̃p,mode < d̃p,bin100 indicates that a certain measurement bin first reaches its maximum concentration and becomes a local maximum at a later time. This is true for collision-controlled nucleation with a decreasing peak concentration but is not necessarily true for other nucleation scenarios. The observed GRs (i.e., slope of curves in Fig. 3a) are shown in Fig. 3b as a function of representative size, with a clear relationship GRm,mode < GRm,sr100 < GRm,tot50 < GRm,sr50. Note that GRm,mode is not available for small sizes, indicating the nucleation mode is yet to form at the early stage of nucleation. Figure 3c shows GRtrue / GRm as a function of representative size, with GRtrue calculated with Eq. (9). Clearly GRtrue accounts for the highest percentage of GRm at the start of nucleation. This is partly due to higher monomer concentrations (see red solid curve in Fig. 2c) and partly due to Eq. (9), which leads to higher true GR for smaller particles: the addition of a monomer leads to a bigger absolute as well as fractional diameter growth for small particles.

Figure 3d–f are counterparts of Fig. 3a–c, but with evaporation constant E set to 1×10-3. Figure 3d shows that d̃p,sr50 and d̃p,tot50 increase relatively slowly at the start of the simulation (see the amplified figure at the lower right corner of Fig. 3d; for reference, the dimensionless sizes of monomer, dimer and trimer are 1.24, 1.56 and 1.79, respectively). Subsequently, a marked change slope of the d̃p=d̃p(τ) curve is observed, indicating accelerated particle growth. This reflects that nucleation occurs with a burst of particle formation following a process of monomer and cluster accumulation. The slow growth of the smallest clusters is an indication that the accumulation process is slow due to the strength of the Kelvin effect.

Figure 3e shows GRm obtained by curve fitting after the nucleation burst and Fig. 3f shows the corresponding GRtrue / GRm values. Different from collision-controlled nucleation, there is a sharp rise in the GRtrue / GRm value at the start of nucleation. This is due to the sharp decrease in the evaporation term in Eq. (9), causing the value of GRtrue to increase sharply. As nucleation progresses, the ratio of GRtrue to GRm,sr100, GRm,tot50 and GRm,sr50 comes close to 1, with GRm,mode not yet available. Eventually, GRtrue / GRm for all representative sizes decreases and falls into the range of 30–50 %, with GRm,mode giving the best estimate of GRtrue. Note the value of GRtrue / GRm,mode significantly exceeds unity for d̃p∈[5,11] due to the distortion of the mode toward smaller sizes by the high flux of freshly nucleated particles into the mode.

Preexisting particles act as particle sinks to decrease the intensity of nucleation. Similarly, in chamber experiments, though loss to preexisting particles is often eliminated by using air that is initially particle free, loss of particles to chamber walls is inevitable. Since wall loss and loss to preexisting particles have a qualitatively similar effect on nucleation (McMurry and Li, 2017), we selectively examine the effect of preexisting particles on GR measurements to qualitatively illustrate the effects of all of these processes. To probe the initial stage of nucleation, we use d̃p,bin50 as the basis for our analysis, with a comparison of representative sizes presented at the end of this section. As to the magnitude of L, we choose L∈[0,0.3] based on previous work. It was shown in Fig. 2b in McMurry and Li (2017) that as L exceeds 0.1, particle size distributions begin to deviate discernably from the collision-controlled case. In addition, L≈0.2 was observed in the ANARChE field campaign carried out in Atlanta for nucleation events with sulfuric acid as the major nucleating species (Kuang et al., 2010).

The influence of preexisting particles on the discrepancy between true and measured GR (GRtrue / GRm) is twofold. On the one hand, preexisting particles can decrease monomer concentration, which leads to a smaller GRtrue. On the other hand, preexisting particles reduce coagulation by scavenging nucleated particles, which could result in a narrower gap between GRtrue and GRm. Therefore, the response of GRtrue / GRm to L depends on the relative magnitude of these two competing effects. Figure 4a shows d̃p,sr50 as a function of time for several L values and Fig. 4b displays the corresponding GRtrue / GRm values. It can be seen that GRtrue / GRm positively correlates with L, indicating preexisting particles are more effective in removing nucleated particles than reducing monomer concentrations. In fact, as further demonstrated by Fig. 4c, monomer concentrations (leftmost point of all the curves) are barely affected: scavenging of monomers by preexisting particles is offset by less condensation of monomers onto nucleated particles. Note that for the range of L values examined, the presence of preexisting particles alter GRtrue / GRm values by no more than 50 % for collision-controlled nucleation.

Figure 4d–f show the same quantities as are shown in Fig. 4a–c, but with E set to 1×10-3 instead of zero. In contrast to collision-controlled nucleation, preexisting particles significantly affect the nucleation process when cluster evaporation is taken into account. As L increases, Fig. 4e shows GRtrue / GRm converges to a value slightly larger than unity. This indicates that the contribution of coagulation to measured GR approaches zero as L becomes large; or equivalently, the concentration of nucleated particles is severely decreased by preexisting particles. Values of GRtrue / GRm,sr50 slightly exceed unity for large sizes (Fig. 4f) due to the slightly higher condensational GRs of smaller particles in the nucleation mode. This shifts values of d̃p,sr50 towards smaller sizes than would occur if all particles were to grow at the same rate, causing GRm,sr50 to be smaller than GRtrue.

The decrease in nucleated particle concentration is further demonstrated in Fig. 4f. From L=0 to L=0.3, the peak concentration of nucleated particles dropped by about 3 orders of magnitude. Such a decrease in concentration of nucleated particles results from the limiting effect of L on monomer concentration. If preexisting particles are absent, then no major loss mechanisms for monomers exist prior to the nucleation burst. Monomers would accumulate until the nucleation energy barrier could be overcome: the higher the energy barrier, the higher the monomer concentration prior to nucleation, as shown in Fig. 2c. The elevated monomer concentration then leads to rapid growth of freshly nucleated particles immediately following the nucleation burst. However, in the presence of preexisting particles (i.e., L≠0), monomer concentration can only increase to the point at which its production and consumption by preexisting particles reach balance, prohibiting its concentration from reaching a high value even prior to the nucleation burst. To facilitate comparison with experimental results, in Appendix B we provide an example of conversion from dimensionless distributions and GRs to dimensional ones.

Finally, Fig. 5 examines the difference between representative sizes used to calculate GRm when loss to preexisting particles is accounted for. Two cases are presented: (1) collision-controlled nucleation (E=0) with L=0.2 (Fig. 5a–c) and (2) nucleation accounting for both cluster evaporation and scavenging by preexisting particles (E=1×10-3 and L=0.2; Fig. 5d–f). For collision-controlled nucleation with L=0.2, the preexisting particles change nucleation only slightly, although GRm decreases and GRtrue / GRm increases to a minor extent compared to collision-controlled nucleation in the absence of a preexisting aerosol (compare Figs. 5a–c to 3a–c). The analysis made in the discussion of Fig. 3a–c still stands for Fig. 5a–c. For nucleation with evaporation and preexisting particles coupled together (Fig. 5d–f), three features are worthy of attention. First, compared to evaporation-only nucleation, GRm is significantly decreased for small particle sizes. For d̃p<10, GRm is no larger than 0.7 with preexisting particles but can be greater than 1.5 without (refer to Fig. 3e). Second, as shown in Fig. 5f, GRtrue / GRm,sr50 and GRtrue / GRm,tot50 come close to unity due to negligible coagulation effects. Third, GRtrue / GRm,mode is between 1.2 and 1.5 and GRtrue / GRm,sr100 is between 1.1 and 1.2 for d̃p>10, indicating the true growth will be slightly underestimated if d̃p,mode or d̃p,sr100 is used to infer GRtrue.

In previous sections, mainly overestimation of the GRtrue by measured GR, GRm, has been discussed. Though we do not quantitatively study underestimation of GRtrue by GRm, in this section we show that in a constant rate system in which particle sink processes (i.e., dilution and loss to preexisting particles) strongly decrease the concentration of nucleated particles, GRm can approach zero and cannot be utilized to estimate GRtrue. Figure 6 shows such nucleation scenarios for (a) collision-controlled nucleation with M = 0.1 and (b) collision-controlled nucleation with L=1.5. In both cases other sink processes were set equal to zero. As shown in both Fig. 6a and b, particle size distributions approach steady state after τ=100. As a result, GRm approaches zero beyond τ=100. At the same time, GRtrue remains finite since monomer concentration remains at steady state after τ=20. Therefore, other methods have to be utilized to infer GRtrue in such situations.